19 Molecular Orbital Theory III

1. VALENCE- BOND METHOD

In this method, atoms are assumed to maintain their individual identity in a molecule and the bond arises due to the interaction of the valence electrons when the atoms come closer. It amounts to considering molecules as composed of atomic cores and bonding valence electrons. Thus, VB method considers bringing the atoms with their associated electrons together and allowing them to interact to form the bond.

HYDROGEN MOLECULE-HEITLER LONDON THEORY

The application to valance problems was first considered by Heitler and London in 1927. The idea was later extended by Slater and Pauling to give a general theory of chemical bonding and the theory is known as the Valence Bond (VB) method.

Consider two hydrogen atoms far apart so that there is no interaction between them. Labeling the electrons as 1 and 2, the nuclei as a and b and the electron nucleus distances by r_al  and r_b2.

Writing the Schrodinger equations:

?_?(1)Ψ_?(1)=E_?Ψ_?(1), H_?(2)Ψ_?(2)=E_?Ψ_?(21)

Where E_?=E_?=E_?, the ground state energy of the hydrogen atoms and Ψ_?(1) and H_?(2) are the 1s hydrogenic wave function.

?_?(1)=−ℎ²/2?∇ 1/ 2 − e² / r?1,   H?(2)=−∇2/2−e²r?1

When the two atoms are brought closer and there is no interaction between the two, the

?=?_?(1)+ H_?(2)

It amounts to assuming that electron 1 is moving about proton ‘a’ and electron 2 is moving about proton ‘b’. The system of two hydrogen atoms can be described by the wave function

Ψ1(1,2)=?_?(1) H_?(2)

With energy eigenvalue E_?+E_?

[?_?(1)+ H_?(2)] Ψ_?(1)Ψ_?(2)=?_?(1) Ψ_?(1)Ψ_?(2)+ H?(2) Ψ?(1)Ψ?(2)

= E?Ψ?(1) Ψ?(2)+E?Ψ?(1) Ψ?(2)

=(E?+E?) Ψ?(1) Ψ?(2)

The electrons are indistinguishable and therefore an equally good description of the molecule with the same energy is given by the structure in which electron 1 is associated with atom ‘b’ and electron 2 is associated with atom ‘a’ , hence the wave function,

Ψ2(2,1)=Ψ?(2) Ψ?(1)

Ψ1(1,2) and Ψ2(2,1) are eigenfunctions of the description H?+H? with eigenvalue (E?+E?). The two functions differ only in the interchange of the two electrons between the orbitals. Hence, this degeneracy is referred to as the exchange degeneracy and the wave functions of the two electron system, must be a linear combination of Ψ1(1,2) and Ψ2(2,1).

Now considering the hydrogen molecule, the Hamiltonian of the system

?=?_?(1)+?₂(2)+?′+?²/?

Where

?′=−?²/??1− ?²/??2+?² /?12

Evidently, the term e2/R is independent of electronic coordinates, its contribution may be incorporated at the final stage as an additional term e2/R.

The problem can be solved either by using the perturbation method or the variation method. Heitler and London followed the perturbation method. As the Hamiltonian H is unchanged on exchange of the two electrons, the wave functions must either be symmetric or antisymmetric with respect to such an exchange. So, the symmetric Ψ? and Ψ?? antisymmetric combinations are:

Ψ?=Ns[Ψ?(1)Ψ?(2)+Ψ?(2) Ψ?(1)]

Ψ??=N?[Ψ?(1)Ψ?(2)−Ψ?(2) Ψ?(1)]

Where Ns and N? are normalization constants and the Normalization conditions gives

?^2/?=1/2(1+?²) , ?^?/2=1/2(1−?²)

Where S is given by the overlap integrals.

The inclusion of electron spin and Pauli’s leads to the Heitler-London wave functions corresponding to a singlet (S=0) and a triplet (S=1)

Ns[Ψ?(1)Ψ?(2)+Ψ?(2)Ψ?(1)]1√2[α(1) β(2)−β(1) α(2)]

N?[Ψ?(1)Ψ?(2)−Ψ?(2)Ψ?(1)]{1√2[α(1) β(2)+α(2)β(1)β(1)β(2)

Since the Hamiltonian does not contain spin terms. The energy is not affected by the inclusion of spin part. The space parts alone can then be taken as the unperturbed wave functions for the evaluation of energy.

Let H’ be the perturbation applied to the system and the first order correction to the energy E’ is the diagonal matrix element of the perturbing Hamiltonian corresponding to the unperturbed wave functions, so that

?′/1=?^2/?⟨Ψ?(1)Ψ?(2)+Ψ?(2)Ψ?(1)|?′|Ψ?(1)Ψ?(2)+Ψ?(2)Ψ?(1)⟩

=(2?+2?)/2(1+?2)=?+?1+?2

Where J and K are called Coulomb and exchange integrals respectively and are

?=⟨Ψ?(1)Ψ?(2)|?′|Ψ?(1)Ψ?(2)⟩

?=⟨Ψ?(1)Ψ?(2)|?′|Ψ?(2)Ψ?(1)⟩

The Ψ′s are hydrogen 1s wave function. The energy of the singlet state corrected to first order

?_?=2?_?+?+?1/1+?2 + ?²/?

The first order correction to the triplet state

?′_2/?+?/1+?2

and the energy of the triplet state

?_?=2?_?+?+?/1+?2+?/2?

