`=>` Molecular orbital (MO) theory was developed by F. Hund and R.S. Mulliken in 1932. The salient features of this theory are :

(i) The electrons in a molecule are present in the various molecular orbitals as the electrons of atoms are present in the various atomic orbitals.

(ii) The atomic orbitals of comparable energies and proper symmetry combine to form molecular orbitals.

(iii) While an electron in an atomic orbital is influenced by one nucleus, in a molecular orbital it is influenced by two or more nuclei depending upon the number of atoms in the molecule. Thus, an atomic orbital is monocentric while a molecular orbital is polycentric.

(iv) The number of molecular orbital formed is equal to the number of combining atomic orbitals. When two atomic orbitals combine, two molecular orbitals are formed. One is known as bonding molecular orbital while the other is called anti-bonding molecular orbital.

(v) The bonding molecular orbital has lower energy and hence greater stability than the corresponding anti-bonding molecular orbital.

(vi) Just as the electron probability distribution around a nucleus in an atom is given by an atomic orbital, the electron probability distribution around a group of nuclei in a molecule is given by a molecular orbital.

(vii) The molecular orbitals like atomic orbitals are filled in accordance with the aufbau principle obeying the Pauli’s exclusion principle and the Hund’s rule.

`=>` According to wave mechanics, the atomic orbitals can be expressed by wave functions (`ψ ’s`) which represent the amplitude of the electron waves.

● These are obtained from the solution of Schrödinger wave equation.

● Since it cannot be solved for any system containing more than one electron, molecular orbitals which are one electron wave functions for molecules are difficult to obtain directly from the solution of Schrödinger wave equation.

● To overcome this problem, an approximate method known as linear combination of atomic orbitals (LCAO) has been adopted.

● Now, apply this method to the homonuclear diatomic hydrogen molecule.

● Consider the hydrogen molecule consisting of two atoms `A` and `B`.

● Each hydrogen atom in the ground state has one electron in `1s` orbital.

● The atomic orbitals of these atoms may be represented by the wave functions `ψ_A` and `ψ_B`.

● Mathematically, the formation of molecular orbitals may be described by the linear combination of atomic orbitals that can take place by addition and by subtraction of wave functions of individual atomic orbitals as shown below :

`ψ_(MO) = ψ_A pm ψ_B`

● Therefore, the two molecular orbitals `σ` and `σ^ast` are formed as :

`σ = ψ_A + ψ_B`
`σ^ast = ψ_A – ψ_B`

● The molecular orbital `σ` formed by the addition of atomic orbitals is called the `text(bonding molecular orbital)` while the molecular orbital `σ^ast` formed by the subtraction of atomic orbital is called `text(anti-bonding molecular orbital)` as depicted in Fig.

`=>` Qualitatively, the formation of molecular orbitals can be understood in terms of the constructive or destructive interference of the electron waves of the combining atoms.

● In the formation of bonding molecular orbital, the two electron waves of the bonding atoms reinforce each other due to constructive interference while in the formation of anti-bonding molecular orbital, the electron waves cancel each other due to destructive interference.

● As a result, the electron density in a bonding molecular orbital is located between the nuclei of the bonded atoms because of which the repulsion between the nuclei is very less while in case of an anti-bonding molecular orbital, most of the electron density is located away from the space between the nuclei.

● Infact, there is a nodal plane (on which the electron density is zero) between the nuclei and hence the repulsion between the nuclei is high.

● Electrons placed in a bonding molecular orbital tend to hold the nuclei together and stabilise a molecule.

● Therefore, a bonding molecular orbital always possesses lower energy than either of the atomic orbitals that have combined to form it.

● In contrast, the electrons placed in the anti-bonding molecular orbital destabilise the molecule.

● This is because the mutual repulsion of the electrons in this orbital is more than the attraction between the electrons and the nuclei, which causes a net increase in energy.

`color{red}("Note ")` (i) The energy of the antibonding orbital is raised above the energy of the parent atomic orbitals that have combined.

(ii) The energy of the bonding orbital has been lowered than the parent orbitals.

(iii) The total energy of two molecular orbitals, however, remains the same as that of two original atomic orbitals.

`=>` The linear combination of atomic orbitals to form molecular orbitals takes place only if the following conditions are satisfied :

(i) The combining atomic orbitals must have the same or nearly the same energy.

● This means that `1s` orbital can combine with another `1s` orbital but not with `2s` orbital because the energy of `2s` orbital is appreciably higher than that of `1s` orbital. This is not true if the atoms are very different.

(ii) The combining atomic orbitals must have the same symmetry about the molecular axis.

● By convention `z`-axis is taken as the molecular axis.

`color{red}("Note ")` Atomic orbitals having same or nearly the same energy will not combine if they do not have the same symmetry.

`color{red}("Example ")` `2p_z` orbital of one atom can combine with `2p_z` orbital of the other atom but not with the `2p_x` or `2p_y` orbitals because of their different symmetries.

