Ionic to molecular transition in AlCl3: an examination of the electronic structure Leonardo Bernasconi, Paul A. Madden and Mark Wilson Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, UK OX1 3QZ. E-mail: madden@physchem.ox.ac.uk Received 29th August 2001, Accepted 3rd December 2001 Published on the Web 2nd January 2002First published as an Advanced Article on the web 25th August 2000 AlCl3 crystallizes as an ionic solid but melts to form a molecular liquid consisting of Al2Cl6 units. In order to see if this transition involves a major change in electronic structure, the dimer and crystal of AlCl3 are examined with the aid of generalized gradient corrected density functional theory (GGC-DFT) calculations and the electronic wavefunctions examined with the aid of a Wannier localisation transformation of the Kohn– Sham eigenfunctions.The change from octahedral to tetrahedral coordination of Al which is observed on melting has been induced by simply rescaling the unit cell parameters, and the variations in the hybridization of the Wannier orbitals across the transition have been analysed thoroughly. The predominantly ionic character of the interactions across the ionic to molecular transition is confirmed, validating the use of ionic interaction potentials to represent AlCl3 and related systems. Dipole moments of the single ions have been estimated from the position of the Wannier function centres and they are found to reproduce with remarkable accuracy the values predicted by a polarizable ionic interaction potential.A similar analysis has been carried over to the AlBr3–Al2Br6 system, which is known to consist of molecular units in both crystal and gas phase. 1 Introduction Aluminium chloride shows an intriguing change in ‘‘bonding’’ upon melting.1,2 In the solid it crystallizes in the 6-coordinate YCl3 structure,3–5 in common with a large number of ionic compounds with larger trivalent cations, but the AlCl3 liquid has a very low conductivity (y1027 V21 cm), in contrast to these other compounds which show typical molten salt conductivities of order 1 V21 cm and melt at much higher temperatures (1100 K vs. 466 K). Melting of AlCl3 results in an enormous increase in the molar volume (y80%), relative to the ionic systems, and the Raman spectrum changes markedly, consistent with a complete change of local structure.6 The Raman spectrum of the liquid is very similar to that of the vapour which is known, from spectroscopy, thermodynamic data and electron diffraction,7 to consist of Al2Cl6 dimers.AlCl3 may therefore be said to make a transition from an ionic solid to a molecular liquid which is held together by simple van der Waals forces.8 Recently, it has been shown that a generic ionic interaction potential, in which the ions carry formal charges, reproduces this behaviour, whilst simultaneously accounting for the properties of the ionic, larger cation systems.9,10 From this ionic perspective, the solid consists of edge-sharing, cation-centred octahedra and on melting the cation coordination number drops to 4 (because of the small radius of Al3z relative to Cl2).Edge-sharing of two of these tetrahedral units produces the charge-neutral Al2Cl6 dimer. The unusual properties of the AlCl3 system are thus associated, from this perspective, with the particular ionic size ratio, which allows both 6- and 4-coordination. For larger cation chlorides, the higher coordination number is retained on melting, and there is no dramatic change of local order. For even smaller size ratios in AlBr3 and AlI3 the coordination number is 4 in the liquid and solid11,6 (the later describable as a molecular crystal of Al2X6 units). Polarization of the anion, which is included in the interaction model, plays an important role in promoting the edge-sharing between the coordination polyhedra responsible for the liquid and solid structures.Akdeniz and Tosi12 have DOI: 10.1039/b107715e PhysChemComm, 2002, 5(1), 1–11 1 also shown that a very similar polarizable ionic interaction model reproduces the structure and vibrational frequencies of the isolated dimer and have used it to discuss the formation energies of ionic species, like Al2Cl27 , which are likely to be responsible for the residual ionic conductivity of the melt. The general implication of these findings is that the ionic to molecular transition is successfully described with a purely ionic interaction model, with no fundamental change in electronic structure, and with polarization effects responsible for the ‘‘covalent’’ interactions13,14 which stabilize the molecular species. It is of interest to see if this conclusion can be accepted at face-value, by a direct examination of the electronic structure across the ionic to molecular transition.This would validate the idea that the ionic potential is representing the true interactions in the system, rather than working as an effective potential which simply reproduces the structures. Such a validation would give confidence that the potential could be transferred to other systems, such as the low temperature ionic melts formed by mixing immidazolium chloride with AlCl3,15 as well as confirming the true nature of the interactions across the full range of MCl3 systems.To carry out this validation by electronic structure calculations is not straightforward. Ab initio methods for periodically replicated systems using plane-wave basis sets now allow for an unbiased way of doing electronic structure calculations on the solid and on isolated molecules, so that their relative energies can be calculated accurately.16 Furthermore, the generalized gradient density functionals17 used in these calculations should be capable of giving a good description of the relevant interactions. Understanding the nature of the binding mechanism between atoms and of the changes which are brought about during a structural transition is, however, a difficult task. In the first place, typical plane-wave codes give Kohn–Sham eigenvectors which are delocalised