摘要:

Faraday Discuss., 1997, 106, 189¡í203 Ab initio study of corundum-like oxides (Me= Ti, V, Me2O3 Cr, Fe, Co, Ni) Michele Catti*¡× and Giovanni Sandrone Dipartimento di Chimica Fisica ed Elettrochimica, di Milano, via Golgi 19, Universita` 20133 Milano, Italy Periodic unrestricted HartreenFock methods, with all-electron basis sets and a posteriori estimate of correlation energy, have been used to study the ground-state total energy of the title compounds with rhombohedral R3 6 c corundum-type structure.and are shown to be stable oxides Co2O3 Ni2O3 with quite regular binding energies at the athermal limit. The relative stabilities of the AF1 (R3c), AF2 and AF3 (R32) antiferromagnetic struc- (R3 6 ) tures have been investigated, showing that adopts the AF1 Cr2O3 con¢�guration, while and prefer AF2; this is Fe2O3, Co2 O3 Ni2O3 accounted for in terms of p- and n-type superexchange interactions.The spin polarization of electron transfer from O2~ to Me3` has been analysed, showing that only minority-spin electrons are transferred for Me\Fe, Co and Ni, while the total transfer is doubled for the other oxides by the majority-spin contribution. 1 Introduction The recent development of very efficient computational schemes and computer codes for band structure and total energy calculations of crystalline solids, both in the Hartreen Fock (HF) and density functional theory (DFT) versions,1 has made it possible to reconsider the structural, electronic and magnetic behaviour of simple transition metal oxides.One of the main families is that of sesquioxides of ¢�rst-row trivalent transition metals with rhombohedral corundum-type crystal structure (two formula units per R3 6 c primitive unit cell).The known, stable compounds of this series are Ti2O3, V2O3 , and has a di¢­erent orthorhombic structure,2 owing to Jahnn Cr2O3 Fe2O3. Mn2 O3 Teller distortion of the Mn3` octahedral environment. and are not Co2O3 Ni2O3 reported to be stable compounds in bulk form:3 attempts to prepare them by thermal dehydration of CoOOH and NiOOH lead, apparently,4 to lower oxides and NiO Co3O4 with loss of oxygen, though their synthesis was claimed in old literature.5 A systematic trend is observed in the electronic behaviour of these solids :6 the early-row compounds and have insulating-to-metal phase transitions driven by temperature, Ti2O3 V2O3 while and are antiferromagnetic insulators and, similarly, insulating Cr2O3 Fe2O3 properties would be expected for and As for the structural and magnetic Co2O3 Ni2O3 .behaviour, the main feature characterizing the corundum-like oxides is the presence of oxygen atoms with distorted tetrahedral coordination, unlike perovskite, rutile and rocksalt-type compounds, where the anion has linear, triangular and octahedral coordination, respectively.The bonding environment of oxygen atoms is a key to understand- ¡× Present address : Dipartimento di Scienza dei Materiali, Universita` di Milano, via Emanueli 15, 20126 Milano, Italy 189190 Ab initio study of corundum-like Me2O3 ing the superexchange interactions leading to di¢­erent magnetic structures and behaviours in the various families of transition metal oxides.In particular, and Cr2O3 have di¢­erent antiferromagnetic spin arrangements, whose relative stabilities Fe2O3 involve subtle balances of contributions to superexchange energy.7 While theories of superexchange with metalnligandnmetal angles of 180¡Æ and 90¡Æ (cf. the rocksalt and perovskite structures) are well established, the e¢­ect of intermediate angles, such as those observed in the corundum and rutile-like structures, has still to be clari¢�ed.Previous theoretical studies of these compounds, based on ab initio periodic approaches, include and by HF method (ref. 8 and 9, respectively), and Fe2O3 Cr2O3 and both by DFTnlinearized augmented plane wave (LAPW)10,11 and by Ti2O3 V2O3 HF.12 The present work has two objectives.First, extending the HF simulations to the hypothetical and terms of the series. The aims are (i) to provide a consis- Co2O3 Ni2O3 tent set of results for the whole ¢�rst-row of transition metals in the corundum structure and (ii) to detect possible anomalies or inconsistencies in the simulated behaviour of these two oxides, which might account for their apparent chemical instability.Second, reconsidering in the light of the above new results the preliminary data obtained previously for the other oxides, so that the e¢­ects of the atomic electron con¢�gu- Me2O3 ration on the trend of electronic and magnetic properties of the whole series of isostructural oxides can be fully explored. 2 Method of calculation Model corundum-type crystal structures were designed for and as Co2O3 Ni2O3 follows.The experimental unit-cell constants a and c and atomic fractional coordinates x(O) and z(Me) of the stable oxides and (ref. 8, 9, 12) were Ti2O3, V2O3, Cr2O3 Fe2O3 plotted against the Me3` ionic radius for octahedral coordination.13 Interpolation for the radii of Co3` and Ni3` yielded the estimated structural parameters of cobalt and nickel sesquioxides to be used in this work.The values reported in Table 1 correspond to high-spin (HS) dimensions of the two ions. For the low-spin (LS) electron con¢�gurations, a and c are reduced to 4.779 and and to 4.819 and 13.180 13.072 ¡± (Co2O3) ¡± respectively, while the atomic coordinates are unchanged. (Ni2O3), All quantum mechanical computations were carried out with CRYSTAL95.1 This code1 is based on a periodic self-consistent ¢�eld (SCF) approach, where the Bloch func- Table 1 Estimated (from ionic radii) and calculated (least-energy) hexagonal unit-cell constants, atomic fractional coordinates and octahedral interatomic distances of HS corundum-like and Co2O3 Ni2O3 Co2O3 Ni2O3 estim.calc. estim. calc. a/¡± 4.945 5.037(]1.9%) 4.921 4.941(]0.4%) c/¡± 13.524 13.413([0.8%) 13.458 13.390([0.5%) c/a 2.735 2.663([2.6%) 2.735 2.710([0.9%) V /¡±3 286.4 294.7(]2.9%) 282.2 283.1(]0.3%) x(O) 0.3056 0.3039 0.3056 0.3032 z(Me) 0.3517 0.3547 0.3517 0.3553 MewO@/¡± 1.932 1.944(]0.6%) 1.923 1.912([0.6%) MewOA/¡± 2.043 2.077(]1.7%) 2.033 2.057(]1.2%) EHF [2987.5673 [3238.3808 EHF`corr [2991.1510 [3242.1209 The percentage deviations are given in parentheses.Total HF and HF]corr. energies per formula are reported (ferromagnetic states). (Eh)M. Catti and G. Sandrone 191 Table 2 Exponents and contraction coefficients of the (individually normalized) Gaussian functions adopted for (a0~2) cobalt and nickel in the present study Co Ni coef–cients coef–cients shell type exponents s p,d exponents s p,d 1s 2.989]105 [2.894]10~4 3.217]105 [2.905]10~4 4.437]104 [2.298]10~3 4.775]104 [2.284]10~3 9.952]103 [1.213]10~2 1.071]104 [1.215]10~2 2.771]103 [5.033]10~2 2.982]103 [5.025]10~2 8.848]102 [1.628]10~1 9.524]102 [1.628]10~1 3.164]102 [3.668]10~1 3.405]102 [3.665]10~1 1.229]10 [4.601]10~1 1.325]102 [4.602]10~1 4.938]10 [1.865]10~1 5.312]10 [1.864]10~1 2sp 1.006]103 [4.273]10~3 8.176]10~3 1.090]103 [4.237]10~3 8.140]10~3 2.402]102 [5.500]10~2 6.124]10~2 2.604]102 [5.467]10~2 6.136]10~2 7.818]10 [1.442]10~1 2.370]10~1 8.463]10 [1.444]10~1 2.377]10~1 2.975]10 2.138]10~1 5.054]10~1 3.231]10 2.117]10~1 5.063]10~1 1.206]10 6.698]10~1 5.130]10~1 1.310]10 6.645]10~1 5.127]10~1 4.417 2.754]10~1 2.041]10~1 4.818 2.747]10~1 2.010]10~1 3sp 4.635]10 9.675]10~3 [2.507]10~2 5.058]10 1.066]10~2 [2.496]10~2 1.561]10 [4.315]10~1 [8.477]10~2 1.711]10 [4.337]10~1 [8.461]10~2 7.498 [8.622]10~1 2.275]10~1 8.246 [8.736]10~1 2.262]10~1 3.368 9.827]10~1 1.010 3.711 9.798]10~1 1.006 4sp 1.514 1.0 1.0 1.655 1.0 1.0 5sp 6.205]10~1 1.0 1.0 6.698]10~1 1.0 1.0 3d 4.798]10 2.583]10~2 5.338]10 2.557]10~2 1.356]10 1.447]10~1 1.503]10 1.456]10~1 4.584 3.776]10~1 5.085 3.760]10~1 1.642 4.828]10~1 1.818 4.836]10~1 4d 5.435]10~1 5.958]10~1 1.0192 Ab initio study of corundum-like Me2O3 tions are expanded in localized basis sets of atomic orbitals ; their radial factors are expressed as linear coGaussian-type functions.HF and KohnnSham (DFT) Hamiltonians are available. In this work the HF method was used, both in its restricted (RHF) and unrestricted (UHF) version, and it was supplemented by an a posteriori correction of the total energy based on a DFT estimate of the correlation energy.14 eAll-electroni basis sets made up of 27 and 18 atomic orbitals, respectively, were used for the transition-metal atoms Me and for O in the whole series of compounds.The radial part of the Me orbitals is represented by an 8-6-411G contraction with two d-type 41G shells, and that of O is denoted as 8-411G*, according to the usual convention.15 Exponents and coefficients of Gaussian functions were optimized variationally for the isolated Me3` ions. The exponents of the outermost sp and d shells of Me, and those of the external sp and polarization d-type shell of O, were re¢�ned so as to minimize the energy of the corresponding solid compound.The basis sets of Me2O3 TiIII, VIII, CrIII and FeIII have been reported in previous papers;8,9,12 that of the oxygen atom is given in a study of The results obtained here for CoIII and NiIII are Al2O3 .16 reported in Table 2. External sp and d shells of oxygen have exponents of 0.1889 (sp, 0.18295 (sp, and 0.3538 (d, both) It should be stressed that all Co2O3), Ni2O3) a0~2.basis sets for the crystals were derived in a consistent way, so that di¢­erences in Me2O3 energy- and wavefunction-dependent properties along the whole series can be considered to be fully reliable. Also computational conditions and numerical approximations used are the same as those described elsewhere.9 3 Results and Discussion Wavefunction and total energy calculations were performed for and both Co2O3 Ni2O3 in their HS total spin moment for Co3`; for Ni3`) and (t2g 4 eg2 , Sz\2 t2g 5 eg2 , Sz\3/2 LS for Co3`; for Ni3`) model corundum structures.(t2g 6 eg0 , Sz\0 t2g 6 eg1 , Sz\1/2 Only the LS case is non-magnetic and has to be treated by the RHF formalism.Co2O3 The other three cases imply magnetic spin ordering within the rhombohedral unit cell and require the UHF approach. The simplest model is ferromagnetic (FM), retaining the space group symmetry; antiferromagnetic structures will be dealt with in the R3 6 c section below. Total energies obtained for the HS FM case are given in Table 1. The LS states turn out to be less stable by 0.1742 (0.1266 including the correlation Eh correction) for and by 0.0898 including correlation) for Co2O3 Eh(0.0686 Ni2O3 .Unfortunately these di¢­erences are not particularly enlightening, as it is well known that the HF approach overestimates the exchange stabilization (Hundis rule) of HS systems with respect to crystal ¢�eld e¢­ects favouring the LS con¢�gurations. DFT-based correlation corrections help little in this respect, and full multi-determinant con¢�guration interactions would be needed to ascertain the correct relative stability of HS and LS states.Therefore, only HS states will be considered in the following for and Co2O3 Ni2O3 . The minimum-energy structural parameters a, c, x(O) and z(Me) were searched for by computing the total energy on a grid of points in the corresponding four-dimensional space.Convergence was attained at the values reported in Table 1. The important result is that the corundum structure is stable to geometry changes for both and Co2O3 Further, the structural parameters estimated on the basis of ionic dimensions Ni2O3 . agree closely to those of least energy. The small deviations appearing in Table 1 ¢�t the decreasing trend with increasing atomic number, which is observed for deviations between least-energy and experimental parameters of and Ti2O3, V2O3, Cr2O3 Fe2O3 . 3.1 Antiferromagnetic structures In the (non-magnetic or ferromagnetic) corundum structure, the primitive rhombo- R3 6 c hedral unit cell contains four symmetry-related Me atoms aligned on the threefold axis,M.Catti and G. Sandrone 193 which we label as 1, 2, 3, 4 with increasing z coordinate (0.15, 0.35, 0.65, 0.85). The pairs 1n2 and 3n4 are closer and related by twofold axes located at z\1/4 and 3/4, respectively; the pairs 2n3 and 4n1 are more distant and related by symmetry centres at z\1/2 and z\0. Atoms 1 and 3 (and similarly 2 and 4) are related by the c glide plane.Hence, magnetic-spin ordering on Me atoms leads to three di¢­erent antiferromagnetic (AF) con¢�gurations with lower symmetry: AF1 (R3c), AF2 and (R3 6 ) AF3 (R32), characterized by spin sequences ][][, ][[] and ]][[, respectively. The ¢�rst structure characterizes and the second below the corre- Cr2O3 Fe2O3 , sponding Nec el temperatures of 310 and 953 K, as proved by neutron di¢­raction experiments.17,18 The AF1 and AF2 con¢�gurations are also equivalent to the lithium niobate and ilmenite crystal structures, respectively, if spin ordering (LiNbO3) (FeTiO3) is replaced by di¢­erentiation of Me into two di¢­erent chemical species.In order to study the relative stability of the three AF structures, and to account for it in terms of MewOwMe@ superexchange interactions, it is necessary to de.ne the geometrical features of the Me@ coordination environment around the Me atom.This is surrounded by 13 metal .rst neighbours, which populate four shells (denoted here as A, B, C, D) of 1, 3, 3 and 6 symmetry equivalent atoms of type 2, 4, 2 and 3, respectively. In Table 3 the MewMe@ distances and MewOwMe@ angles are reported for MexCr, Fe (from experimental structural data19), Co, Ni (from the model structures of Table 1).It appears that the spread of metal¡ímetal distances in the coordination sphere is considerable, and that bond angles are either near to 90¡Æ (closer shells A and B) or to values intermediate between 90¡Æ and 180¡Æ (farther shells C and D), according to the distorted tetrahedral surrounding of the oxygen atom.By considering the three AF structures, it turns out that AF1 and AF2 have a single shell with ferromagnetic ordering (Me and Me@ with parallel spins : D in AF1 and B in AF2), while the other three shells are all antiferromagnetic (Me and Me@ with antiparallel spins). On the other hand, AF3 has two FM (A and C) and two AF (B and D) shells. This simple observation would support the conclusion that the AF3 con.guration is less stable than the other two.It is more difficult to predict the relative stability of the AF1 and AF2 structures because the B shell is less populated (three metal neighbours instead of six) but much closer to the central atom than the D shell, so that the relative strength of the corresponding MewOwMe@ interactions is hard to evaluate. This point will be discussed below after examining the outcome of calculations. In Table 4 the computational results are reported for the four oxides Cr2O3, Fe2O3 , and in terms of E(AF[FM) energy diUerences, i.e.assuming the ferromag- Co2O3 Ni2O3 netic structure as zero-energy con.guration for each compound. All AF and FM magnetic structures considered correspond to maximum spin multiplicity HS con.gurations.Total HF and HF plus a posteriori DFT-based correlation correction (*EHF) (*EHF`corr) energies are given ; further, the kinetic and exchange components of (*Ek) (*Eex) *EHF are also shown, as the most signi.cant energy terms aUecting the superexchange interactions. The concerning and have been reported previously.9 *EHF values Cr2O3 Fe2O3 The following may be noted from the results of Table 4.(i) Out of the three antiferromagnetic structures, AF1 and AF2 are the most and least stable, respectively, for Table 3 MewMe@ distances and MewOwMe@ angles (degrees) in the four shells (A, (¡±) B, C, D) making up the metal¡ímetal coordination sphere of four corundum-like oxides Cr2O3 Fe2O3 Co2O3 Ni2O3 A(]1) 2.650 82.5 2.896 86.5 2.751 84.6 2.737 84.6 B(]3) 2.885 93.2 2.969 94.0 2.897 93.5 2.883 93.5 C(]3) 3.417 121.1 3.361 119.6 3.352 120.3 3.336 120.3 D(]6) 3.644 133.2 3.701 131.6 3.638 132.4 3.620 132.4194 Ab initio study of corundum-like Me2O3 Table y diÜerences (per formula unit) between three antiferromagnetic spin arrangements and the ferromag- R3 6 c netic structure *E(AF[FM)/eV AF1 (R3c) AF2 (R3 6 ) AF3 (R32) Cr2O3 *EHF [0.039 0.001 [0.019 *Ek [0.357 [0.106 [0.234 *Eex 0.115 0.045 0.072 *EHF`corr [0.043 [0.005 [0.022 Fe2O3 *EHF 0.042 [0.053 0.013 *Ek [0.238 [1.005 [0.395 *Eex 0.126 0.351 0.186 *EHF`corr [0.033 [0.142 [0.067 Co2O3 *EHF 0.043 [0.060 0.021 *Ek [0.211 [1.089 [0.281 *Eex 0.122 0.372 0.164 *EHF`corr 0.066 [0.082 0.040 Ni2O3 *EHF 0.109 0.025 0.042 *Ek 0.429 [0.097 [0.115 *Eex [0.043 0.141 0.123 *EHF`corr 0.099 0.008 0.026 while the opposite occurs for the other oxides.AF3 is always intermediate. This Cr2O3 , result includes the successful prediction of the observed antiferromagnetic con–gurations of (R3c) and (ii) For each compound, the most stable AF con–gu- Cr2O3 Fe2O3 (R3 6 ). ration is characterized by the largest kinetic stabilization and exchange destabilization ; the and values decrease for the second and third stable con–gurations. (iii) In o*Ek o *Eex most cases, the correction for correlation energy enhances the diÜerences of stability between magnetic structures given by values. (iv) The most stable magnetic struc- *EHF ture is antiferromagnetic for and but it is ferromagnetic for Cr2O3, Fe2 O3 Co2O3 , (though with a very small value).Data obtained for and Ni2O3 *EHF`corr Ti2O3 V2O3 have been discussed previously,12 and are less signi–cant because the energy diÜerences between the two most stable AF structures are vanishingly small. The most interesting result is that and behave as preferring Co2O3 Ni2O3 Fe2O3 , the to the R3c antiferromagnetic structure, unlike This behaviour would not R3 6 Cr2O3 .be expected on the basis of structural considerations alone, as Table 3 shows that, in this respect, cobalt and nickel oxides resemble rather than Thus the reason Cr2O3 Fe2O3 . would be more likely related to the electron con–gurations of the transition metal atoms involved. Fe3`, Co3` and Ni3` have half-–lled orbitals, while these are empty in eg Cr3`.Hence, superexchange interactions with p-type overlap should pre- p(O)»eg(Me) dominate in the –rst group of oxides, but only interactions with n-type p(O)-t2g(Me) overlap occur in This is con–rmed by the much lower Neç el temperature of Cr2O3 . with respect to according to the weaker n- than p-type superexchange. Cr2O3 Fe2O3 , The p-type overlap decreases strongly as the MewOwMe@ angle decreases from 180° to 90°, while the n-type overlap can be appreciable still at 90°.Now, the AF1 and AF2 structures diÜer by ferromagnetic ordering present in the D or B MewMe@ coordination shell, respectively : the former is characterized by the largest MewOwMe@ angle, the latter by the angle closest to 90° (Table 3). In iron, cobalt and nickel oxides (p-type superexchange) the MewMe@ pairs with a bond angle farther from 90° (D shell) will have a stronger tendency towards antiferromagnetic ordering than MewMe@ pairs with an angle close to 90° (B shell).The opposite is expected for (n-type Cr2O3M. Catti and G. Sandrone 195 Table 5 Calculated and experimental binding energies (eV) of corundum-like oxides HF HF]corr exp.Al2O3 [23.64 [30.08 [32.14 Ti2O3 [21.41 [28.05 [33.45 V2O3 [19.28 [26.23 [31.21 Cr2O3 [18.10 [25.08 [28.05 Fe2O3 [17.46 [22.00 [25.17 Co2O3 [19.80 [24.90 » Ni2O3 [17.41 [22.74 » superexchange). This corresponds exactly to a preference for the AF2 structure by and and for the AF1 structure by chromium oxide, consistent Fe2O3, Co2O3 Ni2O3 , with the results of calculations (Table 4).The situation is not so clear for and Ti2O3 probably because they have partially occupied shells which complicate the V2O3, t2g relations between overlap and geometry of coordination bonds. The leading eÜect in determining the trend of is given by the kinetic term, as *EHF remarked above; this is consistent with Andersonœs theory of superexchange, based on the idea that antiparallel electrons lower their kinetic energy by spreading into nonorthogonal overlapping orbitals,20 and also agrees with similar results on —uorides.21 The result that matches the trend of is not so obvious, however.Electronic *Eex o*Ek o and magnetic eÜects favouring overlap appear also to optimize exchange, though the overall balance is in favour of the kinetic term. Fig. 1 Binding energies of and plotted vs. the Ti2O3, V2O3, Cr2 O3, Fe2 O3, Co2 O3 Ni2O3 , number of d electrons on the Me metal atom. HF, HF]correlation correction calculated (|) ()) data, experimental values from thermochemical cycles (see text). (K)196 Ab initio study of corundum-like Me2O3 Fig. 2 Total density of electronic states a]b (DOS) of antiferromagnetic and Ti2O3, V2O3 (from top to bottom).The Fermi level is assumed at the energy zero. Cr2O3 3.2 Binding energies The binding energies of and were calculated in the Ti2O3, V2O3, Cr2O3, Co2O3 Ni2O3 same way as described for and Atomic energies of Ti, V, Cr, Co and Fe2O3 Al2O3 .8,16 Ni were derived by using the basis sets adopted for solid oxides, but supplemented by two more diÜuse shells, the exponents of which were optimized; the results for Al, Fe and O had been obtained previously.8,16 The binding energy is computed as the diÜerence between the total crystal energy per formula unit (most stable AF con–guration) and the energies of the constituent isolated atoms.Results are reported in Table 5 for both HF and HF]correlation correction energies. For comparison, experimental binding energies of and are also given.These were Al2O3, Ti2O3, V2O3, Cr2O3 Fe2O3 obtained by applying a suitable Born»Haber thermochemical cycle of type EB(expt)\;i [Ha, i 0 ](Hi298[Hi0)]]*fH298[(H298[H0)[Evib 0 the sum is extended to all chemical elements in the formula unit. The formation enthalpy of compounds, sublimation (Me) and dissociation enthalpies *fH298 Me2O3 (O2) Ha, i 0 , and heating enthalpies for all chemical species involved were taken from ref.Hi298[Hi0 22. The zero-point vibrational energies of all oxides were estimated by the Debye Evib 0 model, using an isotropic approximation for the mean acoustic wave velocity derived from the bulk modulus in order to obtain the Debye temperature.8M. Catti and G. Sandrone 197 Fig. 3 Projection onto (d-type) Me states of the DOS of and (from top to Ti2O3, V2O3 Cr2O3 bottom).(»»») a spin, (» » » ») b spin. Thick and thin lines correspond to and states, a1g egn]egp respectively. The results of Table 5 are also plotted vs. the number of d electrons in the Me2O3 series in Fig. 1. The general trend of experimental data (from to is simu- Al2O3 Fe2O3) lated correctly by theoretical binding energies, with deviations of the order of [30% for HF energies, which are reduced to ca.[12% by including the DFT-based correction for electron correlation. A secondary feature of the experimental curve is not reproduced by the calculated ones: the peculiar stabilization energy of with respect to other Ti2O3 oxides. Also, seems to be aÜected by a larger error than the average.This is V2O3 consistent with the difficulties found in achieving SCF convergence to stable insulating states for these two oxides,12 and also with the larger errors shown by their least-energy structures with respect to experimental ones. Oxides of early-row transition metals, with their very diÜuse d orbitals and tendency to metal states, are harder to simulate by HF methods than those of end-row metals.23 The computed binding energies of and Co2O3 are very interesting for two reasons.First, they follow the trend of other oxides Ni2O3 very reasonably, so that these compounds are predicted to be intrinsically stable from the thermodynamic point of view. Second, is predicted to display exactly the Co2O3 same eÜect of larger stabilization energy observed experimentally for in agree- Ti2O3 , ment with their symmetrical electron con–gurations (d1 and d6) showing a single t2g electron which exceeds the empty or half-full d shell.This would con–rm that the later oxides in the row, with their contracted electron shells, are simulated more accurately, as discussed previously. The extra stabilization of and is related to the octa- Ti2O3 Co2O3 hedral crystal-–lled splitting favouring the occupation of levels.t2g»eg, t2g198 Ab initio study of corundum-like Me2O3 Fig. 4 Total density of electronic states a]b (DOS) of antiferromagnetic and Fe2O3, Co2 O3 (from top to bottom). The Fermi level is assumed at the energy zero. Ni2O3 3.3 Electron states and chemical bonding The total density of electron states (DOS), and its projection onto the d(Me) contribution, are shown for and in Fig. 2 and 3, respectively. Similar data Ti2O3, V2O3 Cr2O3 are displayed in Fig. 4 and 5 for the oxides with a more than half-full d shell : Fe2O3 , and Both the valence and conduction bands appear in each diagram, Co2O3 Ni2O3 . separated by an insulating band gap whose width is overestimated by the HF approximation.The possibility of conducting states for and has been discussed in Ti2O3 V2O3 detail previously.12 All these diagrams refer to the stable antiferromagnetic spin con–gurations. Total densities concern the sum of a(spin-up)]b(spin-down) states, while projected densities are distinguished between separate a and b contributions. The triply degenerate levels of Me in an ideal octahedral surrounding (cubic symmetry) are split t2g a by trigonal distortion into an and two degenerate states ; the same occurs for a1g a egna t2g b levels.A striking diÜerence appears between the behaviour of the –rst and second set of three oxides, with open and closed, respectively, da shell on the Me atom. In the Me2O3 former case, the d(Me) and p(O) occupied electron states overlap in the same energy range of the valence band (about 8 eV below while the d(Me) states fall isolated at EF), much lower energy (from [8 to [16 eV) in the latter case.This result shows that and with p(O) states at the Fermi level (and p»d band gap) show Fe2O3, Co2O3 Ni2O3 , a charge-transfer type electronic behaviour, according to the classi–cation of ZaanenM. Catti and G.Sandrone 199 Fig. 5 Projection onto (d-type) Me states of the DOS of and (from top to Fe2O3, Co2O3 Ni2O3 bottom). (»»») a spin, (» » » ») b spin. Thick and thin lines correspond to and states, a1g egn]egp respectively. and Sawatzky.24 The three lighter oxides, on the other hand, are closer to the Mott- Hubbard character [d(Me) states at the Fermi level, d»d band gap]. This is consistent with experimental results from photoelectron spectroscopy on and Ti2O3, V2O3, Cr2O3 which help to de–ne the nature of the valence band.It should be also taken Fe2O3 ,25h28 into account that DFT simulations tend to shift the position of the d(Me) states to energies higher than those of p(O) states,10,11 with respect to HF results, thus emphasizing more the role of Mott»Hubbard than that of charge-transfer systems.Within the group of heavier oxides, the DOSs of and show peculiar Co2O3 Ni2O3 features (Fig. 5). Instead of a single, very narrow d(Me) peak as for several are Fe2O3 , now observed, spread over a larger energy range. Further, an analysis of the diÜerent components of d states in the valence band shows, at increasing energy, the egpa, egna, a1g a and states for and the sequence and for Thus the two a1g b Co2O3, a1g a , egpa, egna egnb Ni2O3 .oxides behave exactly as and respectively, where the occupied d(Me) states Ti2O3 V2O3 , are (titanium) and (vanadium):12 the a and b spins are just exchanged in the a1g a egna upper occupied d subband for each pair of corresponding oxides. The conduction bands of all oxides are made up of db states, with exception of a1g b and states which are lowered down below the Fermi level in the case of and egnb Co2O3 respectively. On the other hand, in and the and states, Ni2O3, Ti2 O3 V2O3 egna a1g a respectively, are also promoted into the conduction band.A useful tool to analyse electron transfer and related chemical bonding eÜects in the series is given by the corresponding Mulliken electron population analysis Me2O3200 Ab initio study of corundum-like Me2O3 Table 6 Mulliken population data of electron charge distribution for Ti2O3, V2O3, Cr2 O3 , and Fe2O3, Co2O3 Ni2O3 Ti2O3 V2O3 Cr2O3 Fe2O3 Co2O3 Ni2O3 AF Me a1g a 0.984 0.053 0.995 0.099 0.995 0.996 b 0.042 0.043 0.030 0.028 0.989 0.028 egn a 0.083 1.969 1.985 1.992 1.995 1.990 b 0.075 0.073 0.055 0.044 0.042 1.922 total t2g a 1.066 2.021 2.980 2.991 2.990 2.986 b 0.116 0.115 0.085 0.072 1.031 1.950 egp a 0.280 0.346 0.324 2.020 2.010 2.007 b 0.254 0.270 0.227 0.213 0.237 0.310 total d a 1.346 2.367 3.303 5.011 5.000 4.993 b 0.370 0.385 0.311 0.285 1.268 2.260 net charge ]2.229 ]2.171 ]2.325 ]2.618 ]2.654 ]2.669 MewO@ a 0.014 0.012 0.010 [0.014 [0.013 [0.010 b 0.018 0.021 0.016 0.019 0.012 0.013 MewOA a 0.002 0.008 0.007 [0.005 [0.008 [0.006 b 0.008 0.014 0.012 0.012 0.009 0.009 FM Me a]b 19.769 20.829 21.677 23.383 24.345 25.335 a[b 1.003 2.012 3.023 4.745 3.745 2.732 O a]b 9.487 9.447 9.549 9.745 9.770 9.776 a[b [0.002 [0.008 [0.015 0.170 0.170 0.179 Results concern a and b spins related to metal (Me) atomic orbital shells and MewO bond overlap for the antiferromagnetic (AF) con–guration, and a]b and a[b overall metal and oxygen populations for the ferromagnetic (FM) case.(Table 6). In particular, as the basis sets have been designed consistently for all oxides, diÜerential eÜects of electron population are expected to be quite reliable and reasonably independent of basis set choice. The d(Me) populations are decomposed into and a1g , egn contributions, separately for a and b spins.Such results are consistent with the eg a scheme of band occupation derived previously from the DOS: and have t2g T2O3 V2O3 and con–gurations, respectively, while and have (a1g a )1 (egna)2 Co2O3 Ni2O3 (a1g a )1 and In and the shell is half full : (egna)2(a1g b )1 (a1g a )1(egna)2(egnb)2. Cr2 O3 Fe2O3, t2g (a1g a )1 (egna)2.The important results mainly concern negative charge-transfer from oxygen to metal, according to the idealized 3/2O2~]Me3`]ye~ process. Most of these electrons (80» 90%) go into the d(Me) shell, actually populating the levels. This is related to the egp covalent interaction with p(O) states, leading to p(Me-O) bond formation and bringing a fraction of and states below the Fermi energy.egpa egpb A clear distinction is observed between the –rst and the (Ti2O3, V2O3, Cr2 O3) second group of oxides, concerning the following points. (i) The (Fe2O3, Co2O3, Ni2O3) total (a]b) amount of charge transferred decreases from 0.7»0.8 o e o (–rst group) to 0.3»0.4 o e o (second group). Consequently, the overall net charge on Me is higher in the second than in the –rst case by ca. 0.4 o e o, on average. (ii) By decomposing the transferred electrons into a and b components, they turn out to be approximately equal for the group of lighter oxides, while the a component vanishes for the heavier ones. This eÜect is shown in Fig. 6, where the numbers of a and b electrons transferred from oxygenM. Catti and G. Sandrone 201 Fig. 6 Number of a and b electrons transferred from oxygen to the d shell of Me atoms in (Ö) (L) the oxides vs. the ideal number of d electrons in the isolated Me atom Me2O3 to the d shell of Me atoms are plotted vs. the formal number of d(Me) electrons in the oxides. Such numbers are obtained by subtracting from the total da and total db Me2O3 values of Table 6 the corresponding ideal integral values.The reason for this eÜect is that all da states are full in the Me3` ions of the second group, and thus they can host no more electrons coming from oxygen. Hence, the sharp increase in net charge on Me, by passing from –rst to second group oxides. This result is reinforced by the amount of transferred electrons decreasing smoothly with increasing atomic number.The type of spin polarization of transferred electrons (nearly vanishing for Me\Ti, V and Cr, full b for Fe, Co and Ni) has remarkable consequences on the overall net spin moment per metal atom. That is signi–cantly lower than the ideal number of na-nb unpaired d electrons for oxides of the second group (4.726 cf. 5, 3.732 cf. 4 and 2.733 cf. 3), but is very close for Ti (0.976), V (1.982) and Cr (2.992).In the ferromagnetic case (last lines of Table 6), the deviation from the ideal value on the cation is compensated by an opposite deviation on the oxide ion, so as to produce very small b-type anion spin polarizations in and and larger a-type ones in and Ti2O3, V2O3 Cr2O3, Fe2 O3, Co2O3 In the AF case, of course, these anion spin polarizations are vanishing, because Ni2O3 .local opposite signs matching those of the spin moments of neighbouring cations give zero sum. It is also interesting to consider the MewO bond overlap populations, which are reported separately for the two independent MewO@ and MewOA bonds in Table 6. For Ti, V and Cr, positive values are observed for a and b electrons, indicating a bonding interaction for both spin states (slightly larger for b spins).This corresponds to the a]b O2~ to Me3` charge transfer discussed above. In the case of Fe, Co and Ni, on the other hand, the overlap population is positive for b spin but negative for the a one, according to an antibonding repulsion of a electrons. The latter is due to the full da shell of Me3` cations of the second group exerting an exchange repulsion towards electrons in the MewO region.The inversion of sign of the a-type bond population appears202 Ab initio study of corundum-like Me2O3 Fig. 7 Number of a and b electrons populating the MewO@ bonding overlap in the (Ö) (L) oxides vs. the ideal number of d electrons in the isolated Me atom Me2O3 clearly in Fig. 7, where this quantity (and the corresponding b) concerning the MewO@ bonding overlap is plotted vs.the ideal number of d(Me) electrons in the oxides. Me2O3 4 Conclusions The ab initio study of and has shown the corundum structure to be a Co2O3 Ni2O3 stable energy minimum, at the athermal limit, for both compounds; further, their theoretical binding energies –t the trend of the series quite reasonably. Thus, experi- Me2O3 mental difficulties in obtaining the two oxides in pure bulk form at ambient conditions should be ascribed to competition with other compounds (hydroxides and lower oxides) or to kinetic reasons.A consistent picture has been obtained for –rst-row corundum-like com- Me2O3 pounds: the groups with an open and and a closed (Ti2O3, V2O3 Cr2O3) (Fe2O3 , and da shell have distinct magnetic and electronic behaviours. All oxides Co2O3 Ni2O3) of the second group show the antiferromagnetic structure to be more stable than the R3 6 R3c one, while the opposite occurs for This is a successful prediction of the Cr2O3 .experimental results of and and can be accounted for on the basis of the Fe2O3 Cr2O3 , diÜerent bonding geometries of p- and n-type superexchange, which are dominant in the second and –rst group of oxides, respectively.DiÜerent behaviour is also found between the two groups of compounds with respect to the oxygen-to-metal electron transfer : that is of minority-spin type only for Me\Fe, Co, Ni, unlike the case of Ti, V, Cr where the additional presence of majority-spin electron transfer leads to a substantial decrease in the net atomic charge.work was supported by the Human Capital and Mobility Programme of the The European Union under contract CHRX-CT93-0155, and by the Ministero Universita` e Ricerca Scienti–ca e Tecnologica, Roma.M. Catti and G. Sandrone 203 References 1 R. Dovesi, V. R. Saunders, C. Roetti, M. Causa` , N. M. Harrison, R. Orlando and E. Apra` , CRY ST AL 95. Userœs Manual, University of Torino (Italy) and CCLRC Daresbury Laboratory (UK), 1996. 2 S. Geller, Acta Crystallogr. B, 1971, 27, 821. 3 Nouveau de chimie Tome XVII, ed. P. Pascal, Paris, 1963, p. 312. traiteç mineç rale, 4 G. F. Hué ttig and R. Kassler, Z. Anorg. Chem., 1929, 184, 279. 5 G. Natta and M. Strada, Gazz. Chim. Ital., 1928, 58, 422. 6 P. A. Cox, T he T ransition Metal Oxides, Oxford University Press, Oxford, 1992. 7 J. B. Goodenough, Magnetism and the Chemical Bond, Wiley, New York, 1963. 8 M. Catti, G. Valerio and R. Dovesi, Phys. Rev. B, 1995, 51, 7441. 9 M. Catti, G. Sandrone, G. Valerio and R. Dovesi, J. Phys. Chem. Solids, 1996, 57, 1735. 10 L. F. Mattheiss, J. Phys. : Condens. Matter, 1996, 8, 5987. 11 L. F. Mattheiss, J. Phys. : Condens. Matter, 1994, 6, 6477. 12 M. Catti, G. Sandrone and R. Dovesi, Phys. Rev. B, 1997, 55, 16122. 13 R. D. Shannon, Acta Crystallogr. A, 1976, 32, 751. 14 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B, 1992, 46, 6671. 15 W. H. Hehre, L. Radom, P. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital T heory, Wiley, New York, 1986. 16 M. Catti, G. Valerio, R. Dovesi and M. Causa` , Phys. Rev. B, 1994, 49, 14179. 17 B. N. Brockhouse, J. Chem. Phys., 1953, 21, 961. 18 C. G. Shull, W. A. Strauser and E. O. Wollan, Phys. Rev., 1951, 83, 333. 19 L. W. Finger and R. M. Hazen, J. Appl. Phys., 1980, 51, 5362. 20 P. W. Anderson, Phys. Rev., 1959, 115, 2; Solid State Phys., 1963, 14, 99. 21 M. D. Towler, R. Dovesi and V. R. Saunders, Phys. Rev. B, 1995, 52, 10150. 22 Handbook of Chemistry and Physics, ed. R. C. Weast, CRC, Boca Raton, FL, 1987. 23 M. D. Towler, N. L. Allan, N. M. Harrison, V. R. Saunders, W. C. Mackrodt and E. Apra` , Phys. Rev. B, 1994, 50, 5041. 24 J. Zaanen and G. A. Sawatzky, J. Solid State Chem., 1990, 88, 8. 25 K. E. Smith and V. E. Henrich, Phys. Rev. B, 1988, 38, 5965. 26 T. Uozumi, K. Okada, A. Kotani, Y. Tezuka and S. Shin, J. Phys. Soc. Jpn., 1996, 65, 1150. 27 X. Li , L. Liu and V. E. Henrich, Solid State Commun., 1992, 84, 1103. 28 A. Fujimori, M. Saeki, N. Kimizuka, M. Taniguchi and S. Suga, Phys. Rev. B., 1986, 34, 7318. Paper 7/01580A; Received 6th March, 1997

