American Mineralogist, Volume 65, pages 746-755, 1980

Stereochemistry and energies of single two-repeat silicate chains

E. P. Mnacsrn Department of Geological Sciences University of British Columbia Vancouver,Brilish Columbia, Canada V6T 284

Abstract

The wide variety of hypothetically possibletwo-repeat chains is discussedand results of cNDo/2 MO calculationson isolatedHsSi3Oro clusters modeling thesechains are presented. Calculations indicate that within the wide spectrum of possiblechains a rather restricted rangeof conformationsis energeticallyfavored. Despite the neglectof M-cationsbetween the chains,calculated conformations of minimum energycompare favorably with observedcon- formations.The chain of lowestcalculated energy is similar to that found in NarSiOr. Chains with conformationsintermediate to NazSiO, and a straight pyroxenechain are predicted to be favored over other conformations.This is in agreementwith observedtwo-repeat chain structures.

Introduction sion of the chain in over fifty different structures.His study revealsthat even-periodicchains becomeless The single-chain silicates comprise an important stretched with higher mean electronegativity and classof mineralsbecause of their great abundancein higher mean valence of the M-cations. In odd-peri- crustal and upper mantle rocks. Members of this odic chains the degree of chain shrinkage is strongly classpossess continuous chains of silicate tetrahedra correlatedwith the meanelectronegativity and lessso linked such that each tetrahedron possessestwo with the mean radius of the M-cations. Such investi- bridging and two non-bridging oxygens,which re- gationsare important for gaining a perspectiveabout sults in a tetrahedral-cation:oxygenratio of 1:3 and the relative importance of thesevariables in influenc- a sharingcoeffi.cient of 1.5(Zoltai, 1960).The classis ing the chain type, but at the present time reliable subdivided on the basisof the number of tetrahedra predictions about chain configurationsbased upon contained within a translational repeat along the the chemistryof M-cations are not possible. chain and although "one-repeatsilicate chains" have High-precision structure refinementsof pyroxene not beenobserved, those with repeatsof 2,3, 4, 5, 6, and 3-repeat pyroxenoids of various compositions 7, 9, znd 12 are known to exist (Liebau, 1972). have prompted several crystal chemical investiga- In the class of single chains, minerals with two-re- tions concerningthe effectM-cations have on the de- peat chains are by far the most abundant, and within tailed stereochemistryof a specificchain type (Clark this subclassall membersare describedas having ei- et al., 1969;Papike et al., 1973;Ohashi et al. 1975; ther the pyroxene type or the sodium metasilicate Ribbe and Prunier,1977;Ohashi and Finger, 1978). (NarSiO.) type chain. Minerals containing single For a given type of chain these investigations clearly chains oftetrahedral repeatsthree or greaterare col- relate the effect various M-cations have on small- lectively referred to as pyroxenoids. scaleinteratomic"adjustments within the chain. All single-chainsilicates possess cations which oc- Since past investigationshave explored the rela- cupy sites between the chains, thereby providing tionships between M-cations and the silicate chain structural continuity. These M-cations are normally configuration, attention in this paper will concentrate six to eight-coordinatedby oxygenand range in size on the bonding and non-bonding interactions be- from 0.53A (Al) to 1.42A (Ba) with formal valences tween the silicon and oxygen atoms in the chain in- ranging from +l to *3. Liebau (1980) has recently dependentof the effectsof the M-cations.With such studied correlations between size and electro- an approachperhaps some insight can be gainedwith negativity of the M-cations and the repeatand exten- regard to the relative influence that packing and 0003-004x/80/0708-0746$02.0o 746 MEAGHER: SILICATE CHAINS bonding requirements of M-cations have relative to where F., is a matrix representation of the Hartree- the influence of Si-O, O-O and Si-Si interactions on Fock Hamiltonian operator, E' is the energy of the i'n the final conformation of the single chain. molecular orbital, Sn,is the matrix of atomic orbital For this reason, a series of molecular orbital calcu- overlap integrals, and c,, is the set of numerical coef- lations have been completed on isolated