’Covalent’ Effects in ’Ionic‘ Systems Paul A. Madden and Mark Wilson Physical and Theoretical Chemistry Laboratory Oxford University South Parks Road Oxford UK 0x1 3QZ 1 Introduction How wide is the range of applicability of an ionic model of con- densed phase structure energetics and dynamics? The question is posed for practical reasons as well as for its intrinsic interest. For our purposes a system which is ‘ionic’ is one whose properties are reproduced by an interaction model based upon discrete closed- shell ions with integer charges. These ions are not simply charged hard-spheres; they may undergo polarization (induction) and dis- persion interactions and may even undergo changes of size and shape (‘compression’ and ‘deformation’) due to interactions with their neighbours. What is excluded is charge transfer and chemical bond formation involving the sharing of pairs of electrons between atoms. It is well-known that the domain of applicability of the sim- plest ionic model as embodied in Born-Mayer type pair potentials -essentially a model of charged hard-spheres -is severely restricted even the structure of a material like MgCl is not explained despite the large electronegativity difference between the elements involved. Our purpose here is to explore the properties of an extended ionic model which allows for the changes in an ion’s properties which are caused by changes in its environment and hence incorporate a many body character in the interactions. Among such effects are polarization compression and deformation. We will show how they may account for many departures from the predictions of the simple ionic model which are conventionally attributed to ‘covalency’. The reason that the question is a practical one is that such an ionic interaction model has two important characteristics. Firstly it may be used as the basis of tractable computer simulation methods which permit the study of large systems for long times. Such studies are often necessary for an understanding of material properties and many materials of technical interest lie in the domain where ‘cova- lent’ effects are prevalent. The MgCI example is the support for the Ziegler-Natta catalyst used in polypropylene synthesis and its crystal surface properties are therefore of interest. More fundamen- tally because it is based upon the properties of individual ions the ionic model is (or should be) transferable -it may be used on dif- ferent phases of the same material on mixtures and furthermore the interaction model for one material should be recognisably related to that of a chemically similar one by a change in ion size or similar property. A transferable model may be tested on one phase and used on another tested in the bulk and used for a surface etc. Because of the relationship between materials the origin of Paul Madden is a Professor of Chemistry at Oxford University and a Fellow of Queen’s College. He was born in Bradford and obtained his BSc and DPhil at the University of Sussex the latter supervised by Prof. J. N. Murrell. After a period of eight years at Cambridge. and two at the Royal Signals and Radar Establishment (Malvern) he took up his present position in 1984. He was the recipient of the Tilden Medal of the Royal Society of Chemistry in 1993. structural trends may be understood and a first guess at an interac- tion model for one material may be constructed from an established model for another. Born-Haber cycles may be constructed to analyse energetics. The long-standing difficulty with examining the applicability of the ionic model (including the many-body effects) is that the indi-vidual ion properties which determine the interionic interactions in condensed phases cannot be determined from experimental data at least not without further assumptions. However it is now possible to perform electronic structure calculations to determine the prop- erties of single ions within their condensed phase environment. This breaks the impasse and allows an ionic model to be parameterised unequivocally. Recently it has been shown how interaction models which allow for an accurate representation of the many-body effects uncovered by such calculations may be constructed and used in tractable computer simulation schemes.I The basic physics included in these models is the same as in a (breathing) shell model the difference being that the shell model3 makes use of a particular mechanical representation of the many-body effects in order to allow the model’s parameters to be determined empirically (effectively an alternative way of breaking the impasse). However in so doing it enormously reduces the flexibility available to accu- rately reproduce the many-body effects a limitation which becomes clear when comparing with the results of electronic structure calcu- lations of ionic properties (such as induced dipoles).’ We begin with an account of the origin of the many-body effects of the way in which they may be characterised in electronic struc- ture calculations and of their representation in the simulation context. We will then illustrate their role in accounting for various phenomena attributed to ‘covalency’. In this we will focus on crystal structure and energetics since this application provides the most readily appreciated sense of progress but note that the simu- lation methods are also applicable to melts and dynamics. 2 Environmental effects and the resulting many-body potentials To understand the interactions it is not sufficient to think of the con- densed phase as a collection of well-defined gas-phase species. A van der Waals material such as a rare gas or molecular solid can be quite accurately modelled this way but in ionic materials the ions themselves are profoundly influenced by their enviroment . The Mark Wilson was born in Derby. He obtained his BA from Keble College Oxford and his DPhil (supervised by Paul Madden) in 1994. He was then awarded an Alexander von Humboldt Fellowship which he took up at the Max-Planck Institute in Stuttgart. At the time of publira-tion of this article he becomes a Royal Society Research Fellow in the Physical and Theoretical Chemistry Laboratory at Oxford. 339 oxide ion for example is unstable in the gas-phase but commonly occurs in condensed matter Because of this the interaction of one ion with another cannot be expressed without reference to the envi- ronment in which each ion is found Consequently interaction potentials must be expected to have an explicit many-body charac- ter -unlike the van der Waals case where pair potentials (which may implicitly include the average effects of many-body interac tions) are the norm To gain more insight into the environmental effects we consider the potential V(r),experienced by an electron in an i0n4-6 due to both electrostatic interactions with the charges of the other ions in the crystal and to its exclusion from the region occupied by the elec- tron density of neighbouring ions As illustrated in Figure 1 in a perfect cubic crystal this environmental potential contains a spher- ical part Vo(r),plus an angularly dependent potential which varies rapidly with the orientation of the electronic position r When expressed as a spherical harmonic expansion this angular part involves terms of angular momentum 1 = 4 or higher see eqn (2 1) V(r) = V&) V/(d y/ (2 1)+/Z4; Since the ground electronic state of a closed shell ion is an S state the angular potential will only be significant to the extent that it can mix in excited states of G symmetry For many ions (especially s-p valence ions) such states are likely to be very far in energy from the ground state so that only the spherical potential plays any role -the electron density of such ions is unable to adjust to the angularly dependent part of the potential and the electron density of the ion remains spherical The spherical potential V tends to compress their electron density relative to that of the free ion as illustrated in Figure 1 It leads to a marked reduction of the polarizability inter and is responsible for the stability of the oxide ion in con-densed matter for cations. the effect is much smaller 2.1 Ion compression in perfect crystals As the crystal is compressed there will be an increase in the overlap between the charge densities of nearest neighbour anions and cations as illustrated in Figure 2 This will lead to an increase in the energy of the system which we will call U0,,because closed shells c---.)R Figure 1 Origin of the spherical confining potential V,,,which acts on the electrons around an anion in a cubic crystal the