5.10 Conclusions

Several energy and variance minimisation schemes for optimising many-body wavefunctions were examined, where the integrals involved are evaluated statistically. Two reasons why variance minimisation techniques are numerically more stable than energy minimisation techniques were identified: (i) In variance minimisation it is allowable to limit the weights or set them equal to unity, which reduces the variance of the objective function while introducing only a ``weak bias'', which disappears as the process converges. Altering the weights in energy minimisation normally leads to a badly behaved objective function. (ii) Variance minimisation, with or without altering the weights, shows greater numerical stability against errors introduced by finite sampling because the positions of the minima of the variance are not shifted by the finite sampling, whereas those of the (properly weighted) energy are.

By investigating a model system, configurations with outlying energies were found to be present and contribute significantly to the variances of the objective functions. The outlying energies result from divergences in the kinetic energy in certain, small, regions of configuration space. Limiting these outlying energies was found to significantly reduce the variance of variance-like objective functions, without significantly altering their value.

The best strategy found for optimising wavefunctions based on these observations is:

1. Minimise the variance of the unweighted local energy (objective function $C$, equation 5.6). This objective function is robust to finite sampling and has the best statistical properties of the all objective functions examined.
2. Limit outlying values of the local energy according to equation 5.15. Limiting the outlying energies greatly reduces the variance of variance-like objective functions, such as the variance of the unweighted local energy.
3. Regenerate the configurations several times with the updated parameter values until convergence is obtained. Regeneration of the configurations reduces the effects of finite sampling, particularly when the parameters are varied substantially during optimisation.

This strategy may be applied to both ground and excited states of atoms, molecules and solids. The behaviour observed in numerous wavefunction optimizations for large systems, presented later in this thesis, is entirely consistent with the analysis and model system results presented in this chapter. Although for small systems with few parameters other methods may be preferable, the above strategy is expected to be optimal for large systems containing many electrons where efficiency is critical.

This strategy is applied in the later chapters of this thesis, particularly in chapter 8 where wavefunctions are optimised for 13 different carbon clusters. In this application, very high efficiency was important due to the number (13) and size of the clusters (up to 128 valence electrons).