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Bonded interactions in silica polymorphs, silicates, and siloxane molecules

¹ Department of Geosciences, Virginia Tech, Blacksburg, Virginia 24061, U.S.A.
² Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24061, U.S.A.
³ Department of Geosciences, University of Arizona, Tucson, Arizona 85721, U.S.A.
⁴ Chemical and Materials Science Division, and the W.R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352, U.S.A.

Correspondence: ^* E-mail: ggibbs{at}vt.edu

	ABSTRACT

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
Experimental model electron density distributions obtained for the silica polymorphs coesite and stishovite are comparable with electron density distributions calculated for various silicates and siloxane molecules. The Si-O bond lengths and Si-O-Si angles calculated with first-principles density functional theory methods as a function of pressure are also comparable with the bond lengths and angles observed for coesite and quartz within the experimental error. The similarity of the topological properties of the Si-O bonded interactions and the experimental and the geometry-optimized structures for the silica polymorphs provide a basis for understanding the properties and crystal chemistry of silica. The agreement supports the argument that the bulk of the structural and physical properties of the silica polymorphs are intrinsic properties of molecular-like coordination polyhedra such that the silica polymorphs can be pictured as "supermolecules" of silica bound by virtually the same forces that bind the Si and O atoms in simple siloxane molecules. The topology of the electron density distribution is consistent with the assertion that the Si-O bonded interaction arises from the net electrostatic attraction exerted on the nuclei by the electron density accumulated between the Si and O atoms. The correlation between the Si-O bond length and Si-O-Si angle is ascribed to the progressive local concentration of the electron density in the nonbonded lone pair region of the O atom rather than to a bonded interaction that involves the d-orbitals on Si.

The accumulation of deformation electron density, (r), in the bonded and nonbonded regions of the Si-O bond, the close proximity of the bond critical point, r_c, of the bond with the nodal surface of the Laplacian and the negative value of the total energy density are taken as evidence that the bond has a nontrivial component of shared character. For M-O bonded interactions for first and second row metal atoms bonded to O, ²(r_c) is positive and increases linearly as (r_c) and G(r_c)/(r_c) both increase and as the value of H(r_c) decreases; the greater the shared character of the interaction, the larger the values of both ²(r_c) and G(r_c)/(r_c). In addition, a mapping of ²(r) serves to highlight those Lewis base domains that are susceptible to electrophilic attack by H, like the O atoms in coesite involved in bent Si-O-Si angles; the narrower the angle, the greater the affinity for H. On the basis of the net charges conferred on the Si and O atoms and the bonded radii of the two atoms, the Si-O bond for stishovite, with six-coordinated Si and three-coordinated O, is indicated to be more ionic in character than that in quartz with four-coordinated Si and two-coordinated O. Unlike the conclusion reached for ionic and crystal radii, it is the bonded radius of the O atom that increases with the increasing coordination number of Si, not the radius of the Si atom. The modeling of the electron density distributions for quartz, coesite, and beryl as a function of pressure suggests that the shared character of the bonded interactions in these minerals increases slightly with increasing pressure. The insight provided by the calculations and the modeling of the electron density distributions and the structures of the silica polymorphs bodes well for future Earth materials studies that are expected to improve and clarify our understanding of the connection between properties and structure within the framework of quantum mechanical observables, to find new and improved uses for the materials and to enhance our understanding of crystal chemistry and chemical reactions of materials in their natural environment at the atomic level.

Key Words: Bonded interactions • electron density distributions • silica • coesite • quartz • stishovite • cristobalite • siloxane

"While science is pursuing a steady onward movement, it is convenient to cast a glance back on the route already traversed, and especially to consider new conceptions which aim at discovering the general meaning of the stock of facts accumulated from day to day in our laboratories." Dmitri Mendeleeff (1889), Faraday Lecture delivered before the Fellows of the Chemical Society

	INTRODUCTION

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
The concept of the chemical bond has played a pivotal role in advancing our understanding of the structures, properties, and reactivity of materials at the atomic level. Since the nineteenth century, interest in the concept has burgeoned as evinced by the large number of papers, review articles and books that have been written on the topic. For example, since Linus Pauling wrote his classic paper entitled "The nature of the chemical bond" in 1931, more than 26 000 papers have been published about one or more aspects of the concept, evidence of the profound impact that it has had on chemical thinking and research. It is generally accepted that a chemical bond is the result of the net electrostatic attraction exerted on the nuclei of a pair of atoms by the electron density (ED) accumulated between the nuclei (Feynman 1939). Nevertheless, given that a chemical bond is not a quantum mechanical observable, the famous mathematician Charles A. Coulson (1955) concluded, in a lecture delivered before the Royal Society, that "A chemical bond is not a real thing: it does not exist, no one has ever seen it, no one ever can. It is a figment of our imagination." Bader (2008) recently stated that "There are no chemical bonds, they simply don’t exist, only bonded atoms exist." Hoffmann (1988) added that physicists do not see chemical bonds, they only see bands. But, Pauling (1939) wrote, in his famous book, The Nature of the Chemical Bond, that quantum mechanics has not only clarified the mystery that has shrouded the nature of the chemical bond, but it has also introduced into chemical theory the concepts of hybridization, resonance, bond character, and a set of rules for the formation of covalent and ionic bonds. But, as countered by Coulson (1953), "Concepts like hybridization, resonance, covalent, and ionic structures do not appear to correspond to anything directly measurable." It also appears that empirical concepts like ionic and crystal radii, bond strength, valence, and bond valence that have worked well for mineralogists and geochemists in the modeling of bond-length variations, properties, acid and base sites and structure likewise do not appear to correspond with anything that is directly measurable. In a recent discussion on the chemical bond quandary, Alvarez et al. (2009) observed that "Chemistry has done more than well in creating a universe of structure ... with just this imperfectly defined concept of a chemical bond. Or maybe it has done well precisely because the concept is flexible and fuzzy." Clearly, as asserted by the chemical physicist Robert Mulliken, the concept of "the chemical bond is not so simple as some people seem to think" (Coulson 1960). Given these inconvenient truths, Cremer and Kraka (1984) reiterated that "There is no way to measure a bond or any bond property." They continued by declaring that a bonded interaction can only be defined within the framework of a given model like Bader’s (1990) topological model of the ED distribution, which provides a basis for defining a bonded interaction, bond strength, net atomic charges, and bonded radii for the atoms in molecules and crystals.

