Vibronic Model for the Relaxed Excited State of theFCenter. I. General Solution

An exact solution is given, using methods familiar from the theory of the dynamic Jahn-Teller effect of orbitally degenerate states, for the vibronic problem posed by a model for the excited state of the $F$ center suggested by recent experiments on Stark effects in $F$-band luminescence of alkali halides. In this model, nondegenerate $2s$ and $2p$ electronic states interact in cubic symmetry via a triply degenerate odd-parity (${\Gamma }_4^{-}$) vibrational mode. From the exact wave functions for the resulting vibronic levels of the coupled electron-lattice system, expressions are derived for the following quantities: radiative lifetimes of the various levels; change in radiative lifetime of the vibronic ground state in an electric field, when this level is a nondegenerate $s$-like state; polarization induced in luminescence from the $s$-like ground state by electric fields, magnetic fields, and applied stress; change in the $g$ factor of the ground state because of the vibronic mixing; and reduction factors for the splitting of degenerate levels by applied magnetic fields and stress. These results take a particularly simple form when the coupling to the ${\Gamma }_4^{-}$ modes is strong. From a comparison of these theoretical results with the experimental data for the alkali halides, it is concluded that, contrary to earlier interpretations, the experimental situation cannot be in or close to the strong-coupling regime. An estimate of this coupling strength to the ${\Gamma }_4^{-}$ modes from a moment analysis of the vibrational broadening of the $F$ band in absorption and of stress-induced linear dichroism indicates that this coupling is probably weak for KCl, and that the Jahn-Teller coupling of $2p$ states to the ${\Gamma }_3^{+}$ and/or ${\Gamma }_5^{+}$ modes is stronger. This estimate is supported by an analysis of the form of the stable distorted configurations given by the adiabatic energy surfaces of the static problem, when simultaneous coupling to all three types of modes ${\Gamma }_3^{+}$, ${\Gamma }_4^{-}$, and ${\Gamma }_5^{+}$ is considered. An interpretation of the experimental data on the basis of this analysis indicates that for KCl the $2s$ state must be below $2p$ by about 0.1 eV om the cubic configuration corresponding to the relaxed position of the symmetric ${\Gamma }_1^{+}$ mode. Also discussed is the relationship of the exact solution of the vibronic problem to an approximate treatment given by Bogan: The adiabatic states introduced by Bogan can be identified with different groups of the exact vibronic states in the strong-coupling limit, but the approximate treatment is shown to have given rise to misleading interpretations of some of the experimental results.