silica bond lenght

7/28/2019 Silica bond lenght

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

CORR.ELATIONS BETWEEN Si_O BOND LENGTH, Si-O-SiANGLE AND BOND OVERLAP POPULATTONSCALCULATED USING EXTENDED HUCKELMOLECULAR ORBITAL THEORYG. V. Gmns,M. M. Hernr,l aNDS.J. LoqrsxlrHaN,D ep'artmentof Geoto'Eiaa.l ciences,Virginia Polytechnic Institute ond State Uniuersitg,B acksburg, V r girui,a406

ANDL. S. Benrcr,r,ANDHsruKANcYow,D epartnzentof Ch,emistrg,Uniu ersity o,fMiehig an,Arm,Arbor, Miahigo"n4810!AssrRecr

Extended Htickel molecr:lar orbital calculations have bee=nompleted fof isolatedand polymerized tetrahedral ions for ten bilicate mineral! in which a(O) s 0.0using ob.servedO-Si-O and Si-O-Si valence angles, assuming all Si-O bond lengthsare 1.63A and including silicon 3s and 3p and oxygen2s and.2p ,tdmic orbitals as thebasis set. The resultant Mulliken bond overlap populations, z(Si-O),,correlate withobserved Si-O bond lengths, the shorter bonds being associatedwith the largern(Si-O). If the five Si(3d) atomic orbitals are also included in the basis set, theobservedSi-O(br) and Si-O(nbr) bond lengths correlate with n(Si-O) as wo separatetrends with overlap populations for the Si-O(nbr) bondsexceeding hose for Si-O(br)by about 20 percent, even if the Si-O-Si angle is 180o.Calculation of z(Si-O) as qfunction of the Si-O-Si angle for the Si-O bo"ds irr Sir6ro- (assumingSi-O : 1.634and { G-Si-O : 109.47o) irst excluding and then including the 3d-orbitals, predictsin each case hat the Si-O(br) bond lengths should decreasenon-linearly and thatthe Si-O(nbr) bond lengths should increaseslightly as the Si-G-Si angle increases.Scatter diagrams of observedSi-O(br) bond lengths to two-coordinated oxygensversus the Si-O-Si angle appear to va,ry non-linearly with the shorter bonds tendingto occur at wider angles. f thesebond lengths a,re eplotted against - 1/cos(Si-O-Si),the scatter diagrams show improved linear trends. Multiple linear and stepwiseregressionanalysesof 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 theSi-O(br) bond length. The conclusionby Baur (1971) that the Si-O(br) bond lengthis independent of Si-O-Si angle n silicates for which Af(O) : 0.0 can be rejected atthe 99 percent confidence level.

IwrnonucnoNPauling (1939)has concludedrom the electronegativity ifferencebetweenSi and O that the Si-O o--bondhas about 50 percent onicl Now at the Geology Department, Rutgers-The State University, NewBrunswick, New Jersey 08903.

1578


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MO CALCULATIONS OE SILICATE IONS 1579character,attributing a charge of about '*2 on the silicon atom.Because his charge exceeds 1, thereby violating his electroneu-trality principle,he (1952)postulated he formation of.d-p.-bondsbetween he two atomsto neutralize he charge.Using an empiricalequatior relating interatornic distance o the amount of "'-bond char-acter,he then computeda Si-O distanceof 1.63A n closeagreementwith reported values. Subsequently, yfe (1954) calculated semi-empiricalbond energiesor the SiO+ etrahedronas a function of theSi-O separation,and concluded hat the relatively short Si-O bondcan be explainedwithout resorting o extensivedouble bonding.Onthe other hand, Cruickshank (1961) has assertedon the basis ofgroup-theoreticarguments,approximatemolecular orbital calcula-tions, and groupoverlap ntegrals ha;t of the five 3d-orbitals, onlythe two e orbitals on silicon form strong z'-bondswith the appropriate2pr lone pair orbitals on oxygen.The Si(3d) orbital involvement nthe Si-O bond has since been substantiated by all-electron a,b rui.tioSCFmolecularorbital calculations or the silicate on and orthosilicicacid (Collins, Cruickshank,and Breeze, g72). These calculationsshow (1) that there is considerableSi(3d,)-orbital participation nthe wavefunctionswith the e orbitals forming "'-bonds and the f2orbitals forming o-bonds with the appropriate valence orbitals onoxygen; (2) that the calculated"e,sX-ray fluorescencepectrausingSd-orbitalsare in remarkably goodagreementwith the experimentalspectra of silica glass; the agreement s notably poorer when the3d-orbitals are not included in the calculation (seeFig. 1); and(3) that the residualchargeon Si is not *4 as assumedn an electro-staticmodelbut is closeo zero n reasonable greement ith Pauling'selectroneutrality rinciple and the resultsof a Si(K)-spectrochemicalshift study by Urusov (1967).

Within the last decade a large number of silicate and siloxanestructures have been carefully refined using precisedifiraction dataobtained by modern techniques. These refinements have revealed anumbet 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 andSi-O-Si angles;and (3) the tetrahedral angles usually decreasen theorderd [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 mbetween silicon and oxygen del that includes3d-orbital involvementcl. Cruickshank, 1961; Lazarev, 1964;


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GIBBg, HAMTL, T,OUISNATHAN,BARTELL, AND YOWFrc. 1. Comparison of (q) the

experimental L." X-raY flu-orescence pectra for silica"glasswith that calculated by Collinset aI, (1972) from rmults ob-tained in all-electron ab ini'tinSFC MO calculations on thesilicate ion (b) using an ex-tended basis set that includesSi(3d) orbitals and (c) using aminimum basis set that ex-cludes Si(3d) orbitals.

E,oF(JlrJFlrJoltJIFFoUJUJ(JlrjE.aFzlo()l!ooz


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MO CALCULATIONSFOR SILICATE IONS 1581McDonald and Cruickshank, 1967a;Cannillo, Rossi, and Ungaretti,1968;Bokii and Struchkov,1968;Noll, 1968;Brown,Gibbs,and Ribbe,1969;Brown and Gibbs, 1970;Louisnathanand Smith, 1971;Mitchell,Bloss,and Gibbs, 1971).On the other hand, consistencywith a covalentmodel hat includes3d-orbital involvementdoesnot, of course,precludeother bonding models. For example, Mitchell (1969) notes that thesame rendsare mplicit both in Gillespie's 1960)valence-shellelectron-pair repulsion model and in a molecular orbital model involving avalence basis set of atomic orbitals (without the 3d-orbitals of Si).Despite the systematicvariations observedbetween Si-O bondlength and the Si-O-Si and O-Si-O angles n silicatesand siloxanes,no attempt hasyet beenmade o rationalize hese rends n terms ofmodernvalence heory.An attempt at sucha classification houldnotonly serve to improve our understandingand interpretation of suchobservabless he elasticconstants Wiednerand Simmons, 972), hebondpolarizabiliiies (Revesz,1977) the diamagneticsusceptibilities(Verhoogen, 958), he Raman and infrared spectra (Griffith, 1969)and the X-ray emissionand fluorescencepectra (Dodd and Glen,1969;Urch,1969;Collinset a1.,1972) f silicatesn general utshouldalso clarify our understanding f the long-range and short-rangeforces nvolved in A/Si ordering (Stewart and Ribbe, 1969; Brownand Gibbs,1970).

Recently,Baur (1970,1971)attempted o rationalize he Si-O bondlength variations for a large number of silicates n terms of an electro-static model by showing that the Si-O bond lengths predicted by hisso-calledextendedelectrostaticvalencerule correlatewell with thoseobserved for structures where LfQ) + 0'. However, he notes thatthe rule fails to predict bond length variations in silicates ike quartzand forsteritewhere Af(O) : 0.0. In spite of this shortcoming,Baur hasrejectedCruickshank'sdoublebonding model n favor of an electrostaticI For Si-O bond length variations in the tetrahedral portion of a silicate, theextended electroStatic valence nrle states that the deviation of an individual Si-Obond from the mean tength, (Si-O), in a silicate ion is proportional to Af(O). Theactual 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)bondstrengths received by oxygen andf(O) is the mean f(O) for the silicate ion underconsideration.Baur (1970,1971)usesp6 instead of f(O) to denote he sum of the bondstrengths receivedby oxygen. However, we prefer to retain Pauling's symbolism fortwo reasons: (1) f(O) is an acceptedsymbol used by many investigators and (2) pdenotes bond-order or bond-number in modern valence theory (c/. Coulson, 1939;Pople el al., 1965; and Mulliken, 1955).


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T582 GIBBS,,AMIL, LOUISNATHAN,,BARTELL,ND YOWmodel (l) becduse e could r;Lot rouea dependence etweenSi-O bond

suggestshat irnoreof the variation in bond length may be explainedin terms of Af(O) than in terms of the angle, it cannot be concludeda priori that fhe bond length variation is independent of the anglewithout appropriatestatisticaltests.According o Baur (1971), he main difference etweenhis electro-static model and Cruickshank'sdouble bonding model is that thelatter stresseshe importance of the intrinsic electronicstructure ofthe tetrahedral ions whereashis electrostaticmodel stresses he ex-trinsic effects from the the non-tetrahedral cations. Although Bauris aware of the empirical nature of his model and the fact that itcerbainly cannot replacea model basedon electronic structure, he hasstated that future models or the Si-O bond (1) should ncorporatethe electronic structure of the constituent atoms as well as the effectsof the non-tetrahedralcationsand (2) shouldbeginwherehis electro-static model breaksdown (for minerals ike forsterile, quarbz, tc.).The Si-O bond length variations in forsterite were recently examinedin terms of extendedHiickel molecular orbital (EIIMO) theory byLouisnathanand Gibbs (1971) who found that the observedbondlength variations show a close correspondence ith bond overlappopulations,n(Si-O), calculated (1) for the silicate ion alone and(2) for the silicate ion with its nearestneighbor non-tetrahedralcationsand their ligands.

r T'he relationship between Si-O distance and Si-O-Si ahgle for the silicapolymorphs was recently re-examined by D. Taylor (lWZ); SeeMineral.:' Mag.38, 62H31. After correcting the $i-O distances in the polymorphs for thermalmotion of the oxygen atoms, Taylor calsulated a regression line for the cor-rected Si-O bond lengths and their associatedSi-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. ourstudy (seeTable 3). Moreover, Taylor notes the probability the slope of the linemight be zero is a little less than 0.05, a result at va.riancewith the findings byBaur (1971) who believes that a dependencebetween Si-O distance and Si-O-Siangles does not exist for minerals like the silica polymorphs where Af(O) - 0.0.


