14.5 Variational Monte Carlo for atoms 487
Secondly, when using the simple symmetric form ofω(rold,rnew), one has to keep in mind
the random walk nature of the algorithm. Transitions will bemade between points that are
relatively close to each other in state space, which also clearly contributes to increase corre-
lation. The seemingly obvious way to deal with this would be just to increase the step size, al-
lowing the walkers to cover more of the state space in fewer steps (thus requiring fewer steps
to reach ergodicity). But unfortunately, long before the step length becomes desirably large,
the algorithm breaks down. When proposing moves symmetrically and uniformly aroundrold,
the step acceptance becomes directly dependent on the step length in such a way that a too
large step length reduces the acceptance. The reason for this is very simple. As the step
length increases, a walker will more likely be given a move proposition to areas of very low
probability, particularly if the governing trial wave function describes a localized system. In
effect, the effective movement of the walkers again becomestoo small, resulting in large
correlation. For optimal results we therefore have to balance the step length with the accep-
tance.
With a transition suggestion ruleωas simple as the uniform symmetrical one emphasized
so far, the usual rule of thumb is to keep the acceptance around 0. 5. But the optimal interval
varies a lot from case to case. We therefore have to treat eachnumerical experiment with
care.
By choosing a betterω, we can still improve the efficiency of the step length versusaccep-
tance. Recall thatωmay be chosen arbitrarily as long as it fulfills ergodicity, meaning that it
has to allow the walker to reach any point of the state space ina finite number of steps. What
we basically want is anωthat pushes the ratio towards unity, increasing the acceptance. The
theoretical situation ofωexactly equal topitself:
ω(rnew,rold) =ω(rnew) =p(rnew)would give the maximal acceptance of 1. But then we would already have solved the prob-
lem of producing points distributed according top. One typically settles on modifying the
symmetricalωso that the walkers move more towards areas of the state spacewhere the
distribution is large. One such procedure is the Fokker-Planck formalism where the walkers
are moved according to the gradient of the distribution. Theformalism “pushes” the walkers
in a “desirable” direction. The idea is to propose moves similarly to an isotropic diffusion
process with a drift. A new positionxnewis calculated from the old one,xold, as follows:
rnew=rold+χ+DF(rold)δt (14.24)Hereχis a Gaussian pseudo-random number with mean equal zero and variance equal 2 Dδt.
It accounts for the diffusion part of the transition. The third term on the left hand side ac-
counts for the drift.Fis a drift velocity dependent on the position of the walker and is derived
from the quantum mechanical wave functionψ. The constantD, being the diffusion constant
ofχ, also adjusts the size of the drift.δtis a time step parameter whose presence will be
clarified shortly.
It can be shown that theωcorresponding to the move proposition rule in Eq. (14.24)
becomes (in non-normalized form):
ω(rold,rnew) =exp(
−
(rnew−rold−DδtF(rold))^2
4 Dδt)
(14.25)which, as expected, is a Gaussian with variance 2 Dδtcentered slightly offrolddue to the drift
termDF(rold)δt.
What is the optimal choice for the drift term? From statistical mechanics we know that a
simple isotropic drift diffusion process obeys a Fokker-Planck equation of the form: