Lie Groups and the Jahn–Teller Effect

After an introduction to the classic theory of the Jahn–Teller effect for octahedral complexes, an account is given of Lie groups and their relevance to the F+ center in CaO. The coincidence of the three-fold and two-fold vibrational modes (both of angular frequency ω) leads to a study of U5 and R5, the unitary and rotation groups in five dimensions. The language of second quantization is used to describe the weight spaces and branching rules. Pairs of annihilation and creation operators for phonons are coupled to zero angular momentum and used as the generators of the noncompact group O(2, 1). This facilitates the evaluation of matrix elements of V, the interaction that couples the oscillations of the octahedral complex to the electron in its interior. Glauber states are used near the strong Jahn–Teller limit, corresponding to [Formula: see text]. The possible extension of the analysis to incorporate the breathing mode is outlined. Correspondences with problems in nuclear physics are mentioned.