Question

According to molecular orbital theory, which of the following will not be a viable molecule?
(A) ${H_2}^{-}$
(B) ${H_2}^{2-}$
(C) ${He_2}^{2+}$
(D) ${He_2}^{+}$

Answer 1

A molecule is viable or not depending upon the bond order of the molecule. Bond order is defined as half of the difference between the number of electrons present in the bonding and antibonding orbitals,i.e.,$Bond \, order=\dfrac{1}{2}({N_b}-{N_a})$. The molecule is stable if ${N_b}>{N_a}$,i.e, if bond order is positive. The molecule is unstable if ${N_b}<{N_a}$ or ${N_b}={N_a}$,i.e, if bond order is negative or zero.Also the molecule does not exists if bond order is zero.

Complete answer:
Molecular orbital theory states that each atom tends to combine together and form molecular orbitals. There exist different types of molecular orbitals viz; bonding molecular orbitals, anti-bonding molecular orbitals, and non-bonding molecular orbitals.The bonding molecular orbital has lower energy and greater stability as compared to anti-bonding molecular orbital.
Electronic configuration in terms of molecular orbitals are:

$[{\sigma}(1s)][{\sigma^*}(1s)][{\sigma}(2s)][{\sigma^*}(2s)][{\pi}(2p_x)][{\pi}(2p_y)][{\sigma}(2p_z)][{\pi*}(2p_x)][{\pi*}(2p_y)][{\sigma^*}(2p_z)]$

Let us discuss the molecules given in options one by one :
(A) ${H_2}^{-}$:
This molecular negative ion is formed by combination atom $(H)$ having one electron in 1s orbital with a hydrogen ion ${H^-}$ having two electrons in 1s orbital. Thus ${H_2}^{-}$ ion has three electrons. The two electrons will enter the bonding $[{\sigma}(1s)]$ molecular orbital and third electron will enter the antibonding $[{\sigma^*}(1s)]$ ,molecular orbital according to aufbau principle and pauli exclusion principle. The electronic configuration of ${H_2}^{-}$ ion is :
$[{\sigma}(1s)]^2 \, [{\sigma^*}(1s)]^1$
Bond order is:
$Bond\, order=\dfrac{1}{2}({N_b}-{N_a})$,where ${N_b}$, ${N_a}$ are the number of electrons present in the bonding and antibonding orbitals respectively.
Here ${N_b}=2 \, {N_a}=1$
$\Rightarrow Bond\, order=\dfrac{1}{2}(2-1)$
$\therefore Bond \,order=\dfrac{1}{2}$
Hence this molecular ion ${H_2}^{-}$ exists and is stable.

(B) ${H_2}^{2-}$:
In a similar way of (A),Electronic configuration of ${H_2}^{2-}$ is :
$[{\sigma}(1s)]^2 \, [{\sigma^*}(1s)]^2$
Here ${N_b}=2 \, {N_a}=2$ so that
Bond order is:
$\Rightarrow Bond\, order=\dfrac{1}{2}({N_b}-{N_a})$
$\Rightarrow Bond\, order=\dfrac{1}{2}(2-2)$
$\therefore Bond\, order=0$
i.e. no bond is formed means ${H_2}^{2-}$ does not exist. Hence it will not be a viable molecule.

(C) ${He_2}^{2+}$:
Electronic configuration of ${He_2}^{2+}$ is : $[{\sigma}(1s)]^2$
Here ${N_b}=2 \, {N_a}=0$ so that
$\Rightarrow Bond\, order=\dfrac{1}{2}({N_b}-{N_a})$
$\Rightarrow Bond\, order=\dfrac{1}{2}$ $(2-0)$
$\therefore Bond \,order=1$
Hence the molecular ion ${He_2}^{2+}$ exists and is stable.

(D) ${He_2}^{+}$:
Electronic configuration of ${He_2}^{+}$ is :
$[{\sigma}(1s)]^2 \, [{\sigma^*}(1s)]^1$
Here ${N_b}=2 \, {N_a}=1$ so that
$\Rightarrow Bond \,order=\dfrac{1}{2}({N_b}-{N_a})$
$\Rightarrow Bond\, order=\dfrac{1}{2}(2-1)$
$\therefore Bond\, order= \dfrac{1}{2}$
Hence the molecular ion ${He_2}^{+}$ exists and is stable.

Thus according to the above explanation option (B) is the correct option.

Note: e main ideas of molecular orbital theory are:
(i) When two atomic orbitals combine or overlap, iey lose their identity and form new orbitals. The orbitals thus formed are called molecular orbitals.
(ii) Only those atomic orbitals can combine to form molecular orbitals which have comparable energies and proper orientation.
(iii) The number of molecular orbitals formed is equal to the number of combining atomic orbitals.
(iv) When two atomic orbitals combine, they form two new orbitals called 'bonding molecular and 'antibonding molecular orbital'.
(v) The bonding molecular orbital has lower energy and hence greater stability than the antibonding molecular orbital.
(vi)The bonding molecular orbitals are represented by ${\sigma},\,{\pi}$ etc. whereas the corresponding antibonding molecular orbitals are represented by ${\sigma},\,{\pi}$ etc.
(vii)The shapes of the molecular orbitals formed depend upon the type of the combining atomic orbitals
(viii)The filling of molecular orbitals takes place according to the same rules as those of the atomic orbitals. These are as follows :—
(a) Aufbau principle, i.e., molecular orbitals are filled in order of their increasing energies.
(b) Pauli exclusion principle, i.e. A molecular orbital can have a maximum of two electrons if these must have opposite spin.
(c) Hund's rule of maximum multiplicity, i.e., pairing of electrons in the degenerate molecular orbitals does not take place until each of them has got one electron each.