is the total number of configurations and is typically chosen to be 8, although varying from 4 to 12 makes no significant difference to the results.
A factor of was included, rather than limiting the energies beyond a given number of standard deviations to incorporate the concept that as more configurations are included, the sampling is improved. In the limit of perfect sampling, , the objective functions are unchanged. For the model silicon system, the percentage of configurations having their local energies limited by this procedure, with = 8, is only 0.024% and 0.047% for = 0.03 and 0, respectively. These values corresponds to those configurations lying beyond 5.7 standard deviations from the mean. The effect of limiting the outlying local energies is illustrated in figure 5.4. In figures 5.4a and c the mean values of the objective functions and versus for configurations generated with = 0.03, with values of the limiting power, , in equation 5.15, of 4, 8, 12 and infinity (no limiting) are shown, while in figures 5.4b and d their variances are plotted.
The mean values of are hardly affected by the limiting, while those of are only slightly altered. The smaller variances of and obtained by limiting the values of the local energy are very clear. If the local energies are not limited then the variances of the objective function are not very accurately determined, even with the large samples of configurations. Similar results hold for configurations generated with = 0. This simple procedure can greatly reduce the variance of the variancelike objective functions without significantly affecting their mean values. Limiting the local energies is even more advantageous when small numbers of configurations are used. Limiting the local energies in this way gives significantly better numerical behaviour for all the variancelike objective functions and therefore all data shown in figures 5.55.7 have been limited with =8, unless explicitly stated otherwise.

Limiting the values of the weights is a crucial part of the variance minimisation procedure for large systems. Comparison of figures 5.4b and d shows that the variance of the unweighted objective function is smaller than that of the weighted objective function, , for all values of , provided one limits the local energies. The variances close to the minimum are similar but away from the minimum the variance of increases much more rapidly than that of . The smaller variance of indicates the superior numerical stability of the unweighted function. Qualitatively similarly behaviour occurs for configurations generated with = 0. A commonly used alternative to setting the weights equal to unity is to limit the maximum value of the weights. In figure 5.5 data for objective function with the largest value of the weights limited to multiples of 1 and 10 times the mean weight is shown, along with data for the weights set to unity. In this graph the standard deviations of the objective functions are plotted as error bars. Figure 5.5 shows that the variance of is reduced as the weights are more strongly limited, but the lowest variance is obtained by setting the weights to unity. In addition, when the weights are limited the curvature of the objective function is reduced, which makes it more difficult to locate the minimum. Setting the weights to unity therefore gives the best numerical stability.