Fast variance optimization

VMC variance optimization usually uses the reweighted variance.

What is reweighting? Consider a set of particle positions obtained from sampling a wavefunction at a given set of parameters. The energy (and the variance of the energy) explicitly depends on the wavefunction parameters by the local energy formula, and implicitly through the particle positions determined by sampling the wavefunction. When optimizing, one would like to vary the parameters without going through all the trouble and expense of sampling new positions. Computing the energy (or variance) change due to the local energy piece is easy - just put the new parameters in and recalculate. Reweighting is the process of correcting the implicit dependence of the energy (or variance) on the wavefunction parameters.


N. D. Drummond and R. J. Needs describe a VMC optimization method in their paper, A variance-minimization scheme for optimizing Jastrow factors, that uses only the explicit dependence of the variance on the parameters (ie, they use the un-reweighted variance). They restrict their scheme to linear parameters in the Jastrow factor, and this makes the resulting parameter surface a quartic. Also, this scheme only requires a single sum over electron positions (the requisite factors can be accumulated during a single run), making the optimization step very fast.

Even though this scheme uses a less accurate approximation for the variance, it appears to give very good results. In fact, the resulting energies for a neon atom are better using the unreweighted variance for optimization than from using the more traditional reweighted variance.