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MO CALCULATIONS ORSILICATE IONS 1583??(Si-O) alculatedwith Si(spd)valencebasis

set for the silicate on in forsterite(1) For the silicate ion (2) With non-tetrahedralalone cationsand their ligandsBond Length

Si-O(1) : 1.6134Si-O(2) : 1.654Si-O(3) : 1.635

n(Si-O)0.9350.8780.892

n(Si-O)1.0030.9310.968

The n(Si-O) calculated or the silica;te on alone (seecol. l above)were made using the observedO-Si-O angles but with all Si-O dis-tancesset equal to 1.634.Despite he assumption n the carculationthat all si-o bond lengths are constant, t is clear that the resurtingn. Si-O) can be used to order the observedSi-O bond lengths nforsterite. Encouragedby this result, the present study was under-taken to learnwhetheror not EHMO theory can serveas a model orclassifyingand orderingsi-o bond length variationswhen the non-tetrahedral cations are ignored and a((O)=0.0'. Moreover, in theevent that thesecalculationsprove successful,hey should improveo'urunderstandingof the nature of the si-o bond in terms of modernvalencetheory and provide plausible reasons or the variation ofindividual si-o bond lengths n silicatesand siloxanesn generar.naddition,newscatterdiagramsof si-o (br) bond engthplottedagainstSi-O-Si angle will be presented or data from Bl carefully refinedsilicate and several siloxanestructures n which A((O) is zero orsmall.The relative contributions f Si-O-Si, -l/cos(Si-O-Si), A((O),and (CN) (the mean coordination number of oxygen bonded tosilicon) to the Si-O(br) bond length in these structureswill be as-sessed sing multiple linear and stepwise egressionmethodsl (seeappendix).The outcomeof the analysiswill show hat we can rejectthe hypothesishat a relationshipbetweenSi-O(br) andSi-O-Si anglemight not exist or silicateswhereA((O) = 0.0.Trrn Cer,cur,ATroNmnSrcNrrrcANcE r Boxl Ovnnr,epporur,lrroysOsrarxnu rN THEExruxopn Hiicrnr, Mor,ncur,enOnsrrAr,Trrrony

SincePauling (1929) ormulatedhis famousset of rules or predict-ing stable atom configurationsn ionic crystals,many silicatestruc-tures haVebeen nvestigatedby X-ray methodsand have been ound'Linear, multiple linear, and stepwise linear regressioncomputations presentedin our paper were completed using the UCLA BIOMEDICAL PROGRAMPACKAGE,1967.


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1584 GIBBS,HAMIL, LOUISNATHAN, ABTELL,AND YO'Wto conform remarkably well. Nevertheless,he steric details of thetetrahedral ons in many of thesesamesilicatesalso appear o con-form with a model hat includescovalentbonding (cf Brown et al',1969;Brown and Gibbs,1970).In an attempt to learn whether hesedetails are indeedconsistentwith covalent heory, extendedHiickelmolecular orbital calculationswere made for the tetrahedral ons inten silicatemineralsspecificallychosen uch hat A((O) is eitherzeroor small. In this way the importanceof the intrinsic electronic truc-ture of the tetrahedral ons can be emphasizedo the reader (Hamil,Gibbs,Bartell and Yow, 1971).Silicates howinga larger variation ina((O) will be considered lsewhere y Louisnathanand Gibbs in astudy of the disilicates(1972d).EHMO calculations, lthoughveryinexpensive nd crudeby ab initio'standards Richardsand Horsley,1970),have beenmoderatelysuccessfuln generatingWalsh-Mullikendiagrams (molecular orbital energiesplotted against bond angle)(c/. Hoffmann, 1963; Gavin, 1969; Allen, 1970; 1972) and' n de-lineating trends in variations of bond length with bond overlap popu-lations or bond.order for a number of relatively large moleculesntheir groundstates (c'1.Dallinga and Ross,1968;Gavin, 1969;Boydand Lipscomb, 1969).Recently,Bartell, Su, and Yow (1970) calcu-lated EHMO bond overlap populations or the tetrahedralbonds nselectedsulfates and phosphatesand found that they show strongcorrelationswith (1) the simple valencebond orders assignedbyCruickshank (1961) and (2) the observed etrahedralbond lengths'To learn whethersi-o bondoverlappopulations or the silicates howsimilar trends, calculationswere undertakenusing an EHMO pro-gramoriginally written by Hoffmann (1963). n the calculations, ne-electronmolecularorbitals,MOs, are constructedas a linear combi-nation of atomicorbitals,LCAO,

vo : t, co,6,where c16i r-the linear .oum.i.ot.- und 41 the Slater type single ex-ponent atomic valence orbitals. The energy associated with an MO,ea, s obtained by diagonalizing the secular determinant lHot - .Stil =0, where Sri are the two center overlap integrals (explicitly evaluatedfrom the positional coordinates of the constituent atoms) , a,nd'Hai a'rethe elements of the Hiickel Hamiltonian matrix. In the rigorousRoothaan-Hartree,Fock method, the Ha1are complicated functions ofthe 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


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MO CALCULATIONS OR SILICATE IONS 1585lhe H15erms are not explicitly evaluated but are chosen o simulatespectroscopically btained quantities, namely the negative of thevalenceorbital ionization potentials (VOIP) whereas he I/ai termsare approximated y the Wolfsberg-Helmholz 1952)parametrization

H t r : B n o ( / / ' , * H , , ) .When one-electronMOs are constructed nd their energies btained,the Pauli exclusion principle is applied as an afterthought and theMOs are filled with electrons pair-wise starbing from the lowesteigenstate. y inserbingeigenvalues,;, , nto the set of secularequa-tions,

i (t,o - els;;)c6,. o,the linear coefficients pi &ra evaluated but they are not in any wayrefinedto self-consistency. t is important to note that trends n theselinear coefficients epend n largepart on the geometryof the tetra-hedral ions modeled n our calculations.Finally, a Mulliken (1955)population analysis s cornpletedo obtain the desiredSi-O bond over-lap 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 ofelectrons,f(k), (0, 1, or 2) in MO rI,7,.he bond overlappopulationsprovide numdrical estimatesof the strength of covalent bonding andanti-bonding between wo atoms. Two atoms are considered o bebondedwhen he overlappopulation s positive,nonbondedwhen t iszero, and antibonded when it is negativel short bonds are usuallyinvolvedwith largeoverlappopulations nd longerbondswith smallerones. f observedSi-O bond lengthsare used in the calculationofn.(Si-O), he shorterbonds,whichusuallyhave argeroverlap ntegrals,S,;y,will tend to have larger bond overlap populat'ions han longerbonds. As it is important to be able to discriminate between his in-ducedcorrelationand that inducedby the geometrical spects f thetetrahedral ons, we have undertaken wo separatecalculations(1)using he observedO-Si-O and Si-O-Si anglesand clampingall Si-Odistancesat 1.634 and (2) using the observeddistances nd angles.By clampingall the Si-O at 1.&34, he correlationbetween (Si-O)and the observedSi-O bond length induced by the overlap integralsis effectively removed and what remains should be related to theintrinsic electronic structure of the ions induced n large part by such


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1586 GIBBS;HAM'IL,LOIIISNATHAN,BARTELL,AND YOWgeometricalactors as the o-si-o and si-o-si angles.Because f thevery crude nature of the EHMO method intrinsic in'the very daringapproximations utlinedaboveand because f the utter neglectof thescreening otential producedby the cotilomb and exchangenterac-tions, little significance an be attached ,oLheactual'nu,rnbersbtainedfor zr,(si-o). However, t is the trendsbetweenhe calculated a(si-o)and the exp,erimentallydetermined si-o bond lengths that ale con-sideredsignificant,not the absolutenumbers or any one tetrahedralion. Thus, EHMO results can be useful in ordering and cl'assifgingobserved ond lengthswith bond overlappopulationsbut it cannotbeusedto proue the observedbond length variations nor can it be usedLopt'o,uehe participation of the 3d-orbitals of si in the wavefunctionscalculatedor the ions studied.The YOIP and orbital exponentsused as input to the program aregiven in Table 1. For the s- and p- orbitals the free-atorn optimizedorbital exponentsof clementi and Raimondi (1963) were chosentogetherwith voIP similar to the zero-chargevaluesof Basch,viste,uoa Gruy (1965).For the 3d orbitals of silicon, an orbital exponentsignificantly higher than the Slater-rule value was adopted or reasonsdiscussedy Bartell et al,. 1970),and'H,s6,saas adjusted o -5'5 eVto yield orbital and overlap populationscomparable,n the caseofSiHa, to ab ini'tio populations ound by Boer and Lipscomb (1969)'Theseparametersgave3d populations n sio+* which turned out tobe of the samemagnitude as Lheab initio poptiations reporbedsub-sequently y Collinse'tal. (1972).A fundamental weaknessof the extendedHiickel scheme(which isnot basedon u b.ormi'd,eHam\ltonian) is that its energiesand wavefunctions are sensitive t0 the somewhat arbitrary parameterizationadoptedand, accordingly, ave only a qualitative significance. n theother hand, he EHMO trends n chargedistributionsand bondpopu-

T.cnr,n 1. Ver,pNcp Onsrrer, IoxrzerronPornNtrar,s (VOIP) eNo OnsttaLExpoNprrs ().

