To see that generally speaking the basic form of the Hamiltonian produces energy degeneracy with respect to spin, but that it is not important for using the Born-Oppenheimer approximation, consider the example of three electrons.
Any three-electron energy eigenfunction
Since the assumed Hamiltonian
It is now seen that given a solution for the first four wave functions, there is an equally good solution for the second four wave functions that is obtained by inverting all the spins. Since the spins are not in the Hamiltonian, inverting the spins does not change the energy. They have the same energy, but are different because they have different spins.
However, they are orthogonal because their spins are, and the spatial
operations in the derivation of the Born-Oppenheimer approximation in
the next note do not change that fact. So they turn out to lead to
nuclear wave functions that do not affect each other. More precisely,
the inner products appearing in the coefficients