The work formalism of electronic structure
Authors: Sahni Viraht 
Published: December 2004

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.