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INTERNATIONAL STUDENT EDITION
McGRAWHILL Boofc Company, Inc. TORONTO NEW YORK SAN FRANCISCO
LONDON
KOGAKUSHA COMPANY,
TOKYO
LTD.
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GORDON
M.
BARROW
Professor of Chemistry Case Institute of Technology
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INTRODUCTION TO MOLECULAR SPECTROSCOPY
INTERNATIONAL STUDENT EDITION
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PREFACE
Molecular spectroscopy cannot be regarded as the private domain of the
physicist or physical chemist
who
describes himself as a spectroscopist.
of infrared
Both organic and inorganic chemists make use spectrometers as if they were standard tools
chemists base
spectroscopy.
and
ultraviolet
of the trade.
Analytical
of
many
of their analytical techniques
all
on the methods
Furthermore, almost
chemists
make
frequent use of
molecularstructure data, and
spectroscopic measurements.
much
of this
has been obtained from
One
with
can,
it is
true,
make very
considerable use of spectroscopy as a
tool for the characterization, identification,
little
is
or no understanding of the
way
in
and analysis of materials which the observed specif
trum
related to the properties of the absorbing, or emitting, molecules.
Likewise, data on molecular properties can be used even
the
which they were obtained
is
not understood.
The tendency
of
way in many
chemists, who are not molecular spectroscopists, to proceed with this lack of appreciation for
the elementary theory of molecular spectroscopy
persists, in part,
because of the lack of suitable introductory material.
This book
gap between the very cursory treatment of spectroscopy generally given in undergraduate
textbooks and the detailed treatments written for the specialist and research
is,
therefore, intended to bridge the
worker that are given in books in the various areas of molecular spectroscopy. The reader
will here be provided with the basic theory which makes understandable the relationship of
the amount and wavelength of
by a sample and the properties of the moleThis introduction should not only allow the
organic, inorganic, or analytical chemist to make surer use of spectroscopy as a
cules of the sample.
radiation absorbed or emitted
no prior knowledge of quantum mechanics is assumed. it is recognized. With this
background the inorganic chemist should. greatly influenced by I have attempted. Herzberg.
are used. considerable in a book which attempts to serve as an introduc tion to a field as
extensive as spectroscopy. It is hoped that the student will be able to under stand and
appreciate advanced and current spectroscopic material if. The problems of selection of
material and depth of treatment are. for example. or results obtained by these techniques.
however. known as quotgroup theoryquot is given Again no methods of group theory. The
level of understanding of the theory of molecular spectroscopy that this book seeks to
achieve is that from which the student can enter into the special ized reference books and
the research literature of molecular spectroscopy.vi PREFACE tool. relative of course. by G.
he pays some additional attention to the details and nomenclature surrounding a particular
spectroscopic subject. after studying this book. No special background is necessary for the
study of this book. is but should also provide that comfortable feeling of knowing quotwhat
happeningquot when spectroscopic techniques. ones particular interests and background.
The level of treatment is main tained by avoiding the many additional avenues that these
subjects open up. This introduction to molecular spectroscopy will show how molecular
spectra can be interpreted in terms of molecular behavior and how. Although the Schrodinger
wave equation is introduced and simple problems to illustrate its relation to quantities that are
important in spectroscopy are solved. molecular properties can be obtained. be able to
appreciate the very important guides to the nature of bonding in coordination compounds that
are now being based. . should be able to make increased use of the information on the
nature and energies of excited states that are being discovered and analyzed by the
spectroscopist. In fact the goal of much of the material presented could be considered to be
that of providing readier access for the student to the two important molecular spectroscopy
books. to use a specific example. is previous knowledge of this subject mechanics and group
theory is carefully confined to areas of specific value In a similar way. on spectroscopic
results. some of the theory in this introduction to spectroscopy. and some analyses are
based on the quantum The use of necessary. The organic chemist. Judgement as to the
importance of the topics to be treated and the depths to which these treatments should be
taken are. from such interpretations. frequently referred to throughout the book. to a large
extent.
Several sections are concerned in some detail with subjects that are perhaps of special
rather than general interest. Barrow . Thompson. Dr. can be omitted without destroying the
continuity of the book. many sections of the manuscript. W. Abrahamson and Dr. and to the
graduate students at Case Institute. I. These sections. Wm. I would like to express my
appreciation to my colleagues. and Mr. Julia Dasch for typing the manuscript. Carol
Haberman. I am also indebted to my wife. for the of the presenta Gordon M. vibrational. Chu.
and electronic spectroscopy that are generally of interest to chemists. Mrs. Miss Kim Vo.
marked with an asterisk. improvements that she made in the style tion and to Mrs.
Harriet.PREFACE vii to maintain a uniform introduction to the areas of rotational. for the
assistance they have given me by carefully working through E.
.
CONTENTS Preface v INTRODUCTION Classifications of Spectroscopy Wavelength. and
Energy of Radiation Chapter INTRODUCTION TO THE THEORETICAL TREATMENT OF
MOLECULAR SYSTEMS The Recognition of Quantum Restrictions The Bohr Theory The
Wave Nature of the Hydrogen Atom of Particles The Timeindependent Schrodinger Equation
The Particleinabox Problem Normalization of Orthogonality Wave Functions Symmetry
Properties of Wave Functions The Quantummechanical Average The Boltzmann Distribution
Chapter THE VIBRATIONAL ENERGIES OF A DIATOMIC MOLECULE The Vibrations of a
Single Particle Classical of The Vibrations Two Particles Connected by a for a Spring
Classical The Potentialenergy Function Bond Chemical . Frequency.
x CONTENTS The Quantummechanical monic Oscillator Solutions of the Har Vibrational
Absorption Spectra of Diatomic Molecules of Molecular The Anharmonicity Vibrations
Chapter THE ROTATIONAL ENERGIES OF LINEAR MOLECULES The Rotation Classical
of a Linear System System Quantum of a Linear The Rotation Mechanical Rotational
Energylevel Populations Rotational Spectra of Rigid Linear Molecules The Nonrigid Rotor
Chapter THE ABSORPTION AND EMISSION OF RADIATION The Timedependent
Schrodinger Equation Induced Quantum Transitions The Interaction of Electromagnetic
Radiation with a Molecular System Comparison with Experimental Quantities The Basis of
Selection Rules The Integrated Absorption Coefficient for a Transition of a Particle in a Box
Induced Emission and Induced Absorption The Integrated Absorption Vibrational Transition
Coefficient for a The Intensities of Absorption Bands due to Electronic Transitions Chapter
ROTATIONAL SPECTRA Rotational Selection Rules for Linear Molecules The Stark Spectra
Effect in Molecular Rotation Molecular Rotation Coupling Nuclearspin The Positive Wave and
Negative Character of the Functions of Linear Molecules .
CONTENTS xi SymmetricAntisymmetric Character and Statistical Weight of Homonuclear
Linear of Molecules The Rotational Energy Molecules Symmetrictop Symmetrictop Selection
Rules and Spectra of Molecules Results from Analyses of Rotation Spectra of Symmetrictop
Molecules The Energy Levels of an Asymmetrictop Molecule Selection Rules. Chapter
MOLECULAR SYMMETRY AND GROUP THEORY Symmetry Elements. Symmetry and
Point Groups Symmetry Operations on Molecular Motions Species and Character Tables of
a Symmetry The Nature Group . Spectra. for and Structure Results Asymmetrictop
Molecules Chapter THE VIBRATIONS OF POLYATOMIC MOLECULES The Number The
Nature Coordinates of Independent Vibrations of a Polyatomic Molecule of Normal Vibrations
and Normal Quantummechanical Treatment of the Vibrations of Polyatomic Molecules The
Symmetry Coordinates Properties of Normal Chapter ROTATIONVIBRATION SPECTRA
and Transitions for the Rigid RotorHarmonic Oscillator Model Selection Rules The Relative
Intensities of the a RotationVibration Absorption Components of Band Coupling of Rotation
and Vibration Parallel Bands of Linear Molecules Perpendicular Bands of Linear Molecules
Parallel Bands of Symmetrictop Molecules Perpendicular Bands of Symmetrictop Molecules
Operations.
xii CONTENTS The Symmetry Operations as a Group Representations of a Group Reducible
and Irreducible Representations The Irreducible Representations as Orthogonal Vectors
Characters of Representations Classes Analysis of a Reducible Representation The
Characters Reducible Representation of Molecular Motion for the Number of Normal Modes
Symmetry Types The The Symmetry of of Various Infrared Active Fundamentals Group
Vibrations Chapter CALCULATION OF VIBRATIONAL FREQUENCIES AND NORMAL
COORDINATES OF POLYATOMIC MOLECULES The Kineticenergy Expressions
Polyatomic Molecule for a The Potentialenergy Expression Polyatomic Molecule for a Use of
Lagranges Equation in Molecular of a Linear vibration Problems The Stretching Vibrations
Triatomic Molecule Use of Symmetry in Vibrational Problems Solution of Vibrational
Problems in Internal Coordinates by the Method of Wilson Formation of the Method G Matrix
by a Vectorial in Wilsons Use of Symmetry Coordinates Method Chapter THE ELECTRONIC
SPECTRA OF DIATOMIC MOLECULES The Vibrational Structure Bands of Electronic
Rotational Structure of Electronic Bands Electronic States of Atoms Electron Orbitals in
Diatomic Molecules Electronic States of Diatomic Molecules .
CONTENTS xiii Potentialenergy Curves for Electronic States of Diatomic Molecules Chapter
ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES Electronic States of Localized
Groups Electronic Transitions and Absorption Bands Conjugated Systems by the
Freeelectron Model Electronic States and Transitions in Aromatic Systems Electron Orbitals
of Coordination Compounds Electronic States of Coordination Compounds Nonradiative
Processes Fluorescence Index .
.
obtained. furthermore. listed at the end of this section. of the radiation and the amount of
radiation emitted or absorbed by the sample. It will and electronic arrangement of a molecule
can be become apparent that spectroscopy offers one of the most powerful tools for a great
variety of molecularstructure studies. There will. be seen. Only brief mention will be made of
the experimental techniques used to obtain the spectra that are discussed. Classifications of
Spectroscopy to When the theory of molecular spectra is treated. spectra can be obtained
and used without a detailed understanding of the behavior of the components of the
instrument. remarkably detailed and exact measurements of the flexibility. as size. it is
convenient to is classify spectra according to the type of molecular energy that being . One
can often understand the nature of the molecular changes that are responsible for the
emission or absorption of the radiation. or wavelengths. primarily from studies of infrared and
ultraviolet spectra. cases. With such equipment. furthermore. A number of
books.INTRODUCTION Molecular spectroscopy is the study of the absorption or emission of
electromagnetic radiation by molecules. be no attempt to deal in detail with the tremendous
body of very useful empirical data that has been accumulated. deal in part or wholly with
these techniques. practice. shape. will In this way. In many commercial spectrometers
operating in the various spectral regions are available. References some treatments and
collections of these data are given below. The experimental data that such studies provide
are the frequencies. In such the experimental spectroscopic data can be used to determine
quantitative values for various molecular properties.
energy of a free molecule as made up of separate rotational. there are two closely related
types that play a comparable role in chemistry. that spectra One can used. vibrational. In
nuclear field is to magnetic resonance spectroscopy the effect of the magnetic orient certain
nuclei in certain directions with respect to the direction of . to treat the to different electron
arrangements It is allowable. photomultiplier detector It electronic spectra of should also be
mentioned that vibrational spectra can be obtained by means Raman spectroscopy and that
this technique uses a visible or ultraviolet spectrometer. from the action of a magnetic field
on the molecules of the sample. These are nuclear magnetic resonance nmr and electron
spin esr resonance spectroscopy. and components as has been implied by this classification.
levels that are studied in these spectroscopic in contrast to those normally studied in
rotational. wave guide. might be given as Microwave spectrometer klystron source. It also
divide spectroscopy according to the instrumentation happens that the categories obtained in
this way are similar to The instrumentation classification those based on molecular energies.
electronic see. Thus an absorption of energy due primarily to a change in the vibrational
energy may show the effects of accompanying rotational energy changes. and crystal
molecular rotation spectra thermocouple detector molecular vibration spectra detector Visible
Infrared spectrometer hot ceramic source. however. We will may result from transitions in
which more than one type of molecular energy changes. rocksalt prism or grating.
INTRODUCTION TO MOLECULAR SPECTROSCOPY altered in the emission or absorption
process. In this way the principal headings for the material that is to be presented can be
arrived at. These are Rotational spectra molecule due to changes the rotational energy of the
Vibrational spectra due to changes in the vibrational energy of the in molecule Electronic
spectra due to changes in the energy of the molecule due generally to a good approximation.
glass or quartz prism or grating. vibrational. categories The energy result. and electronic
spectroscopy. and ultraviolet spectrometer tungsten lamp or hydrogen dis charge tube
source. In addition to the types of molecular spectroscopy listed above.
quot W. and radiation of a the level separations. vibrational. Frequency. and
quotApplicaMagnetic Resonance Spectroscopy in Organic ChemisIn try. A. Figure indicates
the FIG. which are at the same level as the treatments of rotational. can be treated in terms
of electric and magnetic the direction of propagation. New York. and Energy of Radiation
Before the emission or absorption of radiation by molecular systems can be treated. suitable
for the chemist. both by J. Ordinary radiation traveling in the z direction. D. to nuclear
magnetic resonance Of particular interest electronic spin resonance spectroscopy. different
orientations correspond to different energies. York. number of recent books have appeared
which provide excellent introductions. York. and quotAn Introduction to SpinSpin Splitting
Nuclear Magnetic Resonance Spectra. . by M. Roberts.quot Pergamon Press.quot
McGrawHill Book Company. New tions of Nuclear sion of nuclear magnetic resonance and
electronic spin resonance will be included. view of these treatments. Inc. A and are Inc. for
example.. The wave nature oscillating electric description of electromagnetic radiation
associates fields and magnetic with the radiation. X E E cosirvtf .INTRODUCTION These
field.. and electronic spectroscopy given here. suitable frequency can then be used to study
the energySimilarly. .. some of the terms used to describe electromagnetic radiation must be
summarized. L. can be obtained. New in HighResolution Inc. Jackman. . fields perpendicular
to each other and to is Polarized radiation. is oriented one way or the and again the resulting
energylevel separathese two orientations is studied by radiation of suitable frequency. which
more con venient for discussion here. quotNuclear Magnetic Resonance. Wavelength. no
discus Benjamin. in electronic spin resonance spectroscopy the spin of an electron of the
sample molecule other with respect to the tion of field. Planepolarized electromagnetic
radiation.
while at a given value of z the is E with wave respect to time given. the direction of
propagation. W the frequencies of the radiation that are absorbed or emitted in a spectral
study are more directly related to it is the molecular energy changes that cause the
absorption or emission. t. Fig. oscillation of it should be recognized. for completeness. for a
given value of t for example. as will be seen later. wave and corpuscular His equation Ae hv .
electromagnetic radiation con waves of varying electric and magnetic field strength traveling
with the velocity of light and having a given wavelength and frequency. often customary to
deal as well with the wavelength. with a velocity c. The corpuscular description of
electromagnetic radiation views this radiation as a stream of energy packets. As can be
readily verified by considering a position with some fixed value of z and asking how many
cycles pass this position in the time c seconds. INTRODUCTION TO MOLECULAR
SPECTROSCOPY and magnetic fields that are associated with polarized electromagnetic
radiation which has only the xz plane component of the electric electric field. component in
the plane perpendicular to that The oscillation of the electric field. shows that this plane
polarized radia tion has the magnetic field of the electric field. travels out in the z direction.
Basic to an understanding of spectra brings together the is the relation of Planck that
theories of radiation. gives the behavior shown in Fig. field. the relation between the
wavelength and the frequency of oscillation of the wave is quotI Although. With sists of this
picture of radiation. The value of the electric field along the z axis at a given time and at a
given point on the z axis as a function of time can be expressed by the formula E E cos wv t
J This formula. In spectroscopy almost all attention is centered on the electric but. of the
radiation. and also of the associated mag netic field. traveling with the velocity of light. called
photons in the visible region.
. or by the value of Ae hv. for example. It is much on its own. factor c. While it may seem
unfortunate that a consistent set of units. cmquot . such as given by a value of or v. X of
ergsec. differs from by the It is designated as v and is defined as The units of v are invariably
those of It cm . or quanta. . tion. are shown in Table . it is a historical fact that each area of
spectroscopy developed pretty venient set of units. .l X . called reciprocal centimeters or v is
essentially wave numbers. by the constant factor of the velocity of The ranges of
electromagnetic radiation. atoms. such as the cgs wavelength unit of centimeters. with
regard to the entries in Table . the frequency of the radiation. a constant known as Plancks
constant. characterize the particular radiation emitted or absorbed. with the wave nature
concept v.. Conversion factors for all the . TABLE The Spectral Regions of Electromagnetic
Radiation Microwave X cm Infrared Visible Ultraviolet . megacycles/sec V .INTRODUCTION
where h. where the quotparticlesquot may be molecules. quite commonly called a frequency
unit. . ties together the corpuscular quantity Ac. interconversions that are likely to be
encountered are It should shown in Table be pointed out. In the various types of
spectroscopy one makes use of Ae. or quantum. independits ent of developments in other
areas. should be kept in mind that v a measure of frequency. This unit. to another. such as
given by v. with the value . tional unit or X to One addi which v is proportional to v has
convenient numerical values and is often used. XA V X quot X quot . being different from
light.. and each acquired own con frequently necessary to convert from one description of
radiaX. the energy of a radiation energy packet. is not used throughout. X . that a term such
as cal/mole is intended to mean calories per Avogadros number of particles. expressed in
the units usu ally used for each region.
R. W. . London.J.quot Sons. A. . ... X .J. . N. . . Lawson. F. C. quotInfrared Absorption
Publishing Corporation. S. R. . G. Englewood Applications . of Inorganic Substances. . X of
quanta in the four regions of electromagnetic radiation shown in . and E. . . .quot
PrenticeHall. Stern quotAn Introduction to Electronic Absorption Spectroscopy in Organic
Chemistry. Inc. L. quotThe Infrared Spectra York. W. . . Table of Compare RT PRINCIPAL
REFERENCES Experimental Methods . R. . K. Inc. Gillam. John Wiley .quot Edward Arnold
amp Co. X X X .. Englewood Cliffs.. Verify the conversion factors along a row of Table
Calculate typical energies of an Avogadros number . quot quot quot . quotExperimental
Spectroscopy. of Complex Molecules. X lOquot . these values with the classical thermal
kinetic energy cal/mole per degree of freedom at room temperature and with chemical bond
energies of to kcal/mole. quot cal/mole erg/molecule cv X . . Inc. cm cal/mole. Sawyer. Lord.
Bellamy. Cliffs. . Inc. Exercise Exercise Exercise . V...quot John Wiley amp New .
Trambarulo quotMicrowave Spectroscopy... X quot . J. . and J. R.quot PrenticeHall.quot
Reinhold New York. Sons. amp New York. N. Smith. E. X .. A. Harrison. and R. Loofbourow
quotPractical Spectroscopy. .INTRODUCTION TO MOLECULAR SPECTROSCOPY TABLE
Energy Conversion Factors erg/molecule . Gordy.. Complete Table by filling in all the blank
spaces. . E.
radiation from such an oscillating system consists of the and h is . been closely tied to the
study of their spectra. driven to assume. . An appreciation of the restrictions that are placed
on the energies of atomic and molecular systems is basic to an understanding of the spectra
of these systems. In this chapter. Furthermore. since about . The Recognition of Quantum
Restrictions The earliest recognition of discrete energy jumps or quantum restric tions stems
from Plancks studies of the radiation emitted by hot bodies. While standing is many spectra
can be understood to some extent without any further appreciation of the theories of
molecular systems. Plancks constant. INTRODUCTION TO THE THEORETICAL
TREATMENT OF MOLECULAR SYSTEMS The development of theories of atomic and
molecular systems has. that the oscillating atoms of a hot body cannot have any energy of
oscillation but can have only the energies that are integral multiples of hv. as was later
stated. a better under usually obtained if some of the general features of these theories are
known. The principal aspect of these theories that must be introduced here is their
statements about the energy that an atom or molecule can have. where v is a frequency of
He was oscillation a proportionality constant. the general nature of theoretical treatments of
atomic and molecular systems will be introduced to the extent needed for the treatments of
molecular systems given in later chapters.
as a result of the then recent work of Thomson and of Rutherford. each allowed orbit having
a certain energy. by Niels Bohr in . or quantum. The derivation. by the constant h were basic
features of Plancks blackbody radiation deriva They seemed. in particular the hydrogen
atom. since. in brief. that an atom. and the The next major step of spectroscopic interest was
the application of these ideas of quantum restrictions to atomic systems. in some manner out
side the nucleus. in classical to be awkward and troublesome features treatments. in
particular to the hydrogen atom. In the next few years after the ideas of Planck were applied
and extended. they could not be justified. the Bohr theory provides a clear and concrete
illustration of one of the most important quantum rules and is. when an oscillator jumps from
one of the allowed energy The unit. The Bohr Theory of the Hydrogen Atom Although
detailed theories of atomic structure are not pertinent to the discussion here.
INTRODUCTION TO MOLECULAR SPECTROSCOPY energies emitted levels to a lower
one. By it was recognized. however. worthy of study. and his interpretation of the continuous
spectrum provided a hot body was to be followed by analyses of spectra of great detail by
and complexity. a preview of the developments in molecular theory and spectroscopy that
were to take place in the next half century. as heat capacity of in Einsteins theories of the
photoelectric effect solids. or resided. . consisted of a small heavy nucleus carrying the
positive charge and that the electron moved. of energy given out in such is. however.
Plancks blackbody theory was. it way that its angular moved in a circular orbit about
momentum was an integral This statement and the ordinary rules of dynamics and
electrostatics lead to allowed orbits for the electron. Bohrs interpretation of the behavior of
the electron was based on the rather arbitrary assumption that the nucleus in such a multiple
of h/w. a jump from one energy state to the next lowest one Ae therefore. is as follows the
angular momentum . His reluctantly proposed ideas of quantum restrictions were accepted
and expanded into more elaborate theories of molecular behavior. hv idea of discrete
energies and the prominent role played The tion. therefore.
. Use can now be made and the quantum restric tion of Eq. If the potential energy at infinite
separation of the electron and proton is taken as zero. . . the potential energy at separation r
is given by Coulombs law as PE . n jjis n . . unit of angstroms. . defined so that A cm. . the
electron r is allowed to travel as n r me Substitution of numerical values for the constants
gives the radii of the Bohr orbits as r . according to Bohr. angstrom n . by means of the
quantum Eq. . has Of more interest in spectroscopy are the energies of the allowed
orbits.INTRODUCTION TO THE THEORETICAL TREATMENT OF MOLECULAR SYSTEMS
postulate requires that mvr where orbit. to obtain fcivsy w gt lt gt . restriction of Elimination of
igt . m the mass of the electron. the result t KE PE g mu of the relation of Eq.r Addition of the
kinetic energy gives. v its velocity. for the total energy of the electron. The convenient been
introduced. gives the radii of the orbits in which. .n. . . and r the radius of its The requirement
that the electron should have the coulombic force of attraction to the nucleus balanced by the
centrifugal force gives t r r or mv where e is the charge of the electron. .
The will principle of quantized angular momentum used by Bohr is. INTRODUCTION TO
MOLECULAR SPECTROSCOPY of e where the fact that each value of n implies a value
nized has been recog by writing . for example. which is not quite at the nucleus. angular It is
interesting to see that the condition that the momentum Further be quantized in units of h/K
leads to a set of allowed energies. say from that with n n. expression. the orbit radii and
energies for an electron of the He ion. cm This corresponds almost exactly to the empirical
expression known Rydberg atom spectrum. the possible energies that the electron of the
hydrogen atom can assume. as be seen. down to that with n ni. With the Planck relation Ae
hv. to take into made to allow for elliptical account that the electron and nucleus should be
treated as rotating about the center of gravity. and there the possibility of showing the
application of a directed field that this orbit can be inclined in five . velocity. the total angular
momenby tum of the electron is h/r. Elaborations of this result have been orbits. is imposed
on the atom. according to the Bohr theory. must is also be quantized in units of n orbit. as the
which summarizes the observed hydrogen Exercise . and to include the relativistic
dependence of electron mass on These details need not. the Bohr theory predicts that an
amount of energy Ae e. according to the Bohr theory. W has the prediction In terms of the
more often used units of cm one wme /l quotcehrltf gt cm which. gives . in that direction if a
direction field. more. however. the Bohr theory leads to the electron of the hydrogen l the
prediction of emission of radiation with frequencies At T v ir fn lt/ l me/l Wlth ni gt . This result
gives. A more complete statement of the principle applied to the simple Bohr hydrogen atom
would recognize that not only must the total angular momentum momentum Thus for the be
quantized but that. on substitution of numerical values. Derive. if atom is now assumed to
jump from one orbit to another. e must be emitted. as by an applied electric or magnetic the
component of angular h/w. be treated here. a very generally applicable one in atomic and
molecular systems.
particle with He further suggested that the wavelength associated with a mass m and
velocity v. would be given by X mv if it is This interesting relation can be used in a number of
situations. The Wave Nature of Particles In .e. and so forth. One says that the n is orbit.
electron orbit mvr nh/w. as well as radiation quanta. . or state of the electron. while ing the
hydrogenatom spectrum.. Even more shown It indicative of the validity of the de Broglie
relation were the experiments of Davisson and Germer in to give diffraction effects
corresponding to a which an electron beam was wave with wavesuccessful in explain length
given by the de Broglie relation. reasoning from the generally symmetric nature of the
physical world. Instead. has a multiplicity of or fivefold degenerate.INTRODUCTION TO THE
THEORETICAL TREATMENT OF MOLECULAR SYSTEMS different directions
corresponding to angular momentum components The of Vir.e. i. splitting of the energy of the
original n that the original state is state by the application of a field reveals to be thought of
as having five states. proposed that electrons. should have wave properties associated with
with them. and h/w. lh/w. postulated that the For wave associated with an electron in a Bohr
orbit should form a standing wave around the nucleus.. some further general theoretical
developments which led finally to the more powerful and less ad hoc formulation of quantum
restrictions than that of Bohr will be outlined.clear that the Bohr condition. protons. became .
did not recognize in a sufficiently basic way this wave nature of the electron. The importance
of the wave nature led ultimately to a formulation of a general approach to the mechanics of
atomicsized systems. . i. Louis de Broglie. will The details of atomic theory and spectra not
be developed further here. to the same requirement. value to way which atomicscale
problems are handled and by the wave mechanics . the condition that irr n where n is an
integer and irr is the circumference of the is imposed. lh/w. of Schrodinger. example.
momentum mv. It seems best here to present and in in illustrate this approach by means of It
is of great an equation given appreciate the described by Erwin Schrodinger. interestingly. as
imposed by Bohr. all with the same energy. This stipulation together with de Broglies
wavelength relation leads.
equation. the value at a given point of the square of the trigonometric or algebraic function
that solves the Schrodinger equation gives the probability of the particle being found at that
point. it is convenient first to demonstrate the simpler timeindependent features of the
complete equation. of a derivation showing its validity. In a study of spectroscopy there
molecular systems. given only in terms of a probability function. Born. but rather because
wherever properly applied. be extended to more complicated systems.and molecularscale
problems. It will. however. will be a function of x. and the position adopted by the particle.
and the information was in agreement with experiment. The Schrodinger equation might.
yields directly values for the allowed energies of the particle under study. One can then
better appreciate the nature of more general quantummechanical methods and solutions. to
results in agreement with observation. the solution function. be used to solve again the
hydrogenatom problem and predict the position and the energy of the electron exposed to
the coulombic potential of the nucleus. be very helpful to have performed a simple illustrative
calculation. of such classical formulations as Newtons / ma it expression. Bohrs method
could not. calculations of the properties of however. as will be seen. or quantummechanical.
to consider a simple problem in which the particle can move only along one dimension. The
Schrodinger equation accomplishes this extension. is not really derivable and should be
looked applicable to atomic. INTRODUCTION TO MOLECULAR SPECTROSCOPY . The
Bohr theory of hydrogenlike atoms obtained just such information. The Timeindependent
Schrodinger Equation Although the equation originally presented by Schrodinger allowed for
the calculation of timedependent behavior. is little need to perform wave mechanical. For
emphasis one sometimes writes /x. for example. represented by //. The Schrodinger particle
is. The position of the however. is If p is a complex function. Use of the Schrodinger equation
implies that is we are interested in learning about the energy of a particle. The probability of
the particle being at a given value of x along the x axis is at the given value of x. for
simplicity. According to M. which subject to some potential energy. The Schrodinger
equation. not because leads. where is the complex con jugate of . . or eigenfunction. say the
x coordinate. as then equal to the value of is sometimes ltj/ the case. the probability given by
V. The Schrodinger equation upon as the counterpart. is used and accepted. If we choose.
like/ ma.
in fact. Neither of these would be physically reasonable.INTRODUCTION TO THE
THEORETICAL TREATMENT OF MOLECULAR SYSTEMS The Schrodinger equation for a
particle of mass m constrained to one dimension and subject to some potential function Ux is
fe ir m da. Solution of the problem would give the allowed energies. The equation and its
application are more understandable when require the value of for the electron to be used m
e f/ / applied to a specific problem. as mentioned. of while in the second case one would
have two different probabilities at the same position. the second with the potential energy.
the function . Before the equation is illustrated by a simple. Such a function will generally
exist only for certain values of and these values are the allowed energies of the system that
are being sought. The function would give the probability of the electron being at various
positions. three dimensional. there with the total energy. One understands these restrictions
from the fact that the equation involves the second derivative of and that this would go to
infinity at a discontinuity in the slope. lem is. The hydrogenatom problem would and the
expression The hydrogenatom prob/r to be put in as the potential function. the Bohr orbits. go
to infinity or be double valued. for example. example. to be an acceptable solution. and the
sum. and the probability that is found is related to. Solution of a particular problem requires
values of m and an expresfunction ip sion for the potential function to be substituted. first
term can. but important. the probability of the particle being at various positions. dy Ux e The
fore. but not identical with. it is necessary to state some of the limitations that are imposed on
the function f. that solves the resulting differential equation must then be found. which turn
out to be identical with those obtained by Bohr. The e. Briefly. In the first case one would
deduce from the infinite value an infinite probability of the particle being at a given
position.quot It must not. . Further restrictions are that the function must be continuous and
that discontinuities in the slope can occur only at points where the potential energy goes to
infinity. The square of the solution function gives. must be quotwell behaved. be associated
with the kinetic energy of the It is particle. perhaps better at first merely to use the equation to
calculate desired quantities much as one does with / ma. however.
the function yj/ To be in the region between i A and x sin where n factor . is subject to a
potential which for some purposes can be so represented.. Functions which solve the
differential equation and also satisfy these boundary conditions can be seen by inspection to
be well behaved. In the region lt x lt a the potentialenergy function in this region reduces to is
Ux . An electron in a piece of wire. to give T . or t. being in such a region. now necessary to
The function must be It is f/ zero outside the potential well since there the potential is
infinitely high. . and the Schrodinger equation h r dV dx i m find wellbehaved solutions for this
equation. between x and x a. as will be shown in Chap. . . A is some constant The n. and
there is no probability of the particle and prevent a discontinuity and a must be such that it
equals zero at x and at x a. expression rmx/a has. is that of determining the allowed energies
and the position probability function of a particle that can in move This problem implies a
potentialenergy function that has some value. Of more spectroscopic interest. .
INTRODUCTION TO MOLECULAR SPECTROSCOPY The ParticleinaBox Problem A very
simple problem. . in /. Lett side h r OT I nV jA . as can be checked. which in fact has some
counterparts of interest in real molecular problems. is the fact that the doublebonding. and is
is only one dimension and confined to a region of length a. . electrons of a conjugated
system of double bonds in a molecule behave approximately as though the potential which
they experience is such a simple squarewell function. .U . for example. which can
conveniently be taken as zero. . been arranged so that the function goes to zero at x and at
the a for any integral value of That the function satisfies Schrodinger equation can be tested
by substitution in Eq. infinitely high outside this region.. nirx V o sin / o n h ma Right side c I A
sin J .
e A / ma t r/ma ltft J /ma .. A and molecular problems. This leads to a zero probability of a
particle being anywhere in the box and is therefore unacceptable. if mA J ma s n . FIG. are
seen to result much more naturally in Schrodinger approach. No really different solutions can
be found. wove functions. The quantum phenomena. but gives a wave function that is
everywhere zero. UTTX and the expression A sin gives solutions of Eq.box problem. and
probability functions for the particle. Energies. The allowed energies t. . . are equal. which
are represented in Fig. are seen to be quantized as a result of the quite natural intro similar
situation occurs generally in atomic duction of the integers in the solutions of the Schrodinger
equation. provides a solution to Eq. The value n in Eq.in a. which were so arbitrarily
introduced in the Bohr theory.s INTRODUCTION TO THE THEORETICAL TREATMENT OF
MOLECULAR SYSTEMS The left and right sides of Eq. and no energies other than these will
result.
. A sin o and the probability functions are A sin level in Fig. In these cases. i. As an
illustration of this Schrodinger equation problem the one dimensional squarewell. The
allowed . onedimensional region of uniform potential bounded by regions of infinitely high
potential. energies of the ir electrons.quot solution can be applied to the question of the
energies of the doublebonding or r electrons of a These electrons are apparently delocalized
and are relatively free to move throughout the length of the molecule.. are then those given
derivation. sufficient to it will be have the solution functions and allowed energies stated with
out derivation. The success of this simplification of the factors affecting t electrons in
conjugated systems is judged by cule have all their electron can be oriented to give a spin J.
or quotparticleinabox. ignor ing electronelectron repulsions. called a quantum number. .
INTRODUCTION TO MOLECULAR SPECTROSCOPY The ii solution functions. Since the
spin of an quantum number of or two electrons can be accommodated in a state represented
by a single value of n. shown opposite the corresponding energy This simple problem
illustrates many of the characteristics of more difficult quantummechanical calculations on
atomic and molecular systems. remains to recall the Pauli exclusion principle which requires
that no two electrons of a mole quantum numbers the same. by the previous approxima n
/ma /i To apply this squarewell it tion to the w electrons of a conjugated system. One finds
that solutions exist for the Schrodinger equation e only for certain values of and that the
solution functions and the cor responding energies are characterized by an integer. One can
therefore approximate such a system by representing the molecule as a conjugated system.
or eigenfunctions. the comparison of the energy calculated for the promotion of one of the
highest energy electron pair to the next higher energy state with the . The occupancy of the
energy levels by the six k electrons of hexatriene is represented in Fig.e. features of
Schrodinger equation solutions are appreciated. A number of situations will arise in later
chapters where soluthe general tion of the Schrodinger equation will present plexities that
some mathematical comif cannot be dealt with here.
The squarewell model for the w electrons of hexatriene. to arrange it so this function / if/ dx
is For the particleinabox problem. since region quot zero everywhere outside the to a. In
terms of a onedimensional problem this implies that particle. when a wave function f is
obtained for a shows that there is a value of unity for the total probability of the particle being
found somewhere in space. . about . The agreement. occurs The dashed arrow shows the
transition that radiation is when absorbed. Normalization of It is Wave Functions frequently
convenient. . Calculate the wavelength of the radiation that will be Exercise
.INTRODUCTION TO THE THEORETICAL TREATMENT OF MOIECUIAR SYSTEMS
energy of the radiation quanta absorbed in an electronic spectral study. the limits can be
reduced to give / r dx A wave function that normalized. is remarkably good. gives a total
probability of unity is said to be FIG. absorbed in the lowest energy irelectron transition of
hexatriene according The length of the molecule can be taken as to the square. Compare the
observed value of . as Exercise illustrates.well model. A. with A.
a property of considerable importance in spectroscopic problems. / if m is integer. functions
that solve the is Schrodinger equation for a particular problem. sin my dy t/ to give o /o rinfa
o. normalization consists of evaluating A such that o /. or A quotiquot The normalized wave
functions for a particle in a box are. / quot dT .e. therefore. Asmdx a The an integration can
be performed or read from tables which give. i. Z For normalization. lt Orthogonality A
general property of wave functions. are such that / Mm dr . Two wave functions f/i and / m . if
the wave functions are complex. normalized. where I and m imply different quantum
numbers.. wave functions for other molecular problems can. and usually tion are. In the
general threedimensional that case.quot / idr or. INTRODUCTION TO MOLECULAR
SPECTROSCOPY For the particleinabox wave functions. the normalization condi on a wave
function / is . that of orthogonality. if dr represents the differential element of volume. Va Sm
In a similar way. therefore.
. Substitution of the variable y left side Eq. this it should be kept in mind. corresponding to a
single energy of the system. i. lead to considerable simplification in a number of
spectroscopic problems. In this way one obtains zero for the integral and verifies the
orthogonality statement of Eq.quot .. is This property stated without proof. The integral in for
example.dT / if I and m describe states with different energies. . m are integers. real. This is
wave functions. wave functions cor responding to nondegenerate states of a system /. tional
features must be considered when there are several states.INTRODUCTION TO THE
THEORETICAL TREATMENT OF MOLECULAR SYSTEMS or. however. It can. more
generally.e. an example of the general orthogonality result that is applicaSome addible to the
wave functions for states with different energies.. I and form can be shown to be zero by. It is
necessary to show that sin o JC a sin a ax rx/a converts the for of I m. A summary of this gt t
statement that if and the previous section can be given by the and m are normalized.. be
readily illustrated by the particleinabox eigenfunctions. e imy ltr im and performing the
integration on the exponential form of the integral. . I for I a m /o n it will be seen. These
properties. to / quot sin ly sin my dy where. replacing the sine terms by sin ly jp equot
equotquot and sin my .
. INTRODUCTION TO MOLECULAR SPECTROSCOPY . . it is more satisfactory to use the
center of symmetry. n Such symmetry properties are characteristic of wave functions that
arise from a problem with a symmetric potential function. If this is cosines as done in the
squarewell case. antisymmetric i Even function. T. symmetric sin M Odd function. Symmetry
Properties of Wave Functions is A general property of is wave functions that their important
in many form when. The potential of the particleinabox problem. for example. the
potentialertergy function is a symmetric one. the solution functions are sines and shown in
Fig. as is often the case. particular wave function. such as the midpoint of the square well.
When this is the functions shown in Fig. antisymmetric I Even function. The wave function
shown are not normalized. It is apparent from the plots of the wave functions shown in
spectroscopic problems Fig. is symmetric about the point x a/ in Fig. one sees
mathematically that of the replaced by y The symmetry properties of the partideinabox wove
functions. wave functions by investigating what happens in a. It is then possible to investigate
the sym metry properties when y is done for FIG. Generally. as the origin of the coordinate
system. Wave function a Symmetry nature of Odd function. symmetric a/ u y . that those with
even values of n are functions that are antisym metric about the midpoint of the well while
those with odd values of are symmetric.
or symn the function is an odd. function is of an even. i. we have occasion to want to know
the average value of some other properties. The kinetic energy operator.. for example. want
to know the average position of a which we have obtained an eigenfunction of finding a f/
from solution of the appropriate Schrodinger equation. The method be introduced. G g Only a
few functions and these are listed in Table . which will also be encountered later. in some
cases. for kinetic energy can be Gfgt used to illustrate the basic operator relation gf. particle
for We will. their operators will be encountered. and The operator is. or The
Quantummechanical Average of the Although the values quantummechanical will results of
allowed energies of a system are the prime importance for spectroscopy. has associated with
is it momentum. l. by Tx gives r m Xt sin wVa ir w .INTRODUCTION TO THE THEORETICAL
TREATMENT OF MOLECULAR SYSTEMS For n odd Hv Hy For n even My Mv Again one
sees that for odd values of n metric. while for even values antisymmetric. to In quantum
mechanics every variable. then operation on the wave function of the system by one of these
variables G will give. quantummechanical average. allows the concept of operators. denoted
by T for onedimensional motion along the x axis. furthermore. operation on this function
Since A sin nrrx/a. function. Tx h d ampr te The calculation of the average value of the kinetic
energy of a particle in a box can be used to illustrate this operator. the value for the variable
g times .e. If what is called an denoted by g and the operator for this function by G. kinetic
energy. such as position. function. the. and operator.
fridZ vidy h a Angular momentum M x M M v quot VriVaz xiV h / a y I a or ay/ h / sin x a a cot
S cos dgt a I iri V ae . since the potential energy was taken as zero.x. ltfgt Operator x v z x y
z Linear momentum h jr a dx hd Pv P.y./ h / a z h / a J or ax at dgt I cos a cot fl sin dgt a I irt
ae alttgtJ h / a T X A y. are the allowed This result. agrees with that obtained previously.z .
TABLE Some Quantummechanical Operators In terms of cartesian coordinates x.y. such as
re /i /mo in the example. t Lsin e ae h v m i a ae a l sin e aa a J Kinetic energy a ivlvlvl
Potential energy Vx.. Since TYis the kineticenergy operator. momentum operator acting on
the particleinabox Thus the wave function is . y. out. z and polar coordinates Variable
Position . times the wave function. One further feature of the nature of these operators must
be pointed Sometimes an operator acts on and does not lead to a number. titx T h Vi /rar
nwx C S VH T IT r. P / / W Sm . the values of kinetic energies of the particle. a
INTRODUCTION TO MOLECULAR SPECTROSCOPY Here operation on by Tx gives a
quantity nW/Sma n /i /ma times .V x / irt alttgt/ M M I Ml Ml j ri a / quot.z KwW V.
will be spectroscopic problems. n state using the operator Obtain the average position of a
particle in a box in the method of Eq. how many will behave according to the n wave function.
. the average value of a variable is. the average value obtained according to the general rule
/ quot fGdT yidr f For normalized functions the denominator is of course equal to unity. or .
For normalized. therefore. values at various positions. real functions. nh i. nwx a quot . .
Similar applications of Eq. Sm . quotlKtydr this relation as One obtains the average
momentum with P quot fquot J jo ia W fa / .e.INTRODUCTION TO THE THEORETICAL
TREATMENT OF MOLECULAR SYSTEMS is In such cases one is to understand that the
variable being investigated not a constant for one of the allowed states. occupy the n level..
how . Exercise . variable is Rather it has various of the For such functions. made in a number
of The Boltzmann Distribution Spectra can be interpreted in terms of the energies of the
allowed states if one can decide which of the allowed states will in fact be occupied by the
molecules of the sample. i. sin ax . mcx h rmx a / nir IT S V cos j U T CS nvx dX . In the
particleinabox example the question that would have to be asked is If some large number
frequently it is convenient to consider an Avogadros number X of particles are placed in such
a potential well. Jo nh ia / a mr titxIquot a Jo The result of classical picture of the an average
momentum of zero reflects the fact that the motion of the particle would ascribe both positive
and negative values to the momentum since the particle would move in either direction. .
the tacit assumption that the lowest available levels would be occupied was made. If it is
more convenient to measure the energies of the states in calories per mole.IRT t where R. ..
. Sometimes.mr where k is Boltzmanns constant.giNoe ltT Note that the number in each
state at energy e. occupying a energy n will be related to the number N. by the relation ..
There are. the appropriate expression is e E E. and so forth In Sec. there are several states
The population of the energy level is then cor Thus is if gt is the multiplicity of the ith energy .
/ T . INTRODUCTION TO MOLECULAR SPECTROSCOPY many the n level. . the number
of states with energy and as e is usually the case. compared to the number with energy Ni. i.
the number of molecules with is energy . however. cal/mole deg. f e. respondingly greater. is
still N e quot . as was mentioned in Sec.e. The appropriate form of the Boltzmann distribution
is then t Ni JVoelt r with the same energy. equal to Nk. occupying some state with energy ts
which is lower than a. X erg/deg molecule. the lowest energy level a single state. state with
The answer to such questions is given by the Boltzmann distribution which states that the
number of particles N. Most often one is interested in the number of molecules in a state with
an energy t compared to the number in the lowest energy state to.. level. is . g states at
energy a.
will deal with the transition process that takes a molecule from one energy level to another.
to absorption of radiation in the infrared spectral region. . seems desirable. in spectroscopic
studies. It passing will be dealt with in greater detail. but chemically more interesting. We will
see that changes in the vibrational energy of the simple systems dealt with here lead. Chap.
like that of Fig. well be already familiar with the complicated absorption pattern. and Chap.
which large molecules show The rather quotmechanicalquot treatment of simple vibrating
systems in this chapter constitutes an introduction to these more complex. will recognize that
gasphase molecules engage in simultaneous rotation and vibration and that it must be
shown to what extent these motions can be treated separately. The energy levels deduced in
these calculations will then be levels indicated compared with the separation between these
energy by spectroscopic transitions. In later chapters a number of points that are here
mentioned in Thus. First the classical behavior of a single particle and a pair of particles will
be treated according to classical mechanics. systems.THE VIBRATIONAL ENERGIES OF A
DIATOMIC MOLECULE The general introduction to the theory of atomic and molecular
systems of the preceding chapter can now be applied to the particular case of simple
vibrating systems. The chemist may in this region. Then the corresponding problem with
molecularsized units and quantum mechanics will be studied. b.
. l. . INTRODUCTION TO MOLECULAR SPECTROSCOPY however. I x V lt COin CCI n a ft
in microns Wavelength b . general. particles which can be described by classical mechanics
is very helpful when the quantummechanical behavior of molecules is encountered. the
greater the number of atoms the a molecule. The effect of gravity here ignored. .
ordinarysized. .. experiences a restor proportional to its displacement from the equilibrium
the infrared spectral region In FIG. The simplest classical vibrational problem is is that
illustrated in Fig. Wave numbers . . The question that the particle of mass is asked is What
type of vibrational motion does the nature of the particle is m undergo among other things. .
the more ways the molecule can vibrate and the more complex infrared absorption spectrum.
in cm . found that its removed a distance from ing force that is equilibrium position. to present
some introductory material before these questions are raised. be apparent from the next few
sections that an understanding i. on many springs are such that if it The answer the spring.e.
The Vibrations of a Single Particle Classical It will of the behavior of macroscopic. .O . It is
clearly depends. The absorption of radiation b in by a a diatomic is molecule and in a
polyatomic molecule.. .
is a proportionality constant called the force constant. A spring which behaves in this
manner For such behavior one can write lt said to obey Hookes / or x f kx x. and A. force for
unit displacement sign. that k will be positive. written explicitly in i. The displacement is from
the in equilibrium position a measured from the equilibrium position and in b from the point of
attachment of the spring.e.. where is the measure of the displacement from the equilibrium
position. . which will also appear position. The restoring force and potentialenergy function
for a boll andspring system obeying Hookes law.THE VIBRATIONAL ENERGIES OF A
DIATOMIC MOLECULE is position. It should be appreciated that k. it gives the restoring from
the equilibrium so The minus Eq. law. in molecular problems. the displacement coordinate.
measures the stiffness of the spring. enters because FIG. f is the force which the spring
imposes on the particle.
ppii. both corresponding to Hookes law. measuring the stiffness of a spring. k is equal to d
U/dx The force constant. moves in either direcThe work that must be done to disdx.
INTRODUCTION TO MOLECULAR SPECTROSCOPY is as x increases in one direction the
force increases but directed in the opposite direction. dU kxdx If the equilibrium position is
taken as that of zero potential energy. Sometimes a problem is more conveniently set up in
terms of the point than position of the particle measured from some other reference .d dx to
deal with the force / that the spring exerts more convenient particle and. In the particular
case of Hookes law. shows. dx or kx . as differentiation of Eq. .d and dU or / dx dU dx This is
J an important and general relation between force and potential energy. is equal to the
curvature of the potentialenergy function. integration of Eq. where / kx. PIiied and this work is
stored as Thus dU /. is therefore equivalent to that of Eq. place the particle a distance dx
potential energy U. since it will be encountered again . later.pP It is ii. that. . It is well to
recognize here. is /. gives U Px This potential function is illustrated in Fig. the potential
energy derivative is dU j. it is important to point out that Hookes law implies that the potential
energy of the particle increases parabolically as the particle tion from the equilibrium position.
on the one has / /. Before proceeding to the problem of the nature of the vibrations. The
statement of Eq. since it is this force that the applied force acts against.
the equilibrium position. It is often more convenient and potential energies of the about the
kinetic One then writes . Eq. The equation One way of doing obtain describing the motion of
the particle can this is to substitute into now be set up. . since dT ax . has a solution x form A
cos wvt ltp The and correctness of this solution can be verified its by substituting Eq. the
length of the spring is I. Newtons / ma equation to kx mji d x or kx mx to start with statements
particle. is obtained. The differential equation. indicates. is I as Fig.THE . When this is done
one obtains an fr v m k or Ik t m ..One can note that. mx and d dT mx dtTx the same
expression mx as set kx or kx mx of the up by Newtons f ma relation.VIBRATIONAL
ENERGIES OF A DIATOMIC MOLECULE If. Kinetic energy Potential T mx energy U kx An
equation equivalent to the / ma equation that can then be used to is solve for the nature of
the motion of the particle that of Lagrange iS. and the equilibrium length / the previous results
are expressed as kl kl k and U l.. second derivative with respect to time cos tvlt tVA if gt back
into the equality differential equation.
with the frequency v given by Eq.e. mechanical solution retains a considerable similarity to
Eq. Again it is asked What undergo is the nature of the motion that these particles FIG. It
shows that a held by particle with mass according to Eq. The Vibrations of Two Particles
Connected by a Spring Classical It is worth while treating one additional problem in a
completely classical manner before we proceed to molecular systems. but will be seen that
the quantum. Consider the macroscopic system of a spring and two particles. i. the amplitude
A of the vibration. . .. INTRODUCTION TO MOLECULAR SPECTROSCOPY This equation is
the important classical result. Only this frequency is allowed. be allowed to The particles will.
is The displacement coordinate q defined as q r r. . move only along the line of the system.
which provides the counterpart of the diatomicmolecule problem. The shape of the potential
energybond length curve for a diatomic molecule.. for simplicity. The quantummechanical
result will differ it from this not all vibra tional energies will be allowed. The energy with which
the particle vibrates can m a spring with force constant k will vibrate be shown to depend on
the maximum displacement.
Substitution of to be found. which will be important in later work. problem X exist. therefore.
the use of Lagranges equation.. ir v m l kA t .kA kAi rVw kA .dU . two of solution. Since the
simultaneous motion of two particles is being considered.. We A cos COS irvt ltp and Xi A irvt
ltp where the amplitudes i and A tional solution is may and lt/gt be different but where. x i Ai
cos and Xz jr v A. one writes the Lagrange equation as d/dT. the and potential energies can
be written as mx kx. Two methods which are presented here because they introduce
methods that encountered in molecular vibrational problems. can be used. gives. one obtains
miXi fc.r if x and m X kxi X Let us again see solutions of the type found for the singleparticle
try. on rearranging. T and wiX quot U Xl y Although one could solve the problem by writing/
ma for each particle. be Direct Solution. COS wvt ltp and Eqs.. the frequency will be that of
the for particles irvt if a vibrasystem and must be the same . and into Eqs. and kinetic mass
wii and which the particles were separated by their Hookes law is assumed for the spring.
.THE VIBRATIONAL ENERGIES OF If Xi A DIATOMIC MOLECULE and Xi represent
displacements of the particles of m from initial positions in if equilibrium distance. a.. will be
illustrated here. For each particle i. With either Lagranges equations or / ma applied to each
particle. will equations in two unknowns will be obtained.
in view of Eqs. Substitution of substitution of y. yields the equation rV wiim rVfcmi m t k fc
which has the roots v and / t h where n wi mi The quantity p which is introduced here will be
frequently encountered and is given the name the reduced mass. and xi z . The motion
corresponding to /. after rearranging. therefore. There are. by displacing the particles from
their equilibrium position by . . gives. after Ay A. INTRODUCTION TO MOLECULAR
SPECTROSCOPY One can now eliminate Ay or for v. mjm /m rn and rearranging.s/k/y. Ai A
and. to a translational motion of the entire system. and Xi m my TO Xi If my the two masses
are equal. on expansion. therefore. This is A from these two equations to get a relation most
conveniently done by recalling that for such linear nontrivial solutions for the As will exist only
if the is zero. can be generated. The v root corresponds. Xi sists of x . therefore. in either of
the equations of Eqs. gives. The motions that correspond if to these frequencies can be
recognized the relation of the two amplitudes Substitution of v A and A is found. rVmi k k k jr
p m k This determinant. one gets Ay v At. homogeneous equations determinant of the
coefficients of the As That is. Thus the motion corresponding to v con displacement of the
particles by the same amount and in the same direction. one of the frequencies being zero. v
l/x /k/n in either of Eqs. two natural frequencies for the system.
