Application of VBT to the Formation of H₂ Molecule

Valence Bond Theory (VBT) with Special Reference to the Formation of H₂ 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 Ψ₁, Ψ₂, Ψ₃ be the several wave functions, then the true wave function, Ψ is obtained by linear combination i.e.
Ψ = C₁Ψ₁ + C₁Ψ₂ + C₃Ψ₃ + ...
where C₁, C₂, C₃,... 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 (e₁) and (e₂) respectively. Thus the two hydrogen atoms may be represented as H_A(e₁) and H_B(e₁) 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 e₁ belongs to H_A and e₂ belongs to H_B. However, as the electrons are indistinguishable after the overlap of the two atomic orbitals, electron e₁ may be under the influence of the, nucleus of H_B and also the electron e₂ may be under the influence of the nucleus H_A. Thus, the system may exist in two different states as follows-
H_A(e₁) H_B(e₂)      -----Equation-2
H_A(e₂) H_B(e₁)      -----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 = c₁Ψ_I ± c₂Ψ_II
Ψ_VB = c₁Ψ_A(1) Ψ_B(2) ± c₂Ψ_A(2) Ψ_B(1)      -----Equation-4
The coefficient c₁ and c₂ 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. c₁² = c₂²
c₁ = ± c₂
c₁ = c₂ = 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 H₂ 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 H₂ Molecule
Since 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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