Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density

Abstract

Given an exact eigenfunction ψ for some system with N electrons, a procedure is developed for determining ionization potentials of the system to various states of the corresponding system having N−p electrons. The long−range behavior of the electron density and of the natural spin orbitals is shown to involve a set of eigenvalues which are obtained by this procedure. Following is the procedure. First determine the pth−order natural functions for ψ, X_m^(p), and their occupation numbers n_m^(p), by diagonalization of the pth−order density matrix Γ^(p). Calculate the quantities where ? ≡ (x₁x₂⋅⋅⋅x_p) and ?_ξ ≡ (ξ₁ξ₂⋅⋅⋅ξ_p). Then diagonalize the Hermitian matrix μ^(p)_kl = λ^(p)_kl / (n^(p)_kn^(p)_l)^1/2. The resultant eigenvalues μ_α^(p) are approximations to the negative of the p−electron ionization potentials of the system, with successive μ_α^(p) (from the highest to the lowest) being lower bounds to the negative successive p−electron ionization potentials (from the lowest to the highest). For an approximate eigenfunction ?, the same procedure is recommended, except that the Hermitian part of the approximate matrix ?^(p)_kl is diagonalized. For ? a Hartree−Fock approximation to Ψ, the procedure for the case p = 1 is the classic method of Koopmans. It is shown that in general all natural spin orbitals χ_m(x) in a system have long−range expontential fall−off governed by μ⁽¹⁾_max, i.e., χ_m(x) ∼ exp[−(−2μ_max⁽¹⁾)^1/2r], and that the electron density behaves similarly, π (x) ∼ exp[−2(−2μ_max⁽¹⁾)^1/2r]. The behavior naively expected, namely π (x) ∼ exp[−2(2I_min)^1/2r], results if μ_max⁽1) is equal to −I_min.