Molecular orbital theory is one way of explaining how electrons form covalent bonds between atoms in a molecule. It is the most commonly used theory today: virtually all modern quantum chemistry programs use molecular orbital theory as a basis. (Another way of looking at bond formation is valence bond theory. While chemists often use it to describe hybridization, it turns out to not be as easy computationally.)

For molecular orbital calculations, we attempt to solve the Schrodinger equation for the molecule. This will give us a set of molecular orbitals for the molecule, much like the atomic orbitals in an atom. Just like in the atoms, the electrons will fill the lowest energy molecular orbitals first, two electrons per orbital, one spin up and the other spin down. In practice, we cannot do this exactly. Instead, we make the approximation that the molecular orbitals look like linear combinations of the atomic orbitals on the atoms. How do we know what the combinations look like? Remember that the atomic orbitals are wavefunctions: we can add or subtract them like waves. (For more on this, see the constructive/destructive interference page.)

Consider two hydrogen atoms. When far apart, each atom has a single electron in a 1s atomic orbital. The 1s atomic orbitals are spherical: for now we'll assume that both have the same phase.

As we move the two 1s orbitals together, they begin to overlap.

What happens now? There are two possibilities. First, we can have constructive interference, where the two orbitals add. If this happens, we get an area of high electron density near the center of the molecule. Since this is a place where the negatively charged electrons can have favorable interactions with both positive nuclei, this bonding molecular orbital has a lower energy than either of the two separated 1s atomic orbitals.(Truth in lending statement) This is known as a s_1s orbital. The 1s means it's made from 1s atomic orbitals, the s refers to the fact that it looks like a s orbital when we look down the internuclear axis.

However, we can also have destructive interference if we subtract the two orbitals. If this happens, we get an area where there is zero electron density in the center of the molecule. This area is known as a node. This orbital has a higher energy than either the bonding MO or either of the 1s atomic orbitals: it's known as an anti-bonding molecular orbital, the s_1s^*. The superscript * refers to an antibonding orbital.

We can sketch out the relative energies of the atomic, bonding molecular and antibonding molecular orbitals on a molecular orbital energy diagram like the one below. Here, the two atomic orbitals are shown on the outside, and the energies of the molecular orbitals in the center. The antibonding molecular orbital is raised in energy by the same amount the bonding orbital is lowered.

For animations of the formation of the bonding and antibonding molecular orbitals, see the pages below. Areas with high electron density have lots of dots: areas with low density have few. Note especially the formation of the node in between the two nuclei in the antibonding molecular orbital. The animations are fairly large and require the Chime plug in.

Molecular orbitals are like atomic ones in that each can contain exactly two electrons due to the Pauli principle. Because of this, we can determine the bond order of a molecule. Molecular orbitals can be built up from other orbitals as well: see the higher energy orbitals page for details.

Truth in lending: this explanation is almost certainly incorrect, at least for the simple molecule H₂⁺, which can be solved exactly within the Born-Oppenheimer approximation. Scientists are uncertain for other, more complex molecules, but given that bonding MOs always build up electron density between the nuclei it's a possible explanation, and much simpler than the alternatives.

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