On the connection between the Exact Factorization and the Born-Huang representation of the molecular wavefunction

In these Notes we present a new perspective on the exact-factorization expression of a molecular wavefunction, which does not rely of the probabilistic interpretation of the molecular wavefunction as a joint probability amplitude. Instead, we demonstrate a close relation with the traditional Born-Huang representation, as the exact factorization emerges as the representation in the basis the of eigenvectors of the electronic density matrix. In doing so, the partial normalization condition and gauge freedom arise naturally from the formalism. In a second part, we derive the equation of motion in the here introduced exact factorization basis.