Microwave Molecular Spectra

MICROWAVE MOLECULAR SPECTRA WALTER GORDY Duke University Durham, North Carolina ROBERT L.COOK Mississippi State Univer...

Author: Walter Gordy | Robert Lee Cook

64 downloads 1312 Views 12MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

MICROWAVE MOLECULAR SPECTRA WALTER GORDY Duke University Durham, North Carolina

ROBERT L.COOK Mississippi State University Mississippi State, Mississippi

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS New York

Chichester

Brisbane

Toronto

Singapore

Copyright@ 1984 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Catalogue Card Number: 84-40367 ISBN 0-471-08681-9

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

TO THE DUKE MICROWAVERS The more than one hundred graduate students and research associates who have contributed much to the development of microwave spectroscopy.

INTRODUCTION TO THE SERIES

Techniques of Chemistry is the successor to the Technique of Organic Chemistry Series and its companion- Technique of Inorganic Chemistry. Because many of the methods are employed in all branches of chemical science, the division into techniques for organic and inorganic chemistry has become increasingly artificial. Accordingly, the new series reflects the wider application of techniques, and the component volumes for the most part provide complete treatments of the methods covered. Volumes in which limited areas of application are discussed can be easily recognized by their titles. Like its predecessors, the series is devoted to a comprehensive presentation of the respective techniques. The authors give the theoretical background for an understanding of the various methods and operations and describe the techniques and tools, their modifications, their merits and limitations, and their handling. It is hoped that the series will contribute to a better understanding and a more rational and effective application of the respective techniques. Authors and editors hope that readers will find the volumes in this series useful and will communicate to them any criticisms and suggestions for improvements. ARNOLD WEISSBERGER

Research Laboratories Eastman Koahk Company Rochester, New York

vii

PREFACE

Many exciting new developments in microwave spectroscopy have occurred since the earlier edition of this book was published in 1970. The frequency coverage of coherent microwave spectroscopy is still expanding. Although not comparable to the fourfold expansion that occurred in the 17 years preceding the 1970 edition, the millimeter wave range has been extended from 800 GHz to above 1000 GHz since 1970. Subtle new techniques continue to improve the sensitivity and resolving power of microwave spectrometers as well as their applicability to new types of spectra. Detection of the microwave spectra of nonpolar, spherical-top molecules such as CH4 and of “forbidden” AK transitions in symmetric-top molecules such as PH3 has been achieved. Microwave “molecular ion” spectroscopy has become a practical reality since 1970. The observation of rotational spectra of weak, hydrogen-bonded complexes and rare gas atom-molecule complexes (van der Waals molecules) has become widespread. New techniques for observation of molecules in highly excited vibrational states have been devised. Significant advances in the theory of complex microwave molecular spectra have been made. In the last 15 years microwave spectroscopy has advanced from observation o f molecules in laboratory cells to observation of molecules in interstellar space. An important new field, microwave molecular astronomy, has been created. Microwave sources and techniques have been combined with infrared and optical lasers to form the new and rapidly advancing field of microwave-optical double resonance spectroscopy. Although the spectra described in the earlier edition are in no sense out of date, the new developments made the revision of Microwave Molecular Spectra desirable if not necessary. The basic theory and measurements of microwave spectroscopy have a remarkable durability which results from the high resolution and accuracy of measurement that characterize all coherent radiation spectroscopy of sharp line spectra. Because we could not justifiably delete nor significantly reduce the basic material of the earlier edition, a moderate expansion of the volume was necessary to achieve an adequate coverage of microwave molecular spectra in the 1980s. As was true for the earlier edition, we are indebted to many people for assistance in the revision of this book. Again, Vida Miller Gordy graciously assisted with every phase of the manuscript preparation. We have benefited by discussions with Frank De Lucia, Eric Herbst, Paul Helminger, and K. V. L. N. Sastry. The Winnewissers- Manfred, Brenda, and Gisbert-have helped to ix

X

PREFACE

keep us informed about the latest results in the field. M. C. L. Gerry read parts of the manuscript and made helpful suggestions. We are also grateful to other spectroscopists for sending us preprints and reprints describing results obtained in their laboratories. Among them are Lisa Nygaard, E. Tiemann, H. Dreizler, H. D. Rudolph, A. F. Krupnov, and J. L. Destombes. Finally, we wish to express a tribute to the memory of Dr. William West, editor of the earlier edition. He was a great person, a considerate and competent editor, and a treasured friend. WALTER GORDY ROBERT L. COOK

Durham, North Carolina July 1984

PREFACE TO THE SECOND EDITION

In the 17 years since the first book on microwave spectroscopy was written, the field has developed so extensively that it is not possible to give a comprehensive coverage of all its aspects within a single volume of manageable size. Not only have the applications been increased enormously and the instruments and techniques diversfied, but the microwave region itself has also been greatly expanded. The frequency range in which spectral measurements can be made with microwave methods has increased more than fourfold since 1953. Many measurements are now made at submillimeter wavelengths. With the molecular beam maser the already ultrahigh resolution of microwave spectroscopy has been increased by more than an order of magnitude. Rapidly recording, highly sensitive microwave spectrometers have become commercially available. Highspeed computers have taken much of the labor out of the analysis and have made the study of complex molecules more feasible. New theoretical developments have increased the possibility of understanding complex spectra and have also increased the usefulness of such spectra. Microwave spectral measurements on short-lived gaseous free radicals and on substances with vaporization temperatures of the order of a thousand degrees are commonly made. Spectral frequencies are measured to accuracies of the order of one part in lo8. Accurate molecular structures and other properties have been found for molecules numbering into the thousands. In this volume we have sought to provide a basis for the understanding of microwave spectra in the gas phase. In doing this we have developed the theory from what we considered to be the simplest approach consistent with essential correctness and applicability. Although we make no effort to include all the useful information about molecules that has been derived from the spectra-a hardly achievable goal-we have included a variety of types of information about selected molecules. A reasonably complete listing of the molecular structures derived up to 1969 is given in the appendix. Although the book is a member of the Technique of Organic Chemistry series, we have by no means limited the coverage to organic molecules. To do so would have been too wasteful because the theory applicable to organic molecules is generally applicable to inorganic molecules. The book is written for chemical physicists and physical chemists as well as for organic chemists-if indeed such separate classifications are bona $de in this age. We have not included a discussion of the determination of nuclear moments and masses, subjects that xi

xii

PREFACE

are perhaps of more interest to physicists than to chemists. Most nuclear moments and isotopic masses that can be measured to advantage with microwave spectra have already been measured. Rather we have treated nuclear hyperfine structure and isotopic shifts of spectral lines with the aim of using them to gain information about molecules. To achieve more thorough coverage of gases we have omitted solid-state studies, which are made primarily with microwave paramagnetic resonance (included in another volume of this series) and liquid-state studies made chiefly through dielectric dispersion and absorption. For the same reason we have omitted discussion of instruments and techniques of measurement. Nevertheless we have tried to serve the dual purpose of providing a convenient source and reference book for much of the valuable information gained about molecules through microwave spectroscopy and of providing a textbook that explains essential theory for interpretation of the spectra and derivation of information from spectra. Although not written specifically for the purpose, the book can be used as a text for a course or seminar on microwave spectra. A quantitative study of microwave spectroscopy provides numerous, rather elegant examples of the application of quantum mechanics to problems of molecular dynamics. Although the quantum mechanical treatment can in some cases be somewhat complicated in its details, much of the information derivable from microwave spectra of molecules is of such importance that it justifies more than casual attention. Several people have assisted considerably in the preparation of the volume. Vida Miller Gordy has helped persistently and effectively in the preparation and proofreading of the manuscript. Jean Luffman’s typing of the entire manuscript presented us with a beautiful final copy from the many revisions given her in the course of the writing. Janet Jackel drew most of the illustrations. Several research associates and graduate students at Duke University-James Cederberg, Frank De Lucia, Steve Guastafson, Paul Helminger, Sam McKnight, William Oelfke, Edward Pearson, David Straub, Gisbert Winnewisser, Ray Winton, and Fred Wyse-have read sections of the book and made helpful suggestions. To each one we offer our thanks. WALTER GORDY ROBERT L. COOK

Durham, North Carolina April 20, 1969

CONTENTS

CHAPTERS

I.

Introduction

11. Theoretical Aspects of Molecular Rotation

1 11

111. Microwave Transitions-Line Intensities and Shapes

37

IV. Diatomic Molecules

71

V. Linear Polyatomic Molecules

125

VI.

Symmetric-top Molecules

175

VII.

Asymmetric-top Molecules

227

VIII. The Distortable Rotor IX. Nuclear Hyperfine Structure in Molecular Rotational Spectra

297 391

X. Effects of Applied Electric Fields

451

XI. Effects of Applied Magnetic Fields

505

XII. Internal Motions

569

XIII. Derivation of Molecular Structures

647

XIV.

Quadrupole Couplings, Dipole Moments, and the Chemical Bond

725

xv.

Irreducible Tensor Methods for Calculation of Compiex Spectra

803

APPENDIXES

A.

Notes on Matrix Mechanical Methods

B. Calculation of the Eigenvalues and Eigenvectors of a Hermitian Tridiagonal Matrix by the Continued Fraction Method C. The Van Vleck Transformation

841 849

D. Fundamental Constants and Conversion Factors

853 857

E. Isotopic Abundances, Masses, and Moments

859 xiii

xiv

CONTENTS

F.

Bond Radii

873

G. Electronegativities of the Elements

875

H. Computational Procedure for the Vibrational Eigenvalue Problem

879

I. Energies and Relative Intensities of Nuclear Quadrupole Hyperfine Structure

881

J. Nuclear Quadrupole Second-Order Correction Energies for Linear or Symmetric-top Molecules 895 Author Index

897

Subject Index

917

I

Chapter

INTRODUCTION Microwave and radiofrequency spectroscopy are branches of spectroscopy in which the spectral transitions are measured with coherent radiation sources. The radiation sources employed are phase-coherent oscillators which provide energy in a frequency band so narrow that in comparison with most spectral lines they can be considered monochromatic sources. These essentially monochromatic sources are tunable and can be swept over a spectral line to be measured or can be conveniently tuned over wide regions in search of unknown spectral lines. The resolution easily obtainable with these tuned radio electronic oscillators is thousands of times greater than that of conventional infrared spectrometers employing noncoherent sources, with prisms or gratings for dispersion. Figure 1.1 shows the extent of the microwave region of the electromagnetic spectrum. This region is designated as extending from wavelengths of approximately 30 cm to those of 0.3 mm, or from frequencies of 1 GHz to those of 1000 GHz, or 1 THz. The GHz unit (gigacycles per second), now commonly used by microwave spectroscopists, represents lo9 Hz, or a thousand megacycles per second. In the initial period of microwave spectroscopy, measurements were made in the centimeter wave region with oscillators and detectors developed for microwave radar during World War 11. The experimental techniques and extensive results for this period are described in the earlier books on microwave spectroscopy [l, 21. In later years the range of microwave measurements has been extended throughout the millimeter and into the submillimeter wavelengths, to 0.3 mm (1000 GHz or 1 THz). In 1954, coherent microwave spectral measurements by Burrus and Gordy [3] were made to overlap the far-infrared grating measurements of Genzel and Eckhardt [4]. This extension of microwave spectroscopy into the submillimeter wave region was made possible by a crystal harmonic generator and a

ua/

{ I

jgi

i31 ,

I

1

1

1r5

3000A

I

Infrared Waves

10' I

3p.

I

I

Microwaves

Radiowaves

I ~

I

I

I

II O ' * C ~IO"~ IOOOGHz 1 30p. 0 . 3 m m 3rnm 1r3

I

I

10''

I

3cm

I

I

I

I

1

IO'

I

lo6

3m

30m

I 300m

logcps I 10' IOPOMHz

30cm

IdOOKHz

I

)d 3km

Fig. 1.1 Chart showing the extent of the microwave region of the electromagnetic spectrum. 1

2

I NTRO D UCTl ON

crystal diode detector designed by King and Gordy [5]. Later refinement of the harmonic generators and improvement in the sensitivity of the submillimeter wave detectors led to further extension of coherent, tunable, microwave spectroscopy [6-91 to frequencies of 800 GHz (0.37mm) by 1970 [8] and to 1037 GHz (0.38mm) by 1983 (91. Since harmonics are exact multiples of the fundamental frequency of the source power, this type of generator carries with it a precise frequency-measuring chain to the highest detectable harmonic. For example, the J=8+9 transition of CO was detected and measured [9] at 1,036,312.35MHz with a Doppler-limited accuracy of about k0.15 MHz. Among the later improvements of the submillimeter wave spectrometers at Duke was the use of ion-bombarded [lo] silicon crystals in the harmonic generators and replacement of the crystal diode detector with detectors operating at low temperatures, particularly by an indium-antimonide photoconducting detector [8] and by a silicon bolometer [ll]. Very impressive submillimeter wave spectral observations with primary radiation from tunable, submillimeter wave, backward-wave oscillators (BWOs) are now made by A. F. Krupnov [12, 131 and his group working in Gorky, USSR. For detection of the spectra, they employ an acoustic detector that depends on the thermal expansion of the gaseous sample produced by the resonant absorption of the radiation at the frequency of the spectral absorption line. They designate this type of spectrometer by RAD, which signifies “radiospectroscope with acoustic detector.” Their experiments, initiated about 1970, have resulted in high-resolution measurements in the submillimeter wave region to frequencies above 1000 GHz, or to wavelengths of 0.3mm. An illustration of the remarkable performance of this spectrometer [14] is given in Fig. 1.2.A description of the spectrometer with a discussion of its performance is given in reviews by Krupnov and Burenin [12] and by Krupnov [13]. The development of effective microwave instruments and techniques for measurements in the shorter millimeter and the submillimeter wave region

xu)

#a2

x134

1036

~(Gffx)

Fig. 1.2 Part of the submillimeter wave spectrum of NH3 in the 1 THz region (loo0 GHz). From Belov et al. [14].

I NTRO DUCT10 N

3

from 3 mm to approximately 0.3 mm (100 to 1000 GHz) has not only increased the coverage of molecular rotational spectra but has also made possible the use of semioptical techniques [15] for the focusing and direction of radiation without any sacrifice in the resolution and accuracy of the microwave source and detector. It has also made possible the use of free-space cells for corrosive or unstable molecules and ions, the use of molecular-beam, high-temperature absorption spectrometers for molecules having high vaporization temperatures, and the use of precision parallel-plate cells for measurement of Stark components. The increase in the abundance and the strength of microwave molecular lines with increase of frequency insures that the last explored region of the electromagnetic spectrum, the submillimeter wave region, will continue to be a region of interest and value to the chemist and the physicist. Experimental methods and descriptions of the results obtained with millimeter and submillimeter wave spectroscopy are available in several reviews [15-19]. Treatments of selected results will be found throughout this volume. Because of their low intensity, spontaneous emission lines are not observable in the microwave or radiofrequency region. Since the invention of the maser, however, observation of certain microwave spectral lines through stimulated emission has become common. To make such an observation one must in some way upset the Boltzmann distribution so as to obtain an excess population in the upper of two quantum levels involved in the transition. In the first operating maser, Gordon, Zeiger, and Townes [20] accomplished this by removal of the molecules in the lower state through deflection of a molecular beam by an inhomogeneous electric field. Several other methods have since been devised for achievement of excess population in the upper state sufficient for observation of stimulated emission spectroscopy. Perhaps those most widely used are optical pumping [21] and chemical pumping [22]. The exceptional resolution in a beam-maser spectrometer makes it of great advantage in the study of specialized problems. The resolution obtainable in microwave spectroscopy with the molecular beam maser is illustrated [23] by Fig. 1.3, which shows the triplet hyperfine structure due to the deuterium nucleus superimposed upon a hyperfine component of 14N in the J=O+l rotational transition of DCN. The total width of this triplet which occurs at a frequency of 72,414 MHz is only 68.7 kHz, and the two closest components are separated by only 23 kHz. These frequencies are measured to an accuracy of better than a kilohertz, or to one part in lo8. Application of this beam maser has been extended into the upper submillimeter region by Garvey and De Lucia [24] to measure with comparable resolution the hyperfine structure of the J=O-l transition of ND, at 309 GHz and of the l l o ~ l otransition l of D 2 0 at 317 GHz. Developments in techniques and applications of molecular beam masers up to 1975 are described in an excellent review by Lain6 [25]. Since the publication of the earlier edition of this book (1970), microwaveoptical laser double resonance has become increasingly important in measurement of the rotational structure of excited vibrational and electronic levels of molecules [26-301. Harold Jones gives a thorough review of the important

4

I NTRO DUCTION

' - Fig. 1.3 Illustration of the exceptional resolution obtainable with the molecular beam maser. The spectra, observed at 72,414 MHz, represent a part of the hyperfine structure of the J=O-+l transition of DCN. The splitting is due to the nuclear quadrupole coupling of deuterium. From De Lucia and Gordy [23].

infrared-microwave double resonance technique [30]. The development of tunable dye-lasers and the expansion of microwave techniques for the millimeter and submillimeter wave lengths have made microwave-optical double resonance increasingly feasible. An excellent monograph on laser spectroscopy by Demtroder [31] explains experimental techniques and spectral applications. Microwave-microwave double resonance has become a powerful technique for study of special problems such as molecular collision processes. Experimental methods and applications are described in a review by Baker [32]. The use of this technique for identification of spectral transitions in asymmetric rotors is described in Chapter VII, Section 5. Lines of the strong HCN laser source [33,34] in the 0.337 mm wave region have been accurately measured in frequency units by Hocker et al. [35, 361, who detected beat notes between the laser frequencies and the harmonics of a calibrated microwave source. A laser source thus calibrated was later used to give measurable beat notes with a laser operating at a higher frequency [37]; the second laser thus calibrated could be used for measurement of beat notes with a laser operating at a still higher frequency, and so on. With this method, optical laser sources have been measured indirectly, but precisely, in absolute frequency units. By this means the microwave harmonic frequency-measuring chain has been projected to the optical region. Reviews of various types of lasers, with references to numerous monographs on lasers and their various applications, may be found in Quantum Electronics, a 1979 volume of Methods in Experimental Physics [38]. This book is primarily concerned with interpretation of observed spectra.

INTRODUCTION

5

Descriptions of instruments and experimental procedures must be obtained from other sources. An excellent treatment (1979) of modern microwave spectrometers and experimental methods is given by Roussy and Chantry [39]. The pulsed, coherent microwave Fourier-transform spectrometer introduced by Ekkers and Flygare [40] holds much promise for subtle observations requiring exceptional sensitivity and resolution [41]. Interferometric methods for noncoherent (black-body) millimeter and submillimeter spectroscopy described by Fleming [42] are convenient and useful for applications where very high resolution is not required. A technique that has evolved since the earlier edition is the adaptation of the “supersonic free jet” [43] in the study of van der Waals molecules with microwave spectroscopy [41] (Chapter V, Section 9). The method also makes possible observation of very weak hydrogen-bonded complexes that are unstable at ordinary temperatures (Chapter V, section 8) and the study of rotational transitions in normal molecules at very low temperatures, at which they would solidify in a normal absorption cell. The cooling in the supersonic jet is caused by the expansion of a highly pressurized gas as it escapes through a restricting nozzle into an evacuated microwave cavity. The restricting nozzle also collimates the escaping gas into a beam [43]. Applications of microwave spectroscopy to the study of molecular fragments, gaseous free radicals, and molecules in excited electronic states with unpaired electron spins have significantly increased since the earlier edition. Carrington’s monograph on the microwave spectroscopy of gaseous free radicals (1974) describes both spectra and experimental methods [44]. The developments in microwave-optical double resonance have greatly facilitated the measurement of paramagnetic transitions of molecules in excited triplet states [45]. Since the initial observations of the microwave rotational spectra of ionized molecules such as CO’ and HCO’ by R. C . Woods and his associates [46, 471 in 1975, the spectroscopy of molecular ions has become a rapidly developing component of microwave spectroscopy [48]. The extensive observation of microwave spectral lines of molecules in interstellar space is an exciting development, most of which has occurred since the earlier edition. Lines from molecular free radicals and ions, as well as from stable molecules, are being observed with microwave spectral telescopes, operated primarily in the millimeter wave range. Many of these are emission lines originating through a cosmological laser action. The first molecular species to be detected in interstellar space through measurement of its microwave spectral lines was the O H free radical observed in 1963 [49]. The first of the stable molecules thus to be observed [so] in outer space (1968) was ammonia, NH3, which, it is interesting to recall, was the first terrestrial molecule to be observed with microwave spectroscopy, the first to control the frequencies of an atomic clock, and the molecule used in the first operating maser. Within a few years after these initial microwave observations of molecules in outer space, microwave spectroscopy had become a powerful new technique for astronomical investigations. Laboratory measurement and identification of the

6

INTRODUCTION

observed interstellar transitions greatly facilitated the astronomical applications. Late reviews of this rapidly developing field are available [51-531. The interesting microwave spectrum in Fig. 1.4, the J=O+l rotational transition of (NNH)' showing the 14N nuclear quadrupole structure, was recorded from the Orion molecular cloud [54]. This molecular species had not been previously observed in the laboratory. Identification of the species was confirmed by comparison of the observed hyperfine pattern with a theoretically calculated pattern, as shown in the figure. Preliminary observations [ 5 5 ] and a tentative identification [56] had been made earlier. Discrete spectral transitions that are measured primarily in the radiofrequency region up to 1 GHz (lo00 MHz) are nuclear resonances (magnetic and electric quadrupole) and molecular beam resonances (electric and magnetic). Measurable pure rotational spectra of gaseous molecules fall predominantly in the microwave region; electron spin magnetic resonance is most I

'

I

I

I

I -4

1

I -2

I

I

A

in o L - 2

I N,H'

I

0

I

I

I

1 2

I

I

L

4

V - V , (MHr)

Fig. 1.4 The 93.174 GHz lines in OMC-2, compared with theoretical calculations of the hyperfine structure of the J = 1-0 transition of N2H+. From Thaddeus and Turner [54].

REFERENCES

7

advantageously observed in this region. The mathematical methods for treating these types of spectra, the procedures for calculating the energy eigenvalues, line intensities, selectrion rules, and so on, are basically similar. They all include momentum or intrinsic spin operators, dipole moment or direction cosine matrix elements, and state populations that depend on similar statistical mechanisms. The applications described in this volume are primarily rotational spectra of gaseous molecules. Microwave electron spin resonance spectra in condensed matter is treated in a companion volume of this series [57]. The development and increased availability of high-speed computers have paralleled the improvements in techniques and the expanding frequency coverage of microwave spectroscopy. These computers have greatly augmented the contributions in microwave spectroscopy by facilitating the interpretation of complex spectra. We make no attempt to give a complete or comprehensive treatment of microwave spectroscopy. The field is now so large that such a treatment is impossible in a single volume. We shall limit our coverage mainly to problems of interest to chemists and chemical physicists. The tabulations and references we give are not intended to be complete but to be illustrative of what has been done and what can be done in the field. There are other books that describe instruments and techniques as well as spectra [58-631. Extensive tabulations of line frequencies measured with microwave spectroscopy are given in a series of volumes published under the auspices of the National Bureau of Standards [64]. Extensive tabulations of molecular structures and constants derived from microwave spectroscopy are available in the Landolt-Bornstein series [65]. Molecular Spectroscopy, an annual review published by the Chemical Society (London), covers “Microwave Spectroscopy” from 1970 [66]. A book by Mizushima [67] on the theory of rotating diatomic molecules contains many applications to microwave spectra. A tabulation of the constants of diatomic molecules (to 1978) has been provided by Huber and Herzberg [68]. The valuable book Modern Aspects of Microwave Spectroscopy, edited by Chantry, is already referenced several times [13, 30, 32, 39, 42, 521. Advances in the application of microwave spectroscopy to the area of chemical analysis are reviewed elsewhere [69, 701.

