The symbol • | • r indicates an integration over the electronic positions, and for this reason we removed all dependencies on r within the brackets. Equation ( ) is the exact factorization , which expresses the time-dependent molecular wavefunction as a product of the time-dependent nuclear wavefunction χ(R, t) and of the time-dependent conditional electronic amplitude Φ(r, t; R) -with parametric dependence on nuclear positions R. Note that, conversely to the Born-Huang expansion, the electronic sub-system is explicitly time dependent in the exact factorization. One can easily recover Eq. (2) from Eq. (3), by representing the electronic amplitude using a Born-Huang-like expansion, i.e., Φ(r, t; R) = l C l (R, t)ϕ l (r; R), and identifying the nuclear amplitudes as χ l (R, t) = χ(R, t)C l (R, t). When inserted into the TDSE (1), and using the partial normalization condition Φ(t; R)|Φ(t; R) r = 1 ∀ R, t, the coupled nuclear and electronic evolution equations are derived