Molecular geometric phase from the exact electron-nuclear factorization

The Born-Oppenheimer electronic wavefunction ${\mathrm{\Phi }}_R^{BO}(r)$ 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 $e^{i\gamma }$ if the geometric phase $\gamma = \oint $\mathrm{I}\mathrm{m}$ \langle \Phi_R |\nabla_{\mu} \Phi_R\rangle \cdot d$\mathbf{R}$_{\mu}$isevaluatedwiththeconditionalelectronicwavefunction$\Phi_R®$ from the exact electron-nuclear factorization ${\mathrm{\Phi }}_R(r)\chi (R)$ instead of the adiabatic function ${\mathrm{\Phi }}_R^{BO}(r)$. 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.