Molecular orbital theory (MOT) of chemical bonding

Molecular orbital theory, abbreviated as MOT, has a special place among various models of chemical bonding. It is by far the most productive and quantitative explanation for the formation of a new chemical bond. Considering its paramount importance, this article will discuss everything you need to know about the molecular orbital theory. So, continue reading!

What is MOT- Definition

F. Hund, R.S Mulliken and Huckle proposed molecular orbital theory (MOT) in the early 20th century. As per MOT, after chemical bonding, atomic orbitals combine to form molecular orbitals. In this way, the electrons present in a molecule are not confined to the nuclear overlapping region. Rather, the bonded electrons belong to all nuclear centers in the molecule as a whole. Conversely, all the valence electrons strongly influence the stability of a molecule.

The linear combination of atomic orbitals (LCAO) results in an equal number of molecular orbitals. For example, 2 atomic orbitals combine to form 2 molecular orbitals, i.e., a bonding molecular orbital and an antibonding molecular orbital. The valence electrons occupy positions in these molecular orbitals in an ascending energy order.

MOT versus VBT

The molecular orbital theory obsoletes the bonding phenomenon introduced by the valence bond theory (VBT).

VBT says valence electrons stay localized in a fixed region, and the atomic orbitals regain their individuality once a chemical bond is formed. Contrarily, MOT proposes the formation of new molecular orbitals by atomic orbital intermixing after a molecule is formed. Conversely, the bonded electrons are delocalized within the molecule as per MOT.

VBT and hybridization explain how chemical bonding occurs, while MOT explains what happens after a new chemical bond is formed.

Main postulates of MOT

• Molecular orbitals (MOs) are a new set of energy levels which is formed by the linear combination of atomic orbitals (LCAO) after chemical bonding.
• N number of atomic orbitals combine to form N molecular orbitals where N is an integer.
• The combining atomic orbitals must have the same symmetry and comparable energy.
• As per the quantum mechanical treatment, in-phase atomic orbitals combine to form a bonding molecular orbital.
• Out-of-phase atomic orbitals combine to form an anti-bonding molecular orbital.
• A bonding molecular orbital lies at a lower energy than its corresponding anti-bonding molecular orbital.
• Electrons occupy molecular orbitals following the Aufbau principle, Hund’s rule and Pauli Exclusion principle.
• As per the Aufbau principle, a lower energy molecular orbital is filled before electrons occupy higher energy molecular orbitals. In this way, a bonding MO is filled before an antibonding MO.
• According to Hund’s rule, electrons first singly occupy a molecular orbital, and then pairing occurs.
• According to the Pauli Exclusion principle, only a maximum of two electrons are placed in a molecular orbital and that too in opposite spins.

Quantum mechanical treatment of MOT

A homonuclear diatomic molecule such as H2 is formed by the chemical reaction between two identical H atoms. The two H-atoms can be differentiated as HA and HB.

Each consist of a single valence electron only. The wave function for the valence atomic orbital of HA can be represented as ΨA. Similarly, the wave function for HB can be represented as ΨB .

As per MOT, the 1s atomic orbitals of HA and HB combine to form two molecular orbitals, i.e., a bonding MO and an anti-bonding MO in H2.

In quantum chemistry, a bonding molecular orbital is formed by the constructive interference (additive effect) of two wave functions. This gives a new wave function, Ψbonding.

Where;

Ψbonding = ΨA (1s) + ΨB (1s)

Contrarily, the destructive interference (subtractive effect) of the respective wave functions results in forming an anti-bonding molecular orbital. It isrepresented by Ψantibonding.

Where;

Ψantibonding = ΨA (1s) – ΨB (1s)

Zero electron density regions called nodes are present in anti-bonding MOs. These are created where there is a complete cancellation of two wave functions.  The higher the energy of a MO, the more the number of nodes present in it.

The molecular orbital diagram for hydrogen (H2) is given below.

A molecular orbital diagram shows the relative energy levels of atomic orbitals and their corresponding molecular orbitals.

In the MO diagram for H2, the bonding MO is represented as σ, while the antibonding MO is represented as σ*. Out of the 2 valence electrons, 1 electron is first placed in σ; it is then paired by adding another valence electron in it. In this way, both valence electrons are placed in σ. As both the valence electrons get consumed, so there is no electron in σ* of H2.

The bond order for the hydrogen molecule can be determined by the formula given below:

Bond order = [Bonding electrons – Antibonding electrons]/2

Bond order (H2) =

$\frac{2-0}{2}&space;=&space;1$

A bond order of 1 denotes there is a single covalent bond between two H-atoms in H2.

