3.7 Vibrational Structure of Electronic Spectra
95
3.7 Vibrational Structure of Electronic Spectra
The absorption or emission of UV, visible, or NIR light implies a transition between
electronic states, which is often complemented by a certain degree of vibrational
excitation. We shall assume that the exciting light is sufficiently monochromatic
as to promote transitions between vibronic eigenstates. In the Born–Oppenheimer
approximation, the initial and final wavefunctions are the products ϕ l χ lu e ϕ k χ kv ,
respectively. If the molecule is initially in the electronic ground state, i.e., usually the
S 0 singlet state, then l = 0. At room temperature, the vibrational state too is normally
the lowest in energy (u = 0). Exceptions are due to low-frequency vibrations, when
present, having hν of the order of K B T or lower, in which case some of the first
vibrational states can have non-negligible populations. This means that normally the
initial molecular geometry is close to the equilibrium one in S 0 .
From Sect. 3.5 we know that the intensity of an absorption band, i.e., the integrated
area under the corresponding peak of the extinction coefficient ε(ν), is proportional
to ν 00,kv μ
2
00,kv . For emission bands, the relevant factor is ν
3
0u,10 μ
2
0u,10 , where the
electronic index 1 refers to the S 1 state. In fact, as we shall see in Sect. 3.11, the
higher excited states usually decay in a very short time to S 1 , and fluorescence
emission then follows. If the triplet states are populated by ISC, then T 1 is normally
the emitting state (phosphorescence). Moreover, if the excited state lifetime is long
enough and the transfer of vibrational energy to the environment is efficient, as
normal in condensed phase, by far the most populated vibrational state is the lowest
one, just as in S 0 (see Sect. 4.5). Figures 3.3, 3.4, and 3.5 illustrate these concepts:
they show schematically the potential energy curves U 0 and U 1 , belonging to S 0 and
S 1 , the relevant vibrational levels and the transitions giving place to absorption and
fluorescence bands.
The transition dipole for the l, u → k, v transition is the result of two integrations,
one over the electronic coordinates and one over the nuclear ones. The electronic
transition dipole, computed for a given set of nuclear coordinates R, is
µ lk (R) = ϕ l (R) |µ| ϕ k (R) r .
(3.67)
If the ϕ l and ϕ k have different spin multiplicities, the matrix element vanishes and
the transition is “spin-forbidden.” Such transition occur because of the existence
of magnetic couplings, both between radiation and molecules and, more important,
between molecular states of different spin (see Sect. 2.4). µ lk can also be zero by
symmetry, when the product ϕ l ϕ k belongs to an irreducible representation of the point
group that none of the components of the µ vector does match. For instance, in the
C 2v group, x, y, and z belong to the B 2 , B 1 , and A 1 representations, so the transitions
between A 1 and A 2 states are “symmetry-forbidden,” and those between B 1 and
B 2 states as well. Note that the symmetry considerations concern the equilibrium
geometry R eq of the starting state, but in polyatomics a symmetric geometry can
always be modified by molecular vibrations, in such a way that µ lk (R) = 0. The
matrix element of the dipole between vibronic states is
95
3.7 Vibrational Structure of Electronic Spectra
The absorption or emission of UV, visible, or NIR light implies a transition between
electronic states, which is often complemented by a certain degree of vibrational
excitation. We shall assume that the exciting light is sufficiently monochromatic
as to promote transitions between vibronic eigenstates. In the Born–Oppenheimer
approximation, the initial and final wavefunctions are the products ϕ l χ lu e ϕ k χ kv ,
respectively. If the molecule is initially in the electronic ground state, i.e., usually the
S 0 singlet state, then l = 0. At room temperature, the vibrational state too is normally
the lowest in energy (u = 0). Exceptions are due to low-frequency vibrations, when
present, having hν of the order of K B T or lower, in which case some of the first
vibrational states can have non-negligible populations. This means that normally the
initial molecular geometry is close to the equilibrium one in S 0 .
From Sect. 3.5 we know that the intensity of an absorption band, i.e., the integrated
area under the corresponding peak of the extinction coefficient ε(ν), is proportional
to ν 00,kv μ
2
00,kv . For emission bands, the relevant factor is ν
3
0u,10 μ
2
0u,10 , where the
electronic index 1 refers to the S 1 state. In fact, as we shall see in Sect. 3.11, the
higher excited states usually decay in a very short time to S 1 , and fluorescence
emission then follows. If the triplet states are populated by ISC, then T 1 is normally
the emitting state (phosphorescence). Moreover, if the excited state lifetime is long
enough and the transfer of vibrational energy to the environment is efficient, as
normal in condensed phase, by far the most populated vibrational state is the lowest
one, just as in S 0 (see Sect. 4.5). Figures 3.3, 3.4, and 3.5 illustrate these concepts:
they show schematically the potential energy curves U 0 and U 1 , belonging to S 0 and
S 1 , the relevant vibrational levels and the transitions giving place to absorption and
fluorescence bands.
The transition dipole for the l, u → k, v transition is the result of two integrations,
one over the electronic coordinates and one over the nuclear ones. The electronic
transition dipole, computed for a given set of nuclear coordinates R, is
µ lk (R) = ϕ l (R) |µ| ϕ k (R) r .
(3.67)
If the ϕ l and ϕ k have different spin multiplicities, the matrix element vanishes and
the transition is “spin-forbidden.” Such transition occur because of the existence
of magnetic couplings, both between radiation and molecules and, more important,
between molecular states of different spin (see Sect. 2.4). µ lk can also be zero by
symmetry, when the product ϕ l ϕ k belongs to an irreducible representation of the point
group that none of the components of the µ vector does match. For instance, in the
C 2v group, x, y, and z belong to the B 2 , B 1 , and A 1 representations, so the transitions
between A 1 and A 2 states are “symmetry-forbidden,” and those between B 1 and
B 2 states as well. Note that the symmetry considerations concern the equilibrium
geometry R eq of the starting state, but in polyatomics a symmetric geometry can
always be modified by molecular vibrations, in such a way that µ lk (R) = 0. The
matrix element of the dipole between vibronic states is