My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 5.
THE JELLIUM SLAB
reducing the average kinetic energy at the expense of increasing the potential energy
by a larger amount.
The problem occurs both when reweighting of configurations is used and when
it is not. This suggests that the cause is not outliers (configurations with energy far
from the mean). Figure 5.7 shows that the energy distributions are close to being
Gaussian, with only slightly ‘fat’ upper tails.
The configurations sampled initially do not include any with electrons far outside
the slab; the electron density decays sharply to zero here, as can be seen in figure 5.1.
After the first stage of variance minimisation, the Jastrow factor is altered so that
the electron density extends much further outside the slab; therefore the next set
of configurations which are generated will sample a different region of configuration
space to the old set. The effect of pushing many electrons into the vacuum region,
and the consequential increase in the potential energy, cannot be ‘known’ in advance
by the variance minimisation routine: no such configurations have been sampled.
This suggests that the process may be corrected by modifying the initial sampling
distribution to include these configurations. Unfortunately, this approach did not
work; the procedure remains unstable, with the mean and variance of the local
energy usually worsening after each iteration.
Without reweighting, the object of the variance minimisation step is to minimise
the following quantity:
O({X i ; α}) = ∑ i
[
E L (X i ; α) − 1 N
∑
2
E L (X j ; α)]
. (5.8)
j
As before, α denotes the optimiseable parameter, {X i } is the set of configurations
and E L is the local energy. This may be split into kinetic and potential terms; the
potential energy does not depend on α:
O({X i ; α}) = ∑ i
[
T i (α) − ¯T (α) + V i − ¯V
] 2
. (5.9)
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