	OVERVIEW

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
This report begins with an examination of the ED distribution that defines the Si-O bonded interactions for a siloxane molecule in terms of its bond paths, bonded radii, bond critical points, and the associated properties followed by an examination of the Laplacian of the ED distribution, an ideal tool for characterizing and locating Lewis base and acid domains, sites of potential electrophilic and nucleophilic attack, respectively. Next, the topological properties of the Si-O bonded interactions in several silica polymorphs and representative molecules are explored. The results show that the experimentally determined properties of the ED distributions are comparable with those calculated with first-principles quantum mechanical methods, and that model structures for selected framework silicates calculated as a function of pressure are comparable with those observed. A fundamental connection will be established between the topological properties of the ED distribution and Lewis base lone-pair domains on the oxide anions involved in Si-O-Si bonded interactions and the incorporation of water via the hydrogarnet substitution in the coesite structure at high temperatures and pressures. The correlation between the Si-O bond length and the Si-O-Si angle observed for coesite will be examined in terms of a progressive concentration of the ED in the anti-binding lone pair regions of the O atoms. It will be established that the bond critical point properties for the ED distributions calculated for a relatively large number of silicates vary systematically with the experimental Si-O bond lengths, providing a connection between bond length variation and the topological properties of the ED. Likewise, the properties of the bonded interactions for representative siloxane molecules will be shown to be comparable with the calculated and experimental trends in the topology of the ED distributions for the silicates. Together with Si-O bonded interactions, the value of ²(r_c) for intermediate M-O bonded interactions (M = first and second row metal atoms bonded to O) in silicates will be examined in terms of the values of (r_c). The character of the Si-O bond will also be re-examined in terms of a spectrum of bonding models, ranging from closed shell (ionic) to shared (covalent) bonded interactions. The ratio G(r_c)/(r_c) will be examined in terms of the Si-O and M-O bond lengths and the coordination numbers for the Si and M atoms. Ionic and crystal radii, determined by assuming a given size for the oxide anion combined with the additive rule, are considered to be unreal and not measurable, unlike bond radii. The bonded radius of the oxide anion is observed to be highly dependent on the properties of the atoms to which it is bonded. The radii for the Si and O atoms involved in the bonded interactions for the silica polymorphs will be shown to differ from the crystal radii in the way that they respond to changes in coordination number and pressure. The report will conclude with a brief statement examining the connections that exist between the Si-O bonded interactions in crystals and molecules, bonded radii and bond critical point, and local energy-density properties, and the rewards that we believe will be forthcoming by the studying Earth materials and their properties in terms of their ED distributions with first-principle quantum mechanical methods.

	BADER'S TOPOLOGICAL MODEL FOR ELECTRON DENSITY DISTRIBUTIONS

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
Given that there are no chemical bonds, only bonded interactions, Bader and his colleagues (Bader 1985; Runtz et al. 1977) undertook the challenge of probing the topology of the ED distribution in the region between the nuclei of adjacent atoms in a search for a feature that might qualify as an indicator of a bonded interaction. The ED was chosen for study because, unlike the chemical bond, it is a quantum mechanical observable (it is the expectation value of the density operator). It also follows that any property of the ED distribution, including the value of the ED, (r), the gradient field, (r), and the Laplacian of (r), ²(r), at a given point r and the bond lengths and angles of a molecule or crystal, are also observables. In addition, the ED distribution is a robust property that embodies all of the information that can be known about a ground state material, including its kinetic, potential, and total energies (Parr and Yang 1989). Moreover, the distribution adopted by a system at equilibrium (a system in which there are no net forces acting on the nuclei) is one that results in the lowest net energy for the system as a whole (Gatti 2005). In the analysis of the ED distribution, it was discovered that adjacent atoms in molecules are often connected by well-defined continuous pathways of ED along which the ED is a maximum relative to all neighboring paths. As the pathways resembled what were perceived as having the defining characteristic of bonded interactions, they mapped out all of the paths for several molecules and found that the atoms that were perceived to be bonded were connected by continuous pathways of maximum ED. The value of the ED along the paths between a pair of atoms was found to decrease from the nucleus of each pair to a local minimum value at a critical point (a saddle point) along the path where (r) = 0. The path of maximum ED connecting the nuclei of the pair was denoted the bond path and the point along the path where (r) is a local minimum was denoted the bond critical point, r_c. It is the accumulation of the ED between the nuclei of the bonded atoms that is associated with the line of maximum ED (the bond path) connecting the nuclei. As the ED distribution adopted by a ground state system results in the lowest energy, a system with bond paths of maximum ED may be considered to be stabilized relative to one that lacks bond paths (Gatti 2005). As each bond path is also associated with a line of maximally negative potential energy density linking the pair, Bader (1998) concluded that the presence of a bond path together with its topologically equivalent virial path that also connects the pair serve as a universal indicator of a bonded interaction. Moreover, as the ED is topologically determined by the virial field, the presence of the virial path results in a line of maximum ED, a bond path and a state of electrostatic equilibrium between the pair. When these conditions are satisfied, the presence of a bond path linking a pair of atoms suffices as evidence that the pair is bonded. Indeed, all atomic interactions that result in bond paths, including those classified as nonbonded, are responsible for the binding of the atoms in molecules and crystalline materials (Bader 1998). In a study of bond path exchange-correlation energies, Pendas et al. (2007) recently presented evidence that bond paths represent direct electron exchange channels between a bonded pair that necessarily serve to stabilize not only the bonded interaction but also the system.

	BOND PATHS AND BRIDGING Si-O BONDED INTERACTIONS FOR THE H6Si2O7 MOLECULE

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
The ED distribution associated with the Si-O-Si bridging angle for the H₆Si₂O₇ molecule is displayed in Figures 1a and 1b. Both maps were calculated in the plane of the Si-O-Si angle at the B3LYP/6-311(2d,p) level. Figure 1a displays a contoured level line relief map of the ED distribution for the molecule calculated between 0.0 and 0.3 a.u. (1 a.u. = 6.748 e/A³), resulting in extremely large maxima centered on the bridging O and the two coordinating Si atoms of the angle. The two Si atoms are connected to the O atom by well-defined ridges of ED running between the nuclei of the atoms. As the ridges are due to the accumulation of the ED along the bond paths, their presence is consistent with the bonded interactions that exist between the Si and O atoms. The ridges decrease in height from the nuclei of the Si and O atoms to a local minimum value in the ED, (r_c), between the contoured atoms at bond critical points. Figure 1b is a level line contour map of the same ED distribution between 0.0 and 1.0 a.u. with a contour interval of 0.05 a.u., with the exception of one contour that was drawn at the (r_c) = 0.136 a.u. level. Like the relief map, the contour map shows that the Si and O atoms are connected by well-defined ridges of ED that decrease in value along the bond paths toward r_c (denoted by black dots in Fig. 1b). Note that the contour line constructed at the (r_c) level intersects at the local minima along the ridges between the Si and O atoms, locating the positions of the bond critical points for the two bridging Si-O bonded interactions. The separation between the Si and O atoms (the Si-O bond length) is directly related to the value of ED at r_c; with few exceptions, the greater the value of (r_c), the shorter the Si-O bond length and the greater the strength of the bonded interaction. Also, the distances between the nuclei and the bond critical points (measured along the two bond paths) define the bonded radii for the two atoms, denoted r_b(Si) and r_b(O), respectively (Bader 1990). The bonded interaction involving the Si and O atoms of the molecule is considered to be the result of the net electrostatic attraction imposed on the nuclei of the pair by the accumulation of ED along the bond path in the vicinity between the nuclei, balancing the forces of repulsion between the nuclei and stabilizing the structure (Feynman 1939). Furthermore, by counting all the bond paths that radiate from a given atom to adjacent atoms, an unambiguous determination of the coordination number of the atom can be made (Gibbs et al. 1992).

View larger version (49K): [in this window] [in a new window]   FIGURE 1. Electron density, ED, distribution for the H6Si2O7 molecule, calculated at the B3LYP/6-311(2d,p) level through the bridging Si-O-Si angle of the molecule and plotted as a (a) level line relief ED map calculated between (r) = 0.0 and 0.3 a.u. and (b) in a level line contour map calculated between (r) = 0.0 and 1.0 a.u. in intervals of 0.05 a.u. The level contour line in b that is connected by the black dots along the ridge between the Si and O atom was calculated at the 0.136 a.u. level, the value of (rc). The black dots show the locations of the Si-O bond critical points along the Si-O bridging bond paths. The peak heights of the Si and O atoms tower above the ED along the bond paths at heights of ~550 and ~160 a.u., respectively (1 a.u. = 6.748 e/Å3).

