5.6 Further effects of finite sampling

In the previous section the numerical instability arising from the weighting factors was described. The origin of this problem lies in approximating the integrals by the average of the integrand over a finite set of points in configuration space. There is another important issue connected with the approximation of finite sampling, which is whether the positions of the minima of the objective function are altered by the finite sampling itself.

Consider the objective function $A(\alpha)$, in the case where the trial wavefunction has sufficient variational freedom to encompass the exact wavefunction. Approximating equation 5.3 by an average over the set $\{{\bf R}_i\}$ containing $N_s$ configurations drawn from the distribution $\Phi^2(\alpha_0)$ gives

$$A^{N_s} = \frac{\sum_i^{N_s} w({\bf R}_i;\alpha) [E_{{\rm L}... ...bf R}_i\};\alpha)]^2} {\sum_i^{N_s} w({\bf R}_i;\alpha)} \;\;.$$ (5.8)

The eigenstates of $\hat{H}$ give $A^{N_s} = 0$ for any size of sample because $E_{{\rm L}} = E_{{\rm V}}$ for an eigenstate. Clearly this result also holds for $C(\alpha)$. This behaviour contrasts with that of the variational energy, $E_{{\rm V}}$. Consider a finite sampling of the variational energy of equation 5.1, where the configurations are distributed according to $\Phi^2(\alpha_0)$ and properly weighted,

$$E_{{\rm V}}^{N_s} = \frac{\sum_i^{N_s}w({\bf R}_i;\alpha) E_... ...}}({\bf R}_i;\alpha)} {\sum_i^{N_s} w({\bf R}_i;\alpha)} \;\;.$$ (5.9)

The global minima of $E_{{\rm V}}^{N_s}$ are not guaranteed to lie at the eigenstates of $\hat{H}$ for a finite sample. The fact that the positions of the global minima of $A(\alpha)$ and $C(\alpha)$ are robust to finite sampling is a second important advantage of variance minimisation over energy minimisation.