Introduction to Molecular Orbital Theory

The valence bond model is not able to explain all features of bonding. In Valence Bond theory, on the the valence orbitals involved in bonding are modified. An alternative scheme to valence bond theory uses molecular orbitals. A molecular oribital, MO, is a mathematical description of the region in a molecule where there is a high probability of finding electrons. Molecular orbitals are to molecules as atomic orbitals are to atoms. In molecular orbital theory, molecular orbitals are formed by the combination of atomic orbitals.

Recall, an atomic orbital is a wave function whose square gives the probability of finding an electron within a given region of space in an atom. A molecular orbital is a wave function whose square give the probability of finding an electron within a given region of space in a molecule.

Let’s take a look at the hydrogen molecule, H₂. The two s orbitals overlap to form a covalent bond. There are two atomic orbitals therefore there will be two molecular orbitals. The wave functions for the two s orbitals are added, and a bonding molecular orbital is the result. This is called constructive combination. This molecular orbital is lower in energy (see figure below) than the atomic orbitals by themselves. The electron density is between the two nuclei.

In addition to being summed, the wave functions are also combined in a way which cancels the electron density between the two nuclei– this is destructive combination. This is the antibonding molecular orbital, and it is higher in energy (see figure above) than the atomic orbitals. There is a node between the two nuclei and when an antibonding orbital is occupied, it decreases the bond strength. Electrons in bonding molecular orbitals contribute to the bond strength. We indicate the bonding molecular orbital as σ_1s and the antibonding orbital as σ*_1s. The asterisk, *, always indicates an antibonding orbital.

When atoms join to form a molecule, the electrons pair up and occupy the lowest available energy level which in the case of hydrogen is the bonding molecular orbital. Electrons seek lowest energy molecular orbitals that are available to them. A maximum of two electrons can occupy a molecular orbital. Electrons enter orbitals of degenerate energies singly with parallel spins before pairing. The bond order is give by

bond order = \(\frac{1}{2}\) (#e^– in bonding orbitals – #e^– in antibonding orbitals)

A bond order of 1 is a single bond. Bond order of 2 is a double bond, and a bond order of 3 is a triple bond. There are also bond orders of 0.5, 1.5, 2.5, and so on. A bond order of 0.5 means the bond is unlikely to form under normal conditions or it could be a radical species. A bond order of 0, means the bond will not form. For H₂, the bond order is 1. We take the total number of electrons in bonding orbitals and subtract the number of electrons in antibonding orbitals. For H₂ there are 2 electrons in a bonding orbital and 0 electrons in an antibonding orbital.

bond order H₂ = \(\frac{1}{2}\)(2 – 0) = 1

The MO configuration is written as σ(1s²).

Let’s take a look a diatomic helium, He₂. Each He atom has 2 valence electrons. A total of 4 electrons will be distributed into two molecular orbitals, a bonding and an antibonding. We fill the lower energy orbital first. See the figure below.

The last two electrons go into the σ*_1s antibonding oribital.

The bond order is \(\frac{1}{2}\)(2-2) = 0. A bond order of zero means the bond will not form. The MO configuration is written as σ(1s²)σ*(1s²).

In Period 2 of the periodic table, there are 2s and 2p orbitals. For example, the molecular orbital diagram for Li₂ is shown below.

The bond order is \(\frac{1}{2}\)(4 – 2) = 1. Note, the core electrons cancel out when calculating bond order. It is only the valence electrons that participate in bonding. The MO configuration for Li₂ is σ(1s²)σ*(1s²)σ(2s²).

Now we will look at B₂. There are a total of 10 electrons for the two boron atoms with two of the electrons in 2p orbitals. The MO diagram is below.

The bond order for B₂ is \(\frac{1}{2}\)(6 – 4) = 1. It is a single bond.

The filling order of molecular orbitals for Period 2, Li₂ to N₂ follows:

σ(1s), σ*(1s), σ(2s), σ*(2s), π(2p), σ(2p), π*(2p), σ*(2p)

For O₂ to Ne₂ the filling order is:

σ(1s), σ*(1s), σ(2s), σ*(2s), σ(2p), π(2p), π*(2p), σ*(2p)

Note, for O₂, F₂, and Ne₂, the σ(2p) fills before the π(2p) because the the σ(2p) is lower in energy than π(2p) in these three cases.

Below is the MO diagram for O₂.

The bond order is \(\frac{1}{2}\)(10 – 6) = 2. It is a double bond. Another thing to note is O₂ has 2 unpaired electrons. This means that O₂ is paramagetic. A paramagnetic substance is one that is attracted to a magnetic field. A diamagnetic substance has all paired electrons and it is weakly repelled by a magnetic field. O₂ is predicted to be diamagnetic by Lewis dot structures and valence bond theory, but experiment shows it is paramagnetic.

For binary compounds like NO, the MO diagrams are drawn in the same way as previously discussed. The more electronegative atom is lower in energy due to stabilization of the lone pair(s) of electrons.

Worksheet: Molecular Orbital Theory