Molecular Orbital Theory: The molecular orbital theory claims that each atom tends to mix with others to form molecular orbitals. Electrons are found in distinct atomic orbitals as a result of this arrangement, and they are frequently connected with different nuclei. In a nutshell, one electron in a molecule can be found anywhere in the molecule.

One of the most significant consequences of the molecular orbital theory’s development was the opening of a new avenue for understanding the bonding process. According to this idea, molecule orbitals are essentially linear combinations of atomic orbitals. Read this article for more information on molecular orbital theory postulates, molecular orbital theory electronic configuration, and diagram of o2.

Molecular Orbital Theory

An approach, known as Molecular Orbital Theory, was established primarily by Hund and Mulliken in \(1932\) to explain the features of molecules such as their relative bond strengths, paramagnetic and diamagnetic properties, etc.

Molecular Orbital Theory Postulates

Here we have provided the molecular orbital theory postulates:

1. Just as electrons of an atom are present in various atomic orbitals, electrons of a molecule are present in multiple molecular orbitals.
2. Molecular orbitals form by the combination of atomic orbitals of comparable energies and proper symmetry.
3. Only those atomic orbitals can combine to form a molecular orbital, which has comparable energies and proper orientation. For example, \({\rm{1}}\,{\rm{s}}\) can combine with \({\rm{1}}\,{\rm{s}}\) and not with \({\rm{2}}\,{\rm{s}}.\) Similarly, \({\text{s}}\) can combine (overlap) with \({\text{Pz}}\) but not with \({\text{Px}}\) or \({\text{Py}}\), or \({\text{Pz}}\) can combine with \({\text{Pz}}\) but not with \({\text{Px}}\) or \({\text{Py}}\) (taking the \({\text{Z}}\)-axis as the internuclear axis).
4. The number of atomic orbitals formed is equal to the number of combining molecular orbitals. When two atomic orbitals combine, they form two new orbitals called ‘bonding molecular orbital’ and ‘antibonding molecular orbital.’ 
5. The bonding molecular orbital has lower energy and hence greater stability than the corresponding antibonding molecular orbital.
6. In terms of the probability distribution, a molecular orbital gives the electron probability distribution around a group of nuclei just as an atomic orbital gives the electron probability distribution around the single nucleus.
7. The molecular orbitals, like atomic orbitals, are filled by the Aufbau principle obeying Pauli’s exclusion principle and Hund’s rule of maximum multiplicity.
8. Just as the letters \(\sigma ,\;\pi \) and \(\delta \) are used to denote the atomic orbital, the Greek letters , and used to indicate the molecular orbitals.

Formation of Molecular Orbitals

These are formed by the combination of atomic orbitals of the bonded atoms. We have learnt that according to wave mechanics, atomic orbitals can be expressed by wave functions. The wave functions represent the amplitude of the electron waves. These are obtained from the solutions of Schrodinger a wave equation.

Similarly, we can write the Schrodinger wave equation for the whole molecule. However, it isn’t easy to solve the wave equation for molecules. Scientists have used an approximate method known as the linear combination of atomic orbitals abbreviated as the \({\text{LCAO}}\) method to overcome this problem.

Consider the \({\text{LCAO}}\) method can apply to the diatomic homonuclear hydrogen molecule. In a hydrogen molecule, the two hydrogen atoms label as \({\text{A}}\) and \({\text{B}}\). So, each hydrogen atom has one electron in \({\rm{1}}\,{\rm{s}}\) orbital. Each hydrogen atom has one electron in \({\rm{1}}\,{\rm{s}}\) orbital. These atomic orbitals may represent by the wave functions \({\Psi _{\text{A}}}\) and \({\Psi _{\text{B}}}\)

Mathematically, the function of molecular orbitals is described by the linear combination of atomic orbitals that occurs by adding and subtracting wave functions of individual atomic orbitals.

Therefore, two molecular orbitals \(\sigma \) and \({\sigma ^*}\) are formed as shown below.

\({\rm{\sigma }}\,{\rm{ = }}\,{{\rm{\Psi }}_{\rm{A}}}{\rm{ + }}{{\rm{\Psi }}_{\rm{B}}}\)

\({\sigma ^*} = {\Psi _{\rm{A}}} – {\Psi _{\rm{B}}}\)

The molecular orbital \(\sigma \) formed by the addition of atomic orbitals is called the bonding molecular orbital and \({\sigma ^*}\) by subtraction of atomic orbitals is called an antibonding molecular orbital.

