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.”     

Bond length variation in hydronitride molecules and nitride crystals

Bond lengths calculated for the coordination polyhedra in hydronitride molecules match average values observed for XN bonds involving main group X-cations in nitride crystals to within 0.04 Å. As suggested for oxide and sulfide molecules and crystals, the forces that determine the average bond lengths recorded for coordinated polyhedra in hydronitride molecules and nitride crystals appear to be governed in large part by the atoms that comprise the polyhedra and those that induce local charge balance. The forces exerted on the coordinated polyhedra by other parts of the structure seem to play a small if not an insignificant role in governing bond length variations. Bonded radii for the nitride ion obtained from theoretical electron density maps calculated for the molecules increase linearly with bond length as observed for nitride crystals with the rock salt structure. Promolecule radii calculated for the molecules correlate with bonded and ionic radii, indicating that the electron density distributions in hydronitride molecules possess a significant atomic component, despite bond type.     

A molecular orbital study of bonding in sulfate molecules: Implications for sulfate crystal structures

Molecular orbital calculations have been completed on sulfate monomers and a dimer in a determination of minimum-energy geometries and electron density distributions. SO bond lengths calculated for the monomer and dimer correlate linearly with the fractional s-characters of the bonds, as observed for sulfate groups in crystals. With increasing coordination number of S, the bonded radii of S and O, as determined from electron density maps, increase at the same rate. This is at variance with the assumption that the radius of the oxide ion is nearly constant and that bond length variations arise primarily from changes in cation radii. The dimer shows a relatively large change in energy as its SOS angle is deformed from its minimum-energy value (125.6°) to 180°, in conformity with the small variation among observed angles. This is in contrast to the wide variation of bridging angles observed for silicate and phosphate dimers in crystals and molecules, and may imply that significant differences should be expected in the behavior of sulfates with respect to polymorphism and glass formation. The reaction energy of SO₃ + H₂O → H₂SO₄, calculated with second-order Møller-Plesset perturbation theory, agrees with the experimental value. Other properties of H₂SO₄ are also calculated and compared with experimental observations and previous calculations.     

A comparison of procrystal and ab initio model representations of the electron-density distributions of minerals

 The procrystal calculation of the electron density is a very rapid procedure that offers a quick way to analyze various bonding properties of a crystal. This study explores the extent to which the positions, number, and properties of bond-critical points determined from the procrystal representations of the electron density for minerals are similar to those of first-principles ab initio model distributions. The purpose of the study is to determine the limits imposed upon interpretation of the procrystal electron density. Procrystal calculations of the electron density for more than 300 MO bonds in crystals were compared with those previously calculated using CRYSTAL98 and TOPOND software. For every bond-critical point found in the ab initio calculations, an equivalent one was also found in the procrystal model, with similar magnitudes of electron density, and at similar positions along the bonds. The curvatures of the electron densities obtained from the ab initio and the procrystal distributions are highly correlated. It is concluded that the procrystal distributions are capable of providing good estimates of the bonded radii of the atoms and the properties of the electron-density distributions at the bond-critical points. Because the procrystal model is so fast to compute, it is especially useful in addressing the question as to whether a pair of atoms is bonded or not. If the Bader criteria for bonding are accepted, then the successful generation of the bond-critical points by the procrystal model demonstrates that bonding is an atomic feature. The main difference between the critical-point properties of the procrystal and the ab initio model is that the curvature in the electron density perpendicular to the bond path of the ab initio model is sharper than for the procrystal model. This is interpreted as indicating that the electrons that migrate into a bond originate from its sides, and not from the regions closer to the nuclei. This observation also suggests that ab initio optimization routines could see an improvement in speed if the parameters relating to the angular components of atomic wave functions were to vary before the radial components. Received: 6 August 2001 / Accepted: 21 November 2001     

SiO bonded interactions in coesite: a comparison of crystalline, molecular and experimental electron density distributions

