CORR.ELATIONS BETWEEN Si O BOND LENGTH, Si-O-Si ANGLE and BOND OVERLAP POPULATTONS CALCULATED USING EXTENDED HUCKEL MOLECULAR ORBITAL THEORY

Anerican Min*olooist Vol. 57, pp. t57S-1613 (J922')

CORR.ELATIONS BETWEEN Si_O BOND LENGTH, Si-O-Si ANGLE AND BOND OVERLAP POPULATTONS CALCULATED USING EXTENDED HUCKEL MOLECULAR ORBITAL THEORY

G. V. Gmns,M. M. Hernr,l aNDS. J. LoqrsxlrHaN, D ep'artmentof Geot o'Eiaa.l Sciences, Virginia Polytechnic Institute ond State Uniuersitg, B lacksbur g, Vir girui,a2 406 I

AND L. S. Benrcr,r, ANDHsruKANc Yow, D epartnzentof Ch,emistrg, Uniu ersity o,f Miehig an, Arm,Arbor, Miahigo"n4810! AssrRecr Extended Htickel molecr:lar orbital calculations have bee=ncompleted fof isolated and polymerized tetrahedral ions for ten bilicate mineral! in which a(O) s 0.0 using ob.served O-Si-O and Si-O-Si valence angles, assuming all Si-O bond lengths are 1.63A and including silicon 3s and 3p and oxygen 2s and.2p i,tdmic orbitals as the basis set. The resultant Mulliken bond overlap populations, z(Si-O),,correlate with observed Si-O bond lengths, the shorter bonds being associated with the larger n(Si-O). If the five Si(3d) atomic orbitals are also included in the basis set, the observedSi-O(br) and Si-O(nbr) bond lengths correlate with n (Si-O) as two separate trends with overlap populations for the Si-O(nbr) bonds exceedingthose for Si-O(br) by about 20 percent, even if the Si-O-Si angle is 180o. Calculation of z(Si-O) as q function of the Si-O-Si angle for the Si-O bo"ds irr Sir6ro- (assuming Si-O : 1.634 and { G-Si-O : 109.47o)first excluding and then including the 3d-orbitals, predicts in each case that the Si-O(br) bond lengths should decreasenon-linearly and that the Si-O(nbr) bond lengths should increase slightly as the Si-G-Si angle increases. Scatter diagrams of observed Si-O(br) bond lengths to two-coordinated oxygens versus the Si-O-Si angle appear to va,ry non-linearly with the shorter bonds tending to occur at wider angles.If these bond lengths a,rereplotted against - 1/cos(Si-O-Si), the scatter diagrams show improved linear trends. Multiple linear and stepwise regression analyses of the bond lengths as a function of -1/cos(Si-O-Si), Af(O) and (CN) indicate that each makes a significant contribution to the variance of the Si-O(br) bond length. The conclusion by Baur (1971) that the Si-O(br) bond length is independent of Si-O-Si angle in silicates for which Af(O) : 0.0 can be rejected at the 99 percent confidence level. IwrnonucnoN Pauling (1939) has concludedfrom the electronegativitydifference between Si and O that the Si-O o--bond has about 50 percent ionic

l Now at the Geology Department, Rutgers-The State University, New Brunswick, New Jersey 08903. 1578 MO CALCULATIONSFOE SILICATE IONS 1579

character, attributing a charge of about '*2 on the silicon atom. Becausethis charge exceeds*1, thereby violating his electroneutrality principle, he (1952)postulated the formation of.d-p.-bonds betweenthe two atoms to neutralizethe charge.Using an empirical equatio r relating interatornic distanceto the amount of char- "'-bond acter, he then computeda Si-O distanceof 1.63Ain closeagreement with reported values. Subsequently,Fyfe (1954) calculated semiempirical bond energiesfor the SiO+tetrahedron as a function of the Si-O separation,and concludedthat the relatively short Si-O bond can be explainedwithout resortingto extensivedouble bonding.On the other hand, Cruickshank (1961) has assertedon the basis of group-theoreticarguments, approximate molecular orbital calculations, and group overlap integralstha;t of the five 3d-orbitals, only the two e orbitals on silicon form strong z'-bondswith the appropriate 2pr lone pair orbitals on oxygen.The Si(3d) orbital involvementin the Si-O bond has since been substantiated by all-electron a,b irui.tio SCF molecularorbital calculationsfor the silicateion and orthosilicic acid (Collins, Cruickshank, and Breeze, lg72). These calculations show (1) that there is considerableSi(3d,)-orbital participation in the wavefunctions with the e orbitals forming and the f2 "'-bonds orbitals forming o-bonds with the appropriate valence orbitals on oxygen; (2) that the calculatedl"e,s X-ray fluorescencespectra using Sd-orbitals are in remarkably good agreementwith the experimental spectra of silica glass; the agreementis notably poorer when the 3d-orbitals are not included in the calculation (seeFig. 1); and (3) that the residualcharge on Si is not *4 as assumedin an electro- staticmodel but is closeto zeroin reasonableagreement with Pauling's electroneutralityprinciple and the resultsof a Si(K)-spectrochemical shift study by Urusov (1967). Within the last decade a large number of silicate and siloxane structures have been carefully refined using precise difiraction data obtained by modern techniques. These refinements have revealed a numbet of systematic variations in bond lengths and bond angles: (1) the Si-O(nbr) bonds are usually shorter than the Si-O(br) bonds; (2) shorter Si-O bonds are usually involved in wider O-Si-O and Si-O-Si angles;and (3) the tetrahedral angles usually decreasein the order

d [o (nbr)-Si-o (nbr)] > { [O(nbr)-Si-O@r)] > { [o (br)-Si-o(br)]. Several investigators have arg ed that these variations can be ration- alized in terms of a covalent m del that includes3d-orbital involvement between silicon and oxygen cl. Cruickshank, 1961; Lazarev, 1964; GIBBg, HAMTL, T,OUISNATHAN,BARTELL, AND YOW

Frc. 1. Comparison of (q) the experimental L." X-raY fluorescencespectra for silica"glass with that calculated by Collins et aI, (1972) from rmults obtained in all-electron ab ini'tin SFC MO calculations on the silicate ion (b) using an extended basis set that includes Si(3d) orbitals and (c) using a minimum basis set that ex- cludes Si(3d) orbitals.

E, o F (J lrJ F lrJ o ltJ I F F o UJ

UJ (J lrj E. a Fz l ()o ol! o z MO CALCULATIONSFOR SILICATE IONS 1581

McDonald and Cruickshank, 1967a; Cannillo, Rossi, and Ungaretti, 1968;Bokii and Struchkov,1968; Noll, 1968;Brown, Gibbs,and Ribbe, 1969;Brown and Gibbs, 1970;Louisnathan and Smith, 1971;Mitchell, Bloss,and Gibbs, 1971).On the other hand, consistencywith a covalent model that includes3d-orbital involvement doesnot, of course,preclude other bonding models. For example, Mitchell (1969) notes that the sametrends are implicit both in Gillespie's(1960) valence-shell electron- pair repulsion model and in a molecular orbital model involving a valence basis set of atomic orbitals (without the 3d-orbitals of Si). Despite the systematic variations observedbetween Si-O bond length and the Si-O-Si and O-Si-O anglesin silicatesand siloxanes, no attempt has yet beenmade to rationalizethese trends in terms of modernvalence theory. An attempt at such a classificationshould not only serve to improve our understanding and interpretation of such observablesas the elasticconstants (Wiedner and Simmons,1972), the bond polarizabiliiies (Revesz,1977) , the diamagneticsusceptibilities (Verhoogen,1958), the Raman and infrared spectra (Griffith, 1969) and the X-ray emissionand fluorescencespectra (Dodd and Glen, 1969;Urch,1969; Collins et a1.,1972)of silicatesin generalbutshould also clarify our understandingof the long-range and short-range forcesinvolved in A/Si ordering (Stewart and Ribbe, 1969; Brown and Gibbs,1970). Recently,Baur (1970,1971) attempted to rationalizethe Si-O bond length variations for a large number of silicatesin terms of an electrostatic model by showing that the Si-O bond lengths predicted by his so-called extended electrostatic valence rule correlate well with those observed for structures where LfQ) + 0'. However, he notes that the rule fails to predict bond length variations in silicateslike quartz and forsterite where Af(O) : 0.0. In spite of this shortcoming,Baur has rejected Cruickshank'sdouble bonding model in favor of an electrostatic

I For Si-O bond length variations in the tetrahedral portion of a silicate, the extended electroStatic valence nrle states that the deviation of an individual Si-O bond from the mean tength, (Si-O), in a silicate ion is proportional to Af(O). The actual bond length is predicted by the empirical equation

