Quantum Mechanical Foundations of Molecular Spectroscopy 9783527347926

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Table of Contents Cover Title Page Copyright Preface Introduction References 1 Transition from Classical Physics to Quantum Mechanics 1.1 Description of Light as an Electromagnetic Wave 1.2 Blackbody Radiation 1.3 The Photoelectric Effect 1.4 Hydrogen Atom Absorption and Emission Spectra 1.5 Molecular Spectroscopy 1.6 Summary References Problems 2 Principles of Quantum Mechanics 2.1 Postulates of Quantum Mechanics 2.2 The Potential Energy and Potential Functions 2.3 Demonstration of Quantum Mechanical Principles for a Simple, One‐Dimensional, One‐Electron Model System: The Particle in a Box 2.4 The Particle in a Two‐Dimensional Box, the Unbound Particle, and the Particle in a Box with Finite Energy Barriers 2.5 Real‐World PiBs: Conjugated Polyenes, Quantum Dots, and Quantum Cascade Lasers References

Problems 3 Perturbation of Stationary States by Electromagnetic Radiation 3.1 Time‐Dependent Perturbation Treatment of Stationary‐ State Systems by Electromagnetic Radiation 3.2 Dipole‐Allowed Absorption and Emission Transitions and Selection Rules for the Particle in a Box 3.3 Einstein Coefficients for the Absorption and Emission of Light 3.4 Lasers References Problems Note 4 The Harmonic Oscillator, a Model System for the Vibrations of Diatomic Molecules 4.1 Classical Description of a Vibrating Diatomic Model System 4.2 The Harmonic Oscillator Schrödinger Equation, Energy Eigenvalues, and Wavefunctions 4.3 The Transition Moment and Selection Rules for Absorption for the Harmonic Oscillator 4.4 The Anharmonic Oscillator 4.5 Vibrational Spectroscopy of Diatomic Molecules 4.6 Summary References Problems 5 Vibrational Infrared and Raman Spectroscopy of Polyatomic Molecules 5.1 Vibrational Energy of Polyatomic Molecules: Normal Coordinates and Normal Modes of Vibration 5.2 Quantum Mechanical Description of Molecular Vibrations in Polyatomic Molecules

5.3 Infrared Absorption Spectroscopy 5.4 Raman Spectroscopy 5.5 Selection Rules for IR and Raman Spectroscopy of Polyatomic Molecules 5.6 Relationship between Infrared and Raman Spectra: Chloroform 5.7 Summary: Molecular Vibrations in Science and Technology References Problems 6 Rotation of Molecules and Rotational Spectroscopy 6.1 Classical Rotational Energy of Diatomic and Polyatomic Molecules 6.2 Quantum Mechanical Description of the Angular Momentum Operator 6.3 The Rotational Schrödinger Equation, Eigenfunctions, and Rotational Energy Eigenvalues 6.4 Selection Rules for Rotational Transitions 6.5 Rotational Absorption (Microwave) Spectra 6.6 Rot–Vibrational Transitions References Problems 7 Atomic Structure: The Hydrogen Atom 7.1 The Hydrogen Atom Schrödinger Equation 7.2 Solutions of the Hydrogen Atom Schrödinger Equation 7.3 Dipole Allowed Transitions for the Hydrogen Atom 7.4 Discussion of the Hydrogen Atom Results 7.5 Electron Spin 7.6 Spatial Quantization of Angular Momentum References Problems Note

8 Nuclear Magnetic Resonance (NMR) Spectroscopy 8.1 General Remarks 8.2 Review of Electron Angular Momentum and Spin Angular Momentum 8.3 Nuclear Spin 8.4 Selection Rules, Transition Energies, Magnetization, and Spin State Population 8.5 Chemical Shift 8.6 Multispin Systems 8.7 Pulse FT NMR Spectroscopy References Problems 9 Atomic Structure: Multi‐electron Systems 9.1 The Two‐electron Hamiltonian, Shielding, and Effective Nuclear Charge 9.2 The Pauli Principle 9.3 The Aufbau Principle 9.4 Periodic Properties of Elements 9.5 Atomic Energy Levels 9.6 Atomic Spectroscopy 9.7 Atomic Spectroscopy in Analytical Chemistry References Problems 10 Electronic States and Spectroscopy of Polyatomic Molecules 10.1 Molecular Orbitals and Chemical Bonding in the H2+ Molecular Ion 10.2 Molecular Orbital Theory for Homonuclear Diatomic Molecules 10.3 Term Symbols and Selection Rules for Homonuclear Diatomic Molecules 10.4 Electronic Spectra of Diatomic Molecules

10.5 Qualitative Description of Electronic Spectra of Polyatomic Molecules 10.6 Fluorescence Spectroscopy 10.7 Optical Activity: Electronic Circular Dichroism and Optical Rotation References Problems Note 11 Group Theory and Symmetry 11.1 Symmetry Operations and Symmetry Groups 11.2 Group Representations 11.3 Symmetry Representations of Molecular Vibrations 11.4 Symmetry‐Based Selection Rules for Dipole‐Allowed Processes 11.5 Selection Rules for Raman Scattering 11.6 Character Tables of a Few Common Point Groups References Problems Appendix 1: Constants and Conversion Factors Appendix 2: Approximative Methods: Variation and Perturbation Theory A2.1 General Remarks A2.2 Variation Method A2.3 Time‐independent Perturbation Theory for Nondegenerate Systems A2.4 Detailed Example of Time‐independent Perturbation: The Particle in a Box with a Sloped Potential Function A2.5 Time‐dependent Perturbation of Molecular Systems by Electromagnetic Radiation Reference Appendix 3: Nonlinear Spectroscopic Techniques

A3.1 General Formulation of Nonlinear Effects A3.2 Noncoherent Nonlinear Effects: Hyper‐Raman Spectroscopy A3.3 Coherent Nonlinear Effects A3.4 Epilogue References Appendix 4: Fourier Transform (FT) Methodology A4.1 Introduction to Fourier Transform Spectroscopy A4.2 Data Representation in Different Domains A4.3 Fourier Series A4.4 Fourier Transform A4.5 Discrete and Fast Fourier Transform Algorithms A4.6 FT Implementation in EXCEL or MATLAB References Appendix 5: Description of Spin Wavefunctions by Pauli Spin Matrices A5.1 The Formulation of Spin Eigenfunctions α and β as Vectors A5.2 Form of the Pauli Spin Matrices A5.3 Eigenvalues of the Spin Matrices Reference Index End User License Agreement

List of Tables Chapter 1 Table 1.1 Photon energies and spectroscopic rangesa. Chapter 5 Table 5.1 Vibrational modes and assignments for chloroform, HCCl3.

Chapter 8 Table 8.1 Nuclearg‐factors, magnetogyric ratios, and spin moments for some sp... Chapter 9 Table 9.1 Symbols of states for differentl and L values. Table 9.2 Transition, energies, term symbols, and wavelengths of the prominen... Chapter 10 Table 10.1 Symbols of states for differentl and L values.

List of Illustrations Chapter 1 Figure 1.1 Description of the propagation of a linearly polarized electromag... Figure 1.2 (a) Plot of the intensity I radiated by a blackbody source as a f... Figure 1.3 Portion of the hydrogen atom emission in the visible spectral ran... Figure 1.4 Energy level diagram of the hydrogen atom. Transitions between th... Chapter 2 Figure 2.1 Potential energy functions and analytical expressions for (a) mol... Figure 2.2 Panel (a): Wavefunctions for n = 1, 2, 3, 4, and 5 drawn at the... Figure 2.3 (a) Representation of the particle‐in‐a‐box wavefunctions shown i...

Figure 2.4 Wavefunctions of the two‐dimensional particle in a box for (a) n x Figure 2.5 (a) Particle in a box with infinite potential energy barrier. (b)... Figure 2.6 (a) Structure of 1,6‐diphenyl‐1,3,5‐hexatriene to be used as an e... Figure 2.7 Absorption spectra of nanoparticles as a function of particle siz... Figure 2.8 (a) An individual energy well with finite barrier height and a sl... Chapter 3 Figure 3.1 Two state energy level diagrams used for the discussion of time‐d... Figure 3.2 Plot of the PiB ground‐state (trace a) and first excited‐state (t... Figure 3.3 Panel (a): Schematic energy level diagram of a 3‐ level system in ... Figure 3.4 Schematic of a gas laser, consisting of the resonator structure, ... Chapter 4 Figure 4.1 Definition of a diatomic harmonic oscillator of masses m 1 and m 2 ... Figure 4.2 Quadratic potential energy function V = ½ kx 2 for a diatomic mole... Figure 4.3 Schematic of allowed (solid arrows) and forbidden (dashed arrows)... Figure 4.4 Graphical representation of the orthogonality of vibrational wave...

Figure 4.5 Potential energy function of a real diatomic molecule with dissoc... Figure 4.6 Comparison of energy levels for harmonic and anharmonic oscillato... Figure 4.7 (a) Raman spectrum of Br2. (b) Expanded region of the fundamental... Chapter 5 Figure 5.1 Depiction of the atomic displacement vectors q i for the three nor... Figure 5.2 Energy ladder diagram for the water molecule within the harmonic ... Figure 5.3 (a) Observed infrared absorption spectrum of water. (b) Schematic... Figure 5.4 Depiction of the atomic displacement vectors q i for the four norm... Figure 5.5 Gaussian (a) and Lorentzian (b) line profiles. Notice that the ar... Figure 5.6 Dispersion of the refractive index (top) within an absorption pea... Figure 5.7 (a) Energy level diagram for a Stokes and anti‐ Stokes Raman scatt... Figure 5.8 (a) Raman spectrum of chloroform as a neat liquid. (b) Expanded v... Figure 5.9 Atomic displacement vectors for (a) the symmetric –CCl3 stretchin... Chapter 6 Figure 6.1 Definition of spherical polar coordinates. Figure 6.2 Graphical representation for the condition T(φ) = T(φ + b 2π

Figure 6.3 (a) Energy level diagram for linear rotors. (b) Schematic rotatio... Figure 6.4 Simulated rotational spectrum of 35Cl–F at room temperature, usin... Figure 6.5 Schematic of the center‐of‐mass (COM) position in an oblate (a) a... Figure 6.6 Energy level diagram for (a) oblate and (b) prolate top rotors. S... Figure 6.7 (a) Observed rot–vibrational band envelopes in the infrared absor... Figure 6.8 (a) Rot–vibrational energy level diagram for a harmonic oscillato... Figure 6.9 Simulated rot–vibrational spectral band profiles for the deformat... Chapter 7 Figure 7.1 (a) Plot of radial part of hydrogen wavefunctions in units of r/a Figure 7.2 Plot of first few spherical harmonic functions. Notice that the Figure 7.3 Orbital energy eigenvalues and degeneracies for the hydrogen atom... Figure 7.4 Radial part of the wavefunctions (dashed lines) and radial distri... Figure 7.5 Energy level diagram and allowed electronic transitions for the h... Figure 7.6 (a) Energy level diagram of the hydrogen atom orbitals in the pre... Figure 7.7 Spatial, or orientational quantization of the orbital angular mom... Chapter 8

Figure 8.1 (a) Definition of the angular momentum in terms of radius r and l... Figure 8.2 Energy of the α and β proton nuclear spin states as a f... Figure 8.3 (a) Energy level diagram for two noninteracting spins with shield... Figure 8.4 Spectral pattern observed for two interacting spins at lower (a) ... Figure 8.5 Spin–spin coupling patterns for (a) J AXX and (b) J AXXX spin syste... Figure 8.6 Reorientation of magnetization vector following a 90° pulse, view... Figure 8.7 (a) Simulated “free induction decay” (FID) and Fourier transforme... Chapter 9 Figure 9.1 Energy level diagram of multi‐electron atoms, explaining the Aufb... Figure 9.2 Ionization energies (a) and atomic radii (b) for main group eleme... Figure 9.3 Vector addition schemes for (a) the total orbital angular momenta... Figure 9.4 Simplified energy level diagram of the Li atom and transitions in... Chapter 10 Figure 10.1 (a) Overlap of the two 1s orbitals on nuclei a and b. The volume... Figure 10.2 Wavefunctions for the (a) bonding and (b) antibonding molecular ... Figure 10.3 (a) Energy level diagram of the MOs formed from the overlap of 2...

Figure 10.4 Electron and spin populations in the two MOs of the lowest‐ene... Figure 10.5 Observed (a) and simulated (b) vibronic transition of molecular... Figure 10.E1 See Example 10.1 for details. Figure 10.6 Vibronic transition between the ground vibrational state of the ... Figure 10.7 (a) Approximate MO energy level diagram and UV transitions for a... Figure 10.8 (a) Approximate molecular orbital energy level diagram of the hi... Figure 10.9 (a) Energy level (Jablonski) diagram for fluorescence. (b) Energ... Figure 10.10 Schematic diagrams representing fluorescence (a), two‐photon fl... Figure 10.11 (a) Left (top) and right (bottom) circularly polarized light. (... Figure 10.12 Relationship between ORD and CD. Notice that the differential r... Figure 10.13 (a) CD (top) and UV absorption (bottom) spectra of an asymmetri... Figure 10.14 CD (top) and UV absorption (bottom) spectra of (a) α‐helical, (... Figure P.1 Figure P.2 Figure P.3 Figure P.4 Figure P.5 Chapter 11

Figure 11.1 Example of a symmetry operation (C 2). Figure 11.2 (a) Example of one of three σ ν mirror planes in the mo... Figure 11.3 Definition of a center of inversion, located at the coordinate o... Figure 11.4 Description of an improper rotation operation (S6) for ethane. Figure 11.5 Effects of symmetry operations E and σ yz on a Cartesian coo... Figure 11.6 (a) Cartesian displacement vectors for the water molecule. (b) C... Appendix 2 Figure A2.1 Comparison between the unperturbed (a) and perturbed (b) particl... Appendix 3 Figure A3.1 Schematic energy level diagram for degenerate (a and nondegenera... Figure A3.2 (a) Phase matching diagram for frequency doubling (SHG). (b) Pha... Figure A3.3 (a) Schematic energy level diagram for the CARS process. (b) Pha... Figure A3.4 Broadband micro‐CARS spectra of cellular components: (a) nucleol... Figure A3.5 Schematic diagram of FSRS. See text for details (from ref. 7). Appendix 4 Figure A4.1 Representation of data in different domains. (a) Graph of the in... Figure A4.2 (a) Intensity vs. time and (b) intensity vs. frequency plot of a...

Figure A4.3 Approximation of a square wave function (heavy black line) by a ... Figure A4.4 Examples of Fourier transforms (FTs). (a) The FT of a delta func... Figure A4.5 Panel (a): Real part of a reverse transform of a spectrum back t...

Quantum Mechanical Foundations of Molecular Spectroscopy

Max Diem

Author Max Diem, PhD Professor Emeritus Department of Chemistry Northeastern University Laboratory of Spectral Diagnosis Boston, MA USA Cover iStock 965444768 / © StationaryTraveller Supplementary material for instructors, including a Solution Manual, available for download from www.wiley-vch.de/textbooks All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing‐in‐Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2021 WILEY‐VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978‐3‐527‐34792‐6 ePDF ISBN: 978‐3‐527‐82961‐3 ePub ISBN: 978‐3‐527‐82960‐6 Cover Design SCHULZ Grafik‐Design, Fußgönheim, Germany

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 mechanics of molecular spectroscopy, rather than atomic structure, chemical bonding, and what is commonly referred to as “quantum chemistry.” As my own appreciation of many aspects of quantum mechanics evolved, I found it useful to start my lectures in this course with a quote from a famous researcher and Nobel laureate (1995, for his work on quantum electrodynamics), the late Professor Richard Feynman, which – taken slightly out of context – reads [6]: I think I can safely say that nobody understands quantum mechanics. This rather discouraging statement has to be seen from the viewpoint that, when studying quantum mechanics, one realizes that this theory is not based on axioms, but on postulates – a very unusual fact in the sciences. Furthermore, it replaced deterministic results with probabilistic answers. When exposed to these conundrums, students will naturally ask the question: “Why bother studying quantum mechanics, if I will not understand it anyway?” or worse, “Is quantum mechanics for real, or is it the brainchild of some far‐ out mad scientists?” The answer here is also contained in a quote by Feynman: It doesn't matter how beautiful a theory is, …. If it doesn't agree with experiment, it's wrong. This statement could also be formulated to imply that a theory that consistently provides answers that agree with the experiment most

likely is correct. Thus, although nobody may understand quantum mechanics in its entirety, it gives answers that – over and over – agree with experiments and in fact provides a mechanism and framework for explaining the experimental results. Quantum mechanics originated in the early decades of the twentieth century, when it was found that some experiment results just could not be explained by existing laws of physics and, in fact, violated established physical dogmas. It was these results that gave rise to the emergence of quantum mechanics that grew out of a patchwork of ideas aimed at explaining these hitherto unexplainable experimental results. These ideas coalesced into the field we now refer to as quantum mechanics. This newly formulated theory was wildly successful in explaining a myriad of physical and chemical observations – from the shape and meaning of the periodic chart of elements to the subject of this book, namely, the interaction of light with matter that is the basis of spectroscopy. While many aspects of molecular spectroscopy, such as the rotational or vibrational energies of a molecule, can be described in classical terms, the idea that atoms and molecules can exist in quantized, stationary energy states is a direct result of the postulates of quantum mechanics. Furthermore, application of the principles of time‐dependent quantum mechanics explains how electromagnetic radiation of the correct energy may cause a transition between these stationary energy states and produce observable spectra. Thus, the entire field of molecular spectroscopy is a direct result of quantum mechanics and represents the experimental results that confirms the theory. The phenomenal growth of all forms of spectroscopy over the past eight decades has contributed enormously to our understanding of molecular structure and properties. What started as simple molecular spectroscopy such as infrared and Raman vibrational spectroscopy, (microwave) rotational spectroscopy, ultraviolet– visible absorption, and emission spectroscopy has now bloomed into a very broad field that includes, for example, the modern magnetic resonance techniques (including medical magnetic resonance imaging); nonlinear, laser, and fiber‐based spectroscopy; surface and surface‐enhanced spectroscopy; pico‐ and femtosecond time‐ resolved spectroscopy, and many more. Spectroscopy is embedded as

a major component in material science, chemistry, physics, and biology and other branches of scientific and engineering endeavors. Thus, the quantum mechanical underpinnings of spectroscopy are a major subject that need to be understood in the pursuit of scientific efforts.

References 1 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall. 2 Levine, I. (1970). Quantum Chemistry, vol. I&II. Boston: Allyn & Bacon. 3 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon. 4 Kauzman, W. (1957). Quantum Chemistry. New York: Academic Press. 5 Eyring, H., Walter, J., and Kimball, G.E. (1967). Quantum Chemistry. New Yrok: Wiley. 6 Feynman, R. (1964). Probability and Uncertainty: The Quantum Mechanical View of Nature ‐ The Character of Physical Law 1964. Cornell University.

1 Transition from Classical Physics to Quantum Mechanics At the end of the nineteenth century, classical physics had progressed to such a level that many scientists thought all problems in physical science had been solved or were about to be solved. After all, classical Newtonian mechanics was able to predict the motions of celestial bodies, electromagnetism was described by Maxwell's equations (for a review of Maxwell's equations, see [1]), the formulation of the principles of thermodynamics had led to the understanding of the interconversion of heat and work and the limitations of this interconversion, and classical optics allowed the design and construction of scientific instruments such as the telescope and the microscope, both of which had advanced the understanding of the physical world around us. In chemistry, an experimentally derived classification of elements had been achieved (the rudimentary periodic table), although the nature of atoms and molecules and the concept of the electron's involvement in chemical reactions had not been realized. The experiments by Rutherford demonstrated that the atom consisted of very small, positively charged, and heavy nuclei that identify each element and electrons orbiting the nuclei that provided the negative charge to produce electrically neutral atoms. At this point, the question naturally arose: Why don't the electrons fall into the nucleus, given the fact that opposite electric charges do attract? A planetary‐like situation where the electrons are held in orbits by centrifugal forces was not plausible because of the (radiative) energy loss an orbiting electron would experience. This dilemma was one of the causes for the development of quantum mechanics. In addition, there were other experimental results that could not be explained by classical physics and needed the development of new theoretical concepts, for example, the inability of classical models to reproduce the blackbody emission curve, the photoelectric effect, and the observation of spectral “lines” in the emission (or absorption)

spectra of atomic hydrogen. These experimental results dated back to the first decade of the twentieth century and caused a nearly explosive reaction by theoretical physicists in the 1920s that led to the formulation of quantum mechanics. The names of these physicists – Planck, Heisenberg, Einstein, Bohr, Born, de Broglie, Dirac, Pauli, Schrödinger, and others – have become indelibly linked to new theoretical models that revolutionized physics and chemistry. This development of quantum theory occupied hundreds of publications and letters and thousands of pages of printed material and cannot be covered here in this book. Therefore, this book presents many of the difficult theoretical derivations as mere facts, without proof or even the underlying thought processes, since the aim of the discussion in the following chapters is the application of the quantum mechanical principles to molecular spectroscopy. Thus, these discussions should be construed as a guide to twenty‐first‐ century students toward acceptance of quantum mechanical principles for their work that involves molecular spectroscopy. Before the three cornerstone experiments that ushered in quantum mechanics – Planck's blackbody emission curve, the photoelectric effect, and the observation of spectral “lines” in the hydrogen atomic spectra – will be discussed, electromagnetic radiation, or light, will be introduced at the level of a wave model of light, which was the prevalent way to look at this phenomenon before the twentieth century.

1.1 Description of Light as an Electromagnetic Wave As mentioned above, the description of electromagnetic radiation in terms of Maxwell's equation was published in the early 1860s. The solution of these differential equations described light as a transverse wave of electric and magnetic fields. In the absence of charge and current, such a wave, propagating in vacuum in the positive z‐ direction, can be described by the following equations: (1.1)

(1.2) where the electric field and the magnetic field are perpendicular to each other, as shown in Figure 1.1, and oscillate in phase at the angular frequency (1.3) where ν is the frequency of the oscillation, measured in units of s−1 = Hz. In Eqs. (1.1) and (1.2), k is the wave vector (or momentum vector) of the electromagnetic wave, defined by Eq. (1.4): (1.4) Here, λ is the wavelength of the radiation, measured in units of length, and is defined by the distance between two consecutive peaks (or troughs) of the electric or magnetic fields. Vector quantities, such as the electric and magnetic fields, are indicated by an arrow over the symbol or by bold typeface. Since light is a wave, it exhibits properties such as constructive and destructive interference. Thus, when light impinges on a narrow slit, it shows a diffraction pattern similar to that of a plain water wave that falls on a barrier with a narrow aperture. These wave properties of light were well known, and therefore, light was considered to exhibit wave properties only, as predicted by Maxwell's equation.

Figure 1.1 Description of the propagation of a linearly polarized electromagnetic wave as oscillation of electric ( ) and magnetic ( ) fields. In general, any wave motion can be characterized by its wavelength λ, its frequency ν, and its propagation speed. For light in vacuum, this propagation speed is the velocity of light c (c = 2.998 × 108 m/s). (For a list of constants used and their numeric value, see Appendix 1.) In the context of the discussion in the following chapters, the interaction of light with matter will be described as the force exerted by the electric field on the charged particles, atoms, and molecules (see Chapter 3). This interaction causes a translation of charge. This description leads to the concept of the “electric transition moment,” which will be used as the basic quantity to describe the likelihood (that is, the intensity) of spectral transition. In other forms of optical spectroscopy (for example, for all manifestations of optical activity, see Chapter 10), the magnetic transition moment must be considered as well. This interaction leads to a coupled translation and rotation of charge, which imparts a helical motion of charge. This helical motion is the hallmark of optical activity, since, by definition, a helix can be left‐ or right‐handed.

1.2 Blackbody Radiation From the viewpoint of a spectroscopist, electromagnetic radiation is produced by atoms or molecules undergoing transitions between well‐ defined stationary states. This view obviously does not include the

creation of radio waves or other long‐wave phenomena, for example, in standard antennas in radio technology, but describes ultraviolet, visible, and infrared radiation, which are the main subjects of this book. The atomic line spectra that are employed in analytical chemistry, for example, in a hollow cathode lamp used in atomic absorption spectroscopy, are due to transitions between electronic energy states of gaseous metal atoms. The light created by the hot filament in a standard light bulb is another example of light emitted by (metal) atoms. However, here, one needs to deal with a broad distribution of highly excited atoms, and the description of this so‐called blackbody radiation was one of the first steps in understanding the quantization of light. Any material at a temperature T will radiate electromagnetic radiation according to the blackbody equations. The term “blackbody” refers to an idealized emitter of electromagnetic radiation with intensity I(λ, T) or radiation density ρ(T, ν) as a function of wavelength and temperature. At the beginning of the twentieth century, it was not possible to describe the experimentally obtained blackbody emission profile by classical physical models. This profile was shown in Figure 1.2 for several temperatures between 1000 and 5000 K as a function of wavelength.

Figure 1.2 (a) Plot of the intensity I radiated by a blackbody source as a function of wavelength and temperature. (b) Plot of the radiation density of a blackbody source as a function of frequency and temperature. The dashed line represents this radiation density according to Eq. (1.5). M. Planck attempted to reproduce the observed emission profile using classical theory, based on atomic dipole oscillators (nuclei and electrons) in motion. These efforts revealed that the radiation density ρ emitted by a classical blackbody into a frequency band dν as function of ν and T would be given by Eq. (1.5): (1.5) where the Boltzmann constant k = 1.381 × 10−23 [J/K]. This result indicated that the total energy radiated by a blackbody according to this “classical” model would increase with ν2 as shown by the dashed curve in Figure 1.2b. If this equation were correct, any temperature of a material above absolute zero would be impossible, since any material above 0 K would emit radiation according to Eq. (1.5), and the total energy emitted would be unrestricted and approach infinity. Particularly, toward higher frequency, more and more radiation would be emitted, and the blackbody would cool instantaneously to 0 K. Thus, any temperature above 0 K would be impossible. (For a more

detailed discussion on this “ultraviolet catastrophe,” see Engel and Reid [2].) This is, of course, in contradiction with experimental results and was addressed by M. Planck (1901) who solved this conundrum by introducing the term 1/(ehν/kT − 1) into the blackbody equation, where h is Planck's constant: (1.6)

The shape of the modified blackbody emission profile given by Eq. (1.6) is in agreement with experimental results. The new term introduced by Planck is basically an exponential decay function, which forces the overall response profile to approach zero at high frequency. The numerator of the exponential expression contains the quantity hν, where h is Planck's constant (h = 6.626 × 10−34 Js). This numerator implies that light exists as “quanta” of light, or light particles (photons) with energy E: (1.7) This, in itself, was a revolutionary thought since the wave properties of light had been established more than two centuries earlier and had been described in the late 1800s by Maxwell's equations in terms of electric and magnetic field contributions. Here arose for the first time the realization that two different descriptions of light, in terms of waves and particles, were appropriate depending on what questions were asked. A similar “particle–wave duality” was later postulated and confirmed for matter as well (see below). Thus, the work by Planck very early in the twentieth century is truly the birth of the ideas resulting in the formulation of quantum mechanics. Incidentally, the form of the expression

or

is fairly

common‐place in classical physical chemistry. It compares the energy of an event, for example, a molecule leaving the liquid for the gaseous phase, with the energy content of the surroundings. For example, the

vapor pressure of a pure liquid depends on a term

, where

ΔHvap is the enthalpy of vaporization of the liquid, and RT = NkT is the energy at temperature T, R is the gas constant, and N is Avogadro's number. Similarly, the dependence of the reaction rate constant and the equilibrium constant on temperature is given by equivalent expressions that contain the activation energy or the reaction enthalpy, respectively, in the numerator of the exponent. In Eq. (1.6), the photon energy is divided by the energy content of the material emitting the photon and provides a likelihood of this event occurring. Figure 1.2 shows that the overall emitted energy increases with increasing temperature and that the peak wavelength of maximum intensity shifts toward lower wavelength (Wien's law). The total energy W radiated by a blackbody per unit area and unit time into a solid angle (the irradiance), integrated over all wavelengths, is proportional to the absolute temperature to the fourth power: (1.8) (Stefan–Boltzmann law) The irradiance is expressed in units of

or

.

The implication of the aforementioned wave–particle duality will be discussed in the next section.

1.3 The Photoelectric Effect In 1905, Einstein reported experimental results that further demonstrated the energy quantization of light. In the photoelectric experiment, light of variable color (frequency) illuminated a photocathode contained in an evacuated tube. An anode in the same tube was connected externally to the cathode through a current meter and a source of electric potential (such as a battery). Since the cathode and anode were separated by vacuum, no current was observed, unless light with a frequency above a threshold frequency was

illuminating the photocathode. Einstein correctly concluded that light particles, or photons, with a frequency above this threshold value had sufficient kinetic energy to knock out electrons from the metal atoms of the photocathode. These “photoelectrons” left the metal surface with a kinetic energy given by (1.9) where ϕ is the work function, or the energy required to remove an electron from metal atoms. This energy basically is the atoms' ionization energy multiplied by Avogadro's number. Furthermore, Einstein reported that the photocurrent produced by the irradiation of the photocathode was proportional to the intensity of light, or the number of photons, but that increasing the intensity of light that had a frequency below the threshold did not produce any photocurrent. This provided further proof of Eq. (1.9). This experiment further demonstrated that light has particle character with the kinetic energy of the photons given by Eq. (1.7), which led to the concept of wave–particle duality of light. Later, de Broglie theorized that the momentum p of a photon was given by (1.10) Equation (1.10) is known as the de Broglie equation. The wave– particle duality was later (1927) confirmed to be true for moving masses as well by the electron diffraction experiment of Davisson and Germer [3]. In this experiment, a beam of electrons was diffracted by an atomic lattice and produced a distinct interference pattern that suggested that the moving electrons exhibited wave properties. The particle–wave duality of both photons and moving matter can be summarized as follows. For photons, the wave properties are manifested by diffraction experiments and summarized by Maxwell's equation. As for all wave propagation, the velocity of light, c, is related to wavelength λ and frequency ν by (1.11)

with c = 2.998 × 108 [m/s] and λ expressed in [m] and ν expressed in [Hz = s−1]. The quantity is referred to as the wavenumber of radiation (in units of m−1 or cm−1) that indicates how many wave cycles occur per unit length: (1.12) The (kinetic) energy of a photon is given by (1.13) with ħ = h/2π and ω, the angular frequency, defined before as ω = 2πν. From the classical definition of the momentum of matter and light, respectively, (1.14) it follows that the photon mass is given by (1.15) Notice that a photon can only move at the velocity of light and the photon mass can only be defined at the velocity c. Therefore, a photon has zero rest mass, m0. Particles of matter, on the other hand, have a nonzero rest mass, commonly referred to as their mass. This mass, however, is a function of velocity v and should be referred to as mν, which is given by (1.16)

Example 1.1 Calculation of the mass of an electron moving at 99.0 % of the velocity of light (such velocities can easily be reached in a synchrotron).

Answer: According to Eq. (1.16), the mass mv of an electron at ν = 0.99 c is (E1.1.1)

The electron at 99 % of the velocity of light has a mass of about seven times its rest mass. Equation (1.16) demonstrates that the mass of any matter particle will reach infinity when accelerated to the velocity of light. Their kinetic energy at velocity ν (far from the velocity of light) is given by the classical expression (1.17) The discussion of the last paragraphs demonstrates that at the beginning of the twentieth century, experimental evidence was amassed that pointed to the necessity to redefine some aspects of classical physics. The next of these experiments that led to the formulation of quantum mechanics was the observation of “spectral lines” in the absorption and emission spectra of the hydrogen atom.

1.4 Hydrogen Atom Absorption and Emission Spectra Between the last decades of the nineteenth century and the first decade of the twentieth century, several researchers discovered that hydrogen atoms, produced in gas discharge lamps, emit light at discrete colors, rather than as a broad continuum of light as observed for a blackbody (Figure 1.2a). These emissions occur in the ultraviolet, visible, and near‐infrared spectral regions, and a portion of such an emission spectrum is shown schematically in Figure 1.3. These

observations predate the efforts discussed in the previous two sections and therefore may be considered the most influential in the development of the connection between spectroscopy and quantum mechanics.

Figure 1.3 Portion of the hydrogen atom emission in the visible spectral range, represented as a “line spectrum” and schematically as an emission spectrum. These experiments demonstrated that the H atom can exist in certain “energy states” or “stationary states.” These states can undergo a process that is referred to as a “transition.” When the atom undergoes such a transition from a higher or more excited state to a lower or less excited state, the energy difference between the states is emitted as a photon with an energy corresponding to the energy difference between the states: (1.18) where the subscript f and i denote, respectively, the final and initial (energy) state of the atom (or molecule). Such a process is referred to as a “emission” of a photon. Similarly, an absorption process is one in which the atom undergoes a transition from a lower to a higher energy state, the energy difference being provided by a photon that is annihilated in the process. Absorption and emission processes are collectively referred to as “transitions” between stationary states and are directly related to the annihilation and creation, respectively, of a photon. The wavelengths or energies from the hydrogen emission or absorption experiments were fit by an empirical equation known as

the Rydberg equation, which gave the energy “states” of the hydrogen atom as (1.19) In this equation, n is an integer (>0) “quantum” number, and Ry is the Rydberg constant, (Ry = 2.179 × 10−18 J). This equation implies that the energy of the hydrogen atom cannot assume arbitrary energy values, but only “quantized” levels, E(n). This observation led to the ideas of electrons in stationary planetary orbits around the nucleus, which – however – was in contradiction with existing knowledge of electrodynamics, as discussed in the beginning of this chapter. The energy level diagram described by Eq. (1.19) is depicted in Figure 1.4. Here, the sign convention is as follows. For n = ∞, the energy of interaction between nucleus and electron is zero, since the electron is no longer associated with the nucleus. The lowest energy state is given by n = 1, which corresponds to the H atom in its ground state that has a negative energy of 2.179 × 10−18 J.

Figure 1.4 Energy level diagram of the hydrogen atom. Transitions between the energy levels are indicated by vertical lines. Equation (1.19) provided a background framework to explain the hydrogen atom emission spectrum. According to Eq. (1.19), the energy of a photon, or the energy difference of the atomic energy levels, between any two states nf and ni can be written as (1.20)

At this point, an example may be appropriate to demonstrate how this empirically derived equation predicts the energy, wavelength, and wavenumber of light emitted by hydrogen atoms. This example also introduces a common problem, namely, that of units. Although there is an international agreement about what units (the système international, or SI units) are to be used to describe spectral transitions, the problem is that few people are using them. In this book, all efforts will be made to use SI units, or at least give the conversion to other units. The sign conventions used here are similar to those in thermodynamics where a process with a final energy state lower than that of the initial state is called an “exothermic” process, where heat or energy is lost. In Example 1.2, the energy is lost as a photon and is called an emission transition. When describing an absorption process, the energy difference of the atom is negative, ΔEatom < 0, that is, the atom has gained energy (“endothermic” process in thermodynamics). Following the procedure outlined in Example 1.2 would lead to a negative wavelength of the photon, which of course is physically meaningless, and one has to remember that the negative ΔEatom implies the absorption of a photon. Example 1.2 Calculation of the energy, frequency, wavelength, and wavenumber of a photon emitted by a hydrogen atom undergoing a transition from n = 6 to n = 2. Answer:

The energy difference between the two states of the hydrogen atom is given by (E1.2.1)

Using the value of the Rydberg constant given above, Ry = 2.179 × 10−18 J, the energy difference is (E1.2.2) Using Eq. (1.12), ΔE = Ephoton = hν = hc/λ, the frequency ν is found to be (E1.2.3)

The wavelength of such a photon is given by Eq. (1.7) as (E1.2.4)

that is, a photon in the ultraviolet wavelength range. Finally, the wavenumber of this photon is (E1.2.5) This is a case where the SI units are used infrequently, and the results for the wavenumber are usually given by spectroscopists in units of cm−1, where 1 m−1 = 10−2 cm−1. Accordingly, the results in Eq. E1.5 is written as or about 24 380 cm−1.

1.5 Molecular Spectroscopy Example 1.2 in the previous section describes an emission process in atomic spectroscopy, a subject covered briefly in Chapter 9. Molecular spectroscopy is a branch of science in which the interactions of electromagnetic radiation and molecules are studied, where the molecules exist in quantized stationary energy states similar to those discussed in the previous section. However, these energy states may or may not be due to transitions of electrons into different energy levels, but due to vibrational, rotational, or spin energy levels. Thus, molecular spectroscopy often is classified by the wavelength ranges of the electromagnetic radiation (for example, microwave or infrared spectroscopies) or changes in energy levels of the molecular systems. This is summarized in Table 1.1, and the conversion of wavelengths and energies were discussed in Eqs. (1.11)– (1.15) and are summarized in Appendix 1. Table 1.1 Photon energies and spectroscopic rangesa. νphoton λphoton Ephoton Ephoton Ephoton Transition [J] [kJ/mol] [m−1] 5×10−25 3×10−4

2.5

NMRb

Microwave 3 GHz 10 cm

2×10−24 0.001

10

EPRb

Microwave 30 GHz 1 cm

2×10−23 0.012

100

Rotational

105

Vibrational

3×106

Electronic

3×109

X‐ray absorption

Radio

Infrared

750 MHz

3×1013 10 μm 2×10−20 12 Hz

UV/visible 1015 X‐ray a)

0.4 m

1018

300 nm

6×10−19 360

0.3 nm 6×10−16 3.6×105

For energy conversions, see Appendix 1.

b) The resonance frequency in NMR and EPR depends on the magnetic field strength.

In this table, 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 this table can be described by absorption processes although other descriptions, such as bulk magnetization in NMR, are possible as well. As seen in Table 1.1, the photon energies are between 10−16 and 10−25 J/photon or about 10−4–105 kJ/(mol photons). Considering that a bond energy of a typical chemical (single) bond is about 250– 400 kJ/mol, it shows that ultraviolet photons have sufficient energy to break chemical bonds or ionize molecules. In this book, mostly low energy photon interactions will be discussed, causing transitions in spin states, rotational, vibrational, and electronic (vibronic) energy levels. Most of the spectroscopic processes discussed are absorption or emission processes as defined by Eq. (1.18): (1.18) However, interactions between light and matter occur even when the light's wavelength is different from 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 this book 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, and NMR are examples of such resonance interactions. The off‐resonance interactions between electromagnetic radiation and matter give rise to well‐known phenomena such as the refractive index of dielectric materials. These interactions arise since force is exerted by the electromagnetic radiation on the charged particles of matter even at off‐resonance frequencies. This force causes an increase in the amplitude of the motion of these particles. When the frequency of light reaches the transition energy between two states, an effect known as anomalous dispersion of the refractive index takes place. This anomalous dispersion of the refractive index always

accompanies an absorption process. This phenomenon makes it possible to observe the interaction of light either in an absorption or as a dispersion measurement, since the two effects are related to each other by a mathematical relation known as the Kramers–Kronig relation. This aspect will be discussed in more detail in Chapter 5. The normal (nonresonant) Raman effect is a phenomenon that also is best described in terms of off‐resonance models, since Raman scattering can be excited by wavelengths that are not being absorbed by molecules. A discussion of nonresonant effects ties together many well‐known aspects of classical optics and spectroscopy.

1.6 Summary The observation of the photoelectric effect and the absorption/emission spectra of the hydrogen atom and the modifications required to formulate the blackbody emission theory were the triggers that forced the development of quantum mechanics. As pointed out in the introduction, the development of quantum mechanics is based on postulates, rather than axioms. The form of some of these postulates can be visualized from other principles, but their adoption as “the truth” came from the fact that they produced the correct results.

References 1 Halliday, D. and Resnick, R. (1960). Physics. New York: Wiley. 2 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall. 3 Davisson, C.J. and Germer, L.H. (1928). Reflection of electrons by a crystal of nickel. Proceedings of the National Academy of Sciences of the United States of America 14 (4): 317–322.

Problems

1. What is the maximum wavelength of electromagnetic radiation that can ionize an H atom in the n = 2 state? 2. Why is it that any photon with a wavelength below the limiting value obtained in (1) can ionize the H atom, whereas in standard spectroscopy, only a photon with the correct energy can cause a transition? 3. Assume that you carry out the experiment in (1) with light with a wavelength of 10 nm less than calculated in (1). What is the kinetic energy of the photoelectron created? 4. What is the velocity of the electron in Problem 3? 5. Using the de Broglie relation for matter waves, calculate the velocity to which an electron needs to be accelerated such that its wavelength is 10 nm. 6. What percentage of the velocity of light is the velocity in (5)? 7. What is the relativistic mass of this electron? 8. At what velocity is the wavelength of an electron 30 nm? 9. What is the momentum of such the electron in (8)? 10. What is the mass of a photon with a wavelength of 30 nm? 11. What is the momentum of the photon in (10)? 12. Compare and comment on the masses and momenta of the moving particles in Problems (8)–(11). 13. “Frequency doubling” or “second harmonic generation (SHG)” is a little optical trick (Appendix 3) in which two photons of the same wavelength are squashed into a new photon, while the energy is conserved. Calculate the wavelength of the photon created from frequency doubling of two photons with λ = 1064 nm. 14. “Sum frequency generation (SFG)” is another optical trick (Appendix 3) in which two photons of different wavelengths are squashed into a new photon while the energy is conserved. Calculate the wavelength of SFG photon created from combining two photons with λ1 = 1064 nm and λ2 = 783 nm.

15. The value of the Rydberg constant, Ry, can be calculated according to

where e' = e/√4πεo and where mR is the

reduced mass of electron and proton. Perform an analysis of the units of Ry. 16. Which two experiments demonstrate that light has wave and particle character? 17. Which two experiments demonstrate that moving electrons have wave and particle character?

2 Principles of Quantum Mechanics Quantum mechanics presents an approach to describe the behavior of microscopic systems. Whereas 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. (2.1): (2.1) which implies that the uncertainty in the momentum and position always exceeds ħ/2, where ħ is Planck's constant divided by 2π. Mathematically, Eq. (2.1) follows from the fact that the operators responsible for defining position and momentum, and , do not commutate; that is, . (This aspect will be discussed in more detail at the end of Section 2.1.) As we shall see later (Chapter 5), the uncertainty principle also can be rewritten in terms of the uncertainty in energy and lifetime of a spectroscopic state or in frequency and time of a wave. 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, it is possible to predict with certainty where it is going to be found in the future. This principle certainly holds at the macroscopic scale: if the position and trajectory of a macroscopic body, for example, the moon, are known, it is certainly possible to calculate its position six days from now and to send a spaceship to this predicted position. Quantum mechanical systems, on the other hand, obey a probabilistic behavior. Since the position and momentum can never be determined simultaneously at any point in time, 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 at any time is given by the integral of the square of this wavefunction: ∫Ψ(x, t)2 dx. This is, in fact, one of the “postulates” on which quantum mechanics is based to be discussed next. Different authors list these postulates in different orders and include different postulates necessary for the description of quantum mechanical systems [1]. Quantum mechanics is unusual in that it is based on postulates, whereas science, in general, is axiom‐based.

2.1 Postulates of Quantum Mechanics Postulate 1: The state of a quantum mechanical system is completely defined by a wavefunction Ψ(x, t). The square of this function, or in the case of complex wavefunction, the product Ψ*(x, t) Ψ(x, t), integrated over a volume element dτ (= dx dy dz in Cartesian coordinates or sin2θ dθ dφ in spherical polar coordinates) gives the probability of finding a system in the volume element dτ. Here, Ψ*(x, t) is the complex conjugate of the function Ψ(x, t). This postulate contains the transition from a deterministic to probabilistic description of a quantum mechanical system. The wavefunctions must be mathematically well behaved, that is, they must be single‐ valued, continuous, having a continuous first derivative, and integratable (so they can be normalized). Postulate 2: The classical linear momentum expression, p = mv, is substituted in quantum mechanics by the differential operator , defined by (2.2)

operating (or being applied to) the wavefunction Ψ(x, t). In Eq. (2.2), i is the imaginary unit, defined by Equation (2.2) often is considered the central postulate of QM. The form of Eq. (2.2) can be made plausible from equations of classical wave mechanics, de Broglie's equation (Eq. [1.10]) and Planck's equation (Eq. [1.7]), but cannot be derived axiomatically. It was the genius of E. Schrödinger to realize that the substitution described in Eq. (2.2) yields differential equations that had long been known and had solutions that agreed with experiments. In the Schrödinger equations to be discussed explicitly in the next chapters (for the H atom, the vibrations and rotations of molecules, and molecular electronic energies), the classical kinetic energy T given by (2.3) is, therefore, substituted by (2.4) which is, of course, obtained by inserting Eq. (2.2) into Eq. (2.3). The total energy of a system is given as the sum of the potential energy V and the kinetic energy T: (2.5) Postulate 3: All experimental results are referred to as observables that must be real (not imaginary or complex). An observable is associated with (or is the “eigenvalue” of) a quantum mechanical operator . This can be written as (2.6) where a are the eigenvalues and ϕ the corresponding eigenfunctions. The terms “operator,” “eigenvalues,” and “eigenfunctions” are terminology from linear algebra and will be further explained in

Section 2.3 where the first real eigenvalue problem, the particle in a box, will be discussed. Notice that the eigenfunctions often are polynomials, and each of these eigenfunctions has its corresponding eigenvalue. In this book, following generally accepted notations, the total energy operator is generally identified by the symbol and referred to as the Hamilton operator, or the Hamiltonian, of the system. With the definition of the Hamiltonian above, it is customary to write the total energy equation of the system as (2.7) Equation (2.7) implies that the energy “eigenvalues” E are obtained by applying the operator on a set of (still unknown) eigenfunctions ψ that are here assumed to be time‐independent and a function of spatial coordinates x only, ψ(x). Solving the differential equations given by Eq. (2.7) yields the eigenfunctions ψi and their associated energy eigenvalues Ei. Postulate 4: The expectation value of an observable a, associated with an operator , for repeated measurements, is given by (2.8)

If the wavefunctions Ψ(x, t) are normalized, Eq. (2.7) simplifies to (2.9) since the denominator in Eq. (2.8) equals 1. This expectation value may be viewed as an expected average of many independent measurements and embodies the probabilistic nature of quantum mechanics.

Postulate 5: The eigenfunctions ϕi, which are the solutions of the equation , form a complete orthogonal set of functions or, in other words, define a vector space. This, again, will be demonstrated in Section 2.3 for the particle‐in‐a‐box wavefunctions, which are all orthogonal to each other and therefore may be considered unit vectors in a vector space. When evaluating the expectation values (Eq. [2.9]), the functions ψ(x) may or may not be eigenfunctions of because the real eigenfunctions ϕ(x) form a complete vector space. Functions that are not eigenfunctions of can be written as linear combinations of the basis functions ϕ(x). Thus, any arbitrary wavefunction ψ of a system can be written in terms of a series expansion of the true eigenfunctions ϕ(x) as follows: (2.10) The expansion coefficients an indicate how much each wavefunction contributes to, or resembles, the true eigenfunction of the operator. This aspect is particularly important for the approximate methods for solving the Schrödinger equation discussed in Appendix 2. Postulate 6: Time‐dependent systems are described by the time‐ dependent Schrödinger equation (2.11) where the time‐dependent wavefunctions are the product of a time‐ independent part, ψ(x), and a time evolution part: (2.12) We shall encounter the time‐dependent Schrödinger equation mainly in processes where molecular systems are subject to a perturbation

by electromagnetic radiation (i.e. in spectroscopy) and shall develop the formalism that predicts whether or not the incident radiation will cause a transition in the molecule between two states with energy difference ΔE = h ν = ħ ω. Next, a simple operator/eigenvalue example will be presented to illustrate some of the mathematical aspects. Example 2.1 Operator/eigenvalue problem Show that the function operator

is an eigenfunction of the

, that is, show that

Answer: (E2.1.1)

The function

is an eigenfunction of the operator. The

eigenvalue c = −1. Postulate 7: In many‐electron atoms, no two electrons can have identically the same set of quantum numbers. This postulate is known as the Pauli exclusion principle. It is also formulated as follows: the product wavefunction for all electrons in an atom must be antisymmetric with respect to interchange of two electrons. This postulate leads to the formulation of the product wavefunction in the form of Slater determinants (see Section 9.2) in many‐electron systems. The value of a determinant is zero when two rows or two columns are equal; thus, an atomic system where any electrons have exactly the same four quantum numbers would have an undefined product wavefunction. Furthermore, exchange of two rows (or columns) leads to a sign change of the value of the determinant. This last statement implies the antisymmetric property of the product wavefunction that changes its sign upon exchange of two electrons.

Commutation of operators: Although not really a postulate of quantum mechanics (since it follows from well‐defined mathematical principles), a discussion of the effects of operator commutation is included here. In physics, one often wishes to determine several quantities simultaneously, such as the position and momentum of a moving object or the x, y, and z components of the angular momentum. Since Postulate 3 above states that every observable is associated with a quantum mechanical operator, one has to investigate the case of solving for the eigenvalues of two operators simultaneously. Let

and

be two operators such that (2.13)

where a and b are the eigenvalues and φ and ϕ the eigenfunctions of and , respectively. These eigenvalues can be determined simultaneously in the same vector space if and only if the operators commutate, that is, if the order of application of the operators on the eigenfunction is immaterial. This commutator of two operators is written as (2.14) or abbreviated as . If the operators commutate and can be determined simultaneously; if the commutator is not zero, then the eigenvalues cannot be determined simultaneously. This case will be demonstrated in Example 2.2. Example 2.2 Determine the commutator momentum operator and the position operator to a function f(x), i.e. determine

of the when applied

(E2.2.1)

(E2.2.2) Answer: The derivative of the product needs to be evaluated using the product rule of differentiation. Thus, (E2.2.3)

(E2.2.4) Thus, the commutator (E2.2.5) which predicts that the position and momentum of a moving particle cannot be determined simultaneously. This was stated earlier in Eq. (2.1) as the Heisenberg uncertainty principle as (2.1) To show the equivalency of Eqs. (E2.2.5) and (2.1), one has to determine the standard deviations in momentum and position σp and σx that can be related to the uncertainties Δpx and Δx.

Figure 2.1 Potential energy functions and analytical expressions for (a) molecular vibrations and (b) an electron in the field of a nucleus. Here, f is a force constant, k is the Coulombic constant, and e is the electron charge.

2.2 The Potential Energy and Potential Functions In Postulate 2, the kinetic energy T was substituted by the operator (2.4) but the potential energy V was left unchanged, since it does not include the momentum of a moving particle. The potential energy, however, depends on the particular interactions describing the problem, for example, the potential energy an electron experiences in the field of a nucleus or the potential energy exerted by a chemical bond between two vibrating nuclei. The shape of these potential energy curves are shown in Figure 2.1 along with the potential energy equations. When these potential energy expressions are substituted into the Schrödinger equation

(2.7) one obtains a differential equation: (2.15) for the harmonic oscillation of a diatomic molecule and (2.16) for the electron in a hydrogen atom. In Eqs. (2.15) and (2.16), f and k are constants that will be introduced later, and e is the electronic charge, e = 1.602 × 10−19 [C]. Equation (2.16) is not strictly correct since the potential energy is a spherical function in the distance r from the nucleus, but is presented here and in Figure 2.1 as a one‐ dimensional quantity. Also, the mass in the denominator of the kinetic energy operator needs to be substituted by the reduced mass to be introduced later. Due to the difficulties in solving equations such as Eqs. (2.15) and (2.16), a much simpler potential energy function will be used for the initial example of a quantum mechanical system, namely, a rectangular box function. The ensuing particle in a box is an artificial example but is pedagogically extremely useful and presents simple differential equations while offering real physical applications; see Section 2.5.

2.3 Demonstration of Quantum Mechanical Principles for a Simple, One‐Dimensional, One‐Electron Model System: The Particle in a Box

Real quantum mechanical systems have the tendency to become mathematically quite complicated due to the complexity of the differential equations introduced in the previous section. Thus, a simple model system will be presented here to illustrate the principles of quantum mechanics introduced in Sections 2.1 and 2.2. The model system to be presented is the so‐called particle in a box (henceforth referred to as “PiB”) in which the potential energy expression is simplified but still has with wide‐ranging analogies to real systems. This model is very instructive, since it shows in detail how the quantum mechanical formalism works in a situation that is sufficiently simple to carry out the calculations step by step while providing results that much resemble the results in a more realistic model. This is exemplified by the overall similarity such as the symmetry (parity) of the PiB wavefunctions when compared with that of the harmonic oscillator wavefunctions discussed in Chapter 4.

2.3.1 Definition of the Model System The PiB model assumes that a particle, such as an electron, is placed into a potential energy well or confinement shown in Figure 2.2. This confinement (the “box”) has zero potential energy for 0 ≤ x ≤ L, where L is the length of the box. Outside the box, i.e. for x < 0 and for x > L, the potential energy is assumed to be infinite. Thus, once the electron is placed inside the box, it has no chance to escape, and one knows for certain that the electron is in the box. As discussed earlier, the total energy is written as the sum of the kinetic and potential energies, T and V, respectively: (2.17) As before, the kinetic energy of the particle is given by (2.3) where m is the mass of the electron. Substituting the quantum mechanical momentum operator,

(2.4) into Eq. (2.3), the kinetic energy operator can be written as (2.5)

Figure 2.2 Panel (a): Wavefunctions for n = 1, 2, 3, 4, and 5 drawn at their appropriate energy levels. Energy given in units of h2/8mL2. Panel (b): Plot of the square of the wavefunctions shown in (a).

