harris d., bertolucci m. symmetry and spectroscopy 1978
TRANSCRIPT
Symmetry and Spectroscopy AN INTRODUCTION TO VIBRATIONAL AND ELECTRONIC SPECTF~OSCOPY
BY
AND
I
Copyright © 1978 by Oxford University Press, Inc. All rights reserved under Pan American and International Copyright
Conventions.
This Dover edition, first published in 1989, is an unabridged, corrccted republication of the work first published by Oxford University Press, New York, 1978. It is reprinted by special arrangement with Oxford University Press, 200 Madison Avenue, New York, N.Y. 10016.
Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging-in-Publication Data
Harris, Daniel C, 1948- Symmetry and spectroscopy : an introduction to vibrational and
electronic spectroscopy I by Daniel C Harris and Michael D. Bertolucci. p. cm.
Reprint. Originally published: New York: Oxford University Press, 1978.
Includes bibliographies and index. ISBN 0-486-66144-X I. Vibrational spectra. 2. Electron spectroscopy. 3. Molecular orbitals.
4. Symmetry (Physics) I. Bertolucci, Michael D. II. Title. QD96.V53H37 1989 543'.0858--dc20 89-16810
CIP
Contents
I. A chemist's view of group theory, 5
I-I. Introduction, 5 1-2. Symmetry Operations and Molecules, 6 1-3. Groups, IO 1-4. Point Groups, 14 1-5. Classification of Molecules into Point Groups, 29 1-6. Matrix Representation of Symmetry Operations, 35 1-7. Characters and Character Tables, 47 1-8. Decomposition of Reducible Representations and the Direct Product, 55
Additional Problems, 58 Related Reading, 61
2. A skirmish with quantum mechanics, 62 2-1. Introduction, 62 2-2. Light, 62 2-3. The Postulates of Quantum Mechanics, 68 2-4. Some Simple Illustrations from Quantum Mechanics, 80
Related Reading, 92
3. Vibrational spectroscopy, 93 3-1. Introduction, 93 3-2. Infrared and Raman Spectra, 93 3-3. Diatomic Molecules, 100 3-4. Transitions between Stationary States, 130 3-5. The Normal Modes of Vibration of Polyatomic Molecules, 135
I • I I I I I
I I
vi Contents
3-6. Selection Rules and Polarization, lSI 3-7. Symmetry Coordinates and Normal Modes, 170 3-8. Stretching Mode Analysis, 186 3-9. Assignment of Real Spectra, 190
3-10. The Resonance Raman Effect, 198 3-11. Functional Group Analysis, 201
Additional Problems, 217 Related Reading, 224
4. Molecular orbital theory, 225
4-1. Introduction, 225 4-2. Atoms, 225 4-3. Photoelectron Spectroscopy, 242 4-4. The LCAO Molecular Orbital Method, 245 4-5. Diatomic Molecules, 253 4-6. Polyatomic Molecules. 267 4-7. The Hiickel Method, 288 4-8. 'II-ansition Metal Complexes, 299
Additional Problems, 302 Related Reading, 306
5. Electronic spectroscopy, 307
5-1. Introduction, 307 5-2. Another Look at Molecular Vibrations, 307 5-3. Basic Notions, 310 5-4. Selection Rules, 330 5-5. The Electronic Spectra of Some Diatomic Molecules, 343 5-6. The Fate of Absorbed Energy, 357 5-7. Single Bonds, Double Bonds, and Lone Pairs, 368 5-8. Vibronic Analysis, 38 I 5-9. li'ansition Metal Complexes, 395
5-10. Concluding Remarks, 411 Additional Problems, 412 Related Reading, 418
Appendices
A. Character tables, 421 B. Direct products, 479 C. Overtones of degenerate vibrations, 492 D. The shapes of atomic orbitals, 495 E. Physical constants, 497 F. Energy conversions, 498 G. Answers to problems, 499
Index, 543
Preface
Late in the afternoon on a hot, smoggy, Los Angeles, September day in
1970, a well-publicized contest pitting the most pollution-free vehicles in
the country against each other was coming to an end. It was the second
Clean Air Car Race from M.LT. to Caltech. Among the throngs of digni
taries and spectators at the finish line on Greasy Street that day were two
graduate students, George Rossman and mys(~lf. We hoped to capture a
sample of Clean Air Car exhaust in an evacuated glass cylinder I had
wrapped in some rags and carefully protected from the surging crowd.
