CASINO manual - Theory of Condensed Matter

which cannot lie beyond the nucleus.
The drift in the radial direction over one time step is
ρ ′′ = 2¯v ρτz ′′
z ′ . (43)
+ z
′′
The new radial coordinate is approximately ¯v ρ τ when far from the nucleus, but it is forced to go to
zero as the nucleus is approached. Hence, if the electron attempts to overshoot the nucleus, it will
end up on top of it. So time-step errors caused by drifting across nuclei are eliminated.
Let the electron position at the end of the drift process be r ′′ = z ′′ e z + ρ ′′ e ρ .
13.6.3 Diffusion close to a bare nucleus
Close to a nucleus, f is proportional to the square of the hydrogenic 1s orbital (assuming the trial
wave function has the correct behaviour). This cusp cannot be reproduced by Gaussian diffusion at
finite time steps. In fact, starting from the nucleus, we would like our electron to take a random step
w distributed according to exp(−2Z|w|).
However, we only want to diffuse in this fashion when the electron is likely to cross the nucleus. Let Π
be the plane with normal e z that contains the nucleus. For the usual Gaussian diffusion process, the
probability that an electron drifts (assuming that nuclear overshoot is permitted) and diffuses across
Π, is
˜q = 1 − ˜p = 1 ( ) z +
2 erfc ¯vz τ
√ . (44)
2τ
So, with probability ˜p, we sample w from
g 1 (w) = (2πτ) −3/2 exp
) (− |w|2 , (45)
2τ
and set the new electron position to be r = r ′′ + w; otherwise, we sample 21 w from
g 2 (w) = ζ3
π
and set r = R Z + w. We have defined ζ by
ζ =
√
exp(−2ζ|w|), (46)
Z 2 + 1 τ , (47)
which reduces to Z for large time steps, giving the desired cusp; however, this choice of ζ causes the
second moments of g 1 and g 2 to be equal to O(τ). Hence the Green’s function remains correct to
O(τ).
The single-electron Green’s function for the move from r ′ to r is given by
g(r ← r ′ ) = ˜pg 1 (r − r ′′ ) + ˜qg 2 (r − R Z ). (48)
In order to calculate the Green’s function for the reverse move, need to perform all of the steps above
(apart from the random diffusion), starting at point r and ending up at r ′ .
13.6.4 Using the modifications in CASINO
These three modifications to the DMC Green’s function are applied if the nucleus gf mods keyword
is set to T. Note that they can only be used if bare nuclei are actually present!
21 In order to sample w from g 2 (w), we sample the cosine of the polar angle uniformly on [−1, 1], the azimuthal angle
uniformly on [0, 2π] and the magnitude w from 4ζ 3 w 2 exp(−2ζw). This is achieved by sampling r 1 , r 2 and r 3 uniformly
on [0, 1] and setting w = − log(r 1 r 2 r 3 )/2ζ; see reference [29] for further information.
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