4.7 Wavefunction optimisation

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

(4.47) |

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

(4.48) |