electrons in sp valent ions are unable to respond to the rapid angular variation in the potential illus trated by the dashed contour in the upper panel In the lower panel a cross section through the spherical potential V, is shown the dashed line shows the coulombic (Madelung) contribution associated with the point ionic charges This is enhanced by the exclusion from the region occupied by the electron density of the other ions V compresses the free anion charge density (lightly shaded) to the in crystal charge density (heavily shaded) CHEMICAL SOCIETY REVIEWS 1996 may not overlap due to the Pauli exclusion principle If we take electron densities to decay exponentially and the total overlap energy to be the sum of the overlap energies associated with each cation-anion pair then referring to Figure 2 the energy would depend on the total hatched areas associated with the overlapping charge clouds We would therefore have something like eqn (2 2) u<,,= 11 A expl (TIJ q q4p + PJ)l (2 2) 1 /(-I) z e a pair potential of Born-Mayer form to describe the interionic repulsion Here rrJis the separation between i andj u,is a charac- teristic radius for the charge density of ion I and p describes how it falls off with increasing distance from the nucleus To justify this pair potential we have regarded the ions as fixed entities as the crystal is compressed so that the extent of overlap increases as in Figure 2(6) In general what should be anticipated is that as the crystal is compressed the walls of the spherical confining potential will move in and each ion will adjust by shrinking its charge density as illuctrated in Figure 2(c) where the area of overlap is reduced below what would be found with fixed charge densities Hence parameters like p and u,,which reflect the size and shape of the charge density should themselves be regarded as depending on the separation between the ions a,-a,([rNj),etc (where ([rNl)is R' Figure 2 Illustration of the ion compression effect Panel (a)shows the anion charge density (dense shading) in the crystal at lattice spacing R. as confined by V The electron densities of the first cation shell are shown lightly shaded and the region of cation-anion overlap is hatched After shrinking the crystal (to lattice spacing R') the hatched area increases (b) (corresponding to increased repulsion) but the amount of overlap is much reduced las in (c)l if the compression of the anion charge density by the modified confining potential is accounted for The change in shape of the anion charge density gives rise to the self energy ‘COVALENT’ EFFECTS IN ‘IONIC’ SYSTEMS-P A MADDEN AND M WILSON 34 1 intended to imply a dependence on the positions of N other ions in the sample) and the overlap potential acquires a many-body char- acter eqn (2 3) u v = A expI (“ u,IrNl fl,[rNl)h?[rNl+ p,IrNI)I (2 3) This shrinkage of the ions under the influence of the confining potential itself costs energy which might be called a ‘self energy’ or ‘rearrangement energy’ -the energy required to place each ion in the confining potential associated with a particular lattice para meter -and the total energy associated with compressing the crystal written as eqn (2 4) This observation provides a well-defined and practical route for examining the interactions appropriate to ions in the cubic crystal The electronic wavefunctions of an anion and cation confined by the spherical potentials V appropriate to the crystal of interest at a particular lattice constant R is calculated The difference between the energy of this confined ion and the free one gives the self-energy appropriate to lattice constant R The interaction between pairs of these entities is then calculated without further wavefunction relaxation In this way a pair overlap induced-repul- sion potential appropriate to R is obtained Such calculations have been carried out by Pyper5 and his results for Uovand Uselffor an oxide ion in MgO are shown in Figure 3 Results for the MgO in the six-coordinate rocksalt (B l) the eight-coordinate caesium chlo ride (B2) and four-coordinate blende (B3) structures as a function of the lattice parameter R are shown For the oxide ion the self- energy goes to a plateau at large R this is because the free oxide ion is unstable with respect to 0-plus a free electron so that for the oxide ion we must consider the process of placing an 0 ion and an electron in the confining potential rather than simply a free ion The plateau therefore reflects the electron affinity of 0 Note that the self energy is a large component of the total repulsive potential which resists the compression of the crystal Why should we worry about separating the total repulsive energy in this way’ After all the total repulsive energy for say the B1 phase could still be fitted to a sum of pair potentials by dividing the total repulsive energy involving cations and anions U, + Uself amongst the six nearest-neighbour pairs The problem is that this representation would not transfer to ionic environments other than the perfect rocksalt structure Figure 3 shows that U and Usel+.have difSerent dependencies on conrdinatioiz number so that a pair poten- Rlau 101 30 I . 40 . . 50 * . 60 I020 h 1015 08 C 010; (5 06 005 3 ZI 3 000 04 02 00 30 40 50 60 Rlau Figure 3 Ah initlo data for the self energy of an oxide ion (upper family of curves) and for the overlap energy (lower) as a function of the lattice parameter R In four (B3 crosses) six (BI plusses) and eight coordinate (B2,circles) structures for MgO as calculated by Pyper The inset shows the pair potentials calculated from this data for each structure note that these exhibit a coordination number dependence The solid lines in the main figure show the values calculated from the CIM potential which was fit to the B1 data tial calculated with the BI data would be different from that required to fit the B2 or B3 results Hence this pair potential would not be transferable between the different phases of MgO Pair potentials appropriate to each phase obtained from the correspond- ing values of U + Uself,are shown in the inset to Figure 3 and clearly differ However a compressible ion model (CIM) which allows for the dependence of the u values on the coordinates of other ions can be fitted to the data for the rocksalt structure It accu rately predicts the ab initio data for U and Uselfin the other struc- tures? as shown by the solid lines in Figure 3 This model can be used in tractable simulations in place of the pair potential For details the reader should refer to ref 7 The same physical effect of ion compression has traditionally been accounted for in breathing shell models ,2 but these are normally parameterised empirically For halide ions Uselfis found to make a smaller contribution to the total Urepthan is the case for oxidess (reflecting the greater sen sitivity of the oxide ion to environmental effects) Hence for halides we can anticipate that pair potentials will have a wider domain of applicability than is the case for oxides which is consistent with the finding that crystal-parameterised Born-Mayer pair potentials were found to give a good representation of the interactions in alkali halide melts inter alia 2.2 Less symmetrical environments The perfect crystal provides a good reference point for the discus- sion of environmental effects -an ion in a melt is much more like the in-crystal ion than a gas phase one Nevertheless to adequately model melts and surfaces a consideration of environmental effects in much less symmetrical environments than the crystal is required A convenient starting point is to consider how the perfect crystal picture illustrated in Figure 1 is modified when some of the neigh bouring ions are shifted off their lattice sites The spherical har monic expansion of the environmental potential will now in general contain angular momentum 1 = 1,2 components as well as having a modified spherical term As long as the argument -that high angular momentum electronic states are very remote and hence unimportant -remains valid it will be sufficient to consider the effect of the altered l = 0 1 and 2 potentials The l = 0 amounts to a change in the spherical confining potential we discussed above and may result in a change in ion size The 1 = 1 and 2 terms cause deformations of the ionic electron density of dipolar and quadru polar symmetry respectively These may have two consequences Firstly the central ion will acquire a non-zero electric dipole and quadrupole moment which will alter its energy through coulombic interactionswith the charges and multipoles of other