7.2 Silicon

In this section, the silicon test system, the many-body wavefunction and its parameterisation are described. Bulk silicon in the diamond structure was chosen as a test system for several reasons. The material is commonly used as a test system for electronic structure methods and consequently a substantial literature of comparative data exists. Crucially it was considered that calculations of sufficient statistical accuracy to resolve the density matrix would be computationally affordable, partly because the same system had previously been used in studies of the pair-correlation function [14] using QMC techniques.

7.2.1 The supercell

For this study an fcc supercell containing 54 Si$^{4+}$ ions at the experimental lattice constant and 216 electrons was selected. The electron-ion potential, $v_{\alpha}$, was modeled by a norm-conserving non-local pseudopotential[57] obtained from atomic calculations performed within the local density approximation (LDA) to density functional theory.

Energies were computed in VMC using both the Ewald and MPC interactions, using a wave function which was optimized using the MPC interaction (see chapter 6). By using both interactions, some of the finite size effects could be isolated and quantified, providing a useful diagnostic as to the likely size of any finite size errors, without resorting to costly and error prone extrapolation procedures.

7.2.2 The trial wavefunction

A Slater-Jastrow wavefunction of the type described in chapter 4 was used, consisting of a single product of spin-up and spin-down determinants multiplied by a Jastrow factor. The Jastrow factor consisted of a one-body $\chi$ function and two-body correlation factor. A total of 16 parameters were optimized by minimizing the variance of the energy (see chapter 5), obtaining approximately $85\%$ of the fixed-node correlation energy.

The spin-up and spin-down Slater determinants were formed from single-particle orbitals obtained from an LDA calculation employing the same pseudopotential as in the QMC calculations. The LDA orbitals were calculated at the $\Gamma$-point of the simulation cell Brillouin zone using a plane wave basis set with an energy cutoff of 15 Ry. Although the $\Gamma$-point scheme does not give optimal Brillouin zone sampling, it allows comparison with a wider number of established results. The $\Gamma$-point of the simulation cell Brillouin zone unfolds to four inequivalent k-points in the primitive Brillouin zone. These are: (0,0,0) (the $\Gamma$-point), (0,0,$\frac{2}{3}$) $\frac{2 \pi}{a}$ (a point along the $\Delta$ axis, hereafter referred to as the $\Delta$-point), (0,$\frac{2}{3}$,$\frac{2}{3}$) $\frac{2 \pi}{a}$ (a point along the $\Sigma$ axis, hereafter referred to as the $\Sigma$-point), and ($\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$) $\frac{2 \pi}{a}$ (a point along the $\Lambda$ axis, hereafter referred to as the $\Lambda$-point).

7.2.3 VMC and DMC calculations

The VMC and fixed-node DMC calculations were carried out following the methods and procedures described in chapter 4.

In the DMC calculations a time step of 0.015 a.u. was used, which has been shown to give a small time-step error in silicon.[97] This timestep (and wavefunction) gave an acceptance/rejection ratio greater than $99.9 \%$ in the DMC calculations.