Bond critical point properties of electron density distributions calculated for representative Si₅O₁₆ moieties of the structure of coesite are compared with those observed and calculated for the bulk crystal. The values calculated for the moieties agree with those observed to within ∼5%, on average, whereas those calculated for the crystal agree to within ∼10%. As the SiOSi angles increase and the SiO bonds shorten, there is a progressive build-up in the calculated electron density along the bonds. This is accompanied by an increase in both the curvatures of the electron density, both perpendicular and parallel to each bond, and the Laplacian of the electron density distribution at the bond critical points. The cross sections of the bonds at the critical points become more circular as the angle approaches 180o. Also, the bonded radius of the oxide anion decreases about twice as much as that of the Si cation as the SiO bond length decreases and the fraction of s-character of the bond is indicated to increase. A knowledge of electron density distributions is central to our understanding of the forces that govern the structure, properties, solid state reactions, surface reactions and phase transformations of minerals. The software (CRYSTAL95 and TOPOND) used in this study to calculate the bond critical properties of the electron density and Laplacian distributions is bound to promote a deeper understanding of crystal chemistry and properties. Received: 23 February 1998 / Revised, accepted: 16 July 1998     

含不同半径孔洞的颗粒体模型的力学行为数值模拟

本文通过编程建立了非连续介质(颗粒体材料)模型,采用FLAC软件模拟了静水压力条件下不同半径的巷道围岩中的剪切应变增量、最小主应力及最大主应力的分布规律。研究表明,随着孔洞半径的增大,呈圆环形的剪切应变增量与最小主应力的高值区的圈数、呈辐射状的最大主应力的高值区的延伸范围及剪切应变增量的最大值都呈先慢后快的增长趋势。模型中最大的拉应力接近于在模型四周所施加的压应力,而最大的压应力约为所施加的压应力的5～10倍。模型内部的剪切应变增量、最小主应力及最大主应力的分布是高度不均匀的。具有较高的差应力的位置与具有较高的剪切应变增量的位置具有很好的一致性。     

The influence of ionic radii of cations and covalent forces on the unit cell dimensions of spinels

Fe-, Cr- and Al-spinels were synthesized and their unit cell sizes determined by means of X-rays. Differential thermal curves show that the magnetic inversion of Fe₂O₃ at 680° C accelerates the formation of the ferrites when the constituent oxides are heated together.A correlation can be made between ionic radii of cations and unit cell dimensions provided the effect of covalent forces in the lattice is taken into account. The values for ionic radii of cations as given byAhrens (1952) permit a better correlation than those ofGoldschmidt.A shrinkage of 0.01 Å in the unit cell size per 0.01 Å decrease in the ionic radius of the divalent cations was determined when spinels with the same cation arrangement in the same group were compared. A shrinkage of 0.027 Å in the unit cell size per 0.01 Å decrease in the ionic radius of the trivalent cations was determined in spinels having the same divalent cation and cation arrangement when the trivalent cations form the same type of bonds.The half-occupation of the 3d orbits in Mn²⁺ and Fe³⁺ causes abnormally high unit cell dimensions in spinels where these ions are incorporated in octahedral sites. This is attributed to the formation of electrovalent bonds by these ions. Variable forces of contraction in the lattice are revealed when the unit cell dimensions are correlated with the ionic radii of cations. The force of contraction can be satisfactorily explained as being due to covalent forces in the spinel structure. The magnitude of this force or the degree of covalence in the bonds increases in the following order of cations where these are situated in tetrahedral sites:The divalent transition element ions, Fe²⁺, Co²⁺ and Ni²⁺; the B-Sub-group element ions Cd²⁺ and Zn²⁺; Fe³⁺ in tetrahedral co-ordination.     

