Upper- and Lower-Bound Energy Calculations for Atoms and Molecules in the Thomas-Fermi Theory

Using simple functions, upper and lower bounds for the Thomas-Fermi energies of atoms and homonuclear diatomic molecules at various internuclear separations are calculated which differ by less than 0.2% in the atomic case and at most by 0.5% for molecules. The calculation verifies that for this model molecules are not stable for short and intermediate nuclear separations. The possibility of a long-range attraction is also examined, and again the conclusions are found to be in agreement with Teller's theorem.