The Jahn-Teller effect in icosahedral symmetry: ground-state topography and phases

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Abstract
The resurgence of interest in properties of molecules of icosahedral symmetry follows the discovery of the C60 molecule. Because of the high symmetry almost all the electronic and vibrational states are highly degenerate, so in dealing with properties of these systems their Jahn-Teller interactions must almost always be allowed for. In this paper we explore the ground states of the G(X)(g(+)h) coupling scheme and those of the two subsystems, G(X)g and G(X)h. Using a mixture of analytical and numerical methods, we map the lowest adiabatic potential energy surfaces of these systems. The mappings are made in such a way as to facilitate an analysis of the geometrical phase factor acquired by the quantal system on transportation round adiabatic circuits in parameter space. These geometrical phase factors, or Berry phases, depend greatly on the paths followed to complete these circuits. In this paper we introduce parametrizations that elucidate with ease such circuits and lead to simple and easily accessible graphical illustrations of the subsequently induced Berry phases. Finally, we use the information provided by the Berry phase analysis to obtain the correct ordering of the low-lying states at strong coupling, the energies of these states and the Ham factors of the Jahn-Teller-active operators within these states.