# Application of VBT to the Formation of H_{2} Molecule

## Valence Bond Theory (VBT) with Special Reference to the Formation of H_{2} Molecule

Valence Bond Theory is based on two principles-1. If Ψ

_{A}and Ψ

_{B}be the wave functions for any two independent noninteracting systems A and B then the total wave function-

Ψ = Ψ

_{A}. Ψ

_{B}

2. If Ψ

_{1}, Ψ

_{2}, Ψ

_{3}be the several wave functions, then the true wave function, Ψ is obtained by linear combination i.e.

Ψ = C

_{1}Ψ

_{1}+ C

_{1}Ψ

_{2}+ C

_{3}Ψ

_{3}+ ...

where C

_{1}, C

_{2}, C

_{3},... are coefficients which are so adjusted to give a state of lowest energy.

## Formation of Hydrogen Molecule

Let us consider two hydrogen atoms A and B each having one electron marked as (e_{1}) and (e

_{2}) respectively. Thus the two hydrogen atoms may be represented as H

_{A}(e

_{1}) and H

_{B}(e

_{1}) and the wave functions associated with these electrons are represented by-

Ψ = Ψ

_{A}(1) . Ψ

_{B}(2) -----Equation-1

If we plot the value of E as a function of intermolecular distance, a curve I is obtained as show in Figure.

It may be seen that energy of the system decreases with distance till a minimum reached. This represents the formation of the Hydrogen molecule. The decrease in energy corresponding this minimum represents the bond energy and the distance represents the bond length. This curv has been obtained by using the wave function as represented by Equation-1. The bond energy found to be only 24 kJ mol

^{-1}while the actual (experimental) value is 458 kJ mol

^{-1}. Similarly, bond distance is fond to be 90 pm which is much different from the actual bond length of the molecule, i.e. 74 pm. This shows that the wave function used in the above calculations, viz. Ψ

_{A}(1) . Ψ

_{B}(2) is not accurate. Hence, some other interactions between the two hydrogen atoms are takin place and need to be taken into consideration to arrive at a better wave function. The differen possible interactions are discussed below-

**1. Exchange of Electrons**

In writing the above wave function, it was assumed electron e

_{1}belongs to H

_{A}and e

_{2}belongs to H

_{B}. However, as the electrons are indistinguishable after the overlap of the two atomic orbitals, electron e

_{1}may be under the influence of the, nucleus of H

_{B}and also the electron e

_{2}may be under the influence of the nucleus H

_{A}. Thus, the system may exist in two different states as follows-

H

_{A}(e

_{1}) H

_{B}(e

_{2}) -----Equation-2

H

_{A}(e

_{2}) H

_{B}(e

_{1}) -----Equation-2

The wave functions corresponding to these two states may be written as-

Ψ

_{I}= Ψ

_{A}(1) Ψ

_{B}(2) -----Equation-3

Ψ

_{II}= Ψ

_{A}(2) Ψ

_{B}(1) -----Equation-3

The total wave functions of the system will then be a linear combination of the above two wave functions, i.e.

Ψ

_{VB}= c

_{1}Ψ

_{I}± c

_{2}Ψ

_{II}

Ψ

_{VB}= c

_{1}Ψ

_{A}(1) Ψ

_{B}(2) ± c

_{2}Ψ

_{A}(2) Ψ

_{B}(1) -----Equation-4

The coefficient c

_{1}and c

_{2}shows the extent to which Ψ

_{I}and Ψ

_{II}combine to Ψ. Both hydrogen atoms combine with equal strength in the formation of hydrogen molecule. Since Ψ

_{I}and Ψ

_{I}shows state of equal energy. Hence, their weights are also equal. Since, the weight is proportional to the square of the coefficient,-

i.e. c

_{1}

^{2}= c

_{2}

^{2}

c

_{1}= ± c

_{2}

c

_{1}= c

_{2}= 1

Hence,

Ψ

_{VB}= Ψ

_{A}(1) Ψ

_{B}(2) ± Ψ

_{A}(2) Ψ

_{B}(1) -----Equation-5

Thus, there are two possible linear combinations of the wave functions corresponding to wave functions represented by Ψ

_{+}and Ψ

_{-}, i.e.

