The way I understand the Born-Oppenheimer approximation is as follows:
In the general solid Hamiltonian for $N$ electrons at $3N$ coordinates $r$ and $M$ ions at $3M$ coordinates $R$,
$$ H = T_e + T_I + V_{ee} + V_{II} + V_{eI} = H_0 + T_I\,, $$
with kinetic energies of electrons $T_e$ and ions $T_I$ and interaction energies of ions with ions $V_{II}$, electrons with electrons $V_{ee}$ and electrons with ions $V_{eI}$, we consider the ions to be fixed at $R$ and pretend to have solved
$$ H_0\phi_{Rn}(r) = \epsilon_{Rn}\phi_{Rn}(r)\,, $$
for all parameter sets $R$, where $\phi_{Rn}(r)$ is the electronic wave function and $n$ numbers different possible electronic eigenstates, i.e. the electronic ground state $n=0$ and electronic excitations $n>0$. Then we assume we can expand the full solid wave function for electrons and ions $\psi(r,R)$ in the functions $\phi_{Rn}(r)$, because the latter form a basis of $r$-space for each fixed $R$,
$$ \psi(r,R) = \sum_n \chi_n(R)\phi_{Rn}(r)\,. $$
Using this Ansatz in the full solid eigenvalue equation
$$ H\psi(r,R) = E\psi(r,R)\,, $$
and performing the actual adiabatic approximation leads to the equation
$$ (T_I + \epsilon_{Rn})\chi_n(R) = E\chi_n(R)\,. $$
This sure looks like an eigenvalue equation for ionic wave functions $\chi_n(R)$ in the effective potential $\epsilon_{Rn}$, where $n$ now also numbers different possible ionic eigenstates, and most authors seem to interpret it just like that.
I am confused by what this means for the full system wavefunction $\psi(r,R)$. If the Ansatz would look like
$$ \psi_n(r,R) \stackrel{?}{=} \chi_n(R)\phi_{Rn}(r)\,, $$
I would happily interpret it as meaning that, as part of our approximation, electrons and ions like to be excited together and the total solid is in an excited state whenever electrons and ions are both excited. However, since we sum over $n$ in the Ansatz, it looks to me like even though electrons and ions can be excited separately, the full solid does not have excited states (which can not possibly be true, can it?).
Is the interpretation of $\chi_n(R)$ as ionic wave function flawed? Am I missing something?