Probabilistic Functions II

Last time I looked at the sequence of values obtained from sampling a function. I was trying to understand what would happen if used a sum of gaussians, each centered at one of these values.

I was confused because I had a set of sample points, but also a function using those points that could be considered a variational trial function in its own right.
The width of gaussians can be taken as a variational parameter, and optimized.

The idea for a MC scheme would be to generate a set of sample points from a trial function. Then the diffusion algorithm would be stepped forward in time to get a better approximation to the wave function.

But we can go further and consider all the center points as variational parameters as well. And we don't actually care about the diffusion part - it is only a device to relax the wavefunction to the ground state. We could use any optimization method instead. Now this form for the wavefunction is likely to have lots linear dependencies. Some optimization methods may have problems with this.

One could simply do a random walk - try moving a point, evaluate the energy (with VMC), and accept or reject the move based on the energy difference. Could one get DMC-like quality from VMC?

I can see a couple possible problems, so far. Getting the boundary conditions and cusp conditions correct may be difficult. The "basis" functions don't need to be a set of gaussians - just positive functions with a center point and a width. Also, there may be so much variational freedom that convergence takes a long time.