$tomOxygen

Silicon

AtomicOrbital2s2p.JS3p

VOIP (eV)-32.33-r5.79- 14 . 19-8 .15

2.2462.2271.634t .428


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MO CALCALATIONS OE SILICATE IONS 1587lations in a seriesof related moleculeshave been found to be in-sensitive o the exact parameterization;hey are also roughly inde-pendentof whether the VOIP are frozen at plausiblevaluesor &readjusted o self-consistencywith aiornic charges(Baschet at., L}BS).Therefore,since he present nvestigation s a preliminary explorationof. trenils in a seriesof structures, t seemedadequate o choose hesimplestvariant of the extendedHiickel methodwith fixed VOIP andexponents. inally, it is important to note that one of the biggestdeficiencies f the extendedHtickel method s its failure to take ade-quate account of the Coulornb, nteractions. The method works thebest for moleculeswhere the electronegativitydifference,Ax, betweenbondedatoms s small.However, t starts to break down when Ax isgreater han approximately1.3, he breakdownbeingcompletewhenL1a 2.5 (Allen,1970).Thus,whenheteropolar ubstances ith inter-mediatebond type (like the Si-O bond n a silicatewhereAx = 1.2)are studied,EHMO theory may be used o qualitatively diagnosehecovalentaspects f the bonding.Accordingly,t is of particular nterestto apply the EHMO methodto silicate structureswhich have beenfound to conformremarkably well in the past with Pauling's ruleslfor ionic crystalsyet which also conform n many instanceswith amodel hat includes ovalentbonding.

Tnp Rnr,erroNsurpBprwnuN Si-O (br) BoNoOvpm,epPopur,erroxAND HE O-Si-O aNo Si-O-Si ANcr,psIn a study of severalorthosilicates, ouisnathanand Gibbs (1g72a)have found that the tetrahedralvalenceangles n hypotheticallydis-torted SiOe etrahedrahavea pronounced ffecton n(Si-O), the wider

O-Si-O anglesbeing associated ith bondsof larger overlappopula-tions. They also showed or a number of silicates (1g72b) that acorrelationcan be made between he mean of three O-Si-O angles,(O-Si-O)3, and the length of the Si-O bond commo to these threeangles,shorter bonds ending to be associatedwith larger valuesof(O-Si-O)a.McDonald and Cruickshank (l967a) and more recenilyBrown and Gibbs (1970)havealso shown hat shorberSi-O distancesare typically involved n wider O-Si-O angles.AlthoughBaur (1g20)has also indicated that shorter Si-O distances end to be involved inwider O-Si-O angles,his geometricalmodel of a Si atom rattlinglAccording to Bent (1968) Pauline's mles ma5.r ave 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 evento metallic compounds.


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1588 GIBBS,HAMIL, LOAISNATHAN, ARTELL,AND YOVTwithin a rigid" etrahedronof oxygenspheress not realized n generalin the silicates(Louisnathanand Gibbs,L972b)The Si-O-Si angles n silicates (L'iebau,1961) and in siloxanes(Noll, 1956) are usually between130oand 140o,althoughanglesaswide as 18Oo re not,uncornmon. xcept for a,nangleof 93" in Si2Oz(Anderson nd Ogden,1969),anglesess han 120"are rare.A bondingmodel that includes Si(3d) orbital participation predicts that thelength of the Si-O(br) bond should ncrease s the Si-O-Si anglebe-comes arrower (Cruickshank,1961).Moreover,as the Si-O-Si anglenarrows, ncreasing lectrostatic epulsions etween he si-atoms anddecreasing-charactern the O(br)-orbitals (i.e.,a decreasing-bondstrength) are additional factors that are predicted to lengthen Si-O(br) (Cruickshank, 961;Brown et a1.,1969)-

fn a re-investigation of the crystal structure of zunyite, Louisnathanand Gibbs (1972c)havefound that the sizeof the tetrahedralangleshasa pronouncedeffect on zr[Si-O(br)] n Si,Ou- ions with linear Si-O-Silinkages.Thus, if {to(br)-Si-O(nbr)l > {[O(nbr)-Si-O(nbr)], thennlsi-O(br)l{[O(nbr)-Si-O(nbr)], then n'[Si-O(br)] = n'[Si-O(nbr)] whereas f{ tO(br)-Si-O(nbr)ln[Si-O(nbr)] (seeFig. 2). These results calculatedwith all Si-O bondIengths clamped at 1.634 imply that the magnitude of the tetrahedralangles plays an important role in establishing n'[Si-O(br)] in theSi-O(br)-Si linkage. In order to assesshe nature of the relationshipbetween the Si-Oftr) bond length and Si-O-Si angle, we calculatedn[Si-O(br)] as a function of Si-O-Si angle for the three Si,O"z6-onsshown in Figure 2. The results calculatedwith Si-O : 1.63A and aSi(sp) valencebasis set are presented n Figure 3a. Regardlessof thesize of the tetrahedral angles,n'[Si-O(br)] increasesnon-linearly wil}rincreasingSi-O-Si angle, he largestoverlappopulation beingassociatedwith the 1800 angle. Analyses of the linear coefficient, c6;, and theoverlap, 8o;, matrices ndicate that the change n z[Si-O(br)] is relatedto concomitant changes n both the o- and zr-bondingpotentials. Inother words, widening of the Si-O-Si angle increases oth the o- andr-bond. strengths as proposedby Cruickshank (1961), Brown et q,I.(1969), and Brown and Gibbs (1970).Using results provided by thetheory of atomic orbital hybridization, Louisnathan and Gibbs (1972c)have proposed hat the Si-O(br) bond length shouldvary as a functionof -l/cos (Si-O-Si). When the n'[Si-O(br)]arereplottedas a functionof -l/cos (Si-O-S;), a well-defined linear relation (Fig. a) is realized.The trendspredictedby EHMO theory are similar whetheror notthe Si(3d) orbitals are included n the calculation(Figs.3a and 3b),


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MO CALCALATIONSFOR SILICATE IONS 1589

(o) (b) (c)Frc. 2. A comparison of Si-O bond overlap populations (the decimal fractionslisted adjacent to the Si-O(br) and Si-O(nbr) bonds) calculated for three Si:Oz6- ons(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 madeusing 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 thatn[Si-O(br)] < z[Si-O(nbr)] in (c). (SeeLouisnathan and Gibbs, 1972c).

suggesting that the observed steric details of a tetrahedral ion in asilicatemay not beused o establishhe involvement f the 3d-orbitalsin the formationof the Si-O bond.This is in agreement ith Mitchell's(1969)assertion hat "one cannot, . . expectexperimentso establishwhether d-orbitals are (really' used anymore than experiments anshow s and p orbitals definitely occur in bonding of the first rowelements" see lsoBartell et aI., 97A)

Si-O Bono LrNcrn VenrerrolvrN Low-QuARrz,I"ow-cnrsrosAr,rrn,ANDCopsrrpThe silica polymorphsare possibly he most appropriatestructuresfor examining he dependencef Si-O(br) bond length on both theO-Si-O and Si-O-Si angles,because 1) there are no bondsotherthan Si-O(br), (2) al l oxygensare two-coordinatedand bridge twotetrahedra,and (3) A((O) : 0. For our analysis,we havechosen atafrom three precisely efinedstructures,all with e.s.d.'sessor equalto 0.0064: ow-quartz (Zachariasen nd Plettinger,1965), ow-cristo-balite (Dollase,1965),and coesite Araki andZolLai,1969) GroupA


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GIBBS, HAMIL, LOU SNATHAN,( o )

BARTELL; AND YOW(b )

o 7 a

o 5 0

o 4 6

o o ? 4o@

o o 4 8I

o f ?

t40 tss i - o - s i ( . )Fro. 3. Si-O(br) bond overlap populations calculated as a function of theSi-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 withSi-O-Si and accordingly are not shown.

in Table 2). The data for keatite and low-tridymite werenot includedin the analysis for reasonsgiven in the next section.The three struc-tureschosen rovide welve ndividual Si-O(br) bond engthsvaryingfrom 1.598 o 1.631A. igure5c is a scatterdiagramof Si-O(br) bondlength versus (O-S,i-O)3angle; tr'igure 5b shows Si-O(br) versus[-|/cos(Si-O-Si)]. Using the estimatedstandarderror of the slopesfor the lines itted to the distributions n Figures5c and 5b (seeTable3), the null hypothesis an be tested hat the true slope n eachcaseis zero; that is, that the Si-O(br) bond length is independentof(O-Si-O)3 or -1/cos(Si-O-Si). Student's lt l calculates n be 3.3and2.3 for the data in Figures5c and 5b respectively Table3). Sincethesevalues exceedhe 9Opercentconfidenceevel (c/. Draper andSmith,1966)of t(10,0.90) - 1.4,we can ejectthe hypothesishaLarelationship betweenSi-O(br) and -1/cos(Si-O-Si) or (O-Si-O)3might not exist.We haveused he function [-l/cos (Si-O-Si)] rather than the angleitself becausez[Si-O(br)] vary non-linearly with the angle but

./

.t'

s i - o - s i ( .1


14/36

MO CALCULATIONSFON ilLICATE IONS 1501

(o)o.50

t 30 t40 r50 160si-o-s i ( . ) r70 t80

(b)o .50

t . 6 t . 5 t . 4 t . 3 t . 2 t . l l . o- l l cos (S i -O-S i )

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 theconformation in Figure 2b and a Si(sp) basis set.

