Variational principles in quantum Monte Carlo: the troubled story of variance minimization
Abstract
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 statespecific or stateaverage 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 lowestvariance 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.05951
 arXiv:
 arXiv:2002.05951
 Bibcode:
 2020arXiv200205951C
 Keywords:

 Physics  Chemical Physics;
 Physics  Computational Physics
 EPrint:
 doi:10.1021/acs.jctc.0c00147