References 1. W. Gordy, W. V. Smith, and R. F. Trambarulo, Microwave Spectroscopy, Wiley, New York, 1953. Republication by Dover, New York, 1966. 2. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. 3. C. A. Burrus and W. Gordy, Phys. Ret.., 93, 897 (1954). 4. L. Genzel and W. Eckhardt, Z. Physik, 139, 592 (1954). 5. W. C. King and W. Gordy, Phys. Rec., 90,319 (1953). 6. M. Cowan and W. Gordy, Phys. Rev., 111,209 (1958). 7. G. Jones and W. Gordy, Phys. Reo., 135, A295 (1964). 8. P. Helminger, F. C. De Lucia, and W. Gordy, Phys. Rev. Lett., 25, 1397 (1970).

8

9. 10. 11. 12.

13.

14. 15. 16.

17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

INTRODUCTION

P. Helminger, J. K. Messer, and F. C. De Lucia, Appl. Phys. Lett., 42,309 (1983). R. S. Ohl, P. P. Budenstein, and C. A. Burrus, Rev. Sci. Instrum., 30, 765 (1959). W. Steinbach and W. Gordy, Phys. Rev., A8, 1753 (1973). A. F. Krupnov and A. V. Burenin, “New Methods in Submillimeter Microwave Spectroscopy,” in Molecular Spectroscopy: Modern Research, Vol. 2, K. N. Rao and C. W. Mathews, Eds., Academic, New York, 1976, pp. 93-126. A. F. Krupnov, “Modern Submillimetre Microwave Scanning Spectrometry,” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, London, 1979, pp, 217256. S. P. Belov, L. I. Gershstein, A. F. Krupnov, A. V. Maslovskij, S. Urban, V. Spirko, and D. PapouSek, J. Mol. Specrrosc., 84,288 (1980). W. Gordy, “Microwave Spectroscopy in the Region of 4-0.4 Millimetres,” in Pure and Applied Chemistry. Vol. 11, Butterworths, London, 1965, pp. 403-434. G. Winnewisser, M. Winnewisser, and B. P. Winnewisser, “Millimetre Wave Spectroscopy,” in M T P International Review of Science, Physical Chemistry, Vol. 3, Spectroscopy, D. A. Ramsey, Ed., Butterworths, London, 1972, pp. 241-296. F. C. De Lucia, “Millimeter and Submillimeter Wave Spectroscopy,” in Molecular Spectroscopy: Modern Research, K. N. Rao and C: W. Mathews, Eds., Vol. 2, Academic, New York, 1976, pp. 69-92. Y. Morino and S . Saito, “Microwave Spectroscopy,” in Molecular Spectroscopy: Modern Research, K. N. Rao and C. W. Mathews, Eds., Vol. 1, Academic, New York, 1972, pp. 9-28. D. R. Johnson and R. Pearson, Jr., “Microwave Region,” in Methods in Experimental Physics, Vol. 13, Spectroscopy, Part B, D. Williams, Ed., Academic, New York, 1976, pp. 102-133. J. P. Gordon, H. J. Zeiger, and C. H. Townes, Phys. Rev., 99, 59 (1955). J. Brossel and A. Kastler, Compt. Rendu Acad. Sci. (Paris),229, 1213 (1949): F. Bitter, Phys. Rev., 76, 833 (1949). T. A. Cool, “Chemically Pumped Lasers,” in Methods in Experimental Physics, Vol. 15, Quantum Electronics, Part B, C. L. Tang, Ed., Academic, New York, 1979, pp. 95-142. F. C. De Lucia and W. Gordy, Phys. Rev., 187, 58 (1969). R. M. Garvey and F. C. De Lucia, Can. J. Phys., 55, 1115 (1977). D. C. Laink, “Advances in Molecular Beam Masers,” in Advances in Electronics and Electron Physics, Vol. 39, L. Marton, Ed., Academic, New York, 1975, pp. 183-251. T. Shimizu and T. Oka, J. Chem. Phys., 53,2536 (1970). M. Takami and K. Shimoda, J. Mol. Spectrosc., 59, 35 (1976). 1. Botskor and H. Jones, J . Mol. Spectrosc., 81, 1 (1980). F. Kohler, H. Jones, and H. D. Rudolph, J . Mol. Spectrosc., 80, 56 (1980). H. Jones, “Infrared Microwave Double Resonance Techniques,” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, London, 1979, pp. 123-216. W. Demtroder, Laser Spectroscopy, Springer-Verlag, New York, 1980. J. G. Baker, “Microyave-Microwave Double Resonance,” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, London, 1979, pp. 65-122.

33. H. A. Gebbie, N. W. B. Stone, and F. D. Findlay, Nature, 202, 685 (1964). 34. D. R. Lide and A. G. Maki, Appl. Phys. Lett., 11, 62 (1967). 35. L. 0. Hocker, A. Javan, D. R. Rao, L. Frankel, and T. Sullivan, Appl. Phys. Lett., 10, 147 (1967). 36. L. 0. Hocker and A. Javan, Phys. Lett., A25,489 (1967). 37. L. 0. Hocker and A. Javan, Phys. Lett., A26,255 (1968). 38. C. L. Tang, Ed., Quantum Electronics, Vol. 15, Part B of Methods of Experimental Physics, Academic, New York, 1979.

REFERENCES

9

39. G. Roussy and G. W. Chantry, “Microwave Spectrometers,” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, London, 1979, pp. 1-63. 40. J. Ekkers and W. H. Flygare, Rev. Sci. Instrum., 47, 448 (1976). 41. T. J. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare, J . Chem. Phys., 72, 922 (1980). 42. J. W. Fleming, “Interferometric Spectrometry at Millimetre and Submillimetre Wavelengths,” in Modern Aspects of Microwave Spectroscopy, G . W. Chantry, Ed., Academic, London, 1979, pp. 257-309. 43. D. H. Levy, “The Spectroscopy of Very Cold Gases,’’ Science, 214,263-269 (1981). 44. A. Carrington, Microwave Spectroscopy of Free Radicals, Academic, New York, 1974. 45. M. A. El-Sayed, “PhosphorescencerMicrowave Multiple Resonance Spectroscopy,” in M T P International Review of Science, Physical Chemistry, Vol. 3, Spectroscopy, D. A. Ramsey, Ed., Butterworths, London, 1972, pp. 119-153. 46. T. A. Dixon and R. C. Woods, Phys. Rev. Lett., 34,61 (1975). 47. R. C. Woods, T. A. Dixon, J. R. Saykally, and P. G. Szanto, Phys. Rev. Lett., 35, 1269 (1975). 48. R. J. Saykally and R. C. Woods, “High Resolution Spectroscopy of Molecular Ions,” Ann. Ret.. Phys. Chem., 32,403-431 (1981). 49. S . Weinreb, A. H. Barrett, M. L. Meeks, and J. C. Henry, Nature, 200, 829 (1963). 50. A. C. Cheung, D. M. Rank, C. H. Townes, D. D. Thornton, and W. J. Welch, Phys. Rev. Lett., 21, 1701 (1968). 51. W. B. Somerville, “Interstellar Radio Spectrum Lines,” Rep. Prog. Phys., 40, 483-565 (1977). 52. G . Winnewisser, E. Churchwell, and C . M. Walmsley, “Astrophysics of Interstellar Molecules,” in Modern Aspects of Microwaue Spectroscopy, G. ,W. Chantry, Ed., Academic, London, 1979, pp. 313-503. 53. A. Carrington and D. A. Ramsey, Eds., Molecules in Interstellar Space, The Royal Society, London, 1982. 54. P. Thaddeus and B. F. Turner, Astrophys. J . (Letters),201, L25 (1975). 55. B. F. Turner, Astrophys. J. (Letters),193, L83 (1974). 56. S . Green, J. A. Montgomery, Jr., and P. Thaddeus, Astrophys. J . (Letters),,193, L89 (1974). 57. W. Gordy, Theory and Applications of Electron Spin Resonance, Wiley-Interscience, New York, 1980. 58. M. W. P. Strandberg, Microwave Spectroscopy, Methuen, London, 1954. 59 D. J. E. Ingram, Spectroscopy at Radio and Microwave Frequencies, Butterworths, London, 1955. Second Ed., Plenum Press, New York, 1967. 60 T. M. Sugden and C. N. Kenney, Microwave Spectroscopy of Gases, Van Nostrand, London, 1965. 61. J. B. Wollrab, Rotational Spectra and Molecular Structure, Academic, New York, 1967. 62. J. G. Baker gives an excellent treatment of harmonic generators and semiconductor detectors for millimeter and submillimeter waves in Spectroscopic Techniques for Far Infra-red, SubrnillimetreandMillimerre Waves, D. H. Martin, Ed., North Holland Publishing Co., Amsterdam, 1967, Ch. 5. E. H. Putley and D.H. Martin give a thorough treatment of bolometer detectors as well as point detectors. O p cit., Ch. 4. 63. H. W. Kroto, Molecular Rotation Spectra, John Wiley, London, 1975. 64. Microwaue Spectra Tables, National Bureau of Standards, U.S. Department of Commerce, Washington, D.C. From 1964. 65. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, Group 11, Atomic and Molecular Physics, K.-H. Hellwege and A. M. Hellwege, Eds. Vol. 4, Molecular Constants from Microwave Spectroscopy, by B. Starck, Springer-Verlag, Berlin, 1967. Vol. 6, (Supplement and Extension to Vol. 4), Molecular Constants from Microwaoe, Molecular Beam, and Electron Spin Resonance Spectroscopy, Contributors: J. Demaison, W. Hiittner, B. Starck/I. Buck, R. Tischer, and M. Winnewisser. Springer-Verlag, Berlin, 1974.

10

66.

67. 68. 69. 70.

INTRODUCTION

Vol. 7, Structure Data of Free Polyatomic Molecules, J. H. Callomon, E. Hirota, K. Kuchitsu, W. J. Lafferty, A. G . Maki, and C. S. Pote, with assistance of I . Buck and B. Starck. SpringerVerlag Berlin, 1976. Vol. 14 (Supplement to Vols. II/4 and 11/6), Molecular Constants Mostly from Microwave, Molecular Beam, and Electron Resonance Spectroscopy, J. Demaison, A. Dubrulle, W. Huttner, and E. Tiemann. Springer-Verlag, Berlin, 1982. Molecular Spectroscopy: Specialized Periodical Reports, The Chemical Society, London. “Microwave Spectroscopy,” A. C. Ligon and D. J. Millen, Vol. 1 (1973), pp. 1-61; Vol. 2 (1974), pp. 1-99; Vol. 3 (1975), pp. 1-103. J. N. Macdonald and J. Sheridan, Vol. 4 (1976), pp. 1-69: Vol. 5 (1977), pp. 1-59; Vol. 6 (1979), pp. 1-45. M. Mizushima, The Theory of Rotating Diatomic Molecules, Wiley, New York, 1975. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. IV, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. R. L. Cook and G. E. Jones, “Microwave Spectroscopy,” in Systematic Materials Analysis, J. H. Richards and R. V. Peterson, Eds., Vol. 2, Academic, New York, 1974. R. Varma and L. W. Hrubesh, “Chemical Analysis by Microwave Rotational Spectroscopy,” in Chemical Analysis, Vol. 52, P. J. Elving and J. D. Winefordner, Eds., John Wiley, New York, 1979.

Chapter

11

THEORETICAL ASPECTS OF MOLECULAR ROTATION

CLASSICAL ANGULAR MOMENTA AND ROTATIONAL ENERGY

11

ANGULAR MOMENTUM OPERATORS A N D MATRIX ELEMENTS

15

Spin Operators and Matrix Elements The Symmetric-Top Rotor

19

Squared Operators

20 21

MATRIX ELEMENTS OF ROTATIONAL HAMILTONIAN OPERATORS

22

METHODS OF FINDING EIGENVALUES OF HAMILTONIAN OPERATORS

24

EIGENFUNCTIONS OF ANGULAR MOMENTUM OPERATORS

27

MATRIX ELEMENTS OF DIPOLE MOMENTS AND.DIRECTION COSINES

29

1 CLASSICAL ANGULAR MOMENTA A N D ROTATIONAL ENERGY Derivation of the quantum mechanical properties of molecular rotors, including their microwave spectra, begins with the classical expressions for the angular momenta and rotational energy. Likewise, the final derivation of the molecular structure from the observed spectral constants requires a knowledge of classical moments of inertia. Hence we begin with a brief summary of the classical mechanics of rotating bodies. The classical angular momentum of a rigid system of particles

P=I.w

(2.1)

where w is the angular velocity and I is the moment of inertia tensor which in dyadic notation is written as I=Zxxii+Zxyij+ Zxzik

+ Z y j i+ Z y y j + Z,jk + Z,$i + Zzykj+ Z,,kk 11

12


with

1 mxz I,,, = I,, = - 1 myz Izx = Ix, = -

in which m is the mass of a particular particle and x, y, z are its positional coordinates relative to a rectangular coordinate system fixed in the body and with its origin at the center of gravity of the body. The summation is taken over all the particles of the body. The origin of the coordinate system is chosen at the center of mass because this choice allows the total kinetic energy to be written as the sum of the kinetic energy of translational motion of the center of mass plus the kinetic energy of the motion relative to the center of mass. The translational and rotational motions can hence be treated separately. It is always possible to choose the coordinate axes in such a way that the products of inertia vanish, leaving only the diagonal elements, called the principal moments of inertia. The principal moments are the three roots I of the cubic equation

When the notation x, y, z represents the principal axes system, the components of angular momentum become The rotational kinetic energy is

which in the principal axes system becomes

Now suppose that the body is subjected to a torque s =ir,+jr,+ k r ,

CLASSICAL ANGULAR MOMENTA A N D ROTATIONAL ENERGY

13

relative to the space-fixed axes X, X Z. The time rate of change of angular momentum relative to the space-fixed axes is equal to the applied torque,

where X, Y, 2 are space-fixed axes. When no torque is applied, it is evident that P =iPx +jPy+ kPZ=constant (2.10) also that the components, and hence P i , P:, and P i are each constant. Thus p 2 =P:

+ P: + P; =constant

(2.1 1)

The rate of change of the total angular momentum with reference to a system x, y, z, fixed in a rotating body, caused by a torque relative to that system, consists of two parts. One part is due to the time rate of change of the components P,, P,, P,; the other is due to the fact that the body-fixed axes x, y, z are themselves rotating with angular velocities w,, m y , 0,. The latter contributes a term o x P to the rate of change of the angular momentum. Thus (2.12) i -dP, + j - + kdP, - + o x PdP, =r dt dt dt Note that i,j, k are now unit vectors of the body-fixed system. The corresponding component equations are

(2.13)

dPz -+w,P~-oyPx=5z dt Now x, y, z are chosen as the principal axes of the body’so that Px=Zxw,, P,= Zyoy,P,=Z,w,. Substitution of the values of w from these equations, upon the assumption that the body is rotating freely, with no torque applied (r =0) leads to these equations

(2.14)

These are known as Euler’s equations of motion. Multiplication of the first by P,, the second by P,, and the third by P,, followed by addition of the three,

14


yields (2.15) which upon integration and multiplication by 2 yields Pf+P,2+~2=constant=P’

(2.16)

This shows that when no torque is applied, the square of the total angular momentum expressed in the body-fixed axes is constant. Multiplication of the respective equations (2.14) by

followed by addition and integration yields (2.17) which shows that with no torque applied the kinetic energy of rotation remains constant. In angular momentum space with P,, P,, and P, as the coordinates of a point, it is evident that (2.16) is an equation of a sphere with radius P. In the same momentum coordinates, (2.17) represents an ellipsoid with principal semi-axes of (21xEr)1/2,(2I,Er)’l2, and (21zEr)1’2.Since the values of Ps,P y , and P, must satisfy both equations, it is evident that end points of P can only be along the intersection of the sphere and the ellipsoid. If the sphere and the ellipsoid are to intersect, the values of P must be between those of the minimum and maximum axes of the ellipsoid. This is illustrated by the prolate, symmetrictop case (Fig. 2.1) for which the two smaller semi-axes of the energy ellipsoid are equal ( I , = ZJ. The energy ellipsoid is fixed in the body, whereas P is fixed in magnitude and in direction in space. Therefore the body must rotate in such a manner that the fixed vector P continues to terminate on the surface of the ellipsoid. For a symmetric top, this motion is such that the terminal of P traces circles around the symmetrical ellipsoid, as indicated by Fig. 2.1. For the asymmetric

Fig. 2.1 Diagram illustrating the restrictions on the motions of a prolate symmetric top. The allowed values of P are described by the intersection of the momentum sphere (Eq. 2.16) and the energy ellipsoid (Eq. 2.17).

ANGULAR M O M E N T U M OPERATORS AND MATRIX ELEMENTS

15

!

Fig. 2.2

Simulation of the classical motions of a prolate symmetric top.

rotor in which none of the semi-axes are equal, the curve traced out by P on the surface of the ellipsoid is more complicated, but the stable motions are still highly restricted. The rotational motions of a symmetric top are simulated by a cone that rolls without slipping around the surface of a second cone that is fixed in space. The total momentum vector P is along the axis of the fixed cone, and the component P, is along the axis of the rolling cone, as indicated by Fig. 2.2 for the prolate top. That P, is a constant of the motion of a symmetric top is easily proved by the substitution of Z, = Zy in (2.14). One obtains

dp, -=0 dt

hence

PZ=constant

(2.18)

For the asymmetric rotor Z,# Zy f Z, there is, in contrast, no internal axis about which the rotation is constant, although P 2 is a constant for the asymmetric rotor and is independent of the coordinate system in which it is expressed.

2 ANGULAR MOMENTUM OPERATORS AND MATRIX ELEMENTS Most of the problems treated in this volume involve angular momentum operators. The characteristic energy levels required for finding the microwave spectral frequencies are eigenvalues of the Hamiltonian operators which usually can be expressed in terms of the angular momentum operators of a particle or system of particles. The matrix elements of these angular momentum operators are therefore useful in finding the characteristic energies of the system, that is, the eigenvalues of the Hamiltonian operator. The classical angular momentum of a system of particles can be expressed by

P=C TnXPn n

(2.19)

16

THEORETICAL ASPECTS

OF MOLECULAR ROTATION

where pn is the instantaneous linear momentum m,v, of the nth particle and rn is its radius vector from the center of rotation assumed to be fixed in space. Expanded in terms of its components in space-fixed rectangular coordinates X,X 2, it is

P =i Px + j P y + k Pz

=E [i(Ypz-ZpY),+j(Zpx-

XPz)n+k(XpY- Ypx)nl

(2.20)

n

To derive the corresponding quantum mechanical angular momentum operators one substitutes the relations X+X, and so on, and px+(h/i)(Z/ldX), and so on. Thus the component angular momentum operators are PX

=c9[Y (&)- 2 (&)In n

1

P y = En

4[z(-3-x(&)]n

P z = Cn

n [x($)-

l

1

Y

(2.21)

(A)]

where i=(- 1)'l2 and h =h/2n. In spherical coordinates these component operators are

[ [

Px=P1 -sin

I)'( (3 I)+:(

4 (:)-cot

8 cos 4

cos 4 - -cot 8 sin 4 -

Py=: 1

(2.22)

and the important operator conjugate to the square of the total angular momentum is P 2 = - h 2 {('->(2)[sin sin 8 88

B(L)]+h(E)} sin2 8

(2.23)

It is easily shown from the foregoing expression that P 2 commutes with its component operators, for example, P2Pz- PzP2=0

(2.24)

The component operators do not commute among themselves, however. The following commutation rules are easily shown to hold PxPy - PyPx = ihPz

ANGULAR MOMENTUM OPERATORS A N D MATRIX ELEMENTS

PyPz - PzPy = ihPx PzPx - PxPz = ihPy

17

(2.25)

These important commutation relations for the components can be expressed most compactly by the vector equation

P xP=ihP

(2.26)

Also of convenience for some manipulations are the operators defined by

+

P + = P x iPy

(2.27)

P-=Px-iPy

(2.28)

The rules for commutation of these operators are P + P - - P - P , =2hPz

(2.29)

PZP, - P , P z = h P +

(2.30)

PZP- - P - P z = --ZIP-

(2.31)

Since the component operators P x and PY commute with P 2 , it is evident that P , and P - commute with P2. It is one of the principles of quantum mechanics that operators which commute have common sets of eigenfunctions. Therefore P 2 and Pz have common eigenfunctions, which we designate as $ j , M . Thus we can write (2.32) (2.33) where kJ and k M represent temporarily the eigenvalues of P 2 and Pz corresponding to the eigenstate described by $ j , M . By application of P , and P (called raising and lowering operators) to (2.33) and various other manipulations [l] the quantized values k J = h 2 J ( J + 1) and k M = f i M are obtained, where M and J have integral values and IMJ < J. Thus (2.32) and (2.33) can be expressed as P2$J,M=h2J(J+ 1 ) $ J , M

(2.34)

Pz$ J , M = hM$ J ,M

(2.35)

where

J=O, 1, 2, 3 , . . .

and

M = J , J-1, J - 2 , . . , , --.I(2.36)

With the commutation rules it can be shown that (2.37) (2.38) where C+ and C- are scalar constants. Thus P , and P - are raising and lowering operators in the sense that operation with P , on $ j , M raises the M sub-

18


script by one, whereas operation with P - lowers it by one. The constant C can be evaluated upon the condition that the wave function be properly normalized. Multiplication of (2.37) and (2.38) by their complex conjugates and integration over all coordinates with the use of (2.34) and (2.35) and the normalizations

s

$ 7 , +~1$ J , M +

=1

and

yield the values

s

$ 7 , -~I $ J , M

-1

dT= 1

(2.39)

C+ = i h [ J ( J + I ) - M ( M +

(2.40)

c-=- i h [ J ( J + l ) - M ( M - 1 ) ] ” 2

(2.41)

Consequently (2.37) and (2.38) become P + $ J , M = ~ ~ [ J (1 J) -+M ( M + 1 ) ] 1 1 2 $ ~ , ~ + i

(2.42)

P-$J.M= - i h [ J ( J + l ) - M ( M -

(2.43)

1)]1’2*J,M-1

The particular choice of the C’s as imaginary is discussed at the end of this section. Multiplication of (2.42) by $J*,M+ and (2.43) by $J*,M- and integration over all the coordinates yield the nonvanishing matrix elements of P , and Pgiven below, (2.45) and (2.46). For convenience, we shall use in most applications the abbreviated bracket notation for indication of wave functions and matrix elements. For example, the matrix elements $J*,M+ P+$J,MdTof the operator P + in the representation J , M is indicated by

s

/

$Y,M+lP+$J,MdT=(J, M + W + I J , M )

(2.44)

The nonvanishing matrix elements of P , and P - found from (2.42) and (2.43) are

( J , M + lIP,IJ, M ) = i h [ J ( J + l ) - M ( M +

(2.45)

1)]1’2

(J,M - l I P - I J , M ) = - i h [ J + l ) - M ( M - l ) ] ” 2

(2.46)

That these are all the nonvanishing matrix elements of P , and P I can be seen by the fact that multiplication by any other member of the orthogonal set of functions, say by $J*,M+2, would have reduced the right-hand side of the equation to zero because of the orthogonality requirement, $J*,M$J,,Mt dT =O when either J # S or M f M ’ . All eigenfunctions of Hermitian operators having different eigenvalues must be orthogonal, and the functions $.J,~ are eigenfunctions of the Hermitian operators P2 and Pz. From (2.34) and (2.35) it is evident that the nonvanishing matrix elements of P2 and Pz are (J,MIPZIJ, M)=h2J(J+ 1 ) (2.47)

s

(J,MJPZIJ,M ) = h M

(2.48)

ANGULAR M O M E N T U M OPERATORS AND MATRIX ELEMENTS

19

I t is, of course, always true that the matrix of an operator in the representation of its eigenfunctions is diagonal. The operators Px and Pr are not diagonal in the J, M representation, that is, in the eigenfunctions of P2 and Pz, because they do not commute with Pz. Since Px or P y commutes with P2, we could have chosen a set of eigenfunctions common to P2 and Px, or to P2 and P y , but then the matrix elements of Pz would not have been diagonal in either of these representations. The choice of the particular pair P2 and Pz is, of course, completely arbitrary. From algebraic manipulation of (2.27), (2.28), (2.45), and (2.46) and with the condition

( J , MlPijJ‘, M’)=(J’, M’(PilJ,M)*

(2.49)

that the quantum mechanical operators of all physically real quantities must be Hermitian, the nonvanishing matrix elements of Px and P y in the J , M representation are found to be h ( J , MIPyIJ, M + l)=z [ J ( J + l ) - M ( M + 1)]’12

(2.50)

ih ( J , MIPxIJ, M i 1)= TZ [ J ( J + l ) - M ( M i l)I1/2

(2.51)

These elements are seen to be diagonal in J , but not in M . The phase choice here is consistent with that usually selected for the body-fixed angular momentum components [see (2.63) and (2.64)], that is, P y is real and positive and Px is imaginary. The choice made for Px and P y in most books on quantum mechanics is just the reverse of this, since Ci is chosen to be real rather than imaginary as we have done in (2.40) and (2.41). The choice of phase is, however, completely arbitrary and has no effect on physical observables.