FI o.csoI' =ir O.48c

I- 0 .49oI' =

; J O . 4 8c

o.47

o.47


15/36

7592 GIBBS, EAMIL, LOUISNATHAN, BARTELL, AND YOWl inearly with [-1/cos(Si-O-Si)] (Fig. a). However,even if therelationshipbetweenSi-O(br) bond length and Si-O-Si angle s as-sumed o be linear, he hypothesis f no interdependencean againberejectedat the 90 percentconfidenceevel (seeTable 3). When theSi-O(br) lengthsare estimatedusing he interceptand coefficientsb-tained rom a multiple linear regression nalysis, sing both (O-Si-O)3and [-1/cbs(Si-O-Si)] as independentariables, reasonableor-respondence eiween the experimentaland estimated distances sobserved Fig. 5c and Table 2). Finally, two commentson Figures5o and 5b are worth making: (1) the quantity [-1/cos(Si-O-Si)]may be a better definedvariable ihan (O-Si-O[ becausehe formerincludesa more accurateaccountof the hybridizationcharacteristicson O(br) than the latter doesof the hybridizationcharacteristics fSi, and (2) the calculated lope [-0.05(2)] for Figure 5b is statis-tically identical with that [-0.0E4(1)] obtained or n"[Si-O(br)]versus -l/cos(Si-O-Si)] (Fig. ab), an interesting oincidencee-

Table 2

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

G r o u p A r S i - o ( b ! ) b o n d d t . t s n c e w . S l { ( b r ) - S t s n g t e d a t a t o ! s l l l c 'PolForPhe qhere a((o)io 0 and 'z 0

;;;Gi t a


16/36

MO CALCULATIONS FOR SILICATE IONST a b ] e 2 ( c o n t . )

G r o u p C r s 1 - o ( b r ) b o n d d l r t a n c e v . s 1 { ( b r ) - S 1 a q le d a t a t o r e l l t c a r e !aher o(br) 1. ruo coordl@ted8"3s14&5o25 sr -O(r ) 1 ,599 r .599 t6O,O l .mOO t 7 - . lO sbmoo 6!d e t . (1970)Lw rbrte 3ll[3i:3:[:] t:8l i r.610 16r.2 1.0564 2.s -'04 Es e re'4" (re6e):ii[:]-8S[:] i'.211 r.6185 B'.7 L.3et2 2.s ::3XLN Cordlerrce st(3)-o(6) r .603 .02 ctbb. (t965)s r ( 4 ) _ o ( 6 ) 1 . 6 2 0 . o 2tudd& cordlerl tG :l [ i ]*[ : ] i .3l i r .5o5s Lts 5 r @oo 2.7 oz hsher (1e67)Danbur l rc si{( l) 1.614 1.614 r36.E t .37l8 X.7 -.25 phrulp. !. ! . f ! . , ( f97r)KlsuaroPlrte :1[ l ]1[1] i :331 1.644 r3r .e L.4st4 1.4 -:33 ** ! ! 'q ! " (re67)*"lll#,,." 3il[;]:3:3i i:133 r.63r rs'8 r.5304 2.s ::ff "* 6d !r' 'v (re64):i?[:i:333] i:3i3 r'6125 r55'e r'@$ 2.e -'.31(lnort 3i[i]:3[i] i.z:rt r'534 r4o.r 1.303s 2.1 -'os hutid (reIr)EPidrdFlt :1[li:3[1i i:33i r.617r r5r.5 r.r37e z.E ::3i 'o** 'rd !'4 (re7o)s1(r)-o(r) 1.66 r .06 143.E L,2392 .Os1(2)_o(6) r .629 r .oz9 143.0 L,z52L .0 7slG){(E) 1.633 1.613 8t,O r.3250 ,Olkrl tul lr . s1(r ) { (7) 1,621 1.521 141.1 r .2850 3.2 - .13 H&bU !!.q!,, (r9tr)c' l61Eat6rt. s1(r ) { (7) I .613 1.513 ta2.2 L.2622 3.0 _.03:l[i]][;i i:!ii t.5z6s r3e.7 r.3rr2 -:33ctun .r l tc s1( I ) { (7 ) 1 .613 1.613 r44 . ! L .223E 3. 0 -. @ rr { . r (1969):|[]li[;i i:lli r.5re 142.4 1.2622 -:311r@l l r . sr ( I ) { (7 ) 1 .616 r .616 r39. ! r .3 l9o 3. 2 ;. 8 r .p tb ! ! . !1 . , (1969)Gbucopbo. Sr (1) { (7 ) 1 ,6 t1 l .6 r t L17.2 t . IA97 3. 2 - . t lc -c6 te ! .d h - s1( r ) { (7 ) I 616 I 616 la t .O r .2m j .O - .03c@ltrst@lt' :l[i18] i..232 r.628 r&.0 r.ss -:3;?r td t lve b- S l ( r ) { (7 ) 1 .626 t ,52A t39. I 1 .32S 3. 0 - .03c@ln8toa1t. :l$lil[;] i:31 t.622 r3e.8 r.sez -:8351(r)s(7) t.@3 1.603 l4 l .O t .a$ -.03:i[]1i8] i.fll r.6345 136.r r.3373 -:B:pro tryh lbo t . s1( r ) { (7 ) 1 .524 t .624 137.1 1.3651 3. 1 - .Ot c ibbr (1969):l$11[ii i.!ll r.62r 140.6 r.2$r -:Blhthophyl l l te s i ( I ) { (7 ) 1 , f l5 r .6$ l4 l .a L .27% 3, 2 - .07 r lq . r ( t970)3i$ifl[:] i:fli r.6,6 140.6 r.2e6 -.31sr(t){t(7) 1.6t7 r.617 t3r.g 1,3a99 -.I l

:i[ili[3] i:|?i r.6275 r&.3 i.orar 3.5 -'03 brb'c (re6),",.-"""l;i;fifi :.,. ,,. :,- ; :::::..::,,spite the fact that all n(Si-O) were calculated assumingall Si-Oequal 1.634and all O-Si-O angles 109.47" n Figure 4.Tnu Connpr,erroN nrwnnu Si-O (br) BoNoLnNcmr eNo Si-O-SiAxcr,p rN Srnucrunns WHpnna((O) rs ZERo

In structuresof low-quartz, low-cristobalite, coesite,hemimorphite,thortveitite, beryl, emerald, benitoite, YbsSizO,T, r2SisO7,and fivesiloxanes,all the oxygen atoms receive the same bond strengttr, i.e.

1593


17/36

1594 GIBB!, HAMIL, LOAISNATHAN, BARTELL, AND YOW(o)

ao o o

t .t o

r 6 4r 6 4- 1 6 2o

IU'=! ' e z6

r 6 0t 6 0

(b)r50 t6 0

s i - o - s i ( " ) ( o - s i - o ) t(d)

aaa

l o ao o a

r 6 4r 6 4 a.

3 t62ooI

(, ) t6 0

! , e zot 6 0

| 60 | 62 1.64

si- o(.st. l( l)- l l c o s ( S i - O - S i lFrc. 5. Scatter diagrams of si-o6r) bond length data from Table 2, groupA, 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 engthsand si-o-sianglesn thesestructuresare isted n Table 2 (GroupsA and B)' Thescatter diagram of Si-O(br) versusSi-O-Si angleand [-l/cos(Si-O-Si) ] are given in Figures 6o and 6b, respectively.For lhe 27individual si-o(br) bond engths n this data set, he hypothesishatthe true slopesof (l) Si-O(br) versus Si-O-Si arrgleand of (2)Si-O(br) versus -l/cos(Si-O-Si)] arezerowas t'ested. he calcu-lated ntercepts, lopes, artial correlationcoefficients,nd Itl- valuesare given in Table 3. The calculatedltl- values ate 2-5 and 3.6 forFigures6o and 6b, respectively.Becauseheselfls exceedhe criticalvalue of ,(25,0.90) - 1.3,we can reject the null hypothesishatSi-O(br) bond length and Si-O-Si angle might not be correlated(actually2.5alsoexceeds(25,0.90) . This result s contrary o Baur's(1971)conclusionhat a correlation annotbemadebetweenSi-O(br)and Si-O-Si angle or silicateswherea((O) : 0. He reached is con-clusion using data from keatite, high-tridymite, low-quartz, low-

t o254

(c)a

aa '

a 1 Io a ta


18/36

MO CALCULATIONS FOR SILICATE IONS 1595

The correlations presented n our Figures 6a and 6b are somewhatimproved (seeTable 3) when the data from ybrSizOT,Er2Si2O7, ndTable 3

Ln [e rcep ts , a^ r s lopes , b r r and cor re la t i on coe f f i c ieDts , r , fo r . ] - i nea rreg ress lon eqda t lons f i t tad ro dara in TabLe 2 and Flgs , 5 -7 . l t l s ca l c .fo r Ho :B l = O; a l l l t l ' s exceed a two-s lded 90 2 1eve l tes t

Group A of Table 2

S1-0(br) bond length vs .S1-O-S1 angle for sl l lca L,677polynorph data (Fig, 58 )Sl-O(br) bond length v6 .-1 l cos (S i -0 -S1) fo r s l l I ca 1 .539pollnorph data (Flg. 5b)Si-O bond Lengtha vB ,a fo r s1 l1ca 4 ,094polynorpH dats (Ftg. 5c)S1-0 bond length ve.SL-o (es t ) fo r sL l l ca -0 .168pollnorph data (F+g. 5d)

bt-0. ooo42)

0 .06 2 )

-0 .023 6 )

1 1 3 )

-0 .0005(1)-0 00041)0 0s39)0,0479)

I t l sanple size-0 .48

0 . 5 7

- n 7 7

0 . 7 6

1 . 8

2 . 3

3 . 7

S1-0(br) bond length vs .S l -o -S t ang le ; F lg . 6aS1-0(br) bond length v.-Ucos(s l -O-S i ) ; F lg . 6b

Groups A and 82 of Table 2- 0 .0009 (2 ) -o ,72 4 .8-o ,oo04 (2 ) - 0 ,4s 2 . s0 . 0 8 1 ) 0 . 7 s 5 . 30 . 0 6 2 ) 0 . 5 8 3 , 6

and C2 of Table 2

r L . t q )I . O6 J

{ 1 .5161 .545

24 t2 7 '24't2 7 '

78 r81',78 r81 '

Groups A, B,S i -O(b r ) bond leng th vs , 11 ,7008S1-0 -S l ang le ; F ig , 7a '1 .584 IS i -O(b r ) bond leng th ve . 11 .554-1 l cos (S i -o -S l ) ; F le . 7b '1 .561

- 0 , 5 1 5 . 1- 0 . 4 4 4 . 30 . 5 3 5 5o . 4 9 5 . 0

re ,s .d , rs a re g lven 1 o paren theses a nd re fe r to the las t d ig i t .2The data fron thortveiti te, Errsiroi l and ybaslroT rrere not uaed to obtaln theatatlstlcs glven in the flrst iow'of thc dati sEtb enclosed by braces (see text).


