G. V. Gibbs

The tetrahedral framework in glasses and melts — inferences from molecular orbital calculations and implications for structure, thermodynamics, and physical properties

Results of ab initio molecular orbital (MO) calculations provide a basis for the interpretation of structural and thermodynamic properties of crystals, glasses, and melts containing tetrahedrally coordinated Si, Al, and B. Calculated and experimental tetrahedral atom-oxygen (TO) bond lengths are in good agreement and the observed average SiO and AlO bond lengths remain relatively constant in crystalline, glassy, and molten materials. The TOT framework geometry, which determines the major structural features, is governed largely by the local constraints of the strong TO bonds and its major features are modeled well by ab initio calculations on small clusters. Observed bond lengths for non-framework cations are not always in agreement with calculated values, and reasons for this are discussed in the text. The flexibility of SiOSi, SiOAl, and AlOAl angles is in accord with easy glass formation in silicates and aluminosilicates. The stronger constraints on tetrahedral BOB and BOSi angles, as evidenced by much deeper and steeper calculated potential energy versus angle curves, suggest much greater difficulty in substituting tetrahedral B than Al for Si. This is supported by the pattern of immiscibility in borosilicate glasses, although the occurrence of boron in trigonal coordination is an added complication. The limitations on glass formation in oxysulfide and oxynitride systems may be related to the angular requirements of SiSSi and Si(NH)Si groups.Although the SiO and AlO bonds are the strongest ones in silicates and aluminosilicates, they are perturbed by other cations. Increasing perturbation and weakening of the framework occurs with increasing ability of the other atom to compete with Si or Al for bonding to oxygen, that is, with increasing cation field strength. The perturbation of TOT groups, as evidenced by TO bond lengthening predicted by MO calculations and observed in ordered crystalline aluminosilicates, increases in the series Ca, Mg and K, Na, Li. This perturbation correlates strongly with thermochemical mixing properties of glasses in the systems SiO2-M1n/n+AlO2 and SiO2-Mn+On/2 (M=Li, Na, K, Rb, Cs, and Mg, Ca, Sr, Ba, Pb), with tendencies toward immiscibility in these systems, and with systematics in vibrational spectra. Trends in physical properties, including viscosity at atmospheric and high pressure, can also be correlated.

A molecular orbital study of bond length and angle variations in framework structures

Molecular orbital calculations on a variety of silicate and aluminosilicate molecules have been used to explore the bonding forces that govern tetrahedral bond length variations, r(TO), in framework silicates and aluminosilicates. Not only do the calculations provide insight into the variety of structural types and the substitution limits of one tetrahedral atom for another, but they also provide an understanding of the interrelationships among r(TO) and linkage factors, bond strength sum, coordination number, and angles within and between tetrahedra. A study of these interrelationships for a theoretical data set shows that r(SiO) and r(AlO) are linearly correlated with (1) po, the bond strength sum to a bridging oxygen, (2) fs(O), the fractional s-character of a bridging oxygen, and (3) fs(T), the fractional s-character of the T atom. In a multiple linear regression analysis of the data, 92% of the variation of r(SiO) and 99% of the variation of r(AlO) can be explained in terms of a linear dependence on po, fs(O), and fs(T). Analogous regression analyses completed for observed r(Al, SiO) bond length data from a number of silica polymorphs and ordered aluminosilicates account for more than 75% of the bond length variation. The lower percentage of bond length variation explained is ascribed in part to the random and systematic errors in the experimental data which have a negligible effect on the theoretical data. The modeling of more than 75% of the variation of r(Al, SiO) in the framework silicates using the same model used for silicate and aluminosilicate molecules strengthens the viewpoint that the bonding forces that govern the shapes of such molecules are quite similar to the forces that govern the shapes of chemically similar groups in solids. The different regression coefficients calculated for fs(T) indicate that SiO and AlO bond length variations in framework structures should not be treated as a single population in estimating the average Al, Si content of a tetrahedral site.