single-chain ficients in the lceo approximation of equation l. clusters which, in the current study, are limited to The elements of the matrix representation (F',) to two-repeat chains. Within the realm of two-repeat the Hartree-Fock Hamiltonian operator contain chains there is an infinite variety of geometrically terms which relate to the kinetic and potential feasible chains to be considered. The energies of rep- energies of the electrons in a molecule. If we are to resentatives of this continuum of possible two-repeat solve for the linear coefficients, ei, &Ild the corre- chains will be computed and their conformations sponding MO energies, E,, we must have previously compared to observed structures. In a subsequent pa- determined the elements of matrix F.,, which re- per, single chains with repeats greater than two will quires that we know the electron distribution in the be examined. molecule. This, of course, is what we are trying to de- termine. Therefore to solve the Roothaan equations Computations the coefficients cji are initially estimated, a first ap- It is generally believed that the Si-O bond in sili- proximation of F., is made, and a new set of cli values cate minerals is partly ionic and partly covalent in is then computed which, in turn, is used to construct character (Pauling, 1939; Hiibner, 1977). Accord- a new F",. This procedure is continued until the ingly, if one is to evaluate relative energies of silicate change in coefficients between successiveiterations is chains with different conformations, one must use a diminishingly small. The resulting molecular orbitals bonding model which includes both the ionic and are self-consistent with the potential field they gener- covalent contributions to the energy. Therefore, one ate and this procedure is referred to as the Self Con- must consider the variation of electron distributions sistent Field (scr) or LcAo scF molecular orbital between atomic centers as a function of changing method. conformations. Because of the enormous computational effort in- A methodology commonly used to compute elec- volved in evaluation of the Roothaan equations the tron distributions between multiple atomic centers is applications have generally been limited to small mo- the molecular orbital (MO) theory. Basically, one is lecular groups. Application of the Roothaan equa- seeking approximate wave functions for a molecular tions to large molecular groups has been achieved group by assigning each electron to a one-electron through various methods of approximation. One wave function which generally extends over the such method is the approximate seH-consistent field whole molecule. If these wave functions (molecular molecular orbital theory known as cNDo/2 (Pople et orbitals) are approximated as Linear Combinations al., 1965). In this approximate method all two-ele- of Atomic Orbitals (rcao) the computational effort is ment repulsion integrals which depend on the over- considerably lessened. This linear combination is lapping of charge densities of different atomic orbit- most easily expressed as: als are neglected, hence the term Complete Neglect of Differential Overlap. Although the more impor- 9,: ) ci'O, (l) tant electron repulsion integrals are approximated, the neglect of differential overlap significantly re- where {.,, represents the molecular orbital, {, the duces computation time. This is especially evident atomic orbital, and c', the numerical coefficients. when one considers that the number of electron re- If we approximate the molecular orbitals as a lin- pulsion integrals increases as the fourth power of the ear combination of atomic orbitals we must find the number of atomic orbitals used in the LcAo tech- set of coefficients, c,i, for which the total electronic nique. In the cNDo/2 method only the valence elec- energy of our system is minimized. Roothaan (1951) trons are considered explicitly, since all inner elec- formulated the mathematical treatment of this prob- trons are taken as part ofan unpolarizable core. The lem through use of the variational method. The method is semi-empirical in that some of the terms Roothaan equations take the final form: included in the Fu, matrix are approximated with atomic data such as electronegativities or by fitting to (F*,- E,Sr,)c,,:0 (2) accurate non-empirical scn LCAo calculations on di- x atomic molecules (Pople and Segal, 1965). MEAGHER: SILICATE CHAINS

