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Molecular geometric phase from the exact electron-nuclear factorization

The Born-Oppenheimer electronic wavefunction $Φ_R^{BO}(r)$ picks up a topological phase factor $\pm 1$, 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 $e^{iγ}$ if the geometric phase $γ= \oint \mathrm{Im} \langle Φ_R |\nabla_μ Φ_R\rangle \cdot d\mathbf{R}_μ$ is evaluated with the conditional electronic wavefunction $Φ_R(r)$ from the exact electron-nuclear factorization $Φ_R(r)χ(R)$ instead of the adiabatic function $Φ_R^{BO}(r)$. A model of a pseudorotating molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides the first examples of induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current which 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 wavefunction.