over the whole molecule or even the full crystal lattice, and this prevents any direct information on the single atoms to be obtained, or, more generally, chemical pictures of the bonding to be elicited.Second, even if detailed knowledge of the charge density distribution around This journal is # The Royal Society of Chemistry 2002 Papereach ion were made available (e.g. via projection techniques), a distinction between ionic and covalent interactions among atoms or ions would nonetheless remain elusive. A possible way to overcome these difficulties is to switch to a localised representation of the electron density distribution, via a Wannier transformation18,19 of the Kohn–Sham eigenvectors.If the condition of maximal localisation20 is enforced, a set of Wannier (or Boys) orbitals is obtained which provides a picture of the electron distribution around atoms involved in chemical binding or ions polarised by the electrostatic field of the surrounding environment easily interpretable from a chemical point of view. In addition, a complete theory of electric polarisation in crystalline dielectrics has been developed in recent years,21–25 which validates the calculation of the dipole moments of single ions from the centre of charge of the subset of Wannier orbitals which localise in the vicinity of them.26 This approach thus offers an ideal way to compare results, about the degree of polarization of the anions, which are directly available from both the polarisable ionic simulation model and the ab initio calculations, and may be regarded as a stringent test for the validation of an interatomic potential derived from first principle calculations.The paper is organized as follows. In section 2 we give technical details on the ab initio calculations. Results for the structural and electronic properties of crystalline AlCl3 and gas phase Al2Cl6 are described in section 3, which concludes with an examination of the crystal–molecular transition. In section 4, we describe the Wannier function analysis of the electronic wavefunction. The Wannier functions turn out to be localised atomic hybrids, and allow the charge density and total binding energy to be examined within an ionic framework.In section 5, a direct comparison of the ionic dipole moments calculated from DFT via the Wannier analysis are compared with those predicted with the polarizable ionic interaction model (PIM). Section 6 reports, briefly, on similar analyses carried out on the AlBr3–Al2Br6 system. Our findings are briefly summarised in section 7. 2 Electronic structure calculations Kohn–Sham DFT17 total energy calculations27,28 were performed within the Generalized gradient corrected local density approximation29 to the exchange-correlation energy. The Kohn–Sham eigenvectors were expanded in plane-waves up to a cut-off energy of 175–300 eV and integrals in the Brillouin zone were estimated by quadrature30,31 over grids of 1 to 4 k-points.Norm conserving pseudopotentials32 in the Kleinman–Bylander representation33 were used to model the ion–electron interactions. Ionic relaxations were allowed in order to find the minimum energy configurations using forces computed from firstprinciples.34,30 The total energy was self-consistently minimised at each step of ionic dynamics28 via a Pulay density mixing conjugate gradient solver35 up to a convergence in the ground state energy of 5 6 1026 eV atom21. The Kohn–Sham orbitals were updated at each ionic step using a Verlet algorithm.28 In the optimized geometries (determined using the Broyden– Fletcher–Goldfarb–Shanno algorithm as implemented in CASTEP 3.9), the RMS of the forces acting on the ions was lower than 0.05 eV A°21, the RMS of the ionic displacement lower than 0.001 A°, and the energy change per atom lower than 2 6 1025 eV.Maximally localised Wannier functions (MLWF)20 were determined by unitary transforming the Kohn–Sham eigenvectors (1) w UmnjwmT m~1 XJ n(r)~ where the sum runs over all the Kohn–Sham states wi, and the 2 PhysChemComm, 2002, 5(1),, 1–11 unitary matrix U was determined by iterative minimization of the Wannier function spread20 (2) (Sr2Tn{SrT2n) n~1 XJ The Wannier function center (WFC) positions were computed according toref. 36 and 37 (3) ran~{ nn, a~x,y,z, V~ Mbm m~1 X3 (4) nmIm ln ½U{K(m)U n where Mnm ~ (bn ? um)b21 is the normalized projection of the n-th reciprocal lattice vector on the m-th Cartesian versor, and Kij (m)~Swi je{ibm.rjwjT 3 Structure and energies of the crystal and dimer 3.1 The crystal Crystalline AlCl3 belongs to the monoclinic space group C32h (C2/m) with unit cell of dimensions a0 ~ 5.92 A°, b0 ~ 10.22 A°, c0 ~ 6.16 A°and b ~ 108u containing four AlCl3 units.5 The crystal structure of AlCl3 (Fig.1) may be viewed as a cubic close-packed arrangement of chloride ions with Al ions occupying every second plane of octahedral sites.38 The slight distortion in the stacking of the chlorine atoms is mirrored by the presence of two groups of Al–Cl and Cl–Cl distances, with differences in each group of y0.05 A°. The atoms were initially Fig. 1 Structures of crystalline AlCl3, as seen along the c axis, and gas phase Al2Cl6. In the upper panel, the dashed lines define a 3 6 2 6 1 supercell.Black and yellow balls indicate Al and Cl respectively. T and B indicate terminal and bridging Cl ions.3 in the Wyckoff5 and in the Table 1 Geometric parameters for AlCl optimized structures Optimized Wyckoff 2.28, 2.33 3.39 3.14, 3.09 85.7, 84.1 95.9, 95.3 2.29, 2.32 3.41 3.16, 3.10 85.6, 84.6 95.4 Al–Cl Al–Al Cl–Cl Cl–Al-Cl Al–Cl-Al placed in their ideal positions and the geometry was then optimized keeping the unit cell parameters fixed. Results are summarized in Table 1: they show excellent agrrement between the calculated ionic positions and those in the crystallographic structure. Only one value of Al–Al distance is found for all four AlCl3 units in each cell.This is indicative of the fact that, although distortions are present within each AlCl6 group, the octahedral units are