Chapter 9 - Vibrational Rotational Spectroscopy

Abstract

Molecular vibrations are explained by classical mechanics using a simple ball and spring model, whereas vibrational energy levels and transitions between them are concepts taken from quantum mechanics. The quantum mechanics of the translation, vibration, and rotation motions are explored in detail. In addition, the expression for the vibration-rotation energies of diatomic molecule for the harmonic and anharmonic oscillator models is introduced. We then identify the Schrödinger equation for the vibration system of n-atom molecules. We elucidate how to obtain, monitor, and explain the vibrational-rotational excitations, find the quantum mechanical expression of the vibrational and rotational energy levels, predict the frequency of the bands, and compare between harmonic and anharmonic oscillation and between rigid and nonrigid rotor models for the possible excitations. Lagrange’s equation is used to show the change in the amplitude of displacement with time. We also examine how to calculate the relative amplitudes of motion, and the kinetic and potential energies for the vibrational motions of N-atom molecule. The significance of the force constants and how to calculate the force constants using GF-matrix method are discussed. The general steps to determine the normal modes of vibration, and the symmetry representation of these modes, are outlined. We examine the relationship and the differences among the Cartesian, internal, and normal coordinates used to characterize the stretching vibrations. We then show why the normal coordinates are used to calculate the vibrational energy of the polyatomic molecules. In this part, we also focus on how the molecules interact with the radiation and the chemical information obtainable by the measurement of the infrared and Raman spectroscopy. Only the electric waves interact significantly with molecules and are important in explaining infrared absorption and Raman stretching. The requirements and the selection rules for the allowed vibrational and rotational excitations are explored. We investigate the relationship between the center of symmetry and mutual exclusion rule, how to distinguish among isomers, binding modes, and define the forms of the normal modes of vibration, and which of these modes are infrared and/or Raman active.