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     

SiO and GeO bonded interactions as inferred from the bond critical point properties of electron density distributions

The topological properties of the electron density distributions for more than 20 hydroxyacid, geometry optimized molecules with SiO and GeO bonds with 3-, 4-, 6- and 8-coordinate Si and Ge cations were calculated. Electronegativities calculated with the bond critical point (bcp) properties of the distributions indicate, for a given coordination number, that the electronegativity of Ge (∼1.85) is slightly larger than that of Si (∼1.80) with the electronegativities of both atoms increasing with decreasing bond length. With an increase in the electron density, the curvatures and the Laplacian of the electron density at the critical point of each bond increase with decreasing bond length. The covalent character of the bonds are assessed, using bond critical point properties and electronegativity values calculated from the electron density distributions. A mapping of the (3, −3) critical points of the valence shell concentrations of the oxide anions for bridging SiOSi and GeOGe dimers reveals a location and disposition of localized nonbonding electron pairs that is consistent with the bridging angles observed for silicates and germanates. The bcp properties of electron density distributions of the SiO bonds calculated for representative molecular models of the coesite structure agree with average values obtained in X-ray diffraction studies of coesite and danburite to within ∼5%. Received: 18 August 1997 / Revised, accepted: 19 February 1998     

Experimental and theoretical bond critical point properties for model electron density distributions for earth materials

Generalized X-ray scattering factor model experimental electron density distributions and bond critical point, bcp, properties generated in recent studies for the earth materials stishovite, forsterite, fayalite and cuprite with high energy single crystal synchrotron X-ray diffraction data and those generated with high resolution diffraction data for coesite and senarmonite were found to be adequate and in relatively good agreement, ~5% on average, with those calculated with quantum chemical methods with relatively robust basis sets. High resolution low energy single crystal diffraction data, recorded for the molecular sieve AlPO₄-15, were also found to yield model electron density distribution values at the bcp points for the AlO and PO bonded interactions that are in relatively good to moderately good agreement with the theoretical values, but the Laplacian values of the distribution at the points for the two bonded interactions were found to be in relatively poor agreement. In several cases, experimental bcp properties, generated with conventional low energy X-ray diffraction data for several rock forming minerals, were found overall to be in poorer agreement with the theoretical properties. The overall agreement between theoretical bcp properties generated with computational quantum methods and experimental properties generated with synchrotron high energy radiation not only provides a basis for using computational strategies for studying and modeling structures and their electron density distributions, but it also provides a basis for improving our understanding of the crystal chemistry and bonded interactions for earth materials. Theoretical bond critical point properties generated with computational quantum methods are believed to rival the accuracy of those determined experimentally. As such the calculations provide a powerful and efficient method for evaluating electron density distributions and the bonded interactions for a wide range of earth materials.Dedicated to Professor Robert F. Stewart of Carnegie Mellon University on his retirement for his brilliant and original work modeling electron density distributions.     

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.     

A mapping of the electron localization function for earth materials

The electron localization function, ELF, generated for a number of geometry-optimized earth materials, provides a graphical representation of the spatial localization of the probability electron density distribution as embodied in domains ascribed to localized bond and lone pair electrons. The lone pair domains, displayed by the silica polymorphs quartz, coesite and cristobalite, are typically banana-shaped and oriented perpendicular to the plane of the SiOSi angle at ~0.60 Å from the O atom on the reflex side of the angle. With decreasing angle, the domains increase in magnitude, indicating an increase in the nucleophilic character of the O atom, rendering it more susceptible to potential electrophilic attack. The Laplacian isosurface maps of the experimental and theoretical electron density distribution for coesite substantiates the increase in the size of the domain with decreasing angle. Bond pair domains are displayed along each of the SiO bond vectors as discrete concave hemispherically-shaped domains at ~0.70 Å from the O atom. For more closed-shell ionic bonded interactions, the bond and lone pair domains are often coalesced, resulting in concave hemispherical toroidal-shaped domains with local maxima centered along the bond vectors. As the shared covalent character of the bonded interactions increases, the bond and lone pair domains are better developed as discrete domains. ELF isosurface maps generated for the earth materials tremolite, diopside, talc and dickite display banana-shaped lone pair domains associated with the bridging O atoms of SiOSi angles and concave hemispherical toroidal bond pair domains associated with the nonbridging ones. The lone pair domains in dickite and talc provide a basis for understanding the bonded interactions between the adjacent neutral layers. Maps were also generated for beryl, cordierite, quartz, low albite, forsterite, wadeite, åkermanite, pectolite, periclase, hurlbutite, thortveitite and vanthoffite. Strategies are reviewed for finding potential H docking sites in the silica polymorphs and related materials. As observed in an earlier study, the ELF is capable of generating bond and lone pair domains that are similar in number and arrangement to those provided by Laplacian and deformation electron density distributions. The formation of the bond and lone pair domains in the silica polymorphs and the progressive decrease in the SiO length as the value of the electron density at the bond critical point increases indicates that the SiO bonded interaction has a substantial component of covalent character.     