ions this is the polarization energy lo Secondly the ion may become non spherical (‘deformed’) as perceived through the short range overlap interac tions with its near neighbours 2 2 I Polarization efsects Polarization effects can be characterized by examining directly using electronic structure methods the induced multipoles on ions in distorted crystals There are some technical problems associated with assigning the displaced charge to a particular ion but these may be overcome II If an ion in the crystal at a relatively large distance (say greater than next-nearest neighbour separation) from the central ion is dis placed off its lattice site its effect on the potential felt by the elec trons in the central ion is simply that of the electric field (1 = 1 ) and field-gradient(1 = 2) at that site There will be induced dipoles and quadrupoles given by the usual multipole expansion,I0 eqn (2 5) where the superscript ‘as’ means that these moments are appropri ate when the sources of the fields are asymptotically far away from ion i Here (Y and C are the dipole and quadrupole polarizabilities and B is the dipoledipole-quadrupole hyperpolarizability the components of a,C and B are specified by a single number for a spherical ion lo These polarizabilities are those appropriate to the ion in its crystalline environment and as already remarked may be much smaller than the free-ion values6 due to the confinement effects Ea and Em are components of the field and field gradient respecti vel y If a near-neighbouring ion is displaced there is an additional effect Figure 4 shows what happens to the confining potential around an anion familiar from Figure I when one of the first shell of cations is displaced outwards Besides the field and field gradi- ent (related to the gradient and curvature of the potential at the origin) a dent appears in the confining potential Whilst the field and field gradient tend to push the electrons in one direction (away from the displaced cation) this 'dent-in-the-wall'll allows them more freedom to move into the space vacated by the cation Hence there is a short-range contribution to the induced dipole pf" which opposes the 'asymptotic' dipole caused by the electric fields This has been studied in electronic structure calculations The effect is substantial The dipole induced by displacing first neighbour cations is reduced below that expected from the asymptotic term alone by ca SO% For induced quadrupoles the effect is even larger the limited evidence available suggests that the short-range term is as large as the asymptotic one so that the net quadrupole on the anion can be very small or even opposite in sign to the asymptotic quadrupole In principle the experimental manifestation of these two effects could be studied in the far-infrared spectra of disordered ionic systems So far as we know this has not been done quantitatively However analogous effects contribute to the polarizability fluctua- tions responsible for light scattering and good agreement has been demonstrated between calculatedI2 and experimental spectra I Normally we think of the anion as the polarizable entity in ionic systems however in some cases cation polarization can also become important For cation polarization the relative sign of the short-range and asymptotic moments is the same -the short-range effect therefore enhances the dipoles and quadrupoles above the values which would be expected from the coulombically induced moments If we consider Figure 4 but reverse the signs of the charges on the ions so as to make the central ion a cation we can see why As a neighbouring anion is Figure 4 Origin of the 'asymptotic' and 'short range' contributions to the dipole induced in an anion in a crystal which has been distorted by an outward displacement of one of the first shell cations A cross section of the confining potential is shown for the undistorted crystal (dashed) and after the distortion where the 'floor' of the potential well has acquired a gradient (the electnc field) and the wall of the confining potential has been pushed outwards These have opposite effects on the electron density Note the arrows represent the direction of the electron displacement and therefore strictly are antiparallel to the associated dipoles CHEMICAL SOCIETY REVIEWS 1996 displaced outwards the electric field generated will tend to dis- place the cation electrons towards the displaced anion which is also the direction favoured by the displacement of the cation charge cloud Again this consideration is true of the higher order multipoles It makes the role of cation polarization much more substantial than would be suggested by a simple consideration of the relative size of the cation and anion polarizabilities 2 2 2 Deformation of ionic shape The development of the multipoles in the less symmetrical struc- tures is a signal that the ion's charge cloud has been distorted from a spherical shape The polarization energy results from the classical coulombic interaction of the mu1 tipoles with the charges and multi- poles of other ions The deformation of the spherical charge density will also have another manifestation since it will affect its overlap with the charge densities of the neighbouring ions and hence it will charge the short-range repulsive interaction with them This effect is contained within the shell model of the interionic interactions The shell is the centre of repulsive interactions and it may be displaced from the ion's centre of mass (and the site of its formal charge) by repulsive interaction -hence the ion may become anisotropic A potential difficulty with this approach is that this same displacement is closely tied in the shell model to the short-range effect on the induced electric dipole which was discussed above A1though these phenomena are clearly linked we believe on the basis of a limited set of ab inirio results that the connection imposed by the shell model is overly restrictive and that to accurately represent them the short-range induced dipole and the consequence of the non-spherical deformation on the short-range repulsion should be treated as separate phenom- ena This may be done by generalising the treatment of compressible ions indicated in section 2 1 Compression was treated by allowing the ionic radius to depend on the relative position of a number of other ions To allow for a deformation of dipole symmetry a vector property of each ion &[rN]is introduced which again depends upon the positions of the other ions in the sample as illustrated in Figure 5 The overlap energy is now given by eqn (2 6)14 -A expI (rVaflrNlglpl S1lrNl)l(p,lrNl(JI,v =c\' + p,l~l>I (2 6) I J('1) The overlap energy between a particular pair of ions now depends notjust on the distance between them but also on the angle between the inter-ion vector and the internal vector &ofeach of the ions and hence on the configuration of the other ions around I If an ion] is positioned such that rJis parallel to & i e corresponding to the dis- placed cation in Figure 5 its repulsive interaction with i is calcu- lated as if the latter's charge density has expanded (enhanced a') &[rN]is also associated with a charge in the self-energy of an ion Figure 5 Representation of the ion 'deformation effect' in the distorted crystal considered in Figure 4,the electron density ceases to be symmet rically disposed about the anion centre -as indicated by the displaced contours in the figure This affects the repulsive interaction between the anion and its first neighbour cations The vector 6 indicates the direction of the deformation and appears in the modified expression for the repul sive interactions eqn (2 6) ‘COVALENT’ EFFECTS IN ‘IONIC’ SYSTEMS-P A MADDEN AND M WILSON A further refinement would be to include a quadrupolar set of inter-the separation between the more highly charged cations These nal degrees of freedom to allow for the flattening or squashing of crystal structures are illustrated in Figure 6 an ion This deformable ion effect may be studied in the same ab initio calculations on distorted crystals used to study the induced multi- poles Whilst the induced multipoles are obtained directly from the charge distribution of the distorted crystals the ion deformation effect is studied via the energies of these distortions -less the energy which is accounted for by polarization effects 23 Representation of the many-body effects The effects described above give a many-body character to the int-erionic interactions This arises because the expressions for the interaction energy of a particular pair