摘要:

Faraday Discuss., 1997, 106, 1»40 Introductory Lecture Computer modelling as a technique in solid state chemistry C. Richard A. Catlow, Lutz Ackermann, Robert G. Bell, Furio Cora` , David H. Gay, Martin A. Nygren, J. Carlos Pereira, German Sastre, Ben Slater and Phillip E. Sinclair T he Royal Institution of Great Britain, 21 Albemarle Street, L ondon, UK W 1X 4BS We describe the role of computer modelling techniques in solving problems relating to both the structural and dynamical properties and to the reactivity of solids.Recent illustrative examples are taken from the –elds of microporous materials, ionic and transition metal oxides and molecular crystals. 1 Introduction Atomistic computer modelling techniques now play a major role in both physical and biological sciences. In this paper, we aim to show how simulation and modelling methodologies may be used as tools to solve complex problems in solid state chemistry in a way that is complementary to experimental investigations.We illustrate some of the current possibilities in this –eld by recent application from our laboratory, which will describe the modelling of structures and bonding in inorganic materials, of surfaces and morphologies of both oxide and molecular materials, of defect structures of solids and of solid state reactivity with a strong emphasis on catalytic applications.We will also show how modelling is beginning to have an impact on our understanding of the hydrothermal synthesis of silicates»traditionally a very empirical –eld. We aim to exemplify by these applications, together with the many others presented in this volume, the expanding range and scope of this –eld. 1.1 Methodologies Methodological aspects of the –eld are described in several recent reviews1h5 and in many of the articles within this volume; indeed, the methods employed are increasingly the standard techniques of computational chemistry and physics adapted to the particular problems posed by solids.We therefore present only a brief summary. Methods based on interatomic potentials have a wide range of applicability in solid state chemistry. Energy minimisation (EM) methods (often referred to, in the solid state context, as static lattice techniques) have been used for many years, particularly in modelling crystal structures6 and defect energies and structures ;7 in the latter case, methods based on the procedure originally pioneered by Mott and Littleton8 have enjoyed particular success.Despite the simplicity and limitations of minimisation methods, they are robust and widely applicable. The techniques may be extended to include free energy minimisation by including the calculation of vibrational entropies using lattice dynamical theory.7,9 Such methods have been fruitfully applied to studying phase transitions, especially in silicate minerals.9 12 Computer modelling as a technique in solid state chemistry Molecular dynamics (MD) methods have been extensively employed for modelling diÜusion in solids where atomic/ionic transport is rapid, as in fast ion conductors,10 although for slower diÜusion, which is more typical of solids, such methods are inappropriate as the number and range of particle displacements is insufficient in the timescale of an MD simulation, which in current work rarely exceeds 1 ns ; for slower diÜusion we must use methods based on transition state theory which identify saddle points for atomic jump mechanisms. Application to molecular diÜusion within microporous solids has been a particularly fruitful –eld for MD, as will be discussed later in this article, although again the techniques are limited to rapidly diÜusing species, with approaches based on transition state theory again being used for slower diÜusions. MD has also been used to probe rotational dynamics in solids,11 and in a limited number of cases to model directly phase transitions.12 Monte Carlo (MC) methods have proved to be of particular value in studying sorption in solids.13 Grand canonical Monte Carlo (GCMC) methods are being increasingly used in this –eld, although simpler methods which blend MC, MD and EM techniques have proved eÜective in identifying low energy sites for sorption in microporous solids.14,15 A diÜerent and important range of applications relate to kinetic MC studies, in which the MC ìmovesœ are atomic jumps whose frequencies have been calculated using static lattice methods.Several applications have been reported by Murch16,17 to alloy diÜusion; this approach was also used in modelling oxygen diÜusion in the widely used solid state electrolyte, yttrium doped More recently, a successful applica- CeO2 .18 tion of this approach to the diÜusion of benzene in zeolite Y was reported by Auerbach et al.19 The nature of the interatomic potentials employed in such calculations has been extensively reviewed elsewhere,20,21 and further details and discussion will be given in later sections of this article and in other papers in the volume.The type of potential model employed depends on the nature of the bonding in the solid.Molecular mechanics potentials, whose conceptual base is the covalently bonded solid, and which contain terms depending on the deviation of bond lengths and angles from equilibrium values, is obviously appropriate for covalent systems. Representations of polarisability should, wherever possible, be included. The shell model22 has provided a simple and robust way of modelling such eÜects in oxide and halide systems, although the limitations of these potentials, particularly in calculating phase transition energies, has become apparent. 23,24 Parametrisation of potentials can be undertaken using empirical approaches which –t variable parameters to observed properties of model compounds. Alternatively (and increasingly) parameters are calculated directly from electronic structure studies of clusters or periodic arrays of atoms, examples of which are given in ref. 25»27. As the –eld of computational solid state chemistry moves towards increasingly predictive applications, there is a growing need for increased accuracy in interatomic potential models. Explicit electronic structure techniques have of course been used since the early days of solid state theory.The major development in recent years has been the growth, due to the development in both methodology and hardware, of the applicability of high quality ab initio methods, although there is a continuing role for semi-empirical techniques. Local density functional (LDF) methods are being widely used for periodic systems (especially the plane-wave pseudopotential approaches of the type pioneered by Car and Parrinello28), and contemporary work increasingly emphasises the use of gradient corrected density functionals rather than those based on the simpler local density approximation.Hartree»Fock (HF) methods are also extensively used, where the periodic boundary conditions methods, available in the CRYSTAL code,29 have had notable impact.At the time of writing, calculations using ab initio methods on unit cells of ca. 50 atoms are becoming increasingly practicable (although larger systems such as the zeolite ZSM-530 with 192 atoms in the unit cell have been studied). Cluster methods are often more appropriate for modelling defects, although these may be described using periodicC. R.A. Catlow et al. 3 supercells. When clusters are employed, care must be taken, whenever possible, to embed them in a suitable representation of the surrounding lattice. Interfacing of the cluster with its embedding matrix may pose serious problems in such calculations. Recent work in the –eld is described in their proceedings by Sierka and Sauer31 and Sherwood et al.32 With the continuing growth of the applicability of electronic structure techniques, can we see them as replacing interatomic potential based methods? In some cases, this will unquestionably be the case.An example is the case of defect structures in relatively simple ceramic oxides, e.g., MgO, and which could only be modelled by TiO2 Al2O3 , Mott and Littleton8 type methods in the early 1980s, but which are now within the range of good quality electronic structure calculations. However, the more general answer is that there will be a continuing role for interatomic potential based methods as the –eld moves to more complex systems. Moreover, there are many problems, e.g., the modelling of the structures of complex silicates, and of diÜusion in microporous materials, where such methods are the most accurate and appropriate.A –nal comment is needed on hardware, the growth of which constantly expands the horizons of the –eld, and where todayœs workstations are more powerful than yesterdayœs supercomputers. It is hard to predict how the –eld will develop. The growth of usable massively parallel systems is certainly one major trend, which computational solid state chemistry is indeed increasingly exploiting.There is no doubt, however, that the exponential growth in computer power will continue for the foreseeable future. In the sections which follow, we will highlight recent applications in which both interatomic potential and electronic structure methods have been used. 2 Modelling structures Modelling of the structure at the atomic level of a solid is of course a necessary –rst step in any solid state simulation study. If we are unable to reproduce the structure with acceptable accuracy, then we can have little con–dence in subsequent applications to the study of, e.g., lattice dynamical or defect properties.Moreover, when modelling techniques are applied to surfaces or to the structure of glassy phases, it is essential to ensure that they adequately reproduce the structures of the corresponding bulk crystalline phases.Additional incentives for accurate structure modelling are provided by the need for guidance from computationally based methods to the solution of the structures of complex crystals, e.g., microporous materials. Indeed, computational methods are now becoming powerful structure solving tools in the crystallography of complex inorganic materials,5 as shown by the work of Gorman et al.,33 described in this volume.As has been described elsewhere,34 application of modelling methods to crystal structures has passed through three distinct phases: in the –rst, the methods are used to reproduce known structures ; in the second, we re–ne approximate structures ; while in the third (and most ambitious class of applications) we aim to predict structures from little initial knowledge of the arrangements of the atoms within the unit cell. The techniques employed in the –rst two categories are energy minimisation, starting from a known approximate structure ; several versatile computer codes, WMIN,35 METAPOCS, 36 THBREL37 and GULP38 are available for such calculations. Good examples of structure reproduction are found in the work of Bell et al.39 on the monoclinic phase of the microporous zeolite silicalite and the recent detailed exploration of the structure of high silica zeolites by Henson et al.40 who achieved a good measure of agreement between calculated and experimental studies.The work of Shannon et al.41 provides an excellent illustration of the role of modelling in re–ning approximate structures. In this impressive study, modelling techniques were used to generate a structure for the high silica zeolite Nu 87 from an approximate initial structure (which was unable to be re–ned adequately from the available high resolution powder diÜraction data). The –nal,4 Computer modelling as a technique in solid state chemistry energy minimised structure, however, led to a fully satisfactory re–nement of the crystallographic data.Structure prediction is essentially an exercise in global optimisation. The most widely used procedures involve simulated annealing which uses a Monte Carlo algorithm to explore the potential-energy surface of a system and to identify low energy regions, which are then re–ned by EM techniques. Often the simulated annealing calculation uses a simpli–ed cost function to assess the suitability of the structure without recourse to a full lattice energy calculation.An alternative approach uses genetic algorithms which permute (breed) structures in such a way that they evolve towards low cost functions. Recent work, which solved the crystal structure of is a good illustra- Li3RuO4,42 tion of the latter approach.Examples of the use of simulated annealing techniques will be given in the paper of Gorman et al.33 in this volume. A challenging new problem in the –eld of structure modelling is provided by the mesoporous silica structures which will now be described. 2.1 Computer modelling of mesoporous materials Since the discovery of the –rst ordered mesoporous molecular sieves with pore diameters in the range 25»100 there has been a rapid growth in research into these fasci- ”,43,44 nating materials. Much eÜort is currently focused on two areas : (1) rational design of structured inorganic mesopores, with the aim of understanding synthetic mechanisms and of exploiting this knowledge to obtain materials with a desired composition, structure and properties, and (2) the modi–cation of these materials to introduce catalytically active centres into the mesopore structure.Examples of the latter include the substitution of transition metal atoms, such as Ti, into the pore framework of siliceous MCM- 41.45,46 In many cases the short-range structure of these mesopores is not well understood.In particular, the original MCM-41 phases, prepared by liquid crystal templating, form highly regular hexagonal arrays of pores, and were shown to have pore walls as thin as two T atoms in width. However, there was no evidence for any crystallographic order, other than that caused by the two-dimensional ordering of the mesoporous channels, and it has been generally accepted that the pore walls consist largely of amorphous silica, a conclusion which is supported by the presence of a high concentration of terminal silanol groups.It is nevertheless reasonable to suppose that there exists some degree of ordering of tetrahedra. In our initial approach to the problem of meso- SiO4 pore structure, we have therefore followed two approaches: –rst we have designed regular zeolitic mesopore model structures in which, as in conventional zeolites, each T atom is connected to four others by an oxygen bridge.Secondly, we have used a completely amorphous silica surface, still containing four-coordinate T atoms, but replete with terminal SiOH groups. Each model has been subjected to energy minimisation calculations, both as the purely siliceous isomorph, and with titanium substituted into the pore framework. The calculations employed are the standard lattice and defect energy minimisation available in the GULP38 program using the potential parameters reported by Jentys and Catlow.47 2.1.1 Results and Discussion.As –rst postulated by Smith and Dytrych,48 it is possible to construct an in–nite series of hexagonal zeolite structures with uniform channels of 6n T atoms, where n\2 represents the AlPO-5 structure and, as it transpired, n\3 is equivalent to the topology of VPI-5.We have extended this series to obtain structures with eÜective pore diameters of greater than 25 and have so far examined 30-, 36- and ”, 42-ring structures (see Fig. 1). Each framework was constructed from tetrahedra SiO4 and subjected to full energy minimisation using the program GULP.38 The principal features of the optimised structures are summarised in Table 1.C. R. A. Catlow et al. 5 Fig. 1 Energy minimised hypothetical 36-ring hexagonal structure The lattice energies of the hypothetical frameworks, given in Table 2, would seem not to be in—uenced by the density of the material.This result is at variance with trends observed in microporous zeolites where there is a correlation between framework density and lattice energy per unit, and suggests a levelling oÜ in lattice energy TO2 when all, or nearly all, of the tetrahedra lie on the internal surface of the SiO4 materials»a –nding that is consistent with experimental work of Navrotsky et al.49 The incorporation of titanium into the framework was also investigated by substituting a single titanium atom onto a T site in the 36-ring structure and carrying out an energy minimisation of the surrounding lattice.The energy of substitution fell within the range calculated by Jentys and Catlow47 for the various T sites of silicate, in fact within 1 kJ mol~1 of that of the most stable site.The four TiwO bond lengths were in the range 1.74»1.80 ”. Table 1 Summary of energy minimised ìzeolitic mesoporeœ structures T atoms in ring a/” c/” free diameter/” space group 30 27.96 8.36 21.0 P6 6 2m 36 32.40 8.51 25.3 P6/mcc 42 37.03 8.36 30.0 P6 6 2m Table 2 Lattice enthalpies of ì zeolitic mesoporesœ relative to other silica polymorphs material lattice energy/kJ mol~1 per TO2 quartz 0 silicalite 11 30-ring 24 36-ring 27 42-ring 266 Computer modelling as a technique in solid state chemistry Fig. 2 Energy minimised amorphous silica surface, showing the substituted group TiO3OH Amorphous pore walls have also been modelled by taking a dense silica glass structure, obtained by simulated annealing, cleaving out a thin layer, approximately three T atoms in thickness, and satisfying all dangling SiwO bonds with hydroxy groups.Part of an energy minimised structure is shown in Fig. 2 where a Ti atom has again been substituted into the pore wall. In this case the TiwO bond lengths varied from 1.77 to 1.92 (although, because of the presence of the OH group, the substitution energies ” cannot be directly compared with those in the zeolite frameworks). 3 Characterisation of bonding in solids Theoretical investigations aimed at understanding the basic interactions in the solid state chemistry of complex materials, like transition metal oxides, have until recently relied on simpli–ed and parametrised schemes, often based on extended-Hué ckel or tightbinding Hamiltonians.50,51 With the improvement in hardware and software tools now available, ab initio quantum mechanical studies are becoming increasingly feasible, and can be employed to examine in detail the properties of bonding in the solid state, in the absence of empirical parameters and simpli–ed theoretical frameworks.As an example of the kind of insight and detailed information now possible, we report in this section on Hartree»Fock (HF) and density functional (DF) calculations on three binary oxides with stoichiometry i.e., and In both MO3 , MoO3, WO3 ReO3 .schemes, the model adopted to represent the bulk materials is based on periodic boundary conditions. In the HF calculations, the electronic distribution of the solid is described in terms of crystalline orbitals, obtained as a linear combination of localised functions, or atomic orbitals (AOs), associated with lattice sites.The HF code employed is CRYSTAL.52 DF calculations are based on the full-potential linear muffin-tin orbital (FP-LMTO) code developed by Methfessel et al.,53 within the local density approximation (LDA) and the Hedin»Lundqvist exchange-correlation potential ; basis functions are in this case Hankel functions augmented by numerical solutions of the radial Schro é dinger equation in the muffin-tin spheres.In the following discussion we focus on several distinct features of the solid state chemistry of transition metal oxides. 3.1 Nature of the transition metal The three oxides examined have common structural features, all based on the ReO3 structure, which is formed by a network of corner-sharing octahedra, with all the MO6C.R. A. Catlow et al. 7 oxygen ions in a two-coordinate, bridging position between adjacent metal sites. To illustrate the nature of the bonding in these systems, we make reference –rst to the highest symmetry, cubic phase. In Fig. 3 we report the band structure of project- ReO3 , ed onto its components in the Re and O basis sets, as obtained from the DF calculation. 54 The valence band (VB) is formed mainly by O 2p states, although they are mixed with the Re 5d orbitals. The conduction band (CB) has a dominant contribution from the Re 5d orbitals, while the Re 5d orbitals lie at a higher energy; this energy t2g eg dependence is typical of d AOs in an octahedral –eld.It is important to note the presence of three levels along the C»X direction, denoted in Fig. 3 with the labels ìAœ and ìBœ (in the CB) and ì1œ (in the VB), which are pure Re pure Re and pure O 2p t2g, eg levels respectively, with no mixing of orbitals. These levels are —at in k-space, and have been previously denoted as superdegenerate.55 Covalent interactions between the relevant O 2p and Re 5d AOs are symmetry forbidden in the cubic phase. Levels A, B and 1 are therefore non-bonding between metal and oxygens, which interact in the solid via their ionic charge.However, the importance of covalence in the bonding is shown by the bottom levels of the VB, which have a high k-dispersion and contribution from both Re and O. The hybridisation pattern of metal and oxygen levels can be interpreted, in classical terms, as a back-donation of electrons from the –lled O2~ AOs to the empty 5d AOs on the metal.The solution for and is very similar to that described MoO3 WO3 here for ReO3 . The energy gap between the superdegenerate levels A and 1, *E(A[1), is related to the value of the crystal –eld in the metal and oxygen positions : the decrease of the eÜective nuclear charge (taking into account the number of core electrons) of the transition metal ion from Re to Mo and W aÜects the relative energy of levels A and 1.The energy of level A, which as shown in Fig. 3 is a metal d AO, is more sensitive to the Fig. 3 Band structure of cubic projected onto its atomic contributions, along the C»X»M ReO3 , direction of reciprocal space.The bands labelled ìAœ and ìBœ (in the conduction band) and ì1œ (in the valence band) are pure Re pure Re and pure O 2p states respectively. t2g, eg8 Computer modelling as a technique in solid state chemistry nuclear charge of M than the energy of level 1, which is an O 2p state. *E(A[1) is therefore higher in and than it is in Moreover, Mo and W have a MoO3 WO3 ReO3 .similar eÜective nuclear charge, but the orbitals involved are the 4d in Mo and the 5d in W; the gap *E(A[1) is lower in than it is in This feature is important, MoO3 WO3 . because smaller values of the gap *E(A[1) correspond to a higher degree of covalence in the bonding.55,56 Hence, *E(A[1) to an extent measures the tendency towards covalent interactions : the lower *E(A[1), the more covalent the solid.The trend in the balance between covalent and ionic bonding is con–rmed by the population analysis data, as obtained from the HF study, reported in Table 3. A higher value of the bond population corresponds to a higher degree of covalence in the bonding, as is also (qb) con–rmed by the lower values of the net ionic charges on metal and oxygens which accompany the higher values. qb 3.2 Number of valence electrons We now examine the symmetry reductions from the cubic phase: experimental evidence shows that the cubic structure of is stable at all temperatures, while and ReO3 MoO3 are stable in lower symmetry phases, in which the transition metal ion lies oÜ- WO3 centre in its coordination octahedron.For simplicity we focus here on the cubic]tetra- Table 3 Equilibrium bond distances and Mulliken population analysis within each octa- MO6 hedron in bulk and as obtained from the HF study.(The indexing of atoms ReO3 , MoO3 WO3 , is the same as adopted in the text). bond distance bond population atom multiplicity net charge r(MwO)/” qb(MwO) cubic ReO3 Re ]3.31 O 6 [1.10 1.85 ]0.03 cubic b-MoO3 Mo ]2.99 O 6 [1.00 1.89 ]0.01 cubic WO3 W ]4.09 O 6 [1.36 1.88 [0.03 tetragonal b-MoO3 Mo ]2.74 O(1) 1 [0.70 1.70 ]0.17 O(2) 4 [1.02 1.87 [0.05 O@(1) 1 [0.70 2.30 ]0.01 SOTa 6 [0.92 2.86 ]0.03 tetragonal WO3 W ]3.79 O(1) 1 [0.97 1.63 ]0.21 O(2) 4 [1.41 1.90 [0.01 O@(1) 1 [0.97 2.37 [0.00 SOTa 6 [1.26 1.93 ]0.03 orthorhombic a-MoO3 Mo ]2.43 O(1) 1 [0.44 1.64 ]0.29 O(2) 1 [0.75 1.69 ]0.10 O(3) 2 [1.25 1.94 [0.02 O@(2) 1 [0.75 2.21 [0.02 O@(3) 1 [1.25 2.24 [0.00 SOTa 6 [0.81 1.94 ]0.06 a Average M»O value in an octahedron. MO6C.R. A. Catlow et al. 9 gonal distortion, which corresponds to a displacement of the transition metal ion along the [001] crystallographic direction, towards one of its six nearest neighbour oxygen ligands. Fig. 4 shows the band structure of the tetragonal phase of after a ferroelectric ReO3 , movement of the Re sublattice by 3% of the lattice parameter along the [001] direction. The mixing of the superdegenerate levels A and 1 in the distorted structure is no longer symmetry forbidden, and it gives rise to a p bonding level in the VB and to a p antibonding level in the CB.A similar but smaller change also aÜects the levels of r symmetry.In the distortion, therefore, a new covalent bond is formed between transition metal and oxygen, which involves AOs of both r and p symmetry along the MwO directions. The relative stability of the cubic and tetragonal phases depends on the number of valence electrons (n) : in and only the bonding level 1 in the VB is –lled, MoO3 WO3 , and the distortion is energetically stable ; in there is, in addition, one antibonding ReO3 electron in the CB, which opposes the distortion.It has been shown54 that the electronic degrees of freedom in the two phases are isoenergetic for a number of electrons in the CB, When the cubic phase is stable and vice versa for n0\0.98 o e o. n[n0 0\n\n0 where the distortion is favoured, in agreement with experiment.With reference to the population analysis of Table 3, we see that the increase in covalence is restricted to the interaction between the transition metal and only one of its six nearest neighbour oxygens, the one towards which the metal displaces [labelled O(1) in Table 3], while the interaction with the four equatorial oxygens [O(2) in Table 3] is mostly unchanged with respect to the cubic phase.The above feature allowed us to label Fig. 4 Projected band structure of after the Re displacement along the [001] direction. ReO3 Bands ìAœ and ì1œ are the same as in Fig. 3; now they have a completely diÜerent atomic contribution, showing the eÜect of the newly formed p M 2p bond in the distorted structure. t2gwO10 Computer modelling as a technique in solid state chemistry the short MwO pairs in and as tungstyl and molybdenyl groups.57 Fur- MoO3 WO3 thermore, the displacement of the transition metal leaves a longer MwO bond, and creates a more layered bulk structure.57 Mechanical properties can also be shown to depend on n:54 in Table 4 we report the calculated bulk moduli (B) for the oxides examined.The very high B of re—ects the ReO3 population of the RewO p antibonding orbital in the CB: by increasing the applied pressure, the stress acting along the RewO directions is very high.In practice, the solid reacts to this stress by diverting the applied pressure from the RewO towards the OwO direction ; this eÜect will cause a rotation of the oxygen octahedra, and lead to a better exploitation of the empty spaces of the corner-sharing network, in agreement with the experimentally observed pressure-induced phase transition.In and the MoO3 WO3 , eÜect of pressure is absorbed by a change in the interlayer distance ;57 in both cases the eÜect of the structural change is such that the value of B decreases. 3.3 Structural requirements So far, we have examined the in—uence of the transition metal ion on the overall properties of the materials ; this is not the only relevant aspect in describing their solid state chemistry : several transition metal oxides can in fact exist in diÜerent polymorphs, in which the metal ion is the same, but the medium-range arrangement of the ions is diÜerent.To investigate this feature we examine the two polymorphs of the MoO3: (the same structure as discussed in Sections 3.1 and 3.2), and the ReO3-like b-MoO3 layered and thermodynamically stable a-MoO3 .In the crystal structure of (Fig. 5) we identify a direction (a), perpendicular a-MoO3 to the layers, along which the octahedral units are disconnected from one MoO6 another; within the layers, the connectivity of the octahedra occurs in one direc- MoO6 tion (b) by common edges, so as to form zigzag rows, and in the perpendicular (c) by common corners only.Because of this anisotropy, oxygens in have a diÜerent a-MoO3 local environment: the outermost oxygen in each layer has only one nearest molybdenum atom; where the octahedra are connected by corner-sharing, the oxygens are two-coordinated, as in while oxygens along shared edges are three- a-MoO3 , coordinated.In the following discussion, we refer to n-coordinated oxygens as O(n). With reference to the population analysis data reported in Table 3, we clearly see that oxygens with a diÜerent coordination number in the solid behave like diÜerent chemical species.58 In particular, the Coulomb –eld created by only one nearest Mo in the O(1) positions is not strong enough to stabilise an O ion with the formal charge of [2.As a consequence, the interaction between Mo and O(1) is much more covalent than that between MowO(2) and MowO(3); the particularly low value of the net charge on O(1) Table 4 Calculated bulk modulus (B/GPa) for the optimised structures of and ReO3 , MoO3 WO3 , as obtained from our DF and HF study phase DF HF HF]corra cubic ReO3 304 282 320 cubic b-MoO3 » 279 316 cubic WO3 254 257 281 tetragonal WO3 » 137 174 a Results obtained estimating the contribution of electron correlation a posteriori, based on a functional of the HF equilibrium density.C.R. A. Catlow et al. 11 Fig. 5 Network of edge and corner sharing octahedra in showing the layering of MoO6 a-MoO3 , the structure along the a crystallographic direction con–rms this feature.The short MowO(1) pair in behaves again as a single a-MoO3 unit, or molybdenyl group. Another important structural feature to examine in is the connectivity a-MoO3 between layers. In the inter-layer region, we –nd molybdenyl groups facing one another, and connecting in a zip-like manner (see Fig. 6). The low net charge on O(1) contributes to decreasing the repulsion between layers, and stabilises the layered polymorph with respect to As shown above, is more ionic than the net charge on b-MoO3.WO3 MoO3 ; O(1) in a layered polymorph of analogous to would therefore be higher, WO3 a-MoO3 hence destabilising this structure.58 3.4 Long-range Coulombic forces Finally, we show the eÜect of long-range forces, by comparing ferro- and antiferroelectric (FE and AFE respectively) displacements of W in Both structures have WO3 .the same short-range chemical features and medium-range structural requirements, as commented in Sections 3.1»3.3 above, but diÜer in the long-range order in the solid. The movement of one ion in a solid is equivalent to creating a net dipole in the structure ; in the FE phase all dipoles are aligned, while in the AFE the dipoles in neighbouring cells are anti-aligned. Only the Coulombic, long-range interaction will diÜerentiate the two structures.In Fig. 7 we report the energy change for FE and AFE displacements of W in we see there that short- and long-range interactions play a comparable role in WO3; determining the solid state chemistry of WO3 .From all the previous arguments we must conclude that the bonding in the solid state has several components. We may distinguish between a chemical (short-range) contribution, which depends on the nature of the transition metal and on the number of valence electrons, a structural (medium-range) contribution depending on the connectivity of the octahedra in the solid, and a Coulombic (long-range) contribution MO6 which takes into account the ionic interaction.The equilibrium con–guration is due to a12 Computer modelling as a technique in solid state chemistry Fig. 6 DiÜerence electron density map (solid minus isolated Mo6` and O2~ ions), in the plane through the outermost molybdenyl groups [MowO(1)] in adjacent layers of high- a-MoO3 , lighting the zip-like connection of layers and the short O(1)wO(1) equilibrium distance.The zigzag dashed line in the centre of the plot serves as a guideline to separate the molybdenyl groups belonging to each of the two facing layers. The lines correspond to positive (»»), negative (» » ») and zero (» … » …) diÜerence. The interval between the isodensity lines is 0.005 e a0~3.subtle balance of all these forces. Accurate quantum mechanical calculations, of the kind described here, can help us in –nding their relative importance in the structures under investigation, and we can use them to rationalise experimental data on these complex materials. Fig. 7 Dependence of the total energy of tetragonal on ferro- and antiferro-electric WO3 (L) (Ö) displacements of W; the oxygen sublattice is kept –xed in the cubic phase.represents the ZW displacement of W (in fractional coordinates) from the centre of the octahedron. WO6C. R. A. Catlow et al. 13 4 Modelling of the surfaces of complex materials Surfaces play a central role in the chemistry and physics of solids. The details of the surface structure control many of the observed properties of the system, such as growth/ dissolution, catalysis and resistivity as in gas sensors.To understand these phenomena we require knowledge of the surface structure, and the surface structure in diÜerent environments. Modelling plays a complementary role to experiment in this –eld, due to the complexity of making accurate measurements of the surface structure, and difficulty in preparing clean surfaces. 4.1 Techniques The technology of modelling surfaces is very similar to that used in modelling bulk systems, but with diÜerent boundary conditions. The same potential models used for the bulk can generally be used for the surface although results for surfaces tend to be more dependent on the quality of the potential than for those of bulk systems, owing to the reduction of symmetry at the surface.The new boundary conditions are vacuum or solvent over a surface with 2D (two-dimensional) periodic boundaries parallel to the surface normal. These boundary conditions require the use of more parameters to describe the surface uniquely. Specifying the surface normal does not identify a unique crystal face. A position along the surface normal, or shift, is needed as well.The normal and shift are sufficient for surfaces that are planar or smooth cuts. Molecular crystals (to be described later) require special care since molecules should not be fragmented to create a surface. In addition to the structure we can compute the surface energy and the attachment energy. The former is the energy required to cleave the crystal to create the surface.The latter is the energy released when a growth slice is attached to the surface. These quantities can be used to predict the crystal morphology.59 Use of the surface energy implies thermodynamic control of the morphology. The conditions under which use of attachment energy is appropriate are ill de–ned, but they are generally more suited to morphologies controlled by kinetic factors. 4.2 Strategies for modelling surfaces There are three ways to model a 2D system. One is a 3D solution that uses a 3D array of slabs and gaps. The second is to compute the total energy using 2D periodic boundary conditions. The third uses a 3D lattice and computes the energy of the atoms while keeping track of the side of a cutting plane on which they are located.This procedure does not permit the relaxation of the surface and therefore is of limited applicability. While this partitioning technique can be used for quick calculations of the attachment energy, it is not of general applicability for modelling surfaces and hence will not be discussed further. Many simulations, both atomistic and quantum mechanical, are carried out using a 3D array of slabs. The total energy of the system is the energy of the atoms in the cell interacting with their lattice images.However, there are restrictions on when a 3D slabgap approach can be used. The simulation cell must not have a dipole moment normal to the slab or have one created during the simulation, as the dipole moment would interact across the gap distorting the adjacent surface.To guarantee that this problem does not arise, a slab must have a plane of mirror symmetry located in the middle of the slab. The other restriction is that the top and bottom of the slab should be equivalent, otherwise it is not certain to which face the computed properties apply. The other full 2D approach as used in the program MARVIN59 (and in the earlier MIDAS code60) is more general and can be used even when the restrictions on the 3Dz y x region 1 region 2 14 Computer modelling as a technique in solid state chemistry slab-gap apply. The energy of the cell is computed with a 2D lattice summation (with an adaptation of the Ewald sum to 2D).In the 2D approach, the bulk of the crystal can be treated as an array of –xed atoms.This two-region approach is illustrated in Fig. 8. For energy minimisation or dynamics, the region 2 atoms are –xed to reproduce the force exerted by the bulk crystal. The region 1 atoms are allowed to move either to minimise the total energy or dynamics can be introduced using the MD technique. The full 2D, two-region technique also allows for docking calculations. These permit the attachment or incorporation of impurities or molecules at or near the surface.The eÜects of impurities on crystal growth or other surface properties can be determined from this type of calculation. 4.3 Types of surface We can divide surfaces into two classes : polar and non-polar. Non-polar faces are the easiest and most reliable to model, since no corrections need to be made to compute the surface energy or attachment energy.Accurate modelling of polar faces requires special attention, as it is necessary to neutralise the dipole in some way, either mechanical, environmental or chemical. Mechanical neutralisation involves moving some ions from one face to another to counteract the dipole ; chemical neutralisation removes some of the charge by attaching new charged species, such as hydroxy groups, to the surface ; while environmental neutralisation is eÜected by polarisation of the surrounding solvent.We can also distinguish between three diÜerent types of face, depending upon the structure of the unit cell in the direction of the surface normal (Fig. 9). Type 1 faces are non-polar and type 3 faces are polar, regardless of the position of the cutting plane.Type 2 faces are either polar or non-polar depending upon where the cutting plane is. Crystals composed of molecules present special problems to modelling of surfaces. A planar cut through the bulk crystal to create the surface usually passes through molecules located near the surface. Using this plane will leave molecular fragments on the Fig. 8 The two-region surface simulation cell.Region 1 is relaxed to minimise the total energy while region 2 is –xed and represents the bulk crystal.C. R. A. Catlow et al. 15 Fig. 9 The three types of surface. Type 1 has no dipole moment normal to the surface, wherever it is cleaved. Type 2 surfaceœs dipole moment depends upon the cleavage plane, some of which have no dipoles. Type 3 faces have a dipole, regardless of where the cleavage plane is located.surface, which is not chemically or physically realistic. Creation of such fragments is avoided in the MARVIN code. We now give three examples of recent applications of surface modelling. The –rst relates to the key problems of surface hydration of an inorganic material. The second describes recent quantum mechanical studies of transition metal oxide surfaces.The third describes recent surface simulations of molecular materials. 4.4 Hydroxylation of the corundum basal plane surface : modelling of surface chemical reactions Surfaces commonly undergo stabilising chemical reactions. The surface ions and molecules are exposed to the environment and a clean surface will contain coordinatively unsaturated ions and dangling bonds.For surfaces having a net dipole, the dipole must be cancelled to create an electrostatically stable structure, as discussed above. A chemical reaction that reduces the surface charge may stabilise a cut that is unstable as a clean surface. In this section we discuss an important and topical example, namely the hydroxylation of the basal plane surface of corundum. Replacing an oxygen with a hydroxide ion halves the charge; so hydroxylating an oxygen terminated cut reduces the charge by a factor two which, depending on the interplanar spacing, may cancel the net dipole.61 For the oxygen terminated surface this is the case and other examples Al2O3(0001) where hydroxylation cancels a net dipole are oxygen terminated (111) surfaces of rocksalt structured oxides or (001) surfaces of perovskite oxides.The clean, aluminium terminated basal plane surface has been studied by Al2O3 several authors using both atomistic potentials59,62 and periodic Hartree»Fock methods.63 Both types of calculation predict that the top layers collapse inwards in a dramatic fashion. These large relaxations have recently been observed in experiments by Guenard et al.using X-ray scattering at grazing incidence.64 The aluminium terminated surface exposes three-coordinate aluminium ions and half of the aluminium sites are empty, as shown in Fig. 10. A surface exposing both dangling bonds and ions with unsaturated coordination can be expected to be highly reactive. Replacing the topmost oxygen layer with hydroxy ions for the oxygen terminated surface, not only cancels the dipole but gives rise to a surface without any dangling bonds and no low coordinated ions, as shown in Fig. 11. In sharp contrast to the aluminium terminated surface, the hydroxylated surface shows hardly any relaxation, as illustrated in Fig. 12. We now explore the energetics of creating hydroxylated surfaces. For a clean surface, the surface energy only includes the work needed to cleave the perfect crystal.When the16 Computer modelling as a technique in solid state chemistry Fig. 10 Unrelaxed and relaxed aluminium terminated surface of Half of the aluminium Al2O3 . sites are empty and upon relaxation the topmost aluminium relaxes inwards. Small light spheres represent aluminium ions ; larger, darker spheres represent oxygen ions surface undergoes a chemical reaction, the reaction enthalpy must also be included into the surface energy.The ideal way of calculating such energies would of course be to use quantum mechanical methods rather than interatomic potential based procedures. The computational cost of quantum mechanical methods often rules out this possibility and we are faced with the problem of incorporating reaction enthalpies into calculations using interatomic potentials.In the case of hydroxylation, we need to calculate the energy for creation of the surface together with the reaction : Osurf 2~ ]H2O]2OHsurf ~ Such energies of reaction are normally calculated by introducing a hypothetical gasphase reaction, which should then be chosen so that the heat of reaction can be esti- Fig. 11 The hydroxylated surface of where all oxygens have been replaced by hydroxides Al2O3 , leaving a surface with no low coordinated ions or dangling bonds. There is almost no surface relaxation.C. R. A. Catlow et al. 17 Fig. 12 Changes in interplanar spacings upon relaxation. The middle column shows the inter- (”) planar spacing for the unrelaxed surface.The left hand column is the relaxed aluminium terminated surface and the right hand column the hydroxylated surface. mated for all steps, either from calculations or thermochemical data. A Born»Haber cycle commonly used for investigating hydroxylation is therefore Osurf 2~ ]Ogas 2~ H2Ogas]Ogas 2~]2OHgas ~ 2OHgas ~ ]2OHsurf ~ In this reaction we have the problem that O2~ does not exist in the gas phase, which means that we have to de–ne the second electron affinity of oxygen.There have been several previous estimates of this quantity using thermochemical data; for example, a value of 8.27 eV was estimated65 which when used in calculating the surface energy for the hydroxylated surface gives a negative value of [1.12 J m~2.66 A negative surface energy for a clean surface would mean that it would be energetically favourable to maximise the surface area.It is more plausible that negative energies could occur when the surface undergoes a chemical reaction, for which it is also possible that further reaction might be kinetically inhibited. Nevertheless such values must be viewed with considerable scepticism. An alternative procedure is to calculate the enthalpy for the hydroxylation reaction using the experimental heat of formation for the corresponding metal hydroxide67 where this is available.We wish, however, to develop a reliable and general method of calculating the surface energy for reactive surfaces using a procedure that does not rely on hypothetical species, such as gaseous O2~. We will now outline such a scheme together with recent results for the case of the hydroxylated surface.We –rst note that for Al2O3 reactions containing only neutral species it is much easier to calculate the reaction enthalpy using quantum mechanical methods than it is for reactions involving negatively18 Computer modelling as a technique in solid state chemistry charged ions.We therefore choose a gas-phase reaction that is charge neutral. It must also be possible to calculate the enthalpies for surface to gas and gas to surface processes. Starting from the oxygen terminated surface we move half of the oxygens to the other side of the crystal neutralising the dipole. Reacting the resulting surface with water gives rise to the hydroxylated surface.To calculate the surface energy of the hydroxylated surface we begin by calculating the surface energy for this half-layer oxygen terminated surface. Starting from this surface and, keeping the above considerations about the gas-phase reaction in mind, a suitable Born»Haber cycle for calculating the reaction enthalpy is given in Table 5 together with the calculated energies for each reaction step.In the –rst step, we pull oÜ a growth layer from the oxygen terminated surface. To have well de–ned energetics, this layer must have no dipole moment across it ; hence the choice of as the repeat unit. In the next step we separate this layer into clusters. Al4O6 After reacting the clusters with water we put them back together again into a hydroxylated layer.Finally we drop this layer back onto the surface. The enthalpy for reacting the cluster with water has been calculated using gradient corrected density func- Al4O6 tional techniques.68 All other steps have been calculated using a consistently derived set of atomistic potentials.66 If we normalise the total energy given in Table 5 per unit surface area, we obtain a value of [9.95 J m~2.On adding this value to the surface energy for the half-terminated oxygen surface (13.40 J m~2) we calculate a value of 3.45 J m~2. Relaxing the hydroxylated surface we obtain for the –nal surface energy a value of 3.40 J m~2. Comparing this value with that of [1.12 J m~2 obtained using the cycle employing gaseous O2~ illustrates the care that is needed in constructing the appropriate thermochemical cycle.We also note that the relaxed surface energy for the aluminium terminated surface is 2.52 J m~2. We should stress that there still remain uncertainties in our results arising from the chosen interatomic potential parameters and from the geometries used in the DFT calculation on the gas-phase reaction. Nevertheless, the present calculation suggests that creation of the unhydroxylated surface costs less energy than of the hydroxylated.However, our earlier calculations suggest that, once created, the hydroxylated surface will be stable ; indeed we calculated the water desorption energy to be at least 5 eV. There is no inconsistency or paradox here. Removal of water from the hydroxylated surface exposes the high energy oxygen terminated rather than the lower energy aluminium terminated surface and hence is highly endothermic.It would be of great interest to prepare the hydroxylated surface and to examine its structure. 4.5 Quantum mechanical calculations on surfaces : bonding in surface layers Employing similar arguments to those used in Section 3 to examine bulk materials, we can try to understand the eÜect of surfaces on the bonding properties of transition metal Table 5 Calculated energy changes (*E) for the component steps in the Born»Haber cycle for the hydroxylation of the surface of Al2O3 *E/eV Al4O6 (surface)]Al4O6 (layer) 60.97 Al4O6 (layer)]Al4O6 (cluster) 19.05 Al4O6 (cluster)]3H2O]Al2O3Al2(OH)6 (cluster) [13.85 Al2O3Al2(OH)6 (cluster)]Al2O3Al2(OH)6 (layer) [22.62 Al2O3Al2(OH)6 (layer)]Al2O3Al2(OH)6 (surface) [69.33 total energy: [25.78C. R.A. Catlow et al. 19 oxides, a factor of great importance in improving our understanding of the catalytic properties of these materials. In this section we focus on perfect and sur- WO3 MoO3 faces, which we have represented by a slab model, a 2D array of atoms of –nite thickness, described using the same computational conditions employed for the HF study of the respective bulk systems.As we have shown in Section 3, both tetragonal and the two polymorphs of WO3 can be described as layer structures. The layering orientation in bulk provides a MoO3 preferential direction of cleavage for the surfaces : the S001T for the materials, ReO3-like and the S100T for Let us examine the structure –rst : stable surfaces a-MoO3 .ReO3 must be charge neutral and must not have net dipoles perpendicular to the surface.69 In the structure each oxygen is two-coordinated and shared between two transition ReO3 metal ions ; if we imagine the surface as created by cleaving the bulk material along a predetermined plane, the requirements summarised above mean that one half of the oxygens belonging to the cleavage plane must be left in each of the two facing surfaces created.In practice this causes a doubling of the unit cell, and a )2])2 reconstruction of the surface with respect to the bulk structure, as observed experimentally.70 The S001T surfaces of and of are illustrated in Fig. 13(a). The transition metal WO3 b-MoO3 ions on the surface are alternately –ve- and six-coordinated ; in the former case only the longest MwO bond in the bulk arrangement has been cleaved on the surface, thus representing a relatively small perturbation to the bulk arrangement.No major structural and electronic rearrangement occurs on the surface, and the overall bonding properties remain the same as in the bulk materials ; the only relevant diÜerence is the Fig. 13 Schematic representation of the S001T surface of and (a), and of the S102T b-MoO3 WO3 surface of (b). Atoms are labelled here as in the text. WO320 Computer modelling as a technique in solid state chemistry shortening of the surface molybdenyl and tungstyl groups and the corresponding increase in their covalence. Results of the population analysis for the surfaces examined are reported in Table 6.[Note that in Table 6 and in the following discussion we have used the following notation for labelling surface atoms: whenever an index was used in the bulk structure (see Table 3), this has been retained in the surface notation, to highlight the original position of surface species in the bulk; an index ì s œ has been added to the ions directly exposed onto the surface.Metal atoms are further labelled with index 5 or 6, according to their coordination number on the surface.] The –ve-fold coordinated Mo(5s) and W(5s) lie below the plane of the four equatorial oxygens [O(2s)], by which they are eÜectively screened from a direct exposure onto the surface. Similarly, the S100T surface of is cleaved along the direction of layering of a-MoO3 the bulk material, and its bonding remains unchanged.These surfaces are the stable ones for the oxides examined; in cleaving the solid along any other plane we will cut stronger MwO bonds, which will cause a stronger perturbation, and major electronic and structural rearrangements on the surface. To exemplify this feature we examine in more detail the S10nT surfaces of tetragonal these correspond to S001T platelets, WO3: separated by kinks every n unit cells.In particular, we examine one member of this family, the S102T surface [Fig. 13(b)]. For the same requirements of surface stability expressed above, W ions on kinks are four-coordinated (following the notation introduced above, we label atoms on kinks with index ìkœ).Three of their four O ligands are in turn two-coordinated, and continue into the S001T platelets ; they are accordingly labelled O(ls) and O(2s). The fourth oxygen is instead coordinated only to W(k), and we refer to it as O(2k). As we might expect, most of the rearrangement involves W(k) and O(2k): in the unrelaxed structure, O(2k) corresponds to one of the ionically bound equa- Table 6 Equilibrium bond distances and Mulliken population analysis for the surfaces examined of and MoO3 WO3 bond distance bond population atoma multiplicity net charge r(MwO)/” qb(MwO) a-MoO3 S100T Mo(s) ]2.40 O(1s) 1 [0.39 1.65 ]0.31 O(2s) 1 [0.76 1.69 ]0.09 O(3s) 2 [1.25 1.94 [0.02 b-MoO3 S001T Mo(6s) ]2.64 O(1s) 1 [0.44 1.70 ]0.36 O(2s) 4 [1.04 1.86 [0.09 Mo(5s) ]2.67 O@(1s) 1 [0.69 1.70 ]0.18 O(2s) 4 [1.04 1.87 [0.05 WO3 S001T W(6s) ]3.65 O(1s) 1 [0.66 1.59 ]0.35 O(2s) 4 [1.44 1.91 [0.03 W(5s) ]3.73 O@(1s) 1 [0.96 1.63 ]0.20 O(2s) 4 [1.44 1.90 [0.02 WO3 S102T W(k) ]3.31 O(2k) 1 [0.79 1.65 ]0.27 O(1s) 1 [1.10 1.60 ]0.06 O(2s) 2 [1.46 1.87 ]0.01 a Labelling of surface atoms is explained in the text.C.R. A. Catlow et al. 21 torial oxygens, O(2) (compare with Table 3).The relaxation on the surface involves the displacement of O(2k) along the diagonal of the three W(k)wO(s) directions ; the metal ion on the kink, W(k), is therefore in a distorted tetrahedral environment. The energy gained in the relaxation of O(2k) is 2.55 eV, con–rming that the electronic rearrangement accompanying the geometric relaxation is substantial : as we could expect from the analysis of the bulk system, the now singly coordinated O(2k) left free on the surface behaves in a completely diÜerent way from both the corresponding bulk species and from two-coordinated O(2).In particular, covalence in the bond between W(k) and O(2k) is very pronounced (as con–rmed by the bond population reported in Table 6) and grows partially at the expense of the W(k)wO(1s) interaction, which is still covalent, although to a much lesser extent than in the bulk material.The properties of bonding close to the kink are therefore completely diÜerent from those of the bulk species. This –nding is of general validity : the stronger the perturbation created by the surface, the higher the electronic rearrangement and the structural reconstruction on the surface. We expect the chemical reactivity of the exposed surface species to be greatly in—uenced by the above features.Finally, in Fig. 14 we show the electrostatic potential created by the surfaces described above. This is a most important property of surfaces, which controls the adsorption of gas-phase molecules. We stress an important common feature : a negative –eld prevails only above O(1s) species, while the –eld in the region above O(2s) and O(3s) ions is positive. Although the latter have the highest negative charge among the oxygens, the –eld which they create is compensated and dominated by the high positive charge of the neighbouring cations.In adsorption processes, the positive end of molecules would therefore be attracted by the O(1s) end of surface tungstyl and molybdenyl groups, which is very important in determining the chemical properties of the exposed surfaces, for example in catalysis and in gas-sensing devices.The –eld in the region next to kinks is much more intense and structured, and would therefore drive polar or polarisable molecules preferentially towards these surface defects. 4.6 Simulation of the surface of molecular crystals Previous sections have summarised the rapid progress in the –eld of computational surface science, particularly in the area of ionic surfaces and related surface mediated processes. However, in the case of molecular crystals, computational expense precludes all but the simplest molecular crystals from treatment via periodic all-electron or semiempirical methods.71,72 Therefore, the majority of molecular surface or periodic slab simulations reported in the literature are based on interatomic potential techniques.Molecular crystals are distinct from other classes of crystals in that their stability is controlled primarily by dispersive forces, and their packing arrangement by short-range repulsion. This precludes these crystals from treatment via periodic Hartree»Fock methods (except in the case of strongly hydrogen bonded solids : see, for example, ref. 71 and 72) and plane wave based techniques. The majority of electronic molecular surface studies reported are based upon cluster techniques,72 where short-range and polarisation eÜects dominate. We have previously shown how ionic surfaces may reconstruct in order to eliminate permanent perpendicular surface dipoles, giving rise to a divergent surface potential.In molecular crystals, unlike ionic or semi-ionic materials, the molecular fragments often exhibit permanent intrinsic dipoles, giving rise to a surface dipole, but the methods by which the dipole may be eliminated for these materials are less well understood.Molecular surfaces may contain highly polarisable functional groups, which are clearly aÜected by solvents and may undergo substantial relaxation. In order to simulate the polar morphology of urea, Docherty et al.72 reported a study where the potentials and charge22 Computer modelling as a technique in solid state chemistry Fig. 14 Maps of the electrostatic potential generated by three of the surfaces examined: a-MoO3 S100T (a) ; S001T (b) and S102T adjacent to the kink (c).The electrostatic potential b-MoO3 WO3 map for S001T is not reported, due to its close resemblance to plot (b), relating to WO3 b-MoO3 S001T. Positive (»»), negative (» » ») and zero (» … » …) potentials are shown; the interval between isopotential lines is 0.005 au (1 au\27.21 eV).distribution were scaled, based on PHF and cluster studies, in order to distinguish between polar mirror planes. However, their technique does not include surface relaxation, yet the issues of relaxation and potential model modi–cation are interdependent. Relaxation of molecular surfaces is characterised by a more shallow potential-energy surface minimum in contrast to electrostatically dominated ionic surface relaxation.Two classical based codes are commonly employed to explore molecular surface processes : MARVIN (discussed previously59) and HABIT.73 These programs diÜer in their treatment of surfaces, in that HABIT does not consider the eÜects of surface relaxation, which has been shown to be of importance by George et al.74 in the case of urea. In molecular crystals the dispersive and electrostatic forces are required to be computed particularly accurately : MARVIN employs a full 2D Ewald75 sum, an approach that is critical when considering polar molecules.C.R. A. Catlow et al. 23 Several previous studies of the surfaces of molecular crystals have employed the periodic bond chain technique: Berovitch-Yellin et al.76,77 have successfully shown how to manipulate additives to in—uence crystal morphology and Docherty et al.72 have reported a study of the eÜects of solvents on morphology.In order to explore these factors and kinetic eÜects quantitatively and comprehensively, we have developed two extensions to the functionality of MARVIN. MARVIN has previously been used successfully to predict the morphology and surface properties of a number of organic and inorganic crystals59,74,78,79 and has been extended to incorporate automated docking procedures via MC and MD techniques. The addition of these methods represents a signi–cant development in the capabilities of surface simulation.The automated docking scheme allows one to explore efficiently the conformational space of single or multiple additive species. This is of great value in the study of the eÜect of additive/impurities on the surface/morphology.The velocity Verlet algorithm80,81 and Gear82,83 predictor-corrector schemes have been implemented, allowing one to study, for example, kinetic eÜects, surface diÜusion, crystal growth and dynamic interfacial activity. Previously, surface MD studies have generally been carried out using a periodic slab approach (see, for example, Boek et al.84), where, as noted earlier, the simulation box has to be constructed such that a mirror plane is not present, as this may give rise to a dipole perpendicular to the surface. One can combine the MD and MC techniques to enable efficient exploration of conformational space, by using low energy con–gurations generated from MC simulations as initial conformers in MD simulations.We have exploited both these techniques to investigate factors aÜecting the crystal growth of the important molecular crystal aspartame. Aspartame is widely used within the food industry as an arti–cial sweetener, but extraction of robust crystals can be difficult and expensive. Recently, we have been addressing this problem, and carried out simulations using the techniques outlined earlier in this review.Initially, an energy minimisation of the bulk aspartame crystal was carried out, using the CFF91 force –eld,85 extracted from the DISCOVER module supplied by MSI.86 To eliminate unnecessary computational eÜort, we selected only the most signi–cant torsional terms from the force –eld.(Full details of the potentials used and further details regarding the simulations reported here will be given in subsequent publications.) The lattice parameters were reproduced within 1% of those reported in the crystal structure, 87 an excellent agreement for a complex molecular crystal. This relaxed cell was used to generate the four most morphologically signi–cant surfaces : subsequent relaxation of these surfaces showed minimal changes in surface energies and attachment energies, indicating a predicted morphology qualitatively similar to that produced by a Donnay»Harker88 analysis.As in the case of urea, aspartame has polar faces which appear in the morphology. To address the eÜect of polar solvents, on apolar and polar surfaces, we have simulated saturated aspartame, with an overlayer of water.One feature of molecular surfaces that complicates this kind of treatment is the problem of de–ning the solvent surface-accessible volume. This problem is compounded, in the case of aspartame, where the polar surfaces are highly porous and the apolar surfaces are corrugated. Having de–ned the accessible surface, relaxed bulk water was grafted onto the surface.The bulk water was obtained by a 500 ps MD run on a cell containing 1024 atoms at 298 K using an SPC89 potential model. The resulting cell box was relaxed at constant pressure using the static lattice method available in the GULP38 code. Solvent molecules with anomalously small overlaps with the solute of \1.2 ” were eliminated. The resulting surface and interface was relaxed using the MARVIN code, and this structure used as the initial con–guration in a surface MD run.Fig. 15 shows a snapshot taken after 2 ps, including 1 ps equilibration. After 5 ps, the water is seen to diÜuse in well de–ned channels into the aspartame surface. Longer simulations24 Computer modelling as a technique in solid state chemistry Fig. 15 Snapshot of the aspartame(110)/water interface after 2 ps of MD also show dynamic exchange of crystal water with surface water. We are currently carrying out extended MD runs to establish quantitatively the in—uence of the solvent on surface and attachment energies. In summary, the advent of full 2D Ewald surface MD allows one to probe dynamical surface phenomena of crystal surfaces that are intractable or approximate by other methods.This capability has general implications and in this article we have highlighted its particular relevance to simulating solvent eÜects in a complex molecular crystal. 5 Simulation of defects There is a long history in the application of defect modelling techniques in solid state chemistry and physics.5,7 Indeed, the extensive range of studies published in the 1970s and 1980s established Mott»Littleton related techniques as quantitative simulation tools in the –eld.Defect calculations continue to be a fertile and productive application area, as we will illustrate by recent applications90,91 concerning the mechanisms of dissolution of water in upper mantle minerals»a key problem in current mineralogy, asC.R. A. Catlow et al. 25 Table 7 Calculated energies of water dissolution in Mg2SiO4 energy/eV phase reaction (I) reaction (II) olivine 3.71 0.46 b phase 1.18 [1.10 the upper mantle is known to contain substantial quantities of dissolved water.92,93 The broader questions raised by the mechanisms of solution of water in oxides and silicates have been discussed in ref. 94 and 95. The upper mantle minerals are magnesium silicates of general formula and have the olivine structure at lower (Mg0.88Fe0.12)SiO4 temperatures, and a spinel like structure at higher pressures and an intermediate b phase at intermediate pressures. The transition zone in the upper mantle is known to be associated with a phase transformation from the olivine to the b phase.Wright and Catlow90,91 have clearly shown that the dissolution of water in both the olivine and b phase may be eÜected by two mechanisms: MgMg]2OO]H2O][VMg(OH2)]]ìMgOœ (I) 2FeMg~]2OO]H2O]2FeMg]2OH~]12 O2 (II) in which we use the Kroé ger»Vink notation and where the –rst reaction involves the formation of a defect cluster containing a magnesium vacancy with two protonated neighbouring atoms.The second reaction involves reduction of iron(III) (substituting for magnesium) to iron(II), with charge compensation produced by the protonation of neighbouring oxygen ions. The calculated energies for these reactions in both olivine and the b phase are given in Table 7. We note that the energetics of the iron(III)/iron(II) redox reaction is low in both phases, and the calculated energies of water dissolution are lower in the b phase compared with olivine»a result that is in accord with experiment. There is little doubt that the mechanisms shown in reactions (I) and (II) are the main processes responsible for dissolution of water in the upper mantle minerals. 6 Modelling of diÜusion As noted in the Introduction, MD techniques have been widely used in simulating diÜusion in solids and a particularly fruitful –eld of application has concerned the transport of sorbed molecules, especially hydrocarbons within microporous hosts.Many chemical reactions undergone by hydrocarbons when using zeolites as catalysts are very diÜerent when compared to the same reactions carried out in the gas phase.96h98 The hydrocarbons have, of course, to diÜuse through these structures in order to reach the active site.The large number of zeotypes99 allows us, in principle, to select among a wide variety of structures in order to favour the yield of a particular reaction. Indeed, shape selectivity100 is commonly a diÜusion controlled process which in—uences product distribution in reactions such as aromatic isomerisation among others.The understanding of more recently investigated phenomena like secondary shape selectivity and inverse shape selectivity among others illustrates the importance of hydrocarbon diÜusion in in—uencing reactivity in zeolites.101,102 DiÜusion measurements of hydrocarbons in zeolites have been made for several decades,103 although often many experimental variables other than the microscopic diffusion itself in—uence the measured diÜusion coefficients. In addition to direct microscopic experimental techniques like PFG-NMR104 and quasi-elastic neutron scattering (QENS),105 computer simulations can give very good estimates of self-diÜusivities.10626 Computer modelling as a technique in solid state chemistry In this section, we address the study of ortho- and para-xylene diÜusion in CIT-1,107 the –rst microporous material with 10 MR (membered rings) and 12 MR channels.DiÜusion of these two C8 aromatics has been widely studied in 10 MR zeolites like ZSM-5 where the size of the channels precludes the diÜusivity of the ortho isomer.108 DiÜusion of these isomers in a structure also containing 12 MR, through which both isomers can diÜuse, brings new features, in particular the competition of both isomers for the diÜusion path through the 12 MR channels when only one of them can diÜuse through the narrower 10 MR channels.We have, therefore, undertaken atomistic MD simulations to model the diÜusion of ortho- and para-xylene in purely siliceous CIT-1. The calculations require the speci–- cation of a potential-energy function which allows the calculation of its –rst derivatives from which the forces acting on each atom can be obtained.The force –elds employed include four terms as follows : Vtotal\Vzeolite]Vxylene]Vxylenehxylene]Vzeolitehxylene where the zeolite potential includes Buckingham, three-body and Coulombic terms; the intramolecular xylene potential includes two-body, three-body, four-body and Coulombic terms; and the xylene»xylene and zeolite»xylene potentials have Lennard-Jones and Coulombic terms.A more detailed description of the potentials can be found in ref. 109. All the MD simulations were performed using the general purpose DLñPOLY2.0110 code in its parallel version running on a 512 PE CRAY-T3D using 128 processors. The simulations proceed by assigning initial velocities to the atoms and then solving Newtonœs equations of motion using a –nite time step by means of the standard Verlet algorithm.80 Timesteps of 1 fs and an equilibration temperature of 500 K were employed in the simulations.The system comprises a 4]2]4 macrocell of purely silica CIT-1 and a maximum of 32 xylene molecules with a total of 3264 atoms to which periodic boundary conditions are applied throughout the MD simulation.Simulations of 100 ps were carried out within the NVE ensemble at the three diÜerent xylene loadings of 32 ortho-xylene, 8 ortho-xylene and 8 para-xylene molecules in the CIT-1 macrocell. Additionally, activation energies for the diÜusion path of each isomer in each channel were calculated by means of the solids diÜusion path facility implemented in the MSI Catalysis 6.0 software package.111 This simple but eÜective procedure drags a molecule through a channel in a microporous structure, calculating the interaction energy of the molecule with its host.The results of the three simulations yield three basis pieces of information, namely activation energies, the diÜusion path followed by the molecules through the zeolite channels, and self-diÜusion coefficients.The activation energies are shown in Fig. 16. It can be seen that ortho- and para-xylene can diÜuse without important restrictions through the 12 MR channels as shown by the low activation energies (6.21 and 7.03 kcal mol~1 for the para and ortho isomers respectively). Large interactions may be calculated when ortho-xylene diÜuses through the 12 MR channels due to an unoptimised orientation of the molecule in the channel (see Fig. 16) but a reorientation of the methyl groups towards the channel direction produces a low energy diÜusion path.This feature has important implications as to the possibility of rotation of the isomers, a point to which we return later. In the 10 MR channels a diÜerent result arises, as the activation energy for the para isomer of 20.93 kcal mol~1 is still consistent with diÜusion, whereas the 110.31 kcal mol~1 for the ortho isomer make its diÜusion impossible through the narrower channel.It is interesting to note that some penetration of the ortho isomer in the 10 MR channels is allowed at the intersection between the 10 MR and the 12 MR.It can be seen from Fig. 16, bottom right, however, that further penetration into the 10 MR channels by the ortho isomer leads to a very high increase in the interaction energy. Regarding diÜusion paths, we –rst note that in zeolites with parallel and perpendicular systems, visualisation of the channel structure is straightforward, but in a zeolite likeC. R.A. Catlow et al. 27 Fig. 16 Activation energies calculated for the diÜusion of ortho- and para-xylene through the channels of CIT-1 CIT-1 it is more difficult to understand the trajectories of the molecules in the channel system. In order to make the visualisation as clear as possible we have projected both the xylene trajectories and the channel axes on two-dimensional diagrams. The corresponding results for the three simulations are shown in Fig. 17, where only one of the three possible projections (xy, yz, xz) is given which is sufficient for our purpose. The three –gures correspond to the three diÜerent loadings given above. It can be seen [Fig. 17(a) and (b)] that ortho xylene diÜuses exclusively through the 12 MR channel system in both cases regardless of the loading simulated.Some penetration into the 10 MR channels can also be noted as was suggested by the calculated activation energies. The main feature of the diÜusion of ortho-xylene is that it tends to diÜuse in a restricted part of the zeolite with what we denote as extensive local motion. This behaviour is attributable to the high size of the ortho isomer with respect to the channel dimensions which precludes this molecule from travelling more freely through the structure.In the third simulation it can be seen [Fig. 17(c)] that para-xylene can diÜuse through both channel systems. The –gure also shows two additional interesting features : diÜusion through the 12 MR channel seems to be more favourable, as the higher population of this channel indicates ; and diÜusion through the 10 MR channel follows a straight line because the para isomer matches the channel size which precludes the para-xylene from rotating in this channel. Self-diÜusivities have been estimated from the history –les by calculating the mean square displacements of the centre of mass of the xylene molecules in the 100 ps simulation run.The results are illustrated and presented in Fig. 18 from which two main conclusions follow.The –rst concerns the in—uence of loading which can be estimated from the –rst two values corresponding to ortho-xylene. The results show that self-28 Computer modelling as a technique in solid state chemistry Fig. 17 Trajectory plots showing the diÜusion path of the centre of mass of the sorbate molecules in the CIT-1 macrocell: (a) 32 ortho-xylene, (b) 8 ortho-xylene, (c) 8 para-xyleneC.R. A. Catlow et al. 29 Fig. 18 Trajectories and diÜusion coefficients for the diÜerent simulated loadings and trajectory plot (100 ps) showing preferential diÜusion path through 12 MR channels. ortho-Xylene, 1.00 molecules uc~1, D\3.56]10~6 cm2 s~1; ortho-xylene, 0.25 molecules uc~1, D\7.79]10~6 cm2 s~1; para-xylene, 0.25 molecules uc~1, D\25.18]10~6 cm2 s~1.(a) View parallel to [001] (12 MR channels) ; (b) view parallel to [110] (10 MR channels). Trajectories of the centre of mass of the xylene molecule are shown in black. diÜusivities decrease with increasing loading, a feature common to many other zeolites due to the increasing importance of the repulsive guest»guest interactions.On the other hand, for the same loading we see a much higher self-diÜusivity for para-xylene due to its smaller activation energy and to the possibility of diÜusing also through the 10 MR channels. The present simulations have thus yielded semi-quantitative self-diÜusivities, and have yielded a good understanding of the microscopic features of the diÜusion process ; such results will be especially useful when combined with experiments in achieving a better understanding of diÜusion in zeolites. 7 Computer modelling and hydrothermal synthesis One of the most exciting recent developments in the –eld has been the exploitation of computational techniques in understanding the factors controlling the synthesis of microporous materials.A detailed discussion is given by Lewis et al.112 in this volume. Here we concentrate on one aspect of our work in this –eld, relating to the mechanisms of condensation of silica fragments»possibly the most fundamental process in hydrothermal synthesis.Condensation reactions of the type have been studied by Burggraf et al.,113 using semiempirical methods, and the activation energy is estimated experimentally to be 15 kcal mol~1.114 The reverse reaction has been studied by Lasaga and Gibbs115 (to explain the surface dissolution of quartz), who predicted an activation energy of 21.9 kcal mol~1, using ab initio calculations.The condensation reaction is discussed in detail by Iler116 and a comprehensive review is given by Brinker and Scherer.11730 Computer modelling as a technique in solid state chemistry In this work we studied the simplest condensation reaction of two species 2Si(OH)4 to form a dimer, using density functional theory (DFT) coupled with a continuum dielectric model (COSMO) to try to describe the electrostatic conditions found in real silica solutions. We used the Dmol code from MSI,118 with non-local (Becke»Lee» Yang»Parr)147,148 exchange and correlation energy and a DNP double numerical basis set. All atomic arrangements were –rst optimised in the gas-phase and subsequently recalculated with COSMO as a single energy point, without reoptimisation.Acid catalysis was considered throughout the work, corresponding to pH\3, the conditions mostly found in current experimental work. The protonated species are –rst investigated.The Si(OH)3(H2O)`, Si2 OH(OH)6 ` transformation of the protonated reactants into the protonated products is then analysed considering both an and a lateral attack mechanism. Assuming that the SN2 reaction occurs by the attack of a monomer on a second protonated monomer, the proton remaining after the water molecule has left should be attached to the oxygen that initiated the nucleophilic attack, i.e., the bridging oxygen.The reactions to protonate the reactant and the product can then be written as follows, with Si(OH)4 Si2OH(OH)6 ` the calculated energies (*E) given above the arrow: Si(OH)4]H3O` »»»»»»’ 7.4 kcal mol~1 Si(OH)3(H2O)`]H2O (III) Si2O(OH)6]H3O` »»»»»»’ 12.9 kcal mol~1 Si2OH(OH)6 `]H2O (IV) The calculated protonation energies agree with the basicity trends reviewed by Brinker and Scherer.117 The additional charge is more stabilised in the ion than in the H3O` protonated silicate clusters. The protonation energy is smaller for than for Si(OH)4 because the electron-withdrawing eÜect is smaller for OH than for OSi Si2OH(OH)6 ` groups, making the protonated monomer more stable.Assuming that these trends apply for larger clusters, in acid conditions small clusters and chain ends are more likely to be protonated and receive the nucleophilic attack of groups situated in the middle of the chains, forming branched chains.The reaction of the protonated monomer to give the protonated dimer is therefore an endothermic process, given by: Si(OH)4]Si(OH)3(H2O)` »»»»»»’ 5.1 kcal mol~1 Si2OH(OH)6 `]H2O (V) Two mechanisms for this reaction were studied considering both an and a lateral SN2 attack, represented in Fig. 19(a) and (b) respectively. The energy evolution along the reaction path shows –rst a small energy barrier SN2 of 2.5 kcal mol~1 before decreasing to a –ve-silicon intermediate, which is 3.4 kcal mol~1 more stable than the reactants. A second energy barrier of 11.3 kcal mol~1, the largest in the whole reaction process, occurs later, when the water molecule is leaving. In the lateral attack mechanism, after the –rst 2.5 kcal mol~1 energy barrier, a –vesilicon intermediate occurs, 1.9 kcal mol~1 less stable than the reactants. As in the gas phase, a pronounced peak occurs when the water molecule leaves, forming the largest energy barrier in the overall mechanism: 17.3 kcal mol~1, which is 6 kcal mol~1 larger than in the mechanism.SN2 The global energy evolution from the neutral reactants to the neutral products is depicted in Fig. 20. The activation energies calculated in this work for the (11.3 kcal SN2 mol~1) and lateral attack (17.3 kcal mol~1) mechanisms are in good agreement with the value estimated from experiment (15 kcal mol~1).Owing to the strategy utilised in this study, a gas-phase optimisation followed by a COSMO energy calculation, the diÜerence in energy between reactants and products is very small : [0.4 kcal mol~1. OnC. R. A. Catlow et al. 31 (a) and lateral attack (b) mechanisms for the condensation reaction (V) (see text). The Fig. 19 SN2 x axis shows the distance between the Si atoms of the dimer and the O of the (Aé ) Si2 OH(OH)6 ` departing The y axis shows the distance between the attacking O of the mol- H2O.(Aé ) Si(OH)4 ecule and the Si of the attacked monomer Si(OH)4H`. optimising both reactants and products in the COSMO environment, this diÜerence increases to [3.2 kcal mol~1. These results show that both the condensation and the hydrolysis reactions have a small activation energy.For both reactions, the mechanism seems to be a low SN2 energy route for converting the reactants into the products. However, as the results for the lateral attack show, several other mechanisms that are energetically and statistically less favourable should still be possible and may occur simultaneously in the solution.32 Computer modelling as a technique in solid state chemistry Fig. 20 Energy evolution (kcal mol~1) during the condensation reaction, for both Si(OH)4 SN2 and lateral attack mechanisms, from non-local DFT/COSMO calculations The DFT»COSMO implementation clearly oÜers a powerful tool to study the mechanisms occurring in this key condensation reaction. 8 Reactivity In this –nal section of recent applications we consider the use of quantum mechanical techniques in elucidating reaction mechanisms.Two topical examples are discussed. 8.1 The methanol to gasoline reaction The methanol to gasoline process119 in medium pore-sized aluminosilicate zeolites has been the subject of great theoretical120h133 and experimental119,134h146 interest due to its potential industrial importance but also, importantly, because of its status as a prototypical solid state acid catalysed reaction.Indeed, the methanol to gasoline Br‘nsted process is now used as a standard test for the presence of acidity. Despite the Br‘nsted obvious importance of this process we still have limited knowledge of the precise nature of the reaction, particularly with respect to the mechanism for formation of the –rst CwC bond.The past few years has seen a rapid increase in the use of DFT methods to study the electronic structure of chemically relevant systems, particularly since the introduction of accurate non-local corrections147h149 to the local exchange-correlation potentials150,151 that have been so successful in solid state physics. However, there are still questions concerning the validity of such methods in probing transition states of chemical reactions.This section therefore describes the study of a possible pathway for formation of the –rst CwC bond in the methanol to gasoline process with a number of quantum chemical methods in order to understand better the methanol to gasoline process and the limitations of the theoretical methods.Aluminosilicate zeolites consist of corner sharing tetrahedral units where T is TO4 either Al or Si. In the case of T\Al, a charge imbalance of [1 is generated which is compensated for by incorporation of a proton, H`, generating acidity. For Br‘nsted computational tractability only a few of these T atoms can be included in a quantum chemical calculation, a typical cluster model of the active site being a 3T fragment with dangling bonds terminated with hydrogen to preserve charge neutrality.However, a number of authors have used the smaller 1T model for a number of problems associatedC. R. A. Catlow et al. 33 with adsorption and reactivity at the acid site and found that results are sur- Br‘nsted prisingly similar to those obtained from a 3T model.127,152,153 One reason for this result could be that a typical process at an aluminosilicate acid site involves a push» Br‘nsted pull interaction with two of the oxygens attached to Al and although the 1T model is more basic at both oxygens than the 3T model, the diÜerence in the basic properties of the two oxygens in the 1T and 3T models is expected, due to symmetry, to be similar.However, it must be remembered that the electrostatic potential above the 1T model is more negative than that of the 3T model and the latter would therefore stabilise the localisation of negative charge more than the former, and vice versa. We therefore employ a hydrogen terminated 1T model, or ZOH, and (HO)3Al(OH2) its methylated analogue, or ZOMe, to study the formation of an ethyl- (HO)3AlO(Me)H ated surface, or ZOEt, and from that ethene, see Fig. 21. Formation of (HO)3AlO(Et)H the surface methoxy, ZOMe, has been the subject of previous studies123,127,129,130 and will not be considered further except to note that it can be formed at the site Br‘nsted by two methanol molecules with an activation barrier of about 130 kJ mol~1. We have studied this reaction at the HF/6-31G**//HF/6-31G**, the MP2/6»31G**//HF/6- 31G**, the BLYP/DZVP//LDA/DZVP and the BLYP/DZVP//BLYP/DZVP levels of theory (notation indicates the level of theory used for –nal energy calculations//levels of theory used for geometry optimisations).The Hartree»Fock and MP2 calculations were performed using the GAUSSIAN94 code.154 The local density approximation (LDA)150,151 and the gradient corrected Becke147»Lee»Yang»Parr148 (BLYP) DFT calculations were performed with the Cray Research code DGauss 3.0155 using the HF/6- 31G** optimised structures as starting points. Table 8 summarises the results.All structures were similar to the HF/6-31G** structures shown in Fig. 21. Fig. 21 HF/6-31G** reaction path for formation of ethene from a surface methoxy (ZOMe) and adsorbed methanol (MeOH) via a carbene-like transition state (ts-I) and surface ethoxy intermediates (int-I and int-II).The acid site was modelled by a 1T, fragment. Br‘nsted (OH)3AlO(H)H The energy changes are given in Table 8.34 Computer modelling as a technique in solid state chemistry Table 8 Energy changes (kJ mol~1) to Fig. 21. methoda *E(1)b *E(2) *E(3) *E(4)b *E(5) HF/6-31G**//HF/6-31G** 514 [573 39 232 [172 MP2/6-31G**//HF/6-31G** 465 [539 53 194 [126 BLYP/DZVP//LDA/DZVP 386 [414 86 136 [78 BLYP/DZVP//BLYP/DZVP 382 [419 37 140 [88 a Notation indicates method/basis used for calculating the –nal energy//method/ basis used for optimising the geometry.b Activation energy barrier. The reaction path shown in Fig. 21 describes the deprotonation of a surface methoxy group leading to CwC bond formation (in ZOEt) via a transition state with a surface stabilised carbene-like species (ts-I).Fracture of the CwH bond occurs early in the reaction coordinate to form water with the OH of methanol and this bond fracture is probably the main contributor to the activation barrier. Deprotonation of this ZOEt species results in the formation of adsorbed ethene through the transition state ts-II.It is immediately clear that the diÜerent quantum chemical approximations lead to quite diÜerent estimates of the energy changes, especially for the –rst stage of the reaction. It is expected that the HF//HF results should overestimate the activation barriers as the adiabatic transition states are generally formed from an avoided crossing of the ground and one or more excited states156 and therefore the single determinantal HF result for the transition state is expected to be a poorer approximation to the solution of the Schroé dinger equation than are the HF results for the minimum energy structures.Similarly, the MP2//HF activation barrier, although better than the HF//HF result, is still probably an overestimate as the MP2 correction will not recover all of the correlation energy.The BLYP//LDA and BLYP//BLYP results are somewhat surprising as the geometries and the energy changes are remarkably similar (all structures are within 0.2 and 10° of each other) although the calculated activation barriers are prob- ” ably still underestimates.157 This is in contrast to a number of reports suggesting that LDA is particularly poor at reproducing hydrogen bonds (see, for example, ref. 130). Uncertainties in the model and method aside, it is clear that deprotonation of surface methoxy species to form ZOEt will be unlikely, although, as discussed above, use of a larger model will result in a less negative electrostatic potential above the cluster and therefore in a stabilisation of the carbene-like, electron-rich transition state, ts-I.As a comparison with other work, Blaszkowski and van Santen132 have studied a number of reaction pathways with a 1T model at the BP (Becke147»Perdew149)//LDA level of theory in which CwC bond formation occurs in the adsorbate, not the surface carbon species. The resulting calculated activation barriers are substantially lower than those presented here.Calculations on larger clusters will be reported in future work. 8.2 Partial oxidation on oxide surfaces Facilitating CwH bond breakage is one of the typical functions of catalysts. The oxidative coupling of methane (OCM) represents an important process, the initial step of which requires just that : methyl radicals are formed from methane by H-abstraction.One of the factors determining the usefulness of a catalyst for such a process will be the amount by which the energy barrier for the initial CwH bond cleavage is reduced in its presence. A variety of oxides of diÜerent complexity are known to act as OCM catalysts ;158 in some cases their catalytic activity can be promoted by dopants.It has been suggested that these promoters play a role in stabilising electronic defect sites on the catalysts surface.159 For example, localisation of electron hole states preferentially takes place atC. R. A. Catlow et al. 35 anion surface sites adjacent to cation substitutions, where the ionic charge of the dopant is lower than that of the original cation ; the resulting O~ species (known to be present from EPR measurements160) may very well function as active sites in hydrogen abstraction reactions.It has long been proposed that [Li]0 centres constitute such a catalytically active site on the surface of Li doped magnesium oxide surfaces. This would explain why doping is found to have an enhancing eÜect on this very extensively studied OCM catalyst.161 The [Li]0 defect site, consisting of a neighbouring pair of a Li` (substituting for Mg2`) and an electron hole localised on an oxygen ion, is thought to interact with the hydrocarbon molecule in the following process : CH4][Li`O~]]~CH3][Li`(OH)~] (VI) leading to the gas phase radical species observed experimentally.162 ~CH3 Despite continued interest in this system (see, for example, ref. 158 and 161) little is known about reaction (VI) in detail. A number of theoretical studies have been devoted to this question (and related ones) during the last ten years163h176 and even further back.177h179 Most of them are performed at the HF level164,172,177h179 or below.163,165,170,174 On post-HF level studies involving MP2,170,172,174,180 CASSCF,167h169 DFT175 and hybrid176 approaches seem to indicate that electron correlation may well play a signi–cant role in the proper description of the system. Although the eÜect of Li doping is mentioned in most of these studies, only a few incorporate it explicitly in the models used.168,169,174,176,180 In some, relaxation of the substrate is taken into account178h180 but in other cases it is neglected.Finally some of the references deal only with dissociation, not with methane.H2 Taking all this into account it seems desirable to obtain a theoretical description of this system using (a) an explicit model for the [Li]0»methane interaction on a non-rigid MgO and employing (b) a theoretical approach that allows us to estimate the importance of electron correlation in this reaction.It is not easy, however, to satisfy both goals simultaneously at reasonable computational cost. A cluster model of considerable size is necessary to represent substrate relaxation taking place in the course of reaction (VI). On the other hand, to assess whether ab initio calculations at the HF level are sufficient to describe reaction (VI) it is necessary to test ways of including electron correlation to some degree.Using MP2 or MP4 corrections to the HF approach, one is limited to cluster sizes that do not comply with (a) above. Therefore, it is important to validate less expensive alternatives such as, for example, DFT. This has been done, using a very simple model system in which the [Li]0 centre is represented by a minimal cluster Li`O~ embedded in a point charge array.181 Comparing the energy pro–le along the reaction coordinate (Fig. 22) one observes the eÜect of electron correlation : from HF calculations a symmetric energy barrier is obtained; the height of this barrier is substantially reduced as MP2 corrections are added (even more for MP4). DFT (using BLYP gradient corrections147,148) on the other hand reduces the activation barrier to a mere shoulder in the corresponding curve.For future studies employing larger clusters we therefore conclude that while results obtained from HF calculations may be expected to overestimate the barrier, those coming from DFT calculations tend give too low a value and should be used with caution. The next step in reaching a realistic description of reaction (VI) requires the construction of suitable cluster models.To represent ionic relaxation adequately, more than only the nearest neighbours of the active site will have to be included (here it has to be taken into account that the outermost atoms of the cluster must be kept –xed in relation to the surrounding point charges). Enlarging the cluster, however, not only con—icts with computational economy but also leads to SCF convergence problems, especially in the case of DFT.A cluster of the following composition has been used successfully : –rst layer (12 Mg –xed); second layer (8 Mg –xed); third layer (4 Mg16O9 Mg8LiO4 Mg4O36 Computer modelling as a technique in solid state chemistry Fig. 22 Energy pro–le of reaction (VI) as calculated from HF (»»), MP2 (- - -), MP4 (- … - …-) and DFT»BLYP (…………).The sum of the energies of the products in reaction (VI) has been used as an energy oÜset for each of these curves. Mg –xed) and fourth layer Mg (–xed). Preliminary calculations of the [Li]0 centre yield a relaxation energy of 1.9 eV with ionic displacements of up to 0.3 (compared to the ” bulk positions). At the present stage in this study it is clear that both aspects discussed above play a crucial role : (a) the quality of the model (including geometric relaxation) and (b) the quality of the method (beyond HF).Consequently we shall combine the two approaches described here in our future work, leading to a more accurate determination of CwH bond activation by Li/MgO. Further details of the results described above are given in ref. 181. 9 Conclusions The applications described in this article illustrate, we hope, the range and diversity of the growing –eld of computational solid state chemistry. We can expect a continuing increase in the complexity of the systems studied and the accuracy of the predictions made, enhancing the value of computational tools as techniques in solid state chemistry. We thank EPSRC for supporting this work. We also gratefully acknowledge –nancial support from DSM, ICI, and MSI.We are grateful to W. C. Mackrodt, A. M. Stoneham, A. K. 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ISSN:1359-6640    

