Modern Vibrational Spectroscopy and Micro-Spectroscopy: Theory, Instrumentation and Biomedical Applications

By Max Diem

Modern Vibrational Spectroscopy and Micro-Spectroscopy: Theory, Instrumentation and Biomedical Applications unites the theory and background of conventional vibrational spectroscopy with the principles of microspectroscopy. It starts with basic theory as it applies to small molecules and then expands it to include the large biomolecules which are the main topic of the book with an emphasis on practical experiments, results analysis and medical and diagnostic applications.  This book is unique in that it addresses both the parent spectroscopy and the microspectroscopic aspects in one volume.

Part I covers the basic theory, principles and instrumentation of classical vibrational, infrared and Raman spectroscopy. It is aimed at researchers with a background in chemistry and physics, and is presented at the level suitable for first year graduate students. The latter half of Part I is devoted to more novel subjects in vibrational spectroscopy, such as resonance and non-linear Raman effects, vibrational optical activity, time resolved spectroscopy and computational methods. Thus, Part 1 represents a short course into modern vibrational spectroscopy.

Part II is devoted in its entirety to applications of vibrational spectroscopic techniques to biophysical and bio-structural research, and the more recent extension of vibrational spectroscopy to microscopic data acquisition. Vibrational microscopy (or microspectroscopy) has opened entirely new avenues toward applications in the biomedical sciences, and has created new research fields collectively referred to as Spectral Cytopathology (SCP) and Spectral Histopathology (SHP). In order to fully exploit the information contained in the micro-spectral datasets, methods of multivariate analysis need to be employed. These methods, along with representative results of both SCP and SHP are presented and discussed in detail in Part II.

• Language: English
• Publisher: Wiley
• Release date: Jun 16, 2015
• ISBN: 9781118824986

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I would like to dedicate this book to my wife, Mary Jo, who put up with my hiding away in my home office for many an evening and my absentmindedness for 16 months during which this book was written.

I also would like to acknowledge the excellent and enthusiastic crew of postdocs, graduate and undergraduate students in my research laboratory at Northeastern University who was responsible for the majority of the work presented here (listed chronologically).

Miloš, Christian, Susie, Melissa, Brian, Tatjana, Jen (I), Ben, Ellen, Kostas, Antonella, Evgenia, Erin, Jen (II), Christina, Kathleen, and Doug.

Max Diem

Preface

Although this book was conceived as a second edition of Introduction to Modern Vibrational Spectroscopy published by the author in 1993, it really is not a second edition, but a completely rewritten monograph on a subject that has changed so much in 20 years that the old edition appears seriously antiquated. In fact, few other areas of spectroscopy have undergone such radical changes in the past two decades as vibrational spectroscopy has: subjects that then were cutting edge technology have become so common that they have been part of undergraduate physical chemistry core laboratories for quite some time, and areas that were not even thought about 20 years ago are on the verge of commercialization. This enormous progress was spawned by a fortuitous co-incidence of many factors, such as instrumental advances that allow the collection of 10 000 infrared spectra within a few minutes, or the collection of Raman spectra of samples in the picogram regime in a few hundred milliseconds. The ready availability of pico- and femtosecond tunable laser sources has made routine acquisition possible of several nonlinear spectroscopic effects, based on excitation of short-lived vibrational states. Last not least, the enormous increase in computational power over the past 20 years has opened entirely new avenues for data processing and statistical analyses of the plethora of data collected in short times. In fact, the increased computational power is certainly one major enabling factor of an area of vibrational spectroscopic imaging, in which between 10 000 and 100 000 individual pixel spectra are collected through specialized microscopes and converted to pseudo-color images based on the vibrational spectroscopic features. This increase in computational power can easily be felt considering that in 1993, a top-end personal computer (PC) incorporated less than 1 MB of RAM, and was still based on Intel 386/387 processors and much of the scientific programming was carried out using FORTRAN compilers. Twenty years later, workstations with between 16 and 48 GB of RAM and multicore Pentium type processors are routine and available for a similar dollar amount required for the purchase of top-end PCs in 1993. The accessibility of thousands of routines for data processing and analysis, and being contained in the MATLAB environment, have offered sophisticated statistical and analysis routines even to the non-expert. Finally, theoretical developments have solved some puzzling aspects of vibrational spectroscopy; the prime example here is the understanding of the theoretical foundations of surface-enhanced Raman spectroscopy (SERS). In 1993, an overall agreement on the theoretical foundations of this effect had not been reached. Furthermore, the field seemed to be plagued by low reproducibility, which disappeared once the theory of surface enhancement was properly developed.

The need for inclusion of new techniques in vibrational spectroscopy, as well as the shifting research interests of the author, contributed to the very new format of this book, and the material contained herein. Furthermore, the strategies for teaching concepts of vibrational spectroscopy have changed both at the graduate and the undergraduate levels. Finally, the intended readership of this book has changed since the first edition addressed an audience of first year graduate students in chemistry while the present edition addresses, in addition, a readership interested primarily in the medical imaging and diagnostics fields. For these readers, the theory sections have been streamlined, and imaging aspects have been added. The first edition still contained a detailed discussion on how to perform empirical calculation of molecular force fields and vibrational frequencies. This subject has gone the path of dinosaurs, since such calculations are now all performed at the ab initio molecular orbital level. In addition, methods of data handling and analysis have been added that are necessary for both students and researchers in modern vibrational spectroscopy.

All these changes have forced a total rewriting of the book. Part I of the reworked monograph contains the theory of vibrational spectroscopy, presented both from the view of classical mechanics as well as from quantum mechanics. Although the later parts of the book deal mostly with large biological molecules, a short review of the group theoretical foundations of vibrational spectra of small molecules is included, as well as basic instrumental aspects. New techniques – surface and nanostructure-based spectroscopies, nonlinear effects, and time-resolved methods – are introduced as well. Thus, Part I represents a short course into Modern Vibrational Spectroscopy and is somewhat comparable to the first edition.

