Hole Fermi Research Articles

New perspectives of the weighted spin-density approximation: gradient corrections and the valence-only approach

The current state of the weighted spin-density approximation (WSDA) of density functional theory is discussed and new perspective WSDA approaches are put forward. A gradient correction to the exchange pair-correlation function Gxσhom(r12) of the homogeneous electron gas is developed in order to improve its performance for inhomogeneous systems within the WSDA. The correction factor has the form of a Padé-type approximant of the dimensionless gradient |∇nσ(r→)|/nσ4/3(r→) (nσ is the electron spin-density) and of the dimensionless argument k̃Fσ(r→1)r12(k̃Fσ is the weighted spin-density Fermi momentum and r12 is the interelectronic distance). In the zero-gradient limit the corrected pair-correlation function reduces to Gxσhom, while for atoms the present WSDA with gradient correction (WSDA-GC) provides a substantial improvement of the calculated exchange energies compared to the standard WSDA based on Gxσhom. A proposal for the application of WSDA to valence-only calculations is considered and a practical scheme of the valence-only WSDA-GC is developed. Within this scheme, an atomic core is represented with the exchange pseudopotential and an empirical function giving the partial charge of the Fermi hole attributed to this core. The proper choice of the latter function is expected to correct the overestimation of the core–valence exchange produced by the standard WSDA.

Topology of electron correlation in the b [formula omitted] state of the H 2 molecule

Except for the united atom limit, the correlation cage of the b 3 Σ + u state of H 2 is isomorphic to that of the ground state. However, encompassing a genuine cage point rather than a cusp, it is associated primarily with the Fermi hole. Although the spatial extent of the cage evolves continuously with the internuclear separation R, it varies sharply around R=2.0 a.u., indicating a change in the dominant electronic configuration. Unlike in the case of the EF 1 Σ + g state of H 2, the electronic configurations that dominate at small and large internuclear separations share a single topology of the intracule density.

Analytical properties of the Kohn–Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Analytical properties of the Kohn-Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Electronic structures of the bound excited quartet states of the helium anion

The radial and electron-pair density functions of two bound excited states of the helium anion have been calculated at the full configuration-interaction level with an extended basis set. It is shown that the form of the radial density in the ${(1s2s2p)}^{4}P$ and ${(2p}^{3}{)}^{4}S$ states of ${\mathrm{He}}^{\ensuremath{-}}$ is readily interpreted in terms of orbital contributions, even at the full configuration level at which level of theory the notion of an electronic configuration loses its precise meaning. The smaller radial distance of the L-shell maximum in the ${}^{4}S$ state relative to the lower-lying ${}^{4}P$ state is interpreted in terms of a higher effective nuclear charge in the ${}^{4}S$ state. The more compact electron cloud of the ${}^{4}S$ state leads to a narrower intracule density than that observed for the ${}^{4}P$ state. The ${}^{4}S$ state is shown to have a Coulomb hole with the same classic form as the ground state of helium but with a radius that is about six times larger. The depth of the Coulomb hole is substantially greater in the ${}^{4}S$ state than in the ${}^{4}P$ state, as expected from the fact that the correlation energy is greater in the former state. The Coulomb holes of both states exhibit a vanishingly small depth at small interelectronic distances due to the presence of the Fermi hole.

The mapping of the conditional pair density onto the electron density

This paper shows that the Fermi hole of a reference electron can be so strongly localized to a given region of space, as to cause the conditional pair density for same-spin electrons to approach the one-electron spin density outside the region of localization and for a closed-shell system, the conditional pair density for both spins will approach the total density. Correspondingly, the Laplacian of the conditional pair density, whose local concentrations indicate the positions where the density of the remaining electrons are most likely to be found for a fixed position of a reference pair, approaches the Laplacian of the density. The Laplacian of the conditional pair density generated by a sampling of pair space by an α,β pair of reference electrons, exhibits a homeomorphism with the Laplacian of the electron density. This homeomorphism approaches an isomorphic mapping of one field onto the other, as the reference electron pair becomes increasingly localized to a given region of space. Thus the local charge concentrations (CCs) displayed by the Laplacian of the electron density, the local maxima in L(r)=−∇2ρ(r), signify the presence of regions of partial pair condensation, regions with greater than average probabilities of occupation by a single pair of electrons, as has been previously surmized on empirical grounds. This paper establishes a mapping of the essential aspects of electron pairing, determined in six-dimensional space, onto the three-dimensional space of the electron density. The properties of the conditional pair density enable one to determine which CCs of L(r) are coupled and represent the same localized pair of electrons. It is found that the pattern and properties of the electron localization domains predicted by the Laplacian of the conditional pair density differ in important aspects from those predicted by ELF, the electron localization function.

