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We investigate the use of different variational principles in quantum MonteCarlo, namely energy and variance minimization, prompted by the interest in therobust and accurate estimate of electronic excited states. For twoprototypical, challenging molecules, we readily reach the accuracy of the bestavailable reference excitation energies using energy minimization in astate-specific or state-average fashion for states of different or equalsymmetry, respectively. On the other hand, in variance minimization, where theuse of suitable functionals is expected to target specific states regardless ofthe symmetry, we encounter severe problems for a variety of wave functions: asthe variance converges, the energy drifts away from that of the selected state.This unexpected behavior is sometimes observed even when the target is theground state, and generally prevents the robust estimate of total andexcitation energies. We analyze this problem using a very simple wave functionand infer that the optimization finds little or no barrier to escape from alocal minimum or local plateau, eventually converging to the uniquelowest-variance state instead of the target state. While the loss of the stateof interest can be delayed and possibly avoided by reducing the statisticalerror of the gradient, for the full optimization of realistic wave functions,variance minimization with current functionals appears to be an impracticalroute.

Variational principles in quantum Monte Carlo: the troubled story of variance minimization