Vibrational transition probabilities

The transition probability at a particular total energy (E ) from vibrational level i to / may be expressed as the ratio between the coiresponding outgoing and incoming quantities [71]... 

For this reason these vibrations influence tunneling in an entirely different way. For a model in which the reactant and product valleys are represented by paraboloids with frequencies coq and transition probability has been found to be [Benderskii et al. 1991a, b]... 

Jaquet and Miller [1985] have studied the transfer of hydrogen atom between neighbouring equilibrium positions on the (100) face of W by using a model two-dimensional chemosorption PES [McGreery and Wolken 1975]. In that calculation, performed for fairly high temperatures (T> rj the flux-flux formalism along with the vibrationally adiabatic approximation (section 3.6) were used. It has been noted that the increase of the coupling to the lattice vibrations and decrease of the frequency of the latter increase the transition probability. 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

The attractive energies 4D(cr/r)6 and ae2/2 r4 have two important effects on the vibrational energy transfer (a) they speed up the approaching collision partners so that the kinetic energy of the relative motion is increased, and (b) they modify the slope of the repulsive part of the interaction potential on which the transition probability depends. By letting m °°, we have completely ignored the second effect while we have over-emphasized the first. Note that Equation 12 is identical to an expression we could obtain when the interaction potential is assumed as U(r) = A [exp (— r/a)] — (ae2/2aA) — D. Similarly, if we assume a modified Morse potential of the form... 

This study has made no substantial improvement in the original theory of distorted waves. By evaluating the vibrational transition probability explicitly for the inverse (12-6-4) power potential, however, we were able to study some interesting aspects of the ion-molecule collisions. We summarize them here. 

Comparing the vibrational A lifetimes issued from both decay mechanisms (Tables 7 - 8), it is readily seen that the eleetric dipolar transition decay is always slightly favoured. A similar eonclusion holds for the A n state but, as expected, the vibrational transition probabilities are much larger for the dipolar decay which lead to mueh smaller vibrational lifetimes with respect to those via the cascade mode of decay, the differences amounting to five to six powers of ten (Table 7 - 8). 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

Figure 6.1 Nonlinear optical responses, (a) Second-order SF generation, the transition probability is enhanced when the IR light is resonant to the transition from the ground state g to a vibrational excited state V. CO is the angular frequency of the vibration, (b) Third-order coherent Raman scheme, the vibrational coherence is generated via impulsive stimulated...

Note that in the reference model all the interactions of the electron with the medium polarization VeP are included in Eqs. (8) determining the electron states. The dependence of A and B on the polarization and intramolecular vibrations was entirely neglected in most calculations of the transition probability [the approximation of constant electron density (ACED)]. This approximation, together with Eqs. (4)-(7), resulted in the parabolic shape of the diabatic PES Ut and Uf. The latter differed only by the shift... 

This new approach enables us to consider all the physical effects due to the interaction of the electron with the medium polarization and local vibrations and to take them into account in the calculation of the transition probability. These physical effects are as follows ... 

The physical reason for the appearance of the free energies in the formulas for the transition probability consists in the fact that the reactive vibrational modes interact with the nonreactive modes. 

Based on the results obtained in the investigation of the effects of modulation of the electron density by the nuclear vibrations, a lability principle in chemical kinetics and catalysis (electrocatalysis) has been formulated in Ref. 26. This principle is formulated as follows the greater the lability of the electron, transferable atoms or atomic groups with respect to the action of external fields, local vibrations, or fluctuations of the medium polarization, the higher, as a rule, is the transition probability, all other conditions being unchanged. Note that the concept lability is more general than... 

The calculations show that for high, narrow barriers the transitions between unexcited vibrational states of the proton give the main contribution to the transition probability. This result is... 

The height of the potential barrier decreases with the decrease of the transfer distance. Therefore, the contribution of the transitions between excited vibrational states increases and so does the transition probability. However, short-range repulsion between the reactants increases with a decrease of R, and the reaction occurs at an optimum distance R which is determined by the competition of these two factors. In principle, we may imagine the situation when the optimum distance R corresponds to the absence of a potential barrier for the proton. However, we should keep in mind that the transitions between certain excited states may become entirely adiabatic at short distances.40,41 In this case, the further increase of the transition probability with the decrease of R becomes quite weak, and it cannot compensate for the increased repulsion between the reactants, so that even for the adiabatic transition, the optimum distance R may correspond to sub-barrier proton transfer. 

First calculations of the optimum distance between the reactants, R, taking into account the dependence of the probability of proton transfer between the unexcited vibrational energy levels on the transfer distance have been performed in Ref. 42 assuming classical character of the reactant motion. Effects of this type were considered also in Ref. 43 in another model. It was shown that R depends on the temperature and this dependence leads to a distortion of the Arrhenius temperature dependence of the transition probability. 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

The modulation of the charge of the adsorbed atom by the vibrations of heavy particles leads to a number of additional effects. In particular, it changes the electron and vibrational wave functions and the electrostatic energy of the adatom. These effects may also influence the transition probability and its dependence on the electrode potential. 

At not very low temperatures, the excited vibrational states of the tunneling particle make some contribution to the transition probability. Furthermore, at high enough frequencies of the vibrations of the medium atoms, quantum effects may be important for the medium. In the harmonic approximation for the tunneling particle we obtain, in a similar way as in the case of proton transfer, the expression for the transition probability in the Condon approximation48... 

Here (x)t and (x)f denote the mean values of the relative coordinate x over the states of the proton in the first and second potential wells, respectively. Equation (107) shows that the inertia effects lead to a decrease of the activation factor in the transition probability due to an increase of the reorganization energy. The greater the mass, m of the tunneling particle and the frequency of the vibrations of the atom, w0, the greater is this effect. The above result corresponds to the conclusion drawn in Ref. 66. 

Let us assume that all the nuclear subsystems may be separated into several subsystems (R, q9 Q, s,...) characterized by different times of motion, for example, low-frequency vibrations of the polarization or the density of the medium (q), intramolecular vibrations, etc. Let (r) be the fastest classical subsystem, for which the concept of the transition probability per unit time Wlf(q, Q,s) at fixed values of the coordinates of slower subsystems q, Q, s) may be introduced. 

Symmetry-forbidden transitions. A transition can be forbidden for symmetry reasons. Detailed considerations of symmetry using group theory, and its consequences on transition probabilities, are beyond the scope of this book. It is important to note that a symmetry-forbidden transition can nevertheless be observed because the molecular vibrations cause some departure from perfect symmetry (vibronic coupling). The molar absorption coefficients of these transitions are very small and the corresponding absorption bands exhibit well-defined vibronic bands. This is the case with most n —> n transitions in solvents that cannot form hydrogen bonds (e 100-1000 L mol-1 cm-1). 

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