DOI:10.1039/a701964e

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 41»62 Structure and reactivity of silica and zeolite catalysts by a combined quantum mechanicsñshell-model potential approach based on DFT Marek Sierka and Joachim Sauer*§ Humboldt zu Berlin, Institut Chemie, Arbeitsgruppe Quantenchemie, Universitaé t fué r 10/11, D-10117 Berlin, Germany Jaé gerstr. An ion-pair shell-model potential with functional parameters derived from the results of quantum mechanical density functional theory (DFT) calculations on small molecular models is presented.It is used to predict the structure and properties of diÜerent silica and zeolite catalysts. Characteristic diÜerences between the Hartree»Fock and DFT structures of quartz, silica sodalite and silicalite are revealed. A combined quantum mechanics» ion-pair shell-model potential scheme is presented and applied to embedded cluster calculations on catalytically active sites in periodic framework structures. 1 Introduction Understanding the catalytic activity of solids at the molecular level requires assistance from accurate quantum chemical calculations. Advances in computer technology, e.g. the advent of parallel machines and recent developments in quantum chemical codes, permit more complex calculations on more realistic model systems, often involving more than 1000 basis functions and more than 100 atoms, depending on symmetry.Zeolites appear as particularly interesting systems for such studies because of their well de–ned framework structures, the wealth of experimental studies, and their technological importance.Quantum chemical calculations for solids, which, microscopically, are threedimensional in–nite systems, are computationally very demanding, even when applying periodic boundary conditions. Active sites or defects introduce further complications owing to broken space and translational symmetry, which require larger pseudo-unit cells. One solution to this problem is the –nite cluster approach. The treatment is limited to a part of the system and the in—uence of the crystalline environment is neglected or only included approximately.Thus, cluster models are particularly suited to describe local phenomena, e.g. catalytically active sites. There exist, however, classes of problems which require theoretical predictions of the structure of the whole system.The interatomic potential functions (also known as force –elds) serve these needs. They provide an analytical approximation to the potential energy surface of the system. Traditionally, empirical interatomic potentials are used. However, for many but the simplest systems, there is incomplete information for parameter determination. Therefore, the following procedure has been suggested :1h3 quantum chemical calculations are performed for –nite molecular models of the periodic zeolite structures. The results obtained are used as a data base to –t the parameters of the interatomic potential functions.These parameters are assumed to be transferable and § E-mail: js=qc-ag-berlin.mpg.de 4142 Structure and reactivity of silica and zeolite catalysts are used for calculating the energies of the periodic structures and the forces on their atoms.This approach has several advantages. The set of data that may be included in the –t is, in principle, unlimited and there is a one-to-one correspondence between the ìcalculatedœ (by the potential function) and ìobservedœ (by the quantum chemical calculations) data. As ìobservablesœ serve the energy and its –rst and second derivatives with respect to the displacement of all nuclei, ìobservablesœ can be generated, not only for the energy minimum structure of the models but also for an, in principle, unlimited number of distorted structures.A similar procedure has been applied by other authors.4 The disadvantages of interatomic potential functions are that no wavefunction is obtained and that reactions are difficult to model.Therefore, the idea emerges to combine the virtues of the quantum mechanical cluster calculation (QM) with a lattice energy minimization of the periodic solid using interatomic potential functions (Pot). Such a combined scheme, QM»Pot, is particularly useful for studying the properties and reactivity of an active site in diÜerent environments.Our implementation uses the ion-pair shell-model potential,5 which takes into account the polarization of the environment by the active site, to describe the periodic crystal. It accounts for both the long-range crystal potential and the ìmechanicalœ constraints of the position of the atoms of the cluster by the periodic lattice. In principle, any of the quantum chemical methods available can be combined with interatomic potential functions.However, for an accurate description of the active acidic sites in zeolites and of reactions involving such sites, inclusion of electron correlation proved to be important. For example, inclusion of electron correlation was found to be essential in order to correctly describe the interaction of methanol with zeolite Br‘nsted sites.6 Among quantum chemical methods incorporating correlation eÜects, for instance MP2, MP4 or CI, density functional theory (DFT) is particularly attractive since, for large systems, the computational eÜort is much less than for even the simple MP2 approximation.In practice, the computational cost of DFT calculations grows as nF2.6 with the number of basis functions, compared to the nF4.0 dependence of efficient MP2 codes.7 In this paper we –rst derive an ion-pair shell-model potential for silica and zeolites based on DFT calculations using the B3-LYP exchange-correlation functional.8 We then use this DFT parametrized potential to predict the structures and vibrational properties of dense and microporous silica polymorphs.A systematic overestimation of SiwO bond lengths is observed, which is shown to be an inherent feature of the DFT B3-LYP method for the basis set used.This conclusion is based on DFT calculations for octahydridosilasesquioxane, and disiloxane, We then reveal charac- H8Si8O12 H6Si2O. teristic diÜerences between the HF and the DFT potential energy surfaces for quartz, sodalite and silicalite.The former tends to predict the higher symmetry structures incorrectly as the most stable ones. Finally, the performance of the potential for protonated zeolites is discussed and the embedded cluster approach (QM»Pot scheme) is applied to sites in faujasite. Br‘nsted 2 Methods and computational details 2.1 Quantum chemical calculations The cluster calculations use the DFT together with the B3-LYP8 exchange-correlation functional.For the embedded cluster calculations we use the TURBODFT9 program. Unless speci–ed explicitly, we apply the fully optimized basis sets from Ahlrichsœs group,10 double zeta (DZ) for Si, Al and H and the triple zeta (TZ) for O. Polarization functions with the following exponents are added: 0.4 (Si), 0.35 (Al), 1.2 (O) and 0.8 (H).This combination is designated as T(O)DZP basis set. The quantum chemical program package GAUSSIAN9411 is employed for the free cluster calculations. For numericalM. Sierka and J. Sauer 43 integration in GAUSSIAN94 we use the option ì–negridœ which produces a grid formally consisting of 75 radial and 302 spherical points per atom. The DFT calculations in TURBODFT are made with the following grid (see ref. 9 for details) : for elements of the –rst row 5890 points (35 radial and 302 spherical), for the second row 9212 points (40 radial and 434 spherical), and for the third row 10 542 points (45 radial and 434 spherical). The calculations of absolute shielding constants p for the optimum structures of the embedded clusters obtained using the combined QM»Pot scheme are performed within the coupled Hartree»Fock approach using the gauge including atomic orbitals (GIAO)12 implemented in the SHEILA13 module of the TURBOMOLE14 program.The TZP basis sets from Ahlrichsœs group10 are used on all the atoms. The relative 1H chemical shifts are calculated using methanol as internal reference (secondary dTMS standard) :15 dTMS(cluster)\dTMS(CH3OH)]p(CH3OH)[p(cluster) The calculated absolute shielding constants were 32.71 and 31.79 ppm at the p(CH3OH) HF and DFT B3-LYP optimized structure, respectively, while the experimental gas phase value is 0.02 ppm.dTMS(CH3OH) 2.2 Database generation Since four-, –ve- and six-membered aluminosilicate rings are characteristic secondary building units of zeolites, they are adopted as models and saturated with hydrogen atoms (Fig. 1). For a proper description of the aluminium-rich zeolites, we consider all possible Al for Si substitutions permitted by Loé wensteinœs rule, which forbids that Al atoms occupy neighbouring tetrahedral framework sites. The equilibrium structures of Fig. 1 Molecular models used for generating the database using the DFT B3-LYP method and the symmetry constraints applied in structure optimizations.For simplicity the oxygen atoms and terminal OwH groups are omitted.44 Structure and reactivity of silica and zeolite catalysts all the models are determined by imposing symmetry restrictions. This speeds up the calculations and avoids the formation of non-realistic intramolecular hydrogen bonds involving terminal H atoms.The point group chosen is a compromise between a small number of imaginary vibrational frequencies and a high order of the symmetry group. In two cases, cyclotetrasilicic and cyclopentasilicic acid the optimized structures (S4) (D5), prove true minima by frequency calculations. For all the equilibrium structures, the force constant matrix is calculated and used as data in the potential-–tting procedure.The only exception is Al-cyclohexasilicic acid as its low symmetry and large size (36 (Cs) atoms) do not allow computation of force constants analytically. In addition, a set of ca. 60 distorted structures is generated by distorting the internal coordinates, starting from the equilibrium structures. The magnitude of typical distortions is ^10 pm for bond distances and ^4° for bond angles.For the resulting structures the DFT B3-LYP gradients are computed. The entire database used for the –tting procedure consists of a set of 10 equilibrium structures (coordinates and zero gradients) together with their harmonic force constants and a set of 60 distorted structures (coordinates) along with their gradients. 2.3 Parameter –tting We apply the same procedure as used previously by Schroé der and Sauer.1 DiÜerent ion types are assumed for the hydrogen and oxygen ions of bridging and terminal hydroxy groups and O, respectively) and diÜerent short-range parameters are –tted (Hb, Ob Ht , for the interactions between them.To obtain a well balanced parametrization, diÜerent weights are given to diÜerent types of data.The DFT force constants are weighted with a factor 5]10~4 since their number is ca. 2000 times larger than the number of corresponding gradient components. This results in parameters which combine good predictions of structures with good predictions of vibrational spectra. To limit the in—uence of the terminal hydroxy groups, which are not present in the periodic structures, all data connected with them are weighted with a factor 0.1.All potential parameters are –tted simultaneously by the least-squares method. The –nal values are listed in Table 1. 2.4 Lattice-energy minimization Constant pressure minimizations of the lattice energy are performed with respect to positions of the ions, shells and the unit cell parameters. A cut-oÜ radius of 10 is ” chosen for the summation of the short-range interactions.For some of the equilibrium structures we determine properties such as static constants and relative permittivities, as well as the harmonic phonon frequencies. All calculations are performed using the GULP16 and METAPOCS17 codes. 2.5 Combined quantum chemicalñinteratomic potential approach The embedding scheme used18 partitions the entire system (S) into two parts : the inner part (I) usually containing the site in question e.g.the site, and the outer part Br‘nsted (O). The energy of the combined systems can be decomposed as : E(S)\E(I)]E(O)]E(I»O) (1) where E(I»O) is the interaction between the inner and the outer region. The energy of the outer part, E(O), and the interaction between the inner part and the outer part, E(I»O), are approximated by the potential : E(S)BEQM(I)]EPot(O)]EPot(I»O) \EQM(I)]EPot(S)[EPot(I) (2)M.Sierka and J. Sauer 45 Table 1 Parameters of the DFT B3-LYPderived shell-model potential charges/e core shell Si 4.0a » Al 3.0a » O 1.228 58 [3.228 58 Ob 0.817 53 [2.817 53 Hb 1.0a » Ht 1.0a » short-range repulsion A/eV o/” SiwO 1612.459 20 0.299 55 SiwOb 997.880 97 0.332 12 AlwO 1395.774 63 0.304 49 AlwOb 1644.881 77 0.291 39 ObwHb 368.648 03 0.225 11 OwHb 7614.580 03 0.199 13 OwHt 772.068 14 0.185 24 core»shell interaction k/eV ”~2 O 122.478 53 Ob 70.151 23 three-body interaction kb/eV rad~2 H0/rad OwSiwO 0.144 703 109.47a ObwSiwO 0.384 711 109.47a OwAlwO 0.893 930 109.47a ObwAlwO 0.686 678 109.47a a Not adjusted in the –tting procedure.Hence, the energy of the total system is obtained approximately from the QM energy of the internal part and the diÜerence in the energies of the total system and the internal part, calculated by the interatomic potential function. For partially covalent solids, such as zeolites, the partitioning creates dangling bonds which need to be saturated by hydrogen atoms, called link atoms.The inner part (I) together with link atoms (L) form the cluster (C). Now we have: E(S)\EQM(C)]EPot(S)[EPot(C) (3) only if : D\EQM(L)]EQM(I»L)[EPot(L)[EPot(I»L)\0 (4) where and are the QM and potential function interaction energies EQM(I»L) EPot(I»L) between the inner part and link atoms. Hence, the energy of the total system can only be obtained from the subtraction scheme, eqn.(3), if D is small and negligible. This is46 Structure and reactivity of silica and zeolite catalysts expected to be the case if the embedding potential is parametrized using the same quantum mechanical method as used for the cluster. The diÜerentiation of the above expression leads to the following formulae for the energy gradients : Fa(S)\Fa, QM(C)]Fa, Pot(S)[Fa, Pot(C) ; a ½ I (5) Fb(S)\Fb, Pot(S) ; b ½ O (6) For the atoms present in the inner region [a, eqn.(5)] all three components contribute, while forces acting on the atoms of outer region [b, eqn. (6)] are calculated from the interatomic potential function only. The link atoms are not independent variables in the structure optimization but are kept at a –xed distance from atom X on the bond XwY they terminate : E\E(rX , rH)\E[rX , rH(rX)] (7) where the atoms X and Y belong to the inner and outer part, respectively. Hence, the forces on the atoms X and Y have to be modi–ed:18 dE drX, m \ LE LrX, m ] LE LrH, m LrH, m LrX, m (8) The present implementation18 of the embedding scheme describes the outer part by a shell-model ion-pair potential.It couples the GULP program16 for the lattice energy minimization with TURBOMOLE14 and TURBODFT9 programs for the quantum mechanical calculations.When the HF method is applied to the cluster the HF-derived potential is used for the outer part, for the DFT calculations we apply our DFT-based shell-model potential. The hydrogen link atoms of the cluster models are kept at a –xed distance from the oxygen atoms they terminate.These distances are the optimum distances found in free cluster calculations. The (Si)OwH distances are 94.50 pm for the HF and 96.66 pm for the DFT calculations, and the (Al)OwH distances are 94.00 pm for HF and 96.28 pm for DFT calculations. All combined QM»Pot optimizations are performed without any symmetry constraints, i.e. the space group is assumed. P1 However, the unit cell parameters are –xed to values found by a lattice-energy minimization using the embedding shell-model potential alone.In the combined QM»Pot scheme the atoms are relaxed until the largest cartesian gradient component and the RMS gradient are less than 0.0001 Eh a0~1. The OwH stretching frequency is calculated using 5 points (step size 0.3 pm) on the combined QM»Pot potential energy surface along the bridged OwH bond.19 2.6 Deprotonation energy The calculation of deprotonation energies, i.e.the energy of the reaction : ZwOH]ZwO~]H` requires application of the combined QM»Pot scheme to the protonated and deprotonated acidic sites. If the QM cluster is large enough then all short-range terms resulting from the interatomic potential cancel approximately and the total reaction energy can be written as :20 *EQMhPot\*EQM@@QMhPot]*ELR@@QMhPot (9) The –rst term describes the ìmechanicalœ part arising from the fact that the structures of the embedded clusters are diÜerent from free space clusters.The second term describes the correction due to the crystal potential.M. Sierka and J. Sauer 47 Two problems arise when applying periodic boundary conditions to a deprotonation reaction.19 First, the calculation of the energy of the negatively charged faujasite framework is not possible using the traditional Ewald summation technique because of the in–nite Coulomb repulsion of charged unit cells.This can be avoided by applying a neutralizing homogeneous background charge distribution.21,22 Second, by taking the diÜerence between QM»Pot energies of the neutral and deprotonated zeolites, one obtains the energy for removing the proton from every unit cell.We are interested in the proton removal from a single acidic site. This can be obtained by removing the interaction between the charged defects in the anionic unit cells. We follow the method proposed by Leslie and Gillan.21 Details about the implementation are given elsewhere. 19 3 Results and Discussion 3.1 Structure and properties of silicates To test the quality of the DFT-derived potential we determined the equilibrium structures of a selected set of silica modi–cations. Table 2 shows the unit cell parameters obtained and compares them with predictions by two parameter sets derived previously from empirical5 and ab initio HF data.1 The cell parameters predicted by the present DFT-derived potential deviate by 1.4% on average from the observed data.This is a slight improvement compared with the HF-derived shell-model potential (1.9%). However, for the former, the deviations from the observed values are less consistent. In the case of a-quartz, the –rst cell parameter is smaller than the experimental one, while Table 2 Cell parameters of microporous and dense modi–cationsa,b SiO2 empirical HF shell DFT shell modi–cation shell modelc modeld model obsd.e faujasite A\B\C 2423 2463(1.5) 2466(1.6) 2426 sodalite A\B\C 882 895(1.4) 885(0.2) 883 theta-1 A 1382 1414(2.0) 1420(2.5) 1386 B 1739 1777(2.0) 1768(1.5) 1742 C 500 516(2.4) 502(0.4) 504 mordenite A 1802 1830(1.1) 1812(0.1) 1810 B 2004 2049(0.5) 2069(1.5) 2038 C 743 758(1.2) 762(1.7) 749 ZSM-5 A 1998 2043(1.6) 2020(0.4) 2011 B 1974 2021(1.7) 1996(0.4) 1988 C 1332 1363(1.9) 1348(0.8) 1337 90.8 90 90.7 90.7 zeolite RHO A\B\C 1477 1501(1.1) 1505(1.3) 1485 a-quartz A\B 484 499(1.4) 489(0.6) 492 C 535 551(1.8) 545(0.7) 541 b-quartz A\B 500 510(2.0) 511(2.2) 500 C 550 562(2.9) 560(2.6) 546 a-cristobalite A\B 497 513(3.0) 498(0.0) 498 C 701 727(4.6) 717(3.2) 695 b-cristobalite A\B\C 731 739(3.1) 746(4.0) 717 mean dev.f 0.7 1.9 1.4 a Units: pm.b Percentage deviation from observed data is given in parentheses. c Ref. 5. d Ref. 1. e Faujasite : ref. 23, sodalite : ref. 24, theta-1 : ref. 25, mordenite: ref. 26, ZSM-5: ref. 27, zeolite rho: ref. 28, a-quartz : ref. 29, b-quartz : ref. 30, a- and bcristobalite : ref. 31. f Mean percentage deviation from observed data.48 Structure and reactivity of silica and zeolite catalysts the HF-derived potential gives cell parameters slightly larger for all the structures. The predictions of the empirical shell-model potential are in better agreement with experimental data (only 0.7% average deviation).A further test of the potential is the comparison of calculated and measured elastic constants and relative permittivities of a-quartz (Table 3). The calculated elastic constants are of the same quality as those obtained with the empirical potential and the potential –tted to the HF data set. However, relative permittivities seem to be reproduced better with the present potential than with the HF parametrized one.As an example of the structure prediction using the DFT parametrized potential Table 4 compares observed and calculated bond lengths and angles for silica faujasite. The averaged calculated SiwO bond length is ca. 2.7 pm longer than the observed one. Table 3 Elastic constants and relative permittivities of a-quartz at 0 Ka empirical shell HF shell DFT shell modelb modelc model obsd.d C11 9.47 8.45 7.59 8.69 C33 11.61 9.63 10.56 10.60 C44 5.01 4.11 3.81 5.83 C66 3.82 3.65 2.61 3.99 C14 [1.45 [1.37 [0.79 [1.81 C13 1.97 2.39 2.89 1.19 C12 1.84 1.15 2.36 0.70 e11 4.74 4.07 4.47 4.52 e33 5.01 4.42 4.65 4.64 e= 2.12 1.76 1.86 2.40 a Units: elastic constants in 1010 N m~2.b Ref. 5. c Ref. 1. d Ref. 29. Table 4 Bond lengths and SiwOwSi bending angles in pure silica faujasite optimized with present DFT B3-LYP-derived potentiala,b empirical HF shell DFT shell shell modelc modeld model obsd.e r(SiwO1) 161.4(0.7) 162.2(0.9) 163.7(1.9) 160.7 r(SiwO2) 159.9(0.2) 160.9(0.8) 162.8(1.9) 159.7 r(SiwO3) 160.9(0.5) 161.7(0.8) 163.7(2.1) 160.4 r(SiwO4) 160.8(0.6) 161.6(0.1) 163.1(1.1) 161.4 average 160.8 161.6 163.3 160.6 mean dev.f 0.50 0.65 1.8 n(SiwO(1)wSi) 138 141(2.2) 139.1(0.6) 138 n(SiwO(2)wSi) 149 154(3.4) 146.9(0.9) 149 n(SiwO(3)wSi) 146 152(4.1) 144.2(0.8) 146 n(SiwO(4)wSi) 141 145(2.8) 143.2(1.0) 141 mean dev.f 0.0 3.1 0.8 a Units: distances in pm, angles in degrees.b Percentage deviation from observed data is given in parentheses. c Ref. 5. d Ref. 1. e Ref. 23. f Mean percentage deviation from observed data.M. Sierka and J. Sauer 49 The calculated bond angles agree very well with the experimental values. The average deviation, 0.8%, is lower than for the HF-derived potential, which overestimates the angles by more than 3%. Thus, the DFT-parametrized shell-model potential provides an excellent description of the SiwOwSi bond angles, but systematically overestimates SiwO bond distances.In contrast, the HF-parametrized shell-model potential gives good bond distances, but systematically too large bond angles. Do these diÜerences re—ect the eÜect of electron correlation on the potential-energy surface (PES) or are they a result of the representation of the PES by a parametrized shell-model potential ? To answer this question we performed test calculations on the disiloxane, and H6Si2O, octahydridosilasesquioxane molecules, because they contain the character- H8Si8O12 , istic SiwOwSi bond and both experimental and computational data are available for comparison. 3.2 DFT description of the SiwO bond 3.2.1 Octahydridosilasesquioxane. The highly symmetric molecule of octahydridosilasesquioxane (Fig. 2) is a model of the double four ring (D4R) H8Si8O12 found, e.g. in zeolite A. Its structure has been determined experimentally32 in the molecular crystal and the vibrational spectrum is also known.33 The equilibrium structure belongs to the point group32 but shows only small distortions from the symmetry Th Oh due to the crystalline environment. This allows one to perform calculations for the higher symmetry structure and thus to simplify the computational task.Structural (Oh) data predicted at the MP2 and HF level are available34 that used the same basis set as the present DFT study. Appropriate structural parameters are listed in Table 5. The SiwO distance predicted by DFT is ca. 3 pm too long, while the SiwOwSi bond angle deviates by only 1.5° from the observed one.The HF calculations yield an SiwO distance which deviates by less than 1 pm from the observed one, but the predicted angles are too large by ca. 3°. This is the same pattern as observed for the structures of the silica polymorphs predicted with the DFT- and HF-parametrized potentials. This points to an intrinsic eÜect of electron correlation. This is further supported by the MP2 results, which are virtually identical with the B3-LYP results.Fig. 2 Octahydridosilasesquioxane molecule, H8Si8O1250 Structure and reactivity of silica and zeolite catalysts Table 5 Structural parametersa of octahydridosilasesquioxane calculated by various methodsb,c method/basis set ref. r(SiO) n(SiOSi) n(OSiO) obsd. 32 161.7»162.0 147.5»147.6 109.4»109.7 MP2/T(O)DZP 34 164.5(2.8»2.5) 148.8(1.3»1.2) 109.3([0.1 to[0.4) DFT-B3LYP/T(O)DZP 164.8(3.1»2.8) 149.1(1.6»1.5) 119.1([0.3 to[0.6) HF/T(O)DZP 34 162.6(0.9»0.6) 150.5(3.0»2.9) 108.4([1.0 to[1.3) a Units: bond distances in pm, bond angles in degrees.symmetry assumed. c Deviation b Oh from experiment given in parentheses. Fig. 3 shows a simulated IR spectrum, based on DFT B3-LYP results, in comparison with the experimental one.33 Only the six modes belonging to the symmetry group T1u are IR active, however, the small distortion of the molecule from the sym- H8Si8O12 Oh metry gives rise to the additional bands present in the experimental spectra.The predicted intensity of the d(OwSiwH) bending mode is approximately two times lower than the corresponding intensity of the stretching mode, while the experimen- las(SiwOwSi) tal spectrum shows lower intensity for the latter.The intensity of the symmetric mode is also underestimated. Table 6 lists all vibrational frequencies of ls(SiwOwSi) calculated at the DFT B3-LYP level in the harmonic approximation and H8Si8O12 compares them with the experimental IR and Raman frequencies.33 The agreement between observed and predicted frequencies is excellent.The only exceptions are the SiwH stretching modes. The DFT-calculated frequencies are shifted towards higher values by 60»84 cm~1. This is connected with the high anharmonicity of the SiwH bond. We conclude that, despite the too long SiwO bonds, we may expect that the shell-model potential parametrized on DFT data will yield good vibrational spectra.Fig. 3 IR spectrum of octahydridosilasesquioxane. Experimental (upper)33 and calculated with the DFT-B3LYP method in harmonic approximation (lower). A gaussian band shape has been assumed.M. Sierka and J. Sauer 51 Table 6 Experimentala and calculated with the DFT B3-LYP methodb IR- and Raman-active fundamentals of octahydridosilasesquioxane normal mode/type calculated of vibration experimentala (DFT B3LYP) A1g c l(SiwH) 2302 2367 ds(OwSiwO) 580 586 ls(SiwOwS) 456 439 Eg c ds(OwSiwO) 932 916 ls(SiwOwSi) 697 668 ds(OwSiwO) 423 419 d(OwSiwO) 84 61 T2g c l(SiwH) 2296 2357 las(SiwOwSi) 1117 1136 ds(OwSiwO) 890 889 ls(SiwOwSi) 610 590 das(OwSiwO) 414 400 ds(OwSiwO) 171 166 T1u d l(SiwH) 2277 2361 las(SiwOwSi) 1141 1145 das(OwSiwO) 881 879 das(OwSiwO) 566 559 ls(SiwOwSi) 465 456 ds(OwSiwO) 399 397 a Ref. 33. b Units: cm~1. c Raman active. d IR active. 3.2.2 Disiloxane. The question that remains is why including electron correlation deteriorates the agreement with observed SiwO bond distances ? For an answer we consider disiloxane. This is the simplest molecule, containing the H3SiOSiH3 , SiwOwSi bridge (Fig. 4). The SiwOwSi bending potential is very —at, which has consequences both for theoretically predicted and observed structures.For the former, it is not easy to get results which are converged with respect to basis set and method. For the latter, the results obtained will depend on the experimental conditions and the average structures observed make it difficult to infer an equilibrium structure for this molecule.The fact that the angle in disiloxane can apparently assume diÜerent values without a signi–cant increase in the total energy explains the rich variety of the structures found in silicates, silica and zeolites. Fig. 4 Disiloxane molecule, H6Si2O52 Structure and reactivity of silica and zeolite catalysts Baé r and Sauer35 made a careful study of the eÜect that basis set extension has on the disiloxane structure predicted by the HF and MP2 methods.Table 7 summarises these results. We include also DFT B3-LYP results from the present study. For the basis set used in this study (abbreviated ì1dœ in Table 7) we have the already known pattern : the DFT and MP2 bond distances are virtually identical and larger than the HF result (by ca. 2 pm in this case). If we extend the sp basis set and include a 2d1f set of polarization functions, the HF, DFT and MP2 bond distances decrease. Further extension of the basis set makes the HF result slightly smaller, but also reduces the MP2 distance. The MP2 result for the largest basis set (162.5^0.3 pm) is almost identical with the HF result for the standard basis sets [T(O)DZP]. From the parallel change in MP2 and DFT with the basis set size we expect that the same may be true for the DFT method. From these results we learn : (i) the HF/T(O)DZP method yields very good SiwO bond distances owing to a cancellation of neglected correlation eÜects and truncated basis sets ; (ii) methods including electron correlation need very large basis sets for predicting accurate bond lengths.With respect to the structure predictions using DFTparametrized shell-model potentials we conclude that the systematically too long SiwO bond lengths are due to the basis set used for generating the data base and not to representation of the PES by a simple shell-model interatomic potential. Unfortunately, extending the basis set to the extent required to yield accurate SiwO bond lengths is far beyond computational resources. 3.3 Simulations of IR spectra of silica polymorphs A reliable parametrization of the potential should be capable of modelling the dynamical properties of microporous silica. Silicalite, the aluminium-free form of zeolite ZSM-5, and silica faujasite are polymorphs with a broad range of SiwOwSi angles and SiwO bond lengths.Fig. 5 and 6 show their IR spectra calculated with the present DFTderived shell-model potential as well as with the HF-derived one and compare them with observed spectra. Both potentials reproduce the main features of the measured spectrum well. Every band is reproduced. Larger deviations are only found in the range 700»900 cm~1. Table 7 Equilibrium structures and barriers to linearization of disiloxane calculated by various methodsa method/basis set ref.n(SiOSi) r(SiO) *Elin obsd. 36, 37 144^0.8 163.4^1.0 B1.3 MP2(5d4f*)b 35 158.2 162.3 0.12 MP2(5d4f) 35 150.7 162.8 0.88 MP2(2d1f) 35 145.4 163.7 1.86 MP2(2d) 35 139.8 164.6 4.61 MP2(1d) 35 152.9 164.9 0.65 DFT-B3LYP(2d1f) 155.9 163.7 0.52 DFT-B3LYP(2d) 143.8 164.6 3.04 DFT-B3LYP(1d) 170.6 164.5 0.00 HF(4d3f) 35 180.0 160.7 0.00 HF(2d1f) 35 172.0 161.1 0.00 HF(1d) 35 180.0 162.6 0.00 a Units: distances in pm, angles in degrees, energies in kJ mol~1. b Polarisation functions are speci–ed in parentheses, for detailed description see ref. 35.M. Sierka and J. Sauer 53 Fig. 5 Experimental IR spectrum of silica-rich faujasite (a) and IR spectra calculated with the DFT shell-model potential (b) and (c), and the ab initio HF shell-model potential (d) ; (c) and (d) have been generated from the line spectra assuming a gaussian band shape 3.4 Space group change and phase transition The most intriguing diÜerence between the DFT- and HF-parametrized potentials concerns the structures they predict for silica sodalite and silicalite.For silica sodalite, equilibrium structures with diÜerent space groups are predicted by the two potentials.The structure found using the HF potential has a —at four-membered ring, i.e. the four oxygen atoms of the four-membered face of the sodalite cage lie in the same plane, and belongs to the Im3m space group. According to Baur38 such a ìregularœ or ìexpandedœ structure is observed when the sodalite cage is –lled by a template molecule.On the other hand, the structure obtained using our DFT-parametrized potential belongs to the I43m space group and has a ì tilted œ or ìcollapsedœ structure, two of the oxygens belonging to the four-membered face of the sodalite cage are inside and two outside the unit cell. This is the structure observed for the silica sodalite with no template in the sodalite cage.When the Im3m space symmetry group was imposed during optimization using the DFT-based potential the resulting structure proved to be a high-order stationary point with eight negative frequencies in the !-point phonon spectrum. On the other hand, using the HF-derived potential the initially ì tilted œ structure always converges to the higher-symmetry ìexpandedœ one. Silicalite shows a reversible phase transition39 at ca. 340 K from monoclinic symmetry 11)27 to orthorhombic symmetry (Pnma).40 The low-temperature mono- (P21/n clinic structure has been successfully reproduced by the empirical shell-model54 Structure and reactivity of silica and zeolite catalysts Fig. 6 Experimental IR spectrum of silicalite (a) and IR spectra calculated with the DFT shellmodel potential (b) and (c), and the ab initio HF shell-model potential (d) ; (c) and (d) have been generated from the discrete spectra assuming a gaussian band shape potential.41 The HF-–tted potential yields a minimum on the PES only for the hightemperature orthorhombic structure.1 We performed lattice energy minimizations for silicalite using the present DFT-based potential.We obtained several local minima.The two with lowest energy, [12 494.6 and [12 494.1 kJ mol~1 per unit, belong to the SiO2 same space groups as the experimental structures, (monoclinic) and Pnma P21/n11 (orthorhombic), respectively. That these are minima was checked by !-point phonon calculations. To estimate nuclear motion corrections we performed phonon calculations using 27 k-points sampling of the –rst Brillouin zone for the orthorhombic and 30 kpoints for the monoclinic structure. After including zero-point vibrational corrections the energy of the monoclinic structure is lower by 45 kJ mol~1 uc~1 than that of the orthorhombic one. We calculated the free energy, *F, of the reaction We may assume that the phase transition occurs at *F\0, silicalitemono ]silicaliteortho .which we –nd is the case at 865 K. This predicted value is considerably higher than the experimental value of 340 K.27 Quartz at 853 K shows a reversible transition from the a to the b form.42 The PES of quartz has two equivalent minima, which correspond to two twinned con–gurations, and related by a rotation of the tetrahedra around their axes. The a1 a2 , SiO4 C2 corresponding tilt angle, d, is negative in the phase and positive in the phase.a1 a2 b-quartz has a tilt angle of zero, which corresponds to a transition structure on the PES. This means that at high enough temperatures ([853 K) the tilt angle samples positive and negative values and its average is zero (b-quartz). At temperatures below this pointM. Sierka and J. Sauer 55 the system is trapped into one of the minima with either a positive or a negative (a1) (a2) average tilt angle.We performed lattice-energy minimizations for both the a and b form of quartz using three diÜerent parametrizations of the shell-model potential. In all three cases, the a structure has lower energy than the b-structure, in accord with the doubleminimum shape of the PES.Indeed, the !-point phonon frequencies calculated for bquartz using the three parametrizations of the shell-model potential show one signi–cant negative value (transition structure). Based on phonon calculations we can estimate the free energy of the reaction a-quartz]b-quartz. Fig. 7 shows the dependence of the *F value on temperature. Assuming that the transition occurs for F(b)[F(a)\0 the empirical and our DFT-derived potential give a transition temperature of 1400 K.The HF-parametrized potential yields a much lower transition temperature of 540 K. As far as the comparison of the HF- and DFT-parametrized potentials is concerned, a uniform picture emerges for all the three systems studied : the HF gives undue favour to highly symmetric structures. For quartz, the energy diÜerence between b and a is too small ; for silicalite, the low-symmetry monoclinic structure is not a local minimum; for sodalite, an equilibrium structure is predicted that seems to be not even a local minimum on the ìtrueœ PES.It seems that electron correlation makes qualitative changes in the PES of silica polymorphs. It is encouraging that this feature is retained in the DFT-parametrized shell-model potential, even if there is room for quantitative improvement.This follows from the estimated temperatures for the a»b transition in quartz (1400 K) and for the monoclinic»orthorombic transition in silicalite (865 K) which are signi–cantly higher than the observed values (853 and 340 K, respectively). Our naive scheme to get a –rst approximation of the transition temperature may also contribute to this deviation, together with neglected anharmonicity eÜects.We would Fig. 7 Temperature dependence of the free energy of the a to b quartz transition calculated with various shell-model potential parametrizations56 Structure and reactivity of silica and zeolite catalysts like to stress that a qualitatively correct PES is obtained only after including electron correlation. 3.5 DFT shell-model potential results for protonated zeolites The modelling of active protonated forms of zeolites is especially attractive since many details of their structure are not known from experiments and reliable predictions of IR frequencies and NMR chemical shifts can settle assignment problems. In particular, the absolute acidity of the active sites is not known from experiments.Br‘nsted In the silicon-rich H-faujasite all the tetrahedral sites (T atoms) are crystallographically equivalent, but there are four diÜerent oxygen positions associated with each tetrahedron, Only two of the four possible OH groups, and have O1»O4. O1H O3H, actually been detected in the IR and 1H NMR spectra.Table 8 shows the relative energies and OH harmonic stretching frequencies calculated for isolated bridging hydroxy groups in the primitive cell of faujasite (one Al per 48 T sites). Results obtained using other parametrizations of the shell-model potential are also included for comparison. In agreement with observations, all three potentials predict that the and bridg- O1H O3H ing groups are energetically preferred compared to the remaining two.However, the present DFT potential yields the smallest energy gap between the and O1H O2H groups. This is in agreement with neutron diÜraction data which indicate a small amount of groups in protonated zeolite Y.43 Also in agreement with experimental O2H assignments, all three potentials predict the stretch vibration as the low-frequency O3H (LF) IR band and the stretch vibration as the origin of the high-frequency (HF) O1H band (Table 8).Harmonic frequencies calculated with the DFT B3-LYP derived potential underestimate the observed frequencies by ca. 65 cm~1. The experimental44 diÜerence between the HF and LF bands (fundamental, 73; harmonic, 80 cm~1) (O1H) (O3H) is reproduced by the DFT-parametrized potential (79 cm~1), while the other two potentials underestimate it slightly.The empirical and HF potentials predict 36 cm~1 and 22 cm~1, respectively. Table 9 shows the bond distances and angles predicted by diÜerent potentials for the and bridging hydroxy groups in faujasite. The SiwO(H) and AlwO(H) bond O1H O3H lengths calculated with the DFT potential are ca. 1 pm longer for the group than O1H for the group and are ca. 1 pm longer than in structures obtained using the HF O3H potential. From 1H MAS NMR sideband patterns AlwH distances of 248^4 and 240^4 pm for and groups, respectively, have been deduced.45 These O1H O3H compare well with the predictions of our shell-model potential (246 and 240 pm, respectively, Table 9). Lattice-energy minimizations employing the present potential were also performed for a model of H-faujasite with an Si/Al ratio of 2.43, typical for zeolite Y.A sample of Table 8 Relative energies and OH harmonic stretching frequencies of the bridging hydroxy groups in silicon-rich H-faujasitea *Erel lOH emp. shell HF shell DFT shell emp. shell HF shell DFT shell obsd.e obsd.f modelb modelc model model modeld model harm.anharm. O1H 5.3 8.8 11.1 3772 3752 3723 3787 3623 O2H 19.8 18.0 17.1 3702 3628 3602 » » O3H 0.0 0.0 0.0 3736 3694 3644 3707 3550 O4H 23.7 27.9 24.7 3751 3697 3673 » » a Units: energies in kJ mol~1, frequencies in cm~1. b Ref. 5. c Ref. 1. d Frequencies scaled by factor 0.9. e Ref. 44. f Ref. 48.M. Sierka and J. Sauer 57 Table 9 Structural parameters of the four bridging hydroxy groups in silicon-rich faujasitea r[AlwO(H)] r[AlwH] r[SiwO(H)] r(OH) n[SiwO(H)wAl] n(SiwOwH) SAlwOT SSiwOT O1H free cluster (DFT) 194.6 240.0 172.4 97.48 142.2 112.2 178.9 164.1 DFT shell model 189.8 246.0 170.7 97.05 129.8 116.0 176.2 163.8 M4T-1AlN embedded (DFT)b 190.4 250.0 172.0 97.54 123.3 118.5 176.9 164.6 free cluster (HF) 195.4 238.8 167.9 95.51 135.4 118.4 177.2 162.4 HF shell modeld 191.4 247.6 169.6 95.4 130.2 114.5 175.8 162.3 M4T-1AlN embedded (HF)b 190.4 248.2 170.1 95.55 125.6 117.8 175.6 162.8 empirical shell modelc 191.0 238.6 169.4 100.0 131.1 123.0 173.8 160.3 O3H DFT shell model 193.5 240.2 169.3 97.87 136.5 117.1 176.6 163.9 empirical shell modelc 193.0 233.2 169.7 100.2 138.7 120.7 174.2 160.7 HF shell modeld 194.7 239.7 169.1 96.1 140.2 113.8 176.4 162.6 a Units: distances in pm, angles in degrees.b Ref. 55. c Ref. 5. d Ref. 1.58 Structure and reactivity of silica and zeolite catalysts the same module but with a small amount of sodium cations was studied by Czjzek et al.43 in a neutron powder diÜraction experiment. The model was constructed as described by Schroé der and Sauer,1 namely the Al ions were distributed over the lattice sites according to a model suggested by Klinowski et al.,46 and the protons were distributed among the diÜerent sites as which is close to the O1 : O2 : O3 : O4\8 : 2 : 4 : 0, population deduced from the neutron diÜraction data.The space group was Fd36 m assumed. Table 10 shows the mean TwO distances. The average TwO bond length is calculated for each of the four oxygen types, regardless of whether the particular oxygen is protonated or not.This is not the same type of averaging as that involved in the re–nement of the neutron diÜraction data. Nevertheless, a comparison of the neutron diÜraction results with the predictions made by the diÜerent potentials is useful. All the potentials predict the same sequence of average TwO bond distances : O1[O3[ which is caused by the diÜerent occupancies of the oxygen sites.The oxygen O2[O4 , site with the highest occupancy shows the largest mean TwO bond length. This is reasonable, since the protonation of an oxygen lengthens its TwO bonds. The observed increase of the mean TwO bond lengths compared with silica faujasite, between 2.2 for and 7.0 pm for is correctly described by the shell-model potentials.As already O4 O1, discussed (Section 3.2), the present DFT-parametrized potential overestimates the average TwO bond lengths by 2»3 pm. The prediction of the average TwOwT bond angles for the H-Y zeolite model is much better using the present potential than using either of the other two listed in Table 10.This is probably due to a large data set of highly aluminated structures used for the potential –tting. Contrary to the other potentials, the present one predicts correctly the order of mean TwOwT angles : O2[O4[ The diÜerence between observed and calculated values of the average O3[O1. TwOwT angle is least for the present DFT-derived shell-model potential. 