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 E x [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 v x ( r ) and v c ( 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 E x [ρ] and potential v x ( 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 v x ( r ) is explicit, their contribution to the energy E x [ρ] is indirect via the orbitals of the noninteracting fermion or s ‐system. (ii) To leading order O (λ 2 ), the correlation potential v c ,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 v x ( r ) and v c ( 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 v x ( r ) asymptotically. The asymptotic structure of v x ( r ), to exponential accuracy, is due entirely to Pauli correlations and is the work done W x ( r ) to move an electron in the field of the Fermi hole, decaying as −1/ r . (iv) The lowest order O (λ 2 ) Coulomb correlations do not contribute asymptotically to v c ( r ). (v) The leading order O (λ 2 ) correlation potential v c ,2 ( r ) is due entirely to correlation‐kinetic effects decaying as 8κ 0 χ s /5 r 5 , where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χ s an expectation value of the s ‐system ion. (vi) To O (1/ r 5 ), the O (λ 3 ) correlation potential v c ,3 ( r ) to leading order is entirely due to second‐order Coulomb correlations and decays as −α s /2 r 4 , where α s is the polarizability of the s ‐system ion. There are no Coulomb correlation contributions to v c ,3 ( r ) and thus v c ( r ) to O (1/ r 5 ). (vii) For systems for which the N ‐electron atom is orbitally degenerate, the asymptotic structure of v x ( r ) contains in addition terms of O (1/ r 3 ), O (1/ r 5 ), etc. The other remaining asymptotic properties of v c ( r ), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000