of ions now contain variables elrN],pllrN],&[rN]etc ,which themselves depend on the coordi- nates of other particles If we were to express the interaction energy solely in terms of the ionic positions we would find that it involved very complicated expressions containing the coordinates of differ- ent ions simultaneously The key to representing the many-body effects in computer simulations is to treat the extra variables,@[rN] pi[F],&IrNl ,as dynamical coordinates of the system wholly analogous to the positions of the ions and update them along with the ionic coordinates as the ions move In terms of the particle posi- tions and the additional degrees of freedom the interaction energies become a relatively simple function involving the positions and additional degrees of freedom of only pairs of ions For details of how this is done in practice we refer to refs 1 and 7 As described above we can use electronic structure calculations on compressed and distorted crystals to examine how the ion radius etc vary with the ionic environment We can use these calculations to determine suitable functions which allow these properties to be evaluated for an arbitrary ionic configuration These functions then become the input to the simulation procedure The hope is that these functions determined from the crystal calculations will be suffi- ciently robust as to allow the properties to be calculated in the more general environments which will be encountered in the melt at a crystalline defect etc 3 Manifestations of the many-body effects Many-body effects such as those introduced above may affect all aspects of the observable behaviour of ionic systems and influence the properties of melts as well as crystals In order to keep things finite we will focus primarily on the role of these effects in stabi-lizing particular crystal structures As we have stressed the input for the potential models is derived from calculations done on crys- talline environments so that the first step of validating a potential must be to demonstrate that it reproduces and explains observed crystalline behaviour Surveys of the structures of binary materials show that in almost all cases the local structures of the crystal and melt are closely related I5 I6 The backcloth to the rationalization of crystal structures is pro- vided by the simple ionic model effectively a model of charged hard-spheres as embodied in pair potentials of Born-Mayer form For such a model the most stable crystal structure can be under- stood by considering how spheres of appropriate charge and radius may pack together to maximize unlike ion coulombic interactions and minimize like interactions l7 These considerations lead to the prediction of a number of typical ‘ionic’ crystal structures In systems of stoichiometry MX these are the eight-coordinate caesium chloride (B2) six-coordinate rocksalt (B1) and four-coor- dinate blende (B3) or wurtzite (B4)structures which are formed in systems of successively lower cation/anion radius ratio The rock- salt (B1) structure for example may be viewed as a close-packed cubic lattice of one species with the other occupying the octahedral holes this arrangement equalises the nearest-neighbour cation-cation and anion-anion separations r++ = r -and hence minimizes the charge-charge interaction For MX systems the cor- responding sequence is fluorite (eight-coordinate cations) rutile (six-) and ideal crystabolite (four-) Again these crystal structures involve ions symmetrically disposed in such a way as to maximise Crystobalite Rutile Fluorite Layered Figure 6 Illustrations of crystal structures frequently adopted by MX systems the small dark spheres are the cations and the larger pale spheres the anions The fluorite rutile and ideal p crystabolite structures are ‘ionic’ whereas the Cdl IS typical of the layered crystal structures which involve next neighbour anions and a short cation-cation separation CHEMICAL SOCIETY REVIEWS 1996 OBSERVED Figure 7 Structure maps as calculated for alkaline earth halides on the basis of a simple Born-Mayer pair potential (upper) and as observed. Note the agreement in the bottom left corner (large cations/small anions) and the prevalence of layered crystals in the small cation-large anion limit. Departures from the simple ionic model may be recognised qual- itatively in the adoption of crystal structures which do not fit into this pattern. In MX systems SnO has a layered structure with nearest neighbour ions of like charge along certain directions. In ZnO the ions are tetrahedrally coordinated whereas the radius ratio would suggest a rocksalt structure. Such departures are much more common in MX or MX systems. This is related to the excess of both octahedral and tetrahedral holes in the close-packed anion structures over the number of cations available to fill them in this stoichiometry -the cations have a much wider range of choice as to how they will organise themselves in the anion lattice than in MX. ‘Non-ionic’ structures where the cation occupancy does not min- imise the cation-cation interactions are prevalent. For example in MX stoichiometry many systems with small cations crystallize in layered CdI or CdCI structures17 which contain such non-ionic features as nearest neighbour anions see Figure 6. In the following sections we will show how by allowing for envi- ronmental effects on the ionic properties these (and other) anom- alies may be rationalized. 3.1 The role of polarization effects 3.I .I Lqered crystals in MX systems Figure 7 shows that the crystal structures of the alkaline earth halides depart quite markedly from the simple ionic model despite the large electronegativity differences between the elements involved. Figure 7 contrasts the structure map of stable crystal structures predicted by the simple ionic model which depends pri- marily on the radius ratio with those actually observed for these systems. It shows that whilst the structures adopted by the large cation -small anion systems are as predicted there is a large portion of the structure map where ‘non-ionic’ layered crystals are formed whilst the simple ionic model predicts the crystabolite structure. For reference the various structures are illustrated in Figure 6. Note that the ‘ionic’ structures maximize the distance between the highly charged cations and interpose an anion in between a pair of cations whereas the layered CdI structure contains nearest-neighbour ions of like charge short cation-cation separations (i.e. r_-= Y++ despite the higher cation charge). For the halide ion electronic structure calculations reveal that the -2700.0’ 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0 /A Figure 8 The three lower lines show energies calculated for the various MClzcrystal structures from a simple pair potential supplemented with an account of anion polarization showing agreement with the experimental structural trend 7 (a -fluorite; b -rutile; c -layered). In the absence of the polarization term the fluorite and rutile energies are barely shifted whilst the energy of the layered structure is significantly increased (d). ion compressibility effects are quite small (compared to oxides for example); i.e. that the total repulsive energy is dominated by the overlap term. The alkaline earth cations are not very polarizable (compared to the chloride ion) and very similar in the gas and con- densed phase. Consequently to a first approximation we might restrict the consideration of environmental effects to the polariza- tion of the halide ion and use a simple pair potential for the remain- ing interactions. If we consider a given column of the structure map i.e.a series of systems with the same anion and impose a very trans- ferable ionic picture on the nature of the interactions then we would anticipate that the only potential parameter that should vary from one system to another is the cation radius.’