The trial wavefunction should be chosen as close to the
true wave function of the system as possible. If we were
able to approximate the system wavefunction with satisfying
accuracy, then the sampling over the corresponding distribution
(for example with the help of Metropolis algorithm) would give us
all properties of the system. However, the problem is that very
often it is impossible to find the wavefunction of the system using
analytical methods. Here enters the Diffusion Monte Carlo method,
which compensates our lack of knowledge and corrects the trial
wavefunction provided that the projection of the trial wavefunction
on the true system wavefunction of the system differs from zero.
As the use of the trial wavefunction lies at the heart of the
method, it has to be expressed in a way that is fast to calculate
or it has to be tabulated.
The common way to construct many-body wavefunctions is to use the Jastrow
function consisting of the product of an uncorrelated state and
a correlation factor, which is a product of two-body wavefunctions.
(3.30)
The one-body term describes the effect of an external field and is
absent in the case of a homogeneous system. For a homogeneous
system the trial wavefunction can be written in general as
(3.31)
Since we are mainly interested to dilute system a possible way to
obtain the pair function is through the solution of the
two-body Schrödinger equation.
(3.32)
here
is the reduced mass and
is the interparticle distance. Let's search for the
solution in spherical coordinates
(3.33)
The particles are modeled by hard spheres of diameter and the
interaction potential is
(3.34)
The dimensionless Schrödinger equation is obtained by expressing
all distances in units of and energy in units of
(3.35)
So, it is necessary to solve the differential equation
(3.36)
The solution of equation (3.36) is
, where is an arbitrary constant.
In dilute systems for small interparticle distance the
correlation factor is well approximated by the function ,
i.e. by the wavefunction of a pair of particles in vacuum. At large
distances the pair wavefunction should be constant, which
corresponds to uncorrelated particles.
Taking these facts into account let us introduce the trial function
in the following way [24]
(3.37)
This function has to be smooth at the matching point , i.e.
1) the must be continuous
(3.38)
2) its derivative must be continuous
(3.39)
3) the local energy
must be
continuous
(3.40)
The solution of this system is
(3.41)
where we used the notation
and
.
The value of is obtained from the equation
(3.42)
There are three conditions for the determination of five unknown
parameters, consequently two parameters are left free. The usual
way to define them is minimize the variational energy in
Variational Monte Carlo which yields an optimized trial
wavefunction.
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