Franck-Condon factor tunneling

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Suppose now that both types of vibrations are involved in the transition. The symmetric modes decrease the effective tunneling distance to 2Q, while the antisymmetric ones create the Franck-Condon factor in which the displacement 2Qq now is to be replaced by the shorter tunneling distance 2Q, [Benderskii et al. 1991a]... 

Thus, the inertia of the tunneling particle leads to two opposite effects a decrease of the transition probability due to the reorganization along the coordinate of the center of mass and an increase of the transition probability due to the increase of the Franck-Condon factor of the tunneling particle. Unlike the result in Ref. 66, it is found in Ref. 67 that for ordinary relationships between the physical parameters, the inertia leads to an increase of the transition probability. 

Depending on the relationships between the parameters included in eqn. (39), the role of the local vibration can be different. We shall consider the most interesting case when the local vibration is essentially quantum, i.e. > T. Here the multiplier exp( — conJT) in eqn. (39) decreases rapidly with the growth of the quantum number n,. If the shift of the equilibrium position of the local vibration, A, is not too large and the growth of the Franck-Condon factors with the growth of the quantum number n [see eqn. (28)] is not too sharp, then the main contribution to the probability of tunneling, eqn. (39), is made by the transition from the state with the quantum number ri = 0, i.e. 

The possibility of the anomalous isotope effect for electron tunneling reactions was first noted by Ulstrup and Jortner [7]. This effect becomes possible when the reorganization energy is approximately equal to the reaction exothermicity. If, in this case, for example, the relationship Er - J + a) — 0 is satisfied, where co is the vibrational frequency for a heavy isotope, then from the viewpoint of the activation energy [see eqn. (42)1, the transition (0 - 1) is optimal for the heavy isotope. Compared with this transition for the heavy isotope, both the transitions (0 - 0)and(0 - 1) for the light isotope contain the additional activation multiplier. In this situation the anomalous isotope effect will be observed, provided that the Franck Condon factor for the transition (0 -> 1) of the heavy isotope is not too small compared with that of the light isotope. An example of the electron tunneling reaction for which the anomalous isotope effect is observed experimentally will be considered in Chap. 7, Sect. 4. 

If we approximate the dependence of the probability of tunneling on 7 by the function exp( - 2yR), then the dependence of the energy with which the electron tunnels on the quantum numbers of the vibrational degrees of freedom can reveal itself in the appearance of the dependence of the parameter y on R. Let us assume, for example, that at small values of R the values of the Franck-Condon factors are such that it is more favourable to dissipate all the reaction exothermicity on the donor, i.e. the transitions without any changes of the vibrational quantum number of the acceptor,... 

Thus, the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model of (5.56) and (5.68). As an illustration, three-dimensional trajectories for various interrelations between symmetric (vibration frequencies and w0 are shown in Figure 5.2 When both vibrations have high frequencies, wa,s w0 the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, tua s < a>0, the tunneling occurs in the barrier, which is lowered and shortened by the symmetrically coupled vibration qs, so that the position of the antisymmetrically coupled... 

Tunneling Matrix Elements J0 and Franck-Condon Factors for Diffusion of Light Impurities in Metals... 

The nuclear tunneling implicit in the Franck Condon factors is of particular importance in the inverted region (-AG° > 7.). In such cases, the effective barrier is greatly reduced as a result of nuclear tunneling [14], in comparison with the classical barrier, which according to Eq. 27 would rise monotonically with increasing exo-thermicity in the inverted region. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

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