Communication: Explicit construction of functional derivatives in potential-driven density-functional theory

Abstract

We propose a method for imposing an important exact constraint on model Kohn–Sham potentials, namely, the requirement that they be functional derivatives of functionals of the electron density ρ. In particular, we show that if a model potential v(r) involves no ingredients other than ρ, ∇ρ, and ∇2ρ, then the necessary and sufficient condition for v(r) to be a functional derivative is ∂v/∂∇ρ=∇(∂v/∂∇2ρ). Integrability conditions of this type can be used to construct functional derivatives without knowing their parent functionals. This opens up possibilities for developing model exchange-correlation potentials that do not lead to unphysical effects common to existing approximations. Application of the technique is illustrated with examples.

A generalized gradient approximation for exchange derived from the model potential of van Leeuwen and Baerends

“…It is quite feasible, at least at the level of generalized gradient approximations, to develop model potentials that are by construction functional derivatives. 30,31 However, even in that case the presence of r in the Levy-Perdew energy density leads to unappealing artifacts for extended systems. 32,33 What is desired is a method for turning model Kohn-Sham potentials into properly invariant energy functionals with positionindependent energy densities.…”

Construction of integrable model Kohn-Sham potentials by analysis of the structure of functional derivatives

“…Approximate potentials that do not have a parent functional are called stray [27]. There exist a number of analytical and numerical tests for such potentials [2,[28][29][30]. Using these tests, one can show that all semilocal model potentials available today are stray.…”

“…[30], we devised analytic integrability conditions for explicitly density-dependent model Kohn-Sham potentials and showed how to use these conditions to construct functional derivatives without direct reference to parent functionals. Here we explore another approach which is based on direct examination of the analytic structure of functional derivatives.…”

Assessment of density functional methods with correct asymptotic behavior

“…Besides, as these model potentials are not variationally stable, the associated XC energies are not well-defined, and the predicted properties need to be carefully interpreted. The proposals made by Staroverov and co-workers [45,46] are useful for transforming a model GGA exchange potential (which is not a functional derivative) into an exchange potential that has a parent GGA functional. However, for the LB94 potential (an AC model potential with no parent GGA functional) [33], the transformed potential no longer has the correct (−1/r) asymptote [46].…”