Spin Operators and Matrix Elements Although intrinsic spin angular momentum has no classical counterpart, the electronic spin vector S and the nuclear spin vector I are assigned similar angular momentum operators which obey the same commutation rules as those for P : S xS=ihS

(2.52)

I x I=ihI

(2.53)

One can then in a similar way find the nonvanishing matrix elements of S 2 , Z2, and their various components. These have the same form as those for the operator P since they are derived from the same commutation rules. The justification for the assumption of the analogous commutation rules is the test of experience. The consequences of these assumptions are borne out by all measurements of fine or hyperfine structure in atomic or molecular spectra and likewise by electronic spin and nuclear resonance experiments. There is

20


one important difference, however. Whereas the quantum numbers for Pz have only integral values, those for Sz or Zz may take half-integral values also. This difference does not violate in any way the derivations of the angular properties from the commutation rules, which require only that the values of M must differ by integral steps and that the values must be symmetrical about zero. If the smallest numerical value of M is zero, as found for molecular end-over-end rotation, the values of M , hence of J , can be integers only. Note, however, that it is possible to have a ladder of nonintegral values of M separated by integral units and symmetric about zero when-and only when-the nonzero values of M are odd integral multiples of a half. Certain particles or systems of particles are found to have intrinsic spin values which are halfintegrals. I t follows that (+)A is the smallest observable component of this momentum. The most notable of such particles is the free electron which has an intrinsic spin of S= 1/2; hence Ms= 1/2 and - 1/2. Organic molecules are commonly observed in triplet states for which S=l. Nuclear spin values as high as 1=6 (for 50V) have been observed, but the most common ones are 9/2 or less. Matrix elements of the spin operators can be found from (2.47), (2.48), (2.50), and (2.51) by substitution of the Z or S for J and of M I or M s for M .

The Symmetric-Top Rotor The symmetric-top rotor has a component of its angular momentum about the internal symmetry axis of inertia which is a constant of the motion (see Section 1). If the body-fixed coordinate system is chosen as x,y , z with z as the axis of symmetry, then the operator P, will commute with P 2 since both are constants of the motion, that is, are simultaneously defined. Furthermore, Pz still commutes with P2 since the latter is independent of the coordinate system employed

P 2 =P; + Py’+ Pt =P:

+ P’y+ Pi

(2.54)

Thus, both Pz and P, commute with P 2 and hence with each other and have a common set of eigenfunctions, $ J , K , M . Therefore, in the bracket IC/J,K,M E IJ, K, M ) notation (2.55) (2.56) (2.57) The eigenvalues kJ and kM must be the same as those previously determined for the rotor in spaced-fixed coordinates without regard to symmetry, or with k J = A 2 J ( J + 1) and k M = M h . In a similar way, the values of kK can be found from the commutation rules of the angular momentum operators expressed in the internal coordinate system. These rules are similar to those for the space-fixed system except

ANGULAR MOMENTUM OPERATORS AND MATRIX ELEMENTS

21

for a change in sign of i. In this system P,, of course, commutes with P2, and PxP,-P,P,= -ihP, P,P, - P,P, = - ih P, P,P,- P,P,= -ihPy

(2.58)

The change in the sign of i has the additional effect of making P - a raising and P , a lowering operator in opposition to the corresponding space-fixed operators. From these rules the value of kK is found, as for kM, to be Kh, whe;e 14 is an integer equal to, or less than, J. Since P 2 is independent of the coordinate system, kJ is found to be as before, h2J(J+ 1). Thus for the symmetric top the diagonal matrix elements are ( J , K, M(P21J,K, M ) = h 2 J ( J + l )

(2.59)

( J , K, MI P,I J, K, M )= Kh

(2.60)

(J,. K, M(PzIJ, K, M ) = M h

(2.61)

where J=O, 1 , 2 , . . . K = J , J-1, J - 2 , . . . , - J M = J , J-1, J - 2 , . . . , - J

(2.62)

The matrix elements of P, and P, found from the commutation rules in the body-fixed system are independent of M , but we retain the M subscript to indicate the common eigenfunction ( J , K, MlPxl J, K+ 1, M ) =

'Tih [ J ( J + 1 ) -

K ( K + l)]"'

h ( J , K, MIPJJ, K+ 1, M ) = j [ J ( J + 1 ) - K(K+ l)]li2

(2.63)

(2.64)

The nonvanishing matrix elements of Px and P y are the same as those already stated in (2.50) and (2.51) since they are independent of the internal coordinates and hence of K.

Squared Operators In finding the eigenvalues of the Hamiltonian operator we shall often have need of the matrix elements of the squared operators P:, Py',and so on, of the angular momentum operators. These can be found from the matrix elements already given by application of the matrix product rule ( J , K , MIPi(J ' , K ' , M ' ) =

1

(J,K, MIPgIJ", K", M " )

J",K",M"

x (J", K", M"I P g J J ' K', , M')

(2.65)

22


and ( J , K, MIPSIJ, K, M ) = K 2 h 2

(2.67)

However, the matrix elements for P,’ and P: are both diagonal and off-diagonal

( J , K , MIP:(J’, K’, M’)=[(J, K, MlP,JJ, K + 1, M ) ( J , K + 1, MIPJJ, K, M ) + ( J , K, MlPyJJ,K - 1, M ) ( J , K - 1 , MIPyIJ, K, M)]dK’K + [ ( J , K, MIPyIJ, K+ 1 , M ) ( J , K+ 1, M I P y I J , K + 2 , M)]dK’K+2 + [ ( J , K, MIPyIJ, K - 1 , M ) ( J , K-1, MIPyIJ, K-2, M)lhK‘K-.2 (2.68) The first two terms on the right are diagonal, and with the aid of the Hermetian property and (2.64) they can be combined to give (2.69) The last two terms are off-diagonal by two units of K and with (2.64) can be written as

x [J(J+l)-(K+1)(Kf2)]”2

(2.70)

where the upper plus signs are to be taken together and the lower minus signs together. In a similar way, the matrix elements of P,’ are seen to be (2.71)

x [ J ( J + ~ ) - ( K ~ ~ ) ( K + ~ ) ] ”(2.72) ~

3 MATRIX ELEMENTS OF ROTATIONAL HAMILTONIAN 0 PE RAT0 R S The Hamiltonian operator is obtained from the classical Hamiltonian when the momenta are replaced by their conjugate operators. When no torques are applied, the classical Hamiltonian of the rigid rotor consists of only kinetic energy which can be expressed in terms of the components of angular momentum in the principal axes, as in (2.7). To find the corresponding Hamiltonian operator, one simply substitutes for the Ps the conjugate angular momentum operators. In the body-fixed principal axes, x, y , z, this operator is

MATRIX ELEMENTS OF ROTATIONAL HAMILTONIAN OPERATORS

P: Py’ x - -1 -+-+r-2

(

1%

P2

(2.73)

L)

ZY

23

where

(2.74) The eigenvalues of the Hamiltonian operators represent the quantized energies from which the microwave spectral frequencies are determined. Finding these eigenvalues is therefore one of our most important problems. The Hamiltonian operators dealt with in this volume can usually be expressed in terms of angular momentum operators or intrinsic spin operators which are of the same form as angular momentum operators. For this reason one often makes use of the matrix elements of angular momentum operators and spin operators when finding the energy levels involved in microwave spectral transitions. If the Hamiltonian operator: is found to commute with the angular momentum operators, it will be diagonal in the representation in which those operators are diagonal. Its matrix elements can then be readily found from the known diagonal matrix elements of the angular momentum operators. As a simple example, let us consider the rigid spherical-top rotor for which the three principal moments of inertia are equal, 1, = ZY = ZZ = 1. The Hamiltonian operator 8,. becomes

P2

1 2 -- ( P f P; r-21

+ +

(2.75)

Since 1 is a constant, the Hamiltonian obviously commutes with P2. The nonvanishing matrix elements of X,.,in this case the eigenvalues of X,., that is, the quantized energy values, are 1 h2J(J+1 ) M ) = - ( J , MJP’JJ,M ) = 21 21

E J = ( J , Ml*,.lJ,

~

(2.76)

The Hamiltonian of the symmetric-top rotor commutes with P, and Pz as well as with P2 and is therefore diagonal in the J, K, M representation. This can be easily seen if 1, is set equal to 1, in (2.73); the Hamiltonian for the symmetric top can then be expressed as

8--+ P2 - - ‘-21,

(2;z

2 9

Pt

(2.77)

Since P 2 and P,‘ commute with P, and are diagonal in the J, K, M representais also diagonal in the same representation, with matrix elements tion, 2,. Ej,K=(J, K, M\*JJ,

K,

(2.78)

24


which represent the characteristic rotational energies. Note that these diagonal elements of X r do not depend on M , in agreement with the classical principle that the rotational energy in the absence of torques is independent of the direction in which the angular momentum vector points in space. The Hamiltonian operator for the asymmetric-top rotor for which I, f Z, f I , does not commute with the operator P, or with the other component operators P, or P,,. It is thus not diagonal in the symmetric-top J, K, M representation. Its matrix elements may be easily found in the symmetric-top eigenfunctions, as is done later; but the resulting matrix will not be diagonal, and the elements will not represent eigenvalues of X v .However, X v does commute with P2 and Pz, and hence the matrix will be diagonal in the J and M quantum numbers. In principle, and in practice for J not too high, it is possible to diagonalize the resulting matrix and thus to obtain the eigenvalues of X,.. This is equivalent to the setting up and solving of the secular equation as described in Section 4. For this purpose the matrix elements of 2,. in the J, K, Mrepresentation as given below will be needed. By substitution of the matrix elements of P;, Py", and Pt from (2.67) and (2.69) to (2.72) into the Hamiltonian operator for the asymmetric rotor, (2.73), expressed in the coordinates of its principal axes of inertia, the nonvanishing matrix elements of X r are found to be

and h2 (J,q.ui"lIJ,K f 2 ) = - ( J ( J + I ) -

8

x [J ( J

K(Kf1)]"2

+ 1)- ( K fI)(K f2)] '"

(i ); -

(2.80)

Thus there are diagonal elements of X,., (2.79), but these do not represent eigenvalues of 2,.because in this representation there are also nonvanishing, off-diagonal elements, (2.80), in K.

4 METHODS OF FINDING EIGENVALUES OF HAMILTONIAN OPERATORS Certain systems such as symmetric-top rotors have sufficient symmetry that the Schrodinger equation

X*=E*

(2.81)

is solvable for the eigenfunctions II/ and the eigenvalues E of the Hamiltonian operators. Alternately, it is possible to find the eigenvalues of X, also the direction cosine matrix elements for such systems from the commutation rules of

METHODS OF FINDING EIGENVALUES OF HAMILTONIAN OPERATORS

25

the component operators as indicated in Sections 3 and 6, even without a specific knowledge of the eigenfunctions. When the symmetry of the Hamiltonian operator does not allow direct solution of the Schrodinger equation, the customary procedure is to expand the unknown eigenfunction in terms of a known orthogonal set such as those of the symmetric top; for example, let us assume the eigenfunction of (2.81) to be expressed by (2.82) where $, represents a member of a normalized orthogonal set of functions and the c, are weighting constants. Substitution of (2.82) into (2.81) yields (2.83) By multiplication of this equation by the $2, or the conjugate of a member of the orthogonal set, and integration over all coordinates, one obtains (2.84) Since the assumed set is orthogonal S$$$,dr=O except when m=n and since they are assumed to be normalized, this quantity is unity when m=n. For convenience, the matrix of A? may be expressed in the bracket notation S $ : X $ , d ~ = ( m l 2 @ z ) . In many texts it is written simply as .Ye,,,. Thus (2.84) can be written as (2.85)

where 6,,,=1 when m=n and 6,,,=0 when m f n . The expression (2.85) represents a set of 1 linear equations containing 1 unknown coefficients which have a nontrivial solution only if the determinant of the coefficient vanishes, where I is the number of functions in the set. If this determinant is set equal to zero, the secular equation, I(mI.Yi"ln)-E6m,nl =O

(2.86)

is obtained. If the matrix elements ( m l f l n ) are known or can be found, this equation can, in principle, be solved for the values of E. These values, which are the various roots of the secular equation with E considered as the unknown, represent the eigenvalues of X. By substitution of each of these values of E into (2.87) with the known value of (mi q n ) , ratios of the various coefficients cJcn- 1, and so on, can be found; and with the auxiliary equation, (2.87) obtained from normalization of the $ of (2.82), the values of the cn coefficients

26


can be obtained. Thus the eigenfunction t,bj corresponding to the particular root j of the secular equation or particular value Ej can be found in terms of the

assumed functions I),of some other operator. It should be noted that a knowledge of the assumed eigenfunctions t,b, is not actually required for the setting up of the secular equation and hence for the finding of the energy values E j ; only the matrix elements (rnlxln) are required. These matrix elements (Section 2) can often be found from commutation rules without a specific knowledge of the eigenfunctions. For simplicity we have assumed that only the one quantum number n is required for the labeling of the assumed set of functions preceding: but for the problems considered in this volume additional ones will usually be required. It is apparent that operators such as 2 can be represented by matrices and their eigenvalues can be found by diagonalization of the corresponding matrix. The matrix formulation of quantum mechanics has been developed by Heisenberg. In Appendix A some important features of matrix mechanics pertinent to microwave spectroscopy are reviewed. A classical example of the application of the secular equation is in the finding of the eigenvalues of the field-free, rigid, asymmetric rotor. As we have seen in Section 3, the matrix elements of the Hamiltonian operator of the asymmetric rotor in the J, K representation, that is, the representation in which the Hamiltonian of the symmetric-top rotor is diagonal, can be readily found. Because these elements are diagonal in J, the secular equation factorizes into subequations corresponding to the different values of J. Since the values of K range in unit steps from J to - J, the subdetermined equations have the form

in which we have, for convenience, designated (J, I(IX(J, X ) = 2 2 ' . From (2.79) and (2.80) we see that nonvanishing elements of X occur only for X = K and for X = K k 2 . It is evident that the secular determinant has the dimension of 2 J + 1 and will therefore have 2 J + 1 roots or energy values E corresponding to each value of J. Although the required matrix elements and secular equation can be found easily, solution of this equation becomes increasingly difficult as J increases. It can be solved only with approximation methods and most advantageously with modern computers. As a more specific illustration, let us solve the secular equation of a rigid asymmetric rotor when J= 1. From the matrix elements obtained from (2.79)

EIGENFUNCTIONS OF ANGULAR MOMENTUM OPERATORS

27

and (2.80) with A=fi2/21x, B=fi2/21y, and C=fi2/(21z),the secular equation is seen to be

KIK'

1

0

1

r$+C)-E

0

-__

0

0

(A+B)-E

0

-1

A-B 2

-~

-1

A-B 2

=O

(2.89)

Y$+C)-E

0

Solution of this cubic equation yields the three values of E as Eo=A+B E+=B+C E-=A+C

(2.90)

In the treatment of the higher J values of the asymmetric rotor we change the form of the Hamiltonian to take advantage of the symmetric properties in the reduced energies. These more involved levels are treated in Chapter VII.

5 EIGENFUNCTIONS OF ANGULAR MOMENTUM 0 PE RATORS The eigenfunction $ J , M of the angular momentum operators P z and Pz in space-fixed coordinates are most easily found by use of these operators expressed in spherical coordinates. Application of Pz from (2.22) in (2.35) yields (2.91) We assume that $ J , M =a)-

where @ is a function of writ ten

0

(2.92)

4 only and 0 a function of 6 only. Then (2.91) can be (2.93)

The solution is:

O M = N M eiM@

(2.94)

and, with M an integer, 0 is insured of being single valued, that is, @(q5)= @( q5 + 277). Normalization of the eigenfunction requires (2.95)

28


(2.96) To find the 0-dependent part of the function we apply the operator P2 from (2.23) in (2.34) and obtain

which, upon substitution of $J , M from (2.92) and transformation, becomes (2.98) This is the well-known Legendre equation, solution of which is (2.99)

@J , M = NJ,MP>M'(COS 6)

where P!,MMl(cos0) represents the associated Legendre polynomials and N J , M is a constant which is determined by the normalizing condition

Jo

J

~ ~ a= N~: ,, ~ ~[P\MI(COS d ~ e ) ] * [ ~ > ~ ~ ( ce)] O do s =1

(2.100)

to be (2.101) Combination of these various factors gives the normalized eigenfunction of the angular momentum operators P2 and Pz to be 1 $

J

,

M

2J+1 =

Z

[

i

'

1-

(J-M)! (J+M)!

etM+

PJ (cos 6)

(2.102)

which are commonly called the surface spherical harmonics. The operators P2 and Pz commute with the Hamiltonian operator of the rigid linear rotor, and the functions of (2.102) are also rotational energy state functions of linear molecules when centrifugal distortion is neglected. The same eigenfunctions $ J , M are obtained from a solution of the Schrodinger wave equation X,.$= E $ for the rigid linear molecule. Subsequent discussion of spin statistics (Chapter 111, Section 4) will require a knowledge of the effect on the wave function of an exchange of the nuclei in a linear rotor. This operation is equivalent to the transformation (0, 4)(n-0, 4+n). Replacement of q5 by 4 + n in (2.102) multiplies the function by (- l)M.The P>M1(cos 0) is a polynomial of cos 0 of order J - IM1, involving even (odd) powers of cos 0 if J - IMl is even (odd). Replacement of 0 by n - 0 hence multiplies the function by (- l)J-lM'. The net result is that the total function is multiplied by ( - l)J. To obtain the common eigenfunctions $J , K , M for the symmetric-top operators P2, P,, and Pz, one can express the angular momentum operators in

MATRIX ELEMENTS OF DIPOLE MOMENTS AND DIRECTION COSINES

29

Fig. 2.3 Euler’s angles defining the orientation of the body-fixed (x, y , z ) axes relative to the spacefixed (A’,Y, Z ) axes. The line N represents the intersection of the xy plane with the W plane and is called the line of nodes.

