Molecular geometric phase from the exact electron-nuclear factorization

The Born-Oppenheimer electronic wave function Phi(BO)(R) (r) picks up a topological phase factor +/- 1, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in R space. We show that this topological quantity reverts to a geometric quantity e(i gamma) if the geometric phase gamma = closed integral Im <Phi(R) vertical bar del Phi(R)> . dR(mu) is evaluated with the conditional electronic wave function Phi(R)(r) from the exact electron-nuclear factorization Phi(R)(r)x(R) instead of the adiabatic function Phi(BO)(R)(r). A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.