Ab initio calculated geometries and charge distributions for H4SiO4 and H6Si2O7 compared with experimental values for silicates and siloxanes

Ab initio STO-3G molecular orbital theory has been used to calculate energy-optimized Si-O bond lengths and angles for molecular orthosilicic and pyrosilicic acids. The resulting bond length for orthosilicic acid and the nonbridging bonds for pyrosilicic acid compare well with Si-OH bonds observed for a number of hydrated silicate minerals. Minimum energy Si-O bond lengths to the bridging oxygen of the pyrosilicic molecule show a close correspondence with bridging bond length data observed for the silica polymorphs and for gas phase and molecular crystal siloxanes when plotted against the SiOSi angle. In addition, the calculations show that the mean Si-O bond length of a silicate tetrahedron increases slightly as the SiOSi angle narrows. The close correspondence between the Si-O bond length and angle variations calculated for pyrosilicic acid and those observed for the silica polymorphs and siloxanes substantiates the suggestion that local bonding forces in solids are not very different from those in molecules and clusters consisting of the same atoms with the same coordination numbers. An extended basis calculation for H4SiO4 implies that there are about 0.6 electrons in the 3d-orbitals on Si. An analysis of bond overlap populations obtained from STO-3G* calculations for H6Si2O7 indicates that Si-O bond length and SiOSi angle correlations may be ascribed to changes in the hybridization state of the bridging oxygen and (d – p) π-bonding involving all five of the 3d AOs of Si and the lone-pair AOs of the oxygen. Theoretical density difference maps calculated for H6Si2O7 show a build-up of charge density between Si and O, with the peak-height charge densities of the nonbridging bonds exceeding those of the bridging bonds by about 0.05 e Å−3. In addition, atomic charges (+1.3 and −0.65) calculated for Si and O in a SiO2 moiety of the low quartz structure conform reasonably well with the electroneutrality postulate and with experimental charges obtained from monopole and radial refinements of diffraction data recorded for low quartz and coesite.

Applications of quantum mechanical potential surfaces to mineral physics calculations

Ab-Initio quantum mechanical calculations on molecular clusters are used to obtain potential surfaces for the SiO bond in silicates. These potential surfaces form the basis for extracting the key parameters in various commonly employed potential functions. Applications to the usual ionic model demonstates a close relation between the ab-initio derived ionic potential and those empirically dervied. The ionic model is then used to predict structures and elastic properties of orthosilicates and of the silica polymorphs. The deficiencies in the ionic model lead to the application of the quantum results to covalent models. These latter models are then used in theoretical calculations of the properties of silica polymorphs.

Strain-Tensor Components Expressed in Terms of Lattice Parameters

Expressions are developed for the components of the linear Lagrangian, linear Eulerian, finite Lagrangian (Greens), and finite Eulerian (Almansis) strain tensors in terms of a crystals lattice parameters before and after a deformation. The development has been undertaken with the concepts and notations of linear algebra.

Defects in amorphous silica: Ab initio MO calculations

SCF‐MO calculations have been carried out on a number of silicon oxy‐hydroxides chosen to simulate possible defects in amorphous silica. It is concluded that Si=O double bonds and four‐membered SiOSiO rings are not plausible high‐concentration defects. Support is provided for Galeener’s assignment of the D1 and D2 ‘‘defect’’ bands in the Raman spectrum of amorphous silica to eight‐ and six‐membered rings, respectively.

