Computational Physics

386 Quantum Monte Carlo methods

12.2.5 The Fokker–Planck equation approach to VMC

The VMC method described inSections 12.2.1and12.2.2has an important dis-
advantage: typical many-particle wave functions are very small in large parts of
configuration space and very large in small parts of configuration space. This means,
first, that we might have difficulty in finding the regions where the wave function is
large, and second, that attempted moves of walkers from a favourable region (where
the wave function is large) will be rejected when they move out of that region. Hav-
ing a substantial fraction of rejected moves is part of any Metropolis Monte Carlo
scheme, and we could live with that if there did not exist a more efficient approach,
based on the Fokker–Planck equation described in the previous section.
In this method we try to sample the functionρ(R)=ψT^2 (R)rather than the trial
functionψT(R)itself: that is, we use

F= 2 ∇RψT(R)/ψT(R) (12.51)

in the FP equation.
The distributionρ(R,t)can be sampled by simulating a diffusion process. The
algorithm is close to that of ordinary VMC. Now we let a collection of walkers
diffuse with probabilities given by the Green’s function(12.45):

PutNwalkers at random positions;

REPEAT

Select next walker;

Shift that walker from its current positionRtoR+F(R)

t/2;

Displace that walker by an amountηηη

√

t, whereηηηis a

random vector with a Gaussian distribution (see (12.29) and (12.28));

UNTIL finished.

We see that there is no acceptance/rejection step; this causes the gain in efficiency
when using the FP approach.
Note that we have made a time-step error of order(
t)^2. It is possible to eliminate
this error by combining this Langevin approach with a Metropolis procedure. The
point is that we know the form of stationary distributionρ(it is the square of the
trial functionψT), and the Langevin process leads to a distribution which is close
to but not exactly equal to this distribution. The Metropolis algorithm can give
us the desired distributionρby acceptance/rejection of the Langevin steps, which
themselves are considered as trial moves in the Metropolis algorithm. Referring
back toSection 10.3, we call the transition probability of the Langevin equation
ωRR′=G(R,R′; t), whereGis given in(12.48). This is not symmetric inRand
R′asFdepends only onR, and therefore we have to use the generalised Metropolis
algorithm, described at the very end of Section 10.3. The Langevin trial move is