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Bonding comes from balancing kinetic energy and electrostatics

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

$\displaystyle \frac{3 m}{4} \ddot{s}(t) = -\frac{\partial E}{\partial s} = -\frac{\partial}{\partial s}\left(\frac{3}{2} \frac{1}{s^{2}} + V(s)\right)$ (4.2)

The electron size is optimized when the sum of kinetic and potential energy reaches a minimum with respect to variation in $ s$. We see that even an electron whose size and position is stationary has a kinetic energy that varies inversely as the square of its width. This relation may be seen as a consequence of Heisenberg's principle (better localized electrons have a higher momentum spread, and hence kinetic energy) or of the fact that the kinetic energy is $ \int \vert\nabla \phi\vert^{2} dV \propto (1/s)^{2}$.

Consider the case of a hydrogen atom, where the potential energy given by the electrostatic interaction of the electron and nucleus varies as $ -1/s$. More precisely,

$\displaystyle E = \frac{3}{2} \ \frac{1}{s^{2}} - \sqrt{\frac{8}{\pi}} \ \frac{1}{s}.$    

The electrostatic potential attempts to squeeze the electron into a point on top of the nucleus, while the kinetic energy term prevents this collapse. The balance of the two radial forces creates an atom with a stationary size of $ s = 1.88$ bohr, and $ E = -4/3 \pi = -0.424\mathrm{\ hartree}$. The energy is above the variational limit $ E = -0.5\mathrm{\ hartree}$ because the single Gaussian does not have the correct cusp at the nucleus center, or the correct long range drop off; however, it is expected that energy differences in bonding will be more accurately described.

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).

Figure 4.1: $ \mathrm {H_{2}}$ potential energy surface (kcal/mol); eFF properly dissociates $ H_{2}$, but the simplicity of the basis leads to underbinding.

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 $ \mathrm {H_{2}}$ 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 $ \mathrm {H_{2}}$, 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 $ \mathrm {H_{2}}$ molecule relative to H atoms, and its neglect contributes to the $ \mathrm {H_{2}}$ 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.


next up previous contents
Next: Pauli principle causes same Up: General theory of the Previous: Energy expression   Contents
Julius 2008-04-29