Indirect-path methods for atomic and molecular energies, and new Koopmans theorems

Staircase and ladder methods are proposed for atomic and molecular total and dissociation energies. In both methods, the energies are generated by employing indirect paths via information obtained from effective potentials. In the staircase method, the energies are determined in steps by successive alternations of electron and proton removals. Within the Hartree-Fock (HF) staircase formulation, the energy for electron removal is taken as the negative of the highest-occupied orbital energy, and the energy for proton removal is obtained as the difference of conventional HF total-energy expectation values. The HF staircase total and dissociation energies are significantly superior to the traditional HF values. In the ladder method, total energies are obtained by summing successive highest-occupied orbital energies for fixed nuclei. Both methods are useful within more advanced many-body theories and are exact within exact Kohn-Sham density-functional theory where the magnitude of the highest-occupied orbital energy equals the experimental ionization energy. Self-interaction-corrected density-functional results are presented. We assert two new Koopmans theorems: (1) the energy $\mathcal{E}$ of the highest-occupied HF orbital would give the experimental ionization energy $I$ if the exact ground-state wave function were free of single excitations out of this orbital, and (2) $\mathcal{E}=\ensuremath{-}I$ when the exact correlation potential, ${v}_{c}([n];\mathrm{r})$, is added to the Fock potential and self-consistency is achieved in both the HF orbitals and in the density $n$.