The effects of anisotropy in Jahn–Teller systems: application to the icosahedralH  (h g) system

Abstract

Jahn–Teller (JT) coupling between electronic motion and lattice or molecular vibrations results in an adiabatic potential energy surface that contains either wells or troughs of minimum-energy points. When wells are lowest in energy, the system will vibrate about the minimum-energy points. This vibration must be taken into account when describing the quantum mechanical states of the system. In general, the wells will be intrinsically anisotropic. This anisotropy alters the vibrational frequencies and hence the positions of the energy levels, and can be particularly significant when the barriers between wells are shallow. In this paper, we will show how anisotropic states and their energies can be calculated using two unitary transformations. The first locates minima on the adiabatic potential energy surface, and the second accounts for anisotropy in the shape of the minima. The method is developed in a way general enough to allow it to be applied to any linear JT problem. The theory is then applied to the icosahedral H ⊗ (h ⊕ g) JT system. The results obtained will help the understanding of, for example, the effects of vibronic coupling in positively charged fullerene ions.

The p3otimes h Jahn–Teller effect in a distorted environment

“…In previous papers that included anisotropy via the scale transformation, it was found that the energy of the states with no vibrational excitations should be calculated using a wavefunction equivalent to equation ( 19) corrected to second order in perturbation theory via coupling to phonon excited states on the upper APES sheets [42]. Second-order corrections to the states are necessary because the anisotropic corrections to the energy are second order.…”

“…The sum is over µ and ν = θ and with b † and b defined in a similar manner b † θ and b θ . The µν are parameters which form a matrix whose values can be fixed by minimizing the energy of the ground state at a minimum-energy point calculated to second order in perturbation theory [39][40][41][42]. The effect of U s is to scale the Q ν according to the relation…”

“…where the e −2 µν are elements of a matrix e −2 defined in terms of the matrix as a power series expansion. The effect on a harmonic oscillator-like function for vibrational modes Q ν of frequency ω (in units in which the reduced mass and h are set to 1) is thus [42] exp…”

“…In terms of the scale transformation parameters, the (unnormalized) ground-state 'vibrational' wavefunction at a point on the trough including anisotropy and the term to localize the vibrational function can be written (in units h = 1) in the general matrix form [42] ψ vib (0, 0, 0, 0…”

Low-energy excitations of a linearly Jahn-Teller coupled orbital quintet

“…Corrections due to the anisotropy of the wells were introduced in Ref. 21. In the Appendix, Table II reports a few of the vibronic eigenenergies used to draw Fig.…”