CASINO manual - Theory of Condensed Matter

This leads to the single-electron detailed balance condition
s i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N)Ψ 2 (r 1 , . . . , r i−1 , r ′ i, . . . , r ′ N )
= s i (r 1 , . . . , r i−1 , r ′ i ← r i , r ′ i+1, . . . , r ′ N)Ψ 2 (r 1 , . . . , r i , r ′ i+1, . . . , r ′ N),
(27)
where
s i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N) = a i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N)
×t i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N ), (28)
is the effective single-electron transition probability density, once the accept/reject step has been
introduced.
Hence we find that the effective transition probability density for the entire configuration move satisfies
S(R ← R ′ ) =
=
N∏
s i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N)
i=1
N∏
i=1
s i (r 1 , . . . , r i−1 , r ′ i ← r i , r ′ i+1, . . . , r ′ N) Ψ2 (r 1 , . . . , r i , r ′ i+1 , . . . , r′ N )
Ψ 2 (r 1 , . . . , r i−1 , r ′ i , . . . , r′ N )
= S(R ′ ← R) Ψ2 (R)
Ψ 2 (R ′ ) . (29)
And so detailed balance in configuration space is satisfied.
It is more efficient to use an electron-by-electron algorithm than the (perhaps more straightforward)
configuration-by-configuration algorithm in which moves of entire configurations are proposed and
then accepted or rejected. This is because, for a given time step, a configuration will travel further
on average if the accept/reject step is carried out for each electron in turn. For example, it is clear
that it is very unlikely for a configuration not to be moved at all in an electron-by-electron algorithm.
Hence the sampling of configuration space in an electron-by-electron algorithm is more efficient.
After each move of each electron we check whether the configuration has crossed the nodal surface
(by checking the sign of the Slater part of the trial wave function). If it has then the move is rejected.
This has been found to be the least-biased method of imposing the fixed-node approximation [11].
13.4 Branching and population control
The branching Green’s function can be implemented by altering the population of configurations
and/or their weights. At the start of the calculation one chooses a target population, M 0 , and the
actual population M tot (m) [see Eq. (33)] is controlled so that it does not deviate too much from M 0 .
The population control is principally exerted by altering the reference energy, E T (m). Large changes
in E T can lead to a bias and therefore it is varied smoothly over the simulation.
For each move of all the electrons in configuration α the branching factor is calculated as:
[(
M b (α, m) = exp − 1 {
S(Rα,m ) + S(R ′
2
α,m) } ) ]
+ E T (m) τ eff (α, m)
where τ eff is the effective time step for configuration α at time step m (see Sec. 13.5), S is the local
energy (we denote it by S because it is usually a modified version of E L , see Sec. 13.5). Unless weighted
DMC is used (i.e., unless lwdmc=T), the number of copies of this configuration that continue to the
next time step is given by:
M(α, m) = INT{η + M b (α, m)}, (31)
where η is a random number drawn from a uniform distribution on the interval [0,1].
In weighted DMC, each configuration carries a weight that is simply multiplied by M b (α, m) after
each move; only if the weight of a configuration goes outside certain bounds (above wdmcmax or
below wdmcmin) is it allowed to branch or be combined with another configuration.
Throughout this section we denote the best estimate of the ground-state energy at time step α by
E best (m). At the start of a DMC run we set E best (0) = E T (0) = E V , where E V is the average
(30)
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