When the third car came in, and the crowd around the second car (an
entry from the University of California at Berkeley) began to wane, I
pushed my way up to the Berkeley driver's window, still clinging the
precious evacuated cylinder. I explained to the driver that I wanted a
sample of his exhaust and asked if he would please start the car as I
crawled under his tail pipe. Not only did he start the car for me, but when
I crawled underneath and positioned the mouth of the cylinder in the
exhaust pipe, he drove away! The next time I caught that rascal from
Berkeley I did manage to bottle his exhaust. We scurried off to the lab
where George produced the gas phase infrared spectra shown below.
These spectra pit the Clean Air Car against the Harris S mogmobile and
show that the carbon monoxide emission of my car (two humps near 2150 cm- I
) is absent in the Clean Air Car exhaust. This little episode marked
the start of the section of this book dealing with the carbon monoxide rotation-vibration spectrum. Not every part of this text has such a color
ful history, but most have benefited from a similar degree of personal ..
VII
IXPreface
the applications of symmetry in chemistry, as well as for a course in spectroscopy. We cover most of the topics in Cotton's fine book, Chemi cal Applications of Group Theory, but do so in the process of teaching vibrational and electronic spectroscopy and molecular orbital theory.
First and foremost this is a texthook. We have taken great pains not to assume very much background knowledge on the part of the reader. To make the exposition clear and meaningful, each new concept is applied or illustrated with experimental results as quickly as possible. The text includes some 200 problems with solutions in Appendix G. We consider these problems to be an integral part of the tex t and sometimes introduce new material in them. The student is urged to work through as many as time permits.
The present version of this book was written during a two-year period of postdoctoral research in the laboratory of Phil Aisen at the Albert Einstein College of Medicine in New York. The original text was written in collaboration with Mike Bertolucci who taught the course with me for a year at Caltech. Don Titus, Benes Trus, and Harry Gray have made invaluable contributions subsequently. Harry Gray and George Ham mond were instrumental in initiating the course and capturing my interest in it (which was similar to capturing the interest of a hungry monkey in a banana). To keep the price of this volume to a level that students can afford, my wife Sally devoted more than half a year of effort to the pro duction of line drawings. Finally, I cannot overestimate the role my stu dents played in the development of this book. Comments on ways to im prove the book or on errors are solicited from all readers and will be greatly appreciated.
I dedicate this book to the student who is willing to take it to bed with him at night, along with a pencil and occasionally a calculator, and who falls asleep with a smile on his face.
I I I -----'---------,J 1200 1000 800 em-'
1200 1000 800 em-
4000
involvement on the part of myself and my students during the three years the manuscript was used for part of an undergraduate spectroscopy
course at Caltech. This book was written with the goal of introducing the student to vibra
tional and electronic spectroscopy and taking her or him to a rather sophisticated (albeit qualitative) level in some areas. We have tried to write a text most suitable for use on the junior to beginning graduate levels. Taking the approach that group theory is essential to the modern practice of spectroscopy, we devote the first chapter to group theory and then make extensive use of it throughout the text. For this reason we believe that this book may be used as the primary text for a course on
Dan Harris
NOTE (1989): Daniel C. Harris may be reached at Chemistry Division, Research Department, Michelson Laboratory, China Lake, CA 93555.
Note Oil ullits alld COIII'elltiollS •
Xl
Note on units and conventions We use the largest variety of units in measurements of energy. Chem
ists familiar with calories will find the use of ioules not too ditlicult be cause the conversion is simple:
I calorie = 4.184 joule s
In the cgs system, I erg = 10 7 joules. The electron volt (eV) is the kinetic
energy Df an electron accelerated through Olle volt: I electron volt ~ 1.602 x 10 '" joules. The corresponding mol'll' energy is I .602 x 10 '" J x 6.022 X IO:!" mol' = 96.49 kJ mol' = n.o<', kcal mol'.
We very frequently express "energy" as w<lve numbers. The relation between wave number and energy, in cgs units, is
- 11 =
!Ic
where r; is wave number (cm-'). I:' is energy (erg), !I is Planck's constant
(erg s). and c is the speed of light (cm s '). The SI units arc, respectively,
r; (m '), I:' (1), !I (1 s), and c (m s '). To convert wave number in reciprocal
meters to wave number in recriprocal centimeters, we divide by 100
Since "\\'ilFe IIl/miler" is IllliFersill!y C'xpre.\sed ill reciproc{/I cC'lItimeters.
