LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

LM 28

FIRST SEMESTER – APRIL 2007

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date & Time: 25/04/2007 / 1:00 – 4:00         Dept. No.                                                       Max. : 100 Marks

 

 

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

 

1. Show that e^ax is an eigen function of the operator d²/dx² and find the corresponding eigen value.
2. What is a linear operator? Explain it with a suitable example.
3. What is zero-point energy? What is its value for a particle in a 1-D box?
4. Write the time-dependent Schroedinger wave equation for a single particle in 1-D in stationary state.
5. Draw the radial probability density curves for 3s, 3p, 3d orbitals and mention the number of nodes possible for each.
6. Calculate the moment of inertia of ¹H⁸⁰Br, which has an equilibrium bond length of 1.41Å.
7. Explain the following with an example each: (a) a boson (b) a hartree
8. What is quantum mechanical tunneling? Give an example.
9. Identify the point groups for the following molecules:

(a) HCl   (b) CS₂  (d) CHCl₃   (d) SF₆

10. What do x and y represent in the following two irreducible representations and in what way they differ from each other?

C_3v    E    2C₃    3σ_v                                     C_2h           E    C₂         i      σ_h

 

 

E      2      -1     0          (x, y)                     B_u        1     -1     -1     1        x,y

 

 

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

11. Draw the correlation diagrams for F₂ and CO molecules and comment on the nature of bonds.
12. What is a Hermitian operator? Show that the linear momentum operator is Hermitian.
13. Discuss the similarities and the differences between the particle in a box and the Simple Harmonic Oscillator models in terms of its wave function and its energy.
14. Explain with an example: (a) Bohr’s Correspondence Principle (b) Born-Oppenheimer Approximation
15. 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.

16. 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.

17. Ψ = (2a/π)^1/4 exp(-ax²) is an eigen function of the Hamiltonian operator for the 1-D SHO. Find the eigenvalue E and express it in terms of the classical frequency ν, where ν = (1/2π)√(k/m) and the constant a = (π/h)√(km).
18. What is an atomic term symbol? Write the term symbols for the excited configuration 1s²2s²2p¹3p¹ of carbon.
19. The bond length of the C-N bond in ¹²C¹⁴N is 1.17 x 10^-10 m and the force constant is

1630 Nm^-1. Calculate the value of its fundamental frequency and its rotational constant.

20. 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.
21. Evaluate the commuters [y, p_y] and [L², L_z]. What can you say about their significance?
22. The reducible representation obtained using the four C-Cl bonds in CCl₄ as bases is

T_d    E       8C₃        3C₂        6S₄         6σ_d

4     1         0         0         2

Show by reducing this into irreducible representation using the T_d character table given below that the bonds in CH₄ are more likely sp³ hybrids.

T_d          E          8C₃      3C₂      6S₄       6σ_d

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         (R_x,R_y,R_z)

T₂          3          0          -1         -1         1         (x,y,z)              (xy,xz,yz)

 

 

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

 

23. a) Set up the Schrödinger equation for a particle in 1-D box and solve it for its energy and wave function.
24. b) A cubic box of 10Å on the side contains 8 electrons. Applying the simple particle in a box model, calculate the value of ΔE for the first excited state of this system.                (7+3)

 

24. (a) The wave function of 1s orbital of He⁺ 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 (3/4)a₀.

 

 

(b) State and explain the significance of the Variation Principle with an example.

(5+5)

 

25. (a) Discuss the Pauli Exclusion principle applied to electrons in quantum mechanics.

(b) Explain why when an electron is bound to a system it gives a discrete spectrum while a free particle (as in the case of ionization of atoms) gives a continuous spectrum                                                                                                (7+3)

26. a) What are the three important approximations that distinguish the HMO method from other LCAO methods.
27. b) Write down the secular determinant obtained on applying Hückel’s method to the

butadiene molecule and obtain there from expressions for the energy levels. (3+7)

27. Discuss the Molecular Orbital treatment of H₂ molecule and explain how the Valence Bond (Heitler-London) method makes up for what MO theory lacks. (10)

 

28. Find the number, symmetry species of the infrared and Raman active vibrations of NH₃, which belongs to C_3V point group. State how many of them are coincident.

(You may, if you wish, use the table of f(R) given below for solving this).

 

Operation:       E     σ       i       C2      C3     C4     C5         C6     S3       S4      S5       S6     S8               f(R):       3       1    -3      -1       0       1     1.618      2      -2      -1     0.382     0    0.414   For any Cn, f(R) = 1 + 2cos(2π/n),            For any Sn, f(R) = -1 + 2cos(2π/n)

 

 

D_3V      E          2C₃      3σ_v

 

 

A₁        1          1         1              z                      x² +y², z²

A₂        1          1         -1             R_z

E          2          -1         0         (x,y) (R_x,R_y)        (x²-y²,xy) (xz,yz)

 

 

 

 

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