The potential energy inside the box is zero; thus, the total energy of the particle inside the box is (2.18) Since the potential energy outside the box is infinitely high, the electron cannot be there, and the discussion henceforth will deal with the inside of the box. Thus, one may write the total Hamiltonian of the system as (2.19) In the notation of linear algebra, this operator/eigenvector/eigenvalue problem is written as (2.20) Equation (2.20) instructs to apply the Hamiltonian of Eq. (2.19) to a set of yet unknown eigenfunctions to obtain the desired energy eigenvalues. The eigenfunctions typically form an n‐dimensional vector space in which the eigenvalues appear along the diagonal. Thus, Eq. (2.20) implies (2.21)

that is, the Hamiltonian operating on a set of eigenfunctions such that ; ; ; and so forth that is, of course, obtained by carrying out the matrix multiplication indicated in Eq. (2.21).

2.3.2 Solution of the Particle‐in‐a‐Box Schrödinger Equation Rearranging Eqs. (2.19) and (2.20) yields (2.22) which is a simple differential equation that can be used to obtain the eigenfunctions ψ(x): (2.23) Any functions fulfilling Eq. (2.23) must be of the form that their second derivative equals to the original function multiplied by a constant. For example, the function (2.24) could be solution of the differential Eq. (2.23), since (2.25) Here, the term b2 would correspond to 2mE / ħ2, and A is a yet undefined amplitude factor. Similarly, (2.26) or the sum of Eqs. (2.24) and (2.26) could be acceptable solutions. For the time being, and for reasons that will become obvious shortly, Eq. (2.26) will be used as a trial function to fulfill Eq. (2.23):

(2.27)

and (2.28)

At this point, it should be pointed out that the solutions of any differential equation depend to a large extent on the boundary conditions: the general solution of the differential equation may or may not describe the physical reality of the system, and it is the boundary conditions that force the solutions to be physically meaningful. In the case of the PiB, the boundary conditions are determined by one of the postulates of quantum mechanics that requires that wavefunctions are continuous. Thus, if the wavefunction outside the box is zero (since the potential energy outside to box is infinitely high and, therefore, the probability of finding the particle outside the box is zero), the wavefunction inside the box also must be zero at the boundaries of the box. Thus, one may write the boundary conditions for the PiB differential equation as (2.29) Because of these conditions, the cosine function proposed as possible solutions (Eq. [2.24]) of Eq. (2.23) was rejected, since the cosine function is nonzero at x = 0. Because of the required continuity at x = L, the value of the function

must be zero at x = L as well. This can happen in two ways: The first possibility occurs if the amplitude A is zero. This case is of no further interest, since a zero amplitude of the wavefunction would imply that

the particle is not inside the box. The second possibility for the wavefunction to be zero at x = L occurs if (2.30)

Since the sine function is zero at multiples of π radians, it follows that (2.31)

Solving Eq. (2.31) for E yields the energy eigenvalues (2.32) Equation (2.32) reveals that the energy levels of the particle in a box are quantized, that is, the energy can no longer assume any arbitrary values, but only values of and so on. This is the first appearance of the concept of quantized energy levels in a model system and represents a step of enormous importance for the understanding of quantum mechanics and spectroscopy: by substituting the classical momentum with the momentum operator, quantized energy levels (or stationary states) were obtained. This quantization is a direct consequence of the boundary conditions, which required wavefunctions to be zero at the edge of the box. Since the energy depends on this quantum number n, one usually writes Eq. (2.32) as (2.33) Substituting these energy eigenvalues back into Eq. (2.27)

(2.27)

one obtains (2.34)

which are the wave functions for the PiB.

2.3.3 Normalization and Orthogonality of the PiB Wavefunctions In Eq. (2.34), “A” is an amplitude factor still undefined at this point. To determine “A,” one argues as follows: since the square of the wavefunction is defined as the probability of finding the particle, the square of the wavefunction written in Eq. (2.34), integrated over the length of the box, must be unity, since the particle is known to be in the box. This leads to the normalization condition (2.35)

Using the integral relationship (2.36) the amplitude A is obtained as follows:

(2.37)

Thus, the normalized stationary‐state wavefunctions for the particle in a box can be written in a final form as (2.38)

The stationary‐state (time‐independent) wavefunctions and energies are depicted in Figure 2.2, panel (a). Although one refers to these wavefunctions as time‐independent, they may be considered as standing waves in which the amplitudes oscillate between the extremes as shown in Figure 2.3 and resemble the motion of a plugged string. Time independency then refers to the fact that the system will stay in one of these standing wave patterns forever or until perturbed by electromagnetic radiation. The probability of finding the particle at any given position x is shown in Figure 2.2, panel (b). These traces are the squares of the wavefunctions and depict that for higher levels of n, the probability of finding the particle moves away from the center to the periphery of the box. The PiB wavefunctions form an orthonormal vector space, which implies that (2.39)

δmn in Eq. (2.39) is referred to as the Kronecker symbol that has the value of 1 if n = m and is zero otherwise. The wavefunctions' normality was established above by normalizing them (Eqs. (2.36) and (2.37)); in order to demonstrate that they are orthogonal, the integral

(2.40)

Figure 2.3 (a) Representation of the particle‐in‐a‐box wavefunctions shown in Figure 2.2 as standing waves. (b) Visualization of the orthogonality of the first two PiB wavefunctions. See text for detail. needs to be evaluated. This can be accomplished using the integral relationship (2.41) For any two adjacent wavefunction, say, m = 1 and n = 2 or m = 2 and n = 3, the numerator of the first term in Eq. (2.41) contains the sine function of odd multiples of π, whereas the numerator of the second term will contain the sine function of even multiples of π. Since the sine function of odd and even multiples of π is zero, the total integral

described by Eq. (2.41) is zero. This argument holds for any case where n ≠ m. This can also be visualized graphically, as shown in Figure 2.3b for the first two PiB wavefunctions for n = 1 (curve a) and m = 2 (curve b). When multiplied, curve c is obtained. The shaded areas above and below the abscissa of curve c represent the integral in Eq. (2.40) for n = 1 and m = 2 and are equal; therefore, the area under the product curve c is zero. Figure 2.3a also shows that the wavefunctions for the states with quantum number larger than 1 have nodal points, or points with no amplitude. This is familiar from classical wave behavior, for example, for a vibrating string. Since the meaning of the squared amplitude of the wavefunction can be visualized for the particle in a box as the probability of finding the electron, these nodal points represent regions in which the electron is not found. Example 2.3 a. What is the probability P of finding a PiB in the center third of the box for n = 1? b. What is P for the same range for a classical particle? Answer: a. The probability P of finding a quantum mechanical particle– wave is given by the square of the amplitude of the wavefunction. Thus, (E2.3.1)

The integral over the sin2 function can be evaluated using (E2.3.2)

Then the probability is

(E.2.3.3) b. A classical particle would be found with equal probability anywhere in the box; thus, the probability of finding it in the center third would just 1/3. Note that for higher values of n, the probability of finding it in the center third will decrease.

2.4 The Particle in a Two‐Dimensional Box, the Unbound Particle, and the Particle in a Box with Finite Energy Barriers 2.4.1 Particle in a 2D Box The principles derived in the previous section can easily by expanded to a two‐dimensional (2D) case. Here, an electron would be confined in a box with dimensions Lx in the x‐direction and Ly in the y‐ direction, with zero potential energy inside the box and infinitely high potential energy outside the box: (2.42)

The Hamiltonian for this system is (2.43) and the total wavefunction ψx, y can be written as

(2.44) where A as before is an amplitude (normalization) constant. The total energy of the system is (2.45)

Figure 2.4 Wavefunctions of the two‐dimensional particle in a box for (a) nx = 1 and ny = 2 and (b) nx = 2 and ny = 1. For a square box with Lx = Ly = L, the energy expression simplifies to (2.46) The wavefunctions can now be represented as shown in Figure 2.4 for the cases nx = 2 and ny = 1 and nx = 1 and ny = 2. These wavefunctions represent the standing wave on a square drum. Notice that the energy eigenvalues for these two cases are the same: (2.47) When two or more energy eigenvalues for different combination of quantum numbers are the same, these energy states are said to be degenerate. Here, for nx = 2 and ny = 1 and nx = 1 and ny = 2, the

same energy eigenvalues are obtained; consequently, E21 and E12 are degenerate. This is a common occurrence in quantum mechanics, as will be seen later in the discussion of the hydrogen atom (Chapter 7), where all the three 2p orbitals, the five 3d orbitals, and the seven 4f orbitals are found to be degenerate.

2.4.2 The Unbound Particle Next, the case of a system without the restriction of the boundary conditions (an unbound particle) will be discussed. This discussion starts with the same Hamiltonian used before: (2.23) When this differential equation is solved without the previously used boundary conditions (2.29) the new solutions represent a particle–wave that travels along the positive or negative x‐direction. The most general solution of the differential Eq. (2.23) is (2.48) where b is a constant. The second derivative of Eq. (2.48) is given by (2.49) with (2.50) or

(2.51) Equation (2.51) was obtained by substituting (2.52) into Eq. (2.50). Thus, the unbound particle can be described by a traveling wave (as opposed to a standing wave) (2.53) carrying a momentum (2.54) into the positive or negative x‐direction. k is the wave vector defined in Eq. (1.6).

2.4.3 The Particle in a Box with Finite Energy Barriers Finally, the particle in a box with a finite energy barrier, V0, will be discussed qualitatively. This is a situation where the particle is no longer strictly forbidden outside the confinement box and leads to the concept of tunneling, that is, the probability of the electron found outside the box. The shape of the potential function is shown in Figure 2.5b. The potential energy for this case is written as (2.55) and

(2.56) (Notice that the boundaries of the box were shifted from 0 to L to −L/2 to +L/2 for symmetry reasons that will be taken up again in Section 3.2.) The Schrödinger equation is written in two parts: Inside the box, where the potential energy is zero, the same equation holds that was used earlier: (2.23) Outside the box, the Schrödinger equation is (2.57)

Figure 2.5 (a) Particle in a box with infinite potential energy barrier. (b) Particle in a box with infinite potential energy barrier. The solutions of this equation will be of the form (2.58) (2.59) where (2.60)

For

is an exponential decay function, and for is an exponential growth function. This is shown in

Figure 2.5b to the right and left of the potential energy box, respectively. This represents the probability of finding the electron outside the box, a process that is known as “tunneling.” Inside the box, the solutions of Eq. (2.23) resemble the bound wavefunctions of the particle in a box, except that the amplitude at the boundary is no longer zero, but must meet with the wavefunction outside the box. This is depicted in Figure 2.5. Bound states exist for energies E(n) < V0 only; for E(n) > V0, the electron exists as a traveling wave as discussed before for unbound states. The concept of tunneling may seem esoteric at first, but it has interesting consequences. For example, a technique exists that is known as “tunneling electron microscopy (TEM)” where a very sharp metal tip is moved very close (within fractions of a nanometer) to the surface of the analyte, which is at a positive potential with respect to the metal tip. A tunneling current is observed between the tip and the analyte that is due to electrons tunneling from the tip to the analyte. As the substrate is moved laterally under the tip, the tip is lowered or raised to keep the tunneling current constant. In this way, an “image” of the morphology of the analyte can be obtained. Tunneling may also play a role in certain chemical reactions that depend on electron transfer from a donor to a receptor; some of these reactions are faster than expected from computations of the reaction rate from the activation energy. It is thought that in these reactions, the electron may tunnel from donor to receptor at a very fast rate. Finally, in the last example of “real‐world PIBs” in the section below, tunneling plays a major role.

2.5 Real‐World PiBs: Conjugated Polyenes, Quantum Dots, and Quantum Cascade Lasers 2.5.1 Transitions in a Conjugated Polyene

Although the PiB was introduced here as a model to demonstrate quantum mechanical principles in a mathematically manageable system, there are several physical examples that can be treated adequately using the PiB formalism. One of these is frequently incorporated as an experiment in physical chemistry laboratories [2] and involves a conjugated dye such as 1,6‐diphenyl‐1,3,5‐hexatriene, shown in Figure 2.6a. In this molecule, the Lewis structure suggests three double and four single bonds in the link between the two phenyl groups. From the viewpoint of the PiB formalism, one may consider the polyene framework the length of the box, indicated by the straight line connecting the two phenyl groups, and the six π‐ electrons to be delocalized over the entire conjugated length and constituting six electrons in a box in three electron pairs. A schematic of the π‐bonding scheme is shown in Figure 2.6b. The three electron pairs would, in this model, occupy the n = 1, n = 2, and n = 3 levels, as indicated by the up/down arrows in each of these levels. The absorption spectrum in the visible range shows one absorption peak that is, in this approximation, assigned as a PiB transition of one electron from the highest occupied molecular orbital (HOMO) with n = 3 to the lowest unoccupied molecular orbital (LUMO) with n = 4, as indicated by the heavy up arrow. In Example 2.4, the wavelength of this absorption will be calculated. This example may be a bit premature in this chapter, because it introduced aspects of transitions between energy levels, principles of bonding orbitals, and so forth. These subjects will be taken up in later chapters in more detail, but this example shows quite nicely that the PiB formalism can be applied to real systems. In some experiments in physical chemistry laboratory, the dependence of the absorption wavelength on the “length of the box,” that is, the conjugated length, has been described.

Figure 2.6 (a) Structure of 1,6‐diphenyl‐1,3,5‐hexatriene to be used as an example for the PiB calculations. (b) Energy level diagram, based on the PiB formalism, showing the three lowest energy levels occupied by the π‐electrons. Example 2.4 Calculation of the energy difference between n = 3 and n = 4 energy levels for the 1,6‐diphenyl‐1,3,5‐hexatriene system, shown in Figure 2.6, assuming that the electrons obey the particle in a box formalism. What is the wavelength of a photon that causes this transition? Answer: a. Estimation of the conjugated length. Since the single and double bonds, with bond lengths of 154 pm and 130 pm, respectively, are approximately 120o from each other, one can approximate the length of the box as (E2.4.1)

b. Calculation of the energy difference between n = 3 and n = 4. Use me = 9.1×10−31 [kg] and h = 6.6 × 10−34 [Js] for the electron mass and Planck's constant. Since the length of the box was estimated to 2 significant figures, the entire computation is carried out with 2 significant figures:

Analysis of units: (E2.4.2)

ΔE = 3.7 × 10−19 [J]

Figure 2.7 Absorption spectra of nanoparticles as a function of particle size. As expected, the larger particles exhibit lower energy (longer wavelength) transitions. c. ΔE = hc/λ or λ = hc/ΔE (E2.4.3)

2.5.2 Quantum Dots Certain quantum dot structures can also be modeled by a 2D particle in a box. Quantum dots may be manufactured by creating small circular or square semiconductor deposits on a substrate that is an electric insulator. The electrons of the semiconductor spots are free to move over the entire size of the dot, and the energy levels of the free electrons follow a 2D PiB model [3]. Consequently, the color of

electronic transitions can be tuned by varying the size of the quantum dot. Closely related to these 2‐dimensional quantum dots are 3‐ dimensional nanoparticles, such as spheres of metallic or semiconductor materials. The free electrons on the surface of such spheres can assume wave patterns known as “spherical harmonic functions” that are the solutions for particle on a sphere. Similar functions will be discussed in Chapter 7 during the treatment of the hydrogen atom wavefunctions. Again, the optical properties of such nanoparticles can be tuned by adjusting the size of the nanoparticle. This is shown in Figure 2.7.

2.5.3 Quantum Cascade Lasers Finally, an example of a commercial application of the particle in a box will be discussed, namely, that of a solid‐state infrared laser known as the quantum cascade laser (QCL) [4]. In QCLs, the “box” is constructed by creating a semiconductor “superlattice” potential functions that mimics a PiB with finite potential energy barriers. Furthermore, the bottom of the energy well is not flat, but at a slant, as shown in Figure 2.8a. Both the barriers and the slant of the bottom of the energy well can be achieved in the fabrication process by vaporizing different composition of semiconductor materials (doping).

Figure 2.8 (a) An individual energy well with finite barrier height and a sloping energy bottom with the two lowest energy states and wavefunctions. (b) The superlattice structure in a quantum cascade laser modeled by successive PiB potential energy levels. The slant of the energy well bottom has the effect of distorting the PiB wavefunctions as shown exaggerated for the two lowest energy states in Figure 2.8a. (The computation of wavefunctions in the presence of a sloping baseline will be discussed in Appendix 2, perturbation methods.) The distortion causes the amplitude of the wavefunction to shift toward lower potential energy with the consequence that an electron in the ground state of the well has a higher probability of tunneling through the barrier, due to their higher amplitude at the right side of the well. The “superlattice” is formed by having a large number of these energy wells arranged as shown in Figure 2.8b. During the operation of the QCL, electrons are injected, via an electric current, at a potential energy marked by the * symbol in Figure 2.8b into a highly excited energy state and undergo a transition as indicated by the leftmost down arrow. During this transition, an (infrared) photon is emitted. Subsequently, the electron in the ground state may tunnel through the finite‐height barrier and arrives in the next quantum well and undergoes another

transition. The emission and tunneling processes are repeated as many times as there are quantum wells in the superstructure. The term “cascade” in QCL is due to the fact that one electron can undergo many consecutive emission processes in the superlattice structure. By placing the superlattice crystal into an optic cavity, stimulated emission (see Chapter 3) from the excited states into the ground states of each well can be achieved.

References 1 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon. 2 Shoemaker, D.P., Garland, C.W., and Nibler, J.W. (1996). Experiments in Physical Chemistry, 6e. New York: McGraw‐Hill. 3 Banin, U. et al. (1999). Identification of atomic‐like electronic states in indium arsenide nanocrystal quantum dots. Nature 400 (6744): 542–544. 4 Faist, J. et al. (1994). Quantum Cascade Laser. Science 264 (5158): 553–556.

Problems The following trigonometric integral relationships are needed for these problems:

1. Show that the function f(x) = cos(bx) is an eigenfunction of the operator d2/dx2. What is the eigenvalue? 2. Show that the function e−x2/2 is an eigenfunction of the operator (d2/dx2) − x2. What is the eigenvalue? 3. Show that the function e−4ix is an eigenfunction of the operator d2/dx2. What is the eigenvalue? 4. What is the probability P of finding a ground‐state PiB in the center third of the box? What is P for the same range for a classical particle? 5. For the PiB in the ground state, determine the expectation values of x and px. 6. What is the expectation value of the kinetic energy operator T for the ground‐state PiB? 7. What is the probability P of finding a particle in the first excited state in the left half of the box with length L within the PiB approximation? 8. Consider an electron in a one‐dimensional box with a length of 0.1 nm. a. Calculate the energy of the 1st, 2nd, and 3rd energy levels for this electron. b. Calculate the wavelength of a photon required to promote the electron from the 2nd to the 3rd energy level. 9. Describe in your own words why the particle‐in‐a‐box model results in quantized energy levels. 10. What is quantum mechanical tunneling? 11. Calculate the commutator [Tx, px] where Tx is the kinetic energy operator in the x‐direction and px is the momentum operator in the x‐direction. Can the kinetic energy and the momentum be determined simultaneously in a quantum mechanical system?

12. Show by analytical integration that ψ1(x) and ψ2(x) are orthogonal for the one‐dimensional PiB. 13. Using a graphics program such as Excel, plot the first and second wavefunctions for a particle in a box. Show by graphical integration that these functions are orthogonal.

3 Perturbation of Stationary States by Electromagnetic Radiation In the previous chapter, the principles of quantum mechanics were introduced at the level of an artificially conceived model system, the particle in a box. Before investigating real situations, such as the vibration or rotation of molecules, we shall investigate how the simple particle in a box model system will behave when perturbed by electromagnetic radiation and will further develop the ideas of spectral transitions that were introduced qualitatively in Example 2.4. The effect of electromagnetic radiation on a quantum mechanical system is presented in the form of perturbation theory, in general, is discussed in Appendix A2.5. Perturbation theory is an approach used when the quantum mechanical equations to be solved are so complicated that they cannot be evaluated directly. In perturbation theory, one assumes that the perturbation to the system is sufficiently small that the solutions can be approximated by the unperturbed wavefunctions, and an energy correction is computed based on the unperturbed set of basis functions.

3.1 Time‐Dependent Perturbation Treatment of Stationary‐State Systems by Electromagnetic Radiation

Time‐independent quantum mechanics introduced in the previous sections describes the energy expressions and wavefunctions of stationary states, that is, states that do not change with time. Stationary‐state wavefunctions and energies are obtained by solving the appropriate Schrödinger equation. Next, a description is needed that describes how a system can transition from one stationary state

to another when a perturbation, typically electromagnetic radiation, is applied. For this, one needs to invoke the time‐dependent Schrödinger equation, which for the one‐dimensional case is (2.11) When electromagnetic radiation impinges on a system described by the stationary‐state wavefunctions, one describes the perturbation in terms of a perturbation operator such that (3.1) One assumes that there exist exact eigenfunctions for the operator and that the perturbation due to is small. If the perturbation applied to the system is due to electromagnetic radiation, one may write the perturbation operator as (3.2) In Eq. (3.2), the expression introduced previously (Eq. [1.2]) for the electric field in electromagnetic radiation propagating in the z‐ direction is assumed. The electric field will exert a force (3.3) on particles with charge e. If the molecular system consists of i charged particles found at positions xi, one defines the “electric dipole moment” μ according to (3.4) and rewrites Eq. (3.2) as

(3.5) The time‐dependent Schrödinger equation then appears as (3.6) and subsequently is solved with the unperturbed eigenfunctions ψ of the operator (3.7) which may be, for example, the time‐independent (unperturbed) wavefunctions of the particle in a box or the harmonic oscillator (see Chapter 4). The time dependence of each of the wavefunctions is introduced as follows: (3.8) where (3.9) Equation (3.8) is the general definition of a time‐dependent wavefunction that consists of a stationary, time‐independent part ψ(x) and the time evolution of this wavefunction given by φ(t) = eiωt. The time‐dependent wavefunctions Ψ(x, t) of the system undergoing a transition subsequently are expressed in terms of time‐dependent coefficients ck(t) and the time‐dependent wavefunctions Ψ(x, t) according to (3.10)

The coefficients ck(t) describe the time‐dependent response of the quantum mechanical system to the perturbation – in particular, the change in population of the excited state in response to the perturbation. An example may serve to illustrate the procedure invoked so far. Consider a two‐state system in the absence of a perturbation, as shown in Figure 3.1.

Figure 3.1 Two state energy level diagrams used for the discussion of time‐dependent wavefunction during a transition. The system is in the ground state and can be described by cg(t) = 1 and ce(t) = 0 or (3.11) In Eq. (3.11), the subscripts “g” and “e” denote the ground and excited states, respectively. After a perturbation is applied, the coefficients cg(t) and ce(t) change to account for the system undergoing a transition into the excited state that can be described by (3.12)

Thus, the overall time‐dependent Schrödinger equation that accounts for the response of the system is (3.13)

The solution of this equation can be found in many texts on quantum mechanics [1] (vol. II, Chapter 2) and proceeds by taking the necessary derivatives and integrating the resulting terms between time 0 and the duration of the perturbation. This procedure yields an expression for the time evolution of the expansion coefficients cm(t): (3.14)

Equation (3.14) is one of the most important equations for the understanding of spectroscopic processes since it outlines three major features of the response of a molecule when exposed to electromagnetic radiation. First, it implies that the term (3.15) known as the transition moment must be nonzero for a transition between the two states n and m to occur. This statement holds for one‐photon absorption and emission processes only; later, another transition mechanism will be introduced (Raman scattering; see Chapter 5) that is a two‐photon process and obeys different selection rules. The transition moment describes the action of the dipole operator μ defined in Eq. (3.4) on the stationary‐state wavefunctions

n and m between which the transition is induced. The transition moment, in general, determines the selection rules, depending on the exact nature of the wavefunctions and their symmetries. This aspect will be discussed in Section 3.2. Second, the term containing the amplitude of the electric field,

,

indicates that the “light must be on” for a transition to occur. Third, the part in the square brackets in Eq. (3.14) describes how the system responds to electromagnetic radiation of different frequency or wavelength. It is this term that prescribes that the frequency of the light must match the energy difference between the molecular energy levels for the transition to occur. This can be seen from the following discussion. The second term in the square bracket in Eq. (3.14) becomes very large at the resonance condition (3.16) This implies that when the frequency ω of the incident radiation is equal to, or very close to, the energy difference ωnm between states n and m, a transition between these states may occur, and a photon with the corresponding energy ħω may be absorbed, if the transition moment is nonzero. Similarly, the first term in the square bracket in Eq. (3.14) becomes very large if (3.17) This case corresponds to the situation of stimulated emission, where a photon of the proper energy impinges onto a molecular or atomic system in the excited state and causes this state to emit a photon, thereby returning to the lower energy state. This time‐dependent part of Eq. (3.14) also contains explicitly the expressions needed to explain certain off‐resonance phenomena, such as molecular polarizability, to be discussed in Chapter 5. The magnitude of resonance versus off‐resonance effects (see Chapter 1) can be estimated from the expression in square brackets as well.

Equation (3.14) holds for one‐photon absorption and emission situations that include standard infrared (vibrational), microwave (rotational), and visible/ultraviolet (electronic) absorption spectroscopy. The time‐dependent part in the square bracket of Eq. (3.14) can be summarized as (3.18) This equation was first mentioned in the introduction and corresponds exactly to the condition described above by Eqs. (3.16, 3.17): (3.19)

3.2 Dipole‐Allowed Absorption and Emission Transitions and Selection Rules for the Particle in a Box The dipole moment for a transition from state m to state n, given by the expectation value of the dipole operator, (3.20) must be nonzero for a transition to be allowed, and its square is proportional to the intensity of the transition. Whether or not the dipole transition moment is nonzero depends on the symmetry, or parity, of the wavefunctions involved. This will be demonstrated for the model system introduced earlier, the particle in a box. For a one‐dimensional, one‐electron system, the dipole operator is written as (3.4)

where e is the electronic charge. Thus, for the transition from n = 1 to n = 2 for the particle‐in‐a‐box wavefunctions, one needs to evaluate the expression (3.21)

This integral can be solved using the Mathematica or similar software by entering Integrate[Sin[2 Pi x /L]*x*Sin [Pi x/L], [2]] which gave the result −8L2/9π2. After multiplication by the normalization factor, the result given in Eq. (3.21) was obtained. To investigate whether or not the n = 1 to n = 3 transition for the particle in a box is allowed, the integral Integrate[Sin[3 Pi x /L]*x*Sin[Pi x/L], [2]] = 0 was solved and gave a zero transition moment; that is, this transition is not allowed. Here, one encounters, for the first time, the situation that a transition is “allowed” or “forbidden” based on the symmetry of the transition. This can best be visualized graphically. Figure 3.2 shows a plot of the 1st and 2nd wavefunctions for the particle in a box, along with the transition operator that, according to Eq. (3.4), is a straight line. Notice that in Figure 3.2, the center of the box was shifted to be at x = 0 and the length of the box was adjusted to run from −L/2 to L/2. This was done for symmetry reasons. The goal of the zero‐point shift was to make the ground‐state wavefunction “symmetric” with respect to the dotted line and the first excited‐state wavefunction “antisymmetric” to this axis. In this view, the dipole operator is antisymmetric as well.

Figure 3.2 Plot of the PiB ground‐state (trace a) and first excited‐ state (trace c) wavefunctions. Trace (b) represents a plot of the transition operator. Trace (d) represents the transition moment, the product of traces a, b, and c. The integral of this product (shaded area under trace d) is nonzero.

In Figure 3.2, the functions ψ1 (trace a) and ψ2 (trace c) and the transition operator (trace b) are shown. The product 〈ψ2 ∣ μ ∣ ψ1〉 is shown by curve “d,” and the area under the curve when integrated from −L/2 to L/2 is nonzero. This result can also be visualized from a parity or symmetry argument: since ψ2 and the dipole operator are odd functions, their product will be even. This product, multiplied by the ground state ψ1, which has even parity, will result in an overall transition moment with even parity. By the same argument, the transition from n = 1 to n = 3 is forbidden since the n = 3 state has even parity. Thus, the product of the dipole operator and the ground‐state and excited‐state functions will have odd parity, and the integral will be zero. For the particle in a box, this leads to the following selection rules: transitions with Δn = ±1, ±3, ±5, and so on are allowed, whereas transitions with Δn = ±2, ±4, and so on are forbidden because the transition moment integrals are zero. Thus, one encounters here that transitions are allowed or forbidden depending on the symmetry of the wavefunctions. The transition moment is the quantity that needs to be evaluated in order to determine whether or not a transition may occur.

3.3 Einstein Coefficients for the Absorption and Emission of Light When electromagnetic radiation passes from vacuum into a medium, there will be two major interactions that are governed by the properties of the medium: one will be a surface effect, reflection, and refraction at the interface. The other will be an attenuation of the intensity of the light within the medium known as absorption. Both these effects are described by the complex refractive index, η: (3.22) In Eq. (3.22), n is the real refractive index used in classical optics to describe reflection and refraction, and κ is the absorptivity discussed in a moment. The real part of the refractive index n can be used at

conditions far from an absorption band to determine the amount of light being reflected and refracted and determines the angle of reflection and refraction under these conditions. In classical optics, for example, refraction of light to describe lenses can be adequately treated by the real part of the refractive index since glass is transparent in the visible spectrum, and there are no nearby absorption bands. However, even in classical optics, the achromatic behavior of lenses results from the fact that the refractive index is not constant with wavelength, but generally increases toward the short wavelength region of light. This effect will be discussed in more detail in Chapter 5 where absorption and dispersion of light will be covered. The absorptivity κ is related to the molar extinction coefficient ε as follows: (3.23) ε is defined in Beer–Lambert law, which states that the absorbance (attenuation) A of light in a medium is given by (3.24) In Eq. (3.24), C is the concentration, in mol/L, of the absorbing species; l is the path length the light travels through the medium, expressed in cm;1 and I0 and I are the incident light intensity and the intensity after passing a path length l into the sample, respectively. The Beer–Lambert law often is written in the form (3.25) This form better relates to the fact that, if ε is nonzero, the light intensity decreases exponentially with the path the light travels through the medium. For the rest of this section and the next one on laser theory, it may be best to visualize the stationary states as those of an atomic species,

for example, the ground state and first excited states of a noble gas atom, such as neon, although the discussion is true for any atomic or molecular states. Furthermore, for the discussion of the population of the states below, let us assume that the system can only exist in two states, |ψ1〉 and |ψ2〉. The dipole transition moment would then be written as 〈ψ2 ∣ μ ∣ ψ1〉. However, this transition moment does not express the rate of transitions between the two states. This rate depends on the population of the originating state and the number of photons impinging onto the system, in the case here the neon gas exposed to light. The rate of absorption, i.e. the number of photons being absorbed by the gaseous molecules undergoing a transition from the ground state to the excited state |ψ2〉 per unit time, is given by the expression (3.26) Here, N1 denotes the population of the ground state; ρ(ν12) is the radiation density, i.e. the number of photons at frequency ν12 that have the correct energy to cause the transition from the ground state to the excited state; and B12 is the Einstein coefficient for absorption given by (3.27) Eq. (3.26) gives a rate for the absorption of photons or the creation of excited‐state species and therefore the depopulation of the lower energy state. By the same token, photons of frequency ν12 may encounter an atom in an excited state (from a previous absorption process) and may stimulate this excited atom to release its excitation. The rate for this “stimulated emission” process is given in complete analogy as (3.28) where N2 is the population of the excited state and B21 is the Einstein coefficient for stimulated emission. Since the squares of the

transition moments for emission and absorption are equal (3.29) it follows that (3.30) However, the rates of the absorption and emission process are vastly different, since they depend on the populations N1 and N2 of the ground and excited states. The ratio of the populations N2–N1 is given by the Boltzmann equation (3.31) In the following example, this ratio will be calculated for two different energy spacings between the states involved, one for a typical electronic transition where the energy difference corresponds to a visible photon and one for a vibrational transition that corresponds to an infrared photon. Example 3.1 Calculation of population ratios for a two‐state system with an energy spacing between the energy levels of the following: a. ΔE = 15 000 cm−1 (=1.0 × 106 m−1) corresponding to a visible photon with λ = 666 nm. b. ΔE = 1500 cm−1 (=1.0 × 105 m−1) corresponding to an infrared photon with λ = 6.66 μm at room temperature (T = 298 K). Answer: a. Using Eq. (3.31), with a numerical value of R = 0.698 cm−1/K−1 mol−1 (see Appendix 1), one obtains

(E3.1.1)

b.

(E3.1.2)

The two population ratios are obviously different by many orders of magnitude. While in case (a) roughly one atom out of 109 mol (!!) would be in the excited state, in the second case, nearly 0.1 % of the atoms are in the excited state. As demonstrated in Example 3.1, the Boltzmann distribution determines that the population in the lower energy state is always higher than that in the higher energy state, that is, (3.32) Therefore, the rate expression given by Eq. (3.26) will always be larger than the rate expression given by Eq. (3.28), i.e. (3.33) since the two Einstein coefficients are the same (see Eq. [3.30]). The two processes introduced so far, absorption and stimulated emission, depend on the presence of electromagnetic radiation. However, another deactivation process, known as “spontaneous emission,” can happen. The rate of this step depends on the expression (3.34) where A21 is the Einstein coefficient for spontaneous emission that also depends on the transition moment but is about a millionfold smaller than B21 for energy differences corresponding to visible photons. Thus, the rate of light absorption is given by Eq. (3.26), whereas for the combined deactivation process, Eq. (3.35) holds

(3.35) The absorptivity κ, introduced in Eqs. (3.22) and (3.23), is defined in terms of the Einstein coefficients as (3.36) that is, the difference of the absorption and stimulated emission rate equations. Since the first term in the parentheses is larger than the second term (see Eq. [3.33]), κ always will be a positive number for a two‐state energy system discussed here. Consequently, ε also will be positive, and, according to Eq. (3.24), light will always experience an attenuation when passing through a medium within an absorption region.

3.4 Lasers The last sentence of the previous paragraph only is correct for a system with two energy levels. However, light amplification in a medium is possible under certain circumstances, as demonstrated by the existence of lasers, which are very real objects. The term “laser” is an acronym for light amplification by stimulated emission of radiation, and this name implies that there is a way that light, passing through a medium, is amplified, rather than attenuated. For this to be possible, both κ and ε need to be negative such that the exponent of Eq. (3.24) becomes positive, and the intensity of light increases when passing through the medium. According to Eq. (3.36), κ can only be negative if (3.37) that is, if a condition known as population inversion exists. This is thermodynamically impossible in a two‐state system described by Eq. (3.31) since it would require negative (absolute) temperatures (which is a serious no–no).

Figure 3.3 Panel (a): Schematic energy level diagram of a 3‐level system in which population inversion can be achieved. (b) Simplified energy level diagram of the He–Ne laser. The thermodynamic limitations of achieving a population inversion can be bypassed using kinetic principles in a system that incorporates at least three energy states. This is equivalent to certain chemical reactions that should proceed to a thermodynamically most stable product (“thermodynamic control”) but rather proceed to a thermodynamically less stable product because the reaction rate is faster (“kinetic control”). In analogy, a state may be populated by a very fast process, but the deactivation process from this state is slow, leading to a population inversion. This is shown in Figure 3.3a. In the case shown in Figure 3.3a, state 3 is populated by electromagnetic radiation at a frequency at ν31 or any other methods, such as collisional energy transfer to be discussed later. If the Einstein coefficient for spontaneous emission (depopulation) from state 2 to state 1, A21, is much larger than that from state 3 to state 2, A32, the population in state 2 is drained fast, and a population inversion between states 3 and 2 can occur. Laser action may happen between these states; this action is initiated when a photon at ν32 is created to be spontaneous emission and subsequently starts to

stimulate species in state 3. Then, amplification of the radiation field at ν32 occurs. Among the many laser types that now exist – gas lasers, solid state lasers, diode lasers, fiber lasers, etc., gas lasers will be used to elaborate upon the principles of laser operation since the principles here are most easily understood. One of the most commonly produced and used gas lasers is the He‐Ne laser that emits a bright red laser line at 632.8 nm. A laser, in general, consists of a gain medium and a resonator structure, as shown in Figure 3.4. The resonator structure consists of mirrors that reflect the light created in the gain medium back and forth to stimulate excited species in the gain medium to emit. The resonator structure is often a flat and a spherical mirror to focus the light into the gain medium. One of the mirrors may be partially transparent to allow some of the light, typically 1 %, to escape from the resonator and create the laser beam.

Figure 3.4 Schematic of a gas laser, consisting of the resonator structure, defined by two mirrors and the gain medium. In the He–Ne laser, the gain medium is a mixture of helium and neon, at a ratio of about a 10 : 1, contained in a gas cell shown in Figure 3.4 fitted with Brewster angle windows to minimize reflection losses and to linearly polarize the laser output. A DC of 10–20 mA at a few hundred volt is passed through the gas mixture that ionizes the He atoms to produce He ions and electrons. Collisions between electrons and He atoms cause transitions of the He atoms into a

highly energetic state. These excited He atoms, in turn, collide with Ne atoms, promoting one of the 2p electrons into a 5s orbital; see Figure 3.3b. This energy level corresponds to energy level 3 in Figure 3.3a. The deactivation of the 2p53s1 states is very fast, thus allowing the population inversion between the 2p55s1 and 2p53s1 states to build. Laser action occurs, among other transitions, between the 2p55s1 and (still excited) 2p53s1 states. This transition is responsible for the 632.8 nm laser line. He‐Ne lasers produce highly monochromatic light typically in the 0.1–50 mW power regime. Although the gain medium varies between the various laser types, the principle is always the same in that a population inversion between certain states is created. Light amplification occurs when a photon, created by spontaneous emission, encounters excited species that exist in the inverted population and stimulates these species to emit. The photon created in this process is coherent (in phase) with the photon that created it.

References 1 Levine, I. (1970). Quantum Chemistry, vol. I&II. Boston: Allyn & Bacon. 2 Eyring, H., Walter, J., and Kimball, G.E. (1967). Quantum Chemistry. New Yrok: Wiley.

Problems 1. Define the electric dipole transition moment and operator both mathematically and descriptively. 2. Show graphically, using Excel or a similar program, that the dipole moment operator‐mediated transition from n = 1 to n = 2 is allowed for PiB wavefunctions by plotting the PiB wavefunction and the dipole operator. Shift the origin of the potential well to the center of the box.

3. Certain two‐photon spectroscopic effects depend on the square of the dipole moment operator, which for a single charged particle would be e2x2. For a particle in a box (with the origin shifted to the middle of the box), plot the wavefunctions and transition operators and discuss their parity: a. ∫ sin(x) sin(2x) dx

(orthogonality)

b. ∫ sin(x) x sin(2x) dx (one‐photon dipole transition) c. ∫ sin(x) x2 sin(2x) dx (forbidden 2‐photon 1 →2 transition) d. ∫ sin(x) x2 sin(3x) dx (allowed 2‐photon 1 →3 transition) 4. Consider a two‐state quantum mechanical system with two energies E2 and E1 (E2 > E1). The energy difference between the two states is 600 cm−1. a. At what temperature will the ratio of the population of the higher state, n2, and the lower state, n1, be 0.3? (k = 0.698 cm−1/K) b. What will be the ratio n2/n1 at room temperature? 5. What is the fundamental difference between spontaneous and stimulated emission of radiation? 6. Discuss the sine qua non condition for laser action to occur. How is this condition related to quantities that determine absorption or emission of radiation?

Note 1

Although the unit of length should be the meter, molar extinction coefficients are generally reported in units of L/(mol cm).

4 The Harmonic Oscillator, a Model System for the Vibrations of Diatomic Molecules The atoms in molecules are in constant vibrational motion that is a manifestation of the heat content or temperature of matter. These vibrational motions appear to be random but can be decomposed into movement along certain coordinates, which will be discussed in Chapter 5. The amplitude of these motions increases with increasing temperature. Completely stopping this motion is impossible, as we shall see in this and the following chapter, and also would violate the third law of thermodynamics as it could create a complete order and absolute‐zero temperature. This chapter will introduce the next logical model system commonly discussed in quantum mechanics, the vibrational motion of a diatomic molecule obeying the harmonic oscillator approximation. This model, for which the Schrödinger equation still can be solved exactly, already is much closer to real molecular systems than the particle‐in‐a‐box model introduced in Chapter 2 and describes the molecular vibration of diatomic molecules adequately. Further modification to this model (by introducing the anharmonicity) refines it to the point where real diatomic molecules can be described with very high accuracy.

4.1 Classical Description of a Vibrating Diatomic Model System The harmonic oscillator model is usually introduced for a diatomic molecule, as shown in Figure 4.1. Here, two atoms with masses m1 and m2 are connected by a covalent chemical bond along the x‐ direction. This bond can be thought of as a spring with a restoring force F obeying Hook's law:

(4.1) where k is the force constant of the spring, expressed in units of [N/m]. The potential energy V created when extending or compressing this spring by the distant dx is obtained by integrating the force over this distance: (4.2)

Figure 4.1 Definition of a diatomic harmonic oscillator of masses m1 and m2 that are at a distance x1 and x2 from the center of mass (COM). The shape of this potential energy curve was shown in Figure 2.1a. The name of the model system “harmonic oscillator” is derived from the expression in Eq. (4.2) that implies a harmonic or quadratic potential energy function. With Newton's second law of motion, (4.3) one can write the equation of motion for a diatomic molecule as (4.4) In Eqs. (4.2) and (4.3), mR is the reduced mass defined by

(4.5) with m1 and m2 the individual masses of the two atoms. The reduced mass is used to create a vibrational motion in which the center of mass of the diatomic molecule is stationary, that is, when using the reduced mass, the vibrational motion is decoupled from any translational motion of the molecule. One valid solution of the differential equation of motion Eq. (4.4) is (4.6) where (4.7)

or (4.8)

where ω is the angular frequency. Notice that for a classical vibrational problem, the amplitude A is arbitrary but that the frequency is defined by Eq. (4.7). This implies that for larger amplitudes, the velocity of the motion of the particles increases but the frequency remains constant. These equations also express the motion of a mass suspended from a solid beam by a spring. At this point, a quick analysis of magnitudes and units is appropriate. The magnitude of the force constant k acting between the atoms in a diatomic molecule such as gaseous HCl was found experimentally to be about 450 [N/m] = 450 [kg/s2], corresponding to a relatively stiff spring in classical mechanics. This allows us to calculate the classical vibrational frequency of the diatomic molecule HCl as will be demonstrated in Example 4.1.

Example 4.1 Calculation of the classical vibrational frequency of the HCl molecule Answer: The reduced mass of an HCl molecule is, according to Eq. (4.5), approximately (E4.1.1) Thus, the vibrational frequency for the HCl molecule is found to be (E4.1.2)

Using the frequency to wavenumber conversion gives a value close to the observed stretching frequency for gaseous HCl of 2.85 × 103 cm−1 or 2.85 × 105 m−1. Notice that, as pointed out in earlier chapters, this result implies that a molecule such as diatomic HCl has a characteristic vibrational frequency, but this leaves no room for the concept that electromagnetic radiation causes a transition to a more highly excited state. In a classical system, the energy can increase in infinitesimally small increments by increasing the amplitude of the vibration, whereas in the quantum mechanical and experimentally verified situation, the energy can only increase in certain quantized increments, leading to the absorption and annihilation of a photon. This aspect will be discussed next.

4.2 The Harmonic Oscillator Schrödinger Equation, Energy Eigenvalues, and Wavefunctions

Assuming that the chemical bond in a diatomic molecule obeys Hook's law, the vibrational Schrödinger equation for a harmonic oscillator with one degree of freedom (x) is then (4.9) This follows from Eqs. (2.5) and (4.2). (In Eq. [4.9] and the following discussion, the subscript “R” in mR for the reduced mass has been dropped to simplify the notation.) This differential equation is known as “Hermite's” differential equation, in which the wavefunctions ψ(x) are the time‐independent (stationary state) vibrational wavefunctions and E denotes the energy of the vibrational states. Equation (4.9) is a typical operator–eigenvalue equation notation commonly found in linear algebra. This formalism is an instruction to operate with an operator, here, the vibrational Hamiltonian (4.10) on a set of (yet unknown) functions to obtain the eigenvalues. Substituting the eigenvalues into the trial solution and considering the boundary conditions yield the eigenfunctions ψ(x). Detailed methods for solving the vibrational Schrödinger Eq. (4.9) can be found in many books on vibrational spectroscopy or quantum chemistry textbooks [1]. Here, the approach to solve Eq. (4.9) is only outlined to demonstrate how involved such a solution is. Equation (4.9) is reformatted to read (4.11)

Here, the results from Eq. (4.8) were used: (4.12)

Equation (4.10) is solved under the assumption that for large values of x, the following condition holds: (4.13) This results in a simplified equation (4.14) which has the approximate solutions (4.15) that are known as the “asymptotic solutions”where (4.16) Next, one assumes that the final results are of the form (4.17) i.e. Eq. (4.15), multiplied by a yet unknown function f(x). In Eq. (4.17), the Gaussian part of the harmonic oscillator wavefunctions is already in place, but the derivation of the functions f(x) is rather complicated and involves a series expansion of f(x) according to (4.18)

After taken the necessary derivatives required in Eq. (4.11), one obtains

(4.19)

Here, the expansion coefficients “a” for the series expansion of f(x) are given by the recursion formula (4.20)

Equation (4.20) leads to two different infinite series expansions for f(x), one with odd, the other with even expansion coefficients. Next, one has to investigate how the boundary conditions affect these expansion coefficients. This involves rather tricky arguments [1] about the values of successive expansion coefficients, and it appears that the wavefunction will become infinite as x goes to infinity. For the wavefunctions ψ(x) to be finite, the summation in Eq. (4.19) must be terminated at some finite value of n. Then, the numerator of the right side of Eq. (4.20) must be zero: (4.21) This leads to (4.22) With Eq. (4.16), the energy eigenvalues (4.23) are obtained. Quantization of the energy in Eq. (4.22) leads to wavefunctions that are a product of the Gaussian factor and a series expansion in x with the expansion coefficients given by Eq. (4.20). The resulting polynomials are referred to as the Hermite polynomials

These polynomials are related to each other by a recursion formula (see below) that originates from Eq. (4.20). It is very interesting to realize that the solution of a differential equation that appears quite simple requires many steps and quite a few assumptions, as discussed above. The author, as a spectroscopist and not a mathematician, always marvels at the insight and ingenuity of the mathematicians who first proposed solutions to the equations encountered in physical chemistry. In this case, the differential equation was solved by a nineteenth‐century French mathematician Charles Hermite. The vibrational wavefunctions resulting from the discussion above then are of the form (4.24) where N is a normalization constant,

, and

are the Hermite polynomials of order n in the variable

. The

vibrational quantum number n is an integer that may take values from zero to infinity, and α was defined in Eq. (4.16). For simplicity's sake, setting

, the Hermite polynomials

in the variable z can be written as (4.25)

The order n of the Hermite polynomials determines the highest power in which the variable z occurs in each polynomial. Thus, more highly excited states (i.e. those with higher quantum number n) correspond to Hermite polynomials with higher power of z. This

aspect is important since the power of z determines the shape of the wavefunctions. The higher members of the Hermite polynomials can be derived from the recursion formula: (4.26) Thus, the Hermite polynomial of degree n is related to the previous and subsequent polynomial by Eq. (4.26). Example 4.2 Using the recursion formula and the analytical expressions for H2(z) and H3(z), derive the form of H4(z). Answer: We rewrite Eq. (4.24) to read (E4.2.1) Then, H4(z) = 2zH3(z) − 6 H2(z) = 2z(8 z3 − 12 z) − 6(4 z2 − 2) = (E4.2.2) Thus, the first few vibrational wavefunctions are (4.27)

These functions, which are plotted in Figure 4.2, form an orthonormal vector space. Orthonormality implies (see Eqs. [2.35]– [2.41]) that

(4.28) Here, δij represents the Kronecker delta, which equals to one if i = j and zero otherwise. For example, (4.29) and (4.30)

In Eq. (4.29), the integral relationship

was

used. The eigenvalues of the vibrational Schrödinger equation are given by (4.23)

Figure 4.2 Quadratic potential energy function V = ½ kx2 for a diatomic molecule and the resulting quantum mechanical vibrational wavefunctions and energy states (a) and the square of the wavefunctions (b). In Eq. (4.23), the frequency ν is written in units of s−1 such that the term hν has units of energy [J]. Vibrational spectroscopists, however, prefer to use wavenumber (cf. Eq. [1.12]) as a unit of energy; thus, in the remainder of the book, molecular vibrational energies are expressed as (4.31) and the corresponding photon energy as

.

The harmonic oscillator wavefunctions and energy eigenvalues are shown in Figure 4.2a, along with the quadratic potential energy function (Eq. [4.2]) and the energy levels corresponding to Eq. (4.23). There are several interesting facts about the wavefunctions and energy level plot. Firstly, Eq. (4.23) and Figure 4.3a demonstrate

that even in the vibrational ground state with n = 0, the system is not at zero energy, but rather, at an energy (4.32) which is referred to as the zero‐point energy. This zero‐point energy accounts for the fact that even in its vibrational ground state, the atoms in a molecule undergo continuous vibrational motion. This zero‐point vibrational energy also accounts for the 3rd law of thermodynamics, which states that absolute zero temperature is unattainable. This is because atomic “standstill” is impossible due to the residual vibrational energy. This is also in line with Heisenberg's uncertainty principle (2.1), since a vibrational amplitude of zero would define both position and momentum simultaneously. Secondly, Figure 4.2 also indicates some degree of “tunneling,” or a finite probability of the oscillating system to be found outside the potential energy curve. Thirdly, the wavefunctions are symmetric (n = 0, 2, 4,…) or antisymmetric (n = 1, 3, 5,…) with respect to the y‐ axis in Figure 4.2a. In other words, the wavefunctions with even quantum numbers have even parity, that is, f(x) = f(−x), while the wavefunctions with odd quantum numbers have odd parity with f(x) = −f(−x). This aspect will become particularly important in the discussion of the allowed and forbidden transitions in the harmonic oscillator approximation (see below). Fourthly, the spacing between the energy levels is given by the classical expression (4.7)

Figure 4.3 Schematic of allowed (solid arrows) and forbidden (dashed arrows) absorption and emission transitions for the harmonic oscillator. and agrees with what is expected from classical mechanics. Finally, the quadratic potential depicted in Figure 4.3 does not explain bond breakage at sufficiently high energy, since the potential function – the restoring force between the oscillating atoms – increases steadily in the “harmonic approximation.” Therefore, the concept of an anharmonic potential needs to be introduced (see Section 4.4).

4.3 The Transition Moment and Selection Rules for Absorption for the Harmonic Oscillator In Chapter 3, we demonstrated that three conditions are necessary for an absorption transition to occur in an atomic or a molecular system under the influence of a perturbation by electromagnetic radiation. First, radiation must impinge on the molecular system (Eo ≠ 0); second, the radiation must possess the proper energy, or frequency, corresponding to the energy difference between the molecular or atomic states. Third, the dipole transition moment must be nonzero: (4.20) Here, ψn and ψm would be the vibrational wavefunctions defined by Eq. (4.23). Rather than evaluating the integral in Eq. (4.20) for different values of n and m by analytical integration, the recursion formula given in Eq. (4.25) allows a rather painless way to assess whether or not the transition moment is zero for any values of n and m. As established earlier, the vibrational wavefunctions are of the form (4.24)

The Gaussian part of the wavefunction

has even parity; thus, it

does not affect the parity of the integral described by Eq. (4.20), and the following discussion can concentrate on the parity of the Hermite polynomials alone. Thus, the transition moment given by Eq. (4.20) can be simplified to (4.33)

where the factor was set to 1 since it is a constant. Recalling the recursion formula for the Hermite polynomials, (4.26) the term x Hm(x) in Eq. (4.33) can be substituted by the right‐hand side of Eq. (4.26) to yield (4.34)

(4.35) Since the Hermite polynomials are orthogonal, the two integrals in Eq. (4.35) are nonzero if and only if (4.36)

Equation (4.36) implies that transitions are allowed only if the vibrational quantum number n changes by one unit; that is, the selection rule for absorption (and emission) for the harmonic oscillator is (4.37) Equation (4.36) also infers that (4.38)

This selection rule implies that electric dipole transitions are allowed only between adjacent energy levels for the harmonic oscillator. This is shown schematically in Figure 4.3 where solid arrows indicate allowed transitions and dashed arrows indicate forbidden transitions. Incidentally, the transition energy between adjacent energy levels is (4.39) since adjacent energy levels are equidistant and differ by the energy obtained by the classical vibrational frequency. Thus, the classical vibrational frequency of a harmonic oscillator given by Eq. (4.7) is identically the same as the one predicted by the quantum mechanical model. As in the case of the transition moment for the particle in a box, the transition moment (and the orthogonality of the harmonic oscillator wavefunctions) can be demonstrated graphically as well. Figure 4.4a demonstrates by graphical integration that the integral is indeed zero, whereas Panel (b) shows the same result for Thus, it can be seen easily that the vibrational wavefunctions are orthogonal.

Figure 4.4 Graphical representation of the orthogonality of vibrational wavefunctions and the vibrational transition moment. (a) Product of ψ0·ψ1. The light and dark gray regions under the curves have equal areas; thus, integration along x results in zero net area, and the functions are orthogonal. (b) Product of ψ1·ψ2. The same argument demonstrates that the functions are orthogonal. (c) Plot of ψ0 (gray), ψ1 (light gray), and the dipole operator μ = e x (black). (d) Integration of 〈ψ1 ∣ μ ∣ ψ0〉 along x‐axis yields a nonzero transition moment. Panels (c) and (d) show that the transition moment integral 〈ψ1∣μ∣ψ0〉 is nonzero. Panel (C) shows the two wavefunctions and the

transition operator and Panel (D) the product of the three functions. The area under the curve is nonzero.