Part II deals with biophysical, medical, and diagnostic applications of vibrational spectroscopy. It starts with a review of the biophysical applications of macroscopic vibrational spectroscopy, a field that has produced ten thousands of papers on biomolecular structure, dynamics, and interactions. Subsequently, vibrational microspectroscopy (also referred to as vibrational microscopy) will be introduced. Although infrared and Raman microspectroscopy were certainly known and applied in 1993, their relevance was relatively low, and they were not included in the first edition. In contrast, both techniques are so prevalent now that they contribute to about 30% of sales in Raman and infrared spectral instrumentation. Both these techniques present their own challenges in instrumentation, data manipulation, and analysis, and are discussed in detail.

Vibrational microspectroscopy allows the detection and analysis of individual bacterial cells. In 1993, a few brave souls had embarked into this field and found that infrared spectra proved phenomenally sensitive in distinguishing different bacterial species [1]. The analysis of cells and tissue has undergone an explosive expansion during the past decade, and is now on the verge of becoming a major diagnostic and prognostic tool. Specialized journals, such as Biospectroscopy (now part of Biopolymers) and the Journal of Biophotonics are devoted to the application of (mostly) vibrational spectroscopy toward biological sciences and medicine. Many journals that used to concentrate on classical analytical chemistry have devoted entire issues to the emerging field of spectral diagnostics (see, for example, Analyst, Volume 135 and J. Biophotonics, Volumes 3 and 6).

Thus, the author hopes that this book will provide a detailed background, from the quantum mechanical foundation to the specific applications, for researchers in, or entering, the exciting and re-emerging field of vibrational spectroscopy.

In the years since the first edition was published, several new books have appeared discussing in detail several aspects of vibrational spectroscopy, such as D.A. Long's book on Raman theory [2], L.A. Nafie's book on Vibrational Optical Activity [3], volumes on Quantum Mechanics or Group Theory [4], monographs on microspectroscopy, and many more. The present book can impossibly compete with these specialized books on the details presented, since it still was conceived as an Introduction to modern vibrational spectroscopy. Therefore, the subjects discussed here are more appropriate for a researcher entering this field, or for advanced undergraduate students in Chemistry or the Life Sciences.

Finally, the author herewith apologizes categorically for one aspect in this book that is presented inconsistently throughout the chapters: the presentation of spectra from left to right and from right to left. Historically, Raman spectra have been presented mostly from left to right, that is, from low to high wavenumber. Infrared spectra were originally presented from high wavenumber to low wavenumber, or from right to left. Of course, this was due to the fact that the wavelength increases from left to right in this presentation. Although there are some recommendations by IUPAC on the representation of spectra – Raman spectra from left to right and infrared spectra from right to left – too many researchers have not abided by this rule; consequently, spectra are displayed in the literature both ways. Since many figures in this book are taken from the work of many researchers, these figures could not be reversed to a standard theme. Thus, the reader is reminded to pay particular attention to the direction of the wavenumber scale in the figures.

Boston, July 2014

References

1. Naumann, D., Fijala, V., Labischinski, H. and Giesbrecht, P. (1988) The rapid differentiation of pathogenic bacteria using Fourier transform infrared spectroscopic and multivariate statistical analysis. J. Mol. Struct., 174, 165–170.

2. Long, D.A. (2002) The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules, John Wiley & Sons, Ltd, Chichester.

3. Nafie, L.A. (2011) Vibrational Optical Activity: Principles and Applications, John Wiley & Sons, Ltd, Chichester.

4. Cotton, F.A. (1990) Chemical Applications of Group Theory, 3rd edn, John Wiley & Sons, Inc., New York.

Preface to Introduction to Modern Vibrational Spectroscopy (1994)

The aim of this book is to provide a text for a course in modern vibrational spectroscopy. The course is intended for advanced undergraduate students, who have had an introductory course in Quantum Chemistry and have been exposed to group theoretical concepts in an inorganic chemistry course, or for graduate students who have passed a graduate level course in Quantum Chemistry.

There are probably a dozen or so recent books in vibrational spectroscopy, and a few classical texts over three decades old. This large number seems to discourage any efforts to produce yet another book on the subject, unless one is willing to pursue a novel approach in presenting the material. This is, of course, exactly what was attempted with the present book, and the approach taken will be outlined in the following paragraphs.

There are two classical and comprehensive texts on vibrational spectroscopy, Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules by Herzberg [1] and Molecular Vibrations by Wilson et al. [2]. Both of these books are absolutely essential for an in-depth understanding of vibrational spectroscopy, and devote hundreds of pages to theoretical derivations. However, due to the rapid progress in instrumental techniques and computational methods and due to the fact that thousands of molecules have been studied since these two books were written, the practical aspects of these books are certainly limited. However, the value of these classic books for the serious vibrational spectroscopist is immeasurable, since they provide many of the fine points needed for a detailed understanding of the subject.

Among the more recent books, the reader will find either very specialized works dealing with one or a few specific topics of vibrational spectroscopy, or books that are more a compilation of data than a comprehensive text. The more practically oriented books often emphasize correlations of observed spectra with molecular structural features, and may contain large compilations of spectra and group frequencies, and only cursory treatment of theoretical principles. These books are essential for researchers who wish to employ vibrational spectroscopy as a qualitative structural tool.

However, neither of these could be used as a text book in a course, nor could they be used by a researcher who wants to gain insight into modern aspects of vibrational spectroscopy. Thus, the author was faced with the challenging task of writing a text that incorporates some theoretical background material which is necessary for the understanding of the principles of vibrational spectroscopy, in addition to computational methods, instrumental aspects, novel developments in vibrational spectroscopy, and a number of relatively detailed examples for the interpretations of vibrational spectra. Since the scope of this book is much broader than any of the aforementioned specialized texts, some of the theoretical material needed to be adjusted accordingly. Thus, the quantum mechanics of molecular vibrations, time-dependent perturbations, transition moments, and many other topics are only summarized in this text, and detailed derivations are omitted. For details, the reader is referred to specialized books, such as any one of the many available text books on Quantum Chemistry (see, for example, Levine, Quantum Chemistry, Volumes (I) and (II) [3]). For a detailed theoretical background on symmetry aspects, the classical book (Cotton, Chemical Applications of Group Theory [4]) is recommended, and the aforementioned books on vibrational spectroscopy for a more thorough treatment of theoretical aspects of molecular vibrations.