Euler’s angles 0, 4, x, and proceed in a manner similar to that followed previously, but the calculations are much more involved. Dennison [2] has obtained the symmetric-top eigenvalues with matrix methods; Reiche and Radmacher [3] and Kronig and Rabi [4], by solution of the Schrodinger equation for the symmetric top. The resulting eigenfunctions are $J,K,M=@J,K,M

(2.103)

e i K 4 eiMX

where @J,K,M

(

= N J , K , Msln ’ ;)“-“I

F (sin’

(cos

:)

(2.104)

The function F(sin2 0 / 2 ) is a hypergeometric series, and N j , K , M is a normalizing factor determined by the condition N:,K,M $ j , K , M $ J * , K , M dr = 1. Euler’s angles are illustrated in Fig. 2.3. Eigenfunctions of asymmetric rotors can be obtained by expansion of these functions in terms of those for the symmetric top and evaluation of the coefficients in the expansion with the aid of energy values obtained from solution of the secular equation as described in Section 4. However, most of the quantities needed for microwave spectroscopic analysis are obtained from matrix elements derived from commutation rules without a specific knowledge of the eigenfunctions.

s

6 MATRIX ELEMENTS OF DIPOLE MOMENTS AND DI RECTI0 N COS I N ES Particularly for calculating transition probabilities and Stark and Zeeman effects, we shall need the direction cosine matrix elements of body-fixed axes of a rotor relative to space-fixed axes. Microwave spectral transitions are in-

30


duced by interaction of the electric or magnetic components of the radiation fixed in space with the electric or magnetic dipole components fixed in the rotating body. Likewise, the Stark and Zeeman effects arise from interaction of space-fixed electric or magnetic fields with components of the dipole moment fixed in the rotors. These interactions can be considered as occurring between the spaced-fixed fields with components of the dipole moments resolved along the space-fixed axes, but it is evident that the magnitude of the spacefixed components will depend on the state of rotation. In finding these interactions one needs matrix elements of the components of the dipole moment referred to the space-fixed axes. These elements must be expressed in the eigenfunctions of the particular energy states between which the transition occurs. In spinning electrons and nuclei, the body-fixed magnetic dipole moments are along the spin axis and are constants in all magnetic resonance experiments because S and I do not change. Thus the components of p resolve along space-fixed axes as the components of S and Z and can be readily found from the angular momentum matrix elements of the components of S and I . The electric dipole moments matrix elements in space-fixed axes of rotating molecules are more complicated than those of electronic or nuclear spin magnetic moments for a number of reasons, the most important of which is that the electric dipole moment of the rotating molecule does not lie along P as it does along S or I and hence has matrix elements that are off-diagonal in J , whereas the matrix elements of the spin magnetic moments are diagonal in the corresponding quantum number S or I. The electric dipole moment of the linear molecule is perpendicular to P and has no matrix elements diagonal in J. It consequently has no resolvable component along a space-fixed axis and hence no first-order Stark effect. Another complication arises from the lower symmetry possible in molecules, especially in the asymmetric rotor which may have dipole components along each of the principal inertial axes fixed in the rotating body and for which the eigenfunctions of high J transitions are exceedingly complicated. Also, in rotating molecules the body-fixed electric dipole components are not independent of the electronic and vibrational states although these latter states do not generally change with the pure rotational transitions observed in the microwave region. Therefore we can treat the electric dipole moment components in the rotating, body-fixed system as constants. The problem is to find the nonvanishing matrix elements resolved on space-fixed axes in the representation that diagonalizes the rotational Hamiltonian operator. Without loss of generality, we can choose the body-fixed dipole components along the principal axes of inertia. We designate the space-fixed axes by F = X, Y, Z , the body-fixed principal axes of inertia by g =x, y , z, and the cosine of the angle between F and g as @ F , e . For example, a molecule with constant dipole moment components in the body-fixed principal axes of px, p,,, ,uz would have components along the space-fixed Z axes of

+

+

Pz =PLx@zx P’y@zy Pz@Zr

(2.105)

MATRIX ELEMENTS OF DIPOLE MOMENTS A N D DIRECTION COSINES

31

If $,. represents the rotational eigenfunctions, the matrix elements of the dipole moment with reference to the space-fixed axes would be

or, more generally

1

g J

$l?(llF$:.dr=x

Pg

$?@Fg$:.dr

(2.107)

Thus the matrix elements required are those for the direction cosine mFg. For the linear or symmetric-top molecule, the eigenfunctions are known, and the integrals $T@Fg$i.d7 can be evaluated in a straightforward manner. For the linear molecule this procedure is relatively simple, but for the symmetric top it is very tedious because of the complex form of the eigenfunctions. These matrix elements can be found in a simpler way from commutation rules between the angular momentum operators and the direction cosines [5], or between the angular momentum operators and the vector operators [l]. The somewhat involved derivations will not be given here, only the needed results. The nonvanishing direction cosine matrix elements in the symmetric-top representation J , K , M are most useful. They are given in Table 2.1. These elements are separated into factors that depend on the different quantum numbers as follows

s

(J,K,

MI@FgIJ’,

K‘, M’)=(q@FgIJ’)(J, KI@FgIJ’,K‘)(J,Mt@)FgIJ’, M ’ )

(2.108)

Those for the linear molecule may be obtained from them if K is set equal to zero. Those for the asymmetric rotor are obtained from them by methods [5] similar to those described for the finding of the energy eigenvalues of the asymmetric iotor (see Chapter VII). As an illustration of the use of the elements of Table 2.1, let us employ them to find the dipole moment matrix elements that correspond to rotational absorption transitions J - t J + 1 for the symmetric-top molecule in which the permanent dipole moment lies wholly along the axis of symmetry, p = p Z . The accidentally symmetric-top in which Z, = Zy might have p x or p y which is not zero, but an accidentally symmetric-top molecule is exceedingly rare. Using (2.106) with px=O, p y = 0 and the J ‘ = J + 1 column of Table 2.1 we find (J, K,

MIPzIJ+~,K , M ) = p

[(J

ip [(J

=+?

+ 1)’ - K2]”’[(J + 1)2- M’]”’

+ 1)’-

K2]1’2[(Jf M + l)(JfM (J+1)[(2J+1)(25+3)]1’2

- p [ ( J + 1)2- K’]”’[(Jf =+ 2

(2.109)

( J + 1)[(2J+ 1)(25+3)]”’

M + 1)(Jf M ( J + 1)[(2J+ 1)(2J+3)]1’2

+ 2)]’/’

+ 2)]”’

(2.110)

(2.111)

Table 2.1

Factors of Direction Cosine Matrix Elements” of Symmetric-Top Rotors”

Value of J’ Matrix Element Factor

J+ 1

J

J- 1

“Crosset al. [ 5 ] . bThe matrix elements are obtained from the factors with the relation: (J, K . MJ@F,JJ’, K ’ , M’)=(J~@F,JS)(J, KI@F,JJ’, K‘)(J, MI@F,IS, M ) , F = X , Y , Z, and g =x, y , z.

MATRIX ELEMENTS OF DIPOLE MOMENTS AND DIRECTION COSINES

33

Transitions J + J + l and M + M + 1, and M + M - 1 are induced by interactions of the radiation with rotating components of the dipole moment in the XY plane. This is evident from the fact that the nonvanishing matrix elements from these two transitions can be expressed as (J,K,MIPY--PxIJ+~, K , M-1) - p [ ( J + 1)’- K 2 ] 1 ’ 2 [ ( J -M + 1)(J-M+2)]’ ( J 1)[(2J 1)(2J 3)]”’

+

+

( J , K , MIV Y

+

’

(2.112)

+ $xlJ + 1, K , M + 1) =-p

[(J

+ 1)’-

K2I1’2[(J+M + 1)(J+ M+2)]”’ ( J 1)[(2J 1)(2J 3)]’/’

+

+

+

(2.113)

Transitions of the type M + M require radiation with the electric field in the Z direction. The corresponding matrix elements for the linear molecule can be obtained from those for the symmetric top if K is set equal to zero in (2.109)-(2.113) or in Table 2.1. The transition probabilities for induced absorption or stimulated emission are proportional to the squared dipole moment matrix elements with reference to the space-fixed axis (Chapter 111, Section 1) as expressed by the Einstein B coefficient

This equation provides a basis for the selection rules of dipole absorption or emission and is a factor in the calculation of intensities of spectral lines. Dipole transitions occur only between levels for which the matrix elements expressed by (2.109)-(2.113) do not all vanish. The dipole matrix elements for stimulated emission between the same levels can be found in a similar manner from Table 2.1 or simply by a reversal of the quantum numbers in the various elements, J ’ , K’, M ’ for J , K , M . This follows from the fact that the matrices are Hermitian or that the value of B is the same for induced absorption as for stimulated emission. It is of interest that dipole matrix elements corresponding to J+J occur for the symmetric top, but not for the linear molecule. From Table 2.1 it is seen that when p is along the z axis as before, the J - t J elements of the symmetric top are (2.115) (J,K,

MI~y-ipxlJ, K , M - l ) = p

K [ J ( J + 1)- M ( M - 1)I1I2 J ( J 1)

+

(2.116)

34


From the nonvanishing matrix elements thus obtained from Table 2.1, the selection rules for the genuine symmetric top ( p along z ) are found to be AJ=O, + I ,

AK=O,

AM=O, i l

(2.118)

Note, however, that the J + J , K + K , M - M for the rigid molecule gives rise to a constant, space-fixed component, an eigenvalue of p along Z that cannot give resonance coupling with an ac electric field. Also, this transition for the strictly rigid top does not connect different energy states and is therefore trivial. When, however, there is an inversion vibration, as for NH3, the dipole moment does not remain constant but changes with the inversion from + p z to - p z and thus can couple through resonance with an ac electric field component along 2 which has the same frequency as that of the inversion. The inversion gives rise to a splitting of the rotational levels that have opposite symmetry, plus and minus, with transition probabilities proportional to the square of the foregoing matrix elements for AJ=O, AK=O. I t should also be noted that (2.116) and (2.117) indicate that when a field is applied, electric dipole transitions corresponding to J - J , K+ K , M + M + 1 can be observed between the nondegenerate Stark or Zeeman components of a given rotational state, even when there is no inversion or other vibration. For an accidentally symmetric top in which p does not lie exactly along z, the selection rules for K must also include the possibility A K = i1, but such cases are rare. By substitution of K=O in the matrix elements for the symmetric top, the dipole moment matrix elements corresponding to rotational absorption transitions of linear molecules are found to be ( J , MIPzlJ

+ 1, M ) = P

( J , MIPY--iPXlJ+l, M - l ) = p

( J + 1 + M)(J + 1 - M ) (25 1)(2J 3) ( J + 1 - M)(J + 2- M ) (25 1)(2J 3)

[ [

(J, M / P y+ i P xl J+l,M+1 )=-I*

+ +

1

+ +

]

(J+l+M)(J+2+M)

[

+

(2.119)

+

(2.120)

1

(2.121)

(25 1)(2J 3) Note that the J+J transitions are zero when K=O. Those for J + J - 1 are not zero and can be found in a similar way from Table 2.1. From the honvanishing elements the selection rules for the rigid linear molecular rotor are seen to be

AJ= fl,

AM=O, f 1

(2.122)

Transition probabilities for the particular Stark or Zeeman components are proportional to the squares of the foregoing matrix elements. In the field-free molecular rotor, the Stark components are degenerate, and the intensities of the unsplit, pure rotational lines are proportional to the squares of these matrix elements summed over all M values of the final J levels and, if the stimulating radiation is isotropic, over the three coordinates, X, I: Z . The summation can be achieved simply by addition of the squared component

REFERENCES

35

matrices. The M dependency cancels. The result for linear molecules is

= ( J , MIPZlJ+ 1, MI2 + i [ ( J , MIpLy-ipxIJ+ 1, M - 1)2

+ ( J , MI + i P X / J + 1, M + U2]

=p2(J+

1)/(2J+ 1)

(2.123)

Similarly, the dipole moment matrix elements for the field-free symmetric top must be summed over the M‘ degeneracies of the final state when the effective transition probabilities for the rotational absorption lines are obtained. The result for the J , K + J + 1, K transition is

(J+1)2-K2 =P ( J + 1)(2J+ 1)

(2.124)

For the J+J or the inversion type of transition the result is

The (+) and ( - ) symbols represent the two inversion states. When plane-polarized radiation is used, coupling with molecules occurs only along one axis, and the foregoing squared sums must be reduced by 3 for calculation of the line intensities. Dipole moment matrices and selection rules for the asymmetric rotor are described in Chapter VII.

References 1. E. Feenberg and G . E. Pake, Notes on the Quanrum Theory of Angular Momenrum, Addison-

2. 3. 4. 5.

Wesley, Cambridge, Mass., 1953. D. M. Dennison, Phys. Rev., 28, 318 (1926); Reo. Mod. Phys., 3, 280 (1931). F. Reiche and H. Rademacher, Z . Physik, 39,444 (1927). R. de L. Kronig and 1. I. Rabi, Phys. Ret.., 29,262 (1927). P. C. Cross, R. M. Hainer, and G. C. King, J. Chem. Phys., 12,210 (1944).

Chapter

III

MICROWAVE TRANSITIONS -LINE INTENSITIES A N D SHAPES

1

LINE STRENGTHS

37

2

LINE SHAPES A N D WIDTHS

44

Natural Line Width Doppler Broadening Instrumental Distortion

44 45 47 50

POPULATION OF ENERGY STATES

54

Molecular Vibrational and Rotational States

55

SYMMETRY PROPERTIES

58

Inversion of the Coordinates Symmetry of Momenta1 Ellipsoid Effects of Nuclear Spin

58 60 61

Pressure Broadening

3

4

Microwave spectral lines arise from transitions between quantized energy levels of which the separations AE correspond to the microwave quanta hv, where v is the frequency of the microwave radiation. If the molecule, particle, or system of particles has a dipole moment, electric or magnetic, the radiation field will couple with it to induce transitions when the frequency of the radiation corresponds to the Bohr condition hv = AE. An electric dipole will couple with the electric vector: a magnetic dipole, with the magnetic component of the radiation. Electric dipole coupling is generally the mechanism for inducing molecular rotational transitions; magnetic dipole coupling, the mechanism for inducing transitions between states of electron spin resonance. Since electric quadrupole transitions and spontaneous emission are too weak in the microwave region to be of interest, we shall be concerned with induced dipole transitions.

1

LINESTRENGTHS

Consider two discrete, nondegenerate levels of a substance between which microwave transitions can occur. For simplicity, we designate each level by 37

38

MICROWAVE TRANSITIONS-LINE

INTENSITIES AND SHAPES

only one quantum number, m for the lower level and n for the upper one. To be specific, we assume that the substance is made up of molecules, although the theory described applies equally well to spinning electrons, to nuclei oriented in magnetic fields, or to any quantum mechanical system considered in this volume. Now suppose that the substance is exposed to radiation at the resonant frequency

A molecule originally in state m will have a probability (3.2) of absorbing a quantum hvmn and of undergoing a transition m+n in unit time, where p ( v m n ) represents the density of the radiation and B,,, represents the Einstein coefficient of absorption for the particular transition of the substance. A molecule originally in state n will have a probability Pm+n

Pm+n

= P(Vmn)Bm+n

=P ( v m n ) B m + n

+A m e n

(3.3)

of emitting a quantum hv,, and of undergoing the transition m+n in unit time, where Bmen is the Einstein coefficient of induced emission and A,,, is the Einstein coefficient of spontaneous emission for the transition. Einstein originally assumed the coefficients of induced emission and absorption to be equal, an assumption later verified by quantum mechanical calculations and by experiments. From time-dependent perturbation theory it can be shown (see, for example, Pauling and Wilson [I]) that for isotropic radiation the probability of absorption in unit time is (3.4)

where

and where F =,'A Y, Z represents the matrix elements of the dipole moment component of the molecule or other particle referred to space-fixed axes for the transition m-tn. The dipole moment matrix is Hermitian (mlpFln)= (nl pF(rn)*.Thus the Einstein B coefficients are

The quantities (mlpFln)are the matrix elements of the dipole moment components resolved on the space-fixed axes in the representation that diagonalizes the energy matrix of the molecule. For most molecules, the dipole moment components can be considered as constant in the rotating body-fixed axes, and the matrix elements may then be expressed only in molecular rota-

LINE STRENGTHS

39

tional eigenfunctions. Dipole moment matrix elements for molecular rotational transitions are given in Chapter 11, Section 6. Since the B coefficients are equal, we shall hereafter drop the arrow subscripts. The coefficient of spontaneous emission A is relatively inconsequential at microwave and radiofrequencies. A rigorous derivation is difficult, but the coefficient may be found easily from the B coefficients by use of Planck’s radiation law with the assumption that the only mechanism for exchange of energy between levels is through radiation-induced or spontaneous emission. For isolated units in which no collision or like thermal process occurs, this assumption is obviously justifiable, and we assume this idealized condition when finding the coefficient A . It is possible, however, to treat relaxation processes such as collisions as arising from radiation components generated by the collisions that contribute to ~(v,,). If N , represents the number of molecules per unit volume in the lower state, the number per unit volume undergoing transitions to the upper state in time At will be A Nm

+

n=

NmBmn P(vmn)At

(3.7)

The number per unit volume making the reverse transition in time At is A Nm

+

n=

Nn[ B m n P(Vmn) + A m

+

n]

At

(3.8)

If thermal equilibrium is maintained by these processes, (AN,+,,)/(A[)= (AN,+,)/(At), and (3.7) with (3.8) yields _ Nn -

BmnP(Vmn) N m -BmnP(Vmn) + A m + , At thermal equilibrium the Boltzmann law requires:

(3.9)

(3.10) Combination of the Boltzmann relation with (3.9) yields upon transformation Am+, 1 P(vmn)=---B,, exp (hv,,JkT ) - 1

(3.11)

Conformity of this expression with Planck’s radiation law 87th~~

F ) exp ( (hv/kT ) - 1

’(’)=(

requires A,

With B,, from (3.6), this gives

c

=

(F) B,,

(3.12)

(3.13)

40

MICROWAVE TRANSITIONS-LINE

INTENSITIES A N D SHAPES

In the optical region spontaneous emission is of comparable importance with stimulated emission. However, the variation of the A coefficient with the cube of the radiation frequency makes spontaneous emission insignificant in the microwave and radiofrequency regions. In this regions we can neglect it in comparison with other relaxation processes. Suppose now that the radiation density p(vmn)is the coherent radiation from a controlled microwave or radiofrequency oscillator. If I/ is the volume of the sample, the total number of molecules undergoing the transition rn-n per second is VNmpm-tn. Since each of them absorbs an energy hv,,, the total energy per second (power) required for this transition is (3.15) and the total power returned to the radiation field through stimulated emission is (3.16)

The emitted power will be in phase with the coherent radiation that stimulates it and thus will add power of the same frequency and in phase with that from the original source. As a result of these processes, there will be a net change in the coherent power in the radiation field of (3.17) Usually the population of the lower state is the greater, Nm> N,, and the AP of (3.17) is negative, that is, there will be a net absorption of power. If, as in maser spectrometers, the upper state has the greater population, N n > Nm, the AP will be positive, that is, there will be a net gain of power or amplification. It is evident, however, that application of resonant coherent power causes more transitions from the state having the greater population and hence tends to equalize the population of the two states. When N n = N m , there will be no net change in power, AP=O. It is further evident that continuous observation of either absorption or amplification of the applied radiation would require that some other process be operative to preserve a difference in populations, to offset the equalizing tendency of the applied power. Thermal motions that cause interactions between the molecules or particles most commonly serve this function. However, the thermal process always tends to produce a greater population of the lower energy level so that under conditions of thermal equilibrium the states have a difference in population given by the Boltzmann law, (3.10). Clearly, application of coherent resonant power will tend to upset the thermal equilibrium between the two levels so that the relative populations will no longer be given by Boltzmann’s law. If, however, the thermal relaxation is rapid as compared with the rate of exchange of energy of the molecules with the applied radiation field, the thermal processes will dominate and thermal equilibrium will be maintained, or nearly so. Since most microwave absorption spectrometers operate under conditions of approximate thermal equilibrium, we shall calculate the peak absorption coefficients for this condition. We can

LINE STRENGTHS

41

achieve this condition by lowering the applied radiation p ( v m n so ) as to decrease the probability of radiation-induced transitions or by increasing the rate of thermal relaxation through some process such as an increase of temperature or pressure of the sample. Under conditions of thermal equilibrium we can employ Boltzmann’s law with (3.17) to find the power absorbed Pab,= - A p = VNm(1 - e- hvmnikT )BmnP(I1rnn)ht1rnn

(3.18)

A quantity more useful to spectroscopists than the absorbed power is the absorption coefficient s( defined by M=

-(-)(-) 1 P

AP Ax

(3.19)

where AP represents the power absorbed in an element of cell length Ax where the total power is P . The element of volume AV of the sample can be expressed as AV =SAX where S represents the crosssection of the absorption cell; the energy density in this section as p(v)=P/cS where P is the power at the input and c is the velocity of propagation; and the number of particles in the lower state as N , = NF, where N is the number of particles per unit volume of the sample and F, is the fraction of these which are in the state m. Substitution of these relations into (3.18) and transformation give rmn=

1 AP NF, (1- e-hvmnikT)Bmnhvmn c PAX

--

(3.20)

For microwave spectroscopy, where hv >")(N *S)N2= N -S(y")+ 7")N')

(4.61)

is the spin-rotation operator in which the last term accounts for the change in the spin-rotation energy caused by centrifugal distortion. The y") and 7") signify the first-order and second-order spin-rotation coupling constants which are sometimes signified by ,uOand p l . The last term 2 s - s =543s:-SZ)

(4.62)

is the spin-spin operator arising from interaction of the magnetic dipole moments of the two unpaired electrons. This interaction is assumed to be axially symmetric about the molecular axis z. The constant signifies the spin-spin coupling constant. Effects of centrifugal stretching on this term are included later, in the coefficient %,by (4.66)-(4.68). To obtain the characteristic energies of X , one must first find the nonvanishing matrix element (41214')in a suitable representation and then set up and solve the corresponding secular equation, such as (2.85). Since J = N + S (see Fig. 4.7), the logical choice is $)=IJ, M J N , S). The finding of nonvanishing elements of (4.60), and 2s-N, (4.61), in this representation poses no problem for it is evident that both expressions are diagonal in this expansion. However, finding those of 2s-s, (4.62), is difficult because neither S, nor S,' is diagonal in this representation. To solve the problem, one can transform S, to space-fixed axes X, X Z and employ known matrix elements of Sx,Sr, and Sz (Chapter 11, Section 2 ) with those of the transforming direction cosine elements. Probably the simplest procedure for finding these marix elements is to express %or %s-s in spherical tensor notation [57] and use standard formulas already developed in the theories of irreducible tensor operators [59] to obtain the desired matrix elements. This method is described in Chapter XV. The resulting matrix elements are [60] (4.63) I$)=[BN(N f l)-DN'(N + 1)']6N,N' 1 (4'2'?s-j.,l4)=2 [y'O'+Y'"N(N+ 1 ) ] [ J ( J + 1)- N ( N + 1)-s(S+ 1)]6N,N, (4.64)

(4'1*rot

(qyyi"s-sl4)=

-3J30i,(-

S+J+N

1)

J N {2

S

S N(N+1)(2N+l) N } J ( Z N - l ) ( 2 N + ? ) SN,"

MOLECULES WITH ELECTRONIC ANGULAR M O M E N T U M

101

For simplicity, the effects of centrifugal stretching on the spin-spin interaction have not been included in .Xs.s. By comparison with (4.60) and (4.61), we include these stretching effects in the coefficient 2 as i,=).lo)

+ i.(')N2

(4.66)

which, for the term diagonal in N [first term on the right of (4.65)], becomes A=1.'0'+3.'1'N(N+l)

N=N'

(4.67)

For the last two terms of (4.65), which are off-diagonal by 2 in the rotational quantum number N, this coefficient expansion is [54]

E. = %IO'

+

'(J2

+ J + 1)

"=J- 1 N'=J+I

(4.68)

The term in the curly brackets of (4.65) is the six-j symbol [59], which has a definite numerical value for a given set of numbers in brackets. With the restricted values of N, N=J-l,

J,

J+1

the nonvanishing matrix elements of the &of (4.59)are found from (4.63)-(4.65), with (4.67) and (4.68), to be a= .Xj-

1 , j - 1=

B J ( J - 1 ) - D J 2 ( ~ + + + O ' ( J - 1)+ Y " ' J ( J + [+-2J/(2J+ l)][i"+i"'J(J- l)]

(4.69)

b= X j , j = B J ( J + 1)-DJ2(J+1)2-[g'o'+~'1'J(J+1)]+~[i'o'+i"'J(J+1)] (4.70) c = X j + i , j + i = B ( J + 1)(J+2)-D(J+ 1)2(J+2)2- [ + O ' ( J + ~ ) +y"'(J+ 1)(J+2)2] [+-2(J+ 1)/(2J+ 1)][1~"'+3."'(J+ 1)(J+2)]

