Molecular orbital diagrams

Molecular wavefunctions are formed by taking the product of all the individual electron wavefunctions found in both atoms. This can be denoted as ψ = ψ(1)ψ(2)...ψ(N_e) where N_e is the total number of electrons.

While we need computers to calculate these molecular wavefunctions, we can use rules to predict their shape:

1. N atomic orbitals combine to produce N molecular orbitals.
2. The general pattern is one orbital above (in energy) the parent atomic orbitals, one below, and the rest lying in between.

For example, here is the MO diagram of H₂:

So two atomic orbitals (1s from both hydrogen atoms) produce two molecular orbitals, denoted σ and  σ*. We have two total electrons, so these occupy the lower-energy σ bonding orbital. The empty σ* orbital in the diagram is called an anti-bonding orbital, because it is higher in energy then the parent atoms. Having two electrons in the σ* would pretty much cancel out the extra stability gained from filling the σ orbital. This can be observed in an MO diagram of He₂:

This matches the real world, we don't observe He₂.

A pair of electrons being in a σ molecular orbital is the same thing as a "σ bond" from A-level chemistry. The sigma symbol actually means "cylindrical symmetry around the internuclear axis". We know that bonds between s or sp-hybridized orbitals are sigma, while bonds between p orbitals perpendicular to the internuclear axis are called π. This definition means that bonds between "head on" p orbitals are also called sigma.

Sigma bonds

Pi bond

Here is the MO diagram of O₂:

This predicts that oxygen has two unpaired electrons, which isn't predicted by using lewis diagrams. Experiments easily confirm that these two unpaired electrons exist.

Unpaired electrons (or "free radicals") in A-level chemistry are typically taught as being extremely reactive. This is mostly true for other molecules. Wikipedia writes "with some exceptions, unpaired electrons cause radicals to be highly chemically reactive." I don't know if there are any tricks to help spot the exceptions.

Here is the MO diagram of F₂:

Six 2p orbitals (three from each atom) combine to produce one sigma bonding orbital, two pi bonding orbitals, and their corresponding antibonding orbitals. Some intuition about why this happens can be found by considering how the p orbitals are perpendicular to eachother.

Only two p orbitals can combine "head-on", the other four have to form perpendicular pi bonds, since they are pointing in the wrong direction to form sigma bonds.

Finally, MO diagrams can occasionally have non-bonding orbitals, which are at the same energy level as their parent atoms. "Lone pairs" from A-level chemistry are contained in these.