3.6 Embedded cluster calculations for the hydroxy group in H-faujasite O1H Table 9 shows the structural parameters of the bridging hydroxy site in faujasite O1H calculated with diÜerent methods: DFT and HF free clusters, DFT- and HF-derived shell-model potentials, and –nally the combined QM»Pot method presented in Section 2.5.The cluster used in QM»Pot calculations is a four-membered aluminosilicate ring saturated with H atoms, Al-cyclotetrasilicic acid, 4T-1Al (Fig. 8). For both the HF and the DFT methods similar trends are observed on embedding the cluster in its crystal environment. The AlwO(H) distance is shortened and the AlwH non-bonding distance Table 10 Calculated and observed mean bond lengths and angles and unit cell parameter of zeolite H-Y (Si : Al\2.43)a empirical shell HF shell DFT shell modelb modelc model obsd.d Sr(O1wT)T 168.9(7.5)e 169.4(7.2) 170.5(6.8) 167.7(7.0) Sr(O2wT)T 163.0(3.1) 165.0(4.1) 166.4(3.6) 163.2(3.5) Sr(O3wT)T 165.6(4.7) 166.7(5.0) 168.3(4.6) 165.4(5.0) Sr(O4wT)T 162.3(1.5) 164.5(2.9) 165.6(2.5) 163.6(2.2) n(TwO1wT) 136([2) 138([3) 135([3) 136([2) n(TwO2wT) 152(3) 156(2) 149(2) 145([4) n(TwO3wT) 146(0) 150([2) 142([2) 140([6) n(TwO4wT) 143(2) 144([1) 144(1) 144(3) a 2484(61) 2522(59) 2520(54) 2477(51) a Units: distances in pm, angles in degrees.b Ref. 5. c Ref. 1. d Ref. 43. e Deviation from the values for pure-silica faujasite given in parentheses.M. Sierka and J. Sauer 59 Fig. 8 The 4T-1Al cluster model embedded in the faujasite lattice. The oxygen labels correspond to the diÜerent crystallographic positions.becomes longer, –nally giving similar structural parameters: 190 pm for AlwO(H) and 249^1 pm for AlwH. Similarly, for the SiwO(H) distance the diÜerence between the DFT and HF free cluster results of 4.5 pm is reduced to only 1.9 pm in the embedded clusters. The OwH distance changes only slightly when the embedding scheme is applied. The AlwO(H)wSi angle becomes smaller by 10»20° on embedding, probably owing to removal of symmetry constraints and eliminated repulsion between terminal link atoms.In Table 11 we list the deprotonation energies, OwH stretching frequencies and 1H NMR chemical shifts of the bridging hydroxy group in an H-faujasite O1H (Si/Al\47.0, 1 Al atom in the primitive cell) calculated using the combined QM»Pot Table 11 QM»Pot calculations : Propertiesa of the O1H bridging hydroxy group in high-silica H-faujasiteb DFT HF *EDP free cluster » 1212c QM//QM»Pot 1319 1348 LR//QM»Pot [121 [98 QM»Pot//QM»Pot 1198 1250 method correction 0^3 [46^2 1198 1204 vibrational correction [35 –nal value 1166^10 lOH f potential only 3723 4216d QM»Pot//QM»Pot 3646 4016e QM//QM»Pot 3729 4041 1dH 4.2 4.0 a Units: energies in kJ mol~1, frequencies in cm~1, NMR chemical shifts in ppm.b Ref. 55. c Ref. 52. d Ref. 1. e Ref. 19. f Harmonic frequencies.60 Structure and reactivity of silica and zeolite catalysts approach. The deprotonation energies are decomposed into the contribution from the quantum chemical cluster calculation (QM//QM»Pot) and a long-range electrostatic contribution from the shell-model potential (LR//QM»Pot) which includes corrections discussed in Section 2.6. Both are calculated for the structures of the parent and deprotonated systems at their respective equilibrium structures obtained with the combined QM»Pot approach.The DFT method yields lower values than the HF methods. Obviously, inclusion of electron correlation weakens the OwH bond. Sauer has suggested53 removal of the main body of the systematic error from calculated deprotonation energies by adding a constant value for a class of systems which is typical for a given method.This constant value is obtained as follows : the approximation/basis set is used to calculate deprotonation energies for related small molecules, methanol and silanol in this case, and comparison is made with accurate results, in this case obtained by Sauer and Ahlrichs.54 For the HF method with the T(O)DZP basis sets a correction of [46 kJ mol~1 was obtained.53 The DFT B3-LYP method, in combination with this basis set, yields deprotonation energies that are virtually identical with the accurate data.Hence, by fortunate error compensation, the correction is zero in this case.After applying these corrections the results of the two methods agree within 6 kJ mol~1. For comparison with experiment, we have to add nuclear motion corrections which have been estimated previously as [35 kJ mol~1.54 This yields a –nal estimate of ca. 1166^10 kJ mol~1 for both methods, which agrees well with the ìexperimentalœ value of ca. 1190 kJ mol~1 for H-Y and 1200 kJ mol~1 for HNaY.47 The combined QM»Pot approach yields absolute acidities.In another study we use this approach to simulate the dependence of the acidity (deprotonation energy) on the Si/Al ratio55 and the framework type.56 It is well known that the HF method yields stretching frequencies that are systematically too large, while the DFT method nearly reproduces observed values.In both cases, however, comparison is made between values calculated in harmonic approximation and observed frequencies including anharmonicities. Our results show the same pattern, the OH-stretching frequency from the embedded DFT calculations (QM»Pot//QM» Pot), 3646 cm~1, agrees well with the experimental value48 of 3623 cm~1, while the value obtained at the HF level, 4016 cm~1 is too large.When applying, however, the usual scaling factor of 0.9 for the HF frequencies, one obtains a reasonable result of 3614 cm~1. The frequency predicted by the DFT parametrized potential alone is almost equal to the value calculated for the free cluster at the structure found by the embedding calculation. As for the deprotonation energies, the long-range contribution is larger for the DFT method than for the HF method.The 1H NMR chemical shift for the bridging hydroxy group calculated at the DFT level, 4.2 ppm, agrees better with the experimental49 value of 4.4 ppm than the result of the HF calculations, 4.0 ppm. Note, however, that the prediction of converged values of NMR shifts requires larger cluster models for the calculation of the wavefunction. 4 Conclusions We have shown that the ion-pair shell-model potential with parameter sets derived from the data base obtained by quantum mechanical calculations yields reliable predictions for structures and properties of silica and zeolite catalysts. Our parametrization based on DFT calculations yields results which show typical eÜects of electron correlation : longer SiwO bonds and preference for less symmetric structures.The combined quantum mechanics»interatomic potential approach proved to be a powerful tool for simulations of catalytically active sites in periodic framework structures. We thank K-P. Schroé der and M. Braé ndle for providing unpublished results and useful discussion. We are also grateful to R. Ahlrichs (Karlsruhe) and J. Gale (London) forM.Sierka and J. Sauer 61 making available recent versions of the TURBODFT and GULP codes, respectively. This work has been supported by the ìFonds der Chemischen Industrieœ. References 1 K-P. Schroé der and J. Sauer, J. Phys. Chem., 1996, 100, 11043. 2 J-R. Hill and J. Sauer, J. Phys. Chem., 1995, 99, 9536. 3 J-R. Hill and J. Sauer, J. Phys. Chem., 1994, 98, 1238. 4 J. Purton, R.Jones, C. R. A. Catlow and M. Leslie, Phys. Chem. Miner., 1993, 19, 392. 5 R. A. Jackson and C. R. A. Catlow, Mol. Simul., 1988, 1, 207. 6 F. Haase and J. Sauer, J. Am. Chem. Soc., 1995, 117, 3780. 7 F. Haase and R. Ahlrichs, J. Comput. Chem., 1993, 14, 907. 8 D. A. Becke, J. Chem. Phys., 1988, 88, 1053. 9 TURBODFT: O. Treutler and R. Ahlrichs, J. Chem. Phys., 1995, 102, 346. 10 A.Schaé fer, H. Horn and R. Ahlrichs, J. Chem. Phys., 1992, 97, 2571. 11 M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R.Gomperts, R. L. Martin, D. J. Fox, S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, GAUSSIAN 94, Revision B.3, GAUSSIAN, Inc., Pittsburgh PA, 1995. 12 R. Ditch–eld, Mol. Phys., 1974, 27, 789. 13 M. Haé ser, R. Ahlrichs, H. P. Baron, P. Weis and H. Horn, T heoret. Chim. Acta, 1992, 83, 455. 14 TURBOMOLE: R. Ahlrichs, M. Baé r, M.Haé ser, H. Horn and C. Koé lmel, Chem. Phys. L ett., 1989, 162, 165. TURBOMOLE and TURBODFT are available commercially from MSI: San Diego, CA. 15 F. Haase and J. Sauer, J. Phys. Chem., 1994, 98, 3083. 16 GULP written and developed by J. D. Gale, Royal Institute and Imperial College, UK, 1992»1994. 17 METAPOCS, C. R. A. Catlow, A. N. Cormack and F. Theobald, Acta Crystallogr.B, 1984, 40, 195. 18 U. Eichler, C. K. Koé lmel and J. Sauer, J. Comput. Chem., 1996, 18, 463. 19 U. Eichler, J. Sauer and M. Braé ndle, J. Phys. Chem., submitted. 20 M. Braé ndle and J. Sauer, J. Mol. Catal., 1997, 119, 19. 21 M. Leslie and M. J. Gillian, J. Phys. C, 1985, 18, 973. 22 K. Fuchs, Proc. R. Soc. L ondon Series A, 1935, 151, 585. 23 J. A. Hriljac, M. M. Eddy, A. K. Cheetham, J.A. Donohue and G. J. Ray, J. Solid State Chem., 1993, 106, 66. 24 J. W. Richardson Jr., J. J. Pluth, J. V. Smith, W. J. Dytrych and D. M. Bibby, J. Phys. Chem., 1988, 92, 243. 25 B. Marler, Zeolites, 1987, 7, 393. 26 P. R. Rudolf and J. M. Garceç s, Zeolites, 1994, 14, 137. 27 H. van Koningsveld, J. C. Jansen and H. van Bekkum, Zeolites, 1990, 10, 235. 28 R. X. Fisher, W. H. Baur, R. D. Shannon, R. H. Staley, A. J. Vega, L. Abrams and E. Prince, J. Phys. Chem., 1986, 90, 4414. 29 L. Levien, C. T. Previtt and D. J. Weidner, Am. Miner., 1980, 65, 920. 30 H. Grimm and B. Dorner, J. Phys. Chem. Solids, 1975, 36, 407. 31 D. R. Peacor, Z. Kristallogr., 1973, 138, 274. 32 T. P. E. Auf der Heyde, H. B. Bué rgi, H. Bué rgy and K. W. Toé rnroos, Chimia, 1991, 45, 38. 33 M. Baé rtsch, P. Bornhauser, G. Calzaferri and R. Imhof, J. Phys. Chem., 1994, 98, 2817. 34 J-R. Hill and J. Sauer, unpublished results. 35 M. R. Baé r and J. Sauer, Chem. Phys. L ett., 1994, 226, 405. 36 A. Almenningen, O. Bastiansen, V. Ewing, K. Hedberg and M. Acta Chem. Scand., 1963, 17, Trêtteberg, 2455. 37 M. J. Barrow, E. A. V. Ebsworth and M. M. Hardig, Acta Crystallogr. B, 1979, 35, 2093. 38 W. H. Baur, J. Solid State Chem., 1992, 97, 243. 39 C. A. Fyfe, H. Strobl, G. T. Kokotailo, G. J. Kennedy and G. E. Barlow, J. Am. Chem. Soc., 1988, 110, 3373. 40 D. H. Olsen, G. T. Kokotailo, S. L. Lawton and W. M. Meier, J. Phys. Chem., 1981, 85, 2238. 41 R. G. Bell, R. A. Jackson and C. R. A. Catlow, J. Chem. Soc., Chem. Commun., 1990, 10, 782. 42 J. D. Axe and G. Shirane, Phys. Rev., 1970, 1, 342. 43 M. Czjzek, H. Jobic, A. N. Fitch and T. Vogt, J. Phys. Chem., 1992, 96, 1535. 44 K. Beck, H. Pfeifer and B. Staudte, Microporous Mater., 1993, 2, 1. 45 D. Fenzke, M. Hunger and H. Pfeifer, J. Magn. Reson., 1991, 95, 477. 46 J. Klinowski, S. Ramdas, J. M. Thomas, C. A. Fyfe and J. S. Hartman, J. Chem. Soc., Faraday T rans. 2, 1982, 78, 1025.62 Structure and reactivity of silica and zeolite catalysts 47 V. M. Mastikhin, I. L. Mudrakovsky and A. V. Nosov, Bruker Rep., 1989, 2, 18. 48 M. W. Anderson and J. Klinowski, Zeolites, 1986, 6, 455. 49 D. Freude, M. Hunger, H. Pfeifer, Z. Phys. Chem. Neue Folge, 1987, 152, 171. 50 J. Sauer and R. Ahlrichs, J. Chem. Phys., 1990, 93, 2575. 51 B. Winkler, M. T. Dove and M. Leslie, Am. Miner., 1991, 76, 313. 52 M. Braé ndle, unpublished results. 53 J. Sauer, in Modeling of Structure and Reactivity in Zeolites, ed. C. R. A. Catlow, Academic Press, London, 1992, pp. 183»216. 54 J. Sauer and R. Ahlrichs, J. Chem. Phys., 1990, 93, 2575. 55 M. Sierka, J. Datka and J. Sauer, in preparation. 56 M. Braé ndle and J. Sauer, J. Am. Chem. Soc., submitted. Paper 7/01492I; Received 3rd March, 1997

ISSN:1359-6640    

DOI:10.1039/a701492i

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 63»77 Computer simulations of organic solids and their liquid-state precursors A. Gavezzotti Dipartimento di Chimica Strutturale e Stereochimica Inorganica, Universitaœ di Milano, via V enezian 21, 20133 Milano, Italy Representative calculations on 2-pyridone in its crystalline, liquid and solvated form, show the capabilities and the shortcomings of computer simulation techniques for the description of organic molecular systems.Both ì static œ and ìdynamicœ computations, meaning without and with the explicit treatment of thermal motion, are employed. Relationships of liquid and solution systems to the structure and thermodynamics of the crystalline solids are investigated as far as possible, with regards to efficient and systematic crystal structure prediction from molecular structure.The thesis to be discussed here is that a reliable theory for the prediction of the solid-state structure of small organic molecules must depend on an accurate thermodynamic and kinetic evaluation of the pathways for evolution from disorder to order, including liquid structuring and nucleation. Dynamic methods, including thermal motion, are indispensable for the accomplishment of this task, which remains today a faraway goal in structural physical chemistry. 1 Introduction When Johann Joachim Winckelmann wrote his ìThoughts on the imitation of the works of the Greeks in painting and sculptureœ in 1775, he inspired the artistic tastes of his times, exalting the uncontaminated beauty of statues and temples as centuries had delivered them to him, with a polished appearance of crystalline whiteness.It was later discovered, that these works of art were originally adorned with appendages of all kinds, in vivid colours, which had faded with time. In a similar fashion, X-ray crystallography has been producing, over the last thirty years, a ìneoclassicœ view of molecular crystals, whereby a static framework structure is seen, as if impenetrable to the disruptive action of thermal motion.An X-ray diÜraction experiment has a timescale of days or, in more recent times, of hours, but always some 1016 times longer than the timescale of molecular motion. Structural chemistry, as it emerged and is still emerging from the use of crystallographic databases, is often oblivious of the massive time averaging that these data incorporate, and relies on sometimes quite evanescent structure»property relationships.Thermal motion is vital in understanding the thermodynamic and kinetic behaviour of chemical systems, but, surprisingly, thermodynamics and kinetics have been put aside more and more by structural chemists, in their dream of reconstructing equilibrium properties and reactivity from idealized atomic positions alone.Nowadays, several analytical techniques are capable of resolution on a molecular scale in the analysis of crystal properties. Important, if humble and sometimes underestimated, experimental information comes from calorimetry : although no detail is available on the nature of the energies and their distribution among relevant thermal modes, the enthalpies of phase change and the speci–c heats of the phases are directly comparable to calculated energies and their derivative.This paper is an essay in the application 6364 Computer simulations of organic solids Table 1 Structural and energetic parameters of 2-pyridone crystals (a) Calculated structures packing space density energy group a/” b/” c/” a/degrees b/degrees c/degrees /g cm~3 /kJ mol~1 P21 5.35 5.77 7.23 » 74 » 1.47 80.3 6.58 3.35 11.44 » 60 » 1.44 79.7 P1- 6.08 3.38 11.50 93 73 77 1.45 80.2 6.16 3.59 11.48 63 90 80 1.42 79.7 6.09 3.66 11.10 105 73 82 1.41 80.0 6.18 3.41 10.78 96 97 77 1.44 80.0 5.98 3.48 11.46 105 106 81 1.43 79.5 P21/c 11.43 3.36 13.18 » 120 » 1.44 79.5 9.94 3.35 13.18 » 95 » 1.44 79.5 11.32 5.88 6.87 » 71 » 1.46 81.2 P212121 13.78 5.37 5.76 » » » 1.48 81.6 18.89 6.63 3.56 » » » 1.42 79.5 Pbca 11.07 5.19 15.73 » » » 1.40 77.6 11.41 11.03 7.26 » » » 1.38 75.6 (b) Experimental structures, P212121 T a/” b/” c/” a/degrees b/degrees c/degrees room 13.645 5.900 5.692 » » » 1.379 77.6 120 K 13.564 5.795 5.604 » » » 1.434 78.2 relaxed 13.82 5.41 5.75 » » » 1.47 81.4 of some computer techniques being developed for the analysis of molecular crystalline properties and, more generally, of the phase behaviour of organic substances.Beginning with crystal packing analysis,1 and an evaluation of packing energies,2 one may proceed to the construction of possible crystal structures starting from molecular structures,3 given suitably calibrated intermolecular potentials4 (Ref. 1»4 provide an ample historical perspective on these methods.) As will be discussed, the signi–cance of computer studies of organic solids is largely enhanced through coupling with simulations of the concomi- Fig. 1 The crystal structure of 2-pyridone7 (space group Oxygen atoms are in black. P212121). Solid lines denote hydrogen bonds (O… … …H distance 1.811 CwO… … …H and O… … …HwN angles 135 ”, and 160°, CwO… … …HwN torsion 109°).A. Gavezzotti 65 tant liquid (either pure or solution) states.Primarily because of this, molecular dynamics is the most appropriate computational tool,5,6 as in principle it should encompass both the thermodynamics and the kinetics of the bulk systems. However, it has its pitfalls : a macroscopic one being that the entropy is not directly accessible.Merits and other shortcomings will also become apparent from our results presented here, no doubt providing a prominent topic for the ensuing Discussion. A suitable candidate molecule to illustrate the capabilities of the above-mentioned techniques is 2-pyridone [2(1H)-pyridinone], as easily parameterizable system, for which a certain amount of structural and thermodynamic information is available.This paper will hopefully enable the reader to evaluate what can nowadays be done, using a computer, to describe and predict the phase behaviour of small organic molecules, and, at the same time, highlight the (many) points that still have to be clari–ed and systematized before these methods can be con–dently incorporated in the physical chemistœs toolbox. 2 2-Pyridone : structural and thermochemical data 2-Pyridone (mp\380 K, bp\553 K) can exist in either the lactam, 1, or lactim, 2, tautomeric forms, the former being found in the solid state7 and favoured by solvation.8 All our calculations and simulations refer to the lactam form, as the activation energy for isomerization is in the 26»52 kJ mol~1 range.8 The heat of combustion has been measured,9 and the ensuing calorimetry showed that the enthalpy diÜerence between the two gas-phase tautomers (2.6 kJ mol~1) is within the experimental uncertainty of the measured sublimation enthalpy (86.6 kJ mol~1).The vibrational frequencies and ideal gas thermodynamic functions of 2-pyridone are also known.10 Scheme 1 The formation of a hydrogen-bonded, cyclic centrosymmetric dimer seems an obvious option for the lactam form, boundary conditions allowing ; dimerization energies have been calculated (by CNDO) as 43.4 kJ mol~1,11 or measured as 35.9 or 65.3 kJ mol~1 in benzene;12 the discrepancy between these last values re—ecting the awkwardness of the measurement. Despite this, the crystal structure of pyridone7 (Table 1 and Fig. 1) adopts the so-called catemer motif, an in–nite chain of hydrogen-bonded molecules in a non-centrosymmetric space group. 3 Static calculations ; packing energies A molecular model for 2-pyridone was taken from the neutron diÜraction study performed at room temperature7 (Cambridge Structural Database ref. code PYRIDO02), renormalizing the NwH distance (experimental 1.03 at 1.00 as required for many ”) ”, subsequent applications : the molecule is —at.The packing energy calculated by standard potential parameters4 is 6% lower than the experimental sublimation enthalpy. The structure was relaxed13 by optimizing cell parameters and molecular orientation under the action of the potentials ; the only signi–cant drift was a 8% shortening in the b cell parameter (Table 1). 4 Static calculations ; crystal structure ìpredictionœ? The PROMET computer package3,14 was used. Brie—y, the procedure consists of assuming a rigid molecular model, building from it a few basic dimeric or oligomeric66 Computer simulations of organic solids aggregates (using inversion centres or screw axes), and then combining these units using the action of other symmetry operators, eventually packing the molecule into the most common space groups for organic compounds: P1-, Pbca.Aggre- P21, P21/c, P212121, gation energies (including hydrogen bonding) are calculated by standard optimized intermolecular potentials4 in the 6-exp form, i.e., without explicit coulombic contributions. The merits and pitfalls of such a parameterization have by now been discussed ad nauseam.1h4 A centrosymmetric dimer was –rst constructed (Fig. 2). Its binding energy, 38 kJ mol~1, falls within the boundaries of experimental values. This dimer was used as a starting building block for P1-crystal structures, it was combined with a screw axis to generate layers and structures, and it was combined with two screw axes to access P21/c space group Pbca.A molecular ribbon along a screw axis was used as the building block for crystal structures, and, after combination with a second screw axis, for P21 P212121 crystal structures. After extensive searches of the crystal energy hypersurface, a few thousand generated crystal structures were grouped and sorted according to space group symmetry and similarity in packing mode and cell dimensions, after Niggli cell reduction. Table 1 collects the most cohesive.The centrosymmetric dimer packs in two diÜerent space groups [Fig. 3(a) and 3(c)] with distortion to a ladder arrangement with diÜerent step heights (from 0 to ca. 2 ”) ; similar arrangements are obtained along a screw axis [Fig. 3(b) and 3(d)]. The second, radically diÜerent packing mode (Fig. 4) stems from the catemer motif, in space groups and the X-ray crystal structure has the packing mode of Fig. 4(b), and that P21 P212121; structure was promptly identi–ed by the search procedure, the eÜectiveness of which is thus again con–rmed. A third packing mode is the benzene-like structure of Fig. 5, in space group Pbca, where the centrosymmetric dimer packs essentially undistorted from the starting structure of Fig. 2. The energy results (Table 1) are, as usual, puzzling. An optimistic viewer might claim that complete success in structure prediction has been achieved, since the X-ray crystal structure is the most cohesive among those identi–ed in the search. However, the packing energies of the generated crystal structures (Fig. 6) form a continuum (as usual15,16) between 70 and 80 kJ mol~1, and polymorphic forms are experimentally known to diÜer in energy by as much as 5»10 kJ mol~1.17 Besides, energy diÜerences, Fig. 2 The planar centrosymmetric dimer generated in the PROMET search (O… … …H distance 1.832 CwO… … …H and O… … …HwN angles 125 and 175°) ”,A. Gavezzotti 67 Fig. 3 Dimer or pseudo-dimer packing motifs in the crystal structures generated during the PROMET search (Table 1).Solid lines : hydrogen bonds, lengths of which are given (below) in parentheses in (a) P1-structure (1.857), dashed line : translation of 3.59 (b) structure ”. ”; P21 (2.044 and 2.067), screw axis running perpendicular to the molecular planes ; dashed line : translation of 3.55 (c) structure (1.873), CwO… … …HwN torsion angle 137°; (d) struc- ”; P21/c P212121 ture (1.873), dashed line : pseudo-hydrogen bond, 2.54 ”.Fig. 4 Catemer motifs for structures generated in the PROMET search (codes as in Fig. 3). (a) P21 structure (1.868) ; (b) structure (1.859), virtually indistinguishable from the X-ray crystal P212121 structure of Fig. 1.68 Computer simulations of organic solids Fig. 5 Pbca crystal structure generated by the PROMET search (Table 1).Centrosymmetric dimers are nearly coplanar with a O… … …H distance of 1.828 ”. even in the sorted structures of Table 1, are in the range of fractions of 1 kJ mol~1, while the energetic resolution of the entire method is far from being that high. A more realistic interpretation is that at least –ve crystal structures in four diÜerent space groups must be considered equally probable on purely energetic grounds.In such a situation, dynamic (thermal and kinetic) factors at the –rst stages of molecular recognition and of Fig. 6 Crystal structures generated in the PROMET search in –ve space groups. For graphic clarity, structures have been considered equal if the energy diÜerence was less than 0.5 kJ mol~1 and the cell volume diÜerence was less than 0.1 irrespective of symmetry.The subpopulation ”3, at high density and low packing energy is composed of (unrealistic) non-hydrogen bonded crystal structures.A. Gavezzotti 69 crystal nucleation must be of paramount importance in making the –nal decision on which structure is most likely eventually to appear, and that is the reason why simulations of liquid precursor phases must be extremely important.For the above reasons, the whole matter is still in epistemological limbo18 (we donœt know what we do know and what we donœt know). Much thought will be needed before crystal structure prediction can be said to be a realistic possibility, or to reconcile the above results with ideas put forward by Desiraju in a recent paper19 on crystal engineering (ì mischief, thou art afootœ). 5 Molecular dynamics simulations The GROMOS package20 has been used to perform energy minimization (EM) and molecular dynamics (MD) calculations on the pyridone dimer in the gas phase and in solution, and on the crystalline and liquid phases. GROMOS non-bonded CCl4 Lennard-Jones functions were used, with and CwH groups in the united atom CCl4 approximation; CxO and NwH fragments were described as one charge group each, with a charge separation of 0.38 and 0.28 e, respectively.Bond distances were constrained to experimental values (SHAKE algorithm). Angle bending force constants were all set equal to 200 kcal mol~1 rad~2, and equilibrium angles were the experimental values. A combination of standard GROMOS proper and improper dihedral potentials was used.No intramolecular non-bonded contributions were included. The intramolecular force –eld is anyway scarcely relevant, and is only a means of preventing distortions from a rigid planar conformation. Periodic boundary conditions were imposed, with constant temperature and pressure restraints (coupling constants of 0.1 and 1 ps, respectively, unless otherwise speci–ed). The cut-oÜ distance in all energy calculations was 10 presumably just enough to ensure proper convergence of electrostatic terms in ”, the charge-group approximation (such a cut-oÜ would be hopelessly too drastic in the point-charge approximation). 5.1 The gas-phase dimer The gas-phase dimer cohesive energy at 1 K (i.e., in the absence of thermal energy) is 38 kJ mol~1, of which 36 kJ mol~1 are from electrostatic and 2 kJ mol~1 from Lennard- Jones contributions. In spite of the widely diÜering force –elds, this result is remarkably similar to that obtained using PROMET calculations.On increasing the temperature of the simulation, the average hydrogen-bond distance increases to 1.96 and 2.04 at 100 ” and 150 K, respectively, and already at 200 K the hydrogen bonds break.Nevertheless, the two molecules do not drift apart, but form a sort of stacked, non H-bonded dimer; in Fig. 7 one sees the O… … …H distances increasing, and the distance between centres-ofmass decreasing towards the minimum separation between stacked planes. This con–guration still has a substantial cohesive energy [(14]8) kJ mol~1, electrostatic and Lennard»Jones, respectively]. These results are only indicative, since equilibrium with the lactim form was not considered. 5.2 The dimer in carbon tetrachloride solution Starting atomic coordinates for 231 solvent and two solute molecules for the simulation of the dimer in were generated by geometrical considerations, and the system was CCl4 subjected to energy minimization –rst.The output of EM was used as the starting point for MD simulations. The coupling constants for the temperature bath were 0.01 ps for the solute and 0.2 ps for the solvent ; in spite of the short coupling constant, the temperature of the solute was always ca. 100 K higher than that of the solvent. At 300 K,70 Computer simulations of organic solids Fig. 7 Molecular dynamics simulation of the gas-phase dimer of 2-pyridone at 200 K: (a) distance between centres-of-mass ; (b), (c) O… … …H distances quick disruption of the dimer is observed, and the two molecules de–nitely drift apart in the solution.To obtain signi–cant information, it was pragmatic to decide to lower the temperature: at 200 K the system stabilizes with an overall density of 1.53 g cm~3 (experimental value for carbon tetrachloride at 298 K of 1.59 g cm~3) and a cohesive energy of the solvent of 19 kJ mol~1 (experimental heat of vaporization of 21 kJ mol~1 at 298 K).The solute»solvent cohesive energy (roughly, the solvation enthalpy) is [57 kJ mol~1; no experimental counterpart is available for this last quantity ; small amides have solvation enthalpies of between [40 and [80 kJ mol~1 in water.21 At 200 K, the dimerization energy is 29 kJ mol~1.The dimerization energy of caprolactam, computed with the OPLS force –eld,22 is 59 kJ mol~1 in chloroform at 300 K; thus, computed values span about the same range as the experimental ones (see Section 2). More relevant to the present purposes is the dynamic behaviour of the dimer in solution. At 200 K, one O… … …H hydrogen bond is always preserved, while the other exhibits ìcatemer jumpsœ (as also observed in the dimer of tetrolic acid23) characterized by a sudden increase of the O… … …H distance to ca. 6 (see Fig. 8), resulting in a destabi- ” lization by ca. 12 kJ mol~1, or about half the total electrostatic dimerization energy. Fig. 8 Molecular dynamics simulation of the 2-pyridone dimer in carbon tetrachloride at 200 K: (a), (b), (c) as in Fig. 7. Small circles denote instantaneous breaking of one O… … …H bond. The dashed peak around 90 ps is a longer catemer jump (see Fig. 9).A. Gavezzotti 71 Over and above the numerous approximations and the theoretical difficulties inherent to the computational methods, this result should be accepted as both reasonable and informative, since these ì—ying catemerœ solution structures (Fig. 9) are likely to be responsible for the seeding of the catemer nuclei that eventually lead to the observed crystal structure of 2-pyridone. 5.3 Molecular dynamics of the pyridone crystal MD simulations for the pyridone crystal were performed using the experimental X-ray crystal structure (PYRIDO02; 128 molecules, 2]4]4 unit cells) as a starting point.A modi–ed24 version of GROMOS, more —exible for crystal calculations, was employed. After preliminary tests, isotropic pressure scaling was settled upon, because a largely negative y pressure component resulted, and anisotropic scaling brought about a disturbingly large contraction of the b cell parameter, similar to what was observed in static minimizations (Section 3).Admittedly, all this casts a shadow on the overall reliability of the potentials, but we justify the choice of isotropic scaling as one of the (many) restrictions and approximations on what, after all, is but a computer simulation. The starting temperature was 200 K, and inputs for other temperatures were then taken from output of a preceding simulation at the nearest temperature.Typical run times were 50»80 ps, and inspection of diagrams for the time evolution of energy and structural parameters ensured that the system had reached a steady state at each temperature. Table 2 collects the main results. Rather than following in detail the evolution of intermolecular interatomic distances, centre-of-mass oscillation was monitored by computing the displacement of each of the Fig. 9 The dimer structure around 90 ps in Fig. 8: one O… … …H distance 2.05 second H-bond ”, broken. The molecular planes are nearly perpendicular to one another. Table 2 Results of molecular dynamics simulations for the 2-pyridone crystala [Etot density [Eel [E(l.j.) *H(subl) T /K /kJ mol~1 a/” b/” c/” /g cm~3 /kJ mol~1 /kJ mol~1 /kJ mol~1 5 72 13.62 5.89 5.68 1.386 20 53 75 120 56 13.66 5.91 5.70 1.374 19 53 75 140 53 13.67 5.91 5.70 1.371 19 52 74 170 49 13.70 5.93 5.71 1.365 18 52 73 200 45 13.71 5.93 5.72 1.359 18 52 73 250 38 13.78 5.95 5.75 1.342 18 51 73 300 31 13.87 6.02 5.78 1.313 17 49 70 350 23 14.00 6.05 5.84 1.278 16 47 67 a Owing to isotropic pressure rescaling the a : b : c ratio is constant.72 Computer simulations of organic solids 128 from its reference position (i.e., that in the starting structure).At 170 K, motion is roughly harmonic and oscillation times range from 0.2 to 0.4 ps, corresponding to frequencies of 160»80 cm~1, very reasonable for intermolecular vibrations. For temperatures up to 300 K, centres-of-mass oscillate about an equilibrium position no more than 0.5 away from the X-ray position, a physiological deviation almost certainly due ” to inaccuracies in the potential functions.More interesting is the fact that at 350 K the motions begin to show a marked anharmonicity (Fig. 10). At still higher temperatures, centre-of-mass displacements show a steady increase as the simulation time proceeds (Fig. 11), yielding a dynamic picture of a disgregation of the crystalline edi–ce that can be assimilated to melting, or even vaporization. The fact that this happens at a temperature close to the real melting temperature must be mere coincidence, since the true timescale of melting is such that it could never be observed in such short simulation times.Fig. 12 shows frames after 50 ps runs, at 350 (with substantial disordering, but a retention of the basic backbone) and at 400 K (with complete loss Fig. 10 Molecular dynamics simulation of 2-pyridone crystal at 350 K; displacements of the centre-of-mass of a representative molecule from the ìequilibriumœ (X-ray) position (a, b, c and d denote the x, y, z components and the modulus of the displacement). A marked anharmonic jump is evident near 0.6 ps simulation time.Fig. 11 Molecular dynamics simulation of 2-pyridone crystal at 400 K: each vertical bar is the range of displacements of the 128 centres-of-mass, the solid line is the average over the range. The moire-like shadow is a graphic artifact.A. Gavezzotti 73 Fig. 12 Final frames of the simulation of the 2-pyridone crystal at (a) 120 K, (b) 350 K, (c) 400 K. The horizontal dimensions are 27.3, 28.0 and 28.5 respectively.”,74 Computer simulations of organic solids Table 3 Results of molecular dynamics simulations for the 2-pyridone liquid [Etot density [Eel [E(l.j.) *H(vap) T /K /kJ mol~1 /g cm~3 /kJ mol~1 /kJ mol~1 /kJ mol~1 320 19 1.145 10 42 56 340 16 1.142 9.6 42 56 370 9.7 1.093 8.6 39 52 400 4.6 1.063 8.0 38 51 of short- and long-distance order).Although a thorough analysis of preferential displacements with respect to the average crystal structure is, at present, too complicated, a detailed study of these motions could no doubt cast some light on the molecular mechanisms of crystal disruption, the timescale problem being circumvented by strong overheating. In any case, Fig. 12(b) should be regarded as the ì realistic œ view of crystal packing, as opposed to the ìneoclassicœ view of Fig. 1. 5.4 Molecular dynamics of the 2-pyridone liquid phase A starting model for liquid-phase simulations was obtained by building a box with 108 molecules with centres-of-mass that lay at face-centred cubic lattice nodes with a spacing Fig. 13 Histograms of O… … …N distances in (a) crystal at 300 K, (b) liquid at 320 K.Circles denote the spread over simulation time, the solid line is the average over the spread.A. Gavezzotti 75 of 8 and whose orientation, described by three rotation angles, was taken at random; ”, no prearranged hydrogen bonding was present. Extensive energy minimization transformed this tentative structure into a substantially cohesive one (57 kJ mol~1).MD was started from this optimized structure, with a few preliminary steps at 100 K to assign velocities followed by warm-up cycles to higher temperatures, as for the crystal. Typical run times were 80»100 ps. Energetic and structural results are collected in Table 3. Fig. 13 shows histograms for N… … …O distances in the liquid and in the solid. The one for the crystal has obvious peaks corresponding to its H-bonding pattern.Not unexpectedly, the liquid curve (even for a formally highly supercooled liquid 60 K below the melting point) shows a large reduction of the population of the 2.8 peak (with a 2 : 1 ” ratio of the O… … …N to the O… … …H peak height) and disappearance of the 3.8 peak, ” corresponding to second-nearest distances in the catemer chain.Overall, the liquid curve is not too diÜerent from the computed radial distribution function for Nmethylacetamide. 21 Along with simple chemical intuition, these results show that the pure liquid, aside from a generic propensity for H-bonding, has no built-in preference for long- or medium-range structuring (of course, any such structuring is strongly disfavoured on entropy grounds). Thus, the liquid is an average bath where precursors to the crystal structure lay buried among diÜerent structures and pure disorder.Things may change around the freezing temperature, but the problems of timescale and of potential function accuracy con–ne the dynamical simulation of structuring and nucleation within a freezing liquid as a task for the future, although signi–cant results have already been obtained on phase transitions and melting in small molecular clusters.25 6 Thermodynamics The MD results in Tables 2 and 3 may be used to calculate thermodynamic parameters of phase transitions of 2-pyridone.In doing so, it should be remembered that the total kinetic energy and the intramolecular potential energy in the simulation are somewhat ill-de–ned, since stretching motions are frozen, and the number of potential-energy terms does not match the number of normal modes; besides, the temperature is evaluated from the kinetic energy in equipartition over all degrees of freedom, an assumption which, if legitimate in the continuum limit inherent to the classical approach of MD, is hardly valid, at least for the bending motions.This difficulty in the de–nition of temperature is quite general, and has to be somehow overcome if MD is to be used routinely for thermodynamic purposes.In deriving *Hs and speci–c heats, the following assumptions are eventually made: (a) E(cohesion)\E(electrostatic)]E(Lennard-Jones)/0.97, where the numerical factor accounts for the cut-oÜ in the summations; (b) E(kin, crystal)\E(kin, liquid)\E(kin, gas) ; this is appropriate for intramolecular contributions, while kinetic-energy diÜerences for external modes should amount to a few RT units, but have not been considered in a –rst approximation; (c) *H(subl)\[E(cohesion, crystal)]RT , *H(vaporization)\ E(cohesion, liquid)]RT , *H(fusion)\E(cohesion, liquid)[E(cohesion, crystal), as per assumption (b) ; (d) PdV being negligible for Cp\dH/dT \dE(total)/dT , condensed phases.The calculated crystal density is only 4% lower than experiment, and the slope of the density curve is well reproduced (Fig. 14). The overall shape of the liquid»solid dilatometric curve is quite satisfactory, and the density diÜerence between crystal and liquid at the melting point is 12%, a typical value for organic substances.The calculated cohesive energy signi–cantly underestimates the enthalpy of sublimation; in this respect, simple packing energy calculations gave a much better result, presumably because of the more speci–c parameterization of the crystal potentials.4 The crystal as computed Cp , from the slope of the E(total)/T curve, is 141 J mol~1 K~1, and is, unfortunately, almost temperature independent.For comparison, can be estimated by adding the ideal gas Cp76 Computer simulations of organic solids Fig. 14 Dilatometric curves for 2-pyridone. Full circles : experimental (X-ray) crystal ; half-–lled circles : calculated, crystal ; open circles : calculated, liquid. The vertical bar denotes the melting temperature. contribution, which averages 78 J mol~1 K~1 between 200 and 300 K,10 and the external contribution calculated from a lattice-dynamical treatment of the crystal,26 or 60 J mol~1 K~1, totalling 138 J mol~1 K~1; the agreement with the molecular dynamics result is purely fortuitous, and must arise from extensive cancellation of opposite errors.No thermodynamic data are available for the pyridone liquid. The calculated vaporization and fusion enthalpies (56 and 14 kJ mol~1, respectively) are of the correct order of magnitude (compare with N-methylisobutyramide and N-methylacetamide, *vapH\ kJ mol~1,27 and 56 kJ mol~1,22 respectively ; N-methylacetamide and acetanilide, 67 and 22 kJ mol~1, respectively28). and density of the liquid at 370 K (184 *fusH\9.7 Cp J mol~1 K~1 and 1.09 g cm~3) compare favourably with experimental data for Nmethylacetamide22 (169 J mol~1 K~1 and 0.894 g cm~3 at the same temperature). 7 Final comment Using only a computer, a completely independent determination of the molar volumes and of phase transition enthalpies for an organic compound, even before synthesizing it, could proceed through the following steps : (1) preparation of a molecular model, using standard geometrical parameters, this is easily accomplished, given the wealth of structural information available nowadays; (2) choice of a suitable force –eld ; (3) generation of crystal structures from the molecular model, and obtention of a satisfactory close packing (even if the predicted crystal structure were wrong, the crystal density and enthalpy of sublimation would be approximately right, see Table 1) ; (4) simulation of the liquid phase, either by MD or Monte Carlo, and calculation of the molar volume and of enthalpies of fusion and vaporization.Step 2 is indeed critical, but progress has been recently made in the development of force –elds. Careful consideration, with avoidance of mindless use of default parameters, can lead to reasonable guesses for most organic molecules.Step 3 suÜers from the drawback that the molecule has to be rigid or semi-rigid ; but an overview of possible polymorphism would be obtained. Steps 3 and 4 are sensitive to parameterization, but alsoA. Gavezzotti 77 experimental uncertainties in thermochemistry are typically a few kJ mol~1. The cost : quality ratio of the results would probably be as good as or better than that for experimental determinations, given the cheapness of computers with respect to chemical synthesis and instrumentation. The quantitative treatment of kinetic energy is still unsatisfactory, so that prediction of speci–c heats is more problematic.Entropy is not accessible, so that a thermodynamic prediction of melting and boiling points is still impossible. Much more problematic is a treatment of the kinetics of phase transition. While a dynamic treatment is here by de–nition indispensable, present theoretical methods are still a long way from being up to the task.Some guidelines have been explored in this paper. Thanks are due to Prof. W. van Gunsteren for permission to use the GROMOS package, and to B. P. van Eijck for useful discussions and advice, in the framework of EC Human Capital and Mobility grant ERB-CHRX-CT94-0469.References 1 A. Gavezzotti and G. Filippini, Acta Crystallogr. Sect. B, 1992, 48, 537. 2 A. Gavezzotti and G. Filippini, in T heoretical Aspects and Computer Modeling of the Molecular Solid State, ed. A. Gavezzotti, Wiley and Sons, Chichester, 1997, ch. 3. 3 A. Gavezzotti, J.Am. Chem. Soc., 1991, 113, 4622. 4 A. Gavezzotti and G. Filippini, J. Phys. Chem., 1994, 98, 4831. 5 W. F. van Gunsteren and H. J. C. Berendsen, Angew. Chem., Int. Ed. Engl., 1990, 29, 992. 6 W. L. Jorgensen, Chemtracts: Org. Chem., 1991, 4, 91. 7 U. Ohms, H. Guth, E. Hellner, H. Dannohl and A. Schweig, Z. Kristallogr., 1984, 169, 185. 8 C. Adamo and F. Lelj, Int. J. Quantum Chem., 1995, 56, 645. 9 S. Suradi, N. El Saiad, G. Pilcher and H. A. Skinner, J. Chem. T hermodyn., 1982, 14, 45. 10 K. C. Medhi, Bull. Chem. Soc. Jpn., 1991, 64, 1944. 11 A. Fujimoto and K. Inuzuka, Bull. Chem. Soc. Jpn., 1990, 63, 2292. 12 S. Mille–ori and A. Mille–ori, Bull. Chem. Soc. Jpn., 1990, 63, 2981. 13 D. E. Williams, PCK83, QCPE Program 548; Quantum Chemistry Program Exchange, Chemistry Department, Indiana University, Bloomington, IN, 1983. 14 PROMET(5), A Program for the Generation of Possible Crystal Structures from the Molecular Structure of Organic Compounds, University of Milano, 1995 (available from the author upon request). 15 A. Gavezzotti, Acta Crystallogr. Sect. B, 1996, 52, 201. 16 B. P. van Eijck, W. T. Mooij and J. Kroon, Acta Crystallogr. Sect. B, 1995, 51, 99. 17 A. Gavezzotti and G. Filippini, J. Am. Chem. Soc., 1995, 117, 12299. 18 A. Gavezzotti, Acc. Chem. Res., 1994, 27, 309. 19 G. R. Desiraju, Angew. Chem., Int. Ed. Engl., 1995, 34, 2311. 20 W. F. van Gunsteren, GROMOS, Groningen Molecular Simulation Program Package, University of Groningen, 1987. 21 W. L. Jorgensen and C. J. Swenson, J. Am. Chem. Soc., 1985, 107, 1489. 22 W. L. Jorgensen and C. J. Swenson, J. Am. Chem. Soc., 1985, 107, 569. 23 A. Gavezzotti, G. Filippini, J. Kroon, B. P. van Eijck and P. Klewinghaus, Eur. J. Chem., 1997, 3, 893. 24 B. P. van Eijck, personal communication. 25 L. S. Bartell, J. Phys. Chem., 1995, 99, 1080. 26 G. Filippini, personal communication. 27 J. S. Chickos, D. G. Hesse and J. F. Liebman, J. Org. Chem., 1989, 54, 5250. 28 J. S. Chickos, C. M. Braton, D. G. Hesse and J. F. Liebman, J. Org. Chem., 1991, 56, 927. Paper 7/01436H; Received 28th February, 1997

