98
3 Electronic Excitation and Decay
is narrow and shows one or very few peaks. The same occurs in emission, with the
difference that the weaker sub-bands are at lower frequencies than the 0-0 one.
Consider now the case of two quite different potential energy surfaces, with minima at substantially different geometries. In Fig. 3.4, U 0 and U 1 are displaced along
a coordinate Q r , with the equilibrium value of Q r larger in S 1 than in S 0 . In such
a case, the most intense sub-band in the absorption spectrum is due to a transition
to a vibrational level that puts the inner turning point close to the Franck–Condon
point (it would be the outer turning point for a displacement of the minima in the
opposite direction).
1 In fact, the first maximum in the χ 1v wavefunction is close to
that of χ 00 and its first node is sufficiently far on the right (see Fig. 3.4). The overlaps
with other χ 1v wavefunctions, both higher and lower in energy, decrease gradually,
so that overall the electronic band is quite broad. Similarly, in the emission band the
strongest peak corresponds to a χ 0u state with the outer turning point close to the
minimum geometry in the excited state.
In the last example (Fig. 3.5), Q r is a bond stretching coordinate and presents a
very shallow minimum in the excited state. The Franck–Condon point is well above
dissociation, so the states that best overlap χ 00 belong to the dissociative continuum
and will be indicated by χ 0ε , where ε is the asymptotic kinetic energy (see Appendix
C). Then, the Franck–Condon factor χ 00 |χ 0ε
2 is a function of ε and the absorption
band profile is smooth, without peaks corresponding to the quantized vibrational
levels. The maximum of the band occurs again in correspondence to a vibrational
level such that the turning point approximately coincides with the Franck–Condon
point (note that here only the inner turning point exists). After excitation above the
dissociation threshold, the molecule promptly dissociates and no fluorescence can be
observed. This kind of photodissociation is called “direct,” meaning it takes place in
the same electronic state where the molecule was initially excited. Predissociation,
on the contrary, requires a nonadiabatic transition to occur (see Sect. 3.10). In conclusion, we see that the maximum intensity of a band coincides approximately with
the vertical excitation energy, i.e., the energy difference between the two PESs at
the equilibrium geometry in the initial state. Remember, however, that this statement
relies on the hypothesis that only the v = 0 state is populated in the initial electronic
state, plus at most some of the closest lying vibrational states. For fluorescence (or
phosphorescence to which the same considerations apply), this assumption is only
true when the vibrational energy loss of the chromophore occurs in a time much
shorter than the excited state lifetime (see Sects. 4.3 and 4.5). The electronic bands
can be structured, i.e., show one or more “progressions” belonging to discrete vibrational levels, or featureless, because the final states belong to a dissociative continuum
(this can also occur in emission bands of exciplexes, see Sect. 6.3). However, vibrational progressions can be blurred and give place to essentially continuous bands
because of line broadening and the closeness of vibrational levels: this frequently
1 In classical mechanics, the motion of a particle in one dimension has a turning point when the
potential and the total energies are equal, i.e., the kinetic energy vanishes. For a bound state in
quantum mechanics, the potential energy coincides with the eigenenergy in at least two points: the
inner turning point, Q i , and the outer one, Q o , with Q i < Q o . The Franck–Condon point is the
point in the excited PES corresponding to the equilibrium geometry of the ground state.
3 Electronic Excitation and Decay
is narrow and shows one or very few peaks. The same occurs in emission, with the
difference that the weaker sub-bands are at lower frequencies than the 0-0 one.
Consider now the case of two quite different potential energy surfaces, with minima at substantially different geometries. In Fig. 3.4, U 0 and U 1 are displaced along
a coordinate Q r , with the equilibrium value of Q r larger in S 1 than in S 0 . In such
a case, the most intense sub-band in the absorption spectrum is due to a transition
to a vibrational level that puts the inner turning point close to the Franck–Condon
point (it would be the outer turning point for a displacement of the minima in the
opposite direction).
1 In fact, the first maximum in the χ 1v wavefunction is close to
that of χ 00 and its first node is sufficiently far on the right (see Fig. 3.4). The overlaps
with other χ 1v wavefunctions, both higher and lower in energy, decrease gradually,
so that overall the electronic band is quite broad. Similarly, in the emission band the
strongest peak corresponds to a χ 0u state with the outer turning point close to the
minimum geometry in the excited state.
In the last example (Fig. 3.5), Q r is a bond stretching coordinate and presents a
very shallow minimum in the excited state. The Franck–Condon point is well above
dissociation, so the states that best overlap χ 00 belong to the dissociative continuum
and will be indicated by χ 0ε , where ε is the asymptotic kinetic energy (see Appendix
C). Then, the Franck–Condon factor χ 00 |χ 0ε
2 is a function of ε and the absorption
band profile is smooth, without peaks corresponding to the quantized vibrational
levels. The maximum of the band occurs again in correspondence to a vibrational
level such that the turning point approximately coincides with the Franck–Condon
point (note that here only the inner turning point exists). After excitation above the
dissociation threshold, the molecule promptly dissociates and no fluorescence can be
observed. This kind of photodissociation is called “direct,” meaning it takes place in
the same electronic state where the molecule was initially excited. Predissociation,
on the contrary, requires a nonadiabatic transition to occur (see Sect. 3.10). In conclusion, we see that the maximum intensity of a band coincides approximately with
the vertical excitation energy, i.e., the energy difference between the two PESs at
the equilibrium geometry in the initial state. Remember, however, that this statement
relies on the hypothesis that only the v = 0 state is populated in the initial electronic
state, plus at most some of the closest lying vibrational states. For fluorescence (or
phosphorescence to which the same considerations apply), this assumption is only
true when the vibrational energy loss of the chromophore occurs in a time much
shorter than the excited state lifetime (see Sects. 4.3 and 4.5). The electronic bands
can be structured, i.e., show one or more “progressions” belonging to discrete vibrational levels, or featureless, because the final states belong to a dissociative continuum
(this can also occur in emission bands of exciplexes, see Sect. 6.3). However, vibrational progressions can be blurred and give place to essentially continuous bands
because of line broadening and the closeness of vibrational levels: this frequently
1 In classical mechanics, the motion of a particle in one dimension has a turning point when the
potential and the total energies are equal, i.e., the kinetic energy vanishes. For a bound state in
quantum mechanics, the potential energy coincides with the eigenenergy in at least two points: the
inner turning point, Q i , and the outer one, Q o , with Q i < Q o . The Franck–Condon point is the
point in the excited PES corresponding to the equilibrium geometry of the ground state.