Relations among reduction factors in Jahn-Teller systems

A group-theoretical method for construction of vibronic wave functions of required symmetry is developed. The method is applied to investigate relations among Ham's reduction factors in Jahn-Teller systems of cubic (or tetrahedral) symmetry. Three special cases are discussed in detail: (i) in the case of linear Jahn-Teller coupling of an $E$ electronic level with many $E$-type crystal vibrational modes, it is shown, on the basis of a detailed symmetry analysis, that the relation $q=\frac{\mathrm{}(1+p)\mathrm{}}{2}$ between the reduction factors holds for linear coupling to a single mode pair (in agreement with Ham's result) but not in general otherwise; a detailed symmetry analysis made for this system allows dynamic effects to be disentangled from effects of symmetry; (ii) for Jahn-Teller coupling of a $T$ electronic level with many $E$ vibrational modes, it is found that the relations among the reduction factors, as given by Ham for linear Jahn-Teller coupling, are valid under general vibronic coupling; (iii) in the case of coupling of a $T$ electronic level with a single vibrational mode triplet of $T_2$ symmetry, the relation $K\left(E\right)=1-\frac{3\left[K\right(T_2)-K(T_1\left)\right]}{2}$ is shown to be valid in the weak-coupling regime for vibronic states correct to fourth order in the vibronic coupling constant but not for vibronic states correct to fifth order.