The Electron Pair - American Chemical Society

15398

J. Phys. Chem. 1996, 100, 15398-15415

The Electron Pair Richard F. W. Bader,* S. Johnson, and T.-H. Tang Department of Chemistry, McMaster UniVersity, Hamilton, Ontario L8S 4M1, Canada

P. L. A. Popelier Department of Chemistry, UMIST, P.O. Box 88, SackVille Street, Manchester M60 1QD, UK ReceiVed: May 7, 1996; In Final Form: July 5, 1996X

Eighty years have elapsed since Lewis introduced the concept of an electron pair into chemistry where it has continued to play a dominant roˆle to this day. The pairing of electrons is a consequence of the Pauli exclusion principle and is the result of the localization of one electron of each spin to a given region of space. It is the purpose of this paper to demonstrate that all manifestations of the spatial localization of an electron of a given spin are a result of corresponding localizations of its Fermi hole. The density of the Fermi hole determines how the charge of a given electron is spread out in the space occupied by a second same-spin electron, thereby excluding an amount of same-spin density equivalent to one electronic charge. The Fermi hole is an electron’s doppelga¨ngersit goes where the electron goes and vice versa: if the hole is localized, so is the electron. The topologies of two fields have been shown to provide information about the spatial localization of electronic charge: the negative of the Laplacian of the electron density, referred to here as L(r), and the electron localization function ELF or η(r). The measure provided by L(r) is empirical. It is based upon the remarkable correspondence exhibited by its topology with the number and arrangement of the localized electron domains assumed in the VSEPR model of molecular geometry. η(r) is based upon the local behavior of the same-spin probability, and it is shown that the picture of electron localization that its topology provides is a consequence of a corresponding localization of the Fermi hole density. This paper provides a complete determination and comparison of the topologies of L(r) and η(r) for molecules covering a wide spectrum of atomic interactions. The structures of the two fields are summarized and compared in terms of the characteristic polyhedra that their critical points define for a central atom interacting with a set of ligands. In general, the two fields are found to be homeomorphic in terms of the number and arrangement of electron localization domains that they define. The complementary information provided by the similarities in and differences between these two fields extends our understanding of the origin of electron pairing and its physical consequences.

Relation between Electron Pairing and the Pair Density Lewis1

demonstrated that the idea of considering a In 1915, pair of electrons as the basic unit in modeling the electronic structure of a molecule provided a powerful tool for rationalizing and understanding molecular geometry, bonding, and reactivity, an idea that continues to play a dominant role in present day chemistry. Lewis1 originally went so far as to postulate the breakdown of Coulomb’s law to account for the pairing of particles of like charge. We now understand the pairing of electrons to be a consequence of the antisymmetrization requirement imposed on the wave function by the Pauli exclusion principle, as applied to electrons that possess two measurable spin possibilities.2 The oft-quoted result of antisymmetrizing the wave function with respect to the permutation of the space-spin coordinates of every pair of electrons is that no two electrons with the same spin can occupy the same point in space. In chemistry, however, one is most interested in the spatial distribution of the electrons. To determine the manner in which the exclusion principle affects the electron distribution and its properties in real space, one must determine how many pairs of electrons, on the average, contribute to the electron density over the region of interest. This information is given by the electron pair density, alternatively called the pair probability function.3 X

Abstract published in AdVance ACS Abstracts, September 1, 1996.

S0022-3654(96)01297-X CCC: $12.00

Pair Density. The electron density multiplied by an infinitesimal volume element, F(r) dr, gives the number of electrons in dr and integrates to N, the number of electrons. Similarly, the product of the pair density and a corresponding pair of volume elements, F(r1,r2) dr1 dr2, gives the number of electron pairs formed between these two elements, and its double integration yields the number of distinct electron pairs, N(N1)/2. The density of R-spin electrons at r1 is FR(r1). The uncorrelated pair density for simultaneously finding R-spin density at r1 and r2 is given by the product FR(r1)FR(r2). The correct correlated pair density everywhere is less than this, and the Fermi hole determines the difference, a negative quantity, between the correlated and uncorrelated pair densities for samespin electrons.3 This difference is a measure of the degree to which density is excluded at r2 because of the spreading out of the same-spin density originating from position r1, and thus the Fermi hole density determines the manner in which density at r1, in an amount equivalent to one electronic charge, contributes to the pair density at other points in space. Pictorially, one can imagine that as an electron moves through space it carries with it a Fermi hole of ever changing shape, the density of the electron being spread out in the manner described by its Fermi hole and excluding density equivalent to one same-spin electron.4 Since the charge of the electron is spread out in a manner described by the density of its Fermi hole, it follows, as © 1996 American Chemical Society

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demonstrated by Bader and Stephens,4 that the quantum mechanical requirement for the localization of an electron to some particular spatial region in a many-electron system is that the density of its Fermi hole be totally and maximally contained within this region. In this limiting situation, all other samespin electrons are excluded from the region, and since the same behavior is obtained for an electron of opposite spin in a closedshell system, the result is a spatially localized electron pair. The Fermi hole density determines the local extent of exclusion of same-spin density. An electron can go whenever its Fermi hole is different from zero, and if the hole is localized, so is the electron. It is the purpose of the present paper to demonstrate that all physical measures of electron localization or delocalization5 are determined by the corresponding localization or delocalization of the Fermi hole density. The Laplacian of the electron density provides a physical, albeit empirical, measure of localization, a consequence of its topology providing a mapping of the Lewis pairs assumed in the VSEPR model of molecular geometry.6-8 ELF, the electron localization function of Becke and Edgecombe9 that is based upon a modeling of the same-spin probability and thus the Fermi hole,10 is shown to provide an equivalent description of electron localization. This correspondence is demonstrated by showing that the topological structures that characterize these two scalar fields in molecular systems are, in general, either homeomorphic or they are simply derivable, one from the other. Properties of the Density of the Fermi Hole. The principal properties of the Fermi hole that underlie its roˆle in determining electron localization are easily understood in a pictorial manner, and this is done here for the general reader, with the relevant equations appearing in the Appendix and the details in the literature.4,8 The discussion is for the density of electrons of a given spin, say FR, as the Fermi correlation affects only samespin electrons. The density of the Fermi hole, like the pair density itself, is a function of the coordinates of two electrons, those of a reference electron, r1, and those of another, r2. The Fermi hole density determines how the density of the reference electron spreads out from r1 into the space of the second samespin electron. It is a negative quantity, since its value determines the decrease in the same-spin density at each value of r2. The density of the Fermi hole has two important properties: (a) its magnitude equals the total density of same-spin electrons for r1 ) r2. This ensures the complete removal of all other samespin density from the position r1. (b) Its integral over r2 equals -1, corresponding to the removal of one same-spin electron from the density, FR(r2). At the Hartree-Fock level of theory, there is no correlation between electrons of different spin, and the pair density is simply the product of the densities of the R- and β-spin densities multiplied by a factor of one-half (so as not to count the same pair twice), or

FRβ(r1,r2) ) (1/2)FR(r1)Fβ(r2)

(1)

For same-spin electrons, however, the density at r2, FR(r2), is reduced in value by the density of the Fermi hole, and the samespin pair density is given by

FRR(r1,r2) ) (1/2)FR(r1)δR(r2)

(2)

where δR(r2) is the difference between FR(r2), the spin density at r2, and the magnitude of the density of the Fermi hole at r2 (eq A1 of the Appendix). Thus, δR(r2) is a measure of the amount of same-spin density not excluded from the position r2 by the spreading out of the density at r1 and is, therefore, the

Figure 1. Contour plot of the magnitude of the Fermi hole density for LiH, together with its profile and that of the R-spin density along the internuclear axis for three positions of the reference electron: at the Li nucleus (a), at the bond critical point (b), and at the proton (c). The R-spin density for LiH is also shown in (a) along with the boundary defining the atomic basins of Li and H. The shaded area in a profile map denotes the magnitude of the Fermi hole and the extent to which other same-spin density is excluded at each value of the abscissa, |r1 - r2|. A difference exists between the spin density and the shaded area over the entire axis in profile b, except for r2 ) r1. This difference is the quantity δR(r2) of the text. In (b) the hole and the density of the reference electron are delocalized over the entire molecule. In (a) and (c) the same quantities are localized within the basins of Li and H, respectively, and δR(r2) vanishes over the corresponding basin. The contours of the Fermi hole density appearing in a neighboring basin in (a) and (c) are too low in value to appear in the profile plots. The contours are in atomic units, the outer one having the value 0.001 and the remaining ones increasing inward in steps of 2 × 10n, 4 × 10n, and 8 × 10n, with n beginning at -3 and increasing in steps of unity.

conditional probability of finding an R-electron at r2 when another is known to be at r1.9 When δR(r2) vanishes, the samespin pair density vanishes for the pair of coordinates r1 and r2, and only the density of the reference electron contributes to the R-spin density at r2. A plot of the magnitude of the Fermi hole density for a fixed location of the reference electron shows how the density of an electron with coordinate r1 is spread out over the space of the second electron, excluding an equivalent amount of charge, and whether the Fermi density is localized or delocalized relative to a chosen boundary. Such a plot is shown in Figure 1b for a reference electron at the position of the bond critical point (bcp) in LiH. It is to be compared with the plot of the R-spin density FR(r) shown in panel a, a comparison made clearer by the profile

15400 J. Phys. Chem., Vol. 100, No. 38, 1996 of both quantities also shown in panel b. The difference between these two plots is δR(r2). The density of the Fermi hole is delocalized over both atomic basins, as is the electron itself. Aside from the point r1 ) r2, δR(r2) is different from zero and more than a single R-electron contributes to the pair density over both atomic basins when the density of the reference electron spreads out from the bcp. This situation is to be contrasted with that shown in Figure 1a where the reference electron is placed at the position of the lithium nucleus. The Fermi hole density in this case is localized within the basin of the Li atom and over which, except for a region of low density close to the bcp, it equals the total singlespin density. Since δ(r2) vanishes, all other same-spin electrons are excluded from this region and the same-spin pair density also vanishes. The hole is maximally localized in this case, and its integral over the Li atom basin yields over 99% of the charge of one electron. Since a β-electron exhibits the same behavior, a single R,β pair of electrons determines the total electron density within this basin when referenced to the Li nucleus. Fermi correlation does not act directly to “pair up” electrons. Rather, since there is no Fermi correlation between electrons of opposite spin, an R,β pair is obtained as a result of all other electrons of both R- and β-spin being excluded from a given region of space. What is perhaps remarkable, as is demonstrated in the following, is that this same localized behavior and near total exclusion of other same-spin electrons is obtained for almost all positions of the reference electron within the basin of the Li atom, with the result that 95% of the density within this basin comes from a single R,β pair of electrons, a localized electron pair. The density of the Fermi hole is localized nearly equally within the basin of the H atom (Figure 1c), and one has the picture of a molecule wherein the density in real space arises from the presence of a single pair of electrons in each of the atomic basins, with only a small probability of exchange of same-spin electrons between the two atomic basins, the exchange resulting from a delocalized hole as pictured in Figure 1b for reference electrons in or near the interatomic surface. The goal is to find a method or methods to determine the presence of such regions of spatial pairing within any molecule. Measures of Spatial Localization of the Electrons. While displays of the Fermi hole density are useful in determining its basic localized or delocalized nature relative to FR(r),5,7,8,11,12 each display is for a single fixed position of the reference electron. What is needed is a measure of the extent to which some number of electrons are localized to a given spatial region. Daudel and co-workers13-15 reasoned that there should be some “best” decomposition of the real space of a system into a number of mutually exclusive spaces called loges, with the best loges yielding the most probable division of the system into localized groups of electrons. It was proposed that the minimization of the missing information function be used to define the “best” loges,16 with the missing information function being defined in terms of the quantum probabilities of some number of electrons being found in a particular region of space. In the first application of these ideas,17 it was found that the same loge boundary that minimized the missing information function also minimized the fluctuation in its average electron population. This is an important finding, since the calculation of the missing information function requires the full N-particle density matrix while the particle fluctuation is determined by just the pair density. Following on this work, Bader and Stephens18 were able to show that the vanishing of the fluctuation over a region of space requires that the Fermi hole for each of its electrons be totally contained within its