(4.71)

~ = X J - I ,=J* JL + II ,J-~

(4.72)

+

=2[JJ"+1/(2J+1)][E.'o'+i,'1'(J2+J+1)]

1

Because X is diagonal in J, the energy determinant for each value of J , except J=O, factors into a 3 x 3' determinant to give a se.cular equation of the form J;l N Y J-1 a-E b-E =O (4.73) J+ 1 C-E where a, b, c, and d are given by (4.69)-(4.72). For the special case of J=O, N = N = 1, the energy equation reduces to Xll-E=O (4.74) Thus the J=O level is unsplit and has the energy value

E(J=O) = x1 1 = 2(B- 20) - 2(y'0'

+ 2y"') -$(A@' + 21'")J=

0

(4.75)

102

DIATOMIC MOLECULES

Solution of the secular equation (4.73) yields the triplet energy values for each value of J, except zero, to be

Eo(4 =b

(4.76)

+ 4d ]

+

E *(J)=$(a c) ++[(a -

2 1 2

J#O

(4.77)

In some papers, po, p l , Bo, and B 1 are used for y"), y"), B, and -D, respectively, as employed here. Because of admixture of the rotational states for N = J & l with those for N = N f 2 = J = l , N is not a good quantum number except when N = J. For this reason the pseudo-quantum number n is often used to designate the sequence of the triplet rotational levels. However, we shall continue to use N to indicate this sequence with the understanding that it is not, strictly speaking, a good quantum number. The frequencies of the allowed rotational transitions conforming to the selection rules, N+ N + 2, AJ=O, f 1 , and calculated from the energy equations (4.75)-(4.77) may be expressed [61] as

++A('

vR(N)= (2B- 2 ~ ' "

))(N + 1)- 4D(N

+ 3)3

(4.78)

The frequencies of the fine structure transitions conforming to the selection rules AN=O, A J = k 1 may likewise be expressed as v +(N)=i,'O'

+ i ~ '4~ ~ ') "+-(A") +$$")(k2 + k + 2 )

L- { f ( k )- (2k + 1)[B- 2 ~ ( k +' k + I ) - y ( l )+$'''I)

(4.79)

where

f(k)2=F2+k(k+1)G2 in which

+

F =(2k 1)[(B- +p) - ( 2 0 +fr

'

(4.80)

+ 7 k + 4) +k + 1) -;y ' 1 '3 - ,Vo) + 3A(l)(7k2 2k+ 1

')(k2

(4.81) G=

23.'O'

+ 2A'"(k2 + k + 1) 2k+l

(4.82)

and

k=Nf1

(4.83)

The transitions between the lower energy levels corresponding to these frequencies are indicated in the diagrams of Fig. 4.9 for I6O2 and 1 8 0 2 on the left and for l 6 0 l 8 Oon the right. Note that v + ( N )corresponds to transitions of AN=O, ( J = N + l ) + ( J = N ) and that v-(N)corresponds to AN=O, ( J = N - l ) + ( J = N). For reasons of symmetry, explained in Chapter 111, Section 4, the alternate or any diatomic molecule having identical rotational levels of I6O2 and nuclei with zero spins, are missing. Because the ground electronic state of

MOLECULES WITH ELECTRONIC ANGULAR MOMENTUM

103

I6o2 Fig. 4.9

Magnetic dipole transitions of oxygen for the lower N ( J ) levels. From Steinbach [60].

oxygen 3X; is antisymmetric and the ground vibrational state and the nuclear spin state are symmetric, only antisymmetric rotational states can make the overall wave function symmetric, as required. The rotational states having odd rotational numbers, N = l , 3, 5, . . . , are antisymmetric, and thus those states with N even do not exist for l60and l80.Note in Fig. 4.9 that there are no levels corresponding to N=O, 2, . . . , in the diagram on the left. However, l 6 0 l 8 Odoes not have identical nuclei, and hence its rotational levels are not thus restricted. Also note in Fig. 4.9 that there are no missing levels in the diagram to the right. The selection rules and transition probabilities depend on the matrix elements of the transition moments, electric or magnetic, expressed by (3.6), Chapter 111, Section 1. A molecule in a triplet electronic state always’has a magnetic spin moment and may also have an electric dipole moment. Symmetric diatomic molecules such as 1 6 0 2 have only the magnetic dipole moment, whereas an asymmetric molecule in a triplet state, such as SO, has an electric as well as a magnetic dipole moment and may undergo both magnetic and electric dipole transitions. Selection rules for triplet state molecules depend only on the nonvanishing of the matrix elements of the transition moments. For magnetic dipole transitions, the selection rules are: series 1. AN=O,

AJ=&1

series2. A N = + 2 ,

AJ=O, T 1

magnetic dipole transitions

(4.84) (4.85)

Series 1 are transitions between the triplet components of rotational levels.

104

DIATOMIC MOLECULES

Series 2 with AJ=O correspond to pure rotational transitions, and with A J k 1 consist of transitions between both rotational and electron spin states. Electric dipole transitions give rise to two additional series of lines: series 3.

AN= k1,

AJ=

series 4.

AN= & 1,

AJ=O

1

electric dipole transitions

(4.86) (4.87)

Series 3 corresponds essentially to a pure rotational spectrum since the change in rotational state occurs without a flipping of the electron spin. Series 4 consists of combined rotational and electronic transitions. Although obtaining the matrix elements of the transition moments for triplet state molecules is a straightforward operation, the complexity of the wave functions makes it a tedious one which requires evaluation of the coefficients or eigenvectors in the representation which diagonalizes the energy matrix. For the triplet state these coefficients may be found from solution of (2.85) with the matrix elements of 2 given by (4.69)-(4.72) and the energy values E found from solution of the secular equation (4.73).It is evident that the resulting expressions for the eigenvectors will be quite complex and the evaluation of the matrix elements of the squared transition moments will be complicated. By making use of the theory of irreducible spherical tensor operators and computer evaluations, Steinbach [60] has calculated eigenvectors and magnetic dipole transition probabilities for molecular oxygen--'60z, l8O2,and 160180. The resulting quantities, which require too much space for reproduction here, may be found in Steinbachs Ph.D. dissertation. Because of the slight displacement of the center of mass from the internuclear center in l60l8O,there should be a very slight electric dipole moment in this species, which may give rise to rotational transitions corresponding to the selection rules expressed by (4.61) and (4.62). To our knowledge, none of these transitions have yet been detected. Steinbach, however, has also calculated the probable frequencies and relative transition probabilities in terms of the unknown moment ,uzfor transitions up to 6(7)-,7(8). These calculations should' be helpful to anyone trying to detect these transitions. The microwave spectrum of the important oxygen molecule has been extensively investigated. The fine-structure transition of 6O fall primarily in the 5 to 6 mm wavelength range although one transition, N = l , J = O - r l , occurs in the 2.5 mm wave region. This series, particularly for l6OZbecause of its significant influence on atmospheric microwave transmission, was measured many times in the early years of microwave spectroscopy. The results are described in earlier books on microwave spectra. In efforts toward more precise molecular constants, the fine-structure frequencies of -' 60zwere remeasured with high precision by Zimmerer and Mizushima [62] and by West and Mizushima [63]; the results were analyzed by Welsh and Mizushima [64]. Frequencies of the fine-structure series for 1 8 0 2 and l 6 0 l 8 O have been measured precisely by Steinbach and Gordy [57,61]. Table 4.10 shows the close correlation of these measurements with frequencies calculated from (4.78) and (4.79).


105

Table 4.10 Comparison of Observed and Calculated Microwave Frequencies of 1 8 0 2 and 6O" 0

'

Transition

Observed Frequency

Predicted Frequency

(MH4

(MHz)

Dgerence

"02Frequencies" 57,239.907(20) 58,899.771(27) 59,811.414(34) 60,505.782(62) 61,529.854(32) 59,871.473(23) 58,962.067(24) 58,270.727(9) 378,831.51(10)

57,239.952(12) 58,899.732(12) 59,811.404(15) 60,505.840(47) 61,529.864(18) 59,871.464(19) 58,962.047(15) 58,270.794(46) 78,831.5l(2)

- 0.045

+

0.039 +0.010 - 0.058 -0.010 0.009 0.020 - 0.067 0.000

+ +

160180 Frequenciesb

58,656.447(65) 60,094.935(17) 60,801.12l(58) 60,861.117(114) 60,105.612(77) 59,539.420(56) 59,074.760(67) 58,670.780(73) 58,306.003(55) 233,946.179(61)

58,656.313(27) 60,095.012(31) 60,800.64(64) 60,861.240(39) 60,105.615(45) 59,539.445(42) 59,074.722(30) 58,670.743(33) 58,306.058(95) 233,946.178(52)

+0.134 - 0.077 $0.157 -0.123 - 0.003 - 0.025 +0.038 0.037 - 0.055 f0.001

+

"From Steinbach and Gordy [57]. bFrom Steinbach and Gordy [61].

Rotational transitions corresponding to the frequencies of (4.78) are essential for precise evaluation of the molecular parameters. Since these transitions fall in the submillimeter wave region and have rather low transition probabilities, only a few have been measured with microwave methods. The first of these, the N(J)=1(2)+3(2) transition of 1 6 0 2 , was measured by McKnight and Gordy [65] at 424,763.80(20)MHz. This transition has also been measured with microwave techniques for 1 8 0 2 and the N(J)=0(1)+2(1) transition for l60l8O.The frequencies are listed in Table 4.10. Several higher rotational transitions of 602 have been measured with laser magnetic resonance techniques [66, 671. The molecular parameters are listed in Table 4.11 for the three isotopic species 1 6 0 2 , 1 8 0 2 , and l60l8Oin the ground electronic state 3Z; and the ground vibrational state, v=O. For 1802 and l6O1*0,they were obtained from a least-squares, computer fitting of (4.78) and (4.79) to the fine structure frequencies and the one rotational transition as shown in Table 4.10. For

106

DIATOMIC MOLECULES

Table 4.11 Some Derived Molecular Constants of Oxygen in the G r o u n d Stateu

'Xi, v = O

160180

43,l OO.460(6) 0.14501 (27) 59,50 1.34l(4) 0.05848(2) -252.586(1) - O.OOO247( 1) 43,344.7(7.5) 1.207433( 106)

40,707.408(10) 0.129 59,499.097(43) 0.05312(80) - 238.4888(7) -0.000619(116) 40,93 1.7(6.9) 1.207429(103)

38,3 13.730(7) 0.115 59,496.698(11) 0.0521 l(42) - 224.439(3)

-0.000351(54) 3831 8.6(6.3) 1.207427 ( 100)

"From Steinbach and Gordy [61]. Some of the parameter designations are changed from those of the original source.

I6O2,they were similarly calculated from the fine structure frequencies reported by Welsh and Mizushima [64] together with the remeasured N(J)= 1(2)+3(2) rotational frequency, 424,763.21(10) MHz. The equilibrium values of Be given in Table 4.11 were derived from the Bo values with the first-order correction

Be=Bo++CI

(4.88)

for vibration-rotation distortions, (13.103),and with the correction Be = (1 - g r ) &

(4.89)

for electronic-cloud distortions described in Chapter XI, Section 7, (11.115). The cx value used for 1 6 0 2 was 478(15) MHz, derived from a reanalysis by Albritten et al. [68]. of the optical measurements by Babcock and Herzberg [69]. Values of CI for 1 8 0 2 and l6O1*0were obtained from this 1 6 0 2 value by use of isotopic ratio relations. The value of g,.= - 1.25(8)x for I6O2, observed by Evenson and Mizushima [67], was used for all three isotopic species. The fine-structure spectrum of 1 6 0 2 in the first excited vibrational state has been measured by Amano and Hirota [70]. They combined their results with the frequencies for the ground vibrational state measured by Mizushima and associates [62-641 to obtain the equilibrium values of the molecular constants listed in Table 4.12. They also measured some fine structure frequencies for 160180 in the ground vibrational state. Another diatomic molecule having a triplet sigma ground state is sulfur monoxide. It also has been studied extensively. Because SO has an electric dipole moment as well as a magnetic spin moment, it can undergo electric dipole transitions, (4.61)- (4.62), as well as magnetic dipole transitions, (4.59)-(4.60). Unlike 1 6 0 2 , it has no missing rotational levels since its nuclei are dissimilar. Thus, its microwave spectrum is rich in variety. Several of its electric dipole


107

Table 4.12 Equilibrium Constants of I6O2 in the 3E.; G r o u n d Electronic State"

Constant

Value

B,

43,336.2(15) M H z 59,329.08(11)M H z -252.265(11) M H z 1.20748(5) A

'.e I i'

r

1, the 1 splitting is much less than when 111 = 1 and is usually not observable when 111 >2. For the alkali hydroxides, Kuijpers et al. [6] found the 1 doubling to be measurable for 111 = 1 and 2, but not detectable for 111 =3. Bogey and Bauer [7], who measured rotational transitions of very high vibrational states of OCS, excited through active transfer from N2, used the theoretical formalism of Amat et al. [SI for their analysis. They give particular formulas for the frequencies of the ui=3, 4, and 5 degenerate bending modes with the allowed values of I// for each mode. These formulas should be useful in the analysis of the microwave spectra of other linear triatomic molecules in highly excited bending modes, but they require too much space for reproduction here. An observational scan [9] displaying the J=2+3 rotational line of FCP in seven different vibrational states, shown in Fig. 5.2 illustrates the nature of the experimental observations from which vibrational effects on rotational spectra are evaluated.

/-Type Doublet Spectra As Shulman and Townes [lo] proved, direct transitions between 1-type doublet levels lL-lu with no change in J can be observed. The squared dipole moment matrix elements upon which the transition probabilities depend are

(5.29) These elements are obtainable from (2.125) by replacement of K by 1. For light molecules such as HCN, these transitions correspond to frequencies in the microwave region and provide the most precise measurements available of the coupling constant q. When q is measured in frequency units, the /-doublet transitions, according to Eq. (5.23), correspond to frequencies of (5.30) For a given molecule qi is’approximately constant; therefore for a given bending mode ui, the 1-doublet frequencies should be proportional to J ( J + 1). That this is nearly true was shown by the early measurements of Shulman and Townes [lo] on HCN. However, measurable deviations [ l l , 121 from this formula occur as J becomes large. Nielsen and Amat [13] have shown that this deviation can be attributed to a change of qi by centrifugal distortion. To first order, the dependence of qi on J is expressed by

qi=qp-q,(1)J ( J + 1 )

(5.31)

134

THEORETICAL FORMULATIONS: ENERGIES A N D FREQUENCIES

135

so that the I-type doublet frequencies are expressed by

Torring [12] found that the several I-type doublet transition frequencies which they observed for the degenerate bending mode u2 in HCN and DCN when u2 = 1 and 111 = 1, could be fitted to the formulas with vl in kHz v ~ ( "C' H 'N)= 224,47SJ(J+ 1)- 0.002666J2(J+ 1)2

(5.33)

and V I (D

' C ' N) = 186,192J (J + 1) - 0.00220J 2( J + 1

(5.34)

)2

Winnewisser and Bodenseh [14] have found that still higher order terms were required for satisfactory agreement with their precise measurements on I-type doublet transitions of the degenerate bending vibrations u4 = 1 and u5 = 1 of HCNO over a wide range of J values. They employed the expression

(vl)i=qp.qJ+ i ) - q : 1 ) J 2 ( J +

1 ) 2 + q p ~ 3 ( 1)3 ~ +-

q ~ 3 ) ~ 4 ( 1)4+ ~ +

..

(5.35)

which was indicated by the theoretical work of Ramadier and Amat [15]. The fitting of the observed frequencies with those calculated from (5.35) was very close. The values obtained for the various q's are listed in Table 5.2. Figure 5.3 shows plots of the observed I-doublet transition frequencies as a function of J. The v4 and v 5 for HCNO correspond to the HCN and the CNO bending modes, respectively. Later investigations [16-201 of the interesting molecule HCNC (fulminic acid) provide a basis for understanding the anomalies of the doublet splitting described above. Yamada et al. [18] have shown that the abnormal splitting arises partly from accidental resonances among some excited vibrational levels. Near coincidences were found to occur between levels of two bending modes, v4 and v5, and the lowest-lying stretching mode, u3. The accidental resonances were found to be less normal and the I-type splitting more nearly normal [19] Table 5.2 1-Type Doublet Constants of H'2C'4N'60 Obtained by Application of (5.35) to the /-Type Doublet Spectra for the u 5 = l and the u4=1 Vibrational States'. Values are given in MHz units u5=l qi0'=34.6391 kO.0001 qL1)=(0.1623& 0.0002) qp=(1 5 0.1 1) 10- 9

.oo

U4=

1

qL0'=23.6722 +O.O009 q f ) = (0.6139 k0.0032) 10&'= (0.1417 k0.0034) qL3'=(0.199f0.012) 10- l o

"From Winnewisser and Bodenseh [14].

136

LINEAR POLYATOMIC MOLECULES

H12C14N160

Type Doublet Series ( v 5 = I )

J

IO

20

30

40

50

60

70

Frequency GHz Fig. 5.3 Plots of observed frequencies of 1-type doublet transitions of HCNO as a function of J . From Winnewisser and Bodenseh [14].

in DCNO than in HCNO. The abnormalities are also enhanced by the quasilinearity of the molecule (Section 7). Bunker et al. [20] conclude that the equilibrium structure of HCNO is linear but that in the zero-point vibrational state and in the u1 and u2 vibrational states the HCN bending potential has its minimum off the internuclear axis. This would seem to be the cause of its quasilinear behavior. Table 5.3 shows a comparison of the 41') values for the first excited bending modes (vi = 1) for a few linear molecules.

Fermi Resonance In addition to the effects of vibrations on rotational spectra, described earlier, there are smaller and less frequently encountered perturbations caused by Fermi resonance between the two vibrational levels of the same symmetry that have nearly the same energy. As Fermi predicted [21], such an interaction gives rise to a repulsion which further separates the vibrational levels. Such displacements in the vibrational levels will be accompanied by changes in the cli and B, values. Nuclear Effects Practically all stable linear molecules have singlet sigma electronic ground states and hence no molecular magnetic field except a weak one generated by the molecular rotation. For this reason most linear molecules observed with microwave spectroscopy do not have a resolvable nuclear magnetic hyperfine structure. However, hyperfine structure due to nuclear quadrupole

THEORETICAL FORMULATIONS: ENERGIES AND FREQUENCIES

137

Table 5.3 I-Type Doublet Constants for First Excited Bending Modes (vi= 1) of Some Linear Molecules. Values are Given in M H z Units Molecule”

HCN DCN NNO FCN 35C1CN HCCF DCCF

HCCCN DCCCN

q2

224.47 186.19 23.73 19.85 7.459

Ref: b b c

Molecule”

ocs

79BrCN OCsoSe

d

IC N

c

SCTe

q4

q5

19.12 15.33

12.57 13.26

q5

q6

2.56 2.68

3.57 3.10

42

Ref:

6.344 3.915 3.172

e

2.688 0.659

e

C

c

J

d d

q7 6.54 5.97

d d

“Atomic symbols are for the most abundant species where not specified. *T.Torring [12]. ‘C. A. Burrus and W. Gordy, Phys. Rev., 101, 599 (1956). dJ. K. Tyler and J. Sheridan, Trans. Faraday SOC.,59, 2661 (1963). ‘C. H. Townes, A. N. Holden,and F. R. Merritt, Phys. Reo., 74,1113 (1948). IW. A. Hardy and G. Silvey, Phys. Reo., 95, 385 (1954).

coupling is often resolved in the microwave rotational spectra. Nuclear quadrupole moments are electric moments, and their coupling to the molecular frame depends on the molecular electric field gradient. Thus large nuclear quadrupole splittings can occur for molecules in “nonmagnetic” singlet sigma states, but only nuclei with spins greater than 9 can have quadrupole moments. For this reason the rotational spectra of many linear organic molecules do not have resolvable hyperfine structure. When there is nuclear hyperfine splitting of the rotational lines, it must be analyzed, and the hypothetical unsplit rotational frequencies corresponding to those which would occur if there were no nuclear splitting must be found before the preceding formulas are applied for calculation of the spectral constant B and the corresponding moment of inertia. Methods for analysis of the hyperfine structure are given in Chapter IX. Because of its rapid decrease with increase of J, the hyperfine splitting is frequently negligible for transitions observed at millimeter wave frequencies and almost always negligible for those observed in the submillimeter wave region. The effect of nuclear isotopic substitution is to change the moment of inertia of the molecule and hence to shift all the rotational lines. Each isotopic species of a given molecule has a complete rotational spectrum that with microwave spectrometers can be easily resolved from that of other isotopic species except when the isotopic difference occurs only in an atom that happens to occur at,

138


or very near, the center of gravity of the molecule. The different isotopic species are very useful in the determination of the interatomic distances of polyatomic molecules from microwave spectroscopy. See Chapter XIII. The rotational transitions of an isotopic species have their particular rotational spectra and nuclear hyperfine structure, which must be analyzed separately.

2 LINE INTENSITIES The formula for calculation of the rotational absorption coefficients of a linear polyatomic molecule is the same as that for a diatomic molecule (Chapter IV, Section 3). The molecule OCS is often used for tuning or monitoring of spectrometers. Furthermore, its dipole moment, B value, and other constants, upon which the absorption coefficient depends, are in the middle range of those of linear polyatomic molecules. As an example, we shall calculate the peak absorption coefficients of its microwave rotational lines for the ground vibrational state. For this purpose we apply (4.1IO), for which pressure-broadened lines are assumed to be in the range where the peak intensities are independent of pressure. In the computation it is more convenient to express a,,, as a function of J. Hence we substitute vo = 2 B ( J + 1) into (4.110)and obtain

[

amax= X ( J + 113 1 - 0'048:J+

"1

,-0.0488J(J+l)/T

(5.36)

where X=a,,,(O+l)=

3.95 x 10-2icF,p2B3 (AV)1 T 2

(5.37)

In this equation the spectral constant B is expressed in GHz (kilomegahertz), but the line breadth parameter (Av)' is in MHz per mm of Hg at 300°K. For OCS, B=6.08 GHz and ( A v ) ~=6.44 MHz. The isotopic concentration of the species is indicated by i,. We shall choose T as 195"K, the temperature of dry ice. For 1 6 0 1 2 C 3 2 Sin its natural abundance, ic=0.94. The dipole moment p=0.709 D. There are three fundamental vibrational modes, but only the degenerate bending mode w2 =527 cm-' and the CS stretching mode w l = 859 cm- have significant populations at T = 195°K. The CO stretching mode w 3 =2079 cm-'. Therefore the population of the ground vibrational state given by (3.56)for u =O is

'

F,=,=(l

-,-'.44"1/T

)d1 (1-,-1.44%/T )dz (1 -e-1.44"3'T)d3=0.96

(5.38)

With these values am,,(O+1)=1.65 x

cm-'

and r,,,(OCS)=1.65

x 10-5(J+l)3[l-15 x 10-4(J+1)] ~ e - ' ~ ~ ~ (5.39) ~ - ~

LINE INTENSITIES

139

This function is plotted in Fig. 5.4. The graph shows that OCS at dry-ice temperature has its strongest rotational absorption lines in the region of j. =0.77 mm, for which J=31+32, approximately. The frequency for this strongest transition is v,,,=390 GHz, and the absorption coefficient at the line peak rma,(31+32)= 1.15 x l o - ' cm-'. In comparison with such a large absorption, the losses to the cell walls are negligible, and the power absorption fractional in a cell 20 cm in length is Pabs1 - e - a x = 1-_

(5.40) PO Thus, in the optimum region 90% of the power is absorbed in a cell only 20 cm cm-' (that for the in length. Although an absorption coefficient of 1.65 x J=O+ I line) can be detected easily with conventional microwave spectrometers, the J = 0-t 1 line of the isotopic species '80CS in its natural abundance of 0.2% would have a peak absorption coefficient of only 3.3 x cm-' and could be detected with only the more sensitive spectrometers. In the region of optimum absorption, i=0.77 mm, the peak absorption coefficient for this rare isotopic species is 2.3 x cm-', and the lines can be readily detected without isotopic concentration.

