Introduction

We now discuss the complexities that arise for polyatomic molecules. Most of our discussion for diatomic molecules will apply to polyatomic molecules as well, but with two additions. One is the larger number of vibrational degrees of freedom, and the other is the existence of other modes of vibration. For diatomic molecules, the only possible mode of vibration is bond stretching, but the presence of additional internal coordinates (bond angles) makes other kinds of motion, such as bond bending and torsional motion, possible for polyatomic molecules. As in the previous chapters, we will first do an analysis using classical mechanics.

NORMAL MODES OF VIBRATION OF CARBON DIOXIDE

We begin by taking up the normal mode analysis of the linear symmetrical molecule, carbon dioxide, which is easily amenable to both classical and quantum mechanical treatment. From our normal mode analysis for heteronuclear diatomic molecules, we had concluded that the problem is reduced to the diagonalization of the Lagrangian. While the kinetic energy matrix is already in diagonal form, the potential energy matrix is not diagonal because of coupling of motions of the atoms. In order to simultaneously diagonalize both matrices, we had transformed the displacement coordinates to mass-weighted coordinates and then diagonalized the potential energy matrix in this coordinate system to obtain the eigenvalues and eigenvectors. The eigenvalues are related to the natural vibrational frequencies (one in the diatomic case) and the corresponding eigenvectors to the motions in these normal modes. We now apply the method of normal coordinate analysis to another linear, but polyatomic, molecule, carbon dioxide (O=C=O). We again constrain the atoms to move in the bond direction only, which we shall call the z direction.

We label the atoms 1, 2 and 3. Since both bonds are of the same type, i.e., C=O, we would expect both to have identical force constants, say k.

This book is primarily intended for post-graduate students of science, but is simple enough for an undergraduate student to understand. Every chapter begins with simple concepts related to the topic of the chapter, and gradually Quantum Mechanical and Group Theoretical treatments are introduced for a deeper understanding. I have tried to keep the language simple and have introduced new concepts one at a time, so that the reader is not overwhelmed by too many new ideas at the same time.

Spectroscopy is so vast and new concepts are rapidly emerging, so that it is not possible for a book to be complete. This book does not pretend to be complete—but it does try to cover the underlying principles thoroughly so that the reader should not face difficulty in applying these principles to his own problem.

Spectroscopy cannot be understood without a thorough knowledge and understanding of quantum mechanics. Though a number of Quantum Mechanics principles are scattered throughout the book, the reader is advised to first take at least an elementary course on quantum mechanics. The reader must also familiarize himself/herself with Group Theory—at least to the extent of assignment of molecular point groups, calculation of direct products, projection operator techniques, etc. There are a number of excellent texts, namely Cotton, Schonland, Ladd, to name a few. Again, a brief description of these techniques is given as and when required. Though I was advised by a reviewer of the book to include a chapter on Group Theory, it would have added to the volume and cost of the book, because there is nothing I could have removed from the other chapters in order to keep the page number around 400.

One other point—all chapters are arranged in sequence, so that a concept introduced in a certain chapter is used in one of the next chapters—so you cannot expect to understand, say the third chapter, without first reading the first two chapters.

INTRODUCTION

“Science is spectral analysis. Art is light synthesis”, so wrote Karl Kraus, Austrian writer. Light has intrigued both poets and scientists. What exactly is light? What effect does it have on matter? These are questions that have baffled scientists for many years, prompting Einstein in 1917 to say “For the rest of my life I will reflect on what light is”. The branch of science that deals with the study of electromagnetic radiation (of which visible light is a part) and its interaction with matter is called spectroscopy. The word is derived from the Latin: spectron – spectre (ghost or spirit), or the Greek: σκοπειν – to see. This literally means that in spectroscopy, you do not look directly at the molecule – the matter – but what you see is its ‘ghost’ or image. To begin our study, we must, therefore, first discuss the nature of electromagnetic radiation and matter, and then the interaction between the two.

We start this chapter by giving basic formulae and definitions relating to waves, including travelling waves. We then go on to the wave description of electromagnetic radiation and its manifestations, and then discuss the properties emerging from a particulate description of radiation. The entire electromagnetic spectrum, its divisions and sub-divisions, the kind of spectroscopy observed in each region, are the topics of the next section. The populations of energy levels play an important role in the observed intensities. Einstein's coefficients and their interrelation are introduced in this chapter, but the quantum mechanical treatment is reserved for the next chapter. We wind up this chapter with a discussion of line shapes and broadening, followed by a brief introduction to Fourier transform spectroscopy, the almost magical transformation of a time decay to a line width, and the experimental recording of spectra.

INTRODUCTION

As was stated in Chapter 1, spectroscopy is a study of the interaction of matter with radiation. In order to understand their mutual interaction, we must therefore characterize both radiation and matter. The starting point is quantum mechanics, because the origin of quantum mechanics was based on unexplained phenomena related to both the nature of electromagnetic radiation and that of matter. In fact, quantum mechanics owes its birth to some puzzling results of spectral and other experiments. We may say that “spectroscopy is quantum mechanics in action.” We therefore begin our study of quantum mechanics with a recapitulation of its postulates. An understanding of the postulates is also essential for our development of the time-dependent perturbation theory and also forms the basis for the connection between theory and experiment. Indeed, spectroscopy is the experimental verification of the laws of quantum mechanics.

Quantum mechanics is based on a few postulates (from the Latin postulatum), meaning a truth that does not need any further proof, because it is obvious by itself, and its success lies in the fact that no experiment till date has proved it wrong. There is no agreement of how many axioms one needs to describe the machinery of quantum mechanics; however, the six given below is an acceptable number.

INTRODUCTION

We now move on to the next topic, i.e., Raman spectroscopy, discovered in 1928 by the great Indian physicist, Sir C. V. Raman, who was awarded the Nobel Prize in 1930 ‘for his work on the scattering of light and for the discovery of the effect named after him’. All the spectroscopies studied so far were based on the absorption of light. In contrast, Raman spectroscopy is based on scattering of light. Hence, the Bohr condition is not required for observing the Raman effect.

When a molecule is illuminated by visible or ultraviolet radiation, it gets excited to a virtual state from which it may drop to the same, lower or higher energy level. If it drops to the same energy level, the scattered radiation has the same frequency as the incident radiation (elastic scattering), and this is called Rayleigh scattering, responsible for the blue colour of the sky and the red colour of sunset. Lord Rayleigh showed that the intensity of the scattered radiation is inversely proportional to the fourth power of the wavelength, i.e., IRayleigh ∝ 1/λ4, so blue light (shorter wavelength) is scattered more than red light (longer wavelength). When we view sunset through several layers of the atmosphere, what we observe is red light, because the blue part of the visible (white) light is scattered away sideways.

Alternatively, the molecule may drop to a higher energy with absorption of energy from the incident radiation, leaving the scattered light at lower energy (inelastic scattering) and it is then observed at lower wavenumber. Such scattering is called Stokes scattering. In case the molecule is already in an excited internal energy level, it may return to the ground state, and in that case the scattered radiation will be at higher wavenumber than the incident radiation. This kind of scattering is known as anti-Stokes scattering, and collectively Stokes and anti-Stokes scattering account for Raman scattering (Figure 7.1).