LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – NOVEMBER 2012

CH 1814 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

Date : 07/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Part–A

Answer ALL questions:                                                                                                      (10 x 2 = 20)

1. State the order and degree of theSchrödinger equation for a particle in a one dimensional box.
2. Normalize exp(ikx) for 0 ≤ x ≤ π
3. Evaluate p_xx².
4. Define moment of inertia.
5. Identify the perturbation term in the Hamiltonian of Helium atom.
6. Write down the Laugerre polynomial for 1S electron.
7. What are variational integral and variational parameters?
8. Prove that the operation S₁₀⁵ is equivalent to S₆³.
9. Identify the point group of m-dichloro benzene.
10. Mention the condition for an improper axis and its inverse to form a class together.

Part-B

Answer any EIGHT questions:                                                                                          (8 x 5 = 40)

11. Derive the expression for linear momentum operator.
12. Calculate the highest translational quantum number for an oxygen molecule in 1mm length to have its thermal energy kT at 298K(k = 1.38 x 10^-23Jmolecule^-1).
13. Calculate the wave length of π →π* transition in 1, 3, 5-hexatriene
14. Define the following

• Associated Legendre equation
• Associated Legendre polynomials
• Legendre polynomials

15. The internuclear distance of the ¹H³⁵Cl molecule is 0.128 nm. Calculate the spacing of the lines in its rotational spectrum in terms of cm^-1.
16. Obtain all the possible term symbols for a ⁴F state.
17. Prove the commutation relation [p²_x, x] = -2iћp.
18. Illustrate the Pauli Exclusion Principle for the ground state of He atom.
19. At what distance from the nucleus is the probability of finding the electron a maximum for a 1S electron in hydrogen?
20. While the order is the same for both C_3v and C_3h point groups, their classes are different – reason out.
21. Allene belongs to D_2d point group. Identify the order and hence the number and dimensionality of the irreducible representations.
22. Set up the secular determinant for allyl radical and obtain its energy levels.

Part-C

Answer any FOUR questions:                                                                                          (4 x 10 = 40)

23. (a) Define the following:

• Closed interval
• Even function
• Orthonormal set of functions.
• Get the following polynomial functions for a rigid rotor:

(i) P₀⁰(cosθ)                        (ii) P₁⁰(cosθ)

24. Calculate the wave length of π →π* transitionof the following molecule:

 

 

(Bond distances: C-C, 0.154 nm; C=C, 0.133 nm)

25. Set up the Schrodinger equation for a rigid rotor and hence solve for its energy and wave functions.
26. State the principle of Perturbation theory and use first order perturbation theory to calculate the energy of a particle in a one dimensional box from x = 0 to x = a with a slanted bottom, such that V_x = V₀ (x)/a.    Given the wave function   Ψ⁽⁰⁾ = (2/a)^1/2 sin (nπx/a).
27. a)Solve the polar angle dependent equation for Hydrogen atom.
28. b) Highlight the important approximations that distinguish the HMO method from other LCAO methods.

 

 

 

 

28. Work out the hybridization scheme for σ bonding by carbon in CH₄ molecule of T_d point group symmetry, using the character table given below.

Td	E	8C3	3C2	6S4	6σd
A1	1	1	1	1	1		x2+y2+z2
A2	1	1	1	-1	-1
E	2	-1	2	0	0		(2z2-x2-y2, x2-y2)
T1	3	0	-1	1	-1	(Rx, Ry, Rz)
T2	3	0	-1	-1	1	(x, y, z)	(xy, xz, yz)

 

 

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