	THE LAPLACIAN, 2(r), A TOOL FOR LOCATING LEWIS ACID AND BASE SITES

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

View larger version (29K): [in this window] [in a new window]   FIGURE 2. (a) A segment of a nested set of level isosurfaces of the ED distribution calculated at intervals of 0.0275 a.u. at the B3LYP/6-11(2d,p) level around one of the Si-O bridging bond paths of the H6Si2O7 molecule. The outermost green level isosurface was calculated at the (r) = 0.0750 a.u. level, the next inner red one was calculated at the (r) = 0.1025 a.u. level, the blue one at the (r) = 0.1300 a.u. level, and the innermost yellow one was calculated at the (r) = 0.1575 a.u. level. (b) The same figure as a except all of the level isosurfaces have been deleted except the blue one that was calculated at the (r) = 0.13 level, a value close to that at the bond critical point, (rc) = 0.136 e/Å3. The blue isosurface can be described as a single-sheeted hyperboloid around the Si-O bond path (see Fig. 2.1.13, Marsden and Tromba 1976). The red spheres of the molecule represent O atoms, the white ones H, and the yellow ones Si. The red and yellow rods connecting the Si and O atoms and the red and white ones connecting the O and H atoms represent the bond paths of the molecular graph of the molecule.

where (r_c) = 0 and H_ij = ²(r_c)/x_i x_j, i,j = (1,2,3). As H_ij is a real and symmetric Hessian matrix, it can be transformed into diagonal form, , where the diagonal elements define the curvatures of (r_c) along the associated eigenvectors e_i, i = (1,2,3). The trace of , tr() = ₁ + ₂ + ₃ where ₁ = ²(r_c)/x₁², ₂ = ²(r_c)/x₂² and ₃ = ²(r_c)/x₃² is the Laplacian ²(r_c) of (r_c). As observed above, the signs of two principal components ₁ and ₂ are both negative while ₃ is positive. Collectively, (r_c), the three curvatures ₁, ₂, and ₃ of (r_c) and ²(r_c) are referred to as the bond critical point, bcp, properties of a bonded interaction (Bader 1990).

In general, the value of the Laplacian of the ED, ²(r), measures the extent to which the ED is either locally concentrated or depleted at a given point r in the distribution (Bader and Essén 1984). If ²(r) is positive, then the value of (r) is less than all surrounding points lying on the surface of a sphere of radius dr centered at r, and the ED is said to be locally depleted at r. On the other hand, when ²(r) is negative, the converse is true and the ED is said to be locally concentrated at r (Lieb and Loss 1997). To appreciate the topology of the ED distribution for an isolated O atom in terms of the Laplacian (Fig. 3a), an L(r) = –²(r) level line relief map was calculated for the atom at the B3LYP/6-311(2d,p) level in a plane passing through its nucleus with the contribution of one core orbital neglected [note that L(r) is positive in regions where the ED is locally concentrated and negative in regions where it is locally depleted]. The map displays two sets of spherical nodes each centered on the nucleus, representing the shell structure of the atom. For each principle quantum shell of the atom, there is a pair of concentric regions, one positive and one negative. The positive ones define spherical regions of locally concentrated ED and the negative ones define concentric spherical regions of locally depleted ED. Together, the two sets of locally concentrated and locally depleted regions of ED correspond to the K and L quantum shells of the O atom (Bader et al. 1984). The outermost shell of locally concentrated ED defines the chemically important valence shell charge concentration, VSCC, of the O atom. The VSCC is at a distance of 0.66 Å from the nucleus, comparable to both the atomic radius, 0.65 Å (Bragg 1920) and the maximum in the radial charge density of the shell, 0.60 Å, of the O atom (Slater 1964). Upon forming a bonded interaction with Si, for example, the VSCC of the atom is distorted to one degree or another and typically maxima are formed in the bonded and nonbonded domains. Despite the shell structure of an atom, it is noteworthy that, throughout space, the ED distribution decays monotonically away from the nucleus along a radial line in all directions. An L(r) level line relief map, calculated for an isolated Si atom at the B3LYP/6-311(2d,p) level (Fig. 3b), displays a spike-like K shell at its center enclosed by a well-developed L shell that in turn is enclosed by a very diffuse low-lying VSCC M shell. As the VSCC for the Si atom is diffuse and very low-lying and situated substantially further from the nucleus than that of the O atom, it fails to show well-defined distortion features in its valence shell upon chemical combination with O (Bader 1990).

View larger version (32K): [in this window] [in a new window]   FIGURE 3. L(r) = –2(r) (a.u.) level line relief maps calculated at the B3LYP/6-311(2d,p) level in a plane containing the nucleus for an isolated (a) O atom with the contribution of one core orbital neglected and (b) an isolated Si atom. The maps were cut off at the ±15 a.u. level (1 a.u. = 24.099 e/Å5).

View larger version (46K): [in this window] [in a new window]   FIGURE 4. L(r) level line relief map for the H6Si2O7 molecule calculated at the B3LYP/6-311(2d,p) level in the plane containing the Si-O-Si angle with contributions from all of the core orbitals included. The bridging O is located at the center of the figure and the two Si are located at the edges of the figure. The map was cut off at ±5 a.u. level.

As it is relatively difficult to infer from a relief map the spatial features associated with the maxima, a three-dimensional L(r) isosurface map was generated in the immediate vicinity of the bridging O atom of the H₆Si₂O₇ molecule (Fig. 5a). The map shows that the atom is capped by a bracelet-shaped isosurface of locally concentrated ED with its maximum (a Lewis base domain) located on the reflex side of the Si-O-Si angle. Isosurface maps were calculated in an earlier experimental study for the five nonequivalent O atoms for the high-pressure silica polymorph, coesite (Gibbs et al. 2003b). Like the map for the H₆Si₂O₇ molecule, it is noteworthy that the maps displayed little or no features in the bonded regions along the Si-O bonds. Also, like the O atom of the molecule, four (O2, O3, O4, and O5) of the five nonequivalent O atoms in coesite display bracelet-shaped isosurfaces capping the atoms with their maxima also located on the reflex side of the Si-O-Si angles. The map generated for O4 is strikingly similar to those calculated for O2, O3, and O5 (not shown here; see Gibbs et al. 2003b, column 1, Fig. 5) and the bridging O atom of the H₆Si₂O₇ molecule, evidence that the properties of the ED for O atoms in coesite and in H₆Si₂O₇ are comparable. The map calculated for the molecule (Fig. 5a) is compared with those calculated for the O4 and O1 atoms in coesite in Figures 5b and 5c, respectively (the O4 atom and the O atom in the molecule are both involved in Si-O-Si angles of ~145°, whereas O1 is involved in an 180° angle). In contrast, the isosurface calculated for O1 is a ring-torus shaped feature that encircles the atom unlike the bracelet-shaped isosufaces that cap the remaining O atoms in the coesite structure and in the H₆Si₂O₇ molecule. With the Si-O-Si angle for the H₆Si₂O₇ molecule clamped at 180.0°, a L(r) level line map generated for the bridging O also displays ring-torus isosurface like that displayed about the O1 atom in coesite (Gibbs et al. 2003a). A similar torus-shaped feature has also been reported to exist on the O atom of the wide 170.2° Si-O-Si bonded interaction for beryl (Prencipe and Nestola 2007).

