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 ν xc ( r )=δ E xc KS [ρ]/δρ( r ), where E xc KS [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 κ 0 2 /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/ r 3 ) and O (1/ r 5 ) due to Pauli correlations. We show further that there is no O (1/ r 5 ) contribution due to Coulomb correlations. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 671–680, 1998