On the connection between the exact factorization and the Born–Huang representation of the molecular wavefunction

From the above discussion, we notice that the density matrix ρ can be diagonalized by unitary matrices, ρd = U†ρU, consisting of orthonormal eigenvectors, U=(v̲1,v̲2,…,v̲N). Then, the density operator ρ̂(R̲,t) at position R̲ can be written in terms of its eigenvalues and eigenvectors, namely ρ̂=∑n,m=1N|vn〉[ρd]nm〈vm|. Since only λ1 = ρ ≠ 0, then ρ̂=|v1〉ρ〈v1| and we haveρ̂(R̲,t)=|v1(R̲,t)〉χ(R̲,t)χ*(R̲,t)〈v1(R̲,t)| (3) =|Ψ(R̲,t)〉BH〈Ψ(R̲,t)|BH.(4)In Eq. (3), the nuclear density is ρ(R̲,t)=|χ(R̲,t)|2, with χ(R̲,t)=|χ(R̲,t)|eiS(R̲,t), a nuclear wavefunction with an arbitrary phase S(R̲,t), and in Eq. (4), we introduced the EF form of the BH representation as |Ψ(R̲,t)〉BH=χ(R̲,t)|v1(R̲,t)〉. Note that we interpret |Ψ(R̲,t)〉BH as a molecular state; since ⟨R̲|Ψ(t)⟩=|Ψ(R̲,t)〉, it is the molecular state vector projected onto the nuclear position eigenstates. Projecting |v1(R̲,t)〉 onto the electronic position eigenstates, |r̲〉, we identify the electronic conditional amplitude, namely ΦR̲(r̲,t)=⟨r̲|v1(R̲,t)⟩. Note that this derivation completely bypasses the “usual” justification of the EF form of a molecular wavefunction based on marginal and conditional probability amplitudes.