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Why Minimise the Variance of the Energy?

Perhaps the most intuitive method of wavefunction optimisation is to vary a set of parameters within the wavefunction so as to minimise the energy with respect to the values of those parameters. Proper application of this method for a parameterised wavefunction gives the best (lowest) value for the energy of the system, but it may give poor values for other expectation values. If the energy is minimised then the local energy may be too high in some regions of configuration space and too low in others, so that the overall quality of the wavefunction is poor. This type of behaviour contributes to the variance of the energy and this therefore suggests that minimisation of the variance may give a better fit for the wavefunction as a whole, so that satisfactory results are obtained for a range of quantities, including the energy[22]. Furthermore, the variance of the energy is zero for an eigenfunction and positive for an approximate wavefunction, and therefore the quantity to be minimised has a well defined minimum value.

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