As in the original Roothaan method, the cNoo/2 l(SiOSi) angles must also be given for each plot. In calculation is iterative in that successiveapproxima- the general case two symmetry non-equivalent tions of the molecular orbitals are made through ad- l(SiOSi) angles are possible although the over- justment of coefficients in the linear combination of whehning majority of observed two-repeat chains atomic orbitals until convergence of the electronic have equal l(SiOSi) values. The notable exception is energy is achieved (i.e., a minimized energy is ob- omphacite with non-equivalent l(SiOSi) angles of tained). The total energy of the system is given rela- 140.0'and 135.4' (Matsumoto et a|.,1975).The con- tive to isolated atom cores and valence electrons. tinuum of all possible 4 and l(OOO)r. values is The computer program cNINDo (Dobash, 1974) plotted in Figure 2 for chains with both l(SiOSi) an- was used for all cNDo/2 calculations with bonding gles equal to 135" (solid line) and 140' (dashed line). parameters for H and O and for Si taken from Pople Only half the polar coordinates are plotted, since - and Segal (1965) and from Santry and Segal (1967), chains with 4 and 360" O are equivalent. Similarly, respectively. The atomic basis set was limited to s chains along the line from A to E are equivalent by and p orbitals for Si and O. All calculations were reflection symmetry to those along the line A to J. To made on an Amdahl 470/V6 computer at the Uni- illustrate how the chains plotted in Figure 2 differ, a versity of British Columbia. series of these chains is shown in Figures 3 and 4. A cluster of HrSirO,o composition consisting of The chain in Figure 3A corresponds to the plot of three silicate tetrahedra linked through corners was point A in Figure 2 with l(OOO)b. : 180o,Q: 97", used in this study to model two-repeat chains. All tet- and lSiOSi : 135"; Figure 38 is the chain related to rahedra possessTo point symmetry with OSiO angles point B, etc. equal to 109.47" and all Si-O distances, d(Si-O), are Figure 3 illustrates that chains which plot in the set equal to the sum of the Shannon and Prewitt general region between points A and D resemble py- (1969) ionic radii of l.6lA. The cluster is neutral in roxenes whereas the chains plotting about point E in charge as a result of the addition of hydrogen atoms Figure 2 resemble the NarSiO, chain (McDonald l.0A from each non-bridging oxygen with an SiOH and Cruickshank, 1967). The chains in Figure 4 (E angle of 180". through J) refer to points E through J in Figure 2. Chains 4E and 4J are the Na,SiO. type chain while Discussion the intermediate chains G and H have not as yet From a purely geometrical point of view it is pos- been observed in natural or synthetic compounds. sible to construct an infinite number of unique two- Note that chain E in Figure 4 is viewed from a ditrer- repeat chains. Because the chains in this study are ent direction from that in Figure 3, but is otherwise constructed of ideal tetrahedra, they can be described identical. by three angular values. The first ofthese angles, p, is In the preceding discussion an l(SiOSi) of l35o defined to be the angle between the Si,-O, and Sir- was chosen because it is close to those values ob- O, vectors projected on the plane containing Si, and served in two-repeat single-chain compounds. For the two non-bridging oxygens (Fig. l). The remain- comparison, the dashed line in Figure 2 illustrates all ing angles are l(SiOSi) (the valence angle Si,-O,-Si, possible f and /(OOO)0, values for chains with both in Fig. l) and l(OOO)b. (the angle O,-O,-O| be- l(SiOSi) equal 140o. In the course of this investiga- tween bridging oxygens). Figure I illustrates the link- tion, however, energies of over 200 different chains age between adjacent tetrahedra for chains with 6 : within the allowable range of angular values have 0o, 149", and 180'. For two-repeat chains the Si, tet- been computed by the cNoo/2 method. rahedron will be related by translational periodicity : to a third tetrahedron (not shown) which is linked to Energies of straight chains with l(OO O) * I 80o the Si, tetrahedron through oxygen Or. The chain in The first in a series of calculations is concerned Figure la has a special angular relationship I(SiOSD with straight chains of I(OOO)'. : l80o but with : : 109.47" and l(OOO)o, 180"1,which results in a variable 0 and l(SiOSD angles. Figure 5 illustrates one-repeat chain; however, it will be included in the six of a continuum of possible straight chains which discussion of two-repeat chains. vary from the one-repeat chain (chain K) to a chain Geometric relations between the chains can be with l(SiOSi) : 180' (chain P). The total energies of compared graphically by plotting @and l(OOO)o, as these chains reveal a minimum at an l(SiOSi) of ap- polar coordinates (Fig. 2). Because these two angles proximately 140", corresponding to chain M, which do not uniquely define a two-repeat chain, the has the conformation of a straight pyroxene chain. MEAGHER: SILICATE CHAINS

9=o' c. I (OOO)br= 180' 9'ug" =1Bo" Z (O0o)6r-1a9' I 1O947" I (ooo) = 18o" 1(SiOSi)= I (siOsi)=1so" br 1(siosi)=180" Fig. l. Segmentsofsinglechainsillustratingtheangularvaluesofp, l(OOO)b.(angleO2-O,-O!),andl(SiOSi)(angleSir-Or-Si2) which can be used to describe chain conformations.