ISSN:1359-6640    

DOI:10.1039/a701436h

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 79»92 Computer simulation of zeolite structure and reactivity using embedded cluster methods Paul Sherwood,a* Alex H. de Vries,a Simon J. Collins,a Stephen P. Greatbanks,b Neil A. Burton,b Mark A. Vincentb and Ian H. Hillierb* a CCL RC Daresbury L aboratory, Daresbury, W arrington, UK WA4 4AD b Department of Chemistry, University of Manchester, Manchester, UK M13 9PL The use of bare cluster models to understand the nature of zeolite»substrate interactions may be improved to take account of the environment of the acid site.We consider two models for introducing the electrostatic Br‘nsted eÜects of the zeolite lattice. The –rst involves generating a specialised correction potential by –tting a non-periodic array of ca. 60 point charges to the diÜerence between the bare cluster and periodic potentials.The second starts by –tting a periodic array of atomic charges to the potential of the in–nite lattice and then builds up a classical cluster of ca. 2000 atoms into which the QM cluster is embedded. Such embedded cluster calculations, employing a T3 cluster, with electron correlation at the density functional theory level, are described, to model the interaction of water at a acid site.Br‘nsted Structures of the water»zeolite complex, and associated vibrational frequencies and 1H NMR shifts are calculated and compared with calculations of bare clusters of varying size and with experimental data. We then describe a mixed quantum mechanical»molecular mechanical (QM»MM) model derived by combining charges from the second model with a standard aluminosilicate force –eld.We report preliminary results on the eÜect of embedding on the energetics of a prototypical hydrocarbon cracking reaction ; the methyl-shift reaction of a propenium ion coordinated to the acid site. The importance of zeolites in many industrial catalytic processes has prompted continuing studies to understand the relationship between their structure and reactivity.A particular area of study has been the interaction of substrates with a acid site, Br‘nsted central to many reactions catalysed by zeolitic aluminosilicates, such as the cracking of hydrocarbons and the conversion of methanol to gasoline.1 Recently, there have been many examples of the functionalisation of zeolites by the exchange of protons by metal ions, leading to a range of catalytic applications such as the decomposition of nitrogen oxides over Cu-exchange zeolites.2 Interactions occurring at acid sites have been explored by a range of Br‘nsted experimental techniques, particularly NMR and IR spectroscopy, in an attempt to clarify the nature of the adsorbed substrate, with particular emphasis on the degree of proton transfer from the zeolite to the substrate.In parallel with these experimental studies, there have been continuing theoretical and computational developments in order to achieve the necessary realism to model the often quite subtle features of the zeolite»substrate interaction. The accurate modelling of these interactions is particularly challenging as the catalytic activity and selectivity arise from a number of diÜerent factors, including local 7980 Simulation of zeolites using embedded cluster methods structural eÜects on acidity, the level of aluminium substitution, the zeolite pore structure, chemical impurities and extra-framework ions.The most commonly used computational strategy is to employ clusters of varying sizes to represent the active site of the zeolite,3h5 and to use standard electronic structure methods, available in a number of commercial packages, to model the adsorbate»zeolite system.These models typically include two or three tetrahedral (T) atoms (Si, Al) and the bonds cleaved to generate the cluster are terminated by adding additional monovalent atoms, typically hydrogen.A drawback of these bare cluster models is that they omit long-range electrostatic eÜects and fail to include the geometric and electronic environment of the site speci–c Br‘nsted to a particular zeolite. These problems can, in principle, be addressed by the use of substantially larger clusters and studies using clusters up to T46 have been carried out.6,7 However, the routine use of ab initio quantum chemical methods is computationally prohibitive for systems of this size.A more realistic way of modelling the catalytic reaction, which avoids these problems, is to employ methods that can treat periodic systems, and to use these to model periodic supercells involving the active site and substrate. Such a strategy will often necessitate the use of quite large unit cells and hence may also be computationally prohibitive.However, such a strategy has been employed using periodic calculations based upon both a gaussian basis8 and a plane wave expansion within a density functional theory (DFT) formalism.9 The latter approach has been particularly useful in view of the ease of calculation of energy gradients allowing, in principle, both stationary structures to be obtained and molecular dynamics (MD) simulations to be performed.Embedded cluster models In other areas of condensed-phase modelling, particularly solvation studies and enzyme catalysis, hybrid methods are widely employed.10h12 Here, the reactive centre is described using a full QM treatment, and the more distant environment, which takes no direct part in the chemical reaction, is described at a much lower level of theory, often with a quite simple force –eld.The success of such hybrid methods naturally depends upon the use of an appropriate level of theory to describe the reactive centre, and the accurate description of both the electrostatic and other non-bonded interactions between the two regions, as well as the proper treatment of the junction between the two regions.In view of the polar nature of zeolites and other micro- and meso-porous materials, it is particularly important to model correctly the electrostatic environment of the active site resulting from the surrounding three-dimensional structure. We have suggested that a —exible and computationally economic way of achieving this is to use an embedded cluster model to study zeolite»substrate interactions.13 Here the cluster is embedded in an array of point charges which are chosen to model the electrostatic environment of the periodic lattice.However, since the formal atomic charges used in aluminosilicate force –elds are chosen to reproduce structural rather than electrostatic data, such charges may be inappropriate for use in embedded cluster models of zeolites.Indeed, oxygen charges vary from [0.26 for the CFF force –eld of Hill and Sauer,14 and [1.1 in the rigid ion model of van Beest et al.,15 to the formal [2.0 charge of the shell model of Catlow et al.16 One approach that we have used both to model the environment of the cluster and to correct for termination eÜects, is to derive a set of formal point charges by a leastsquares –t involving the electrostatic potential (ESP) of the in–nite 3D lattice, calculated at the Hartree»Fock (HF) level.13,17 Here the diÜerence between the ESP of the cluster and of the in–nite lattice in the region of the acid site is –tted to a point charge Br‘nsted potential to obtain point charges (potential-derived charges, PDC).This approach, which we henceforth label as model 1, has been validated by comparison with adsorp-P.Sherwood et al. 81 tion energies calculated at the full, periodic supercell level. Furthermore, when compared to full periodic results, indications are that our simpler description is competitive with more sophisticated embedding techniques.18 Although these charges which are derived from such a –tting technique, do re—ect the periodic nature of the lattice they need not be atom centred, and the individual values have no physical signi–cance.Indeed, even if they are atom centred, atoms of the same type will have diÜerent formal charges associated with them, depending upon the actual cluster chosen in the –tting procedure. Thus, they are unsuitable for inclusion in a general hybrid QM»MM method, where a complete force –eld is used to model the non-QM region.Furthermore, in the spirit of using general force –elds, it may be advantageous if such atomic charges were transferable between zeolites. We have, therefore, explored the use of an alternative strategy for obtaining PDC.19 Here formal atomic charges are derived for a periodic siliceous structure which reproduce the ESP for the same periodic structure calculated at the HF level.We have derived PDC for eight periodic siliceous zeolite systems by reference to ESPs from periodic HF calculations. Using a separate charge for each crystallographically unique atom, it was possible to –t the potential outside the van der Waals surface to within an RMS deviation of ca. 4 kJ mol~1. Fitted charges spanning [0.8 to [1.0 for oxygen, and 1.6 to 2.0 for silicon, were obtained when –tting to STO 3G calculations. Subject to the constraint that a common charge be assigned to all atoms of a given type, the RMS error in the point charge potential may rise as high as ca. 10 kJ mol~1 and a signi–cant reduction in the magnitude of the charges, relative to the unconstrained case, is observed.For –tting to an STO 3G ESP, a single assignment of an oxygen charge of [0.78 is suggested, whereas for a larger basis (6-21G*) a reduction in the compromise oxygen charge to ca. [0.6 is found. Once these periodic atomic charges, capable of reproducing the quantum periodic potential have been obtained, they may be used to construct a –nite set of point charges which, after treatment for termination eÜects, form the basis for embedded QM cluster models.We will describe a simple scheme of this type, referred to below as model 2. The advantage of both such embedded models is that the electrostatic eÜect of the PDC may be directly included in the one-electron Hamiltonian of conventional electronic structure codes, allowing for the location and characterisation of stationary structures, and for the calculation of molecular properties.However, there are some additional complications that need to be addressed. In view of the lack of van der Waals interactions with the non-QM region, geometric constraints must be imposed on the QM cluster to prevent the generation of unrealistic stationary structures.This has implications for the subsequent optimisations and calculation of vibrational frequencies, to be discussed later. Here, we describe embedded cluster calculations, based upon both approaches, to obtain PDCs. We discuss their use to model the interaction of water with a acid site, focusing on the prediction of substrate binding energies, vibrational Br‘nsted frequencies, and 1H chemical shifts. Hybrid QMñMM models Since the atomic charges from model 2 have been chosen to generate accurate electrostatic interactions, it is reasonable to assume that they may serve as a suitable atomic charge model for use in a classical force –eld for zeolite modelling, allowing the development of QM»MM models that can treat both the electrostatic interactions and the mechanical —exibility of the zeolite lattice.To date, no zeolite force –eld based on such a PDC charge model has been derived. We discuss mixed QM»MM results using the following three models for the QM»MM coupling, similar to those of Bakowies and Thiel.2082 Simulation of zeolites using embedded cluster methods (1) Mechanical embedding, where the QM»MM interactions are the classical bonded and non-bonded forces. (2) Electrostatic embedding, where the electrostatic potential due to the MM region is included in the electronic Hamiltonian of the QM region, allowing QM polarization.(3) MM polarization, where the electronic response of the MM region is modelled by coupled atomic polarizabilities in part of the MM region. We have explored these schemes based on combining –tted atomic charges with a standard aluminosilicate force –eld (CFF),21 and used them in a preliminary study reported here, of the so-called methyl»shift reaction in a propenium ion over an acid site in zeolite Y.The non-classical, cyclic propenium transition state structure in this reaction has been proposed by Sie22 as an alternative to earlier suggested pathways in hydrocarbon conversion reactions over zeolite sites.Computational details In the models used here, H-terminated T3 clusters based upon zeolite Y were chosen in view of the number of experimental studies involving the interaction of small substrate molecules with this zeolite. All PDC models were obtained using a periodic HF calculation, at the 3-21G level, of zeolite Y in which the aluminium sites were replaced by silicon, allowing the sodium counterions to be neglected.PDC model 1 A hydrogen-terminated T3 cluster, was excised from the zeolite Y structure, Si3O4H8 , the diÜerence in the ESP between the periodic structure and the cluster was –tted to obtain the PDC, as described in ref. 13. PDC model 2 The PDC used in model 2 were again obtained using an HF 3-21G periodic calculation on a purely siliceous zeolite Y structure.The oxygen and silicon charges which best reproduce the quantum mechanical ESP were found to be [0.92 and 1.84, respectively. However, these charges, which are appropriate for an in–nite periodic system must be modi–ed for use in our cluster model. This is accomplished by introducing, and treating, two boundary regions, the inner QM»MM boundary, and the outer boundary of the MM cluster.Initial modi–cations to the atomic charges are obtained by considering the charge on a given centre to arise from a sum of terms, one for each bond formed. The atomic charges are then assigned automatically, in a bond increment manner, based on connectivity, such that a silicon with only two bonds would have a charge of 1.84/2, i.e. 0.92. To correct for the loss of the long-range Madelung potential, we consider a wholly classical cluster, compare the ESP in the interior of the cluster with that from the in–nite array of periodic charges, and correct the former by addition of further charges outside the cluster. The values of these additional charges are derived by least-squares –tting to the diÜerence potential, by direct analogy with the PDC scheme of model 1.Fig. 1 illustrates the steps involved, comparing the periodic HF results [Fig. 1(a)] with the potential from the periodic point charges [Fig. 1(b)]. When the –nite cluster is terminated as described above the resulting potential in the interior of the cluster [Fig. 1(c)] is essentially identical to that of the periodic point-charge array ; the residual error is shown in Fig 1(d). To treat the boundary between the QM and MM regions, we –rst adjust the charges on the MM junction atoms using the bond charge-increment model described above.However, the problem of unrealistic electrostatic interactions between the MM junction and QM termination atoms remains. To overcome this we have chosen to redistributeP.Sherwood et al. 83 Fig. 1 Electrostatic potentials of the silicate : (a) periodic HF 3-21G calculation ; (b) –tted periodic point-charge potential ; (c) potential due to the corrected classical cluster (model 2) ; (d) diÜerence between the two potentials (b) and (c). Contours for (a), (b) and (c) at 12.5 kJ mol~1, (d) at 2.1 kJ mol~1. the charge of the junction atom amongst the adjacent MM sites, and to correct for the dipole of these charge shifts by adding a point dipole, implemented in this study by a pair of point charges, at the MM sites that receive the charge.The result of this procedure is illustrated in Fig. 2(a), which should be compared with the periodic HF result [Fig. 1(a)]. For comparison, the potential due to the bare cluster is illustrated in [Fig. 2(b)]. Energetics and spectroscopy of water adsorption Electronic structure calculations were carried out on the T3 cluster shown in Fig. 3 using these two sets of PDC. A 6-31G** and a larger basis [6- were 311]]G(2d,2p)] employed and electron correlation was included using a DFT method employing a hybrid functional (B3LYP).23 In this study, we utilise a modi–ed form of the redundant coordinate optimisation technique, implemented within the Gaussian9424 package to perform constrained optimisations of the cluster within the point-charge –eld.In particular, we freeze the positions of the cluster-termination hydrogen atoms to lie along the84 Simulation of zeolites using embedded cluster methods Fig. 2 (a) Model 2, electrostatic potential for the QM»MM cluster ; (b) potential for the bare cluster.Contours at 12.5 kJ mol~1. crystallographic SiwO bonds at 1.48 from the Si. Furthermore, the two Si atoms of ” the cluster are also kept –xed in their crystallographic positions. Thus, the unit, AlO4 acidic proton and substrate are allowed to relax fully within a rigid zeolite framework. These constraints included modi–cation to the cartesian to internal coordinate transformations by increasing the relative mass of the frozen atoms.Thus the vibrational frequencies re—ect an immobile framework and the constrained optimisations are, indeed, true minima in the reduced coordinate space. For PDC model 2, it was also necessary to freeze the two oxygen atoms at their crystallographic positions to maintain a reasonable SiO orientation and prevent collapse to We found that this AlO3]SiOH.further constraint results in a very similar bare cluster geometry. NMR shieldings were evaluated at the optimised geometries using the HF» GIAO25,26 approach with the TZP basis set of Ahlrichs,27 and PDC, where appropriate. Absolute shieldings can be compared with experimental shifts using the equation dTMS(complex)\dTMS(internal reference)]p(internal reference)[p(complex) Fig. 3 Acidic cluster Si2AlO4H9P. Sherwood et al. 85 Fig. 4 Methyl-shift reactions over a zeolite acid site As is usual, the substrate molecule is the internal reference from which the hydrogen shifts are measured. We use the experimental gas-phase shift of 0.73 ppm for water.28 Methyl-shift reaction We use a T3 cluster, as in the embedding calculation, but with hydrogen atoms terminating the aluminium atom (see Fig. 4) and without the geometric constraints. These calculations were carried out at a fairly low level, HF/3-21G, to explore the important features of the model. At the mechanical embedding level, we have used the aluminosilicate consistent force-–eld (CFF) due to Hill and Sauer21 for all zeolite interactions. The hydrocarbon fragment was uncharged, and all atoms within the QM fragment were given zero charge.Thus, there is no electrostatic interaction between the excised QM system and the MM system. The QM fragment includes terminating H atoms to replace the O atoms, bonded to the junction Si and Al atoms, constrained to lie along the QM»MM bond.The OwSiwO angle bend is assumed to be described correctly by the QM OwSiwH force, and the Si/AlwO(MM) stretches are described by the CFF terms, constraining the Si/AlwH distances to a –xed length. During optimisation of the QM cluster, the surrounding MM region was allowed to relax up to three bonds from the junction region. Non-bonded framework»hydrocarbon interaction parameters were the MM2 Lennard- Jones parameters taken from an MD study of absorption of propane in all-silica zeolites by Nicholas et al.29 In calculations at the electrostatic embedding level we utilised PDC (model 2) rather than the CFF charges, with cluster-termination eÜects on the potential at the boundary of the QM region corrected using the procedures described above for model 2.Computation of the eÜects of MM polarization were computed using the wavefunction and geometry from the electrostatic embedding calculations. The representation of the MM polarizability consisted of distributed polarizabilities at both Si and O atoms. (6.434 for Si, 6.074 for O).30 The number of MM atoms bearing a polarizability a0 3 a0 3 was limited to ca. 200 to save computer time; however, enough polarizabilities are taken into account to have converged the polarization energy (using 500 polarizabilities yielded less than 1 kJ mol~1 extra stabilization). The calculations were performed using a hybrid QM»MM code (ChemShell) coupling the GAMESS-UK QM code31 with the MM code, DL-POLY32, modi–ed to support the CFF force –eld. The MM polarisation calculations were carried out at –xed geometries utilising the direct reaction –eld version of HONDO8.1.33 We have investigated the eÜects on the reaction barrier of the methyl-shift reaction in the propenium ion of a single, isolated acid site in zeolite Y at the three progressive levels of QM»MM embedding described above.Hydrocarbon zeolite clusters of 35 radius from the single a0 Al-substituted T site were employed.86 Simulation of zeolites using embedded cluster methods Computational results Zeoliteñwater clusters Structures and binding energies.The calculated structures are summarised in Table 1. Both PDC models and diÜerent basis sets reveal the same trends. For the bare cluster there is a small increase in the OwH length due to the PDC.All calculations show a similar structure for the 1:1 complex with water. The water is bound as the neutral species, with a quite short hydrogen bond involving the acid site, and a longer Br‘nsted secondary hydrogen bond involving a framework oxygen atom. The eÜect of the PDC is to increase the zeolite OwH bond length, increasing the primary hydrogen-bond interaction and decreasing the secondary interaction. A similar eÜect has been found from bare cluster models, on increasing the size of the cluster.Thus, Krossner and Sauer34 studied both T3 clusters and a substantially larger model which is typical of the faujasite lattice. They found that, as the size of the model increases, the larger intermolecular distance [O(7)wH(11)] becomes longer and the shorter one [O(9)wH(3)] shortens. A comparison of the two PDC models at the 6-31G** level reveals an encouraging level of agreement between the predicted structures, with the intermolecular hydrogen bond lengths being well within 0.1 The eÜect of increasing the basis set size, investi- ”.gated for PDC model 1, is to increase the hydrogen-bond lengths, an eÜect observed previously in calculations on simpler hydrogen-bonded systems.The calculated binding energies of the water molecule to the acid site are Br‘nsted given in Table 2. These values re—ect the diÜerent intermolecular distances predicted by the various treatments. Thus, the eÜect of the PDC is to increase the binding energy by ca. 15 kJ mol~1 for all models considered. The binding energies are considerably smaller for the large basis set.It is of interest to note that our value for the large basis set, in the Table 1 Cluster bond lengths and 1:1 complexes with watera (”) cluster complex O(2)wH(3) O(2)wH(3) O(9)wH(3) H(11)wO(7) O(9)wH(10) O(9)wH(11) model 1 6-31G**/noneb 0.970 1.039 1.482 1.717 0.967 0.994 6-31G**/PDC 0.974 1.089 1.352 1.924 0.968 0.979 6-311]]G(2d,2p) 0.965 1.019 1.543 1.753 0.962 0.987 /none 6-311]]G(2d,2p) 0.970 1.056 1.424 2.164 0.964 0.968 /PDC model 2 6-31G**/none 0.969 1.042 1.479 1.713 0.967 0.995 6-31G**/PDC 0.973 1.098 1.350 1.835 0.968 0.987 a See Fig. 3 for atom labelling.b Bare clusters with no point-charge –eld. Table 2 Binding energies (kJ mol~1) of water model 1 6-31G**/none 92.6 6-31G**/PDC 105.1 6-311]]G(2d,2p)/none 64.8 6-311]]G(2d,2p)/PDC 79.4 model 2 6-31G**/none 102.0 6-31G**/PDC 114.5P.Sherwood et al. 87 presence of the PDC (79.4 kJ mol~1) is close the value found for the faujasite model of Krossner and Sauer34 (73.3 kJ mol~1). Vibrational frequencies. There have been a number of calculations of vibrational frequencies of bare water»zeolite clusters.35h37 The vibrational frequencies that we have calculated at the various levels are shown in Table 3.The changes in these values follow the structural changes already discussed. For the bare cluster, the OwH-stretching frequency is noticeably reduced by the presence of the PDC. The eÜect of the PDC on other hydrogen atom motions (d, c) is less readily explained. Considerable changes in the vibrational frequencies of both the zeolite and substrate water are found on binding.The OwH stretching frequency of the zeolite is reduced and the other modes (d, c) are at higher energy, in line with the increase in the cluster OwH bond length and restrictions on OwH deformation upon hydrogen bonding. The eÜect of the PDC is to increase all these vibrational shifts, particularly that of the OwH-stretching frequency, which correlates with the increased binding energy predicted by the PDC models.The energies of the OwH deformation modes are relatively insensitive, both to the basis set level and to the inclusion of the PDC, being at ca. 1100 cm~1 [c(OH)] and ca. 1500 cm~1 [d(OH)]. The predicted value of the OwH-stretching frequency in the complex diÜers by nearly 1000 cm~1 for the various models used.All models predict a substantial lowering of this frequency on the inclusion of the PDC, in line with the increased hydrogen-bond strength. It is probable that the values at the 6-31G** level are too small, owing to the overestimation of the intermolecular interactions. Thus, the most realistic value that we have obtained is probably 2192 cm~1 at the level.It has been 6-311]]G(2d,2p)/PDC observed that a number of functionals used in DFT calculations predict too great a red shift in the OwH-stretching frequency of a hydrogen-bond donor on hydrogen bond formation.38 However, the functional that we have used here, B3LYP, gives a value for this frequency close to the MP2 value for the case of the water dimer,38 so that our calculation may not give a value that is unduly low.There has been considerable discussion on the interpretation of the IR spectra of water»zeolite complexes. The adsorption of water onto H-ZSM-539 yields two new bands at 3700 and 3600 cm~1 and three bands at 2885, 2457 and 1630 cm~1. These bands were previously interpreted in terms of an ion-pair complex involving a hydroxonium cation.39 An alternative assignment, consistent with the general theoretical conclusion that the energetically most favourable complex involves neutral water,1,34h36 with no proton transfer from the zeolite, explains the observed bands in terms of the so»called A»B»C pattern found for strong H»bonded complexes.40,41 This characteristic triplet of bands is caused by resonant interactions between the OH-stretching vibration, strongly broadened by interaction with low-frequency O… … …O modes, and overtones of the bending modes of the strongly perturbed bridging OH group.The strong double maximum (A»B) observed in the experimental spectrum39 at 2885 and 2457 cm~1 may be attributed to the zeolitic OH vibration, downshifted by the H-bond interaction and strongly broadened by coupling O… … …O modes.The splitting of this band at 2675 cm~1 arises by interaction with the second overtone of the dOH bending mode, placing the fundamental at 1350 cm~1. This can be compared with our highest level result of 1513 cm~1. The considerable red shift of the OH-stretching mode that we predict, giving a frequency of 2192 cm~1, is perhaps a little low when compared with the experimental estimate in the region 2700»2300 cm~1.More controversy, however, concerns the assignment of the experimental peak at 1630 cm~1, viewed by some workers34,35,37 as a third part of the A»B»C triplet. The splitting at 2000 cm~1 may be due to interaction with the second overtone of the cOH bending mode or with the combination. However, an alternative cOH]dOH explanation36 considers such a resonance process unlikely and assigns this peak to the HwOwH deformation mode of the absorbed water. Our highest level results place the88 Simulation of zeolites using embedded cluster methods Table 3 Harmonic vibrational frequencies for cluster and 1:1 complex with water (cm~1) cluster complex l[O(2) H(3)] d[O(2) H(3)] c[O(2) H(3)] l[O(2) H(3)] d[O(2) H(3)] c[O(2) H(3)] l[O(9) H(10)] l[O(9) H(11)] model 1 6-31G**/none 3803 1057 321»334 2516 1516 1113 3845 3346 6-31G**/PDC 3736 1051 308»421 1861 1551 1167 3843 3629 6-311]]G(2d,2p)/none 3790 1045 276»378 2735 1470 1010»1019 3863 3391 6-311]]G(2d,2p)/PDC 3725 1088 313»418 2192 1513 1101»1109 3852 3738 model 2 6-31G**/none 3819 1050 245»349 2477 1490 1108 3851 3324 6-31G**/PDC 3766 1110 231»394 1823 1558 1215 3846 3494P.Sherwood et al. 89 mode at 1105 cm~1 and the HwOwH deformation at 1634 cm~1, consistent with cOH both explanations. The two high-energy bands, observed at 3700 and 3600 cm~1 can be assigned to the OH stretch of the free proton and the H-bonded proton, respectively, calculated to occur at 3852 and 3738 cm~1 at our highest level.The corresponding values calculated in the absence of the PDC, 3863 and 3391 cm~1 are quite close to the MP2 values of Krossner and Sauer34 (3740, 3328 cm~1) and show the improved agreement obtained by inclusion of the PDC. 1H NMR shifts. The experimental NMR data, for loadings of one water molecule per bridging hydroxy group, are reported to be 6.2 ppm (H-Y),42 4.3»5.8 ppm (H-Y),43 7.1 ppm (H-ZSM-5)44 and a broad peak at 7 ppm (H-rho).45 For model 1, all calculations predict an average shift in the range 7»9 ppm, with the value being larger when the PDC is included (see Table 4).Thus, on inclusion of the PDC, the shift of the zeolitic hydrogen increases, and that of the water proton involved in H bonding decreases, in line with the increase in the primary H-bonding interaction, and the decrease in the secondary H bonding upon inclusion of the PDC.A similar though considerably smaller eÜect was observed by Krossner and Sauer,34 when bare clusters of diÜerent sizes were studied. Thus, since comparison with experiment is not helpful in assessing the relative quality of the results from model 1, results for model 2 are not presented.Methyl-shift reaction Table 5 summarises the results for the methyl-shift reaction on zeolite Y at the three diÜerent levels of embedding. Activation barriers were obtained for the propoxide(I)» transition state (TS) (barrier I) and propoxide(II)»TS (barrier II) processes in zeolite Y, propoxide(I) and propoxide(II) being the minima in which a CwO bond is formed on either side of the Al (see Fig. 4). The O atoms involved are the O(1) and O(4) zeolite sites. For the gas-phase calculations, the barrier is 316 kJ mol~1, and no distinction can be made between the two minima. The eÜects of mechanical embedding on the geometries of the minima and the TS connecting them (see Fig. 5) are shown in Table 6. Looking closely at these gas-phase and embedded structures it can be seen that the con–ning of the acid site reduces the OwAlwO angle and pushes the propenium ion into the cage.The propenium ion is further encouraged into the cage by the Van der Waals forces of the surrounding cage (zeolite hydrocarbon non-bond interactions). The eÜects of the relaxation of the zeolite cage (MM energy) is seen to also reduce the barrier. The p-bond in the minima puts more constraints on these structures than on the loose TS structure, thus putting the minima at a disadvantage.If the QM contribution to the energy of the embedded clusters is compared with the energy of the corresponding gas-phase systems it can be seen that the energy cost of deforming the QM cluster to –t the zeolite backbone is greater for the minima than for the TS.The resulting stabilisation of the TS (by 55 kJ mol~1 relative to minimum I and 93 kJ mol~1 relative to 1H NMR shieldings Table 4 H2O d(ZOH) da(H2O) db(H2O) ave. model 1 6-31G**/none 14.0 8.1 1.6 7.9 6-31G**/PDC 17.9 6.0 3.3 9.1 6-311]]G(2d,2p)/none 12.4 7.5 1.4 7.1 6-311]]G(2d,2p)/PDC 16.0 4.4 3.2 7.9 a Hydrogen bonded. b Free hydrogen.90 Simulation of zeolites using embedded cluster methods Table 5 Energies, activation barriers (kJ mol~1) and dipole moments (D) of the QM cluster for methyl-shift reaction in propene on a zeolite acid site at diÜerent embedding levels model energy propoxide I TS propoxide II barrier I barrier II gas phase totala 0 316 0 316 316 mechanical totala 0 247 55 247 192 QMa 0 261 38 261 223 MMa,b 0 [6 9 [6 [15 Z»(C,H) non-bond [12 [20 [4 [8 [16 l (D) 6.3 10.9 6.1 electrostatic totala 0 253 68 253 185 QM»MM elecc [93 [103 [100 [10 [3 polarised QM»MM pold [30 [45 [33 [15 [12 a Energies for propoxide I, propoxide II and TS given relative to propoxide I.b Excluding the zeolite»hydrocarbon non-bonded interaction [Z»(C,H) non-bond]. c Electrostatic interaction between the DPC representation of the embedded QM cluster and the MM point charges.d Interaction of the DPC representation of the embedded QM cluster with the distributed polarizabilities on the MM atoms. minimum II) is the main contribution to the lowering of the barrier in the embedded case. To get an impression of the eÜects to be expected by electrostatic embedding, the dipole moments of the clusters at the mechanical embedding level are also presented in Table 5.Analysis of the diÜerent contributions to the barrier at the electrostatic embed- Fig. 5 Structural parameters for the methyl-shift reaction Table 6 Optimised geometries (in and degrees) for the methyl-shift ” reaction (see Fig. 5) using the mechanical embedding scheme. propoxide TS structural coordinate gas phase embedded gas phase embedded O(1)AlO(4) 110.6 92.4 106.4 99.5 SiO(4)Al 170.6 133.5 139.6 133.8 SiO(1)Al 114.2 124.7 139.2 130.1 hCXC 130.1 128.5 90.1 88.6 rXAl » » 3.37 3.44 rXC 1.95 1.90 1.79 1.80P.Sherwood et al. 91 ding level is not as straightforward as at the mechanical embedding level. Here, we give the interaction between the QM system and the surrounding point charges by representing the QM charge distribution by the so-called dipole preserving charges (DPCs).46 The computed interactions are all attractive (by 93, 100 and 103 kJ mol~1 for the two minima and the TS, respectively) but do not diÜer strongly for the diÜerent points along the reaction path.The additional eÜects of electrostatic embedding on the barrier are, therefore, not nearly as large as those resulting from the mechanical constraints, lowering barrier I by 10 kJ mol~1, and lowering barrier II by 3 kJ mol~1.This barrier change is relatively insensitive to changes in the MM charges, changing by no more than 5 kJ mol~1 when the bond contribution to the oxygen charge is changed from ^0.46 to ^0.30. The build-up of dipole moment on going from minimum to TS is also expected to aÜect the MM polarization contribution to the barrier.We can estimate this eÜect by computing the interaction energy for the DPC representation of the QM region and distributed atomic polarisabilities at the MM atom sites.47 The eÜect on the barrier of the MM polarization is found to be at least as big as that of the electrostatic interaction, even though the magnitude of the interaction is signi–cantly smaller (30»45 kJ mol~1).Discussion We have described two alternative strategies for obtaining PDC, which can be used in a embedded cluster model of zeolite reactivity to include the electrostatic environment and to take account of termination eÜects. When used in a model of water bound to a hydrogen-terminated T3 cluster, both give very similar eÜects, as judged by predicted structures, substrate binding energies and vibrational frequencies.The inclusion of the PDC results in substantial changes in all of these three properties compared with the bare cluster results. The trends we –nd on inclusion of the PDC mirror those found in bare cluster calculations on increasing the size of the cluster, although the changes are in general larger than those from the cluster calculations.It is clearly computationally prohibitive to use sufficiently large bare clusters that eÜectively reproduce the results of a full periodic calculation. For example, although quite large clusters have been used to predict structures and substrate binding energies, the determination of vibrational frequencies is more computationally demanding.A computational strategy that involves a hybrid QM»MM approach has obvious advantages and the consideration of the electrostatic potential for such a model, described in this paper, is a necessary step towards the construction of such a model. The water absorption and methyl-shift reaction show an interesting contrast in the in—uence of the charge –eld in the zeolite pore.Whereas the structural changes of the water adsorbed on the site induce large vibrational frequency shifts, the structures and barriers in the methyl shift appear rather insensitive to the electrostatic –eld. The latter do show, however, that electronic polarization of the surroundings must also be taken into account to complete a realistic modelling of zeolite chemistry. thank EPSRC and Shell Research and Technology Center in Amsterdam for We support of this research.We would like to thank A. M. Rigby and M. V. Frash for providing gas-phase structures for, and discussions on, the methyl-shift reaction. References 1 R. A. van Santen and G. J. Kramer, Chem. Rev., 1995, 95, 637. 2 M. Iwamoto, H. Yahiro, N. Mizuno, W. X. Zhang, Y. Mine, H. Furukawa and S.Kagawa, J. Phys. Chem., 1992, 96, 9360. 3 R. A. van Santen, Stud. Surf. Sci. Catal., 1994, 85, 273. 4 U. Fleischer, K. Kutzelnigg, A. Bleiber and J. Sauer, J. Am. Chem. Soc., 1993, 115, 7833.92 Simulation of zeolites using embedded cluster methods 5 L. A. Curtiss, H. Brand, J. B. Nicholas and L. E. Iton, Chem. Phys. L ett., 1991, 184, 215. 6 H. V. Brand, L. A. Curtiss and L.E. Iton, J. Phys. Chem., 1992, 96, 7725. 7 M. S. Stave and J. B. Nicholas, J. Phys. Chem., 1993, 97, 9630. 8 E. H. Teunissen, C. Roetti, C. Pisani, A. J. H. de Man, A. J. P. Jansen, R. Orlando, R. A. van Santen and R. Dovesi, Model. Simul. Mater. Sci. Eng., 1994, 2, 921. 9 R. Shah, J. D. Gale and M. C. Payne, J. Phys. Chem., 1996, 100, 11688. 10 J. Gao, in Reviews of Computational Chemistry, ed.K. B. Lipkowitz and D. B. Boyd, VCH, New York, 1995, vol. 7, p. 119. 11 M. J. Field, P. A. Bash and M. Karplus, J. Comput. Chem., 1990, 11, 700. 12 U. C. Singh and P. A. Kollman, J. Comput. Chem., 1986, 7, 718. 13 S. P. Greatbanks, P. Sherwood and I. H. Hillier, J. Phys. Chem., 1994, 98, 8134. 14 J. R. Hill and J. Sauer, J. Phys. Chem., 1994, 98, 1238. 15 B. W.H. van Beest, G. J. Kramer and R. A. van Santen, Phys. Rev. L ett., 1990, 64, 1955. 16 K. P. Schroé der, J. Sauer, M. Leslie, C. R. A. Catlow and J. M. Thomas, Chem. Phys. L ett., 1992, 188, 320. 17 S. P. Greatbanks, I. H. Hillier, N. A. Burton and P. Sherwood, J. Chem. Phys., 1996, 150, 3770. 18 S. P. Greatbanks, I. H. Hillier and P. Sherwood, J. Comput. Chem., 1997, 18, 562. 19 S.J. Collins, A. H. de Vries, P. Sherwood, S. P. Greatbanks and I. H. Hillier, J. Comput. Chem., submitted. 20 D. Bakowies and W. Thiel, J. Phys. Chem., 1996, 100, 10580. 21 J. R. Hill and J. Sauer, J. Phys. Chem., 1995, 99, 9536. 22 S. T. Sie, Ind. Eng. Chem. Res., 1992, 31, 1881. 23 A. D. Becke, J. Chem. Phys., 1993, 98, 1372; P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M.J. Frisch, J. Phys. Chem., 1994, 98, 11623. 24 M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. A. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W.Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, GAUSSIAN 94, Revision A.1, Gaussian Inc., Pittsburgh, PA, 1995. 25 R. Ditch–eld, Mol. Phys., 1974, 27, 789. 26 K. Wolinski, J. F. Hilton and P. Pulay, J. Am. Chem. Soc., 1990, 112, 8251. 27 A. Schaé fer, H.Horn and R. Ahlrichs, J. Chem. Phys., 1992, 97, 2571. 28 W. G. Schneider, H. J. Bernstein and J. A. Pople, J. Chem. Phys., 1958, 28, 601. 29 J. B. Nicholas, F. R. Trouw, J. E. Mertz, L. E. Iton and A. J. Hop–nger, J. Phys. Chem., 1993, 97, 4149. 30 E. de Vos-Burghart, V. A. Verheij, H. van Bekkum and B. van de Graaf, Zeolites, 1992, 12, 183. 31 GAMESS-UK is a package of ab initio programs written by M. F. Guest, J. H. van Lenthe, J. Kendrick, K. SchoeÜel and P. Sherwood, with contributions from R. D. Amos, R. J. Buenker, M. Dupuis, N. C. Handy, I. H. Hillier, P. J. Knowles, V. Bonacic-Koutecky, W. von Niessen, A. P. Rendell, V. R. Saunders and A. J. Stone. The package was derived from the original GAMESS code due to M. Dupuis, D. Spangler and J. Wendoloski, NRCC Software Catalog, vol. 1, Program QG01 (GAMESS) 1980. 32 DL POLY is a package of molecular simulation routines written by W. Smith and T. R. Forester, copyright The Council for the Central Laboratory of the Research Councils, Daresbury Laboratory, nr. Warrington, 1996. 33 M. Dupuis, A. Farazdel, S. P. Karma and S. A. Maluendes, in MOT ECC-90, ed. E. Clementi, ESCOM, Leiden, 1990, p. 277. 34 M. Krossner and J. Sauer, J. Phys. Chem., 1996, 100, 6199. 35 A. G. Pelmenschikov and R. A. van Santen, J. Phys. Chem., 1993, 97, 10678. 36 S. A. Zygmunt, L. A. Curtiss, L. E. Iton and M. K. Erhardt, J. Phys. Chem., 1996, 100, 6663. 37 J. Sauer, P. Ugliengo, E. Garrone and V. R. Saunders, Chem. Rev., 1994, 94, 2095. 38 P. E. Sinclair and C. R. A. Catlow, J. Chem. Soc., Faraday T rans., 1997, 93, 333. 39 A. Jentys, G. Warecka, M. Derewinski and J. Lercher, J. Phys. Chem., 1989, 93, 4837. 40 M. F. Clydon and N. Sheppard, Chem. Commun., 1969, 1431. 41 A. G. Pelmenschikov, J. H. M. C. van Wolput, J. Janchen and R. A. van Santen, J. Phys. Chem., 1995, 99, 3612. 42 M. Hunger, D. Freude and H. Pfeifer, J. Chem. Soc., Faraday T rans., 1991, 87, 657. 43 P. Batamack, C. Doremieux-Morin and J. Fraissard, J. Chim. Phys., 1992, 89, 423. 44 P. Batamack, C. Doremeiux-Morin, J. Fraissard and D. Freude, J. Phys. Chem., 1991, 95, 3790. 45 Z. Luz and A. J. Vega, J. Phys. Chem., 1987, 91, 374. 46 B. T. Thole and P. Th. van Duijnen, T heoret. Chim. Acta, 1987, 63, 209. 47 B. T. Thole and P. Th. van Duijnen, T heoret. Chim. Acta, 1980, 55, 307. Paper 7/01790A; Received 14th March, 1997

