The Franck-Condon approximation for second-order Jahn-Teller vibronic reduction in icosahedral T⊗h systems

Abstract

The Franck-Condon (FC) approximation is used to obtain analytical values for the second-order factors describing vibronic reduction for the case of the strongly coupled icosahedral T1u⊗hg Jahn-Teller (JT) system in which an electronic triplet ground state is coupled to vibrations of fivefold degeneracy. This is believed to be applicable to the ground state of the C60- ion and so the calculations should help in providing valuable physical information on this fullerene ion. The procedures for determining vibronic reduction effects depend critically on the particular JT system and upon the vibronic coupling strength. In the FC approximation, the major contributions to these reduction factors originate from virtual vertical transitions between the ground electronic adiabatic state and excited vibronic states associated with upper sheets in the adiabatic potential energy surface. At strong coupling, other transitions may be neglected. The FC results are compared with those obtained from an alternative approach involving a shift transformation. Details are given for perturbations containing orbital operators of threefold (T1) and fivefold (H) symmetry. The discussion emphasizes the inherent high symmetry of the system.

Accurate calculations of second-order vibronic reduction factors for C60ions

“…Thus the FC approximation is an accurate analytical technique in the strong coupling regime. As emphasized in [5] and [11] for the octahedral T ⊗ t 2 and icosahedral T ⊗ h JT systems respectively, the underlying physics of soRFs is clearly exposed by the FC analysis.…”

“…This applies therefore to the systems involving strong vibronic coupling that are of interest here. The Born-Oppenheimer approximation can be then used with the factored adiabatic wavefunctions to calculate the soRFs as described in [5] for the icosahedral T ⊗ h JT system.…”

“…The definition of the soRFs K (2) M ( l ⊗ m ) involving two perturbations, H (1) ( l ) of symmetry l and H (1) ( m ) of symmetry m , can be found in the general theory summarized in [10]. Their evaluation within the FC approximation for the icosahedral T 1 ⊗ h JT system has been described in detail in [5] and is summarized below.…”

“…The basic ideas can be best understood in terms of WKB wavefunctions. Detailed calculations given in [5] show that this leads to the so-called FC Green operator G ( p) FC for well p in which each well is considered separately. The effective virtual transitions are vertical in an FC diagram.…”

“…Another property of soRFs is that they can introduce additional terms into the effective Hamiltonian. The major problem is that soRFs are generally difficult to calculate with high accuracy, except in the limit of infinitely strong coupling when the Franck-Condon (FC) approximation [5] is valid. However, in order to overcome many of these difficulties, a new procedure has been devised involving non-Condon corrections to the standard FC method [6].…”