Electron-pair densities of group 2 atoms in their $\mathsf{{}^1P}$ and $\mathsf{{}^3P}$ terms

Electron-pair intracule (relative motion) and extracule (center-of-mass motion) densities are studied in both position and momentum spaces for the 1 P and 3 P terms of the group 2 atoms Be (atomic number Z =4 ), Mg (Z = 12), Ca (Z = 20), Sr (Z =3 8), Ba (Z = 56), and Ra (Z = 88). In position space, the 1 P − 3 P difference in the intracule densities shows that the probability of a small interelectronic distance is larger in the triplet for all the six atoms, as reported for the lightest Be atom in the literature. The position-space extracule density clarifies that the triplet electrons are more likely to be at opposite positions with respect to the nucleus than the singlet electrons for all the atoms. In momentum space, the singlet generally has a larger probability of a small relative momentum between two electrons as a na¨ ive manifestation of the Fermi hole in the triplet. The extracule density in momentum space shows that the 1 P term has a distribution larger in a large center-of-mass momentum region than the 3 P term.

Electron-pair momentum densities of group 14, 15, and 16 atoms in their low-lying electronic states

Electron-pair intracule (relative motion) h̄(υ) and extracule (center-of-mass motion) d̄(P) densities in momentum space are studied for the 3P, 1D, and 1S terms of five group 14 atoms with p2 configurations, the 4S, 2D, and 2P terms of five group 15 atoms with p3 configurations, and the 3P, 1D, and 1S terms of five group 16 atoms with p4 configurations. Common to all fifteen atoms, the intracule densities show that a low energy term has a greater probability of finding a pair of electrons with a large relative momentum υ than a high energy term. The Fermi hole effect in a high spin term appears naively in momentum space, and the average relative momentum 〈υ〉 is larger in a high angular momentum term. For the terms arising from the pm electronic configurations (m=2–4), the differences in the radial extracule densities are found to be almost isomorphic with the corresponding intracule ones. In a term with a high angular momentum, the average center-of-mass momentum 〈P〉 of an electron pair is always larger, and two electrons are less likely to have opposite momenta. The major origin of these differences in the electron-pair densities lies in the valence np orbitals, and the contribution of different electron-pair motions to the relative stability of terms can be explained by a rigorous relation between the kinetic energy Tnp and the second intracule 〈υ2〉np and extracule 〈P2〉np moments of the valence np subshell.

Electron-pair densities of group 14, 15, and 16 atoms in their low-lying multiplet states

Electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are studied for the 3P, 1D, and 1S multiplet states of five group 14 atoms with p2 configurations, the 4S, 2D, and 2P multiplets of five group 15 atoms with p3 configurations, and the 3P, 1D, and 1S multiplets of five group 16 atoms with p4 configurations. For all 15 atoms, it is confirmed that a lower total energy E in a high angular momentum state is always accompanied with the decrease of the electron–nucleus attraction energy Ven which is greater than the increase in the electron–electron repulsion energy Vee, as discussed in the literature for a few light atoms. In all the cases, the difference in the radial intracule densities 4πu2h(u) clarifies that for a multiplet with a higher angular momentum, the probability of finding a pair of electrons separated by a short/long distance is larger/smaller, leading to a greater Vee and a smaller average interelectronic distance 〈u〉. The Fermi hole effect is not important for the relative stability of the multiplets examined. For the multiplets arising from the pn electronic configurations, the differences in the radial extracule densities 4πR2d(R) are found to be almost isomorphic with the corresponding intracule ones: In a multiplet with a high angular momentum, the average center-of-mass radius 〈R〉 of an electron pair is always smaller, and two electrons are more likely to be at opposite positions with respect to the nucleus. Accordingly, the valence p orbital in a more stable multiplet is more tight, resulting in a lower electron–nucleus attraction energy Ven for all the 15 atoms.