Thus, the present book does not supersede any of the classical texts, but is a further extension of them, and intends to bring the reader to a more practical and up-to-date level of understanding in the field of vibrational spectroscopy. Subjects such as Raman spectroscopy, which has become a major area of research in vibrational spectroscopy, is not treated as an afterthought, but experimental and theoretical aspects are discussed in detail. Items of historical significance, such as the Toronto arc for excitation of Raman spectra (which was actually mentioned as a viable source for Raman spectroscopy in a recent text), or the manual solution of the vibrational secular equation, have been banished from this book. Instead, modern experimental aspects, such as multichannel Raman instrumentation, time-resolved and resonance Raman techniques, nonlinear Raman effects, and Fourier transform infrared and Raman techniques are introduced. In addition, computational methods for the calculation of normal modes of vibration are treated in detail in this book.

One chapter is devoted to the biological applications of vibrational spectroscopy. This is a rapidly developing field, and perhaps the most fascinating, for the molecules are often very large and difficult to study due to low solubility and solvent interference. It is in this area that the enormous progress of modern vibrational spectroscopy can best be gauged, since this field is not even discussed in the books of 30 or 40 years ago. The final chapter is devoted to a new branch of vibrational spectroscopy carried out with circularly polarized light. The new techniques introduced here combine principles of vibrational spectroscopy with those of optical activity measurements of chiral molecules. Applications of these techniques to biological molecules, and to simple chiral systems, are presented.

The author would like to thank his colleague, Prof. John Lombardi from the Department of Chemistry, City University of New York, City College, for his encouragement about this book, and for correcting a large number of errors in the original manuscript. The Graduate Spectroscopy class (U761) at the City University of New York in Spring, 1992, also was instrumental in pointing out inconsistencies in the manuscript, when an early version was used for the first time. The author is grateful for the input he received from these students.

New York, January 1993

References

1. Herzberg, G. (1945) Molecular Spectra and Molecular Structure, Van Nostrand Reinhold Company, New York.

2. Wilson, E.B., Decius, J.C. and Cross, P.C. (1955) Molecular Vibrations, McGraw-Hill.

3. Levine, I. (1970) Quantum Chemistry, vol. 1 and 2, Allyn & Bacon, Boston.

4. Cotton, F.A. (1990) Chemical Applications of Group Theory, 3rd edn, John Wiley & Sons, Inc., New York.

Part I

Modern Vibrational Spectroscopy and Micro-spectroscopy: Theory, Instrumentation and Biomedical Applications

Introduction

I.1 Historical Perspective of Vibrational Spectroscopy

The subject of this monograph, vibrational spectroscopy, derives its name from the fact that atoms in molecules undergo continuous vibrational motion about their equilibrium position, and that these vibrational motions can be probed via one of two major techniques: infrared (IR) absorption spectroscopy and Raman scattering, and several variants of these two major categories. IR spectroscopy experienced a boon in the years of World War II, when the US military was involved in an effort to produce and characterize synthetic rubber. Vibrational spectroscopy, for which industrial application guides were published as early as 1944 [1], turned out to be a fast and accurate way to identify different synthetic products. The rapid growth of the field of vibrational spectroscopy can be gauged by the fact that by the mid-1940s, the field was firmly established as a scientific endeavor, and the volume Molecular Spectra and Molecular Structure by Herzberg (one volume of a trilogy of incredibly advanced treatises on molecular spectroscopy [2]) reported infrared and Raman results on hundreds of small molecules. Similarly, the earliest efforts to use infrared spectroscopy as a means to distinguish normal and diseased tissue – the subject of Part II of this book – were reported by Blout and Mellors [3] and by Woernley [4] by the late 1940s and early 1950s. In the 1960s, the petroleum industry added another industrial use for these spectroscopic techniques when it was realized that hydrocarbons of different chain lengths and degrees of saturation produced distinct infrared spectral patterns, and for the first time, computational methods to understand, reproduce, and predict spectral patterns were introduced [5]. By then, commercial scanning infrared spectrometers were commercially available.

The 1970 produced another boon when commercial Fourier transform (FT) infrared spectrometers became commercially available, and gas lasers replaced the mercury arc lamp as excitation source for Raman spectroscopy. Before lasers, Raman spectroscopy was a somewhat esoteric technique, since large sample volumes and a lot of time were required to collect Raman data with the prevailing Hg arc excitation sources. Yet, after the introduction of laser sources, Raman spectra could be acquired rapidly and as easily as infrared spectra. After the introduction and wide acceptance of interferometry, infrared spectroscopy became the method of choice for many routine and quality control applications. During the ensuing decade, the field of vibrational spectroscopy blossomed at a phenomenal rate, and it is safe to state that no other spectroscopy grew at such a pace than vibrational spectroscopy, perhaps with the exception of nuclear magnetic resonance techniques. Other spectroscopic methods, such as ultraviolet/visible (UV/vis), microwave, or electron paramagnetic resonance (EPR) spectroscopy certainly profited from theoretical and technical advances; however, in vibrational spectroscopy, the sensitivity of the measurements increased by orders of magnitude while the time requirements for data acquisition dropped similarly.

After the introduction of tunable, high power, and pulsed lasers, not only were faster and more sensitive techniques developed (for example, resonance Raman and time-resolved techniques), but also, entirely new spectroscopic methods were discovered, among them a number of non-linear Raman techniques (such as Hyper-Raman and Coherent Anti-Stokes Raman Scattering, CARS) in which the effect depends non-linearly on the laser field strength. Dramatic progress was also achieved in IR spectroscopy, due to the advent of infrared lasers and further refinements of interferometric methods.