Thus T im. and the problem can be set Q.THE VIBRATIONAL ENERGIES OF A DIATOMIC
MOLECULE equal and opposite amounts. J. Thus if Wi is less than w Eq. existence of a
translational The mode and a vibrational one can be recog up in terms of coordinates which
are especially suited to describe these motions. It should be recognized from Eq. . b. and X
is a measure of the displacement of the center of mass of the system. Thus. rather than
displacements xi and x one could choose to use the coordinates . . Exercise . of Eq. m z This
expression must be converted to one involving q and X. nized at the outset. one obtains by
rearrangement. particle . In terms of these coordinates one can immediately write down the
potential energy as q is a Now particles U The zi xt is ikq kinetic energy always easily written
in terms of simple cartesian coordinates. that if the masses of the two particles are not equal.
shows that the amplitude of particle will be correspondingly greater than that of one of
vibration and . x Solution in Terms of an Internal and a Center ofmass Coordinate. From Eqs.
Xi Xl X mampi mzXz mi m measure of the displacement of the distance between the from the
original equilibrium value. Xi A A j q and Xi mi m These allow the kinetic energy to be written
as r. or by forming the inverse transformation.J fc l. . will The motion that will occur will clearly
be have the frequency v /t s/k/ii.. the lighter one will move with a greater amplitude than the
heavier one. Obtain the relation between A and A for the two roots by comparing the
appropriate minors of the determinant.
The solution to Eq. Previously. is. therefore. . two simultaneous equations were obtained. .
Exercise . The advantage translational and vibrational modes initially is that one ends up with
one equation which determines the vibrational motion and another which determines the
translational motion. For problems of much complexity it is of great value to eliminate the
translational and rotational motions before the vibrational problem is developed.
INTRODUCTION TO MOLECULAR SPECTROSCOPY The Lagrange equation. for the
vibration of a single Only a change in notation from m to n and x to q has occurred. the study
of the vibrations of a molecule. the same we can write down the solution as q and A cos wvt
ltp and for The Lagrange equation and leads to mi or X retains only the kineticenergy term
miX X A solution which is formally like that obtained previously is X A with v cos rvt ltp .
obtain the expressions of Eq. as with a is a consideration of the nature of . then gives i kq or
nq kq particle. t Alternatively one can write the solution velocity as X const representing a
uniform velocity of the center of mass of the of this procedure of recognizing the existence of
system. The Potentialenergy Function for a Chemical Bond A preliminary to macroscopic
mass and spring system. and these had to be solved to extract the desired information on
the vibrational motion. . xs in By finding the inverse of the transformation matrix for the terms
of q and X. as that for Eq. This can be recognized as identical to Eq. applied q to the
coordinate q.
W . D e is the dissociation energy measured from the equilibrium position. in much the same
way as was done for the squarewell energies problem.and D e . cmDetermine using . and
that bonds strongly resist compression. to look for functions that solve the resulting
differential equaThe pattern of allowed energies. k X dyne/cm. as before. . has been given by
P. The value of / in the Morse equation can be conveniently calculated from the relation . The
quantity we curves for H as has been done for HC in Fig. one asks about the change in
potential energy of the molecule as a function of the distortion of the of the molecule bond
from its equilibrium distance. for H for which ltb e .. for a diatomic molecule. A straightforward
procedure for deducing the allowed vibrational would be to substitute the Morse function into
the Schrodinger equation and. . known in terms of e and /. curves of Morse. ideas. D e and j
that are appropriate to HC is shown in Fig. q measures the distortion of the bond from its
equilibrium length. M. Exercise . of the potential well. however. Such qualitative Although no
exact mathematical expression for the potential energy all molecules is known. or The Morse
curve with the values of curvature. More particularly. tion. that will Fig. i. as incompressibility
of solids. . from the minimum of the curve. for example.THE VIBRATIONAL ENERGIES OF
A DIATOMIC MOLECULE the variation of the potential energy with the position of the atoms.
by the relative and more exact ones be developed. Uq He has suggested the relation Dl
eltquot where. a molecule can be dissociated. is related to the vibrational level spacing as
will be shown in Sec. cm. a simple and often convenient one that yields curves of the shape
of that in Fig.. it has not been possible to calculate the energy of the molecule as a function
of internuclear distance from the The interactions between the bonding electrons and the
atomic nuclei. . for example.e. D . for i. far be sketched. and / is a constant any given
molecule and can be said to determine the narrowness. general shape of the potentialenergy
versus bondlength curve can. One knows. that if a bond is stretched enough it will break.e.
lead to a potential curve of the form shown in is revealed. Except for the case of H . and
Morse Plot the harmonic oscillator.
If only the next higher term in the expansion is The usual choice of U at r FIG.ltgt /dHJ dq Q
. dU dq. sets t/ g o equal to zero. It will be seen shortly that the vibrations of a molecule result
in only its small distortions of the bond from equilibrium length. If q is the displacement of the
bond from equilibrium length. . particularly an expression for the potential energy near the
minimum This suggests that a suitable expression for the potential energy can be obtained
by a series expansion about the its minimum. . therefore. at q the potential energy is a
minimum. E u c i e .p re y . we are. In a study of the vibrational motion of a molecule
interested only in in the potentialenergy curve. or q . i . V . and therefore dU/dq g o must be
zero. and a simpler and more fruitful procedure is followed. INTRODUCTION TO
MOLECULAR SPECTROSCOPY could then be compared with the observed spectrum.
Some mathe matical difficulties arise in this procedure. The harmonic oscillator dashed line
and Morse solid line potentialenergy functions for HO. a Maclaurin series expansion about q
q gives Uq . re Furthermore.
The Schrodinger equation for a single atom of mass m subject is identical in to a potential of
U kx is while that for the vibrations of a diatomic molecule with reduced mass ju is. For a true
Hookes law spring. V The deduction of solution functions will not be followed through here
but can be found in any textbook on quantum mechanics. and those which be given as
solutions to the har monicoscillator problem should be compared with those previously
obtained for the squarewell problem.THE VIBRATIONAL ENERGIES OF A DIATOMIC
MOLECULE retained. It is only necessary to replace m by /i and x. d U/dq parabola. by q Xi
X. Figure shows the parabola that fits the Morse curve for HC near the minimum. . the
displacement of the single particle from its equilibrium position. amp I A n dq fcQ. i. by
comparison with Eq. we have the approximation. is equal to k at all values of q. the distortion
of the distance between the two particles from the equilibrium bond length. U is a For a
chemical bond the curve can be said to approximate a parabola only near the minimum. the
system said to be treated as a harmonic .e. The Quantummechanical Solutions of the
Harmonic Oscillator The solution of the twoparticle problem by method b in Sec. The same
situation is true in the quantummechanical problem. the parabolic potential function is used
to approximate the potential of a chemical bond. valid near the equilibrium position. . showed
that the vibrational equation that arises for the vibration of the twoparticle system form to that
for a single vibrating particle. The procedure should be recognized to be essentially that
followed in the simpler squarewell problem. is When oscillator. correspondingly. The solution
functions for the two problems will are in fact similar. as identical to that which would have
been obtained if it had been assumed that a chemical bond behaves as a spring following
Hookes law. This result should be recognized.. The force constant is then the value of d U/dq
near the minimum.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
It is found, as
would be expected from the similar squarewell probe and that these functions, and the
values of t, are characterized by a quantum number. Thus, there are again certain allowed
states, and each of these states has
lem, that solution functions exist for certain values of
a specified energy.
represented by
v.
The
It
integer that describes these states
,,,
....
is
usually
can take on the values
The
energies of the allowed states are given
by the expression
sgt
where
v
,,,
These allowed energies are shown, along with the potentialenergy function, in Fig. . The
appearance in Eq. of the same quantities as
in the classical calculation of
a vibrating system should be noted.
This
p
igt
v
l
FIG.
The harmonicoscillator
wave
v
functions
and the funda
mental vibrational transition.
THE VIBRATIONAL ENERGIES OF
A DIATOMIC MOLECULE
relation to the classical
problem
is
sometimes made use of by writing
.
where w
ifo
is
/t y/k/p
written for the frequency with which, according
In to Eq. , the system would vibrate if it behaved classically. what follows, to will be used for a
molecular vibrational property and v
will
be reserved for the frequency of radiation.
The form
in Fig. .
of the solution functions, or eigenfunctions,
is
also
shown
Although little will be done with the solution wave functions themselves, it can be mentioned
that each involves a set of series terms known as a Hermite polynomial. If one introduces a
as
a
r
h
vv
the solution functions can be written as
gt v
N elltgtH
Vq
where
v
,,,
v.
.
.
.
and
H
v is
the Hermite polynomial of degree
The normalizing
factor
Vis
VV.VV
quot
J
For most purposes
functions
it
will
be enough to recognize the form of the solution
shown
it
in Fig. .
polynomials
first
may
For those unfamiliar with the Hermite be informative, however, to write down a few of the
solution functions.
o
i
/
e iaq
ft
qeM V aquotV K
a
gti
yY
ei quot
fa
Vi
f
aiq
Zaiqe
iaq
ag
ag
equot
The
spectral absorptions or emissions attributable to changes in the
vibrational energy of a molecule will
now be
discussed in terms of the
energy levels of Fig.
.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
. Vibrational
Absorption Spectra of Diatomic Molecules
section has
The previous
tion that
is
shown that the allowed energy
by the
relation
levels of
a
vibrating molecule form a pattern of equally spaced levels with a separarelated to the
molecular properties
A
,.
A
The
energylevel pattern can be predicted
be expected to arise from such an if one also knows what transitions can occur with the
absorption or emission of radiation and what energy
spectral transitions that are to
levels are appreciably populated at the
temperature of the experiment. Although the interaction of molecules with radiation will be
dealt with in Chap. , some of the results that will be obtained can be mentioned here.
First of all, it will be shown that a vibrating molecule cannot interact with electromagnetic
radiation unless a vibrating, or
oscillating, dipole
litatively,
moment accompanies the molecular vibration. Quaone can picture an oscillating dipole as
coupling with the
energy can be exchanged between the
electric field of the radiation so that
This stipulation that the vibration be accompanied by a dipolemoment change implies that all
homonuclear diatomic molecules, which necessarily have zero dipole moment for all
molecule and the radiation.
bond
lengths, will fail to interact with radiation
and
will exhibit
no
vibrational spectral transitions.
On
the other hand, a heteronuclear
diatomic molecule will generally have a dipole
the dipole
moment
and, generally,
moment
will
be dependent on the internuclear distance.
will, therefore,
The
vibration of a heteronuclear diatomic molecule
be
generally accompanied
can interact can thereby change their vibrational state to one of higher energy. They can,
also, emit radiation energy and thereby change their vibrational state to one of lower energy.
by an oscillating dipole moment. Such molecules with radiation and can absorb energy of the
radiation and
,
One further transition restriction, which will be discussed in Chap. must be mentioned. Even
for those molecules that have an oscillating
dipole
ble
moment
there
is
the further restriction, which
is
rigorously applica
only to harmonic oscillator type systems, that the vibrational
unit.
quantum number can change only by one
therefore, only
if
Transitions are allowed,
Av
THE VIBRATIONAL ENERGIES OF A DIATOMIC MOLECULE
This restriction is an example of a selection rule. For absorption experiments, which are
those most commonly performed, the applicable part of the selection rule is, of course, An .
Now that the allowed transitions have been stated, it is only necessary to decide in which
state, or states, the molecules are likely to be found at the beginning of an experiment. It will
be shown that the
energy spacing between vibrational energy levels is typically of the order erg/molecule or
cal/mole. With Boltzmanns disof X tribution one can calculate the number of molecules in
the.v state
compared with the number
quot
in the v
state at
C as
/
X S /l.SX
Thus,
less
.
than
per cent of the molecules are in the v
still
state,
and
negligibly small numbers will be in
that, in experiments not
higher energy states.
It follows
much above room
temperature, the transitions
that begin with the
v
state will be of major importance.
The
selection rule
arrow of Fig. to be
transition for
and population statements allow the vertical drawn to indicate the expected spectral
vibrational
an absorption experiment with a heteronuclear diatomic
molecule.
found experimentally that such molecules do lead to the absorption of radiation quanta in the
infrared spectral region and that this absorption can be attributed to the vibrational transition.
Thus, HC
It is
absorbs radiation with
v
v
,
.
cmquot
or
cycles/sec
is
X
hv
,
X
The energy
Ae
of the
quanta of this radiation
v
..
XX
quot. erg
X
.
It is this
energy that must correspond to the energy difference between
the
v
Ae
and
h
levels, i.e.,
to
lie
equating these two results for Ae and using
Mhci
We
.
Wei
X
g
INTRODUCTION TO MOLECULAR SPECTROSCOPY
one obtains
k
.
X
dynes/cm
In this way it is seen that the spectral absorption is understood terms of a
molecularenergylevel pattern and that new quantitative information about the molecule is
gained.
in
The frequencies of radiation absorbed by some diatomic molecules and the deduced force
constants are shown in Table . One should
recall that the force constant
measures the
stiffness of the
bond and
should see that the force constants of Table are to some extent understandable in terms of
qualitative ideas about these chemical bonds.
It should be mentioned finally that it is not customary to convert from radiation frequency to
quantum energies in comparing spectral
absorptions with energylevel patterns.
It is
more convenient
i.e.,
to use
the conversion factor h between energy and frequency,
Ae
hv,
and to convert the energies
of allowed energy levels to frequency units.
TABLE
Frequencies of Fundamental Vibrational
Transitions v
to v
and the Bond Force Con
stants Calculated
from These Data
Values for homonuclear molecules are from
spectral studies
Raman
Molecule
pcm
,. ,. ,. ,. ,. ,. ,. ,. . . ,. ,. . .
k dynes/cm
H D HF
HC
HBr
HI
CO
NO
F,
CI
Br
I
o.
N
a
lis
Nas
NaCl KC
.................
.
X
THE VIBRATIONAL ENERGIES
OF A DIATOMIC MOLECULE
Thus the energy
levels of the
harmonic
oscillator
can be written as
ltv
ss
ergs
J
cycles/sec
The symbol
are written as v
I
will
be used for energies expressed in units of
cm
to be consistent with frequencies or
if
expressed in
cm
.
quantum energies With the last of
of radiation that
these expressions
one can compare the energy separation between two levels directly with
the
wave number
P
of the radiation absorbed or emitted in a transition
i.e.,
between these
Exercise .
levels
At will be the same as
v for
the radiation.
compared
to the
Calculate the numbers of molecules in the v state number in the v state at C C for Br which
has the
,
rather small vibrational energylevel spacing of
cm and
,
for H,
which has a vibrational spacing of about ,
molecule.
cm
,
the largest for any
.
The Anharmonicity of Molecular Vibrations
That the harmonic
oscillator expression
U
kq
is
only an approx
imate representation of the correct potentialenergy function shows up in the presence of
weak spectral absorption corresponding to Av ,
,
.
.
.
,
in violation of the Av
,
selection rule,
is
and
in the fact
,,
.
.
that the frequencies of these overtone absorptions
not exactly
.
times that of the fundamental, Ay
absorption.
The
One
retention of one additional term in the Maclaurin series expan
sion of Eq. provides a better approximation to the potential energy.
has, then,
This expression for the potential can be used in the Schrodinger equation
to deduce the energy levels of the allowed states of the
oscillator.
anharmonic
tion,
The solution is obtained by an approximation, or perturbamethod and leads to an energylevel
expression that can be written
as
.
.
J.
v
i
J
position. This quantity w e is. shows. therefore. the separation between successive
vibrational levels is not constant but much less rather decreases slightly with increasing
values of v. . . . as a bond stretches. Harmonic a Anharmonic b . The vibrational energy
levels calculated for HCI. that would occur quantumnumber term always is known as the
anharmonicity constant. Pattern a is based on the transition. and . It is With the minus sign
than the principal term gt e written explicitly in Eq. that quantity. as shown in Fig. . lt B . or
equilibrium. the spacing of the if the potential curve were a parabola with the curvature that
the actual curve has at the minimum. therefore. . shown in Figs. . ./i where d U/dq q o. . it is
found that w e x e is always a positive The energylevel expression of Eq. The coefficient
oieXe of the squared energy expressed in cm . It should be recognized that the broadening
out of the potential curve. levels. confines the FIG. harmonicoscillator approximation and the
frequency of the fundamental Pattern b is based on the anharmonic values of Table .
INTRODUCTION TO MOLECULAR SPECTROSCOPY ogt e Here fc has been written for
convenience in place of /tc y/k.
transitions from one can check the success of the and determine the constants ogt. levels. i.
. provides a very satisfactory to the observed frequencies of HC. . . i The distinction is that
ampgt is a measure of the curvature of the potential v curve at the very bottom of the curve.
X dynes/cm Comparison v TABLE lator Frequencies of the Vibrational Transitions of HCI....
and Table shows the data obtained for the fundamental and first These data can be
compared with those derived from Eq..THE VIBRATIONAL ENERGIES OF A DIATOMIC
MOLECULE vibrating particles less closely than would a parabolic curve. . term in the
expression based on a harmonic potential. i. . to v . . as If v an effect which decreases the
spacing in Fig.... .. cm . .v . . . therefore. Thus for HC v and . . of the Observed Frequencies
with Those Calculated from the Harmonic Oscil Approximation and with Those from the
Anharmonic Expression o . four overtones of HC. cm and fit ux e . energylevel expression of
Eq. The v harmonicoscillator approximation takes the difference in energy of the potential
curve u.. Such loosening of the restrictions on the motion of particles always leads to closely
spaced allowed energy levels.. ... where a hypothetical level of the would be. .... cm K . that
Eq. for the energies of the transitions from v to v v.. for agt .. One notices that is considerably
larger than the quantity which would have been identified with the coefficient of the v f. and
so forth.e.... cteX. of the higher energy shown one observes some of the overtone bands.
levels as a measure of the curvature and therefore gets a lower value. . It follows that the
force constants calculated from these two quantities will be different.vv Soto cm Av
Description fobs cm Harmonic oscillator Anharmonic oscillator gt Fundamental First overtone
gt Second overtone Third overtone Fourth overtone .e.. more intro The anharmonicity term
duces. iv e ii e v WeXiv One finds. . . v .
For polyatomic molecules. . this information always available. instead of the observed
fundamental energy in a number not more satisfactory. Princeton.. and . Herzberg.. . . X
dynes/cm is. . The use of ways.. cm fc . D. What value of u e can be deduced PRINCIPAL
REFERENCE .. quotSpectra of Diatomic Molecules. . . cm.. Fit these frequencies with a
suitable equation that allows for anharmonicity. Van Nostrand Company. N.quot chap.. ..J.
G.. INTRODUCTION TO MOLECULAR SPECTROSCOPY while of i Jj e . Exercise ..
stretching vibration of The fundamental and overtone frequencies of the CH CHC are
reported by Herzberg to be at . Inc. but its use requires information on the overis tone
frequencies.
The rotational effects will be considered by themselves. is It is now natural to ask whether
the rotational energy of a molecule of radiation. is A treatment that assumes that the
molecular dimensions are independent of molecular vibrations and undisturbed by molecular
rotation known as the rigidrotor approximation. or emitted. can be used to determine the ness
of the bond holding the atoms together. so. of radiation in the infrared spectral region and
that information on the frequency stiff of the radiation absorbed. the theory that is is of the
examples of linear first developed is equally applicable to linear polyatomic molecules. A final
section will consider the slight . The nature of the rotational energies of molecules can be
introduced by a consideration of the particularly simple systems of linear molecules. No
attempt be made in this introductory treatment to allow molecules undergo simultaneous
rotation and for the fact that the vibration. classical It again convenient to consider the
behavior of a linear mass system and then to see the difference is in behavior that imposed
when the system is of molecular dimensions and the quantum restrictions will become
important. also quantized. whether changes in the rotational energy can lead to the
absorption. or emission. if what molecular property can be deduced from a study of the
frequencies of radiation emitted or absorbed.THE ROTATIONAL ENERGIES OF LINEAR
MOLECULES In the previous chapter it has been shown that changes in the vibrational
energy of simple molecules can lead to absorption. or emission. Although diatomic molecules
provide most molecules. and.
for instance. usually a second. For a diatomic molecule. Location of the center of gravity of a
diatomic molecule. INTRODUCTION TO MOLECULAR SPECTROSCOPY modifications that
occur as a result of the centrifugal distortion the rotation produces. of inertia of a The
moment I system is defined as n Y.G. rotor. are given by v/vr and the angular The rotation of
ltgt velocity by H number of revolutions per unit time. Such a treatment is said to deal with
the nonrigid . i. is v/vr or lta/v. the value of / can be worked and then applying Eq. The
Rotation of a Linear System Classical any system is most conveniently treated in terms and
the moment of inertia /. . first For any particular system. lead to the center of gravity being
located so that FIG. the masses and distances of Fig. mr i where out by is the distance of the
ith particle from the center of gravity of locating the center of gravity the system. the
revolutions per unit time. For a particle moving with a linear velocity v a distance r from the
center of gravity. mi .e. C.. i It follows also that the the dis tance traveled per unit time divided
by the circumference of the particle path. of the angular velocity It will be recalled that agt is
defined as the number of radians of angle swept out in unit time by the rotating system.
is placed on the angular velocity with which the system can rotate. since r x OF LINEAR
MOLECULES r m n mi m r r. m t m m main y m r mim mi iur m j r . the moment of inertia now
gives . where m n. respectively.THE ROTATIONAL ENERGIES or. furthermore. It can now
be expected that when the same problem is treated for a molecular will appear.y. . If the
system performs a rotational motion.m r kinetic energy of a system of wtim The given by
moving macroscopic particles is i m. system some restrictions on the allowed rotational
energy Exercise . defined as in the previous chapter as . No restriction. more conveniently
treated in terms of the angular One then can write the kinetic energy T as m. and . Calculate
the linear moment of inertia of a HC HC and . A. A. mfm nil m . is the reduced mass.coV t
WX i quotW i/quot Equation shows that a system of particles that obeys classical mechanics
has a kinetic energy that is dependent on the moment of inertia of the system and on the
angular velocity. DC . Eq. which which have an equilibrium bond length of . all of N . the
particle velocities are velocity w of the system. for Application of the denning equation. and is
b and has NN and N bond lengths of . to the relations and r mi mi r r wi . .
. first. INTRODUCTION TO MOLECULAR SPECTROSCOPY . only slightly different. the
wave functions and energies of the allowed states. following customary notation. in view of
the definition of I as niirf and w as Vi/r i is Thus. condition requires that Iw j with/ . has been
designated by J.. the quantized angular momentum . being ej jJJ D The is correct
quantummechanical treatment turns out to allow. it is. for example. /JJ gt . where the
quantum number. if J an integral quantum number. . angular momenta given by rather than
in early by the simpler form Jh/w introduced by Bohr and used If this quantummechanical
treatments. obtained. The angular momentum J rnivdu i of a rotating system is which. . i. is
by solution of the Schro dinger equation. The Rotation of a Linear System Quantum
Mechanical Although a complete quantummechanical description. at more informative to see
what result is obtained by applying Bohrs defined as condition that the angular momentum
be quantized in units of h/w. This stipulation leads to the allowed rotational energies of Si
The correct result. term.e. . can be obtained by solving the Schrodinger equation for a
rotating linear system.
tional states allowed For spectroscopic work. orbitals. . as given by Eq. and hence derivation
by application of the Schrodinger equation will not be carried out here. using the convenient
polar coordinates For a free gaseous molecule. potentialenergy function can be set equal to
zero. The problem consists of setting up the Schrodinger and the equation for three
dimensions. in these units we have the expression . The wave functions themselves need
not now can be mentioned. . These allowed energies. that they are identical to the angular
part of the solutions for the behavior of the electron of the concern us. Polar coordinates. sAc
The group i gt cm is of terms h/nr i Ic sufficiently frequently encountered in FIG. . Solution of
the molecularrotation problem. however. For J . . for the resulting differential equation only if
Solutions are found the energies have the values indicated in Eq. Thus for J . . and so forth.
Again it is more convenient to express these energies in wavenumber units. of Fig. Any
textbook on quantum mechanics can be consulted for such a derivation. obtained. are shown
schematically in Fig. It hydrogen atom. . the wave function is the same as that for the px py
and pz . the expression for the energies of the rotaby the quantum restrictions is of prime
importance.THE ROTATIONAL ENERGIES is OF LINEAR MOLECULES introduced for Iu in
the classical energy expression. . the rotational wave function j has the same form as the
angular part of an s orbital wave function. rotation is unhindered. the correct quanis
tummechanical result of Eq.
molecule can now be written as allowed rotational energies of a linear Again the barred
symbol lj BJJ cm FIG. each rotational level is The degeneracy. or multiplicity. and one uses
the symbol defined as B B h Sx/c em is . Jamp/t J energies E U B JJ B B OB . of if indicated
by the number of levels that would appear an electric field were applied to the rotating
molecule. Allowed rotational / Energy pattern if I degeneracy were removed schematic
Multiplicity . The energylevel pattern for a rotating linear molecule. INTRODUCTION TO
MOLECULAR SPECTROSCOPY molecular spectroscopy to merit a symbol. The used to
emphasize the units of cm.
. N has an N N bond length of . AVo Rotational quantum number. as Fig. . times h/r that
would appear if an external field were applied. . This can most easily be seen from the
condition that the comIt follows. A. y. as suggested in Exercise . there are. / w be seen later
that the allowed rotational energies typically have small energy spacings compared to the
roomtemperature value of For example. the Boltz mann distribution expression for the
populations of the rotational energy levels is Nj It will iJV equot gt . . . there are arises states
with energy BJJ .. The moment of inertia can be calculated. . or . The relative populations of
rotational energy levels of r N at C. that ponent of the angular momentum in a direction
imposed on a rotating molecule he quantized in units of h/ir.. . as with the hydrogen atom
treated in Sec. in gen a number of rotational states corresponding to a given rotational
energy. Now eral. Rotational Energylevel Populations The next tional states step in laying the
foundation for an understanding of rotational spectra involves a discussion of the populations
of the rota whose energies are given by Eq. . . J . The number from the states with angular
momentum components of . as a function of in Fig.J lt J gt is shown The population of the
levels of N FIG. indicates. X lO g cm and .THE ROTATIONAL ENERGIES OF LINEAR
MOLECULES . A and an N O bond length of . . to be . kT. at C the energylevel population
expression is Nj lJV e. In view of this degeneracy of the th energy level.
previous section. for but also to higher HC a population versus at and C. The rotating dipole
can be thought of as interacting with the electric field of the radiation and allowing energy to
be exchanged between the moleand the radiation. Rotational Spectra of Rigid Linear
Molecules be obtained It is again necessary to anticipate the results that will in the following
chapter when the transitions that electromagnetic radia tion can induce will be studied. an
absorption experiment. The selection rule for rotational transitions of linear molecules that
occur with the absorption or emission of electromagnetic radiation is AJ For ably done
studies of the absorption of radiation. There is a further restriction on rotational transitions
even for those molecules which have dipole moments. the expected transitions can be
indicated by the vertical arrows of Fig. The requirement is seen to be equivalent to that
stated in the previous chapter where the vibration of a molecule had to be accompanied by
an oscillating dipole for such interaction to cule occur. we must expect transitions starting
from energy levels corresponding not only to J J values to be important. the appropriate part
is of the selection rule AJ . that quite a is few of the allowed rotational generally valid. in
result of this example. . general expression for the energies that are absorbed in the
transitions indicated by arrows in Fig. . v is BJ i BJ . If the quantum number for the lower level
involved in a transition is J and that for the upper level is and therefore the value of f for the J
. the experiment that is invari when rotational spectra are obtained.BJJ l gt . It follows energy
levels are appreciably populated. can be obtained from the The allowed energies given in
Eq. Exercise . Make J value diagram like Fig. . and the population results obtained in the
With this selection rule. INTRODUCTION TO MOLECULAR SPECTROSCOPY The that. the
energy difference in cm . absorbed radiation. The first requirement for the absorption of
radiation by a rotational energy transition is that the molecule must have a permanent dipole.
The symbol B used to represent the quantity h/T Jc. attributable to rotational transitions. .
FIG. Table shows the frequencies of absorption. absorptions of radiation at frequencies of B.
ampB. be . . Energy B Ai Spectrum . .THE ROTATIONAL ENERGIES OF LINEAR
MOLECULES We thus expect. . . Rotational is energy levels and transitions for a rig id rotor
linear molecule. One observes in the microwave spectral region such series of absorption
lines when linear molecules with dipole moments are studied. observed for the Unear
molecule OCS. . This amount can then be identified with . since the levels J . will generally
JB. . populated. and should therefore observe a set of lines spaced by a constant amount.
. A A one will discover.. cm lc i s .. more detailed analyses and C S reported by Strand berg.
and Kyhl Phys.. lOquot g/sq these momentofinertia data... Kyhl. megacycles/sec and S X Z
of inertia of the Therefore. In such cases. what is done whenever possible is to obtain the
rotational spectrum of different isotopic of the rotational spectra of species. as from the
moments of inertia. Measurements of the spacing of rotational spectral frequencies have
Absorptions of TABLE the Microwave Region Due Transitions C S in to Rotational Transition
Frequency v J J gt megacycles/ sec . C S Thus. emquot From i c i S and and / . lgt . and the
assumption that the bond lengths are independent of the isotopic species. X . . .. and
Strandberg. INTRODUCTION TO MOLECULAR SPECTROSCOPY From line these data
one can obtain an average value of the absorption spacing of . gt From Wentink. the moment
OCS mole cule can be calculated to be /ocs It . Wentink. X X . g cm should be apparent that
from this single experimental result it is not possible to deduce both the CO and CS
distances of the molecule.. give i cl lcls n . gt . g/sq cm cm . one deduces that in the OCS
molecule rCO rCS . since B h/STr Ic. Rev. . . Rev. Phys. to verify that these distances are
conmomentofinertia data than to derive the bond lengths the sistent with It is easier. .
and therefore on the moment of inertia of a rotating system. and can be expected to stretch
under the influence of the centrifugal forces that arisampwhen they rotate. Gordy. The
constant spacing expected from the treatment given here is that obtained from the rigidrotor
approximation. .quot John Wiley amp Sons. . Smith. in fact. The rotational constant term B
cB is given by Gordy. Some Bond Lengths Obtained from wave Spectroscopy of Linear
Molecules TABLE Micro Molecule Bond Bond distance A . . Smith. . Consider. New York. F.
exactly constant. as . . Close examination of data such as those of Table reveals that the
spacing between adjacent spectral lines is not. . .H c a THE ROTATIONAL ENERGIES OF
LINEAR MOLECULES led to very accurate values for the cases. DC N . . The Nonrigid Rotor
The stretching effect of the centrifugal forces on the bond lengths. Some of these results are
shown in Table . calculate the H C and C N bond lengths. in many bonds of molecules.
barulo. . Exercise . and Trambarulo for HC as . . megacycles and for N Deduce moments of
inertia for these molecules.. Inc. . V. megacycles. CO NaCl C NaCI HCN CCN HCCC
HCCCN C CN CCI CN C CC c C cc c CN From W. can first be calculated on the basis of
classical behavior. for simplicity. TramquotMicrowave Spectroscopy. Now we can see
although this is a minor matter that can be passed over what alterations to the theory arise
from the recognition that molecules are not rigid. W.. . for the lengths of the moments of
inertia and. Assuming that the bond lengths are independent of isotopic substitution.. and R. .
. but are flexible.
INTRODUCTION TO MOLECULAR SPECTROSCOPY a single particle of mass velocity of
u. Assume when about a fixed point with an angular there is no rotation. therefore. minor
term of Eq. with Eq. The kr centrifugal force of r stretching www is balanced by a restoring
force of that accompanies bond stretching. In this way. and approximating r by r in the
second. to r by means of Eq. The extent of bond that occurs for an angular velocity of w is.
becomes t Iugt I o IW fcr The quantum tized according to restriction that the angular
momentum Ilta be quan y/jJ Vt will convert this classical result to a quantummechanical
result. . major importance in Eq. is m rotating that. the total energy can be written as e Jw ikr
r which. the particle a distance r from the fixed point and that this length increases to a value
of r when the particle rotates. . therefore. given by the expression kr r www On
rearrangement. this gives the distorted bond length as r Jr k rrua is Since the energy of the
rotating system made up of kinetic and potentialenergy contributions. which of It is finally
necessary to relate the distorted distance r in the first term.JV armVJIb l . deduced to be The
correct allowed energies are. . one obtains is tj D gSSFT quot . .
known as the centrifugal distortion constant. can be related to other spectroscopic molecular
parameters relation D difference Bgt/tf of The value D is always very between the rigid than
that of B. less much J ft FIG. as Fig. It is customary to write this expression for the rotational
energy levels as tj. B h/WIc. and the and nonrigid rotor treatments can. this result becomes
BJJ where ogt JJ is the vibrationalenergy term l/xc s/k/m and. as before. BJJ tJJ where D.
Rigid rotor Nonrigid rotor . THE ROTATIONAL ENERGIES OF LINEAR MOLECULES In
terms of wavenumber units. distortion The effect of centrifugal shown by the comparison of
the allowed rotational energy levels of a rigid solid lines and nonrigid dashed lines rotor. is a
quantity that can be evaluated from spectral results and. according to the above by the
derivation.
. . according to Eq. Gordy. cmquot . Inc. . . . . Pcalc B BJ .J. Herzberg. Van Nostrand
Company. . Table illustrate the improved to experimental for centrifugal absorption
frequencies that can be obtained by allowing PRINCIPAL REFERENCES . . . AiJ J of BJ DJ l
fit The data distortion. . G. em . and R. . . . Trambarulo quotMicrowave amp Sons... The
expression for the energies of the rotational transitions AJ for a nonrigid rotor are. V.. .
Princeton. . .quot D. . Herzberg.quot Company. Princeton. New York. suggests. JJ gt gt gt gt
gt cmquot . D . N. quotSpectra of Diatomic Molecules. W.DJ l s . . F.. Inc. .. . of From G.
INTRODUCTION TO MOLECULAR SPECTROSCOPY Absorptions Due to Rotational
Transitions of TABLE Transition HQ gtb . .J. with BJ B . N. . Inc. . . be expected to show up
only when high J value levels are involved... . . . John Wiley W. quotSpectra Diatomic
Molecules.quot chap. Van Nostrand Spectroscopy. D.. Smith. .
For one dimension the equation pendent. selection rules governing the transitions that an
thus occur have been quoted. it is approprie to investigate the theoretical basis gte
understood. tion is. The Timedependent Schrodinger Equation trated The form previously
given by the particleinabox for the Schrodinger equation. Now that some introduction to
lowed energy levels and to simple spectra has been given. A considerable amount on which
spectral transitions can of manipulation with quantum mechanical expressions will be
necessary. equation. the intensity of the absorption or emission will be observed. an
expression that can often be used in a qualitative. system and. The net result of the deriva
however. or amplitude. that transitions between allowed energylevels can occur with the
absorption or emission of electromagnetic radiation. or diagrammatic. without adequate
explanation or proof. Furthermore. is and illus known as the timeinde A more complete
expression for Schrodingers equation allows the calculation of atomicscale phenomena as a
function of time as well as of space.t Uxnxt jgM m . calculation. moreover. written for x and t
is v az. THE ABSORPTION AND EMISSION OF RADIATION In the two previous chapters it
has been necessary to state. way to understand what transitions will occur in a particular
band that and . The reader will notice that in the previous chapters the emphasis has been
on the frequencies of absorption bands little attention has been paid to their intensities.
and . The solution .e. ent part of the complete Schrodinger equation can be readily solved.
involve a quantum number which can be denoted by by Eq. In a similar way. x. in the
onedimensional squarewell potential.e. ti The x and since t. Substitution is of this relation in
Eq. i. the solution lttgtt is elt quotgt where is written because the values of that are required n.
left side of this equation t.t. gives ampampgt quotgtampamp or x ltKx L ir m dx J dx v h ltgt dlt
dt x. is a function of x and and that is a function only of Previously it was shown that /
described the system. is a function only of and the right side is a function only of only if each
side is it can be identified The equation can be valid for aH values of equal to a constant. To
reduce tion to into equations such as the timedependent Schrodinger equa more
manageable forms. . respectively. it is customary to try to separate them two equations each
involving only one of the variables. describes the system in space and time. With this tries
the general substitution approach one where f gtx and ltfgtt are functions of x and t. gave
information on the position of the particle.. equal to the constant are jja and . The constant is
called e with the energy of the system. Uxf A dV ampw mdx i e The second of these
equations can be recognized as the timeindeThe timedependpendent Schrodinger equation
introduced in Chap. with the supposition that U a function only of x. not be continued. t be
understood that x.. The will INTRODUCTION TO MOLECULAR SPECTROSCOPY explicit
indication that It will is a function of x and t. i. The two equations that are obtained by setting
Eq.
THE ABSORPTION
AND EMISSION OF RADIATION
to the complete equation for a general onedimensional problem can,
therefore, be written as
,
If
relt quot
solutions that
one thinks of the particleinabox problem as an example, the would have been obtained from
the timedependent Schro
dinger equation would have been
,x,
JinU
nn
ma
quotquotquot
where
The implications of the solution in x and t can be seen if the quantummechanical description
of the system, as given by is obtained.
r
,
This
is
given in general by
j/
e Ti quot
,
lt
e
M quot
V
of the solution to the space
is
The timedependent part
tion
is
and time equaSystems, like
seen, therefore, to
drop out, and the system
again described
simply in terms of the timeindependent wave functions.
the particleinabox system studied in Chap.
,
which lead to solutions
that are time independent, are said to have allowed stationary states. These are the allowed
states that are obtained as solutions to the timeindependent Schrodinger equation.
For a spectroscopic process to occur, a procedure must
time dependence
is
exist
which
causes a state to change from one stationary state to another.
How
a
introduced into a quantummechanical problem will
now be
.
investigated.
Induced Quantum Transitions
Much of spectroscopy is concerned with the absorption of radiant energy as a system goes
from one stationary state to another under the
influence of incident electromagnetic radiation.
tigate, therefore,
It is necessary to inves
how
radiation can disturb, or as one says perturb, a
system so that a transition is induced. In general any atomic or molecular system will have
many allowed energy levels corresponding to the stationary states of the system. It
is sufficient,
INTRODUCTION TO MOLECULAR SPECTROSCOPY
and the notation is simplified, if a system is treated that is assumed to have only a lower
energy state, with a wave function and an upper energy state, with a wave function m where
I and m are two different values of the quantum number for the system. The complete time
and space Schrodinger equation for one dimen/i,
f/
,
sion
is,
as before,
w m dx
Ux K
.wi dt
symbol H,
It is convenient to introduce the
called the Hamiltonian, to
represent the terms
h
d
w m dx
Although
that
it is
,
TT , Ux
,
it is sufficient
to treat
H
as a symbol, one should recognize
the quantummechanical operator for the total energy of the
system.
With
this notation,
Eq. becomes
The
supposition that only the two stationary states described
by
and
yp
m exist for the system implies that the only two solutions to
Eq. are
and
Since such solutions to the Schrodinger equation, as pointed out in Sec. , form an orthogonal
set of functions, a general solution to the equation
can be written as a
series of these functions.
In this case, where only
two solutions
exist,
the general solution
is
simply
t
aampi
am r m
where a and am are weighting coefficients which are not functions of x The two allowed
stationary but, as we will see, can be functions of time. states of the system are described by
the general expression with Oj ,
am
or a
t
,
am
.
The potential energy of the system before it is disturbed, or pert turbed, by electromagnetic
radiation is now represented, more specifically,
THE ABSORPTION
AND EMISSION OF RADIATION
i.e.,
by
Uo,
and the corresponding Hamiltonian by Ho
Ha
as
ff
case
is
The Schrodinger equation for this initial, unperturbed
then written
quotaSStwo solutions fy and
ty m
.
It is this equation that has the
Now
let
us assume that
It
it is
the lower energy state
initial state,
with the
known that initially the system is in wave function j and energy ei. The
by Eq.
,
at time
.
,
is
therefore described
with
and am
As was seen from the previous
discussion, the solu
tion will lead to a description of the system as being time independent
with the wave function
and energy
is
.
This state of
i.e.,
affairs will con
tinue as long as the system
undisturbed,
as long as
U
Jo.
Now
tem,
let
us suppose that electromagnetic radiation
falls
on the
sys
As
will
radiation
may
be shown later in more detail, the electric field of the act on the system and change the
potential energy. The
U and that in
H.
change in
U is denoted by
is
H is denoted by H.
and i
The new
will
Hamiltonian
satisfy the
then
Hq
The
is
differential
equation corresponding not
to the initial state of the system
therefore changed,
new
equation.
The more
general expression
ait t
am tm
where O and am are timedependent parameters, can be used to provide a solution to the
new Schrodinger equation that is applicable to the perturbed system.
This wave function, with undetermined a and o m can be substit
,
tuted into this
new Schrodinger equation
to give
o
HHaflr,
am m
AA
,,
om
Expansion of
a
this equation gives
t
H .,
am
Hm
amHm Zrm , at fedflUfcd, h
a,ff
A
quot
dt
wi
l
dt
ri
am
dm
dT
.
.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
first
The
fied
two and
last
two terms cancel because j and
Wm
are solutions
is
of the
unperturbed wave equation.
multiplied through
The remaining
expression
simpli
if it is
from
x gttox
.
by V and integrated over all space, i.e., The orthogonality of m and fa eliminates
one of the integrals, and the timedependent part drops out of the , integral as in Sec. . On
rearranging the remaining terms, and
substituting Eqs. , one has
daquot
dt
fljerlt
h
, H/A
m
,
f
H h dx
,
It is
m
h
/quot
known
that initially a
t
not
initially contribute.
It
and a m so that the can be shown also that the
final final
term does term does
not contribute appreciably at longer times, and one has
dam
dt
e
h
riMej i f Hlx f, idx
,
This important equation gives the rate at which a system can be changed from one stationary
state to another under the influence of a
perturbing
rate at
which a m increases corresponds to the system changes from fa to m To proceed it is
necessary to be more specific about the perturbation H. Some features of the interaction
between the electric field of the electromagnetic radiation and the molecular system that
absorbs
effect.
The
rate with
which the description
of the
/
.
the system are
.
now
treated.
The Interaction of Electromagnetic Radiation with a Molecular System
falls on a molecule, the oscillating can in some cases disturb the potential energy of the
molecule and allow it to escape from its initial stationary
When
electromagnetic radiation
electric field of the radiation
state, here
will
assumed to be characterized by the quantum number I. It be recalled from the discussion
presented in the introduction that
field of
the electric
the radiation oscillates at the point occupied by the
v.
,
molecule with a frequency
radiation can be described at the position occupied
For example, the x component of the by the molecule by an
equation which
is
usually written as
E.
EX
cos wvt
It is here
more convenient to use the exponential form
for the time
THE ABSORPTION
AND EMISSION OF RADIATION
dependency and write
Ex
The
e quotquot
e
derivation of the influence of the electric field on the molecule
can, for simplicity, be followed through in terms of the x
of the interaction.
component
At
the end of the derivation, the effects of the inter
actions in the y
can be added on. can act on a dipolemoment component / This term adds to the potential to
produce a change in energy of E x n x that occurs energy of the system and is responsible
for the change in
and
z directions
The
electric field
Ex
.
H
when
radiation falls on the system.
One can
write, therefore,
H
If
ExP x
,
the frequency dependence of
Ex is
shown
explicitly, this
becomes
H
e quotquot
ei
M nx
Substitution of this perturbation in Eq. gives
t
E
th
J
quot f tltxi
daelt
i quotgtAgtlt
e r/ .,lt
The
integral over x that appears here is customarily represented
by
n x im i.e.,
Im.
/quot
Knh dx
and integration
to
With
this notation,
of the timedependent functions of
t,
Eq. over the time interval
one obtains
p.Tilhe m eihvt
m
ti
xr hv
is
.
e
/
gtirtM,fcgtl
m
e
hv
r
J
The
process of interest here
that in which the system goes from a
lower energy level u to a higher one e m For such an arrangement of energies, the
denominator of the second term in the square brackets
will
go to zero when the radiation frequency
hv
is
such that
t
m
e,
For such conditions this second term can take on large values and be of major importance in
determining am t. On the other hand, if tm were lower than t t the first term in the square
brackets would be of major
,
importance in determining am t, for then hv equal to the energy
differ
INTRODUCTION TO MOLECULAR SPECTROSCOPY
It follows that the first
I
ence would lead to a zero denominator.
in the square brackets is important
if
the transition from
to
m
is
term one
of emission, whereas the second term is important if the transition is one of absorption. Since
we are here investigating an absorption process, the energy of the to state is greater than
that of the I state, and it is only
necessary to retain the final term of Eq. . the assumed energylevel pattern
We
have, therefore, for
tgtiirilhii m T,t
described
As the system changes from the description ty to m it can be by the product amp, where
aampi am rm and the as are
,
time dependent.
tion,
as a function of time
This quantummechanical description can be given if it is integrated over all values of x. This
integraall
moreover, removes
the spacedependent
wave functions because
only with
of their orthogonality
and normalization.
l
One
is left
/quot
dX
a
ai
am a
is is
The importance
given, therefore,
of the
m
state in the description of the system to Eq. ,
by the product aam which, according given as a function of time by
alta m t
mm HEIY
L
,
em
ei
hvy
o gt
rrtwi
The above
Integration over
all
expression shows the effect of a given frequency of radiation.
frequencies, treating
E x as a constant
since the
absorption will usually occur over a narrow frequency range, gives, with the definite integral
quot
sin x
/
the result
ax
,
a m ta m t
ii m
KKH
This
radiation
is
is
the desired result except for the characterization of the
its electric field
by
strength instead of
its
energy density.
It
the latter measure of the
are
when comparisons
amount of radiation that is more convenient made with experimental determinations of the
THE ABSORPTION AND EMISSION OF RADIATION amount of absorption. moment. wV
This rate of change of the system as a result of absorption of radiation under the perturbing
written with effect of the electric field of the radiation is usually B tm called Einsteins
coefficient of induced absorption. rate of transition is The fn m . . becomes J OrnOm t B lmP
with ir B tm gp Wm This expression obtained for the time dependence of the coefficient of for
a system originally described by j shows the factors that are responsible for the change from
one stationary state to another. is The necessary relation.e. and on the energy density of the
radiation. p For isotropic radiation the three components interaction are equal. and one writes
of the radiationdipole J oOm t jjj Hxlm HylmY Glmp . by this relation gives the rate daZam /dt
with which the importance of state increases as m in the description Jt w ft. which is here
given with out derivation. P KY the energy density. intro duced so that Eq.. radiation. where p
is the energy per unit volume that is irradiated by the electromagnetic Replacement of E x in
Eq. it remains only to compare this derived expression seen to depend on the term called the
transition with the quantities usually encountered in the experimental determinations of the
absorption of radiation. i. To complete the general study of the process by which radiation
energy is absorbed.
Comparison with Experimental Quantities The quantities used in reporting the experimental
results for the absorption of radiation can be introduced by considering the derivation of
Beers law for an absorbing solute in a it nonabsorbing solvent. proportional to the radiation
dl. C moles/liter . and see that the frequencies that contribute most to the integral are those
close to the Bohr condition em e hv. is. intensity. . Plot the function sin ir/h e m . The
dependence of a on the frequency is here emphasized cell av. versus frequency near hv e m
n. or the intensity with no absorbing material. Integration of this equation over the length . as
the proportionality constant dl by writing avIC dl to be written. according to Beers law. The as
in decrease in intensity of the radiation as Fig. to C. i hvY . and the intensity of the radiation
after passing through the the solution. the absorption allows the equation coefficient. t . as
cell containing The integrated form of Beers law is obtained in this way aOOijln FIG. hv which
must be integrated over all frequencies to obtain Eq. . incident intensity. INTRODUCTION
TO MOLECULAR SPECTROSCOPY Exercise . I allows the absorption coefficient to be
measured in terms of the I. and to the path length Introducing av. the molar concentration.
Spectrometer ir dl Solution of concentrate. dl. Source The absorption of radiation by a
Sample solution. penetrates a distance I.
The theoretical result of the previous discussion is that the rate of transfer of molecules from
state I to state m is B . gadros number is N. the integrated absorption coefficient A. since / of
sq is the energy flowing through a crosssection area related to the energy density cm in sec.
or energy. or A. over band or av dv A J over band av dv A c It is now necessary to relate the
experimental quantities av. this relation is N NC . it is by I cp where c is the velocity of
propagation of the radiation. it is If Avofinally necessary to relate N to the molar concentration
C.. /. v . usually by graphical integration. absorption band for a given transition usually
extends over a The total intensity of the band is obtained by measuring av in the region of
the absorption and determining. can therefore be written as Equation dl Wm hv lmN dl For
comparison with the experimental expression.e. i. to the theory of the previous section.p If iW
P one introduces the symbol N to represent the number of molecules in a cubic centimeter of
the sample and recognizes that an amount of energy hv tm is removed from the radiation by
each transfer. the decrease in intensity.THE ABSORPTION AND EMISSION OF
RADIATION An range of frequencies. Eq. of the beam passing through a length dl of the
sample is given by the theory of the previous section as dl dl p Wm VJV Furthermore. . .
and therefore fail to absorb radiation. by means of Eq. and the identification should be made
with A jau dv. yjmlWml where v tm is the center of the absorption band and. will In
subsequent work. the total absorption for the transition rather that for a given frequency.
From this relation. can be performed. iMtml / T quot. be used. some very important
qualitative statements can be made about which transitions can be induced by radiation. In
this way we obtain /icl.dx result given The important for a transition if by Eq. Eq.. Such
general statements are called selection rules. both procedures . to be symmetrically
distributed about the midpoint. INTRODUCTION TO MOLECULAR SPECTROSCOPY
Substitution for N in Eq. The deduction of selection rules for the particleinabox system will
tion of radiation often conveniently demonstrate the deduction of selection rules. as it will be
recalled. . something about the known and wave functions of the states involved in the
transition. The dipole moment . and thus lead to the absorption of radiation. or simply the
band intensity. allows the terms corresponding to av to be identified. and which transitions
cannot be so induced. .d SwS a differential Comparison with the form of Beers law. Let us
consider the particle to be an electron and the balancing positive charge to be located at the
midpoint of the well or. however. The Basis of Selection Rules The results of the previous
section provide a quantitative relation between the transition moment of the absorbing
species and the absorp by a given sample. at least. m and of /xim and thereby learn. one can
use the measured value of A to obtain a value of ju. is that the integrated absorption
coefficient. can be calculated the wave functions for the initial and final states are
Conversely. and for a given system they can be deduced by deciding for which transitions
the integral m m must necessarily have a zero value. gives . furthermore. if the integration of
Eq. The theory than predicts.
the integral W The / MejMm ltz must he nonzero. the distance of the electron from the center
with n m For a transition from a state with ItO one to be induced by electromagnetic radiation.
. . and in Eq. . is clearly an odd. of Fig. or antisymmetric. y of the potential well. when a One
can here illustrate that for the deduction of selection rules it is often sufficient that appear to
consider the symmetry of the three functions if/. . All the wave odd function FIG. The
coordinate system. as Fig.THE ABSORPTION will AND EMISSION OF RADIATION then
have the form ft ey is where. ey. function since it changes sign at the origin. . with y at the
midpoint is convenient for the symmetry discussions that are necessary here. The symmetry
proper ties of the porticleinobox wave tranrule function and some of the sitions allowed by
the . integration will be carried out in the next section is quantitative result for the absorption
intensities desired. The coordinate y. shows. and therefore the function ey.
one writes. .e. state with i. is an odd function. for For the dipole moment term p. one must
work out the integral involved in /j. are either about this origin. and the selection rules will
usually metry A. The symmetry of the system will be important in this regard. using the
coordinate system which the wave functions of the square well were given in Sec. in Sec. it
will often be possible to deduce selection rules without a detailed working out of the integrals.
allowed by these selection rules are indicated in Fig. in of the have some simple direct
relation to the symwave functions corresponding to the energy levels. The transi tion for a
particle in a box can be easily treated. The selection rule can be conveniently expressed as
where ltgt . connect a state the transitions for which m m is not necessarily with an even
value of the quantum number to a an odd value.e. INTRODUCTION TO MOLECULAR
SPECTROSCOPY functions. As for this particleinabox example. / I ex ea The wave functions
are .. . symmetric or antisymmetric they are either even or odd functions. from the integration
over the right Thus m m will necessarily be zero for a transition from n I to n m if I and m are
both odd numbers or both even numbers. recognize that the selection rule stating that
transitions One should can occur between energy levels only if one level has an odd
quantum number and the other an even quantum number is equivalent to the statement that
only levels of different symmetry can combine.. . zero. If the product of the three functions of
the integral of Eq. The allowed transitions. Particle The Integrated Absorption Coefficient for
a Transition of a a Box To evaluate the theoretical integrated absorption coefficient for a
transition. as was shown i. wave wave function and The transitions indicates a state with a
symmetric indicates one with an antisymmetric function. sin ja nwx a . the result of the
integration over the left half of the potential well will exactly cancel the contribution half.
