Singlet-triplet exchange splitting in helium ($E_S-E_T=2K_{ab}$)

For fixed orthonormal orbitals $a,b$, symmetrizing or antisymmetrizing $a(1)b(2)$ gives electron-repulsion expectations $J_{ab}+K_{ab}$ and $J_{ab}-K_{ab}$. Antisymmetry of the complete electronic state therefore places the singlet above the triplet by $2K_{ab}$, with $K_{ab}$ the positive Coulomb exchange integral. Orbital relaxation and further correlation refine this comparison. Exchange splitting here is a consequence of Fermi statistics and the electrostatic interaction, not a separate classical spin-spin force.