Symmetric Nonlocal Weighted Density Approximations from the Exchange-Correlation Hole of the Uniform Electron Gas

Abstract

Nonlocal exchange-correlation energy functionals are constructed using the accurate model exchange-correlation hole for the uniform electron gas developed by Gori-Giorgi and Perdew. The exchange-correlation hole is constrained to be symmetric and normalized, so the resulting functionals can be viewed as symmetrized versions of the weighted density approximation; we call them two-point weighted density approximations. Even without optimization of parameters or functional forms, the exchange-correlation energies for small molecules are competitive with those of the best generalized gradient approximation functionals. Two-point weighted density approximations seem to be an interesting new direction for functional development. A more general version of the conditions that define the energy for fractional electron number and fractional spin are presented. These "generalized flat-planes" conditions are closely linked to the normalization of the spin-resolved exchange-correlation hole at noninteger electron number. This and many other properties of the exact exchange-correlation functional can be imposed straightforwardly and directly in symmetrized weighted density approximation.

Communication: Two types of flat-planes conditions in density functional theory

“…In this paper we provide definitive experimental evidence that type 2 flat planes exist, thereby disproving the strong convexity postulate in Ref. 16. To find examples of type 2 flat planes, we built a database containing the lowest energy states of each multiplicity for every neutral atom and atomic ion of elements hydrogen (Z = 1) to fermium (Z = 100) using the NIST Atomic Spectra Database.…”

“…Although the key issues were recognized much earlier, 4 much recent interest has focused on manyelectron self-interaction errors (fractional electron number errors), [5][6][7][8] fractional spin errors, 9,10 and their combination into flat-planes conditions. 2,[11][12][13][14][15][16] These constraints are critical (although not enough to ensure correct behavior for systems with spatial degeneracies 4,[17][18][19][20][21] ) for describing bond dissociation and formation with density-functional theory. 2,3,[9][10][11]16,[22][23][24] The flat-planes conditions are designed to ensure that density functionals have the correct behavior when one adds a fraction of an electron of each spin, 0 ≤ ∆N α ≤ 1 and 0 ≤ ∆N β ≤ 1, to a reference state with an integer number of electrons of each spin.…”

“…25,26 ) There are only two possible types of flat-planes conditions. 16 Type 1 flat planes occur when…”

“…(See Figure 1(b).) The mathematical characterization of type 2 flat planes was first given by Cuevas-Saavedra et al 16 Specifically, if ∆N α ≤ ∆N β then,…”

“…16, it seems that no one has considered the possibility of type 2 flat planes. However, those authors hypothesized that type 2 flat planes do not exist, calling this the strong convexity postulate.…”