Figure 5 also reveals that cNDo/2 predicts the one- Energies of chains with qt: 169o repeat chain (K) ro be the least stable of all the straight chains. To date no such silicate chain is A second series of calculations were completed on known to occur in crystalline materials. chains along the line 4, : l80o for l(OOO)b. values In the initial stages of this investigation it was de- between 120' and 180" (Fig. 2). The results illus- cided to test whether a larger cluster would yield sig- trated in Figure 6 show a total energy minimum at nificantly different results; therefore, the above calcu- approximately 135'. This chain (equivalent to both E lations were repeated with a two-repeat chain, five and J of Fig. a) with l(OOO)b, : (SiOSi) : 135o, is tetrahedra in length. Alternate tetrahedra were ro- very similar in conformation to that of the NarSiO, tated, once again rnaintaining a straight chain, and chain with l(ooo)b. : 134.0' and l(SioSi) : 133.7" the resulting energy minimum was observed within (McDonald and Cruickshank, 1967). This con- lo of the value for the HrSi,O,o cluster. Although a formation has also been reported in LirSiO, (Hesse, cluster with more than three tetrahedra adds consid- 1977), high-temperature BaSiO, (Grosse and Till- erably to the expense of the calculation it does not manns, 1974) and NarBaSirOu (Gunawardane et al., appear to significantly change the results. 1973).

o o o N o o-L{ooo;or-- 50 3eE o tSoo+ I J

Fig. 2. A polar coordinate plot ofd and /(OOO)5, for all possible chains with both l(SiOSi) : 135. (solid line) and for those chains : with both l(SiOSD 140' (dashed line). Points labeled A-J r€present the polar coordinates for chains illustrated in Figs. 3 and 4. 750 MEAGHER: SILICATE CHAINS

t\A\M

Fig. 3. Chains with both l(SiOSD angles equal l35o which correspond to the d and I(000)6' A-E of Fig. 2. Note ttre similarity of chain C with a pyroxene chain and chain E with the chain in Na2SiO3.

Energies of chains with variable Q, l(OOO)u, and E (Figs. 2 and3) is the lowest-energychain of all pos- t(siosil sible chains with equivalent l(SiOSi) angles.If we 135o To this point the discussion has been restricted to changeone of the two l(SiOSi) anglesfrom to a of chains with special types of conformations. In the non-equivalent value, say 130", the total energy the chain increases.This is illustrated (Fig. 7) by a general case all possible combinations of 4, l(OOO)o,, and l(SiOSi) must be considered, which series of calculations where one angle is locked at is varied. results in a formidable task. This becomes apparent 135" while the remaining l(SiOSi) angle the con- when one considers that a continuum of chains with Each point plotted in Figure 7 represents formation of minimum energy of all the chains which varying 4 and I(OOO)". values exists for every pair given pair values.An in- of non-equivalent l(SiOSi) angles chosen. The efort can possessthat of l(SiOSi) both is simplified somewhat if we consider only those creasein energy will also result if we change l(SiOSi) angle of chains for which l(SiOSi) angles are equivalent, as l(SiOSD valuesso that an average occurs in practically all the chains observed to date. 135' is maintained. In doing so, however, it is important to understand This does not necessarily mean that a chain with how chains with non-equivalent l(SiOSi) values re- non-equivalentl(SiOSi) angleis alwayshigher in en- late, energetically, to chains with equivalent l(SiOSi) ergy. For example,the minimum-energychain with angles. both l(SiOSi) anglesequal to l50o will have a higher It will be shown in a subsequent section that chain energy than one with l(SiOSi) angles of l50o and

MMMWMM Fig. 4. Chains with both l.(SiOSi) angles equal l35o which correspond to the d and I(000)6, values of points E-J of Fig. 2. Chains E and J are Na2SiO3 chains rotated 180o relative to each other. MEAGHER: SILICATE CHAINS I

- t97.150

-197.158

j < -197.166 F IIJ -197.'174

z(siosi) yr. Fig. 5. Total energy l(SiOSi) for straightchains with l(OOO)b. : 1g0..The chainsK_p correspondro points plotted in the energy curve,with chain M representinga minimum energyconformation at an l(SiOSi) of approximately 139".Chain M has a conformation nearly identical to that found in the pyroxeneLiFesi2ou. (l A.u. : 62i.51 kc,ar/mole)

