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

Abstract

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