Bader et al. boundaries, work that leads directly to the demonstration that the spatial localization of some number of electrons is determined by the corresponding localization of their Fermi hole density.4 Establishment of the relation between the localization of the electron and the density of its Fermi hole leads to a direct physical measure of the extent to which some number of electrons are localized to a given spatial region. One first determines the total possible Fermi correlation contained within a region Ω, a quantity denoted by FRR(Ω,Ω), by integrating the density of the Fermi hole weighted by FR(r1) over Ω for all possible positions of the reference electron within Ω.4,8 This procedure corresponds to the double integration of the exchange density in Hartree-Fock theory. If NR(Ω) electrons are completely localized within Ω, then the Fermi correlation attains its limiting value of -NR(Ω), that is, a corresponding number of same-spin electrons are excluded from Ω. The ratio |F(Ω,Ω)/ N(Ω)| for electrons of either spin is thus the fraction of the total possible Fermi correlation per particle contained within a region Ω, and this ratio, when multiplied by 100, is l(Ω), the percent localization of the electrons in Ω. The method of defining the contained Fermi correlation is given an atomic basis by identifying the regions Ω with the basins of the topologically defined atoms in the theory of atoms in molecules.8 The boundary enclosing an atomic basin, a quantum mechanical proper open system,19a is defined as one of local zero flux in the gradient vector field of the electron density. Such proper open systems are identified with the functional groups of chemistry,19b since they fulfill the observational requirement of exhibiting characteristic and transferable sets of properties. The Fermi correlation over an atomic basin and other atomic properties are calculated by using PROAIMV.20 All SCF calculations, including geometry optimization, are performed by using GAUSSIAN 92 and the basis set 6-311++G(2d,2p) with six d functions.21 The Appendix lists the atomic charges q(Ω) and percent localizations l(Ω). Also listed are the total energies, geometrical parameters, and properties of the electron density at the bond critical point. Extensive investigation of the contained Fermi correlation shows that, aside from core regions and including ionic systems such as LiH or BeH2, electrons are not, in general, spatially localized to yield individual pairs4,17 (Table 6), the same systems wherein it is possible to define loges that minimize the missing information function.17 Unsurprisingly, the electron correlation energy in such systems is found to be dominated by the intrapair contributions.22 It is possible for some number of pairs of electrons to be spatially localized within an atomic basin in ionic systems. In NaF and MgF2, for example, the populations of the atoms approach the limiting value of 10 for an LM core and the localizations are in excess of 95% (Table 7). Summary and Relation to the Orbital Model. The general conclusion drawn from the spatial pairing exhibited by the pair density in many-electron systems is that, aside from core regions, the density is not characterized by the presence of strongly localized pairs of electrons. This observation is not at variance with the possibility of obtaining an equivalent description of a system by a transformation to a set of localized molecular orbitals, the pair density being invariant to the required unitary transformation.4 Maximizing the contained Fermi correlation for a region Ω requires that each of the orbitals be localized to a separate spatial region Ω, so that the overlap of the orbitals between two different regions vanishes.4,8 Thus, to achieve localization of N electrons into N/2 distinct spatial pairs requires that each orbital be separately and completely localized to one of N/2 spatially distinct regions. While orbitals can appear

The Electron Pair

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Figure 2. Contour maps of L(r) and η(r) for ClF5 (a) and SF6 (b) with η(r) on the RHS. The contour values for L(r) in this and succeeding maps begin at zero and increase (solid contours) and decrease (dashed contours) in steps of (2 × 10n, (4 × 10n, and (8 × 10n, with n beginning at -3 and increasing in steps of unity. The first dashed contour in η(r) adjacent to a solid contour has the value 0.50 and decreases in steps of 0.05 away from the solid contours. The first solid contour adjacent to a dashed contour has the value 0.55 and increases in steps of 0.05. The atomic boundaries, as determined by the zero-flux surface condition, along with the bond paths are indicated in each diagram, with the bond cps being denoted by dots. The values of the functions at the maxima and other cps evident in the contour maps are given in the corresponding tables. The equatorial maxima in η(r) for ClF5 are of limited extent and are nearly coincident with the corresponding bond cps. In L(r) they occur within the basin of the Cl atom. For a third-row atom, the inner core concentration in L(r) is spikelike at the nucleus, while the innermost region of solid contours in η(r) is reduced to a point in the scale of these diagrams.

localized in their displays, they actually exhibit a considerable degree of absolute overlap23 and they fail to satisfy the requirement of separate localization. This must be the case in the face of the properties determined for the pair density. The ability to define a set of doubly occupied localized molecular orbitals does not imply physical localization of the electrons into spatial pairs. In ionic systems, even the canonical set of orbitals is strongly localized to the separate atomic basins yielding atomic sets of localized electrons. If the canonical set is not localized, a unitary transformation to a localized set will leave the properties of the pair density and its predicted lack of electron pairing unchanged. The Lewis Model As Displayed in the Laplacian of the Electron Density Clearly, electrons are not spatially paired to the extent perhaps anticipated in view of the success of the Lewis model. The topology of the electron density, while providing a faithful mapping of the concepts of atoms, bonds, and structure,8,24 does not give any indication of the bonded and nonbonded electron pairs anticipated on the basis of the Lewis model. However, as is well documented, the topology of the Laplacian of the electron density, the quantity ∇2F(r), does provide a physical basis for the Lewis model.6,7,8,25 The Laplacian of a scalar field such as F(r) determines where the field is locally concentrated, ∇2F(r) < 0, and where it is locally depleted, ∇2F(r) > 0, with the corresponding local minima and maxima providing features that are absent from the topology of F(r) itself. Since electronic

charge is concentrated in regions where ∇2F(r) < 0, it is convenient to define the function L(r) ) -∇2F(r), with a maximum in L(r) then denoting a position at which the electron density is maximally concentrated. Empirically, one finds that the local maxima in L(r) provide a remarkably faithful mapping of the localized electron domains that are assumed to be present in the valence shell of the central atom in the VSEPR model of molecular geometry.26 Not only is there is a one-to-one correspondence in their number but also, as importantly, in their angular orientation within the valence shell of the central atom, as well as in their relative sizes. Thus, one may make the assumption that the pattern of local charge concentrations defined by the Laplacian of the electron density denotes a corresponding pattern of partial condensation of the electron pair density to yield regions of space dominated by the presence of a single pair of electrons. The Topology of L(r). The typical pattern of electron localization revealed by the topology of the Laplacian distribution through the presence of bonded and nonbonded charge concentrations (CCs) in the valence shell of a main group element is illustrated in Figure 2a for the ClF5 molecule. One notes that L(r) exhibits a shell structure for F and Cl wherein each shell is characterized by a shell of charge concentration (which appears as a spike at the position of the nucleus for the inner shell) followed by one of charge depletion, behavior that is characteristic of elements with Z < 40.8,27 For the remaining elements, L(r) exhibits the number of maxima and minima required by the appropriate number of shells, but the other

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Bader et al.

TABLE 1: Critical Points in L(r) and η(r) for Molecules with Five and Six Electron Domainsa molecule ClF5 [6,8,4]

ClF3 [5,7,4] SF6 [6,12,8] SOF4 [5,6,3]L, [6,10,6]η SF4 [6,8,4]L, [5,9,6]η

n×(r,s)

L(r)

r

F(r)

η(r)

r

F(r)

typeb

1×(3,-3) 1×(3,-3) 4×(3,-3) 4×(3,-1) 4×(3,-1) 4×(3,+1) 2×(3,-3) 1×(3,-3) 2×(3,-3) 6×(3,-3) 12×(3,-1) 8×(3,+1) 1×(3,-3) 2×(3,-3) 2×(3,-3) 1×(3,-3) 2×(3,-3) 2×(3,-3)

2.095 0.689 0.421 0.358 0.125 0.008 1.764 0.398 0.101 0.401 -0.137 -0.163 0.915 0.414 0.266 1.104 0.330 0.088

1.108 1.218 1.221 1.198 1.239 1.262 1.129 1.256 1.268 1.357 1.368 1.413 1.284 1.356 1.362 1.233 1.360 1.357

0.3944 0.2767 0.2400 0.2362 0.1924 0.1691 0.3646 0.2525 0.1964 0.2537 0.1316 0.1086 0.3999 0.2691 0.2254 0.2609 0.2586 0.1843

0.9882 0.8235 0.7209 0.6324 0.5238 0.4502 0.9860 0.7905

1.570 1.420 1.418 1.317 1.368 1.385 1.618 1.582

0.1926 0.2728 0.2246 0.2092 0.1656 0.1432 0.1718 0.2788

0.8467 0.3326 0.2703 0.8380 0.8438 0.8278 0.9961 0.8348 0.8178

1.595 1.453 1.482 1.483 1.574 1.804 1.801 1.651 2.093

0.3166 0.1214 0.0998 0.3706 0.3348 0.3718 0.1050 0.3546 0.2375

n b to Fa b to Fe n-be ba-be n-be-be-ba n b to Fe b to Fa b to F b-b b-b-b b to Oc b to Fe b to Fa n b to Fe b to Fa

a n×(r,s) denotes the number and type of critical point; r is radial distance from nucleus to cp. Values of L(r) and F(r) at the cp are in atomic units, as is r. b n denotes a nonbonded maximum, b a bonded one. Subscript a (e) denotes an axial (equatorial) ligand. Maxima linked by (3,-1) cp are joined by a-, as in b-b. Maxima linked in forming the face for a (3,+1) cp are indicated. c There are two such maxima in η(r).

