On a fundamental difference between energy functionals based on first- and on second-order density matrices

Abstract

The Euler equations and kernel F[γ] of an energy functional of the first-order density matrix are compared to the corresponding quantities which result from Löwdin’s treatment of the extended Hartree–Fock equations (the latter are based on an energy functional ▪v dependent on the second-order density matrix). Comparison of the functionals Ev and ▪v, facilitated by transformation of Löwdin’s kernel to the Hermitian kernel ▪[γ] which is central to the extended Koopmans’ theorem, leads to a clarification of the fundamental difference between ionization and chemical potentials. A definition of chemical potential (electronegativity) appropriate to Hartree–Fock theory is proposed. Denoting the Fock operator by the symbol FN[γ;x′,x], this definition is μ=−χ=ℱℱdxdx′ FN[γ;x′,x] [∂γ (x,x′)/∂N]. This reduces, in the special case of a system with a single valence electron, to a measure of the Hartree–Fock electronegativity proposed originally by Mulliken and by Moffitt; namely χ=−ε−J/2, where ε is an eigenvalue of the Fock operator, and J is a Coulomb integral evaluated for the canonical valence orbital χN.