THE ABSORPTION AND EMISSION OF RADIATION and the . as Fig. values in Eq.
transition.my I cos I J wr m where the fact that I m and m are integers has been used to elim
inate the sine terms. e I Mi Since the orthogonality condition eliminates the integral arising
from the a/ term. for which . The selection rules deduced from symmetry considerations in
the I previous section can be obtained analytically by the insertion into Eq. ea ea is The
integrated absorption coefficient then given. making use of the integral y cos quotJ cos dy y y
sin y. A. specific The intensity of the ireleetron transition of hexatriene.lquot a Jo x cos I I m a
x cos l m dx y Integration. moment tu m is. quotquot e a NeWvlm SwhclflOO Using X lB . .
therefore. one is left with the expression Iquot T /quot x sin sin quot dx of the relation lt/gt
Making use sin B sin Mcos cos lt one gets ImiI . can be used as a The . corre sponds to
gives . gives ea Ml Iquot I cos I i i mir . transition . as N Acl. by Eq. indicates. of the various
even and odd combinations for and m. . A and a . the frequency was calculated in Exercise
application of Eq. . one gets A X sec cm mole liter . I and m Substitution of these.
The interaction of the quadrupole with the electric field of the radiation is smaller than the
dipole interaction by a factor of about and one can therefore almost always ignore these
quadrupole effects. the order of magnitude agreement quite satisfactory. as was mentioned
term for induced emission is equal to that for induced if I. It can ii now be recognized that this
implies that only for Av can tm for a vibrational transition be nonzero. used the selectionrule
result that for a vibrational transition to be induced by electromagnetic radiation Av . there are
the same number of molecules in the state to as in the state the effect of subjecting the
system to electromagnetic radiation will be to cause just as many molecules to go from from I
to to. I m to as bility of In addition to induced absorption and emission. using the squarewell
potential model. The probability of such emission is usually here. problem of the important
relations of Eqs. that the absorption. tional The application to a vibrawill. Emission or
absorption can also occur as a result of the quadrupole of the atom or molecule. where v is
the vibrational quantum number. there is the possispontaneous emission. A net absorption of
energy is then observed. Induced Emission and induced Absorption In some experiments it
is important to recognize. . one generally deals with a system where each state of lower
energy has a larger population than each upper state. and how . and it need not be treated
Finally it can be mentioned that the treatment of induced absorption has been based on the
interaction of the electromagnetic radiation with the dipole moment of the atom or molecule.
This means that. . much less than for the induced process. The Integrated Absorption
Coefficient for a Vibrational Transition The treatment in of vibrational energies and vibrational
spectra given Chap. Exercise . .mole liter In view of the simplicity of the model. Calculate the
frequency and the integrated absorption coefficient for the lowest irelectron transition of
octatetraene. when one studies absorption spectra. in Sec. In practice. INTRODUCTION TO
MOLECULAR SPECTROSCOPY is The observed value A is Jav dv . This result can be
readily obtained by those familiar with the general properties of Hermite polynomials. X
secquot cm. . No net absorption of energy will be observed.
and m . the expression for A is hlfi igt lM l The dipole moment. of a heteronuclear diatomic
molecule must be expected to be some not easily determined. In Sec. necessary for the
evaluation of moi. function of the internuclear distance. . lt Mr. the integrated absorption
coefficient A is Rlfj v W are I nudcr liter For the fundamental transition for which the values of
the vibrational quantum number Ao .e displacement coordinate rr. the result was obtained
that the integrated absorption coefficient y l dv Jo A dv see cm mole liter is related to the
transition moment according to the relation A vM SeC Cm mle Uter In the experimental study
of vibrational transitions one usually deals with frequencies v. In terms of q th. It is possible.
be demonstrated. with units of sec . An important result be obtained by a derivation of the
relation between molecular properties and the amount of to v vibrational absorption caused
by the fundamental v transition. with units of cm instead of v. is. however. if it is required
thatquot the dipole moment must be expressed only for small displacements of the molecule
from its equilibrium configuration.JHE ABSORPTION AND EMISSION OF RADIATION will
ever. or even expressed. . the quantity usually referred to as the perma nent dipole moment.
to represent the dipole moment by the series expansion m . where essentially. In terms of v.
is it. . as . IMoi if only the first and x the transition moment p. A nonzero value of moi can.
two terms of Eq.jic and /i d . to be expressed in terms of the displacement coordinate as w e
aqm and where t /iwfc x iu d . however. / gto g g o and i leads to a zero contribution from the
and shows that a constant dipole moment is not sufficient to lead to a vibrational transition
being induced by electromagnetic radiation.ldg iL vr. written to distinguish the reduced mass
from the dipole expressions for moment With these can be set up. are retained.quot aa The
necessary integration can be performed with the tabulated integral i o Xi quot dx ita lt gt One
obtains Wl i . INTRODUCTION TO MOLECULAR SPECTROSCOPY is the dipolemoment
expression quotquot The wave functions for the v and the v states that are involved in the
fundamental transition are shown in Sec. . This leads to the result of o The orthogonality
constant term /j g M l o/. result from the second term of Eq.
has been reported by W. X esu. Brooks and B. . X g. M With this be on one atom of the
bond and model the bond dipole moment is r be on the other Ser r and. rather unsatisfactory
to . One can. one calculates dg/o X . Phys. esu Although distortion of a chemical bond from
its equilibrium distance undoubtedly generally accompanied by a complicated redistribution
of the electrons of the bonded atoms. Crawford J. mole liter this value and the reduced mass
of .Mred. the order of magnitude that is to be expected for values of dn/dq q o can be
obtained by assuming a is constant charge of atom. however.. desired to obtain the . by
means of Eq.A N important result that shows is Equation ties Ma. it is The transposition is
that which is used when dipolemoment derivative from the measured absorption intensity.o /
c l. A From cm. . attempt to express the dipole moment of a bond. in fact.Mred.THE
ABSORPTION AND EMISSION OF RADIATION and with this value for the transition
moment the integrated absorption coefficient A is found to be related to molecular properties
according to inSl V A c l. V. The result obtained for BrCl It is consistent with this expectation.
dq qa This result suggests that values of dn/dq q o should be of the order of magnitude of the
electron charge. d m or /djA dq q. is the how the absorption intensity of a fundamental
vibrational transition related to the proper and dii/dq qa of the molecule under study. F. to be
given in Eq. or a diatomic molecule. since g dr rr . recognize that at r the dipole moment /i is
necessarily is. The measured value of A for the fundamental transition of BrCl at cm for
example. Chem.
With this model. and . Mobs. n is also zero. Types of ft versus r curves that can be drawn
from measured values of fi and dn/dq. It is. usually in the visible or ultraviolet regions. lead to
the accumulation of data that are of con urement . with a Hookes law type of force. however.
The information available. One assumes that the is responsible for the absorption of the
radiation is attracted to the center of the molecule. convenient to obtain here an oftenused
guide to the intensities of absorption bands that result from electronic transitions.
Nevertheless. The Intensities of Absorption Bands Due to Electronic Transitions Detailed
discussion of the changes in the electronic structure of molecules that lead to the absorption
of radiation. including a deduced value of of A. . siderable potential value in consideration of
the electronic structure of molecules. It is at present a difficult matter dn/dq qa from the
measured value to decide whether the curve with a positive value of dn/dq q o or a negative
value is correct for a given molecule. the FIG. lation is based on a simple model for the
behavior of an electron in electron that molecule or ion. will not be given until Chaps.
INTRODUCTION TO MOLECULAR SPECTROSCOPY zero and that at r tially all o. can be
illustrated by the curves for possible dipolemoment versus bonddistance dependence shown
in Fig . since in inert solvents or gaseous media essen bonds are expected to break to give
neutral particles. the measof values of A and the deduction of values for dn/dq a q according
to Eq. but even this determination does not lead directly to a determination of the sign of this
equilibrium dipole moment. g . . The calcur. here assumed to be spherical. The value of n at
the equilibrium distance can be determined.
one can replace d/u/dx x o according to Eq. . This is denoted by / and is denned as the ratio
of the observed integrated absorption coefficient Thus one has to the value predicted by Eq.
To do one introduces the term oscillator strength. by m. .cm sec cm mole liter Ne n n . X cm
mole liter This result this often used as a reference value against which the actual intensities
of electronic absorption bands are compared. ez. . for the integrated absorption coefficient
predicted by this model. and p. l. that many /. since the mass of the electron is much less
than that of the remainder of the molecule. One also finds electronic absorption bands have /
bands with will much smaller values of and in these cases one expects to assign the
absorption band to some quotforbidden transition. With these variations the transition
moments for this simple model of a bound electron are. y. X /ai. dv One very finds. iVe /l. the
mass of the electron. since p. Each cartesian coordinate contributes a dipolemoment integral
like Now. Likewise.c m . mole liter Insertion of numerical values gives the prediction as A Jav
dv is . electronic absorption values near unity. IMoii ImJ S Addition of the equal cartesian
components then gives . however.quot These ideas be enlarged upon in Chap.x ex. one
replaces /tw. and the expression for a. . the result A or A iVe ttf. by e.c wi cm . . that of
Eq.THE ABSORPTION AND EMISSION OF RADIATION is behavior of the electron given by
the harmonic oscillator wave func tions used in the previous section.. e ft ir TOvC Moi
Substitution of this expression in Eq. . Uav digtU.y ey. and so forth. in fact. gives.
. PrenticeHall. chap.. Pauling.quot .quot . Slater. New York. quotIntroduction to chap.
INTRODUCTION TO MOLECULAR SPECTROSCOPY PRINCIPAL REFERENCES .
McGrawHill Book Company. Inc.ll. L. Quantum Mechanics.. . New York. Mc GrawHill Book
Company. Inc.J. C quotQuantum Theory of Atomic Structure. and E. chap.. quotQuantum
Chemistry. . Cliffs. .. Engle wood . S. Pitzer. Wilson. N. . Jr. J.quot vol. . Inc. B. . K.
. interacts with the molecular rotation and for commenting on the effect that any nuclear spin
has. . on the rotational states that exist. as a result of the statistics that are followed. LINEAR
MOLECULES The selection rules for transitions between rotational energy levels and the
multiplicity of these levels have been stated in Chap. the chapter divided into the three
principal topics linear. if present. On this basis. The rotational energies between these and
rotational spectra of general polyatomic molecules are conveniently treated under headings
that specify the relative value of the three principal is moments of inertia of the molecule. and
asymmetrictop molecules. Now the heading will be used principally to introduce a number of
finer features of rotational spectra that were not mentioned in the earlier introductory chapter.
tional states of diatomic molecules and have been discussed in Chap.ROTATIONAL
SPECTRA Some states of the principal features of the energy levels of the allowed rotaof the
transitions . Before proceeding to molecules of general geometry. symmetrictop. simple
situation of linear molecules. . . of linear molecules A number of aspects of the rotational
spectra have already been dealt with in Chap. it is perhaps advisable to look in somewhat
more detail at these two features for the relatively These molecules also provide a is
convenient basis for mention of the fine structure that rotational spectra observed in when
nuclear spin.
the other involves only the distance between the electron and the would be variables. /.
nucleus. M is less of the lowest The Schrodinger equation solution to the rotating linear
molecule problem leads to allowed energies that are identical to those given in Sec.
Rotational Selection Rules for Linear Molecules The requirement that a molecule must have
a permanent dipole moment and that the rotational quantum number can change only by one
unit for a rotational transition to be induced by electromagnetic radiation must first be
investigated. These are where both the J quantumnumber notation appropriate to molecular
rotation problems and the s. . notation familiar angular part of the illustrated in Fig. Those
familiar with the hydrogenatom wave functions that the angular part of these solutions
associated is will recall related to functions solution is known as Legendre functions. . The
problem is. . . . . d. and Table . Solution of this problem will not somewhat intricate and can
be found in any of the standard texts on quantum mechanics. to do this it is only necessary
to investigate the value of the transition moment for the transition under consideration.
generally solved when the hydrogen atom is be given here since it is studied. The
threedimensional nature of the problem suggests the use of the polar coordinate system of
Fig. with the provision that or equal to. to describe the wave functions and solution of the the
dipole moment that appear in the transitionmoment integral. The rotational wave functions
can be obtained by Schrodinger equation expressed in polar coordinates for a molecule
freely rotating in threedimensional space. . p. It turns out that the Schrodinger equation for
the hydrogen into lttgt atom can be separated angular coordinates two parts.
INTRODUCTION TO MOLECULAR SPECTROSCOPY . appropriate to atomic problems are
given. One part involves only the and of the electron considered to move about a fixed
nucleus. as was mentioned in Chap. of course. The wave functions corresponding to some
energy states are given in Table . As was shown in Chap. M and which than. would of these
equations involve only the angles describing the orientation of the molecule as The
correspondence means that the rota tional wave functions that are of importance here are
identical to the hydrogenatom wave functions. set The former part is identical to the equation
that up for a freely rotating rigid molecule which. The solution functions illustrated in Fig.
Each characterized by two quantum numbers which are here denoted by J and can be
assigned integral values.
HI .CM C o i V gt quot o o B e a. a S E o i o r quotlquot.
IT Af amp. / /sin sm J M M . . . .. . ir / I sin /l . cos gt r i S r . A sin x / cos /l ./i. cos or ir ./ e i or
.... . / / sm cos sin . the same as that obtained previously from the quantized restriction. lttgt /
./sin quotquot . momentum Furthermore. INTRODUCTION TO MOLECULAR
SPECTROSCOPY if satisfy the Schrodinger equation only in that equation has the values
wU This result angular is JJ / .iCOS mee lr /i /. I OT . / I sin IT cos / sin sin cgt jr . . average
angular it is momenta from the wave found that Total angular momentum /JJ h Component
TABLE Molecule of angular momentum along an axis Wave r Some of the Functions for a
Rigid Rotating Linear J Af ltP Vi / M Af . if quantummechanical operators for angular
momentum given in Table are used to compute functions given in Table .. . / sin cos cos .
After the the average angular wave functions is is of the states involved are specified. of
appl ied field Angular momentum of rotating molecule KA/T ft/T . The number of angular
momentum components along an axis that can be obtained from a given value of J can be
seen from such a diagram to be J . sometimes convenient to illustrate the angular
momentum components implied by the rotational wave functions by means of a vector
diagram. of The diagrams that result for quantization in terms and are illustrated in Fig. M
Apply the operator for total angular momentum to the J and wave functions. . momen The
first of these the while the second corresponds to the stipulation that the angular tum
component It is in a direction in space be quantized. . The total and component angular
momentum vectors for components. and confirm that momentum is y/JJ lh/ir in these cases.
Angular momentum in direction component . . / . /JJ Exercise . permanent dipole moment of
the molecule is represented by the components along the three axes FIG. the remaining term
that necessary for the evaluation of the transitionIf the moment integral the dipole moment. is
J now familiar total angular momentum stipulation.ROTATIONAL SPECTRA where M . J .
For an applied field reveals five More generally for the Jth rotational level states would be
revealed.
for any orientation of the molecule. that the transition moment for J nonzero while that for the
J to J will be necessarily zero. and will not be given here. requires a more detailed
investigation of the integrals to see that they also vanish unless J Jquotl that is. M and one
characterized by Jquot and integrals JJfJquotMquot Mquot gj n fl is investigated. .
Herzberg.quot of this result is outlined in quotSpectra of Diatomic by G. and the
antisymmetric nature of the dipole moment. t mo sin cos /xo gt sin sin ltfgt mo cos of Table
and the dipoleone can investigate the contributions of the transitionmoment components to
the total transition moment. moment expressions of Eq. The verification of this selection rule
for particular values of J. . are given. such verifications are suggested as exercises. A/ The
deduction JMolecules. Mquot can. Exercise . Deduce from the symmetry of the wave
functions of Fig. be carried out in a straightforward manner. INTRODUCTION TO
MOIECUUR SPECTROSCOPY of Fig. one obtains the M MQ Mo f JM J CQS JMquot gm df
d jMjjfi Jltjquotltjfquot o Jj x o jquot o fw sin sin Wquotquot sin de dtf dfl M Mo jquot /Af cos
jquotKquot sin It will be recalled that the element of solid angle is sin dd dlttgt in polar
coordinates. transition These components allow the calculation as of the total moment
lJquotAfJquotikfquot /JV/JquotMquot l/JlfJquotJlfquot JJtf/quotJfquot It is immediately clear
from Eq. M and Jquot. In this way. as can be verified to J will be from Eq. that the transition
moment for It rotational transitions will be zero unless mo has a nonzero value. if a transition
between a state characterized by J and With the rotational wave functions . by Mx nv y. how
ever.
in a classical picture. the selection rule for J is of prime importance dependent . the energies
of these states depend only by The J the on total angular momentum quantum number J.
When. and understood that they are based tions. the of Table . of the rotational spectra of
generally The treatments other selection rules. can be made apparent if an external field is
applied M to the molecules while their rotational spectrum field is is observed. shaped moleit
is cules that will be presented in the remainder of the chapter will introduce These will be
stated. and the shifting and splitting of rotational spectral lines that results is known as a
Stark effect. indicated here for linear molecules. of these selection rules . it will not be given.
The quantum number M gives the component field. The Stark Effect in Molecular Rotation
Spectra Although the rotational states of linear molecules are characterized two quantum
numbers. due to the various values of for a given value of J. Verify the AJ and the AM
selection rules for J to J transition. next section. An electric used. since the general proof of
this requires a knowledge of the behavior One can. the states with different values of and the
selection rule for M M becomes significant.ROTATIONAL SPECTRA Since the energy of a
freely rotating molecule is. Since the molecule has a dipole moment. as will be discussed in
the in rotational spectroscopy. it can be expected that these different orientations relative to
the applied electric field will cor respond to the different energies. however. degeneracy of
these rotational levels. It is found that the integrals of Eq. of the angular momentum in an
applied is direction. however. vanish unless AM Again. an external electric field imposes a
direction on the rotating have different energies molecule. to different orientations of the
rotating molecule relative to the applied field. only on J. can be found in ref . verify of
spherical harmonics. according to Eq. therefore. that this selection rule operates for
transitions between states described by the wave functions Exercise . and in a Stark effect
experiment states with this direction that of the applied electric The different values of M
correspond. on the same considerations as those Derivations. The calculation of these
energy . or reference to deriva. without derivation.
The derivation of the expression beyond the scope .M . and the calculation of the energy
requires a secondorder perturbation calculation to be made. J levels of the molecule are
responsible for the In Chap. however.IgjS fe JJ JJ J . The effect of the applied field as will be
very small. and this is done in Fig. AJ . are important. in ized Se. These energylevel rules
show up when spectral transitions are obtained for molecules subject to an electric field. The
effect on the energy shifts levels given by these equations is shown for a particular case in
Fig. both selection AJ or. for AM AM absorption experiments. . . not simply done. for the
energy shifts is . INTRODUCTION TO MOLECULAR SPECTROSCOPY shifts that the states
with given J and various values of M experience is. Observation of the number of
components that rotational line splits into when an it electric field is applied is often a great
help in deciding which transition. With these rules as mentioned in the previous section. of
this treatment and is outlined. is. was assumed that a simple if series of equally spaced lines
could be expected is obtained. the allowed transitions can be added to the energylevel
diagram. seen. Now. the unit usually used in microwave spectroscopy. sufficient complexi
ties are introduced into the rotational spectrum of generally shaped . with references. ref The
result of the calculation is that the energy of a state characteris shifted in an electric field of
volts/ cm by by values of J and M where for / . While this is a rotational spectrum of a linear
molecule true for linear molecules.J e g while for / quot P er amp can be converted to
frequency shifts of These energy shifts in ergs cm or megacycles per second.
The effect of a Stark field of . the determination . moment /x The W. X y M X / E u e M X E
so J . E. In addition to illustrating the space quantization of Stark effect splittings
allow.ROTATIONAL SPECTRA molecules that a simple matching of the spectral lines with
the expected pattern is seldom possible. and FIG. Good.Q volts/cm on the rotational energy
levels of HCI. . of often rather precise values of the molecular dipole first such determination
was made by T. measurement with the aid of Eq. volts/cm X . Dakin. effect. W. M l M No
applied field .
. as we will see. Rev.. for the molecule OCS. . CH F CH C CH. in fact. INTRODUCTION TO
MOLECULAR SPECTROSCOPY D. . such of rotational states both the molecular and
nuclearspin angular to the total momentum contributions angular momentum. One finds.Br
CH. PH. oscilloscope pattern of the The J to J line is shown for different electric field
strengths in Fig. NF. made using the Stark effect in rotational spectra are given in Table N
energy level a Calculate the shift in the J . volts/cm.. . . experimental evi dence in the
hyperfine structure of rotational transitions of some moleThe basis and nature of this fine
cules for the effect of nuclear spin. F. multiple of tum. Trambarulo. . of that have been . From
W. K. . Coles Phys. . The angular momentum of a nucleus results from the spinning of the
nucleus and is a characteristic of the nucleus. Smith. . detail of rotational spectra will here is
be only briefly outlined. quotMicrowave Spectroscopy. . Molecular Rotation It M . Some
dipolemoment determinations Exercise . Its introduction to appreciate the many rotational
student intended to allow the spectra studies reported in the literature where this effect
occurs. Gordy. V. . It is quantized in units of TABLE Some Molecular Dipole Moments from
Stark Splittings of Rotational Transitions Molecule Dipole moment Debye Molecule Dipole
moment Debye FC FBr BrCl OCS HCN NO s NH. W. must take into account molecules. . . .
the dipole moment of the OCS molecule. Nuclearspin Coupling has been recognized that the
allowed rotational st ates of a mole yP e cule are those for which the angular momentum is a
/JJ momenangular rotation molecular this to addition In h/. CHF.quot John Wiley amp Sons. .
Inc. and R.I CH CCH BH . b Estimate from Fig. . as a result of an applied electric field of .
some molecules have angular momentum because of the nuclear A complete
characterization of the spin of one or more of their nuclei. .
E Good. volte/cm Frequency . interaction between the orientation of the nucleus and that of
the molecule.e. absorption line of OCS. Cotes. This dependence can be expressed by
introducing a quantum number for the total angular momentum of the system. ROTATIONAL
SPECTRA h/w. Phys. the spin angular momentum of a particular nucleus has one of the
values VlI where I . W. . . megacycles/sec. In such a case the energy of a given molecular
rotation state. there is an energy of inter action the energy of the system will depend on the
orientation of the nuclear spin relative to that of the molecular rotation. on the other hand.
and D.. and. Rev. the molecule will rotate and leave the spinning nuclei unchanged in
orientation. The line for no field centered at about . A molecuFIG. Dakin. f If there is no
coupling.. if the nuclearspin quantum number is designated by /. From T. . i. K. designated by
J. and the total angular momentum of the system is then y/FF h/ir. would be unaffected by
the nuclear spin /. This is usually denoted by F. No electric field volts/cm . If. and the two
markers on each curve are spaced megacycles/sec apart. is The Stark splitting of the J W..
For almost all molecules. JI . . that the J I type terms are not always realized for states with
low J numbers. Something must be said about the source of the interaction between the
orientation of the molecule and that of the nuclear spin. therefore. of course. with by
attributing the extent of coupling.// with the provision that the total angular momentum cannot
be negative and. have a charge asymmetry that gives them quadrupole moments. Those
with / greater than . this interaction occurs between the electric field that the molecule exerts
at the spinning nucleus All nuclei lack dipole and the quadrupole moment of the nucleus.
would be imposed on the nucleus. however. F J hh / / / / / / / / . The hyperflne spin / splittings
schematic of rotational levels resulting from coupling a nuclear s.. Although s orbits.
moments... INTRODUCTION TO MOLECULAR SPECTROSCOPY lar rotation state with a
given value of J can then lead to states with various values of F according to . It is. The
splittings of the J levels that result from a nuclear spin of f are illustrated schematically in Fig.
Some inter esting results have been obtained FIG. and no directional effect Again. a
completed p shell has spherical symmetry. bring their electrons close to the nucleus. F J I.
they are ineffective in orienting the nucleus because of their spherical symmetry. the
exceptions being odd molecules such as NO. the directional asymmetry is of the electric field
of the molecule quot acting on the nucleus that now important. The next most effective
electron orbits in producing an electric field at the nucleus are the p orbits. however.
Simmons.. symmetric. W.. with the extent to which the p subshell molecule . Considerable
complexity can be introduced into rotational spectra by this effect. provided by the molecule
DCN. Rev. is The absorption spectrum from W. Gordy. Frequency F FIG. The J gt y i ab F F l
sorption and transitions for DCN showing the effect of the nuclear spin / of the nitrogen
nucleus. Rev. E. and Gordy Phys. if the nucleus is embedded in an asymmetric electric field
of the molecule. in which The energy levels and J gt transition. of the atom is more or less
complete in a particular The spectral effect produced by the coupling of J and I to produce
like that a resultant of Fig. F can be understood with an energylevel diagram selection rules
and the AJ AF . Phys. In summary any molecule containing a nucleus with a spin greater
than will. are shown in Fig. No detailed treatment of this quadrupole coupling be attempted in
the following treatments of fine structure will. show such hyperfine structure due to
nuclearspin simple illustration is A N has a spin of unity. . observed by Simmons.
ROTATIONAL SPECTRA for a given nucleus with a given quadrupole moment.. therefore.
and W. / Fl . Anderson. interaction.and asymmetrictop molecules. J.Anderson. .
It was found. as by the is substitu x. important weights of of a i. gt by electromagnetic
radiation could be deduced in the on the basis transition. gt . v. the very simple system of a
particle in a box were either unchanged or changed in sign the origin.e. in Sec. this function
is inverted f/ the coordinates involved in at . does not alter . and the result of this reflection
depends on the particular wave function that describes the electronic state of the The nature
of the many different electronic waye functions that occur in the ground and excited states of
diatomic molecules will be considered in some detail in Chap. through the origin. . that the
wave functions for. . when the wave function was subjected to an inversion through On the
basis of this behavi. when y was replaced by y. that the selection rules for the transitions
induced lt operation.e. Mvfr where the subscripts e. by deg about an axis perpenand then
performing a reflection through rotation axis and passing through the interfirst Since the step
in this process leaves the it coordinates of the electrons unchanged with respect to the
nuclei. The Positive and Negative Character of the Wave Functions of Linear Molecules It
has been shown. ltIgt . vibrational. We can now investigate the effect of inversion on each
factor and then combine these results to see the effect on Mta Inversion of the electronic part
of the wave function can be accomi. depending on their behavior with regard to this
furthermore. be assigned to the wave functions that describe diatomic molecules..
considered in the following section. For the present it is enough to molecule. The second
step reflects the electronic wave function through the designated plane. z. . in connection
with selection rules and with the statistical rotational levels. plished by rotating the entire
molecule dicular to the internuclear axis a plane perpendicular to this nuclear axis see Fig.
INTRODUCTION TO MOLECULAR SPECTROSCOPY . but adequate separated function
lAtatal for the present argument. diatomic molecule when y. y. of the plus or minus character
of the states involved We will now see that plus and minus signs can.i are reversed. . and r
stand for electronic.. and rotational. We now when tion all consider the effect on the total
wave function t. in an entirely This positive and negative character is analogous manner.
symmetry the allowed states were labeled with a plus or a minus. ior. The effect of this on
tota i best deduced by writing the approximate. z x.
must now be conThe polar coordinate equivalent to the inversion implied by x Vgt z x V.
Most groundstate wave functions. . r . will be discussed in Chap. . shown in Sec. either basis.
j/ The vibrational component v of the total wave function. With the recognition that
FIG.ROTATIONAL SPECTRA concern ourselves with the electronic wave functions that
occur in the ground state of almost all diatomic and linear molecules that have even numbers
of electrons. It follows that fa will also be unaffected by an inversion through the origin since
this operation leaves the magnitude of the internuclear distance as unchanged. is obtained
by replacing r. discussed in Sec. Electronic functions that are symmetric with respect to such
planes are labeled with a plus sign. fgt e we conclude that an inversion of through the origin
leaves unchanged. and by r. . and The effect of this inversion on fa can be seen either
analytically by this replacement in the expressions for fa given in Table or diagrammatically
from the wave functions depicted in Fig. while for odd values sign. . On gt lttgt ltjgt. one sees
that for even values of unchanged by inversion. . ltp e therefore. are are of this type. on the
magnitude of the internuclear distance. and one finds that most groundstate electronic wave
functions fact. This common groundstate function is positive both above and below any plane
passing through the internuclear axis. and fa are unchanged by axis by deg around an and
reflection through a plane perpendicular to the axis correspond to inversion through the
origin. The remaining factor fa. After rotation Initial position . . of J the function fa remains J
the function fa changes inversion. depends. in The basis for the use of the letter For these
often encountered states. Illustration that rotation fa. sidered. designed as states.
. of the rotational levels for those molecules with a S ground electronic state. such as CO or
HCCH. The positivenegative character. INTRODUCTION TO MOLECULAR
SPECTROSCOPY for the molecules considered here. We are concerned now with the effect
of an interchange of identical nuclei on the wave function of the molecule. Exercise . .
Mention of this operation for the two electrons of the Behavior of . is perhaps best known to a
chemist in terms of an interchange of electrons. Deduce that the interaction of a molecular
dipole with electromagnetic radiation can lead only to transitions of the type . consistent with
that of of the rotational with electromagnetic radiation in the same way as illustrated for the
This result can be recognized to be A J obtained from a more functions. . ltgt . This feature.
detailed analysis wave We now will consideration and will proceed to another important type
of symmetry make use of the plus and minus assignments obtained above. An immediate i gt
result of this assignment the selectionrule result . one deduces that to tai behaves in a is
manner dictated by fn as shown gt in Fig. which must be considered whenever a system
contains like particles. SymmetricAntisymmetric Character and Statistical Weight of
Homonuclear Linear Molecules Another rather nized different when the molecule under type
of symmetry behavior must be recogconsideration is a homonuclear diatomic molecule with
identical nuclei. show ing the behavior of totai on inversion. that can be deduced for
interaction onedimensional example in Sec. or a linear molecule with identical nuclei. otill on
inversion FIG.
to the orbital functions.e. and one should The electronic write the more complete expression
tfWl lM. It follows that a suitable antisymmetric combining the orbital and spin factors to give
can be written by lllal this /l One deduces by means the familiar result. pi when the electrons
are interchanged.. the total wave function contains no vibrational and rotational factors and
might be written as lt/w .ROTATIONAL SPECTRA helium atom may clarify the parallel
treatment that will be given for like nuclei of molecules.pia Each electron has a spin quantum
number . and since the angular momentum associated say by the symbol a. designated fgt n
by the symbol p. with this spin can be directed one way. given immediately by the . two elec
lslls Thus far our above discussion of the helium factor of the spin of the electrons. . clearly
unchanged by the interchange lslsl. Bin will go to . i. designated or the other way. wave
functions. one can write various possible spin functions for alj as a/Jl a/Sl all al These have
been written so that f. properly adjusted Thus one would write lslls for the increased . one
can write a suitable approximate expression for . or any atom. For the helium atom.. To
attach correct spin functions. atom by regarding each electron as behaving according to Is
hydrogenlike nuclear change. one must make use of the fact that the behavior of if electrons
in nature can be accounted for only wave functions that toui are antisymmetric with respect
to the exchange of any pair of electrons are used. describes only the orbital atom has lacked
the important wave function given behavior of the electrons. from these four possibilities. for
the helium Furthermore. is of the This function trons.
whereas if the spin quantum number is zero or integral. the particles. lyzed..said to follow
BoseEinstein statistics. The first step of this process is nothing more than the symmetry
operation treated in the previous section. reflection through a plane containing the molecular
axis. electrons and through the origin. We therefore on the total wave function.
antisymmetric. or must be changed in sign. . are labeled. with a g for the German gerade. for
states. those of the second type. The property plus and minus assigned to electronic wave
functions in the previous section depends on states. the wave function must change sign.
way the total for the interchange of like of the spin of the nucleus. i. NaCl. Such behavior can
be considered even for molecules like HC. wave function for a molecule must behave. or any
particle. as is that of the electron. if the electronic state is . and so forth. Particles of the first
type.. and then of the inversion of the electrons back through the origin. The net effect of the
twostep process is. such as the lowest particle inabox state. Some electronic wave functions.
the center of symmetry of the molecule. dealt with in another connection in Sec. are
symmetric with respect to this inversion through These symmetric electronic which constitute
the most often encountered ground states. must have opposite spins.e. electrons and
protons are the most often encountered examples. i. The effect of an interchange of like
nuclei on toU will now be anaThen the nuclear spin. that the electrons of the helium atom.
are. INTRODUCTION TO MOLECULAR SPECTROSCOPY usual form of the Pauli exclusion
principle. are said to follow FermiDirac statistics. . will be introduced in a way so that the
complete wave function behaves in a manner suitable to the statistics followed by the nuclei.
If the spin quantum number of the particle is half integral. or linear. The second step of the
process has an effect that depends only Vgtn the electronic wave already know that its effect
function. nuclei. It is informative to analyze the effect of an interchange of identical this nuclei
of a diatomic. and so forth. such as Of with zero spin andH with unit spin. is plus or minus
depending on whether J is even or odd. in a manner dictated by the nature It turns out that
the spin of the nucleus. the wave function must remain unchanged. molecule by imagining all
exchange to occur as a result of an inversion of nuclei. with respect to an exchange of
identical particles. is the index which determines whether the wave function must be
unchanged. Those electronic states which are antisymmetric with respect to inversion
through the center of symmetry are labeled with aw for ungerade. Note that electronic
functions only have g or u properties when there is a center of symmetry as there is in H
CO.e. in In a similar its ground state. symmetric.
i. For molecules. In a the like nuclei have . this Symmetric or antisymmetric behavior with
regard to by an s or an a. we have found the behavior of the total wave function. as a result
of this exchange. rise to rotational The rotational energylevel diagrams for linear molecules
with weight factor due to the nuclear spin / nuclei. A U .. no spin function is exchange is
indicated . The / statistical is shown.e.a ra . the above treatment is complete. tu / a i y a ft
Ortho f Para a a ft rtho ft Para a a ft ft quot Ortho Para O CO type H type a b . gt . that only
the states with even values of J. be observed in absorption like rotational spectra because
such symmetric molecules with identical nuclei have zero dipole moments and do not give
FIG. in b they have if. exclusive of the effect of nuclear spin. only the states with even values
of The absence of the odd levels cannot. since the twostep process equivalent to the
exchange of the identical nuclei. such as O u C that have identical nuclei with zero spin. It
follows. of course. since nuclei with zero spin behave according to BoseEinstein statistics
which require sym metric behavior as a result of nuclear interchange. The rotational
energylevel diagram should show. Thus. .ROTATIONAL SPECTRA of the g type. involved in
the total wave function. For u states this conis clusion is reversed. to leave the total of J and
to change its sign for wave function unchanged for even values odd values of J. J can exist.
as illustrated in Fig.
the odd J levels have a statistical weight due to the spin function of while the even J levels
have a levels odd J weight of . spectra. three symmetric spin functions aa. As electrons of
the helium atom. Again this alternation in multiplicities. A . cannot be observed in rotational
spectra but. illustrated levels odd J in Fig.lMva/ /a and for the o states. transitions between s
states and a states. nuclei with a spin quantum number of follow FermiDirac statistics that
require a total wave function which is antisymmetric with /a respect to the interchange of the
nuclei. the have an additional multiplicity factor of while this factor is only for even J levels.
Now states with both even and odd values coupling f/ of toU J can be brought to the required
antisymmetry by suitably with spin functions. in addition to the J multiplicity that all levels
have. Furthermore. which have J odd. confirm the absence of the odd /levels for such
molecules. which are those having nuclear spins. the absence of these levels which be
noticed in rotationvibration spectra that will be studied in Chap. for molecules like HCCH is
that will be discussed in Sec. One should notice that the selection rule s lt ated. as .
acetylene have zero spin and can here be ignored. an alternation in the intensities of the
rotational components of rotationvibration absorption bands observed. by any mechanism.
which have opposed nuclear very strong selection rule exists which prohibits. which have J
even. Thus. Furthermore. and the para states. which leads to an alternation in populations.
Let us now consider molecules. antisymmetric functions for the s states. The C nuclei of will
H CC for the //. . Thus ortho and para states cannot be interconverted. gt o. unless the
molecules are dissocialt s. . for g electronic functions. that rotational will Raman not be
discussed here. should be mentioned. however. as Since three spin functions are available
to provide the correct for the symmetry whereas only one spin function makes the total wave
function antisymmetric when J is even. INTRODUCTION TO MOLECULAR
SPECTROSCOPY It spectra. such as and H H. a/ two and and one antisymmetric function a/
/a can be written. . that have like nuclei with spin quantum number . Thus one writes.
recognize that the analysis given above has led us to One should parallel spins. recognize
the ortho states of RYlike molecules.
Benzene and the methyl hal Figure shows the principal axes of these two examples.
symmetric A typical example is cyclo butane. those with U In to. A gt l B . . FIG.
..ROTATIONAL SPECTRA s lt which applies to homonuclear molecules. which has its
unique axis perpendicular to the plane of the carbon atoms. the s and a states. . however.
elimdiatomic any gt applies to which inates the possibility of pure rotational spectra of
diatomic homonuclear gt a. oblate symmetric lopj .e. Symmetrictop molecules are defined as
those having two equal principal moments of inertia. with even and odd J values. prolate
symmetric top. and the two axes with equal moments of inertia pass through either the
carbon atoms at opposite corners of the square or through the opposite sides of the square.
one finds that all / levels can be combined with spin functions to give total symmetry required
by the nuclei. i. I lt / . ides also are symmetrictop molecules. nuclei with spins of and so forth.
statistical weights. have different SYMMETRICTOP MOLECULES wave functions with the
Again.Most symmetrictop molecules that will be treated belong to this class because of a
arrangement of some of the like atoms of the molecule. t molecules. Examples of
symmetrictop molecules. A similar treatment can be applied to molecules containing like .
The cules will classical first treatment of the rotation of such symmetrictop mole be given so
that the quantized angular momentum restricA statetion can be applied to give the allowed
rotational energy levels. Again. the third being different from these two. The unique moment
of inertia is generally represented by IAl while the two equal moments of inertia are I B Ic.
and lt molecule.
. For a symmetrictop molecule I B Ic. correct quantummechanical solution for symmetrictop
molecules obtained by further assuming that the component of the angular momentum about
the unique axis is quantized and that this quantization is is K K X gt gt gt gt J lt gt where the
plus and minus values correspond to the possibility of clockwise and anticlockwise rotation
about the unique axis. These quantum restrictions lead to and Pi P c gt jj iAyAy . ft that the
total angular quot The energy levels of a quantummechanical symmetrictop system can be
obtained from Eq. ilA Ibltb Icltc IB . with the quantum number J. is given by VJJ The J . and
this general expression becomes quotamp. The rotation of a system with and Ic is written
classically as AAA IA . to be The Rotational Energy of Symmetrictop Molecules As with linear
molecules. . . As for linear molecules. the allowed rotational energy levels classical can be
obtained from the equations by the appropriate angular principal momentum moments e
restrictions. . of inertia IA Ib. . INTRODUCTION TO MOLECULAR SPECTROSCOPY ment
of the selection rules will then allow the spectral transitions drawn in and a comparison with
observed spectra to be made.c as. and p c are the angular momenta about the three axes. .
it is assumed momentum is quantized and. where pA Pb. . .
one would also have found that the quantum number J determines the total angular
momentum of the system while the number K determines the component of this angular
momentum along the unique axis of the molecule. corresponds to the assumption of a rigid
molecule and that additional terms must be intro duced to allow for centrifugal stretching.. is
positive. ROTATIONAL SPECTRA These relations convert Eq. and for a given value of J the
higher the value of K. . The significance of Eq. . . rgt and cmquot It is again convenient to
introduce rotational constants B and A defined by cm B iJcT B A sn cm With this notation.
and for a given J value the higher the value of K. . By the application of the appropriate
quantum mechanical operators. the shifting of the energy levels due to this effect always very
small. the lower K . inertia IA less J and K. Two different Molecules that have the unique
moment of than I B Ic are said to be prolate molecules. . J This expression gives the same
set of energy levels that would have been obtained if solutions to the Schrodinger equation
for a rotating symmetrictop molecule had been sought. An example is is For this class of
symmetric tops. to the quantummechanical result and tJK amp cIb . For such molecules the
coefficient of ple is CH CN. . and K . the coefficient of negative. greater than I B Molecules
that have the unique moment of inertia IA Ic are said to be oblate molecules. is As for linear
molecules treated in Sec. . . It will only be mentioned here that Eq. . benzene. is best
illustrated by a diagram of the energies that are predicted for various values of situations are
recognized. An exam in Eq. . I EK . the higher K the energy. . the energies of the allowed
states of a rigid rotating symmetrictop molecule are given by ijk with BJJ J .
. the part of the selection rule that corresponds to an increase in the energy of the system
and is appropriate is AJ The AK fact that electromagnetic radiation cannot induce a transition
between different values can be understood from the fact that. . . . CN CH methyl CC . . .. or
unique. Microwave Spectroscopy. New York. Smith. Trambarulo. K K Exercise . . rotation
about the figure axis contributes K no rotating dipole moment. Thus all levels with are doubly
degenerate. CC sCH CC NH .quot . NH. John Wiley V. inertia. CH CH CH . axis of the
molecule. The energylevel diagrams for these two cases are shown schematically in Fig. R. .
It should be pointed out that the occurrence of in Eq. INTRODUCTION TO MOLECULAR
SPECTROSCOPY the energy. as a squared term leads to the same energy for a state with
positive or negative values of K. Selection Rules and Spectra of Symmetrictop Molecules
dipole Most symmetrictop molecules that have a this dipole moment have moment in the
direction of the figure. AK AK for Z AJ for K For an absorption experiment. From W.F CH C
CH Br CH CN HCH HCH HCH HCH HCH CCC CH. Gordy. .CCH CHC. . CF CC CBr CC . .
and F. Inc. for the molecules considered here. and calculate the three principal moments of
Also calculate the rotational constants and plot to scale the first levels. few rotational energy .
Some Molecular Structure Results for Symmetrictop Molecules Molecule Bond angle Bond
distances CH. W. . CH . TABLE This component of the rotation cannot. CH . In such cases
the selection rules for the absorption or emis sion of electromagnetic radiation are AJ and .
HNH amp Sons. locate the center of gravity Given the geometry of the CH C molecule in
Table .
to in u V IT gt ogt J J J JJJJ o o E . Q. t Q .f. c e gt . o O E o IS IO J gt a c T gt o Pgt OS E o
C SI .
F. however. Part of the J The diagram R. Inc. Trambarulo. W. therefore. The alterna
FIG.quot D. the centrifugal distortion effects are resolved. is transition of CF CCH showing
the effect of centrifugal centered at about . . Gordy. distortion. . to the expression v BJ . for a
symmetrictop molecule was found a linear molecule. From G. f.. megacycles/sec. Anderson.
. Inclusion of centrifugal stretching effects leads. J. From . INTRODUCTION TO
MOLECULAR SPECTROSCOPY I KAK i i l v vl/ l A/ z z j i I FIG. and the value of K does not
change. therefore.J. the absorption spectral frequencies are obtained as BJ for is cm The
same pattern as expected. will provide a distinction between linear and symmetrictop
spectra. v selection rule AJ and the energylevel expression of . Sheridan.D KJ KJ If DjJ l for
the transition frequencies. Phys. Princeton. Rev. the splittings indicated in Fig. interact with
the electromagnetic radiation. Herzberg. I I I l transition absorptions The effect of centrifugal
distortion on the rotational of a symmetrictop molecule according to Eq. and W.. . N. With the
Eq. The details of part of the J gt transition of CF CCH are shown in Fig.. Van Nosfrand
Company. quotInfrared and Raman Spectra.
such a procedure requires the assumption that bond lengths and angles are independent of
isotopic substitution. siderable accuracy. of course. Again it is make use of As we will see.
The rotational energy of an asymmetric system can be expressed by the general. to provide
data on the internuclear distances between the heavy atoms and to supplement these data
with the momehtof inertia result from the rotational spectrum. many structures have been
determined with conof these structures are listed in Table . fix the positions of the hydrogen
atoms of a molecule. diffraction data. Another procedure is to make use of different isotopic
species to obtain further momentofinertia data. measurement of the frequencies of the rota
tional transitions of a symmetrictop molecule leads only to information about the two equal
moments of inertia about the axes of the molecule perpendicular to the figure axis. classical
formula Va A . Some ASYMMETRIC MOLECULES A molecule with three different It will
moments of inertia is known as an asymmetrictop molecule. . i B Vl c . In this way one can
often deduce additional bond length and angle data and. while this would undoubtedly be
true to a very good approximation if the atoms were at their equilibrium positions. P . Results
from Analyses of Rotation Spectra of Symmetrictop Molecules Accordingto Eq. This one
determined quantity is. the rotational spectrum of such a molecule are expressed with much
more difficulty than was encountered for linear and symmetricThe Energy Levels of an
Asymmetrictop Molecule top molecules. . however. be apparent that the rotational energies
and. In spite of the inadequacy of the data provided by the rotational spec trum of a given
molecule.ROTATIONAL SPECTRA tion in intensity of the different statistical K components
is due to the different weights of the levels involved. . insufficient for the determination of the
bond lengths and bond possible to angles of a polyatomic molecule. it introduces some
uncertainty when data for the v states are used. for example. either xray or electron
diffraction. therefore.
is drawn at the left of the diagram. although the problem is one of some difficulty. which has.
The nature of the energy pattern can. One can now schemat ically connect the energy levels
of the symmetrictop patterns to show what would happen if the moment of inertia along the B
axis were grad . with A B and C . J for the is lowest energy of the set to J for any but the
lowest few energy levels of an asymmetrictop molecule. even for an asymmetrictop moleA
difficulty cule. closed general expression available for No at the right of the diagram.
therefore. be illustrated by a diagram of the type shown in Fig. the J states J. . however. We
are brought to this result by the fol lowing statement. in characterizing these states arises in
the asymmetrictop case because no quantized component of this total angular momentum.
no available to characterize the rotational energy states. This quotadiabaticquot argument
shows that. its given integers going from the highest energy. usual agreement to have the
Notice that this procedure violates the A axis unique. with A and B C . i. it quantum number
other than J As for symfor each value of metrictop molecules. K . K is not a quotgoodquot
quantum number for an asymmetrictop molecule. . for example. INTRODUCTION TO
MOLECULAR SPECTROSCOPY Conversion to a system which obeys quantum mechanics
requires the stipulation that the total angular momentum. characterized A small change of the
moments of inertia from a sym metrictop system. the / states will still exist for each value of J.
however. is quantized according to the relation Total angular momentum y/JJ Unlike the
symmetrictop case. and for each value of J it is found that there are J solution functhere is
tions. Solution of the Schrodinger equation for an asymmetrictop molecule has been
accomplished. it is customary to keep This index t is of them by adding a subscript to the J
value. no component of this total angular momentum is is quantized. can be expected only to
. . Since these J solutions for a given not characterized by quantum numbers.e. shift the
energy levels and would not eliminate any of the allowed states. by . for example. The
energylevel pattern for an oblate symmetrictop molecule. . for a given value of J. . ignoring
nuclearspin angular momentum. each with J are track own energy.. must be expected that J
there will be J states. and the corresponding symmetrictop diagram for the prolate
symmetrictop molecule.
rotors. . quotMicrowave Spectroscopy. Prolate symmetric top Asymmetric top symmetric top
. Oblate New York. and the index t can be entered as J value as shown.ROTATIONAL
SPECTRA ually changed so that B varied from the value of to the value of . sugt. Inc. Smith.
are drawn. the interconnecting lines of Fig. W. The labeling of the asymmetrictop energy
levels in Fig. V. a subscript to each The J value of any line remains unchanged. Gordy. gests
an alternative to the use of One can notice from that figure that the value of t is related to the
K values of the symmetrictop moleenergylevel pattern for asymmetrictop FIG. In this way.
Trambarulo. F.quot John Wiley amp Sons. A correlation diagram illustrating the From W. and
R..
k lies and therefore A gt B between and . The expressions that have been obtained by B. . .
molecules of various degrees of asymmetry. Using the expressions of Table several
correlation lines of Fig. Phys. Physik. as Fig. applicable also to asymmetric tops. For states
with higher J values. . See. k . W.quot A convenient K quantity is the asymmetry parameter
defined as BUA C A C C are the rotational constants where A. Hainer. encountered with
symmetrictop molecules. for example. . energies have been tabulated for Chem. prolate
encountered. . while for a prolate symmetric top with lt Ib . Ray Z. and P. and Structure
Results for Asymmetrictop Molecules The selection rule is AJ . readily verify that for an
oblate symmetric top with Iagt Ia as Ib Ic C. Hainer. Both types of notation may be . since. k
C. necessary to retain all three possible changes in J shows. even in absorp tion
spectroscopy. King. Phys. k and therefore A ltB C . as J. S. and Ic. k it is convenient to have
an abscissa scale to show the quotdegree of asymmetry. For quantitative treatments of
asymmetrictop molecules.k Thus a level with the with the K notation. Spectra.
INTRODUCTION TO MOLECULAR SPECTROSCOPY cules according to T Kprol. B. R. .
the energy levels are not necessarily ordered . such A B .. Cross for the energy levels of
some of the lower J states of asymmetric molecules are listed in Table . M. plot to scale the
first Exercise . Selection Rules. C. . tops. Chem. and by G. J. K notation as Jk . and A JV
One can cm etc. . it is Now. For all asymmetric for the most asymmetric case. or with the
notation is i.te KobUte With t this recognition one can identify an energy level with the
notation oblate . King. and Cross J..
The this work is the splittings of the the J and and the relative intensities of the components
are t numbers involved in the transition.B B C . to a corresponding complexity in the
observed rotational spectrum. A B C VA . VAA . The most helpful aid in when a Stark field is
applied. i. ViB .i o B C VB BC A IB C A B C AB C VB A A C s A CA B C A CA B C C C .
moment of the molecule along the The selection rules themselves. S i i o.CA .C A .BB . The
complexity of the energylevel diagram of an asymmetric top leads.C A .C B IOC . Selection
rules for the t values have also been given.B B C VA .CB . A A A m A A iC B C .BB .
difficulties characteristic of In spite of the encountered in analyzing the rotational spectra of
asymmetrictop molecules. quite a few molecules have been TABLE metric Tops Some
Closed Solutions for the Rotational Energy Levels of Asym Jku iKlti Energy ero t.A . Ooo lio
ln loi A B A C B C M i .C B bC .. VA Ci .CB .C .ROTATIONAL SPECTRA in the order of J.C
A . with the selection rules for induced transitions.C A .BA .C B C V B .C . .e.BB A B C VB .C
B IOC VA . and their derivation.C A .C B C .BA A A A A A S .B A .Cfi . are sufficiently
elaborate so that it is adequate here merely to indicate that they are treated in ref.CA .By A
.B A . These depend on the components of the dipole directions of the principal axes.A .CB A
B C VA .C B C . an energy level with a higher value of J may occur at lower energy than that
corresponding to a lower value of J. V A .A . The assignment of each of the observed lines to
the energy levels involved in the transition is now a problem of some splitting observed lines
difficulty.B A . V B . VA .BB .C .