and since any equation for energy in this book is in joules, unless other
wise specified, the conversion iJ = EIIOO!lc must be used to obtain wave numbers in reciprocal centimeters. When we want to emphasize that a quantity is in cm ' units, we will write a bar over the symbol (e.g., E and (/),.). The use of cm-' units is discussed further in Chapter Three. The
1,000 cm ' unit is a kilokayser (kK). Units of length often encountered are centimeters and Angstroms (I
A- 10 '" m). Millimicrons (mfJ.) have been largely replaced by nano
meters (I mfJ. =~ I nm = 10 " m). Temperature is generally expressed in
Kelvins. written K (not OK). Concentrations arc always expressed as
moles per liter (1 moll '- 1 M). The molar extinction coefficient (E) used in spectroscopy is uni\'ersally exprcssed in M 'em' units, and we dare not
tamper with them.
The choice of coordinate systems and symmetry elements can be a
major problem in the literature. We recommend that coordinate systems
and symmetry elements always be defined at the outset of a research paper, homework problem, blackboard example, or anything else. We
adhere to this policy faithfully. Conventions on coordinate systems and
Although the Systeme International d' Unites (S 1 units) are the primary
units in this book,t a number of other standard units are also used. It is suggested that the student become familiar with the different units we employ because all are encountered in practice. Unless otherwise stated,
however, all calculations and all equations here employ SI units. When we wish to convert an answer to a unit other than an 51 unit, the conver
sion is the last step of the calculation. In the SI system, the units of mass, length, time, and charge are the
kilogram (kg), meter (m), second (s), and coulomb (C), respectively. Force is expressed in newtons (I N = I kg m s 2) and energy in joules (1 J = 1 N m = 1 kg m2 S-2). Coulomb's law is generally written
where F is force, q is charge, r is distance, and k is a constant. In the
centimeter-gram-second (cgs) system, the unit of charge, the electrostatic unit (esu), is such that k is dimensionless and has the magnitude unity. In SI units, k is written 1/47TEn , where En(= X.85419 x 10 '2 C2N'm 2) is
called the dielectric constant (or permittivity) of free space. The factor 47TE n appears in several equations in this book and .should alert you to the fact that coulombs, meters, and kilograms are useu in such equations.
+ f;or discussions of Sf unih. sec A.C. Norris . .I. Chern. f~·d.. 4R, 797 (1(71); (;. Socrates. Ihid., 4{J,
711 (1%9); T.l. Quickcnden and R.l". Mar<hall. lhid .. 49. 114 (1972); J.l. Hoppe", lhid .. 49. 505
(1972); and G. Pass and H. Sutcliffe. Ii"d .. 411.180 (1971).
x
') h(J s) dm s ')
" •
•, the applications of symmetry in chemistry, as well as for a course in spectroscopy. We cover most of the topics in Cotton's fine book, Chemi cal Applications of Group Theory, but do so in the process of teaching vibrational and electronic spectroscopy and molecular orbital theory.
First and foremost this is a textbook. We have taken great pains not to assume very much background knowledge on the part of the reader. To make the exposition clear and meaningful, each new concept is applied or illustrated with experimental results as quickly as possible. The text includes some 200 problems with solutions in Appendix G. We consider these problems to be an integral part of the text and sometimes introduce new material in them. The student is urged to work through as many as time permits.
The present version of this book was written during a two-year period of postdoctoral research in the laboratory of Phil Aisen at the Albert Einstein College of Medicine in New York. The original text was written in collaboration with Mike Bertolucci who taught the course with me for a year at Caltech. Don Titus, Benes Trus, and Harry Gray have made invaluable contributions subsequently. Harry Gray and George Ham mond were instrumental in initiating the course and capturing my interest in it (which was similar to capturing the interest of a hungry monkey in a banana). To keep the price of this volume to a level that students can afford, my wife Sally devoted more than half a year of effort to the pro duction of line drawings. Finally, I cannot overestimate the role my stu dents played in the development of this book. Comments on ways to im prove the book or on errors are solicited from all readers and will be greatly appreciated.
I dedicate this book to the student who is willing to take it to bed with him at night, along with a pencil and occasionally a calculator, and who falls asleep with a smile on his face.
1000120014001600leDO2000
--....- C'O
3000
HARRIS SMOGMOBILE
involvement on the part of myself and my students during the three years the manuscript was used for part of an undergraduate spectroscopy
course at Caltech. This book was written with the goal of introducing the student to vibra
tional and electronic spectroscopy and taking her or him to a rather sophisticated (albeit qualitative) level in some areas. We have tried to write a text most suitable for use on the junior to beginning graduate levels. Taking the approach that group theory is essential to the modern practice of spectroscopy, we devote the first chapter to group theory and then make extensive use of it throughout the text. For this reason we believe that this book may be used as the primary text for a course on
Dan Harris
NOTE (1989): Daniel C. Harris may be reached at Chemistry Division, Research Department, Michelson Laboratory, China Lake, CA 93555.
