Valence bond theory: sigma and pi bonds

From Lewis to orbitals

Lewis structures say that a covalent bond is a shared pair of electrons. They do not say why sharing gives out energy, or why bonds point in definite directions. Valence bond theory answers both. Heitler and London first worked it out for the hydrogen molecule. It uses the atomic orbitals you already know.

Two hydrogen atoms come together

Take two hydrogen atoms, A and B, with nuclei $N_A$ and $N_B$ and electrons $e_A$ and $e_B$. Far apart, they do not affect each other. As they come closer, new forces appear.

• Attraction: each nucleus pulls its own electron, and also the electron of the other atom.
• Repulsion: the two electrons repel each other, and so do the two nuclei.

At first the attractions are larger, so the energy falls. It reaches a minimum at a distance of $74\ \text{pm}$, the bond length. Closer than that, the repulsion of the nuclei takes over, and the energy rises steeply.

At the minimum, $435.8\ \text{kJ}$ is released for every mole of $\text{H}_2$ formed. That is the bond enthalpy of hydrogen. To break a mole of $\text{H}_2$ molecules into atoms, you must supply the same $435.8\ \text{kJ}$.

Orbital overlap

When the atoms approach, their atomic orbitals start to occupy the same space. This partial merging is called overlap. The shared electron pair lives in the overlap region, between the nuclei, attracted by both.

• Each orbital brings one electron, and the two electrons must have opposite spins.
• The greater the overlap, the stronger the bond.

So a covalent bond forms by the overlap of two half-filled orbitals. Filled orbitals cannot overlap to form a bond, as they have no room for another electron.

Overlap can be positive, negative or zero. When lobes of the same sign meet, the overlap is positive and a bond forms. When lobes of opposite sign meet, or orbitals meet at right angles so that their effects cancel, no bond forms.

Sigma bonds

When two orbitals meet head on, along the line joining the nuclei, the bond is a sigma bond, written $\sigma$. Three kinds of overlap give one:

• s with s, as in $\text{H}_2$.
• s with p, as in $\text{HF}$: the $1s$ orbital of hydrogen meets a $2p$ orbital of fluorine end on.
• p with p, end to end, as in $\text{F}_2$.

The electron cloud of a sigma bond lies along the axis between the nuclei, and is symmetrical around it.

Pi bonds

Two p orbitals that stand side by side, parallel to each other, can also overlap. They meet above and below the axis, not on it. This gives a pi bond, written $\pi$. Its electron cloud has two parts, one on each side of the axis.

A pi bond never forms on its own. The atoms must first be held together by a sigma bond. So:

• a single bond is $1\sigma$;
• a double bond is $1\sigma + 1\pi$;
• a triple bond is $1\sigma + 2\pi$.

Which is stronger?

Head-on overlap is large, because the orbitals point straight at each other. Sideways overlap is smaller. So a sigma bond is stronger than a pi bond between the same two atoms.

That is also why a double bond is stronger than a single bond, but less than twice as strong. Its second part is a weaker pi bond.

Counting sigma and pi bonds

Write out every bond, then count.

• $\text{N}_2$, $\text{N} \equiv \text{N}$: $1\sigma + 2\pi$.
• $\text{CO}_2$, $\text{O}{=}\text{C}{=}\text{O}$: two double bonds, so $2\sigma + 2\pi$.
• $\text{C}_2\text{H}_4$: four C-H single bonds and one C=C, so $5\sigma + 1\pi$.
• $\text{C}_2\text{H}_2$: two C-H single bonds and one triple bond, so $3\sigma + 2\pi$.

Practise with new numbers

Try it in Python

The program works out the energy of two hydrogen atoms at many distances, finds the lowest point, and prints a few values.

Common mistakes

• Thinking the energy keeps falling as the atoms approach. Below the bond length the nuclei repel, and the energy rises.
• Making a pi bond from s orbitals. An s orbital has no sideways lobes. Pi bonds come from parallel p orbitals.
• Counting a double bond as two sigma bonds. It is one sigma and one pi.
• Calling a pi bond stronger. Sideways overlap is smaller, so the pi bond is weaker.
• Bonding with a filled orbital. Each orbital that overlaps must bring one unpaired electron.

Remember these

• A covalent bond forms when two half-filled orbitals overlap, with the electron spins opposite.
• $\text{H}_2$: bond length $74\ \text{pm}$, bond enthalpy $435.8\ \text{kJ mol}^{-1}$.
• Head-on overlap gives a sigma bond: s with s, s with p, p with p end to end.
• Sideways overlap of p orbitals gives a pi bond, above and below the axis.
• Single $1\sigma$, double $1\sigma + 1\pi$, triple $1\sigma + 2\pi$. Sigma is stronger.

Check yourself

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

1. Two hydrogen atoms are pushed closer than $74\ \text{pm}$. What happens to the energy, and why?

2. Which orbitals overlap to form the bond in hydrogen chloride, and what kind of bond is it?

3. How many sigma and how many pi bonds hold the two atoms of a nitrogen molecule together?

4. Can two atoms be joined by a pi bond alone, with no sigma bond?

5. Count the sigma and pi bonds in carbon dioxide.