ISSN:1359-6640    

DOI:10.1039/a701580a

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 205»217 First principles calculations on crystalline and liquid iron at Earthœs core conditions Lidunka Vocó adlo,a Gilles A. de Wijs,b Georg Kresse,b,c Mike Gillanb and GeoÜrey D. Pricea a Research School of Geological and Geophysical Sciences, Birkbeck College and University College L ondon, Gower Street, L ondon, UK W C1E 6BT b Physics Department, Keele University, Keele, StaÜordshire, UK ST 5 5BG c Institute for T heoretical Physics, T echnical University of V ienna, W iedner Hauptstrasse 8-10/136, A-1040 V ienna, Austria Ab initio electronic structure calculations, based upon density functional theory within the generalised gradient approximation using ultrasoft nonnorm- conserving Vanderbilt pseudopotentials, have been used to predict the structure and properties of crystalline and liquid iron and solid FeSi at conditions found in the Earthœs core.The quality of the pseudopotentials used was assessed by calculating well documented properties of the solid phase: we have accurately modelled the equation of state of bcc and hcp Fe and FeSi, the bcc]hcp phase transition, the magnetic moment of bcc Fe, the elastic constants of bcc Fe, the bcc]bct distortive phase transition and the phonon frequencies for fcc Fe; the results show good agreement with both theory and experiment. Simulations were also performed on liquid iron and we present the –rst ab initio quantum molecular dynamics calculations on the structure and transport properties of liquid iron under core conditions.Our calculations show that the structure of liquid iron at the conditions to be found in the outer core is highly compressed with a –rst-neighbour coordination number inferred from the radial distribution function of ca. 12. We have also predicted a diÜusion coefficient of 0.5]10~4 cm2 s~1 indicative of a core viscosity of ca. 0.026 Pa s, in line with current estimates. The physical properties of iron and its alloys are of considerable interest to both material scientists and geophysicists.Geologically, the properties of solid and liquid iron are fundamental to our understanding of the behaviour of the Earthœs core, including the generation of the dynamo causing the Earthœs magnetic –eld and the structure of the solid phase in the inner core. The physical properties of solid and liquid iron at these conditions are poorly understood; this is because direct observation of the properties and structure of solid and liquid iron and its alloys is very difficult at the extreme conditions that exist at the Earthœs centre, and the considerable pressure and temperatures involved prohibit de–nitive experimentation. Computer simulations can provide an accurate means of calculating the thermoelastic properties of materials at high pressures (P) and temperatures (T ) via a variety of techniques.Unfortunately, when using many of the computational methods it is often necessary to adopt a number of approximations because of computational limitations. More recently, the growing capacity of high-performance supercomputers has enabled large calculations on increasingly complex materials to be attempted; consequently the study of the full ab initio electronic structure of such phases has now become possible. 205206 First principles calculations on crystalline and liquid iron However, current CPU limitations still make such calculations highly time consuming and expensive. An alternative approach to all-electron calculations is the use of pseudopotentials, where approximations are made to describe the potential associated with the core electrons, and the electron density of only the valence electrons is explicitly calculated ; this results in a signi–cant reduction in the CPU requirements, thereby making the calculations more efficient.This approach is only eÜective if the pseudopotentials used are able to describe the electronic and energetic state of the materials as accurately as the all-electron calculations.In our study of core phases, it is therefore essential to perform test calculations on iron phases with well documented characteristics before embarking on the technically more advanced liquid iron simulations. In this paper we shall –rst review the current status of the experimentally determined iron phase diagram and the complementary computer simulations which have been carried out on this system.We then describe the methodology used in our pseudopotential calculations, and assess the quality of the pseudopotential by calculating some of the static properties of body-centred cubic (bcc), body-centred tetragonal (bct), facecentred cubic (fcc) and hexagonal-close packed (hcp) iron, and FeSi using Vanderbilt non-norm-conserving pseudopotentials within the generalised gradient approximation (GGA).We compare these results with both experiment and theoretical calculations. For these preliminary calculations we then apply this methodology to pure liquid iron, recognising that the outer core is not pure iron but has ca. 10% lighter alloying elements. We present the results from the –rst ever quantum mechanical molecular dynamics simulation of this phase at two high-P,T states representative of the boundaries of the Earthœs outer core with the inner core (ICB) and the lower mantle (CMB). Finally, we discuss the geophysical signi–cance of our initial calculations. Review of past work Experimental data The status of our knowledge of the phase diagram of solid iron up to 1993 has been summarised by Anderson1 and is shown in Fig. 1. Up to ca. 20 GPa and 2000 K the phase diagram is universally accepted. However, Anderson1 has highlighted some of the areas of uncertainty associated with the high-P,T regions of the iron phase diagram where the exact structure is uncertain. Recent studies2,3 have shown a new phase, b, which is suggested to exist near 90 GPa and ca. 2000 K, and another phase, #, inferred to exist above 200 GPa and ca. 4000 K. Experiments suggest that the b phase may have the dhcp (double hexagonal close-packed) structure,2 and the # phase might be a high-T ,P bcc phase;4 the latter is supported by molecular dynamics calculations,5 but is in apparent con—ict with ab initio studies which suggest that a bct phase may be favoured.6 Therefore, to date, there is much uncertainty as to the exact phase stability and structure of high-P,T iron.The equation of state of liquid iron has been recently summarised by Anderson and Ahrens.7 From a thermodynamic analysis of available ultrasonic, pulse heating and shockwave compressional data, they provide a self-consistent dataset to 10 Mbar.From this, they calculate thermodynamic properties (o, K, V ) of pure liquid Fe along isentropes associated with the ICB temperatures of 5000, 6000, 7000 and 8000 K. The viscosity of the Earthœs liquid outer core is probably close to that of pure iron at ambient pressure and has been discussed by Poirier,8 who has analysed the transport properties of liquid metals and suggests, on the basis of empirical systematics, that the viscosity in this region takes a value of ca. 0.06 Pa s.L . V ocó adlo et al. 207 Fig. 1 Current understanding of the high-pressure phase diagram of iron ; the well established low-pressure bcc, fcc and hcp phases and the uncon–rmed possible dhcp and bcc»bct hightemperature phases Simulations Despite recent experimental progress in determining the existence of uncharacterised high-P,T phases of iron, the exact nature of their structure is still unknown. However, we can further our understanding of the structural and thermoelastic properties of solid iron phases through computer modelling.Using simulation techniques, such as pair potentials, all-electron calculations and tight-binding methods, it is possible to obtain the equation of state (EOS) as a function of pressure and temperature in an attempt to constrain the possible candidates for these high-P,T phases.If a suitable pair-potential can be found, then a study of the high-P,T behaviour of Fe would be relatively straightforward. Matsui5 used a potential model –rst described by Ross et al.9 to model the EOS of the hcp and bcc phases.This simulation reproduced the EOS of hcp Fe very well ; however, the problem with this type of study is that the transferability of the model potential used to diÜerent polymorphs of iron has not yet been proven. Therefore this work suÜers, as do other calculations based upon empirical potentials, from the lack of reliability that can be placed on using potentials beyond the range of empirical –tting and, therefore, the con–dence with which one can predict properties outside the experimental range.As a result, ab initio methods must be adopted. Some of the –rst ab initio full linear augmented plane wave (FLAPW) calculations on iron were carried out by Jansen et al.10,11 They concluded that their results depended critically upon the nature of the non-local corrections to the exchange and correlation, i.e., use of the local density approximation (LDA) and local spin density approximation (LSDA) were insufficient to describe the electron density and hence structure and magnetic properties of iron.A signi–cant improvement in this area was reported by Stixrude et al.12 where they showed that, by using the GGA13 in the LAPW calculations, they obtained excellent agreement with measured EOS of bcc and hcp Fe to [350 GPa.In addition, they reproduced the bcc»hcp phase transition and the bcc magnetic moment. This work was later elaborated upon by Stixrude and Cohen6 who also reported the behaviour of fcc Fe and showed that the bcc structure becomes mechanically unstable208 First principles calculations on crystalline and liquid iron with respect to tetragonal strain at 150 GPa undergoing a distortive phase transition from bcc]bct.Thus far, iron had been studied using all-electron calculations, which are intrinsically CPU intensive. To study energy»geometry space more eÜectively, a more approximate wavefunction is desirable. Hence, in 1995, Sasaki et al.14 reported the use of pseudopotential calculations combined with LSDA for bcc iron and fcc nickel.They showed that these calculations reproduced structural and magnetic properties, as well as the all-electron calculations within the LSDA. However, although these calculations showed how the pseudopotential method could give results comparable with FLAPW, they con- –rmed the view, mentioned earlier,11 that the use of the LSDA in these calculations does not give results in agreement with observation and GGA calculations.More recently, the high-pressure phases of iron have been investigated by Soé derlind et al.15 using full-potential linear muffin tin orbitals (FP-LMTO) within the GGA. They con–rmed previous theoretical calculations and obtained the magnetic states and EOS of the bcc, fcc, bct, hcp and dhcp polymorphs.They predict a bcc»hcp transition at 10 GPa, a metastable dhcp phase which might be accessible at high T up to 50 GPa, and con–rm the mechanical instability of bcc above 100 GPa. They then went on to predict the elastic constants and phonon frequencies of fcc and hcp iron, using ab initio calculations within the LDA, at three volumes for both high (400 GPa) and low pressures. In addition, they –tted a many-body interaction potential to their ab initio calculations and used that to calculate the phonon spectra of fcc and hcp Fe.To date, no ab initio studies of liquid iron have been reported, although ab initio studies of liquid metals in general are now becoming possible using Car»Parrinello methods.16 The use of ultrasoft pseudopotentials has great advantages in modelling the dynamics of transition metals as shown by Pasquarello et al.17 and Kresse and Hafner.18 In the following section we outline the methodology adopted in our study of solid and liquid iron.Calculation methodology and pseudopotentials The simulations presented here have been carried out using –rst-principles density functional theory.The valence orbitals were expanded in a basis of plane waves, and the interaction between the core and the valence electrons was described by means of pseudopotentials. Until recently most pseudopotentials have been norm-conserving.19 The pseudopotential concept requires that the scattering properties of the pseudo-atom and of the exact atom are the same at a speci–ed matching radius, Inside the matching Rmatch .radius the pseudo-wavefunctions are nodeless and only approximate. The matching radius controls the overall accuracy and ì transferability œ of the pseudopotential. It was shown by Hamann et al.19 that a good description of the scattering properties requires that the pseudo-wavefunction ful–ls a norm-conservation condition : /lm PP P0 Rmatch (o/lm exact(r) o2[o/lm PP(r) o2) d3r\0 (1) where is the exact all-electron wavefunction, and l,m are the angular momentum /lm exact quantum numbers.This restriction poses a serious problem, especially for –rst row transition metals, because it forces the pseudo-wavefunction to vary rapidly. The only compromise one can make is to increase the matching radius, which makes the pseudo-wavefunction ì softer œ (i.e., a smaller basis set might be used), but this always decreases the quality of the pseudopotential.In general, the minimum basis set size for transition metals is around 500 plane waves per atom, making calculations very expensive.L . V ocó adlo et al. 209 A solution to this problem was proposed by Vanderbilt20 who showed that it is possible to give up the norm-conservation condition if one corrects for the resulting diÜerence between the exact and the pseudo-charge density [i.e., the terms between the brackets of eqn.(1)].This is achieved using localised augmentation functions centred on each atom. Giving up the norm-conservation constraint results in signi–cantly smoother pseudo-wavefunctions.The new ultrasoft pseudopotentials give results which are very close to, or even indistinguishable from, results obtained with the best all-electron –rst principles methods currently available. Details of the construction of our pseudopotentials can be found in Kresse and Hafner21 and speci–c information about Fe pseudopotentials from Moroni et al.22 Another point which requires some care is the description of the exchangecorrelation energy.As discussed above, it is now well known that GGA must be used in order to get an accurate description of the ground-state properties of Fe at ambient pressure.23 It has also been shown that the neglect of GGA strongly aÜects results at high pressure.15 Here we have used a GGA based on the LDA of Ceperley and Alder24 and the gradient corrections according to Perdew et al.13 All the pseudopotentials have been constructed with non-linear partial core corrections.25 Another advantage of ultrasoft pseudopotentials is that it is possible, at least in principle, to describe lower lying ìpseudo-coreœ states as valence states.In the case of Fe, treating 3p states as valence turned out to be vital to get an accurate description of the pressure at conditions found in the Earthœs core.However, the description of the highly localised 3p states as valence states is relatively expensive and such a description is therefore unsuitable for the more time consuming calculations on the liquid. This problem can be overcome since the eÜect of the 3p states can be described by a pairwise additive repulsive potential (similar to the approach Ballone and Galli26).Therefore, to decrease the computational costs of the liquid state simulations, we have decided to keep the 3p states in the core and to describe their eÜect with this repulsive pair potential. The pseudopotential calculations presented below have been done with VASP (Vienna ab initio simulation package).In VASP the ground state is calculated exactly for each set of ionic positions using an efficient iterative matrix diagonalisation scheme and a Pulay mixer.27 A smearing method was used to avoid problems with level crossing, and the electronic free energy was taken as variational quantity. More details concerning the computational method can be found in Kresse and Furthmué ller.28,29 The results for the solid Fe/FeSi calculations described in the following section were obtained using the GGA pseudopotential with the 3p electrons treated explicitly as valence electrons, i.e., with an [Ne]3s2 core.The liquid calculations employed the pseudopotential with [Ar] core together with the pair potential correction. The pseudopotentials are labelled according to their frozen core con–guration, i.e., [Ne]3s2 and [Ar].Results Solid iron and FeSi To con–rm fully the validity of the [Ne]3s2 pseudopotential, calculations were performed on the bcc, hcp, fcc and bct structures of iron, and also on FeSi. Energy convergence as a function of k-point sampling was investigated, and all calculations reported converged to 0.01 eV in total energy per atom; as a result the calculations were performed with wavefunctions being sampled at 35 k-points in the irreducible Brillouin zone for bcc Fe and bct Fe, and 36 for hcp Fe, 344»567 for the fcc Fe phonon calculations and 11 for FeSi.210 First principles calculations on crystalline and liquid iron Fig. 2 Calculated pressure»volume curve at T \0 K for bcc Fe and hcp Fe.The bcc Fe was simulated with the [Ne]3s2 pseudopotential ; the hcp Fe with both the [Ne]3s2 and the [Ar] pseudopotentials. The hcp is compared with theory15 and experiment.30 Fig. 2 shows the equation of state for both bcc and hcp Fe using both the [Ar] and the [Ne]3s2 pseudopotential. The optimal c/a ratio for hcp Fe was found to be to 1.575 over a range of volumes between 14 and 24 The results show excellent agreement Aé 3.with both theory15 and experiment,30 thus showing that the degree of precision one can get from ultrasoft pseudopotentials is more than satisfactory. Table 1 shows the results of –tting these data to a second-order Birch»Murnaghan equation, using K@\4 to enable comparison with theory and experiment. The transition pressure for bcc]hcp was obtained from the enthalpy»pressure curves of the two phases and is predicited to occur at ca. 10 GPa, in good agreement with the FP-LMTO calculations15 and close to the experimental value of 10»15 GPa. Until recently, the use of the LDA to predict the magnetic properties of the ground state of iron incorrectly destabilised bcc Fe structure in favour of fcc Fe.12 However, incorporating the GGA into the calculations stabilises the observed bcc structure and predicts a magnetic moment, k, for bcc Fe which is in excellent agreement with the Table 1 Atomic volumes and incompressibilities of bcc and hcp iron, and the magnetic moment of the ferromagnetic bcc phase; comparisons with all-electron calculations and experiment FLAPW FLAPW VASP VASP LSDAa GGAa VASP GGA GGA K@D4 K@D4 LSDAb [Ar]b [Ne]3s2 b expt.c bcc V0/Aé 10.45 11.4 10.5 11.7 11.55 11.80 K/GPa 245 189 235 156 176 162»176 k/atom 2.04 2.17 2.05 2.32 2.25 2.12d hcp V0/Aé 9.6 10.2 9.6 10.3 10.4 11.2 K/GPa 344 291 309 270 290 208 a Ref. 12. b This study. c Ref. 31. d Ref. 32.L . V ocó adlo et al. 211 ambient experimental value (Table 1) ; we also predict a suppression of the magnetic moment with pressure from 2.25 at V \11.55 to 0.9 at V \7 in very good Aé 3 Aé 3 agreement with the FP-LMTO calculations.15 The elastic constants for fcc and hcp iron have been calculated theoretically by Stixrude and Cohen6 and by Soé derlind et al.,15 the former using a tight binding –t to ab initio GGA and the latter using ab initio LDA.In order to con–rm further the validity of the ab initio pseudopotential GGA method with the [Ne]3s2 core, we have obtained the elastic constants for bcc Fe. The elastic constants (Table 2) were calculated for bcc Fe via very small distortions (\3%) of the ideal bcc structure in order to keep within the harmonic approximation.The incompressibility, given by was calcu- K\13 (c11]2c12), lated from a Birch»Murnaghan –t to compressional data; the and elastic c11, c12 c44 constants were obtained from a second-order polynomial –t through the energy» distortion curve using the following elastic strain matrices : for the tetragonal distortion : a1]d 0 0 0 1]d 0 0 0 1 (1]d)2b (2) for the shear distortion : a1 d 0 d 1 0 0 0 1 1[d2b (3) and the elastic constants are given by: AL2U Ld2 Btetr \6c@\6 2 (c11[c12) (4) AL2U Ld2 Bshear \4c44 (5) where U is the energy per unit volume.At ambient conditions, the bcc structure is the most stable ; however, tetragonal distortions in this structure destabilise this idealised c/a\1 structure under compression, therefore making the bct structure a possible candidate for the Earthœs core. Fig. 3 shows how the c/a ratio varies as a function of compression; the ideal bcc structure is the most stable close to but as the volume is reduced, the structure undergoes a V0 , transformation through a metastable state at V \8.5 to a bct structure with a c/a Aé 3 Table 2 Elastic constants of the bcc Fe phase; pseudopotential calculations and experiment calculations experimenta experimentb K/GPa 176 173 c11/GPa 289 243 226»232 c12/GPa 118 138 130»136 c44/GPa 115 122 116»117 a Ref. 33. b Ref. 34.212 First principles calculations on crystalline and liquid iron Fig. 3 Calculated energies of bct Fe as a function of compression where c/a\1 is the special case of the bcc Fe structure ratio of 0.87»0.88 at ca. 150 GPa, in excellent agreement with the all-electron calculations. 6,15 To illustrate further the ability of the [Ne]3s2 pseudopotential to model iron phases accurately, we have performed calculations on fcc Fe.Table 3 lists the phonon frequencies of several fcc zone-boundary phonons at a strongly compressed volume. Once more the pseudopotential is in good agreement with the FP-LMTO results.15 Table 3 also illustrates how the pair-potential correction to the pseudopotential with the [Ar] core con–guration accounts well for the change in phonon frequencies due to the compression of the 3p electrons.Silicon is thought to be a possible alloying element in the outer core so, for our –nal test on core forming phases, we have performed compression calculations on FeSi. Fig. 4 shows the predicted EOS for FeSi, which is in excellent agreement with experimental data. A third-order Birch»Murnaghan –t over the full PV range gives predicted EOS parameters in good agreement with highest pressure experiments (Table 4).Liquid iron Having established that the pseudopotential method works extremely well for solid iron phases under compression, we now have sufficient con–dence in our pseudopotentials to Table 3 Phonon angular frequencies (1012 rad s~1) for non-magnetic fcc Fe with volume\6.17 atom~1 Aé 3 LDAa LDA GGA GGA GGA FP-LMTO [Ne]3s2 [Ne]3s2 [Ar] and corr.[Ar] L[100] 123 125 127 127 114 T[100] 85.6 84.9 88.7 86.4 78.4 L[111] 138 142 145 142 130 T[111] 60.1 59.6 60.8 59.8 55.1 a Ref. 15.L . V ocó adlo et al. 213 Fig. 4 Calculated pressure»volume curve for FeSi at T \0 K compared with experiment35,36 perform simulations on liquid iron. First principles molecular dynamics simulations of non-magnetic liquid Fe were carried out at two selected thermodynamic state points representative of the ICB and the CMB.These were selected to lie on the 6000 K adiabat, which is probably representative of the temperature at the ICB7 (see Table 5). From experimental observations and the inferred EOS of liquid Fe along the 6000 K adiabat the pressures, (where the subscript refers to the 6000 K adiabat) can be P6000 determined for these state points,7 which we can compare with the pressures obtained from our simulations, Pcalc .We have employed a 64 atoms supercell with periodic boundary conditions, sampling wavefunctions at the C-point of the Brillouin zone only. The pseudopotential with an [Ar] core was used together with the pair potential correction for the additional repulsion due to compression of the 3p electrons.We used an integration time step of 1 fs and the ionic temperature was controlled with the Noseç scheme.37 The simulations were started from a liquid sample obtained in previous simulations at ambient pressure with the positions rescaled to the required density.The ICB and CMB liquids were equilibrated for 0.7 and 1.2 ps, respectively, and statistics were gathered from 1.7 and 0.7 ps runs. The quality of the calculations was assessed by studying the total energy conservation which showed the drift corresponding to a temperature change of 40 K ps~1 for Table 4 Birch»Murnaghan equation of state parameters for FeSi; pseudopotentials calculations, time-of-—ight neutron diÜractiona and X-ray diÜraction experimentsb Knittle and Wood et al.b Williamsc PP[Ne]3s2 V0/Aé 90.25 89.02 89.92 K/GPa 147 209 218 K@ 7.7 3.5 4.2 a Ref. 34.b Ref. 35. c Ref. 36.214 First principles calculations on crystalline and liquid iron Table 5 Temperature (T ) and density (o) at which simulations were carried out, the corresponding pressure the simulated pres- (P6000),7 sure and the self-diÜusion coefficient (D) (Pcalc) as obtained from the simulations of liquid Fe at the state points representative of the ICB and the CMB ICB CMB T /K 6000 4300 o/g cm~3 13.3 10.7 P6000/Mbar 3.30 1.35 PU/Mbar 3.05 1.10 Ppair/Mbar 0.38 0.13 Pideal gas/Mbar 0.12 0.07 Pcorr/Mbar 0.03 0.02 Psim/Mbar 3.58 1.32 D/]10~4 cm2 s~1 0.4»0.5 0.5 the ICB simulations, but was less stable for the CMB simulations where the drift was ca. 100 K ps~1. The resulting simulated pressure was calculated as the sum of four contributions : Pcalc\PU]Ppair]Pideal gas]Pcorr (6) is the potential-energy contribution, which is the only term aÜected by k-point sam- PU pling. To check whether there might be an eÜect due to the limited k-point sampling we calculated the pressure for a (ICB) con–guration with four k-points instead of the C-point only.This resulted in a change of of less than 10 kbar and the electronic PU density of states was found to be hardly aÜected, therefore validating the use of our restricted sampling. is the contribution caused by the pair potential that mimics the Ppair correction originating from the compression of the 3p electrons.the kinetic Pideal gas , contribution, is the ideal gas pressure at the temperature and density of our calculation. is a correction to due to the incompleteness of our plane-wave basis set and it Pcorr PU was obtained from calculations on hcp Fe, and is found to be very small (34 kbar for ICB, 23 kbar for CMB). The pressure, and its breakdown into various contribu- Pcalc tions are listed in Table 5.The calculated pressures agree within less than 10% with the values expected from the EOS of Anderson and Ahrens,7 which gives us con- P6000 –dence that our simulations are capable of describing liquid Fe at Earthœs core conditions. For crystalline solids the highest coordination, 12, arises for atoms in closely packed arrangements, like hcp and fcc.In liquids, the coordination number cannot be de–ned completely unambigously, but a reliable number can be obtained by integration of the pair distribution, g(r), to the minimum beyond the –rst peak. Fig. 5 shows the g values for both high-P,T simulations ; integration to the –rst minimum yields coordination numbers of 13.8 and 13.5 for ICB and CMB, respectively, indicative of a very closepacked liquid.In the liquid phase, atoms exhibit a diÜusive behaviour which can be characterised by a self-diÜusion coefficient, D. This coefficient is most easily calculated from the mean squared displacement (MSD) via the Einstein relation : SoR(t)[R(0) o2T\6Dt for t]O where R stands for an atomic position and the brackets denote a thermal average.After a sufficiently long lapse of time the MSD should increase linearly with time, allowing DL . V ocó adlo et al. 215 Fig. 5 Radial distribution functions obtained from simulations at the selected ICB and CMB conditions (full and dashed curves, respectively) to be calculated. Fig. 6 shows the MSD for both the ICB and CMB runs. The diÜusive behaviour and linear increase with time are obvious.The diÜusion coefficients extracted are listed in Table 5 and are of the same order of magnitude as those of many liquid metals at ambient pressure. As discussed by Poirier,8 a possible explanation for this is that, for liquid metals under compression, the activation energy for diÜusion is a constant close to the melting curve and so the viscosity at high pressure might be expected to be similar to that at ambient pressure.Fig. 6 Mean-squared displacements obtained from simulations at the selected ICB and CMB conditions (full and dashed curves, respectively)216 First principles calculations on crystalline and liquid iron Using these results we may make an estimate for the viscosity of liquid iron at core conditions via the Stokes»Einstein relation : Dg\ kB T 2na where D is the diÜusion coefficient, g is the viscosity coefficient, is Boltzmannœs con- kB stant, T the temperature and a is an atomic size parameter.With DB0.4»0.5]10~4 cm2 s~1, T B6000 K and aB1 this leads to a value for the viscosity of gB0.026» Aé , 0.03 Pa s, which is not dissimilar to the estimates8 based upon empirical systematics for the viscosity of the outer core of 0.06 Pa s.Conclusions First-principle calculations using the non-norm-conserving ultrasoft pseudopotential method within the GGA have proven to be very successful at predicting the static properties and equations of state of solid iron and FeSi. We have used this method to predict the structure and properties of liquid iron. Our calculations show that, at the conditions of the outer core, liquid iron is very close packed with a coordination number of P12.First estimates of the viscosity of liquid iron under these conditions give a value of 0.16»0.2 Pa s, close to the expected value for the outer core of ca. 0.06 Pa s. These are the –rst ab initio calculations on liquid iron and our preliminary investigations have led to excellent results.There is much scope for using the ultrasoft pseudopotential methodology to simulate both solid and liquid phases in the future, and to predict ultimately the full phase diagram and melting curve of iron and its alloys. We are currently focusing on simulating the high-temperature properties of crystalline iron using the frozen-phonons approach, and on performing further calculations and analysis of the melt structure of iron at core conditions.References 1 O. L. Anderson, AIP Conference Proceedings, American Institute of Physics, New York, 1993. 2 S. K. Saxena, L. S. Dubrovinsky and P. Haé ggkvist, Geophys. Res. L ett., 1996, 23, 2441. 3 R. Boehler, Nature (L ondon), 1993, 363, 534. 4 W. A. Bassett and M. S. Weathers, J. Geophys.Res., 1990, 95, 21709. 5 M. Matsui, AIP Conference Proceedings, American Institute of Physics, New York, 1993. 6 L. Stixrude and R. E. Cohen, Geophys. Res. L ett., 1995, 22, 125. 7 W. W. Anderson and T. J. Ahrens, J. Geophys. Res., 1994, 99, 4274. 8 J. P. Poirier, Geophys. J., 1988, 92, 99. 9 M. Ross, D. A. Young and R. Grover, J. Geophys. Res., 1990, 95, 21713. 10 H. J. F. Jansen, K.B. Hathaway and A. J. Freeman, Phys. Rev. B, 1984, 30, 6177. 11 H. J. F. Jansen, K. B. Hathaway and A. J. Freeman, Phys. Rev. B, 1985, 31, 7603. 12 L. Stixrude, R. E. Cohen and D. J. Singh, Phys. Rev. B, 1994, 50, 6442. 13 J. P. Perdew, J. A. Chervary, S. H. Voska, K. A. Jackson, M. R. Perderson, D. J. Singh and C. Fiolhais, Phys. Rev. B, 1992, 46, 6671. 14 T. Sasaki, A. M.Rappe and S. G. Louie, Phys. Rev. B, 1995, 52, 12760. 15 P. Soé derlind, J. A. Moriarty and J. M. Wills, Phys. Rev. B, 1996, 53, 14063. 16 R. Car and M. Parrinello, Phys. Rev. L ett., 1985, 55, 1471. 17 A. Pasquarello, K. Laasonen, R. Car, C. Lee and D. Vanderbilt, Phys. Rev. L ett., 1992, 69, 1982. 18 G. Kresse and J. Hafner, Phys. Rev. B, 1993, 48, 13115. 19 D. R. Hamann, M. Schlué ter and C. Chiang, Phys. Rev. L ett., 1979, 43, 1494; G. B. Bachelet, D. R. Hamann and M. Schlué ter, Phys. Rev. B, 1982, 26, 4199. 20 D. Vanderbilt, Phys. Rev. B, 1990, 41, 7892. 21 G. Kresse and J. Hafner, J. Phys. : Condens. Matter, 1994, 6, 8245. 22 E. Moroni, G. Kresse, J. Furthmué ller and J. Jafner, Phys. Rev. B., 1997, submitted. 23 D. J. Singh, W. E. Pickett and H. Krakauer, Phys. Rev. B, 1991, 43, 11628. 24 D. M. Ceperley and B. J. Alder, Phys. Rev. L ett., 1980, 45, 566; we use the parametrization by J. P. Perdew and A. Zunger, Phys. Rev. B, 1981, 23, 5048.L . V ocó adlo et al. 217 25 S. G. Louie, S. Froyen and M. L. Cohen, Phys. Rev. B, 1982, 26, 1738. 26 P. Ballone and G. Galli, Phys. Rev. B, 1989, 40, 8563. 27 P. Pulay, Chem. Phys. L ett., 1980, 73, 393. 28 G. Kresse and J. Furthmué ller, Comput. Mater. Sci., 1996, 6, 15. 29 G. Kresse and J. Furthmué ller, Phys. Rev. B, 1996, 54, 11169. 30 H. K. Mao, Y. Wu, L. C. Chen and J. F. Shu, J. Geophys. Res., 1990, 95, 21737. 31 E. Knittle, in Mineral Physics and Crystallography : A Handbook of Physical Constants, ed. T. J. Ahrens, American Geophysical Union, Washington, 1995, p. 131. 32 G. G. Lonzarich, Electrons at the Fermi Surface, ed. M. Springford, Cambridge University Press, Cambridge, 1980, p. 225. 33 G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties : A Handbook, MIT Press, Cambridge, MA, 1971. 34 D. G. Isaak and K. Masuda, J. Geophys. Res., 1995, 100, 17689. 35 I. G. Wood, T. D. Chaplin, W. I. F. David, S. Hull, G. D. Price and J. N. Street, J. Phys. : Condens. Matter, 1995, 7, L475. 36 E. Knittle and Q. Williams, Geophys. Res. L ett., 1995, 22, 445. 37 S. Noseç , J. Chem. Phys., 1984, 81, 511. Paper 7/01628J; Received 7th March, 1997

ISSN:1359-6640    

DOI:10.1039/a701628j

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 219»232 Semi-empirical methods as a tool in solid-state chemistry Julian D. Gale Department of Chemistry, Imperial College of Science, T echnology and Medicine, South Kensington, L ondon, UK SW 7 2AY A new method for performing periodic calculations using the MNDO, AM1 and PM3 semi-empirical methods, with treatment of the long range interactions by lattice summation techniques, has been devised and implemented.The performance of the existing parametrisations are examined for the bulk and surface properties of alumina, the prediction of the relative stability of silica polymorphs, and for the modelling of ice crystals. While the results for the more ionic materials suggest that some reparametrisation will be needed for solid-state problems, the method shows promise for molecular crystals, particularly those where hydrogen bonding is the dominant interaction.Semi-empirical methods have a long and useful history in solid-state chemistry, spanning several decades.1h4 Indeed, approximate molecular orbital calculations laid the foundation for subsequent ab initio methods at a time when computing resources did not permit more accurate techniques to be applied.5 As computers become ever more powerful, we may wonder if there is any need to pursue semi-empirical methods.For now it is, arguably, worthwhile, as there are always problems which are still too large for –rst principles techniques or situations where qualitative answers are needed on a short timescale. From a chemistœs point of view, three semi-empirical methods have been predominantly applied to solid-state problems: extended Huckel theory,6 CNDO (complete neglect of diÜerential overlap)7 and INDO (intermediate neglect of diÜerential overlap).8 Despite the dramatic simpli–cations that are made to the Hamiltonian, these approaches have been successful in tackling many problems in oxide and halide chemistry.However, in many cases this has only been achieved by tuning some of the adjustable parameters for speci–c materials, thus reducing transferability.Given that –tting can often be time consuming in its own right, this surrenders some of the advantage gained over more accurate methods. Ideally, we would like to use a method which avoids the parametrisation of pair-wise interactions and maximises transferability.Currently, there is an increasing interest in being able to simulate molecular crystals, often involving combinations of both the organic and inorganic components. Unfortunately, the above three semi-empirical methods will do far worse, on the whole, when it comes to reproducing molecular crystal structures as opposed to simple ionic materials. Hence, some development is needed if approximate quantum mechanical techniques are to be applied more generally to solids. If we study the literature within the organic community then this –eld is dominated by methods based on the modi–ed neglect of diÜerential diatomic overlap (MNDO).9 In addition to the original formulation, there have been some modi–cations of the core» core repulsions which have led to two subsequent reparametrisations, AM110 and 219220 Semi-empirical methods in solid-state chemistry PM3.11 These methods are generally superior in the reproduction of molecular properties and have a number of desirable features, such as the ability to model the stereochemical activity of lone pairs, as demonstrated by calculations on the water dimer.12 A desirable attribute of the MNDO scheme and its subsequent modi–cations is that the parameters are all based around one-centre terms and, therefore, there is no need to reparametrise the method for each pair of elements, unlike in the CNDO/INDO formalisms.Of course it is preferable to include compounds which involve as many diÜerent combinations of elements as possible in the original database for –tting.However, it does not exclude the study of compounds where there is no existing experimental data for a particular pair of elements. Because of the larger number of parameters involved in the MNDO scheme, parametrisation is also a more complex aÜair, requiring a sizeable body of experimental or theoretical data as a starting point. Consequently, for many years, there has been only a limited range of elements for which parameters were derived and this has tended to centre around organic compounds and non-metallic elements where there is no shortage of molecular data.Recently, however, there has been an increasing amount of work on extending the parametrisation to main group metallic elements.13,14 With the extension of the theory to include d orbitals15,16 it has now also become possible to tackle transition-metal systems.Part of the reason for the initial limited parametrisation lies in the lack of accurate gas-phase data for many metals. However, if it were possible to perform calculations on solids as well as molecules, within the same approach, then a whole new database of information would become accessible. The aim of this work is explore the extension of the popular MNDO family of methods to the solid state.This will provide a serious test of the underlying approximations, to see if the parameters determined in the gas phase will allow the successful extrapolation of properties to periodic systems or whether some modi–cation is necessary. Method Within the NDDO approximation, all integrals are retained provided they involve pairs of orbitals on the same atom. Hence, all three- and four-centre integrals are neglected, but the majority of the largest two-centre integrals are retained.The elements of the Fock matrix are thus given by the following expressions ;9 Fkk\Ukk];B Vkk, B];l A PllC(kk, ll)[1 2 (kl, kl)D];B ; j, p B Pjp(kk, jp) Fkl\;B Vkl, B]1 2 Pkl[3(kl, kl)[(kk, ll)]];B ; j, p B Pjp(kl, jp) Fkj\bkj[1 2 ;l A ;p B Plp(kl, jp) where k and l are orbitals on atom A and j and p are orbitals on atom B.If we consider the application of the above expressions to a periodic system we need to handle correctly the two-centre terms in the long-range limit. Taking –rst the terms that arise in the oÜ-diagonal blocks between two distinct atoms, the two-centre Fkj , one-electron core resonance integrals are calculated from the overlap integral multi- Fkj plied by a combination of one-centre terms: bkj\12 (bk]bj)SkjJ. D.Gale 221 The overlap integrals are calculated analytically for Slater orbitals and, therefore, this term decays exponentially in the long-range limit and can be readily converged in real space within ca.depending on the exponents. 10 ”, The two-centre exchange term that appears in the oÜ-diagonal blocks should only be calculated for one pair of atoms out of each set of images, as the density matrix element will be diÜerent for each pair, yet only one can be included in the equations as given above. Hence, this term must be truncated at half the unit cell length in each direction to satisfy this requirement. This restriction on the range of the two-centre exchange term can be overcome either by integration across the Brillouin zone17 or through the large unit cell (LUC) approach.1 Since any semi-empirical calculations are likely to target systems too large to tackle in the required time by ab initio methods then, initially, we have implemented the method for the C-point only and supercells are created as necessary to examine the convergence of the electronic properties according to the LUC method.For the on-diagonal blocks, the sum of the two-centre one-electron core attraction and two-electron repulsion integrals represents the electrostatic-potential contribution. These contributions are slow to decay and cannot be truncated readily as, in the longrange limit, it takes the form of a Coulomb interaction.Consequently, care must be taken to evaluate these terms using a suitable lattice sum. In an earlier attempt to produce a solid-state program for MNDO calculations, called MOSOL,18 these terms were handled by summing over complete unit cells. While this approach of the Evjen type may lead to approximate convergence it has a number of undesirable characteristics.One of the primary difficulties is that atoms which are equivalent by symmetry may not end up with the same charge distribution, as they may be in diÜerent positions relative to the cut-oÜ boundary. Also, difficulties will be encountered if the unit cell has a non-zero dipole moment. In order to discuss how these terms can be handled in the long-range limit it is necessary to describe how the integrals are evaluated in the MNDO scheme.Rather than using exact integrals based on the Slater orbitals, which is expensive, the integrals are approximated by a set of multipole»multipole interactions : (kl, jp)\;l 1 ;l 2 ;m [Ml1m A , Ml2m B ] For an sp basis, all charge distributions can be represented by combinations of suitable monopoles, dipoles and quadrupoles, where each multipole moment is generated by a series of point charges distributed about the atomic centre.Full details of these con–gurations can be found elsewhere.9 The interactions between the point charges are evaluated using the Dewar»Sabelli» Klopman (DSK) formula: I(rAB)\ 1 [rAB 2 ](ol1A]ol2 B )2]1@2 This has the correct behaviour, in that it tends to a pure Coulomb potential in the long-range limit and, by suitable choice of the o parameters, can be made to tend to the one-centre limit as well.At intermediate distances, the magnitudes of integrals calculated in this fashion are lower than for the exact expression, which re—ects the empirical inclusion of correlation eÜects.The DSK formula for the integrals can be directly manipulated into a long range sum by performing a Laplace transform in the same way that is used for the Coulomb operator in the Ewald sum.19 Unfortunately, the outcome of the Laplace transform is a Bessel function which leads to series expansions in both real and reciprocal space which are not guaranteed to be rapidly convergent.A simpler approach to the problem is to222 Semi-empirical methods in solid-state chemistry start from a series expansion in the limit of large distance : I\ 1 (r2]a2)1@2 ; where a\(ol1A]ol2 B ) For rAa; IB1 r[ a2 2r3]3a4 8r5[ 5a6 16r7]O(r9) Based on the this series expansion, we can calculate the integrals in the limit of large distance, using the more straightforward transforms of inverse powers of the distance.The leading term is just the standard Ewald summation, while the term involving the third power of inverse distance is given below (higher-order terms can be found elsewhere) :20 V (3, r)\1 2 ;i ;j @Cij ;d 1 r3 Cerfc(g1@2r)]2Ag nB1@2r exp([gr2)D ]1 2 A2n V B;G ;i ;j Cij exp(iGr)E1AG2 4gB[1 2A4n 3 BAg nB3@2 ;i Cii Unfortunately, the lattice sum for the inverse third power of the distance is still a conditionally convergent series.Although there is no restriction that the coefficients must be separable into one-centre terms which sum to zero, as is the case for charges in the Ewald sum, there is the condition : ;i ;j Cij\0 In this case the coefficients are functions of the a parameters from the DSK equation and the charges on the atoms: Cij\aij 2 qi qj\(oi]oj)2qi qj K; i oi qi\0 There is no unique solution to the above condition. However, since the sum of the charges must already equal zero, then a straightforward way of satisfying this equality is to make o a constant in the long-range limit.This can be achieved by averaging the values over all atoms within the unit cell and corresponds to a mean-–eld approach to the long-range potential.By making this choice, the potential tends smoothly to the proper limit as the cut-oÜ between the explicit integral calculation and the lattice sums increases. A further crucial step in the evaluation of the long-range potential is to use point multipoles for each atom, rather than the explicit point charge con–gurations. The multipole»multipole interactions can then be determined by taking the –rst and second derivatives of the potential about each atomic site.The strategy for integral evaluation then becomes: to calculate the lattice sums using the series expansion and, subsequently, to subtract the terms for distances, where the integrals are calculated explicitly. Hence, the accuracy with which the true MNDO integrals are evaluated can be controlled by a single-distance cut-oÜ.This cut-oÜ needs to be sufficiently large so that only a few of the leading terms of the series expansion are needed and so that the point multipole approximation is valid. For typical values of aij , all terms in the series expansion beyond the inverse –fth power of distance have become negligible within to an accuracy of eight signi–cant –gures.Hence, we have only 10 ”, found it necessary to implement the –rst three terms of this series.J. D. Gale 223 As a simpli–cation, it is generally the case that, when the above criteria are satis–ed, the contribution of the quadrupoles to the long-range potential is small, at the level of the accuracy of the method, and their contribution to the forces is even smaller.Hence, initially, we have only fully implemented the method for terms up to and including the dipole»dipole interaction. Quadrupolar terms can be included for the energy, but not yet for the derivatives. The number of integrals generated within the MNDO-type methods can be large for a semi-empirical method and so it is not always feasible to store them in core.When memory is limited then it is found to be signi–cantly faster to use direct SCF rather than store the integrals on disk, given the simplicity of the integral evaluation compared to the equivalent in an ab initio calculation. Analytical –rst derivatives have been derived for this periodic implementation of the NDDO Hamiltonians, including internal and strain derivatives. Unlike the standard molecular code, the overlap integral derivatives are determined directly from the Slater basis functions, rather than approximating them with an STO-type Gaussian combination.Because second derivatives are not yet included, it is important to ensure a continuous energy surface, if geometry optimisation is to be performed readily with only –rst derivatives. To this end, the transition between the region for exact integral calculation and the long-range lattice sum approximation is smoothed by the use of a –fthorder polynomial taper function.All calculations in this work have been performed with a cut-oÜ of for the exact calculation of integrals and tapering over a range of 12 ” 2 ”. Increasing the cut-oÜ beyond this leads to a small change in the absolute energies but an almost negligible change in relative energies, with which we are normally concerned.The parameters for all methods used are the standard ones derived from the geometries, enthalpies of formation, dipole moments and other properties of gas-phase species. As yet, we have made no attempt to reparametrise for the solid state and one of the issues to be examined in this work is whether this is necessary or not.Based on the above formulation we will now examine a number of problems in the solid state to investigate the utility of the method. Results and Discussion Bulk and surface properties of corundum There are two prototypical oxides which have been studied heavily using theoretical methods: magnesium oxide and alumina.Both have been examined with periodic localised orbital methods at the Hartree»Fock level and beyond,21,22 as well as with totalenergy pseudopotential planewave techniques.23 In addition, there have been many sets of interatomic potentials derived to reproduce their structural and mechanical properties, ranging from dipolar shell models,24 through breathing shell models and also including more general compressibility treatments.25 In this study we shall, for a number of reasons, consider results for alumina, rather than MgO, using the NDDO Hamiltonians.First, there are only parameters for magnesium at the PM3 level, whereas there are parameters for aluminium in all cases. There are also parameters for magnesium within the MNDO/d formulation,26 however, this involves some modi–cations to the above scheme, which have yet to be made, for the treatment of integrals. Secondly, the simplicity of the rock salt structure makes alumina a more challenging case.Although alumina exists in a number of polymorphs, depending on the conditions, we shall concentrate here on corundum, the ground-state phase at low pressures, which has been best characterised.The structure of corundum has been optimised to constant224 Semi-empirical methods in solid-state chemistry pressure using all three NDDO Hamiltonians as a 2]2]2 supercell of the rhombohedral unit cell which contains a total of 80 atoms. In addition, selected elastic constants have been calculated by central –nite diÜerences of the analytical –rst derivatives using a strain of ^0.005.At each point all fractional coordinates are re-minimised to allow for the internal strain contribution. The results of these calculations are summarised in Table 1. Considering –rst the geometrical aspects, the original MNDO method yields quite reasonable cell parameters with an error of 1»2%, which is comparable with the performance of many interatomic potential models when speci–cally –tted to a structure. Unfortunately, the supposedly improved method of AM1 gives cell parameters which are, in this case, too small and more in error than at the MNDO level.For PM3 the results are even worse, but this time with the unit cell far too large. The errors in the structure under the AM1 and PM3 Hamiltonians also clearly show themselves in the mechanical properties, as the elastic constants are too high and too low, respectively.Even in the case of MNDO, where the structure is not too bad, we –nd that the elastic constants are systematically too large. As with most interatomic potential models, we observe that the apparent isotropic nature of the on-diagonal elastic constants, seen in experimental measurements, is not reproduced.One diÜerence from the typical results of potential models is the relatively large magnitude of some of the oÜ-diagonal elements, such as and This is partly to be expected, because of the changes that result in C12 C13 . the charge density at each atomic site leading to added coupling between Cartesian directions. However, the magnitude seems to be exaggerated. In the periodic Hartree»Fock31 and density functional32 studies of alumina, the structural parameters are reproduced with much better accuracy that at the semiempirical level, as would be hoped for.However, a full set of elastic constants is yet to be determined, because of the number of calculations involved when using numerical methods. If we compare the electronic structure predicted by the three theoretical methods then we can see a number of diÜerences. The periodic Hartree»Fock calculation systematically overestimates the band gap for alumina with a value in the region of 16 eV, whereas the local density functional calculations are likely to seriously underestimate this quantity.Experimental values for the band gap of corundum are between 8.5 and 9.9 eV.28,29 Both AM1 and PM3 give values which lie within this range, but the gap at the MNDO level is too high, though still lower than the Hartree»Fock value.Table 1 Comparison of calculated and experimental values for some of the structural and electronic properties of based on the MNDO, AM1 a-Al2O3 and PM3 Hamiltonians experiment MNDO AM1 PM3 a/” 4.7602 4.845 4.686 5.279 c/” 12.9933 13.123 12.328 15.473 *fH/kJ mol~1 [1676 [1115 [1012 [989 band gap/eV 8.5»9.9 12.12 9.52 9.29 charge Al/e » ]1.25 ]1.13 ]1.33 C11/GPa 496.9 703 905 165 C12/GPa 163.6 310 391 119 C13/GPa 110.9 282 394 180 C33/GPa 498.0 536 681 407 C44/GPa 147.4 104 245 116 C66/GPa 166.7 186 257 34 Experimental data from ref. 27 for the crystal structure, ref. 28 and 29 for the band gap and ref. 30 for the elastic constants.J. D. Gale 225 The Mulliken charges obtained from the semi-empirical methods for aluminium all suggest ionic character of ca. 40% with charges of ca. ]1.2. This is very close to the result obtained from a minimal basis set in an ab initio calculation,33 but considerably lower than the values of between ]2.0 and ]2.25 yielded by a double zeta basis set (without polarisation functions) optimised for the crystalline environment.X-Ray crystallography27 suggests a charge of ]1.32 for aluminium, closer to the semiempirical result. However, little signi–cance should be read into this, as the charge will depend on the method of population analysis, with the Mulliken approach not being the most ideal. In addition to studies of the bulk properties of corundum, there has been much interest in the surfaces of this material because of the use of alumina, though often in its gamma-polymorph, as a catalyst support.Interatomic potentials have been used to calculate the surface energies for corundum and thereby to predict the morphology of the crystals.34,35 Again there have been both Hartree»Fock36 and density functional32 studies of the 0001 surface, which is one of the predominant crystal faces.All methods show that there is substantial relaxation of the terminal aluminium ions towards the layer of oxygens below, though there are quantitative diÜerences as to the precise magnitude of the displacement. Despite the limitations of the bulk elastic constants, we decided to examine how well the MNDO Hamiltonian, which is the best of the three for bulk corundum, would perform for the 0001 surface.As we have yet to implement two-dimensional sums for the long-range contribution to the interaction energy, the surface calculations were performed using a three-dimensional system with a gap being introduced to create slabs of alumina. In all the systems studied there is no dipole perpendicular to the surface and it is found that a gap of is sufficient for there to be no signi–cant interaction between 15 ” neighbouring slabs.The slab of alumina is six formula units deep (18 atomic layers) and is based on a 2]2 supercell of the hexagonal unit cell within the plane of the surface. The calculated surface energy for the 0001 face is found to be 3.93 J m~2 for the unrelaxed case, which reduces substantially to a value of 1.03 J m~2 on relaxation of the atomic positions.The unrelaxed surface energy compares favourably with the value obtained from local density approximation calculations which yielded 3.77 J m~2, though is lower than the corresponding Hartree»Fock result of 4.95 J m~2.37 The results of interatomic potential calculations can lead to even higher unrelaxed surface energies.While the unrelaxed surface energy appears to be very reasonable from the semi-empirical method, the relaxed energy seems to be a little too low when compared to the local density approximation result of 1.76 J m~2. The slab model used in the present work was larger, which will allow a greater amount of relaxation to occur, accounting for some of the diÜerence.However, it is still likely that the MNDO value represents an underestimate. One of the most widely reported parameters of the surface structure of corundum is the displacement of the outermost aluminium towards the upper layer of oxygens, as this is the largest eÜect of relaxation at the 0001 face. Both interatomic potentials, based on electron gas calculations, and periodic Hartree»Fock calculations, suggest that the perpendicular distance to aluminium decreases by between 59 and 68%, with the value obtained from the MNDO optimisation lying in this range, at 60%.All these methods con—ict with the local density functional result, which suggests that the magnitude is much larger, at 85%, which leads to the aluminium being almost in the oxygen layer.It has been argued that the surface geometry from periodic Hartree»Fock calculations reproduces the results of surface spectroscopic investigations better than that obtained from the density functional study.36 So far, most of the results reported for alumina could have been readily obtained with interatomic potential methods, with equal or better accuracy.Where the potential methods have greater difficulty is in handling combinations of diÜerent species, for226 Semi-empirical methods in solid-state chemistry which there is little or no empirical information from which to derive parameters, without resorting to the use of combination rules to transfer constants from other environments. Alternatively, higher level calculations could be performed as a basis for potential derivation.38 Either approach, however, involves some difficulties.Such a case is presented if we wish to consider the adsorption of molecules at the surface of alumina. As an example of how the semi-empirical methods would extend to this problem, we have modelled the adsorption of hydrogen —uoride on the above slabs of alumina.To preserve the zero dipole of the system we have adsorbed two molecules, one on each side of the slab, with a centre of inversion between them. Results for this adsorption process are given in Table 2. There are, potentially, several local minima for this system, but in this preliminary study to demonstrate the application of the method we have only treated the case where the HF molecule is perpendicular to the surface, with the —uorine atom positioned directly above the aluminium cations. We have modelled the adsorption of HF in two stages.In the –rst, the molecule is adsorbed onto a rigid surface, while, in the second, the surface is allowed to relax in response to the adsorbate. In this way, we can see that the surface relaxation makes an important contribution to the adsorption energy of the molecule, as the value almost doubles when the alumina slab is allowed to optimise.It is hard to assess the magnitude of the adsorption energy. There have been earlier periodic Hartree»Fock calculations37 on the same geometry which gave a binding energy of ca. 60 kJ mol~1 and an AlwF distance of However, there is a high degree of uncertainty in these numbers 1.93 ”.because of the limited relaxation possible for the surface and basis set eÜects. The main eÜect of the relaxation is the displacement of the aluminium ions, which are coordinated to —uorine, by out of the plane of the uncoordinated alu- 0.163 ” miniums. Correspondingly, there is a contraction in the AlwF distance to compensate. Interestingly, the HwF bond contracts slightly on adsorption, rather than weakening as might be expected from the fact that this molecule is experimentally known to dissociate on the surface of alumina.However, there is no guarantee that this is the active local minimum. There also appears to be an appreciable amount of charge transfer from —uorine to the surface, though the excess charge appears to be only partially localised on the aluminium to which it is directly coordinated.To investigate the in—uence of coverage, we have also modelled the system in which twice as many molecules of HF are adsorbed, the extra molecules being placed with a shift of half of the supercell in both the a and b cell directions so as to maximise the intermolecular distance. For this con–guration the binding energy per molecule is substantially lowered, by 15 kJ mol~1.If the repulsion between HF molecules is estimated by repeating the calculation with the alumina removed and comparing this to the sum of Table 2 Binding energies (kJ mol~1), geometries and charges (e) for HF adsorbed (”) on a slab model for the 0001 surface of corundum calculated using the MNDO Hamiltonian, with and without relaxation of the surface for two molecules per unit cell and also for twice this coverage with relaxation two molecules two molecules four molecules gas phase unrelaxed relaxed relaxed binding energy » 21.3 38.6 23.3 r(HwF) 0.9563 0.9525 0.9529 0.9515 r(F… … …Al) » 1.8197 1.7866 1.8001 q(H) ]0.284 ]0.337 ]0.348 ]0.332 q(F) [0.284 [0.129 [0.121 [0.118 Note that the gas-phase values quoted refer to an isolated molecule of HF.J.D. Gale 227 the energies of the isolated molecules then this accounts for 4.7 kJ mol~1 of the destabilisation. The rest of the loss of the binding energy must therefore be a cooperative eÜect. The above results for the adsorption of HF on alumina only represent a preliminary study, since there are several other minima on the energy surface.For instance, an alternative con–guration is found when the molecules are tilted away from the surface normal, with an AlwFwH angle of 136°, which is 5 kJ mol~1 more stable than the upright orientation. However, it demonstrates the potential of the method for the study of molecular adsorption at surfaces. Polymorphs of silicon dioxide One of the most important classes of mineral in solid-state inorganic chemistry is the zeolite family, as a result of their many applications in molecular sieving, heterogeneous catalysis and ion exchange, to name but a few.In addition, because of the geological relevance of silicates, there is a strong interest in modelling their behaviour under pressure and at elevated temperatures not readily accessible by experiment. Consequently, there have been many studies of silicates, again using both interatomic potentials39,40 and ab initio/–rst principles techniques.41,42 In this section we shall examine the application of the semi-empirical methods to the high-density forms of silica.These represent a situation where the bonding is likely to be intermediate between covalent and ionic.At ambient temperature and pressure the most stable polymorph of is known to be a-quartz and this is the starting point for most SiO2 interatomic potential derivations by empirical –tting, as all its properties are well characterised. 43 There are a number of other relatively dense metastable forms, also based on corner-sharing tetrahedra, including a-cristobalite and coesite.There is also an idealised form of cristobalite which has a cubic structure with linear SiwOwSi bonds. In addition to these tetrahedral forms, there exists one polymorph containing octahedral silicon based on the rutile structure. This material, stishovite, is only formed at elevated temperatures and pressures. The enthalpies of formation have been calculated for the above polymorphs and are listed in Table 3.All the structures were fully optimised for supercells chosen to ensure that there were no cell parameters shorter than ca. 8 ”. Subsequently, to check the convergence of the exchange terms and their in—uence on the relative energies, single-point calculations were performed on increasingly large supercells. Based on this, the relative energies were found to be well converged using unit cells containing 48 formula units for all polymorphs.Both of the semi-empirical methods tried, MNDO and AM1, systematically over estimated the unit cell dimensions of the silicate polymorphs, as can be seen from Table 4. For MNDO, the error is typically ca. 5%, while for AM1 it is normally twice this. Calculations using PM3 showed the same characteristic errors seen for alumina and, therefore, have not been included here.The general overestimation of the unit cell volume can probably be ascribed to the fact that the minimal basis cannot accommodate changes in the size of atoms relative to their typical gas-phase state. Table 3 Calculated enthalpies of formation (kJ mol~1) for silica polymorphs relative to quartz experimenta MNDO AMI a-cristobalite 2.84 [5.99 [3.22 idealised cristobalite » [7.16 [4.45 coesite 2.93 18.86 13.52 stishovite 51.88 [125.85 [7.57 a Ref. 44.228 Semi-empirical methods in solid-state chemistry Table 4 Unit cell parameters for silica polymorphs from experiment and semi-empirical calculations using the MNDO and AM1 Hamiltonians experiment MNDO AM1 a-quartza a/” 4.902 5.282 5.420 c/” 5.400 5.762 5.881 a-cristobaliteb a/” 4.957 5.383 5.511 c/” 6.890 7.606 7.838 coesitec a/” 7.137 7.527 7.569 b/” 12.370 12.982 13.392 c/” 7.174 7.542 7.815 b/degrees 120.3 120.5 120.5 stishovited a/” 4.177 4.198 4.341 c/” 2.665 2.803 2.873 a Ref. 45. b Ref. 46. c Ref. 47. d Ref. 48. From the data in Table 3, it is clear that the semi-empirical Hamiltonians have difficulty in correctly ordering the thermodynamic stability of the phases. Both forms of cristobalite come out lower in energy than a-quartz when they should be higher and, furthermore, the idealised form of cristobalite with linear SiwOwSi bonds is more stable than the lower-symmetry equivalent.This latter fact indicates that the polarisability of oxygen is too low to be appropriate for this environment, which is not surprising given the diÜerences between the properties of oxygen as a constituent of a molecular system and when it is formally an oxide anion.Hence, this suggests that reparametrisation is necessary speci–cally for the solid state for oxygen when oxide ions are involved. Obtaining the correct order for the closely related high-density tetrahedral forms of silica is quite difficult and diÜerent interatomic potential models have had variable success in this area also,49 though comparison with other less dense microporous polymorphs is more favourable.50 Again, proper treatment of polarisation eÜects appears to be the key to success.51 Equally difficult to predict correctly is the lower stability of stishovite at atmospheric pressure than that of quartz because of the change in coordination number.Some interatomic potentials get the order right by virtue of the use of three-body potentials with an equilibrium angle equal to that for a perfect tetrahedral site. However, the transition between a tetrahedral- and octahedral-based structure would then contain a discontinuity in the energy. Not surprisingly, the semi-empirical methods at the NDDO level favour stishovite over quartz as the ground state, since hypervalent compounds are a known weakness of the standard parametrisations. This problem has been partly addressed through the development of the MNDO/d modi–cation in which d orbitals are included for second-row main-group elements, including silicon.Although we have yet to test this Hamiltonian for this particular problem, it has been shown that the MNDO/d formalism reduces the mean absolute error in the enthalpy of formation of a standard set of hypervalent compounds from 143.2 to 5.4 kcal mol~1 when compared to MNDO. Both AM1 and PM3 seek to achieve a similar eÜect by modi–cation of the nuclear repulsion term.This can be seen to be an improvement, by the fact that although AM1 gets the order wrong the magnitudes of the errors are greatly reduced. From the results of ab initio calculations52 we know that obtaining the correct ordering of lattice energies for quartz and stishovite is not straightforward. Only when the basis set is made sufficiently —exible, through the addition of polarisation functions, does stishovite become more stable.Hence, this gives further support to the idea that the MNDO/d parametrisation should oÜer a major improvement. A clue as to why theJ. D. Gale 229 energetics of stishovite are wrong in the semi-empirical methods with a restricted basis comes from the charge distributions. In the periodic Hartree»Fock calculations, stishovite is found to be signi–cantly more ionic than quartz, whereas at both the MNDO and AM1 levels it is marginally less so.This leads to a lower increase in repulsion between the oxygens that surround silicon on going from four- to six-fold coordination. Although the predictions for the relative stability of high-density silicate phases are rather poor, preliminary results on zeolites and their defects compare more favourably with accurate calculations.Polymorphs of ice One type of system where we might expect the MNDO family of methods to work well is for molecular crystals, given the widespread application of these techniques to isolated molecules. However, the treatment of crystalline forms of the same molecules is more demanding as it also probes intermolecular forces as well as intramolecular ones.While it might be unreasonable to expect good results for crystals where the dominant interaction is the dispersion force, it may be possible to obtain an adequate treatment of materials where hydrogen bonding and electrostatic interactions are the key to the structure. Because of the much weaker perturbation of the atomic states caused by molecular crystals, as compared with ionic materials, the limitations of the parametrisation and the single zeta basis set should be less apparent.The treatment of molecular crystals also represents a more challenging situation for interatomic potential studies, since representing the charge distribution by atomiccentred monopoles may no longer be sufficient. Recent work, involving the use of distributed multipoles, has shown that a much better description of the intermolecular forces can be obtained in this way.53 An advantage of using an NDDO-based Hamiltonian is that the atomic-centred charges, dipoles and quadrupoles arise naturally in response to the crystalline environment and there is no need to assume that the multipole moments for the isolated molecule are transferable to the solid state.For this preliminary study, we shall investigate the properties of a reasonably simple in structure, yet much studied, molecular crystal»that of ice. There exist several polymorphs of ice, some of which are better characterised than others. For the purposes of this study we shall examine ice XI (which will subsequently be referred to as C-ice after its space group which is the only know form to be proton ordered,54 and an Cmc21), alternative ordering in space group (P-ice).The latter of these two is a hypotheti- Pna21 cal structure proposed to be slightly more stable than the experimentally known C-ice based on ab initio-derived interatomic potential calculations.55 Both have also been studied at the periodic Hartree»Fock level.56 There have been many attempts to derive interatomic potentials for water.Most have involved the use of rigid water molecules as a simpli–cation and the intermolecular forces have been tuned to the properties of liquid water. The introduction of —exibility into the water molecule is highly problematical with simple two-body interatomic potential models, as calculations show the wrong qualitative features in many respects.For instance, if comparisons are made with high-level quantum mechanical calculations for the water dimer, it is observed that, in a two-body interatomic potential model, the bond angle of the water to which the hydrogen bond is made decreases relative to the monomer, whereas it increases in more accurate studies. This is simply due to electrostatic repulsion between the hydrogens.In the quantum mechanical case, the polarisation of oxygen resulting from the hydrogen bond counters this eÜect and leads to an increase in the angle. Furthermore, simple interatomic potential models fail to yield the result that the binding energy per molecule of water is higher in ice than it is for the water dimer. This is because the increased binding energy in ice is a many body eÜect resulting from the increased ionicity of water in the crystal.230 Semi-empirical methods in solid-state chemistry Table 5 Comparison of the calculated properties for the water monomer, dimer, C-ice and P-ice according to the PM3 Hamiltonian monomer dimer C-ice P-ice r(OwH)/” 0.951 0.951 0.965 0.965 to 0.960 n(HwOwH)/degrees 107.53 107.40 108.7 108.4 108.09 r(O… … …H)/” » 1.808 1.776 1.778 binding energy/kJ mol~1 » 7.2 38.0 38.2 charge (O) [0.358 [0.367 [0.484 [0.485 [0.402 In Table 5 we present results for the water monomer, dimer and the two polymorphs of ice discussed above, calculated with the PM3 Hamiltonian, which is known to yield the correct local minimum for the structure of the water dimer.12 From this data, we can clearly see that this method shows all the correct behaviour, at least qualitatively.Although the bond angle in water is predicted to be ca. 3° too large, the angle of the molecule which acts as the Lewis base in the water dimer increases by 0.5°, as compared to a change of 0.4° at the CCSD(T)/TZ2P level.57 The bond angle increases further in the structure of ice, while the OwH bonds are lengthened.Although the PM3 method underestimates the OwO distance for the water dimer by the reproduction of the crystallographic orthorhombic unit cell for C-ice is 0.02 ”, quite good, with cell parameters of 4.452, 7.889 and as compared with experi- 7.303 ” mental values of 4.502, 7.798 and Note that the calculations were actually run 7.328 ”.as a 2]1]1 supercell to increase the range of the exchange interaction in the x direction. Doubling the unit cell in the other two directions also only changes the total energy by less than 0.001 eV. The experimental binding energies of the water dimer and ice are 11.4 and 56 kJ mol~1 per molecule of water.58,59 While the PM3 method underestimates both values, it is a systematic underestimation, as both are in error by approximately the same scale factor, which may make it possible to improve empirically the quantitative prediction of this quantity.However, the key point is that the binding energy for ice is higher than that of the dimer, owing to the inclusion of many-body eÜects. From the charges for the oxygen atoms we can observe that there is roughly a 35% increase in the ionicity of water in ice, thus accounting for this result. Previous studies have debated the relative stability of the two polymorphs of ice considered here.Results from ab initio-derived interatomic potentials suggest that the antiferroelectric arrangement of protons found in P-ice should be ca. 7.5 kJ mol~1 more stable than that in the experimentally observed C-ice.60 Periodic Hartree»Fock calculations56 predict a negligible energy diÜerence between the two at the HF/6-31G**// HF/6-31G level.Using the PM3 Hamiltonian, a diÜerence of 1.56 kJ mol~1 is obtained in favour of P-ice, thus placing it between the other estimates, but much nearer to the higher-level quantum mechanical result. Conclusions In this study we have presented the –rst results for the application of the popular MNDO family of methods to solid-state problems, using a new algorithm which properly handles the long-range Coulomb terms. A variety of diÜerent materials, varying from traditional ionic solids through to molecular crystals, have been taken as examples so as to make a preliminary assessment of the performance of this approach.J.D.Gale 231 For ionic oxides, such as alumina, the results obtained for the bulk structure and curvature properties are rather mediocre and generally inferior to interatomic potential models, though when it is remembered that no parameters have had to be –tted speci–- cally to alumina and the only data utilised come from gaseous aluminium species then perhaps the results should be viewed as being not so bad.Also, they clearly oÜer some advantages when considering more complex situations and bonding changes, such as the adsorption and subsequent dissociation of a molecule at the surface of the oxide. Furthermore, they have the bene–t of oÜering some insight into the electronic structure of the material, which cannot be obtained from interatomic potentials.The inclusion of correlation through the modi–ed integrals appears to lead to band gaps which are superior to those obtained from Hartree»Fock calculations for the materials tested. When it comes to the study of molecular crystals then the NDDO methods show potential, particularly where the system is composed of a hydrogen-bonded network. Even in cases where the interactions are weaker, any de–ciencies may possibly be corrected for by the superposition of interatomic potential terms, for example, a C6 term to mimic the inclusion of dispersion forces.Using semi-empirical methods as a basis for the treatment of the molecules has the desirable features that the electrostatic interactions are far more accurately modelled than for an interatomic potential, because of the presence of atomic-centred multipoles and the self consistency leading to many-body contributions to the binding energy.All the results in this study are based upon the standard parameters for the Hamiltonians developed for gas-phase species. However, in the same way that it is necessary to optimise the exponents of a Gaussian basis set for use in an ab initio solid-state calculation, signi–cant improvements could be made through the determination of solid-state optimised parameters.This is because all the models are fundamentally based around a single zeta basis set which cannot accommodate changes in eÜective ionic radii. A further improvement would be, therefore, to generate a double zeta based parametrisation. In the past this would not have been feasible because of the limitations of gasphase data available.However, if solid-state information were now to be included, then it may be possible to do this for certain key elements, such as oxygen, to handle situations where a multiply charged anion is present. 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Phys., 1995, 102, 310. 58 D. Eisenberg and W. Kauzmann, T he Structure and Properties of W ater, Oxford University Press, Oxford, 1969. 59 B. Y. Yoon, K. Morokuma and E. R. Davidson, J. Chem. Phys., 1985, 83, 1223. Paper 7/01331K; Received 25th February, 1997

