To determine the optimal choice of 
for diamond-structure
germanium, a series of LDA calculations were performed using
-point meshes with different offsets from the
origin[33]. The offset from the origin determines the
value of of the Slater determinant constructed from the
orbitals. The results of these calculations are illustrated in
figure 
. The results show the convergence of the total
energy with supercell size for four different values of .
For the 
-point ( =0) the energy converges slowly as
the size of the supercell is increased. A far more rapid convergence
is observed when L-point sampling is used. This means the
-point grid is offset from the origin by 
,
where 
and the 
are
the primitive reciprocal lattice translation vectors of the supercell.
The best convergence is obtained when using an offset equal to the
Baldereschi mean value point[53] of the supercell BZ.The
Baldereschi mean point was not chosen for the QMC calculations, as
wavefunctions at that -point are necessarily complex and the
use of complex arithmetic would slow down the code considerably. It
is possible to construct a real wavefunction using Baldereschi-point
states by taking a linear combination of 
and its
complex conjugate, 
, as both
and have the same energy eigenvalue.
Unfortunately, this combination is no longer a Bloch function, and
this results in a local energy function that does not have the
periodicity of the supercell.
Although it would be too computationally expensive to repeat all the
LDA calculations from figure within the QMC formalism,
VMC and DMC calculations for supercells with n = 2 and 3
[33] show that QMC calculations follow the LDA trend
very closely. The calculations described in
chapters and use Slater
determinants where the one-electron orbitals have -points
chosen on a mesh offset from the origin at the L-point. This
corresponds to Monkhorst-Pack (MP)[54] sampling for n
even supercells and better than MP sampling for n odd supercells.
Figure: Convergence of the total
energy with simulation cell size for different -point
sampling schemes. The black line shows the results of 
sampling. The green line shows 
sampling, where the are
the primitive reciprocal lattice translation vectors of the
supercell. The red line shows 
sampling, and
the blue line shows 
sampling, where 
is the Baldereschi mean value point.