Wavepacket Dynamics and Geometrical Relaxation

Abstract

In this chapter we shall present the dynamics that takes place after electronic excitation, under the influence of the new potential energy surface (PES). Nonadiabatic transitions to other electronic states will be assumed to be slow enough as to be neglected, so we shall qualify this topic as “adiabatic dynamics.” We shall examine the basic features of quantum wavepacket dynamics, and we shall find that some details of the excitation process affect the nature and the time evolution of the excited state even after the end of the radiation pulse. We shall see how, in certain conditions, quantum dynamics can be described with classical concepts, that are easier to grasp and provide the common language of qualitative arguments about reaction dynamics. The effects of the chemical environment on the dynamics of excited molecules will also be considered, trying to distinguish between interactions that change the PES and energy flow processes, i.e., the “static” and the “dynamic” effects, respectively.

Problems

4.1

Calculate the averages and uncertainties of x and $\hat{p}$ for the $\chi _v$ eigenfunction of the harmonic oscillator with mass M, frequency $\omega$, and equilibrium position $x_e$. Verify that for $v=0$ we have the minimum uncertainty product $\varDelta \!\, x \varDelta \!\, p = \hbar /2$. 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:

$$ U_1 = \frac{1}{2} M \omega _1^2 (R-R_1)^2 ~~~~~ \mathrm{and} ~~~~~ U_2 = \varDelta \!E_{adia} + \frac{1}{2} M \omega _2^2 (R-R_2)^2 $$

Make use of the relationships listed in Appendix F.

4.3

Prove the relationship (4.17).

4.4

Prove the relationship (4.6).

4.5

Prove the relationship (4.21).

4.6

Find all the ways the vibrational energy of 1000 $\mathrm{\;cm^{-1}}$, in excess of the ZPE, can be distributed among the three normal modes of a triatomic molecule with frequencies 200, 300, and 500 $\mathrm{\;cm^{-1}}$. 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 $n\rightarrow \pi ^*$ type and $S_2$, $\pi \rightarrow \pi ^*$, 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 $\pi \rightarrow \pi ^*$ state is about 20 times larger than that of the $n\rightarrow \pi ^*$ state. The molecular dipole in the $n\rightarrow \pi ^*$ state is 0.79 a.u. and in the $\pi \rightarrow \pi ^*$ 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 $\varepsilon /\varepsilon _0 = 2.3$ 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 $\mathrm{\;cm^{-1}}$.

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 $\omega =2\pi /T$ do they correspond? Compare these frequencies with the vibrational frequencies of the two oscillators.