Si-op*a. : (si-o) + 0.091 ^f(o) where af(O) : f(O) - f(O), l(O) is the sum of the Pauling (1929; 1960)bond strengths received by oxygen andf(O) is the mean f(O) for the silicate ion under consideration.Baur (1970, 1971) usesp6 instead of f(O) to denote the sum of the bond strengths received by oxygen. However, we prefer to retain Pauling's symbolism for two reasons: (1) f(O) is an accepted symbol used by many investigators and (2) p denotes bond-order or bond-number in modern valence theory (c/. Coulson, 1939; Pople el al., 1965; and Mulliken, 1955). T582 GIBBS,,HAMIL, LOUISNATHAN,,BARTELL,AND YOW model (l) becdusehe could r;Lotproue a dependencebetween Si-O bond

suggeststhat irnoreof the variation in bond length may be explained in terms of Af(O) than in terms of the angle, it cannot be concluded a priori that fhe bond length variation is independent of the angle without appropriatestatistical tests. Accordingto Baur (1971),the main differencebetween his electrostatic model and Cruickshank'sdouble bonding model is that the latter stressesthe importance of the intrinsic electronic structure of the tetrahedral ions whereas his electrostatic model stressesthe ex- trinsic effects from the the non-tetrahedral cations. Although Baur is aware of the empirical nature of his model and the fact that it cerbainly cannot replace a model basedon electronic structure, he has stated that future models for the Si-O bond (1) should incorporate the electronic structure of the constituent atoms as well as the effects of the non-tetrahedralcations and (2) shouldbegin where his electrostatic model breaks down (for minerals like forsterile, quarbz,etc.). The Si-O bond length variations in forsterite were recently examined in terms of extendedHiickel molecular orbital (EIIMO) theory by Louisnathan and Gibbs (1971) who found that the observedbond length variations show a close correspondencewith bond overlap populations,n(Si-O), calculated (1) for the silicate ion alone and (2) for the silicate ion with its nearest neighbor non-tetrahedral cationsand their ligands.

r T'he relationship between Si-O distance and Si-O-Si ahgle for the silica polymorphs was recently re-examined by D. Taylor (lWZ); SeeMineral.:' Mag. 38, 62H31. After correcting the $i-O distances in the polymorphs for thermal motion of the oxygen atoms, Taylor calsulated a regression line for the cor- rected Si-O bond lengths and their associated Si-O-Si angles. Ile found the re- sulting line to be less steep than those calculated earlier by Brown et aI. (1969) for the framework silicates but statistically identical to the one calculated in. our study (see Table 3). Moreover, Taylor notes the probability the slope of the line might be zero is a little less than 0.05, a result at va.riance with the findings by Baur (1971) who believes that a dependence between Si-O distance and Si-O-Si angles does not exist for minerals like the silica polymorphs where Af(O) - 0.0. MO CALCULATIONSFOR SILICATE IONS 1583

??(Si-O)calculated with Si(spd) valencebasis set for the silicate ion in forsterite (1) For the silicate ion (2) With non-tetrahedral alone cations and their ligands Bond Length n(Si-O) n (Si-O) Si-O(1) : 1.6134 0.935 1.003 Si-O(2) : 1.654 0.878 0.931 Si-O(3) : 1.635 0.892 0.968 The n(Si-O) calculatedfor the silica;teion alone (seecol. l above) were made using the observedO-Si-O angles but with all Si-O distances set equal to 1.634. Despite the assumptionin the carculation that all si-o bond lengths are constant, it is clear that the resurting n.(Si-O) can be used to order the observedSi-O bond lengths in forsterite. Encouraged by this result, the present study was under- taken to learn whetheror not EHMO theory can serveas a modelfor classifyingand ordering si-o bond length variations when the non- tetrahedral cations are ignored and a((O)=0.0'. Moreover, in the event that these calculationsprove successful,they should improve o'ur understandingof the nature of the si-o bond in terms of modern valence theory and provide plausible reasonsfor the variation of individual si-o bond lengthsin silicatesand siloxanesin generar.rn addition,new scatterdiagrams of si-o (br) bond lengthplotted against Si-O-Si angle will be presented for data from Bl carefully refined silicate and several siloxane structures in which A((O) is zero or small.The relative contributionsof Si-O-Si, -l/cos(Si-O-Si), A((O), and (CN) (the mean coordination number of oxygen bonded to silicon) to the Si-O(br) bond length in these structureswill be as- sessedusing multiple linear and stepwiseregression methodsl (see appendix).The outcomeof the analysiswill showthat we can reject the hypothesisthat a relationshipbetween Si-O(br) and Si-O-Si angle might not existfor silicateswhere A((O) = 0.0. Trrn Cer,cur,ATroNemn SrcNrrrcANcEor Boxl Ovnnr,epporur,lrroys Osrarxnu rN THEExruxopn Hiicrnr, Mor,ncur,enOnsrrAr, Trrrony SincePauling (1929)formulated his famousset of rules for predict- ing stable atom configurationsin ionic crystals,many silicate structures haVe been investigated by X-ray methods and have been found 'Linear, multiple linear, and stepwise linear regression computations presented in our paper were completed using the UCLA BIOMEDICAL PROGRAM PACKAGE,1967. 1584 GIBBS,HAMIL, LOUISNATHAN,BABTELL, AND YO'W to conform remarkably well. Nevertheless,the steric details of the tetrahedral ions in many of thesesame silicatesalso appearto conform with a model that includescovalent bonding (cf. Brown et al', 1969;Brown and Gibbs, 1970).In an attempt to learn whetherthese details are indeed consistentwith covalenttheory, extendedHiickel molecular orbital calculationswere made for the tetrahedral ions in ten silicateminerals specifically chosen such that A((O) is either zero or small. In this way the importanceof the intrinsic electronicstruc- ture of the tetrahedralions can be emphasizedto the reader (Hamil, Gibbs,Bartell and Yow, 1971).Silicates showing a larger variation in a((O) will be consideredelsewhere by Louisnathanand Gibbs in a study of the disilicates (1972d).EHMO calculations,although very inexpensiveand crude by ab initio'standards(Richards and Horsley, 1970),have beenmoderately successful in generatingWalsh-Mulliken diagrams (molecular orbital energiesplotted against bond angle) (c/. Hoffmann, 1963; Gavin, 1969; Allen, 1970; 1972) and' in de- lineating trends in variations of bond length with bond overlap populations or bond.order for a number of relatively large moleculesin their ground states (c'1.Dallinga and Ross,1968; Gavin, 1969;Boyd and Lipscomb, 1969).Recently, Bartell, Su, and Yow (1970) calculated EHMO bond overlap populationsfor the tetrahedral bonds in selectedsulfates and phosphatesand found that they show strong correlationswith (1) the simple valence bond orders assignedby Cruickshank (1961) and (2) the observedtetrahedral bond lengths' To learn whethersi-o bond overlappopulations for the silicatesshow similar trends, calculationswere undertakenusing an EHMO program originally written by Hoffmann (1963).In the calculations,one- electronmolecular orbitals, MOs, are constructedas a linear combi- nation of atomic orbitals,LCAO,

vo : t, co,6,

where c16i&r-the linear .oum.i.ot.- und 41 the Slater type single exponent atomic valence orbitals. The energy associated with an MO, - = ea,is obtained by diagonalizing the secular determinant lHot .Stil 0, where Sri are the two center overlap integrals (explicitly evaluated from the positional coordinates of the constituent atoms) , a,nd'Hai a're the elements of the Hiickel Hamiltonian matrix. In the rigorous Roothaan-Hartree,Fock method, the Ha1are complicated functions of the linear coefficients and of two-, three-, and four-centered integrals, and this fact requires an iterative solution of the secular equation (Richards and Horsley, 1970). On the other hand, in EHMO theory MO CALCULATIONSFOR SILICATE IONS 1585

lhe H15terms are not explicitly evaluated but are chosento simulate spectroscopicallyobtained quantities, namely the negative of the valenceorbital ionization potentials (VOIP) whereasthe I/ai terms are approximatedby the Wolfsberg-Helmholz(1952) parametrization

