5.2 The importance of wavefunction optimisation

Wavefunction optimisation is one of the most critical, time consuming and important stages of a QMC calculation. Several strategies can be adopted depending on the intended use for the optimised wavefunction, but in all cases the stability and efficiency of the process is critical.

Typical solid-state QMC calculations currently use of order parameters for -dimensional functions. The number of parameters and dimensionality of the problems tackled by QMC methods can only be expected to increase. The development of reliable and efficient methods for wavefunction optimisation is therefore important. Although many optimisation methods are present in the literature (some of these are reviewed in the next section), most have only been applied to small molecules where the number of parameters and problem dimensionality are small.

In VMC calculations, the accuracy of a trial wavefunction limits the statistical efficiency of the calculation and the final accuracy of the result obtained. It is desirable to use wavefunctions which are as accurate as possible yet may be rapidly computed. In practice, this means that the parameterised form of a many-body wavefunction should be relatively easy to compute, and that each parameter in the wavefunction should significantly improve the accuracy (quality) of the wavefunction. In extreme cases, such as when the best possible variational wavefunction is required, hundreds of parameters may be used. Optimisation of these parameters, and hence the wavefunction, can be difficult, particularly when there is only a small energetic dependency on some of the parameters. In this case, very long Monte Carlo calculations are required in order to obtain statistically significant data. It is clearly desirable, in all cases, that any optimisation method makes efficient use of data and is stable, delivering consistent improvements to the wavefunction.

In DMC calculations, the results obtained are less dependent on the
trial wavefunction than in VMC calculations. Typically, the nodes of
the wavefunction are not optimised and only the Jastrow part of the
wavefunction are parameterised. Speed of evaluation is a little more
important than in VMC calculations because derivatives of the Jastrow
function are always required. The overall
efficiency of a DMC calculation is determined by how well the
trial wavefunction approximates an eigenstate over *all* of
configuration space, as the accuracy of the local energy determines
the branching of configurations.