Whack A Mole

In QMC, a central role is given to the local energy, defined as (1/psi)H psi, where psi is the trial wave function. If psi is an eigenstate, then H psi = E psi by definition, and the local energy becomes a constant. Just as lower energy trial functions are deemed better than higher energy ones, those with fewer fluctutations are better than those with more fluctutations.

One could imagine looking at the fluctuations in the local energy, and using those to determine where best to adjust the trial wave function to reduce the fluctutations (side note: one would want to use psi^2 times the local energy, since fluctuations are weighted by psi^2 in the integrals). This seems similar to the game of Whack-A-Mole, where one attempts to hit mechanical moles that pop up randomly for a brief period of time. Maybe the method could be called Whack-A-Local-Energy (WHALE).

Unfortunately for my naming scheme (and fortunately for everyone else), the method described already exists, and it's called the Energy Fluctutation Potential (EFP) method. It was first described by Stephen Fahy in "Quantum Monte Carlo Methods in Physics and Chemistry" [NATO ASI Ser. C 525 101, 1999; M. P. Nightingale and C. J. Umrigar, eds]. Further papers describing the method:

• Optimal orbitals from energy fluctuations in correlated wave functions, by Fahy and Filippi
  
• Optimization of inhomogeneous electron correlation factors in periodic solids by Prendergast, Bevan, and Fahy
  
• Optimized Jastrow-Slater wave functions for ground and excited states: Application to the lowest states of ethene by Schautz and Filippi
  

The EFP method involves an iteration that starts with a one-body solver (HF or LDA). The resulting orbitals are used in a VMC calculation, and the fluctuations in the local energy form a extra potential that is fed back into the one-body solver to get new orbitals, and the iteration repeats until a self-consistent solution is reached.

(Name note: this paper by Umrigar and Filippi calls it the Effective Fluctuation Potential method. This paper also gives another reference I didn't put on the above list)