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**Subsections**

#

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.

For this study an fcc supercell containing 54 Si ions at the
experimental lattice constant and 216 electrons was selected. The
electron-ion potential, , 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.

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 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 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 -point of the simulation cell Brillouin
zone using a plane wave basis set with an energy cutoff of 15 Ry.
Although the -point scheme does not give optimal Brillouin
zone sampling, it allows
comparison with a wider number of established results. The
-point of the simulation cell Brillouin zone unfolds to four
inequivalent ** k**-points in the primitive Brillouin zone. These
are: (0,0,0) (the -point), (0,0,)
(a point along the axis, hereafter referred to as
the -point), (0,,)
(a point along the axis, hereafter referred to as the
-point), and
(,,)
(a point
along the axis, hereafter referred to as the
-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 in the DMC
calculations.

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** Previous:** 7.1 Introduction
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© Paul Kent