Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 467

Tˆ(R) =−h ̄

2

2 M

∇^20 −

N

∑

i= 1

̄h^2

2 m

∇^2 i. (14.11)

Here the first term is the kinetic energy operator of the nucleus, the second term is the kinetic
energy operator of the electrons,Mis the mass of the nucleus andmis the electron mass.
The potential energy operator is given by

Vˆ(R) =−

N

∑

i= 1

Ze^2

( 4 π ε 0 )ri

+

N

∑

i= 1 ,i<j

e^2

( 4 π ε 0 )ri j

, (14.12)

where theri’s are the electron-nucleus distances and theri j’s are the inter-electronic dis-
tances.
We seek to find controlled and well understood approximations in order to reduce the
complexity of the above equations. TheBorn-Oppenheimer approximationis a commonly used
approximation. In this approximation, the motion of the nucleus is disregarded.

14.5.1The Born-Oppenheimer Approximation

In a system of interacting electrons and a nucleus there willusually be little momentum
transfer between the two types of particles due to their differing masses. The forces between
the particles are of similar magnitude due to their similar charge. If one assumes that the
momenta of the particles are also similar, the nucleus must have a much smaller velocity
than the electrons due to its far greater mass. On the time-scale of nuclear motion, one
can therefore consider the electrons to relax to a ground-state given by the Hamiltonian of
Eqs. (14.10), (14.11) and (14.12) with the nucleus at a fixed location. This separation of the
electronic and nuclear degrees of freedom is known as the Born-Oppenheimer approximation.
In the center of mass system the kinetic energy operator reads

Tˆ(R) =− h ̄

2

2 (M+Nm)

∇^2 CM−h ̄

2

2 μ

N

∑

i= 1

∇^2 i−h ̄

2

M

N

∑

i>j

∇i·∇j, (14.13)

while the potential energy operator remains unchanged. Note that the Laplace operators∇^2 i
now are in the center of mass reference system.
The first term of Eq. (14.13) represents the kinetic energy operator of the center of mass.
The second term represents the sum of the kinetic energy operators of theNelectrons, each
of them having their massmreplaced by the reduced massμ=mM/(m+M)because of the
motion of the nucleus. The nuclear motion is also responsible for the third term, or themass
polarizationterm.
The nucleus consists of protons and neutrons. The proton-electron mass ratio is about
1 / 1836 and the neutron-electron mass ratio is about 1 / 1839. We can therefore approximate
the nucleus as stationary with respect to the electrons. Taking the limitM→∞in Eq. (14.13),
the kinetic energy operator reduces to

Tˆ=−

N

∑

i= 1

̄h^2

2 m∇

(^2) i
The Born-Oppenheimer approximation thus disregards both the kinetic energy of the cen-
ter of mass as well as the mass polarization term. The effectsof the Born-Oppenheimer ap-
proximation are quite small and they are also well accountedfor. However, this simplified
electronic Hamiltonian remains very difficult to solve, andclosed-form solutions do not exist