CASINO manual - Theory of Condensed Matter

In the UNR [19] scheme the modified local energy, S(α, m), is given by
S(α, m) = E best (m) − [E best (m) − E L (α, m)] |Ṽ|
|V| , (36)
where Ṽ(α, m) = (ṽ 1, . . . , ṽ N ). Note that we define S slightly differently from Ref. [19]. S is used only
in the branching factor. When evaluating the average energy, we sum the unlimited local energies,
E L . In practice, even though the local energy is calculated at the end of the configuration move, the
limiting scheme of Eq. (36) is applied to the single-electron drift velocities. The ratio of drift velocities
is therefore calculated as
2 ¯V (R)
(¯v
V (R) = 1 (R) + . . . + ¯v N 2 (R)) 1/2
. (37)
V (R)
Finally, a very simple option is simply to cut off the local energy drift vector when their magnitudes
becomes large using the method of Depasquale et al. [18],
S(R) = E V + sign[2/ √ τ, E L (R) − E V ] for |E L (R) − E V | > 2/ √ τ
ṽ i (R) = sign[1/τ, v i ] for |v i | > 1/τ . (38)
In the paper of Depasquale et al. [18] they recommend using the variational energy for E V , but we
use the best estimate of the energy. We prefer the UNR scheme.
The limited single-electron drift velocities are calculated at the same time as the local energy of the
configuration, after all of the electron positions in the configuration have been updated.
13.6 Modifications to the DMC Green’s function at bare nuclei
13.6.1 Modifications to the limiting of the drift velocity
The limiting of the drift velocity is intended to remove the divergence at the nodal surface; interference
with the cusps at bare nuclei is an undesirable side-effect, which may introduce bias. In order to
distinguish between nodes and nuclei, UNR make the a-parameter in their limiting scheme dependent
on electron position. Immediately before the limited drift velocity of an electron at r ′ is calculated, a
is evaluated as
a(r ′ ) = 1 2 (1 + ˆv · e Z 2 z 2
z) +
10(4 + Z 2 z 2 ) , (39)
where ˆv is the unit vector in the direction of the unlimited drift velocity, e z is the unit vector from
the closest bare nucleus to the electron, z is the distance of the electron from the nucleus and Z is
the atomic number. This formula makes a small (and hence the limiting weak) if the electron is both
close to the nearest nucleus and drifting towards it.
13.6.2 Preventing electrons from overshooting nuclei
In its immediate vicinity, the single-electron drift velocity is always directed towards a bare nucleus.
Therefore, drifting particles should never cross the nucleus; rather, they should end up on top of it.
In order to impose this condition at finite time steps, we work in cylindrical polar coordinates with
the z-axis lying along the line from the nucleus to the electron. Let the position of the closest nucleus
be R Z .
The position of the electron relative to the nucleus is
while the limited drift velocity can be resolved as
where e ρ is a unit vector orthogonal to e z .
The new z-coordinate after drifting for one time step is
r ′ − R Z = z ′ e z , (40)
¯v = ¯v z e z + ¯v ρ e ρ , (41)
z ′′ = max{z ′ + ¯v z τ, 0}, (42)
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