Aside from small molecule applications of vibrational spectroscopy, the past three decades have seen an ever increasing use of vibrational spectroscopy in biophysical, biochemical, and biomedical studies. This field has enormously enhanced the ability to determine solution conformations of biological molecules, their interaction and even reaction pathways. All major classes of biological molecules, proteins, nucleic acids, and lipids, exhibit vibrational spectra that are enormously sensitive to structure and structural changes, and thousands of research papers have been published demonstrating the value of these methods for the understanding of biochemical processes.

With the advent of vibrational micro-spectroscopic instruments, that is, infrared or Raman spectrometers coupled to optical microscopes, cell biological and medical application of vibrational spectroscopy was practical since the spatial resolution in Raman and infrared absorption microscopy is sufficient to distinguish, by spectral features, parts of cells and tissue. These methods are poised to enter the medical diagnostic field as inherently reproducible and objective tools. This monograph will provide an introduction to many of these fields mentioned above.

I.2 Vibrational Spectroscopy within Molecular Spectroscopy

Molecular spectroscopy is a branch of science in which the interactions of electromagnetic radiation and matter are studied. While the theory of these interactions itself is the subject of ongoing research, the aim and goal of the discussions here is the elucidation of information on molecular structure and dynamics, the environment of the sample molecules and their state of association, interactions with solvent, and many other topics.

Molecular (or atomic) spectroscopy is usually classified by the wavelength ranges (or energies) of the electromagnetic radiation (e.g., microwave or infrared spectroscopies) interacting with the molecular systems. These spectral ranges are summarized in Table I.1.

Table I.1 Table of photon energies and spectroscopic ranges

In Table 1.1, NMR and EPR stand for nuclear magnetic and electron paramagnetic resonance spectroscopy, respectively. In both these spectroscopic techniques, the transition energy of a proton or electron spin depends on the applied magnetic field strength. All techniques listed in Table 1.1 can be described by absorption processes (see below) although other descriptions, such as bulk magnetization in NMR, are possible as well.

The interaction of the radiation with molecules or atoms that was referred to above as an absorption process requires that the energy difference between two (molecular or atomic) stationary states exactly matches the energy of the photon:

equation

where h denotes Planck's constant (h = 6.6 × 10−34Js) and ν the frequency of the photon, in s−1. As seen from Table 1.1, these photon energies are between 10−16 and 10−25 J/molecule or about 10−4 to 10⁵ kJ/(mol photons). In an absorption process, one photon interacts with one atom or molecule to promote it into a state of higher excitation, and the photon is annihilated. The reverse process also occurs where an atomic or molecular system undergoes a transition from a more highly excited to a less highly excited state; in this process, a photon is created. However, this view of the interaction between light and matter is somewhat restrictive, since radiation interacts with matter even if its wavelength is far different than the specific wavelength at which a transition occurs. Thus, a classification of spectroscopy, which is more general than that given by the wavelength range alone, would be a resonance/off-resonance distinction. Many of the effects described and discussed in spectroscopy books are observed as resonance interactions where the incident light, indeed, possesses the exact energy of the molecular transition in question. IR and UV/vis absorption spectroscopy, microwave spectroscopy, or EPR are examples of such resonance interactions. However, interaction of light and matter occurs, in a more subtle way, even if the wavelength of light is different from that of a molecular transition. These off-resonance interactions between electromagnetic radiation and matter give rise to well-known phenomena such as the refractive index of dielectric materials, and the anomalous dispersion of the refractive index with wavelength. The normal (non-resonant) Raman effect also is a phenomenon that is best described in terms of off-resonance models. A discussion of non-resonance effects ties together many well-known aspects of classical optics and spectroscopy.

This interplay between classical optics and spectroscopy will be emphasized throughout the book, and the study of vibrational or other fields of spectroscopy exposes students to a more unified picture of physical phenomena than individual courses in chemistry or physics provide. As such, vibrational spectroscopy has enormous pedagogical values, because it provides the link between different scientific fields: data can be collected easily by students in the laboratory, often on compounds synthesized or prepared by them. The spectral results are tangible, qualitatively interpretable and can be used to identify compounds. A deeper exposure to the material can be used to explain eigenvector/eigenvalue problems and demonstrate quantum mechanical principles such as allowed and forbidden transitions, breakdown of first order approximations and many more. Symmetry and group theory can be introduced logically when discussing vibrational spectroscopy, since the symmetry of atomic displacements during a normal mode of vibrations can be visualized easily and provide an intuitive approach for teaching the concepts of symmetry. Furthermore, vibrational spectroscopy is a useful probe for the structure of small molecules, because vibrational spectra can be predicted from symmetry considerations (group theory) and group frequencies. In fact, microwave (rotational) and vibrational spectroscopies were instrumental in determining the structure and symmetries of many small molecules. Rules for predicting the structures and shapes of small molecules, such as the VSEPR (Valence Shell Electron Pair Repulsion) model taught routinely in introductory chemistry classes are partially based on vibrational spectroscopic results of the 1950s and 1960s. This aspect of the importance of vibrational spectroscopy becomes apparent when one reads the chapter on Individual Molecules in Molecular Spectra and Molecular Structure [2] because in the 1940s, many structures were not known for certain. In fact, for many molecules that are now known to exhibit tetrahedral shapes, alternative structures were offered then, and the implications of other structures on the observed vibrational spectra were discussed.

Thus, vibrational spectroscopy cannot be regarded as a static field which has outlived its usefulness. Quite contrary, it is a very dynamic and innovative field, and there is no reason to believe that the progress in this field is slowing down. Thus, this book emphasizes many of these new developments. As mentioned in the foreword, this volume is not intended to replace the previous books on vibrational spectroscopy, but to combine in one volume the necessary background to understand vibrational spectroscopy, and to introduce the reader to the many new and fascinating techniques and results.

References

1. Barnes, R.B., Gore, R.C., Liddel, U., et al. (1944) Infrared Spectroscopy. Industrial Applications and Biography, Reinhold Publishing Co, New York.