4.4 The Anharmonic Oscillator The potential energy used up to this point was the harmonic potential, which is a crude approximation to the real potential function in a diatomic molecule. At elevated temperatures, all molecules tend to fall apart into atoms; thus, when molecules are in highly excited states, bond breakage will occur. The harmonic potential used so far does not predict bond breakage ever to happen, since the quadratic potential function increases steadily with increasing values of x, whereas a real potential function must predict that at a sufficiently high value of energy, bond breakage will occur. The real potential function can be obtained by detailed quantum mechanical calculations, in which the electronic energy is computed as a function of the internuclear distance, and is shown schematically in Figure 4.5. This potential energy can be approximated by the Morse potential, given by (4.40) with

Figure 4.5 Potential energy function of a real diatomic molecule with dissociation energy De. The function has a minimum at the bond equilibrium distance xo. When compressing the bond beyond xo, the potential energy rises sharply due to 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 xo 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 4.5 would be difficult. Thus, one approximates the shape of the potential function V(x) in the vicinity of the potential energy minimum by a power series expansion about the equilibrium distance: (4.41)

V(xo) 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 since the equilibrium geometry corresponds to an energy minimum. The quadratic expression in Eq. (4.41) (4.42)

is the harmonic potential energy function used so far. The cubic term gives the next level of approximation, and an anharmonic force constant is defined as follows: (4.43)

The anharmonic vibrational Schrödinger equation thus is written as (4.44) and is solved by perturbation methods (see Appendix 2.3). The perturbed energy eigenvalues for an anharmonic diatomic molecule

are given by (4.45)

with the anharmonicity constant given by (4.46) χ is always a positive number; thus, the energy levels of the anharmonic case are always lowered as compared with the harmonic oscillator; this lowering increases with the square of the quantum number n. This is depicted in Figure 4.6, which shows a comparison between harmonic and anharmonic oscillator energy levels.

Figure 4.6 Comparison of energy levels for harmonic and anharmonic oscillators. One further effect of the anharmonic potential is the fact that the observed fundamental transition from n = 0 to n = 1 is no longer the harmonic frequency. Within the harmonic approximation, the energy of the n = 0 to n = 1 transition was given by (4.47) Taken into account the anharmonicity, the energy difference between the ground and first excited state is given by (4.48) In addition to lowering the energy values, the wavefunctions – although similar in shape to the harmonic oscillator wavefunctions –

will be shifted toward longer internuclear distances; consequently, the symmetry of the wavefunctions changes; thus, they are no longer symmetric or antisymmetric with respect to the xo position. The direct consequences of the asymmetry of the wavefunction are that the selection rules change somewhat. Whereas in the harmonic oscillator approach transitions with (4.49) are strictly forbidden, these transitions are weakly allowed in the case of an anharmonic oscillator. That implies that a weak absorption band is observed at (4.50)

i.e. just under twice the frequency of the fundamental transition. The n = 0 to n = 2 transition is also referred to as the overtone (or the 1st harmonic) of the fundamental. In addition, the n = 0 to n = 1 transition and the n = 1 to n = 2 transition will no longer have the same energy, since the spacing between adjacent energy levels is no longer constant: (4.51)

Thus, the n = 1 to n = 2 transition will appear on the low frequency side of the n = 0 to n = 1 transition with a much lower intensity, which is determined primarily by the low population of the n = 1 state at room temperature. At elevated temperatures, the n = 1 state becomes more populated, and therefore, the n = 1 to n = 2 transition will become more intense. This transition is therefore referred to as a “hot band.”

4.5 Vibrational Spectroscopy of Diatomic Molecules The discussion above will now be illustrated using the vibrational spectra of selected molecules, namely, bromine, Br2, and iodine bromide, I‐Br. The choice of good examples is actually quite hard, since many diatomic molecules do not exhibit an infrared absorption spectrum due to the lack of a permanent dipole moment, which is a prerequisite for diatomic molecules to have an infrared absorption spectrum. Since all homonuclear diatomic molecules such as H2, N2, F2, Cl2, O2, etc. are nonpolar, the vibrational transitions are not allowed in absorption. However, they are allowed in an alternate method of vibrational spectroscopy, Raman scattering, which will be discussed in detail in the next chapter. The vibrational energy levels are the same in both forms of vibrational spectroscopy, but the Raman process has different selection rules. Thus, homonuclear diatomic molecules do exhibit a Raman vibrational spectrum. Therefore, one of the examples presented below will use Raman spectra to illustrate the energy levels and transitions for a homonuclear diatomic species, Br2. These data are taken from the literature [2]. Due to the high mass of the bromine atoms, the molecule has low vibrational transition frequencies, and the fundamental stretching frequency is observed at c. 315 cm− 1 . Since the frequency is so low, the higher harmonics can readily be observed and appear in the spectrum as a series of nearly equidistant peaks (see Figure 4.7a) marked by the change in quantum number Δn = 1, 2, 3,…. The spectrum shown in Figure 4.7 further demonstrates a number of important features of vibrational spectroscopy. The inset (b) shows an expanded region of the fundamental transition that is split into several peaks. These are, in part, due to the fact that Br2 is a mixture of isotopic species: 79 Br2, 79 Br‐, 81 Br, and 81 Br2 with an abundance ratio of 1:2:1. Thus, the fundamental transitions exhibit these peaks at 322, 320, and 318 cm− 1 (features f, e, and d in the insert of Figure 4.8b). The effects of isotopic species will be discussed further

in Example 4.3. Furthermore, the spectrum contains the “hot bands” of the three isotopic species, since the population of excited states is quite high for molecular species with low‐lying energy states. These hot bands (features a, b, and c) are superimposed for different isotopic species and for different levels of “n” from which they originate. The energy difference between the fundamental and the hot bands is only a few wavenumbers, since anharmonicity effects are small for strongly bonded covalent molecules. This will be demonstrated in Example 4.3.

Figure 4.7 (a) Raman spectrum of Br2. (b) Expanded region of the fundamental with overtones and hot bands (adapted from [2]). See text for details. Example 4.3 Heteronuclear diatomic molecules such as HF, HCl, ClF, ClBr, CO, NO, CN−, etc. do exhibit infrared absorption spectra. Many of these are gaseous at standard conditions and are

complicated by the interaction between vibrational and rotational transitions, which, in turn, complicates the observed spectra. This aspect will be discussed in Section 6.5. Thus, we shall discuss the infrared spectrum of I‐Br, which is a solid with a melting point of 27 °C, has low‐lying vibrational energy levels, and has a similarly strong bond as Br2 discussed before. The vibrational constants for I‐Br are given in literature [3] in wavenumber [cm−1] units: . Here, is the

harmonic transition energy in wavenumber units for the 127I‐79Br isotopic species. Since the mass of iodine is even larger than that of Br, the vibrational frequency is significantly lower than for Br2. First, we shall compute the frequency of the observed fundamental. For this, we reformat Eq. (4.47) by dividing by the velocity of light to get the transition energy in wavenumber units: (4.47)

(E4.3.1)

(E4.3.2)

This result shows that the observed transition energy is close to the harmonic value, indicating that the molecule vibrates in a steep and deep energy well for which the anharmonic perturbation is small.

This is in agreement with the reported bond dissociation energy value of 177.8 [kJ/mol] for the homolytic bond breakage of I‐Br. This energy, when expressed in wavenumber per molecule, comes out to be 14 656 cm−1 (see Appendix 1 for energy conversion factors). Thus, the energy well is many vibrational quanta deep, and the fundamental transition occurs far away from the anharmonic perturbation. The partially allowed 2←0 overtone will be observed at (E4.3.3)

and the 3←0 overtone at (E4.3.4)

These frequencies correspond to the overtone spectrum shown in Figure 4.8a. The 2←1 hot band frequency is obtained by a similar approach: (E4.3.5)

In Eq. (E4.3.2), the observed fundamental frequency was calculated to be 267 cm−1; therefore, the hot band will appear as a low wavenumber shoulder at 265.3 cm−1. Finally, we shall investigate the isotopic splitting for this molecule. To this end, we assume that the force constant is the same for all

isotopic species. This is a fair assumption, since the isotopes of atoms differ only by additional neutrons in the nucleus. There are two dominant isotopic species, 127I‐79Br and 127I‐81Br with molar masses of 206 and 208, respectively. Using (4.41)

we form the ratio of the vibrational harmonic frequencies of the two isotopic species: (E4.3.6)

where mR 206 and mR 208 are the reduced masses of the two species. These are given by

Thus, the isotopic shift observed for the two I‐Br species, 2.3 [cm−1], is of the same magnitude as the anharmonicity correction. This fact was seen before in the example of Br2, where the hot bands and the isotopic species appeared under one overlapping band envelope.

4.6 Summary

The discussion in Chapter 4 bridged from a very simplistic example, the particle in a box, to a model where the potential function closely resembles a real potential function, namely, the anharmonic diatomic oscillator. The mathematics becomes rather involved when using realistic potential function, but the results are in excellent agreement with experimental data. Thus, one can say that the approach taken and alluded to in the introduction (“It doesn't matter how beautiful a theory is…. If it doesn't agree with experiment, it's wrong”) about the interplay between theory and experiment is in line with the philosophy of science, in general: one starts with a theory as simplistic as possible and continues a refinement process of the theory until agreement between experiment and theory is achieved. In the case of the harmonic oscillator, many aspects can be derived even for this approximate model, for example, the odd/even parity of wavefunctions and its effect on allowed transitions and the transition frequency that approximately equals that of the classical model (Eqs. [4.6, 4.7]). The concept of bond dissociation, as well as the observation of overtones and hot bands, requires refinement of the model, and the introduction of the anharmonicity provides this refinement. The visualization of the meaning of the wavefunctions plotted in Figure 4.3 requires a little more thought. In the case of the particle in a box, the squared amplitude of the wavefunction simply implied the probability of finding the electron at a given value of x inside the box. In the case of the harmonic oscillator, we are dealing with two atoms vibrating about a fixed center of mass. Thus, the square of the amplitude of the wavefunction shown in Figure 4.2b indicates that in the ground state, the harmonic oscillator vibrates about x0 or, in other words, the most likely distance between the two atoms is x0. This is harder to visualize for the first excited state, where there are two most likely distances between the atoms and there is a node at the distance x0. In the case of the particle in a box, the number of nodal points was given by n−1, whereas it equals to n in the case of the harmonic oscillator. Further refinements, to be introduced in later chapters, will take into account that during rotational motion of a diatomic molecule, the bond between the atoms will stretch due to centrifugal forces. This

effect is observed in the pure rotational and rot‐vibrational spectra and demonstrates that further refinements of any model may be necessary the more sophisticated the experimental methods become. This does not invalidate prior models, but rather enhances them. Unfortunately, anti‐scientific opinions often do not understand this interplay between more detailed experimental methods and enhanced scientific descriptions.

References 1 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon. 2 Baierl, P. and Kiefer, W. (1975). Hot band and isotopic structure in the resonance Raman spectra of bromine vapor. The Journal of Chemical Physics 62: 306–308. 3 NIST. Chemistry WebBook, SRD 69. National Institute of Standards and Technology https://webbook.nist.gov/cgi/cbook.cgi? ID=C7789335&Mask=1000.

Problems 1. From the recursion formula for the Hermite polynomials, determine H5(z) and H6(z). 2. Plot the functions H5(z) and H6(z) between z = −3 and z = +3 (Excel will work fine). 3. Plot unnormalized harmonic oscillator wavefunctions ψ5(z) and ψ6(z) between z = −5 and z = +5. 4. Plot unnormalized harmonic oscillator wavefunctions ψ52(z) and ψ62(z) between z = −5 and z = +5. 5. What features (parity, nodal points, intensity distributions, etc.) can you observe?

6. Show graphically (analog to Figure 4.4) that ψ2(z) and ψ3(z) are orthogonal and that the transition from n = 2 to n = 3 is electric dipole allowed in the harmonic oscillator approximation. You can use unnormalized wavefunctions. 7. In analogy to Problem (6) in Chapter 2, solve for the expectation value of the total energy operator for the harmonic oscillator for n = 0. The following problems deal with the vibration of the carbon monoxide molecule. Assume that the vibration of the carbon– oxygen triple bond in CO follows the anharmonic oscillator formalism, unless stated otherwise. 8. Calculate the force constant k [in N/m] for CO from the observed stretching frequency, which is 2169 cm−1. 9. What is the stretching frequency [in cm−1] for the isotopic species 13C18O, assuming that the potential energy does not depend on the mass. 10. What is the energy [in cm−1] of the n = 0 vibrational state of CO (i.e. what is the vibrational zero‐point energy)? 11. Calculate the anharmonicity constant χ from the bond dissociation energy (1077 kJ/mol). 12. Given that the observed frequency (2169 cm−1) is the anharmonic transition and using the anharmonicity constant from Problem (11), calculate the transition frequencies (in cm−1) for a. n = 1 ← n = 0 harmonic transition b. n = 2 ← n = 1 hot band c. weakly allowed n = 2 ← n = 0 overtone

5 Vibrational Infrared and Raman Spectroscopy of Polyatomic Molecules Vibrational spectroscopy as an analytical method was established in the late 1940s mostly by researchers in polymer chemistry when it was discovered that vibrational (then mostly infrared [IR]) spectroscopy provides important information on polymer chain crosslinking. Later, the oil industry further developed the methodology for the analysis of oil products, in particnular the dependence of the spectra of hydrocarbons on chain length and saturation. These efforts produced the first software, then confined to mainframe computers, to perform what was referred to “normal coordinate analysis” in which the normal modes of vibration were calculated. This approach uses a strictly classical method in which the force field, that is, a matrix of all the forces acting between the atoms in a molecule, was calculated by empirical fitting. In addition, these calculations revealed the atomic motions during the normal modes of vibration. These empirical calculations have now been superseded by quantum mechanical computations in which the force field is calculated as the partial derivatives of the total energy E of a molecule with respect to the Cartesian displacement coordinates (the Hessian matrix [1]). The total energy is obtained by detailed molecular orbital calculations using extended basis sets or density functional theory (DFT). This example again shows the interplay between spectroscopy and quantum mechanics: the observed vibrational frequencies can be used for the refinement of the computational methods of molecular energy. Although these calculations are still computationally involved, they have produced excellent agreement between observed and computed vibrational frequencies and intensities.

5.1 Vibrational Energy of Polyatomic Molecules: Normal Coordinates and Normal Modes of Vibration As in the case of the harmonic oscillator, the treatment of vibrational spectroscopy of polyatomic molecules starts with a classical description of the vibrational energy. However, this description requires the derivation of the “normal modes of vibration” of a molecule. This derivation was well established in classical mechanics for masses connected by springs with known force constants but is quite involved for a set of atomic masses where the forces acting between atoms are unknown. Thus, the derivation of the classical normal modes of vibration of a molecule requires many steps and will be presented here in a highly abbreviated form. For step‐by‐step derivations, the reader is referred to more specialized resources on vibrational spectroscopy [2]. The classical description is based on the assumption that a molecule consisting of N atoms can be described by N point masses connected by springs. In the original efforts in vibrational analysis, the force constants of these springs were transferred between different molecules, or fitted to experimental data, first by manual computation, later by a set of early programs developed for normal coordinate analysis. As a starting point for the computation of the vibrational energies of a polyatomic molecule, Cartesian displacement coordinates xi are attached to all atoms, since the motions of atoms during a normal mode can be decomposed into these Cartesian displacement coordinates. Furthermore, these coordinates are “mass‐weighted” according to (5.1) In mass‐weighted Cartesian displacement coordinates, the amplitudes of vibration of each atom are properly accounted for by

weighting them by the atomic masses. In terms of these mass‐ weighted coordinates, the kinetic energy is written as (5.2)

where denotes the time derivative of the coordinate. Notice that Eq. (5.2) is just the classical definition of kinetic energy (Eq. [2.3]), summed over all particles. The potential energy for motions along each Cartesian displacement coordinate is given by (5.3)

where the terms fij are the Cartesian components of the force constants. Again, this expression corresponds to the definition of the harmonic potential energy introduced by Eq. (4.2) but decomposed along the 3N Cartesian displacement directions. The sums in Eqs. (5.2) and (5.3) are overall 3N Cartesian displacement components. Note that Eqs. (5.2) and (5.3) correspond exactly to Eq. (4.4) for a one‐dimensional case: in diatomic molecules, there is just one degree of vibrational freedom – the stretching of the bond connecting the two atoms. In a polyatomic molecule with N atoms, there will be 3N degrees of freedom. This number of degrees of vibrational freedom is based on the concept that each of the N atoms can move in three independent directions – the x, y, and z directions. Furthermore, one assumes that every bond stretching or angle deformation in a polyatomic molecule obeys a harmonic oscillator formalism or, in other words, all atoms may be assumed to be connected by springs which obey Hook's law. When the expressions for the kinetic and potential energy given by Eqs. (5.2) and (5.3) are substituted into Lagrange's equation of motion

(5.4) one obtains a set of 3N differential equations (5.5)

Here, denotes the second derivative of q with respect to time. Equation (5.5) is a short form for a set of 3N simultaneous differential equations, with the index i is running from 1 to 3N. Following steps described in the literature [2], one can show that there are 3N solutions to these simultaneous, linear differential equations which are given by (5.6) where the Ai are amplitude factors, and λi are quantities related to the vibrational frequency and determined by the force constants. The amplitude factors Ai give relative magnitude of the displacement vectors that provide a view of the relative amplitudes of all atomic motions. Notice that Eq. (5.6) is the equivalent of Eq. (4.5) in the case of a diatomic molecule. Furthermore, six of these equations will have zero eigenvalues, as will be discussed next. Molecular vibrations depend on a restoring force to bring the atoms of a molecule back to their equilibrium position, as discussed before, for diatomic molecules. Thus, if 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. Therefore, the total number of degrees of freedom is reduced by three degrees. Similarly, a rotation of a molecule without any changes in bond length or bond angles would not change the potential energy. Thus, one needs to eliminate three more degrees of freedom, and one arrives at 3N–6 vibrational degrees of freedom for a nonlinear polyatomic molecule.

Next, the solutions described in Eq. (5.6) may be recast in a new system of 3N–6 coordinates, the so‐called normal coordinates Q. Normal coordinates are linear combinations of the mass‐weighted Cartesian displacement coordinates, as shown in Figure 5.1, given in matrix notation as (5.7)

Figure 5.1 Depiction of the atomic displacement vectors qi for the three normal modes Q1 (antisymmetric stretching mode), Q2 (symmetric stretching) and Q3 (symmetric deformation mode) of the water molecule. The magnitude of the displacement vectors is not known, but the relative displacements are drawn approximately to scale. There are 3N–6 normal coordinates that are corresponding to the 3N–6 degrees of vibrational freedom and to the 3N–6 normal modes of vibration that are associated with the quantum mechanical observables for the system. In a normal mode of vibration, all atoms oscillate with the same frequency and in‐phase but with different amplitudes. This definition 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. The normal modes for a simple molecule, such as water, are shown in Figure 5.1. The vibrational frequencies and depictions of the normal modes are obtained in the classical calculations by transforming the vibrational problem into normal coordinate space. For that, one recasts Eqs. (5.2) and (5.3) into matrix notation:

(5.8)

and (5.9)

Here, the superscript T denotes the transpose of a matrix; thus, the column vector becomes a row vector upon transposition. The dot implies, as before, the time derivative of the coordinates. F denotes the matrix of mass‐weighted Cartesian force constants. Subsequently, one applies the transformation matrix between mass‐ weighted Cartesian displacement coordinates and normal coordinates (Eq. [5.7]) and obtains

and (5.10) Here, the diagonal matrix Λ = LT F L is obtained by numerically diagonalizing the force constant matrix F (in mass‐weighted Cartesian displacement coordinates). The (eigenvector) matrix L that diagonalizes the potential energy matrix, F, is the matrix that transforms from the mass‐weighted Cartesian displacement space into normal coordinate space according to (5.7) Λ is the diagonal matrix of the values λk that are the vibrational frequencies for each normal mode:

(5.11) A more detailed description of the process of classical normal coordinate analysis can found in the literature [3]. These computations are based strictly on a classical model but use the concept discussed in Chapter 4 that the natural frequencies of vibration are the same as the quantum mechanically allowed transition frequencies (see discussion with Eq. [4.39]).

5.2 Quantum Mechanical Description of Molecular Vibrations in Polyatomic Molecules The vibrational Schrödinger equation for a polyatomic molecule is written in terms of the normal coordinates, since in this coordinate system, the vibrational frequencies are the quantum mechanical observables. As in Chapter 4, where the vibrational Schrödinger equation in the harmonic oscillator approximation was written as (4.9) one writes the vibrational Schrödinger equation for a polyatomic molecule (in the harmonic approximation) in terms of the normal coordinates as (5.12) where ψvib is the total vibrational wavefunction of the molecule which is a product of the wavefunctions along each of the 3N–6 normal coordinates: (5.13)

This definition of the total vibrational wavefunction as products of wavefunctions associated with one and only one normal coordinate succeeds since the expressions for kinetic and potential energy are both diagonal in normal coordinate space (cf. Eq. [5.10]). Substitution of Eq. (5.13) into Eq. (5.12) yields the Schrödinger equation in terms of the 3N–6 normal coordinates: (5.14)

Since the normal coordinates Qk are orthogonal functions, Eq. (5.14) can be separated into 3N–6 individual differential equations of the form (5.15)

Equation (5.15) is the 3N–6‐ dimensional equivalent of Eq. (4.8) of the harmonic oscillator discussion, and assumes a harmonic potential for each of the 3N–6 normal coordinates. The solutions of Eq. (5.15) for each normal coordinate are the eigenfunctions (15.17) in complete analogy with the harmonic oscillator situation. The total vibrational wavefunction of an N‐atomic molecule is then given by Eq. (5.13). A similar situation was encountered before in the two‐ dimensional particle in a box, where the wavefunctions were the product of the two one‐dimensional wavefunctions, and the total energy simply the sum of the energies given by the respective one‐ dimensional energy expressions. The energy eigenvalues (in wavenumber units) for each normal mode (within the harmonic approximation) is given by

(5.18) and the total vibrational energy is (5.19)

The results of this quantum mechanical description are graphically shown in Figure 5.2 for the water molecule. Each of the three normal modes of vibration has its own energy ladder, and the vibrational transitions occur (within the harmonic approximation) within each ladder, that is, the selection rules Δnk = ±1 apply within each energy ladder. The vibrational energy spacing for the water molecule (expressed in wavenumber units; see Eq. [1.11]) associated with each normal coordinate are approximately

Note that each ladder starts at the zero‐point energy (5.20) that is, at approximately 1875, 1825, and 810 cm−1 for Q1, Q2, and Q3, respectively. Thus, the total zero‐point vibrational energy of water is given by (5.21) or approximately 4510 cm−1. In any of the energy ladders, a transition to the more highly excited vibrational state may occur if a photon of proper wavenumber (3750, 1650, or 1620 cm−1) interacts with the sample, and certain requirements are fulfilled (see also Figure 5.3). Whether or not a transition occurs depends not only on

the presence of photons with the correct energy but also on some symmetry considerations discussed in Chapter 11.

Figure 5.2 Energy ladder diagram for the water molecule within the harmonic oscillator approximation.

Figure 5.3 (a) Observed infrared absorption spectrum of water. (b) Schematic of the three fundamentals observed in panel (a). (c) Energy level diagram of the observed transitions. Linear molecules exhibit 3N–5, rather than 3N–6, degrees of vibrational modes. The additional degree of vibrational freedom results from the fact that linear molecules are assumed to have one less rotational degree of freedom, since rotation along the direction connecting all atoms has a zero moment of inertia (assuming the atoms can be described as point masses). Thus, only three translational and two rotational degrees of freedom need to be subtracted to arrive at the number of vibrational degrees of freedom. At this point, a comment about the actual atomic motions in a molecule is appropriate. These motions are random and increase with temperature. The familiar “thermal ellipsoids” observed in X‐ ray crystallography of molecular crystals are manifestations of this random motion. Keeping the temperature low during X‐ray diffraction data acquisition reduces the size of the thermal ellipsoids. The random atomic motion, however, can be decomposed into contributions from the normal modes of vibration. When a vibrational transition into one of these normal modes occurs, the random motion along this coordinate increases in amplitude. The rate of chemical reactions that depend on an initial breakage of a bond can be enhanced by increasing the vibrational amplitude of the

corresponding bond, either by a non‐selective method – increasing the reaction temperature – or by illuminating the molecule with light that is absorbed and increases the amplitude of a highly anharmonic excited state. This method worked well for the photo‐decomposition of SF6, where illumination with an IR laser into one of the antisymmetric S–F stretching vibration actually accelerated the decomposition [4]. The goal of this work was the development of an optical method for the separation of uranium isotopes by illuminating a mixture of UF6 isotopic species with narrow‐band laser radiation such that only one of the species' decomposition was accelerated by the IR light. While this method worked for the separation of 34SF6 from 32SF6, it did not work for uranium hexafluoride, presumably due to the rapid deactivation of the excited vibrational states, or the overall density of available states in a complicated reaction mixture which makes selective excitation of one highly excited state very difficult.

5.3 Infrared Absorption Spectroscopy In IR spectroscopy, absorptions of IR photons by the sample are observed, which cause excitation along one normal coordinate of vibration into a higher vibrational energy level. Such a process can be described by Eq. (5.22) which states that the quantum mechanical transition moment determines the molar extinction coefficient as follows (5.22) which, in turn, determines the likelihood that a photon of wavenumber is absorbed by the sample. A plot of the molar extinction coefficient against the wavenumber of radiation gives the desired IR spectrum.

5.3.1 Symmetry Considerations for Dipole‐Allowed Transitions Similar to the case of the diatomic harmonic oscillator, the transition moment given in Eq. (5.22) must be determined for each of the 3N–6 normal coordinates for a polyatomic molecule. In the case of the harmonic diatomic molecule, the necessary condition for a vibrational transition to occur was that the molecule must possess a permanent dipole moment. In a polyatomic molecule, this rule is modified to read that the dipole moment along a normal coordinate must change for a vibration to be allowed to occur in IR absorption spectroscopy: (5.23) This will be discussed in more detail in Chapter 11 on molecular symmetry but can be viewed as a special condition arising from parity considerations. At this point, a simple qualitative discussion of the symmetry requirements will be presented. Transitions along the three vibrational coordinates shown for water in Figure 5.1 all are allowed since the water molecule possesses a dipole moment, and this dipole moment changes for each of the three vibrational coordinates. Therefore, the water molecule exhibits an IR absorption spectrum shown schematically in Figure 5.3. The intensity of each transition shown reflects the magnitude of the dipole change induced by the corresponding normal mode. A symmetric linear molecule such as liquid carbon disulfide presents quite different a situation. Its 3N–5, or four normal modes, are shown in Figure 5.4. Since the molecule is linear, the two bond dipole moments of each carbon–sulfur double bond cancel, and the molecular is non‐polar. The vibrational mode designated as Q2, the symmetric stretching mode, does not change the dipole moment of the molecule, and therefore is forbidden in IR absorption. The mode designated as Q1, the antisymmetric stretching mode, on the other hand, creates a dipole moment since the two C=S band lengths and bond dipole moments are different in the vibrationally excited state.

Thus, this transition is allowed. Similarly, the two degenerate bending modes create a structure that has a dipole moment in the excited state.

Figure 5.4 Depiction of the atomic displacement vectors qi for the four normal modes Q1 (a, antisymmetric stretching mode), Q2 (b, symmetric stretching) and Q3 and Q4 (c, degenerate deformation modes) of the CS2 molecule. The magnitude of the displacement vectors is unknown, but the relative displacements are drawn approximately to scale. As in the case of diatomic molecules, a further refinement of the model presented so far would be the inclusion of anharmonicity into the vibrational force field. Anharmonicity relaxes the selection rules, as seen before, and weakly allows overtones and combinations and difference bands. Combination bands are those in which the quantum number in two different normal modes change simultaneously; that is, the normal modes of vibration of these modes interact. There are certain symmetry requirements for this to happen: the two vibrational transitions must belong to the same symmetry species (see Chapter 11). Thus, in the water molecule, a combination band Q2 + Q3 would be allowed, and is observed at about 5200 cm−1. Incidentally, the slight blue color of water, observed at long pathlength of visible light through water, is due to a high overtone 3Q1 + Q2, which occurs at about 14 300 cm−1 or about 700 nm, in the red. Thus, blue light is absorbed less, and water appears blue at long pathlength. More examples of vibrational IR spectra will be presented after the discussion of Raman spectra of polyatomic molecules.

5.3.2 Line Shapes for Absorption and Anomalous Dispersion

5.3.2.1 Line Shapes and Lifetimes The width and shapes of the observed IR bands will be discussed next. A spectral “band” is a plot of the variation of molar extinction coefficient, or the absorbance of the sample, in the neighborhood of the band center, as shown in Figure 5.3a. However, the observed water vibrational spectrum shown in Figure 5.3 has exceptionally broad IR bands, measured as the full width at half maximum (FWHM). For the deformation mode Q3 the FWHM is nearly 200 cm−1. This halfwidth is unusually large for a small molecule because water exists as relatively poorly defined hydrogen‐bonded clusters. Monomeric water, for example, as a diluted solution in non‐polar organic solvents, exhibits much narrower peaks, with an FWHM of about 20 cm−1, and characteristic band shapes shown schematically in Figure 5.5. The width and shape of observed bands in spectroscopy is quite intriguing and will be discussed next.

Figure 5.5 Gaussian (a) and Lorentzian (b) line profiles. Notice that the areas under the line profiles are unequal; however, both bands have a full width “a” at half maximum (FWHM) of 5 and an intensity I = 1.0. As we have seen in the discussion of the perturbation treatment of stationary states by electromagnetic radiation (ending in Eq. [3.14]), an absorption transition occurs if the radiation incident on the sample has a frequency of

(3.16) However, if this condition is fulfilled exactly, the denominator in the expression (5.24) (Eq. [3.14]) would become zero, and the expression would be undefined. In reality, the denominator does not become exactly zero since it contains a damping term that depends on the lifetime of the excited state. When the damping term is included, the denominator of the expression in Eq. (5.24) is usually written as (5.25) This expression can be derived from classical oscillator theory [5, 6], with γ the damping term, related to the lifetime of the excited state. When a photon is absorbed to create an excited vibrational state, the molecules remain in this excited state for a certain time, referred to the lifetime of this state. In IR vibrational spectroscopy, this lifetime is about 10 ps = 10−11 [s]. Given that a vibration with a wavenumber of 1000 cm−1 has a frequency (cf. Eq. [1.11]) of ν ≈ 3 × 1013 [Hz], or a period of about 3 × 10−14 [s], the molecule remains in the excited state for hundreds of complete vibrational cycles, before it deactivates, either by spontaneous emission, by collisional deactivation, or by internal energy conversion into other vibrational energy modes. The finite lifetime of the excited state determines the width of an observed absorption peak as follows. The uncertainty principle introduced in Chapter 2 (Eq. [2.1]) can also be written in terms of energy and time as follows: (5.26) Thus, if the uncertainty in the lifetime by spontaneous emission is about 10 ps, the corresponding uncertainty in the energy of the state

would be about 5 × 10−23 [J], or about 2.5 cm−1. This uncertainty in the energy of the state leads to an inherent minimal linewidth of a few wavenumbers, also referred to as “natural broadening” of a transition, and the deactivation process is also referred to as the longitudinal relaxation. Collisional deactivation and internal energy conversion have similar inherent lifetimes and follow damping equations of similar form to Eq. (5.25). The shape of an absorption band with the denominator given by Eq. (5.25) is referred to as a Lorentzian band that can be described by the general equation (5.27) In Eq. (5.27), x0 is the peak position, “I” the intensity at the band maximum, and “a” is the FWHM. A Lorentzian band shape is shown in Figure 5.5b. Thus, the inherent band shape of a transition will be a Lorentzian band with an FWHM of a few wavenumbers. Thermal motion of the molecules, particularly in the gas phase, contributes to another mechanism of band broadening via the Doppler effect. This effect, which influences the transition frequency of a molecule, depends on whether it moves toward or away from the detector of the radiation. This will cause a band broadening that has a Gaussian profile since the thermal motion itself follows a Gaussian distribution. The broadening of the band shape by a Gaussian mechanism produces a line shape given by (5.28) shown in Figure 5.5a. The symbols used in Eq. (5.28) are the same as in Eq. (5.27). Thus, there are several distinct mechanisms that give raise to the broadening of spectral bands. Depending on which mechanism dominates, Gaussian, Lorentzian or mixed bands shapes will be observed in the IR spectra. These mixed band shape functions often are described by another function that is a convolution of Gaussian

and Lorentzian band shapes in Fourier space, known as the Voigt function. Due to the complexity of computing this band shape, it is often approximated by a “pseudo‐Voight” function that is just a mixture of Gaussian and Lorentzian band shapes. In addition, to these “physical” causes of line broadening, there are several more “chemical” mechanisms, such as the aforementioned hydrogen bonding, solvation in general, the presence of molecular interactions, and so forth. Asymmetries in the observed band shapes at the low wavenumber side can be due to the presence of low‐ intensity hot bands (see Section 4.4 4.4). Many other causes of line broadening and distortion can be found in the literature on vibrational spectroscopy [7]. 5.3.2.2 Anomalous Dispersion In the last section, the band shapes created during an absorption process – that is, a change in the extinction coefficient over a wavenumber interval – was discussed. The extinction coefficient is only one part of two quantities that change within an absorption band. The other quantity that changes simultaneously is the refractive index. Both of them are combined in the complex index of refraction η, defined as (3.22) in which n is the real refractive index familiar from classical optics, and κ is known as the absorption index which is related to the extinction coefficient ε by (3.23) Equation (3.22) demonstrates one of the fundamental aspects of spectroscopy: that the refraction and absorption of light are coupled processes. That absorption of light always is accompanied by changes in the refractive index of the medium. Interestingly, the connection expressed in Eq. (3.22) is often ignored in basic courses in geometric optics since optical materials from which lenses and prisms are

produced generally do not have any absorptions in the spectral region for which they are manufactured. Therefore, the refractive index often is treated as independent of wavelength, or at best, slowly varying with wavelength. However, at the wavelength at which any absorption process occurs, the refractive index undergoes what is known as “anomalous dispersion”, indicated by a deflection point in the otherwise smooth n(λ) curve. This is shown schematically in Figure 5.6. At the center of the absorption curve (or at the deflection point of the dispersive curve), the phase between the electromagnetic radiation and the response of the medium changes as well. These effects are well understood and discussed in books on classical optics, such as the monograph by Born and Wolf [5]. Notice that the refractive index is always a positive number, even within the regions of anomalous dispersion. The coupling of absorption and dispersion is expressed mathematically by the Kramers–Kronig transform, written here in terms of the angular frequency ω of the light, rather than its wavelength λ: (5.29)

(5.30)

Figure 5.6 Dispersion of the refractive index (top) within an absorption peak (bottom). In these equations, ω0 describes the frequency of light at which a transition occurs. Eqs. (5.29) and (5.30) indicate that knowing one of the quantities, for example n(ω), uniquely defines the other, ε(ω), and vice versa. The integrations in Eqs. (5.28) and (5.29) are over a singularity at ω = ω0, which requires that the principal value of this Cauchy integral is evaluated. The relationship between the refractive index and the extinction coefficient is also well‐known in other areas of spectroscopy: optical rotatory dispersion, which is based on the differential refractive index of a sample toward left and right circularly polarized light is related by the Kramers–Kronig transform

to circular dichroism, which is the differential absorption of a sample toward left and right circularly polarized light (see Section 10.7.2).

5.4 Raman Spectroscopy 5.4.1 General Aspects of Raman Spectroscopy As indicated in Sections 4.5 and 5.3, vibrational spectral information can be collected by direct absorption of IR photons or by light scattering mechanism known as the Raman effect. Light scattering at a microscopic (molecular) level can easily be visualized: A strong, focused visible light laser beam travelling through a clean, transparent liquid sample can be seen with the naked eye since the molecules in the sample scatter photons in the direction of the observer. This scattering occurs at the same wavelength as the incident light and involves a change in the direction, but not the momentum and energy of the scattered photon. Therefore, this process is often termed elastic scattering. The scattering cross‐ section (efficiency) of molecular scattering depends on the fourth power of the frequency of the incident light. Scattering is a relatively weak process, and only about 1 in 106 photons traveling through a medium will undergo elastic scattering that is also referred to as Rayleigh scattering. In addition to this molecular form of elastic scattering, inelastic scattering also may occur, but with even lower efficiency than elastic scattering. In this inelastic scattering, both the direction and energy of the photon change. This scattering process is known as Raman scattering, named after Sir Chandrashekhar Raman, who first observed this effect experimentally in 1928 after A. Smekal had predicted it in 1923.

5.4.2 Macroscopic Description of Polarizability Whereas IR spectroscopy is caused by direct absorption of a photon to promote the molecule into a vibrationally excited state and requires a change of the dipole moment along a vibrational coordinate, Raman spectroscopy requires the polarizability of a molecule to change along a vibrational coordinate. The polarizability

of a molecule is its response to the incident radiation far from an electronic transition. This can be visualized as follows. In a clear, colorless liquid, for example, there are no electronic transitions between ca. 400 and 750 nm, or ca. 25 000 and 13 000 cm−1 (hence the material is colorless). Visible electromagnetic radiation, however, still can interact with the molecule by setting in motion the electron clouds (particularly those in multiple bonds) by inducing a dipole moment, μind, mediated by the polarizability α: (5.31) Here, E denotes the strength E of the electromagnetic field. Since both the induced dipole moment and the electric field are vectors, the polarizability is actually a tensor. Thus, Eq. (5.31) can be written as a vector equation: (5.32) where both μ and E are vectors, and matrix:

is represented by a 3 × 3 (5.33)

This polarizability varies as the molecule oscillates along its normal coordinates, Qk, since the polarizability (that is, the ease with which electrons can be moved around) depends very much on the nuclear coordinates, and thereby on the vibrational modes of the molecule. Thus, one may expand the polarizability in a Taylor series about the equilibrium position according to (5.34) where ωk is the vibrational frequency of normal mode Qk.

The electric field exciting the Raman scattering is usually provided by laser radiation at ωL, one can represent this radiation by (5.35) Any oscillating dipole, whether induced or permanent, emits radiation of intensity I into all space according (5.36) Equation (5.36) shows that the scattered light intensity depends on the fourth power of the frequency, and on the square of the electric field strength and the polarizability. The polarizability itself is modulated by the molecular vibrational motion according to Eq. (5.33); thus, the induced dipole moment contains two frequency components, ωk and ωL. The radiation emitted by the induced dipole contains frequency components at the frequency of the exciting light ωL, and at the beat frequencies between the molecular frequency and the frequency of the incident radiation. The beat frequencies ωL + ωk and ωL − ωk arise from the product of the two cosine functions in Eqs. (5.33) and (5.34) and are given by a well‐known trigonometric identity (5.37) The beat frequencies are the anti‐Stokes and Stokes Raman frequencies, respectively. Thus, a simple, classical description of the off‐resonance interaction of light with a polarizable molecular system can explain some aspects of Raman scattering.

5.4.3 Quantum Mechanical Description of Polarizability A basic quantum mechanical description of the polarizability can be found in reference [8]. In short, this derivation proceeds as follows.

One writes the induced dipole moment, μind, as the expectation value of the operator, μ: (5.38)

where Ψ denotes the time‐dependent wavefunction defined in Eq. (3.10): (5.39) As before, the wavefunctions ψ(x) are the stationary state wavefunctions, φ(t) the time‐dependent wavefunctions, and ck(t) the time‐dependent expansion coefficients that describe how the system transitions from one state to another. These coefficients are given by Eq. (3.14). Substituting Eqs. (3.14) and (5.38) into Eq. (5.37), and assuming that the time dependence of the electric field can be written as (5.35) the induced dipole moment is given by (5.40)

Here, one assumes that the molecule is in the ground state |ψ0〉 before any interaction with the radiation occurs. The first term in Eq. (5.40) describes the permanent dipole moment, the second term a component of the induced dipole that oscillates at the same frequency as the incident light, and the third term an induced dipole

that oscillates at the transition frequency ω0m and is clearly not in phase with the frequency of the light inducing the dipole moment. The second term is called the polarizability, and a comparison between Eqs. (5.35) and (5.40) reveals that (5.41) Equation (5.40) implies that any medium exposed to electromagnetic radiation will undergo some, albeit small, change, even when the frequency of the radiation is far from any transition frequency, ω0m. This change can be viewed as a small, induced oscillatory motion of the electrons in a molecule; tightly bound inner electrons will respond to a lesser degree than more loosely bound electrons such as π‐electrons in a double bond. This interaction also determines the refractive index, n, and the relation between polarizability and refractive index is given by (5.42) Here, N is Avogadro's number. Thus, one finds that off‐resonance spectroscopic properties, such as the dielectric constant or the refractive index, are related to the transition moment via the polarizability. Thus, the proximity of an electronic transition will increase the refractive index. In the visible region of the spectrum, one can, therefore, predict that acetone has a higher refractive index than water, since its closest UV transition occurs around 280 nm, whereas the closest transition for water lies below 200 nm in the far UV region. Raman scattering can then be described as a process in which an incident photon, typically from a laser and with an angular frequency ωL interacts momentarily with the sample by creating a virtual state, indicated by the dashed line in Figure 5.7a. When it is scattered from this state, a photon of the same frequency is emitted (Rayleigh scattering), or a photon with angular frequency ωL – ωM is scattered

(the Stokes Raman photon) where ωM is the frequency of a molecular vibration. The Raman scattering leaves the molecule in the same excited state that is reached by direct absorption of a photon with angular frequency ωM. It is also possible that the original laser photon encounters the molecule in the vibrationally excited state. The virtual state now is higher in energy, and a photon with an angular frequency of ωL + ωM is scattered, a so‐called anti‐Stokes Raman photon. Anti‐Stokes Raman intensities are lower than those of the Stokes spectrum since the former depends on the population of the vibrationally excited state from which they originate. This population depends, via the Boltzmann equation (Eq. [3.31]) on the absolute temperature and the energy of the vibrational level. Thus, anti‐Stokes intensities are highest at high temperature, and for those vibrational energy levels with the lowest vibrational frequencies.

Figure 5.7 (a) Energy level diagram for a Stokes and anti‐Stokes Raman scattering process involving a virtual state (dashed line). (b) Energy level diagram to model polarizability in terms of sum of excited state transitions. See text for details. The virtual state depends on the energy (frequency) of the incident photon, and not on a real molecular energy level since this level is created by the photon itself via the polarizability. According to Eq. (5.40), the polarizability can be viewed as the sum of all the electronic transition moments in a molecule, each one weighted by

an energy term in the denominator. This is shown in Figure 5.7b for a hypothetical molecule with two real electronic excited states defined by the time‐dependent vibronic wavefunctions Ψe ′ v and Ψe " v. Here, as well as in the discussion below, we use the subscript “e” for an electronic and “v” for a vibrational state. A prime or double prime on either subscript indicates an excited state. As shown in Figure 5.7b, the transitions into the first real excited state can happen from the ground vibrational state of the ground electronic state, Ψev or from a vibrationally excited state from the ground electronic state . The polarizability matrix elements for both these transitions are given, divided by the energy difference between the transition energy ωee' minus the energy of the laser photon. Equivalent expressions would be written for the matrix elements into the other electronic state. This is summarized by the definition of the polarizability tensor elements as follows: (5.43) The subscripts α and β in Eq. (5.43) indicate the Cartesian coordinates of the polarization tensor. This equation also indicates that Raman spectroscopy is a two‐photon process since there are two electronic transition moment expressions contained in the definition of the polarizability. Thus, the selection rules for the Raman effect, depending on the binary components x·x, x·y, x·z, etc. of the Cartesian displacement coordinates, which are listed in the character tables (see Chapter 11). Equation (5.43) holds for the so‐called “far from resonance” (FFR) case in which the photons of the incident light do not have energies close to those of an electronic transition. Therefore, the contribution of a given transition moment is small. However, if the frequency of the exciting light approaches the energy difference between two real stationary states, that is, if , the corresponding term in the sum expressed by Eq. (5.42), gets very large. In this case, one

observes an enormous increase in Raman scattered intensities. This phenomenon is referred to as “resonance Raman” scattering that will be discussed next. Resonance Raman spectroscopy was experimentally verified in 1972 [9] and has had a profound impact on the study of the structure and dynamics of biophysical systems. In addition to the enhancement of the scattering cross‐section by several orders of magnitude, Resonance Raman spectroscopy offers the enormous advantage that only groups on which the electronic transition is located will experience the resonance enhancement. The basic theory of resonance Raman intensities usually is discussed in terms of the scattering tensor, rather than the polarizability, which has the form (5.44)

As in Eq. (5.43), the subscripts α and β of the scattering tensor α refer to all permutations of the Cartesian coordinates x, y, and z. In Eq. (5.44), r is the intermediate (real or virtual) state, and each element of the scattering tensor is defined as the sum over all vibronic states of the molecule. The subscripts n and m denote the final and original states of the system. The scattering tensor equation explicitly contains a damping term iΓ in the denominator, which was neglected in the previous discussion of non‐resonant Raman spectroscopy. This damping term physically is the lifetime of the intermediate state and prevents the denominator from becoming exactly zero at the resonance condition. In the “far‐from‐resonance” case, and if the molecule is initially in its ground state, Eq. (5.44) assumes the form of the polarizability tensor introduced in Eq. (5.43). Details on the discussion of the scattering tensor can be found in the literature [10]. Within the Born–Oppenheimer approximation (see Section 10.4.2), the transition moments in the numerator of Eq. (5.44) can be

separated into pure electronic transition moment between states r and m, and the Franck–Condon overlap integrals between the vibrational wavefunctions: (5.45) Here, states i and j represent the vibrational states of the ground electronic state; v is a vibrationally excited state of the resonant excited state. Equation (5.45) thus represents how much the resonance excited state is displaced along the vibrational coordinate. For the discussion of resonance enhancement, all states involved are written as the products of vibrational and electronic wavefunctions, and the dipole transition moments are evaluated separately for the purely electronic and vibrational wavefunctions. This allows the scattering tensor to be written as the sum of two terms, referred to as the A and B terms: (5.46)

(5.47)

The equation for the A term describes the resonance enhancement in totally symmetric modes, whereas the B term dominates when the vibrational modes mix the two excited electronic states. The resonance enhancement due to the A and B terms have different frequency responses, which determine the onset of resonance enhancement as the laser wavelength approaches, in energy, an electronic transition. This aspect is particularly important when discussing the resonance Raman spectra of inherently symmetric moieties, such as the iron–porphyrin groups in heme proteins. Resonance Raman studies also focused on the intermediates and the conversation dynamics in the light‐harvesting proteins such as bacteriorhodopsin and similar molecules. Here again, the prosthetic

group and its interaction with the protein bundles spanning the cellular membrane could be studied.

5.5 Selection Rules for IR and Raman Spectroscopy of Polyatomic Molecules In Chapter 4, the selection rules for the harmonic and the anharmonic oscillator were derived to be (4.37) and (4.49) respectively. A diatomic harmonic oscillator has but one normal mode of vibration, Q; and the only condition for a transition to be allowed in absorption is the requirement of the molecule being polar; otherwise, the expression (5.48) For polyatomic molecules, the general rule holds that for a transition along a normal coordinate Qk to occur in absorption – both for polar and non‐polar molecules – the condition (5.49) must hold. In Eq. (5.49), the subscript α implies that any x, y, or z component of the dipole transition moment must be non‐zero. This was mentioned before in Section 5.3.1 for the non‐polar symmetric CS2 molecule. In Chapter 3, the general form of the transition moment in absorption between two states was given as

(3.20) For a vibrational absorption transition, we accordingly define the transition moment for the 1 ← 0 transition along normal coordinate Qj as (5.50) However, for polyatomic molecules, it may not be obvious which vibrations change the dipole moment, and we need to resort to group theory and symmetry to determine which transitions are allowed in absorption. This is presented in more detail in Chapter 11. The condition corresponding to Eq. (5.49) for a transition to be active in Raman scattering is (5.51)

where the subscripts αβ denote permutations of the coordinates x, y, and z. The expression for the transition moment contains two terms, since Raman scattering is a two‐photon process, see Figure 5.7b. Thus, terms such as (5.52) are involved where ψint is the intermediate state from which the scattering takes place. Since there are two transitions involved, the selection rules will depend not just on the x, y, and z components, but on binary combinations such as x2, xy, xz, and so forth. This also will be discussed in Chapter 11.

5.6 Relationship between Infrared and Raman Spectra: Chloroform

In this section, a comparison between the observed IR absorption and Raman spectra for the same molecule, chloroform, will be presented. Raman and IR absorption spectroscopy are two complementary techniques in the sense that for some molecules (those with a center of inversion symmetry element; see Chapter 11), transitions that are allowed in Raman scattering are forbidden in IR absorption, and vice versa. However, even for many other molecules, certain vibrational modes – those that strongly change the dipole moment – are strong in absorption and weak in scattering. This is shown in Figure 5.8 for chloroform, CHCl3. This highly symmetric, pentatomic molecules should exhibit 3N–6 = 9 vibrational modes of freedom. However, some of these modes are degenerate (see Section 2.4 and Chapter 11); thus, only six bands are observed in both Raman and infrared spectra shown in Figure 5.8 and summarized in Table 5.1. The modes observed include the C–H stretching motion that is observed at 3034 cm−1, typical for CH stretching modes (cf. Example 4.1), and the C–H bending motion at 1220 cm−1 in which H–C–Cl bond angle changes. This mode is doubly degenerate and is quite strong in infrared absorption and weak in Raman scattering. The symmetric ‐CCl3 stretching mode at 680 cm−1 is the strongest band in the Raman spectrum but quite weak in absorption, whereas the antisymmetric stretching mode at 774 cm−1 is extremely strong in the absorption spectrum but weak in Raman scattering. The atomic displacement vectors of these two modes are shown schematically in Figure 5.9. The two remaining normal modes are the symmetric and the antisymmetric ‐CCl3 deformations. In the former, all three Cl–C–Cl angles decrease by the same amount and in phase. This mode is nicknamed the “umbrella” mode and occurs at 363 cm−1. The degenerate antisymmetric deformation is observed at 261 cm−1.

Table 5.1 Vibrational modes and assignments for chloroform, HCCl3. Designation Observed transition wavenumber (cm−1) ν6 261

Mode description

ν5

363

ν4

680

ν3

774

ν2

1220

CCl3 antisym deformation CCl3 sym deformation CCl3 sym stretching CCl3 antisym stretching CH bending

ν1

3034

CH stretching

As in the case of diatomic molecules, the potential energy function that determines the molecular vibration of polyatomic molecules is not quadratic along the normal coordinates. Thus, anharmonicity corrections need to be included for the detailed understanding of the vibrational transitions. The anharmonicity, as before, permits the occurrence of overtone and hot bands. In addition, transitions between different energy ladders (see Figure 5.3) are now weakly allowed, which leads to combination and difference bands, such as ν2 + ν3, that is, the normal coordinates mix in such transitions. Moreover, an effect known as Fermi resonance can increase the intensity of such transitions. Detailed discussions of these effects can be found in more specialized books on vibrational spectroscopy [1].

Figure 5.8 (a) Raman spectrum of chloroform as a neat liquid. (b) Expanded view of (a) showing the C–Cl stretching and the C‐H deformation modes. (c) Infrared absorption spectrum in the same region as spectrum (b) showing a number of overtone and combination bands.

5.7 Summary: Molecular Vibrations in Science and Technology Vibrational spectroscopy is a widely used analytical technique, and it also is an active field of research to probe vibronic states and very fast molecular dynamic processes. Also, the field of non‐linear spectroscopy – where molecular properties that depend on the square or cube of the electric field strength (see Appendix A3) are probed – requires a detailed understanding of the principles of vibrational spectroscopy discussed in this chapter. This field of nonlinear spectroscopy is a hot topic in modern optical research, and results from this research have trickled down into consumer products, such as the green laser pointer available for under $50. This device uses a frequency‐doubled Nd:YAG laser at 532 nm, and is the size of a pen. This field of nonlinear optics and non‐linear spectroscopic effects will be touched upon in Appendix 3. Molecular vibrations play a major role in many other branches of science and technology. At standard thermodynamic conditions (1 atm pressure, 25 °C), the energy levels that are mostly populated are vibrational energy levels. The thermal energy, RT, at room temperature, is given approximately by (5.53) (For conversion of the energy units, see Appendix 1). Thus, at room temperature, the vibrational energy levels are most populated, according to the Boltzmann distribution, whereas the electronic energy level population is very low (electronic transitions in the visible and ultraviolet spectral regions have energies upward of ca. 15 000 cm−1), therefore, are barely populated. Rotational states, with

much lower energy spacing, are highly populated at room temperature only for gaseous molecules since molecular rotation depends on molecules rotating freely in space (see Chapter 6). Thus, molecular vibrations are the most populated energy states under standard conditions. According to statistical thermodynamics, quantities such as the heat capacity of materials are mostly determined by the population of the vibrational energy levels via the expressions of the partition function. The heat capacities of gases, a favorite subject in engineering courses are also highly dependent on the rotational and vibrational degrees of freedom, since each degree contributes the amount of ½ R.

Figure 5.9 Atomic displacement vectors for (a) the symmetric – CCl3 stretching and the two degenerate antisymmetric –CCl3 stretching modes (b, c).

References 1 Diem, M. (2015). Modern Vibrational Spectroscopy and Micro‐ Spectroscopy: Theory, Instrumentation and Biomedical Applications. Chichester, UK: Wiley. 2 Wilson, E.B., Decius, J.C., and Cross, P.C. (1955). Molecular Vibrations: The Theory of Infrard and Raman Vibrational Spectra. New York: McGraw–Hill Co. 3 Diem, M. (1993). Introduction to Modern Vibrational Spectroscopy. New York: Wiley‐Interscience. 4 Grant, E.R. et al. (1977). The Extent of Energy Randomization in the Infrared Multiphoton Dissociation of SF6. LBL Report 6201.

