Molecular Orbital Theory & MO Diagrams | Inorganic Chemistry PDF Download

Molecular orbital Diagrams: Homonuclear Diatomic Molecules up to N₂ (i.e. having Z ≤ 14)

Homonuclear diatomic molecule with Z > 14 (O₂ onwards)

If the energy differences between 2s and 2p atomic orbitals are small, there is a possibility of mixing of molecular orbitals σ(2s) with σ(2pz) as they are of the same symmetry. Similarly, σ*(2s) with σ*(2p_z)  can also mix resulting that σ(2s) and σ*(2s) MOs do not retain the pure s character and the σ(2p_z) and σ*(2p_z) MOs do not retain the pure p character. In fact, all the four MOs acquire a mixed sp character. Due to this s-p mixing, the energies of all the four orbitals change in such a way that the resulting MOs σ(2s) and σ*(2s), which also contain some p character, become more stable and are thus lowered in energy whereas the MOs σ(2p_z) and σ*(2p_z) which contain some s character also now become less stable and are thus raised in energy.

Electron Configurations in MO: (Filling of bonding orbitals in order of increasing energy)

• Molecular electron configurations can be written similar to atomic electron configurations.
• Each molecular orbital can hold 2 electrons with opposite spins.
• In these molecular orbitals electrons are filed according toAufbau principle.
• Energy Level Order

From H₂ to N₂: σls < σ * ls < σ2s < σ * 2s < π2p_y = π2p_z < σ2p_x < π * 2p_y = π * 2p_z < σ * 2p_x For O₂ & F₂: sls < σ * ls < σ2s < σ * 2s < π2p_x < π2p_y = π2p_z < σ * 2py < π * 2py = π * 2pz = σ * 2p x

• Bond order = 

N_b = NO. of electron in bonding Mo’s; N_a = No of electron in antibonding Mo’s.

• Bond order (BO) is a measure of net bonding interactions.
• BO must be greater than 0 for a stable molecule to form. If bond order = 0, it means species does not exist.
•  Higher is the BO, stronger is the bond.

 Summary of Electronic Configurations of Molecular Orbitals of 1^st and 2^nd row diatomic Molecules

Determination of Magnetism Behaviour: If any molecule has unpaired electron in molecular orbital diagram, then it is paramagnetic. In case of paired electrons the molecule is diamagnetic.

 Table. Molecular Orbital Summary of Second Row Diatomic Molecules

Illustration: Determine the electron configuration and bond order for the following. if it exists, discuss its magnetism.
 Solution:

Illustration: Determine the electron configuration and bond order for the following. if it exists, discuss its magnetism.
 Solution:

Molecular Orbital Treatment for Hetronuclear Diatomic Molecules

• The difference in energy between orbitals on different atoms leads to reduced mixing – The reduced mixing does not imply weaker bonding
• When two atomic orbitals with different energies overlap, the lower molecular orbital is primarily composed of the lower atomic orbital, and vice versa. Moreover, the shift in energies of the two levels is less than if they had the same energy in the atoms.

NO Molecule: 

It has 13 electrons therefore its molecular orbital diagram matches to that with Z > 14.

Bond order in NO =  
NO⁺: Removal of an electron from NO gives NO⁺, nitrosonium ion. The electron is removed from π* antibonding MO to given NO⁺, which resembles the electronic configuration of CO.
Bond order in NO⁺ =

Bond order is electronic with N² and there is shortening of bond length to 10⁶pm

NO^–: In contrast addition of an electron to π* orbital causes a decrease in bond order and an increase in bond length.
Bond order in NO^– =   ((8-4)/2) = 2

Out of NO, NO⁺ and NO^–, NO^– has the largest bond and is also paramagnetic.

Note: CN Molecule:

It has 13 electrons thus its molecular orbital diagram is similar to that of Z < 14.
CO Molecule (Coulson Method): The electronic configurations of C and O atoms are 1s²2s²p² and 1s²2s²2p⁴, respectively. There are 4 electrons in the outer shell of carbon and 6 electrons in the outer shell of oxygen. Thus, a total of 10 outer electrons are to be accommodated in the molecular orbitals of CO molecule. Because of higher electronegativity of oxygen, its atomic orbitals would be of lower energy than the corresponding atomic orbitals of carbon. Due to this energy difference, the bonding molecular orbitals receive more contribution from atomic orbitals of lower energy, i.e., the atomic orbitals of oxygen and antibonding molecular orbitals would have more characteristics of atomic orbitals of carbon.
The 2s and 2p_z atomic orbitals of C atom mix to produce two atomic orbitals of mixed spz character. The higher
energy mixed sp_z orbital of C atom is closer in energy to its pure 2p_z orbital and has major contribution from it whereas the lower energy mixed sp_z orbital of C is closer in energy to its pure 2s orbital and has major contribution from it. Similarly, 2s and 2p_z atomic orbitals of oxygen atom also mix with each other (though not to the same extent to which the orbitals of C atom mix because the energy difference between 2s and 2p_z atomic orbitals of oxygen atom is higher than that between the similar atomic orbitals of C atom) to form two atomic orbitals of mixed sp_z character. The higher energy mixed spz orbital of O is very close in energy to its pure 2p_z orbital and has contribution mostly from 2p_z orbital whereas the lower energy mixed spz orbital, being very close in energy to pure 2s orbital of oxygen atom, has contribution mostly from 2s orbital of oxygen atom.