19/36

1596 GIBBSTHAMII" LOU-I&NATHAN'BARTELL, AND YOW

.cloIa

t . 6 6

t 6 4

r . 6 2

r .60

r . 5 8

r . 66

r .54

1 . 6 2

t . 6 0

t . 5 8

(o)' a

a-\ oa

t--.1 't-a- --t a a \ -

oo

. O\ .-= _ _la

t 4 0 r5 0 1 6 0 1 7 0si-o-si(") r80(b)\ - o-- - --oo - \

a. o

ao\ . ! r 'O \oa o - ' - r - - O' . - - - t f -a - \ ., .

a1 . 6 1 . 5 t . 4 1 , 3 t . 2 l . l t o- l l cos (S i -O-S i ) : ,

Frc. 6. Scatter diagrams of Si-O(br) bond'length data from Table 2; groupsA 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 aquaxes ine. The dashed line in (a) is trans-generated from (b). The open circle plots are data for thortveitite, Ybosirozand ErgSLOn.

r30

L-ooI6


20/36

. MO CALCALATIONS ONSILIAATE .ONy I8S7thorbveititeare excluded open'symbolsn Figs.,6aa,nd.'6b).n-these

Rm,errvp ConrnrsurroNsol. -1/cos(Si-O-Si), a((O) exn (Cff)To rHE Venr,lnrr,rryoF TrrESi-O(br) BomuLnxcruAs' a((O) and the. mean coordinationnumber, (CN), of oxygenbonded o silicon have also beensuggested s determining factors ofthe Si-O(br) bond length (Baur, lg7l), an examinationoieach in the

The resultsare listed in Table 4 in the order ranked by the stepwiseTable 4

Analysls of varlance: Mult iple l inear regression analysls fo rthe data used to prepare Fig. 7 wl th Si_O1try bond length asth e dependenr var iabte; t (74, O.9O) = 1,7, The order of th eiodependent variable was found by scepvise reglessioD oethocls, l

fndependeot Varlable lt l Par t ia l rAddeal Regres6ion

Su of Squares Sdple Slze-1lcos (Si-O:Sl)

AE o) t 4 ' 2 F,2

t 1 ' o 0 ' 6 3' 6 . 4 0 ; 5 9o .440 .41U . J )0 ,34

0 ,0020 78 rq..0020 81,0.0005 78,0.0006 81 '

0 ,00430 .0038 81 ' ,

l lhe data f rom thortveit i te, ErrSi ,Oa, an d yb^Si"o, were not uaed to obtaln th estat . ls t ies glven ln th e f l rs t iow'of the datf s6tC e.rclosed by braces_ (see text).


21/36

1598 GIBBS,IIAMIL, LOAISNATHAN, ARTELL,AND YO'Wregressionmethod (c/. Draper and Smith, 1966).The ltl- values n-dicate that all three variables contribute significantly to the varia-

a random sample n the sensehat the data are omitted for structureswhereA((O) is relatively large and whereO(br) is coordinatedbymore than two atoms. Therefore the regressioncoefficientcalculatedfor our dataprobablywill not applyto an expanded ataset.Howevet,sinceA((o) and actual coordination umberof o(br) aremoderatelycorrelated,we cannot,concludea priori whether the magnitudeof theslope associatedwith (cN) will be larger or smaller than 0.006 orwhether (cN) witl make a significant contribution to the regressionsum of squaresn the presence f a((O).Plots of Si-O(br) bond length versus Si-O-Si angle for a widevariety of silicates (Cannillo et a1.,1968;Brown and Gibbs, 1970)exhibit the same rend as Figures7,aor 7b (seeTable 3) irrespectiveof the coordination number of the bridging oxygen' Accordingly, itappears hat the length of the si-o (br) bond is related in part to the$i-O-Si angle regardless f the value of a((O) or (CN), with theshorter bondsbeing involved in the wider angles.Although this resultis consistentwith a covalentmodel hat includessi (3d) orbital par-ticipation, one cannot concludeas indicatedearlier that theseorbitalsare definitely used n the compositionof the si-o bond (Bartell et al.,1970).

Tnn Rnr,erroNSHIPntwnnN n(Si-O) ANDTHEOssunvED i-OBoNp LnNcrnAs both O-Si-O and Si-O-Si anglesappear to exert a pronouncedrole in establishingz(si-o), we calculated he bond overlappopula-

tions for isolated and polymerized tetrahedral ions in ten silicates(Table 5) usingthe observedO-Si-O and Si-O-Si anglesand clampingal l si-o at 1.63a.The calculation or a valencebasisset consistingof silicon 3s and 3p and oxygen2s and 2p atomicorbitals yields bondoverlap populations which, when plotted against the observedsi-odistances Fig.8), showan apparent inear relationwith longerbondlengthsbeingassoeiatedwith smalleroverlappopulations.As expected,


22/36

3 r.ezoIan

t .66

t .64

t.60

r . 5 8r30

(b)

ta

r50 160s i -o -s i ( " )

r70 r80

lloI6

r .66

t .64

1.62

r.60

r 5 8t l

- l l c o s ( S i - O - S i )Fra. 7. Scatter diagrams of Si-O(br) boad length data from Table 2, groupsA, 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 andEr"SLO". The reported e.s.d.'s of ihe Si-O bond lengths for the structures givenin Table 2 are all less or equal to 0O1A.

t .o. 23456

(o)ao

o f o o . .a o a aa

oo

a oI


23/36

1600 GIBBS, HAMIL, LOUISNATHAN, BARTELL, AND YOWTable 5

mortv. lr tr . sl-o(l)s t -0(2)sr-0(t)tosilltn. sl-0(4)sr_0( t )s t_0(6)ar_0(7)hrrld Sl-O(r)st-o(l) l8r -0(2)l.ryl

l .6sl . t8 t 1 , 6 2 1l . !94t .59 t 1 . 6 2 0r .6031.6 t6r .6x1 .646 1.60t

1 .507l .a2 lI 6 t6L ,612t . 6 t 9 1 .596I 6061.4601 . 6 7 8 1 . 5 1 7r .5951 . 6 9 8

l .6o t1 . 6 0 31.5031.6261 . 7 0 1 1.589t .609L65t1 ,669 t . 6 0 0I t5 l1 5611,504L 6 l rt .649 t . 6 t lL , 6 4 1

b!' !.!:!,(lrlr1-0( l ) 1.66s r { ( r ) , l . t 4 tsi4(2)sa-0(3)st -0( t )si-o(2)sr-o(4t Le21

an b..t. .!d b..!tlLeLit t ="* "=","t

t* .4!so.(8! 0,4t5 O,rLl 0.734 h! l l , c ! !b. .nd i lbbao.as 0.478 0,155 0,739 (r97r)0.t06 0.515 0 942 0,9550 , 5 1 1 0 . 5 3 0 0 , 9 4 6 0 , 9 7 E

C.st0l

Fo.rt. ! tr . sr-O(t)sr-o(2)sr-0(3)D.tolrr. sr -o( l )s t -0(2)s1-0(!)sr-o(4)

0.494 0.506 0 .?610.501 0. t0 I 0 .9? t0 . !03 0 .502 0 .926o.{91 o.( i2 o ?5 Eo,a6E 0.405 0 ,7 t10 ,510 0 .515 0 ,9410,50E 0 . t2 t 0 ,9440.505 0.5r r 0 , t t80 ,50E 0 . t30 0 7420.50! 0 .503 0,9 !90 .505 0 . t22 0 ,r r70 .50 t 0 .51 t 0 ,1820. t0 ! 0 , t0 { 0 9 t90 ,56 0 , t16 0 1A2o; t i l 0 . t06 0 ,1620.4s 0 .499 o ,?750.4 t6 o .a? l 0 .7 t to . !o r 0 .521 0 ,9{o0.50 ! 0 :531 0 ,9410.509 0 514 0 .9400 , 4 6 4 0 . 4 t 0 o , t t t0 4! 6 0 ,165 0 .7610 , 1 1 5 0 . 4 1 ! O 1 L t0 .52r 0 .545 0 ,9590.506 0.523 0 .9340,411 0 .454 0.7410.48! 0 .462 0. t450.512 0 . t23 0.9{ t0 ,501 0. t16 0 .919o . a t t 0 4 ( 5 0 . 7 5 1

0 . 5 2 10 52!o .4620.4880.549o 5220 4690 . 4 6 1o 525o 5550 . 4 5 !0 . 5 2 10 . 4 9 90 . 4 5 10 . 5 1 5o . a 6 E

0 515 0 .525 0 9350 , 4 0 0 0 , 4 t 0 0 8 7 80,469 0 487 0 .8980.506 0 .545 0,9180.{93 0 484 0.9000,492 0 4 !2 0 .8990 , 4 8 7 0 , { 6 9 0 . 8 9 0

C!!rct.h.d (1967)

Ttpp. .Ed fldllton(19?1)

h8h. ! (1968)

b . ib . r (1968)

0 .78 t acbr t . . . r rl d0 ,7aE Pl . rcrnr . r (1965)0 ,162 l t .ch .r ( !969)o tx '0 9n{0.98t lr.d .!d rx.o! (rt69)o 1420 , 7 t 00 . t r t0 .9990 9600 7 l !o , t 2 t0 .9600 . 9 8 t0 69!0 ,96 ! r ro l . r (1969)o 9640 7030 . 1 7 L0.6931 00r0 95 90 . 1 80 , 7 2 40 9 7 rLOl60 torb 950 Donily rtrd Allqn0.918 (1968)0 . t 0 20 . 9 5 10 7 2 3

8 1 0 6 . d C l l b . ( 1 9 7 2 ) ;but.n.rh.n .nd c1bb.( r972.)For t , Ph lu tp . .n d Orb6.(r n Dr .pr . . ! ron)

t .60t

0 . 5 @0.5050.495

1 . 5 1 71,500| ,6261,63r

t ,627l a28

1.616r ,st

o,9220 , 9 3 !o .?560 . 1 1 50 , 9 2 5o.9260 , 1 6 1o,7520.9600,96!0 .7E20 , 8 r 50 , 9 1 20, t9 50 800

1.623l .a2E

sr-o(I)sr-0(I) ls t -0(2)3r-0(r)sr -o(3)sr{( l}s!-o(r) l8r{(2}sr(r){(r)st(r)-o(2)sr(r)-0(3)sr(r)-0(a)sr(2)-0(4)sr(2)-0(5)s1(2){ (6)s1(2){ (7)sr(3)-0(r)51(l)-0(6)st(3)-o(9)st(t)-0(3)sr r)-0(l)s I ( r ) - 0 ( 2 )s r ( ! ) - o ( 3 )s1(l)-0(9)sr(2)-0(r)s!(2)-o(.)sr(2)-0(5)sr(2)-0(a)s!(3)-o(5)s r ( r ) - 0 ( 7 )s!(3)-0(6)sro)-o(9)sr(2)-0(2)sr(2)-0(3)sr(2) '0(8)sr(3)-0(6)sr(3)-0(3)

