A new approach for calculating the electronic structure

Section 1.Synthesis,structure and properties of glassesA new approach for calculating the electronic structure and vibrational properties of non-crystalline solids:Effective charges for infrared-active normal mode vibrations in oxide and chalcogenide materialsG.Lucovskya,*,L.S.Sremaniak b ,T.Mowrer a ,J.L.WhittenbaDepartments of Physics,Electrical and Computer Engineering,Materials Science and Engineering,North Carolina State University,Raleigh,NC 27695,USAbDepartments of Chemistry,North Carolina State University,Raleigh,NC 27695,USAAbstractThis paper applies ab initio calculations to small clusters with short range bonding order for oxide and chalcogenide dielectrics that form continuous random networks (CRN),including,respectively,SiO 2and GeO 2,and As 2S 3and GeS 2.The calculations have been used to determine the dependence of the total energy on the bond-angle at the twofold coordinated O-and S-atom sites,and the infrared effective charges of the normal mode network vibrations associated with symmetry determined asymmetric bond-stretching,symmetric bond-stretching (bending)and out-of-plane bond rocking displacements,including decomposition of effective charges in equilibrium distribution,and dynamic charge redistribution components.The results of these calculations are in very good to excellent agreement with experiment.The approach is extended to silicates and their chalcogenide analogs in which pseudo-binary alloy formation causes a disruption of the CRN hosts through formation of ionic bonding arrangements of the metal atoms.This is accom-panied by increases in the coordination of O-,S-,Se-and As-atoms of the respective oxide and chalcogenide host CRNs,including SiO 2and As 2Se 3,and leads to a breakdown of 8ÀN bonding coordination.Ó2003Published by Elsevier B.V.1.IntroductionEmpirical force constant analyses of infrared (IR)and Raman scattering spectra have focused on significant differences between non-crystalline ox-ides and chalcogenides in spite of the fact that Si,As and Ge oxides,and As and Ge sulfides and selenides form continuous random networks (CRNs)with equivalent 8ÀN rule bonding coor-dination [1,2].This paper demonstrates through ab initio electronic structure calculations that differ-ences between oxides and chalcogenides are Ôa di-rect result of differences in their electronic structures Õ.In particular differences in total energy as a function of the bond-angle at the twofold co-ordinated O-,S-and Se-atom bonding sites corre-late directly with the Pauling bond ionicity [3].The*Corresponding author.Tel.:+1-9195153468;fax:+1-9195157331.E-mail address:gerry_lucovsky@ (G.Lucovsky).0022-3093/$-see front matter Ó2003Published by Elsevier B.V.doi:10.1016/S0022-3093(03)00369-7Journal of Non-Crystalline Solids 326&327(2003)1–14IR effective charges of O-and S-atoms also reflect quantitative differences between equilibrium charge distributions,and dynamic changes in these dis-tributions during normal mode motions.The nor-malized equilibrium charge distributions obtained from both experiment and the ab initio theory also show scaling trends with bond-ionicity.The ab inito calculations have been extended to non-crystalline oxides in which the atomic coordi-nations do not conform to the8ÀN rule[4].Non-crystalline alloys comprised of group IIIB and IVB transition metal oxides,and rare earth oxides, such as Y(La)2O3and Zr(Hf)O2,and Gd(Dy)2O3, respectively,and SiO2and Al2O3,have been pro-posed as candidate alternative high-k gate dielec-trics for aggressively scaled advanced Si devices with decreased lateral dimensions and increased levels of integration[5].This has lead to the de-velopment in Ref.[4]an empirical classification scheme for non-crystalline oxide dielectrics that includes these oxides as well the CRN oxides identified above.This classification scheme of Ref.[4]separates non-crystalline oxides into three different bonding groups[6],where the average Pauling electroneg-ativity differences,D X,bond-ionicities,f i,provide convenient metrics for this separation.Non-crystalline oxides with D X up to about1.5form covalently bonded CRNs,in which the constituent atoms have a coordination that reflects their pri-mary chemical valence;e.g.,two for O,three for P and As,and four for Si and Ge[6,7].The bonding in these CRN oxides obeys the8ÀN rule independent of any p-bonding contributions that arise from back donation into d-like polarization states of the P,As,Si or Ge atoms.The glass forming character of these materials,as well as their low defect den-sities in thinfilms and bulk glasses,is correlated with the number of bonding constraints per atom and has been