ISSN:1359-6640    

DOI:10.1039/a701331k

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 233»251 Ab initio simulations of non-stoichiometric lithiumñoxygen clusters Fabio Finocchi, Tristan Albaret and Claudine Noguera* L aboratoire de Physique des Solides (URA 2), 510, Paris-Sud, Ba� t. Universiteç 91405 Orsay, France Bonding and electronic properties of non-stoichiometric lithium oxide clusters (nO10) are studied by means of ab initio simulations. We focus LinO2 on the –rst stage of lithium enrichment of stoichiometric where the Li4O2 , formation of additional LiwO bonds is favoured.The bonding con–guration of the lowest-energy isomers and their stability are analysed in detail, and their structures compared to those found in bulk non-stoichiometric alkali-metal suboxides. As a function of the increasing number of excess Li atoms, coexistence of an ionic and a delocalized character of the electronic density takes place, accompanied by a progressive LiwO bond weakening.These issues, and the existence of odd»even staggering of electronic properties, are discussed, in relation to recent experiments and other anionde –cient systems such as defective oxide surfaces and alkali-metal halide clusters. 1 Introduction Non-stoichiometry in oxide materials is known to change drastically their intrinsic properties and give rise to a variety of optical, catalytic and electrical conductivity phenomena which are absent in the pure material. On most oxide surfaces, the predominant type of defects formed when samples are heated or exposed to particle beams consist of oxygen vacancies.1,2 It has been shown that their presence may induce surface reconstructions, with surface cell sizes which depend upon the annealing temperature, i.e.upon the vacancy density. They are also responsible for enhanced reactivity, for example in water dissociation or other catalytic reactions. The understanding of oxide clusters is by far less advanced than their bulk or surface counterparts, whether they are stoichiometric or non-stoichiometric.However, owing to their importance in chemistry and physics, it is essential to understand the variations in their physico-chemical properties as a function of their size. Unsupported, mass-selected Mg, Ca and Ba oxide,3h6 caesium oxide,7h9 and lithium oxide clusters10,11 have only recently been obtained and studied. In particular, a thorough study of small massselected oxidized lithium clusters10,11 has evidenced interesting systematic trends in the relative abundance of species as a function of n (0OnO10) at constant number LinOm m of oxygen atoms (0OmO6).It has been shown that the odd»even oscillations which are generally seen in abundance spectra and ionization potentials in metallic clusters, progressively weaken and vanish when the oxygen content increases.On more general grounds, the question of whether electronic or geometric eÜects drive the odd»even oscillations has been raised, together with the question of the degree of localization of the excess electrons in these clusters. From a theoretical viewpoint, several studies of stoichiometric oxide clusters have been performed, either using simple ionic models and a global optimization strategy able 233234 Simulations of non-stoichiometric lithium»oxygen clusters to –nd the many local minima,12 or using ab initio,13h15 or semi-empirical Hartree» Fock16 approaches without a search for global minima.Focusing on stoichiometric small clusters, we have shown17 that ab initio molecular dynamics (AIMD) Li2O simulations18 are an eÜective tool for the investigation of such systems, in which the mixed iono»covalent character of the metal»oxygen bond and the details of the charge distributions are sensitive functions of the actual coordination of the atoms.In nonstoichiometric clusters, the need for an accurate description of the ground state is even more crucial, since the removal of oxygen atoms is expected to induce important charge redistributions as well as structural modi–cations, as it does in the bulk and on the surfaces of metal oxides.The possibility of strong electron redistributions raises doubts on the usefulness of empirical ionic models, whose parameters, and especially the charge values, will no longer be transferable between systems of diÜerent stoichiometry.In this work, we report the results of AIMD simulations for small clusters. Li4`pO2 Starting from the stoichiometric parents, we consider the progressive growth of Li4O2 the cluster by enrichment of lithiums, in its –rst stages (p ranging from 0 to 6). We will only discuss the properties of clusters in which all the lithium atoms are bound to oxygens and discard isomers in which bonding to other lithiums occur.The thorough investigation of the –rst stages of adsorption permits us to analyse trends in the evolution of the cluster morphology and electronic structure and is a necessary premise to understanding the large-p regime. The paper is organized as follows : we brie—y describe the computational method used in the simulations, in Section 2.The results for the morphology and the bond characteristics of the lowest energy isomers are presented in Section 3. In Sections 4 and 5 trends in the evolution of the electronic structure and attachment energy of lithiums on are discussed as a function of the excess number, p, of Li atoms. Section 6 Li4`pO2 ends the paper with a comparison between the oxygen-de–cient lithium oxide clusters and other non-stoichiometric systems.It also contains a critical discussion of the theoretical approaches designed for these systems. 2 Computational details The electronic structure calculations have been performed within the density functional theory, by using both local density approximation (LDA) and local spin density approximation (LSDA) for the exchange and correlation energy.In particular, LSDA is used for clusters with an odd number of electrons, and for some even-numbered clusters that could present a non-spin-paired ground state. We have thus checked the stability of the calculated electronic structure with respect to spontaneous spin polarization. The Kohn»Sham orbitals are expanded in a plane-wave basis set, and soft, normconserving pseudopotentials are used to describe the interaction between the ionic cores (1s atomic states both for Li and oxygen), and the valence electrons.The choice of the supercell is the same as in our previous study of stoichiometric lithium oxide clusters.17 The pseudopotentials are used in the Kleinman»Bylander representation19 including s, p and d components for oxygen, and s and p components for Li.A local reference potential is chosen (d-component for O, s-component for Li). In order to obtain an optimal total energy convergence as a function of the size of the basis set, as determined by the cut-oÜ kinetic energy, we followed the prescriptions given by Troullier and Ecut , Martins20 in the generation of the O pseudopotential. The core radii chosen for O are 1.38 a.u., 1.75 a.u.and 1.38 a.u. for the s, p and d components, respectively. For this pseudopotential, the total energy of the molecule is converged within 0.5 and 0.1 Li2O eV at 39 and 60 Ry cut-oÜ, respectively. Since the Li pseudopotential converges faster than that of oxygen as a function of we choose rather small cut-oÜ radii, equal to Ecut , 1.75 a.u.for both s and p components, in order to have better transferability, from the stoichiometric molecule and bulk to pure metallic clusters. Li2O Li2 O, LinF. Finocchi et al. 235 The pseudopotentials have already been tested in the study of stoichiometric lithium oxide clusters.17 Further test calculations have been performed on pure Li systems. For the Li dimer, excellent agreement is found with experiments and other calculations.21 The calculated equilibrium distance is equal to 2.69 and the dissociation energy to ” 1.02 eV.For clusters, the planar structure turned out to be less stable than the Li5 C2v double-tetrahedra con–guration by 0.25 eV, at variance with a previous Hartree»Fock calculation,22 which predicted that the plar isomer is 0.34 eV lower in energy.However, the a posteriori inclusion of correlations reduces the energy diÜerence to 0.17 eV. The importance of a self-consistent and accurate description of correlation eÜects in clusters had already been pointed out.23 In the case of it was shown that the Lin Li6 , stability of the three isomers, of point symmetry and can be completely C2v , C5v D3h reversed, depending upon the treatment of correlation eÜects.Fully self-consistent calculations of exchange and correlation energy in the generalized-gradient approximation unambiguously give the sequence, from the most to the least stable isomer. C2v»C5v»D3h The ordering agrees with our LDA results, apart from the fact that the precise values of the energy diÜerences that we found between the isomers are overestimated by ca. 0.1 eV. This quantitative discrepancy is attributed to the use of LDA, but it is found to have little in—uence on our results on clusters. LinO2 In order to –nd out the equilibrium geometries of clusters, we performed Li4`pO2 some simulated annealing runs. Moreover, from the geometries resulting from the dynamic runs, we obtain other starting con–gurations by changing manually the local topology of the clusters. The resulting geometries are then optimized until the atomic forces do not exceed 0.005 eV and the calculated electronic structures thus refer to ”~1, the stable con–guration optimized in a fully self-consistent way.The stability of all the con–gurations issued from the local minimizations has been checked against a random displacement of the atomic coordinates. 3 Morphology We have focused on clusters (0OpO6), containing two oxygen atoms and an Li4`pO2 increasing number of attached lithiums, starting from the stoichiometric compounds In a previous study,17 we showed that the two most stable isomers have Li4O2. Li4 O2 and symmetries, as shown in Fig. 1. The isomer has a planar struc- D2h C3v D2h-Li4O2 ture with the oxygen atoms connected by two bridging lithiums, denoted Li(2).The two remaining lithiums are singly coordinated and will be denoted Li(1). The central part of the cluster presents a rhombohedral ring as in the bulk anti—uorite structure. By Li2O2 contrast, the isomer is not planar and contains three Li(2) atoms and one C3v-Li4O2 Li(1). Those two isomers are very close in energy.and isomers (from left to right). Li atoms are drawn in light grey, O Fig. 1 D2h , C3v C2v-Li4O2 atoms in dark grey, in all ball and stick representations. Only LiwO bonds are drawn.236 Simulations of non-stoichiometric lithium»oxygen clusters The small energy diÜerence between the and isomers results from a partial D2h C3v cancellation between nearest-neighbour attractive Li»O and repulsive Li»Li interactions (6 LiwO and 1 LiwLi shortest bonds in compared with 7 LiwO and 3 D2h-Li4O2 , LiwLi bonds in It can be accounted for, by using a simple pairwise poten- C3v-Li4O2).tial of the type : Vij(r)\ qi qj r ]Aij exp ([bij r) (3.1) whose parameters are adjusted to reproduce the stretching frequencies and of the las lss molecule, and the lattice parameter and the bulk modulus of the anti—uorite Li2O a0 B0 This atomistic approach, although very rough, correctly accounts for the rela- Li2O.17 tive stability of the two isomers and gives a good estimate of their 0.16 eV energy diÜerence. A similar ionic approach, applied to clusters,24 predicted the same Cs4O2 topology for the two lowest-energy isomers.Interestingly, the energy ordering of the C3v and isomers was reversed, an indication of the delicate balance between attractive D2h and repulsive Coulomb interactions.The pair model also predicts that the competing structure (see Fig. 1), which has 6 LiwO and 2 LiwLi short bonds, is much C2v-Li4O2 higher in energy, as con–rmed by ab initio calculations. As a result of the AIMD simulations, we found that the structures of nonstoichiometric isomers are related to those of the two most stable parents, with Li4O2 attachment of additional lithiums on the oxygens, without topological change of the central part of the cluster.We will thus denote, by the isomers in which the II-LinO2 , two oxygens are bridged by two Li(2), as in and, by those in D2h-Li4O2 , III-LinO2 , which the two oxygens are bridged by three Li(2), as in The former contain C3v-Li4O2 .(n[2) one-fold coordinated Li(1) lithiums and the latter (n[3), as shown in Fig. 2. Other geometries, such as isomers with a single Li(2), or con–gurations built from a pure lithium cluster with the two oxygens outside, are much higher in energy or spontaneously rearrange to more stable isomers.However, a second mechanism of growth exists, yielding clusters in which some lithiums are bound to other lithium atoms, rather than to oxygens. Such a process will undoubtedly dominate the growth to the limit of a very large number of excess lithiums. We –nd that the crossover between the two regimes occurs at n\8. Up to n\7, the most stable isomers have all their lithiums bound to oxygens.Beyond this size, metastable (nO10) and (nO9) II-LinO2 III-LinO2 clusters are, nevertheless, found, at energies only slightly larger than the most stable isomers. As already said in the Introduction, we will focus here on the clusters which contain only Li(1) and Li(2) lithiums and which are shown in Fig. 2. It is of interest to understand how their properties evolve as a function of their size over the whole range of n values where they present a (meta)stable structure.isomers 3.1 II-LinO2 In the family derived from the clusters with n even have a mirror D2h-Li4O2, Lin O2 symmetry plane, perpendicular to the oxygen»oxygen axis and containing the two Li(2) atoms. When n is odd, this symmetry operation does not exist, since the two oxygens have inequivalent local environments.has a point group symmetry. The oxygens are bound to one and two Li(1) Li5O2 C2v atoms, respectively. The two Li(1) which are located on the same side of the cluster form a triangle with their –rst-neighbour oxygen, in a plane perpendicular to the rest of the cluster. Their distance to the two Li(2) atoms is of the same order of magnitude as in bulk Li (3.05 cf. 3.023 ”). In the point group symmetry is The four Li(1) atoms are located in a Li6O2 , C2v . plane perpendicular to the central rhombus. In addition, a large displacement of Li2O2F. Finocchi et al. 237 most stable isomers, obtained by attachment of lithiums on oxygens (1OpO6). Fig. 2 Li4`pO2 The clusters drawn in the left and right columns belong to the II- and III-families, respectively.Li atoms are drawn in light grey, O atoms in dark grey, in all ball and stick representations. Only LiwO bonds are drawn.238 Simulations of non-stoichiometric lithium»oxygen clusters two Li(1) located on opposite sides of the cluster (see Fig. 2) is found, suggesting the formation of an LiwLi bond. To con–rm that this is the most stable structure, we performed a few simulations starting from initial con–gurations in which the four Li(1) are equivalent by symmetry.In all cases, a spontaneous atomic rearrangement took place, leading to the structure. This spontaneous distortion is similar to a Jahn» C2v Teller eÜect and leads to the largest energy stabilization (see Section 4.2). has a con–guration very close to with an additional Li(1) bound to one Li7O2 Li6O2 , oxygen.This latter becomes –ve-fold coordinated and its local environment is a truncated octahedron. The overall point group symmetry of the cluster is Cs . In two LiwLi bonds are formed between two pairs of Li(1) atoms belonging Li8O2 , to opposite ends of the cluster, probably to lower the total energy as the (see Li6O2 Section 4.2). As in the latter cluster, the point group symmetry is C2v .In and an octahedral environment of lithium is progressively com- Li9O2 Li10O2 , pleted around the oxygens. The point group symmetries of the clusters are and Cs D2h , respectively. can be viewed as resulting from the association of two octa- Li10O2 Li6O hedra sharing an edge made of two Li(2) atoms. Formation of six LiwO bonds around each oxygen marks the –nal step of stable lithium attachment that we found.We will see in Section 3.2 that the same is true for isomers. III-LinO2 isomers 3.2 III-LinO2 In the family derived from the central unit has a three-fold symmetry C3v-Li4O2, Li3 O2 axis. In addition, a symmetry plane exists for odd values of n, at variance with II-LinO2 isomers. This plane is perpendicular to the oxygen»oxygen axis and contains the three Li(2) atoms.When n is even, the local environments of the two oxygens are inequivalent. In the two oxygens and the two Li(1) atoms are aligned along a normal to the Li5O2 , Li(2) plane, which results in a point group symmetry. C3h and clusters have a symmetry and display an increasing number of Li6O2 Li7O2 Cs short Li(1)wLi(2) interatomic distances, of ca. 2.6 as in the dimer (2.69 This ” Li2 ”).suggests that Li(1)wLi(2) bonds are formed. In and clusters, progressive completion of an octahedral environment Li8O2 Li9O2 for the oxygens is achieved. The point group symmetries of the clusters are and Cs C2v , respectively. However, in the distortion which prevents the invariance by rota- Li9O2 , tion of 2n/3 around the oxygen»oxygen axis is very small (ca. 0.5%). results from Li9O2 the association of two octahedra sharing a face. No (meta)stable isomer of this Li6O family could be found by attachment of additional lithiums. 3.3 Analogy with alkali-metal suboxides Simulation thus provides complementary information to experiment, especially as regards the equilibrium geometry of the clusters, which cannot be directly observed. Since many metastable isomers exist, the reliability of our results has to be assessed.A good con–rmation comes from a study of bulk alkali-metal suboxides performed two decades ago.25 Crystalline suboxides of caesium and rubidium, such as Rb6O, Rb9O2 , and were synthesized. Their structures were discussed in terms of Cs7O, Cs4O Cs11O3 , discrete clusters of composition and intercalated with purely metallic Rb9O2 Cs11O3 , regions in the more metal-rich compounds (e.g.The clus- Rb6O\[Rb9O2]Rb3). Rb9O2 ters are two face-sharing coordination octahedra, each consisting of an oxygen atom surrounded by six Rb atoms. They are isostructural with the clusters that we III-Li9O2 have obtained in our simulation. The units were recognized as also built from Cs11O3 two face-sharing coordination octahedra upon which is added a third octahedron, which shares adjacent faces with the others.These are the only units serving as building blocks in the crystalline suboxides. Further addition of metal atoms results in a phase separa-F. Finocchi et al. 239 tion rather than addition to the unit itself. Our –ndings support this conclusion.First, we were not able to –nd stable (or metastable) clusters in which the oxygen coordination number was larger than six. Moreover, the stability of the local octahedral environment for the oxygens is re—ected in the fact that the simulation produced the cluster, III-Li9O2 the cluster and an isomer, in which six-fold coordinated oxygens are II-Li10O2 Li11O2 bridged by three, two and one Li(2), respectively.These represent the three possible ways to connect two octahedra (by a face, an edge or a vertex). It can be hypothesized that the eight-fold coordination found in the anti—uorite structure is not stable as a local environment of an isolated oxygen, but requires a larger ratio of oxygen to metal atoms. 3.4 Analysis of interatomic distances According to the –rst models by Born and Madelung, ionic solids are described as a pile of positively and negatively charged spheres, held together by attractive Coulomb interactions between anions and cations. These models were sustained by the observation that an approximately constant radius, the so-called ionic radius, could be assigned to each ion, the anion»cation shortest interatomic distance being in most cases equal to the sum of ionic radii.Later, it was recognized that small variations in the ionic radii have to be introduced when the local environment of the ions changes.26 As a qualitative trend, the larger the coordination number, the larger the ionic radius. In our previous study of stoichiometric clusters,17 we analysed the oxygen»lithium –rst- Li2nOn neighbour distances, over a wide range of diÜerent local environments.We pointed out that the mean bond lengths and around a p-fold coordinated oxygen or d 6 (O(p)) d 6 (Li(p)) lithium, respectively, are increasing functions of p. The two families of nonstoichiometric clusters also display a wide variety of LiwO bonds, owing to Li4`pO2 the increasing coordination of the oxygens in each family.In addition, extra bond distortion could be induced by the non-stoichiometry. Fig. 3 shows the variations in the –rst-neighbour LiwO distances (in as a function ”) of both the lithium and the oxygen coordination numbers. As in the stoichiometric clusters, the Li(1)wO bond lengths are systematically shorter than their Li(2)wO counterparts for a given p value, i.e.for a given oxygen environment. Moreover, a large bond stretching takes place when the coordination of the oxygen increases, for both Li(1) and Li(2) atoms. The eÜect is ca. 12»13% when one compares LiwO(3) and LiwO(6) bonds. Part of the dilatation eÜect may be assigned to the Li»Li repulsion which increases as a function of the oxygen coordination number.However, quantitatively, the slope of the LiwO distances vs. p curves is much larger than in the stoichiometric clusters. In the latter, the maximum expansion of Li(1)wO and Li(2)wO bonds only amounts to 2% and 6%, respectively. In the non-stoichiometric clusters, the LiwO bond lengths are, thus, not only –xed by the local environment of the atoms but also by the overall electronic structure of the cluster.Starting from the Li4O2 parent, the addition of each neutral lithium brings one excess electron, which is redistributed in the cluster, thus changing the ionic charges. It may be anticipated from observation of LiwO bond stretching that the excess electrons are (at least partially) borne by the lithiums, thus reducing their attractive interaction with the oxygens.The following section is devoted to a more thorough investigation of this eÜect. We conclude this section by some additional remarks on the geometry of the IIIisomer. As mentioned above, this isomer is isostructural with the cluster Li9O2 Rb9O2 which serves as a building block in crystalline and suboxides. It was noted Rb9O2 Rb6O by the author of ref. 25 that the characteristic distances in clusters are close to Rb9O2 those observed in bulk and that the oxygens are shifted from the octahedron Rb2O, centre towards the outer Rb atoms, so as to increase the OwO distance. Both conclusions hold in The shift of the oxygens is due to the inequivalence of Li(1) and III-Li9O2 .240 Simulations of non-stoichiometric lithium»oxygen clusters Fig. 3 Lithium»oxygen interatomic distances in the symmetric clusters (II-isomers when Li4`pO2 p is even and family III-isomers when p is odd).Full and dashed lines refer to Li(1)wO and Li(2)wO bond lengths, respectively. Li(2), the Li(2)wO bonds being longer than the Li(1)wO bonds. The mean LiwO interatomic distances in (1.99 are also close to those observed in bulk III-Li9O2 ”) Li2 O (1.97 This is the result of the competition between bond contraction associated with ”).the decrease in coordination (O(6), Li(1) and Li(2) in cf. O(8) and Li(4) in bulk III-Li9O2 , and bond expansion induced by the non-stoichiometry in the cluster. Li2O) 4 Electronic structure Bulk is a very ionic insulating oxide whose electronic structure displays a valence Li2O band, mainly built from oxygen orbitals, and a conduction band of mainly lithium character.These features are shared by the stoichiometric clusters and, in particular, Li2nOn by Clusters actually possess a discrete set of electronic levels, that we will refer to Li4O2 . as valence and conduction states. When neutral lithium atoms are added, the number of conduction states increases and the lowest energy ones get –lled.As a function of the number, p, of excess lithiums in clusters, the electronic structure displays sys- Li4`pO2 tematic changes in the position of the valence states, the nature of the conduction states, the charge distribution and the HOMO (highest occupied molecular orbital) energy. 4.1 Valence states The oxygen-derived valence states form two groups (equivalent to the lower and upper valence bands in oxides) associated, respectively, to the oxygen 2s and 2p orbitals.When the number of additional lithiums increases, a systematic shift of these levels towards lower energies is found, as shown in Fig. 4. The variations are better described as a function of the oxygen coordination number which is equal to 3, 4, 4, 5, 5, 6 and 6 in the series. The shift is of the order of 1 eV per unit change in the oxygen coordination number.It may be assigned to an enhancement of the Madelung potential acting on the oxygens, which is created by the positively charged surrounding lithiums. Interestingly, the value of the shift is equivalent to that created by a point charge of ]0.12 electronF. Finocchi et al. 241 Fig. 4 Mean energy position of the upper (left panel) and lower (right panel) valence states, as a function of p, for the symmetric clusters (family II when p is even and family III when p is Li4`pO2 odd) located at a distance of 1.7 (the average LiwO distance) from the oxygen.This very ” small value suggests that the lithium atoms in non-stoichiometric clusters cannot be simply interpreted as Li` entities. It also suggests that the additional lithium does not remain in a neutral state, otherwise there would be no change in the Madelung potential.A careful analysis of the charge distribution for the higher occupied states is therefore necessary to understand the non-trivial modi–cation of the electronic structure induced by Li adsorption. 4.2 Conduction states In non-stoichiometric clusters, above the valence states, some conduction states get –lled.Plate 1 (a)»(d) represent electron density maps of these states in the two Li6O2 isomers. Several features concerning (i) the delocalization of the wavefunction, (ii) the relative weight of Li(1) and Li(2) orbitals, (iii) the electron redistribution and (iv) the energy of –lled conduction states are worth noting.(i) The conduction states are highly delocalized over the whole cluster. By comparison with the valence states, their maximum density is ca. two orders of magnitude smaller. The iso-density maps shown in Plate 1 clearly indicate that the wavefunctions have a complex character in which coexist anti-bonding between the oxygen 2s and 2p orbitals and the lithiums as well as collective hybridization between lithium orbitals.(ii) Quite systematically [see e.g. Plate 1(c)], in a given cluster, the lowest conduction states have a larger weight on Li(1) than on Li(2) atoms. This is consistent with the relative values of the Madelung potentials exerted by their oxygen neighbours. This potential raises the lithium eÜective atomic levels and is stronger on the Li(2) than on the Li(1).For the same reason, when inequivalent Li(1) are present in an isomer, those located in the most compact part of the cluster have a larger contribution to the lowest conduction states. (iii) Each excess lithium brings one electron which is distributed in the whole cluster. All eÜective lithium charges are decreased, the eÜect being stronger on the Li(1) atoms.The consequence is a weakening of the LiwO bonds, as noted in the previous section. (iv) We have reported in Fig. 5 the dependence of the energies of the –lled conduction states of both spin directions, as a function of p, in non-magnetic clus- II-Li4`pO2 ters. The most noticeable feature is the negative slope of these curves which results both242 Simulations of non-stoichiometric lithium»oxygen clusters Plate 1 Iso-density surfaces of the HOMO in isomers: (a) and (b) at high and low Li6O2 II-Li6O2 electronic density (25% and 12%, respectively of the maximum density) ; (c) and (d) at III-Li6O2 high and low electronic density (25% and 12%, respectively, of the maximum density).Atoms are represented by –lled spheres, oxygens in red and lithiums in yellow.(a) and (c) evidence the antibonding LiwO character of the conduction states and the relative contributions of inequivalent lithiums ; (b) and (d) illustrate the high degree of delocalization of the conduction states. from a shift of the Li eÜective levels towards lower energies (same Madelung potential eÜect as for the valence states) and from an overall increase of the bonding character of the state as it gets delocalized on more lithium atoms.The behaviour is similar for the clusters of the second family. A modi–cation of these curves around p\4 takes place when the magnetic ground state of is taken into account (see below), but the Li8O2 trend remains unchanged. The energy diÜerence between the vacuum level and the cluster HOMO is report- Ei ed in Fig. 6, as an indication for the variations of the ionization potential of these clusters. The calculation does not take into account self-consistent –eld relaxation eÜects, so that only qualitative trends can be discussed. displays its largest value in Ei the stoichiometric clusters. This is because their HOMO is a valence state, while the HOMO of non-stoichiometric clusters is a conduction state.also displays odd»even Ei oscillations as a function of p. Whenever a new conduction state starts being –lled (for p even), a decrease in takes place, which is followed by an increase, resulting from the EiF. Finocchi et al. 243 Fig. 5 Energy position of the conduction states, for both spin directions, in non-stoichiometric non-magnetic clusters, with respect to the vacuum level II-Li4`pO2 negative slope of the level energy curves as a function of p (Fig. 5). The oscillations disappear around p\4, which is the stoichiometry for which a spin-polarized ground state is found. In Fig. 7, we have reported the HOMO»LUMO (lowest unoccupied molecular orbital) gap, G as a function of p. Great care should be taken in the interpretation of G, since DFT-LDA theory is, in principle, not designed to yield correct band gaps.27 Fig. 6 Energy diÜerence between the vacuum level and the HOMO, in non-stoichiometric Ei clusters, as a function of p. (»») family ; (» ») family. Li4`pO2 II-Li4`pO2 III-Li4`pO2244 Simulations of non-stoichiometric lithium»oxygen clusters Fig. 7 HOMO»LUMO energy diÜerence, G, in non-stoichiometric clusters, as a function Li4`pO2 of p.Only values for p even are reported. (»») family ; (» ») family. II-Li4`pO2 III-Li4`pO2 However, it gives information on the nature of the conduction states and on the overall stability of the clusters. The stoichiometric clusters display the highest G values owing to the valence character of their HOMO. In G ranges from 1.3 to 2.3 eV depending Li4O2 , upon the isomer, while the LDA gap in bulk is close to 5 eV.More sophisticated Li2O ab initio calculations of the self energy in the bulk give 7 eV.28 Gap variations come partly from diÜerences in the Madelung potentials acting on the atoms in diÜerent environments. In the bulk, a large value of the Madelung potential on Li(4) and O(8) atoms, strongly shifts their eÜective levels towards higher and lower energies, respectively, thus opening a wide gap.In clusters, the shifts are weaker than in the bulk Li4O2 because of the lower coordination numbers. In non-stoichiometric isomers, G Li4`pO2 decreases as a function of p in the –rst stages of adsorption. In family II, for example, the low gap values at p\2 and p\4 may be understood as resulting from a quasidegeneracy of lithium-derived bonding states, involving, respectively, lithium orbitals from one or the other side of the clusters.This degeneracy is lifted by a structural distortion which brings either one, or two pairs of Li(1), belonging to diÜerent ends of the cluster, at a distance characteristic of a dimer. It is a Jahn»Teller type distortion. Li2 G reaches a minimum value for Although the HOMO»LUMO gap is not a Li8O2 .reliable measure of the actual gap in the excitation spectrum of the clusters, the low G value in suggests the existence of a quasi-degeneracy of the electronic levels close Li8O2 to the Fermi level. It is well known, in Peierlsœ-type systems for instance, that a small perturbation in the atomic or spin degrees of freedom may in many cases suppress the degeneracy and induce a lowering of the electronic energy.This led us to run LSD calculations for the two isomers to know whether the ground state is spin pol- Li8O2 arized. In both cases, the answer turned out to be positive with a tiny energy gain of 0.03 and 0.09 eV for the II- and isomers, respectively. III-Li8O2 5 Attachment energy and cluster stability Understanding the mechanism of the growth of the clusters by attachment of lithium atoms is of key importance, because it can shed some light on the reverse process ofF.Finocchi et al. 245 oxygen vacancy formation in oxide materials. Among the two mechanisms of growth, by Li attachment on oxygens or on other lithiums, we will restrict ourselves to the former, and we will show that its probability of occurrence decreases as the number of excess lithiums grows.We consider two energetic parameters characterizing this growth. The –rst one is the energy of attachment of a one-fold coordinated Li(1) on a cluster, equal to dp Li4`pO2 minus the change in energy in the processes : dp(II) : II-Li4`pO2]Li]II-Li4`p`1O2 (5.1) dp(III) : III-Li4`pO2]Li]III-Li4`p`1O2 (5.2) where we have distinguished the growth of clusters belonging to the II- or the III-family.The second parameter is the attachment energy of a two-fold coordinated Li(2) on a dp @ isomer. It is equal to minus the change in energy in the transformation : II-Li4`pO2 II-Li4`pO2]Li]III-Li4`p`1O2 (5.3) De–ned in this way, positive values of and mean that the process of adsorption of dp dp @ the lithium is energetically favoured. Table 1 gives the calculated values of dp(II), dp(III) and as a function of p. dp @ Note, in Table 1, that and are of the same order of magnitude.This could have dp@ dp been expected because of the small energy diÜerence between the parent clusters. Moreover, the attachment energy is only weakly dependent on the site of adsorption (one-fold or two-fold coordination).The values of these energetic parameters compare well with the heat of evaporation of an extra metal atom from with p[1, determined Lip `(Li2O)n experimentally10 to be equal to 1.3 eV. The overall decrease in both and as a function of p, shows that the Li enrich- dp dp @ , ment of the cluster through the formation of new LiwO bonds becomes less and less favoured as the cluster size increases.This process is intimately related to the weakening of LiwO bonds, as discussed in Section 4.2, and to the strengthening of Li»Li repulsion. This latter is also responsible for the attachment energies being larger on III-Li4`pO2 than on clusters, at equal coordination of the oxygens, which arises from the II-Li4`pO2 smaller number of Li(1) in the former.The relative stability of the II- and isomers for a given value of p is III-Li4`pO2 illustrated in Fig. 8. The II- and III-isomers are alternatively lower in energy, respectively, for even and odd values of p, up to p\3, and the energy diÜerence decreases as p grows. Beyond p\3 the III-isomers are slightly more stable. However, the disappearance of the oscillation at p\4 is not intrinsic to the geometry or to the charge Table 1 Attachment energy, of Li(1) on dp , a cluster (in eV) and attachment Li4`pO2 energy of Li(2) on a isomer dp @ II-Li4`pO2 to give a cluster (see text) III-Li4`p`1O2 p dp(II) dp(III) dp @ 0 2.19 2.69 2.56 1 2.27 1.83 2.20 2 1.32 1.43 1.36 3 1.30 1.32 1.36 4 1.26 1.23 1.29 5 1.32 The numbers in bold represent the largest attachment energy for a given value of p.246 Simulations of non-stoichiometric lithium»oxygen clusters Fig. 8 Energy diÜerence between the III- and isomers as a function of p, taking into II-Li4`pO2 account (»»), or not (» »), the magnetic ground state of the isomers Li8O2 distribution. It results from the spin-polarized nature of the ground state of (see Li8O2 on Fig. 8 the dashed line which refers to the energies in the non-magnetic state). The oscillating behaviour cannot be explained on the basis of changes in the oxygen local environments. Rather, it is a non-local eÜect, for which the whole cluster shape is important. We propose that the existence of a symmetric environment for the two oxygens (for even values of p in II-isomers, and odd values for III-isomers, see Fig. 2) allows a larger delocalization of the conduction states in the cluster, and thus an increased LiwLi hybridization, which results in a larger stability of the cluster. From the analysis of the dynamical behaviour of the clusters, one can expect, furthermore, that, at non-zero temperatures, entropic eÜects will enhance the stability of isomers belonging to the II-family. We thus conclude that the attachment of lithium atoms on oxygens is energetically favoured in the –rst stages of lithium adsorption.In order to have a qualitative estimate of its competition with the attachment on other lithiums, two energies may be considered : the cohesive energy eV21 of the dimer and the cohesive Ecoh(Li2)\1.03 Li2 energy eV per lithium atom in metallic bulk lithium.29 In pure Li Ecoh(Li)\1.63 systems, the attachment energy thus increases when the number of LiwLi bonds increases, starting from 1 eV when a single bond is formed, up to 1.6 eV in a bcc environment.Although the charge state of lithiums in the non-stoichiometric clusters is not similar to that in pure lithium systems, we can, nevertheless, anticipate that a change of growth mechanism is likely to occur when becomes smaller than a critical energy dp bracketed between 1 and 1.6 eV.We will show in a forthcoming publication that, indeed, this transition takes place for p[3. 6 Discussion Here, we consider the similarities of clusters to other non-stoichiometric Li4`pO2 systems, and stress their peculiar features. We successively discuss the degree of delocalization of the –lled conduction states, the origin of odd»even oscillations found in diÜer-F. Finocchi et al. 247 ent physical quantities, the weakening of LiwO bonds and, –nally, we consider the numerical approaches potentially designed for such systems. 6.1 Character of the conduction states The degree of delocalization of the conduction states has been discussed in the context of other non-stoichiometric clusters.In alkali-metal halide or clus- NanCln~x NanFn~x ters, for example, it is recognized30h34 that the excess-electron wavefunction is highly localized and forms an F centre when x\1. The degree of delocalization increases as x grows. At low density of oxygen vacancies, a similar conclusion can be drawn for the bulk and the surfaces of some oxides.35,36 By analogy with theoretical arguments proposed to explain the lowering of metal surface work functions upon oxidation,37 the degree of localization of the HOMO has implications for the value of the ionization potential Low values of are found Ei . Ei whenever the HOMO is strongly localized, owing to a high kinetic contribution to the energy.Conversely, a large delocalization of the electrons in the HOMO is generally related to a large ionization potential.We –nd that the conduction states in are highly delocalized and possess Li4`pO2 antibonding LiwO character. The electrons do not completely avoid the oxygen region, even though the latter represents a weak contribution to the total wavefunction. In addition, owing to the shape of the central part of the clusters, the charge density associated with the excess electrons does not have a spherical extension, especially at low values of p.The image of excess electrons avoiding the oxygens and moving in a spherical eÜective potential is thus not supported by our calculations on clusters. LinO2 We have found no sign of F centre formation in the –rst stage of lithium attachment, neither in the II- nor in the III-family. We believe that this result is speci–c to oxide clusters, in which the creation of an oxygen vacancy in a stoichiometric species requires large topological changes.This conclusion is easily sustained by the examination of the lowest energy and isomers found in our previous study.17 In Fig. 9, we Li6O3 Li8O4 have reported the structure of which yields upon oxygen removal.C2v-Li6O3 , III-Li6O2 In this example, little overall change in the bond topology takes place. However, some lithiums experience large displacements and no sign of the vacancy location remains, neither in the geometry nor in the electronic structure of This is at variance III-Li6O2 . with bulk or surface oxides or alkali-metal halide cuboid clusters.Fig. 9 Stoichiometric parent (left) and the non-stoichiometric isomer (right). C2v-Li6O3 III-Li6O2 Li atoms are drawn in light grey, O atoms in dark grey, in all ball and stick representations. may be obtained by removal of the oxygen located at the bottom of and III-Li6O2 C2v-Li6O3 displacements of the neighbouring lithiums.248 Simulations of non-stoichiometric lithium»oxygen clusters As far as ionization potentials are concerned, the energy diÜerences between the vacuum level and the HOMO that we have found in non-stoichiometric clusters are lower than those in the stoichiometric clusters and also lower than those obtained by a similar calculation in pure and clusters (which are ca. 3 eV). This is in agreement Li6 Li8 with the arguments of ref. 37 although, in clusters, the space available for electron delocalization is not a well de–ned concept. 6.2 Oddñeven oscillations Metallic clusters present odd»even oscillations in the values of the ionization potentials and in abundance patterns obtained in mass spectra, as a result of electron pairing.38 In non-stoichiometric compound clusters, observation of similar oscillations is considered to be a signature of the presence of a metallic component in the cluster, whether segregation takes place or not.7,11 We have found odd»even staggering in the variations of with p in the two cluster Ei families, for p\4.Both the presence of oscillations for p\4 and their absence for p[4 –t well with the general scheme proposed in ref. 38. This is supported by the similarity between Fig. 6 of the present paper and Fig. 1 of ref. 38, which stresses the role of the spin degeneracy. It is, furthermore, reinforced by the fact that oscillations disappear around p\4, a stoichiometry at which the spin degeneracy is lifted and the ground state of the clusters becomes magnetic. The energetic quantity most directly comparable to abundance spectra is the second diÜerence expressed as a function of the energies of the clus- Dp\Ep`1]Ep~1[2Ep ters containing, respectively, p]1, p[1 and p excess lithiums.We have reported in Fig. 10, the values of in the two families. There is no well de–ned signature of odd» Dp even staggering, either in the II- or III-family. The existence of delocalized conduction states is thus not sufficient to induce odd»even oscillations in We believe that our Dp .result is due to the absence of a segregated lithium part in the clusters and that it Fig. 10 Second diÜerence (in eV) for the two cluster families, as a func- Dp\Ep`1]Ep~1[2Ep tion of p. refers to the total energy of an isomer. (»») family ; (» ») Ep Li4`pO2 II-Li4`pO2 family. III-Li4`pO2F. Finocchi et al. 249 stresses the importance of geometrical eÜects in the physics of non-stoichiometric clusters, as opposed to purely electronic eÜects. 6.3 Bond weakening Understanding the mechanism of bond weakening in non-stoichiometric systems gives information on the process of fragmentation, as well as on the interaction strength between vacancies. The calculated values (Section 5) of the energy required to detach dp , a lithium, suggest that the evaporation of lithium is easier when the overall degree of non-stoichiometry of the cluster is high.It can be related to the weakening of the LiwO bonds, as evidenced by their stretching. Moreover, if one looks at the strength of a particular LiwO bond, Li detachment turns out to be easier from the oxygen with the largest coordination number.This is exempli–ed by considering the isomer, II-Li5O2 which can yield the with symmetry if an Li(1)wO(4) bond is broken or the II-Li4O2 D2h with the symmetry (see Fig. 1) if an Li(1)wO(3) bond is broken. In the –rst II-Li4O2 C2v case, 2.32 eV are required, while in the second case, the detachment costs 3.74 eV. The diÜerence is related to the weaker character of the Li(1)wO(4) bond, on which the excess electron is more localized.This result may be compared to the process of vacancy formation in oxides like MgO. There are suggestions from electron-spin resonance, optical absorption or highresolution electron energy loss spectroscopic experiments,39h41 as well as theoretical predictions39,42 that vacancies may cluster. This means that the formation of a second vacancy is easier close to the site where a –rst one has already been created, as a result of the weakening of neighbouring MgwO bonds. 6.4 Description of the non-stoichiometry In the –eld of insulating materials, pair potential approaches have proven to be a very useful and —exible tool to describe the cohesive properties, morphologies, growth processes, defect diÜusion, surface relaxation, segregation etc.They rely on an ionic picture of the materials, in which the two driving types of interactions are the Coulomb charge» charge interactions between ions and the short-range repulsive interactions. To re–ne the description, van der Waals terms are sometimes added as well as terms accounting for the ion polarisation, usually in the so-called shell model.43 Parameters are required to make the method operative, for example the ionic charges.They are either –tted to reproduce some given measurable quantities (e.g. the cohesive energy, the bulk phonon dispersion spectrum etc.) or they are derived from ab initio calculations. In both cases, the simplicity of the approach relies on the concept of transferability of the parameters in a variety of systems characterized by diÜerent short- and long-range order.Since these methods have been so widely used for describing diÜerent physical eÜects in the –eld of ionic materials, it is useful to have a perception of how they apply in the case of non-stoichiometric oxide systems. In our previous study of stoichiometric clusters, we had found that the simple Li2nOn pair potential given in eqn.(3.1) was very helpful in exploring the con–gurational space and making a selection of low-energy isomers, whose atomic and electronic structures were subsequently re–ned by means of local ab initio minimizations. In many cases, we found that the energy ordering of the isomers was correctly given by the empirical approach, although the energy diÜerences were unreliable.We have searched the maximum degree of non-stoichiometry for which the same potential (with –xed parameters) gives qualitatively good information. Surprisingly, we have found that it correctly predicts the isomers to be the most stable, 0.5 eV below the II- III-Li5O2 isomer (0.4 eV in the ab initio approach). However, at the next stage of lithium addition, most con–gurations generated during the annealing cycles led to the detachment of a lithium ; a single low-energy isomer of the II-type was found.We also tried to simulate250 Simulations of non-stoichiometric lithium»oxygen clusters species but none were stable. The reason for this failure is very obvious. Keeping Li8O2 the ionic charges constant induces a progressive charging of the clusters. The attachment energy of the lithiums decreases drastically as p grows: eV; eV; for d1\2.1 d2\0.5 pP3, a too-large repulsion between positively charged lithiums takes place, preventing further growth.This is a physical situation in which the charge distribution is far from that present in the model system on which the –t was performed (bulk and the Li2O triatomic molecule).Very likely, a pair potential could be built to describe nonstoichiometric clusters, but the parameters should be changed for each lithium content. The method would lose its attraction which comes from the transferability of parameters. 7 Conclusions Ground-state properties of non-stoichiometric lithium oxide clusters (pO6) Li4`pO2 have been studied by means of AIMD simulations.The present study is restricted to the investigation of the low-energy isomers in which all lithiums are bound to oxygens. The structure, bonding properties and stability have been analysed as a function of the number, p, of excess lithiums. We have found two families of isomers for each lithium content, in which the two oxygens are bridged by either two or three lithiums.Their energies are very close to each other, the most stable isomer being the one in which the two oxygens have the same coordination number. Formation of six LiwO bonds around each oxygen marks the –nal step of stable lithium attachment that we –nd. The largest size clusters are two octahedra sharing an edge (II- or a face (III- the latter con–gu- Li6O Li10O2) Li9 O2), ration being isostructural with the entity, found to be the building block of Rb9O2 and suboxides.25 Rb9O2 Rb6O The electronic structure of clusters is characterized by the progressive –lling Li4`pO2 of conduction states, of LiwO antibonding character, that we have found to be highly delocalized.The relative contributions of inequivalent lithium orbitals to the conduction states may be related to the Madelung potentials acting on these atoms, which raise up to a diÜerent degree their eÜective atomic levels. An LiwO bond expansion is induced by the –lling of conduction states, which is revealed by the progressive decrease of the Li attachment probability as p grows. The presence of even»odd staggering in electronicrelated quantities (the vacuum»HOMO energy diÜerence) and their absence in energetics quantities (the second diÜerence in total energies) reveals the subtle interplay between electronic and geometric eÜects in these families of non-stoichiometric clusters. Relying upon the relative values of the attachment energies that we –nd, and those expected in pure lithium systems, we argue that a cross-over takes place at intermediate values of p (p[3) towards another growth mechanism by attachment of lithiums on other lithiums.The thorough investigation of this mechanism is the subject of a forthcoming paper. Calculations were performed on the Cray C98 at the IDRIS computational centre in Orsay (project 960109). We thank the IDRIS staÜ for helpful collaboration and technical assistance and in particular T. Goldmann for his help in the use of the AVS application (Plate 1).Fig. 1, 2 and 9 were drawn using the RasMol program developed by Roger Sayle. Useful discussions with C. Breç chignac and M. De Frutos are gratefully acknowledged. References 1 V. E. Henrich and P. A. Cox, T he Surface Science of Metal Oxides, Cambridge University Press, Cambridge, 1994.F. Finocchi et al. 251 2 C.Noguera, Physics and Chemistry at Oxide Surfaces, Cambridge University Press, Cambridge, 1996; Physique et Chimie des Surfaces dœOxydes, Eyrolles, 1995, collection Alea. 3 W. A. Saunders, Phys. Rev. B, 1989, 37, 6583. 4 T. P. Martin and T. Bergmann, J. Chem. Phys., 1989, 90, 6664. 5 P. J. Ziemann and A. W. Castleman, J. Chem. Phys., 1991, 94, 718. 6 A. Nakajima, T. Sugioka, K.Hoshino and K. Kaya, Chem. Phys. L ett., 1992, 189, 455. 7 T. Bergmann, H. G. Limberg and T. P. Martin, Phys. Rev. L ett., 1988, 60, 1767. 8 H. G. Limberg and T. P. Martin, J. Chem. Phys., 1989, 90, 2979. 9 T. Bergmann and T. P. Martin, J. 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B, 1987, 35, 9437. 23 V. Bonacic-Koutecky, I. Boustani and J. Koutecky, Int. J. Quantum Chem., 1990, 38, 149; and personal communication. 24 T. P. Martin, Physica B, 1984, 127, 214. 25 A. Simon, in Crystal Structure and Chemical Bonding in Inorganic Chemistry, ed. C. J. M. Rooymans and A. Rabenau, North Holland Publishing, Amsterdam, 1975. 26 R. D. Shannon, Acta Crystallogr. Sect. A, 1976, 32, 751. 27 R. W. Godby, M. Schlué ter and L. J. Sham, Phys. Rev. L ett., 1986, 56, 2415; Phys. Rev. B, 1988, 37, 10159. 28 S. Albrecht, G. Onida and L. Reining, Phys. Rev. B, 1997, 55, 10278. 29 C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976. 30 U. Landman, D. Scharf and J. Jortner, Phys. Rev. L ett., 1985, 54, 1860. 31 E. C. Honea, M. L. Homer, P. Labastie and R. L. Whetten, Phys. Rev. L ett., 1989, 63, 394. 32 G. Rajagopal, R. N. Barnett and U. Landman, Phys. Rev. L ett., 1991, 67, 727. 33 P. Xia and L. A. Bloom–eld, Phys. Rev. L ett., 1993, 70, 1779. 34 R. N. Barnett, H. P. Cheng, H. Hakkinen and U. Landman, J. Phys. Chem., 1995, 99, 7731. 35 L. N. Kantorovich, J. M. Holender and M. J. Gillan, Surf. Sci., 1995, 343, 221. 36 A. M. Ferrari and G. Pacchioni, J. Phys. Chem., 1995, 99, 17010. 37 M. G. Burt and V. Heine, J. Phys. C: Solid State Phys., 1978, 11, 961. 38 M. Manninen, J. Mansikka-aho, H. Nishioka and Y. Takahashi, Z. Phys. D, 1994, 31, 259. 39 K. C. To, A. M. Stoneham and B. Henderson, Phys. Rev., 1969, 186, 1237. 40 B. Henderson and D. H. Bowen, J. Phys. C, 1971, 4, 1487. 41 M. C. Wu, C. M. Truong and D. W. Goodman, Phys. Rev. B, 1992, 46, 12688. 42 E. Castanier and C. Noguera, Surf. Sci., 1996, 364, 17. 43 W. Cochran, Crit. Rev. Solid State, 1971, 2, 1. Paper 7/02171B; Received 1st April, 1997

