  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. 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 the unique
lowest-variance state instead of the target state. While the loss of the state
of interest can be delayed and possibly avoided by reducing the statistical
error of the gradient, for the full optimization of realistic wave functions,
variance minimization with current functionals appears to be an impractical
route.
