σ-SCF: A direct energy-targeting method to mean-field excited states

The mean-field solutions of electronic excited states are much less accessible than ground state (e.g., Hartree-Fock) solutions. Energy-based optimization methods for excited states, like Δ-SCF (self-consistent field), tend to fall into the lowest solution 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 solutions that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. We present results for this new method for small atoms (He, Be) and molecules (H₂, HF). We find 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 our results suggest that this method could be further improved with spin projection.