CH 1800 – QUANTUM CHEMISTRY

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

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

The peak in the sun’s emitted energy occurs at about 480 nm. Assuming it to behave as a black-body emitter, find the temperature of the sun’s surface.
What is the de Broglie wavelength of a body of mass 1g traveling at a speed of 100 Kms^-1?
Give the Laplacian operator in spherical polar coordinates. What are their limits of integration?
Show that e^-ax is an eigen function of the operator d²/dx². What is the eigen value?
What is a node? Draw the radial distribution plot for 2s orbital of H-atom and indicate where the nodes are.
Define a ‘fermion’ and give an example.
The energy of a particle moving in a cubic box of side ‘a’ is 12(h²/8ma²). What is its degeneracy?
Indicate the difference in the nature of the spectrum of an electron in an atom and the same electron when it is freed from the atom by ionization.
What is Born-Oppenheimer approximation?
Write the Hamiltonian operator for the H₂ molecule in atomic units defining each term involved in it.

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

What is UV catastrophe? Explain how Planck explained the black body radiation.
Draw the correlation diagrams for F₂ and CO molecules and explain the nature of bonds.
What is a hermitian operator? Show that the eigen value of a hermitian operator is real.
Explain Bohr’s correspondence principle with a suitable example.
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.

Show that [d/dx, x] = 1

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

What are quantum mechanical postulates? With an example discuss the postulate regarding average or expectation value of an observable?

Show that the zero point energy for a particle in a 3-D box is three times that of the one in 1-D box.
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.
The force constant of ⁷⁹Br⁷⁹Br is 240 Nm^-1. Calculate the fundamental vibrational frequency and the zero-point energy of the molecule.
(a) What is a term symbol?

(b)Taking an example of an atomic term symbol explain what information you get from the three

parts of the symbol.

Explain quantum mechanical tunneling with a suitable example.

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

(a) Discuss Einstein’s theory photoelectric effect

(b) The work function for Cesium is 2.14 eV. What is the kinetic energy and the speed of the

electrons emitted when the metal is irradiated with light of 300 nm? (7+3)

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.

25 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.

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

(b) An aliphatic conjugated diene (linear) has end to end distance of 8.67 Å and it absorbs at 3540 Å. Determine the number of double bonds and hence predict the structure of the molecule. (7+3)

What is Variation theorem and its usefulness? 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. Give your comment on the result. (4+6)
(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 allyl cation and hence

obtain its energy levels. (3+7)

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