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Variational Monte Carlo methods have recently been applied to the calculationof excited states; however, it is still an open question what objectivefunction is most effective. A promising approach is to optimize excited statesusing a penalty to minimize overlap with lower eigenstates, which has thedrawback that states must be computed one at a time. We derive a generalframework for constructing objective functions with minima at the the lowest$N$ eigenstates of a many-body Hamiltonian. The objective function uses aweighted average of the energies and an overlap penalty, which must satisfyseveral conditions. We show this objective function has a minimum at the exacteigenstates for a finite penalty, and provide a few strategies to minimize theobjective function. The method is demonstrated using ab initio variationalMonte Carlo to calculate the degenerate first excited state of a CO molecule.

Ensemble variational Monte Carlo for optimization of correlated excited state wave functions