Molecules as models for bonding in silicatesr G. V. Grsss ... · (SiOSi) group in a solid silicate and in a siloxane molecule (Table 1). For example, the disiloxy group in the silicate

American Mineralogist, Volume 67, Pages 421450, 1982

Molecules as models for bonding in silicatesr

G. V. Grsss

Department of Geological SciencesVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

Abstract

Computational quantum chemistry is a powerful method developed by the chemist toelucidate bond length and angle variations, electron density distributions, spectra, reac-tions and energetics of small molecules. Because little difference exists between the shapesand sizes of SiOSi groups in gas phase siloxane molecules and solid silicates, the methodhas been employed to generate potential energy surfaces and deformation maps for avariety of molecules especially designed to model the bonding properties of complexsilicate minerals. The use of computational quantum chemistry in studying the chemicalbonding in these minerals is motivated by a need to understand the physical laws thatgovern their structures and stabilities beyond that afforded by Pauling's rules. Jhe ability ofthis ab initio method to mimic a priori bond length and angle variations, chargedensity distributions and force constants of silicates and other solids indicates that theseobservables are governed in large part by the local atomic arrangement.

A survey of these computations shows that the quantum mechanically derived SiO bondlengths for hydroxyacid molecules and 4- and 6-coordinate Si are in good agreement withthose in comparable silicates. Also, theoretical and experimental deformation maps ofthese bonds show modest accumulations of charge density between Si and O in conformitywith the partial covalent character of the SiO bond. Potential energy curyes generated forthe molecules H6Si2O7, H6SiAlO+-, H6SiBO+- and H6SiBeO?- yield equilibrium bridg-ing bond lengths and angles in close agreement with those in silicates with SiOT(T:Si,Al,B,Be) groups. The relatively narrow range of angles exhibited by the SiOB andSiOBe groups and the wide range of angles exhibited by SiOSi and SiOAI groups conformwith the shape of each curve. Bond lengths calculated for hydroxyacid moleculescontaining first and second period metal atoms are linearly correlated with coordinationnumber as observed in studies of crystal radii. By reproducing the bond strength versusbond length trends and the correlation between bond strength sum and bond length, thecalculations have provided a quantum mechanical underpinning of Pauling's electrostaticvalence rule. The assertion that the electrostatic bond strength, s, can be equated withbond number is corroborated by a strong correlation between s and bond overlappopulation. In addition to providing a good account ofbond length and angle variations insilicates, thiosilicates and aminosilicates, the calculations have also afforded valuableinsights into the compressibilities, the polymorphism and the glass forming tendencies ofsilica. The achievements of the calculations examined in this survey indicate thatcomputational quantum chemistry is destined to become an important tool in mineralogyfor identifying systematic relationships between structural and physical properties and forfurnishing new insights into the many types of reactions involving minerals.

Introduction

Since the development in the early sixties ofmodern single crystal diffractometers, gas phase

' Presidential Address, Mineralogical Society of America.Delivered at the 62nd Annual Meeting of the Society, November3, r9E1.

0003-004)v82l0506-0421$02.00 42r

electron diffraction apparatus and high speed digitalcomputers, the structures of a large number ofsilicates and gas phase and molecular crystal silox-anes have appeared in the literature, providing awealth of accurate bond length and angle data. Oneof the most important results afforded by these datahas been the discovery that very little difference

GIBBS: MOLECULES AS MODELS FOR BONDING IN SILICATES

exists between the size and shape of a disiloxy(SiOSi) group in a solid silicate and in a siloxanemolecule (Table 1). For example, the disiloxy groupin the silicate a-qtartz has an angle of 144' and aSiO bond length of 1.61A which is in close agree-ment with an angle of 144" and a bond length of1.63A in the gas phase molecule disiloxane(H3Si)2O (Almenningen, Bastiansen, Ewing, Hed-berg and Traetteberg, 1963; Gibbs, Hamil, Louis-nathan, Bartell and Yow, 1972; Tossell and Gibbs,1978; Barrow, Ebsworth and Harding,1979; Gibbs,Meagher, Newton and Swanson, 1981). Despite theintermolecular forces, crystallization of this gasphase molecule into a molecular solid has virtuallyno effect on the shape ofthe group (Table 1). Thus,as observed by Barrow et al. (1979), the packing ofthe molecule in the solid seems to conform with theshape of the group in the free molecule rather thanvice versa. Also, Table I shows that the shape ofthe group is virtually the same, on the average, asthat in solid siloxanes, silicates and silica glass.Thus, the forces that control the size and shape ofthe disiloxy group in disiloxane and in o-quartz aswell as other silicates seem to be nearly the same,notwithstanding the absence in the gas phase mole-cule of the long range Madelung potential peculiarto a solid. Thus, on the one hand. a mineral like a-quartz can be portrayed as a molecule of megascop-ic dimensions held together by essentially the sameforces that hold the SiOSi group together in themolecule. On the other hand, disiloxane can beportrayed as a tiny hydrogenated SiOSi moiety ofthe a-quartz structure. Because of the relativelysmall size of the molecule, computational quantumchemistry can be used to evaluate its wave func-tions and to probe the properties of its disiloxygroup (Newton and Gibbs, 1979, 1980; Newton,1981; Sauer and Zurawski, 1979; Meier and Ha,1980; Oberhammer and Boggs, 1980; Ernst, Allred,

Table l: Comparison of experimental bond lengths, R(SiO), andangles for the SiOSi groups in gas phase and solid state molecular

crystal siloxanes and silicates.

R ( s i o ) A / s i o s i ( ' ) R e f e r e n c e

( H a s i ) r 0 ( g a s ) 1 . 6 3

( H 3 s i ) 2 0 ( s o l i d ) 1 . 6 3

S i l o x a n e s ( s o l i d ) a 1 . 6 3

S i l i c a t e s ( s o l i d ) a I . 6 3

S i l i c a g l a s s a I . 6 2

Ratner, Newton, Gibbs, Moskowitz and Topiol,1981). Moreover, because the bonding in the mole-cule appears to be similar to that in o-quartz, wemight expect that molecular orbital (MO) calcula-tions on small molecules with SiOSi groups can beapplied to the groups in o-quartz thereby improvingour understanding of the length and stiffness of itsSiO bond, the energetics and compliance of itsSiOSi angle, its isothermal bulk modulus, and itselectron density distribution (Newton and Gibbs,1980; Newton, O'Keeffe and Gibbs, 1980; Gibbs etal., l98l).

Since 1979 computational quantum chemistry hasbeen used to generate potential energy surfaces fora variety of siloxane and silicate molecules especial-ly designed to model the bonding in a number ofsolid silicates. The equilibrium geometries and en-ergetics provided by these surfaces are in accordwith the experimental geometries and force con-stants recorded for a number of solid state silicates,thiosilicates and aminosilicates (Newton and Gibbs,1979, 1980; Meagher, Swanson and Gibbs, 1980;Swanson, Meagher and Gibbs, 1980; Downs andGibbs, 1981; Geisinger and Gibbs, 1981a; Naka-jima, Swanson and Gibbs, 1980). In addition, defor-mation electron density maps generated for the SiObonds in molecules with one-, four- and six-coordi-nate Si are in qualitative agreement with experimen-tal deformation distributions recently obtained for anumber of silicates (Hill, Newton and Gibbs, 1981;Downs, Hill, Newton, Tossell and Gibbs, 1982;Stewart, in prep.; Newton and Gibbs, in prep.).Additional calculations have been completed incollaboration with Dr. R. J. Hill, CSIRO, Divisionof Mineral Chemistry, Port Melbourne, Australia,Professor E. P. Meagher of the Department ofGeological Sciences at the University of BritishColumbia, Dr. M. D. Newton of the ChemistryDepartment at the Brookhaven National Labora-tory, Professor Michael O'Keeffe of the Depart-ment of Chemistry at Aizona State University,Professor J. A. Tossell of the Department of Chem-istry, University of Maryland (Gupta, Swanson,Tossell and Gibbs, 1980), and with my talented andbright students B. C. Chakoumakos, J. W. Downs,K. L. Geisinger, R. C. Peterson and D. K. Swanson(Chakoumakos, 1981; Chakoumakos and Gibbs,1981a; Downs, 1980; Swanson, 1980; Peterson,1980). The objective of this report is to examine,compare and summarize the results of these manycalculations.

This paper is divided into five sections. In the

L 4 4 A l n e n n i n g e n e t a l . ( 1 9 6 3 )

L42 Barrou ec aL (1979)

t 4 0 N e w t o n a n d C i b b s ( 1 9 8 0 )

t 4 5 c e i s i n g e r a n d c i b b s ( 1 9 8 1 b )

L 5 2 D a S i l v a e t a 1 . ( 1 9 7 5 )

a A v e r a g e d v a l u e s ,


first section. the essentials of molecular orbitaltheory will be examined as it is used to estimate thetotal energy of a closed-shell molecule, to constructmolecular potential energy surfaces, and to opti-mize molecular geometries. This will be followed bya discussion of Gaussian-type basis functions andtheir use in simplifying the two-electron integrals.Mulliken's recipe for calculating bond overlap pop-ulations will then be reviewed and related to thetotal electron density of the molecule. In the secondsection, molecular-orbital-generated geometriesand deformation maps of the SiO bonds in silicatemolecules with four- and six-coordinate Si will becompared to experimental bond lengths and defor-mation maps reported for solid silicates. Next, thecontribution of the d-type polarization functions ofSi to the wavefunctions of the molecules will bediscussed. The third section of the report examinesthe geometry and force constants of the disiloxygroup in the silica polymorphs in terms of a poten-tial energy surface generated for the group in thedisilicic acid molecule. It also compares static de-formation maps of the SiOSi groups in a-quartz,disiloxane and silicon monoxide. The fourth sectiondiscusses the bridging bond lengths and anglesreported for a number of silicates in terms ofpotential energy curves calculated on moleculeswith SiOSi, SiOAl, SiOB and SiOBe groups. Thestiffness of the bridging angle in the thiosilicates andtheir limited assortment of tetrahedral topologiescompared to those of silicates will next be discussedin terms of a potential energy curve generated forthe disilathia (SiSSi) group. Then multiple linearregression methods will be employed to estimatethe dependence of SiO and AIO bond length varia-tions on the bond strength sum and the fractional s-character ofthe bridging oxygen. In the fifth sectionof the report, bond strength versus bond lengthcurves generated with molecular orbital theory forhydroxyacid molecules containing first and secondrow atoms will be compared to empirical onesreported by Brown and Shannon (1973), and theelectrostatic strengths of the bonds in the moleculeswill be correlated with the Mulliken bond overlappopulations provided by the calculations.

Computational quantum chemistry

The molecular orbital method

In the last decade, computational quantum chem-istry (Pople,1973) has become an important tool fordetermining the structure and electron density dis-

tributions of a molecule and for providing insightinto why molecules have their characteristic sizes,shapes and topologies. In addition, the method isamenable to simple interpretative algorithms thatallow the information content of the total wavefunc-tion to be distilled into a relatively small number ofchemically informative quantities such as orbitaland bond-overlap populations, atomic charges anddeformation densities (Newton, 1981).

In the theory of molecular orbitals, the total wavefunction of a closed-shell molecule, ry', is approxi-mated by an antisymmetrized product of one-elec-tron wavefunctions or molecular orbitals, @i,

1 nv & f f iAn 0, ( l ). Y n !

i = t . a

where n is the total number of electrons in themolecule and Aiis the antisymmetrization operator.If r/ is normalized, then the total energy of themolecule, E-o1, is the expectation value of theSchrodinger Hamiltonian

Emor : | *u* r , (z )J

where I/ is the many-electron Hamiltonian operatorwhich, if the nuclear kinetic energy terms areneglected, is the sum ofthe kinetic energy operatorsfor the electrons together with the various Coulom-bic contributions to the potential energy. Minimiza-tion of the energy using the method of undeter-mined Lagrangian multipliers results in the effectiveone-electron Hartree-Fock equation:

FQi: ei6t , (3)

where F is the effective one-electron Hartree-FockHamiltonian operator for an electron in the environ-ment of the molecule, and e1 is the one-electronenergy.

Since atoms are the building blocks for mole-cules, equation (3) may be solved approximately byconstructing each molecular orbital as a linear com-bination of m atomic orbital (AO) basis functions,1,, each centered on the atoms of the molecule,

m

0t - ) x ,C^,v : l

where C,iare the appropriate linear weighting coef-ficients.

