With the new expression in Eq. () for the electron-electron interaction proving so successful for calculations on the HEG, it was decided to attempt to produce a more general expression for use in inhomogeneous solids. Now the long range interaction term is non-trivial because the charge density is no longer uniform. In a VMC calculation the electronic charge density, , appearing in Eq. () could be accumulated during the simulation and the interaction energy evaluated afterwards. In a DMC calculation this is not possible because the total energy needs to be evaluated at the end of each step of the simulation. Therefore, one must know the charge density before starting the calculation. To overcome this problem calculations were performed with the LDA charge density, , as the reference density in Eq. () i.e.
It would be possible to update the input charge density afterwards and perform a self-consistent calculation. However, LDA charge densities are normally remarkably good and, moreover, the interaction energy is insensitive to the quality of the charge density used because differs significantly from zero only when is large. This approximation was found to be so successful that for convenience it was also used for VMC calculations and in the variance minimisation procedures for optimising wavefunctions.
To implement the new interaction into the QMC code for solids described in chapter the following changes were made
This extra one-body potential is stored in reciprocal space and accumulated on the same set of primitive cell reciprocal lattice vectors that the function is evaluated on and the density is accumulated on.
The interaction leading to the energy expression in Eq. () is illustrated in Fig. . This depicts a rhombohedral simulation cell containing two electrons, on one of which is centred a hexagonal window corresponding to the Wigner-Seitz cell of the simulation cell. The (red) electron at the centre of the window experiences a 1/r interaction with all the other electrons within the window (one (blue) in this case) and an electrostatic interaction with the electronic charge density of the shaded region outside of the window.
Figure: An illustration of the new interaction for a rhombohedral simulation cell containing two electrons (crosses). The hexagonal clear window centred on one of the electrons has the shape of the Wigner-Seitz cell of the simulation cell.