135o. It appears,therefore, that for the particular ure will either possessnon-equivalent l(SiOSi), or model used here there is a preferencefor both the l(SiOSi) and l(OOO)o, values will be geometri- l(SiOSi) angles to be equal at values of approxi- cally impossible combinations for two-repeat chains mately l35o or, lesspreferably, to haveone l(SiOSi) composedof tetrahedrawith point symmetry [. For angleapproaching 1350. example, a two-repeat chain with l(SiOSD : l80o To compare chains with equivalent l(SiOSi) an- cannotpossess an I(OOO)* of I l0o. gles the energiesof such chains for various 0 and In the construction of Figure 8, 66 data points l(OOO)b, valueshave beencomputed and contoured were used as input to a computer routine for con- (Fig. 8). Chains within the hachuredarea of this fig- touring. The map is contoured with an interval of

L (ooo)br 140 150 WW$"$' -197.174

j -197.178

< -197.182 F UJ - 197.186

130 160 170 180 r.(siosi) Fig. 6. Total energyys. l(SiOSi) for chainswith E = 180.. Chain E possessesthe conformationof minimum energyfor this seriesat approximatelyl(SiOSi) : 135o,which is nearly identical to that of the Na2SiO3chain. MEAGHER: SILICATE CHAINS

shorterthan to bridging oxygens,and in most pyrox- -197.183 enesthe averaged(Si-O) values are closerto 1.63A. A seriesof cNoo/2 calculations on straight chains E(OOO)". : 180"1with tetrahedra of 4 symmetry values equal to 1.63A yielded a -197.184 but with d(Si-O) j minimum energy l(SiOSi) angle of about 132o, as comparedto the value of l38o for chain M (Fig. 8). In a secondseries of calculationson straight chains ut -197.185 with non-bridging Si-O bonds equal to 1.62A and bridging Si-O bonds equal to 1.64A (average1.63A) an energyminimum occurredat approximately 130", - 197.186 which is a shift of 8o from the ideal chain with d(Si- O) : l.6lA. It appears that the basic trend in 120 130 140 150 energiesof conformation shown in Figure 8 will not Z (SiOSi)1 changewith thesetypes of adjustments.Nevertheless, Fig. 7. Minimum total energiesfor chainswith non-equivalent future calculations,of a more rigorousnature, should l(SiOSD angles.Energies are for minimum energy conformation include the Si-O bond length as an additional vari- of each seriesof chains with l(SiOSi), set at the indicated value able. while l(SiOSi)r is locked at 1350. In addition to the correlation between the pre- dicted minimum energy trough and the observed 100 energyunits in the low-energyregion, 200 units trend in angular values for the various two-repeat in the intermediate region, and 400 units in the high- chains,Figure 8 revealsthere is no clear break in ob- energyregion, to highlight the generalfeatures of the servedl(OOO)o, and l(SiOSi) anglesupon progress- plot. The figure is only intended to give general ing from the nearly ideal straight pyroxene chain of trends of energies as a function of l(SiOSi) and LiFeSi,Ou to Na,SiO, (Fig. 8). To further illustrate l(OOO)o. anglesand is not consideredto be particu- this point, the chains of LiFeSLO., MgrSirOu,and larly accurate in detail, considering the small number NarSiO, are compared(Fig. 9). The fact that a con- of data points used in its construction.Nevertheless, tinuous gradation appearsto exist betweenthe classic the plot is informative in that it illustrates the regions pyroxenetype chain and the NarSiO' chain is consis- where preferred conformations are predicted by tent with cNDo/2 calculations. cNDo/2 MO calculations.The lowest energiesoccur Tetrahedrallinkage and M-cationpacking in the areal(SiOSi) : l(OOO)o, : 135o,from which a low-energyregion extendsto l(SiOSi) : l40o and In the silicateminerals SiOotetrahedra are polym- l(OOO)b.: 180". eraed into a wide variety of linkages such as dimers, The crosseslabeled with bold letters refer to those tri.rrers, rings, chains, sheets,and frameworks. Al- chainspreviously discussed and illustrated in Figures though individual l(SiOSi) values range from ap- 3, 4, 5, and 6. In addition, the positions of observed proximately l25o to 180o,it has been realized for pyroxene chains are plotted as squares,and those some time that an overall averagel(SiOSD value is chains from compoundsreported to be isostructural close to 140" (Liebau, 196l). Tossell and Gibbs with NarSiO. are plotted as circles.As indicated by (1978), in a compilation of data for silicateswhose this plot, there appearsto be a good correlation be- structures were published prior to 1976,found the tween the predicted low-energy conformations of most frequently encounteredl(SiOSi) value to be in two-repeatsilicate chains and those observedin nat- ttLe 140-142orange, and the averagevalue for their ural and synthetic compounds.Nevertheless, if one total data set to be 144.3". recalls the simplifications built into the model, it This observed preference in silicates for a bent would seemthat only the generaltrends indicated in l(SiOSD angle appearsto reflect local bonding and Figure 8 are meaningful and the exact anglesof min- nonbonding interactions among the silicon and oxy- imum energy are less signfficant. For example, the gen atoms in the polymerized silicate group, inde- model includesideal tetrahedrawith all d(Si-O) val- pendent of the influence of non-tetrahedralcations. uesequal to l.6lA, which is the sum of the ionic radii In the silica polymorphs,where non-tetrahedralcat- for Si and O. In observed5ingle-chain silicates Si-O ion packing is not a factor, the preferencefor a bent bond lengths to non-bridging oxygens are usually l(SiOSD is evident. In quartz, tridymite, cristobalite, MEAGHER: SILICATE CHAINS 753