If we substitute Eq. (4) into equation (3), multiplyon the left by X* and integrate, we obtain the

(4)


l r lF*,: Ht", + ) p^.1 etvt\o) - |{rxlre:)l nl) ,oL l

where

r | | z lH*u: I x,(r t l - ;v? - > 3l x,( l )drr , (8)

j I

t A r t o j

pxo :25c^ ,c r , (9 )

Roothaan equations for a closed-shell molecule (seeRoothaan (1951), Parr (1964), Pople and Beveridge(1970) and Dewar (1969) for a more rigorous andinformative derivation) :

m

2 ( F * , - e i S r , ) c , i : 0 , ( 5 )v : I

where ei are the orbital energies, S' are the overlapintegrals,

s: f .-pv I x"(1)x"(1)drr, (6)J

where dr1 is the volume element of electron I and

and

t l

Qtvllto) : | | xr(l)x"(1)llrpy-1,(2)y,(2)drirz (10)J J

Subscripts lL, v, )t and o refer to basis functions, thesymbols Xp(l) and x1(2) imply, for example, thatelectrons I and 2 are in atomic orbitals 1, and 1;r,respectively, V2 is the Laplacian operator, Ze is thecharge on the nucleus A and r* is the separationbetween particles p and q. The two-electron inte-grals defined in Eq. (10) represent electron interac-tions between orbital products XrX, (1) and 111. (2),f1r, represents the kinetic energy and the potentialenergy of attraction to the nuclear field of theorbital product X,X, (1) and P1o is the densitymatrix element involving the coefficient products ofthe AO's N1 and Xo summed over each of theoccupied molecular orbitals, fi.

Since each element of the Fock Matrix, Fr,, isdependent upon P1o, equation (5) cannot be solveddirectly. In the absence of such a solution, aniterative procedure is usually adopted. For exam-ple, a minimum basis Huckel-type calculation willbe performed to provide an initial set of densitymatrix elements and then the appropriate atomic

integrals are accurately evaluated. With these re-sults, the F' elements of the Fock matrix and theS' elements of the overlap matrix are constructedto formulate an m x m determinantal equation

lFp, - e;Sr,l : 0 , ( l 1 )

which may be solved for m values of e;. Each newvalue of e; is then substituted in succession intoequation (5) to give m new sets of coefficients, oneset for each e1. The coefficients of the occupiedMO's are next normalized and used to generate anew density matrix which in turn is used to computea new Fock matrix and a new secular determinantwhose non-trivial solutions yield an improved set ofsi values. The procedure is repeated over and overagain, each time starting with the results of theprevious calculation until the elements of the densi-ty matrix converge to a final set of steady valuesthat do not differ by some threshold value (- 10-6)and self-consistency is obtained. This gives the"best set" of MO's from which the molecule's totalenergy can be estimated. When the MO's are writ-ten in LCAO form (Eq. 4), then the total SCF (selfconsistent field) electronic energy can be written as(Pople,1953):

Pp,(Hp, + Fp,) (r2)

Finally, an estimate of the molecule's total ener-gy is found by adding the repulsion energy,2azsZaZslras, between the clamped nuclei of themolecule to equation 12.

Gaussian expansions of Slater type atomicorbitals

In the types of calculations described in thisreport, the expansion in equation (4) generallyinvolves at least one basis function for each AO upto and including the valence shell on each atom inthe molecule. Such a basis set is called a minimumbasis set. For example, the minimum basis set forthe molecule H6Si2O7 would be the hydrogen lsatomic function plus the silicon ls , 2s , 2p, 3s and 3pfunctions and the oxygen ls, 2s and 2p functions.As the size of the basis set is increased. the model-ing of each molecular orbital improves and the SCFtotal energy becomes progressively lower. In theHartree-Fock limit, where in principle an infinitenumber of basis set functions is required, the mod-eling of the MO's can neither be improved nor canthe SCF energy be lowered further. Because the

Ie":i)


number of terms in each LCAO must necessarily befinite for a large molecule like H6Si2O7, it is impor-tant that rapidly convergent basis functions be usedin the calculations. As Slater-type orbitals (STO)seem to best satisfy this requirement, @; in equation(4) is often expanded as a finite sum of STO's(functions with exponential dependence exp(-(r)).But, on the other hand, the two-electron integrals inequation (7) arc extremely dfficult and time con-suming to evaluate when STO's are used as basisfunctions. This is particularly true when these inte-grals range over three or four atomic centers. Asimplification found to alleviate this difficulty is toexpand each STO as a sum of least-squares fittedprimitive Gaussian functions (functions with expo-nential dependence exp(-of)). Since the productof two Gaussian functions is itself a Gaussianfunction on line between the two functions (Boys,1950; Shavitt, 1963), each four-centered integral,for example, may be replaced by a representativetwo-centered integral which may be rapidly evaluat-ed with the computer. Thus, it is common practicein molecular orbital calculations to use a least-squares fitted combination of Gaussian functions(usually up to six) to mimic each Slater-type orbital,thereby speeding up the time of the calculationconsiderably (Hehre, Stewart and Pople, 1969;Binkley, Whiteside, Hariharan, Seeger, Hehre,Lathan, Newton, Ditchfield and Pople, 1978). Asmight be expected, the larger the number of primi-tive Gaussians used in the linear combination, thebetter we can mimic each STO, but, as might alsobe expected, the larger the expense ofthe increasedcomputer storage. Thus, a trade-off must be madebetween the two, depending on the available funds,the quality ofthe trends sought, and the size ofthemolecule.

In this report, STO-3G and STO-3G* basis setswere employed to calculate the bond lengths andangles of the molecules studied. In the case of theSTO-3G basis set, each occupied atomic orbital isrepresented by a single STO basis function which inturn is approximated by a sum of three primitiveGaussian functions. Hence, the abbreviation, STO-3G is used. On the other hand, when a manifold offive 3d-type polarization functions (each approxi-mated by an individual primitive Gaussian function)is added to the STO-3G basis set of Si, the resultingbasis is denoted as a STO-3G* basis (Collins,Schleyer, Binkley and Pople, 1976). Thus, by con-vention, the addition of a star to the abbreviationimplies that the basis has been augmented by d-typewave functions.

The so-called "split valence" 66-31G* basis wasused to calculate the bond lengths and angles anddeformation density maps of the disiloxy group indisiloxane and deformation maps of the SiO bondsin the molecules H+SiO+ and HsSiO6. The K and Lshell electrons in this basis are each represented bya minimal basis (represented by a combination ofsix Gaussian functions each) while the valenceelectrons are each represented by a double set ofAO's (approximated by a sum of three Gaussiansplus an additional Gaussian). The 3d AO's added tothe sp basis of Si serve as polarization functions andappear to be necessary to mimic experimental de-formation density distributions for the SiO bond(Newton and Gibbs, 1979).

C alculat e d mole c ular prop ertie s

The atomic nuclei and electrons comprising theatoms of a molecule strive to adopt a configurationwherein the total energy of the resulting structure isminimized.2 By applying the principles of computa-tional quantum chemistry, the energy of this struc-ture can be estimated regardless of the type ofbonding forces (ionic or covalent) that obtains be-tween the constituent atoms. The optimized geome-tries of the molecules described in this report werefound by minimizing the molecule's total energy bysystematically adjusting the bond lengths and an-gles among the constituent atoms. In addition tooptimizing the geometry, the topography of theresulting energy surface can provide considerableinsight into a molecule's force constants as well asthe expected variability of the molecule's bondlengths and angles. Further, a meaningful explana-tion ofthe bonding characteristics and charge distri-butions of the molecule can be obtained from apopulation analysis whereby the total number ofelectrons in the molecule is partitioned into thevarious atomic and bond contributions which areused to estimate atomic charges, bond overlappopulations and deformation density distributions.

According to Mulliken (1955), the total molecularorbital electron densitv.

n

p(r) : ) df(")dt("),i : 1

( 13)

of a molecule can be expanded in terms of the AO

2 Strictly speaking, this is true only at 0'K, but the efects ofentropy, the TS term, are generally negligible compared to thetotal binding energy at all temperatures of geologic interest.

426

basis as

p(r): ) P,,x[(r)1,(r), (r4)tL,v

which when integrated over all space yields thetotal number of electrons:

(15)l L , v

The bond overlap population between the p",v pairof atomic orbitals is defined bv

n(pv) : 2Pp,Sp, , (16)

which when summed over all the p atomic orbitalson center M and all the z atomic orbitals on centerN, is defined as the bond overlap population n(MN)for the atom pair MN. As discussed by Mulliken(1932,1935) in his original formulation, a positive ornegative value for n(MN) corresponds to net bond-ing (attraction) or antibonding (repulsion) betweenatoms M and N.

For the calculations presented in this report, theGaussian 76 computer program (Binkley et al.,1978) was employed to calculate the Fock andoverlap matrix elements and to solve the appropri-ate equations as discussed above to obtain thelinear coefficients for the best set of molecularorbitals and an estimate of the total energy of themolecule as well as other molecular properties(Schaefer, 19771' Carsky and Urban, 1980).

The SiO bond

Tetrahedral silicate group

The most important structural moiety in manysilicates is the tetrahedral silicate group. The SiObond in this group is observed in silicates to range inlength from 1.55 to 1.76A with a mean value of1.6264, while the SiO bond in the silanol (SiOH)group in four-coordinate SiO3(OH)3- andSiOr(OH)3- groups in hydrated silicates shows asmaller range in lengths from 1.63 to 1.70A with asomewhat larger mean value of 1.67A (Gibbs et al.,1981).

The monosilicic acid molecule, Si(OH)a, is ofparticular interest in that it is the simplest entity tocontain the silicate tetrahedral group. Although themolecule occurs in almost all aqueous environ-


n: ) Pp,Sp,

ments on earth, it polymerizes spontaneously whenconcentrated in amounts greater than about thesolubility of amorphous silica (-115 ppm SiO2 at25'C) and hence has never been isolated for astructural analysis (Iler, 1979). As computationalquantum chemistry has become a standard methodfor determining molecular structure, Newton andGibbs (1980) and Gibbs et al. (1981) undertookstructural analyses of the molecule using molecularorbital STO-3G and Pulay (1969) eradient methods.Assuming that the topology of the molecule in-volves a silicon atom bonded to four hydroxylgroups, the structural analyses indicate the mole-cule to possess Sl (:4) point symmetry with fourSiO bond lengths of 1.65A, four OSiO angl^es of107.1", two of 114.2", four OH bonds of 0.98A andfour SiOH angles of 108.8" (Fig. l). Not only is thisgeometry in close agreement with that of silicategroups in the monosilicates (Fig. 2) and in hydratedsilicates, but also the fundamental breathing fre-quency (8.14 m-t) calculated from the SiO stretch-ing force constant (665 Nm-t) of the moleculeagrees with that observed (8.19 m-t) for tetrahedralsilicate groups in aqueous solution (Gibbs et al.,1981) .

Fig. 1. A drawing of the molecular structure of monosilicicacid, HaSiOa, as determined by molecular orbital methods. Theintermediate-sized sphere at the center of the moleculerepresents a silicon atom, and the four large spheres disposed atthe corners of a tetrahedron and attached to Si represent oxygenatoms. The small sphere attached to each oxygen atomrepresents hydrogen. No significance is attached to the relativesizes of the spheres. (After Gibbs et al., l9El.)


-583.32

- 5 8 3 . 3 3

^ -593.34=o

Ft tJ -Sgg.gS

- 5 8 3 . 3 6

orbitals, they also observed that the L4 X-rayfluorescence spectra of solid and amorphous silicacould be reproduced only when the d-orbitals wereincluded in the basis set. However, in spite of theirsuccess in rationalizing and reproducing the Lz,r X-ray fluorescence spectra of silica, there has been arather protracted discussion in the literature regard-ing the significance of these orbitals (Baur, 1971,1977, 1978; Gilbert, Stevens, Schrenk, Yoshimineand Bagus, 1973; Griscom, 1977). For example,Gilbert et al. (1973) observed in a study of anextended basis calculation for molecular SizO thatthe five d-orbitals on Si make a relatively unimpor-tant contribution to the wave functions of the linearmolecule and that the large population obtained inthe Collins et aI. (1972) calculations for H+SiO+ maybe an artifact of a poor basis set.

In an effort to clarify the extent to which theseorbitals might contribute to the structure of the SiObond in H+SiO+, Newton and Gibbs (1980) complet-ed STO-3G* and "split valence" s,p basis 66-3lG*calculations for the molecule. As both calculationsyielded essentially the same 3d-orbital populations,they suggested that the populations are qualitativelysignificant and not an artifact of a poor basis asindicated by Gilbert et ql. (1973). Optimization ofthe SiO bond lengths of the molecule with a STO-3G* basis, yielded an equilibrium bond length of1.604, well within the range of recorded values inthe silicate tetrahedra of the monosilicates. Thus,the STO-3G and STO-3G* calculations yield bondlengths that bracket the observed SiO bond lengthsin these minerals, notwithstanding the neglect of aMadelung potential in the calculations. Also, de-spite the lack of constraints in the calculations, theratio of the charges on Si and O was found to bealmost exactly 2 to l, the expected ratio for asilicate. As predicted by Pauling, the charge on Si isreduced from +1.36 to *0.81 when the minimalbasis set for the molecule is augmented with themanifold of five d-orbitals on Si.