• • XII Note on units and conventions
symmetry elements have been recommended by the Joint Commission for Spectroscopy of the International Astronomical Union and the Inter national Union of Pure and Applied Physics [1. Chern. Phys., 23, 1997 (1955)]. We generally adhere to these conventions, with some notable exceptions. We accidentally adopted a coordinate system for ethylene that differs from the system in common use. We apologize for this but chose not to change our coordinate system for fear of introducing errors into the text in the process of making changes. We intentionally disre garded the recommended choice of symmetry elements for XeF4 (D 4h
symmetry), but not for benzene (D fih ). We recommend that C 2 ' and (T,o
axes always be colinear and C 2 " and art axes always be colinear and that C 2 ' and a v go through as many atoms as possible and that C 2" and art go through as few atoms as possible. We further recommend the use of our convention for all point groups, instead of adopting different conventions for each point group.
With regard to the naming of symmetry operations (e.g., C2 , C2 ', and C 2 ") and other conventions in character tables, we adopted the widely used tables of Cotton (F.A. Cotton, Chemical Applications of Group Theory, John Wiley & Sons, New York, 1971); we hope that this set of tables will become standard.
Symmetry and Spectroscopy
O· Opening remarks
Spectroscopy is the study of the interaction of electromagnetic radiation
(light, radio waves, x-rays, etc.) with matter. In this book we will de,d with
a central portion of the electromagnetic spectrum (Fig. 0-1). spanning the
infrared (ir), visible (vis), and ultraviolet (uv) wa velengths.
Molecules, consisting of electrically charged nuclei and electrons. Illay
interact with the oscillating electric and magnetic fields of light and ahsorh
the energy carried by the light. The molecule doc', not interact with ;i1llight
that comes its way, but only with light that carries the right amount of
energy to promote the molecule from one discrete energy level to another.
For example, the diatomic molecule 127)79 Sr in its lowest vibrational state
(ground state) vibrates with an energy 01'2.662 )< 10- 21 J. The next lowest
vibrational energy available to the molecule is "1.961 x 10- 21 .I. Suppose
that far infrared light of energy (7.961 - 2.662) x 10 21 .1 = 5.299 x 10 21
J (= 266.8 cm 1) is shined on a sample of 127 1-() Hr. The light ean he
absorbed and a ground state molecule can be promoted to its first excited
vibrational state. When this happens we say that the molecule has Illade a
transition hetween the ground state and the first excited state. The 1wo energy levels we have been discussing and the absorptIon specl rUIll or
127 j7Q Hr in the far infrared arc shown schematically in Fig. 0-). light or
energy other than 5.299 x 10 21 .I would not be absorbed by the salllpk'
because the energy carried hy such light docs not precisely span two enngv
levels of the molecule.
Light of infrared frequencies can generally prolllote lllolecules frolll onl'
- _.
.......................--
! , ! U• >'(m) -12 -10 -8 -7
1.00xl0 l.oox10 1.00x10 (3.80 7.8Olx10
v(Hz) 20 3.00x10 18 3.00.10 16
(7.89 3.84xI0 14
3.00x10
>. -- ~
.~- acc- -- - --- -- --- - --- ---c: ~ ~~ ~ -~ -.-- Q) bc ac be ab ae be- -0a.. -
I ]I III -....
parallelepipeds in Fig. 0-3. Each of these solid ligures has six stable resting positions in the earth's gravitational field, whell it is resting on its two ah,
two ac, or two be faces. The potential energy, V, associated with each position can be calculated from the formula V 0'" mgh, when~ m is the mass of the parallelepiped, g is the acceleration of gravity, and h is the height of the center of mass above the surface on which the object rests. The potential energies of the six resting positions of paralldepiped I arc divided. into three groups, depending on which face it res ts. The potential energy is greatest when its center of mass is highest, i.e., when it rest:; on its ,,17 faces (Fig. 0-3). The parallelepiped II has only two energetically distinc1 resting positions because the lengths of sides a and b are equal. ;\\1 six (aces of the cube, III, are equal and hence only a single potential energy is obtained, regardless of which face it rests on. In this example, as the symmetry of the solids increases in the order I < II < III, the number of differenl energy levels decreases. The degeneracy (number of states which have…
BY
AND
I
Copyright © 1978 by Oxford University Press, Inc. All rights reserved under Pan American and International Copyright
Conventions.
