Molecular mimicry of the bond length-bond strength variations in oxide crystals

Minimum energy theoretical bond lengths R _t obtained with robust split-basis molecular orbital calculations for 27 hydroxyacid molecules containing first- and second-row cations X ⁿ⁺ reproduce XO bond lengths in crystals. Plots of ln(R _t ) vs. ln(s), where s is the Pauling bond strength, define two different but essentially parallel trends (for first- and second-row cations, respectively) as observed for crystals. A new bond strength parameter p=s/r is defined where r=1 for first- and r=2 for second-row main-group cations. When a ln(R _t ) vs. ln(p) plot is prepared with these theoretical bond lengths, a single trend is obtained. A regression analysis of this data set shows that more than 99 percent of the variation of ln(R _t can be explained in terms of a linear dependence on ln(p), yielding R=1.39 p ^?0.22 as an estimator of the bond lengths. A comparison of 153 mean XO bond lengths compiled by Shannon (1976) for main-group closed-shell X-cations from all 6 rows of the periodic table with those estimated with this formula for r=1, 2, ..., 6, respectively, shows that these bond lengths are estimated within 0.05 Å on average with nearly 85 percent estimated within 0.10 Å of the observed value. More than 97 percent of the variation of these observed bond lengths can be ranked in terms of a linear dependence on the estimated bond lengths. The success of these calculations is further evidence that the forces that govern bond length variations in oxide crystals behave as if they are short-ranged.     

Power law relationships between bond length, bond strength and electron density distributions

The strength of a bond, defined as p=s/r, where s is the Pauling bond strength and r is the row number of an M cation bonded to an oxide anion, is related to a build-up of electron density along the MO bonds in a relatively large number of oxide and hydroxyacid molecules, three oxide minerals and three molecular crystals. As p increases, the value of the electron density is observed to increase at the bond critical points with the lengths of the bonds shortening and the electronegativities of the M cations bonded to the oxide anion increasing. The assertion that the covalency of a bond is intrinsically connected to its bond strength is supported by the electron density distribution and its bond critical point properties. A connection also exists between the properties of the electron density distributions and the connectivity of the bond strength network formed by the bonded atoms of a structure. Received: 20 August 1997 / Revised, accepted: 3 November 1997     

Pauling bond strength, bond length and electron density distribution

The power law regression equation, <R(M–O)> = 1.46(<ρ(r _c)>/r)^?0.19, relating the average experimental bond lengths, <R(M–O)>, to the average accumulation of the electron density at the bond critical point, <ρ(r _c)>, between bonded pairs of metal and oxygen atoms (r is the row number of the M atom), determined at ambient conditions for oxide crystals, is similar to the regression equation R(M–O) = 1.41(ρ(r _c)/r)^?0.21 determined for three perovskite crystals at pressures as high as 80 GPa. The pair are also comparable with the equation <R(M–O)> = 1.43(<s>/r)^?0.21 determined for oxide crystals at ambient conditions and <R(M–O)> = 1.39(<s>/r)^?0.22 determined for geometry-optimized hydroxyacid molecules that relate the geometry-optimized bond lengths to the average Pauling bond strength, <s>, for the M–O bonded interactions. On the basis of the correspondence between the equations relating <ρ(r _c)> and <s> with bond length, it seems plausible that the Pauling bond strength might serve a rough estimate of the accumulation of the electron density between M–O bonded pairs of atoms. Similar expressions, relating bond length and bond strength hold for fluoride, nitride and sulfide molecules and crystals. The similarity of the expressions for the crystals and molecules is compelling evidence that molecular and crystalline M–O bonded interactions are intrinsically related. The value of <ρ(r _c)> = r[(1.41)/<R(M–O)>]^4.76 determined for the average bond length for a given coordination polyhedron closely matches the Pauling’s electrostatic bond strength reaching each the coordinating anions of the coordinated polyhedron. Despite the relative simplicity of the expression, it appears to be more general in its application in that it holds for the bulk of the M–O bonded pairs of atoms of the periodic table.     

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) p _o, the bond strength sum to a bridging oxygen, (2) f _s(O), the fractional s-character of a bridging oxygen, and (3) f _s (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 p _o, f _s (O), and f _s (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 f _s (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.     