Htr:Bno(//',*H,,). When one-electronMOs are constructedand their energiesobtained, the Pauli exclusion principle is applied as an afterthought and the MOs are filled with electrons pair-wise starbing from the lowest eigenstate.By inserbingeigenvalues, e;,, into the set of secularequa- tions,

i (t,o - els;;)c6,.: o, the linear coefficientscpi &ra evaluated but they are not in any way refined to self-consistency.It is important to note that trends in these linear coefficientsdepend in large part on the geometryof the tetrahedral ions modeledin our calculations.Finally, a Mulliken (1955) population analysis is cornpletedto obtain the desiredSi-O bond overlap populations,

n(Si-o) : 2 X N(/r) t I co,coiS,i, using the overlap integrals, ,ri. ,rrr"urt"to"*lr"n* and the number of electrons,ff(k), (0, 1, or 2) in MO rI,7,.The bond overlappopulations provide numdrical estimates of the strength of covalent bonding and anti-bonding betweentwo atoms. Two atoms are consideredto be bondedwhen the overlappopulation is positive,nonbonded when it is zero, and antibonded when it is negativel short bonds are usually involvedwith large overlappopulations and longerbonds with smaller ones.If observedSi-O bond lengths are used in the calculation of n.(Si-O),the shorterbonds, which usuallyhave larger overlap integrals, S,;y,will tend to have larger bond overlap populat'ionsthan longer bonds. As it is important to be able to discriminate betweenthis induced correlationand that inducedby the geometricalaspects of the tetrahedral ions, we have undertakentwo separatecalculations (1) using the observedO-Si-O and Si-O-Si anglesand clampingall Si-O distancesat 1.634 and (2) using the observeddistances and angles. By clamping all the Si-O at 1.&34,the correlationbetween z(Si-O) and the observedSi-O bond length induced by the overlap integrals is effectively removed and what remains should be related to the intrinsic electronic structure of the ions induced in large part by such 1586 GIBBS;HAM'IL, LOIIISNATHAN,BARTELL, AND YOW geometricalfactors as the o-si-o and si-o-si angles.Because of the very crude nature of the EHMO method intrinsic in'the very daring approximationsoutlined aboveand becauseof the utter neglectof the screeningpotential producedby the cotilomb and exchangeinterac- tions, little significancecan be attached!,o Lheactual' nu,rnbers obtained for zr,(si-o). However,it is the trends betweenthe calculatedra(si-o) and the exp,erimentallydetermined si-o bond lengths that ale con- sideredsignificant, not the absolutenumbers for any one tetrahedral ion. Thus, EHMO results can be useful in ordering and cl'assifging observedbond lengthswith bond overlappopulations but it cannotbe used to proue the observedbond length variations nor can it be used Lo pt'o,uethe participation of the 3d-orbitals of si in the wave functions calculatedfor the ions studied. The YOIP and orbital exponentsused as input to the program are given in Table 1. For the s- and p- orbitals the free-atorn optimized orbital exponentsof clementi and Raimondi (1963) were chosen together with voIP similar to the zero-chargevalues of Basch, viste, uoa Gruy (1965). For the 3d orbitals of silicon, an orbital exponent significantly higher than the Slater-rule value was adoptedfor reasons -5'5 discussedby Bartell et al,.(1970), and'H,s6,sa was adjustedto eV to yield orbital and overlap populationscomparable, in the caseof SiHa, to ab ini'tio populationsfound by Boer and Lipscomb (1969)' Theseparameters gave 3d populationsin sio+* which turned out to be of the same magnitude as Lheab initio poptiations reporbedsub- sequentlyby Collins e'tal. (1972). A fundamental weaknessof the extendedHiickel scheme (which is not based on u b.ormfi'd,e Ham\ltonian) is that its energiesand wave functions are sensitive t0 the somewhat arbitrary parameterization adoptedand, accordingly,have only a qualitative significance.On the other hand,the EHMO trendsin chargedistributions and bond popu-

T.cnr,n 1. Ver,pNcp Onsrrer, Ioxrzerron PornNtrar,s (VOIP) eNo OnsttaL ExpoNprrs (€).

(eV) $tom Atomic Orbital VOIP Oxygen 2s -32.33 2.246 2p -r5.79 2.227

Silicon .JS - 14.19 1.634 3p -8 .15 t.428 MO CALCALATIONSFOE SILICATE IONS 1587

lations in a seriesof related moleculeshave been found to be in- sensitiveto the exact parameterization;they are also roughly independent of whether the VOIP are frozen at plausiblevalues or &re adjusted to self-consistencywith aiornic charges (Basch et at., L}BS). Therefore, since the present investigation is a preliminary exploration of. trenils in a series of structures, it seemedadequate to choosethe simplestvariant of the extendedHiickel methodwith fixed VOIP and exponents.Finally, it is important to note that one of the biggest deficienciesof the extendedHtickel methodis its failure to take ade- quate account of the Coulornb,interactions. The method works the best for moleculeswhere the electronegativity difference,Ax, between bondedatoms is small. However,it starts to break down when Ax is greaterthan approximately1.3, the breakdownbeing completewhen L1a2 2.5 (Allen, 1970).Thus, when heteropolarsubstances with inter- mediatebond type (like the Si-O bond in a silicatewhere Ax = 1.2) are studied, EHMO theory may be used to qualitatively diagnosethe covalentaspects of the bonding.Accordingly, it is of particular interest to apply the EHMO method to silicate structureswhich have been found to conform remarkably well in the past with Pauling's rulesl for ionic crystals yet which also conform in many instanceswith a modelthat includescovalent bonding.

Tnp Rnr,erroNsurpBprwnuN Si-O (br) BoNo Ovpm,epPopur,errox ANDrHE O-Si-O aNo Si-O-Si ANcr,ps In a study of severalorthosilicates, Louisnathan and Gibbs (1g72a) have found that the tetrahedralvalence angles in hypotheticallydis- torted SiOetetrahedra have a pronouncedeffect on n(Si-O), the wider O-Si-O anglesbeing associatedwith bondsof larger overlap populations. They also showedfor a number of silicates (1g72b) that a correlationcan be made betweenthe mean of three O-Si-O angles, (O-Si-O)3, and the length of the Si-O bond commor to these three angles,shorter bonds tending to be associatedwith larger values of (O-Si-O)a.McDonald and Cruickshank (l967a) and more recenily Brown and Gibbs (1970)have also shownthat shorberSi-O distances are typically involved in wider O-Si-O angles.Although Baur (1g20) has also indicated that shorter Si-O distancestend to be involved in wider O-Si-O angles,his geometricalmodel of a Si atom rattling

lAccording to Bent (1968) Pauline's mles ma5.rhave a more universal applica- tion as structural principles than heretofore realized because when suitably ra- tionalized they may apply to covalent as well as ionic compounds, perhaps even to metallic compounds. 1588 GIBBS,HAMIL, LOAISNATHAN,BARTELL, AND YOVT

within a rigid"tetrahedron of oxygenspheres is not realizedin general in the silicates (Louisnathanand Gibbs, L972b). The Si-O-Si angles in silicates (L'iebau,1961) and in siloxanes (Noll, 1956) are usually between130o and 140o,although anglesas wide as 18Ooare not,uncornmon. Except for a,nangle of 93" in Si2Oz (Andersonand Ogden,1969), angles less than 120" are rare.A bonding model that includes Si (3d) orbital participation predicts that the length of the Si-O(br) bond shouldincrease as the Si-O-Si anglebe- comesnarrower (Cruickshank,1961). Moreover, as the Si-O-Si angle narrows,increasing electrostatic repulsions between the si-atoms and decreasings-character in the O(br)-orbitals (i.e.,a decreasingo-bond strength) are additional factors that are predicted to lengthen Si- O(br) (Cruickshank,1961; Brown et a1.,1969)- fn a re-investigation of the crystal structure of zunyite, Louisnathan and Gibbs (1972c)have found that the sizeof the tetrahedral angleshas a pronouncedeffect on zr[Si-O(br)] in Si,Ou- ions with linear Si-O-Si linkages.Thus, if {to(br)-Si-O(nbr)l > {[O(nbr)-Si-O(nbr)], then nlsi-O(br)l {[O(nbr)-Si-O(nbr)], then n'[Si-O(br)] = n'[Si-O(nbr)] whereas if { tO(br)-Si-O(nbr)l n[Si-O(nbr)] (seeFig. 2). These results calculated with all Si-O bond Iengths clamped at 1.634 imply that the magnitude of the tetrahedral angles plays an important role in establishing n'[Si-O(br)] in the Si-O(br)-Si linkage. In order to assessthe nature of the relationship between the Si-Oftr) bond length and Si-O-Si angle, we calculated n[Si-O(br)] as a function of Si-O-Si angle for the three Si,O"z6-ions shown in Figure 2. The results calculated with Si-O : 1.63A and a Si(sp) valence basis set are presentedin Figure 3a. Regardlessof the size of the tetrahedral angles,n'[Si-O(br)] increasesnon-linearly wil}r increasingSi-O-Si angle,the largestoverlap population being associated with the 1800 angle. Analyses of the linear coefficient, c6;, and the overlap, 8o;, matrices indicate that the changein z[Si-O(br)] is related to concomitant changesin both the o- and zr-bonding potentials. In other words, widening of the Si-O-Si angle increasesboth the o- and r-bond. strengths as proposed by Cruickshank (1961), Brown et q,I. (1969), and Brown and Gibbs (1970). Using results provided by the theory of atomic orbital hybridization, Louisnathan and Gibbs (1972c) have proposedthat the Si-O(br) bond length should vary as a function of -l/cos (Si-O-Si). When the n'[Si-O(br)]are replotted as a function of -l/cos (Si-O-S;), a well-defined linear relation (Fig. a) is realized. The trends predictedby EHMO theory are similar whetheror not the Si(3d) orbitals are includedin the calculation (Figs. 3a and 3b), MO CALCALATIONSFOR SILICATE IONS 1589

