374 Quantum Monte Carlo methods
The loop stops when the minimum energy is reached according to some criterion.
It is the second step in this algorithm that we consider in this section. However,
below, we shall describe a variational method in which the parametersαααare adjusted
according to some numerical scheme within the Monte Carlo simulation.
It turns out that in realistic systems the many-body wave function assumes very
small values in large parts of configuration space, so a straightforward procedure
using homogeneously distributed random points in configuration space is bound to
fail. This suggests that it might be efficient to use a Metropolis algorithm in which
a collection of random walkers is pushed towards those regions of configuration
space where the wave function assumes appreciable values. Suppose that we can
evaluateHψTfor any trial functionψT, which we shall always assume to be real,
and let us define
EL(R)=HψT(R)
ψT(R)(12.2)
(we omit theααα-dependence ofψT).EL(R)is called thelocal energy: it is a function
that depends on the positions of the particles and it is constant ifψTis the exact
eigenfunction of the Hamiltonian. The more closelyψTapproaches the exact wave
function (apart from a multiplicative constant), the less strongly willELvary withR.
The expectation value of the energy can now be written as
〈E〉=
∫
dRψT^2 (R)EL(R)
∫
dRψT^2 (R). (12.3)
Let us now construct a Metropolis-walk in the same spirit as in ordinary Monte
Carlo calculations, but now with a stationary distributionρ(R)given by
ρ(R)=ψT^2 (R)
∫
dR′ψT^2 (R′). (12.4)
The procedure is now as follows.
PutNwalkers at random positions;
REPEAT
Select next walker;
Shift that walker to a new position, for example by moving one
of the particles in the system within a cube with a suitably
chosen sized;
Calculate the fractionp=[ψT(R′)/ψT(R)]^2 , whereR′is the new and
Rthe old configuration;
Ifp<1 the new position is accepted with probabilityp;
Ifp≥1 the new position is accepted;
UNTIL finished.