ISSN:1359-6640    

DOI:10.1039/a702171b

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss. 1997 106 253»272 General Discussion Prof. Hillier opened the discussion by addressing Prof. Gillan Prof. Pacchioni and Prof. Dovesi Periodic calculations in both a plane wave basis and atom-centred Gaussian functions should give the same predictions for sufficiently large basis sets. What considerations determine which method should be used for a particular application ? Prof. Gillan answered The advantages of plane waves are freedom from the bias of the user robustness of convergence with respect to the size of the basis set and ease of calculation of forces. (The last is mainly responsible for the widespread use of plane wave basis sets in dynamical ab initio calculations.) The advantages of atom-centred basis functions are the possibility of doing all-electron calculations economy in both CPU time and memory and ease of chemical and physical interpretation.The method chosen depends on the importance that the user attaches to these advantages. Prof. Pacchioni responded to Prof. Hillierœs question One of the major advantages in the use of localized basis functions in quantum mechanical calculations is the possibility to interpret the results in terms of atomic and molecular orbitals. This provides immediately the natural language of chemistry the language of orbitals and makes the analysis of the results very transparent. For instance one can consider the importance of high angular momentum basis functions for the calculation of a given property (e.g. the d orbitals on S or P) and easily see if these orbitals are directly involved in the chemical bond or rather act simply as polarization functions.The use of localized orbitals is also convenient from a practical point of view since a few properly optimized Gaussian functions are usually sufficient to describe the characteristics of core orbitals quantitatively. Prof. Dovesi also replied As stated in the question at the limit of a complete set the two basis sets are equivalent but this is not very useful practically. In discussing this point there are many aspects to be taken into account and many variables that can in—uence the choice. For example a plane wave (PW) basis set is much more ì friendly œ than a Gaussian-type orbital (GTO) basis evaluation of integrals and implementation of algorithms is much easier ; part of the success of PWs is related to this aspect.A GTO basis requires some experience in the choice of contractions shell type (s p d f ) exponents of the Gaussians etc. ; this experience is quite diÜuse in molecular quantum chemistry ; to many solid-state physicists on the contrary this matter appears in some ways magic. In my opinion a GTO basis is to be preferred in many cases the exception being metals where very delocalized functions are required. Let me summarize some of the positive features of a localized basis (i) They can be transferred from system to system so that the optimization eÜort is not so large as it might appear. For example it is possible to describe Si and O atoms in all silicates where Si is four-fold coordinated by essentially the same basis set with a minor re–nement eÜort for just the most diÜuse functions.(ii) Both all-electron and pseudopotential basis sets are easily prepared. Moreover in codes such as CRYSTAL the cost of an all-electron calculation is only marginally greater than for a pseudopotential calculation. (iii) The interpretation of the wavefunction coefficients in terms of chemical concepts is natural and easy. (iv) When the unit cell is large the dimension of the matrices to be handled remains reasonably small. Consider for example silico-faujasite (144 atoms in the primitive cell). 253 254 General Discussion If a GTO basis is used 13»18 basis functions (atomic orbitals) per atom are sufficient to describe the system very accurately.This means that the basis set size is between 1500 and 3000; with these numbers the cost of the calculation remains reasonable both in terms of CPU time (ca. 5 h on a RISC 6000 workstation) and disk requirements. Prof. Sauer made a general comment I would like to comment on the use of diÜerent ab initio approaches for solid-state problems. One type uses basis functions localized on atoms (Gaussians as in the molecular codes GAUSSIAN9x CADPACK GAMESS or TURBMOLE and the solid-state code CRYSTAL numerical atomic functions as in DMOL and DSOLID). The other type uses PW basis sets (CASTEP Parrinelloœs CPMD) which have the major advantage that one gets the forces on the nuclei needed for structure optimizations or molecular dynamics simulations basically ì for free œ.However the computational efficiency of the PW codes is intimately connected with the use of pseudopotentials and the density functional. For example dispersion energy is not covered by existing functionals and eÜects which crucially depend on dispersion as does hydrocarbon adsorption within zeolite catalysts cannot be described. In contrast dispersion is already included in the second order of Moé ller»Plesset perturbation theory (MP2). The computational efficiency of MP2 calculations using Gaussian basis sets has recently been improved by almost one order of magnitude using the so-called RI approach (resolution of identity see the work of Marco Haé ser1 from Karlsruhe and similar developments at other places). It is also known that even gradient corrected functionals underestimate reaction barriers and overestimate the bond elongation on the formation of hydrogen bonds.This can be recti–ed to an appreciable extent by mixing some Hartree»Fock (HF) exchange into the functionals as done in the B3LYP functional. However using plane waves HF exchange cannot be calculated with the same efficiency as the density functional (DF) term. The two approaches are complementary and have their speci–c limits and merits. Both will continue to be needed in the future. 1 F. Weigaud and M. Haé ser T heor. Chem. Acc. 1977 97 331. Dr Mackrodt directed the discussion towards Prof. Gillanœs paper Since pseudopotentials in the various forms that they now appear are the key to so many density functional approaches to the calculation of electronic structures could you possibly give a few explanatory words as to the number of adjustable parameters contained in such potentials how they are generally determined and the sensitivity of the –nal solutions to these parameters? Prof.Gillan responded Nowadays pseudopotentials are generated in one of a small number of standard ways from ab initio calculations on free atoms. Two main choices have to be made (i) Which atomic states are treated as valence states and which as core states (the latter by de–nition are not allowed to respond to their environment). (ii) The degree of transferability (as described usually by the energy dependence of the logarithmic derivatives of all-electron and pseudo-orbitals). Through these choices the user controls the quality of the pseudopotential description i.e.the precision with which it reproduces an all-electron description. Prof. Jansen addressed Prof. Gillan and Prof. Dovesi I am not sure whether it is a good idea to establish and use a standardized set of pseudopotentials. Strictly speaking separating core electrons and valence electrons is always arbitrary. Where one draws the dividing line depends on the problem to be considered. DiÜerent physical probes record the in—uence of the bonding situation on the core electrons with diÜerent sensitivity or as an example from chemistry consider silver for which in K AgAlF6 including the ten 4d electrons in the core would allow for an appropriate description whereas in 2 255 General Discussion Ag2O NaAg3O2 the d electrons are heavily involved in chemical bonding.Thus or when –xing the parameters for the pseudopotentials the intended application needs to be considered. Prof. Dovesi responded I agree that in some cases the separation between core and valence electron is not so sharp; in alkali alkaline earth and transition metal atoms ìsub-valanceœ electrons although belonging to an inner shell can be perturbed by the chemical environment. For this reason in many cases (see e.g. ref. 1) two families of pseudopotentials (ìsmall coreœ and ìlarge coreœ) are proposed for the same atom; small core pseudopotentials are obviously recommended as more precise. I hope we agree that a pseudopotential should reproduce as far as possible the all-electron results.Now it happens frequently that diÜerent values are reported in the literature for simple properties (e.g. the lattice parameter or bulk modulus) of a given system (e.g. MgO) as obtained with the same method basis set and type of pseudopotential the explanation for the diÜerence being that ìdiÜerent parametersœ have been used in the pseudopotential. In these cases the pseudopotential appears to be a sort of empirical parameter inserted in the method which then loses its ab initio character. It is surprising that in solid-state physics where pseudopotentials are so widely used there are no official well tested standard complete sets of modern pseudopotentials available to the scienti–c community; the situation being on the contrary that each group/researcher builds pseudopotentials in-house ; in this situation cross-checks and comparisons are very difficult or impossible.1 P. J. Hay and W. R. Wadt J. Chem. Phys. 1985 82 270; 1985 82 284; 1985 82 299. Prof. Gillan added I agree with Prof. Jansen though I would add that even in rather ionic compounds the d electrons of silver will respond fairly strongly to their environment and this can only be properly described by including the d electrons in the valance set. I am thinking here particularly of the well known quadrupolar polarizability of the silver ion in materials like AgCl. Prof. Catlow commented The question of plane wave vs. localized basis sets may depend at least partially on the structure of the solid. For example it would seem intuitively more sensible to model a low-density framework structure using a localized basis set which avoids using large numbers of plane waves to describe the empty regions of the structure.Plane wave basis sets might be more appropriate for more dense structures. Prof. Gillan added Prof. Catlowœs comment would be correct if the density of the structure and the economy of the calculation were the only considerations. However they are not. For example in some cases it might be crucially important to demonstrate complete convergence with respect to the size of basis set. Alternatively it might be highly desirable to perform dynamical ab initio calculations. In both cases I would suggest that plane waves would have a decisive advantage. Of course in other cases localized basis functions might well be the best choice.Dr Shluger commented I would like to draw attention to a diÜerent aspect of the relationship between the DF-based techniques and methods starting from the HF approximation. A wealth of information about the electronic and geometric structures of crystals and of defects in crystals is obtained using spectroscopic methods based on electron excitation and ionization. Periodic DF calculations are great for the ground state and prediction of geometries. However as the results presented by Prof. Pacchioni clearly show only the HF based methods are presently capable of predicting the optical 256 General Discussion absorption and luminescence energies. I think that in many cases only a combination of diÜerent techniques that are most appropriate for a particular aspect of a complex problem can allow one to –nd answers that could be thoroughly compared with experiment.Prof. Klein addressed Prof. Gillan Would you like to comment on the convergence of calculated surface relaxation vs. slab thickness. From rather general principles one would expect long-range oscillatory behaviour. Prof. Gillan responded You are quite correct in saying that ionic relaxations can decay slowly as one goes from the surface into the bulk. The magnitude of this eÜect depends very much on the system and this governs the slab thickness required. For example MgO(001) is a case where surface relaxations are very small and rather thin slabs are adequate. In contrast TiO (110) is a case where strong oscillatory eÜects are found and thick slabs are needed.For further details see ref. 1. 2 1 S. Bates and M. J. Gillan Surf. Sci. in press. an oxygen vacancy model rather than the correct defect model involving crystallo- Prof. Cheetham asked In your calculations on reduction of bulk TiO2 you focus on graphic shear (CS) planes. Is the treatment of the CS planes beyond the scope of the method? It would certainly be of interest to explore the behaviour of Ti3` at the CS planes. Prof. Gillan responded As you rightly say we did our oxygen vacancy calculations on the wrong defect. This was deliberate and was done to focus in a simple way on the consequences of reduction. However the correct defects namely shear planes are certainly not beyond the scope of ab initio methods and I agree that it would be interesting to use these methods to explore Ti3` ions at shear planes.Prof. Catlow commented Following Prof. Cheethamœs remark I would like to emphasise that there is a very rich solid-state chemistry of TiO2~x to which computational methods can make a direct contribution. There is good evidence1 that in the near-stoichiometric region point defects are in equilibrium with shear planes although the nature of the point defects (oxygen vacancies vs. metal interstitials) is controversial. Calculations on both these basic point defects and on the shear plane structure would be of great value. It would also be of interest to calculate the heat of reduction of TiO using your 2 vacancy energy as experimental data are available for this quantity.1 J. F. Baumard D. Panis and A. M. Anthony J. Solid State Chem. 1977 20 43. 2 Prof. Gillan responded I agree that it would be of interest (and relatively straightforward) to calculate the heat of reduction of TiO from our work. We have not yet studied this. Prof. Woodcock said Many of the complex ionic solids and solid surfaces that are now being modelled have transport and structural relaxation times which exceed even the times for formation and observation accessible in the laboratory to the real experimentalist. To perform a proper scienti–c experiment on such a solid material in addition to temperature and pressure one really needs to specify the process by which this non-equilibrium solid was produced i.e.to characterise the ì state œ of the specimen. Likewise we can easily generate a computer model of a complex solid or solid surface on ultrafast timescales of at best nanoseconds with a prescribed structure. In 257 General Discussion doing so however the relaxation processes that might bring the computer model solid to thermodynamic equilibrium are frozen out. Is it not all the more essential to characterise the ì state œ of the ìcomputer solid œ by also specifying the process by which it was produced? We know that very subtle changes in the potential models can lead to quite diÜerent structures. How do we know the structure of a computer model of a complex ionic solid in thermodynamic equilibrium ? Or to be more speci–c how do you know that you are investigating the equilibrium structure of the surface of TiO or hydrated TiO2? 2 Prof.Gillan replied You raise some important points. If there are relaxation times that greatly exceed the duration of the simulation then the simulation is not sampling correctly the states characteristic of thermal equilibrium. In the speci–c case of water on TiO (110) there are indications that the relevant relaxation times are rather short (of the order of 1 ps except at very low temperatures). However we should certainly be cau- 2 tious about this. Prof. Pacchioni asked There is no doubt that an important issue in the future will be the determination of surface reconstruction from –rst principles. However in order to do this we need reliable methods to describe surface relaxation.The calculations on the surface relaxation of TiO (110) show good agreement between DFT and HF but a substantial deviation from experiment. What is in your opinion the origin of the discrep- 2 ancy? Prof. Gillan answered I cannot be sure about the origin of the discrepancy but a likely explanation is that we are not calculating exactly what is measured. The calculations yield relaxed nuclear positions. The X-ray diÜraction measurements probe the electron distribution from which ionic relaxations are inferred via a model. We are not comparing like with like. With a bit of eÜort the correctness of this explanation could be tested by using the calculations to predict exactly what is measured. Dr Islam asked A detailed theory»experiment comparison of the ionic displacements of the TiO (110) surface is given in the paper.You note that there is a substantial discrepancy for the bridging oxygen from both DFT and HF calculations. As in the case 2 of Al2O3 have you made a comparison with simulation studies of TiO surfaces using 2 shell-model potentials ? It is interesting to note that recent atomistic simulations of TiO2 surfaces1 –nd comparable ionic relaxations to your results. 1 P. M. Oliver G. W. Watson E. T. Kelsey and S. C. Parker J. Mater. Chem. 1997 7 563. Prof. Gillan replied We have not done shell-model calculations on TiO though closely related shell-model work on TiO is being carried out at Daresbury Laboratory. 2 2 Dr Shluger said Your calculations predict dissociation of water molecules on the surface of TiO only if the concentration of water is large.What is the role of water» water interactions in this eÜect ? It is known for instance that the dipole moment of the 2 water molecule increases considerably in bulk water and even in small water clusters.1 Do you observe a similar eÜect in your calculations ? Can this change in the dipole moment of water molecules play some role in the dissociation of water adsorbed on the TiO surface ? 2 1 J. K. Gregory D. C. Clary K. Liu M. G. Brown and R. J. Saykally Science 1997 275 814. Dr Lindan responded to Dr Shluger First dissociation is predicted at the lower coverage we have studied which is when every second surface cell contains H2O (half-258 General Discussion monolayer).At monolayer coverage water»water interactions are crucial in determining the adsorbate geometry and can be discussed in terms of hydrogen bonding. The situation can be summarised as follows. itself dissociate. Instead it is favourable for it to hydrogen bond to the OH already on (i) A water molecule adsorbed on a Ti site adjacent to a dissociated H2O does not the surface. (ii) We can force all H2O to be dissociated or molecular and these states are metastable. The diÜerences in energy compared to the mixed state are small (ca. 0.02»0.10 eV per molecule). (iii) We can calculate the binding of the water molecules alone in the absence of the TiO surface. This is 0.12 eV per molecule. (iv) The dissociative adsorption energy is identical (0.91 eV per molecule) at both 2 coverages.Therefore there is no eÜective interaction between OH groups. Gathering these facts together it is clear that intermolecular interactions stabilise the mixed state produce metastability in the fully molecular state and do not contribute when all H2O is dissociated. The latter explains why in our calculations only one of the two molecules dissociates since in this case some hydrogen bonding is preserved outweighing the energy gain to be had by complete dissociation. 2O) There is clear evidence of the strong intermolecular binding in the geometry shown in Fig. 8 of our paper. The OH group in the centre of the –gure is hydrogen bonded to the water molecule which results in a contraction of their separation. The O»O distance is 2.70 ” and the O (OH)»H (H separation is 1.76 ”.Contrast this to the values of 3.28 and 2.62 ” on the non-bonded side. The same symmetry breaking is seen to a lesser is molecular but when all is dissociated there are no symmetry- degree when all H breaking relaxations. 2O Finally one further eÜect of hydrogen bonding is in the calculated vibrational spectra. For the half-monolayer case the high frequency modes are broadened by the OH»H interactions. In this respect one could say that the molecule has a highly stretched OH bond. These arguments demonstrate that intermolecular interactions are very important in this system. Whilst we have not interpreted our results explicitly in terms of dipoles and their changes they are implicitly accounted for in the calculations.and surface OH? Prof. Klein asked The oxygen»oxygen separation for adsorbed water molecules interacting with surface OH is likely to be a crucial factor in determining the resulting properties and behaviour. Can you comment on the value of this O»O separation ? Also do you see any evidence of proton transfer between the adsorbed H2O to OH. Dr Lindan replied I agree with your observation. In this system the O»O separation is 2.7 ” rather less than for example in the water dimer. However the OH bond lengths in the H2O remain close to their value in the free molecule. At the temperature studied (150 K) we –nd no evidence of proton transfer from H2O Mr Muscat communicated Subsequent to writing this paper we have improved the quality of our HF stoichiometric (110) slab calculations by doubling the thickness of our slabs.We –nd that the agreement with our DFT calculations is improved as a result but discrepancies with experimental observations (particularly the position of the bridging oxygens) are still apparent. (See Table 1.) Dr Schoé n opened the discussion of Prof. Pacchioniœs paper Referring to Table 4 of your paper is the accuracy and the agreement between theory and experiment for the hyper–ne coupling constants typical for current calculations at the UHF or DFT level ? Prof. Pacchioni responded The similarity of the computed hyper–ne coupling constants at the UHF and DFT levels found for an unpaired electron trapped in an MgO Table 1 ” Ionic displacements (in from the bulk terminated positions) due to relaxation of the (110) 1]1 surface labela 12 3 4 5678 9 10 11 12 13 a Labels refer to Fig.1 in our paper. b M. Ramamoorthy D. Vanderbilt and D. King-Smith Phys. Rev. B. 1994 49 16 721. c G. Charlton P. Howes C. Nicklin P. Steadman J. Taylor C. Muryn S. Harte J. Mercer J. McGrath D. Norman T. Turner and G. Thornton Phys. Rev. L ett. 1997 78 495. DFT calculations in our Discussion Ramamoorthy DFT calculationsb [110] [1 6 10] [110] 0.23 [0.11 [0.02 0.18 0.18 0.03 0.12 0.13 [0.16 [0.06 0.13 0.13 0.00 0.00 0.00 [0.04 0.04 0.00 0.00 0.00 0.00 [0.07 0.06 [0.08 0.02 [0.06 0.03 0.00 0.00 0.03 0.00 [0.05 0.05 0.00 » [0.03 [0.03 [0.01 » subsequent HF calculations using thicker slabs paper [1 6 10] [110] [1 6 10] 0.25 [0.17 [0.01 0.12 0.12 0.03 0.15 0.00 0.00 0.00 0.00 0.00 0.00 [0.07 0.07 0.00 0.00 0.00 0.00 0.03 [0.05 0.05 0.00 0.00 0.00 0.00 [0.12 [0.01 0.02 0.02 [0.02 0.02 0.00 0.00 [0.03 0.00 0.00 [0.03 0.04 Charlton surface X-ray diÜractionc [110] ^0.05 ^0.05 ^0.08 0.12 [0.16 [0.27 0.05 0.05 0.05 0.07 ^0.05 ^0.05 ^0.08 ^0.04 ^0.04 ^0.08 [0.09 0.00 0.02 0.02 ^0.06 ^0.06 ^0.08 [0.09 [0.12 ^0.07 [1 6 10] 0.00 0.00 0.00 0.16 0.16 0.00 0.00 0.00 0.00 0.07 0.07 0.00 0.00 » » » ^0.08 ^0.08 »» » » ^0.06 ^0.06 » » 259 General Discussion 260 General Discussion oxygen vacancy is not a general feature.It has been shown that correlation eÜects act to reduce substantially the hyper–ne coupling constants in molecular radicals.1 This is also the reason why we wanted to compare the two approaches. The fact that in the case of MgO F centres the two methods give similar answers can be easily explained with the high localization of the electron in the centre of the vacancy and the very small interaction with the neighbouring Mg ions. On the other hand we also found that for SiO2 paramagnetic defects where the unpaired electron is more localized on an Si atom UHF and DFT give similar answers.It seems that for this class of inorganic materials the inclusion of correlation for the description of the hyper–ne interactions is less crucial than for organic molecules but there is insufficient experience to draw a general conclusion. 1 V. Barone J. Chem. Phys. 1994 101 6834. Dr Shluger addressed Prof. Pacchioni and Prof. Gillan One of the main outcomes of theoretical calculations is a defect model. One of the models of electron centres in ionic crystals associated with anion vacancies which has been around for years is that of the F centre. In this model the electron is believed to be strongly localized in the anion vacancy. Conceptually it is interesting to understand how far one can stretch this model especially for less ionic systems.In Prof. Gillanœs paper the authors compare the clusion seems to be very unclear. Could you please comment on how you visualize the results of HF and DFT calculations for the anion vacancy in TiO2 . However the conmodel of this centre ? How strongly is the excess electron localized in the vacancy? Is the F centre model applicable ? Prof. Pacchioni replied The idea of F centres i.e. of electrons trapped in an anion vacancy is valid for very ionic materials such as NaCl or MgO. In this case the driving force for the localization is the large energy gain due to the electrostatic interaction of the trapped electron(s) with the Madelung –eld of the crystal. It is much more favourable to localize the electronic charge in the centre of the vacancy than to distribute it over the empty 3s orbitals of the surrounding cations.The situation is completely diÜeroxygen atom results in the formation of a very localized electron pair between two Si ent in materials with a high covalent character e.g. SiO2 where the removal of an atoms with formation of a direct SiwSi bond. In the charged version of the same defect the paramagnetic V0 ` centre the unpaired electron is fully localized on one Si atom. In this respect EPR turns out to be an extremely powerful technique since it provides a direct measure of the degree of localization and hence of the nature of the defect. Prof. Gillan responded to Dr Shluger As discussed in our paper we have not arrived at any –nal conclusions about the degree of localization of the excess electrons in the vacancy.More work is needed. Dr Noguera addressed Prof. Pacchioni You mentioned in your presentation that the degree of localization of the two excess electrons in the site of a neutral oxygen vacancy is dictated by the ionicity of the oxide under consideration. The more general criterion in my opinion is the strength of the Madelung potential at the vacancy site which depends not only on the charge values but also on the geometry around the vacancy. As I show in my paper in small lithium oxide clusters there is no localization at all of the excess electrons despite the high ionic character of Li2O. Prof. Cheetham asked Prof. Dovesi You have demonstrated that the relative energies of antiferromagnetic (AFM) and ferromagnetic (FM) structures can be predicted by using the CRYSTAL95 code.The spin structures are of course assumed in these calculations. I have two questions : 261 General Discussion (i) Can we predict more features of the magnetic structure e.g. the spin direction ? (ii) Could we predict the spin structure a priori ? (This has been done at a less rigorous Monte Carlo level by Feç rey and co-workers.1) 1 P. Lacorre J. Pannetier M. Leblance and G. Feç rey J. Magn. Magn. Mater. 1991 92 366. Prof. Dovesi responded (i) In our calculations we impose a priori the total number of unpaired electrons (a[b) in the unit cell ; no constraint at all is imposed on the partition of the spin density between the transition metal atoms and between the d orbitals of a given atom.The –nal spin structure is then the result of a variational calculation with the constraint of an a priori de–ned number of (a[b) electrons. (ii) It is possible to evaluate the relative stability of diÜerent magnetic structures (see for example the three diÜerent spin structures reported in Fig. 1 of our paper) and then establish (within the limits of the theory) if the system is ferromagnetic ferrimagnetic antiferromagnetic or non-magnetic and within a given category which is the preferred spin structure. (iii) The results have been obtained with the unrestricted Hartree»Fock (UHF) method whose wavefunctions are also wavefunctions of the Så z operator (projection along a selected direction) but not of Så 2. It is impossible for us to know the total spin direction or the direction of the spin of individual atoms.The same consideration applies to other methods (DFT for example). Dr Mackrodt added Ignoring the question of correlation eÜects the UHF method since it is Ising-like i.e. only spin-up and spin-down cannot give a full description of spin-wave patterns or spectra other than for spin-1 systems. 2 CoF63~ for octahedrally coordinated Co3` in a high-spin con–guration is Prof. Jansen opened the discussion of Prof. Cattiœs paper Considering that only one example known and all ternary oxides containing Co3` in an octahedral environment are diamagnetic or show temperature-independent paramagnetism your results concerning the magnetic properties of Co3` seem rather surprising to me.Can you comment on this issue with respect to the quality of your calculations ? Prof. Catti responded The UHF approximation on which our calculations are based accounts for electron exchange energy exactly but neglects correlation eÜects. Thus high-spin (HS) electron con–gurations (for which the exchange energy is maximum) are generally predicted to be more stable than low-spin (LS) ones for openshell systems. The a posteriori correction for correlation which may be applied to the HF wavefunction reduces the energy diÜerence between HS and LS states but is not accurate enough to change its sign. Therefore the method used is not able to predict a LS con–guration (and diamagnetic behaviour) for Co2O3 (and possibly Ni2O3). However as all other M2O3 oxides investigated have M3` in HS con–guration as expected from experimental evidence and Co2O3 and Ni2O3 (not observed compounds) were considered primarily to investigate the eÜect of increasing the number of d electrons within homogeneous conditions of structure and electron con–guration in the M2O3 series the hypothetical HS FM or AFM states of cobalt and nickel oxides are those of real interest in this context.Co Prof. Cheetham asked Do you believe that the oxides Co2O3 have never been made experimentally) are thermodynamically stable ? If the oxidation of and Ni2O3 (which 3O4 to Co2O3 has a positive free energy surely the compound is inaccessible ? Prof. Catti responded The results of our calculations show that the binding energies oxides. Also are negative and comparable to those of other M2O3 of Co2O3 and Ni2O3 262 General Discussion the corresponding formation enthalpies at T \0 K are negative indicating that the compounds may be thermodynamically stable in some range of temperature and oxygen partial pressure unless the entropy contribution is particularly unfavourable.In this respect I believe that the oxidation of Co may also be feasible thermody- 3O4 to Co2O3 major problems in obtaining Co namically (with a negative free energy) in some domain of the (T pO2 ) space. However 2O3 are probably the competition with CoOOH formation and unfavourable kinetics. Prof. Dovesi commented The simulation of solid-state reactions and of the relative stability of diÜerent polymorphs is a delicate problem that requires high numerical accuracy as often energy diÜerences are very small (a few kcal mol~1).Accurate ab initio periodic codes can however tackle this problem providing quite accurate results. I would like to mention two recent papers referring to the following reactions MgAl2O4 (spinel)¢MgO]Al2O3 (ref. 1) Mg3Al2Si3O12 (garnet)¢3MgO]Al2O3]3SiO2 (quartz) (ref. 2) 1 M. Catti G. Valerio R. Dovesi and M. Causa` Phys. Rev. B Condens. Matter 1994 49 14 179. 2 Ph. DœArco F. Freyria-Fava R. Dovesi and V. R. Saunders J. Phys. Condens. Matter 1996 8 8815. 2O3 Dr Schoé n said The free energy aspect mentioned by Prof. Cheetham is very important but the distinct possibility exists that the proposed Co2O3 structure corresponds to a metastable minimum.How high do you suppose is the energy barrier separating Co from the competing Co3O4 compound? Prof. Catti replied I con–rm that the kinetics may be hindering the synthesis of 2O3. C3O4 Competition may occur not only withobut also particularly with Co CoOOH. In order to calculate energy barriers between multiple minima corresponding to these cobalt compounds one would have to study by ab initio methods the crystal structures of Co3O4 and CoOOH to devise reasonable atomistic paths for phase transpaths. I am not able to estimate a priori the results of this very complex work. formation into Co2O3 and to compute the pro–le of the energy function along such Prof. Sauer asked In your calculations of FM and AFM couplings of HS states do you observe signi–cant spin contamination? How large is SS2T? If the contamination is large how does this eÜect the conclusions ? Prof.Catti answered The calculation of the SS2T value and thus of the extent of spin contamination of spin-polarized periodic UHF wavefunctions has not been implemented yet in the code we are using. However I would make two comments. First HS con–gurations (such as those considered in the paper) invariably show the lowest degree of spin contamination. Second it has been shown very recently by Moreira and Illas1 that similar energy diÜerences between FM and AFM con–gurations are obtained either by periodic UHF or by cluster-model CASCI (complete active space con–guration interaction) methods on perovskite-like KNiF and K NiF4 which are ionic systems as 3 2 function it seems plausible to conclude that a possible spin contamination of the UHF are our M2O3 oxides.As the CASCI approach gives an uncontaminated spin eigenwavefunction should not aÜect the (AF»FM) energy diÜerences signi–cantly. 1 I. de P. R. Moreira and F. Illas Phys. Rev. B 1997 55 4129. Dr Johnston asked I have two questions concerning your DFT estimate of electron correlation. (i) Could you tell me how this is done? 263 General Discussion (ii) More generally what do you think are the prospects for performing a direct correlation calculation (CI/MP2) in real space in systems where the electrons are quite localized (e.g. organic molecular solids) ? Prof. Catti responded (i) The electron correlation energy is estimated a posteriori by straightforward integration of a convenient correlation functional of the electron density using the electron density derived from the SCF-converged HF wavefunction.The correlation functional used in this work is that of Perdew et al.,1 it is derived from electron gas behaviour and includes the generalized gradient approximation (GGA) accounting for the non-local character of the dependence of correlation energy on the electron density. (ii) As far as I know CI is presently hopeless for a periodic solid while attempts to implement the MP2 approach for very simple periodic systems are being carried out by some groups. However the case of organic molecular crystals is quite difficult because the crystal structures are complex and it is not clear yet whether dispersive intermolecular forces are accounted for satisfactorily by say the MP2 approximation to correlation.1 J. P. Perdew J. A. Chevary S. H. Vosko K. A. Jackson M. R. Pederson D. J. Singh and C. Fiolhais Phys. Rev. B 1992 46 6671. Prof. Hillier asked Within the CRYSTAL code the DFT corrections are applied to the converged HF density. How reliable is this procedure especially when small energy diÜerences are being predicted ? Prof. Catti replied The reliability of the a posteriori DFT correction for correlation energy can be assessed by considering the results obtained so far for say mainly ionic crystals. Its eÜect on the total energy is generally to reduce the underestimation of binding energies from ca. 30% (typical of the HF error) to ca.10%. As for small energy diÜerences in addition to the results quoted in the present paper concerning diÜerent AFM structures I can refer to phase transitions and solid-state chemical reactions. For instance in the study of the decomposition of spinel MgAl2O4 into MgO and Al2O3 at high pressure,1 it turned out that applying the DFT correction for correlation to the energy diÜerence between spinel and the assembly of simple oxides was essential in order to reproduce the observed decomposition pressure correctly. 1 M. Catti G. Valerio R. Dovesi and M. Causa` Phys. Rev. B 1994 49 14 179. Dr Moore communicated LS Ni3` should be subject to a Jahn»Teller distortion of for a tetragonal distortion ? Such a distortion would lower the energy of the LS state its environment.When calculating the energy for the LS state of Ni2O3 did you allow and could result in the LS state being the more stable. Prof. Catti communicated in reply No we did not consider the possibility of a tetragonal distortion of the LS state of Ni2O3. e DiÜerentiation of the orbitals of Ni3` by the Jahn»Teller eÜect would decrease the symmetry of corundum dramatically g raising the computational cost very much. However I agree that a sophisticated study of this system would probably require such a step. Dr Lindan opened the discussion of Prof. Priceœs paper I would like to ask three questions. (i) Is DFT capable of describing Fe? In particular are self-interaction corrections needed? (ii) You use the Stokes»Einstein relation to obtain an estimate of the viscosity of liquid iron.How good an approximation is this ? What is the uncertainty in your estimate? 264 General Discussion (iii) Calculations of the viscosity via the autocorrelation function of the stress tensor have often been performed within classical MD calculations. Have you attempted to do this ? Prof. Price responded (i) Yes DFT has been used successfully to model Fe for some time. See for example the reference to Stixrude et al.1 for more details. (ii) Our unpublished work on the Stokes»Einstein relation shows that it describes experimentally determined systems to within ca. 10%. (iii) We did not study the autocorrelation functions because our run times did not give sufficient data to make these statistically robust.1 L. Stixrude R. E. Cohen and D. J. Singh Phys. Rev. B 1994 50 6442. Prof. Gillan added In principle it would be possible to calculate the viscosity of liquid iron (or other liquids) using –rst principles MD. This has been completely standard for many years using empirical interaction models. One way is to use the Green» Kubo relation which involves the calculation of the stress autocorrelation function. The only problem is that rather long dynamical simulations (perhaps 100 ps) would be needed to get adequate statistical accuracy. It is only a matter of (a fairly short) time before this kind of direct calculation becomes possible. Dr Yashonath addressed Prof. Price and Prof. Gillan Our recent estimation of the diÜusion coefficient (D) from long MD simulations1 suggests that 1.0 ns long simulations might be required to get an accuracy of ca.10%. Errors increase rapidly with shorter run lengths (see Fig. 1) so the value of D obtained from a 1 ps run may be highly unreliable. Also the ballistic regime sometimes extends to 1»2 ps before the diÜusive regime is obtained. Hence estimating D from the –rst 1»2 ps of the U2(t) may be overestimating the value of D. Could you please comment? It is necessary to be warned that the error in D does not decrease linearly with the duration of the simulation T . If the correlation decays within 1 ps and if the simulation Fig. 1 Dependence of the percentage error in the self-diÜusion coefficient D on the duration of the MD simulation for a pure —uid with 100 particles.The solid curve is obtained from the best least squares –t to the points. The run was performed at 123 K using 2.21 ” sized particles. Details of error analysis are given in ref. 1. 265 General Discussion is 100 ps it does not mean that we could have a hundred points over which to average. This is because even to obtain D during that 1 ps a much longer run is called for since D is a —uctuation-dependent property. In order to obtain U2(t) with some accuracy for a duration of even 1 ps considerably longer runs are necessary. The difficulties related to calculation of D are discussed well by Allan and Tildesley2 and in our paper mentioned above.1 1 R. Chitra and S. Yashonath J. Phys. Chem. 1997 101 5437. 2 M. P. Allan and D. J. Tildesley in Computer Simulation of L iquids Clarendon Press Oxford 1987.Prof. Price responded The diÜusion behaviour of a system becomes well de–ned when the mean-square displacement becomes linear. For liquid metals at T [1000 K this happens in much less than 1 ps. The diÜusion coefficients inferred from such studies are likely to be accurate to within ca. 20%. Prof. Catlow added The periods of ca. 1»5 ps are probably acceptable in this system (compared with hydrocarbons in zeolites) owing to the much higher diÜusion coefficients (ca. 3»4 orders of magnitude higher in the system discussed here). In addition it is worth emphasizing that 100 ps runs on this system will become quite feasible in 1»2 years with the developments that are taking place in high-performance computing technology.Prof. Klein addressed Prof. Gillan Based on the g(R) shown in Fig. 5 of your paper (with Prof. Price) the ìaverageœ Fe atom has not diÜused out of its cage of neighbours on the 1 ps timescale. So what is the likely error bar on your estimate of the diÜusion coefficient ? Prof. Gillan responded The question assumes that the rms displacement of an atom must become comparable with the nearest neighbour separation before the diÜusion coefficient can become well de–ned. This is incorrect as a general statement. (One has only to think of diÜusion in a crystal due to the hopping of a low concentration of vacancies.) In fact the diÜusion coefficient becomes well de–ned when the mean square displacement becomes linear in time. In many liquids (including our simulated liquid iron) this happens in much less than 1 ps.However to answer the question our diÜusion coefficient is probably good to ca. 20%. Dr Yashonath communicated We would like to share some of our recent results which relate to the estimation of the diÜusion coefficient from molecular dynamics simulations. We have carried out one of the longest runs ever reported in the literature 0.12 ls and the analysis of these results shows that the diÜusion coefficients (D) obtained from rather short runs of a few hundred picoseconds or less (for argon) overestimate the value of D. The runs were carried out for monatomic species diÜusing in the zeolite (in NaCaA by employing the velocity Verlet scheme in the microcanonical ensemble). It is not clear whether pure —uids show a similar behaviour.Table 2 lists the value of D calculated over blocks of diÜerent lengths. The total length L of 54 ns is split into m blocks of length p where L \m]p. The correct value estimated by obtaining the mean squared displacement curve from the total run of 54 ns yielded a value of 0.122 911]10~8 m2 s~1. It is evident from the table that for the shorter blocks the overestimation is usually greater. (For further information see ref. 1.) These results suggest that a shorter run usually overestimates the value of D. Hence I suspect that the value of D for Fe is likely to be somewhat overestimated. 1 R. Chitra and S. Yashonath J. Phys. Chem. 1997 101 5437. 266 General Discussion Table 2 DiÜusion coefficientsa obtained from 54 ns run and –tting to the mean squared displacement curve over 200»500 ps p/ns D/10~8 m2 s~1 0.149 671 0.133 690 0.136 488 0.138 357 0.122 911 54 1369 a L (54 ns)\m]p; the sorbate diameter is 2.21 ”.Prof. Hillier asked The modelling of iron at the Earthœs core requires Fe»Fe interactions for quite small interatomic separations. Is the pseudopotential employed reliable in this respect ? Prof. Gillan replied This raises an important point. As explained in our paper it becomes essential at high pressures to include the Fe 3p electrons in the valence set. This is precisely because of the short-range interactions you mention. As we also explained once this is done comparison with all-electron calculations at Earthœs core pressures shows that our pseudopotential performs extremely well.Prof. Bliek addressed Prof. Price The numerical values obtained for both the viscosity and diÜusion coefficient are very close to the values one would expect for liquid phases at ambient conditions. Would you have any idea of the sensitivity of the numerical values obtained on the system pressure ? Prof. Price answered The work outlined by Poirier,1 which suggests that the viscosity in the core is similar to the viscosity of liquid Fe at ambient pressure explains this as being due to the fact that viscosity scales with melting temperature. Thus liquid Fe under core conditions is close to its high-pressure melting point and so is to be expected to have a viscosity similar to liquid Fe close to its ambient-pressure melting temperature.1 J. P. Poirier Geophys. J. 1988 92 99. Prof. Woodcock asked To what extent can the properties of liquid metals such as iron at very high temperatures and pressures be represented by simple classical onecomponent plasma (OCP) models? One of the earliest MC studies of charged systems1 simulated fully ionized —uid iron that exists in galactic bodies. 1 S. G. Brush H. L. Sahlin and E. Teller J. Chem. Phys. 1966 45 2102. Prof. Gillan replied Some liquid metals can be well represented by the OCP. For this to be true the atomic cores must be small compared with the nearest neighbour distance as in the alkali metals. However in transition metals such as iron core repulsion is an important eÜect and the OCP will not be good especially at high pressures.Dr Johnston said A few years ago we1 developed an empirical many-body potential 267 General Discussion for iron by parametrizing to the bcc and the fcc phases. We have not studied liquid iron yet. 1 F. Gao R. L. Johnston and J. N. Murrell J. Phys. Chem. 1993 97 12 073. Dr Line opened the discussion of Dr Galeœs paper As an experimentalist who has recently1 carried out work on ice XI (a\4.465 b\7.859 c\7.292 ”) which you refer to as C-ice I would like to ask a few questions as to possible reasons for which you may have failed to show that C-ice is more stable than P-ice as we have con–rmed that ice XI indeed has the C-ice structure and very de–nitely not the P-ice structure.1,2 (i) What symmetry constraints did you impose other than the unit cell ones? I refer to the fact that you report only one value for r(OH) and n(HOH).Did you relax all coordinates (including oxygen)? (ii) Did you consider the eÜect of the K` dopant as was considered by Pisani et al. ?3 (iii) What is the eÜect on the dipole moment of the water on considering diÜerent cell sizes ? One would perhaps expect that considering a larger cell would increase the collective eÜect of the dipole moment alignment. (iv) Why did you consider only a 2]1]1 cell rather than 1]2]1 or 1]1]2? 1 C. M. B. Line and R. W. Whitworth J. Chem. Phys. 1996 104 10 008. 2 S. M. Jackson V. M. Nield R. W. Whitworth M. Ogwa and C. C. Wilson J. Phys. Chem. B 1997 101 6142. 3 C. Pisani S. Casassa and P. Ugliengo Chem.Phys. L ett. 1996 253 201. Dr Gale responded The key point concerning the relative stability of the two ordered ice polymorphs is that the diÜerence is small as also found by earlier ab initio calculations and it would be beyond the accuracy of the method to state categorically which one is more stable. In response to the speci–c points raised (i) Because the calculations were run as a supercell the space group used was P1 and therefore no symmetry constraints other than the unit cell were imposed. The fact that only one value for the bond lengths and angles was reported is purely for brevity as the scatter of values is small about the numbers quoted. All unit cell strains and fractional coordinates were allowed to optimise freely in the calculations except for one atom which is –xed to remove the translational invariance.(ii) So far we have yet to consider the eÜect of doping with potassium. (iii) The eÜect of increasing cell sizes on dipole alignment is actually found to be negligible. However there is an important question relating to the treatment of dipoles in the lattice sums. By default it is assumed that the crystal surfaces reconstruct so as to remove any total dipole. However if this is not the case then the energy of a polymorph will depend on the morphology of the crystal. (iv) The reason for choosing a 2]1]1 supercell is that in the large unit cell method the exchange interactions are limited to half the length of the supercell along each axis and in each direction. Hence it is important to construct a supercell so that there are no short cell vectors.In the case of the ice polymorphs considered the a cell vector is signi–cantly shorter than the others and therefore this was doubled. Large supercells were also checked to examine the convergence of the energy. Dr Schoé n asked How fast is a single total-energy evaluation using the semiempirical method say for a system containing 10»100 atoms in the simulation cell ? In particular how do the times for diÜerent approximations compare with each other and with the time required for a calculation at the ab initio level or for one using typical empirical potentials ? Dr Gale replied As one would expect the cost of a semi-empirical calculation on a solid is intermediate between that of an empirical potential calculation and that of an ab initio calculation.To give a typical example of timings a single point energy/gradient 268 General Discussion evaluation for chabazite (a 36 atom per unit cell silicate) requires 1 min of CPU time on a Silicon Graphics 150 MHz R5000 though this can be accelerated by adjusting the convergence method and summation parameters. For comparison an equivalent shellmodel calculation requires 0.45 or 0.27 s depending on whether symmetry is used or not. A –rst principles plane wave calculation on the same machine would take several hours. It should be noted that the program is currently far from optimized for performance and signi–cant savings could be made through the use of symmetry (both space group and translational in supercells).For large systems containing several hundred atoms the dominant computational expense is matrix diagonalisation as with all standard QM methods. Hence the implementation of linear scaling methods to overcome this bottleneck will be crucial in the future. Dr Price asked Your paper mentions that it would be unreasonable to expect your solid-state semi-empirical calculations to give good results for crystals where the dominant interaction is the dispersion force a comment that would be equally applicable to HF or DFT ab initio methods. Unfortunately the dispersion forces are important in most organic crystal structures because of the large number of molecules in the coordination shells where this long range (R~6) attractive energy term is non-negligible.This produces a signi–cant contribution to the lattice energy and a compression force on the structure. What are the possibilities for including the dispersion energy in the calculation for example could it be crudely absorbed in reparametrization ? Dr Gale answered The inclusion of some representation of dispersion is indeed essential for the modelling of molecular crystals and so far our studies have been restricted to those cases where hydrogen bonding or electrostatics are the dominant interactions. Incorporating the eÜects of dispersion into the parametrization would be fraught with difficulties and has no guarantee of being successful. Given that we know from many years of experience with empirical potentials that we can do a reasonable job of representing the dispersion with potentials of the form [(C opinion is that the best way forward is to simply add on the dispersion as a pair poten- 6]R~6)[(C8]R~8) (with short-range damping where necessary) my personal tial while utilising the semi-empirical Hamiltonians to model the static multipolar interactions.Dr Schoé n asked What is the status of the semi-empirical methods with regard to the values of the parameters employed in the calculation ? In particular can these parameters be transferred from the systems used for –tting to not-yet-synthesized systems? Dr Gale responded One of the merits of the MNDO AM1 and PM3 parametrizations of the NDDO approximation is that the parameters are based on one-centre terms. Hence there is no need to derive further parameters for individual pairs of atoms.The reliability of the parametrization obviously depends on two main factors ; –rst the quality and variety of data used in the –tting procedure and secondly on the ability of the method to be transferable between varied environments. There are clearly limitations of the parametrizations»for example hypervalent compounds have long proved difficult to reproduce accurately»however improvements are being made such as the MNDO/d scheme. As to the speci–c question of whether the parameters are likely to be reliable for as yet unsynthesised compounds»the answer is that it will depend on whether the materials contain similar coordination environments and oxidation states to those molecules in the –tting database as with any empirical method.There will always be a need to check results against high-level calculations if we need greater certainty. 269 General Discussion Prof. Pacchioni said A potential application of semi-empirical methods is in the study of extended and localized defects. There are already several examples of important structural information on the defects in SiO using semi-empirical wavefunctions and cluster models. Do you think it is possible to generate a sufficiently reliable set of 2 parameters to study various types of defects in oxide materials ? Dr Gale replied There is no reason why it should not be possible to generate a parameter set which is suitable for the study of oxide materials including their defects. Existing parameters for the NDDO-based Hamiltonians have already been used to study defects in silicate systems as noted in your question.For the more ionic oxides the parametrization should be modi–ed to re—ect the nature of the oxide anion rather than neutral oxygen as a single-zeta basis is unable to accommodate such large changes. 2 Mr Persson asked You discuss both adsorption on surfaces and the possible need for reparametrizations for the solid state. In many adsorption systems speci–c atom types may be present in both the adsorbed molecule and the surface one example being oxygen in water adsorption on TiO surfaces. In your opinion will it be possible to –nd parameters that work for both adsorbed molecules and surfaces ? Dr Gale answered Because the parameters are one-centred it would be possible to use diÜerent parameters for oxygen in diÜerent environments within the same calculation.This is analogous to using a solid-state optimised basis set for oxygen in TiO2 while using a molecular basis set for oxygen in water within the context of an ab initio calculation. If we require the same parameter set to work equally well for an atom in both ionic and covalent environments then it would be necessary to reformulate the method in terms of a double-zeta basis set. In the past this may have been too expensive to contemplate or there may have been insufficient data to derive the increased number of parameters. However by extending the method to the solid state the amount of data available for many elements has increased dramatically so this would now be feasible.Dr Shluger said You are using several semi-empirical methods with diÜerent parametrization strategies. Most of the methods are parametrized for molecules and in many cases with the emphasis on organic systems. Therefore it is difficult to expect that they will give reliable data for inorganic crystals. Some parameter optimisation is clearly needed. What would be your parametrization strategy ? Do you think that these methods should reproduce the results of crystal calculations by the HF method which they approximate? We have found that when the parametrization is made to reproduce the experimental data which are beyond the HF approximation the scheme appears to be too twisted and can give some parameters absolutely incorrectly.Could you please comment on this ? Dr Gale responded There have been two main philosophies in the –eld of semiempirical quantum mechanics through the years. The –rst was to try to reproduce the results of the HF calculations to which they are approximations. The second which is in the spirit of MINDO/3 MNDO AM1 and PM3 is to try to reproduce experimental data instead of other theoretical results which are known to have systematic errors due to neglect of correlation. Arguments can be made in favour of both approaches. However in the case of the Hamiltonians that have been implemented in this work there is no reason to abandon their original philosophy when considering the solid state. In fact it is fundamentally more difficult to approximate HF results using the above schemes than it is in CNDO or INDO as the two-centre integrals are calculated using 270 General Discussion an empirical expression with the correct limiting behaviour and modi–cation to allow for the eÜect of correlation rather than by using the exact expression for the integral over two Slater orbitals.Prof. Sauer commented I welcome the use of semi-empirical NDO-type methods in quantum chemical solid-state calculations using periodic boundary conditions. This is the proper way of taking advantage of the low computational cost of these approximations. However I am rather sceptical as far as the description of intermolecular interactions in general and the hydrogen bond in particular is concerned. The basic approximation»zero diÜerential overlap»seems to be a balanced approximation for chemical bonds in organic molecules.In contrast the small overlap region of intermolecular interactions cannot be properly described. When the proper hydrogen-bond structure is studied meaningful distances are frequently found. But when a full scan of the potential surface is made it is sometimes found that other structures appear to be the global minimum. The reason is that these methods tend to simulate a chemical bond even if there is an intermolecular interaction. I see the potential of these methods in the –eld of partially ionic solids such as silica and zeolites. Although with the current parametrizations the errors of predicted structures are too large I am very optimistic that a specialized parametrization based on ab initio data (as those used by us to parametrize the shell-model potential) would result in a useful ZDO-type method.Its most interesting application would be in a combined ab initio and semi-empirical ZDO-type approach. The implementation within our embedding scheme would be easy. It would provide an interesting tool for studying hydrocarbon reactions in zeolites. The reaction site would be described by the ab initio method and the periodic zeolite structure by a specially parametrized semi-empirical method. Prof. Hillier addressed Prof. Sauer The embedding technique that you have described requires a force –eld description for the QM cluster. How easy is it to obtain this especially for structures far from equilibrium ? Prof.Sauer responded When using empirical data for parametrizing force –elds the parametrization may fail far from equilibrium structures since no data are available. This is diÜerent when the parametrization is based on ab initio data which is key to our approach. Ab initio data can easily be generated for all parts of the potential-energy surface. However there is another problem for transition states which involve the breaking and creating of bonds. Most force –elds rely on a given connectivity. If this connectivity changes during the reaction one would like to simulate the solution is to use diÜerent force –elds for reactants and products and to connect them within a valencebond-type scheme (cf. ref. 1). We are implementing such schemes.1 J. Aqvist and A. Warshel Chem. Rev. 1993 93 2523. Prof. Klein opened the discussion of Dr Nogueraœs paper (i) Do you have any evidence of —uxional behaviour occurring in your clusters ? (ii) Can you understand the structural motifs in the clusters in terms of simple ideas such as ionicity three-centre two-electron bonds etc. ? Dr Noguera responded (i) Generally for a given stoichiometry several isomers exist which are rather close in energy i.e. within a few tenths of an electronvolt. These isomers may transform into one another if energy is provided to overcome activation barriers. We occasionally met this situation during dynamical runs at high temperature. In such cases the activation barrier is expected to be lower than the average energy required for the detachment of an Li atom and/or cluster fragmentation and the simulated annealing strategy to global optimization is found to perform especially well.271 General Discussion (ii) The common feature to all LinO2 ment of atoms which enhances the electrostatic attractive O»Li interactions and miniclusters discussed here is a geometric arrangemizes the Li»Li repulsion. Actually the competition is quite subtle and this is the reason why isomers belonging to families II and III are so close in energy. In addition spontaneous distortions such as those found in II-Li6O2 and II-Li8O2 can happen which result in a lower symmetry and in the formation of a strong covalent LiwLi bond. This exempli–es the coexistence of bonds of a diÜerent nature (ionic and covalent) in the same physical system.Dr Shluger asked (i) Is it possible in your calculations to deduce information regarding approximate eÜective charges of the oxygen ions ? Are they in highly ionic states ? (ii) Since the electron density of oxygen ions is so inhomogeneous is it correct to use just LDA or LSDA approximations? Would you expect considerable changes in cluster con–gurations if you could use gradient corrections ? show that ìvalence states œ are highly localized on oxygens. A convolution of the charge Dr Noguera replied (i) In the stoichiometric clusters Li4O2 charge density maps density by Gaussians centred on the atomic sites gives a tentative value of [1.6 for the oxygen charge. This is in agreement with a Mulliken charge analysis in bulk Li While these estimates should be considered with care there is no doubt that the oxygens 2O.are in a highly charged state and that the LiwO bonds are very ionic in the stoichiometric species. (ii) We were able to compare favourably several of our results with the results of more elaborate treatments of electron correlations such as generalized gradient approximation (GGA) (Li clusters) con–guration interaction (Li5) and HF plus correlations (Li O). 6 As far as a comparison between LDA and GGA is concerned the order of stability of the three most stable isomers of Li is the same for LDA and GGA while the 4 energy diÜerences between them are larger for the LDA than for the GGA by one or 6 two tenths of an electronvolt at most. We are thus rather con–dent that the use of GGA would yield only marginal changes in our results.Prof. Pacchioni said We have heard a discussion on the problem of de–ning what is core and what is valence in pseudopotential plane wave calculations on solid-state problems. Do you have an idea of the importance of core polarization in your calculations on lithium»oxygen clusters ? Dr Noguera responded We addressed the question of how well we treat the lithium 1s electrons (only 55 eV binding energy) not in LinO2 clusters but in separate work on LiH. It turned out that all-electron and pseudopotential calculations yield results on the equilibrium distance and vibration frequency in remarkable agreement. This is probably to be expected also for the LinO2 clusters especially those in which the lithiums are in a highly ionic state.On the other hand the Li dimer structural properties are also well described in our approach. It is however clear that we are unable to predict reliably the 2 precise spatial dependence of the electronic density in the core to compare for example with experimental Compton pro–les. I presume that this would depend on 2O? Dr Lindan asked Do you think that the bond weakening ideas described in Section 6.3 of your paper are valid for reduced bulk Li the extent of delocalization of the conduction states in the bulk. I would comment that bond weakening is signi–cant in the context of superionic conduction in the material. 272 General Discussion Dr Noguera answered The bond weakening that we –nd in our clusters is clearly related to a delocalization of some electronic states on the lithiums. This is not what occurs for isolated vacancies in ionic bulk oxides where the Madelung potential is able to trap the excess electrons in the vacancy sites. However if vacancy clusters are present (and there is experimental evidence that oxygen vacancies do form clusters in some oxides) an eventual overlap of vacancy states may result in an anionwcation bond weakening. One can therefore expect strong implications on diÜusion mechanisms and possibly on ionic conduction. Prof. Catlow asked Following the success of your calculations on Li2O do you intend to extend your approach to other oxide clusters ? There is as you note a growing interest in oxide cluster chemistry and it is a –eld where high quality calculations could have a major impact. Dr Noguera replied Oxide clusters are species in which a subtle interplay between various forces takes place such as ionic vs. covalent contributions to bonding electronic vs. geometric eÜects on the properties of non-stoichiometric clusters etc. Elaborate numerical calculations are thus necessary to predict some behaviour interpret experimental results or give complementary information which cannot yet be obtained experimentally. Whether they are free or supported oxide clusters will certainly attract much attention in future.

