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)

 

 

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DB 23

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – November 2008

    CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date : 08-11-08                   Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

ANSWER ALL QUESTIONS                                                                                                (10 ´ 2 = 20 marks)

 

1. Show that the function y(x,y,z) = cosax cosby coscz (where a,b,c are constants) is an eigen function of the Laplacian Operator Ѳ. What is its eigen value?
2. Define a ‘well-behaved function’ in quantum mechanics.
3. In the far infrared spectrum of ³⁹K³⁵Cl, an intense absorption line occurs at 378.0 cm^-1. Calculate the force constant.
4. What is the difference in the nature of the spectrum when an electron is bound to a system and when it is free? Justify your answer quantum mechanically.
5. Show that <x> for v=0 for a 1D-SHO. ψ₀ = (a/π)^1/4 exp(-ax²/2).
6. Explain with an example: (a) a boson (b) a hartree
7. What is a node? Sketch a rough graph of y² for quantum numbers n=3 for a particle in a

1D box model and indicate how many nodes are present in it.

8. Write the Hamiltonian operator for the H₂⁺ molecule defining each term involved in it.
9. Identify the point groups for the following molecules:

(a) H₂              (b) CHCl₃          (c) CH₂Cl₂          d) [Co(NH₃)₆]³⁺

10. Give the meaning of the following letters with their subscripts, which represent the irreducible representations in the character table: (a) A_g (b) E_u

PART-B

ANSWER ANY EIGHT QUESTIONS                                                                                  (8 ´ 5 = 40 marks)

 

11. What is a hermitian operator? Show that the wave functions corresponding to two

different eigen values of a Hermitian operator are orthogonal.

 

12. The high temperature microwave spectrum of KCl vapor shows an absorption at

7687.94 MHz that can be identified with J=0 to J=1 transition of ³⁹K³⁵Cl molecules in

the lowest v=0 vibrational state. Calculate the bond length of KCl and the spacing between

adjacent rotational lines in the spectrum.

 

13. Explain briefly with a suitable example: (a) quantum mechanical tunneling

(b) Principle of Mutual Exclusion                                                                                                         (3+2)

 

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

(b) What are the values of [y,p_y] and [L²,L_y]? What is their physical significance?                                          (2+2+1)

 

15. What is Slater determinant? Taking He atom in its excited state (1s¹, 2s¹) write the

four Slater determinants.

 

16. y = (2a/p)^1/4exp(-ax²) is an eigen function of the hamiltonian operator

H = – (h²/8p²m) d²/dx² + (1/2) kx² for the 1-D Simple Harmonic Oscillator.

1. a) Find the eigenvalue E of HY = EY
2. b) Show that the above obtained eigen value in terms of the classical frequency

n = (1/2p)Ö(k/m) and the constant a = (p/h)(km)^1/2 is E = (1/2)hn.                                                          (4+1)

 

17. With a neat diagram explain Bohr’s Correspondence Principle.

 

18. 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.

 

19. (a) The point group of staggered confirmation of ethane is D_3d. What do these letters

D, 3 and d in this point group represent?

(b) Formulate the term symbols for the ground state configuration of F atom.                                         (3+2)

 

20. Draw the radial probability distribution curves for 3s, 3p and 3d orbitals and discuss the

meaning and significance of the number of  maxima and minima found there.

 

21. Discuss the Pauli Exclusion Principle applied to electrons taking He atom in its ground state as example.

 

22. The reducible representation obtained using the four Mn-O bonds in MnO₄ 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 interpret the nature of the bonds in MnO₄ using group theory.

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 marks)

 

23. a) Explain the use of Born-Oppenheimer approximation with a suitable example.
24. b) 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.                                                                               (6+4)

 

24. a) Derive the wave function and energy for a particle in 1-D box.
25. b) A cubic box of 12 Å on the side contains 10 electrons. Applying the simple

particle in a box model calculate the value of ΔE and the corresponding wave

length for the first excited state of this system.                                                                                   (7+3)

 

25. (a) State the Variation Theorem. Apply it to the problem of particle in a 1-D box of

length a, by using the trial function ψ = x(a-x). Compare your result with the true

value.

(b) Define and explain the overlap, coulomb and resonance integrals which are

found in solving H₂⁺ problem using the LCAO method?                                                                    (6+4)

 

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

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

theory.                                                                                                                                                     (10)

 

27. a) What are the three important approximations that the Huckel MO method uses for the

treatment of π-orbitals in conjugated systems?

1. b) Write down the secular determinant obtained on applying Huckel’s method to

allyl  anion. Obtain there from expressions for the energy levels and the wave

functions.                                                                                                                                              (3+7)

 

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

Boron trichloride (BCl₃), 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)

 

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