Molecular orbital theory

Why another theory?

The oxygen molecule is drawn with a double bond, and every electron in its Lewis structure is paired. Yet liquid oxygen is pulled towards the poles of a strong magnet. Pull towards a magnet needs unpaired electrons. Lewis structures cannot explain this, and neither can the idea of a bond between two atoms alone.

Molecular orbital theory solves it. It treats the electrons of a molecule as belonging to the whole molecule, not to one atom or one bond.

Atomic orbitals combine

When two atoms come close, their atomic orbitals overlap and combine. This method is called the linear combination of atomic orbitals, LCAO.

• Adding the two wave functions, $\psi_A + \psi_B$, piles up electron density between the nuclei. The nuclei are held together. This is a bonding molecular orbital, lower in energy than the atomic orbitals.
• Subtracting them, $\psi_A - \psi_B$, leaves a node between the nuclei, with almost no electron density there. This is an antibonding molecular orbital, marked with a star, and higher in energy.
• The number of molecular orbitals always equals the number of atomic orbitals that combine. Two atomic orbitals give two molecular orbitals.

Atomic orbitals combine well only under three conditions:

1. They have similar energies. A $1s$ orbital does not combine with a $2s$ orbital of the other atom.
2. They have the same symmetry about the line joining the nuclei. A $2p_z$ orbital combines with $2p_z$, not with $2p_x$.
3. They overlap as much as possible. The greater the overlap, the lower the bonding orbital.

Sigma and pi molecular orbitals

• Orbitals that combine along the line of the nuclei, such as $1s$, $2s$ and $2p_z$, give sigma orbitals: $\sigma$ and $\sigma^*$. They are symmetrical about that line.
• Orbitals that combine side by side, $2p_x$ with $2p_x$ and $2p_y$ with $2p_y$, give pi orbitals: $\pi$ and $\pi^*$. Their density lies above and below the line, and the line itself is a nodal plane.
• The two $\pi$ orbitals have the same energy, and so do the two $\pi^*$ orbitals. They are degenerate.

The energy order

For molecules up to nitrogen, the $\pi 2p$ orbitals lie below $\sigma 2p$:

• $\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < (\pi 2p_x = \pi 2p_y) < \sigma 2p_z < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$

For oxygen and fluorine, $\sigma 2p_z$ moves below the $\pi 2p$ pair:

• $\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < \sigma 2p_z < (\pi 2p_x = \pi 2p_y) < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$

Filling the orbitals

Electrons go into molecular orbitals by the same three rules as into atomic orbitals.

• Aufbau: the lowest energy first.
• Pauli: at most two electrons in one orbital, with opposite spins.
• Hund: in two orbitals of the same energy, one electron goes into each before any pairs up.

Nitrogen has $14$ electrons. Its configuration is $\sigma 1s^2$ $\sigma^* 1s^2$ $\sigma 2s^2$ $\sigma^* 2s^2$ $\pi 2p_x^2$ $\pi 2p_y^2$ $\sigma 2p_z^2$. Every electron is paired.

Practise with new numbers

Try it in Python

The program fills the molecular orbitals of every molecule from $\text{H}_2$ to $\text{F}_2$, using the right energy order for each. It prints the bond order and the number of unpaired electrons.

Common mistakes

• Making more molecular orbitals than atomic ones. Two atomic orbitals always give exactly two molecular orbitals.
• Thinking an antibonding orbital is empty by nature. It holds electrons whenever the bonding orbitals below it are full.
• Using one energy order for every molecule. Up to nitrogen, $\pi 2p$ is below $\sigma 2p$; for oxygen and fluorine it is the other way round.
• Pairing electrons in degenerate orbitals too early. Hund's rule puts one electron in each $\pi$ orbital first.

Remember these

• Bonding orbital: $\psi_A + \psi_B$, lower energy. Antibonding orbital: $\psi_A - \psi_B$, higher energy, marked with a star.
• Conditions: similar energy, same symmetry, maximum overlap.
• Sigma orbitals lie along the line of the nuclei; pi orbitals lie above and below it.
• Up to $\text{N}_2$: $\pi 2p$ below $\sigma 2p$. For $\text{O}_2$ and $\text{F}_2$: $\sigma 2p$ below $\pi 2p$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many molecular orbitals form when the $2s$ and three $2p$ orbitals of one atom combine with those of another?

2. Why does a $2p_z$ orbital of one atom not combine with a $2p_x$ orbital of the other?

3. Write the configuration of $\text{Li}_2$, which has six electrons.

4. In which orbitals does the twelfth electron of $\text{C}_2$ go?

5. Why is an electron in an antibonding orbital bad for the bond?