View larger version (21K): [in this window] [in a new window]   FIGURE 5. L(r) level line isosurface maps generated for the nonbonded domains for the bridging O atoms for (a) the H6Si2O7 molecule and for the (b) O4 and the (c) O1 oxygen atoms of coesite, respectively (Gibbs et al. 2003b), each constructed in the immediate vicinity of the plane that bisects the Si-O-Si angle. The Si-O-Si angles involving the bridging O of the molecule and the O4 atom of coesite are both ~144°, whereas O1 is involved in a 180.0° angle. The maxima of the bracelet-shaped isosurfaces are located on the reflex side of the angle, while the maxima encircle the O1 atom in a continuous ring. The dark gray spheres at the center of each figure represent O atoms and the rods radiating from the spheres represent segments of the Si-O bond paths.

	Si-O BOND CRITICAL POINT PROPERTIES

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
In an earlier study of the ED distributions associated with the Si-O bonded interactions for a relatively large number of silicates, the bond critical point properties were calculated (Gibbs et al. 2001) and compared with model experimental properties (Gibbs et al. 2008a). The calculated properties are plotted in Figure 6 with respect to the experimental Si-O bond lengths, R(Si-O). With the accumulation of the ED at the bond critical points, the Si-O bond lengths generally decrease, with a few exceptions, in a roughly linear trend (Fig. 6a). At r_c, the curvatures, ₁ and ₂, of the ED, are both negative and decrease in value linearly as R(Si-O) decreases. As ₁ and ₂ measure the extent to which the ED is locally concentrated perpendicular to the bond path, the systematic decrease of the two curvatures with decreasing bond length indicates that the ED becomes progressively more locally concentrated toward the bond path at r_c as the bond length decreases. As ₁ ~ ₂ for the bulk of the bonded interactions in the silicates, we may conclude that the distributions of the ED in cross sections of the bond paths passing through r_c are generally circular, a result that indicates that the Si-O bonded interactions are relatively stable and are primarily -type bonded interactions with little or no character (Bader et al. 1983). The average magnitude of the two, |_1,2| = (|₁ + ₂|), is plotted in Figure 6b with respect to R(Si-O) where |_1,2| increases slightly with decreasing bond length. The value of the third curvature, ₃, measures the extent to which the ED is locally concentrated along the bond path at r_c; the larger the value of ₃, the greater the local concentration of the ED into the basins of the two atoms (Bader 1985; see below) and the greater the shielding of the nuclei. Figure 6c shows that the value of ₃ increases substantially more, relative to |_1,2|, with decreasing bond length, indicating that the decrease in Si-O bond length is related in part to the extent to which nuclei of the Si and O atoms are progressively shielded with decreasing bond length. As the value of ₃ is substantially larger than that of either ₁ or ₂, the sign of ²(r_c) is necessarily positive. Also, inasmuch as the value of ₃ increases at a faster rate than ₁ and ₂ both decrease, ²(r_c) increases substantially as the bond length decreases and (r_c) increases. As 1/4²(r_c) = 2G(r_c) + V(r_c) increases, the negative definite local potential energy, V(r_c), decreases and the positive definite local kinetic energy, G(r_c), increases, with V(r_c) decreasing at a faster rate than G(r_c) increases such that the total local energy density, H(r_c) = G(r_c) + V(r_c), is negative and decreases as R(Si-O) decreases and (r_c) increases at the bond critical point of the bonded interaction. For first and second row metal atoms (Li to B and Na to S) bonded to O, ²(r_c) is positive in value and increases linearly with the value of (r_c) as the bond lengths decrease (Gibbs et al. 2006b). Concomitant with these changes, the total energy density is positive for the bonded interactions involving the more electropositive atoms (Li, Na, and Mg), whereas it is negative for the remaining more electronegative atoms, with the value of H(r_c) becoming progressively more negative as the bond length decreases and ²(r_c) increases in value. In contrast, the H(r_c) values for the bonded interactions involving the Li, Na, and Mg atoms become progressively more positive with decreasing bond length and the increasing value of (r_c).

View larger version (19K): [in this window] [in a new window]   FIGURE 6. Calculated bond critical point properties for the Si-O bonded interactions for the experimental structures determined for several silicates and model structures for geometry optimized siloxane molecules (Gibbs et al. 2001), plotted with respect to their experimental Si-O bond lengths, R(Si-O) Å. (a) (rc) e/Å3, the value of the electron density at the bond critical point, rc, plotted with respect to R(Si-O), (b) |1,2| = |(1 + 2)| e/Å5 plotted with respect to R(Si-O), where 1 and 2 measure the local concentration of (rc) perpendicular to the bond path, (c) 3 e/Å5, plotted with respect to R(Si-O), where 3 measures the local concentration of (rc) parallel to the bond path in the directions of the Si and O atoms (d) 2(rc) e/Å5, the Laplacian of the electron density at rc, plotted with respect to R(Si-O).

	OPTIMIZED STRUCTURES AND EXPERIMENTAL ELECTRON DENSITY DISTRIBUTIONS

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

View larger version (23K): [in this window] [in a new window]   FIGURE 7. (a) A comparison of <R(Si-O)>, the average of the two Si-O bond lengths comprising each of the nonequivalent Si-O-Si angles and (b) the five nonequivalent Si-O-Si angles for coesite plotted with respect to pressure. The black dots represent data obtained by optimizing the coesite structure as a function of pressure, red symbols represent experimental data determined by Levien and Prewitt (1981), and the blue ones represent experimental data determined by Angel et al. (2003). The red and blue circles represent Si-O1 bonded interactions, the red and blue diamonds represent Si-O2 bonded interactions, red and blue triangles represent Si-O3 bonded interactions, red and blue squares represent Si-O4 bonded interactions, and red and blue pentagons represent Si-O5 bonded interactions. (b) A comparison of the Si-O-Si angles calculated for coesite plotted with respect to pressure (black dots). The experimental Si-O-Si angles are plotted in the figure as red and blue symbols. The data plotted as red symbols were determined by Levien and Prewitt (1981), and the ones plotted as blue symbols were determined by Angel et al. (2003). Solid lines were arbitrarily drawn through the calculated data (black dots) for coesite.

The modeling of the aspherical ED distributions for the two silica polymorphs was undertaken with the elegant software VALRAY (Stewart et al. 2000), which applied the nucleus centered, flexible pseudoatom multipole expansion of the ED to each of the nonequivalent atoms. The experimental population parameters P_ic and P_ilm were determined in a least-squares refinement in terms of the observed structural amplitudes. Upon completion of the multipole refinement, the resulting model multipole representation of the ED distribution was subjected to a topological analysis in the determination of the experimental model bcp properties for the Si-O bonded interactions for stishovite and coesite.

