5.5 Analysis of objective functions

In this section, different objective functions are considered and their suitability for wavefunction optimisation is evaluated. First, the procedure of setting the weights to unity is considered. This gives a new objective function, $C(\alpha)$, where

$$C(\alpha) = \frac{\int \Phi^2(\alpha_0) \; [E_{{\rm L}}(\alp... ...lpha)]^2 \, d{\bf R}}{\int \Phi^2(\alpha_0) \, d{\bf R}} \;\;.$$ (5.6)

This objective function has the property that its absolute minimum is zero and that this value is obtained if and only if $\Phi(\alpha)$ is an exact eigenstate of $\hat{H}$, i.e., the minima have the same properties as those of $A(\alpha)$. A similar analysis can be applied to the case where the weights are subject to an upper limit, and we will refer to all such expressions with modified weights as variants of $C$.

It is interesting to consider the behaviour of the expectation value of the energy calculated without weighting. The minima of

$$E_{C} = \frac{\int \Phi^2(\alpha_0) \; [\Phi^{-1}(\alpha) \h... ...\alpha)] \, d{\bf R}}{\int \Phi^2(\alpha_0) \, d{\bf R}} \;\;,$$ (5.7)

are not located at the eigenstates of $\hat{H}$ unless $\Phi(\alpha_0)$ is an eigenstate. The energy $E_C$ is therefore not a satisfactory objective function. Given this result it is easy to see that if we replace the energy $E_{{\rm V}}(\alpha)$ in equation 5.6 by $\bar{E}$, then unless $\bar{E}$ is equal to an eigenenergy then the minima of this new variance-like objective function are not located at the eigenstates. Therefore, if the weights are to be limited or set to unity, then one should use the reference energy $E_{{\rm V}}$ in the variance rather than a value of $\bar{E}$ which is below $E_{{\rm V}}$.

The above analysis applies for wavefunctions with sufficient variational freedom to encompass the exact wavefunction. In practical situations we are unable to find exact wavefunctions and it is important to consider the effect this has on the optimization process. Although the objective functions $A(\alpha)$ and $C(\alpha)$ are unbiased in the sense that the exact ground state wavefunction corresponds to an absolute minimum, $C(\alpha)$ is biased in the sense that for a wavefunction which cannot be exact the optimized parameters will not exactly minimise the true variance. This may be considered a ``weak bias'' because it disappears as the wavefunction tends to the exact one. In practice this is not a problem because in minimising $C(\alpha)$ the configurations are regenerated several times with the updated distribution until convergence is obtained, so that minimisation of $A(\alpha)$ and $C(\alpha)$ turns out to give almost identical parameter values. On the other hand, the unweighted energy, $E_C$, shows a ``strong bias'' in the sense that the nature of its stationary points are very different from those of the properly weighted energy. In section 5.7 it is shown that the exact wavefunction may not even correspond to a minimum of $E_C$. The ability to alter the weights while not affecting the positions of the minima is an important advantage of variance minimisation over energy minimisation. This is one of the factors which leads to the greater numerical stability of variance minimisation over energy minimisation.