A new gradient expansion of the exchange energy to be used in density functional calculations on atoms

Abstract

A gradient expansion of the exchange energy density functional for an N electron atom is obtained by imposing the following conditions: (i) The functional scaleslike potential energy. (ii) Nonlocality is introduced through powers of electron position and powers of the gradient of the density. (iii) The atomic cusp condition is exactly satisfied. The resulting functional is K[ρ] = C(N)ℱρ4/3 dτ+D(N)ℱr2ρ−2/3 (∇ρ⋅∇ρ) dτ = ℱρ4/3 [C(N)+D(N) y2] dτ, where y = ∂ ln ρ/∂ ln r. Imposing conditions on the functional and its functional derivative which must be satisfied when evaluated with the exact one-electron density, we obtained the following estimates of the coefficients: C(N) = −(3/4)(π/3)1/3 −(3/4)(π1/3/N2/3)[1−(3/π2)1/3] and D(N) = π1/3/729 N2/3. Evaluating the present functional with 1785 Hartree–Fock densities and comparing the resulting energies to the Hartree–Fock energies gave an average 0.04% error. Solving the Euler–Lagrange equation associated with our functional gave energies which differed from Hartree–Fock energies by about the same percentage error as when the energy density functional was evaluated with Hartree–Fock densities. Unlike the usual gradient expansion, our expansion does not prevent the virial theorem or the atomic cusp condition from being exactly satisfied.