.o r50.o o o I !

\o

160.o tto.o z(siosi) Fig. 8. A contouredtotal-energy map for two-repeatchains over a range of l(SiOSi) and l(OOO)b, values.Labeled crossesare for chains depicted in Figs. 2-6, squaresare observedangular values for representativesof the pyroxene group, and circles are for materials associated with the Na2SiO3 structure. Contours r€present energy differcnces based on the lowest-energy chain (E and J) being arbitrarily assigneda value of 40 energyunits. 100energy units is approximatelyequal 0.001A.U. The observedstructures are: LiFe, LiFeSi2Ou;Zn-A,ZnSiOt chain A; Jd, Jadeite;Pig.-A, Pigeonitechain A; NaIn, NaInSi2O5;Sp, spodumene; Fe-A, orthoferrosilitechain A; Di' diopside;Zn, cltno ZnSiO3;En-A, orthoenstatitechain A; Pig.-B,pigeonite chain B; Zr-B, ZnSiOl chain B; Fe-B, orthoferrosilite chain B; En-B, orthoenstatitechain B; NaBa, Na2BaSi2O6;Na, Na2SiO3;Ba, high{emperature BaSiO3;Li, Li2SiO3. and coesitethe rangeof meanl(SiOSi) anglesis from and angleswithin and betweenthe SiOntetrahedra. 143.7"to 150.7'with an overallmean l(SiOSi) angle If there is an inJrinsic preference with regard to the of 147.6" for all four structures (Meagher et al., angular relationship between adjacent SiOo tetra- 1979).This intrinsic preferencefor a bent l(SiOSD hedra, it would seemthat in the attempt to accom- angle approaching 140" has been reproduced in modate M-cations certain adjustmentswithin a chain cNDo/2 calculationson Hr2Si5Or6framework clusters will be energetically favorable while others will be (Meagher et al., 1979\ and,ab initio SCFcalculations improbable.The energymap (Fig. 8) indicatesquali- on HuSirOand H6Si2O?groups by Newton and Gibbs tatively that angular distortions in the chains are (1e80). more restrictedin the range of l(SiOSD values than In compounds with polymerized silicate groups I(OOO)". values, which is consistentwith observed such as single chains, the M-cations are an integral two-repeatsilicate chain structures. part of the structure and the final atomic arrange- An alternate reason for the limited range in ment will be one in which coordination of these cat- l(SiOSD valuesobserved in chain structuresmay be ions is achievedthrough adjustmentof bond lengths that the high-energy chains in Figure 8 simply do not 754 MEAGHER: SILICATE CHAINS

c.