Since the orbital energies are lower for more electronegative O atom compared to the energies of similar orbitals of C atom, the lower energy mixed sp_z orbital of C atom is comparable in energy to the higher energy sp_z orbital of O atom, as shown in MOEL diagram of CO. These two atomic orbitals of comparable energy and of same symmetry combine to give s bonding and antibonding molecular orbitals. The lower energy sp_z, orbital on O and the higher energy sp₂ orbital on C atom are so far apart in energy that they hardly influence each other with the result that their energies remain almost unchanged.
In other words, these orbitals remain almost non bonding, the non bonding sp_z orbital on C(σ_3nb in the figure) will have more characteristics of 2p_z orbital of C atom and the non bonding sp_z orbital on O atom C(σ_3nb in the figure) will have more characteristics of 2s orbital of O atom. The degenerate set of 2p_x2p_y orbitals of C and the similar degenerate set of 2p_x2p_y orbitals of oxygen are of p symmetry to form a set of degenerate π₁,π₂ bonding molecular orbitals and another set of degenerate antibonding molecular orbitals. The electronic configuration of CO molecule is, therefore,

and the bond order of CO molecule is obviously 3 (due to two σ bonding and four p bonding) electrons.
The electrons in the non bonding orbitals do not contribute towards bonding.
Since σ_3nb is the highest placed occupied non bonding orbital localised on C atom, its electron pair can easily be donated for forming dative (or coordinate) bond.
NOTE: Bond order of CO molecule

Bond order of CO⁺ ion 

Bond order remains same as that of CO as electron is withdrawn from non bonding orbital rather than from anti bonding orbital.
Hydrogen Flouride Molecule, HF

The two 2s and 2p_z of F, being of the same symmetry, can mix with each other. However, the extent of mixing of 2s and 2p_z orbitals will be small because the energy gap between in these two orbitals in high.
The 1s orbital of hydrogen atom is more comparable in energy to the higher energy mixed sp_z orbital (which is practically 2p_z orbital) of F. The two combine to give σ₁ bonding and σ₃* antibonding molecular orbitals. The lower energy mixed sp_z orbital of F cannot effectively combine with 1s orbital of H atom due to large energy difference between this mixed orbital and 1s orbital of H atom and, therefore, remain almost non bonding (σ_2nb) orbital. The molecular orbital energy level diagram of HF is shown in following figure.

The electronic configuration of HF molecule is thus,

and the bond order is obviously 1 since only two electrons occupy the bonding molecular orbital σ₁.
 

Term Symbols for Diatomic Molecules

Term symbols are used to designate electronic energy levels which arise out of an electronic configuration because of electrostatic repulsions amongst the electrons of a molecule. These terms are very helpful in understanding electronic spectra of molecules.

The term symbols employed for diatomic molecules are written in the following manner:

(i) The symbols ∑ , π and Δ are used to represent values 0, 1 and 2 respectively for |M_L| or ∑ m_L is 0,  1 and 2 for an electron in σ, π and δ molecular orbitals, respectively.

(ii) Since all homonuclear diatomic molecules have a centre of symmetry in the middle of their nuclear axis, the overall symmetry to inversion is indicated by u or g and is written as a right subscript of ∑ , π,  Δ.
(iii) The electron spin multiplicity in indicated by 2S + 1 as the left superscript of ∑ , π and Δ (S is the total spin of electrons in the system).
iv) The term symbol is written as
For example, consider a H₂ molecule whose ground state electronic configuration is σ²(1s).
In this case, S(i.e. s₁ + s₂) is 0 and |M_L|i.e.   is also 0. The σ_1s molecular orbital is symmetric with respect to the centre of inversion located in the middle of the nuclear axis, Therefore the term symbol for  should be g × g = g type. Hence, the term symbol    for H₂ molecule in its ground state, is ¹∑_g
Consider now hydrogen ion, H⁺₂ its ground state electronic configuration is  so that S = 1/2 and 2S + 1 = 2. Since |M_L| = 0, hence the term symbol for its ground state is ²∑_g .
Let us now work out the term symbols for O₂ molecule. The ground state electronic configuration of O₂ is
 . The electrons in the filled sigma molecular orbitals and the completely filled degenerate set of pi orbitals contribute nothing to the overall component of angular momentum ML along the nuclear axis. Since both the   molecular orbitals are symmetric with respect to the centre of symmetry of the oxygen molecule, the term symbol will be g × g = g type.
Let us first consider the case when both the unpaired electrons in  molecular orbitals have opposite spins. The ml for each electron in  molecular orbital will be +1 or -1 so that |M_L| = 0 or 2 and 2s + 1 = 1 (because   ). The term symbols which arise from this situation are ¹∑_g and ¹Δ_g .
Now consider the case when the spins of both the electrons are in the same direction so that 1 1
  and the spin multiplicity is 3. Since spins of both the electrons in π* orbitals are in the same direction, their ml values have to be different (Paulis exclusion principle). That is, m_l for one electron is +1 and for the other electron it is -1 (The + and - signs indicate whether the component of that the term is ³∑_g .
Out of ³∑_g , ³∑_g and ¹Δ_g the first term, i.e., ³∑_g , having maximum multiplicity will evidently represent the ground state term of the molecule.
Terms for Li₂, Be₂, N₂, F₂, F₂₊ molecules try yourself.
The term symbols for heteronuclear diatomic molecules are derived in a similar manner. Since a heteronuclear diatomic molecule has no centre of symmetry, we do not write the subscript g or u in the term symbols of such molecules.
The ∑ term are sometimes further codified by writting + or - as a right superscript depending upon whether the system is symmetric or asymmetric with respect to a plane of symmetry passing through the nuclear axis.