2. Herzberg, G. (1945) Molecular Spectra and Molecular Structure, Van Nostrand Reinhold Co, New York.

3. Blout, E.R. and Mellors, R.C. (1949) Infrared spectra of tissues. Science, 110, 137–138.

4. Woernley, D.L. (1952) Infrared absorption curves for normal and neoplastic tissues and related biological substances. Cancer Res., 12, 516–523.

5. Schachtschneider, J.H. (1965) Vibrational Analysis of Polyatomic Molecules: FORTRAN IV Programs for solving the vibrational secular equation and least-square refinement of force constants, The Shell Development Company, Emeryville, USA.

Chapter 1

Molecular Vibrational Motion

The atoms in matter – be it in gaseous, liquid, or condensed phases – are in constant motion. The amplitude of this motion increases with increasing temperature; however, even at absolute zero temperature, it never approaches zero or perfect stillness. Furthermore, the amplitude of the atomic motion is a measure of the thermodynamic heat content as measured by the product of the specific heat times the absolute temperature. If one could observe the motion in real time – which is not possible because the motions occur at a timescale of about 10¹³ Hz – one would find that it is completely random and that the atoms are most likely to be found in ellipsoidal regions in space, such as the ones depicted in X-ray crystallographic structures. Yet, the random motion can be decomposed into distinct normal modes of vibration. These normal modes can be derived from classical physical principles (see Section 1.2) and are defined as follows: in a normal mode, all atoms vibrate, or oscillate, at the same frequency and phase, but with different amplitudes, to produce motions that are referred as symmetric and antisymmetric stretching, deformation, twisting modes, and so on. In general, a molecule with N atoms will have 3N − 6 normal modes of vibrational normal modes.

At this point, a discrepancy arises between the classical (Newtonian) description of the motion of atoms in a molecule and the quantum mechanical description. While in the classical description the amplitude of the motion, and thereby the kinetic energy of the moving atoms, can increase in arbitrarily small increments, the quantum mechanical description predicts that the increase in energy is quantized, and that infrared (IR) photons can be absorbed by a vibrating molecular system to increase the energy along one of the normal modes of vibration.

In the discussion to follow, the concepts of normal modes of vibration will be introduced for a system of spring-coupled masses, as shown in Figure 1.1, a typical mechanical model for a molecular system. Through a series of mathematical steps, the principle of normal modes will be derived from Newtonian laws of motion. Once this set of normal modes is defined, it is relatively trivial to extend these coordinates to a quantum mechanical description that results in the basic formalism for stationary vibrational states in molecules, and the transitions between these stationary states that are observed in IR and Raman spectroscopies.

Figure 1.1 Mass-and-spring model of Cartesian displacement vectors for a triatomic molecule. The gray cylinders represent springs obeying Hook's law

Sections 1.1–1.7 are aimed at presenting a summary of the physical principles required for understanding the principles of vibrational spectroscopy. They do not present the subject with the mathematical rigor presented in earlier treatments, for example, in Wilson et al. [1] for the classical description of normal modes nor the quantum mechanical detail found in typical texts such as those by Kauzman [2] or Levine [3]. However, sufficient detail is provided to expose the reader to the necessary physical principles such as normal modes of vibration and normal coordinates, the basic quantum mechanics of vibrating systems, and the transition moment, but in addition to the aforementioned texts, it will introduce the reader to many practical aspects of vibrational spectroscopy, as well as branches of vibrational spectroscopy that were not included in the earlier treatments.

For the remainder of this book, a standard convention for expressing vibrational energies in wavenumber units will be followed. Although energies should be expressed in units of Joule (1 J = kg m² s−2), these numbers are unyielding, and wavenumber units are used throughout. The following energy unit conversions apply:

equation

Here, h has the value of 6.6 × 10−34 J s. Using c01-math-0002 , one finds that 1 cm−1 ≈ 30 GHz = 2 × 10−23 J (see also the table in the introduction and comments after Eqs. 1.31 and 1.63). Herewith, the author categorically apologizes for referring to transitions, expressed in wavenumber units, as energies.

1.1 The concept of normal modes of vibration

Consider a set of masses connected by springs that obey Hook's law, as shown in Figure 1.1. Two of these springs act as restoring force when the bonds between atoms 1 and 2 and between atoms 1 and 3 are elongated or compressed, whereas one spring acts to restore the bond angle between atoms 2–1–3. Furthermore, we assume that the force required to move one atom along a coordinate depends on the momentary position of all other atoms: the force required to extend bond 1–2 may decrease if bond 1–3 is elongated. In a mechanical system, this situation is referred to as a coupled spring ensemble, where the stiffness of a spring depends on all coordinates. In a molecular system, this corresponds to electron rearrangement when the molecular shape changes, with a concomitant change in bond strength. In order to describe the atomic motions in a vibrating system, one attaches a system of Cartesian displacement coordinates to every atom, as shown in Figure 1.1. A normal mode of vibration then can be described as a combination of properly scaled displacement vector components.

1.2 The separation of vibrational, translational, and rotational coordinates

Based on Figure 1.1, one may expect a system of N atoms to exhibit 3N degrees of vibrational freedom: a degree of freedom for all three Cartesian displacement coordinates of each atom, or any linear combinations of them. However, vibrational spectroscopy depends on a restoring force to bring the atoms of a molecule back to their equilibrium position. If, for example, all atoms in a molecule move simultaneously in the x-direction, by the same amount, no bonds are being compressed or elongated. Thus, this motion is not that of an internal vibrational coordinate but that of a translation. There are three translational degrees of freedom, corresponding to a motion of all atoms along the x, y, or z axes by the same amount. Another view of the same fact is that these modes have zero frequencies as there is no restoring force acting during the atomic displacements. This view will be favored in the derivation of the concepts of normal modes presented in Section 1.3, which is carried out in (mass-weighted) Cartesian displacement coordinates. Later on, a different and simpler coordinate system will be introduced as well.

