LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – NOVEMBER 2010

    CH 1808  – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date : 03-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part-A

ANSWER ALL QUESTIONS                                                                                                   (10 ´ 2 = 20)

 

1. Give the Laplacian operator in spherical polar coordinates. What are their limits of integration?
2. Show the wave function Ae^-2x is an eigen function of the operator d²/dx². What is the eigen value?
3. The energy of a particle moving in a cubic box of side ‘a’ is 3h²/2ma². What is its degeneracy?
4. CO absorbs energy in the microwave region of the spectrum at 1.93 x 10¹² This is attributed to the J=0 to J=1 transition. Calculate the moment of inertia of the molecule.
5. What is the value of [y,p_y]? What is its physical significance?
6. Simple Harmonic Oscillator has zero as one of the quantum numbers while the particle in a box model does not have. Why?
7. What is a boson? Give an example.
8. Write the Hamiltonian operator for the H₂⁺ molecule in atomic units defining each term involved in it.
9. Prove that the operations S₂ and i have the same effect.
10. Predict the trace of C₃

 

Part-B

ANSWER ANY EIGHT QUESTIONS                                                                                        (8 ´ 5 = 40)

 

11. What is a hermitian operator? Show that the wave functions corresponding to two different eigen values of a Hermitian operator are orthogonal.
12. State the variation theorem. Apply it to the problem of particle in a 1-D box of length a, by taking ψ = x(a-x) as a trial function for 0≤x≤a.
13. Write the Schrödinger equation for 1-D harmonic oscillator. Verify ψ = (2a/π)^1/4exp(-ax²) is an eigen function of the Hamiltonian operator for the 1-D harmonic oscillator.
14. The force constant of ⁷⁹Br-⁷⁹Br is 240 Nm^-1. Calculate the fundamental vibrational frequency and the zero-point energy of the molecule.
15. What is a Slater determinant? Write the four Slater determinants for the excited state of He atom (1s, 2s).

 

16. Explain briefly with a suitable example:

(a) Bohr’s Correspondence Principle (b) Born-Oppenheimer Approximation.

17. The wave function of 1s orbital of Li²⁺ is Ψ_1s = (1/√π) (Z/a₀)^3/2 exp(-Zr/a₀), where a₀ is the most probable distance of the electron from the nucleus. Show that the average distance is a₀/2. [Help: ₀ò^¥ xⁿe^-qx =  n!/qⁿ⁺¹]
18. Write the Hamiltonian in atomic units and explain briefly how the Valance Bond (Heitler-London) treatment of H₂ molecule makes up for what MO theory lacks.
19. Explain the following with a suitable example:

• Spherical Harmonics (b) Quantum mechanical tunneling. (2+3)

20. What is a term symbol? Explain the origin of the fine structure of the emission spectrum of sodium vapor used in street lighting using term symbols.
21. List down the symmetry elements and symmetry operations of trans-1,3-dibromo cyclobutane and Furan.
22. Identify T₁, T_2, X and Y of the following partially constructed character table.

 

X	E	i
T1	1	1
T2	1	Y

 

 

                     Part-C

ANSWER ANY FOUR  QUESTIONS                                                                                       (4 ´ 10 = 40)

 

23. a) Derive the wave function and energy for the particle in a 1-D box.
24. b) For butadiene CH₂=CH-CH=CH₂, take the box length as 7.0Å and use the particle in 1-D box as model to estimate theoretically the wavelength of light absorbed when a pi electron is excited from the highest-occupied to the lowest vacant box level. If the experimental value is 2170Å, comment on your theoretical model. (7+3)
25. (a) Derive the time-independent Schroedinger equation from the time-dependent and prove that the

property such as electron density is time independent although the   wave function describing an

electron is time dependent

1. b) The microwave spectrum of the CN radical shows a series of lines spaced by a nearly constant

amount of 3.798 cm^-1. What is the bond length of CN?         (6+4)

1. a) What are the three approximations Hückel employs in defining the integrals of the  secular

determinant in the case of π electrons?

1. b) Write down the secular determinant by applying Huckel’s method to the allyl cation and obtain the

expressions for the energy levels of the π electrons.    (3+7)

 

1. (a) In solving the H₂⁺ problem using the LCAO method, the lowest energy obtained is  given

by E₊ = (H_AA + H_AB) / (1+S_AB) where A and B refer to the two hydrogen  nuclei. Explain each of the integrals in the above equation and their significance.

1. Show that the wave function describing the 1s orbital of H-atom is normalized,

given:  Y_1s = (1/Öπ) (Z/a₀)^3/2 exp(-Zr/a₀). [Useful integral: ₀^µòxⁿe^-axdx = n!/aⁿ⁺¹]

(6+4)

27. What is a permutation operator? State and illustrate the Pauli Exclusion Principle for the ground state of

He atom that wave functions must be antisymmetric in the  interchange of any two electrons.

 

28. Work out the hybridization for sigma bonding by Boron in BCl₃ molecule using the

following character table.

 

 

	E	2C3	3C’2	σh	2S3	3σv
A’1	1	1	1	1	1	1		x2+y2, z2
A’2	1	1	-1	1	1	-1	Rz
E’	2	-1	0	2	-1	0	(x, y)	(x2-y2, xy)
A”1	1	1	1	-1	-1	-1
A”2	1	1	-1	-1	-1	1	z
E”	2	-1	0	-2	1	0	(Rx, Ry)	(xz, yz

 

 

 

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