ffiMM Fig. 9. Projections along and normal to the chains of (a) LiFeSi2Oo (Ctark et al.,1969); (b) orthoenstatite chain B (Hawthorne and Ito, 1977); (c) Na2BaSi2O6 (Gunawardane et al., 1973); (d) Na2SiO3 (McDonald and Cruickshank, 1966).

pack together in a way that M-cations can be prop- (L2-tepeat:146") is consistentwith this observation. erly coordinatedor that a crystal-chemicallyfeasible A general computational approach of the type used structure will result. This doesnot negatethe above in this investigation becomesincreasingly difficult to premise regarding the high energy of these chains; follow as the number of tetrahedrain the repeatdis- however,it doespoint out the inportance of the re- tance increases;nevertheless, molecular orbital calcu- quirements of M-cations in forming a stable struc- lations on specific chain types are currently under ture. The packing argumentdoes not hold, however, way. for chains with a conformation similar to chain K in Conclusions Figure 8. Although no such chain is observedin sili- cNDo/2 MO calculationson isolated atomic clus- cates,the chain doespack togetherso that M-cations 1s1smsdsling portions of a complex three-dimen- can be accommodatedas is demonstratedin the com- sional solid, ofer a semi-empirical means of eval- pound CuGeO. (Vcillenkle et al., 1967).The Ge-O- uating short-range bonding and non-bonding Ge angle is I l3o in this material, in keepingwith the interactions and their effect on structural con- generalpreference for Ge compoundsto possessnar- formation. Calculations on over 200 different con- row T-O-T angles(Vdllenkle et al.,1968). formations of an HrSi'O,o cluster modeling two-re- Single chainswith repeatsgreater than two peat chains indicate that chains with l(SiOSi) ranging from approximately l25o to l50o and ln an investigationof the influenceof cation prop- l(OOO)b. from l25o to l80o are energetically fa- erties on the shapeof silicate chains, Liebau (1980) vored over other possible conformations. This re- found no correlation between the periodicity of stricted range in predicted angular values is observed chainsand the radius,electronegativity, or valenceof in natural and synthetic chain silicates, suggesting the M-cations. Nevertheless,bonding requirements that Si-O, Si-Si, and O-O bonding and non-bonding of the M-cations will influence the periodicity of a interactions play an important role in determining chain and one can consider the periodicity, structur- chain conformations.The investigation is of a pre- ally, as an additional degree of freedom in estab- liminary nature, and more rigorous MO calculations, lishing a configuration of minimum energy.In addi- which can evaluate equilibrium interatomic distances tion, if short-rangeSi-O, Si-Si, and O-O bonding as well as angular relations,will be required to nore and non-bonding interactions are important in the accuratelyevaluate the relationshipsbetween M-cat- two-repeat chains they must also play an important ions and chain conformations in single-chain sili- role in chains with greater periodicities.The re- cates. strictedrange in averagel(SiOSi) valuesin chain sil- icates such as wollastonite(3-repeat: 143"), Acknowledgments CurNarSLO,, (4-repeat 136"), rhodonite (5-repeat: Financial support for this research through NSERC Operating 137o),pyroxferroite (7-repeat: 136"), and alamosite Grunt 67-7061 is gratefully acknowledged. I thank Professors G. MEAGHER: SILICATE CHAINS