Molecular orbital studies of geometries and spectra of minerals and inorganic compounds

Extended Hückel molecular orbital theory (EHT) and simple, approximate Self-Consistent-Field MO methods are employed to explain the geometries of nontransition metal bearing minerals and inorganic compounds. The spectra of such minerals and the electronic structure of transition metal oxidic minerals are explained using the Self-Consistent-Field X α MO method.EHT provides an objective algorithm for rationalizing and correlating bond length and angle data for insular and polymerized TO4−ntetrahedral oxyanions where T=Be, B, Al, Si, P, S, Ge, As and Se. Calculated bond overlap populations n(T-O), correlate linearly with the observed T-O bond lengths with shorter bonds tending to involve larger n(T-O) values. Such calculations show that n(T-O) is strongly dependent upon the average of the three O-T-O angles associated with a common bond, larger n(T-O) values involving wider angles. Calculations of n(T-O) as a function of the T-O-T angles in T2O7−nions, indicate that the n(T-O) values for the bonds to the bridging oxygen atoms increase nonlinearly with increasing T-O-T angle whereas those the nonbridging oxygens decrease slightly as the angle widens. In agreement with the experimental data, these results predict that shorter T-O bonds should involve wider O-T-O and T-O-T angles.The SCF-X α MO cluster model is then applied to silica and FeO. The calculations yield a satisfactory interpretation of the visible, UV and X-ray emission and X-ray photoelectron spectra of these materials. Theoretical and empirical MO diagrams are constructed and the electronic structures of the materials are discussed.

Ab initio calculation of interatomic force constants in H6Si2O7 and the bulk modulus of α quartz and α cristobalite

Ab initio force constants calculated for Si-O stretch and Si...Si non-bonded interactions in H6Si2O7 are found comparable with experimental values derived from the lattice dynamics of α quartz. The bulk moduli of α quartz and α cristobalite are calculated using the molecular Si...Si force constant and assuming rigid regular SiO4 tetrahedra. In the case (α quartz) where data are available the calculation agrees well with experiment.

Mechanisms of silica dissolution as inferred from the kinetic isotope effect

The rate of dissolution of many rock-forming minerals is controlled by the hydrolysis of the bridging silicate bonds at the mineral surface. This hydrolysis can be studied at a detailed level by combining experiments in isotopically distinct solutions with molecular-orbital calculations of the reaction energetics and geometry. These calculations are linked to the macroscopic process of dissolution via the transitionstate theory. Rates of silicate leaching and dissolution in D2O and H2O differ for several reasons. First, hydrolysis involves transfer of hydrogen from the solution to a bridging siloxane bond at the mineral surface. The transition state equilibrium describing this reaction varies with the vibrational properties (and, hence, isotopic composition) of the reactant and the transition state. Secondly, the equilibrium acid-base properties of the oxide surface in H2O and D2O are not identical. These differences are important because rates of hydrolysis reactions are enhanced by adsorbed hydrogen and hydroxyl ions. On the basis of the molecular orbital calculations and an assumed mechanism of hydrolysis, quartz dissolution is predicted to be roughly a factor of four slower in D2O than in H2O at pH = pD = 3. The activation energy is predicted to be ≈20 kcal /mol, which is in agreement with values measured at hydrothermal conditions. At pH/pD conditions near the isoelectric point of quartz and in the temperature range 20–70°C, the measured rate in D2O is only about 15% slower than in H2O and the activation energy is ≈8 kcal/mol. The activation energy is slightly higher at pHpD = 11, but the kinetic isotope effect remains small. The discrepancy suggests that the hydrogen transfer to bridging oxygen from water at the experimental conditions is more rapid than modeled and may proceed early in the overall reaction.

Quantum mechanical potential surfaces and calculations on minerals and molecular clusters

Recently, ab-initio quantum mechanical potential surfaces calculated for silicate hydroxyacid molecules were used to extract covalent potentials for use in mineral physics calculations (Lasaga and Gibbs 1987). The calculations showed that these potentials are capable of generating the structure and physical properties of silicate minerals. In this paper we explore in more detail the suitability of various covalent potentials in mimicking the topography of the ab-initio potential surfaces. We also extend the use of such quantum-derived potentials in generating the structures of hydroxyacid dimers, trimers, and pentamers of silicate tetrahedra and in studying the structure and the dynamical properties of minerals and glasses.