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Generating Configurations

  The configurations used in in the optimisation procedure Eq.(gif) are generated by periodically storing the positions of all the electrons (a configuration) after a certain number of moves during a VMC calculation. To keep the computation time for generating the configurations to a minimum, the number of moves required to produce independent configurations (i.e. the correlation length) was investigated by a procedure based on that used by Jacucci and Rahman [65] which works as follows.

A series of N sequential configurations and their energies were generated by a VMC calculation. The N energies were divided up into b blocks each containing tex2html_wrap_inline7027 energies. The average value for each block and the variance of the block averages are then given by


When the block size is large enough such that the individual block averages tex2html_wrap_inline7029 can be considered as being independent, the value of tex2html_wrap_inline7031 might be expected to be inversely proportional to tex2html_wrap_inline7027 . This is because the individual error in the mean, tex2html_wrap_inline7035 is proportional to tex2html_wrap_inline7037 . In an attempt to calculate this constant of proportionality, the statistical inefficiency, s, is defined as


It is possible to calculate a value for s from the N energies by plotting the value of s for a series of block sizes. Finally, from the definition of the variance in the mean for a series of N values and a correlation length tex2html_wrap_inline6435 , we have


It is clear that for the case of a single block that constitutes the whole sample (i.e. tex2html_wrap_inline7051 ) s is equivalent to tex2html_wrap_inline6435 in (gif). For the energies produced in a VMC calculation the value of s proved to be much smaller than expected, about 2.8. This meant that every third move of all the electrons could be written out as an independent configuration. Where results are presented later in this report for fitting with 1000 configurations this required a 3000 move VMC calculation, and 30,000 moves were required to generate 10,000 configurations.

next up previous contents
Next: Optimising the function Up: Optimisation Method Previous: Choice of Average Local

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