5 Born, M. and Wolf, E. (1970). Principles of Optics. New York: Pergamon Press. 6 Tatum, J. (2019). Natural Broadeninf (Radiation Damping). Physics Libretexts. https://phys.libretexts.org/. 7 Bradley, M.S. (2015). Lineshapes in IR and Raman spectroscopy: a primer. Spectroscopy 30 (11): 42–46. 8 Kauzman, W. (1957). Quantum Chemistry. New York: Academic Press. 9 Spiro, T.G. and Strekas, T.C. (1972). Resonance Raman spectra of hemoglobin and cytochrome c: inverse polarization and vibronic scattering. Proceedings of the National Academy of Sciences of the United States of America 69 (9): 2622–2626. 10 Koningstein, J.A. (1972). Introduction to the Theory of the Raman Effect. Dordrecht: D.Reidel Publising Co.

Problems Advise: The interpretation of vibrational spectroscopy of polyatomic molecules requires an understanding of the symmetry properties of the molecules and their infrared‐ and Raman‐active vibrations. Thus, the problems in this chapter are doable only after studying Chapter 11. 1. Acetylene, H–CΞC–H is a linear, centrosymmetric molecule. This molecule exhibits a strong Raman band at ca. 1975 cm−1, assigned to the CΞC stretching vibration. Do you expect this band to be observed in the infrared spectrum as well? Explain. 2. Acetylene exhibits the antisymmetric C–H stretching mode at 3295 cm−1. Is this mode Raman active? Treating this molecule as a diatomic species H–X, predict the vibrational frequency of the carbon–deuterium stretching frequency (in cm−1) for D–CΞC–H 3. Some of the early applications of vibrational spectroscopy were in the determination of molecular shapes from observed spectra.

Such structural information is nowadays obtained from simple models like the valence shell electron pair repulsion theory (VSEPR) that predicts that SF4 has a seesaw structure of C2v symmetry shown on page 92, rather than a tetrahedral structure. Predict the spectral differences you would expect between a Td and C2v structure.

4. For the following molecules, determine the symmetry groups, the infrared and Raman allowed transitions and their symmetry species. Use the literature to find as many of the observed vibrational spectra and assignments. a. Carbon tetrafluoride, CF4 b. Fluoroform, CHF3 c. Difluoromethane, CH2F2 d. Fluoro–chloro–methane, CH2FCl 5. How does the symmetry of a molecule affect the number of infrared and Raman transitions observed in a spectrum? Specifically, methane shows only two absorptions in infrared spectroscopy, whereas another pentatomic molecule, CH2FCl shows nine absorption fundamentals.

6 Rotation of Molecules and Rotational Spectroscopy Whereas vibrational spectroscopy discussed in the previous chapter is a widely applied technique for qualitative analysis via group frequencies and extensive search libraries containing tens of thousands of spectra, rotational spectroscopy is used less frequently in analytical science, mostly because of sampling issues – the sample has to be in the gaseous phase – instrumental complexity, and the fact that a simple qualitative interpretation as in nuclear magnetic resonance (NMR) or infrared spectroscopy is not possible for rotational spectroscopy. This is unfortunate since much of today's detailed chemical structural information was obtained from rotational spectroscopic data, since this technique reveals structural information like no other spectroscopic technique. In introductory chemistry courses, for example, a predictive model is introduced, the so‐called valence‐shell electron‐pair repulsion (VSEPR) model that allows qualitative prediction that the H–N–H angle in ammonia is smaller than the H–C–H angle in methane, since the free electron pair in ammonia occupies more space than a bonding pair in methane. Such information was originally deduced from microwave data, since it allows the measurement of the moment of inertia extremely accurately, and therefore, bond distances and bond angles can be calculated to five or six significant figures. As in the case of vibrations of di‐ and polyatomic molecules, a classical model allows the calculation of the moment of inertia of a molecule from structural data. This aspect will be covered first in Section 6.1. The quantum mechanical treatment of the rotation of a rigid molecule (Sections 6.2 and 6.3) introduces the angular momentum operator that leads to two differential equations that give us the rotational wavefunctions, eigenvalues, and selection rules. These differential equations also form the basis of solving the Schrödinger equation for the hydrogen atom – after all, the hydrogen

atom could be thought of as a “diatomic molecule” consisting of a proton and an electron rotating in space. The concept of the angular momentum also is required for the understanding of spin spectroscopy. Thus, the mathematics presented in Section 6.3 is the basis of several subjects in this book.

6.1 Classical Rotational Energy of Diatomic and Polyatomic Molecules Classically, any rotatory motion can be described by the rotational kinetic energy (6.1) where L is the angular momentum, and I is the moment of inertia. This definition is in complete analogy to the definition of the (linear) kinetic energy (1.17) in which the kinetic energy, in both cases, is proportional to the square of the corresponding momentum divided by a quantity measuring the inertia, or “resistance” to this motion. The angular momentum in Eq. (6.1) is defined as the following: (6.2) that is, the vector product of the linear momentum and the radius of the circular motion. Thus, L is a vector quantity, indicated in bold face. Multiplying out the vector components r = ix + jy + kz and p = ipx + jpy + kpz in Eq. (6.2) reveals that (6.3)

with (6.4) and similar expressions holding for the other components of L. The (scalar) moment of inertia, as pointed out above, depends very much on the shape (symmetry) of the molecule. In the simplest case of a linear, diatomic molecule, the moment of inertia is given by (6.5) where m1 and m2 are the masses of the two atoms and r1 and r2 their distances from the center of mass (see Figure 4.1). The center of mass condition is given by m1r1 + m2r2 = 0. For a diatomic molecule, one moment of inertia is zero (the one along the bond axis) because it is assumed that the atoms are point masses, and rotation about the bond connecting the two atoms experiences no inertia. The other two components of the moment of inertia, Ix and Iy, are equal. The same argument holds true for linear molecules with more than two atoms. For a nonlinear molecule in an arbitrary coordinate system, the moment of inertia is a tensor defined as (6.6)

In Eq. (6.6), the summation is over all atoms in the molecule, mi is the mass of the individual atoms, and x, y, and z are their Cartesian coordinates [1]. If the inertial tensor is written in arbitrary coordinates x, y, and z, its values will depend on the choice of the origin of the coordinate system; i.e. different numerical values for the tensor terms Ixx and others are obtained when the origin is shifted. However, the rotational motion of a molecule, or for that matter any

freely rotating body, can be described such that the moment of inertia is independent of the choice of the coordinate origin. This is the case when the overall rotation and translation of the molecule is uncoupled and was accomplished for diatomic molecules by introducing the reduced mass, which is based on a coordinate system whose origin lies at the center of mass. If the center of mass of the molecule does not translate, the molecular motion is pure rotational motion. Under these conditions, the inertial tensor can be written in the diagonal form: (6.7)

Here, T is a coordinate transform matrix between an arbitrary coordinate system and the principal axes of inertia; in this latter coordinate system, the inertial tensor is diagonal, and the diagonal terms IA, IB, and IC are known as the principal moments of inertia and are related to the observables in rotational spectroscopy. The superscript in TT denotes the transpose of the matrix T. The square of the total angular momentum, L2, required in Eq. (6.1) can be written as a sum of the individual principal components according to (6.8) since squaring the vector quantity in Eq. (6.3) eliminates all cross‐ terms. Thus, the rotational kinetic energy is (6.9)

Since the potential energy of free rotational motion is zero, the total energy of the system is given by the kinetic energy, i.e. E = T. The kinetic energy is generally expressed in terms of the rotational

constants A, B, and C, rather than IA, IB, and IC. The rotational constants are defined as (6.10) with the convention that IA ≤ IB ≤ IC. Notice that the rotational constants are expressed in units of 1/s = Hz. To convert to energy units, these constants need to be multiplied by Planck's constant. In terms of the rotational constants, the rotational kinetic energy is written as (6.11) The rotational constants are related to the quantum mechanical energy eigenvalues observed in rotational spectroscopy. They contain all the structural parameters of a molecule and allow the determination of molecular structures within picometer and millidegree accuracy. Depending on the shape of molecules, they can possess either one, two, or three distinct moments of inertia and are referred to as spherical and linear top rotors, symmetric top rotors, or asymmetric top rotors. These have different energy eigenvalues as we shall see in Section 6.2 that depend on the rotational constants as follows: a. If all rotational constants A, B, and C of a molecule are equal, it is referred to as “spherical top rotor.” A molecule has to belong to one of the spherical point groups (cf. Chapter 11) of Td or Oh symmetry. The total rotational energy can be expressed in terms of L2 only. By convention, the rotational constant B was selected to represent the rotational energy (cf. Eq. [6.11]) as (6.12)

A spherical top molecule cannot have a permanent dipole moment and therefore will not exhibit a rotational absorption (microwave)

spectrum. Its polarizability also will not change during a rotation, and spherical top molecules will not exhibit a pure rotational Raman spectrum either. They will, however, exhibit a rotational–vibrational absorption spectrum in some bands (see Section 6.6) since some vibrational transitions distort the molecule such that it shows a temporary dipole moment. b. For linear molecules (obviously including diatomic molecules), one rotational constant is zero (the one along the bond axis), and the other two are equal. This leads to an energy equation for linear molecules identical to the one for spherical molecules (Eq. [6.12]). However, a linear molecule must possess a permanent dipole moment to exhibit a rotational absorption (microwave) spectrum. Thus, N2 and Cl2 will not show a rotational spectrum, but they will exhibit a pure rotational Raman spectrum due to the different selection rules in Raman spectroscopy. c. For symmetric tops, two moments of inertia are equal: either IA = IB < IC (oblate symmetric top rotor) or IA < IB = IC (prolate symmetric top rotor). For the former case, (6.13)

since (6.14) (6.15) (6.16)

(6.17) For the prolate top rotor, this corresponding equation is (6.18) In both equations, the two operators, L2 and , commutate and have eigenvalues in the same vector space. Thus, the problem can be solved. d. For the asymmetric top rotor, all three moments of inertia are unequal, and their operators do not commutate. Therefore, the problem cannot be solved explicitly. The solutions of the rotational Schrödinger equation will be presented in Section 6.3, first for the case of linear or spherical top molecules for which the rotational energy depends on the eigenvalues of the L2 operator according to Eq. (6.12).

6.2 Quantum Mechanical Description of the Angular Momentum Operator In analogy to the discussion of vibrational spectroscopy, a classical model for the rotational energy was derived above. The classical models for both the rotation and the vibration of molecules predict the molecular energies properly in a steady‐state situation, that is, without interactions with light. In order to predict these interactions and the possibility of observing a spectrum due to the molecular system undergoing transitions between stationary state energy levels, quantum mechanics needs to be invoked. Since there is no change in the potential energy of a molecule during molecular rotations, the equation that governs the total energy during molecular rotations is just Eq. (6.11). To obtain the rotational Schrödinger equation, the angular momentum components in Eq. (6.11) need to be replaced by the corresponding quantum mechanical

operators. This is accomplished by writing the angular momentum components given by Eq. (6.4) (6.4) and subsequently substituting the linear momentum components by (2.2) The component of the angular momentum operator given in Eq. (6.4) then becomes (6.19) with equivalent expressions holding for the other two Cartesian components of the angular momentum. Because of the differentiation required when operating with the , and operators on a function f(x, y, z), the commutator of these operators is defined as (6.20)

Figure 6.1 Definition of spherical polar coordinates. is nonzero. As discussed in Section 2.1, the consequences of Eq. (6.20) are quite far‐reaching: for a molecule with three different moments of inertia along the three principal axes (a so‐called asymmetric top rotor), it is impossible to determine the three principal moments of inertia. However, the total angular moment operator, L2, does commutate with all the components of the angular momentum operator and thus, it is possible to determine the total moment of inertia and one of the three components, normally assumed to be the Lz operator (although any of the three Cartesian components could be selected). (6.21) The problem of solving the quantum mechanical expressions for the rotational energy of a molecule rotating about its center of mass is best tackled in spherical polar coordinates, rather than Cartesian coordinates. Spherical polar coordinates are defined as shown in Figure 6.1

In spherical polar coordinates, the form

and

operators take the

(6.22) and (6.23) The transformation of the operators from Cartesian to spherical polar coordinates is by no means trivial and described in detail in [2]. The functions that simultaneously are eigenfunctions of the two operators in Eqs. (6.22) and (6.23) are the so‐called spherical harmonic functions Y(θ, φ): (6.24) with eigenvalues c and b. The angular momentum operator components have been defined above in a space‐fixed coordinate system as x, y, and z, whereas the classical moment of inertia has been described in a coordinate system that rotates with the molecule, the principal axes of inertia. The relation between these two coordinate systems is given by Eq. (6.7) and is just a simple coordinate system transformation.

6.3 The Rotational Schrödinger Equation, Eigenfunctions, and Rotational Energy Eigenvalues To obtain the eigenfunctions and eigenvalues for a rotating molecule, the Schrödinger equation for the total angular momentum

(6.1) with (6.25)

and one of its Cartesian components, for example,

, with (6.26)

need to be solved simultaneously. As we shall see below, the solutions of these two operator/eigenvalue problems are the spherical harmonic functions Y(θ, φ), which can be separable into two functions in the variables θ and φ according to (6.27) To find the eigenvalues and eigenfunctions of the operators, we start with the

and

operator: (6.28)

Since

only contains the variable φ and T(φ) depends on φ only,

Eq. (6.28) simplifies to (6.29) Integrating Eq. (6.29) yields (6.30)

where A is a constant (the amplitude). Since T is a function of φ only, the boundary conditions for T(φ) are such that the function has to have the same value for multiples of 2π; otherwise, the function would self‐destruct by interference, as shown in Figure 6.2. This boundary condition can be written as (6.31) from which follows that (6.32)

Figure 6.2 Graphical representation for the condition T(φ) = T(φ + b 2π), where b is an integer. Defining (6.33) Eq. (6.30) can be rewritten as (6.34)

with K = 0, ± 1, ± 2, ± 3, …, which implies that the eigenvalues of the z‐component of the angular momentum are quantized, and K is a rotational quantum number associated with the

operator.

Normalization of Eq. (6.34) by integrating over φ( 0 ≤ φ ≤ 2π) results in the final form of the eigenfunctions of the operator as (6.35)

Next, the θ‐dependent part of Eq. (6.27) or (6.28) needs to be solved to give the eigenfunctions S(θ) of the operator. Since this operator contains derivatives with respect to both θ and φ, we utilize the second derivative of Eq. (6.35) in Eq. (6.36): (6.36)

which accounts for the factor K2 in Eq. (6.37): (6.37) Equation (6.37) shows why the solutions of S(θ ) contain the quantum number K, which accounts for, as we shall see later, the degeneracy of the eigenvalues of S(θ ). The differential equation, Eq. (6.37), is solved [3], as in the case of the harmonic oscillator, by a power series expansion in cos(θ). Again, this power series has to be broken off at some value of the expansion coefficient; otherwise, the wavefunctions wouldn't be finite. This condition, along with the recursion formula for the expansion coefficients, leads to the energy eigenvalues “c” in Eq. (6.37):

(6.38) where J can assume values J = 0, 1, 2, 3, … and K can be between −J and +J: (6.39) Substituting the energy eigenvalues (Eqs. [6.34] and [6.38]) back into the power series expansion for S(θ ) gives the final form of the rotational wavefunctions: (6.40)

The functions are referred to as the spherical harmonic functions that are polynomials in the variables θ and φ and contain two indexes, J and K, that determine the highest power of the variables θ and φ, respectively, in the polynomials. Notice that the index K in the form of these polynomials PJ |K| can have values form −J to +J, but cannot exceed | J|. The K‐values between −J and +J account for the 2J + 1‐fold degeneracy of the energy levels determined by J. The functions

in Eq. (6.40) are known as the “associated

Legendre polynomials” in the variable (cosθ) only but parametrically contain the integer index K from φ‐dependent part (Eq. [6.34]). The associated Legendre polynomials form an infinite orthogonal vector space, as did the Hermite polynomials discussed in Chapter 4. The first few of these polynomials are given below: (6.41)

with z = (cos θ). Higher‐order polynomials can be computed from the recursion formula given below. The first few spherical harmonic functions follow directly from Eqs. (6.40) to (6.41), as demonstrated in Example 6.1: Example 6.1 Derivation of the form of the first few spherical harmonic functions from

and Equations (6.40) and (6.41) Answer: (E6.1.1)

(E6.1.2)

(E6.1.3)

Eq. (E6.1.3) is often written as a linear combination that eliminates the imaginary function as

The remaining functions can be constructed from the recursion formula for the associated Legendre polynomials and Eq. (6.42). The spherical harmonic functions Y(θ, φ) are commonly encountered in physics and describe, among other situations, the tides on a planet covered by an ocean of uniform depth or the modes of electrons on a spherical particle (the electron‐on‐a‐sphere). This latter model is useful to describe effects such as surface plasmon resonances that lead to several forms of interesting spectroscopic effects [4]. Like the Hermite polynomials, the functions of degree (J + 1) can be obtained via a recursion formula from polynomials of degree J and (J − 1): (6.42) Thus, the selection rules for rotational transitions (see Section 6.4) may be derived in analogy to the derivation of the harmonic oscillator selection rules. Notice that in the recursion formula for each subsequent polynomial, the index K does not change. The eigenvalues of the rotational Schrödinger equation are quite simple expressions: (6.43) where J assumes integer values, J = 0, 1, 2, 3,…. Since the rotational energy of linear and spherical top molecules depend on the eigenvalues of the operator only, Eq. (6.43) contains all the information needed for these special cases. However, for symmetric top molecules, the eigenvalues of the operator are needed as well, which are given by (6.44)

Readers familiar with the quantum mechanics of the hydrogen atom (see Chapter 7) will recognize that the eigenfunctions and eigenvalues presented here are the same for the nonradial part of the hydrogen atom, and the limitations of the indexes of the spherical harmonics determine the shape and the splitting of the s, p, d, f, and other orbitals into the sublevels determined by the magnetic quantum number. K can assume integer values from −J to +J. Equations (6.12) and (6.43) lead to the surprisingly simple equation for the rotational energy of spherical and linear molecules: (6.45) where B was defined above as (6.10) and J = 0, 1, 2, 3,…. The consequences of Eq. (6.40) on the appearance rotational spectra observed for linear molecules will be discussed in Section 6.5, after the discussion of selection rules for these species. The commutation rules of the Cartesian components of the angular momentum with each other (see Eq. [6.20]) and the fact that the total angular momentum operator, commutates with lead to some interesting aspects of what is known as the “spatial quantization” of the angular momentum. This aspect will be introduced in Section 7.6 for the spatial quantization of the electronic angular momentum in the hydrogen atom. The spatial quantization affects many aspects of spectroscopy dealing with electronic and spin angular momenta. Example 6.2 Analysis of the units of the rotational constants In rotational spectroscopy of heavy molecules, it is customary to express the rotational constants in units of Hz (or MHz), since microwave spectroscopy is carried out using instruments with radiofrequency generators that are typically calibrated in frequency units. Answer:

According to Eqs. (6.5) and (6.6), the units of the moment of inertia are given as [kg m2]. Therefore, the units of B are obtained as follows (E6.2.1)

Since E = h ν, the expression in Eq. (6.10) is multiplied by h for energies to be expressed in units of Joule. Thus, the expression for B would be (E6.2.2) For light molecules, such as H‐F, the rotational constant is conventionally expressed in wavenumber units. This conversion is obtained by dividing the frequency by the velocity of light.

6.4 Selection Rules for Rotational Transitions The selection rules for rotational transitions are derived from the recursion formula of the associated Legendre polynomials [5]. The derivation of these selection rules is carried out in analogy to the derivation of the selection rule for the harmonic oscillator where the recursion properties of the Hermite polynomials were employed (cf. Eqs. [4.34]–[4.36]). The transition moment between two rotational energy levels J and J′ for the z‐component of the electric dipole transition operator is written as (6.46) As in any form of absorption spectroscopy, one requires this transition moment to be nonzero: (6.47)

for a transition to occur. In analogy to the arguments for the transition moment between harmonic oscillator wavefunctions, one substitutes the expression in Eq. (6.46) by the recursion formula (6.48) This yields an expression that is written in simplified form (omitting the fractions containing the parameters J and K) as (6.49) (6.50) Since the associated Legendre polynomials are orthogonal, the two integrals in Eq. (6.50) are nonzero if and only if (6.51)

(6.52) in complete analogy with the arguments presented for the harmonic oscillator transition moment in Eqs. (4.33)–(4.34). This leads to the selection rules for a diatomic rigid rotor: (6.53) Thus, transitions between adjacent energy levels are allowed, both up and down the energy ladder. Although this selection rule is the same as that for the harmonic oscillator, the spectra are vastly different. In the latter case, only one peak was predicted due to the equidistant energy levels. In rotational spectroscopy, a set of spectral peaks is observed due to the fact that

the energy levels are spaced quadratically, and the transition energy depends on the J‐level from which the transition originates. This will be demonstrated in the next section.

6.5 Rotational Absorption (Microwave) Spectra 6.5.1 Rigid Diatomic and Linear Molecules According to Eq. (6.45), the energy levels for a spherical or linear molecule are given by (6.45) which leads to energy levels

(6.54) etc. With the selection rule given by Eq. (6.53), this leads to transition energies of 2B, 4B, 6B, etc. between adjacent energy levels. This is shown in Figure 6.3a by the vertical arrows. In general, the spacing between rotational transitions can be expressed as follows. Let us assume that transitions occur from state J to a more energetic state J' = J + 1. Such transitions occur at transition frequencies of (6.55)

Thus, the rotational spectrum for a linear molecule consists of equidistant spectral lines spaced by 2B. This is shown schematically in Figure 6.3b. The intensity of the spectral lines shown in Figure 6.3b will be discussed next. (Recall, however, that that spherical top molecules, although they have exactly the same energy levels, do not exhibit a rotational absorption spectrum because they are, by definition, devoid of a dipole moment. The same holds for homonuclear diatomic molecules.) The transition moments for the rotational transitions differ by the factors (6.56) in Eq. (6.48) that were omitted in the derivation of the dipole transition moment (Eqs. [6.49]–[6.52]) and are found to differ only slightly. Thus, one may argue that the intensity of the observed spectral lines depends mostly on the population of each of the energy levels, rather than the differences in transition moments between different J‐levels. Assuming that the rotational energy differences (a few wavenumbers) are small compared to the thermal energy (208 cm−1 at room temperature), the Boltzmann distribution (Eq. [3.31]) predicts that many rotational energy levels have similar populations. The probability PJ of finding a molecule in a given rotational level J is given by

Figure 6.3 (a) Energy level diagram for linear rotors. (b) Schematic rotational spectrum for linear rotors. (6.57) In this equation, the exponential expression is the normal Boltzmann distribution. The pre‐exponential expression contains the factor (2J+1), which results from the K‐fold degeneracy of the rotational energy (see Section 6.3). The factor B/kT results from the total number of occupied rotational states, given by

(6.58) Here, one assumes that the energy levels are so closely spaced that J becomes a continuous variable. The intensity distribution, as a function of the rotational transition, is shown schematically in Figure 6.4. Using Eqs. (6.45) and (6.58), the rotational spectrum for a diatomic molecule can be predicted, as demonstrated in 6.3. Example 6.3 Calculation of the expected rotational spectrum of 35Cl–F at room temperature, given

Figure 6.4 Simulated rotational spectrum of 35Cl–F at room temperature, using B = 0.369 cm−1 and Eq. (6.58) to compute the populations of each of the J‐levels. The spacing between adjacent spectral bands is ca. 0.74 cm−1; thus, the spectral envelope shown covers about 37 cm−1. and

Answer: For a diatomic molecule, the equation for the moment of inertia is (6.5) or (E6.3.1) This relationship is the subject of problem (1) at the end of the chapter. (E6.3.2)

(E6.3.3) (6.10)

(E6.3.4)

(E6.3.5) The rotational spectrum consists of equidistant lines, spaced by 2B = 0.738 [cm−1]. Next, the intensity profile as a function of J given by Eq. (6.58) needs to be computed. For this, EXCEL software can be used advantageously. The result of this simulation is shown in Figure 6.4. Here, it was assumed that the transition moments for all J‐levels are equal. The calculations for the 35Cl–F molecules were carried out using the rigid rotator approximation. The observed spectrum is slightly different since the molecule rotates faster at higher values of J and

centrifugal effects increase the bond distance. This, in turn, increases the moment of inertia, which reduces the rotational constant B; consequently, the rotational line spacing becomes smaller at increasing J‐levels.

Figure 6.5 Schematic of the center‐of‐mass (COM) position in an oblate (a) and prolate (b) symmetric top rotor such as chloroform and methyl chloride, respectively.

6.5.2 Prolate and Oblate Symmetric Top Molecules It was pointed out earlier (Section 6.1) that for some molecules, two of the moments of inertia are equal and different from the third one. These molecules belong to the C3v point group (see Chapter 11) or higher (but not to spherical point groups) and are referred to as symmetric top rotors. Examples of such molecules are chloroform (CHCl3), ammonia (NH3), or methyl chloride (H3CCl). Depending on the masses of atoms, the center of mass may lie on either side of the central atom: in methyl chloride, for example, the center of mass lies along the C–Cl bond, and the molecule is said to be a prolate top (see Fig. 6.5b). On the other hand, in chloroform, the center of mass lies on the symmetry axis below the central carbon atom. This species is referred to as an oblate top; see Figure 6.5. For oblate top rotors, the classical rotational energy was found to be

(6.17) With the eigenvalues of the and operators given in Eqs. (6.43) and (6.44), the quantum mechanical energy of the oblate top rotor is given by (6.59) For the prolate top rotor, the corresponding equations are (6.18) and (6.60) An energy level diagram for the oblate and prolate symmetric top rotor is shown in Figure 6.6a and b, respectively. For symmetric top rotors, the rotational energy depends on both J and K, as shown. From the aforementioned convention IA ≤ IB ≤ IC, it follows that A ≥ B ≥ C; therefore, (C–B) is always negative, and (A–B) is always positive. Thus, the energy levels for increasing K‐values are decreasing for the oblate and increasing for the prolate top. This increase/decrease occurs with the square of K.

Figure 6.6 Energy level diagram for (a) oblate and (b) prolate top rotors. See the text for detail. A comparison between Figures 6.3 and 6.6 demonstrates the loss of degeneracy of the rotational energy levels for symmetric top rotors. In Figure 6.3, for linear or spherical top rotors, the energy levels are (2J + 1)‐fold degenerate, because K can assume values from −J, −J+1, 0, …J−1, J. The discussion of the selection rules based on the recursion properties of the associated Legendre polynomials (Eqs. [6.49] and [6.52]) indicated that the quantum number K stays constant when applying the recursion formula. This leads to selection rules for symmetric top rotors: (6.61) These selection rules require that transitions occur within a given K‐ level, as indicated by the gray arrows in Figure 6.6a. Therefore, the transition energies for the J = 1 to J = 2 transition for K = −1, K = 0, and K = 1 are still the same, since the energy levels are

reduced by the same amount, namely, (C − B)K2, yet this transition is composed of three separate components. Only in the presence of an external electric field (Stark splitting) will these components be observed separately. Under these conditions, the transitions will exhibit multiplet splitting. In the absence of an electric field, symmetric top rotors exhibit equidistantly lines spaced at 2B, just as linear molecules.

6.5.3 Asymmetric Top Molecules Asymmetric top rotors, i.e. molecules with symmetry point groups lower than C3v, as mentioned above, cannot be treated in a simple model described for the other tops above. The water molecule that belongs to the C2v point group (cf. Chapter 11) has three different moments of inertia and, therefore, is an asymmetric top rotor. Since the components of the angular momentum operator do not commutate, approximate methods are used to interpret the microwave spectrum of asymmetric top rotors. This is accomplished by describing the molecule by its “asymmetry parameter κ,” which is defined as (6.62) This parameter takes the value of −1 for the prolate top (with B = C) and +1 for the oblate top (A = B). Thus, transition energies E(κ) are interpolated for an asymmetric top rotor between the oblate and prolate top limiting cases, and the overall rotational energy is written as E(J, κ): (6.63) The rotational spectra of asymmetric top rotors are quite complicated and beyond the intent of this book. Interested readers are referred to specialized books such as Townes and Schawlow [5].

6.6 Rot–Vibrational Transitions

Infrared and Raman vibrational spectra of gaseous molecules exhibit broad wings on the low‐ and high‐frequency side of the corresponding vibrational transitions, as shown in Figure 6.7a. These wings are due to rotational transitions superimposed on vibrational transitions and show the distinct rotational–vibrational (rot– vibrational) fine structure at high spectral resolution, see Figure 6.7b. To describe the origin of these transitions, one writes the rot–vibrational wavefunctions as products of the pure rotational and the pure vibrational wavefunctions: (6.64) and the corresponding energy levels simply as the sum of rotational and vibrational energies: (6.65) For a diatomic molecule obeying the harmonic oscillator and rigid rotor approximation, Eq. (6.65) would read as (6.66) In Eq. (6.66), both the vibrational and rotational energies are expressed in units of energy; that is, B is expressed in Hz and the vibrational energies in terms of the frequency (rather than the wavenumber) of the transition. The energy level diagram for such a diatomic molecule is shown in Figure 6.8a.

Figure 6.7 (a) Observed rot–vibrational band envelopes in the infrared absorption spectrum of gaseous methane, CH4. (b) High‐ resolution spectrum of the region shown in (a).

Figure 6.8 (a) Rot–vibrational energy level diagram for a harmonic oscillator/rigid rotor. (b) Rot–vibrational transitions with Δn = ± 1; ΔJ = ± 1 selection rules. (c) Spectral lines corresponding to these transitions. The selection rules for rot–vibrational transitions are (6.67) This leads to rot–vibrational transitions from the rotational sublevels of the ground vibrational state to rotational sublevels of the vibrationally excited state with either lower or higher rotational quantum numbers, as shown in Figure 6.8b. Inspection of Figure 6.7b also reveals that the spacing between the rotational transitions becomes smaller from lower to higher wavenumber. This is due to the centrifugal distortion effect: as the molecule rotates faster, centrifugal forces stretch the bond, thereby increasing the moment of inertia and decreasing the rotational

constants, resulting in more closely spaced lines. This effect, as well as the contribution of anharmonicity, is contained in Eq. (6.68): (6.68)

The anharmonicity term

in Eq. (6.68) was defined

before (cf. Eqs. [4.45] and [4.46]). The centrifugal distortion constant D accounts for the change of the moment of inertia at higher rotational energy levels. The rot–vibrational transitions for which ΔJ = −1 (the low‐frequency progression of rot–vibrational bands) are referred to as the P‐ branch, those with ΔJ = 0 are referred to as the Q‐branch, and those with ΔJ = +1 as the R‐branch. Symmetry rules sometimes prohibit the occurrence of the Q‐branch, and typical rot–vibrational absorption spectra, for example, for gaseous H‐Cl, may appear as shown in Figure 6.8c. Notice that there are many transitions in the Q‐branch, since the Δn = 1, ΔJ = 0 transitions can originate from J = 0, J = 1, J = 2, etc. In linear polyatomic molecules, vibrational transitions that do not change the dipole moment are forbidden in absorption; however, transitions that change the dipole moment, such as the antisymmetric stretching vibration of CO2 (cf. Chapter 5), do exhibit a rot–vibrational spectrum. For this vibration, the transition dipole lies along the molecular axis; consequently, the band envelope is referred to as a “parallel” envelope, characterized by the absence of a Q‐branch, with an appearance similar to the rot–vibrational spectrum of a heteronuclear diatomic molecule. This is shown in Figure 6.9b. In the CO2 deformation mode, on the other hand, the dipole moment change is perpendicular to the molecular axis; consequently, the resulting band envelopes are referred to as

“perpendicular” bands. Perpendicular bands have a pronounced Q‐ branch, broadened by the overlap of the n = 0 to n = 1 transitions for different J‐values (cf. Figure 6.9a).

Figure 6.9 Simulated rot–vibrational spectral band profiles for the deformation mode (a) and the antisymmetric stretching mode of CO2 (b). [From vpl.astro.washington.edu/spectra/co2/htm]. Although spherical top molecules such as CH4 do not exhibit a pure rotational absorption spectrum, the triply degenerate modes such as the antisymmetric stretching mode exhibit a rot–vibrational spectrum that permits the determination of the rotational constant B. This vibration exhibits a rot–vibrational spectrum of the perpendicular type, with a distinct Q‐branch at ca. 3019 cm−1 as shown in Figure 6.7.

References 1 Kwon, Y.H. Mechanical Basis of Motion Analysis: Inertia Tensor. http://www.kwon3d.com/theory/moi/iten.html. 2 Levine, I. (1970). Quantum Chemistry, vol. I&II, 83. Boston: Allyn & Bacon. 3 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon Chapter 5. 4 Zeng, S. et al. (2014). Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications. Chemical Society Reviews 43 (10): 3426–3452. 5 Townes, C.H. and Schawlow, A.L. (1975). Microwave Spectroscopy. New York: Dover Publications.

Problems 1. Demonstrate that [L2, Lz] = 0. 2. The classical definition of the moment of inertia for a diatomic molecule is (6.5) where r1 and r2 are the distances of masses m1 and m2 from the center of mass. Show that Eq. (6.5) is equivalent to where mR is the reduced mass of atoms 1 and 2 and r12 is the bond length. 3. In the linear, centrosymmetric molecule acetylene C2H2, the two carbons are at ±60 pm, and the two hydrogens are at ±67 pm from the center of mass. a. Calculate the moment of inertia for acetylene. b. Calculate the rotational constant.

c. What is the spacing of the rot–vibrational lines observed for the antisymmetric, infrared active C–H stretching mode of at 3295 cm−1? 4. The spacing of rotational lines in the spectrum of H–35Cl is 21.173 cm−1. Do you expect the spacing of lines in H–37Cl to be lower or higher? Justify your answer. 5. The rotational constant for deuterium fluoride, 2D–19F, is 11.007 cm−1. The atomic masses for 19F and 2D are 18.9984032 and 2.0141018 amu, respectively. Calculate the bond length of deuterium fluoride to the maximum number of significant figures consistent with this information. 6. Given the selection rule for a rigid rotor, ΔJ = ±1, show that its rotational spectrum consists of equidistant lines spaced by 2B. 7. Although the rotational Schrödinger equation cannot be solved exactly for an asymmetric rotor such as water, experimental observation of its microwave spectrum gives the rotational constants, using the approximate methods outlined in Section 6.5.3. a. Use the definition of the inertial tensor (Eq. [6.6]) and the following coordinates of the atoms in a water molecule to calculate its rotational constants. b. According to the results in part (a), where does the water fall in terms of its κ‐value? Atomic coordinates (in pm) X Y Z O 0.0 6.52 0 H1 −75.8 −54.5 0 H2 75.8 −54.5 0 mO = 16.0 amu mH = 1.00 amu

7 Atomic Structure: The Hydrogen Atom Although this book was conceived as an introduction to the quantum mechanical foundations of molecular spectroscopy and the hydrogen atom hardly qualifies as a molecule, this chapter is included for several reasons. First, a quantum mechanical textbook just isn't complete unless the hydrogen atom is given some detailed discussion. Second, the work presented in Chapter 6 on the solution of the rotational Schrödinger equation is very applicable to solving the hydrogen atom Schrödinger equation; in fact, the graphical representation of the spherical harmonic functions makes more sense in the context of orbital shapes than for the rotational states of molecules. This is so at least in the eyes of the author who cannot imagine a rotating molecule to have rotational energy states shaped like spherical harmonic functions. Third, the concept of electronic orbitals – and particularly their shapes – is of importance in the discussion of chemical bonding that needs to be introduced in order to discuss molecular electronic spectroscopy. Finally, it was the hydrogen atom to which quantum mechanics was first applied, and it cannot be overemphasized that it was the ingenious insight of E. Schrödinger who realized that by substituting the classical momentum by its quantum mechanical equivalent (see Eq. [2.2]), a set of differential equations was obtained that described the hydrogen atom. Schrödinger, trained originally as a classical wave mechanicist, knew these differential equations before applying them to the hydrogen atom and intuitively knew that the form of their solutions would explain atomic electronic structure. This discovery forever changed science and brought order into the periodic chart of elements that, at this point, was a purely empirical correlation of chemical properties of elements without rhyme or reason. This latter aspect is in line with the philosophy of this book: rather than approaching the subject of quantum mechanics as a theoretical concept, the view taken here is to connect experiment and theory. The experimental observations underlying the discussion in this chapter are the previously discussed atomic emission and absorption

spectra (see Section 1.4) that were fit to an empirical, yet unexplained equation (cf. Eq. [1.19]) that explains the now well‐ established form of the periodic chart that is the basis of all chemistry. As we shall see in the present chapter, the organization of the periodic chart is a direct consequence of the solutions of the angular part of the Schrödinger equation and explains the number of elements in each row in the periodic chart. Again, for the author, it is nearly miraculous how the form of the spherical harmonic functions and the rules that control the indexes in these functions – the quantum numbers in the parlance of scientists – are responsible for all aspects of chemistry.

7.1 The Hydrogen Atom Schrödinger Equation As indicated in Section 2.2, an electron in the potential field of a nucleus experiences a potential energy that is given by Eq. (7.1): (7.1) Equation (7.1) is the integrated form of Coulomb's law. Here, “e” is the electronic charge in Coulomb (C) (see Appendix 1), and ε0 is the dielectric constant (permittivity) of vacuum. (Notice that the electronic charge “e” is written as a non‐italicized quantity, whereas the basis of the natural growth function ex is written in italics.) A quick analysis of the units in Eq. (7.1) is appropriate at this point. The permittivity has the numerical value of ε0 = 8.854 × 10−12 [F/m = C2/(J m)]. Thus, the right‐hand side of Eq. (7.1) has units of (7.2) which are indeed units of energy.

The potential given in Eq. (7.1) has spherical symmetry, that is, for any constant value of r, the potential energy is equal for all values of the angles θ and ϕ (see Figure 6.1). Thus, the Schrödinger equation for the electron in the potential field of the nucleus is best defined in spherical polar coordinates. In Chapter 2, the Schrödinger equation for a one‐dimensional case of an electron in a (negative) hyperbolic potential of a nucleus was written as (2.16) With the condition of the potential having spherical symmetry discussed in the previous paragraph, Eq. (7.1) needs to be substituted into Eq. (2.16), and the total equation recast into spherical polar coordinates: (7.3) Here, mR is the reduced mass of the proton–electron system, and ∇2 is the Laplace operator (or the Laplacian) given by (7.4) in Cartesian or (7.5)

in spherical polar coordinates [1]. The angle‐dependent part of Eq. (7.4) is just the operator that was discussed in Chapter 6:

(6.22)

and was obtained there by assuming a constant value of r for the rigid rotor. Thus, the ∇2 operator will have eigenfunctions that can be separated into distance‐ and angle‐dependent functions according to (7.6) where the θ‐ and ϕ‐dependent parts are the eigenfunctions of the operator that are the spherical harmonic functions Y(θ,ϕ). Furthermore, the reduced mass of the proton–electron system is nearly equal to the electron mass, since mp + me ≈ mp. Thus, Eq. (7.3) then can be written as follows: (7.7)

where we used the eigenvalues ħ2l(l + 1) of the operator. For historical reasons, the quantum number of the total angular momentum operator that was referred to as “J” in the discussion of rotational spectroscopy (cf. Eq. [6.38]) is called “l” in the discussion of the hydrogen atom. Since this quantum number describes the component of the angular momentum of the electron orbiting the nucleus, this quantum number l is usually referred to as the “orbital angular momentum” quantum number. Similarly, the quantum number of the operator that was referred to as “K” in rotational spectroscopy (Eq. [6.34]) is designated “m” or “ml” in the discussion

of the hydrogen atom and referred to as the “magnetic” quantum number for reasons we shall see in Section 7.6. The fact that the angular part of the hydrogen atom Schrödinger equation is the same solutions as the rigid rotor problem is not too surprising: after all, we may envision the proton–electron pair as just a rigid rotor tumbling in space. Defining a new constant “a” (which happens to be the “Bohr radius”; see Example 7.1) (7.8)

we may rewrite Eqs. (7.5) and (7.6) as (7.9) where R″ and R′ are the second and first derivative of R(r) with respect to r. Equation (7.9) is known as the LaGuerre differential equation, and its solutions (to be presented below) are the LaGuerre polynomials. The fact that the eigenvalues of the operator are explicitly contained in Eq. (7.8) leads to the appearance of the quantum numbers l and ml in the hydrogen atom wavefunctions.

7.2 Solutions of the Hydrogen Atom Schrödinger Equation The process of solving Eq. (7.9) follows a similar but even more complicated approach that was introduced for the Hermite and Legendre differential equations. This approach involves a power series expansion: (7.10)

where (7.11)

and “s” a yet undefined integer value. The expansion coefficients are given by the recursion formula (7.12) where n is an integer (see Eq. [7.13]). Complicated arguments on the value of R(r) for large values of r in the trial function, Eq. (7.10), lead to the requirement of terminating the power series expansion after a finite number of terms. As in the case of the Hermite and Legendre differential equations, this leads directly to the allowed energy eigenvalues for the hydrogen atom: (7.13)

Here, Ry is the Rydberg constant introduced in Eq. (1.19) (see Example 7.1 in which the numerical value of Ry and a are calculated) and “n” the main quantum number that is restricted to the values n ≥ l + 1. The eigenvalues of the angle‐dependent part of the hydrogen atom Schrödinger equation are exactly the same as those for the rigid rotor, given by Eqs. (6.43) and (6.44): (6.43) (6.44) where the symbols for the quantum numbers have been changed from J and K to l and m, as pointed out in Section 7.1.

Like Eq. (6.39) that restricted the values of the J and K, the quantum numbers determining the eigenvalues, the hydrogen atom eigenfunctions are restricted as follows:

The solutions of Eq. (7.5) (7.5) then can be written as (7.14) where (6.40)

and Rnl(r) are the radial part of the LaGuerre polynomials given by (7.15)

A plot of the radial parts of the wavefunctions is shown in Figure 7.1a. Remember that these radial parts have spherical symmetry. Thus, in any direction from the nucleus, the radial wavefunction drops off as shown in Figure 7.1, and the wavefunction has a cusp at the nucleus as shown in Figure 7.1b. The first few spherical harmonic functions given (see Example 6.1) by (7.16)

are shown in Figure 7.2. The hydrogen atom wavefunctions are then given by the product of Eq. (7.15) and (7.16). Using the convention of designating l = 0 wavefunctions as “s,” l = 1 wavefunctions as “p,” and l = 2 as “d” orbitals, the complete wavefunctions for the hydrogen atom are obtained as follows:

(7.17)

Figure 7.1 (a) Plot of radial part of hydrogen wavefunctions in units of r/ao. (b) Cusp of the “s” orbital at the origin. Since the radial part of the wavefunction has spherical symmetry, the decrease in the wavefunction occurs in all directions from the origin.

The orbital energies and degeneracies due to the rules governing the allowed quantum numbers n, l, and ml are shown in Figure 7.3. The hydrogen wavefunctions presented in Eq. (7.17) form an orthonormal vector space, which is an important consideration for the later discussion of many‐electron systems such as molecules, where the molecular orbitals are approximated by linear combination of the hydrogen‐like wavefunctions (see Postulate 5 in Section 2.1). The hydrogen wavefunctions in Eq. (7.17) also hold for all one‐electron ions, such as He+, Li2+, etc., with the atomic number Z entering Eq. (7.1): (7.18)

Consequently, every occurrence of the expression needs to be changed to

in Eq. (7.17)

for all other one‐electron systems.

Accordingly, the ψ200 = ψ2s wavefunction, for example, reads (7.19) for a one‐electron ion with an atomic number Z.

Figure 7.2 Plot of first few spherical harmonic functions. Notice that the and the functions have the same shape but differ only in phase. The same holds for the functions. https://math.stackexchange.com/.

Modified

and the from

Source: Modified from Stack Exchange, plot of first few spherical harmonic functions.

Figure 7.3 Orbital energy eigenvalues and degeneracies for the hydrogen atom. Equation (7.17) and Figure 7.1 demonstrate that for “s” orbitals, the hydrogen wavefunctions have a maximum at r = 0, namely, at the cusp shown in Figure 7.1b. This may (erroneously) be construed as suggesting that the highest probability of finding the electron in a hydrogen 1s orbital is at the nucleus. This will be discussed next. The probability P of finding an electron in the 1s orbital in the volume element dτ, where is (7.20) is expressed as the square of the wavefunction times the volume element: (7.21)

Figure 7.4 Radial part of the wavefunctions (dashed lines) and radial distribution functions (solid lines) for 1s (black) and 2s (gray) orbitals. To determine the probability of the electron to be found in a thin spherical shell of thickness dr, independent of θ and ϕ, one needs to integrate Eq. (8.21) over θ and ϕ to obtain (7.22)

(7.23)

since the spherical harmonics are normalized. The expression r2 (R10)2 is called the radial distribution function and presents the probability of finding the electron in a shell with thickness r + dr, normalized with respect to the volume of the shell. At r = 0, the volume of this shell is zero; thus, the probability of finding the electron at the nucleus is zero. The argument presented here holds for all orbitals, not just the “s” orbitals. A plot of the radial distribution function is given in Figure 7.4, which shows that the 1s orbital has its maximum at the Bohr radius (see Example 7.2). The orbital shapes rendered in every chemistry textbook are created by drawing a (usually a 90 %) contour of the probability of finding the electron in this region of space. Although these orbital shapes have been predicted decades ago and have become an integrated part of many branches of chemistry, it was not until the last two decades that experimental verification of these shapes was achieved [2], although some of the interpretation of this work has to be taken with a grain of salt (see [3]). Example 7.1 Computation of numerical values and analysis of the units of the Bohr radius and the Rydberg constant from Eqs. (8.7) to (8.12) Answer: The Bohr radius a, as defined in Eq. (8.7), was just a number of constants. Here, we'll show that a, indeed, has units of inverse length and determine its numerical value: (7.7)

(E7.1.1)

(E7.1.3)

(E7.1.4) Now, for the Rydberg constant: (7.12)

(E7.1.5)

(E7.1.6)

(1.19) Thus, the quantum mechanically derived energy eigenvalues perfectly agree with experimental data from the hydrogen absorption/emission spectra. We shall now turn to another example, namely, the determination of the maximum of the radial distribution function that should, for the 1s orbital, just be the Bohr radius. This is accomplished as follows in Example 7.2:

Example 7.2 Calculation of the maximum probability of finding a 1s electron. Answer: The probability of finding the electron at any distance r is given by (E7.2.1)

Since the integration over the angle‐dependent part of the wavefunction gives a factor of 1 due to the normalization condition of the spherical harmonics, we may use here just the radial distribution function of the 1s electron, which is (E7.2.2)

To determine the maximum of the probability of finding the electron, we differentiate the expression in Eq. (E7.2.2) with respect to r and set the derivative to zero: (E7.2.3)

(E7.2.4) from which follows that r = a, as expected.

7.3 Dipole Allowed Transitions for the Hydrogen Atom Next, the selection rules that determine which transitions are allowed for the hydrogen atom will be discussed. As introduced before, an electric dipole transition is allowed if

(3.20) Equation (3.20) implies for the hydrogen atom: (7.24)

The selection rules for the angle‐dependent part of the wavefunction were calculated previously: (6.61) and are the same here. There are no restrictions to the changes in the main quantum number n: (7.25) Equation (7.25) can be proven rigorously by calculation of the transition moment using just the radial part of the wavefunctions. However, it can also be visualized easily by considering that the distance between nucleus and electron increases for increasing values of n. Such an increased distance between the nucleus and electron is, of course, a manifestation in the change of magnitude of the dipole moment. All changes in n are allowed and give rise to the different spectral series (Lyman, Balmer, Paschen, Brackett) that originate from the states n1, n2, n3, n4… with Δn = ± 1, ± 2, ± 3…. values, respectively. However, since the quantum number l has to change according to Eq. (6.61), the energy level diagram for the hydrogen atom transitions appears as shown in Figure 7.5

7.4 Discussion of the Hydrogen Atom Results At this point, it may be appropriate to pause and summarize what was learned so far in the discussion of the hydrogen atom.

Figure 7.5 Energy level diagram and allowed electronic transitions for the hydrogen atom, demonstrating the selection rules Δl = ± 1. The task of solving the hydrogen atom problem was attacked using the same approach that was described earlier for the particle in a box, the harmonic oscillator, and the rigid rotor: first, the Schrödinger equation was set up with the appropriate potential energy function. The differential equation was solved easily in the first scenario, but from the harmonic oscillator on, a power series expansion was required to solve the Hermite and Legendre differential equation. In both these cases, rather involved arguments were required to ensure that the resulting eigenfunctions were finite, which led to the requirement of truncating of the power series expansions. The recursion equation for the expansion coefficients of the power series led to the rather simple equations of the energy eigenvalues in both the harmonic oscillator and the rigid rotor. In the hydrogen atom, the power series expansion was required for both the radial and the angular part of the wavefunction, the latter of which was the same for the proton–electron system and the rigid rotor. Finally, substituting the energy eigenvalues into the trial solution of the differential equation resulted in the final form of the wavefunctions. The order of the progression presented in this book follows the complexity of the Schrödinger equation for each problem. When one excludes the pure model case – the particle in a box – one finds the validation of the approach by comparison of theory with the experimental results. The harmonic oscillator presents a single peak in the vibrational spectra, the rigid rotor a progression of peaks in the rotational spectra, and the hydrogen atom the hydrogen emission/absorption spectra, as predicted from the energy eigenvalues and selection rules. Small deviations between observed and the idealized models could readily be explained and corrected for, by next‐level approximations such as the anharmonic oscillator and the nonrigid rotor models. In the hydrogen atom, the quantum mechanical solutions match perfectly with the experimental results and support the quotation by Feynman about the necessity of agreement between experiment and

theory. Finally, the argument raised in Section 7.1 that the quantum mechanical results for the hydrogen atom brought order into chemistry (that is, the periodic system) is exemplified by Figure 7.3. Although not quite in its final form for many‐electron systems, this figure predicts the number of elements in each row of the periodic chart and, as we shall see later, contains the concept that certain elements will exhibit similar chemical traits. It is awe‐inspiring that the mathematical conditions that relate the quantum numbers, namely, the allowed values of l and m for each n‐level, dictate the appearance of and the order within the periodic chart and thereby the entire field of chemistry. Perhaps, the true genius of Schrödinger's work was the realization that the formalism presented in Sections 7.1–7.3 determines the structure of matter like no other discovery.

7.5 Electron Spin The discovery of electron spin is another chapter in the history of science that is fascinating. It emphasizes not only the connection between theory and experiment but also how the opinion of older, well‐established scientists can suppress new and ultimately correct ideas of young scientists proposing new concepts. This aspect is nicely demonstrated in two review articles on the discovery and theoretical formulation of electron spin [4, 5]. It was stated above that the quantum mechanical description of the hydrogen atom brought order into the field of chemistry in the mid‐ 1920, when this work was performed and published. However, it was again an unexplainable experimental observation that required refinements of this model. This experimental work was based on the observation of atomic emission spectra in the presence of a magnetic field that needs a short introduction. When the hydrogen atom emission spectrum was carried out in the presence of a static magnetic field, it was found that some of the spectral lines split into multiple lines or “multiplets.” This is known as the Zeeman effect that had been well‐established by the mid‐1920,

and at first, the splitting seemed to be in perfect agreement with the new quantum mechanical interpretation of the hydrogen atom, as can be seen from the following argument. The presence of a magnetic field lifts the degeneracy of the 2p, 3p, 3d, and other sublevels that have different “magnetic” quantum numbers ml, as shown in Figure 7.6a: p orbitals split into three and d orbitals into 5 levels, according to their ml value. The energy splitting of these sublevels is given by (7.26) where B is the magnetic field (in Tesla, T), “e” the charge, and me the mass of the electron. Furthermore, in the presence of a magnetic field, the selection rule stated above (Eq. [6.61]) needs to be augmented to read (7.27)

Figure 7.6 (a) Energy level diagram of the hydrogen atom orbitals in the presence of a magnetic field. (b) Energy level diagram for the sodium 589 nm doublet. See text for detail.

The energy level diagram shown in Figure 7.6a is obtained, and the Balmer series transition from 3d into the 2p orbitals are split into triplet with transition frequencies ν + Δν, ν, and ν − Δν, where ν is the frequency of the transition in the absence of a magnetic field and Δν is the splitting between the energy levels of different ml values (see Eq. [7.26]). This is shown by the vertical lines in Figure 7.6a. The splitting into three components is due to the selection rule Δl = ± 1 for the electron's angular momentum quantum number and Δml = 0, ± 1 for the magnetic quantum number. Thus, there are only three observed spectral lines although there are nine different transitions. However, in other atoms, the splitting pattern of spectral lines cannot be explained by the discussion above. A prime example for this was found in the so‐called sodium D line (at 589 nm) of the sodium emission spectrum, which results from a transition between the excited sodium state (1s2 2s2 2p6 3p1) to its ground state (1s2 2s2 2p6 3s1). The exact meaning of this notation – the atomic electron configuration – will be explained in Section 9.3. Since this transition occurs from a 3p state with threefold degeneracy to a 3s state with no degeneracy and because of the selection rule Δl = ± 1, two transition are observed, namely, the famous sodium doublet at c. 589 nm (see Figure 7.6b). This will be discussed in more detail using atomic term symbols in Chapter 9 (see Example 9.1). However, in the presence of a magnetic field, one of the sodium doublets splits into four and the other into six components. The only explanation for this “anomalous Zeeman” effect was that the electron undergoing this transition itself had two energy states. At the point in time when the anomalous Zeeman effect was first reported, the results of the Stern–Gerlach experiment were known, which also had suggested that electrons could have two inherent energy states. Since these energy states seemed to be caused by two different orientations of the electron's magnetic moment and since a rotating charge causes a magnetic moment, these two energy states were (somewhat misleadingly) referred to as “spin states” of the electron, implying that the electron was actually spinning about an axis. Thus, one may be tempted to visualize the two possible energy states as an electron spinning either clockwise or counterclockwise to create a magnetic

moment that aligns parallel or antiparallel to an external magnetic field. However, this classical view is strictly a visualization, and the electronic spin should be viewed as an inherent property of fundamental particles. Other approaches to quantum mechanics, noticeably those by Heisenberg and Dirac, directly predict the spin quantum number as a fourth required parameter to specify an electron in a hydrogen atom [5]. Since the electron exhibits two distinct energy states, there must be an operator that has these two eigenstates as solutions. This operator is referred to as the spin angular momentum operator. In analogy to the orbital angular momentum operator and its Cartesian components operator

, one defines a spin angular moment and its components

with the same

commutation properties as the orbital angular momentum operator (see Eqs. [6.20] and [6.21]).1 For electrons, the eigenvalues of the

operator are (7.28)

With the the allowed spin quantum number

, the eigenvalues

become (7.29) For the

operator, the eigenvalues are (7.30)

in analogy to m that can have values from −l to +l. The corresponding electron spin eigenfunctions are referred to as α and β such that

(7.31) Because of the commutation rules between the

and the

operator, the eigenfunctions described by Eq. (7.31) are eigenfunctions of the operator as well, with eigenvalues

(see

Eq. [7.29]). The consequences of the discussion in the last section are that the electronic wavefunctions for the hydrogen atom need to be augmented by a spin function, that is, (7.32) The electron spin results presented in this section do not change the outcome of the quantum mechanical treatment of the hydrogen atom that can be adequately described by three quantum numbers, but it directly affects any system containing more than one electron, both in atoms such as the He atom and up, as well as any molecules containing more than one electron, such as H2 and up. However, it is interesting to note that the relativistic quantum mechanical approach by Dirac includes the electron spin quantum number explicitly. The electron's spin states lead to the splitting of atomic spectra in a magnetic field. Again, experimental results led to the refinement of the earlier theories.

