138
4 Wavepacket Dynamics and Geometrical Relaxation
energy distributions between vibrational modes at equilibrium have been illustrated
in Sects. 4.3 and 4.5. Of particular concern is the role of the zero-point energy in
classical dynamics, because due to IVR this amount of energy can be used to reach
regions of the PES (transition states or high energy conical intersections) that in
quantum dynamics are not accessible. In part, such differences can be reduced by
freezing the high-frequency nonreactive modes that contribute most to the problem.
A last observation concerns the excitation process, which deeply affects the photophysical and photochemical dynamics as shown in Sect. 4.2. In the computational
simulations of excited state dynamics, this aspect is often drastically simplified,
by assuming a certain form of the “initial” excited wavepacket: either the exciting light is supposed to be close to monochromatic and the simulation starts with
a vibronic eigenstate in the Born–Oppenheimer approximation; or, more often, a
Franck–Condon excitation is postulated and the ground-state v = 0 wavefunction is
translated to the excited PES. Similar assumptions can be applied to create the initial
swarm of trajectories in classical dynamics, by vertical excitation from the groundstate distribution of coordinates and momenta, or by imposing a narrow range of
energies in the excited state. Quantum mechanical methods can be complemented
with a light–molecule Hamiltonian term, normally in the form of Eq. (3.10), in order
to deal correctly with the excitation process. The same can be done in classical trajectory treatments, but some problematic issues arise, mainly because of the unphysical
interference between the initial and final state long after a partial transfer of population has taken place. This is due to a general failure of classical (independent)
trajectories in multistate systems to reproduce the phenomenon of quantum decoherence (see Sect. 5.5 and Ref. [20]).
Problems
4.1 Calculate the averages and uncertainties of x and ˆ
p for the χ v eigenfunction of
the harmonic oscillator with mass M, frequency ω, and equilibrium position x e . Verify
that for v = 0 we have the minimum uncertainty product ΔxΔ p = /2. Compare
the uncertainty of x in state v with the classical amplitude of the oscillation for the
same energy. Make use of the relationships listed in Appendix F.
4.2 Calculate the vibrational energy in the final state for the Franck–Condon excitation between one-dimensional harmonic PESs defined as:
U 1 =
1
2
Mω
2
1 (R − R 1 )
2
and U 2 = ΔE adia +
1
2
Mω
2
2 (R − R 2 )
2
Make use of the relationships listed in Appendix F.
4.3 Prove the relationship (4.17).
4.4 Prove the relationship (4.6).
4 Wavepacket Dynamics and Geometrical Relaxation
energy distributions between vibrational modes at equilibrium have been illustrated
in Sects. 4.3 and 4.5. Of particular concern is the role of the zero-point energy in
classical dynamics, because due to IVR this amount of energy can be used to reach
regions of the PES (transition states or high energy conical intersections) that in
quantum dynamics are not accessible. In part, such differences can be reduced by
freezing the high-frequency nonreactive modes that contribute most to the problem.
A last observation concerns the excitation process, which deeply affects the photophysical and photochemical dynamics as shown in Sect. 4.2. In the computational
simulations of excited state dynamics, this aspect is often drastically simplified,
by assuming a certain form of the “initial” excited wavepacket: either the exciting light is supposed to be close to monochromatic and the simulation starts with
a vibronic eigenstate in the Born–Oppenheimer approximation; or, more often, a
Franck–Condon excitation is postulated and the ground-state v = 0 wavefunction is
translated to the excited PES. Similar assumptions can be applied to create the initial
swarm of trajectories in classical dynamics, by vertical excitation from the groundstate distribution of coordinates and momenta, or by imposing a narrow range of
energies in the excited state. Quantum mechanical methods can be complemented
with a light–molecule Hamiltonian term, normally in the form of Eq. (3.10), in order
to deal correctly with the excitation process. The same can be done in classical trajectory treatments, but some problematic issues arise, mainly because of the unphysical
interference between the initial and final state long after a partial transfer of population has taken place. This is due to a general failure of classical (independent)
trajectories in multistate systems to reproduce the phenomenon of quantum decoherence (see Sect. 5.5 and Ref. [20]).
Problems
4.1 Calculate the averages and uncertainties of x and ˆ
p for the χ v eigenfunction of
the harmonic oscillator with mass M, frequency ω, and equilibrium position x e . Verify
that for v = 0 we have the minimum uncertainty product ΔxΔ p = /2. Compare
the uncertainty of x in state v with the classical amplitude of the oscillation for the
same energy. Make use of the relationships listed in Appendix F.
4.2 Calculate the vibrational energy in the final state for the Franck–Condon excitation between one-dimensional harmonic PESs defined as:
U 1 =
1
2
Mω
2
1 (R − R 1 )
2
and U 2 = ΔE adia +
1
2
Mω
2
2 (R − R 2 )
2
Make use of the relationships listed in Appendix F.
4.3 Prove the relationship (4.17).
4.4 Prove the relationship (4.6).