The model experimental bcp properties for stishovite and coesite are compared in Figure 8 with the properties calculated for the silicates used to prepare Figure 6. The experimental properties for the two polymorphs follow the trends, for the most part, established by the calculated data with the experimental (r_c), ₃, |_1,2|, and ²(r_c) values each increasing with the decreasing bond length. The experimental (r_c) values for both polymorphs fall along and largely within the scatter of the calculated data, whereas the |_1,2| values are somewhat larger than the calculated values. The ₃ values fall within the scatter of the calculated values for the most part, whereas the ²(r_c) values follow the trend of the calculated data but several of the coesite values depart slightly from the trend. Model experimental (r_c) and ²(r_c) properties determined for the Si-O bonded interactions in several olivines (Gibbs et al. 2008c), diopside (Bianchi et al. 2005), and scolecite (Kuntzinger et al. 1998) with synchrotron and high-resolution single-crystal data are comparable with the calculated values with the bulk of the experimental values agreeing, for the most part, within ~10%. In addition, model experimental bcp properties for the Mg-O, Fe-O, Mn-O, and Co-O bonded interactions, determined with single-crystal synchrotron X-ray diffraction data sets for the olivines, are also comparable with calculated values (Gibbs et al. 2008b). It is noteworthy that the experimental model bcp properties determined for Fe-O, Mn-O, and Co-O bearing organometallic polymers scatter along the experimental and calculated trends observed for the olivines Fe₂SiO₄, Mn₂SiO₄, and Co₂SiO₄ (Gibbs et al. 2008b).

View larger version (21K): [in this window] [in a new window]   FIGURE 8. The experimental Si-O bond lengths, R(Si-O), for the silicates displayed in Figure 6, plotted with respect to (a) (rc), (b) |1,2|, (c) 3, and (d) 2(rc), all plotted as red squares. Superimposed upon the calculated data are model experimental bond critical point properties determined for stishovite with high-resolution single-crystal synchrotron diffraction data by Kirfel et al. (2001) plotted as blue circles, and model properties determined for coesite high-resolution single-crystal data by Gibbs et al. (2003b) plotted as black circles.

View larger version (20K): [in this window] [in a new window]   FIGURE 9. The (rc) values calculated for silicate crystals (diamonds) and the model experimental (rc) values determined for coesite (solid triangles) and stishovite (solid squares), plotted with respect to the Brown and Shannon (1973) empirical bond strength, s, calculated with the experimental Si-O bond lengths (Gibbs et al. 2004).

	SILOXANE MOLECULES AND SILICATES

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

View larger version (15K): [in this window] [in a new window]   FIGURE 10. A comparison of the bond critical point properties calculated for silicate crystals displayed in Figure 4 (open red squares) with the bond critical point properties calculated for a variety of geometry optimized siloxane molecules (blue spheres) (Gibbs et al. 1998). In preparing these scatter diagrams, it was discovered that the bcp properties for the disiloxane molecule were incorrectly reported in Table 1 of Gibbs et al. (1998). They should read (rc) = 0.906 e/Å3, rb(O) = 0.966 Å, |1,2| = –6.212 Å5, 3 = 30.88 e/Å5, 2(rc) = 14.46 e/Å5, and H(rc) = 0.238 a.u.

	BONDED RADII OF Si AND O

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

For the silica polymorphs, r_b(O) = ~0.95 Å for the two-coordinated ^IIO atoms in coesite and ~1.10 Å for the three-coordinated ^IIIO atom in stishovite. However, these radii are not only substantially smaller than the Shannon and Prewitt (1969) two- and three-coordinated crystal radii, 1.21 and 1.22 Å, respectively, for the oxide anion, but also the difference between the crystal radii is substantially smaller than the difference between the bonded radii. In contrast, the bonded radius of the Si atom in coesite is ~0.66 Å, while it is ~0.68 Å in stishovite compared with the crystal radii for four- and six-coordinated Si, 0.40 and 0.54 Å, respectively. On the basis of the ED distribution, it is the bonded radius of the O atom, not the crystal radius of the Si atom that changes substantially with the increasing coordination number of Si.

Furthermore, the bonded radius of an individual O atom is not single-valued. A given O atom may display two or more different radii, depending on the kinds of metal atoms to which it is bonded. For example, the three nonequivalent O atoms in danburite, CaB₂Si₂O₈ (Downs and Swope 1992) are each bonded to three different kinds of metal atoms, B, Si, and Ca, with average experimental M-O bond lengths of 1.47, 1.62, and 2.44 Å, respectively. Rather than displaying a single radius, each O atom displays a bonded radius of ~1.00 Å along the B-O bond path, a radius of ~ 0.94 Å along the Si-O bond path and a radius of ~1.23 Å along the Ca-O bond path, a result that substantiates the claim of O’Keeffe and Hyde (1981) that the radius of an atom like O is not single-valued. As they point out, the use of the term "radius" for an O atom implies that the size of the atom is the same in all directions, which they consider to be "clearly absurd." The bond path closely parallels the line between two bonded atoms in the bulk of the materials studied such that the bonded radius is effectively measured along the bond length, a result that reflects on the stability of the bonded interactions (Bader 1990). The variability of the r_b(O) with the type of atom to which it is bonded is also consistent with the observation that r_b(O) correlates with the electronegativity difference between bonded M and O atoms; the greater the difference and the greater closed-shell interaction, the larger the bonded radius for the O atom (Feth et al. 1993). As is well known, the Shannon and Prewitt (1969) crystal radii have been used with considerable success in the generation of bond lengths, structure field diagrams, and the modeling of ion conductivity, defects, diffusion and site preferences, among other things, in advancing the understanding of oxide crystal chemistry (Prewitt 1985). As observed by O’Keeffe and Hyde (1981), this is expected, given that the Shannon and Prewitt (1969) table of radii is in effect a table of average M-O bond lengths. After all, experimental bond lengths were used in the compilation of the radii, assuming a given radius for an oxide anion with a given coordination number and the Bragg (1920) additive rule. Given that average bond lengths were used to construct the radii, assuming a given radius for the oxide anion, it is clear that one could use any radius of interest for the oxide anion and compile a set of workable radii. For example, if one used the strategies devised by Shannon and Prewitt (1969) and Bragg’s (1920) atomic radius for the O atom (0.65 Å), for example, one would obtain another set of empirical radii that would be comparable with Slater’s (1964) universal radii, and it would work as well as the Shannon and Prewitt (1969) radii in predicting bond lengths. But, unlike the Shannon and Prewitt (1969) radii, the resulting radii would be universal in their application, serving to predict bond lengths for oxides, sulfides, metals and molecules alike, regardless of whether bonded interactions were ionic, covalent, or metallic in nature. Clearly, it goes without saying that the fact that a set of ionic radii works and provides a basis for understanding properties and crystal chemistry says little about the actual sizes of the atoms themselves. As the radius of an ion is not constant, it can be argued that it is the bond length (an observable), not the radii of the atoms that should be considered when conjecturing whether one bonded interaction (rather than a given atom) in a structure can be replaced by another or whether one structure is favored over another in a structural field map based on bond length. After all, it is the bond length that is connected to the energy and the stability of a bonded interaction, not the radii of the atoms comprising the bonded interaction. It is clear, however, that the main reason that crystal radii work and that they have been so useful in crystal chemistry in understanding a wide range of properties is that radii are correlated on a one-to-one basis with the average experimental bond lengths used in their derivation.

In the modeling of the structures of quartz and coesite as a function of pressure, r_b(Si) and r_b(O) were found to decrease slightly with increasing pressure. Over the pressure range studied, r_b(O) was found to decrease from 0.95 to 0.90 Å, while r_b(Si) decreased substantially less from 0.66 to 0.65 Å. Given that r_b(O) is more dependent on the Si-O bond length and the coordination number of Si than the bonded radius of the Si atom, the greater decrease of the r_b(O) with increasing pressure may be expected. The decrease in r_b(Si) atom with increasing pressure, albeit small, suggests that the O atom is more compressible than the Si atom, at least along the Si-O bond path (Gibbs et al. 1999, 2000; Prencipe and Nestola 2007). The greater compressibility of the O atom is consistent with Ida’s (1976) conclusion that the compressibilities of Mg, Fe, and Si are negligibly small and that the compressibilities of silicates in the mantle are dictated in large part by the greater compressibility of the O atom. Miyamoto and Takeda (1980) and Matsui et al. (1982) reached a similar conclusion on the basis of the molecular dynamics determined for selected Mg-silicates.

