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The Variance Minimisation Method

We begin by writing the variance of the energy as


where tex2html_wrap_inline5881 is the Hamiltonian as defined in chapter gif and tex2html_wrap_inline6049 is the trial/guiding wavefunction which is to be optimised. The sum is over a set of 3N-dimensional electron configurations, tex2html_wrap_inline6975 , tex2html_wrap_inline6499 is an average energy,


and the reweighting factors, tex2html_wrap_inline6979 , are given by


The electron configurations are sampled from the starting distribution tex2html_wrap_inline6981 and then kept fixed throughout the optimisation. This ``correlated sampling'' approach gives a good estimate of the difference in variance between wavefunctions corresponding to different sets of parameters. The process can be used iteratively by using the optimised set of parameters to regenerate a new set of configurations which are then used to perform a new optimisation. This is useful when the reweighting factors differ significantly from unity (see section gif). The non-linear optimisations over the multi-dimensional parameter spaces were performed using the NAG\ routine E04FCF. This works by finding the unconstrained minimum of a sum of squares, as in Eq.(gif), using a modified Newton algorithm that requires the function values only.

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