Critical point properties of electron density distributions for oxide molecules containing first and second row cations

 Minimum energy geometries and electron density distributions, ϱ(r), for ∼40 polyatomic oxide molecules containing first and second row M-cations have been calculated at the Hartree-Fock level with a 6-311++G** basis set. The nature of the bonded interactions in these molecules is examined in terms of the relative electronegativities, χ _M , of the M-cations and the properties of the electron density distribution, ϱ(r _c ), evaluated at the bond critical points, r _c , along each MO bond. As ϱ(r _c ) and the Laplacian of ϱ(r _c ) increase, χ _M increases indicating an increase in the covalent character of the bonded interactions between M and O. The ratios of the curvatures of ϱ(r _c ) indicate that the NO bond is predominantly covalent, that the CO and SO bonds are of intermediate type and that the remaining MO bonds are indicated to be predominantly ionic in character. A comparison of the critical point properties of ϱ(r _c ) and χ _M indicates that the minimum energy MO bond length is an important determinate of the properties of ϱ(r _c ) and the character of the MO bonds. On the other hand, values of the local energy density, H(r _c ), indicate that the LiO, BeO, NaO, MgO and AlO bonds are predominantly ionic and that the BO, CO, NO, SiO, PO and SO bonds are predominantly covalent in character. The χ _M -values provided by the properties of ϱ(r _c ) indicate that the covalent component of a bond increases with decreasing bond length, coordination number and increasing bond strength. Each MO bond seems to represent a unique entity and to possess a distinct set of ϱ(r _c ) properties, the distinction being greater for the more electronegative cations. The bonded radius of the oxide ion, r _b (O), and the χ _M -values determined from ϱ(r _c ) correlate with values determined from promolecule electron density distributions. In addition, r _b (O) and χ _M -values determined from experimental electron density distributions for crystals correlate with values determined from procrystal electron density distributions. The number of critical points and bond paths are modeled rather faithfully by procrystal and promolecule electron density distributions, despite the neglect of the binding forces in their constructions. Received: October 15, 1996/Revised, accepted: February 10, 1997     

Thioarsenides: a case for long-range Lewis acid–base-directed van der Waals interactions

Electron density distributions, bond paths, Laplacian and local-energy density properties have been calculated for a number of As₄S _n (n = 3, 4 and 5) thioarsenide molecular crystals. On the basis of the distributions, the intramolecular As–S and As–As interactions classify as shared bonded interactions, and the intermolecular As–S, As–As and S–S interactions classify as closed-shell van der Waals (vdW) bonded interactions. The bulk of the intermolecular As–S bond paths link regions of locally concentrated electron density (Lewis-base regions) with aligned regions of locally depleted electron density (Lewis-acid regions) on adjacent molecules. The paths are comparable with intermolecular paths reported for several other molecular crystals that link aligned Lewis base and acid regions in a key–lock fashion, interactions that classified as long-range Lewis acid–base-directed vdW interactions. As the bulk of the intermolecular As–S bond paths (~70%) link Lewis acid–base regions on adjacent molecules, it appears that molecules adopt an arrangement that maximizes the number of As–S Lewis acid–base intermolecular bonded interactions. The maximization of the number of Lewis acid–base interactions appears to be connected with the close-packed array adopted by molecules: distorted cubic close-packed arrays are adopted for alacránite, pararealgar, uzonite, realgar and β-AsS and the distorted hexagonal close-packed arrays adopted by α- and β-dimorphite. A growth mechanism is proposed for thioarsenide molecular crystals from aqueous species that maximizes the number of long-range Lewis acid–base vdW As–S bonded interactions with the resulting directed bond paths structuralizing the molecules as a molecular crystal.     

