Exact exchange potential evaluated solely from occupied Kohn-Sham and Hartree-Fock solutions

TL;DR

The paper introduces a new iterative algorithm to accurately compute the exact exchange potential in density functional theory using only occupied orbitals and energies, avoiding differential equation solutions.

Contribution

A novel method for determining the exact exchange potential solely from occupied Kohn-Sham and Hartree-Fock solutions, enhancing efficiency and accuracy in DFT calculations.

Findings

Rapid convergence to the exact exchange potential within few iterations.

Achieved high-precision orbital energies with minimal computational steps.

Validated method on several atoms with results matching theoretical expectations.

Abstract

The reported new algorithm determines the exact exchange potential v_x in a iterative way using energy and orbital shifts (ES, OS) obtained - with finite-difference formulas - from the solutions (occupied orbitals and their energies) of the Hartree-Fock-like equation and the Kohn-Sham-like equation, the former used for the initial approximation to v_x and the latter - for increments of ES and OS due to subsequent changes of v_x. Thus, solution of the differential equations for OS, used by Kummel and Perdew (KP) [Phys. Rev. Lett. 90, 043004 (2003)], is avoided. The iterated exchange potential, expressed in terms of ES and OS, is improved by modifying ES at odd iteration steps and OS at even steps. The modification formulas are related to the OEP equation (satisfied at convergence) written as the condition of vanishing density shift (DS) - they are obtained, respectively, by enforcing its satisfaction through corrections to approximate OS and by determining optimal ES that minimize the DS norm. The proposed method, successfully tested for several closed-(sub)shell atoms, from Be to Kr, within the DFT exchange-only approximation, proves highly efficient. The calculations using pseudospectral method for representing orbitals give iterative sequences of approximate exchange potentials (starting with the Krieger-Li-Iafrate approximation) that rapidly approach the exact v_x so that, for Ne, Ar and Zn, the corresponding DS norm becomes less than 10^{-6} after 13, 13 and 9 iteration steps for a given electron density. In self-consistent density calculations, orbital energies of 10^{-4} Hartree accuracy are obtained for these atoms after, respectively, 9, 12 and 12 density iteration steps, each involving just 2 steps of v_x iteration, while the accuracy limit of 10^{-6}--10^{-7} Hartree is reached after 20 density iterations.