476 14 Quantum Monte Carlo Methods
V(r 1 ,r 2 ) =−
2 ke^2
r 1−
2 ke^2
r 2+
ke^2
r 12,
with the electrons separated at a distancer 12 =|r 1 −r 2 |. The Hamiltonian becomes then
Ĥ=−h ̄(^2) ∇ 2
1
2 m
− ̄
h^2 ∇^22
2 m−
2 ke^2
r 1−
2 ke^2
r 2+
ke^2
r 12,
and Schrödingers equation reads
Ĥψ=Eψ.
Note that this equation has been written in atomic units (a.u.) which are more convenient
for quantum mechanical problems. This means that the final energy has to be multiplied by a
2 ×E 0 , whereE 0 = 13. 6 eV, the binding energy of the hydrogen atom.
A very simple first approximation to this system is to omit therepulsion between the two
electrons. The potential energy becomes then
V(r 1 ,r 2 )≈−
Zke^2
r 1−
Zke^2
r 2.
The advantage of this approximation is that each electron can be treated as being indepen-
dent of each other, implying that each electron sees just a central symmetric potential, or
central field.
To see whether this gives a meaningful result, we setZ= 2 and neglect totally the repulsion
between the two electrons. Electron 1 has the following Hamiltonian
̂h 1 =−h ̄(^2) ∇ 2
1
2 m
−
2 ke^2
r 1,
with pertinent wave function and eigenvalueEa
̂h 1 ψa=Eaψa,wherea={nalamla}are the relevant quantum numbers needed to describe the system. We
assume here that we can use the hydrogen-like solutions, butwithZnot necessarily equal to
one. The energyEais
Ea=−Z^2 E 0
n^2 a.
In a similar way, we obtain for electron 2
̂h 2 =−h ̄(^2) ∇ 2
2
2 m
−
2 ke^2
r 2,
with wave functionψb,b={nblbmlb}and energy
Eb=Z^2 E 0
n^2 b.
Since the electrons do not interact, the ground state wave function of the helium atom is
given by
ψ=ψaψb,
resulting in the following approximation to Schrödinger’sequation
(
̂h 1 +̂h 2
)
ψ=(
̂h 1 +̂h 2)
ψa(r 1 )ψb(r 2 ) =Eabψa(r 1 )ψb(r 2 ).