W. Smith. Smith.. . CHCI S HCH CCH SO OO CH / CH HCH COC S FSF S F . Princeton.
Inc...quot amp Sons.quot D. N. Van Nostrand Company. G.quot D.quot John Wiley amp
Sons. W. . .. energy studied and the observed transitions identified with the participating
When this is done.. CH . Herzberg. quotInfrared and Raman Spectra. CC . Trambarulo
quotMicrowave Spectroscopy. Herzberg. CC . V. and R. Gordy. CH . or the tables of King.
Inc. F. Inc. V. New York. John Wiley . Some of the results that have been obtained from such
studies are shown in Table . INTRODUCTION TO MOLECULAR SPECTROSCOPY TABLE
Some Molecular Structure Results for Asym metrictop Molecules Molecule Bond angle Bond
distance S . From W.J. . especially moments These when supplemented by isotopic studies.
Princeton. Trambamlo. G. and Cross to deduce the three rotational of inertia of the molecule.
W. one can use the equations of Table levels. . F. Van Nostrand Company. . PRINCIPAL
REFERENCES . SO . Hainer. New York. constants and thus the three three data. Gordy. N...
SF . Inc.J. quotSpectra of Diatomic Molecules. CO . . provide a good basis for the deduction
of the structure of the molecule. quotMicrowave Spectroscopy. and R.
The Number of Independent Vibrations of a Polyatomic Molecule If a molecule containing n
atoms is imagined to have these atoms held together by extremely weak bonds. of solutions
and pure In the follow ing chapter. nearly independent individual atoms. . in fact. For actual
molecules the n cartesian coordinates are not. beyond imposed on an atom. is . the motions
of the system might best be described in terms of the motion of the n. or on the molecule.
This theodevelopment provides the background for an understanding of the liquids
containing polyatomic molecules. which show the effect and rotational energies. and rotation
of molecules will be and the absorption spectra of gases. Each of these n coordinates
represents a degree of freedom of the system in that any arbitrary displacement and velocity
can be given along each these w. of changes in both vibrational . like that of Fig. will be
studied. One can recognize that if the bonding of the hypothetical molecule with very weak
bonds were to be gradually strengthof these coordinates. could be described in terms of the
w cartesian coordinates. absorption spectrum in the infrared region.THE VIBRATIONS OF
POLYATOMIC MOLECULES An introduction to the classical and quantummechanical nature
of the vibrations of polyatomic molecules will retical now be presented. there will be a total of
n such coordinates required for a description of the set of the n atoms. If three cartesian
coordinates are used to describe the position and motion of each atom. that Any additional
displacement or velocity. simultaneous vibration dealt with. very convenient.
For any molecule there furthermore. and. a particular set of coordinates. For such
molecules. translational It follows that if three and three rotational degrees of freedom are
recognized. . The Nature of Normal Vibrations and Normal Coordinates A collection of many
mass initial units interconnected by springs can be expected to undergo an infinite variety of
internal motions. internal degrees of freedom can be further described. as the previous
section indicates. which are very important in vibrational problems. to use three coordinates
to describe the motion of the center of mass of the molecule and. and there remain m
internal degrees of freedom. one will see that none of the w degrees of freedom is destroyed.
The nature of normal coordinates. independent One likes. independent internal coordinates.
It is now necessary to see if these n or. If one imagines a gradual strength ening of the bonds
between the atoms as one goes from a set of inde pendent atoms to the rather firmly held set
that constitutes a molecule. which especially convenient for the description of the vibrations
of a system. can be illustrated by a If Consider a particle of mass m held by springs. The
degrees of freedom are merely reclassified. . INTRODUCTION TO MOLECULAR
SPECTROSCOPY it ened. molecular rotations can occur only about the two axes that can
be drawn perpendicular to the molecular axis. for example. The number of internal degrees
of freedom of a molecule corresponds to the number of ways in which the atoms of the
molecule can be given independent displacements and velocities relative to one another.
called is normal coordinates. in accordance with these n coordinates constitute vibrations of
the molecule. three coordinates to describe the rotation of the molecule about this center of
mass. as will be shown in the following section. of the molecule rather would become
increasingly important to recognize the existence than to treat the system as consisting of n
rather particles. three overall translations and two overall rotations are subtracted from the
total of n degrees of freedom. depending on what system. for linear molecules. the number of
vibrations of the molecule. They are. or vibrations. there will be left a total of n degrees of
freedom which must be accounted Displacements and velocities for by internal coordinates
of the system. there are n or n is. as in Fig. simple example. for nonlinear molecules. If the
molecule is linear. displacements are given to the particles of the The general motion of such
a system can be analyzed in terms of a set of internal coordinates. n .
arises only from a displacement in the x direction and. . x and y coordinates are set up as
indicated. One can see that the potentialenergy expression of Eq. Kv can The motion of the
particle now be deduced by applying La granges equation to the coordinates of the system.
nate. A twodimensional. can be deduced from the potential and kinetic energies which are
written as U iampa ik yy T mx ltmy where the springs have been assumed to obey Hookes
law and to have and kv It should be noticed that the spring system. is such that the restoring
force in the x direction force constants k x . forces in the y direction only from y
displacements. for small displacements. the motion of the particle. oneparticle spring system.
similarly. k and /.THE VIBRATIONS OF POLYATOMIC MOLECULES in the plane of the
spring system. the equation For the x coordi dtdx dx gives mx This is fcjX the now familiar
differential form which indicates vibrational FIG. leads to this result by forming J jr v.
therefore.e. consist of one atom of the system. will execute the displacement coordinates of
more than The normal coordinate will.. lead to Displacement in any other simple sinusoidal
vibrations along this axis. Lagranges equation gives my kyy which leads to the solution y Ay
sin wv y t ltp v where v / displacements along the x or y axis will. therefore. The x and y
coordinates allow the motion simply described. and will simultaneously apply. i. In such
cases a normal coordinate will. Similar normal coordinates and normal vibrations will exist for
more complicated vibrating systems. and a displacement of the system according to the
normal coordinate mil lead to a simple motion in which all the particles move in phase with
the same frequency and. often For some systems. and such motion is said to be a normal
vibration or Motion of the normMl mode. . system along a normal coordinate consists of
simple harmonic motion. in general. if Hookes law is applicable. In terms of this coordinate.
as we will see. that can be interpreted in motion direction will lead to a complicated Initial
terms of sinusoidal components in the x and y directions. be a complicated combination of
atomic displacements. the vibrational motion of the system will be simply described.
INTRODUCTION TO MOLECULAR SPECTROSCOPY motion with the displacement as a
function of time given by x Ax sin nvJ ltp x where T P Similarly. and the motion is described
when the appropriate values of A x and A v for the displacement are used. Eqs. however.
and they are the normal of the particle to be most coordinates for the system. for the y
coordinate. it will be necessary to modify this description of normal coordinates so that it is
lead stated that displacement according to the normal coordinate will simple harmonic
motion.
described by an awkward coordinate system initially such as might be used the directions of
the springs were not . . side by side. therefore. anticipate the problems that will arise with
molecular systems where one generally does not initially know what coordinates are the
normal coordinates and. . Let us suppose that the spring system of Fig. the allowed energies
for the two normal modes treated quantum mechanically. The spring system of if Fig. can be
recognized by considering the restoring force that would act for displacements along the
coordinate axes. M Calculate the classical natural frequencies of this system. What would be
the wave numbers of the radiation absorbed in the fundamental transia b Draw a diagram
tions of this system the It is informative to consider also to a description of the vibrating
system of Fig. values that are of the order of magnitude encountered with chemical bonds. k
x X and k y X dynes/cm. The axes relative to this orientation of the system are labeled x
difference and y as in Fig. The simple calculation that is necessary will. in fact. and that of
Fig. did know the convenient directions in which to choose coordinates. what atomic
displacements will lead to normal modes of vibration. Restoring force components and
resultant for a displacement along x X . . in way in which one would proceed if one did not
know not advance the directions along which the springs lay and. known. The essential
between the axis choice of Fig.THE VIBRATIONS OF POLYATOMIC MOLECULES t all to
motion for which the cartesian coordinates of the atoms of the molecule will execute simple
harmonic motion.. showing. therefore. it is clear that displacement FIG. In Fig. Exercise
Consider a spring system like that of Fig. has been turned through an angle as shown in Fig.
with atomic mass unit equivalent to a hydrogen atom.
the x and y directions. will where the normal coordinate directions are not recognized is not
just the sum of squared lead to a potentialenergy expression which between the coordinates.
It follows form u KWY KWKv iW the restoring forces This potential expression can be seen to
lead to one has Thus . coordinate now gives. and ma kj y . in vibrations of this oneparticle Let
us proceed now to study the The potential is written in . Fig. or the y axis. Thus. with Eqs.
and since the coordinates is therefore. we can again look for vibrationtype frequency of
vibration x is v. the coordiFor the coordinate system of Fig. displacement along lead to
restoring forces nate axes will not. as the force vectors suggest. convenient than those of Eq.
coordinates. using the coordinates the kinetic energy orthogonal. products cross addition. of
orientation expected for the spring id A ments in both in advance. where the by trying the
functions rrf A x sin and y sin A y irvi ltp . INTRODUCTION TO MOLECULAR
SPECTROSCOPY force acting along along the x axis. terms but contains. from
displacerestoring force in the a direction can result. . . and. function has the potentialenergy
the that along these axis. that axis and directed toward the form of the potential written in Eq.
Fig. .Ktf coordinate. for the my KX Kv o Although these equations are clearly less and Eq.
solutions. a selection of t imx iMyy x Lagranges equation for the . . results in a restoring the
basis for the was This origin. of system. still are Eq.
Finally. tVto k y Expansion of the determinant gives rVm The k x w v m k y k xv Y roots of
this quadratic in v are found to be v / K kk k y y Thus the two natural calculated. same values
are obtained as This rather detailed treatment of a very simple system will later be
recognized as an illustration of the procedure that . the appropriate minors of Eq.is necessary
when the relation treated. . the potentialenergy function . and see that the in Exercise . the
roots given or. Exercise . vibrational frequencies of the system can be by Eq.THE
VIBRATIONS OF POLYATOMIC MOLECULES Differentiation twice with respect to time
gives functions for x and y and. is V a X z VI X lWxy . the directions in which the particle
moves in each vibration. and values for the ratio of A can be obtained for each vibration x to
A y frequency. with Eqs. between the force constants and vibrations of a molecule is The
immediate purpose of the example is to illustrate that .mA. b Calculate the relative values of
A and A for each vibration.. now labeled x and y. for other than the trivial solution with .
alternatively. from the given potential function. When the spring and particle system of
Exercise is rotated counterclockwise by deg with respect to the axes. can be calculated. i.
and . leads to the expressions pmA m k x x A kA. the determinant of the coefficients of the
amplitudes A x A y must be zero. X B y Calculate the natural frequencies of the system.e.
and x y see that one can deduce. k x xv A k A y y For these equalities to hold. and . rrVm k
xy k k. can be substituted into Eq.
In general one is in terms of the relative motions of the not initially able to describe these
normal coordinates atoms of the molecule. for example. separation of external from internal
coordinates is carried out will not can be found in many texts on quantum mechanics.
however. be written for n as particles. vibrational. and the normal coordinates can then be
deduced from the amplitudes that are obtained for the natural. INTRODUCTION TO
MOLECULAR SPECTROSCOPY is there a convenient set of coordinates. . treat separately
can. It Let us now proceed to investigate the form that the solutions of the Schrodinger
equation will take when it is applied to the internal. in the classical discussion of the previous
section. called normal coordinates. or will proceed. from the simple example of the preceding
section the important fact that normal coordinates are such that both the kinetic and potential
energies of a system are expressed by the sum of terms each involving the square of a
normal coordinate. as in the the three coordinates that describe the motion of the center of
mass of the system and the three or two for a linear molecule that describe the rotation of the
system about the The derivation that shows how this principal momentofinertia axes. Pauling
and E. the position of each being described by three is cartesian coordinates. Furthermore
we formally. be given here. of can be carried through initially solution are not with any other
set of coordinates. Quantummechanical Treatment of the Vibrations of Previous applications
of Polyatomic Molecules the Schrodinger equation have been restricted to onedimensional
problems. for the kinetic treating the vibrations of a system. sunt squared terms. classical
case. vibrations of the system.quot by L. We know. In this way the Schrodinger equation
written f l quot where the function will involve gtlt As One lt gt the n coordinates. coordinates
of a molecule. . as. as the of the problem known. however.. Inc. Wilson McGrawHill Book
Company. The equation can. New York. . such that they allow both Furthermore these
coordinates are and potential energies to be written even if these convenient coordinates
Finally. designated by Q of the molecule. with the normal coordinates. quotIntroduction to
Quantum Mechanics. B. the n cartesian coordinates are not convenient for describing the
molecular motions. or normal.
in n h Y d J/ where the solution function normal coordinates. the kinetic energy a function of
squared terms.Qn i X where the molecule. be a function of the n This manydimensional
equation is greatly simplified because of the fact that. is in the denominator of this generally
absorbed in the normal coordinate The operator itself. one can write Since. We write lQllQ or
. X H X It is profitable to try to separate this equation into w equations. but so also is the
potential energy. in general.is given by an expression of the type h a ampr dQ The mass
term that normally appears kineticenergy operator Qi.lt. each involving only one of the normal
coordinates. for the potential energy is the potential energy func tion is terms of the normal
coordinates. will.lt . not only is the kinetic energy a sum of squared terms. Xs are coefficients
that Xlt amp depend on the force constants of the Equation can now be written as n n x. The
form of the potentialenergy function can be shown by writing UiQtQt Q n e iQ n iXzQ iA .THE
VIBRATIONS OF POLYATOMIC MOLECULES The correct form of the SchrQdinger
equation for treatment of the internal motion of a molecule in terms of the n or n normal
coordinates can be deduced from the fact that the kineticenergy operator for a coordinate Q.
in terms of normal coordinates..
For some of the simpler molecules they have been deduced. into Eq. . the
quantummechanical problem cf . the selection rules for transitions between the vibrational
states for each normal coordinate are are the same Aigt For absorption experiments. and the
observed spectrum of H It are shown schematically perhaps should be pointed out that in
general we do not know the normal coordinates of a molecule. one gets n expressions of the
type or Each such expression is identical to that already encountered in onedimensional
vibrational problems. With this knowledge one deduces . can expect. the pertinent rule to is
gt again obtained. know that there are coordinates in terms of which T and U are both sums
of squared terms. INTRODUCTION TO MOLECULAR SPECTROSCOPY and substitute Eq.
these coordinates. therefore. The important result has been obtained that. The solution
functions for each of the n equations like Eq. the classical solution at the beginning of the
previous section is separable in terms of molecule containing n atoms will have. in fact. We
corresponding to the transition from the energylevel pattern. each pattern cor responding to
one of the classical normal vibrations of the system. is When this is done and the entire
equation divided through by . since normal coordinates allow both the kinetic and potential
energy to be written as the sums of squared terms. an absorption of radiation to the v level of
each v will see that. the harmonic Furthermore. . . Since the wave functions written in terms
of normal coordinates as those discussed in Sec. A n or n vibrational energylevel patterns. .
however. each normal oscillator wave functions listed in Sec. therefore. but their deduction is
often rather difficult. are. . require us to be able to describe the It is sufficient to normal
coordinates. coordinate Q will lead to a set of energy levels with a constant spacing.
therefore. The previous derivation of the separation of the vibrational problem does not. for
We symmetric molecules some of these transitions are forbidden. The energylevel diagrams
in Fig. t such as shown for the onedimensional case in Fig.
cm A .. like COj.e. A . The Symmetry Properties of Normal Coordinates Normal coordinates
molecule. in fact. widespread and profound. d. vibrations is a plane of symmetry. . and to
recognize that a plane perpendicular to the molecular axis FIG. spectroscopy are. One way
in which symmetry enters into treatments of molecular systems will here be illustrated by a
discussion of the nature of the normal coordinates of a When the molecule has symmetric
molecule. i. are often complicated functions in that they involve the displacement coordinates
of many. or all. of the atoms of the some symmetry. Here it will be enough to consider the
molecule in a linear system. as many do. a great The consequences of symmetry in
molecular simplification occurs. A complete classification of the ways in which a molecule
can be will be given in symmetric Chap. its The energylevel patterns and schematic
representations of the corresponding and a schematic infrared absorption spectrum of
HsO.THE VIBRATIONS OF POLYATOMIC MOLECULES is that the vibrational energy of a
molecule to be treated in terms of n energylevel patterns.Scm cm . em it l.
symmetry elements. It should The symmetry operation for an axis of is symmetry will is
rotation about the axis. That symmetry and normal vibrations are related can be readily If one
were illustrated by reference to the spring system of Fig. Symmetry operations corresponding
to these elements transform the solid arrows into the dashed arrows. one introduces Thus
the symmetry operation corthe idea of a symmetry operation. one would. . distort the system
symmetrically. that the coordinates along which the system must be distorted in order for the
particles to vibrate in phase and with the same frequency are either symmetric or
antisymmetric with respect to the plane of symmetry. It is customary to such a plane an
element of symmetry. only the displacement vectors will undergo the symmetry operation.
FIG. INTRODUCTION TO MOLECULAR SPECTROSCOPY is equilibrium configuration
identical on both sides of the plane. as in Fig. unaltered asked to set this system vibrating
according to its natural or normal modes of vibration. Two symmetry elements of COi o plane
and an axis of symmetry. Fig. placement of an oxygen atom of relation of such
displacements to CO This operation transforms the displacement vector of Fig. . The axis of
the molecule of COj is another such symmetry element call of the molecule. particularly if
one has a model to work with. fc. A similar conclusion can also be drawn for the less easily
visualized normal coordinates of molecular systems. or antisymmetrically. in the manner
shown. In vibrational spectroscopy we are concerned with displacements of the atoms of a
molecule from their equilibrium positions. as shown for a general dis To deal with the s in Fig.
. Plane of symmetry Axis of symmetry . These displacements can be depicted by vectors. a.
and this operation also shown in Fig. as in In this simple example it is obvious. without any
detailed analysis. be pointed out that the numbering of the atoms be kept by symmetry
operations. responding to the plane of symmetry is a reflection through the plane.
In a similar way. and b antisymmetric with respect to the plane of symmetry. will is This have
the same potential energy as the original illustrated in Fig. since kinetic energy of it does not
alter the potential or a molecule. contain only squared terms. does not alter the total
energy.e. since the normal coordinates represent the displacements of the atoms during a
vibration. it is first necessary to point out that both the kinetic and potential energy of a
molecule must be unaltered by a symmetry operation. the distorted molecule that is obtained
by the symmetry operation distorted one.THE VIBRATIONS OF POLYATOMIC
MOLECULES To do this. therefore. ticles that The arrows indicate the initial motions of the
par would follow such displacements. they can be represented by displacement FIG. A
twoparticle spring system with a plane of symmetry perpendicular to the Distortions are
indicated that are a symmetric axis of the springs.. Q iX. . a. molecules. This is apparent
where the velocities of the atoms are represented vectorially and the the magnitude of the
velocities of atoms of given masses. some higher potential energy than the equilibrium
configuration. If a symmetry operation is performed on a distorted molecule.QJ iQ amptQi T
quotB y Sn e Now. Plane of symmetry LwwB/SAA/ IaA/S/BVW a h . a symmetry in Fig. which
has. it In the previous section is was shown that the vibrational energy most simply
expressed in terms of the normal coordinates of the In these coordinates both the kinetic and
potential energies i. if a its molecule has kinetic energy as a result of motion of operation
cannot change this energy. symmetry It follows operation changes the directions of these in
space but does not affect then that a symmetry operation. atoms.
it is In view of Eq. change sign. CO molecule. seen that for the energy to remain constant. is
unchanged as a result of a symmetry operation Q Qgt This important conclusion or is that
normal coordinates are either symmetric to antisymmetric with respect a symmetry operation.
as in Fig. . in the general case all where Thus. for example. each normal coordinate must
remain unchanged lt most. Distorted INTRODUCTION TO MOIECULAR SPECTROSCOPY
J position Reflection Diane of symmetry through fnj Reflection through plane of symmetry
CM FIG. One needs only to draw arrows. vectors such as those of Fig. Arrows indicating the
relative motions of the atoms of COa in two normal vibrations with displacements along the
molecular axis. at if is performed. which keep the center of gravity of the molecule fixed.
when a symmetry operation or. . or . lead to no FIG. kinetic Illustration that a symmetry
operation does not change the potential or the energy of a molecule. however. With this
result one can. of Eq. immediately represent two of the normal coordinates of the . Xlt are
different.
of direction. one has IQ iQl iQl of iiQl where the equal potential energy coefficients have
been designated Xi. Rearrangement lt Eq. quantum mechanically. supposition corresponds
to cases in which all the normal vibrations have different frequencies. as for C one can often
write down immediately the form In general. and are such that they are result in a simple
unchanged or change of sign. i. as symmetry operation does not change the value of Q Q.
however. accordIf the normal coordiing to Eq. ties of a molecule can be used to classify the
normal coordinates accordoperation . gives HQt QI MQ i Now long as the FIG.. Such
vibrations are said to be degenerate. since degrees of freedom of the molecule COs in the
are involved and. the vibration said to be doubly degenerate. . The previous discussion has
been based on the general case where all This the normal coordinates have different values
of X in Eq. When such degeneracy occurs. as far as Qi and Q are concerned. to the
restriction that Qi gt Qi. two is identical sets of energy levels will exist. symmetry. frequency
expected for each vibration is therefore identical. . ing to whether they transform
symmetrically or antisymmetrically with the various symmetry operations of the molecule.
nates of the degenerate vibration are Qi and Q. .THE VIBRATIONS OF POLYATOMIC
MOLECULES left rotation of the molecule. .e. The two components of the doubly degenerate
bending vibration of COs. is In later chapters we will sec that such a classification a
tremendous aid in studies of the vibrations of polyatomic molecules. the symmetry properof
the normal coordinates. when a symmetry For simple molecules with a high degree of is
performed. provide an The illustration of normal coordinates that have equal values of X. The
bending vibrations of two perpendicular planes. the requirement that the energy remain
unchanged during a symmetry operation does not lead. as shown in Fig. and in this case. the
energy remains constant.
C.. The behavior of deThe figure Q Qz Q cos e sinfl generate motions under a symmetry
operation.sin Q cos corresponds to an end view of Fig.. ..J. the normal coordinates remain
unchanged or change sign. B. New York. Jr. Princeton. E. subject to a restriction imposed by
the particular sym metry operation. operation consisting of a rotation of considered.quot Inc.
PRINCIPAL REFERENCES . G. INTRODUCTION TO MOLECULAR SPECTROSCOPY
FIG. Herzberg. Within this restriction. Cross quotMolecular Vibrations. into combinations of
the coordiset. Van Nostrand Company. Draw. . . each normal coordinate of the set nates of
the can be transformed. C about its axis where the symmetry by an angle is respect to In
summary. N. For degenerate vibrations. For nondegenerate vibrations. it is allowed that a
symmetry operation produce the transformations Qi aQ x bQ t and An illustration of this
behavior is given in Fig. Exercise . arrows showing the relative displacements of the atoms of
the linear molecule CS and the bent molecule HS in the normal vibrations of each molecule. .
. McGrawHill Book Company. by the operation.. to the extent that one can on the basis of
symmetry and zero translation and rotation. Decius. J. Inc. Wilson. and P. .lt. with only the
arrows on the oxygen atoms shown. C. quotInfrared and Raman Spectra.quot chap. D.. . the
behavior of normal coordinates with symmetry operations can be stated as .
the previous chapter it was indicated that each absorption and la. but rather. absorption
regions of considerable complexity. The infrared absorption spectrum of a simple molecule in
the gas if phase does not. We will now investigate in some detail the nature of
rotationvibration bands. are patterns energy various vibrational spectra of fairly simple
molecules are shown in Figs. The structure of such absorption bands can be attributed to
changes in the rotational energy of the molecule accompanying the vibrational transition.
welldefined regions of absorption which can. The general features of the rota . the nature of
the rotational structure of a rotationvibration absorption band will be seen to reveal much
about the type of vibration that is occurring. as Fig. the resolution of the instrument is
adequate. structureless absorption bands. be of great aid when we attempt to match up the
observed vibrational absorption bands with the normal and this information will vibrations of
polyatomic molecules. to v . In band can be associated with a fundamental transition in an
energylevel pattern corresponding to a specific normal mode of vibration of the molecule.
Furthermore. In the liquid phase no welldefined rotational energy levels
exist.ROTATIONVIBRATION SPECTRA When the infrared radiation absorbed by a pure
liquid or by a material in solution is measured. be attributed to the v for Typical liquidphase
observed. and rotational structure is not observed on a vibrational absorption band. trates.
show such illus smooth. and we will see that molecularstructure information can be deduced
from an analysis of the structure of such bands. transitions in the the most part.
UOtSStlUSUBJ tUampiJfyj . .
.S UOjSSjUISUej U Jcj S u.
harmonicoscillator energy expression is deduced. will be dealt with separately.e. With this
approximation. of will less importance than a rotationvibration coupling term that be
introduced after the rotationvibration spectrum cor responding to the simple rigidrotor. are
important. and fairly satisfactory. harmonicoscillator system. absorption lines to be
calculated. linear. energylevel expression is The simplest. together with an expression for the
rotational and allow the energies. This electron contributes angular momentum have an
about the molecular and this in turn allows AJ transitions to occur. the selection rule Ay is
appropriate and the fundamental transitions. and now. transition. For absorption spectra. The
complexities that arise in the rotational energylevel pattern of asymmetrictop molecules lead
to corresponding complexities in their rotationvibration bands. diatomic. These will not be
analyzed here. and J J These correction terms however. .. the anharmonic and the nonrigid v
rotor. of the vibrational energy levels. i. is again often studied. as before. The selection rule
for J was given AJ . Selection Rules and Transitions for the Rigid RotorHarmonic Oscillator
Model and vibrational and The selection rules governing the allowed rotational . remain
applicable when both types of energy are changed in a given transition. from v as to v .
obtained by treating the molecule as a rigid rotor. Thus the Av selection rules for a
rotationvibration absorption band are AJ Exceptions to this rule are provided by molecules
like NO that odd electron. axis. one writes The two improved models oscillator treated earlier.
or frequencies. both unlike the situation in pure possibilities rotational absorption
spectroscopy. given in Chaps. DIATOMIC MOLECULES . These rules. could be introduced
and would lead l . INTRODUCTION TO MOLECULAR SPECTROSCOPY tionvibration bands
of the three molecular types. and symmetric top. to terms involving are.
BJ J gt J i. the lowfrequency set of lines as the P branch It is customary For of the rotation
vibration band and the highfrequency side as the R branch. given in Fig. With and the
selection rules of Eq. are best shown on a diagram such as that of Fig. . .e. central
components of the P and R branches can be identified with the molecular this quantity. the
central AJ and it is known as the Q branch. . transitions. the factor the symbol ogt /t Wm by v
this notation. . For AJ . . which occurs at v u. . some molecules. These transitions. There it is
clear. further more.e. all of There should. the energylevel expression of Eq. of the v and or
Fig. . as the above expressions show. that can be attributed to the v is. is One should note
that in these expressions the value to be inserted for J the rotational quantum number for the
lower vibrational state. branch is allowed. BJ J . The general shape of this experimental
curve in fact. and the resulting absorption lines. values for the quantity .J J . the separation of
the tional energy levels of the molecule. to v transition. l. For AJ igt .JJ where J . It is clear
that the frequency . . spaced by the constant amount .Jtov in the J transition can be deduced
from the gap band and that this value gives the spacing of the vibraFurthermore. gt J l/ . . be
a gap in the band center corresponding to the absent to label AJ which would have had a
frequency lta. . BJ where J . . / . ... on the highfrequency side of the band center. The infrared
absorption spectrum of HBr shows the absorption. . that there should be a set of absorption
lines.ROTATIONVIBRATION SPECTRA It is convenient to replace. and a corresponding set
on the lowfrequency side with a similar constant spacing. give the allowed transition
frequencies v v At for the transitions as i. amp w J . . as was done in Sec. as we will see.
From moment of inertia of the molecule can be deduced as they were when the spacing of
pure rotational lines was determined from studies in the and the bond length microwave
region. and l/rc y/kh by the symbol ugt. gt S. that predicted on the basis of Eqs.
linear molecule based on the assumption independent of Two features illustrated yet been
accounted for intensities of the by the HBr example in and must now be considered. The
Relative Intensities of the Components of a RotationVibration Absorption Band relative
intensities of the components of bands such as that of can be understood on the basis of the
population of the various The approximarotational energy levels of the ground vibrational
state. second. The transitions that lead to rotational structure that B is in a vibrational band
for a y. Fig. The tion can be made that the intensities of the components of the rotation .k u l .
have not First. the components are not found to be exactly constant must be dealt with. .
INTRODUCTION TO MOLECULAR SPECTROSCOPY Energy . II igt II iamp II lt B lt l lt i lt i lt
lltu l lt lt lt B B IB fi Spectrum lt P branch A branch FIG. Fig. the relative fact that the spacings
of the components remain to be investigated and.i h t Energy levels quot a iamp ii to iamp ii
iamp ii lamp II II la.
o II S uojsniusuej jure Jd o E .
. in fact. As was shown in Sec. The fact that the spectral transitions derived i. according to
the Jth rotational energy level is J Boltzmann distribution. . Much of the disagreement that
will be found because of the experimental problems that arise when very sharp absorption
lines are measured. can be attributed to coupling between the rotation and the vibration of
the molecule. . on this basis. HBr at C. on this basis are not in complete accord with the
observed bands. will correspond rather closely to that observed. and energy levels of
Calculate the relative populations of the J .. the spacing of the components is not constant. is
based on the assumption that the energy of a molecule can be treated in terms of separate
contributions from the rotation of the molecule and the vibration of the molecule. the popula
tion of the Jth / level is Nj lN e. v il/rc VVm BJJ is where S v implies that the rotational term a
function of the vibrational is quantum number. and the value of Nj/N can be calculated for
various J values. . with the rigidrotor approximation. . and compare these results with the
relative values of log j for the components of the P and R branches is shown in Fig. The
intensities of the proportional to the values of components starting from various J values will
be Nj/N The absorption band predicted . Thus one usually the coupling by writing . iquotltT
The momentofinertia value for the molecule being studied and the temperature of the
experiment can be substituted into this expression. since involves the effective treats bond
length r. the multiplicity of the and it follows that. . that the dependence pretty well . Coupling
of Rotation and Vibration The energylevel expression of Eq. One finds.e. . This coupling
introduces a cross term in the energy expression for a rotatingvibrating molecule that
involves a term of the type v JJ .jr J h. . INTRODUCTION TO MOLECULAR
SPECTROSCOPY vibration band are proportional to the populations of the rotational energy
levels from which the components originate. Exercise . it This term is usually introduced by
recognizing that B depends. on the vibrational energy.
Similarly. It is these terms that are primarily responsible for the nonconstant spacing of the
components of a rotationvibration band and for the asymmetry of the expressions for the B
ltB P and R x . lead to values of Analysis of the rotationvibration band can. and as a
consequence this branch extends t The high J components of the frequencies than would be
lower somewhat are at branch similarly R expected for B x B B. Such behavior is observed
only in electronicvibrationrotation bands.JJ B J i . and the high J components of the R branch
It can be mentioned that this effect tend.BoJJ . . difference in the frequency of the component
of the R J branch and that of the P branch that start from the same value of gives information
on the rotational spacing in the upper vibrational level.ROTATIONVIBRATION SPECTRA
represented by an expression of the type B.J . The coefficients of the J terms in the band that
of the is apparent in the band of Fig.o/ x P branch. to bunch up. a.v i where B is the rotational
term corresponding to the equilibrium bond length and a.J Since the higher energy
vibrational state has a greater vibrational amplitude. It follows that Ii gt Jo and that branches
are therefore negative. The high J components on P branch are at frequencies that are lower
than would be expected the basis of So B. vp J J xJ S ib B . that the fii and B . branch start
moving back to lower frequencies for very high J values. therefore. It should be clear from
Fig. suggests. . The band is then said to have a band head. With this energy expression the
selection rules give the allowed transition frequencies P Ae for the v to v transition as R
branch. J J gt vB i gt SJ J . B. is a small. J JSi . positive constant. the difference in the
frequency of the component of the R branch and that of the P branch that end at the same
value of / in the v level gives . important so that the components of the R sufficiently can
become farther down to the lowfrequency side. it will have a larger effective value of r. as
Fig.
ences between the frequencies of FIG.HO INTRODUCTION TO MOLECULAR
SPECTROSCOPY information on the rotational spacing in the lower vibrational level. or. The
difference frequency of the components shown as heavy frequency of the state. the
rotational structure of a vibration An illustration that the rotational spacing and. From the
rotational spacings in the v and v levels. from various pairs of transitions as a function of J.
one can deduce values of B and B u Furthermore. to values of The combinations that lead by
considering Fig. while the difference in the dashed lines gives the spacing in the v J .
analytically. therefore. and any effect of centrifugal distortion can be recognized. the moment
of inertia the v and v states can be deduced from in rotation absorption band. solid lines
gives the spacing in the v state. values of B and S t can be obtained. t I i i t I i I I I i I i i I i i i i I
i i I I o i i i i . in and B can be deduced by taking the appropriate differthe P and R branch
components.
J From these differences. . i. nA . . . . j Molecule . . as was mentioned Chap. . . by means of
the formula B v . which are deduced. . . r. and r. . by writing J in the expression for the R
components and J in the expression for the P components and performing the subtraction.
one has bJ pJ J starting from the component An evaluated. of I The dependence and
therefore and r. vrJ v P J of B. . the bond length are given in Table . . . polyatomic molecules
from which additional bond lengths or angles can be determined. . . which is tion for the
equilibrium bond length. . . makes in itself known when one uses isotopic substitution. a
substitution usually what is studied in pure rotational isotopic substitution of. corresponding to
the minimum in the potentialenergy curve. . The assumption of constant bond length with to
obtain additional momentofinertia data for would appear to be valid to a very good
approximaFor the v state.. . . for example. values of B can be obtained. . for at least two
vibrational levels. . . . . These results correspond to the equilibrium bond length. . . . . One
This should particularly notice that difference r t is significantly different from r . spectral
studies. o. are for the formula . .e. . cm a. B. H D HD HC DC CO N . D for H lowers the v
energy level in the potential curve and moves the effective bond length part TABLE way from
the value r Dependence of and the Bond Distance on the Vibrational State of a Molecule
Values of a. Some results for diatomic molecules Also shown in the table are values for B. . a
e v i and the measured values of . . on the vibrational level turns out to be
appreciable.ROTATIONVIBRATION SPECTRA Ml P and R components that start from the
same J value. . cm A roA . One then can obtain. R be which B can from on level end the
same the J starting from P component J level and a Thus for X x J level in the upper state.
. as far as rotationvibration selection rules concerned. that could be used to calculate B and
two that could be used to calculate Si. . an equilibrium bond length The molecule DC has a
reduced mass of about . and the cules. . b Draw vertical arrows to represent the transitions
that would occur in an absorption study of the fundamental band. AJ . P and R branches and
no Q branch. for the Hcontaining INTRODUCTION TO MOLECULAR SPECTROSCOPY be
analyzed. and a of . is apply and the rotationvibration band the vibration is said to be a
parallel band. as discussed here. results for isotopic species. the situation. Parallel Bands of
Linear Molecules When dipole the vibration of a linear molecule results in an oscillating that
is parallel moment is to the molecular axis. as is done in Fig. force constant of first five X
dynes/cm. If such that the oscillating dipole moment is perpendicular to the molecular axis.
The treatment of the previous bands of linear molecules and leads.e. bands can can be
obtained for different and these can be safely assumed to remain constant with re compound
to r e If rotationvibration isotopic substitution. is parallel which will be discussed in the
following section. to values of S x and S a Values of Si and would be expected to depend
somewhat on the particular vibration being studied. and therefore absorption lines. Exercise .
Place the arrows so that the one corresponding to the lowest energy change is at the left and
the highest one is at the right. X g. . selection rules are Av the diatomic molecules. band is
said to be a perpendicular band. for the fundamental vibrational transition. the value deduced
for B e should be section can also be applied to the . c Pick out two transitions. in no way
different and resulting band shapes are from the simpler linear molecules. a different set of
selection rules apply and the rotationvibration . A. a Draw to scale rotational levels of the v an
energylevel diagram showing the and v vibrational states. This is as that observed for
diatomic mole illustrated in Fig. . rotationvibration band has the same appearance. a certain
connected with the vibration set of selection rules. LINEAR MOLECULES Two molecules.
principal types of dotationvibration bands occur for linear If the vibration is such that the
oscillating dipole moment to the molecular axis. Thus the and i.
E u c a o ee o c o q z o o c o lt UOISSIIUSUampll S IU JJ d .
as discussed in Sec. From analysis of such bands. are examples of vibrations which lead to
rotationvibration absorption . J.I It I quotIquot . CC bands. the same as that encountered in
studies of pure rotational spectra. T . It is of interest to point out that. B or /. of course. The
problem of deducing bond lengths from the single experimental result. a The energylevel dia
gram and like transitions of a molecule C . . H H. and that of acetylene. linear symmetric
molecules can if exhibit rotationvibration absorption bands even the molecule possesses no
permanent dipole moment. momentofinertia data for these nonpolar molecules can be
obtained. is. .. The statistics followed by the like atoms of molecules such as CO and CH
lead. to the decrease in statistical weight of some of the rotational levels or to the absence of
some of these .T f h INTRODUCTION TO MOLECULAR SPECTROSCOPY the same for all
vibrations of the molecule.. U FIG. in contrast to the situation that exists in pure rotational
spectroscopy. The antisymmetric stretching vibration of C C. antisymmetric stretching
vibration Nmhpf J. It is only necessary for the vibration to be such that there is an oscillating
dipole moment..
a. asymmetric stretching vibration is also shown in Fig. one finds that the spacing of the
components is twice B and that the central gap is half again as large as the spacing in the P
and R branches. do not have an effect.ROTATIONVIBRATION SPECTRA levels. The
absorption band that results from the is shown in Fig. b A parallel band of the linear molecule
statistics. . The carbon nuclei have no spin and. The molecule acetylene illustrates the case
where Fermi statistics apply since the hydrogen nuclei have spin . The appropriate
energylevel diagram for COJ an example . The alternation in statistical weight of the
rotational levels is like that of H . Although the appearance of the band does not immediately
indicate the effect of the missing rotational levels. therefore. in which the like atoms have
zero spin and obey BoseEinstein statistics. CO which has a center of symmetry and like
atoms which obey BoseEinstein E M C s . and this FIG.
in R Q branches that are similar to those found in parallel branch corresponding to the AJ
transitions. If the rotational spacing in the upper vibrational state were the same as that in the
lower state. the Q branch. a The energylevel diagram and transitions i of a molecule like H
CsC H. therefore. Perpendicular Bands of Linear Molecules Bending vibrations of linear
molecules have oscillating dipole moments that are At. For these vibrations the selection
rules for rotationvibration transitions P and AJ . components shown in Figs. The
perpendiculartype band addition to of a linear molecule has. The bands. are ina direction
perpendicular to the molecular axis. to the alternation in intensity of the . in the case of
acetylene. . a central principal new feature. The appearance of such bands is illustrated by
the absorption band corresponding to the bending vibration of N shown in Fig. I alternation
leads. corresponds to the overlap of the AJ transitions. the / J transitions . INTRODUCTION
TO MOLECUUR SPECTROSCOPY A j r t o quot lift l j j FIG. a and b.
a I E E i o .c s i X o o E o lt uoissiujsuej u Jd .
The two satellite state. The Q branch usually. . . and the two perpendicular motions of the
molecule can be coupled to show that in this vibrational state the molecule has angular
momentum as a result of this vibrational motion. . Pcmquot . as mentioned doubly
degenerate. INTRODUCTION TO MOLECULAR SPECTROSCOPY all would have the same
frequency Po The fact that Si is smaller than B leads to a slight decrease in frequency of the
somewhat Q com ponents with high slight / values. . This The perpendicular angular
momentum is tures of perpendicular bands. about the figure axis leads to some additional
feaThe lowest J value for the v state. for example. Q branches can be attributed to NjO
molecules that are in a vibrational excited . has a asymmetry. vibrations of a linear molecule
are. . FIG. J because the one unit of angular momentum that A perpendicular band of the
linear molecule N . . in Sec. therefore. .
if K AK in AJ if K These rules can be combined with the energylevel expression. As for linear
molecules. a. All these vibrations have an oscillating dipole molecule. band can then be
represented on the energy diagram of Fig. . Parallel Bands of Symmetrictop Molecules
Vibrations that lead to parallel bands can be illustrated by the displacements indicated in Fig.
moment along the unique axis. these two types of vibrations have different rotationvibration
selection rules and therefore different rotationvibration . . best They will occur again in our
study of the rotationvibration spectra of symmetrictop molecules and will be discussed at
greater length there. J / and K quantum numbers. Chap. for the unique axis. . which are
symmetric tops because of the symmetric arrangement of the atoms. . K where .
.VIBRATION SPECTRA U the vibration contributes disallows the total angular of zero. . .
have their oscillating dipole moment either parallel to the unique axis of the molecule or
perpendicular to this axis. be passed over here. for the rotational energies of a symmetrictop
molecule ijK S JJ are l I SKgt J . can be attributed to rotation of the These finer details can.
SYMMETRICTOP MOLECULES The infrared active normal vibrations of molecules like C
and benzene.ROTATION. . . If the dependence of the moments of inertia on the value of K is
small. . . . furthermore. and A is the rotational constant is the rotational constant for the axes
perpenThe transitions that constitute the parallel dicular to the figure axis. the spacing within
each set of levels with a given value of K will be almost . momentum value rotation is split
The angular momentum. . Eq. however. CH band shapes. the figure axis of the For such
vibrations the selection rules for the rotational changes in a rotationvibration transition are AK
and AJ . couples with the overall of the molecule in such a way that each J level of the v
state levels that result and the two molecule in opposite directions.
// J FIG. Vibrations of symmetrytop molecules that lead to a parallel absorption b anas and b
perpendicular absorption bands. INTRODUCTION TO MOLECULAR SPECTROSCOPY V . .
This follows from the requirement that. . and JJ B terms of the energy expression. .e. J .
since determines a component of the angular momentum while / determines of FIG. as both
and show. i.ROTATIONVIBRATION SPECTRA the same since they are governed by the
same term. f quot K K K Kl K . J do K The energy levels of a symmetrictop molecule and the
transitions that are allowed for a parallel band. that for the higher K values the low values not
occur. . the JJ lBi The contributions to the total given value of Figs.. One should notice. band
from the transitions within each set of levels with a K can be shown as in Fig.
energylevel diagram of Fig. Ml i I I i L i i I I I i i K K I I i L Jh. consists of simple Q. figure axis
is obtained. J must be equal K K contribution which shows no Q branch. . These . for the
except branches P.iLilUUu FIG. The components of a parallel band showing the
contributions from each of the K levels of the v state. and R in Fig. b. or at least an aver age
value of B. the components of the P and R branches of the parallel band can be resolved.
the total angular momentum. iii J I I I I I i T K nULUlh. L I I lliii I I I i Ki l I . to or greater than
contribution shown feature each this Outside of for any allowed state. INTRODUCTION TO
MOLECULAR SPECTROSCOPY JL I . one can deduce values for i and B . Perpendicular
Bands of Symmetrictop Molecules Vibrations that have oscillating dipole moments
perpendicular to the unique axis of the molecule are shown in Fig. tions. I K . the selection
rule for rotational transitions For such vibra accompanying the vibrational transition is AK The
ampJ . in the same way as previously indicated for diatomic With such data the moment of
inertia perpendicular to the molecules. . shows the transitions that undercan occur in a
perpendicular band. The appearance expected for the total band is seen to be borne out by
the absorption band of CH Br shown If in Fig. The complete band can be of consist subbands
stood in terms of a summation of subbands.
.ROTATIONVIBRATION SPECTRA S FIG. while only the contour of the ft branch is
obtained. S . partly resolved. lim Jd The P uoisnuKueji branch is A parallel absorption band
of the symmetrytop molecule CHsBr.
AJ transitions that occur for a given change in one gets two subbands corresponding to the
sets of transitions that occur within AK and AK . K . The total band. and so forth. for K bands
result from the AJ . as the experimental FIG. The energy levels of a symmetrytop molecule
showing the / transitions that are allowed for a perpendicular band. the P. transitions. . for K
gt . Q. and type subR . Similar subbands occur For each initial K value. Thus. the . except K .
all INTRODUCTION TO MOLECULAR SPECTROSCOPY K. as shown in Fig.
C M o II e II II II I h gt A O u . E S . II II I II I II II II II X lt lt lt lt lt lt lt fetj o I. a o o a. c O .
of the ing to this result. usually reveals the Q branches of the subbands superimposed on an
unresolved background. S. one has For Q branches. . The spacing methyl halides. The
alternation in intensity of the Q branches of the CH C band of Fig.y . The spacing level
expression of these Q branches can be calculated from the energy h. indicate the transitions
that lead to the branches in a perpendicular type band.BK that result in the of v A and the
changes AK . . ForAK Sflu. like that of Fig.nch ampA l . page .K . easily resolved.
INTRODUCTION TO MOLECULAR SPECTROSCOPY curve of Fig. The basis for the
strong. c On . weak. . B C a . . strong variation and the intensity ratio of is the same as that
dis cussed for linear molecules in Sec. accord B. If and if one again designates the AK P ta .
Draw the rotational spacings to scale for both the v and v states. . K .K wABABK BK iorK . to
v energy difference. in Detailed analysis of the situation will. CH Br are A . cm and Make two
energylevel diagrams. . weak. For molecules like the larger where A is than B.quot but
reduce the v b On one diagram. The rotational constants of cm. Exercise . .B . be identified
with A branches of a perpendicular band can. indicate the transitions that occur in a vibration
in which the dipolemoment oscillation is along the figure axis. AJ and B are taken as
independent band origin by to. K BJJ A . See ref. the Q branches are well sep arated and.BK
for K Q . however.. shows. as Fig. shows. A A .. which there are three rather than two
identical nuclei not be given here. can be attributed to the different statistical weights of the
levels with different K values. Q the other diagram. for CHsBr.
UOISSIIUSUeJI UD J .
Obtain the infrared spectrum of a gaseous methyl halide. which can be combined to show
an angular momentum. Van Nostrand Company.quot chap. Princeton. The coupling occurs
through Coriolis forces.. which leads to the expectation of a constant spacing of A B. G. The
treatment given here. and the basis and effect of such forces branch spacing is are
discussed in ref. quotInfrared and Raman Spectra. angles reported in the literature. . but the
calculation requires a knowledge of the form of the normal vibrations of the molecule. . one
might find the rather disturbing fact that the Q nowhere near the same in the different bands.
and see that the values obtained are not identical. . in error because of neglect of a coupling
that occurs between the doubly degenerate vibrations. Compare with the value of A B
calculated from values of the bond lengths and bond Exercise . The effect of Coriolis
coupling can be calculated. is. INTRODUCTION TO MOLECULAR SPECTROSCOPY If one
investigates in detail a number of perpendicular bands of a given molecule. Determine A B
for as many of the perpendicular bands as possible. Inc.. Herzberg. that lead to
perpendicular bands and the rotation of the molecule. D.J. and in general the analysis of a
perpendicular band does not yield molecular structure information. PRINCIPAL
REFERENCES . N. for some molecules. . Only simple molecules can therefore be handled.
in view of our of general of the vibrations of studies of the previous chapters. . the ways in
which symmetry elements of a molecule are described and the ways in which the symmetry
nature of molecular motions is determined and t . and that the symmetry properties of the
electronic states an important and helpful characterization of the A now be systematic study
of the symmetry properties of molecules will undertaken. convenient illustrations of the
applica tion of symmetry arguments. for a symmetrictop molecule. It should be emphasized.
MOLECULAR SYMMETRY AND GROUP THEORY The previous two chapters have
suggested the importance of. of the molecule of a molecule provide states. Furthermore. we
will see that the wave functions that describe the behavior of the electrons of a molecule
must conform to the symmetry is band Chap. Vibrational modes merely provide. applications
methods and results will continually be made to the study symmetric molecules. in this
connection. of the molecule either must be symmetric or antisymmetric with respect to any
symmetric element. This study can be divided into two major parts. that deductions based on
the methods presented here are applicable to many areas of molecular spectroscopy and
molecularstructure studies. molecular symmetry for spectroscopy. the previous chapter has
shown that. the appearance of a rotationvibration absorption determined by the symmetry
nature of the vibration. First. To avoid a rather abstract treatment. or vibration. In moreover. It
has been shown that for nondegenerate vibrations each normal coordinate.
and It is Point Groups . All molecular symmetries can be listed. Second. element has a
symmetry operation associated with it. such as a plane of cules drawn so as to exhibit and
.elements that the molecule has its equilibrium configuration. INTRODUCTION TO
MOIECULAR SPECTROSCOPY tabulated will be given. Symmetry Elements. such as CO
and H metric. Various moleincludes a brief statement of these operations. such as a
reflection through a plane of symmetry. symmetry element E. operations of a molecule form
what be shown that the symmetry known in mathematics as a quotgroup. SYMMETRY
PROPERTIES OF MOLECULES . Symmetry Operations. and some of these applications are
illustrated.quot To proceed with a it is systematic study of the consequences of such
symmetry. which all molecules Its inclusion as an element of symmetry is TABLE The
Symmetry Elements and Symmetry Operations Symmetry elements Symmetry Symbol
Description operations E a i Identity Plane of symmetry Center of sym No change Reflection
through the plane Inversion through the center metry C. would be trivial. in Table . treated in
terms of the five symmetry elements along with the symbols used to denote them. then
developed to the extent that it is needed for application to problems of molecular
spectroscopy. and symmetry operations.quot is it will The mathematical treatment that can
be given group is for a general. Axis of symmetry Rotationreflection axis of symmetry
Rotation about the axis by /p deg Rotation about the axis by /p deg followed by reflection
through the plane . or abstract. are quotsym apparent that some molecules. is The symmetry
of a mole i discussed in terms of the symmetry. symmetry elements. Illustration of the have.
These symmetry elements Each symmetry are best explained in terms of symmetry
operations. symmetry elements are shown in Figs. cule in necessary to be able to describe in
more detail the nature of the symmetry of a given molecule. and Table The distinction
between symmetry. is necessary for an orderly treatment of the consequences of symmetry.
to a space group. . The term point group is used because. The onlycomment that need be
made about the symmetry elements is that. it will be found that only a few different
combinations of symmetry elements occur. it is customary to set up the principal axis of
symmetry vertically and. the symmetry operations that are associated with the symmetry
elements of the molecule leave a point of the molecule fixed in space. and aliens. as is This
is in contrast found in a crystal.MOLECULAR SYMMETRY AND GROUP THEORY become
apparent later. where there to it. If a great variety of molecules investigated. for the various It
is worth while to memorize the symbols used is sym metry elements. if a coordinate system
is dictated by considerations that will further used. operations result in a translation of a
molecule. to have the z axis in this vertical direction. . as we will see. in diagrams such as
those of Fig. is is a sixfold axis and also twofold axes perpendicular the sixfold axis
considered to be the principal axis. cbc. In a case such as benzene. where some of the
symmetry cell. or a unit location in the crystal. carbon dioxide. to a new RG. Some of the
symmetry element in term of the examples nansdichloroethane. Each combination of
symmetry elements that can occur is known as a point group.
S t coincident with one of the C .. be noticed that the boldface symbols used for the point
group are based. i. .. not necessary to commit to memory the symbols and symmetry TABLE
Some of the Point Groups That Are Important in Molecular Problems The number of times a
symmetry element occurs is indicated by a number before the symbol for that symmetry
element Point group Symmetry elements Examples c. C . perp..j cyclohexane Za ltr DV E. ltr
E. to the C axis. a. a. of the important combinations of symmetry elements i. the possible
point groups.. Cj mutually perp. r an. CHjClj NH HCC. HC CCH CeH D.CHClBr t Cs H . Cs.
Cj perp. C CS C H. axis. INTRODUCTION TO MOLECULAR SPECTROSCOPY A summary
that can occur. a amp through the S. t mu XlC CH D sk Da D. . cs C. t Egt C. CH PtCl. axis.
Ct. St coincident with the Cs axis. to the C. ir.. E E CH. S coincident with the C axes. trans.
