5.9 Other objective functions

The effect of using the objective function $B(\alpha)$ (equation 5.5), in which the variational energy, $E_{\rm V}$, is replaced by a fixed reference energy, $\bar{E}$, which is chosen to be lower than the exact energy is considered. In figure 5.6 objective function $B(\alpha)$ versus $\alpha$ is shown for configurations generated with $\alpha_0$ = 0. The overall shapes of the curves are hardly changed as $\bar{E}$ is decreased, although the variance of the objective function slowly increases.

If $\bar{E}$ is chosen to be too low then a significant amount of energy minimisation is included and the numerical stability deteriorates. Using a value of $\bar{E}$ slightly below $E_{\rm V}$ appears to offer no significant advantages.

Figure 5.6: Objective function $B$ versus $\alpha$ with $\alpha_0$ = 0 and $\bar{E} < E_{\rm V}$.

Figure 5.7: Comparison of variance-like objective functions with $\alpha_0$ = 0 as a function of $\alpha$.

A direct comparison of the different variance-like objective functions is made in figure 5.7. The behaviour of the following objective functions are displayed: (i) $A$, (ii) $B$ with $\bar{E} = E_{\rm V} -0.3750$ a.u., (iii) $C$, and (iv) a variant of $C$ with the maximum value of the weights limited to 10 times the mean weight. Limiting outlying values of the local energy improves the behaviour of all the objective functions, so in each case we have limited them according to equation 5.15 with $p$=8. The mean values of the objective functions are plotted in figure 5.7a, which shows them to behave similarly, with the positions of the minima being almost indistinguishable. However, the curve for the variant of $C$ with limited weights is somewhat flatter, which is an undesirable feature. The standard deviations of the objective functions are plotted in figure 5.7b, and here the differences are more pronounced. The unweighted variance, $C$, has the smallest variance, which is slightly smaller than that of the variant of $C$ with strongly limited weights. The variances of the objective functions which include the full weights increase rapidly away from $\alpha$=0. This rapid increase is highly undesirable and could lead to numerical instabilities.