3.7 Vibrational Structure of Electronic Spectra
97
Fig. 3.5 Potential energy
curves and electronic
spectra. The Franck–Condon
point in the excited PES is
above the dissociation limit
along the coordinate Q r . The
red dot indicates the
Franck–Condon point
energy
•
abs
intensity
absorption
Q r
U 0
U 1
µ lu,kv =
χ lu
µ lk (R)
χ kv
R
.
(3.68)
The completeness of the basis of vibrational states χ kv leads to the sum rule:
v
μ
2
lu,kv =
χ lu
μ
2
lk
χ lu
.
(3.69)
If the transition is symmetry-allowed, i.e., µ lk (R eq ) = 0, one can apply the Franck–
Condon approximation, i.e., consider µ lk (R) as constant and replace it by its value
at the equilibrium geometry. Then
µ lu,kv µ lk (R eq ) χ lu |χ kv
(3.70)
and
v
μ
2
lu,kv μ
2
lk (R eq ) .
(3.71)
If the relative intensities of the vibrational sub-bands are approximately proportional
to the squared overlap integrals χ lu |χ kv
2 , the strongest band in the absorption
spectrum is due to the χ kv state that best overlaps with χ 00 .
In Fig. 3.3 we show the potential energy curves of S 0 and S 1 and the related absorption and emission spectra. We represent the case where the U 0 and U 1 are similar
and their minima approximately coincide. Then, the largest overlap is obtained for
v = 0 (0-0 band), while all the upper states, with one or more nodes, give place to
much weaker sub-bands at higher frequencies. Overall, the whole electronic band