Deformation density maps for four-coordinate Si

Hlll et sI. (1981) have calculated a deformationdensity map through the OSiO group in HrSiO+ toassess the net transfer of charge from Si and O intothe SiO bond for four coordinate Si' Calculatedusing a 66-3lG* basis with d-type polarization func-tions, the theoretical deformation density was ob-tained by subtracting the superimposed densities ofthe component atoms of the molecule (i'e., the

-583.37r . 5 5 1 . 6 0 r . 65 r .70

R(sio)iFig. 2. The total energy, E1, of the monosilicic acid molecule

(S+ point symmetry) calculated as a function of its SiO bondlength, R(SiO). Data used to prepare this plot were taken fromTable I of Newton and Gibbs (1980). The histogram superim-posed on the plot is a frequency distribution of the experimentalSiO bond lengths in the monosilicates. The average bond lengthfor this group- of minerals is 1.635A compared with a minimumvalue of 1.65A. Er is given in atomic units (a.u.) where I a.u. :

6.275 x 102 kcal mole-r.

Si3d orbital population

More than forty years have passed since Pauling(1939) made the controversial proposal that siliconmay utilize its relatively unstable 3d-orbitals informing double bonds with the oxygen atoms of asilicate tetrahedral group. Assuming that each ofthe bonds has an ionic character of49 percent and abond number of 1.55, he calculated a charge of+0.96 (: 4 - 4 x 0.49 x 1.55) on Si in conformitywith his electroneutrality principle (Pauling, 1948,1952). About ten years ago, Collins, Cruickshankand Breeze (1972) completed calculations for themolecules SiOX- and H4SiO4 with a minimal basisplus d-type AO's on Si and concluded that the d-orbitals may make a significant contribution to themolecular wave functions. In addition to finding arather large number of electrons occupying these

428 GIBBS: MOLECULES AS MODELS FOR BONDING IN SILICATES

promolecule), from the molecular charge density,each atom being located at its appropriate position.The resulting map, displayed in Figure 3a, shows amodest accumulation of electron density in the SiObond with the peak appreciably shifted from themidpoint of the bond in the direction of the oxygenatom as expected from electronegativity consider-ations. The peak has a height of 0.32 eA-3 and issituated 0.65A from the oxygen in close agreementwith the Bragg-slater atomic radius of the atom(Slater, 1972).In contrast, the minimum in the totalelectron density distribution along the bond (Bader,Keeveny and Cade, 1967) indicates an oxide ionradius of 0.954 compared with the somewhat larger"ionic radius" of the ion in MgO and CaO (1.12 and1.194, respectively) determined by Bukowinski

(1980, 1981) and a crystal radius of 1.224 reportedby Shannon and Prewitt (1969). On the other hand,Johnson (1973) has proposed a variable oxide ionradius which decreases linearly from 1.4 to 0.84 asthe field strength of the cation bonded to it in-creases from K+ to Be2+. Clearly this topic war-rants further study.

Despite the large number of structural analysescompleted for the monosilicates, only a few havebeen undertaken to record the deformation densityof the tetrahedral silicate group with which a map ofthe theoretical density may be compared. A surveyof these studies reveals a fluctuation in the SiObonding peak heights ranging from 0.15 to 0.80eA-3 with an average height of 0.35 eA-3. In addi-tion, the distances between the oxygen atoms andthe bonding peaks range between 0.60 and 1.054with an average separation of 0.754. The mainfeatures in the experimental maps conform, on theaverage, with those displayed in the theoreticalmap. For sake of comparison, the experimentalmap for the silicate group in the monosilicate anda-lusite (Peterson, 1980) is displayed in Figure 3b withthe theoretical one calculated for the monosilicicacid molecule. Despite the dissimilar bonding envi-ronments, the two maps show a resemblance whichlends credence to the applicability and transferabili-ty of the molecular wave functions and propertyfunctions to analogous solid materials. In iddition,both theory and experiment indicate a charge densi-ty distribution like that expected from a bondmanifesting significant covalent character. As thetheoretical map was obtained from static wavefunc-tions calculated at absolute zero without zero pointmotion, the comparison in Figure 3 is perforcequalitative. For the comparison to be quantitative,the theoretical density must be averaged over theappropriate vibrational states of the molecule.Nonetheless, since the Debye temperature of amonosilicate is large and since the zero point mo-tion is not expected to have a gross effect on thedistribution, the thermally smeared distribution of atheoretical map should not depart significantly fromthat of the unsmeared distribution displayed inFigure 3a (Spackman, Stewart and LePage, 1981).

Octahedral silicate group

By far the vast majority of silicates contain four-coordinate silicon. Nonetheless, Si is six-coordi-nate in such high pressure silicates as SiO2 (stisho-vite) with the rutile structure, MgSiO3 with the

Fig. 3. A comparison of a theoretical difference deformationmap (after Hill et al., 1981) (a) of the OSiO group in rhemonosilicic acid molecule with an experimental map (b) of thegroup in the monosilicate andalusite, Al2SiO5. The contourinterval is drawn at 0.07 electrons A-3 in (a) and 0.10 e A-3 in(b). The region about the nucleus of each atom in the theoreticalmap represents the core region where the theoreticaldeformation map is not expected to be accurate.

/ a

, , / \/ r \

I - -

r /"\t . 7 ' - ' - - - ] \ . J- :; ;;": 1:.;:1. _\ i : -,- : : _oot . . . \ , , "5- ' )


ilmenite and perovskite structures and KAlSi3O8with the hollandite structure and in such low pres-sure silicates as SiPzOz and thaumasite, Ca:Si(OH)oSO4CO312H2O. Also, in the moderately high pressurephase K2SivISilvOe (Swanson and Prewitt, inprep.), Si occurs in both four-coordination in 3-membered rings and in six-coordination in octahe-dra, linking the rings together into a three-dimen-sional structure. Earlier, with the first synthesis ofstishovite. it was assumed that six-coordinate Siwas adopted only in phases formed at high pres-sures. But, with the discovery of six-coordinate Siin the molecular crystals tris(O-phenylenedioxy)siliconate (Flynn and Boer, 1969), tris(acetylaceton-ato)silicon(IV)perchlorate and tristhylammonium-tris (pyrocatochloro)silicate (Adams, Debaerde-macker and Thewalt, 1979), it is now recognizedthat high pressure is not essential for the synthesisof phases with octahedrally coordinate Si. In fact,an examination of the low pressure phases shows(unlike the high pressure phases) that the oxygenatoms bonded to Si are also bonded to atoms like H,C and P whose electronegativities are greater thanthat of Si. According to Edge and Taylor (1971) andLiebau (1971), these atoms drain electrons from theSiO bonds, thus weakening and lengthening themand reducing their mutual repulsions to the extentthat four-coordinate Si tends to be destabilizedrelative to six-coordinate Si. Under these condi-tions, the oxygen atoms are believed to behave likethe fluoride atoms in the SiF?- ion which is readilyformed in phases produced at low pressures. On theother hand, Shannon, Chenavas and Joubert (1975)have pointed out that six-coordinate Si in thauma-site and SiP2O7 can be explained equally well interms of the sum of bond strengths, po, reaching theoxygen atoms bonded to Si. For example, as poincreases, particularly in excess of 2.0, the SiObonds in a tetrahedral group will tend to weakenand lengthen until a critical value is reached and six-coordinate Si is adopted, thereby reducing the bondstrength sum to a value in better agreement with thesaturated value of 2.0.

In his pioneering work on the chemistry of silica,Iler (1955, 1979) made the proposal that the hexoxo-silicate ion, Si(OH)!,-, forms in aqueous environ-ments as an intermediate product in the polymeriza-tion of silicic acid. Since data are lacking for thisproposal, Gibbs et al. (1981) undertook an analysisof the anion to explore whether there is any theoret-ical reason why it might not form and to see howwell the anion might model the SiO bond lengths of

the hexoxosilicate ion in the mineral thaumansite(Edge and Taylor, 1971). For the calculation, theanion was assumed to consist of a Si atom bondedto six hydroxyl groups each disposed at the cornersof an octahedron with the SiOH angles set at109.47" and with hydrogen fixed at 0.96A from eachoxygen atom (Fig. 4a). The minimum energy SiObond length of the anion was obtained from apotential energy curve calculated with a STO-3Gbasis set. As the calculations converged rapidly ̂ andyielded a minimum energy bond length of lJ7Ainclose agreement with the average bond length(1.784) of the anion in thaumasite, there is noapparent reason why the anion might not form as anintermediate in an aqueous environment containingthe monosilicic acid molecule. But, on the otherhand, Raman and 2eSi NMR studies of the constitu-tion of various aqueous sodium silicate solutionsindicate that the predominant form of the silicategroup is probably tetrahedral (Earley, Fortnum,Wojcicki and Edwards, 1959; Freund, 1973;Engle-hardt, Zeigan, Jancke, Hoebbel and Wieker,1975)'

Fig. 4. Drawings of the molecules (a) Si(OH)Z-' (b)

S(OH)4(OH2), and (c) HsSiOe. The silicon atom centering each ofthese molecules is represented by an intermediate-size sphere,the six coordinating oxygen atoms disposed at the corners of aperfect octahedron are each represented by a large sphere' and

the hydrogen atoms attached to the oxygen atoms are each

represented by a small sphere . The point symmetries of Si(OH)eand Si(OH)a(OHi2 are both C1 whereas that of HeSiOo is H6. No

significance is attached to the relative sizes of the spheres used

to represent the atoms. (After Hill et al., l9El.)


Since the hexoxosilicate ion discussed above isnegatively charged, its minimum energy SiO bondlength may not be comparable with that of a solidlike stishovite.3 Thus, in their study of the crystalchemistry of stishovite , Hill et al. (1981) optimizedthe SiO bond lengths of the neutral Si(OH)4(OH)2molecule. They formed this molecule by adding twoprotons to Si(OH)?-, converting two hydroxylgroups on opposite ends of the molecule to watermolecules (Fig. ab). Optimization of the SiO bondlength, R(SiO), assuming that all the bonds areequivalent, yielded a slightly shorter minimum en-ergy value of 1.75A compared with the averagebond length (1.774) observed for a number ofsilicates and silicate molecular crystals with six-coordinate Si (Fig. 5). The quadratic stretchingforce constant calculated from the curvature of thepotential energy curve is 540 Nm-r, about l0 per-cent less than that calculated for four-coordinate Siin monosilicic acid. The larger force constant of thebond in the latter molecule is consistent with itsshorter bond length and smaller coordination num-ber.

Si3 d orbital population

In a study of six-coordinate second row cations,Tossell (1975) undertook an Xo Scattered Wavecalculation for the highly charged holosilicate ion,SiOS-, when investigating variations in the elec-tronic structure of the oxyanions MgOAo-, AIOS-and SiOS-. As expected from electronigitirrity

"on-siderations, the calculations indicate a substantialincrease in covalency in the series as evinced by anincrease in the width of the valence band. However,the paucity of experimental data prevented compar-ing these calculated spectral data to those of stisho-vite. In addition, as Tossell's X. results for the ionindicate that the higher energy lt2"-type Si3d AO'sare associated with a smaller fraction of electrondensity in the interatomic region than the 2e"-type

3In a bond length study of solids and chemically similarmolecules, Gibbs er a/. (1981) found that the calculated bondlengths for neutral molecules agree to within 0.02A, on theaverage, with those in chemically similar solids. On the otherhand, those calculated for highly charged molecules were foundto depart from experimental values, on the average, by 0.07A inlength. Gupta and Tossell (1981) ascribe this poor agreement toelectron--€lectron repulsions between the extra electrons. Aselectrostatic neutrality generally obtains within the fundamentaldomain of a solid, the values for highly charged molecules arenot expected to be comparable with those in chemicallv relatedsolids.

;o

Flrl

- 7 3 3 . t 7

- 7 3 3 . r 8

- 7 3 3 . r 9

- 7 3 3 . 2 0r . 70 r . 7 5 t . 8 0

R(sio)aFig. 5. A potential energy curve calculated for Si(OH)r(OHJz

(see Fig. 4b) as a function of the SiO bond length, R(SiO). In thecalculation of the total energy, Er, all R(SiO) values wereconsidered as equivalent. A frequency distr ibution ofexperimental SiO bond lengths in silicates and siloxanemolecular crystals with six-coordinate silicon is superimposedon the drawing. The average bond length for the distribution is1.77A compared with a minimum energy bond length of 1.754.(Modified after Hill et al., 1981.)