extrema do not undergo a sign change, with L(r) remaining negative over the outer shell.28,29 The sign change reappears for the noble gas elements at the close of each period. This property of L(r) enables one to identify the shell of interest. For example, in molecules containing an atom from the first or second transition series, the ns electrons are transferred to the ligands and the chemistry is determined by the distortions of the outer shell of the core, which is clearly identified as the (n - 1)th shell of charge concentration for which L(r) > 0. The topology of a scalar field and its stability are succinctly summarized in terms of its critical points,30 points where ∇F(r) or ∇L(r) ) 0, for example. The application of these ideas to the topology of the electron density, coupled with the theorem of structural stability of Palis and Smale31 and the catastrophe theory of Thom,32 yields a theory of molecular structure and structural stability.24 The same topological analysis can be applied to the critical points in L(r) found in the outer shell of charge concentration, generally the valence shell charge concentration (VSCC) of a main group element.8 A critical point (cp) at position rc is labeled by the duo (r,s), where r, the rank, is the number of non-zero curvatures of L(rc) and where s, the signature, is the algebraic sum of their signs. The local maxima or (3,-3) cps, whose presence denotes a bonded or nonbonded charge concentration (CC), are linked one to another by the unique pair of trajectories that originate at an intervening (3,-1) cp, thereby generating the atomic graph, the analogue of the molecular graph defined by the corresponding set of cps in F(r). The atomic graph, in general, assumes the form of a polyhedron bounding the nucleus in question. In general, each critical point in L(r) for a given shell exhibits a negative radial curvature, and its topology is consistent with the presence of a (3,+3) or cage cp, a minimum in L(r), positioned at the nucleus within the shell. The Poincare´-Hopf relationship30 for the cps defining an atomic graph is then readily transformed into the more familiar polyhedral formula of Euler:

V-E+F)2

(3)

The vertices are defined by the (3,-3) cps or maxima in -∇2F and the edges by the unique pairs of trajectories that originate at (3,-1) cps and terminate at neighboring vertices, while the (3,+1) or ring cps define the resulting faces of the polyhedron. In some ionic or very polar systems the ring cps, at which L(r) attains its lowest values, occur in regions where the electron

density is diffuse and small in value, with the result that certain associated curvatures of L(r) approach zero and the cps are topologically unstable. In these systems the ring cps can bifurcate and be linked to further structure in a region of low density that is of no physical significance. This instability is not found for the (3,-3) or (3,-1) cps, and the form of the polyhedron is, therefore, stable with respect to any bifurcation occurring in its faces. The atomic graph for Cl in the pseudooctahedral geometry of ClF5 is illustrated in Figure 2a, and its critical points are summarized in Table 1. It is an irregular octahedron, with the positions of the six vertices corresponding to the presence of one nonbonded and five bonded CCs. All of these cps are found within the VSCC of Cl with characteristic radii of 1.1-1.2 au. The geometry is dominated by the large nonbonded CC with L(r) ) 2.10 au, whose presence causes the Fe-Cl-Fa bond angle formed by the equatorial Fe with the unique axial Fa to be less than 90° (Table 5). This feature is made particularly clear in the envelope map of L(r) illustrated in Figure 3a. Also in accord with the VSEPR model, the bonded CC facing Fa is larger than those facing the equatorial fluorines (Table 1). Each of the latter bonded CCs is linked to the bonded CC directed at the axial F and to the nonbonded CC to yield a polyhedron with eight edges and four faces (Figure 3), with the corresponding critical points satisfying eq 3 with 6 - 8 + 4 ) 2. The number of each kind of critical point that satisfies Euler’s equation, the characteristic set,8 is denoted by [V,E,F] (Table 1). We shall find in the following that Cl exhibits the same characteristic set [6,8,4] in ELF and the topologies of ELF and L(r) are homeomorphic. The properties of L(r) recover not only the geometrical aspects of Lewis’s model, but also his definition of an acidbase reaction. A nonbonded CC is a Lewis base serving as a nucleophilic center, while a ring cp, since it defines the point where electronic charge is least concentrated in its atomic graph, is a Lewis acid serving as an electrophilic center.8 L(r) is usually negative at a ring cp, and this corresponds to the presence of a “hole” in the VSCC of the acid, that is, in a face of the polyhedron representing its atomic graph. A Lewis acidbase reaction corresponds to aligning a CC on the base with a “hole” on the acid, with the alignment of the two cps providing a guide to their relative angle of approach. In an ionic crystal, the vertices of the anion are directed at the faces of the cation,33

The Electron Pair

J. Phys. Chem., Vol. 100, No. 38, 1996 15403 The Lewis Model As Displayed in ELF ELF as a Measure of Localization of the Fermi Hole. The electron localization function (ELF) defined by Becke and Edgecombe9 is based upon the properties of a function ∆ introduced by Tal and Bader42 in a study of functionals for the kinetic energy density. The function ∆ is a measure of the difference between two kinetic energy densities:

∆(r) ) G(r) - g(r)

(5)

G(r) is the positive definite form of the kinetic energy density that can be expressed as43a

G(r) ) (p2/8m)∑λi∇Fi(r)‚∇Fi(r)/Fi(r)

(6)

i

where Fi ) |φi|2 is the density of orbital φi (natural or HartreeFock) with occupation number λi. The function g(r), the Weisza¨cker functional,43b is expressed in terms of the total electron density F(r): Figure 3. Displays of L(r) and ELF or η(r) in terms of isovalued surfaces or “envelope” plots. (a) The surfaces of L(r) ) 0.38 au isolate the six maxima in the VSCC of Cl in ClF5: the upper broad nonbonded maximum, the axial bonded maximum, and the four smaller equatorial bonded maxima surrounding the inner core of Cl. (b) The surfaces of η(r) ) 0.65 showing the inner core and the same set of maxima that exhibit the same relative sizes as found in L(r). The bonded maxima in this case are linked to the surfaces of similar value enveloping the fluorine ligands. The polyhedron defined by the characteristic set [6,8,4] is shown as an inset. Panels c and d are corresponding displays of L(r) and η(r) for SF6 for envelopes equaling 0.0 and 0.8, respectively. The ligands appear in both displays, in addition to the six bonded maxima and the inner core of S. The characteristic polyhedron in this case is [6,12,8], as displayed in the inset. The surfaces in L(r) encompass regions within which L(r) > 0 and within which the electron density is concentrated.

and the same interactions account for the layered geometry of solid chlorine.34 The activation of an oxygen atom in the surface of MgO toward the adsorption of CO is the result of the formation of a hole in its VSCC by the introduction of vacancies generated by doping with cations of reduced charge35 or by electronic excitation.33 Numerous studies have shown a correlation of the magnitude of the nonbonded CC with experimental proton affinities.36-39 Base strengths and preferred sites of protonation have been similarly rationalized.40,41 The correlation has been shown to break down in cases where charge dispersal in the products dominates the reaction, as found in the protonation of fluoroand chlorophosphines.39 There is an energetic basis underlying these correlations, a result of L(r) appearing in the local expression for the virial theorem:

(p2/4m)∇2F ) 2G(r) + ν(r)

(4)

where G(r) > 0 is an electronic kinetic energy density defined in the following section and ν(r) < 0 is the electronic potential energy density defined by the virial of the forces exerted on the electrons. A local CC, a Lewis base, is thus a region of space where the local energy is stabilized by a dominant potential energy contribution. A Lewis acid-base reaction is the combination of a region with excess kinetic energy with one of excess potential energy, with the excesses being measured relative to the values required for the local satisfaction of the virial relation obtained where L(r) ) 0.

g(r) ) (p2/8m)∇F(r)‚∇F(r)/F(r)

(7)

It is clear from eq 6 that g(r) is the correct form for the kinetic energy density only for a single electron or a two-electron Hartree-Fock ground state, that is, for a system described by a single orbital. The difference between these two functions for any other system vanishes at a nuclear position, r ) 0, and for positions far removed from nuclei, r w ∞. Tal and Bader42 demonstrated two other important properties of ∆: (a) that ∆(r) g 0 and thus g(r) underestimates the kinetic energy density for 0 e r e ∞ and (b) that the requirement for ∆(r) to approach zero is for each orbital and, thus, each orbital density Fi to be localized to its own spatial region. They noted that this is the very condition required for the spatial localization of the Fermi hole density and for the creation of electron pairs.4 Thus, one has the important result that the kinetic energy density contains information regarding the spatial localization of the electrons.42 Consequently, they found g(r) to be a good model of the kinetic energy density in LiH,42 where the orbitals are spatially localized and where, as discussed with reference to Figure 1, electron pairing is very pronounced in each atomic basin with l(Li) ) l(H) ) 95% (Table 6). The function ∆(r) can be defined for each set of same-spin electrons, with ∆R(r) ) ∆(r)/2 for a closed-shell system. In a region of space over which ∆R(r) approaches zero, the Fermi hole density thus is correspondingly localized, approaching -FR(r) in value, and a single electron of given spin will determine the kinetic energy density. If each orbital in a many-electron system is so localized to a distinct spatial region, then the expression for G(r) becomes a sum over separate spatial regions. There will be one region for each Fi(r) over which it will equal the density F(r) appearing in the expression for g(r) and hence, in this limit, G(r) will equal g(r). This situation is to be contrasted with the ground state of a system of non-interating Bosons in which all of the particles occupy the same orbital and every particle is equally delocalized over the entire system. A Bose-Einstein condensate has recently been observed in the cooling of a cloud of integerspin 87Rb atoms that closely approach the non-interacting condition.44 When the de Broglie wavelength becomes larger than the mean spacing between the particles, the wave functions for the atoms overlap to such an extent that individual atoms can no longer be distinguished and Bose statistics favors the condensation of all of the particles into a single quantum state, which is just the opposite extreme to the situation required for

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the spatial pairing of electrons. Thus, g(r) describes the kinetic energy density in both limiting situations: for Fermions, F(r) is given by a sum of spatially separated orbital densities, one for each electron pair, while for Bosons, F(r) is given by the density of a single orbital extending over the entire system and occupied by all of the particles. In a modeling of the exchange energy, Becke10 showed that the leading term in the Taylor expansion of the spherically averaged conditional pair probability, the quantity δR in eq 2, is proportional to ∆R(r). Becke’s finding is understandable, for the vanishing of ∆(r) corresponds to the Fermi hole density attaining its limiting form of -FR(r2), the same requirement for the vanishing of δR(r2). What we emphasize here is that the degree of electron exclusion determined by the smallness of ∆R(r) is a consequence of the spatial localization of the density of the Fermi hole. Becke and Edgecombe9 note that ∆R(r) itself is not suited for a direct determination of the degree of electron localization; a low value of ∆R(r) denotes a high localizability and ∆R(r) is not bounded from above. They, therefore, propose the function called ELF, denoted here by η(r), where

