σ-SCF and HP σ-SCF: A unified approach to both ground and excited mean-field electronic states

Abstract

The mean-field solns. of electronic excited states are much less accessible than ground state (e.g., Hartree-Fock) solns. Energy-based optimization methods for excited states, like Δ-SCF, tend to fall into the lowest soln. consistent with a given symmetry -- a problem known as "variational collapse". In this work, we combine the ideas of direct energy-targeting and variance-based optimization in order to describe excited states at the mean-field level. The resulting method, σ-SCF, has several advantages. First, it allows one to target any desired excited state by specifying a single parameter: a guess of the energy of that state. It can therefore, in principle, find all excited states. Second, it avoids variational collapse by using a variance-based, unconstrained local minimization. As a consequence, all states -- ground or excited -- are treated on an equal footing. Third, it provides an alternate approach to locate Δ-SCF solns. that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. Numerical results on small atoms and mols. show that σ-SCF is very effective at locating excited states, including individual, high energy excitations within a dense manifold of excited states. Like all single determinant methods, σ-SCF shows prominent spin-symmetry breaking for open shell states and we show how the results can be imrpoved by a spin-projection procedure.