ISSN:1359-6640    

DOI:10.1039/a701790a

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 93»104 Simulation of adsorption and diÜusion of hydrocarbons in zeolites Berend Smit,*§a L. Danieé l J. C. Loyensîb and Guy L. M. M. Verbist°c a Department of Chemical Engineering, Universiteit van Amsterdam, Nieuwe Achtergracht 166, 1018 W V Amsterdam, T he Netherlands b Shell International Exploration and Production B.V ., Research and T echnical Services, V olmerlaan 8, 2288 GD Rijswijk, T he Netherlands c Shell International Oil Products B.V ., P.O.Box 38000, 1030 BN Amsterdam, T he Netherlands Molecular simulations are used to investigate the energetics and siting of linear and branched alkanes in the zeolite silicalite. The calculated heats of adsorption of the branched alkanes are in good agreement with the experimental data.The simulations show a striking diÜerence between the behaviour of linear and branched alkanes. The linear alkanes are relatively free to move in all channels of the zeolites. The branched alkanes are trapped with their CH group in the intersection of the zig-zag and straight channels of silicalite. This trapping of the branched alkanes suggests that diÜusion of these molecules is an activated process ; most of the time the molecule is located in the intersection but, occasionally, it hops from one intersection to another.The straight and zig-zag channels form a barrier for the diÜusion. We present some preliminary calculations of this hopping rate, from which the diÜusion coefficient can be calculated. These preliminary results are in reasonable agreement with experimental data. 1 Introduction The catalytic conversion of molecules inside the pores of a zeolite can be seen schematically as a three-step process ; adsorption and diÜusion of the reactants in the pores of the zeolites, catalytic conversion at the active site and, –nally, diÜusion and desorption of the products. Each of these steps contributes to the overall activity of a zeolite.To understand the shape selectivity of a given zeolite it is, therefore, important to have a detailed understanding of the sorption and diÜusion of the molecules in the pores of a zeolite. Experimentally, it is difficult to obtain this type of information under reaction conditions and therefore computer simulations could be a possible alternative. In principle, the conventional simulation techniques, such as molecular dynamics (MD) or the Monte Carlo (MC) method, can be used to obtain this type of information.However, in practice, it turns out that these techniques are limited to the sorption and diÜusion of relatively small molecules. The diÜusion of these small molecules or atoms is sufficiently high, such that within a reasonable amount of CPU time a representative part of the zeolite is sampled (for a review, see ref. 1). For hydrocarbons, this implies that § E-mail: smit=chemeng.chem.uva.nl î E-mail: D.LOYENS=siep.shell.com ° E-mail: verbist1=siop.shell.nl 9394 Adsorption and diÜusion of hydrocarbons in zeolites standard MD can be used efficiently to simulate the diÜusion and sorption of methane, ethane and propane.2,3 If we increase the number of carbon atoms the CPU time becomes too great.4,5 To simulate the thermodynamic properties of long-chain alkanes, it is necessary to use alternative simulation techniques. For example, Smit and coworkers have used the con–gurational-bias Monte Carlo (CBMC) technique to compute the energetics and siting of linear alkanes in various zeolites.6h9 Similar methods have been used by Maginn et al.10 In this work we use the CBMC technique to simulate the behaviour of branched hydrocarbons. Branched hydrocarbons are of importance for catalytic dewaxing and alkane isomerisation.We compare the sorption properties of linear and branched alkanes in silicalite. In particular, we show that the siting of the branched alkanes diÜers signi–cantly from the siting of the linear alkanes.It is argued that this diÜerence in siting has consequences for the diÜusion mechanism of branched alkanes and we present some preliminary results for the diÜusion coefficients of these molecules. 2 Model and computational details We focus on alkanes with a single chain-end branch with the structure The branched alkanes are described with a united-atom (CH3)2wCHw(CH2)nCH3 .model, i.e. and CH groups are considered as single interaction centres. We CH3, CH2 have used the model of Wang et al.11 The pseudo-atoms in diÜerent molecules, or belonging to the same molecule, but separated by more than three bonds, interact with each other through a Lennard-Jones potential uij LJ\4eijCApij rijB12[Apij rijB6D (1) where is the distance between sites i and j.The Lennard-Jones potentials were trun- rij cated at 9.626 and the usual tail corrections have been applied.12 The Lennard-Jones ”, parameters used are shown in Table 1. The pseudo-atoms in a given chain are assumed to be connected by rigid bonds Bond bending is modelled by a harmonic (dCC\1.53 ”). Table 1 Parameters for the Lennard- Jones potential describing the interactions between pseudo-atoms of a branched alkane as developed by Wang et al.11 (e/kB)/K p/” CH2wCH2 59.38 3.905 CH3wCH3 88.06 3.905 CHb3wCHb3 80.51 3.910 CHwCH 40.25 3.850 CH3wCH2 72.31 3.905 CH3wCHb3 84.20 3.9075 CH3wCH 59.53 3.8775 CH2wCHb3 69.14 3.9075 CH2wCH 48.89 3.8775 A group connected to a CH CH3 group is denoted by This wCHb3 .group is given a diÜerent set of interaction parameters.The interactions are truncated at Rc\9.626 ”.B. Smit et al. 95 Table 2 Parameters for the torsion potential of the branched alkanes11 C0 C1 C2 C3 CH3wCH2wCHwCHb3 373.0512 919.0441 268.1541 [1737.216 CH2wCH2wCHwCHb3 373.0512 919.0441 268.1541 [1737.216 CH2wCH2wCH2wCH 1009.728 2018.446 136.341 [3164.52 CH3wCH2wCH2wCH 1009.728 2018.446 136.341 [3164.52 CH2wCH2wCH2wCH2 1009.728 2018.446 136.341 [3164.52 CH3wCH2wCH2wCH2 1009.728 2018.446 136.341 [3164.52 CH3wCH2wCH2wCH3 1009.728 2018.446 136.341 [3164.52 A group connected to a CH group is denoted by For a CH group the CH3 CHb3 .total torsion potential is the sum of two contributions. potential ubending(hi)\(1/2)kh(hi[heq)2 (2) with as the equilibrium angle and with a force constant equal to heq\112.4° kh\ 63 390.976 K rad~2.Changes in the torsional angles are controlled by: utorsion(/i)\C0]C1 cos(/i)]C2 cos2(/i)]C3 cos3(/i) with parameters shown in Table 2. In our calculations, we focus on all-silica zeolites. Following Kiselev and co-workers,13 the zeolite lattice was assumed to be rigid. For alkane adsorption the energetics will be dominated by dispersive interactions. Since the Si atoms are much smaller than the O atoms, they make a very small contribution to the energetics and can be ignored in the calculations.In fact, the interactions of the guest molecules with the Si atoms are implicitly accounted for in the eÜective potential for the interactions with the O atoms. The dispersive interactions of the O atoms of the zeolite with the host molecules are described with a Lennard-Jones potential, eqn.(1). The parameters used are shown in Table 3. 3 Energetics and siting In Fig. 1 the calculated heats of adsorption as a function of the total number of carbon atoms, in silicalite are compared with experimental data of Calvalcante and Nc Ruthven14 and Eder.15 The simulations show that the temperature dependence of the heat of adsorption is very small.Only for is a small decrease in the heat of adsorp- C12 tion observed. The agreement of the experimental data with the simulation results is surprisingly good. In Fig. 2 the structure of silicalite is shown schematically. Silicalite has two types of channels, zig-zag and straight, that cross each other at the intersections. Fig. 3 compares Table 3 Parameters for the Lennard-Jones potential describing the interactions between the alkane pseudo-atoms and the O atoms of the zeolite (e/kB)/K p/” OwCH3 87.5 3.64 OwCHb3 87.5 3.64 OwCH2 54.4 3.64 OwCH 51.3 3.64 The interactions are truncated at Rc\13.8 ”.96 Adsorption and diÜusion of hydrocarbons in zeolites Fig. 1 Heats of adsorption of the branched alkanes as a function (CH3)2wCHw(CH2)N~4(CH3) of the total number of carbon atoms at various temperatures.The experimental data are from Nc ref. 14 for at T \398 K and from ref. 15 for at T \372 K. Nc\6 Nc\8 the distribution of the CH group of the heat of 2-methylbutane with the distribution of the middle segment of pentane in the pores of silicalite at T \498 K. It shows that the distribution of the linear alkanes is very diÜerent from the distribution of the branched alkanes.Whereas pentane has an equal probability of being in the straight channels, zig-zag channels or intersections, the branched alkanes have a strong preference to be with the head group in the intersections. These results are in very good agreement with the MC integration results of June et al.16 For the other branched alkanes we also a –nd a preference for the head group to be in the intersections.If the head group is localized in the intersection, the tail of the molecule can either be in the straight or zig-zag channels or when the molecule is sufficiently small, in the intersection. In Fig. 4 we compare the distribution of the tails of the branched alkanes over the various channels of silicalite with the distribution of the linear alkanes at T \298 K.For the branched alkanes, a nearly identical distribution is found at T \398 and 498 K. For the linear alkanes, the distribution is relatively simple; Fig. 2 Schematic drawing of the pore structure of silicalite, the straight channels are in the y»z plane and the zig-zag channels in the x»z plane.The channels cross at the intersection.B. Smit et al. 97 Fig. 3 Distribution of alkanes in the channels of silicalite. The lines represent the zeolite lattice. At regular intervals a dot, representing the position of the CH pseudo-atom of the head group for 2-methylbutane or the middle segment for pentane, is drawn. The density of the dots is a CH2 measure of the probability of –nding a molecule in a particular section of the zeolite.The top –gures give a projection along the straight channels (z»x plane) and the bottom –gures along the zig-zag channel (x»y plane). Fig. 4 Distribution of the alkanes over the zig-zag and straight channels and intersections, as a function of at T \298 K. Left-hand –gure is for linear and right-hand –gure for branched Nc alkanes.98 Adsorption and diÜusion of hydrocarbons in zeolites the short alkanes are equally likely to reside in the straight or zig-zag channels, the long alkanes have a preference for the straight channels (see ref. 6 and 17 for more details). The siting of the branched alkanes is more complex. The small branched alkanes (Nc\ 5, 6) are nearly spherical and can ìrotateœ freely in the intersection.For these molecules it is not very favourable to put their tail into one of the channels. In fact, for these small molecules it is difficult, because of the bulky head, to reach the entrance of the zig-zag channel, therefore they prefer the straight channel. If we increase the tail length, the molecules become too big to be completely in the intersection and they have to put their tail in one of the channels.For these molecules the tail is sufficiently long so that it can be in the zig-zag channel while the head remains in the middle of the intersection. For these tail lengths we observed, therefore, a nearly equal probability of being in the straight or zig-zag channel. A further increase in the tail length makes these tails longer than the period of the zig-zag channel.As for the linear alkanes, this is not a favourable con–guration and therefore the long branched alkanes prefer the straight channels. 4 DiÜusion It is interesting to discuss the consequences of the results of the previous section for the diÜusion of these molecules in the pores of the zeolite. Comparison of the siting of the linear and branched alkanes shows that the (short-chain) linear alkanes have a uniform distribution whereas the branched alkanes prefer to be at the intersection.This suggests that these linear alkanes can move ì freely œ in the channels and therefore their diÜusion coefficient can be obtained from MD simulations within a reasonable amount of CPU time. The branched alkanes, however, are pinned with their head group at the intersections and have a very small probability of being in the channels connecting the intersections.These straight and zig-zag channels, therefore, form a barrier to diÜusion. If this barrier is much higher than the diÜusion of such an alkane is an activated process ; kBT , most of the time the molecule resides at an intersection but occasionally a molecule hops from one intersection to another.If the diÜusion of these branched alkanes is an activated process, we can use the simulation techniques developed by Bennett18 and Chandler19 to simulate rare events.20 The basic idea behind these calculations is that the rate at which a barrier crossing proceeds is determined by the product of a static term, namely the probability of –nding the system at the top of the barrier, and a dynamic term that describes the rate at which systems at the top of the barrier move to the other valley.To compute the diÜusion coefficients of a branched alkane in a zeolite we have to determine a reaction coordinate for which we can compute the free energy. For diÜusion, a natural reaction coordinate is the position of one of the atoms of the adsorbed molecules.For branched alkanes it is convenient to take the position of the CH group (i.e. the group for which the distribution is shown in Fig. 3). Let us assume the concentration of hydrocarbons is sufficiently low, such that the probability that two hydrocarbons are in neighbouring intersections is very small. In this limit, the jumps from one intersection to another are independent.In silicalite, a molecule can jump from one intersection to another via the straight channel or zig-zag channel (see Fig. 5). We have to calculate the jump rates for each of these paths. Because of the symmetry of the crystal, the two diÜerent paths via the straight channels (jumping up or down) and the paths via the zig-zag channels are equivalent.The calculation can therefore be limited to computing the jump rates via these two paths. For the straight channel the reaction coordinate is de–ned as the projection of qstr(z) the head group on the line connecting two intersections via a straight channel. For the zig-zag channel the reaction coordinate is y) de–ned as the projection of the qzz(x,B. Smit et al. 99 Fig. 5 Schematic drawing of the silicalite pore structure. An alkane can jump from one intersection to another. The dotted lines show the paths via zig-zag channels and the solid lines those via straight channels. head group of the line connecting two intersections via a zig-zag channel (see Fig. 6). Both reaction coordinates are normalized in such a way that q ½ [0; 1]. In practice, the computation of a rate constant consists of two steps.The expression of the rate constant is given by20 kA?B(t)\ Sq5 (0)d[q*[q(0)]h[q(t)[q*]T Sd[q*[q(0)]T ] Sd(q*[q)T Sh(q*[q)T (3) where A and B are neighbouring intersections, q(t) is the reaction coordinate, h(x) is the Heavyside step-function, h(x)\1 for x[0 and h(x)\0 otherwise, and q* is the top of the free energy barrier separating the states A and B.is reaction coordinate along the straight channel, obtained by projecting the y coor- Fig. 6 qstr(y) dinate of the molecule on the line indicated. z) is obtained by projection of the x»z coordi- qzz(x, nate on the line indicated.100 Adsorption and diÜusion of hydrocarbons in zeolites The –rst part on the left-hand side of eqn. (3) is a conditional average, namely the average of the product given that the initial position of the reaction q5 (0)h[q(t)[q*], coordinate is q(0)\q*.An assumption in transition-state theory is that all trajectories that start on top of the barrier with a positive velocity will end up in state B. If this assumption holds, we have q5 (0)h[q(t)[q*]B1 2 o q5 o\SAkB T 2nmB (4) It is important to note that it is possible to test the validity of the above approximation and to compute this ensemble average exactly.This conditional average can be calculated from MD simulations. In these simulations we start with an initial con–guration taken from a Boltzmann distribution on top of the barrier. Such a distribution can be obtained from constrained MD or, if the constraint is sufficiently simple, from an MC simulation.In this work, we focus on the calculation of the second term on the right-hand side of eqn. (3), i.e. Sd(q*[q)T/Sh(q*[q)T, the probability density of –nding the system at the top of the barrier, divided by the probability that the system is on the reactant side of the barrier. This ratio, can be calculated from the free energy as a function of the order parameter.We can use the CBMC algorithm to compute this free energy as a function of the order parameter. Details of this calculation are given in the Appendix. A typical result is presented in Fig. 7. The free energies of 2-methylhexane as a function of order parameter in the straight and zig-zag channels are calculated. This –gure indicates that in the straight channel there are three barriers.The height of the –rst barrier (q\0.29) is ca. 14 which demonstrates that a jump over this barrier is kBT , indeed a rare event. In addition this –gure shows two additional barriers at q\0.5 and q\0.68. Because of the symmetry of the crystal the barriers at q\0.68 and q\0.29 are of equal height. Within the accuracy of the calculation, the barrier at q\0.5 is also of the same height.For the zig-zag channel we observe four barriers, the highest barrier has a height of 18 kBT . For both the zig-zag and straight channels, the middle barriers have a height of several therefore crossing of these barriers is also a rare event on the timescale of kBT , an MD simulation. Thus, the jump from one intersection to another consists of three consecutive jumps over the three barriers shown in Fig. 7 for the straight channel or over four barriers for the zig-zag channel. Fig. 7 Free energy of 2-methylhexane as a function of the position of the head group in the straight (left) and zig-zag (right) channels. For q\0 and 1 the head group is in the intersections. T \398 K.B. Smit et al. 101 If we assume that transition-state theory can be applied to this system, Fig. 7 is sufficient to compute the crossing rate. If we combine the results of the free energy calculation with those of the transmission rate, as obtained from transition-state theory, eqn. (4), we can compute the crossing rates. The results of this calculation are shown in Table 4. For the straight channel we –nd that the highest barrier is crossed 1.4]105 times s~1.This implies that a molecule resides in the intersection for ca. 7 ls, which is a very long time on the timescale of an MD simulation. Since, for both the zig-zag and straight channel, there is one barrier which is much higher than the others, we can assume that these barriers determine the hopping rates. With this assumption we obtain : ]105 and events s~1. wstr\1.37 wzz\5.0]104 Having computed the hopping rates from one intersection to another either via a straight channel or via a zig-zag channel, we have to relate these crossing rates to the diÜusion coefficients. In the limit of in–nite dilution the molecules perform a random walk on a lattice spanned by the intersections.The unit cell of this lattice is shown in Fig. 8. Since this lattice is anisotropic, we have three diÜerent diÜusion coefficients for the x, y and z directions21 Dxx\ 1 12 wzz a2, Dyy\ 1 12 wstr b2, Dzz\ 1 12 wzz wstr wzz]wstr c2 (5) where a, b and c are the unit vectors of the diÜusion lattice (see Fig. 8), and are wzz wstr the hopping rates via the zig-zag and straight channels, respectively. The formula for the diÜusion in the z direction re—ects that for a molecule to diÜuse in this direction it has to jump via a straight channel followed by a jump via a zig-zag channel.For the overall diÜusion coefficient, we can write D\ 1 12 wzzAa2] wstr wzz]wstr c2 2 B] 1 12 wstrAb2] wzz wzz]wstr c2 2 B (6) Numerical values for the diÜusion coefficient of 2-methylhexane are given in Table 5. Experimentally, diÜusion coefficients of branched alkanes are found in the range 10~9» 10~11 cm2 s~1.2 Comparison with our result : 8.5]10~10 cm2 s~1 shows that our –rst estimate of the diÜusion coefficient has the same order of magnitude as the experimental results. In the previous calculations, we have assumed that transition-state theory holds.We have performed some MD simulations with con–gurations that start on top of the Table 4 Hopping rates kTST/ [bF(qmin) [bF(q*) events s~1 str (1]2) [22.9 [6.8 1.4]105 str (2]3) [12.3 [6.9 4.3]1010 str (3]1) [11.9 [6.5 2.6]1010 zz (1]2) [23.8 [5.8 5.0]104 zz (2]3) [9.6 [5.3 1.3]1011 zz (3]4) [10.5 [5.2 1.0]1011 zz (4]1) [20.2 [9.0 1.4]109 bF(q) is the free energy for the order parameters, qmin the bottom of the well and q* the top of the barrier, kTST is the hopping rate as approximated with transition-state theory.102 Adsorption and diÜusion of hydrocarbons in zeolites Fig. 8 DiÜusion unit cell of silicalite ; the intersections are represented by dots and the channels by lines. a\20.1, b\19.9 and c\2]13.4 for this cell (where 20.1, 19.9 and 13.4 are the ” ” vectors of the unit cell of silicalite). barrier, to test whether or not transition-state theory is a reasonable approximation.These preliminary calculations indicate that transition-state theory may overestimate the diÜusion coefficients by a factor of 5»10. Unfortunately, these calculations were not sufficiently accurate to compute the crossing rate accurately. 5 Concluding remarks We have used the CBMC technique to investigate the behaviour of linear and branched alkanes in the pores of the zeolite silicalite.We –nd that the calculated heats of adsorption for both the linear and the branched alkanes are in good agreement with the experimental data. The simulations indicate that siting of the branched alkanes is very diÜerent from the siting of the linear ones. The linear alkanes can move ì freely œ in the channels of silicalite, Table 5 DiÜusion coefficients of 2-methylhexane in silicalite at T \398 K Dxx Dyy Dzz D /cm2 s~1 /cm2 s~1 /cm2 s~1 /cm2 s~1 1.7]10~10 4.7]10~10 2.1]10~10 8.5]10~10B. Smit et al. 103 the branched alkanes, however, are trapped with their CH group in the intersections of the zig-zag and straight channels. This trapping suggests that the diÜusion of the branched alkanes is an activated process ; most of the time the molecules are in the intersections but once in while a molecule hops from one intersection to another via a straight or zig-zag channel. These straight and zig-zag channels form a barrier for the diÜusion.We demonstrate that the CBMC technique can be used to compute the free energy of these diÜusion barriers. From these free energy barriers an estimate of the diÜusion coefficient can be made, if we assume that transition-state theory is valid for this system.The resulting diÜusion coefficient is in reasonable agreement with experimental data. In the future we will extend these calculations to test whether the transition state is valid for this system. At this point it is important to note that these calculations have been performed for a rigid zeolite lattice ; one can expect that allowing the zeolite atoms to move can have signi–cant consequences for the height of the free energy barrier.It is therefore, important to repeat these calculations with a —exible zeolite lattice. Appendix: free energy calculation One part of the calculation of barrier crossing rate is the computation of the free energy as a function of the order parameter.For the diÜusion of branched alkanes in zeolites, we use the position of the head as the order parameter. Here, we demonstrate how to calculate the free energy as a function of this order parameter. In the CBMC algorithm the Rosenbluth scheme is used to generate new conformations of the hydrocarbons. A molecule is grown atom by atom using the algorithm of Rosenbluth and Rosenbluth.23 During the growing of an atom several trial positions are probed, the energy of each of these positions is calculated, and the one with the lowest energy is selected with the highest probability according to : Pi( j)\ exp[[bui( j)] ;l/1 k exp[[bui(l)] \exp[[bui( j)] w(i) , where is the energy of atom i at trial position l.When the entire chain is grown, the ui(l) normalized Rosenbluth factor of the molecule in con–guration C can be computed: W (C)\ < i/1 l w(i)/k In ref. 20 it is shown that the average Rosenbluth factor is related to the chemical potential of the molecule: Sexp([bk)T\CSW T where C is a constant de–ning the reference chemical potential. One can also calculate the Rosenbluth factor as a function of the order parameter.This gives the chemical potential or free energy as a function of the order parameter. The number of samples for a given value of the order parameter is determined by the way we grow the molecule. For example, if we start the growing procedure by giving the CH group a random position in the zeolite, we obtain a uniform sampling of all values of the order parameter, irrespective of whether we sample the top or the bottom of the barrier.The method does not rely on the acceptance of the con–guration on top of the barrier. The fact that we do not need to rely on sampling con–gurations in which the molecule is on top of the barrier may cause difficulties when a —exible zeolite is used. During the simulation, the zeolite atom will never ì seeœ an alkane molecule on top of the barrier.As a consequence, one would never sample those con–gurations in which the zeolite104 Adsorption and diÜusion of hydrocarbons in zeolites lattice would ìrespondœ to the presence of a molecule on top of the barrier. Such —uctuations of the zeolite lattice may change the height of the barrier signi–cantly. Therefore, it is important to use a scheme in which we force the system to sample con–gurations on top of the barrier.A method which allows us to do this is, for example, the multiple histogram technique.20 References 1 Modelling of Structure and Reactivity in Zeolites, ed. C. R. A. Catlow, Academic Press, London, 1992. 2 S. D. Pickett, A. K. Nowak, J. M. Thomas, B. K. Peterson, J. F. Swift, A. K. Cheetham, C.J. J. den Ouden, B. Smit and M. Post, J. Phys. Chem., 1990, 94, 1233. 3 A. K. Nowak, C. J. J. den Ouden, S. D. Pickett, B. Smit, A. K. Cheetham, M. F. M. Post and J. M. Thomas, J. Phys. Chem., 1991, 95, 848. 4 R. L. June, A. T. Bell and D. N. Theodorou, J. Phys. Chem., 1992, 96, 1051. 5 E. Hernaç ndez and C. R. A. Catlow, Proc. R. Soc. L ondon, Ser. A, 1995, 448, 143. 6 B. Smit and J. I. Siepmann, Science, 1994, 264, 1118. 7 B. Smit and T. L. M. Maesen, Nature (L ondon), 1995, 374, 42. 8 S. P. Bates, W. J. M. van Wel, R. A. van Santen and B. Smit, J. Am. Chem. Soc., 1996, 118, 6753. 9 S. P. Bates, W. J. M. van Wel, R. A. van Santen and B. Smit, J. Phys. Chem., 1996, 100, 17573. 10 E. J. Maginn, A. T. Bell and D. N. Theodorou, J. Phys. Chem., 1995, 99, 2057. 11 Y. Wang, K. Hill and J. G. Harris, J. Phys. Chem., 1994, 100, 3276. 12 M. P. Allen and D. J. Tildesley, Computer Simulation of L iquids, Clarendon Press, Oxford, 1987. 13 A. G. Bezus, A. V. Kiselev, A. A. Lopatkin and P. Q. Du, J. Chem. Soc., Faraday T rans. 2, 1978, 74, 367. 14 C. L. Cavalcante Jr. and D. M. Ruthven, Ind. Eng. Chem. Rev., 1995, 34, 177. 15 F. Eder, PhD thesis, Universiteit Twente, The Netherlands, 1996. 16 R. L. June, A. T. Bell and D. N. Theodorou, J. Phys. Chem., 1990, 94, 1508. 17 B. Smit and J. I. Siepmann, J. Phys. Chem., 1994, 98, 8442. 18 C. H. Bennett, in DiÜusion in Solids : Recent Developments, ed. A. S. Nowick and J. J. Burton, Academic Press, New York, 1975, pp. 73»113. 19 D. Chandler, J. Chem. Phys., 1978, 68, 2959. 20 D. Frenkel and B. Smit, Understanding Molecular Simulations : from Algorithms to Applications, Academic Press, Boston, 1996. 21 B. Smit, L. D. J. C. Loyens, G. L. M. M. Verbist and D. Frenkel, in preparation. 22 J. Kaé rger and D. M. Ruthven, DiÜusion in Zeolites and other Microporous Solids, Wiley, New York, 1992. 23 M. N. Rosenbluth and A. W. Rosenbluth, J. Chem. Phys., 1955, 23, 356. Paper 7/01559C; Received 5th March, 1997

ISSN:1359-6640    

DOI:10.1039/a701559c

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 105»118 Temperature dependence of the levitation eÜect Implications for separation of multicomponent mixtures§ Subramanian Yashonath*abc and Chitra Rajappaa a Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012, India b Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore-560012, India c Condensed Matter T heory Unit, Jawaharlal Nehru Centre for Advanced Scienti–c Research, Jakkur, Bangalore-560064, India Sufficiently long molecular dynamics simulations have been carried out on spherical monatomic sorbates in NaY zeolite, interacting via simple Lennard-Jones potentials, to investigate the dependence of the levitation eÜect on the temperature.Simulations carried out in the range 100»300 K suggest that the anomalous peak in the diÜusion coefficient (observed when the levitation parameter, c, is near unity) decreases in intensity with increase in temperature.The rate of cage-to-cage migrations also exhibits a similar trend. The activation energy obtained from Arrhenius plots is found to exhibit a minimum when the diÜusion coefficient is a maximum, corresponding to the cB1 sorbate diameter.In the linear or normal regime, the activation energy increases with increase in sorbate diameter until it shows a sharp decrease in the anomalous regime. Locations and energies of the adsorption sites and their dependence on the sorbate size gives interesting insight into the nature of the underlying potential-energy surface and further explain the observed trend in the activation energy with sorbate size.Cage residence times, show little or no change with temperature for the qc , sorbate with diameter corresponding to cB1, whereas there is a signi–cant decrease in with increase in temperature for sorbates in the linear regime. qc The implications of the present study for the separation of mixtures of sorbates are discussed.A number of phenomena and processes in nature involve diÜusion of atoms, ions or molecules in con–ned regions.1 Hence, it is necessary to understand diÜusion in porous media. One of the best known and most widely used class of porous solids are the zeolites.2h7 Two, frequently interdependent, principal processes occur in zeolites viz. reactions and diÜusion.8 DiÜusion in zeolites, when no reactions occur, is used in the separation of mixtures of atoms and molecules owing to the molecular sieve properties of zeolites.Recently, it has been observed that diÜusion is in—uenced not only by geometrical factors such as the relative size and shape of the guest and the void, but also by the sorbate»host interaction.9 Under certain conditions, the in—uence of the non-geometrical factors could predominate over the geometrical factors.9 Studies carried out recently suggest that, over a wide range of sorbate diameters, the diÜusion coefficient is proportional to the reciprocal of the square of the sorbate diameter.This is termed the § Contribution No. 1000 from the Solid State and Structural Chemistry Unit. 105106 T emperature dependence of the levitation eÜect Fig. 1 Eight a-cages of zeolite NaY.Note that each a-cage is tetrahedrally connected to four other a-cages through 12-membered rings. The spheres indicate the cation positions. linear regime. There exists a range of sorbate diameters for which the diÜusion coefficient is not a simple function of the sorbate diameter. This is termed the anomalous regime.10 A signi–cant peak is observed in the diÜusion coefficient at a particular value of the sorbate diameter, for a given zeolite.More generally, the peak occurs when the Table 1 Potential parameters for sorbate»sorbate and sorbate»zeolite interactionsa pss psO psNa /” /” /” 6.8 4.67 5.08 6.3 4.42 4.83 6.0 4.27 4.68 5.5 4.02 4.43 4.96 3.75 4.16 4.1 3.32 3.73 3.48 3.01 3.42 3.07 2.81 3.22 2.67 2.61 3.02 and are 1.837 485, a ess , esO esNa 1.539 079 and 0.268 502 kJ mol~1, respectively.S.Y ashonath and C. Rajappa 107 levitation parameter, c, approaches unity c\2.21@6psz pw (1) where is the sorbate»zeolite interaction parameter in the Lennard-Jones potential psz and is related to the diameter of the narrowest part of the void space. This eÜect may pw be termed the levitation eÜect, for reasons pointed out elsewhere.10 The eÜect appears to be quite general and is observed in all sorbate»host systems investigated to date.9h11 Thus, the eÜect appears to manifest itself irrespective of the geometrical and topological features of the porous system.All studies reported hitherto have been carried out at low sorbate concentrations close to the in–nite dilution limit and at a single temperature.9,10 The dependence of the diÜusion coefficient, D, on the sorbate concentration has been investigated recently.12 Here, we report a study of the levitation eÜect and its dependence on temperature.We have carried out sufficiently long molecular dynamics (MD) simulations of spherical monatomic sorbates in zeolite NaY at several diÜerent temperatures.Properties such as the diÜusion coefficient, rate of cage-to-cage migration and cage residence times have been calculated. Activation energies and their dependence on c or sorbate diameter are also reported. Implications of these results for the separation of mixtures of two or more components will be discussed. Structure of zeolite and potential functions Zeolite Y belongs to the space group as shown by the neutron diÜraction work of Fd3 6 m Fitch et al.13 The cubic unit cell has a length of 24.85 and consists of 144 Si, 48 Al, ” 384 O and 48 Na atoms, corresponding to an Si/Al ratio of 3.The sodium atoms occupy all the SI and SII sites completely and there is zero occupancy of the SIII sites. The network of voids consists of eight a-cages of diameter ca. 11.8 in one unit cell, each of ”, which is tetrahedrally interconnected with four other cages via 12-membered rings of diameter ca. 7.4 (see Fig. 1). ” Both sorbate»sorbate and sorbate»zeolite interactions were assumed to be of the well known (6, 12) Lennard-Jones form, Only the interactions between the sorbate and /(rij). the O and Na atoms of the zeolite have been considered, Si atoms having been rendered inaccessible to the sorbate by the bulkier oxygen atoms.The self-interaction parameters for the zeolite atoms were taken from the work of Kiselev and Du.14 Cross-interaction parameters for the sorbate»zeolite interactions were computed using the Lorentz» Berthelot combination rules.15 Table 1 lists the sorbate»sorbate and the sorbate»zeolite potential parameters.The total potential energy of the system is given by U\Uss]Usz\1 2 ; i/1 N ; j/1, jEi N /(rij)] ; i/1 N ; z/O, Na /(riz) (2) Computational details The system consisted of 64 sorbate atoms in 2]2]2 unit cells of zeolite Y. The dimension of the cubic simulation cell is 49.7 The atoms of the zeolite framework and the ”. extraframework cations were not included in the MD integration.All sorbate atoms were of the same size in any particular simulation. Simulations were carried out at several sorbate sizes between 2.67 and 6.8 (Table 1). The sorbate»sorbate interaction ” parameter, was chosen to be 221 K for all sizes. The sorbate atoms were assigned a ess , mass of 40 u. A spherical cut-oÜ of 12 for both guest»guest and guest»zeolite inter- ” actions was used.Cubic periodic boundary conditions were imposed.108 T emperature dependence of the levitation eÜect All the simulations were carried out in the microcanonical ensemble using the MD technique. The velocity Verlet scheme was used to carry out the integration. A timestep of 10 fs was found to be adequate for a good conservation of the total energy.Most properties have been calculated from positions and velocities stored every 0.2 ps in the production phase of the MD run but the mean-square displacement was obtained from positions of particles at intervals of 0.1 ps. The starting con–guration consisted of one sorbate in each a-cage. All sorbates were assigned initial velocities from a Maxwell» Boltzmann distribution appropriate to that temperature.Simulations were performed with an equilibration of 150 ps. Data from a period of 150 ps immediately after equilibration were not used but this was followed by a production run of 4 ns. Mean-square displacements were calculated from data accumulated over 4 ns. However, cage residence times, rate of cage-to-cage crossings, and rate qc , kc of cage visits, were calculated from data accumulated over the last 2 ns, as discussed kv in ref. 16. Recent calculations on sorbates in zeolite A suggest that the error in D for a system of 10 particles and a run of ca. 12 ns is ca. 30%.17 Since zeolite NaY closely resembles zeolite NaCaA in structure, the errors involved in the two systems are likely to be similar. Here, we have a larger system (64 particles) and so a run length of 2 ns is expected to be sufficient to obtain D to the same accuracy.We carried out runs at 100, 150, 200 and 300 K for various sizes of sorbates. Results and Discussion Temperature dependence of diÜusion anomaly Table 2 lists the average values of sorbate»sorbate, and sorbate»zeolite SUssT SUszT interaction energies for various values of the sorbate diameter together with the pss average temperature for each of the runs.It is evident that, at any given temperature, the average values of and generally increase in magnitude with sorbate size, SUssT SUszT pss , Table 2 Average sorbate»sorbate, sorbate»zeolite potential energies and average temperature for runs with various sizes of sorbates pss SUssT SUszT STT pss SUssT SUszT ST T /” /kJ mol~1 /kJ mol~1 /K /” /kJ mol~1 /kJ mol~1 /K 100 K: 200 K: 2.67 [0.059416 [10.2111 97.9 2.67 [0.177874 [7.08402 214.9 3.07 [0.112130 [11.6637 101.4 3.07 [0.242538 [9.13442 203.7 3.48 [0.153722 [13.6013 101.4 3.48 [0.347867 [10.7193 205.7 4.1 [0.510877 [16.4659 112.7 4.1 [0.730173 [14.4345 192.7 4.96 [1.055040 [21.9666 119.6 4.96 [1.148860 [20.1186 205.2 5.5 [1.091600 [25.5386 101.4 5.5 [0.955524 [23.7426 196.6 6.0 [1.396790 [29.8292 97.7 6.0 [1.087670 [28.8276 201.3 6.3 [1.619580 [33.8558 114.3 6.3 [1.295490 [32.9441 192.6 6.8 [1.820640 [40.5764 114.5 6.8 [1.365260 [39.5742 193.9 150 K: 300 K: 2.67 [0.156873 [8.60395 145.3 2.67 [0.175078 [5.95413 314.4 3.07 [0.216168 [10.04810 153.1 3.07 [0.251729 [7.53546 303.2 3.48 [0.308058 [12.21220 143.5 3.48 [0.352864 [9.29703 310.9 4.1 [0.753267 [15.35860 151.5 4.1 [0.584873 [12.6788 312.4 4.96 [1.408530 [21.20690 159.1 4.96 [0.830921 [18.5899 299.3 5.5 [1.042260 [24.43570 150.5 5.5 [0.828279 [22.7609 301.4 6.0 [1.231750 [29.33750 144.7 6.0 [0.917660 [28.0115 313.5 6.3 [1.426180 [33.40470 151.4 6.3 [1.018940 [31.9473 296.5 6.8 [1.665620 [40.20140 147.3 6.8 [0.979625 [38.2838 302.7S.Y ashonath and C. Rajappa 109 Fig. 2A: D vs. at temperatures as shown. The least-squares –t to the points in the linear 1/pss 2 , regime i.e. with up to 4.96 is also shown. B: without correction for recrossings, vs. pss ”, kc, 1/pss 2 at temperatures shown. C: with recrossings taken into account, vs. at temperatures kv, 1/pss 2 shown. leading to stronger interaction of the sorbate with the zeolite.As we shall see, this behaviour is responsible for the observed trend in D and other dynamical quantities. Fig. 2A shows a plot of D vs. for the four diÜerent temperatures. At 100 K, it is 1/pss 2 , clear that there is a pronounced peak in the diÜusion coefficient. At 150 K the peak is110 T emperature dependence of the levitation eÜect still signi–cant, but at 200 K the peak begins to decrease in intensity and by 300 K, there is a signi–cant attenuation of the peak, leading to only a small rise near cB1.Similar trends are also observed in and (see Fig. 2B and C). By 300 K, the peak in also kc kv kc shows some attenuation, though not to the same extent. These results suggest that at higher temperatures the anomalous peak is comparatively less pronounced.Activation energy and its dependence on Ea r ss The Arrhenius behaviour of sorbates whose sizes lie in the anomalous linear regime and the dependence of the activation energy, on the sorbate size was investigated. Fig. 3 Ea , shows the Arrhenius plot of the diÜusion coefficient D for diÜerent sizes of sorbates. The least-squares –t straight line is also shown.It is seen that the –t obtained is quite good in most cases. The are plotted vs. in Fig. 4. For the smallest kJ Ea s pss pss\2.67 ”, Ea\4.79 mol~1. With increase in D decreases and the minimum in D in the linear regime is at pss , . The shows an increase with increase in until 4.96 when pss\4.96 ” Ea pss ”, Ea\5.89 kJ mol~1. Thus the maximum in occurs when D is a minimum.In the linear regime Ea increases with in the anomalous regime, kJ mol~1 for Ea pss; Ea\3.26 pss\6.0 ”, corresponding to the peak position. This value of is the lowest for all values of Ea pss : even lower than that obtained for the smallest sorbate, viz. We have pss\2.67 ”. avoided choosing values of signi–cantly lower than 2.67 since there is a likelihood pss ” that the smaller sorbates may then enter the sodalite cages.This would result in additional complexity in the diÜusion process. Note, also, that decreases sharply as we Ea enter the anomalous regime, when increases from 4.96 to 6.0 pss ”. Fig. 5 and 6 show Arrhenius plots for and respectively. The obtained from kc kv , Ea s these are also shown in Fig. 4. The trend in the obtained from is similar to that Ea kv obtained for D.The obtained from shows a trend similar to that obtained from the Ea kc Arrhenius plots of D in the linear regime but shows no increase beyond pss\6.0 ”. Fig. 3 Arrhenius plots of ln D vs. 1/T shown for diÜerent sorbate sizes. The solid lines represent the least-squares straight lines –tted to the data points, which are represented by the solid squares.Note the lowering of for Ea pss\6.0 ”.S. Y ashonath and C. Rajappa 111 obtained from Arrhenius plots of (a) D, (b) and (c) vs. Fig. 4 Ea s kc kv pss Fig. 5 Arrhenius plots of vs. 1/T for diÜerent sorbate sizes. The lines indicate the least- ln kc squares –t to the data points. Note the change in slope with sorbate size.112 T emperature dependence of the levitation eÜect Fig. 6 Arrhenius plots of vs. 1/T for sorbate sizes as indicated. The lines are the least-squares ln kv –t to the data points. Further, from Arrhenius plots of are negative in the anomalous regime. This is Ea s kc consistent with earlier –ndings where a study of Ar in NaCaA showed that when c is near 1, the derived from the rate of cage-to-cage migration, is negative.18 Not e Ea kc , that re—ects short-time behaviour which is, in turn, dependent on the local potential- kc energy surface near the window.Implications of these results for the separation of mixtures will be discussed later. Subprocesses of overall diÜusion DiÜusion of the spherical sorbates in the zeolite consists of two subprocesses : (i) site-tosite migration within a given a-cage and (ii) cage-to-cage migration or diÜusion from one cage to another via the 12-membered window.Site-to-site migration. Before we try to understand the site-to-site migration process, it is essential to know the location and number of sites, the associated energy and how these vary with the sorbate size. We have carried out detailed energy-minimization studies in which the energy of the sorbate was minimized assuming a rigid zeolite framework.These calculations were carried out as a function of the sorbate size. Table 3 lists the coordinates of one of the sites located within one of the eight [rs4(xs , ys , zs)] a-cages. We refer to this site as a surface site for reasons that will become evident shortly. By symmetry, there are six such sites in each of the a-cage and these can be obtained by applying appropriate symmetry operations to the coordinates listed in Table 3.Also listed are the sorbate»zeolite interaction energies at such sites. For sorbates with up to 4.96 there are only six of these sites and the energies at these sites pss ”, correspond to the global minima. Beyond 4.96 additional sites begin to appear.(es) ”, Again, by symmetry, the number of such additional sites is four and they are located near the 12-membered rings. These are termed the window sites. All the window sites are located on the window plane with the exception of the largest sorbate size (pss\6.8 ”). The energies at these window sites are somewhat higher than the surface sites for (ew)S. Y ashonath and C.Rajappa 113 Table 3 Coordinates of surface sites, and window sites, and rs4(xs , ys , zs) rw4(xw , yw , zw) associated energies for diÜerent sorbate sizes surface site window site pss es ew /” (xs , ys , zs) /kJ mol~1 (xw , yw , zw) /kJ mol~1 2.67 (0.4005, 0.6411, 0.6089) [12.03 » » 3.07 (0.4113, 0.6371, 0.6129) [13.48 » » 3.48 (0.4224, 0.6250, 0.6250) [15.42 » » 4.1 (0.4392, 0.6250, 0.6250) [18.66 » » 4.96 (0.4631, 0.6250, 0.6250) [23.82 » » 5.5 (0.4781, 0.6250, 0.6250) [27.65 (0.5107, 0.5107, 0.4793) [23.91 6.0 (0.4923, 0.6250, 0.6250) [31.68 (0.5003, 0.5003, 0.5003) [30.94 6.3 (0.5007, 0.6250, 0.6250) [34.34 (0.5003, 0.5003, 0.5003) [35.39 6.8 (0.5159, 0.6250, 0.6250) [39.27 (0.5244, 0.5244, 0.5244) [41.82 kJ mol~1 and kJ mol~1.By 6.0 kJ pss \5.5 ”: es\[27.7 ew\[23.9 ”, es\[31.7 mol~1, comparable to kJ mol~1.For larger sorbate sizes, the energy at the ew\[30.9 window is the lower of the two (see Table 3). The location of the surface site shifts with change in size of the sorbate. When the sorbate diameter is small, the location of the surface site is near the centre of the central of the three four-membered rings interconnecting any two six-membered rings.With increase in sorbate size, the location of the surface site shifts away from the inner surface of the a-cage towards the cage centre. Fig. 7 shows a single a-cage together with the location of the surface and the window sites for several sorbate sizes. All six surface sites move towards the cage centre with increase in sorbate size, leading to a reduced distance between two neighbouring surface sites. The distance of the surface site from the cage centre, seems to vary linearly with the sorbate size, as shown in Fig. 8: rcs , pss , rcs\apss]b (3) It is possible to predict the location of a surface site, for any size of rs4(xs , ys , zs), sorbate where a\[0.69697 and b\7.472897. The following relation gives the position, of the surface site for any given rs , pss: rs\rc]n� rcs (4) where is the unit vector from the cage centre to one of the surface sites and is the n� rc position vector of the centre of the a-cage.This relation assumes an Si/Al ratio of 3.0 and a simple Lennard-Jones potential to model the sorbate»zeolite interaction, as described in the previous section. For sizes below 5.5 there are no local minima in the ” potential-energy surface near the window. Above and including 5.5 such local minima ”, begin to appear and, for 5.5 the location of this minimum is slightly oÜ the window ”, centre, while for and 6.3 the window site coincides with the centre of the pss \6.0 ” window.Thus, only surface site positions show considerable shift with sorbate size. The diÜerence in energy between the surface site and the window site becomes insigni–cant by the time the sorbate diameter approaches a value that corresponds to cB1, viz.for and beyond. pss\6.0 ” The data presented in Table 3 for the surface and window sites of a sorbate in the anomalous regime suggests a potential-energy surface (PES) which is less undulating for sorbates of large size. For a smaller sorbate, the PES is more undulating and, consequently, during a migration from one site to another, the sorbate particle is likely to encounter an energy barrier.This explains the dependence of the activation energies obtained from Arrhenius plots shown in Fig. 4, where it was found that the smaller sized sorbate had a higher activation energy than the larger sized sorbates that lie in the114 T emperature dependence of the levitation eÜect Fig. 7 (a) Surface site positions have been shown inside an a-cage of zeolite NaY for 1, 2.67 ; 2, pss: 3.07 ; 3, 3.48 ; 4, 4.1 ; 5, 4.96 ; 6, 5.5 ; 7, 6.0 ; 8, 6.3 and 9, 6.8 These are close to the inner surface of ”. the a-cage for small sorbates but move closer to the cage centre (C) with increase in The site pss .positions for the two smallest sizes deviate from the line de–ned by the site positions for the larger sorbates. (b) A single 12-ring window is shown with the window site positions for the four largest sorbate sizes (sorbates with less than 5.5 do not have window sites). Site positions for pss ” and 6.3 coincide with the window centre, while that for 5.5 is in the window plane pss \6.0” ” but oÜ centre whereas that for 6.8 is oÜ the window plane.” anomalous regime. Thus, there is no facile path, i.e. one with a small barrier, from one site to another, which could be traversed by a sorbate particle in the linear regime during its sojourn. As pointed out in the lattice model calculations of van Tassel et al.19 the lowering of the activation energy for cage-to-cage migration (which represents a vs.The line represents the least-squares –t to the data points. Fig. 8 rcs pss .S. Y ashonath and C. Rajappa 115 vs. T for as indicated. Note that is nearly independent of temperature for the larger Fig. 9 qc pss qc sorbate. migration from a site located in the parent cage to a site in a neighbouring cage) results in an increased mobility which is more than what would be obtainable by a similar lowering of the intracage site-to-site activation energy.Thus, for a sorbate in the anomalous regime, where the cage-to-cage barrier is much lower, a higher diÜusion coefficient is observed. Cage-to-cage migration. This process is somewhat slow and may be the ratedetermining step in the overall diÜusion. Since eliminates all cage-to-cage migration kv events from cage j to cage i which were preceded by cage-to-cage migration from cage i to j, this will underestimate the actual number of cage-to-cage migration events.This is because the dynamical correction for the rate would eliminate only a fraction of these recrossings.20h22 While is likely to lead to an overstimate of actual cage-to-cage kc migration events, it is still worthwhile to look at how this quantity changes with temperature and sorbate size.The inverse of is the cage residence time. Fig. 9 shows a kc plot of the cage residence times at various temperatures for and 6.0 The pss\3.07 ”. cage residence time, changes signi–cantly with temperature from ca. 16.6 ps at 150 K qc , to ca. 7.1 ps at 300 K for In contrast, for changes from 3.5 ps pss\3.07 ”.pss\6.0 ” qc Table 4 Cage residence times, for diÜerent qc , sorbate sizes at various temperatures qc/ps ca. 150 K ca. 200 K ca. 300 K pss /” 2.67 16.9 9.6 6.5 3.07 16.6 10.6 7.1 3.48 16.7 11.9 7.5 4.1 27.2 17.1 8.9 4.96 34.4 21.1 10.1 5.5 13.8 9.2 6.4 6.0 3.5 3.5 3.7 6.3 2.5 2.6 3.0 6.8 2.9 3.1 3.6116 T emperature dependence of the levitation eÜect at 150 K to 3.7 ps at 300 K.Thus, there seems to be only a marginal change in for a qc particle in the anomalous regime. Table 4 lists the cage residence times for various sorbate sizes at diÜerent temperatures. Conclusions The diÜusion anomaly is pronounced at lower temperatures and the intensity of the observed anomalous peak decreases with increase in temperature; in zeolite NaY, at ca. 300 K, the peak almost vanishes. This will have considerable implications for the use of the diÜusion anomaly for separation of mixtures. increases with increase in the size, of the sorbate initially, when lies in the Ea pss , pss linear regime. It is highest foe particle whose diameter is at the junction between the linear and anomalous regimes, when the diÜusion coefficient is a minimum.Further increase in takes it into the anomalous regime and now shows a sharp decrease pss Ea with Corresponding to this minimum in we observe a peak in the diÜusion pss . Ea , coefficient. One of the subprocesses of diÜusion is cage-to-cage migration. It is found that the cage residence time, exhibits a strong dependence on the temperature for a sorbate qc , whose diameter is signi–cantly smaller than that corresponding to cB1.For a sorbate with cB1, the cage residence time is only weakly dependent on the temperature. It would be interesting to see how site-to-site migration depends on temperature for sorbates with diÜerent c values. These results have important consequences for the separation of mixtures of sorbate atoms.There are two principal methods of separation of mixtures: (i) equilibrium separation methods and (ii) kinetic based methods. The process we propose here is based on the diÜerence in kinetic properties, in particular, on the diÜerence in the diÜusion coefficients of the constituents of the mixture. Fig. 10 shows plots of ln D vs. 1/T for sorbates with and 6.0 corresponding to the linear and anomalous regions.pss\4.1 ”, The plots intersect at ca. 204 K. Below this temperature, D for the larger sorbate is greater than that of the smaller sorbate. Above this temperature the reverse is true. From Fig. 2A it is seen that the ratio of the diÜusion coefficients of the sorbates of size Fig. 10 Arrhenius plots of ln D vs. 1/T for sorbate sizes as indicated, belonging to linear (4.1 ”) and anomalous (6.0 regimes.The lines represent the least-squares –t to the data points for each ”) sorbate size. Note that the two lines intersect at ca. 204 K. It is expected that the larger sorbate would diÜuse faster at temperatures lower than 204 K, whereas the reverse would be true at higher temperatures.S. Y ashonath and C. Rajappa 117 4.1 and 6.0 is 2.1 at 100 K with the larger sorbate having the higher diÜusion coeffi- ” cient.At 300 K, this ratio is 0.70, which suggests that the separation of mixtures of these sorbates may be more eÜectively achieved at 100 K than at 300 K. Note, however, that the diÜusion coefficients are ca. 1 to 2 orders of magnitude lower at 100 K than at 300 K. It might still be possible to circumvent this problem by reducing the length of the zeolite column or by using a zeolite membrane.It is possible to use the levitation eÜect to separate the components of mixtures. If there is a binary mixture with two components of sizes and then a kinetic-based p1 p2 , separation of mixtures could be achieved. (1) By using the diÜerence in the diÜusion coefficient in the linear regime to separate the components of the mixture.Here a zeolite is chosen such that c values for both the components lie in the linear regime. (2) By choosing a zeolite for which one of the components has a c value close to unity so that D for this component is in the anomalous regime while the other component has a c value that lies in the linear regime. We propose a possible method of separation of the latter type which makes use of the levitation eÜect.From Fig. 10, it is clear that, for such a system, temperature plays a crucial role. Above 204 K, the smaller component has a higher D and so is likely to be the –rst component to emerge from a column packed with the appropriate zeolite. Below 204 K, it is likely that the larger sorbate will emerge –rst as it has a higher value of D.Thus, temperature can be used as a switch, depending on which component is desired. From Fig. 2 it is evident that the diÜerence in D between a sorbate in the linear regime and one in the anomalous regime is larger at lower temperatures. Hence, a separation process based on the levitation eÜect would best be carried out at relatively low temperatures.On the contrary, a separation method based purely on the linear regime would be more efficient at higher temperatures. However, other factors may also have to be considered during the actual implementation. Recent studies have shown that the diÜusion anomaly persists even at higher concentrations of the sorbate.12 Therefore, it may be possible to use the diÜusion anomaly to separate mixtures eÜectively, even under industrial conditions. The authors acknowledge the Supercomputer Education and Research Centre, Indian Institute of Science for a generous grant of computational resources and the Department of Science and Technology, New Delhi, for partial –nancial support of this work.References 1 J. Chem. Soc., Faraday T rans., 1991, 87(13). 2 R. M. Barrer, Zeolites and Clay Minerals as Sorbents and Molecular Sieves, Academic Press, New York, 1978. 3 J. Karger and D. M. Ruthven, DiÜusion in Zeolites and Other Microporous Solids, Wiley, New York, 1992. 4 D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. 5 R. L. June, T. A. Bell and D. N. Theodorou, J. Phys. Chem., 1990, 94, 1508. 6 J. O. Titiloye, S. C. Parker, F. S. Stone and C. R. A. Catlow, J. Phys. Chem., 1991, 95, 4038. 7 N. J. Henson, A. K. Cheetham, B. K. Peterson, S. D. Pickett and J. M. Thomas, J. Comput. Aided Mater. Des., 1993, 1, 41. 8 D. W. Breck, Zeolite Molecular Sieves, Wiley, New York, 1974. 9 S. Yashonath and P. Santikary, Mol. Phys., 1993, 78, 1. 10 S. Yashonath and P. Santikary, J. Phys. Chem., 1994, 98, 6368. 11 S. Bandyopadhyay and S. Yashonath, J. Phys. Chem., 1995, 99, 4286. 12 R. Chitra and S. Yashonath, unpublished results. 13 A. N. Fitch, H. Jobic and A. Renouprez, J. Phys. Chem., 1986, 90, 1311. 14 A. V. Kiselev and P. Q. Du, J. Chem. Soc., Faraday T rans 2, 1981, 77, 1. 15 M. P. Allen and D. J. Tildesley, Computer Simulations of L iquids, Clarendon Press, Oxford, 1987. 16 S. Yashonath and P. Santikary, J. Phys. Chem., 1993, 97, 3849.118 T emperature dependence of the levitation eÜect 17 R. Chitra and S. Yashonath, J. Phys. Chem., 1997, in press. 18 S. Yashonath and P. Santikary, J. Phys. Chem., 1993, 97, 13778. 19 P. R. van Tassel, S. A. Somers, H. T. Davis and A. V. McCormick, Chem. Eng. Sci., 1994, 49, 2979. 20 D. Chandler, J. Chem. Phys., 1978, 68, 2959. 21 J. A. Montgomery, D. Chandler and B. J. Berne, J. Chem. Phys., 1979, 70, 4056. 22 T. Mosell, G. Schrimpf, C. Hahn and J. Brickmann, J. Phys. Chem., 1996, 100, 4571. Paper 7/01278K; Received 24th February, 1997