η(r) ) (1 + χ2)-1

(8)

where χ ) ∆R/∆Ro and ∆Ro is the quantity corresponding to ∆R but for a uniform electron gas with the same spin density as the system in question. The range of values for η(r) is thus restricted to 0 e η(r) e 1, with η(r) ) 1 denoting complete localization and η(r) ) 1/2 corresponding to uniform electron gaslike pair probability. It is important to note “the more-orless arbitrary”9 nature of the transformation used to obtain a function with these bounded values, a feature that will require comment in the comparison of the topologies of L(r) and η(r). Becke and Edgecombe9 find that η(r) defines atomic shell structure up to and including the shell for n ) 6 and offers remarkably clear three-dimensional realizations of core, bonded, and nonbonded regions in molecular systems. They note that the Laplacian of the electron density provides alternative definitions of the same features, and this raises the question as to whether the two functions L(r) and η(r) complement one another or provide alternative but equivalent descriptions of the spatial localization of the electron density. Savin and co-workers have made extensive applications of ELF for solids with the diamond structure45a and for a wide variety of molecules45b using computer color graphics to display the results. More recently, Silvi and Savin46 have demonstrated how the local maxima in η(r) may be isolated through a study of the topological behavior of isovalued surfaces47 of η(r). This topological approach to identifying the (3,-3) cps or attractors in a scalar field is most easily understood in terms of an example. Consider two attractors in F(r) situated at the nuclear positions of a diatomic molecule and linked, as they necessarily are, by a (3,-1) or bond cp where the value of the density is denoted by Fb. This is a saddle point in a two-dimensional display of F(r) in a plane containing the internuclear axis. A single isovalued surface in the electron density will be obtained for all values of the density F(r) < Fb, with the envelope retaining its generic shape as the value of the density in the surface approaches the value Fb but becoming increasingly pinched in the vicinity of this critical point. The isovalued surface with density equal to Fb is, however, unstable, with any further increase in F(r) causing it to bifurcate into two surfaces, each enclosing a single attractor. This pattern of two isolated attractor domains persists for all further increases in the density. Silvi and Savin characterize the topology of the isovalued surfaces of ELF by defining localization domains:46 a surface

in η(r) containing more than one attractor is called a reducible domain, whereas a domain containing only one attractor is called irreducible. Thus, as the value of η(r) in the surface defining a reducible domain is increased from some initial value it undergoes bifurcations at critical values of η(r) (the values of its (3,-1) cps) into two or more domains containing fewer attractors. This process may be continued until all domains attain their irreducible form. The irreducible localization domains, since they isolate the local maxima in ELF, correspond to core, bonded, and nonbonded attractors, as originally identified by Becke and Edgecombe.9 Still more recently, Fa¨ssler, Ha¨ussermann, and Nesper48 use displays of ELF to visualize free surface bonds and bonds connecting surface atoms for various models for the Si(100) surface. Topology of ELF. This paper presents a full topological analysis of ELF, one that determines all of its critical points and compares the structure so defined with the corresponding structure exhibited by the Laplacian of the electron density. Critical points other than the (3,-3) or local attractors of the associated gradient vector field are of importance not only in determining the structure of a scalar field but also in relating the field to chemical properties, as demonstrated earlier for L(r). Location of the cps of rank 3 in η(r) and their signatures is accomplished by using a newly developed version of the program MORPHY 96.49 This is a new semiautomated program for the application of the theory of atoms in molecules that uses a superior technique to locate cps.50 The ELF distribution for the ClF5 molecule is illustrated in the form of a contour map of η(r) in Figure 2a, and its electron domains are displayed in an isovalued surface map in Figure 3a. Its critical points are listed in Table 1. In all contour maps, values of η(r) > 0.5 are denoted by a solid contour and those with η(r) e 0.5 by a dashed contour, with the solid contours denoting a localization greater than that for a uniform electron gas with an identical spin density. The chlorine atom exhibits an atomic graph consisting of the set [6,8,4] in both fields, and they are homeomorphic for this atom. The similarity extends to the relative values of the maxima, with the values of the (3,-3) cps in η(r) and L(r) decreasing in the order nonbonded n, axial bonded b to Fa, and equatorial bonded b to Fe, in agreement with the VSEPR model. Note the small size of the equatorial bonded compared to the axial bonded domain in η(r). The VSEPR model relates the displacement of the equatorial ligands toward Fa to yield an acute value for the bond angle ∠Fe-Cl-Fa (Table 5) to the dominant size of the nonbonded domain, an effect that is accounted for in the interdomain angle ∠n-Cl-be being obtuse in both fields, equaling 93.2° and 98.3° in L(r) and η(r), respectively, as determined by the coordinates of the cps. A comparison of Figure 2a,b illustrates the general features that the two fields have in common, as well as their principal differences. Both fields exhibit the same shell structure for both atoms, three for Cl and two for F. The nonbonded domains are, in general, of greater spatial extent in η(r) than they are in L(r), and the corresponding critical points are located farther from the nucleus, 1.11 au in L compared to 1.57 au for η. The bonded domains are of more equal size. The same disparity in size of the nonbonded domains is found for the ligands. Both fields exhibit two nonbonded domains on each Fe, while Fa, as a consequence of symmetry, exhibits four, those in η(r) having the greatest extent and being farthest removed from the nuclei. Each Fe also exhibits a bonded domain in η(r), but not in L(r). The spread in the radial distances of the critical points for a given atomic graph is, in general, smaller for L(r), where the

The Electron Pair

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TABLE 2: (3,-3) Critical Points in L(r) and η(r) for Hydridesa molecule borane [3,3,2] methane [4,6,4] ammonia [4,6,4] water [4,5,3] hydrogen fluoride silane [4,6,4] phosphine [4,6,4] hydrogen sulfide [4,5,3] hydrogen chloride

L(r)

r

F(r)

η(r)

r

F(r)

typeb

24.980 1.268 2.632 2.194 5.349 3.383 4.703 9.343 22.978 0.340 22.900 0.602 0.657 0.828 0.858

2.243 1.012 0.748 0.831 0.649 0.717 0.635 0.576 2.781 1.443 2.649 1.297 1.500 1.281 1.185

0.4178 0.2991 0.5562 0.4805 0.9427 0.7445 1.1169 1.4736 0.3879 0.1319 0.3865 0.1973 0.2211 0.2807 0.2730

>0.9500 >0.9900 0.9567 0.9994 0.9213 0.9981 0.9960 0.8896 1.0000 0.9954 1.0000 0.9694 0.9998 0.9994 0.9245

2.240 2.083 1.388 1.921 1.109 1.810 1.730 0.955 2.837 2.327 2.685 1.867 2.531 2.412 1.625

0.4170 0.4162 0.1584 0.4170 0.3242 0.4074 0.3856 0.5292 0.3694 0.0420 0.3758 0.0900 0.3692 0.3532 0.1404

b b n b×3 n×2 b×2 b n (ring) b×4 n b×3 n×2 b×2 b n (ring)

a See footnote a to Table 1. b A > sign before η(r) means that the maximum is flat to the extent that its location could not be determined precisely, and the values of F(r) and r refer to the position of the proton, its approximate location in other hydrides.

cps occur in a given shell of charge concentration with its own characteristic radius, than it is for η(r). The L and η atomic graphs for the sulfur atom in SF6 are again homeomorphic with six domains, this time all bonded to yield the characteristic set [6,12,8]. The same general comparative observations apply (Table 1 and Figures 2 and 3). In this molecule the bonded domains, which in L possess a radial distance characteristic of the VSCC of the free sulfur atom, fall within the atomic boundaries of the F atoms in both fields, a reflection of the more polar nature of the S-F interaction, with q(F) ) -0.7e, as opposed to Cl-F, with q(F) ) -0.5e (Table 5). In general, we shall be concerned with the atomic graph associated with the central atom only, one that can include maxima lying within the atomic basins of the ligands as found in SF6. Only data for the maxima in the two fields are tabulated for the remaining molecules. Comparative Morphology of L(r) and η(r) Second- and Third-Row Hydrides AHn. The L(r) and η(r) fields for atom A are homeomorphic for all members of the second- and third-row hydrides AHn (Table 2 and Figures 4 and 5). The general forms of the two fields for members of both series with q(H) < -0.7e, Li w B and Na w Si (Table 6), are represented by BH3. Note, as illustrated for BH3 and as also found up to PH3 in the third row, that the interatomic surfaces for these molecules fall on the natural boundaries defined by the L(r) and η(r) fields, a reflection of the high degree of localization of the electrons within the basins of the A atoms in these cases (Table 6). In all cases of an A atom bonded to a hydrogen, the bonded domains in η(r) are localized on the the protons, behavior that is found for L(r) only in those molecules with significant hydridic character, Li w B and Na w P. In the remaining members, L(r) exhibits bonded maxima within the basin of the A atom in addition to the maxima at the positions of the proton that are necessarily present in all molecules. The hydridic members illustrating bonded maxima only on the protons in L(r) are those whose l(H) values are in the range 70-95%, that is, in those cases where the hydridic nature is reflected in a corresponding localization of an electron pair within the basin of the H atom. The bonded domains in η(r) are also significantly larger than those in L(r). In LiH the 0.95 contour almost envelops the whole of the basin of the H atom, closely mimicking the interatomic surface, a reflection of the localization of the Fermi hole illustrated in Figure 1. The same behavior is found for the other ionic members of both series, and the association of high l(H) values for atomic basins

with η(r) > 0.95 demonstrates that electron localization as determined by ELF is a consequence of the localization of the Fermi hole. In methane, ammonia, silane, and phosphine, the four maxima define a tetrahedral atomic graph with six edges and four faces. For both water and hydrogen sulfide, however, the same number of maxima now defines a polyhedron that possesses one less edge, the one linking the bonded domains, and hence one less face in both fields. This same atomic graph is found for the oxygen atom in an ether and in the OH fragment of a carboxylic acid. Sulfur exhibits the same atomic graph in thioethers and, as discussed in the following, in SF2. Aside from maxima associated with the protons in ionic systems, the critical points in L(r), unlike those in η(r), exhibit values and radii characteristic of the atom and the quantum shell in which they occur, with the values of the maxima increasing and their radii decreasing across a row of the periodic table. The values of η(r), on the other hand, provide a measure of the absolute degree of localization associated with each attractor domain. The VSEPR model assigns a larger size to nonbonded than to bonded domains, as is generally observed in the L(r) field. In H2O and H2S, the angle formed with the A nucleus by the two nonbonded charge concentrations in L(r) exceeds the tetrahedral angle, equaling 128° and 121°, respectively, while that between the two bonded charge concentrations is less than this value, equaling 106° and 95°, respectively, all as anticipated on the basis of the VSEPR model. In η(r) the same pairs of angles are 109° and 121° for the nonbonded domains and 106° and 96° for the bonded domains. The two fields are again homeomorphic for the A atom in HF and HCl, where one encounters nonisolated critical points.24 An off-axis critical point in a linear molecule must necessarily belong to a set of nonisolated critical points, i.e., a connected set, which encircles the axis, each member of the set exhibiting a single zero curvature tangent to the ring. Such critical points are thus of rank 2. The radial curvature directed toward the axis is negative, while the remaining one, the one parallel to the axis, may be negative to yield a ring attractor, a connected set of (2,-2) cps, or it may be positive to yield a connected set of (2,0) cps. A (2,0) cp is equivalent to an edge or (3,-1) cp, as it links neighboring attractors. Both HF and HCl possess an axial bonded attractor and a nonbonded ring attractor with an intervening set of (2,0) cps. Gradient paths of the corresponding vector fields originate at both the axial and ring attractors and terminate at the intervening ring of (2,0) cps. The remaining trajectories originating at the ring attractor terminate at an axial (3,+1) cp on the nonbonded side of the nucleus. The nonbonded

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Figure 5. Contour plots of L(r) on the left and η(r) on the right for third-row hydrides: (a) SiH4, (b) PH3, (c) SH2, and (d) ClH. The plots for SH2 are for the plane containing the two nonbonded domains. The bonded domains are localized on the protons in all cases except for L(r) in SH2 and ClH.