ISSN:1359-6640    

DOI:10.1039/FD106253

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 273»289 Ab initio molecular dynamics study of crystal hydrates of HCl including path integral results Tycho von Rosenvinge,a Mark E. Tuckermanb and Michael L. Kleinb a Center for Molecular Modeling and Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6323, USA b Center for Molecular Modeling and Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA The hydrates of hydrogen chloride are ionic crystals, which contain hydronium.The hydronium in the monohydrate has been reported to be statistically disordered between two possible sites related by inversion symmetry. Ab initio molecular dynamics (MD) calculations are presented for the mono-, di- and tri-hydrates of hydrogen chloride using the density functional based Car»Parrinello technique.The simulations were carried out with the goal of investigating proton disorder in these crystals. The possible role of nuclear quantum eÜects has been explored via path integral MD simulations. The results suggest that the proposed disordered sites in the monohydrate are dynamically unstable and therefore unlikely to be responsible for the reported disorder.Unfortunately, little information was obtained for the dihydrate because the large unit cell leads to difficulties in carrying out the simulations. Nuclear quantum eÜects are shown to be important for characterizing the proton distributions in the trihydrate. I Introduction Crystal hydrates of strong acids provide an excellent forum for observing the manifestations of proton disorder in molecular crystals.As these systems contain a measure of hydrogen bonding, depending on the acid to water ratio, proton disorder correlates with the quantum nature of these hydrogen bonds. Indeed, the quantum nature of the shared proton in hydrogen bonds has received considerable attention recently, both experimentally and theoretically, owing to signi–cant advances in current technology on both fronts.Low-temperature spectroscopic techniques are sensitive to changes in the local environment as a result of individual proton tunnelling events, giving direct access to such phenomena. At the same time, improvements in electronic structure theory, and its combination with MD techniques, allow hydrogen-bond breaking and proton-transfer events to be followed with a degree of microscopic detail not yet accessible to experiments. The result is that our understanding of the nature of hydrogen bonding and its dependence on the local environment has improved substantially.There are, however, still many unresolved issues surrounding, for example, the importance of the coupling between thermal and quantum eÜects and —uctuations in the local environment.On these questions, considerably less is known about crystalline than liquid systems. For this reason, we have undertaken to study the signature of proton disorder in the crystal hydrates of hydrochloric acid as the degree of hydration is increased. Hydrogen chloride will form hydrates with the ratios 1, 2, 3, and 6 to 1,1 with the crystal structures of the –rst three having been determined.The crystal structure of the monohydrate can 273274 Molecular dynamics study of crystal hydrates of HCl best be described as an ionic crystal of hexagonal close-packed layers of Cl~ ions sandwiched between hexagonal sheets of hydronium ions with hydrogen bonding (H3O`), between the layers. The monohydrate has been reported to have random disorder2,3 as a result of a possible umbrella inversion of the hydronium ions, which would allow them to form hydrogen bonds with either the upper or lower sandwiching layers of Cl~ ions.The crystal structures of the di- and tri-hydrates are both monoclinic with Cl~ hydrogen bonded to units. In the trihydrate, water molecules connect diÜerent H5O2 ` H5O2 ` units with each other, forming continuous hydrogen-bonded –laments, which span the entire crystal.In the di- and tri-hydrate crystals, the manifestation of disorder is the proton transfer motion within the complex. The importance of this complex in H5O2 ` mediating proton transfer phenomena in water has been demonstrated using ab initio MD techniques.4,5 Very recent path integral calculations, taking into account –nite temperature nuclear quantum eÜects,6,7 suggest that, in an acidic solution, proton transfer in the is not strongly in—uenced by nuclear quantum eÜects.It is expected that H5O2 ` in a crystalline environment, the coupling of thermal and quantum —uctuations could lead to a very diÜerent conclusion. The methodology employed in this work8h10 combines the Car»Parrinello (CP) ab initio MD method11,12 with the path integral (PI) formulation of quantum statistical mechanics.13,14 In this scheme, quantum equilibrium properties at –nite temperature are computed on a potential surface that is determined ìon the —yœ from an accurate electronic structure calculation, thus circumventing the need to input an empirically determined potential function.As will be discussed below, the algorithm for implementing the PICP scheme contains intrinsic parallelism and can thus be implemented on a parallel computer with nearly perfect scaling. The PICP methodology has already been applied to studies of charged water complexes and protonated acetylene.6,15 Although this article deals speci–cally with the hydrates of HCl, the methodology will likely be useful in a wide range of applications such as catalysis involving acidic sites within crystals.16 This paper is organized as follows.In Section II, the PICP methodology is reviewed, including a discussion of how the problem of inefficient sampling is circumvented. Section III gives the salient computational details. In Sections IV»VI, the results of the simulations for each of the three crystal systems are described.A general discussion and conclusions are given in Section VII. II Theoretical models The simulations in this study fall into two categories : CP and PICP. The CP method is an efficient method for carrying out ab initio MD with the forces predicted by density functional theory (DFT). Similarly the PICP method is a convenient way of obtaining nuclear quantum distribution functions.The CP-MD and PI-MD methods are discussed independently in the next two subsections, followed by a discussion of the details of eÜectively combining the two schemes to form the PICP-MD method. II.A CP method In the CP method the electronic structure is described using the Kohn»Sham (KS) formulation of DFT.17h19 This formalism provides a method for calculating the unique ground-state electron density and energy of an electron, ion system, given only Nelec Nion the positions and types of the ions.A system can therefore, in principle, be evolved in time, within the Born»Oppenheimer approximation, by repeating the following process : calculate the ground-state electron density for the current set of positions, –nd the forces on the ions from the electron density and from other ions, move the ions a small distance according to the classical equations of motion.In practice, this is difficult becauseT . von Rosenvinge et al. 275 of the complexity of calculating the ground-state electron density rapidly and with sufficient accuracy. The CP method, which utilizes a plane-wave basis set and pseudopotentials for the electron»ion interactions, provides a way to evolve dynamically both the ions and the electron density simultaneously.In the KS formalism the electron density is expressed as the sum of the squared moduli of linearly independent electron wavefunctions. Nelec n(r)\ ; i/1 NelecS/i o rTSr o/iT\ ; i/1 Nelec o/i(r) o2 (1) The states the KS orbitals, are treated as dynamical variables in the CP method.The /i , ground-state energy of the system is expressed as E[X]\E[X, U]\[1 2 ; i/1 NelecS/i oZ2 o/iT]1 2 Pdr dr@ n(r)n@(r@) o r[r@ o ]Exc[n] ]Pdr Vloc(r)n(r)] ; i/1 NelecS/i o Vå nl o/iT]U(X) (2) where . . . , is a vector of the ion positions, X\X(x1, xNion ) 3Nion-dimensional U\U(/1, . . . , represents the KS orbitals, is the exchange and correlation functional, /Nelec ) Exc[n] and are the local and non-local parts, respectively, of the pseudopotentials de- Vloc(r) Vå nl scribing the electron»ion interaction, and U(X) is the ion»ion interaction energy.In order to reduce the number of electrons treated, and thereby the computational expense, the core electrons are usually incorporated into the pseudopotential.The CP method treats the KS orbitals as dynamical variables with a –ctitious mass. The following Lagrangian can be written, from which the equations of motion, for both the ion positions and the KS orbitals, can be derived L\k ; i/1 NelecS/ 5 i o/ 5 iT]1 2 ; I/1 Nion mI x 5 I2[E[X, U]] ; i, j/1 Nelec Kij(S/i o Så (X) o/jT[dij) (3) Here, k is a –ctitious mass parameter, and the last term is a set of generalized orthonormality constraints multiplied by Lagrange multipliers, The equations of motion are Kij .therefore mI xé I\[ LE[X, U] LxI ]Fc(xI) (4) k o/ é iT\[ dE[X, U] dS/i o ] ; j/1 Nelec Kij o/jT\[Hå KS o/iT] ; j/1 Nelec Kij Så (X) o/jT (5) where and the KS Hamiltonian, are given by Fc(xI) Hå KS , Fc(xI)\ L LxI ; i, j/1 Nelec KijS/i o Så (X) o/jT Hå KS\[1 2 Z2]Pdr@ n(r@) o r[r@ o ] dExc[n] dn(r) ]Vloc(r)]Vå nl (6) Given a reasonable initial nuclear con–guration and its ground-state KS orbitals, ab initio MD can be performed using these equations of motion in combination with the orthonormality constraint equations.In order for the KS orbitals to remain in their ground state during the time evolution of the system they must adapt quickly to the motion of the ions.This can be accomplished by setting k to a value, which is small in comparison to the ionic masses but large compared to the electron mass.12276 Molecular dynamics study of crystal hydrates of HCl The equations of motion require knowledge of an operator that depends on the Så (X), types of pseudopotentials used. For norm-conserving pseudopotentials making Så (X)\Iå The use of non-norm-conserving pseudopotentials adds some difficulty.For Fc(xI)\0. the Vanderbilt ultrasoft pseudopotentials20 used in this work the Hermitian overlap operators is given by where is composed of projectors onto Så Så (X)\Iå ]På (X), På (X) functions that characterize the pseudopotential. Details of the implementation of ultrasoft pseudopotentials for the CP method can be found in the literature21,22 and are therefore not repeated here.II.B PI method The PI method provides a way to calculate the equilibrium properties of a system in which quantum dispersion may play a role.13 The basis of this method exploits the isomorphism between a quantum particle and a classical polymer.14 In this way, the approximate position distribution function of the quantum system can be obtained by simulating the relevant classical system using the MD method.The quantum partition function for a system of N particles can be written as Q\Tr[exp([bHå )]\Pd3N xSx1, x2 , . . . , xN oexp([bHå ) o x1, x2 , . . . , xNT \PdX SXoexp([bHå ) oXT (7) This integral can be calculated by making use of the Trotter formula exp([bHå )\ < i/1 n exp([bKå /n)exp([bVå /n)]O(1/n) (8) For n\1, eqn.(7) is reduced to the classical partition function. Although formally exact only as n]O, n does not have to be very large for the approximation in eqn. (8) to be valid for treating protons.15 Combining eqn. (7) and (8) and inserting complete sets of position states yields.Q\PdXn SX1 oexp([bKå /n)exp([bVå /n) oX2T ]SX2 oexp([bKå /n)exp([bVå /n) oX3T … … … SXn oexp([bKå /n)exp([bVå /n) oX1T \PdXn < i/1 n SXi oexp([bKå /n)exp([bVå /n) oXi`1T \PdXn < i/1 n exp([bV (Xi)/n)SXi oexp([bKå /n) oXi`1T (9) where The n terms of the form are equivalent to X1\Xn`1\X. SXi oexp([bKå /n) oXjT the unnormalized density matrix of a free particle at temperature T n.For an individual particle Sxi oexp([bKå /n) o xjT\Pdp Sxi oexp([bp2/2mn) o pTSp o xjT \P dp exp([bp2/2mn)exp[ip Æ (xi[xj)/+] (2n+)3 \A nm 2nb+2B3@2 exp[[nm(xi[xj)2/2b+2] (10)T . von Rosenvinge et al. 277 where m is the particle mass. Inserting eqn. (10) into eqn. (9) and generalizing for N particles using Boltzmann statistics gives : Q\C PdXn < i/1 n expG[bCV (Xi)/n] ; I/1 N nmI(xi, I[xi`1, I)2/2b2+2DH (11) where C is a constant.Then Q is in a form that is proportional to the Nn particle classical partition function Z\Ncl PdXn PdPn expG[bC; i/1 n ; I/1 N Pi, I 2 N2Mi, I]V @(X1, X2 , . . . , Xn)DH (12) where is a constant, is the momentum conjugate to is an arbi- Ncl pi, I xi, I , Mi, I trary mass associated with the particle at and the potential .. . , xi, I , V @(X1, X2 , Xn)\ Because these two partition functions ;i/1 n [V (Xi)/n];I/1 N nmI(xi, I[xi`1, I)2/2b2+2]. have the same con–gurational integrals, a simulation of the classical system produces the desired quantum position distribution functions.14 Evolution can proceed using either Monte Carlo or MD schemes.23,24 II.C Combining PI-MD and CP-MD The CP-MD method and the PI-MD method can be combined by using V (X)\E[X].Thus, for each time step has to be calculated for each i. This is accomplished by V (Xi) performing an individual CP calculation for each with eqn. (4) changed to read Xi mi, I xé i, I\[ LE[Xi , Ui] Lxi, I ]Fc(xi, I)[ nmI b2+2 (2xi, I[xi`1, I[xi~1, I) (13) Note that a combined PICP-MD calculation using the DFT scheme is computationally expensive.Classical ab initio CP-MD simulations using DFT are already, themselves, very expensive and when combined with the PI method the computational cost is increased by a factor of n. Because of this high cost, every eÜort needs to be made to improve the efficiency of these algorithms. The staging method24 used in conjunction with Noseç »Hoover chain thermostats25 provides a signi–cant improvement over the straightforward application of eqn. (13).In the staging method a coordinate transform has the eÜect of diagonalizing the forces due to the stiÜ harmonic potentials between successive beads in the polymer. u1, I\x1, I , us, I\xs, I[1 s [(s[1)x(s`1), I]x1, I] (s\2, . . . , n) (14) Inserting this transformation into eqn. (12) gives Z\Ncl PdUn PdPn expG[bC; s/1 n ; I/1 N ps, I 2 N2Ms, I]V @(U1, U2 , .. . , Un)DH (15) where the momenta are now conjugate to and ps, I us, I V @(U1, U2 , . . . , Un)\V @(X1, X2 , . . . , Xn)\ ; s/1 n MV [Xs(U)]N/n] ; I/1 N nMs, I st 2b2+2 us, I 2 (16) where and (s\2, . . . , n) are the ìstaging massesœ, and M1, I st \mI Ms, I st \mI s/(s[1) can be calculated through use of the recursive inverse staging transform.Xs(U) x1, I\u1, I , xs, I\us, I] s[1 s x(s`1), I]1 s x1, I (s\2, . . . , n) (17)278 Molecular dynamics study of crystal hydrates of HCl The equations of motion of the staging transformed system are then Ms, I ué s, I\1 n L ; i/1 n E[Xi , Ui] Lus, I ]Fc(us, I)[ nMs, I st b2+2 us, I (18) where and can be computed directly from knowledge of L ;i/1 n E[Xi , Ui]/Lus, I Fc(us, I) and with the transform LE[Xi , Ui]/Lxi, I Fc(xi, I) Lf (X1, .. . , Xn) Lu1, I \1 n ; i/1 n Lf (X1, . . . , Xn) Lxi, I , Lf (X1, . . . , Xn) Lus, I \(s[2) (s[1) Lf (X1, . . . , Xn) Lu(s~1), I ] Lf (X1, . . . , Xn) Lx(s), I (19) By setting the –ctitious masses equal to a constant times the staging masses, the Ms, I normal modes of a chain can be sampled at approximately (exactly for a free particle) the same timescale.The result being more efficient sampling of phase space.24 The last problem to surmount is that the motion of an isolated polymer is not ergodic. This causes the phase space to be sampled inefficiently, even in the presence of an external –eld. Coupling a Noseç »Hoover chain thermostat25 to each of the staging coordinates makes the motion of each polymer ergodic, thereby increasing the phase space sampling efficiency.With this new coupling the ion equations of motion become Ms, I ué s, I\1 n L ; i/1 n E[Xi , Ui] Lus, I ]Fc(us, I)[ nMs, I st b2+2 us, I[Ms, I g5 s, Ixu 5 s, I QA gé 0s, I, l\cs, I, l[QA g5 s, I, lxg5 s, I, l`1 (l\1, . . . , L [1) QA gé s, I, L\cs, I, L[QA g5 s, I, L~1xg5 s, I, L (20) where is the thermostat mass, L is the number of thermostats on any u, the QA\(b+2/n) symbol x is de–ned by axb\ax bx i � ]ay by j � ]az bz k � (21) and cs, I, 1\Ms, I u 5 s, Ixu 5 s, I[1 b cs, I, L~1\CQA g5 s, I, L~2xg5 s, I, L~2[1 bD]CQA g5 s, I, Lxg5 s, I, L[1 bD cs, I, l\QA g5 s, I, l~1xg5 s, I, v~1[1 b (v\2, 3, .. . , L [2, L ) (22) The KS orbital equation of motion, eqn.(5) can be thermostatted in a similar manner as used for the ions. III Computational details DFT-based classical CP-MD simulations of each of the three hydrates were performed using Noseç »Hoover chain thermostats on each ionic degree of freedom to facilitate comparison with the PICP-MD simulations. Constant energy simulations of the monohydrate were also performed so that rough comparisons with experimental IR absorption studies3 can be made.A Bachelet»Hamann»Schlué ter-type pseudopotential26,27 was used for Cl, and Vanderbilt pseudopotentials20 were used for O andT . von Rosenvinge et al. 279 H. With these pseudopotentials an energy cut-oÜ of 30 Ry was sufficient. The exchange functional of Becke28 and the correlation functional of Lee et al.29 were used, as well as a time step of 5 au and a –ctitious electron mass parameter of 600 au.In the constant temperature simulations, groups of electronic degrees of freedom were also thermostatted. The thermostat chains were of length 4 on each ionic degree of freedom and also of length 4 on the chains connected to groups of electronic degrees of freedom. The thermostat masses were determined with characteristic frequencies of 3500 and 10 000 cm~1 for the ions and electrons, respectively.The PICP-MD simulations were conducted with the same parameters as the classical CP-MD constant energy simulations. The –ctitious masses of the staging variables, the so-called ìstaging massesœ, were calculated using twice the true ion masses. Additionally, each PICP-MD simulation was conducted using eight beads per ring polymer.While this is not large enough number to ensure a completely converged result, it is large enough to make semi-quantitative comparisons with the classical simulations. All simulations, performed using the program CPMD,30 were of more than 10 000 and 20 000 steps for each PICP-MD and CP-MD calculation, respectively.IV Monohydrate The monohydrate structure, which can be thought of as being composed of and H3O` Cl~ ions, has a rhombohedral unit cell, of dimensions a\4.05 and a\73° 30@, con- Aé taining one formula unit.2 X-Ray crystallographic data suggests that this crystal is disordered, namely the can occupy either of two positions in the cell with equal H3O` probability. These two positions are, with Cl~ at the origin, related by an inversion about the centre point of the cell.Each cell has R3m symmetry itself but the crystal is said to have symmetry with the atoms having only half weight. An individual cell R3 6 m has a Cl~ anion at the origin and an cation at either 0.44 or 0.66 of the body H3O` diagonal with the hydrogens pointing in the general directions of the S001T, S010T, and S100T Cl~ positions for the –rst case and S011T, S101T, and S110T Cl~ positions for the second (see Fig. 1). The structure can be pictured as hexagonal close-packed layers of Cl~ anions stacked in an ABC manner alternating with hexagonal layers of H3O` cations stacked in a CAB manner, e.g. the C layer is in between the A and B Cl~ H3O` layers. In each of the layers, the can individually bond to either the H3O` H3O` Fig. 1 Rhombohedral unit cell of the HCl monohydrate, drawn from the coordinates of ref. 2, illustrating the proposed disordered sites for hydronium280 Molecular dynamics study of crystal hydrates of HCl Fig. 2 Instantaneous con–gurations taken from the CP-MD simulations of the HCl monohydrate based on an eight unit cell system. (a) Classical CP-MD for a system with one inverted hydronium.(b) PICP-MD for a system with one inverted hydronium. (c) Classical CP-MD for the experimental (X-ray) system, which began with half of the hydronium sites inverted. Cl~ layer above or below, e.g. an in the C layer can bond to either the A or B H3O` Cl~ layer. The proposed disorder2 corresponds to a mirror inversion of an about H3O` the plane midway between its two adjacent Cl~ layers and a 60° rotation about the axis of symmetry. H3O`T .von Rosenvinge et al. 281 We have performed CP-MD and PICP-MD simulations of the monohydrate crystal with the intent to investigate the nature of the proposed disorder. Speci–cally, we seek to understand if the disorder static or if there is the possibility that hydronium could disorder via an rotation and inversion about its axis of symmetry.Dynamical events H3O` such as these may well be in—uenced by quantum eÜects because of the small mass of the proton. The crystallographic evidence that disorder exists rests on the measured electron diÜerence map.2 The conventional discrepancies, R, for the ordered R3m and the disordered models are 5.4% vs. 5.3%, respectively. In itself, this diÜerence is marginal R3 6 m and barely sufficient to distinguish between the two models. The difficulty of distinguishing between the two models is that is the Patterson symmetry of R3m. R3 6 m Other evidence that this crystal may be disordered comes from IR absorption spectra.3 The IR absorption bands of the OH-stretch modes are extraordinarily broad.3 This leads to the conclusion that there exists some type of frozen-in disorder, which accounts for the width of these bands.The OH-stretching IR absorption bands in the HBr monohydrate are just as wide as those of the HCl monohydrate. Moreover, the structure of the HBr monohydrate is now thought to be isomorphic to the ordered R3m H3O`Cl~ structure.31 IV.A MD With the overall goal of investigating the nature of possible disorder in crystal hydrates, we began with an ab initio study of eight unit cells of the monohydrate.PI and classical CP-MD simulations were performed, each with the same two initial con–gurations. The –rst initial con–guration was with all but one of eight unit cells of the same type. The odd cell hydronium we take as bonded to the Cl~ plane above while the other seven as bonded to the plane below [see Fig. 2(a) and (b)]. In the other case, four cells of one type are randomly intermixed with four of the other [see Fig. 2(c)]. With the –rst initial con–guration, in both the CP-MD and PICP-MD simulations, the odd hydronium —ipped to the same position as found in the other seven cells, thereby generating the fully ordered crystal.Moreover, in both cases the transformation occurred with the same mechanism. The in question remained bonded to the plane above for a short H3O` period until it rotated by 60° about its axis of symmetry while simultaneously pivoting about a perpendicular axis [see top two panes of Fig. 3(a) and (b)]. At this stage two of the protons have completed the transition to the fully ordered structure, while the third proton is in an intermediate state, bound simultaneously to two Cl~ ions in the upper plane.This situation persisted for ca. 300 fs of simulation time before an inversion of the hydronium completes the transition to the fully ordered state [see bottom of Fig. 3(a) and (b)]. We continued both the quantum and classical simulations in order to obtain data for comparison of the ordered system with the results of the X-ray diÜraction experiment.Part of this remaining classical CP-MD run was done at constant energy so that information about the crystal vibrational frequencied be obtained. The two simulations with four randomly inverted hydroniums as an initial con–guration [Fig. 2(c)] behaved diÜerently from the simulations begun with only one inverted cation.In this case both the PICP-MD and CP-MD simulations resulted in structures that very quickly changed from their initial con–guration. Although, some of the hydroniums remained in their initial bonding patterns, others changed to the half-—ipped state mentioned above or became bound to non-close-packed chlorine planes, misaligned with the symmetry axis of the crystal.This reorganization occurred at the beginning of the MD run and persisted with the same bonding patterns for the remainder of the simulation. The simulation results for systems with the single and four inverted hydroniums do not correspond well with the predictions of the disordered model proposed by experiment. 2 For example, in our simulations the chlorine atoms each prefer to bond with282 Molecular dynamics study of crystal hydrates of HCl Fig. 3 Hydronium transition from its inverted state to the fully ordered structure. (a) Top view along the rhombohedral S111T axis. (b) Side view, perpendicular to the S111T axis. three protons whereas, in a randomly con–gured system, the chlorine atoms can bond to anywhere between zero and six protons.Thus, in the eight unit cell simulation system with one inverted hydronium there are three chlorines bound to two protons, three to four protons, and the remaining two to three protons. The simulation of four of each cell type resulted in a serious distortion of the positioning of the ions [see Fig. 2(c)]. As will be seen below, this lattice distortion was so large that the disordered cannot be correlated with X-ray data.T .von Rosenvinge et al. 283 IV.B X-Ray structure factor The experimental X-ray data were reported as observed structure factors, which were compared with the predictions of a disordered model.2 Also presented were the results of an ordered model, although only the particle positions and R factor for this model were given. The temperature correction factors were omitted.In our analysis of the experimental data we used analytical –ts to the scattering form factors from the IC tables,32 while ref. 2 used less accurate factors from an earlier table, which did not have a separate entry for Cl~. Using the proposed disordered model in a comparison with the observed structure factor values gave an R factor of 5.75%. The discrepancy of our value with the published value (5.3%) is accounted for in the diÜerences in the scattering form factors used.In fact, a minimization of R, with respect to only the thermal parameters and scaling factor, also gave a value of 5.34%, which is consistent with the reported value.2 The fully ordered model gave an R of 6.61%, using the same thermal parameters reported for the disordered case and a minimized scaling factor.A minimization of the same parameters as above gave an R of 5.12%. Note that these minimized R values do not indicate what the actual R values are, they only set a minimum for the possible values of the reported structures. The actual R values cannot be calculated by us because the standard deviations of the experimentally measured structure factors were not reported in ref. 2. The R factor between the ordered and disordered models, using the same temperature correction factors and a minimized scaling factor, was only 3.61%. In fact, there even exists an ordered model that has an R between it and the disordered model of 2.61%. With this in mind, it is not surprising that the two models give extremely close results. The structure factors calculated from the classical CP-MD trajectory of the fully ordered model had a discrepancy from the observed values of 6.67%.The temperature factors calculated from this trajectory were (in the same form as presented by ref. 2) B\1.74^0.17, a(O)\0.025^0.005, b(O)\[0.009^0.009, a(Cl)\0.024^0.006, and b(Cl)\[0.006^0.008.Except for the hydrogen parameter, which has very little eÜect, these values are all within a standard deviation of the parameters needed for the minimum possible R value of 5.4% for the average ionic positions in this trajectory. The PICP-MD calculation gave very similar results to the classical simulation. However, the statistics for the quantum calculation are not as good as those for the classical simulation.The temperature factors calculated from the quantum trajectory were B\2.06^0.022, a(O)\0.019^0.01, b(O)\[0.0012^0.008, a(Cl)\0.017 ^0.010, and b(Cl)\[0.004^0.014. For the simulation with four inverted hydroniums the R factor was ca. 30%. Since this value is so far oÜ the measured value we can draw the conclusion that this is not likely to be the correct structure.IV.C Vibrational analysis Information about the monohydrate vibrational spectrum was obtained from Fourier transforms of the velocity autocorrelation functions calculated from constant energy CP-MD runs. The information obtained in this manner can be correlated with the positions, although not the magnitudes of the experimentally determined IR absorption bands.Fig. 4 shows the vibrational spectra obtained from both the ordered and fully disordered simulations. The calculated spectra correlate well with the observed IR absorption spectra.3 While the bands in the spectrum from the disordered system are wider than those from the ordered system they do not account for the widths seen in the observed absorption spectrum. The positions of the peaks of both spectra match the positions of the peaks seen in the IR absorption spectrum, except for the peak attributed to an overtone band, when the error due to the use of DFT and pseudopotentials is taken into account.This error is manifested mainly as a red shifting of the high-284 Molecular dynamics study of crystal hydrates of HCl Fig. 4 Power spectra taken from the classical CP-MD simulations.(a) Fully ordered structure. (b) Proposed X-ray structure with half of the hydronium sites disordered. (c) Projection of the l1 symmetric stretch in the ordered structure. frequency modes.33 The peaks above 400 cm~1 are all attributed to the motions of the hydronium ions, with the lowest peak being a libration about the axis of symmetry, the next being the symmetric bending mode followed by the doubly degenerate bend l2 , l4 , and then the doubly degenerate stretch and –nally the symmetric stretch Fig. 4 l3 , l1. also shows the spectrum of the symmetric stretch con–rming that it is the highest freqency absorption band; a –nding which agrees with a supposition based on an analysis of the IR absorption spectrum.3 V Dihydrate The dihydrate crystal of HCl is monoclinic, belongs to space group and has four P21/c, formula units per unit cell.34 The cell has dimensions: a\3.991, b\12.055, c\6.698 Aé and b\100.58°.The structure can be thought of as We performed both H5O2 `Cl~. quantum and classical simulations on a single unit cell of this crystal to study the diÜerences (if any) in the distribution of the hydrogen-bonded proton in the cation.H5O2 ` Unfortunately, this was not possible. During the course of the classical simulation, the con–guration of the system changed. Two of the four cations altered their H5O2 ` bonding con–gurations. Instead of bonding to four chlorines, as in the correct unit cell, in both of these cations there was a small rotation of the oxygens causing one of the oxygen»chlorine hydrogen bonds to break and a new bond to form between that oxygen and the periodic image of the other oxygen in the same cation.Thus, two in–n- H5O2 ` ite chains of oxygen atoms connected by hydrogen bonds were formed.T . von Rosenvinge et al. 285 The formation of a structure diÜerent from the correct crystal structure is most likely attributable to one or more of three eÜects.First, the unit cell has one very short dimension, a fact that gives rise to severe –nite size eÜects when just one unit cell is considered. Increasing the size of the system to include even one more unit cell (thus doubling the length of the short dimension) increases the computational eÜort roughly eightfold, which becomes prohibitive for PICP simulations.Such a simulation would be feasible if the calculation were to be parallelized over both imaginary time slices and reciprocal space. Secondly, there is the imposition of constant volume. Relaxing this constraint involves carrying out a constant stress calculation, in which the cell parameters are allowed to adjust dynamically. We have recently devised MD algorithms for path integrals at constant stress35 and will, in future investigations of this system, apply this new methodology to explore quantum eÜects on the shape of the unit cell.Finally, there is always the possibility that our functionals or pseudopotentials are inadequate. VI Trihydrate Like the dihydrate, the trihydrate is monoclinic and has four formula units in a unit cell. It belongs to the space group Cc and has dimensions a\7.584, b\10.154, c\6.715 Aé and b\122.96° 36 at T \83 K.The unit cell is shown in Fig. 5. The trihydrate contains long chains of oxygen atoms bonded together by protons. This chain structure contains an that hydrogen bonds to a water molecule, which in turn bonds to H5O2 ` another continuing in this manner to form an in–nite chain. The problem with H5O2 ` Fig. 5 Monoclinic unit cell of the trihydrate crystal, drawn from the coordinates of ref. 36286 Molecular dynamics study of crystal hydrates of HCl Fig. 6 (a) Time-averaged con–guration of one (b) Corresponding result from the H5O2 `…H2O. PICP simulation. the simulation of the dihydrate, i.e. that there is another structure, which for our model and system size, has a lower minimum energy, did not occur in the trihydrate simulation. Therefore, this crystal provides an opportunity to investigate cations in a H5O2 ` crystalline state.These chains are not known to be conduits for proton conduction, unless possibly by a defect mechanism.T . von Rosenvinge et al. 287 CP and PICP simulations were carried out at T \200 K with unit cell parameters a\7.811, b\10.458 and c\6.916.Fig. 6(a) and (b) show time-averaged con–gurations of an individual complex from the CP-MD and PICP-MD simulations, H5O2 ` …H2O respectively. The position distribution function of the excess proton is aÜected by its quantum nature. While the distribution for the classical simulation is bimodal for oxygen»oxygen distances greater than 2.53 the distribution is unimodal in the Aé , quantum simulation (Fig. 7) showing that, in the present situation, quantum eÜects play a role in proton transfer within the complex in a crystalline environment. H5O2 ` The notion that proton transfer within in the trihydrate crystal exhibits sig- H5O2 ` ni–cant quantum eÜects contrasts sharply with what one might have expected from the character of in the gas and liquid states.Recent PICP simulations of this H5O2 ` complex in the gas phase predict that the quantum nature of the shared proton does not strongly aÜect the spatial distribution functions.6 In the gas phase, it was observed that the OwO distance rarely increased beyond ca. 2.53 In the crystal, the —uctuations in Aé . the OwO separation are larger, despite the fact that the temperature at which the gasphase studies were carried out is greater than that used here.In addition, our previous studies of the solvation of hydronium in water,4,5,37 which have emphasized the importance of in the structural diÜusion process, suggest that, in an aqueous acidic H5O2 ` environment, the anomalously high proton mobility is also not strongly in—uenced by quantum eÜects. These –ndings accord well with experiment, where the kinetic H/D isotope eÜect has been shown to be small for proton transfer between and H2O H3O` (ca. 1.6^0.2).38 Again, the —uctuations in the OwO separation appear to be smaller in the liquid at 300 K than found here for the crystal at 200 K.It is worth noting that each Fig. 7 Contour plot of the oxygen»oxygen distance vs.the hydrogen atom asymmetric stretch (Aé ) for cations in the trihydrate crystal. (a) Classical ; (b) quantum model. (Aé ) H5O2 `288 Molecular dynamics study of crystal hydrates of HCl of the water molecules in the complex in the crystal is, on the average, only H5O2 ` threefold coordinated, while the third water molecule is fourfold coordinated. This suggests that there may be a ìcompetitionœ between the two water molecules in for H5O2 ` the shared proton, as each of these waters favours being hydronium, corresponding to its coordination state. Nevertheless, the hydrogen bond in this complex is not symmetric, as the environment surrounding each water molecule is diÜerent.This accounts for the asymmetric nature of the distribution functions observed, but also explains the frequent excursions of the proton between the two oxygen atoms in the classical simulation.In the quantum simulation, by contrast, the manifestation of this competition is a fairly broad distribution, which is only slightly asymmetric. For the distribution functions of the protons in the longer hydrogen bonds (not shown), quantum eÜects only act to broaden them somewhat, but the protons remain localized near a given oxygen atom.The average distances between oxygen atoms along the in–nite chains are nearly the same for the quantum and classical simulations and have the values 2.72, 2.48, and 2.72 for the classical simulation and 2.73, 2.50, and 2.72 Aé for the PICP case. The experimental OwO bond lengths are 2.65, 2.43, and 2.75 Aé Aé .VII Discussion and Conclusion The techniques of CP-MD and PICP-MD (based on a DFT description of the electronic structure) have been used to study proton disorder in three crystal hydrates of hydrochloric acid. For the monohydrate crystal, our results suggest that the interpretation of the crystal structure as containing randomly disordered hydronium is most likely not correct.However, there may be other kinds of disorder, which account for the X-ray data and the band widths of the IR absorption spectrum. For example, grain boundaries may cause a twinning between ordered domains of opposite parity. From the simulations of the trihydrate crystal, it can be concluded that protons in the complex- H5O2 ` es, situated in the hydrogen bonded –laments, exhibit strong quantum (zero point) eÜects.These are manifest by the broad, unimodal spatial distribution functions obtained when the quantum nature of the proton is accounted for. From this, it can be inferred that the proton is not well localized in this bond, leading to a certain degree of disorder in this crystalline system. These –ndings are in stark contrast to what is observed for proton transfer in acidic solutions.In the liquid state, proton transfer through the hydrogen bond in the complex appears not to be strongly eÜected H5O2 ` by quantum mechanics.6 The observed qualitative diÜerence between the liquid and crystalline environments correlates well with the observed hydrogen bond lengths. Although it was not possible to stabilize the correct structure for the dihydrate system, it is expected that the same eÜect will be observed for this crystal also, since the hydrogenbond length is also longer for this system than it is in the liquid.Given the dearth of experimental data on the trihydrate crystal, we must view our –ndings as theoretical predictions of the behaviour of this system at –nite temperature. Since –nite size eÜects are not as severe as they are in the dihydrate crystal, it is not expected that the simulations of the trihydrate crystal are beset with the same difficulties encountered in the dihydrate.Indeed, we observed that the experimental crystal structure was preserved throughout both the classical and quantum simulations. Thus the qualitative –ndings presented here are not expected to be aÜected.References 1 Gmelins Handbuch der Anorganischen Chemie, Verlag Chemie, Weinheim, 1969, vol. 6, p. 215. 2 Y. K. Yoon and G. B. Carpenter, Acta Crystallogr., 1958, 12, 17.T . von Rosenvinge et al. 289 3 C. C. Ferriso and D. F. Hornig, J. Chem. Phys., 1955, 23, 1464. 4 M. E. Tuckerman, K. Laasonen, M. Sprik and M. Parrinello, J. Phys. Chem., 1995, 99, 5749. 5 M. E. Tuckerman, K. Laasonen, M. Sprik and M. Parrinello, J. Chem. Phys., 1995, 103, 150. 6 M. E. Tuckerman, D. Marx, M. L. Klein and M. Parrinello, Science, 1997, 275, 817. 7 D. E. Sagnella and M. E. Tuckerman, J. Chem. Phys., in press. 8 D. Marx and M. Parrinello, J. Chem. Phys., 1996, 104, 4077. 9 M. E. Tuckerman, D. Marx, M. L. Klein and M. Parrinello, J. Chem. Phys., 1996, 104, 5579. 10 M. E. Tuckerman, P. Ungar, T. von Rosenvinge and M. L. Klein, J. Phys. Chem., 1996, 100, 12878. 11 R. Car and M. Parrinello, Phys. Rev. L ett., 1985, 55, 2471. 12 G. Galli and M. Parrinello, in Computer Simulations in Material Science, ed. M. Meyer and V. Pontikis, Kluwer, Boston, 1991. 13 R. P. Feynman, Statistical Mechanics: A Set of L ectures, Benjamin/Cummings, Reading, MA, 1982. 14 D. Chandler and P. Wolynes, J. Chem. Phys., 1981, 74, 4078. 15 D. Marx and M. Parrinello, Science, 1996, 271, 179. 16 A. Trare, F. Buda and A. Pasolino, Phys. Rev. L ett., 1996, 77, 5405. 17 W. Kohn and L. Sham, Phys. Rev. A, 1965, 140, 1133. 18 R. Parr and W. Yang, Density Functional T heory of Atoms and Molecules, Oxford University Press, New York, 1989. 19 W. Kohn, A. Becke and R. G. Parr, J. Phys. Chem., 1996, 100, 12974. 20 D. Vanderbilt, Phys. Rev. B, 1990, 41, 7892. 21 K. Laasonen, A. Pasquarello, R. Car, C. Lee and D. Vanderbilt, Phys. Rev. B, 1993, 47, 10142. 22 J. Hutter, M. E. Tuckerman and M. Parrinello, J. Chem. Phys., 1995, 102, 859. 23 M. Parrinello and A. Rahman, J. Chem. Phys., 1984, 80, 860. 24 M. E. Tuckerman, B. J. Berne, G. J. Martyna and M. L. Klein, J. Chem. Phys., 1993, 99, 2796. 25 G. J. Martyna, M. E. Tuckerman and M. L. Klein, J. Chem. Phys., 1992, 98, 1990. 26 G. B. Bachelet, D. R. Hamann and M. Schlué ter, Phys. Rev. B, 1982, 26, 4199. 27 D. R. Hamann, Phys. Rev. B, 1989, 40, 2980. 28 A. D. Becke, Phys. Rev. A, 1988, 38, 3098. 29 C. Lee, W. Yang and R. G. Parr, Phys. Rev. B, 1988, 37, 785. 30 J. Hutter, P. Ballone, M. Bernasconi, P. Focher, E. Fois, S. Goedecker, M. Parrinello and M. E. Tuckerman, CPMD Version 3.0, MPI fué r Festkoé rperforschung and IBM Research, 1990»1996. 31 J-O. Lundgren, Acta Crystallogr. B, 1970, 26, 1893. 32 International T ables for Crystallography, Kluwer Academic Press, Boston, 1992, vol. C, p. 500. 33 M. Sprik, J. Hutter and M. Parrinello, J. Chem. Phys., 1996, 105, 1142. 34 J-O. Lundgren and I. Olovsson, Acta Crystallogr., 1967, 23, 966. 35 G. J. Martyna, A. Hughes, M. E. Tuckerman and T. von Rosenvinge, Mol. Phys., in press. 36 J-O. Lundgren and I. Olovsson, Acta Crystallogr., 1967, 23, 971. 37 K. E. Laasonen and M. L. Klein, J. Phys. Chem., 1997, 101, 98. 38 B. Halle and G. Karlstroé m, J. Chem. Soc., Faraday T rans. 2, 1983, 79, 1031. Paper 7/02374J; Received 7th April, 1997