discussed in detail in a series of sem-inal papers by Phillips and his coworkers[8–11].The classification scheme has also provided in-sights into the local chemical bonding arrange-ments in the small clusters used in ab initio calculations.These calculations have established correlations between the energies of transition metal atomic states,and the conduction band offset energies between(i)transition metal oxides,and silicate and aluminate alloys and(ii)Si[12].These offset energies determine(i)the barrier for direct tunneling between the Si substrate and the transi-tion metal dielectric,and(ii)the activation energy for injection into localized transition metal anti-bonding states in the silicate and aluminate alloys. In addition the scheme has lead to a new under-standing of the fundamental electronic structure of transition metal and rare earth oxide non-crystalline alloys,which can be transported and applied to non-crystalline chalcogenides as well. For example,it establishes a theoretical approach for electronic structure of amorphous transition metal chalcogenide alloys in which the coordina-tions of constituent atoms fall outside of the con-fines of the8ÀN rule that applies to amorphous Se,and As2S(Se)3,GeS(Se)2.This establishes that atomic coordination is a property of alloy compo-sition,and differences with respect to8ÀN coor-dination can be understood with ab initio quantum chemistry calculations[13].The classification scheme also has provided the stimulus for revisiting the infrared and Raman response of non-crystalline oxides and chalcogenides,and identifying the mi-croscopic origin of differences in the infrared(ir) effective charges of the respective normal mode O-atom,and S-and Se-atom motions.The specific objectives of this paper are twofold: (i)to apply ab initio calculations to relatively small clusters that characterize the bonding of O-and S-and Se-atoms in dielectrics with continuous ran-dom structures in order to determine(a)the total energy as a function of the bond-angle at the twofold coordinated oxygen and chalcogen atoms sites,and(b)the IR effective charges of the normal mode vibrations of the CRNs,and(ii)to extend the concepts developed in(i)to the metal–atom modified/disrupted silicate and metal chalcogenide pseudo-binary alloys with As2S(Se)3and GeS(Se)2.2.Analysis of IR spectra of oxides and chalcoge-nides with CRN morphologiesAnalyses of the IR reflectance and absorption spectra of fused silica glass(SiO2),thermally relaxed thinfilms of SiO2annealed at900°C [14,15],and bulk glasses of As2S3[16],As2Se3[16],2G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–14and GeS2[17,18],have yielded the optical fre-quency dielectric functions,e1and e2,from which the vibrational frequencies and IR effective charges in Table1have been determined for the normal mode vibrations displayed in Fig.1.The effective charges for the normal mode vibrations of GeO2 and GeSe2,not included in Table1,are similar to those of SiO2and GeS2,respectively,but the normal mode vibrational frequencies reflect the increase mass of Ge with respect to Si,and Se,with respect to O and S[14,19].Table1specifically in-cludes the experimentally determined frequencies for the asymmetric bond-stretching,symmetric bond-stretching(some times also referred to as bending),and out-of-plane bond-rocking modes. It is important to note that the IR effective charges in Ref.[16]were determined with respect to a particular empirical force constant model that does reflect the correct mode masses associated with the normal mode vibrations of the CRN structure.This has been remedied in Table1.The values of the effective charges for As2S3in Table1 are therefore different from those reported in Ref.[16].The effective charges of Table1were obtained directly from the experimentally determined values of e2by through the usual definition of the ma-croscopic IR effective charge,eÃ,are given by Eq.(1),eÃi¼1:43Â10À6m iðS iÂm i=N iÞ0:5;ð1Þwhere S i is the oscillator strength,m i is the mode mass and m i is the vibrational frequency,all of the i th vibrational mode,and N i is the density of os-cillators per unit volume and is equal to the oxygen or chalcogenide atom density is this normal mode representation[20].The mode masses reflect the respective motions of(i)the Si-,Ge-or As-atoms, and(ii)the O-,S-or Se-atoms in the network normal mode vibrations in Fig.1that satisfy conservation of momentum constraints.The mode masses for all of the Si–O,Ge–S and As–S normal mode vibrations in SiO2,GeS2and As2S3,respec-tively,are the same,and are given in Eqs.