As the expectation value of the non-local pseudopotential taken with respect to the trial wavefunction is a small fraction of the total energy, one option is to store the non-local energy associated with each configuration at the beginning of the optimisation procedure and keep it fixed. Care has to be taken when adopting this method to be sure to evaluate the non-local energy to sufficient accuracy at the beginning of the optimisation procedure. During a VMC calculation, the non-local pseudopotential is evaluated by the method proposed by Fahy et al. . The non-local Hamiltonian for the i electron is given by
where the ion is at the origin. is the angular momentum l projection operator acting at a distance r from the origin;
The non-local energy is then evaluated along the Monte Carlo random walk of points sampled from according to
The z-axis is chosen along to use the fact that for to simplify such that
The integral over in Eq.() is performed statistically with the QMC code. A series of points are sampled on a grid surrounding each ion and the ratio of the value of the wavefunction with all other electrons fixed, and the electron at and at each of the grid points is used to evaluate Eq.().
During a VMC calculation, it is not necessary to sample enough points from the spherical grid to evaluate the non-local integral to high precision. Any variance in the value of Eq.() will be averaged out over the duration of the run. However, if the non-local energy is to be kept fixed throughout a variance minimisation run, it is important to ensure the sampling in the non-local integral is sufficient for each individual value of the non-local energy to have a small variance. Typically, up to 8 times as many sampling points are used to ensure accurate individual values of the non-local potential energy compared with a normal VMC calculation.