This Dover edition, first published in 1989, is an unabridged, corrccted republication of the work first published by Oxford University Press, New York, 1978. It is reprinted by special arrangement with Oxford University Press, 200 Madison Avenue, New York, N.Y. 10016.
Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging-in-Publication Data
Harris, Daniel C, 1948- Symmetry and spectroscopy : an introduction to vibrational and
electronic spectroscopy I by Daniel C Harris and Michael D. Bertolucci. p. cm.
Reprint. Originally published: New York: Oxford University Press, 1978.
Includes bibliographies and index. ISBN 0-486-66144-X I. Vibrational spectra. 2. Electron spectroscopy. 3. Molecular orbitals.
4. Symmetry (Physics) I. Bertolucci, Michael D. II. Title. QD96.V53H37 1989 543'.0858--dc20 89-16810
CIP
Contents
I. A chemist's view of group theory, 5
I-I. Introduction, 5 1-2. Symmetry Operations and Molecules, 6 1-3. Groups, IO 1-4. Point Groups, 14 1-5. Classification of Molecules into Point Groups, 29 1-6. Matrix Representation of Symmetry Operations, 35 1-7. Characters and Character Tables, 47 1-8. Decomposition of Reducible Representations and the Direct Product, 55
Additional Problems, 58 Related Reading, 61
2. A skirmish with quantum mechanics, 62 2-1. Introduction, 62 2-2. Light, 62 2-3. The Postulates of Quantum Mechanics, 68 2-4. Some Simple Illustrations from Quantum Mechanics, 80
Related Reading, 92
3. Vibrational spectroscopy, 93 3-1. Introduction, 93 3-2. Infrared and Raman Spectra, 93 3-3. Diatomic Molecules, 100 3-4. Transitions between Stationary States, 130 3-5. The Normal Modes of Vibration of Polyatomic Molecules, 135
I • I I I I I
I I
vi Contents
3-6. Selection Rules and Polarization, lSI 3-7. Symmetry Coordinates and Normal Modes, 170 3-8. Stretching Mode Analysis, 186 3-9. Assignment of Real Spectra, 190
3-10. The Resonance Raman Effect, 198 3-11. Functional Group Analysis, 201
Additional Problems, 217 Related Reading, 224
4. Molecular orbital theory, 225
4-1. Introduction, 225 4-2. Atoms, 225 4-3. Photoelectron Spectroscopy, 242 4-4. The LCAO Molecular Orbital Method, 245 4-5. Diatomic Molecules, 253 4-6. Polyatomic Molecules. 267 4-7. The Hiickel Method, 288 4-8. 'II-ansition Metal Complexes, 299
Additional Problems, 302 Related Reading, 306
5. Electronic spectroscopy, 307
5-1. Introduction, 307 5-2. Another Look at Molecular Vibrations, 307 5-3. Basic Notions, 310 5-4. Selection Rules, 330 5-5. The Electronic Spectra of Some Diatomic Molecules, 343 5-6. The Fate of Absorbed Energy, 357 5-7. Single Bonds, Double Bonds, and Lone Pairs, 368 5-8. Vibronic Analysis, 38 I 5-9. li'ansition Metal Complexes, 395
5-10. Concluding Remarks, 411 Additional Problems, 412 Related Reading, 418
Appendices
A. Character tables, 421 B. Direct products, 479 C. Overtones of degenerate vibrations, 492 D. The shapes of atomic orbitals, 495 E. Physical constants, 497 F. Energy conversions, 498 G. Answers to problems, 499
Index, 543
Preface
Late in the afternoon on a hot, smoggy, Los Angeles, September day in
1970, a well-publicized contest pitting the most pollution-free vehicles in
the country against each other was coming to an end. It was the second
Clean Air Car Race from M.LT. to Caltech. Among the throngs of digni
taries and spectators at the finish line on Greasy Street that day were two
graduate students, George Rossman and mys(~lf. We hoped to capture a
sample of Clean Air Car exhaust in an evacuated glass cylinder I had
wrapped in some rags and carefully protected from the surging crowd.
