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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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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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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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

WD 27

FIRST SEMESTER – April 2009

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date & Time: 28/04/2009 / 1:00 – 4:00   Dept. No.                                                    Max. : 100 Marks

 

 

PART-A

 

ANSWER ALL QUESTIONS          (10 ´ 2 = 20)

 

1. Show that e^ikx is an eigenfunction of the operator p_x = -iħδ/δx. What is the eigenvalue?
2. Give the Laplacian operator in spherical polar coordinates. What are their limits of integration?
3. What is a node? Draw the ψ² plot for the first two lowest energy of the particle in a 1-D box and indicate the nodes.
4. What are the atomic units for length and energy?
5. What are degenerate states? Find the degeneracy of a state having

E = 26h²/8ma² in a 3D cubic box of length a.

6. Explain why an electron in an atom shows a discrete spectrum while a free electron shows a continuous spectrum.
7. What is a ‘fermion’? Give an example.
8. Write the Hamiltonian operator for the H₂⁺ molecular ion defining each term involved in it.
9. Identify the point groups for the following molecules:

(a) C₆H₆    (b) DCH₃         (c) Br₂              (d) [Co(NH₃)₆]³⁺

10. Give an example for the point groups C_s. What are its symmetry elements?

PART-B

ANSWER ANY EIGHT QUESTIONS        (8 ´ 5 = 40)

 

11. Derive the time-independent Schroedinger equation from the time-dependent equation.
12. What is a hermitian operator? Show that the linear momentum operator is Hermitian.
13. When do we say two symmetry operations are in the same class? Illustrate with

a suitable example.

14. The spacing between adjacent rotational lines in the spectrum of HI molecule is 3.924 x 10¹¹s^-1.

Calculate the moment of inertia of HI molecule and the  internuclear spacing if the atomic

masses are H = 1 and I = 127.

15. Explain quantum mechanical tunneling with a suitable example.
16. (a) Show that [d/dx, x] = 1

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

(2+2+1)

17. State the Variation Theorem and prove it with a suitable example. (2+3)
18. Prove that y₀ = (b/p)^1/4exp(-bx²/2) is a normalized wave function. [You may make use of the standard integral ₀ò^¥ exp(-bx²) dx = (1/2)(p/b)^1/2.]
19. Define or explain the three parts that make an atomic term symbol and

formulate the term symbols for the ground state configuration of F atom.

20. 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.

 

21. 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]

 

 

 

 

 

 

22. Applying group theory, find out the nature of the three C-H bonds in CH₃Cl

which belongs to C_3V point group.

 

C_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)

 

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.

8. b) An aliphatic conjugated diene (linear) has an end to end distance of 8.67 Ǻ and it absorbs at

3540Ǻ.

1. Determine the number of double bonds and hence predict the structure of the molecule.
2. If the bond length of C-C is 1.48 Ǻ and C=C is 1.34 Ǻ, calculate the penetration effect.                                     (7+3)

 

24. Discuss the Pauli Exclusion Principle in quantum mechanics applied to He atom in its ground state.                                    (10)
25. Discuss the Molecular Orbital treatment of H₂ molecule and explain the contribution made by Heitler-London and its significance.                                          (10)
26. a) What are the three important approximations that distinguish the HMO method from other

LCAO methods.

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

Obtain expressions for the energy levels and the wave functions.              (3+7)

27. a) Explain Bohr Correspondence Principle with respect to particle in 1-D box model
28. b) Explain the Born-Oppenheimer approximation with an example. (6+4)

 

28. Find the number, symmetry species of the infrared and Raman active vibrations of CCl₄, which

belongs to T_d 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)

 

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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – NOVEMBER 2010

    CH 1808  – QUANTUM CHEMISTRY & GROUP THEORY

 

 

 

Date : 03-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part-A

ANSWER ALL QUESTIONS                                                                                                   (10 ´ 2 = 20)

 

1. Give the Laplacian operator in spherical polar coordinates. What are their limits of integration?
2. Show the wave function Ae^-2x is an eigen function of the operator d²/dx². What is the eigen value?
3. The energy of a particle moving in a cubic box of side ‘a’ is 3h²/2ma². What is its degeneracy?
4. 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.
5. What is the value of [y,p_y]? What is its physical significance?
6. Simple Harmonic Oscillator has zero as one of the quantum numbers while the particle in a box model does not have. Why?
7. What is a boson? Give an example.
8. Write the Hamiltonian operator for the H₂⁺ molecule in atomic units defining each term involved in it.
9. Prove that the operations S₂ and i have the same effect.
10. Predict the trace of C₃

 

Part-B

ANSWER ANY EIGHT QUESTIONS                                                                                        (8 ´ 5 = 40)

 

11. What is a hermitian operator? Show that the wave functions corresponding to two different eigen values of a Hermitian operator are orthogonal.
12. State the variation theorem. Apply it 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.
13. Write the Schrödinger equation for 1-D harmonic oscillator. Verify ψ = (2a/π)^1/4exp(-ax²) is an eigen function of the Hamiltonian operator for the 1-D harmonic oscillator.
14. The force constant of ⁷⁹Br-⁷⁹Br is 240 Nm^-1. Calculate the fundamental vibrational frequency and the zero-point energy of the molecule.
15. What is a Slater determinant? Write the four Slater determinants for the excited state of He atom (1s, 2s).

