# Molecular orbital theory

### Molecular orbital theory

#### Lessons

In this lesson, we will learn:

• To understand the forming of molecular orbitals using the LCAO method.
• To understand the bonding and antibonding nature of molecular orbitals.
• To apply MO theory when explaining the existence and nonexistence of chemical substances.
• To use molecular orbital diagrams and bond order to explain the type of bonding observed in molecules.

Notes:

• We now know how electrons are held in atomic orbitals of different energy levels and shape. In the same way that atoms combine to make molecules, atomic orbitals (AOs) combine to form molecular orbitals (MOs).
This is called the linear combination of atomic orbitals (LCAO) and when applied, it predicts the stability of molecules that we know exist, and the instability of molecules that we don't know exist.
• Before we go further with electrons in atomic orbitals (AOs) making MOs, remember that atomic orbitals – the electron 'houses' that show where electrons 'probably are' - are wave functions. They mathematically describe how likely it is an electron will be in a certain place at a certain time. These atomic orbitals can combine like waves can combine, either constructively (mathematically adding them together) and destructively (subtracting them and just cancelling each other out).
• Whenever two different atomic orbitals combine, two different molecular orbitals are made.
• One is made when atomic orbitals overlap (think mathematically + and +, wave functions combining, or waves in the same phase) and is called a bonding molecular orbital.
• One is when the atomic orbitals cancel out (think mathematically + and –, or waves in opposite phases) and is called an antibonding molecular orbital. Here the two wave functions have cancelled each other out, and a node is created.
Just like with atomic orbitals, molecular orbitals can be drawn using an energy level diagram and in terms of energy, these MOs are positive and negative versions of each other – the energy level diagram should look symmetrical.
Drawing MOs when 1s orbitals combine looks like this:
The two MOs created from AOs will be of different energy because of the effects of where the electrons will be 'spending most of their time' in the molecule:
• In the bonding molecular orbital, the constructive overlap means in this MO any electrons will most likely be found between the two nuclei of the atoms involved.
Between the two nuclei, any electrons have more nuclear charge to be attracted to than in one individual atom with just one of those nuclei. Also, with both atoms providing electrons in forming the MO, there will be more electrons for the nuclei to be attracted to as well. This is what a chemical bond is.
In short, using the wave analogy, two in-phase waves combine to create a larger sum than as individual waves. Therefore this MO is of lower energy than the individual AOs that combine to make it.
• In the antibonding molecular orbital, the destructive overlap (cancelling out of the wave functions) means there is zero probability that any electrons occupying this MO will be found between the two nuclei – it is a node. This leaves the two positive nuclei exposed to each other with no mutual negative charge to be attracted to; the nuclei will just repel one another in a destabilizing interaction. Therefore this MO is of higher energy than the individual AOs that combine to make it, where individually in the AO, no such repulsion occurs.
You can apply MO theory to real molecules to explain their stability – why they exist – and to 'imaginary' molecules to explain their instability – why they don't exist! We will do this for hydrogen and helium as examples when their electrons fill in molecular orbitals.
• WORKED EXAMPLE: Hydrogen, 1H
A hydrogen atom has only one electron occupying the 1s orbital. Using MO theory, we can show that a diatomic hydrogen molecule would be of lower energy than an individual atom of hydrogen:

If two hydrogen atoms interact, their combined two electrons (one each) fill up the bonding MO, which is lower energy (more stable) than their individual AO as a lone hydrogen atom.
Since hydrogen atoms only carry one electron each and orbitals can hold up to two electrons, the bonding MO is full and the antibonding MO is empty. The H-H interaction is a stabilizing, attractive interaction (a chemical bond) with no destabilizing interactions.
This is supported by empirical evidence; the H2 molecule is stable and is observed in nature. Individual H atoms on the other hand are unstable; they are not observed in nature.
• WORKED EXAMPLE: Helium, 2He
A helium atom has two electrons both occupying its 1s atomic orbital. If two atoms of helium were to try and form a chemical bond, then the following molecular orbitals would be made:

