FUNDAMENTAL DIFFERENCE BETWEEN ENERGY FUNCTIONALS BASED ON 1ST-ORDER AND ON 2ND-ORDER DENSITY MATRICES

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 ℰv dependent on the second-order density matrix). Comparison of the functionals Ev and ℰv, facilitated by transformation of Löwdin's kernel to the Hermitian kernel ℱ [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 = ∫∫dxdx′ 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. © 1979 American Institute of Physics.