12.3 Diffusion Monte Carlo 393
Evaluateq=exp{−
τ[ELocal(R′)+ELocal(R)]/ 2 −ET};
Eliminate the walker or create new ones atR′,
depending ons=q+r, whereris random,
uniform between 0 and 1;
END IF;
END FOR
UpdateETusing(12.55);
UNTIL finished.Let us first apply the importance sampling method to the one-dimensional har-
monic oscillator. We use the same trial (or guide) functionT(x)=e−αx
2
as in the
VMC simulation. In that case the quantum force is given by
F(x)=− 4 αx, (12.62)and the local energy byEq. (12.6). Indeed, the local energy is a constant ifα= 1 / 2
and it will be slowly varying ifαis close to 1/2. Forα=0.4, a target number of 6000
walkers and 4000 steps, we find for the ground state energyE=0.5002±0.0003
and withα=0.6,E=0.4998±0.0003.
We can now do the hydrogen and the helium atom problems. For hydrogen we
use a guide function exp(−αr)and a target number of 2000 walkers performing
4000 steps. The local energy is given by (12.10). Obviously, forα=1 we find the
exact ground state energy of−0.5 Hartree as the local energy is constant and equal
to this value. Forα=0.9, we find a ground state energy of−0.4967( 5 )and for
α=1.1 we findEG=0.5035( 5 ). Neither of these values agrees with the exact
value. The reason is that the guide function should solve the divergence problem at
r=0, but it can do this only if the cusp conditions are satisfied. Forα=1 this is
not the case. This shows the importance of the cusp conditions being satisfied for
the trial function.
Finally we present results for the helium atom. We use the Padé–Jastrow wave
function (12.11). Varying the parameterαgives values above and below the exact
energy. If we monitor the variance of the energy, we find a minimum atα≈0.15 and
an energyEG=−2.9029( 2 )for 1000 walkers performing 4000 steps. Remember
the exact energy is−2.903 and the variational energy for the uncorrelated wave
function (the Hartree–Fock energy) is−2.8617 atomic units.
programming exercise
Modify the DMC programs of the previous section to include a guide function
and compare the results with those given in this section.