(2a)–(2c),mðSiO2Þ¼2mðSiÞmðOÞ=ð2mðSiÞþmðOÞÞ;ð2aÞmðGeS2Þ¼2mðGeÞmðSÞ=ð2mðGeÞþmðSÞÞ;andð2bÞmðAs2S3Þ¼3mðAsÞ2mðSÞ=ð3mðAsÞþ2mðSÞÞ;ð2cÞTable1Experimentally determined normal mode frequencies and IR effective chargesMode SiO2(f i¼0:45)frequency(±4cmÀ1)Effectivecharge(±0.1e)GeS2(f i¼0:08)frequency(±3cmÀ1)Effectivecharge(±0.1e)As2S3(f i¼0:05)frequency(±3cmÀ1)Effectivecharge(±0.1e)Asymmetricstretch1075 2.29367 1.53309 1.30Symmetric stretch8100.933280.523400.61 Out-of-plane rock4600.921490.281620.29 Ratio m AS=m SS 1.33±0.02– 1.12±0.02–0.91±0.02–Ratio eÃAS =eÃSS– 2.45±0.4– 2.94±0.7– 2.14±0.6Ratio eÃAS =eÃR– 2.48±0.4– 5.52±1– 4.48±1G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–143where the chemical symbols denote either the designated compound mode mass in atomic units, or the designated atomic mass in the same units.In spite of large Pauling bond-ionicity(f i)dif-ferences between Si–O,and the As–S(Se)and Ge–S bonds,there are relatively small differences in their respective IR effective charges.Equally impor-tantly these charges show significant differences for the different normal modes.This has previously been noted in Ref.[14]where it was proposed that dynamic charge transfer,or effectively dynamic rehybridization of bonding orbitals[21]occurred during normal mode motions and was responsible for the relatively large IR effective charges for the bond-stretching normal modes of As2S3and As2Se3.Table1also includes ratios of the vibra-tional frequencies of the asymmetric and sym-metric stretching modes,m AS=m SS,and ratios of the effective charges for the asymmetric and symmetricstretching modes,eÃAS =eÃSS,and the asymmetricstretching and bond rocking modes,eÃAS =eÃR.Thesignificance of these ratios for the effective charges will become apparent as the electronic structure is determined from the ab initio calculations pre-sented below.The ratio of m AS=m SS for SiO2is smaller than anticipated from central and valence forcefield empirical force constant calculations, but is in better agreement with values that can be determined directly from ab initio calculations like those reported below.This paper an outline of the ab initio quantum chemistry calculations of the electronic structure [22,23]that have been used for the determination of the IR effective charges of the normal mode motions of the twofold coordinated O-,S-and Se-atoms as they occur in these CRNs.The theory has also been applied to disrupted CRNs including pseudo-binary silicates such as SiO2and ZrO2,and the pseudo-binary chalcogenides such as As2Se3–NiSe[24–26]and As2Se3–Cu2Se[27].3.Ab initio calculationsThe ab initio calculations arefirst applied to IR effective charges(also sometimes designated as dynamic dipoles)of SiO2to illustrate the method and new insights that can be obtained from this approach[28].In particular,a decomposition of these IR effective charges into equilibrium charge and charge redistribution contributions is demon-strated.The electronic structure calculations are ab initio in character,and employ variational methods in which an exact Hamiltonian is used. They are qualitatively different from the empirical tight-binding and valence forcefield approaches used for the respective CBL electronic and vibra-tional calculations of Joannopoulos and cowork-ers[29,30],and those of Ref.[19].In addition no core potential,or exchange approximations are assumed[22,23].The Hamiltonian,H,is given in Eq.(3),H¼R iðÀ1=2r2iþR kðÀZ k=r ikÞÞþR i<j1=r ij;ð3Þcontaining kinetic energy(1=2r2i),nuclear-elec-tron attraction(ÀZ k=r ik)and electron–electron re-pulsion(1=r ij)contributions.The calculations are done at two levels,initially through a self-consistentfield(SCF)Hartree–Fock calculation with a single determinant wave func-tion of the general form,W¼C detð/að1Þ/bð2Þ/cð3Þ/dð4Þ.../xðNÞÞ;ð4Þwhere functions/are molecular orbitals contain-ing spatial and spin components.This approach does not include effects due to electron correlation. Following this,there is a configuration interaction (CI)refinement of the bonding orbitals that is based on a multi-determinant expansion wave function,W¼R i c i U i;ð5Þwhere the U i denote determinants of the form specified in Eq.(4)with different choices of occu-pied molecular orbitals.Expansions of this type include electron correlation effects.The expansion coefficients c i are determined by energy minimi-zation.Since the Hamiltonian is exact,through electrostatic interactions,the energy variational theorem is satisfied.The SCF and CI calculations have been applied to the clusters in Figs.2and3.The local cluster in Fig.2is embedded mathematically in a larger net-work structure through a1-electron embedding potential VðrÞ,and basis functions S1and S2, which are represented by SiÃin the diagram.The4G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–14functions S1and S2are Gaussian expansions of long and short range functions representing an sp 3hybrid orbital of Si.The mixing parameter,k ,in S1þk S2,is set such that all Si core energy levels are correct.This also assures a correct overall charge distribution in the cluster,so that the clus-ters in Fig.2in 3have zero dipole moments.The approach is conceptually similar to the cluster Bethe lattice (CBL)approach of Ref.