3.11 Excited State Decay to a Quasi-continuum
115
rigid molecules, as for instance naphthalene (see Sect. 1.6.2). For such molecules, S 1
lifetimes are at least 1 ns or much longer. However, when the molecular geometry
can undergo large rearrangements, as for instance by twisting a double bond, the S 1
and S 0 PESs can get much closer and the S 1 lifetime decreases dramatically. A clue
about different lifetimes in analogous compounds is the fluorescence quantum yield,
which is inversely proportional to the S 1 lifetime. Figure 3.10 shows two examples of
the effect of structural rigidity imposed to otherwise flexible molecules. In Chap. 5
we shall analyze the fast decay of excited states and we shall see that it is often
related to the intersection of potential energy surfaces.
The energy gaps between excited singlet states are usually much smaller than the
gap between S 1 and S 0 , and similarly for triplets. As already discussed in Chap. 2, the
“crowding” of states beyond S 1 is due to the many ways one can promote an electron
from bonding or nonbonding orbitals to the virtual ones, for approximately the same
energetic cost, especially in large molecules. It follows that the internal conversion
between excited electronic states of the same spin is usually very fast, with typical
times of 1 ps or less. No matter which state the molecule was initially excited to, in
a very short time it will be found in S 1 or, if ISC is very efficient, in T 1 . Kasha’s rule
states that fluorescence is related to S 1 → S 0 transitions, irrespective of the exciting
wavelength. In a similar way, phosphorescence normally starts from T 1 . Other slow
processes, such as bimolecular reactions between partners that need to collide in gas
phase or meet by diffusion in liquid phase, also occur only after the excited molecule
has decayed to S 1 or T 1 . Moreover, as already anticipated, in condensed phase the
vibrational energy loss to the environment is also fast, so in a short time only the
lowest vibrational level(s) of S 1 or T 1 are populated. In this way, the exciting photon
wavelength does not affect the fluorescence and phosphorescence spectra, nor their
quantum yields and those of slow photoreactions. Exceptions to Kasha’s rule are very
fast processes, such as direct photodissociations or the twisting of double bonds, that
may outcompete the decay of higher excited states. Other interesting exceptions are
molecules such as azulene, where the S 2 − S 1 and S 1 − S 1 energy gaps are both large
and almost equal: as a result, fluorescence from S 2 can be observed.
3.12 Computational Note: Franck–Condon Factors and
Coupling Matrix Elements
The calculation of absorption and emission spectra is a common task, which aims
at assigning the experimental bands and at predicting optical properties of new
molecules and materials. We have reviewed in the previous chapter how the electronic
problem is tackled by deploying several quantum chemistry methods. The comparison with recorded electronic spectra or their prediction is often done by resorting to
simplifying assumptions. For instance, vertical energy differences are put in relation
with the intensity maxima of absorption and emission bands. More accurately, adi-
115
rigid molecules, as for instance naphthalene (see Sect. 1.6.2). For such molecules, S 1
lifetimes are at least 1 ns or much longer. However, when the molecular geometry
can undergo large rearrangements, as for instance by twisting a double bond, the S 1
and S 0 PESs can get much closer and the S 1 lifetime decreases dramatically. A clue
about different lifetimes in analogous compounds is the fluorescence quantum yield,
which is inversely proportional to the S 1 lifetime. Figure 3.10 shows two examples of
the effect of structural rigidity imposed to otherwise flexible molecules. In Chap. 5
we shall analyze the fast decay of excited states and we shall see that it is often
related to the intersection of potential energy surfaces.
The energy gaps between excited singlet states are usually much smaller than the
gap between S 1 and S 0 , and similarly for triplets. As already discussed in Chap. 2, the
“crowding” of states beyond S 1 is due to the many ways one can promote an electron
from bonding or nonbonding orbitals to the virtual ones, for approximately the same
energetic cost, especially in large molecules. It follows that the internal conversion
between excited electronic states of the same spin is usually very fast, with typical
times of 1 ps or less. No matter which state the molecule was initially excited to, in
a very short time it will be found in S 1 or, if ISC is very efficient, in T 1 . Kasha’s rule
states that fluorescence is related to S 1 → S 0 transitions, irrespective of the exciting
wavelength. In a similar way, phosphorescence normally starts from T 1 . Other slow
processes, such as bimolecular reactions between partners that need to collide in gas
phase or meet by diffusion in liquid phase, also occur only after the excited molecule
has decayed to S 1 or T 1 . Moreover, as already anticipated, in condensed phase the
vibrational energy loss to the environment is also fast, so in a short time only the
lowest vibrational level(s) of S 1 or T 1 are populated. In this way, the exciting photon
wavelength does not affect the fluorescence and phosphorescence spectra, nor their
quantum yields and those of slow photoreactions. Exceptions to Kasha’s rule are very
fast processes, such as direct photodissociations or the twisting of double bonds, that
may outcompete the decay of higher excited states. Other interesting exceptions are
molecules such as azulene, where the S 2 − S 1 and S 1 − S 1 energy gaps are both large
and almost equal: as a result, fluorescence from S 2 can be observed.
3.12 Computational Note: Franck–Condon Factors and
Coupling Matrix Elements
The calculation of absorption and emission spectra is a common task, which aims
at assigning the experimental bands and at predicting optical properties of new
molecules and materials. We have reviewed in the previous chapter how the electronic
problem is tackled by deploying several quantum chemistry methods. The comparison with recorded electronic spectra or their prediction is often done by resorting to
simplifying assumptions. For instance, vertical energy differences are put in relation
with the intensity maxima of absorption and emission bands. More accurately, adi-