1 , 6 1 41 654r . 6 1 5r t70r 64 8r , 5 5 11.661

0 . 9 5 20 . 8 6 2o a90o 98 to 6 50 s l0 . 8 6 1

the overlap populationsobtainedusing observed etrahedral anglesand bond distancesn the calculations how a similar but better de-veloped rend whenplotted against he observed ond ength (Fig. 9).Theslopes,ntercepts, orelation,coeffi ients,and I | statistics alcu-lated:for the data in"Figures8 and g are given in Table 6 and Showthat both rendsarehighly significant.The curvesof Bartel"let at. (1970)which relatebond ength o bondoverlappopulationare similar to those obtained.,inFigures 8 and 9


24/36

E r,63Io-l - | 62U'

t 7 0

r . 6 9r 6 8

t 6 7

r 6 6

r 6 5

1 6 4

t 6 l

r 6 0

1 5 9r 5 8r 5 7r 5 6

r 5 5

WITHOUTd ORBITALS;oS i - O = 1 . 5 3 A

tr : MorgorosonileA Beni lo i leD BcrylV EmeroldO Quorlzo Tourmolinec 0ioplose,O Thorfveilile+ Forslerilex Dolo l i le

v

>oTv

o 4 7 0 4 8 0 4 9 o 5 0 0 5 1 o 52 0.55n s i -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 excludilgSi(3d) orbitals for the minerals Iisted in the upper right. Si-O(br) bonds aie iqdicated by open symbols and Si-O(nbr) bonds by bolid ones.'Data takbn fromTable 5.


25/36

GIBBS, HAMIL, LOAISNATHAN, BARTELL, AND YOW

s oo

wrTHOUt d OREIIALSiS i - O ( o b e

O Nigh 9r.nur. CaSiOr

d \ ,- - " . . \

" o o

o \ . 'O e O

D D \ - '

n ( S i - O lFrc. 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.lsfor 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. Datataken.from Table 5.suggesting hat the overlap populations calculated for the sulfates'phosphates, nd the silicatesare comparablewhen the 3d basis unc-tions are neglected.Correlations can also be made between he Si-Obond lenglh, the net non-bonded geminal populations (c/. Barbellet o,1., 970), and the net chargedifierences n Si and O; however,inasmuch as the non-tetrahedralcations were not included n thecalculations,such correlations are not appropriatehere.When the valenceorbital basisset of silicon s extendedo includethe five Si(3d) orbitals, the EHMO calculationgivesn,(Si-O) thatagain correlatewith observedSi-O bond lengthsbut as two separatetrendswith the overlappopulationsor the Si-O(nbr) bondsexceedingthose for the Si-O(br) bonds by about 20 percent (Figs. 10 and 11,Table 5). The overlap populationscalculatedusing the observed

l ? o

t 6 9

t 6 t

r 5 7

t 6 6

t 6 5

1 6 4ooo r 6 so

t 6 2

l 5 l

t 6 0

r 5 9

r 5 6

r 5 c


26/36

MO CALCULATIONS FOR SILICATE IONS 1603tetrahedralanglesand all Si-O = 1.63AareplottedagainstSi-O(obs)in Figure 10 whereas hose calculatedusing observeddistancesandanglesare plotted againstSi-O(obs) in Figure 11.Thetrends in bothscatterdiagramsare highly significantas evincedby the ltl -values(Table 6). The correlationsare much better developedn Figure 11wherez(Si-O) were calculatedusing he observedSi-O distances ndangles.n addition, he slopes alculated or the Si-O(br) bond rends(Figs.10 and 11) are steeperhan thoseof the Si-O(nbr) bonds.TheSi-O(br) bond rends or the data given n Figures8 and9 alsoappearto be steeper,ndicating hat the Si-O(br) and Si-O(nbr) bondover-lap populationscharacteristically onstitute wo distinct populationsevenwhena Si(qp) valencebasisset s used(Table4). The differencecalculatedn n. Si-O) for the Si-O(br) and Si-O(nbr) bonds s due n

Table 6

In te rcep ts , a^ , s lopes , b i l and co r re la t i on coe f f i c ien ts , r ,reg ress lon e{ua t lons f i t ted ro da ra ln F lgs . 8 -11 ; l t l fo r t tfo r linearo : B l = o

b l

- z . z d L

- t 9 . ,

- L , t t 6

-7.024

- 2 . O 7

-0 .816

-0 . 906

- 0 . 5 4 8

l . lobserved S1-0 bond length vs .n (S i -O) ca lc . f ,o r Si -0 = 1 ,63 f r ;F 1 g . 8Observed S1-0 bond length vs .g l s l -0 (b r )1 ca l c . fo r S i -O =1 . 6 3 ; F l g . 8Observed Sl-O bond length vs.n l s i -o (nb r ) l ca l c , fo r S1 -0 =1 , 6 3 ; I ' t g . 8Observed S1-0 bond length vs.n (S i -o ) ca lc . fo r obs . S1 -0 ;F ig . 10Observed S1-O bond length vs.n tS t -O(b r ) l ca1c , fo r S i -O =1 , 6 3 ; F i g . 8observed S1-0 bond length vs .n [S l -O(nb r ) ] ca l c . fo r S1 -0 =1 , 6 3 ; F 1 g . 8observed Si-o bond length vs.n I S 1 - 0 ( b r ) ] c a l c . f o r o b s .s i - O ; F 1 g , I lobse rved S l -0 bond leng th vs ,n I S 1 - O ( n b r ) ] e a l e , f o r o b s .s t -0 ; F ig . 11

o2 . 7 6 3

3 , 0 2 5

2 .L39

3 ,220

2 .3L9

- 0 . 8 4 9 ,

- 0 , 8 9 8 . 3

- 0 . 7 7 5 . 3

-0 ,94 20 .0

- 0 , 7 7 6 . 0

* 0 . 7 0 5 . 5

- 0 . 9 4 1 3 . 4

-0 ,94 L0 .2


27/36

1604 GIBBS, HAMIL, LOAISNATHAN, BARTELL, AND YOW

r 7 0 WITH d OREITALS;S i 0 . | 6 l l

oo

r 5 8

r 5 6

1 5 4

t 7 2

j ! r64ooI t 6 2v,t 60

o 7 4 0 7 6 0 7 8 0 8 0 0 8 2 0 8 4 0 S O 8 a O 9 0 0 9 2 0 9 4 0 9 6 0 9 8n ( s i - 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. nd includingSi(3d) orbitals for minerals listed in upper right of Figure 8. Data taken fromTable 5.

part to the very different environments mposedon these bonds byour modelingof the tetrahedral on in each silicate as an isolatedmolecule. ccording o Bartell et aL.(1970),part of the differencemayalsobe stericand part may be due o electrostaticorcesnot reflectedin the bond overlap populations.The a(Si-O) valuesused o prcpareFigures8 and 10werecalculatedwith the Si-O distances lampedat 1.68Abut with observedO-Si-Oand Si-O-Si angles.Clampingal l Si-O distances t l.68A may elim-inate the bias inherent n using the observedpositionalparameters,but it can ntroducebias becausehe bond lengthsare not allowed orespond o differencesn the bond overlappopulationsestimatedbyHiickel theory. Moreover, or many complexpolymerizedSiOa ons,the Si-O bond lengthscannot,always be clampedat 1.68Aand stillretain the observed nglesand the continuity of the polymerized ni tat the same ime. In thesecases,t may be possible o obtain a rea-sonableestimateof the ra(Si-O) by usingthe observed istances ndanglesn the calculations. his is because ell-developed orrelationsexist betweenn (Si-O) ealculated or Si-O = 1.68Aand thosecalcu-


28/36

fl"t"i

1605O CAI,CULATIONS FON yLICATE IONSht)

N Fq EsE

@

o =O Po . E ;- xr X Qc 6 fo a r. 9 t rb 0 A8 EEO a a Y A> , r 4B HSo . o d

o Qd . E qx d

a ! E6 ; i 6o 3 c o

* Q .o /a :q b . -o ?

I t 9 a- ^ t so v Ff h v

P ' H oc t : Sd a i! . i

- uor,F do ho

.P

oHd

Do

Do

o/o

+.

oJ00

0o$

al,JF -E 8E O o9 lt Ots'

iEg_EEq_EiEi(sqo)- !s


29/36

1606 GIBBS,HAMIL, LOAISNATHAN, AR,TELL,AND YO'WIatedusing he observed ositionalparameters Fig. 12) (seeCameron,1971) .

Concr,usroNsAllen (1970) has implied that EHMO results may be useful indetermininga relationshipbetween ond overlappopulationand bondlength relations ike that obtained or the hydrocarbons c/. Schug,1972)but this is Lhemost he believeshat it can do. Our calculationsappear o have been successfuln demonstrating uch a relationship

and suggesthat at leastpart of the Si-O bond length variationsob-served n the silicatescan be rationalizedn termsof a covalentbond-ing model.Our results also mply that the steric detailsof a silicatecannotbe used ,oprouethe participationof the Si(3d) orbitals in aSi-O bond formation because imilar structural trends are predictedwhen hey are omitted and a valencebasisset s used.Nevertheless,covalentbonding model including d-orbital participation permits anunderstanding f the X-ray emission Dodd and Glen, 1969;O'Nionsand Smith, 1971;Collinset aI., L972) and luorescencepectra(Urch,1969, 971;Collinset aL,,1972) nd he bondpolarizabilitiesRevesz,1971). n addition, t givesa reasonablygoodaccountof the trendsin Si-O bond length variations. Consequentlya covalent bondingmodel that includes Si(3d) orbital involvementshould not be dis-missedas chemically nsignificant.SinceEHMO theory fails to takeexplicit accountof electrostatic ffects,he seniorauthor and his stu-dents are presentlyundertakingMO calculationsat the next level ofsophisticationwhich involve he inclusionof the two electronntegralsand refi.nement f the linear coefficientso self consistency s em-bodied in the CNDOI2 approximationof Pople,Santry, and Segal(1965). These calculationsdo take ordinary Coulornb nteractionsinto account. t will be of interest, herefore,o comparehe CNDO/2results with those obtained in the presentstudy as well as with thoseobtained by the ab iruitiomethod.Finally, the ultimate goal of any bonding heory for the silicatesis to provide some nsight into the physical laws that governatomarrangements, hysical propertiesand stabilities.Molecular orbitaltheory in its simplest orm has been usedwith moderatesuccess ythe organicchemist o rationalize eactionmechanisms,pectra,bondlength variations,and shapesor a largenumberof organicmolecules(Streitwieser, 961). The trends obtained n our study suggest hatextendedHiickel molecularorbital theory, unlike the extendedelec-trostatic valence ule, may be usedto classify and order the bondlength variations n the tetrahedralportion of a silicatewhen A((O)


