Theory of Activated Radiationless Processes: Orientation Relaxation of Polar Molecules

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Abstract
A first‐principles theoretical expression for the rate of a thermally activated process is derived. The treatment is specialized to the orientation relaxation of dipolar molecules, which is viewed as a radiationless transition between electronic states corresponding to the dipole in discrete orientations. The dipolar electronic states are calculated in the Born‐Oppenheimer approximation and the associated adiabatic potential‐energy surfaces are taken to be displaced, undistorted harmonic potentials. Transitions between the different electronic states arise through the nonadiabatic kinetic‐energy operator associated with the librational degree of freedom of the reorienting dipole. The explicit expression for the transition rate is calculated to first order by evaluating the integral over the time‐correlation function of the Kubo representation of the rate constant in the strong‐coupling limit using the method of steepest descent. It is shown that the transition rate is actually a sum of five contributions arising from phonon scattering processes in which 0, ±1, ±2 quantum changes take place in the librational degree of freedom of the dipole. When kT is large compared with a quantum of librational energy, the expression for the transition rate simplifies to the functional form of the Arrhenius equation in which the pre‐exponential factor and the activation energy are given in terms of specific molecular properties. The theory is applied to a quantitative treatment of the relaxation of dipolar impurities in ionic crystals. The dielectric behavior of polar organic molecules in liquids is also qualitatively examined, and the explicit dependence of the relaxation time on the molecular dimensions in a homologous series of compounds is obtained. Even though the theory is specialized to dipolar relaxation, the basic expressions are generally applicable to any thermally activated process involving the migration of ions or atoms, e.g., ionic conductivity, mobility, viscous flow, and diffusion.