(iii) The combining atomic orbitals must overlap to the maximum extent.

● Greater the extent of overlap, the greater will be the electron-density between the nuclei of a molecular orbital.

`=>` Molecular orbitals of diatomic molecules are designated as `σ` (sigma), `π` (pi), `δ` (delta), etc.

`=>`In this nomenclature, the sigma (`σ`) molecular orbitals are symmetrical around the bond-axis while pi (`π`) molecular orbitals are not symmetrical.

`color{red}("Example ")` (i) The linear combination of `1s` orbitals centered on two nuclei produces two molecular orbitals which are symmetrical around the bond-axis. Such molecular orbitals are of the `σ` type and are designated as `σ1s` and `σ^(ast)1s`.

(ii) If internuclear axis is taken to be in the `z`-direction, it can be seen that a linear combination of `2p_z`- orbitals of two atoms also produces two sigma molecular orbitals designated as `σ2p_z` and `σ^(ast)2p_z` .

`=>` Molecular orbitals obtained from `2p_x` and `2p_y` orbitals are not symmetrical around the bond axis because of the presence of positive lobes above and negative lobes below the molecular plane.

● Such molecular orbitals, are labelled as `π` and `π^(ast)` . A `π` bonding MO has larger electron density above and below the inter -nuclear axis. The `π^(ast)` anti-bonding MO has a node between the nuclei.

`=>` We have seen that `1s` atomic orbitals on two atoms form two molecular orbitals designated as `σ1s` and `σ^(ast)1s`.

● In the same manner, the 2s and 2p atomic orbitals (eight atomic orbitals on two atoms) give rise to the following eight molecular orbitals :

`tt ( ( text{Antibonding} , MOs , σ^(ast)2s , σ^(ast)2p_z , π^(ast)2p_x , π^(ast)2p_y ) , ( text{Bonding} , MOs , σ2s , σ2p_z , π2p_x , π2p_y ))`

`=>` The energy levels of these molecular orbitals have been determined experimentally from spectroscopic data for homonuclear diatomic molecules of second row elements of the periodic table.

● The increasing order of energies of various molecular orbitals for `O_2` and `F_2` is given below :

`σ1s < σ^(ast)1s < σ2s < σ^(ast)2s < σ2p_z < (π 2p_x = π 2p_y) < (π^(ast)2p_x= π^(ast)2p_y) < σ^(ast)2p_z`

`=>` But, this sequence of energy levels of molecular orbitals is not correct for the remaining molecules `Li_2`, `Be_2`, `B_2`, `C_2`, `N_2`.

`=>` For instance, it has been observed experimentally that for molecules such as `B_2`, `C_2`, `N_2` etc. the increasing order of energies of various molecular orbitals is

`σ1s < σ^(ast)1s < σ2s < σ^(ast)2s < (π 2p_x = π 2p_y) < σ2p_z < (π^(ast)2p_x= π^(ast)2p_y) < σ^(ast)2p_z`

`=>` The important characteristic feature of this order is that the energy of `σ2p_z` molecular orbital is higher than that of `π 2p_x` and `π 2p_y` molecular orbitals.

`=>` The distribution of electrons among various molecular orbitals is called the electronic configuration of the molecule.

`=>` From the electronic configuration of the molecule, it is possible to get important information about the molecule as discussed below :

`color{green}("Stability of Molecules ")` If `N_b` is the number of electrons occupying bonding orbitals and `N_a` the number occupying the anti-bonding orbitals, then

(i) the molecule is stable if `N_b` is greater than `N_a`, and

(ii) the molecule is unstable if `N_b` is less than `N_a`

● In (i) more bonding orbitals are occupied and so the bonding influence is stronger and a stable molecule results.

● In (ii) the anti-bonding influence is stronger and therefore the molecule is unstable.

`color{green}("Bond Order ")` Bond order (b.o.) is defined as one half the difference between the number of electrons present in the bonding and the antibonding orbitals i.e.,

Bond order (b.o.) `= ½ (N_b–N_a)`

● The rules discussed above regarding the stability of the molecule can be restated in terms of bond order as follows :

(i) A positive bond order (i.e., `N_b > N_a`) means a stable molecule

(ii) A negative (i.e., `N_b < N_a`) or zero (i.e., `N_b = N_a`) bond order means an unstable molecule.

`text(Nature of the Bond :)` Integral bond order values of `1`, `2` or `3` correspond to single, double or triple bonds respectively as studied in the classical concept.

`color{green}("Bond-length ")` The bond order between two atoms in a molecule may be taken as an approximate measure of the bond length. The bond length decreases as bond order increases.

`color{green}("Magnetic Nature ")` (i) If all the molecular orbitals in a molecule are doubly occupied, the substance is `color{red}("diamagnetic")` (repelled by magnetic field).

(ii) If one or more molecular orbitals are singly occupied it is `color{red}("paramagnetic")` (attracted by magnetic field), e.g., `O_2` molecule.