The Lewis Model and Beyond

The electron pair density, in conjunction with the definition of an atom in a molecule, enables one to determine the average number of electron pairs that are localized to each atom and the number that are formed between any given pair of atoms. Thus, it is through the pair density that the Lewis model of electronic structure finds physical expression. The pairing of electrons is a consequence of the Pauli principle whose effect is made manifest through the creation of the Fermi hole. The density describing the spatial distribution of the Fermi hole for an electron of given spin determines how the density of that electron is spread out in space, excluding an equivalent amount of same-spin density. The averaging of the Fermi density over single atoms or pairs of atoms determines the corresponding contributions to the total Fermi correlation. It is these terms that yield the localization and delocalization indices that determine the intra- and interatomic distribution of electron pairs that enables one to compare the pairing predicted by theory with that of a Lewis structure. The agreement is best at the Hartree−Fock level, where the Fermi hole is the sole source of correlation between the electrons. The introduction of the remaining correlation, the Coulomb correlation, disrupts the sharing of electron pairs between the atoms, and its effect is therefore, most pronounced for shared interactions. For example, Coulomb correlation reduces the number of shared pairs in N2 from the Hartree−Fock value of three to just above two. In ionic systems, the electrons are strongly localized within each atomic basin and the effect of Coulomb correlation on the atomic pairing is minimal, approaching zero over each of the atomic basins, as it does for the total molecule.

Structure of the Pauli and Correlation-Kinetic Components of the Kohn–Sham Exchange Potential at a Metal Surface

According to the rigorous physical interpretation of Kohn–Sham (KS) density-functional theory in terms of the components of the true wavefunction, the KS exchange potentialνKSx(r)=δEKSx[ρ]/δρ(r), whereEKSx[ρ] is the exchange energy functional, is the work done to move an electron in a conservative fieldR(r). This field comprises a component EKSx(r) representative of Pauli correlations and anotherZ(1)tc(r) that constitutespartof the correlation contribution to the kinetic energy. The field EKSx(r) is derived via Coulomb's law from the KS Fermi hole charge, and the fieldZ(1)tc(r) from the kinetic-energy-density tensor involving the first-order correction to the KS single-particle density matrix. For systems in which the curls of these component fields separately vanish, the potentialvKSx(r) is the sum of the work doneWKSx(r) andW(1)tc(r) in the fields EKSx(r) andZ(1)tc(r), respectively. In this paper we study the structure of the workWKSx(r) andW(1)tc(r) at a simple-metal surface as represented by the jellium and structureless-pseudopotential models for which the workWKSx(r) andW(1)tc(r) are separately path-independent. A general expression for the field EKSx(r) is derived in terms of momentum-space integrals of the electron orbitals. This enables its easy determination, and thereby determination of the potentialWKSx(r). The field expression further allows for the derivation of theexact analyticalasymptotic structure of the potentialWKSx(r) in the vacuum region, a result valid for thefully self-consistentlydetermined orbitals of both models. With the exact analytical asymptotic structure ofvKSx(r) in the vacuum known, that of the potentialW(1)tc(r) in this region is then determinedanalytically. As is the case forvKSx(r) which decays asymptotically in the vacuum as −αKS/x, the potentialsWKSx(r) andW(1)tc(r) also decay as −αW/xandα(1)tc/x, respectively, the decay coefficients depending upon the metal Fermi energy and barrier height. It is further shown that for metallic densities,WKSx(r) does not approach thevKSx(r) asymptotic metal-bulk value of (−kF/π), so thatWtc(1)(r) is also finite in this region. Thus, at a metal surface, the KS exchange potentialvKSx(r) comprises principally its Pauli componentWKSx(r), with the correlation-kinetic partW(1)tc(r) beingfiniteandlong-rangedin both the vacuum and metal-bulk regions, its contribution diminishing with increasing metal density.

Electron-electron counterbalance hole in Hartree–Fock theory

The electronic extracule density E(R) and its spherical average d(R) are probability densities of finding the center-of-mass vector (rj+rk)/2 and its magnitude |rj+rk|/2 of any two electrons j and k to be R and R, respectively. Theoretical analysis of spin–orbital-pair contributions Ejk(R) and djk(R) to the extracule densities shows that within the Hartree–Fock framework, there exists an “electron-electron counterbalance hole” Ejk(0)=djk(0)=0 between two electrons in spin–orbitals j and k with the same spin and the same spatial inversion symmetry, which implies that the center-of-mass vector (or distance) of these electrons cannot be zero or these electrons cannot be exactly at the opposite positions in space with respect to the inversion center. The same is also true in momentum space. The electron-electron counterbalance hole may be considered as a complement to the familiar electron-electron coalescence or Fermi hole for two electrons with the same spin. We also point out that the electron-electron counterbalance and coalescence densities are rigorously proportional for a pair of spin–orbitals with different spins.

