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AM65 746.Pdf 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.
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