We may exactly write the wavefunction of a molecular system as $$ \phi(r, R, t) = \sum_{k = 0}^{\infty} \chi_k(R, t) \psi_k(r, R) $$ where $r$ is the coordinates of electrons, $R$ is the coordinates of nuclei, $\phi_k(r, R)$ is the $k$-th eigenfunction of the electronic Hamiltonian for the electrons and $\chi_k$ is a corresponding nuclear wavefunction.
We then get that the time-dependent Schrodinger equation for the nuclear wavefunctions are $$ i \hbar \frac{\partial \chi_k}{\partial t} = \left[ - \sum_I \frac{\hbar}{2M_I} \Delta_I + E_k(R) \right]\chi_k + \sum_l C_{kl} \chi_l $$ where $I$ denotes a specific nucleus, $E_k$ is the $k$-th eigenvalue of the electronic hamiltonian and $C_{kl}$ is a coupling function defined by $$ C_{kl} = \int dr \psi_k^* \left[- \sum_I \frac{\hbar^2}{2M_I} \Delta_I \right] \psi_l + \sum_I \frac{1}{M_I} \left[\int dr \psi_k^* \left[ -i\hbar \nabla_I \right] \psi_l \right] \left[-i\hbar \nabla_I \right] $$ We may make the adiabatic approximation and assume that the coupling matrix is diagonal such that $$ C_{kk} = - \sum_I \frac{\hbar^2}{2M_I} \int dr \psi_k^* \Delta_I \psi_k $$ I am curious in what real life situations this approximation causes an issue. I know with the full Bohn-Oppenheimer approximation where the coupling is completely ignored there are many cases where this causes issues and I know there are many cases where issues arise when the approximation is made to only consider the ground state $k=0$ but I am not sure what possible issues could arise by ignoring the off-diagonal coupling functions. Are there any chemical reactions in non-extreme, non-relativistic situations that are not adequately described by just a diagonal coupling matrix?