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 estimation 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 estimation
of total and excitation energies. We analyze this problem using a
very simple wave function and infer that the optimization finds little
or no barrier to escape from a local minimum or local plateau, eventually
converging to a lower-variance state instead of the target state.
For the increasingly complex systems becoming in reach of quantum
Monte Carlo simulations, variance minimization with current functionals
appears to be an impractical route.