Vibronic terms

The anharmonic potentials, the second term in equation (13), transform as At representation, while the /" in the nonlinear vibronic term, the second term in equation (14), transform as E2 representation. The product representations are... 

Occupied Virtual As (eV) Vibronic terms 2uniVni (mdyne/A)... 

In the next subsection we restrict the consideration by Rayleigh and hyper-Rayleigh scattering in the limit of low temperatures when the ground vibronic term only is populated. In this case, the summation in Eq. (188) has to be performed over m, m, and m" appropriate to the states... 

Third, in most cases in this paper, only the linear vibronic terms are considered. Meanwhile, it is known (Bersuker and Polinger, 1982, 1983) that the quadratic terms of vibronic interactions may be of primary importance in vibronic problems. Therefore the evaluation of the electric properties of molecules considering also the quadratic vibronic terms (in the cases where it has not been done yet) may result in new effects and regularities in this area. 

A quantum-mechanical tunnelling model of non-radiative decay 492 has been proposed. Several papers have been concerned with the choice of basis set for treatments of non-radiative decay,498 and it has been shown in the paper of Sharf that higher-order vibronic terms are comparable in magnitude with those of the first-order term. It is thus not sufficient to make the usual assumption that interactions describing non-radiative decay are independent or linearly dependent on nuclear co-ordinates. The other papers reinforce this point, and... 

Figure 2-37. Efficiency of vibrational energy in a simple exchange reaction A + BC —> AB + C. (A) Solid curve - reaction profile dashed line represents a vibronic term, corresponding to an atom A interaction with a vibrationally excited molecule BC (Ey) (B) part of the reaction profile near the summit of the exchange reaction barrier.

Figure 6-7. System of electronic-vibrational (vibronic) terms for linear configuration of N2-O interaction. Arrows (1,2) point out the most probable transitions leading to formation of the N2O complex (1) transition in the case of lower temperatures, To /5 ln( 1 /y r)< 1 (2) transition in the case of higher temperatures, ro/51n(l/y u) > 1.

Transition Probabiiity Between Vibronic Terms Corresponding to Formation of Intermediate N20 ( S+) Complex... 

Thus, the formation of the intermediate complex N20 ( S+) takes place through the non-adiabatic transitions between the vibronic terms (see Fig. 6-7). The probability of transition in an intersection can be calculated assuming separation of electronic and nuclear motion in the molecular system. Perturbation matrix elements can be factorized into electronic and vibrational parts the transition probability between electronic (n n ) and vibrational (pi V2) states can be presented as... 

Integral (6-18) is calculated for the intersection point of the vibronic terms. In integral (6-18), n and n correspond to the terms n and E+ r" and r" are relevant equilibrium interatomic N-N distances in these states. The Frank-Condon integral (6-18) can be calculated using the harmoiuc oscillator -functions (Frank-Kamentsky Lukashin, 1975) ... 

As shown in Fig. 6.2, within the purely electronic terms S (u > 0) and T (u>l) there are superposed vibronic terms, that is terms with excited molecular vibrations. In the crystal, the intramolecular vibrations are only slightly different from those in the free molecules (cf. Chap. 5). Electronic transitions between levels with zero vibrational quantum numbers are called 0,0 transitions. 

This correction is responsible for the warping of the adiabatic potential surface near the nuclear configuration with electron degeneracy. The analytic form of the adiabatic potential surface becomes a non-linear (not only polynomial) function of nuclear displacements in which potential constants of two kinds occur. The force constants K , Ku, Kjik, etc. contain the pure nuclear term Fj, Fa, Fak, etc. as well as the electron-nuclear (vibronic) term X, Xa, Xak, etc. The vibronic constants Xj, Xy, etc. enter into the vibronic correction term e. They determine a warping of the adiabatic potential surface and they can couple vibration modes of different symmetry, e.g. Xy for i j. 

With the results of Appendix 1, a complete third-order vibronic term can be obtained... 

Assume now that one of the partners is in a degenerate electronic state. Electronic and vibrational states form the so-called vibronic terms. Those correlating at R -> oo with different vibrational states of BC can cross or pseudocross. In these regions (one shown in Fig. 18), non-adiabatic transitions are probable. They represent an additional path of changes in the vibrational quantum number of BC (dashed arrows near R2). There are several types of interactions... 

For atoms of a medium atomic number the spin-orbital splitting is sufficiently weak to be treated by perturbation (see Section III.9). Then the transition probability between vibronic terms will be low. However, since these paths do not involve tunnelling, the probability of a combined electronic-vibrational transition might be higher than that of purely vibrational transition [339, 344]. 

As to the temperature dependence of the vibronic transition probability, it obeys the Arrhenius rather than the Landau-TeUer equation. The activation energy is defined by the crossing point of the relevant vibronic terms. Thus, for both the vibrational and vibronic mechanisms the transition probability (Pi,o) may be expressed by... 

Note that the widths Fj are very small as the levels of tp do not carry oscillator strength. In the statistical limit when the density Q of vibronic states t j is sufficiently large to exceed the reciprocal of the vibronic terms Vsi between the BO states, Q(V 1, considerable configuration mixing occurs. In this limit, we may replace the sum over Iby an integral using J EiQi Ei). Thus, (6.23a) becomes... 

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