Energy relaxation rate

Such a construction is not a result of perturbation theory in <5 , rather it appears from accounting for all relaxation channels in rotational spectra. Even at large <5 the factor j8 = B/kT < 1 makes 1/te substantially lower than a collision frequency in gas. This factor is of the same origin as the factor hco/kT < 1 in the energy relaxation rate of a harmonic oscillator, and contributes to the trend for increasing xE and zj with increasing temperature, which has been observed experimentally [81, 196]. 

Keeping the above considerations in mind it is relatively easy, starting from an a priori correlation function approach, to derive successive approximate expressions for binary and nonbinary terms in vibrational relaxation in liquids, to define the limits of validity of binary dynamics, and to obtain easily evaluated analytical expressions for energy relaxation rates that can be directly compared to experiment. 

Since the absolute value of the vibrational energy relaxation rate is very difficult to calculate precisely, even in the gas phase, it is preferable, for comparison with experiment, to eliminate this variable using measured gas-phase relaxation rates. 

Thus the model predicts that the adiabatic effect is to reduce the energy relaxation rate by a factor of 4. Using the estimates of the zero-frequency friction discussed above and setting S= in Eq. (5.19) gives... 

Relative Values of Energy Relaxation Rate for Brj in Argon at 300 K with Initial Energy of 0.935 De... 

In this form the rate is given as the inverse of the sum of two times, the barrier crossing time and a time characteristic to the energy relaxation rate within the reactant well. 

A simplified model, which illustrates these effects and may be solved analytically if IVR is slow relative to the total energy relaxation rate, is given by (for simplicity we use the Markov limit for the following demonstration) ... 

This is a final quantum expression, which can be interpreted as an energy relaxation rate and be used to estimate the VER rate.1... 

In this Section we apply the general formalism developed in Section 13.3 together with the interaction models discussed in Section 13.2 in order to derive explicit expressions for the vibrational energy relaxation rate. Our aim is to identify the molecular and solvent factors that determine the rate. We will start by analyzing the implications of a linear coupling model, than move on to study more realistic nonlinear interactions. 

Equations (13.26) and (13.29b) now provide an exact result, within the bilinear coupling model and the weak coupling theory that leads to the golden rule rate expression, for the vibrational energy relaxation rate. This result is expressed in terms of the oscillator mass m and frequency ca and in tenns of properties of the bath and the molecule-bath coupling expressed by the coupling density A ((a)g (a) at the oscillator frequency... 

This rate has two remarkable properties First, it does not depend on the temperature and second, it is proportional to the bath density of modes g(ct>) and therefore vanishes when the oscillator frequency is larger than the bath cutoff frequency (Debye frequency). Both features were already encountered (Eq. (9.57)) in a somewhat simpler vibrational relaxation model based on bilinear coupling and the rotating wave approximation. Note that temperature independence is a property of the energy relaxation rate obtained in this model. The inter-level transition rate, Eq. (13.19), satisfies (cf. Eq, (13.26)) k = k (l — and does depend on temperature. 

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum hannonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. 

We can make some estimations of this time using, e.g. data from the vibrational energy relaxation rate of electronic states of organic molecules (see (75) and references therein). In the majority of cases however the question of the competition between vibrational (our case) and electronic relaxation unfortunately arises. 

Fujisaki, H. Straub, J. E., Vibrational energy relaxation of isotopically labeled amide I modes in cytochrome c Theoretical investigation of vibrational energy relaxation rates and pathways. J. Phys. Chem. B 2007, 111, 12017-12023. 

We can carry this comparison with experiment a little further by measuring the rate of the cyclopropane isomerisation at sufficiently low pressures that the mean free path exceeds the diameter of the vessel [63.K2]. A 11 spherical flask was used, and the results are shown in Figure 5.2 the solid theoretical line is calculated on the assumption that the internal energy relaxation rate is given by... 

C RELINT collisional energy relaxation rate (/sec/Torr) ... 

The validity of Eq. (15.11) even at the limit Na Nm means in fact that for harmonic oscillators relaxing in a heat bath there exists a closed equation for the mean energy. This is intimately connected with the linear dependence of transition probabilities on the vibrational quantum number (see Eq. (14.1)) and implies that the energy relaxation rate is independent of the initial distribution of harmonic oscillators over vibrational states. Still another peculiarity of this system is known the initial Boltzmann distribution corresponding to the vibrational temperature Tq =t= T relaxes to the equilibrium distribution via the set of the Boltzmann distributions with time-dependent temperatures. If (E ) is explicitly expressed by a time-dependent temperature, this process is again described by Eq. (15.11). 

Marianer et al. [18] assume that The energy relaxation rate does not depend exponentially on the energy difference. Consequently, we assume that relaxation from the hot state is independent of energy ... 

To focus on the question of surface effects on vibrational energy relaxation rate, without the complications of intramolecular energy flow, Benjamin and coworkers studied the vibrational relaxation of a diatomic solute molecule (single vibrational mode) at various liquid/vapor and liquid/liquid interfaces." " The solute is modeled using the Morse potential ... 

The energy relaxation time, Eq. [50], as a function of the magnitude of the solute s dipole and its location exhibits an opposite trend." The increased collision rate (which slows down reorientation) enhances the rotational energy transfer to the solvent molecules. Thus, for a solute with a small dipole, the energy relaxation at the interface is much slower than in the bulk. However, the difference between the bulk and surface energy relaxation rates decreases as the dipole is increased because preserving the solute hydration shell makes the interfacial friction similar to that of the bulk, despite the fact that the average solvent density just outside the solute hydration shell is smaller than in the bulk. 

Nonequilibrium energy relaxation rates, calculated using nonequilibrium MD, are somewhat slower than the equiHbrium rates in the case of a nonpolar solute but are almost the same for polar solutes." However, the trends in relaxation times as a function of solute dipole and location are essentially the same as the equilibrium trends discussed above. The difference between equilibrium and nonequilibrium rotational relaxation reflects, in part, the difference between the equilibrium structure of the solute-solvent complex." " The insight gained from studying a simple diatomic solute has been useful for understanding the rotational behavior of large dye molecules." " ... 

I. Navrotskaya and E. Geva,/. Chem. Phys., 127, 054504 (2007). Comparison between the Landau-Teller and Flux-Hux Methods for Computing Vibrational Energy Relaxation Rate Constants in the Condensed Phase. 

J. L. Skinner and K. Park,/. Phys. Chem. B, 105,6716 (2001). Calculating Vibrational Energy Relaxation Rates from Classical Molecular Dynamics Simulations Quantum Correction Factors for Processes Involving Vibration-Vibration Energy Transfer. 

Skinner, ).L. and Park, K. (2001) Calculating vibrational energy relaxation rates from classical molecular dynamics simulations quantum correction factors for processes involving vibration-vibration energy transfer. J. Phys. Chem. B, 105 (28), 6716 6721. 

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