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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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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