A scale transformation in Jahn - Teller systems

Abstract

The Jahn - Teller (JT) system has been studied previously by many authors. It is well known that the potential energy surface for this system contains four equivalent wells in strong coupling. The wells are not isotropic. In the strong coupling limit, the vibrational t-mode splits into an -mode of frequency and an e-mode of frequency . However, it is difficult to incorporate this anisotropic effect into analytical models. Previously, the current authors have used a unitary shift transformation and energy minimization procedure to model many moderately to strongly coupled JT systems. However, the part of the Hamiltonian which produces the anisotropy was not treated fully. We now present a modification of this procedure for the system in which a scale transformation is applied in addition to the shift transformation. This is shown to introduce anisotropy automatically into the problem. We show that the correct frequencies are obtained in the infinite coupling limit. Symmetry-adapted combinations of the states associated with the wells are taken to obtain expressions for the ground state and inversion level. The inversion splitting between them is compared with existing results. We then discuss how the scale transformation method can be applied to other JT systems (for which the limiting frequencies are unknown), such as those in the symmetry which applies to the molecule.