Bond length and radii variations in fluoride and oxide molecules and crystals

Molecular orbital calculations completed on fluoride molecules containing first and second row cations have generated bond lengths, R, that match those observed for coordinated polyhedra in crystals to within ～0.04 Å, on average. The calculated bond lengths and those observed for fluoride crystals can be ranked with the expression R=Kp ^?0.22, where p=s/r, s is the Pauling strength of the bond, r is the row number of the cation and K=1.34. The exponent -0.22 (≈ -2/9) is the same as that observed for oxide, nitride and sulfide molecules and crystals. Bonded radii for the fluoride anion, obtained from theoretical electron density maps, increase linearly with bond length. Those calculated for the cations as well as for the fluoride anion match calculated promolecule radii to within ～0.03 Å, on average, suggesting that the electron density distributions in the vicinity of the minima along the bond paths possess a significant atomic component despite bond type. Bonded radii for Si and O ions provided by experimental electron density maps measured for the oxides coesite, danburite and stishovite match those calculated for a series of monosilicic acid molecules. The resulting radii increase with bond length and coordination number with the radius of the oxide ion increasing at a faster rate than that of the Si cation. The oxide ion within danburite exhibits several distinct radii, ranging between 0.9 and 1.2 Å, rather than a single radius with each exhibiting a different radius along each of the nonequivalent bonds with B, Si and Ca. Promolecule radii calculated for the coordinated polyhedra in danburite match procrystal radii obtained in a structure analysis to within 0.002 Å. The close agreement between these two sets of radii and experimentally determined bonded radii lends credence to Slater's statement that the difference between the electron density distribution observed for a crystal and that calculated for a procrystal (IAM) model of the crystal “would be small and subtle, and very hard to determine by examination of the total charge density.”     

A method for calculating fractional s-character for bonds of tetrahedral oxyanions in crystals

A method for calculating fractional s-character, f _s , for TO bonds has been devised to apply to TO₄ tetrahedral oxyanions in crystals. These f _s -values rank bond lengths with the better correlations obtained for T atoms associated with larger bond strengths and larger electronegativities. As a simple formula, it is found that 2cot²〈?〉₃ does a good job of estimating f _s where 〈?〉₃ is the triple angle average of the three angles common to a given bond.     

A physical basis for Pauling's definition of bond strength

 The average strength, s, of the bonded interactions comprising a cation containing oxide anion coordination polyhedron and the value of the electron density, ρ(r _c ), at the bond-critical points are inversely correlated with bond length. In each case, the observed bond lengths, R, were modeled with power-law expressions defined in terms of s/r and ρ(r _c )/r, respectively, where r is the Periodic Table row number of the cation involved in the bonded interaction. On the basis of the close connection between bond strength and the value of the electron density at the bond-critical point, we conclude that bond strength is a direct measure of bond type; the greater its value, the greater the localization of electron density in the binding region and the greater the shared–electron covalent character of the bonded interaction. Received: 15 October 2002 / Accepted: 17 February 2003 Present address:G. V. Gibbs in care of M. Spackman Department of Chemistry, University of New England, Armidale 2351, Australia Acknowledgements The NSF is thanked for supporting this study with grant EAR–9627458. The paper was written while GVG was a Visiting NSF Scholar at The University of Arizona. The faculty and graduate students of the Department of Geosciences and Bob Downs and Marelina Stimpf in particular are thanked for making the visit great fun.     

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-M ₁ ^n/n+ AlO₂ and SiO₂-M ⁿ⁺O _n/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.     

Tetrahedral bond length variations in sulfates

Observed S—O bond lengths for sulfate tetrahedral oxyanions isolated from crystalline solids correlate with the tetrahedral angles impressed upon these ions by their environments. An algorithm is presented which can account for more than 90 percent of the S—O bond length variations from a consideration only of the average of the three angles common to each bond. Extended Hückel molecular orbital theory serves to rationalize these correlations by relating changes in bond overlap population (i.e., bond strength) to angular distortions of sulfate ions observed in a large number of natural and synthetic compounds.     

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 AO's of Si and the lone-pair AO's 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.     

