IGNOU MCH-18 Quantum Chemistry and Group Theory - Latest Solved Assignment

1. Answer any five of the following in brief.

(a) Give the significances of de Broglie’s idea of matter waves and Heisenberg’s uncertainty principle with reference to the description of behaviour of electron in an atom.
(b) Why can’t the ground state energy of a particle confined to a one-dimensional box be zero?
(c) What is accidental degeneracy?
(d) Calculate the number of possible orientations of the orbital angular momentum for an electron with l = 3.
(e) Differentiate between probability density and probability of finding the electron in an atom.
(f) Why do we need to transform the SWE for the relative motion of electron in hydrogen like ions to spherical polar coordinates?
(g) Why does the expression for energy levels of a hydrogen-like atom get modified in multielectron atoms?
(h) The average distance of the electron in the ground state of hydrogen atom is more than its most probable distance in the same state. Justify the statement qualitatively.

2. (a) Define blackbody and discuss how classical mechanics was inadequate in explaining the results of black body problem.

(b) i) What are the characteristics of a well behaved wavefunction?
ii) Define Hermitian operators and show that they give real values of the observables.

3. (a) i) Devise the Schrödinger wave equation for a particle of mass m confined in one dimensional box of length L.
ii) Calculate the energy of the first excited state of an electron confined in a onedimensional box of length L = 2 × 10-10 m.

(b) Define degeneracy and calculate the degeneracies of first five energy levels for a particle of mass m confined in a cubic box of edge length=L.

4. (a) i) Derive the expression for the rotational energy levels of a rigid rotor.
ii) Calculate the ground state energy of a diatomic molecule behaving as a quantum harmonic oscillator.
(Given: Force constant, k = 500 N/m and reduced mass µ = 1.67 × 10-27 kg.)

(b) Formulate the Schrödinger wave equation for the hydrogen atom. Discuss the strategy to solve it. You are expected to outline different stages involved in solving the SWE and not to solve it.

5. (a) i) Calculate the most probable distance of the electron from the nucleus in the ground state of the hydrogen atom. You may use the normalised radial wavefunction for hydrogen atom from Table 7.1.
ii) Calculate the wavelength of light emitted when an electron in a helium ion (He+) makes a transition from the n = 4 energy level to the n = 2 energy level.

(b) Evaluate, [Ly, Lz] and [L2, Lz] and outline the significance of the outcome.

6. Answer any five of the following in brief.

(a) What is the difference between a symmetry element and a symmetry operation?
(b) A symmetry operation can be expressed in terms of a transformation matrix. Give the transformation matrix for a C2 rotation.
(c) State Born-Oppenheimer’s approximation and outline its significance.
(d) What are Coulomb’s and exchange integrals?
(e) What are the basic requirements of linear combination of atomic orbitals-molecular orbital (LCAO-MO) approach?
(f) What determines whether the 2s and 2p orbitals of the elements of period 2 would undergo mixing or not?
(g) VBT completely ignores the ionic terms in the trial wavefunction for hydrogen molecule whereas the MOT over emphasises them. Comment
(h) Give the assumptions of Hückel molecular orbital theory.

7. (a) Using the variational method, approximate the ground-state energy of a particle in a onedimensional box with a trial wavefunction, ψ(x) = Ax (L – x). Normalise the wavefunction and calculate the expectation value of the energy.
(b) Formulate the Schrödinger wave equation for helium atom. Find the approximate ground state energy of helium atom by ignoring the electron-electron repulsion.

8. (a) What is perturbation method? Formulate the expression for the energy of helium atom by considering the electron-electron repulsion as a first order perturbation. You need to just formulate the expression and not evaluate it.
(b) i) Identify the symmetry elements and assign point group for water molecule
ii) Generate group multiplication table for water molecule

9. (a) State Great Orthogonality theorem. Using the consequences of Great Orthogonality theorem derive the characters of irreducible representation of C3v point group.
(b) Illustrate LCAO-MO approach to chemical bonding by taking the example of hydrogen molecule ion. Give detailed solution to get the expressions for the wavefunctions.

10.(a) Discuss the key differences in the Valence Bond and Molecular Orbital approach to chemical bonding by taking the example of bonding in hydrogen molecule.
(b) Give the assumptions of Hückel molecular orbital (HMO) theory for conjugated molecules. Determine the expressions for the energy levels for ethylene molecule by using HMO.