ISSN:1359-6640    

DOI:10.1039/a701278k

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss. 1997 106 119»133 General Discussion Dr Schoé n opened the discussion of the Introductory Lecture We have been using global optimisation methods to –nd the local minima of the potential-energy hypersurface in order to predict structure candidates for not-yet-synthesized compounds.1 Information about neither the exact composition nor the parameters or symmetry of the unit cell of the hypothetical compound is available and thus these quantities need to be determined during the global optimization»in contrast to the case where the compound has already been synthesized and e.g. powder diÜraction data or information about local coordination of the atoms/ions are available. Therefore it is crucial to employ potentials that can describe reasonably well as large a region of the energy landscape of the chemical system as possible in order to encompass a large range of potential structure candidates.However we often –nd that e.g. the transferability of potentials derived for known binary compounds to the unknown ternary compound under investigation is beset with problems. Thus I wonder (i) whether you have encountered similar problems and (ii) what route you would suggest for dealing with this problem when deriving the parameters for empirical potentials ? 1 J. C. Schoé n and M. Jansen Angew. Chem. Int. Ed. Engl. 1996 35 1286. Prof. Catlow replied You are right that transferability of potentials is a key issue. It is well known that in inorganic materials there are systematic changes in parameters on varying coordination numbers.However there are a number of cases where it has proved possible to derive reasonably transferable sets of parameters for oxide materials as in the work of Bush et al.1 I think that the best procedure to try to ensure transferability when using empirical potentials is to build into the potential functions systematic variations with key structural features e.g. coordination number as was done in the four-fold coordination. recent study of Takada et al.2 on B2O3 where boron can show both three-fold and 1 T. S. Bush J. D. Gale C. R. A. Catlow and P. D. Battle J. Mater. Chem. 1994 4 831. 2 A. Takada C. R. A. Catlow and G. D. Price J. Phys. Condens. Matter 1995 7 8659. Prof. Gillan commented I am a strong believer in the use of empirical models but it is worth commenting on the way that the role of these models is changing.With the rapid increase in the power of ab initio methods [both Hartree»Fock (HF) and density functional theory (DFT)] it is becoming possible to do with ab initio what would have been done before with empirical models. It is now possible to do accurate ab initio calculations (both static and dynamic) on systems of over 100 atoms. Good examples of this are recent simulations by Schwarz and co-workers,1 and Payne and co-workers2,3 on methanol in zeolites. 1 E. Nusterer P. E. Bloé che and K. Schwarz Chem. Phys. L ett. 1996 253 448. 2 J. D. Gale R. Shah M. C. Payne and I. Stich Abst. Pap. Am. Chem. Soc. 1997 213 20. 3 R. Shah M. C. Payne M-H. Lee and J. Gale Science 1996 271 1395.Prof. Catlow added I agree. Many problems that were only tractable a few years ago by using interatomic potential methods can now be investigated by electronic structure techniques and the range and scope of the latter methods will continue to expand. Nevertheless I am convinced for three reasons that there is a continuing role for the former techniques. First as we push simulation techniques to increasingly large and 119 120 General Discussion complex systems the interatomic-potential based approaches will continue to be needed as there will remain a domain of complexity which is beyond the reach of electronic structure techniques. Secondly there will be a need for such methods to provide approximate structures and models which can then be re–ned by electronic structure techniques.Thirdly there are a number of problems that are handled more appropriately by interatomic-potential based methods including for example the modelling of microporous crystal structures and the diÜusion of hydrocarbons within these structures. In summary there will be an on-going need to use methods employing interatomic potentials which are increasingly complementary to those based on ab initio techniques. Prof. Klein commented Both speakers (Prof. Gillan and Prof. Catlow) are correct. There is going to be a continuing need for empirical potentials to deal with large aperiodic systems. The current limitation of a few hundred atoms is inadequate to handle many interesting systems. Hence the continuing need for embedding methods which in turn will require improved empirical models.Sir John Meurig Thomas commented The point raised by Prof. Klein concerning the need for massive computational eÜort if one wishes to address the question of the electronic and structural properties of complicated zeolitic structures merits further comment. When one is interested (as I am1,2) in the role of transition metals (in the framework of open-structure molecular sieves) as selective oxidation catalysts it is quite realistic to take metal silsesquioxane (molecular) analogues3 because these have been shown4 to perform (as epoxidation catalysts) in a manner closely akin to the performance of titanosilicate catalysts. I know from Fig. 2 of his paper that Prof. Sauer has looked at polyhedral oligomeric silsesquioxanes [molecular structure (XSiO1.5)n with n\6 or 8] but more needs to be done computationally on these molecular entities with ions such as TiIV located at one of their vertices to which is attached a hydroxy group (just as exists in single-site titanosilicate solid oxidation catalysts).1 J. M. Thomas Philos. T rans. R. Soc. L ondon A 1990 333 17. 2 T. Maschmeyer F. Rey G. Sankar and J. M. Thomas Nature (L ondon) 1995 378 159. 3 H. W. Roesky R. Murugavel and V. Ahandrasekhar Accs. Chem. Res. 1996 29 183. 4 T. Maschmeyer M. Klunduk C. M. Martin J. M. Thomas B. F. G. Johnson and D. S. Shephard Chem. Commun. 1997 1847. Prof. Martin addressed Prof. Catlow In your paper you have discussed mainly isolated defects and their formation energies.However we also need calculations of the concentrations of defects. This means that we also need entropies or Gibbs energies of formation of defects to be able to use the law of mass action. In addition the interaction of defects (resulting possibly in defect clusters) and its in—uence on the energies and entropies of formation is of great interest. Could you comment on the actual possibilities in calculating such quantities and the agreement with experiment e.g. in systems where defect clusters are formed. Prof. Catlow responded It is certainly possible to calculate defect entropies using methods developed by Jacobs Gillan and Harding. Such methods were used by e.g. Tomlinson et al.1 to study defect clustering in transition metal oxides using a mass action formalism.Calculations on defect supercells which are increasingly feasible may also be used to investigate defect»defect interactions. 1 S. M. Tomlinson C. R. A. Catlow and J. H. Harding J. Phys. Chem. Solids 1990 51 477. Dr Allan communicated I have a brief comment on the calculated enthalpy of mixing for NiO»MnO presented by Prof. Catlow. Calculations on such mixtures have 121 General Discussion generally involved determining the energy of the smallest supercell consistent with the composition of interest (e.g. Ni MnO for an overall Mn content of 25%). If so care 3 may be required when comparing calculation and experiment owing to the arti–cial 4 periodicity and the possible neglect of clustering eÜects. It is worth referring to a few of our results.1 We have recently been studying the enthalpy of mixing of MnO and MgO.Using a combination of lattice statics and quasiharmonic lattice dynamics with two-body potentials,2 then the lowest energy supercell * of formula Mg6Mn2O8 indicates an enthalpy of mixing mixH at 1300 K of 4.8 eV. The larger supercell Mg12Mn4O16 [with the same Mn content (25%)] with more internal * degrees of freedom suggests a value of mixH smaller by ca. 25% i.e. 3.6 eV. A value * for mixH of 3.6 eV is also obtained in preliminary calculations using the same potentials and a new hybrid Monte Carlo»molecular dynamics method currently under development at Bristol for this type of problem. 1 J. A. Purton G. D. Barrera and N. L. Allan in preparation. 2 G. V.Lewis and C. R. A. Catlow J. Phys. C. Solid State Phys. 1985 18 1149. Prof. Cheetham opened the discussion of Prof. Sauerœs paper You describe embedded cluster calculations on the pure silica polymorph of faujasite. Have you been able to compare these calculations with periodic HF or DFT calculations on this system? The symmetry is very high and there are only –ve atoms in the asymmetric unit. Such a calculation would give us a higher level of con–dence in the embedding procedure. Prof. Sauer replied When we completed our study periodic HF and DFT results on silica faujasite were not available. At this meeting Prof. Dovesi presents a poster which shows that periodic HF calculations on silica faujasite are feasible using an improved CRYSTAL code. The use of symmetry-adapted crystal orbitals results in a blockdiagonal Fock matrix at each k point in reciprocal space.Table 1 shows the results obtained for faujasite.1 Our paper describes DFT not HF calculations. However we have published embedded cluster calculations using the HF method previously ;2 the results are also shown in Table 1. Although there are some diÜerences in the basis sets used the two sets of bond distances agree within 0.01 ”. The largest deviation of the bond angle is 5° but we should take into account that the bending potential of the SiwOwSi bond angle in silica is very —at. The sequence of observed bond angles3 is reproduced by both sets of calculations. Moreover the deviation between the two sets of calculated data is of the same magnitude as the remaining deviation from experiment owing to systematic errors connected with the HF approximation and the basis set used.In ref. 2 comparison has been made with HF calculations on the ìexpandedœ structure of sodalite using exactly the same basis set. For the SiwO bond distance and the Table 1 Observeda and calculatedb,c bond lengths and bond angles of silica faujasite SiwOwSi/degrees r(SiwO)/” QM-Potc obs.a QM-Potc obs.a QMperiodicb QMperiodicb 137 155 149 142 154 149 138 149 146 143 145 141 1.622 1.608 1.622 1.612 24.63 1.61 1.60 1.62 1.61 24.53 1.607 1.597 1.604 1.614 24.26 O(1) O(2) O(3) O(4) cell a Ref. 3. b Ref. 1. c Ref. 2. 122 General Discussion Si ” wOwSi bond angle diÜerent embedded cluster calculations yield 1.607 to 1.608 and 161.5 to 160.7° while the periodic calculation (CRYSTAL code) yields 1.612 ” and 160.6°.1 C. M. Zicovich-Wilson and R. Dovesi Chem. Phys. L ett. submitted. 2 U. Eichler C. M. Koé lmel and J. Sauer J. Comput. Chem. 1996 18 463. 3 J. A. Hriljac M. M. Eddy A. K. Cheetham J. A. Donohue and G. J. Ray J. Solid State Chem. 1993 106 66. Prof. Dovesi added Recent improvement of the CRYSTAL code has permitted a study of the silicon-only faujasite where the full symmetry of the system is maintained by using an all-electron basis set. These data together with a full optimization of the structure have recently been reported.1 1 C. M. Zicovich-Wilson and R. Dovesi Chem. Phys.L ett. submitted. Dr Hill addressed Prof. Sauer In your paper you mention the calculation of NMR chemical shifts for embedded systems. It is clear that the embedding method you use does only a mechanical and not an electronic embedding. NMR chemical shifts are clearly an electronic eÜect. How important is the inclusion of electronic eÜects in the embedding to calculate NMR chemical shifts and how large is the error you estimate for your mechanical embedding? Prof. Sauer responded Indeed in our embedding scheme the wavefunction of the cluster does not ìseeœ the electron distribution of the outer part. However the electrostatic interaction and mutual polarization between the inner and outer parts are described at the level of the interatomic potential function (shell model potential).This changes the structure of the embedded cluster and indirectly aÜects its wavefunction since a diÜerent structure will lead to a diÜerent wavefunction. As far as the calculation of NMR shifts is concerned we know that NMR shielding constants are structure-sensitive parameters and we can get them right by calculation only if the structure is right. Our calculation proceeds as follows we use the structure from the embedded cluster calculation de–ne a (generally diÜerent) cluster around the nucleus the shielding of which we are interested in and add an increasing number of coordination shells to this cluster until the calculated value of the shielding constant is converged. Speci–c information is given in ref. 1 and 2. The remaining error due to the –nite-size model is smaller than the errors connected with the CPHF method used and the basis sets employed.The latter errors are minimized in our approach by using secondary internal standards. 1 U. Eichler J. Sauer and M. Braé ndle J. Phys. Chem. B in press. 2 B. Bussemer and J. Sauer Solid State NMR in press. Prof. Sauer continued I should like to add some general comments on diÜerent embedding techniques. Embedding is an obvious idea but the diÜerent approaches are very diÜerent in their aims methods and implementations. There are theoretically very appealing ì electronic œ embedding techniques which start from the known periodic solution of the ideal crystal and look for the solution of a defect crystal. One example of a working implementation is the EMBED code of Pisani et al.1 It has been successfully used in several applications but a severe limitation of the approach is the assumption that the density matrix for the defect crystal outside the internal part is identical with the density matrix of the ideal crystal.This means that charge transfer between the internal part and the environment is forbidden. Although Pisani et al. found a method to make an a posteriori correction this is not satisfying for cases where the correction is as large as the eÜect itself. An example is the application of the EMBED code to the proton transfer from the acidic zeolite to an adsorbed ammonia molecule yielding 123 General Discussion ammonium cations.2 Comparison is made between a supercell calculation using periodic boundary conditions and the embedded cluster calculation.After applying the corrections the adsorption energies still have an error of ca. 20 kJ mol~1. Moreover the computational demands of the periodic calculation neither allow the use of an appropriate basis set (STO-3G was employed) nor can all degrees of freedom be optimized. The result is that contrary to experiments the neutral adsorption complex of ammonia is more stable than the ion-pair complex of the ammonium cation with the negatively charged framework. The error is ca. 200 kJ mol~1 but most of it is due to the poor description of the anionic framework with the STO-3G basis set (see the discussion in Section C.1 in ref. 3). Our embedding scheme is much more pragmatic.We would like to avoid periodic ab initio calculations since we are interested in very large supercells and we consider it vital to relax the structures fully. It is a lesson from gas-phase calculations that reliable reaction energies will not be obtained unless all degrees of freedom are relaxed. We only describe the cluster quantum mechanically and the outer part and the interaction between the internal part and the outer part are described by an ab initio parametrized interatomic potential function. We include mutual polarization of the inner and outer parts by use of the shell model potential. We have applied our approach to the same problem adsorption of and proton transfer to ammonia for the same zeolite.4 We used a double-zeta polarization basis set and fully relaxed all structure parameters.We –nd that the ammonium form is more stable than the neutral adsorption complex and we calculate meaningful values for the heat of adsorption. In addition we have made explicit comparison with a periodic calculation using the same basis set. We employed the CRYSTAL code in single-point calculations on the structure obtained by our combined QM-Pot approach. The error is 6»9 kJ mol~1 less than half the error of the theoretically more appealing ì electronic œ embedding scheme mentioned above. 1 C. Pisani F. Cora` R. Nada and R. Orlando EMBED93 user documentation University of Torino Italy 1993. 2 C. Pisani and U. Birkenheuer Int. J. Quant. Chem. 1995 29 221. 3 J. Sauer P. Ugliengo E. Garrone and V.R. Saunders Chem. Rev. 1994 94 2095. 4 M. Braé ndle J. Sauer R. Dovesi and N. Harrison in preparation. Prof. Hillier commented We need benchmark calculations to assess the validity of various embedding techniques. Perhaps a large plane wave calculation should be carried out to provide this information. Prof. Sauer commented There is the question of what criteria we have to assess the quality of an embedded cluster calculation. The best criteria are internal ones since they do not depend on the systematic deviation from observed data which every quantum mechanical method in combination with any choice of basis set shows. One such criterion is that the result should be stable with respect to the choice of diÜerent clusters for the same problem in particular with respect to a series of clusters of increasing size.Another criterion is agreement with fully periodic calculations using the same quantum method and basis set as used for the embedded cluster calculations. Prof. Klein said Linear scaling will enable the study of several thousand atoms. The prospects are excellent that this will be possible in the near future. However larger system often have important dynamical processes that occur in the nanosecond time domain. What are the prospects for achieving this length of simulation with ab initio methods? I suggest that at present it doesnœt look too hopeful unless some form of embedding technique is employed. Prof. Gillan responded The problems of going to larger systems and longer times are completely diÜerent.For large systems the problem is that for traditional ab initio 124 General Discussion methods the computer time is proportional to at least N2 (where N is the number of atoms) and the challenge is to develop methods in which the eÜort is proportional to N. It is now virtually certain that this will be achieved (as it has been already within tightbinding methods). However for long times the eÜort is already proportional to time and there doesnœt seem to be anything straightforward that can be done to speed up calculations. Dr Line commented Although we obviously cannot calculate the dynamics over such long timescales for large systems we surely can calculate quantities such as the activation energy for diÜusion which is an experimentally measurable quantity in quasielastic neutron scattering.Dr Schoé n asked Prof. Sauer Have you tried to go beyond the harmonic approximation in your calculation of free energies of a- and b-quartz ? Prof. Sauer replied No. This is feasible but would be computationally very demanding. For example Tsuneyuki et al.1 simulated the a»b phase transition of quartz by molecular dynamics. A 4]4]3 supercell (432 atoms) was used and 6000 time steps each for 18 diÜerent temperatures were found to be necessary. 1 S. Tsuneyuki H. Aoki M. Tsukada and Y. Matsui Phys. Rev. L ett. 1990 64 776. Prof. Hillier opened the discussion of Prof. Gavezzottiœs paper The stabilization of the 2-pyridone tautomer compared to 2-hydroxypyridine in a polar environment arises to a signi–cant extent from the preferential electronic polarization of the 2-pyridone.Would the inclusion of such polarization aÜect the conclusions of your simulation studies ? Prof. Gavezzotti responded The pyridone tautomer is stable in condensed phases and this justi–es our assumption. On the other hand polarization certainly aÜects all force –elds but there are no simple methods to account for it. Dr Line asked Have you carried out calculations of the polarization of the pyridine complexes and of the solvent eÜects of surrounding water molecules? These eÜects surely must be signi–cant for pyridine since it is polar. Prof. Gavezzotti responded The answer is unfortunately no for reasons alluded to in my reply to Prof. Hillier. Dr Price addressed Prof.Gavezzotti One very signi–cant result in your paper is that it clearly demonstrates the limitations of the assumption that organic crystal structures can be predicted by searching for the global minimum in the lattice energy. This assumption is the basis of several methods aiming at crystal structure prediction,1 including a commercial package. I have done a brief study of the possible crystal structures of 2-pyridone using a diÜerent model intermolecular potential and a diÜerent search method and obtain complementary results. I used an accurate model for the electrostatic component of the intermolecular forces using atomic multipoles up to hexadecapole to describe the atomic charge distribution derived by a distributed multipole analysis (DMA)2 of an MP2 6-31G** wavefunction for the molecule.The other intermolecular contributions were described by a diÜerent set of 6-exp atom»atom potential parameters which have been found to reproduce successfully the crystal structures of related molecules3 when used in conjunction with a DMA-based electrostatic model. Thus the two model potentials one with an accurate electrostatic model the other without any explicit representation of the electro-125 General Discussion static forces but with empirically –tted hydrogen-bonding potentials are very diÜerent in their philosophy. As shown in Table 2 the DMA-based potential also gives a minimum in the lattice energy reasonably close to the room-temperature structure with a reasonable lattice energy. The two potentials diÜer in which cell lengths are most signi–cantly in error con–rming that they are far from equivalent but yet reasonable approximations to the intermolecular potential for 2-pyridone.Using the DMA-based model potential I minimized the lattice energy starting from 100 hypothetical crystal structures for 2-pyridone. These were generated by a MOLPAK4 search for close-packed crystal structures in the 10 commonly observed packing coordinations in the –ve space groups in Table 1 of your paper. For each coordination type the 10 lowest-energy of the 25 most-dense structures were minimized using the DMA-based potential giving a mere 100 minimizations. The results in Fig. 1 (cf. Fig. 6 of your paper) con–rm that there are a large number of crystal structures within the energy range of possible polymorphs.The experimental structure was found easily with an exact match of the in–nite hydrogen-bonded chain structure as found in the structure obtained in minimizing from the experimental structure. However a P21/c structure involving hydrogen-bonded dimers was [1.3 kJ mol~1 lower in energy. Hence there are de–nitely two qualitatively very diÜerent types of crystal structure for 2-pyridone based on either dimers or catemers whose static energy diÜerence is even smaller for the more elaborate theoretically based potential than for the empirical crystal potentials. This is not an unusual case we have observed similar small energy diÜerences between minimum-energy structures with diÜerent hydrogen-bonding motifs for uracil and 6-azauracil.5 Hence the need to go to more elaborate dynamic simulation methods even to predict qualitatively which type of crystal structure might be found.My question based on the molecular dynamics simulations is what sort of simulation results might have led to a fairly con–dent prediction that a catemer would be found rather than a dimer in the experimental structure ? For example if there had been no ì—ying catemersœ in the carbon tetrachloride solution simulation would that have Fig. 1 Predicted minima in the lattice energy for 2-pyridone using a DMA-based potential showing the exact coincidence (superimposed square and diamond in the top left-hand corner) of the lowest-energy P212121 minimum with the minimum found from the P212121 experimental structure.All minima (except min from experiment) obtained using MOLPAK/DMAREL from 100 close-packed starting structures. Table 2 Results of static minimization starting from the experimental b/” a/” *a/a (%)a 5.900 5.795 5.41 5.64 » » ]1.3 ]5.3 13.645 13.564 13.82 expt 295 K expt 120 K empirical 14.37 DMA based a The percentage errors are relative to the room-temperature structure. P212121 crystal structure of 2-pyridone *c/c (%)a *b/b (%)a c/” 5.692 5.604 5.75 5.80 » » ]1.0 ]1.8 » » [8.3 [4.4 lattice energy /kJ mol~1 *V /V (%)a [86.6 (*Hsub) » » [6.1 ]2.5 » [81.4 [83.3 126 General Discussion 127 General Discussion implied that a dimer-based crystal structure would crystallize from carbon tetrachloride ? 1 H.R. Karfunkel and R. J. Gdanitz J. Comput. Chem. 1992 13 1171; A. M. Chaka R. Zaniewski W. Youngs C. Tessier and G. Klopman Acta Crystallogr. Sect. B 1996 52 165; B. P. van Eijck W. T. M. Mooij and J. Kroon Acta Crystallogr. Sect. B. 1995 51 99. 2 A. J. Stone and M. Alderton Mol. Phys. 1985 56 1047. 3 D. S. Coombes S. L. Price D. J. Willock and M. Leslie J. Phys. Chem. 1996 100 7352. 4 J. R. Holden Z. Du and H. L. Ammon J. Comput. Chem. 1993 14 422. 5 S. L. Price and K. S. Wibley J. Phys. Chem. A 1997 101 2198. Prof. Gavezzotti replied Molecular dynamics simulations for precursors to crystal structures are just tentative so far. I am afraid the question cannot be answered before (much) more experience is gained.Prof. Klein commented Given the large number of possible structures for your system the crystallization pathway may be complex. You seem to be dealing with competition between hydrogen bonding and van der Waals interactions. Perhaps you could look at an analogy to the protein folding problem to understand what is going on. There the idea of a ìrugged energy landscapeœ has evolved and the idea of funnelling down to the correct structure. In proteins the van der Waals (and entropy) terms are extremely important. That is also likely to be the case here. Prof. Cheetham said As a point of clari–cation I would like to point out that the number of hydrogen bonds per molecule is the same for both dimer and chain structures of the pyridone.Dr Schoé n commented Concerning the ruggedness of energy landscapes we have been studying the energy landscape of small polymers on a periodic lattice interacting via simple potentials1 using the lid-algorithm2 to determine the growth in the number of states and local minima as a function of the energy barriers one is able to cross. We –nd (i) many nearly degenerate deep-lying minima (ground states) and (ii) a highly rugged landscape where the number of accessible local minima grows exponentially with the height of the barrier that can be crossed starting from one of the ground states. Thus the large number of local minima found during the work presented here really should not be too surprising. Finally even for very simple ionic systems like Na`/Cl~ there exists a multitude of local minima representing very distinct structures that while not having been found experimentally in NaCl are often observed in other ionic or partly ionic AB compounds.3 1 J. C. Schoé n Habilitationsschrift Universitaé t Bonn 1997. 2 P. Sibani J. C. Schoé n P. Salamon and J-O. Andersson Europhys. L ett. 1993 22 479. 3 J. C. Schoé n and M. Jansen Comput. Mater. Sci. 1995 4 43. Prof. Jansen commented When comparing the molecular packings obtained by computer simulations with those found experimentally e.g. by crystallization from solution it is not sufficient just to discuss the energies of the –nal solids one must also consider kinetic and solvent eÜects governing nucleation and growth of the respective polymorphs. Prof.Cheetham said Dr Priceœs comment»that one typically –nds only one polymorph even though calculation predicts very large numbers»is rather depressing. It is also true for inorganic species (as Kapustinski showed many decades ago). Is it safe to ignore the entropy and base these calculations solely on enthalpy? Prof. Gavezzotti replied Apparently yes,1 but this conclusion is based on lattice vibrational entropy only. The question is one that still awaits a more complete answer. 128 General Discussion 1 A. Gavezzotti and G. Filippini J. Am. Chem. Soc. 1995 117 12 299. Dr Freeman commented Dr Price and Prof. Gavezzotti report that although the experimentally observed polymorphs are among those generated in their studies of the possible packings of molecules using potential-energy functions the experimentally observed structures are not necessarily of lowest energy.This implies of course that when powder diÜraction data can be obtained (as is the case for the majority of organic compounds) polymorph prediction (see for example ref. 1»5) can yield crystal packing information through the match of calculated and observed diÜraction pro–les. Indeed a number of studies have made use of pro–le match as an additional discriminant in simulated annealing and Monte Carlo studies.6h10 It is reasonable to conclude that as a complement to traditional methods and particularly where powder diÜraction is the only possible structural probe polymorph prediction methods are of immediate value. 1 R. J. Gdanitz Chem. Phys.L ett. 1992 190 391. 2 H. R. Karfunkel and R. J. Gdanitz J. Comput. Chem. 1992 13 1171. 3 H. R. Karfunkel and F. J. J. Leusen Speedup 1992 6 43. 4 H. R. Karfunkel B. Rohde F. J. J. Leusen R. J. Gdanitz and G. Rihs J. Comput. Chem. 1993 14 1125. 5 H. R. Karfunkel F. J. J. Leusen and R. J. Gdanitz J. Comput. Aided Mater. Des. 1993 1 177. 6 J. M. Newsam M. W. Deem and C. M. Freeman in Accuracy in Powder DiÜraction II NIST Special Publication No. 846 ed. E. Prince and J. K. Stalick National Institute of Standards and Technology Bethesda MD 1992 pp. 80»91. 7 T. M. NenoÜ W. T. A. Harrison G. D. Stucky J. M. Nicol and J. M. Newsam Zeolites 1993 13 506. 8 M. W. Deem and J. M. Newsam J. Am. Chem. Soc. 1992 114 7189. 9 C. R. A. Catlow J. M. Thomas C. M. Freeman P.A. Wright and R. G. Bell Proc. R. Soc. L ondon A 1993 442 85. 10 K. D. M. Harris M. Tremayne P. Lightfoot and P. G. Bruce J. Am. Chem. Soc. 1994 116 3543. Dr Freeman asked Could you comment on the possible role of crystal structure databases in determining packing rules ? Is it possible that energy functions could provide a convenient route to structural models and their –nal rankings could also draw on empirical relations obtained from known structures ? Prof. Gavezzotti responded I agree with your comment and did so in print.1 To answer your question The use of databases as a help in –nding the correct structure is still more of an art than a systematic exercise because ranking based on geometrical patterns is even more awkward than ranking based on energies.1 A. Gavezzotti and G. Filippini J. Am. Chem. Soc. 1996 118 7153. Dr Schoé n asked Are you familiar with the work by Hofmann and Lengauer1 on the structure prediction of molecular compounds using simple potentials ? 1 D. W. M. Hofmann and T. Lengauer Acta Crystallogr. Sect. A 1997 53 225. Prof. Gavezzotti responded I have not seen Hofmann and Lengauerœs paper. Thank you for pointing it out to me. I will consider it carefully. Dr Schoé n asked How did you calculate the contribution of the potential energy (as a function of temperature) to the total energy when calculating the heat capacity Cp ? Prof. Gavezzotti answered The C was roughly approximated by –nite diÜerences p over total energies at diÜerent temperatures. Clearly a better method is needed here.Dr Line opened the discussion of Dr Sherwoodœs paper In the section on 1H NMR shifts you report the shift of the OH stretch frequencies due to the inclusion of the point charge array relative to that with the Br‘nsted acid cluster alone. We have performed ab initio calculations1 on the water cluster found in the zeolite natrolite. We considered 129 General Discussion only one water molecule and the two neighbouring sodium atoms at 2.4 ” with the zeolite framework completely omitted. We –nd that our OH vibrational frequencies are remarkably similar to yours obtained for the point charge array (3754 and 3842 cm~1 for 6-31G** with MP2). This is therefore from the cation bonding eÜect alone since we did not consider hydrogen bonding at all. Have you tried calculating the OH shift for the point charge array alone without including the Br‘nsted acid site ? 1 C.M. B. Line and G. J. Kearley unpublished work. Dr Sherwood responded No we havenœt tried this and it isnœt clear how such a calculation would be interpreted. In considering the electrostatic environment of the water molecule it must be borne in mind that the strongest eÜects will arise from the closest zeolite framework atoms which are part of the ab initio cluster rather than being included through the point charges. To remove the cluster would leave an arbitrary rather meaningless partial contribution to the actual –eld. There is the additional problem that the structure would no longer be at equilibrium and the computation of IR vibrational frequencies would not be possible.Prof. Sauer commented It is certainly possible to calculate frequency shifts for water molecules adsorbed on metal cations in zeolites. Already very simple calculations (HF Mini-1 basis set) on free Mn`»OH complexes explain some qualitative features. For example for water adsorbed in Y-type zeolites containing diÜerently charged cations 2 cations but to lower values if trivalent cations are present. Accordingly the calculations the bending frequency of H2O is shifted to higher values for monovalent and divalent 2O»Li` and for H 2O»Mg2` yield bending frequencies higher than that of H2O while H for the H2O»Al3` complex a lower frequency than that of free H2O is calculated.1 Much better calculations are possible with todayœs computer codes.1 J. Sauer and R. Zahradnik Int. J. Quant. Chem. 1984 26 793. Dr Islam opened the discussion of Prof. Smitœs paper In your Monte Carlo simulations of hydrocarbon diÜusion in zeolites the host lattice was assumed to be rigid. How important is the incorporation of framework —exibility particularly for diÜusion of large hydrocarbon molecules? How would such a —exible model be included in your calculations ? Prof. Smit responded Since for large hydrocarbons there are no simulations reported on diÜusion in a —exible zeolite I can only speculate on the eÜects of a —exible lattice. Our simulations indicate that the diÜusion of linear alkanes is diÜerent from the diÜusion of branched alkanes. Compared with the branched alkanes the linear alkanes move freely in the channels.The —exibility of the zeolite lattice can have some in—uence on the diÜusion but I do not expect a large eÜect. The diÜusion of the branched alkanes however is an activated process in which the bulky head of the molecule has to squeeze itself through a narrow channel. In a —exible lattice this may be much easier and as a result in a —exible zeolite one would observe a lower free energy barrier. In principle the Rosenbluth free energy calculations can be done with a —exible lattice. In these calculations the hydrocarbon is a ìghost particle œ i.e. it does not in—uence the conformation of the zeolite. In the case of a rigid zeolite this is exactly what one wants to simulate. However in the case of a —exible zeolite this may give inaccurate results.If the hydrocarbon is introduced in the zeolite as a real particle the zeolite will respond to the presence of this particle. If this response is sufficiently small such that linear response theory is valid then one can compute the eÜects from —uctuations of the empty lattice and one can still use the Rosenbluth scheme. However if the perturbations 130 General Discussion of the zeolite caused by the presence of the molecule are large one has to use alternative free energy schemes in which the presence of the particle is explicitly taken into account (for example the blue moon ensemble umbrella sampling or histogram methods).1 1 D. Frenkel and B. Smit in Understanding Molecular Simulations From Algorithms to Applications Academic Press Boston 1996.Dr Yashonath asked (i) You have stated that 2-methylbutane preferentially places the tail in the entrance to the straight channel as compared to the zigzag channel. Why is this ? The dimensions of the straight and zigzag channels are about the same. (ii) You have mentioned the rotation of 2-methylbutane. Did you –nd any preference for rotation around the long molecular axis ? Prof. Smit answered (i) The reason that the tail of 2-methylbutane prefers the straight channel is that because of the bulky head group the zigzag channel is more difficult to reach than the straight channel. (ii) We have not looked at preferences for rotations. Prof. Bliek addressed both Prof. Smit and Dr Yashonath Obviously calculations on the self-diÜusivity of species in zeolitic structures are easier than those on diÜusivities involving counter-diÜusing species.How can such a situation be adequately modelled given the necessity to account for both sorbent»sorbate and sorbate»sorbate interactions. Dr Yashonath replied It is certainly easier to perform calculations on the selfdiÜusivity in zeolites as compared to the situation where a counter-diÜusing species is present. The difficulty is greater if the adsorbed —uid and the counter-diÜusing species are in the liquid phase instead of the vapour phase. The sorbate»zeolite interactions for many species are well known. In particular these are well known for hydrocarbons (see the recent work of Siepmann et al.1). Existing models for the sorbent»sorbate interactions are inadequate even for simple guest species such as rare gas atoms; see for example the recent work by Pellenq and Nicholson2 where they found that the potential of Kiselev and Du,3 used extensively in the literature yielded a signi–cantly higher value for the volume of channels.Signi–cantly more work needs to be done before one is able to obtain fairly accurate potentials for the sorbent»zeolite interactions of other molecules. More important seems to be the fact that the counter-diÜusion normally occurs signi –cantly more slowly than normal diÜusion. Experiments on counter-diÜusion indicate that the diÜusivities in these systems are signi–cantly smaller than those in previously evacuated zeolite samples.4 Molecular dynamics might prove to be inadequate in modelling counter-diÜusion since the diÜusivities are expected to be smaller and one may have to resort to other methods such as smart Monte Carlo or other more efficient sampling techniques.1 J. I. Siepmann M. C. Martin C. J. Mundy and M. L. Klein Mol. Phys. 1997 90 637. 2 J. M. Pellenq and D. Nicholson J. Phys. Chem. 1994 98 13 339. 3 A. V. Kiselev and P. Q. Du J. Chem. Soc. Faraday T rans. 2 1981 77 1. 4 C. N. Satter–eld and J. R. Katzer Adv. Chem. Ser. 1971 102 193; J. Karger and D. M. Ruthven in DiÜusion in Zeolites Wiley New York 1992 p. 433. Prof. Smit responded The simulations we have performed so far describe a single molecule at in–nite dilution. These simulations give a hopping rate of a molecule jumping from one intersection to another.Extending these calculations to higher concentrations is not straightforward. A possible way to study the eÜects of concentration and multi-component systems is to use a lattice model and use hopping rates as com-131 General Discussion puted from our MC scheme. If we assume that these hopping rates are not in—uenced by the presence of other molecules we can use a lattice model in which molecules are allowed to jump from one site to another. With such an approach it may be possible to study concentration eÜects. Prof. Catlow addressed Prof. Smit Your paper comments that the transition state theory approach exaggerates the diÜusion coefficient of the hydrocarbons by a factor of ca. 5»10 compared with corresponding MD simulations. Could you amplify this comment; are there possibilities for reducing this discrepancy ? Prof.Smit replied The hopping rate is a product of a term which includes the height of the free energy barrier and the so-called transmission coefficient.1 This product should give the same answer as a corresponding MD simulation provided someone has the patience to simulate sufficiently long to obtain reliable statistics with such an MD simulation (which is for our system of the order of microseconds!). Transition state theory assumes that all trajectories that start on top of the free energy barrier with a positive initial velocity will end up in the ìproductœ state. This implies that the transmission coefficient is 0.5 and the hopping rate can be estimated from the free energy pro–le only.Some preliminary tests have shown however that in our case far fewer trajectories end up in the product state. This implies that transition state theory for our choice of order parameter is not a good approximation and we have to perform MD simulations to compute the transmission coefficient. It may be possible that a diÜerent choice of order parameter gives a lower free energy barrier and hence a transmission coefficient that is closer to 0.5. 1 D. Frenkel and B. Smit in Understanding Molecular Simulations From Algorithms to Applications Academic Press Boston 1996. Prof. Jansen asked You have been entering the hydrocarbons into the zeolite channels by a non-physical mechanism. Does this aÜect the results ? For example the heights of the saddle points seem to be derived from a sequence of optimized con–gurations i.e.from local minima and therefore might be underestimated. How did you make sure that the hydrocarbons as a whole really can pass through the bottlenecks ? Prof. Smit answered It is correct that the growing scheme that is used in the con–gurational-bias Monte Carlo (CBMC) calculations is non-physical and hence the molecules enter the zeolite channel in an arti–cial way. It is important to note that the resulting distribution of con–gurations is the correct Boltzmann distribution only the way in which we obtain these con–gurations is not physical. If we were to use a physical scheme e.g. MD it would take many billion years of CPU time since experimentally it may take days before these systems are equilibrated.There are some zeolites however in which some of the cages are not accessible from the outside. The CBMC scheme like any other grand-canonical simulation technique would introduce molecules into these cages which would not be correct. If one knows this such artifacts can be avoided by –lling these cages with hard sphere molecules. For the systems that we have studied it is known from experiment that the molecules are adsorbed. Dr Freeman asked Could you comment on the feasibility of calculating free energy pro–les for activated events which account for sorbate»sorbate interactions ? Prof. Smit responded It is not straightforward to extend our calculations in which we assume in–nite dilution to cases in which sorbate»sorbate interactions are impor-132 General Discussion tant.An alternative approach would be the use of a lattice on which molecules can hop using the hopping rates calculated by our approach. My earlier response to Prof. Bliek is also relevant to your question. Prof. Gillan commented For simple defects in solids there are well established methods for calculating free energy pro–les and barriers. These involve doing constrained simulations with the reaction coordinate –xed at a sequence of values and hence calculating the potential of mean force. Is there any reason why such methods should not be applied to molecules in zeolites ? Prof. Smit responded No there is no reason why the methods you mentioned could not be applied to these systems. My earlier response to Prof.Islam is also relevant to your question. Prof. Sauer said Years ago Klaus Fiedler from the Central Institute of Physical Chemistry in Berlin performed MC simulations on adsorption of alkane chains and aromatic hydrocarbons in zeolites.1 He used a special importance sampling procedure. How does your con–gurational-bias MC diÜer from his approach? 1 B. Grauert K. Fiedler H. Stach and J. Jaé nchen in Zeolites Facts Figures Future ed. P. A. Jacobs and R. A. van Santen Elsevier Amsterdam 1989 pp. 701»713. Prof. Smit replied The starting point of both methods is the same molecules adsorbed in a zeolite and the interactions of these molecules with each other and with the zeolite are described with a given classical potential. The behaviour of this system is fully described by the partition function i.e.once this partition function is known all ensemble averages can be computed from this function. For most systems however this partition function cannot be computed exactly and Fiedler and co-workers have developed an MC scheme to approximate this partition function. In our CBMC scheme we cannot compute a partition function we can only calculate ensemble averages. However the calculation of these ensemble averages does not involve additional approximations. Prof. Cheetham opened the discussion of Dr Yashonathœs paper You have presented a large number of MD simulations relating to the so-called levitation eÜect. Is there any experimental validation of this eÜect ? Do you have candidate systems for the separations that you believe might be facilitated by the levitation eÜect ? Dr Yashonath answered Experimental evidence for the existence of this eÜect is something that is of considerable importance.To the best of my knowledge there have not been any experiments carried out subsequent to 1994 when the existence of an anomalous peak in the diÜusion coefficient was –rst reported based on molecular dynamics simulations.1 Among the problems in designing experiments to verify this eÜect are the in—uence of the grain boundary which can be overcome by the use of single crystals. Again there is the in—uence of the extra-framework cations and water but most of these can be circumvented. Assuming that these have been overcome it is possible to use two diÜusing species of diÜering size and then study their diÜusion in a zeolite or any other porous media of known structure.There are many porous solids whose structures are well known and this should not be a problem. However one diÜusing species should have a size with c@1 which corresponds to the linear regime while the other should have a size with cB1 (these shall be referred to as type I experiments). For simplicity we take two species which are spherical in shape. The problem is that in practice it is rarely possible to –nd two diÜerent sized sorbates without some change in other characteristics such as 133 General Discussion the mass or the interaction strength. Corrections for the change in mass could be applied but it is not clear how the changes in the interaction strength would alter the diÜusion though we have recently reported some preliminary results.2 For these reasons this design of the experiment does not seem appropriate.There is an alternative method which is probably better suited for laboratory experiment (termed type II experiments). This involves choosing one diÜusing species and two diÜerent hosts with diÜering void diameters in such a way that one void size corresponds to c@1 and the other cB1. Zeolites (preferably highly siliceous) and nanotubes may be ideal candidates. Note that the mass and even the interaction strength are unaltered if both hosts are of the same material. Yet another alternative method is to introduce one guest in a host material in which there are channels of diÜering diameters along diÜerent directions.If one could obtain from experiment the components of the diÜusion coefficient tensor then it would be possible to see if the levitation eÜect exists. The calculation on xenon3 in which you found that xenon diÜuses faster in the 10-membered channels of ferrierite as compared to the 12-membered channels of zeolite L provides a clue. If one could –nd a zeolite consisting of 10-membered channels along one direction and 12-membered cylindrical channels along another direction this would provide a good system to test the eÜect. Finally the problems with measuring the self-diÜusion coefficient have to be tackled since many of the techniques such as NMR neutron scattering etc. yield information about the diÜusion coefficient over a narrow range of D and place additional restrictions on the nature of the sorbate that can be used.It is gratifying to note that the range over which anomalous diÜusion is observed is not that narrow it extends from 5 to 7 ” in zeolite Y. However it should be noted that the intensity of the peak falls oÜ rapidly and hence one prefers to choose a sorbate»host system so as to be as close to the peak as possible. This places more stringent restrictions. In the case of zeolite Y if one were to be in the top half of the intensity then this range is narrowed to 5.3»6.3 ”. 1 S. Yashonath and P. Santikary J. Chem. Phys. 1994 100 4013. 2 B. Bhattacharjee and S. Yashonath Mol. Phys. 1997 90 889. 3 N. J. Henson A. K. Cheetham B. K. Peterson S.D. Pickett and J. M. Thomas J. Comput. Aided Mater. Des. 1993 1 41. Prof. Smit asked The levitation eÜect occurs if the size of the molecule matches the size of a channel. This occurs especially at low temperatures and with a narrow range of the diameter. Does this eÜect still exist if a —exible zeolite lattice is used? Dr Yashonath responded Yes the eÜect manifests noticeably only at low enough temperatures and in a fairly narrow diameter range. Please see the answer to Prof. Cheethamœs question (above) where we have pointed out that the range is not really that narrow. The eÜect still exists when the framework of the zeolite is modelled as a —exible one and the evidence for this comes from simulations that we have done in at least two zeolitic systems. First in the case of zeolite A it was found that the eÜect persisted when the framework of the zeolite was modelled as a —exible cage.1 There was however some decrease in the intensity of the anomalous peak. Secondly we studied a silicalite system. Here results for the rigid cage have also been reported.2 Even in this silicalite system the peak in the diÜusion coefficient persisted. These results seem to suggest that the levitation eÜect does exist when the zeolite framework is modelled as —exible. 1 P. Santikary and S. Yashonath J. Phys. Chem. 1994 98 9252. 2 S. Bandyopadhyay and S. Yashonath J. Phys. Chem. 1995 99 4286.