Figure 4. Contour plots of L(r) on the left and η(r) on the right for second-row hydrides: (a) BH3, (b) CH4, (c) NH3, (d) OH2, and (e) FH. In η(r) the bonded domains are localized on the protons, while in L(r) this is true only in the ionic and polar cases such as BH3. The upper axial cp in water is the (3,-1) cp linking the two nonbonded domains in the perpendicular plane, a plane illustrated in Figure 5 for SH2. Electron domains in both fields contract in size with increasing nuclear charge on A.

domains in both fields are necessarily ringlike for linear molecules and only slightly depart from this behavior in situations with high axial symmetry. The pronounced nature of the nonbonded ring attractor on Cl is typical and found in Cl2 as well. Its presence imparts a very anisotropic shape to the nonbonded axial hole or (3,+1) cp. It is the interaction of the nonbonded attractor on one Cl with the hole on another that accounts for the dominant intermolecular interaction found in the layered structure of solid chlorine.34

Second- and Third-Row Fluorides AFn. The behavior of the two fields for fluorides AFn is illustrated by the examples given in Table 3 and Figures 6 and 7. The BF3 molecule provides an example of a connectivity not observed in F(r), wherein a pair of attractors is linked by two (3,-1) cps. The three maxima in both fields for this molecule are so linked by three pairs of (3,-1) cps, these being placed symmetrically on each side of the symmetry plane to generate three faces. Both fields possess the same polyhedron with three vertices and six edges, but the set [3,6,6] does not satisfy Euler’s formula because of connectivity to outer structure. The atomic graphs for C and N in their fluorides are regular tetrahedrons for both fields, as found in their hydrides. The bonded maxima in η(r) for the molecules NF3, CF4, and BF3 appear shared between A and F, while for L(r) they appear separate from the valence shell on F, becoming increasingly smaller (N w B) and being entirely absent from the ionic members MgF2 and LiF. In the less electronegative third-row

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TABLE 3: (3,-3) Critical Points in L(r) and η(r) for Fluorides and PCl3a molecule BF3 [3,6,6] CF4 [4,6,4] NF3 [4,6,4] PCl3 [4,6,4] OF2 [2,3,3]L, [4,5,3]η SF2 [4,6,4]L, [4,5,3]η ClF2+ [4,5,3] F2 a

L(r)

r

F(r)

η(r)

r

F(r)

type

0.182 1.078 5.359 1.561 0.417 0.182 10.017

1.141 0.979 0.691 0.842 1.411 1.557 0.607

0.3155 0.3382 0.7515 0.4034 0.1476 0.1274 1.2195

0.8621 0.8824 0.9935 0.8515 0.9969 0.8643 0.9718 0.7732

1.330 1.154 1.333 1.126 2.076 1.979 1.064 1.157

0.4366 0.3824 0.2072 0.3774 0.0634 0.3422 0.4196 0.3710

1.006 0.088 1.724 0.496 -0.154 12.540

1.253 1.396 1.130 1.245 1.256 0.560

0.2489 0.2123 0.3594 0.2768 0.3669 1.6348

0.8178 0.9873 0.7922 0.7101 0.9207

1.864 1.639 1.496 1.256 0.941

0.3946 0.1604 0.2948 0.3669 0.5826

b×3 b×4 n b×3 n b×3 n×2 b×2 n×2 b×2 n×2 b×2 b b (ring)

See footnote a to Table 1.

Figure 6. Contour plots of L(r) and η(r) for second-row fluorides: (a) BF3, (b) CF4, and (c) NF3 and for PCl3 (d).

Figure 7. Contour plots for L(r) and η(r) for second- and third-row fluorides: (a) OF2, (b) SF2, (c) ClF2+, and (d) F2.

atoms, no bonded CCs appear until SF2 (Figure 7). In PF3, η(r) has the set [4,6,4] characteristic of a tetrahedron as found for NF3. L(r) exhibits the same tetrahedral structure, but while the nonbonded CC is quite marked, the bonded maxima now occur in regions where the density is depleted rather than concentrated, and, in addition, no bonded CCs are found on the fluorines. SiF4 exhibits similar behavior, with η(r) having

the set [4,6,4], but in L(r) the four maxima occur in regions where L(r) < 0. Thus, in very polar fluorides as well as in the corresponding hydrides, only nonbonded CCs appear within the VSCC the A atom in L(r). The situation is different for the less polar third-row chlorides, as evidenced by PCl3 where the net charge on P is decreased to +1.8e from +2.5e in the fluoride (Table 7). Both fields exhibit

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TABLE 4: (3,-3) Critical Points in L(r) and η(r) for Single, Double, and Triple Bondsa molecule C2H6 [4,6,4] C2H4 [3,5,4]L, [4,6,4]η C2H2 N2 CO O C H2CO [3,3,2] O [3,5,4] C dimethyl ether [4,5,3] O [4,6,4] C SO2 [3,4,3] SO3 [3,6,3]L, [6,9,5]η

L(r)

r

F(r)

1.034 1.282 1.478 1.405 1.560

0.976 1.010 0.977 0.994 1.115

0.2795 0.3004 0.3769 0.3122 0.4407

1.515 3.963 2.922 5.483 4.901 1.310 5.878 3.794 1.537 1.452 2.706 5.688 0.837 1.457 1.389 0.866 0.743 0.811

0.982 0.824 0.733 0.688 0.649 0.863 0.644 0.710 0.970 0.955 0.717 0.646 1.023 0.987 0.992 1.257 1.322 1.299

0.3255 0.7738 0.5779 0.9811 0.9165 0.3174 0.9741 0.8259 0.3198 0.4826 0.7034 0.9700 0.2805 0.3170 0.3095 0.2283 0.3866 0.3902

r

F(r)

type

0.9685 0.9999 0.9412 0.9999

1.440 2.088 1.332 2.063

0.2496 0.4206 0.3044 0.4224

0.8906 0.9999 0.8854 0.9811 0.8786 0.9379 0.9980 0.9320 0.8801 >0.9500 0.8801 0.9069 0.9256 0.9069 0.9500 0.9500 0.9939 0.8529 0.8348

1.443 2.015 1.008 1.342 1.002 1.148 1.645 1.103 1.104 2.063 1.123 1.331 1.116 1.299 2.040 2.040 1.819 1.532 1.506

0.2250 0.4072 0.7648 0.1656 0.6914 0.2760 0.0834 0.3340 0.5424 0.4387 0.5424 0.3198 0.3270 0.3198 0.4000 0.4000 0.1022 0.4630 0.3772

b to Cb b to H b to Cc b to H b to Cd b (ring) b to H b×1 in η n b to C n n n b b to H b to O b to C n b to O b to H b to H n b×2 b×3 in L b×6 in η

η(r)

a See footnote a to Table 1. b A central maximum in η. c Two central maxima in η. d Single central maximum in L. Ring attractor in η is centered at midpoint.

a [4,6,4] tetrahedral structure for P with the bonded domains in L(r) appearing as distinct CCs, as found for the A nuclei in CF4 and NF3 (Table 3). The atomic graph for oxygen in OF2 for η(r) is isomorphic with that for both fields in H2O, but while the L(r) field appears similar (Figure 7), what appear to be the two bonded charge concentrations are actually (3,-1) cps. This behavior for singly bonded oxygen is found for ligands more electronegative than C in the second row and for S and Cl in the third. In SF2 there are four attractors in both fields, and that for η(r) is homeomorphic with that found for S in SH2 with three faces while that for L(r) is a regular tetrahedron. The two fields are homeomorphic for the molecule ClF2+, and the pattern is as found for the oxygen and sulfur nuclei in the hydrides and for sulfur in SF2. The two fields exhibit the same basic structure in F2; both possess a central bonded maximum and a ring of nonbonded attractors (Figure 7). In η(r) the bonded and nonbonded attractors are linked by a (2,0) ring and in L(r) by a corresponding axial (3,-1) cp. Both fields exhibit an axial (3,+1) cp, which serves as the terminus for the trajectories from the nonbonded ring of attractors. The central maximum in F2 is atypical. It is found in a region where L(r) is negative and it is not a CC. However, a very large basis set does yield a negative Laplacian at the bcp in F2.34 The l(F) values in Table 7 show that, regardless of its net charge, the electron density on a fluorine atom not only is tightly bound but is physically localized within its own basin. While this leads to stability in ionic fluorides, the lack of exchange with the electrons in a neighboring atomic basin results in weak shared interactions, as found in F2. The separate accumulation of density in each of the atomic basins in F2 is reflected in ∇2Fb being close to zero (it is slightly negative with the use of a very large basis set), with the final distribution being dominated by the positive curvature and resulting stress parallel to the bond path. While the Hartree-Fock description of F2 does not account for its binding, the properties of the electron density are relatively insensitive to any added correlation.

Single, Double, and Triple Bonds. The ethane, ethene, and ethyne molecules provide a comparison of the two fields for single, double, and triple bonds (Table 4, Figure 8). The bonded maxima associated with the protons in L(r) in these molecules are, like those in the covalent and polar hydrides, present as separate charge concentrations in the VSCC of the carbon atom, while in η(r) the corresponding bonded domains are localized on the protons. A carbon atom of ethane has the characteristic set of a regular tetrahedron [4,6,4] in both fields, with the difference that in L(r) each carbon has four bonded maxima while in η(r), in addition to the three domains associated with the protons, only a single bonded domain midway between the carbons is shared by both atoms (Figure 8). The [4,6,4] set is characteristic of a saturated carbon for both L(r) and η(r) being found in methane, ethane, and also dimethyl ether (Table 4). In the latter molecule, one again finds separate bonded maxima on the carbon and the oxygen in L(r), while in η(r), a single maximum is shared between them. With this understanding, then, the two fields are also homeomorphic for the oxygen atom, which has the same characteristic set found in H2O. In ethene each carbon exhibits a bonded domain within its own VSCC for the C-C link in L(r), while in η(r) the bonded domains, two in number, are located midway between the carbon atoms, one on each side of the molecular plane. Thus, η(r) indicates the presence of a carbon-carbon double bond by defining a corresponding number of bonded domains,46 but this is not a general result for double bonds between other atoms. This behavior is not observed in L(r), which, like F(r), indicates the presence of a double bond through an increase in the value of L(r) or F(r) at the critical point and by exhibiting marked ellipticity.51 In ethene the value of L(r) at the charge concentration for the C-C interaction is 1.5 compared to 1.0 au in ethane, while the two curvatures of L(r) at the critical point perpendicular to the C-C axis, which are equal in ethane, differ by a factor of 2, the lesser one being directed perpendicular to the molecular plane leading to an enhanced concentration of electronic charge in the “π” plane as evident in Figure 8.