When the third car came in, and the crowd around the second car (an
entry from the University of California at Berkeley) began to wane, I
pushed my way up to the Berkeley driver's window, still clinging the
precious evacuated cylinder. I explained to the driver that I wanted a
sample of his exhaust and asked if he would please start the car as I
crawled under his tail pipe. Not only did he start the car for me, but when
I crawled underneath and positioned the mouth of the cylinder in the
exhaust pipe, he drove away! The next time I caught that rascal from
Berkeley I did manage to bottle his exhaust. We scurried off to the lab
where George produced the gas phase infrared spectra shown below.
These spectra pit the Clean Air Car against the Harris S mogmobile and
show that the carbon monoxide emission of my car (two humps near 2150 cm- I
) is absent in the Clean Air Car exhaust. This little episode marked
the start of the section of this book dealing with the carbon monoxide rotation-vibration spectrum. Not every part of this text has such a color
ful history, but most have benefited from a similar degree of personal ..
VII
IXPreface
the applications of symmetry in chemistry, as well as for a course in spectroscopy. We cover most of the topics in Cotton's fine book, Chemi cal Applications of Group Theory, but do so in the process of teaching vibrational and electronic spectroscopy and molecular orbital theory.
First and foremost this is a texthook. We have taken great pains not to assume very much background knowledge on the part of the reader. To make the exposition clear and meaningful, each new concept is applied or illustrated with experimental results as quickly as possible. The text includes some 200 problems with solutions in Appendix G. We consider these problems to be an integral part of the tex t and sometimes introduce new material in them. The student is urged to work through as many as time permits.
The present version of this book was written during a two-year period of postdoctoral research in the laboratory of Phil Aisen at the Albert Einstein College of Medicine in New York. The original text was written in collaboration with Mike Bertolucci who taught the course with me for a year at Caltech. Don Titus, Benes Trus, and Harry Gray have made invaluable contributions subsequently. Harry Gray and George Ham mond were instrumental in initiating the course and capturing my interest in it (which was similar to capturing the interest of a hungry monkey in a banana). To keep the price of this volume to a level that students can afford, my wife Sally devoted more than half a year of effort to the pro duction of line drawings. Finally, I cannot overestimate the role my stu dents played in the development of this book. Comments on ways to im prove the book or on errors are solicited from all readers and will be greatly appreciated.
I dedicate this book to the student who is willing to take it to bed with him at night, along with a pencil and occasionally a calculator, and who falls asleep with a smile on his face.
I I I -----'---------,J 1200 1000 800 em-'
1200 1000 800 em-
4000
involvement on the part of myself and my students during the three years the manuscript was used for part of an undergraduate spectroscopy
course at Caltech. This book was written with the goal of introducing the student to vibra
tional and electronic spectroscopy and taking her or him to a rather sophisticated (albeit qualitative) level in some areas. We have tried to write a text most suitable for use on the junior to beginning graduate levels. Taking the approach that group theory is essential to the modern practice of spectroscopy, we devote the first chapter to group theory and then make extensive use of it throughout the text. For this reason we believe that this book may be used as the primary text for a course on
Dan Harris
NOTE (1989): Daniel C. Harris may be reached at Chemistry Division, Research Department, Michelson Laboratory, China Lake, CA 93555.
Note Oil ullits alld COIII'elltiollS •
Xl
Note on units and conventions We use the largest variety of units in measurements of energy. Chem
ists familiar with calories will find the use of ioules not too ditlicult be cause the conversion is simple:
I calorie = 4.184 joule s
In the cgs system, I erg = 10 7 joules. The electron volt (eV) is the kinetic
energy Df an electron accelerated through Olle volt: I electron volt ~ 1.602 x 10 '" joules. The corresponding mol'll' energy is I .602 x 10 '" J x 6.022 X IO:!" mol' = 96.49 kJ mol' = n.o<', kcal mol'.
We very frequently express "energy" as w<lve numbers. The relation between wave number and energy, in cgs units, is
- 11 =
!Ic
where r; is wave number (cm-'). I:' is energy (erg), !I is Planck's constant
(erg s). and c is the speed of light (cm s '). The SI units arc, respectively,
r; (m '), I:' (1), !I (1 s), and c (m s '). To convert wave number in reciprocal
meters to wave number in recriprocal centimeters, we divide by 100
Since "\\'ilFe IIl/miler" is IllliFersill!y C'xpre.\sed ill reciproc{/I cC'lItimeters.
and since any equation for energy in this book is in joules, unless other
wise specified, the conversion iJ = EIIOO!lc must be used to obtain wave numbers in reciprocal centimeters. When we want to emphasize that a quantity is in cm ' units, we will write a bar over the symbol (e.g., E and (/),.). The use of cm-' units is discussed further in Chapter Three. The
1,000 cm ' unit is a kilokayser (kK). Units of length often encountered are centimeters and Angstroms (I
A- 10 '" m). Millimicrons (mfJ.) have been largely replaced by nano
meters (I mfJ. =~ I nm = 10 " m). Temperature is generally expressed in
Kelvins. written K (not OK). Concentrations arc always expressed as
moles per liter (1 moll '- 1 M). The molar extinction coefficient (E) used in spectroscopy is uni\'ersally exprcssed in M 'em' units, and we dare not
tamper with them.
