Exchange integral ($K_{ab}=\iint a^{\ast }(1)b(1)b^{\ast }(2)a(2)r_{12}^{-1}{d}^3{r}_1{d}^3{r}_2$)

For two orthonormal spatial atomic orbitals, the normalized symmetric and antisymmetric two-particle factors have Coulomb energies $J_{ab}+K_{ab}$ and $J_{ab}-K_{ab}$. With common orbitals, the fermionic singlet state and triplet state therefore differ by $2K_{ab}$. This is an exchange interaction resulting from the state symmetry and the Coulomb operator, rather than a separate classical force.