The Electron Pair

Figure 8. Contour plots for L(r) and η(r) for (a) ethane, (b) ethene in the plane of the nuclei, (c) ethene in the plane perpendicular to (b) and containing the C-C axis, and (d) ethyne. The central contour in η(r) in plot c of value 0.9 envelops two off-axis maxima of value 0.94 (Table 4). Plot d for η(r) shows a cross section of the ring attractor.

The field for η(r) is readily derived from that for L(r) in ethene by replacing the bonded domain for the C-C interaction in the atomic graph for L(r) by a linked pair of maxima in η(r), thereby accounting for their characteristic sets differing by one (3,-3) and one (3,-1). The remainder of the atomic graphs is identical with a split pair of (3,-1) cps, as found in BH3, linking the bonded maxima to carbon with those associated with the protons. Of particular importance is the presence of two (3,+1) cps situated on each side of the molecular plane above and below each carbon atom. Their presence as “holes” in the VSCC of carbon, along with their relative size, has been used to account for the point and ease of nucleophilic attack in a series of substituted ethenes.52 In ethyne a single symmetrically placed bonded charge concentration links the two carbon atoms in L(r), while in η(r) this is replaced by a ring attractor,46 a connected set of (2,-2) cps. Both fields have a bonded domain associated with each proton, and the central attractor is linked to these by a ring of connected (2,0) cps whose presence of also clear in Figure 8. The charge concentrations associated with the C-C interactions

J. Phys. Chem., Vol. 100, No. 38, 1996 15409

Figure 9. Contour maps for L(r) and η(r) for (a) N2, (b) CO, (c) H2CO in the plane of the nuclei, and (d) in the perpendicular plane containing the C-O axis. Note the pronounced nonbonded domain on C in CO evident in both fields in (b) and the very marked “holes” or (3,+1) cps in the VSCC of the carbonyl carbon in (d).

in L(r) increase in the order ethane < ethene < ethyne, with the electron localization domains of η(r) exhibiting the reverse order. The isoelectronic molecules N2 and CO are also triply bonded within the Lewis model, but neither exhibits a ring attractor as found in ethyne. In N2 the two fields are homeomorphic, both exhibiting an axial nonbonded domain on each atom that is linked via a ring of (2,0) cps to a central axial bonded domain (Figure 9). The fields are again homeomorphic in CO, with the oxygen atom exhibiting the same structure as N in N2. The carbon atom exhibits a very pronounced axial nonbonded domain in both fields that is linked to the central bonded domain by a ring of (2,0) cps. In the H2CO molecule, η(r) exhibits only a single axial bonded domain between the C and O atoms, as opposed to the two found in L(r). If the single domain in η(r) is considered shared by both atoms, the two fields are homeomorphic for the C and O atoms. Only axial cps are found for the formally doubly bonded CsO interaction. The atomic graph for carbon in this molecule is homeomorphic with that for the L(r) field

15410 J. Phys. Chem., Vol. 100, No. 38, 1996

Figure 10. Contour maps of L(r) and η(r) for (a) SO2, (b) SO3 in the plane of the nuclei, and (c) SO3 in a perpendicular plane containing an S-O axis. The central bonded contour of value 0.8 for η(r) in (c) encompasses two off-axis maxima of value 0.83 (Table 4).

for carbon in ethene. Most importantly, as noted previously, L(r) for the carbonyl carbon exhibits very pronounced “holes” or (3,+1) cps on each side of the molecular plane, which, as for the C atoms in ethene, are the sites of nucleophilic attack.6 The holes are larger than in ethene (Figure 9), and their critical points form an obtuse angle with the CsO bond axis and in L(r) it equals 110°, the angle found by Burgi and Dunitz53 to be the most favored angle of approach of a nucleophile to a carbonyl carbon as determined by crystallographic data. The carbon atom of the CdO group in a carboxylic acid exhibits the same atomic graphs as found in formaldehyde and is characteristic of the carbonyl carbon. The two fields are homeomorphic for the oxygen atom in the carbonyl group (Table 4). It exhibits one bonded domain, as it does in carbon monoxide, but the single axial nonbonded domain found in CO is replaced by two pronounced in-plane nonbonded domains. It is observed that when CO acts as a terminal ligand in transition metal complexes, the oxygen VSCC possesses a single axial nonbonded CC as in isolated CO, but when acting as a bridging ligand, its nonbonded pattern is that found in the carbonyl functional group.54 In a Lewis structure for SO2 that satisfies the octet rule for both atoms, the average bond order linking them is 1.5, which increases to 2.0 for a valence shell expanded to 10 on S. The two fields are homeomorphic for S in this molecule and they exhibit a single bonded maximum directed at each oxygen (Table 4, Figure 10). This is shared with the oxygen in η(r), while in L(r) a separate bonded maximum is located in the VSCC of each oxygen. Each bonded maximum of S is, however, connected to the nonbonded one by (3,-1) cps

Bader et al. symmetrically placed on each side of the plane of the nuclei in the manner found for the carbon in the carbonyl group. The double-bond character is also reflected in the significant ellipticities of 1.4(S) and 3.1(O) for the bonded CCs, with the value for the shared bonded maximum in η(r) being 2.3. The oxygen exhibits two nonbonded maxima in both fields. In SO3 the corresponding Lewis structure predicts a bond order of 1.33, increasing to 1.67 or 2.0 by allowing for an expanded valence shell of 10 or 12 on S. The S-O link in L(r) for this molecule has a single CC in each VSCC, as found in SO2. η(r), however, exhibits two bonded maxima symmetrically placed on each side of the plane of the nuclei, as found for the C-C link in ethene, and because of this the two fields are not homeomorphic. The bonded CCs in L(r) again exhibit significant ellipticities of 2.2 on S and 2.5 on O (Figure 10), and the values of L(r) at the bonded maxima on S exceed those found in SO2 (Table 4). The greater double-bond character found in SO3 than in SO2 is reflected in the larger ellipticity found for F(r) at the bond cp in the former: 0.261 compared to 0.155. In summary, in some molecules such as ethene and SO3, a double bond is represented in η(r) in the form of two shared bonded domains, while in others, such as the carbonyl group and SO2, η(r) exhibits a single bonded domain. Similarly, in ethyne, the triple bond is represented by a ring of (2,-2) attractors, while in N2 and CO only axial bonded and nonbonded attractors are present. Double and triple bonds are represented by axial cps in L(r), one on each atom in most cases, and by a single shared maximum in others, as in ethyne and CO. A single bond is, in general, represented by a single bonded domain in η(r), which appears to be shared by both atoms except when bonded to hydrogen, in which case the bonded domain is localized about the proton. In L(r), a single bond is represented by a maximum in the VSCC of each atom in other than hydrides and ionic or very polar species where a single maximum appears in the VSCC of the electronegative atom. The cps in L(r) exhibit radii characteristic of the corresponding valence shell of the atom. Molecules with Five Electron Pair Domains. The two fields appear to be homeomorphic for the Cl atom in ClF3 with the characteristic set [5,7,4]. The T-shaped geometry of this molecule is a consequence of the two large nonbonded domains, which, along with the equatorial bonded domain, are evident in both fields in Figure 11a. By occupying the less crowded equatorial positions, the angle formed between the two nonbonded cps can expand to a value well in excess of 120°, equaling 147° in L(r) and 144° in η(r). The three bonded and two nonbonded domains in L(r) exhibit the relative angular positions and sizes anticipated on the basis of VSEPR, with the axial ones being the smallest (Table 1). In spite of the fact that both fields exhibit the number of edge and face cps requiring five vertices, MORPHY is unable to locate the two axial bonded domains in η(r). The plots for both fields in the plane of the three ligands are very similar in the regions of the Cl-F bonds to the corresponding plots for ClF5 in Figure 2, with the roles of the equatorial and axial ligands reversed. The absence of the axial domains in η(r) for ClF3 is a mathematical anomaly. The remaining two examples, SOF4 and SF4 (Figure 11, Table 1), are molecules that, according to VSEPR theory, should possess five electron pair domains as does ClF3, with the one associated with the S-O linkage in SOF4 being large and asymmetric since it represents two electron pairs. This is indeed the case for L(r) in SOF4, but for η(r) the domain for the S-O linkage, as it is in SO3, is bifurcated in two symmetrically placed about the C2 axis in the plane containing the equatorial ligands.

The Electron Pair

J. Phys. Chem., Vol. 100, No. 38, 1996 15411 5). These geometrical angles can be considered consequences of the differences in the two related interdomain angles: b(O)S-b(e) ) 125° and b(O)-S-b(a) ) 98.3°. There are secondary differences between the two fields in the manner in which the bonded domains are linked to one another. The SF4 molecule has a single nonbonded domain in place of the oxygen in SOF4. Both L(r) and η(r) do indeed exhibit one large nonbonded domain spread out in the equatorial plane and four bonded domains. The nonbonded domain is linked to the four bonded domains in both fields, and the fields would be homeomorphic but for the appearance of an axial nonbonded pseudomaximum in a region where L(r) is negative and the density low in value. It does not appear as a maximum in the diagrams for L(r), and it possesses one near-vanishing curvature. It and the four (3,-1) cps linking it to the fluorines are readily transformed by a bifurcation catastrophe into five corresponding (3,-1) cps linking the fluorines and the two associated ring cps found in the η(r) field. The structure derived from these five cps in L(r) is transformed into the structure defined by the seven cps in η(r) by the bifurcation of the central (3,-3) cp into a (3,-1) and two ring cps. Thus, the [5,9,6] set of η(r) is derivable from the [6,8,4] set of L(r) by a bifurcation catastrophe involving an unstable cp in a region of relatively low density. Transition Metal Complexes. The Lewis model is not well defined for heavy metal atoms: those from the third row onward and the VSEPR model can fail.26 The topology of L(r) has recently been used to determine the relative positions and sizes of the electron domains in numerous examples of such molecules, using large basis sets and electron correlation where needed, as described in the original references.55,56 In all cases, except for hydrides, q(M) > +2.0 and the outer ns electrons of the metal atom M are transferred to the ligands, and one finds the outer shell of the core of M containing the (n - 1)d electrons to be significantly distorted. The distortions, which remarkably are apparent in contours of the electron density with values as high as 2.0 au,56 are made most evident in the terms of the local CCs present in the Laplacian distribution of the outer shell of the core of M, distortions that are already evident in the alkaline earth dihydrides and dihalides beginning with Ca. The pattern of electron localization determined in this manner is indeed different from that observed for the main group elements, in that the distortions are such as to produce electron domains that are opposed to rather shared with or directed at the ligands. This behavior is found even in molecules that apparently follow the VSEPR rules, such as the trigonal bipyramidal geometry found for VF5. While one cannot distinguish between bonded or opposed CCs for the axial ligands in this molecule, the three equatorial CCs are all ligand opposed. VSEPR apparently fails for VH5 and V(CH3)5 in that calculations predict these molecules to possess square-based pyramidal geometries,57 and one finds the electron domains to be larger

Figure 11. Contour maps of L(r) and η(r) for ClF3 in the equatorial plane containing the two nonbonded domains (a), for SOF4 in the plane of the axial fluorines (b), for SOF4 in the plane of the equatorial fluorines (c), and for SF4 in the plane of the equatorial fluorines (d). The bonded contour shared with oxygen in η(r) of plot c encompasses two maxima of value 0.84 (Table 1). This maximum exhibits considerable ellipticity in both fields, as does the corresponding nonbonded domain on S in SF4 shown in (d), as described in the text.