geometrically equivalent. Al2Cl6 dimer The structure of the free dimer was optimized using cubic periodic supercells of size a~10 A°(with cutoff energies of 300 and 450 eV in the plane-wave expansion of the Kohn–Sham eigenvectors) and a ~ 12 A°(300 eV). Results are summarised in Table 2. The optimized parameters are already well converged with the smaller simulation cell and low cutoff, and they are typically in very good agreement with both Hartree–Fock6 calculations and with experimental data.39 The only two ‘‘critical’’ parameters, for which deviations from experiment may be as large as 0.06 A°, are the Al–Al and ClB–ClB distances (B denotes ‘‘bridging’’—see Fig.1). Errors are nevertheless comparable with the (uncorrelated) Hartree–Fock results. Since the Al2Cl6 dimer can be formulated as a complex between two AlCl3 neutral species (in which each molecule may be thought of as acting simultaneously as a Lewis acid and base), the origin of these discrepancies may reside in the poor treatment of non-local correlation effects in DFT,17 which prevents weak interactions like van der Waals forces to be correctly dealt with, at least when standard treatments of the exchange-correlation energy are employed. We also determined the equilibrium structure for an isolated AlCl3 molecule, and we found, in excellent agreement with experiment,40 that this system is trigonal planar with Al–Cl distances of 1.066 A°.From the total energies thus computed we estimated a dissociation energy for the dimer (Al2Cl6 A 2(AlCl3)) of 113 kJ mol21, which compares favourably with the experimental value of 126 kJ mol21.41 3.3 The crystal to molecular transition Based on the information available on the structure of the crystal and the molecular liquid,8 March and Tosi42 have Table 2 Optimized geometric parameters for the Al2Cl6 dimer. The first three columns refer to different supercell sizes and cutoff energies in the plane-wave expansion of the Kohn–Sham eigenfunctions. Ab initio results (HF) are from ref. 6, PIM from ref. 9 and experimental data from ref. 39 a ~ 12 A°a ~ 10 A° Exp.PIM 300 eV 450 eV 300 eV HF 3.26 2.28 2.05 3.19 3.15 2.25 2.07 3.21 3.61 88.9 3.21 2.25 2.06 3.16 3.64 91.0 123.4 3.26 2.29 2.08 3.21 3.64 89.1 121.8 3.16 2.25 2.07 3.22 3.61 91.1 121.3 3.15 2.25 2.07 3.21 3.61 91.1 121.3 Al–Al Al–ClB Al–ClT ClB–ClB ClT–ClT ClB–Al–ClB 91.1 ClT–Al–ClT 121.3 proposed a model for the transition from the crystalline to the molecular regime in AlCl3. If the solid lattice is expanded, it becomes energetically more favourable for each Al to satisfy its own valence requirements by appropriating a halogen ion from each of the two pairs which it shares with neighbouring Al ions (other two Cl ions being shared between octahedral/tetrahedral units before and after the change in coordination), rather than to maintain the equilibrium configuration. The transition to the molecular regime is particularly favourable in systems containing polyvalent cations, in which the number of nearest neighbour anions in the molecule equals the cation valency.We adopt a simplified approach to modelling the structural (end electronic) changes accompanying the melting process. Starting from the optimized crystal structure, we gradually increase the volume of the unit cell, performing a full optimization with respect to the ionic positions at each increment of the cell volume. Following the argument of March and Tosi, the gradual transition from the crystal to the extended phase representative of the liquid brings about a competition between the sharing of some pairs of halogen ions between different octahedral units and the splitting of these pairs between Al ions.It might be imagined that these two situations correspond to two energy vs. volume curves for the relevant ionic structures. Initially the system resides on the curve of the crystalline structure and the change in volume increases its energy until a crossing takes place with the ‘‘molecular’’ curve, at which point the tetrahedral coordination becomes energetically more favourable and the molecular transition takes place. This is illustrated in Fig. 2, where (pair distribution functions) PDF’s for the optimized ionic configurations corresponding to energy minima are plotted for several values of the volume change D~(V 2 V0)/V0, where V is the actual value of the cell volume and V0 is the equilibrium cell volume.Initially, increasing the volume does not affect the octahedral coordination of the Al ions, and Al–Cl distances are simply shifted to larger values, up to y0.2 A°with respect to the optimized geometry. The transition to the tetrahedral coordination (Fig. 3) is marked, at D ~ 0.72, by the appearance of the two peaks corresponding to Al–ClT and Al–ClB in the dimer, essentially at the same distance from Al as in the gas-phase species. Despite the rather unphysical approach we have used to force the crystal–molecular transition, the change in the Al coordination occurs at a value of D (0.72) which is not far from the experimental estimate for the change in volume on melting (D ~ 0.88).The abrupt character of the transition is reflected in the dependence of the total energy on the volume change (Fig. 4). Initially, the energy rises in the expected monotonic fashion as the crystalline structure is expanded (y2 eV), after which a sudden drop occurs corresponding to the change of Al coordination. The energy then starts to behave rather smoothly (though oscillations may be present, as large as 0.3 eV) keeping below (and possibly slowly converging to) the free dimer energy. This is indicative of the presence of weak interactions between distinct dimers, the oscillations in energy being determined by changes in their mutual orientation and distance, rather than in their internal structure.The evolution of the HOMO–LUMO energy gap Eg (Fig. 5) is similarly characterized by an abrupt change from an initially smooth decreasing trend with increasing volume, followed by a jump of y1.9 eV at the transition, and