AO's, he concluded that evidence is lacking for the&sp3 hybidization model proposed for hexacoor-dinate Si compounds by Voronkov, Yuzhelevskiiand Mileshkevich (1975). In a later CNDO/2 MOstudy of the X-ray emission and absorption spectraof stishovite, Brytov, Romashchenko and Shchego-lev (1979) reported 0.9 electrons in the lt2"-typeAO's and 1.0 in the 2e"-type AO's of the holosili-cate ion. An examination of the STO-3G* wavefunctions described above for the S(OH)4(OH2)2molecule indicates a somewhat smaller populationof 3d electrons, with 0.6 electrons in the 2e"-typeand 0.3 in the ltzs-type AO's. A 66-31Gx basiscalculation for the molecule indicates 0.4 electronsin the 2e"-type and 0.2 in the lt2"-type AO's, for atotal of 0.6 electrons in the 3d AO's of Si (Hill et al.,1981). Also, a total of 0.6 electrons was obtained inthese AO's in similar calculations for SiO2 andH4SiO4 molecules (Pacansky and Hermann, 1978;Newton and Gibbs, 1980). Thus, the 3d type polar-ization functions on Si seem to make about thesame contribution to the wave function of a mole-cule regardless of the coordination number of Si.

A population analysis of the STO-3G wave func-tions of Si(OH)4(OH)zyielded a charge of * 1.40 onSi compared with +1.36 obtained for four-coordi-nate Si in HrSiO+. However, the charge on the six-

GIBBS: MOLECULES AS MODELS FOR BONDING IN SILICATES 43r

coordinate Si is reduced to +0.74 when the 3d typeAO's of Si are added to the basis set. This chargecompares favorably with that (+0.97 : 4 - 6 x 1.03x 0.49) calculated with Pauling's (1952) method fora six-coordinate SiO bond length of 1.76A. It isnoteworthy that Pauling's method yields a charge of- +1.0 on Si regardless of whether it is bonded toeither four or six oxygen atoms. Nonetheless, as thecharge on an atom in a molecule or a solid isarbitrary and not an observable, there is no experi-mental technique that can be used to verify thesenumbers (Stewart and Spackman, 1981).

Deformation density maps for six-coordinate Si

Of the silicates with six-coordinate Si, stishoviteis the only one for which deformation density mapshave been produced from X-ray diffraction data.Actually, because of its chemical and structuralsimplicity, the mineral is ideally suited for such acharge density study. The structure of the mineralconsists of a framework of corner and edge sharingsilicate octahedral groups. As the octahedral shar-ing coefficient is 3.0, each oxygen atom in themineral is bonded to three Si atoms lying at thecorners of a triangle. The Si atoms occupy octahe-dra that share opposite edges with other octahedrato form interconnected octahedral chains along c.

A deformation map calculated through a SiOaplane of one of these octahedra is displayed inFigure 6a (Hill et al., 1981). The most prominentfeatures in the map are peaks of electron density ineach bond located about 0.684 on the average fromoxygen. As may be expected, the peaks in theshorter equatorial bond are significantly larger(0.47e A-3; than those in the longer apical bonds(0.29 eA-3). Interestingly, these peaks have aboutthe same heights as those recorded for the SiObonds in the monosilicates. On the other hand, thebonding peaks in stishovite are significantly smallerthan those (-0.85 eA-r; in a-qruarlz, where thebonds are -0.2A shorter in length. However, weare unable to offer a satisfactory explanation of whythe peak heights in the monosilicates are significant-ly smaller on the average than those in a-quartz.

To assess the net charge transfer of electronsfrom Si and O into the SiO bond for six-coordinateSi, Hill et al. (1981) completed "split valence" 66-31G* calculations for two different molecules withsix coordinate Si. In devising the topology of thesemolecules, they considered three factors: (1) thesimulation of the bonding requirements of both Siand O; (2) the flexibility of an adequate AO basis

set; and (3) computational feasibility. Earlier stud-ies by Newton and Gibbs (1979) on the disiloxanemolecule indicate that a 66-3lG* basis is required toyield qualitatively meaningful deformation densi-ties. Because of the very large size of this basis,computational feasibility limited it to moleculeswith a few atoms, thus placing severe constraints onits ability to model the Si and O environment instishovite. With these limitations in mind, Hill et al.(19S1) completed calculations for the two followingmolecules: (1) the S(OH)4(OH2)2 molecule (Fig. ab)and (2) an octahedral molecule with the same for-mula (HrSiO6), but with the eight H atoms symmet-rically arranged on the 3-fold axes of the SiO6octahedral group in conformity with Or G 4lm32lm) point symmetry (Fig. 4c).

The deformation map for the S(OH)4(OH2)2 mol-ecule was deficient as a model for the six-coordi-nate Si in stishovite in that it lacked any significantbuild up of electron density in the SiO bonds of themolecule. At first glance, the HaSiOo molecule mayalso seem to be deficient as a model, but a carefulscrutiny of its topology shows that it satisfies thebond strength sum requirements in stishovite, i.e.,the bond strength sum to each oxygen atom isexactly 2.0. Moreover, the features of a deforma-tion map calculated for a SiOa plane of the moleculeare similar to those in the experimental map forstishovite (Fig. 6b). Not only are the bonding peaksin the theoretical map (0.31e A-3) similar in heightto those in stishovite, but also the peaks are locatedabout 0.654 from the oxygen atom compared with abond peak to oxygen atom separation of 0.684 inthe mineral. Clearly, both theory and experimentindicate that the bonding in stishovite has apprecia-ble covalent character, but its smaller bondingpeaks relative to those in o-qruarlz suggest that thecovalent character of the bond may be less for six-than for four-coordinate Si.

The disiloxy (SiOSi) grouP

Silicates exhibit a wide variety of polymerizedtetrahedral silicate groups of differing topologiesranging from the oligosilicates with oligomers ofcorner sharing silicate tetrahedra to the tectosili-cates with continuous three dimensional extensionsof corner sharing tetrahedra (Liebau, 1980). Thepolymerized tetrahedral groups in this latter class ofsilicates are held intact by the important disiloxygroup in contrast with the monosilicates whichcontain insular silicate tetrahedral groups. In the

i*iil;;::NiwD:-',... :ji'* ,"""" _

'.-Lr -Z


Fig. 6. An experimental deformation map (a) calculatedthrough the apical and equatorial bonds of the holosilicate anionin- stishovite, a very high pressure polymorph of SiO2, comparedwith a theoretical map (b) calculated through a SiOa plane of theH8SiO6 molecule displayed in Figure 4c. The contoui interval isdrawn at 0.05 e A-3 in (a) and 0.07 e A-3 in (b). In (a), the zerocontour is drawn as a long-dashed line, the negative contours areshort-dashed lines, and the positive contours are as solid lines.(After Hill et al., 1981.)

last section, we examined a series of calculationsfor the SiO bond in molecules with four- and six-coordinate Si and observed that the optimized bondlengths are in good agreement with those in compa-

rable silicates. In this section, geometries and de-formation density distributions generated for themolecules H6Si2O7 and H6Si2O will be comparedwith those observed for the silica polymorphs. Thenthe bond lengths and angles generated for rings ofdisiloxy groups will be compared with those insilicate and siloxane solids.

Disiloxy groups in the silica polymorphs

The disiloxy groups in the silica polymorphsexhibit a relatively wide range of SiOSi angles,varying from about 135 to 180". As the angle wid-ens, the bridging bonds tend to shorten by a smallyet significant amount. In fact, they appear toshorten nonlinearly when plotted against the anglebut linearly when plotted against either the hybrid-ization index, \t (: - l/cosZSiOSi defined by thesuperscript spn), or the fraction s-character, /l :(1 + \1-r, of the bridging oxygen of the disiloxygroup (Gibbs et al.,1972; McWeeny, 1979; Newtonand Gibbs, (1980). For a brief but very informativediscussion of f, as provided by the concept ofhybridization, the reader is referred to an excellentpaper by Newton (1981) who examines the roleplayed by the hybrid AO's of the bridging oxygen inthe disiloxy group and provides a quantum mechan-ical underpinning of the hybrid concept in terms ofab initio hybrid AO's obtained with localized mo-lecular orbital theory.

In an attempt to elucidate the nature of thedisiloxy group, Tossell and Gibbs (1978), Meagher,Tossell and Gibbs (1979), DeJong and Brown(1980a, b) and Lasaga (1982) have employed semi-empirical CNDO/2 theory to calculate a series ofpotential energy curves for disiloxane and variousmolecules containing the disilicate group and havefound that the minimum energy angle for a fixed setof SiO bond lengths is in good agreement with atypical angle in the silica polymorphs. It was alsofound that the broad range of angles recorded forthese minerals can be interpreted in terms of theshape of the curve for the disiloxane molecule(Tossell and Gibbs, 1978). In a recent study of thecorrelation between bond length and angle, the SiObond lengths in disilicic acid were optimized as afunction of the SiOSi angle using STO-3G theory(Newton and Gibbs, 1979, 1980). The resultingcorrelation displayed in Figure 7 agrees to withinabout 0.03A with that observed for coesite, a highpressure polymorph of silica (Gibbs, Prewitt andBaldwin, 1977a). Also, as observed for the silicapolymorphs, the mean SiO bond lengths for the

/.:-,/ -

t'..::

si2

'<'}

-/.t

\ \ ̂ A

\( b ) \ .


o 4 l 6 o

Ia(r t .58

r . 56

t s ios i f sFig. 7. A comparison of the experimental SiO bond lenghs,

R(SiO), in coesite, a high pressure polymorph of SiO2,(uppermost curves in (a) and (b)) with those calculated for thebridging bonds in the disilicate molecule, H6Si2O7, with itsbridging angle being fixed in succession at 140, 160 and 180"(lowermost curves in (a) and (b)). The bond lengths in bothcoesite and the molecule vary nonlinearly when plotted againstzSiOSi and linearly when plotted against/" = l/(l + \2) where )'2: - l/coszSiOSi is called the hybridization index of the bridgingoxygen atoln because its state of hybridization is given by thesymbol spr'. lAfter Newton and Gibbs, 1980.)

molecules decrease linearly with the hybridizationindex of the bridging oxygen (Meagher and Gibbs,1976).

The energetics of the disiloxy group

In a study of the quadratic bending and stretchingforce constants of the disiloxy group, Newton andGibbs (1980) used STO-3G basis functions to calcu-late three potential energy curves for the disilicicacid molecule, H6Si2O7, as a function of the SiOSiangle. To maintain the average SiO bond length ofthe molecule at 1.624, the bridging bond lengthswere set in succession at 1.59, 1.62, and 1.65A,whereas the nonbridging bonds were set at 1.63,1.62 and 1.614, respectively. For simplicity, theOSiO and SiOH angles were each set at 109.47" andthe neutralizing protons were placed at 0.96A fromeach of the nonbridging oxygen atoms. The threecurves that resulted from the calculations haverespective minima becoming progressively broaderwith each curve rising gradually with increasingangle to 180". In contrast, the curves rise steeply (inthe region of -120') with decreasing angle from theminima. When a three point parabola was fit to eachof these curves, Newton et al. (1980) found that theresulting symmetric SiOSi bending constant is inrough agreement with that reported for a-quartz.They interpreted this to mean that the bonding

forces in the two systems are comparable and thatthe force field for the group in the molecule may beapplicable and transferable to a-quattz' Moreover,since the SiO stretching (-850 Nm-t) and the OSiObending (-100 Nm-t) force constants are signifi-cantly larger than that of the SiOSi bending forceconstant (-10 Nm), the bulk modulus of the mineralwas concluded to be controlled in large part by thecompliant nature of the SiOSi angle. When it wasassumed that the dominant contribution to theisothermal bulk modulus, K (:VdPl|Vli, is thebending force constant of the SiOSi angle, thecalculated bulk modulus (0.397 megabars) wasfound to be in good agreement with the experimen'tal value (0.393 megabars) extrapolated to 0'K. Thebending force constant, when examined as a func-tion of the shape of the disiloxy group, was found toincrease as the bond lengthened and the anglenarrowed. Since the bulk modulus appears to becontrolled by the compliance of the SiOSi angle, theisothermal pressure derivative of the bulk modulusshould also decrease as the mineral is compressedand narrower equilibrium angles are adopted (Ross

and Meagher, 1981). The isothermal pressure deriv-ative of the bulk modulus, K' (: AIVAPIT), shouldalso be relatively large inasmuch as the SiOSi angleshould narrow at a relatively rapid rate upon initialappl icat ion of pressure (Bass, Liebermann,Weidner and Finch, 1980). Eventually, when theangle is compressed to about 120o, nonbondedSi ' ' ' Si and O ' ' 'O, repulsions should dominateand the value K' should approach a constant value.Finally, temperature and pressure changes shouldhave a relatively large effect on the soft SiOSi anglewhile they should have a moderate effect on theOSiO angle and little effect on the stiff SiO bond ino-quartz (Levien, Prewitt and Weidner, 1980; Rossand Meagher, 1981; Lager, Jorgensen and Rotella,1982).

