New method for the direct calculation of electron density in many‐electron systems. I. Application to closed‐shell atoms

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Abstract
A new density‐functional equation is suggested for the direct calculation of electron density ρ(r) in many‐electron systems. This employs a kinetic energy functional T2 + f(r)T0, where T2 is the original Weizsäcker correction, T0 is the Thomas–Fermi term, and f(r) is a correction factor that depends on both r and the number of electrons N. Using the Hartree–Fock relation between the kinetic and the exchange energy density, and a nonlocal approximation to the latter, the kinetic energy–density functional is written (in a.u.) where \documentclass{article}\pagestyle{empty}$ C_k = {\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {10}}}\right.}\!\lower0.7ex\hbox{${10}$}}(3\pi ^2)^{2/3} $ . Incorporating the above expression in the total energy density functional and minimizing the latter subject to N representability conditions for ρ(r) result in an Euler–Lagrange nonlinear second‐order differential equation where μ is the chemical potential, we have ρ(r) = |ϕ(r)|2, and g(r) is related to f(r). Numerical solutions of the above equation for Ne, Ar, Kr, and Xe, by modeling f(r) and g(r) as simple sums over Gaussians, show excellent agreement with the corresponding Hartree–Fock ground‐state densities and energies, indicating that this is likely to be a promising method for calculating fairly accurate electron densities in atoms and molecules.