ALBERT

Notes: In this paper we show that atomic shell structure is exhibited throughout the periodic table, and accurate core–valence separations thereby obtained, via the radial probability density determined from the uncorrelated wave functions of Hartree theory. Further, essentially equivalent results are obtained via Hartree-theory-level quantal density functional theory in an approximation in which the correlation contributions to the kinetic energy are also neglected. Thus, accurate atomic shell structure can be obtained solely via electrostatic fields determined from charge distributions that are derived from wave functions which neither obey the Pauli exclusion principle nor incorporate Coulomb correlations. © 2001 American Institute of Physics.

Notes: The local many-body potential of density-functional theory is thus far understood in its mathematical context as the functional derivative of the exchange-correlation energy functional of the density. In recent work we have attempted to provide a physical interpretation for this potential. We interpret it as the work required to move an electron against the electric field of its Fermi-Coulomb hole charge distribution. Implicit in this interpretation is that the potential is path-independent. For symmetric systems this is rigorously the case. For systems where this may not be the case, the potential may be derived from an effective charge distribution given by the divergence of the field, thus ensuring its path independence. Also implicit as a consequence of the total Coulomb hole charge being zero is that the asymptotic structure of the potential is entirely due to the Fermi hole charge distribution, and thus known precisely. The potential lies explicitly within the rubric of density-functional theory in that within the exchange-only approximation it satisfies the exchange energy virial theorem sum rule and all scaling properties that the exact exchange potential must satisfy. The potential does not satisfy the virial theorem sum rule for the correlation energy, and consequently a term proportional to the difference between the interacting and noninteracting system kinetic energies must be added for the sum rule to be satisfied exactly. The formalism differs from density-functional theory in that it is not derived from the variational principle for the energy, thus obviating the requirement of determining functional derivatives, as well as allowing for the study of excited states. The interpretation also leads to insights into the exact Slater exchange potential, and other approximation schemes such as the Xα method, and local density and gradient expansion approximations. The results of application to few-electron atomic and many-electron metallic surface inhomogeneous electronic systems are remarkably accurate when compared with other theoretical calculations and experiment.

Notes: To understand the density-gradient expansion approximation for the exchange-correlation energy of density-functional theory from a fundamental viewpoint, we have performed an analysis of the corresponding expansion of the Fermi-coulomb hole charge distribution. The Fermi-Coulomb hole represents the correlations between electrons resulting from the Pauli exclusion principle and Coulomb's law. The analysis is performed in the exchange-only approximation by considering the expansion for the Fermi hole to terms of O(▽3) as applied to atoms. Our study shows that the expansions to O(▽), O(▽2), and O(▽3) all severely violate the constraint of positivity, becoming progressively worse with increasing orders of ▽. Further, the expansion to O(▽2) also severely violates the constraint of charge neutrality. (Terms of O(▽) and O(▽3) do not contribute to this constraint or to the exchange energy.) Thus the description of the physics of Pauli correlations in atoms as given by this approximation is highly unphysical. In spite of this, the exchange energy to O(▽2) is superior to the local density approximation because the expansion hole better approximates the exact Fermi hole in the interior of atoms from which arise the principal contributions to the energy. However, the improvement is not substantial, as the oscillations in the expansion Fermi hole occur within the atom itself. For asymptotic positions of the electron, the expansion holes to each order neither approximate the local density approximation nor the exact Fermi hole. Thus we understand why the expansion cannot lead to accurate highest occupied eigenvalues. The oscillations of the expansion Fermi hole also demonstrate why the Slater potential and electric field that result from these hole charge distributions are singular. On the other hand, we show that the expansion approximation is mathematically consistent in that the coefficient of the gradient correction term for screened Coulomb interaction to O(▽2) as obtained from the approximate Fermi hole is the same as that derived from linear response theory. We conclude with remarks on the Coulomb hole as obtained within this gradient expansion approximation scheme.