Figure 7.7 Spatial, or orientational quantization of the orbital angular momentum for (a) l = 1 and (b) for l = 2. Source: Goudsmit [4]; Commins [5].

As will be shown in Chapter 8 on nuclear magnetic resonance spectroscopy, nuclei themselves can have an intrinsic spin angular momentum, which however is much smaller than that of the electron. Thus, the interaction between nuclear and electronic spin can be ignored for the hydrogen atom.

7.6 Spatial Quantization of Angular Momentum Since the eigenvalues of the

operators cannot be

determined simultaneously, the exact location the eigenvalue of the operator, cannot be determined either, although its length is know exactly. This can be seen from the following argument. Consider a Cartesian coordinate system oriented along the axes of the operators, as shown in Figure 7.7a. Since the eigenvalue of

, namely, ±mħ, is known and

lies along the z‐axis, the |l| vector must lie on a cone of loci with radius l(l + 1) − m2 (in units of ħ). This follows directly from Pythagorean's theorem. In Figure 7.7, the possible values of m (−1, 0, +1) for l = 1 and (−2, −1, 0, 1, 2) for l = 2 are shown along the z‐axis. In the former case, the length of |l| is

and in the latter

. This

spatial quantization of the angular momentum will be encountered again in spin spectroscopy, such as electron paramagnetic or nuclear magnetic resonance (NMR) spectroscopy. The net magnetization encountered in NMR is a sum of the spin angular momenta of each nuclear spin “s” (see Eq. [7.26]), which can be depicted to precess along a conical pattern about the z‐direction that is the direction of the applied external magnetic field.

References 1 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon. 2 Wang, S.G. and Eugen Schwarz, W.H. (2000). On closed‐shell interactions, polar covalences, d‐shell holes, and direct images of orbitals: the case of cuprite. Angewandte Chemie International Edition 39 (10): 1757–1762. 3 Scerri, E.R. (2000). Have orbitals really been observed. Journal of Chemical Education 77 (11): 1492–1495. 4 Goudsmit, S.A. (1971). De ontdekking van de electronenrotatie. Nederlands Tijdschrift voor Natuurkunde 37: 386–394. 5 Commins, E.D. (2012). Electron spin and its history. Annual Review of Nuclear and Particle Science 62: 133–157.

Problems Qualitative Questions

1. What are the four quantum numbers of an electron in a hydrogen‐like orbital called? 2. With which variable of the hydrogen atom Schrödinger equation is each of the quantum numbers associated? 3. What are the allowed values for the four quantum numbers for an electron in a hydrogen‐like orbital, and what physical properties are designated by each of the four quantum numbers? 4. What is the degeneracy of the hydrogen orbitals with the quantum number l? 5. Which two parameters determine the size of a hydrogen‐like orbital in one‐electron systems? 6. The wavefunctions of all s orbitals all have peak amplitudes at the nucleus. What mathematical procedure was invoked to arrive at a spherical electron distribution that confirms the experimental observation of the electron found most likely at the Bohr radius? 7. Describe briefly the integral that had to be solved to answer question 6. Quantitative Questions. For the following problems, use . 1. Normalize the H‐atom 2s orbital, for which ψ200 = N (1−r/2ao) e−r/2ao. 2. In Example 7.2, the maximum of the radial distribution function for the 1s orbital was found to occur at the Bohr radius. Perform the same calculations for radial distribution function of the 2s orbital. This will result in a quartic equation that can be solved graphically. Discuss the results. 3. For the H‐atom 1s orbital, the maximum of the radial distribution function occurs at ao (see problem 2). Compare this

result with another measure of the electron's most probable location, the expectation value 〈r〉 of the electron's position. 4. Show that the H 1s and 2s wavefunctions, ψ100(r) and ψ200(r), are orthogonal.

Note 1 The qualitative discussion of the spin operators

and

and the

corresponding spin functions is sufficient at this point. However, a more detailed background of the spin operators and functions is given in Appendix 5.

8 Nuclear Magnetic Resonance (NMR) Spectroscopy 8.1 General Remarks There are two major spectroscopic techniques that are based on the observation of transitions due to reorientation of spins in a magnetic field when exposed to electromagnetic radiation. These are electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) spectroscopies. In the former, the transitions are between energy states that an unpaired electron spin experiences in an external magnetic field. Some principles of the electron spin were introduced in the previous chapter. NMR spectroscopy, on the other hand, is due to nuclear spin energy states that have not yet been introduced in previous discussions. Whereas EPR requires the presence of a radical molecule or ion (i.e. one with an unpaired electron), NMR is observed for any atom with a nonzero nuclear spin and has been developed mostly for 1H and 13C atoms. One can safely say that no spectroscopic method used in chemical and biochemical research has seen as explosive a growth over the past five decades as has NMR spectroscopy that has become a cornerstone for researchers for identification of molecules and to establish molecular structure in solution and in the solid state. A major reason for this growth was the introduction of pulse Fourier transform (FT) methodology in the 1970s (to be discussed in Appendix 4) over continuous excitation that was common before that time. The advent of FT methodology itself was the result of the availability of low cost and dedicated instrument control computers and software to perform the FT rapidly. One other advantage of pulse FT methodology is the ability to program pulse sequences that excite specific aspects of the magnetization of the sample that lead to several new forms of spin interactions that reveal structural details not available in continuous excitation. One further reason for the

growth of NMR spectroscopy was the available of high (magnetic) field superconducting magnets that enormously improved spectral resolution and sensitivity of NMR methods. In this chapter, the principles of spin spectroscopy will be introduced for 1H and 13C NMR spectroscopy. Many of the principles of 1H NMR can be applied to EPR spectroscopy that will not be discussed any further. Even 1H NMR will be treated at an introductory level only since entire volumes are necessary to treat all aspects of NMR spectroscopy.

8.2 Review of Electron Angular Momentum and Spin Angular Momentum Spin resonance spectroscopy deals with the “spin angular momentum” of an atomic nucleus or an electron and, therefore, with the “spin Hamiltonians.” In classical physics, the angular momentum L is used to describe the rotational kinetic energy of a spinning body: (6.1) where I, the moment of inertia, is defined as (6.5) Equation (6.1) relates angular momentum and moment of inertia just as the linear momentum and mass are related for linear motion: (1.17) The angular momentum is defined in classical physics as (6.2)

which is shown in Figure 8.1a. Equation (6.2) leads to the classical definition of the Cartesian components of L according to (6.3) with (6.4) and similar expressions holding for the other components of L. The quantum mechanical expression for the component of the angular momentum operator given in Eq. (6.4) then becomes (6.19)

Figure 8.1 (a) Definition of the angular momentum in terms of radius r and linear momentum p. (b, c) Depiction of spin angular momentum eigenvalues of the operator (solid arrows) and operator (dashed arrows) for spin wavefunctions α (panel B) and β (panel C). As discussed before, the Cartesian components

,

, and

do

not commutate, and, therefore, their eigenvalues cannot be determined simultaneously. They do commutate, however, with the operator for the total angular momentum, which has eigenvalues (8.1)

The eigenvalues of one of the Cartesian components (usually

) are

given by (8.2) with allowed values of K between –J and +J. As we have seen, the quantization of the orbital angular momentum, or “spatial quantization” (see Section 7.6), resulting from Eq. (8.1) has a major influence on the quantum mechanical treatment of the H atom. Furthermore, we saw that an electron has its own angular momentum, referred to as its spin angular momentum. Incidentally, the interaction of orbital and spin angular momentum determines the energetics and spectral behavior of multi‐electron systems, as we shall see in Chapter 9. We now return to the discussion of the intrinsic spin angular momentum (see also Appendix 5). For an electron we defined the spin angular momentum operator and its components and with the same commutation properties as the orbital angular momentum operator

and its components

,

, and

, that is, (8.3)

In analogy to Eq. (6.43) (6.43) for the total angular momentum Schrödinger equation, one writes the eigenfunctions of the operator as (8.4) Here, α and β are the spin wavefunctions (see below), and the eigenvalues are

(8.5) or (8.6) For electrons, the spin quantum number only can assume the value of (8.7) Thus, the eigenvalues of the are

operator for an electron are (8.8)

For the

operator, the eigenfunctions are (8.9)

with eigenvalues (8.10) since the quantum number m can have values from −s to +s, in analogy to Eq. (8.2). The corresponding electron spin eigenfunctions are referred to as β and α, respectively. Because of the commutation rules between the operator and the operator, the eigenfunctions described by Eq. (8.9) are eigenfunctions of the operator as well. Often, the “spin” is described in terms of the electron spinning around an axis, similar to the earth spinning around its axis. However, this view is a coarse simplification and does not imply that the electron or nucleus really “spins.” The spin is an inherent quantity of an electron that implies that it has an angular momentum.

However, the idea of the electron actually “spinning” produces the correct consequence, namely, that the electron (or a nucleus) has a magnetic moment, that is, it may behave like a small permanent magnet. In the absence of a magnetic field, the energy states of the spin functions given in Eq. (8.9) are indistinguishable (degenerate). However, if an electron is placed in a magnetic field – either an externally applied magnetic field or the magnetic field of another electron or nucleus – the energy states of the spin momentum split into “parallel” and “antiparallel” spin states. When exposed to electromagnetic radiation of the appropriate energy, transitions between these spin states may occur. This gives rise to a spectroscopic technique referred to as “electron paramagnetic resonance,” also sometimes known as “electron spin resonance” (ESR). EPR signals can only be observed for molecular or atomic systems that have unpaired electronic spin, that is, for “radical” atomic or molecular systems. Although EPR is a mature and very useful technique, many of its principles are transferable from proton magnetic resonance spectroscopy, and it will not be covered here in favor of “nuclear” magnetic resonance spectroscopy, which is one of the most widely applied spectroscopic methods in chemical research.

8.3 Nuclear Spin What was elaborated upon in Eqs. (8.3)–(8.10) for electrons holds equally well for any nucleus that has a nuclear spin quantum number , such as 1H and 13C. The concept of nuclear spins arises from the fact that nucleons themselves have an – albeit much smaller – intrinsic spin moment. In atomic nuclei that have even numbers of protons and neutrons, the spin moment is zero. Examples for such nuclei are 12C and 16O. If the number of protons and neutrons in a nucleus are both odd, the spin quantum number can be either of the following: (8.11) For example, 14N has a nuclear spin quantum number of 1, whereas for 10B, this value is 3.

Finally, if the sum of protons and neutrons is odd, such as in 1H, 13C, and 19F, the spin quantum number can be any odd multiple of : (8.12) 1H, 13C, 17O

and 19F all have nuclear spin quantum number of , whereas

has a spin of

All nuclei with a spin quantum number s > 0 have a magnetic moment or create a magnetic moment along an axis. The magnetic moment is proportional to the total nuclear angular momentum S (see Eq. [8.6]) and is given by (8.13) In Eq. (8.11), e is the electronic charge unit, mP is the mass of the proton (see Appendix 1), and gN is the unitless “nuclear g‐factor” (see below). The expression

(see Appendix 1) also is referred to as the

“nuclear magneton” and has the value (8.14) (see Example 8.2 for units and unit conversions). Here, the magnetic field strength is given in units of T (tesla). Thus, Eq. (8.13) can be rewritten as (8.15) In Eq. (8.15), γ is the magnetogyric ratio defined by (8.16)

The magnetogyric ratio is the ratio of a nucleus' magnetic moment to its angular momentum, cf. Eq. (8.16). Its units (and its name) suggest that it is an indication of how fast a nucleus spins (in rad/s) for a given magnetic field. Equation (8.15) shows that the nuclear g‐factor is just a nucleus‐ specific proportionality constant between the observed magnetic moment and the nuclear angular momentum S. The value of gN for the proton is 5.5854. Values of other nuclei (with spin of ½) commonly used in NMR spectroscopy are listed in Table 8.1, along with the magnetogyric ratio and magnetic moments, expressed in units of nuclear magnetons. The quantities in this table are related to each other by Eqs. (8.15) and (8.16). When a magnetic moment μ interacts with a static magnetic field B0 (also known as the magnetic flux density) along the z‐direction, the energy of interaction is given by (8.17) Table 8.1 Nuclear g‐factors, magnetogyric ratios, and spin moments for some spin ½ nuclei. gN

γ [rad/(T s)] μ [units of βN]

1H

5.5854 26.752 × 107

2.7927

13C

1.4048 6.728 × 107

0.7024

19F

5.2537 25.162 × 107

2.6289

Combining Eqs. (8.15) and (8.17) yields (8.18) For a proton the value for s is given by

(see Eq. [8.10]). Thus,

the energy difference between the α and β states of a proton in a magnetic field along the z‐axis is given by

(8.19) Example 8.1 Calculation of the proton resonant frequencies and energy differences for the field strength achievable a) with a room temperature electromagnet (B0 = 2.34 T) and b) a very high field superconducting magnet (B0 = 21.1 T) Answer: (E8.1.1)

(E8.1.2)

(E8.1.3) (E8.1.4) (E8.1.5) Case a) corresponds to a “100 MHz NMR” spectrometer that can be operated with a room temperature electromagnet. Case b) corresponds to a high field, commercially available “900 MHz” machine that require a liquid helium cooled, superconducting magnet Example 8.2 Analysis of the units of the magnetic moment: (8.13)

where

and gN and S are unitless.

Thus, the magnetic moment μ has units of (E8.2.1) The unit conversion

in Eq. (E8.2.1) results from the

Lorentz force law that states that a particle that carries a charge of 1 coulomb and passes through a magnetic field of 1 T at a speed of 1m per second experiences a force of 1 N. Equation (8.18) leads to a very plausible explanation of NMR spectroscopy. The two nuclear spin states α or β of a proton in a magnetic field lead to two energy states that we can visualize as the two spins magnetic moments being either parallel or antiparallel to the external field (more on this spin alignment below). If electromagnetic radiation with energy equal to the energy difference between the two spin states impinges on the sample, the spin function changes from α to β, photons of this frequency are absorbed, and an NMR signal is generated. This is shown schematically in Figure 8.2. This view treats the NMR effect strictly as an electric dipole‐mediated transition, and, therefore, electric dipole selection rules apply (see below). However, we shall see that a different interpretation for NMR is possible (Section 8.4.3). For a more detailed quantum mechanical representation of nuclear spin, the reader is referred to the literature [1].

Figure 8.2 Energy of the α and β proton nuclear spin states as a function of the external magnetic field. The vertical lines indicate the transition frequencies (100, 400, and 900 MHz) for common commercial magnets.

8.4 Selection Rules, Transition Energies, Magnetization, and Spin State Population 8.4.1 Electric Dipole Selection Rules for a One‐Spin Nuclear System In keeping with the approach followed so far in this book, the next step in discussing allowed interactions between nuclear spin systems and electromagnetic radiation is to consider the selection rules for a single proton spin to undergo a transition between spin states. Interestingly, in many descriptions of NMR spectroscopy, this aspect is not discussed, and it is tacitly assumed that these transitions are always electric dipole allowed. As we know from the discussion of molecular rotation (see Chapter 6), the selection rules for the total angular momentum operator and its component along the z‐axis, , respectively, are (6.53) and

using the nomenclature of atomic systems, rather than that of molecular rotations. The corresponding operators for nuclear spins are spin quantum number of

and

, with a

(8.7) The two spin states α or β can have values between –s and +s or With Δs = ± 1 for a one‐spin system, the transition between the two spin states α or β is allowed. For spin systems

consisting of more than one spin, different selection rules will hold [2].

8.4.2 Transition Energies From Eq. (8.19), it follows that the allowed transition between states α and β requires a photon of frequency (8.20) This equation is novel with respect to the fact that the molecular energy levels are no longer molecule‐specific quantities (such as a rotational, vibrational, or electronic energy levels discussed in previous chapters) but depend on the strength of the external magnetic field. Therefore, NMR can, in principle, be carried out at a fixed magnetic field by scanning the photon energies or at fixed photon energies by scanning the magnetic field until the resonance condition is reached. Before the advent of pulse FT NMR spectroscopy, nearly all NMR experiments were performed at a fixed photon frequency by fine‐tuning the magnetic field. The reasons for that will become clear from the following consideration. In NMR spectroscopy, one of the desired prime pieces of experimental information is a quantity known as the “chemical shift” (see Section 8.5) that expresses how the electron density around a nucleus shields the nuclear spin from the external magnetic field. This shielding effect is quite small; consequently, the absorption frequencies of differently shielded protons in a molecule may differ only by a few parts per million. For an NMR experiment carried out at a magnetic field of 2.34 T (see Example 8.1), the resonance frequencies for a typical organic molecule may differ by about 1000 Hz due to the different shielding the proton nuclei experience. Technically, it is very difficult to build a radio frequency (RF) generator that can be tuned from 100 000 000 to 100 001 000 Hz (for a 100 MHz NMR spectrometer) with sufficiently low bandwidth and high‐frequency accuracy. Thus, NMR spectrometers, in the past, used a fixed RF signal and varied the magnetic field

slightly to achieve the resonance conditions for differently shielded nuclear spins. With pulse FT NMR spectrometers, the observation of the signal proceeds quite differently as will be discussed in Section 8.7.

8.4.3 Magnetization Similar to the electric field of electromagnetic radiation exerting a force on electrically charged particles (see Eq. [3.3]), the magnetic field will exert a torque on the individual spin magnetic moments. The effect of this torque will be an alignment of the magnetic moments to assume a minimum energy configuration. The resulting alignment of the magnetic moments will cause a macroscopic magnetization. This magnetization can be viewed from the principles of nuclear spins introduced earlier. Since only one Cartesian component of the spin angular momentum, , and the square of the total spin angular momentum, can be determined simultaneously, the direction of the eigenvalues of the in the external magnetic field cannot be determined. Therefore, a situation similar to the spatial quantization discussed in Section 7.6 will occur, namely, that the total spin angular moment lies on a cone about the z‐axis and “precesses” around the central axis of the cone, as shown in Figure 8.1b and c, at a frequency known as the “Larmor” frequency. The sum of the magnetic moments of all individual nuclear spins is referred to as the magnetization. When electromagnetic radiation (in form of a broadband radio‐frequency pulse) is applied to a system of aligned spins, the direction of this magnetization changes, as will be discussed in Section 8.7. After the perturbation due to the electromagnetic field ends, the spins will relax into the original state, and energy will be released in this process. This “free induction decay (FID)” is the signal picked up in pulse FT NMR spectroscopy. It is interesting to note that NMR spectroscopy can be described in two different ways, namely, at a molecular levels where electromagnetic radiation provides the energy to cause a transition between different energy states of individual spins or, in terms of a

macroscopic quantity, the net magnetization. In this respect, NMR spectroscopy is similar to some of the nonlinear optical effects discussed in Appendix 3. The hyper Raman effect, for example, can be described either by the hyperpolarizability (on the microscopic or molecular level) or the second‐order dielectric susceptibility (on the “bulk” level). The corresponding macroscopic magnetization will be used later in more detail in the discussion of pulse FT NMR spectroscopy (Section 8.7.2).

8.4.4 Spin State Population Analysis As we have seen in Example 8.1, the energy difference between the α and β spin states is much smaller than the energy differences between vibrational or rotational energy states discussed before and only amounts to about 10−25 J, or about 0.01 cm−1. In Example 8.3, the population ratios for energy differences in two‐state systems with energy differences of 1000, 1, and 0.01 cm−1 is calculated at room temperature. The results represent population ratios typical for vibrational, rotational, and NMR spectroscopies (see also Example E3.1). Example 8.3 Calculation of population differences for typical vibrational (ΔE = 1000 cm−1), rotational (ΔE = 1 cm−1), and nuclear spin energy states (ΔE = 0.01 cm−1) at room temperature and at −100 °C. Use R = 0.698 [cm−1 K−1 molecule−1]: Answer:

For the vibrational energy difference, the population of the excited state is only 0.82 % or 0.025 % at room temperature or −100 °C, respectively; thus, there is no danger of ever saturating the excited state, particularly, since the lifetime of the excited state is very short (see Section 5.3.2). In rotational spectroscopy, the population of the two states is much closer to unity, particularly at room temperature. In NMR spectroscopy, the two spin states have nearly the same population, leading to the possibility of saturation (equalization) of the two energy states. This is particular so since the spin states have a much longer deactivation, or relaxation, times (0.1 s and higher) than the other states discussed above. Working with magnetic fields as high as possible and at temperatures below room temperature reduces the saturation effect.

8.5 Chemical Shift In Eqs. (8.19) and (8.20) and Example 8.1, the energy states of an isolated proton spin in a magnetic field B0 were defined as (8.19) However, in chemistry and biochemistry, isolated protons are rather uncommon, and hydrogen atoms either are attached to a molecule by a covalent bond or, even in the case of a H+ cation in an aqueous environment, are surrounded by solvent molecules and, therefore, electrons. In response to an external magnetic field, the electrons

surrounding all nuclei create a secondary magnetic field that opposes the external magnetic field, which is known as the diamagnetic response. This induced magnetic field, Bind, is proportional to the external magnetic field (8.21) where σ is a shielding constant. Thus, the local field at a nucleus under the shielding effect of the electronic distribution is given by (8.22) Accordingly, Eq. (8.20) needs to be modified: (8.23) The frequency shift Δν exhibited by a nuclear spin then can be expressed as (8.24) The resulting frequency shift is generally reported as a “chemical shift” δ: (8.25) where νref is a reference signal due to an internal standard added to the sample (see below). Chemical shifts depend on the local electron density and present an exquisitely sensitive probe of the chemical environment (i.e. the electronic distribution) around a nucleus. Groups with high electronegativity adjacent to a given proton will reduce the shielding, thereby increasing the chemical shift. In contrast, electron‐donating groups will increase the electron density around a nucleus, thereby decreasing the chemical shift. In proton NMR spectroscopy, the

internal standard used most often is tetramethylsilane, Si(CH3)4, abbreviated as TMS, and chemical shifts are generally reported in ppm downfield from the TMS proton signal. The protons in methyl groups in organic compounds have chemical shifts very similar to that of TMS and, therefore, appear between 0.5 and 1 ppm from TMS. Protons adjacent to carbon–carbon double and triple bonds show higher chemical shifts, and protons adjacent to ketones exhibit chemical shift around 10 ppm. Aside from the different signals created by the shielding/deshielding effect of adjacent groups, another effect provides even more information on the local surrounding of a probe nucleus. This effect is due to the local magnetic fields created by the nuclear spin of adjacent nuclei. This effect will be discussed next.

8.6 Multispin Systems 8.6.1 Noninteracting Spins Let us consider two individual nuclei 1 and 2 in a molecule that experience a different chemical environment and, therefore, have different chemical shifts. Such nuclei are said to be nonequivalent. The magnetic field perturbation due to other nuclei shows up in the NMR spectrum when the nuclei are nonequivalent and if the distance between nonequivalent nuclei is less than or equal to three bond lengths. From Eqs. (8.19),

and (Eq. [8.22]), Blocal = (1

− σ) B0, we can write the Hamiltonian for a system of two distinguishable, noninteracting spins, labeled 1 and 2, as follows [3]: (8.26) The eigenfunctions for the two‐spin system are the products of the eigenfunctions of the individual spin operators and . There are four possibilities to write these product functions:

(8.27) With these eigenfunctions, the following energy eigenvalues are obtained (see Example 8.4): (8.28)

Example 8.4 Find eigenvalues of the operator for the wavefunctions ψ1 = α(1)α(2) and ψ2 = β(1)α(2): Answer: a. For ψ1 one obtains

(E8.4.1) For the two‐spin wavefunction ψ1 = α(1)α(2) (see Eq. [8.9]), and . Thus,

(E8.4.2)

b. For ψ2 one obtains

(E8.4.3) For the two‐spin wavefunction ψ2 = β(1)α(2) (see Eq. [8.9]), and . Thus,

(E8.4.4)

Figure 8.3 (a) Energy level diagram for two noninteracting spins with shielding constants σ1 and σ2. Energies are given in units of ħ γ B0 (cf. Eq. [8.28]). (b) Simulated NMR spectrum for two noninteracting spins. These energy levels are depicted in Figure 8.3a. The selection rule (see Section 8.4.1) Δs = ± 1 implies that only one of the two spins can change in a transition. Thus, four transitions can occur, with the following frequencies: (8.29) and (8.30) These transitions are indicated by the gray up arrows in Figure 8.3a. Since two each of the transition frequencies are the same, only two spectral peaks are observed, as shown in Figure 8.3b. These peaks are split by (σ1 − σ2).

8.6.2 Interacting Spins: Spin–Spin Coupling

For two noninteracting spins, as we have seen in the previous section, the NMR spectrum would show two peaks at (1 − σ1) and (1 − σ2), in accordance with Eq. (8.24). This rather unexciting result would seriously undermine the importance of NMR spectroscopy, if the two spins would just produce two signals at their individual chemical shift. Fortunately, for the structural sensitivity of NMR spectroscopy, spins in close proximity to each other interact via spin–spin coupling. There are two mechanisms for such coupling to occur: “through‐space” and “through‐bond” (also referred to as scalar) coupling. Of those, the former is influencing solid‐state NMR, whereas the latter mechanism is of major importance for the assignment of peaks in NMR studies in liquids and for structural analysis. This effect is also called J‐coupling and is described as follows. Since each spin acts like a magnet, it will affect neighboring spins by an interaction such that the Hamiltonian given in Eq. (8.26) needs to be modified to take into account this interaction: (8.31) In Eq. (8.31), J12 is known as the coupling constant that indicates the strength of interaction between the spins. This interaction acts over relatively short distances and drops off rapidly with distance. Thus, spin–spin coupling is observed mostly between spins separated by less than 3 or 4 bonds. This perturbed Hamiltonian is solved, using the perturbation methodology described in Appendix 2, using the eigenfunctions of the unperturbed Hamiltonian (cf. Eqs. [8.26]–[8.28]) to compute the perturbation energies ΔE (to the unperturbed wavefunctions) as follows: (8.32)

where m1 and m2 have the values

for the spin states α and

β, respectively. The transition frequencies listed above for the noninteracting spins (8.29) and (8.30) are modified to give (8.33)

This results in a quartet of peaks as shown in Figure 8.4 for two cases of external magnetic fields. As can be seen from Figure 8.4 and Eq. (8.33), the splitting between the nuclei with different chemical shift depends on the external magnetic field, whereas the splitting to the spin–spin interaction depends only on J12. Spin–spin coupling produces distinct spectral patterns in NMR spectra if more than two spins interact. This splitting pattern aids in the assignment of peaks and is taught in course in organic chemistry and spectral identifications. These splitting patterns can be summarized as follows.

8.6.3 Interaction of Multiple Spins

For systems containing multiple interacting spins, a generalized form of Eqs. (8.30) and (8.32) can be written for protons: (8.34)

Figure 8.4 Spectral pattern observed for two interacting spins at lower (a) and higher (b) external magnetic field. The centers of the doublets in each case are split by (σ1 − σ2).Notice that the J‐coupling does not depend on the external magnetic field. In this equation, σA is the shift for nucleus A; mA and mX, as before, are the spin quantum numbers that have the values

for

the spin states α and β; and JAX is the spin–spin coupling constant

between spins A and X, where X denotes the number of additional spins that interact with spin A. Equation (8.3) collapses to Eq. (8.33) in the case of two spins, and the splitting pattern of four peaks with a 1:1:1:1 intensity ratio is obtained (Figure 8.4). For the case of two identical spins X interacting with spin A (an AXX system), the first spin interacting with A creates two states with a splitting JAX. Further interaction of the two resulting states creates four states, two of which are degenerate. This leads to a triplet of peaks, all spaced at JAX, with a 1:2:1 intensity pattern. This is shown in Figure 8.5a. By the same token, we can predict that an AXXX spin system will create a quartet of peaks, all spaced at JAX, with an intensity pattern of 1:3:3:1, as shown in Figure 8.5b. This explains, for example, the observed spectrum of an ethyl (–CH2CH3) group. Here, the methyl protons exhibit a chemical shift of about 1 ppm from TMS, and the methylene protons a higher shift, between 2 and 4 ppm, depending on the exact chemical environment. The two methylene protons split the three equivalent methyl protons into a triplet with 1:2:1 intensity pattern according to the AXX coupling scheme, whereas the three methyl protons split the two equivalent methylene protons into a quartet with 1:3:3:1 intensity pattern, according to the AXXX coupling scheme [2].

Figure 8.5 Spin–spin coupling patterns for (a) JAXX and (b) JAXXX spin systems. See text for details.

8.7 Pulse FT NMR Spectroscopy 8.7.1 General Comments In this text book, experimental approaches generally are neglected in favor of the theory of the spectroscopic method. However, here in the case of NMR spectroscopy, a few experimental details will be presented at a rather introductory level in order to point out how the experimental method development has spawned entirely new methods in NMR spectroscopy that were either impossible or impractical to carry out before the methodology known as “pulse FT NMR spectroscopy” was developed. As indicated toward the end of Section 8.4.2, most modern NMR experiments are being carried out in a manner quite differently from the “conventional NMR” methods that were prevalent before the 1980s. In these conventional methods, the sample was illuminating by monochromatic electromagnetic radiation in the RF range (typically 100 MHz) while the magnetic field was varied by small amounts to achieve the resonance condition. Conversely, one could have kept the magnetic field constant and scanned the radio frequencies over an appropriate range, but the former method proved to be more feasible. The disadvantage of the “conventional NMR” described above is that only one spectral element is measured at a given time, where a spectral element can be visualized at a small band of chemical shift frequencies. In order to collect a spectrum over a band of 10 ppm from the TMS signal at a spectral resolution of 0.1 ppm, more than 100 spectral elements would need to be collected. However, in FT methods, all the spectral elements are collected at the same time via a relaxation phenomenon known as the free induction decay (see below). The FID represents the same signal as observed in conventional NMR experiments but is collected in a different data domain. By performing an operation known as a Fourier transform on the FID, the same spectrum is obtained as in the conventional NMR experiment. However, the FT methodology affords enormous time savings in spectral data acquisition or in a substantial increase of the data's signal‐to‐noise (S/N) ratio.

In this respect, pulse FT NMR spectroscopy is similar to FT‐IR spectroscopy, where an interference pattern (an “interferogram”) of all infrared frequency bands is collected simultaneously. This interference pattern again represents a different data domain, and the infrared spectrum is obtained by FT of the interferogram. Both FT techniques require that a computer is interfaced to the spectrometers to render the data in the form researchers are accustomed to see. Details of the actual mathematical procedure known as Fourier transform are presented in Appendix 4.

8.7.2 Description of NMR Event in Terms of the “Net Magnetization” In the presence of an external magnetic field, the nuclear spins experience a torque and align such that the z component of the spin aligns with the external field (remember that the x and y component cannot be determined and lie on a cone as shown in Figure 8.1b and c). Thus, there is no magnetization in these two directions, and all magnetization is along the z‐direction. This magnetization is referred to as MZ. When such a system of spins is exposed to RF electromagnetic radiation along the y‐direction of the appropriate energy, spins can flip from the lower energy α state into the β state. In this process it is possible to saturate the spin system, i.e. equalize the population of the α and β states, and reduce the magnetization along the z‐ direction to zero (MZ = 0). Remember that the excess of the α state over the β state is small to begin with, as was demonstrated in Example 8.3. An RF field that achieves the net magnetization to become zero is known as a 90° pulse. After the perturbation ceases, the relaxation back into the lowest energy state occurs with the magnetic vector leaving the x–y plane and gyrating back into the z‐direction. This relaxation process follows Eq. (8.35):

(8.35) In Eq. (8.35), M0 is the equilibrium net magnetization, and T1 is referred to as the spin–lattice relaxation. In this process, energy is released since the system is relaxing back into its lowest energy state. In order to describe this energy emission, one must consider that the magnetic moment, after absorption of the RF pulse, still precesses in the x–y plane. Thus, when the magnetization vector flips back into the z‐direction, it does so by its tip describing a spiral pathway as shown in Figure 8.6a. This process of reorientation of the magnetization can be viewed in two ways. If one views this situation in a framework of laboratory coordinates, the magnetization will describe a spiral pathway. However, if one describes this movement of the magnetization vector in a coordinate system that spins about the z‐axis at the Larmor frequency, the movement of the magnetization vector can be described by a simple arc in the y–z plane from the x–y plane into the direction of the z‐axis, as shown in Figure 8.6b. A detector mounted along the x‐direction detects a periodic signal due to the energy loss of the system, known as the free induction decay. This periodic signal contains all the dephasing information of all the spins of the system. Therefore, it is a superposition of all the spin frequencies, as shown in Figure 8.7. Consequently, an FT needs to be carried out to extract individual frequency components. The principles of Fourier transformation and the fast FT will be elaborated upon in Appendix 4.

Figure 8.6 Reorientation of magnetization vector following a 90° pulse, viewed in a laboratory coordinate system (A) and in a coordinate system rotating about the z‐axis.

Figure 8.7 (a) Simulated “free induction decay” (FID) and Fourier transformed signal (b).

References 1 Cavanagh, J. (2007). Theoretical description of NMR spectroscopy. In: Protein NMR Spectroscopy (eds. J. Cavanagh et al.). New York: Academic Press.

2 Keeler, J. (2010). Understanding NMR Spectroscopy, 2e. Chichester, UK: Wiley. 3 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall.

Problems 1. Why is the J‐coupling in an NMR experiment independent of the overall field strength? 2. In NMR spectroscopy, two different effects are observed: chemical shift and spin–spin coupling. a. Discuss the physical phenomena responsible for these effects. b. Which of these effects depends on the magnitude of the applied magnetic field? c. Why is it advantageous to perform NMR spectroscopy at high magnetic fields and low temperature? 3. Why is the NMR signal reported with respect to an internal standard and not in absolute photon energies as in most other spectroscopic techniques? 4. In a 500 MHz proton NMR instrument, what is the shift, in Hz, experienced by a proton whose signal is observed at 8.0 ppm from TMS? 5. Repeat the calculations performed in Example 8.4, namely, to find the eigenvalues of the two‐spin Hamiltonian: for the remaining two wavefunctions ψ3 = α(1)β(2) ψ4 = β(1)β(2). NO PEEKING in Example 8.4! 6. Propionic acid has a pKa value of about 4.9. In a 0.1 M aqueous solution, one mostly will find the propionate anion. Predict the 1H NMR spectrum of the propionate anion.

9 Atomic Structure: Multi‐electron Systems 9.1 The Two‐electron Hamiltonian, Shielding, and Effective Nuclear Charge In analogy to the hydrogen atom, the Schrödinger equation for the first atom containing more than one electron, helium, is written as: (9.1)

or (9.2) In Eq. (9.1), the first two terms in the bracket are the kinetic energy expressions for electrons 1 and 2; the third and fourth terms are the attractive forces between the helium nucleus with nuclear charge +2 and electrons 1 and 2, and the fifth term is the electron–electron repulsion. This equation cannot be solved analytically because each electron's wavefunction depends simultaneously on r1, r2, and r12. Thus, one has to resort to an approximate method, known as the orbital approximation, in which the total electronic wavefunction of a multi‐electron atom is written as the product of independent one‐ electron wavefunctions. For helium, the orbital approximation can be formulated as (9.3)

That is, one assumes that the total wavefunction of multi‐electron atom can be written just as the product of each electron's individual wavefunction. This approach ignores the electron correlation, which is basically an attempt of the electrons to stay out of each other's way by coordinating their motion. Instead, one assumes that each electron experiences the presence of other electrons just by an “effective nuclear charge ζ"due to the time‐averaged position of all other electrons. The effective nuclear charge the 2nd electron experiences in the He atom is 1.688, indicating that the first electron shields the nucleus to such a degree that the second electron does not perceive a nuclear charge of 2.0, but 1.688 charge units. For the Li atom, the effective nuclear charges are 2.691 for the 1s and 1.279 for the 2s electrons [1]. Thus, the wavefunction of the 1s orbital, in the presence of another electron, is no longer expressed as (7.17) but as (9.4)

where ζ is the shielded nuclear potential (see below). Similarly, the orbital energies for a multi‐electron atom are no longer given by Eq. 7.12 (7.12)

but by an equation that includes the shielded nuclear potential: (9.5)

In order to describe a multi‐electron atom, the so‐called Hartree– Fock self‐consistent method is used. In it, each electron is described by the orbital approximation (Eq. (9.3)), in which the electron correlation is ignored and approximated by an averaged position of all other electrons expressed via the effective potential, Veff that is calculated by averaging the inner electron wavefunctions over all angular coordinates. Each of the one‐electron wavefunctions in Eq. (9.3) has the form (9.6) where the effective nuclear charge experienced by each electron depends on the attraction between the nucleus and electron, and the shielding by the other electrons. Hartree–Fock calculations are carried out by minimizing the energy of the multi‐electron system by numerically varying ζ for the hydrogen‐like wavefunctions. Furthermore, the spin properties of all electrons in the atom need to be considered in a multi‐electron atom, according to the Pauli principle. Thus, the total Hamiltonian is written in the form of an antisymmetrized wavefunctions φj(rj), given by the Slater determinant (see next Section).

9.2 The Pauli Principle When writing the total wavefunction for multi‐electron atom, the electron spin wavefunctions need to be included. For the He atom, for example, the assumption was made that the second electron is in the same 1s orbital as the first electron, but has an opposite spin function, as compared to the first one. This is achieved by rewriting the total electronic wavefunction (Eq. (9.3) of the He atom by including the spin functions, as follows:

(9.7) which often is abbreviated to (9.8) In the description so far, it appeared that we can put labels on the electrons and call one of them “electron1” and the other “electron2.” However, since electrons are indistinguishable, Eq. (9.8), as presented above, is incorrect since it would imply that “electron2” has the spin function β. Instead, we need to present Eq. (9.8) in such a way that both electrons can have either spin functions. This is accomplished by writing the total wavefunction as a superposition of both possibilities (9.9) Equation (9.9) often is further abbreviated, for simplicity of notation, as (9.10) The Pauli principle (see Postulate 7 in Chapter 2) states that the antisymmetric combination (9.11) is the correct way to write the superposition of the two states, and implies that the wavefunction changes sign when electrons 1 and 2 are exchanged. Equation (9.11) can be represented as a normalized determinant known as the Slater determinant: (9.12)

For the Li atom, for example, the Slater determinant assumes the form

(9.13)

Here, we have written all possibilities to have two electrons in the 1s and 1 electron in the 2s orbital, with all different permutations of the spin functions. Since a determinant is zero if two rows or two columns are equal, this formulation accounts for the Pauli principle: the wavefunction will be zero (forbidden) if any two electrons have identically the same set of four quantum numbers n, l, m, and ms. This formalism is included in the Hartree–Fock method in that Eq. (9.6) needs to be reformulated to include the normalized, antisymmetric Slater determinant of the spin wavefunctions. This method then produces the best one‐electron approximation for the orbital energies in multi‐electron atoms and yields orbital energies that are summarized in Figure 9.1

9.3 The Aufbau Principle The energy eigenvalues of the orbitals resulting from the reduced nuclear charges expressed in Eq. (9.6), combined with the results from the discussion of the Pauli principle, lead to a revised energy level diagram for multi‐electron atom, as compared to Figure 7.3. This new energy level diagram is shown in Figure 9.1. It is responsible for the shape of the periodic chart, in particular, the existence of main group elements, the transition metals, and the lanthanides and actinides.

Figure 9.1 Energy level diagram of multi‐electron atoms, explaining the Aufbau principle, and the form of the periodic chart. In the periodic chart, elements in a given row are listed in order of increasing atomic number, Z. The number of electrons in a neutral atom also equals Z. These electrons for each atom are filled in the orbitals according to the orbital energy scheme shown in Figure 9.1, starting with the lowest energy orbitals, and two electrons with opposite spin quantum numbers to each orbital (according to the Pauli principle discussed above). The occupation of electrons in orbitals is abbreviated by the following example for the element carbon: 1s2 2s2 2p2, indicating that there are two electrons in the 1s orbital, two electrons in the 2s orbital, and 2 unpaired electrons in two 2p orbitals (Hund's rule, see below). In Figure 9.1, the energy of the 4s orbital is lower than that of the 3d orbitals. Consequently, the order of filling the orbitals, as Z increases from element to element, is 1s, 2s, 2p, 3s, 3p, 4s, 3d,…, and not 1s, 2s, 2p, 3s, 3p, 3d, 4s,…. This accounts for the form of the periodic chart

with the transition metal elements (Sc through Zn) inserted into the fourth row between Ca and Ga. This is repeatesd for the 14 lanthanide elements that are inserted after the 6s2 5d1 configuration is reached. If orbitals are degenerate, such as the 2p orbital, electrons are filled with parallel spins into l = 1 orbitals with different m‐values. Thus, the C atom in its ground state has two, and the N atom has three unpaired and parallel spins. This further rule within the Aufbau principle is known as “Hund's rule” that states that degenerate orbitals are filled in such a way as to maximize spin multiplicity or maximum number of unpaired spins. This is due to the fact that pairing the spins of two electrons requires a certain amount of energy, known as the spin‐pairing energy. The number of unpaired spins directly affects the magnetic properties of atoms (as well as of molecules). Species with two unpaired, parallel spins, such as the C atom in its lowest energy state, are paramagnetic. If its spins were paired, the atom would be diamagnetic. Thus, one finds that the shape of the periodic chart of elements, and many properties of the atoms, are determined by the energy level diagram shown in Figure 9.1 that can be obtained from Hartree–Fock calculations. The spin‐pairing energy mentioned above is also responsible for a few minor exceptions in the Aufbau order, for example, in the case of the chromium atom. According to the energy level diagram shown in Figure 9.1, one should expect the Cr atom to have a [Ar]4s23d4 electronic configuration, but the actual configuration is [Ar]4s13d5, that is, an arrangement devoid of paired electrons in the outer shells. (Here, [Ar] denotes filled energy shells up to the 18th electron: [Ar] ≡ 1s2 2s2 2p6 3s2 3d6). This demonstrates that the energy difference between the 4s and 3d orbitals is so small that it can be overridden by the spin‐pairing energy.

9.4 Periodic Properties of Elements Many experimentally and theoretically obtained periodic properties of elements can be explained by the energy level diagram shown in

Figure 9.1 and are discussed in detail in many introductory texts on general chemistry and introductory physical chemistry. Here, just two of these properties will be discussed further to illustrate the information available from this energy level diagram. One of these is the ionization energy (also known as the ionization potential) that expresses the energy required to ionize an element M to give a positively charged cation M+ and an electron, e− : (9.14) All alkali metals, for example, have a single electron in the “s” orbitals (Li: 1s2 2s1, Na: 1s2 2s2 2p6 3s1, etc.). Since these electrons experience a nucleus that is highly shielded by the inner shell electrons (see Eq. (9.6)), it is relatively easy to remove these electrons, either by subjecting gaseous atoms to an external potential (hence the term ionization potential) or to light (the previously discussed photoelectric effect, see Section 1.3). When plotted against the atomic number, a graph of the ionization energies, as shown in Figure 9.2, is obtained. This figure demonstrates that the ionization energy is low for all alkali metals and the alkali earth metals but increases toward the right (with increasing atomic number) for each row in the periodic table. The graph even shows that for elements filling the “p” orbitals, there is a dip in the ionization energy when a half‐filled shell is reached, in agreement with the discussion of the spin‐pairing energy above. In this plot, the transition metals were omitted, since for these elements, inner electrons (for example, the 3d orbitals in the elements from Sc to Zn) are filled that occupy space within the radius of the 4s orbital and for which the ionization energy varies only a little.

Figure 9.2 Ionization energies (a) and atomic radii (b) for main group elements. Similarly, atomic radii can be predicted from an analysis of Figure 9.2. Since the nuclear charge increases for elements within a row of the periodic chart, the attraction an electron experiences (despite the shielding effect) increases, and the atomic radius decreases within a row of elements in the periodic chart. However, when the next element in a new row is reached, a dramatic increase in the atomic radius is observed; see Figure 9.2.

9.5 Atomic Energy Levels We used the hydrogen atom emission/absorption spectra as a vehicle to introduce the stationary‐state energy levels in the simplest atom, and the quantized energy packets required for transitions between these states. As it turns out, atoms with more than one electron also exhibit rich spectral features that have been used for centuries (think Chinese fireworks!) due to the bright color emissions exhibited by certain elements. Also, in modern analytical science, the detection and quantification of many elements by atomic absorption spectroscopy (down to part‐per‐billion concentrations) are accomplished by the absorption of a specific wavelength of light by atomic species.

Atomic spectroscopy will be briefly discussed here to introduce some principles of electronic transitions in atoms that contain more than one electron. In particular, the formalism of combining total angular and spin angular momenta into a new quantity that determines the selection rules for atomic electronic transitions will be introduced, since this formalism also allows the classification of electronic transitions in small (diatomic) molecules.

9.5.1 Good and Bad Quantum Numbers and Term Symbols For the hydrogen atom, all operators required to solve the Schrödinger equation, i.e., , , and , commutated with the total Hamiltonian ; thus, the eigenvalues l, ml, and s, as well as the total energy, can be determined simultaneously. Since the Schrödinger equation for a multi‐electron atom, given by Eq. (9.1):

Figure 9.3 Vector addition schemes for (a) the total orbital angular momenta and (b) the total spin angular momenta. (9.1)

cannot be solved explicitly (even for the case of two electrons), one cannot obtain good quantum numbers any longer. Good quantum numbers are defined as those whose operators commutate with the total Hamiltonian. Although the eigenvalues of the individual orbital angular momenta do not commutate with the total Hamiltonian, their sum (i.e. the total orbital angular momentum of all electrons) does commutate with the total Hamiltonian. This total orbital angular momentum is obtained by the vector sum of the orbital angular momenta li of all i electrons: (9.15) In Eq. (9.15), “s” electrons never contribute to the vector sum, since their angular momentum is zero due to the spherical nature of “s” orbitals. Furthermore, paired electrons in the same orbital do not contribute either since their spin moments add to zero. Allowed values (for a two‐electron system) of the vector addition in Eq. (9.15) are (9.16) Equation (9.16) is also referred to as Clebsch–Gordan expansion that will be used here without proof. For example, in an excited atom or ion that has one electron in a “p” orbital (l = 1) and another in a “d” orbital (l = 2), the vector addition leads to three different total angular momenta, namely 3, 2, or 1 (see Figure 9.3a). In accordance with the nomenclature of individual electronic orbital angular momenta, the sum of the angular momenta is denoted by the capital letters S, P, D, F… as shown in Table 9.1: Table 9.1 Symbols of states for different l and L values.

Similarly, one defines a total spin angular momentum S as (9.17) where S commutates with the Hamiltonian. Again, for the case of two spins with s = ½, there are two possibilities, S = 1 if the spins are parallel, or S = 0 if they are antiparallel. The vector addition schemes described by Eqs. (9.16) and (9.17) are depicted for a two‐electron system in Figure 9.3b. The spin multiplicity is defined as 2S + 1. Thus, for S = 0, the multiplicity is one (a “singlet” state) and for S = 1, the multiplicity is 3 (a “triplet” state). The multiplicity is written as a left superscript of the total orbital angular momentum, e.g. 1D or 3F. These symbols are called “term symbols” and specify the spin states of a multi‐electron system (cf. Example 9.1). Finally, the total orbital angular momentum and the total spin momenta can interact via spin‐orbit coupling to produce a total angular momentum that is specified by (9.18) This spin‐orbit quantum number J is not to be confused with the rotational quantum number discussed in Chapter 6. The allowed values of J are again given by the Clebsch–Gordan expansion rule (9.19) This total spin‐orbit angular momentum quantum number J is written as a right subscript to the term symbol, such as 3P0. Such an expression is known as a “level.” Example 9.1 What are the term and level symbols of the ground and excited states of the Na atom referred to in Section 7.5 and Figure 7.6b?

Answer: The ground‐state Na atom has a 1s2 2p6 3s1 electron configuration. The filled shells do not contribute to the term symbol. The lone electron in the 3s orbital has an orbital angular momentum of zero (since it is in an s orbital) and a spin angular momentum of . The spin multiplicity is 2; therefore, the term is 2S. The total angular quantum number Thus, the ground state can be described as 2S1/2. The excited state is a 1s2 2p6 3p1 configuration. The electron in the p‐ orbital has L = 1, and the term is 2P. The total angular quantum number is or ; thus, the excited states can be described as 2P3/2 or 2P1/2. These two states differ slightly in energy; thus, the transition (the 589 nm sodium line) is split into a doublet. Atomic spectra are very rich in spectral lines since the atomic species exhibit a large number of excited states due to the existence of many unoccupied atomic orbitals into which electrons can be promoted. Transitions into or from these states are either dipole allowed (for the discussion of selection rules, see next section) or are caused by radiationless processes, such as collisions with other highly excited species. The former case of radiation‐induced transitions was discussed before for the hydrogen atom emission and absorption spectra (see Section 7.3). The latter case was mentioned for the excitation process in the He–Ne laser where the Ne atom excited states are populated via collisions with electrons and/or He ions (cf. Section 3.4). The number of excited states is further increased by the interaction of angular and spin momenta, as indicated by Eqs. (9.15)–(9.19). Just how many states can be produced from one electronic configuration is shown in Example 9.2. Population or depopulation of some of these states may not be allowed by radiation‐induced transitions; nevertheless, these states are real and can contribute to the observed atomic spectra.

Example 9.2 What are the possible term and levels of an excited C atom with an electronic configuration of 1s2 2s2 2p1 3d1? Answer: As we have seen in Example 9.1, any electrons in filled shells do not contribute to the total orbital angular momentum. The electron in the p orbital has an angular momentum l = 1, and the electron in the d orbital has l = 2. According to Eq. (9.16), the allowed L values are: L = 3, 2 or 1, corresponding to D, P and S symbols. The total spin angular momentum S can be 1 or 0; thus, the spin multiplicity will be 3 or 1. Consequently, the possible term symbols are 3D, 3P, 3S, 1D, 1P, 1S. According to Eq. (9.19), (9.19) the total angular momentum J can have values from 4, 3, 2, 1; thus, each of the term symbols can have the subscripts from 4 to 1, e.g., 3D , 3D , 3D and 3D 4 3 2 1

9.5.2 Selection Rules for Transitions in Atomic Species The selection rule for atomic species, in general, follows closely the selection rule for the H‐atom in that it is determined by the condition Δl = ± 1. With the coupling of individual angular momenta into a total angular momentum (9.15) and the spin‐orbit coupling (9.18)

Figure 9.4 Simplified energy level diagram of the Li atom and transitions indicated in Table 9.1. additional selection rule arises, given by (9.20) and (9.21) There is an additional condition, that the total spin angular momentum cannot change, that is, (9.22) This latter selection rule prohibits, for example, singlet to triplet transitions, and will be particularly important in electronic spectroscopy of di‐ and polyatomic molecules.

9.6 Atomic Spectroscopy The large number of atomic energy levels, discussed in Section 9.5.1, makes for rich atomic spectra in both absorption and emission. We shall investigate in some detail the atomic spectrum of one of the simplest atoms, Lithium, which has a 1s22s1 electronic configuration. A tabulation of the transitions of neutral Li can be found in the literature [2] and lists 21 transitions between 200 and 2700 nm wavelengths. The most relevant of these are listed in Table 9.2 and shown in Figure 9.4. Similar to the situation in sodium, the major transition, marked “C” in Table 2, is between the ground state and an excited state in which the 2s electron transitions into the 2p orbital. This configuration results in two levels, 2P3/2 and 2P1/2, that differ in energy by about a third of a wavenumber. Consequently, all transitions involving these two energy levels are split into doublets. The splitting is only about 0.02 nm for the transition marked “C.” We saw before (see Example 9.1) that the same two levels involved, namely 2P3/2 and 2P1/2 were

responsible for the splitting observed in the sodium 589 nm line. Incidentally, the bright red emission at 670.8 nm gives Li its deep red color in firework displays. It is also easily visible when Li compounds are aspirated into a flame (see below).

Table 9.2 Transition, energies, term symbols, and wavelengths of the prominent Li atomic lines. Source: Adapted from National Institute of Standards and Technology [2], Transition, energies, term symbols and wavelengths of the prominent Li atomic lines.

