Multiple groups in the symmetry classification of adiabatic electronic wavefunctions

It was originally shown by Longuet-Higgins and colleagues that when the ele ctronic Schrodinger equation is solved as a function of the nuclear coordin ates the adiabatic electronic wavefunction can undergo a change of sign aft er completing a closed circuit. This geometric phase occurs for a circuit a round a conical intersection, and in particular around a conical intersecti on corresponding to a linear Jahn-Teller effect. The adiabatic wavefunction s are classified here under a group called the adiabatic multiple group, wh ich is a generalization of the 'vibronic double group' of C-3v introduced b y Hougen, and is distinct from the familiar electron-spin double group. Alt hough the real electronic wavefunctions can be only double-valued, the grou ps can have higher multiplicity because of the possibility of different cir cuits. For a number of symmetric- and spherical-top point groups, the adiab atic multiple group is shown to be the direct product of the point group wi th a phase group. The adiabatic multiple group can be applied to individual adiabatic orbitals, and so to configurations built from these orbitals. Th is leads to the rule that the linear Jahn-Teller effect vanishes in the sin gle-configuration approximation for configurations containing non-degenerat e electrons plus an even number of e electrons. There does not appear to be any cancellation effect for electron configurations of cubic molecules con taining f electrons.