ISSN:1359-6640    

DOI:10.1039/FD106119

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 135»154 Ab initio simulation of molecular processes on oxide surfaces Philip J. D. Lindan,a* Joseph Muscat,a Simon Bates,b Nicholas M. Harrisona and Mike Gillanb a DCI, CCL RC Daresbury L aboratory, W arrington, UK WA4 4AD b Physics Department, Keele University, StaÜordshire, UK ST 5 5BG Ab initio calculations based on both density functional theory (DFT) and Hartree»Fock (HF) methods are used to investigate the energetics and equilibrium structure of the stoichiometric and reduced (110) surface, the TiO2 adsorption of potassium on the (100) surface and of water on the (110) surface.It is shown that DFT and HF predictions of the relaxed ionic positions at the stoichiometric surface agree well with each other and fairly well with recent X-ray diÜraction measurements.The inclusion of spin polarisation is shown to have a major eÜect on the energetics of the reduced surface formed by removal of bridging oxygens. The gap states observed to be induced by reduction are not reproduced unless spin polarisation is included. Static and dynamic DFT calculations on adsorbed water on TiO2 (110) con–rm that dissociation of leads to stabilisation at low cover- H2O ages, but suggest a more complex picture at monolayer coverage, in which there is a rather delicate balance between molecular and dissociated geometries.I Introduction Until recently, the understanding of oxide surfaces lagged far behind that of metals and semiconductors. This was partly because of experimental difficulties, and partly because the complexity of many oxide materials made them difficult to study with accurate theoretical methods.The situation is now changing rapidly, and the last few years have seen important new experimental measurements1h3 as well as accurate quantum-mechanical calculations on a variety of oxide surfaces and on adsorbed molecules.4 Titanium dioxide is of particular interest, both because of its important and varied applications, and because of the challenges it poses to ab initio theory. The theoretical problems arise from the treatment of exchange and correlation in partially occupied d states when the material is reduced.These problems make it interesting to compare diÜerent ab initio methods. We have therefore used both DFT and HF approaches using plane-wave and local basis sets in order to explore the sensitivity of the computed answers to the approximations made. Our calculations address three issues : the equilibrium structure of the perfect surface, the atomic and electronic structure of the reduced surface obtained both by removal of oxygen and by addition of alkali metal and the energetics and structure of water adsorbed on the surface.Early ab initio calculations on the energetics and relaxed structure of oxide surfaces were made on MgO (001) using HF theory,5 and HF calculations have since been reported on several other oxide surfaces, including The HF approach has TiO2 .6h10 also been used in ab initio studies of molecular adsorption, work on adsorption at H2O MgO (001)11 being particularly relevant to the results to be reported here.DFT methods began to be applied to oxide surfaces somewhat later, but progress has been rapid, and a number of detailed studies have been reported on perfect and defective oxide surfaces, as well as on the adsorption of simple molecules.12h23 135136 Simulation of molecular processes on oxide surfaces The equilibrium positions of the ions at oxide surfaces can diÜer greatly from their perfect-lattice positions, and the comparison of predicted and measured positions is an important way of testing the reliability of the theoretical methods. Recent progress in the application of glancing angle X-ray diÜraction to oxide surfaces provides detailed surface structural information for the –rst time. We shall compare DFT and HF predictions of the relaxed positions at the perfect (110) surface with the very recent TiO2 diÜraction measurements of Charlton et al.3 The comparisons demonstrate both the good accord of the two theoretical approaches and their semi-quantitative agreement with experiment.Departure from stoichiometry is a characteristic of transition-metal oxides. Reduction by removal of oxygen is particularly important, since experiments are often performed under ultra-high vacuum, which tends to induce oxygen loss.In addition, many experiments require signi–cant surface conductivity, which is often produced by deliberate reduction. Reduction is also important in practice, since is often sub- TiO2 stoichiometric in real applications or under processing conditions. We shall present DFT and HF calculations on the relaxed structure of the reduced (110) surface TiO2 and on its electronic structure.A major issue here is that of spin polarisation, since reduction from Ti4` to Ti3` necessarily leads to unpaired electrons, and we shall show that the inclusion of spin polarisation in the calculations is essential. Reduction can be produced not only by oxygen removal but also by the adsorption of reducing species such as alkali metals, and we shall present HF calculations on the eÜects of K adsorption on the (110) surface.TiO2 Finally, we shall address the problem of water adsorption. Recent static20 and dynamical calculations22 indicate that dissociates rather readily on (110). We H2O TiO2 present new calculations which con–rm this, but which also suggest that the balance between molecular and dissociative adsorption is quite delicate.II Techniques The DFT-pseudopotential and HF techniques for calculating the energetics of solids and their surfaces have been reviewed extensively,24,25 and here we give only a brief summary. Both techniques work with single-electron orbitals which satisfy a ti(r) Schroé dinger-like equation: Hti\A[ +2 2m Z2]Ven]VH]V effBti\viti (1) where the potential operator is separated into the electron»nucleus interaction the Ven , Hartree potential and an operator representing the eÜects of exchange and VH , Veff correlation. However, the meanings of the orbitals and the operator are diÜerent ti Veff in the two approaches.In DFT, the are not real single-electron wavefunctions, but are merely auxiliary ti(r) quantities used to represent the electron density n(r).The exchange-correlation operator has the form of a local potential which is de–ned to be the functional derivative Veff(r) where is the exchange-correlation energy. The major issue in DFT is the dExc/dn(r), Exc approximation used for Exc . Until recently, most DFT work on solids was based on the local-density approximation (LDA):26 Exc\Pdr n(r)vxc[n(r)] (2) where is the exchange-correlation energy per electron in a uniform electron gas of vxc(n) density n.This works well for many bulk solids, but is not accurate enough for molecular dissociation energies, and most of the work to be described later is based on gener-P. J. D. L indan et al. 137 alised gradient approximations (GGAs),27,28 in which the dependence of on Exc gradients of n(r) is included. A widely used implementation of DFT combines a plane-wave basis set with the pseudopotential method.29,30 The pseudopotential method assumes that the core electrons are in the same states as in the free atoms, and the pseudopotential represents the interaction between valence electrons and the ionic cores. The pseudopotential is generated via DFT calculations on free atoms.Errors can arise from the choice of pseudopotential, but these errors are usually under satisfactory control. In the pseudopotential approach, it is common practice to use supercell geometry (i.e. periodic boundary conditions) and plane-wave basis sets, so that the occupied orbitals are expanded ti(r) as : ti(r)\; G ciG exp(iG … r) (3) where the sum goes over reciprocal lattice vectors G of the supercell lattice.This is an in–nite sum in principle, but in practice it is restricted to those G for which +2G2/2m\ where is a chosen cut-oÜ energy. The size of governs the completeness of Ecut , Ecut Ecut the basis set. The practical determination of the ground state is performed by minimising the total energy of the system with respect to the plane-wave coefficients ciG .For some of the work on it is essential to include electronic spin. The gener- TiO2 , alisation of DFT to include spin polarisation is formally straightforward, and GGAs in which depends on the densities and of up-and down-spin electrons are well Exc nè(r) né(r) established.In practice, we perform the calculations by –xing the separate numbers of spin-up and spin-down electrons in advance. Comparison of ground-state solutions with diÜerent spin excesses is then made to determine the lowest-energy solution. In contrast to DFT, the HF approximation attempts to describe the many-electron wavefunction . . . , The orbitals appearing in eqn. (1) are the factors in the W(r1, rN).ti antisymmetrized product representing W: W(r1, . . . , rN)\AA< i/1 N ti(ri)B (4) The operator now represents the non-local exchange interaction between single- Veff electron orbitals : Veffti(r1)\Pdr2 V (r1, r2)ti(r2) \[;j Pdr2 e2 o r1[r2 o tj p (r2)tj(r1)ti(r2) (5) By de–nition, electron correlation is neglected in this approach, though approximations are available for reintroducing correlation.31h33 It is common practice to represent the in a local basis set of Gaussian orbitals, in which all the required integrals ti are analytic.Spin-polarised calculations are readily performed in the HF approximation. Because the energy expression is based on analytic integrals, the numerical stability is very high, and very small magnetic coupling energies can be studied.DiÜerent spin states can be explored by initialising the calculation with a superposition of spinpolarised ions. Our DFT-pseudopotential calculations have been performed using two separate codes: the CASTEP code24 (or its parallel version CETEP34) and the VASP code.35 The two codes employ essentially the same overall strategy, namely a ground-state search by global minimisation of the total energy with respect to the plane-wave coefficients.The diÜerences are that VASP uses more efficient search techniques, and is also able to use Vanderbilt-type ultrasoft pseudopotentials,36 which means that a given accuracy can be138 Simulation of molecular processes on oxide surfaces Fig. 1 (110) surface of Light and dark spheres indicate titanium and oxygen ions, respec- TiO2 .tively. (a) Perspective view showing the slab geometry used. The simulation cell is extended for display purposes. The –ve-fold and six-fold coordinated titanium sites and the bridging oxygen site are labelled 5f, 6f and BO, respectively. (b) The 1]1 surface unit cell used in the simulations. (c) Side view of the relaxed stoichiometric geometry. achieved with much smaller plane-wave basis sets.The HF calculations described later were carried out using the widely used CRYSTAL package,25,37 which is also capable of performing DFT calculations. III TiO (110) surface Several research groups have recently used ab initio methods to investigate the relaxed structure of the (110) surface. Surface X-ray diÜraction measurements of the TiO2P.J. D. L indan et al. 139 relaxed ionic positions at the surface have given crucial evidence that the theoretical predictions are essentially correct. We compare here the experimental values of ionic displacements with published DFT predictions,17 and with our own previously unpublished HF and DFT-pseudopotential predictions. The structures of the perfect crystal and of the perfect stoichiometric surface are shown in Fig. 1. A prominent feature of the surface is the rows of ìbridgingœ oxygens which stand proud of the surface plane of Ti and O ions. The latter plane contains two kinds of Ti ions : six-fold coordinated Ti ions lying beneath bridging oxygens, and exposed –ve-fold Ti ions ; the O ions in this plane are referred to as in-plane O.Experiment and previous calculations show substantial displacements of these surface ions away from their perfect lattice positions, and they also show that six-fold Ti and inplane O move out of the surface, while –ve-fold Ti and bridging O ions move in. Our new DFT-pseudopotential calculations were performed with Vanderbilt ultrasoft pseudopotentials using the VASP code.35 Exchange and correlation were included using the Perdew-91 GGA functional.27 We employed the usual supercell method, and the surface was treated using periodically repeated slab geometry (see Fig. 1). In this scheme, it is absolutely crucial to ensure that the slab thickness, L , and the width, L @, of the vacuum layer between neighbouring slabs are large enough so that interactions between surfaces are negligible. We have systematically studied the convergence of the equilibrium surface structure as L and L @ are increased.We –nd that convergence with respect to L @ is very rapid, and a vacuum width of 4 reduces errors in relaxed posi- ” tions to ca. 10~2 Convergence with respect to L is more complex, and we –nd oscil- ”. latory eÜects due to interactions through the body of the slab.However, when L is increased beyond ca. 20 the errors in relaxed positions again fall below ca. 10~2 ”, ”. A detailed theory»experiment comparison of the ionic displacements is given in Table 1. We note the following points : –rst, independent DFT-pseudopotential calculations are in quite satisfactory agreement with each other ; second, DFT and HF calculations are also in reasonable accord; third, the theoretical and experimental displacements are in semi-quantitative agreement, except for the bridging oxygens.The reason for the substantial discrepancy for bridging oxygens is unclear, but it is evidently signi–cant, since all the calculations show roughly the same discrepancy. Table 1 Ionic displacements due to relaxation of the (110) 1]1 surface (a) (b) (c) (d) label [110] [16 10] [110] [16 10] [110] [16 10] [110] [1 610] 1 0.13 0.00 0.23 0.00 0.13 0.00 0.12 ^0.05 0.00 » 2 [0.17 0.00 [0.11 0.00 [0.13 0.00 [0.16 ^0.05 0.00 » 3 [0.06 0.00 [0.02 0.00 [0.11 0.00 [0.27 ^0.08 0.00 » 4 0.13 [0.04 0.18 [0.05 0.09 [0.07 0.05 ^0.05 [0.16 ^0.08 5 0.13 0.04 0.18 0.05 0.09 0.07 0.05 ^0.05 0.16 ^0.08 6 [0.07 0.00 0.03 0.00 [0.04 0.00 0.05 ^0.08 0.00 » 7 0.06 0.00 0.12 0.00 » » 0.07 ^0.04 0.00 » 8 [0.08 0.00 [0.06 0.00 » » [0.09 ^0.04 0.00 » 9 0.02 0.00 0.03 0.00 [0.01 0.00 0.00 ^0.08 0.00 » 10 [0.03 [0.05 0.00 [0.02 » » 0.02 ^0.06 [0.07 ^0.06 11 [0.03 0.05 0.00 0.02 » » 0.02 ^0.06 0.07 ^0.06 12 [0.01 0.00 0.03 0.00 » » [0.09 ^0.08 0.00 » 13 » » 0.00 0.00 » » [0.12 ^0.07 0.00 » Labels refer to Fig. 1(c). The displacements are in and are from the bulk terminated positions. ”, The results are from: (a) DFT calculations by Ramamoorthy et al. ;17 (b) the present DFT calculations ; (c) the present HF calculations and (d) surface X-ray diÜraction experiments.3140 Simulation of molecular processes on oxide surfaces Overall, the present understanding of the equilibrium structure of the stoichiometric surface seems to us fairly satisfactory, and we think it provides an adequate basis for the work on molecular adsorption and surface reduction reported below.IV Reduced TiO2 We now turn to the reduced (110) surface. As explained in the Introduction, TiO2 reduction can be achieved either by removal of oxygen or by adsorption of reducing species such as alkali metals.We begin by considering oxygen removal. Two questions arise when considering a transition-metal oxide which is oxygen de–- cient. The –rst is, what is the defect structure, both physical and electronic, associated with the reduction ? The second, what are the consequences of the reduction for surface structure, reactivity, transport etc. For it turns out that the defects associated with TiO2 reduction are in themselves very complex, and here we shall con–ne ourselves to the –rst of these questions.Speci–cally, we will look at oxygen vacancies in the bulk and on the (110) surface. The examination of computed results from both DFT theory implemented using pseudopotentials and a plane-wave basis set and HF theory using local basis functions allows us to examine the sensitivity of these results to the main theoretical and numerical approximations made.A familiar feature in experimental studies of reduced is the presence of ìTi3`œ ions. The evidence for this description is discussed in ref. 38, and we summarise the key features here. The UV photoemission spectroscopy (UPS) spectra contain a bandgap feature for reduced samples which is interpreted in terms of occupied states formed from Ti(3d) orbitals,38,39 a view supported by the resonant behaviour of UPS across the range of photon energies corresponding to the Ti 3p»3d excitation threshold.The shift of the Ti(2p) core levels seen in X-ray photoelectron spectroscopy (XPS)40 and the lack of surface conductivity also evidence the localised nature of the additional electrons. Regardless of the location of the oxygen vacancies (i.e.in the bulk or at a surface) the energy levels of the excess electrons lie in the upper half of the bulk bandgap, though their position depends on the degree of reduction. The exact nature of these states, and their relation to the oxygen vacancy, have not been established by experiment. There have been previous theoretical studies of the reduced (110) surface of TiO2 .Most of these predict gap states, but there is no accord on the surface defect structure responsible for these states. Wang and Xu41 (tight-binding extended Hué ckel), and Tsukada et al.42 (DV-Xa cluster methods) both –nd the states 0.7 eV below the conduction-band minimum. However, Munnix and Schmeits,43 using a tight-binding model, found that gap states only occurred after the removal of sub-surface oxygen atoms. It is, of course, highly desirable that –rst-principles methods be able to describe the electronic structure of reduced transition-metal oxides, being a particular case.The TiO2 fact that the Ti4` ion in rutile is in the d0 state, and that upon moderate reduction Ti3` ions in the d1 state are produced, has important consequences for the use of DFT theory.The failure of the LDA approximation to produce the correct ground state in many transition-metal oxide systems is now well known.44 For example, is described La2CuO4 as a non-magnetic metal when in reality it is a magnetically ordered wide band-gap insulator. Although for many years this failure was attributed to an incorrect treatment of electron correlation we now know that a qualitatively correct ground state is obtained if the description of the exchange interaction is improved.45h47 This understanding underpins the alternative corrections to the LDA such as the self-interaction correction (SIC-LDA)46,47 and the inclusion of explicit on-site potentials (LDA]U).45 In the HF approximation, the exchange interactions are treated exactly, and this is therefore of great interest in these systems.48h50 One might expect such corrections to beP.J. D. L indan et al. 141 rather small for the d1 state expected in reduced because the orbital is not exces- TiO2 sively contracted (W. Temmerman, personal communication). It is, therefore, interesting to compare the results of LSDA calculations and HF calculations in some detail.As will become clear from our results, a qualitatively correct description of excess-electron gap states in can be obtained from calculations performed within either HF or DFT approximations. IV. A Oxygen vacancy in bulk TiO2 It is natural to approach the question of reduced through calculations on the bulk TiO2 material, but this immediately raises the question of the defect structure of This TiO2~x .is a surprisingly complex issue, since a family of stable Magneç li phases51 is found to occur between and It is beyond the scope of this work to calculate the Ti2O3 TiO2 . most stable structure for a given stoichiometry, and this is not necessary for the present purposes. Here, we investigate the predictions of spin-polarised DFT and HF for the electronic structure of the reduced bulk, the general features of which should not depend on the exact defect structure.One caveat is that the defect concentrations accessible to simulation are much higher than those encountered in experiment, a fact which should be borne in mind when comparisons are made. Using plane-wave pseudopotential DFT we have treated systems consisting of one, two and eight unit cells, each containing a single vacancy.23 In each case we –nd that the lowest energy state is spin-polarised, with the two excess electrons occupying states formed from Ti(3d) orbitals.For all three defect densities studied, these states are localised on the three Ti ions nearest to the oxygen vacancy. The energies of these states lie in the gap between occupied and unoccupied levels : for the largest system they were 1.3» 1.9 eV above the valence band maximum, and occupied similar positions in the other systems.This behaviour is in contrast to spin-paired calculations on reduced TiO2 . These do not give gap states, and instead yield metallic solutions in which the excess electron states, although still formed from Ti (3d) orbitals, remain in the conduction band, and are not localised.The energy gained through spin polarisation (i.e. the energy drop in going from spin-paired to spin-polarised solutions) ranged from 0.2 to 0.5 eV. For the two unit cell system we performed full structural relaxation. This did not change the character of the excess-electron states, but did raise their energies in the gap slightly.Unrestricted HF calculations on the reduced bulk have also been made, using a system of four unit cells formed by doubling in the M001N plane.10 To make explicit allowance for the possibility that the excess electron density resulting from oxygen removal might localise, at least in part, at the vacancy site, as it does at the MgO (001) surface,12,52 we have carried out calculations using basis sets both with and without the complete set of oxygen functions at the vacant (oxygen) site.For the bulk vacancy, i.e. both basis sets lead to insulating, triplet-spin ground states, with that derived Ti8O15 , from the larger basis set (vacancy functions included) lower in energy by 1.4 eV. The Mulliken population analysis of bulk indicates site charges of Ti(]2.8) and TiO2 O([1.4) and thus, when an O vacancy is created, an excess electron density of [1.4 is generated.In this state [0.8 of the excess electron density is localised at the vacancy in a diÜuse s-like orbital, [0.4 at the nearer of the two Ti sites and [0.1 at the other Ti, while the net spin populations at the three sites are 0.8, 1.0 and 0.2, respectively. Both solutions are triplet-spin polarised, with gap states 1»2 eV above the valence band upper edge.States of this kind were only found in spin-polarised (unrestricted) solutions. The reason for the diÜerences between spin-paired (restricted) and spin-polarised (unrestricted) solutions lies in the nature of the states occupied by the excess electrons.The removal of a neutral oxygen ion leaves two electrons which previously occupied O(2p) levels in the valence band. These states are no longer available, and the electrons must go into the conduction band, the bottom of which is formed from Ti(3d) orbitals.142 Simulation of molecular processes on oxide surfaces Without spin polarisation the excess-electron states are forced to be doubly occupied.The spin-polarised solution also has the excess electrons occupying Ti(3d) orbitals, but now these electrons are unpaired, which is the origin of the large energy diÜerence between the two solutions. The occupation of spin-unpaired orbitals rather than the partial occupancy of spin-paired orbitals reduces the on-site exchange energy but increases the kinetic energy.In these narrow d-bands it is clear that the exchange term dominates and, therefore, the ground state has spin-polarised electrons which are localised on the Ti sites.53 The energy gained lowers these states. A further consequence is that the spin-unpaired orbitals contract closer to the ionic core, which is apparent through comparison of the excess-electron charge density distributions from the spinpaired and spin-polarised calculations (not shown here).Whilst yielding results that are qualitatively similar, there are notable diÜerences between the DFT and HF solutions for the reduced bulk. HF predicts that at least some of the excess-electron charge occupies the oxygen vacancy site, whilst in all three DFT calculations, negligibly small charge density was located there.This diÜerence indicates that the nature of the electronic state is sensitive to the approximations made. We are currently examining the cause, which may be due to the diÜerent one electron potentials or to the numerical approximations (basis set, pseudopotential). The variation in the position of the excess-electron levels above the valence band is of less concern: the signi–cant points are that in both cases they are separated from unoccupied levels, and that this separation arises from spin-polarisation.IV. B Reduced (110) surface The surface calculations we report here have used periodically repeating slab geometry, as indicated in Fig. 1 and discussed previously. The reduced 1]1 surface is formed by removing all the bridging oxygen ions, giving a density of surface vacancies of one monolayer, and a total of four excess electrons in the slab.We have performed both DFT23 and HF10 calculations on this surface, and in both studies we investigated solutions with spin-polarisation and full structural relaxation. Here, we make direct comparison of the predictions of the methods.In addition, the plane-wave study includes spin-paired solutions, allowing us to assess the eÜects of spin-polarisation, both at –xed geometry and after full relaxation. Both methodologies yield ground-state solutions that have insulating spin-polarised states above the valence band maximum, and these states localise the excess charge roughly equally on the two surface titanium ions, in bands formed from Ti(3d) orbitals.According to HF, these states lie 2 to 3 eV above the valence band maximum, while DFT places them 0.71 to 1.8 eV above; the lowest unoccupied states are ca. 1.9 eV above the valence-band edge, less than the experimental bandgap (3.1 eV)39 as is usual for DFT eigenvalues. These latter results are illustrated in Fig. 2, which shows the calculated density-of-states for up-and down-spin electrons.The excess-electron states are clearly visible in the up-spin density, and some spin-polarisation of lower levels is apparent. In these calculations, we did not attempt to –nd the magnetic ground state, since the energy diÜerence between ferromagnetic and anti-ferromagnetic solutions is expected to be no more than a few meV.However, the HF calculations show that the antiferromagnetic solution is the most stable. Comparison of the spin densities yielded by the two theories (not shown here) shows them to be very similar, underlining the agreement as to the qualitative nature of the electronic ground state. This picture is in good accord with experiment: UPS measurements54,55 on the Ar-ion-bombarded (110) surface show that at low defect concentrations, a band of gap states exists, centred 2.3 eV above the valence band maximum. The gap state lies in the upper half of the gap, and moves up towards the bottom of the conduction band as the defect concentration increases.P.J. D. L indan et al. 143 Fig. 2 Densities-of-states, g(E) for spin-up (SC) and spin-down (SB) electrons, calculated using DFT, for the reduced 1]1 surface after structural relaxation.The energy zero is set at the topmost occupied state. In Table 2 we report the ionic relaxations. From DFT calculations, the stoichiometric results of Lindan et al.23 are reproduced for reference, along with the displacements on the reduced surface calculated without and with spin polarisation. To complete the table we show results for the reduced surface using unrestricted HF.10 Reduction of the surface produces very diÜerent relaxations compared with the stoichiometric case : the –ve-fold Ti now remains close to the bulk terminated position, the six-fold Ti relaxes into, rather than out of, the surface, and the in-plane oxygens move much further out of the surface.The results with spin polarisation are in good agreement for the two methods, and it is notable that the inclusion of spin polarisation makes little diÜerence to the relaxed geometry.An interesting feature is that the spin polarised DFT solution has small displacements of the in-plane oxygens along [001], thereby lowering the symmetry of the surface. This surprising result is a direct consequence of the nature of the excess-electron states, and is caused by the interaction of these states with the oxygen ions.The Ti(3d) orbitals rotate about [110] which, along with the oxygen displacements, reduces the overlap of the charge density on the oxygen ions with that of the Ti(3d) electrons. Table 2 Ionic displacements due to relaxation of the (110)1]1 surface (110)1]1 reduced (110)1]1 (a) (b) (c) (d) label [110] [16 10] [110] [16 10] [110] [16 10] [001] [110] [16 10] 1 0.09 0.00 [0.13 0.00 [0.11 0.00 0.00 [0.10 0.00 2 [0.12 0.00 0.00 0.00 0.01 0.00 0.00 [0.02 0.00 3 [0.09 0.00 » » » » » » » 4 0.11 [0.05 0.40 0.09 0.39 0.10 [0.04 0.39 » 5 0.11 0.05 0.40 [0.09 0.39 [0.10 0.04 0.39 » 6 [0.05 0.00 [0.02 0.00 [0.05 0.00 0.00 » » Labels refer to Fig. 1(c). The displacements are in and are from the bulk terminated positions.”, (a) stoichiometric surface ; (b) reduced surface without spin-polarisation ; (c) and (d) reduced spinpolarised surface. The results are from plane-wave pseudopotential DFT calculations (a)»(c) and HF calculations (d).144 Simulation of molecular processes on oxide surfaces Fig. 3 Plan view of the unrelaxed (100) surface. The 1]1 unit cell is represented by the rectangular box whilst the diamond box marks the c(2]2) supercell used.The c(2]2) adsorption geometry for the K ion is also shown. Non-irreducible atoms (those symmetry related to atoms in the asymmetric unit of the surface unit cell) are labelled with the suffix ìaœ. Perhaps the most striking feature of the DFT results is the energetics : for identical ionic con–gurations (the relaxed, stoichiometric positions minus the two bridging oxygen ions, a 16-ion system) we –nd that a spin-polarised solution with four more electrons spin-up than spin-down is 2.77 eV (1.1 J m~2) lower in energy than the spinpaired solution.After structural relaxation of both systems, the diÜerence in energy is 3.98 eV (1.6 J m~2).For the spin-polarised system the energy gained by relaxing the ions was 2.8 eV. IV. C Reduction by K adsorption on the (100) surface Controlled reduction of oxide surfaces may be achieved by adsorbing alkali-metal overlayers. Because of the intrinsic interest in the surface electronic structure and the technological importance of such overlayers a number of experimental studies have recently been performed on these systems.The adsorption of potassium on NiO56 and ZnO,57 caesium on NiO58 and sodium on MgO59 have been reported. The surface geometry of the adsorbed species is found to be strongly dependent on the nature of the substrate. Potassium on ZnO leads to an ordered p(2]2) overlayer whereas potassium or (00016 ) caesium on NiO and sodium on MgO forms islands around surface defects. On these metal oxides a strong interaction between the alkali metal and surface oxygen atoms is seen.Theoretical studies of these systems have been inhibited by the complex nature of the electronic structure. To our knowledge, no theoretical investigation of the nature of alkali-metal adsorption exists, although LMTO-DFT methods have recently been used60 to study transition metal overlayers on (110).TiO2 A number of recent experimental investigations have probed the geometry and electronic structure of potassium and sodium adsorption on surfaces.38,61h66 Low- TiO2 energy electron diÜraction (LEED) data indicate that K adsorption on the (100) surface at half-monolayer coverage results in an ordered c(2]2) geometry.61 This is con–rmed by surface-extended X-ray absorption –ne structure (SEXAFS) studies which also probe the local geometry of the K site.63 The electronic structure has been studied using photoelectron spectroscopy62 which revealed the existence of K-induced band gap statesP.J. D. L indan et al. 145 on titanium atoms some 1»2 eV above the valence band edge, as for the substoichiometric surface (see above).These states are apparently populated by charge transfered from the K overlayer to the substrate. In the present work, we have performed HF calculations on the adsorption of K on the (100) surface. The c(2]2) surface unit cell is shown in Fig. 3 and 4 in the TiO2 geometry obtained by relaxing the K position and top two layers of atoms , (O(1) O(2) , and The symmetry of the surface was retained ; symmetry equivalent atoms Ti(3) Ti(4)).are labelled with suffix a in Fig. 3. A number of adsorption sites for the K atom on the unrelaxed surface were studied. At the short bridge site (bridging and the O(1) O(2a)), ì 4-fold hollowœ site (equidistant from and and the atop site O(1), O(1a), O(2) O(2a)) (directly above one of the oxygen atoms), the energy was signi–cantly higher than that obtained for adsorption at the long bridge site (bridging Relaxation of the K O(1), O(2)).ion and the top two layers of the substrate results in the geometry displayed in Fig. 4. The optimal adsorption site is with the K atom 1.16 above the surface oxygen plane ” and equidistant (2.59 from the and atoms. This agrees well with the KwO ”) O(1a) O(2) bond length of 2.62 found in SEXAFS studies.63 The K is 0.06 nearer than ” ”Ti(4) The surface oxygen layer shows signi–cant relaxation as and move Ti(3).O(1) O(2a) closer together along the [001]-direction to increase the eÜective coordination of Ti(4) (see Fig. 4). The surface oxygen and the titanium layers (as de–ned by and Ti(3a) Ti(4)) move in opposite directions along the [010]-direction so that there is further increase in the eÜective oxygen coordination of the surface Ti sites.Similar relaxations have been observed on clean surfaces,17,21 the main diÜerence here being the tendency of TiO2 to have closer coordination to the K than Ti(4) Ti(3) . Some understanding of this rather curious adsorption site can be obtained by examining the charge and spin distributions at the surface which were estimated using a Mulliken population analysis,67 Table 3.The K atom is almost completely ionised, with 0.97 o e o being transferred to the surface. This charge is distributed over the (0.34 Ti(4) o e o) and the two surface O-sites (0.32 o e o). The site is almost unperturbed. The Ti(3) nature of this electronic state is even clearer in the spin populations.A single unpaired spin is generated on the site. The adsorption process can therefore be considered to Ti(4) involve charge transfer from the K atom to a speci–c surface Ti site generating a spin polarised Ti3` ion which, being negative compared to the lattice Ti4` ions, is less repulsive to the K. The density of states for this surface is compared with that in the bulk crystal and clean surface in Fig. 5. The broad valence band between [7.5 eV and the valence band maximum at 0.0 eV is dominated by orbitals and varies only in detail between the O2p bulk, clean surface and K-adsorbed surface. The clear signature of K adsorption is the band-gap state 1.4 eV above the valence band maximum, which is similar to the states Fig. 4 Relaxations of the substrate upon K adsorption. The K ion is shown in its optimised (”) position.146 Simulation of molecular processes on oxide surfaces Table 3 Mulliken population analysis of the charge (Qo e o ) and spin density ( o e o) of the c(2]2)- (100) K surface TiO2 atom Q Q[Qclean spin O(1) 9.49 0.32 0.00 O(2) 9.49 0.32 0.00 Ti(3) 19.34 0.06 0.01 Ti(4) 19.62 0.34 0.99 O(5) 9.48 0.02 [0.01 O(6) 9.48 0.02 0.00 O(7) 9.43 0.00 0.00 O(8) 9.44 0.02 [0.01 Ti(9) 19.19 0.00 0.01 Ti(10) 19.19 0.00 0.01 K 18.03 » 0.00 Fig. 5 HF densities of states projected onto Ti and O sites for (a) bulk and (b) the clean TiO2 , (100) and (c) (100)c(2]2) K surfaces. The shading indicates states derived from the induced Ti3` ion.P. J. D. L indan et al. 147 observed at 2»3 eV on the reduced (110) surface discussed above.As mentioned TiO2 above, this spin polarised state is almost entirely localised on the site. K-induced Ti(4) band-gap states are observed in photoelectron spectroscopy61h63 at 0.9 eV binding energy or 1»2 eV above the valence band maximum. V Water on (110) TiO2 Water interacts quite strongly with many oxide surfaces, and there have been many experiments on adsorbed water.In the case of MgO (001), there is now rather –rm evidence from experiment and theory that water is adsorbed in molecular form on the perfect surface, and that dissociation occurs only if defects are present.13 The situation on (110) is not so clear.68 Thermal desorption and UPS69 experiments have been TiO2 interpreted as showing that water is adsorbed dissociatively at sub-monolayer coverage.But very recently, high-resolution electron energy loss (HREELS) measurement70 have given evidence that dissociative adsorption occurs only at somewhat lower coverages, with molecular adsorption becoming dominant as monolayer coverage is approached. Recently we reported exploratory DFT-pseudopotential calculations of water adsorption, in which fully relaxed adsorption energies were obtained for a number of diÜerent con–guration.20 The simplest molecular candidate has the molecule coor- H2O dinated by its O atom to a –ve-fold surface Ti site ; the simplest dissociated con–guration has the water OH~ ion at the –ve-fold site (we call this a terminal hydroxy group), and the H] ion attached to a bridging O ion to form a second type of hydroxy ion (a bridging hydroxy group).With monolayer coverage and with the constraint of full symmetry (i.e. the molecular and OH~ axes point along the surface normal), a comparison of these two showed that the molecular con–guration is more stable, with Eads\ eV, compared with eV for the dissociated con–guration. However, the 0.82 Eads\0.45 symmetrical dissociated con–guration is only metastable, and by allowing the axes of the OwH groups to vary in a plane normal to the surface, distortion occurs, to produce hydrogen bonding between the two kinds of OH~ group; the resulting stabilisation increases to 1.08 eV.The conclusion is that for the con–gurations examined, disso- Eads ciation is energetically favoured. Support for dissociative adsorption at lower coverage comes from our recent dynamical DFT simulations22 in which the time evolution is obtained when a molecule coming from the gas phase reacts with the surface.We examined the adsorption of a single molecule on a surface doubled along [001] (i.e. with a 2]1 area), which corresponds to half-monolayer coverage. From several initial conditions studied we could not –nd a molecular adsorption site : once near one of the –ve-fold Ti sites, the molecule spontaneously dissociated on the surface, and the resulting con–guration was similar to that identi–ed in the monolayer-coverage static calculations.As an illustration of the time evolution, we show, in Fig. 6, three snapshots from one of the simulations, showing the spontaneous formation of the two types of surface hydroxy group mentioned above.After relaxation of the –nal dynamics con–guration, the distance between the O and H in the broken water bond was 1.8 . This is the typical length of a hydrogen bond and, ” indeed, there are clear signs of interaction between these atoms, as we describe shortly. One way to analyse the adsorbed species on a surface is to use an experimental technique which probes the vibrational spectra of those species.Henderson70 used HREELS techniques to do this for water adsorbed on (110). In interpreting these results, certain assumptions must be made as to the signals expected for the adsorbates. In particular, it is assumed that hydroxy groups contribute sharp, high-frequency peaks to the spectra.Thus, dissociation of water should be signalled by two such sharp peaks (the two hydroxy groups are in diÜerent environments) and an absence of the bond-bending frequency corresponding to the molecular vibrational mode in which the HwOwH angle varies.148 Simulation of molecular processes on oxide surfaces Fig. 6 Three snap-shots from a dynamical simulation of an molecule being dissociatively H2O adsorbed on the (110) surface.Large, medium and small spheres represent O, Ti and H TiO2 atoms respectively. Time proceeds from left to right, with an interval of ca. 0.2 ps between successive frames. Reproduced from ref. 22 with permission. We have performed equilibrium molecular dynamics (MD) simulations at low temperature, from which we have calculated the vibrational spectra of the hydrogen ions.Surprisingly, we –nd that only the terminal hydroxy group gives a well-de–ned high frequency signal. The bridging hydroxy group has a broad vibrational spectrum. These features are shown in Fig. 7. Direct examination of the vibrational motion shows that this broadening is due to the hydrogen-bonding interactions along the broken water bond: the thermal motion results in considerable variation in the length of this hydrogen bond, which consequently alters the vibrational frequency of the hydrogen ion.The high-frequency features we calculate appear to agree well with the HREELS spectrum in the corresponding frequency range. (We note that for technical reasons, these simulations employed an arti–cially large hydrogen mass of 3 u.This lowers the vibrational frequencies, but leaves the shape of the spectrum unchanged.) The conclusion from these calculations is that the vibrational signals of adsorbates may not be as anticipated from simple arguments. However, according to Henderson,70 a clear bond-bending signal is present at this coverage, a fact which our calculations do not explain. Of course, since we have only performed calculations at half-monolayer coverage in which a single molecule is treated explicitly, we cannot investigate intermolecular interactions which may be important at high coverage.Therefore, our most recent calculations aim to investigate the eÜect of intermolecular interactions on adsorption. We have taken the hydroxylated surface structure yielded by our previous MD studies of water adsorption22 and explored the adsorption of a second molecule on the surface.Again, we used several initial con–gurations, but the only adsorption site we found was over the second –ve-fold Ti site, i.e. the one not occupied by a hydroxyl group. At this site, adsorbs molecularly, and lies almost —at on the surface. This H2O con–guration, shown in Fig. 8 allows the formation of hydrogen bonds to both the surface bridging oxygen ion and the OH~ group which was produced by dissociation of the –rst molecule. Thus, from these calculations, we –nd that the most stable con–guration at monolayer coverage is one containing both dissociated and molecular water. Note that in this state, the separation of the molecule H and the terminal-hydroxy O, ca. 1.8 is very similar to the length of the hydrogen bond in the water dimer.72 In fact, ”, the geometry of the adsorbates is closely related to the dimer geometry. A metastable state, in which both molecules are intact, lies ca. 0.2 eV above this state. This remarkable result may provide an explanation of experiment, since it is clear that the adsorption state depends crucially on the interactions between molecules, and hence on the cover-P.J. D. L indan et al. 149 Fig. 7 Power spectra for the hydrogen ions in (a) the terminal hydroxy and (b) bridging hydroxy groups after water dissociation. The simulation was at 120 K. Note that the frequencies are shifted because of the use of a large hydrogen mass (see Section V). Note also the diÜerent y-scales used.Reproduced from ref. 22 with permission. age. In addition, the vibrational spectrum calculated from this surface contains a clear bond-bending signal, and is thus in good agreement with the HREELS data. The interaction of water with this surface is clearly a rather subtle matter, and the hydrogen bonds between water molecules and hydroxy groups are of crucial importance. Explicit examination of these interactions is vital if the correct adsorption state is to be determined.Also, since some of these states are of rather similar energy (within a few tenths of an eV), further re–nement and re-examination of our static calculations is warranted. We are, therefore, performing additional static-relaxation calculations using improved pseudopotentials. These pseudopotentials are of the ì ultrasoft œ type due to Vanderbilt.36 Their advantage is that smaller plane-wave cut-oÜs (i.e.smaller basis sets) can be used without sacri–cing accuracy, and this will allow us to go to larger supercells. We have used these pseudopotentials to investigate the wider range of adsorbate geometries shown in Fig. 9. In addition to the symmetric molecular (SM) and symmetric and unsymmetric dissociated (SD and UD) con–gurations studied before, we now include the unsymmetrical molecular con–gurations (UM1, UM2 and UM3) shown in the –gure.These new con–gurations were chosen on electrostatic grounds, which suggest that it should be favourable for the water O to coordinate to –ve-fold Ti and for both the H atoms to coordinate to O ions.This can be achieved in three ways: both Hs to the150 Simulation of molecular processes on oxide surfaces Fig. 8 Relaxed ionic con–guration for two water molecules adsorbed at monolayer coverage on the (110) surface. Ti and O are represented by light and dark large spheres, respectively, and the small spheres represent H. The relaxation was performed on the con–guration resulting from the dynamical simulation described in Section V.This is the lowest-energy con–guration found at monolayer coverage. (a) View looking down onto the surface. The 2]1 area of the simulation cell is outlined, and the dashed line indicates the boundary between the two 1]1 cells that form the surface. (b) View along Note the contraction along the H bond joining the molecule and the (116 0).OwH group. same bridging O (UM1); both Hs to two diÜerent in-plane Os (UM2); and one H to a bridging O and the other to the O of the water molecule in the neighbouring cell, which is one of its periodic images (UM3). We have calculated fully relaxed adsorption energies for these six con–gurations at monolayer coverage, using the three-layer slab.The values are systematically ca. 0.3 eV higher than those found in our previous static calculations, and we believe that part of the reason may be that the previous calculations were performed at a slightly larger lattice parameter. However, the adsorption energies for SM, SD and UD are in the same order as before, and the diÜerences between them are similar. The new molecular con- –gurations are all stable, with values of over 0.5 eV, and the most stable case, UM3, Eads has an which is only slightly less than that of the most stable dissociated con–gu- Eads ration.It is interesting to note that the molecular con–guration UM3 is very similar to the con–guration of the molecule in the mixed adsorption state found via the dynamics calculations at full coverage.P.J. D. L indan et al. 151 Fig. 9 Schematic representation of adsorbate geometries for water investigated using DFT staticrelaxation calculations. The labels are de–ned in the text. The 1]1 surface unit cell was used in all calculations, thus –xing the coverage at one monolayer. If one takes the static and dynamic calculations together, they suggest several things. First, dissociation is clearly favoured at half-monolayer coverage, but at full coverage there are signi–cant intermolecular interactions which make the molecular and dissociative adsorption energies similar.Another consequence of these interactions is that the molecule lies —at in the most stable molecular adsorption geometry, thereby forming a hydrogen bond along [001] with its neighbour.Our larger, two-molecule calculations allow additional stabilisation to occur through the contraction of this bond and, in fact, show that the lowest-energy geometry is the mixed dissociative/molecular state shown in Fig. 8. By analogy with the water dimer we may expect that this hydrogen bond contributes ca. 0.2 eV.71 Having demonstrated the importance of intermolecular interactions for water on this surface, we must note that the real adsorption geometry may have a periodicity other than the 2]1 case we have studied, but it is certain that it is not of 1]1 symmetry.Also, our results indicate that, at –nite temperatures, a given molecule will —uctuate between a dissociated and molecular state. Finally, we note that the lattice parameters of the substrate material may be crucial in determining the mode of adsorption.which also has the rutile structure, has larger a and c parameters and there- SnO2 , fore neighbouring molecules or OwH groups would be further apart than on the152 Simulation of molecular processes on oxide surfaces surface. The eÜect of this on the strength of the hydrogen bonds may result in very diÜerent behaviour. VI Conclusions Our ab initio calculations on have been used to study: the equilibrium structure of TiO2 the stoichiometric (110) surface ; the atomic and electronic structure of surfaces TiO2 reduced by removal of oxygen and by addition of potassium; and the molecular and dissociative adsorption of water at the (110) surface.We have shown that DFT and HF calculations give very similar predictions for the ionic displacements at the stoichiometric (110) surface, and that these are in semi-quantitative agreement with recent experimental measurements. The most important conclusion from our calculations on oxygen vacancies in both the bulk and at the surface is that spin unpairing is a dominant eÜect.Our DFT calculations show that the unpairing of spins leads to a substantial lowering of the total energy and to the appearance of the localised states in the band gap which are observed experimentally.Unrestricted HF calculations yield gap states with the same general characteristics. The two theoretical methods predict ionic displacements at the reduced (110) surface which agree very closely with each other. Our HF calculations show that K bonds to the (100) surface via charge transfer of an electron into a TiO2 particular surface Ti site.The resulting geometry is in excellent agreement with recent X-ray adsorption data. The gap states induced by surface reduction are similar to those observed on the non-stoichiometric surface. Finally, our static and dynamic DFT calculations on the adsorption of water on (110) are consistent with earlier –ndings that TiO2 dissociation gives stabilisation at half-monolayer coverage. However, our new dynamical calculations at monolayer coverage indicate a more complex picture in which water is present in both molecular and dissociated forms, which is consistent with recent HREELS measurements.The work of P.J.D.L. was done as part of a CCP3/CCP5 joint project supported by a grant from Cray Research Inc., and the work of S.P.B.and J.M. was supported by EPSRC grants GR/J34842 and 95563789 respectively. We also used local computer equipment provided by EPSRC grant GR/J36266. We are grateful for an allocation of time on the Cray T3D at EPCC provided by the High Performance Computing Initiative through the U.K. Car-Parrinello consortium and the Materials Chemistry consortium. Useful discussions with Prof.G. Thornton and Dr W. Temmerman are also acknowledged. This work has bene–ted from the collaboration within the Human Capital Mobility Network on ìAb initio (from electronic structure) calculation of complex processes in materialsœ. References 1 H. J. Freund, H. 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Cox, T he Surface Science of Metal Oxides, Cambridge University Press, Cambridge, 1994. 69 M. B. Hugenschmidt, L. Gamble and C. T. Campbell, Surf. Sci., 1994, 302, 329. 70 M. A. Henderson, Surf. Sci., 1996, 355, 151. 71 S. S. Xantheas and T. H. Dunning, J. Chem. Phys., 1993, 99, 8774. Paper 7/02103H; Received 26th March, 1997