(o) (b) (c)

Frc. 2. A comparison of Si-O bond overlap populations (the decimal fractions listed adjacent to the Si-O(br) and Si-O(nbr) bonds) calculated for three Si:Oz6-ions (Da,, point symmetry with Ca along the Si-O-Si linkage) where (o) {[O(br)-Si-O (nbr)l > dlo(nbr)-Si-O(nbr)1, (b) *[O(br)-Si-O(nbr)] : d[O(nbr)-Si-O(nbr)] and (c) {[O(br)-Si-O(nbr)] < {lO(nbr)-Si-O(nbr)1. The calculation was made using a Si(sp) valence basis and with all Si-O : 1.63,i. Note that z[Si-O(br)] > n[Si-O(nbr)] for the conformation in (o), nlsi-O(br)l - z[Si-O(nbr)] for (b) and that n[Si-O(br)] < z[Si-O(nbr)] in (c). (See Louisnathan and Gibbs, 1972c). suggesting that the observed steric details of a tetrahedral ion in a silicatemay not be usedto establishthe involvementof the 3d-orbitals in the formationof the Si-O bond.This is in agreementwith Mitchell's (1969) assertionthat "one cannot,. . . expectexperiments to establish (really' whether d-orbitals are used anymore than experimentscan show s and p orbitals definitely occur in bonding of the first row elements"(see also Bartell et aI., l97A). Si-O Bono LrNcrn Venrerrolv rN Low-QuARrz, I"ow-cnrsrosAr,rrn,AND Copsrrp The silica polymorphsare possiblythe most appropriatestructures for examiningthe dependenceof Si-O (br) bond length on both the O-Si-O and Si-O-Si angles,because (1) there are no bonds other than Si-O(br), (2) all oxygensare two-coordinatedand bridge two tetrahedra,and (3) A((O) : 0. For our analysis,we have chosendata from three preciselyrefined structures,all with e.s.d.'sless or equal to 0.0064: low-quartz (Zachariasenand Plettinger,1965), low-cristobalite (Dollase,1965), and coesite(Araki andZolLai,1969)(Group A GIBBS, HAMIL, LOU ISNATHAN, BARTELL; AND YOW

(o) (b)

o7a

o50

oo?4 o o48 o I ./ @

of? o46 .t'

t40 ts si-o-si(.) si-o-si(.1 Fro. 3. Si-O(br) bond overlap populations calculated as a function of the Si-O-Si angle for the three pyrosilicate ions depicted in Figure 2. (a) Si(sp) basis with atl Si-O = 1.63A and (b) Si(spd) basis with all Si-O - 1.63A. Si-O(nbr) bond overlap populations for each ion increase only slightly with Si-O-Si and accordingly are not shown. in Table 2). The data for keatite and low-tridymite werenot included in the analysis for reasonsgiven in the next section.The three structures chosenprovide twelve individual Si-O(br) bond lengthsvarying from 1.598to 1.631A.Figure 5c is a scatterdiagram of Si-O(br) bond length versus (O-S,i-O)3 angle; tr'igure 5b shows Si-O(br) versus [-|/cos(Si-O-Si)]. Using the estimatedstandard error of the slopes for the lines fitted to the distributionsin Figures5c and 5b (seeTable 3), the null hypothesiscan be testedthat the true slopein eachcase is zero; that is, that the Si-O(br) bond length is independentof -1/cos(Si-O-Si). (O-Si-O)3 or Student's ltl calculatestn be 3.3 and 2.3 for the data in Figures5c and 5b respectively(Table 3). Since these values exceedthe 9O percent confidencelevel (c/. Draper and Smith,1966) of t(10,0.90) - 1.4,we canrejectthe hypothesisLhaLa relationship between Si-O(br) and -1/cos(Si-O-Si) or (O-Si-O)3 might not exist. We haveused the function [-l/cos (Si-O-Si)] rather than the angle itself becausez[Si-O(br)] vary non-linearly with the angle but MO CALCULATIONSFON ilLICATE IONS 1501

(o) o.50

F I o.cs o '=I ir O.48 c

o.47

t30 t40 r50 160 r70 t80 si-o-si (.) (b) o.50

I - 0.49 o '=I ;J O.48 c

o.47

t.6 t.5 t.4 t.3 t.2 t.l l.o -llcos(Si-O-Si)

Frc.4. Plots of ztSi-O(br)l as a function of (a) Si-G-Si angle and (b) -1/cos(Si-O-Si) where the bond overlap population was calculated for the conformation in Figure 2b and a Si(sp) basis set. 7592 GIBBS, EAMIL, LOUISNATHAN, BARTELL, AND YOW linearly with [-1/cos(Si-O-Si)] (Fig. a). However, even if the relationshipbetween Si-O (br) bond length and Si-O-Si angle is as- sumedto be linear,the hypothesisof no interdependencecan again be rejected at the 90 percent confidencelevel (seeTable 3). When the Si-O(br) lengthsare estimatedusing the interceptand coefficientsob- tained from a multiple linear regressionanalysis, using both (O-Si-O)3 and [-1/cbs(Si-O-Si)] as independentvariables, a reasonablecor- respondencebeiween the experimental and estimated distancesis observed(Fig. 5c and Table 2). Finally, two commentson Figures 5o and 5b are worth making: (1) the quantity [-1/cos(Si-O-Si)] may be a better definedvariable ihan (O-Si-O[ becausethe former includesa more accurateaccount of the hybridization characteristics on O(br) than the latter doesof the hybridization characteristicsof Si, and (2) the calculatedslope [-0.05(2)] for Figure 5b is statistically identical with that [-0.0E4(1)] obtainedfor n"[Si-O(br)] versus [-l/cos(Si-O-Si)] (Fig. ab), an interestingcoincidence de-

Table 2

truober' ' Sr-o(bt) bond rengths, s14(br)-sr angles, orygen tr€an .ooldldtlon ad A!(O) to. 3111c6 PolrrPhs, sUleates and Etldtne'

Group Ar Si-o(b!) bond dt.tsnce w. Sl{(br)-St sngte data to! slllc' qhere 'z 0 PolForPhe a((o)io 0 and

s1-0(b!) s1-GSt ;;;Gi t a<(o)

sl-o(r) 1.601 6045 146,8 t.r95l 20 0.0 Dorrade (1965) sr-o(2) r.608 1

r @00 2.O 00 sr(1)4(r) r.600 1.600 $00 (1969) sr(2){(2) r.6oa l. @8 145.E L 2392 20 O O Arakl and zoltar s1(r)-0(3) I:607 145-0 r,2204 20 00 s1(2){(3) r.619 I 611 sr(r){(A) r.598 150.4 1.1501 2.0 0.0 sr(2)-o(4) r.625 l.6115 sI(r)4(5) 1.63r 1!6. r 1.3874 20 00 51(2)-0(5) 1,511 I 5240

sr-o(l) 1.603 I 6095 r41.5 I 246C 2.O 0 0 Zacbrkden t?lettlnger sr-o(l)' 1.616 (1965)

Cloup Dr S1{(br) bond dl.tdc. v!. Sl{(bt)-Sl .41. d.t. for .Ulc.t.. ad .tlosE6 vh.r. ar(o)-0.0 tor au oqg€u

s14(4) L.627 l-527 rto 3 l.l5l2 2.8 0.0 &|'oald a crulck.Mnk(1967b)

s1-o(l) r32.e r.4690 2.1 00 rr.ch.r(1969) s1-o(t)l l:3i3 r.6re sr4(r) r.6052 1.6052 UO.0 LO 2.9 0 0 Prdtt (P.r.oul cd.)

