; Variational Quantum Monte Carlo and Diffusion
Monte Carlo. Recent work [8] has been carried out to
calculate the total energy of solid germanium in the diamond structure
for a 2x2x2 supercell containing 16 Ge atoms and 64 valence electrons
using both the VMC and DMC technique. The results are illustrated in
Figure .
Figure: Difference in the energy of VMC and DMC results.
Where the VMC energy is defined by
and the DMC energy is defined in the long time approximation as
where 
is the wavefunction with the lowest energy consistent
with the nodal structure of 
.
As mentioned in section , the DMC energy is in principle exact
(to within the fixed node approximation), i.e. for a given guiding
wavefunction 
, the DMC energy is the lowest possible energy for any
wavefunction with the same nodal surface as 
.
Now the DMC and VMC calculations share the same trial wavefunction, 
and hence have the same nodal surface. Therefore, the difference in
energy 
is due only to the difference
between the VMC trial wavefunction 
and the wavefunction to which the
DMC calculation converges to in the long time approximation, 
.
This difference will be referred to as the quality of 
where the
quality of 
describes how closely it matches the
converged DMC ground state wavefunction, 
throughout all space.
Comparison of the DMC and VMC variance of the energy is more complicated. The VMC variance of the energy is
and the DMC variance of the energy in the long time approximation is
Now, as 
throughout any domain bounded by the nodes of
, the DMC variance of the energy, 
is zero.
However, due to approximations made during the DMC calculations such as
the finite time step used and not moving all the electrons at once,
will not be identically zero.
Even including these effects the DMC variance of the
energy is still considerably smaller than the VMC variance of the
energy and this can again
be attributed to the quality of 
. For the VMC calculations
illustrated in Figure the VMC variance of energy,
was 2.58 units.
In summary, the difference in quality of the VMC trial wavefunction and
the wavefunction to which the DMC result converges, 
,
has three main effects; it raises the VMC energy with respect to the DMC
energy, it increases the variance of the energy in a VMC
calculation and it increases the time for a DMC calculation to reach a
converged result. Therefore, the motivation to try and improve the
quality of 
can be summarised by the following advantages.
should reduce the time taken for a DMC
calculation to reach a converged energy. This is because if 
is
closer to 
the probability distribution 
(see section
) starts closer to its converged form.
(see section ).
Therefore, if the quality of 
improves,
i.e. it moves closer to 
then the branching will be
reduced and this makes the DMC calculation more efficient.
, the number of moves in a VMC calculation required to achieve a
specific variance of the mean 
decreases linearly with
the reduction in variance. i.e. 
, where N
is the number of moves.
improves significantly, the gap between the VMC and DMC energies
might be reduced enough to remove the need to do a DMC calculation at all.
This not only greatly reduces the computational expense of calculating
ground state energies but it also allows the use of techniques such as
non-local pseudopotentials and correlated sampling that cannot be used
in DMC calculations.