Computational Physics - Department of Physics

488 14 Quantum Monte Carlo Methods

∂f

∂t

=∑

i

D∂

∂xi

(

∂

∂xi

−Fi(F)

)

f (14.26)

wherefis the continuous distribution of walkers. Equation (14.24) is a discretized realization
of such a process whereδtis the discretized time step. In order for the solutionfto converge
to the desired distributionp, it can be shown that the drift velocity has to be chosen as follows:

F=

1

f

∇f

where the operator∇is the vector of first derivatives of all spatial coordinates. Convergence
for such a diffusion process is only guaranteed when the timestep approaches zero. But in the
Metropolis algorithm, where drift diffusion is used just asa transition proposition rule, this
bias is corrected automatically by the rejection mechanism. In our application, the desired
probability distribution function being the square absolute of the wave function,f=|ψ|^2 , the
drift velocity becomes:

F= 2

1

ψ

∇ψ (14.27)

As expected, the walker is “pushed” along the gradient of thewave function.
When dealing with many-particle systems, we should also consider whether to move only
one particle at a time at each transition or all at once. The former method may often be more
efficient. A movement of only one particle will restrict the accessible space a walker can move
to in a single transition even more, thus introducing correlation. But on the other hand, the
acceptance is increased so that each particle can be moved further than it could in a standard
all-particle move. It is also computationally far more efficient to do one-particle transitions
particularly when dealing with complicated distributionsgoverning many-dimensional anti-
symmetrical fermionic systems.
Alternatively, we can treat the sequence of all one-particle transitions as one total transi-
tion of all particles. This gives a larger effective step length thus reducing the correlation.
From a computational point of view, we may not gain any speed by summing up the individual
one-particle transitions as opposed to doing an all-particle transition. But the reduced corre-
lation increases the total efficiency. We are able to do fewercalculations in order to reach the
same numerical accuracy.
Another way to acquire some control over the correlation is to do a so called blocking
procedure on our set of numerical measurements. This is discussed in chapter 16.

14.6 Exercises.

14.1.The aim of this problem is to test the variational Monte Carloapppled to light atoms.
We will test different trial wave functionΨT. The systems we study are atoms consisting of
two electrons only, such as the helium atom, LiIIand BeIII. The atom LiIIhas two electrons
andZ= 3 while BeIIIhasZ= 4 but still two electrons only. A general ansatz for the trial wave
function is
ψT(R) =φ(r 1 )φ(r 2 )f(r 12 ). (14.28)

For all systems we assume that the one-electron wave functionsφ(ri)are described by the an
elecron in the lowest hydrogen orbital 1 s.
The specific trial functions we study are

ψT 1 (r 1 ,r 2 ,r 12 ) =exp(−α(r 1 +r 2 )), (14.29)