Optimum Regions for Observations Now that the microwave region has been extended well into the submillimeter wave region, it covers the frequency range where most molecules have their strongest rotational lines. With such coverage available, it is of advantage to know in what regions the different molecules have their optimum detectability. This is particularly desirable when one is searching for lines of molecules

ocs T = 195" K

V E

oo*;

.E 006

002

12

10-+11 I34

2c

1+31 377

4c 499

620

742

u in GHz

Fig. 5.4 Theoretical peak absorption coefficients of rotational lines of OCS in the ground vibrational state.

140


with very small dipole moments, for lines of rare isotopic species, or for rotational lines of molecules in sparsely populated vibrational states. To find the J value of the strongest J + J + 1 transition we differentiate,,A,' with respect to J in (5.36), set the derivative equal to zero, and solve for J. With the omission of some very small terms, the result is

Jopt (for strongest line) 2 5 . 5 Having obtained the relation

(;EJ1'2 ~

(5.41)

Jopt, one can readily find the optimum frequency region from

+

v (for strongest line)=2B(Jopt 1)

(5.42)

I t is apparent from examination of Table 5.4 and the similar one for symmetric-top molecules in Chapter VI that the optimum region for observation of most molecular rotational spectra is already accessible to the highresolution, high-precision methods of microwave spectroscopy. Only the very lightest linear polyatomic molecules, HCN and HCP, have their greatest absorption coefficient in the infrared region, beyond the range of microwave spectrometers, and by observing these molecules at reduced temperatures one can shift their region of optimum absorption to the edge of the microwave region. Although one can obtain better resolution of nuclear hyperfine structure on low J transitions, which usually fall in the lower frequency microwave region, the optimum region for observation of microwave rotational spectra is that from about 5 to 0.5 mm wavelength.

Table 5.4 J Transitions and Frequencies for Regions of Greatest Absorption Line Strengths of Linear Molecules at T=300"K

Molecule HCN HCP NNO HCNO FCN HCCF

ocs

35C1CN HCC3'C1 HCCCN ''BrCN OC7*Se ICN SCTe

B (GHz)

J- J + 1

vopt(GHz)

44.31 19.97 12.56 11.47 10.55 9.70 6.08 5.97 5.68 4.55 4.12 4.04 3.22 1.56

14-15 21-22 27-28 28-29 29-30 30+31 39-40 39-40 40-41 44-45 47-48 47-48 53-54 76-77

1330 877 703 665 634 600 496 478 465 400 394 388 348 240

jbopt(rnrn)

0.23 0.34 0.43 0.45 0.47 0.50 0.60 0.63 0.65 0.75 0.76 0.77 0.86 1.25

MEASUREMENTS WITH SPECIAL HIGH-RESOLUTION TECHNIQUES

141

3 MEASUREMENTS WITH SPECIAL HIGH-RESOLUTION TECH N IOU ES The Doppler-broadened line width, which increases directly with the transition frequency [see (3.39)], becomes an important factor in limiting the accuracy of transitions measured with conventional spectrometers at high frequencies. Three effective devices for circumventing effects of Doppler broadening are the molecular beam maser, the collimated molecular beam spectrometer, and the Lamb-dip method. A section of the DCN spectrum in the shorter millimeter wave region, shown in Fig. 1.3, demonstrates the remarkable resolution of the beam maser. In an ordinary spectrometer, the well-resolved hyperfine structure of the deuterium, shown in this figure within a span of 100 kHz, would have been completely masked by Doppler broadening. The values of Bo and Do for H’’Cl4N that are obtained from these beam measurements [22] are listed later, in Table 5.7. The frequencies of the J = O t l and l t 2 transitions from which they are derived were measured to a precision of one part in lo8.The first HCN maser operation was achieved on the J=O -1 transition by Marcuse [23], who did not use it for accurate measurement of the molecular constants. A collimated molecular beam absorption spectrometer was developed [24, 251 in the early years of microwave spectroscopy for reduction of line broadening in the alkali halides at their high vaporization temperatures. Since that time, Dymanus and his associates at Nijmegen [26-28) have developed a Starkmodulation, molecular-beam absorption spectrometer that is operable for gases at ordinary or lowered temperatures. It has resolution comparable to that of the beam maser and offers the advantage of being simpler to operate. Another effective means of combating the limitations to measurement caused by Doppler broadening is the Lamb-dip method [29-311. It does not require collimated molecular beams, but it does require sufficient source power to saturate the molecules in the center of the Doppler-broadened line and an absorption cell in which the source power is so reflected as to pass through the absorbing molecules in opposite directions. A relatively simple qualitative explanation for the dip can be given. Because the molecules absorbing at the center of the Doppler band have zero velocity relative to that of the transmitted radiation, they absorb, at the same resonant frequency vo, the power traveling in either the forward or the reverse direction. In contrast, molecules having a velocity t‘ relative to the forward direction will have a velocity - v relative to the reverse direction and will thus absorb the forward and the reverse radiation at different frequencies, vo + A v and vo - Av, on opposite sides of the center frequency vo. Thus the molecules absorbing at the center frequency v o are subjected to twice as much resonant power as are those absorbing at slightly higher or slightly lower frequencies. If the power can be critically adjusted to saturate the molecules absorbing at the center of the band without complete saturation of those absorbing at frequencies on either side, the result is a “holeburning” at the exact center of the band. The “burned hole,” or Lamb dip, can

142


be used for precise measurement of the resonant frequency vo. This effect is illustrated in Fig. 5.5 by the Lamb dip in a Doppler-broadened OCS line. Several rotational frequencies of OCS as measured with the method are listed in Table 5.5, together with the rotational constants derived from them. These measurements were made with a klystron-driven harmonic generator, a source

Fig. 5.5 Lamb dip in the J = 15+16 rotational line of OCS at 194,586.4333 MHz. The broad downward curve is the Fabry-Perot cavity resonance; the broad upward curve is the Dopplerbroadened line (width about 310 kHz); the small, sharp depression at the center of the line is the Lamb dip (width about 10 kHz). From Winton and Gordy [31].

Table 5.5 Ground-state Rotational Frequencies and Constants of I6O1'C3'S from Measurements of Lamb Dips"

Frequencies in MHz

Jb

Measured

Calculated

Difference

8 9 10 11 12 13 14 15 16 19 20

109,463.063 121,624.638 133,785.900 145,946321 158,107.360 170,267.494 182,427.198 194,586.433 206,745.161 243,218.040 255,374.461

109,463.064 121,624.638 133,785.900 145,946.818 158,107.361 170,267.498 182,427.198 194,586.433 206,745.161 243,218.042 255,374.460

- 0.001

0.000 0.000 +0.003 -0,001 - 0.004 0.000 0.000 0.000 -0.002 +0.001

Molecular Constants in MHz

Bo =608 1.49205 0.0002 DJ=(1.3008~0.0006)x HJ=(-0.85k0.8)x "From Winton and Gordy [31]. 'Rotational quantum number for lower level,

OBSERVATIONS ON LINEAR TRIATOMIC MOLECULES

143

of relatively low power: the saturation power was achieved through amplification with a high-Q, Fabry-Perot cell. Cazzoli et al. [32] have used Lamb-dip spectroscopy to measure the first two rotational transitions of D12C15N.The Lamb dips of the D hyperfine structure are well resolved. The value they obtained for (eQq)D,0.207 MHz, compares favorably with 0.194 MHz, that from the beam maser experiments [221. Microwave-microwave double resonance is becoming an increasingly important method, not only for ultra-high resolution, but also for special types of measurements, particularly for study of line shapes and molecular collision processes. Notable studies by Oka [33] and by Lees [34] have been made on the linear molecules HCN and OCS. These studies, the experimental techniques, and their potentialities are described in an excellent review by Baker [35].

4


Alkali Hydroxides Through measurements of rotational transitions of KOH and CsOH vapors in a high-temperature microwave spectrometer, Kuczkowski et ai. [36] gained the first evidence of the linear configuration of these molecules. This preliminary finding was confirmed by more complete measurements on RbOH and CsOH by Lide and his associates [37, 381, who obtained, with isotopic substitution, accurate values for the internuclear distances. The ground-state rotational constants and structures are given in Table 5.6. The dipole moments of these Molecular Constants of Alkali Hydroxides

Table 5.6

ro and re (1) Hydroxide MOH NaOH 3

9

~

0

~

3

9

~

0

~

Bo (MHz)

Do (kHz1

MO

OH

12,567.054(10)"

28.72(5)b

ro 1.95b

(0.96)b

8,208.679(10)c 7,494.827(10)'

12.19(6)c 9.46(6)c

ro 2.212' re 2.200'

0.912' 0.968'

85RbOH 85RbOD

6,290.10' 5,120.11'

ro 2.316' r, 2.301(2)f

0.9 13' 0.957(10)'

CsOH CsOD

5,501.08g 4,996.83g

ro 2.403g re 2.391(2)1

0.9209 0.960(10)'

"Pearson and Trueblood [39]. bKuijpers et al. [42]. 'Pearson and Trueblood [40]. dPearson et al. [43]. 'Matsumura and Lide [38]. lLide and Matsumura [41]. gLide and Kuczkowski [37].

144


structures are very high, 7.1 D for CsOH for example, indicating a completely ionic structure of the form Cs’O-H, as would be expected for the electronegativity differences of Cs and OH. Pearson and Trueblood [39, 401 made similarly thorough measurements on NaOH, KOH, and KOD. The accurate values they obtained for the ground-state constants are listed in Table 5.6. Other measurements on alkali hydroxides in several vibrational states have led to accurate values for the equilibrium structures and rotation-vibration interaction constants [41-431. The re values are listed with the yo values in Table 5.6 for all except NaOH. Generally, the re distances are found to be shorter than ro for M O and longer than that for OH. A preliminary study with molecular-beam electric resonance [44] has been made on LiOH, but details apparently are not yet published. Its structure may not be linear.

Boron Compounds The millimeter wave spectra of several isotopic species of the unstable thioborine molecule HBS have been measured by Pearson and McCormick [45], who proved that the molecule has a linear structure. They obtained the accurate substitutional structure, r,(HB)= 1.1692 8, and r,(SB)= 1.5994 8,. They produced the HBS by causing the H2S (or D2S) to flow over solid boron at 1100°C in a “free-space” hot cell. Afterward, microwave spectra were measured for four isotopic species of ClBS by Kirby and Kroto [46], who showed that it also has a linear structure, with r,(CIB)= 1.681 8, and r,(BS)=1.606 A.

Cyanides and Phosphides Table 5.7 displays the bond lengths and ground-state rotational constants for the most abundant isotopic species of some related cyanides and phosphides. Since the first observation [47] in 1948, hydrogen cyanide has been observed many times with millimeter wave spectroscopy; its ground-state rotational constants for the various isotopic combinations [48], including the tritium species [49], are accurately known. Those for its most abundant species, listed in Table 5.7, were measured precisely with a molecular beam maser. It was one of the earlier molecules to be detected with microwave spectroscopy in interstellar space [SO], and it has proved to be very useful in various types of interstellar observations. It is responsible for the strong, much used, submillimeter maser source [51, 521 at 0.33 mm. It is interesting to compare the lengths of the same bonds in the cyanides and phosphides of Table 5.7. The C F bond is significantly shorter in FCN than in FCP, primarily because of the greater conjugation of the single and triple bonds formed with the first-row elements. This conjugation, which may be represented as contribution of valence structures of the form F+=C=N-, has much more influence in shortening the single bond than in lengthening the triple bond. The C N length of FCN is only slightly longer than that of HCN. Note that the lengths of the C N bonds in all the cyanides listed are quite close, as are the C P lengths also. There is, of course, considerable conjugation of the bonds in FCP, although less than in FCN. With no such conjugation, the FC


145

Table 5.7 Rotational Constants and Bond Lengths of Some Linear Cyanides and Phosphides for the Ground Vibrational State and the Most Abundant Isotopic Species

1.063

1.155

0.987

1.171

H-C-N

H-N-C H- 1 . 0 6 7 c-P1 . 5 4 2

F-C-N1 . 2 6 2

1.159

F-C-P1 . 2 8 5

1.541

1.159

1.378

N-C-

C

1.544

P

44,315.9757(4)

87.24(6)

45,332.005(40) 19,976.005(9)

101.9(50) 21.23(10)

c.d

5.3(5) 1

I

10,554.20(2) 5,257.80(3) 2,704.4803(19)

.o

(I

b

e

0.216(13)

“De Lucia and Gordy [22]. bCreswell et al. [53]: also, Pearson et al. [48]. ‘J. W. C. Johns, J. M. R. Stone, and G. Winnewisser, J . Mol. Spectrosc., 38,437 (197 1). dJ. K. Tyler, J . Chem. Phys., 40, 1170 (1964). ‘J. K. Tyler and J. Sheridan, Trans. Faraday Soc., 59,2661 (1963). lKroto et al. [9]. BT.A. Cooper, H. W. Kroto, J. F. Nixon, and 0. Ohashi, J . Chem. Soc., Chem. Commun., No. 8, 333 (1980).

bond would be much longer than 1.285 A ; the C F single-bond length in CH3F is 1.38 A. Without reduction by contributions from structures of the form F+=C=P-, the dipole moment of FCP would be expected to be appreciably larger than the observed value [9] of 0.279 D.

Hydrogen Isocyanide HNC Not until 1976 was the microwave rotational spectrum of HNC, hydrogen isocyanide observed in the laboratory [53-551, and that only after this elusive species had been discovered six years earlier in interstellar space through observation of its J=O-1 transition [56, 571 with a rest frequency of 90,663.9 MHz and tentative identification of the line by’use of a theoretically calculated structure [58]. Although the series CH3NC, C2H5NC,and so on, are commonly known chemicals studied in the gas phase by microwave spectroscopy, the first evidence for the existence of HNC was the infrared spectral observation by Milligan and Jacox [59] made in 1963 on photolized methylazide in an argon matrix at 4°K. Creswell et al. [53] measured the J=O+1 and 1-2 rotational transitions of HNC, DNC, and HN13C: they obtained precise values of the ground-state rotational constants and structural parameters. The J=O- 1 line was measured at 90,663.602 MHz, thus providing experimental confirmation of the correct assignment of 90,663.9(5) MHz as the rest frequency of HNC in the earlier radio-astronomical observation. Pearson et al. [48] extended the laboratory observations to the J = 2+3 transitions and included species containing the 13C isotope. The constants of HNC are recorded in Table 5.7.

146


5 LONG LINEAR MOLECULES O N EARTH AND IN OUTER SPACE Much interest in long linear molecules has been generated by the discovery with microwave spectroscopy of carbon-chain cyanides in interstellar space. The ground-state rotational constants and bond lengths for these cyanides, as derived from spectral measurements in the laboratory, are listed in Table 5.8. The constants of HCN are repeated for comparison. All molecules listed in Table 5.8 have been observed in interstellar space. The longest of them, HC9N and HCllN, have been observed only in interstellar space. Detection of their microwave rotational lines and their assignment were achieved by means of a theoretical model based on the projections of the measured constants of the preceding members of the HC,N series recorded in Table 5.8. Because of the orderliness of this sequence, the projections proved reliable and provided a close fitting of the interstellar lines observed. The lines of HC9N and H C l l N from outer space are very weak. To this time, it is the longest molecule to be observed in outer space. Whether either of these moleTable5.8 Rotational Constants and Bond Lengths of Some Linear HC,N Molecules for their G r o u n d Vibrational States and M o s t Abundant Isotopic Species

HCN

44,315.9757(4) 4,549.0579(4) HC5N 1,331.332714(46) H C 7 N 564.OOO74(16) HC9N (Detected only in interstellar space) H C l l N (Detected only in interstellar space) HC3N

87.24(6) 0.54311(45) 0.0301016(58)

a

b c,d

0.00382 l(87)

e

r 9

Bond Lengths in A 1.0631

H-C=N

1.0569

1.378 1.205

H-C-C-C-CP1.2087

H-

1.0569

h

1.155

1.058

H-C-C---C=N

L

1.159

1.3623

1.3636 1.2223

1.3623

(1.348)

c*-

C ~ C - C ~ C 1-. 2 2C2 3 = C - C ~1 . 2 N 223 1.2087

d

1.3636

e

1.1606

“De Lucia and Gordy [22]. bR.A. Creswell, G. Winnewisser, and M. C . L. Gerry, J . Mol. Spectrosc., 65,4201 (1977). ‘M. Winnewisser, J. Chem. Soc., Furudq Discussions 71, 1 (1981). dAlexander et al. [63]. ‘Kirby et al. [64]. fBroton et al. [60]. gM. B. Bell. P. A. Feldman. S. Kwok. and H. E. Matthews, Nurure, 295, 399 (1982). hG.Winnewisser, A. H. Maki, and D. R. Johnson, J. Mol. Spectrosc., 39, 149 (1971). ‘J. K. Tyler and J. Sheridan, Puns. Furuduy SocierJ,,59, 266 (1963).

EQUILIBRIUM STRUCTURES

147

cules is sufficiently stable under achievable earth conditions to be observed in an absorption cell in a laboratory seems problematical. Broten et al. [60], who detected the HC9N, speculate on the possibility of detection in interstellar space of still longer members of the chain, to HCI3N, and conclude that this would be extremely difficult with the equipment presently available. The highly significant discoveries of the long molecules, HC5N, HC7N, and HC9N, in outer space [60-621 have resulted from a fortunate collaboration between Oka et al., of the Herzberg Institute in Ottawa, and Kroto et al., at the University of Sussex. The laboratory synthesis and measurement of HC5N and HC7N by the Kroto group greatly aided the astronomical observations. The shorter members of this series, HCN and HC3N, were among the first molecules to be discovered in interstellar space [50,65]. Their discovery was aided by the frequencies that were accurately known from previous measurements in the laboratory. Although rotational transitions in highly excited bending vibrational states of the long, linear molecules HC,N are observed in the laboratory, only transitions of the ground vibrational state were detected for molecules in the cold interstellar spaces. Like most other lines of interstellar molecules, those of the long-chain cyanides were observed through maser emission. The mechanisms of the required population inversion of the rotational states are not certain, but collisional interactions and radiative transfer are possible causes [66]. The mechanisms of formation of such long-chain molecules in outer space and their role in the history of the evolving universe are challenging problems. The microwave rotational spectrum of the rather long molecule 0 =C =C =C =S (tricarbon oxide sulfide) has been measured for the first time by Winnewisser and Christiansen [67]. Their measurements prove that its structure is strictly linear, in contrast to the similar molecule OCCCO, which is quasilinear (see Section 7). Rotational constants for the ground vibrational state and most Abundant isotopic combination are Bo=1,413.898 MHz and D o = 0.046 kHz. A later study [68] of the molecule in excited bending states yielded the equilibrium constants: Be= 1,407.230 MHz and 0,=0.0347 kHz. Values for q, y , and xu were also obtained. The dipole moment for the ground state was found to be 0.662 D. Knowledge of these constants should be helpful in the search for this long molecule in interstellar space.

6

EQUILIBRIUM STRUCTURES

An important objective for measurement of the rotational structure of excited vibrational states is to obtain the equilibrium structures of the molecule. Descriptions of the various types of structures of polyatomic moleculesequilibrium, r e , ground state, y o , substitutional, r s , and so on-and methods for their calculation from the spectral measurements are described in Chapter XIII. Although it is not particularly difficult to obtain equilibrium structures for diatomic molecules, the measurements required for even the simplest polyatomic molecules are an order of magnitude greater. Nevertheless, many

148


accurate equilibrium structures have now been obtained for linear triatomic molecules, most of them since the earlier edition of this book in 1970. The progress is accelerating partly because of the improved sensitivity and the increased frequency range of microwave spectrometers and partly because of new techniques for excitation of the higher vibrational states. Since there are two independent structural parameters of nonsymmetrical linear triatomic molecules, measurements on two isotopic species are required for determination of their structures. For calculation of re structures, the Be value for each of these species is required. To obtain each Be value, one must measure rotational transitions in excited levels of each of the three fundamental vibrational modes and over a sufficient frequency range for evaluation of the centrifugal distortions. Theoretically, only the lowest excited levels must be measured, but for greater accuracy, especially where there are Fermi resonances or other complications, it is desirable to make more extended measurements. Also, it is desirable for a consistency test to include more than one isotopic pair. The minimum of data for giving accurate Be values of the two isotopic species necessary for evaluation of the equilibrium structures of a linear triatomic molecule is illustrated by the constants of ICN in Table 5.9. Although accurate measurements on excited v1 and u2 states of ICN were measured in 1972 by Simpson et ai. [69], the necessary measurements on an excited level of u3, which corresponds to the CN vibration, were accomplished much later, in 1978, by Cazzoli et ai. [70], who excited the v 3 states by energy transfer from activated NZ. Equilibrium structures for ICN and a few other linear triatomic molecules are given in Table 5.10 as an illustration of the accuracy obtainable. Even though considerable microwave data were available, some data from infrared rotationvibration spectra were included in the solutions for HCN and OCS. The degree of consistency of the derivations for different isotopic combinations of OCSe is illustrated in Table 5.11.

Table 5.9 Equilibrium Constants for ICN" Constants

I 'C14N 3229.159(6) 11.895(4)b -9.497(2)b 14.332(7) 0.587(27) 0.022* 0.01 I b - 0.040

"Cazzoli et al. [70]. *Simpson et al. [69].