* Figure 8 shows the energy minima cal~ulated’~ for the different chloride crystal struc- tures with a family of interaction models consisting of Born-Mayer pair potentials with formal ionic charges supplemented with a description of the halide dipole polarization which uses an ab initio calculated value for the chloride ion polarizability in the condensed phase and a description of the short-range dipole also derived from ab initio calculations. In this family the only parameter which varies from one system to another is the cation radius; the anion parameters are conserved from one substance to another and the parameters describing the cations are linked in a well defined and chemically meaningful way. Further calculations show (Figure 8) that the polarization effects have lowered the energy of the layered crystal structures relative to what would be expected from the pair potential alone by a huge amount -ca. 500 kJ mol-I for MgCI -whereas they cause little change in the energies of the ‘ionic’ fluorite rutile and crystabolite structures. Figure 6 shows why this has occurred. In the layered crystals the planes of highly polarizable anions are unsymmetrically sandwiched between a plane of cations and anions and are thus strongly polarized in such a way that the negative ends of their induced dipoles are dragged down into the layer of cations. This negative charge ‘screens’ the repulsion between the positive cations and therefore reduces the unfavourable coulombic energy which results from placing them in such close proximity in the layered crystal. In the ideal ionic structures by contrast the anions sit in sites of such high symmetry that no dipoles are introduced and the energies calculated with the polarizable model are very close to those obtained with the simple pair potential. Molecular dynamics simulations with the polarizable model confirm that the CdI struc- ture is indeed the globally stable structure for MgCI,.” We therefore see a general pattern emerging:- the simple ionic model favours highly symmetric structures whereas polarization favours pushing highly polarizable ions into unsymmetrical sites. Another manifestation of this occurs in the structures of MX melts. A snapshot of the structure in a ZnCI melt,2O simulated with a potential model of the same type as discussed above is shown in ‘COVALENT’ EFFECTS IN ‘IONIC’ SYSTEMS-P A MADDEN AND M WILSON A Figure 9 Snapshot of the ionic positions in a simulation of a ZnC1 melt with a polarizable ion model Cations are small and dark and the anions are larger and more lightly shaded Note the predominant tetrahedral coordination and the corner sharing of tetrahedra involving a bent Zn-C1-Zn bridge The extent of the network is limited by the fact that ‘bands’ to C1 ions not inside the region visualised are omitted Figure 9 It can be seen that the local structure consists of a tetrahe- dral arrangement of C1- ions around each Zn2+ this is an ion size effect contained within the pair potential and is a property of the structures obtained when the polarization effects are included or omitted The polarization effects influence the way these tetrahedral units are linked Whereas the pair potential predicts linear Zn-CI-Zn bridges on average the polarizable model gives a bent ‘bond’ with a bond angle of ca 110” Figure 10 shows that polar- ization is the driving force for the occurrence of this non-trivial bond angle a phenomenon often attributed to ‘covalency’ If the bond were linear the CI would be symmetrically located between two cations and no induced dipole could result If on the other hand the C1 is displaced off the line of centres of the cations a dipole is induced which serves to screen the cation repulsion and lowers the total energy A more open structure containing voids results so that the price paid for the polarization energy is an increase in the ID Coulomb or Madelung energy Numerous observable effects in MX melts may be explained through this phenomenon in particu-lar the relationship between like and unlike radial distribution func- tions,2O and the ‘pre-peak’ which is a characteristic feature of network-forming liquids The specific consequences of the polarization effects depend on a subtle interplay between them the straightforward coulombic interactions between the ionic charges and excluded volume effects as determined by the ion size ratio The polarization effects become more important for highly polarizable ions and when the cation radius is significantly smaller than the anionic one For large cations the simple ionic structures with an anion interposed between the cations emerge as most stable At intermediate size this bridge becomes bent as illustrated above resulting in corner linked polyhedra For very small cations and polarizable anions the bending is such as to give edge-linked polyhedra in which the induced dipoles on two anions screen the cation-cation repulsion as indicated in Figure 10 For example BeCl forms several crystal structures based on chains of edge-linked tetrahedra and edge- sharing octahedra and tetrahedra become the dominant local struc tures in MX melts Both of these phenomena are reproduced with potentials of the type described above 3 I 2 Polarization in silica isomorphs Amongst systems of stoichiometry‘ MX silica (SiO,) is known to exhibit a particularly exotic range of crystalline isomorphs l7 Whilst there are good reasons to expect that the simple model of a pair potential plus polarization will not suffice for oxides (see below) it is of interest to see how polarization phenomena con tribute in these structures by pushing ahead with the simple model described in the previous section At atmospheric pressure all silica polymorphs are based on corner-linked SiO tetrahedra With increasing pressure there is a transition to a six-coordinate cation rutile-like phase (stishovite) and a possible higher pressure transi tion to a fluorite-based or a-PbO structure which may have geo- physical significance We will discuss results obtained with a Born-Mayer potential with full formal charges as parameterised by Woodcock Angel1 and Cheeseman (WAC) and supplemented by an account of anion polarization using the oxide polarizability obtained from the experimental refractive index and the short-range dipole function used in MCI work but scaled by the ratio of sums of ionic radii The limitations of this model will become apparent when we consider the relationship between the energies of struc- tures with different coordination numbers Figure 11 shows the energyholume curves for the important polymorphs calculated using the WAC pair potential2’ (a)without and (b)with induced dipoles on the anions In the absence of polar- ization the lowest energy structures are the ideal-P-crystobalite and tridymite These follow near identical curves (only the ideal-P-crys- tobalite is shown) as they differ only in the packing of the oxide sub- lattice ideal-P-crystobalite as illustrated in Figures 6 and 12 has an fcc oxide lattice whereas the tridymite is hexagonally packed These structures have linear Si-0-Si triplets As discussed above in the absence of polarization effects the driving force for the linear Figure 10 Illustration of the induced dipole as a driving force for bond bending and how wlth decreasing cation size (increasing polarization effects) this eventually favours a polyhedral edge sharing motif 100 20.0 30.0 40.0 50.0 60.0 70.0 80.0 VIA^ -11300.0 -11500 0 -11700 0--z -119000 A25 -12100.0 -1 2300 0 -1 2500.0 -12700 0 10.0 200 300 40.0 500 60.0 700 800 VIA^ Figure 11 Energy vs volume curves for various isomorphs of SiO calcu lated (a)with the simple pair potentialz7 and (6)with the additional effect of anion polarization Note the stabilization of the experimentally observed P-crystabolite and quartz structures relative to ideal crystabo lite and stishovite which occurs because of polarization effects bond is the repulsive cation-cation coulombic interaction which is minimized by interposing an anion between adjacent cations The structures predicted by the pair potential do not agree with experi- ment ideal crystabolite and tridymite are less stable than the form of crystabolite which is illustrated in Figure 12 and less than the a and p forms of quartz In all these structures the Si-0-Si bond is bent The pair potential also overestimates the stability of the denser six-coordinate stishovite structure From Figure 12 it can be seen that stishovite is predicted to be stable with respect to both quartz and the experimentally observed form of crystabolite The inclusion of the polarization effects [Figure 1 l(b)J radically improves this picture Polarization stabilizes the bent Si-0-Si triplets found in the real-crystobalite structures in the same manner as found in the MX systems described above The high symmetries of the oxide sites in the ideal crystabolite tridymite and stishovite structures preclude such polarization effects The result of allowing for polarization is that the real-crystabolite energy minimum is con- siderably reduced and now lies significantly below that of the ideal structure and also that of stishovite The same induction stabilization is seen in the a-quartz structure resulting in the interesting pattern illustrated