S. ltr. on the principal symmetry element of the group and that the subscripts added to some
It is of the pointgroup symools further tie in the symbol with the elements in the group. for the
most part. E. axis E.. S axes. i T O E. t E. and the symbols used to It will denote these are
given in Table . OCS CHCCHC trans.. HCCH ltgta ah. tually perp. NOQ H. C. perp. Ct and St
both coincident with C .i CH. C C . cyclobutane CH benzene H.a E. Za. E. to the C axis.. to
the C.CHCCHCCH. E. gt C s perp. CsSW Cj Tgtu HCN. c Cg Cj E. Ei C.e. BC. D. to the C
axis.. C mutually perp. ff an. coincident with the C. ff. C t and C. CO.. i E. E. C mutually perp..
C t C. C. mutually perp. Ct. C. C. perp. a t. and So all coincident with the C axis. and ZCi
coincident with the C CoNH. E.
illustrations. One is counterclockwise rotation . and the symmetry elements they conIt tain.
Symmetry Operations on Molecular Motions The symmetry elements etry of a molecule in its
of a molecule are determined by the geomDisplacements of the equilibrium configuration. of
the H molecule are so described Application of a first done by describing the motion . It is.
such as occur in a translation or rotation of the molecule. or a vibration of the atoms of the
molecule.. at this stage. The symmetry elements of H .MOLECULAR SYMMETRY AND
GROUP THEORY C C FIG. . symmetry implies different symmetry operations. elements for
all the point groups. twothirds a counterclockwise rotation by one of onethird of a revolution.
can be related to these symmetry elements by means of the symmetry operation associof
ated with each element. are illustrated in Fig. point group Ci. One of the rotations and one of
the translations. point group The coordinate systems are drawn in ways that will d and CHjCI.
will be used in later These molecules. and the third of a by a clockwise or a of revolution.
symmetry operation to a molecular motion is best by means of arrows along the cartesian
coordinates associated with each Atom of the molecule. be convenient for later treatments.
which belong to these point groups. sufficient to notice particularly those of the C and C dv
groups since molecules of the type H and CH C. atoms a molecule. should be pointed out
that the C axis of a two is a other revolution. The operation symbol C designates the first of
these and C the second.
Note that the coordinate system of Fig. since the features associated with degenerate vibra
tions are then encountered. is used. Also. by for the symmetric result of These two no
change or for the antisymmetric result of a reversal in direction. CI y s . Let us consider the
effect of the symmetry operations on the is molecular motions of the to the various H
molecule. which an example of a molecule with no degenerate motions. similar manner.S T.
INTRODUCTION TO MOLECULAR SPECTROSCOPY Translation in the direction Y T
Rotation about the principal axis in the Y direction R FIG. one sees that the displacement
arrows either are left possibilities are represented unchanged or are reversed in direction. the
molecule tration CH C is also used as an illusand the three translational motions are
represented by vectors first along the atomic cartesian coordiriates in Fig. a or In a can be
associated with the result of each to the remaining translations symmetry operation acting on
the displacement vectors that correspond and rotations of the H molecule. The symmetry
operations corresponding symmetry elements can be performed on the displacement vectors
such as those of Fig. . Representation of the translation T and the rotation R of the vectors.
Representation of the translational motions of CHjCI by means of displacement vectors. . As
the FIG. When this is done for the illustrated motions Ty and Ry for H . . . HO molecule by
means of displacement in Fig.
. and these introduce a number of In CH C. DB.fl.quot The result. represent displacements
that have suitable symmetry behavior for molecular vibrations. will be recognized that the
and trans formation factors that appear in the H example are in fact onebyone as
transformation matrices. for example. . The vectors of Fig. one recognizes the reason for the
use of With this procedure and to represent symmetric and antisymmetric behavior. . CTt to
the and C ffv These symbols are to be interpreted as quotthe operation corresponding C
symmetry element acting on the molecular displacements Ty and tion. R v .rr iT i and
C.MOLECULAR SYMMETRY AND GROUP THEORY discussion of normal coordinates in
Chap.Or H H H n H H . known Draw the cartesian displacement vectors that correspond
molecule in the x and z directions and show that to the symmetry elements of the molecule
result in either no change or a simple reversal of these vectors. When molecules have a
threefold or higher axis of symmetry. it is found that degenerate motions occur. the vibrations
of the H molecule will similarly transform. it When degenerate motions are considered. FIG.
for is the motions of Fig. matrices and are examples of what are Exercise the . one
recognizes that rotations about the x axis and the y axis will lead to identical sets of rotational
complications. Do the same for rotations about axes in the x and z directions and for the to
translation of the H symmetry operations corresponding vibrations of Fig. The results of such
diagrammatic analyses can be put in a form that more convenient for mathematical extension
by writing the operation of a symmetry operation on a molecular displacement as. that the
vectors change direction as a result of the opera can then be indicated by writing c. for
example. Displacements that correspond to possible vibrations of the HO molecule.
indicated.
Similar arrows can be drawn on the H atoms. or CI. Let us now consider the operations C T
X and C T y . . . Similarly. A vector displacement indicating the motion of the C. from the top.
x T t. atom in the translation Tx and T is shown in projection in Fig. that such and that.
Operation C quot x XT Operation C Operation nquotirv r x Y. and since the effect of
operations on them is a little harder to imagine and draw. The diagrams here correspond to
end views. This behavior can be illustrated by considering the effect of the symmetry
operations of the CH C molecule on the degenerate set T x and T v . as a result of the
molecular tical energylevel patterns. they are not included. when symmetry will considered.
The effect of the symmetry operations on the translations Tx and Ty of the Cj. iden It follows.
These will behave in an identical manner to the vectors located on the axial atoms. of Fig.
OT.gttEtx . molecule CHCI. certain vibrations of such molecules will necessarily have. they
cannot be treated separately.XTt TX Operation oj x gtTx ampTt jT. HTX ltr. The FIG.Y T r x T
u T .or r. INTRODUCTION TO MOLECULAR SPECTROSCOPY energy levels because the
moments of inertia about the axes in these two directions must be identical. as we
degenerate motions must be treated as a set properties are now see.. symmetry.T rx ir.
The student with no background if follow the developments be able to he learns matrix
multiplication enough to understand in this subject should equations like Eq. Similarly. the
vectors of Tx are not simply unchanged or reversed of in direction. the result of the
transformation can be described in terms of the vectors T x and Ty by W. as i T. As the
geometry of Fig. are obtained. I r. It is therefore con venient to treat their transformation
properties together and write the result of the operation corresponding to the symmetry
element C on the pair of motions Tx and Ty by the matrix equation . Now. . with the aid of
Fig. and this is the char acteristic feature of degenerate motions. . the vector displacements
shown by dashed vectors in Fig. the result of rotating the Ty vector by deg can be described
CTy T x T y One sees that T x and Tv cannot be kept separate. for the operations of the
group Only a few features of matrix methods are necessary for the treatments given in this
chapter. group by transformation matrices.MOLECULAR SYMMETRY AND GROUP
THEORY C operation consists of a rotation by deg. writes. and when the operation acts on
the vectors of Tx and Ty . . shows.. In a similar if describe the effect of the way one can
corresponding to the remaining symmetry elements of the symmetry operations C . all Thus
one C.
Some of the reasons for setting up the system that is actually used will. Symmetry Species
and Character Tables A brief introduction can now be given to the ways in which the
behavior of molecular motions with respect to symmetry operations is usually tabulated. and
a and so forth. is . INTRODUCTION TO MOLECULAR SPECTROSCOPY and. with respect
to symmetry operations. Rot. R. The transformation matrices For the present it work. T. . V
will play an important role in later enough to recognize that they allow the effect of the
symmetry operation to be represented in mathematical rather than diagrammatic forms. as
we will see later. not be completely understandable until later material is studied. symmetry
elements E and C but for a. The behavior of the motions of . and trans. for R they are for the
. is given in Table for . a number of different molecular motions. v Since. TABLE Summary of
the Transformation Matrices for the Symmetry Operations for the Translations and Rotations
of the C Molecule H ltr Csu E C. Tz This table summarizes the fact that the transformation
matrices are for each symmetry element. or in The transformation matrices that H fact any C
molecule. arise when the motions of a molecule belonging to a given point group are
investigated can be arranged in tabular form. including for completeness the C operation
given above. Rv Tx T Rx . . however.
as can be readily checked from the matrices of Table . The procedure is to designate a row
by A if the transformations are symmetric with respect to the principal axis of symmetry and
B if antisymmetric with respect to this axis. as illustrated for to tabulate. as we will see later.
to be sufficient for most purposes a group which. not the complete transformation matrices. it
is convenient to introduce symbols for each row. . the two B rows are distinguished by Bi and
B . point group. T Rx. and a as TABLE The Transformation Matrices for Cj. T. Rot. of the
symmetry behavior for C . with this added notation. and also the various electron orbitals for
the molecule. as Ai and the second row as A. Ry. where the completely symmetric set of
transformations written. Tz and T in the previous section. are summarized in Table .
molecules shown in doubly degenerate symmetry species is generally labeled Table E.
Similarly. Bx B. A slightly different notation is used if the molecule contains a center Then
one uses. In order to distinguish the two A rows one labels the is first row. a subsymmetry
species that is symmetric with respect to inversion at the center of symmetry and u if it is
antisymmetric with respect to this of symmetry. It turns out. script g for a operation. leads to
degenerate sets of translations and rotations. The transformation matrices that are found for
the C . Thus the first two rows of Table would be labeled A and the second two B. The rows
of such a table are said to be symmetry species to which motions of a molecule or electron
orbitals of a molecule belonging to the point group of the table can be assigned. and trans.
Ty . E Ct a.MOLECULAR SYMMETRY AND GROUP THEORY including vibrations as well
as translations and rotations. Ax A. instead of the subscripts and . the sets of transformation
C r symmetry are given in Table . R. matrices of a molecule with Thus. Molecule with
Symbols for the Different Behaviors with Regard to A Symmetry Operations c. to the
summary . not to be confused with the identity symmetry element. transform with the
transformation matrices given in one row of such a table. but merely the sums of the diagonal
elements of the transformation matrices. This procedure leads.
a a a w h IN ICC bquot gt i gtr i w Q. a . rH H V O a CO lt o . s i s as . rt N i ICO gtr i o Kl . lti
win b h I IN ICO o e s i gtr quot .quot a amp. o h j iH i b o N r t in i i o i gtr O gtl lt l IIN o o
quot.
C. and trans. J. T..Ry. Princeton. A. and P. quotInfrared and Raman Spectra. G. and G. and
trans.. Wilson. really to be interpreted as the If sum of the diagonal elements of the is
transformation matrix. in fact.T v . itself. point group one abbreviates Table and writes the
information as in Table . At E T. therefore possible to abbreviate such tables only one column
for each type of symmetry element. Eyring. Decius. properties. B. or . by showing The
number of The and basis.quot McGrawHill Book Company. Inc. R. Herzberg. T. is Their is
introduction here used. symmetry elements in the class is then indicated at the top of the
column. R. of the remainder of the chapter. the matrix a onebyone matrix. Thus for the C .
intended principally to establish the notation that One will notice in many advanced texts on
spectroscopy and quantum mechanics that such tables are given for all possible point
groups. ff Rot. quotMolecular Vibrations. New York. be seen later that each entry for the
nondegenerate species in tables such as Tables is and is. J.quot D. New York. identical with
the transformation matrix.. TABLE The Sums of the Diagonal Elements of the in Con
Transformation Matrices of a Cgt Molecule densed Form Cg E C. E. for example.. The and
identical transformation matrices for the two rotation elements in for the three planes of It is
symmetry that can be noted Table are typical. Inc. Kimball. Inc. A. . See.J. and H.quot Wiley
amp Sons. this of course. CI ltr Rot. Walter. .Ty E X triply degenerate set as F or T. and uses
of tabulations such as Tables will be the subject. R.Rvt Tx.MOLECULAR SYMMETRY AND
GROUP THEORY TABLE The Sums of the Diagonal Elements of the Transformation
Matrices of a Cj Molecule Cj. E c. K. Cross. N. Rx. quotQuantum Chemistry. Van Nostrand
Company. C. It will.
that are of great importance in molecular spectroscopy. which are important for molecular
spectroscopy. a set of elements. which . for example. be developed later. nondegenerate
character tables by this method. usually To be a mathematical group. some of the general
grouptheory The Nature of a Group To be a mathematically useful term. The character table
for any point group can be deduced by finding the transformation matrices for various
motions of a molecule belonging to that point group. with examples. INTRODUCTION TO
MOLECULAR SPECTROSCOPY The first of these references also gives a detailed
discussion. and it will then be easier to understand the larger character tables containing
degenerate types. of symmetry elements and point groups. since the sum of the diagonals of
a transformation matrix is known as the character of the transformation . The theory can. . an
extensive digression into some of the general mathematical properties of the transformation
matrices and character tables will be made. and deductions of properties of such groups. a
collection of items. We will then see how the character table can be used to deduce some
results completed. The introductory summary of molecular symmetry has now been Before
making use of the quantities introduced here. given a precise definition than the meaning.
many matrix. It can also be mentioned here that. Some of the mathematical features that can
be ascribed to general. however. groups will be pointed out. A number of general properties
of character tables will. GROUP THEORY An apparently rather abstract mathematical
subject called group theory turns out to be directly applicable to investigations of a number
symmetric molecules. symmetry operations that have been treated in the preceding section
and can be used to draw a number of important deductions regarding the nature of the
vibrations of symmetric molecules and the electronic properties and electronic transitions of
such molecules. these tables summarizing character tables for the symmetry behavior are
known as the symmetry point groups. or deal with the abstract. The previous treatment of
symmetry elements and symmetry of properties of operations provides material with which
results can be illustrated. the word group must be more associated with it. Little difficulty will
be encountered in checking some of the smaller. will be made.
therefore. the procedure need not be the familiar multiplication. for is example. When the
elements P and Q are combined one writes PQ. An inverse is defined so that if S. combined
and that the element they form then combined with It should here be mentioned that the
commutative law will not be stipulated as a requirement for a group. must conform to four
requirements. It should. moreover. operations. makes PQ equal to an element of the group.
Q. S. all elements of the group. Thus PQ will not necessarily be equal to QP. . is also For
every element of the group there must exist an inverse which an element of the group.
following conditions are satisfied a rule for combining any two elements of the group.quot
That they make the term group a useful mathematical concept will be apparent as our study
of the theory that can be built up for groups develops. There is the combination procedure
multiplication although. An identity element E is an element with the properties such that EP
PE P EQ QE Q ER RE R and does so forth. quantities. combined accordand then P
multiplies the element formed rule. One must.e. for all elements of the group. and so forth.
Commonly one calls . and the combination of any two elements according to this rule leads
to the formation of one of the elements of the group.. The first requirement for a group is that
there be a rule which makes the combination PQ meaningful and. means that Thus if PQR Q
and R are multiplied. be pointed out immediately that these requirements may appear
quotunreal.. . . R. be careful to preserve the order in which elements are written. is said to
form a group if the A set of elements P. ing to the appropriate requires law from QR. Note
that the element E commute with . the associative The associative law PQR PQR The
righthand side implies that is P and Q are first R. perhaps. The set must contain an identity
element. . which we will designate by E.MOLECULAR SYMMETRY AND GROUP THEORY
can be objects. the inverse of Q QS SQ E . must hold for the combination of elements. i. as
we will see. .
. We find Left side ltr v Cr v ltt v Right side C v E gt E ltrri gt In a similar manner it can be
verified that various ways of associating the elements in a sequence of operations lead to the
same element. .rC.. .e. that successive operations are always equivalent to the single
operation of one of E. group. . i.. C ltf v and c These operations will be recogt . has been
included in the set of symmetry operations apparent that these operations will constitute a
group. i. is a member to be a group. they obey the associative law. ltr. INTRODUCTION TO
MOLECULAR SPECTROSCOPY The inverse of the element Q will for the set of elements of
which Q sometimes be designated as Q l and. the combination CV. The associative law
tested by investigating such combinations as ltriltV. Q.ltri C CW Cr. by operating on various
rotational and translational motions of H . EC dE C etc.e.. nized as forming a group if the rule
of combination is successive operation on the molecule by the symmetry operations. One
can verify. and the symmetry operations for such a molecule are those corresponding to the
symmetry elements E.e. and then with the element on the left. The Symmetry Operations as
a Group The rules that must be obeyed by a set of elements if they are to constitute a group
can now be illustrated by the symmetry operations belonging to a given point for a molecule
of a particular symmetry. inverse do commute. Let us of ments . is a. . i. EC C ltr.. in which
no change It is is imposed on the so that molecule. ltt v etc. for example. is written to operate
first with the element on the right. . One generally agrees if. C ltr or ltr Thus t C. must Notice
that an element and its be one of the elements of the group. The identity operation. now see
if the symmetry operations conform to the requirea group. i.e.. . The H molecule was seen to
have the symmetry C . ..
verify that the elements of the Using T. number which represents the element R. Any
deduction that can be made about the elements of a group is. and the fourth requirement
Exercise . binding on the symmetry operations of a molecule. have shown. i. C . C v Using T
x and T y as a basis. can be represented erally matrices.MOLECULAR SYMMETRY AND
GROUP THEORY . or of a group and can . if PQ R. C v group own inverse. postulates. It will
be found that these sets of symmetry operations similarly obey the group requirements. C
group do. obey the group Exercise . every element is that there some other element of the
must be such group which. To satisfy the inverse requirement. involving molecular problems
theory to group that ties abstract result symmetry.. they are a group. written before or For the
after the first element. the multiplication of the For the representation to be trice. produces
the identity. tion. . such as the symmetry operations of a molecule. numbers representing
elements P and lead to the Q of the group must. in fact. Verify by drawing the appropriate
displacement diagrams that two successive symmetry operations on a translation or rotation
of molecule are equivalent to a single symmetry operation of the the molecule and that this
single symmetry operation is a member of the H C iv point group. We Exercise . each
element of the is satisfied.e. which can be assigned to the elements A set of numbers. the
symmetry operations associated with other point groups can be tested. E oW. It is this
therefore. or more genand these can then be combined by ordinary multiplicaor faithful. by
numbers. In a similar manner. verify that the elements of the group obey the group
postulates. Representations of a Group Of particular importance to be put is for the purposes
to which group theory is the fact that the elements of a group. matrices. E tbis will Thus. and
is its not generally be so. therefore. point group we see that CC E w. as a basis on which to
apply the group operations. that the set of symmetry operations conforms to the
requirements for a group.
c / a.. D D D r D D It can easily be verified that these representations all properly represent
the set of the symmetry operations of the point group C . tant representation of the An
constructed by assigning C t to apparently trivial but. said C point group can be used
Customarily. in fact. for example. r r and r can be given as . To do this one writes. Thus the
representations Ti. to illustrate this feature. they are onebyone matrices. representations are
designated by T and the components of a particular V are written under the group Again the
symmetry operations of the elements that they represent. Thus one would write E The
parentheses are placed about each number because. this is properly represented by the
multiplication Three other onebyone representations the way in which they were deduced will
be shown shortly can also be written and a table of representations can be constructed. as
we will see. . E Ti Vt r. group correspond to this equality. INTRODUCTION TO MOLECULAR
SPECTROSCOPY is properly represent the multiplications of the elements of the group to
constitute a representation of the group. and investigates the ways in which the various
representations rpu Thus of the For the Ti representation For the r representation For the r s
representation For the Ti representation Dl . Since any combination of symmetry operations
leads to one of the symmetry operations of the group. imporgroup of symmetry operations
can be each element.
It follows. of the transformation matrices of C and ltr v and this is an illustration of a general
result. write C. will be properly represented . r are represented by trans formation matrices.
We must now see why such transformation matrices should form a representation group of
symmetry operations. of the Cr v Tx c v Tx U The operation CrTx of ltr on Tx will give lTx and
one can . This implies that and C can be written in either order. D the direction of the
Furthermore.r sees that.lr. A similar test of other combinations will show that multiplication of
the symmetry operations is always faithfully represented. and one can write The operation C
c. for example. they commute. if C s and a. one represents rv Tx by Tx .e. that the
transformation matrix for CV. since as Table shows. it is interesting to see that they are
nothing more than the transformation matrices discovered for H in Sec. In this ic. Test that
the representations of the C group faithfully represent the multiplication of several pairs of
symmetry operations. i. in each case. While these simple representations might have been
discovered by trial and error. This example shows that. that the combination of symmetry
operations corresponds to simple multiplication of the corresponding transformation matrices
and that this combination is faithfully represented by transformation matrices. Although the
transformation matrices dealt with so far do form . is if iiVgt ir.T. to give the number which
that representation attaches to a. by the product therefore. the operation ltr v on T x changes
Tx displacement vectors.. C is ltr way one represented the combination Cr v represented by
and by the simple multiple by . acts only on Tx and cannot affect the pure number . is equal
to the simple product of the transformation matrices To do this. the product CV. Exercise . It
is necessary to show. one might investigate the operation equation for C and
ltr.MOLECULAR SYMMETRY AND GROUP THEORY The result of the four multiplications is
seen.. .
in view of Fig. A particularly important one for a given molecule is that based on the Sn
cartesian displacement coordinates that can be associated with the atoms of the Such a
representation can be set up by writing the n carcolumn matrix and deducing. There are.
effect of the ltr coordinates ofquot that x H H . tesian coordinates as a FIG. Some of these
that These genmight be built up from those of the previous table are E c o GO lit / G /l G /l G
/l O / o o / o / G and so forth. J t G G faithful representa Again one can verify that these form
tions of the group C v . by looking at a diagram of the molecule and the n displacement
vectors. For example. they do not form the only representations. Other large matrix
representations can be found. an infinite number of representations. the operation ltr on H
can be seen to lead to the transmolecule. that xh transforms as shown into xlt and Operation
ZB SO. xii quotH . erally are square matrices of various orders. in fact. The xhgt.
INTRODUCTION TO MOLECULAR SPECTROSCOPY representations of the group. One
sees. for symmetry operation on the n cartesian displacement example. . the Zn X n
transformation matrix for the particular operation.
there will be only a . if . is broken down into representations which dimension. We will find
that for any given group. C.T V or R X R y . with an inverse /J . Verify that the transformation
matrices for a given molecular motion provide faithful representations of the symmetry
operations of the C . matrix multiplications can be performed to give . could be built up to
give any number of representations involving matrices of larger Of much more importance is
the process by which a largedimension matrix representation. consist of matrices with as
small a dimension as possible. . forms a representation of the group. . Xo . y zo The
corresponding transformation matrices of the for the other operations group can similarly be
found. Thus. there is a square matrix /. although there are an infinite number of reducible few
irreducible representations. such as those involving onebyone matrices. Do this for the
onebyone matrices that are obtained when T z or R is considered and also for the twobytwo
matrices that are obtained on the basis of T X . and those which involve matrices whose
dimensions cannot be further decreased are said to be irreducible representations.
Reducible and Irreducible Representations In the preceding section it was shown that
representations of small dimension. B. . representations.MOLECULAR SYMMETRY AND
GROUP THEORY formation matrix Xh Vh zh xH Vh zh Xh xH Vhgt zh Vh Zh x y o . A.
Exercise . Suppose that the set of largedimension matrices E. New matrices can be obtained
from these by what are called similarity transformations. group. In this way one would obtain
a representation of the C group consisting of four n X n matrices. . Representa tions that can
be so broken down are said to be reducible representations. such as the n X n one based on
cartesian displacements.
. and so forth. also form a representation of the group. from Eq. on the by p Since //. where .
are square matrices of the same dimension. this D. and so forth. . . . B. o . will jSCjS The
new matrices ment that then AB to A.e. connected to the old ones by a similarity
transformation. C oi.l AWBp / / p l D each other. ai might be possible to find a transformation
matrix such that are of the form . c h If such transare square matrices of the same dimension.
B ft. B the eleif B did. and since EB B. which led we substitute the expressions necessary to
show that and D into the equation AB D and obtain p. left ptABp PDP Now we on the right
can multiply both sides of this equation.. i. . now be shown It is to be also a representation of
the group with A representing the group element that A did. B. Now all it the matrices A. p l P
E. INTRODUCTION TO MOLECULAR SPECTROSCOPY A B C r l A p Bp l etc. and the new
matrices. one has by / and AB D Thus AB D is equivalent to AB D. one has. Now since and
are the inverse of fl E Q I J . To do AB D. /ampi . fti. . etc. . a o a.
groups in Sec. . . C. Any other matrix representation can be reduced to one or more of these
irreducible representations . . the only irreducible representations for these
groups.MOLECULAR SYMMETRY AND GROUP THEORY formations can be performed. . a
b . Vr /jr. since matrices are equal only their corresponding elements are di ds lt It follows that
ai. and so forth. . the previous multiplication result AB D equal. . The orthogonality property
requires that. if IiJ is the matrix which represents the . As the heading ible representations of
this section suggests.ffth symmetry operation of the group in the jth irreducible
representation. respectively. the repre said to have been reduced by thequot similarity
transformation. it should be apparent that their orders are and . aiampi amp or AB D if
becomes. . are irreducible representations. sentation is . Since they are composed of
matrices of . C. The symbol g will be used to represent the order of a group. the
representation is said to be irreducible. . . rifliyfl X B o o etc. also form representations of the
group. . . Before proceeding to a general statemight be helpful to illustrate these properties
representations of the C and C point First it is necessary to define the order of a group as the
number of elements in the group. Ci. the elements of the irreduc behave like the components
of orthogonal vectors. . . or A. The transformation matrices deduced for the C and C r point
groups in Sec. From the treatment of the C and C . . as will be shown later. c . . B. If it is not
possible to find a matrix / that will further reduce all the matrices of a given representation. by
a suitable similarity transformation. . it by reference to the irreducible groups. They are. The
Irreducible Representations as Orthogonal Vectors We come now to some of the most
important properties common to all irreducible representations. ment of these properties. B. .
smaller dimension than A.
for example. can be expressed as Ir R i i m rJ i m v o ij I R r i r R mv i m m. These results for
the C and C groups are examples of an important row. Let us now see if these orthogonality
and normalization properties also apply to the C irreducible representations group where
degeneracy occurs and one of the One can test the is a twobytwo matrix. notice that if the
representations only by treating Thus one can first the elements of the representation listed
in Table . which have mnth component of the tth representation for the ation been illustrated.
entries in the column. n iA n I URUURU . the first row. group corresponds to the components
of a vector normalized to a value equal to the order of i.B R for i j C Furthermore. for any
component. of However. can be represented as rltflr. t or of T gives the value of the squares
of first .e. first general property of irreducible representations.. all orthogonality condition for
separately each component of the twobytwo matrix. in general. If YiR mn designates the R
symmetry operand r. multiply. for example. TRMR I r X R RT R etc. the sum any component
of the matrices r the . each representation of the the group. the order of the group. gives the
value . the order of the group divided by .mv designates the mn component of the jth
representation for the R symmetry operation. first column elements of the third
representation r the sum of the products Likewise the sum of the squares of the elements of
T is again zero. INTRODUCTION TO MOLECULAR SPECTROSCOPY Inspection of the
irreducible representation table for C confirms this property which. . which we will now label
as r . the general properties.
group. Verify that the elements of the transformation matrix C group behave like orthogonal
vectors. If I is not neces is the th irreducible representation. therefore. or their components in
a g dimensional space. one writes XiR X r. This fact introduces a considerable simplification
into the applications of group theory. The character is of a representation of the Rth
symmetry operation denoted by xR The definition of the character can then be written as xR
T R X m where r without a subscript implies a representation that sarily irreducible. presented
in some detail in characteristic of the set of irreducible representations that exist for a point
group can be immediately drawn from the if fact that the irreducible representations. only be
g orthogonal vectors. relations are always It is obeyed by irreducible representation will not
be given here.and threedimensional it is space. One important .quot given in Table . Do the
same for the matrices of the representations of the C group. behave as orthogonal vectors
By analogy with two. a complete set for this fourthorder group. the two the Thus in g
dimensional space there can The four onebyone representations of onebyone and one
twobytwo matrices. C r group given in Table form. One sees that we now have a a proof that
there are no irreducible representations other than those that have already been reported for
these point groups. for the sixthorder C. they are twobytwo or threebythree
matrices..MOLECULAR SYMMETRY AND GROUP THEORY is where g is the order of the
group and U the dimension of the tth irreduc ible representation. Characters of
Representations The characters of the matrices of a representation are defined as the sums
of the diagonal terms of the matrices. The proof that these Appendix II of ref. Exercise .
representations of the . the latter contributing four quotvectors. It turns out that the characters
rather than the complete matrices can be used for many important deductions. Likewise.fi
mm . form a complete set. One has way of determining when one has found all the
irreducible representa tions of a group. apparent that one can have only as many orthogonal
vectors as one has dimensions in space.
. .lP is If the process of matrix multiplication involved in written out explicitly. in Sec. The
derivation. The student can proceed directly to the result.l Q is obtained as X i Qu J kl G p
KflGl p i amptiPw ZP S. Now we will show it that. The Kroeniker defined such that Su equals
for k I and for k I. involving more matrix manipulation than previous work. Eq. the sum i.
Equation can now be written as Qu I kl GkP of the diagonal elements of Finally. although the
similarity transformation if changes the matrices. the characters of the matrices unaltered.
need not be followed through. INTRODUCTION TO MOLECULAR SPECTROSCOPY We
the have already seen. We wish to show that if P ity transformation and Q are matrices
related by the similar Q pWe then xq xp . since is presumed to be an orthonormal matrix. one
expresses the tth diagonal element of Q by Qu X u / Gquot Wi/ Now. k /S T where / indicates
the transposed matrix. product of two rows of the orthonormal matrix . the similarity
transformation matrices and normal. that is if a set of matrices that constitutes a
representation subjected to a similarity transformation new set of matrices that is obtained is
also a representation for the group. are orthogonal leaves. has been used to express the
result of i j/Sk which corresponds to the .k.
on irreducible representations. one maintains the orthog onality result of the expression
because of the orthogonality of the components that are summed when the characters are
formed. in view of Eq. of the partially diagonalized matrix will be the same as that of the
original matrix. R The proof from the results ff of these character relations is easily given. We
have. C and groups will show that the characters. where k TiR nm T.Rw for i j TiiRViiR f is
the order of the matrices of the ith representation. . Alternatively. behave as orthogonal
vectors. in terms of the diagonal components . their behavior can be written as The general
expression for XiRxAR for i j R and X. the product of the characters of one representation by
itself can be written out. one can say that the sum of the characters of the irreducible
representations that can be obtained from a reducible representation will be the
representation. If the sums of the diagonal elements are first now taken. Inspection of the
characters of the representations of the C. MOLECULAR SYMMETRY AND GROUP
THEORY We xq have now reached the desired result xp This important result will be used
when we investigate the irreducible representations that can be obtained from a reducible
one.. like the representation matrices themselves. same as the character of the reducible
This result will be of great value. We see that if the breaking down of the large reducible
representation is depicted by o the character. Similarly.
e. as we will be possible to make use of the characters of the representations and the
relations involving the characters that have been obtained here rather than the
representations themselves. and give properties of the characters of two representations as
one goes across a character table. It is interesting to notice also that one can relate the
number of irreducible representations to the characters that appear in the column under the
identity element of the group. In view of the discussion at the end of Sec. . one sums over the
various operations of the group. INTRODUCTION TO MOLECULAR SPECTROSCOPY that
are involved as / / TiR mm TiR mm I k m g In this way the result of Eq. . lts and ltrquot are
also identical. i. This follows from the fact that the character of the repreis sentation of the
identity operation tion. a v identity operation. simpli grouptheory applications. Inspection of
the character table for the C group shows that the characters of the representations for
identical C and C are operations tions. belongs to a class by . is obtained. . symmetry
operaany other operasymmetry The and the operations ltr. . Exercise . behave in the manner
indicated by Eqs. The expressions of Eqs. and Verify that the characters of the C and C t
groups. and . that the representation contributes. Exercise . The C z and Cjj are said to
belong to one class of . Classes One fies further concept. and those for a v .. equal to the
order of the representagives the The square of this number number of vectors. that of
classes of symmetry operations. therefore. Verify that the C tgt and C group characters
behave according to Eq. Similarly. one can write. belongs to a class by itself. it will In almost
grouptheory applications. X E the X E xKE g where c is number all of representations in the
group. in Tables . given . see. having characters unlike that of ltrquot and are said to belong
to another class. tion.quot as discussed in Sec. each itself. since all four operations of the C
group have different columns of characters.
when subjected to the operation X PX where and XQX which is X is in turn all the elements
of the group. X as ltr. In a similar way one could verify that E by itself and a v a and aquot
form classes. MOLECULAR SYMMETRY AND GROUP THEORY P and Q. . The detailed
working out of the combination X PX is not usually necessary in order to decide which
symmetry operations belong to a class. Thus C and C constitute a class. indicated as was
done in the and C v character tables. l In abstract group theory a class consists of a set of
elements. say of the group which. and one will find that rotations such as C and C and
reflections through similar planes such Cj. When the character table is written with the
characters for the different classes exhibited rather than the character for all symmetry . P or
Q. C and C to show that they C l . is a member of the set. ltr lttquot constitute classes.
however. the result is either C or C. The following combinations are is and the operation to
which they are equivalent noted EC E C Cj C C C CflCCJ PKr. The number of elements in
the class is. The matrix of each member of a class transformed into another member of the
class by at least one of the transformation matrices. It is this result that allows character
tables to include only typical elements of each class rather than all the elements of the class.
when C and C are subject to a similarity type operation by all the elements of the group.
gives a result i.. the same character. When the multiplication of the representation matrices
replaces the corresponding combinations of symmetry operations. C C Ocv c oigtC C OCX c
wycy c We see that. This general definition can be applied to constitute a class in the group
investigated.e. It follows that these this In way all two matrices must have members of a
class can be seen to have the same characters for all irreducible representations. the
previous result that a similarity transformation leaves the character of a matrix unchanged
can be used of is to deduce that the representations of all members a class have the same
characters. Geometric consideration will usually be sufficient. aE CC E i a C CI c CIVCICI
ffCJir.
l aiKR W . of its class. the character table a square array. say n. and so forth. be reduced it
contains by a similarity trans to the irreducible representations that formation. a simple and
important procedure for dis covering in how many times each of the irreducible
representations occurs a given reducible representation. which leaves the character
unchanged. recalled. Thus we conclude that there can be only as many representations as
there are classes. to the previous This conclusion is parallel. in each the characters behave
as orthogonal vectors in a space with dimension equal to that of the ber of different classes.
counting one for a onebyone representation. it was pointed out that. Since any reducible
representation can. in principle. allowance is made for the num of elements. one can write xR
. as there are symmetry elements. in principle. Analysis of a Reducible Representation The
final aspect of abstract group theory that must be dealt with we can proceed to some
applications is the question as to whether. If R now denotes a typical symmetry element and
can be written as Over all lVnkxiRVnixiR for i j and Over all classes X VrxtRlVnixiR Thus the
/nXiR terms behave as components of orthogonal vectors. formation to convert the reducible
representation to a form in which the In practice there is no convenient this. . if That this must
be number so can be shown from the fact that. but this does not alter the argument given
here. one could find a similarity transbefore irreducible representations appeared explicitly.
four for a twobytwo. there is any convenient way to deduce what irreducible representations
make up the reducible one. INTRODUCTION TO MOLECULAR SPECTROSCOPY is
operations exhibited. way is. also be It should mentioned that the factors n are usually placed
at the top of the character table rather than as part of the characters themselves. and there
can be only as many such vectors as the dimensionality of the space allows. of finding the
transformation matrix that accomplishes There however. class. In Sec. given a reducible
representation. Eqs. it should be one that there can be only as many representa tion vectors.
For expression requires the operations. summation over all symmetry more convenient to
have the corresponding expression where the summation is over all classes of symmetry
operations. In this way we obtain xRxiR I R micmxim lt R jl and . that Now the results of Eqs.
It is The above this one writes V Over all classes nnxRxiR where n R operation is is the R. .
to convert Eq. the characters of a reducible representation are known. ltgtltgt j. x R are the c
irreducible representations of the group. The methods of application will be illustrated in the
remainder of the chapter with problems based on molecular vibrations. tit w can be used on
the right side of Eq. and the a/s are the number of times the jth irreducible representation
occurs in the reducible representation. of the irreducible representations are available and
the characters from a character table. number of elements in the class for which a typical We
now have the basic results from abstract group theory and can proceed to apply them to
problems in molecular spectroscopy and molecular structure. when these subjects are
studied.MOLECULAR SYMMETRY AND GROUP THEORY where xR is a reducible
representation. the number of times each irreducible representation occurs in the reducible
representation can be readily calculated. Similar applications to electronic structure and
electronic spectra will be made in Chap. We over first multiply both sides of this equation by
xgtR and then sum all symmetry operations R. to xRxiR or at ckg X if xRxi R o Thus.
These transformation matrices would constitute a reducible representation. Ci. . etc. .
however. The character of this reprethe sentation. dimensional representations. d v and
Exercise . which is found rather easily. Cz. and The Characters for the Reducible
Representation of Molecular Motion The re effect of the symmetry operations on the motions
of a molecule could be represented by the transformation matrices that show how the normal
coordinates including translations and rotations are changed when the various symmetry
operations are performed. the irreducible the Zn dimensional representation are of make up
It follows that it is only necessary to see how many times the various irreducible
representations occur in the reducible representation to discover how many of the n normal
coordinates belong to the various symmetry species. or a degenerate set must transform
according to two. We cannot. Deduce the irreducible representations contained in the
reducible representation of the C group which has the characters ltr. representation on the
basis of the n reducible the of constructing Such transformation matrices were cartesian
displacement coordinates. construct the reducible representation on the basis of the normal
coordinates because these are generally is initially unknown except for the translations that
overcomes this difficulty differ and that which rotations.e. transform according to the various
irreducible representations. group are of. deduced for the H molecule in Sec. .
representations that particular interest. and . irreducible Deduce the representations that the
reducible representation is made up for the classes of E. ra The normal coordinates are said
to form a basis for the representation Since each normal coordinate must transform
according of the group. The characters of a reducible representation of the .. normal the n on
representation based . a v a v . the two linear are of one base representations are said to be
equivalent and the characters of the two The result is based on the fact that two such to be
related by a similarity transformashown be can representations can resort to the much
simpler procedure one this theorem With tion. . to an irreducible representation. three. for the
classes of E.. states if A helpful result two representations only in their basis coordinates and
if the coordinates of a group combinations of those of the other base. will be the same as that
for coordinates. . representations are equal. i. INTRODUCTION TO MOLECULAR
SPECTROSCOPY Exercise . .
which constitute the reducible representation. on the diagonals done. Zo can be found by
inspection of diagrams such as that shown for the symmetry operation ct in Fig. The four
transformation matrices for the H molecule are found to be E . . . It is is not necessary to
write out the complete only necessary to deduce the elements that appear of these matrices.
matrices. for vectors are indicated as H The n cartesian coordinate The matrices which type
XH yn describe the transformations by equations of the Xh Vh Zh .MOLECULAR
SYMMETRY AND GROUP THEORY To obtain the character of the n dimensional
representation for the it molecule being studied. this can easily be we again consider the
transformation matrices. Xfj /o Transformation matrix Zh x /o J . To show how . XH Xh z.
therefore. one can write . lt o One should first notice that diagonal elements occur only if the
sym metry operation leaves the position of an atom unchanged. respectively. not atoms that
change positions as a result of the symmetry operation. INTRODUCTION TO MOLECULAR
SPECTROSCOPY T . along the diagonal. by concerning oneself only with the cartesian
displacement vectors of the do. One can deduce the character of the representation matrix.
With this recognition and the fact that the vectors of unchanged atoms in molecules like
effects lead to H are left unchanged or are reversed and that these and .
will contribute to the appropriate a character. to the character of the n dimensional
representation for the operations C and S.MOLECULAR SYMMETRY AND GROUP
THEORY down. it if it if will contribute a net contribution any atom is unchanged a in position
by a reflection through a plane. the result E C av The sums of the diagonal elements of the
complete transformation matrices given above confirm these values. cos sin sin cosd
ltRolatiol. three cartesian coordinate vectors will be reversed. its position will be unchanged
all symmetry and by inversion at the center of symmetry. FIG. rotation tributed by atoms
when. Furthermore. Slightly will be made to the more difficult to visualize are the characters
conso forth. Figure shows the geometry that leads to the transformation matrices and the
contributions to the character in such cases. their positions are unchanged. general
statements can be made about such characters. a molecule has a center of an atom is
situated at this center. each atom on the axis. as a result of a C d. One might further notice
that simple. i. Similarly. and a contribution of character. The contributions for each unshifted
atom. and . operation. by inspection of the displacement vector diagram.gt quot Rotation fay
cos gt and reflection through plane perpendicular to axis COS v . When an atom is
unchanged in position as a result of a to the C C character.e..
Now one calculates the number of normal modes that transform first according to the
representation. i. for molecules of the type . i. The H example can again be used. to be
determined. c . The characters of the irreduc ible representations are given in the character
table. n dimension representation Number of Normal Modes it of Various Symmetry Types In
the previous section sentation for a molecule could easily be set up.. the contributions that
make is to the characters of the reducible representation This done in With these and
recipes. c u a. and in Sec. C ci. of the Exercise .. shown that the number of times the various
irreducible representations are contained in a reducible representation could be deduced.
including translation and rotation as well as vibration.IM INTRODUCTION TO MOLECULAR
SPECTROSCOPY With these unshifted atoms of order w. allow the number of normal
modes of the different symmetry types.e. one can readily write down the characters of the
reducible representation without visualizing the effect making the cartesian displacement
diagram of the various symmetry operations. was shown that the n dimensional repreit was
These results. Table . along with the fact that each normal mode. generalizations we can
summarize Table . belong to the symmetry TABLE Contributions to the Characters of the n
Dimensional Reducible Representation Operation per Unshifted Atom Character contribution
E a i C C. belonging to the different reducible representations. Deduce the characters CH X.
transforms according to one of the irreducible representations.e. The characters previous
section as of the reducible representation were given in the B Ci av it.
.. Thus M gt i n RX RxiR Over all classes Similarly. played for H in Fig. The Infrared Active
Fundamentals apparent that some vibrations of symmetric molecules. indicated in the last
column of Table .MOLECULAR SYMMETRY AND GROUP THEORY S type A.e. molecules
of the type CH X because of the previous discussion of rotaof the different types of
vibrations. quotquot C gt. are removed.e. to decide for i. therefore. the necessarily zero
oscillating dipole It is possible. a a. such as It is the symmetric stretching oscillating dipole
mode of CO. perhaps more apparent for difference. have no moment and therefore cannot
interact with electro magnetic radiation. Exercise . i. of the vari band contours which pointed
out the great and perpendicular bands.e. in fact. two vibrations of the symmetry type A and
one symmetry type B One can verify that the displacements dis. Determine the number of
normal vibrations ous symmetry types for the molecule . i i and a These results can be
summarized by saying that the reducible representation contains representations of the to
symmetry types according r If Aj A Bi B the assigned translations and rotations. Furthermore.
by application of Eq. are. we will now will see. ai . In a similar way one can of classify the The
importance tionvibration parallel such classifications is normal modes of any molecule. for .
transform according to these symmetry types. do. CH C. which symmetry types the
fundamental transition be active.. i. one has r vlb There of the Ai B. as moment is a result of
the symmetry of the molecule and the symmetry type of the vibration.
. moment will not necessarily As was pointed out in Sec. A nonzero integral leads to a
nonzero value for integrated absorption coefficient. INTRODUCTION TO MOLECULAR
SPECTROSCOPY oscillating dipole which symmetry type the be zero. the value of the
integral must be unchanged for all symmetry operations on the molecule.. on the quantities in
the integrals and then on the complete The dipole components can be represented by the
charge separations shown in Fig. Tii y . only if one of the integrals fMQltnMQi dr dr dr
JMQivUQi or SMQinMQi is nonzero. . The dipolemoment vector of its a molecule and
cartesian axes. as indicated intensity is an observable quantity. Since this band must have
the same value for all the molecule. It follows that the representations T/x x . a fundamental
vibrational transition. It should be clear that these dipole components will transform under the
various symmetry operations in exactly the same way as do the displacement arrows
representing Tz Ty and T . FIG. It follows that for any it moment integrals to have a nonzero
value. from v to v . indistinguishable orientations of of the three transition by Eq. It is now
necessary to see the effect of symmetry operations integrals. and ry will be the same as the
representations Igt T T and Tt. The lAvib vibrational wave function for a molecule can be
described as vQivQtMQ state. ltp all For the ground where vibrational quantum numbers are
zero. components along the . can occur with the absorption of electromagnetic radiation in
the vibrational energylevel pattern corresponding to the normal coordinate Q. .
for example. and therefore ertQevM A . For nondegenerate vibrations Qi symmetry
operation on gt Qi. therefore.MOLECULAR SYMMETRY AND GROUP THEORY one has ltA
gt ltPoQiltPoQzltPoQ quot quot quot ltPoQine For harmonic oscillator wave functions this
total wave function has the form. the vibrational wave function of the molecule will transform
as does the degenerate pair of normal coordinates. YWO .ero gtgt where lty k Now the effect
of a can be investigated.. according to the normal coordinate vibrations are excited to the v
Similarly. In this case.i wave function is of the form const QieTe v.. and since only the
squares of Qs occur in the expression. have the same value of vt. These results can now be
summarized according to the effect of the symmetry operation R R Hy yx R as V Tx Rn x r r/
it/Xj/ gt p. degenerate pair contributes terms of the type i n Sec. ltAo const wvk/h.
Degenerate vibrapair. it e yt. i. if a degenerate pair of level.Q t Qiy was shown that a
symmetry left operation on a degenerate set of vibrations the value of Q Q unchanged. . in
view of the wave functions of Eq. lead to fo. It follows. where . the wave function for the
molecule transforms Qi.. yk . Tt. operations. in o are tions.e. R tfrO T Qi Ri Vi i result of the
The R symmetry operation on the transitionmoment . the nondegenerate terms a doubly
degenerate of unchanged by the symmetry operation. eTii equot Q .Rp. that all symmetry
operations on under all o the ground vibrational state is invariant symmetry The wave
functions responding to Qi i i is for a molecule in which the wave function corthe form excited
to the v state is of const Q fc v which follows because the ltp.
and therefore the symmetry. It will only be mentioned that whether or not a fundamental
transition can lead to a The requirement of the for a Raman Raman shift can be deduced in a
similar manner. the polarizability components a xx a xy It is for this etc. product of T Tl R and
T Qi R be unity for every R. C vibrations will lead to absorption of these will The Symmetry of
Group Vibrqtions molecules have groups of atoms that vibrate rather inde Many pendently of
the remainder of the molecule. one can determine the number fundamental transitions that
can occur for a molecule symmetry. shift is that at least one of the integrals type JoQltiQ dr
SMQiltMQi dr etc. it should be mentioned. of the Exercise . reason that . normal coordinate
coordinates With this result and an analysis of Tz how many normal belong td the various
symmetry species. be written as Jrtii dr T. to v transition for a given normal coordinate can
belongs to the be different from zero only same symmetry species as Tx Ty or . since . can
be shown to transform as the product of the representations for the translations indicated in
the subscript on a. INTRODUCTION TO MOLECULAR SPECTROSCOPY integrals can. of a
molecule is determined from the number of infrared absorption bands of infrared active of a
given that are observed. In a manner similar to that used to investigate transitionmoment inte
can show that for a fundamental transition to be active the normal coordinate must belong to
the same of symmetry species as one TX T Z Tx Ty and . so forth. The argument can. the
transition It follows that moment integral for the v . . The hydrogen atoms. sometimes be
turned around so that the geometry. character tables sometimes show the grals one
symmetry species of TX T X Tx Ty Raman etc.RQltRNvwJv i dr i The previous result that all
symmetry operations must leave the integral unchanged if it has a nonzero value means that
TtARqXR must be unity for each and every symmetry operation. The normalization and
orthogonality characteristics of representations show that only if Y Tx and T Qi are identical
representations can the and so forth. . if the . and which be parallel and which perpendicular
vibrations . . be nonzero. Which CH bands corresponding to fundamental transitions.
therefore.. . Furthermore.
Let us first calculate the types of the vibrations of the molecule without the two hydrogen
atoms. or at least visualized. in treated. A discussion of the molecule CC CH CC will illustrate
the procedure. the normal coordinates can . The characters of the reducible representation
for tljis nineatom residue can be worked out.MOIECULAR SYMMETRY AND GROUP
THEORY their W than that of the other atoms of the mass is usually so much less It turns out
that in such a case molecule. at H i Removal of the translations for the vibrations is and
rotations shows that the representation composed of X r h At A B B i. Ax A Bi B This same
result could have been obtained tributions to the reducible representation that the by writing
down the contwo hydrogen atoms make and then sentations that resolving this
representation into the irreducible repreit contains.... The molecule probably belongs to the C
point group. often lead to such vibrations. This separation is of great aid in assigning the
observed infrared absorption bands to normal vibrations of a molecule. gives entire molecule.
The same treatment with the hydrogen atoms. now be drawn to represent Exact indications
of how the atoms move in Pictures can the six hydrogen vibrations. including the two r vib A
B B The additional two hydrogen atoms introduce. can be molecule the vibrations of the
terms of separate vibrations of the molecular skeleton and vibrations of the hydrogen atoms
of the molecule. to be x The number of vibrations of the different symmetry types atom
fragment can now be calculated as a of this heavy a a. therefore.e. vibrations of the types
PH. with the aid of Table .
group.H rU . however. .cm.cm .Jj . With this approach the six diagrams of Fig.CH Z bending
v .. In a similar manner one can represent the vibrations characteristic of a it is CH group.
When such group vibrations are recog often possible to reach an initial understanding of
some of the absorption bands of the absorption spectrum of quite complicated molecules
containing CH and CH groups.H H H . more or less disturbed and altered.CH. Vibrations of
the the actual vibrations CH group classified according to symmetry. CH nized. helpful to
recognize that each hydrogen atom introduces one vibration that is essentially a C stretching
vibration and two vibrations that are essentially bending ones. The form of and the
frequencies of the fundamental absorption bands depend the molecule containing the
somewhat on the nature of CH group.. in any molecule that contains a in the following
chapter. can be drawn for the methylene vibrations.cm. f jCH stretching v . These vibrations
can be expected to occur. . twisting v .H INTRODUCTION TO MOLECULAR
SPECTROSCOPY not be made without the lengthy calculations that will be introduced It is.
cm.cm Hi . FIG. l.CH T stretching I fljtCHj nuking v h CHj gt wagging .
New York. Inc. J.quot John Wiley Complex Molecules. E. CH PRINCIPAL REFERENCES .
amp New ..MOLECULAR SYMMETRY AND GROUP THEORY Analyze the symmetries of
the vibrations of a Classify the group. L. J.quot Academic Press. .. Wigner. .Inc. .quot John
Wiley amp Sons. Inc. Walter. Sons. and G. New York. P. quotGroup Theory. . Kimball
quotQuantum Chemistry. H. Exercise .... quotThe Infrared Spectra York. E. diagrams
according to symmetry type and bending or stretching nature. . Eyring. of Bellamy. and draw
diagrams to represent these vibrations.
cule. related to the force constant and the reduced mass of the mole It is now necessary to
see how the frequencies of the n fundamental transitions of a polyatomic molecule are
related to the force constants and atomic masses of the molecule. v transition in that
Furthermore. . said to be analyzed or understood. The frequency of this transition was. due
to overtones and combinations of fundamentals.. which may look like those of Figs.. that
certain natural. an to v to a vibrational energylevel pattern and.. . or normal. its infrared
absorption is spectrum. weaker bands.. in each one of the normal vibrations. absorption
band corresponding to the energylevel pattern will be observed. quot Rgtv tquotquot J . and .
only one vibraand only one fundamental transition observed spectroscopically. what the form
of the normal coordinate is. tional energylevel pattern exists. .V quotquot. J lt quot ampigt
quott . is For a diatomic molecule only one vibration occurs. When all this can be
accomplished for a particular molecule. In a molecular system each normal vibration leads
the selection rules permit. We will also be interested in determining how the molecule
vibrates. TVi V l CALCULATION OF VIBRATIONAL FREQUENCIES AND NORMAL
COORDINATES OF POLYATOMIC MOLECULES It will has been shown in Chap. may be
observed. in Chap. vibrations occur in a manyparticle system and that these normal
vibrations are characterized by particle motions that are in phase and have the if same
frequency.e. i.
kineticenergy expression of Eq. some typical. yj. This requirement is imposed by requiring
that no movement of the center of gravity of the molecule and that no angular momentum be
imparted to the molecule by the velocity components Xj. Finally. i This expression tional
problems. expressions for the kinetic energy T and the potential energy U of the After an
initial discussion of how this can be done for a polymolecule. gt and ti n X miVfr ziV. will be
worked through.CALCULATION OF VIBRATIONAL FREQUENCIES The treatment of
diatomic molecules in Chap. is the basis for the treatment of kinetic energy in vibra One
should note that the orthogonality of the cartesian displacement coordinates results in an
expression that contains squared terms and no cross products. by ately written rotational x.
and the kinetic energy due to the vibrations of the molecule can be immedi down as r . mfcjXj
. can be used along with the relations ji mjXj ji mampgt gti X mamp XjZ.... yj and In this way
the . due to the vibrations This contribution to the kinetic energy of the molecule can be
treated in terms of coordinates that measure the positions of the atoms relative to a set of
axes that motion. the form of the equations that result from substitution of Lagranges
equation rather than the of these functions in Lagranges equation will be considered. It is
important to remember that a motion described as a vibration must not involve a net
translation or rotation of the molecule. The Kineticenergy Expression for a Polyatomic
Molecule The of the kinetic energy that is here of interest is that of the molecule relative to
atoms each other. oped by Wilson for performing such calculations . Zj. in terms of these
axes. anticipated the use f ma relation in calculations With this approach it is necessary to
set up for polyatomic molecules.. simple examples the more systematic methods develwill be
outlined. move with the molecule as it undergoes translational and If the jth atom is located.
