Essentials of Computational Chemistry

510 14 EXCITED ELECTRONIC STATES
is absorbed in the transition between the two states, although it takes a more sophisticated
theoretical treatment to demonstrate this. However, this term fails to differentiate any one
state m from another, all states being predicted to undergo transitions with equal probability
at their respective frequencies.
It is the last term that accounts for differences in absorption probabilities. This term is the
expectation value of the dipole moment operator (see Section 9.1.1) evaluated over different
determinants. Its expectation value is referred to as the transition dipole moment.
The matrix elements rm0 are quite straightforward to evaluate. Before leaving them,
however, it is worthwhile to make some qualitative observations about them. First, the
Condon–Slater rules dictate that for the one-electron operator r, the only matrix elements
that survive are those between determinants differing by at most two electronic orbitals.
Thus, only absorptions generating singly or doubly excited states are allowed.
In addition, group theory can be used to assess when transition dipole moments must
be zero. The product of the irreducible representations of the two wave functions and the
dipole moment operator within the molecular point group symmetry must contain the totally
symmetric representation for the matrix element to be non-zero (note that, if the molecule
does not contain an inversion center, the operator r does not belong to any single irrep, except
for the trivial case of C1 symmetry; see Appendix B for more details). A consequence of this
consideration is that, for instance, electronic transitions between states of the same symmetry
are forbidden in molecules possessing inversion centers.
The derivation above may be generalized to wave functions other than electronic ones.
By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator
rotational and vibrational wave functions, respectively, one arrives at the well-known selection
rules in those systems that absorptions and emissions can only occur to adjacent levels,
as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many
‘forbidden’ transitions being observable in the laboratory as weakly allowed, both in the
electronic case and in the rotational and vibrational cases.
As a final point, let us consider the transition not simply between electronic states, but
between wave functions described as products of (decoupled) electronic and vibrational
states. That is, we consider wave functions of the form
= (14.34)
where is the electronic wave function of Eq. (14.22) and is a vibrational wave function,
e.g., as defined by Eq. (9.40). If we carry out the same analysis as above, for radiation of
wavelengths that are far from regions associated with vibrational transitions (as UV/Vis is
from IR), then we find that Eq. (14.31) generalizes to
cm,n(τ) = 1
2i¯h e0
i(ωm0+ω)τ e − 1
ωm0 + ω − ei(ωm0−ω)τ
− 1
〈m|r|0〉〈
ωm0 − ω
m n |00 〉 (14.35)
where n indexes the vibrational wave functions of electronic state m, and we have assumed
that the ground electronic state is also in its ground vibrational state. We now ask the question,
when is the overlap between the vibrational wave functions (the so-called Franck–Condon