Notes: In recent work, we have provided a rigorous physical interpretation for the exchange energy and potential (or functional derivative) as obtained within the local-density approximation via the Harbola-Sahni formulation of many-electron theory. In this article, we analyze the gradient-expansion approximation (GEA) for these properties from the same physical perspective. The source charge distribution in this approximation is the GEA Fermi hole to O(▽3). This charge distribution is unphysical, so that the resulting force field and work done cannot be defined in a physically meaningful manner, and the exchange energy is singular. Thus, when viewed from the perspective of a source charge, the existence of the gradient expansions for the potential and energy is questionable. We next discuss the conventional method of employing a screened-Coulomb interaction to eliminate the singularities due to the GEA source charge, and show that it leads to inconsistent results. These inconsistencies are also intrinsic to a proof of the inequivalence of the Harbola-Sahni and Kohn-Sham exchange potentials within the GEA. Thus, although the inequivalence of these potentials has been established by other analyses, this proof is shown not to be rigorous. Finally, we demonstrate that when the physics of the GEA exchange source charge is corrected by the satisfaction of sum rules, the modified charge distribution then leads to a well-behaved local exchange potential and exchange energy density, and to a finite exchange energy. The consequences of our analysis on the gradient expansions for the correlation and exchange-correlation potential and energy are also noted. © 1992 John Wiley & Sons, Inc.

Notes: In this article we calculate the asymptotic structure of the exchange potential for the Fermi level electron in the vacuum region outside a metal surface, and show it to be the image potential. In this asymptotic region the exchange potential due to the Fermi level electron is equivalent to the optimized potential of exchange-only density-functional theory. Thus, the asymptotic Kohn-Sham effective potential at a metal surface is the image potential, and arises soley due to correlations which result from the Pauli exclusion principle. © 1993 John Wiley & Sons, Inc.

Notes: This article is a brief review of the work formalism of electronic structure, its recent developments, and the results of its application to spherically symmetric and nonspherical density atoms. The formalism, which is founded in Schrödinger theory, is derived by physical arguments based on Coulomb's law. The fundamental quantity in the formalism is the pair-correlation density that constitutes the nonlocal quantum-mechanical source charge distribution giving rise to both a local potential representing electron correlations as well as the electron interaction energy. The potential is the work done to move an electron in the force field of the pair-correlation density and the energy the interaction energy between the electronic and pair-correlation densities. (For systems for which the curl of the force field may not vanish, the potential is obtained from the irrotational component of the field, the solenoidal component being neglected). The differential equation governing the system is a Sturm-Liouville equation, and as such, the exact wave function can in principle be obtained as an infinite linear combination of Slater determinants of the self-consistently determined spin-orbitals of the occupied and virtual states. The correctness of the interpretation for the local potential representing electron interaction is evidenced as follows: In the Pauli-correlated and central field approximations, ground-state energies of atoms (2 He -86Rn) lie within 50 ppm of those of Hartree-Fock theory, differing by less than 10 ppm for atoms with Z 〉 35. The densities thus generated clearly exhibit atomic shell structure and also satisfy the Kato-Steiner electron-nucleus cusp condition to 2 ppm. Another attribute of the formalism is that the asymptotic structure of the potential (when both Pauli and Coulomb correlations are considered) is that of the Pauli-correlated approximation. This is rigorously the case as shown for the He atom for which the potential vanishes in the classically forbidden region, the potential there being the exchange potential. As such, it is meaningful to compare the highest occupied eigenvalue of the differential equation in the Pauli-correlated approximation to experiment. A comparison for atoms and atomic ions of this eigenvalue to experimental ionization potentials and electron affinities show them to be consistently superior to the corresponding eigenvalue of Hartree-Fock theory. Transition energies determined from eigenvalue differences are also superior to those obtained from total energy calculations via Hartree-Fock theory when compared to experiment. Further, by considering the carbon atom in one of its degenerate ground states for which the curl of the field due to the Fermi hole does not vanish, it is shown that the solenoidal component of the field is negligible and two orders of magnitude smaller than is the irrotational component. Thus, the approximation of obtaining a path-independent potential for nonspherical density systems from the irrotational component of the field is accurate. Finally, Coulomb correlation effects can be incorporated within the work formalism in practice via the configuration interaction approximation. The self-consistent orbitals thus obtained explicitly incorporate the effects of both Pauli and Coulomb correlations in their structure because the source charge from which they are generated is a pair-correlation density. Furthermore, these orbitals possess the correct asymptotic structure since they are also generated by a potential that is local. The work formalism also provides a physical interpretation for the local potential representing electron correlations of Kohn-Sham density functional theory. Further, the exchange potential of the work formalism satisfies analytically two requisite conditions of the Kohn-Sham theory exchange potential. These are the scaling requirement and the sum rule relating the exchange energy to its functional derivative. The work formalism also leads to a deeper understanding of electron correlations in various approximations within Kohn-Sham theory. For example, it can be rigorously shown that the pair-correlation density in the local density approximation contains a term proportional to the gradient of the density. Thus, in contrast to the Kohn-Sham theory interpretation that electron correlations in this approximation are those of the uniform electron gas assumed valid locally, we learn that the nonuniformity of the electronic density is, in fact, explicitly accounted for by the approximation. This then explains the accuracy of the approximation. © 1995 John Wiley & Sons, Inc.