Following consideration of these general features. atomic molecule.
INTRODUCTION TO MOLECULAR SPECTROSCOPY here .. for. find a way in as the
molecule which the change in the potential energy of the molecule is distorted can be
described without recourse to this most general quadratic potential function expression.
Many more parameters are introduced than can be evaluated. from experimental data. to
attempt to A. The central force field assumes that the change in potential energy . follows
that displacement coordinates which show the change in the geometry of a molecule from its
equilibrium configuration will form a suitable base for a potentialenergy expression. Two
different points of view as to the dimensions of the molecule that are most impor tant in
determining the potential energy have been used to approach this goal. as Eq. V for
example. are seldom satisfactory. The Potentialenergy Expression for a Polyatomic Molecule
The of the potential energy of a molecule its is a function of the distortion It molecule from
equilibrium configuration. except in favorable cases. from the equilibrium value of the
distances between the and j. The separation of vibration from translation and rotation
suggested is treated in a more rigorous manner in ref.i X Kr l H O H. of the potentialenergy
function. One approach would be nuclei i to base the description of the geometry on the
changes. With this coordinate system and the harmonicoscillator to keep only the quadratic
approximation that assumes it is sufficient terms in the potentialenergy function. This might
be done in a number of ways. Bry. while complete. and the valencebond force more
frequently used by chemists. . one might write u For i i. It is necessary. ampr ho r Bn ampHo.
and these ould not be determined from the w fundamental transition frequencies that are
observed spectroscopically. These ideas led to the early use of the central force field. ik B
oSrao o an fc Such formulations. this would lead to the potential function /c H oSrVo IWOhh
ampHO. field.HHSrH or H H where the fact that H o Wo has been recognized. Thus.HoSr H
oar H kno. generally favored by physicists..H there would be four force constant terms in the
potentialenergy function. . therefore.
librium internuclear distances. One can. When such calculations are performed it is clear
that neither simplified potential function is com pletely satisfactory. as k a roa so that like fc H
o. will have the units of dynes per centimeter.e. therefore. for H H the simple centralforcefield
expression for the potential energy would be U where ik B r B ikvoiSrno itHHrHH r H o. H h. if
the available spectroscopic data warrant. one can determine the two force constants in the
previous potentialenergy expressions so that the calculated frequencies are close to two of
the observed absorption tial band frequencies. however. As we will see later in the
chapter.HO is i/c H or Ho iMM the change in the angle between the two bonds from their
equilibrium values. equilibrium value and. Thus. the absorption band frequencies can be
used to corresponding to the fundamental transitions of HO deduce values of the constants
in the potentialenergy expression.CALCULATION OF VIBRATIONAL FREQUENCIES that
occurs when a molecule is distorted from its equilibrium position can best be described by
terms in the potentialenergy expression. In fact. dr B and r S o are changes of the HO and
HO bond lengths from Again only two k a force constants appear in this simplified
potentialenergy expression. Frequently one writes the final term of Eq. reduces the The
assumption i. The results suggest. k a The two force constants k a and are related by k a k a
/r. With this approach one would attempt to describe the potentialenergy function of HO by
the expression its U where a i/cHor. The success of the poten function can then be judged by
how well the calculated value of the third frequency agrees with that observed. each of which
involves a force constant and a square of an internuclear dis O tance. r H o.. add additional .
fc number of terms in the description of the potential energy of the H molecule to two. H o
and A. as before. that the valence bond field is usually the preferable approach to a
satisfactory potential function. and r H H represent changes from the designated equiof a
central force field. The valencebond force field assumes that the potential energy of a
distorted molecule can best be described in terms of the changes in length of chemical
bonds and changes in angles between chemical bonds from their equilibrium values.
one makes use of the spectra of isotopically substituted species. species can be studied.
INTRODUCTION TO MOLECULAR SPECTROSCOPY quadratic terms to improve on the
simple force fields. not only as the result of these displacements. NH Write expressions for
the potentialenergy function for on the basis of a the simple valencebond force field and the
field. a Sa iJc B Sr B y a lt r BO Sa Sr H o This function will certainly provide a better
description for the way in which the potential energy varies with molecular distortion. Unless.
however. Exercise . it should be mentioned that a potential function. simple central force a
diagram showing a displacement of the atoms from their equilibrium positions that would lead
molecule to a calculated increase in the potential energy that would be different for the
simple valencebond and centralforcefield approximations. It has. Thus one might add a bond
stretchingangle bending interaction constant to the simple valencebond potential and write U
iampHoSrHo ik B o. The significance of the various force constants can best be seen by
determining the force that would act to restore a bond distance or angle to its equilibrium
distance. is now often used when fairly simple molecules are dealt with and several different
isotopically substituted This field combines the features of both the and the valencebond
field. but also as a result of changes in the bond lengths from their equilibrium distances.
therefore. it positive signs. Thus. one might investigate the force acting to restore the angle
between the bonds of H to its equilibrium value and obtain Restoring force on Sa ST I .
seems reasonable. that interaction terms. k a a k BO a r BO . . and turns out to be so. a
should generally be small and need not always have Furthermore. Draw of the H Finally.
there is no way of verifying its success. central force field . terms involving the distortion of
chemical bond lengths and angles and also terms depending on the distances between
nonbonded atoms. based on what is called a UreyBradley field. r B gt One sees that the
added terms allow for a restoring force to act on angular displacements. such as k H o.
Exercise .
it is convenient up T in cartesian displacement coordinates i. as we will see. If a typical
coordinate is q. Rt. . generally be tji The values of amp will. be determined by application of
Newtons / ma relation or. To proceed from procedure. i. and the cartesian displacement
coordinates are represented by x n one usually sets up the expressions . subject to a
particular potentialenergy function. will be. is generally The first when an example is worked
out in the following easier to set up since equations of the type . . bond stretching and Thus..
as Eqs.CALCULATION OF VIBRATIONAL FREQUENCIES . . i. Lagranges equation is
applied to each coordinate needed to describe the system. to a system. internal coordinates.
either angle bending or Ri.. Rz ne represent the internaldisplacement coordinates. . .. x. This
requirement creates considerable complexity in molecularvibration problems to set since. ..
of course. it is vibra tional frequencies In order to apply Eq. can. the coordinates internuclear
distances which describe the displacements of the atoms from their equilibrium and the bond
angles from their equilibrium values. conveniently. unknown at the beginning of the
calculation. xi.e.e. Use of Lagranges Equation in Molecularvibration Problems The motion of
particles constrained by certain forces. and if U in what are called . and one must be able to
express either the Rs in terms of the xs or the xs in terms of the Rs. Sn T l yi I mfbj U i k/RiB.
we will see section. one obtains the ith equation as For a vibrating molecule there tions.Xtn
Ri fiXl. . . more by Lagranges equation. In the later method. They generally are evaluated
such that the observed frequencies are obtained. . as pointed out in Chap. necessary that
both T and U be expressions involving the same set of coordinates. internuclear distance
type coordinates. w such equa The set of these equations can be solved for the m and
modes of vibration. as we have seen in the previous two sections. .e.
In molecules of any complexity it is usually better to obtain. however. even for fairly small
molecules. iin. Transformation of Eqs. be written. More systematic ways of handling the
problem will be introduced later. It is true. and so forth. i/c Sr generally we let is will use Rs to
indicate internal displacement coordi. lem can be immediately written down as T frnampl m x
m x . of which there are n. . solution . and to cartesian coordinates.. x h x and x represent the
positions of the three atoms along the onedimensional line of the molecule. in the simple
valence bond approach. OCS. The Stretching Vibrations of a Linear Triatomic Molecule A
tions is suitable HCN. If the equilibrium bond lengths are denoted by r and r and changes
from these values are denoted by dr u and r the potential energy can . will To illustrate . by
one means or another. and one translatype i. which will involve stretching of the bonds. and
specific example by these methods becomes rather troublesome. however. INTRODUCTION
TO MOLECULAR SPECTROSCOPY can often be immediately written down.R n . a be
worked out in the following section by this approach. the problem in terms of the n . The
potential energy is best expressed in terms of changes from the equilibrium values of the
distances between the atoms of the molecule.e. . along the axis of the degrees of freedom
along this axis will lead to two vibrations. leads. . . to n equations when Lagranges equation
is applied. differ general problem of a threedimensional molecule will only in complexity. as
indicated in Fig.how this can be done. transformations of the type xs fjR h .t and to proceed
to solve internal coordinates. that even n equations become difficult to handle. and if we Ri
represent r u and R represent r the potential energy function written as U ihRl faRl The
kinetic energy expression can be If immediately written with . the kinetic energy for the
onedimensional prob respect to a set of axes fixed in space. The The more example to
illustrate the discussion of the previous secthat which considers motion of a linear triatomic
molecule of the molecule. tion. as U More nates.
that no net linear momentum be involved. and . if one imposes the condition that the motions
to be studied be vibrations and. In the convenient matrix notation. Thus one has the FIG. and
can be written as are written typical. In the matrix expression. x Br R R a.CALCULATION OF
VIBRATIONAL FREQUENCIES . . can be written in terms of the internal coordinates .. . The
atoms of a molecule sional in location of the type moving onedimen space by means of a set
of axes fixed in space.GM MRiA Ri and jR so that the kinetic energy expression of Eq. and
are Ri Ri Xi x ii x The ease with which these equations. rj r and r i x x rg. for i. therefore. . or
Eq. x and x in terms of Ri and Ri because of the occurrence of three cartesian coordinates
and two internal coordinates. since rj and are molecular properties that are not time
dependent. Eqs. must now proceed to obtain expressions of the type x x fiampuampt /. We i..
. as will be seen. One cannot immediately solve Eqs. is required to obtain the desired
expressions which are of the type x fRiR . Furthermore. however. some further effort. The
desired inverse relations can be obtained. Ti v x X. . while . one . sees this in the impossibility
of taking the inverse of the nonsquare matrix. the time derivatives of Eqs. The two types of
coordinates are related by the t expressions. . of the type is R fxi l .
the expresand x in terms of Ri and ij to be obtained. Xi. and . or Eq. the desired result m zA
Xi J M m m M M mi xj M TOl M or one can obtain. INTRODUCTION TO MOIECUIAR
SPECTROSCOPY additional relation ffiiXi nnxz miXs sion for i. .m. Xl m mi M m m Rx gt m
M Ri X Ki M Kl MJ x M R i M R gt Substitution of these expressions into Eq. for T gives. after
rearrange ment. with Eqs. M M mi m M M m by means of the necessary algebraic
manipulations. f mf . the algebraic equations equivalent to this . This expression allows. In
matrix notation one can add this additional relation to give Now we can take the inverse of
the square matrix to get. mRl m m EiiJ Jk m p m This expression for mim iim m wm mitn R T
together with U ihR fan . T ml m. with the notation M mi m wi .
along with the values of the atomic masses. cmquot. to a quadratic in The two roots of the
equation only X v / . This determinantal equation leads. One can now choose numerical one
were studying HCN. when Lagranges equation ml is applied successively to Ri and R it to jp
m m m miwt mim fli and j ml s m s mii m m S ampii mim m m m m miR x One can again Ri
see if m mim m m R k R solutions of the type Ai A cos Kvt and Ri cos tjltlt exist for this pair of
equations. on expansion. cmquot. give the desired frequencies. positive values are
appropriate for It is ple.CALCULATION OF VIBRATIONAL FREQUENCIES leads. and . yield
roots of the equation with frequencies corresponding to the observed values of . and leads to
the expressions I V ml m m m mm mm. for exam one would know from spectroscopic studies
that the frequencies of the absorption bands corresponding to the two stretching modes are .
Substitution of these trial functions and their second derivatives in Eqs. now perhaps better
to consider a particular case. kA Ai and quotquot I quot IJ m l wim m m A rV I Jgp m s ml m
m m m x A. when inserted in the determinantal equafrom Eq. ttV mim m m m wma ifc A
Again. If. nontrivial solutions will exist for the As only if the determinant of the coefficients is
zero. and . values of tion arising fc x and k which. In this way one deduces that the force .
Calculate the positions of the absorption bands due to to be expected for DCN. X dynes/cm .
. whereas in two previous the of diagrams the draw to necessary Exercise the corresponding
vibrations of the CS molecule could be drawn with less calculation. one can take force
constants from some molecules and carry them over to other molecules and thereby reduce
frequencies from a calculation of the It type illustrated here. . JW fc HC . i. cm A A i . when v
is . and For v . X and v is . atoms of given to the molecule translation is no net arrows to the
atoms suoh that Eqs. Eq. X dynes/cm Exercise . Sometimes. Draw diagrams HCN and so
obeyed. remains for us to see how a molecule. the displacements of the Do this by attaching
in the two stretching modes. Exercise . however. lead. HCN example one has For v . again
the HON example can be considered. Verify that the force constants of Eq. .. given by and
are amplitudes relative the that Exercise . Frequently one must make the calculation in this
direction. Deduce relative displacement amplitudes and draw suitable diagrams for the two
stretching vibrations of DCN. the relative values of A x and A that solve the set of equations.
vibrations the stretching Exercise . vibrates in the vibrational modes that have the
frequencies used in the calculation. X cycles/sec. One can recognize that the first is vibration
predominantly one of C N stretching while the second predominantly one of illustrating CH
stretching. therefore. to the observed stretching frequencies of the HCN molecule. . deduce
values of the ks to fit known frequencies. The relative values of Ai and A and thus the form of
the vibrations. One needs. formed by the coefficients of the As. For the determinant the of
tors . . can be immediately obtained by More generally one takes the ratio of the cof
acrearranging Eq. when substituted in Eq. calculation was Exercise . cm A A x .e. Consider
the basis for the fact that some exercises. INTRODUCTION TO MOLECULAR
SPECTROSCOPY constants for the bonds in HCN are . is .
For example. cm. illustrate the proIt cedures that can be used in the general case. benzene
molecule. the solution will require the imposition of zero angular momentum as well as zero
linear momentum. The above examples. The fundamental stretching mode transitions of the
C molecule correspond to frequencies of . only needs to be men is considered. when force
constants Considerable simplification results. and one would be led to a X determinant of the
type that would arise from Eq. tioned that. the The basis for the simplifications that symmetry
introduces is. feasible only when the simplifications that result of the molecule are
recognized. the conditions of Eqs. Some of these frequencies and must determine what
features will . obtain a value for the force constant from these data a on the basis of a
potential function like that of Eq. Thus one imposes. Exercise . inear i. This is particularly
troublesome when one knows the nately. and . if full use is made of the symmetry of the . the
problem factors down to one X . and draw diagrams to represent the nature of the vibrations.
as . XYZ type molecule solution for the stretching modes of a in internal coordinates. and .
what values of the stretching force constants for the and bonds are calculated Exercise . and
cm If a simple potential function like that of Eq. internal coordinates. On the other hand. is
assumed. in the general when a more than onedimensional problem case. C CS Calculate
the relative distortion amplitudes for the two bonds in each vibration. one X . . fortumolecules
have elements of symmetry. now be considered. and on the basis of a potential function that
includes a cross term between the two bonds of the C molecule. By the procedure outlined in
Sec.. in Use of Symmetry Vibrational Problems of the relation between the force constants
and the vibrational frequencies of all but the simplest molecules is frequently The
determination from the symmetry vibrational problem of benzene involves w . lead to these
frequencies. OCS . and several X and X determinants. The calculation is then perfectly
feasible. the former is not infrared active.e.CALCULATION OF VIBRATIONAL
FREQUENCIES The fundamental transitions of the stretching modes of lead to absorption
bands at . It should be apparent from the simple example treated here that the analysis of the
vibrations of a molecule with many atoms will involve the solution of equations of high order.
vibrational block can be set equal to zero and can be solved for the will frequencies of that
symmetry is determinants to be solved benzene. internalcoordinate of . the necessary
absence of cross if and potentialenergy expressions these expressions are set terms in the
kineticup in terms symmetry coordinates. the of energies potential the would alter the kinetic
and symmetry that a in Sec.U mentioned of INTRODUCTION TO MOLECULAR
SPECTROSCOPY in Sec. and S p occur must not S S term cross the P ticular operation.
then gt Si p Sp in for a par T and a term S SP symmetry operation would convert SiSp lt. of
high This reduction in the order of the very great in the case of molecules. The com if
binations are such that each symmetry coordinate is symmetric or antisymmetric with respect
to each symmetry operation. such as type. The use of the H symmetry coordinates can be
illustrated by the example problem can molecule.lt and SiSp SiSp and This molecule. of
different symmetry types means that the determinant resulting from will application of
Lagranges equation to the symmetry coordinates have blocks of zeros in the offdiagonal
positions that correspond to the Thus crossterm positions between the different symmetry
coordinates. symmetry. the determinant has the form a o o o o a o o where each shaded
block be a subdeterminant of order equal to the number of coordinates of that symmetry
type. the operation cannot result in any change of the kinetic or potential energy The effect of
the absence of cross terms between elements of a molecule. These coordinates consist of
linear combina tions of the cartesian or internaldisplacement coordinates. reached
conclusion would be contrary to the involving must not occur in U. If such cross terms did
occur. The n . perhaps Si gt S. Since the total determieach nant will be zero if the
determinant of any diagonal block is zero. One recalls that and S are symmetry coordinates
that behave differently with respect to any symmetry operation. .
Three symmetry coordinates Si. for motion or rotation. in the directions These coordinates
are convenient for solving the vibrational H type molecule. and S that are convenient for the
calculation of the vibrational frequencies and normal coordinates of the H molecule. Suitable
coordinates are represented dia grammatically in Fig. T im B xl and / moxl y m a x tf. . of type
A and one antih symmetric coordinate. molecule of Fig. to one involving two totally symmetric
coordinates. . The proper relative lengths of the arrows in each figure can. according to the
discussion of Sec. Again one must write the expressions for T and U in the coordinates in
which they can be immediately formulated. problem of the of the molecule in the plane of the
molecule. and by mu/m units as shown. V . We consider the H ..CALCULATION OF
VIBRATIONAL FREQUENCIES be expected to reduce. displacement according to S
consists of motions of H and H by S units. and we set up coordinates which not only have the
symmetry appropriate to the normal coordinates. and so forth. The relative atomic motions
that constitute S S and S are then h compatible with the symmetry requirements and involve
no translation . by the three figures. We can write. S . i. two Aj and one B but also are such
that they involve no translation or rotation of the molecule.be designated by indicating that
displacement according to Sj consists of motion of and H by Si units H shown.e. of type B . U
PoHr OH i/b oa SroH PM m H m S H FIG.
to the desired relations r H sin S cos a l cos a l S. a. and Sr necessary to have the
coordinates x B h.S t It is first necessary to ask . ltS and S s so that we can obtain .St.S h S
U uSi. . gives. . j/h. . after rearrangement. Xw ltgti S a . x cos a. for the time element dt. can
be used to convert Eqs. how displacements described by the sym metry coordinates Si. dXo
ampXn sin a Sy dy Q J/h dXo J Sxw sin a a COS a Syw cos a /h sin The Sxw Sxk cos /hlt a j/
sin a expressions for the effect of changes in the symmetry coordinates on the cartesian
displacements. INTRODUCTION TO MOLECULAR SPECTROSCOPY it is Now expressed
in terms of Si. l r OH. S t and ltS can lead to motions of the atoms equivInspection of Fig. T
and tSi. relation f H r OH . allows one to write alent to xs. written in Eq. T mS The m H l SI m
H l sin a Sj between the internal coordinates and the symmetry coordinates can best be
found by first writing the cartesian displacements Thus of the atoms that result in bond
stretching and angle bending. Vh .cos . sin a too .S V S S cos a Substitution in Eq.sin a l sin
a Si sin a S sin aquot S and These transformation equations between internal coordinates
given V symmetry coordinates can be substituted into the expression for . y m H ltS sin a .
and so forth. l S. . immediately the velocities that the atoms would have along the cartesian
directions in terms of velocity contributions based on the symmetry coordinates as xB a Si Si
Si sin a. . sin aSi aSi . .
cos a S J . . fc . and S h Lagranges equation can be applied to each of these
coordinates.quot Now of the that T and C/ are expressed. and . HO cos A m H l j a cos Wo /
ka . A. and this leads to expressions comparable with those shown as Eq. and it is
convenient to introduce the notation T and a n ampl a Sl owSj l U where the buS amp S S SI
b n Sl as and bs refer to the coefficients of the terms of Eqs. . The elements of the
determinant are rather cumbersome. H o sin A. The determinant of the coefficients of the
amplitudes A h A h and A is set equal to zero to give the equation u Xan Xfl amp Xltl It is
clear that the symmetry coordinates have led to the desired breakblock. to obtain.
Furthermore. Again three equations are and solutions of the irrf form Si Si Sz Ai Ai A z cos
COS rvt cos icvt can be sought. u k BO sin a r . after rearrangement. . if Nontrivial solutions
of this form are then seen to exist the determinant composed of the coefficients of the As is
zero. H sin a cos a lo. by Eqs. and . . fe. in terms symmetry coordinates S h Si. . down of the
X determinant toaXandalXl The two .CALCULATION OF VIBRATIONAL FREQUENCIES in
Eq. the customary procedure of designating the term irV by X will be followed. in the internal
coordinate treatment of Sec. sin H i m smajjs . obtained.
Chem. X dynes/cm the lead to the calculation of frequencies in quite good agreement with
and modes observed values of .. Repeat the calculations of Exercise illustrated here. Wilson
J. It should be pointed out that the on a familiarity with matrix methods.At. The original
presentation of The Wilson method sets up both the kinetic and potential energies . for O
bonds has been determined example. Solution of Vibrational Problems in Internal
Coordinates by the Method of Wilson The treatment illustrated in the previous section can be
systematized by an approach introduced by E. . but use the method of symmetry coordinates
. One of the roots of Eq. and . the ratio At. noth Exercise . and for this value of X.
INTRODUCTION TO MOLECULAR SPECTROSCOPY roots corresponding to the symmetric
coordinates result of the from expansion X subdeterminant. or v. cm for the antisymmetric
mode. now being done. This procedure vibrations is the basis for many of the calculations of
molecular Wilson will will be approach Additional material on this therefore be elaborated. the
angle between the two One can then verify that the force constants to be deg. depends
procedure found in ref. When this is done for each of the three modes can be described by
linear combinations of Si and antisymmetric normal coordinate is. for the three if H type
molecule values of the atomic masses. Phys. is seen immediately to be X / The The above
equations can be frequencies of an solved. The form of each of the three normal coordinates
can now be determined.A roots. one sees that the two symmetric ltS while the factor. cmquot
for the symmetric . apart from a constant ing other than the antisymmetric symmetry
coordinate ltS . corresponding to the antisymmetric coordinate. is substituted into the
determinant. . For the H molecule. X and . and force constants are inserted. is calculated.
inter bond angle. in a particular case. On I rearrangement one has amp X hi i I hi hl a / an an
A bl anffl a / third root of the determinantal equation. . B. fcno H .
are set up to correspond to the general equations given here. of Sec. As has been pointed
out. W RFR where R is the transpose of R.CALCULATION OF VIBRATIONAL
FREQUENCIES in terms of the internal coordinates Rk . Thus Eq.s. in matrix notation. g. can
be written as and the matrix F. the potential energy can be easily formulated in these
coordinates. We U fhiR k Ri kl or. is recognized to be ki set To up the n kineticenergy
expression. or the are related by a . the time set derivatives of the cartesian coordinates.
becomes n I ti in the As we saw molecule example of Sec. The general discussion of this
section ticular equations appropriate to the may be clarified if the par problem studied in Sec.
one must start with the cartesian coordinate expression T It is I mltx defining convenient to
absorb the mass coefficients into the coordinates by new coordinates g that are proportional
to the cartesian coordi nates and are related to them by miixi With these coordinates Eq. for
this example. . and one can immediately write the general expression as U k R k R kl t kl will
The notation used by Wilson designates the now follow this notation and write force
constants by /..
in effect. i i R Dq The principal problem now is to proceed from T ti mi ii X l and the
transformation equation R Dq to an expression for sion for is T in terms of R that is U in
terms of R given by Eq. the inverse transformation of Eq. The general expression of Eq.
comparable with the expresConsiderable manipulation necessary to impose the conditions of
zero translation and zero angular momentum so that. for the particular of the example
molecule. INTRODUCTION TO MOLECULAR SPECTROSCOPY of linear equations to the
time derivatives of the internal coordinates. in matrix form. The the ltji expressions
corresponding to Eqs. can be found and inserted into the kinetic energy expression. with Eq.
the six conditions of zero translational and zero angular momentum. . as six additional The
matrix rows. but involving coordinates. can be made square by writing. are n Rh and J D Mq
where D B ki ki m. writes With Wilsons notation one or. comis pares. compared with those
which arose Again is suggested that the expressions here be in Sec. D In this way the matrix
D where . . and . . R Bx where with R is a column matrix with n rows. for instance. which
appropriate to the f problem. x is a column matrix ft rows. and B is a transformation matrix
consisting of m it and ft columns.
matrix. D and D . Furthermore. and one could calculate. desired reduction to It n matrices
has the matrix now been formally accomplished. ar QQ where and D are square w X n
matrices. remains to see how Q can be conveniently . dimensions. D has been arbitrarily
divided into Q. a n X w matrix. the last three terms are zero and one with T RQQR The
deduced. as also does Q . and Q a n X . Note that Q depends on both gt. be written down
SCrCR With as this relation one can express T in terms of the internal coordinates T qq
XrOSCr Further manipulations allow the problem to be reduced to n rather than ra. In terms
of the square matrix ltR one has gtg where as q r Igto is a column matrix consisting of six
zero elements. Since quotf and and and r GO . Now the desired inverse of Eq. formally. . in
principle. can. SLV Oigto QQo ori to/ S the expression for T can be expanded as T IT Rr ltQ
QQ a f RQ Since r is iiQQB rQQB Q f RQQor rQ QR rQoQo is left a zero
matrix.CALCULATION OF VIBRATIONAL FREQUENCIES could be formed.
allows us to recognize that The second result is conDQ. we can . which is composed n X m
n X n matrix D i. is a w X n identity matrix. . i. . . term of Eq. matrix. n rows and n columns.
INTRODUCTION TO MOLECULAR SPECTROSCOPY The matrix Q useful relationships
arises in the inverse of the matrix X A number of become apparent if one writes ST QQ and
QD QoDo J X wrgt venient in that of the it gt S lt where / represents a diagonal unit matrix.. n
It will be . and leaves This relationship eliminates the second DDQQ I The product QQ QQ
desired for T is then obtained as DDT G important matrix. . and the . . . matrix Q.e. It is first
relation can transposed to give DQ D Q X left and then multiplied on the by D and on the an
right by Q to give DDQQ DD Q Q DQ X Now one must recognize that the internal
coordinates will be the basis for descriptions of the pure vibrations of the molecule and that
these coordinates will therefore be orthogonal to the overall translational and rotational
coordinates. It follows that D and D a are orthogonal matrices and that DD is a zero matrix..
The first now be made to yield a valuable result. . where the usually used symbol G DD has
been introduced for this In view of the relation between the components of D and B given in
Eq.e. . . one can express the elements of G DD as an Q From . . n and I . Gki for k . this
result calculate each element of the G .
section is presented. we can solve for the w values of X that molecular type example can be
satisfy the equation. . quotExercise . the secular determinant formed by the coefficients Of
the has the form Fu F iiX . by the method of Sec. this equation can be written as F Finally. or
. One now has B matrix T RGR and RFR is When Lagranges equation applied to each of the
Rk coordinates and vibrationaltype solutions Rh A k A k s cos rvt are sought. The used to
illustrate Eq. Xs ffA G it is sometimes more convenient to rearrange this determinant by to
give multiplying through by G F GiA GF A A GF With either Eq. The . If A is used to represent
a diagonal matrix with on the diagonals. Calculate the frequencies of the the given force
constants. and the procedure obtained for setting up the F and G matrices. or
.CALCULATION OF VIBRATIONAL FREQUENCIES recalled that the internal coordinates is
the transformation matrix between the and the cartesian coordinates and that it can b e
written down from a consideration of a diagram of the molecule. from Formation of the G
Matrix by a Vectorial Method In a second paper by Wilson a convenient method for setting up
the G matrix introduced in the previous. iX F GOiiX GUX where X rV. . HCN molecule. In
more complicated cases the systematic nature of quotWilsons FG methodquot is of real
value. Probably most of the molecularvibration calculations that have been published have
made use of this procedure.
This is st su B B lXl B zt ky . INTRODUCTION TO MOLECULAR SPECTROSCOPY
previous treatment required use of the relation e / where the ZT BkiBli terms that k. rather
than on the running index if The formulation of Eq. n B ki arise are determined by the
relations ra Rk il BA It is now suggestive to write out the atomic cartesian coordinates more
specifically to show Eq. expressing the s k vectors. With this notation Rk is written as Rk lt st
p i and. zt and sjtt with components B kx B ky B kZl where t numbers off the n atoms of the
molecule. i. .gi form Rk B kXl i Byt These B kXl xt B kXn x n B kVl B y B kyJ n B kz J
products of components can be more neatly expressed by introducing the vectors Qt many
with components x t. there except for the notation based on x y is t.Bi Vl B/cBu. . gives Gkt
as . t helpful only is a convenient way of This can be done by introducing unit vectors e a/s
along the bonds in the molecule. I . The subscripts a and / indicate . y. . in the i . .. as will be
illustrated. the n G matrix elements are simply gt bm ti B This result for Gu can its be verified
by considering atom contribution to kXl t and recognizing that Eq. is expected from Eq. the
same contribution as t.
from the atom labeled a to that labeled e vectors. the s vector contributions of each atom to
a particular internal coordinate Rk. one draw the e vectors directed away from the apex angle
thus If agrees to e OH// WCoh More generally. one might number the three atoms that fix the
value of the interbond angle and draw the vectors as . and are directed agreed that the unit
vectors . If Rk involves the stretching of a bond.e. the s a and s vectors must show how the
terminal atoms must move to lead to a bond stretching. say between atoms labeled a and /. i.
In this Ska way one has simply lt and Thus if molecule one were dealing with the internal
coordinate Ri H O H and the vector had been indicated as Sr on for the H one would have /
H and SlO BO Rk involves an angle bending.CALCULATION OF VIBRATIONAL
FREQUENCIES it is the terminal atoms of the bond. it is si With these possible to set up
convenient recipes for determining and vectors.. Sa for H . for example.
the s vectors and the e vectors. allow. Calculate the frequencies predicted on the basis of
the force constants kno k D . Compare with the observed absorption band frequencies of .
Apply the Wilson FG method to the molecule HOD. introduce the molecular geometry into
the G matrix. which can be expressed as linear combinations of internal coordinates. finds
that. Exercise . r cos gt Sk On r n cos ltpe nn mji . r r sin ltp These relations between the
molecules. . the latter intro ducing the molecular geometry into the problem. X dynes/cm and
lc a /r H . allow the secular determinant to be factored into a of determinants of lower order.. .
in Sec. . Use of Symmetry Coordinates in Wilsons Method It has already been shown. if R k
is an angle bending coordinate. S r sin r i . Some geometric manipulation is now necessary
to arrive at the general expressions which result show how. Sl One cos ygt ltp t r sin ci cos ye
ltp . the three atoms must move to in an increase in the angle with no accompanying bond
stretching. cm. two Si sections are applied to symmetric The symmetry coordinates can be
set up in terms of the internal .. way cosines of angles between bonds of the molecule will
arise and will. X dynes/cm. for very many G matrix elements to be readily calculated
according to the expression G quot w ZT s e lt squot The dot products between the various
between the In this s vectors will lead to dot products vectors that are fixed along the
directions of the bonds. in a very convenient way. and . that the use of symmetry
coordinates. .. number This reduction can be performed when the methods of the previous
molecules. INTRODUCTION TO MOLECULAR SPECTROSCOPY Furthermore one lets r i
and r denote the equilibrium bond lengths and ltp the equilibrium value of the interbond
angle. in terms of the vectors.
U will hold. and the relation. Thus. as Jhis /a/ illustrates. three matrices are nonsingular. for
example. one might write S S r Sa Sr r symmetric Ai symmetric Ai antisymmetric B or /r.
Since. In Wilsons procedure the kinetic energy is expressed as T RGiR The corresponding
expression involving symmetry coordinates can be obtained by inserting the identity matrices
UU. we can l write T SUGWS Samp. li which be used below.S where the notation gi UGW
Since all has been introduced. and C/ C to give T RUWGUWR is Now UR recognized as S.
since U and are known from the way in which the symmetry coordinates were set up and the
elements of G V . and RlJ. example will the up to be orthogonal to one
another.CALCULATION OF VIBRATIONAL FREQUENCIES coordinates Rk according to JI
S t U R lk h i or For UR ri H S. RU is recognized as S. the symmetry coordinates can be set
U matrix is orthogonal. . again with U l U. the matrix can be expressed by / UGU W This
matrix equation allows.
.kUi. along with the appropriate atomicmass terms in Eq. . the calculation of Eq.
INTRODUCTION TO MOLECULAR SPECTROSCOPY g. Thus one calculates the various S
vectors from Eq. as shown in Sec. With Eqs.quot except for numerical values sents no
difficulty. potentialenergy expression based on these coordinates preOne can write down.
and the kineticenergy matrix g of Eqs. R uW and RU US RU. of the components of the F
matrix. to obtain the desired kineticenergy matrix based on the symmetry coordinates of the
problem. shows that the elements of are given by i Expansion of n u. gives U SUFUS or SS n
. .h s quot i X kik akt sk v This cumbersome expression can be used more conveniently if
symmetry displacement vectors S that are comparable to the s k t vectors are introduced.
can be calculated from Eq. and The the relation of Eq. and sub stitutes these. S or R SU
Substitution in Eq. obtained from Eq. and can be set up. If Sj is defined as n S I Uun the
elements of the kineticenergy matrix can be calculated from n . iSSS ii S . the potential The
energy expression RFR internal coordinate matrices R and R can be replaced by symmetry
coordinate matrices by use of the relations. . ..
B. Cross quotMolecular Vibrations. making use of symmetry S molecule. cm . McGrawHill
Book Company. Compare with the observed frequencies of .. as indicated in Sec.. C. Jr. X
dynes/cm and k a . Apply the Wilson FG method. to the . . constants reported for a simple
valence force field are k a . PRINCIPAL REFERENCES . . ...quot Inc. Wilson. Decius. . the
vibrations of a symmetric molecule can be analyzed in terms of symmetry coordinates and.
New York. The force. and and the methods for obtaining g and ff given by
Eqs.CALCULATION OF VIBRATIONAL FREQUENCIES where the potentialenergy matrix
been introduced as ff S based on symmetry coordinates has UFU With Eqs. E. and . The
molecule is bent and has a bond angle of deg. the convenient factoring of the secular
determinant will occur. X dynes/cm. and P. and Exercise . . C. J. as seen in Sec.
generally adopted pattern for the energies of these excited states makes. or emit. of these
states can also be diatomic molecules can a reasonably complete diagram of the energies of
the excited electronic states be drawn. The energy changes involved in a transition from one
electronic state of a molecule to another are usually relatively large and correspond to
radiation in the visible or ultraviolet regions. will see. since all materials. and. in fact.
therefore. and that these bands contain a of inertia large amount of fine Analysis of this
structure often leads to a wealth of informa tion on the moments and the potentialenergy
function for the electronic states involved in the transition. radiation not only as a result of
changes in their rotational and /ibrational energies but also as a result of changes in their
electronic arrangement and. In the course of such high energy transitions. . that electronic
transitions result in We broad absorption or emission bands structure. The that is analysis of
electronic bands of diatomic molecules is perhaps Follow best introduced by considering the
vibrational and rotational structure found in typical electronic absorption or emission bands.
cules exhibit Furthermore. The lack of a simple. diatomic mole many different excited
electronic states. as we will see. their electronic energy. the rota tional energy of the
molecule will also change in the transition. and the energies deduced from studies of
electronic transitions. IV ELECTRONIC SPECTRA OF DIATOMIC MOLECULES Molecules
can absorb. it must be expected that the vibrational energy considered in this chapter are
gaseous. the assignment of transitions to For only a few particular states a problem of
considerable difficulty.
i. The question immediately arises as to which transitions are to be expected between the
various vibrational levels. states. . are. v . . and assigned to a transition between two of these
states. as a function of the internuclear distance for two different The way arrangements of
the electrons of the molecule. of the . The curves shown in Fig.e. horizontal lines are drawn
to represent the energies in Fig. state. labeled vquot . or ions. lower electronic state and the
various vibrational levels. Much of the detailed analysis of the rotational structure observed in
the gasphase spectra of diatomic molecules does not carry over to the analysis of these
polyatomic solution spectra. The potentialenergy curves of Fig. . . of the upper electronic
question. are shown on the energy levels to which they will refer. . typical of a ground and an
excited electronic state. and.e. labeled . the probabilities of the molecule being found at
various internuclear distances. imply that for each of the two electronic arrangements the
molecule will vibrate. We now consider an absorption or emission of radiation that changes
the molecule from one electronic state to the other. . We will. . . . The Vibrational Structure of
Electronic Bands in which the potential energy of a diatomic molecule might vary.
Furthermore. perhaps. . as in Chap. the visible and ultraviolet spectra of these species are
likely to be obtained on solution samples. treat only the principal features of the electronic
spectra of diatomic molecules and will for those make reference to the very complete
treatment given in ref. . two different electronic on such diagrams and with various shapes for
the different electronic states of a particular molecule. the square of the harmonic oscillator
wave functions. a consideration can occur for a particular molecule mention will be made of
how an observed band can be will be given. will see later that shown We potentialenergy
curves occur variously placed of the allowed vibrational states. who wish to pursue further
the detailed study of the electronic spectra of diatomic molecules.. be mentioned here that
the analysis of the is electronic spectra of diatomic molecules a rather specialized study and
that most chemists are likely to encounter only electronic spectra of polyatomic molecules.
therefore.ELECTRONIC SPECTRA OF DIATOMIC MOLECULES ing this analysis of the fine
structure of typical bands. As observation of the vibrational structure of electronic . Three
factors must be kept bands in mind to answer this . . of the kinds of electronic states that It
should. however. . is i.. Furthermore.
Internuclear distance . and vibrational prob ability functions for two typical electronic states
of a diatomic molecule. vibrational energy levels. Potentialenergy functions.
INTRODUCTION TO MOLECULAR SPECTROSCOPY FIG.
. the expected transitions for the potential curves of Fig. . . Draw potentialenergy curves for
ground and excited electronic states such that. in view of the transition moment integral
defined in Eq. . follows that on a diagram. An electronic transition must be expected. The fact
that the probability of the molecule being at a particular internuclear distance is a function of
the distance. Exercise . as shown by the probability curves of Fig. This characteristic of
electronic and nuclear motions leads to the FranckCondon principle that an electronic
transition in a molecule takes place so rapidly compared to the vibrational motion of the
nuclei that the internuclear distance It can be regarded as fixed during the transition. in view
of the shapes of the probability functions sidering transitions to be relatively at the middle of
the v level in Fig. as is done in Fig. only the transition to high vibrational levels of the is
spectrum of upper observed. there are transition going no general restrictions on the
changes in v for a from one electronic state to another. as observed in the absorption
electronic state Fig. With these three statements one can draw. therefore. electronic
transitions must be represented by essentially vertical lines connecting the initial and final
states at some fixed internuclear distance. This result is in contrast to the rule of Av that is
operative in vibrational transi tions within a given electronic state. This rule is usually
simplified. . from the fact that an electron in a Bohr orbit of an atom completes a revolution
around the nucleus in about sec whereas a typical molecule vibrates with a period of about
nuclear distance. One expects. It must be kept in mind that electrons can move and
rearrange themselves much faster than can the nuclei move to alter the inter The relative
times for electronic and nuclear motion can be recognized. often near . . must be taken into
account. a. quotquot sec. .ELECTRONIC SPECTRA OF DIATOMIC MOLECULES will
confirm. by conmore probable if they begin or end or either end of any of the higher vi shown
brational levels. . to be most favored if it occurs while the molecule has an internuclear
distance such that the transition connects probable states of the molecule. with vquot and v
varying perhaps from to large values. to see a progression of absorption lines in the
electronic absorp band due to transitions. In an absorption experiment the temperature is
room temperature. such as that of Fig. for instance. and the vquot tion v level is most
populated. Figure shows absorption bands of CO and I which illustrate this behavior.
in such a ence in the frequencies in adjacent columns is way that the differ approximately
constant and varies uniformly. . i. to various lower levels. . ... relatively o Some of the most
probable transitions. band are usually analyzed The components of such a
vibrationalelectronic by arranging the frequencies of the compo nents in a table. many from
transitions from various upper i. that for a given value of v transitions to states with two
different values of v are preferred. of the v levels are appreciably populated. lines arising v In
emission spectra. generally lead to a band of exhibited in Pig.e. for an absorption experiment
at a Fig. are to be expected. as that of the molecule N shown in Fig. b low temperature. as
Table shows. various values. . The FIG.e. INTRODUCTION TO MOLECULAR
SPECTROSCOPY In emission spectra the temperature is usually high enough so that many
levels. between vibrational levels of two Note. for the potentialenergy curves of . and the
difference in the frequencies in adjacent rows is likewise approximately constant and varies
uniformly. called a Deslandres table. various vquot The many probable transitions that occur
such complexity. that without considerable experience one cannot recognize how the band
structure is related to transitions such as those values.mission transitions Examples
electronic of the most probable states. state therefore.
It should now be clear that data from electronic absorption or vibrational energy emission
bands provide much more information on the than can be obtained from direct One obtains
information on the enerstudies of vibrational transitions. preferred transitions are apparently
those going to vquot and boliclike vquot or . in Fig. and the Each frequency v quantum
number vquot of the lower elecrows with the quantum numbers v of the upper in the table
can then be identified as a transiit tion between the its value of the row occupies and the
vquot value of column. while the differences between the values in the columns give the
vibrational spacing in the lower electronic state. This pattern is understandable in terms of
the emission transitions of Fig. can be . From the data on these vibrational spacings. and one
can learn. much about the potentialenergy function. The most intense lines. Thus for a given
value of v two different vquot levels to be preferred. The Morse curve is generally accepted
as a good detail. if the columns are labeled with the vibrational tronic state state. bers
approximation to the potential function of a diatomic molecule. It is on the shape and often on
the dissociation energy. arranged to conform to these requirements. some of which are
summarized in Table .ELECTRONIC SPECTRA OF DIATOMIC MOLECULES components
of an electronic emission band of the molecule PN. gies of levels with high vibrational
quantum numbers. are given in Table . The vibrational structure of this band has been
measured in considerable and transitions involving groundstate vibrational quantum numup
to vquot have been reported. as shown by the frequency differences also exhibited.
therefore. and the dissociation energy for the molecule in that electronic state particularly
important to recognize that this information of the potential function. tend to follow a para
curve as would be expected from the potential curves of Fig. and the three statements given
earlier in this section. such as the I transition mentioned in the previous paragraph. for
example. moreover. can be deduced by a suitable extrapolation. for comparison. a This is
shown welldefined potential energy function can be deduced. the pattern of vibrational levels
almost up to the limit of dissociation can be obtained. an electronic transition of I that leads to
an absorption band at about A connecting the ground electronic state with an excited state.
there level pattern of a diatomic molecule is . For example. The one expects transitions to
differences between the rows of the Deslandres table can now be recognized as giving the
spacing of the vibrational levels in the upper electronic state. the Morse curve that was
introduced in Sec. In favorable cases. . along with. For v .
O g c z C C C JQ O a.ltk O S o o gt X j io o o m oquot. u to ja o t E .F I LI V O I E o lt O u Ji
Q . S lt . Kl P .
s SE O H IS to LLS a S E o a o SSIE Of S D e i lt i O D j S ZZZ II og w N ii n o M H C O rr a
e O .BeS UOJp jaddf JAU o J .
. S .. in a lower electronic state and lj the rotational energy of the molecule and the /quot
rotational state. . . for excited electronic states.quot D. . .. .. the rotational TABLE The
Components of cm Emission Band of PN Arranged in a eslandres X . INTRODUCTION TO
MOLECULAR SPECTROSCOPY also. Herzberg.. . . does not consist of a single line but is
rather a subband of considerable detail. . . . An example provided in Fig. . . . . . . .. This
additional detail can be attributed to changes in the rotational energy of the molecule that
accompany the vibrationalelectronic transition. .. . quotMolecular Spectra and Molecular
Structure.. studied in the previous section. . From G. . . . . ... . .. . If igto represents the
frequency of a particular vibrational transition.. . in contrast to studies in the infrared region. .
Van Nostrand Company. . Rotational Structure of Electronic If Bands an electronic band is
high resolving power... .. Some molecules have been studied sufficiently so that a rather
detailed diagram obtained not only for the ground electronic state but showing the variation is
of potential energy with internuclear distance for various electronic states of the molecule can
be drawn. lj component of an electronic the rotational energy of the excited molecule in the J
rotational state. it is studied with a spectrograph with sufficiently observed that each of the
vibrational com ponents. .
Princeton. . . Although many excited electronic states have electronic angular momentum
contributions that couple with the rotational angular momenwill tum of the molecule. prime
represents a lower energy state. in cm . . . which lead to considerable complexity.. a single
prime. .. . . ..ELECTRONIC SPECTRA OF DIATOMIC MOLECULES details of the subband
can be discussed ij in terms of the expression h eJ It should be mentioned that. .. . . . it not
be necessary to change over to this standard notation. a double its rotational energy. i. N.. it
is here Table The wave numbers. .J. a higher energy Since detailed analyses of electronic
bands will not be given here.. . . . the energy that the molecule would have if it were not rotat
ing and remained at the state. T is used to denote the total energy T is used to denote the
electronic e energy.. while F represents used to represent the vibrational energy of the
Finally... of a given state of the molecule. . particularly in studies of the rotational and
vibrational structure of electronic bands.. of the spectral lines are given. .... . . ...e. minimum
of the potentalenergy curve for that The symbol G is molecule. state. . . use is generally
made of a different notation than has been used here to designate the various types of
energy that a molecule can have. . Inc. ..
BJJ B V JquotJquot where B v and Bquot are the rotational constants for the vibrational and
electronic states involved in the transition. J B J V B v B VJ Q branch Jquot igt J c h Bjj . J. In
general. rigid rotor expressions. . J./ TABLE Some of the Vibrational Energylevel l Spacings
for the Ground Electronic State of tained from Studies of the Electronic Band at . For these
rotational branches is one has. J J B v . . . further complica tions arise in that the selection
rules states involved in the transition. b. tions in which ampJ . if the value of Jquot simply
labeled as /. INTRODUCTION TO MOLECULAR SPECTROSCOPY sufficient to consider the
simpler cases in which the rotational energies for the two electronic states can be expressed
by the simple. . on J depend on the types of electronic However.. From R. . Chem. D. Phys.
P branch Jquot v J. In such cases the subband structure can be expressed as . . one
frequently has situa are allowed. . Verma. . gtquot ObA Ai cmquot ..
the minima of the potentialenergy curves for different The potentialenergy curve for the
ground electronic state of h. The solid computed from the vibrational energy levels deduced
from an electronic emission From band that is observed at about . . where vibrationrotation
bands were treated. . is line l .. J J . . The dashed line is a Morse curve for R. . .
.ELECTRONIC SPECTRA OF DIATOMIC MOLECULES R branch Jquot v J. Phys. I. D.
There. J. ./ B v B VJ These expressions are similar to those encountered in Chap. Chem.
can lengths. be very the bond and therefore the moments of inertia and The principal
difference that is different for different electronic states of the molecule. FIG. . A. tional states
equal and to neglect the terms equivalent to B v it apparent when the rotational structure of
vibrationalelectronic transitions are analyzed is that B v and Bquot. it will be recalled. . . was
a good approximation to set the B values for the different vibra Bquot.. As can be seen in Fig.
Verma.
is shown in Fig. . therefore. at high J values. known as a Fortral parabola. as shown for the v
. an example of which shown in Fig. and ponent of an electronic band. Cose of Technology.
In fact. dominate the terms linear in J and the branch components will quotturn aroundquot
and move to higher. FIG. Such a parabola. i. A . frequencies as . INTRODUCTION TO
MOIECUIAR SPECTROSCOPY electronic states can occur at quite different bond lengths.
vquot band of a CN transition in Fig. for which the rotational selection rule is A . the linear
terms and the high J components of the band can turn around and move This behavior is to
lower. and the assignment of J values to individual lines can be made so that all points fall is
on a smooth parabola. now apparent Insfjfufe lines o Fig. is then observed. . can be quite
large. as this is particularly necessary when the branch is overlapped or forms a band head
and moves back on itself. of spreading out the Q branch components and thereby preventing
the sharp Q branch that is often a dominant feature of vibrationrotation spectra. if. A
vibrational component of an electronic transition of CuH. . move out linearly from the band
origin. . if BJ is greater than B the J terms of the Q branch can dominate J increases.
Cleveland.. Enlargement of a component of the emission bond of Ns showing thot the are in
fact band heads. Marquises. further effect. instead of lower. B is greater than quot. by
plotting the line frequencies against a quantum number. instead of higher. A. . the J terms of
the P branch can. Courtesy of J. frequencies as leads to the often striking feature of a band
head. for example. It follows that the coefficient of the J term of Eqs. in the rotational
structure of each vibrational com A large difference in B and BJ has the shown by the second
of Eqs.e. Ohio. increases. The parabolic behavior that is to be expected on the basis of Eqs.
and the components of the P and R branches will not. first recognized by Fort rat. Similarly. .
The components of the branches of such bands are usually sorted out.
Van . GoHrrow. FromH.. energy electronic Nostrand Company. Princeton. i L J . Schuter. a
component of the electronic band of Curl at . of the A Fortrot component of a CN band state.
L . . quotSpectra of Diatomic Mofecules. Hohn. moreover.J. parabola formed by plotting the
frequencies of the rotational lines against the rotational quantum number in the lower From
G. H. and H. Analysis of the structure of the rotational branches leads. excited electronic
states.quot D. J J . are often obtained with FIG. with Eqs.ELECTRONIC SPECTRA OF
DIATOMIC MOLECULES /j Rn Kl FIG. Z. Inc. jjcm . to values of the rotational constants E v
and Bquot for the two states involved in the transition. The rotational fine structure of . N.A.
Herzberg. Physik. In this way one obtains information on the equilibrium bond lengths not
only for the ground but also for various These data..
Electronic States of Atoms of A review of some features of the way in which the electronic
states atoms are described is necessary before the corresponding descriptions Application of
the Schrodinger equation to a hydrogenlike atom. . the energy and probability function for the
electron are given. of states of molecules are attempted. When this can be done. . d. i.l. . . . .