	Si AND O NET ATOMIC CHARGES

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
Bader (1985) postulated that an atom in a material consists of a nucleus and its associated electrons that are collectively enclosed within a surface, S(r), that exhibits a zero flux property, a surface that is not crossed by any trajectories of the gradient field vectors of (r). This condition is satisfied when the inner product (r) · n = 0 holds, where n is a unit vector oriented perpendicular to S(r) (see Fig. 1 in Bader 1985). The region of space enclosed by S(r) is referred to as the basin of the atom. By integrating the ED over the range of an atomic basin, the number of electrons in the basin can be determined. The net atomic charge of an atom, q, is obtained by simply summing the nuclear charge of the atom and the electronic charge associated with the number of electrons in the basin.

A careful study of the structures and topologies of the ED distributions for disiloxane and several related molecules (Gillespie and Johnson 1997), resulted in a geometry optimized disiloxane molecule with an Si-O bond length of 1.621 Å, compared with the experimental length of 1.634 Å, and an Si-O-Si angle of 148.6°, compared with the experimental angle of 144.1°. Considering the very compliant nature of the angle, the agreement between the experimental and model structures is considered to be good. The short Si-O bond compared with the sum of the single covalent radii for Si and O, 1.92 Å, the large net atomic charges conferred on the Si and O atoms, q(Si) = +3.05 e and q(O) = –1.52 e, respectively, and the wide Si-O-Si angle were concluded to be a direct consequence of the highly ionic character of the Si-O bond. However, given the compliant nature of the Si-O-Si angle, it is difficult to understand, in light of the large net charges on the Si atoms, why the angle is bent rather than straight (Geisinger and Gibbs 1981; O’Keeffe and McMillan 1986).

An integration of the ED distribution over the basins of the ^IVSi and ^IIO atoms for quartz resulted in even larger net charges of q(^IVSi) = +3.20 e and q(^IIO) = –1.60 e (Gibbs et al. 1999). The net charges calculated for the ^VISi and ^IIIO atoms in stishovite are even larger, q(^VISi) = +3.39 e and q(^IIIO) = –1.69 e, as expected, given the larger coordination numbers of the Si and O atoms in stishovite (Kirfel et al. 2001). Prencipe and Nestola (2007) found that the ^IVSi atom in beryl, has a q-value of +3.2, in close agreement with that found for quartz. The large net charges obtained in these studies are consistent with charges inferred from maxima in the experimental deformation [(r)] map reported for quartz and stishovite (Cohen 1994), charges of ~ +4 on Si and ~ –2 on O. The large charges are also consistent with the fully ionic electrostatic potential energy model used by Smyth (1989) to locate potential sites for hydroxyl substitution and to estimate oxygen isotope fractionation in silicate and oxide minerals. Clearly, a highly ionic model for the Si-O bond contradicts Pauling’s (1939) long held view that the bond is intermediate in character between ionic and covalent, rather than highly ionic. These large charges conform with the convictions of Prencipe et al. (2002) and Prencipe and Nestola (2007), who believe that an ionic picture of the Si-O bond is more appropriate than a covalent one.

It is well known that the net atomic charges conferred on the atoms of quartz, coesite, and stishovite are substantially larger than those obtained by X-ray diffraction methods. For example, spherical -refinements of single-crystal X-ray diffraction data sets resulted in a –0.74 e net charge on the ^IIO atoms of coesite (Downs 1995) and –0.86 e net charge on the ^IIIO atoms of stishovite (Hill et al. 1983), substantially smaller than the charges obtained in the virial partitioning of the ED. A careful multipole study of accurate structure factors for synthetic quartz by Stewart et al. (1980) yielded smaller charges with a net residual charge of +1.0 e conferred on the pseudoatom Si atom and charge of –0.5 e conferred on the pseudoatom O atom. On the basis of these charges and electronegativity considerations, it was concluded that the Si-O bond in quartz is 25% rather than 100% ionic.

In a recent assessment of Voronoi deformation ED density charges, Guerra et al. (2004) concluded that net atomic charges are much too large in certain cases, conferring excessive ionic character on a variety of covalent bonds. (See also Haaland et al. 2000.) As a case in point, the net atomic charges (~3.5 e) conferred on the P atoms of PO₄ tetrahedral oxyanions in the AlPO₄-15 molecular sieve framework structure, determined in a careful ED study, are even larger than those conferred on the Si atom in quartz (Aubert et al. 2003). In a rebuttal of the criticisms of the large net charges, Bader and Matta (2004) pointed out that the net charge conferred on an atom is the expectation value of the number operator that counts the average number of particles within a given spatial region, and as such, the number of electrons in the basin of an atom is an observable. This is true, but the question remains whether all of the electrons "belonging" to a given atom are always contained within the basin of the atom (Parr and Yang 1989). Furthermore, it is difficult to rationalize the experimental Si-O-Si angle of cristobalite, 146.5° (Gibbs et al. 1999) and the bent 148.6° angle for the disiloxane molecule with net charges of +3.05 and –1.72 e conferred on the Si and O atoms. Nonetheless, the larger net charges conferred on the ^VISi and ^IIIO atoms of stishovite relative to those conferred on the ^IVSi and ^IIO atoms in quartz are consistent with the greater ionic character of the ^VISi-^IIIO bond. This result suggests that the relative magnitudes of the net charges are correct but perhaps their absolute magnitudes may be too large, as concluded by Guerra et al. (2004).

In the study of the quartz structure as a function of pressure, the magnitude of net charge conferred on O was found to decrease slightly from –1.600 to –1.595 e with increasing pressure over the pressure range studied, 0.0 to 18 GPa. The magnitudes of q(O) for the zero pressure model structure of coesite also decrease from –1.606 to –1.595 e with decreasing Si-O bond length and Si-O-Si angle. In contrast, the magnitudes of the net charges conferred on the O atoms involved in the narrower Si-O-Si angles (O2 and O5) actually increase with increasing pressure, whereas the magnitudes of net charges on the remaining O atoms, (O1, O3, and O4) actually decrease as observed for the O atoms of quartz. These conflicting results suggest that little can be said about how q(O) changes with increasing pressure in the case of coesite. The magnitude of the experimental atomic charge for the O atom in stishovite is slightly larger, q(O) = –1.69 e, than that for either quartz or coesite, a result that is expected given the longer Si-O bond length, the smaller value of (r_c), and the larger coordination number of the Si atom in stishovite (Kirfel et al. 2001). The bcp properties, the larger q(Si) and the smaller q(O) values, the larger bonded radius of the oxide anion and the larger coordination number of the Si atom in stishovite, are consistent with the argument that the Si-O bond in stishovite is more ionic than it is in either quartz or coesite. The ionicities, Si-O = |q(Si)/_Si – q(O)/_O| where _Si and _O are the valences of Si and O, respectively (Zwijnenburg et al. 2002), calculated for the Si-O bonds in quartz (_Si-O = 0.80) and stishovite (_Si-O = 0.84) are likewise consistent with the assertion that the bonds in stishovite are more ionic than those in quartz.