 

16. Explain briefly with a suitable example:

(a) Bohr’s Correspondence Principle (b) Born-Oppenheimer Approximation.

17. 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]
18. 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.
19. Explain the following with a suitable example:

• Spherical Harmonics (b) Quantum mechanical tunneling. (2+3)

20. What is a term symbol? Explain the origin of the fine structure of the emission spectrum of sodium vapor used in street lighting using term symbols.
21. List down the symmetry elements and symmetry operations of trans-1,3-dibromo cyclobutane and Furan.
22. Identify T₁, T_2, X and Y of the following partially constructed character table.

 

X	E	i
T1	1	1
T2	1	Y

 

 

                     Part-C

ANSWER ANY FOUR  QUESTIONS                                                                                       (4 ´ 10 = 40)

 

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

1. 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)

1. a) What are the three approximations Hückel employs in defining the integrals of the  secular

determinant in the case of π electrons?

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

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

 

1. (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.

1. Show that the wave function describing the 1s orbital of H-atom is normalized,

given:  Y_1s = (1/Öπ) (Z/a₀)^3/2 exp(-Zr/a₀). [Useful integral: ₀^µòxⁿe^-axdx = n!/aⁿ⁺¹]

(6+4)

27. 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.

 

28. Work out the hybridization for sigma bonding by Boron in BCl₃ molecule using the

following character table.

 

 

	E	2C3	3C’2	σh	2S3	3σv
A’1	1	1	1	1	1	1		x2+y2, z2
A’2	1	1	-1	1	1	-1	Rz
E’	2	-1	0	2	-1	0	(x, y)	(x2-y2, xy)
A”1	1	1	1	-1	-1	-1
A”2	1	1	-1	-1	-1	1	z
E”	2	-1	0	-2	1	0	(Rx, Ry)	(xz, yz

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – APRIL 2012

CH 1808 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

Date : 30-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Part–A

Answer ALL questions (10 x 2 = 20)

1. How does ahorizontal plane differ from a vertical plane?
2. Represent the complex number (1 – i) in the Euler form.
3. For what value of A the function Ax²is normalized for 0 ≤ x ≤ 1
4. Find equivalent operator for (A+B)² if the operators A and B commute.
5. Show that [x, d/dx] = -1.
6. How will you apply Born-Oppenheimer approximation to simplify the Hamiltonian for H₂⁺?
7. Account for the origin of the fine structure of the emission spectrum of sodium vapor.
8. State the variation theorem.
9. Prove that the operation S₆³ is equivalent to an inversion operation.
10. What are proper and improper axes of rotation? Give an example for each.

Part-B

Answer any EIGHT questions (8 x 5 = 40)

11. Define Hermitian operator. Give an example.
12. Calculate the wave length of π →π* transition in 1, 3, 5-Hexatriene (C-C and C=C distances are 1.54 and 1.33 Ả , respectively).
13. The inter nuclear distance of D₂ is 0.74Ả. Determine its rotational constant in cm^-1.
14. Get the following normalized polynomial functions:

(i) P₀⁰(cosθ)          (ii) P₁⁰(cosθ)

15. Set up the Hamiltonian for a simple harmonic oscillator
16. Show that [L²,L_x] = 0. Mention its significance.
17. Explain the importance of the integrals H_AAand H_BB obtained for the lowest energy of H₂⁺using LCAO method.
18. Find the radius of the shell where there is a maximum probability of finding the electron. Given: The probability, P = 4πr²e^-2ar.

 

 

 

19. The term symbol of a particular atomic state is ⁶S_5/2. Suggest a possible electronic configuration.
20. Obtain the Pauli antisymmetric wave function for the excited state He atom.
21. Identify the point groups of biphenyl and chloroform molecules mentioning their symmetry elements and operations.
22. The order and the classes of a molecule are 20 and 8 respectively. Identify the number and the dimensions of the irreducible representations.

 

 

 

Part-C

Answer any FOUR questions (4 x 10 = 40)

23. (a) Write expressions for the third levels for Ψ_nand E_n for a particle in 3D

box.