. is. a oneelectron atom or ion. each identi fied by the values of three quantum numbers. by
values for the Furthermore. The remainder of the chapter will be devoted primarily to this in
end. magnetic quantum number An electronic state of a oneelectron atom is defined. if he is
to have access to this information. The material of Sec. . of additional interest in that the
molecular orbitals discussed there will be used again in connection with studies of the
electronic states of polyatomic molecules in the following chapter. . be characterized.
quantum number p. as a function of the internuclear distance. . used to describe electronic
states. . It is essential for the chemist. . are available to the electron.e. ate operator would
show.. . . and on the internuclear distance at the i. for various electronic states It is of a
diatomic molecule. the potentialenergy functions can be assigned to particular electronic
arrangements and a vast amount of information on molecular bonding is made available. . . n
or s. . . as application of the approprithree quantum numbers. minimum of this
potentialenergy curve.e. the value of the quantum number I implies an . the equilibrium
internuclear distance. to learn something of the notation. principal . and the implications of
the notation. however. now necessary to see if the states involved an electronic transition
can. shows that various orbitals. that is. I . .. angular momentum quantum number and m I. .
It has now been shown how typical electronic bands of diatomic molecules can be analyzed
to yield information both on the shape of the potentialenergy curve. . . INTRODUCTION TO
MOLECULAR SPECTROSCOPY an accuracy comparable to that provided by direct studies
of rotational transitions in the microwave region. These three quantum numbers are n I . in
some way. .
for example. the important angular for momentum implications of shapes. and d orbits. an
outer electron are not destroyed. do not greatly FIG. possible orientations. for the lighter
elements.quot pp. s. Walter. Kimball. John Wiley amp Sons. Inc. . By such an extension.
quotQuantum Chemistry. or ions. however see. J. n tn i One electron Af Ad Ap d p n l atom Is
One outer electron atom . The energy of the outer electron This is shown scheI as well as on
n. Eyring.ELECTRONIC SPECTRA OF DIATOMIC MOLECULES orbital angular momentum
of /ll lfc/ir and the value of the of quantum number m implies a component of this angular
momentum mh/ir along a specified direction. p. Schematic representation of the effect of
filled inner shells on the allowed energies of an outer electron. that. and G. our ability to
predict a precise energy distribution for the outer electron is and spatial lost.. . . consisting of
one electron outside one or more filled inner shells by using an effective nuclear charge in
place of the actual charge. With the theoretically and experimentally established result that
inner shells do not upset the angular momenta associated with the quantum numbers I and
m and. since becomes dependent on It matically in Fig. . can be shown. The and magnetic
quantum numbers m for the which are important in spectroscopic studies. New York. This
procedure for describing the electronic state of a oneelectron system in terms of quantum
numbers can be extended to atoms. H. E. however. are shown and m in Fig. closed inner
shells provide spherically symmetric screening effects and no angular momentum I
contribution.
one can begin to describe an electronic state by assigning each outer electron to an
available electron orbit. Note that. one can proceed outer electrons. and for the assumed to
behave relatively independently. one can assign values of n. i. field direction is greater the
more the orbital projects out from the axis Symbol z r Orbitals angular factor Angular
momentum in direction of field lt . rather classical rotation situation. like the The shapes and
orientations of the s. If to describe atoms in terms of their if there are several outer electrons.
and d orbitals. p. the angular momentum along the in this direction. and m are moment they
FIG.e.. m l g m p lt Jr mJCw ml m m . I. INTRODUCTION TO MOLECULAR
SPECTROSCOPY upset the approximate energy and spatial implications of the three
quantum numbers. d .
giving a description of an electronic state in terms of the orbitals occupied by the individual
electrons. for example. quotQuantum Chemistry. Such an abbreviation. however. the inner
closed shells are set off with brackets. shells. for example.ip. of the individual. which need
not be given here. two p electrons outside the inner closed the possible values of L are L .
One would indicate the ground state of the oxygen atom.ELECTRONIC SPECTRA OF
DIATOMIC MOLECULES to each electron. In the next section we will see that the writing of
an electronic configuration can also be the in describing an electronic state of a molecule. for
example. orO .y The number of electrons in each orbit is indicated by the superscript. . the
Pauli exclusion principle allows two electrons to be assigned identical values of n. The
quantummechanical treatment further shows that there is a simple relation between the
values of /.quot chap. and as is customary. . in part. L h If h. The effect of a magnetic field on
a manyelectron atom will be treated later. The possible electronic states. . by ltUap. h h . It
turns out. as can be seen from a rather lengthy quantummechanical argument see. I. Walter.
electrons are noninter acting cannot be made. for example. and Kimball. For an actual atom.
. h l t the electronic configuration under consideration has. the supposition that the It is not
even clear what is meant by the contribution of the individual electrons of the atom when they
interact with each other to give some net effect. . since each electron can be assigned a of
quantum number i or . Eyring. presumed noninteracting outer electrons and the values of L
for the atom as a whole. and m. that the electronic an atom is determined. fluorine Write the
groundstate electron configurations for the configuraelectronic tions atom and for the
phosphorus atom. h U . . and then the counterpart of the individual electron quantum
numstate of ber m will be introduced. Suggest electron of these two atoms that would
correspond to excited states. of an atom with two outer electrons in orbits with quantum
numbers h and l are described by values of L calculated according to . however. or molecule.
is said to specify the electronic configuration of the atom. by the quantum number L which is
the counterpart of the individual electron quantum number I. first step Exercise . spin
Furthermore.p.
Just as the value of implies an orbit in which an electron has an angular h/ir. however. . For
the former case the spins must be outer electron example. or by the for individual electrons.
the value of L. P. to indicate L . . which two outer electron system. since the spin quantum
number has the fixed value of . so also momentum of /ll does Limply an angular momen . two
electrons were paired up in a single p orbital. . I values of . P. for example. P.. D. way that
one uses s. rather than. therefore. trons. for the two it would be expected that states with ltS
would have lower energies than those with S . states of the Thus. d.e. designate the
electronic states corresponding to L . There is.. . Thus. . This rule can be easily remembered
in terms of the obvious electron repulsion and. ltS . The energy of the electronic is
configuration. to indicate . . again. If they occupy different p orbitals. . p.. also applies both to
atomic lie and molecular systems. a general rule formulated by Hund. is of primary
importance in determining the energy of the electronic state. . with a given electron
dependent on the net electron spin. . S . the values that S can have S Si are calculated
according to s or S Si s and. i. bearing on the energy.. paired. the way in which the two
electrons are arranged in the p orbitals. say. . the possible S.e. one can have S for a or S
state of an atom. i. The relative energies of these states cannot readily be calculated but can
be deduced from spectroscopic studies. In spectroscopy angular it is often helpful to focus
ones attention on the momentum I implications of the quantum numbers L and S. and D p
configuration would have somewhat different energies.e. this atomic quantity can be
deduced from the spins of the individual outer electrons. in this example. Exact calculations
of the relative energies of states with different net spins cannot be made. or D. . has some
letters S. . INTRODUCTION TO MOLECULAR SPECTROSCOPY In the same . The net
electron spin of the atomic state is described by a quantum number S and. one Although the
fact that the two electrons are p elecp and one d. increased energy that would set in if. one
uses S. For the present example with two p electrons one would . i. the repulsion will be
much less. For two outer electrons. and for the latter they may be parallel.. . that states with
greater spin at lower energies than those with smaller spin.
Instead of using the previous equations to deduce the possible values of the quantum
numbers L and S from the quantum numbers of the individual electrons. Since it is the . can
be separately treated. h/w. idea that the quantum number L implies an angular momentum of
Lh/ic rather than LL A/ir and I an angular those which give turns out to be much L from the
values of h. momentum of lh/ir rather than y/lQ. such as if one returns to the
early.ELECTRONIC SPECTRA OF DIATOMIC MOLECULES turn of /LL h/w for the
electronic state of the atom. In fact this is not so. atom would make the a vector L represents
the magnitude s/LL h/ir and the direction it is of the orbital angular momentum The of the
atom when in the electronic state specified possible values of the vector L are correctly given
by L. by the vector combinations i which lead to vectors It L that correspond to integral values
of L. is what y/ll known convenient to introduce vectors and to deal with as the vector model
of the atom. and not really correct. how the angular momentum con tributions of the
individual electrons could lead to net angular To combine the momenta angular momen tum
contributions. The vector model assigns it is an orbital angular momentum vector to
represent the magnitude h/w and the direction of the orbital angular momentum if
contribution that each of the electrons of the electrons were noninteracting. In this way one
would draw vector diagrams such as those of Fig. one can ask for the electronic state of the
molecule. s implies an electronic spin angular momentum of ih/ir on the part of the individual
electrons. and in terms of the angular momentum contributions of the individual spinning
electrons one has The previous discussion has assumed that the orbital and spin quantum
numbers. easier to draw the vector diagrams. Further more. one describes the total angular
momentum due to the spin of the electrons of the atom by a vector S. although in some
systems it turns out to be a satisfactory approximation. and the corresponding angular
momentum contributions. Similarly. Similarly. to represent the various possible combinations
of to form the net vector L. and S implies a net spin angular momentum of Sh/ir.
momentum of the atom that is really quantized. and the value of the total angular momentum
of the atom is given by s/JJ h/ir. in which the and S vectors be combined J Ss . ir ft i r JL T D
FIG. L L . It turns total angular ingful out again. that the correct vectors J are obtained by
vectorially combining FIG. can L and S.. L . the meanquantum number of the atom is that
which determines this total amount. lOll PS is L trons to give the total The vector additions of
the angular momentum vectors of two porbit elecan angular momentum vector for the atom.
INTRODUCTION TO MOLECULAR SPECTROSCOPY quot i ir i LA. Again the vector model
of the atom introduces a vector J to represent this magnitude and the direction of the total
angular momentum. T JT L. . In terms of t The vector diagrams that represent the ways to
give resulting vectors. and this another illustration of the reason for using the vector model. .
This is illustrated in Fig. The quantum number / is used. fK L J The three components ot a D
term resulting from The two components of a D term resulting from S .
in a manner by the values of the quantum numbers and J. . . as mentioned
earlier.ELECTRONIC SPECTRA OF DIATOMIC MOLECULES quantum numbers. . . atom
can now be described.L S. To indicate that L has the value . S. . results for such atoms
indicate that the atomic orbitals must orient themselves so that the electric field. . the value of
J is given correctly by . D. forth. . multiple of Qi/r. Electron Orbitals in Diatomic Molecules it is
generally impossible to calculate the energies and detailed electronic arrangements for the
possible elec As for polyelectronic atoms . Just as for the case of a rotating momentum a
quantummechanical treatment shows of the atom is y/JJ h/ir. we indicate this number for L. .
treated in Chap. J L The L.L Sl. S. We are now in . . . as well as theoretical. . in many
respects. customary to indicate the values of these three angular momentum quantum
numbers by a term symbol. where M J. . . . one writes. component of the angular momentum
field. . . It is . if the total angular . ways that the spin quantum number can be combined with
the orbital angular momentum quantum number is often an Since the of number important
feature energy levels with a given value of L are sometimes split into sublevels. .L S
electronic state of a free suitable for spectroscopic studies. and so Finally. half integral. to
those of atoms under the influence an Experimental. . J Again the vector model is convenient
in that it suggests that the angular momentum vector J can take up various orientations
relative to the angular applied field and that these orientations are such that the component
momentum along the field is an integral. P. for the electronic state of an atom. of ways S as a
left superscript on the symbol for L. of the atom must be quantized along the direction of the
electric molecule. diatomic molecules are of We will see that the electronic states of similar.
the value of J is written as a right subscript descriptions such as Thus one encounters ltSj. or
half integral if J is This is illustrated in Fig. J . on the symbol Pj. the component along the
direction of the applied field will be M h/. that. . . . P. S. depending on the coupling between L
and S. a position to proceed to a similar description of possible electronic states of molecules
and then to assign these states to the observed electronic transitions.
whose quantization can be immediately recognized. the internuclear distance diminished to
zero.. is given. and the separated atoms. not only the orbital angular momentum. i. . . first
consider the united atom of some diatomic molecule and investigate how the electron orbits
are to be described as the nucleus of the united atom is imagined to be subdivided and the
separate nuclei that correspond to those of the diatomic molecule are formed. .e. Either
diagram shows that the J state splits into three components. M ir J lt M M l ir . Just as for the
case of atoms. to Vector diagrams field illustrating that an applied by the angular three
orientations can be taken up momentum vector corresponding to a J Jh/ir is relative state.
The effect on the orbits is comparable to that of an external electric field applied along the
direction in which the bond is being formed. states Again it is necessary to describe the
allowed of the properties. but also the component of this along the internuclear Thus. axis. .
as indicated by the quantum number I . On the right the correct diagram.e. therefore. initial
effect The on the atomic orbits can be understood on the basis of the fact that the divided
nucleus presents a field of axial rather than spherical symmetry. this can best be done by
first considering the allowed individual electron orbits and then investigating principally
angular the molecular states that result when certain of these orbits are occupied by
electrons.. . T . if the orbits of such a deformed atom are to be described. with h/r vectors. .
On the left the simpler diagram based on vectors of length JJ shown. The electron orbits of
diatomic molecules are best described in terms of the orbits to which they would go in the
limits of the united atom. by means of values of the quantum numbers momenta. although an
orbit of the united atom is characterized by its total angular n momentum. to specify.
INTRODUCTION TO MOLECULAR SPECTROSCOPY tronic states of molecules. FIG. It
becomes necessary. Let us the internuclear distance increased to infinity. i.
. This distinguishes between orbitals of the given atom I that have the same value of
internuclear axis that is but different orientations relative to the The three p orbitals. which
has zero angular momentum in this direction and is therefore labeled as ltrp. . It is well to
note the geometric arrangement of the electron orbit associated with each energy level as
well as their angular momentum contributions. and values of X . . these orbits will be
characterized also and for large distortions better axis.e. . . designate the quantum number
for the component along the interhuclear are usually indicated by saying axis. the principal
orbital angular is momentum quantum feature the component along the axis of the molecule.
when the two separated atoms that correspond to the diatomic molecule are allowed to
approach one another. the orbital angular momentum quantum number is of major
importance and is designated for each electron by I. and . . .. for example. is when the
diatomic molecule It is heteronuclear. . The nomenclature principle that the orbit is a a. by the
component of the angular momentum along the molecular For a given value of I of the initial
atomic orbit. . a. i. the two orbitals that project perpendicularly from the axis. number initial for
this therefore. . This order is maintained in the diagrams for initial effects in Figs. . as the
atom is deformed toward a diatomic molecule.ELECTRONIC SPECTRA OF DIATOMIC
MOLECULES or by the letters s. in the spherical field of an atom. Thus. have one unit of
angular orbitals. In a similar way. is differently affected than being formed. . momentum along
this axis. ir. I. is followed consists of using Greek letters for molecular quantities and Roman
letters for atomic properties. and are designated as irp .p. . . A second atom affects the
orbitals of a given atom by imposing a direction in space. For the cyclindrical that field of a
diatomic molecule. With this notation the step in the formation of a diatomic molecule from
the united atom can be illustrated as in Fig. the electron orbits are affected perturbation that
this causes by the imposition of an axial direction. . .d. . When an axial direction is imposed.
which might be labeled p x pS and pz are identical in the spherical symmetry of an isolated
atom. is illustrated in Fig. . orbit. the quantized momentum components along the bond
direction can have the The symbol X is usually used to quantum numbers . designated by X.
the orbit projecting along this direction. . and the is. . . The when the separated atoms are
different. angular . customary to represent the unitedatom limit at the left of the page and the
separatedatom limit at the right of the page.
The splittings of the atomicenergy levels that result from the initial step in breaking apart a
united atom.FIG. dS d /JT p pir JFquotquot ppltr sa UUnited atom lsu United atom
quotdeformedquot nucleus with Angular parts of orbitals . tations of the orbitals The notation
based on angular momenta and the orien are shown.
E n quot quothquotquot Oquot o Ik CO . O IB isg quotI Hi b N o o a a o b fcfcfci I JE S o o .
from studies of the Heitler. designated by g. or as The FIG. . when molecular orbitals are
formed from like atoms. or antisymmetrically. two like wave functions can be combined either
symmetrically. INTRODUCTION TO MOLECULAR SPECTROSCOPY In the case of
homonuclear diatomic atoms. familiar As should be the H molecule. . states deduced for the
perturbed limits of the united and A squarewell analogy to show the importance of the g and
u character of wave functions resulting from the uniting of two similar systems.London
description of is evident from the simple onedimensional illustration of Fig. as is done in Fig.
designated by u. These symbols must be attached to the description of the orbitals. one
additional feature must be recognized in order to see how the orbits of the separated atoms
interact as they go over into molecular orbitals of the diatomic molecule. Infinite separation
Slight interaction Large interaction comparable to bond formation . to higher Notice that the
wave function with the node goes energy as the bond is formed while that without a node
goes to lower energy.
Thus. This can be done by joining orbitals designated by u for separated atoms with. for
example. Princeton. when it is realized that the quantum number X is a good one even when
the axial field is very important as it is at intermediate internuclear distances. orbitals of
heteronuclear diatomic molecules. a a with a a state.quot Van Nostrand Company. as shown
by the wavefunction sketches in Fig. for homonuclear molecules it is necessary to preserve
the g and u character of the orbital.KTRONIC SPECTRA OF DIATOMIC MOLECULES
separated atoms must now be connected to give the desired molecular orbitals appropriate
to intermediate nuclear distances. have the correct antisymmetry character. The correlation
diagrams of Figs. Herzberg.e. From G. and allow the electron orbits of a diatomic molecule to
be described and the order of their FIG. A correlation diagram describing the energies of the
electron ... The vertical broken lines indicate the approximate positions which give the correct
ordering of the energies of the molecular orbitals for the indicated molecules. and so forth. to
give a correlation diagram. ltrp states which. quotSpectra of Diatomic Molecules.
SEPARATED ATOMS . states with the same values of X. D. Fur thermore..H. Inc. and . The
two limits can be correlated with each other. .J. as in Figs. a jt with a w state. N. one joins up.
i.
Princeton. . are always the orbitals two of that to form a high electron density between one
overlaps possible. equivalent to same basis for describing the elecdoes the use of the
hydrogenlike wave func tions for a manyelectron atom. in Fig. A correlation diagram
describing the energies of the electron Inc. From G. Herzberg. some information on the
Something of the shape of the corresponding orbitals can also be deduced by looking at the
shapes This is illustrated for the case of ap of the limiting atomic orbitals.. D.. orbitals of
homonuclear diatomic molecules. The correlation diagram gives immediately relative
energies of molecular orbitals.quot . The the nuclei is of lower energy recall the ordering of
particleinabox FIG. therefore. and provide essentially the tronic state of a molecule as They
are.J. when the molecule is formed. Van Nostrand Company. INTRODUCTION TO
MOIECULARSPECTROSCOPY energies to be roughly estimated. N. quotSpectra of
Diatomic Molecules. One should notice that two arrangements irp orbitals and that overlap.
Correlation diagrams are resorted to because of the relative ease with which the general
shapes and energies of electron orbits of atoms can be described and the great difficulty of
doing the same for molecules. Fig. UNITED ATOMS .
Exercise . figuration for the and S aid of Fig. Note that each w energy level can
accommodate a total of four electrons. . orbitals These qualitative pictures of electron are
often used.ELECTRONIC SPECTRA OF DIATOMIC MOIECULES FIG. suggest an
electronic conground state of the molecule NO. crp. r u p. wpA. one can draw the bonding
and antibonding orbitals that are formed from the coming together of other atomic orbitals.
wave functions according to the number of nodes and is said to be a bonding orbital. These
shapes indicated far up. Diagrams indicating something of the shape of some molecular
orbitals. and ttpb . is of higher energy and is known as an antibonding orbital. . in which the
function must change sign and have a node between the nuclei. With the . ap. In a similar
way. The second arrangement. and c c p. and jrp in Fig. and wp apply. particularly when
polyatomic molecules are studied. xp. as a basis for describing electron changes that occur. .
in Fig. for example. to the energy levels labeled apA laps.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
Exercise .
With the
aid of Fig. suggest
.
an electronic con
figuration for the molecule
. Electronic States of
Diatomic Molecules
The arrangement
tions that are based
of the individual electrons in a given electronic
by the individual electron descripon the correlation diagram. Some examples are shown in
Table . If the molecule is composed of like atoms, the separated atom designation for the
orbitals is more revealing and is used. For heteronuclear molecules, the united atom
designation is used. As for atoms, the electronic state of the molecule depends on the
state of a molecule can be described
net or total electronic arrangement. Various electronic states can arise from a given
electronic configuration, i.e., from a given description of the electrons considered one at a
time. The electronic arrangement of the molecule as a whole can best be characterized by
the net orbital angular momentum component along the internuclear axis and the net
electronic spin angular momentum along this axis. The orbital component along the
molecular axis, designated by A since this quantity depends on the individual electron
contributions designated by X, is easily obtained by summing up the contributions of the
separate atoms, for a, for w, for S, and so forth. The value of A , , , is indicated by writing a
term symbol , II, A, The spin angular momentum along the axis is, as for the atomic case,
indicated by a superscript giving the
......
.
multiplicity of the
state.
If
the molecule
is
is
homonuclear, the net
symmetry,
i.e.,
the g or u property,
significant for the molecule as
is
well as for the individual orbitals.
The electronic state
is
of.
even and
is
labeled with a g
if
the number of u electrons
if
even, and the state
is
odd
and labeled with a u
the
number
u electrons
odd.
Finally,
TABLE Some Examples of the Descriptions Used for Electronic States K is used to denote
electrons in Is orbits, L to denote electrons in the s and p orbits
Molecule
Ground configuration
ltrl
First excited configuration
HN
l
ltr
Li
KKltr,s
KK
ltrs ltrs wp
t XK
K
n
c lsals
J, J
tf,
KKltr g sltrs
ffl
S
,s ltrTp
quotn,, n
ltr,phr a p
LiH
iCff
Ksltrpltr
i,
s
CH
sa sltr pir
Ksvpltrpw
, A, ,
S
ELECTRONIC SPECTRA OF DIATOMIC MOLECULES
if
a molecule lt or
is
in a
state as a result of the opposition of
two w
or
angular
momentum vectors,
lt
these vectors can have either the orientation
gt
gt.
It
turns out, by an argument not easily given, that these
alternatives correspond to molecular orbitals that are either symmetric
or antisymmetric with respect to a plane through the molecular axis.
this, and the discussion of Sec. , we label states, which from two v or two S orbitals, as and It
should be clear, in view of Fig. , that antisymmetry with respect to any plane through the
internuclear axis cannot be achieved from ltr orbitals. The
In view of
arise
.
notation used for molecular electronic states
is illustrated,
along with
the indicated orbital occupancy, in Table .
We are now at a stage where we can describe, in terms of angular momenta, the different
electronic states that can be expected for a given molecule, and with the aid of the
correlation diagrams, we can tell, although only approximately, the order of these states on
an energy
scale.
. Potentialenergy
Curves for Electronic States of
Diatomic Molecules
Although the energies
of states arising
from
different electronic
arrangements for diatomic molecules cannot yet be calculated, the
electronic states of diatomic molecules can, as indicated
by the lengthy
It
discussions of the previous sections, be described
and
labeled.
can
now be
pointed out that a given electronic transition can, in some cases,
be associated with a particular pair of states.
vibrational structure of the spectral
When
this
can be done,
the detailed information obtained from analysis of the rotational and
band can be used along with the
electronic configurations of the states involved so that diagrams, such as
that of Fig. , can be constructed.
The way
here.
in
resulting in an electronic absorption or emission
which electronic states are assigned to the transition band cannot be treated
is
The
subject
covered in considerable detail in the references
It
given at the end of this chapter.
need only be mentioned here that
the principal basis for the assignment stems from a comparison of the
observed rotational structure with that expected for transitions between
various types of electronic states.
tions
is
strongly affected
The nature of the rotational transiby the coupling between the electronic angular
one observes such features as
momenta
and the rotational angular momentum P and R branches with or without a Q branch, P and R
branches with each line
of the states involved of the molecule.
As a
result
INTRODUCTION TO MOLECULAR SPECTROSCOPY
FIG.
The potentialenergy curves for various electronic states of the molecule Cj.
Inc.,
From
G
is
Herzberg, quotSpectra of Diatomic Molecules,quot D. Van Nostrand Company,
Princeton, N.J., .
More recent data indicate that some revision
The notation
is
necessary.
.
.
The
,
symbol X
C,
is
inserted to identify the ground state.
a, b, c,
.
and A,
...
added
to identify the different states of the
same
multiplicity.
,
CD
,
CS
CP
C SJ cr a
croj
cm
,
CP
CD
e.v.
CP
CP
,
K cm
ELECTRONIC SPECTRA OF DIATOMIC MOLECULES
split into
two or three
lines,
missing early
members
of the P, Q,
and
R
branches, and so forth.
For some diatomic molecules, a fairly complete analysis of the ground and lower excited
states can be made. For many more molecules, only
specific
a few of the observed bands can be definitely attributed to
electronic states.
Much
of the wealth of chemical
bonding
is
information that has arisen from studies of electronic transitions
summarized
results
.
Appendix of ref. . A summary of some more recent has been given by P. G. Wilkinson J.
Molecular Spectroscopy,
in the
PRINCIPAL REFERENCES
.
Herzberg, G. quotAtomic Spectra and Atomic Structure,quot Dover Publications,
New
.
York, .
Herzberg, G. quotSpectra of Diatomic Molecules,quot D.
Inc., Princeton, N.J., .
Van Nostrand Company,
.
Gaydon, A. G. quotDissociation Energies and Spectra of Diatomic Molecules,quot Dover
Publications, New York, .
.
Mulliken, R. S. Rev. Mod. Phys., .
ELECTRONIC SPECTRA
OF POLYATOMIC MOLECULES
The wealth
of
experimental data and detailed analyses that characterize
is
the spectroscopy of electronic states of diatomic molecules
in sharp
contrast to the situation encountered with most polyatomic molecules.
Since the energy of an electronically excited polyatomic molecule can
usually be distributed so as to break a
cule,
weak chemical bond
of the mole
bound highenergy
electronic states are not so
diatomic molecules.
A
further consequence of this
abundant as in is that a sample
transitions
usually cannot be heated to a temperature at which emission spectra
can be obtained without risk of decomposition.
excited electronic state that
The only
that are normally observed are, therefore, those from the ground to an
show up
in absorption spectra.
Only a
few bands arising from such transitions are usually obtained, and since
the samples under study are often solutions or solids, the absorption
bands are often broad and relatively structureless. Spectra of compounds that will be studied
later in this chapter are shown in Figs. , , and . The revealing rotational structure of diatomic
absorption bands is, therefore, absent, and a band does not immediately
reveal the nature of the electronic states involved in the transition.
In spite of the relatively greater
difficulty
involved in the analysis
of the electronic spectra of polyatomic molecules,
much work
has been
and
is
being done in this area.
Spectral data on the nature and energies
than the ground state of the molecule provide important testing data for theories of chemical
bondThe geometry of molecules in excited states is not necessarily the ing.
of electronic configurations of states other
ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES
same
as that of the ground state formaldehyde
and ethylene
are, for
example, nonplanar in one or more of their excited states. In studies of reaction mechanisms,
particularly those resulting from photochemical
excitation, such information
on excited electronic states
is
basic to a
detailed understanding of the mechanism. It should also be mentioned that the absorption
bands due to electronic transitions, which usually occur in the ultraviolet or visible
spectral regions,
have long played an important role in the analysis and A number characterization of both
organic and inorganic compounds. been developed which predict generalizations have of
important empirical and intensity of the on the frequency effect structural changes the of
absorption bands of a parent molecule. The relation between the electronic structure of the
absorbing molecule and its absorption spectrum that will be introduced in this chapter
provides some basis for the understanding of such empirical relations and for the use of
ultraviolet and
visible spectra as a tool for the
study of electronic structures.
chapter
The material
falling within the scope of the title of this
can be conveniently divided into four topics.
The
first
topic concerns itself with molecules that absorb radiation
is
because of an electronic transition that
or group, of the molecule.
essentially localized in a bond,
Most studied
of such systems,
and suitable
to illustrate
the procedures used, are molecules containing a carbonyl
group
/
C.
The
analysis of the spectra of such systems
is
similar
to that for diatomic molecules, but important differences arise because
of the experimental limitations
mentioned previously and the
be quantized.
loss of the
axis along
which angular
momentum would
The second molecular type
to be studied consists of molecules con
taining conjugated relectron systems.
Aromatic molecules constitute
the largest and most studied group in this category, and after a brief
discussion of linear conjugated systems, the
way
in
which one attempts
to describe the various electronic states resulting from excitation, of the
jr
electrons in these molecules will be introduced.
The next topic deals with the absorption spectra produced by systems that contain a
transition metal ion in a coordination compound. Many such systems are colored i.e., they
have electronic transitions that result in absorption in the visible region. The spectrum of
such a compound is, therefore, an often referred to characteristic of the material. Analyses of
such absorption bands in terms of the electronic states
of the bonding in these coordination compounds. uses dot diagrams to indicate the number
of bonding and nonbonding electrons in a molecule. Lewis. or s and p hybrid. Thus. the
diagrams of the type used in the previous chapter. these diagrams could be refined to show
in somewhat more arrangement of the electrons. There will also be. Thus. N. In view of the
notation used for diatomic molecules. for formaldehyde. Finally. due to G. this approach
would give a diagrammatic description of the giound electronic state as electronic state of a
molecule. and some features of the processes of fluores cence and phosphorescence will be
introduced. i. HjCO.. orbitals projecting in the direction of the and the bonding orbital resulting
bond are said to be a orbitals. detail the spatial could be drawn. bonding p. INTRODUCTION
TO MOIECUUR SPECTROSCOPY involved lead to further understanding. These atomic
orbital type diagrams suggest a method for describing the available electron orbits in
molecules of this type. problem.e. from the overlap of such orbitals is can be seen by
inspection of known as a a bond. as . H HC where the placement of the dots represents the
role of the outer electrons With the advent of quantummechanical descriptions based on the
Schrddinger equation solutions for the hydrogenatom in the molecule. the process electronic
state can get rid of its excess energy state will by which a molecule or an ion in an excited
and return to its ground be investigated. TRANSITIONS LOCALIZED N BONDS OR
GROUPS Electronic States of Localized Groups Chemists have devised a number of ways
for describing the ground One of these.
if hybridization of the atomic orbitals on oxygen is not considered. be located in the available
oxygenatom orbitals. Furthermore. momentum values can be used to charac We will see. as
for diatomic molecules. This antibonding usually denoted is. the diagrams given in Fig. the
The principal distinguishing feature of a node in the ltr orbital. that the shapes of the orbitals
can be drawn well enough so that the symmetry of the orbitals can be deduced. molecules.
They can. orbits for a given molecule are recognized. however. the exact spatial distribution
corresponding to these orbitals cannot be calculated.ELECTRONIC SPECTRA OF
POLYATOMIC MOLECULES the correlation diagrams of Figs. lSc lSo s will ltr CH ltr C H rco
ttco o The three ltr bonds to the carbon atom are thought of as being formed from sp trigonal
hybrid orbitals on the carbon atom. a corresponding higher energy antibonding a orbital.
should be recognized that. Fig. not occupied in the ground state. it is considered to be
involved in the observed can be designated simply as a nonbonding orbital and the symbol n
can be used It for it. The atomic p orbitals that project perpendicularly from the bond direction
and form a bond are said to be it orbitals. and . . produced by the overlap of the two orbital is
ltr atomic orbitals. The ir bond between the carbon and oxygen atoms then is formed from
the remaining carbon atom p orbital and one of the oxygen p orbitals. which will be occupied
in the ground state of H CO. Once the available individual electron electronic configuration of
the molecule. no exact angular terize the orbitals. the ground state of H CO can be expected
to correspond to the designation. are important in electronic for the orbitals of the carbonyl
bond that transitions of that group should be noted. and a higher energy antibonding orbital
designated by ir. The remaining electrons that are of interest in the study of electronic
transitions of such groups are quotlonepairquot or quotnonbondingquot electrons on the
oxygen atom. The brackets enclose the lowlying inner and bonding electron orbitals that are
not normally expected to be involved in the transitions studied. The overlap of such atomic
orbitals leads to a bonding w orbital. . . . If only the higherenergy p orbital transitions. and ltr
orbitals as by a shown in . electrons can be assigned to these orbitals to specify an be Thus.
which later abbreviated. It is enough. and the behavior under the symmetry operations of the
molecule will be seen to provide the best characterization of orbitals in polyatomic In
anticipation of this. which are a lowlying s orbital and the remaining p orbital.
some of the excited states that might be expected to be impor tant can be indicated by Inner
electrons r mjr Inner electrons x n a it Inner electronsx iff FIG. for the groundstate electron
arrangement. .fi INTRODUCTION TO MOLECULAR SPECTROSCOPY therefore. Inner
electrons jrC o s n or simply Inner electrons r n s Similarly . to write. Some spectroseopically
important orbitals of HCO and their symmetry properties.
It is this symmetry behavior that best characterizes electronic states of polyatomic along an
axis. for a given assignment of electrons to the individual orbitals. For some purposes the
electronic state electron descriptions.. belongs to the point group C . Write the electronic
configurations for the ground and several excited states of the acetonitrile CH CN molecule.
As previously menon the basis of a net tioned. to an approxi TABLE CgB The C c Character
Table E ltr v Ai T. as for diatomic molecules. by such individual more thorough understanding
of the electronic transitions of polyatomic molecules requires. First one orbitals. recognize
that the total electronic wave function must have a symmetry that is. as is customary. of the
electronic state of the molecule as a whole. Exercise . to symmetry species can readily be
made by investigating whether the orbital is left unchanged or changes sign as a In this way
the result of the four symmetry operations of the group. In the above. however. shown in Fig.
or any symmetric ketone.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES and so
forth. Assignment of the orbitals of Fig. of a molecule is adequately described by such
electronic configurations. Expressions such as those given above constitute electronic
configurations for the various electronic states of the H CO molecule considered here. a
superscript is understood and is not written when an orbital is singly occupied. symmetry
classification of the individual orbitals. momentum effectively H CO.e. is obtained. one
recognizes that. compatible with the symmetry of the molecular skeleton. A i. a description. It
is must consider the symmetry types of the The molecule being used as an illustration here.
The character table for this group is shown in Table . individual now necessary to assign the
total electronic First wave function of the molecule to a symmetry type. A B Bi Tx Ty . this
cannot be done for nonlinear molecules orbital angular One can. molecules. .
all of the orbitals are occupied by two electrons and that the product of one nondegenerate
representation and itself necessarily leads to the totally symmetric representation. the net
char usually acter for all the doubly occupied orbitals is the set of ls that constitute of the the
totally symmetric representation. INTRODUCTION TO MOLECULAR SPECTROSCOPY
TABLE The Effect of Symmetry Operations on ltp. is The task greatly when recognized that
many in the ground state. The effect of the given in the third row of the table. wave function
wave functions Thus one writes . the total for the molecule can be written as the product of
the describing the individual behavior of the electrons. investigations of gral. if electron is in
an orbital of type A and electron is in an orbital of type B h the result of symmetry operations
on ltpl and ltp could be tabulated as shown in Table . . mation sufficient for symmetry
considerations. to transform according to the symmetry type this result B . one total
electronic wave function is of the molecule with given electronic assignments. electronic
states of The symmetry types H CO. according to the results of Table . and so forth. R V fl.
can calculate simplified and the symmetries the symmetry type of the it of the individual
orbitals. for example. Presumed to Be of the Type A it and ltp. Thus.gtM . DCMM M.
corresponding to the configurations previously displayed. of the Type B u and on the Product
C. The results are shown in Table .gt M X MM . as we did in the symmetry behavior of the
transitionmoment inteis equal to the fact that the character for such a product of functions
direct product for the product of the characters of the individual functions. gtM where gt ltpl is
the wave function for the orbital occupied by electron the wave function results of The by
electron . symmetry operations on the product is We have used again. MDtgt i. The product
With ltplp is found.D gt M MDgt M gtM . In view of this. the various symmetry operations on
the ltps for that occupied are known. can then be easily deduced from the symmetries of the
singly occupied orbitals. In group theory one uses the term such a product of two functions.
. The designa tion singlet or triplet must. This rule states that. It can be mentioned in this
connection that the rule based on atomic spectral results. .. A B A. Now we see if the
observed absorption bands can be correlated with transitions between the various electronic
states. It follows that the ground state of H CO must be a singlet state. the net spin must
equal zero. when orbitals are occupied by single electrons. Electronic Transitions and
Absorption Bands A schematic absorption spectra of a carbonyl compound such as
formaldehyde or methyl ethyl ketone. is also applicable to molecular systems. the triplet state
will state. state. the net spin can be either zero. Exercise .e. For the excited states
considered above. howmentioned. ignored the relative directions of spin of the electrons.
therefore. ImI . have a lower energy than the corresponding singlet Thus. The object of such
absorption spectra can be understood on the . lower energy electronic states are to be
expected when the above excited configurations are occupied by electrons with parallel spins
than when they are occupied with paired spins. i. and known as Hunds rule. Verify the
symmetry assignments shown in Table for the various electron configurations. /i. . . i. The
Pauli principle requires that the spins of two electrons occupying the same orbital be
opposed but allows.e. since all the electrons are paired and each pair must have opposite
spins. W kl kl fy nltr M .e. . this section is to see if is shown in Fig. if both singlet and triplet
configurations are possible with the same set of orbitals.. either spin orientation. This
qualitative discussion is as far as we need go must in attempting to describe the electronic
states that are involved in the electronic transitions with which we will be concerned. or unity.
so far. One further feature of the electronic states of molecules must be The description has.
be added to the description of a molecular state. i. ever.ELECTRONIC SPECTRA OF
POLYATOMIC MOLECULES Electronic States of TABLE Some Electronic Configurations
and Formaldehyde Configuration Symmetry of elec Transition moments from tronic state
ground state Inner electrons x n Inner electrons irnr Inner electrons irnir Inner electrons ir A.
the molecule can be in a singlet the molecule can be in a triplet state.
a S a E i a . X in A . . According to the discussion of Sec. for example. INTRODUCTION TO
MOLECULAR SPECTROSCOPY basis of the electronic states of molecules as described. if
It is first necessary. . Knowledge of the A schematic diagram for the ultraviolet absorption
spectrum of a carbonyl compound. the relative energies of the various electronic
arrangements of the molecule can be deduced. to see ground state and each of these
excited states can be expected to be induced by electromagnetic radiation. it is necessary for
one of the integrals transitions between the Kl Mx JVexitdMf ground dr JMiitodftdrVoimd
Aquot or M MsTcitodMground quot to be nonzero for the transition to be induced. therefore.
FIG. . in the previous section. If assignments of the absorption bands to transitions between
given states can be made. . The excited states of a carbonyl compound that are usually sus
pected of being involved in electronic spectral transitions are those described in the previous
section. .
near unity. such as formaldehyde. altered. . use approximate wave functions for the orbitals
shown diagrammatically in the previous section and perform the integrations in order to a
transition is some idea. a completely allowed transition. has the symmetry type ili. but also of
how intense the absorption will The integrals. or eicited and ju s belong to the same
symmetry ii .. the presence of a nucleus with a large nuclear charge results in a strong
coupling of the spinning electron with the orbital motion of the electron and allows the relative
spins of electrons to be. i z . Even in simple molecules.. Again. just as in vibrational
transitions. or / value. an analysis applied to CH are shown in Table . Similar statements can
be made for the transitionmoment integral in the y and z directions. Tx Ty . tion studies. The /
values for such bands will be much less than unity. x .ELECTRONIC SPECTRA OF
POLYATOMIC MOLECULES symmetry of the wave functions be nonzero. or T z. i. It follows
that the entire integrand has the symmetry Ai if Excited and x lilted and y. must not be
changed be.e. is expected to have an intensity fav dv which corresponds to an oscillator
strength. on the basis tion bands. unless the symmetry species tion to an indistinguishable
one. by the symmetry operations which take the molecule from one orientaThus. comparison
of the observed band intensities with those expected of the transitionmoment integrals
provides the most frequently used approach to the assignment of electronic states to absorp
A As mentioned in Sec. . such as that from the ground state to the A x excited state of
formaldehyde. to which WitedMiground belongs is A h the value of the integral must be zero.
like all observable properties. not only of whether allowed or not.e. transitions that are
predicted to be forbidden on the basis of the symmetry treat ment illustrated above can occur
to such an extent that weak absorption bands are observed. we see that they play an
interesting and important role in fluorescence and phosphorescence. and for all molecules
with no unpaired electrons. The ground state for the molecules considered here. One
mechanism that allows forbidden transitions to occur . type. triplet states are relatively
unimportant in absorpwill they lead to very weak absorption bands. belongs to a symmetry
type that contains n y or . Thus. however. On the other hand. an electronic transition from a
totally symmetric ground state is allowed if the excited state y. transitions to an excited triplet
state cannot be entirely ruled out. is sufficient to decide whether or not the integrals may One
can. in get fact. While i. The selection rule prohibiting changes in spin that is operative in
small atoms and diatomic molecules appears to be fairly well obeyed in The results of such
simple polyatomic molecules.
A. The first of these is concerned with the direction of the transition moment. encountered in
align ing the absorbing molecules so that studies with polarized radiation can be used to
determine the direction of the transition moment. Symmetry forbidden Symmetry forbidden
Symmetry and spin forbidden . the nonbonding electrons on the oxygen.. Cf.
INTRODUCTION TO MOLECULAR SPECTROSCOPY involves vibrations of the molecule
that lead to nuclear configurations symmetry as does the molecule in its equiAs a result. . . Bi
Allowed regt a re x re i A. state involved in the transition.. to some extent. for the H CO
example. molecule is increased. . Upper state symmetry . TABLE Assignments The
Ultraviolet Absorption Bands of Carbonyl Compounds and Suggested Values of X and / are
Typical Values for a Number of Compounds Assignment quotband max / osc. i. In some
cases. . One expects. This correlation theoretical basis. Two other general methods for
aiding in the assignment of absorption bands are used. analyses given in Table indicate that
a determination of the direction of polarization would identify the upper is. principally on the
basis of empirical correlations. io T IT M. McConnell. Phys.e. or dielectric environment. that
absorptions due to n gt t ir lengths while those due to gt transitions are shifted to shorter
wavex transitions are shifted to longer wave lengths as the solvation. . a suitable crystalline
form can be found that permits such studies. not strictly C . for example. a hydrogen bond to
the carbonyl group to This tie up. The generally accepted assignment for the observed
carbonyl bands is shown in Table . the skeletal symmetry to which the electronic wave
functions must conform is. The that do not have the same librium configuration. J. for
example.orption band that accompanies solvation of the molecule. Considerable
experimental difficulty however. Chem. . of the absorbing is frequently useful and has some
H. Another assignment aid comes from the shift in the wavelength of the ab. It is found.
strength A Configuration change . and the prohibitions based on that symmetry group are not
strictly adhered to. the polarization of the absorption.
if so. since the symbols for electronic it is states do not immediately reveal the relative
energies of the states.e. To what symmetry types do the ground and first excited irjr states
belong Would a transition between these states be allowed. both absorption and emission
spectra are dealt with. for a transition that promotes one of these electrons. in what direction
would it be polarized Exercise to ELECTRONIC TRANSITIONS OF CONJUGATED AND
AROMATIC SYSTEMS In molecules containing conjugated double bonds and in aromatic it
is now clear that the k electrons. of the . becoming standard. is be dealt with in the the
process by which molecules in excited electronic states lose their excess energy and revert
to groundstate molecules. as is l . when transitions between two electronic states are
considered. Thus. therefore. i. always to write the higherenergy state first.. An interesting and
important aspect is of electronic transitions that has not been considered but which ized
absorbing groups. A very simple but surprisingly successful approach. will be introduced for
linear . as we will see. designations according to this system avoid much of the confusion
that However. known as the freeelectron model. the band in the H CO absorption spectrum
at about . and. Draw the tt and it orbitals of ethylene. often the case. when the transition is
discussed in terms of the configuration change of an electron of the molecule. It should be
mentioned that. for example. If the same transition were studied in emission. In studying
such systems we can. those in the p orbitals systems that project perpendicularly from the
plane of the trigonal sp orbitals forming a bonds. two configurations are usually evident. This
process. are responsible for the absorption found in the nearultraviolet or even the visible
region. and one writes the initial electron orbit first and then an arrow pointing to the new
orbit reached by the electron.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES
would lower their energy and lead to a greater energy change. A would be referred to as due
to a B X lt Ai transition. or a shorter wavelength band. the relative energies can arise.
concern ourselves with the possible orbitals that can arise from the atomic pw orbitals. can
involve fluorescence and phosphorescence emission of radiation. and assign them symmetry
types in the point group to which ethylene belongs. one would write Bi A x When. will
pertinent to a discussion of local and to the conjugated and aromatic systems that final
section of this chapter.
The carbon atoms can be considered to form a bonds with sp trigonal A total of eight orbitals
and x bonds with the remaining p orbitals. . can be understood on the model for the x
electrons of the molecule. the electrons are free to move. Each carbon atom forms three
Each remaining p orbital. electrons will occupy the x orbital system. INTRODUCTION TO
MOLECULAR SPECTROSCOPY conjugated systems. The structure. shown os with fp
hybrid orbitals. the molecular orbit als that it leads In the following to are spread out over the
entire molecular skeleton. . as indicated in Fig. . i. aromatic be set up for benzene by means
of the classical molecular orbital approach which combines the px atomic orbitals into
appropriate molecular orbitals. In fact. Conjugated Systems by the Freeelectron Mode at
longer known that conjugated systems have absorption bands and longer wavelengths as
the number of conjugated double bonds gets greater and greater. orbitals will will be systems
section. derealization of the electrons. an atomic orbital here. . is FIG. CaHjo.. A treatment of
the x electrons can be based on the idea. on the simplest model. the absorption band is in
the visible region. Indicated by The bonding and structure of octatetraene.e. enters into trie
delocalized x bonding. as well as the wavelength and intensity of the absorption It is well
band for a given molecule. long held by chemists. basis of a very simple Let us consider the
specific case of the molecule octatetraene. ltr bands. general features.e. This suggests that
one might treat the molecular skeleton as a region of roughly uniform potential throughout
which. subject to the wave functions that can be calculated from the Schrodinger equation.
that these electrons are delocalized and are not confined to a given atom or even to a given
bond position. long conjugated systems are These often colored. consists of a planar nuclear
array. This treatment puts obvious emphasis on the i. Outside the molecule the potential
energy.. and molecular considered.
ELECTRONIC SPECTRA
OF POLYATOMIC MOLECULES
assumed to be
infinitely high.
This freeelectron or partieieinabox
for octatetraene
is
model, although clearly ratheT crude, leads to quite satisfactory results.
The potentialenergy function The exact width
usually assumes that the
jt
is
shown
in Fig. .
set.
of this potential well
not unambiguously
electrons are free to
move about
is
half a
One bond
length beyond each end carbon atom.
Likewise, one usually assumes
that an endtoend measurement of the molecule
appropriate rather
than one that follows the zigzag of the chain.
With these assumptions,
and ignoring repulsion between the r electrons, one can deduce the allowed orbitals and their
energies. The quantum mechanical problem has already been solved in Sec. . There it was
found that the wave functions for the particleinabox
problem are
IA
sin
n
,,,
.
.
.
and that the energies
of these solutions are
e
quot
ma
The Pauli exclusion stipulation that no two electrons can have the same quantum numbers
requires that no more than two electrons, one with spin i and one with spin , be assigned to a
wave function with a
FIG.
Oecuponcy of molecular orb
transition
i
tali
by
to
the
electrons of octatetraene In the ground state.
The
Hie
arrow indicates the
that
leads
observed absorption band.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
n.
given value of
The occupation
of the available
it
molecular orbitals
that which
for octatetraene can then be illustrated as in Fig. .
The
longest wavelength transition that
is
expected
is
takes one of the
arrow in Fig.
n electrons to the n . The energy involved in
orbital as indicated
this transition is given
by the by
.gt
would cause
The frequency
with a
v
of radiation that
this transition is calculated,
. A, to be
,
cm
This calculated result
is
to be
compared with the absorption band
maximum found
at ,
cm
.
The agreement
is
surprisingly good.
that, as discussed in
A
further advantage of this simple model
is
Sec. , the intensity with
which a sample absorbs radiation can be
calculated.
The
prediction, for octatetraene, that the integrated absorp
tion coefficient be, according to Eq. in Chap. ,
iVe
aV
sec
ir/icl,
.
X
cm molequot
liter
can be compared with the experimental value of approximately
fav dv
.
X
sec
cm molequot
liter
Again the agreement
is
satisfactory.
A
number
of elaborations of this simple
method
for calculating the
electronic states of conjugated systems
have been introduced.
The
chief
merit of this approach, which clearly oversimplifies the potential experi
enced by the
ir
electrons,
is,
however,
its simplicity.
Exercise . Note the similarity between the orbitals of Exercise and the n and n particleinabox
wave functions that would be drawn for the ground and first excited orbitals for ethylene
using the
particleinabox model.
Exercise . What are the calculated and observed / values for the longest wavelength
irelectron transition of octatetraene Exercise . Lycopene is a linearly conjugated hydrocarbon
conIt is responsible sisting of all double bonds alternating with single bonds.
for the red color of tomatoes.
Calculate the wavelength expected for
ELECTRONIC SPECTRA OF POIYATOMIC MOLECULES
v transition and compare with the observed band which occurs at X , A.
the w
. Electronic States
maximum
and Transitions
in
Aromatic Systems
description of the ground electronic state of aromatic goal that occupied the attention of
many early organic was systems a Following the advent of quantum organic chemists.
physical and
An adequate
mechanics, attention could be extended to the excited electronic states
and the spectral transitions that occur between these
states,
A number
of approaches to the description of the excited relectron states of aromatic systems have
been used. An introduction to the electronic spectra of aromatic systems can, perhaps, best
be given in
terms of a simple
classification
molecular orbital treatment, based on pr atomic
the example benzene.
orbitals, applied to
The importance
of the
symmetry
of the molecular orbitals and the electronic states will be brought out,
and
it
will
be shown that some assignments of observed absorption bands
to the states involved in the transitions can be made.
The treatment
can be extended to other aromatic systems, but this will not be done here. As in the previous
section, attention will be centered on the prr
orbitals,
shown
for
benzene
in Fig. .
It is first
necessary to see
how
these atomic orbitals can be combined to provide suitable molecular
orbitals.
The guiding
principle here
is
that an electronic wave function,
must conform to the That is, by an argument analogous to that used in discussing the
symmetry of normal coordinates in Sec. and illustrated for locabzed orbitals in Sec. , each
molecular orbital must transform, under the various symmetry operations, according to one
or molecular orbital, of the benzene molecule
symmetry
of the molecule.
of the irreducible representations of the group.
Let us
now
see
if
such
molecular orbitals can be obtained
prr
by
suitable combinations of the six
atomic orbitals.
FIG.
The p atomic orbitals of
the carbon atoms of benzene.
INTRODUCTION TO MOLECULAR SPECTROSCOPY
TABLE
The
D
Character Table
D,
A,
E
II
Ce
C
C
C
c
I
i
A,
B,
I
X
B
Bi E,
ii
i
T,
T,T y
The
point group to which the nuclear skeleton of the benzene molefi
.
is D The treatment can be somewhat simplified if the symmetry about the plane of the
molecule is, for the time being, ignored. It is enough to treat the wave function above the
horizontal plane of the molecule or below that plane. Later the fact that all px orbitals are
antisymmetric with respect to this plane can be added in. With this
cule belongs
problem can be analyzed in terms of the simpler which the character table is given in Table
and the symmetry elements are illustrated in Fig. .
simplification, the
point group
D
,
for
The
six
px atomic
i.e.,
orbitals
form a basis
are
for a reducible representa
tion of the group,
the transformation matrices that are obtained applied to these atomic
when the various symmetry operations Thus
orbitals will lead to representation matrices that can later be reduced.
to obtain the reducible representation, one investigates the oper
ations on the individual s orbitals
and
writes the transformation equations
FIG.
The symmetry elements of the
Da
point group.
Note
that, unlike the
is
ease for
the Dbi point group, symmetry with regard to the plane of the molecule
Cfin
not considered.
C gT Cj
ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES
as
p /l
pquot
E
character for
E
character for
d
and
so forth.