V. Gibbs, F. Liebau, and J. Tossellfor critical reviewsof the man- compared to experimental values for silicates and siloxanes. uscript and many helpful suggestions.Professors Gibbs and Phys. Chem. Minerals, in press. Liebau are also thanked for allowing the author to read preprints Ohashi,Y. and L. W. Finger (1978)The role of octahedralcations of their publications. in pyroxenoid crystal chemistry.I. Bustamite,wollastonite, and the pectolite-schizolite-seranditeseries. lm. M ineral.,63, 27+ 288. References C. W. Burnhamand L. W. Finger (1975)The effectof Ca- Fe substitutionon the clinopyroxenecrystal structur€. Am. Min- Clark, J. R., D. E. Appleman and J. J. papike (1969) Crystal- eral.,60,423434. chemicalcharacterization ofclinopyroxenes based on eight new Papike,J J., C. T. Prewitt, S. Suenoand M. Cameron(1973) Py- structurerefinements. Mineral. Soc.Am. Spec.pap., 2,31_50. roxenes:comparisons ofreal and ideal structuraltopologies. Z. Dobash,P A. (1974) ChemistryProgram Exchange No. Quantum Kristallogr., I 38,254-27 3. I4l favailable from Indiana University, Bloomington, QCPE, Pauling, L. (1939)The Nature ofthe Indiana474011. ChemicalBond. CotrrelllJrri- versity Press,Ithaca, New York. Grosse, H. and E. Tillmanns (1974) Bariummetasilicate, Pople, J. A. and G. A. Segal(1965) Approximate self-consistenr BaSiO3(h).Cryst. Struct. Comm., 603-605. i molecular orbital theory. II. Calculations with complete neglect Gunawardane,R. P., M. E. Cradwick and L. S. Dent (1973) Crys- of differential overlap.J. Chem.Phys., 43, 5136-515l. tal structureof Na2BaSi2O6. J. Chem.Soc. Dalton Trans.,2j97- 2400. consistentmolecular orbital theory. I. Invariant procedures.,L Hawthorne,F. C. and J. Ito (1977)Synthesis and crystal-structure Chem.Phys., 43, Sl29-S135. refinementof transition-metalorthopyroxenes. I: orthoenstatite Ribbe, P. H. and A. R. Prunier, Jr. (1977)Stereochemical system- and (Mg,Mn,Co) orthopyroxene.Can. M ineral.,I 5, 321-j38. atics of ordered A / c sllicate pyroxenes.Am. M ineral., 62, 7 I0- Hesse,K. F. (1977)Refinement of the crystal structureof lithium 720. polysilicate.Acta Crystallogr.,83 3, 901-902. Roothaan,C. C. J. (1951)New developmentsin molecularorbital Hiibner, K. (1977)Chemical bond and related properties of SiO2 theory. Rev. Mod. Physics,23,69-89. I. Character of the chemical bond. Pi7s. Status Solidi, (A)40, Santry, D. P. and G. A. Segal(1967) Approximate self-consistent t33-140. molecular orbital theory. IV. Calculations on molecules includ- Liebau, F. (1961) Unrersuchungeniiber die Grossedes Si-O-Si ing the elements sodium through chlorine. J. Chem. Phys., 47, Valenzwinkels.A cta Crystall o gr., I 4, | 103-l lO9. t58-t74. Liebau, F. (19'12)Crystal chemistryof silicon. In K. H. Wedepohl, Shannon,R. D. and C. T. Prewitt (1969)Effective ionic radii in Ed., Handbook of Geochemistry,YoI. ll/3, p. 14-32. Springer- oxidesand fluorides.Acta Crystallogr.,825,925-946. Verlag, New York. Tossell,J. A. and G. V. Gibbs (1978)The useof molecularorbital - (1980)The influenceofcation propertieson the shapeof calculations on model systems for the prediction of bridging silicatechains. Z. Kristallogr.,in press. bond angle variations in siloxanes,silicates, silicon nitrides and McDonald, W. S. and D. W. J. Cruickshank(1967) A reinvestiga- silicon sulfides.Acta Crystallogr.,A34, 463472. tion of the structure of sodium metasilicate,Na2Si03. laa Vcillenkle, H., A. Wittmann and H. Nowotny (1967) Zur Kris- Crystallogr.,22,3742. tallstruktur von CuGeO3.Monatsh. Chem., 98, 1352-1357. Matsumoto,T., M. Tokonami and N. Moromoto (1975)The crys- - - and - (1968) Die Kristallstruktur von tal structureof omphacite.Am. Mineral., 60, 63+641. Lir(Sior5Geo',r)zOs. Z. Kristallogr.,I 26, 3745. Meagher,E. P., J. A. Tosselland c. V. Gibbs (1979)A CNDO/2 Zoltai, T . ( I 960) Classification of silicates and other minerals with molecular orbital study of the silica polymorphs quartz, cristo- tetrahedralstructures. Am. M ineral.,4 5, 960_973. balite and coesite. Pftys. Chem. Minerals, 4, ll-2I. Newton, M. D. and G. V. Gibbs (1980)Ab inrTrocalculated geom- Manuscript received, October 9, 1979; etries,charges and chargedistributions for tLSiOo and H.SirO, acceptedfor publication, January 4, 1980.