420 Quantum Monte Carlo methods
(a) Show that for a two-dimensional table containing values of a functionf(x,y)for
integerxandy, the valuef(x,y)for arbitraryxandywithin the boundaries set
by the table size is given as
f(x,y)=( 2 −x−y+[x]+[y])f([x],[y]) (12.107)
+( 1 +x−[x]−y+[y])f([x]+1,[y]) (12.108)
+( 1 +y−[y]−x+[x])f([x],[y]+ 1 ) (12.109)
+(x−[x]+y−[y])f([x]+1,[y]+ 1 ).
Here[x]denotes the largest integer smaller thanx.
(b) Find analogous expressions for a table with a noninteger (but equidistant)
spacing between the table entries and also for a three-dimensional Table.12.5 [C]
In this problem we consider applying variational Monte Carlo to the hydrogen
molecule. There are two complications in comparison with the helium atom. One is
the calculation of the local energy which is quite cumbersome, although
straightforward. The second one is the cusp condition.
To specify the trial wave function we take the nuclei at positions±s/2. A
one-particle orbital has the form (in atomic units):
φ(r)=e−|r−sxˆ/^2 |/a+e−|r+sxˆ/^2 |/a
whereais some parameter. The two-electron wave function is given as
ψ(r 1 ,r 2 )=φ(r 1 )φ(r 2 )f(r 12 )
withfthe Jastrow factor
f(r)=exp(
r
α( 1 +βr 12 ))
.
(a) Show that the Coulomb-cusp condition near the nuclei leads to the relation
1
1 +exp(−s/a)
=a.
For a given distances, this equation should be solved numerically to give you
the valuea.
(b) Show that the electron–electron cusp condition leads to the requirementa=2.
This leaves a single parameterβin the wave function.
(c) Now you can implement the hydrogen molecule in VMC. Calculate the energy
as a function of the parametersβandsand find the minimum.
(d) You may also evaluate the ground state by applying the method of Harjuet al.
[6]which was described inSection 12.2, in order to update the values ofβands
simultaneously.
(e) What would you need in order to calculate the molecular formation energy from
this? Note that this is the difference between the energy of the hydrogen
molecule and that of two isolated hydrogen atoms. Consider in particular the
contribution arising from the nuclear motion.