1 1 3 ~ 1 4 ~

3180.674(6) 11.634(5) -8.861(3) 13.353(5) 0.570(9) 0.028 0.008 - 0.021

QUASI-LINEAR MOLECULES

149

Table 5.10 Examples of Equilibrium Structures of Linear Triatomic Molecules

Equilibrium Distance Molecule XYZ HCN I CN

ocs

OCSe

re(X Y )

re(Y Z )

1.06549(24) 1.99209(22) 1.15446 1.1535(1)

1.15321(5) 1.16044(33) 1.56295 1.7098(1)

Ref: (1

C

d

"G. Winnewisser, A. G. Maki, and R. D. Johnson, J . Mol. Spectrosc, 39, 149 (1971). *Cazzoli et al. [70]. 'Y.Morino and T. Nakagawa, J . Mol. Spectrosc, 26,496 (1968). dA. G. Maki, R. L. Sams, and R. Pearson, Jr., J . Mol. Spectrosc, 64,452 (1977).

Table 5.11 Equilibrium Structures for Carbonyl Selenide as Derived from Various Isotopic Pairs'

Isotopic Pairs 3C80Se)b re("O1 zC80Se- 1601 re(180'2C80Se-'6012C80Se)b re(I 6 0 2C80Se-16013C80Se)b

re(C-o)

1.15347(4) 1.15333(21) 1.15372(36) re(1sO'ZC'oSe-'6013C80Se)c 1.15341(4) 1.15356(23) re("O'ZC80Se-'6012C80Se)c re(1601 2C80Se-1 6013C80Se)' 1.15313(40)

re(C-Se)

(A)

1.70978(3) 1.70989(16) 1.70960(26) 1.70971(3) 1.70959(18) 1.70991(29)

"From A. G. Maki, R. L. Sams, and R. Pearson, J . Mol. Spectrosc., 64,452 (1977).

bExcited states used: (10'0, 01'0, 00'1). 'Excited states used: (10°0,02'0, 00'1).

Equilibrium structures have now been obtained for most of the alkali hydroxides. Their r, values are given in Table 5.6, where they are compared with the ground state structures, yo.

7


The quasi-linear characterization of certain molecules apparently was made first by Thorson and Nakagawa [71] in 1960. They based their criteria for quasi linearity on a theoretical model of a two-dimensional harmonic oscillator with a superimposed barrier at the minimum of the harmonic potential. Specifically, the model was designed to simulate deviation in the behavior of a triatomic

150


Next Page

linear molecule when barriers of different heights are superimposed at the minimum of the bending potential. A computer program was made for calculation of the energy levels, eigenfunctions, dipole transition moments, and other significant properties as a function of the imposed barrier heights. Not surprisingly, Thorson and Nakagawa found that in the high barrier limit the calculated properties are those of a nonlinear, or bent, triatomic molecule. However, the calculations for the low-to-moderate barriers and those for anharmonic vibrations in a well with no barrier, revealed properties that could not be assigned either to bent or to strictly linear molecules. They concluded that certain molecules cannot be correctly classified as either bent or linear and are best classified as quasi-linear molecules. Their theory has been applied by Johns [72] to HCN in an excited bending state and to certain bent triatomic molecules such as H 2 0 in their lowest bending state. He found that the effects of quasi linearity give rise to large anharmonicities in the bending vibrations. Yamada and M. Winnewisser [73] have proposed a relatively simple test for quasi linearity which is based on the degree of separation of the coordinates of the rotational and vibrational motions in the Born-Oppenheimer approximation [74]. The degree of separation required by the approximation is based on the ratio of the differences of the rotational energy levels AEr and those of the vibrational levels AE, and is approximately (5.43) where me and mp signify electron and proton masses. Yamada and Winnewisser label the energy ratio as (5.44) and show that the parameter y o as defined by ?0=1-4?

(5.45)

is a measure of quasi linearity when the AE,. and AE, are properly chosen. The proposed relationship is suggested by consideration of the limiting cases of the ideal linear molecule and the ideal bent molecule. In an ideal linear molecule the coordinates of one of the three rotational degrees of freedom is not separable from those of the bending vibrations, with the result that a completely linear molecule has only two rotational degrees of freedom. In contrast, an ideal bent molecule has three rotational degrees of freedom; that is, the rotational coordinates are completely separable from the coordinates for vibration. The parameter yo, defined by (5.45), gives a measure of the allowed departure of "well-behaved" molecules from these ideal cases. An ideal bent molecule has A 4 v k n dand y 2 0 ; consequently, from (5.46), y o z 1. An ideal linear molecule, which conforms to the harmonic approximation of a two-dimensional isotropic oscillator, has E(02°0)=2E(01' 0 ) ; thus from (5.47)y =f.Likewise, (5.46) gives y =+for this case. With this y value, (5.45) gives y o = - 1 for the idealized linear molecule.

Previous Page


151

Upon the assumption that a well-behaved bent molecule conforms to the Born-Oppenheimer approximations, a deviation of 0.01 from y =O is allowed by (5.43). If a comparable deviation of 1% is allowed in the value y = f for linear molecules, the well-behaved linear molecule could have y =0.495 to 0.505, and hence y o = - 1.02 to 0.98. With these considerations, linear and bent molecules are classified by Yamada and Winnewisser [73] in terms of the y o parameter, as shown in Table 5.12. Yamada and Winnewisser listed numerous molecules on an ordered scale according to their y o values. Most of those listed fall within, or close to, the limits of the well-behaved linear or bent molecules. Certain ones, however, fall far from the limits of either. Those falling outside the limits of the wellbehaved linear and well-behaved bent molecules are classified as quasi-linear. Figure 5.6 shows an abbreviated version of this scale as given by Winnewisser and Christiansen [67]. This plot includes the newly observed CJOS(see Section 5) which, according to its position on the scale is a well-behaved linear molecule. Note, however, that the closely related C 3 0 2molecule is definitely quasi-linear, as is the interesting HCNO molecule discussed in Section 1. For evaluation of the bent triatomic molecule, Yamada and Winnewisser [73] chose the fundamental bending frequency V k n d for AE, and the rotational constant A for AE,. and set A y =-(5.46) v bend Conventionally, the spectral constant A is a measure of the energy of rotation about the axis of least moment of inertia, or the a axis. In slightly, or moderately, bent molecules the a axis would correspond to the figure axis in the limiting case of a completely linear molecule. Thus, in tests for quasi-linearity, the constant A should be usable for AE,. in all cases, as indicated by Yamada and Winnewisser. For a linear triatomic molecule the level E(0 1' 0) represents the energy of the lowest excited, degenerate bending state for which the energy is rotational in form, and E(O2OO) represents the energy of the lowest excited bending state for which the energy is vibrational in form. Both levels are measured relative to that of the ground state, E(OOoO). Therefore, Yamada and Winnewisser set

Table 5.12 Limiting yo Values for Well-Behaved Linear Molecules" Molecule

Linear Bent

Ideal Molecule yo=-l yo=+l

"From Yamada and Winnewisser [73].

Well-Behaved Molecule

- 1.02 to -0.98

+ 1.00 to +0.96

152

LINEAR POLYATOMIC MOLECULES MOLECULES

+ 1 .o

CF,, SO,,

YO

BENT

S,O, ONCl

HNCS

+k - c3 OZ

-0.5

HCNO

Fig. 5.6 Ordering of some chain form molecules in the sequence of quasi linearity according to 1 and - 1. From Winnewisser and Christiansen [ 6 7 ] .

yo = 1 -4y, where y is defined by (5.48). The quadratic scale is chosen to expand around

+

for this case y=-----E(01'0) E(02'0)

(5.47)

or, more generally,

Y=

E(1owest state with K, or I, = 1) E(1owest excited state with K, or I, =O

(5.48)

where E represents the energy of the indicated state above that of the ground state. The molecules HNCO and HNCS are also classified as quasi-linear although they fall nearer to the limits of the bent rather than the linear molecules on the y o scale. The quasi-linear carbon suboxide molecule 0 =C =C =C =O has no observed rotational spectrum, but transitions of its vibration-rotation spectrum have been observed in the 545 to 595 GHz (0.55 to 0.50 mm) submillimeter wave region by Burenin et al. [75]. The part of the spectrum observed is the Q-branch, with some R- and P-branch lines of the u7 bending vibrational transition, (oooOoOoOo)+(oooOoOol'). Except for the inversion-type transitions

HYDROGEN-BONDED DIMERS

153

of NH3, this appears to be the first observation of a vibration-rotation spectrum with microwave spectroscopy. Other observations of this type will no doubt follow as further expansion of the submillimeter wave coverage with coherent microwave techniques occurs.

8


Since the early thirties hydrogen-bonded systems have been studied with infrared vibration spectroscopy. Most of these studies were made on liquids or solids, but the vibration bands of a few dimers in the gaseous state were observed. Notable among them are dimers of hydrogen fluoride, hydrogen cyanide, and acetic acid. Structural information has also been obtained about hydrogenbonded dimers from studies with gas-phase electron diffraction. As is well known, hydrogen bonding has great significance in biological and chemical processes, and the quantitative information about this bonding now being gained from microwave spectroscopy is likely to be of much importance. The first radiofrequency and microwave investigation of gas-phase, hydrogenbonded complexes appears to be that on the dimers of hydrogen fluoride, (HF),, (DF),, and HFDF, by Dyke et al. [76] with the molecular-beam, electric resonance method. They concluded that these dimers are not linear, that the H F units are 60 to 70" from the FF axis. Although this interesting system is not a suitable subject for elaboration in this chapter, it is relevant to the linear dimers studied with microwave spectroscopy. The dipole moment was found to be 2.987(3) D for (HF), and the FF distance, to be 2.79(5)w. Perhaps the most interesting discovery in this study is a hydrogen-tunneling motion which involves the breaking and reforming of the hydrogen bond in (HF), and (DF),, but not in HFDF. This tunneling was revealed by a splitting of the rotational energy levels of the symmetrical dimers. This MBER study was followed by conventional microwave spectral investigation of the nonlinear dimers (H202),, by Dyke and Muenter [77] in 1974, and H 2 0 . . . H F by Bevan et al. [78] in 1975. The first microwave spectral observation of a linear, hydrogen-bonded dimer was made by Legon et al. [79] on HCN . . . H F in 1976. They measured three ground-state rotational transitions, J = 1+2, 3+4, 4+5, and established the linear structure of the dimer. Later, they carried out more thorough investigations [SO] which included dimers of four different isotopic combinations and measurements on several excited vibrational states. The additional vibrational modes resulting from the dimer formation were labeled v u , uDp,and us, as indicated in Fig. 5.7. Excited states were observed for the parallel vibrational mode u,, and the lower frequency degenerate bending mode u p . For each of the four isotopic species they obtained the rotational constants Bo and D,,the rotation-vibration constants E, and q, and the 1-type doubling constants qa. They also derived the fundamental vibrational frequencies, vu = 197 cm- ' and vp=91 cm-', for the most abundant species. The vu represents the vibrational frequency of one molecular unit against the other and corresponds to

154


Fig. 5.7 Estimated equilibrium structure of C0,-HF. The broken lines represent the average H F position. From Baiocchi, Dixon, Joyner, and Klemperer [89].

the hydrogen-bond stretching vibration. The up mode, corresponding to hydrogen-bond bending, is doubly degenerate. They also calculated the force constants for the normal vibrations. Values for the other force constants may be found in the original paper [80]. The rotational constants and lengths of the hydrogen bridge, ro(N * * F ) found for the H and D species are given in Table 5.13. It is surprising that no significant difference in length is found for the hydrogen and the deuterium bridges. From Stark-effect measurements [81] on two rotational transitions of the HCN . . . HF, an electric dipole moment of 5.612 D was obtained for the dimer. It is of interest that this moment is greater by 0.80 D than the sum of the separate HCN and H F moments, thus indicating that the dipole moment is significantly enhanced by the formation of the hydrogen bond. A comparison of this dipole enhancement with that in other hydrogen-bonded complexes is given in Table 5.14. From intensity measurements of the rotational lines, Legon et al. [82] derived the dissociation energy of the hydrogen bond for the ground vibrational state and for the equilibrium configuration of the HCN * H F dimer. The final values obtained [SO] are g0=18.9 and 9,=26.1 in kJ/mole. The method described is applicable for measurement of dissociation energy for other hydrogenbonded complexes. It gives a measure of the hydrogen-bond strength for the particular complex. The hydrogen cyanide dimer HCN . * HCN has been observed with microwave spectroscopy by Legon et al. [83]. Four rotational transitions, J=7+8 to J = lo-, 11, were measured for (HCN)2 and for (DCN)2.The regularity of the spectrum and the changes in spacing caused by the isotopic substitutions verify the structure as linem. However, rather large differences appear between the rotational constants for (HC'4N)2 and (HC'5N)2 and the corresponding ones measured later by Buxton et al. [84], who concluded that the ground-state constants reported earlier for these species were likely those for an excited vibrational state. The constants reproduced for (HCN)2 in Table 5.13 are the later ones, whereas those for (DCN)2 are from the earlier work. Note that the row.. . C) distance for (DCN)2 is 0.055 8, less than that for

Table 5.23 Rotational Constants and Structural Parameters of Some Linear, Hydrogen-bonded Complexes Complex

Bo ( M H z )

DJ (kHz)

yo(

Y

. . )‘A (A)

O(deg.)”

Ref:

ro(TV.. . Hal) HCN . . . HF DCN . . . DF NCCN . . . H F HCN . . . H79Br NN , . . H F NN . . . H35C1

3591.11 3351.87 1195.0616 1396.5958 3195.353 1745.664

5.2 5.5 0.34 1.89 17.2 8.56

2.796 2.192 2.862 3.610 3.082 3.701

18.7 20.1 21.36 25.5 26.0

ro(C. . . Hal) OC . . . H F OC . . . H35C1 OC . . H79Br

3063.899 1671.723 1150.512

9.77 4.802 2.703

3.068 3.696 3.9 17

21.9 23.0 23.2

ro(N. . . C) HCN . . . HCN DCN . . . DCN

1745.8097 1698.37

2.133

k

3.287 3.232

I

ro(O . . . Hal) OCO . . H F OCO . . . H35C1 SCO . . . H F

1951.170 1109.886 1302.462

10.69 4.83 2.46

2.84 3.42 2.87

25.1 25.0 25.0

m

n rn

“0 is the vibrationally averaged angle of HXwith the inertial axis a of the complex, approximately with the Y . X bridge. Some workers designate this angle as y and use 8 to designate the angle between the averaged H Xdirection and the line between the mass centers of the two components of the complex. In complexes with linear equilibrium structures like those in this table, the two angles are equal, or nearly so. bLegon et al. [go]. ‘Legon et al. [91]. dE. J. Campbell, A. C. Legon, and W. H. Flygare, J. Chem. Phys., 78, 3494 (1983). ‘P. D. Soper, A. C. Legon, W. G. Read, and W. H. Flygare, J. Chem. Phys., 76,292 (1982). fR. S. Altman, M. D. Marshall, and W. Klemperer, J. Chem. Phys., 79, 57 (1983). BLegonet al. [85]. hLegon et al. [88]. ‘Soper et al. [87]. ’Keenan et al. [86]. kBuxton et al. [84]. ‘Legon et al. [83]. “‘Baiocchi et al. [89]. “R. S. Altman, M. D. Marshall, and W. Klemperer, J. Chem. Phys., 77,4344 (1982).

155

156


Table 5.14 Dipole Moment Enhancement due to Hydrogen Bonding

HCN . . . H F OCO . . . HF

4.812 1.827 1.827 2.542 1.109 1.109

5.612b 2.2465' 2.3024' OCO . . . DF SCO . . . HF 3.2085' NN . . H35CI 1.244d OCO.. . H35C1 1.451'

0.80Ob 0.420 0.475 0.667 0.14 0.34

"Sum of moments of separated monomers. bLegon et al. [SO]. 'Baiocchi et al. [89]. dR. S. Altman, M. D. Marshall, and W. Klemperer, J . Chem. Phys., 79, 57 (19831. 'R. S . Altman, M. D. Marshall, and W. Klemperer, J . Chem. Phys., 77,4344 (1982).

(HCN)2. This is consistent with the difference recorded for O C . . . DF and O C * * . HF. If real, these differences indicate a slightly strqnger bonding by D than by H, resulting from the lower zero-point vibrational energy of D. Weakly bonded dimers comprised of carbon monoxide hydrogen-bonded to the hydrogen halides, HX(X=F, C1, or Br), have been discovered with microwave spectroscopy by Legon et ai. [85-881. Detectable dimers were produced by pulsed injection of mixtures of C O (4%), the halide (4%), and argon into a Fabry-Perot absorption cavity of a highly sensitive spectrometer. From its rotational spectrum, each dimer was shown to have a linear, or nearly linear, structure, with the hydrogen bonding to the carbon, OC * H X. For the OC . . . HCI dimer, this structure was verified by measurements on dimers of four different isotopic combinations. Additional evidence for collinearity of the nuclei at equilibrium was obtained from the observed nuclear quadrupole couplings of the C1 and Br. The ground-state rotational constants for the OC . . . H X dimers and the hydrogen bond lengths r(C . . . H) obtained from them with known monomer structures, assumed unchanged, are listed in Table 5.13. By subtraction of the H X length, one can obtain the r(C . . H) distance. With re values for the hydrogen halides, this gives r(C . . . H)=2.132 8,, 2.430 8,, and 2.503 8, for X=F, C1, and Br, respectively. This suggests a decreasing strength of the bonding from F to Br. It is of interest that r(C . . . D)=2.410 for the O C . . . DCI complex, 0.01 A less than that for the O C . . . HCI complex. The later, more complete, study of OC . HBr includes four isotopic combinations [SS]. Although the vibrationally averaged structure is that of a prolate symmetric top with the averaged Br-H bond axis 23" off the linear 0-C . * . Br axis, the data, like those for the other OC * HXcomplexes in Table 5.13, are consistent with a linear equilibrium structure. 1

9


157

Hydrogen-bonded complexes formed by C 0 2 and by OCS with the hydrogen fluoride have been studied by Klemperer's group [89] with microwave and radiofrequency spectroscopy. Their rotational spectra show that these molecules have linear equilibrium structure with the chemical form OCO . . . HF. The geometrical structure is like that indicated for OCO . . . H F in Fig. 5.8. Although the equilibrium structure is linear, the zero-point vibrational energy causes the vibrationally averaged H F angle to deviate from the molecular axis by approximately 25", as indicated by the dotted lines in Fig. 5.8. This angular deviation, as for the other H-bonded structures described in this section, is revealed and measured by the nuclear hyperfine structure. This rotational spectrum is like that of a linear molecule or of a symmetric-top rotor with K=O. Rotational constants Bo and Do obtained from the measurements are given in Table 5.13. The 0 . . . H or 0 . . . F distances are derived from Bo with the assumption that the structures of the component molecules are the same as those for the separated monomers. The 0 . . . H distance for SCO . . . H F is 1.94 A, somewhat larger than the 1.91 A for O C O . . . HF. Distances of 0 . .. F are given in Table 5.13. I t is surprising that the N N O . . . H F complex, which is isoelectric with the linear OCO . . . HF, was found not to have a linear structure [90]. Instead, the N N O axis makes an angle of about 47O.with the center-of-mass line. The complexes Ar . . . C 0 2 and Ar . . . N 2 0 have almost identical T-shaped structures [90]. The rotational spectrum of the linear hydrogen-bonded dimer NCCC . . H F has been observed by Legon et al. [91]. Their measurements included three isotopic species with 15N substituted in different positions. Rotational constants obtained for the most abundant isotopic species are reproduced in Table 5.13 with the length of the N . . . F hydrogen bridge derived from Bo. No significant difference from this length was observed for the "N-substituted species. The averaged angle, 6 in Table 5.13, derived from the nuclear magnetic complex of the HF, is 5" less than that for OCO . . . HF. 9

vu

's

VB

~os ~1)

(5.49)

where xz is the coupling value along the H X bond axis and 6z,ais the instantaneous value of the angle between the H X bond axis z and the principal inertial axis a. The average is taken over all the vibrational motions. The ratio of the observed coupling for the dimer to that observed for the free H X monomer, Xa/xz, gives, with (5.49), the averaged value (cos’ e,,) and hence the vibrationally averaged value for 6. The magnetic coupling constant C, of H or of X measured in the free HXmonomer has the same formal relationship as that of (5.49), and the ratio of Ca/Cz can in a similar manner be used to determine the value for the vibrationally averaged 6. Accurate values of the magnetic coupling constants of the H F monomer are given by Muenter [92]. Quadrupole coupling constants for the hydrogen halides are given in Chapter XIV. In Table 5.15 we have recorded for several dimers fundamental hydrogenbond stretching frequencies v, and related force constants k , corresponding to the parallel mode v, indicated at the top of Fig. 5.7. The stretching force constants k , or k,, depend directly on the strength of the hydrogen bonds and provide a comparison of the relative energies of these bonds for the molecules in Table 5.15. The frequencies were not directly measured because they fall outside the microwave range, in the far-infrared region. Instead, they were calculated from the centrifugal stretching constants listed in Table 5.13 by simple approximate methods.


159

Table 5.15 Vibrational Frequencies and Hydrogen-bond Force Constants Frequency ( c m - ' ) Complex HCN . . . H F HCN . , , HCN OC . . . H F OC . . DF OC . . H35C1 OC . . D35C1 OC . . . H7'Br OCO . . H F OCO . . . DF SCO . . H F

Force Constant (mdynelW) kdko)

VS(V0)

0.24 0.11 0.108 0.118 0.0446 0.049 1 0.0330 0.021 0.020 0.028

197 120 125 127 69 72 52 51 49 56

RC$ (1

b 5

c

d d e

f f f

"Legon et al. [80]. bBuxton et ai. [84]. 'Legon et al. [88]. dSoper et al. [87]. eKeenan et al. [86]. 'Baiocchi et al. [89].

The v,, mode corresponds closely to the vibration of one molecular unit of the dimer against the other. For calculation of v, and k, one can neglect the distortions of the molecular units and treat the dimer as a diatomic molecule with the masses of the two weakly bonded molecular units concentrated at their respective mass centers. The fundamental frequency v,( =v,) is then calculated from (4.16) with the measured Bo and DJ values in Table 5.13 used for Be and D e . Thus, 4B: (5.50) DJ The force constants k, are cal'culated from the harmonic oscillator formula --

s-

,,=LJ" 271

pdimer

(5.51)

where the reduced mass of the dimer is (5.52) in which M 1 and M 2 are the masses of the two molecules linked by the hydrogen bond. Values for the fundamental bending frequencies v b and force constants kb corresponding to the V B mode of Fig. 5.7 have been estimated [86,87,89] from

160


the vibrationally averaged 6 values listed in Table 5.13. The theory used is that described in Section 9, (5.53)-(5.59). However, the simplifying assumptions required are more uncertain than those for the stretching vibration v,. Later, more complete measurements on the rotational structure of excited vibrational states, or direct measurements of the bending frequencies in the infrared region, should provide accurate values for the bending constants.

