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10.2 Nuclear motion in diatomic molecules 269

electronic energy level, the Born±Oppenheimer approximation and especially
the adiabatic approximation are quite accurate for the electronic ground state.
The in¯uence of the coupling terms for the ®rst few excited electronic energy
levels may then be calculated using perturbation theory.

10.2 Nuclear motion in diatomic molecules
The application of the Born±Oppenheimer and the adiabatic approximations to
separate nuclear and electronic motions is best illustrated by treating the
simplest example, a diatomic molecule in its electronic ground state. The
diatomic molecule is suf®ciently simple that we can also introduce center-of-
mass coordinates and show explicitly how the translational motion of the
molecule as a whole is separated from the internal motion of the nuclei and
electrons.

Center-of-mass coordinates
The total number of spatial coordinates for a molecule with Ù nuclei and N
electrons is 3(Ù ‡ N), because each particle requires three cartesian coordi-
nates to specify its location. However, if the motion of each particle is referred
to the center of mass of the molecule rather than to the external spaced-®xed
coordinate axes, then the three translational coordinates that specify the
location of the center of mass relative to the external axes may be separated out
and eliminated from consideration. For a diatomic molecule (Ù ˆ 2) we are
left with only three relative nuclear coordinates and with 3N relative electronic
coordinates. For mathematical convenience, we select the center of mass of the
nuclei as the reference point rather than the center of mass of the nuclei and
electrons together. The difference is negligibly small. We designate the two
nuclei as A and B, and introduce a new set of nuclear coordinates de®ned by
M A M B
X ˆ Q A ‡ Q B (10:22a)
M M
R ˆ Q B ÿ Q A (10:22b)
where X locates the center of mass of the nuclei in the external coordinate
system, R is the vector distance between the two nuclei, and M is the sum of
the nuclear masses

M ˆ M A ‡ M B
^
The kinetic energy operator T Q for the two nuclei, as given by equation
(10.2), is