n χ n |φ n ⟩. In order for the molecular state to be physical, it has to be independent of the total phase. 10 However, this is only possible if S(R, t) = -θ(R, t). Thus we have just shown how the U (1) gauge freedom emerges from requiring the molecular state to be physical. Furthermore, the normalization condition follows from the property that |v 1 (R, t)⟩ is a normalized eigenvector, 1 = ⟨v 1 (R, t)|v 1 (R, t)⟩ ∀R, t. Hence, while the adiabatic basis is defined as the set of eigenstates of the electronic Hamiltonian operator at positions R, Ĥel (R), the EF basis is the set of eigenstates of the electronic density operator at positions R at time t, ρel (R, t). We now propose to derive the equations of motion for the nuclear and electronic wavefunctions that comprise the EF. Before doing this, we would like to stress that up to this point we have been treating the EF within the BH representation. However, it is important to recognize that diagonalizing the electronic density operator can also be interpreted as moving from the BH representation to the representation in its eigenbasis, which we refer to as the EF representation, |Ψ(R, t)⟩ EF = χ(R, t) |v 1 ⟩. There, the electronic basis {|v α ⟩} α=1,...,N is independent of R and t, its dependence being transferred to Û (R, t) = α |v α (R, t)⟩ ⟨v α | which diagonalizes the electronic density operator. This transfer of R, t-dependence from the basis to the operators is akin to transforming the electronic degrees of freedom from the Schrödinger picture to the Heisenberg picture.