State-Targeted Energy Projection: A Simple and Robust Approach to Orbital Relaxation of Non-Aufbau Self-Consistent Field Solutions

Orbital optimization is crucial when using a non-Aufbau Slater determinant that involves promotion of an electron from a (nominally) occupied molecular orbital to an unoccupied one, or else ionization from a molecular orbital that lies below the highest occupied frontier molecular orbital. However, orbital relaxation of a non-Aufbau determinant risks variational collapse back to the Aufbau solution of the self-consistent field (SCF) equations. Algorithms such as the maximum overlap method (MOM) that are designed to avoid this collapse are not guaranteed to work, and more robust alternatives increase the cost per SCF iteration. Here, we introduce an alternative procedure called state-targeted energy projection (STEP) that is based on level shifting and is identical in cost to a normal SCF procedure, yet converges in numerous cases where MOM suffers variational collapse. Benchmark calculations on small-molecule reference data suggest that Delta SCF calculations based on STEP are an accurate way to compute both ionization and excitation energies, including core-level ionization and excited states with significant double-excitation character. For the molecule 2,4,6-trifluoroborazine, Delta SCF calculations based on STEP afford excellent agreement with experiment for both vertical and adiabatic ionization energies, the latter requiring geometry optimization of a non-Aufbau valence hole. Semiquantitative agreement with experiment is obtained for the absorption spectrum of chlorophyll a. Finally, the importance of asymptotic exchange and correlation is illustrated by application to Rydberg states using spin-scaled Moller-Plesset perturbation theory with a non-Aufbau reference determinant. Together, these results suggest that STEP offers a reliable and affordable alternative to the MOM procedure for determining non-Aufbau solutions of the SCF equations.