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- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A concise textbook bridging quantum theory and spectroscopy!
Designed as a practical text, Quantum Mechanical Foundations of Molecular Spectroscopy covers the quantum mechanical fundamentals of molecular spectroscopy from the view of a professional spectroscopist, rather than a theoretician. Written by a noted expert on the topic, the book puts the emphasis on the relationship between spectroscopy and quantum mechanics, and provides the background information and derivations of the subjects needed to understand spectroscopy including: stationary energy states, transitions between these states, selection rules, and symmetry.
The phenomenal growth of all forms of spectroscopy over the past eight decades has contributed enormously to our understanding of molecular structure and properties. Today spectroscopy covers a broad field including the modern magnetic resonance techniques, non-linear, laser and fiber-based spectroscopy, surface and surface-enhanced spectroscopy, pico- and femtosecond time resolved spectroscopy, and many more. This up-to-date resource discusses several forms of spectroscopy that are used in many fields of science, such as fluorescence, surface spectroscopies, linear and non-linear Raman spectroscopy and spin spectroscopy. This important text:
- Contains the physics and mathematics needed to understand spectroscopy
- Explores spectroscopic methods the are widely used in chemistry, biophysics, biology, and materials science
- Offers a text written by an experienced lecturer and practitioner of spectroscopic methods
- Includes detailed explanations and worked examples
Written for chemistry, biochemistry, material sciences, and physics students, Quantum Mechanical Foundations of Molecular Spectroscopy provides an accessible text for understanding molecular spectroscopy.
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Veröffentlichungsjahr: 2021
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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 H
2
+
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 ranges
a
.
Chapter 5
Table 5.1 Vibrational modes and assignments for chloroform, HCCl
3
.
Chapter 8
Table 8.1 Nuclear
g
‐factors, magnetogyric ratios, and spin moments for some sp...
Chapter 9
Table 9.1 Symbols of states for different
l
and
L
values.
Table 9.2 Transition, energies, term symbols, and wavelengths of the prominen...
Chapter 10
Table 10.1 Symbols of states for different
l
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 Br
2.
(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 –CCl
3
stretchin...
Chapter 6
Figure 6.1 Definition of spherical polar coordinates.
Figure 6.2 Graphical representation for the condition
T
(φ) =
T
(φ +
b
2
π
Figure 6.3 (a) Energy level diagram for linear rotors. (b) Schematic rotatio...
Figure 6.4 Simulated rotational spectrum of
35
Cl–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...
Guide
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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
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Preface
When the author took courses in quantum mechanical principles and chemical bonding in graduate school in the early 1970s, the course materials seldomly covered the fascinating interplay between spectroscopy and quantum mechanics, and textbooks of these days devoted the majority of space to derivations and mathematical principles and the discussion of the hydrogen atom and chemical bonding. While an understanding of these subjects is, of course, a necessity for further study, this book emphasizes a slightly different approach to quantum mechanics, namely, one from the viewpoint of a spectroscopist. In this approach, the existence of stationary energy states – either electronic, vibrational, rotational, or spin states – is considered the fundamental concept, since spectroscopy exists because of transitions between these states. Quantum mechanics provides the theoretical framework for the interpretation of experimental data. On the other hand, spectroscopic results provide the impetus for refining theories that explain the results. Classical physics cannot provide this framework, since the idea of stationary energy states violates the laws of classical physics.
Thus, the approach taken here in this book is to present early on, in Chapter 2, how the application of quantum mechanical principles leads necessarily to the existence of stationary energy states using the particle‐in‐a box model system. The third chapter then introduces the concept of spectroscopic transitions between these stationary states, using time‐dependent perturbation theory.
The following chapters are presented in order of mathematical complexity of the Schrödinger equation that describes the problem. The simplest case, the particle in a box, is discussed in Chapter 2. The next subject is the simple harmonic oscillator, for which the eigenfunctions resemble those of the particle in a box, and transitions can be visualized in terms of the discussion in Chapters 2 and 3. In the following discussions (Chapters 5–10), vibrational, rotational, atomic, molecular electronic, and spin spectroscopies will be introduced. These discussions, if possible, start with a classical description, followed by the quantum mechanical equations for wavefunctions and eigenvalues, and the derivation of the selection rules. These selection rules determine the form and information content of the respective spectroscopic techniques. Although space limitations prevent in‐depth discussions of spectroscopic applications to complex molecular systems, all efforts have been made to include molecular systems larger than diatomic molecules (the level of molecular complexity where many textbooks capitulate), since the world we live in mostly consists of more complicated molecules than diatomics.
