LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – CHEMISTRY

AD 11

FIRST SEMESTER – NOV 2006

         CH 1800 – QUANTUM CHEMISTRY

 

 

 

Date & Time : 26-10-2006/1.00-4.00          Dept. No.                                                              Max. : 100 Marks

 

 

 

Part-A   ANSWER ALL QUESTIONS (10 x 2 = 20)

 

1. Explain the Ultraviolet catastrophe.
2. The size of an atomic nucleus is 10^-14 Calculate the uncertainty in momentum of the electron if it were to exist inside the nucleus.
3. The accepted wave function Φ for a rigid rotor is Nexp(±imφ) for 0≤φ≤2π.

Determine N.

4. Show that the function Ψ(x,y,z) = cosax cosby coscz (where a,b,c are constants) is an

eigen function of the Laplacian Operaator ∇². What is its eigen value?

5. Show that the energy E = 14h²/8ma² of a particle in a cubic box of side ‘a’ is triply degenerate.
6. What is a well-behaved or acceptable wave function in quantum mechanics?
7. Explain the radial plot and the radial probability density plot for 2s electron.
8. What is Born-Oppenheimer approximation?
9. Write the Hamiltonian operator for the H₂⁺ molecular ion in atomic units defining

each term  involved in it.

10. Write the Slater determinant for the ground state of He atom which takes into account the antisymmetric condition of Pauli Exclusion Principle for electrons.

 

 

Part-B   ANSWER ANY EIGHT QUESTIONS (8 x 5 = 40)

 

 

11. Discuss the failures of classical mechanics and the success of quantum theory in the explanation of black body radiation. .
12. What are quantum mechanical postulates? Explain briefly any two of the postulates.
13. The work function for Cesium is 2.14 eV. What is the kinetic energy and the speed of the electrons emitted when the metal is irradiated with the light of 300 nm?
14. What is a hermitian operator? Show that the eigen value of a hermitian operator is real.                                                                                                                 (2+3)
15. The normalized wave function for the 1s orbital of Hydrogen atom is

ψ = (1/√π)(1/a₀)^3/2exp(-r/a₀). Show that the most probable distance of the electron is    a₀.

16. Explain Bohr’s correspondence principle with a suitable example.
17. Write the Schroedinger equation for 1-D harmonic oscillator. Verify ψ = (2a/π)^1/4exp (-ax²) is an eigenfunction of the Hamiltonian operator for the 1-D harmonic oscillator.

 

 

18. What is a node? Draw the first three wave functions and probability plots for the Particle in a 1-D box and Simple Harmonic Oscillator models and compare them.

(1+4)

 

 

 

19. The spacing between adjacent rotational lines in the spectrum of HCl molecule is

6.33 x 10¹¹s^-1. Calculate the moment of inertia of HCl molecule and the internuclear spacing  if the atomic masses are H = 1.008 and Cl = 34.97.

20. (a) Explain the conditions under which an electron may give a continuous or a discrete spectrum.
    • The continuous probability distribution Gaussian function is given by

f(x) = Aexp(-x²/2a²) with interval (-∞, +∞). Show that <x> = 0.                (2+3)

21. 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.
22. With a suitable example explain the quantum mechanical tunneling.

 

 

Part-C   ANSWER ANY FOUR QUESTIONS (4 x 10 = 40)

 

23. (a) Write the Schroedinger equation to be solved for H atom and solve it for its energy using a simple solution, which assumes the wave function to depend only on the distance r and not on the angles θ and φ

(b) In the Compton experiment, a beam of x-rays with wave length 0.0558 nm is scattered through an angle of 45˚. What is the wavelength of the scattered beam?

(7+ 3)

24. (a) Apply the variation method to get an upper bound to the ground state energy of hydrogen atom using the trial function φ(r) = exp(-αr²). Compare your result with the true value.

(b) Discuss the Pauli Principle of antisymmetric wave function.             (6+4)

25. Set up the Schroedinger equation for a rigid rotor in polar coordinates and solve it for its energy and wave function.

26  (a) Set up and solve the Schroedinger wave equation for a 1-D particle in a box model for its energy and wave function.

(b) For the hexatriene molecule, calculate λ_max on the basis of particle in a one dimensional box of length equal to 7.3Ǻ                                                     (7+3)

27. Discuss the Molecular Orbital treatment of H₂ molecule and explain how the Valence

Bond (Heitler-London) method overcomes some of the difficulties of MO theory.

28. (a) What are the three important approximations of Huckel-LCAO-MO theory?

(b) Using this theory set up the secular equation and secular determinants for

allyl radical and hence obtain its energy levels.                                            (3+7)

 

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