CASINO manual - Theory of Condensed Matter

local energy of the initial configurations. During equilibration E best is updated after each iteration
as the average local energy over the previous ebest av window moves. During accumulation E best
is set equal to the current value of the mixed estimator of the energy, given by Eq. (51), with  =
E L (α, m). The algorithms differ because we wish to discard data from the start of the simulation
during equilibration. E T is updated after every iteration as
( )
E T (m + 1) = E best (m) −
g−1
τ EFF (m) log Mtot (m)
, (32)
M 0
where g −1 = min{1, τc ET }, c ET is a constant which must be set in the input file (cerefdmc), but
is usually set equal to one, M 0 =dmc target weight is the target number of configurations (in our
implementation it is allowed to take non-integer values) and
M tot (m) =
N config (m)
∑
α=1
w α (m), (33)
where N config (m) is the number of configurations and w α is the weight of configuration α. Note that
E best (m) is the best energy at time step m while E T (m + 1) is the trial energy to be used in the
next time step. τ EFF is the current best estimate over all configurations and time steps of the mean
effective time step, calculated using Eq. (51) with  = τ eff(α, m). Note that g is the time scale (in
terms of time steps) over which the population attempts to return to M 0 .
13.5 Modifications to the Green’s Function
13.5.1 The effective time step
Time-step errors can be reduced and the stability of the DMC algorithm improved by modifying the
Green’s function. An important modification is to introduce an effective time step, τ eff , into the
branching factor [28]. When the accept/reject step is included, the mean distance diffused by each
electron each move (which should go as the square root of the time step) is reduced because some
moves are rejected. When calculating branching factors it is therefore more accurate to use a time
step appropriate for the actual distance diffused. Umrigar and Filippi [26] have suggested using an
effective time step for each configuration at each time step. The effective time step is given by
∑
i
τ eff (α, m) = τ
p i∆rd,i
2 ∑ , (34)
i ∆r2 d,i
where the averages are over all attempted moves of the electrons i in configuration α at time step
m. The ∆r d,i are the diffusive displacements [i.e., the distances travelled by the electrons without the
drift-displacement: see Eq. (22)] and p i is the acceptance probability of the electron move [see Eq.
(26)]. The values averaged over the current run are written in the output file.
We calculate τ EFF (m) using Eq. (51) with  ≡ τ eff(α, m).
13.5.2 Drift-vector and local-energy limiting
The drift vector diverges at the nodal surface and a configuration which approaches a node can exhibit
a very large drift, resulting in an excessively large move in the configuration space. One can improve
the Green’s function by cutting off the drift vector when its magnitude becomes large. The total drift
vector is defined in Eq. (21).
We use the smoothly cut-off drift vector suggested by UNR [19]. For each electron with drift vector
v i , we define the smoothly cut-off drift vector:
ṽ i = −1 + √ 1 + 2a|v i | 2 τ
a|v i | 2 v i , (35)
τ
where a is a constant that can be chosen to minimize the bias. The value of a = alimit can be entered
by the user if nucleus gf mods is set to F (the default is a = 1); otherwise a will be calculated as
described in Sec. 13.6.
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