[29,30];however,for these ab initio calculations,the Hamiltonian is exact and the calculation is ab initio rather than empirical.The CBL and the embedding potential approach of this paper are each used to emulate the properties of the CRN amorphous morphology of non-crystalline SiO 2[4,6].3.1.Total energyThe total energy for the cluster in Fig.2has been calculated as a function of the Si–O–Si bondangle a (see Fig.4)at three different computational levels:(i)SCF with d-polarization on the Si atoms,(ii)SCF +CI with no d-polarization,and (iii)SCF +CI with d-polarization.Since the Hamilto-nian is exact,any increase in the total energy through a variational process represents an im-proved solution to that calculation,and addition-ally to the IR effective charge calculations that use the equilibrium cluster geometry as a point of reference.This variational approach to binding energy optimization unequivocally establishes that the SCF +CI with d-polarizations is the best ap-proach for calculating the total binding energy,and therefore for other properties of the cluster to be addressed as well.It is significant to note that the calculated minimum in total energy occurs at a Si–O–Si bond angle of approximately 145–150°,very close the average bond angle determined experimentally for vitreous SiO 2,144°[31].In addition,the broad minimum in total energy is consistent with radial distribution functions studies on fused silica that indicate that the Si–O–Si bond angle distribution is broad,extending from approximately 120–180°.It is also consistent with the unusually small value of the empirical Si–O–Si three-bodybond-bendingG.Lucovsky et al./Journal of Non-Crystalline Solids 326&327(2003)1–145force constant[6,8–11].The spread in bond-angle also contributes significantly to the configurational entropy that promotes the high metastability of non-crystalline SiO2.The calculated total energy dependence on the Si–O–Si bond angle a adds additional validity of this local cluster approach for other properties which depend primarily on the local Si–O–Si bonding geometry.It also indicates that the terminal O-atom groups and embedding terminal atom SiÃfunctions emulate the connec-tivity of the Si–O–Si group to SiO2CRN.3.2.IR effective chargesThe IR effective charges,D l j=D q i,q i¼x;y and z,j¼x;y;z have been calculated for the asym-metric bond-stretching(AS)vibration,m y,symmet-ric bond-stretching(SS or B),m z,and out-of-plane bond-rocking(R),m x,infrared active vibrations of non-crystalline SiO2[14].The normal mode mo-tions of O-atoms of these vibrations are defined with respect to the coordinate system displayed in Fig. 1.For the asymmetric bond-stretching vi-bration,m y,the normal coordinate is y,and therelative motion of the O-atom is parallel to a line joining the two Si-atom neighbors.For the sym-metric bond-stretching vibration,m z,the normal coordinate is z,and the relative motion of the O-atom is in the direction of the bisector of the Si–O–Si bond angle,a.Finally,for the bond-rocking vibration,m x,the normal coordinate is x,which is perpendicular to the plane of the Si–O–Si bonding group.The IR effective charges or dynamic dipoles D l y=D y,D l z=D z and D l x=D x have been calculated from SCF and CI calculations,with and without d-polarizations,and as a function of a.Hereafter the subscripts x,y and z will not be used for the normal mode dynamic dipoles defined above.The displacements x,y and z for the calculations were <5%of a nominal Si–O bond length of0.16nm,or $3atomic units(1a.u.¼0.053nm).The dynamic dipoles in Fig.5are in units of the electronic charge(e)to facilitate comparisons with other theoretical and experimental studies.The inclusion of the d-symmetry polarizations increases the magnitude of the dynamic dipole for the asym-metric stretching mode with the normal coordinate displacement,y.The dynamic dipoles for the other two normal modes are each significantly smaller than that of the asymmetric stretching mode,and are decreased in magnitude when the d-polariza-tions are included.For the calculations including the d-symmetry polarizations,D l=D x is essentially independent of a,whilst D l=D y,D l=D z and dis-play opposite dependencies on a.D l=D y decreases (becomes more negative)with increasing a, whereas D l=D z