Hierarchical VMC (was Probabilistic Functions III)

A couple comments from the last post.

I had wondered whether it was possible for VMC to get as good of results as DMC. In theory, of course, the answer is "yes". Provided there is enough variational freedom. This is the hard part.

On to the second point.

I had proposed a variational wavefunction consisting of the sum of gaussians. The centers and widths were then the parameters to be optimized. The starting center points were to be generated by sampling from a more traditional wavefunction. Why bother? Why not start from a uniform distribution?

My speculative answer is that it would be horribly inefficient, and very likely to follow wrong paths (since the wavefunction is so "floppy"). So starting from a traditional wavefunction is a form of accelerating the convergence of the optimization procedure.

The idea seems to be one of hierarchical VMC, or of a hierarchical wavefunction that can be optimized in steps. The coarse parts are optimized first, and then the finer details are added later. Vaguely like a wavelet decomposition or a multigrid method deals with spatial resolution.

It still remains to be seen if the sum-of-gaussians wavefunction survives contact with reality.