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Control of the Reweighting Factors


The variance minimisation procedure in Eq.(gif) is stable for systems containing small numbers of electrons, but gradually becomes unstable as the numbers of electrons increases. The problem arises because the reweighting factors, tex2html_wrap_inline6979 , in Eq.(gif) deviate greatly from unity at the beginning of the optimisation process when the changes in the parameters are large due to the poor quality of the initial guess of the trial wavefunction. Note that for a small change, tex2html_wrap_inline6985 , in the function, the local energy tex2html_wrap_inline6987 changes by an amount proportional to tex2html_wrap_inline6989 , where N is the number of electrons, but the reweighting factor is multiplied by a factor which is exponential in tex2html_wrap_inline6989 . For large N, this behaviour of the reweighting factors dominates the optimisation process. One way to counteract this is to increase the number of configurations used, but this rapidly becomes impractical and it was found that it is better to reduce the importance of the reweighting factor. For tex2html_wrap_inline6997 and above, the reweighting factors were set to unity and several sets of configurations were generated with successively better functions, until the procedure converged. This works because as the procedure converges the reweighting factors approach unity. At each iteration only limited variations of the parameters were allowed, which promotes the smooth convergence of the process. This optimisation is perfectly stable even up to the largest number of electrons studied (N=338).

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996