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

• Published in: The Journal of chemical physics 2017-12, Vol.147 (21), p.214104-214104
• Main Authors: Ye, Hong-Zhou, Welborn, Matthew, Ricke, Nathan D., Van Voorhis, Troy
• Format: Article
• Language: English
• Subjects: Accessibility; Broken symmetry; Collapse; Optimization; Physics
• ISSN: 0021-9606; 1089-7690
• DOI: 10.1063/1.5001262

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.