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Preface
When the author took courses in quantum mechanical principles and chemical bonding in graduate school in the early 1970s, the course materials seldomly covered the fascinating interplay between spectroscopy and quantum mechanics, and textbooks of these days devoted the majority of space to derivations and mathematical principles and the discussion of the hydrogen atom and chemical bonding. While an understanding of these subjects is, of course, a necessity for further study, this book emphasizes a slightly different approach to quantum mechanics, namely, one from the viewpoint of a spectroscopist. In this approach, the existence of stationary energy states – either electronic, vibrational, rotational, or spin states – is considered the fundamental concept, since spectroscopy exists because of transitions between these states. Quantum mechanics provides the theoretical framework for the interpretation of experimental data. On the other hand, spectroscopic results provide the impetus for refining theories that explain the results. Classical physics cannot provide this framework, since the idea of stationary energy states violates the laws of classical physics.
Thus, the approach taken here in this book is to present early on, in Chapter 2, how the application of quantum mechanical principles leads necessarily to the existence of stationary energy states using the particle‐in‐a box model system. The third chapter then introduces the concept of spectroscopic transitions between these stationary states, using time‐dependent perturbation theory.
The following chapters are presented in order of mathematical complexity of the Schrödinger equation that describes the problem. The simplest case, the particle in a box, is discussed in Chapter 2. The next subject is the simple harmonic oscillator, for which the eigenfunctions resemble those of the particle in a box, and transitions can be visualized in terms of the discussion in Chapters 2 and 3. In the following discussions (Chapters 5–10), vibrational, rotational, atomic, molecular electronic, and spin spectroscopies will be introduced. These discussions, if possible, start with a classical description, followed by the quantum mechanical equations for wavefunctions and eigenvalues, and the derivation of the selection rules. These selection rules determine the form and information content of the respective spectroscopic techniques. Although space limitations prevent in‐depth discussions of spectroscopic applications to complex molecular systems, all efforts have been made to include molecular systems larger than diatomic molecules (the level of molecular complexity where many textbooks capitulate), since the world we live in mostly consists of more complicated molecules than diatomics.
Thus, in Chapter 5, the concept of the harmonic oscillator (Chapter 4) will be extended to vibrational (infrared and Raman) spectroscopy of polyatomic molecules. This chapter introduces concepts of band shapes, lifetimes, and a quantum mechanical description of molecular polarizability. Next in complexity are the differential equations for a rotational molecule that leads to rotational spectroscopy (Chapter 6). These equations will introduce the quantum mechanical description of the angular momentum and the energy levels of simple and more complicated molecules. The results from the rotational Schrödinger will also be used to solve the radial part of the hydrogen atom Schrödinger equation (Chapter 7). The principles learned from the rotational Schrödinger equation will also be used to introduce the spin eigenfunctions and eigenstates, a subject that leads directly to spin spectroscopy such as nuclear magnetic resonance (NMR), which is discussed in Chapter 8.
Next, the structure of atoms and ions containing more than one electron will be presented. This discussion includes an introduction to atomic spectroscopy and term symbols of electronic states. However, since the main theme of this book is molecular spectroscopy, this chapter only serves as an introduction to these subjects.
Chapter 10 is devoted to electronic spectroscopy of di‐ and polyatomic molecules. Again, as in previous chapters, it is necessary to define the states between which electronic transitions occur. This leads necessarily to the discussion of chemical bonding in terms of molecular orbital theory. Chemical bonding will be discussed to the level that electronic spectra of simple molecules can be explained, but the interaction between vibrational and electronic wavefunctions to produce vibronic states will be discussed in more detail to explain fluorescence phenomena as well as some Raman effects that rely on transitions into vibronic energy levels. Finally, Chapter 11 introduces group theory and the symmetry properties of molecules and the influence of symmetry on the appearance of molecular spectra.
The approach taken here in this book was strongly influenced by an excellent textbook Physical Chemistry by Engel and Reid [1] that was used as a required text in undergraduate physical chemistry courses at Northeastern University. This book emphasizes the unconventional approach taken by the early theorists who are responsible for the field of quantum mechanics as we know it. I gained substantial understanding of the philosophical background of quantum mechanics from this book. What is presented here in Quantum Mechanical Foundations of Molecular Spectroscopy is a similar approach but with much more emphasis on molecular spectroscopy.
Although the present book emphasizes the relationship between spectroscopy and quantum mechanics more than other texts, the author wishes to point out the importance of following up on some proofs and derivations (omitted here) by studying books on “real” quantum mechanics or quantum chemistry. In particular, the one‐ and two‐volume treatments by I. Levine [2, 3] are highly recommended, as well as many other old and new books [4, 5].
The mathematical requirements for understanding this book do not exceed the level achieved after a three‐semester sequence of calculus, and all efforts have been made to provide examples and problems that will illuminate the mathematical steps. Most importantly, although some derivations are presented, the goal is not to lose sight of what quantum mechanics does for spectroscopy in the mathematical complexities.
Boston, August 2019
Literature references for the Preface are at the end of the Introduction.
Introduction
This book, Quantum Mechanical Foundations of Molecular Spectroscopy, is based on a graduate‐level course by the same name that is being offered to first‐year graduate students in chemistry at the Department of Chemistry and Chemical Biology at Northeastern University in Boston. When I joined the faculty there in 2005, I revised the course syllabus to emphasize the philosophical underpinnings of quantum mechanics and introduce much more of the quantum