The Jahn-Teller theorem

Abstract

A compact proof of the Jahn-Teller theorem is given. The method distinguishes between two cases: (1) If the irreducible representation Γ to which the wavefunction belongs is reducible under one of the subgroups G s which leave one atom invariant; the proof is trivial and would apply in space of any number of dimensions. (2) If Γ remains irreducible under all Gs, more detailed attention is necessary. In two or three dimensions, however, it is straightforward to establish the theorem for this case. It is further shown that in composite-dimensional spaces, some violations of the theorem would be found under (2).