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

AD 16

FIRST SEMESTER – NOV 2006

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

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

 

 

 

 

 

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

 

1. Show that Ae^-ax is an eigen function of the operator d²/dx². What is the eigen value?
2. Define an orthonormal function.
3. The energy of a particle moving in a cubic box of side ‘a’ is 3h²/2ma². What is its degeneracy?
4. What is a node? Draw the radial distribution plot for 3s orbital of H-atom and indicate where the nodes are.
5. What is the value of [x,p_x]? What is its physical significance?

1. What is a fermion? Give an example.
2. Set up the Schroedinger wave equation to be solved for the rigid rotor in spherical

polar coordinates.

1. Define the Variation Principle and mention its significance.
2. Identify the point groups for the following molecules:

(a) POCl₃         (b) SF₆   (c) Br₂            (d) Ni(CN)₄^2- (square planar)

10. The following irreducible representations are parts of the C_4v and C_2h character

tables. What do x and y in these tables mean and in what way they differ from

each other?

 

C_4v    E    2C₄    C₂     2σ_v   2σ_d                   C_2h           E     C₂           i      σ_h

 

 

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

 

 

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

 

11 Apply variation theorem 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.

 

12. What is a hermitian operator? What is its significance in quantum mechanics?

Show that the eigen value of a hermitian operator is real

 

13. Show that the wave functions describing 1s and 2s electron for He atom are

orthogonal to each other, given Ψ_1s = (1/4π)^1/2 2(2/a₀)^3/2 exp(-2r/a₀)   and

Ψ_2s = (1/4π)^1/2 (1/a₀)^3/2 (2 – 2r/a₀) exp (-r/a₀).

 

14. What is a Slater determinant? Write the four Slater determinants for the excited state

of He atom (1s,2s)

 

 

 

 

15. The force constant of ⁷⁹Br is 240 Nm^-1. Calculate the fundamental vibrational

frequency and the zero-point energy of the molecule.

 

 

16. Explain briefly with a suitable example:

(a) Bohr’s correspondence principle (b) Born –Oppenheimer approximation.      (3+2)

 

17. The wave function of 1s orbital of Li²⁺ is given by 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^¥ xⁿ e^-qx dx = n! / qⁿ⁺¹]

 

18. (a) Show that [x, d/dx] = -1

(b) What are the values of [x,p_x] and [L²,L_x]? What is their physical significance?

(2+2+1)

 

19. 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.

 

20. Explain the following with a suitable example:

(a) A ‘Class’ in group theory (b) Principle of mutual exclusion                  (3+2)

 

21. Explain ‘quantum mechanical tunneling’ with a suitable example.

 

22. Find the number, symmetry species of the infrared and Raman active vibrations of

CCl₄, which belongs to T_d point group by reducing the following reducible

representation. State how many of them are coincident.

T_d          E       8C₃            3C₂             6S₄            6σ_d

GCCl₄         15        0              -1                 -1             3

 

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) Derive the wave function and energy for the particle in a 1-D box.

(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.

 

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

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

ethylene and hence obtain its energy levels and corresponding wave functions.

 

25. 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.

 

26. (a) Derive the time-independent Schroedinger equation from the time-dependent

equation and show that the property such as electron density is time independent

although the wave function describing an electron is time dependent.

(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)

 

27. (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

(b) Outline the salient features of VB (Heitler-London) theory as applied to H₂

                 molecule                                                                                                         (5+5)

 

28. The MoO₄ belongs to the T_d point group. Find the reducible representation for the

molecule using the four Mo – O bonds as bases and reduce this into irreducible

representation using the T_d character table given below and show that the bonds in

MoO₄ are more likely to be sd³ 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)

 

 

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