Temperature quenching of luminescence for linear and derivative nuclear operators

An earlier study of the thermal quenching of luminescence using the single-configurational-coordinate model is extended from Condon-approximation overlap integrals 〈u_n|v_m〉² to the linear and derivative integrals 〈u_n|z_v|v_m〉² and 〈u_n|?/?z_v|v_m〉². For non-radiative transitions, the thermally weighted nuclear factor in the transition rate is, for the linear and derivative integrals, the corresponding factor for 〈u_n|v_m〉² integrals multiplied by 2E_Xv/?ω_v and $\text{2E}{}_{\mathrm{Xv}}\text{- E}{}_{\mathrm{p}}{}_{\mathrm{U}}\text{(T)]/}\text{h}\text{?}\text{ω}{}_{\mathrm{v}}$, respectively. E_Xv is the energy of the crossover above the initial- v-parabola minimum, and E_pU(T) is the single activation energy fitted to the nuclear factor's temperature dependence for 〈u_n|v_m〉² integrals. These multiplying factors are exact for equal parabola force constants and good approximations for unequal force constants. These multiplying factors will be difficult to distinguish experimentally. The more important considerations for fitting the model to thermal-quenching data are the parabola placement and the Condon-approximation integrals described previously.