基于不同半径分段组合的微动探测方法研究

空间自相关法是一种传统的微动勘查方法。笔者在双重圆形观测台阵中不同半径组合方式下,利用空间自相关法分别求出自相关系数以及频散曲线,对比不同半径的频散曲线,结合前人研究成果,总结出大半径组合方式在低频部分的可靠性更高,小半径组合方式在高频部分的可靠性更高,进而提出了分频段拟合贝塞尔函数、分半径平均频散曲线的方法,一定程度上扩宽了可用频率范围,提高了探测精度。通过场地实验证实了方法的有效性。     

Rare earth and high field strength element partitioning between iron-rich clinopyroxenes and felsic liquids

Rare earth elements are commonly assumed to substitute only for Ca in clinopyroxene because of the similarity of ionic radii for REE³⁺ and Ca²⁺ in eightfold coordination. The assumption is valid for Mg-rich clinopyroxenes for which observed mineral/melt partition coefficients are readily predicted by the lattice strain model for substitution onto a single site (e.g., Wood and Blundy 1997). We show that natural Fe-rich pyroxenes in both silica-undersaturated and silica-oversaturated magmatic systems deviate from this behavior. Salites (Mg# 48–59) in phonolites from Tenerife, ferrohedenbergites (Mg# 14.2–16.2) from the rhyolitic Bandelier Tuff, and ferroaugites (Mg# 9.6–32) from the rhyolitic Rattlesnake Tuff have higher heavy REE contents than predicted by single-site substitution. The ionic radius of Fe²⁺ in sixfold coordination is substantially greater than that of Mg²⁺; hence, we propose that, in Fe-rich clinopyroxenes, heavy REE are significantly partitioned between eightfold Ca sites and sixfold Fe and Mg sites such that Yb and Lu exist dominantly in sixfold coordination. We also outline a REE-based method of identifying pyroxene/melt pairs in systems with multiple liquid and crystal populations, based upon the assumption that LREE and MREE reside exclusively in eightfold coordination in pyroxene. Contrary to expectations, interpolation of mineral/melt partition coefficient data for heavy REE does not predict the behavior of Y. We speculate that mass fractionation effects play a role in mineral/melt lithophile trace element partitioning that is detectable among pairs of isovalent elements with near-identical radii, such as Y and Ho, Zr and Hf, and Nb and Ta.     

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.     

硅酸盐矿物中Fe^2＋离子同质异能位移与平均有效离子半径之间的关系

同质异能位移是矿物的重要穆斯鲍尔参数之一，本文首次讨论了硅酸盐矿物中硅酸盐矿物中Ｆｅ＾２＋离子的同质异能位移ＩＳ与离子的平均有效半径ｒ↑－之间的关系，发现它们之间的关系可以用一简单线性方程ＩＳ（ｍｍ／ｓ）＝０．５８ｒ↑－＋０．６９５来表示，其相关系数Ｒ＝０．９３。本文还根据同质异能位移与原子核处ｓ电子密度│ψＡ（０）│＾２之间的关系，定性地解释了上述线性方程。     

Rare earth element distribution in some hydrothermal minerals: evidence for crystallographic control

Rare earth element (REE) abundances were measured by neutron activation analysis in anhydrite (CaSO₄), barite (BaSO₄), siderite (FeCO₃) and galena (PbS). A simple crystal-chemical model qualitatively describes the relative affinities for REE substitution in anhydrite, barite, and siderite. When normalized to ‘crustal’ abundances (as an approximation to the hydrothermal fluid REE pattern), log REE abundance is a surprisingly linear function of (ionic radius of major cation—ionic radius of REE)² for the three hydrothermal minerals, individually and collectively. An important exception, however, is Eu, which is anomalously enriched in barite and depleted in siderite relative to REE of neighboring atomic number and trivalent ionic radius. In principle, REE analyses of suitable pairs of co-existing hydrothermal minerals, combined with appropriate experimental data, could yield both the REE content and the temperature of the parental hydrothermal fluid.The REE have only very weak chalcophilic tendencies, and this is reflected by the very low abundances in galena—La, 0.6 ppb; Sm, 0.06 ppb; the remainder are below detection limits.