There are well-established mathematical relationships for the energy of a diatomic molecule (the exact energy, but also both the Hartree-Fock and Thomas-Fermi energies) in the limit of large nuclear charge of the atoms and small internuclear distances. We present calculated energies for a number of homonuclear diatomic molecules at short internuclear distances for numerical comparison with these established relationships. Intriguingly, they hold in practice for relatively small nuclear charges and for relatively long internuclear distances.