The choice of coordinate systems and symmetry elements can be a
major problem in the literature. We recommend that coordinate systems
and symmetry elements always be defined at the outset of a research paper, homework problem, blackboard example, or anything else. We
adhere to this policy faithfully. Conventions on coordinate systems and
Although the Systeme International d' Unites (S 1 units) are the primary
units in this book,t a number of other standard units are also used. It is suggested that the student become familiar with the different units we employ because all are encountered in practice. Unless otherwise stated,
however, all calculations and all equations here employ SI units. When we wish to convert an answer to a unit other than an 51 unit, the conver
sion is the last step of the calculation. In the SI system, the units of mass, length, time, and charge are the
kilogram (kg), meter (m), second (s), and coulomb (C), respectively. Force is expressed in newtons (I N = I kg m s 2) and energy in joules (1 J = 1 N m = 1 kg m2 S-2). Coulomb's law is generally written
where F is force, q is charge, r is distance, and k is a constant. In the
centimeter-gram-second (cgs) system, the unit of charge, the electrostatic unit (esu), is such that k is dimensionless and has the magnitude unity. In SI units, k is written 1/47TEn , where En(= X.85419 x 10 '2 C2N'm 2) is
called the dielectric constant (or permittivity) of free space. The factor 47TE n appears in several equations in this book and .should alert you to the fact that coulombs, meters, and kilograms are useu in such equations.
+ f;or discussions of Sf unih. sec A.C. Norris . .I. Chern. f~·d.. 4R, 797 (1(71); (;. Socrates. Ihid., 4{J,
711 (1%9); T.l. Quickcnden and R.l". Mar<hall. lhid .. 49. 114 (1972); J.l. Hoppe", lhid .. 49. 505
(1972); and G. Pass and H. Sutcliffe. Ii"d .. 411.180 (1971).
x
') h(J s) dm s ')
" •
•, the applications of symmetry in chemistry, as well as for a course in spectroscopy. We cover most of the topics in Cotton's fine book, Chemi cal Applications of Group Theory, but do so in the process of teaching vibrational and electronic spectroscopy and molecular orbital theory.
First and foremost this is a textbook. We have taken great pains not to assume very much background knowledge on the part of the reader. To make the exposition clear and meaningful, each new concept is applied or illustrated with experimental results as quickly as possible. The text includes some 200 problems with solutions in Appendix G. We consider these problems to be an integral part of the text and sometimes introduce new material in them. The student is urged to work through as many as time permits.
The present version of this book was written during a two-year period of postdoctoral research in the laboratory of Phil Aisen at the Albert Einstein College of Medicine in New York. The original text was written in collaboration with Mike Bertolucci who taught the course with me for a year at Caltech. Don Titus, Benes Trus, and Harry Gray have made invaluable contributions subsequently. Harry Gray and George Ham mond were instrumental in initiating the course and capturing my interest in it (which was similar to capturing the interest of a hungry monkey in a banana). To keep the price of this volume to a level that students can afford, my wife Sally devoted more than half a year of effort to the pro duction of line drawings. Finally, I cannot overestimate the role my stu dents played in the development of this book. Comments on ways to im prove the book or on errors are solicited from all readers and will be greatly appreciated.
I dedicate this book to the student who is willing to take it to bed with him at night, along with a pencil and occasionally a calculator, and who falls asleep with a smile on his face.
1000120014001600leDO2000
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3000
HARRIS SMOGMOBILE
involvement on the part of myself and my students during the three years the manuscript was used for part of an undergraduate spectroscopy
course at Caltech. This book was written with the goal of introducing the student to vibra
tional and electronic spectroscopy and taking her or him to a rather sophisticated (albeit qualitative) level in some areas. We have tried to write a text most suitable for use on the junior to beginning graduate levels. Taking the approach that group theory is essential to the modern practice of spectroscopy, we devote the first chapter to group theory and then make extensive use of it throughout the text. For this reason we believe that this book may be used as the primary text for a course on
Dan Harris
NOTE (1989): Daniel C. Harris may be reached at Chemistry Division, Research Department, Michelson Laboratory, China Lake, CA 93555.
