Requirements for correlation energy density functionals from coordinate transformations

Abstract

The behavior of the density-functional correlation energy under general linear coordinate transformations is considered. It is demonstrated that it is uniform and nonuniform scaling that yield new conditions which are fulfilled by the exact correlation energy functional, besides the well-known requirements of rotational, reflectional, and translational invariance. By considering various types of nonuniform scaling in the limit of the scale factor λ going to zero and infinity, a set of relations for the correlation energy is derived. These relations, which are easy to test on approximate functionals, are of the type ${\lim }_{\lambda \to 0}$(1/λ)${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda \lambda }^{\mathit{x}\mathit{y}}$]=0 and ${\lim }_{\lambda \to \mathrm{\infty }}$ ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]=0. In addition, a previous proof is simplified for ${\lim }_{\lambda \to \mathrm{\infty }}$ ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$]=const, which is bounded from below, and we identify the constant as a second-order energy. Here ${\mathit{n}}_{\lambda }$(r)=${\lambda }^3$n(λx,λy,λz), ${\mathit{n}}_{\lambda }^{\mathit{x}}$(r)=λn(λx,y,z), and ${\mathit{n}}_{\lambda \lambda }^{\mathit{x}\mathit{y}}$(r)=${\lambda }^2$n(λx,λy,z) where n is the electron density. The limiting relations are not all met by any known approximate correlation energy functional. These relations apply to a correlation energy for addition to a Hartree-Fock calculation as well as to a correlation energy for use as part of a full density-functional calculation.