Notes: 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 $\nu_{xc}({\bf r})\mathrel{\mathop{\sim}\limits_{r\to\infty}}-1/r-\alpha/2r^4+8\kappa_0\chi/5r^5$, 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

Notes: In this article we have determined the structure of the exchange potential v(0)x(r) at a jellium metal surface previously derived by restricted functional differentiation of the exchange energy functional. The potential, which depends on the Slater potential due to the Fermi hole, the density, and their gradients, is obtained analytically for the orbitals of the infinite barrier model. We have also determined the exchange potential Wx(r) of the work formalism, which is the work done to move an electron in the forcefield of the Fermi hole, for the same model effective potential, the field being derived analytically. A comparison of these potentials shows them to be close approximations. The functional derivative v(0)x(r) is further provided a physical interpretation by rewriting it in Slaterpotential form. The corresponding effective Fermi hole charge distribution, also determined analytically, has a dynamic structure as a function of electron position similar to that of the Fermi hole but smaller in magnitude. Finally, proofs are provided of the satisfaction by vX(0)(r) of the virial theorem sum rule, the second functional derivative condition, and the sum rule relating the exchange potential to its functional derivative. © 1995 John Wiley & Sons, Inc.

Notes: In this article, we derive and there by reinterpret various approximations in Schrödinger theory and Kohn-Sham density-funtional theory via a hierarchy within the work formalism of electronic structure due to Harbola and Sahni. In the work formalism, which is based on Coulomb's law, the local potential representing electron correlations as well as the electron correlation energy both arise from the same quantum mechanical source charge distribution that is the pair-correlation density. The potential is the work done to move an electron in the force field of the pair-correlation density, and the energy is the energy of interaction between the electronic and pair-correlation densities. The differential equation governing the system is a sturm-Liouville equation so that the system wave function can, in principle, be obtained as an infinite linear combination of Slater determinants of the spin-orbitals corresponding to the occupied and virtual states. The hierarchy is achieved by improvement of the pair-correlation density either by systematic improvement of the wave function or, as is the case of Kohn-Sham theory, by an expansion of the pair-correlation density in gradients of the density about the uniform electron gas result. The derivations of the approximations of Kohn-Sham theory via the work formalism, in turn, exhibit the existence of additional correlations that are not evident through the Kohn-Sham prescription, whereby the potential is obtained by functional differentiation. The approximations considered within Schrödinger theory are the Hartree, Hartree-Fock, and configuration-interaction approximations. Those within Kohn-Sham theory are the density functional theory Hartree, local density, and gradient expansion approximations. © 1995 John Wiley & Sons, Inc.

Notes: 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 Wtc(1)(r) in the field Ztc(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 Ztc(1)(r) and work Wtc(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