Topological phase in molecular bound states: Application to the E⊗e system

We extend recent works of Berry, Simon, and others on the evolution of adiabatic wave functions in parameter spaces with nontrivial global geometry, to show the interesting ways in which wave functions can acquire nonintegrable phase (commonly termed Berry’s phase, geometric phase, or topological phase) upon transport along paths in the parameter space. We emphasize the case of arbitrary paths on the Born–Oppenheimer potential energy surfaces (the parameter space of the electronic states in an isolated molecule) of the linear plus quadratic E⊗e Jahn–Teller system. It is found that these surfaces are degenerate not only at the origin but also at three other, equivalent points, which lie on a radius ρ=2k/g. Here k and g are the linear and quadratic vibronic coupling constants, respectively. This radius is then shown to mark a sharp transition between Jahn–Teller behavior, characterized by half-odd-integral vibronic angular momentum, and Renner–Teller behavior, which has integral angular momentum. Finally, we examine the conditions necessary for adiabatic evolution in the E⊗e system, and the observable consequences of the geometric phase.