The rate of multiphonon relaxation of rarehyphen;earth ions in crystals has been calculated using the Kubo representation of the rate constant in terms of linear response time correlation functions. The nonadiabatic electronhyphen;phonon operator arising from the effect of the kinetic energy of the ions on the Bornhyphen;Oppenheimer basis functions is employed in the firsthyphen;order perturbation theory. The calculations yield the explicit dependence of the decay rate on the temperature, the transition energy gap, the phonon energy, and the displacements of the adiabatic potentials. Excellent agreement between theory and experiment has been obtained for the multiphonon relaxation phenomenon observed in LaF3:Er3plus;, LaCl3:Dy3plus;, LaCl3:Nd3plus;, and LaBr3:Dy3plus;. Although the comparison of theory and experiment is specialized to the nonradiative relaxation of rare earth ion excited states in crystals, the treatment given in the present paper is also generally valid for the relaxation of excited states by internal conversion in large molecules.
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