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5.3 Objective functions

In order to optimize a wavefunction we require an objective function, i.e., a quantity which is to be minimised with respect to a set of parameters, $\{\alpha\}$ . The criteria that a successful objective function should satisfy for use in a Monte Carlo optimization procedure are that (i) the global minimum of the objective function should correspond to a high quality wavefunction, (ii) the variance of the objective function should be as small as possible, and (iii) the function is suitable for numerical optimisation. Ideally the global minimum of the objective function should be as sharp and deep as possible and there should be no local minima, i.e. as best adapted to numerical optimization algorithms as possible.

One natural objective function is the expectation value of the energy,

$\begin{displaymath} E_{{\rm V}} = \frac{\int \Phi^2(\alpha) \; [\Phi^{-1}(\alpha... ...\Phi(\alpha)]\,d{\bf R}}{\int \Phi^2(\alpha) \,d{\bf R}} \;\;, \end{displaymath}$

(5.1)

where the integrals are over the 3-dimensional configuration space. The numerator is the integral over the probability distribution, $\Phi^2(\alpha)$ , of the local energy, $E_{{\rm L}}(\alpha) = \Phi^{-1}(\alpha) \hat{H} \Phi(\alpha)$ .

The energy is in fact not the preferred objective function for wave function optimization, and the general consensus is that a better procedure is to minimise the variance of the energy, which is given by

$\begin{displaymath} A(\alpha) = \frac{\int \Phi^2(\alpha) \; [E_{{\rm L}}(\alpha... ...\alpha)]^2 \, d{\bf R}}{\int \Phi^2(\alpha) \, d{\bf R}} \;\;. \end{displaymath}$

(5.2)

Optimising wavefunctions by minimising the variance of the energy is actually a very old idea, having being used in the 1930's. The first application using Monte Carlo techniques to evaluate the integrals appears to have been by Conroy, [80] but the present popularity of the method derives from the developments of Umrigar and coworkers. [37,29]

A number of reasons have been advanced by many workers in the field for preferring variance minimisation over energy minimisation, including: (i) it has a known lower bound of zero, (ii) the resulting wavefunctions give good estimates for a range of properties, not just the energy, (iii) it can be applied to excited states, (iv) efficient algorithms are known for minimising objective functions which can be written as a sum of squares, and (v) it exhibits greater numerical stability than energy minimisation. The latter point is very significant for applications to large systems.

Minimisation of $A(\alpha)$ has normally been carried out via a correlated sampling approach in which a set of configurations distributed according to $\Phi^2(\alpha_0)$ is generated, where $\alpha_0$ is an initial set of parameter values. $A(\alpha)$ is then evaluated as

$\begin{displaymath} A(\alpha) = \frac{\int \Phi^2(\alpha_0) \; w(\alpha)\; [E_{{... ...{\bf R}}{\int \Phi^2(\alpha_0) \; w(\alpha) \, d{\bf R}} \;\;, \end{displaymath}$

(5.3)

where the integrals contain a weighting factor, $w(\alpha)$ , given by

$\begin{displaymath} w(\alpha) = \frac{\Phi^2(\alpha)}{\Phi^2(\alpha_0)} \;\;. \end{displaymath}$

(5.4)

$A(\alpha)$ is then minimised with respect to the parameters $\{\alpha\}$ . The minimum possible value of $A(\alpha)$ is zero. This value is obtained if and only if $\Phi(\alpha)$ is an exact eigenstate of $\hat{H}$ . The ensemble of configurations is normally regenerated several times with the updated parameter values so that when convergence is obtained no change in parameters $\{\alpha\}$ occurs on optimization. A variant of equation 5.3 is obtained by replacing the energy $E_{{\rm V}}(\alpha)$ by a fixed value, $\bar{E}$ , giving

$\begin{displaymath} B(\alpha) = \frac{\int \Phi^2(\alpha_0) \; w(\alpha)\; [E_{{... ...{\bf R}}{\int \Phi^2(\alpha_0) \; w(\alpha) \, d{\bf R}} \;\;. \end{displaymath}$

(5.5)

Note that if $\bar{E} \leq E_0$ , where is the exact ground state energy, then the minimum possible value of $B(\alpha)$ occurs when $\Phi$ = $\Phi_0$ , the exact ground state wavefunction. Minimisation of $B(\alpha)$ is equivalent to minimising a linear combination of $E_{\rm V}$ and $A(\alpha)$ . [23] The absolute minima of both $E_{\rm V}$ and $A(\alpha)$ occur when $\Phi$ = $\Phi_0$ . If both of the coefficients of $E_{\rm V}$ and $A(\alpha)$ in the linear combination are positive, which is guaranteed if $\bar{E} \leq E_0$ , then it follows that the absolute minimum of $B(\alpha)$ occurs at $\Phi$ = $\Phi_0$ . Using this method with $\bar{E} \leq E_0$ allows optimization only of the ground state wavefunction, and it has been claimed that this method yields more accurate wavefunctions than variance minimisation alone (for example, Ref.[53]).

Next: 5.4 Numerical instabilities Up: 5. Wavefunction optimisation Previous: 5.2 The importance of Contents