Transition Term

Energy [cm−1] Wave‐ excited ‐ ground length [nm]

Relative intensity

1s24d1 ← 1s22p1

2D 3/2 2 ← P1/2

36 623.30 – 14 903.62

460.28

15

A

1s24d1 ← 1s22p1

2D 5/2 ← 2P1/2

36 623.31 – 14 903.62

460.29

30

A

1s23d1 ← 1s22p1

2D 3/2 ← 2P1/2

31 283.02 – 14 903.62

610.35

300

B

1s23d1 ← 1s22p1

2D 5/2 ← 2P1/2

31 283.05 – 14 903.62

610.36

400

B

1s22p1 ← 1s22s1

2P 3/2 ← 2S1/2

14 903.96 ‐ 0.00

670.77

500

C

1s22p1 ← 1s22s1

2P 1/2 ← 2S1/2

14 903.62 ‐ 0.00

670.79

1000

C

1s23s1 ← 1s22p1

2S 1/2 ← 2P1/2

27 206.07 – 14 903.62

812.62

150

D

1s23s1 ← 1s22p1

2S 1/2 ← 2P3/2

27 206.07 – 14 903.96

812.64

300

D

1s23p1 ← 1s23s1

2P 3/2 ← 2S1/2

30 925.61 – 27 206.07

2687.76

10

E

1s23p1 ← 1s23s1

2P 1/2 ← 2S1/2

30 925.51 – 27 206.07

2687.78

5

E

A Li atom energy level diagram is shown in Figure 9.4 with the transitions listed in Table 9.1 indicated by the capital letters. In this energy level diagram, the splitting by the spin‐orbit coupling is not indicated since it contributes only fractions of a wavenumber.

9.7 Atomic Spectroscopy in Analytical Chemistry The abundance of atomic transitions in metal makes atomic spectroscopy, specifically, atomic absorption spectroscopy, one of the most widely and cost‐effective methods to analyze for the presence of metal ions in analytes down to the ppm‐level. In atomic absorption spectroscopy, the analyte is aspirated into the flame of a burner, typically a very hot flame from an acetylene–oxygen mix. Under these conditions, all compounds are broken down into atoms and ions that – due to the high temperature in the flame, exhibit absorption spectra with broadened lines. The flame is illuminated by light from a special source, known as a hollow cathode lamp (HLC). This lamp is specific for the element to be analyzed: to determine the arsenic concentration in a sample, an arsenic hollow cathode lamp must be used. In an HCL, the analytes emission spectrum (in this case, the arsenic emission spectrum) is produced at low temperature by sputtering atoms off the As cathode in a carrier gas. These atoms collide with electrons in the electric discharge, get excited and emit narrow atomic emission lines. The absorption of the HCL radiation by the atoms/ions in the flame is observed and converted to an analyte concentration via the Beer–Lambert law (see Eq. [3.25]). The narrow lines emitted by the HCL makes this method particularly useful to eliminate interference from other elements. Furthermore, the sheer number of strong and weak transitions in an atomic emission allow the user to select lines that are most suitable for the concentration range to be analyzed.

References

1 Clementi, E. and Raimondi, D.L. (1963). Atomic screening constants from SCF functions. The Journal of Chemical Physics 38 (11): 2686–2689. 2 National Institute of Standards and Technology. Basic Atomic Spectroscopic Data. https://physics.nist.gov/PhysRefData/Handbook/Tables/lithiumt able2.htm.

Problems 1. Write electronic configurations for Sc, Sc+2, and Sc+3 2. State Pauli's principle in at least two different ways. 3.

a. State Hunds' rule, and give a physical explanation for it. b. What are the implications of Hund's rule given the magnetism of transition metals and rare earth metals?

4. Discuss the trend in the 1st ionization energies of the alkali metals 5. Discuss the trend in atomic radii of the alkali metals 6. Why is the 3rd ionization energy of Ca much larger than the 1st and 2nd? 7. Explain why shielding is more effective by electrons in a shell of lower principal quantum number than by electrons having the same principal quantum number (i.e., why does a 2s electron shield a 3p electron more effectively than a 3s electron would?) 8. Write the normalized Slater determinant for the ground‐state configuration of Be. 9. In analogy to the Li and Na emission spectra, discuss the observation of a doublet at 393.37 and 396.85 nm in the emission spectrum of potassium.

10 Electronic States and Spectroscopy of Polyatomic Molecules In electronic spectroscopy, electrons are promoted into more highly excited electronic or vibronic (a concatenation of vibrational and electronic; see Section 10.4) states that can either be centered at metal atoms in coordination compounds or unoccupied molecular orbitals (MOs) such as antibonding σ* or π* molecular orbitals. In general, the electrons originate from the highest occupied molecular orbitals (HOMOs). There are other techniques in which core electrons are excited or even ejected from the molecular systems, but these techniques usually require photon energies beyond the wavelength limit discussed here, namely, below about 190 nm. In this chapter, spectroscopy in the more classical ultraviolet–visible (UV‐vis) spectral range will be discussed, again from the viewpoint introduced earlier, namely the interplay between quantum mechanical theory and the experimental results that can be and have been used for decades to refine the theory. Furthermore, chemical structural information can be obtained from UV‐vis spectra, although this information is not as direct as that obtained from rotational spectroscopy (Chapter 6) or NMR spectroscopy (Chapter 8). In order to discuss electronic or UV‐vis spectroscopy, we need to understand the electronic structure and bonding of the molecular samples. Whereas classical models are available for rotational and vibrational energy states of molecules (see Sections 6.1 and 5.1), such simple models do not exist for electronic structures and energy states of molecules; thus, a quantum mechanical description is necessary. This will be presented in Section 10.1 in a somewhat cursory fashion, because a detailed treatment of molecular orbital theory is far beyond the scope of this book, in particular the computational methods that have been developed in this field. The discussion

follows the development presented in a textbook Physical Chemistry by Engel and Reid [1].

10.1 Molecular Orbitals and Chemical Bonding in the H2+ Molecular Ion The first step in understanding electronic spectroscopy is a description of the chemical bond in molecules, in particular the formation and energetics of molecular orbitals. This discussion starts with the simplest of all molecules, the H2+ molecular ion that consists of two protons and one electron. In analogy to the Hamiltonian that was introduced for a system consisting of a nucleus and two electrons (Eq. [9.1]) (9.1)

one defines the Hamiltonian for the H2+ molecular ion with two nuclei and one electron as (10.1)

This equation will need to be augmented for electron–electron repulsion terms in a H2 molecule with two electrons. Equation (10.1) cannot be solved exactly due to the interdependence of the spatial variables R, ra, and rb. However, it can be solved explicitly after invoking the so‐called Born–Oppenheimer approximation (see Section 10.4): since the nuclei are about 2000 times heavier than

electrons, they move much more slowly than the electrons. Thus, one may treat the nuclear coordinates as fixed in their equilibrium position while the electron carries out its motion independently of the nuclear position. This removes the dependence of the total energy on R. Next, the electronic wavefunction of the single electron is written to be delocalized over the two nuclei. One may view the molecular system in the following discussion as a hydrogen atom with a nucleus referred to as “a,” to which a proton, without an electron (nucleus “b”), is added. Then, the wavefunction ψ of the one electron is written as a linear combination of two hydrogen atom wavefunctions ϕ located at nuclei a and b as follows: (10.2) Because of symmetry arguments and the fact that the electron density must be invariant to interchange of the two nuclear positions, the condition (10.3) holds. This leads to the formation of two molecular orbitals that are, due to the symmetry (see Chapter 11) of the species, referred to as the gerade (g, or even) and ungerade (u, or odd) states: (10.4a)

(10.4b)

The values of the expansion coefficients cg and cu can be determined by normalizing the wavefunctions. For ψg,this yields

(10.5)

Since the hydrogen 1s orbital is a normalized wavefunction, the first two integrals in Eq. (10.5) have a value of one. Thus, (10.6) We define the overlap integral (10.7) that is, the area that lies under both the 1sa and 1sb atomic orbitals (cf. Figure 10.1a). With this definition, the normalized coefficient cg becomes (10.8) and (10.9) Next, the energy expectation values for the “u” and “g” states will be discussed. As usual, the energy expectation value is given by (see Postulate 4, Section 2.1) (10.10)

Equation (10.10) is one of two energy eigenvalues of the Hamiltonian that was defined in Eq. (10.1) but subsequently was simplified to (10.11) after invoking the Born–Oppenheimer approximation.

Figure 10.1 (a) Overlap of the two 1s orbitals on nuclei a and b. The volume obtained by rotating the shaded area about the r‐axis represents the overlap integral. (b) Energy level diagram for bonding and antibonding orbitals in the H2+ molecular ion. Taking into account the normalization condition for the denominator of Eq. (10.10), the energy of the “g” state becomes (10.12)

or

(10.13) where the abbreviations Haa and Hba denote (10.14)

(10.15) Thus, we may write (10.16) and (10.17) Next, the expression for Haa is evaluated: (10.18)

The first integral in Eq. (10.16) can be solved exactly, since it denotes the energy ground state, E1s, of the hydrogen atom. The second term

is a constant for a fixed value of R and evaluates to just e2/4πεoR, since the atomic orbital wavefunctions are normalized. The third term is referred to as the Coulomb integral and expresses the interaction energy of a positively charged nucleus “b” with the electron in the 1s orbital of nucleus “a.” Thus, Eq. (10.18) is rewritten as (10.19) Haa is the energy of a hydrogen atom perturbed, in the absence of a chemical bond, by a naked proton at a distance R. Thus, one may consider Haa the energy of a nonbonded state. The strength of the bond formed between the nuclei by delocalization of the electron (see Eq. [10.4a]) will be given by Eg − Haa. Similar to the integral expression in Eq. (10.18)), the terms Hba = Hab are evaluated: (10.20)

(10.21) With the definitions of Haa and Hba, the energy changes for the lowered “bonding” state ΔEg and the raised “antibonding” state ΔEu

are [1] (10.22)

(10.23) The results in Eqs. (10.16, 10.17, 10.22, 10.23) can be summarized as follows for the H2+ molecular ion (and similarly for more complicated molecular systems, see Section 10.2): two atomic orbitals, , are combined to form two molecular orbitals, the bonding σ molecular orbital and the antibonding σ* molecular orbital. The energy for the bonding orbital is lowered by the amount, ΔEg, given in Eq. (10.22) that accounts for the formation of the chemical bond. The absolute value of ΔEg is less than that of ΔEu since (1 + Sab) > (1 − Sab); thus, the energy lowering of the bonding orbital is less than the energy increase of the antibonding orbital (see Figure 10.1b). The wavefunctions of the resulting bonding and antibonding orbital are similar to those shown in Figure 10.2 for the H2 molecule.

Figure 10.2 Wavefunctions for the (a) bonding and (b) antibonding molecular orbitals in H2. Notice the larger overlap of the 1s orbitals in H2, as compared with H2+ (Figure 10.1a), due to the shorter bond length in H2 (74 pm vs. 105 pm).

10.2 Molecular Orbital Theory for Homonuclear Diatomic Molecules For a molecule with more than one electron, one proceeds in analogy with the discussion presented in Section 10.1 for the H2+ molecular ion. For the H2 molecule, for example, one could form molecular orbitals σi by combining the atomic orbitals ϕi of each of the hydrogen atoms according to (10.24) assuming that a set of interacting electrons in a molecule may be represented by a sum of single‐electron Hamiltonians. Next, one minimizes the energy of the system by varying the expansion coefficients that form the molecular orbitals. This is illustrated for the case of two atomic orbitals in a H2 molecule. Again, one starts by writing the energy expectation value (see Eq. (10.9))

(10.25)

(10.26)

(10.27)

Next, the values of c1 and c2 corresponding to the minimum value of E need to be found. This is accomplished by calculating ∂E/∂c1 and ∂E/∂c2 and setting the derivatives to zero. This leads to two equations: (10.28)

These equations have nontrivial solutions when (10.29)

The solutions of this determinant are (10.30) in analogy to the solutions for the H2+ molecular ion (Eqs. [10.16] and [10.17]). Details of the intermediate steps can be found in [1]. The resulting molecular orbital wavefunctions are shown in Figure 10.2.

The one‐electron orbitals used in these calculations are of the form (9.4)

that is, spherical hydrogen‐like orbitals with a parameter ζ that accounts for the size of the orbital. Notice that the mathematical form of these orbitals was introduced earlier (Eq. [9.4]). The expressions for H11, H12, and S12 contain the overlap, Coulomb, and exchange integrals defined earlier (see Eqs. [10.7], [10.18], and [10.20]): (10.7)

(10.18)

(10.20) written here in a more general form of atomic orbitals. The integrals in Eqs. (10.7), (10.18), and (10.20) are generally solved by approximating the orbital function by a sum of 3 or 5 Gaussian functions. Gaussian functions are used to facilitate the integration necessary in the computation of the Coulomb, exchange, and overlap integrals, although Gaussian functions do not exhibit the cusp at r = 0, as the true wavefunctions do (cf. Figures 7.1 and 7.4). This introduces a small error, but the orbital interactions described in Eqs. (10.7), (10.18), and (10.20) are most significant at values of r closer to the bond distance, rather than at very small values of r. The formalism introduced so far neglected electron correlation that was discussed in Section and restricted the molecular orbital expansion (Eq. (10.24)) to just two atomic orbitals. In typical

molecular orbital calculations, the expansion given in Eq. (10.24) includes a much larger number of orbitals even if they are not occupied: (10.31) This approach is referred to as the linear combination of atomic orbital (LCAO) method. In addition, there exist more sophisticated molecular orbital calculations that explicitly include electron correlation, which was neglected in this approach described here. Second‐row diatomic molecules are treated in a similar fashion by adjusting the expansion coefficients in Eq. (10.31) to minimize the energy of the molecule. Since these calculations seek to establish the minimum energy of the molecular ground state, the variation method (see Appendix 2) can be employed. These computations result in molecular orbital energy schemes similar to the one shown in Figure 10.3 for the O2 molecule. The “inner” (1s) electronic orbitals form bonding σ1s and antibonding molecular orbitals that are completely filled with two electron pairs and resemble the 2s, σ2s, and orbitals shown at the bottom of Figure 10.3a. These electrons do not contribute to the bonding since the energy increase of the antibonding orbitals is larger than the energy reduction of the bonding orbitals; see Eqs. (10.22) and (10.23) and Figure 10.1b. In Li2, the 2s electrons form molecular orbitals that are described as bonding σ2s and antibonding molecular orbitals.

Figure 10.3 (a) Energy level diagram of the MOs formed from the overlap of 2s and 2p orbitals in diatomic homonuclear molecules such as N2, O2, and F2. (b) Visualization of the “head‐on” overlap of the 2pz orbitals to form the σ2p and the lateral overlap of the 2px and 2py orbitals to form the two π2p orbitals. The elements that have electrons in 2p orbitals form two types of molecular orbitals: if we define the z‐axis as the direction of the chemical bond, the overlap of the 2pz orbitals forms bonding and antibonding molecular orbitals that are referred to as σ2p and , respectively. The lateral overlap of the 2px and 2py atomic orbitals of the two atoms is responsible for the double‐ or triple‐bond molecular orbitals that are referred to as the π2p and orbitals, as shown in Figure 10.3b. Notice that in these orbitals, the highest electron density is not along the direction of the chemical bond. The MO energy diagram for a homonuclear diatomic molecule is given in Figure 10.3a. The order of the orbital energies shown holds for N2, O2, and F2. Since the molecular orbitals are degenerate, the oxygen molecule, with 12 valence electrons, has two unpaired electrons in these orbitals. Notice that the (inner) 1s atomic orbitals, and the resulting σ1s and molecular orbitals, are not shown in Figure 10.3a, since they are much lower in energy than the orbitals formed by the valence electrons.

The energy level diagram for the oxygen molecule shown in Figure 10.3a and elaborated upon in Problem 10.3 explains very impressively two facts about this species: first, the molecule has a bond order of two1, and second, it is paramagnetic. The bond order of two is expected from a naïve attempt to write a Lewis structure for O2 and from the fact that the bond stretching vibrational frequency is typical for a double bond. However, such a Lewis structure would have no unpaired electrons, which contradicts the fact that molecular oxygen is paramagnetic and, therefore, must possess unpaired electrons. These two electrons occupy the two orbitals with parallel spins. These aspects will be discussed in more detail in later sections, see also Problems 2–4 of Chapter 10.

10.3 Term Symbols and Selection Rules for Homonuclear Diatomic Molecules Based on the energy level diagram in Figure 10.3, O2 in its ground state has an electronic configuration of (10.32) This ground‐state configuration obeys Hund's rule (see Section 9.3) in that the degenerate orbitals are filled with two electrons with parallel spins. When writing the term symbols of molecular species such as the O2 molecule, one proceeds the same way as one would in the assignment of the term symbols of multi‐electron elements (see Eq. [9.15]) by summing orbital angular momenta for electrons in unfilled subshells according to (10.33) The resulting Λ values are designated as given in Table 10.1.

Here, electrons in a σ orbital have l values of zero, whereas electrons in π orbitals have l = ± 1. Thus, in oxygen, the Λ value is either 0 (for the ground state) or ±2,leading to either Σ or Δ terms for the oxygen molecule; see Figure 10.4. The total spin angular momentum is obtained as before (see Eq. [9.17]) according to (10.34) Since there are two unpaired electrons in the oxygen ground state, the total spin moment S = ± 1. The spin multiplicity 2S + 1 is added as a left superscript to the term, resulting in a 3Σ designation for the oxygen ground state (a triplet state). In order to uniquely define the electronic configuration of a homonuclear diatomic molecules, two further indices are necessary. A right subscript g or u indicates whether the orbitals in which the two electrons are found have even or odd symmetry with respect to the center of inversion (see Chapter 11). The two MOs formed from the lateral overlap of 2p orbitals with the same phase have even parity; thus, the term for the ground‐state oxygen molecule is written as 3Σg. The final symbol, written as a right superscript + or –, indicates whether MOs change sign when reflected by a plane that contains the molecular bond direction. The orbitals do so, and, therefore, the final designation of the molecular oxygen ground state is electrons in two Figure 10.4.

. The

MOs are shown schematically in panel (A) of

Table 10.1 Symbols of states for different l and L values. L 0 1 2 3 (For multi‐electron atoms) S P DF Λ 0 1 2 3 (For molecules) ΣΠΔ Φ

Figure 10.4 Electron and spin populations in the two

MOs of

the lowest‐energy configurations of the oxygen molecule. See text for details. There are several excited‐state species of the oxygen molecule that are either singlet or triplet states. The next lowest energy state is a 1Δ species in which the two antibonding electrons are, with opposite g spins, in one of the MOs; see Figure 10.4b. Since the spins are paired, it is obviously a singlet state. Another singlet state is due to a configuration with both orbitals occupied by one electron, but the electrons have opposite spins (or do not obey Hund's rule). The total spin angular moment is zero, but the total orbital angular momentum is 2, as in the ground state. Thus, the term symbol for this state is . This state is shown in Figure 10.4c. The three states described so far have the same bond order, since they differ only in the way the two highest orbitals are populated, as shown in Figure 10.4a–c. The next state in energy is one where one electron from the doubly occupied π2p orbital (see Eq. (10.32)) is promoted into one of the MOs, leading to two electronic configurations: (10.35) one of which is shown schematically in Figure 10.4d. These two states represent a quite different situation, since the bond order in this species is no longer 2. Some of the consequences of the changes

in bond order will be discussed in more detail in Section 10.4. The term symbol for these states are and . The selection rules for electronic transitions are as follows: (10.36) Thus, Σ↔ Σtransitions will be allowed, as will be the Π↔ Σ transitions. However, singlet to triplet transitions are forbidden. For homonuclear diatomic molecules, additional rules hold: (10.37)

With these rules in place, we can start to interpret observed electronic spectra of diatomic species like O2, for which the only dipole‐allowed transition is from the ground state to the state, that is, the transition, for which all selection rules listed above are fulfilled. The energy difference between these two states is about 6.2 eV. The resulting UV absorption spectrum is presented in the next section.

10.4 Electronic Spectra of Diatomic Molecules 10.4.1 The Vibronic Absorption Spectrum of Oxygen Based on the discussion in the previous section, one may expect a single transition peak in the UV spectrum of molecular oxygen for the transition at just above 200 nm. Instead, one observes a number of transitions, as shown schematically in Figure 10.5 [2, 3]. The appearance of the spectral results immediately

suggests that, in addition to the electronic transition, one observes excitation into vibrational and even rotational states. The major sawtooth‐like features shown in Figure 10.5a are due to vibronic transitions from the ground vibrational state of the electronic ground state into vibrationally excited state of the electronic state. As can be seen from the band heads in Figure 10.5b, the energy spacing between the vibrationally excited states of the decreases, as expected from a strongly anharmonic oscillator. The spacing of the band heads, however, is much less than what one would expect from the spacing of the vibrational energy levels of the electronic ground state. The reason for this can readily be explained as follows. O2 in its ground state has a bond order of two. Therefore, the O–O stretching frequency is 1580 cm−1, typical for double‐bonded species. This transition is not allowed in absorption but detected in Raman scattering. However, the spectral progression shown in Figure 10.5a corresponds to a much lower vibrational energy level spacing of c. 700 cm−1. This is due to the fact the state that is accessed by the electronic transition at c. 200 nm has a bond order of one. This follows from an inspection of the MO energy diagram shown in Figure 10.4d. Since oxygen in the state has a single bond, one expects the much lower vibrational frequency. The rotational constant of the excited‐state oxygen species is 0.819 cm−1, resulting in very closely spaced (c. 0.01 nm) transitions for low J values (“J” here refers to the rotational quantum number; see Section 6.3 and Eq. [6.38]), but up to 20 cm−1 for high J values when considering centrifugal interactions. Inspection of Figure 10.5b shows the vibrational bands [2] with decreasing energy splitting, whereas the splitting of the rotational energy levels increases for increasing J values.

Figure 10.5 Observed (a) and simulated (b) vibronic transition of molecular oxygen. See text for details. Source: Trentmann et al.[2]; Stamnes [3].

The observation of the progression of vibrational sub‐bands in the observed UV absorption spectrum of the oxygen molecule needs further discussion. As pointed out earlier (cf. Sections 4.5 and 6.5), oxygen does not exhibit neither a vibrational nor rotational absorption spectrum, being a nonpolar molecule. Obviously, the electronic selection rules discussed above (Eqs. (10.25, 10.26)) overwrite the vibrational and rotational selection rules. This is discussed in Section 10.4.2 and an approximate energy level diagram for all these transitions is given in Example 10.1. Example 10.1 Schematic energy level diagram for O2, approximately drawn to scale. The energy difference between the two electronic states is ca. 50 000 cm−1; the vibrational energy levels are split by ca. 1500 cm−1 in the ground electronic state and by about 700 cm−1 in the electronically excited state. They are represented here without the anharmonic contributions. The rotational energy levels are spaced so closely that they only would be visible at this scale for J levels larger than ca. 25. Remember that at room temperature, all O2 species are in the electronic ground state, and nearly all of them are in the n = 0 vibrational state. However, the rotational states of the vibrational ground state are highly populated.

Figure 10.E1 See Example 10.1 for details. Furthermore, the appearance of a broad absorption continuum below ca. 175 nm (see Figure 10.5a) suggests that the excited oxygen molecule has reached its dissociation limit after about 15 vibrational energy levels. Any photons of energy corresponding to 175 nm or less no longer must have the exact energy of a molecular transition; rather, photons with energies higher than the highest bound state will cause photodissociation of the excited species, with the excess energy being converted into kinetic energy of the two oxygen atoms.

The intensity pattern observed for the procession of vibrational transitions needs further explanation, and a further discussion of the consequences of the Born–Oppenheimer approximation is introduced earlier. This leads to the so‐called Franck–Condon rules of vibronic spectroscopy that we shall encounter in the next section and in the discussion of fluorescence spectroscopy as well.

10.4.2 Vibronic Transitions and the Franck–Condon Principle The Hamiltonian of the Schrödinger equation for a multinuclear, multi‐electron molecule or molecular ion contains the following terms: (10.38)

This Hamiltonian, in the words of I. N. Levine, is “formidable enough to strike terror in the heart of any quantum chemist” [4] (Chapter 13). Benzene, for example, has 12 atomic nuclei and 42 electrons. The time‐independent Schrödinger equation, which must be solved to obtain the energy and wavefunction of this molecule, is a partial differential eigenvalue equation in 162 variables, the spatial

coordinates of the electrons and the nuclei. Invoking the Born– Oppenheimer approximation that uncouples the motion of nuclei and electrons reduces the problem to 126 dimensions, but it still is a very complex calculation. We shall use the Born–Oppenheimer to further explain the vibronic spectral pattern observed for oxygen. The Born–Oppenheimer approximation can mathematically be formulated as follows: (10.39) In Eq. (10.39), the expression ψ(ri, Rα) denotes the wavefunctions in all variables defined by the Hamiltonian in Eq. (10.38), namely, the coordinates Rα of all nuclei α, as well as the coordinates ri of all electrons i. Because of the fact that the nuclei are much heavier than the electrons, one argues that their motion is much slower than that of the electrons; consequently, the wavefunctions are approximated in Eq. (10.39) as a product of the electronic wavefunctions at fixed nuclear positions and the purely vibrational wavefunctions that have been described in Chapters 4 and 5. The dipole transition moment for the observed spectrum shown in Figure 10.5 is given by (3.15) where n and m denote the states between which the transition occurs. Thus, the transition moment for the (the lowest‐ energy allowed electronic transition in oxygen) must be nonzero: (10.40)

Since the position of the nuclei is fixed for the electronic wavefunctions , Eq. (10.40) can be separated into the product of an electronic transition moment for fixed nuclear

positions and the overlap of the vibrational wavefunctions in any of the vibronic states as follows: (10.41)

The transition moment 〈μ〉 therefore can be written as a product of the electronic transition moment (10.42) and the overlap S of the vibrational wavefunctions (also known as the Franck–Condon factor) (10.43) Within the Born–Oppenheimer approximation, the transition moment therefore is given by the product of the electronic dipole operator and the vibrational overlap (the Franck–Condon factor): (10.44) Equation (10.44) determines whether or not transitions between vibronic levels are allowed, as alluded before in the statement that the electronic selection rule modifies both the rotational and vibrational selection rules. However, it is the overlap of the vibrational wavefunctions that determine the intensities of the particular vibronic transitions, as shown in Figure 10.6. This figure introduces a scheme commonly used in depicting vibronic processes. In this picture, the lower part of the two anharmonic potential curves represents the electronic ground‐state potential energy in a diatomic molecule, such as O2, or any typical bond stretching coordinate in a polyatomic molecule. The squares of the vibrational wavefunctions for the four lowest energy states of this

vibrational coordinate are also shown. The upper potential curve represents the excited electronic state. In the case of the O2 molecule, it was pointed out that this electronically excited state has a lower bond order than the ground state. Therefore, the vibrational force constant will be lower and the bond is weaker. The weaker bond is indicated by the fact that the potential energy curve is broader and shallower and the vibrational energy levels are more closely spaced. Furthermore, the excited potential function is shifted along the internuclear axis (the x‐axis) toward a slightly longer bond distance. Therefore, the maxima of the ground‐state vibrational wavefunctions in the ground and excited electronic states no longer line up. The Franck–Condon factor predicts, for the example shown, that the most intense transition would occur between the ground vibrational state of the electronic ground state and the fourth vibrational state of the electronically excited state. A transition into the third vibrational state of the electronically excited state would have nearly the same intensity. In these considerations, it is assumed that only the ground vibrational state of the electronic ground state is populated (see Example 3.1). The vibronic spectrum, therefore, appears as a series of nearly equidistant bands, as shown in Figure 10.5a where the highest intensity band corresponds to the maximum overlap of the vibrational wavefunctions of the ground and excited electronic states. This discussion of vibronic transitions following the Franck– Condon rules will be picked up again in Section 10.6.

Figure 10.6 Vibronic transition between the ground vibrational state of the electronic ground state and the third vibrational state of the electronically excited state, according to the Franck–Condon principle.

10.5 Qualitative Description of Electronic Spectra of Polyatomic Molecules In the previous section, the electric transitions in a simple diatomic molecule were discussed. For larger molecules, a different, simplified approach is generally used, since the treatment of polyatomic molecules by the same approach would become very cumbersome, for several reasons. First, the reduced symmetry of larger molecules makes the selection rules less stringent (see Chapter 11). Second, larger molecules often will be in condensed phases, and rotational transitions cannot be observed. Furthermore, the density of vibrational states (remember, there are 3N‐6 normal modes) often prevents progressions like the one shown in Figure 10.5b from being observed. This simplified and more qualitative approach pursued here utilizes the concept of localized electronic transitions in “chromophores” (literally “color carriers”), that is, groups in which the transition occurs. This is based on the observation that molecules that contain a carbonyl group, for example, exhibit similar UV spectral features regardless of the exact environment the carbonyl group; however, this concept contradicts the fact the electronic states between which the transitions occur are delocalized molecular orbitals which vary from molecule to molecule. This is somewhat analogous to the approach using group frequencies in vibrational spectroscopy although we know that a “group vibration” always extends over the entire molecule.

10.5.1 Selection Rules for Electronic Transitions For diatomic molecules, we had the selection rule (10.36)

For polyatomic chromophores in larger molecules, it is not possible to define good quantum numbers; therefore, the first condition in Eq. (10.25) no longer applies, and transitions need to be evaluated on an individual basis, using group theory (Chapter 11) and the symmetry of the ground and excited states. What remains is the condition

which implies that singlet to triplet transitions remain forbidden. This particular aspect will determine many aspects of fluorescence and phosphorescence and will be discussed in Section 10.6.

10.5.2 Common Electronic Chromophores The term “chromophores” applies to any part of the molecules, generally double‐bonded atoms or metal atom that exhibit a somewhat “localized” electronic transition somewhere in the ultraviolet to visible part (190–800 nm) of the electromagnetic spectrum. Such chromophores include carbonyl groups, the peptide linkage, aromatic groups, or metal ions in coordination compounds such as myoglobin. 10.5.2.1 Carbonyl Chromophore A carbonyl “chromophore,” for example, may be visualized by inspection of an approximate MO energy scheme of the orbitals involved in a carbonyl group in a molecule such as formaldehyde, H2CO, see Figure 10.7a. These orbitals can be described, in order of increasing energy, as the σC − O bond, the πC − O bonding orbitals, the nonbonding orbitals of the oxygen nO, and the antibonding and orbitals, as shown in Figure 10.7a. The transition occurs at about 195 nm and is dipole allowed with a molar extinction coefficient of about 103 [L/mol cm]. The transition, on the other hand, is electric dipole forbidden and is observed as a weak band at about 275 nm with a molar extinction coefficient of c. 20 [L/mol cm], resulting in an absorption spectrum shown schematically in Figure 10.7b. The values of the transition

wavelength and dipole strengths are reported here as approximate numbers, since they depend on the exact nature of the molecule and its physical state, the chemical environment such as solvation, etc. Yet, most ketones larger than formaldehyde exhibit quite similar UV absorption spectra.

Figure 10.7 (a) Approximate MO energy level diagram and UV transitions for a carbonyl chromophore. no denotes a nonbonding orbital on the oxygen atom. (b) Simulated UV absorption spectrum of a carbonyl chromophore with extinction coefficients ε of 1000 and 20 for the π*←π (at 195 nm) and π*← nO (at 275 nm) transitions, respectively. The peptide linkage shown below incidentally has a very similar UV absorption spectrum with a high‐energy (c. 195 nm) transition and a lower‐energy transition that contribute prominently to the protein circular dichroism (CD) spectra discussed in Section 10.7.

10.5.2.2 Olefins The C=C double bond exhibits the π* ← π transition around 180 nm, depending on the molecular species and chemical environment. This transition is dipole allowed with a molar extinction coefficient of >1000 [L/mol cm]. As discussed in Section 2.5, molecules containing conjugated double bonds have lower‐energy π* ← π transition but with higher extinction coefficients. In 1,3‐butadiene, for example, the transition occurs at c. 220 nm with a molar extinction coefficient of ∼20,000 [L/mol cm]. With increasing number of double bonds, the transition energy is lowered even more. In carotene, where the conjugated chain incorporates nine double and ten single bonds, the transition has been shifted all the way into the visible range, with two prominent peaks at c. 450 and 500 nm and a molar extinction coefficient of 140 000 [L/mol cm]. All four DNA bases adenine (A), guanine (G), thymine (T), and cytosine (C) have highly conjugated C=C, C=O, and C=N

chromophores that absorb in the 240–280 nm region. Double‐ stranded DNA has a broad, featureless band at 260 nm that exhibits sensitivity to base‐stacking interaction: upon undergoing a transition from double‐ to single‐strand forms, the absorption intensity increases by about 20%. The reduction of the absorption upon base stacking is known as hypochromicity, which is an interaction that can be explained by the exciton theory discussed in Section 10.7.4. 10.5.2.3 Benzene Another common chromophore in UV spectroscopy are aromatic benzene derivatives such as phenyl groups. Benzene itself has an orbital energy diagram for the six most energetic electrons as shown in Figure 10.8a. The lowest energy transition from the HOMO to the lowest unoccupied molecular orbital (LUMO) occurs at 255 nm with a molar extinction of 180 [L/mol cm]. As described before for the electronic absorption spectrum of molecular oxygen, the spectrum of benzene shows a progression of bands as shown in Figure 10.8b. This progression is due to vibronic transitions into the symmetric ring breathing mode in benzene. This transition is symmetry forbidden in infrared (IR) absorption but produces an extremely strong Raman peak at c. 992 cm−1. The transition into vibrationally excited states of the excited electronic state, however, is dipole allowed and produces the progression of peaks that are spaced by vibrational quanta. 10.5.2.4 Other Aromatic Molecules Toluene and phenylalanine exhibit UV spectra quite similar in wavelength and intensity to benzene. In both cases, vibronic effects are observed, although not as well resolved as in benzene. Aromatic amino acids (tyrosine and tryptophan) exhibit similar UV features, but these are shifted toward longer wavelength, presumably due to more delocalized aromatic systems, particularly in the case of tryptophan.

Figure 10.8 (a) Approximate molecular orbital energy level diagram of the highest occupied and lowest unoccupied molecular π orbitals of benzene. The gray vertical arrow corresponds to the electronic transition shown in panel (b). (b) Vibronic structure of the transition shown in panel (a). The vibrational structure is due to Franck–Condon transitions into the symmetric ring breathing vibration of benzene. 10.5.2.5 Transition Metals in the Electrostatic Field of Ligands Metal atoms or ions in tetrahedral or octahedral ligand environments also constitute chromophores that are generally described in terms of ligand field theory or in terms of crystal field theory for solids. Since many transition metal complexes have either four or six ligands, the symmetry of the resulting complexes either belongs to the Td or Oh point groups (see Chapter 11). In both cases, the ligands provide a perturbation in space that effects the energetics of the 3d orbitals. In octahedral ligand fields, the energy of the dxy, dxz, and dyz orbitals that transform as the t2g irreducible representation is lowered by the interaction with the ligand orbitals. The energy of the and orbitals that transform as the eg irreducible representation is raised, and the degeneracy of the five original 3d orbitals is split into a triply degenerate and doubly degenerate set of orbitals. The colors of transition metal complexes are primarily due to electronic transition between these two sets of orbitals and depend of course on the energy difference Δo between these orbitals. Here, the subscript “o” denotes an octahedral ligand filed. Typical values of Δo correspond to

visible or near‐IR transition energies and depend, in turn, on the electron density of the ligand. The energy splitting Δo also determines whether the complex is “high spin” or “low spin”: in Fe2+ complexes, for example, the six iron valence electrons can either all occupy the three t2g orbitals with completely paired spins (a “low‐ spin” situation) or distribute in all five orbitals (“high‐spin” case). In the latter situation, the two electrons in the eg orbital would occupy two different orbitals with parallel spin, and the other four electrons are in the three t2g orbitals. The total number of unpaired spins is four in this case. In tetrahedral ligand fields, the situation is opposite in that the energy of the doubly degenerate orbitals that transform as the e irreducible representation is lowered, whereas the triply degenerate orbitals of the t2 representation are raised in energy. Due to the different interaction geometry of the 3d orbitals with a tetrahedral ligand field, the energy splitting Δt is only about one half of Δo. For more details on the electronic states and transitions in transition metal complexes, the reader is referred to textbooks on inorganic chemistry, for example, Cotton and Wilkinson [5].

10.6 Fluorescence Spectroscopy Fluorescence is a direct consequence of the Franck–Condon principle discussed in Section 10.4.2 and involves vibronic transitions to be discussed in the next section. Fluorescence spectroscopy is a highly useful spectroscopic method that has found broad applications in cellular biology in the form of fluorescence microscopic imaging, fluorescence resonance energy transfer, and elucidation of very fast chemical reaction mechanisms. While fluorescence is a process that occurs to some extend in many molecular systems (intrinsic fluorescence), its main applications involve the use of fluorescent probes or dyes that have been chemically modified to bind to particular sites in a sample such that the location and abundance of the binding site can be determined with high accuracy. A particularly useful fluorescent probe is one referred to as “green fluorescent protein” (GFP). The gene to produce

this protein is known and available. Thus, this gene can be spliced into the DNA of a host, for example, into a gene that is responsible for the formation of a specific organ, say, the liver, in a mammal. When such a tagged animal grows, it produces an organ that incorporates GFP, and one literally has an animal that has a green fluorescent liver. One can only imagine how important such tools are in developmental biology and cancer research.

10.6.1 Fluorescence Energy Level (Jablonski) Diagram The discussion of the fluorescence process starts with the same energy level diagram that was introduced in the discussion of the Franck–Condon principle in Section 10.4.2 and Figure 10.6 but includes other pathways that give rise to fluorescence. In Figure 10.9a, the lower anharmonic potential curve, marked S0, represents the (singlet) electronic ground‐state potential energy of a vibrational coordinate in a polyatomic molecule. The upper potential curve, marked S1, represents the excited electronic state with a broader potential function shifted along the internuclear axis toward a slightly longer bond distance. The fluorescence process consists of a very fast electronic absorption processes with ΔS = 0 (see Eq. (10.25)) into the excited singlet state S1 indicated by the black up arrows that are governed by the Franck– Condon principle. In the case shown, three vibronic states in the S1 manifold are excited. These states may deactivate back to their original state by spontaneous emission or deactivate by collisional energy loss, mostly to solvent molecules, into the vibrational ground state of the S1 state. The collisional deactivation processes are indicated by the dashed lines in Figure 10.9a. Emission of redshifted photons from this state into various vibrational states of the electronic ground state returns the molecule back into the electronic ground state, indicated by the gray down arrows. This last step again follows the Franck–Condon principle in that the transitions occur between states of highest overlap of the vibrational wavefunctions. The collisional deactivation does not necessarily follow dipole selection rules; thus, electronic states can be populated that cannot be excited directly by absorption. Whereas the original excitation

process is very fast, the fluorescent emission occurs at much slower timescale, about 10−7–10−9 [s] after the excitation. Measurement of the fluorescent lifetime gives information on molecular dynamics and solvent exposure of the excited vibronic state.

Figure 10.9 (a) Energy level (Jablonski) diagram for fluorescence. (b) Energy level diagram for intersystem crossing and phosphorescence.

10.6.2 Intersystem Crossing and Phosphorescence Another deactivation process may occur that is shown schematically in Figure 10.9b. The excitation process into the S1 state is the same as in Figure 10.9a. What is different is the existence of a triplet state, marked T1 in Figure 10.9b and indicated by the gray excited‐state potential function. The direct transition into the triplet state from the S0 ground state is forbidden by the ΔS = 0 selection rule. If one of the vibrational energy levels of the T1 and the S1 states are near degenerate (in Figure 10.9b, the third vibrationally excited state of T1

and the fourth of S1), the excitation energy can be transferred from the singlet state to the triplet state. This state deactivates through collisional energy transfer to the ground vibrational state of T1 and is trapped there. Transitions back into the S1 state are unlikely because the triplet state is lower in energy than the S1 state and those to the ground state of the S0 manifold are forbidden. The consequence is that the excitation energy is trapped in the T1 state and may persist there for milliseconds or even longer. Weak and slow emission will occur eventually, which is observed as phosphorescence.

10.6.3 Two‐Photon Fluorescence Another application of fluorescent methods is that where the excitation source utilized is an extremely powerful laser with photon energies half of the amount required to excite a vibronic state in the S1 manifold. This so‐called two‐photon fluorescence (TPF) is a nonlinear optical process in which two (near‐IR) photons, typically from a Ti : sapphire laser, are combined in the medium by the first‐ order non‐linear susceptibility into one photon of half the wavelength or twice the energy (for discussion of non‐linear spectroscopic effects, see Appendix 3). This process is represented in Figure 10.10b. In this case, the fluorescent emission is blueshifted with respect to the excitation wavelength. The advantage of TPF spectroscopy is that the near‐IR excitation wavelength has a much better penetration depth into turbid or heterogeneous media, for example, into biological tissue. Thus, TPF can be used to detect fluorescence from depth of many millimeters inside a sample of human or animal tissue. This technique, combined with second‐harmonic generation (SHG) (sensitive to collagen) and coherent anti‐Stokes Raman (CARS) (sensitive to lipids) imaging modalities in a one microscope setup, has created images of biological specimens of unprecedented compositional details [6]. The two nonlinear processes, SHG and CARS, will be discussed in Appendix 3.

10.6.4 Summary of Mechanisms for Raman, Resonance Raman, and Fluorescence Spectroscopies Fluorescence and Raman spectroscopies have similar vibronic origins and may occur together in an experiment, albeit at quite different time scales. The following discussion is presented for a simple diatomic molecule with one degree of vibrational freedom, although one could consider Figure 10.10 to hold for any vibrational coordinate, for example, a stretching coordinate in a polyatomic molecule. As discussed above in Figure 10.9a, the lower row of potential functions in Figure 10.10 represents the electronic ground state with its associated anharmonic vibrational wavefunctions. The upper row depicts the potential surface for an electronically excited state. A typical fluorescence process is depicted in panel (A) of Figure 10.10. Here, the system is promoted into excited vibronic states, as indicated by the gray arrows. The specific excited vibrational level of the electronically excited state is determined by the Franck–Condon principle. Thus, the most likely transitions shown in the example here are into the n = 2 and n = 3 vibrational states of the electronically excited state. After slow collisional deactivation (in the nanosecond regime, shown by dashed lines in panel A) into the vibrational ground state of the electronically excited state, transitions occur rapidly into the ground electronic state manifold, again determined by the Franck–Condon factor. The fluorescence emission is redshifted with respect to the exciting wavelength and often exhibits vibronic fine structure, i.e. the fluorescent bands are split by the vibrational quantum (see Problem 10.12).

Figure 10.10 Schematic diagrams representing fluorescence (a), two‐photon fluorescence (b), spontaneous Raman (c), and resonance Raman (d) processes. See text for details. Panel (B) of Figure 10.10 shows the TPF process. Two photons with half the energy required to reach the excited vibronic state combine in a nonlinear process (see Appendix 3) to create one photon with sufficient energy to reach the excited vibronic state. The process proceeds via a virtual state (dotted energy level) that is created by the second‐order nonlinear susceptibility. High laser fields are required to make this process practical. Collisional deactivation leads to population of the same state as in standard fluorescence spectroscopy. The fluorescence emission is blueshifted with respect to the exciting wavelength. This technique, as pointed out above, offers the advantage of using low‐energy photons with better penetration into turbid media to excite the fluorescence. In contrast to fluorescent processes, the spontaneous, nonresonant Raman process (far from resonance) is shown in Figure 10.10c. Here, the dashed horizontal line represents the virtual state that is created by the polarizability of the molecule. In the resonance Raman process, the virtual state coincides or is energetically very close to a real electronic state, as shown in Figure 10.10d. In this case, the

fluorescent photons and the Raman‐scattered photons, indeed, have the same energy and may overlap in the observed spectra. Furthermore, the Raman‐scattered intensities may be amplified thousandfold or more due to the small energy difference in the denominator of the expression for the polarizability tensor elements (see Eq. [5.43]). Interestingly, fluorescence and Raman spectra can be separated by the different timescales at which they occur. The emission of photons from the virtual state in Raman spectroscopy takes place in femtoseconds, whereas the fluorescent photons are emitted nanoseconds or even microseconds after the excitation. Using pulsed lasers and time‐gated detectors, it is therefore possible to eliminate fluorescent contributions from the Raman spectra.

10.7 Optical Activity: Electronic Circular Dichroism and Optical Rotation 10.7.1 Circularly Polarized Light and Chirality There are molecules that possess a property that allows them to interact differently with left and right circularly polarized light. These molecules are said to be chiral or possess “handedness,” and such molecules exist in two forms known as enantiomers. Left and right circularly polarized light itself is chiral and can be viewed as a form of light in which the electric (or magnetic) vector proceeds along a left‐ or right‐handed helix, rather than oscillates in a plane. The two forms of circularly polarized light are shown in Figure 10.11 and can be created from two orthogonal, linearly polarized waves with wavelength λ that are shifted by ±λ/4 with respect to each other (see Figure 10.11a). The different interaction of a medium with left and right circularly polarized light is referred to as optical activity.

Figure 10.11 (a) Left (top) and right (bottom) circularly polarized light. (b) Examples of the two enantiomeric forms of an asymmetric molecule, fluorochlorobromomethane (top), and a dissymmetric chiral molecule, 2,3‐pentadiene. The planes in which the methyl groups lie and the symmetry axis are shown. Handedness is a property found in macroscopic entities as well as in molecules. Think about a pair of shoes or a regular bold and nut you can purchase in a hardware store, which both normally are right‐ handed. A left‐handed nut will never fit on a right‐handed bolt, even after turning it around, since the handedness is invariant to all symmetry operations (see Chapter 11) except reflection by a mirror plane. Thus, your left shoe will be superimposable on the mirror image of your right shoe. This last sentence actually defines chirality, in general: an item is chiral if it is nonsuperimposable on its own mirror image (or superimposable on the mirror image of its enantiomer). Chiral molecules fall into two generally classes: asymmetric molecules that belong to a symmetry group that has no symmetry element except the identity element (see Chapter 11) and dissymmetric molecules that belong to purely rotational point groups. An example of an asymmetric molecule would be any tetrahedral carbon atom surrounded by four different groups, such as glyceraldehyde or fluorochlorobromomethane, shown in Figure 10.11b, whereas an example of a dissymmetric molecule would be 2,3‐pentadiene (also known as 1,3‐dimethylallene), which has a C2‐axis through the center C atom and at 45o with respect to the two planes that contain the methyl groups and the hydrogen atoms. Helical molecules also belong to the class of dissymmetric molecules. For a chiral molecule to exhibit optical activity, one of the enantiomers must be in excess over the other. A 50:50 mixture of enantiomeric molecules is known as the racemate, and a racemic mixture does not exhibit optical activity. Optical activity was discovered in the mid‐nineteenth century by L. Pasteur who separated mirror image crystals of ammonium sodium

tartrate, a salt of tartaric acid, under a microscope into “left‐ and right‐handed” forms. When these mirror image crystals were separately dissolved in water, the resulting solutions rotated the plane of incident linearly polarized light in opposite directions. The “optical rotation” is one of several manifestations of electronic optical activity, that is, an unequal response of the medium toward left or right circularly polarized light.

10.7.2 Manifestation of Optical Activity: Optical Rotation, Optical Rotatory Dispersion and Circular Dichroism The rotation of plane polarized light described in the previous paragraph can be explained as follows. Linearly polarized light can be described as being composed of two co‐propagating circularly polarized light waves, one of which is left circularly polarized and the other is right circularly polarized. If the refractive index of the sample toward right circularly polarized light, nR, differs from that for left circularly polarized light, nL, the two circularly polarized waves propagate with different velocity through the medium; consequently, the polarization plane of linearly polarized incident light will be rotated upon exiting the sample solution. In the following discussion, the subscripts L and R represent left and right circular polarizations. Thus, optical rotation can be described by the difference in refractive indices toward circular polarization as (10.45) Optical rotation was first observed in the visible part of the spectrum and was found to generally increase in magnitude with decreasing wavelength of light. Furthermore, it was found that the optical rotation underwent anomalous dispersion at the wavelength of an absorption maximum as shown by the top trace in Figure 10.12. The wavelength‐dependent variation of the optical rotation of a medium is known as optical rotatory dispersion (ORD). In addition, it was found that within an absorption peak, there exists a nonzero

difference between extinction coefficients toward left and right circularly polarized lights: (10.46)

Figure 10.12 Relationship between ORD and CD. Notice that the differential refractive index changes sign within a region of maximum CD signal. or (10.47)

This effect is known as circular dichroism. The relationship between ORD and CD is shown in Figure 10.12 that is analogous to Figure 5.6 with a minor modification, namely, that the differential refractive index actually undergoes a sign change, whereas the refractive index shown in Figure 5.6 always stays positive. ORD and CD are often introduced during the discussion of optical activity and stereochemistry in introductory courses in organic chemistry. In the following two sections, the optical activity of asymmetric molecules and dissymmetric molecules will be presented separately, although this conceptual separation is somewhat arbitrary but serves as a pedagogical tool to point out different origins of natural optical activity.

10.7.3 Optical Activity of Asymmetric Molecules: The Magnetic Transition Moment The transitions discussed so far in this book, giving rise to rotational, vibrational, atomic, and molecular electronic spectroscopies, have one feature in common: they are caused by the electric field of the incident light, mediated by the electric dipole operator, according to (3.20) The exceptions to this statement are the transitions observed in NMR that can be attributed either to the electric field and electric dipole moment or in terms the magnetic dipole operator acting on net magnetization, as described in Section. One defines the dipole strength D01 for a transition from the ground state |0〉 to an excited state 〈1| as (10.48) D01 represents the integrated area under an absorption band. Up to now, we have assumed that only the electronic dipole moment

contributes to the transition, and the effect of the magnetic field of electromagnetic radiation has been ignored. There is a class of molecules, however, in which the magnetic transition moment also contributes to observed transitions, namely, optically active molecules. For these molecules, one defines a quantity known as the rotational strength R01 in analogy to Eq. (10.48): (10.49) This quantity corresponds to the integrated area under a CD curve for a pure enantiomer for a given transition. In Eq. (10.49), μ is the electric dipole operator defined earlier (Eq. [3.4]) as (3.4) and m is the magnetic dipole transition operator defined as (10.50) where e, m, r, and p are the charges, masses, positions, and momenta of particle i and the summation is over all particles. The vector cross product in Eq. (10.50) is the same that was encountered before for the definition of the angular momentum operator; see Eq. (6.2). Thus, the magnetic dipole operator expresses a torque exerted by the magnetic field on the motion of electrons. In most manifestations of natural optical activity, the desired observables – i.e. Δε, ΔA, or R01 – arise through the interference of electric and magnetic dipole transition moments. The components of the electric dipole operator transform like translation along the X‐, Y‐, and Z‐directions, whereas the magnetic moment transforms like rotation about the X‐, Y‐, and Z‐axis (such as RX, RY, and RZ; see Chapter 11). Only in molecules that belong to the purely rotational

point groups do the magnetic and electric dipole operators transform in the same irreducible representation and thereby allow for optical activity. Molecules in these molecules possess “chirality” or handedness as introduced above in Section 10.7.1. The sensitivity of chiroptical techniques toward the handedness of the molecule results directly from the form of the magnetic momentum operator: it contains the vector product of momentum and position vectors, the result of which is another vector. The sign of this vector is determined by the handedness of the coordinate system and will change sign upon converting from a left‐handed to a right‐handed coordinate system. By changing the configuration of the molecule from one to the other enantiomer, the sign of the magnetic transition moment changes. Similarly, by keeping the configuration of the molecule fixed and changing from left to right circularly polarized light, the sign of the magnetic transition moment is reversed.

Figure 10.13 (a) CD (top) and UV absorption (bottom) spectra of an asymmetric molecule, camphor. (b) CD (top) and UV absorption spectra (bottom) of two achiral but dissymmetrically arranged chromophores (oval structures) to produce an “exciton couplet”. Source: Redrawn from Woody [10].

The consequences of the last paragraph are that the enantiomeric forms of a chiral molecule give exactly equal and opposite CD spectra, as shown in Figure 10.13a for (R) and (S) camphor. This result shows the standard form of electronic CD spectra in which an electronic transition – in this case, the carbonyl π* ← n transition at ca. 290 nm – is perturbed by a nearby chiral center and exhibits equal and opposite CD for the two enantiomeric forms. CD is a relatively weak effect, as measured by the ratio of Δɛ/ɛ, which is about 10−3 for many chiral molecules. In the absence of an electronic transition, both CD and optical rotation of a molecule will be very small since the magnitude of the magnetic transition moment alone is much smaller than the electric transition moment. Thus, a chiral hydrocarbon such as 3‐

methylheptane or 2‐deuterobutane does not exhibit any observable CD above 200 nm and shows minimal optical rotation at 589 nm (the standard wavelength at which optical rotation is reported). However, in the presence of a chromophore, the chiral perturbation of its electronic spectrum has been extremely useful in assigning the stereochemistry of many compounds. In chiral molecules whose structures are derived from the androstane skeleton and contain a carbonyl chromophore (such as testosterone and estrogen), the observed CD patterns correlate with the location of the chiral center within the reference framework of the C=O group. These resulting “quadrant” or “octant” rules were used to establish the stereochemistry of many of these substituted androstane structures [7].

