5.1: Formation of Molecular Orbitals from Atomic Orbitals
Molecular orbital theory extends from quantum theory and the atomic orbital wavefunctions (\(\psi\)) described by the Schrödinger equation. While the Schrödinger equation defines a \(\Psi\) for electrons in individual atoms, we can approximate the molecular wavefunction (what \(\Psi\) would look like if we combined the \(\psi\) of individual atoms). The addition or subtraction of wavefunctions is termed linear combination of atomic orbitals (LCAO). Molecular orbital theory is applied to LCAO to describe bonding.
The LCAO for the wavefunction of two atoms (\(\psi_a\) and \(\psi_b\)) is represented by the general expression below. The coefficients \(c_a\) and \(c_b\) quantify the contribution of each atomic \(\psi\) to the molecular \(\Psi\).
\[\Psi=c_{a} \psi_{a}+c_{b} \psi_{b} \nonumber \]
For two atomic \(\psi\)s to form a bond, three conditions must be satisfied:
- First, the distance between the atoms must be small enough to provide good overlap. This is because the two orbitals must be able to overlap using regions of \(\psi^2\) where the probability of finding electrons is significant. If atoms are not close enough, \(\psi_a\) and \(\psi_b\) cannot interact productively. At the same time, the atoms must be far enough apart that their nuclei do not repel each other.
- The symmetry of the orbitals must be compatible such that regions of \(\psi_a\) and \(\psi_b\) with the same sign constructively interfere more than regions with opposite sign destructively interfere. In other words, for a productive interaction, there must be a relatively high probability of finding an electron between the two nuclei, and this depends on the symmetry (and sign of the \(\psi\)) of the atomic orbitals.
- Third, the energies of the atomic orbitals must be similar. Orbitals with similar energy combine to make the most stable bonding molecular orbitals. Atomic orbitals with very different energies form less stable bonding interactions, and the larger the difference in the atomic orbitals, the weaker the bonding interaction will be.
Atomic orbitals combine to form molecular orbitals when these three conditions are met. The result is a set of bonding molecular orbitals that are lower in energy than the original atomic orbital energies.