a gradual approach to the free dimer value (which is only y0.3 eV higher than the crystal) afterwards. The initial decrease of the HOMO–LUMO gap with increasing volume, can presumably be explained (see e.g. ref. 43) by the different response of the occupied and empty electronic states to smooth changes in the electron–ion potential, brought about by the volume expansion. As long as the crystalline structure is conserved, the more localised PhysChemComm, 2002, 5(1), 1–11 3Fig.2 Al–Cl PDF’s for the minimum energy ionic configuration at selected values of D ~ (V2V0)/V0 during the expansion of the AlCl3 unit cell. Dashed lines correspond to the running coordination number c(r) ~ br0dr’ 4p (r’)2 g(r’). B and T refer to the bridging and terminal Cl ions, as in the lower panel of Fig. 1. nature of the occupied states results in a larger destabilization with respect to the empty ones (i.e., they follow the effective Madelung potential), giving an overall decrease of the energy gap with increased volume. The large variation in the gap at D ~ 0.72 suggests that major changes take place in the electronic charge distribution as soon as the expanded molecular phase is reached, which restore an Eg very close to the optimised crystal value.In Fig. 6 we show the bandstructure and the EDOS of the crystal just before the molecular transition and of the expanded phase just after it. The large increase in the HOMO–LUMO gap clearly results from a large upward shift of the empty Fig. 3 Octahedral to tetrahedral (ionic to molecular) transition in AlCl3 induced by the unit cell expansion. See text for details. 4 PhysChemComm, 2002, 5(1), 1–11 Fig. 4 Total energy variation during unit cell expansion in AlCl3. The line is just a guide to the eye. The fact that the energy does not seem to converge to exactly the free molecule value is due to the different quality of the Brillouin zone sampling in the crystal and in the molecular dimer (the last treated within the C point approximation).Fig. 5 HOMO–LUMO energy gap variation during unit cell expansion in AlCl3.Fig. 6 Bandstructure and EDOS (with Gaussian smearing of 0.25 eV) for AlCl3 at D ~ 0.65 and D ~ 0.72. G and F indicate the reciprocal lattice points k ~ (0.0, 0.0, 0.0) and k ~ (0.0, 0.5, 0.0) in the Brillouin zone corresponding to a real primitive unit cell of parameters a0 ~ b0~5.98 A°, c0~6.34 A°, a~b~98u, c~120u. Energies are referred to the Fermi energy EF. bands, while the majority of the occupied states, expecially in the vicinity of the Fermi energy, are more or less unaffected. The energy of the last occupied band (averaged over the Brillouin zone) is shown in Fig. 7 as a function of the volume change.In this case, the molecular transition results only in a smooth decrease of this curve, which gradually converges to the Fig. 7 Dependence of the HOMO energy on the expansion parameter D. The inset shows the charge density isosurface (corresponding to 0.05 e A° 23) obtained from this orbital. The arrow indicates the value of D at which the octahedral–tetrahedral coordination takes place. The line is just a guide to the eye. free dimer limit. This is explained by the fact that this band has predominantly 3p non-bonding character (see below), and is thus affected by the surrounding environment almost exclusively by repulsive interactions with nearby non-bonding orbitals and, possibly, Coulombic attraction with the Al3z ions screened by other Cl valence electrons. The smooth variation of this energy is to be contrasted with the abrupt change of the total energy.The energies quoted here are with respect to a zero which changes smoothly with the cell volume and should thus not be assigned a direct chemical significance. In order to confirm the reliability of the picture of the molecule electronic structure which emerges from the DFT calculations, we have performed a conventional Hartree–Fock MP2 calculation (with a 6-311zG(2d,p) basis set) using the GAUSSIAN package. This gives a band gap of 13.77 eV, as expected larger than the DFT value (6.05 eV) shown in Fig. 5, and a charge density for the HOMO which is virtually indistinguishable from that shown in Fig. 7. 4 MLWF examination of the charge density The calculations described above show that the ab initio methodology provides a good description of the ionic crystal, the isolated molecule and the ionic to molecular transition which occurs upon volume expansion.However, it is clear that no deep insight into the nature of the chemical interactions can be gained simply from examination of Kohn–Sham orbital energies and charge densities. In this section we therefore examine whether more specific information can be gained from an examination of the localised orbitals. 4.1 Wannier orbitals for the dimer The results of the Wannier localisation transformation on the occupied valence orbitals of the dimer are illustrated in Fig. 8 and 9. The predominantly ionic nature of the Al–Cl interactions is immediately demonstrated by all the WFC’s being located in the vicinity of Cl, indicating that the Al atom is fully ionised and that a full set of 8 valence electrons is found in the immediate around each Cl ion.The Wannier centres are displaced from the nuclei of the Cl ions and approximately tetrahedrally disposed. In an isolated Cl ion, the MLWF’s would correspond to the atomic 3s and 3p orbitals and the WFC’s would all be located at the nucleus. In the presence of an external electrostatic field, the ion would polarise as a consequence of mixings being induced between atomic orbitals, and the centre of charge of the resulting hybrid orbitals would move away from the nuclear position. In conventional valence terms they should be regarded as ysp3-like orbitals formed by the hybridization of the 3s and 3p orbitals on the Cl centres.It is worthwhile to emphasise that, since the basis sets used in the electronic structure calculations Fig. 8 WFC positions in gas phase Al2Cl6. The ionic core positions are indicated by black (Al3z) and