The factors influencing cation site-preferences in spinels a new mendelyevian approach

Pseudopotential orbital radii r _s , r _p are used to construct an index, r _σ=r _s +r _p , which characterizes the average potential experienced by atomic valence electrons. A plot of r _A ^σ verses r _B ^σ for 172 chalogenide spinels (AB₂X₄, X=O, S, Se, Te) leads to two well defined areas, which separate normal and inverse spinels, with only four errors (a predictive success rate of 98%). The gross sorting is achieved without recourse either to the number of d-electrons or an orbital radius r _d , from which it is inferred that it is the energies and extents of the cation s and p-orbitals which primarily determine coordination number in these systems. This approach to the problem of cation distribution in spinels is contrasted with the less generally applicable, traditional, crystal field ideas. The relevance of both r _σ and crystal field stabilization energies to the thermodynamics of spinel reactions is also discussed.     

The variation of tetrahedral bond lengths in sodic plagioclase feldspars

Multiple linear regression analysis has been applied to the geometric and chemical variables in sodic plagioclases in order to determine their relative effects on individual T-O bond lengths in the Al_1+xSi_3?xO₈ tetrahedral framework. Using data from crystal structure analyses of low and high albite, An₁₆ and An₂₈, and assuming that low albite is completely ordered, 1 $$\begin{gathered} {\text{T}} - {\text{O = 1}}{\text{.568}} + {\text{[(0}}{\text{.122) x (Al content of the T site)]}} \hfill \\ {\text{ }} - {\text{[(0}}{\text{.037) x (}}\Delta {\text{{\rm A}l}}_{{\text{br}}} )] + [0.063){\text{ x }}(\Sigma {\text{[}}q{\text{/(Na,Ca}} - {\text{O)}}^{\text{2}} ])] \hfill \\ {\text{ }} + {\text{[(0}}{\text{.029) x (}} - {\text{1/cosT}} - {\text{O}} - {\text{T)]}} \hfill \\ \end{gathered}$$ where the Al content of a particular tetrahedral (T) site can be estimated from empirically-derived determinative curves, where Δ Al_br is a linkage factor to account for the Al content of adjacent tetrahedral sites, where the formal charge on the (Na_1?xCa_x) atom is q=1+x, and where T-O-T is the inter-tetrahedral angle involving the T-O bond. For sodic plagioclases it is essential to know only the anorthite content and the 2Θ₁₃₁-2Θ_1¯31 spacing (CuK _α radiation) in order to determine the independent variables in this equation and thus to evaluate the individual T-O distances. The 64 individual T-O distances predicted for the four sodic plagioclases by this equation agree well with the observed T-O bond lengths (σ=0.004 Å; r=0.994), and the method has been used by way of example to rationalize the T-O bond lengths in analcime (cf. Ferraris, Jones and Yerkess, 1972).     

Optimization of CNDO for molecular orbital calculation on silicates

The use of approximate molecular orbital (MO) calculations [particularly complete neglect of differential overlap (CNDO)] as a tool in understanding chemical bonding in silicates is investigated. This requires first a detailed analysis of the parametrization employed by the CNDO theory when third row atoms are involved. The accuracy of the CNDO calculations is tested by calculations on the equilibrium bond lengths, orbital energies, and bond stretching force constants of simple third row molecules, for which we have experimental data and/or ab initio results. The effects of an optimization of the parameters in the theory on the calculated properties are then analyzed. The theory is subsequently applied to a sequence of silicate prototypes: silicic acid, H₄SiO₄, disiloxane, (SiH₃) - O - (SiH₃), and disilicic acid, (SiO₃H₃) - O - (SiO₃H₃). With proper tuning of the parameters, the CNDO method can be useful in further elucidating the details of the bonding in silicates.     

Structural relaxation in the MnCO3–CaCO3 solid solution: a Mn K-edge EXAFS study

Mn K-edge EXAFS spectroscopy of solid-solution samples encompassing the complete MnCO₃–CaCO₃ series shows that first-shell Mn–O distances deviate little from the 2.19-Å distance observed in pure MnCO₃. Very slight lengthening is observed only in the limiting case of dilute Mn(II) calcite solid solutions, where the Mn–O distance is 2.21 Å. The observed nearly complete structural relaxation and the composition independence of the Mn–O distance are consistent with the Pauling model behavior of solid solutions, and agree with previous studies showing a high degree of relaxation around hetero-sized substituents in the calcite structure. Strain occurs through bond bending, which is facilitated by the exclusively corner-sharing topology of calcite. Observed distances from Mn to more distant neighbors show significant variation across the solid-solution series that resembles Vegard's law-type behavior but reflects averaging. The high degree of relaxation suggests modest enthalpies of mixing in the solution, consistent with calorimetric studies.     

