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

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, Xm(p), and their occupation numbers nm(p), by diagonalization of the pth−order density matrix Γ(p). Calculate the quantities where ? ≡ (x1x2⋅⋅⋅xp) and ?ξ ≡ (ξ1ξ2⋅⋅⋅ξp). Then diagonalize the Hermitian matrix μ(p)kl = λ(p)kl / (n(p)kn(p)l)1/2. The resultant eigenvalues μα(p) are approximations to the negative of the p−electron ionization potentials of the system, with successive μα(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 χm(x) in a system have long−range expontential fall−off governed by μ(1)max, i.e., χm(x) ∼ exp[−(−2μmax(1))1/2r], and that the electron density behaves similarly, π (x) ∼ exp[−2(−2μmax(1))1/2r]. The behavior naively expected, namely π (x) ∼ exp[−2(2Imin)1/2r], results if μmax(1) is equal to −Imin.