Similarly, one can argue that certain combinations of Cartesian displacements correspond to a rotation of the entire molecule, where there is no change in the intermolecular potential of the atoms. There are three degrees of rotational freedom of a molecule, corresponding to rotations about the three axes of inertia. Subtracting these from the remaining number of degrees of freedom, one arrives at 3N − 6 degrees of vibrational freedom for a polyatomic, nonlinear molecule. Linear molecules have one more (3N − 5) degree of vibrational freedom, because they have only two moments of inertia. This is because one assumes a zero moment of inertia for a rotation about the longitudinal axis.

Mathematically, the separation of rotation and translation from the vibration of a molecule proceeds as follows: in order for the translational energy of the molecule to be zero at all times, one defines a coordinate system that translates with the molecule. In this coordinate system, the translational energy is zero by definition. The translating coordinate system is defined such that the center of mass of the molecule is at the origin of the coordinate system at all times. This leads to the condition

1.1 equation

where ξ denotes the X, Y, and Z coordinates of the α'th atom of a molecule with N atoms. Similarly, the rotational energy can be reduced to zero by defining a coordinate system that rotates with the molecule. This requires that the angular moments of the molecule in the rotating coordinate frame are zero, which leads to three more equations of zero frequencies. These six equations are needed to define a coordinate system in which both translational and rotational energies are zero. Details of this derivation can be found in Wilson et al. [1, Chapters 2 and 11].

1.3 Classical vibrations in mass-weighted Cartesian displacement coordinates

The concept of normal modes of vibration, necessary for understanding the quantum mechanical description of vibrational spectroscopy and obtaining a pictorial description of the atomic motions, can be introduced by the previously described classical model of a molecule consisting of a number of point masses, held in their equilibrium positions by springs. This kind of discussion is treated in complete detail in the classic books on vibrational spectroscopy, for example, in Chapter 2 of Wilson et al. [1].

The treatment starts with Lagrange's equation of motion:

1.2 equation

where T and V are the kinetic and potential energies, respectively, the xi are the Cartesian displacement coordinates, and the dot denotes the derivative with respect to time. Equation 1.2 is another statement of Newton's equation of motion (Eq. 1.3) expressed in terms of the kinetic and potential energies, rather than terms of force and acceleration:

1.3 equation

In Eq. 1.3, F represents the force, which is related to the potential energy by

1.4 equation

Equation 1.4 is Hook's law, which states that the force F needed to elongate or compress a spring depends on the spring's stiffness, expressed by the force constant k, multiplied by the elongation of the spring, x. The acceleration (d²xi/dt²) in Eq. 1.3 can be related to the kinetic energy as follows:

1.5 equation

Thus,

1.6 equation

Substituting Eqs. 1.4 and 1.6 into Eq. 1.3 yields Lagrange's equation of motion in which the expressions for kinetic and potential energies appear separately. This formulation is advantageous for writing the quantum mechanical Hamiltonian, cf. the following section.

Next, one rewrites Eq. 1.2 in terms of mass-weighted displacement coordinates, qi. The reason for this is that the amplitude of a particle's oscillation depends on its mass. When mass-weighted coordinates are used, all amplitudes are properly adjusted for the different masses of the particles. In addition, the use of mass-weighted coordinates simplifies the formalism quite a bit. Let

1.7 equation

Then, the kinetic energy can be written as

1.8 equation

Note that Eq. 1.8 contains only diagonal terms; that is, no cross terms qij appear in the summation. (The term diagonal here refers to a matrix notation to be introduced shortly.) Next, the potential energy of the particles needs to be defined. For masses connected by springs obeying Hook's law, one may assume that the potential energy along each Cartesian displacement coordinate is given by

1.9 equation

or

1.10 equation

with

1.11 equation

The fij are known as mass-weighted Cartesian force constants and differ from the force constant k defined in Eq. 1.4 by the fact that latter does not explicitly contain the masses of the atoms. The constants fij express the change in potential energy as an atom or group is moved along the directions given by qi and qj. For small displacements about the equilibrium positions, Eq. 1.10 can be written as

1.12 equation

In contrast to the kinetic energy expression in mass-weighted Cartesian coordinates (Eq. 1.8), the potential energy depends on diagonal (fiiqi²) and off-diagonal (fijqiqj) terms as pointed out earlier. Taking the required derivatives and substituting the expressions for c01-math-0015 and c01-math-0016 into Lagrange's equation of motion (Eq. 1.2) yields

1.13 equation

Here, c01-math-0018 denotes the second derivative of q with respect to time. Equation 1.13 is a short form for a set of 3N simultaneous differential equations, with the index i running from 1 to 3N. Note that the double summation in Eq. 1.12 disappears when the derivative with respect to one of the displacements, ∂V/∂qi, is taken. In expanded form, Eq. 1.13 can be presented as:

1.14

equation

In each equation, only one term in the summation c01-math-0020 has the same index as the term containing the time derivative. Thus, these equations can be simplified to read

1.15 equation

where C is a constant. There are 3N solutions to these simultaneous, linear differential equations:

1.16 equation

where the Ai are amplitude factors, ϵ are phase angles, and λ is a quantity related to the frequency and determined by the force constants (cf. below). Following standard practice in solving linear differential equations, one takes the solution given by Eq. 1.16, differentiates twice with respect to time,

1.17 equation

and substitutes Eq. 1.15 back into Eq. 1.14. One obtains, after canceling the terms c01-math-0024 from each equation:

1.18

equation

or

1.19

equation

Thus, 3N simultaneous homogeneous linear equations were obtained from 3N simultaneous linear differential equations. Homogeneous equations have two kinds of solutions. One of them is the so-called trivial solution in which all coefficient Ai are zero. This condition indeed fulfills Eq. 1.19 but is of no interest here because it implies that all particles are at rest; that is, there is no vibrational motion at all. The other solution for Eq. 1.19 is obtained when the determinant of the coefficients of A is zero in order for the left-hand side of Eq. 1.19 to be zero:

1.20 equation

This is called the nontrivial solution, which requires

1.21 equation

Equation 1.21 is known as the vibrational secular equation. The solution of this equation gives the eigenvalues λ, which are related to the vibrational frequencies for each of the normal modes. The amplitude factors Ai in Eq. 1.19 are not determined, but the relative magnitude of the displacement vectors can provide a view of the relative amplitudes of all atoms during a normal mode of vibration.

