Page 277 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

268 Molecular structure
U k (Q) ˆ å k (Q) ‡ c kk (Q) (10:19)

The ®rst term on the left-hand side of equation (10.18) has the form of a
È
Schrodinger equation for nuclear motion, so that we may identify the expansion
coef®cient ÷ k (Q) as a nuclear wave function for the electronic state k. The
second term couples the in¯uence of all the other electronic states to the
nuclear motion for a molecule in the electronic state k.
^
If the coef®cients c kk (Q) and c kë (Q) and the operators Ë kë are suf®ciently
small, the summation on the left-hand side of equation (10.18) and c kk (Q)in
(10.19) may be neglected, giving a zeroth-order equation for the nuclear
motion
(0)
^
(0)
[T Q ‡ å k (Q) ÿ E ]÷ (Q) ˆ 0 (10:20)
kí
kí
(0)
where ÷ (Q) and E (0) are the zeroth-order approximations to the nuclear wave
kí
kí
functions and energy levels. The index í represents a set of quantum numbers
which determine the nuclear state. The neglect of these coef®cients and
operators is the Born±Oppenheimer approximation and equation (10.20) is
identical to (10.8). Furthermore, the molecular wave function Ø(r, Q)in
equation (10.10) reduces to the product of a nuclear and an electronic wave
function as shown in equation (10.9).
^
When the coupling coef®cients c kë for k 6ˆ ë and the coupling operators Ë kë
are neglected, but the coef®cient c kk (Q) is retained, equation (10.18) becomes
^
(1)
(1)
[T Q ‡ U k (Q) ÿ E ]÷ (Q) ˆ 0 (10:21)
kí kí
(1)
where ÷ (Q) and E (1) are the ®rst-order approximations to the nuclear wave
kí kí
functions and energy levels. Since the term c kk (Q) is added to å k (Q) in this
approximation, equation (10.21) is different from (10.20) and, therefore,
(0)
(1)
(0)
÷ (Q) and E (1) differ from ÷ (Q) and E . In this ®rst-order approximation,
kí
kí
kí
kí
the molecular wave function Ø(r, Q) in equation (10.10) also takes the form of
(1)
(10.9). The factor ÷ (Q) describes the nuclear motion, which takes place in a
kí
potential ®eld U k (Q) determined by the electrons moving as though the nuclei
were ®xed in their instantaneous positions. Thus, the electronic state of the
molecule changes in a continuous manner as the nuclei move slowly in
comparison with the electronic motion. In this situation, the electrons are said
to follow the nuclei adiabatically and this ®rst-order approximation is known
as the adiabatic approximation. This adiabatic feature does not occur in
higher-order approximations, in which coupling terms appear.
An analysis using perturbation theory shows that the in¯uence of the
^
coupling terms with c kë (Q) and Ë kë is small when the electronic energy levels
å k (Q) and å ë (Q) are not closely spaced. Since the ground-state electronic
energy of a molecule is usually widely separated from the ®rst-excited