yellow (Cl7z) balls and red balls have been placed on the centres of the Wannier functions. Note the proximity of the WFCs to the Cl nuclei and their approximately tetrahedral disposition around them. PhysChemComm, 2002, 5(1), 1–11 5Fig. 9 Charge density isosurfaces (corresponding to 0.05 e A°23) from selected b (‘‘bonding’’), directed towards adjacent Al centres, and nb (‘‘non-bonding’’) MLWF’s centered on ClB (above) and ClT atoms.are plane-waves, these atomic orbitals appear entirely naturally in the calculation. The orientation of the tetrahedra of Wannier centres around each atom and their distance from the nucleus reflect the polarization of the electron density of the Cl ion by neighbouring cations, and therefore differs between the bridging (B) and terminal (T) ions in the dimer (Fig. 1). ClB are characterized by two of the MLWF’s (‘‘bonding’’, b, at 0.60 A°from Cl), pointing towards Al and the other two (‘‘nonbonding’’, nb, at 0.41 A°) directed away from the center of the dimer. In the ClT ions the WFC corresponding to the b MLWF is at 0.65 A°from Cl, and those corresponding to the three nb MLWF’s at 0.42 A°.The orientation of these functions and their degree of displacement from the nucleus thus reflect just what would be expected for a Cl ion in the potential field of the neighbouring cations. 4.1.1 Orbital energies. Besides examining the charge density in the Wannier representation, it is possible also to assign energies to the Wannier functions and therefore to use them as a basis to discuss the interaction energies which stabilize the molecule. The orbital energies of the MLWF’s associated with a given ion may be assigned from (5) n~1 XK ei as (6) U ECl~ inUinei ECl~ n~1 XK SwnjH ^KSjwnT where H�KS is the Kohn–Sham Hamiltonian. This expression can be cast in terms of the Kohn–Sham single particle energies n~1 i~1 XKXJ ½U{LUnn~ which can be deduced from eqn.(1) by taking into account that U is unitary. Here L is the matrix of the Lagrange multipliers which is obtained by minimizing the Kohn–Sham total energy functional under the assumption of orthonormality between the single particle orbitals, and enforcing the diagonal-gauge condition Lmn ~ dmn em.44 ECl can be interpreted as the energy of the valence charge density surrounding the Cl ion estimated within the DFT framework. 6 PhysChemComm, 2002, 5(1), 1–11 Fig. 10 EDOS of gas phase Al2Cl6 in the Kohn–Sham (solid) and MLWF representation. A Gaussian smearing of 0.5 eV has been introduced. The electron density of states (EDOS) computed from the Kohn–Sham energies and from the MLWF energies as in eqn. (6), with the sum running over all the MLWF’s, is shown in Fig.10. The group of states at lower energy in the Kohn– Sham EDOS plot corresponds to orbitals with s symmetry associated with ClB, ClTor both, whereas only p states contribute to the higher energy bands. The MLWF plot splits into two main bands, at y2.3 eV from each other. The origin of this feature can be easily interpreted as the separate contributions of B and T chloride ions to the total MLWF EDOS plot and, in addition, the relative weight of each type of MLWF (b and nb) can be singled out for each class of anion. Results of this analysis are summarised in Fig. 11. The lower energy peak arises from b and nb MLWF’s of the bridging ion, ClB, and from b MLWF’s of ClT, while the sharper peak at higher energy originates from nb ClT states only.The splitting in energy between b and nb orbitals is y1.4 eV in ClB andy2.2 eV in ClT ions. This is indicative of the fact that the difference in the nature of the b and nb orbitals is more pronounced in ClT than in ClB and may, qualitatively, be associated with the different degrees of polarization interactions between the electrons in these orbitals and the adjacent Fig. 11 Decomposition of the MLWF EDOS for ClB and ClT in Al2Cl6. The thin dot-dashed line represents the total EDOS, the solid line corresponds to the separate contributions of the ions, and the dotted and dashed lines are the b and nb contributions for each ion.Al3z centre. The total energy of the valence electrons around ClT is thus 6.44 eV higher than for those around the bridging ion. 4.2 Wannier functions for the crystal The electronic structure of the crystal was analysed in terms of MLWF’s.The localisation algorithm we have adopted introduces errors in the value of the total MLWF spread of the order of L22,36,37 where L c V1/3 is a measure of the linear size of the simulation cell. For this reason, we performed a Wannier localisation transformation on a supercell of 2616 2 unit cells (with cell parameters 11.84 6 10.22 6 12.32 and V ~ 1417.81 A°3), which gives an error of only ¡0.008 A°. The positions of the WFC’s in the crystal are shown in Fig. 12. Again, all the WFC’s are located in the close vicinity of Cl ions, with the Al atoms fully ionised, consistent with a strong ionic component in the Al–Cl interactions.The degree of displacement of the WFC’s from the nuclei tends to be smaller than for the dimer, reflecting the fact that in the crystal the coordination environment is more symmetrical than in the dimer, so that the polarising fields are smaller, and also the fact that the effective polarizabilities of the Cl ion in the crystal are lower than those of the dimer due to the confining effect of the local coordination environment.13 The distribution of the WFC’s varies for ions in different sites of the unit cell, and it is possible to regroup inequivalent Cl ions into two classes (which will be labelled Cl1 and Cl2) according to the symmetry with which they arrange themselves around the Cl centre and to their distances from it.Cl1 ions are characterised by all WFC’s being confined in the plane defined by Cl and the two neighbouring Al ions, with distances of 0.06, 0.44, 0.60 and 0.61 A°from the Cl centre, whereas the WFC’s near Cl2 ions have a roughly tetrahedral sp3-like arrangement, similar to that found in the dimer, and distances of 0.30, 0.38, 0.48 and 0.61 