Potential energy surface of the disiloxy group

In a recent examination of the energetics of thedisiloxy group, Meagher et al. (1980) calculated thepotential energy of HeSizOz for more than 70 differ-ent pairs of bridging bond lengths and angles. Theresulting total energies, plotted as a function ofR(SiO) and ZSiOSi and contoqred, generate thesurface displayed in Figure E. A study of the surfaceshows for the most part that the potential energy ofthe group varies relatively slowly with angle in thevicinity of the minimum but relatively rapidly with

r 8 0t 6 0t 4 0


t . 65

t20 r40 r60 r80tsiosi(.)

Si2O7, plotted as a function ofirs bridging bond length, R(SiO), andto increments of 0.001 a.u. (= 0.6257 kcal mole r) relative to the

'r sake of convenience, the 0.005 a.u. increments are indicated bye data points plotted on the energy surface are experimental bond:oesite, tridymite, low cristobalite and a-qnartz. (After Meagher er

"19reo9E,

change in bond length. The minimum energy config-uration is located at the bottom of a relativelynarrow energy valley blocked at one end andbounded laterally by steeply rising energy barriers.In contrast, the other end of the valley is boundedby a gradually rising surface. It is also observed thatthe valley shows a slight curvature which conformswith the curvilinear bond length-angle trend ob-served for coesite (Fig. 7). Superimposed upon theenergy surface are the experimental bond lengthand angle data for the disiloxy groups in the silicapolymorphs (Hill and Gibbs, 1979; Gibbs et al.,1981). The data follow the general trend of thesurface, but the observed bond lengths are about0.024 longer on the average than thaidefined by thevalley bottom (Fig. 8). Despite this difference,which may be related to lattice vibrations at roomtemperature, the fact that the bond length and angledata for the silica polymorphs fall close to the valleybottom indicates that the energetics of the disiloxygroup in the disilicic acid molecule may be trans-ferred and applied as a model to account for theenergetics and geometry of the group in the silicapolymorphs.

The barrier to linearity is defined to be thedifference between the total energy of the moleculeevaluated for a straight bridging angle and thatevaluated at the minimum energy angle. It is small(- 3kT at room temperature) and indicates that a

relatively small amount of energy will be expendedin deforming the SiOSi angle from its minimumenergy value to 180". If the bonding forces indisilicic acid and the silica polymorphs are similar,then the SiOSi angles in the latter may be readilydeformed from their equilibrium values to form aperiodic extension of silicate tetrahedra withoutunduly destabilizing the resulting structure. Thus, abroad continuum of SiOSi angles is expected tooccur in agreement with the relatively large range ofobserved angles. Also, because ofthe steep energybarrier blocking the valley at narrow angles, SiOSiangles less than -120" are indicated to be unstable.Finally, the glass forming tendencies and the abilityof silica to exist in a relatively large variety ofpolymorphs depending on the ambient temperatureand pressure may also be ascribed in part to theflexible nature of the disiloxy group and the easewith which its angle can be deformed from itsequilibrium value without excessively destabilizingthe final structure.

Rings of disiloxy groups

A silicate ring of difunctional units differs from acomparable siloxane ring in that each Si atom in thesiloxane ring is bonded to two bridging oxygenatoms and two nonbridging organic groups, where-as in the silicate ring each Si atom is bonded to fouroxygen atoms (Noll, 1968). In spite of this impor-

GIBBS: MOLECULES AS MODELS FOR BONDING IN SILICATES 415

tant difference, Chakoumakos et al. (1981) havefound that a close correspondence exists in theshape of the disiloxy group in silicate and siloxanerings. The SiOSi angles in 4-, 5- and 6-memberedsilicate rings exhibit a wider range of angles (130 to180') and a wider angle on average (145') than the 3-membered silicate rings which exhibit a range ofvalues from 120 to 140" with an average angle of130'. Despite a somewhat smaller range of angles(125 to 170") and a paucity of 5- and 6-memberedrings, the average angles exhibited by siloxane ringsare practically the same as those observed forsilicate rings (Fig. 9). Also, as observed for thesilica polymorphs and other silicates (Brown e/ c/',1969; Brown and Gibbs, 1970; Gibbs et aI., 1972:'Gibbs et al.. 1974; Gibbs et al., 1977a; Hill andGibbs, 1979), the bridging SiO bond lengths in bothsiloxane and silicate rings appear to shorten withincreasing SiOSi angle (Ribbe, Gibbs and Hamil,1977).

In their STO-3G calculations, Chakoumakos et

Siloxones6 - Membered

c/. (1981) optimized the SiO bond lengths and SiOSiangles for several molecules, including cyclotrisi-loxane and cyclotetrasiloxane which consist of a 3-and 4-membered ring, respectively' To reduce com-puter costs, the calculations were made assumingnfSiUl : 1.50A, zOSiH and zHSiH : 109.47'andvarious point symmetries for the molecules. As

cyclotrisiloxane has yet to be isolated, no compari-son could be made between its calculated and

observed bond lengths and angles. On the otherhand, the shape of the disiloxy group in cyclotetra-siloxane determined for a gas phase molecule(Glidewell et al., 1970), shows a fairly close corre-spondence with that calculated (Chakoumakos et

c/ . , 1981):

obs. calc.R(SiO) 1.63A 1.614zsiosi 149" 142"zosio lr2" 111'

Chakoumakos et al. (1981) also discovered that the

r ings

) 1 6 0

ls iost(")t20

5- Membered r ings c+ '

C 3 "Dgt t

Fig. 9. The SiOSi angle frequency distributions for rings ofdisiloxy gtoups in silcates and siloxanes. The average angle for each

distribution is represented by "

UotAfu." arrow while regular arrows mark the SiOSi angles of cyclotrisiloxane and cyclotetrasiloxane

molecules optimized for various point symmetries. The point symmetries assumed in optimization of the angles are stated at the

origin of each regular anow. (After Chakoumakos, 19E1.)

Dsn

S i l icotes

4 - Membered

ls iosl(")

GIBBS: MOLECULES AS MODELS FOR B)NDING IN SILICATES

bridging bond lengths and angles calculated forthese molecules reproduce the correlations dis-played in Figure t0 between R(SiO) and /l forsiloxanes and silicates. Note that the calculatedSiOSi angle frequency distributions plotted in Fig-ure 9 correspond fairly well with those observed. Itwas also observed that the maximum SiOSi anglethat a 3-membered ring of regular tetrahedra canadopt is 130'. Inasmuch as this is about l5o narrow-er than the minimum energy angle of the SiOSigroup, they went on to suggest that the bond in a 3-membered ring is strained and less stable than thatin a larger ring which may adopt a wider, morestable angle by embracing a nonplanar configura-tion. However, they did note that a 3-memberedring may be favored in a high pressure environmentwhere nalrower SiOSi angles are indicated to bestabilized (Ross and Meagher, 1981). Chakoumakoset al. (1981) also suggested that 3-membered ringsshould be relatively uncommon in silica glass andmelts where more flexible 4-, 5- and 6-memberedrings should predominate.

Deformation maps for the disitory groupIn an earlier section of this report, we examined

the tetrahedral distribution ofcharge density in the

monosilicic acid molecule and noted a net transferof density from Si and O into the SiO bond region informing a polar covalent bond. In this section, adeformation map of the disiloxy group in disiloxanewill be compared with a static deformation map ofthe group in o-quartz (Fig. 11).

The map of the SiOSi group in disiloxane wascalculated for the observed geometry of the mole-cule, using a 66-31Gx "split valence" basis wavefunction (Newton and Gibbs, in prep.). The result-ing map displayed in Figure llb indicates that therehas been a transfer of electron density from the Siand O atoms to a peak in each bond and also to thelone pair region on the back side of the bridgingoxygen atom. Interestingly, these peaks are offsetfrom the line between Si and O toward the interiorside of the SiOSi angle in conformity with thenarrower angle between the hybrid orbitals on thebridging oxygen atom and with the suggestion(Newton, 1981) that these orbitals play a crucialrole in modeling the electron density distribution ofthe disiloxy group. Also, in their study of disilox-ane, Newton and Gibbs (in prep.) optimized themolecule's geometry using a "split valence" basisand obtained an equilibrium SiOSi angle of 149", aSiO bond length of 1.6424 and a dipole moment of0.32 I compared with experimental values of 144a,1.6344 (Table 1), and 0.24D (Varma, MacDiarmidand Miller, 1964).

In a mapping of the electrostatic properties fromX-ray data for a-qaartz, Spackman et al. (1981)computed a static deformation map for the disiloxygroup in the mineral. This map displayed in Figure1la represents the deformation density of the bondat absolute zero and, as such, may be compareddirectly with the theoretical map of the bond dis-played in Figure 1lb. As observed in the theoreticalmap for disiloxane, the bonding peaks are arcuate inshape and are situated closer to the more electro-negative oxygen atom. However, the electron den-sity goes to zero on both sides of each bondingmaxima in the theoretical map in contr'ast with themap for a-qurtz which shows the bonding peakssuperimposed on a continuum of charge densitybetween the bonded centers. In fact, the chargedensity distribution in a-quartz appears to besmeared more than one would expect for a localizedbonding model. Also, the heights of the bonding

4 Durig et at. (1977) report a SiOSi angle for disiloxane of149+2o.

S i l oxo n es

U)

t

0 3 0 0 4 0 0 5 0 0 3 0 0 4 0f s = l O l ( l + ) , 2 )

o 5 0

Fig. 10. Experimental SiO bond lengths, R(SiO), for rings ofdisiloxy groups in siloxanes (a) and silicates (b) plotted againstf: ll(l + )r2) where )t2 = -llcosZSiOSi. The solid linessuperimposed on both figures were determined by linearregression analyses. The five data points plotted as solid circlesrepresent theoretical values obtained by optimizing the bridgingbond lengths and angles of several cyclotrisiloxane andcyclotetrasiloxane molecules with various point symmetries.The dashed lines denote regression lines fit to the theoreticaldata. The SiO bonds used to prepaxe (a) each have a bond

++ + +

GIBBS: MOLECI]LES AS MODELS FOR BONDING IN SILICATES

Fig. 1 l. Comparison of an experimental static deformation map (a) of the disiloxy group in o-qnartz with a theoretical map (after

Spackman et al., (1981) (b) of the bond in disiloxane calculated with a 66-3lG* basis (after Gibbs and Newton, in prep.). The contours

are drawn at intervals of 0. 10 e A-3 in (a) and 0.07 e A 3 in (b).

peaks in the mineral are significantly larger (0.85eA-3) than those calculated for disiloxane. Thesepeaks are even larger than those (0.6e A-3) in adeformation map of the silicon monoxide moleculewhich was calculated at the Hartree-Fock limit(Stewart, in prep.). Nonetheless, the bonding peaksin SiO are arcuate in -shape as in o-quartz and aresituated closer (0.60A) to the oxygen atom. Asrecorded for the disiloxane molecule, the electrondensity associated with the bonding peak in SiOalso drops to zero between Si and O. There is,however, a conspicuous absence of charge densityin a-quartz on the back side of the oxygen atomsthat may be ascribed to lone pair density. To ourknowledge, such an accumulation in the lone pairregion has yet to be observed in a silicate.

In addition to the static maps for a-quartz, defor-mation maps have also been reported for the SiOSigroups in pectolite (Takeuchi and Kudoh, 1977),coesite (Gibbs et al., 1978) and enstatite (Ghose,Wang, Ralph and McMullan, 1980). The chargedensity distributions recorded for the bonds in theseminerals are in closer agreement with the theoreti-cal map in that the bonding peaks between Si and Oaverage 0.35eA-3. The electron density between Siand O drops to zero in each of these minerals ratherthan showing peaks superimposed on a continuumof charge density between the bonded centers as ina-quartz. The modest peaks of charge density in thebond region give the same picture of a covalentbond as was inferred from aspects ofthe bond suchas overlap population (Gibbs et al., 1972; Newton

( b)

/

Il il l

\II

I

/ - -))


and Gibbs, 1980). However, the bond picture issomewhat obscured in that the theoretical mapsexhibit nodes between the bonding peaks and Si.This feature is observed in all the theoretical mapscalculated and will be discussed at a later time(Newton and Gibbs, in prep.).

