next up previous contents
Next: DMC Calculations Up: Motivation Previous: Motivation

VMC Calculations

The importance of the trial/guiding wavefunction in determining the accuracy of VMC calculations is illustrated by recent work [33] to calculate the total energy of solid germanium in the diamond structure using a 2x2x2 supercell containing 16 Ge atoms and 64 valence electrons within both the VMC and DMC formalisms. The results are illustrated in Figure gif.

Figure: Difference in the energy of VMC and DMC results for 2x2x2 bulk germanium in the diamond structure.

The VMC energy and DMC energy are defined as



where tex2html_wrap_inline6875 is the wavefunction that tex2html_wrap_inline6049 propagates to during the diffusion process.

It has already been stressed in section gif that the DMC energy is in principle exact (apart from the fixed-node approximation), i.e. for a given trial/guiding wavefunction tex2html_wrap_inline6049 , the DMC energy is equivalent to the lowest variational energy for all wavefunctions with the same nodal surface as tex2html_wrap_inline6049 . As the DMC and VMC [50] calculations share the same trial wavefunction, tex2html_wrap_inline6049 and hence have the same nodal surface, the difference in energy tex2html_wrap_inline6885 is due only to the difference between the VMC trial wavefunction tex2html_wrap_inline6049 and the wavefunction, tex2html_wrap_inline6875 , to which the DMC calculation converges to in the long time approximation. This energy difference acts as a measure of the quality of tex2html_wrap_inline6049 , where the quality describes how closely tex2html_wrap_inline6049 matches the converged DMC ground state wavefunction, tex2html_wrap_inline6875 , throughout all of configuration space. If one is able to optimise the trial/guiding wavefunction and therefore improve its quality, then the difference tex2html_wrap_inline6885 will be reduced and the accuracy of the VMC calculations is improved.

It is not only the value of the total energy which reflects the quality of a trial/guiding wavefunction. The intrinsic variance of the energy estimator, as defined by


is also important. If one is able to reduce the intrinsic variance, tex2html_wrap_inline6899 , of the VMC energy by improving the quality of tex2html_wrap_inline6049 , then the number of VMC moves required to achieve a specific variance of the mean, tex2html_wrap_inline6903 , decreases linearly with the variance. i.e. tex2html_wrap_inline6905 , where N is the number of moves.

next up previous contents
Next: DMC Calculations Up: Motivation Previous: Motivation

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