Thus, in Chapter 5, the concept of the harmonic oscillator (Chapter 4) will be extended to vibrational (infrared and Raman) spectroscopy of polyatomic molecules. This chapter introduces concepts of band shapes, lifetimes, and a quantum mechanical description of molecular polarizability. Next in complexity are the differential equations for a rotational molecule that leads to rotational spectroscopy (Chapter 6). These equations will introduce the quantum mechanical description of the angular momentum and the energy levels of simple and more complicated molecules. The results from the rotational Schrödinger will also be used to solve the radial part of the hydrogen atom Schrödinger equation (Chapter 7). The principles learned from the rotational Schrödinger equation will also be used to introduce the spin eigenfunctions and eigenstates, a subject that leads directly to spin spectroscopy such as nuclear magnetic resonance (NMR), which is discussed in Chapter 8.
Next, the structure of atoms and ions containing more than one electron will be presented. This discussion includes an introduction to atomic spectroscopy and term symbols of electronic states. However, since the main theme of this book is molecular spectroscopy, this chapter only serves as an introduction to these subjects.
Chapter 10 is devoted to electronic spectroscopy of di‐ and polyatomic molecules. Again, as in previous chapters, it is necessary to define the states between which electronic transitions occur. This leads necessarily to the discussion of chemical bonding in terms of molecular orbital theory. Chemical bonding will be discussed to the level that electronic spectra of simple molecules can be explained, but the interaction between vibrational and electronic wavefunctions to produce vibronic states will be discussed in more detail to explain fluorescence phenomena as well as some Raman effects that rely on transitions into vibronic energy levels. Finally, Chapter 11 introduces group theory and the symmetry properties of molecules and the influence of symmetry on the appearance of molecular spectra.
The approach taken here in this book was strongly influenced by an excellent textbook Physical Chemistry by Engel and Reid [1] that was used as a required text in undergraduate physical chemistry courses at Northeastern University. This book emphasizes the unconventional approach taken by the early theorists who are responsible for the field of quantum mechanics as we know it. I gained substantial understanding of the philosophical background of quantum mechanics from this book. What is presented here in Quantum Mechanical Foundations of Molecular Spectroscopy is a similar approach but with much more emphasis on molecular spectroscopy.
Although the present book emphasizes the relationship between spectroscopy and quantum mechanics more than other texts, the author wishes to point out the importance of following up on some proofs and derivations (omitted here) by studying books on “real” quantum mechanics or quantum chemistry. In particular, the one‐ and two‐volume treatments by I. Levine [2, 3] are highly recommended, as well as many other old and new books [4, 5].
The mathematical requirements for understanding this book do not exceed the level achieved after a three‐semester sequence of calculus, and all efforts have been made to provide examples and problems that will illuminate the mathematical steps. Most importantly, although some derivations are presented, the goal is not to lose sight of what quantum mechanics does for spectroscopy in the mathematical complexities.
Boston, August 2019
Literature references for the Preface are at the end of the Introduction.
Introduction
This book, Quantum Mechanical Foundations of Molecular Spectroscopy, is based on a graduate‐level course by the same name that is being offered to first‐year graduate students in chemistry at the Department of Chemistry and Chemical Biology at Northeastern University in Boston. When I joined the faculty there in 2005, I revised the course syllabus to emphasize the philosophical underpinnings of quantum mechanics and introduce much more of the quantum 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.
1Transition 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:
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
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):
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):
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:
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:
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:
(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
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
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
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:
The (kinetic) energy of a photon is given by
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,
it follows that the photon mass is given by
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
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
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
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:
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
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
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
Using the value of the Rydberg constant given above, Ry = 2.179 × 10−18 J, the energy difference is
Using Eq. (1.12), ΔE = Ephoton = hν = hc/λ, the frequency ν is found to be
The wavelength of such a photon is given by Eq. (1.7) as
that is, a photon in the ultraviolet wavelength range. Finally, the wavenumber of this photon is
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.
ν
photon
λ
photon
E
photon
[J]
E
photon
[kJ/mol]
E
photon
[m
−1
]
Transition
Radio
750 MHz
0.4 m
5×10
−25
3×10
−4
2.5
NMR
b
Microwave
3 GHz
10 cm
2×10
−24
0.001
10
EPR
b
Microwave
30 GHz
1 cm
2×10
−23
0.012
100
Rotational
Infrared
3×10
13
Hz
10 μm
2×10
−20
12
10
5
Vibrational
UV/visible
10
15
300 nm
6×10
−19
360
3×10
6
Electronic
X‐ray
10
18
0.3 nm
6×10
−16
3.6×10
5
3×10
9
X‐ray absorption
a) 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
