next up previous contents
Next: The Variance Minimisation Method Up: Optimisation Method Previous: Why Minimise the Variance

Previous Applications of Variance Minimisation

The method of variance minimisation was first applied to quantum mechanical problems in the 1930's. It was first used in QMC calculations by Coldwell [61], and some of the most impressive QMC applications have been by Umrigar and coworkers [22, 30]. Umrigar developed the variance minimisation technique [22] to calculate wavefunctions for use in VMC and DMC calculations on the Be atom. He took the standard atomic trial wavefunction


where tex2html_wrap_inline6927 is equal to 1/2 for antiparallel-spin electrons and 1/4 for parallel-spin electrons, to satisfy the cusp condition. He then optimised the value of the parameter b using variance minimisation.

He then attempted to generalise the Jastrow part of the wavefunction to take account of the individual positions of the electrons as well as the electron-electron separation.


where tex2html_wrap_inline6931 and P is a complete tex2html_wrap_inline6935 order polynomial in r, s and t with sets of coefficients [a] and [b]. The sets of coefficients [a] and [b] were then optimised using the same variance minimisation technique. The error in the expectation value of the energy was reduced from 0.001 to 0.000003 Hartrees by the optimisation process.

Recently, Umrigar and Filippi[62] have extended their work to study first row diatomic molecules with QMC. They have used multiconfigurational wavefunctions and used the variance minimisation method to not only optimise the Jastrow and functions but also to optimise the wavefunctions with respect to some variational parameters present in the one-electron wavefunctions that make up the Slater determinants.

Mitas and Martin have also optimised wavefunctions for use in atomic QMC calculations [41]. They also chose a two-body correlation function that depends on the electron-electron separation and an additional electron-ion term, tex2html_wrap_inline6951




and tex2html_wrap_inline6953 is the separation between the tex2html_wrap_inline6451 electron and the tex2html_wrap_inline6957 ion, and tex2html_wrap_inline6959 . The largest range of k,l,m used was from 0 to 5 but only a subset of values were actually used.

When performing VMC calculations on atomic and molecular nitrogen [63], Mitas and Martin used 21 variational parameters in tex2html_wrap_inline6951 . These were optimised using the Umrigar minimisation of variance technique.

Mitas and Martin have also begun optimising wavefunctions for use in QMC calculations on solids. They used the same correlation function, Eq.(gif) from the nitrogen atom calculations, but with only 6 optimised parameters, to perform calculations on solid nitrogen.

A few QMC trial/guiding wavefunctions have also been generated by methods other than the Umrigar minimisation of variance technique [22]. Tanaka [64] has performed VMC calculations on the cohesive energy of NiO using wavefunctions generated by minimising the total energy of the wavefunction. He used a trial wavefunction where the Jastrow factor took its standard form


and the Chi function was expressed as a sum of Gaussians


where tex2html_wrap_inline6965 is the electron position and tex2html_wrap_inline6967 are the positions of the ionic cores.

next up previous contents
Next: The Variance Minimisation Method Up: Optimisation Method Previous: Why Minimise the Variance

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