CASINO manual - Theory of Condensed Matter

the move rejection probability is high, which obviously leads to serial correlation. On the other hand,
for low time steps the configuration does not move very far at a given step, and so serial correlation
is again large.
A rule of thumb for choosing an appropriate dtvmc is that the average acceptance probability should
be about 50%. If you use at least 500 steps 20 in the VMC equilibration phase, then casino will automatically
optimize the VMC time step to achieve an acceptance ratio of 50% (provided opt dtvmc
is set to 1, which is the default).
Notice that one could determine the optimal time step by analysing the correlation length of the
local-energy sequence, but this is just too expensive and complicated and would provide little (if any)
benefit over the simple, inexpensive ‘50% rule’.
13 Detailed information: the DMC method
See, e.g., Refs. [11], [12] and [13] for general information about DMC. Here we present some more
detailed information about the implementation of DMC within casino. Our algorithm follows that of
Umrigar, Nightingale and Runge [19] (referred to simply as UNR), with some additional developments
due to Umrigar and Filippi [26]. Throughout this section we assume for simplicity that the trial wave
function is real, though casino is perfectly capable of dealing with complex wave functions.
13.1 Imaginary-time propagation
Let R be a point in the configuration space of an N-electron system. The importance-sampled DMC
method propagates the distribution f(R, t) = Ψ(R)Φ(R, t) in imaginary time t, where Ψ is the trial
wave function and Φ is the DMC wave function. The importance-sampled imaginary-time Schrödinger
equation may be written in integral form,
∫
f(R, t) = G(R ← R ′ , t − t ′ )f(R ′ , t ′ ) dR ′ , (15)
where the Green’s function G(R ← R ′ , t − t ′ ) satisfies the initial condition
G(R ← R ′ , 0) = δ(R − R ′ ) . (16)
The Green’s function used in DMC is an approximation to the exact form which is accurate for short
time steps, τ = t − t ′ (dtdmc),
G DMC (R ← R ′ , τ) = G D (R ← R ′ , τ) G B (R ← R ′ , τ) , (17)
where
G D (R ← R ′ , τ) =
(
)
1
(2πτ) exp − (R − R′ − τV(R ′ )) 2
3N/2 2τ
(18)
is the drift-diffusion Green’s function and
G B (R ← R ′ , τ) = exp
(
− τ )
2 [E L(R) + E L (R ′ ) − 2E T ]
(19)
is the branching Green’s function. E T is the reference energy, which acts as a renormalization factor
(see Sec. 13.4). E L is the local energy,
where Ĥ is the Hamiltonian and V is the drift vector,
Note that V = (v 1 , . . . , v N ), where v i is the drift vector of electron i.
20 2000 steps are required for configuration-by-configuration VMC.
E L (R) = Ψ −1 ĤΨ, (20)
V(R) = Ψ −1 ∇Ψ. (21)
123