Analytical asymptotic structure of the Pauli, Coulomb, and correlation-kinetic components of the Kohn-Sham theory exchange-correlation potential in atoms

In this article, we derive the analytical asymptotic structure in the classically forbidden region of atoms of the Kohn–Sham (KS) theory exchange–correlation potential defined as the functional derivative νxc(r)=δExcKS[ρ]/δρ(r), where ExcKS[ρ] is the KS exchange–correlation energy functional of the density ρ(r). The derivation is via the exact description of KS theory in terms of the Schrödinger wave function. As such, we derive the explicit contribution to the asymptotic structure of the separate correlations due to the Pauli exclusion principle and Coulomb repulsion, and of correlation–kinetic effects which are the source of the difference between the kinetic energy of the Schrödinger and KS systems. We first determine the asymptotic expansion of the wave function, single-particle density matrix, density, and pair–correlation density up to terms of order involving the quadrupole moment. For atoms in which the N- and (N−1)-electron systems are orbitally nondegenerate, the structure of the potential is derived to be , where α is the polarizability; χ, an expectation value of the (N−1)-electron ion; and κ02/2, the ionization potential. The derivation shows the leading and second terms to arise directly from the KS Fermi and Coulomb hole charges, respectively, and the last to be a correlation–kinetic contribution. For atoms in which the N-electron system is orbitally degenerate, there are additional contributions of O(1/r3) and O(1/r5) due to Pauli correlations. We show further that there is no O(1/r5) contribution due to Coulomb correlations. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 671–680, 1998

Bond Characterization of Chromium−Fischer Carbene Complexes: A Combined Study of Experiment and Theory

Four pentacarbonyl chromium−carbene complexes, (CO)5CrC(XR‘)R, have been studied via X-ray diffraction and molecular orbital calculations. One of the carbene complexes (XR‘ = OCH3, R = −(C⋮CPh)) has been investigated extensively at 110 K by X-ray diffraction using Mo Kα radiation. The electron density distribution of this compound as well as the asphericity in electron density around the Cr atom is clearly demonstrated by deformation density and the Laplacian of electron density. The comparison between experiment and theory is made in terms of deformation density, topological properties, and d orbital populations of Cr. Further chemical bond characterization is based on quantum mechanical molecular orbital calculation and properties associated with bond critical points. The π bond character of a metal carbene can be best represented by a Cr−C−X three-centered four-electron bond with the π electron density mainly located at either the dyz orbital of Cr or the pz orbital of X in the carbene ligand. This makes the carbene carbon an electrophilic site in the pπ direction. The resemblances and differences between amino−(XN) and alkoxy−(XO) carbene complexes are of special interest. Because the energy of π*C-N orbital is fairly close to that of the π*C⋮O (carbonyl on Cr), the π bond character is delocalized toward the M−Ccarbonyl at the trans position i.e., OCM−CN in the amino carbene case; this is in accord with the shortening of bond lengths of the trans M−Ccarbonyl for many amino−carbene complexes. The π bond-delocalization is also illustrated by the Fermi hole distribution. The bond dissociation energies of these carbene complexes are calculated at the CASSCF level and with the density functional method (DFT). The relative orbital energies are also compared with photoelectron spectroscopy (PES) data. The values based on DFT using the transition-state approximation give the best agreement with the experimental results.

Analytical Asymptotic Structure of the Slater Exchange and Kohn–Sham Exchange and Correlation Potentials at a Metal Surface

In this paper we consider the semi-infinite jellium and structureless pseudopotential models of a metal surface and derive the exact analytical asymptotic structure in the classically forbidden region of (i) the Slater exchange potentialVsx(r), (ii) the Kohn–Sham exchange potentialνx(r), and thus of the optimized potential method exchange potentialνOPMx(r), and show these latter potentials to decay as 12Vsx(r). With the assumption that the Kohn–Sham exchange-correlation potentialνxc(r) decays asymptotically as the image potential, we thereby derive (iii) the analytical asymptotic structure of the Kohn–Sham correlation potentialνc(r). These potentials all decay as −x−1with coefficients which depend upon the Fermi energy and surface barrier height. However, it is onlyνx(r) that is image-potential-like, the coefficient being 14for stable jellium. Thus, with exchange and correlation effects considered separately, the principal contribution to the asymptotic image potential structure arises due to Pauli correlations, the Coulomb correlation contribution being weak. We also show (iv) analytically that for a slab-metal geometry, the potentialνx(r) decays as −x−2and explain why this is the case. Finally, we show (v) analytically that two approximate Kohn–Sham exchange potentials in the literature possess the correct 12Vsx(r) asymptotic structure in the vacuum. With exchange and correlation considered together, the assumption that the asymptotic image potential structure ofνxc(r) is independent of metal parameters provides an alternate “classical” interpretation of the physical origin of this structure. It is due to those Coulomb correlations which contribute to that part of the Coulomb hole (of charge −e) localized to the surface region, the component (of charge +e) delocalized in the metal bulk screening out the corresponding delocalized Fermi hole distribution (of charge −e).

