This ansatz is not antisymmetric under the exchange of electrons with opposite spins but it can be shown (show this) that it gives the same expectation value for the energy as the full Slater determinant.

As long as the Hamiltonian is spin independent, the above approach gives us the same expectation value for the energy. It is rather straightforward to see this if you go back to the equations for the energy. We leave this as an exercise to the eager reader.

Can you think of observables where not respecting the symmetry can have consequences?