GH 28

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – APRIL 2008

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date : 03-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

                                              ANSWER ALL QUESTIONS                      (10 ´ 2 = 20)

1. For the wave function Ψ(φ) = Ae^imφ, where m is an integer, for 0≤φ≤2π. Determine A so that the wave function is normalized.
2. Give the Laplacian operator in spherical polar coordinates. What are their limits of integration?
3. 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.
4. The energy of a particle moving in a 3-D cubic box of side ‘a’ is 26h²/8ma². How many degenerate energy levels are there in this state?
5. Simple Harmonic Oscillator has zero as one of the quantum numbers while the particle in a box model does not have. Why?
6. What is the value of [y,p_y]? What is its physical significance?
7. Write the Hamiltonian operator for the H₂⁺ molecule in atomic units defining each term involved in it.
8. Explain ‘mutual exclusion principle’ with an example.
9. Identify the point groups for the following molecules:

(a) HBr              (b) Cl₂       (c) IF₅        (d) C₆H₆

10. Explain the meaning and significance of xy, yz, and zx in the T₂ representation of

the T_d point group as shown below

T_d      E          8C₃      3C₂      6S₄       6σ_d                                               

T₂      3          0          -1         -1         1          (xy,yz, zx)

 

PART-B

                                      ANSWER ANY EIGHT QUESTIONS               (8 ´ 5 = 40)

11. What are quantum mechanical postulates and their significance? Explain any two of them in detail.
12. Derive the time-independent Schroedinger equation from the time-dependent

equation.

13. What is a hermitian operator? Show that the wave functions corresponding to two different eigen values of a Hermitian operator are orthogonal.
14. 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.
15. If the fundamental frequency of ⁷⁹Br⁷⁹Br is 9.634 x 10¹² Hz, calculate the zero-point energy and the force constant.
16. What are nodes? With a neat diagram explain Bohr’s Correspondence Principle.
17. What is a well-behaved function? The continuous probability distribution Gaussian function is given by f(x) = A exp(-x²/2a²) with the interval (-¥, +¥).

Show that <x> = 0.                                                                                        (2+3)

18. 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]
19. 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.
20. Explain the following with a suitable example:

• Spherical Harmonics (b) Atomic term symbol                                 (3+2)

21. Explain the concepts of ‘groups’ and ‘classes’ in group theory with suitable example.
22. The reducible representation obtained using the four C-H bonds in CH₄ as bases is

T_d    E       8C₃        3C₂        6S₄         6σ_d

4     1         0         0         2

Reduce this into irreducible representation using the T_d character table given below and show that the bonds in CH₄ are more likely to be 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 ´ 10 = 40)

 

23. a) Set up the Schroedinger equation for a particle in 1-D box and solve it for its energy and wave function.
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) 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) Apply the variation method to get an upper bound to the ground state energy of particle in a 1-D box using the trial function ψ = x(a²-x²), where a is the length of the box. Compare your result with the true value.                                                  (6+4)

25. (a) State and illustrate the Pauli Exclusion Principle for the ground state of He atom.

• Write the four Slater determinants for the excited state of He (1s¹, 2s¹). (7+3)

26. a) What are the three important approximations that distinguish the Huckel MO

method from other LCAO methods?

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

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

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) Explain quantum mechanical tunneling with a suitable example                 (6+4)

28. Find the number, symmetry species of the infrared and Raman active vibrations of Boron trifluoride (BF₃), which belongs to D_3h point group.

(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_3h   E          2C₃      3C₂      σ_h         2S₃       3σ_v

A₁‘   1          1         1         1         1         1                          x² +y², z²

A₂‘   1          1         -1         1         1         -1         R_z

E’     2          -1         0         2         -1         0         (x,y)         (x²-y²,xy)

A₁”  1          1         1         -1         -1         -1

A₂”  1          1         -1         -1         -1         1         z

E”    2          -1         0         -2         1         0         (R_x,R_y)     (xz, yz)

 

 

Go To Main Page

Latest Govt Job & Exam Updates:

View Full List ...