A°from the ion centre. The planar distribution of the WFC’s in Cl1 suggests that orbital hybridization is restricted to the 3s and the two 3p atomic orbitals lying in the plane containing Cl and the two nearest Al ions, with the unmixed p orbital extending perpendicular to it. This is confirmed by plotting the charge density distribution of the MLWF corresponding to the WFC which is closest to Cl (Fig.13, upper panel). One of the ysp2 hybrids, corresponding to the ‘‘non-bonding’’ orbital at 0.44 A° from Cl, is also shown for comparison. The distribution of the Cl1 and Cl2 in the crystal lattice is Fig. 12 WFC positions in AlCl3, indicated by red balls. See text for details. Fig. 13 Charge density isosurfaces (corresponding to 0.09 e A°23) from selected MLWF’s centered on Cl1 ions. 1) shown in Fig. 14. Cl1 ions define parallel layers containing Cl1, Al and Cl2 which are shared with Al ions outside the plane. Each Cl1 has a nearest neighbour Cl1 at 3.16 A°(shared between the same two Al ions), a second neighbour Cl2 at 3.32 A°(which belongs to the same octahedral unit) and a third Cl1 at 3.95 A° (belonging to another octahedral unit and shared between different Al ions).Although the electrostatic field on this ion is likely to be shaped almost exclusively by the Al3z ions nearby, contributions to the electronic distribution around it may arise from correlations between the valence charge of neighbour Cl ions. Further calculations confirm that this is the case, suggesting that Cl1 ions adopt the sp2 z p arrangement to minimize exchange repulsion14 with the valence charge of second neighbour Cl1 ions. sp2 hybridization (with the ‘‘nonbonding’’ hybrid pointi towards the second neighbour Cl decreases the destabilization arising from the overlap of two sp3 hybrids competing for the free space between Cl1 ions of different octahedral units.Despite the qualitatively different arrangement of the WFCs around Cl1 and Cl2 ions in the crystal, these ions are much more similar to each other than are the bridging and terminal ions in the dimer. Differences in the total energy of the valence charge densities ECl between Cl a and b are only of the order of PhysChemComm, 2002, 5(1), 1–11 7Fig. 14 Distribution of the Cl1 (blue) and Cl2 (yellow) ions in the crystal. Dashed lines indicate the unit cell boundaries. 1 eV (compared to 6.44 eV between ClB and ClT in the dimer), with the lowest value for Cl2. The total spread associated with the valence charge density of a Cl ion can be estimated as (7) v1=2 (Sr2nT{SrnT2)1=2 Cl ~n~1 XK where the sum is restricted to the four MLWF’s localised in the vicinity of the ion.This gives essentially the same value for both types of Cl ion, though the valence charge density around Cl1 is slightly more compressed ([vCl1/vCl2]1/2 ~ 0.98). 5 Ionic dipole moments The Wannier analysis of the charge densities and orbital energies is qualitatively consistent with the picture emerging from the simulation studies,9,10,12 of ionic bonding in both the crystal and dimer, with strong polarization effects contributing in an important way to the binding. In this section we will attempt to use the Wannier analysis to support this picture quantitatively, by comparing the actual induced dipole moments on the ions, obtained from the electronic structure calculations, with those predicted by the polarizable ion potential used in the simulations.5.1 The ab initio ionic dipoles The Wannier function approach to a rigorous definition of the polarization in crystalline insulators is well known and documented in the literature.22,24,25 However, the possibility of using MLWF’s to gain quantitative information on the contribution of distinct ions or of subsets of ions to the total dipole moment of a molecule or to the polarisation of a periodic system has not been fully explored so far. The components of the electronic contribution to the total dipole moment p (or, in a periodic system, to the total polarization P ~ p/V, where V is the volume of the unit cell) can be estimated from the WFC positions, under the assumption that the system is not spin-polarized, as (8) e rn n~1 XJ p~2q where qe is the electronic charge, and the sum runs over the total number of electronic states.In addition, since the 8 PhysChemComm, 2002, 5(1), 1–11 Table 3 Ionic dipole moments in AlCl3 and Al2Cl6 computed from DFT–MLWF (DFT) and from PIM, with (PIM) and without (PIM*) inclusion of short-range effects AlCl Al 3 2Cl6 ClB ClT Cl2 Cl1 1.66 1.64 2.25 1.79 1.72 2.57 1.41 1.34 1.74 1.51 1.57 2.15 DFT PIM PIM* MLWF’s localise in discrete and possibly disconnected regions of space, partial dipole moments may be associated with subsets of states. This allows for a simple and intuitive separation of the charge density into the contributions of the single ions, and to define ionic partial dipole moments, mI for each ion I, as (9) rnzZIRI n~1 XK I~2qe where K is the number of WFC’s associated with the ion of pseudopotential core charge ZI in the position RI.The dipole moments estimated for the crystal and the gas phase dimer are reported in Table 3. In the crystal, the presence of two classes of Cl ions results in the appearance of two values of the dipole moment, differing by y0.10 au, with the Cl1 ions carrying the larger value. Similarly, ClT and ClB ions in the free dimer differ by y0.13 au, and the ClTs carry the larger dipole. On average, dipole moments in the gas phase dimer are y0.3 au larger than in the crystal, consistent with a more pronounced deviation from the ideal ionic behaviour in the Al–Cl interactions.The orientation of the ionic dipoles is represented in Fig. 15 for the dimer. In the crystal, all the dipoles are arranged as for the B ions, with pairs of ions shared between the same Al ions pointing their negative end approximately towards each other. This distribution of the dipoles corresponds to what would be expected from a fundamentally ionic picture of the Al–Cl interactions. In particular, the reciprocal orientation of the