30/36

o

lf)(D

Iao

!oJo

U)c

o 5 4

o 5 2

o 4 8

o 4 6

o 8 0

o78

o76

o74

o72

o 9 6

o,94

o92

o . 4 8 0 5 0 0 5 2

bO 1o

a '

0 7 0

cl a a . ) . - ' aa

o 88 0,90 0.92 0 94 0 9 6 0 9 8 r O On ( S i - O 6 6 e

Frc. 12. Bond overlap populations, rr(Si-O), calculated with Si-O = 163A us. ra(Si-O) calculated with observed distances, both with observedvalence angles.(a) For Si-O(br) and Si-O(nbr) bonds without Si(3d) orbitals; cor-relation coefficient - 0.94 (Data from Figs.8 and 10).(b) For Si-O(br) bonds with Si(3d) orbitals; r := 0.90 (Data fromFies.9 and 11).(c) For Si-O(nbr) bonds with Si(3d) orbitals; r '= 0.90 (Data fromFigs.9 and 11).


31/36

1608 GIBBS,HAMIL, LOAISNATH.A.N,ARTELL,AND yOW- 0. Moreover,similar calculationsby Urch (1971) have providedvaluable insight into the L2,s X-ray fluorescence pectrarecorded orsilica glass.As the rewards are great, earth scientists are urged toconsider he use of MO theory in their interpretation of spectra,physicalproperties, rder-disordermechanisms,nd phaseransforma-tions of minerals.Even if applicationof the theory provesonly mod-erately successful, t should, for example, improve our ability toevaluateevidencebearingon the physical conditions hat prevailedwhen a mineral or a mineral assemblageorrned.

AcxNowtgocupNtsWe are grateful to Drs. G. E. Brown, Jr., R. G, Burns, J. R. Tossell, andD. R. Waldbaum for reviewing the manuscript and making a number of helpfulsuggestions.ProfessorsF. D. Bloss, D. W. J; Cruickshank, P. H. Ribbe, and J. C.Schug, and Dr. M. TV'.Phillips are thanked for very helpful discussionsand forcritically reading the manuscript. G.V.G. gratefully acknowledges the NationalScience 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.rrrxStepwise regression analysis is a systematic and objective procedure forrecognizing those variables that contribute most significantly to the variationin the response, egardlessof their actual entry points into the model. As a firststep, a correlation matrix is computed and the va.riable most highly correlatedwith the response variable is used to compute a linear regression. The secondvariable to be entered into the regression is the one whose pa.rtial correlationwith the response s the largest, i.e., the one that makes the greatest reduction

in the error sum of squares. Next, the method examines the contribution thefirst variable would have made if the second one had been entered first andthe first variable second. If the partial F' of the first variable is statisticallysignificant at some preselected confidence level, the fust variable is retained;otherwise it is rejected. Assuming that both the first and second variables makea 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 thatthe first and second variables a,re in the regression. A regression equation iscalculated 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 toleanr whether they should be retained in the regression model. If their partial F'sexceedthe preselected evel, they are retained; otherwise they are rejected. Theprocedure is continued and the next most highly correlated variable is enteredinto the regression. Significance tests a.re made as before and the method con-tinued with each variable incorporated into the model in a previous step beingtested. The method is terminated when all the variables have been consideredand no more are rejected (cl. Draper and Smith (1966), p. 178-195 or numericalexample).


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MO CALCULATIONSFOR SILICATE IONS 1609Rnnnnpwcns

Ar,r,nw,L. C. (1970) Why three-dimensional Hiickel theory works and where itbreaks down, 1z O. Sinanoglu and K. B. Wiberg, eds.,SigmntMolecular OrbitalTheory. Yale Univ. Press, New Ifaven, Connecticut, ZZ7-245.- (1972) The shape of molecules. Theoret. Chim. Acta 24, ll7-l}l.Al,unwurwourv, A, O. BesrreNsn, V. EwrNc, K. Ilnonnnc, eNn M. Tneorrnsnnc(1963) The molecular structure of disiloxane, (SiIL),O. Acta Chem. Scand.r7,2455-2t60.Awnnnsow, J. S., aNo J. S. Oconw (1g6g) Matrix isolation studies of Group-IVoxides. I. Infrared spectra a:rd structuresof SiO, SirOr, and SLO".,1. Chem.Phgs. 5r,4189-4196.Anerr, T., ewn T. Zor,rar (1g69) Refinement of a coesite structure. Z. Kristal,Iogr.129,381-387.Benrnr,l, L. S., L. S. Su, enn H. Yow (1g70) Leneths of phosphorus-oxygen andsulfur-oxygen bonds. An extended Hiickel molecular orbital examination ofCruickshank's d,rpn piclure. Inorg. Chem.g, 1908-1912.Bescu, 8., A. Vrsrn, exo H. B. Gney (1965) Valence orbital ionization potentialsf rom atomic ryectra dala. T heoret. Chi,m.Ac t a 3, 459-464.Beun, Wnnxun (1970) Bond length variation and distorted coordination polyhedrain inorganic c-rystals.Trans. Amer. Crgstallngr.,4.ss. 6, 12b-15b.- (1971) The prediction of bond length variations in silicon-orygen bonds.

Arner. M'ineral. s6, lSZz-1599.Bnwt, H. A. (196S)Taqgent-sphere models of molecules: VI. Ion-packing modelsof covalentcompounds. . Chem.Ed.45,Z6g-728.Bonn, F. P., ervn W. N. Lrpscorus (196g) Molecular SCF calculations for SiII. andH"S. J. Chem. Plrys. So,98g-992.Borrr, N. G., arn Y. T. Srnucnrov (1g68) Structural chernistry of organiccompounds of the nontransition elements of Group IV (Si, Ge, Sn, Pb).Zh. Struht.: Kluinu9,63M72 [transl. J. Struct. Chem. 9, 618-6Z2].Boro, D. R., exo W'. N. Lrpscorrs (1g6g) Electronic structures for energy-richphosphates.J. Theoret. BioI. 2s, 40Z4ZO.Bnoww, B. E., aNn S. W. Barr,ny (1964) The structure of maximum microcline.Acta C ystall.ogr. t7, l}gl-1400.Bnowr, G. E. , eNn G. V. Grsss (196g)Oxygen coordination and the Si-O bond.Amer. Mi,neral. s4, l128-l1zg.-, AND- (1970) Stereochemistry and ordering in the tetrahedral por-tion of silicates.Amer. Mineral.55, 15gZ-1607.aNn P. H. Rrnnu (1969) The nature and variation in length of theSi-O and Al-O bonds in framework silicates. Amer. Minerar. 54, 1044-1001.BusrNo, W. R., aNn II. A. Lnw (1964)The effectof thermal motion on the esti-mation of bond lengths diffraction measurements. cta Crystallogr. 17, 142-146.CeruunoN, M. (1971) The Crystal Clrcmi,strg ol Tremolite anil Richteri,te: AStudg of Selected, Anion and, Cati,on &tbstituti.on. Ph.D. thesis, VirginiaPolytechnic Institute and State University, Blacksburg, Virginia.CaNwrr,r,o,E., G. Rossr, eNn L. Urvcennru (1968) The crystal structure of mac-donaldite. Accad,.N,az.Lincei 4s,gW4L4.CmuoNrr, E., aro D. L. R"rruornr (1968) Atomic screening constants fromSCF functions. .I. Chem. Phgs. 38,2686-2689.


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1610 GIBBS, HAMIL, LOAISNATHAN, BARTELL, AND YOWCooe, A., A. Der, Nncno, ewn G. Rossr (1967) The crystal structure of krau-

skopfite. .dccod.Naa. Dei, Lincei 42,859-873.Corr,rr.rs,G. A. D., D. W. J. Cnurcrsnewr, eNn A. Bnnuzn (1972) Ab i,nitiocalculations on the silicate ion, orthosilicic acid and their Zr, a X-ray spectra.J. Chern. Soc., Faradny Trans. I I, 68, I 189-1195.CmtoN, A. F., awo G. Wrr,rrNsor.r (L969) Ad.uam,ed,rcrgoni'c Chemistra. JohnWiley and Sons,New York, N. Y.Corrr,sor.r,C. A. (1939) The electronic stmcture of some polyenes and aromaticmolecules. VII. Bonds of fractional order by the molecular orbital method.Proc.Rogal Soc.,Ser.A,169,413-428.CnurcrsnaNr, D. W. J. (1961) The role of 3d-orbitals in r-bonds between (a)silicon, phosphorus, sulfur, or chlorine and (b) oxygen or nitrogen. J. Chem.Soc.7077,5486-5504.Der,r,rxca,G., eNo P. Ross (1968) Semi-empirical prediction of bond lengths andconformations . R ec. Trau. Chi,rn.87,906-915.Donn, C. G., ewo G. L. Gr,nll (1969) A zurvey of chemical bonding in silicateminerals by X-ray emission spectroscopy.Amer. Mi.neral. 54, 1299-l3ll.Dor,rAsn, W. A. (1965) Reinvestigation of the structure of low cristobalite. Z.Kristallo gr. 721,369-377- (1968) Refinement and comparison of the structures of zoisite andclinozoisite. Amer. M,i,nerar. , 1882-1898.

