Page 274 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

10.1 Nuclear structure and motion 265
^
time and the kinetic energy operator T Q in equation (10.2) vanishes. The
Schrodinger equation (10.5) under this condition becomes
È
^
(H e ‡ V Q )ø k (r, Q) ˆ å k (Q)ø k (r, Q) (10:6)
where the coordinates Q are no longer variables, but rather are constant
parameters. For each electronic state k, the electronic energy å k (Q) of the
molecule and the eigenfunction ø k (r, Q) depend parametrically on the ®xed
values of the coordinates Q. The nuclear±nuclear interaction potential V Q is
now a constant and its value is included in å k (Q).
We assume in this section and in Section 10.2 that equation (10.6) has been
solved and that the eigenfunctions ø k (r, Q) and eigenvalues å k (Q) are known
for any arbitrary set of values for the parameters Q. Further, we assume that the
eigenfunctions form a complete orthonormal set, so that

ø (r, Q)ø ë (r, Q)dr ˆ ä kë (10:7)
k
In the second step of the Born±Oppenheimer approximation, the energy
å k (Q) is used as a potential energy function to treat the nuclear motion. In this
case, equation (10.5) becomes
^
[T Q ‡ å k (Q)]÷ kí (Q) ˆ E kí ÷ kí (Q) (10:8)
where the nuclear wave function ÷ kí (Q) depends on the nuclear coordinates Q
and on the electronic state k. Each electronic state k gives rise to a series of
nuclear states, indexed by í. Thus, for each electronic state k, the eigenfunc-
^
tions of [T Q ‡ å k (Q)] are ÷ kí (Q) with eigenvalues E kí . In practice, the nuclear
states are differentiated by several quantum numbers; the index í represents,
then, a set of these quantum numbers. In the solution of the differential
equation (10.8), the nuclear coordinates Q in å k (Q) are treated as variables.
The nuclear energy E kí , of course, does not depend on any parameters. Most
applications of equation (10.8) are to molecules in their electronic ground
states (k ˆ 0).
1
In the original mathematical treatment of nuclear and electronic motion, M.
Born and J. R. Oppenheimer (1927) applied perturbation theory to equation
^
(10.5) using the kinetic energy operator T Q for the nuclei as the perturbation.
The proper choice for the expansion parameter is ë ˆ (m e =M) 1=4 , where M is
the mean nuclear mass
Ù
1 X
M ˆ M á
Ù
áˆ1
2
When terms up to ë are retained, the exact total energy of the molecule is
1 M. Born and J. R. Oppenheimer (1927) Ann. Physik 84, 457.