In this
way one
finds that the character of the reducible
is
representation corresponding to the six pir orbitals
tl
C
C
C
C
C
As
it.
in the
normal vibration problem,
this representation
can be
analyzed in terms of the irreducible representations that correspond to
Thus, application of Eq. in Chap. shows, as can readily be
checked, that the reducible representation contains representations of the types
Ah B E
x,
x,
and
E
l
, i.e.,
A
l
BEE
l
t
We
type
Ei,
see
from
this that the
atomic orbitals should be combined to give
one pair according to the degenerate class
.
molecular orbitals, one of which transforms according to the symmetry
A
,
one according to
Bh
and another pair according to E We must now see what combinations of the atomic orbitals
have these transformation properties.
A
systematic group theoretical procedure
is
available for obtaining
Rather than develop this, we will present the results and merely check that they have the
correct transformation properties. The two nondegenerate orbitals can, in fact, be
constructed
the suitable combinations.
by
inspection.
The molecular
orbitals that
have the correct symmetry
properties are given in Table .
The degenerate pairs can be written
INTRODUCTION TO MOLECULAR SPECTROSCOPY
TABLE
Molecular Orbitals for the
tt
Electrons of
Benzene
Symmetry
type
Molecular orbital
fa
iAigti
Al
Bi Ei El
ltAn e, H e
ltls
ltPb ltPc ltPd ltPb ltPf ltPa ltPb ltPc .ltPd ltPe ltPf gtx ltPb ltPc ltp D ps ltPf ltPA ltPB ltPC fgtD
PB ltPF ltPA ltPB ltPC ltfD ltPE ltPF ltPD ltPE ltpF ltPA
ltPa
fiB
ltfiC
Lowercase letters are used as subscripts to identify the molecular orbitals, just as lowercase
letters were used for individual electron orbitals in atoms and diatomic Capital letters are
reserved for the symmetry type and the net electronic molecules.
state of
an atom or molecule.
and it may be more revealing to consider the sum and the difference of the two functions
given in Table . The form of the molecular orbitals can be seen by drawing the wave One
should recognize that a corresponding functions as in Fig. . function, but with opposite signs,
will project out on the other side of the benzene ring. Also one should keep in mind that each
molecular orbital can accommodate two electrons.
in various ways,
The order
of the energies of these individual molecular orbitals
can be recognized by recalling that, as can readily be seen from the
particleinabox problem, the
the higher
is its
energy.
a,i
more nodes there The wave functions
are in a
wave
function,
of Fig.
ordered so that the
function at the bottom of
have been the diagram has the
lowest energy and the ampi function at the top has the highest energy.
Calculations of the energies of these orbitals can be made, and the
simplest calculation leads to the spacing
shown
in Fig.
.
energy spacings are usually expressed in terms of a quantity
represents an integral that
is difficult
These which
to evaluate.
The symmetry
are also
designations for the orbitals on the basis of the
is
De point group, for which the character table
given in Table ,
shown in Figs. and . The fact that the pr wave functions, or any linear combination of them,
have opposite signs on
opposite sides of the plane of the molecule allows, after inspection of the wavefunction
diagrams of Fig. , these designations to be given to the
orbitals previously labeled a,
ei,
e,
and
ampi in
accordance with the De
group.
Exercise .
, verify the
classified
By
D
inspection of the molecular orbital diagrams of Fig. classifications of a lg , e lg , e iu , and b ig
for the orbitals
on the basis of
D
symmetry
as
oi, ex, e ,
and
fei,
respectively.
given in terms of D and. Da w z FIG. FIG.bibi eie iu Clquotl B J a. tions are given The
relative energies of the individual Symmetry designaon the basis of the De the. on the basis
of Ds group. molecular orbitals of benzene. symmetry type is The parts. in parentheses. ltai
Dark gray Schematic representation of the molecular orbitols for bemene. in parentheses.
negative outlines gray light of outlines positive parts . . point group and.
unlike the tions.e. It will be important. . i i I I i i T T . of orbitals of given resulting electrons.
situation encountered with nondegenerate representaa reducible representation is obtained.
Thus. acters of the Ei s and C E iu representations is C C C oh ltr ltr d S e S XJS la XB. since
the ground state involves pairs of electrons occupying the same orbitals. is promoted to an e
orbital. It is necessary to analyze TABL Ell D. the total symmetry of this electronic state must
be A lg The symmetries of the excited states can be obtained by analyzing the direct product
of the representations of the orbitals e lg and e u occupied by the lone electrons.
furthermore. Now.. that are involved in spectroscopic transitions. the symmetry symmetry by
the six w these electronic states. The direct product of the char. Aiu Bi Biu J j Bt a Bi Ela E lu
Eig Eju I I J I I J i i i T. INTRODUCTION TO MOLECULAR SPECTROSCOPY Now it is
necessary to consider the assignment of the six prr electrons to the available orbitals in ways
that correspond to the states. from the occupancy The electronic configurations that need to
be considered remembering that two electrons can occupy a given orbital and four a
doubledegenerate orbital are that of the ground state and that of the excited states which
result when one electron These are indicated in Fig. The symmetries of the electronic states
of these configurations can be derived from the products of the characters for the symmetry
species of the orbitals occupied by each electron as described for localized groups in Sec.
ground and excited.i Alg TheDe Character Table E C C CI C C Th ampT V tTd S e i A iu At.
to determine the symmetry of each of i. .
in a purely theoretical way. xs. Furthermore. and the result is obtained. . that /CBtg A. Xtf
two The electronic states of benzene corresponding to the lowest excited electronic
configuration are. two electrons in each of these excited states need not have their spins
paired. to have both singlet It is possible. therefore. This can be done by application this in of
Eq.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES terms of its component
irreducible representations in order to determine the symmetries of the states arising from
singleelectron occupancy of the e ig and e orbitals. and triplet states. correspond to the same
It is a matter of considerable difficulty to total electronic energy. Distribution of the r electrons
of benzene in the ground and first excited elec tron configurations. deduce. calculations
indicated that the triplet states lower than the corresponding singlet states. Many attempts
have been made to calculate the energy pattern for the excited states of the benzene FIG. P
Groundstate configuration Excitedstate configuration . expected to consist of states with
symmetries Bm and B iu and a degenerate pair of states with sym metry Eu . as can be
readily checked. therefore.E t Xb. and as suggested by Hunds lie rule cited previously. the
relative energies of these states and the ground state. To a first approximation these four
states. since they involve occupancy of the same set of individual orbitals. in Chap.
. Thus the jnost intense band in the absorption spectrum of of benzene should be assigned
to the completely allowed transition to the E Xu state. would be allowed. more absorpis most
intense band that assigned to the E lu A lg transition. . . one needs to ask . Inc. A . r B ta or .
transitions to the states with sym. for a D h molecule. therefore. A. Englewood Cliffs. The
absorption spectrum of benzene in the ultraviolet spectral region. Pitzer. is attributed to a
FIG. molecule. One must. which upper states . X in . and just as for localized groups.. around
. but so far basis.quot PrenticeHall. make what deductions one can from the observed
spectrum. INTRODUCTION TO MOLECULAR SPECTROSCOPY it has not been possible to
make. Rydberg series . K. about the states involved in the various absorption bands.
Furthermore. It is first necessary to consider which transitions will be allowed. on a
theoretical a certain assignment of energies to the various electronic states. shown in Fig.
The very short wavelength band. changes in multiand one expects only singletsinglet
transitions to lead to strong absorption bands.A. Some assignments of the remaining
absorption bands can be made. metries ij. S. From quotQuantum Chemistry. . The
absorption spectrum tion bands than the single l benzene shows. . ... N. belong to a
symmetry type that contains Tx T or Tz One sees from y Table that.J. however. and E lu
plicity are expected to be forbidden.
the energies of the various electronic states of the molecule. Exercise . . although the
electronic it states of benzene can be classified as to symmetry.ELECTRONIC SPECTRA
OF POLYATOMIC MOLECULES quotRydbergquot type transition one of the pr electrons is
in which the principal quantum number of increased. H H H in a similar manner to that in
which benzene was treated above. treatment given here. . transition. forbidden transition B u
. The This band near . On the basis of the assignments of the absorption bands draw an
energydiagram showing. aromatic systems can be approached have been discussed by R. A
is attributed to the transition B iu lt A ig . A or by coupling with a vibration of appropriate
symmetry. N. . . Soc. Attach electron volt and kcal per mole energy scales to the diagram.
prohibited on the basis of the selection rule derived from the electronic states. It should be
apparent from these comments that. A is attributed to the i . Jones J.l A it highly . to
electronic states of the benzene molecule. As in between electronic states studies of
molecules with localized absorb by orientedmolecule and the Some of the ways in which
more complicated effect of substituents. by means of horizontal lines. included in the
spectrum of Fig. aids to the assignment are provided spectra studied with polarized radiation.
the effect of solvents.. It has the in included transitions have not been been suggested that
the band near . Similar attempts can be made to assign the ultraviolet absorption bands of
other aromatic systems to transitions of certain symmetries. Chem. remains a task of
considerable difficulty definitely to identify each absorption band with a The assignments
suggested here are transition to a particular state. Am. as in the Rydberg series usually Such
highenergy discussed in connection with the hydrogen atom. Treat the molecule
cyclobutadiene of Fig. A is due to either the transition lBiiAig or E ig lt A u These forbidden
transitions could occur as a result of quotborrowingquot intensity from the nearby intense
band at . H c / c II II H H cc/ / cc I I H H /. ing groups. can again be allowed as a result of
coupling with a The very weak absorption at about vibration of suitable symmetry. Exercise
level . .
as these familiar examples do. CuNH . NH H . What are the symmetry species and
multiplicities of the electronic states that arise from these configurations e a Deduce the
number of molecular orbitals of the various symJ metry types. Coordination compounds.
called ligands such as ion. when a more detailed treatment the of . CI . all bands in the
visible. Other absorptions. as the crystalfield theory and. the familiar. these molecular
orbitals. c Arrange these orbitals in order of their energy. Almost the transition metals have
absorption coordination and Ni dimethylglyoxcompounds of become clear that the principal
absorption of these compounds can be part. that occur are attributed to what are called
chargetransfer transitions. These chargetransfer The treatment of the d orbitals that has
been used and that allows the absorption spectrum of a coordination compound to be related
to the properties of the metal ion and the ligands is known. for example. transferred from the
ligands interpreted. usually more intense and farther out in the ultraviolet. in which a number
of electronrich groups. to these systems. by inspecting the way in which the orbitals must
trans What transitions would be allowed between the ground and these excited states
ELECTRONIC TRANSITIONS OF COORDINATION COMPOUNDS The absorption spectra
of compounds or ions containing transition metals have long been used to characterize these
species empirically. comprise the most extensively studied type of transition metal In this
class one has. b Draw. d Assign the electrons to these orbitals in accordance with the lowest
and the first excited electronic configurations. in terms of transitions involving the electrons in
the incomplete outer d shell of the metal ion. for the most to the metal or from the metal to
the ligands. . to some extent. colorful ions gives an intense blue color. coordinate to the
metal species. transitions will not be dealt with here. In recent years an earlier theory
developed by Bethe and by Van Vleck has been applied. In these transitions electrons are.
and extended. INTRODUCTION TO MOLECULAR SPECTROSCOPY form. in its simplest
form. and it has ime which forms a red crystal. which . or in the nearinfrared or nearultraviolet
spectral regions.
. The procedure of is tant example of octahedral complexes. E C C c. The effect of the ligand
field is to break this degeneracy. the theory supposes that the ligands impose on the central
metal ion an electric field whose symmetry depends on the number and arrangement of the
ligands and whose strength depends on the electrical nature of the ligands and. Since all six
ligands are equivalent. i. the relative energies of these orbitals can be investigated and the
available electrons can be assigned to these orbitals.e. d orbitals have identical energies.
The character table for this TABLE The given as Table . to change the energies of orbitals
relative to other d orbitals. In its simplest form. Coordination Compounds the effect of the
ligand field on the be recalled that in a spherical potential . Eu Tu Ti u i l l Tn Tzu l J X yi i z .
to find orbitals that have symmetries compatible with that of the system. which is all that will
be introduced here. more particularly..ELECTRONIC SPECTRA OF POLYATOMIC
MOLECULES is participation of the ligands made.e. as the Ugandfield theory. It will such as
exists in the free ion the i. When this is . with the arrangement of the ligands. such ions have
the symmetry is of the point group O.done. Electron Orbitals of It is first necessary to
investigate d orbitals of the metal ion. and some of the O h Character Tabl e C o Ai. Le.
adequately illustrated by treating only the imporAn illustration of the geometry such
complexes is shown group in Fig. by the attractive interaction primarily held together being
pictured as is of the charge of the metal ion with the charges or dipoles of the ligands. some
of the d To analyze these changes we must proceed. . on the repulsion which those
electronrich groups The coordination compound exert on the d electrons of the metal ion. i S
S e o avj A lu A ig Azu E.. they constitute a degenerate five set. as in the benzene analysis of
the previous section.
. A radial factor that depends on the principal to determine the quantum number must also
be introduced wave function completely. T NH. . furthermore. O. benzene were combined to
give this molecular orbitals consistent with the skeletal symmetry of the benzene Again the
details of the way in which is done five will be passed over. for . and only the results will be
given. and the complex belongs to the symmetry point group symmetry elements of the Oj.
The d orbitals of an I atom or ion are those corresponding to the solution functions with for an
electron of a hydrogenlike atom. basic functions from which orbitals can be constructed that
have symmetries compatible with the octahedral arrangement of the ligands that are now
imagined symmetry. b Some representative NH.. a to Cj and C . define only the angular
factors of the orbitals. . They provide. the magnetic quantum number m can have the values
.c. The five d orbital wave functions that These. it is to be located about the central ion. of
course.. also be given for complexes with other symmetries. I Five functions occur since.
Treatments similar to that given here for octahedral complexes can. arise in this way were
shown in terms of J in Table . point group. of course. It is found that the reducible
representation that the d orbital wave functions lead to under the operations of the octahedral
point FIG. c . The dorbital functions of Table are appropriate to a system with spherical
symmetry. INTRODUCTION TO MOLECULAR SPECTROSCOPY symmetry elements are
shown in Fig. a The geometry of an octahedral complex. For considerations of enough to
concern ourselves with angular factors. The procedure in is similar to that in which the
separate yw atomic orbitals molecule. All six ligonds are equivalent.
ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES can be recognized as containing
the doubly degenerate irreA set ducible representation E g and the triply degenerate one V
latter representathe according to transform of three orbital s that group O h tion can be
directly constructed from the original wave functions. d. orbits. . the two orbitals that
transform according to the E g representation by drawing three pictures and imagining linear
combinations of these to be made the als. zgt d z Z . These are indicated diagram matically.
to illustrate of these diagrams to yield the two orbit shown . keep in mind that no orbitals that
project along orbitals with E g symmetry consist of a pair of lowercase symbols. . . orbitals by
combining two in Fig. corresponding to the notation dgi axis is and dn.quot d lti. . jt. to
describe individual electron e g will be used to describe orbits that transform FIG.
notaDiagrams can be drawn to represent tion usually used. however. . . along with the d zv d
a dg. p. . . the symbols tig and and ltr. such as s. so that Eg two orbitals result. the of use To
be consistent with the cartesian axis. . Diagrammatic representations of the angular factors
of the d orbitals. . . It is customary. in Fig. unique and that the One must. however. .
cm . . One supposed that the six ligands occupy sites along the cartesian axes of Fig. The
extent of splitting is customarily discussed either in A or lODq. We now consider the and e
orbitals that must be occupied by the ion forms a outer d electrons of a transition metal ion
when the metal sets of octahedral complex. lie Thus the two e g orbitals can be expected.
between the electron of the mentioned that it has become customary to use either the
quantity e A or lODq sets. therefore. to designate the energy splitting between the and is t g
orbital For bivalent ions one generally finds A Dq to be of the order of . more repulsion than
one in a fa g orbital. Both notations are included . that a metalion electron in an e g orbital will
since such orbitals point in the direction of the ligand electrons. These ligands present
electronfilled lonepair orbitals in the direction of the metal ion and. The energy difference
between the two d orbitals can. to the deduction that the energies of the molecular orbitals of
benzene are arranged as in Fig. Finally it should be spectroscopic studies often allow this to
be done. The removal of d orbital degeneracy by an octahedral terms of field.
INTRODUCTION TO MOLECULAR SPECTROSCOPY according to the symmetry types
relative energies of the l g T g and E g . It follows effect along these experience. t g as
indicated in Fig. . to at a higher energy than do the sets of orbitals orbitals. . might be
mentioned. in a it qualitative way. depends on the details of the interaction metal ion and
those of the ligands. be immediately recognized. while for trivalent ions this splitting
increased to FIG. here. produce an electronrepulsing axes. The exact amount of splitting of
the It cannot be satisfactorily calculated. and as we will see. The amount of splitting must be
deduced empirically. This step corresponds. .
When ligands are now placed in octahedral positions about such an ion. this is a correlation
diagram between the situation for the spherical field of the ion at the left and the strong
crystal field at the right.e. Various ligands will have crystal fields of various strengths and will
correspond to positions along the abscissa. there are two different electronic states. .
Electronic States of Coordination Compounds d electron. The quantity A. however. the net
orbital angular momentum quantum number L must . . or lODg. which can be identified with
the transition . Since there is only one electron.. be expected to be characteristic of both the
metal ion and the ligands. of interesting correlations number We splits have now obtained the
symmetrycorrect orbitals for a metal ion in an octahedral complex and have recognized that
the ligand field the original fivefold degeneracy. one E. i. one d electron species must be
expected to give a D state. the electronic states are letter designations similarly described
and are labeled with the capital T ig and E. . cm . the electronic state of the Ti ion a D state.
For no crystal therefore. For example. Furthermore. cmquot Alterna. which will be more
evident later. A. therefore. be discussed first. must. indicates the spin and writes the two
states as T and One sometimes indicates the effect of the crystal field for this one d electron
example by a diagram like that of Fig. in which the single d electron split to give e g orbitals.
or ionic. configurations corresponding to these The electronic fo. be and the net spin S must
be field. accommodated. The effect of the crystal field on the states of such an ion will. cm.
The electronic states of an ion with only one outer are in Sec. and thus S is . as for the
original D atomic state.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES around .
at a position where the splitting is . A can be made between this quantity and chemical
properties of both the transition metal ions and the ligands that have been studied. the
aquated Ti ion TiH s has an absorption band at . depending on which set of orbitals the
electron is assigned to. electron states indicated as Ti . or . In view of the survey of atomic. lt
V . such most easily treated. the is degenerate d orbitals. two states are indicated as and e.
Now we must consider the electronic states that arise when the d electrons of the transition
metal ion occupy the available orbitals. In a sense. This observation allows the vertical
dashed line to be drawn in Fig. . a set of lowerenergy Ug orbitals and a set of higherenergy
Now.
cm of the observed absorption band. the accordance with the Pauli exclusion principle. so
that the energylevel pattern for any strength of crystal two I field be combined to . is example
since it has two outer d electrons and We wish to draw a diagram. . Fig. octahedral forms
The Now let us consider ion V serves as an the cmsituation when . S D. An illustration of how
The energylevel diagram for a metal ion with one d electron. of the crystalfield splitting factor
A. . in Sec.. . F. . the assignment of the two electrons to the available d orbitals must be
made in can be picked off.. discussed can. FIG. . G. tion of TiOHj is located and the value A
determined by the frequency of . A cm Zero crystal field Large crystal field . . showing the
effect of increasing crystal field The posiarbitrarily assigned the value zero for the ground
state of the undisturbed ion. Now. or lODq. like that of complexes. For the zero crystalfield
limit.e. i. these features were not mentioned in the earlier discussion. P. or i. . TiltOH P State
E g configuration e State T configuration t g . there are two d electrons. d conis energy scale
effect. . give L values of . The figuration.e. . INTRODUCTION TO MOLECULAR
SPECTROSCOPY tively we can say that for TiH e the value . as electronic contributions and
or states.
. Herzberg. form the tion is of each electron in a given configuration. S. illustrated in Analysis
of the lt configura Table . and will not be repeated here. that must be considered are. In a
similar way the states for the other two configurations can be found. D. Analysis in terms of
characters of irreducible representations XAjg XB. and these results for all nine
configurations of an incomplete d shell subject to octahedral symmetry TABLE Illustration of
the Determination of the Electronic States Arising from the Configuration hg Oh E Ct C Se
Formation of direct product XT. the electronic states that arise from other electronic
configurations can be found from the appropriate direct products of irreducible
representations. and the results of these analyses are shown in Table . as in the previous
treatments of localized groups direct product of the representations and conjugated systems.
XA U XB. quotAtomic Spectra and Atomic Structurequot pp. Similarly.. One finds by this
means that the d configuration can only lead to states P. to a strong crystal The three
possible electronic configurations t g . XT. The energy Z pattern of these states can be is not
easily predicted but can be determined from In this analyses of atomic spectra. New York.
Dover Publications. drawn for the V ion. respond to Now. would be expected for a d ion
subjected Let us now move over to the far righthand side of the diagram and consider the
states that field. and e . way the pattern at the origin of Fig. k g e g . . XT. F.. X XT. to find
what electronic states these configurations corwe must. XT la XT. or G... XT U . in order of
increasing energy.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES these
restrictions operate to allow only certain spin arrangements of the two electrons for certain
values of ing to the detailed discussion given L can be followed through accord by G.
lti. T gt.. . Aip. one assignments included in the electronic states listed in Table .. . .. A l A o .
d lt.. A .. . . i ltA . T i A ln M. . . Verify that the e configuration leads to the states configuration
leads to two states A ig Ai g and Eg and each of the type that the ti g e g T ig and T g fields.
T. For large crystal mined by the number the energy of an electronic state t ig is deter of
electrons in the lowenergy orbitals and the .... l.. . . Tip.. A o .. r. d d U dd e WW . T U r lln .
as listed in Table ... A.quot and it is states for d n systems are similar to those for d systems..
r s . Again. .. .. T s T r. gtT. gt T. ia. M. VE B. . Ai. T lg T lg . necessary to obey Paulis
exclusion principle in assigning electron spins when. .. T Ti. A. T d. T. . r l r l .. Ai. T.. . t T.. A.
r. T A lg . T T o r are given in Table . T T . two electrons have the would arrive at the spin
same orbital description. T o ... gtT ig T e iA. . T r... A o M.. . as in the and e configurations.
folfe. gtA. We will see later that the configurations . d e W WW WW W d TU . .
INTRODUCTION TO MOLECULAR SPECTROSCOPY Electronic States Arising TABLE
Symmetry from an Electron System Subject to Octahedral Electronic configuration Electronic
states Free ion Ion subject to octahedral symmetry dgt.. r lln r A.. j gtr WW WW WW WW W
T r l . ir. . . gt. . gtT I E g .. IT. T. Exercise . By a procedure similar to that which allows spins
to be assigned to the atomic states.
For intermediate field strengths the tendency for the electrons to occupy the lowenergy t ig
orbitals as well as their tendency to avoid one another. The righthand limit of Fig. A
schematic correlation diagram lines for the states of a is d ion. Free ion Strong crystal field
No repulsions between the two d electrons . On the left is the energylevel pattern for the free
ion. the pattern for very strong crystal multiplicities The correlation are drawn with regard to
the of the levels and the correlations of Table . with the states of the d configuration. can be
drawn. are important. on the right fields. which produces the different states of the free atom.
on this basis.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES number in the
highenergy e orbitals. The energylevel pattern for such situations can be drawn FIG.
S. It follows that. It is now appropriate to point out that the often used spectroscopic trick of
recognizing that the treatment for an atom with two d electrons.. T g A le Ea T lg T lg . When
this is done. consequence. the correlation of atomic states and the octahedral crystalfield
descriptions are determined by the atomicstate symbol. one must also preserve the spin.
schematic energylevel diagram that would be drawn on this basis for Ni is shown in Fig. with
the States Occurring in a Field of Octahedral Symmetry Atomic state symbol Octahedral
symmetry states S Alg P D F G Tu Ea T H A a Ti. or multiplicity. Phys. . a d system. for
Which Spherical Symmetry Exists.. . . In practice one does not encounter ligands that have a
sufficiently strong crystal field completely to overshadow the electronrepulsion factors that
operate in the atom to produce the different atomic states TABLE Correlation of Atomic
States. i. the ordering of the configurations with regard to energy The at the right of a
diagram such as that of Fig. such as the d configuration shown in Fig. INTRODUCTION TO
MOLECULAR SPECTROSCOPY by joining up the appropriate in states of the right and left
sides of Fig. which has the configuration d has a set of atomic states arranged in Fig. Bethe
Ann. . The way which such correlations must be made was first shown by H. The derivation
shows that it is the net angular momentum of the atomic state that determines the
correlation. for any atom.e. is reversed. by means of a grouptheoretical derivation of the
relation between the atomic states of the spherically symmetric problem and those states
described by the crystalfield notation appropriate to the potential field of octahedral
symmetry. of the electronic states. P.. The appropriate correlations are given in Table .
Furthermore like those drawn for the d example of V . and so forth. a d configuration. the
correlation lines shown in Fig. i. D. the crystalfield configurations will be the same except that
we will be As a dealing. Thus s Ni. i. is in many respects identical to that for an atom lacking
two d electrons from a completed d shell. can be drawn. as it were.e.e. In a particular case.
with positive holes in the completed d shell. .
The energy FIG. One usually. Free ion Strong crystal field No repulsions between the d
electrons .ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES for a given number of
d electrons. shown taken of the fact that like which is often studied in Here account is also
quantummechanical states tend to quotmixquot ion. focusses ones attention on the lefthand
part of the diagrams that have just been discussed. . and to form somewhat modified from
one another. The lefthand part for the is coordination chemistry. Ni in Fig. states whose
energies remain different levels of like states are said to repel The schematic correlation
diagram between the free ion and the strong octahedral field complex for d ions. therefore.
Ltd. . What are the differences in the limit of strong crystal . . quotMethuen amp Co. such as
that of Fig. . rules governing transitions it remains for us to consider the selection energy
states. Exercise . .Metal Chemistry. INTRODUCTION TO MOLECULAR SPECTROSCOPY
one another. possible configurations of a d ion in an octahedral complex. Arrange according
to energy. fields of . Before the predicted energylevel pattern of a Ni complex. cmquot .g l e g
y. ltv . quotquot is located along the abscissa to fit the observed spectrum. to the higherOne
can immediately carry over the rule that forbids FIG. the two In Fig. hgYieg t. A g state. such
as Ni . state are The energylevel diagram for a d ion. . London. and Illustrate the answer by
an energy diagram and show the energy e g scale in terms of both the parameter A and the
parameter D Exercise . the energies of the configurations h Y. NiH /ltV A . E u c i . in a field
of octahedral The energies of the important states with the same multiplicity as the ground
lines. as in Exercise . shown as heavy The position of NilHjOlequot From . Orge/. can be
used as a basis for the interpretation of the observed spectral transitions. . E energylevel
lines. quotAn Introduction to Transition. the . .. T S and the two . symmetry. . this effect the
two is noticed in the curvatures of g T lg .. from the ground. .
all symmetric and. to very weak bands. in two regards..Y. and the positions of these bands
match up rather well with the spectrum. much more intense AltS bands occur. in this case.
well known in atomic spectra and referred to as the Laporte This rule. be mentioned that the
AltS selection rule can also be violated. consideration of the transitionmoment integral leads
to the conof the other triplet states. for not too strong a crystal the groundstate configuration
is U l elt. is based on the symmetry result that. .ELECTRONIC SPECTRA OF POLYATOMIC
MOLECULES changes in the net spin of the system and can therefore. the clusion.
Furthermore.e. transitions between energy tion levels such as those of Fig. that no
transitions between states arising from a given configuration such as d are allowed. The d
configuration. according to the discussion of Sec. the integral would necessarily be zero. cm
for the NiH ion. Fig. . the product of the terms in the transitionmoment integral would be anti
shows. lead to relatively weak aigt absorption bands. and since the dipolemoment term itself
is antisymmetric. . One finds. because some of the vibrations of the ligands with respect to
the metal ion destroy the symmetry i. is different from the previous examples field. however.
expect the important spectral bands to be interpretable on the basis T ig state and the two Ti
g states. It should Tranchanges lead. that dd transitions. however. to some extent. The three
absorption bands can be attributed to the states indicated above each absorption band. The
spectrum of the aquated ion NiH is shown in Fig. the energylevel pattern for a given
octahedrally coordinated metal ion can be used as a base for the interpretation of the
observed Now The example of Ni is still suitable. found in Mn and Fequotquot. is First. if one
assumes a value of A . For a coordination compound this rule is broken down. as d orbitals
are symmetric with respect to their origin. of the system. therefore. spacing between the
energy levels of Fig. One final illustration of the crystal field interpretation of the delectron
energy levels and the spectrum resulting from transitions between these levels can be given.
with and these are not generally observed when the / values of perhaps sitions involving spin
. and this configuration has an energy that independent of the value . Typical values of is at
the maximum of the absorp band for such absorptions quotWm ylog j cm mole liter is The
corresponding typical oscillator strength about .
one would now normally focus ones attention primarily on the higher energy states with the
same spin as the ground state. the low extinction coefficients. this ground state configuration
greater than can be achieved has a spin of /. INTRODUCTION TO MOLECULAR
SPECTROSCOPY of A. states when radiation is absorbed low crystal is shown in the region
of relatively field energy in Fig. in line with First. and this by any of the other possible electron
configurations. is Secondly. . would be made on the basis of Fig. should be noticed. .
Although. e This leads to the horizontal groundstate energy level for the S state shown in Fig.
a net spin of / and quartet states. to interpret the electronic spectrum. and this again can be
attributed to the spin change involved in the visible region transitions. The reader may. The
number and positions of the absorption bands of various Mn coordination of to the
suggestion that a value of A about . The highest spin found in the excited configurations
corresponds to two electrons being paired with opposite spins. as compared for example with
the cor responding spectrum of the aquated Ni ion. The spectrum of the predictions that MnH
shown in Fig. FIG. . From Holmes and McClure. . . cmquot complexes have led for MnH . .
Chem. .. in fact. are and this situation produces The energy levels for these states. J. . The
absorption spectrum due to the octahedral ion NiH . . . The oscillator strengths are less for
Mn by a factor of about one hundred and this can be attributed to the violation of the AS rule
that must accompany the dd transitions of the MnH ion. which now must serve as the excited
in a dd transition. . Phys. have noticed the generally pale colors displayed by manganese
solutions. . such states do not exist. .
. An . . . i . state. . The absorption spectrum due to the ion MnH . Only the ground
designated as S and e T ig and quartet states are shown.. . K. . .. FIG. Jorgensen. in cm The
energylevel diagram for a a ion such as Mnquot quotquot quot. A . Orgel. i . . ltquotVltV . A
FIG. Chem. .quot Methuen amp Co.ELECTRONIC SPECTRA OF POLYATOMIC
MOLECULES . From L. Introduction to TransitionMetal Chemistry.. From C. . Acta. gtltcm .
ltd. . quotAn . Scand. XinA . London. E.
can be studied. cm gives the best correlation between the energylevel diagram and the
observed spectrum. i. struct energylevel diagrams for different way one can con octahedral
complexes of ions with numbers of tures of the spectra of these ions. This ligand field
approach will not. contribute to the characteristic colors often observed with transition metal
complexes. tude of stood. INTRODUCTION TO MOLECULAR SPECTROSCOPY which
predicts a group of three absorptions around . square planar. can be Other spectra of other
octahedral ions can also be underof A determined. other geometries. tetrahedral. They do
not. energylevel diagrams for these geometries can be constructed. it might be added. types
of energydissipation processes can be recognized. EMISSION AND DECAY PROCESSES
Various types of electronic transitions that lead to the absorption of electromagnetic radiation
and raise the absorbing molecule or ion to some highenergy electronic state have been dealt
with in the preceding parts ion. they usually occur farther out in the ultraviolet region than do
the dd bands treated here. and the spectra of ions with these geometries can also be treated.
and the value spectra with the energylevel diagram. which we will call nonradiative
processes. of the chapter. measuring the magnithe interaction between the ligands and a
metal ion. energy is transferred to molecules of this energy that collide with the excited
molecule and carry as translational. Then one constructs suitable orbitals from these two
types of localized orbitals. Again it should be pointed out that other electronic transitions.e. In
one type. The above examples illustrate the chief goals and results of the dis. and so forth.
leading to chargetransfer bands. however. It should.. Now we will investigate the ways in
which a molecule. d electrons and can understand the principal feaFurthermore. or can get
rid of its excess energy and return to the ground state after Two general it has been raised to
such a highenergy electronic state. cm and another group of three around . be mentioned
again that a more complete and detailed study must take into account the orbitals of the
ligands as well as those of the metal ion. Although these bands are generally very intense.
and the parameter A. The positions of the principal rfelectron absorp tion bands are
understood. or rotational away some and vibrational. however. cussion of this section.
energy. from comparisons of the observed In a similar evaluated. be dealt with here. occur
with coordination compounds. No emission of .
the excess higher vibrational states of the solution it vibrational energy. . effect a slowing
down of this process consists under study in a solvent which. and perhaps It appears. it is
seen that many the excess vibrational energy Since the principal being lost. Such radiation
can be detected and analyzed. vibrational. This energy goes into the thermal. can be
expected that restricting process. probably by one vibrational level step at a time. and is not
detected from ultrasonic dispersion measure ments and theoretical of sec. motion of the
molecules of the solvent as emitted radiation. another is a mixture of ether. to of dissolving
the material sets to a rigid glass. do occur while is sec. calculations. for reasons which will be
made clear in the remainder of the chapter. mechanism for the vibrational deactivation of a it
molecule involves molecular collisions. the potential energy for a given electronic in the
region covered state will be a function of all the internal coordinates of the molecule. that a
typical time required for the dissipation of excess vibrational energy When this is by such a
process is of the order compared with a typical vibrational period of vibrations. One solvent
often used is boric acid. and illustrated in Fig. The simple diagram of Fig. Furthermore. and
as we will see. . The second type of dissipation on the other hand.ELECTRONIC SPECTRA
OF POLYATOMIC MOLECULES radiation processes. isopentane. A number of different
electronic configurations must be expected to have energies that lead to potentialenergy
curves by this diagram. showing the potential curve of the ground electronic configuration
configuration is and that of a single excited state not typical of the situation that exists for
polyatomic molecules and is inadequate for a complete discussion of radiationless energy
dissipation. In this regard Fig. involves an emission of radiant energy. say a thousand.
Nonradiative Processes As was discussed in Sec. For molecules in appears that the
molecular collisions are very effective in removing. is observed in such processes. such
analysis leads to the further classifications of fluorescent emission and phosphorescent
emission. the transi tion associated with an electronic absorption band produces a higher
energy electron configuration and often leaves the molecule in one of the new electronic
state. and ethanol. is more typical. The net . when cooled. or eliminating such collisions will
slow down the A procedure that has been often used. referred to as EPA. and for polyatomic
molecules the potential energy would have to be represented by a surface in this
manydimensional space.
moreover. two potential curves cross as in Fig. This type can. is that shown in Fig. The
crossing of the potential curves facilitates this process. more properly. reached by the
absorption process its potential surface with that leads to a second type of radiationless
process. effect of these INTRODUCTION TO MOLECULAR SPECTROSCOPY two features
is that potentialenergy curves or. and a representative arrangement of potential curves . it is
When two potential surfaces or. initially The occurrence of a crossing. . or nearby. surfaces
of different electronic states will cross one another. state. possible for the molecule. lead to
the return of the excited molecule to ground to simplify the discussion. to change over into
the configuration corresponding to the curve labeled S. originally in the excited electronic
configura tion corresponding to the curve labeled S. FIG. Energy dissipation by vibrational
deactivation and internal conversion. The dashed curve suggests the course of these
nonradiative processes. Ground state potential curve Representative internal coordinate .
known as internal conversion.
and we will now investigate how this happens. For polyatomic molecules. electronic states in
. various emission processes can occur. than the absorption band suggests that vibrational
deactivation within the potential curve of the upper electronic state is essentially FIG. is
consistent with observed emission is illustrated . before any emission process has had a
chance to occur. some of the The observation that the emission band generally appears at
longer wavelengths. The simplest mechanism that in Fig.e.. Fluorescence If the efficiency of
internal conversion processes that return the molecule to its ground state is not too great.
either do not have such crossing either state which the molecule becomes trapped. potential
curves often exist with suitable relative positions so that combinations of vibrational
deactivations and internal conversions can return the molecule to its ground state Other
molecules. i. lower energy. curves or have potential however. In such cases emission does
occur.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES because at the crossover
point the potential energies of the two electronic states are equal and the vibrational kinetic
energy of the molecule in would be zero. Representative internal coordinate . Fluorescent
emission from the same electronic state as initially reached by the excitation process.
INTRODUCTION TO MOLECULAR SPECTROSCOPY complete before transition. although
other conventions are used. . in contrast to up to seconds after the absorption process is
ended. usually a singlet state. Absorption / Emission J i i i i I I . therefore. . Transitions
connecting states of different multiplicities. Z. We will now see that fluorescent transitions. It
is now becoming customary. as the ground state. . FIG. Finckh. XA . as suggested by the
transition arrows of Fig. often continue for periods be defined as phosphorescence and.
Kortum and B. . that the emission transition of Fig. . to use the term fluorescence for
emission transitions that occur between states of the same multiplicity. . is a transition with
no multiplicity change. the emission process of Fig. From G. . rather than the exact reverse
of the absorption is to be drawn. usually triplet to singlet transitions. This definition includes. .
. much emission occurs and. can and generally do occur within about sec after excitation.
The absorption and fluorescent emission spectrum of anthracene dissolved in dionane
showing the occurrence of the emission band at longer wavelengths. . . Since most fairly
intense absorption bands that are observed occur from the ground state to an excited state
with the same multiplicity as the ground state. . physik Chem. emissions from states reached
from the initial excited state by internal conversion as long as the second state has the same
multiplicity. will fluorescence. as defined above. Absorption and emission bands which
illustrate this behavior are shown in Fig. therefore. B. .
A h. . At equilibrium the ratio of the number g of molecules N h in the higher energy state to
the number N in the ground state will be given by Boltzmanns distribution as Ei. or
absorption.e h. about the process of spontaneous emission.iO A h gt g where Ah g of is
Einsteins coefficient of spontaneous emission. therefore. between the two states. from the
discussion of Chap.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES Let us now
consider the rate. downward can occur by either induced emission or sponOne can write the
net rate of these processes as h Rate of gt g transitions iV. been turned Information is
needed. h B h. .h introduced in Eq. for an emission process the type illustrated in Fig. Let us
consider molecules with a ground electronic state designated as g and a higher electronic
state designated as h. Molecules can be excited from state g to state h by transitions induced
by the radiation in the cavity. radiation law P . Recalling the Einstein coefficient of induced
absorption B g. Rate of g h transitions N B hpy g g g h Transitions taneous emission. e.
Consider the sample containing these molecules to be in a blackbody cavity in which the
radiation density pv is related to the frequency and the temperature by Plancks . . of Chap.
We will then be in a position to use the measurable absorption coefficient to deduce the
behavior of the spontaneousemission process.g The nature We know.. must investigate the
mechanism by which this equilibrium is established and maintained. however. The
experimental results that of might try to understand would consist of measurements of the
one change of intensity of the emitted radiation with time after the radiation beam causing the
excitation to the higherenergy electronic state had off. coefficient of A brief derivation will
show that a it is coefficient for spontaneous emission can be deduced and that related to the
.B. g .rt /r is the energy difference. will be determined by relations established here. . derived
in Chap. one writes where hv g h Now we .lkT whvquot i A quot . pgt. or probability. that B g .
induced emission. per molecule.
. The time it lifetime of is exciting radiation an excited state that spontaneously emits after
the turned off can be calculated as a half life. by Eqs. with the desired relation . and one can
obtain from the above rates the relation N Now tity. rhv gh C irhv gh B h g We can further
relate to the observed. Thus quot . h . the value of gf d v . absorption band.cft J With this
result. the rate equation takes for the number of molecules in the excited state to decrease to
half its value.. rivM. g B gP Vgh A g one can replace N h /N a by the Boltzmann expression
for this quanand one can also replace pvhg by the blackbody expression of Eq.
INTRODUCTION TO MOLECULAR SPECTROSCOPY At equilibrium the two opposing rates
must be equal. Rearrangement of the resulting equation gives. One can write the or d In N h
.A kg dt I Integration over the time interval to t t in which N h goes from N h to iNh gives or In
AT . A hg can be calculated from the observed integrated intensity of the corresponding
absorption band.integrated intensity of the /lt dv.e. i.l. . and in Chap. .
view of Eq. Emissions corresponding to these bands occur. Eq.. therefore. or .e. that the
most . collisional deactivation. perhaps. due X quot X Q x gec Such section. halflife. occur
with a smaller integrated absorption coefficient. Many absorption bands are less intense.
absorption should. . i. transition has av dv The emission from the excited state reached by
this A. It is now recognized. the early work of G. N. in halflife. typical nonthat any decrease in
the efficiency of freezing in a glass or cooling to liquid radiative processes involving internal
conversion treated in the previous It is clear. X quot iVJk /aP dv Alternatively one can relate
the halflife to the oscillator strength / If one inserts the relation of of the transition discussed
in Sec. gives h Vln ircl.. is small enough and nonradiative processes are relatively
ineffective.P /aP dv . different net electron spin angular states with primarily as a result of
momenta. according to the above analysis. lifetimes are comparable with.
Lewis.ELECTRONIC SPECTRA OF POLYATOMIC MOLECULES Finally. substitution of the
expression for A h g obtained in Eq.. cm and a . . of . . that connect electronic states of
different multiplicities. or shorter than. . as by nitrogen temperature.. have a to the emission
process.e. with a correspondingly longer One finds in practice that fluorescent emission can
be prolonged to halflives of as much as sec in cases where the value of Ah. than that cited
for typical completely allowed transitions. one has lt gt ltS A typical strong absorption band
due to an allowed electronic band maximum at. Phosphorescence is Phosphorescent
emission here defined as resulting from transitions i. . will allow the fluorescent emission to
occur to a significant extent. of Chap.
The potentialenergy diagram that represents a situation that can lead to phosphorescence is
shown in Fig.. Although internal conis version between states of different multiplicities this
process can occur apparently not easy. INTRODUCTION TO MOLECULAR
SPECTROSCOPY tions between a triplet excited state important phosphorescent bands of
organic compounds arise from transiand a singlet ground state.e. Arrangement of
potentialenergy curves of singlet S and triplet T states that can lead to phosphorescence.
Emission from this phosphorescence. Collisional deactivation can then drop the energy past
the point where the potential curve of the singlet state crosses that of the triplet state. if the
not carry the molecule too quickly past the Once a triplet electronic state is formed. S T V gtv
. singlet Absorption of radiation from the ground singlet to an excited can occur to an
appreciable extent. triplet state to the ground state will constitute Since such transitions will
be forbidden by the pro hibition against spin changes. the spontaneousemission process will
FIG. S / Representative coordinate .y / / / gt/s. tripletstate molecules further vibrational
energy will be lost and the molecules will occupy the lowlying vibration levels of the triplet
electronic state. . and can lead to collisional deactivation does potential crossing point. i. the
transition can be allowed.
generally referred to as inter system crossing. often provides the means whereby
photochemical reactions can occur..e. and the phosrelatively long periods of time. ticularly In
this fluorescence way the study of electronic spectra. or longer.ELECTRONIC SPECTRA OF
POLYATOMIC MOLECULES occur with low probability. It should be mentioned that the
lifetimes of excited states is a subject of great importance in photochemical studies. that is
seldom observed in liquids at ordinary temperatures. With this procedure many organic
compounds undergo appreciable phosphorescence excited singlet to triplet internal
conversions. One should recognize that the relation between the . such as the triplet states
that lead to phosphorescence. are not A uncommon.amp and Aj coefficients and the relation
between A h g and ij require a roundabout mechanism. The occurrence of a longlived
highenergy species. The state is process of internal conversion between a singlet sufficiently
difficult. such as that of Fig. and parand phosphorescence. Again the process of forming a
glass with the material as a solute and cooling the system to liquid nitrogen temperatures is
often resorted to. i. continue over phorescent emission will Halflives for phosphorescence of
the order of seconds. in order to populate a longlived excited state. . and photochemistry are
intimately related. a will be very small. and are observed to emit radiation as
phosphorescence. and a triplet even if the potential curves cross. .
Correlation diagrams. BoseEinstein Angstrom unit. M. . Character table. Operators for.
electronic. Bohr condition on. Amplitudes of vibration. internal. energy levels structures of. for
spring systems. L. references to. definition Band head. in linear molecules. quantum
restrictions on. B. for vibrational transitions. . table of. . electronic states of. Broglie. . in
atoms. Bohr orbits. displacement. . ultraviolet spectrum of. de. Centralforce field. . Angular
momentum. energy levels. for particleinabox model. Degenerate vibrations. rotation
Centrifugal distortion. Configurations. . . Conversion factors. in assymmetric top mole for
electronic transitions.B. Crystalfield theory..INDEX Absorption coefficient. in symmetric top
molecules. Bohr hydrogen atom model. . Asymmetric top molecules. . Coordination
compounds. for rotational for vibrational energy levels. . . N. Conjugated systems. Boltzmann
distribution. Angular velocity. Anharmonicity. Davisson and Germer. Degeneracy. .
Coordinates. . rotational spectra of. Applications of spectroscopy. symmetry. of. Bohr. . d
orbitals. normal. Bond lengths. Chargetransfer transitions. of. cules. Born. Beers law. in
molecules. Coupling of rotationvibration. in group theory. Benzene.. statistics. . Characters of
representations. Average value. quantum mechanical. in symmetric top molecules. Classes.
Induced emission. Ligandfield theory. of carbonyl compounds. Eigenfunction. in aromatic
systems. for benzene. of diatomic molecules. Experimental methods. electronic states of.
Displacement coordinate. of benzene. oscillator strength. of a rotating molecule. Dipole
moments. . Fortrat parabola. of spontaneous emission. . Hookes law. Deslandres table. table
of. of carbonyl compounds. . Electron spin resonance spectroscopy. . . Freeelectron model. .
Diatomic molecules. . . of benzene. Infrared spectra. symmetry Electromagnetic radiation. . .
. cal solution for. effect . rotation of. Hunds rule. . Electronic orbitals. G matrix. Induced
absorption. . Einstein. in conjugated systems. of diatomic molecules. Emission spectra of
diatomic molecules. Electronic transitions. vibrational. rotationvibration bands of. rotation of.
Kinetic energy.. . vibration of. rotational selection rules for. of a vibrating particle.
rotationvibration bands of. Gerade states. of HC. Formaldehyde. Linear molecules.
references to. on energy levels. . of atoms. principle. Degrees of freedom. Group. of a
vibrating molecule. of coordination compounds. in diatomic molecules. Force constant.
Dissociation energy. Lagranges equation. . Einsteins coefficient. A. FranckCondon
Frequency. . statistics. of coordination compounds. Emission. of. for carbonyl group. table of.
of electromagnetic radiation. Electric field. Group theory. for atoms. Hexatriene. in
coordination compounds. Electronic configurations. interaction with molecules. Harmonic
oscillator. . electronic states of. Electronic spectra. for diatomic molecules. Interaction force
constants. Intersystem crossing. electronic configurations of. Internal conversion. regions of.
electronic spectrum of. . intensities of. Electronic states. FermiDirac Fluorescence. for
polyatomic molecules. INTRODUCTION TO MOLECULAR SPECTROSCOPY /. . . Laporte
rule. definition of. of induced absorption. from rotational spectra. . quantum mechani Hermite
polynomial. . Group vibrations.
reducible. Overtones. Parallel bands. of vibrational energy levels. of symmetric top
molecules. of a linear triatomic molecule. of linear molecules.. Perpendicular bands.
Operators. of linear molecules. of linear molecules. Reduced mass. for polyatomic
molecules. symmetry of. . symmetry of. Rigid rotor. M. of asymmetric top molecules. of
symmetric top molecules. . Rotationvibration spectra. Reciprocal centimeters. Morse
function. Octatetraene. Rotational selection rules. Oscillator strength. Nonbonding orbitals.
Particleinabox problem. table of. . . .INDEX Linear molecules. Rotational energies. .
irelectron transition. . . . Radiative processes. . Prolate symmetric top molecules. Quantum
restrictions. . Quantum numbers of electrons in atoms. Plancks constant. for springs. . for
excited electronic states. rotational Stark effect for. Nuclearspin coupling. Normal
coordinates.. of diatomic molecules. . of a linear system. Ortho states. of symmetric top
molecules. Normal vibrations. Raman spectroscopy. . of symmetric top molecules.
Multiplicity. Molecular orbitals for benzene. Pauli exclusion principle. Orthogonality of wave
functions. for HC. Lonepair electrons. Point groups. Representations. Telectrons. . Potential
energy function. M. . of rotational levels. . Population. Normal modes. Quadrupole coupling.
Nuclear magnetic resonance spectroscopy. Nonrigid rotor. . irreducible. Nonradiative
processes. . tt bonds. . for chemical bonds. . Rotational structure of electronic bands.
intensity of. relation to Hookes law. . . Probability function. Normalization. of linear molecules.
from rotationvibration spectra. Phosphorescence. Moment of inertia. P. Planck. . vibrations
of. Octahedral symmetry. Oblate symmetrictop molecules. Rotation. Newtons equation. from
rotation spectra. of symmetric top molecules. of rotational energy levels. table of. Morse. for I
Maclaurin series expansion. Polyatomic molecules. Para states.
United atom.. Thomson. of rotation of. H . E. rotationvibration bands of. . Stark effect.
induced. Timedependent Schroedinger equation. H vibra of degenerate vibrations. .
Symmetric top molecules. of linear triatomic molecules. . . of linear molecules. for rotating
linear molecules. Vibrational transitions. J. structures of. Symmetry. timedependent.
Vibrational energies. in vibrational calculations.. . harmonic oscillator. intensity . Transitions.
symmetric top molecules. ltr bonds. of electronic states. . of diatomic molecules. Separated
atoms. of Vibrations. Rydberg equation. functions. Symmetry operations. of polyatomic
molecules. Spectrometers. . . classical. wave functions. of asymmetric top molecules.
application to tions. Wave Wave for equation. electronic. vibrational. for atomic orbitals. . . .
Sehroedinger equation. Ungerade rotational. spontaneous. Vibrational structure of electronic
bands. INTRODUCTION TO MOLECULAR SPECTROSCOPY Rutherford. Valenceforce
field. Spontaneous emission. timeindependent. states. Statistical weight of rotational levels.
Structure. basis of. Wavelength. Transformation matrices. . J. of normal coordinates. E.
Wilsons method for vibrational calcula tions. Selection rules. moment. . of. Vector model of
the atom. Transition Schroedinger. of Wave numbers.. Symmetry elements.
rd Ed. nd Ed he Modern Theory of Solids Chemistry. Modern Physics. . Eiperimental
Physical Chemistry. th Physical Organic Chemistry. th Ed Physical Chemistry of Metals
Instrumental Methods of Chemical Analysis. rd Ed fundamentals of Electricity and Magnetism
nemistry of Engineering Materials. Principles of Mathematical Analysis T Quantum
Mechanics. th Ed rmciples ol Modern Physics w o c gt It hemodynamics.cs. nd Ed I
Electronics. College Physics. nd Ed Fundamentals of Optics. nd Ed Electromagnetism
Modern Physics Mathematics of Physics and Modern Engineering Principles of
Mechanics.Other McGrawHill INTERNATIONAL STUDENT EDITIONS in gt Related Fields O
Physical Chemistry for PreMedical Students. n SYNGE WEBER WHITE O cn o o Production
to Atomic Spectra Principles of Biochemistry. nd Ed. nd Ed hys. ieat and Thermodynamics.
th Ed. nd Ed as Physical Chemistry Organic Chemistry. PIPES Applied Mathematics
itroduction to . rd Ed. th Ed. nd Ed. PAULING Introduction to Quantum Mecha for Physicists.
Engineers and nd Ed. nd Ed. nd Ed Fundamental Concepts ol Inorganic Chen iiiclear Pliv
IMING Numencal Methods For Scientists And Engini Ed ilculations of Analytical Chemistry.