SiOT groups (T = Si, Al, B, Be)

The shape and variability of the SiOT groups insilicates

Assuming that the chemical interactions in gas-phase molecules and solids are essentially the sameand that carefully selected molecules can serve asmodels for describing the SiOT bridging angles insolids, Tossell and Gibbs (1978), Meagher et al.(1979) DeJong and Brown (1980a, b) and Lasaga(1982) computed bridging angle potential energycurves for several siloxane and silicic acid mole-cules, using the CNDO/2 method. Not only werethe average bridging angles in various silicatesreproduced, but also their ranges seem to conformwith the shapes of the curves. In the last section, weexamined the nature of the disiloxy groups in disili-cic acid and various ring silicates and showed thattheir equilibrium geometries conform with the bondlength and angle variations in comparable silicates(within the experimental error). Then we found thatthe group comprises significant accumulations of

bonding density in conformity with its covalentcharacter. In this section. we will examine a similarset of curves generated for the SiOT groups in themolecules whose optimized bridging bond lengthsand angles are given in Table 2. As observed in theTossell and Gibbs (1978) study, the range of anglesin solids conforms for the most part with theenergetics of the bridging bonds in various silicatemolecules (Downs and Gibbs, 1981; Geisinger andGibbs, 1981b). Equally important, we will show thatthe bond length and angle trends provided by thesemolecules mimic experimental trends betweenR(TO), /i and the bond strength sum to the oxygenatom.

The tetrahedral bond lengths and angles of themolecules listed in Table 2 were optimized withtheir OTO and TOH angles fixed at 109.47" andtheir OH bond lengths fixed at 0.964 (Fig. t2a). Thehydrogen atoms bonded to the bridging oxygens ofthe H6T2O6(OH) molecules in Table 2b were locat-ed in the plane of the bridging angle, bisecting itsexterior angle (Fig. 12b). An examination of Table 2shows, despite the relatively large variation of theindividual SiO bond lengths (-0.104), that themean SiO bond length for each molecule deviates,on the average, by only 0.014 from the grand meanSiO bond length. By the same token, the individualAlO, BO and BeO bond lengths show a similarvariation, but again the individual mean bond

Table 2: Optimized bond lengths (A) and angles (') for T,OT2 bonds in H6T,T2O7 and H7T1T2O7 molecules (see Fig. l2);po is the bondstrength sum to the bridging oxygen and f" : l(l + 12) where )r2 : -secZTrOTz

( a ) H 6 T l T 2 o 7 m o l e c u l e s

H6T, tTzO7 n(T to) t . n (TzO)u , R(T tO) .u ,

H 6 S i S i O T 1 . 6 0

H6SiA1OT 1 59

uus i ro r l - 1 .60

Hus iBeof - 1 . s8

uuaurof - 1 .6suurnof- 1.4r

1 . 6 0

R ( T ^ 0 ) Z T " o T ^ p F R e f .z n o t L z o s

I . 6 6 1 4 2 2 . 0 0 O . 4 4 I a , b , c

L . 1 2 1 , 3 6 r . 7 5 0 . 4 1 8 b , c

1 . 4 8 1 2 6 I . 7 5 0 . 3 6 4 c

r . 6 2 1 3 1 1 . 5 0 0 . 3 9 6 d

I . 7 4 1 5 0 1 . 5 0 0 . 4 6 9 c

1 . 4 9 L 3 3 1 . 5 0 0 . 4 0 5 c

1 . 6 9 r . 6 7

I . 4 4 1 . 6 4

1 . 6 0 1 . 6 8

1 . 6 5 7 . 7 4

r . 4 t ) - . 4 9

HTTtr20 j

nrsrs io )+ r .71- r .7 rH 7 S 1 A 1 O 7 I . 6 7 1 . 8 0

H T S i B O T L . 6 7 1 . 5 1

n r s i n e o ) - 1 . 6 5 L . 6 6

H T A r A l o + - r . 7 6 L . 7 6

ErBBor l - r .4g L .Ag

(b) H7TIT2O7 molecu les

1 . 6 5 I . 6 5 1 3 2

1 . 6 5 r . 7 0 1 3 3

1 . 6 5 1 . 4 5 r 2 8

1 . 6 6 r , 5 8 I 2 9

1 . 7 1 r . j L 1 4 0

L . 4 6 r . 4 6 I 3 7

3 . 0 0 0 . 4 0 1 c

2 , 7 5 0 . 4 0 r c

2 , 1 5 0 . 3 8 1 c

2 , 5 0 0 . 3 8 6 d

2 . 5 0 O . 4 3 4 c

2 . 5 O 0 . 4 2 2 c

a Newton aod Glbbs (1980)

b M . . g h . . e t a 1 . ( 1 9 8 0 )

" C . i " i n g . . a n d c l b b s ( I 9 8 J )

d Do*" . .d cibbs (1981)


that the minimum energy SiO, AlO, BO and BeObond lengths (Table 2) agree within the expectederror with experimental values obtained for a num-ber of silicates (Fig. 14). When the frequency distri-butions of the SiOSi, SiOAl, SiOBe and SiOBangles in solid silicates were superimposed on thegraphs in Figure 13, they found that the averageangle obtained for each population agrees to withina few degrees of the minimum energy angle ob-tained in the calculations:

Fig. 12. Drawings of the molecular structures of HeSiTOz (a)and HTSiTOT (b). The intermediate-sized spheres represent thetetrahedrally coordinated Si and T (: Si, Al, B, Be) atoms, thelarge spheres represent oxygen and the small spheres representhydrogen. No distinction is made in the drawings between Si andT atoms. The bridging oxygen is only bonded to a Si and a Tatom in HrSiTOT, whereas it is bonded to Si, T and H in HTSiTO?.Hence, the po value of HzSiTOz is one unit larger than that ofI+SiTO7. No significance is attached to the relative sizes of thespheres in the molecules. (After Geisinger and Gibbs, 1981.)

lengths deviate by only 0.014 from their grandmeans. Interestingly, Smith and Bailey (1963) re-ported a similar variation in bond distances in animportant review of AIO and SiO tetrahedral bondlengths. With the equilibrium TO bond lengthsgiven in Table 2, potential energy curves werecalculated for the molecules H6Si2O7, H6SiAlO+-,H6SiBO+- and H6SiBeOT- as a function of thebridging angles between 110 and 180'. The resultantcurves (Fig. 13) are similar in that each shows aminimum in the region between 125 and 145" andeach rises steeply with decreasing angle in thevicinity of 110-120". This agrees with hybridizationarguments by Coulson (1973) which imply that thedirections of the hybrid orbitals on the bridgingoxygen, and hence the SiOT angle, must exceed90'. Clearly, the calculated angles having minimumenergy decrease in the series SiOSi, SiOAl, SiOBeand SiOB with a concomitant increase in the barrierto linearity. It is noteworthy that the barrier tolinearity seems in general to vary directly with theequilibrium angle, the narrower the angle, the largerthe barrier. In addition, Geisinger and Gibbs(1981b) and Downs and Gibbs (1981) have found

Group

SiOSiSiOAISiOBeSiOB

Minimumenergy angle(")

1421361 3 1126

Average SiOTangle(")

145138r27r29

Unlike the potential energy-angle curves generat-ed for H6Si2O7 and H6SiAlOl-, both curves forH6SiBO+- and H6SiBeO]- show relatively largerbarriers to linearity, indicating that the range ofSiOB and SiOBe angles in solids and moleculesshould be somewhat less than that observed formaterials with SiOSi and SiOAI angles. An exami-nation of the frequency distributions in Figure 13shows that their angles conform with the energeticsof the curves, i.e., LSIOB and ZSiOBe range from120 to 142' and 118 to 152", respectively, whileZSiOSi and ZSiOAI range from 120 to 180" and 115to 180o, respectively. Thus, not only are molecularorbital generated potential energy curves capable ofreproducing the shapes of SiOT groups in silicates,but they also provide important insight into thevariability of their angles in solid silicates contain-ing other tetrahedral atoms.

When a proton is attached to the bridging oxygenof an H6T2O7 molecule, the barrier to linearityincreases significantly, suggesting that the variabili-ty of a given angle should be reduced with increasedcoordination number of the bridging oxygen (Gei-singer and Gibbs, 1981b; Downs and Gibbs, 1981).A study of the SiOSi groups for the disilicatestabulated by Baur (1971) shows that the SiOSi anglefor two-coordinate oxygen ranges between 130 and180o whereas the angle for three-coordinate oxygenis less variable and ranges between 124 and 137".Thus, if each bridging oxygen atom in a structure isthree-coordinate, then the range of bridging anglesis indicated to be relatively narrow. In contrast,when some oxygens in a structure are two-coordi-nate, then the range of angles is relatively large


a - O O 4

t

UJ

o 0 6

r s iosi

loo r20 r40 t60 t80 80 too t20 t40 t60 taoI S | O B I S i O B e

The energetics of the disilathia and disiloxygroups-Q comparison

The basic moiety in the thiosilicates is the tetra-hedral thiosilicate group consisting of a Si atombonded to four sulfur atoms disposed, on the aver-age, at the corners of a tetrahedron. Even thoughthe SiS bond is -0.54longer than its SiO counter-part, the range of bond lengths (0.19A) in thiosili-cates is the same as that observed in silicates.However, in contrast with the silicates, not only dothe thiosilicates show a much narrower range ofbridging angles (106-115), but they also show amore limited assortment of tetrahedral topologies.In a study of these differences, Geisinger and Gibbs(1981) found in a series of STO-3G calculations thatthe barrier to linearity of the disilathia group is morethan a magnitude larger than that calculated for thedisiloxy group. Their calculations also yielded aminimum energy SiSSi angle of 111" and bridgingbond length of 2.llA compared with averaged ex-periment?l values in molecules and solids of 110.3"and 2. 13 A, respectively.

Geisinger and Gibbs (1981a) have indicated thatthe assortment of tetrahedral topologies exhibitedby a compound should be related in part to thebarrier to linearity and to the relative stiffness of thebridging angle, the greater the barrier and stiffnessof the angle, the more restricted the assortment of

r .75

1 .65

r . 55

t .45t . 50 t . 60 t . 70

( n{ro))oo,Fig. 14. The averaged tetrahedral bond lengths, (R(TO))"et.,

optimized for the molecules in Table 2 versus the averagedbridgrng bond lengths, (R(TO))"b"., involving the angles used roprepare the histograms in Figure 13. The 45. line represents thepoints ofperfect agreement. (After Geisinger and Gibbs, l9Elb.)

o

-o 02

o 0 4

-o 04

- o 0 6 Lao

o{

CLo

o

E.

Fig. 13. Potential energy curves for the [I6SiTOT and HTSiTOTmolecules (T = Si, Al, B, Be) displayed as a function of (a)zSiOSi, (b) zSiOAl, (c) zSiOB and (d) zSiOBe (after Downsand Gibbs, l98l). Each curve was calculated using the optimizedbridging bond lengths given in Table 2; the solid and dashedcurves were calculated for H5SiTOT and HzSiTOr, respectively.The histogram superimposed above each pair of curves is afrequency distribution of the experimental SiOT bridging anglesrecorded for various silicates. (After Geisinger and Gibbs,l98lb.)

whether or not other three-coordinate oxygens arepresent. In addition, the typical angle forthe three-coordinate oxygen atoms in the disilicates is signifi-cantly narrower on the average (-132") than that forthe entire population (-144\. The theoreticalcurves for H6Si2O7 and H7Si2O)* lGeisinger andGibbs, 198lb) corroborate these results. Not only isthe barrier to linearity of H7Si2O|+ (-0.02 a.u.)with a three-coordinate bridging oxygen larger thanthat for.H5Si2O7, but the minimum energy angle forHiSi2Ol* is about l0o narrower than thatlor fLSirO,(Table 2).

l s ioA l

R(Ar0)

R ( S i O ) o

on( geo)

lR( Bo)

GIBBS: MOLECULES AS MODELS FOR BONDING IN SILICATES 44r

stable tetrahedral topologies. As the SiOSi angle iscompliant and has a small barrier to linearity, theconstituent tetrahedra in a silicate may bond togeth-er in a wide range of angles, allowing a largeassortment of polymeric groups of similar energiesto form. On the other hand. the more limitedassortment of tetrahedral topologies in the thiosili-cates may be related to the larger barrier to linearityand to the greater stiffness of the SiSSi angle.Geisinger and Gibbs (1981a) also suggested that thenarrow distribution of bridging angles in the thiosili-cates may inhibit the formation of SiS2 as a glassinasmuch as nearly the same SiSSi angle should beadopted and repeated again and again as in a solid.

The effect of po and f, on the variation of R(TO)(T : Si, Al) in the tectosilicates

Few rules have had a more profound effect on ourthinking in mineralogy than Pauling's (1929) electro-static valence rule, proposed more than half acentury ago but still used today to characterize therelative strengths ofbonds and local charge balancein silicates. The rule postulates that the sum ofstrengths of the electrostatic bonds, po, to eachoxygen atom from the adjacent metal atoms in asilicate should exactly or nearly equal 2.0 in orderto saturate the valency of the atom. Through theyears, not only has this rule been useful in restrict-ing plausible structure types for a substance, but ithas also been found to be an important aid in thedetermination of crystal structures. When first pro-posed, it was found to be exactly or nearly com-pletely satisfied in a variety of silicates. However,as more and more silicate structures were deter-mined, more and more violations were reported.Smith (1953), for example, found in a reexaminationof the melilite structure that the ps value of one ofthe oxygen atoms departs by as much as 20 percentfrom 2.0. Following the discovery that a positivecorrelation exists between R(SiO) and p6, he wenton to make the important postulate that the varia-tions in R(SiO) can serve to alter the bond strengthsso that the valence of each oxygen atom in thestructure is effectively saturated by the strengths ofthe bonds reaching it.