in Figure 12 Looking down on a plane of SiO tetrahedra as in Figure 12 (corresponding to the crystallographic ab plane) we see alternating up/down spirals of clockwise and anticlockwise induced dipoles which are linked to the piezoelectric properties of quartz The transition pressure from the a-quartz to stishovite rises to around 50 GPa when polarization effects are included This is a direct result of the greater polarization energy in the four-coordinate structures w'th respect to the 'Ix *lthough there Is Some uncer-tainty as to the actual Owing to the metastab''-ItY ofthe four-coordlnate Structure the quoted (5 GPa22)Is well below our calculated value though the quoted reduction In molar volume on going from quartz to stishovite of 38% is in good CHEMICAL SOCIETY REVIEWS 19% accord with the calculations Thus although the inclusion of polar- ization effects has qualitatively improved the description of the silica systems it is clear that more is required than adding polariza- tion effects to a pair potential in order to quantitatively account for the phase behaviour Latz and Gordon23 have argued that the effects of the different environments in these isomorphs on the actual size and shape of the oxide ion must be accounted for to recover the phase transitions accurately we will discuss such effects in the next section 3.2 The role of compressible-ion effects So far we have simply examined the consequences of adding polarization to a heuristic pair potential It is unclear whether such shortcomings as the poor quantitative description of the phase behaviour of silica are due to an inadequacy of the description of Ideal crystobalite p-Crystobalite J-a-Quartz Figure 12 Illustration of the polarization effects in the SiO isomorphs A projection of a single plane of SiO tetrahedra in each structure is shown the Si ions are small dark circles and the 0 ions are larger and grey The ideal crystabolite (shown in '3D' in Figure 6) is compared with P crys tabolite and a-quartz where the Si-0-Si bond is bent The negative ends of induced dipoles on the oxide ions are indicated by small open circles ‘COVALENT’ EFFECTS IN ‘IONIC’ SYSTEMS-P A MADDEN AND M WILSON the short-range interactions between the ions contained in the pair potential or due to a more general limitation of the underlying ionic model To proceed further it is clearly necessary to examine the short-range interactions at an ab initio level As discussed in section 2 1 such an examination in cubic crystals reveals the phe- nomenon of ion compression We therefore now consider the char- acteristic effects of generalising the description of short-range repulsion from that afforded by a pair potential to allow for the many-body effects which arise from the environmental effects on the ion size To demonstrate the role of spherical compression we expand on the MgO example discussed earlier Figure 13 shows the energy/volume curves for the B 1 B2 and B3 crystal structures of MgO calculated using (a)the compressible-ion model (CIM) dis- cussed above and (b)a pair potential fit to the same rocksalt (B 1) data used to parameterise the CIM potential The CIM has been shown to predict both the four- and eight-coordinate ab initio data also shown in Figure 3 wlthout further modtjcatton -it is transfer- able whereas the pair potential is not (inset to Figure 3) Two fea- tures are evident Firstly the value of the Bl -B2 pressure transition is greatly reduced in the CIM (the transition pressure between the two phases is obtained from the gradient of the line which is a common tangent to the corresponding energy vs volume curves) Secondly and perhaps more strikingly the predicted ground state for the pair potential is the four-coordinate B3 struc-ture rather than the experimentally observed Bl (rocksalt) The CIM predicts the correct ground state and gives excellent values for its lattice energy lattice constant and bulk modulus A similar story holds for CaO for which the B 1 -B2 transition pressure is known experimentally and found to be predicted accurately by the CIM potential Thus allowing for the coordination number dependence of the ion compression effects leads to a transferable potential which appears to favour high coordination structures relative to the pre- dictions of a pair potential This appears to be a general observation -2600 0I 1 -2700.0k r ‘5 -2800.0 E 5 -2900.0 -3000.0 -3100 O1 I 10 0 20.0 300 40 0 -3100 0’ I 10.0 20.0 300 40 0 VIP Figure 13 Energy vs volume curves for MgO calculated (a)with the rock salt pair potential (from the inset to Figure 3) and (6)from the CIM poten tial which represents Uouand Use,,separately Key plusses -B2 crosses -B1 circles B3 Note that in (a)the B3 structure (erroneously) emerges as of lowest energy whilst the CIM stabilizes the higher coordination structures for example a single pair potential with sensible ionic radii cannot reproduce the seven-coordinate ground state of ZrO instead pre- ferring a lower six-coordinate option The use of a CIM corrects this by stabilizing the higher coordinate structure 24 This effect could also help to stabilize the stishovite structure relative to quartz in SiO and thus improve the predicted transition pressure as dis- cussed in the last section How important are the compressibility effects for other anions? The oxide ion being unstable in the gas-phase is particularly sus- ceptible to environmental effects Ab initio data are available for CsCl and for CaF28 and indicate that Uselfis a smaller component of the total repulsive energy than in the oxides Nevertheless Pyper8 has shown that a full account of the coordination number depen dence of Uovand Uselfis necessary to obtain the correct ground-state (B2) structure for CsCl When the same ab initio data are used to fit a pair potential this potential overestimates the stability of the lower coordination number BI structure For the fluoride ion in CaF it does seem that a pair potential representation of the repul- sive interactions is sufficient 2s The transition energies from the ground-state fluorite structure to a denser a-PbC1 structure pre- dicted by CIM and pair potential fit to the same ab initio data differ by only 8 kJ mol I in a total lattice energy of cu 2500 kJ mol I and both are in good agreement with experiment The ab initio pair potential (plus polarization) gives a good account of numerous other properties of CaF including its superionicity In the Introduction we stressed that one of the touchstones for assessing the validity of an ionic model should be the transferabil- ity of the potential Not only should the potential work in different phases of the same material but similar potentials in which the parameters change in a chemically reasonable way should describe other chemically related materials The CIM potentials for MgO and CaO are evaluated from specialised electronic struc- ture calculations which focus on the properties of single ions within an idealised representation of the crystalline environment Nevertheless the ab initio potentials which describe the short- range repulsion between an Mg2+ and O2 ion look very similar to those which describe the Ca2+ -02 interaction if they are scaled to allow for the different radii of the Mg2+ and Ca2’ ions Furthermore these potentials may be ‘transmuted’26 into poten- tials for other oxides of stoichiometry MO simply by replacing the values of well-defined ionic radii in the expression for U The self-energy Uselfis a property of the oxide ion and should not be changed from one oxide to another It has been shown that these potentials successfully account for the lowest energy crystal struc- tures energies and lattice parameters of the other alkaline earth oxides and accurately predict the transition pressures to higher density structures Figure 14a shows the energyholume curves for the smallest cation system studied Be0 The observed ground state is now the wurtzite (B4) structure in line with experiment The energy difference between the B4 and B3 structures is entirely consistent with the slight preference for the B4 indicated by the Madelung constant Figure 14(b) shows the same curves for SrO Here the tangent to the B1 and B2 curves predicts a transition pressure of 27 GPa which may be compared to an experimental value of 36 GPa Although the CIM potentials are more complex than pair poten- tials they appear to contain an essential aspect to describe phase transitions in oxides but are nonetheless transmutable in a chemi- cally meaningful manner between materials with different cations 33 Ion deformation In