ISSN:1359-6640    

DOI:10.1039/a702374j

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 291»306 Monte Carlo simulations of pattern formation at solid/solid interfaces Gert Schulza and Manfred Martinb* a Institute for Physical Chemistry and Electrochemistry, University of Hannover, Callinstr. 3-3A, D-30167 Hannover, Germany b Institute for Physical Chemistry, Darmstadt University of T echnology, Petersenstr. 20, D-64287 Darmstadt, Germany Pattern-formation processes at solid/solid interfaces are investigated by Monte Carlo simulations using an appropriate two-dimensional model system AX/BX with an initially planar, coherent interface.The motion of cations A and B occurs via vacancies in the regular cation sublattice, and the jump frequencies of both A and B are described by a simple Boltzmannansatz. Therefore, the jump frequency is a function of temperature and the nearest neighbourhood of each cation, whose in—uence is determined by repulsive pairwise interaction energies and Using appropriate eAA , eAB eBB .boundary conditions, a directed vacancy —ux and, therefore, the growth of one or both phases is caused. In this way an external force, e.g. an external electric –eld, in a real experiment is simulated. Below a critical temperature (limited miscibility between AX and BX), the phase boundary roughens TC but remains morphologically stable if the less mobile phase is the growing phase.In comparison, a critical parameter is observed, *eC\ eAA[eBB above which the phase boundary becomes morphologically unstable if the more mobile phase is the growing one. In this case is dependent on *eC temperature, the boundary conditions and the external driving force.With increasing *e a transition from –nger-like to branched structures is observed. In exceptional cases the latter can be described as fractals. Similar results are obtained at temperatures above (complete miscibility of AX TC and BX) where we –nd instabilities of diÜusion fronts. 1 Introduction Morphological variations at interfaces in systems with a solid/liquid phase boundary have been observed in many experiments.The formation of dendritic patterns during crystal growth in a supercooled melt1 or the formation of fractal structures during the electrochemical deposition of zinc from an aqueous zinc sulfate solution2 are well known examples, however, these phenomena are rarely found in experiments with solid/ solid interfaces.There are, for instance, precipitations during the internal reduction of (Mg,Ni)O3 or moving solid/solid phase boundaries in the system AgCl/KCl.4 Applying an external electric –eld to the electrochemical cell Ag o AgCl o KCl o Ag a —ux of silver ions in AgCl and potassium ions in KCl occurs, and experimental parameters can be found at which the initially planar phase boundary becomes morphologically unstable.Fine branched structures of AgCl growing into the KCl crystal arise at a temperature of 290 °C, where both crystals are totally immiscible.4 In experiments on the system AgCl/ NaCl, where no miscibility gap exists, moving interdiÜusion fronts are observed which become morphologically unstable at sufficiently large driving forces, i.e.external electric 291292 Simulation of pattern formation at solid/solid interfaces –elds5. All these phenomena have, in common, a transition from stable growth (i.e. the development of stationary, stable interfaces) to unstable growth on varying the external parameters. Of very great interest in these experiments was the relation between the driving forces, e.g.concentration, temperature or electric potential gradients, and the resulting morphologies. Because of the enormous complexicity of the processes, even in very simple systems, an accurate mathematical solution is impossible in most cases. Monte Carlo (MC) simulation investigates processes on a microscopic scale, performing changes in systems exclusively on an atomic scale with a well-considered probability.One of the –rst examples using the MC simulation method was performed by Flinn and McManus6 investigating order»disorder phenomena in binary systems with a bcc lattice. In this paper, pattern formation in cation-conducting crystalline solids (A,B)X due to instabilities of phase boundaries is investigated by computer simulations using the MC method and an appropriate two-dimensional, two-phase model system AX/BX in which a directed vacancy —ux causes the growth of one of both phases. 2 Model The two-dimensional model system consists of the ionic crystals AX and BX with the same cubic structure, in which the anions and the cations A or B form two interpenetrating cubic lattices.7 The lattice constant, a, of both crystals is identical resulting in a coherent and initially planar phase boundary.Only vacancies in both sublattices occur, i.e. the intrinsic disorder is assumed to be exclusively of the Schottky-type. The mobility of the anion vacancies is assumed negligible compared to the cation vacancies, since this is the case for silver and alkali-metal halides. The interaction energies of a cation with its ionic surrounding in the ionic crystals are given by the electrostatic energy and the short range interaction energy both Eij C eij , of which are assumed to be pairwise additive.Concerning the coulombic interaction, the ions are considered to be point charges located exactly at the above described lattice sites. Therefore, in a perfect crystal, cations and anions possess the same coulombic potentials.The short-range interaction energies (i\A,B) with the anions X are the eiX same for both A and B, while merely the short-range interaction energies between the cations A and B are dependent on the cation nature. Considering the potential minima, of the cations (equilibrium position), they diÜer from each other only by their Emin , nearest-neighbour interaction energies (i, j\A,B), simply denoted by and eij eAA , eAB and are given by where is the sum of all interaction eBB , Emin\E0, min]; eij , E0, min energies apart from The potential maximum of a cation jumping into a neighbouring eij .vacancy lies exactly halfway between two cation sites where the repulsive forces of the large anions should be largest. Its value, is assumed to be constant as well as Emax , independent of the cation nature.The jump frequency of a cation i into a neighbouring vacancy is given by a simple Boltzmann-ansatz, where the activation energy *E is determined by the diÜerence of and (sum over the three nearest neighbours, since the inter- Ei, max Ei, min\E0,min]; eij action of cations with vacancies is set to zero, i.e.eAV\eBV\0) : *Ei\Ei, max[E0, min[ ; j/1 3 eij\*Ei, 0[ ; j/1 3 eij (1) The absolute jump frequency, for a given cation A, for instance, with three nearest CA cation neighbours, can be described as CA\C0, A expC[A*EA, 0[ ; j/1 3 eAjBNkT D (2)G. Schulz and M. Martin 293 Fig. 1 Simulation matrix, characterized by extensions in x- and y-direction, *X and *Y , respectively, and the initial position b of the phase boundary.The vacancy —ux is directed from top to bottom. where k is the Boltzmann constant, T the absolute temperature and the vibrational C0, A frequency of the cation A in its equlibrium position.This is an often used ansatz in simulations.8h11 The vibrational frequencies and are assumed to be identical : C0, A C0, B C0, A\C0, B\C0 .For the following simulations the jump activation energy is –xed to eV. *E0\1.0 Because it is well known that, in a regular solution, all thermodynamic properties depend only on e is –xed to e\0.04 eV, the interaction energy e\eAB[0.5(eAA]eBB), between A and B is –xed to eV, and only is varied in such a eAB eAB\0.2 *e\eAA[eBB way that e remains constant. 3 Boundary conditions For the choice of one of the four cations to exchange the site with the vacancy the so-called relative jump frequency, representing a probability is used.7 Ci, rel\Ci/; Ci , The simulation matrix is characterised by its extensions in the x- and y-directions (the height *Y and the width *X) and by the position of the initially planar phase boundary which is determined by b, the width of the considered section of the AX crystal (Fig. 1).The left and the right matrix boundaries are at and respectively. Periodic x\xL x\xR , boundary conditions are assumed at these boundaries. To generate a vacancy —ux the vacancy is always re—ected reaching the upper boundary of the simulation matrix while reaching the lower boundary it is re—ected with the probability or taken out of PL½(0,1) the matrix with the corresponding probability In this case the ìoldœ vacancy is 1[PL .replaced by an A cation and a ìnewœ vacancy is placed at a random position at the upper boundary. At any time, only one cation vacancy is in the matrix, the corresponding anion vacancy is not considered because it does not in—uence the motion of the cation. For a vacancy —ux from BX to AX results, for interdiÜusion takes PL\1 PL\1 place. 4 InterdiÜusion simulations InterdiÜusion simulations have been carried out for several temperatures between 300 and 700 K to observe the morphological development of the phase boundary in the absence of a vacancy —ux and to compare the resulting solubilities of AX in BX and vice versa with the miscibility gap of the phase diagram.The determination of the phase diagram of the quasi-binary system (A,B)X with demixing below a critical temperature was carried out by MC simulation7 using the grand canonical ensemble.12 The TC results of the simulations with e\0.04 eV are shown in Table 1 and are compared to294 Simulation of pattern formation at solid/solid interfaces Table 1 Solubilities, of AX in BX obtained from phase diagram and xA , interdiÜusion simulations in comparison to the calculated values using the Bragg»Williams approximation (symmetric miscibility gap) xA (equilibrium) temperature/K Bragg»Williams MC simulation MC interdiÜ. 300 0.00210 0.00250 0.00232 350 0.00522 0.00704 0.00655 400 0.01051 0.01640 0.01470 450 0.01844 0.03520 0.03370 500 0.02944 0.08420 0.08280 520 0.03480 0.14100 » 530 0.03769 0.29000 » 800 0.19605 » » calculations based on the Bragg»Williams approximation for regular solutions.13 The simulations lead to a critical temperature K while the Bragg»Williams approx- TCB530 imation yields K.The miscibility gap is symmetric and independent of TC\2e/k\923 *e, as expected. For each temperature, a simulation matrix of width *X\250 a and height *Y \180 a was chosen, consisting of two initially pure crystals AX and BX with a —at phase boundary at with b\90 a.Periodic boundary conditions were y\yL]b assumed at the left and the right matrix boundaries at and respectively. x\xL x\xR , was set to one, *e to 0 eV. At the beginning of each simulation the vacancy was set PL to a randomly chosen position directly on the phase boundary.Results The con–gurations of the simulation matrices were analysed and stored after 108 Monte Carlo steps (MCS), i.e. after 108 vacancy jumps. Fig. 2 shows the area of the matrices from to in a distance of 30 a around a for temperatures of x\xL x\xR y\yL]90 300, 400, 500 and 700 K after 4.0]109 MCS, i.e. after 8.89]105 MCS per matrix site (mcs).The solubilities of AX in BX and vice versa were determined, i.e the number of dissolved A(B) particles in BX(AX). Table 1 shows the mole fractions in BX calcu- xA lated in this way for temperatures of 300, 350, 400, 450 and 500 K after reaching a stationary state where only —uctuates around a mean value.Comparing these results xA Fig. 2 Area around the initially —at phase boundary at a distance of ^30 a after 8.89]105 mcs for T \300 K (M1), 400 K (M2), 500 K (M3) and 700 K (M4).Initially the simulation matrix consisted of two pure crystals AX (black) and BX (white) of the same size.G. Schulz and M. Martin 295 with the results of the phase-diagram simulation one observes lower solubilities in the case of interdiÜusion, for all temperatures. These deviations become smaller with increasing temperature and amount to 8% at 300 K but only 4% at 450 K.This deviation is due to the presence of the phase boundary which is missing in the simulations of the phase diagram. It consists of AwB bonds with relatively high energy and results in an important contribution to the total system energy. The in—uence of the phase boundary energy on the total system energy decreases with increasing temperature owing to the increasing solubility of AX in BX and vice versa at the same time.In addition to this observation, the phase boundary becomes more irregular and rougher with increasing temperature. For T P550 K the phase boundary disappears during interdiÜusion, as expected.The results of the interdiÜusion simulations show that the chosen boundary conditions, including the above described ansatz for the jump frequencies and probabilities, lead to stationary and, furthermore, to equilibrium states.Characterisation of the phase boundary The description or characterization of the phase boundary requires –rst a de–nition of the phase boundary, considering the speci–c characteristics of the model system (A,B)X, especially the kind of interactions between A and B.Sapoval et al.14 have given an appropriate proposal for the description of a diÜusion front in a system (A,B). The diÜusion front is described as the line given by the set of A particles being next neighbours and dividing the set of A particles from the B particles. This de–nition of the geometry of a phase boundary is based on its physical meaning, i.e.the phase boundary is formed by those particles of the same type being nearest neighbours. This de–nition may be applied to the sytem (A,B)X, since the phase boundary is merely determined by the cation sublattices owing to the geometrically and energetically identical anion sublattices of both crystals AX and BX. Fig. 3 shows a particular part of a simulation matrix with the phase boundary printed in bold.The isolated A cation is not part of the phase boundary but is considered as an A particle dissolved in BX. De–ning the phase boundary considering the frontmost line of B particles may lead to diÜerent shapes, especially at temperatures just below the critical temperature. However, at 300 K this eÜect becomes negligible. The phase boundary determined in this way was characterized by two diÜerent characteristic numbers, namely the so-called atomic roughness and the fractal dimension RA after Haussdorf»Besicovitch.Df , The atomic roughness, The atomic roughness, is de–ned as the ratio of the RA . RA , number of A particles at the phase boundary with at least one AwB bond and the total number of AwB bonds at the phase boundary.For the part of the phase boundary shown in Fig. 3, generally For a totally —at phase RA\0.714, 0\RA\1. boundary RA\1. Fig. 3 Atomic shape of the phase boundary between AX and BX. The phase boundary is de–ned as the line given by the set of A particles being next neighbours and dividing the set of the A particles from the B particles (. . . . . .). Determination of Df .296 Simulation of pattern formation at solid/solid interfaces Table 2 Atomic roughness, and Haussdorf»Besicovitch RA , dimension, of the phase boundaries shown in Fig. 2 and 4»7 Df , matrix temperature M Fig. /K RA Df 1 2 300 0.75^0.01 1.05^0.02 » » 350 0.72^0.01 1.07^0.02 2 2 400 0.70^0.01 1.10^0.02 » » 450 0.68^0.01 1.17^0.02 3 2 500 0.67^0.01 1.27^0.03 4 2 700 0.64^0.01 1.50^0.03 5 4 300 0.74^0.01 1.08^0.02 6 4,7 300 0.74^0.01 1.16^0.02 7 5 300 0.75^0.01 1.13^0.02 8 5 300 0.73^0.01 1.23^0.03 9 6 500 0.68^0.01 1.35^0.02 10 6 500 0.67^0.01 1.37^0.02 11 7 300 0.74^0.01 1.08^0.01 HaussdorfñBesicovitch dimension, To determine the ìlengthœ M, of the phase Df .Df boundary is measured by segments of diÜerent length m as shown in Fig. 3 [dashed line with segments of the length m\(17 a2)1@2].If the relation MDm~DfFDf\[d ln(M) d ln(m) (3) is valid is called the Haussdorf»Besicovitch dimension15 of the phase boundary. Con- Df cerning the simulations described below, this applies generally to a certain interval of m. For a totally —at phase boundary, the phase boundary is exactly one- Df\1, dimensional, while corresponds to a fractal phase boundary, whose dimension Df\1.34 lies between the two euclidian dimensions one and two, respectively.14 For all simulation matrices discussed below the results for and are given in RA Df Table 2. 5 Transference ìexperimentsœ The potential energy E(x,y) determined by the interactions of the ions is superposed by a constant one-dimensional electric –eld +/(x,y) with L//Lx\0. As a consequence, the jump frequencies of the cations parallel to the electric –eld change to Ci, ’\Ci exp[(<)*E’/kT ] (4) resulting in a favoured motion of the vacancy towards the negative y-direction (from BX to AX).+/, where a is the distance between two neighbouring cation sites *E’\ea/2 (lattice constant) and e the elementary charge. For all following simulations the parameter b was set to b\20 a, both crystals AX and BX were initially pure crystals and the temperature was set to T \300 K (\0.57 TC). 5.1 Variation of De For eV the simulations gave the same results, independent of the mag- *E’O5]10~5 nitude of so, for these simulations, was set to zero. *e was varied systematically *E’ *E’ in the range [0.04O*e/eVO0.06. was set to zero. Fig. 4 shows some results for PLG.Schulz and M. Martin 297 Fig. 4 Morphology of the phase boundary from transference simulations with *e\0.01 eV (after 3.56]105 mcs, M5) and 0.04 eV (after 4.0]105 mcs, M6), respectively, eV. Matrix size : *E’\0 250]180 a2 (width]height). Vacancy —ux from top to bottom. varying *e. It was observed that for *eO0.004 eV the phase boundary roughened but remained stable during its motion through the considered matrix.For *eP0.0046 eV the phase boundary became morphologically unstable. For relatively small *e\0.006 eV the disturbances can be described as wave-like, while with increasing *e more –nger-like structures appear. By further increasing *e, these –ngers became wider at their top and sometimes split, as shown in Fig. 4. The number of –ngers increased while their width decreased. For *e\0.06 eV the thickness of the –ngers had decreased to a few lattice constants. 5.2 Variation of De and DE’ In the described range of the electrical energy diÜerence was systematically *E’ , *E’ , varied between 10~4 and 5]10~2 eV. For a\5 this corresponds to a variation of ” +/ between 4]105 and 108 V m~1 which can be compared with real systems.Fig. 5 shows some representative results from simulations with a given set of *e and For *E’ . *e\0 eV, no instabilities of the moving phase boundary could be detected, even for relatively large values of +/ (108 V m~1). For *e\0 eV and the (eAA\eBB) *E’\0 Fig. 5 Morphology of the phase boundary from transference simulations with *e\0.02 eV and eV after 5.1]104 mcs, M7, (left : part of a matrix 180]350 a2) and with *e\0.04 *E’\10~3 eV and eV after 2.6]104 mcs, M8, (right : matrix 180]300 a2).Vacancy —ux *E’\5]10~3 from top to bottom.298 Simulation of pattern formation at solid/solid interfaces phase boundary remains planar, while the formation of a smooth wave-like phase boundary could be observed for the –rst time for eV. Increasing *E’P2.5]10~4 *E’ to 10~3 eV the structures became more –nger-like. These –ngers split with increasing simulation time and, –nally, branched. Repeating the simulations several times some critical parameter sets were found, where the phase boundary remained stable in one simulation but became unstable in some other for a given parameter set.This observation shows how important stochastic —uctuations in the position of the phase boundary can be for its stability and instability, respectively.Generally, the number of the growing disturbances (–ngers) increases with increasing The same applies for the tendency *E’ . of the disturbances to split and branch. At the same time the width of these branches became smaller. With increasing the growth of the disturbances, especially of the *E’ split and tree-like shapes, became more and more parallel to the potential gradient +/.The same applies for decreasing *e at constant Fig. 6 shows two matrices of *E’ . simulations with the same parameters as the simulations shown in Fig. 5, except for the temperature which is now T \500 K. The observed morphologies are quite similar to those obtained at 300 K.However, beside the formation of more or less –nger-like disturbances a distinct interdiÜusion takes place owing to the increasing mutual solubilities (see Table 1). 5.3 Variation of De and PL Setting eV the re—ection probability at the lower boundary of the simulation *E’\0 PL matrix was varied systematically for some particular values of *e for which the phase boundary became morphologically unstable with Fig. 7 shows some results for PL\0. *e\0.04 eV. Generally, the density of the disturbances decreased with increasing while their PL extension in the x-direction increased. The same tendency could be observed for decreasing *e at constant A critical parameter dependent on *e, could be found, PL\1. PL, crit , above which the phase boundary remained morphologically stable during the simulation.The phase boundary remained stable for (*e\0.01 eV), PLP0.999 PLP0.99995 (*e\0.02 eV) and (*e\0.04 eV). In the simulations, represents a PLP0.99999 PL qualitative measure for the magnitude of the vacancy —ux. For interdiÜusion PL\1, takes place (no vacancy —ux), while decreasing simulates an increasing vacancy —ux. PL Setting eV the critical parameter increased for a –xed *e.For *E’P10~4 PL, crit Fig. 6 Morphology of the phase boundary from transference simulations with *e\0.04 eV and eV (after 1.73]105 mcs, M10), respectively, at a temperature of T \500 K. Matrix *E’\10~3 size : 250]300 a2. Vacancy —ux from top to bottom.G. Schulz and M. Martin 299 Fig. 7 Morphology of the phase boundary from transference simulations with (after PL\1 0.244]105 mcs, M6) and (after 2.889]105 mcs, M11), respectively (*e\0.04 eV, PL\10~3 eV.Matrix size : 250]180 a2). *E’\0 *eP0.02 eV the phase boundary became morphologically unstable, independent of the magnitude of PL . 5.4 Simulations with initially disturbed phase boundaries In these simulations the planar phase boundary was replaced by a periodically disturbed phase boundary.These simulations correspond to a –nite amplitude stability analysis, in contrast to the previous sections where small —uctuations of the phase boundary caused the instabilities (linear stability analysis). The morphology of these disturbed phase boundaries is described by a sine shape and given by the function y(x)\Z sinA2n j xB]b (5) where Z is the amplitude, j the wavelength and b the mean distance of the phase boundary from the lower boundary of the simulation matrix. b was set to 40 a at the beginning of each simulation of this kind.The amplitude, Z, and the wavelength, j, were varied systematically in the range 2 aOZO20 a and 5 aOjO300 a, respectively. The widths *X of the matrices were chosen in such a way that periodic boundary conditions could be assumed at the left and right boundary of the matrices.Therefore *X was always a multiple of the wavelength j of the sine-shaped disturbance, at least twice the value of j. was always set to zero, the temperature remained constant at PL T \300 K. 5.4.1 Simulations with eV. The con–gurations of the simulation matrices DE’= 0 were analysed and stored after 109 MCS and the shape of the phase boundary was –tted to the above sine function with b and Z as –t parameters, to observe the development of the amplitude with continuing vacancy —ux j was found to remain constant for a certain time.To compare simulations with diÜerent matrix sizes the dependence of the amplitude from the fraction of the A particles in the whole simulation matrix was investi- xA @ gated since this fraction is a measure of the real time.Fig. 8 shows the development of the shape of the phase boundary obtained from a simulation with j\93.3 a, Z(start)\10 a and *e\0.004 eV after 1, 4, 7, 12 and 17]108 mcs, respectively. For comparison the –tted sine function is also shown (dotted line). The deviations of the phase boundary from the ideal sine shape increase with increasing simulation time.Nevertheless, the goodness of –t remains good up to a proportion of for most xA @ \0.6 of the simulations. Simulations have been carried out in the range of 5 aOjO300 a for four diÜerent values of *e (*e\0.002, 0.0026, 0.003 and 0.004 eV) at which the initially planar phase boundaries remained stable during the simulations. The starting amplitudes have been varied from Z\2 a to Z\20 a.For a given starting amplitude Z(start)300 Simulation of pattern formation at solid/solid interfaces Fig. 8 Morphology of the phase boundary (»») from disturbance simulations with j\93.3 a, Z(Start)\10 a and *e\0.004 eV after 1, 4, 7, 12 and 17]108 mcs, respectively, with the corresponding –ts (. . . . . . .). Matrix size : 280]180 a2.a critical wavelength was found, below which the initially given disturbance disap- jcrit peared during the simulation. is dependent on Z(start) as well as on *e. For wave- jcrit lengths the amplitudes increased with continuing vacancy —ux. Fig. 9 shows the j[jcrit critical wavelengths, for diÜerent *e in dependence on the starting amplitude. Gen- jcrit , erally decreases with increasing Z(start).The same applies for increasing *e at con- jcrit stant Z(start). The ìerror barsœ in Fig. 9 indicate the interval of j in which the amplitude either disappeared or grew, repeating a simulation with a particular set of parameters several times. Setting Z(start)\4 a the disturbances always disappeared during the simulation, independent of their wavelengths, i.e. the phase boundary is totally stable against disturbances of –nite amplitude Z\4 a.Setting Z(start)[15 a and jO50 a the phase boundaries lost their sine like shape after relatively short simulation times. Fig. 9 Dependence of critical wavelength, on starting amplitude of the disturbance for jcrit , *e\ 0.002, 0.0026, 0.003 and (S) 0.004 eV; eV. The solid lines have no physical (L) (|) ()) *E’\0 relevance.G.Schulz and M. Martin 301 5.4.2 Simulations with eV. For *e\0.004 eV, the dependence on DE’P10ó4 jcrit on the starting amplitude varying the electric –eld was investigated. Simulations with three diÜerent values of 2]10~4 and 3]10~4 eV) have been carried *E’ (*E’\10~4, out. The results are illustrated in Fig. 10 which shows the dependence of the starting amplitude on these particular simulation parameters.The curve describing the transition from stable to unstable behaviour of the disturbed phase boundaries shifts to smaller wavelengths with increasing This transition from stable to unstable behaviour *E’ . could already be observed for starting amplitudes Z(start)P2 a and eV. *E’[0 5.4.3 Growth velocity of the disturbances.Setting the starting amplitude Z(start)\10 a and *e\0.004 eV, the development of the amplitude Z of the disturbances has been investigated for diÜerent wavelengths measuring the size of Z with increasing Setting b\40 a [see eqn. (5)] and *Y \180 a for all considered simula- xA @ . tions yields at the beginning of each simulation, independent of the width, *X, xA @ \0.22 of the simulation matrix and the chosen wavelength.An almost linear dependence of the amplitude Z on was observed over the whole range of wavelengths. This linearity xA @ extends over diÜerent intervals, depending on the wavelength, but is generally observed for 0.22OxA@ O0.6. The amplitude decreases immediately for wavelengths of j\60 a. The smaller j, the faster this decrease occurs, thus the so-called growth velocity, de–ned as vZ , LZ/LxA @ , decreases with decreasing j.For j\60 a, Z remains initially constant but decreases with continuing vacancy —ux. Setting j[60 a, the amplitude grows exclusively, vZ[0. In Fig. 11 is plotted vs. the wavelength, shows a maximum at jB125 a and vZ vZ decreases afterwards but remains positive. At the wavelength, a transition from jcrit , negative to positive values of occurs.vZ 5.4.4 Variation of For two particular wavelengths, j\70 and 125 a, respec- PL . tively, setting Z(start)\10 a, *e\0.004 eV and eV the re—ection probability *E’\0 of the vacancy at the lower boundary of the simulation matrix was varied. The PL wavelength j\70 a is only slightly larger than the critical wavelength a while jcritB60 j\125 a corresponds to the maximum of The objective was to investigate whether vZ .an increase in i.e. a decrease in the vacancy —ux, would lead to stable phase bound- PL , aries, i.e. to a decrease of the disturbances in comparison to the case of where PL\0, the considered disturbances grew for both j\70 and 125 a. was varied in the range PL Fig. 10 Dependence of critical wavelength, on starting amplitude of the disturbance for jcrit 0, 10~4, 2]10~4 and 3]10~4 eV at constant *e\0.004 eV.The solid *E’\(S) (|) ()) (L) lines have no physical relevance.302 Simulation of pattern formation at solid/solid interfaces Fig. 11 Mean values of the slope (growth velocity of the amplitude with continual LZ/LxA @ vZ vacancy —ux) from simulations repeated several times for each wavelength with Z(start)\10 a and *e\0.004 eV.For j\70 a, as for j\125 a, values of were found above which 0.9OPLO0.99999. PL the disturbances disappeared during the simulations. In the case of j\70 a this occurred at –rst for for j\125 a a relatively sharp transition from positive to PL\0.9, negative values of was found between and As a consequence, vZ PL\0.999 PL\0.9995.the stability curve in Fig. 11 moves to the right and downwards, at the same time, with increasing re—ection probability of the vacancy at the lower matrix boundary, i.e. with a decreasing vacancy —ux. At sufficiently large values of corresponding to sufficiently PL , small vacancy —uxes, the phase boundary becomes stable, independent of j. 5.5 Simulations above TC For particular values of *e and simulations were carried out for temperatures *E’ , above the critical temperature (900 and 1100 K) to check whether interdiÜusion pro–les might become unstable, as observed experimentally.5 Throughout all these simulations, a distinct interdiÜusion took place, as expected for temperatures above the critical tem- Fig. 12 InterdiÜusion sections for simulations above (T \900 K) with eV, (a) TC *E’\10~2 *e\0.06 eV (after 4.4]103 mcs) and (b) *e\0.1 eV (after 8.8]103 mcs).Starting conditions : pure crystals AX and BX with a planar phase boundary at y\20 a. Matrix size : 250]300 a2.G. Schulz and M. Martin 303 Fig. 13 Fraction, as function of the position x at diÜerent y-positions of the matrix in Fig. xA , 12(b) (height : 300 a).for a particular matrix side is de–ned by determining the fraction of A in a xA square with the edge size of 5 a around the considered matrix site. perature of K. It was observed that, for *eO0.08 eV and eV, TCB530 *E’O5]10~3 the moving interdiÜusion front remained ì stable œ, i.e. a relatively homogeneous distribution of A particles was found in the x-direction while a gradient of the mole fraction xA was detected only in the y-direction, comparable to results of interdiÜusion simulations in the absence of a vacancy —ux.Increasing *e and/or the mole fraction pro–le *E’ of the A particles became inhomogeneous in special parameter regions. Fig. 12 xA(x) shows the results of simulations with eV and *e\0.06 eV and 0.1 eV, *E’\10~2 respectively. While, for *e\0.06 eV the interdiÜusion front remains stable, i.e.is xA constant in the x-direction, is no longer constant in the x-direction for the latter case, xA Fig. 14 Migration of initially round AX precipitates of diameter d\20, 30, 40, 50, 60, 70, 80 a in BX with *e\0.02 eV in an electric –eld eV). Vacancy —ux from top to bottom. (*E’\10~3 Matrix size : 600]300 a2. (a) Initial matrix, (b) after 7.22]103 mcs.304 Simulation of pattern formation at solid/solid interfaces Fig. 15 Simulations with initially randomly con–gured (A,B)X crystals and 0.4). (xA, start\0.1 Vacancy —ux from top to bottom. *e\0.02 eV, eV, matrix size : 250]300 a2. At the *E’\10~3 lower boundary the vacancy was replaced exclusively by B particles. Matrix with xA, start\0.1 : after (a) 233 and (b) 6660 mcs; and matrix with after (c) 233 and (d) 6660 mcs.xA, start\0.4 : i.e. the pro–le shows more or less extended regions with large deviations from a mean xA value. This is demonstrated in Fig. 13 where the fraction pro–les are shown for xA(x) diÜerent y-positions of the simulation matrix. for a particular matrix site was de–ned xA by determining the mole fraction of the A particles in a square with edge size 5 a around the considered matrix site. 6 Migration of AX precipitates in an electric –eld Placing some round AX precipitates of diÜerent diameters into a pure BX crystal one observes a migration of these precipitates in a direction opposite to the vacancy —ux. The migration velocity, as well as the morphological development, of these initially round ì particles œ depends on their diameter.Fig. 14 shows (a) the initial state of the simulation matrix of the size of 600]300 a2 with precipitates of the diameters, d\80, 70, 60, 50, 40, 30 and 20 a, respectively, centred at the same y-position and (b) the shape of the matrix after sufficient simulation time. The precipitates of diameters d\20 and 30 a keep their more-or-less round shape during the migration.For precipitates with diameters dP50 a, one observes the formation of –nger-like structures at their upper side which branch in the case of the largest precipitates. It is quite obvious that the lower boundaries of the larger precipitates develop into a smooth shape, perpendicular to the direction of the electric potential gradient. Chosing squares instead of round precipitates, the same phenomena were observed, independent of their initial orientation.In every case, the upper side of the precipitates became morphologically unstable while the lower side developed into a —at and stable phase boundary.G. Schulz and M. Martin 305 7 Precipitating processes in supersaturated crystals Pure BX crystals were doped with diÜerent amounts of A particles distributing them randomly on the matrix sites.A vacancy —ux in an electric –eld eV) was (*E’\10~3 then generated at T \300 K. Taking the vacancy out of the matrix at the lower matrix boundary it was replaced by a B particle. Setting *e\0.02 eV diÜerent starting matrices with diÜerent fractions 0.2, 0.3 and 0.4 were investigated. Since the misci- xA, start\0.1, bility gap of the system (A,B)X is found at for T \300 K (see Table 1), xAB2.5]10~3 these simulations correspond to experiments where the crystals are quenched from temperatures above (where the particles are distributed nearly randomly) down to tem- TC peratures far below (spinodal decomposition).Fig. 15 shows some examples at TC diÜerent states of the simulations for and 0.4, respectively.The con–gu- xA, start\0.1 rations of the matrices after 333 and 6660 mcs are shown. Initially, a large number of —uctuating seeds arise which then grow till they form stable precipitates. Owing to the applied electric –eld, these precipitates grow preferentially parallel to the –eld, as observed in the previous simulations. In summary, this computer experiment shows qualitatively that our model also describes spinodal decomposition in an external electric –eld. 8 Conclusion Using a simple model of a cation-conducting ionic system AX/BX with identical, perfect anion sublattices and a Boltzmann-ansatz for the cation jump frequencies, several real experiments such as transference experiments with varying vacancy —uxes, interdiÜusion experiments above and below the critical temperature, or transference experiments TC , with quenched and supersaturated crystals have been simulated.The starting point of each simulation experiment was a system of two pure crystals AX and BX with an initially planar phase boundary. Concerning the transference simulations the following observations have been made. Below the critical temperature, TC , (limited miscibility between AX and BX) the phase boundary roughens but remains morphologically stable if the less mobile phase (with the smaller value of is the eii) growing one, independent of temperature and almost independent of the external driving force.In comparison, one observes a critical parameter above *eC\eAA[eBB which the phase boundary becomes morphologically unstable if the more mobile phase (with the larger value of is the growing one.In this case, is dependent on the eii) *eC temperature, the boundary conditions (especially the vacancy re—ection probability at the lower side of the simulation matrix, i.e. the vacancy —ux) and the external driving force. With increasing *e and one observes a transition from –nger-like to *E’ , branched structures. In exceptional cases the latter can be described as fractals.For temperatures above the critical temperature, instabilities of diÜusion fronts could be TC , observed by increasing the applied electric –eld. These results describe qualitatively the observations in real systems. In the totally immiscible system, AgCl/KCl,4 stable phase boundaries as well as branched structures of AgCl in KCl were found, while in the completely miscible system, AgCl/NaCl,5 instabilities of diÜusion fronts occurred on application of sufficiently large electric –elds.Recently, instabilities have also been found in the system AgCl/CuCl,16 which is an example of a system with a miscibility gap but small mutual solubilities and therefore directly comparable to our simulations.In addition to these simulations, the stability of periodic sine-shaped disturbances was investigated by varying the wavelength, amplitude, vacancy —ux and external driving force (–nite amplitude stability analysis). No experiments have been made on this subject owing to the difficulties in preparing appropriate samples with the corresponding disturbed phase boundaries.Some investigations of the development of the306 Simulation of pattern formation at solid/solid interfaces solid/gas phase boundary during the chemical reduction of CoO17 have been performed, but the attempt to predict the observed morphologies was limited to the stability of the planar phase boundary to in–nitesimal disturbances, using linear stability analysis.The growth velocity of the disturbances observed in the simulations shows similar behaviour to the results of the linear stability analysis, although the ìexperimentalœ parameters of both models are somewhat diÜerent. The linear stability analysis calculates a critical wavelength, (below which the disturbances disappear jcrit , during the vacancy —ux and above which they grow) as well as a maximum of the growth velocity at The same behaviour was found as a result of the MC simula- jmax . tions. Decreasing the vacancy —ux, the transition from stable to unstable migration of the disturbed phase boundary no longer occurred, i.e. the phase boundary became stable during its migration, independent of the wavelength of the disturbances. References 1 Y. Saito, G. Goldbeck-Wood and H. Mué ller-Krumbhaar, Phys. Rev. A, 1988, 33, 2148. 2 D. Grier, E. Ben-Jacob, R. Clarke and L. M. Sander, Phys. Rev. L ett., 1986, 56, 1264. 3 D. L. Ricoult and H. Schmalzried, J. Mater. Sci., 1987, 22, 2257. 4 S. Schimschal-Thoé lke, H. Schmalzried and M. Martin, Ber. Bunsen-Ges. Phys. Chem., 1995, 99, 1. 5 S. Schimschal-Thoé lke, H. Schmalzried and M. Martin, Ber. Bunsen-Ges. Phys. Chem., 1995, 99, 7. 6 P. A. Flinn and G. M. McManus, Phys. Rev., 1961, 124, 54. 7 G. Schulz and M. Martin, Solid State Ionics, in press. 8 H. Bakker, N. A. Stolwijk, L. van der Meij and T. Zuurendonk, Nucl. Metall., 1976, 20, 96. 9 R. Kikuchi and H. Sato, J. Chem. Phys., 1969, 51, 161. 10 R. Kikuchi and H. Sato, J. Chem. Phys., 1972, 57, 4962. 11 G. E. Murch, in DiÜusion in Crystalline Solids, ed. G. E. Murch and A. S. Nowick, Academic Press, Orlando, FL, 1984. 12 W. Schweika, in Structural and Phase Stability of Alloys, ed. J. L. Moraç n-Lopez et al., Plenum Press, New York, 1992. 13 G. H. Findenegg, Statistische T hermodynamik, Dietrich SteinkopÜ Verlag, Darmstadt, 1985. 14 B. Sapoval, M. Rosso and J. F. Gouyet, in T he Fractal Approach to Heterogeneous Chemistry, ed. D. Avnir, Wiley, New York, 1989. 15 P. Meakin, Prog. Solid State Chem., 1990, 20, 135. 16 H. Teuber and M. Martin, to be published. 17 M. Martin and H. Schmalzried, Ber. Bunsen-Ges, Phys. Chem., 1985, 89, 124. Paper 7/02170D; Received 21st March, 1997

ISSN:1359-6640    

DOI:10.1039/a702170d

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 307»323 Grand canonical Monte Carlo simulations of adsorption of mixtures of xylene molecules in faujasite zeolites Veç ronique Lachet,a Anne Boutin,a Bernard Tavitianb and Alain H. Fuchsa*§ a Department of Physical Chemistry, Chimie-Physique des Amorphes (URA Mateç riaux 1104 CNRS), 490, de Paris-Sud, 91405 Orsay Cedex, France Ba� timent Universiteç b Department of Applied Physical-Chemistry and Analysis, Institut du 1 Franc ” ais Peç trole, et 4 avenue de Bois BP 311, 92506 Rueil-Malmaison Cedex, France.Preç au, The selective adsorption of xylene isomers in sodium Y faujasite has been studied by grand canonical Monte Carlo simulation. Biased particle insertions and deletions were implemented to allow simulation of the adsorption of such complex molecules.For mixtures, a new method including orientational biased swap moves was proposed. Such algorithms were found to improve the efficiency signi–cantly. m-Xylene was found to be adsorbed preferentially and the p-xylene/m-xylene selectivity predicted from the simulations was in good agreement with experiment. A detailed structural analysis of the adsorbed phases is presented. I Introduction The adsorption of aromatic molecules in zeolitic microporous materials is of great scienti –c interest because of applications in separation technology and catalysis.For instance, the separation of p-xylene from aromatic mixtures is performed on an industrial scale C8 by using selective adsorption in synthetic faujasite type zeolite. Many microscopic and macroscopic measurements, such as crystallographic1h5 and IR studies,5h7 thermodynamic8h15 and diÜusion coefficient16h20 measurements have been carried out on aromatic molecules adsorbed in faujasite, in order to understand their sorption and diÜusion properties.The need to gain a detailed insight into the behaviour of zeolite/ sorbate systems on the molecular scale has also inspired some molecular simulation studies.21h23 The technique of grand canonical Monte Carlo (GCMC) computer simulation is well suited to adsorption studies : adsorption isotherms are obtained directly from the simulation by evaluating the average number of adsorbed molecules.In the case of multicomponent adsorption, attempts were made to predict the coadsorption isotherm from the single component isotherms using adsorbed solution theories24 (IAST) or statistical models (Ruthven et al.15). However, the exact coadsorption isotherm can be obtained directly from a GCMC simulation of a mixture in which the chemical potential of each component in the bulk phase is speci–ed during the simulation.This direct mixture simulation has the advantage of giving information on the possible diÜerence between the behaviour of the single component adsorbed phase and that of the multicomponent adsorbed phase.Such information can be very helpful in understanding molecular selectivity. To our knowledge, no work has been undertaken until now to investigate the adsorption properties of aromatic compounds in faujasite-type zeolites by means of C8 § E-mail: fuchs=cpma.u-psud.fr 307308 Adsorption of xylene in faujasite zeolites GCMC simulation.This is presumably because the conventional GCMC simulation technique fails to predict sorption equilibria in the case of complex adsorbates such as the xylene isomers. In the present paper we discuss our eÜorts to use biased GCMC algorithms to address these more difficult systems.Biased GCMC simulations have been performed at a temperature of 423 K, to study the adsorption properties of p- and m-xylene in the sodium Y faujasite (NaY). Single species and binary mixture adsorption isotherms have been calculated and compared with the available experimental data. Three binary mixtures of diÜerent gas-phase compositions were studied. The experimental selectivity of NaY for the meta isomer has been reproduced by these coadsorption simulations.A detailed analysis of the adsorbed phase structure is presented. II Simulation model II.1 Structural details Zeolites are porous crystalline aluminosilicates. The framework consists of tetrahedral aluminium and silicon atoms bridged by oxygen atoms. The presence of aluminium atoms introduces charge defects, which are compensated by some non-framework cations.The X and Y faujasite zeolites display cubic crystalline lattices. The microporous network is made of cuboctahedral sodalite cages with a diameter of ca. 6.5 ”. These cages are linked together in a tetrahedral arrangement by six oxygen atom rings and form large cavities, called supercages. The supercages have a diameter of ca. 12.5 ”. They are interconnected in a tetrahedral arrangement by windows of diameter ca. 7.5 ” (Fig. 1). One cubic unit cell contains eight sodalite cages and eight supercages. The ratio of silicon to aluminium atoms and the number of cations vary from one faujasite to another. A faujasite is named Y when it has a Si/Al ratio greater than 1.5. Several cations sites have been determined both from neutron diÜraction1 and X-ray scattering25 studies.Sodium cations are found to occupy mainly sites I, I@ and II (see Fig. 1). In order to avoid introducing periodic crystal defects into the simulation model, we have chosen a full occupancy of sites I and II by sodium ions. This corresponds to a Si/Al ratio of 3, with 48 sodium cations per unit cell. No distinction was made between Si and Al atoms in the model.The composition is thus where T indi- T192O384Na48 , cates tetrahedrally coordinated atoms (Si or Al). The framework structure was taken from neutron diÜraction studies1 of NaY with a Si/Al ratio of 2.43. The crystalline structure is described in the Fd3m space group.1 The cubic lattice parameter is 24.8536 ”. In a previous study,22 we used another model of faujasite NaY taken from the work of Uytterhoeven et al.26 The NaY structure was described in the F23 space group, and a distinction was made between Si and Al atoms.The proposed distribution of the 48 Al atoms yielded two types of supercages alternating throughout the crystal ; the –rst type had 12 Al atoms and the second had none. The problem with such a model is that the inhomogeneous distribution of Al atoms yields an inhomogeneous –lling of supercages which has no physical counterpart.Hence, we have chosen here a zeolite model with no distinction between Si and Al atoms. In this work, the zeolite structure, including the cations, was considered as rigid. The eÜect of framework expansion1,2 upon loading and cation mobility will be the subject of a future study.One unit cell of zeolite NaY was used to construct the simulation box and periodic boundary conditions were applied. II.2 Interaction potentials II.2.1 Adsorbateñzeolite potential function. The adsorbate»zeolite potential function used in this work is based on the usual partition of the total interaction energy derived from perturbation theory applied to intermolecular interactions. It considers the inter-V .L achet et al. 309 Fig. 1 Schematic view of a supercage of zeolite NaY. The sites I for sodium cations are located at the centres of the hexagonal prism, and the positions of the sites II are in front of the six T rings inside the supercage. The T atoms are located at the vertices of the cavity and the oxygen atoms occupy positions close to the centres of the lines.actions between each atom of the adsorbate molecule and each zeolite species. The general expression for the adsorbate»zeolite potential function is Utotal\Ucoul]Uind]Udisp]Urep (1) The –rst term is the Coulombic term and is calculated using the Ewald-sums technique. The partial charges of all interacting species are given in Table 1.The second term is the induction energy and is given by the –rst term of the multipole expansion Uind\ ; i | ads [12 aiE2 (2) where is the dipole polarizability of atom i of an adsorbed xylene molecule and E is ai the electrostatic –eld, at the position occupied by atom i, due to the partial charges carried by all the framork species. Table 1 Partial charges and dipole polarizabilities of zeolite species and xylene atoms O Si, Al Na Car H CH3 q / o e o33,43 [0.7 ]1.2 ]0.8 [0.115 ]0.115 ]0.115 a/”3 1.11 0.44 0.15 1.90 0.64 2.00310 Adsorption of xylene in faujasite zeolites The third term of eqn.(1) for the adsorbate»zeolite potential function corresponds to the dispersion energy and is given by Udisp(rij)\[Af6 C6 ij rij 6 ]f8 C8 ij rij 8 ]f10 C10 ij rij 10 B (3) with f2n\1[ ; k/0 2n (bijrij)k k! exp([bijrij) ; n\3, 4, 5 (4) Pellenq and Nicholson28,29 have recently shown that the dispersion coefficients (C6 ij, C8 ij, can all be estimated from a knowledge of the dipole polarizabilities and the C10 ij .. .) partial charges of all the interacting species. These values are given in Table 2. The f2n functions are damping functions used to represent possible electronic exchange between the two species in interaction.Their eÜect is to attenuate the dispersion part of the potential function at short distances.27 These functions are parametrized with the single repulsive parameter bij corresponding to each interacting atomic pair. The last term in eqn. (1) concerns repulsion between pairs and is represented with an exponential Born»Mayer term Urep(rij)\Aij exp([bijrij) (5) The subscripts i and j describe the atoms of the adsorbate molecule and the zeolite species, respectively.The fact that the faces of a supercage cavity are covered with oxygen atoms (the silicon and aluminium atoms being further away in the crystal) and sodium cations implies that the repulsive interaction between the adsorbate and T atoms (silicon or aluminium) can be neglected, since there is no electron cloud overlap.Boé hm and Ahlrichs30 have reported some combination rules to estimate the repulsive parameters Aij and bij for a pair of heteroatoms ij using those of the ii and jj pairs Aij\)(AiiAjj) 2 bij\ 1 bii] 1 bjj (6) The values of the repulsive parameters for each pair ii and jj are reported in Table 3.The oxygen»oxygen and sodium»sodium parameters were –rst estimated from values Table 2 Dispersion coefficients used for adsorbate» zeolite interactions O T Na C6/K C 1.33]107 5.36]106 2.32]106 H 4.05]106 1.64]106 6.91]105 CH3 1.63]107 6.61]106 3.02]106 C8/K C 2.99]108 1.08]108 3.97]107 H 7.15]107 2.49]107 8.87]106 CH3 3.51]108 1.26]108 4.72]107 C10/K C 6.86]109 » 7.40]108 H 1.31]109 » 1.25]108 CH3 7.79]109 » 8.34]108V .L achet et al. 311 Table 3 Repulsive parameters used for the adsorbate»zeolite interaction pair A/108 K b/”~1 O»O 4.87 4.16 T»T » » Na»Na 1.46 4.35 C»C 0.45 3.61 H»H 0.29 5.48 CH3»CH3 1.08 3.36 found in the literature31,32 and then readjusted in order to reproduce the isosteric heat of adsorption at zero coverage. The carbon»carbon, hydrogen»hydrogen and methyl» methyl repulsive parameters were obtained from the –t of the repulsive part of the short range Lennard-Jones potential function used for the xylene»xylene interaction (see next Section). II.2.2 Adsorbateñadsorbate interaction.m- and p-xylene are assumed to be rigid, planar molecules, consisting of carbon atoms, hydrogen atoms and methyl groups. Carbon»carbon, carbon»hydrogen and carbon»methyl lengths are –xed at 1.40, 1.08 and 1.51 respectively.The carbon»carbon»carbon, carbon»carbon»hydrogen and carbon» ”, carbon»methyl angles are set to 120 degrees. Xylene molecules were treated atom by atom, except for the methyl groups which were taken as united atoms centred on the carbon atoms. The potential energy between two adsorbed molecules is the sum of a dispersion»repulsion term, modelled by a Lennard-Jones type potential, and a Coulombic term Uij\[4eijCApij rijB6[Apij rijB12D] qi qi rij (7) The Lennard-Jones parameters and used in this work (see Table 4) are taken from pii eii the optimized potentials for liquid simulations (OPLS) proposed by Jorgensen.33 The cross terms, and are obtained using the Lorentz»Berthelot combination rules.pij eij , II.3 Biased GCMC calculations The technique of GCMC computer simulation is well suited to adsorption problems. The volume of the pores and the temperature are –xed as well as the chemical potential of the adsorbed phase, which is assumed to be in equilibrium with a gas reservoir. The adsorption and coadsorption isotherms can be obtained directly from the simulation by evaluating the ensemble average of the number of adsorbate molecules.The Monte Table 4 Lennard-Jones parameters for xylene»xylene interaction p/” e/K C»C 3.55 35.24 H»H 2.42 15.08 CH3»CH3 3.80 85.47312 Adsorption of xylene in faujasite zeolites Carlo process implies, at equilibrium, the following detailed balance equation: PoBoltzPo?n gen P0?n acc \PnBoltzPn?o gen Pn?o acc (8) where PBoltz is the Boltzmann probability of a microscopic state, subscripts ìoœ (for old) and ìnœ (for new) refer to con–gurations of the system before and after a move, respectively, and Pgen and Pacc are the probabilities of generating and accepting this move.In a standard GCMC simulation, three types of moves are performed.The –rst consists of a displacement and/or rotation step, handled by the usual Metropolis method.34 In the second type of move, a new molecule is inserted into the system (transferred from the gas to the zeolite) at a randomly chosen position with a randomly chosen orientation. In the third type of move, a molecule is randomly chosen and removed. For a two component system, it is possible to use an additional type of trial (swap) which consists in changing the identities of adsorbed particles without changing their positions and orientations.35 When these Monte Carlo moves are randomly generated, the probability of generating a speci–ed move is equal to the probability of generating the inverse move, and the acceptance probabilities depend only on the Boltzmann factor : Pdisplacement@rotation acc \minM1, exp[[b(Un[Uo)]N (9) Pinsertion acc \minG1, zi V Ni]1 exp[[b(Un[Uo)]H (10) Pdeletion acc \minG1, Ni zi V exp[[b(Un[Uo)]H (11) Pswap i?j acc \minG1, zjNi (Nj]1)zi exp[[b(Un[Uo)]H (12) The subscripts i and j refer to the two components in a binary mixture, N is the number of molecules in the system before the MC move, V is the volume of the simulation box, b\1/kT where k is the Boltzmann constant and T is the temperature, and are Un Uo the energies of the con–guration after and before the move, respectively, and is the zi absolute activity given by zi\exp(ki/kT ) Ki3 (13) where is the thermal de Broglie wavelength.Ki GCMC simulations have proven to be successful in predicting adsorption thermodynamics of a variety of simple sorbates (argon, methane31 etc.) in zeolites.Attempts to apply the normal GCMC technique to larger sorbates (benzene, xylene,36 chain molecules37 etc.) are bound to be frustrated by low acceptance rates of the insertion and deletion steps. In our study of xylene isomers in zeolite NaY, these rates fell to 10~3% at high loading. Thus, in order to sample the GC ensemble correctly, the use of biased insertions and deletions is necessary.The –rst bias consists in reducing the volume V in which insertions are attempted. In normal GCMC, these insertions are attempted uniformly throughout the volume of a unit cell of NaY zeolite. However, much of this volume (sodalite cages, volume –lled by atoms of the framework etc.) is inaccessible to xylene molecules. The accessible volume, has a complex shape.It is estimated using an energetic criterion. The total volume Vacc , V is discretized into elementary volumes. An adsorbate molecule is placed at the centre of each elementary volume and the corresponding adsorbate»zeolite interaction is calculated. is the union of all elementary volumes corresponding to an interaction energy Vacc lower than an arbitrary positive value (1000 K in this work).For non-spherical mol-V . L achet et al. 313 ecules such as xylenes, this energy is calculated using a –ctitious one-centre-of-force model placed at the moleculeœs centre of mass. The hydrogen potential parameters are attributed to this centre of force. This procedure overestimates but includes all Vacc possible adsorption sites. Xylene molecules can, indeed, be adsorbed at a distance from the zeolite wall smaller than the mean size of the molecule, because of a strong orientation eÜect.In the case of xylenes in NaY, the volume of the simulation box (one unit cell) is V \15 352 and the accessible volume is reduced to In order to ”3 Vacc\6919 ”3. ensure microscopic reversibility, the acceptance rules for insertions and deletions must be modi–ed by replacing the volume V by Vacc .The second bias is the cavity-bias proposed by Mezei.38 In this method, insertions are only attempted in regions of the accessible volume where cavities of radius RPRc exist. Such cavities can accommodate a particle insertion. In the case of xylene molecules, which corresponds to the nearest-neighbour distance in the xylene Rc\3.5 ”, centre of mass pair distribution function.In the acceptance probabilities, V must be replaced by the volume of the subspace which is formed by the union of all points which are centres of a cavity of radius This volume is estimated during the simulation RPRc . by evaluating the probability of –nding a suitable cavity in a con–guration of N Pc(N), molecules (N is the total number of molecules).The volume in the acceptance probabilities has then to be multiplied by the factor The third bias used here is an orienta- Pc(N). tional bias.39 In the case of a particle insertion, the method consists in generating k trial orientations (k\10 in our work), and choosing the new con–guration ìnœ out of these k trials with an energetic criterion.The new con–guration is then generated with probability Pgas?zeo, n gen \ exp([bUn) ;m/1 k exp([bUm) (14) In order to calculate the acceptance probability of this biased move, the probability of generating the inverse move must also be calculated using the same process [see eqn. (8)]. In the case of a particle insertion, the inverse move consists in transferring a molecule from the zeolite to the gas reservoir. The gas phase is assumed to be ideal in the low-pressure domain with zero interaction energy.Following eqn. (14), the probability of generating a speci–ed orientation out of k trial orientations in the reservoir is simply Pzeo?gas gen \1 k (15) Following eqn. (10), the acceptance probability of a particle insertion has to be corrected by the probability of generating the move [eqn.(14) and (15)]. The same procedure is applied to the deletion move, which consists in transferring a molecule from the zeolite to the gas phase. The probability of generating such a move is given by eqn. (15). The probability to realize the inverse move is the probability to generate a molecule into the zeolite with the orientation ìoœ of the molecule to be deleted.Thus (k-1) trial orientations have to be generated and the probability of generating the transfer of a molecule from the gas to the zeolite with a speci–c orientation ìoœ is calculated as follows : Pgas?zeo,o gen \ exp([bUo) exp([bUo)];m/1 k~1 exp([bUm) (16) In our GCMC simulations, we combine the three types of bias described above.Therefore the acceptance probability for a particle insertion is : Pinsertion acc \minA1,G[Vacc Pc(N)]zi Ni]1 exp[[b(Un[Uo)]HC Wn exp([bUn) 1 kDB (17)314 Adsorption of xylene in faujasite zeolites with Wn\ ; m/1 k exp([bUm) (18) The term in the –rst brackets in eqn. (17) corresponds to the standard Boltzmann probability ratio [eqn. (10)], where the volume is reduced because of the –rst two types of bias.The second term is the correction due to orientational bias. is the energy of the Uo con–guration before the move, i.e. when the molecule is in the reservoir phase and thus Eqn. (17) can be rewritten as follows : Uo\0. Pinsertion acc \minA1,G[Vacc Pc(N)]zi Ni]1 Wn k HB (19) Similarly, the acceptance probability of a particle deletion is Pdeletion acc \minA1,G Ni zi[Vacc Pc(N[1)] k WoHB (20) with Wo\exp([bUo)] ; m/1 k~1 exp([bUm) (21) Finally, we have also introduced an orientational bias in the identity swap move.Since the aromatic ring planes have roughly the same orientation in adsorption sites for both isomers (perpendicular to the [111] axis), we have used an orientation bias which retains the aromatic ring orientation.The orientation of a xylene molecule with respect to the zeolite coordinate axes is speci–ed by three Eulerian angles, /, h and r, and the desired rotations (retaining the aromatic ring orientation) can be easily achieved by modifying only the r angle. This procedure is analogous to the orientational bias for particle insertion and deletion except that the k trial orientations are all in the same plane. The acceptance probability of the i for j swap is Pswap i?j acc \minG1,C zjNi zi(Nj]1) Wn WoDH (22) and are calculated using eqn.(18) and (21) for molecules of type j and i respec- Wn Wo tively. III Simulation results III.1 Efficiencies and convergence tests for biased algorithms We –rst performed unbiased GCMC simulations. Such simulations were characterized by low acceptance rates of the particle insertions and deletions and the corresponding amount of adsorbed molecules was underestimated at high density.We tested our biased algorithms with simulations at low loading (less than one xylene per supercage of NaY). These calculations were performed to verify that the standard GCMC and biased GCMC led to the same results and to compare the efficiency of the two methods. Simulations have been performed on the FUJITSU VPP500 computer of the Institut du Peç trole, in Rueil-Malmaison, France.The computer code was optimized for Franc ” ais vectorization. Results are shown in Table 5. All methods yield the same average number of adsorbate molecules. By reducing the insertion volume to the accessible volume, the insertion/deletion acceptance rate is doubled. The addition of cavity bias does not signi –cantly improve the efficiency, as expected for low-density regime.The introduction of orientational bias multiplies by a factor of seven the acceptance rate and only doublesV . L achet et al. 315 Table 5 Comparison of standard GCMC and biased GCMC algorithmsa low loading high loading P\0.01 Pa P\10 Pa SNT Pacc (%) SNT Pacc (%) standard I/D 0.44 0.1 2.0 \10~3 insertion using Vacc 0.40 0.2 (p-xylene) using Vacc]CB 0.42 0.2 using Vacc]CB]OB 0.43 0.7 2.68 0.15 swap swap 1.12 11.7 2.80 5.6 (50»50 mixture) OB-swap 1.00 27.1 2.88 14.5 a SNT refers to the average number of molecules adsorbed per supercage, is the Pacc acceptance probability for particle insertion/deletion (I/D) or for particle identity change (swap) moves. The results correspond to the standard GCMC scheme and to the use of the diÜerent bias described in the text is the accessible volume, CB is the cavity (Vacc biased scheme and OB is the orientational biased scheme).the computing time. Tests have also been performed at high loading (three molecules of xylene per supercage of NaY).Without using any biased algorithm, the insertion/ deletion acceptance rate is less than 10~3%. This rate increases to 0.15% when using all types of bias previously described. Even after more than 7]106 MC steps using standard GCMC, the average number of molecules is signi–cantly less than the average value obtained after 400 000 steps using biased algorithms.Therefore, careful attention must be paid to convergence when performing GCMC simulations of such complex systems at very high densities. In order to validate the maximum loading value, we have constructed a test con–guration with one molecule in each possible adsorption site. The localisation of each individual molecule has then been optimized using canonical MC to obtain a realistic low-potential-energy con–guration with a loading greater than the expected maximal loading. GC simulations performed, starting from both this con–guration and a lower density con–guration, led to the same maximal number of adsorbed molecules. This indicates that our biased algorithm allows us to obtain the stable maximum of loading value for this system.In the case of the adsorption of mixtures, the introduction of identity change moves has been found to be crucial for convergence.In addition, the use of the biased scheme described in the previous section improves signi–cantly the efficiency of such a move, the acceptance rate being increased by a factor of three at high loading. III.2 Adsorption of pure components The calculated adsorption isotherms of pure m- and p-xylene at 423 K are shown in Fig. 2. Simulations were performed at 423 K because there are experimental data8h10,13h15,40,41 available at this temperature. Both simulated curves are type I isotherms. The amount of adsorbed molecules is lower for p-xylene than for m-xylene over the whole pressure range, in agreement with experiment.8 The maximal loading is 3.1 molecules per supercage for p-xylene and 3.9 for m-xylene.These values can be compared with the experimental data of Bellat et al.8 at the same temperature. They found an adsorption capacity of 3.3 p-xylenes per supercage and 3.6 m-xylenes per supercage. The diÜerence between the experimental and the simulated high-pressure loading lies within the intrinsic statistical error of the simulation.In the present work, the uncertainty in the average number of particles in the system is 0.25 molecule per supercage.316 Adsorption of xylene in faujasite zeolites Fig. 2 Simulated adsorption isotherms at 423 K for pure m-xylene pure p-xylene 20% (Ö), (=), p-xylene»80%m- m-xylene xylene (L), 50% p-xylene»50%m-xylene (K) and 80% p-xylene»20% (|) As in experiments, no hysteresis was found when performing simulations by increasing the pressure, starting from empty cavities, or by decreasing the pressure, starting from –lled cavities.The isosteric heat of adsorption was calculated during the simulation using the usual expression Qst\RT [ SNUT[SNTSUT SN2T[SNT2 (23) where S T refers to the average over the simulation run, N is the number of molecules and U is the energy.The simulated zero-coverage isosteric heat for m-xylene is equal to 95 kJ mol~1. This value lies in the range of reported experimental data: 75^6,9 84.515 and 125 kJ mol~1.40 For p-xylene, the zero-coverage isosteric heat of adsorption obtained from simulation is 80 kJ mol~1. This value is also in the range of experimental data: 70^69, 76.5,15 114,40 89.3 kJ mol.41 Despite the discrepancy between these experimental results, all authors have found a zero-coverage heat of adsorption for pxylene of ca. 10 kJ mol~1 smaller than that for m-xylene. This diÜerence in behaviour between the two isomers can be accounted for by the fact that the dipole moment of m-xylene interacts more strongly with the zeolite framework than the non-dipolar pxylene molecule.Additional information can be obtained by examining the histograms of the adsorbate»zeolite potential experienced by individual molecules during the simulation [see Fig. 3(a)]. At low coverage, the average interaction energies with the zeolite framework are [11 400 K ([94.8 kJ mol~1) and [9800 K ([81.5 kJ mol~1) for one molecule of m-xylene and p-xylene, respectively.These results are of the same order of magnitude as the recent values of Kitagawa et al.23 obtained from canonical MC simulations with a Lennard-Jones type potential ([125 kJ mol~1 for m-xylene and [110 kJ mol~1 for p-xylene). In both studies, the m-xylene/zeolite interaction is 25 kJ mol~1 stronger than the p-xylene/zeolite one. No change in the energy distribution is observed when increasing the pressure, indicating that xylene molecules lie in the same adsorption sites whatever the loading.V .L achet et al. 317 Fig. 3 Xylene»zeolite potential-energy distribution for pure components (a) and for an equimolar mixture (b) at maximum loading In order to characterize the location of the adsorbed molecules obtained from our simulations, several distance and angle histograms have been calculated.Both xylene isomers are found to be adsorbed in supercages in front of the sodium cations in site II. No molecule was ever observed in the circular twelve-membered ring windows. Fig. 4(a) shows the sodium»xylene centre of mass distance averaged over 200 000 con–gurations at high pressure (P\100 Pa). The average sodium»m-xylene distance is 2.8 and the ” average sodium»p-xylene distance is 3.1 This is in agreement with the neutron diÜrac- ”.tion study of Czjzek et al.,2 who have reported a decrease in the centre dis- Na`»ring tance in going from p-xylene to m-xylene, indicating a decrease in the Na`»xylene interaction. This diÜraction study also reveals that the plane of the aromatic ring of both isomers is perpendicular to the [111] axis and that the centre of the ring lies on the axis.Fig. 4(b) shows histograms of the distance between the xylene centre of mass and the [111] axis and Fig. 4(c) presents the angle between the aromatic ring and the [111] axis. All these histograms are averaged over 200 000 con–gurations. Centres of mass of adsorbed molecules are not perfectly localized on the [111] axis and aromatic rings are not exactly perpendicular to this axis : the average distance to the axis is ca. 0.5 and ” the average angle is ca. 10°. Broad distance and angular distributions are observed, which reveal an important disorder of the adsorbed phases. p-Xylene molecules always seem to be more delocalized than m-xylene. This is in agreement with the diÜraction study of Czjzek et al.,2 who report that they managed to localize three quarters of the adsorbed m-xylenes but only half of the adsorbed p-xylenes.A higher disorder of pxylene molecules is also observed experimentally by Bellat et al.9 who found a higher molar entropy of the adsorbed p-xylene molecules than the adsorbed m-xylene molecules. All the histograms reported in Fig. 4 were calculated from con–gurations at high loading (P\100 Pa).The corresponding curves at low loading, not shown in this work, are very similar. No speci–c ordering has been observed when increasing the density. A snapshot of a m-xylene molecule adsorbed in front of a site II sodium cation is shown in Fig. 5. The energy and geometry of this con–guration are close to the average values reported for previous histograms.This snapshot illustrates the disorder of the adsorbed318 Adsorption of xylene in faujasite zeolites Fig. 4 (a) Distance between the xylene centre of mass and the nearest cation in site II. The bold line corresponds to pure m-xylene and the plain line to pure p-xylene. (b) Distance to the [111] crystallographic axis of the m-xylene (bold line) and p-xylene (plain line) centre of mass. (c) Angle between the [111] crystallographic axis and the aromatic ring normal vector of an adsorbed m-xylene (bold line) and of an adsorbed p-xylene (plain line).All these histograms are averaged over 200 000 con–gurations at high loading (P\100 Pa). phase, in agreement with the large peaks observed on the histograms of Fig. 4. We have calculated the distance between methyl groups of an adsorbed xylene and the diÜerent atoms of the adjacent hexagonal window, averaged over 200 000 con–gurations.In the case of m-xylene, the average nearest distances of methyl groups to oxygens labelled 1 (see Fig. 5) is 3.5 to oxygens labelled 2 it is 4.5 and to Si/Al atoms it is 4.0 We ”, ” ”. may infer from these results that the most favourable con–guration for m-xylene molecules is a con–guration with short distances between the two methyl groups and oxygens 1 (as shown in Fig. 5, for instance). Such orientations are only possible for the m-xylene isomer which has an angle of 2n/3 between the two methyl groups. In the case of the p-xylene isomer, the two methyl groups have diÜerent surroundings and no similar conclusions can be drawn.Fig. 6 shows the distribution of supercage occupancy for p-xylene averaged over 200 000 con–gurations. The histograms are calculated by assigning a molecule to a supercage whenever the distance between the xylene centre of mass and the centre of the supercage is lower than the estimated radius of a supercage, taken to be equal to 6.25 ”. At all loadings, the supercage occupancy is homogeneous.The same behaviour has beenV . L achet et al. 319 Fig. 5 Snapshot of an adsorbed m-xylene in front of a sodium cation in site II and of (+). K = the hexagonal window correspond to silicon/aluminium and oxygen atoms, respectively. Only carbon atoms and methyl groups of xylene are represented. observed for the m-xylene isomer. The molecular dynamics study of Schrimpf et al.21 revealed a homogeneous supercage occupancy for an average loading of three molecules per supercage, but at lower loadings (one and two molecules per supercage) these authors reported an inhomogeneous distribution.They consider that this behaviour is due to a stronger adsorption energy in the case of an aggregate of three molecules inside Fig. 6 Average number of adsorbed molecules in each of the eight supercages of a unit cell, for p-xylene at 423 K at low loading (P\1 Pa) and at high loading (P\100 Pa)320 Adsorption of xylene in faujasite zeolites a supercage than in the case of three molecules in three diÜerent supercages. In our simulations, we did not observe such a tendency for aggregation. III.3 Adsorption of binary mixtures The coadsorption of m- and p-xylene in NaY has been studied by biased GCMC at a temperature of 423 K.Simulations were carried out for binary mixtures of various gasphase compositions: an equimolar mixture, a mixture with 20% p-xylene and 80% mxylene and a mixture with 80% m-xylene and 20% p-xylene. Total adsorption isotherms (total amount of adsorbed m-xylene plus p-xylene per supercage) are shown in Fig. 2 for the three mixtures and are compared with the pure component simulated isotherms. All these isotherms are of type I. As expected, isotherms corresponding to the mixtures are located between the two curves of the pure components over the whole pressure range. This is in agreement with experiments of Cottier.13,14 Total and partial adsorption isotherms for equimolar mixture are shown in Fig. 7. Fig. 3(b) shows the distribution of the xylene»zeolite interaction in the case of an equimolar mixture. Comparison with Fig. 3(a) (pure components) reveals that the average adsorbate»zeolite interaction energies are unchanged. However, p-xylene molecules become more localized when they are mixed with m-xylene, as evidenced by a sharper energy distribution.When analysing the various histograms previously described, we have found that both isomers are adsorbed in the same sites as in the case of pure components (histograms have very similar shapes and exactly the same position of the maxima). The adsorbed phase still displays some disorder. However, p-xylene molecules in the mixture are somewhat more localized than they were in the single component study.The adsorption selectivity for p-xylene is calculated using the equation: apx@mx\ xpx xmx ymx ypx (24) where and are the molar fractions of component i in the adsorbed phase and in the xi yi gas phase, respectively. Fig. 8 shows the selectivity as a function of the total loading for the three mixtures. Whatever the gas composition, the zeolite NaY appears to be selec- Fig. 7 Total and partial simulated adsorption isotherms for an equimolar p-xylene (Ö) mixture at 423 K (=)»m-xylene (>)V . L achet et al. 321 Fig. 8 Variation of selectivity with loading for binary mixtures of various gas-phase com- (apx@mx) positions : 20% p-xylene»80% m-xylene 50% p-xylene»50% m-xylene and 80% p-xylene» (Ö), (=) 20% m-xylene (>).tive for m-xylene This is in agreement with experimental results of Cottier et (apx@mx\1). al.13,14 at high loading and of Iwayama et al.42 The selec- (apx@mx\0.5) (apx@mx\0.34). tivity is found to depend on the gas-phase composition: the selectivity for m-xylene increases with the molar fraction of p-xylene in the mixture. No drastic change in the selectivity is observed as the total amount of adsorbed molecules increases.The same behaviour has been observed experimentally13,14 for a 20% p-xylene»80% m-xylene mixture whatever the number of adsorbed molecules). In the case of an (apx@mx\0.5 equimolar mixture, experimentalists13,14 have found no selectivity at low loading and a selectivity of 0.5 at high loading. This discrepancy might be due to our potential model but we suggest that this may also be due to some kinetic eÜect in the experiments during the adsorption process.In any case, it is important to mention that the coadsorption results obtained from one mixture simulation can not be directly compared with the corresponding experimental data.13,14 Indeed, during experimental measurements, the Fig. 9 Molar fraction of p-xylene in the adsorbed phase vs.molar fraction of p-xylene in the (Xpx) gas phase at low loading and at high loading The experimental13,14 curve at high (Ypx) (K) (L). loading is also shown for comparison. (Ö)322 Adsorption of xylene in faujasite zeolites gas-phase composition at equilibrium varies with the loading whereas, in GCMC simulations of mixtures, the gas-phase composition is constant (equal to the imposed value) whatever the pressure.Fig. 9 shows the gas composition vs. the adsorbed phase composition at low and high loadings. The third simulated points correspond to the third mixtures studied (20% p-xylene»80% m-xylene, 50% p-xylene»50% m-xylene and 80% p-xylene»20% m-xylene). Such diagrams can be directly compared with experiments, 13,14 the agreement is fairly good and especially good at maximum loading.IV Conclusion Molecular simulations of the adsorption and coadsorption of m- and p-xylene in faujasite NaY have been presented. The simulations were performed using a GCMC technique in which particle insertions and deletions are biased. These biased algorithms result in an improvement in the efficiency of the simulations compared with traditional GCMC, making it possible to study the adsorption of systems such as xylenes in NaY at high loading, that could not otherwise be investigated with GCMC.The use of a detailed adsorbate»zeolite potential function, combined with a simple zeolite structure model, unables us to reproduce, with good agreement, the experimental adsorption data and, especially, the selectivity for the meta-isomer.Molecules are found to be adsorbed in supercages near sodium cations in sites II. The p-xylene molecules are more delocalized around the adsorption sites than the mxylene ones. Both m- and p-xylene isomers exhibit the same behaviour (same average locations and adsorption energies) in both the single component and the binary mixture studies.Therefore, the selectivity cannot be explained in this case by a modi–cation of the sorption properties of one isomer due to the presence of the other isomer. Single component studies might have been sufficient to predict the coadsorption selectivity of the xylenes/NaY system. No attempt was made here to use phenomenological models such as the ideal adsorption theory (IAS) to calculate the selectivity for a given mixture from the single species adsorption isotherms.However, direct coadsorption simulations or measurements are needed for the study of other faujasite selectivities, such as the barium Y faujasite for instance. For this system, the pure m-xylene and p-xylene isotherms are identical, whereas BaY exhibits a selectivity in favour of p-xylene.13,14,44 GCMC studies of other xylenes/faujasite systems are now in progress.Preliminary results obtained from the coadsorption of an equimolar mixture of xylene isomers in the potassium Y faujasite are very encouraging. The experimental selectivity of KY in favour of p-xylene has been reproduced by our simulations using the same type of potential model. wishes to thank the Institut du Peç trole for –nancial support through a V.L.Franc” ais BDI/CNRS grant and for a generous allocation of FUJITSU computer time. Professor M-H. Simonot-Grange is gratefully acknowledged for fruitful discussions, for reading the manuscript and for communicating experimental results prior to publication. References 1 A. N. Fitch, H. Jobic and A. Renouprez, J. Phys. Chem., 1986, 90, 1311. 2 M. Czjzek, H. Fuess and T. Vogt, J. Phys. Chem., 1991, 95, 5255. 3 C. Mellot, D. 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Process Des. Dev., 1985, 24, 78. 41 Y. Gorkey, M. Sc. E. T hesis, PhD thesis, University of New Brunswick, Fredericton, Canada, 1985. 42 K. Iwayama and M. Suzuki, Stud. Surf. Sci. Catal., 1994, 83, 243. 43 P. Demontis, S. Yashonath and M. L. Klein, J. Phys. Chem., 1989, 93, 5016. 44 J-P. Bellat, V. Cottier, M-H. Simonot-Grange, X. Alain and A. Methivier, en des Reç cents Progre` s Geç nie 1995, 9, 159. Proceç deç s, Paper 7/01490B; Received 3rd March, 1997

