Quantum Chemistry and Chemical Bonding — DU B.Sc (hons) Chemistry Previous Year Question Papers

Physical constants Atomic mass unit 1.66 x 10-27 kg Planck's constant 6.626 x10-34 J s Velocity of Light 3x108 ms' Boltzmann constant : 1.381 ×103JK~ Mass of Electron : 9.1 × 10-31 kg Standard Integrals n! 2ntiar = form >1 1. (a) (i) Write the properties of a function that make it acceptable as a wave function solution of Schrodinger wave equation. (i) A particle in a one-dimensional box moving under a constant potential in a space has no boundaries. Will the energy be quantized in such a case? Explain. (b) Apply the variation theorem to the wave function f(x) =x L? - x2), where 0 ≤× ≤ L to a particle of mass m in a one-dimensional box and estimate the ground state energy. Is the wave function a suitable choice for a particle in a one-dimensional box problem? (c) Naphthaiene may be considered to be a rectangle of length 0.8 nm and breadth 0.4 nm. Write the wave function corresponding to the highest occupied level for naphthalene. Using free electron model calculate the energy required by the electron to go to the first excited state. 2. (a) A particle of mass 'm', attached to a flexible spring, undergoes simple harmonic oscillations. (1) Give the expression for energy level based on quantum mechanical treatment. (ii) What is the energy difference (AE) between two successive energy levels? (ill) What is the zero point energy of the system? Explain its significance. (b) Given that the ground state normalized wave function for the harmonic oscillator is va ½ (i) Determine the value of r at which the ground-state function of the harmonic oscillator exhibits the maximum. Predict the value of x at which the probability of finding the particle is maximum. Compare this value with that predicted by the classical treatment. (ii) Evaluate the expectation value of kinetic energy for harmonic oscillator in ground state. 3. (a) What are Hermitian operators? Show that the eigenvalues of a Hermitian operator are always real. (b) Determine whether the following functions are eigen functions of the operator, Lz = -it • Evaluate the eigen values if applicable. (c) Find the commutator of position operator (f) and momentum operator (Px). What is its physical significance? State the name of the principle it verifies. 4. (a) Write the simplest trial wave functions for H2 molecule using VBT and MOT. Compare the merits and demerits of the above two methods. (b) Calculate the bond order of the following molecular ions and compare the strength of their bonds with their respective neutral molecules. (c) Explain the molecular orbital formation in the Lit molecule using the LCAO approach with the help of MO diagram. Discuss the overlap of atomic orbitals involved and indicate which combinations lead to bonding and antibonding orbitals. 5. (a) A rigid rotator consists of two particles of mass m, and mr joined by a rigid rod of length, r. Based on classical treatment, show that the total energy of a rigid rotator is L'/21, where