- (1969) Crystal structure and cation ordering of piedmontite. Amer. Min-eral. 54,71tu_-717.- (1971) Refinement of the crystal structures of epidote, allanite and han-cockite. Amer. M ineral. 56, 447464.DoNNev, G., rwo R. Ar,r,rvrew 1968) SLOrc groups in the crystal structure ofardennite. Acta Crystallngr. B24, 845-855.Dnarnn, N. R., exr I[. Surrn (19ffi) AWIieiI Regression Analysis. John Wileyand Sons, nc., New York.Frrcnn, L. W. (1969) The crystal strusture of grunerite. Mineral. Soc. Amer.Spec.Pap.2, 95-100.- (1970) Refinement of the crystal structure of anthophyllile. Carnegie Inst.W aslui,ngon Year Booh, 68, 283-288.Frscnnn, K. (1969) Yerfeinerung der Kristallstruktur von Benitoit BaTitSilOJ.Z. Kristall,ogr. 729,369-377Fnnno, R. L., eNr D. R. Poecon (1969) Determination and refinement of thecrystal structure of margarosanite, PbCaoSi*;On.. Kristallngr. 728,213-228.Fyru, W. S. (1954)The problem of bond !,ype.Amer. M'ineral.39, S1-1004.Gevrw, R. M., Jn. (1969) Simplified molecular orbital approach to inorganicstereochemistry. . Chetn.Ed. 46,413422.Gruus, G. Y. (1966) The polymorphism of cordierite. I, The crystal structure oflow cordierite. Amer. Mi,nerar. 51, 106&1087.- (1969) Crystal structure of protoamphlbole. Mi'neral. Soc. Amer. Spec,Pap. z, l0l-109.D. W. Bnocx, eNo E. P. Mnecnnn (1968) Structural refinement of hydrousand anhydroussynthetic beryl, AL(Be"Si")Oreand emerald, Al'.nCr".'(Be"Si')Or.Lithos, 1,275-285.Grr.r,nsrrc, R. J. (1960) Bond angles and the spatial correlation of electrons.J. Amer. Ch.em.Soc.82,5978-5983.Gnrrrrtn, W. P. (1969) Raman studies on rock-forming minerals. Part 1. Ortho-silicatesand cyclosilicates.J . Chem.Soc. (A), 1372-L377


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MO CALCULATIONS FOR SILICATE IONS 1611Eeuu,, M. M., G. Y. Grsss, ewo P. H. R"rsse(1971) The location of the hydro-

gen atoms in the water molecule of dioptase. [abstr.]. Geol. Soc. Amer.Abstr. Programs,3, 315.-, G. V. Grsns, L. S. Benrnrr,, eNn I[. Yow (1971) A comparison of Si-Obond lengths wiih EIIMO bond overlap populations. [abstr.]' Geol. Soc.Amer" Abstr. Programs, 3, 589.Horrrvreww, R. (1963) An extended Eiickel theory. I. Ifydrocarbons. J. Ch'em.Phgs.39, 3W-L412.Laucnorv, R. B. (1971) The crystal stmcture of kinoite. Amer. MhwraL 56,193-200.Lezennv, A. N. (1964) Polymorphism of molecules and complex ions in oxygencompounds of silicon and phosphoms. Report 1. Nature of the Si-O-Si bondsand values of the valence angles of oxygen. Izu. Akail. iVauk ,$SSE, Ser.Khi/m.2.235-241.Lmueu, F. (1961) Untersuchungen iiber die Griisse des Si-O-Si Yalenz-winkels.Acta Crgstallogr.14,1103-1109.LoursxerneN, S. J., awo G. V. Grsns (1971) A comparison of Si-O distances inthe olivines with the bond overlap populations predicted by the extendedHiickel molecular orbital (EHMO) theory. [abstr.]. GeoI.$oc. Amer. Abstr.Prog:rams,3,686.AND (1972a) The efrect of tetrahedral angles on Si-O bond over-lap populations for isolated tetrahedra. Amer. M,i,neral s7,16l+-1642.AND - (1972b) Variation of Si-O distances in olivines, soda meliliteand sodium metasilicate as predicted by semiempirical molecular orbitalcalculations. Amer. M ineral. 57,1643-1663.AND - (1972c) Aluminum-Silicon distribution in znnyite. Arner.Mineral. s7, 1068-1087.- AND - (1972d) The Si-O bond length variation in pyrosilicates.labstr.l GeoI.Soc.Amer. Abstr. Prograrns,4,580-581.aNu J. V. Surrn (1971) Crystal structure of tilleyite: Refinement andcoordination. Z. Kri*tallo gr. 132,288-306.McDoNer,o, W. S., exn D. W. J. Cnurcrsrrexr (1967a) A reinvestigation of thestructure of sodium metasilicate, Na"SiO". Acta CrystalX.ogir, 2, 3743._ AND _ (1967b) Refinement of the structure of hemimorphite. z.Kris tallogr. 124, 180-191.Meagher, E. P. (1967) Polym,orphism ol Cord:ieri,te.Ph.D. thesis, Penn. StateUniversity, University Park, Pennsylvania.Mrtcnnll, J. T., F. D. Br,oss, ervn G. V. Grsss (1971) Examination of theactinolite structure and four olher C2/m a.mphiboles in terms of doublebonding. Z. Kri*tall,ogr. 133,?73-300.Mrrcnnrr,, K. A. R. (1969)The use of outer d orbitals in bonding.Chem. Rea.69,157-r78.

Mur,r,rrow, R. S. (1955) Electronic population analysis on LCAO-MO molecularwave function s. l. J. Chem. Phgs. 23, 1833-1846.Nor,r,, W. (1956) Characteristika der Siloxan-bindung in polymeren Anionen(Silikaten) und polymeren molekiilen (Siliconen). Fortschr. Mirrc,ral. 34,63-&1.- (1968) Chemistry and, Technologa of Si.Itcunes, nd ed.. Academic Press,Ine., New York, N.Y.OmnueMrtnn, I{. (1972) The molecular structures of cyclosiloxanes. latrstr.]Fowth Austi,n Sym.posium on Gs,s Phase Molewlar Structure, Ll-12.


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t6l2 GIBBS, HAMIL, LOAISNATHAN, BA&TELL, AND yOWO'IrTroxs, R. K., exo D. G. W. Snrrrn (1971) Bonding in SiO, and Feno" by

oxygen KaX-ray emission pectra.Natwe,231, 130-133.Partxo, J. J., ewl J. R. Cr,enr (1968) The crystal structure and cation distribu-ti,onof glaucophane.Amer. Mineral.s3, 1156-1173.-:, M. Rnss, ewn J. R. Cunr (1969) Crystal chemical characterization ofclinoamphiboles based on five new structure refinements. Mineral Soc.Amer.Spec.Pap. 2,177-186.Petrr,rrvc,LrNus (1929) The principles determining the structure of complex ioniccrystals. J. Amer. Chem. Soc. 51, 101G-1026.- (1939) The Natwe of the Chemical Bond, lst ed,. Cornell Univ. Press,Ithaca, N. Y.- (1952) Interatomic distances and bond character in the oxygen acids andrelated substances. . Plrys. Chern. 56,361-365.- (1960) Th.e Nattne ol tlte Ch.emical Bond", 9rd ed. Cornell Univ. Press,Ithaca, N. Y.Punr,res, M. W., P. H. Rrssp, ewn G. V. Grsss (1971)Refinement of the crystalstructure of danburite. [abstr.] Geol. Soc. Amer. Abstr. Programs 3, 338.Ponr,n,J. A., D. P. S,rxrnr, aNl G. A. Sncer, (1965) Approximate self-consistent'molecular orbital theory. I. Invariant procedures.J. Chem. Phgs,43, L29-135.Rnvnsz, A. G. (1971) r-bonding and delocalization effects in SiO, polymorphs.

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MO CALCALATLONS OR SILICATE IONS 1613YnnuoocnN, J. (1953) Physical properties and bond type in Mg-Al oxides andsilicateir.Amer. M'ineral. $, 5EZ-879.Wrnnren, D. J., elln G. SrrmoNs (lg72) Elastic properties of alpha quartz andthe alkali halides basedon an interatomic force model J. Geophus.Res.77,826-856.Wor,rsnunc, M., eNn L. Hnr,lrnor,z (19b2) The spectra and electronic structureof the tetrahedral ions MnOr-, CrOn- and CIO;. J. Ch,em.Ph,ys. 20,837-843.Zecnanresow, W. H., aNn H. A. Pr,orrrxcEn (1965) Extinction in quartz, ActaC ystallo gr. tB, 7 0-7 14.Manuscrtpt recei,ued,Janunry 11, 197P; accepted, or publi,cati,on,Jw,e 6, 1n2.

Note add,eil i,n proof ba GVG: The electrostatic valence rule proposed byPauling (1929) and extended by Baur (1970) was originally.conceivdd to chaftirc-terize strengths of ionic-type bonds and local charge balance in stable iohiccrystals. It is evident, however, that the model seems to apply equally well tocompounds in which the bonds have a relatively large arriount of covalent char-acter. For these compounds,Pauling (1960,S4Z-548, oohrote 64) has indicatedthat "if the bonds resonate among the alternative positions, the valence of themetal atom will tend to be divided equally among the bonds to the coordinatedatoms, and a rule equivalent to the electrostatic valence rule would express thesatisfaction of the valences of the nonmetal atoms.', Pauling,s statement implies[1] that the strength of an electrostatic bond, s, can be equated with bondnumber, rz (Pauling, 1947,J. Amer. Clrcm. Soc. 69, 542-559); t2l that the correla-tions 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-numbercurve for carbon-carbonbonds; and t3l ihat the chargeson bonded atoms aresmall in agreement with his electroneutrality principle (Pauling, 19ffi, GeneralChemistry, W. II . Freeman and Co.,286-287). The inference that z and s areequivalent is consistent with the observation ihat f(O) and z(Si-O) are inverselycorrelated (Louisnathan and Gibbs, l972b). Although Baur calls his model theextended electrostatic valence rule, he is careful to note that the correlation be-tween observed Si-O bond length and f(O) could be interpreted in terms ofCruickshank's (1961) double bonding model as well as the Born model for ioniccrystals.

silica bond lenght

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