17 Molecular Orbital Theory I

Contents:

1. Molecular Orbital Method

2. Molecular Orbital Treatment of Hydrogen Molecule Ion( + ion)

3. Molecular Orbital Theory for Hydrogen Molecule

4. Assignment

The students will be able to learn about

Molecular orbital theory of Hydrogen atom and Hydrogen ion

In order to describe the electronic structure of molecules: two models have been developed (i)the molecular orbital (MO) method developed by Mulliken and (ii) the valence bond (VB) method by Heitler and London.

1. MOLECULAR ORBITAL METHOD

Molecular orbitals have the same significance as the atomic orbitals and also the molecular quantum numbers are associated with these orbitals in the same way.

With a quantum mechanical approach, the molecular orbital are associated with molecular wave functions that describe molecular energy states that are then extended to include all the nuclei in it and hence will be polycentric.

However, the filling the orbital with system of electrons is done in accordance with Paul’s Exclusion Principle. Therefore, first construct reasonable molecular orbitals that can be done in number of ways. Molecular orbitals are taken as a function of atomic orbitals centered on the individual atoms as molecules consist of atoms. The trivial approach for obtaining the MOLECULAR ORBITAL (MO) is the linear combination of atomic orbitals.

The molecular orbital Ψ is written as a linear combination of the atomic orbitals as

? = ?1?1 + ?2?2 + …

Here ?_i are the individual atomic orbitals. The constants ?_i are to be selected so that the energy given by Ψ is minimum. Further the combining atomic orbitals must have

(i) Energies of comparable magnitude

(ii) Considerable overlapping and

(iii) the same symmetry.

The electron has tendency to remain most of the time in the lowest energy atomic orbital if the orbitals have defferent energies.

2. Molecular Orbital Treatment of Hydrogen Molecule Ion( H2⁺ ion)

Considering an electron of charge e- associated with two nuclei and b each with positive charge separated by a distance R. The electron in the neighbourhood of is to be described by the atomic orbital ?_a centred on a and by the atomic orbital ?_b centred on b when it is in the nieghourhood of b,
Let us find firstly a set of molecular orbitals for the molecule so that it is close to ?a in the neighbour hood of a and close to ?_b in the neighborhood of b. As MO has to be a linear combination of ?a and ?b

Therefore, ? = ?1?? + ?2?b

c1 and c2 are constants to be selected so that the wave function ? corresponds to a minimum value of energy.

Writing the Schrodinger equation and the energy E of the system

The Hamiltonian of the system is

m is the mass of the electron and the fourth term represents the electrostatic repulsion between the nuclei and the electron.

Therefore,

as Overlap integrals.

These simultaneous homogeneous equations in c1 and c2 have non-trivial solutions only if

and the two roots for E are the allowed energy values of the system.

Noticing that for the Homonuclear diatomic molecule H2⁺, the nuclei and b are identical, ??? = ??b and therefore the above equation becomes

??? − ? = ±(??? − ??) ————–B

This gives the two value of energy

Combining Eqs. (A) and (B),

Now From this equation, c1=c2 for energy E1 and c1=-c2 for the energy E2. The wave functions, corresponding to energies E1 and E2 are respectively,

The first term is simply the ground state energy of the hydrogen atom EH since the operator in it is the hydrogen atom Hamiltonian and Ψ is the one electron wave function. Note that the nuclear repulsion term, e2/R is independent of the electronic coordinates.

The value of the quantities Vaa, Vab, S and the nuclear repulsion energy e²/R depend on the inter-nuclear distance R and are always positive. If the two nuclei are infinitely separated then overlap S = 0, and S = 1 if these are together. Using Equations C and D , E1 and E2 can be written as

Ψ1 (Corresponding to MO) has an energy E1 lower than-that of the atomic orbitals from which it is formed as is clear from the figure. In case of Ψ2 the energy E2 is higher than that of the atomic orbitals.

Fig.2 depicts the relative energies of molecular orbitals and their constituent atomic orbitals.

The wave function Ψ1 corresponds to the situation where there is a build up of electron density between the two nuclei and a more effective screening of one nucleus from the other. This suggests that a bond has been formed that is described by a bonding molecular orbital.

The other possibility Ψ2 corresponds to the situation where there is a depletion of charge between the two nuclei and a larger nuclear repulsion that results in an anti-bonding orbital.

Figure 3 depicts the formation of bonding and anti-boding orbitals from two 1s atomic orbitals. Both these orbitals are symmetrical about the internuclear axis. Molecular orbitals that are symmetrical about the interuclear axis are designated by σ (sigma) and those which are not symmetrical about the interuclear axis are designated by π (pi).

The bonding orbitals are denoted by the symbol 1?σ as it is produced from two 1s atomic orbitals.

The anti-bonding state is given the symbol 1?σ^∗ , σ^∗ representing higher energy. The figure 4 demonstrates the Probability density for the bonding and antibonding states

3. Molecular Orbital Theory for Hydrogen Molecule

The treatment of hydrogen molecule in MO theory is essentially the same as that of H₂⁺ molecule. The Hamiltonian operator for H₂ molecule is

1 and 2 represent electrons and & b protons that has been represented in the figure

Now compute the expectation value of H with a trail wave function. It is noteworthy that the term –e2/R is independent of electronic coordinates.

The Hamiltonian of the above equation is the same as the one solved for H₂⁺ molecule. Therefore, in the ground state of hydrogen molecule both the electrons occupy the bonding orbital Ψ1 of H₂⁺ that is symmetric with respect to the interchange of nuclei and b and therefore considering the trial wave function for the hydrogen molecule

Ψ_MO = Ψ1(1)Ψ1(2)

Also include the electron spin and Pauli’s principle into the formalism. Spin functions for the two electron system are

a(1) means the first electron is in a ‘spin up’ state, b(1) means the first electron is in a ‘spin down’ state and likewise a(2) means the first electron is in a ‘spin down’ state, b(1) means the first electron is in a ‘spin up’ state. As the Pauli’s principle dictates that the total wave function must be antisymmetric with resptect to the interchange of the two electrons.

Therefore, the symmetricΨ_MO , has to combine with the antisymmetric spin part to give the wave function

Ψ1(1) Ψ1(2) √12 [ (1) (2) − (1) (2)]

This corresponds to a singlet state as its spin S=0.

The energy is not affected by the inclusion of spin part because the Hamiltonian does not contain spin terms and therefore the space part is to be considered for the MO for the evaluation of energy.

The total energy is to be minimized with respect to the internuclear separation R One gets a binding energy of about -2.68 eV and equilibrium internuclear distance of 0.85 Å. However the experimental values are-4.75 eV and 0.74 Å, respectively.

The above equation can also be written as

The first two terms correspond to the situation when both the electrons are associated with the same proton. These represent the ionic structures ?_?⁻ ?_?⁺ and ?_?⁺ ?_b^– respectively.

However, the third and fourth terms represent the situation where the electrons are shared equally by both the protons and hence they correspond to covalent structures of the hydrogen molecule.

Assignments:

1. What will be the the spin functions for a system of two indistinguishable electrons.

Ans. As the spin states of electrons labeled as 1 and 2:

Obviously, the first and the last are symmetric with respect to an interchange of electrons (1) and (2). However, the second and third are neither symmetric nor antisymmetric and by linear combination of the two these can be made symmetric or antisymmetric.

So, the following spin functions for a two electron system are: