Computational Physics - Department of Physics

14.3 First Encounter with the Variational Monte Carlo Method 461

as much as possible of the pertinent physics since they form the starting point for a variational

calculation of the expectation value of the HamiltonianH. Given a HamiltonianHand a trial

wave functionΨT, the variational principle states that the expectation value of〈H〉

〈H〉=

∫

d∫RΨT∗(R)H(R)ΨT(R)

dRΨT∗(R)ΨT(R) , (14.2)

is an upper bound to the true ground state energyE 0 of the HamiltonianH, that is

E 0 ≤〈H〉.

To show this, we note first that the trial wave function can be expanded in the eigenstates

of the Hamiltonian since they form a complete set, see again Postulate III,

ΨT(R) =∑

i

aiΨi(R),

and assuming the set of eigenfunctions to be normalized, insertion of the latter equation in

Eq. (14.2) results in

〈H〉=∑mn

a∗man

∫

dRΨm∗(R)H(R)Ψn(R)

∑mna∗man

∫

dRΨm∗(R)Ψn(R) =

∑mna∗man

∫

dRΨm∗(R)En(R)Ψn(R)

∑na^2 n

,

which can be rewritten as

∑na^2 nEn

∑na^2 n

≥E 0.

In general, the integrals involved in the calculation of various expectation values are multi-

dimensional ones. Traditional integration methods like Gaussian-quadrature discussed in

chapter 5 will not be adequate for say the computation of the energy of a many-body sys-

tem.

We could briefly summarize the above variational procedure in the following three steps:

1. Construct first a trial wave functionψT(R;α), for say a many-body system consisting
   ofNparticles located at positionsR= (R 1 ,...,RN). The trial wave function depends
   onαvariational parametersα= (α 1 ,...,αm).


2. Then we evaluate the expectation value of the HamiltonianH

〈H〉=

∫

d∫RΨT∗(R;α)H(R)ΨT(R;α)

dRΨT∗(R;α)ΨT(R;α)

.

3. Thereafter we varyαaccording to some minimization algorithm and return to the
   first step.

The above loop stops when we reach the minimum of the energy according to some speci-

fied criterion. In most cases, a wave function has only small values in large parts of configu-

ration space, and a straightforward procedure which uses homogenously distributed random

points in configuration space will most likely lead to poor results. This may suggest that some

kind of importance sampling combined with e.g., the Metropolis algorithm may be a more

efficient way of obtaining the ground state energy. The hope is then that those regions of

configurations space where the wave function assumes appreciable values are sampled more

efficiently.