Structure of the correlation-kinetic component of the Kohn-Sham exchange potential in atoms and at metal surfaces

The Kohn–Sham density functional theory “exchange” potential vx(r)=δExKS[ρ]/δρ(r), where ExKS[ρ] is the “exchange” energy functional, is composed of a component representative of Pauli correlations and one that constitutes part of the correlation contribution to the kinetic energy. The Pauli term is the work done WxKS(r) in the field ℰxKS(r) obtained by Coulomb's law from the Fermi hole charge distribution constructed from the Kohn–Sham orbitals. The correlation–kinetic term is the work done W(1)(r) in the field Z(1)(r) derived from the kinetic-energy–density tensor involving the first-order correction to the Kohn–Sham single-particle density matrix. The sum of these fields is conservative, so that the total work done is path-independent. There is no explicit correlation–kinetic contribution to the “exchange” energy ExKS[ρ]. Its contribution is manifested via the Kohn–Sham orbitals generated via the potential vx(r). The functional ExKS[ρ] is thus expressed in virial form entirely in terms of the Pauli field ℰxKS(r). In this article, we determine and study the structure of the correlation–kinetic component field Z(1)(r) and work W(1)(r) for the nonuniform electron density system in atoms and at metal surfaces. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 893–906, 1997

Parallel electron correlation effect and Fermi hole structure

Parallel electron correlation effect and Fermi hole structure

The Electron Pair

Eighty years have elapsed since Lewis introduced the concept of an electron pair into chemistry where it has continued to play a dominant rôle 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 doppelgängerit 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.

Barrier to Rotation in Thioformamide: Implications for Amide Resonance

We investigate the energetics of rotation about the C−N bond in thioformamide at the molecular and atomic levels using the HF/6-311++G**//HF/6-311++G** level of theory. The barrier to rotation is 19.9 kcal mol-1 and is dominated by the increase in the C−N distance and the consequent loss in attractive energies between carbon and nitrogen. The origin of the barrier to rotation is shown to be the same as that found in formamide. There is a large transfer of charge from nitrogen to carbon as the system moves away from planarity, but, unlike the essentially unchanged oxygen in formamide, sulfur also transfers charge to carbon upon rotation. It is the preference of the amide nitrogen for planarity, making it more electronegative and better able to stabilized itself by withdrawing charge from its bonded neighbors, that dictates the barrier to rotation about the C−N bond. The Fermi hole is used to quantitatively demonstrate that there is little delocalization of the π charge density from nitrogen to sulfur. The larger barrier to rotation in thioamides and the negligible delocalization of π charge from nitrogen to sulfur is inconsistent with the expectations of the resonance model but is consistent with a the view that (thio)amides behave as ‘(thio)formylamines'.

Electron Delocalization and the Fermi Hole

The delocalization of electrons is controlled by the Pauli exclusion principle through a corresponding delocalization of the density of the Fermi hole for electrons of given spin. It is proposed that the spatial distribution of the Fermi-hole density be used to provide a common, quantitative basis for the concept of electron delocalization, as it is used throughout chemistry. To this end, the following correspondences are made: The atomic patterns of delocalization of the Fermi hole recover the resonance structures corresponding to different possible spin-pairings and their relative importance, in terms of both energy and reactivity. The π-electron density of benzene distorted into a Kekulé-like structure is shown to be less delocalized than in the symmetrical structure. The effects of a hetero ring atom and of substituents on the delocalization of the π-electrons in benzene are quantified. To illustrate the range of phenomena that are determined by the delocalization of the Fermi-hole, it is shown that the measure of the extent of delocalization of the spin density from one hydrogen to another vicinal to it correlates with the variation in their spin−spin coupling with torsion angle, as predicted by the Karplus equation.