380 Quantum Monte Carlo methods
Forl>0, the radial wave function is written in the formrlρ(r)whereρdoes
not vanish atr=0. Analysing this in a way similar to thel=0 case leads to the
cusp condition
1
ρ(r)dρ(r)
dr=−
Z
l+ 1. (12.19)
Note that this form is the same as(12.18)if we putl=0.
Another cusp condition is found for two electrons approaching each other. Con-
sidering the trial wave function of the helium atom, Eq. (12.11), we see that it is the
dependence on the separationrbetween the two electrons which must incorporate
the correct behaviour in this limit. The resulting radial equation for therdepend-
ence is the same as for the electron–nucleus cusp except for the−Z/rpotential
being replaced by 1/r(the Coulomb repulsion between the two electrons), and the
kinetic term being twice as large (because the reduced mass of the two electrons is
half the electron mass):
[
2
d^2
dr^2+
4
rd
dr−
2
r−
l(l+ 1 )
r^2]
R(r)=0. (12.20)The cusp condition, written in terms ofρ(r)=r−lR(r), is therefore
1
ρ(r)dρ(r)
dr=
1
2 (l+ 1 ). (12.21)
The right hand side reduces to 1/2 in the usual case of an s-wave function (l=0).
For like spins, the value of the wave function must vanish if the particles approach
each other; therefore the wave function with lowest energy is a p-state and the
right hand side will reduce to 1/4. For a general system, containing more than two
electrons, we have this cusp condition for each electron pairij. It is recommended
to have a look at Problem 12.5 to see how cusp conditions are implemented in
practice.
12.2.4 Diffusion equations, Green functions and Langevin equationsIn the following sections we shall discuss several QMC methods in which the
ground state of a quantum Hamiltonian is found by simulating a diffusion process.
In the next section for example, we shall use such a simulation to improve on the
variational method described above. In this section, we give a brief overview of
diffusion and the related equations.
Consider a one-dimensional discrete axis with sites located atna, with integer
n. We place a random walker on a site, and this walker jumps from site to site with
time intervalsh. The walker can only jump from a site to its left or right neighbour.
Both jumps have a probabilityα, and the walker remains at the current position with