Temperature dependence of non-radiative transitions

Here we apply the method described above (see also Ref. [13]) for the investigation of the temperature dependence of non-radiative transitions between electronic levels in a center. The method [13] is based on the calculation of the large t asymptotic of the phonon correlation function D,(f, r) = (l,0lx (f + T)xI(t)l 1,0) with t averaged over the vibrational period here 11,0) = 1110) is the product of the initial electronic state and the zero-point vibrational state. 

We consider a non-radiative transition caused by linear diagonal and quadratic non-diagonal vibronic interactions. If the states belong to different representations of the point group of the center, then the Hamiltonian of the system reads  

Let us consider first the T = 0 case. If the non-diagonal interaction Hint is weak then the rate of the non-radiative transition is determined by the Fermi golden rule 

To apply described above non-perturbative method to the non-radiative transitions caused by the quadratic non-diagonal vibronic interaction qfli, one needs to find the phonon correlation functions Du(t, r) in the initial (non-stationary) state 11,0) for l 0. To this end, we use the equation of motion xu + (ojxu + Veuqt j = 0 il, l 0 / V l1) the integral form of it for t 0 reads 

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The temperature dependence of non-radiative transitions, caused by linear diagonal and quadratic non-diagonal vibronic interactions, is also investigated on the basis of the non-perturbative quantum theory. It was found that the usual increase of the transition rate with temperature does not hold near some critical values of the non-diagonal interaction and temperature. At these critical values the rate is high (comparable to the mean phonon frequency) and its temperature dependence has a maximum. The results may be important for understanding the mechanisms of catalysis in chemical reactions. 

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