Ensemble optimization of excited states in variational Monte Carlo*

Variational Monte Carlo methods have recently been applied to the calculation of excited states.[1] It is still an open question what the best objective function should be for excited states. A promising approach is to optimize excited states using a penalty method, such as the one introduced by Pathak et al,[2] which has the drawback that the states must be computed one at a time. The challenge with optimizing multiple excited states simultaneously is that the sum of the energies of the states is invariant to any unitary rotation of the eigenstates.

Drawing on the idea from Gross et al,[3] we propose an objective function for an ensemble of states using a weighted average of the energies and an overlap penalty. We show this objective function has a minimum at the exact eigenstates for a finite penalty. We derive conditions on the weights and penalty parameters of the objective function and demonstrate its application on the degenerate first excited state of a CO molecule. We believe this method surpasses the previous penalty method in efficiency and convenience.

1. L. Otis and E. Neuscamman, WIREs Comput Mol Sci. 13, e1659 (2023).

2. S. Pathak et al, J. Chem. Phys. 154, 034101 (2021).

3. E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A 37, 2805 (1988).