ISSN:1359-6640    

DOI:10.1039/a701490b

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 325»338 Computation of the free energy for alternative crystal structures of hard spheres Leslie V. Woodcock Department of Chemical Engineering, University of Bradford, Bradford, UK BD7 1DP A single-occupancy cell (SO-cell) method has been applied to calculate the free energies of diÜerent crystal structures of hard spheres via molecular dynamics (MD). The objectives are (i) to examine the nature of the phase transition in the SO-cell model, (ii) to determine the thermodynamic stability of the bcc crystal phase relative to the fcc and the free —uid, (iii) to establish the relative stability of the fcc and hcp crystal structures, and (iv) to investigate hybrid structures of the unit stacking type ìwABCABwœ.MD computations are reported for the pressures of the SO-cell models of all these structures.The SO-cell phase transition for fcc and hcp is –rst-order, with ordered and disordered phases coexisting at the same p, V and T . The transition is much weaker for bcc. The metastable —uid»bcc phase transition parameters are determined; the bcc phase is everywhere unstable compared with fcc. The bcc solid melts to the metastable —uid at a pressure of 14.5 and has a melting volume of 0.95 Np3, i.e., very close to that of the kBT /p3, fcc crystal.A more precise numerical estimate for the fcc»hcp entropy diÜerence is reported. At close packing the fcc phase is the more stable by 0.0026(1) the Gibbs and Helmoltz energy diÜerences are the same at NkBT ; close packing. For expanded volumes close to melting, the hcp crystal has a slightly higher pressure than the fcc ; the enthalpy diÜerence at melting is 0.0030(5) Consequently the Gibbs energy diÜerence approaching NkBT .melting becomes less than the uncertainty in the computations, i.e. \0.001 NkBT . Introduction Hoover and Ree1 devised a method for calculating the relative chemical potential of a crystal and —uid by reversibly connecting the two phases via the low density ideal gas.The method involves calculating the properties of a single-occupancy cell-model —uid (hereafter referred to as the SO-cell system) in which the centre of each spherical atom is constrained to move within its own dodecahedral cell. At low density the Gibbs energy diÜerence between the SO-cell —uid and the unconstrained —uid is known analytically.At high density, the SO-cell system and the unconstrained crystal phase are indistinguishable. Thus, a knowledge of the whole SO-cell equation-of-state establishes a reversible path between —uid and crystal, and hence enables the coexistence state parameters to be determined. The method was applied successfully by Hoover and Ree to the then longstanding question of the hard-sphere phase transition.1 It was subsequently applied to determine the melting point of the Lennard-Jones model2 and also to the two-dimensional Lennard-Jones model,3 and to the high-temperature limit of the Lennard-Jones model, the inverse 12th power soft-sphere model.4,5 Implicit in the Hoover»Ree technique are three basic assumptions. 325326 Computation of free energy The –rst is that the most stable crystal arrangement is known a priori.In the original Monte Carlo (MC) study of hard spheres it was taken to be fcc.1 For soft-spheres and Lennard-Jones molecules, this assumption is probably sound, since the Gibbs energy diÜerence between competing crystal arrangements is determined largely by enthalpic contributions and these can be established by straightforward lattice sums, at least at low temperatures.For hard spheres, however, there is no potential energy in the system; the relative stability of competing crystal phases is determined by the entropy term alone. If there exists a detectable entropy diÜerence between two alternative crystal structures, it follows that there must be a diÜerence in the SO-cell p»V »T equations-of-state, somewhere along the path from close-packed crystal to ideal gas.The fcc and hcp arrangements have the same number of 1st and 2nd neighbours; one would not expect to see a signi–cant pressure diÜerence in the stable crystal region where only 1st neighbour collisions contribute to the pressure. This has been con–rmed by Alder et al.6,7 Likewise, at low density, in the SO-cell model —uid, there exists a density below which 3rd neighbours cannot reach each other to collide.Thus, if there were to be a pressure diÜerence it was to be found in the intermediate density range, i.e. the region where there are 3rd-neighbour collisions contributing to the pressure. In this respect, the molecular dynamics (MD) method has the advantage over the MC method.In MD the pressure is obtained directly from the collision rate, or the virial theorem, and can more readily be resolved into contributions from collisions between successive shells of neighbours. The second assumption implicit in the Hoover»Ree SO-cell method is that the SO-cell equation-of-state can be simulated reversibly along the whole path from high to low density.The system is seen in the Hoover»Ree computations to exhibit a cusp; a curious kind of phase transition at the point where the walls are necessary for even transient stability of a crystal-like phase. All the early studies showed a clear singularity1h5 but the nature of this phase transition has remained obscure. If it were to be –rst order, and a theoretical study by Honda8 indicated that this may be so, there is the possibility of metastability on either side of the transition, of irreversibility in ìcomputer experimentsœ and of pronounced –nite size eÜects.A second objective of the present study, therefore, is to establish the nature and a more precise location of this phase transition in the various SO-cell model structures of hard spheres.The third assumption in the Hoover»Ree method is that the SO-cell model and the unconstrained crystal phase have identical thermodynamic properties in the stable crystal range. This implies that up to melting, contributions to the crystal free energy from vacancy and dislocation defects is negligible. The vacancy defect concentration in hard-sphere crystals has not been reported but an estimate, based upon the Lennard- Jones model, however, would put it around 1 in 10 000 at melting.For the present, therefore, this assumption is implied. There is evidence from MD studies, however, that in two dimensions, defects do contribute signi–cantly to the structure and hence free energy of the crystal in the vicinity of melting.9 MD computations A conventional hard-sphere MD program has been modi–ed so that the sphere centres are contained within the walls of their own primary (–rst neighbour) Voronoi polyhedra, of any prescribed initial lattice structure, by elastic collisions with the containing walls.For fcc, hcp and hybrid close-packed structures, the containing cells are irregular dodecahedral in shape (12 faces), whereas for bcc the cells are octahedral (8 faces). The program proceeds as with an unconstrained hard-sphere —uid MD,10 except that, in addition to the list of the time to the next particle»particle collision a list of times for (tp), each particle to the next particle»wall collision is also kept.(tw)L . V . W oodcock 327 The value is obtained by solving for the positive real root of the quadratic tp p2[rij\2rij Æ øij tp]øij tp2 (1) and the particle»wall collision time is obtained from tw\(rio2 [rjo 2 )/2vw (2) where and are the distances of particle i to the centres of its own cell and a rio rjo neighbouring cell (with a shared wall) that the sphere might collide next with, respectively ; is the particle velocity in the direction of the wall, obtained from vw vw\øi Æ (ri[rio)/ro (3) and are the vector positions of particle i and its own cell centre, respectively, and ri rio ro is the –rst-neighbour lattice parameter.At close packing, the sphere diameter at ro\p, expanded volumes is related to the density, o, by where o is the ro ro/p\(21@2/o)1@3 reduced number density, Np3/V . Computations have been carried out over a range of densities and the raw data for the pressures computed are collected in Table 1, together with the duration in millions of collisions of each MD run.The pressure (p) is calculated both from the virial theorem, by summing the momentum exchange over the collisions in time *t (nc) pV NkT \1];t rij øij 3N*t (4) and also directly from the collision rate pV NkT \1] nc(t)p1@2 9N*t (5) The collision-rate pressure and the virial pressure —uctuate independently and the pressures quoted in Table 1 are the mean of the two in all runs.Only the particle» particle collisions contribute to the pressure. The program further resolves the collisional contribution to the total pressure into contributions from successive shells of neighbours. Particle»wall collisions do not contribute to the total pressure.The total number of wall collisions, however, is monitored, and the ratio of particle»particle to particle»wall collisions determined. MD computations have been carried out for four diÜerent crystal structures. The most extensive data has been obtained for fcc and hcp structures with a system size of N\2048. In these computations the primary box volume and shape was determined by taking 8]8]8 unit cells, each containing four lattice sites, of each crystal structure.For bcc N\2000 (i.e., 10]10]10]2) there are two sites per unit cell. The fcc and bcc primary boxes are cubic in shape, but the hcp primary box shape for N\2048 computations has the side ratios 1 : 31@2 : (8/3)1@2. In the vicinity of the SO-cell phase transitions, the number and shape dependence might be signi–cant ; a much larger model system was therefore investigated, in which layers could be stacked up in such a way that the fcc, hcp and hybrid structures all have the same size and shape.This measure was intended to eliminate small diÜerences in pressure and hence free energies arising from the diÜerence in size or shape of the primary MD box. The model SO-cell systems of N\12 000 are constructed as follows.In a 2D hexagonal lattice there are two sites per unit cell ; the side ratio X/Y of the unit cell is 1 : 31@2. Hexagonal sheets with dimensions 20]1 by 10]31@2 each containing 20]10]2 (\400) atoms are then stacked on top of each other as follows to construct the primary lattice box. If the –rst layer is designated A, the second layer B, and successive AB layers are placed directly on top of each other, the hcp structure is328 Computation of free energy Table 1 MD results to date for the SO-cell model pressures as a function of density for hard-spheres in diÜerent crystal structures ; the integers refer to the number of collisions in millions for each state point fcc fcc hcp hcp bcc hybrid o N\2048 N\12000 N\2048 N\12000 N\2000 N\12000 0.20 0.2772 5 0.30 0.5244 5 0.40 0.8848 20 0.8847 20 0.9038 5 0.50 1.4568 25 1.4564 20 1.4855 5 0.55 1.8501 20 1.8488 10 0.60 2.3394 10 2.3369 19 2.3626 5 0.62 2.6050 5 0.64 2.8559 5 0.65 2.9481 10 2.9443 10 0.66 3.1295 5 0.68 3.4228 5 0.70 3.7059 20 3.7004 20 3.7495 5 0.71 3.8784 20 3.8733 20 3.9198 5 0.72 4.0546 20 4.0476 20 4.0998 5 0.73 4.2466 20 4.2392 20 4.2902 5 0.74 4.4447 20 4.4374 20 4.4842 5 0.75 4.6508 20 4.6424 20 4.6898 5 0.76 4.8666 20 4.8565 20 4.9054 5 0.77 5.0906 20 5.0806 20 5.1308 5 0.78 5.3264 20 5.3151 20 5.3596 5 0.79 5.5743 20 5.5614 20 5.6013 5 0.80 5.8312 20 5.8160 20 5.8594 5 0.81 6.1030 20 6.0847 20 6.1216 5 0.82 6.3821 20 6.3641 20 6.4016 5 0.83 6.6723 20 6.6507 20 6.6861 5 0.84 6.9751 20 6.9547 20 6.9808 5 0.85 7.2935 20 7.2683 20 7.2810 5 0.86 7.6176 20 7.5891 20 7.5954 5 0.87 7.9377 20 7.9132 20 7.9205 5 0.88 8.2489 100 8.2229 10 8.2336 110 8.2378 10 8.2330 5 0.89 8.4962 40 8.4748 10 8.5148 40 8.5105 20 8.5405 20 8.5175 5 0.90 8.5608 20 8.5298 20 8.6741 40 8.6827 20 8.8075 5 8.5841 5 0.91 8.5149 10 8.5373 20 8.6802 20 8.6894 15 9.0649 5 8.5835 10 0.92 8.5674 15 8.5745 10 8.6265 15 8.6873 10 9.3021 5 8.6264 10 0.93 8.6588 10 8.6935 5 8.7179 10 8.7636 10 9.5020 5 8.6982 15 0.94 8.8139 10 8.8245 10 9.7582 5 0.95 8.9768 20 9.0038 20 10.0042 5 0.96 10.3223 5 0.97 10.6543 5 0.98 11.0105 5 0.99 11.4196 5 1.00 10.2434 20 10.2521 20 1.05 12.0759 20 12.0791 20 1.10 14.5915 20 14.5929 20 1.15 18.1241 20 18.1248 20L .V . W oodcock 329 obtained.For 30 layers we have hcp ABABABABABABABABABABABABABABAB If, on the other hand, the third layer (C) is placed in the alternative position, displaced from A, and repeated, the fcc structure is obtained fcc ABCABCABCABCABCABCABCABCABCABC Hybrid crystal structures may also be produced by the regular placing of C-layers, for example, ABCABABCABABCABABCABABCABABCAB This method of construction also permits other hybrid structures with the same number of spheres and geometry, to be simulated, such as ABABCABCABABABCABCABABABCABCAB The vertical distance between sheets is (2/3)1@2.The third box dimension (Z) is therefore 30](2/3)1@2. All these model systems with 30 sheets have the same shape and number of sites (N\12 000). Periodic boundary conditions are applied in all computations.The SO-model phase transition The SO-cell model undergoes a phase transition where the walls would become necessary to maintain even transient crystal-like stability of the otherwise free crystal. Up to this pressure, the equation-of-state can be accurately represented by pso V NkBT \1]nJ2AV0 V B4@3](pJ2)2CAV0 V B3]AV0 V B6D (6) where is the pressure, is Boltzmannœs constant, V is the volume and is the pso kB V0 minimum volume at close packing.Beyond the –rst two terms, eqn. (6) is empirical ; it is accurate for volumes up to the structural disordering phase transition. At pressures above this transition point (see Fig. 1) the equation-of-state of both the SO-cell model, and the free crystal, can be represented from up to in–nite pressure pi* (close packing), by the expansion6,1 V Np3\ 1 J2 ]3AkBT pp3B[4 3 J2AkBT pp3B]4AkBT pp3 B]36J2AkBT pp3B4 (7) Fig. 1 shows the MD pressure data for the fcc and hcp SO-cell models in the vicinity of the phase transition. The transition pressures diÜer signi–cantly in both cases. This diÜerence has been integrated and shown largely to account for the stability of the fcc phase of the hard-sphere crystal compared with the hcp.11 A revised estimate of the free energy diÜerences at close packing and melting is obtained below. Both transitions in the respective SO-cell models show characteristics of a –rst-order phase transition ; an inversion loop for the smaller systems, and transient metastability on either side of the transition.The data in Fig. 1 have been obtained mainly for a system of N\12 000 spheres in the transition region. The transition pressures are found to be 8.54(^0.02) and 8.69(^0.02) for fcc and hcp, respectively.kBT /p3 kBT /p3 The SO-cell phase transition is an unusual –rst-order phase transition between a crystal phase on the one side, with a periodic singlet density distribution, where the particles are ordered and centred on the lattice sites, and a ì—uidœ side in which the singlet density probability function within each cell is uniform.At any point along the330 Computation of free energy Fig. 1 MD pressures for the SO-cell model fcc and hcp structures in the vicinity of the (=) ()) phase transition ; the transition is weak –rst-order, with the ordered and disordered phases apparently coexisting at the same p»V »T state point over a range of densities.The transition pressures diÜer signi–cantly between the fcc and hcp structures ; the fcc having the lower transition pressure. The SO-cell equation-of-state data can be represented fairly well (to 3% accuracy) by a semiempirical expression [eqn. (6) (»»»)] for all densities up to the phase transition.tie lines drawn in Fig. 1, there coexist two distinct phases, each with the same density, temperature, pressure and Gibbs energies, but with diÜerent structures. Further information about the nature of this SO-cell phase transition can be gleaned from the particle»wall collision frequency. This ratio, relative to the particle»particle collision frequency, has been determined over the whole density range and is shown in Fig. 2 as a function of pressure. The number of wall collisions required to bestow thermodynamic stability on the SO-crystal, above the free crystal melting volume, is seen to Fig. 2 The ratio of the number of particle»wall collisions to particle»particle collisions as a function of pressure for the fcc SO-cell model at the pressure of the phase transition (8.5 (=); kBT /p3) there is seen to be a vertical increase in the wall collision frequency whilst, across the transition, the particle»particle collision rate remains constant as the pressure is constant.L .V . W oodcock 331 be miniscule right up to the SO-cell order»disorder phase transition. At this point, the wall collisions increase dramatically and vertically at constant pressure, until ìmeltingœ is complete.Fluidñfcc coexistence In order to determine a —uid»crystal transition pressure in the hard-sphere model, Hoover and Ree1 used an MC method to get the pressure of the fcc (SO)-cell model, as –rst de–ned by Kirkwood.12 For both the fcc and the hcp crystal structures, all the primary cells are regular dodecahedral, each of the 12 faces representing the perpendicular plane bisecting the straight line connecting –rst-neighbour sites.For N spheres there are N space-–lling dodecahedra, each containing a single sphere, and all of which are identical. For pressures above freezing, the unconstrained crystal is stable without the cell walls, and its thermodynamic properties are deemed to be indistinguishable from those of the SO-cell model.At very low, ideal-gas, pressures the diÜerence in entropy between the free —uid phase and the SO-cell model is exactly equal to The follow- NkB . ing statistical mechanics argument is the proof. For a hard-sphere system with no intermolecular energy, the excess Helmholtz energy and entropy (S) are essentially equivalent, and determined by the con–gurational partition function (Q) A\T S\kB ln Q for the ideal gas Q\V N/N!\(NV1)N/N! where v is volume per particle, i.e., the cell volume in the SO model.For the SO-cell ì ideal gasœ QSO\V 1N (since the particles are now distinguishable by their cells there is no division by N!) The communal entropy *com ST, V \kB ln(Q/QSO) \kB[N ln N[(N ln N[N)] \NkB (use has been made of Stirlingœs approximation for large N, i.e., ln N!\N ln N[N).Kirkwoodœs communal entropy at non-ideal —uid densities is the entropy (*com ST, V) diÜerence between the SO-cell system and the free —uid at the same T and V . Physically, communal entropy represents the additional entropy a model —uid acquires when its molecules are free to explore the whole available volume, compared with a crystal or glassy state in which the molecules are naturally constrained.One could equally de–ne an alternative communal entropy as the entropy diÜerence between the SO-cell model and the free —uid at the same temperature and pressure The two communal (*com ST, p). entropies are then exactly related by *com ST, V\*com ST, p]NkB ln Z (8) (where Z is pV /RT ). In the low density limit, Z]1, ln Z]0, and therefore *com ST, V\*com ST, p\NkB The corresponding ìcommunalœ Gibbs energy is Gfluid[GSO\*comGT, p\p*VT, p[T *com ST, p (9)332 Computation of free energy Freezing of the —uid occurs at a pressure when this communal Gibbs energy is zero.can be regarded as a ìcommunal volumeœ, i.e., the excess volume of the free *VT, p —uid over the SO-cell model at the same temperature and pressure.It approaches the constant value of 2pNp3/3 (where p is the sphere diameter) in the ideal gas limit, that p]0. At constant T , and vanishing p, therefore, the Gibbs energy diÜerence between the SO-cell model and the free —uid is also exactly NkBT . For any reversible process, from the –rst and second laws of thermodynamics, Gibbs energy changes with temperature and pressure according to dG\V dp[SdT (10) where V is the volume and S is the entropy. The pressure at which the hard-sphere —uid crystallizes is obtained by integrating eqn.(10). Along any p»V isotherm the SdT term is zero and the coexistence pressure can be obtained from the communal volume *Gfluid~SO\P0 coexistencep *com V dp[NkBT \0 (11) Using the the Carnahan»Starling (C»S) equation13 for the stable —uid from freezing to the ideal gas, the SO-cell model from low density to crystal close packing, the —uid» crystal (fcc) coexistence pressure of the hard-sphere —uid is determined analytically by integrating the communal volume as shown in Fig. 3, and found to be 12.06(5)kBT /p3. The coexistence percentage packing fractions at melting and freezing are 48.9(1)% and 54.3(1)% [from eqns.(6) and (7), and the C»S equation, respectively]. These revised parameters agree with the original results of Hoover and Ree1 to within the uncertainties but are probably more precise. Fig. 3 Communal entropy (T , p) (» » » ») and communal volumes (T , p) (»»») of the hard-sphere —uid de–ned with reference to the SO-cell model at the same T and p; the communal entropy at zero pressure is exactly and since the enthalpy is zero the communal Gibbs energy is NkB , NkBT .The crystallisation coexistence parameters of hard spheres can be calculated by integration of the communal volume as shown; this is the diÜerence in volume between the SO-cell model (fcc) and the unconstrained —uid at the same T and p. The communal volume changes discontinuously at the SO-cell phase transition.The low density limit of the communal volume is a constant 2pNp3/ 3. The reduced crystallisation pressure (or temperature) is calculated by matching the shaded area of integration, which represents a Gibbs energy diÜerence, with the communal molar Gibbs energy diÜerence at zero pressure shown as the square area. (NkBT )L .V . W oodcock 333 Fig. 3 also shows that, over the whole range of existence of the stable —uid, its communal entropy at constant pressure hardly varies from the low density (*com ST, p) limit value of to the entropy of crystallisation (1.2 over the whole —uid range. (NkB) NkB) Previous MD computations on the metastable —uid region and glassy states of the hardsphere —uid show that this communal entropy actually persists up to amorphous close packing, and that the residual entropy, i.e., the communal entropy of hard- *com ST, p , sphere glasses is also of the order NkB .14 The bcc phase Studies of nucleation phenomena in metastable compressed hard-sphere —uids suggest that there may be a region where the bcc phase, although always less stable than the fcc, is more stable than the ìsupercooledœ metastable hard-sphere —uid.A knowledge of the metastable —uid»bcc phase transition might help to explain why the bcc phase seems to predominate in initial nucleation of the metastable —uid, even when it is thermodynamically less stable than the fcc phase.15 Another reason for investigating the Gibbs energy of the bcc phase is that computer simulation studies of sheared hard-sphere —uids in the high density range have revealed a phase-diagram, with a de facto Gibbs phase rule in operation.16 The predominant sheared crystal-like phase turns out to be the 1,1,1 slip-plane of the bcc structure.As this behaviour can be seen as a shear perturbation of the hard-sphere phase diagram, a knowledge of the relative stability of the unperturbed bcc crystal is required to complete the picture. Recent theory combines these two phenomena with the suggestion that nucleation homogeneous, initially to bcc, is naturally induced by incipient local —uctuations in shear stress.17 Accordingly, the pressure of the bcc SO-cell model has been determined from MD calculations over the whole density range and the raw MD pressure data are included in Table 1.The data are plotted in Fig. 4 alongside the corresponding pressure data for the fcc SO-cell model, and the free —uid. The results are quite surprising. For pressures Fig. 4 Equation-of-state data for the bcc SO-cell model (N\2000) compared with the (»L») fcc SO-cell model and the unconstrained HS —uid (»»») ; the bcc system also shows a (»=») phase transition, around the volume, which is somewhat weaker than fcc.For densities lower than the phase transition the SO-cell pressures of the two structures are very similar ; consequently, the coexistence parameters of the bcc»metastable HS —uid coexistence can be roughly seen by matching the two shaded areas as indicated.334 Computation of free energy below the SO-cell phase-transition pressure there is hardly any diÜerence between the bcc and the fcc pressures.The phase transition pressures occur at about the same pressure, but because the bcc crystal is more akin to the free —uid, the bcc SO-cell transition is much weaker. One consequence of this similarity, is that the metastable —uid»bcc transition pressure can be roughly measured by matching the two areas as shown in Fig. 4. A proper numerical integration shows that the metastable —uid is in thermodynamic coexistence with the bcc crystal phase at p\14.5(^0.5) i.e., surprisingly close to kBT /p3, the equilibrium —uid»fcc crystallisation transition. At this pressure, the coexistence volumes are, melting and freezing Bayens18 has Vm\0.955(5)Np3, Vf\0.990(5)Np3. recently applied a variation of scaled particle theory (SPT) to the solidi–cation of the hard-sphere —uid. Taking Bayensœ SPT equation-of-state with two-parameters, gives for the bcc»metastable —uid melting and freezing volumes,19 0.969Np3 and 1.007Np3, respectively, in quite good agreement with the present result.Fccñhcp diÜerences Fcc, hcp and other hybrid crystal structures diÜer in the third and successive-shell coordination numbers (n).Up to –fth neighbours for fcc and hcp are neighbour 1st 2nd 3rd 4th 5th fcc n 12 6 24 12 24 dist./r0 1 21@2 31@2 41@2 51@2 hcp n 12 6 2 18 12 dist./r0 1 21@2 (8/3)1@2 31@2 (11/3)1@2 The two crystal structures also diÜer in the shapes of the primary Voronoi dodecahedra. In the crystal region at high density approaching close packing, only –rst neighbours collide, one would not expect to –nd a signi–cant pressure diÜerence in the stable crystal range up to the melting transition because of near-neighbour collision dominance.Likewise, in the lower density region of the SO-cell model, the fcc and hcp structures have exactly the same equation-of-state and hence have identical free energies. Lengthy MD calculations, in excess of 40 million collisions for each state point, of the pressure in the SO-cell model, in the intermediate density range, however, have revealed that there are signi–cant contributions to pressure from second, third, and, for hcp, fourth and –fth neighbour collisions, which diÜer between fcc and hcp.The contributions to the pressures from third and higher neighbour collisions have also been determined and are presented in Fig. 5. It is in this intermediate density region around the phase transition, that this small, but signi–cant, diÜerence in pressure between the two structures was –rst found.11 Extensive MD computations in the SO-cell model, for both fcc and hcp structures, with systems of 500, 2 048 and 12 000 spheres, averaging between 10 and 100 million collisions for each data point, were reported in the volume range V /(Np3)\1.00»1.25.This study indicated that the previously unknown small Gibbs energy diÜerence between these two close-packed structures is accessible on modern workstations, and the fcc structure to be the more stable. In particular, a large area of pressure diÜerence, which arises because the two structures have diÜerent transition pressures, was determined and integrated between volumes 1.00 and 1.25Np3.The result was a Gibbs energy diÜerence in 0.005(1)NkBT favour of fcc of with a large uncertainty, estimated around 20%. The present data (seeL . V . W oodcock 335 Fig. 5 Contribution to the pressure from collisions between 3rd neighbours (fcc) and 3rd, (»=») 4th, and 5th neighbours (hcp) (- - as a function of density ; the range of these contributions is )- -) of the same extent as the pressure diÜerences ; the actual pressure diÜerence itself (Fig. 6) is not accounted for by 3rd neighbour collisions alone Table 1) include much longer computations at lower densities and reveal an unforeseen long weak tail in the pressure diÜerence for V [1.25Np3, in favour of hcp.There is also a weak tail in the pressure diÜerence for volumes below the melting volume. More accurate pressure diÜerence data for these weak tails, on both side of the SO-cell phase transition, have now been obtained. The present data cover the wider range until the tails become imperceptible, and are shown in Fig. 6. The pressure diÜerence data points can be integrated numerically, using the simple integration algorithm ;n *pCV (n]1)[V (n[1) 2 D over the whole range of volume.The following result is obtained for the molar Helmholtz energy diÜerences between the fcc and hcp structures (fcc having the lower free energy). At close packing PV0 =(phcp[pfcc) dV \0.0026(1)NkBT At the melting volume PVm =(phcp[pfcc) dV \0.0023(1)NkBT These estimates remain subject to uncertainties arising from both the MD data for the pressures, i.e., ca.^0.0010]2 for any state point, and to the numerical kBT /p3 integration of the graphical data in Fig. 6. The change in Helmholtz energy diÜerence between close packing and the melting point amounts to only as evi- 0.0003(1)NkB , denced by the tiny area in pressure diÜerence data up to the melting volume in Fig.(Vm) 6. Because of this non-negligible pressure diÜerence at melting, however, the Gibbs and Helmholtz energies diÜer very slightly, by ca. 10~5 The fcc structure, therefore, is kBT . everywhere more stable than hcp.336 Computation of free energy Fig. 6 MD results for the pressure diÜerence between the fcc and hcp SO-cell models (»Ö») from close packing to the density below which there is no detectable diÜerence.A numerical (V0) integration of the pressure diÜerence with respect to volume yields a Helmholtz energy diÜerence between the two structures at the volume of close packing of (V 0\0.7071Np3) 0.0026(1)NkBT . The Helmholtz energy diÜerence at melting is obtained by integrating the small (Vm\0.964Np3) area up to the and subtracting from the latter.It is clear from the data that the (0.0003NkBT ) Vm enthalpy diÜerence at melting (V *p) is approximately 0.003NkBT . Near melting, there is a signi–cant pressure diÜerence. These pressure diÜerence data for the stable crystal region agree well with some very recent results of Speedy,21 who additionally reports a very small (0.01%), almost imperceptible, pressure diÜerence in favour of fcc that persists to close packing.For much of the crystal range the enthalpy diÜerence is negligible compared with the entropy diÜerence and the fcc structure is therefore also the more stable at constant pressure. The search for the most stable crystal phase of hard spheres has a long history, going back to the early cell-cluster theory predictions of Stillinger and co-workers.22,23 The original cell-cluster theory predicts that hcp is more stable by The problem 0.0008NkBT .is clearly many-bodied and not amenable to theory involving gross approximations. By using MD simulation methods, Alder et al.24 –rst reported (without giving details) a pressure diÜerence from a 216 particle MD computation at the melting point of pfcc The present data at V \0.95Np3 for N\2048 give this pressure [phcp\0.01kBT /p3.diÜerence as Alder et al. assumed that the absolute diÜerence decreased 0.003kBT /p3. linearly with density and estimated the Gibbs energy diÜerence at close packing (relative to that at melting) to be in favour of fcc. The present data show that the 0.002NkB T pressure diÜerence found at melting actually decreases to negligible values much more rapidly, and that the change in Helmholtz energy diÜerence between close packing and melting is only The closeness of Alder et al.œs ìguesstimateœ of *S to the 0.0003(1)NkBT .actual value at close packing appears to have been largely fortuitous. Other attempts to quantify the diÜerence from free volumes have also indicated that fcc is more stable but with widely diÜering estimates. Hoover25 found that the diÜerence between fcc and hcp in the nearest neighbour harmonic oscillator model was in favour of fcc.Kratky26 calculated the free volume from the mean square 0.0015NkBT displacement for both structures. By using an approximate equality between the logarithm of the free volume and the entropy, Kratky obtained the result *G\ again in favour of fcc.This is in the right direction but is more than 10 times too 0.05NkBT , large. It seems there must be other many-body subleties aÜecting the elusive diÜerence.L . V . W oodcock 337 The only comparable extensive study to the present, in recent years, is that of Frenkel and Ladd.27 As in the the present method, they also connect the two structures via a reversible path to a state where the diÜerence is zero and the Gibbs energy is analytic, i.e., an Einstein crystal with the same crystallographic structure.They were unable to establish a signi–cant diÜerence within the uncertainty of their computations, but bounded the molar Gibbs energy diÜerence at melting between and [0.0001NkBT the balance favouring fcc.The present data are consistent with that result. ]0.002NkBT , wABCABw hybrid structures The fcc and hcp lattice structures are but two extreme cases of an in–nite number of possible ways of stacking 2D hexagonal layers of spheres to form an in–nite 3D structure. The ìCœ layers can be either regularly or randomly placed every third layer. The simplest hybrid, repeating –ve-fold layers of has been studied and the (wABCABw)n MD data obtained so far are included in Table 1.For this particular hybrid structure, the pressures in the vicinity of the order»disorder transition only have been computed. The pressure of the transition is found to be intermediate between fcc and hcp. Although, a hybrid structure could, in theory, have a lower Gibbs energy than the fcc crystal, these early MD data for this particular hybrid structure suggest that is unlikely.Further computations are still required to assess the eÜect of imposing isotropy on the hcp and hybrid structures. All except the fcc have only two-fold symmetry compared with the six-fold fcc. Conclusions The SO-cell model, originally introduced by Hoover and Ree 30 years ago to establish and determine the hard-sphere freezing transition is the subject of an in-depth MD investigation to calculate the relative Gibbs energies of various alternative crystal structures.These computations are on-going, but the results to date lead to the following conclusions. (i) The SO-cell model with the crystallographic structure, of fcc, hcp and other hybrid structures, exhibits a –rst-order phase transition from order to disorder.The pressure at which this transition occurs diÜers appreciably from one structure to another with a substantial diÜerence between fcc and hcp. (ii) The nature of the SO-cell transition is unusual; in the transition range, two diÜerent phases, ordered and disordered, coexist, while p, V and T are all the same. (iii) The bcc SO-cell model also shows an order»disorder transition but it is much weaker and appears second-order for a system of 2000 particles.For densities below the transition, the SO-cell pressure of the bcc structure is very similar to that of the fcc. This enables a simple estimate of the melting pressure of the bcc phase to the metastable hard-sphere —uid to be 14.5(5) not much greater than the —uid»fcc coexistence kBT /p3, pressure.(iv) The fcc close-packed crystal structure is found to have lower Gibbs and Helmholtz energies than the hcp by 0.0026(1) at close packing, and 0.0023(1) at NkBT NkB T melting, and is hence the more stable of the two structures for hard spheres over the whole crystal phase range at constant volume, and also at constant pressure.(v) Preliminary results for a hybrid structure indicate an intermediate pressure between fcc and hcp, in the SO-cell transition range, implying that none of these hybrid crystal structures are likely to be more stable than fcc.338 Computation of free energy Finally, a correction to experiment: it has been pointed out by Car,20 that the conclusion, fcc is the more stable phase, is consistent with the experimental transformation of crystalline helium at 4 K from an hcp to an fcc structure ; if the fcc is the more stable classical phase as the atomic motions go from quantum mechanical to classical, and the classical diÜerence increases with temperature. It may be also be noted that very recent experimental crystal growth studies of another type of hard-sphere, i.e., of monodisperse colloidal particles, have been reported very recently.28 These novel experiments appear also to con–rm the preferred stability of the fcc phase for hard-sphere crystals.I wish to thank Dr R. Speedy for supplying unpublished results.21 References 1 W. G. Hoover and F. H. Ree, J. Chem. Phys., 1968, 49, 3609. 2 J-P. Hansen and L. Verlet, Phys. Rev., 1969, 184, 151. 3 S. Toxvaerd, J. Chem. Phys., 1978, 69, 4750. 4 J-P. Hansen, Phys. Rev. A: Gen. Phys., 1970, 2, 221. 5 W. G. Hoover, M. Ross, K. W. Johnson, D. Henderson, J. A. Barker and E. C. Brown, J. Chem. Phys., 1970, 52, 4931. 6 B. J. Alder, W. G. Hoover and D. A. Young, J. Chem. Phys., 1968, 49, 3688. 7 B. J. Alder, D. A. Young, M. R. Mansigh and Z. W. Salzburg, J. Comput. Phys., 1971, 7, 361. 8 K. Honda, Prog. T heor. Phys., 1976, 55, 1024. 9 F. van Swol, L. V. Woodcock and J. N. Cape, J. Chem. Phys., 1980, 73, 913. 10 B. J. Alder and T. W. Wainwright, J. Chem. Phys., 1960, 33, 3813. 11 L. V. Woodcock, Nature (L ondon), 1997, 385, 141. 12 J. G. Kirkwood, J. Chem. Phys., 1950, 18, 380. 13 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1979, 51, 635. 14 L. V. Woodcock, Ann. NY Acad. Sci., 1981, 371, 274. 15 S. Alexander and J. P. McTague, Phys. Rev. L ett., 1978, 41, 702. 16 D. B. Nicolaides and L. V. Woodcock, J. Phys. A: Math. Gen., 1997, 30, 345. 17 R. A. Gray, P. B. Warren, S. Chynoweth, Y. Michopoulos and G. S. Pawley, Proc. R. Soc. L ondon A, 1995, 448, 113. 18 B. Bayens, PhD Thesis, University of Ghent, 1996. 19 B. Bayens, 1996, personal communication. 20 R. Car, Nature (L ondon), 1997, 384, 115. 21 R. Speedy, 1997, personal communication. 22 F. H. Stillinger Jr. and Z. W. Salsburg, J. Chem. Phys., 1967, 46, 3962. 23 W. G. Rudd, Z. W. Salzburg, A. P. Yu and F. H. Stillinger Jr., J. Chem. Phys., 1968, 49, 4857. 24 B. J. Alder, B. P. Carter and D. A. Young, Phys. Rev., 1969, 183, 831. 25 W. G. Hoover, J. Chem. Phys., 1968, 49, 1981. 26 K. W. Kratky, Chem. Phys., 1981, 51, 167. 27 D. Frenkel and A. J. C. Ladd, J. Chem. Phys., 1984, 81, 3188. 28 A. Van Blaarderen, R. Ruel and P. Wiltzius, Nature (L ondon), 1977, 385, 321. Paper 7/01761H; Received 13th March, 1997

