Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 473

depends only on the dimensionless radiusρ. It is the solution of a one-dimensional differential
equation, as is the case for the harmonic oscillator as well.The latter has the trial wave
function

ΨT(x) =

√

α

π^1 /^4

e−x

(^2) α (^2) / 2
.
However, for the hydrogen atom we haveρ∈[ 0 ,∞), while for the harmonic oscillator we have
x∈(−∞,∞). In the calculations below we have used a uniform distribution to generate the vari-
ous positions. This means that we employ a shifted uniform distribution where the integration
regions beyond a given value ofρandxare omitted. This is obviously an approximation and
techniques like importance sampling discussed in chapter 11 should be used. Using a uni-
form distribution is normally refered to as brute force Monte Carlo or brute force Metropolis
sampling. From a practical point of view, this means that therandom variables are multiplied
by a given step lengthλ. To better understand this, consider the above dimensionless radius
ρ∈[ 0 ,∞).
The new position can then be modelled as
ρnew=ρold+λ×r,
withrbeing a random number drawn from the uniform distribution ina regionr∈[ 0 ,Λ],
withΛ<∞, a cutoff large enough in order to have a contribution to the integrand close to
zero. The step lengthλis chosen to give approximately an acceptance ratio of50%for all
proposed moves. This is nothing but a simple rule of thumb. Inthis chapter we will stay with
this brute force Metropolis algorithm. All results discussed here have been obtained with this
approach. Importance sampling and further improvements will be discussed in chapter 15.
In Figs. 14.1 and 14.2 we plot the ground state energies for the one-dimensional harmonic

0

1

2

3

4

5

0.2 0.4 0.6 0.8 1 1.2 1.4

E 0

α

MC simulation with N=100000

Exact result

Fig. 14.1Result for ground state energy of the harmonic oscillator asfunction of the variational parameter
α. The exact result is forα= 1 with an energyE= 1. See text for further details.

oscillator and the hydrogen atom, respectively, as functions of the variational parameterα.
These results are also displayed in Tables 14.4 and 14.5. In these tables we list the variance
and the standard deviation as well. We note that atα= 1 for the hydrogen atom, we obtain