	THE ELUSIVE CHARACTER OF THE Si-O BOND

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
Given the vast amount of literature written about silicates and siloxanes, it may come as a surprise that a general consensus has yet to be reached regarding the character of the Si-O bond. Workers like Smyth (1989), Cohen (1994), Gillespie and Johnson (1997), Prencipe et al. (2002), and Prencipe and Nestola (2007) believe that the bond is largely ionic; others like Harrison (1978) and Stewart et al. (1980) believe that it is predominately covalent, whereas others like Pauling (1939) and Gibbs et al. (1999) believe that it is intermediate in character between ionic and covalent. Given these divergent viewpoints, it is clear that the character of the Si-O bond is an elusive subject (Catlow and Stoneham 1983; Gibbs et al. 1994), shrouded in mystery and contradictory results. A fully ionic model for quartz, consisting of Si⁺⁴ and O^–2 ions, gives a reasonably good fit to the dielectric properties and the cohesive energy (Sanders et al. 1984; Jackson and Gordon 1988; Burnham 1990); yet, an intermediate model with Si⁺² and O^–1 ions gives a better fit to the photon frequencies and to Phillip’s (1970) dielectric model (Vempati and Jacobs 1983; Jackson and Gordon 1988) and a covalent model with residual charges of +1 on Si and –0.5 e.u. is consistent with the ED distribution observed for quartz (Stewart et al. 1980).

With the advent of the robust theory of Bader (1990) and his colleagues for characterizing the topology of ED distributions for molecules and crystals, it has been demonstrated that a bonded interaction can be classified in terms of the bond critical point properties, the local potential density, V(r_c), and the local kinetic energy density, G(r_c). As a basis for the classification, Bader and Essén (1984) observed that the kinetic density, G(r) in a.u., and its components may be related directly to the inner product of the gradients of the orbital electron densities, _i, and their occupation numbers, n_i, by the expression:

where the summation is completed over the natural spin-orbital components (Parr and Yang 1989). With a determination of G(r), V(r) can be determined once ²(r) is known, using the local form of the virial theorem, V(r) = ²(r) – 2G(r) (Bader 1990). With values (r_c), ²(r_c), and G(r_c), Bader and Essén (1984) introduced a classification of bonded interactions based on: (1) the sign of ²(r_c); (2) whether the local kinetic energy per electronic charge, G(r_c)/(r_c), is greater or less than unity; and (3) whether the value of (r_c) is > or < 1.0 e/ų, each evaluated at the bond critical point of a bonded interaction. They asserted that a bonded interaction is shared (covalent) when the value of ²(r_c) is negative, (r_c) is greater than ~1.0 e/ų and G(r_c)/(r_c) is less than unity, and that it is closed shell (ionic) when ²(r_c) is positive, (r_c) is smaller than ~1.0 e/ų, G(r_c)/(r_c) is greater than unity and r_c is located relatively far from the nodal surface [²(r) = 0 r] of the Laplacian. But when r_c is located in close proximity to the surface, the bonded interaction is asserted to be an intermediate interaction, the closer r_c is to the surface, the more shared the interaction. It is noteworthy that the net atomic charges conferred on the bonded atoms were not expressly considered in the classification.

As the ²(r_c) values for the silicates considered in Figure 6 are all positive, each of the Si-O bonded interaction qualifies either as a closed-shell or an intermediate interaction, depending on the distance that r_c is from the nodal surface of the Laplacian. In an earlier study of several siloxane molecules, Gibbs et al. (1997) measured the distances between the nodal surfaces and r_c for the molecules and found that they decrease systematically as the Si-O lengths decrease and (r_c) and ²(r_c) each increases in value. The distance between r_c and the nodal surface was found be ~0.60 Å for the ^VIIISi-O bonded interaction of the H₁₂^VIIISiO₈ molecule, ~0.40 Å from the surface for the ^VISi-O bonded interaction of the H₈^VISiO₆ molecule to ~0.15 from the surface for the ^IVSi-O bond interaction of the H₄^IVSiO₄ molecule. On the basis of the Bader and Essén classification, the ^VIIISi-O bonded interaction qualifies as a closed-shell ionic interaction [(r_c) = 0.47 e/ų; ²(r_c) = 6.16 e/Å⁵], the ^VISi-O bonded interaction qualifies as an intermediate interaction with a marginal component of closed-shell character [(r_c) = 0.74 e/ų; ²(r_c) = 15.78 e/Å⁵] and the ^IVSi-O bonded interaction [(r_c) = 0.94 e/ų; ²(r_c) = 26.00 e/Å⁵] qualifies as an intermediate interaction. Consistent with these results, the G(r_c)/(r_c) ratio increases from ~1.0 for the ^VIIISi-O interaction to ~2.7 for the ^IVSi-O interaction (Fig. 11) as the shared character of the bonded interactions increases in conformity with the decrease in the distance between the bond critical point and the Laplacian nodal surface and the decrease in the coordination number of Si (Gibbs et al. 2006b). Concomitant with this trend, ²(r_c) increases from 6.16 to 26.00 e/Å⁵ as the coordination numbers of the Si atoms of the siloxane molecules decrease from eight to four and as (r_c) increases from 0.47 to 0.94 e/ų. The decrease in the coordination number and the increase in (r_c) suggest that ²(r_c) actually increases as the shared character of an intermediate bonded interaction increases. To establish whether this connection holds in general, the ²(r_c) values, calculated for a large number of M-O bonded interactions involving first and second row M atoms, were plotted with respect to the (r_c) values calculated for the interactions (Fig. 12) (Gibbs et al. 2001, 2008a). As displayed by the figure, ²(r_c) increases linearly with (r_c) for each of the bonded interactions, the larger the value of (r_c), the smaller the coordination number of the M atoms and the larger the value of ²(r_c). Furthermore, when the bond lengths for the bulk of these interactions are plotted with respect to the ratio G(r_c)/(r_c) (Fig. 11), the ratio increases linearly as the bond length decreases, (r_c) increases and the coordination numbers of the M atoms decrease, as observed for Si-O bonded interactions. The Bader-Essén (1984) assertion that a bonded interaction qualifies as a shared interaction when the value of G(r_c)/(r_c) is less than unity and a closed-shell one when it is greater than unity is not borne out by the trends displayed for the non-transition M-O bonded interactions (Fig. 11) or those displayed for a number of transition M-O bonded interactions observed for carboxylate Fe bridged and butterfly complexes and Mn and Co containing organometallic coordination polymers (Clausen et al. 2008; Gibbs et al. 2008b). The trends suggest, at least for M-O bonded interactions, that relatively large positive G(r_c)/(r_c) ratios are compatible with a non-trival component of shared character, the larger the ratio for a given M-O bonded interaction, the more shared the interaction. The trends in Figure 12 show, at least for the M-O bonded interactions, that when the value of ²(r_c) is positive, (r_c) increases as the shared character of the bonded interactions increases, particularly given that the local total energy density values for the bulk of the bonded interactions are negative (except Li-O, Na-O, and Mg-O bonded interactions as discussed above) and that each decreases with the increasing value of (r_c). It also suggests that ^VISi-O and ^IVSi-O are both intermediate bonded interactions, despite their large ²(r_c) values, with the ^IVSi-O bonded interaction possessing a slightly greater component of shared character than the ^VISi-O bonded interaction.