A normal mode of vibration is defined to by one of the 3N solutions of Eq. 1.21, where all atoms oscillate with the same frequency and in-phase but with different amplitudes. This definition is one of the most important ones in vibrational spectroscopy. It implies that all atoms are in motion during a normal mode of vibration, which is required to maintain the center of mass of the molecule. If 3N mass-weighted Cartesian displacement coordinates are defined, six rotational and translational modes will appear in these calculations as eigenvalues with zero frequencies, as discussed earlier. The displacement vectors will confirm that these motions are, indeed, translations and rotations.

Before continuing the discussion of the normal modes in polyatomic molecules, the simpler case of the vibration of diatomic molecules will be presented. For a diatomic molecule (which, of course, must be linear), Eq. 1.21 described a set of six equations, five of which have zero frequencies (namely the three translational and the two rotational coordinates). That leaves one equation,

1.22 equation

or λ = f11. Next, it is instructive to visualize that Eq. 1.22 actually represents a vibrational frequency. For a diatomic molecule, the vibrational frequency can also be derived, starting with Newton's second law,

1.3 equation

and assuming a harmonic restoring force obeying Hook's law:

1.23 equation

Thus, one can write the equation of motion for a diatomic molecule as

1.24 equation

Here, k is the spring's (bond's) force constant, as discussed earlier (Eq. 1.4) that corresponds to the terms fij in Eq. 1.19, and m is the reduced mass defined as

1.25 equation

with m1 and m2 the individual masses of the two atoms.

One valid solution of the differential equation of motion (Eq. 1.24) is

1.26 equation

where

1.27 equation

where ω is the angular frequency and c01-math-0036 is a phase angle. Note that for a classical vibrational problem, the amplitude A is arbitrary, but that the frequency is defined by Eq. 1.25. This implies that for larger amplitudes, the velocity of the motion of the particles increases, but the frequency remains constant.

Rewriting Eq. 1.27 as

1.28 equation

and comparing Eqs. 1.22 and 1.28, one finds that

1.29 equation

when using mass-weighted Cartesian displacement coordinates. This relationship is true for diatomic and polyatomic molecules. At this point, a quick analysis of magnitudes and units is appropriate. The force constant k acting in a diatomic molecule such as gaseous H—Cl typically is about 500 N m−1 = 500 kg s−2, corresponding to a relatively stiff spring in classical mechanics. The reduced mass of an H—Cl molecule is, according to Eq. 1.25, approximately

1.30 equation

and thus, the vibrational frequency for the H—Cl molecule is found to be

1.31

equation

Using the frequency/wavenumber conversion c01-math-0041 gives a value close to the observed stretching frequency for gaseous H—Cl of 3 × 10³ cm−1. When working in mass-weighted Cartesian displacement coordinates, the reduced mass in Eq. 1.27 disappears, and the frequency of the vibration is given by

1.32 equation

Thus, Eq. 1.22, indeed, denotes the vibrational frequency of diatomic or polyatomic molecules.

Returning to the secular equation 1.21 for a triatomic molecule, one finds that one needs 81 force constants to describe the problem. Of these 81 force constants, only 9 are nonzero, because there are 3N − 6, or 3 degrees of vibrational freedom. However, it is clear that even this reduced problem cannot be solved, because there are three equations with nine unknowns. Symmetry arguments reduce this number of unknown force constants even further, but the number of unknowns still exceeds the number of equations. In the past, this problem was alleviated by transferring the diagonal force constants, that is, those with common indexes (fii) from similar molecules or from isotopic species, assuming that the diagonal force constants should not depend on isotopic substitution. Off-diagonal force constants were fitted to reproduce observed frequencies. More recently, all force constants – diagonal and off-diagonal – are determined computationally via ab initio methods (cf. Chapter 6). However, the meaning of the off-diagonal force constants needs to be pointed out in more detail. Referring to Figure 1.1, the diagonal force constants f11, f22, and f33 refer to the force required to move atom 1 in the x, y, and z directions, respectively. The off-diagonal constant f12 describes the force required to move atom 1 in the y direction after displacement in the x-direction.

To obtain vibrational frequencies and a depiction of the normal modes, the system of simultaneous homogeneous linear equations (Eq. 1.21) needs to be solved. As described earlier, values for force constants are substituted into the force constant matrix (also referred to potential energy matrix) that is subsequently diagonalized numerically according to:

1.33 equation

Here, the matrix c01-math-0044 corresponds to the Cartesian force constant defined in Eq. 1.21, and the eigenvector matrix c01-math-0045 that diagonalizes the potential energy matrix is also the matrix that transforms from the mass-weighted Cartesian coordinate system to a new coordinate system Q, known as the normal coordinates. These normal coordinates are defined such that each of the 3N − 6 normal modes of vibration is associated with one and only one normal coordinate Q. For the discussion of the relationship between mass-weighted displacement coordinates q and the normal coordinates Q, it is advantageous to cast the previously obtained equations into matrix notation. In the following discussion, bold quantities imply matrices or vectors. The two relationships derived earlier, expressing the potential and kinetic energies in terms of mass-weighted Cartesian coordinates, are written in matrix notations as

1.8 equation

and

1.12

equation

Here, the superscript T denotes the transpose of a matrix; thus, the column vector c01-math-0048 becomes a row vector upon transposition. The dot implies, as before, the time derivative of the coordinates. c01-math-0049 denotes the matrix of mass-weighted Cartesian force constants, as defined in Eq. 1.33.

Normal coordinates are defined such that

1.34 equation

In normal coordinate space, both the kinetic and potential energy matrices are diagonal. As the kinetic energy is diagonal in both q and Q space, the problem simplifies to finding the transformation (eigenvector) matrix c01-math-0051 that diagonalizes the potential energy matrix c01-math-0052 . This matrix also transforms from the mass-weighted Cartesian displacement space into normal coordinate space according to

1.35 equation

Thus, the diagonalization of the potential energy matrix provides the vibrational frequencies of the system, according to

1.36 equation

as well as the transformation to visualize each normal mode of vibration in terms of a normal coordinate and therewith, in terms of the displacement vectors. The displacement vectors for the normal modes of vibration of the water molecule are shown in Figure 1.2.