Qualitatively, we can understand the formation of molecular orbitals in terms of the destructive or constructive interference of the electron waves of the combining atoms.

 In the formation of bonding molecular orbital, the two electrons waves of the bonding atoms reinforce each other due to constructive interference. On the other hand, while forming an antibonding molecular orbital, the electron waves cancel each other due to destructive interference.

The antibonding orbital energy is raised above the parent atomic orbitals energy that has combined. As a result, the energy of the bonding orbital is lower than the parent orbitals. However, the two molecular orbitals’ total energy remains the same as that of the two original atomic orbitals.

The main differences between atomic orbital and Molecular Orbital are given below. The order of energies of the orbitals is

Bonding < Non- bonding < Anti- bonding

Atomic orbitals	Molecular orbitals
Their electron cloud extends around the nucleus of a single atom, i.e., the atomic orbital is monocentric.	Their electron cloud extends around all the nuclei of bonded atoms in the molecule, i.e., a molecular orbital is polycentric. They are obtained by combining atomic orbitals of comparable energy.
They are less stable.	They are more stable.
They have simple shapes.	They have complex shapes.
They are represented by \({\text{s, p, d, f}}\), etc.	They represented by, \(\sigma ,\,{\sigma ^*},\,\pi ,\,{\pi ^*}\), etc. It may be bonding or antibonding.

Conditions for the Combinations

1. The combining atomic orbitals should have comparable energies.
2. The combining atomic orbitals must have the exact symmetry about the molecular axis. \({\text{Z}}\)-axis is taken as a molecular axis by convention.
3. The extent of overlapping should be large. The greater the overlap, the greater will be the electron density between the nuclei.

Types of Molecular Orbitals

Molecular orbitals are of two types– bonding and antibonding.

The two types of bonds are \(\sigma – \) bond and \(\pi -\) bond.

The \({\text{s}}\)-orbitals of one atom can overlap with the \({\text{s, p, d, f}}\), orbital of another atom such that the overlapped region is symmetrical about the internuclear axis. Similar symmetrical overlaps are also possible among \({\rm{p,d}}\) and \({\rm{f}}\) orbitals. The bond formed with cylindrical symmetry is known as \(\sigma \) -bond. \(\sigma \) -bonds due to \({\text{s-s, s-p}}\) and \({\text{p-p}}\) axial overlaps.

When atomic orbital overlap in a sidewise fashion, the molecular orbitals formed have a nodal plane and lobes of orbitals are on either side of the intermolecular axis. The bond formed with no cylindrical symmetry is known as \(\pi -\) bonds.

The sequence of energy levels of molecular orbitals for \({\text{L}}{{\text{i}}_{\text{2}}}{\text{,}}\,{\text{B}}{{\text{e}}_{\text{2}}}{\text{,}}\,{{\text{B}}_{\text{2}}}{\text{,}}\,{{\text{C}}_{\text{2}}}\) and \({{\text{N}}_2}\) is \(\sigma 1\,{\rm{s}} < {\sigma ^*}1\,{\rm{s}} < \sigma 2\,{\rm{s}} < {\sigma ^*}2\,{\rm{s}} < \left( {\pi 2\,{{\rm{P}}_{\rm{x}}} = \pi 2\,{{\rm{P}}_{\rm{y}}}} \right) < \sigma 2\,{{\rm{P}}_z} < \left( {{\pi ^*}2\,{{\rm{P}}_{\rm{x}}} = {\pi ^*}2\,{{\rm{P}}_{\rm{y}}}} \right) < {\sigma ^*}2\,{{\rm{P}}_{\rm{z}}}\)

The molecular orbital energy level diagram for lithium is shown below.