View larger version (33K): [in this window] [in a new window]   FIGURE 11. Scatter diagram of the bond lengths, R(M-O) Å, determined for many silicates, oxides, and representative molecules plotted with respect to G(rc)/(rc) a.u./e for first and second row M atoms bonded to O. For each of the bonded interactions, as G(rc)/(rc) increases with decreasing bond length, the coordination numbers of the M atoms decrease and (rc) increases. For example, as the G(rc)/(rc) ratio increases for the B-O bonded interactions from ~1.5 to ~2.0 a.u./e, the coordination number of B decreases from 4 to 3 and the B-O bond lengths decrease from R(IVB-O) = ~1.45 Å to R(IIIB-O) = ~1.35 Å. For the Al-O bonded interactions, as the ratio increases from ~1.5 to ~2.0 a.u./e, the coordination number decreases from 6 to 4 and the Al-O bond lengths decrease from R(VIAl-O) ~ 1.95 Å to R(VAl-O) ~ 1.85 Å to R(IVAl-O) ~ 1.75 Å. For the Si-O bonded interactions, as the ratio increases from ~1.0 to ~2.7 a.u./e, the coordination number decreases from 8 to 4 and the Si-O bond lengths decrease from R(VIIISi-O) ~ 2.01 Å to R(VISi-O) ~ 1.77 Å to R(IVSi-O) ~ 1.62 Å. As the ratio for the C-O bonded interactions increases from ~0.5 to ~2.5 a.u./e and the coordination number decreases from 4 to 1 as the C-O bond lengths decrease from R(IVC-O) ~ 1.39 Å to R(IIIC-O) ~ 1.29 Å to R(IIC-O) ~ 1.16 Å to R(IC-O) ~ 1.12 Å. With exceptions of R(Na-O) and R(Mg-O), as R(M-O) decreases, the local energy density is negative and H(rc) = G(rc) + V(rc) decreases, and the shared character of the bonded interactions are indicated to increase (see Gibbs et al. 2006b).

View larger version (24K): [in this window] [in a new window]   FIGURE 12. Scatter diagram of (rc) plotted with respect to 2(rc), calculated for first and second row M atoms bonded to O. The bcp properties used to construct these plots were taken from the properties of structures used to construct Figures 17a and 17b published by Gibbs et al. (2008a). Li-O is plotted as blue circles, Na-O as red circles, Mg-O as light green diamonds, Be-O as brown triangles, Al-O as blue triangles, Si-O as black squares, P-O as orange pentagons, B-O as dark green triangles, and S-O as violet hexagons.

	CONCLUDING REMARKS

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
The optimization of the structures for the silica polymorphs as a function of pressure with first-principles density functional theory and the generation of model structures that rival the accuracy of experimental ones have demonstrated that theory and software have evolved to the level where it is now possible to generate accurate structures for a wide range of minerals as a function of pressure and composition, and to determine their equations of state (Demuth et al. 1999). In addition, it is also possible to model physical and surface properties, templating, and catalysis and the reactivity of materials (Panero and Stixrude 2004; Brodholt and Vocadlo 2006; Catlow et al. 2005; Woodley and Catlow 2008). Also, as the ED distribution for a ground state mineral suffices for the determination of all of its properties (Parr and Yang 1989), it is an ideal medium for studying bonded interactions. Bond paths and the bond critical point properties calculated for the ED distributions for siloxane molecules are comparable with those observed for the silica polymorphs, a result that indicates that the forces that govern the structures and properties of the silica polymorphs are largely short-ranged and molecular in nature rather than being long-ranged and strongly ionic.

The success of first-principles computational quantum mechanical calculations in the determination of structures and ED distributions that are in close agreement with experimental structures and distributions for the silica polymorphs bodes well for future first-principles studies of Earth materials, their structures, properties, and reactivity. Also, as observed by Gibbs et al. (2008a), the calculations are not only expected to assist in the development and interpretation of experimental results, but they will also provide a deeper understanding of crystal chemistry, properties, mineralogical processes, and chemical reactions beyond the unobservables embodied in Pauling’s rules, bond strength relationships, and ionic and crystal radii, thereby enriching the fields of mineralogy and geochemistry, particularly when a given property or reaction is studied for a variety of chemically related materials like the silicates (Gibbs et al. 2008a, 2008c), transition metal sulfides (Gibbs et al. 2005b, 2007) and arsenites (Gibbs et al. 2009). In short, if mineralogists and geochemists persist in their study of minerals, their properties and relationships within the framework of empirical parameters like ionic radii, bond strength and electrostatic potential and forces and do not include first-principles quantum mechanical calculations and the study of ED distributions, then it questionable whether our understanding of the crystal chemistry and the properties of minerals in their natural environments will advance much beyond that of last century.

As a final point, software is currently being written in our laboratories to evaluate the potential, kinetic, and the total energy densities globally with the goal of obtaining a more comprehensible understanding of the bonded interactions throughout a crystal rather than just at its bond critical points (Cremer and Kraka 1984). Furthermore, energy density isosurface maps will be generated for crystals that have been determined over a range of pressures as a means of monitoring how the densities vary globally throughout a crystal as a function of pressure. By locating those regions where H(r) < 0, that are stabilized relative to those where H(r) > 0, we may expect to pinpoint those regions in the structure where the bonded interactions are stabilized relative to those that are destabilized (Tsirelson 2002). This information is expected to provide important insight into how the structure of the crystal responds to pressure and how the bonded interactions might be disrupted at the atomic level upon approaching a phase transformation. The information may also improve our understanding of the connection between the bonded interactions and the equation of state of the crystal at the atomic level. Also, calculations for topologically equivalent but chemically different structures like forsterite and fayalite are expected to shed light on the impact on the stability of a structure at the atomic level accompanying the replacement of one atom by another.

	ACKNOWLEDGMENTS

Top Abstract Introduction Overview Bader's topological model for... Bond paths and bridging... The Laplacian, {nabla}2{rho}(r),... Si-O bond critical point... Optimized structures and... Siloxane molecules and silicates Bonded radii of Si... Si and O net... The elusive character of... Concluding remarks Acknowledgments References cited

 
The National Science Foundation and the U.S. Department of Energy are thanked for supporting this study with Grants EAR-0609885 (N.L.R. and G.V.G.), EAR-0609906 (R.T.D.), and DE-FG02-97ER14751 (D.F.C.). K.M.R. acknowledges a grant from the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Geoscience Division and computational facilities and support from the Environmental Molecular Sciences Laboratory (EMSL) at the Pacific Northwest National Laboratory (PNNL). The computations were performed in part at the EMSL at PNNL. The EMSL is a national scientific user facility sponsored by the U.S. DOE Office of Biological and Environmental Research. PNNL is operated by Battelle for the DOE under contract DEAC06-76RLO 1830. G.V.G. owes a debt to his good friend and colleague Richard Bader for forging his elegant theory on the topology of electron density distributions and in particularly for his patience in responding to his many emails seeking advice and help in understanding the science. We are pleased to acknowledge Mauro Prencipe of the University of Torino for reviewing the manuscript and for making several important suggestions that clearly improved the manuscript. We also thank a second reviewer who urged that we make a statement at the beginning of the manuscript, giving the reader an overview of the contents of the paper. It was a good idea and we thank the reviewer for the suggestion. Color in figures courtesy of MSA Color Fund.

	Footnotes

 
MANUSCRIPT HANDLED BY G. DIEGO GATTA

MANUSCRIPT RECEIVED February 9, 2009; MANUSCRIPT ACCEPTED April 23, 2009

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