In L(r) the corresponding single S-O domain is the largest of the five bonded domains and exhibits a large ellipticity with the charge concentration spread out toward the equatorial ligands, a picture entirely consistent with η(r) and one that accounts for the angle formed by O with the axial ligands being considerably smaller than that with the equatorial ligands (Table

TABLE 5: Bond Critical Point and Atomic Data for Molecules with Five and Six Electron Domainsa molecule ClF5, E ) -956.345 74, ∠FeClFa ) 85.5 ClF3, E ) -757.626 88, ∠FaClFe ) 86.9 SF6, E ) -994.229 80 SF4, E ) -795.314 32, ∠FaSFa ) 171.0, ∠FeSFe ) 102.2 SOF4, E ) -870.171 39, ∠OSFa ) 97.2, ∠OSFe ) 124.2

a

bond

RA-B

Fb

∇2Fb

Hb

q(A)

q(B)

l(A)

l(B)

Cl-Fa Cl-Fe Cl-Fa Cl-Fe S-F S-Fa S-Fe S-O S-Fa S-Fe

2.962 3.076 3.173 2.955 2.905 3.087 2.871 2.621 2.971 2.855

0.2716 0.2264 0.1859 0.2519 0.2341 0.1821 0.2350 0.3635 0.2127 0.2427

-0.5154 -0.1303 +0.0652 -0.3802 +0.2333 -0.0155 +0.3458 +0.9570 +0.1552 +0.3998

-0.283 -0.185 -0.127 -0.274 -0.288 -0.182 -0.278 -0.550 -0.248 -0.292

+2.526 +2.526 +1.629 +1.629 +4.389 +2.914 +2.914 +4.313 +4.313 +4.313

-0.420 -0.527 -0.595 -0.440 -0.732 -0.722 -0.736 -1.395 -0.736 -0.723

0.86 0.86 0.91 0.91 0.88 0.90 0.90 0.88 0.88 0.88

0.93 0.94 0.95 0.93 0.94 0.94 0.95 0.92 0.95 0.94

Bond angles in degrees; all other quantities in atomic units except for l(Ω) which is dimensionless. The 6-311++G(2d,2p) basis was used for geometry optimization and determination of the electron density using GAUSSIAN92.21 See Appendix for explanation of symbols.

15412 J. Phys. Chem., Vol. 100, No. 38, 1996

Bader et al.

TABLE 6: Bond Critical Point and Atomic Data for Hydridesa AHn

energy

RA-H ∠HAH

Fb

∇2Fb

Hb

q(A)

q(H)

l(A)

l(H)

LiH BeH2 BH3 CH4 NH3

-7.986 16 -15.771 36 -26.399 18 -40.212 32 -56.218 58

0.0398 0.0978 0.1864 0.2894 0.3604

+0.156 +0.189 -0.258 -1.124 -1.889

-0.001 -0.050 -0.210 -0.326 -0.536

+0.912 +1.729 +2.112 +0.175 -1.047

-0.912 -0.865 -0.704 -0.044 +0.349

0.95 0.88 0.74 0.66 0.83

0.95 0.90 0.77 0.47 0.29

OH2

-76.057 43

3.307 2.515 2.243 2.044 1.886 107.9 1.776 106.3 1.696 3.622 3.225 2.981 2.785 2.656 95.4 2.506 94.1 2.390

0.3964

-3.106

-0.853

-1.254

+0.627

0.93

0.16

0.4043 0.0246 0.0545 0.0838 0.1220 0.1651

-4.006 +0.088 +0.220 +0.251 +0.179 -0.107

-1.091 +0.003 -0.006 -0.030 -0.082 -0.164

-0.779 +0.816 +1.614 +2.361 +2.895 +1.694

+0.779 -0.816 -0.807 -0.787 -0.724 -0.565

0.98 0.98 0.96 0.94 0.91 0.91

0.10 0.90 0.89 0.84 0.77 0.65

0.2210

-0.657

-0.228

+0.284

-0.142

0.93

0.45

0.2571

-0.787

0.254

-0.222

+0.222

0.97

0.33

FH NaH MgH2 AlH3 SiH4 PH3

-100.056 47 -162.380 56 -200.731 93 -243.640 96 -291.257 25 -342.481 92

SH2

-398.706 38

HCl

-460.098 80

a

See footnote a to Table 5.

TABLE 7: Bond Critical Point and Atomic Data for Fluorides and PCl3a AFn

energy

RA-H ∠HAH

Fb

∇2Fb

Hb

q(A)

q(H)

l(A)

l(H)

LiF BeF2 BF3 CF4 NF3

-106.978 94 -213.760 38 -323.308 00 -435.787 77 -352.661 26

0.0763 0.1498 0.2917 0.3186 0.3773

+0.768 +1.480 +0.443 -0.153 -0.859

+0.024 -0.014 -0.241 -0.528 -0.453

+0.940 +1.807 +2.576 +2.964 +1.086

-0.940 -0.904 -0.859 -0.741 -0.362

0.96 0.90 0.83 0.70 0.74

0.99 0.99 0.97 0.95 0.93

OF2

-273.550 44

0.3697

-0.079

-0.318

+0.332

-0.166

0.84

0.93

F2 NaF MgF2 AlF3 SiF4 PF3

-198.747 55 -261.356 86 -398.691 03 -540.580 79 -687.119 44 -639.264 51

0.3669 0.0518 0.0834 0.1215 0.1628 0.1793

+0.154 +0.453 +0.859 +1.210 +1.351 +0.936

-0.306 +0.015 +0.021 +0.003 -0.054 -0.122

0.000 +0.941 +1.826 +2.647 +3.428 +2.510

0.000 -0.941 -0.913 -0.882 -0.857 -0.837

0.93 0.99 0.98 0.96 0.94 0.93

0.93 0.99 0.99 0.98 0.97 0.96

SF2

-596.412 71

2.953 2.574 2.444 2.447 2.496 102.9 2.526 103.5 2.512 3.647 3.279 3.050 2.912 2.922 97.3 2.970 97.2 2.882 100.2 3.889 100.4

0.1952

+0.344

-0.201

+1.427

-0.714

0.94

0.95

0.2768

-0.496

-0.344

+1.764

-0.382

0.92

0.93

0.1270

-0.164

-0.106

+1.776

-0.592

0.90

0.97

ClF2

-657.853 56

PCl3

-1719.342 60

+

a

See footnote a to Table 5.

TABLE 8: Bond Critical Point and Atomic Data for Single, Double, and Triple Bondsa molecule C2H6, E ) -79.257 33, ∠CCH ) 111.2 C2H4, E ) -78.061 42, ∠CCH ) 121.6 C2H2, E ) -76.846 95 N2, E ) -108.981 24 CO, E ) -112.776 38 H2CO, E ) -113.908 29, ∠HCO ) 121.8 (CH3)2O, E ) -154.120 20, ∠HCO ) 107.8,b ∠HCO ) 111.4c SO2, E ) -547.271 32, ∠OSO ) 118.8 SO3, E ) -622.118 46 a

bond

RA-B

Fb

∇2Fb

Hb

q(A)

q(B)

l(A)

l(B)

C-C C-H C-C C-H C-C C-H N-N C-O C-O C-H C-O C-Hb C-Hc S-O S-O

2.881 2.048 2.485 2.029 2.229 1.992 2.015 2.085 2.227 2.063 2.628 2.040 2.054 2.644 2.625

0.2496 0.2908 0.3710 0.3000 0.4407 0.3080 0.7649 0.5332 0.4475 0.3003 0.2730 0.3029 0.2964 0.3343 0.3492

-0.614 -1.132 -1.342 -1.216 -1.560 -1.316 -3.766 +0.739 +0.139 -1.227 -0.343 -1.232 -1.178 +1.205 +1.041

-0.205 -0.329 -0.475 -0.343 -0.709 -0.360 -1.600 -1.041 -0.833 -0.335 -0.426 -0.345 -0.333 -0.461 -0.507

+0.183 +0.183 +0.035 +0.035 -0.136 -0.136 0.000 +1.357 +1.292 +1.292 +0.776 +0.776 +0.776 +2.712 +4.105

+0.183 -0.061 +0.035 -0.017 -0.136 +0.136 0.000 -1.357 -1.271 -0.010 -1.288 -0.022 -0.054 -1.356 -1.368

0.65 0.65 0.67 0.67 0.68 0.68 0.78 0.89 0.66 0.66 0.65 0.65 0.65 0.90 0.87

0.65 0.47 0.67 0.45 0.68 0.39 0.78 0.92 0.91 0.47 0.89 0.47 0.48 0.91 0.92

See footnote a to Table 5. b H in plane of C-O-C. c H out of plane.

and more diffuse in these cases than those found in VF5.56 The ligand-opposed nature of the CCs in the outer core, as defined by L(r) for the V atom, is displayed in Figure 12 for VH5 where q(V) ) +1.7e. Also shown is a corresponding display of η(r) for this molecule, and the atomic graphs for V are clearly homeomorphic in terms of the electron domains that they define.

The same homeomorphism is evident in corresponding displays for CrOF4, q(Cr) ) +2.9e, with the large ligand-opposed electron domain generated by the presence of the oxygen atom being evident in both fields. Molecules with nonbonded CCs behave similarly, as illustrated in the previously given display55 of the four tetrahedrally arranged CCs present in the outer core

The Electron Pair

Figure 12. Envelope maps for the minimum-energy geometries: VH5 L(r) (a) and η(r) (b); CrOF4 L(r) (c) and η(r) (d). The function values are 0.15 au in (a), 0.85 in (b), 21 au in (c), and 0.83 in (d). The corresponding geometries for the same orientation of the nuclei indicated in the insets make clear the ligand-opposed nature of the electron domains in the outer shell of the cores of the transition metal atoms. Note that the domain induced by and opposed to the oxygen in CrOF4 is larger in both fields than are those induced by the fluorines.

of Ba in BaH2 with η(r) exhibiting a similar pattern of two nonbonded and two ligand-opposed CCs. The closed-shell core of Zn in ZnCl42-, on the other hand, is more nearly spherical, and the four maxima that are present in L(r) are interior to the outer limits of the Zn atomic surface, are of limited spatial extent, and are tetrahedrally disposed in the manner of bonded CCs found in main group elements.58 This parallel behavior of the two fields for the transition metal atoms is general and firmly establishes that the Lewis model, as implemented in terms of spatially localized electron pair domains, is different for atoms with vacancies in the (n - 1)d shell from that found for main group elements. Summary. An overview of the figures comparing the L(r) and η(r) fields convinces one of the essential similarity in their structures. A wide spectrum of atomic interactions is represented, and, in nearly all cases, one finds a homemorphism in the number of electron pair domains that they define for a central atom interacting with a set of ligands. In general, the atomic graph of L(r) is more characteristic of a given atom than is that for η(r). In ionic and very polar interactions, both fields in general exhibit a single bonded domain on each ligand, this behavior being obtained for the η(r) field in all hydrides. In shared interactions, a single bonded domain of η(r) is, in general, replaced by a pair of domains in L(r), one in each VSCC of the bonded atoms. Multiple bonds in L(r) are represented by single domains, the multiplicity of bonding being reflected in their magnitudes and ellipticities, similar to the topology exhibited by F(r). A double bond in η(r) can be represented by two domains and a triple bond in a linear system by a ring attractor, but this behavior is not general. The homeomorphism exhibited by the two fields for transition metal molecules demonstrates the presence of a new pattern for the spatial pairing of electrons in the form of ligand-opposed domains in the outer shell of the metal atom core. By accepting the presence of the doubling of bonded domains in some instances and the replacement of a shared domain by