The sequence of increasing order of energy of various molecular orbitals for \({{\text{O}}_{\text{2}}}{\text{,}}\,{{\text{F}}_{\text{2}}}\) is \(\sigma 1\,{\rm{s}} < {\sigma ^*}1\,{\rm{s}} < \sigma 2\,{\rm{s}} < {\sigma ^*}2\,{\rm{s}} < \sigma 2\,{{\rm{P}}_{\rm{z}}} < \left( {\pi 2\,{{\rm{P}}_{\rm{x}}} = \pi 2\,{{\rm{P}}_{\rm{y}}}} \right) < \left( {{\pi ^*}2\,{{\rm{P}}_{\rm{x}}} = {\pi ^*}2\,{{\rm{P}}_{\rm{y}}}} \right) < {\sigma ^*}2\,{{\rm{P}}_{\rm{z}}}\)

Molecular Orbital Diagram of O2

The molecular orbital energy level diagram for dioxygen molecule  is shown below:

Differences between \(\sigma \) and \(\pi \) Molecular Orbitals:

\(sigma\) Molecular Orbitals	\(\pi \) Molecular Orbitals
The overlap of atomic orbitals forms it.	It is formed by the sideways overlap, perpendicular to the internuclear axis.
Due to head-on overlap, the overlap is maximum.	Due to sideway overlap, overlapping is less.
It consists of one electron cloud.	It consists of two-electron clouds, one lying above and the other lying below a plane passing through the nuclei.
Its electron cloud is symmetrical about the internuclear axis.	Its electron cloud is not symmetrical about the internuclear axis.

Practice Exam Questions

Molecular Orbital Theory Electronic Configuration

Check the Molecular Orbital Theory Electronic Configuration in detail below:

Stability of Molecules 

Let \({{\text{N}}_{\text{b}}}\) be the number of electrons occupying the bonding orbitals and \({{\text{N}}_{\text{a}}}\) be the number of electrons occupying the antibonding orbitals.

1. If \({{\text{N}}_{\text{b}}}\) is greater than \({{\text{N}}_{\text{a}}}\) is stable.
2.  If \({{\text{N}}_{\text{b}}}\) is less than \({{\text{N}}_{\text{a}}}\) or \({{\text{N}}_{\text{b}}} = {{\text{N}}_{\text{a}}}\), The molecule is unstable.

Bond Order

The bond order defines one half of the difference between the number of electrons present in bonding and antibonding orbitals.

 Bond order \( = \frac{1}{2}\left( {{{\text{N}}_{\text{b}}} – {{\text{N}}_{\text{a}}}} \right)\)

A positive bond order means a stable molecule, while a negative bond order means an unstable molecule. The integral values of bond order \(1, 2\), or \(3\) correspond to single, double or triple bonds, respectively.

Bond Length

The bond order between two atoms is taken as an approximate measure of the bond length. As the bond order increases, the bond length decreases.

Magnetic Behaviour

If all the molecular orbitals in a molecule occupy two electrons each, the substance is diamagnetic. However, if electrons singly occupy one or more molecular orbitals, it is said to be paramagnetic.

The more unpaired electrons present in the molecule/ion, the greater its paramagnetic nature. It is because it expresses the magnetic moment of the substance according to the expression Magnetic moment \( = \sqrt {{\text{n}}({\text{n}} + 2)} \) \({\text{B}}{\text{.M}}{\text{.}}\). (\({\text{B}}{\text{.M}}{\text{.}}\) = Bohr Magneton) where \({\text{n}}\) is the number of unpaired electrons present in the molecule or the ion.

Bonding in Some Homonuclear Diatomic Molecules

As already explained, the filling of molecular orbitals is governed by the following principles (a) Aufbau principle. (b) Pauli exclusion principle, and (c) Hund’s rule of maximum multiplicity. Applying these rules to write the electronic configuration and the limitations discussed above to predict their characteristics, we shall now discuss examples of homonuclear diatomic molecules and the molecular ions of some elements.

Hydrogen Molecular Positive ion \(\left({{\text{H}}_2^ + } \right)\): This is the simplest molecular species containing one electron only. Its existence had been detected spectroscopically when an electric discharge passed through a discharge tube containing hydrogen gas at low pressure. 

The only electron presents in \({\text{H}}_2^ + \) ion enter the \(\sigma \left({{\text{Is}}} \right)\) molecular orbital, which has the lowest energy. Thus, the electronic configuration of \({\text{H}}_2^ + \) is \(\sigma {\left( {{\text{Is}}} \right)^1}\).