Gtbb., E!cc& .od s1{r(2) r,0194 sr-or(r) l:?31 r.5e4s r6s.6 e.sh.! (196!)

sr-0r(2) r5e.! I 01t7 2.7 00 s14I(l) i:S r 5e4r (1963) H3S1-O-SIB3 1,6s. I 634 1a{.1 l.2t4t 2.0 0.0 rl4Dlg@ 4.r!.,

(q3)6(sro)3 1.63t 1.63t 13r.6 r.s62 2.0 00 ob.r@t(1e72)

(cB3)E(sro)4 L.622 L.522 144,0 1.223E 2-o o.o

(cH3)ro(sio)5 L 620 I 620 t46.2 r.2034 2.o o.o

(cff3)r2(sio)6 r,622 t.622 149.6 l.1594 zo o'o lbzs12o7 sr4(1) 1.626 1.626 1E0.0 l0 2.9 0.0 srolh r shepel.v (1970)

Er2Sr207 sr{(r) I 6t2 r 632 180.0 l0 29 0.0 MO CALCULATIONS FOR SILICATE IONS 1593

Tab]e 2 (cont.) Group Cr s1-o(br) bond dlrtance v. s1{(br)-S1 aqle data tor elltcare! aher€ o(br) 1. ruo coordl@ted

(1970) 8"3s14&5o25 sr-O(r) 1,599 r.599 t6O,O l.mOO t 7 -.lO sbmoo 6!d et. Lw rbrte -'04 Ese re'4" (re6e) 3ll[3i:3:[:] t:8li r.610 16r.2 1.0564 2.s :ii[:]-8S[:] i'.211 r.6185 B'.7 L.3et2 2.s ::3X LN Cordlerrce st(3)-o(6) r.603 .02 ctbb. (t965) sr(4)_o(6) 1.620 .o2

tudd& cordlerltG oz hsher (1e67) :l[i]*[:] i.3li r.5o5s Lts 5 r @oo 2.7

Danburlrc si{(l) 1.614 1.614 r36.E t.37l8 X.7 -.25 phrulp. !.!.f!., (f97r) KlsuaroPlrte -:33 ** !!'q!" (re67) :1[l]1[1] i:331 1.644 r3r.e L.4st4 1.4 *"lll#,,." 6d !r''v (re64) 3il[;]:3:3i i:133 r.63r rs'8 r.5304 2.s ::ff "* :i?[:i:333] i:3i3 r'6125 r55'e r'@$ 2.e -'.31 (lnort€ hutid (reIr) 3i[i]:3[i] i.z:rt r'534 r4o.r 1.303s 2.1 -'os EPidrdFlt€ 'o** 'rd !'4 (re7o) :1[li:3[1i i:33i r.617r r5r.5 r.r37e z.E ::3i s1(r)-o(r) 1.66 r.06 143.E L,2392 .O s1(2)_o(6) r.629 r.oz9 143.0 L,z52L .07 slG){(E) 1.633 1.613 8t,O r.3250 ,Ol

krltullr. s1(r){(7) 1,621 1.521 141.1 r.2850 3.2 -.13 H&bU !!.q!,, (r9tr) c'l61Eat6rt. _.03 s1(r){(7) I.613 1.513 ta2.2 L.2622 3.0 -:33 :l[i]][;i i:!ii t.5z6s r3e.7 r.3rr2

ctun.rltc s1(I){(7) -.@ (1969) 1.613 1.613 r44.! L.223E 3.0 -:31 rr{.r :|[]li[;i i:lli r.5re 142.4 1.2622

1r@llr. sr(I){(7) ;.8 1.616 r.616 r39.! r.3l9o 3.2 r.ptb !!.!1., (1969)

Gbucopbo. Sr(1){(7) 1,6t1 l.6rt L17.2 t.IA97 3.2 -.tl

c-c6te!.d h- s1(r){(7) I 616 I 616 j.O -.03 c@ltrst@lt' lat.O r.2m -:3; :l[i18] i..232 r.628 r&.0 r.ss

?rtdtlve b- Sl(r){(7) 1.626 t,52A -.03 c@ln8toa1t. t39.I 1.32S 3.0 -:83 :l$lil[;] i:31 t.622 r3e.8 r.sez 51(r)s(7) t.@3 1.603 l4l.O t.a$ -.03 -:B: :i[]1i8] i.fll r.6345 136.r r.3373

protryhlbot. s1(r){(7) 1.524 t.624 137.1 1.3651 3.1 -:Bl-.Ot cibbr (1969) :l$11[ii i.!ll r.62r 140.6 r.2$r hthophylllte si(I){(7) 1,fl5 r.6$ l4l.a L.27% 3,2 -.31-.07 rlq.r (t970) 3i$ifl[:] i:fli r.6,6 140.6 r.2e6 sr(t){t(7) 1.6t7 r.617 t3r.g 1,3a99 -.Il -'03 brb'c (re6) :i[ili[3] i:|?i r.6275 r&.3 i.orar 3.5 ,",.-"""il;i;fi ;fi :.,.:,,. ::,- .; .::: :::.. ,::,,

spite the fact that all n(Si-O) were calculated assumingall Si-O equal 1.634 and all O-Si-O angles: 109.47"in Figure 4.

Tnu Connpr,erroNBnrwnnu Si-O (br) BoNo LnNcmr eNo Si-O-Si Axcr,p rN Srnucrunns WHpnn a((O) rs ZERo In structures of low-quartz, low-cristobalite, coesite,hemimorphite, thortveitite, beryl, emerald, benitoite, YbsSizO,T,Er2SisO7, and five siloxanes,all the oxygen atoms receive the same bond strengttr, i.e. 1594 GIBB!, HAMIL, LOAISNATHAN, BARTELL, AND YOW

(o) (c) r64 r64 a a a = - 162 'ez o a' ! ooo I a 6 U' 1I o t. r60 at to a t60

r50 t60 (o-si-o)t si-o-si(") (b) (d) r64 r64 a. a 3 t62 o ! ,ez a a o o I lo a (,) t60 oo t60 a | 60 | 62 1.64 t4 r5 t2 to si- o(.st.l(l) -llcos(Si-O-Sil

Frc. 5. Scatter diagrams of si-o6r) bond length data from Table 2, group A, plotted as a function of (a) Si-O-Si, (b)-1lcos(Si-O-Si)' (c) (O-Si-O)s: the average of the three o-si-o angles involving a common si-o bond and (d) si-o(est.)A using the results of a multiple linear regression analysis with -1rlcos(Si-O-Si) and (O-Si-O)" as independent variables'

((o) = 2.0 and a((o) = 0.0.The si-o(br) bond lengthsand si-o-si anglesin thesestructures are listed in Table 2 (GroupsA and B)' The scatter diagram of Si-O(br) versus Si-O-Si angle and [-l/cos(Si- O-Si) ] are given in Figures 6o and 6b, respectively.For lhe 27 individual si-o(br) bond lengthsin this data set,the hypothesisthat the true slopes of (l) Si-O(br) versus Si-O-Si arrgle and of (2) Si-O(br) versus [-l/cos(Si-O-Si)] are zerowas t'ested.The calculated intercepts,slopes, partial correlationcoefficients, and Itl- values are given in Table 3. The calculatedltl- values ate 2-5 and 3.6 for Figures6o and 6b, respectively.Because these lfls exceedthe critical value of ,(25,0.90) - 1.3,we can reject the null hypothesisthat Si-O (br) bond length and Si-O-Si angle might not be correlated (actually 2.5 alsoexceeds t (25, 0.90) ) . This resultis contraryto Baur's (1971)conclusion that a correlationcannot be madebetween Si-O (br) and Si-O-Si anglefor silicateswhere a((O) : 0. He reachedhis conclusion using data from keatite, high-tridymite, low-quartz, low- MO CALCULATIONS FOR SILICATE IONS 1595

The correlations presentedin our Figures 6a and 6b are somewhat improved (seeTable 3) when the data from ybrSizOT,Er2Si2O7, and

Table 3

Ln[ercepts, a^r slopes, brr and correlation coefficieDts, r, for.]-inear regresslon eqdatlons fittad ro dara in TabLe 2 and Flgs, 5-7. ltls calc. = for Ho:Bl O; all ltl's exceed a two-slded 902 1evel test

Group A of Table 2

bt t Itl sanple size

S1-0(br) bond length vs. S1-O-S1 angle for slllca L,677 -0. ooo4(2) I -0 .48 1.8 polynorph data (Fig, 58)

Sl-O(br) bond length v6. -1lcos(Si-0-S1) for sllIca 1.539 0.06(2) 0.57 2.3 pollnorph data (Flg. 5b)

Si-O bond Lengtha vB, a for s1l1ca 4,094 -0.023(6) -n 77 polynorpH dats (Ftg. 5c)

S1-0 bond length ve. SL-o(est) for sLllca -0.168 1.1 (3) 0.76 3.7 pollnorph data (F+g. 5d)

Groups A and 82 of Table 2 -0.0009(2) S1-0(br) bond length vs. rL. tq) -o,72 4.8 24t Sl-o-St angle; Flg. 6a I. O6J -o,oo04(2) -0,4s 2.s 27'

S1-0(br) bond length v€. 0.08(1) 0.7s 5.3 24't -Ucos(sl-O-Si) {1.516'1.545 ; Flg. 6b 0.06(2) 0.58 3,6 27'

Groups A, B, and C2 of Table 2

Si-O(br) bond length vs, -0.0005(1) -0,51 5.1 78r 11,7008'1.584I S1-0-Sl angle; Fig, 7a -0,0004 (1) -0.44 4.3 81',