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 TO ₄ ^?n tetrahedral 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 T ₂O ₇ ^?n ions, 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.     

Neutron diffraction study of aluminous hydroxide δ-AlOOD

We have determined the position of deuterium atoms in δ-AlOOD by neutron powder diffraction at ambient pressure. As previously reported by theoretical and experimental studies, the deuterium atoms are located in the tunnel formed by the chains of AlO₆ octahedra. The data are best fit with the P2₁ nm structure, producing bond lengths of D–O1 of 1.552(2) Å, O2–D of 1.020(2) Å and O1–O2 of 2.571(2). This study confirms that the hydrogen bond is asymmetric at ambient conditions in agreement with recent single-crystal synchrotron study for δ-AlOOH.     

The crystal and magnetic structure of Li-aegirine LiFe3+Si2O6: a temperature-dependent study

Single crystals of Li-aegirine LiFe³⁺Si₂O₆ were synthesized at 1573?K and 3?GPa, and a polycrystalline sample suitable for neutron diffraction was produced by ceramic sintering at 1223?K. LiFe³⁺Si₂O₆ is monoclinic, space group C2/c, a=9.6641(2)?Å, b= 8.6612(3)?Å, c=5.2924(2)?Å, β=110.12(1)^° at 300?K as refined from powder neutron data. At 229?K Li-aegirine undergoes a phase transition from C2/c to P2₁ /c. This is indicated by strong discontinuities in the temperature variation of the lattice parameters, especially for the monoclinic angle β and by the appearance of Bragg reflections (hkl) with h+k≠2n. In the low-temperature form two non-equivalent Si-sites with 〈SiA–O〉=1.622?Å and 〈SiB–O〉=1.624?Å at 100?K are present. The bridging angles of the SiO₄ tetrahedra O3–O3–O3 are 192.55(8)° and 160.02(9)° at 100?K in the two independent tetrahedral chains in space group P2₁ /c, whereas it is 180.83(9)° at 300?K in the high-temperature C2/c phase, i.e. the chains are nearly fully expanded. Upon the phase transition the Li-coordination changes from six to five. At 100?K four Li–O bond lengths lie within 2.072(4)–2.172(3)?Å, the fifth Li–O bond length is 2.356(4)?Å, whereas the Li–O3?A bond lengths amount to 2.796(4)?Å. From ⁵⁷Fe Mössbauer spectroscopic measurements between 80 and 500?K the structural phase transition is characterized by a small discontinuity of the quadrupole splitting. Temperature-dependent neutron powder diffraction experiments show first occurrence of magnetic reflections at 16.5?K in good agreement with the point of inflection in the temperature-dependent magnetization of LiFe³⁺Si₂O₆. Distinct preordering phenomena can be observed up to 35?K. At the magnetic phase transition the unit cell parameters exhibit a pronounced magneto-striction of the lattice. Below T _N Li-aegirine shows a collinear antiferromagnetic structure. From our neutron powder diffraction experiments we extract a collinear antiferromagnetic spin arrangement within the a–c plane.     

High temperature X-ray study of single crystal stishovite synthesized with Li2WO4 as flux

Single crystal stishovite with a square prismatic habit and maximum length 0.8 mm was grown from α-quartz at 120 kbar and ～1,300° C. Li₂WO₄, chosen as a result of a previous experiment in growing coesite, was also successful as flux for stishovite. Single crystal X-ray structure analysis of the crystals thus obtained has been carried out at high temperatures under ambient pressure. Lattice constant measurements give a larger thermal expansion coefficient along the a-axis than along the c-axis. The bond distances and bond angles show a decreasing distortion of the SiO₆ octahedron with increasing temperature. The increasing amplitude of thermal vibrations of oxygen atoms with increasing temperature results in increasing O-O repulsion in the basal plane, which explains the observed crystallographic changes.