ISSN:1359-6640    

DOI:10.1039/a702103h

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 155»172 Cluster model calculations of oxygen vacancies in and SiO2 MgO Formation energies, optical transitions and EPR spectra Gianfranco Pacchioni,*§ Anna Maria Ferrari and Gianluigi Ierano` Dipartimento di Scienza dei Materiali, INFM, di Milano, via Emanueli 15, Universita` 20126 Milano, Italy The electronic structure of neutral and charged oxygen vacancies in SiO2 and MgO, two oxide materials with diÜerent structural and electronic properties, has been studied with cluster models and all-electron wavefunctions. DiÜerent embedding schemes have been used to account for the eÜect of the surrounding.The geometrical structure of these point defects has been determined by full geometrical optimisation. Starting from the minimum structures, a series of observable properties have been computed, in particular, formation energies, hyper–ne interactions in paramagnetic centres, and optical transitions. All these properties, in order to be correctly described, need the extensive use of correlation eÜects.They have been introduced at various levels : second-order perturbation theory (MP2), con–guration interaction (CI) or density functional theory (DFT).However, the quantitative description of the observable properties requires ìcorrectœ structural models of the oxygen vacancies. The interplay between experimental and theoretical information allows the unambiguous determination of the structure of oxygen vacancies in metal oxides. 1 Introduction Metal oxides are of great technological importance in catalysis, electrochemistry, optical –bres, sensors etc.1h4 The accurate theoretical description of their electronic structure is essential for a detailed understanding of the structure»properties relationship.Much attention in recent years has been directed to the theoretical study of non-defective bulk oxides. On the other hand, perfect, defect-free, oxides are simply an idealisation and do not exist in nature.It is, therefore, important to describe also with the same accuracy the point and extended defects present in the bulk or on the surface of a metal oxide.5 In fact, the properties of a material are often determined by the defects. For example, while the regular MgO(100) surface is very unreactive,5h7 high-surface-area polycrystalline MgO exhibits a rich and not completely understood chemical reactivity.8h10 From a computational viewpoint the treatment of point defects is particularly challenging.The electronic properties of a point defect are often determined by the local environment and require a localised description. On the other hand, it is essential to take into account solid-state eÜects by ìembeddingœ the local system in the host crystal.Several approaches have been proposed to treat neutral and charged defects, especially in ionic or partially ionic materials like MgO and SiO2 .11h21 In this work, we have investigated the electronic structure of oxygen vacancies in the bulk of and in the bulk and on the surface of MgO, two oxide materials with SiO2 § E-mail pacchioni=mi.infn.it 155156 Cluster model calculations of oxygen vacancies rather diÜerent structural and electronic properties.An oxygen vacancy in can SiO2 either be neutral, or positively charged, This latter defect is paramagnetic and VO, VO ` . is also called the E@ centre. Oxygen vacancies in MgO are known as F centres ; if the vacancy is localised at the surface a subscript s is added, F centres exist in three Fs .states, depending on their electronic charge.21 Diamagnetic oxygen vacancies can be either neutral, F, or doubly charged, F2`. In the former case two electrons are associated with the vacancy. Of particular interest for the characterisation of oxygen vacancies are the paramagnetic F` centres, consisting of a single electron trapped in the cavity formed by removing an oxygen anion, O~.21h23 The peculiar electronic structure of MgO surface and centres, with ì free œ electrons in the cavity, is the origin of their Fs Fs ` high chemical reactivity.8h10 We have determined cluster model all-electron Hartree»Fock (HF) and post-HF wavefunctions to study the electronic and geometrical structure of point defects in SiO2 and MgO.The cluster model approach is particularly appropriate to study localised phenomena, such as defects and impurity states, local excitations or core level ionisations, 24 provided that the eÜect of the rest of the crystal is included.The diÜerent nature of and MgO implies the use of diÜerent approaches to take into account the SiO2 surroundings. is a material dominated by covalent polar SiwO bonds;25 MgO can SiO2 be classi–ed as an almost fully ionic oxide.26 These characteristics suggest that diÜerent strategies must be adopted to ìembedœ the cluster.It is the aim of this paper to show that these approximate embedding schemes are sufficiently accurate for the description of a series of surface and bulk properties. Advantages and limitations of the approach will be discussed in the perspective of the potential use of quantum-chemical calculations as a practical tool to solve problems in solid-state chemistry. 2 Computational approach 2.1 Clusters and embedding Two completely diÜerent approaches have been followed to ìembedœ the and MgO SiO2 clusters. In the cluster dangling bonds have been saturated by H atoms, a common SiO2 procedure to terminate clusters of covalent materials.27,28 The positions of the cluster atoms were initially –xed to those of a-quartz29 and the H atoms were –xed at a distance of 0.96 from the respective O atoms along the OwSi directions.The position of Aé all the Si and O atoms of the cluster has been reoptimised. The –xed H atoms provide a simple representation of the mechanical embedding of the solid matrix. This is an important aspect of the embedding, but not the only one.In this way we neglect the Madelung –eld arising from the charge asymmetry of the SiwO bond. In principle, this term can be important, in particular for the description of charged defects.30 All geometry optimisations have been performed by computing analytical gradients of the total energy. Given the low local symmetry of all the clusters are computed SiO2 , without any symmetry element symmetry group).Clusters of various sizes were (C1 used, from to for non-defective structures, Fig. 1 and 2; with these Si2O7H6 Si8O24H18 clusters the computed long and short SiwO bond lengths, 1.630 and 1.620 respec- Aé , tively, are close to the experimental values of 1.614 and 1.605 The models of an Aé .29 oxygen vacancy have been obtained by removing one oxygen atom, Fig. 1»5. Also, for the defective structures, the cluster geometries have been fully optimised with the boundary condition that the position of the H atoms is –xed. This means that only local relaxation eÜects have been considered ; long-range lattice relaxation is not included. The ionic nature of MgO implies that the Madelung potential must be included in the models.Indeed, several properties of MgO are incorrectly described if the long-range Coulomb interactions are not taken into account.31 To provide a simple representation of the Madelung potential we used large arrays of ^2 point charges (PC). UnfortunatelyG. Pacchioni et al. 157 Fig. 1 (a) and (b) cluster models of a regular site and of a centre in a-quartz Si2O7H6 Si2O6H6 VO Fig. 2 (a) and (b) cluster models of a regular site and of a centre in Si5O16H12 Si5O15H12 VO a-quartz Fig. 3 (a) and (b) (b) cluster models of a centre in a-quartz [Si2O6H6]` [Si5O15H12]` VO `158 Cluster model calculations of oxygen vacancies cluster models of (a) a ìnear-perfect crystal œ and (b) a ìpuckeredœ centre Fig. 4 [Si8O24H18]` VO ` in a-quartz the PCs polarise the oxide anions at the cluster border causing an incorrect behaviour of the surface electrostatic potential (EP).32h34 The problem can be substantially reduced by placing at the position of the ]2 PCs around the cluster an eÜective core potential, ECP, representing the –nite size of the Mg2` core.33,34 No basis functions are associated with the ECP35 which accounts for the Pauli or exchange repulsion of the O2~ valence electrons with the surrounding. This is a very simpli–ed approach compared with more rigorous embedding techniques.36,37 On the other hand, it is computationally simple and, as we will show below, quite reliable.Furthermore, the use of clusters allows an extensive treatment of correlation eÜects.A discussion of the role of the ECPs in the description of the centres is given below. Fs To describe the MgO vacancies we used various clusters, Fig. 6. In general, they are derived from clusters by removing one oxygen atom. The clusters can be neutral OnMgn or positively charged to represent F, F`, or F2` bulk or surface vacancies, respectively and are surrounded by ca. 700 PCs.If ECPs have been added we denote the cluster as When possible, the non-negligible geometrical relaxation of the charged OnMgn]ECP. defects has been taken into account; otherwise, the geometry is that of the bulk or of the truncated MgO(100) surface. Fig. 5 (a) ìringœ cluster model of a ìpuckeredœ centre in a-quartz with a three- [Si8O24H18]` VO ` coordinated O. (b) minimum model of the same defect.[Si2O6H6]`…H2OG. Pacchioni et al. 159 Fig. 6 (a) and (b) cluster models of oxygen vacancies in the bulk and on the Mg6O12 O20Mg21 surface of MgO 2.2 Basis sets HF, self-consistent –eld (HF-SCF), wave functions for clusters have been con- SiO2 structed using double-zeta (DZ) basis sets.38 One d polarisation function (a\0.4) has been added to the Si basis set for the computation of optical transitions and formation energies. The formation energies have been determined by also adding a d polarisation function (a\0.74) on the O atom involved in the defect formation.In this case, the basis sets are [12s8p1d/6s4p1d] (Si) and [10s5p1d/3s2p1d] (O). The d functions are essential for the calculation of the formation energies and of the optical transitions but they have no eÜect on the geometrical structure. For paramagnetic defects or open shell excited states we used the restricted open HF (ROHF) and unrestricted HF (UHF) formalisms.The basis sets used for the oxide and magnesium ions of the MgO clusters are [8s4p/ 4s2p]39 and [13s8p/6s3p],40 respectively. The 3s and 3p orbitals have been removed from some of the Mg2` ions at the cluster border.21 A problem connected with the description of vacancies in oxide materials is the possible localisation of trapped electrons in the cavity.A plane-wave basis set automatically includes the possibility of electron localisation at sites other than nuclei. Using atomic basis sets, the same result can be obtained if the set of basis functions is —exible enough, e.g.when it includes diÜuse functions at the atomic positions. In this work, we placed —oating Gaussian functions in the centre of the vacancy. In particular, we have left the entire set of basis functions at the position of the missing O atom. In this way the basis set superposition error in the computation of the defect formation energies is strongly reduced. For the case of a bulk F centre we have carefully compared calculations with and without —oating Gaussians in the centre of the vacancy.A diÜerent approach has been followed for Starting SiO2 . from the hypothesis that the removal of an oxygen atom leads to the formation of160 Cluster model calculations of oxygen vacancies dangling bonds or localised SiwSi covalent bonds, no basis functions have been placed at the position of the missing oxygen.The previous discussion of the embedding approach and of the basis sets raises the question of the assumptions which are behind a cluster model. In fact, although the calculations are of ab initio type (no parameters are –t to the experiment), some basic assumptions about the nature of the material lead to diÜerent treatments of the same problem, the description of the oxygen vacancy.In ultimate analysis, it is the comparison of the results, in particular for observable properties, with the available experimental data which provides a direct validation of the approach used. 2.3 Formation energies The formation energies of defects in oxides are difficult to determine experimentally. While the HF cluster wavefunctions are sufficiently accurate for the description of geometrical parameters, dissociation energies require the inclusion of electron correlation, in particular, when the bonding has a partial covalent character like in The SiO2 .importance of correlation is expected to be smaller for materials like MgO where electrostatic eÜects dominate. To determine the formation energies, of point defects in we have included Ed , SiO241 electron correlation at the second-order perturbation theory level, MP2.In general we have performed a single-point MP2 calculation at the HF optimal geometry but we also considered fully optimised MP2 structures ; the deviations are always smaller than 0.2 eV. We use the notation HF/MP2 for –nal MP2 energies evaluated on the optimal HF geometry and MP2/MP2 when the energy is that of a fully optimised structure at MP2 level.Furthermore, we indicate the inclusion of d functions on Si and O basis sets by (d, d@). Therefore, HF/MP2(d,d@) refers to a single point MP2 calculation including d functions on Si and O atoms involved in bond breaking based on the HF optimal geometry; MP2/MP2(d) refers to an MP2 geometry optimisation with d functions on Si but not on O.41 The formation energies of the MgO surface centres, determined at the HF level, Fs have been discussed previously.21 Here, we will brie—y outline some aspects of the calculations in order to compare MgO with In particular, given the difficulties inherent SiO2 .with the description of charged defects, we will concentrate on the formation energy of a neutral centre. Fs 2.4 Hyper–ne interactions The determination of the hyper–ne coupling constants of unpaired electrons with the 29Si or the 25Mg nuclides is of fundamental importance for the spectroscopic and electronic characterisation of paramagnetic defects in and MgO.The hyper–ne Hamil- SiO2 tonian, H\S Æ A Æ I, is given in terms of the hyper–ne tensor A which describes the coupling of the electron with the nuclear spin.The components of A can be represented by the following matrix notation : A\;A1 0 0 0 A2 0 0 0 A3 ;\aiso];B1 0 0 0 B2 0 0 0 B3; (1) The isotropic part of the coupling constants, is related to the spin density at the aiso , nucleus (the Fermi contact term) : aiso\(8/3n)gN bN g0 b oWs(0) o2 (2)G.Pacchioni et al. 161 where and are the nuclear and electronic g-factors, and and b are the nuclear gN g0 bN and Bohr magnetons. The anisotropic contribution B results from the dipolar interaction. The comparison of the computed with the experimental hyper–ne constants provides, therefore, a direct validation of the computational approach. We have determined the components of the Atensor for the paramagnetic oxygen vacancies, in and VO ` SiO2 F` or in MgO, from UHF calculations.These values are therefore obtained at the Fs ` one-electron level. To estimate the importance of correlation, we have performed spinpolarised gradient-corrected DFT calculations using exactly the same clusters and basis sets. The hybrid exchange-correlation potential of Becke42 and Lee et al.43 (B3LYP) was employed in the DFT calculations. 2.5 Optical transitions (SiO2) The calculation of transition energies very important for the characterisation of (Te), point defects in crystalline and amorphous silica, requires an extensive inclusion of correlation eÜects. We performed multireference single and double excitations CI calculations, MRD CI,44h46 for ground and excited states of the clusters.In the MRD CI scheme the generation of single and double excitations with respect to more than one reference con–guration allows the inclusion of higher excitation classes with respect to the leading con–guration in the –nal CI results. Only those con–gurations with an estimated energy contribution to the total CI energy larger than a given threshold (E) are included in the secular determinant; the contribution of the remaining con–gurations is estimated perturbatively, based on an extrapolation technique.44 Excitations from the highest occupied levels to all virtual orbitals have been allowed. 24 or 25 valence electrons have been correlated, depending on the defect, and ca. 3»5000 con–gurations have been included in the secular problem, while the entire CI space consists of more than 106 con–gurations.The reported CI energies are extrapolated to this larger CI space. All con–gurations contributing more than 0.2% to the –nal CI wavefunction are used as main (M) con–gurations. Two roots (R) have been determined for each case. In the CI calculations we used a 6-31G basis set47 augmented by a d function on Si and an MIDI]d function48 for the O atoms directly involved in the electronic transition, while reduced MINI-1 basis sets49 have been used on the peripheral oxygen atoms to reduce the computational eÜort.DiÜuse s and p functions have been added to describe high excited states with Rydberg character. Absorption intensities have been estimated by means of the oscillator strength, f.Allowed transitions exhibit oscillator strengths of 0.1 or larger. Three quantum chemical program packages have been used for the calculations : HONDO 8.5 for HF and MP2,50 Gaussian 94 for DFT51 and GAMESS-UK for MRD CI52 calculations. 3 Oxygen vacancies in SiO2 3.1 The neutral oxygen vacancy, VO 3.1.1 Formation energies. The removal of a lattice oxygen in a-quartz gives a VO centre where two SiwO bonds are broken and replaced by a direct two-electron twocentres SiwSi bond.After relaxation, the SiwSi distance decreases from 3.06 to 2.52 Aé (computed with the minimum cluster, Fig. 1). Clearly, the breaking of two Si2O6H6 SiwO bonds requires substantial energies. Calorimetric experiments have been performed to provide measures of the formation energies of dominant defects in nonstoichiometric oxides,53 but experimental data for a-quartz or amorphous silica are not available. Theoretical estimates diÜer signi–cantly from method to method.54h57 Cluster model extended Hué ckel calculations gave ca. 10 eV for the energy associated with the162 Cluster model calculations of oxygen vacancies process ySiwOwSiy]ySiwSiy]O.54 Higher values have been obtained by more recent semiempirical quantum mechanical studies.55 In 1990, the –rst ab initio value, eV, was reported, based on local density functional (LDF) band structure calcu- Ed\7.9 lations,56 but it is well known that LDF overestimates binding energies. Using a formally exact embedding scheme a much smaller eV, was obtained by ab initio HF EdB5 calculations.57 Very recently, it has been observed58 that the ab initio HF estimate of 5 eV57 is not consistent with thermodynamic criteria, which suggest 7.3 eV as a lower bound for the formation energy of an oxygen vacancy in Thus, not only are the SiO2 .data scarce, but also inconsistent. We have computed using HF and MP2 wavefunctions. Since we are particu- Ed(VO) larly interested in the SiwO and OwO bonds, for comparison we have computed the SiO and diatomic molecules whose experimental dissociation energies, are 8.26 O2 De , and 5.11 eV, respectively.59 The MP2/MP2(d,d@) 8.09 and 5.04 eV, respectively, are De s, in excellent agreement with the experiment.At the HF/MP2(d,d@) level, i.e. the same approach used for the determination of in the computed 8.01 eV (SiO) Ed SiO2 , De s, and 4.80 eV are also reasonably close to the measured ones.(O2), We –rst discuss the eÜect of basis set and cluster size on by considering the Ed(VO) minimum cluster, Table 1. At the HF/HF level (no d on O and Si) eV. Si2O7H6 Ed\5.5 This value compares rather well with the eV obtained with the EMBED EdB5 approach and a similar basis set.57 By adding a d function on Si, HF/HF(d), becomes Ed 6.6 eV; when the d function on O is also included, HF/HF(d,d@), is 6.7 eV.Thus, the d Ed function on Si is considerably more important than that on O. For our study we used both small clusters and larger models like Fig. 2. In this case we described Si5O16H12 , the ySiwOwSiy unit involved in bond breaking with the same basis set as in while all the other atoms were treated with a smaller 3-21G basis set.60 With Si2O7H6 this larger, fully optimised, cluster, at the HF/HF(d) level, 6.7 eV, is almost coincident Ed with that obtained with the smaller model.Therefore, we can safely conclude that the size of the cluster has little eÜect on the binding energies, at least for neutral defects (charged defects can induce substantial polarisation of the lattice which is not taken into account by a –nite cluster).We discuss now the role of electron correlation. The computed value of with the Ed cluster at the HF/MP2 level is 7.0 eV, Table 1, i.e. 1.5 eV larger than at the Si2O7H6 one-electron level. With a d function on Si, HF/MP2(d), becomes 7.9 eV, Table 1 (the Ed same value is found after a full MP2 geometry optimisation). Finally, adding a d function on O, HF/MP2(d,d@), is 8.5 eV.This is our best estimate of the formation energy Ed Table 1 Formation energy, of an oxygen vacancy in a- Ed , quartz as a function of the theoretical levela Si basis set O basis set theoretical levelb Ed/eV DZ DZ HF/HF 5.5 DZ]d DZ HF/HF(d) 6.6 DZ]d DZ]d HF/HF(d,d@) 6.7 DZ DZ HF/MP2 7.0 DZ]d DZ HF/MP2(d) 7.9 DZ]d DZ MP2/MP2(d) 7.9 DZ]d DZ]d HF/MP2(d)(d@) 8.5 experimentc [7.3 and clusters, Fig. 1, have been used for a Si2O7H7 Si2O6H7 the regular and defective structures, respectively. b See text for de–nitions. c Ref. 58.G. Pacchioni et al. 163 of and is perfectly consistent with the recent proposal of Boureau and Carniato VO eV).58 These data also explain the discrepancy between the other ab initio (Ed[7.3 value reported in the literature, ca. 5 eV,57 and the experimental guess ;58 this is due to the lack of d functions on Si and O and of correlation eÜects. Their simultaneous inclusion is essential in order to obtain reliable formation energies of defects in silica. 3.1.2 Optical transitions. Now we report the MRD CI calculations of the optical spectrum of a centre.The presence of an O vacancy in results in an impurity VO SiO2 state in the band gap. In MO terms, there is a localised level, the cluster HOMO, above the valence band. The lowest empty level, or LUMO, is below the conduction band. The –rst excited state in corresponds to the transfer of one electron from the p bonding VO HOMO to the p* antibonding LUMO with consequent breaking of the SiwSi bond and formation of a dangling bond on each Si atom, The high-spin, triplet ySi~ ~Siy.coupling of the unpaired electrons is lower in energy but the 3A(p)1(p*)1^X1A(p)2 transition, computed at 6.3 eV,61,62 is spin forbidden. The –rst allowed transition is 1 1A(p)1(p*)1^X1A(p)2; in the 1 1A excited state the two open-shell electrons are coupled singlet. The description of high-energy transitions with Rydberg character requires large, diÜuse, basis sets and is very challenging when the number of valence electrons is large.With the DZ]d basis set described previously, we obtain an MRD CI of 8.80 eV with an oscillator strength, f\0.27, typical of a strong absorption, Te Table 2. Experimentally, an optical transition at ca. 7.6 eV has been found in aquartz. 63h65 Therefore, our computed absorption is overestimated by ca. 15%. We considered, for comparison, the molecule which has an optical transition at 7.56 eV,66 Si2H6 and we computed an value of 8.66 eV, i.e. with the same error found for This Et VO . relatively large error is not uncommon for highly excited states. On the molecule, SiF2 which exhibits a transition at 5.5 eV, we obtain a much better agreement with a computed of 5.7 eV.61 Using a basis set with diÜuse s and p functions on Si (a\0.03), Te [7s5p1d], the –rst allowed vertical transition in occurs at 7.5 eV in excellent agree- VO ment with experiment, Table 2.This agreement is partly fortuitous since the uncertainities connected to the use of cluster models, to the size of the CI, or to the correct representation of solid-state eÜects introduce an error of ca.^5% in the computed Te . In any case, the calculations –rmly support the assignment of the 7.6 eV absorption band in oxygen-de–cient silica63h65 to centres. VO 3.2 Eº centre (VO ë) 3.2.1 Formation energy. The E@ centre is one of the few well characterised (VO `) defects of It gives rise to an optical transition around 5.8 eV.67 The E@ centre can SiO2 .be schematically represented as where one Si atom has a pyramidal tri- ySi~ `Siy, gonal coordination while the second is almost —at, because of the removal of the dangling bond. Using a small model of we compute a formation energy for Si2O7H6 VO , of 7.9 eV. This energy corresponds to the breaking of the ySiwSiy bond and the VO ` Table 2 MRD CI transition energies, of neutral, Te , and charged, oxygen vacancies of a-quartz VO, VO `, vertical Te/eV defect transition theory experiment VO ` 1 2A^X2A 6.3 5.8a VO 1 1A^X1A 8.8 (7.5)b 7.6c a Ref. 67. b Including diÜuse s and p functions in the basis set. c Ref. 63»65.164 Cluster model calculations of oxygen vacancies consequent ionisation of one unit. The process can be decomposed in two steps, ySi~ eV).Therefore, the ySiwSiy]ySi~ ~Siy (Ed\2.6 eV)]ySi~ `Siy (Ed\5.3 strength of the SiwSi bond is 2.6 eV only, while the ionisation of a fragment uses ySi~ more than 5 eV. Unfortunately, there are no experimental data for comparison. While one can con–dently assume that the computed formation energy for a neutral O vacancy is close to the real one, for a charged defect the energy of the ionisation could be substantially diÜerent because of the neglect of long-range electronic and geometric relaxation. 3.2.2 Hyper–ne interactions.The electronic structure of the E@ centre has been characterised by means of EPR spectroscopy.68 The calculation of the coupling constant, A, provides direct information on the spin distribution and an indirect test of the validity of the structural model used.The experimental spectrum shows the presence of a large coupling constant, 411 G, and of two very small values, of ca. 8»9 G each.68 This indicates an almost complete localisation of the unpaired electron on an Si atom. The two small values suggest a weak interaction of the unpaired electron with two neighbouring Si atoms. On this basis, various models have been proposed.The most accepted is the model mentioned above. More complex models have been formulated to ySi~ `Siy explain the other small hyper–ne interactions.27 For calculation of the A values we –rst considered the minimum model of Fig. 3(a). The geometry opti- [Si2O6H6]` VO ` , misation leads to the preferential localisation of the unpaired electron on but the A Si1 values computed with UHF wavefunctions, G and G, Table 3, are Si1\172 Si2\75 substantially diÜerent from the measured ones.We carefully analysed the basis set dependence but even the use of fully uncontracted inner s and p functions does not improve the result signi–cantly, Table 3. We performed DFT calculations using the Table 3 Dependence of hyper–ne coupling constants, A, for bulk (E@) centres of a-quartz on VO ` model, basis set, and theoretical treatment A/G model geometry method Si basis set Si(1) Si(2) [Si2O6H6]`, Fig. 3(a) ySi` ~Siy UHF DZP [172.1 [75.4 DFT DZP [192.2 [104.7 UHF DZP]3s@]3p@ [169.3 [74.9 UHF DZP (core [167.7 [73.5 uncontracted) UHF 3-21G [135.6 [54.4 DFT 3-21G [142.6 [73.3 UHF 6-31G [190.6 [83.1 DFT 6-31G [195.5 [105.5 UHF 6-31G* [162.8 [74.2 DFT 6-31G* [171.5 [94.2 UHF 6-31G** [160.7 [73.8 DFT 6-31G** [169.7 [93.6 [Si5O15H12]`, Fig. 3(b) ySi` ~Siy UHF 3-21G [160.6 [28.5 DFT 3-21G [142.6 [73.3 [Si8O24H18]`, Fig. 4(a) NPC UHF DZP [100.4 [89.8 [Si8O24H18]`, Fig. 4(b) puckered UHF DZP [322 0 [Si8O24H18]`, Fig. 5(a) puckered UHF DZP [386 0 [Si2O6H6]` OH2, Fig. 5(b) puckered UHF DZP [377 0 experimenta 411 » a Ref. 68.G. Pacchioni et al. 165 B3LYP exchange-correlation functional to assess the importance of correlation eÜects and we found only minor changes in the hyper–ne constants, Table 3. This may indicate the inadequacy of the model owing to the small cluster size. In fact, Sim et al.30 using an cluster embedded in PCs, found A values of 190 [Si5O15]13~ and 9 G. Although the largest value is only 50% of the experimental one, these data show a higher degree of localisation.We performed a calculation with a larger cluster, Fig. 3(b), identical to that used by Sim et al.30 apart from the embed- [Si5O15H12]`, ding: we used H atoms instead of PCs. With the same 3-21G basis set of ref. 30, we found Avalues of 161 and 29 G, respectively, with only little improvement compared to the smaller model, Table 3.Furthermore the use of DFT at the B3LYP level does not change the situation : the deviation from experiment is even larger, Table 3. We considered a larger cluster, Fig. 4(a). This cluster is ìsymmetricalœ with [Si8O24H18]`, respect to the central oxygen atom which has been removed, in the sense that it contains two units around the vacancy (the actual symmetry of the cluster is as for Si4O12H9 C1 the other clusters).Starting from the a-quartz structure and performing a full geometry optimization we found a local minimum for the centre where the unpaired electron VO ` is almost equally shared between the two Si atoms; also the relaxation of the two fragments is similar. The –nal structure is similar to that of a-quartz with an O atom removed and it has been classi–ed as ìnear perfect crystal œ (NPC) in previous works.69 The Avalues, and G, Table 3, re—ect the absence of spin localisation. Si1\100 Si2\90 This is clearly in contrast with the experimental evidence and rules out the computed structure as the lowest structure of the E@ centre. All models considered so far represent local minima on the potential energy hypersurface, but none of them accounts for the measured hyper–ne constants.A few years ago, Rudra and Fowler27 found a structure where one Si atom is oriented towards a lattice oxygen which becomes eÜectively three-coordinated. The distance between this oxygen and the Si` of the defect is ca. 2 Semiempirical calculations Aé .27 indicated that this ìpuckeredœ structure is the most stable one and gave hyper–ne constants in much better agreement with the experiment.27 We repeated the minimum search with the model, starting from a distorted structure where one Si [Si8O24H18]` atom has been displaced outside the cavity, as suggested by Rudra and Fowler.We found a second minimum similar to the puckered structure of ref. 27, Fig. 4(b). The calculations of the hyper–ne constants for this second structure give values which are in much better agreement with experiment, A\322 G, Table 3. However, with this cluster, Fig. 4(b), the bond between the ìpuckeredœ atom and the three-coordinated lattice Si2 oxygen cannot be represented. Therefore, we considered an cluster with a [Si8O24H18]` diÜerent structure, Fig. 5(a). In particular, this cluster includes a complete ring of Si and O atoms which provides the correct representation of the surrounding of the puckered With this cluster in the minimum-energy con–guration the bond distance between Si2 . and the three-coordinated O is 1.88 as found in previous studies.27 With this Si2 Aé , model we computed a hyper–ne constant in close agreement with experiment, A\[386 G, Table 3.We considered also a minimum model of the ìpuckeredœ con–guration with the cluster where a water molecule has been added to [Si2O6H6]`…H2O represent the three-coordinated oxygen, Fig. 5(b). The distance has been ySi`… … …OH2 –xed to that obtained with the larger ìringœ model and the geometry of [Si8O24H18]` the resulting complex has been fully reoptimised with the boundary condition that the H atoms are –xed.With this small model A\[377 G is also in very good agreement with the experiment, Table 3. This provides a strong indication that the proposed structure27 is the actual conformation of the E@ centre in a-quartz. 3.2.3 Optical transitions. Using the model of the ìpuckeredœ [Si2O6H6]`…H2O structure, Fig. 5(b), we found an MRD CI of 6.3 eV for the 2 2A^X2A excitation.Te The transition corresponds to the transfer of the unpaired electron from to ySi`. ySi~166 Cluster model calculations of oxygen vacancies The computed is ca. 8% larger than the experimental value, 5.8 eV.67 The oscillator Te strength, f\0.1, indicates that this transition carries sufficient intensity, Table 2. The theoretical results are therefore consistent with the experimental observation (within the ca. 5»10% error inherent in the approach used). Using other structural models of E@ we computed of 4.6»5.3 eV.61 Thus, not only the hyper–ne constants but also the Te s optical transitions do not coincide with the experimental values for the ìwrongœ models. The fact that an almost quantitative agreement is obtained between theory and experiment for both EPR and optical spectra with the ìpuckeredœ model provides compelling evidence for the structure of this defect. In conclusion, we have shown that hyper–ne coupling constants and optical transitions can be extremely useful to verify the validity of a structural model. Discrepancies between computed and measured properties are not due, as one could think, to the limitations of the cluster approach or to solid-state eÜects not included in the calculations.Rather, they provide a strong indication of the inadequacy of the model used. It is only from the combined use of experimental and theoretical information that reliable models of point defects in silica can be proposed. 4 Oxygen vacancies in MgO 4.1 Formation energies HF cluster and band structure calculations fail to reproduce the wide gap of MgO, 7.8 eV, mainly because of the limits of Koopmansœ theorem (but, for clusters, also because of the neglect of long-range polarisation eÜects70).With the largest clusters considered in this work the HOMO-LUMO gap is ca. 13»14 eV. DFT, on the other hand, gives too small a gap, ca. 50% of the experimental value.71 When a neutral O atom is removed from the bulk or the surface of MgO to form an F centre, a doubly occupied electronic state appears in the gap, about midway between the top of the O 2p valence band and the bottom of the Mg 3s and 3p conduction band.21 If one of the two electrons is promoted to the conduction band, it may move away from the defect leaving a charged defect behind, F` (or The singly occupied impurity state moves toward the top of Fs `).the valence band, but keeps its distinct character. If the electron occupying this state is also removed or promoted to the conduction band, one forms a doubly charged vacancy, F2` (or The occupied impurity states due to MgO oxygen vacancies give Fs2`). rise to typical absorptions in the region of 4.95 eV which are responsible for the blue colour of the polycrystalline material.72 A question related to F and F` centres, and their surface counterparts, is related to the electron distribution.The trapped electron(s) can either be localised in the centre of the vacancy or delocalised over the 6 (or 5) Mg2` ions around the cavity. Attempts to identify the degree of electron localisation with common computational techniques, like the Mulliken population analysis, lead to frustrating results.21 Depending on the exponents of the —oating basis functions or of the 3s Mg orbitals, one can obtain completely contradictory results, ranging from almost full localisation to complete delocalization over the Mg ions.This is a typical case of failure of the Mulliken analysis due to the strong basis set dependence.73 A much more reliable description of the electron distribution comes from charge density and spin density plots.21 They show, in an unambiguous way, that the electron(s) in and centres (as well as in bulk vacancies13,17,74) are Fs Fs ` trapped in the cavity with little delocalization over the neighbouring Mg ions.Before discussing the experimental evidence supporting this picture, we brie—y comment on the formation energy of a neutral O vacancy on the MgO surface.Using various HF cluster models of an centre, and Fs [O12Mg5]16~, [O12Mg13], [O20Mg21], Fig. 6, we obtained formation energies between 7.2 and 7.7 eV.21 This latter value, computed as is the most accurate one since it includes the O21Mg21 ]O20Mg21]O,G.Pacchioni et al. 167 local geometry relaxation following the oxygen removal. This is not large (the MgwO distances around the cavity change by ca. ^1%)21 but has some eÜect on Recently, Ed . the formation energy of an O vacancy on the MgO surface has been estimated by means of band structure HF calculations using the CRYSTAL program.74 The computed value of 7.4 eV is very close to our value of Although the basis sets are not identical and Ed .21 the geometry relaxation has been accounted for in a diÜerent way, one can safely conclude that the response of a MgO cluster embedded in PCs is similar to that of an extended system.What is missing from these calculations is electron correlation. Orlando et al.74 have estimated the correlation eÜects using a functional of the HF density and found that, at the correlated level, is 8.4 eV.Recent LDA calculations Ed gave eV,75 i.e. ca. 2 eV larger than the HF values. If one considers that the Ed\9.8 computed cohesive energy of MgO is overestimated in LDA by ca. 10%,75 it is possible to conclude that should be close to 9 eV. Ed These results have been obtained with clusters embedded in PCs but without ECPs.The ECPs strongly reduce the arti–cial polarisation of the O2~ ions at the cluster borders and change the electrostatic potential at the centre of the cavity. This has little impact on the formation energy of a neutral defect, but has substantial eÜect on ionisation processes or on the formation energies of charged defects. To show the importance of the embedding we have compared the ionisation potentials, of models of Ei, Fs centres surrounded by PCs only and by PCs and ECPs.76 The of an centre com- Ei Fs puted with (no ECP embedding) is 7.0 eV; the addition of the 16 ECPs lowers O8Mg9 this value to 3.7 eV.Similarly, the of an centre is 13.9 eV without ECP and 10.9 Ei Fs ` eV with ECP. This shows the large correction introduced by the ECPs.Of course, other important eÜects are the cluster size and the geometrical relaxation (not included in the model). The geometrical relaxation reduces considerably the With an O8Mg9 Ei . ECP cluster where all the ions in the –rst layer are free to move the O12Mg9]20 Ei decreases from 3.8 to 2.1 eV for and from 10.9 to 7.5 eV for These values Fs Fs `.76 compare relatively well with those obtained recently by Pisani77 with the EMBED approach,37 1.05 and 6.23 eV for and centres, respectively.The change in the Fs Fs ` Ei is a direct consequence of the oscillations in the electrostatic potential due to the diÜerent embedding. In general, it is desirable to use cluster models of ionic materials where the O ions are properly coordinated. 4.2 Hyper–ne interactions We restrict our analysis to the isotropic part, of the hyper–ne coupling constants. aiso , The experimental for an F` centre in bulk MgO is 3.94 G.78 We will use various aiso models, methods, and basis sets to check the accuracy of the computational approach, Table 4. With an cluster, Fig. 6(a), with —oating functions placed at the [Mg6O12]13~ centre of the cavity G is in excellent agreement with experiment, Table 4.aiso\4.4 Notice that, with the —oating functions, the Mulliken population indicates small localisation of the unpaired electron on the Mg ions, consistently with the small value of the Fermi contact term. We repeated the calculations without —oating functions ; aiso becomes slightly larger, 5.2 G, but still close to the experimental value.On the other hand, the spin population according to the Mulliken analysis indicates full localisation of the unpaired electron on the Mg ions. Clearly, the Mulliken analysis is unreliable. We also performed a calculation without —oating functions but with additional diÜuse 3s and 3p functions on the Mg ions (3s@,3p@), but the results do not change, Table 4. Having established that the use of —oating functions does not introduce a bias in the description of the electron distribution, we consider now the eÜect of the embedding in ECPs.The previous calculations have been repeated surrounding the external shell of O ions by 32 ECPs, Table 4. We found a small decrease in by ca. 1 G. These results have been aiso , obtained with UHF wavefunctions (the value of SS2T\0.75 indicates no spin168 Cluster model calculations of oxygen vacancies Table 4 Dependence of isotropic hyper–ne coupling constants, for bulk F` centres of MgO on model, basis set, and theoretical aiso , treatment Mg spin spin model geometry method Mg basis set density aiso/G population Mg` » UHF 3s,3p 2.09 203.8 1.00 [Mg6]11` bulka UHF 3s,3p] —oating Gaussiansb 0.11 10.6 0.05 bulka UHF 3s,3p 0.17 16.6 0.17 [Mg6O12]13~ bulka UHF 3s,3p]—oating Gaussiansb 0.045 4.4 0.08 bulka DFT 3s,3p]—oating Gaussiansb 0.041 4.1 0.21 bulka UHF 3s,3p 0.053 5.2 0.19 bulka DFT 3s,3p 0.043 4.2 0.19 bulka UHF 3s,3s@,3p,3p@ 0.052 5.1 0.19 [Mg6O12]13~]32ECP bulka UHF 3s,3p]—oating Gaussiansb 0.030 3.0 0.05 bulka DFT 3s,3p]—oating Gaussiansb 0.030 3.0 0.19 bulka UHF 3s,3p 0.039 3.8 0.17 [Mg6O18]25~]32ECP bulka UHF 3s,3p 0.034 3.3 0.16 relaxedc UHF 3s,3p 0.035 3.4 0.21 [Mg6O18]25~ no PCs bulka UHF 3s,3p 0.075 7.3 0.23 experimentd 3.9 » a Atomic positions –xed at unrelaxed bulk values.b Placed in the centre of the vacancy. c The position of 6 Mg and 12 O atoms has been fully optimized. d Ref. 78.G. Pacchioni et al. 169 contamination). We have performed spin-polarised DFT-B3LYP calculations using the same clusters and basis sets but we found virtually no diÜerence from the UHF values, Table 4.This is similar to what we observed for the paramagnetic defects of Table SiO2, 3. Thus, we can conclude that the spin distribution in these inorganic materials is properly described already at the UHF level using cluster models.All these results have been obtained for unrelaxed bulk geometries. With an cluster we have optimised the position of the six Mg ions and [Mg6O18]25~]32ECP of the 12 O ions nearest and second-nearest neighbours to the vacancy. The Mg ions move outwards by 0.04 and the O ions move inwards, by ca. 0.2 Despite this Aé , Aé . relatively large relaxation, virtually no change occurs in Table 4.We –nally con- aiso , sidered an cluster without embedding in PCs, to verify the importance of the Mg6O18 Madelung –eld in the electron localisation. The results, Table 4, show that, without external –eld, the spin density is about twice as large as that in the analogous calculation with the Madelung potential and show a substantial deviation from experiment.To summarise, all the models used give similar correct descriptions of the spin density in a bulk F centre except those where the Madelung –eld is not included. In contrast to the case, the neglect of the Madelung potential in MgO can give unphysical results. SiO2 This re—ects the much more ionic nature of the alkaline-earth-metal oxide. In general, the hyper–ne interactions in MgO F centres are small. This means that the unpaired electron is interacting only to a minor extent with the Mg ions around the vacancy.To prove this fact, let us consider a octahedral cluster with MgO bulk Mg611` geometry where one unpaired electron is delocalized over the Mg 3s and 3p orbitals, Table 4. The corresponding B17 G, is four times larger than in reality. Localisation aiso , at the centre of the cluster is not possible in this case because of the absence of the electrostatic –eld of the ionic crystal.Notice that for a free Mg`(3s)1 ion is very aiso large, 204 G, Table 4. This shows the role of the 3s»3p mixing in reducing the Fermi contact term. The computed constants for models of a bulk F centre, 4 G, are much smaller than in 17 G, indicating that the unpaired electron is localised in the Mg611`, centre of the vacancy and not on the Mg ions.It is also important to note that this result is independent of the basis set. We –nally discuss the paramagnetic surface F centres. With an model of O8Mg9 Fs ` we compute an of 6 G (UHF). As for the bulk, DFT results give the same hyper–ne aiso constants, 6.2 G. Larger clusters or clusters embedded in ECPs give values in the same range, 4»6 G.Thus, the coupling constants for the surface are similar to the bulk. EPR spectra for paramagnetic surface F centres exist only for high-surface-area polycrystalline MgO. No data are available, to date, for single crystals in UHV. The original measurements of 10»11 G by Tench and Nelson79 have been con–rmed by very recent data obtained by Giamello et al.22 These values are about two times larger than in our calculations.Furthermore, Tench and Nelson observed an inhomogeneous distribution of the electron spin in the cavity.79 According to their interpretation the spin is preferentially localised on the apical atom in the second MgO layer, see Fig. 7. The experiments of Giamello et al.22 also indicate the existence of an inhomogenous distribution of the unpaired electron.Our calculations, however, do not support the Tench and Nelson model. We found similar values, ca. 5 G, for all –ve Mg ions of a centre, inde- aiso Fs ` pendent of the cluster size, see Fig. 7.22 Therefore, not only are the absolute values of the constants diÜerent in the experiment and in the calculations, but also the distribution of the unpaired electron seems to be diÜerent.This is in contrast with the result obtained for the bulk, where the computed values are almost coincident with the experimental ones. Again, the discrepancy between theory and experiment can be due to limitations of the quantum chemical approach or, more likely, to the fact that the models used do not re—ect the ì real œ situation.Better resolved EPR spectra have shown the presence of a hyper–ne interaction between the unpaired electron on a centre and the H atom of an OH group on the Fs `170 Cluster model calculations of oxygen vacancies Fig. 7 Proposed models of the hyper–ne interaction in surface centres of MgO. (a) Computed Fs ` values for a fully dehydroxylated surface.(b) Tench and Nelson79 model. (c) Giamello et al.22 model of a hydroxylated surface. surface.22 The models of F centres considered so far do not include the presence of hydroxy groups. Thus, we have considered a series of cluster models of centres on Fs ` terraces, steps and edges of a hydroxylated MgO surface.22 We found that, when an OH group is close to a centre on a terrace, the unpaired electron becomes polarised by Fs ` the positively charged H atom, Fig. 7. This polarisation leads to an inhomogeneous distribution and to larger values for one or two Mg ions in the cavity (typically 9»13 aiso G).22 The new model completely reconciles theory and experiment, Fig. 7. The analysis of similar models on steps and edges suggests that what is observed in the experiment are F centres on MgO terraces with an adjacent OH group, also known as Fs(H) centres.22 5 Conclusions Cluster models can be used successfully to interpret local phenomena in solid-state chemistry.Modern quantum chemistry allows the solution of the Scrhoé dinger equation at a level of sophistication which makes possible, in principle, and in practice, the computation of many properties of solids from –rst principles.Here we have focused on two important observables, hyper–ne interactions and electronic transitions, but one shouldG. Pacchioni et al. 171 not forget the success of this method in many other applications, such as the interpretation of vibrational and photoemission spectra, the simulation of STM images, chemisorption and surface reactivity, etc.80 The method has the great advantage of being computationally simple (and cheap) but, in order to be used correctly, requires a continuous interplay with the experimental side and, to some extent, recourse to chemical intuition.It is mainly from this combined theoretical»experimental eÜort that a more detailed knowledge of phenomena in solids or at interfaces can be obtained.References 1 V. E. Heinrich and P. A. Cox, T he Surface Science of Metal Oxides, Cambridge University Press, Cambridge 1994. 2 Adsorption on Ordered Surfaces of Ionic Solids and T hin Films, ed. H. J. Freund and E. Umbach, Springer, Berlin 1993. 3 Chemisorption and Reactivity of Supported Clusters and T hin Films, ed. R. M. Lambert and G. Pacchioni, NATO ASI Series E, Kluwer, 1997, vol. 331. 4 E. A. Colbourn, Surf. Sci. Rep., 1992, 15, 281. 5 H. J. Freund, H. Kuhlenbeck and V. Staemmler, Rep. Prog. Phys., 1996, 59, 283. 6 G. Pacchioni, G. Cogliandro and P. S. Bagus, Surf. Sci., 1991, 255, 344. 7 J. W. He, C. A. Estrada, J. S. Corneille, M-C. Wu and D. W. Goodman, Surf. Sci., 1992, 261, 164. 8 L. Marchese, S. Coluccia, G. Martra, E. Giamello and A.Zecchina, Mater. Chem. Phys., 1991, 29, 437. 9 E. Giamello, D. Murphy, L. Marchese, G. Martra and A. Zecchina, J. Chem. Soc., Faraday T rans., 1993, 89, 3715. 10 G. Pacchioni, A. M. Ferrari and E. Giamello, Chem. Phys. L ett., 1996, 255, 58. 11 W. C. Mackrodt and R. F. Stewart, J. Phys. C, 1979, 12, 431. 12 R. W. Grimes, C. R. Catlow and A.M. Stoneham, J. Chem. Soc., Faraday T rans. 2, 1989, 85, 485. 13 R. Pandey and J. M. Vail, J. Phys. : Condens. Matter, 1989, 1, 2801. 14 R. Pandey, M. Seel and B. A. Kunz, Phys. Rev. B, 1990, 41, 7955. 15 B. M. Klein, W. Pickett, L. 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ISSN:1359-6640    

DOI:10.1039/a701361b

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 106, 173»187 Structural, electronic and magnetic properties of KMF3 (M = Mn, Fe, Co, Ni) Roberto Dovesi,a* Federica Freyria Fava,a Carla Roettia and Victor R. Saundersb a Department of Chemistry IFM, University of T orino, via P. Giuria 5, I-10125 T orino, Italy b Daresbury L aboratory, Daresbury, W arrington, UK W A4 4AD The structural, electronic and magnetic properties of the perovskite systems (M\Mn, Fe, Co, Ni) have been investigated with CRYSTAL95, a KMF3 periodic ab initio Hartree»Fock program.An all-electron Gaussian basis set has been used. The equation of state has been determined –rst for the cubic structure ; then deviations from cubic symmetry have been explored, with the result that the Mn, Fe and Co systems are found to be slightly more stable in a tetragonal geometry.The systems are almost fully ionic, with net charges for K and M of ]1, ca. [0.9 and ca. ]1.8 o e o , respectively. The antiferromagnetic (AFM) is always more stable than the ferromagnetic (FM) phase; the energy diÜerence *E\E(FM)[E(AFM) is shown: (a) to be additive with respect to the number of MwM –rst neighbours; (b) to increase with decreasing lattice parameter according to an inverse power law; and (c) to become zero when the angle approaches 90°.The MwFå wM super-exchange coupling constants, evaluated from *E by using an Ising model hamiltonian, are in qualitative agreement with the experimental data (from 30% to 45% of the latter). Mulliken population data, charge and spin density maps and density of states are used to illustrate the electronic structure. 1 Introduction Substantial progress towards a full ab initio account of the structural and electronic properties of important classes of crystalline compounds such as simple metals, semiconductors, ceramics and silicates has been achieved in recent years. For other classes of compounds and properties ab initio methods have been less successful.A typical area where additional eÜort is required is that of transition metal (TM) ionics (oxides, sul- –des, halides) and their magnetic properties ; the latter are of intrinsic interest and also because of the relationship between magnetic order and superconductivity.1 The perovskites (M\Mn, Fe, Co, Ni) represent ideal prototypes for the investigation of KMF3 the magnetic super-exchange interaction in a large class of ionic compounds, and of the relationship between structural, electronic and magnetic properties.Their highly ionic character and high symmetry, and the low coordination of the anions, are expected to allow the interpretation of their electronic and magnetic properties in terms of simple models. In the past they have been the object of intensive experimental and theoretical investigation.Much experimental data concerning the MwM magnetic interaction have been collected by de Jongh et al.,2,3 and interpreted in terms of simple model spin hamiltonians. Values of J, the super-exchange coupling constant, are proposed for the compounds with Mn, Co and Ni;2 their dependence on the MwM distance is discussed by comparing results obtained from diÜerent compounds (A\K, Rb, Tl).AMF3 173174 Structural, electronic and magnetic properties of KMF3 As regards theoretical ab initio methods, both cluster and periodic approaches have been adopted in the study of these systems. The super-exchange magnetic interaction in has been investigated with the cluster model by Mejiç as and Fernaç ndez Sanz4 KNiF3 and Illas and co-workers in a series of papers.5h10 The main advantage of the cluster scheme is related to the possibility of using sophisticated many-body techniques able to take into account electron correlation eÜects ; pure eigenstates of the spin operators can be obtained.Limitations are related to the –nite size of the cluster (usually only two TM atoms are considered) : the wavefunction is not correctly periodic, and border eÜects are expected to in—uence the obtained energies.Even so, J values close to those obtained with the present periodic method have been obtained by the above authors when the same level of theory (unrestricted Hartree»Fock, UHF) is adopted.4,10 Periodic calculations have been performed on the systems under study within the local density approximation (LDA);11 band structures and density of states are produced, but the total energies of the FM and AFM states were not provided.Density functional theory based computational schemes very often describe as metallic, TM systems which are known to be large-gap insulators.12 Corrections to the LDA formalism, such as the self-interaction correction (SIC)13,14 or the on-site Coulomb repulsion (U)15h17 do not seem to be able to improve the description of the ground-state total energy of the FM and AFM states.The present calculations are based on the periodic ab initio UHF method as implemented in the CRYSTAL code.18 This method has been applied in recent years to many TM oxides (NiO, MnO;19 and —uorides Fe2O3 ;20 Cr2O321) (FeF2 ,22 NiF2 , MnF223).In all cases, the systems are large-gap insulators ; the sign of the AFM»FM energy diÜerence is correctly reproduced and its magnitude is proportional to the values of the Neç el temperature. The UHF solutions are eigenfunctions of the spin operator, not of however, Så z Så 2; owing to the localized nature of the unpaired spins and the distance between the M ions, the eÜect of spin contamination on the energies of the magnetic states is expected to be small.The main limitation of the present scheme is related to the neglect of electron correlation, which may account for about half of the FM»AFM energy diÜerence.4,10 There are, however, many advantages in the present scheme: periodicity is fully exploited and border eÜects are absent ; basis set eÜects, additivity with respect to the number of neighbours, the relative importance of –rst- and second-neighbour interactions, and the eÜect of distance and angles on J can all be investigated.Preliminary results concerning have been presented in a previous paper.24 KNiF3 Here, the study is extended to and The structural, electronic KMnF3 , KFeF3 KCoF3 . and magnetic properties of which exhibits a considerable Jahn»Teller distor- KCuF3 , tion, have been presented elsewhere.25 The present work is organized as follows : in Section 2 computational details concerning the method and the basis set are presented.Section 3.A is devoted to the discussion of the geometry and, in particular, to the cubic]tetragonal distortion of the Mn, Fe and Co compounds.In Section 3.B, the electronic structure is discussed in terms of Mulliken population data, charge and spin density maps, and density of states. Section 3.C is devoted to the discussion of the FM»AFM energy diÜerence, additivity of the super-exchange interactions, and eÜects of geometry on J. 2 Computational details In the present work, we have used the CRYSTAL9518 code, which is based on the ab initio periodic Hartree»Fock method.26,27 All the systems have been investigated within the UHF approximation, to allow a description of the spin polarization due to the unpaired d electrons of the TM ions.Bloch functions are constructed from local functions (ìatomic orbitals œ, AOs), which are linear combinations (ìcontractionsœ) of Gaussian-type functions (GTFs); these areR.Dovesi et al. 175 Table 1 Exponents and coefficients of the contracted Gaussian-type basis functions used for Co2` in the present work coef–cient type exponents s/d p s 341 701 0.000227 48 850.0 0.001929 10 400.9 0.0111 2718.99 0.0501 819.661 0.1705 283.878 0.3692 111.017 0.4033 46.4757 0.1433 sp 855.558 [0.0054 0.0088 206.504 [0.0684 0.062 69.0516 [0.1316 0.2165 27.2653 0.2616 0.4095 11.5384 0.6287 0.3932 4.2017 0.2706 0.225 sp 51.5542 0.0182 [0.0287 18.9092 [0.2432 [0.0937 7.7251 [0.849 0.2036 3.5428 0.8264 1.4188 sp 1.4914 1.0 1.0 sp 0.6031 1.0 1.0 d 29.9009 0.0617 8.1164 0.2835 2.6433 0.529 0.8869 0.4976 d 0.3011 1.0 Coefficients multiply individually normalized basis functions.Fig. 1 Cubic unit cell of systems. The FM (left) and two possible AFM (AFM, centre and KMF3 AFM@, right) structures are shown.Open and –lled small circles indicate spin up and down M ions, respectively. In the central –gure, the dashed lines connect M ions with the same spin in the (111) plane.176 Structural, electronic and magnetic properties of KMF3 the product of a Gaussian and a real solid spherical harmonic. Extended all-electron basis sets are used containing 27, 17 and 13 AOs for the TMs, —uorine and potassium ions, respectively.The —uorine basis set can be denoted as 7-311G (the –rst shell is of s type and is a contraction of 7 GTFs; there are then three sp shells). For the TMs an 8-6- 411-(41 d)G basis is used, with two d shells. The potassium basis set is an 8-6-511G. The basis set is as used in ref. 19 (Mn), 24 (Ni, K and F) and 22 (Fe) ; the Co basis is given in Table 1. The FM unit cell contains one formula unit (5 atoms, 83 AOs in the basis) ; for the AFM phases (see Fig. 1) a double cell (10 atoms) is required. As regards the computational conditions, high numerical accuracy is required to investigate the relative stability of the FM and AFM phases,24 whose energy diÜerence is ca. 0.5]10~3 (cell)~1; for Eh this reason the following values have been used for the truncation tolerances in the evaluation of the Coulomb and exchange series :18,26 7 7 7 7 14. A shrinking factor of 4 has been used to de–ne the reciprocal net, corresponding to diagonalization of the Fock matrix at 10 points belonging to the irreducible Brillouin zone. At this level of accuracy the energy diÜerence between the single and double cells in the FM state is smaller than 10~6 (cell)~1.The use of larger sampling nets con–rms the uncertainty due to this Eh factor in the total energy to be less than 10~6 (cell)~1. Eh 3 Results and Discussion 3. A Equilibrium geometry The four systems under consideration are cubic (space group at room tem- Pm3 6 m) perature.The experimental lattice parameters are given in Table 2; the unit cell is shown in Fig. 1. Each M ion is surrounded by six second-nearest-neighbour M ions with the same (FM phase, Fig. 1, left) or opposite spin (AFM, Fig. 1, middle). At very low temperature, small deviations from the cubic symmetry have been observed by Okazaki and Suemune28 for the –rst three compounds of the series.was found to be mono- KMnF3 clinic (but the three lattice parameters were very close to each other, 4.168, 4.171 and Table 2 Calculated and experimental lattice parameters (a and c, in fractional coordinate (x), ”), bulk moduli (B, in GPa) and cell volumes (V , in for cubic and tetragonal systems, and ”3) KMF3 energy diÜerences between the two structures [*E(c»t), in kcal mol~1] cubic tetragonal system parameter calc.expt. ref. parameter calc. expt. ref. *E(c»t) KMnF3 a 4.28 4.19 35 a 4.267 4.168 0.43 B 65 65 35 c 4.298 4.174 36 V 78.4 73.6 x 0.279 0.273 V 78.3 72.51 KFeF3 a 4.22 4.12 37 a 4.237 0.31 B 69 c 4.139 V 75.2 69.9 V 74.30 KCoF3 a 4.16 4.07 28 a 4.139 4.057 28 0.48 B 74 c 4.225 4.049 V 72.0 67.4 V 72.37 66.64 KNiF3 a 4.10 4.01 28 » » » 0.00 B 79 85 38 » » » V 68.9 64.5 » » »R.Dovesi et al. 177 4.185 and b\89° 51@), rhombohedral (a\4.108, a\89° 51@) and ” KFeF3 KCoF3 tetragonal (a\4.057, c\4.049 Two subsequent studies29,30 indicate, however, that ”). is tetragonal (space group I4/mcm; see Table 2). The small diÜerence between a KMnF3 and b and the small deviation of b from 90° found in ref. 38 for are, thus, KMnF3 probably attributable to experimental error.A similar inaccuracy might aÜect the and structural data. KFeF3 KCoF3 The results of the present study are shown in Table 2. When cubic symmetry is enforced, the lattice parameter is overestimated by ca. 2% for all the systems, in agreement with previous Hartree»Fock results for TM compounds. When a lower symmetry is allowed (for the Co and Fe compounds we considered only the tetragonal subgroup P4/mmm of the ideal cubic perovskite) a small deformation is observed for KMnF3 , and as shown in the Table.In the case, the distortion, which is KFeF3 KCoF3 , KMnF3 attributed to anion»anion repulsion, is a nearly rigid rotation of the octahedra (see sketch in Fig. 2) by an angle d\6° 19@ (or 5° 43@, according to experiment).The total energy of the system as a function of d is reported in Fig. 3; the curve is symmetric with respect to the cubic situation (d\0). The barrier separating the two minima is very low (0.69]10~3 as expected because of the low experimental transition temperature Eh), between the cubic and the tetragonal phases (180»187 K29,30). In the case of the Fe and Co compounds, the deformation involves the octahedra, because of Jahn»Teller eÜects.Fe2` has a d6 high-spin con–guration; and up-spin t2g eg levels are fully occupied; the remaining down-spin electron occupies one of the three t2g levels (say a shortening of the octahedra along the z and elongation along the x and dxy ; y axes is then expected because of non-bonded repulsion between the TM d orbitals and the anion) and the triple degeneracy is lifted.Co2` has a d7 con–guration, so that t2g two down-spin electrons occupy, say, the and orbitals, with a consequent elon- dxz dyz gation of the octahedra along z. This Jahn»Teller eÜect for Co and Fe is small, because the radius of the d orbitals is small compared with the size of the octahedra, the latter being largely determined by anion»anion non-bonded repulsions, and because the lobes of the d orbitals are not directed towards the anions.For all three systems the t2g stabilization energy with respect to the cubic structure is very small (a fraction of 1 kcal mol~1). The tetragonal deformation determined experimentally for is much KCoF3 Fig. 2 Visualization along the z axis of the relative rotation of the octahedra in the tetrago- MnF6 nal structure of KMnF3178 Structural, electronic and magnetic properties of KMF3 Fig. 3 Total energy of tetragonal as a function of d, the angle (see previous KMnF3 MnwMn’ wF –gure) smaller in magnitude and of opposite sign with respect to that determined here (a»c of ]0.01 to be compared with [0.09 and is also contradictory with respect to the ”), explanation given above.More accurate low-temperature X-ray experimental data are required for both and KFeF3 KCoF3 . 3. B Electronic properties The electronic properties (which are similar for the FM and the AFM solutions) were evaluated at the experimental room-temperature geometry (cubic). The four systems are nearly fully ionic insulators ; the net charges, evaluated according to a Mulliken partition of the electron density, are ca.]0.99, ]1.8 and [0.93 o e o for K, M and F, respectively (see Table 3), very close to the formal charges]1,]2 and [1 o e o of completely ionic compounds. The unpaired electrons are almost completely localized on the transition metal 3d orbitals, whose spin populations are 4.93 (Mn), 3.92 (Fe), 2.93 (Co) and 1.93 (Ni).The strongly ionic character of these systems is also con–rmed by the bond population data (see Table 4) : these are null for MwK (no interaction) ; small and nega- Table 3 Electron population data (in o e o units) according to a Mulliken analysis Q Ns q3d ns system M K F M M K F M KMnF3 ]1.77 ]0.99 [0.92 5.19 4.94 [0.00 0.02 4.93 KFeF3 ]1.83 ]1.00 [0.94 6.13 3.93 0.0 0.0 3.93 KCoF3 ]1.82 ]1.00 [0.94 7.15 2.93 0.0 0.0 2.92 KNiF3 ]1.85 ]1.00 [0.95 8.10 1.95 0.0 0.02 1.94 Q and are the net charges and the 3d orbital populations, respectively ; and are q3d Ns ns the corresponding spin quantities.Numbers refer to the FM solutions (AFM data are very similar).R. Dovesi et al. 179 Table 4 Bond population data (in electrons) at the experimental cubic geometry according to a Mulliken analysis system MwFap MwFeq MwK KwF KMnF3 FM [0.010 0.0 [0.001 AFM [0.009 0.0 [0.001 KFeF3 FM [0.021 [0.019 0.0 0.0 AFM 0.001 [0.006 0.0 [0.003 KCoF3 FM [0.016 [0.018 0.0 0.0 AFM [0.007 0.001 0.0 [0.003 KNiF3 FM [0.004 0.0 [0.005 AFM [0.003 0.0 [0.005 and indicate apical and equatorial —uorine ions of the octa- Fap Feq hedra.tive, as a consequence of the short-range repulsion, for MwF and KwF.This picture is supported by the diÜerence density maps, obtained by subtracting the superposition of the isolated spherical ion electron distributions from the bulk electron density (the same basis set as for the bulk has been used), reported in Fig. 4 for Only minor KMnF3 . modi–cations arise in going from the superposition of ions to the bulk; the most total (top) and diÜerence (bottom) electron density maps in a (001) plane (left) Fig. 4 KMnF3 through the Mn and F atoms, and in a (110) plane (right) through the three type of atoms. The diÜerence maps are obtained by subtracting, from the bulk density, the superposition of the isolated spherical ion distributions obtained with the basis set used for the bulk.For the total maps, the separation between two contiguous isodensity lines is 0.01 the innermost curves in o e o … a0~3; the atomic region correspond to 0.08 For the diÜerence maps, the separation between o e o … a0~3. contiguous line is 0.005 the function is truncated in the core regions at ^0.03 o e o … a0~3; o e o … a0~3; continuous, dashed and dot»dashed lines correspond to positive, negative and zero values, respectively.180 Structural, electronic and magnetic properties of KMF3 obvious feature is the usual shrinking of the electron charge density of both cations and anions, which is a consequence of the Madelung –eld and short-range repulsion ; this contraction is spherical for K, whereas it is directed along the MwF bonds for M and F; on the latter a small outward displacement of charge is observed in directions orthogonal to the MwF bond.The picture of the electronic properties of these systems can be completed by analysing the projected density of states (DOS). The four systems are large-gap insulators ; the gap ranges from 0.25 to 0.35 The highest occupied states are predominantly of TM d Eh . character with small contributions from —uorine p orbitals ; for the lowest unoccupied states, the DOS shows small contributions from —uorine p orbitals ; at slightly higher energies a sharp peak due to TM d orbitals appears. In Fig. 5 the valence DOS of is reported. The valence states are largely KMnF3 constructed from —uorine p and TM d orbitals. The potassium 4s states, and the Mn 4s and 4p states are almost completely unoccupied; Mn 3sp states are much lower in energy.The DOS of the AFM structure presents relatively narrow and well separated peaks, that can be interpreted in terms of molecular levels perturbed by the MF6 Fig. 5 Valence bands projected density of states of the AFM (top) and FM (bottom) phases of In the top –gure, only the up-spin Mn atom projection is given.KMnF3 .R. Dovesi et al. 181 environment. Assume that M is up-spin, and consider the interaction of the M d orbitals with the up-spin p orbitals on neighbouring F atoms. The six p —uorine orbitals oriented along the MwF bonds generate hybrids of and symmetry; the other 12 p A1g, Eg T1u orbitals combine to give hybrids of and symmetry; the M atoms T1g, T2g, T1u T2u contribute only with d orbitals (4sp orbitals are empty), with symmetry and The Eg T2g .valence DOS therefore shows states which exhibit MF hybridization with and Eg T2g symmetry. The bonding»antibonding splitting for the former is larger, because the interaction in this case is of p character. The other states generated by F p orbitals do not hybridize with M d orbitals for symmetry reasons, giving rise to a large peak at the centre of Fig. 5. For the FM solution the peaks in the up-spin DOS are wider as a consequence of the greater number of orthogonality constraints. Atomic like peaks then overlap, and the simple interpretation given for the AFM DOS is not appropriate. In Fig. 6 the DOS for in the AFM state is reported ; the most evident feature is the KCoF3 appearance of a down-spin sharp band in the valence DOS.As a consequence dxz]dyz Fig. 6 Valence bands projected density of states of the AFM phase of The d orbital KCoF3 . projection is shown in the bottom –gure. Only one type of Co atom is shown; the projection for the other type is symmetric with respect to the horizontal line.182 Structural, electronic and magnetic properties of KMF3 of the symmetry reduction and degeneracy removal, four d peaks are now easily identi- –ed for up-spin states, corresponding to and from low to high dx2~y2, dz2, dxy dxz]dyz energies.The reason for this ordering is clear : the presence of two down-spin electrons in the and orbitals destabilizes the (with respect to and (with dxz dyz dz2 dx2~y2) dxz]dyz respect to up-spin levels because of on-site Coulomb interactions.dxy) 3. C Magnetic properties Two AFM cells are shown in Fig. 1 (centre and right) ; they correspond to a sequence of (111) and (001) TM planes, alternatively with up- (open circles) and down-spin (–lled circles) ; in the –rst case (AFM) each M ion has the six –rst neighbours with opposite spin ; this structure corresponds to the experimentally observed situation.In the second case (AFM@) only two –rst neighbours have opposite spin ; this structure will be used for discussing the additivity of the super-exchange interactions. The AFM is more stable than the FM phase for all four systems considered, even when the unit cell undergoes relatively large geometrical modi–cations (tetragonal or trigonal deformation, isotropic expansion and compression).The energy diÜerence, *E (per formula unit), between the FM and the AFM structures is reported in Table 5. *E has been obtained at the experimental high-temperature (cubic) geometry; it is always less than 1]10~3 and increases nearly linearly with the atomic number. Eh , The experimental data are usually expressed in terms of magnetic coupling constants, J, of simple spin hamiltonians, such as the Heisenberg or Ising models.According to the Ising model, under the hypothesis of additivity of the MwM interactions and taking into account only –rst-neighbour spin interactions, the energy diÜerence (per formula unit) between the FM and AFM solutions is related to J through the following equation (see ref. 10 for a more explicit discussion of the relationship between UHF and Ising states ; see also ref. 24 for the diÜerence between the present and previous results for KNiF3) : *E\2zJS2 k (1) where z is the number of nearest-neighbour TM ions of opposite spin with respect to the TM selected ion, S is its conventional spin (5/2 for Mn, for example) and k\3.1577]105 is the ratio between the conversion factor from to J (1 Eh Eh\ 4.359 748]10~18 J) and the Boltzmann constant (1.380 655 8]10~23 J K~1).The calculated J values (see Table 5) are always smaller than the experimental ones (they range from 30% to 45%). The discrepancy can, in principle, be attributed to many factors : (i) Table 5 Calculated and experimental coupling constants (in K) at the experimental geometry J 102 J2/J system calc.expt. 102 calc./expt. ref. calc. expt. ref. T KMnF3 1.24 3.65 34.0 2 0.34 1»3 39, 40 295 3.70 33.5 2 200 KFeF3 2.54 6.0 42.3 32 KCoF3 5.48 19.1 28.9 33 0.52 114 19.2 28.5 33 KNiF3 14.91 44.5 33.5 32 0.30 0.5 41 300»700 50.8 29.4 42 4.2 J and are the coupling constants between the nearest-neighbour and next-nearest-neighbour J2 TM ions ; T is the temperature (K) to which the experimental determination refers.R.Dovesi et al. 183 inadequacy of the model (Ising) adopted in converting the calculated *E to J and (ii) the experimental data to J; (iii) the UHF solutions are not eigenfunctions of (iv) corre- Så 2; lation eÜects are disregarded at the HF level. As regards point (iv), according to cluster calculations5h8 performed at the UHF level and including correlation treatments of various degrees of sophistication, interelectronic correlation is expected to increase J by about a factor two.An estimate of correlation energy through a density functional correction scheme,31 based on the HF charge density, increases *E by 10% to 40% (see Table 6, last column). The data of Table 7 exclude large basis set eÜects on *E. Presently, we are unable to estimate the importance of the other factors listed above.The additivity hypothesis, which is implicit in the z factor in eqn. (1), can be checked by using the energy of the AFM@ structure (see Fig. 1). Under the hypothesis that only –rst-neighbour interactions are relevant, *E/*E@ should be equal to 6/2 ; this is the case, with very minor deviations, for Mn and Ni (only 1.7%).For Fe and Co the deviation is larger (17%), because the two interactions along z (which are the only ones considered in AFM@) are diÜerent from the four in the xy plane as a consequence of the anisotropic spin density on the M ions. In these cases (Fe and Co) it is more appropriate to use the following expressions, instead of eqn. (1) : g*E\2S2 k (4Jxy]2Jz) *E@\2S2 k 2Jz (2) The resulting values for Jxy and Jz are 2.78 and 2.16 (Fe) and 5.11 and 6.22 (Co).Comparison with experiment has been performed by using (see Table 5), J1 \(4Jxy]2Jz)/6 because only a single J value is provided.32,33 Table 6 Energy diÜerence (in 10~3 per formula unit) between the FM *EHF Eh , and AFM Hartree»Fock total energies system *EHF *EC *Etot dEHF dX dK dC KMnF3 0.293 0.122 0.419 1 [3.84 14.98 [10.14 KFeF3 0.385 0.088 0.479 1 [3.56 13.05 [8.49 KCoF3 0.468 0.070 0.542 1 [3.39 12.22 [7.84 KNiF3 0.569 0.051 0.619 1 [3.28 13.09 [8.81 is the correlation contribution to this diÜerence (evaluated a posteriori *EC according to Perdewœs formula31,43).In the second half of the *Etot\*EHF]*EC . table the exchange (X), kinetic (K) and Coulomb (C) contributions to are *EHF reported.d quantities have been obtained after the division by so that *EHF dX ]dK]dC\dEHF\1. Table 7 Total energy diÜerence between the FM and AFM phases (*E) of as a function of the basis set KMnF3 case basis set *E a as described in Section 2 0.293 b case (a)]d on F (a\0.7) 0.293 c case (a) but 5-1d on Mn instead of 4-1d 0.292 d case (c)]sp (a\0.25) on Mn 0.334 e case (d)]d (a\0.4) on K 0.325184 Structural, electronic and magnetic properties of KMF3 *E and *E@ can be used to estimate the importance of magnetic interactions between second-nearest neighbours magnetic constants).From inspection of Fig. 1, and (J2 assuming that the interaction is additive, the following equations hold for AFM and AFM@, respectively : g*E\2S2 k 6J *E@\2S2 k (2J]8J2) (3) By solving for J and the latter is found to be 200»300 times smaller than the former, J2 , in qualitative agreement with experimental –ndings (see Table 5).The present model can be used to investigate two other aspects of the superexchange interaction, namely its dependence on R, the MwM distance, and h, the MwFwM angle. When the system is isotropically compressed, *E increases very rapidly ; if it is assumed that *E\cRn, as proposed by de Jongh and Miedema,3 the calculated n values are [12.2 (Ni) and [13.9 (Mn, see Fig. 7), in reasonable agreement with the experimental values (n\12^2) determined by measuring J for various AMF3 systems, with A\K, Rb, Tl.3 The variation in *E as a function of d (h\n[2d) has been explored in the interval 0OdO30° and the results are shown in Fig. 8 for If the data are –tted with a KMnF3 . parabola and extrapolated, we –nd that *E\0 when dB45°; this result supports the simple model of super-exchange proposed, for example, in ref. 34 (Fig. 3.22) and discussed below. The reason for the higher stability of the AFM with respect to the FM phase can be discussed with reference to the spin density maps (see Fig. 9 and 10). These maps are very similar for the two magnetic structures (apart, obviously, from the inversion of the spin density on half of the TMS in the AFM structure). Potassium ions appear to be unpolarized. In the —uorine region, on the contrary, a small but extremely important spin polarization appears, which is responsible for the (very small) energy diÜerence Fig. 7 Energy diÜerence per formula unit between the FM and AFM phases as a func- KMnF3 tion of the lattice parameterR. Dovesi et al. 185 Fig. 8 Energy diÜerence per formula unit between the FM and AFM phases as a function of d (see Fig. 3) between the two phases. In the FM –gure, owing to the Pauli repulsion (which is re—ected in our calculation by the orthogonality constraints on the crystalline orbitals) between up-spin F electrons and the up-spin (unpaired) electrons of the two neighbouring TMS, the up-spin density slightly contracts along the MwFwM direction in the neighbourhood of the F ion ; down-spin F electrons, not directly involved in the MwF spin interactions, are slightly more diÜuse.This gives rise to the spin polarization of the F ion shown in the –gure.In the AFM case the F spin polarization is much smaller (two isodensity lines instead of –ve), because the —uorine ion is surrounded by two Mn2` ions whose d electrons have opposite spin. In this situation a (very small) up-spin and downspin electron shift in opposite directions is sufficient to account for the Pauli repulsion between electrons with the same spin.Thus, the spin polarization on F is antisymmetric with respect to the plane orthogonal to the MwFwM direction, and much smaller than for the FM phase. Maps for the Mn and Fe compounds look very similar, apart from the obvious diÜerence that Mn (d5 con–guration) is spherical [so that the spin density in Fig. 9 Spin density maps for the FM (left) and AFM (right) solutions of in the (001) KMnF3 plane.The separation between contiguous isodensity lines is 0.005 the function is trun- o e o … a0~3; cated in the core region at ^0.03 Continuous, dashed and dot»dashed lines correspond o e o … a0~3. to up-spin, down-spin and zero values of the spin density, respectively.186 Structural, electronic and magnetic properties of KMF3 Fig. 10 Spin density maps for the FM (top) and AFM (bottom) solutions of in the (001) KFeF3 (left) and (110) (right) planes.Scale and symbols as in Fig. 9. the (001) and (110) planes is the same, and only the –rst is given in Fig. 9], whereas Fe is not, as the spin density map in the two sections shows; as a consequence the polarization of the two types of F ions is diÜerent, as is also indicated by the bond population data given in Table 4.There is a –nal point that deserves comment. In Table 6, *E, the HF energy diÜerence E(FM)[E(AFM), is decomposed into three contributions, namely Coulomb (C), exchange (X) and kinetic (K) ; it turns out that, in going from the FM to the AFM solution, a large reduction in kinetic energy takes place (*K is 10»15 times larger than *E) as a consequence of the removal of the symmetry constraint requiring the two M atoms to be equivalent.Coulomb and exchange contributions favour the FM solution, and cancel more than 90% of the kinetic energy gain ; *C is 2 to 3 times larger than *X. As anticipated, correlation eÜects, as estimated by a density functional scheme, increase the AFM stability with respect to the FM solution. 4 Conclusions The UHF method, as implemented in the CRYSTAL95 program, has been shown to be a useful tool for understanding the electronic and magnetic properties of crystalline compounds.Such a delicate quantity as the FM»AFM energy diÜerence is always qualitatively reproduced; its dependence on the MwM distance and MwFwM angle is easily obtained and described. The super-exchange interaction is shown, at least for these simple cases, to be additive with respect to the number of neighbours.The distortion of the octahedra in the and cases is easily explained in terms of KFeF3 KCoF3 Jahn»Teller eÜects. In the case the rotation of the octahedra is well reproduced. 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ISSN:1359-6640    

DOI:10.1039/a701528c

出版商:RSC    

年代:1997

数据来源: RSC