MOT example problems

As another example, let’s see how MOT explains important properties such as the paramagnetism of oxygen.  O2 is not a magnetic substance, but it collects between the magnetic poles when you pour liquid oxygen over a strong magnet. The attraction of a substance to a magnetic field without being a magnet itself is known as paramagnetism.

VSEPR theory explains that O2 is a linear molecule made up of two identical O-atoms. VBT tells us that both O-atoms are sp2 hybridized in oxygen. In contrast, MOT explains that there is a double covalent bond between two O-atoms in O2. It exhibits a paramagnetic behavior as it has two unpaired electrons in its molecular orbitals.

Thus, MOT gives more comprehensive information about an O2 molecule than the previous two theories.

The MOT diagram for O2 is given below.

The electronic configuration of an O-atom is 1s2 2s2 2p4.

Two 1s atomic orbitals combine to form two MOs, σ1s and σ*1s. Similarly, two 2s atomic orbitals combine to form two MOs, σ2s and σ*2s. Two electrons are placed in each of the four molecular orbitals. The electrons present in these molecular orbitals have no significant effect on the chemical nature and properties of O2. Thus, most of the MO diagrams show only the molecular orbitals formed by combining valence shell atomic orbitals of oxygen, i.e., 2p orbitals.

Three 2p orbitals of each O-atom combine to form a total of six MOs comprising three bonding molecular orbitals (σ2pz, π2px and π2py) and three anti-bonding molecular orbitals (π*2px, π*2py and σ*2pz).

Now let us see in the next section how we have placed electrons in the above molecular orbitals following Aufbau, Hund’s and Pauli’s exclusion principle.

Molecular orbital filling order

Four 2p electrons from each O-atom make a total of 4 + 4 = 8 electrons available to be placed in the six molecular orbitals of oxygen discussed above.

As per the Aufbau principle, σ2pz lies at the lowest energy, so it is filled first with 2 electrons.

The next two electrons singly fill π2px and π2py (Hund’s rule), and then pairing occurs (Pauli exclusion principle).

The remaining two electrons singly occupy anti-bonding molecular orbitals, i.e., π*2px and π*2py. As all 8 electrons are already consumed, therefore, no pairing occurs in the anti-bonding MOs. 2 unpaired electrons in the molecular orbitals of an O2 molecule account for its paramagnetic character.

Bond order (O2) =

$\frac{10-6}{2}&space;=&space;\frac{4}{2}&space;=&space;2$

Now that we know how to draw the MO diagram for a homonuclear diatomic molecule let’s see how we can do so for a heteronuclear diatomic molecule such as HCl.

MO diagram for a heteronuclear diatomic molecule

HCl is the chemical formula for hydrogen chloride. It consists of two atoms from two different elements, i.e., an H-atom and a Cl-atom.

The electronic configuration of hydrogen (H) is 1s1.

The electronic configuration of chlorine (Cl) is 1s2 2s2 2p6 3s2 3p5.

We will only consider valence shell atomic orbitals for drawing this MO diagram that are actually involved in chemical bonding.

The 1s valence electron of H overlaps with the 3p electrons of Cl to form an H-Cl sigma bond.

After chemical bonding, the linear combination of the 1s atomic orbital of hydrogen with one of the three 3p orbitals of chlorine form two molecular orbitals. These include a bonding MO (σ3pz) and an anti-bonding MO (σ *3pz). Meanwhile, the 3px and 3py atomic orbitals of the Cl-atom maintain their parent energy level. This is known as the non-bonded state

This example illustrates that as per MOT, only those AOs combine to form MOs that have a uniform symmetry and a nearly equal energy content, i.e., 1s of H-atom and 3pz of Cl-atom in this case.

This is because the electron wave of the same symmetry orbitals ensures maximum overlap to facilitate chemical bonding.

One 1s electron of hydrogen and five 3p electrons of chlorine make a total of 6 electrons. 2 electrons are placed in σ3pz (these are the electrons involved in chemical bonding). 2 electrons stay un-bonded in each of 3px and 3py AOs. Consequently, there is no electron in the antibonding MO (σ *3pz) of HCl. Additionally, as there are no unpaired electrons in the molecular orbitals of HCl thus it is diamagnetic in nature.

Bond order (HCl) =

MOT diagram for a polyatomic molecule

The molecular orbital diagram for a polyatomic molecule such as water (H2O) is shown below.

Practice some questions to find out how well you know MOT.

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