4.7 Wavefunction optimisation

The principles of wavefunction optimisation are straightforward: optimise (vary) a number of free parameters in a wavefunction in order to improve it. The preferred objective function used for optimisation is the variance of the local energy, $E_{L}=\hat{H}\Psi/\Psi$, and the reasons for this are developed in chapter 5. However, in order to evaluate any objective function related to the local energy, efficient methods for calculating this quantity are required. In this section, methods of evaluating the value of the wavefunction and its local energy for a fixed set of electron positions are presented.

During wavefunction optimisation, a representative sample of electron configurations is taken. This sample is then used for optimisation. As the parameters are varied during optimisation, the local energy is repeatedly evaluated for the fixed set of electron configurations. At a certain point, such as when parameters have been varied beyond a certain threshold, the new parameters are used to generate another set of electron configurations. This process is repeated until convergence is obtained. Typically, several thousand values of the parameters are tested during optimisation, so complete recalculation of all terms in the local energy would be very costly.

As detailed in section 4.4, the local energy consists of kinetic and potential energy terms. If the electron positions are fixed, only the kinetic energy and non-local pseudopotential energy change when the parameters are varied. These energies may be rapidly evaluated by taking advantage of the form of the Jastrow factor.

Equation 4.27, 4.28 and 4.29 show that the gradient and laplacian of the Slater determinants are also required, but the values of the actual orbitals in the determinants are not. The determinant dependent terms in the kinetic energy are independent of the Jastrow factor, with the exception of a cross term in the gradient of the determinants and Jastrow factor. This greatly simplifies the optimisation process, as it is easily made independent of the specific orbitals used.

4.7.1 Linearity of the Jastrow factor

A useful property of the Jastrow factor outlined in section 4.3.2 is the linearity of Jastrow factor with respect to the parameters. With the exception of the $A$ parameter in the cusp-satisfying part of the function, all of the parameters are linear. For a fixed set of electron positions, the value of the Jastrow function is given by

$$J=f_{0}+\sum_{k=1}^{N_{p}}\alpha_{k}f_{k} \;\;\;,$$ (4.47)

where the $N_{p}$ variable parameters, $\alpha$, are multiplied by terms $f_{k}$ that are dependent on the specific Jastrow factor and electron positions. The three Cartesian components of gradient and the value of the laplacian are similarly calculable in terms of products of parameters and ``basis functions''. By storing the $f_{k}$ and similar terms for the gradient and laplacian, the value of the wavefunction, its gradient and laplacian are computable in terms of $N_{p}$, $3NN_{p}$ and $N_{p}$ variables respectively. Recalculation of inter-particle distances and terms in the Jastrow factor is avoided, reducing the cost of the calculation by many orders of magnitude.

4.7.2 The non-local energy during wavefunction optimisation

The non-local energy is dependent on the wavefunction. During wavefunction optimisation the non-local energy changes, and should strictly be recalculated as parameter values are optimised.

For most systems, the non-local pseudopotential energy represents only a small fraction of the total energy, and the change in non-local pseudopotential energy throughout wavefunction optimisation is small. A reasonable approximation for the non-local energy is to consider it fixed throughout each cycle of wavefunction optimisation. This approximation biases the optimisation during any one cycle, but the non-local energy is recalculated when the configurations are regenerated. At worst, provided the non-local energy is only weakly dependent on the wavefunction parameters, this approximation will only slow convergence of the optimisation procedure. The approximation reduces the computational cost and memory requirements of optimisation.

The non-local energy may also be computed efficiently during wavefunction optimisation by taking advantage of the linearity of the parameters in the Jastrow factor, as is done for the evaluation of the kinetic energy.

Following equation 4.36, the non-local energy may be written

$$E_{\mathrm{nl}}=\sum_{i} \sum_{I} \sum_{g} \sum_l v_{Igl}(r^... ...iI}) \frac{\Psi({\bf r}^{g}_{iI})}{\Psi({\bf r}_{iI})} \;\;\;,$$ (4.48)

where the sums run over electrons, $i$, ions, $I$, integration grid points, $g$, and angular momenta, $l$. $\Psi({\bf r}^{g}_{iI})$ denotes the value of the many-body wavefunction with electron $i$, which is in the non-local range of ion $I$, with the electron at ${\bf r}_{i}$ moved to the integration grid point ${\bf r}^{g}_{i}$ and all other electrons unmoved. These sums may be simplified due to the short range of the non-local pseudopotential, so that only a small proportion of electrons and ions need be included. If only the linear parameters in the wavefunction are varied, the storage required for the evaluation is of order $n_p n_h n_g$, where there are $n_p$ variable parameters, $n_h$ electron-ion ``hits'' over all electrons and ions, and $n_g$ integration grid points.