Si-O(br) bond length ve. 0.0s3 (9) 0.53 5.5 78r 11.554'1.561 -1lcos(Si-o-Sl); Fle. 7b 0,047(9) o.49 5.0 81'

re,s.d,rs are glven 1o parentheses and refer to the last digit. 2The data fron thortveitite, Errsiroil and ybaslroT rrere not uaed to obtaln the atatlstlcs glven in the flrst iow'of thc dati sEtb enclosed by braces (see text). 1596 GIBBSTHAMII" LOU-I&NATHAN' BARTELL, AND YOW (o) t.66 'a t64 a- o \o t o .cl --.1 ' r.62 a o I t-a- -- a taa\- \.-= .O r.60 _ _l

a r.58 r30 t40 r50 160 170 r80 si-o-si(") (b) r.66

r.54 \-o -- - .o --o-\ L o -o ao 1.62 \. !r' o a -O\ I oa o-'-r 6 ' - -O . --tf--\. t.60 a ,. a t.58 1.6 1.5 t.4 1,3 t.2 l.l t o -llcos(Si-O-Si) : , Frc. 6. Scatter diagrams of Si-O(br) bond'length data from Table 2; groups A and B, plotted as a function of (a) Si-O-Si angle and (b) -1lcos(Si-O-Si)' The dashed line in (b) is a least aquaxesline. The dashed line in (a) is trans- generated from (b). The open circle plots are data for thortveitite, Ybosiroz and ErgSLOn. . MO CALCALATIONSFON SILIAATE I.ONy I8S7

thorbveititeare excluded(open'symbols in Figs.,6aa,nd.'6b). In-these

Rm,errvp ConrnrsurroNs ol. -1/cos(Si-O-Si), a((O) exn (Cff) To rHE Venr,lnrr,rryoF TrrESi-O (br) BomuLnxcru As' a((O) and the. mean coordination number, (CN), of oxygen bonded to silicon have also been suggestedas determining factors of the Si-O(br) bond length (Baur, lg7l), an examination oieach in the

The results are listed in Table 4 in the order ranked by the stepwise

Table 4

Analysls of varlance: Multiple linear regression analysls for the data used to prepare Fig. 7 wlth Si_O1try bond length as the dependenr variabte; t(74, O.9O) = 1,7, The order of the iodependent variable was found by scepvise reglessioD oethocls,l

Addeal Regres6ion fndependeot Varlable ltl Partial r Su of Squares Sdple Slze

-1lcos (Si-O:Sl) t1'o'6.4 0'63 0,0043 0;59 0.0038 81',

AE (o) t4'2 o.44 0,0020 78r 0.41 q..0020 81, F,2 U.J) 0.0005 78, 0,34 0.0006 81'

llhe data from thortveitite, ErrSi,Oa, and yb^Si"o, were not uaed to obtaln the stat.lsties glven ln the flrst iow'of the datf s6tC e.rclosed by braces_ (see text). 1598 GIBBS,IIAMIL, LOAISNATHAN,BARTELL, AND YO'W regressionmethod (c/. Draper and Smith, 1966).The ltl- valuesin- dicate that all three variables contribute significantly to the varia-

a random sample in the sensethat the data are omitted for structures where A((O) is relatively large and where O(br) is coordinatedby more than two atoms. Therefore the regressioncoefficient calculated for our data probablywill not apply to an expandeddata set.Howevet, sinceA((o) and actual coordinationnumber of o(br) are moderately correlated,we cannot,conclude a priori whether the magnitude of the slope associatedwith (cN) will be larger or smaller than 0.006 or whether (cN) witl make a significant contribution to the regression sum of squaresin the presenceof a((O). Plots of Si-O (br) bond length versus Si-O-Si angle for a wide variety of silicates (Cannillo et a1.,1968; Brown and Gibbs, 1970) exhibit the sametrend as Figures7,a or 7b (seeTable 3) irrespective of the coordination number of the bridging oxygen' Accordingly, it appearsthat the length of the si-o (br) bond is related in part to the $i-O-Si angle regardlessof the value of a((O) or (CN), with the shorter bonds being involved in the wider angles.Although this result is consistentwith a covalentmodel that includessi (3d) orbital participation, one cannot concludeas indicated earlier that these orbitals are definitely used in the compositionof the si-o bond (Bartell et al., 1970). Tnn Rnr,erroNSHIPBntwnnN n(Si-O) ANDTHE OssunvED Si-O BoNp LnNcrn As both O-Si-O and Si-O-Si angles appear to exert a pronounced role in establishingrz(si-o), we calculatedthe bond overlap populations for isolated and polymerized tetrahedral ions in ten silicates (Table 5) using the observedO-Si-O and Si-O-Si anglesand clamping all si-o at 1.63a.The calculationfor a valencebasis set consisting of silicon 3s and 3p and oxygen2s and 2p atomic orbitals yields bond overlap populations which, when plotted against the observedsi-o distances(Fig.8), show an apparentlinear relation with longerbond lengthsbeing assoeiatedwith smaller overlap populations.As expected, (o)

t.66 a o

t.64 o o 3 r.ez o I . an o foo . a a o ao a t.60 a I

r.58

r30 r50 160 r70 r80 si-o-si(") (b) r.66

t.64

ll 1.62 o I 6 r.60 t

a r58

t6 r5 t4 t3 1.2 tl t.o -llcos(Si-O-Si)

Fra. 7. Scatter diagrams of Si-O(br) boad length data from Table 2, groups A, B and C, plotted as a function of (a) Si-O-.Si angle and (b) -1lcos(Si-O-Si)- The dashed line in (b) is a least squares line. The dashed line in (a) is trans- generated from (b). The open circles are data for thortveitite, Yb"StOr and Er"SLO". The reported e.s.d.'s of ihe Si-O bond lengths for the structures given in Table 2 are all less or equal to 0O1A. 1600 GIBBS, HAMIL, LOUISNATHAN, BARTELL, AND YOW

Table 5

an b..t. .!d b..!t b!' !.!:!,(lrl lLeLit t ="* t* .4!s r1-0(l) 1.66 o.(8! 0,4t5 "=","tO,rLl 0.734 h!ll, c!!b. .nd ilbba sr{(r), l.t4t o.as 0.478 0,155 0,739 (r97r) si4(2) 1.517 0.t06 0.515 0 942 0,955 sa-0(3) 1,500 0,511 0.530 0,946 0,97E

st-0(t) |,626 0.5@ o,922 si-o(2) 1,63r 0.505 0,93! C!!rct.h.d (1967) sr-o(4t Le21 0.495 o.?56

mortv.lrtr. sl-o(l) t.60t 0.494 0.506 0.?61 0.115 st-0(2) t,627 0.501 0.t0I 0.9?t 0,925 sr-0(t) l a28 0.!03 0.502 0.926 o.926

tosilltn. sl-0(4) 1.623 o.{91 o.(i2 o ?5E 0,161 Ttpp. .Ed fldllton sr_0(t) l.a2E o,a6E 0.405 0,7t1 o,752 (19?1) st_0(6) 1.616 0,510 0.515 0,941 0.960 ar_0(7) r,st 0,50E 0.t2t 0,944 0,96!

hrrld Sl-O(r) l.6s 0.505 0.5rr 0,tt8 0.7E2 st-o(l)l l.t8t 0,50E 0.t30 0 742 0,8r5 h8h.! (1968) 8r-0(2) 1,621 0.50! 0.503 0,9!9 0,912

l.ryl sr-o(I) l.!94 0.505 0.t22 0,rr7 0, t95 sr-0(I)l t.59t 0.50t 0.51t 0,182 0 800 b.ib.r (1968) st-0(2) 1.620 0.t0! 0,t0{ 0 9t9

3r-0(r) r.603 0,56 0,t16 0 1A2 0.78t acbrt...r rld sr-o(3) 1.6t6 o;til 0.t06 0,162 0,7aE Pl.rcrnr.r (1965)

sr{(l} r.6x 0.4s 0.499 o,?75 0,162 lt.ch.r (!969) s!-o(r)l 1.646 0.4t6 o.a?l 0.7tt o tx' 8r{(2} 1.60t o.!or 0.521 0,9{o 0 9n{

sr(r){(r) 1.507 0.50! 0:531 0,941 0.98t lr.d .!d rx.o! (rt69) st(r)-o(2) l.a2l 0.509 0 514 0.940 sr(r)-0(3) I 6t6 0,464 0.4t0 o,ttt o 142 sr(r)-0(a) L,612 0 4!6 0,165 0.761 0,7t0 sr(2)-0(4) t.6t9 0,115 0.41! O 1Lt 0. trt sr(2)-0(5) 1.596 0.52r 0.545 0,959 0.999 s1(2){(6) I 606 0.506 0.523 0.934 0 960 s1(2){(7) 1.460 0,411 0.454 0.741 0 7l! sr(3)-0(r) 1.678 0.48! 0.462 0.t45 o,t2t 51(l)-0(6) 1.517 0.512 0.t23 0.9{t 0.960 st(3)-o(9) r.595 0,501 0.t16 0.919 0.98t st(t)-0(3) 1.698 o.att 0 4(5 0.751 0 69!

