Computational Physics

12.3 Diffusion Monte Carlo 387

accepted with probability min(1,qRR′), where

qRR′=

ωR′Rρ(R′)

ωRR′ρ(R)

. (12.52)

Note that the fractionωR′R/ωRR′is in equilibrium approximately equal to the ratio
ρ(R)/ρ(R′)– if no time step error was made in constructingωRR′, they would have
been exactly equal – soqRR′is always close to 1. The acceptance rate is therefore
always high when tis taken small, and the method is very efficient. The Metropolis
acceptance/rejection step is merely a correction for the time step discretisation error
made in the Langevin procedure.
The implementation of the algorithm is straightforward. The resulting energies
must be the same as for the standard VMC method, but the error bars are smaller. As
an example, an MC simulation for the harmonic oscillator using 300 walkers which
perform 3000 steps andα=0.4 yields for the energy expectation in the ordinary
VMC program valueE=0.51±0.03, to be compared withE=0.515±0.006 in
the Fokker–Planck program.
Variational Monte Carlo has the advantage that it is simple and straightforward.
An important disadvantage is that it relies on the quality of the trial function, hence
subtle but important physical effects are sometimes neglected when they are not
taken into account when constructing the trial function.

12.3 Diffusion Monte Carlo

12.3.1 Simple diffusion Monte Carlo

The second quantum Monte Carlo method that we consider is the so-calleddiffusion
orprojectorMonte Carlo method, abbreviated as DMC. This method does not use
variational principles for obtaining ground state properties, but as we shall see,
the convergence rate of the practical version of this method relies heavily on the
accuracy of the trial functions. The idea of this method has already been sketched in
Section 12.2.4. We use the imaginary time form of the time-dependent Schrödinger
equation. This is a diffusion equation with a potential. We use the Green’s function
in the ‘normalised’ form, i.e. with the normalisation factor exp(−
τET)present:

G(R,R′;

τ )=e−

τ[V(R)−ET]

1

√

2 π

τ

e−(R−R

′) (^2) /( 2 π
τ)
+O(
τ^2 ). (12.53)
This Green’s function is a short-time approximation of the imaginary-time operator
exp[−τ(H+ET)]. If we resolve this operator in its eigenstates|φn〉, we obtain
e−τ(H−ET)=

∑

n

|φn〉e−τ(En−ET)〈φn|. (12.54)