Ψ

_{+}= Ψ

_{A}(1) Ψ

_{B}(2) + Ψ

_{A}(2) Ψ

_{B}(1) -----Equation-6

Ψ

_{-}= Ψ

_{A}(1) Ψ

_{B}(2) − Ψ

_{A}(2) Ψ

_{B}(1) -----Equation-7

If the energies are calculated as a function of intermolecular distance by employing the wave function Ψ

_{-}, the curve II is obtained which shows that energy increases when the atoms are brought close together. This represents a

*nonbonding state*. On the other hand, if energies are calculated as a function of intermolecular distance by using the wave function Ψ

_{+}, the curve III is obtained. It represents decrease of energy with minimum at particular distance and hence represents a

*bonding state*.

Using this wave function, the minimum in the curve is found to be at 303 kJ mol

^{-1}at an internuclear distance of 80 pm. Thus, both the values have come closer to the experimental values.

*The extra lowering of energy is due to the exchange of electrons between the two hydrogen atoms and is called exchange energy*. Thus, exchange energy for H

_{2}molecule = 303 - 24 = 279 kJ mol

^{-1}.

**2. Screening Effect of Electrons**

When the atoms come close together, the electron of one atom shields the electron of the other atom from the nucleus. Thus, the electron does not feel full attraction by the nucleus. Thus, its effective nuclear charge is smaller than the actual nuclear charge. Taking this fact into consideration, the wave function was further improved. Using this wave function, energy versus internuclear distance curve IV is obtained as shown in Figure. The minimum in this curve lies at 365 kJ mol

^{-1}which shows a further improvement in the value.

**3. Ionic Structure for H**

_{2}MoleculeSince the calculated value of 365 kJ mol

^{-1}is still much lower than the experimental value of 458 kJ mol

^{-1}, it requires further improvement in the wave function. It was suggested that when the atoms come closer together, both the electrons may lie close to the nucleus H

_{A}or close to the nucleus H

_{B}, giving rise to two possible ionic structures as-

Ψ

_{A}(1) Ψ

_{A}(2)

H

_{A}

^{-}H

_{B}

^{+}

Ψ

_{B}(1) Ψ

_{B}(2)

H

_{A}

^{+}H

_{B}

^{-}

Though the possibility of existance of these states is very little because repulsions between the electrons will be much larger than the attractions between H

_{A}

^{−}and H

_{B}

^{+}and H

_{A}

^{+}and H

_{B}

^{−}. Taking these two structures into considerations, the wave functions could be further improved. The wave functions corresponding to states III and IV may be written as-

Ψ

_{III}= Ψ

_{A}(1) Ψ

_{A}(2) -----Equation-8

Ψ

_{IV}= Ψ

_{B}(1) Ψ

_{B}(2) -----Equation-9

The total wave function due to these ionic structures (Ψ

_{ionic}) may be obtained by a linear combination of Ψ

_{III}and Ψ

_{IV}.

Thus,

Ψ

_{ionic}= Ψ

_{III}+ Ψ

_{IV}

Ψ

_{ionic}= Ψ

_{A}(1) Ψ

_{A}(2) + Ψ

_{B}(1) Ψ

_{B}(2)

Now combining the covalent and ionic wave functions, the complete wave function for the system may be written as-

Ψ = Ψ

_{covalent}+ λ Ψ

_{ionic}

Ψ = [Ψ

_{A}(1) Ψ

_{B}(2) + Ψ

_{A}(2) Ψ

_{B}(1)] + λ[Ψ

_{A}(1) Ψ

_{A}(2) + Ψ

_{B}(1) Ψ

_{B}(2)] -----Equation-10

where, λ gives the extent of contribution of the ionic structure owards bonding and is usually called mixing coefficient. Its value for hydrogen molecule is found to be 0.17.

Usuing the above wave function, the minimum in the curve shown at V is obtained at 388 kJ mol

^{-1}. Thus, bringing the value still closer to the experimentl value of the bond energy of 458 kJ mol

^{-1}. The bond distance is found to be 75pm which agrees fairly well with the experimental value of 74pm.

Further improvement in the wave function have been done to find more closer value.

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