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

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We investigate the use of different variational principles in quantum Monte Carlo, namely energy and variance minimization, prompted by the interest in the robust and accurate estimate of electronic excited states.

• For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively.
• On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state.
• This unexpected behavior is sometimes observed even when the target is the ground state, and generally prevents the robust estimate of total and excitation energies.