Figure 1.2 Display of the atomic displacement vectors and the symmetries (see Chapter 2) for the three normal modes of the water molecule. The magnitude of the displacement vectors is not known, but the relative displacements are drawn approximately to scale. The terms A1, and B1 refer to the symmetry species of the coordinates (see Chapter 2).

1.4 Quantum mechanical description of molecular vibrations

1.4.1 Transition from classical to quantum mechanical description

Next, a connection between the classical normal mode picture and the quantum mechanical description will be presented. The approach here starts with a simple case of a diatomic molecule for which the classical equation of motion was derived earlier (Eqs. 1.27) and for which the solution of the Schrödinger equation is relatively straightforward. Once the (one-dimensional) situation of the diatomic molecule has been introduced, the transition to polyatomic molecules is fairly simple because the concept of the normal coordinates can be used. As pointed out earlier, the classical description of the vibrations of a diatomic or polyatomic molecule predicts the vibrational coordinates (the normal modes) and their frequency, but not the transitions that are observed in IR absorption or Raman spectroscopy. In order to explain the observed spectra, quantum mechanics has to be invoked.

Quantum mechanics presents an approach to the behavior of microscopic particles very different from that in classical mechanics. While, in classical mechanics, the position and momentum of a moving particle can be established simultaneously, Heisenberg's uncertainty principle prohibits the simultaneous determination of those two quantities. This is manifested by Eq. 1.37 for the one-dimensional case:

1.37 equation

which implies that the uncertainty in the momentum and position always exceeds ħ/2. Mathematically, Eq. 1.37 follows from the fact that the operators responsible for defining position and momentum, c01-math-0056 and c01-math-0057 , do not commutate; that is, c01-math-0058 . The incorporation of this uncertainty into the picture of the motion of microscopic particles leads to discrepancies between classical and quantum mechanics: classical physics has a deterministic outcome, which implies that if the position and velocity (trajectory) of a moving body are established, and it is possible to predict with certainty where it is going to be found in the future. Quantum mechanical systems obey a probabilistic behavior. As the position and momentum can never be determined at the starting point, the position (or momentum) in the future cannot be precisely predicted, only the probability of either of them. This is manifested in the postulate that all properties, present or future, of a particle are contained in a quantity known as the wavefunction Ψ of a system. This function, in general, depends on spatial coordinates and time; thus, for a one-dimensional motion (to be discussed first), the wavefunction is written as Ψ(x, t). The probability of finding a quantum mechanical system is given by the integral of the square of this wavefunction: c01-math-0059 . Any property c01-math-0060 one wishes to observe for the system is expressed as the expectation value of the operator c01-math-0061 associated with the property, where the expectation value is defined as

1.38 equation

As discussed earlier, a diatomic molecule possesses only one degree of vibrational freedom, the periodic elongation and compression of the bond connecting the two atoms that will be designated the x coordinate in the following discussion. Thus, the total energy, in analogy to Eq. 1.2, of an oscillating diatomic molecule can be written as the sum of kinetic energy T and potential energy V:

1.39 equation

The kinetic energy is written in terms of classical physics as

1.40 equation

where the momentum is given by p = mv. Here, v is the velocity and m is the reduced mass of the oscillating diatomic molecule, defined earlier (Eq. 1.25). In quantum mechanics, the classical momentum is substituted by the momentum operator c01-math-0065 ,

1.41 equation

where ħ is Planck's constant, divided by 2π, and i is the imaginary unit, defined by c01-math-0067 . This substitution of the classical momentum by a differential operator is often considered the central postulate of quantum mechanics because it cannot be derived, although it can be visualized from the classical wave equation. Equation 1.41 is a mathematical instruction that requires taking the spatial derivative of the wavefunction c01-math-0068 is operating on, and multiplying the results by ħ/i = −iħ to obtain the equivalent of the classical momentum. Examples of the use of such an operator are given in the following sections.

1.4.2 Diatomic molecules: harmonic oscillator

The potential energy for a diatomic vibrating system is discussed next. This potential function is shown schematically in Figure 1.3, and can be obtained by detailed quantum mechanical calculations, in which the electronic energy is computed as a function of the internuclear distance. This potential energy can be approximated by the Morse potential, given by

1.42 equation

with

equation

Figure 1.3 Graph of the potential energy function for a diatomic molecule. Parameters are specified in Eq. 1.58

The function has a minimum at the bond equilibrium distance x0. When compressing the bond beyond x0, the potential energy rises sharply because of the repulsion of the two atoms. When the bond is elongated toward large interatomic distances, the potential function eventually levels out, and the bond breaks. One normally defines the potential energy at very large interatomic distances as the zero energy (no bonding interaction takes place at large distances); thus, the potential energy of the bond is at a negative minimum at the equilibrium distance. The energy difference between zero potential energy and the minimum potential energy at point x0 is referred as the bond dissociation energy, De.

Solving the quantum mechanical equations for the vibrations of a diatomic molecule with the potential function shown in Figure 1.3 would be difficult. Thus, one approximates the shape of the potential function in the vicinity of the potential energy minimum by a more simplistic function by expanding the potential energy V(x) in a power series about the equilibrium distance:

1.43

equation

V(x0) is an offset along the Y-axis and does not affect the curvature of the potential energy. The term containing the first derivative of the potential energy with respect to x is zero because the equilibrium geometry corresponds to an energy minimum. Terms higher than the quadratic expression in Eq. 1.43 are ignored at this point. Thus, one approximates the potential energy V(x) by

1.44 equation

which also could have been obtained by integrating

1.23 equation

for a system obeying Hook's law.

Thus, to a first approximation, one assumes that the chemical bond in a diatomic molecule obeys Hook's law, just as the motion of two spring-coupled masses. Combining Eqs. 1.41, and 1.43, the vibrational Schrödinger equation for a two-particle