A Gaussian wave packet automatically satisfies the Heisenbserg uncertainty principle by virtue of its functional form -- in fact, it is a minimum uncertainty wave packet. That leaves as a free parameter the size of the wave packet, which is propagated using the equation of motion
Consider the case of a hydrogen atom, where the potential energy given by the electrostatic interaction of the electron and nucleus varies as . More precisely,
The same logic can be used to explain the stability of the two electron covalent bond. In the eFF description of ground state hydrogen molecule, two electrons lie at the midpoint between two protons. The electrons shrink to interact more strongly with the protons (s = 1.77 bohr versus 1.88 bohr in the atoms), and the decreased potential energy of having each electron interact with two protons drives the formation of the bond (Figure 4.1).
Pulling the protons apart causes the electrons to interact with the protons less strongly, and the bond weakens. As the bond length is increased past 2.1 bohr, it becomes more favorable for the electrons to become atom-centered. The energy varies smoothly as each electron associates with one proton, and the wavefunction goes from a closed shell to an open shell description. In Hartree-Fock theory, the analogous transition between RHF and UHF occurs at 2.3 bohr. The eFF bond energy is found to be 67 kcal/mol at a bond length of 0.780 bohr (versus 104 kcal/mol exact at 0.741 bohr).
There are some features missing from the eFF picture. First, in the true molecule, the electron density is a doubly peaked function that reaches a maximum at the sites of the protons. Because the single Gaussian wavefunction cannot become multiply peaked, the bond energy is underestimated. Second, there is a measure of static correlation that is missing; in dissociating , there is a resonance stabilization between having the spin up electron on the right and the spin down electron on the left, and vice versa. This neglect makes the energy fall to zero too quickly. Finally, dynamic correlation is missing; electron-electron repulsion should be diminished when two electrons are placed in the same orbital, as they have a tendency to avoid each other. This correlation effect stabilizes molecule relative to H atoms, and its neglect contributes to the underbinding.
Issues of underbinding aside, it remains remarkable that a floating Gaussian description of electrons can give a potential energy curve for hydrogen molecule dissociation that has a plausible inner wall, bonding region, long range tail, and a correct transition between closed and open shell wavefunctions.