Since an orbital can only hold two electrons maximum, the four electrons from the two helium atoms fill up both the bonding MO and the antibonding MO.
This results in the bonding and antibonding MOs cancelling out completely – in an "He2" molecule, there is the same amount of bonding as there is antibonding.
In short, there is no 'chemical bond' here at all.
This is supported by the evidence. He2 is not observed in nature; He only exists as single elemental atoms.
• With higher energy (2s and above) orbitals, because now there are differently-shaped s and p orbitals, we name the types of bonds with MOs based on their symmetry. Despite the s orbitals "2s" and above having nodes, we can still draw MOs with them like we would the 'node-less' 1s orbitals.
• When s orbitals (and p orbitals, when head-on) combine, they make cylindrical MOs with symmetry around an axis. These are sigma molecular orbitals and when electrons fill MOs with this symmetry, we call it a sigma bond and give them the symbol σ.
This is true for antibonding MOs too – if it is symmetrical when rotating on an axis, it is a sigma MO! Antibonding sigma MOs are given the symbol σ*, the * to show antibonding nature.
• σ orbitals from s orbitals can be drawn like in the energy-level diagrams of hydrogen and helium above.
• σ molecular orbitals can be made from p orbitals too. See below:
• When other p atomic orbitals combine, they can form MOs with symmetry through a plane because the p AOs are planar too. MOs with planar symmetry are called pi molecular orbitals (given the symbol π) and when electrons fill pi MOs we call it a pi (π) bond, as we do with the antibonding pi orbitals, π*.
• These two pi MOs are orthogonal and of equal energy to each other.
• Because these orbitals are out at a plane perpendicular to the two nuclei and not in line with them, there is less interaction with the nuclei so π bonding orbitals are slightly higher energy than their counterpart σ bonding MOs.
• As bonding and antibonding MOs are symmetrical in energy, π antibonding orbitals are slightly lower energy than their counterpart σ antibonding MOs.
• Drawing pi molecular orbitals (especially antibonding orbitals) is quite hard to do accurately so they are normally left drawn as if they were still their lone atomic orbitals.
• With π MOs, drawing energy level diagrams has become more complicated, since π and σ orbitals have different energies.
Here are some guides for drawing energy level MO diagrams correctly with the O2 molecule as an example:
• Two MOs are made of two AOs coming together, so you need to draw the AOs of the two atoms making the MOs on either side of the diagram. The MOs form in the 'middle' which you can show on your diagram.
See the green "1" marks on the diagram below.
• Atomic orbitals of identical atoms will have identical energy – so draw them level with each other!
See the green "2" marks on the diagram below.
• For any MO, the bonding and antibonding forms are symmetrical in energy; compared to their AOs, the bonding MO will be as low as the antibonding MO is high. You should see SYMMETRY! (We will get to exceptions later…)
See the green "3" marks on the diagram below.
• Just like s atomic orbitals are lower in energy than p atomic orbitals, MOs made from s atomic orbitals will be lower in energy than MOs made from p atomic orbitals.
• Fill in electrons using the lowest energy MO first (this is the Aufbau principle). Fill in the electrons in the AOs too; this should help you avoid any mistakes in the number of electrons you put in the MOs.
• When filling incomplete π orbitals, place electrons in separate orbitals first, only pairing them up when they have to be – this is obeying the Pauli principle.
See the green "4" mark on the diagram below.
A completed molecular orbital diagram for the O2 molecule would look like this:
• Molecular orbital diagrams help to explain the 'number of bonds' that atoms in a molecule make to each other:
• Why is the bond in O2 a double bond?
• Why is the bond in N2 a triple bond?
We calculate the bond order to find this out.
$Bond\;order = \frac{(\#\;bonding\;electrons) - (\#\;antibonding\;electrons)}{2}$
Bond order is found by subtracting electrons in bonding MOs ('bonding electrons') by electrons in antibonding MOs ('antibonding electrons') and then dividing by two because you need two electrons make a covalent bond. If we apply this to O2 for example, you will see why using MO theory, we say O2 has a double bond. (Note: In bond order calculations, ignore the lower energy levels because they are much lower in energy and do not bond).

Counting from 2s and above, we can see that there are eight bonding electrons (two in the 2s σ bonding MO, two in the 2p σ bonding MO and four in the 2p π bonding MO) and only four antibonding electrons (two in the 2s σ* antibonding MO and two in the 2p π* antibonding MO). Putting these into the equation we get:
$Bond\;order = \frac{(\#\;bonding\;electrons) - (\#\;antibonding\;electrons)}{2} = \frac{8-4}{2} = 2$
A bond order of 2 is found – this is why we say the O-O bond in O2 is a double bond according to MO theory. There is a sigma bond, and a pi bond.
• Another reason to always look out for symmetry (equal bonding/antibonding MOs) in the MO diagrams is because it will help you predict the number of lone pairs in a molecule or on an atom.
When bonding MOs and antibonding MOs cancel out, you are left with non-bonding electrons – these are the lone pairs in a molecule.
• Look at the O2 MO diagram above. The full 2s σ and σ* MOs cancel four electrons, and two π electrons cancel with two π* electrons. This is eight electrons cancelling out, or four lone pairs over two oxygen atoms – two lone pairs for each atom!
• 2nd WORKED EXAMPLE: F2

F has 9 electrons, so an F2 molecule has 18 electrons in total.
Filling the molecular orbitals from the lowest energy level first gives us the setup above.
Note that the π bonding and antibonding MOs completely cancel each other out – always try and look for symmetry in these MO diagrams to see what is not cancelled out. What do you think the bond order will be?
We can calculate it using the formula – remember you don't need to add the 1s electrons to this.
We have eight bonding electrons (two in the 2s σ bonding MO, two in the 2p σ bonding MO, four in the two π bonding MOs) and six antibonding electrons (two in the 2s σ* antibonding MO, four in the 2p &pi* antibonding MO).
$Bond\;order = \frac{(\#\;bonding\;electrons)-(\#\;antibonding\;electrons)}{2} = \frac{8-6}{2} = 1$
The bond order in F2 is 1 – we say the F-F chemical bond is a single bond!
We can also show the number of lone pairs as the bonding/antibonding MOs cancel out for non-bonding electrons – the 2s σ and σ* MOs cancel for four electrons, as do all of the π and π* electrons, eight electrons in total. We are just left with a σ bonding MO with the 2p electrons and twelve electrons, or six lone pairs over two atoms – three each.
• Introduction
How do atomic orbitals become molecular orbitals?
a)
Linear combination of atomic orbitals (LCAO).

b)
Making molecular orbitals from atomic orbitals.

c)
Using MO theory to predict: Hydrogen

d)
Using MO theory to predict: Helium

e)
Sigma and pi MOs

f)
Drawing correct MO diagrams: example.

g)
Bond order and lone pairs.

h)
Worked example: F2