CASINO manual - Theory of Condensed Matter

13.2 The ensemble of configurations
The f distribution is represented by an ensemble of electron configurations, which are propagated
according to rules derived from the Green’s function of Eq. (17). G D represents a drift-diffusion
process while G B represents a branching process. The branching process leads to fluctuations in the
population of configurations and/or fluctuations in their weights.
We will introduce labels for the different configurations α present at each time step m. From now
on R represents the electron coordinates of a particular configuration in the ensemble and i labels a
particular electron.
In the casino implementation of DMC, electrons can moved one at a time (electron-by-electron,
dmc method= 1, default) or all at once (configuration-by-configuration, dmc method= 2). The
former is much more efficient and is the standard procedure for large systems, but most of the algorithms
described in the literature are for moving all electrons at once.
13.3 Drift and diffusion
We now discuss the practical implementation of the drift-diffusion process using the electron-byelectron
algorithm in which electrons are moved one at a time, as used in casino.
To implement the drift-diffusion step, each electron i in each configuration α is moved from r ′ i (α) to
r i (α) in turn according to
r i = r ′ i + χ + τv i (r 1 , . . . , r i−1 , r ′ i, . . . , r ′ N), (22)
where χ is a three-dimensional vector of normally distributed numbers with variance τ and zero mean.
v i (R) denotes those components of the total drift vector V(R) due to electron i.
Hence each electron i is moved from r ′ i to r i with a transition probability density of
(
t i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ 1
N) =
(2πτ) exp (ri − r ′ i − τv i(r 1 , . . . , r i−1 , r ′ i , . . . , )
r′ N ))2 .
3/2 2τ
(23)
For a complete sweep through the set of electrons, the transition probability density for a move from
R ′ = (r ′ 1, . . . , r ′ N ) to R = (r 1, . . . , r N ) is simply the probability that each electron i moves from r ′ i to
r i . So the transition probability density for the configuration move is
T (R ← R ′ ) =
N∏
t i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N ). (24)
i=1
In the limit of small time steps, the drift velocity V is constant over the (small) configuration move.
Evaluating the product in this case, we find that the transition probability density is
T (R ← R ′ ) = G D (R ← R ′ , τ), (25)
so that the drift-diffusion process is described by the drift-diffusion Green’s function G D .
At finite time steps, however, the approximation that the drift velocity is constant leads to the violation
of the detailed balance condition.
We may enforce the detailed balance condition on the DMC Green’s function by means of a Metropolisstyle
accept/reject step introduced by Ceperley et al. [27]. This has been shown to greatly reduce
time-step errors [28]. The move of the ith electron of a configuration is accepted with probability
{
min 1, t 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 )
t i (r 1 , . . . , r i−1 , r i ← r ′ i , r′ i+1 , . . . , r′ N )Ψ2 (r 1 , . . . , r i−1 , r ′ i , . . . , r′ N ) }
{ [ (
= min 1, exp r ′ i − r i + τ )
2 (v i(r 1 , . . . , r i−1 , r ′ i, . . . , r ′ N ) − v i (r 1 , . . . , r i , r ′ i+1, . . . , r ′ N )
· (v
i (r 1 , . . . , r i−1 , r ′ i, . . . , r ′ N ) + v i (r 1 , . . . , 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 )
≡ a i (r 1 , . . . , r i−1 , r i ← r ′ i, r ′ i+1, . . . , r ′ N). (26)
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