An approximate analytical treatment of the T⊗(ε⊕τ₂) Jahn-Teller effect

The octahedral Jahn-Teller system T(X)( epsilon g (+) tau 2g ) is considered, in which the epsilon g and tau 2g modes correspond to the same frequency and are equally coupled to the electronic state T. A correspondence is drawn between the energy matrix for the J=1 states of this system and the energy matrix for the m=0 states of a three-dimensional harmonic oscillator displaced along the z axis. It is shown that the two sets of matrix elements become asymptotically proportional to each other as one proceeds towards the far interior of each infinite matrix. This permits a perturbation procedure to be set up for the Jahn-Teller system. Orthonormal basis states are established and a number of relations involving Laguerre polynomials are exploited to calculate to first order the energies of the J=1 levels, the Ham reduction factors and the intensities for the vibronic structure of the transition s to p. Analytic expressions are obtained for these quantities as a function of the coupling strength, and good agreement is obtained with the numerical calculations that are available at present.