In terms of energy level diagram, it represents as

Here, \({{\text{N}}_{\text{b}}} = 1,\,{{\text{N}}_{\text{a}}} = 0\) so that

 Bond order \( = \frac{1}{2}\left( {{{\text{N}}_{\text{b}}} – {{\text{N}}_{\text{a}}}} \right) = \frac{1}{2}(1 – 0) = \frac{1}{2}\)

Thus, two results follow

(a) Stability: The bond order is positive. Hence \({\text{H}}_2^ + \) ion should be stable. However, the low value indicates that the stability is not very high. 

(b) Paramagnetic character: One unpaired electron is present. Hence, \({\text{H}}_2^ + \) ions should be paramagnetic. 

Both the above results are correct experimentally.

Beryllium Molecule \(\left( {{\text{B}}{{\text{e}}_2}} \right)\) – Electronic configuration of beryllium is \({\rm{I}}{{\rm{s}}^2}2\,{{\rm{s}}^2}.\) The total number of electrons in \({\text{B}}{{\text{e}}_2}\) should be \(8\), two from each beryllium atom. Keeping in view the Aufbau principle and Pauli exclusion principle, the electronic configuration of \({\text{B}}{{\text{e}}_2}\) will be \(\sigma 1\,{{\rm{s}}^2} < {\sigma ^*}1\,{{\rm{s}}^2} < \sigma 2\,{{\rm{s}}^2} < {\sigma ^*}2\,{{\rm{s}}^2}\) or in terms of energy level diagram, it is represented as below.

Here, \({{\text{N}}_{\text{b}}} = 4,{{\text{N}}_{\text{a}}} = 4\) So that

Bond order \( = \frac{1}{2}\left({{{\text{N}}_{\text{b}}} – {{\text{N}}_{\text{a}}}} \right) = \frac{1}{2}(4 – 4) = 0\)

i.e., no bond is formed between two beryllium atoms or in other words, in \({\text{B}}{{\text{e}}_2}\) does not exist.

Summary

We have learned how the sigma and pi bonds are formed. How to calculate bond length, stability of molecules, bond order using molecular orbital theory, and bond in homonuclear diatomic molecules. The Molecular orbital theory is also used to predict the reactivity and properties of UV spectra.

FAQs on Molecular Orbital Theory

Q.1: What are the main points of molecular orbital theory?
Ans: Molecular orbitals are formed by the combination of atomic orbitals of the bonded atoms. Hence, the arrangement of electrons is found in various atomic orbitals and they are usually associated with different nuclei.

Q.2: How do you use the molecular orbital theory?
Ans: The molecular orbital theory characterizes the electronic structure of molecules using quantum mechanics. It is used to calculate bond order, bond length, bond strength, paramagnetic and diamagnetic character.

Q.3: What does the molecular orbital theory explain?
Ans: Molecular orbital \(\left( {{\rm{MO}}} \right)\) theory uses a linear combination of atomic orbitals \(\left( {{\rm{LCAO}}} \right)\) to represent molecular orbitals resulting from bonds between atoms. Chemical substances will form bonding interactions if their orbitals become lower in energy when they interact with each other.

Q.4: Explain the molecular orbital diagram in \(\left({{\text{H}}_2^ + } \right)\) molecules?
Ans: Hydrogen molecular positive ion \(\left({{\text{H}}_2^ + } \right)\): This is the simplest molecular species containing one electron only. Its existence has been detected spectroscopically when an electric discharge is passed through a discharge tube containing hydrogen gas at low pressure. 

The only electron presents in \({\text{H}}_2^ + \) ion enter the \({\text{(Is)}}\) molecular orbital, which has the lowest energy. Thus, the electronic configuration of \({\text{H}}_2^ + \) is \({\left( {{\rm{1}}\,{\rm{s}}} \right)^1}.\)

In terms of the energy level diagram, it represents as

Here, \({{\text{N}}_{\text{b}}} = 1,\,{{\text{N}}_{\text{a}}} = 0\) so that

 Bond order \( = \frac{1}{2}\left( {{{\text{N}}_{\text{b}}} – {{\text{N}}_{\text{a}}}} \right) = \frac{1}{2}(1 – 0) = \frac{1}{2}\)

Thus, two results follow

(a) Stability: The bond order is positive. Hence \({\text{H}}_2^ + \) ion should be stable. However, the low value indicates that the stability is not very high. 

(b) Paramagnetic character: One unpaired electron is present. Hence, \({\text{H}}_2^ + \) ions should be paramagnetic. 

Both the above results are correct experimentally.

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