The Coulomb integral represents the interaction of the classical electron charge clouds about one nucleus with the charge in the other nucleus and the interaction of the two charge clouds with one another. Exchange integral represents a non-classical interaction. It is a consequence of the inclusion of both Ψ1(1,2) and Ψ2(2,1) in the unperturbed wave function. The two functions differ only in the interchange of the electrons between the orbitals Ψ? and Ψ?.

The overlap integral S is zero, when the two protons are far apart and when these are in contact is unity. K and J both tend to zero for large R, while both are negative for intermediate values.The magnitude of K is several times larger than that of J. Thus Es can have a value less than 2E_H, whereas ??is always shows a minimum for the Ψ_? combination corresponding to the formation of the stable molecule. The state characterized by the Ψ?? combination corresponds to repulsion for all values of R. The equilibrium internuclear distance r0 is the one corresponding to the minimum of the Ψ? curve. The theoretical value of 0.85 Å for r0 is high compared to the experimental value of 0.74 Å. The binding energy corresponding to 0.85 Å is about 72 kcal/mol whereas the experimental value is 109 kcal/mol.
In case, the exchange degeneracy is not considered one would have got a binding energy of about 6 kcal/mol. The additional binding energy of 66 kcal/mol is referred to as exchange energy.

Figure Energies E’1 and E’2 versus internuclear distance R.

The structures given by Ψ_? and Ψ_?? are referred to as covalent structures as one electron is associated with one nucleus and the second one with the other nucleus. The agreement between experimental and theoretical values can be improved further by considering two additional structures in which both the electrons are associated with only one nucleus. These are the ionic structures. ??− ??+ and ??+ ??− with wave functions Ψ?(1) Ψ?(2) and Ψ?(1) Ψ?(2) respectively. Inclusion of these structures gives a binding energy that agrees fairly well with the experimental value.

2. Valance Bond METHOD OF HYDROGEN MOLECULE ION

The Hydrogen molecule ion is the system consisting of a single electron and two nuclei and the Schrodinger equation for the ?^+/2 ion is

−ℏ²/2?∇² Ψ+(−e²/r?−e²/r?+e²/R)Ψ=EΨ

Consider the system initially with the two nuclei very far apart with the electron associated with the nucleus ?. This leads to the structure of H_??_?+ and the wave function for this structure be assumed as Ψ?.

When the electron is associated with the nucleus b, the structure is described by ?_?+ H_? with wave function Ψ_?.

Both these structures correspond to the same energy and the electron wave function will be the same as the hydrogenic 1s wave function.

Further when the two nuclei are bought closer, they repel one another but the electron gets attracted to both the nuclei thereby creating a bond. A hydrogen molecule ion is formed when the energy of the system is minimum. That is, the wave function of the system is described by a linear combination of Ψ? and Ψ?.

Ψ=c1Ψ?+c2Ψ?

c₁=±c₂ As both the states have the same energy they contribute equally to Ψ.

The two possible linear combinations are

Ψ?=1/√2+2S(Ψ?+Ψ?),   Ψ2=1/√2−2S(Ψ?−Ψ?)
?=⟨Ψ?|Ψ?⟩

The charge density is proportional to Ψ².

The increased electron charge density in the case of Ψ₁ is concentrated in the region between the nuclei. However, in the case of Ψ₂ the electron charge density in pushed away from the internuclear region.

In Figure, the variation of energy of the states Ψ₁ and Ψ₂ is plotted against the internuclear distance. The calculations lead to energy and bond length values same as that obtained on the basis of simple Molecular Orbital theory. This is understandable since Ψ_? and Ψ_? are 1s atomic orbitals.

3. Difference between Molecular orbital (MO) method and valence bond method(VB):

The MO method considers that the electron pairs forming the bond gets delocalized and not concentrated between the nuclei. While, in the Valence Bond picture the electrons in a molecule occupy atomic orbitals and the overlap of atomic orbitals results in the formation of a bond.

Larger the overlap stronger is the bond. Thus, the VB theory represents a localized picture of the chemical bond. However, the localized and delocalized pictures of the chemical bond are equivalent as the charge distribution in each molecular orbital when added up over the region where the bond is extended gives the same total charge density as that given by the VB theory.

In this localized picture, the direction in which maximum overlap occurs will be the direction of the bond.

4. Visualizing the Shape of Polyatomic Molecules:

It is easier to visualize the shape of polyatomic molecules with the electron pair concept.

Consider the structure of water molecule. The electron configuration of the oxygen atom is

1?² 2?² 2?^2/? 2?^1/?2?1/?

The two unpaired electrons in the 2?_? and 2?_? orbitals that are at right angles are available for bonding with the 1s orbital of hydrogen atoms. The s-orbital can overlap to the same extent in all directions as it is spherically symmetric whereas the p orbitals have certain preferred directions. Therefore, one gets a water molecule with HOH angle 90⁰.

However, the actual value is 104.5⁰ . When the water molecule is formed, the electrons in the OH bond are drawn towards the oxygen atom making the hydrogen atom slightly positive.That means the mutual repulsion between the hydrogens is partly responsible for the increases in the angle. Note that the angles are 91⁰ and 93⁰ in the case of H₂Se and H₂S, respectively.
NH₃ molecule, the central nitrogen atom has the electron configuration

1?² 2?² 2?^1/? 2?1/? 2?1/? .

The maximum overlapping of the three p orbitals with the 1s hydrogen orbitals are possible along the x,y and z direction.

The bond angle in this case is 107.3⁰ that is again partly due to the mutual repulsion between the hydrogen atoms. Note that in the related molecules PH₃, AsH₃ and SbH₃ the observed angles are 93.3⁰ ,91.8⁰ and 91.3⁰ respectively which is close to 90⁰.