order to illustrate the role of aspherical deformation of ions insofar as these affect the repulsive interactions we depart from a consideration of crystal structures and consider the lattice vibra- tional frequencies in MgO How well does the CIM potential which provides an excellent description of the perfect crystal and its phase transitions do in predicting the lattice vibrations? As illustrated in Figure 15 the answer is ‘very badly’’ We know that since the lattice vibrations involve the movement of ions off their lattice sites polarization effects must be added to the straight CIM potential and that this will lower the energy of the lattice distortions and hence CHEMICAL SOCIETY REVIEWS 1996 -3100. -3500.01 -3700.0I I 5.0 10.0 15.0 20.0 25.0 VIA^ -2300.0 I I20.0 40.0 60.0 80.0 VIA^ Figure 14 Energy vs. volume curves for (a)BeO (6) SrO calculated by scaling the CIM potentials for MgO and CaO to allow for the change in cation radius.33 Key plusses -B2; crosses -B 1 circles -B3 triangles -B4 (wurtzite). For Be0 the wurtzite structure is now (correctly) predicted to be the lowest energy. The B 1 -B2 phase transition in SrO is predicted to occur at a pressure of 27 GPa -in reasonable agreement with experi- ment. reduce the vibrational frequencies. However the calculated phonon frequencies even when the CIM is supplemented with a description of the oxide ion dipole polarization (i.e.with variable ion radius and dipole) are very poor as shown in Figure 15(a).Figure 15(b)I4 shows the additional effect of allowing for a deformation of the ion shape of dipolar symmetry 1i.e.by introducing the variable 51 as suggested in eqn. (2.6)).Even when the self-energy associated with the deformation is introduced on an ad hoc basis as was done in these calculations the shape of the phonon dispersion curves is brought into much better agreement with the experimental ones although the absolute frequencies of the optic modes is still too high. A better parametrization of this term and the inclusion of quadrupolar effects is required to obtain the MgO phonons accu- rately.I4 4 More complex crystals In illustrating the characteristic consequences of each of the many- body effects in the last section we have tried to provide examples which draw attention to the role of an individual effect. In the halides compressibility effects are small and the effects of polar- ization are clearly seen; in the cubic phases of the alkaline earth oxides there is no polarization and the compressibility effects are exemplified. We have seen that each of the effects has a character- istic influence on which crystal structure is adopted -polarization favours unsymmetrical sites for highly polarizable ions neglecting compression leads to an underestimate of the stability of high coor- dination number structures. In general the observed structure will be a consequence of competition between opposing tendencies. In this section we consider some more complex examples than those discussed above in which the observed structure reflects this com- petition. w-L 2oomom0.0 0.0 0.2 0.4 0.6 0.8 1 .o 800.0 1 0.0 0.2 0.4 0.6 0.8 1 .o to 0951 Figure 15 Experimental phonon dispersion curves for Mg034 along the k = (Ook) direction are compared with phonon frequencies calculated (a) with the compressible ion potential plus polarization and (6) with the additional deformation effect. The type of phonon mode is indicated by the linestyle in both cases (long-dash -LO short-dash -TO dots -TA solid -LA) and the calculated points are indicated by the lines with circles. 4.1 A1,0 -higher-order multipoles The competition between the different effects becomes marked when considering the more complex M20 stoichiometry since almost all possible crystal structures for such systems give site sym- metries which are capable of sustaining induced multipoles. Attempts to model Al20 with shell models which allow for dipole polarization cannot account for the higher stability of the observed corundum structure over the less dense bixbyite phase unless unrea- sonably large values for the dispersion interaction between pairs of aluminium ions are included.28 This artificially stabilizes the close approach of a pair of A13+ions a characteristic feature of the corun- dum structure. In both the bixbyite and corundum structures the oxide ions are fourfold coordinated but whereas the coordination is almost tetrahedral in bixbyite in corundum the tetrahedra are twisted towards a planar D geometry. Although neither of these site symmetries can support a significant induced dipole moment in the corundum structure there is an appreciable field gradient at the oxide site which can induce a quadrupole. This suggestion is corroborated by findings from nuclear quadrupole resonance studies29 of the magnitude of the field gradients at '80 and Al nuclei which had been used to estimate the magnitudes of induced quadrupoles on the 02-ions and which are consistent with a value for the quadrupole polarizability Cof 02-of 5-7 a.u.30 Apart from supporting different polarization effects the oxide sites in corun-dum and bixbyite will lead to different degrees of compression of the oxide ions. Simulations of A120331have been carried out with a CIM poten-tial derived from the MgO by a small change in the cation radius ‘COVALENT’ EFFECTS IN ‘IONIC’ SYSTEMS-P A MADDEN AND M WILSON and including dipole and quadrupole polarization [using the asymp- totic model of eqn (2 5)j A C value of 6 a u predicts the corundum structure to be stable with respect to bixbyite with an energy differ- ence between the structures consistent with results fi-om ab initio calculations as well as good structural parameters The dipole polar- ization plays a very minor role because of the relatively high sym- metries of the oxide site in both lattices This is not quite the end of this story however In MgO the ab initio O2 quadrupole polarizability has a value of 26 a u ?2 and its value in A120 would be expected to be similar (in-crystal oxide polarizabil ities are found to depend largely on the cation-anion separation which are very similar in MgO and A1,0,) Such a large polarizability would give a much larger value for the energy differ- ence between corundum and bixbyite and would lead to nuclear field gradients considerably larger than those observed in NQR We would interpret this as a good example of the influence of the short range contribution to the induced multipoles as discussed in section 2 2 1 The effect of these terms is to substantially reduce the induced quadrupoles on anions from the values predicted with the asymp- totic model The influence of the short-range terms is mimicked in the calculations by introducing a much reduced value for the quadrupole polarizability 4.2 SnO (and PbO) -cation polarization As described above a family of CIM potentials can be obtained for the alkaline earth oxides which differ substantially only in the value of a cation radius and which quantitatively account for the phase transitions observed in this series between the ‘ionic’ B2 (CsCl-eight-coordinate) B 1 (rocksalt-six-coordinate) and B3/B4 (blende/wurtzite-four-coordinate)crystal structures We might therefore expect to be able to predict the structures of other oxides of stoichiometry MO simply from the cation radius Amongst the exceptions to the general pattern of ‘ionic’ structures in MO systems is SnO which together with PbO crystallizes into the litharge structure shown in Figure 16 l7 In the ideal structure (sketched the true structure involves a slight distortion) the anions occupy tetrahedral holes in a cubic cation lattice whereas radius considerations would suggest a rocksalt structure as formed by SrO (qyrI2+1 12 8 compared with asr2+ 1 18 A18) in which the = = larger octahedral holes are occupied More strikingly the pattern in which the tetrahedral holes are occupied is different from the four- coordinate cubic blende structure In the latter the occupancy is strictly charge-ordered and r = r++ In the SnO structure since alternate planes of holes are occupied (see Figure 16) r = r++/d2 and nearest-neighbour cations occur an arrangement which clearly does not minimise the charge4harge interactions Further evidence regarding the difference between SnO (and PbO) and the alkaline earth oxides comes from the lattice formation ener gies,Is which are more negative than those of the alkaline earth oxides with similar cation radii The analogy between the SnO structure and the layered crystals formed by MCI systems seems clear There are however two important differences Firstly