Following the method of Ref., we choose to represent the ionic cores in our germanium and silicon supercell calculations with pseudopotentials. This enables the number of valence electrons that are explicitly handled by the QMC algorithm to be reduced to four per atom in both cases.
The pseudopotential used to represent the Ge ions in the germanium calculations described in chapter was a local pseudopotential of the Starkloff-Joannopoulos form. The pseudopotential used to represent the Si ions in the silicon calculation described in chapters and is a norm-conserving, non-local pseudopotential generated using the method described by Kerker. In this pseudopotential, the s and p potentials were generated from an atomic groundstate and the d potential was generated from an atomic configuration as in Ref.. In our calculations we chose the p potential to be the local potential as this results in a smaller contribution from the remaining non-local potential to the total energy than choosing either s or d to be local. A small non-local energy is desirable as the non-local energy is evaluated by a statistical integration within the QMC code. This integration is expensive to evaluate and can be evaluated more approximately (and cheaply) if the overall contribution from the non-local potential is small. Also, in DMC calculations, we would like the non-local energy to be as small as possible to reduce the effect of the ``locality approximation''.
Both these pseudopotentials feature a cutoff radius, beyond which the pseudopotential reduces to the full Z/r potential due to a +Z point charge, where Z is the valence of the ion. To deal with the long ranged tails of the ionic potential the Ewald prescription, as described in the previous section, is used to evaluate the interaction energy between the lattice of charges representing the ionic core and all its periodic images and the lattice representing an electron and all its periodic images. This is illustrated in figure . For each electron-ion pair, if the electron is outside the cutoff radius of the pseudopotential (position 1 in figure ), then the Ewald interaction is directly applied to calculate the Coulomb energy between the two corresponding lattices of charged particles and their screening background charges. If the electron falls within the cutoff radius of the pseudopotential, (position 2 in figure ), then the Ewald interaction is still used to evaluate the interaction between the electron and all the periodic images of the ion, but a correction is applied to include the effect of the pseudopotential from the ion in the simulation cell on the electron in the simulation cell and identical ``in cell'' effects in all the periodic images of the simulation cell.
Figure: Schematic representation of the electron-ion interaction. The red point represents an ionic core with a cutoff radius, , marked by the red circle. The blue circles marked 1 and 2 represent two different positions for an electron in the simulation cell. Only a central simulation cell and one periodic image in each direction is shown.