Relationships between the terms in the gradient expansion: kinetic and exchange energy functionals

Abstract

Rigorous lower bounds for all bound-state systems, for the first-gradient corrections to the kinetic and exchange energy functionals, viz., (i) T₂[ρ]=1/72∫|∫(r)|²/ρ(r)dr=π^4/32^2/3/24N^2/3∫ρ^5/3(r)dr=10/72[2/3]^2/3T₀[ρ]/N^2/3 and (ii) |K₂[ρ]|=∫|ρ ρ(r)|²/ρ^4/3(r)dr≥27(π/2)^4/3/N^2/3∫ρ4/3(r)dr=3π(6π)^2/3/N^2/3|K₀[ρ]| have been derived [N is the number of electrons and ρ(r) is the electron density]. Numerical investigations on these bounds employing Hartree-Fock atomic-electron densities have been carried out. An empirical relationship between the Hartree-Fock T₂ and T₀ for neutral atoms has also been presented.