In 1970, Baur reexamined this correlation andfound that about half the variation in R(SiO) fornearly 300 SiO bonds from a wide variety of sili-cates could be described in terms of a linear depen-dence onps. Later, Phillips, Ribbe and Gibbs (1973)also found that about half of the variation of ROO)

(T : Si, Al) in the feldspar, anorthite, could besimilarly described. But they also found that about40 percent of the R(TO) variations in the feldsparcould be described in terms of the SiOAI angle withshorter bonds tending to involve wider angles (seealso Brown, Gibbs and Ribbe, 1969).

To ascertain whether both p6 and /l make asignificant contribution to the regression sum ofsquares when considered in the presence of eachother, Geisinger, Swanson, Meagher and Gibbs(unpublished) completed a multiple linear regres-sion analysis of R(SiO) as a function of po and/" formore than 130 SiO bond lengths in eleven tectosili-cates, including anorthite. The analysis yielded theequation

R(SiO) : 1.49 + 0.10p0 - 0.16^ (17)

Not only do statistical tests indicate that both poand/. make a significant contribution to the regres-sion sum of squares, but also the resultant multiplecorrelation coefficient indicates that about 70 per-cent of the total variance in R(SiO) may be account-ed for by its regression on ps andf,. When the AIObond length data for the tectosilicates were includedin the regression analysis, the calculation yieldedthe equation

R(TO) : 1.49 * 0.10ps - 0.16n + 0.132x (18)

where.r was set equal to 0.0 and 1.00 for SiO andAIO bonds, respectively. Note that the regressioncoefficients for pq and f; are identical in both (17)and (18), indicating that both data sets may becommingled and treated as a single population. Thefact that they are identical suggests that the bondingforces that govern R(TO) in the tectosilicates arevery similar irrespective of whether a tetrahedralgroup contains either Si or Al. As expected, thecoefficient of x in (18) is equal to the differencebetween the crystal radii of four-coordinate Al andSi published by Shannon and Prewitt (1969).

The observed tetrahedral bond lengths,R(TO)ou"., used in the regression analysis are plot-ted in Figure 15b against R(TO).4.., calculated with(18) using the ps, /" and x values observed for themore than 200 bond lengths employed in the analy-sis. Although the agreement is not perfect (some ofthe bond lengths depart by as much as 0.03A fromthe 45' line of perfect agreement), equation (18)orders the experimental bond lengths rather well inconformity with the line drawn in Figure 15b.

As the bond lengths and angles generated for the


o

I t 7 o

t E . ^ -

t 6 0

o

oFE

( o )

:y,'

,{'

R(TO)opf

( b )

t 6 t ? t 8

R( TO)s6s1 6 t 7 t 8

R ( TO)sg5

Fig. 15. Tetrahedral TO (T = Si, Al) bond lengths, R(TO)"d. ,calculated as a linear combination of bond strength sum, po, thefraction of s-character of the bridging oxygen atom, i, and theAl-content, x, of the tetrahedron containing the bond. (a) Thetetrahedral bond lengths, R(TO)"d", calculated using equation(19) and the po andf, values for a number of molecules versastheir optimized bridging bond lengths, R(TO)opt.t (b) Thetetrahedral bond lengths, R(TO)."r"., calculated with equation(18) using the observed po and[ values for eleven tectosilicates(c-quartz, low cristobalite, coesite, anorthite, low albite, lowmicrocl ine, low cordieri te, paracelsian, slawsonite,reedmergnerite and danburite) versus the bridging bond lengths,R(f0)ou" , observed for these minerals (The Al and Si atomswere assumed to be ordered in each of these phases); (c) Thetetrahedral bond lengths, R(TO).d., calculated with equation(19) using the observed po and /l values for the eleventectosilicates versus their observed bridging bond lengths,R(TO)"b".. The solid lines drawn on each scatter diagramrepresent lines ofperfect agreement. (After Geisinger and Gibbs,l9Elb.)

TOT groups in the silicate and siloxane moleculesdescribed in this report and elsewhere (Meagher etaI., 1980; Swanson et al., 1980; Gibbs er at., l98l)agree rather well with those in solid silicates andsiloxanes, Geisinger et al. (trnpublished) completeda multiple linear regression analysis of R(TO) as afunction of po, -/l and -r for more than 25 moleculeswhose bond lengths and angles had been optimizedwith STO-3G basis functions. As in the case of thetectosilicates, each variable was indicated to makea significant contribution to the regression sum ofsquares. Also, the optimized bond lengths,R(TO)opt., used in the regression analysis wereobserved to be highly correlated (R2 : 0.94) withthose calculated for the molecules as evinced bv

Fig. 15a. As a matter of fact, when the equation

R(TO) : 1.64 + 0.08p6 - 0.41A + 0.11.r (19)

obtained in the regression analysis is evaluatedusing the p0,/l and r values for each TO bond in thetectosilicates, the resulting R(TO)"4.. values serveto rank the observed bond lengths fairly well (Fig.15c). The ability of this equation to rank the ob-served bond lengths is additional evidence that theforces governing the shape of the SiOT group in amolecule and a tectosilicate are virtually the same,notwithstanding the long range Madelung potentialof the solid.

An examination of the regression coefficients of(18) indicates that R(TO) increases with decreasing/l (with narrowing of the SiOT angle) and withincreasingp6. Inasmuch as the expected distance ofa 2p-electron from the oxygen atom nucleus isgreater than that of a corresponding 2s-electron, theincrease of R(TO) with decreasing/" is anticipated.In other words, R(TO) should increase with de-creasing/" because the effective radius of the oxy-gen atom is expected to increase linearly with itsstate of hybridization in the order sp I spz I sp3(Dewar and Schmeising, 1960; Brown and Gibbs,1969; Newton and Gibbs, 1980).

The linear dependence between bond length andpo is well-documented for a number of oxides,including phosphates, sulfates, borates, silicatesand aluminates (Baur, 1970; Swanson et al., 1980aGeisinger and Gibbs, 1981b). Interestingly, eventhough Si is more electronegative and effectivelysmaller than Al, both R(SiO) and R(AIO) exhibitabout the same linear dependence on po within theexperimental error. This is in contrast with theobservation by Lager and Gibbs (1973) that thedependence seems to be better developed for themost electronegative metal atoms. The dependenceof R(SiO) and R(AIO) on p0 has been analyzed byMeagher et al. (1980) and Swanson et al. (1980) bothof whom optimized the geometries of a number ofmolecules with bridging SiOSi, SiOAl, and AIOAIgroups and with p6 values ranging between 1.50 and3.00. Their analyses indicate that the bond lengthvariations in the molecules are linearly dependentupon ps and that the R(TO) versus p0 correlationsgenerated for the molecules duplicate the trendsrecorded for various silicates (Fig. 16) by Baur(1970) and Geisinger et al. (unpublished). Popula-tion analyses for several of these molecules indicatethat the bond-length dependence on p6 is related toa redistribution of the charge density in the bonds

t 5 5

t a 5

r 8 0

t 7 0

t 6 5

t 6 0

i 7 5oo

oF

E

( c )


O B S E R V E D T R E N D SR (A lO) - Ge is inger e f o l . ( 1982 )R ( S i O ) - - - B o u r ( 1 9 7 0 )

^ l

--L--4"I

a

a t 3 - - . '--?-"1-

- ' - t '1.55 lz :

t . 2 0 f 8 0 2 4 0 3 0 0Pe

Fig. 16. Bridging tetrahedral TO (T = Al, Si) bond lengths,R(TO), optimized for a number of molecules verszs the bondstrength sums, ?o, of the bridging oxygen atoms. The R(SiO) andR(AIO) data are plotted as bullets and triangles, respectivelyThe lines drawn through the two trends were obtained inregression analyses of observed tetrahedral AIO and SiO bondlength data versus po for a large number of silicates. (AfterGeisinger and Gibbs, l98lb.)

(Geisinger and Gibbs, 1981b). In particular, &S poincreases, the charge density build-up in the bondsreaching the bridging oxygen may be reduced as thecharges on the Si atoms increase, implying anincrease in the overall ionicity of the system. This isexpected to lead to an overall weakening and aconcomitant lengthening of the bonds to the oxygenatoms. In addition, when the electrostatic bondstrengths for each of the SiO and AIO bonds aremultiplied by 2.0lps, a well-developed correlationobtains between the resulting bond strengths andthe Mulliken overlap populations calculated for theoptimized bond lengths of the molecules (Fig. l7).Thus, it is apparent that po is a measure of bondstrength irrespective of whether the bonds are con-sidered to be ionic, covalent or some resonatingstructure in between the two extremes.

The Pauling concept of bond strength

As we have seen in the earlier sections of thisreport, the application of molecular quantum chem-istry to bonding problems in silicates has providednew and valuable insights into bond length andangle variations, charge density distributions andthe elastic properties of the bonds. In this section,we will examine two sets of first- and second-periodbond strength-bond length curves generated for thehydroxyacid molecules in Table 3 (Gibbs and New-ton, in prep.). In addition to mimicking the wellknown Brown-Shannon curves (1973), the theoreti-cal curves will be found capable of reproducing the

t . 40

r .20

r .oo

o.80

o .60

o .40o.30 0.40 0.50 0.60

n(TO)Fig. 17. A scatter diagram of the normalized bond strengths,

2s/po, for the bridging SiO and AIO bonds in more than 25molecules versus the Mulliken bond overlap populationscalculated with the optimized bridgrng bond lengths and angles.The quantity s is the Pauling electrostatic bond strength, and pois the bond strength sum of the bridging oxygen atom. (After

Geisinger and Gibbs, l9Elb.)

Pauling (1929) bond strengths, s (: the valence ofametal atom divided by its coordination number), ofthe bonds in the molecules. Also, a Mulliken popu-lation analysis will show that s is linearly correlatedwith bond overlap population in agreement withearlier assertions that s may be equated with bondnumber (Gibbs et al., 1972;Brown and Shannon,1973;Lager and Gibbs, 1973).

Bond strength-bond length curves

A number of relations have been proposed be-tween bond strength and bond length (see Brownand Shannon (1973) for an excellent review oftheserelations). One of the simplest ones is presented inFigure 18a where ln(s) is plotted against ln(R(XO))for first and second-period metal atoms, X (Gibbsand Newton, in prep.). The R(XO) values used inpreparing these plots were generated with the Shan-non-Prewitt (SP) crysta-l radii (1969), assuming anoxide ion radius of l.22A.Inasmuch as the SP radiiwere obtained from accurate bond length data froma large variety of crystal structures, the R(XO)values used in the plots may be taken as accurate

r .85

C\o

^ l 7 qo " -F

(rt 6 5

oo.

tnN

o/

-a

44

Table 3: Optimized bond lengths, R(XO), for various hydroxy-acid molecules; s is the electrostatic bond strength of the XObond and s."1e. ?ro the bond strengths calculated with equation(21) and the theoretical constants given in the final section of this

repon

Molecule R(XO) s 6 ,c a 1 c .

First per iod XO bonds


1 . 0 0

l . 0 0

o , 7 5

o , 6 7

o . 6 7

0 . 5 0

0 . 5 0

0 . 3 3

0 . 3 3

o . 2 5

0 , 1 7

estimates of the XO bond lengths for various coor-dination polyhedra in oxide solids. In an effort toreproduce the correlations in Figure 18a, Gibbs andNewton (in prep.) undertook a series of molecularorbital calculations for a variety of hydroxyacidmolecules (Table 3) with three-, four- and six-coordinate metal atoms to explore the extent towhich the ln(R(XO)) values for the molecules mightbe linearly correlated with ln(s) and to see whetherthe overlap populations, n(XO), of the optimizedXO bonds correlate with the electrostatic bondstrengths. The protons used to neutralize the mole-cules in Table 3 were placed at 0.964 from theoxygen atoms and the XOH angles were fixed at109.47" as described earlier. for instance. for theSi(OH)4(OH2)2 molecule displayed in Figure 4b. Inthe calculations, the XO bonds of each moleculewere treated as though they were equivalent whilethe optimized R(XO) values were found by fitting athree-point parabola to each potential energy curvein close proximity to the energy minimum. WhenGibbs and Newton (in prep.) plotted the resultingln(R(XO)) values against ln(s), two linear trendsemerged (Fig. 18b) like those displayed in Figurel8a. Unlike the two empirical trends, which parallelone another, the theoretical trends possess slightly

steeper slopes and show a slight divergence. On theother hand, the empirical ones are identical in bothslope and intercept with those reported by Brownand Shannon (1973) which were obtained by requir-ing the bond strength sums reaching the metalatoms in more than 400 crystals to equal the formalvalences on these atoms.

