LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – APRIL 2012

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

Date : 30-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Part–A

Answer ALL questions (10 x 2 = 20)

1. How does ahorizontal plane differ from a vertical plane?
2. Represent the complex number (1 – i) in the Euler form.
3. For what value of A the function Ax²is normalized for 0 ≤ x ≤ 1
4. Find equivalent operator for (A+B)² if the operators A and B commute.
5. Show that [x, d/dx] = -1.
6. How will you apply Born-Oppenheimer approximation to simplify the Hamiltonian for H₂⁺?
7. Account for the origin of the fine structure of the emission spectrum of sodium vapor.
8. State the variation theorem.
9. Prove that the operation S₆³ is equivalent to an inversion operation.
10. What are proper and improper axes of rotation? Give an example for each.

Part-B

Answer any EIGHT questions (8 x 5 = 40)

11. Define Hermitian operator. Give an example.
12. Calculate the wave length of π →π* transition in 1, 3, 5-Hexatriene (C-C and C=C distances are 1.54 and 1.33 Ả , respectively).
13. The inter nuclear distance of D₂ is 0.74Ả. Determine its rotational constant in cm^-1.
14. Get the following normalized polynomial functions:

(i) P₀⁰(cosθ)          (ii) P₁⁰(cosθ)

15. Set up the Hamiltonian for a simple harmonic oscillator
16. Show that [L²,L_x] = 0. Mention its significance.
17. Explain the importance of the integrals H_AAand H_BB obtained for the lowest energy of H₂⁺using LCAO method.
18. Find the radius of the shell where there is a maximum probability of finding the electron. Given: The probability, P = 4πr²e^-2ar.

 

 

 

19. The term symbol of a particular atomic state is ⁶S_5/2. Suggest a possible electronic configuration.
20. Obtain the Pauli antisymmetric wave function for the excited state He atom.
21. Identify the point groups of biphenyl and chloroform molecules mentioning their symmetry elements and operations.
22. The order and the classes of a molecule are 20 and 8 respectively. Identify the number and the dimensions of the irreducible representations.

 

 

 

Part-C

Answer any FOUR questions (4 x 10 = 40)

23. (a) Write expressions for the third levels for Ψ_nand E_n for a particle in 3D

box.

(b) Draw the MO diagram for the π-electrons in 1, 3-butadienyl anion radical.

24. Set up the Schrodinger equation for a rigid rotor and hence solve for its energy and wave function.
25. (a) Define the following.

• Hermite equation
• Hermite polynomials
• Get the normalized functions for the simple harmonic oscillator for its third vibrational level.

26. (a) Show that for anhydrogen like atom, in its ground state, the average distance of the electron from the nucleus is 3/2 times the most probable distance. Given: Ψ_1s = 1/(π)^1/2(Z/a₀)^3/2exp(-Zr/a_o).

(b)  Highlight the features that distinguish the Huckel method from other LCAO methods.

27. (a) Obtain the value of [x, H]. Mention its significance.

(b) Apply Huckel’s method to allyl cation and obtain expressions for the

energy levels.

28. Arrive at the IR and Raman active modes of vibrations of trans-1,2-dibromo ethylene molecule, using the following character table and verify the relevance of mutual exclusion principle.

C2h	E	C2	i	h
Ag	+1	+1	+1	+1	Rz	x2, y2, z2, xy
Bg	+1	-1	+1	-1	Rx, Ry	xz, yz
Au	+1	+1	-1	-1	z	–
Bu	+1	-1	-1	+1	x, y	–

 

 

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