• • XII Note on units and conventions
symmetry elements have been recommended by the Joint Commission for Spectroscopy of the International Astronomical Union and the Inter national Union of Pure and Applied Physics [1. Chern. Phys., 23, 1997 (1955)]. We generally adhere to these conventions, with some notable exceptions. We accidentally adopted a coordinate system for ethylene that differs from the system in common use. We apologize for this but chose not to change our coordinate system for fear of introducing errors into the text in the process of making changes. We intentionally disre garded the recommended choice of symmetry elements for XeF4 (D 4h
symmetry), but not for benzene (D fih ). We recommend that C 2 ' and (T,o
axes always be colinear and C 2 " and art axes always be colinear and that C 2 ' and a v go through as many atoms as possible and that C 2" and art go through as few atoms as possible. We further recommend the use of our convention for all point groups, instead of adopting different conventions for each point group.
With regard to the naming of symmetry operations (e.g., C2 , C2 ', and C 2 ") and other conventions in character tables, we adopted the widely used tables of Cotton (F.A. Cotton, Chemical Applications of Group Theory, John Wiley & Sons, New York, 1971); we hope that this set of tables will become standard.
Symmetry and Spectroscopy
O· Opening remarks
Spectroscopy is the study of the interaction of electromagnetic radiation
(light, radio waves, x-rays, etc.) with matter. In this book we will de,d with
a central portion of the electromagnetic spectrum (Fig. 0-1). spanning the
infrared (ir), visible (vis), and ultraviolet (uv) wa velengths.
Molecules, consisting of electrically charged nuclei and electrons. Illay
interact with the oscillating electric and magnetic fields of light and ahsorh
the energy carried by the light. The molecule doc', not interact with ;i1llight
that comes its way, but only with light that carries the right amount of
energy to promote the molecule from one discrete energy level to another.
For example, the diatomic molecule 127)79 Sr in its lowest vibrational state
(ground state) vibrates with an energy 01'2.662 )< 10- 21 J. The next lowest
vibrational energy available to the molecule is "1.961 x 10- 21 .I. Suppose
that far infrared light of energy (7.961 - 2.662) x 10 21 .1 = 5.299 x 10 21
J (= 266.8 cm 1) is shined on a sample of 127 1-() Hr. The light ean he
absorbed and a ground state molecule can be promoted to its first excited
vibrational state. When this happens we say that the molecule has Illade a
transition hetween the ground state and the first excited state. The 1wo energy levels we have been discussing and the absorptIon specl rUIll or
127 j7Q Hr in the far infrared arc shown schematically in Fig. 0-). light or
energy other than 5.299 x 10 21 .I would not be absorbed by the salllpk'
because the energy carried hy such light docs not precisely span two enngv
levels of the molecule.
Light of infrared frequencies can generally prolllote lllolecules frolll onl'
- _.
.......................--
! , ! U• >'(m) -12 -10 -8 -7
1.00xl0 l.oox10 1.00x10 (3.80 7.8Olx10
v(Hz) 20 3.00x10 18 3.00.10 16
(7.89 3.84xI0 14
3.00x10
>. -- ~
.~- acc- -- - --- -- --- - --- ---c: ~ ~~ ~ -~ -.-- Q) bc ac be ab ae be- -0a.. -
I ]I III -....
parallelepipeds in Fig. 0-3. Each of these solid ligures has six stable resting positions in the earth's gravitational field, whell it is resting on its two ah,
two ac, or two be faces. The potential energy, V, associated with each position can be calculated from the formula V 0'" mgh, when~ m is the mass of the parallelepiped, g is the acceleration of gravity, and h is the height of the center of mass above the surface on which the object rests. The potential energies of the six resting positions of paralldepiped I arc divided. into three groups, depending on which face it res ts. The potential energy is greatest when its center of mass is highest, i.e., when it rest:; on its ,,17 faces (Fig. 0-3). The parallelepiped II has only two energetically distinc1 resting positions because the lengths of sides a and b are equal. ;\\1 six (aces of the cube, III, are equal and hence only a single potential energy is obtained, regardless of which face it rests on. In this example, as the symmetry of the solids increases in the order I < II < III, the number of differenl energy levels decreases. The degeneracy (number of states which have…