Vibrational spectroscopy quantum-mechanical treatment

Molecular spectroscopy is now a mature field of study. It is, however, difficult to find references superior to the classic treatise written by Herzberg nearly 50 years ago (1). The origin of vibrational spectra is usually considered in terms of mechanical oscillations associated with mass of the nuclei and interconnecting springs (9). Vibrational spectroscopy considers the frequency, shape, and intensity of internuclear motions due to incident electromagnetic fields. In the harmonic approximation, the vibrational bands are associated with transitions between nearest vibrational states. When higher order transitions, resonance, and coupling between vibrational motions require analysis, quantum mechanical treatment is mandated (1). Improvements and advancements in poljuner spectroscopy are driven by the many problems of interest in the polymer community. 

One noticeable exception is molecular spectroscopy, more specifically infrared spectroscopy, where, almost by definition, quantization of the vibrational states cannot be neglected since it concerns the measurement of the transitions between the quantized vibrational states [15]. Due to its importance in chemistry, for instance for the detection of functional groups in organic chemistry, infrared spectroscopy is one of the very few domains where the students in both physics and chemistry experience the application of a full-quantum mechanical treatment for both the electrons and the nuclei in a molecular system. However, there is growing evidence that a significant number of various chemical reactions are impacted by strong quantum-mechanical effects involving nuclei [16]. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

The major changes in the new edition are as follows There are three new chapters. Chapter 1 is a review and summary of aspects of quantum mechanics and electronic structure relevant to molecular spectroscopy. This chapter replaces the chapter on electronic structure of polyatomic molecules that was repeated from Volume I of Quantum Chemistry. Chapter 2 is a substantially expanded presentation of matrices. Previously, matrices were covered in the last chapter. The placement of matrices early in the book allows their use throughout the book in particular, the very tedious and involved treatment of normal vibrations has been replaced by a simpler and clearer treatment using matrices. Chapter 7 covers molecular electronic spectroscopy, and contains two new sections, one on electronic spectra of polyatomic molecules, and one on photoelectron spectroscopy, together with the section on electronic spectra of diatomic molecules from the previous edition. In addition to the new material on matrices, electronic spectra of polyatomic molecules, and photoelectron... 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

Conveniently, the transitions that involve the different types of energy levels (electronic, vibrational, and rotational) usually occur in different parts of the electromagnetic spectrum. This makes it even easier to consider them separately. We begin our multichapter treatment of spectroscopy by considering rotations and vibrations of molecules. Such motions are considered first for a reason. Both types of motions can be understood in terms of relatively simple quantum mechanics. We will also introduce some tools that we can apply to other forms of spectroscopy. 

The reason for the delay is that a detailed discussion is slightly more complicated for electronic spectra than for rotations or vibrations. Some new ideas will have to be developed in order to begin to understand the electronic spectra and structures of many-electron systems. (We should recognize that the electronic spectrum of hydrogen, even in the formalism of quantum mechanics, will be relatively simple.) fiowever, as with rotational and vibrational spectroscopy, our treatment of electronic spectroscopy in this chapter is limited by necessity. Entire books are written on the subject, and we can only introduce some basic ideas here. 

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