sr(r)-0(l) l.6ot 0.521 0,96! rrol.r (1969) C.st0l sI(r)-0(2) 1.603 0 52! o 964 sr(!)-o(3) 1.503 o.462 0 703 s1(l)-0(9) 1.626 0.488 0.17L sr(2)-0(r) 1.701 0.693 s!(2)-o(.) 1.589 0.549 1 00r sr(2)-0(5) t.609 o 522 0 959 sr(2)-0(a) L65t 0 469 0. t18 s!(3)-o(5) 1,669 0.461 0,724 sr(r)-0(7) t.600 o 525 0 97r s!(3)-0(6) I t5l o 555 LOl6 sro)-o(9) 1 561 0.45! 0 tor

sr(2)-0(2) 1,504 0.521 b 950 Donily rtrd Allqn sr(2)-0(3) L6lr 0.499 0.918 (1968) sr(2)'0(8) t.649 0.451 0.t02 sr(3)-0(6) t.6tl 0.515 0.951 sr(3)-0(3) L,641 o.a6E 0 723

Fo.rt.!tr. sr-O(t) 1,614 0 515 0.525 0 935 0.952 8106.d Cllb. (1972); sr-o(2) 1 654 0,400 0,4t0 0 878 0.862 but.n.rh.n .nd c1bb. sr-0(3) r.615 0,469 0 487 0.898 o a90 (r972.)

D.tolrr. sr-o(l) r t70 0.506 0.545 0,918 o 98t Fort, st-0(2) Phlutp. .nd Orb6. r 648 0.{93 0 484 0.900 o €65 (rn Dr.pr..!ron) s1-0(!) r,551 0,492 0 4!2 0.899 0sl sr-o(4) 1.661 0,487 0,{69 0.890 0.861

the overlap populationsobtained using observedtetrahedral angles and bond distancesin the calculationsshow a similar but better de- velopedtrend when plotted againstthe observedbond length (Fig. 9). - Theslopes,intercepts, corelation, coeffi cients, and It | statisticscalcu- lated:for the data in"Figures 8 and g are given in Table 6 and Show that both trendsare highly significant. The curvesof Bartel"let at. (1970)which relatebond lengthto bond overlap population are similar to those obtained.,inFigures 8 and 9 t70

WITHOUTd ORBITALS; o r.69 Si-O = 1.53A

tr: Morgorosonile r68 A Beniloile D Bcryl V Emerold t67 O Quorlz o Tourmoline c 0ioplose r66 ,O Thorfveilile + Forslerile x Dololile r65

164

E r,63 I o v -l- | 62 U'

t6l

r60 >o 159 T v r58

r57

r56

r55 o47 048 049 o50 051 o 52 0.55 n(si-o) Fro. 8. Scatter diagram of Si-O bond length data plotied against; z(Si---O), calculated using observed valence angles, assuming'Si-O = !63 A; aud excludilg Si(3d) orbitals for the minerals Iisted in the upper right. Si-O(br) bonds aie iq dicated by open symbols and Si-O(nbr) bonds by bolid ones.'Data takbn from Table 5. GIBBS, HAMIL, LOAISNATHAN, BARTELL, AND YOW

l?o

wrTHOUt d OREIIALSi t69 Si-O (obe) o t6t s

O Nigh 9r.nur. CaSiOr r57 o

t66

d\, t65 - -" ..\

164

o o o r6s "o o o t62 o\.'

l5l OeO t60 ' DD \ - r59

r56

r5c

n(Si-Ol Frc. 9. Scatter diagram of Si-O bond length data plotted against n(Si-O)' ealculated using the observed distances and angles and excluding Si(3d) orbita.ls for the minerals listed in the upper right of this figure and Figure 8. Si-O(br) bonds are indicated by open symbots and Si-O(nbr) bonds by solid ones. Data taken.from Table 5. suggestingthat the overlap populations calculated for the sulfates' phosphates,and the silicatesare comparablewhen the 3d basisfunc- tions are neglected.Correlations can also be made betweenthe Si-O bond lenglh, the net non-bonded geminal populations (c/. Barbell et o,1.,1970), and the net chargedifierences on Si and O; however, inasmuch as the non-tetrahedral cations were not included in the calculations,such correlations are not appropriate here. When the valenceorbital basisset of silicon is extendedto include the five Si(3d) orbitals, the EHMO calculation gives n,(Si-O) that again correlate with observedSi-O bond lengths but as two separate trendswith the overlappopulations for the Si-O(nbr) bondsexceeding those for the Si-O(br) bonds by about 20 percent (Figs. 10 and 11, Table 5). The overlap populations calculated using the observed MO CALCULATIONS FOR SILICATE IONS 1603 tetrahedralangles and all Si-O = 1.63Aare plotted againstSi-O(obs) in Figure 10 whereasthose calculatedusing observeddistances and anglesare plotted againstSi-O(obs) in Figure 11.Thetrends in both -values scatter diagramsare highly significantas evincedby the ltl (Table 6). The correlationsare much better developedin Figure 11 wherez(Si-O) were calculatedusing the observedSi-O distancesand angles.In addition,the slopescalculated for the Si-O (br) bond trends (Figs. 10 and 11) are steeperthan thoseof the Si-O(nbr) bonds.The Si-O(br) bondtrends for the data givenin Figures8 and 9 alsoappear to be steeper,indicating that the Si-O(br) and Si-O(nbr) bond overlap populationscharacteristically constitute two distinct populations evenwhen a Si(qp) valencebasis set is used (Table 4). The difference calculatedin n.(Si-O) for the Si-O(br) and Si-O(nbr) bondsis due in

Table 6

Intercepts, a^, slopes, bil and correlation coefficients, r, for linear regresslon e{uatlons fitted ro dara ln Flgs. 8-11; for tt = o ltl o:Bl

o bl l.l observed S1-0 bond length vs. n(Si-O) calc. f,or Si-0 = 1,63fr; 2.763 -z.zdL -0.84 9,I F1g. 8

Observed S1-0 bond length vs. glsl-0(br)1 calc. for Si-O = 3,025 -t 9., t -0,89 8.3 1.63; Flg. 8

Observed Sl-O bond length vs. nlsi-o(nbr)l calc, for S1-0 = -L,tt6 -0.77 5.3 1,63 ; I'tg. 8

Observed S1-0 bond length vs. n(Si-o) calc. for obs. S1-0; 2.L39 -7.024 -0,94 20.0 Fig. 10

Observed S1-O bond length vs. ntSt-O(br)l ca1c, for Si-O = 3,220 -2 .O76 -0,77 6.0 1,63; Fig.8 observed S1-0 bond length vs. n[Sl-O(nbr)] calc. for S1-0 = -0.816 *0.70 5.5 1,63; F1g.8 observed Si-o bond length vs. nIS1-0(br)] calc. for obs. 2.3L9 -0. 906 -0.94 13.4 si-O; F1g, Il observed Sl-0 bond length vs, nIS1-O(nbr)] eale, for obs. -0.548 -0,94 L0.2 st-0; Fig. 11 1604 GIBBS, HAMIL, LOAISNATHAN, BARTELL, AND YOW

t72

r70 WITH d OREITALS; Si-0. | 6ll

j! r64 o o I t62 v, o o t60

r58

r56

154 o74 076 078 080 082 084 0S O8a O90 092 094 096 098 n(si-o)

Frc. 10. Scatter diagram of Si-O bond length data plotted against ??(Si-O), calculated using observed valence angles, assuming Si-O - 1.68,4.and including Si(3d) orbitals for minerals listed in upper right of Figure 8. Data taken from Table 5.

part to the very different environmentsimposed on these bonds by our modeling of the tetrahedral ion in each silicate as an isolated molecule.According to Bartell et aL. (1970),part of the differencemay also be steric and part may be due to electrostaticforces not reflected in the bond overlap populations. The a(Si-O) valuesused to prcpareFigures 8 and 10were calculated with the Si-O distancesclamped at 1.68Abut with observedO-Si-O and Si-O-Si angles.Clamping all Si-O distancesat l.68A may elim- inate the bias inherent in using the observedpositional parameters, but it can introducebias becausethe bond lengthsare not allowedto respondto differencesin the bond overlap populationsestimated by Hiickel theory. Moreover, for many complexpolymerized SiOa ions, the Si-O bond lengthscannot, always be clampedat 1.68A and still retain the observedangles and the continuity of the polymerizedunit at the sametime. In thesecases, it may be possibleto obtain a rea- sonableestimate of the ra(Si-O) by using the observeddistances and anglesin the calculations.This is becausewell-developed correlations exist betweenn (Si-O) ealculatedfor Si-O = 1.68Aand those calcu- fl"t"i

MO CAI,CULATIONS FON yLICATE IONS 1605

ht)

qENF

sE @

o= OP o .E; -x r XQ c 6f o ar D .9tr o b0A 8 EE al, Oaa J YA F- >,r4 E8 B HS EO +. o .od o 9l tO oQ ts d .Eq ' xd a !E 6 ;i6 o 3co

*Q. o/ a: q b.- o ?€

D o/o o

9a It-^ts oJ o vF fhv 'Ho 0 P 0 o ct :S 0 dai - uor,!.i $ Fd o€

0 h o .P iEg_EEq_EiEi o (sqo)o - !s H d 1606 GIBBS,HAMIL, LOAISNATHAN,BAR,TELL, AND YO'W

Iatedusing the observedpositional parameters (Fig. 12) (seeCameron, 1971).