10.7.4 Optical Activity of Dissymmetric Molecules: Transition Coupling and the Exciton Model The discussion of CD, so far, has emphasized asymmetric species (belonging to point group C1 that has only the identity symmetry element) whose optical activity is attributed to the interaction of electric and magnetic dipole transition moments; see Eq. (10.49). However, optical activity is also exhibited by dissymmetric species of pure rotational point groups. These molecules are exemplified by twisted or helical structures, as shown in Figure 10.11b. Inherently achiral chromophores or groups can produce optical activity by a coupling mechanism of the electric dipole moments that is described by an “exciton” model of optical activity (see Tinoco [8] and Bayley et al. [9]). In the exciton model, one considers a chiral polymer, such as peptide α‐helix composed of n identical peptide units in a fixed chiral geometry, for example, in a helix. The peptide electronic transitions, notably the electronically allowed π* ← π transitions, have near‐ degenerate energy eigenvalues En and interact with each other via dipolar coupling to form a delocalized excited state. A one‐photon excitation into the manifold of coupled peptide groups results in an “excitonic” state in which the excitation is highly delocalized between

the coupled excited states. The energies of these excited states are defined by the eigenvalues of the interaction energies: (10.51)

where the diagonal values are the degenerate or near‐degenerate energy eigenvalues of unperturbed transitions and the off‐diagonal elements Vij are the interaction or coupling energies between the transition moments, given to a first approximation by dipole–dipole interactions: 10.52

In Eq. (10.52), μi and μj are the transition dipole moments of groups i and j, and Tij is the distance vector between them. For the case of n coupled transitions, Eq. (10.51) is diagonalized numerically to yield the energy eigenvalues νk of the k'th exciton component and the eigenvector matrix C, from which the transition dipole strength Dk for each of the k coupled transitions can be computed according to 10.53

where cik are the appropriate eigenvector matrix elements. Such molecular systems containing chirally arranged transitions exhibit rotational strengths for the k'th exciton component that are given by

10.54

where c is the velocity of light. For the dimeric case (n = 2), Eq. (10.54) simplifies to the well‐known “coupled transition” equation [8] 10.55 in which R± denotes the rotational strengths of the symmetric ∣ +〉 and antisymmetric ∣ −〉 combination states of the two transitions. The resulting CD features are often referred to as an “exciton couplet,” shown in Figure 10.13b, in which the absorption spectrum shows two (often overlapping) peaks split by the “exciton” energy. The CD of the symmetric and antisymmetric states shows a couplet of peaks with opposite signs, as described by Eq. (10.55). The sign pattern can be correlated with a right‐ or left‐handed twist. The exciton formalism conveys a simple model for the optical activity produced by the dissymmetric interaction of achiral transitions. Notice that in the presentation here, the magnetic transition moment was omitted, although it is nonzero as well, but the electronic coupling contributions are assumed to dominate the exciton rotational strength. The most prominent application of the exciton formalism is for the interpretation of the large CD signals observed for different secondary structural motifs of peptide chains in proteins, depicted in Figure 10.14. Here, the large CD signal with a zero crossing at just over 200 nm for the α‐helix is assigned as the π* ← π exciton component. Although model calculations are hampered by the uncertainty in the direction of the transition moment for this transition, good agreement between observed and computed CD spectra was achieved [10]. It is interesting that the β‐sheet conformation exhibits a CD couplet of the same sign pattern as the α‐ helix that leads to the conclusion that the sheet conformation has an

overall twist and is not planar. What is often referred to as the “random coil” conformation should be rather called the poly‐L‐Pro (II) structure that appears to have an overall twist opposite to that of the sheet structure. These peptide secondary structures and models to reproduce the observed CD patterns have been the subject of numerous research papers (see [10] and references therein).

10.7.5 Vibrational Optical Activity In a chiral molecule such as camphor (see Figure 10.13a), one observes the perturbation of the chromophore – in this case, a carbonyl group – by the chiral environment. There are, however, other methods that probe chirality directly at the chiral center. These methods are (IR) vibrational circular dichroism (VCD) and Raman optical activity (ROA). In these forms of vibrational optical activity (VOA), the effects of the chirality on vibrational, and not electronic, transitions are observed. VCD is an analogue to electronic CD in that Eq. (10.49) describes VCD, if vibrational wavefunctions are substituted. ROA, like Raman spectroscopy, is a scattering, rather than an absorption phenomenon where the scattered intensity differential contains cross terms between the polarizability tensor elements defined earlier

Figure 10.14 CD (top) and UV absorption (bottom) spectra of (a) α‐helical, (b) β‐sheet, and (c) polyproline II (also sometimes referred to as “random coil”) peptide conformations. (5.42) and the magnetic polarizability (10.56)

Thus, both ROA and VCD are truly forms of vibrational spectroscopy but take into account the magnetic transition dipole moment that was omitted in the discussion in Chapter 5. As indicated earlier, both techniques offer the advantage that they can sample the chirality right at the chiral center and do not require an electronic transition in a chromophore. The theory and applications of these techniques are beyond the scope of this book, and the reader is referred to abundant literature in the field [11].

References 1 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall. 2 Trentmann, J. et al. (2003). Impact of accurate photolysis calculations on the simulation of stratospheric chemistry. Journal of Atmospheric Chemistry 44 (1): 225–240. 3 Stamnes, K. (2015). Radiation Transfer in the Athmosphere: Ultraviolet Radiation, in Encyclopedia of Athmosphereic Scince (eds. G.R. North, J. Pyle and F. Zhang). Academic Press. 4 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon.

5 Cotton, F.A. et al. (1999). Advanced Inorganic Chemistry, 6e. Wiley. 6 Pope, I. et al. (2013). Simultaneous hyperspectral differential‐ CARS, TPF and SHG microscopy with a single 5 fs Ti:Sa laser. Optics Express 21 (6): 7096–7106. 7 Crabbe, P. (1972). ORD and CD in Chemistry and Biochemistry. New York: Academic Press. 8 Tinoco, I. (1962). The exciton contribution to the optical rotation of polymers. Radiation Research 20: 133–139. 9 Bayley, P.M., Nielsen, E.B., and Schellman, J.A. (1969). The rotatory properties of molecules containing two peptides groups: theory. The Journal of Physical Chemistry 73 (1): 228–243. 10 Woody, R.W. (2005). The exciton model and the circular dichroism of polypeptides. Monatshefte für Chemie 136: 347–366. 11 Nafie, L.A. (2011). Vibrational optical activity: principles and applications. Chichester, UK: Wiley. 12 Pescitelli, G. Electronic Circular Dichroism. https://encyclopedia.pub/212

Problems 1. Figure P.1 shows the molecular orbital energy level diagram for first row (H, He) diatomic molecules. Which of the following molecules are expected to exist, and what would be the bond order? a. H2+ b. H2 c. H22− d. He2+

Figure P.1 2. Figure P.2 shows the molecular orbital energy level diagram for the valence electrons of N2. Based on this diagram, answer the following questions. a. What is the bond order in N2? b. Does this bond order agree with the Lewis structure? c. Is the molecule paramagnetic or diamagnetic?

Figure P.2 3. Figure P.2 also holds for the MO scheme for homonuclear diatomic molecules O2 and F2. For each of the four species listed below, draw the MO diagram. From these diagrams, indicate orbital occupancy, bond order, and magnetic properties (paramagnetic vs. diamagnetic). Predict the vibrational stretching frequency, assuming that a single bond contributes a stretching force constant of 570 [N/m]: a) O2+ b) O2 (

)

c) O2 ( d) F2 4. In the second‐row diatomic molecules B2 and C2, the MO energies of the σ2p and π2p orbitals are exchanged as shown in Figure 3. Predict the electron configuration, bond order, vibrational stretching frequency (see previous problem), magnetic properties, and term symbol for the lowest energy state of B2.

Figure P.3 5. List the two electronic transitions observed in the 180–300 nm UV spectral region of ketones. 6. Describe a spectroscopic method discussed in the chapter to monitor the “melting” of DNA from double‐ to single‐stranded structures. 7. What conditions must be fulfilled for a medium to exhibit circular dichroism (CD)? 8.

a. Name everyday items that possess chirality. b. What is circularly polarized light?

9. A sample molecule dissolved in a transparent solvent and contained in a cuvette with 1 cm path exhibits an absorbance of 1.22 in a transition that has a molar extinction coefficient of 1490 [L/(mol cm)] at 280 nm. a. What is the concentration of the sample? b. What percentage of the incident light is transmitted at 280 nm? 10. For polyelectronic atoms or ions, write the orbital approximation, and explain the symbols, notation, and

implications of this approximation. 11. What transitions are responsible for protein absorptions at the following? a. About 200 nm b. About 270 nm c. Between 270 and 290 nm d. About 480–600 nm e. About 1650 cm−1 12. Draw a vibronic energy level diagram and explain the relationship between the vibronic absorption spectrum and the corresponding fluorescence spectrum shown in the Figure P.4; in particular, discuss the origin of the near‐equidistant bands in both spectra.

Figure P.4 13. Consider the hypothetical MO energy scheme shown in Figure P.5. Fill in the resulting MOs, and discuss the bond order and the spin multiplicity of the resulting MO scheme, if each atom contributes a. 2 electrons b. 3 electrons

Figure P.5

Note 1 Bond order is defined as (number of electrons in bonding orbital – number of electrons in antibonding orbital)/2.

11 Group Theory and Symmetry In earlier chapters, selection rules for infrared (IR), Raman, and electronic spectra were discussed, and general rules were developed to determine whether or not a given transition is allowed or not. For IR spectroscopy, we found that for dipole‐allowed transitions, the integral (5.45) must be nonzero. This led to the conclusion that a normal mode of vibration must change the dipole moment of a molecule: (5.44) In analogy, for a mode to be Raman active, the polarizability (5.46)

must change during a normal mode. For a transition between electronic states to occur, the expression (10.29)

which holds within the Born–Oppenheimer approximation, must be nonzero. In the following discussion, the principles of electric dipole‐ allowed transitions will be developed for vibrational spectroscopy

but hold as well for electronic transitions, except that electronic (or vibronic) rather than vibrational wavefunctions need to be utilized. In small molecules, such as CO2, simple arguments help determine which modes are IR and/or Raman active. In more complex molecules, particularly those of high symmetry, one needs to establish how the molecular symmetry influences their spectra. This is the subject of group theory, introduced in this chapter. We shall see that each normal mode of vibration can be described by a symmetry representation and that the transformation properties of these representations will determine whether or not a given normal mode of vibration is symmetry allowed. Group theory is a topic that is much too broad to be treated in full detail in just one chapter; thus, the reader is referred to specialized textbooks such as the classic Chemical Applications of Group Theory [1] for derivations and more examples. However, the basic concepts of group theory as applied to vibrational and electronic spectroscopies will be presented in this chapter, and a connection between this of mathematics to multidimensional vector spaces and linear algebra will be made. The aim of this chapter is to present sufficient information for the discussion of spectroscopy and the derivation of symmetry‐based selection rules.

11.1 Symmetry Operations and Symmetry Groups Molecules are classified into symmetry‐related categories, called “symmetry groups,” according to the number and nature of symmetry operations that can be carried out on a molecule. A symmetry operation is a procedure applied to a molecule that leaves it in an arrangement that is indistinguishable from the arrangement before the operation was carried out. An example of such a symmetry operation is shown in Figure 11.1: when the water molecule is rotated by 180° about the axis shown, it appears identical to and indistinguishable from its original arrangement, since the two H atoms are indistinguishable. This particular symmetry operation is called a proper twofold rotation about a symmetry axis and referred

to as a C2 symmetry operation. The symbol “C” will be used for all proper rotations, which will be discussed below in more detail. Investigating the symmetry properties of a number of arbitrary molecules, or even macroscopic objects such as a snow crystal, a cube, certain letters, and many other items which one intuitively associates as being “symmetric,” one finds that there exist only a few distinct and independent symmetry operations. These can be combined and repeated to create other symmetry operations. For the discussion of the symmetry properties of small molecules, there are five symmetry operations of interest: the identity element, proper axes of rotation, the reflection by a mirror plane, the center of inversion, and rotation–reflection axes, also known as improper axes of rotation. These operations will be discussed next. 1. The Identity Element, designated E. This operation leaves the molecule unchanged, and the necessity for such an operation appears superfluous at first. However, the identity operation is necessary for two reasons. The first of these is purely mathematical and will become clear later during the discussion of symmetry groups. The other reason can be viewed intuitively as follows: if one defines a symmetry operation as a procedure that will return the molecule into a state that is indistinguishable from the state before the operation, a formalism is needed to include the possibility that the molecule was, indeed, left unchanged, that is, the molecule was not operated on at all.

Figure 11.1 Example of a symmetry operation (C2).

Figure 11.2 (a) Example of one of three σν mirror planes in the molecule chloroform. Notice that the major symmetry axis, marked C3, is contained in the symmetry plane. (b) Example of a σh plane that is perpendicular to the C3 axis in the molecule boron trifluoride.

2. Proper Axes of Rotation (Cn). This operation was introduced above for a twofold rotation. In general, one designates Cn operations as rotations about an axis where “n” describes how often the operation has to be repeated until a rotation by 360° has occurred. Thus, Cn indicates a rotation about an axis by 360/n degrees (see Figure 11.1). 3. Reflection by a Mirror Plane (σ). Reflection operations can be visualized as a reflection by a mirror plane that can be parallel or perpendicular to a symmetry axis. The former is called σν and the latter σh. Such mirror planes are shown in Figure 11.2. 4. Center of Inversion (i). When pairs of atoms are at positions for which the coordinates are identically the same except that all their signs are reversed, then these atoms are related by a center of inversion. Planar ethene (C2H4) is a molecule that incorporates a center of inversion (see Figure 11.3), located halfway between the two C atoms. The hydrogen atoms numbered 1 and 4 in Figure 11.3 have equal and opposite coordinates, as do hydrogen atoms 2 and 3 and the two C atoms. 5. Rotation–Reflection Axes or Improper Axes of Rotation (Sn). This operation consists of a rotation about an axis by 360°/n, followed by a reflection by a plane perpendiculars to the axis. The hydrogen atoms in ethane, for example, are related by an S6 operation (Figure 11.4): rotation of the molecule about the direction of the C–C bond by 60° brings the top three hydrogen atoms into a position opposite to the original position of the lower three hydrogen atoms. Reflection of the three upper hydrogen atoms by a plane bisecting the C–C bond brings the upper three hydrogen atoms into the positions originally occupied by the lower three hydrogen atoms.

Figure 11.3 Definition of a center of inversion, located at the coordinate origin, in planar ethene (C2H4). The coordinates of H1 and H4, for example, are equal in value but opposite in sign.

Figure 11.4 Description of an improper rotation operation (S6) for ethane. It is a matter of practice to identify all the symmetry elements of a molecule. Benzene, with a perfect hexagonal structure, belongs to the D6h point group and has the following symmetry elements: two C6 axis, two C3 axes coincident with the C6 axis, one C2 axis coincident with the C6 axis, three C2 axes that bisect opposing sides, three C2 axes that contain opposing atoms, two S6 and two S3 axes, a horizontal symmetry plane σh, three σν planes that contain opposing atoms, and three σd planes that bisect opposing bonds. When one hydrogen atom is substituted by deuterium, or one carbon by a 13C atom, the symmetry is lowered enormously to just contain one each of the following elements: E, C2, σν, and σν′. Certain molecules, such as water, benzene‐d1, and formaldehyde, H2CO, possess the same symmetry elements, namely, E, C2, σν, and σν′. (The prime denotes that there are two mutually perpendicular planes both containing the major axis. These symmetry elements are also denoted as σ[yz] and σ[xz], respectively.) The occurrence of the same symmetry elements in different molecules suggests that the grouping of the four symmetry elements above is, indeed, special, and it can be shown that the four symmetry elements form a group.

A group in the mathematical sense is a collection of elements that are connected according to certain rules: 1. Successive applications of any two or more operations of the group produce another symmetry operation of the group. For example, successive applications of the C2 operation bring the molecule back to its original orientation. Thus, (11.1) with E being another operation of the group. Similarly, (11.2) 2. There exists an identity element E such that for any operation A: (11.3) As pointed out above, the E operation leaves the molecule unchanged; thus, any operation A followed by the E operation is equivalent to performing the operation A only. 3. The associative law holds. Let A, B, and C be symmetry operations in a group. Then, the order in which they are applied to a molecule does not matter: (11.4) 4. Every operation must have an inverse operation. The inverse operation A−1 for a symmetry operation A is defined such that AA−1 = E, i.e. the inverse operation negates the original operation A and leaves the molecule unchanged (or operated on by the E operator). For the discussion of molecular symmetry relevant to molecular vibrations, one needs to be concerned with about 40 different symmetry groups, of which only about a dozen or so are common in molecules. The groups are given special designations depending on the symmetry elements they contain, and the particular group with the symmetry elements discussed above (E, C2, σv, and σv′) is known

as the C2v point group. In this symmetry, or point group, there are four symmetry elements, each occurring only once. Thus, one defines the “order h” of this point group to be 4. Ammonia, NH3, belongs to a symmetry group known as C3v, which contains the following elements:

The two C3 axes denote clockwise and counterclockwise rotation by 120° (see Eq. (11.19)), which are two different operations since they produce different orientation of the hydrogen atoms. The three symmetry planes are all parallel to the C3 axes and contain one of the N–H bonds each. Thus, in C3v, there are six symmetry elements in three classes, and some elements occur more than once in this group. The order of the group is h = 6. The order of a group and the number of classes will be used in the next section for the derivation of the irreducible representations. A few of the common symmetry groups are listed at the end of this chapter.

11.2 Group Representations Next, a scheme will be presented to describe the effect of a symmetry operation mathematically. This is accomplished most easily by considering a Cartesian coordinate system and following the effect of a symmetry operation on this coordinate system. This is shown for an identity (E) and a reflection by a mirror plane, (the yz‐plane) in Figure 11.5. To avoid confusion about the definitions of the symmetry planes σν and σν′, these symmetry planes will henceforth be referred to as σxz and σyz. For the discussion in this chapter, the following conventions will be adopted. When a symmetry operation is performed, the original coordinates are referred to as the x, y, and z coordinates, whereas the coordinates after the transformation are designated as x′, y′, and z′ coordinates. Then, each symmetry operation can be represented by a transformation matrix between the old and the new coordinate

system. For example, the E operation in Figure 11.5 can then be described by (11.5)

whereas the σyz reflection is described by (11.6)

These 3 × 3 transformation matrices can be verified by multiplying out, for example, Eq. (11.6), which yields (11.7) This is what one expects by inspection of Figure 11.5. Thus, the transformation matrices of a Cartesian coordinate system under the four symmetry operations of the point group C2ν can be summarized as (11.8)

The matrix representation introduced in Eq. (11.8) can also be used to demonstrate, by simple matrix multiplication, that the successive application of two symmetry operations of a group will produce another operation of the group. For example, the successive

application of σyz and σxz in the C2ν point group can be represented as follows:

Figure 11.5 Effects of symmetry operations E and σyz on a Cartesian coordinate system. (11.9)

This result implies that the product of the transformation matrices for two consecutive operations must equal the matrix representing the product operation, which is another member of the group, namely, C2.

The traces of the transformation matrices within a group for given objects can be used to construct what is known as “representations” of a group. For example, the traces of the transformation matrices for a Cartesian coordinate system for the E, C2, σxz, and σyz operations in the C2v point group are 3, −1, 1, 1, respectively (see Eq. [11.8]). The “vector” 3, −1, 1, 1, therefore, is a “representation” for the transformation properties of a Cartesian coordinate system for the four symmetry operations that define the point group C2ν: (11.10) where the symbol Γ designates a representation and the numeric values in the “representation vector” are given the symbols X. Thus, the notation X(Γi) implies any of the numeric values listed in Eq. (11.10) for a given representation. Next, we show that this particular representation, [3, −1, 1, 1], can be interpreted as a vector in the four‐dimensional space defined by C2v that can be decomposed into the contributions along four one‐ dimensional unit vectors. These unit vectors are referred to as “irreducible representations” of a group. These irreducible representations can be considered a way of representing the transformation properties of the very simplest objects, such as a general point. Under C2v, a point lying along the z‐axis (the rotation axes) would transform symmetrically (+1) under all symmetry operations of the group and thus have a representation of {1, 1, 1, 1} for the four operations E, C2, σxz, and σyz. A point on the x‐axis, however, would transform antisymmetrically (−1) under C2 and σyz and thus would have a representation {1, −1, 1, −1}. Similarly, a point on the y‐axis will transform antisymmetrically under C2 and σxz and has a representation {1, −1, −1, 1}. It can be shown (see below) that these irreducible representations are the equivalents of unit vectors in a space whose dimension is given by the number of symmetry elements in a group. Thus, C2ν, which has four symmetry elements, will have four irreducible representations, three of which have been visualized so far:

The theoretical derivation of the irreducible representations is quite complicated and follows directly from the orthogonality theorem of group theory (see, for example, [1]), but will not be discussed here in detail. Some of the consequences of this theorem are as follows: 1. The number of irreducible representations (unit vectors) in a group is equal to the number of symmetry classes. Thus, C2v will have four irreducible representations, and the C3v point group, introduced in Section 11.1, has three irreducible representations, although it has six symmetry operations. 2. If one defines a number li to be the dimensions of the ith irreducible representations (the largest number X[Γ]), then (11.11)

In Eq. (11.11), h denotes the order of the group, which was defined in Section 11.1 to be the total number of symmetry operations in a group. Thus, for C2v (11.12) and (11.13) Thus, C2v has four one‐dimensional representations, three of which are listed above. The fourth irreducible representation can be derived from the orthogonality theorem, which postulates that the four representations (being unit vectors) must be orthogonal to each other. Thus, in addition to the three irreducible representations listed above, one writes a fourth orthogonal vector as

which results in a (reorganized) table: (11.14)

Irreducible representations for a few selected other point groups will be discussed later. Next, the orthogonality of the four representations shown in Eq. (11.14) will be demonstrated. For this, an analogy in three‐dimensional space, with unit vectors i, j, and k will be used. In three‐dimensional space, defined by a Cartesian coordinate system, these unit vectors can be represented by the matrix (11.15)

In the matrix defining the i, j, and k vectors, the orthogonality of the unit vectors can be demonstrated by summing the products of the elements corresponding any two unit vectors, for example, i and j, over the three directions x and y: (11.16)

In complete analogy, orthogonality in the four‐dimensional space defined by C2ν is established by summing over all symmetry operations of a group the products of the X′s for any two unit vectors, i and j: (11.17)

For the table of irreducible representations for C2v listed in Eq. (11.14), the vector dot product, designated by the symbol ⊗, of Γ1 and Γ2, is (11.18) i.e. the two unit vectors Γ1 and Γ2 are orthogonal. It is easy to demonstrate that any binary combination of the unit vectors designated Γ1 × Γ4 are, indeed, orthogonal. Next, it is advantageous to discuss some more complicated symmetry point groups, such as C3ν, which was introduced earlier:

This dimension (order) of this point group is h = 6, since there are six symmetry elements in three classes, and some elements occur more than once in this group. First, the matrix representation of the C3 operation (a clockwise rotation by an angle of θ = 120°) will be introduced: (11.19)

Equation (11.19) shows that for a rotation by 120°, the trace of the transformation matrix is zero. According to Eq. (11.11), C3v will have three irreducible representations, but in order to fulfill Eq. (11.12), one of these, Γ3, must be two‐dimensional: (11.20)

since Eq. (11.12) requires that (11.21) The dimensionality of a representation was alluded to in Eq. (11.11). If the largest number in an irreducible representation is two, this representation is referred to as “doubly degenerate,” if it is three, the representation is “triply degenerate.” The concept of two‐ dimensional representations, for example, can be visualized by a coordinate system that rotates in space about the z‐axis. In this case, the x–y plane would be a two‐dimensional representation. Similarly, one can argue that a representation that has elements larger than one represents two one‐dimensional directions that cannot be separated. In groups where the number of symmetry elements per class is larger than one, such as the two C3 elements or the three σν planes in C3ν, the orthogonality condition shown in Eq. (11.17) needs to be modified to include a factor g that indicates the number of times an operation occurs in a group. In C3ν, for example, this factor g would be two for the C3 operations and three for the reflection planes. The orthogonality condition then reads: (11.22) Equation (11.22) is the more general form of the orthogonality condition that was defined previously (Eq. (11.17)). To demonstrate the orthogonality of the representation listed in Eq. 11.20, one needs to determine the products (11.23)

and finds, indeed, that the three irreducible representations are orthogonal. In Eq. (11.23), the bold numbers indicate the “g” factors, and the following digits the values Xα(Γi) and Xα(Γj). In general, one reserves the designations Γ1 × ΓN for molecular representations (such as the symmetry properties of molecular vibrations, or molecular orbitals; see below) and utilizes a different nomenclature for the irreducible representations of a group. This nomenclature is as follows: the totally symmetric representation of a group, i.e., the irreducible representations whose X(Γ) values are all +1, is generally given the designation Ax, where the subscript x can be “g,” “1,” or “1g” (depending on whether or not a group includes a center of inversion operation). In both C2ν and C3ν, the totally symmetric representation is designated A1. The subscripts g and u denote symmetric (gerade) and antisymmetric (ungerade) with respect to a center of inversion. There is always a totally symmetric representation in a group that is always written as the first row in the character table. The irreducible representation that transforms symmetrically (+1) with respect to the highest axis of symmetry will be given the designation A2, or A2g. Representations that transform antisymmetrically (−1) with respect to the major axis of symmetry are given the symbol B. Doubly degenerate (two‐dimensional) representations are referred to as E and three‐dimensional representations as T. With these designations, the irreducible representations (Eq. [11.14]) of the symmetry group C2ν can be written as (11.24)

The construct given by Eq. (11.24) is referred to as a “character table.” Each symmetry group has its own characteristic character

table, some of which are listed in at the end of this chapter. Similarly, the character table for C3ν is given by (11.25)

Any representation Γ for a given symmetry group, such as the one given in Eq. (11.10), can be decomposed into irreducible representations just as a vector in three‐dimensional space can be decomposed into its Cartesian components along the unit vectors. This is accomplished by projecting the vector Γ onto the directions of the irreducible component unit vectors to determine its component along this direction. The representation derived in Eq. (11.10) for the transformation properties of a Cartesian coordinate system under the operations of the C2v point group will serve as an example of how this decomposition is accomplished. The representation of the transformation properties of a Cartesian coordinate system (11.10) henceforth will be referred to as ΓCCS, where the subscript CCS stands for “Cartesian coordinate system.” The decomposition of ΓCCS into contributions from the four unit vectors A1, A2, B1, and B2 is carried out via the reduction (projection) formula: (11.26) In Eq. (11.26), ni indicates the number of times an irreducible representation i occurs in the reducible representation, here ΓCCS. This can be summarized in the following set of equations:

(11.27)

In these equations, the factor “g” is indicated in boldface, the X(Γi) in gray typeface, and the X(ΓCCS) in regular typeface. The results presented in Eq. (11.27) indicate that the “reducible representation” ΓCCS can be decomposed into the following unit vectors: (11.28) One can easily show that 1 A1 + 1 B1 + 1 B2 yields the original reducible representation:

The same procedure will be utilized, in Section 11.3, to derive symmetry species of the individual vibrational modes of polyatomic molecules. At this point, a few more comments about the character tables introduced above are appropriate. In addition to the

information given in Eqs. (11.24) and (11.25), character tables of molecular point groups contain two more columns, as shown below: (11.29)

The first of these additional columns contains entries of the form Tα and Rα, where α stands for x, y, or z, and T and R for translation and rotation, respectively. Thus, this column indicates how a translation or rotation of the coordinate would transform under the symmetry operations of the group. It can be visualized easily that a translation of a Cartesian coordinate system along the z‐direction transforms symmetrically under all operations of the group. Often, the translational components Tx, Ty, and Tz are just written as x, y, and z in character tables. The reason for this will be appreciated later (Section 11.4) since the Cartesian components of the dipole operator, μx, μy, and μz, have the same transformation properties as do the translations in the x‐, y‐, and z‐directions; thus, this column in the character table determines whether or not a given representation is allowed in any spectroscopy that depends on the change in the molecular dipole moment or one of its components. A rotation of the coordinate system about the z‐axis transforms as a C2 operation, which transforms as A2 (see Eq. (11.9)). The last column in the table denoted as Eq. (11.29) indicates the transformation properties of binary combinations of Cartesian coordinates. This column, therefore, determines whether or not a given symmetry representation is allowed in Raman spectroscopy, since in Raman spectroscopy, one or more of the polarizability tensor components must change during a vibration and these tensor components transform as the binary combinations of the Cartesian coordinates.

11.3 Symmetry Representations of Molecular Vibrations As pointed out in the introduction to this chapter, the main goal of the discussion of group theory is to determine which transitions in a molecule are allowed in vibrational or electronic spectroscopy. This will be demonstrated here for vibrational spectroscopy to determine whether a normal coordinate's representations transform like x, y, or z for IR spectroscopy or any binary combinations of x, y, or z for Raman spectroscopy. Thus, one needs to determine how a given normal coordinate transforms under the symmetry operations of a group. In order to accomplish this, one assumes that the normal modes of vibration can be written as linear combinations of x, y, or z Cartesian displacement coordinates. Thus, one attaches a Cartesian displacement coordinate system to each atom and follows how each of these coordinate components transforms under the symmetry operations of the group. This is shown for the normal modes of the water molecule (see Section 5.2) in Figure 11.6. As before, when deriving the reducible representation of a coordinate system under the symmetry operations of a group, one needs to establish the traces of the transformation matrices for each of the atomic displacement coordinate systems. Figure 11.6b shows the molecule after applying a C2 operation. As before, one designates the new coordinates (after the symmetry operation) as the primed coordinate and the original coordinates as the unprimed ones, and one obtains the transformation matrix for the C2 operation as shown in Equation 11.30:

(11.30)

Figure 11.6 (a) Cartesian displacement vectors for the water molecule. (b) Cartesian displacement vectors after a C2 operation. The trace of this transformation matrix is −1. Similarly, the trace of the transformation matrix under the identity operation is 9, since each Cartesian displacement vector component is unchanged under the E. From Eq. (11.30), it becomes obvious that the “molecular” transformation matrix (i.e., the 9 × 9 element matrix 11.30) for the water molecule contains three transformation sub‐matrices, one for each individual Cartesian displacement coordinate system, and that these transformation sub‐matrices are located at the diagonal of the overall transformation matrix if the particular atom does not change its position with another atom during the symmetry operation. If the atoms change position during a symmetry operation, the sub‐ matrices will occur at off‐diagonal positions.

The σν′ (=σxz) operation exchanges the positions of atom 1 and 3, and the transformation is (11.31)

with a trace of 1. For the σν (=σyz) operation, all atoms remain in place; therefore, all displacement coordinate sub‐matrices appear on the diagonal, and the trace of the overall transformation matrix is 3. Thus, the (reducible) representation of the nine displacement coordinates, from which the normal modes of vibration can be constructed, is (11.32) Next, the reduction formula discussed (Eq. (11.26)) will be used to determine the contributions of the four irreducible representations to the reducible representation of the displacement coordinates for water: (11.26) which reveals

(11.33)

Thus, the reducible representation ΓH2O = [9 −1 3 1] can be decomposed into or reduced to (11.34) At this point, one should check that these contributions, indeed, add up to nine degrees of freedom (3+1+3+2 = 9), since nine displacement coordinates were attached to the water molecule. Furthermore, one may wish to ascertain that the sum of the four representations, multiplied by their abundance, reproduces the original reducible representation:

This decomposition presented in Eq. (11.33) revealed nine degrees of freedom, since three Cartesian degrees of freedom were assigned to each atom (cf. Figure 11.6). However, since there are only 3N − 6 or three degrees of vibrational freedom for the water molecule, the

other six degrees are three translational and three rotational degrees of freedom. These translational and rotational modes originate from certain combinations of displacement coordinates: for example, if all atoms are displaced simultaneously in the z‐direction, the combined motion is a translation of the entire molecule in the z‐direction. Similarly, if atom 1 moves in the positive x‐direction and atom 3 moves in the negative x‐direction, the entire molecule rotates about the z‐axis. As pointed out above, these translational and rotational degrees of freedom are listed in the character table of the symmetry group C2v (Eq. (11.29)) in the column containing the entries Tx, Ty, Tz, Rx, Ry, and Rz. This table reveals that the translation of the water molecule along the positive z‐axis transforms as the A1 representation, while the Tx and Ty translation transform as B1 and B2, respectively. In order to obtain the symmetry properties of the three vibrational modes of water, one subtracts the translational and rotational representations from the total decomposition of the nine displacement coordinates given by Eq. (11.36) and obtains (11.35)

Thus, one ends up with two A1 and one B1 irreducible representations for the symmetry species of the three normal modes of water, which were depicted in Figure 5.1. This procedure is generally applicable for any molecule. For ammonia with C3v symmetry, for example, the reducible representation of the Cartesian displacement coordinates is (11.36) This reducible representation can be derived easily using Eq. (11.19) for the rotation matrix applied to each of the Cartesian displacement coordinate sets. The trace of Eq. (11.19), with θ = 120°, is zero, and the trace for each of the σν operation is +1. This trace enters into the

overall transformation matrix only for the atoms that do not change position during the reflection operation. Using the reduction formula Eq. (11.26) discussed above, the reducible representation shown in Eq. (11.36) can be decomposed into (11.37) Since each of the degenerate E representations accounts for 2°of freedom, Eq. (11.37), indeed, represents 12° of freedom. Next, the translational and rotational degrees of freedom will be subtracted. These are available from the complete character table for C3ν: (11.25)

Notice that the two‐dimensional representation E accounts for two degrees of translation (Tx, Ty) and two degrees of rotation (Rx, Ry). The vibrational degrees of freedom, therefore, are composed of 2 A1 and 2E irreducible representations, accounting for six (3N‐6 = 12−6 = 6) degrees of freedom. Examples for the analysis of more complicated systems are given in the literature ([2], Section 2.6.1)

11.4 Symmetry‐Based Selection Rules for Dipole‐Allowed Processes

Next, the symmetry rules for IR absorption based on the symmetry species for given vibrational transitions will be discussed. As discussed earlier, the vibrational energy of a molecule is determined by the (time‐independent) vibrational Schrödinger equation that

yields the vibrational energy eigenvalues, which are independent of symmetry, that is, a symmetry operation performed on the vibrational wavefunction will not affect the energy. This last statement, in mathematical terms, implies that the vibrational Hamiltonian, , and any arbitrary symmetry operator, , commutate: (11.38) that is, the same eigenfunctions ψvib are solutions both for the vibrational energy as well as symmetry operations. The total vibrational wavefunction for a polyatomic molecule was introduced earlier: (5.13) which may be written in an abbreviated form as (11.39)

Here, denotes the ground‐state vibrational wavefunction of normal coordinate Qi. If a transition along coordinate Qj is excited, the excited‐state total wavefunction can be written as (11.40)

For a transition described by (11.41)

only the properties of the jth normal mode need to be considered since all other modes remain unchanged: (11.42) For this coordinate, the transition moment (11.43) needs to be evaluated, which can be written in terms of the Cartesian components of the transition operator μ: (11.44)

Equation (11.44) implies that for a transition to occur, the integral over the excited‐ and ground‐state wavefunctions and at least one component the dipole operator must be nonzero. The ground‐state wavefunction for any normal coordinate always transforms as the totally symmetric representation of the symmetry group since the ground‐state wavefunction of any normal coordinate is given by (11.45) which yields (11.46) Equation (11.47) describes a Gaussian distribution in the coordinate Qj that is totally symmetric. The first excited state of the same coordinate has the symmetry of the normal coordinate Qj since the Hermite polynomial for the first excited state:

(11.47) and thus has symmetry of the excited state that is determined by the irreducible representation of the particular normal mode of vibration. Integrals of the form ∫ψ1j(Qj) μz ψ0j(Qj) dz will henceforth be abbreviated as ∫ f1 f2 f3 dτ, where f3 denotes the ground‐state wavefunction that always transforms as the totally symmetric representation (see Eq. (11.46)). f1 and f2 represent the excited‐state wavefunction and transition operator, respectively. Thus, for the total integral ∫ f1 f2 f3 dτ to be nonzero or even, the product f1 f2 must be even as well. This is the case if f1, the excited‐state wavefunction, transforms as one of the dipole moment components of the group. Then, its product with f2 contains the totally symmetric representation of the group. The transformation properties of the components of the dipole operator were discussion before (see Eq. (11.29)). To illustrate the points in the last paragraph, the example of the representations of the vibrations of the water molecule will be discussed. Equation (11.37) demonstrates that the vibrations of the water molecule belong to the irreducible representations A1 + B1: the symmetric stretching mode and the deformation mode transform (see Figure 5.1) as A1 and the antisymmetric stretching mode as B1. The question now arises which of these modes is allowed in IR absorption. According to the character table for C2v, reproduced below, (11.29)

the A1 irreducible representation transforms like a translation along the z‐direction. Thus, the dipole operator component μz also transforms as A1. For either of the transitions of A1 symmetry (i.e., the deformation and the symmetric stretching mode), the ground vibrational state is totally symmetric and transforms as A1. The excited vibrational state for either of these modes also transforms as A1, as pointed out before. Since the transition moment μz also transforms as A1, the product of the excited‐state and the dipole operator component (both A1) certainly contains the totally symmetric representation, and both transitions are allowed in absorption. For the 1 ← 0 transition of the antisymmetric stretching mode of B1 symmetry, one proceeds as follows. Again, the ground‐state vibrational mode transforms as A1 (see Eq. (11.46)). The excited state transforms as B1, as does one of the components of the dipole operator, μx. Thus, the product f1 f2 transforms as B1B1, and it is easy to see that this product contains the totally symmetric representation (or transforms as the totally symmetric representation) of the group, and the antisymmetric stretching vibration of B1 symmetry is allowed as well. In summary, the discussion in Section 11.4 demonstrated that any vibrational transition will be allowed in absorption if its irreducible representation contains a component of the electric dipole moment, μx, μy, or μz, (or the translational directions Tx, Ty, or Tz).

11.5 Selection Rules for Raman Scattering The principles discussed in the previous section can be applied to Raman scattering as well, but the Cartesian components of the dipole transition moment in Eq. (11.46) need to be replaced by the polarizability elements, since Raman spectra arise from the changes in polarizability α during a normal coordinate (see Eq. [5.46]). Thus, whereas IR transitions require

(5.44) the corresponding condition for a transition in Raman spectroscopy is (5.46)

As elaborated upon in Chapters 4 and 5, a direct transition from a lower to a higher vibrational state requires absorption of exactly one photon; this absorption process is mediated by the transition dipole operator. Raman spectroscopy, on the other hand, is a process that involves the interaction of two photons. Thus, the transition moment is determined by expressions of the form (11.48) where ψ0 and ψ1 are the vibrational ground and excited state, respectively, of a given normal coordinate Qk and ψint is a virtual (vibronic) intermediate state (see Figure 5.7). Since two transition moments are formally involved in the transition process, different selection rules apply for Raman spectroscopy. The irreducible representations that support Raman transitions, therefore, have binary combinations of the Cartesian coordinates listed, e.g. x2, y2, z2, xy, xz, yz, or others. The character table for the symmetry group C2v, shown in Eq. (11.29), has these binary combinations listed in a column to the right of the column containing the translational and rotational components. In complete analogy to the selection rules for absorption, a vibration transforming as a given irreducible representation will be allowed in Raman scattering if at least one of the binary combinations of the Cartesian coordinates is listed. Inspection of the character tables below reveals that symmetry groups that contain the inversion symmetry element, i, never have irreducible representations that simultaneously contain elements of the dipole operator (i.e. x, y, and z) and any binary combinations of these coordinates. This observation leads to what is known as the

“mutual exclusion rule” that states that in a point group that contains a center of inversion, i, vibrational transitions cannot be simultaneously active in Raman scattering and IR absorption. Thus, in molecules such as CO2 or octahedral SF6, vibrations that are Raman allowed are IR forbidden, and vice versa. Isomeric molecules such as cis‐dichloroethene (C2ν) and trans‐dichloroethene (C2h) can be identified easily by the fact that for the latter molecule, the mutual exclusion principle holds.

11.6 Character Tables of a Few Common Point Groups

References 1 Cotton, F.A. (1990). Chemical Applications of Group Theory, 3e. New York: Wiley. 2 Diem, M. (2015). Modern Vibrational Spectroscopy and Micro‐ Spectroscopy: Theory, Instrumentation and Biomedical

Applications. Chichester, UK: Wiley.

Problems 1. For the C4ν point group for which the character table is given below, demonstrate the orthogonality of the irreducible representation, A1, A2, B1, B2, and E:

2. For the C2h point group: a. Write the 3 × 3 transformation matrices for a Cartesian coordinate system for each of the four symmetry operations. b. Show by matrix multiplication that C2 ⊗ C2 = E. c. Show by matrix multiplication that C2 ⊗ σh = I. d. Show that Au and Bu are orthogonal. 3. Generate a 3 × 3 matrix representation for the following operators: (a) C6, (b) S4, (c) i. 4. a. For the C3v point group, define the 3 x 3 transformation matrices that describe the E and any of the three σν operations for a Cartesian coordinate system. b. Determine by matrix multiplication the symmetry operation that results from consecutive application of two σν

operations. c. Show that the A2 and E irreducible representations are orthogonal. 5. Consider the planar molecule trans‐diazene, trans‐H–N=N–H. a. What is the symmetry group to which this molecule belongs? b. How many vibrational normal modes do this molecule exhibit? c. Between 3000 and 3300 cm−1, this molecule exhibits two vibrations, one in Raman, the other in IR spectroscopy. Describe these two vibrations in terms of their atomic displacements and symmetry species (irreducible representation). 6. For the Td point group with the following symmetry elements: E 8C3 3C2 6S4 6σd a. Decompose the reducible representation 15 0 −1 −1 3 into the contributions of the irreducible representations. b. Ascertain that these representations account for 15° of freedom and that the contributions of these irreducible representations produce the reducible representation. c. Which are the translational and rotational degrees of freedom?

Appendix 1 Constants and Conversion Factors Avogadro's constant NA = 6.022 × 1023 [mol−1] Boltzmann's constant

k = 1.381 × 10−23

[J/K]

Electron charge

e = 1.602 × 10−19

[C]

Electron mass

me = 9.109 × 10−31

[kg]

Gas constant

R = 8.314

[J K−1mol−1]

= 0.698

[cm−1 K−1 molecule−1]

=5.0508 × 10−27

[J/T] = [A m2]

Permittivity

ε0 = 8.854 × 10−12

(F/m = C2/[J m])

Planck's constant

h = 6.626 × 10−34

[Js]

ħ = 1.054 × 10−34

[Js]

Proton mass

mp = 1.673 × 10−27

[kg]

Rydberg constant

Ry = 2.179 × 10−18

[J]

Velocity of light

c = 2.998 × 108

[m/s]

Nuclear magneton

Energy Conversion 1. [eV] = 8065.5 [cm−1] = 1.602 × 10−19 [J] Since

Appendix 2 Approximative Methods: Variation and Perturbation Theory A2.1 General Remarks Aside from the simplest quantum mechanical systems – the particle in a box, the rigid rotor, the harmonic oscillator and simple one‐ electron systems such as the H, He+, Li2+, etc. atoms and ions – most of the quantum mechanical problems cannot be solved analytically since the differential equations describing these systems cannot be solved explicitly. Thus, approximate methods must be used to solve any problems of higher complexity. At first sight, this mathematical limitation seems to seriously affect the usefulness of quantum mechanics. However, there exist methods to solve the differential equations by approximation methods, which are the subject of this appendix. Approximate methods can be applied since one of the postulates (Postulate 5) of quantum mechanics (see Chapter 2) stated that the real eigenfunctions φ(x) of an operator form a complete vector space. Functions that are not eigenfunctions of can be written in terms of a series expansion of the true eigenfunctions φ(x) as follows: (2.10) where the φ(x)'s are the true eigenfunctions, and the expansion coefficients an indicate how much each wavefunction looks like the true eigenfunction of the operator. Typical examples of these approximate methods were encountered in the discussions of earlier chapters, for example the orbital approximation described in Chapter 7: since the multi‐electron problem encountered in the He

atom cannot be solved, due to the electron–electron correlation, we assumed that the true wavefunctions of the He atom may be approximated by a linear combination of the individual hydrogen‐ like wavefunctions, approximating the electron–electron correlation by the effective charge. Subsequently, the expansion coefficients in Eq. 2.10 above need to be established that produce the best agreement with experimental results. This can be accomplished in two ways via variational or perturbational methods.

A2.2 Variation Method Let us assume we have a system that obeys the Schrödinger equation (A2.1) where the Hamiltonian is so complicated that the explicit solution of Eq. (A2.1) does not exist. However, there exists a somewhat simpler Hamiltonian that provides eigenfunctions φ that form a complete vector space. Hence, we write the solutions ψ(x) as a series expansion in a set of trial functions φ(x) as given by Eq. (2.10). The variation method establishes that the energy expectation value of the operator in the space of the trial functions, φ is always larger (or equal at best) than the true energy. Thus, one needs to show that: (A2.2) where (A2.3) that is, that the approximate functions have higher energy than the true lowest energy eigenvalue of the original problem (notice that this variational formalism works for ground states only). Thus, we need to establish that

(A2.4)

(A2.5) Here, ∫ψ*ψ dτ = 1 because the functions ψ are normalized. Next, we substitute the series expansion given by Eq. 2.10 into Eq. A2.4: (A2.6)

(A2.7)

(A2.8)

(A2.9) The last simplification was again due to orthonormality of the eigenfunctions φk. It follows that Eq. (A2.4), can be written as (A2.10)

Since the ak in Eq. (2.10) is positive, and since (Ek − E0) is positive as well (since E0 is the ground state), we have shown that by substitution of trial function that obey the same boundary

conditions, we can be assured that the approximate energy for the ground state is never below the real energy value. Then we may vary the expansion coefficients in the series expansion

until the observed energy is a minimum closest to the (experimentally observed) energy. For the application in this book, the variational method has less importance than, for example, for molecular orbital calculations since it can only be applied to ground state energy systems. In spectroscopy, by nature of the transitions involved, we need to consider at least one excited state. Thus, the perturbation method has much higher importance here and will be discussed in more detail.

A2.3 Time‐independent Perturbation Theory for Nondegenerate Systems Again, let us assume we have a system that obeys the Schrödinger equation (A2.11) Furthermore, let us assume that the Hamiltonian is so complicated that explicit solution of Eq. (A2.11) is impossible, but that there is a similar Hamiltonian for which exist explicit solutions (A2.12) At this point, one defines a “perturbation Hamiltonian”

such that (A2.13)

where λ is a scaling parameter so that one may gradually and incrementally apply the perturbation. An example would be the (one‐ dimensional) anharmonic oscillator Schrödinger equation: (A2.14)

where f and g are anharmonicity constants. It is a reasonable assumption that both the perturbed wavefunctions ψn and the perturbed eigenvalues En in Eqs. (A2.11) and (A2.22) depend on how much perturbation is applied: (A2.15) Thus, one expands the wavefunctions and eigenvalues in a power series in λ; the first‐order perturbations to the wavefunctions and energies are (A2.16)

and (A2.17)

where the first‐order perturbed energies are given by (A2.18) or

(A2.19)

Equation (A2.19) implies that the first‐order correction to the energy is found by calculating the expectation value of the perturbation Hamiltonian using the unperturbed wavefunctions. It can be shown ([1], Section 9.2) that the perturbed wavefunctions are: (A2.20)

For example, the first‐order perturbed wavefunction ψ1 would be (A2.21)

Assuming that the perturbed energies

get smaller, as compared

to , we may truncate the summation in Eq. (A2.21) after a few terms, as demonstrated in the next section.

A2.4 Detailed Example of Time‐independent Perturbation: The Particle in a Box with a Sloped Potential Function In the following section, an example of a typical perturbation problem is presented. This example is somewhat lengthy and best presented as its own section. The example to be discussed was mentioned before in Chapter 2, Figure 2.8, and is based in the quantum cascade laser that may be described as a particle in a box (PiB) with finite energy barriers and a sloped baseline, as shown in Figure 2.8. The sloped baseline in a quantum well can be achieved by suitable doping of the semiconductor materials.

For this example, it is assumed that the potential energy barrier is still infinitely high, but that inside the energy well of length L = 4, the potential energy is given by

(A2.22) that is, a straight line, as shown in the sketch at left. Thus, the potential energy for this particular problem is (A2.23)

The first perturbed energy level is given by (see Eq. [A2.19]):

(A2.24)

where the

are the unperturbed PiB wavefunctions. For a box

with length 4, these are (cf. Eq. [2.38]): (A2.25) Thus, within the box, (A2.26)

(A2.27)

These integrals can be solved by hand, or in this case here (which was a take‐home exam question in one of the authors classes) by using the Mathematica software by typing Integrate[Sin[Pi x/4]*Sin[Pi x/4], [2]] and Integrate[x*Sin[Pi x/4]*Sin[Pi x/4], [2]] and obtaining the values 2 and 4, respectively. Thus,

and

. Rather than integrating Eq. A2.26 in parts, Integrate[(2‐(x/2))* Sin[Pi x /4]* Sin[Pi x/4], [2]] will give the same result. Higher‐order perturbed energies are obtained in an equivalent manner:

(A2.28)

by evaluating Integrate[(2‐(x/2))* Sin[2Pi x /4]* Sin[2Pi x/4], [2]] for and Integrate[(2‐(x/2))* Sin[3Pi x/4]*Sin[3Pi x/4], [2]] for

.

In all cases, the perturbation integral has the value 1; thus, the perturbed energy levels are just raised by one unit as compared to the PiB with a flat bottom. Next, the perturbed wavefunctions will be evaluated according to Eqs. (A2.20) and (A2.21): (A2.20)

(A2.21)

To evaluate these, we need to calculate integrals of the form: (A2.22) We again use Mathematica to solve, for example, the integral Integrate[(2‐(x/2))* Sin[Pi x /4]* Sin[2Pi x/4], [2]]; (A2.23)

The following summarizes these results. For the first‐order perturbed ground state wavefunction ψ1, we find: (A2.24)

(A2.25) (A2.26)

(A2.27) Thus, (A2.28)

(A2.29) The first‐order perturbed first excited wavefunction ψ2 is: (A2.30)

(A2.31)

(A2.32)

(A2.33) (A2.34) The first‐order perturbed second excited wavefunction ψ3 is: (A2.35) (A2.36) (A2.37)

(A2.38) (A2.39) The gist of this example is that we have expanded the perturbed wavefunction in a series expansion in the unperturbed wavefunctions, scaled by the energy differences between the perturbed eigenvalues. The results of these calculations are presented in Figure A2.1. The left panel shows the standard PiB energy eigenvalues and (squared) wavefunctions. Panel (B) shows the PiB with the sloped potential energy barrier and the resulting energy eigenvalues. Notice that the ground state E1 is no longer at energy value 1, as in the unperturbed PiB, but at E1 = 2. (All energies given in units of [h2/8πmL2], see Eq. [2.33]). That is, the electron in the ground state avoids the sloped energy barrier by moving up one notch. This is true for all energy eigenvalues that are increased by one unit. Notice furthermore that the maxima of the wavefunction is shifted to the right to avoid the potential energy perturbation. This is

particularly obvious for the n = 2 and n = 3 states where the probability of finding the electron shifts toward the right.

Figure A2.1 Comparison between the unperturbed (a) and perturbed (b) particle‐in‐a‐box eigenvalues and wavefunctions. The squared wavefunctions are shown. See text for details This example demonstrates the inner workings of perturbation theory, and the results make intuitive sense since the electron tends to avoid the perturbed side of the energy well and escapes to the right. For the quantum‐cascade laser discussed in Chapter 2, this implies that electron has a higher probability density at the side of the well with the lower potential energy, and consequently, has a better chance of tunneling through the (right) barrier.

A2.5 Time‐dependent Perturbation of Molecular Systems by Electromagnetic Radiation Time‐dependent perturbation theory was introduced in Chapter 3 to define the response of a quantum mechanical system to a perturbation that acts for a limited time, and in most cases, is due to electromagnetic radiation impinging on a sample. Here, we assume that the system is originally in a stationary state, often the ground state, and responds by making a transition to an excited state if certain conditions are fulfilled. Mathematically, one describes the time dependent perturbations as follows. The quantum mechanical system, in the absence of a perturbation, is described by the time independent Schrödinger equation (A2.40) where E0 and ψ0 are the stationary state and energies and wavefunction of the system. When the time‐dependent perturbation is applied, the system needs to be described by the time dependent Schrödinger equation

(A2.41) Here, Ψ(x, t) is a wavefunction that depends both and spatial and temporal coordinates: (A2.42) and again is the perturbation operator. In the absence of the perturbation, the solutions of the time dependent Schrödinger equation (A2.43) are (A2.44) The functions are the eigenfunctions given in Eq. (A2.40). When the perturbation is present, the functions given in Eq. (A2.44) are no longer the eigenfunctions of Eq. (A2.41) but need to be replaced by an expansion (A2.45)

(A2.46)

where the coefficients ck themselves are time‐dependent to account for the changes the system will undergo as a response to the perturbation. The wavefunctions given in Eq. A2.46 then are substituted into Eq. A2.41 and the dependence of the coefficients ck on time is obtained

(see [1], p 227). When the perturbation is explicitly written as being due to electromagnetic radiation (see Chapter 3), the perturbation operator can be written as (3.2) The expansion coefficients are integrated from time t = 0 when the perturbation starts to a later time t = t'. The time‐dependent expansion coefficients are obtained (after an additional number of steps described in the literature [1]) to be (3.14)

This equation was previously described as one of the fundamental equations governing the interaction between electromagnetic radiation and matter. It contains the conditions that a) the electric field must be present, the transition moment 〈ψn ∣ μ ∣ ψm〉 must be nonzero, and the energy matching condition between the photon and the molecular system must be fulfilled.

Reference 1 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon.

