4.2 Vibrational Wavepacket Dynamics
129
the chirality is conserved till the achiral product is formed, while in the latter case
a chirality inversion occurs. Simulations show that the chirality conserving pathway
is preferred when exciting in the n → π
∗ band, while the more energetic π → π
∗
excitation overrides this bias [2, 3].
4.3 Intramolecular Vibrational Energy Redistribution
As we have seen, the electronic excitation is normally associated with a certain degree
of vibrational excitation. Depending on the light wavelength, bandwidth, and polarization, a single vibrational eigenstate or a time-dependent wavepacket may be produced. In both cases, the vibrational energy in the excited state is hν − (E k,0 − E 0,u ).
Here E 0,u is the energy of the starting state and E k,0 is the energy of the lowest vibrational state in the excited PES k. This relationship is exact in the case of monochromatic light and is usually a good approximation of the mean value of the wavepacket
energy when using broadband light pulses. In polyatomics, if the vibrational energy
excess is a few thousands cm
−1 or more, it is hard to populate a chosen eigenstate,
because the huge density of states (see Table 2.1) prevents to resolve single levels.
However, even when the outcome is a nonstationary wavepacket, i.e., a superposition
of eigenstates, the initial distribution of the vibrational excitation is not statistical but
can privilege one or a few vibrational modes. This bias depends on the characteristics of the exciting light and on the transition dipoles between the initial vibronic
state ϕ 0 χ 0,u and the possible final states ϕ k χ k,v . The transition dipoles in turn depend
very much on the shapes of the initial and final PESs through the Franck–Condon
factors χ 0,u
χ k,v ; see Eq. 3.70. As a consequence, excitation will mostly concern
vibrational coordinates along which the two PESs are sharply different, as we have
seen in Sect. 3.7.
Franck–Condon excitation and its classical analogue, vertical excitation, provide the simplest illustration of the above considerations. The center of the excited
wavepacket approximately lies, in the normal coordinate system Q
(ex) of the excited
PES, in the position ΔQ, which is determined by the difference between the equilibrium geometries of ground and excited state. Classically, for each normal mode,
ΔQ α is the maximum elongation of the oscillatory motion that follows the excitation. E α = ω
2
α ΔQ
2
α /2 is the classical normal mode vibrational energy and is a good
approximation for the quantum mechanical expectation value too. The short-time
dynamics is determined by the ΔQ α elongations and E α energies, so for instance if
E α for a bond stretching coordinate exceeds the dissociation energy, bond breaking
is likely to occur promptly. However, the E α distribution of energies in the vibrational modes does not remain unaltered for many oscillations, mainly because of
anharmonicity. This phenomenon is called “internal” or ‘intramolecular vibrational
energy redistribution” (IVR). The IVR is important in photochemistry, because several elementary processes can happen only when enough energy is concentrated in
one or few vibrational modes. IVR can provide or subtract energy to a particular
vibrational mode, so triggering or inhibiting such “activated” processes. Examples
129
the chirality is conserved till the achiral product is formed, while in the latter case
a chirality inversion occurs. Simulations show that the chirality conserving pathway
is preferred when exciting in the n → π
∗ band, while the more energetic π → π
∗
excitation overrides this bias [2, 3].
4.3 Intramolecular Vibrational Energy Redistribution
As we have seen, the electronic excitation is normally associated with a certain degree
of vibrational excitation. Depending on the light wavelength, bandwidth, and polarization, a single vibrational eigenstate or a time-dependent wavepacket may be produced. In both cases, the vibrational energy in the excited state is hν − (E k,0 − E 0,u ).
Here E 0,u is the energy of the starting state and E k,0 is the energy of the lowest vibrational state in the excited PES k. This relationship is exact in the case of monochromatic light and is usually a good approximation of the mean value of the wavepacket
energy when using broadband light pulses. In polyatomics, if the vibrational energy
excess is a few thousands cm
−1 or more, it is hard to populate a chosen eigenstate,
because the huge density of states (see Table 2.1) prevents to resolve single levels.
However, even when the outcome is a nonstationary wavepacket, i.e., a superposition
of eigenstates, the initial distribution of the vibrational excitation is not statistical but
can privilege one or a few vibrational modes. This bias depends on the characteristics of the exciting light and on the transition dipoles between the initial vibronic
state ϕ 0 χ 0,u and the possible final states ϕ k χ k,v . The transition dipoles in turn depend
very much on the shapes of the initial and final PESs through the Franck–Condon
factors χ 0,u
χ k,v ; see Eq. 3.70. As a consequence, excitation will mostly concern
vibrational coordinates along which the two PESs are sharply different, as we have
seen in Sect. 3.7.
Franck–Condon excitation and its classical analogue, vertical excitation, provide the simplest illustration of the above considerations. The center of the excited
wavepacket approximately lies, in the normal coordinate system Q
(ex) of the excited
PES, in the position ΔQ, which is determined by the difference between the equilibrium geometries of ground and excited state. Classically, for each normal mode,
ΔQ α is the maximum elongation of the oscillatory motion that follows the excitation. E α = ω
2
α ΔQ
2
α /2 is the classical normal mode vibrational energy and is a good
approximation for the quantum mechanical expectation value too. The short-time
dynamics is determined by the ΔQ α elongations and E α energies, so for instance if
E α for a bond stretching coordinate exceeds the dissociation energy, bond breaking
is likely to occur promptly. However, the E α distribution of energies in the vibrational modes does not remain unaltered for many oscillations, mainly because of
anharmonicity. This phenomenon is called “internal” or ‘intramolecular vibrational
energy redistribution” (IVR). The IVR is important in photochemistry, because several elementary processes can happen only when enough energy is concentrated in
one or few vibrational modes. IVR can provide or subtract energy to a particular
vibrational mode, so triggering or inhibiting such “activated” processes. Examples