dipoles on the ClB ions in the dimer and on all the Cl ions in the crystal is found to agree with the mechanism of polarisation proposed by Madden and Wilson to explain the occurrence of polyhedral edge-sharing units in crystalline systems characterized by strongly polarizing cations.13 5.2 Dipoles from the PIM for AlCl3 In the polarizable-ion model (PIM),13 the induced dipole on each ion, for a given set of ionic positions (ri), is obtained by Fig. 15 Orientation of the ionic dipoles in gas phase Al2Cl6.The red portion of the cylinders indicates the negative end (conventionally chosen to correspond with the average position of the four WFC’s in the vicinity of the ion). The positive ends have been put in the ionic positions.minimizing the potential, Upol~ iT(1)(rij )qj fij (rij )z12 iT(2)(rij )j i,j X (10) z 2ki2i i e{b fij (r)~1{cij a (r)~ra=r3 T(2) ab (r)~(3rarb{r2dab)=r5 In addition to the moments induced on an ion by the electric field from (point) charges and dipoles of the other ions, extensive ab initio calculations45–55 have demonstrated the presence of additional moments which can be associated with short-range interactions of an ion with its nearest neighbours.In order to account for these shor-range moments, the charge– dipole interaction energy in eqn. (10) is modified by an additional r-dependent term, fij(rij), which depends on the identities of the ions i and j. Ab initio calculations in which the nearest-neighbour ion shell is distorted about a central ion, show that a suitable (‘‘damping’’) function is of the (Tang and Toennies56) form ijr)k k~0 X1 with respect to all dipoles mi. The set of calculated (selfconsistent) dipoles gives the polarization energy, Upol, associated with that configuration, and the force on each ion is obtained from derivative with respect to the position of that ion with the dipoles taking their self-consistent values.The final term in eqn. (10) is a Drude-like term to represent the energy required to polarize each ion. The force constant, ki, is directly related to the ion polarizability, ai, by ki~1/2ai. qi are the (formal) ionic charges on each ion and the T(n) tensors are the standard charge–dipole and dipole–dipole interaction tensors, T(1) (11) (12) ijr (13) The ability of the PIM to reproduce the moments on the ions in both the crystalline and isolated dimer geometries, without modification of the model parameters, is further confirmation of the previous (empirical) observation that these environments could be accounted for using a single polarizable model.In particular, the occurrence of two values of dipoles in the crystal, which would not be foreseeable on the basis of simple considerations based on structural information alone, is indicative of the fact that the account of ion polarization effects given by the PIM is fully capable to reproduce the deformations in the ionic valence charge density brought about by the crystalline environment, in a form which can be easily extracted from conventional DFT calculations. (bk! ! X4 The function switches from its large r limit of 1 (corresponding to a charge–dipole interaction of a purely Coulombic form) to 1-cij at zero separation, controlled by a range parameter bij (the short-range damping parameter—srdp).This srdp has been determined ab initio for a range of alkali halides and alkaline– earth oxides45–55 and has found to transfer between systems as a function of cation radius according to k b~sizsj where si(j) is the radius of ion i(j) and k is a constant for each anion. The AlCl3 potential model has been described previously and shown to reproduce a number of experimental structural and dynamical properties.9,10 Here, we are only concerned with the polarization aspect of the model. A dipole polarizability of 20.0 au is used for the chloride anion consistent with the results of ab initio calculations in the condensed environments.57,58,45–53 The cation polarizability is very small compared with that of the anion (aAl3z ~ 0.265 au57) and so is taken as zero.In addition, the stoichiometry means that the cations sit in highly symmetry environments which precludes the formation of low order moments. The damping parameters used in the present work are bz2 ~ 1.58 au, cz2 ~ 1.0 au, scaled from ab initio calculations on alkali halides as described above. The self consistent induced dipoles are calculated in both the crystalline and dimer forms using configurations optimized by the DFT calculations. Table 3 lists the dipoles for both the PIM and DFT calculations. The agreement in both geometries is excellent. For the isolated dimer ther percentage differences are 4% and 1% for the terminal and bridging anions, respectively, with the trend in the magnitude of each moment in the two sites successfully reproduced.The differences are slightly larger for the crystal moments, although the trends in the two sites are again reproduced. To highlight the effect of the short-range damping in the polarization model, Table 3 also lists the pure Coulombically-induced moments (that is, the dipole moments in the absence of any short-range effects). The short-range damping acts to reduce the moments by between 23 and 33% in both the crystal and the dimer. Exp. HF 2.414 2.222 2.46 2.25 92.3 122.8 91.4 120.8 6 The AlBr3–Al2Br6 system Crystalline AlBr3 can be viewed as a close packed anion lattice in which the cations occupy tetrahedral holes.The similarity between the positions of the Raman bands in the crystal, liquid and solid phases11 suggests that this structure can alternatively be formulated as an assembly of molecular Al2Br6 units. Unlike AlCl3, the melting of this system is not associated with an ionic–molecular transition; for example, the Raman bands of liquid and solid agree well.6 The crystal belongs to the space group P21/c (C52h) with a tetramolecular unit cell of size a0 ~ 10.20 A°, b0 ~ 7.09 A°, c0 ~ 7.48 A°, b ~ 96u.5 Calculations were performed on AlBr3 and Al2Br6 using nonconserving pseudopotentials and a plane