My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
When the trial wave function is equal to an eigenfunction of Ĥ, this variance is
reduced to zero. Minimising the variance, rather than the energy, has several advantages,
the most important being that it is much more numerically stable.
Variance minimisation first appeared in the 1930s [82, 7] and was used in numerical
optimization of trial wave functions by Conroy in the 1960s [16, 14, 15]. It
was originally applied to VMC by Coldwell [13], but became popular largely thanks
to the efforts of Umrigar and coworkers [80]. Several modifications of the original
scheme exist and are in use [40].
The usual way to carry out the minimisation involves correlated sampling. A set
of configurations is generated by a VMC calculation, using an initial set of parameter
values α 0 . The variance for a new set of parameter values α (close to the original
set) is then
σ 2 E L
(α) =
∫
Ψ 2 (X; α 0 )w(α, α 0 )(E L (X; α) − E V (α)) 2 dX
∫
Ψ2 (X; α 0 )w(α, α 0 ) dX
(3.84)
where the variational energy is now calculated as
E V (α) =
∫
Ψ 2 (X; α 0 )w(α, α 0 )E L (X; α) dX
∫
Ψ2 (X; α 0 )w(α, α 0 ) dX
(3.85)
and a weighting factor has been introduced:
w(α, α 0 ) = Ψ2 (X; α)
Ψ 2 (X; α 0 ) . (3.86)
The number of configurations is, of course, finite, and the integrals indicated here
are approximated by finite sums.
The advantage of the correlated sampling approach is that, in theory, only one
set of configurations needs to be generated. In practice, it is almost always necessary
to generate more than one set of configurations. This is because the minimisation
process may become numerically unstable; this instability is characterised by a few
configurations acquiring very large weights, and leads to incorrect results [40]. Thus,
once the variance of the weights reaches a certain level, it is normal to regenerate the
configurations (and therefore reset all weights to unity) [22]. It is often preferable
(particularly in large systems) simply to set all the weights equal to unity [76, 83],
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