Concr,usroNs Allen (1970) has implied that EHMO results may be useful in determininga relationshipbetween bond overlappopulation and bond length relations like that obtained for the hydrocarbons(c/. Schug, 1972) but this is Lhemost he believesthat it can do. Our calculations appearto have been successfulin demonstratingsuch a relationship and suggestthat at least part of the Si-O bond length variationsob- servedin the silicatescan be rationalizedin terms of a covalentbond- ing model. Our results also imply that the steric details of a silicate cannot be usedf,o proue the participation of the Si (3d) orbitals in a Si-O bond formation becausesimilar structural trends are predicted when they are omitted and a valencebasis set is used.Nevertheless, a covalent bonding model including d-orbital participation permits an understandingof the X-ray emission(Dodd and Glen, 1969;O'Nions and Smith, 1971;Collins et aI., L972) andfluorescence spectra (Urch, 1969,1971; Collins et aL,,1972)and the bondpolarizabilities (Revesz, 1971). In addition, it gives a reasonablygood accountof the trends in Si-O bond length variations. Consequentlya covalent bonding model that includes Si(3d) orbital involvement should not be dis- missedas chemicallyinsignificant. Since EHMO theory fails to take explicit accountof electrostaticeffects, the seniorauthor and his stu- dents are presentlyundertaking MO calculationsat the next level of sophisticationwhich involve the inclusionof the two electronintegrals and refi.nementof the linear coefficientsto self consistencyas em- bodied in the CNDOI2 approximationof Pople, Santry, and Segal (1965). These calculationsdo take ordinary Coulornb interactions into account.It will be of interest,therefore, to comparethe CNDO/2 results with those obtained in the present study as well as with those obtained by the ab iruitio method. Finally, the ultimate goal of any bondingtheory for the silicates is to provide some insight into the physical laws that govern atom arrangements,physical properties and stabilities. Molecular orbital theory in its simplest form has been used with moderatesuccess by the organicchemist to rationalizereaction mechanisms, spectra, bond length variations,and shapesfor a large numberof organicmolecules (Streitwieser,1961). The trends obtained in our study suggestthat extendedHiickel molecularorbital theory, unlike the extendedelec- trostatic valencerule, may be used to classify and order the bond length variations in the tetrahedralportion of a silicate when A((O) o54 o

o52

o48

o46 o.48 050 052

o80 lf) (D b

I o78 a o O1 ! o76 o o

J a' o o74

U) c o72 070

o96 c

a o,94 a laa .).-'

o92

o 88 0,90 0.92 0 94 096 098 rOO n(Si-O66e) Frc. 12. Bond overlap populations, rr(Si-O), calculated with Si-O = 163 A us. ra(Si-O) calculated with observed distances, both with observed valence angles. (a) For Si-O(br) and Si-O(nbr) bonds without Si(3d) orbitals; correlation coefficientr - 0.94 (Data from Figs.8 and 10). (b) For Si-O(br) bonds with Si(3d) orbitals; r := 0.90 (Data from Fies.9 and 11). (c) For Si-O(nbr) bonds with Si(3d) orbitals; r '= 0.90 (Data from Figs.9 and 11). 1608 GIBBS,HAMIL, LOAISNATH.A.N,BARTELL, AND yOW

- 0. Moreover, similar calculationsby Urch (1971) have provided valuable insight into the L2,s X-ray fluorescencespectra recordedfor silica glass. As the rewards are great, earth scientists are urged to considerthe use of MO theory in their interpretation of spectra, physicalproperties, order-disorder mechanisms, and phasetransforma- tions of minerals.Even if applicationof the theory provesonly moderately successful,it should, for example, improve our ability to evaluate evidencebearing on the physical conditionsthat prevailed when a mineral or a mineral assemblageforrned.

AcxNowtgocupNts We are grateful to Drs. G. E. Brown, Jr., R. G, Burns, J. R. Tossell, and D. R. Waldbaum for reviewing the manuscript and making a number of helpful suggestions.Professors F. D. Bloss, D. W. J; Cruickshank, P. H. Ribbe, and J. C. Schug, and Dr. M. TV'.Phillips are thanked for very helpful discussions and for critically reading the manuscript. G.V.G. gratefully acknowledges the National Science Foundation for its generous support in the form of grants GU-3192, GA-L?D?, and GA-30864X, and Virginia Polytechnic Institute and State Uni- versity for providing funds to defray the relatively large computer costs in- curred in the study. M.M.E. and SJ.L. gratefully acknowledge the NSF de- partmental $art (GU-8192) support in the form of research associateships.

Appor.rrrx Stepwise regression analysis is a systematic and objective procedure for recognizing those variables that contribute most significantly to the variation in the response,regardless of their actual entry points into the model. As a first step, a correlation matrix is computed and the va.riable most highly correlated with the response variable is used to compute a linear regression. The second variable to be entered into the regression is the one whose pa.rtial correlation with the response is the largest, i.e., the one that makes the greatest reduction in the error sum of squares. Next, the method examines the contribution the first variable would have made if the second one had been entered first and the first variable second. If the partial F' of the first variable is statistically significant at some preselected confidence level, the fust variable is retained; otherwise it is rejected. Assuming that both the first and second variables make a significant contribution, the method selects the next variable to be entered, i.e., the one that now has the largest partial correlation with the responsegiven that the first and second variables a,re in the regression. A regression equation is calculated with all three variables and the third variable is tpsted for significance. Moreover, partial F' tests are again made for the first and'second variables to leanr whether they should be retained in the regression model. If their partial F's exceed the preselected level, they are retained; otherwise they are rejected. The procedure is continued and the next most highly correlated variable is entered into the regression. Significance tests a.re made as before and the method continued with each variable incorporated into the model in a previous step being tested. The method is terminated when all the variables have been considered and no more are rejected (cl. Draper and Smith (1966), p. 178-195for numerical example). MO CALCULATIONSFOR SILICATE IONS 1609

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Manuscrtpt recei,ued,Janunry 11, 197P; accepted,tor publi,cati,on,Jw,e 6, 1n2.

Note add,eil i,n proof ba GVG: The electrostatic valence rule proposed by Pauling (1929) and extended by Baur (1970) was originally.conceivdd to chaftirc- terize strengths of ionic-type bonds and local charge balance in stable iohic crystals. It is evident, however, that the model seems to apply equally well to compounds in which the bonds have a relatively large arriount of covalent character. For these compounds,Pauling (1960,S4Z-548, Foohrote 64) has indicated that "if the bonds resonate among the alternative positions, the valence of the metal atom will tend to be divided equally among the bonds to the coordinated atoms, and a rule equivalent to the electrostatic valence rule would express the satisfaction of the valences of the nonmetal atoms.', Pauling,s statement implies [1] that the strength of an electrostatic bond, s, can be equated with bond number, rz (Pauling, 1947,J. Amer. Clrcm. Soc. 69, 542-559); t2l that the correlations between f (O) and the length of the Si-O bond (Smith, 1953,Amer. Mineral. 38, 643-661; Ba-ur, 1970) are similar to the well known bond-length bond-number curve for carbon-carbonbonds; and t3l ihat the chargeson bonded atoms are small in agreement with his electroneutrality principle (Pauling, 19ffi, General Chemistry, W. II. Freeman and Co.,286-287). The inference that z and s are equivalent is consistent with the observation ihat f(O) and z(Si-O) are inversely correlated (Louisnathan and Gibbs, l972b). Although Baur calls his model the extended electrostatic valence rule, he is careful to note that the correlation between observed Si-O bond length and f(O) could be interpreted in terms of Cruickshank's (1961) double bonding model as well as the Born model for ionic crystals.