About the difference between density functionals defined by energy criterion and density functionals defined by density criterion: Exchange functionals

Abstract

The difference between density functionals defined by energy criterion and density functionals defined by density criterion is studied for the exchange functional. It is shown that Slater potentials are exact exchange potentials in the sense that they yield the Hartree-Fock electron density if all operators are given by local expressions.

The exact Fermi potential yielding the Hartree–Fock electron density from orbital‐free density functional theory

“…For groundstate systems the formal exact expression for the Pauli potential has been derived in Refs. [15 and ] and the Slater potential has been shown to be the corresponding local potential yielding the HF electron density, when the kinetic energy is equally expressed as a local operator (as it is the case in an orbital‐free ansatz according to Equation ). The strategy for deriving Equations is the same as the strategy that has been used by Staroverov and his colleagues to derive the exact KS exchange potential and the exact KS exchange‐correlation potential associated with a specified electron density …”

A simple model for the Slater exchange potential and its performance for solids

“…It has been shown recently that the exact exchange potential yielding the HF electron density from a purely local density functional formalism (not a Kohn–Sham formalism) is the Slater potential: where the first order reduced density matrix is given by: and the sum runs over all occupied HF orbitals . Therefore, the Euler equation yielding the HF electron density as minimizing density from a local density functional formalism can be written as: …”

“…Recently, it has been shown that the Slater potential numerically corresponds to the exact exchange potential yielding the Hartree–Fock electron density from Hohenberg–Kohn principle when the kinetic energy is equally expressed by a local operator, as for example realized in orbital‐free calculations . This approach for the exact exchange potential must be distinguished from the optimized effective potential method and the method of obtaining Kohn–Sham wavefunctions and potentials from given constraint density in which case the kinetic energy operator is nonlocal.…”

“…Several successful potentials [6,7] belonging to this family have been derived from semilocal approximation proposed by Becke and Roussel [8] for the nonlocal Slater exchange potential. [9] Recently, it has been shown that the Slater potential [9] numerically corresponds to the exact exchange potential [10] yielding the Hartree-Fock electron density from Hohenberg-Kohn principle [1] when the kinetic energy is equally expressed by a local operator, as for example realized in orbital-free calculations. [11][12][13] This approach for the exact exchange potential must be distinguished from the optimized effective potential method [14][15][16] and the method of obtaining Kohn-Sham wavefunctions and potentials from given constraint density [17][18][19] in which case the kinetic energy operator is nonlocal.The direct evaluation of Slater potentials is computationally demanding as it involves spatial integration over the reduced oneelectron matrix.…”

“…and E i are the eigenvalues for the corresponding set of eigenfunctions / i . It has been shown recently [10,23] that the exact exchange potential yielding the HF electron density from a purely local density functional formalism [1,24] (not a Kohn-Sham formalism) is the Slater potential: v S ð½n;rÞ52 1 2nðrÞ ð jgðr;r 0 Þj 2 j r2r 0 j dr 0 ;…”