4.6 Computational Note: Quantum Wavepacket Dynamics and Classical Trajectories
139
4.5 Prove the relationship (4.21).
4.6 Find all the ways the vibrational energy of 1000 cm
−1 , in excess of the ZPE, can
be distributed among the three normal modes of a triatomic molecule with frequencies
200, 300, and 500 cm
−1 . What is the microcanonical energy distribution in the three
modes for this molecule? Same question if each frequency is sixfold degenerate, as
in the case of six weakly interacting identical molecules.
4.7 Quinoline has a much higher fluorescence quantum yield in polar solvents than
in the apolar ones, just as 1-pyrenecarboxaldehyde (see Sect. 2.8) and for the same
reason. A TD-DFT calculation for the isolated molecule shows that the two lowest
excited states are S 1 of n → π
∗ type and S 2 , π → π
∗ , with an energy difference of
18 kJ/mol. The respective oscillator strengths for transitions from and to the ground
state are 0.0019 and 0.0435, so the emission rate of the π → π
∗ state is about 20
times larger than that of the n → π
∗ state. The molecular dipole in the n → π
∗ state
is 0.79 a.u. and in the π → π
∗ state, 1.79 a.u. Estimate the free energy difference
between the two states in benzene and ethanol, using Onsager’s formula, Eq. (4.24)
and neglecting the polarizability term. The relative permittivities of benzene and
ethanol are ε/ε 0 = 2.3 and 24.5, respectively. The R
3 parameter can be evaluated
from the molecular volume, using the density of quinoline, 1.093 g/cm
3 and its
molecular weight, 129.16.
4.8 Compute how many states have an equilibrium population larger than 1%,
according to the canonical distribution at T = 300 K, Eq. 4.25, with mode frequencies of 100, 400, and 1000 cm
−1 .
4.9 Watch Animations 4.1 and 4.4 and obtain from them the two oscillation periods, T h for the harmonic potential and T m for the Morse one. To which frequencies
ω = 2π/T do they correspond? Compare these frequencies with the vibrational frequencies of the two oscillators.
References
1. Zewail, A.H.: Femtochemistry: atomic-scale dynamics of the chemical bond using ultrafast
lasers. Angew. Chem. Int. Ed. 39, 2586–2631 (2000)
2. Ootani, Y., Satoh, K., Nakayama, A., Noro, T., Taketsugu, T.: Ab initio molecular dynamics
simulation of photoisomerization in azobenzene in the n → π ∗ state. J. Chem. Phys. 131,
194306/1-10 (2009)
3. Cantatore, V., Granucci, G., Persico, M.: Simulation of the π → π ∗ photodynamics of azobenzene: decoherence and solvent effects. Comput. Theor. Chem. 1040, 126–135 (2014)
4. Tomasi, J., Persico, M.: Molecular interactions in solution: an overview of methods based on
continuous distributions of the solvent. Chem. Rev. 94 2027–2094 (1994)
5. Tomasi, J., Mennucci, B., Cammi, R.: Quantum mechanical continuum solvation models.
Chem. Rev. 105, 2999–3094 (2005)
6. Ladanyi, B.M.: Solvation dynamics. In: Cammi, R., Mennucci, B. (eds.) Continuum Solvation
Models in Chemical Physics: From Theory to Applications, pp. 366–388. Wiley, Chichester
(2007)
139
4.5 Prove the relationship (4.21).
4.6 Find all the ways the vibrational energy of 1000 cm
−1 , in excess of the ZPE, can
be distributed among the three normal modes of a triatomic molecule with frequencies
200, 300, and 500 cm
−1 . What is the microcanonical energy distribution in the three
modes for this molecule? Same question if each frequency is sixfold degenerate, as
in the case of six weakly interacting identical molecules.
4.7 Quinoline has a much higher fluorescence quantum yield in polar solvents than
in the apolar ones, just as 1-pyrenecarboxaldehyde (see Sect. 2.8) and for the same
reason. A TD-DFT calculation for the isolated molecule shows that the two lowest
excited states are S 1 of n → π
∗ type and S 2 , π → π
∗ , with an energy difference of
18 kJ/mol. The respective oscillator strengths for transitions from and to the ground
state are 0.0019 and 0.0435, so the emission rate of the π → π
∗ state is about 20
times larger than that of the n → π
∗ state. The molecular dipole in the n → π
∗ state
is 0.79 a.u. and in the π → π
∗ state, 1.79 a.u. Estimate the free energy difference
between the two states in benzene and ethanol, using Onsager’s formula, Eq. (4.24)
and neglecting the polarizability term. The relative permittivities of benzene and
ethanol are ε/ε 0 = 2.3 and 24.5, respectively. The R
3 parameter can be evaluated
from the molecular volume, using the density of quinoline, 1.093 g/cm
3 and its
molecular weight, 129.16.
4.8 Compute how many states have an equilibrium population larger than 1%,
according to the canonical distribution at T = 300 K, Eq. 4.25, with mode frequencies of 100, 400, and 1000 cm
−1 .
4.9 Watch Animations 4.1 and 4.4 and obtain from them the two oscillation periods, T h for the harmonic potential and T m for the Morse one. To which frequencies
ω = 2π/T do they correspond? Compare these frequencies with the vibrational frequencies of the two oscillators.
References
1. Zewail, A.H.: Femtochemistry: atomic-scale dynamics of the chemical bond using ultrafast
lasers. Angew. Chem. Int. Ed. 39, 2586–2631 (2000)
2. Ootani, Y., Satoh, K., Nakayama, A., Noro, T., Taketsugu, T.: Ab initio molecular dynamics
simulation of photoisomerization in azobenzene in the n → π ∗ state. J. Chem. Phys. 131,
194306/1-10 (2009)
3. Cantatore, V., Granucci, G., Persico, M.: Simulation of the π → π ∗ photodynamics of azobenzene: decoherence and solvent effects. Comput. Theor. Chem. 1040, 126–135 (2014)
4. Tomasi, J., Persico, M.: Molecular interactions in solution: an overview of methods based on
continuous distributions of the solvent. Chem. Rev. 94 2027–2094 (1994)
5. Tomasi, J., Mennucci, B., Cammi, R.: Quantum mechanical continuum solvation models.
Chem. Rev. 105, 2999–3094 (2005)
6. Ladanyi, B.M.: Solvation dynamics. In: Cammi, R., Mennucci, B. (eds.) Continuum Solvation
Models in Chemical Physics: From Theory to Applications, pp. 366–388. Wiley, Chichester
(2007)