Generalized Koopmans' Theorem

Koopmans' theorem states that if the wave function of a many-electron system is approximated by a Slater determinant of Hartree-Fock one-electron wave functions, with one-electron energies defined as the difference in energy of ($N+1$)- and N-particle systems, then these one-electron energies are given by the expectation value of the Hartree-Fock Hamiltonian with respect to the one-electron wave functions. Koopmans' theorem is here generalized to include correlation effects by using Hubbard's expression for the total energy of a free-electron gas. The resulting one-electron Hamiltonian contains in first-order screened exchange. Hubbard's lowest polarization diagram gives, in addition, part of the screened second-order Coulomb interactions, which is small for metallic densities. Collective terms are also obtained. Comparison with the Bohm-Pines Hamiltonian shows a one-to-one correspondence, but with different cutoff functions in each term. Following Hubbard, we extend the method to include the effects of a periodic potential to first order. The resulting one-electron Hamiltonian provides a convenient and accurate basis for self-consistent energy band calculations including exchange and correlation in metals and semiconductors.