As the trends in Figure 18 are highly linear, eachmay be modeled by the linear equation

ln(s): a + bln(R(XO)). (20)

And, if we set a : Nln(R) and b = -N, where Nand R are constants, (20) becomes

ln(s): Nln(R) - Nln(R(Xo)),

ln(s) : In(R(XO)/R)-N'

and we have

s: (R(XO)/R)-N (2r)This equation corresponds with an expression pro-posed by Donnay and Allman (1970) and indepen-dently refined by Pyatenko (1973) and Brown andShannon (1973) for modeling bond strength versusbond length variations. An analysis of the regres-sion equations fitted to the data in Figure 18 yields

c0 (0H) 2

B (0rr) 3

B(0H) 3 (0H2)

Be (0H), (oHr)

Be (on) 2

(oH2) 2

L i (0H) (0H2)3

L i (oH) (oH2)s

s i (0H) 4

A1 (OH) 3

Ar(oH) 3(0H)

s i (oH)4(oH2) 2

Mg (oH), (oH, )

41 (0H) 3

(oH2 ) 3ug(0H)

r (oHr ) ,Mg(oH)

r (oHr )aNa(0H) (0H2)

2Na(0n) (oHr ) ,

Na(0H) (oHr ) ,

Second period XO bonds

1 . 3 3

1 . 3 6

1 . 4 6

1 . 5 1

1 . 5 8

1 . 8 0

1 . 9 7

1 . 3 3

1 . 0 0

0 . 7 5

o , 6 7

0 . 5 0

0 . 2 5

0 . 1 7

I . 2 6

r . 0 8

o , 7 7

0 . 6 5

0 . 5 1

0 . 1 6

1 . 0 5

o . 9 6

o . 7 7

0 . 6 5

0 . 6 0

0 . 5 8

o , 4 9

0 . 3 6

0 , 2 7

O .2 t+

0 . 2 0

1 . 6 5

1 . 6 7

L . 1 2

r , 7 6

1 . 7 8

r , 7 9

1 , 8 3

1 . 9 1

1 . 9 9

2 , 0 2

2 . O 7

.3 -o eoc

(b )

cn

.,Ji i\"'0 3 0 0 5 0 0 7 0

I n ( R ( x O ) )o 9 0 0 3 0 0 5 0 0 7 0 0 9 0

In ( R(xO))Fig. 18. Plots of ln(s) uerszs ln(R(XO)) where s is the Pauling

bond strength and R(XO) is the interatomic separation betweenthe first and second row cation X and an oxygen anion. (a) ln(s)verszs In(R(XO)) where R(XO) is the sum of the crystal radii ofan X cation and the oxygen anion. (These plots were preparedwith those X cations whose radii were considered to be accurateby Shannon and Prewitt (1969); cation radii tagged with aquestion mark were omitted); (b) ln(s) versus ln(R(XO)) whereR(XO) is the bond length optimized for each of the H2,_,X'O.molecules in Table 3. (After Gibbs and Newton, in preparation.)


the following least-squares estimates of R and N(Gibbs and Newton, in prep.).

FIRST SECONDPERIOD PERIODs

R NEmpirical SP-R(XO): 1.38 4.2Theoretical R(XO): 1.39 5.2

R N1.62 4 .3r .66 7 .3

When equation (21) is mapped out for each R and Npair above, it is apparent that the theoretical bondstrength versus bond length curves are somewhatsteeper than those obtained with the SP radii (Fig.19). Despite this diference, when the s values forthe hydroxyacid molecules in Table 3 are calculatedwith the theoretical equations, they agree to within0.04 units, on the average, with Pauling's values(Table 3). Finally, when the bond overlap popula-tions calculated for the molecules are plottedagainst Pauling bond strength (Fig. 20), a strongcorrelation (r2 : 0.98) is observed in agreementwith the assertion that s can be equated with bondnumber (Gibbs et a1.,1972). Thus, we consider thebond length versus bond strength curves of Donnayand Allman (1970), Pyatenko (1973) and Brown andShannon (1973) to have the same meaning andsigniflcance as the well known carbon-carbon bondlength versus bond number curves published for thehydrocarbons by Pauling (1960) and Coulson (1961).Even though the Pauling bond strength has its originin the ionic model, the trend in Figure 20 indicatesthat is a direct measure of the strength of a bondregardless of whether the bond is considered to beionic or covalent, the larger the value of s, theshorter the bond. Moreover, when the bond lengthsin an oxide are taken into account. the Brown-Shannon curves have been shown to generate bondstrengths that sum to values close to the formalvalences on the atoms in conformity with Pauling'selectrostatic valence principle. Finally, the repro-duction of the bond strength versus bond lengthcurves with molecular orbital theory has estab-lished a connection between experiment and theory

s Employing a STO-3G* basis set with d-type AO's on Si,Chakoumakos and Gibbs (l98lb) have since derived the theoreti-cal equation s : (R(XOyl.59;-r s, using the SiO bond lengthsoptimized for three hydroxyacid molecules with 4-, 6- and 8-coordinated Si. Although this equation is based on only threebond lengths, it agrees very well with an empirical equation s =(R(XOyl.62)-4'3 derived by Brown and Shannon (1973). Notethat the constants derived by them are virtually the same asthose derived here with the Shannon-Prewitt crystal radii.

l .o

Fig. 19. Theoretical and experimental bond strength-bondlength curves. (After Gibbs and Newton, in preparation.)

1 .5 2 .O 2 .5

R(xo)i

t . 5

r .o

so.5

o.o

s

t .oo

o.7 5

o .50

o . 2 5

siEl-4,n

!o,*

+(.'o.oo o.25 o.50

n(xo)Fig. 20. The Pauling bond strength s veffrs the average

Mulliken bond overlap population of the XO bonds for themolecules listed in Table 3 (after Gibbs and Newton, in prep.).


and has provided a quantum mechanical underpin-ning for Pauling's second rule (Pauling,1929).

Some concluding remarks

Important advances have been made in the lastfew years in generating structures, heats offorma-tion, electron density distributions and equations ofstate for a variety of simple solids, using the princi-ples and methods of quantum mechanics.Bukowinski (1980, 1981), for example, has success-fully calculated equations of state and electronicdensities for the minerals periclase, MgO, and lime,CaO, using a self-consistent symmetrized augment-ed plane wave method. Clearly the success of thesecalculations augurs well for establishing new stan-dards for future high pressure studies of minerals ofgeophysical interest and for furthering our under-standing of the deep interior of the earth. Equallyimportant in the field of solid state physics, phillips(1973) and Cohen (1973), for example, have usedthe pseudopotential method to improve our under-standing of structures, stabilities, electron chargedistributions and compressibilities of a variety ofsimple solids, consisting of either one, two or threeatom types (see Chapters 1-6, O'Keeffe and Nav-rotsky, 1981). In addition, Chelikowsky andSchluter (1977) have used the method to generatethe band gap, the photoemission spectra and thepseudocharge density distribution for a-quartz.Notwithstanding the efrcacy of these powerfulmethods, they have, for the most part, been limitedby computational effort (i.e., computer costs) to arelatively simple class of structures like diamond,rock salt and wurtzite and as such have yet to beapplied globally in calculating bond length and anglevariations, deformation densities and force con-stants for such important rock-forming minerals asthe silica polymorphs, the feldspars, and the biopyr-iboles. Until these calculations are affordable andforthcoming, we suggest that molecular orbital cal-culations on representative molecules can be usedto improve our understanding of the solid stateproperties and the nature of the bonding forces inthese rock-forming minerals.

As described in this report, a variety of recent abinitio calculations on various siloxane and silicatemolecules has given a fairly good account of thebond length and angle variations in solid siloxanesand silicates. Besides providing valuable data onthe charge density distribution, the compressibility,the polymorphism and the glass-forming tendenciesof silica, the calculations have also shed light on the

question of why silicates exhibit such a large assort-ment of structure types. They have also provenuseful in interpreting the bond length and anglevariations and the glass-forming tendencies of sili-cates with SiOSi, SiOAl, SiOB and SiOBe groups.By reproducing the Brown-Shannon bond length-bond strength curves, the calculations have alsoprovided a quantum mechanical underpinning ofPauling's electrostatic valence principle by showingthat the strength of a bond is dependent on bondlength to the extent that the valence on each oxygenatom in a structure is effectively saturated by theelectrostatic valences of the bonds reaching it aspostulated by Smith (1953).

In light of the successful application of thesecalculations to silicates, we are forced to concludethat the bond length and angle variations, the localforce field and the deformation densities in silicatesare governed in large part by the local atomicarrangement of a solid. Moreover, the success of alarge number of semi-empirical molecular orbitalcalculations in ranking the bond lengths and anglesfor a variety of phosphates, sulfates, germanates,etc., indicates that this conclusion may hold for arelatively large class of inorganic solids (Lager andGibbs, 1973; Tossell and Gibbs,1977; Louisnathan,Hill and Gibbs, 1977;Hill, Louisnathan and Gibbs,1977).In fact, Bullett (1980) has reached a similarconclusion in a review of the tight-binding methodand its ability to generate equilibrium cell dimen-sions, directional Compton profiles, and local den-sities of state for a variety of semiconductors andtransition metals.

During the coming years, we believe that molecu-lar orbital theory will provide a great deal of newand valuable mineralogical information about theprinciples governing the formation of various struc-ture types as well as identifying systematic relation-ships between structure and physical properties. Inparticular, we suggest that the theory will providenew insight into such physico-chemical processesas sorption, diffusion and catalysis in the zeoliteminerals, ion exchange and order-disorder transfor-mations in the feldspars, chemical reactions andreaction paths in silicate minerals and glasses, thecleavage mechanism of the disiloxy group by waterand its bearing on the reaction of silicate mineralsand glasses with water,dissolution mechanisms andthe self-condensation and hydrolysis of such mate-rials as monosilicic acid (Gibbs, Meagher, Smithand Pluth, 1977b; Mortier, Geerlings, Van Alsenoyand Figeys, 1979; DeJong and Brown, 1980b;


Sauer, Hobza and Zahradnik, 1980; Hass and Me-zey, l98l; Chakoumakos and Gibbs, 1981a).

In concluding this report, we suggest that miner-alogists can no longer look upon the application ofquantum mechanical theory to mineralogical prob-lems as an object of mere curiosity. The insightafforded by the theory, the understanding providedby its application as well as its ability to generatedata for unknown quantities not amenable to directmeasurement are compelling and cogent reasons foracquiring some knowledge of the theory. However,if we persist in studying minerals solely by conven-tional methods, then we can expect to forfeit asegment ofour field ofresearch and expertise to thechemists and physicists who are being attracted intothe field of quantum mineralogy by its many impor-tant and challenging problems.

AcknowledgmentsI owe a debt of thanks to Professor E. P. Meagher, Dr.

Marshall D. Newton, my students and colleagues for sharingwith me their ideas on the meaning and significance of thecalculations and the results discussed in this report. I amparticularly indebted to Marshall Newton for teaching me anumber of important concepts in theoretical quantum chemistryand for showing patience with me while I struggled to assimilatethe subject. I wish to thank Professors F. D. Bloss, G. E. Brown,Jr., M. S. T. Bukowinski, J. K. Burdett, G. C. Grender, A.Lasaga, A. Navrotsky, P. H. Ribbe, J. D. Rimstidt, M. A.Spackman, J. W. Viers and D. R. Wones, as well as each of mycollaborators for their valuable remarks in reviewing this report.However, they are neither responsible for my errors nor myinfelicities. I am very grateful to VPI for promoting the growthand development of an excellent University Computer Centerand for contributing generous sums of monies to help defray thecomputing costs incurred in our molecular orbital calculations.Ramonda Haycocks, a secretary of exceptional talent and inge-nuity, prepared the script file of the report, transforming myhandwritten rough drafts into immaculate typescript. ProfessorRobert F. Stewart and Dr. Mark A. Spackman of the ChemistryDepartment at Carnegie Mellon University are thanked forproviding me with a deformation map for SiO calculated withHartree-Fock wavefunctions and for calculating the map forstishovite in Figure 6a. Professor Alan Clifford of the ChemistryDepartment at VPI is thanked for helpful advice on the nomen-clature of the molecules examined in the study. I thank SharonChiang and Martin Eiss for drafting the figures. I am especiallyindebted to the National Science Foundation for supporting thiswork with Grant EAR 77-23114 awarded to Professor P. H.Ribbe and myself to study bonding in minerals.

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Manuscript received, December 23, l9El;acceptedfor publication, January 4, 1982.

Documents

Molecules as models for bonding in silicatesr G. V. Grsss ... · (SiOSi) group in a solid silicate and in a siloxane molecule (Table 1). For example, the disiloxy group in the silicate