I've been doing some research on singlet-triplet molecular states, and one things I can't perfectly settle is a rigorous demonstration of why the triplet state is a higher energy than the singlet. One way I can qualitatively understand why the singlet state has lower energy is this: If the atoms are far apart then it does not matter much whether the spin state is triplet or singlet. But when the atoms come close together to form a molecule then we can view the two valence electrons to have almost similar position co-ordinates. It is then necessary that Pauli exclusion be strictly obeyed, and for molecule formation it's would thus be much more favorable to have spins aligned oppositely. For a triplet state with spins pointing in the same direction it is impossible for the atoms to form a molecule due to Pauli exclusion principle. That, however, is an insufficient answer for me. It's too much "it makes sense if we want it too, but not a priori". Anyone have a good handle on this?