the greater number of anions in the MX stoichiometry and the greater cation charge lend themselves to the anion-dipole stabilization mechanism -the excess of octahedral holes over cations means that asymmetric structures can arise even for intermediate size cations Only symmetric structures which do not allow dipole induction can arise from occupying octahedral holes in an fcc lattice at MO stoichiometry because of the equal number of ions and holes Secondly previous examples of such dipolar stabilization have been confined to systems in which it is the anion that sits in the asymmetric environment whereas in SnO it is the cation Simple electronic structure arguments suggest that the dipole polarizability of the Sn2+ cation could also be very large Its ground-state configuration is 5s2 which means that there are low energy dipole-allowed transitions (5s + 5p) which could make a very large contribution to the polarizability Ah initio results confirm this,?? with the value of a = 15 a u very similar to that expected for the anion in this system and much larger than the Sr2+ polarizability (5 a u ) Calculations show that polarization energies associated with such a large cation polarizability are sufficient to Figure 16 The SnO crystal structure illustrating the unsymmetrical occu pation of the unit cell by oxide ions Cations are the small dark spheres and oxide ions the larger pale ones overcome the charge ordering tendency and to drive the O2 ions into the tetrahedral holes in the cubic Sn2+ lattice despite the large cation/anion radius ratio in such a way as to generate the layered structure which allows the cation polarization 43 ZnO -the preference for a tetrahedral site From the perspective of the alkaline earth oxide CIM another ‘anomaly’ may be recognised Zn2+ has a slightly larger crystal radius (0 74 Al8) than the Mg2+ ion (0 68 A) and yet the lowest energy isomorph of ZnO is the four-coordinate B4 (wurtzite) struc- ture,I7whilst MgO is rocksalt In the alkaline earth oxides only the tiny Be2+(0 45 Ai8)falls in the four-coordinated B4 domain This preference of the Zn2+ ion for a tetrahedral site relative to Mg2+ is general -seen in the crystal structure of the chlorides for example The application of pressure to ZnO does result in a phase transition to the higher coordinate B 1 between 9 O4I and 9 5 GPa 74 Thus the energy of the B 1 phase is relatively close to that of the B4 in ZnO whilst the former has a smaller molar volume Dipole (and quadrupole) polarisation effects which we have con- sidered previously cannot contribute to the energetics of these structures because the site symmetry is too high to permit such mul- tipoles to be induced Mahan has noted the potential significance of octupole-induction at tetrahedral sites 35 The octupole polarizabil- ity which would control the magnitude of this effect might be large for a post-transition metal ion like Zn2+ because of the possibility of octupole-allowed d + p transitions This is confirmed by elec tronic structure calculations which show that the octupole polariz ability CR of Zn2+ (ca 29 a u ) is ca 50 times greater than Mg2+ suggesting that cation octupole polarization could be responsible for the different coordination preferences of the two ions However the same calculations show that CR for the O2 ion in these systems should be about 161 a u -so that if the octupole polarization were simply that driven by the field gradient from the ionic charges (1 e the asymptotic model of section 2 2) the octupole polarization of the anion would swamp that of the cation so that any discrimination between Zn2+ and Mg2+ from this mechanism would be lost Furthermore were the oxide ion octupole polariza tion energy obtained with the asymptotic model to be included in the energetic considerations for the alkaline earth oxides a B3/B4 ground-state structure would be predicted for MgO as well’ The item missing from these consideration is the short-range contribu tion to the polarization As discussed in section 2 2 I for anions this dramatically reduces higher-order polarization effects such as oxide octupole induction but for cations it enhances it Detailed considerations show that the crystal structure of ZnO can be ratio- nalised on the octupole induction mechanism 26 This finding is a significant one in trying to evaluate the limits of validity of the ‘extended’ ionic model at least in a practical sense The starting point of this model as expounded in section 2 2 IS that ions are basically spherical in high symmetry crystal structures and that only distortions of low multipolar order need to be included to account for the changes in their properties as they sample the envi- ronments typically encountered in the condensed phase The ZnO finding however points to the fact that higher order distortions of post-transition metal cations need to be incorporated even to account for the lowest energy crystal structure -the Zn2+ ion is intrinsically aspherical in a condensed phase This certainly poses a challenge for a computationally tractable implementation of the ‘extended’ ionic model 5 Conclusion In the article we have described an approach to the representation of interionic interactions in which formal ionic charges are used and in which the many-body effects have been broken down into dis- tinct physical effects which may be separately characterized in electronic structure calculations and separatefv represented in tractable computer simulation models We have shown that each of these effects can exert a distinctive influence on condensed phase structures and account for phenomena which have conventionally been regarded as non-ionic (or ‘covalent’) The potentials have been shown to be transferable not only between different phases of a given material but ‘transmutable’ between chemically related systems by substituting parameters with a physically transparent significance We believe that the work has shown that an ionic model where the ions carry formal charges has a wider domain of applicability than has sometimes been supposed This has opened the way for more wide-ranging simulation work with well-founded potentials We have tended to use the expression ‘covalent’ in a somewhat pejorative sense as a catch-all term to ‘explain’ anything not pre- dicted by the simplest ionic picture Enderby and Barnesi6 describe covalency as ‘interactions which change the charge dis- tribution of the valence electrons relative to some conceptual extreme represented in our case by the ionic model’ this seems to sanction the catch-all useage As chemists we would prefer a less catholic use of the term and to retain it for the description of interactions arising from chemical bond formation involving the sharing of pairs of electrons between atoms If we then attempt to find more specific explanations of the most spectacular ‘covalent’ anomalies (layered crystals bent ‘bonds’ etc ) it would seem that many can be attnbuted quantitatively to zunzc polarization -covalency in the chemical sense is not involved Detailed con- siderations of the relative energies of different structures necessi- tate an allowance for the fact that an ion’s size and shape may change in different environments but this may still be encom- passed within a fully ionic picture of closed shell species carry- ing formal charges The invocation of each of these aspects of the ‘extended’ ionic model in a particular material is driven by a con- sideration of the electronic structure of the ions involved Thus the concept of ions as sphencal rests on the remoteness of elec- tronic states of high angular momentum the oxide ion is particu- larly compressible compared to halides due to its instability in the gas-phase cation polarization becomes especially important CHEMICAL SOCIETY REVIEWS 1996 when low-energy dipole-allowed transitions are possible as in Sn2+ Quite how far this picture can be carried beyond the examples we have given at a quantitative level remains to be seen Acknowledgements We are grateful to several colleagues for their work and discussions on the topics we have covered in particular we thank John Harding Nick Pyper Adrian Rowley Nick Wilson and Malcolm Walters We also thank Emily Carter for her com- ments on the manuscript and John Freeman for help with the figures References 1 M Wilson and P A Madden J Phys Condens Matter 1993,5,2687 J Phys Chem 1996,100,1227 2 U Schroder Sol Stat Commun ,1966,4,347 3 B G Dick and A W Overhauser Phys Rev 1958.112,90 4 G D Mahan and K R Subbaswamy ‘Local 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