ISSN:1359-6640    

DOI:10.1039/a701761h

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 339»366 Relationship of crystal structures to interionic interactions in mono-, di-, tri- and ter-valent metal oxides Mark Wilson and Paul A. Madden Physical and T heoretical Chemistry L aboratory, Oxford University, South Parks Road, Oxford, UK OX1 3QZ The ability of an extended ionic interaction model to account for the observed crystal structures of a very wide range of binary oxides is assessed.The model includes the many-body eÜects which result from ionic polarization and their changes in eÜective size and shape of ions in diÜerent coordination environments. Starting from potentials which have been fully obtained for certain substances on the basis of electronic structure calculations, systematic modi–cations in the values of key parameters (such as ionic radii) are made, so as to derive potentials for other substances on the basis of an assumed transferability of an ionic interaction model. Comparison of the predicted structures with observation enables the identi–cation of the key parameters which control the evolution of structure type and the physical nature of the interactions responsible for the adoption of ìcovalentœ structures by some materials.I Introduction Recent work has shown how an extended ionic interaction model can account for many of the crystal morphologies adopted by metal oxides.1 The ionicity of the model is enforced through the use of formal charges and the restriction of the non-coulombic terms in the potential to those appropriate to closed-shell interactions, i.e.overlapdriven repulsion at short-range, dispersion and induction (polarization). The ìextendedœ nature of the model arises from the allowance for the dependence of the ionic properties on the instantaneous coordination environment in which an ion –nds itself : properties such as the ionic size, shape and charge distribution, which enter the expressions for the potential, respond to the con–guration of neighbouring ions.As a consequence, the potential acquires a many-body character. It has been shown how such a potential can be successfully incorporated in a simulation methodology: this builds upon earlier work done with the shell model,2 which also incorporates many-body eÜects in a computationally tractable scheme. The signi–cant diÜerence from the shell model is that the representation of the many-body eÜects is more —exible than allowed for by the restrictive mechanical model on which the shell model is based.Furthermore, the parameters and functional forms required to model the many-body interactions are derived, where possible, from well directed ab initio electronic structure calculations which focus on the properties of individual ions in idealized representations of the condensed phase environment.3h5 A model of this sort should be transferable as long as the signi–cant aspects of the underlying physics have been correctly accounted for.Transferability can imply the ability of a single potential to describe several phases of the same material equally well. 339340 Relationship of crystal structures to interionic interactions It can also mean that the potentials for two substances should diÜer only through chemically motivated alterations to certain parameters.One might imagine, for example, that potentials modelling the short-range oxide»cation repulsive interaction might transfer within a given stoichiometry simply by changing some cation radius parameter.Whilst transferability of a potential (with only ìchemically motivatedœ parameter shifts) between diÜerent substances within the same stoichiometry is a reasonable expectation when the cations all belong to the same group of the periodic table, and hence have the same valence electron con–guration, it is less obvious that the same potentials can be transferred to other cations with the same charge but from a diÜerent group.To what extent, for example, can Sn2` and Pb2`, which have s2 outer electron con–gurations, be treated as analogues of Sr2` and Ba2`, which have similar ionic radii ? Even the potentials for diÜerent stoichiometries should be related, although to do this successfully requires some insight into how to allow for the change in the ionic properties in between necessarily diÜerent coordination environments. This paper is concerned with an exploration of transferability of the extended ionic model for oxides in this general sense.II Overview II. A Interionic interactions II. A. 1 Simple ionic model. The simplest ionic interaction model is a Born»Mayer» Huggins eÜective pair potential (EPP): UEPP\;i ; j;i A exp[[(rij[pi[pj)/(oi]oj)]]QiQj/rij[C6/rij 6 (2.1) where is the separation between ions i and j, is the radius of the ion and rij pi(j) oi(j) describes the range of the repulsive interaction, which re—ects how each ionœs electron density decays radially with separation from the nucleus.The remaining terms are the coulomb and dispersion interactions. Within this EPP, the response of the ionic properties etc.] to the ionic environment is ignored, is a –xed parameter.[pi(j) pi(j) The structural predictions of such a model are dominated by two factors. T he ion sizes (radius-ratio rules). The coulombic nature of the interactions means that anions will tend to surround cations. If we consider a crystal as simply generated from hard spheres of appropriate charge and size then the number of anions that can pack around a central cation simply depends on the relative size of the two species.For a small cation only a relatively small number of anions will be able to pack around the central cation (a favourable interaction) whilst minimizing the repulsive anion»anion coulombic interactions (unfavourable). It is clear that this consideration will tend to aÜect the short-range order (SRO) of the structure.Another (equivalent) viewpoint is to consider the holes occupied by one species in the sublattice occupied by the other species. In an fcc lattice, for example, the ions have the choice of the tetrahedral or octahedral holes (or a mixture), of which there are twice as many of the former. Ion charges. The attractive anion»cation and repulsive anion»anion coulombic interactions largely govern the SRO about a cation.However, if one species has a greater charge then there may be an additional, more intermediate-ranged, eÜect. For example, in the —uorite crystal stoichiometry) the nearest-neighbour cation»cation separa- (MO2 tion is )2 times the anion»anion analogue, as a direct result of the greater cation charge.Crystal structures which follow this pattern are termed charge-ordered and tend to have high site symmetries.M. W ilson and P. A. Madden 341 As is well known, departures from such simple rules are legion. At a qualitative level, many important crystal structures are not charge ordered and contain ions at highly unsymmetrical sites. Even for charge-ordered systems, quantitative studies, where an attempt is made to derive an EPP from an electronic structure calculation, fail to give the observed coordination number.Such failings are often attributed to ìcovalencyœ : it is our objective to establish to what extent the failings of the simple ionic model can be attributed to the many-body eÜects contained within the extended ionic model. The most important eÜects for a consideration of crystal morphology and their consequences are outlined below; a fuller account has recently been given elsewhere.1 II.A. 2 Manyñbody eÜects. To visualize correctly the many-body eÜects it is important to realise that ions in a condensed phase are profoundly diÜerent from gas-phase ions.1,4,5 The electron densities of anions, in particular, are stabilized by a con–ning potential and, in part, by the exclusion of the anionœs electrons from the space occupied by the charge clouds of neighbouring cations.Thus, we always consider an ionœs properties with respect to their values in some reference state, often the perfect crystal, and whenever the ion is found in some other state we must consider how the change in the con–ning potential will change these properties.Polarization eÜects. Polarization eÜects involve the interactions of the multipoles induced in an ion with the charges and multipoles of other ions in the system. Induced multipoles can only occur on an ion which is found at a low-symmetry site. Polarization interactions necessarily lower the energy of such a con–guration, relative to what would be predicted from the simple ionic model.Polarization eÜects are, therefore, the dominant cause of the adoption of low-symmetry, non-charge-ordered structures and, in this sense, oppose the coulomb charge-ordering tendency of the simple ionic model. Polarization eÜects can be characterized quantitatively by examining directly, using electronic structure methods, the induced multipoles on ions in distorted crystals.3 The reference point is the high-symmetry environment in a cubic crystal ; the polarization eÜects are associated with the electrostatic consequences of the deformation of the ionœs electron density when its site symmetry is lowered.If an ion in the crystal at a relatively large distance (say, greater than next-nearest neighbour separation) from the central ion is displaced oÜ its lattice site, its eÜect on the potential felt by the electrons in the central ion is simply that of the electric –eld and –eld-gradient at that site.There will be induced dipoles and quadrupoles given by the usual multipole expansion.6 ka i , as\aabEb(ri)]13 Bab, cdEb(ri)Ecd(ri)] … … … hab i, as\12 Bab, cd Ec(ri)Ed(ri)]Cabcd Ecd(ri)] … … … (2.2) where the superscript as means that these moments are appropriate when the sources of the –elds are asymptotically far away from ion i.Here a and C are the dipole and quadrupole polarizabilities and B is the dipole»dipole»quadrupole hyperpolarizability : the components of a, C and B are speci–ed by a single number for a spherical ion.6 These polarizabilities are those appropriate to the ion in its crystalline environment and may be much smaller than the free-ion values7 owing to the con–nement eÜects.and Ea are components of the –eld and –eld-gradient, respectively. Eab If a cation next to an anion is displaced, there is an additional eÜect on the anionœs charge density. Besides the –eld and –eld-gradient, a deformation of the con–ning potential occurs. Whilst the –eld and –eld-gradient tend to push the electrons in one direction (away from the displaced cation), this ìdent-in-the-wall œ3 allows them more freedom to move into the space vacated by the cation.Hence there is a short-range contribution to the induced dipole on an anion, li,sr, which opposes the ìasymptoticœ dipole caused by the electric –elds. This has been studied in electronic structure calculations.The eÜect is342 Relationship of crystal structures to interionic interactions substantial. The dipole induced by displacing –rst-neighbour cations is reduced below that expected from the asymptotic term alone by ca. 50%. For induced quadrupoles, the eÜect is even larger : the limited evidence available suggests that the short-range term is as large as the asymptotic one, so that the net quadrupole on the anion can be very small or even opposite in sign to the asymptotic quadrupole.For the complementary case of a cation surrounded by anions the short-range moment induced on the cation acts in the same sense as the asymptotic; the short-range moment eÜectively enhances the asymptotic moment on the cation.The technique by which this short-range polarization eÜect may be included in simulations is described in ref. 8. Ion compression (and deformation). In general, and for the oxide ion in particular, it is important to recognize that when an ion moves from one coordination environment to another (in passing from the reference crystal structure to some other structure, for instance) the ionœs size and shape will also change and this will aÜect the degree of overlap, and hence the repulsive interactions, with its neighbours.If an ionœs electron density is only weakly changed by the changes in its coordination environment of interest it may be sufficient to model the changed degree of overlap between the frozen charged clouds of the ions with an EPP.This representation seems adequate for many halide systems.9,10 If the environment changes are between two high-symmetry crystals it may be sufficient to treat the ion as spherical and simply to account for the change in size. This might be done, schematically, by modifying the Born»Mayer representation of the repulsive interactions in the following way: UCIM\;i ; j;i A expM[[rij[(p 6 i]di)[(p 6 j]dj)]/(oi]oj)N];i ure(di) (2.3) where now the –rst (exponential) term is associated with the overlap of the charge clouds of a pair of ions whose radii are appropriate to their instantaneous environment. represents the ion radius in the reference environment and is the change in radius in p 6 i di the new environment.The second term, (d), is the ìrearrangementœ energy required to ure compress or expand the ion from its size in the reference environment to radius (p 6 i]di).In practice, is found by minimizing the total energy, with respect to all for the di uCIM di new con–guration. Eqn. (2.3) is dubbed a compressible ion model (CIM) potential ; the same idea is embodied in the ìbreathingœ modi–cation of the shell model.11 In principle, the range parameter could also change in the same way.If the con–guration of inter- oi est is very distorted from the high-symmetry reference state, it may become necessary to allow for the change in shape of the ion as well as its size (deformation).12 The way in which the overlap and rearrangement contributions to the total repulsive energy of a crystal may be separated in electronic structure calculations for diÜerent crystal structures has been demonstrated by Pyper.5 Such calculations may be used to parametrize a CIM potential directly.13 Fig. 1 shows the results of such a series of ab initio calculations (the points) performed on MgO in three crystal structures : the six-coordinate rocksalt (B1), eightcoordinate CsCl (B2) and four-coordinate blende (B3).The two sets of curves show the rearrangement energy of a single oxide ion, and the overlap energy between an oxide ion and one of its cation neighbours in each crystal at the lattice parameter R. The ure curves tend to a –nite value at large separation, re—ecting the second electron affinity of the O atom, i.e. the energy is given with respect to O~]e~. The results indicate a strong dependence of the potential on the coordination number of the crystal, owing to the diÜerence in the oxide ion electron density in each crystal.The solid lines represent the CIM –t. This is obtained by requiring that, at each R and at the minimum of UCIM with respect to d, the overlap and rearrangement terms in eqn. (2.3) match the respectiveM. W ilson and P. A. Madden 343 Fig. 1 Ab initio data for the self-energy of an oxide ion (upper curves) and for the overlap energy (lower), as a function of the lattice parameter R in four- (B3, ]), six- (B1, ]) and eight-coordinate (B2, structures for MgO, as calculated by Wilson et al.13 The inset shows the pair potentials L) calculated from this data for each structure ; note that these exhibit a coordination number dependence. The solid lines in the main –gure show the values calculated from the CIM potential,13 which was –t to the B1 data.energies for the six-coordinate ab initio data (i.e. this is the reference structure in these considerations). The excellent prediction by the model of both the four and eightcoordinate data demonstrates that the CIM representation recaptures the dependence of the oxide ion charge density on its coordination environment.However, one can still construct an EPP by simply combining the calculated overlap and rearrangement energies and –tting them with a sum of pair potentials. The insert in Fig. 1 shows the resultant EPPs for the three structures. The crucial point is that this procedure produces three diÜerent EPPs, one for each structure.From the insert, we can anticipate the eÜect of using a single EPP on the relative energetics of the crystal structures. Use of, for example, the six-coordinate EPP will tend to systematically underestimate the four-coordinate short-range repulsive energies whilst over-estimating the eight-coordinate. This observation applies generally and can be summarized by saying that the compressible ion potential will exhibit a greater degree of transferability than an EPP and will favour high-coordination number structures relative to a pair potential –tted on the same reference state.This ì principle œ is nicely illustrated by calculations on MgO.13 In general, the result of allowing for ion compressibility is some blurring of the simple radius ratio criterion for assessing coordination number, since an ion does not have a single, well de–ned radius.III Rationalization of crystal structures with the extended ionic potentials In the following sections we will survey the extent to which extended ionic potentials, which incorporate the many-body eÜects described above, can account for the crystal structures of the oxides of the elements highlighted in the reduced periodic table in Table 1 The factors aÜecting the preferred crystal structures will be introduced by considering each stoichiometry in turn.For each stoichiometry there is a set of possible crystal structures. How the chemical properties of the metal ion in—uence the chosen344 Relationship of crystal structures to interionic interactions Table 1 Reduced periodic table considered in this work Li Be Na Mg Al Si K Ca Sc Ti … … … Cu Zn Ga Ge Rb Sr Y Zr … … … Ag Cd In Sn Cs Ba La Hf … … … Tl Pb crystal structure will be highlighted.This will be done by attempting to impose a ìgenericœ potential model for each stoichiometry and by changing only the minimum number of potential parameters from one substance to another on the basis of the change in the chemical nature of the cation.In this way we identify the role of the many-body eÜects in causing departures from simple radius-ratio and charge-ordering rules, and assess whether the potentials contains sufficient ìphysicsœ to achieve the goal of transferability. Trends and patterns which link the diÜerent stoichiometries will be brought together in the discussion.Work on the MO stoichiometry, with which we begin, has been largely described previously13h15 and is included for orientation and completeness. The comparative study of the other stoichiometries represents new work. The ìgenericœ potentials are of somewhat greater complexity than has been anticipated above. In particular : the functional forms of the CIM potentials are multiexponentials13,14 but only oxide ion compression is incorporated (no deformation) and –xed cation radii are used; the pair dispersion terms are damped at short-range, in accordance with ab initio –ndings;5,14 cation polarizabilities are taken from electronic structure calculations5,7 and the oxide ion polarizability is varied from one substance to another in accord with the change in lattice parameter in the crystalline compound; dispersion coefficients are calculated from these polarizabilities via the Slater»Kirkwood formula. Full details of the new potentials described in this work are relegated to the Appendix (Section IX).The oxide»oxide interactions are handled via a simple EPP parametrized from ab initio calculations using the anion electron density frozen at the equilibrium lattice parameter values.The use of the full size-dependent oxide»oxide potential energy function is more complicated, with recent work showing how such interactions may be important in the eÜective modelling of the more subtle aspect of the dynamics.12 IV Systems of stoichiometry MO The alkaline-earth-metal oxide series presents an excellent opportunity to assess the importance of the many-body compressible-ion eÜect.The highly symmetric nature of the preferred structures, coupled with the relatively small cation polarizabilities typical of cations at the left-hand-side of the periodic table, means that polarization eÜects do not have to be invoked to explain these structures. The question is, therefore, whether the simple models in which the anion electron density is eÜectively frozen (the EPPs) can account for the structural trends in this series.Qualitatively, the structural eÜects can be interpreted with a simple hole-–lling argument. The smallest (Be2`) cation in the series can most eÜectively –t into half the tetrahedral holes available in the close-packed anion sublattice (maximizing the Be2`»O2~ whilst minimizing the O2~»O2~ interactions).This gives the four-coordinate zinc blende (B3) or wurtzite (B4) structures, which diÜer only in the packing of the anion sublattice, the latter being hexagonally close packed. For the largest (Ba2`) cation the tetrahedral hole is too small to satisfy these conditions which are best met by the cations occupying all the (larger) octahedral holes, resulting in a six-coordinate rocksalt (B1) structure.For still larger cations, a switch to an eight-M. W ilson and P. A. Madden 345 coordinate CsCl (B2) structure would be anticipated ; such a phase is known to occur for some of the heavier alkaline-earth-metal oxides at elevated pressure. IV. A MgO Although the qualitative trend is represented by the simple ionic picture, which implies that some EPP should adequately describe the ground-state crystal of each system, it does not follow that an ab initio parametrized EPP will work quantitatively.Illustrated in Fig. 2 are the energy»volume curves for diÜerent crystal structures with an EPP and a CIM potential –t to the ab initio data on MgO in the rocksalt phase. It is worth stressing that both these models give identical results (to within the quality of the potential model –t) for this phase, which are in excellent agreement with experimental properties of the perfect crystal.The diÜerences between the two models arise when we consider the relative energies of the other structures, that is, consider how the two models transfer under a change of coordination number.Two particular features are evident. First, the EPP actually predicts the four-coordinate B3 structure to be the ground state rather than the experimentally observed B1. The B4 structure was not considered as its energy Fig. 2 Energy»volume curves for MgO calculated (a) from the CIM potential and (b) with the rocksalt pair potential [from the inset to Fig. 1.(]) B2; (]) B1; B3]. (L)346 Relationship of crystal structures to interionic interactions Fig. 3 Experimental lattice energies for three stoichiometries : (a) MO, (b) and (c) M2O3 M2O will be very close to that of the B3. Secondly, the pressure for the B1]B2 transition (related to the common tangent to the B1 and B2 energy»volume curves) appears greatly reduced in the CIM. The CIM-predicted transition pressure is in good agreement with ab initio data for MgO.16 These –ndings illustrate the quantitative consequences of the general principle that the CIM potentials favour the higher coordination structures.M. W ilson and P.A. Madden 347 IV. B Other alkaline-earth-metal oxides Given the ab initio CIM potential for MgO we now turn to the question of how readily this potential can be transformed into a potential suitable to describe the other alkalineearth- metal oxides, i.e.we examine transferability between chemically similar substances. It is found that by simply changing the cation radius parameter in the cation»oxide and cation»cation repulsive pair potential [cf. eqn. (2.3)], in accord with tabulated crystal radii for the cations involved, good potentials for the other substances are obtained.14 Some further, relatively minor, –ne-tuning produces potentials which are in excellent accord with experiment.The –ne-tuning involves scaling the dispersion parameters (C6 etc.) to allow for the increase in cation polarizabilities going down the alkaline-earthmetal series, using the Slater»Kirkwood formula.Less easily anticipated a priori, is that a slightly diÜerent (15% larger) range parameter (i.e. oi) in the short-range overlap potential for the heavier ions Ca2`, Sr2` and Ba2`, to that used for Mg2` and Be2`, gives an improved agreement with experiment. The use of a larger-range parameter was directly supported by a –t to new ab initio data for CaO. It may re—ect the fact that the shape of the electron density in the heavier elements, which precede a transition series, is somewhat diÜerent to that in Mg2` and Be2`.With these few parameter changes, the ìgenericœ alkaline-earth-metal oxide potential, whose functional form was uncovered in the electronic structure calculations on MgO, gives excellent agreement with the experimental properties of all the alkaline-earth-metal crystals. In particular, the ground-state crystal lattice parameters and lattice energies are very good; the predicted pressures for the B1]B2 transition pressures for CaO and SrO are in good agreement with experiment. 14 In addition, the CIM for BeO does predict the B4 structure as the ground state and predicts a B4]B1 transition pressure in good agreement with electronic structure calculations.17 IV.C Other MO systems Fig. 3(a) shows the experimental lattice energies plotted against the cation crystal radius for nine systems of MO stoichiometry, including oxides of metals from other groups together with the Group 2 (alkaline-earth-metal) systems. As with their crystal morphologies, the Group 2 oxide energies behave as expected from considerations of the simple ionic model, becoming less negative as the cation size increases.Two subgroups stand out in this –gure, their lattice energies are lower than for the Group 2 system with a similar cation radius and, in some cases, they exhibit very diÜerent ìnon-ionicœ crystal structures. We now examine what additional eÜects, within the extended ionic, model might be responsible for this non-alkaline-earth-like behaviour. Attention focuses on the polarization eÜects.IV. C. 1 SnO and PbO. SnO and PbO both adopt a four-coordinate litharge structure (Fig. 4) in which the O2~ ions occupy half the available tetrahedral sites in the cubic close-packed cation sublattice. Simple application of the radius-ratio rules would indicate a preference for the larger octahedral holes, the analogous Group 2 systems would be SrO and BaO, which are both rocksalt.Furthermore, as is clear from Fig. 4, the result of this particular occupation pattern is to leave alternating layers of fully occupied and empty tetrahedral sites in the cation sublattice, leading to a structure which must have a large charge»charge interaction energy. Previous experience with layered crystal structures in halides18 suggests that MX2 polarization eÜects can play a major role in overcoming this unfavourable energy.In the present context, it must be the cation polarization which plays the crucial role, as the348 Relationship of crystal structures to interionic interactions Fig. 4 Litharge crystal structure, illustrating the unsymmetrical occupation of the unit cell by oxide ions.Cations are the small light spheres and oxide ions the larger dark ones. same (oxide) anion is present in SrO and BaO. Simple electronic structure arguments indicate that both Sn2` and Pb2` should be very dipole polarizable as a result of their s2 ground-state electronic con–guration which leaves a low-energy, dipole-allowed s]p transition.Ab initio values for Sn2` and Pb2` polarizabilities are around 15 and 18 au, respectively.5,15 In MO stoichiometry, the polarization eÜects can only come into play when the anions are forced into tetrahedral sites in such a way as to create an unsymmetrical cation site. Only symmetrical structures can arise from occupation of octahedral holes (since there are equal numbers of holes and ions).It has already been shown that the Sn2` polarizability is sufficient for the cation polarization eÜects to overcome the unfavourable short-range repulsions associated with the occupation of the tetrahedral site and the coulomb energy associated with layering. 15 In order to construct a model for PbO we scale the ab initio CaO CIM potential for the cation»anion interactions data using a value for the cation radius of Pb2` of 1.20 (similar to that of Ba2`) giving the parameters listed in Table 2.We supplement the ”19 alkaline-earth-metal potential with an account of cation polarization and use several values for the polarizability of the Pb2` ion in order to see how polarization aÜects the interplay between hole occupancy and layering. Fig. 5(a) shows the energy»volume curves for PbO for the four structures considered in the absence of polarization eÜects. As expected from the simple radius-ratio arguments the B1 structure is predicted to be the ground state. The litharge structure lattice energy is ca. 550 kJ mol~1 more positive than the B1 structure. The litharge energy is also ca. 500 kJ mol~1 less negative than the B3 structure in which the tetrahedral holes are half occupied but in a charge-ordered Table 2 CIM parameters for the models introduced in the present work system a`~ A`~ B`~ C`~ D c PbO 1.5 79.0 119 000 9.0]107 0.142 1.3 Y2O3 1.7 190.0 83 200 5.3]108 0.146 1.7 La2O3 1.7 397.0 364 000 4.8]109 0.146 1.7 TiO2 1.7 109.0 138 900 0 0.1505 1.56 See Section IX for details.M.W ilson and P.A. Madden 349 Fig. 5 Calculated lattice energies»volume curves for the PbO model with (a) no polarization eÜects and (b) au: (]) B2, (]) B1, B3, litharge aPb2`\12.5 (L) ()) arrangement. Fig. 5(b) shows the eÜect of increasing the Pb2` polarizability. For au the litharge structure becomes favoured over the B1. At au the aPb2`B8 aPb2`B12.5 lattice energy is around the experimental value.The equilibrium volume is 33.5 per ” molecule which is ca. 15% less than the experimental 39.7 per molecule.20 The total ”3 dipole polarization energy is ca. 950 kJ mol~1, ca. 35% of the overall lattice energy, compared with ca. 31% for SnO. Whilst this –nding strongly indicates that the formation of the litharge structure in PbO can indeed be ascribed to polarization eÜects, and that the PbO and SnO interactions are related to those for the Group 2 oxides, despite the dissimilarity of their crystal structures, it also throws up a conundrum.As was discussed earlier, one might expect to have to enhance the cation polarization to model the short-range polarization eÜect. This is not found to be the case here with the ab initio polarizabilities actually over-stabilizing the litharge structure. We believe that the source of the problem, at this quantitative level, is the use of the same form of CIM potential for the short-range cation»anion interaction as was derived for the Group 2 systems, with a simple shift of ion radius.It seems reasonable that to model this interaction accurately one would need to recognize the diÜerence between the interaction between an oxide ion and a cation with an s2 rather than inert-gas outer electron con–guration.350 Relationship of crystal structures to interionic interactions IV.C. 2 ZnO and CdO. The Group 12 oxides have experimental lattice energies that are signi–cantly more negative than the alkaline-earth-metal counterparts. Both ZnO and CdO adopt ground-state structures which are charge-ordered, but for ZnO the coordination number is not that predicted by the radius-ratio rules.ZnO adopts a B4 structure (like BeO) despite having a slightly larger cation radius than Mg2`, the latter adopting the B1 structure. ZnO does transform into the six-coordinate B1 structure under the application of modest pressure (ca. 9»9.5 GPa21), indicating the similar energies of these two structures.In the B4 structure the cation sits in a tetrahedral site. In the B4 structure the lowest order-induced multipole moment allowed by the symmetry is an octupole. The possible signi–cance of such eÜects was –rst mooted by Mahan and Subbaswamy, 4 the idea being that the post-transitional nature of the cation leads to large octupole polarizabilities via the octupole-allowed d]p transitions.Indeed, free-ion calculations show the octupole polarizability for the Zn2` cation to be ca. 50 times that of Mg2`.22,23 Although the oxide ion octupole polarizability is nominally much larger than either of these cation values, the opposition of the short-range induction mechanism greatly reduces its eÜect. An estimate based on the energies of BeO, which has a very unpolarizable cation, indicates that the short-range mechanism may reduce the anion octupole polarization eÜect by as much as 80%.On the other hand, the cooperative nature of the short-range and asymptotic induction eÜects for the cations leads to an approximate doubling of the moment over that predicted from purely coulombic considerations. Adding the eÜects of octupole polarization, with a suitably enhanced value for the cation polarizability to account for this eÜect, to a CIM potential derived by scaling the MgO potential for the change in cation radius, allows the B4 structure to become stabilized with respect to B1.14 Furthermore, this potential reproduces the lattice parameter and lattice energy and gives the correct value for the transition pressure to the B1 structure.CdO adopts a B1 structure, as does its analogue CaO (which has a similar cation radius), despite having the same post-transitional nature as Zn2`. The larger Cd2` cation has a greater preference for the six-coordinate structure than does Zn2`, and it is found that an implausible enhancement of over 500% of the cation octupole polarizability would be required to stabilize the B4 structure in this system.In both cases, a large proportion of the ìextraœ lattice energy (compared with the alkaline-earth-metal values) arises from the increase in dispersion energy resulting from the more polarizable cations. For example, in MgO the total dispersion energy is dominated by the oxide»oxide interactions as the polarizability of the cation is so small.In ZnO the larger cation dipole polarizability leads to a contribution to the total dispersive energy of the same order as the oxide»oxide term. V Systems of stoichiometry M2O3 In terms of the cation : anion charge ratio, the stoichiometry is the most similar M2O3 to MO and, in so far as this dictates that immediate environment of a given oxide ion, should normally involve a complete shell of cations, the appropriate reference state for the oxide ion in MO systems might not be too far removed from that appropriate to However, the stoichiometry is particularly ìawkwardœ in terms of simple M2O3. M2O3 hole-–lling arguments.Thus, if the anion sublattice adopts a close-packed arrangement then the cations must occupy two thirds of the octahedral holes or one third of the tetrahedral or some combination of the two.Such occupation ratios necessarily lead to a greater range of much less symmetric crystal structures than in the MO stoichiometry and to considerable lattice distortions away from idealized hole-–lling structures. Consequently, we can anticipate that, whilst the ìbasic physicsœ of the important interactions uncovered in MO might apply in much greater attention will need to be paid to M2O3 those interactions which come into play when the coordination site is asymmetric, in particular, to polarization eÜects.M.W ilson and P. A. Madden 351 V. A Al2O3 The most studied system, adopts a variety of complex structures. The ground Al2O3 , state is a six-coordinate corundum structure in which two thirds of the octahedral holes in the hexagonal close-packed sublattice are occupied by Al3` cations.24 An alternative structure, also based on six-coordinate cations, is the bixbyite (C-type lanthanum oxide) structure which can be thought of as derived from the —uorite structure (MO2 stoichiometry) by removal of one quarter of the anions.25 Both structures have anions in relatively asymmetric sites. In corundum the Al3` ions do not sit in the centre of an octahedral hole but are pushed to one side by the combination of a near-neighbour cation across an adjacent oxide layer on one side and an unoccupied cation site across the oxide layer on the other.This leads to the two characteristic anion»cation lengths seen in the crystal.Similarly, the removal of the anions from the —uorite lattice to form the bixbyite structure leaves two types of cation coordination, neither of them octahedral. 25 Previous shell model potentials26 for have always given the bixbyite as Al2O3 the lowest-energy structure, except where an arti–cially enhanced (by a factor of 104) dispersion parameter has been used in order to stabilize the relative close C6AlAl approach of nearest-neighbour Al3` ions in corundum.There are no electronic structure calculations, analogous to those carried out for MgO and CaO, on which to base a –rst-principles CIM potential for (though Al2O3 there are numerous ab initio calculations of total crystal energies). Indeed, for this stoichiometry, it is difficult to see how to form a high-symmetry reference crystalline environment on which to base such calculations.However, Al3` is isoelectronic with Mg2` and this provides us with a useful point of comparison: hence we have a constructed a potential by appealing strongly to the concept of transferability and using the CIM potential constructed for MgO as the starting point for an potential.The Al2O3 observant reader will note that the ionic radius of Al3` is smaller than that of Mg2` and we will discuss why there has been no adjustment for this fact in Section VIII, Conclusions. The ì secret œ of the stability of the corundum structure is to include anion quadrupole polarization eÜects.27 Although both the corundum and bixbyite structures have nominally tetrahedrally coordinated anions, the local symmetry of the former is far more distorted towards the square-planar con–guration (bond angles are in the range 84»132° compared with 102»123° in the bixbyite).This distortion leads to a non-zero electric –eld gradient at the anion site, precluded by symmetry for the perfect tetrahedral symmetry, and hence the induction of a quadrupole. An oxide quadrupole polarizability of 6 au (see Appendix), is found to give ca. 2.5% of the total lattice energy and to be sufficient to stabilize the corundum structure relative to bixbyite. The predicted properties of the crystal are in excellent agreement with experiment. The value of 6 au for the oxide ion quadrupole polarizability requires further comment. This value is signi–cantly smaller than the value obtained from electronic structure calculations on MgO (26 au).On the other hand, some reduction would be anticipated because of the somewhat tighter coordination of the O2~ ion in and also because the asymptotic quadrupole induction Al2O3 eÜects are opposed by the short-range polarization. Support for the value of 6 au comes from experimental values for the nuclear quadrupole coupling of Al3`.28,29 The electric –eld gradient at the Al3` site is very sensitive to the induced quadrupole on the O2~ ion and its value is consistent with the 6 au value for the polarizability.In this particular example, the crystal symmetry results in an almost complete vanishing of the electric –eld at the anion sites, resulting in zero dipoles, and projects the quadrupoles into a greater level of signi–cance.In the liquid, where the anions are found in far more disordered instantaneous con–gurations, the dipole induction eÜects would be expected to swamp the quadrupoles.352 Relationship of crystal structures to interionic interactions Fig. 6 Energy»volume curves for the CIM: bixbyite, corundum, Y2O3 (L) (K) (») h[Al2O3 , (]) A-type lanthanum oxide, (]) B-type lanthanum oxide V.B Y2O3 Y3` (and Sc3`) have a similar electronic structure to Al3` but with slightly larger ionic radii, and it seems reasonable to attempt to generate suitable potentials for them by scaling the with the cation radius, as previously done for the Group 2 oxides. Al2O3 forms a bixbyite structure. Although the coordination number of this structure is Y2O3 nominally the same as for the corundum, we can understand the preference of the intermediate-sized cations for this structure in terms of the relative hole sizes in the anion sublattice. In the corundum structure, the holes occupied by the cations are octahedral in a close-packed anion sublattice (although the cations do not sit in the centre of the hole).In the bixbyite structure, the hole is eÜectively that for an eight-coordinate cation (as in the parent —uorite structure) in a non-close-packed sublattice and, as a result, can accommodate larger cations.A CIM potential was generated by scaling the cation radius parameter in the cation» anion CIM potential for discussed above (Y3` cation radius 0.93 Al3` 0.50 Al2O3 ”, ”). Modi–cations were also made to the dispersion parameters in order to allow for the Fig. 7 Energy»volume curves for the CIM: bixbyite, (]) A-type lanthanum oxide, La2O3 (L) (]) B-type lanthanum oxideM. W ilson and P. A. Madden 353 more polarizable nature of Y3` relative to Al3`. The description of polarization eÜects was exactly as for with the same polarizabilities being used. Polarization of the Al2O3 , cation was found to have a very small eÜect on calculated properties in other Y3` salts.30 Fig. 6 shows the energy»volume curves for this CIM model. As anticipated Y2O3 above, the bixbyite structure is now seen to be favoured. V. C La2O3 The larger cation (lanthanide) systems (e.g. La3`, 1.15 in this stoichiometry favour ”) the A- and B-type lanthanum oxide structures, as would be expected on radius-ratio grounds, since both of these structures have an average cation coordination number greater than six.We can attempt to construct potentials for these systems by further scaling the cation radii in the cation»anion short-range interactions in the successful alumina potential, in the manner described above for and simultaneously chang- Y2O3 , ing the dispersion parameters in accord with the cation polarizabilities.For for La2O3 , example, the potential parameters are listed in Table 2. Fig. 7 shows the energy»volume curves for the model for the three lanthanum La2O3 oxide structures, the other structures of interest, such as corundum, have a much less negative lattice energy. The B-type lanthanum oxide structure is now predicted to take over from the C-type (bixbyite) as the most stable crystal.However, it is the A-type which is observed experimentally. Further work is currently in progress to assess whether this is a shortcoming of the CIM transferability or an additional (perhaps cation) polarization eÜect. Following the CIM work on the MO stoichiometry we might anticipate that a change in the range parameter is required for the larger cations.V. D Post-transition systems M2O3 When we contrast the properties of the post-transition, trivalent metal oxides, with those oxides in which the metal ion has an inert-gas electronic con–guration considered above, there are strong echoes of the comparison between the Group 2 and Group 12 oxides in the MO stoichiometry. As is highlighted by Fig. 3(b) the experimental lattice energy for is actually Ga2O3 more negative than that of alumina, despite the larger cation.Additionally, is b-Ga2O3 anomalous, in that it adopts a crystal structure in which the anion sublattice is cubic close-packed but with the cations occupying equal numbers of octahedral and tetrahedral holes. This structure is only observed for alumina as a transient precursor (the h-phase) to the corundum in the dehydration of the mineral hydrates. Thus, the larger cation system appears to favour a lower average coordination number structure in contravention of radius-ratio rules.We can, however, draw a clear analogy with the MO stoichiometry. In Section IV we saw how the induced cation octupoles could account for the preferred four-coordination adopted by ZnO over the six-coordinate system favoured by MgO.Ga3` sits one place to the right of Zn2` in the Periodic Table and has an octupole polarizability of ca. 6 au.22,23 This bare octupole polarizability is signi–- cantly smaller than that of Zn2` (ca. 30 au22). However, the reduction in octupole polarizability may be oÜset by the reduction in cation radius (and hence in anion»cation separation) which leads to a larger –eld gradient at the cation site and hence a larger stabilizing octupole.The octupole polarization energy is particularly sensitive to the nearest-neighbour anion»cation separation as it scales as R~8.23 As a result we can speculate that the same arguments used to explain the ZnO structure can be invoked for estimates suggest that this requires an octupolar contribution to the lattice Ga2O3 ; energy of ca. 2%, which is a smaller proportion than in ZnO. As with its Group 12 counterpart Cd2`, the oxide of In3` (and Tl3`) is ìnormalœ, forming a six-coordinate bixbyite structure, like Thus, although both In3` and Y2O3 . Tl3` would be expected to have large octupole polarizabilities (larger than Zn2` for354 Relationship of crystal structures to interionic interactions In3` 22) the potential octupole stabilization energy is not large enough to force these ions, which are larger than Ga3`, into the tetrahedral sites.VI Systems of stoichiometry MO2 The stoichiometry is likely to present the most difficult set of systems to which to MO2 apply an ionic interaction model.To some, an M4` ion is intrinsically implausible : from our perspective, the most worrying aspect is that, simply because of the excess of anions over cations, the oxide ion is likely to be found in sites with a low number of cation nearest neighbours. Consequently, the environmental potential for the oxide ion in an system is likely to be very diÜerent from that in the MO systems on which our MO2 models for the oxide ion interactions discussed so far have been founded.There are two further anxieties. Because of the large cation charge and the small size of many of the cations to be considered, the polarization eÜects are likely to be very large. Our polarization model, built upon the idea of correcting a high-symmetry crystal description to allow for some distortion of a spherical electron cloud, may be too far from reality.Furthermore, as we have seen, polarization pushes polarizable species into asymmetric sites and, coupled with the low cation coordination number, this means that we are likely to encounter situations where oxide ions appear as near neighbours. Although we have learnt how to deal with the consequences of overlap between cation and anion electron densities, we have not yet discussed the consequences of anion»anion overlap.The simple ionic model applied to systems of stoichiometry predicts a progres- MX2 sion of charge-ordered crystal structures as the cation : anion radius ratio is decreased from —uorite (with a four-coordinate anion, eight-coordinate cation), through rutile (3 : 6), to the ideal cristobalite (2 : 4).At least for the —uorite structure, in which the anions occupy all of the available tetrahedral sites in the fcc cation sublattice, the anxieties discussed above should not be too severe, since the oxide ion is surrounded by cations and the site symmetry is sufficiently high (tetrahedral) as to preclude the induction of dipole and quadrupole moments. In reality, the —uorite structure is observed for very large cations, and so this provides us with a useful point of contact with the well understood MO systems.For the lower coordination rutile and ideal cristobalite structures, the site symmetry is so low that induced quadrupoles may contribute to the crystal energetics even for the ideal charge-ordered crystals. For small cations, such as Si4`, the observed crystal structures are of much lower symmetry than ideal cristobalite, re—ecting still more dramatic polarization eÜects.VI. A Large (actinide) cations Harding and Pyper31 have carried out electronic structure calculations of the rearrangement and overlap energies and dispersion energy parameters for the case of in its ThO2 experimentally observed —uorite crystal structure.32 They have already shown that the properties (energy, lattice constant, etc.) of the —uorite crystal of obtained from ThO2 , their calculations, are in excellent agreement with experiment.A CIM potential for the Th4`»O2~ short-range interaction, may be constructed, in the same way as for MO systems, from the electronic structure data. It has a similar shape to that obtained previously for MgO and CaO, except that the eÜective cation radius is correspondingly larger.The change of stoichiometry does not appear to have had any dramatic consequence for the functional form of the interactions. If the CIM potential is supplemented by an account of anion quadrupole polarization, which is necessary to assess the stability of the rutile structure, and values of the oxide ion quadrupole polarizability compatible with those gleaned from calculations on and (see below) are used, it is Al2O3 ZrO2 found that the —uorite structure is stable with respect to rutile.Such a large cation as Th4` is unable to –t into the six-fold cation site in the rutile crystal and the energy penalty for forcing it to do so is not overcome by the polarization eÜects.M. W ilson and P.A. Madden 355 Fig. 8 Energy»volume curves for the CIM: (]) —uorite, rutile (no quadrupole TiO2 (|) polarization), rutile (with quadrupole polarization) (L) VI. B Intermediate-sized M4ë cations For cations of size between Ge4` and Ce4` (Table 1) structures based on linked MO6 octahedra are observed. The general preference for six-coordination can be understood in hole-–lling terms.Cations such as Ti4` are too large to –t eÜectively into a tetrahedral hole and too small to –t into the eight-coordinate site in the —uorite structure. The charge-ordered arrangement of octahedra gives rutile : in this case the almost planar three-coordinate nature of the local anion environment precludes large anion dipoles by symmetry.There is, however, a large –eld gradient at each anion site leading to the possibility of large anion quadrupole polarization eÜects. We have already seen that such eÜects can be important in the prediction of ground state crystal structures in the stoichiometry (Section V) when, as here, the anion dipoles are precluded by sym- M2O3 metry. Less symmetrical linkages of the polyhedra, which further lower the oxide site symmetry, will be favoured by dipole polarization eÜects.The importance of the anion quadrupoles in this stoichiometry has already been high-lighted for Zr4` has a cation radius intermediate between the rutile- and ZrO2 .33 —uorite-forming cations. As a result the (monoclinic symmetry) ground state is made up of seven-coordinate cations with alternating layers of rutile-like (three-coordinate) and —uorite-like (four-coordinate) anions.As for the stoichiometry, straightforward M2O3 shell models have been unable to predict this structure as the ground state, instead preferring the structure observed as a high-pressure phase in many rutile a-PbO2 systems and diÜering from the rutile only in terms of the linking of the octahedra.MO6 Calculations have been done with a CIM potential derived directly from ab initio calculations. 34 Without polarization eÜects, this stabilizes both the higher-coordinate —uorite and monoclinic structures over the but erroneously predicts the eight- a-PbO2 coordinate —uorite as the ground state. However, the inclusion of the anion quadrupoles stabilizes the monoclinic over the —uorite.An ìeÜectiveœ quadrupole polarizability of 10 au is found to give excellent agreement with additional ab initio calculations focusing on the cubic to tetragonal distortion.33 The stabilization mechanism can be traced back directly to the large quadrupoles induced on the three-coordinate anions. The value of 10 au for the eÜective quadrupole polarizability is larger than for consistent with Al2O3 , the larger size of the oxide site in and the known dependence of polarizabilities on ZrO2 lattice parameter.356 Relationship of crystal structures to interionic interactions Fig. 9 Energy»volume curves for various isomorphs of for (a) the simple pair potential37 and SiO2 (b) for the additional eÜect of anion polarization The large anion quadrupoles responsible for the stabilization of the experimentally observed structure may also play an important role in the eÜective modelling of ZrO2 the rutile-forming systems in which all the anions sit in such three-coordinate sites.The problems associated with modelling itself are well known.35 The central problem TiO2 revolves around reproducing the experimental structure (for example the c/a unit cell length ratio) whilst simultaneously reproducing the dielectric properties.It is tempting to speculate that such problems may be overcome by a model with a proper description of the anion quadrupoles and work is underway to this end. Fig. 8 shows the energy» volume curves for a preliminary model derived from the CIM for by a TiO2 ZrO2 further reduction of the cation radius.In the absence of anion quadrupoles the —uorite structure is predicted to be the ground-state structure (as was the case for and ZrO2 The inclusion of quadrupoles, with the same quadrupole polarizability as used ThO2). for successfully stabilizes the rutile structure. It should be stressed that the –gure ZrO2 , represents the early stages of the model development.For example, the c/a ratio is –xed at the experimental value for all volumes. VI. C Small M4ë cations, silica and germania The smallest 4] cations highlight perfectly the problems associated with constructing truly transferable potential models for such low anion coordination environments. AtM. W ilson and P. A. Madden 357 Fig. 10 Illustration of the polarization eÜects in the isomorphs. A projection of a single SiO2 plane of tetrahedra in each structure is shown: Si ions (9) O ions.The negative ends of SiO4 (Ö) induced dipoles on the oxide ions are indicated byL. atmospheric pressure all the observed silica and germania polymorphs are based on corner-linked tetrahedra. As a result, the anions are exclusively two-coordinate.In MO4 ideal cristobalite the MwOwM ìbondœ is linear in order to minimize the repulsion of the highly charged cations. In all the observed polymorphs, the bond is bent. The bending of the MwOwM triplet controls the intermediate-range order (i.e. the nextnearest neighbour level) with subtle changes in this angle responsible for the exotic nature of the crystalline isomorphs. As we have already noted, there are considerable problems associated with systematically transferring the CIM potentials which have been uncovered on other systems to these structures with very low anion coordination numbers.Nevertheless, there is a considerable history of simulations of with empirically parametrized ionic pair SiO2 potentials with formal charges which do succeed in recapturing the basic structural features, such as tetrahedral coordination of Si4` and the SiwO bond lengths.Some progress can be made in unravelling the interactions in the low-pressure polymorphs, and amorphous by using these EPPs as a starting point. SiO2 36 Within the extended ionic model the bending of the MwOwM bond is directly facilitated by the anion (dipolar) polarization eÜects.Fig. 9 highlights this point for a selection of silica polymorphs generated with a simple EPP37 supplemented with anion dipole polarization. In the absence of anion polarization the idealised b-cristobalite [Fig.358 Relationship of crystal structures to interionic interactions 9(a)], in which the SiwOwSi triplets are linear, is the preferred structure. The inclusion of the dipolar polarization eÜects [Fig. 9(b)] stabilizes the experimentally observed structures which have the non-linear triplets. Fig. 10(b) and (c) show two such structures : b-cristobalite and a-quartz with the role of the induced dipoles highlighted. These structures diÜer only in the intermediate range order of the tetrahedra (the ring SiO2 structure). Although such calculations give some insight into the factors controlling the stability of four-coordinate structures, at a quantitative level, and especially when a change of coordination number occurs, both spherical compression and aspherical anion deformation eÜects should be signi–cant, as also argued by Lacks and Gordon.38 This is illustrated by considering the pressure transition to the six-coordinate cation stishovite structure (having the rutile structure).The simple EPP plus polarization predicts a transition at ca. 50 GPa compared with the experimental value of 5.5 GPa. This overestimation of the transition pressure is consistent with that obtained with an EPP for the B1]B2 transition in the MO stoichiometry (Section IV) and so the naié ve assumption might be that the use of the CIM (with the anion dipolar polarization eÜects) will reproduce the experimental transition pressure.Preliminary calculations (using the MgO-derived CIM) show that the CIM does indeed stabilize the six-coordinate stishovite as expected. However, even with the inclusion of the anion polarization eÜects the stishovite is now predicted to be the ground state. It seems clear that simply modifying existing potentials is not an appropriate way forward; a de novo approach to based SiO2 on new calculations is required.VII Systems of stoichiometry M2O In the MO stoichiometry, the high-symmetry reference state, which was typically the rocksalt crystal, has the anion symmetrically surrounded by a large number of cations. This environment stabilized the oxide ion sufficiently well so that ionic properties calculated in this reference state could be used to model the interactions in the other crystal structures of interest.The stoichiometry should involve the smallest step from the M2O MO systems, in that, by virtue of the excess of cations over anions, each oxide ion will tend to –nd itself surrounded by a coordination shell containing a large number of cations. The oxide ion should thus –nd itself stabilized in a similar way as in MO and allow for the establishment of reasonably transferable potentials.VII. A Alkali-metal oxides From to crystallize in the charge-ordered anti-—uorite struc- Li2O Rb2 O (r~~\)2r``) ture with the anions eight-coordinated.25 on the other hand, adopts a layered Cs2O, or an structure (these structures diÜer only in the stacking of anti-CdCl2 anti-CdI2 25 the close-packed cation layers and so are energetically very close).In, for example, the structure, cations form a cubic close-packed sublattice with the anions occu- anti-CdCl2 pying half of the octahedral holes, in such a way as to leave alternating layers of fully occupied and unoccupied octahedral holes.The resultant (ideal) structure has equal r~~ to and has adjacent cation layers. On purely coulombic grounds, this kind of r`` arrangement cannot be energetically favourable. There is always an alternative chargeordered structure in which the more highly charged anions can be kept further apart. We anticipate a cation dipole polarization stabilization for these layered structures analogous to that already illustrated for SnO and PbO (Section IV).To date, there are no electronic structure calculations, comparable to those done on MgO and CaO, on which to base a –rst-principles parametrization of the interaction potentials in systems. We have therefore adopted an EPP for the short-range M2O repulsive cation»anion potential from work of Bush et al.,39 in which a series of pairpotentials were empirically derived for the whole range of alkali-metal oxides.In orderM. W ilson and P. A. Madden 359 Fig. 11 Crystal energy vs. volume for (a) and (b) (»») with the full PIM, Li2O Cs2 O. Anti-CdCl2 (» ») with the RIM, (…»…) anti-—uorite with the RIM. anti-CdCl2 to retain consistency, we also take the MwM and OwO (including dispersion) terms from the same work.For MgO, their potential was very similar to the EPP potential for the rocksalt structure which we obtained as described above. The anion coordination in the and anti-—uorite structures is diÜerent (six and eight, respectively) and so, anti-CdCl2 as in the MO stoichiometry, we would expect to have to incorporate an explicit account of the anion compressibility eÜect to account quantitatively for the location of the phase transition in these systems.The use of a consistent set of EPPs, which are linked through a change in the cation radius parameter, at least allows us to assess whether the role of cation dipoles is sufficient to explain the transition to the non-charge-ordered, low-coordination structure for the larger cations. Calculations have been performed on the energies of the competing crystal structures with these EPPs and with the EPPs supplemented with an account of cation dipole polarization (in these crystal structures the high symmetry of the oxide site precludes anion polarization).Free-ion ab initio polarizabilities5,7 were used for the cation polarizabilities (for alkali-metal cations, the crystalline environment has little eÜect on the polarizability7).An allowance is made for the enhancement of the cation induced dipole360 Relationship of crystal structures to interionic interactions Fig. 12 Energy»volume curves for the EPP: (]) anti-—uorite, cuprite (no cation quad- Ag2O (|) rupole polarizability), cuprite (with cation quadrupole polarizability) (L) by the short-range polarization eÜect, as was –rst done in calculations on This is CaF2 .9 described in the Appendix.Fig. 11(a) and (b) show the energy vs. volume plots for and respectively, Li2O Cs2 O, using the EPP with and without the inclusion of polarization eÜects. The anti-—uorite curves are unaÜected by the inclusion of dipolar induction eÜects due to the high symmetry of both ion sites.For the system the anti-—uorite structure is favoured over Li2O the despite the inclusion of the cation polarization eÜects. The anti-CdCl2 anti-CdCl2 energy»volume curves with the polarizable and unpolarizable models are very similar as the lithium cation is relatively unpolarizable (0.1 au). On the other hand, the Cs` ion is very polarizable (15 au) ; in fact, its polarizability is greater than that of the O2~ ion in the crystal.The polarization energy now makes a very large contribution to the stability of the structure, which is now the predicted ground state. Note that, within anti-CdCl2 the model described, the enhancement of the cation polarization by the short-range eÜect does play a role ; it increases the polarization energy by ca. 20% and is, therefore, responsible for the order of the crystal energies. We note that the interplay of the energetics of the layered and charge-ordered systems and the role of cation polarization here is somewhat diÜerent to the SnO and PbO cases. In the latter the polarization has to overcome both the desire of the system to remain charge ordered and the desire for ions of this size to adopt a six-coordinate structure as dictated by the radius-ratio rules.In the systems the increase in ion M2O size and polarizability descending the group act in concert, as the change from the eight-coordinate anion site in anti-—uorite to the six-coordinate is what anti-CdCl2 would be expected from the increase in cation : anion size ratio. VII. B Other systems M2O Both and adopt a cuprite structure,25 which is very diÜerent to the crystal Ag2O Cu2 O structures observed for the alkali-metal oxides.This structure can be thought of as constructed from two interpenetrating cristobalite sublattices. The stability of this structure appears to be related to the stability of linear OwMwO triplets. The symmetry of these units precludes dipole polarization eÜects on the central cation but allows for potentially large induced quadrupoles.Such low-order moments are precluded on the anion sites which sit in tetrahedral holes. Induced quadrupoles have been invoked to explain aM. W ilson and P. A. Madden 361 number of anomalous features of the relationship between Group 1 and 11 halides, e.g. the diÜerent character of the phonon dispersion curves in AgCl and NaCl.8 Fig. 12 shows the energy»volume curves for an model, constructed by scaling Ag2O the alkali-metal oxide potentials, described above, for the change in cation radius, for the anti-—uorite and cuprite structures. The model includes cation quadrupole polarization, with the quadrupole polarizability taken from the AgCl potential.8 As anticipated, the cuprite structure is less stable than the anti-—uorite in the absence of cation quadrupoles but becomes stabilized when they are added in. We note that the lattice energy is ca. 150 kJ mol~1 too positive, indicating that a more careful parametrization is required to model these compounds accurately. We have also checked that the dipole polarization of the Ag` ion does not make the structure more stable than those anti-CdCl2 discussed.VIII Conclusions This survey has built upon work on the limited range of oxide materials (MgO, CaO, for which sufficient information was available from electronic structure ZrO2 , ThO2) calculations for the development of potentials from –rst principles. We have transmuted these potentials into potentials for other substances by systematically changing the values of certain parameters in accord with the changing identity of the cation involved.The survey indicates the possibility of constructing a transferable, extended ionic interaction model for a wide range of oxide materials. Furthermore, the survey has identi–ed the nature of the interactions responsible for the adoption of non-ionic, ìcovalentœ structures by the oxides of certain metals.We should stress that, where a new eÜect has been invoked to explain such a change in behaviour (such as the introduction of cation dipole polarization to explain the diÜerence between Group 2 and SnO and PbO), we have gone back to check that this eÜect did not change the pattern of stable structures in the original main group. We should also stress that if, say, a CIM model without polarization is sufficient to account for the crystal structures of the Group 2 oxides, it does not follow that such a potential would be sufficient to account for all aspects of the behaviour of these materials.Polarization eÜects, for example, which cannot play a role in the perfect crystal because of symmetry, would come into play in the melt and even in getting the phonon frequencies and other properties of the distorted crystal correctly.In the absence of a complete characterization by electronic structure methods of all the substances considered, success can, so far, only be regarded as qualitative. For the most part, changes in parameter values, from a substance for which such a characterization has been made to one where it has not, have been justi–ed by appeal to some independent source of information and not simply assigned new values to obtain a best –t to the observed crystal structure.For the oxides of one group of the periodic table, for example, simply changing the ionic radii and polarizabilities according to established values has been sufficient to locate the transition from one crystal structure to another and to account, quantitatively, for lattice energies and densities. In jumping from one group to another within the same stoichiometry (e.g.MgO]ZnO) or between stoichiometries (e.g. however, less well controlled changes have been introduced MgO]Al2O3), with an eye on the predicted structures. In closing, therefore, we summarize below those points where we feel that the necessary parameter changes have not been fully justi–ed.These points are the ones on which further ab initio work would be most pro–tably directed. Short-range cation polarization. Elementary consideration of the shape of the environmental potential leads one to expect that the short-range and asymptotic multipole-induction mechanisms on cations act in concert.However, the electronic structure calculations necessary to con–rm this have not been performed. We have seen362 Relationship of crystal structures to interionic interactions empirical evidence of the eÜect in our rationalizations of the structure of (dipoles) Cs2O and ZnO (octupoles) (and, elsewhere, in considerations of and AgCl8), where CaF2 9 enhanced values of the relevant polarizability are necessary for agreement with experiment.However, we have found that the cation polarization obtained with ab initio polarizability for PbO, when used with the MgO/CaO derived CIM, gives too large a polarization energy in litharge. Short-range contributions to higher-order multipoles. We have shown that quadrupole and higher-order multipole induction on the anion can play an important role in stabilizing certain crystal structures, in the technologically important cases of and Al2O3 inter alia.In these cases, we have dealt with the short-range induction mecha- ZrO2 , nism, which opposes the asymptotic contribution to polarization of the anion, by simply reducing the value of the relevant polarizability below the ab initio value, and using the asymptotic model for induction.Clearly, the short-range mechanism should change the dependence of the induced moment on the interionic separation, in a similar manner to the dipole-induction case, and not simply reduce the magnitudes of the induced moments. Furthermore, the factors by which the polarizability has been reduced have been quite large, and we would wish for some independent way of estimating the eÜect.Scaling the ion radius between stoichiometries. Within a given stoichiometry, we have found that the cation»anion CIM potentials can be satisfactorily ìtransmutedœ from one material to another by simply scaling the cation radius parameter in eqn. (2.3) in p 6 i accord with the crystal radii taken from standard tabulations. However, to change the potential to one appropriate to a diÜerent stoichiometry, it would appear that a more subtle change is required.For example, we found that the MgO CIM potential worked very satisfactorily for without further modi–cation, despite the fact that the stan- Al2O3 , dard crystal radius of the Al3` ion is signi–cantly smaller than that of Mg2`. Similar eÜects are noticed when comparing the directly –tted potentials for MO systems with those for This may indicate that the linking of the ion size variable (d) in the CIM MO2 .simply to the short-range repulsion is not complete and that some coupling to the strength of the Madelung potential is appropriate. Further work is under way to clarify the nature of these remaining shortcomings. We are grateful to several colleagues for their work and discussions on the topics we have covered; in particular we thank Mike Finnis, John Harding, Nick Pyper, Adrian Rowley, Uwe Schoé nberger and Martin Exner.We also thank Malcolm Walters for reading the manuscript. M.W. is grateful to the Royal Society for –nancial support. IX Appendix IX. A Short-range cationñanion interactions The basis of the description of the cation»anion interactions is the CIM, which was developed in work on MgO and CaO13 from an ab initio standpoint.5,40,41 The anion» cation total repulsive energy is decomposed into a ìrearrangement energyœ, the selfenergy of the ionic wavefunction appropriate to a particular crystalline environment, and the ìoverlap energyœ, the energy associated with the overlap of this wavefunction with the wavefunctions of the surrounding ions.5 The rearrangement energy of an ion is represented in the simulation potential by a function of an internal variable (di) which may be thought of as the departure of the radius of that ion from some mean, uncompressed value The ab initio calculated values for the rearrangement energy in MgO p 6 ~.and CaO were well represented by the following form: ure(di)\D[exp([cdi)]exp(cdi)] (9.1) As a result, 2D plays the role of the energy of the oxide ion in some uncompressed,M.W ilson and P. A. Madden 363 d\0, reference state and is linked to the electron affinity of the anion.31 c re—ects the stiÜness of the ion to being compressed by its environment. The ab initio calculations also give a value for the overlap energy between an oxide ion at radius and a p 6 ~]di cation wavefunction.This is represented in the simulations with a pair potential for which the following functional forms were found to give a good representation of this data: uoverlap `~ (rij, di)\A`~ exp[[a`~(rij[di[p 6 `)] ]B`~ exp[[2a`~(rij[di[p 6 `)] ]C`~ exp[[3a`~(rij[di[p 6 `)] (9.2) where, the cation radius has been shown explicitly.For a given ionic con–guration p 6 ` the value of d for each ion is to be found by minimizing the total energy with respect to all di simultaneously. The short-range cation»anion energy is then given by the sum of rearrangement energies for each anion plus the sum of over all pairs. uoverlap `~ , For a given material, seven parameters (A`~, B`~, C`~, D, a`~, and c) are p 6 ` involved.In forming our ìgenericœ oxide model we wish to minimize the number of free parameters and link the values for diÜerent materials together in a chemically meaningful way. In particular, we will determine the cation radii from pre-ordained tabulations, in the expectation that this parameter will express most of the dependence of the potential of the identity of the cation.Several tabulations of ionic radii exist : after some experimentation, the comprehensive compilation of radii due to Shannon19 were chosen with radii taken for cations at the same coordination number. The parameters A`~, B`~ and C`~ give the relative weights of the three exponentials used to –t the dependence of the ab initio overlap energies for MgO and CaO on the cation»anion separation, and a`~ gives the exponential decay rate.These parameters must be linked to the shape and rates of decay of the cation and anion electron densities. For a set of cations with the same valence electron con–guration we would expect little variation of these parameters from one material to another. IX. B Polarization The polarizable-ion model (PIM) representation of polarization eÜects works by introducing the components of the dipoles, … … … on each ion as additional degrees of kx i , ky i freedom, alongside the ionic positions, in the molecular dynamics procedure.At each time step, these dipoles take the values which minimize the dipole interaction energy, which for pure coulombic dipoles (no short-range eÜect) is U\ ; i/1 Nt ; j/i`1 Nt MT a ij(Qjka i [Qika j )[T ab ij ka i kb j N] ; j/1 Ns k1 j ; i/1 Nt ki2 (9.3) for a system of ions and species (here two).Here, and are components of Nt Ns T a ij T ab ij the charge»dipole and dipole»dipole interaction tensors.6 The –nal term is a Drude-like self-energy, giving the energy required to polarize an ion : the parameter is related to k1 j the ionic polarizability of species j It has been shown that it is possible to [k1 j \(1/2aj)].generalize this expression to allow for the short-range induced dipoles on anions, by modifying the distance dependence of the part of the above expression in which an anion dipole moment interacts with a cation charge to the form: where f 4(rij) is a Tang»Toennies function ;(i | anions);(j | cations) f 4(rij)T a ijQjka i , fn(rij)\1[exp([brij) ; k/0 n (brij)k k! (9.4) which, in practice, only signi–cantly aÜects the interaction for nearest-neighbour ions.364 Relationship of crystal structures to interionic interactions The single parameter which this function contains, was originally uncovered in the b4 , ab initio study of LiF but is successfully transferred to other materials by scaling with the sum of ion radii.This is the procedure used in this work. For cation enhancement, the anion procedure is adapted by replacing those terms in the energy in which a cation dipole interacts with an anion charge by where is precisely the same function ;(i | cations);(j | anions)(1[cM f4(rij)[1NT a ij)Qjka i , f4(rij) used in the anionic term.This re—ects the fact that the cation enhancement and anion diminution arise from the same short-range cation»anion interaction. c is a parameter which controls the extent of the enhancement. The eÜect of this term is best illustrated by considering the dipole induced in an ion in a crystal which has been distorted by displacing some other ion oÜ its lattice position.If the displaced ion is at a large separation from the ion under consideration, the induced dipole will be that resulting from the electric –eld caused by the displacement acting on the in-crystal polarizability. If, on the other hand, the displaced ion is in the –rst coordination shell, the short-range f4(rij) function will come into play and diminish the dipole induced on an anion but enhance that induced on a cation, to an extent which is determined by c.The full model requires a quadrupole polarizability, C, a dipole»dipole»quadrupole hyperpolarizability, B and an overlap function f (2)(r).8 Following previous work on AgCl,8 B can be approximated via the relationship BB[6C. Even with this approximate C and B parameter set we are still at present lacking real information to derive the function f (2) eÜectively.A suitable range of ab initio calculations would allow the f (2) function to be parametrized from distorted crystal calculations in an analogous fashion to the dipole model.8 In the absence (as yet) of such a range of ab initio calculations one option is to set the f (2) function to unity and to use an ìeÜectiveœ C (and hence B) smaller than that predicted for the purely electrostatic interactions, eÜectively mimicking the opposing coulombic and overlap eÜects.Typical values are in the range 6»12 au compared with an ab initio value of 26 au for O2~ in MgO.42 IX. C Remainder of the potential The remaining terms in the total interaction energy are described by simple pair potentials. Formal ionic charges are used in calculating coulomb or polarization terms.The oxide»oxide short-range repulsion is obtained by considering the interactions of the frozen ion charge densities appropriate to the equilibrium lattice parameters, –tted to a simple EPP. For the systems of MO and stoichiometry the frozen potential M2O3 appropriate to MgO13 is used whilst a potential for is used for the stoi- ZrO2 31 MO2 chiometry.These terms are listed in Table 3. The short-range cation»cation term, as obtained from ab initio calculations for the Ca2`»Ca2` potential41 by scaling the cation radius, is always insigni–cant at the range of interaction separations actually sensed in the structures of interest and can, therefore, be omitted. The total dispersion energy is given by Udisp\ ; n/6,8,...; i/1 Nt~1 ; j/i`1 Nt Cn ij rijn f n ij(rij) (9.5) where is the appropriate dispersion coefficient acting between species i and j and Cn ij f n ij is a damping function representing the eÜect of nearest-neighbour overlap on the disper- Table 3 Oxide»oxide Born»Mayer terms ion pair a/au B/au OwO (MO) 1.22 2.84 OwO (MO2) 1.40 7.72M. W ilson and P. A. Madden 365 sion interaction.The series is cut oÜ at n\8 (dipole»quadrupole dispersion) as it is known that the damping becomes more efficient along the series making terms beyond n\8 relatively insigni–cant.43 Previous work has demonstrated that the damping terms are necessary in order to reproduce experimental properties13,40 with reliable (ab initio) and values. Tang»Toennies-like dispersion damping functions44 are used as for C6 C8 the induction damping [eqn.(9.4)]. The dispersion damping coefficients, b, are taken from those derived for MgO13 and scaled by the ratio of ionic radii (pM2`]pO2~)/(pMg2` ]pO2~). Dipole»dipole and dipole»quadrupole dispersion coefficients are derived (C6) (C8) from the polarizabilities of the two species as described by Pyper.5 The polarizability of the oxide ion is known to have a strong dependence on the lattice constant ; it has been shown45 that the relationship log aO2~\A[ B R2 (9.6) where A\1.706 and B\[10.31, reproduces the oxide polarizability in systems with the MO stoichiometry, where R is the experimental lattice parameter.The cation polarizabilities are taken from ref. 46. The and coefficients can be derived from the C6 C8 Slater»Kirkwood formula47 and Starkschall»Gordon formula,48 respectively. The required electron number for O2~ is 4.455 5 with the alkaline-earth-metal values taken from the same source.Values for other cations (Zn2`, Cd2`, Ag` and Cu`) are taken from ref. 49. References 1 P. A. 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ISSN:1359-6640    

DOI:10.1039/a702305g

出版商:RSC    

年代:1997

数据来源: RSC