Computational Physics

4.2 The Born–Oppenheimer approximation and the IP method 45
Therefore, important approximations must be made, and a first step consists of
separating the degrees of freedom connected with the motion of the nuclei from
those of the electrons. This procedure is known as the Born–Oppenheimer approx-
imation [1] and its justification resides in the fact that the nuclei are much heavier
than the electrons (the mass of a proton or neutron is about 1835 times as large as the
electron mass) so it is intuitively clear that the nuclei move much more slowly than
the electrons. The latter will then be able to adapt themselves to the current config-
uration of nuclei. This approach results also from formal calculations (see Problem
4.9), and leads to a Hamiltonian for the electrons in the field generated by a static
configuration of nuclei, and a separate Schrödinger equation for the nuclei in which
the electronic energy enters as a potential. The Born–Oppenheimer Hamiltonian
for the electrons reads

HBO=

∑N

i= 1

p^2 i

2 m

+

1

2

1

4 π 0

∑N

i,j=1;i
=j

e^2

|ri−rj|

−

1

4 π 0

∑K

n= 1

∑N

i= 1

Zne^2

|ri−Rn|

. (4.2)

The total energy is the sum of the energy of the electrons and the energy resulting
from the Schrödinger equation satisfied by the nuclei. In a further approximation,
the motion of the nuclei is neglected and only the electrostatic energy of the nuclei
should be added to the energy of the electrons to arrive at the total energy. The pos-
itions of the nuclei can be varied in order to find the minimum of this energy, that
is, the ground state of the whole system (within the Born–Oppenheimer approx-
imation with static nuclei). In this procedure, the nuclei are treated on a classical
footing since their ground state is determined as the minimum of their potential
energy, neglecting quantum fluctuations.^1
Even with the positions of the nuclei kept fixed, the problem of solving for the
electronic wave functions using the Hamiltonian (4.2) remains intractable, even on
a computer, since too many degrees of freedom are involved. It is the second term
containing the interactions between the electrons that makes the problem so difficult.
If this term were not present, we would be dealing with a sum of one-electron
Hamiltonians which can be solved relatively easily. There exist several ways of
approximating the eigenfunctions of the Hamiltonian (4.2). In these approaches, the
many-electron problem is reduced to an uncoupled problem in which the interaction
of one electron with the remaining ones is incorporated in an averaged way into a
potential felt by the electron.

(^1) Vibrational modes of the nuclei can, however, be treated after expanding the total energy in deviations of
the nuclear degrees of freedom from the ground state configuration. A transformation to normal modes then
gives us a system consisting of independent harmonic oscillators.