Connections between variation principles at the interface of wave-function and density-functional theories

Abstract

A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn-Sham effective potentials is examined and extended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb's convex-conjugate functional, which allows for the determination of a potential associated with a given electron density by maximization, yielding the Kohn-Sham potential for a non-interacting system. The mathematical structure of the two functionals is shown to be intrinsically related; the variation principle put forward by Gidopoulos may be expressed in terms of the Lieb functional. The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework.

Improving the exchange and correlation potential in density-functional approximations through constraints

“…8-10 we solved the OEP equation for the CDFA method, using nite orbital and auxiliary basis sets, with l ¼ 0. The indeterminacy of the effective potential was restricted by expressing y(r) in terms of the screening density r scr (r) in eqn (6) and then constraining the screening charge Q scr (7) as well as the sign of r scr (r) (8).…”

“…In the rest of this section, we show indicative results for the CDFA method applied to the LDA functional, henceforth denoted by CLDA, where the minimisation was performed under just the constraint for the screening charge, Q scr ¼ N À 1 (7). In order to determine y(r) and r scr (r), we employ the extended response function c l y (r,r 0 ), in the limit of small l. We use l/D ¼ 0.01, but the results seem to converge and do not change if we reduce l/D by an order of magnitude.…”

“…Previously, we explored the minimisation of potential functionals dened by an energy difference, instead of density-functionals of the total energy, as a means of improving the quality of the KS potential. [4][5][6][7] The advantage of this approach is that the energy difference is bound from below, even in approximations from nite-order (second) perturbation theory; the latter can then be employed directly to derive accurate exchange-correlation (xc) potentials without the risk of variational collapse. 4,6 In this paper, we review briey and expand on our work [8][9][10] to improve the performance of local and semi-local density-functional approximations (DFAs), by imposing physical constraints on the single-particle, local, effective (KS) potential, whose orbitals minimise the total energy functional.…”

Improving the exchange and correlation potential in density functional approximations through constraints

“…In the rest of the section, we show indicative results of CLDA, where the minimisation was performed under just the constraint for the screening charge, Q scr = N − 1 (5). In order to determine v(r) and ρ scr (r), we employ the extended response function χ λ v (r, r ), in the limit of small λ.…”

“…A way we explored to improve the quality of the KS potential was to define appropriate potential-functionals of an energy difference, instead of density-functionals of the total energy, and to minimise the aforementioned energy difference rather than the total energy [4,5,7,28]. The advantage of the approach is that the energy difference is bound from below, even in approximations from finite-order (second) perturbation theory; the latter can then be employed directly to derive accurate XC potentials without the risk of variational collapse [4,7].…”