J. Phys. Chem., Vol. 100, No. 38, 1996 15413 a ring attractor in another, the two fields offer a consistent view of the regions in a molecule that are dominated by the presence of electron pairing and their angular distribution in the vicinity of a single atom, as envisaged in the VSEPR model or as found for transition metal atoms. In general, the radial distances at which the localization domains are found are larger in η(r) than in L(r). The cps in L(r) are determined entirely by the properties of the electron distribution. The agreement between L(r) and η(r) in their predicted angular distribution of the localization domains would suggest that the element of arbitrariness introduced into the definition of η(r)9 by measuring the smallness of ∆ relative to that for a uniform electron gas, while affecting the radial distances, does not affect the number or angular orientation of the domains. From the examples given previously and here, it appears that L(r) is superior to η(r) in predicting the location and magnitude of the “lumps” and “holes” created in real space by the partial pairing of electrons, which underlies the Lewis model of acidbase reactivity. The Laplacian of the electron density is the only property of F(r) directly related to an energy density. It provides, through eq 4, a measure of the local potential and kinetic energy densities of the electrons, thereby yielding the physical basis underlying an acid-base reaction: a base site is a spatial region exhibiting an excess of potential energy in which the extent of electron pairing is a maximum whereas an acid is one exhibiting an excess of kinetic energy for which electron pairing is a minimum in the quantum shells of interest. The single electron domain shared by two bonded atoms that is exhibited by η(r), on the other hand, is more useful for comparing the bonding through a series of similar compounds a series of substituted X-O-X molecules, for example. That the homeomorphism is not complete means that the two fields complement one another. L(r), in general, reflects the VSEPR model of associating a set of electron domains with a central atom, while η(r) generally exhibits a single shared domain between two bonded atoms. Thus, L(r) associates a bonded CC in the VSCC of each carbon in a C-C bond, while η(r) yields a single shared domain (Figure 8). Clearly, one cannot in such cases associate a pair of electrons with every CC in L(r), but must instead do so for the linked pair of CCs. What is represented in η(r) as a single bonded domain appears as a bifurcated set of CCs in L(r) linked by a ridge of charge concentration (Figure 8). There are counterexamples. In ClF3 the equatorial fluorines in η(r) exhibit a bonded maximum in addition to the one found on Cl, while L(r) exhibits only the CC within the VSCC of Cl. Neither field yields an electron count in terms of the number of local maxima it exhibits. The description provided by L(r) is, however, nonarbitrary and indicates that the pairing phenomenon associated with bonding extends well into the valence shell of each atom, as indicated by the radial values listed in tables. Conclusions The pairing of electrons is a consequence of the spatial localization of an electron of a given spin, as determined by a corresponding localization of its Fermi hole.4 While the quantitative measures of electron localization made possible by this understanding show that the pairing of electrons is less pronounced than envisaged in the original Lewis model, there is evidence of partial condensation of the pair density to yield spatial domains with pair populations approaching unity.4,8 Further evidence is provided by the success of the VSEPR model in predicting the geometry of main group elements by using arguments of electron pairing based upon the operation of the exclusion principle.26 The recovery of this model in terms of

15414 J. Phys. Chem., Vol. 100, No. 38, 1996

Bader et al.

the topology of the Laplacian of the electron density is symbiotic: it provides a physical basis for the VSEPR model and, because of the success of this model, it establishes an empirical correspondence between the localized charge concentrations defined by L(r) with the electron domains assumed in the VSEPR model. Thus, it establishes a connection between L(r) and the pair density. The measure of electron localization provided by Becke and Edgecombe’s9 ELF, the function η(r), is directly related to the pair density. The function ∆ appearing in η(r), the difference between the many- and one-electron kinetic energy densities, provides a measure of the conditional pair probability for an electron of given spin. Electron localization is related to the smallness of ∆,9 a condition which, as previously demonstrated,42 is once again a direct result of the localization of the Fermi hole. Thus, the general homeomorphism between the η(r) and L(r) fields demonstrated here establishes a physical link between the protrayal of electron localization in terms of the charge concentrations of L(r) and the associated properties of the pair density. By extension, it provides justification for the qualitative arguments used to rationalize the VSEPR model of molecular geometry. Acknowledgment. We thank Professor R. J. Gillespie for useful discussions concerning this paper. Appendix Definition of the Fermi Hole Density. The motions of the same-spin electrons are correlated, and the pair density written for the R-spin electrons obtained from a single-determinant wave function is

FRR(r1,r2) ) 1/2FR(r1){FR(r2) + hR(r1,r2)}

(A1)

where hR(r1,r2) is the density of the Fermi hole for an R-spin electron. It is the negative of Slater’s exchange charge density60 and expressed in terms of the spin orbitals φi is R

R

i

j

R hR(r1,r2) ) -∑∑{φ* i(r1)φi(r2)φ* j(r2)φj(r1)}/F (r1) (A2)

Unlike the pair density for electrons of opposite spin in eq 1, the same-spin pair density in eq A1 is decreased from the simple product of single particle densities in a manner described by the term in curly brackets. This term, δR(r2) in eq 2 of the text and displayed in Figure 1, is usually interpreted as a conditional probability.3,9 It has two important properties: (a) It equals zero for r1 ) r2 because hR(r1,r1) equals -FR(r1), corresponding to the complete removal of same-spin density from the position of the reference electron. The density of the Fermi hole may approach -FR(r1) for values of r1 * r2, and the conditional probability will then approach zero over the region of space where this near-equality is obtained, leading to a correspondingly near-exclusion of all other same-spin electrons, as illustrated in Figure 1a,c. (b) The integral of the conditional probability over r2 equals NR - 1, the integration of the Fermi hole density resulting in the removal of one R-spin electron, reducing their total number NR by 1. In this manner, the product of single particle densities is corrected for the self-pairing (and self-repulsion) of the electrons to yield 1/2NR(NR - 1) pairs. A bound electron cannot be localized, and r1 and r2 cannot be strictly interpreted as electronic coordinates. Interpretation of the vanishing of the conditional probability for r1 ) r2

requires that the charge of the reference electron at r1 be equal to FR(r1) dr1 rather than to δ(r - r1) dr1, as it would be if indeed the whole electronic charge was localized at r1. Thus, r1 and r2 denote the coordinates of two points in a six-dimensional space, and the term in curly brackets describes the manner in which the remaining charge of an electron with density FR(r1) at r1 is spread out in the space described by r2, decreasing the same-spin density at each point in this space by an amount determined by the Fermi hole. The Fermi correlation is so demanding and so pronounced that it is well described at the single-determinant approximation to the wave function in Hartree-Fock theory, the level of theory used here. All of the ideas, however, apply to the exact wave function as well. Tables of Atomic and Bond Critical Point Data. Tables 5-8 compile the energy, geometrical parameters, bond critical point data, atomic populations q(A), and electron localizations l(A). Quantities subscripted with a b refer to their value at the bond critical point. Hb is the energy density at the bond critical point61 and is equal to the sum of the kinetic and potential energy densities, G(r) and ν(r), appearing in the expression for the local virial theorem (eq 4). According to this same theorem, Hb(r) is equal to -K(r), the alternative expression for the kinetic energy density.8 Unlike the sign of ∇2Fb, which is determined by the local virial expression in eq 4 wherein the potential energy is compared with twice the kinetic contribution, Hb is determined by the energy density itself and is thus negative for all interactions that result from the accumulation of density at the bond cp. The bond path has been given added physical significance with the demonstration of a structural homeomorphism between the topology of F(r) and that of the virial field ν(r), the potential energy density appearing in the local virial theorem eq 4. This homeomorphism means that a molecular graph defining a molecualr structure is mirrored by a corresponding virial graph and the lines of maximum density linking bonded nuclei in F(r) are matched by a set of lines of maximally negative potential energy density in ν(r).62 References and Notes (1) Lewis, G. N. J. Am. Chem. Soc. 1916, 38, 762. (2) If one could increase the spin angular momentum quantum number for an electron from S ) 1/2 to S ) 3/2, or in a universe where this was so, there would be four observable spin components. One could construct an antisymmetrized orbital wave function with up to four electrons occupying each space orbital, allowing for the possibility of observing the spatial quadrupling of electrons. The rows of the periodic table would be correspondingly longer and Be would be the first of the inert gases. (3) McWeeny, R. ReV. Mod. Phys. 1960, 32, 335. McWeeny, R.; Sutcliffe, B. T. Methods of Molecular Quantum Mechanics; Academic Press: London, 1969; pp 100. (4) Bader, R. F. W.; Stephens, M. E. J. Am. Chem. Soc. 1975, 97, 7391. (5) Bader, R. F. W.; Streitwieser, A.; Neuhaus, A.; Laidig, K. E.; Speers, P. J. Am. Chem. Soc. 1996, 118, 4959. (6) Bader, R. F. W.; MacDougall, P. J.; Lau, C. D. H. J. Am. Chem. Soc. 1984, 106, 1594. (7) Bader, R. F. W.; Gillespie, R. J.; MacDougall, P. J. J. Am. Chem. Soc. 1988, 110, 7329. (8) Bader, R. F. W. Atoms in MoleculessA Quantum Theory; Oxford University Press: Oxford, UK, 1990. (9) Becke, A. D.; Edgecombe, K. E. J. Chem. Phys. 1990, 92, 5397. (10) Becke, A. D. Int. J. Quant. Chem. 1983, 23, 1915. (11) Maslen, C. W. Proc. Phys. Soc. (London) 1956, A69, 734. (12) Luken, W. L. Croat. Chim. Acta 1984, 57, 1283. Luken, W. L.; Beratan, D. N. Theor. Chim. Acta 1982, 61, 265. Luken, W. L.; Culberson, J. C. Int. J. Chem.: Symp. 1982, 16, 265. (13) Daudel, R. C. R. Acad. Sci. (Paris) 1953, 237, 601. (14) Daudel, R.; Brion, H.; Odiot, S. J. Chem. Phys. 1955, 23, 2080. Odiot, S. Cahiers Phys. 1957, 81, 1. (15) Aslangul, C.; Constanciel, R.; Daudel, R.; Kottis, P. AdV. Quant. Chem. 1972, 6, 93. (16) Aslangul, C. C. R. Acad. Sci. (Paris) 1971, 272B, 1.

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