CASINO manual - Theory of Condensed Matter

The vmc method input parameter selects which is to be used: a value of 1 means electron-byelectron,
whereas a value of 3 means configuration-by-configuration. 17
The local energy does not have to be calculated every configuration move. In particular, energies are
not calculated during the first vmc equil nstep moves of a VMC simulation when the Metropolis
algorithm has yet to reach its equilibrium. Furthermore, because the configurations are serially correlated,
the expense of calculating the energy at every configuration move is unjustified: it is more
efficient to calculate the energy once every vmc decorr period moves, where typically the input
parameter vmc decorr period might be in the range 4–20.
Note that it is not necessary to write out every local energy calculated. vmc ave period successive
energies can be averaged over before being written out to vmc.hist: this helps ensure that the
vmc.hist file is not excessively large when long production calculations are carried out. Furthermore,
on parallel machines the local energies computed on each processor are averaged over before being
written out.
The input parameter vmc nstep gives the number of energy-calculating steps in the VMC simulation.
In parallel machines, each processor goes over vmc nstep/nproc energy-calculating steps.
12.3 Two-level sampling
The Metropolis acceptance probability for a move from R ′ to R in the standard algorithm 18 is
} {
}
p(R ← R ′ ) = min
{1, |Ψ(R)|2
|D↑ 2
|Ψ(R ′ )| 2 = min 1,
(R)D2 ↓
(R)| exp[2J(R)]
|D↑ 2(R′ )D↓ 2(R′ )| exp[2J(R ′ . (12)
)]
It is straightforward to show that if detailed balance [11] in configuration space is satisfied then the
resulting ensemble of configurations will be distributed according to the square of the trial wave
function.
However, casino employs a two-level sampling algorithm, which has been shown to be considerably
more efficient [25].
Let us define the first-level acceptance probability
{
p 1 (R ← R ′ |D↑ 2 ) = min 1,
(R)D2 ↓ (R)|
}
|D↑ 2(R′ )D↓ 2(R′ )|
and the second-level acceptance probability
p 2 (R ← R ′ ) = min
(13)
{
1, exp[2J(R)] }
exp[2J(R ′ . (14)
)]
In the two-level algorithm we accept trial moves from R ′ to R with probability p 1 (R ← R ′ )p 2 (R ←
R ′ ). It can be shown that, provided detailed balance in configuration space is satisfied, this procedure
also results in an ensemble of configurations distributed according to the square of the trial wave
function.
The two-level approach is computationally advantageous because the Metropolis accept/reject step
can be carried out in two stages: if the ‘first level’ is accepted (with probability p 1 ) then we compute
the Jastrow factors of the new and old configurations and determine whether the ‘second level’ is
accepted (with probability p 2 ). Thus, if a move is rejected at the first level then we do not have to
compute the Jastrow factors for the new and old configuration 19 .
12.4 Optimal value of the VMC time step
Ideally, the VMC time step dtvmc should be chosen such that the correlation period (as determined
by reblocking analysis) of the resulting configuration local energies is minimized. For large time steps
17 There used to be a vmc method=2, but after extensive testing we concluded that it did not offer any advantage
over the other methods and was hard to support.
18 In the electron-by-electron algorithm, R and R ′ differ only in the coordinates of a single electron.
19 It is better to use the Slater wave function to compute the first-level acceptance probability because p 1 is much
lower (in general) than the corresponding acceptance probability for the Jastrow part (p 2 ); hence we are less likely to
need to compute the second level acceptance probability than if the situation were reversed.
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