A self-consistent field formulation of excited state mean field theory

We show that, as in Hartree-Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator condition, a one-electron mean-field working equation, and acceleration via direct inversion in the iterative subspace. When combined with a configuration interaction singles Davidson solver for the excitation coefficients, the self-consistent field formulation dramatically reduces the cost of the theory compared to previous approaches based on quasi-Newton descent.