increases,but with a total positive change in D l=D z that is about a factor of7smaller than the total negative change in D l=D y,$0.1e for D l=D z as compared to$0.7e for D l=D y.The influence of next nearest neighbor Si–O–SiÃbond angles on the dynamic dipoles for a¼150°is displayed in Fig.6.Changes in the dynamic di-poles are about5%of their150°values for±30°excursions of the Si–O–SiÃbond angle.This dem-onstrates that the dynamic dipoles are determined primarily by nearest neighbor bonding within the Si–O–Si portion of the cluster,and are effectively insensitive to the details of the next-nearest neighbor O-atom bonding arrangements with embedding SiÃterminations.This establishes that6G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–14the IR effective charges for the infrared active normal modes of SiO2are a local property of the CRN amorphous morphology.3.3.Decomposition of IR effective chargesThese calculations compare the IR effective charges for two different electronic structures.The total IR effective charges for the normal mode displacements have been determined by comparingthe equilibrium structure x,y and z dipole moments for a given bond angle a with the dipole moments of dynamic structures that include respectively the x,y and z displacements of the normal modes.This was done at(i)SCF with d-polarizations,(ii) SCF+CI without d-polarizations,and(iii)SCF+ Ci with d-polarization levels.For the determina-tion of the equilibrium charge density contribu-tion,D l=D q ieq,the change in the dipole moment was calculated for normal mode displacements of the O-atom orbitals,with all other orbitalsfixed. The contribution due to the orbital displacements, i.e.,theÔdynamic rehybridization of the bondingorbitalsÕ,D l=D q ioc,is then the difference betweenthe total IR effective charge,D l=D q itotal,and the contribution from the equilibrium charge densityshift,D l=D q ieq,as in Eq.(5),D l=D q itotal¼D l=D q ieqþD l=D q ioc:ð6ÞThese calculations have been performed for two different Si–O–Si bond angles,a¼150°,the opti-mum bond angle,and for a¼180°.The results of these calculations are also included in Table2.The average value of the equilibrium charge density contribution is about35%larger(a¼150°)for the asymmetric stretching mode with the normal co-ordinate y,than the symmetric stretching mode and rocking modes with respective normal mode coordinates,x and z.These equilibrium charge density contributions are negative,consistent with the Pauling electronegativities of1.90for Si,andTable2Decomposition of IR effective charges aa¼150°Effective charges(e)Mode Equilibrium charge Orbital change Totalx,rocking)1.0+0.1)0.9z,symmetric stretch)1.0+0.1)0.9y,asymmetric stretch)1.3)1.2)2.5a¼180°Mode Equilibrium charge Orbital change Totalx,rocking)1.0+0.1)0.9z,symmetric stretch)1.0+0.1)0.9y,asymmetric stretch)1.4)1.3)2.7a The calculations have precision of at least one part in a hundred;however,a realistic assessment of the model is consistent with uncertainty of the order of at least0.1e.G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–1473.44for O[32].Finally the orbital change contri-butions for the symmetric stretching and rocking modes are relatively small and positive,and de-crease the magnitude of the total dynamic dipole, whilst the orbital change contribution for the asymmetric mode is more than an order of mag-nitude larger,and is negative,so that it increases the magnitude of the total dynamic dipole.There is also an increase of about10%in the orbital change contribution when the Si–O–Si angle is increased from150°to180°,whilst the equilibrium density contribution increases about7%.Finally,these decompositions demonstrate a significant charge redistribution contribution for asymmetric bond-stretching modes in which the two bonds of the respective bridging-atoms change in different directions,one increasing and the other decreasing(Fig.1).In contrast,the charge redis-tribution contributions of the symmetric stretching and out-of-plane rocking modes,in which both Si–O bond lengths change in the same sense,are sig-nificantly smaller.3.4.Total energy calculationThe total energy ab initio calculations of this paper represent an improvement over the more empirical approaches in previous publications. The calculations of Pantelides and Harrison[33], based on empirical tight-binding theory gave an optimum a of90°,which is a significant depar-ture from the experimentally determined average Si–O–Si bond angle of approximately145–150°[31].However,attempts to improve the tight-binding empirical calculations by Harrison through the addition of an empirical Si–Si repul-sion term resulted in a small improvement,in-creasing a to approximately114°[34].Revesz