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\( \let\bibref\eqref %make-command_bibref-equivalent-to-eqref\)\(\bibref{Cite1} %in-text-citation\)Vibrational relaxation of polyatomic molecules in solids or in solution involves anharmonic coupling of energy between internal vibrations of the molecule, also called IVR (internal vibrational energy redistribution). Mechanical interactions between multiple modes of vibration of the molecule act to rapidly scramble energy deposited into one vibrational coordinate and lead to cascaded energy flow toward equilibrium.
For this problem, the bilinear coupling above doesn't capture the proper relaxation process. Instead, we can express the molecular potential energy in terms of well-defined normal modes of vibration for the system and the bath, and these interact weakly through small anharmonic terms in the potential. Then we can extend the perturbative approach above to include the effect of multiple accepting vibrations of the system or bath. For a set of system- and bath coordinates, the combined potential energy for the system (\(V_S\)) and the system-bath interaction (\(V_{SB}\)) can be expanded as
\[\begin{aligned}V_S+V_{S B}&=\frac{1}{2} \sum_a \frac{\partial^2 V}{\partial Q \vphantom{X}_a^2} Q \vphantom{X}_a^2+\frac{1}{6} \sum_{a, \alpha, \beta} \frac{\partial^3 V}{\partial Q_a \partial q_\alpha \partial q_\beta} Q_a q_\alpha q_\beta + \frac{1}{6} \sum_{a, b, \alpha} \frac{\partial^3 V}{\partial Q_a \partial Q_b \partial q_\alpha} Q_a Q_b q_\alpha \cdots \end{aligned}\]
Focusing explicitly on the first cubic expansion term, for one system oscillator:
\[V_S+V_{S B}=\frac{1}{2} m \Omega^2 \underset{^\sim}{Q}^2+V^{(3)} \underset{^\sim}{q}\vphantom{j}_\alpha \underset{^\sim}{q} \vphantom{j}_\beta\]
Here, the system-bath interaction potential describes the case for a cubic anharmonic coupling that involves one vibration of the system \(Q\) interacting weakly with two vibrations of the bath, \(q_\alpha \) and \(q_\beta \), so that \(\hbar \Omega \gg V^{(3)}\). Energy deposited in the system vibration will dissipate to these two vibrations of the bath in a three-quantum process. Higher-order expansion terms would describe interactions involving four or more quanta.
Working specifically with the cubic example, we can use the harmonic bath model to calculate the rate of energy relaxation. This picture is applicable if a vibrational mode of frequency \( \Omega \) relaxes by transferring its energy to another vibration close in energy (\( \omega_\alpha\)), with the energy difference \( \omega_\beta \) being accounted for by a continuum of intermolecular motions. For this case, one can show: \[ k_{b a}=\frac{1}{\hbar^2} \left[\left(\bar{n}(\omega_\alpha\right)+1) \left(\bar{n}(\omega_\beta)+1\right) \rho_{b a}(\omega_{a b}) + \left(\bar{n}(\omega_\alpha\right)+1) \, \bar{n}(\omega_\beta) \, \rho_{b a}(\omega_{a b}) \vphantom{X^2} \right]\]
where \(\rho(\omega) \equiv W(\omega)\left(V^{(3)}(\omega)\right)^2\). Here we have taken the frequencies of the initial vibration \(\Omega\) and accepting mode \(\omega_\alpha\) to be much greater than the bath mode \(\omega_\beta\). The two terms in equation (18.3.3) describe two possible relaxation pathways, shown in Figure 18.3.1. In the first, annihilation of a quantum of frequency \(\Omega\) leads to the creation of two quanta, one each of \(\omega_\alpha\) and \(\omega_\beta\). The second term describes the dissipation of a vibration through coupling to a higher energy vibration, with the excess energy being absorbed from the bath—annihilation of a quantum of \(\Omega\) leads to the creation of one quantum of \(\omega_\alpha\) and the annihilation of one quantum of \(\omega_\beta\). Naturally, this latter pathway is only allowed when there is adequate thermal energy present in the bath.
Rate calculations using classical vibrational relaxation
In general, we would like a practical way to calculate relaxation rates, and calculating quantum correlation functions is not practical. How do we use classical calculations for the bath, for instance drawing on a classical molecular dynamics simulation? Is there a way to get a quantum mechanical rate?
The first problem is that the quantum correlation function is complex \(C \vphantom{S}_{a b}^*(t)=C_{a b}(-t)\) and the classical correlation function is real and even \(C_\text{Cl}(t)=C_\text{Cl}(-t)\). To connect these two correlation functions, one can derive a quantum correction factor that allows one to predict the quantum correlation function based on the classical one. This is based on the assumption that at high temperature it should be possible to substitute the classical correlation function with the real part of the quantum correlation function \[C_\text{Cl}(t) \Rightarrow C_{b a}^{\prime}(t)\]
To make this adjustment we start with the frequency domain expression derived from the detailed balance expression \(\tilde{C}(-\omega)=e^{-\beta \hbar \omega} \tilde{C}(\omega)\) \[\tilde{C}(\omega)=\frac{2}{1+\exp (-\beta \hbar \omega)} \tilde{C} \vphantom{X}^{\prime}(\omega)\]
Here, \(\tilde{C} \vphantom{X}^{\prime}(\omega)\) is defined as the Fourier transform of the real part of the quantum correlation function. So the vibrational relaxation rate is
\[k_{b a}=\frac{4}{\hbar^2\left(1+\exp \left(-\hbar \omega_{b a} / k T\right)\right)} \int_0^{\infty} d t \, e^{-\mathrm{i} \omega_{b a} t} \operatorname{Re}\left[\left\langle V_{a b}(t) V_{b a}(0)\right\rangle \! \vphantom{X^2}\right]\]
Now we will assume that one can replace a classical calculation of the correlation function here, as in eq. (18.3.4). The leading term out front can be considered a "quantum correction factor" that accounts for the detailed balance of rates encoded in the quantum spectral density.
In practice, a classical calculation might be done with molecular dynamics simulations. Here one has an explicit characterization of the intermolecular forces that would act to damp the excited vibrational mode. One can calculate the system-bath interactions by expanding the vibrational potential of the system in the bath coordinates
\[\begin{aligned}
V_S+V_{S B} & =V_0+\sum_\alpha \frac{\partial V^\alpha}{\partial Q} Q+\sum_\alpha \frac{\partial^2 V^\alpha}{\partial Q^2} Q^2+\cdots\\
& =V_0+F Q+G Q^2+\cdots\end{aligned}\]
Here \(V^\alpha\) represents the potential of an interaction of one solvent coordinate acting on the excited vibrational system coordinate \(Q\). The second term in this expansion \(FQ\) depends linearly on the system \(0\) and bath \( \alpha \) coordinates, and we can use variation in this parameter to calculate the correlation function for the fluctuating interaction potential. Note that \(F\) is the force that molecules exert on \(Q\)! Thus, the relevant classical correlation function for vibrational relaxation is a force correlation function
\[\begin{align}
C_\text{Cl}(t) &=\langle F(t) F(0)\rangle \\[4pt]
k_\text{Cl} &=\frac{1}{k_B T} \int_0^{\infty} d t \cos \omega_{b a} t\langle F(t) F(0)\rangle \end{align}\]
References
| \[\tag {1} \label {Cite1}\] |
Kenkre, V. M.; Tokmakoff, A.; Fayer, M. D. Theory of vibrational relaxation of polyatomic molecules in liquids. Journal of Chemical Physics 1994, 101 (12), 10618-10629. DOI: 10.1063/1.467876 |
18.3: Multiquantum relaxation of polyatomic molecules is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.
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