Fast vibrational relaxation limit

As a further assumption, we adopt the sovibrational relaxations in the excited electronic states are more efficient than electronic relaxations. Provided that vibrational relaxation rates considerably exceed the noru adiative decay probability, thermal equilibri-... 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

In obtaining this result we have taken proper account of the conservation of energy manifested through the delta function 6(co + Wc(COa— oSco ). If we wish the rate at which phonons are emitted into frequency range of the lattice including resonance and local impurity modes, we adopt the validity of the limit of fast vibrational relaxation, defined in terms of the relation EaY f > 1 with t being the time scale. The method of carrying out the calculation of (4.130) and the MID ria- i ( ai Si been described before quite explicitly. 

Fast concentration and sample injection are considered with the use of a theory of vibrational relaxation. A possibility to reduce a detection limit for trinitrotoluene to 10 g/cnf in less than 1 min is shown. Such a detection limit can by obtained using selective ionization combined with ion drift spectrometry. The time of detection in this case is 1- 3 s. A detection technique based on fluorescent reinforcing polymers, when the target molecules strongly quench fluorescence, holds much promise for developing fast detectors. 

The relevance of adiabatic electron transfer to the primary charge separation reaction has been the subject of considerable discussion, mainly due to the observation of undamped low-frequency nuclear motions associated with the P state (see Section 5.5). More recently, sub-picosecond time-scale electron transfer has been observed at cryogenic temperatures, driven either by the P state in certain mutant reaction centres (see Section 5.6) or by the monomeric BChls in both wild-type and mutant reaction centres (see Section 5.7). These observations have led to the proposal that such ultra-fast electron transfer reactions require strong electronic coupling between the co-factors and occur on a time-scale in which vibrational relaxation is not complete, which would place these reactions in the adiabatic regime. Finally, as discussed in Section 2.2, evidence has been obtained that electron transfer from QpJ to Qg is limited by nuclear rearrangement, rather than by the driving force for the reaction. 

The competition between intramolecular vibrational relaxation and chemical reaction has been discussed in terms of the applicability of transition state theory to the kinetic analysis [6], If the environment functions mainly as a heat bath to ensure thermalization among the vibrational modes in the excited complex, then transition state theory is a good approximation. On the other hand, when the reaction is too fast for thermalization to occur the rate can depend upon the initial vibronic state. Prompt reaction and prompt intersystem crossing are, by definition, examples of the latter limit. 

Molecular dynamics is a true first principles dynamic molecular model. It simply solves the equations of motion. Given an intermolecular potential, MD provides the exact spatial and temporal evolution of the system. The stiffness caused by fast vibrations compared with slow molecular relaxations demands relatively small time steps and challenges current simulations. As an example, the time scale associated with vibrations is a fraction of a picosecond, whereas those associated with diffusion or reaction may easily be in the seconds to hours range depending on the activation energy. Consequently, MD on a single processor is usually limited to short time and length scales (e.g., pico- to nanoseconds and 1-2 nm). 

The exciton transfer is coherent when the transfer time is fast compared to dissipative relaxation times, where the dissipative relaxation is usually inter or intramolecular vibrational relaxation and the transfer time is proportional to the inverse of the exciton transfer integral (described in the next section). In this limit the exciton is described by a wavefunction, whose time d3mamics are controlled... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

For the moment, assume that the VE picture is correct and inertial solvent motion causes negligible dephasing. Diffusive motion must be the primary cause of coherence decay. In the VE theory, the diffusive motion is the relaxation of stress fluctuations in the solvent by viscous flow. The VE theory calculates both the magnitude Am and lifetime z0J of the resulting vibrational frequency perturbations. A Kubo-like treatment then predicts the coherence decay as a function of the viscosity of the solvent. Figure 19 shows results for typical solvent parameters. At low viscosity, the modulation is in the fast limit, so the decay is slow and nearly exponential. Under these conditions, the dephasing time is inversely proportional to the viscosity, as in previous theories [Equation (19)]. As the viscosity increases, the modulation rate slows. The decay becomes faster and approaches a... 

Figure 20. The 5j 12j dispersed fluorescence of ultracold anthracene in a supersonic beam. The available vibrational energy is 1792 cm1. The parameters of the optically active modes are given in Table II. The top figure is the experimental spectrum.60 The bottom figure is the emission in the harmonic approximation (y6.6 = 0). The calculation clearly fails to reproduce the broad redistributed emission. The middle figure was calculated with IVR [Eqs. (131)]. Only one b ) state (the ground vibrational state f/> = 0 was used. yt.t/y, = 40. The relaxed emission was calculated in the fast modulation limit [Eq. (116a)], with f0 = 2f = 75 cm-1.61...

As these remarks indicate, chemical lasers employ infrared chemiluminescence. As a method for obtaining kinetic information, they have to be looked at in relation to other spectroscopic techniques having the same goal. The study of spontaneous vibrational-rotational emission has been most fruitfully applied to fast reactions in the gas phase. This method has experimental limitations due to the relaxation processes competing with spontaneous emission. A very authentic discussion of this method has been given in a recent review by J. C. Polanyi 3>. As opposed to this steady-state technique, chemical lasers permit observations in the pulsed mode. 

Contribution of high levels into the VT losses is usually related to the highest vibrational levels before the fast decrease in the distribution function due to the VT relaxation or chemical reaction. These losses in the fast reaction limit can be calculated from (3-183) as (To fico, SviT < 1) ... 

This kinetic approximation assumes a single vibrational temperature 77 for CO2 molecules and, therefore, is sometimes referred to as quasi equilibrium of vibrational modes. As one can see from (5-20), most of the vibrationally excited molecules can be considered as being in quasi continuum in this case. Vibrational kinetics of polyatomic molecules in quasi continuum was discussed in Chapter 3. The CO2 dissociation rate is limited not by elementary dissociation itself, but via energy transfer from a low to high vibrational excitation level of the molecule in the W-relaxation processes. Such a kinetic situation was referred to in Chapter 3 as the fast reaction limit. The population of highly excited states with vibrational energy E depends in this case on the number of vibrational degrees of freedom 5 and is proportional to the density of the vibrational states p E) a. The... 

There have been limited attempts to correlate the spectrum of the random force responsible for vibrational dephasing with induced spectra. In the fast modulation limitthe dephasing time (t) is determined by the amplitude (< 5vp>)2and relaxation time of the solvent-induced fluctuations in the oscillator frequency... 

---