wave cutoff energy of 175 eV. The unit cell for the crystal was optimized starting from the Wyckoff structure for AlBr3 (with negligible changes in cell lengths and angles) and from the YCl3 structure.Cell parameters obtained in the latter case were a0 ~ 6.23 A°, b0~11.39 A°, c0~6.39 A°, b~103u. This structure turned out to be y0.40 eV less stable than the former, confirming the AlBr3 crystal adopts preferentially the molecular Al2Br6 structure. For this reason, the Wyckoff structure was used in the electronic structure calculations, while the optimized structure was adopted for the gas phase dimer (Table 4). The distribution of the WFC’s in the dimer is closely related to the Al2Cl6 molecule, though distances of the WFC’s from Br are larger: 0.69 and 0.46 A°for BrB (b and nb), and 0.76 and 0.47 A°for BrT. The ionic dipole moments (Table 5) are also larger with respect to Al2Cl6, consistent with the larger polarizability of the Br ion, though the splitting between B and T dipoles is only slightly smaller (0.03 au) than that observed in Al2Cl6. In the crystal, the WFC’s are tetrahedrally arranged around Br ions, though notable distortions from the ideal geometry are Table 4 Geometric parameters for crystal AlBr3 (Wyckoff structure5) and gas phase Al2Br6 Al2Br6 DFT AlBr3 Al–Al Al–BrB Al–BrT BrB–BrB BrT–BrT BrB–Al-BrB 3.31 2.41 2.22 3.50 3.90 93.0 122.2 3.29 2.38, 2.41 2.22 3.49 3.85 93.4 120.1 BrT–Al-BrT PhysChemComm, 2002, 5(1), 1–11 9Table 5 Ionic dipole moments in crystalline AlBr3 and gas phase Al2Br6 from DFT–MLWF and PIM.See text for details AlBr3 Al2Br6 ClT ClB Cl3 Cl2 Cl1 2.18 2.20 2.00 2.11 2.17 2.33 1.90 2.01 2.30 2.63 DFT PIM very frequent (particularly on BrT ions), which tend to flatten their distribution, similar to Cl1 in AlCl3. The occurrence of numerous slightly differing configurations of the WFC’s results in the presence of several values of Br dipole moments. These can be conveniently grouped in three classes, as is shown in Table 5. It is however interesting to notice that the average value of the dipole on BrB (2.18 au) increases and that of BrT (2.08) decreases in passing from the crystal to the gas phase dimer. The larger dipole on BrB is easily interpreted in the Madden Wilson model in terms of the larger Al–BrB-Al angle and shorter BrB–BrB distance (induced by stronger dipole–dipole repulsion).The smaller value of the BrT dipole is probably explained by the removal of dimer–dimer interactions in the gas phase. 6.1 Comparison of ab initio and PIM dipoles for AlBr3 In order to construct a polarization model for AlBr3, scaling arguments are used (eqn. (13)) to generate a damping parameter from the AlCl3 value. The ion radii used are 0.50 A°, 1.81 A°and 1.95 A°for Al3z, Cl2 and Br2 respectively, to give a damping parameter of 1.49 au for the Al–Br ion pair. The dipole polarizability is taken as 30.0 au, again consistent with values from ab initio calculations.57 Table 5 lists the values of the induced dipole moments compared with those from the DFT calculations.The agreement between the two is again good. For the isolated dimer the values agree closely with the correct trend between the bridging and terminal anion sites reproduced. The agreement in the crystal is somewhat less impressive with the moments generated from the PIM all slightly higher than those from the DFT calculations. Again, however, the trend in the site moments is reproduced. 7 Summary and conclusion We have described results concerning the structural and electronic properties of crystalline AlCl3 and gas phase Al2Cl6, based on generalized gradient corrected Kohn–Sham DFT calculations and a Wannier orbital representation of the ground state charge density distribution. Consistent with the recent successful attempts at modelling these systems by means of polarizable ionic simulation models, we found that the fundamentally ionic nature of the Al–Cl interaction in the solid is retained in the gas phase dimer, though changes in the orbital hybridization can be brought about by the structural transition which is known to take place on melting.The WFC’s are always located in the vicinity of Cl atoms, and the localised orbitals do not have nodes at the cation positions, which is indicative of the absence of off-site hybridization between Cl orbitals and empty states of Al3z. Dipole moments of the Cl ions computed from the WFC’s have been used as a quantitative estimate of the anion polarization induced by the presence of the nearby Al3z ions, and they have been found in excellent agreement with the predictions of the PIM.The octahedral–tetrahedral transition in the Al3z coordination has been induced on the crystal by expanding the unit cell and allowing full ionic relaxation. Despite the simplicity of this approach, information on the ionic distortions could be obtained which supports the cooperative transition model 10 PhysChemComm, 2002, 5(1), 1–11 proposed by March and Tosi for the change of coordination. The variation in the HOMO–LUMO energy gap, total energy and HOMO orbital energy during the expansion suggest that the electronic properties of the expanded tetrahedral phase (and ultimately of the free dimer) are closer to those of the crystal in its equilibrium configuration than to the expanded– distorted lattice in the vicinity of the transition, thus supporting the idea of a close analogy between the Al–Cl interactions in the crystal and in the dimer in liquid or gaseous phase.Finally, an analysis of the structural and electronic properties of the AlBr3–Al2Br6 system has been carried out. Despite the approximations implicit in this case in the ionic interaction model, comparison with the ab initio dipole moments is reasonably good, and, in particular, the trends which would be expected in passing from the condensed phase coordination environment to the free dimer on the basis of a polarizable ionic prescription are fully reproduced by the ab initio calculations. 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