(b) Draw the MO diagram for the π-electrons in 1, 3-butadienyl anion radical.

24. Set up the Schrodinger equation for a rigid rotor and hence solve for its energy and wave function.
25. (a) Define the following.

• Hermite equation
• Hermite polynomials
• Get the normalized functions for the simple harmonic oscillator for its third vibrational level.

26. (a) Show that for anhydrogen like atom, in its ground state, the average distance of the electron from the nucleus is 3/2 times the most probable distance. Given: Ψ_1s = 1/(π)^1/2(Z/a₀)^3/2exp(-Zr/a_o).

(b)  Highlight the features that distinguish the Huckel method from other LCAO methods.

27. (a) Obtain the value of [x, H]. Mention its significance.

(b) Apply Huckel’s method to allyl cation and obtain expressions for the

energy levels.

28. Arrive at the IR and Raman active modes of vibrations of trans-1,2-dibromo ethylene molecule, using the following character table and verify the relevance of mutual exclusion principle.

C2h	E	C2	i	h
Ag	+1	+1	+1	+1	Rz	x2, y2, z2, xy
Bg	+1	-1	+1	-1	Rx, Ry	xz, yz
Au	+1	+1	-1	-1	z	–
Bu	+1	-1	-1	+1	x, y	–

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – CHEMISTRY

FIRST SEMESTER – NOVEMBER 2012

CH 1814 – QUANTUM CHEMISTRY & GROUP THEORY

 

 

Date : 07/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Part–A

Answer ALL questions:                                                                                                      (10 x 2 = 20)

1. State the order and degree of theSchrödinger equation for a particle in a one dimensional box.
2. Normalize exp(ikx) for 0 ≤ x ≤ π
3. Evaluate p_xx².
4. Define moment of inertia.
5. Identify the perturbation term in the Hamiltonian of Helium atom.
6. Write down the Laugerre polynomial for 1S electron.
7. What are variational integral and variational parameters?
8. Prove that the operation S₁₀⁵ is equivalent to S₆³.
9. Identify the point group of m-dichloro benzene.
10. Mention the condition for an improper axis and its inverse to form a class together.

Part-B

Answer any EIGHT questions:                                                                                          (8 x 5 = 40)

11. Derive the expression for linear momentum operator.
12. Calculate the highest translational quantum number for an oxygen molecule in 1mm length to have its thermal energy kT at 298K(k = 1.38 x 10^-23Jmolecule^-1).
13. Calculate the wave length of π →π* transition in 1, 3, 5-hexatriene
14. Define the following

• Associated Legendre equation
• Associated Legendre polynomials
• Legendre polynomials

15. The internuclear distance of the ¹H³⁵Cl molecule is 0.128 nm. Calculate the spacing of the lines in its rotational spectrum in terms of cm^-1.
16. Obtain all the possible term symbols for a ⁴F state.
17. Prove the commutation relation [p²_x, x] = -2iћp.
18. Illustrate the Pauli Exclusion Principle for the ground state of He atom.
19. At what distance from the nucleus is the probability of finding the electron a maximum for a 1S electron in hydrogen?
20. While the order is the same for both C_3v and C_3h point groups, their classes are different – reason out.
21. Allene belongs to D_2d point group. Identify the order and hence the number and dimensionality of the irreducible representations.
22. Set up the secular determinant for allyl radical and obtain its energy levels.

Part-C

Answer any FOUR questions:                                                                                          (4 x 10 = 40)

23. (a) Define the following:

• Closed interval
• Even function
• Orthonormal set of functions.
• Get the following polynomial functions for a rigid rotor:

(i) P₀⁰(cosθ)                        (ii) P₁⁰(cosθ)

24. Calculate the wave length of π →π* transitionof the following molecule:

 

 

(Bond distances: C-C, 0.154 nm; C=C, 0.133 nm)

25. Set up the Schrodinger equation for a rigid rotor and hence solve for its energy and wave functions.
26. State the principle of Perturbation theory and use first order perturbation theory to calculate the energy of a particle in a one dimensional box from x = 0 to x = a with a slanted bottom, such that V_x = V₀ (x)/a.    Given the wave function   Ψ⁽⁰⁾ = (2/a)^1/2 sin (nπx/a).
27. a)Solve the polar angle dependent equation for Hydrogen atom.
28. b) Highlight the important approximations that distinguish the HMO method from other LCAO methods.

 

 

 

 

28. Work out the hybridization scheme for σ bonding by carbon in CH₄ molecule of T_d point group symmetry, using the character table given below.

Td	E	8C3	3C2	6S4	6σd
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	(Rx, Ry, Rz)
T2	3	0	-1	-1	1	(x, y, z)	(xy, xz, yz)

 

 

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