In the book K. Huang and M. Born, Dynamical Theory of Crystal Lattices (1954, Appendix VIII) and also in the Wikipedia article https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation#Derivation, the derivation of the Born-Oppenheimer approximation is given. In these derivations, the following expansion of the exact electron-nuclear wave function is used \begin{equation} \Psi\left(r,R\right) = \sum_{m} \phi_{m}\left(R\right) \chi_{m}\left(r,R\right). \label{eq:BornHuangExpansion} \end{equation} Here, $r$ and $R$ refer to all electronic and nuclear variables, respectively. Further, $\Psi$ is the wave function satisfying the exact time-independent Schrödinger equation of the exact Hamiltonian of electrons and nuclei (comprising the kinetic energies and Coulombic interactions) and $\chi_{m}$ the wave function satisfying the Schrödinger equation for the exact Hamiltonian minus the nuclear kinetic energy. Please check some details from the Wikipedia article, if necessary.
I have not found any discussions from the literature in which this expansion is discussed in detail from a mathematical point of view. Does anyone see how to prove that this expansion is exact or see that it can't be exact?