Appendix 3 Nonlinear Spectroscopic Techniques The advent of tunable, high power, and ultrashort pulse lasers has spawned the development of new optical techniques that have truly revolutionized the field of spectroscopy. Perhaps with the exception of FT–NMR techniques, using various pulse sequences to create multidimensional NMR processes, no other spectroscopic method has experienced such an explosive expansion during the past 30 years as has nonlinear spectroscopy. The term “nonlinear” implies a dependence of the induced effects on the square or cube of the laser intensity. The first of the nonlinear Raman effects – the hyper‐Raman effect, a noncoherent three‐ photon effect, and coherent anti‐Stokes Raman scattering – were carried out at the Ford Motor Company Research laboratory in 1965 using a ruby laser for excitation [1]. This scientific achievement occurred a mere five years after the first experimental verification of a visible laser in 1960 (which was, incidentally, also a ruby laser). In the present times of financial hardship for any scientific endeavor, it may come as a surprise that this research was performed in the laboratory of a private company. Subsequently, several other nonlinear phenomena have been reported, some of which will be introduced in this chapter.

A3.1 General Formulation of Nonlinear Effects With the advent of high‐power pulsed lasers, several novel spectroscopic effects were discovered, where the size of the effects depend nonlinearly on the strength of the electric field of the exciting electromagnetic radiation, and therefore, referred to as “nonlinear spectroscopies.” The electric field in a pulsed laser can exceed 1010 V/m, which is about 100,000 times stronger than the field

strength of a common continuous wave laser. At these high laser fields, the induced dipole moment is no longer represented by Eq. (5.31) (5.31) but needs to be rewritten as a series expansion to included higher‐ order contributions to the induced electric dipole: (3.1) where is known as the first hyperpolarizability tensor of rank 3 (a 3 × 3 × 3 matrix). Equation (A3.1) is often written in terms of macroscopic dielectric susceptibilities: (3.2) where χ(n) is a tensor of rank n + 1, known as the dielectric susceptibility, which relates the induced electric macroscopic polarization P to the electric field E of the exciting radiation. In Eqs. (A3.1) and (A3.2), the first terms on the right‐hand side, describe the polarizability and macroscopic polarization, respectively, which are responsible for the effects discussed previously in Chapter 5. The second term in Eq. (A3.2) is responsible for hyper Raman scattering, whereas the second and third terms in Eq. (A5.2) account for nonlinear effects such as frequency‐doubling and nonlinear Raman effects such as coherent anti‐Stokes Raman scattering (CARS).

A3.2 Noncoherent Nonlinear Effects: Hyper‐Raman Spectroscopy Hyper‐Raman scattering results from the second term in Eq. (A3.2), and is a nonlinear, noncoherent form of nonlinear Raman spectroscopy. Hyper‐Raman and hyper‐Rayleigh spectroscopies are

three‐photon processes depicted schematically in Figure A3.1a. Two photons of frequency ω1 (up arrows) create two virtual states, shown by the dashed lines. A hyper‐Rayleigh photon at frequency 2ω1, or a hyper‐Raman photon at frequency 2ω1 – ωm is created from the upper virtual state, where ωm is the frequency of a molecular vibrational quantum. This is shown by the downward arrows in Figure A3.1b. The two laser photons need not necessarily have the same frequency: the hyperpolarizability can also mix photons of different frequencies, ω1 and ω2, resulting in what has been referred to as “nondegenerate” hyper‐Rayleigh and hyper‐Raman scattering [2] with frequencies ω1 + ω2 and ω1 + ω2 − ωm, respectively. This is shown in Figure A3.1b. Just as the hyper‐Rayleigh effect can be viewed as the incoherent form of frequency doubling, the nondegenerate hyper‐Rayleigh effect is the noncoherent analog of sum‐frequency generation (see below). In analogy to the Raman scattering tensor, (5.48)

the hyperpolarizability for initial and final and intermediate states i, f, r, and s can be defined in terms of the transition moments Mfr, Mrs and Msi (in the far‐from resonance approximation) as

(A3.3)

Figure A3.1 Schematic energy level diagram for degenerate (a and nondegenerate (b) hyper Rayleigh and hyper‐Raman scattering. See text for details. In Eq. (A3.3), the subscripts f and i denote the final and initial states, and s and r the two intermediate virtual states. Comparison of Eqs. (5.48) and (A3.3) suggest that the selection rules should be different for Raman and hyper‐Raman spectroscopy since two or three photons are involved in these processes, respectively. In fact, low

symmetry vibrations often are allowed, and more intense, in hyper‐ Raman spectroscopy, whereas totally symmetric vibrations may be very weak or entirely forbidden. Since the selection rules are all different for Raman, hyper‐Raman, and infrared spectroscopy, the hyper‐Raman effect complements the two other forms of vibrational techniques in the sense that it allows the observation of vibrational modes that cannot be observed in Raman or infrared spectroscopy. The torsional vibration in tetrachloroethene, C2Cl4, for example, which transforms as the Au irreducible representation of the D2h point group, is not active in either infrared absorption or Raman scattering, but is observed in hyper‐Raman spectroscopy at 110 cm−1. As pointed out above, the hyper‐Rayleigh effect is the noncoherent analog of the second harmonic generation, also known as frequency doubling that will be introduced in the next section.

A3.3 Coherent Nonlinear Effects There is an important distinction between the hyper‐Raman effect, the first form of nonlinear Raman spectroscopy discussed here, and the coherent forms of nonlinear spectroscopy, such as the stimulated Raman, inverse Raman, Raman gain, and coherent anti‐Stokes‐ Raman effects. In the former technique, the hyper‐Raman scattered photons are scattered noncoherently (spontaneously) into all 4π steradians, just as in the case of spontaneous Raman scattering. Also, in both Raman and hyper‐Raman effects, all normal modes are excited simultaneously when a molecule is exposed to the exciting laser radiation. In the coherent, nonlinear Raman effects, on the other hand, the scattered light exits the sample as a coherent beam with the properties of laser light. Since some of the nonlinear Raman techniques require two input laser beams, only one normal mode, determined by the frequency difference between the two laser beams,

is excited, leading to a much higher scattering efficiency. Thus, some of the nonlinear effects are very strong (albeit still not easily observed), whereas the incoherent hyper‐Raman effect is so weak that it is still not a particularly practical technique. In the remainder of this section, some of the theory underlying the coherent nonlinear techniques will be introduced. The equation most commonly used as a starting point for the discussion of nonlinear effects is Eq. (A3.2).

A3.3.1 Second Harmonic Generation A very common nonlinear optical effect will be discussed first, namely coherent frequency doubling. This process, which is also known as second harmonic generation (SHG), is mediated by the second‐order dielectric susceptibility χ(2). Two photons of angular frequency ω are combined in a non‐centrosymmetric crystal into one photon with angular frequency 2ω. Since the molecular system involved in this frequency‐doubling process is left in the original state after the new photon is created, the momenta and energies of the photons involved must be conserved. The momentum conservation is indicated by writing the electromagnetic fields as (A3.4) where k is the wave (or momentum) vector of the photon, defined by (A3.5) and (A3.6) The conservation of momenta requires that k3, the momentum of the frequency‐doubled photon, is given by (A3.7)

where k1 and k1 are the momenta of the original photons. Eq. (A3.7) is often referred to as the phase‐matching condition in nonlinear optics. Figure A3.2 depicts the phase‐matching condition for frequency doubling. Here, Eq. (A3.7) and Figure A3.2a predict that the frequency‐doubled photon emerges from the (nonlinear) crystal material collinearly with the incident beam. Typical nonlinear materials used for second harmonic generation are LiIO3, KNbO3, LiNbO3, KH2PO4 (KHP), KD2PO4 (KDP), LiB3O5·(LBO), β‐BaB2O4 (BBO), GaSe, KTiOPO4 (KTP), and (NH4)H2PO4 (ADP), where the commonly used (engineering) abbreviations, given in parentheses, strike horror into the hearts of chemists. Frequency doubling has become such a commonplace technique that green laser pointers, available for under $40, contain a frequency‐doubled diode‐pumped solid‐state laser, which gives a nice green (532 nm) spot on a reflective screen.

Figure A3.2 (a) Phase matching diagram for frequency doubling (SHG). (b) Phase matching diagram for sum‐frequency generation (SFG). (c) Directions of the incident beams 1 and 2 to create beam 3 in SFG. When the photons incident on the nonlinear crystals do not have the same frequency, they still can interact and combine to form a new photon according to (A3.8) and (A3.9)

This process is known as sum‐frequency generation (SFG); second harmonic generation or frequency doubling can be considered a special case of SFG. Since the vectors k1 and k2 have different lengths, the phase‐matching condition appears, as shown in Figure A3.2b, with the direction of the incident and emitted photons given by the arrows. This is shown in Figure A3.2c that depicts that the incident beams 1 and 2 have to intersect at the angle derived from the vector addition shown in Figure A3.2b to fulfill the phase‐ matching criterion. Since the direction of the incident and emitted photons are different, and the refractive indices within the nonlinear crystal may differ along the different directions, Eq. (A3.8) should be written as (A3.10) To account for the phase matching conditions of nonlinear processes, the wave vector notation of the incident and emitted fields is included in the equations for the induced polarization. For sum‐ frequency generation, for example, the second‐order susceptibility term is written to include the phase‐matching condition as (A3.11) Here, the sign convention in the exponent indicates an emitted photon at ω3 and incident photons at ω1 and ω2.

A3.3.2 Coherent Anti‐Stokes Raman Scattering (CARS) For the further discussion of nonlinear Raman effects, the following conventions will be used. The medium is exposed to various electromagnetic fields at frequencies ωa, traveling in the z‐direction. Such a field is represented by (A3.12)

where the subscript a denotes any of the individual radiation fields. In CARS, which is probably the most commonly used nonlinear Raman technique, there are three such radiation fields incident on the molecule to create a fourth photon, the CARS photon, according to the energy level diagram shown in Figure A3.3. In the following paragraphs, the four events necessary for the creation of a coherent anti‐Stokes photon are described as if they occurred consecutively. In reality, the four processes do not occur as separate events but are a four‐wave mixing phenomenon mediated by the third‐order nonlinear susceptibility χαβγδ (see below). In CARS, the sample is illuminated by two lasers, one of them with a fixed wavelength, usually referred to as the pump laser ωP or ω1, and a second tunable laser referred to as the Stokes frequency ωS or ω2. A photon ħωP at the pump frequency promotes the system into a virtual state, shown by the lower dashed line in Figure A3.3a. A photon ħωS from the laser at the (Stokes) Raman frequency causes the system to populate the vibrationally excited state, shown by the upper solid line. The vibrationally excited state in turn interacts with a second pump photon, ħωP, to populate another virtual state that undergoes a transition back to the ground state. The energy released in this last step is carried off by a photon of frequency (A3.13) where ħωM is energy of one of the molecule's vibrational modes. Thus, the wavelength of the emitted photon is that of an anti‐Stokes Raman process, and the emission of the anti‐Stokes photon occurs only if the wavelength of the tunable Stokes laser fulfills the condition (A3.14) For this process, in analogy to the discussion of SHG (Eq. [A3.11]), the term responsible for the CARS process can be written as:

(A3.15)

Figure A3.3 (a) Schematic energy level diagram for the CARS process. (b) Phase matching condition. θ denotes the angle between the pump and Stokes beams. θ′ is the angle between the pump and the CARS beam that is emitted along the dotted line. where the sign associated with each term in the exponential indicates whether a photon is annihilated or created. This exponential expression contains the four wave vectors of the interacting electromagnetic fields. Since the molecular system is left in the original state after the creation of the CARS photon, the wave vectors need to add up to zero. This leads to the phase‐matching condition for CARS, which can be written as (A3.16) and visualized in Figure A3.3b. It implies that the pump and Stokes beams must intersect at an angle given by the vector addition in Figure A3.3b for CARS photons to be generated. The CARS intensity scattered at ωAS is given by (A3.17)

where all symbols have their usual meaning and z is the distance over which phase matching is valid. At this point, it is appropriate to investigate the form of the third‐ order susceptibility tensor of rank 4, used in Eq. (A3.15). This tensor has 81 (=34) elements, of which only 21 are nonzero in isotropic media. In fact, only tensor elements for which all four indices are the same (e.g., χxxxx) and those for which there are two pairs of identical indices (e.g., χxxyy, χxyyx or χxyxy) are nonzero. The 21 nonzero elements exhibit only four different numeric values, commonly referred to as χ1111, χ1122, χ1221, or χ1212. With that, Eq. (A3.15) can be rewritten as (A3.18) where D is an integer factor between 1 and 6 that indicates how often each susceptibility term must be counted [3]. This equation assumes that the fields are polarized along the x‐axis. The anti‐Stokes photons leave the sample as a collimated, coherent laser beam, for which the spectral resolution is given by the line width of the exciting lasers. As pointed out before, spectral information on only one normal mode at a time is obtained if the Stokes laser is scanned to cover the spectral range. CARS spectroscopy has taken a huge step forward when it was demonstrated in microscopic measurements [4]. In the large solid angle of the light cone in a microscope objective, the phase‐matching angle is always fulfilled [5]. In an elegant optical arrangement reported by Kano [6], femtosecond laser pulses from a Ti:Sapphire laser were split to obtain pulses that were used as CARS pump pulses after filtering them to ca. 20 cm−1 bandwidth. The other part of the split beam was directed to a photonic crystal fiber to create a coherent super‐continuum pulse for Stokes excitation. In this way,

CARS spectra of 2000 cm−1 width could be collected simultaneously in a microscope set‐up (cf. Figure A3.4).

Figure A3.4 Broadband micro‐CARS spectra of cellular components: (a) nucleolus, (b) chromosome, (c) cell membrane, and (d) background (from ref. 6).

A3.3.3 Stimulated Raman Scattering (SRS) and Femtosecond Stimulated Raman Scattering (FSRS) Another nonlinear Raman effect due to the third‐order susceptibility is stimulated Raman spectroscopy. In stimulated Raman scattering, the sample is illuminated with only one laser at the frequency ωP. If the intensity of this pump laser increases past a threshold level, the intensity of the Stokes–Raman scattering becomes sufficiently large that nonlinear mixing of the Raman radiation field with that of the pump laser causes coherent laser output at ωS to occur, where ħωS = ħ(ωP − ωM) (see Eq. [A3.14]). This effect can convert up to 50% of the incident photons into Raman scattering, compared to an efficiency of spontaneous Raman spectroscopy on the order of 10−10– 10−12. SRS also has taken on a new direction with the development of femtosecond stimulated Raman spectroscopy, referred to as FSRS, which is a logical extension of SRS. Here, broadband femtosecond pulses are mixed coherently with a narrow pump laser frequency, and the same frequency mixing described before for SRS takes place. The major difference in FSRS is that all frequencies contained in the broadband pulse simultaneously can mix with the pump pulse; thus, the entire Raman spectrum can be probed at once. This is shown schematically in Figure A3.5, taken from [7]. This Figure depicts the narrow Raman pump pulse and the broadband femtosecond probe pulse. Typically, this pulse is about 20 fs long and has a natural linewidth of about 1600 cm−1. In the presence of a sample that exhibits allowed Raman transitions, some of the pump photons are transferred into the probe beam at the frequencies of the Raman modes. Ratioing the probe beam profile collected with and without the pump pulse yields the desired Raman spectrum, shown as the trace on top of Figure A3.5. This spectrum corresponds to a single laser pulse data acquisition (ca. 20 fs).

Figure A3.5 Schematic diagram of FSRS. See text for details (from ref. 7). This time resolution is about the same as the low wavenumber vibrational frequency (a vibration at 500 cm−1 has a vibrational period of 60 fs); thus, FSRS can probe the time evolution of a vibrational mode and vibrational dephasing. This time scale is of significant interest since the corresponding low‐frequency vibrations sample nuclear motion along the reaction coordinates, which may be described in terms of the normal modes of the molecular systems during chemical reactions. Thus, FSRS offers very fast access to assess molecular reaction dynamics, and an entire branch of Raman spectroscopy has evolved around these concepts, and interested readers are referred to review articles by the Mathies group [7]. A number of items from these reviews are summarized below: first, the FSRS signal is phase‐matched according to

(A3.19) which implies that the FSRS photons are emitted collinearly with the probe pulse. Second, the FSRS appears to violate the Heisenberg uncertainty principle in that the time‐frequency product Δν Δt is about an order of magnitude better in FSRS than expected from the uncertainty principle that predicts [7] (A3.20) This may be understood in terms of disentanglement of energy and time resolution because the broadband femtosecond pulse provides a molecular polarization with extremely high time resolution, whereas the not‐time resolved detection of the FSRS photon provides independent and very high wavelength (frequency) resolution. Third, FSRS spectra appear very similar to the spontaneous Raman spectra and are devoid of the line shape distortions observed in CARS. Furthermore, the spectra are linear in the concentration of the chemical to be analyzed.

A3.4 Epilogue This chapter explored some of the techniques that make nonlinear spectroscopy one of the most versatile optical techniques to study molecular structure and dynamics. Several of these techniques have experienced explosive growth during the past decade, mostly due to the availability of pulsed lasers with extremely short pulses, high repetition rates, and high power. Results of novel techniques and improvements appear at every major conference on advanced spectroscopic methods, and at present, new developments seem unlimited.

References 1 Maker, P.D. and Terhune, R.W. (1965). Study of optical effects due to an induced polarization third order in the electric field strength. Physical Review 137: A801–A818. 2 Ziegler, L.D. (1990). Hyper Raman Spectroscopy. Journal of Raman Spectroscopy 21: 769–779. 3 Harvey, A.B. (ed.) (1981). Chemical Applications of Non‐linear Raman Spectroscopy. New York: Academic Press. 4 Cheng, J.‐X. et al. (2001). An epi‐detected anti‐Stokes Raman scattering (E‐CARS) microscope with high spectral resolution and high sensitivity. The Journal of Physical Chemistry B 105: 1277– 1291. 5 Evans, C.L. et al. (2005). Chemical imaging of tissue in vivo with video‐rate coherent anti‐Stokes Raman scattering microscopy. Proceedings of the National Academy of Sciences of the United States of America 102 (46): 16807–16812. 6 Kano, H. (2008). Molecular vibrational imaging of a human cell by multiplex coherent anti‐Stokes Raman scattering microspectroscopy using a supercontinuum light source. Journal of Raman Spectroscopy 39: 1649–1652.

7 Lee, S.‐Y. et al. (2004). Theory of femtosecond stimulated Raman spectroscopy. The Journal of Chemical Physics 121 (8): 3632– 3642.

Appendix 4 Fourier Transform (FT) Methodology A4.1 Introduction to Fourier Transform Spectroscopy Infrared (IR) spectroscopy can be carried out experimentally in two quite different ways. One uses a color sorting device, such as a monochromator, to isolate one particular IR “color” band and sends it into the sample where it may be partially absorbed. A spectrum is collected by sequentially isolating different colors and plot the absorbance at each color. This is a time‐consuming process since about 1800 “color bands” need to be analyzed sequentially for a standard IR spectrum extending from 400 to 4000 cm−1, assuming a spectral resolution of 2 cm−1. Furthermore, certain aspects of detector noise make this approach less favorable than an approach where all frequency bands are sampled simultaneously via an “interferogram” from which the spectrum is constructed by what is known as a Fourier transform (FT). Both data acquisition methods give the same spectral information, but the FT‐methodology is nearly 1000‐fold faster. The same is true for nuclear magnetic resonance (NMR) spectroscopy. The spectral information can be obtained by scanning the external magnetic field at constant radio frequency (or scanning the radio frequency at constant external magnetic field) to obtain a spectrum. This may be viewed as the “classical” NMR experiment and corresponds to the first data acquisition mode in infrared spectroscopy, described above, where a monochromator is used. In NMR spectroscopy, FT methodology is now used nearly exclusively in which the external magnetic field is kept constant, but instead of varying the frequency of the radiofrequency excitation, a short pulse of radio frequencies is applied to the sample, which changes its overall magnetization. When returning to its original magnetization,

the sample emits the energy difference as a radio frequency signal containing all frequency components known as the “free induction decay” (FID). This signal is collected by a radio frequency receiver and subject to FT to give the same spectral information that would have been observed in the classical NMR experiment. However, the “pulse FT” method is hundreds times faster and, furthermore, allows for different NMR experiments that are impossible or near impossible to carry out in classical NMR mode. So, what is this magical “Fourier transform” procedure? Basically, it is a mathematical method that allows signals to be collected in a different (and more suitable) “domain” and transform them back into the domain that spectroscopists are more familiar with. No information is gained or lost in the process, and it is possible to analyze the interferogram (in IR spectroscopy) or the free induction decay (in NMR spectroscopy) directly. However, it will be shown that the information required in spectroscopy, such as chemical shifts or group frequencies, are much more easily discerned after FT. To indicate qualitatively how Fourier Transform methodology works, we resort to an example from music that demonstrates how data can be represented in different domains.

A4.2 Data Representation in Different Domains Consider, for example, the musical note “A,” the 440 Hz sound produced by a violin's or guitar's “A” string. A plot of such a sound wave is shown in Figure A4.1a which shows the amplitude of the sound wave as a function of time. Since this signal varies periodically with time, it can be represented as a sine or cosine wave with frequency ν = 440 Hz and an intensity (amplitude). However, one could depict the same information by plotting the intensity in a graph that has frequency on the abscissa. This results in a graph shown in Figure A4.1b. This panel indicates that all frequencies, except 440 Hz, have no amplitude or are not represented, while the frequency at 440 Hz appears with a given amplitude. The “frequency domain” representations shown in Figure A4.1b has the advantage that a sound signal that contains more than one frequency can be

interpreted more easily. This is shown in Figure A4.2. Here, a signal consisting of two sine waves, at 440 and 880 Hz, is shown in (a). This more complex and harder to interpret amplitude vs. time representation is much clearer when depicted in the amplitudes vs. frequency plot shown in (b). The two representations depicted above are mathematically related by a Fourier transform. To introduce the concept of Fourier transforms, it is best to first discuss the concept of Fourier series and transition to Fourier transforms from there.

A4.3 Fourier Series Fourier series expansion or harmonic analysis extracts appropriately weighted harmonic components from a general periodic waveform. Any function f(x), which is periodic between −π and +π (or L to +L, or 0 to 2π), can be expanded in this interval by a Fourier series. The Fourier series expansion of the function f(x) is defined by (A4.1)

Figure A4.1 Representation of data in different domains. (a) Graph of the intensity vs. time of a sound wave (the musical note “A,” top trace) and its first (middle trace) and second harmonic (bottom) or octaves. (b) Representation of the same information in an intensity vs. frequency display.

Figure A4.2 (a) Intensity vs. time and (b) intensity vs. frequency plot of a sound signal consisting of two notes at 440 and 880 Hz. where the expansion coefficients cn are given by (A4.2)

In the case of real functions, the expansion given in Eq. (A4.1) takes the form (A4.3)

Figure A4.3 Approximation of a square wave function (heavy black line) by a scaled sum of harmonic frequencies (n = 1 to n = 9). See text for details. with the real expansion coefficients given by

and (A4.4)

An example of Fourier series expansion will be presented next. Consider a square wave, given by Eq. (A4.5): (A4.5) This function is shown as the heavy black trace in Figure A4.3 and is assumed to repeat periodically. Substituting Eq. (A4.5) into the equations for the expansion coefficients an and bn (Eq. [A4.4]) and integrating from 0 to 2π, one finds that all the terms an will be zero: (A4.6)

(A4.7)

whereas the terms bn assume the values: (A4.8)

for even values of n, and

(A4.9)

Thus, a square wave can be expanded into an infinite series of all odd harmonics, scaled by 1/n: (A4.10)

This is shown in Figure A4.3 for n = 1 to n = 9. It is obvious from this graph that an increasing number of higher harmonics improves the fit between the square wave and the sum of all the co‐added harmonics. Similar expansions can be carried out to approximate any periodic function (such as a saw tooth function and other, more complex functions) by a sum of harmonics.

A4.4 Fourier Transform Next, the standard concepts of the Fourier transform will be introduced. A logical connection between the principles of Fourier expansions and Fourier transforms can be made by substituting (A4.11) into Eqs. (A4.1) and (A4.2) to obtain (A4.12)

(A4.13)

and letting the interval, L, over which the function is expanded, go to infinity. Thus, k gets very small, and one can substitute the sum in Eq. (4.12) by an integral: (A4.14)

(A4.15) where C is a normalization constant. Thus, one may view the process of taking a Fourier transform as a harmonic analysis with infinitely small increments in the frequency intervals. Equations A4.14 and A4.15 are called a “Fourier pair” and each equation in such a pair completely defines the other. Therefore, no information is lost when the signal is transformed from one domain to the other.

Figure A4.4 Examples of Fourier transforms (FTs). (a) The FT of a delta function is a cosine function (black: real part, gray: imaginary part). Note that the real and imaginary parts are 90° out of phase, (b) The FT of a broad spectral distribution is a sharp, symmetric interferogram. The FT's shown here were calculated via the Microsoft EXCEL fast FT implementation. The interferogram shown in (b) was unfolded (see text).

Next, some important properties of Fourier transforms will be discussed. Equations (A4.14) and (A4.15) show that a Fourier transform is a complex operation; thus, the output of FT calculation, in general, have a real and an imaginary part. This is shown, for example, for a shifted δ‐function,

, which

one may think of an infinitely narrow‐band (monochromatic) light source, in Figure A4.4a. Its FT is a periodic cosine function with frequency , and the imaginary part is phase‐shifted (or sine) function shown by the gray trace in Figure A4.4a. The Fourier transform of a broad spectral distribution gives a very sharp interferogram, shown in Figure A4.4b.

A4.5 Discrete and Fast Fourier Transform Algorithms The equations given so far for Fourier transform pairs describe the process and concepts but not a practical method to carry out Fourier transforms of a set of data. This transformation between the two data domains is performed by an algorithm known as fast Fourier transform (FFT) algorithm. FFT is based on the principles of the discrete Fourier transform since the interferogram is not sampled continuously, but at discrete times, the interferogram is obtained as a one‐dimensional vector of digital values. For such a situation, the process of taking the Fourier transform can be written as (A4.16)

(A4.17)

In Eqs. (A4.16) and (A4.17), N is the total number of data points, T is the sampling interval, and n and k are the running indexes in g and G space, respectively. These equations are the discrete (point‐by‐ point) versions of Eqs. (A4.14) and (A4.15). It is interesting to note that Eq. (A4.16) resembles very much the equation at the starting point of Fourier series expansion (Eq. [A4.1]). Setting n/NT = m and kT = p, Eq. (A4.17) can be written as (A4.18)

where (A4.19) Equation (A4.19) can be cast into matrix notation: (A4.20) Thus, the computation of a discrete Fourier transform from g (interferogram) to G (spectrum) space is reduced to computing a (complex) transformation matrix Wmp and multiplying the vector of discrete points with this matrix. For typical spectroscopic applications, a sample set may consist of 8K data points (i.e. 8192 points in the g[p] vector). The Fourier transform operation, according to Eq. (A4.20), requires for each of the 8192 points in G space an 8K × 8K matrix to be multiplied by an 8K vector. Such matrix manipulations are slow, since for each data point, 8K multiplications and 8K additions are required. This problem was alleviated by the FFT algorithm, developed by Cooley and Tukey [1], which avoids the problem of a large number of multiplications and additions by factoring the W matrix into sparse matrices that have many zero elements. It can be shown that such factoring is always possible, but the factoring will require reordering the entries in the G and g vectors. Furthermore, the FFT algorithm only works for data vectors that have integer powers of 2 (256, 512, 1024, etc.) entries. A detailed discussion of the FFT algorithm is

beyond the scope of this chapter, and the reader is referred to the literature [1, 2]. With the increased computational power of modern desktop machines and the implementation of FFT routines in the MATLAB environment, FFT computations can be carried out for a 1024 point data vector in a few milliseconds.

A4.6 FT Implementation in EXCEL or MATLAB A final comment is appropriate about the presentation of the FFT results, for both MATLAB and EXCEL implementations. As pointed out in the Caption to Figure A4.4, all Fourier transform examples shown in this Appendix were carried out using Microsoft EXCEL. In both FT–IR and FT–NMR, data are collected in the time‐domain (time after the broadband excitation pulse in NMR, or time of the interferometer mirror travel after passing the “zero‐path difference” point). These time points will be referred as the “time zero point” in the time domain. The FT transforms the intensity vs. time interferogram into intensity vs. frequency space.

Figure A4.5 Panel (a): Real part of a reverse transform of a spectrum back to interferogram domain. Notice the unfolded representation of the interferogram. Panel (b): Imaginary part of interferogram shown in (a). Panel (c): Interferogram shown in (a) folded to demonstrate the symmetry of the real part of the interferogram. Panel (d): Interferogram shown in (b) folded to demonstrate the antisymmetric nature of the imaginary part of the interferogram.

A spectroscopist, therefore, expects an interferogram similar to Figure 8.7a, that is, with a starting point at “time zero” point (in FT– NMR), or at the center burst or “zero‐path difference” (in FT–IR spectroscopy). However, if a spectrum is reverse transformed into the time domain using the FFT algorithm as included in MATLAB or EXCEL, both the real and imaginary parts are presented in a “folded” form, as shown in Figure A4.5, Panels (A) and (B). Here, the “time zero” or the “center burst” is shown on the far left of the graph, with the next data point at the far right of the graph; that is, the reverse transform is unfolded such that points 1–256 are moved to positions 257–512, and the points 257–512 are moved to positions 1–256 in a 512 point transform. After un‐folding, the interferograms shown in Panel (C) and (D) are obtained. Notice that the real part of the interferogram is symmetric about the “time zero” point, whereas the imaginary part is antisymmetric about this point.

References 1 Cooley, J.W. and Tukey, J.W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics and Computation 19: 297–310. 2 Brigham, E.O. (1974). The Fast Fourier Transform. Englewood, NJ: Prentice Hall.

Appendix 5 Description of Spin Wavefunctions by Pauli Spin Matrices The introduction of electronic spin in Chapter 7 followed an argument that compared the eigenvalue of the total angular momentum operator (6.43) and the eigenvalue of a Cartesian component, Lz (6.44) to the total spin operator, , and its Cartesian components , and it was argued that, in analogy, the eigenvalues of the total spin operator for an electron should be (7.28) and the eigenvalues of the

operator

Here, the angular momentum quantum number J can have integer values ≥0, and K from –J to +J, the spin quantum numbers “s” for an electron are confined to the value ½. Thus, the eigenvalues for and

operators are (7.29)

(7.30) In Eqs. (7.28) and (7.29), the spin functions α and β were not further defined, except that they were postulated to be orthonormal functions:

and (A5.1) and described as “spin‐up” and “spin‐down” or “clockwise” and “anti‐ clockwise” motion. This latter description is somewhat simplistic since it implies a truly spinning particle which the electron is not. Rather, it was pointed out that the spin is an inherent quantity that is much more difficult to visualize. Historically, it is interesting to note that the wave mechanical approach pursued by Schrödinger did not include the spin quantum number s (or ms). However, the alternative formulation (based on matrix mechanics) of quantum mechanics by Heisenberg, developed nearly simultaneously to Schrödinger's work, incorporated all four quantum numbers n, l, ml, and ms for the hydrogen atom. At this point in time, experimental work had demonstrated that there exist two spin states and that each spin state has an (experimentally verifiable) eigenvalue of , leading to an energy difference of the two states of (A5.2) (see Appendix 1) Since there are two eigenvalues, a matrix‐based theoretical formalism, based on Heisenberg's matrix mechanics, was devised by Pauli [1] in which the spin Hamiltonians consist of 2 × 2 matrices, and the spin functions of 2 × 1 column vectors, as discussed in the next section. These following equations are presented without proof

and should be construed as a short introduction into matrix mechanics and spin matrices.

A5.1 The Formulation of Spin Eigenfunctions α and β as Vectors In matrix mechanics, the wavefunctions are represented by vectors and the operators as matrices. We start by discussing the spin wavefunctions that we know have two eigenvalues (see Eq. 7.29 above). However, the Stern–Gerlach experimental (Section 7.5) revealed that any linear combination of these eigenfunctions α and β are also valid eigenfunctions. Thus, any linear combination (A5.3) is also a solution of the spin Hamiltonian. This insight is due to the results of the Stern–Gerlach experiment that revealed that any number of mutually perpendicular magnetic fields split a beam of silver atoms (with a single unpaired electron) passing through them into two beams, since the components of the spin momentum, cannot be determined simultaneously. In the matrix formulation of electron spins, the spin wavefunction ψαβ is represented by a column vector: (A5.4)

Arguments about the orthonormality of the spin wavefunctions (see Eq. A5.1 above) leads to the final form of the spin wavefunctions α and β as (A5.5)

A5.2 Form of the Pauli Spin Matrices

The spin operators

and its Cartesian components

are defined by the Pauli spin matrices as (A5.6)

The derivation of these spin matrices is beyond the scope of this discussion. To simplify the calculations below, the normalization factor often is omitted, and the matrices are written as (A5.7)

Since the total spin operator,

is given by (A5.8)

it can be evaluated by squaring each of the spin matrices and summing them up:

(A5.9)

Thus,

(A5.10) and (A5.11) in agreement with Eq. (7.29) above. Following the argument cited above that the spin operators follow the same commutation rules as the angular momentum operator, we evaluate the commutators: (A5.12) (A5.13)

Similarly, it can be shown easily that the other two commutators of the Cartesian components are (A5.14) Thus, it was shown that the spin matrices, indeed, reproduce the commutation properties of the Cartesian components of the angular momentum operator discussed in Chapter 6. By similar arguments, it can be shown that

(A5.15)

The other two Cartesian components also commute with the σ2 operator.

A5.3 Eigenvalues of the Spin Matrices At this point, the eigenvalues of the spin operators can be evaluated easily. For example, the relationships (7.29) can be verified by simple matrix multiplication: (A5.16)

(A5.17)

and (A5.18)

(A5.19)

Finally, the orthonormality conditions

and (A5.1) can be verified as follows. To form the inner products of two spin vectors, one has to transpose the first vector (in fact, one has to form the adjoint vector, where all terms are the complex conjugates. However, since in this example the vectors are real, the transpose equals the adjoint vector): (A5.20)

Thus, the results of the matrix mechanical treatment of the spin functions provide the same results that were obtained by just transferring the results from the angular momentum operators and , but it has been demonstrated here how the matrix mechanics formalism works.

Reference 1 Pauli Matrices (2020). Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Pauli_matrices.

Index a absorption 1, 3, 7–12, 32, 33, 39–45, 51, 56–59, 62, 63, 75–81, 84, 87–89, 96, 104–113, 115, 123, 125, 138, 147, 156, 159–161, 172–174, 179, 180, 182, 183, 187, 189, 190, 192, 193, 214, 216, 217, 235 absorptivity 42, 45 adenine (A) 180 ammonium sodium tartrate 187 androstane 190, 191 angular momentum 19, 93–95, 97–100, 103, 110, 117, 127–129, 132–135, 137, 139, 157–160, 171, 172, 189, 253, 255–257 anharmonic force constant 60 anharmonic oscillator 59–62, 87, 125, 173, 225 anharmonic potential 56, 61, 176, 182 anharmonicity 49, 61–63, 65, 77, 89, 112, 225 anomalous dispersion 12, 77–81, 187 anomalous Zeeman 127 anti‐Stokes Raman process 83, 84, 238, 239 associated legendre polynomials 101, 102, 104, 109 associative law 203 asymmetric molecule 186–191 asymmetric top rotor 96–98, 110 atomic absorption 3, 7–10, 161 atomic number 120, 154, 155 atomic radii 156

atomic spectra 2, 159, 160 atomic spectroscopy 10, 157, 160–162 Aufbau principle 153–155

b beat frequencies 82, 83 Beer–Lambert law 43, 162 benzene 175, 180, 202 blackbody radiation 3–5 blueshift 183, 185 Bohr radius 117, 122, 123 Boltzmann distribution 4, 44, 84, 90, 105, 106, 221 bond order 170, 172, 173, 176 Born–Oppenheimer approximation 86, 164, 165, 174–176, 199 boundary conditions 23, 24, 28, 51, 53, 99, 225 Brewster angle 47 1,3‐butadiene 179

c camphor 190, 192 carbon disulfide 76 carbonyl chromophore 178–179, 190 carotene 180 Cartesian component 70, 97–99, 103, 128, 132, 133, 139, 209, 210, 215, 217, 253, 255, 256 Cartesian displacement coordinates 69–72, 85, 211–214 Cauchy integral 81

center of inversion 88, 171, 200, 201, 208, 217 centrifugal distortion 112 centrifugal effects 107 character table 85, 208–210, 213, 214, 216–218 characteristic vibrational frequency 51 chemical bonding 115, 163–167 chemical shift 138, 140–141, 143–146, 244 chirality 185–187, 189, 192, 193 chloroform 88–90, 108, 201 chromophore 178–181, 190–193 circular dichroism (CD) 81, 179, 185–193 circularly polarized light 81, 185–187, 190 Clebsch–Gordan expansion 157, 158 coherent anti‐Stokes Raman scattering (CARS) 184, 233–235, 237– 239 nonlinear Raman effects 233, 234, 236, 237, 240 collisional energy loss 182–183 color band 243 combination band 77, 89 commutation 18, 103, 128, 133, 134, 255, 256 commutators 19, 97, 255, 256 conservation of momenta 236 constructive and destructive interference 2 C3 operation 207 Coulomb's law 116 C2ν point group 203 crystal field theory 181

cytosine (C) 180

d de Broglie equation 6, 16 degeneracy 120, 121 detector noise 243 2‐deuterobutane 190 diamagnetic response 140 dichloroethene 217 dielectric susceptibility 139, 234, 236 differential operator 16 1,3‐dimethylallene 187 dipolar coupling 191 dipole‐allowed absorption 40–42, 124 dipole‐allowed transitions 76–77, 124, 172, 199 dipole moment 38, 40, 62, 76, 77, 81–83, 87, 88, 96, 105, 112, 113, 124, 189, 191, 193, 199, 210, 216, 233 dipole strength 179, 189, 191 Dirac 1, 128, 129 discrete Fourier transform 248, 249 disentanglement 241 dissymmetric molecules 187, 188, 191–192 DNA bases 180 doubly degenerate 88, 181, 208

e

effective nuclear charge 151–152 Einstein coefficients

42–45

elastic scattering 81 electric and magnetic dipole transition moments

189

electric dipole moment 38, 189, 191, 216 electric dipole operator 188, 189 electric field 2, 3, 38, 40, 82, 83, 90, 109, 138, 188, 189, 231, 233, 234 electric transition moment 3, 190 electromagnetic radiation 2, 3, 10, 11, 18, 25, 37–46, 51, 56, 78, 80, 81, 83, 131, 134, 136–139, 146, 147, 189, 230–231, 233 electron configuration 127, 158 electron correlation 151, 152, 169 electron–electron correlation 223 electron–electron repulsion 151, 164 electronic circular dichroism and optical rotation 185–193 electronic spectra 173 of diatomic molecules 173–177 of polyatomic molecules 177–181 electronic spectroscopy 115, 160, 163, 211 electronic transition moments

84, 85, 176

electron paramagnetic resonance 131, 134 electron spin 11, 126–129, 131, 134, 152, 254 enantiomeric forms 186, 190 enantiomers 185–187, 189, 190 energy eigenvalues 17, 22, 24, 28, 51–56, 60, 74, 95, 96, 99–103, 118, 121, 123, 125, 142, 153, 165, 191, 214, 224, 229, 230 energy expectation value 165, 168, 224

energy states 3, 8–10, 28, 34, 40, 43–45, 55, 63, 90, 115, 127, 128, 131, 134, 136, 139, 140, 147, 155, 163, 176, 195 estrogen 190 exciton couplet 190, 192 exciton formalism 192 excitonic state 191 exciton model 191–192 expectation value 17, 40, 83, 165, 168, 224, 226 external magnetic field 128, 129, 131, 137–140, 144, 145, 147, 243

f fast Fourier transform (FFT) algorithms 248–249 femtosecond stimulated Raman scattering (FSRS) 240–242 first order non‐linear susceptibility 183 fluorescence microscopic imaging 181 fluorescence resonance energy transfer 181 fluorescence spectroscopy 174, 181–185 fluorochlorobromomethane 186, 187 formaldehyde 178, 179, 202 Fourier series 244–247, 249 Fourier transform 131, 146–147, 243–244, 247–249 Franck–Condon factor 176, 184 Franck–Condon principle 174–177, 181–184 free induction decay (FID) 139, 146–148, 243, 244 frequency doubling 234–237 FT‐IR spectroscopy 147, 249, 250 full width at half maximum (FWHM) 77

g gain medium 46, 47 gas laser 46 Gaussian distribution 79, 169, 215 Gaussian profile 79 glyceraldehyde 187 green fluorescent protein (GFP) 182 group frequencies 93, 178, 244 group representations 204–211 guanine (G) 180

h half‐filled shell 155 Hamiltonian 17, 22, 23, 27, 28, 51, 132, 141, 143, 144, 151–152, 157, 158, 164, 165, 168, 175, 214, 224–226, 254 Hamilton operator 17 handedness 185, 186, 189 harmonic analysis 244, 247 harmonic oscillator 38, 49–66, 69, 70, 73, 74, 76, 87, 101, 102, 104, 110, 111, 125, 173, 223, 225 harmonic oscillator Schrödinger equation 51–56 harmonic oscillator wavefunctions 21, 52, 55 harmonic potential energy function 60 Hartree–Fock method 152, 153, 155 He–Ne laser 46, 47, 159 heat capacity 91 Heisenbergs matrix mechanic 254

Heisenberg's uncertainty principle 15, 19, 55, 241 helical molecules 187 α‐helix 192 Hermite's differential equation 51, 118, 125 Hermite polynomials 53, 54, 57, 101, 102, 104, 215 high spin 181 highest occupied molecular orbital (HOMO) 32, 163, 180 hollow cathode lamp (HCL) 3, 161 homonuclear diatomic molecule 62, 105, 168–172, 195 hot bands 62–65, 79, 89 Hund's rule 154, 171, 172 hydrogen atom 2, 7–10, 12, 20, 28, 33, 93, 103, 115–129, 151, 156, 159, 164, 166, 168, 187, 201–203, 254 hydrogen‐like orbitals 169 hyper‐Raman scattering 234 hyper‐Raman effect 139, 233, 235, 236 hyperpolarizability tensor 234

i identity element 186, 200, 203 improper axes of rotation 200 induced dipole moment 82, 83, 233 induced electric dipole 233 inelastic scattering 81 inertial tensor 95 infrared absorption spectroscopy 62, 63, 76–81, 88, 111, 214, 235

infrared spectroscopy 93, 235, 243 intersystem crossing 182, 183 intrinsic fluorescence 181 inverse operation 203 inversion symmetry element 88, 217 ionization energy 6, 155 ionization potential 155 irreducible representations 181, 189, 204–209, 212–214, 216, 217, 220 IR vibrational circular dichroism (VCD) 192 isotopic species 63–65, 75 isotopic splitting 64

j Jablonski diagram 182–183 J‐coupling 143, 145

k Kramers–Kronig transform 12, 80, 81 Kronecker symbol 25, 54

l Lagrange's equation of motion 71 LaGuerre differential equation 117 LaGuerre polynomials 117, 119 Laplace operator 116 Larmor frequency 139, 147

laser theory 43 Legendre differential equations

118, 125

Legendre polynomials 101, 109 lifetimes 77–79 ligand field theory 181 line shapes 77–79 linearly polarized light 187 linear molecules 75, 76, 94, 96, 103, 105–109 lithium 160 Lorentz force law

136

Lorentzian band 78, 79 lowest unoccupied molecular orbital (LUMO)

32, 180

low spin 181

m magnetic dipole transition operator 189 magnetic moment 128, 134–136, 139, 147, 189 magnetic transition moment 3, 188–192 magnetic quantum numbers 103, 117, 126, 127 magnetization 11, 129, 131, 137–139, 147, 148, 189, 243 magnetogyric ratio 135 mass‐weighted Cartesian displacement coordinates 70–72 Maxwell's equation 1, 2, 5, 6 3‐methylheptane 190 mirror image 186, 187 molar extinction coefficient 42, 43, 76, 77, 178–180

molecular orbital theory 163, 164, 168–170 moment of inertia 75, 93–95, 98, 103, 107, 108, 112, 132 momentum conservation 236 Morse potential 59 multidimensional NMR processes 233 multi‐electron systems 133, 151–162 multiple spin interaction 144–145 multispin system 141–146 mutual exclusion rule 217

n natural broadening 79 Nd:YAG laser 90 net magnetization 129, 139, 147–148, 189 Newtonian mechanics 1 Newton's second law of motion 50 noncoherent nonlinear effects 234–235 noncoherent three‐photon effect 233 nondegenerate hyper‐Rayleigh effect 234 nonlinear spectroscopic techniques 233–242 nonlinear spectroscopy 90, 233, 235, 242 normal coordinate analysis 69, 70, 72 normalization 25–27, 41, 53, 100, 123, 166, 247, 255 normal modes of vibration 69–72, 74, 75, 77, 211, 212 nuclear magnetic resonance (NMR) 93, 129, 131–148, 243 nuclear magneton 135, 221

nuclear spin 129, 134–139, 141, 147 nuclear spin energy states 131, 139

o oblate symmetric top rotor 96 oblate top rotor 108 olefins 179–180 one‐dimensional representation 206 optical activity 185, 187 asymmetric molecules 188–191 dissymmetric molecules 191–192 manifestation of 187–188 vibrational 192–193 optical rotation 185–193 optical rotatory dispersion (ORD) 81, 187 orbital angular momentum quantum number 117 orbital approximation 151, 152, 223 orthogonality condition 208 orthogonality theorem 206 orthonormal functions 253 orthonormality condition 254, 256 orthonormal vector space 25, 54, 120

p parallel envelope 113 paramagnetic 155, 170

parity 21, 41, 42, 56, 57, 65, 76, 171 particle‐in‐a‐box 17, 21, 23–26, 29–31, 33, 37, 38, 40–42, 49, 58, 65, 66, 125, 223, 226–230 particle‐in‐a‐2D‐box 27–28, 33, 73 particle–wave duality 5, 6 partition function 91 Pauli exclusion principle 18, 152–154 Pauli spin matrices 253–256 P‐branch 112 2,3‐pentadiene 186, 187 periodic chart of elements 115, 155 perpendicular 2, 113, 201, 202, 254 perturbation Hamiltonian 225, 226 perturbation method 34, 37, 60, 144, 225 perturbed wavefunction 225, 226, 228, 229 phase‐matching 236–239 phenylalanine 180 phosphorescence 178, 182, 183 photoelectric effect 1, 2, 5–7, 12, 155 photon mass 7 Planck's constant 4, 5, 15, 16, 32, 95, 221 polarizability 40, 81–86, 96, 185, 193, 199, 217, 234 polarizability tensor 85, 86, 185, 193, 210 polyenes 31–32 poly‐L‐Pro (II) 192

population inversion 45–47 potential energy 16, 20–22, 24, 27, 29, 30, 33, 34, 49, 50, 55, 56, 59–60, 70–73, 89, 95, 97, 116, 125, 176, 182, 226, 227, 230 principal axes of inertia 95, 99 prolate symmetric top rotor 96, 108 prolate top rotor 97, 108, 109 proper axes of rotation 200–201 pseudo‐Voight function 79 pulse Fourier transform (FT) NMR spectroscopy 138, 146 pulse FT method 131, 243

q Q‐branch 112, 113 quadrant or octant rules 190–191 quantized energy level 24 quantum cascade lasers 31–34 quantum dots 31–34 quantum mechanics 1–12, 15–21, 24, 28, 37, 39, 49, 69, 97, 103, 115, 128, 223, 254 quantum number 8, 18, 24, 26, 28, 53, 56, 57, 61, 62, 77, 100, 101, 103, 107, 109, 112, 116–118, 120, 124, 126–129, 133–135, 145, 153, 154, 157–160, 173, 178, 253, 254

r racemate 187 racemic mixture 187 radial distribution function 122, 123

Raman optical activity (ROA) 192 Raman scattering 12, 39, 62, 81, 82, 84, 87, 88, 173, 217, 233–236, 240 Raman spectroscopy 81–86, 96, 185, 192, 210, 211, 217, 234, 235, 240, 241 random coil 192, 193 Rayleigh scattering 81, 84 R‐branch 112 recursion formula 52–54, 57, 101, 102, 104, 109, 118 redshift 183, 185 reducible representation 209–214 reflection 42, 47, 201, 204, 208, 214 reflection by mirror plane

186, 200, 201, 204

refraction 42, 79, 80 refractive index 11, 12, 42, 79–81, 84, 187, 188 resonance condition 40, 86, 138, 146 resonance/off‐resonance 11 resonance Raman process

185

resonance Raman spectroscopy 85 resonator structure 46, 47 rigid rotator 107 rotation–reflection axes 200, 201 rotational constants 95, 96, 103, 108, 112, 113, 173 rotational kinetic energy 94, 95, 132 rotational quantum number 100, 112, 158, 173 rotational Raman spectrum 96 rotational Schrödinger equation 97, 99–103, 115

rotational spectroscopy 93–113, 117, 140, 163 rotational strength 189, 192 rotational wavefunction 93, 101 rot–vibrational transitions 110–113 ruby laser 233 Rydberg constant 8, 10, 118, 221

s scattering tensor 85, 86, 234 Schrödinger equation 16, 18, 20, 23–24, 29, 37, 49, 73, 93, 99, 115, 116, 125, 151, 157, 175, 224, 225 second harmonic generation (SHG) 184, 235–237 second‐order dielectric susceptibility 139, 236 second‐order nonlinear susceptibility 185 second‐row diatomic molecules

169

selection rules 40, 57, 85, 102, 104, 109, 124, 137, 160, 172, 199, 217, 235 for absorption 57 for dipole‐allowed processes 214–216 for electronic transition 178 for harmonic oscillator 56–60, 102 for homonuclear diatomic molecules 171–172 for IR and Raman spectroscopy 87–88 for one‐spin nuclear system 137–138 for particle in a box 40–42 for Raman effect 85 for Raman scattering 217 for rotational transitions 104 for transitions in atomic species 160 series expansion 17, 52, 53, 223–225, 229, 233 SF6 75 β‐sheet 192 shielding effect 138, 140, 141, 156 singlet state 158, 172, 182, 183 singlet to triplet transition 160, 172, 178, 183 Slater determinant 18, 152, 153 sloped baseline 226 sodium D line 127 sodium doublet 127 sodium 589nm line 159, 161 spatial quantization 103, 129, 133, 139

spectral transitions 3, 9, 37 spectroscopic transitions xi spherical harmonic functions 33, 98, 99, 101, 102, 115, 117, 119, 121 spherical polar coordinates

16, 98, 116, 117

spherical top rotor 96, 109 spin angular momentum 103, 128, 129, 132–134, 139, 156, 157, 159, 160, 171 spin functions 128, 134, 136, 152, 153, 253, 254, 257 spin–lattice relaxation 147 spin multiplicity 154, 158, 159, 171 spin‐orbit coupling 158, 161 spin‐pairing energy 155 spin spectroscopy 93, 129, 131 spin–spin coupling 143–146 spin state population 137–140 spin states 11, 128, 134, 136–140, 144, 145, 158, 254 spin wavefunctions 132, 133, 142, 152, 153, 253–254 spontaneous emission 45–47, 78, 182 spontaneous, nonresonant Raman process

185

spontaneous Raman 184, 236, 240, 242 square wave 246, 247 Stark splitting 109 stationary states

3, 8, 24, 37–40, 43, 51, 78, 85, 97, 230

stationary‐state wavefunctions 25, 37, 40, 83 statistical thermodynamics 91 Stefan–Boltzmann law 5

stereochemistry 188, 190, 191 Stern–Gerlach experiment 127, 254 stimulated emission 34, 40, 43–45, 48 stimulated Raman scattering (SRS) 240–242 Stokes Raman frequencies 83, 84, 238 successive application 203–205 sum‐frequency generation 234, 237 superimposable image 186 symmetric top rotors 96, 97, 108 symmetric tops 96, 108, 109 symmetry group 186, 200–204, 209, 213, 215, 217 symmetry operation 186, 200–207, 210–212, 214 symmetry representation 199, 210–214

t Taylor series 82 term symbols 127, 156–161, 171–172, 195 testosterone 190 tetrachloroethene 235 tetramethylsilane 141 thermal ellipsoids 75 thermal energy 90, 105 3rd law of thermodynamics 49, 55 third‐order nonlinear susceptibility 238 third‐order susceptibility tensor 239 thymine (T) 180

time‐dependent perturbation 37–40, 230–231 time‐dependent Schrödinger equation 18, 37–39, 230 time‐dependent wavefunction 18, 38, 39, 83 time‐independent perturbation theory 225–226 toluene 180 total orbital angular momentum 157–159, 172 total spin angular momentum 139, 159, 160, 171 trace 25, 41, 42, 187, 205, 207, 211, 212, 214, 241, 245, 246, 248 transformation matrix 72, 204, 207, 211, 212, 214, 249 transition coupling 191–192 transition metals 154, 155, 181 transition moment 3, 39–45, 47, 56–59, 84–87, 104, 105, 107, 124, 175, 176, 188–192, 215–217, 231, 234 translational and rotational degrees of freedom 213, 214 trial function 23, 118, 224 triplet state 158, 171, 172, 183 triply degenerate 113, 181, 208 tunnelling 29–31, 34, 56, 230 two‐photon fluorescence 183–184 two‐photon process 39, 85, 87

u unbound particle 27–31 uncertainty principle

15, 19, 55, 78, 241

v valance‐shell electron‐pair repulsion (VSEPR) model 93 variation method

169, 224–225

vector space 17, 19, 22, 25, 54, 97, 101, 120, 200, 223, 224 vibrational dephasing 241 vibrational infrared and Raman spectroscopy 69–91 vibrational optical activity

192–193

vibrational Schrödinger equation 51, 52, 54, 60, 73, 214 vibrational spectroscopy 52, 62–65, 69, 70, 78, 79, 90, 93, 97, 178, 193, 199, 211 vibronic absorption spectrum 173–174 Voigt function 79

w wave vector 2, 29, 237, 239 Wien's law 5 work function 6

z Zeeman effect 126, 127 zero‐point vibrational energy 74

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