Help me understand HF Exchange
I work through Szabo Ostlund and on page 88 (2.3.7 Pseude-Classical Interpretation of Determinantal Energies) it says that "The exchange Interactions between electrons with parallel spin are not real physical interactions but a convenient way of representing the energy of a system described by a single determinant. The physical interaction between two electrons [...] does not depend on the spin of the electrons"
This is the part where they introduce the idea that the energy of a single determinant can be summed up with H, J and K Terms.
Now consider a single determinant with two spacial orbitals. Both are occupied with one electron each. In case 1), both electrons have the same spin. In case 2), both electrons have opposite spins.
From Chemical knowledge I know that 1) is singulet and 2) is triplet and energy wise the triplet state is lower in energy. This is measurable.
Now the theory with this "adding up H, J and K terms to get determinantal energies" also predicts the same as:
Energy of 1) = H11 + H22 + J12
Energy of 2) = H11 + H22 + J12 - K12
So the - K12 lowers the energy of the triplet state which we measured and therefore proved correct.
But on the other hand Szabo Ostlund says that this exchange interaction is not physical.
How can this be not physical if we can measure it and the theory correctly predicts the relative energies of those two determinants?