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Choice of Average Local Energy,

Several different definitions of the average local energy, , in Eq.() were experimented with. The choice which most accurately describes the average local energy for a given set of variational parameters is the reweighted local energy, summed over all the configurations,

If one adopts this definition, then the optimisation process can be pictured as minimising the variance of the energy about a mean energy which is also slowing decreasing as the quality of the trial wavefunction improves. This is illustrated in figure . It was found that this choice of is the most unstable due to fluctuations in the reweighting factors, . The overall stability of the optimisation process was improved by choosing the simpler definition,

Figure: Schematic representation of the reduction in the energy and variance of the energy during the optimisation process. The blue line shows the mean local energy, summed over the ensemble of configurations. The black line schematically represents the variance of the local energy of the configurations.

Both definitions of are equivalent at the end point of the optimisation where the reweighting factors are all unity. The simple definition of in Eq.() has been used in all the optimisations described in this thesis. It is worth noting at this point that alternative choices for have also been proposed. Umrigar[22] often chooses to keep fixed, throughout the optimisation process, to the original VMC energy produced by the VMC calculation used to generate the configurations. Presumably, this is to improve the stability of the optimisation process still further. However, in our calculations, this choice of produced a lower quality of trial wavefunction because as the optimisation proceeds and the average local energy decreases, one is no longer minimising the variance of the energy about the mean value. Finally, one could think of attempting to 'guide' the optimisation process in the right direction by choosing to minimise the variance about a fixed energy that was lower than the original VMC energy, such as the DMC energy. This also produced a lower quality trial wavefunction in our tests.

Next: Generating Configurations Up: Optimisation Method Previous: Control of the Reweighting

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