and Gibbs performed ab initio calcula-tions on a pyrosilicate acid molecule,(HO)3Si-OSi(OH)3or equivalently H6Si2O7,at the SCF level[35].This molecule terminates the Si–O–Si arrangement of interest with terminal OH groups. Their calculation specifically included Si3d-states in the basis set,as opposed to the two sets of Gaussians used in these calculations.They ob-tained results that are qualitatively similar to those presented in this paper in Fig.4;i.e.,their results also display a shallow minimum in binding energy, but at approximately140°rather than145–150°as in our calculations[35].The energy of this mini-mum of about5%lower than the value of the binding energy at180°,whilst in Fig.4,it is lower by<1.5%for the SCF+CI with d-polarization calculations.The calculations of Lucovsky and Yang in Ref.[36]at the SCF+CI level,but without the Si d-polarizations,give essentially the same results as in corresponding curve of Fig.4.These calcula-tions show a continuous improvement in binding energy between120°and180°,with no shallow minimum in the vicinity of150°or elsewhere.As evident in Fig.4,the inclusion of the Si d-polar-ization gives a large,$2.7eV,increase in the total energy,and also produces a shallow minimum in binding energy close to the experimentally deter-mined Si–O–Si value of145–150°.From a comparison between the calculations of Lucovsky and Yang[36],Revesz and Gibbs[35], and those presented in this paper,it can be con-cluded that the inclusion of Si3d states as in Ref.[35],and Si d-symmetry polarizations as in this paper,appear to be the determinant factor in generating the shallow minimum in the binding energy curve.3.5.Calculations of total energy and IR effective charges for chalcogenidesSimilar ab initio calculations have been per-formed for GeS2and As2S3based on clusters that are the analogs of those in Fig.2.The primary differences between these calculations and those for SiO2are quantitative in nature and relate the differences in the basis sets for the oxygen and sulfur atoms.Differences between the atomic basis sets required for energy minimization for O-and S-atoms contribute very little to differences in the Si–O–Si,and the Ge–S–Ge and As–Ge–As bond angles,and the angular dependence of the total effective charges;whilst differences in bond ionicity between the oxides($30–50%)and sul-fides(<10%)the determinant factor.The basis sets for the As-,Ge-,Si-,O-,and S-atoms include the core levels,and respective valence bonding states,4s and4p for As and Ge,3s and3p for Si8G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–14and S,and2s and2p for O.In addition,the basissets for Si,S,As and Ge also include compactGaussian functions with d-like symmetries.Thisreflects the increased size of these atoms with re-spect tofirst row atoms such as O,and C and N.The inclusion of these Gaussian d-states differ-entiates S(and also Se)from O and is quantita-tive in nature.Table3compares the IR effective charges ofSiO2,GeS2and As2S3,including the decomposi-tion into equilibrium and charge redistributioncomponents.Fig.7display energy as a function ofbond angle for calculations at the SCF and CIlevels with and without d-state symmetry contri-butions to the basis for S,for Ge–S–Ge andAs–S–As bonding arrangements,respectively,andthese should be compared with Fig.4for Si–O–Si.The marked differences between the total energyversus bond angle for SiO2,and GeS2and As2S3derive from two effects:(i)predominantly fromdifferences in bond-ionicity,and(ii)weakly fromthe inclusion of the Gaussian d-state polarizationsfor the sulfur-atoms.Fig.8is a plot of bond angleversus Pauling bond-ionicity that demonstratesthat the inclusion of d-symmetry contributions tothe basis sets for Si,Ge and As,and S yields acalculated bond angle that agrees with experiment,and furthermore,additional experimentally deter-mined bond angles for GeO2and As2O3combinedwith the calculations indicate a linear scaling be-tween bond angle and Pauling bond ionicity.Similar considerations apply to the variation of IReffective charges,where the results for Ge–S–Geand As–S–As are presented in Fig.9and should becompared with Fig.5for Si–O–Si.Table3Calculated effective charges for O-atoms in SiO2and S-atoms in GeS2and As2S3Mode SiO2GeS2As2S3Total (±0.1e)Equili-brium(±0.1e)Charge re-distribution(±0.1e)Total(±0.1e)Equili-brium(±0.1e)Charge re-distribution(±0.1e)Total(±0.1e)Equili-brium(±0.1e)Charge re-distribution(±0.1e)Asymmetricstretch)2.51)1.24)1.27)1.30)0.80)0.50)1.31)0.59)0.72Symmetricstretch)0.93)0.98+0.05)0.54)0.56+0.02)0.37)0.36)0.01Out-of-plane rock )0.89)0.96+0.07)0.48)0.39)0.09)0.15)0.35+0.20G.Lucovsky et al./Journal of Non-Crystalline Solids326&327(2003)1–149。