9

RARE GAS ATOM-MOLECULE COMPLEXES

Rare gas atoms-neon, argon, krypton, and xenon-form weakly bonded van der Waals complexes with a variety of molecules. Microwave and radiofrequency spectra of a number of them have been measured with molecular beam, electric resonance techniques by Klemperer and his group at Harvard [93-1001 and with special microwave spectral techniques by Flygare and his group at Illinois [101-1061. Many of the complexes in which the molecular component is diatomic are found to have linear equilibrium structures although the wide-angle bending vibrational motions they undergo, even in the ground state, produce an averaged structure that conforms to that of a slender prolate symmetric top. Similar features are found for the hydrogen-bonded dimers described in Section 8. In fact some of these complexes may be classified as weak, hydrogen-bonded complexes. These atom-molecule complexes are usually produced by the expansion of a highly pressurized mixture of the constituents when the mixture is passed through a restricting supersonic nozzle into the evacuated cavity of the spectrometer. The molecular constituent is the more dilute, only a few percent of the mixture. The Illinois group use a pulsed microwave Fourier-transform spectrometer and pulse the supersonic nozzle to admit the gas at a frequency synchronized with the pulse of the spectrometer. Descriptions of their highly sensitive system are given in some of their papers [104, 1071. The rapid expansion of the pressurized gas as it escapes into the evacuated cavity cools the molecular complexes to a very low temperature so that thermal motions do not dissociate the very weak van der Waals bonds. Furthermore, at this low temperature, only the lower rotational states are populated. Although the number of molecular complexes in the escaping beam is very small compared to the molecules in a conventional microwave absorption cell, they are detectable with the exceptionally sensitive, pulsed Fourier-transform microwave spectrometer just mentioned [104, 1071 and with the comparably sensitive molecular-beam, electric-resonance method. Favoring the detection is the high population of the lowest rotational state. The first observations of the microwave spectrum of a linear complex of this kind appears to be those on ArHCl by Novick et al. [93,97], who obtained the rotational constants, nuclear couplings, and electric dipole moments for the ground vibrational states of ArH35C1 and ArD3'C1. They found that the complex has the atomic arrangement Ar-HCl with the averaged Ar . . * CI distance of 4.006 A. The large, vibrationally averaged bending angle 6 of 45" for ArHCl and 34" for ArDCl shows that the complex is highly flexible.

RARE GAS ATOM-MOLECULE COMPLEXES

161

The vibrationally averaged angle 8, sometimes designated y, is obtained from the experimentally measured nuclear quadrupole coupling or the nuclear magnetic couplings, as described in Section 8. See (5.49). Rotational constants and derived structural constants ro and Q are recorded in Table 5.16 for a number of rare gas atom complexes with diatomic halides. These complexes include argon, krypton, and xenon, but not neon. Evidences have been obtained [99,108] that similar complexes are formed with Ne but that they are nonpolar, probably because of a very large vibrationally averaged angle Q which leads to restricted internal rotation of the molecular components [99]. The references cited in Table 5.16 are those which give the latest, or most complete, information about these van der Waals complexes. Further information and references to earlier or preliminary work on them may be found in the cited papers. Bond-stretching and bond-bending frequencies and force constants which provide a comparison of the relative strength of the van der Waals bonds in some Table 5.16

Rotational and Structural Constants for Some Rare Gas Atom-Molecule

Complexes Bo ( M H z )

DJ (kHz1

r o ( X . . . Hal) (A)

e(&)

3065.7099(4) 3039.8529(4) 1678.511 1657.627 1106.6695(2) 1087.5089(3)

70.90(6) 59.53(6) 20.0 17.1 12.397(4) 8.497(6)

3.5445(2) 3.5349(2) 4.006 4.025 4.146 4.182

41 32.7(2) 45 34 42.1 34.4

2392.4152(7) 2353.1407(7) 1200.6255(7) 1183.7735(5) 684.2295(2) 675.8486(2)

31.94(10) 27.31(10) 7.389(16) 6.530(12) 2.597(1) 2.190(1)

3.647 3.640 4.111 4.125 4.257 4.281

39 30.8 37.8 30.5 38.0 31.1

994.1448(1) 979.4096(1)

3.813(2) 3.452(1)

4.275 4.287

34.5 27.9

4.66(20) 1.86(20)

3.330 (Ar . . . Cl) 3.388 (Ar . . . C1)

1327.113(5) 925.186(5) "Keenan et al. [106]. *Dixon et al. [lOO]. 'Novick et al. [97]. dNovick et al. [93]. 'Keenan et al. [103]. fBuxton et al. [105]. gBalle et al. [102]. hKeenan et al. [104]. 'Harris et al. [95]. JNovick et al. [96].

8.69 8.1 1

Ref: a,b

n3b c,d c,d e

e

I f 9 9

e e h h L

j

162


of the complexes are given in Table 5.17. The bond-stretching frequencies are calculated from the bond-stretching constants DJ in Table 5.16 with (5.50). The bond-bending frequencies are calculated from the 8 values in Table 5.16 with the assumption of a harmonic bending potential V ( 0 )=+kbe2

(5.53)

which leads to the bending frequency [I031 h vb=2*p*(02)

(5.54)

For the triatomic complexes in Table 5.16, the reduced mass of the bending vibration may be obtained from the general formula for the binding mode v 2 of a linear XYZ molecule for which [I091 (5.55)

Table 5.17 Derived Vibrational Frequencies and Force Constants for Rare Gas AtomMolecule Complexes v, ( c m - ’ )

k , (mdyne/W)

Ar . . . H F Ar . . . D F Ar . . . H3’CI Ar . . . D35C1 Ar . . . H”Br Ar . . . D7’Br

42.5 45.6 32.4 34.4 22.1 26.0

0.0142 0.0168 0.0117 0.0134 0.0076 0.0106

79.8 65.5

30.8 23.4

0.0031 0.0039 0.0015 0.0018 0.00114 0.00129

84Kr. . . H F 84Kr.. . DF 84Kr.. . H35C1 84Kr.. . D35C1 84Kr.. . H79Br 84Kr.. . D’79Br

43.7 43.1 32.3 33.6 23.4 25.0

0.0187 0.0210 0.0155 0.0171 0.0132 0.0152

87.5 74.7 52.9 39.8 37.7 28.7

0.00378 0.00533 0.00128 0.00142 0.00170 0.00195

d

1 2 9 ~ e. .. H35Cl 129Xe. . . D35C1

33.9 34.8

0.0190 0.0205

56.8 44.3

0.00311 0.00370

1

Ar . . . 35C1F 84Kr. . . 35CIF

47.2 43.6

0.0301 0.0369

41.0 48.4

0.0215 0.0311

Complex

“Dixon et al. [IOO]. bNovick et al. [97]. ‘Keenan et al. [103]. dBuxton et al. [IOS]. ‘Balk et al. [102]. fKeenan et al. [104]. gNovick et al. [96].

vb(cm-1)

kb

(mdyne/A)

R~J: a

a

b,o b,a e c

d

e e c c

f 9

4

LINEAR MOLECULAR IONS: INTERSTELLAR AND LABORATORY

163

in which mx, my, and mZ are the atomic masses of ,'A I:and 2,and the 1's are the indicated atomic distances. For the hydride or deuteride complexes, m y =mH or mD, mZ=mHal,and m x is the mass of the rare gas atom. In these, my < m x E m Z , and approximately, 1

1

2

(5.56)

The I H may ~ be taken as the ro(H-Hal) of the free hydrogen halide and I x ~ = r o ( X .. . Hal)-ro(H-Hal)

(5.57)

where the r o ( X . * . Hal) values are those recorded in Table 5.16. Because ro(H-Hal) < 1XH, this expression is sometimes reduced further pb =mHrHZ 2

(5.58)

which is simply the moment of inertia of the free diatomic molecule. With the calculated pb, the bending frequency can be obtained from (5.54), and the bending force constant, from the harmonic oscillator formula, (5.5I), kb =4?T2Vzpb

(5.59)

In Table 5.1 7 we have recorded vibrational frequencies and force constants derived from experimental DJ and 0 values. The constants provide a comparison of the relative bonding forces in these van der Waals complexes. Note that the bond-stretching constants k, are generally smaller than those for the hydrogenbonded dimers listed in Table 5.15.

10 LINEAR MOLECULAR IONS: INTERSTELLAR AND LABORATORY Observation of molecular ions is relatively new in microwave spectroscopy. The first laboratory observation of a microwave rotational transition of an ion was that of CO', reported in 1975 by Dixon and Woods [110]. Before that time, only fine-structure transitions in the hydrogen molecule ion had been observed [lll], in the low-frequency range of 4 to 1276 MHz. Microwave astronomers have led in the detection of microwave lines of linear polyatomic ions. Rotational constants of all three linear triatomic ions observed to date are listed in Table 5.18. Lines of each of these ions were first detected from interstellar space and were later identified by laboratory measurements. The most extensive studies have been made on HCO', both in the laboratory and in outer space. It was the first molecular ion to be detected with microwave spectroscopy in interstellar space. In 1970, Buhl and Snyder [112] observed from several interstellar sources an unknown line at 89,190 MHz. Because they were unable to assign it to any known species, they tentatively labeled the unknown species X-ogen. Soon after this discovery, Klemperer [113] on theoretical grounds suggested HCO' as the source of the unknown line and predicted that its J=O-+1 transition would occur at 89,246 MHz. Later, Herbst

164

LINEAR PO LYATO M IC M0 LECU LES

Table 5.18 Rotational Constants of Some Linear Triatomic Ions

(H"C' 60)+ 44,594.420(2) ( H I ~ c ~ ~ o ) + 43,377.32(4) (H12C'80)+ 42,581.21(4) 36,O 19.776(3) (D 1 2 C 1 6 0 ) + ( ~ 1 3 ~ 1 6 0 ) t 35,366.71(1) 34,413.798(1) (D' zC'80)t (H16 0 1 2 C ) + 44,743.924(14)

82.39(7) 786) 65(8) 55.875(5) 53.2(5) 50.29(4) 114.9(5)

(H I4N14N)+ (D I 4 N l 4 N ) +

46,586.863(15) 38,554.71 7( 14)

(H12 C 3 z S ) +

v(J= 1+2)=85,347.90(3) MHz

87.5(5) 60.8(4)

o,b b

b a,b

b b c d d e

"K. V. L. N. Sastry, E. Herbst, and F. C. De Lucia, J . Chem. Phys., 75,4169 (1981). *Bogey et al. [122]. 'Blake et al. [131]. dSastry et al. [126]. 'Gudeman et al. [128].

and Klemperer [114] proposed a theory for formation of molecules in interstellar clouds in which HCO' has a significant function as a reactive intermediary. It is believed that the failure to observe CO' in interstellar molecular clouds where CO is relatively abundant is due to conversion of these ions to HCO' by reaction with H atoms [114,1151. Support for HCO' as the unknown X-ogen came from observation by Snyder et al. [116] of an interstellar line at 86,754 MHz which could be assigned to H13CO+ and by more detailed theoretical predictions [117, 1181 of the structure of HCO'. Experimental proof of the identification came with the laboratory detection of HCO' by Woods et al. [119], who measured the rest frequency of the J=O+l transition at 89,188.545(20) MHz, quite close to some of the more recent astronomical measurements [120, 1211. The laboratory measurements of the important HCO' ion have been extended to include several isotopic species and several other rotational transitions. The accurate ground-state rotational constants listed in Table 5.18 should be useful to microwave astronomers in detection of other lines and other isotopic species. The references cited in Table 5.18 give the rest frequencies of the measured transitions. Bogey et al. [122] have used their observed constants of several isotopic species of HCO' to calculate the interatomic distances of the ion, as listed in Table 5.19. The equilibrium structure predicted by Wahlgren et al. [I171 and by Kraemer and Dierksen [1181 were found to agree closely with the experimental ro and rs values. Generally, rs should be somewhat closer than ro to re (see Chapter XIII, Section 8).

LINEAR MOLECULAR IONS: INTERSTELLAR AND LABORATORY

165

Table 5.19 Observed and Theoretical Structures of the Formyl Ion, HCO'

Experimental" Theoreticalb

HCO' r(H-C) 8, r(C-0) 8,

ro

rs

1.0913(4) 1.1093(1)

1.1070(2)

1.0930(1)

re

1.095' 1.1045'

1.091' 1.103'

"Bogey et al. [122]. *Ab initio C.I. calculation of equilibrium structure. 'Wahlgren et al. [117]. dKraemer and Diercksen [118].

The beautiful hyperfine pattern shown in Fig. 1.4 is that for the J=O-+l transition of interstellar HNN'. Nuclear quadrupole coupling by the terminal 14N gives the gross triplet with total spread of about 4 MHz. The substructure partially resolved in the center line is due to quadrupole coupling by the central 14N.This interstellar transition was first observed by Turner [I231 in 1974, but its identification, like that for HCO ', remained uncertain until later calculations and observations could be made. The proposal that the observed triplet comes from HNN' was quickly made by Green et al. [124]. The close agreement of the theoretical and observed hyperfine patterns in Fig. 1.4 confirms their assignment. The first laboratory observation of this ion was made in 1976 by Saykally et al. [125], who resolved the gross triplet of the J=O+1 transition but could not detect splitting by the central 14N because of the weakness and broadness of the lines. They mention that the HNN' spectrum proved to be considerably more difficult to study than that of HCO'. In 1981, Sastry et al. [126] measured several higher millimeter wave transitions for both HN: and DN: and from them derived the accurate values for ground-state rotational constants listed in Table 5.18. In 1981 Thaddeus et al. [127] observed from several astronomical sources a line at 85,348 MHz which they assigned to the J = 1 -+2 transition of the linear ion HCS'. Their assignment was confirmed shortly afterward by Gudeman et al. [I281 with a laboratory detection of a line at 85,347.90 MHz in a glow discharge under conditions expected to produce HCS'. A laboratory observation of the J =O+ 1 transition of HOC' has been made by Gudeman and Woods [I291 and used by Woods et al. [130] for a tentative identification of interstellar HOC' in the constellation Sgr B2. In the Duke laboratory, Blake et al. [131] measured the higher transitions, J=1+2, 2+3, and 3+4, and derived the accurate rotational constants for this ion listed in Table 5.18. Note the significantly higher stretching constant Doof HOC' over that of its isomer HCO'. The principal cause for the long delay in the development of microwave spectroscopy of molecular ions is, of course, their instability relative to the neutral singlet C molecules commonly studied in the past. The "free-space''

166


absorption cell usable with semioptical techniques at the higher frequencies of the millimeter wave region made possible the observation of spectra of molecular ions directly as they are produced in a glow discharge enclosed in glass. Nevertheless, many spectroscopists were probably reluctant to attempt spectral detection and identification of short-lived ions in a discharge column composed of mixed constituents. One group evidently not reluctant is that led by R. C. Woods at Wisconsin University. They made the first successful laboratory observations on all the molecular ions discussed in this section. A strong impetus to the development of techniques for microwave spectroscopy of molecular ions within the laboratory has been the detection of molecular ions in interstellar space. One factor contributing to the success of the Woods group was their lowering of the temperature of the discharge column, usually to that of liquid air. In this way they both increased the stability of the ions and decreased the collision broadening. Line widths observed for these molecular ions in a glow discharge mixture at liquid air temperature proved to be surprisingly small, a fortunate circumstance aiding their detection. Anderson et al. [1321made careful measurement of the line-breadth parameter for the J=O+l transition of HCO' broadened by H 2 at a temperature near 100°K. Measurements were taken over a pressure range of 4 to 40 mTorr. At that temperature and pressure range, ~ found to be 29.3 MHz/Torr. This paramthe line-breadth parameter ( A V ) was eter is comparable in magnitude to the self-broadening parameters of many polar molecules at room temperature (see Table 3.1).

11 LINEAR FREE RADICALS: LABORATORY AND INTERSTELLAR Since the detection of spectral lines of molecular free radicals in interstellar space, microwave spectroscopists have become increasingly interested in the study of these unstable species in the laboratory. As already mentioned in Chapter I, the first molecular species observed in interstellar space was the O H radical. The microwave lines of this light radical as well as those of CH observed in interstellar space are A doublet transitions of their 211 ground states (see Chapter IV, Section 2). Rotational transitions of linear polyatomic free radicals are now observed from interstellar space and from molecules in the laboratory. Microwave magnetic resonance spectra of such free radicals are treated in Chapter XI, Section 9. Discovery of the interesting linear free radical CCH in the interstellar medium was announced by Tucker et al. [133] from detection of its lowest rotational transition, N=O+l. Several years later the assignment was confirmed by detection of the N=2+3 transition of the interstellar radicals by Ziurys et al. [134] and by laboratory measurement of the N=1+2, 2+3, and 3+4 transitions by Sastry et al. [135]. The CCH radical has the 2C ground state, and its spectrum is analyzed by the theory described in Chapter IV, Section 2 for diatomic molecules. The only hyperfine splitting is that due to the magnetic

LINEAR FREE RADICALS: LABORATORY A N D INTERSTELLAR

167

interaction of the H nucleus, and this is smaller than the electron spin-rotation interaction. Therefore the coupling scheme used in the analysis is

N+S=J

and

J+I=F

There is no nuclear quadrupole interaction, and the C7,N-I term is negligible. The observed spectra from outer space [134] and in the laboratory [135] were analyzed satisfactorily with the Hamiltonian

Af= [Bo - DoN(N+ I)] N2 + yo( N .S)+ b(I . S ) + cZ,S,

(5.60)

The energy level diagram of Fig. 5.9 is that for the N=2 and N = 3 states of CCH calculated with the X o f (5.60)by Ziurys et al. [134]. The allowed emission transitions are indicated by arrows, with those observed in bold face. The signals observed from the interstellar source, OMC-1, are shown in Fig. 5.10. These consist of the J = $ - t , &+$ fine structure doublet with a partially resolved proton hyperfine structure. Table 5.20 shows the spectral constants of the Hamiltonian of (5.60) as derived from the laboratory measurement compared with those derived from the interstellar spectra. The agreement between the two sets proves, without doubt, the existence of the CCH radical in outer space. Microwave lines observed from interstellar sources have been tentatively assigned to the radicals CCCCH and CCCN on the basis of predicted frequencies of theoretical models. From structural considerations both radicals are expected to be linear and to have 2C electronic ground states. N o laboratory F=2

3

4

3

I

2

3 2

CCH Fig. 5.9 Energy level diagram of the N = 2 and 3 levels of the 'X state of CCH including the fine and hyperfine splittings, which are drawn to scale. The N levels are separated by approximately 262 MHz. Allowed emission transitions are indicated by arrows. Arrows in bold face indicate observed interstellar transitions shown in Fig. 5.10. From Ziurys, Saykally, Plambeck, and Erickson [1341.

168


3 ,

I

-

I

I

J = 5/2

I

I

I

I

-

J = ?/2 *5/2

+3/2

2T;(K)

-

-

-

I -

-

-

0-

I

I

I

I

I

I

I

I

Fig. 5.10 Interstellar lines of the N=2+3 rotational transition of CCH, observed in OMC-1. The bars indicate the theoretically predicted lines indicated in Fig. 5.9 by arrows in bold face. From Ziurys, Saykally, Plambeck, and Erickson [1341.

Table 5.20 in MHz Constant

BO DO Yo b C

Spectroscopic Constants of CCH

Laboratory"

Interstellarb

43,674.514(5) 0.1052(3) - 62.66(2) 40.40(19) 12.33(11)

43,674.7(10) 0.1137(6) - 62.62(5) 40.39(14) 12.32(9)

"Sastry et al. [135]. *Ziurys et al. [134].

measurement of rotational lines for either radical has, to our knowledge, been reported. Four successive rotational transitions, N = 8+9 to 1l+12, observed from an astronomical source, have been assigned by Guelin et al. [136] to the linear C4H radical in the *Cstate. The measured frequencies fit closely those predicted from the theoretical model of Wilson and Green [137]. The observed Bo is 4758.48(10) MHz compared with the theoretically predicted value of 4753 MHz.

REFERENCES

169

The evidences for identification of C3N in interstellar space, although strong, are somewhat less than those for C4H. Only two observed rotational transitions, the N = 8+9 and 9+10, each with a doublet structure, are reported in the original paper by Guelin and Thaddeus [138], but GuClin et al. [136] mention later observations of a third, the N = l O t l l . The frequencies of the observed lines fit closely those predicted from the Hartree-Fock theoretical model calculated by Wilson and Green [137]. The observed Bo is 4947.66(10) MHz compared with the theoretical value of 4955 MHz, and the observed magnitude of the spin-rotation constant, lyol = 18.7(5)MHz, is within the range of that expected for the linear CCCN radical. So far this molecule has not been detected on earth by any method. Rotational spectra of linear polyatomic molecules having electronic orbital momentum, II,A, or higher states are, except for bending vibrational effects, similar to those for diatomic molecules in the same electronic states as described in Chapter IV, Section 2, Theory for the ll and A states is reviewed in Chapter IV, Section 2 and Chapter IX, Section 9. Formulation of the theory and the nature of the solution for the energy eigenvalues for either the diatomic or the linear polyatomic molecule having. both electronic spin and orbital angular momentum depend sensitively on the relative magnitudes of the couplings of the various spin and angular momentum vectors, including the nuclear spins. N o generally applicable formulation can be given. Hougen [1393 has given a theoretical treatment for linear triatomic molecules in 211 states which takes into account the effects of bending vibrations. An instructive application of Hougen’s theory is made by Saito and Amano [140], who measured and calculated rotational transitions for both the 211112 and 2 1 1 3 / 2 states of the linear radical NCO.

References 1. 2. 3. 4. 5.

A. H. Nielsen, J. Chem. Phys., 11, 160 (1943). R. S. Mulliken, J . Phys. Chem., 41, 159 (1937). F. C. De Lucia and P. A. Helminger, J. Chem. Phys., 67,4262 (1977). H. H. Nielsen and W. H. Shaffer, J. Chem. Phys., 11, 140 (1943). H. H. Nielsen, Rev. Mod. Phys., 23,90 (1951). 6. P. Kuijpers, T. Torring, and A. Dymanus, Z. Naturforsch., 32a, 930 (1977). 7. M. Bogey and A. Bauer, J . Mol. Spectrosc., 84, 170 (1980).

8. G. Amat, H. H. Nielsen, and G. Tarrago, Rotation Vibration of Polyatomic Moleculrs, Dekker, New York, 1971. 9. H. W. Kroto, J. F. Nixon, and N. P. C. Simmons, J. Mol. Spectrosc., 82, 185 (1980). 10. R. G. Shulman and C. H. Townes, Phys. Rev., 77, 421 (1950). 11. J. F. Westerkamp, Phys. Rev., 93, 716 (1954). 12. T. Torring, Z . Phj,sik, 161, 179 (1961). 13. H. H. Nielsen and G. Amat, J. Phys. Radium, 15, 601 (1954).

14. M. Winnewisser and H. K. Bodenseh, Z. Naturforsch., 22a, 1724 (1967). 15. J. Ramadier and G. Amat, J. Phys. Radium, 19, 915 (1958). 16. M. Winnewisser and B. P. Winnewisser, J. Mol. Spectrosc., 41, 143 (1972).

170


17. B. P. Winnewisser, M. Winnewisser, and F. Winther, J. Mol. Spectrose., 51, 65 (1974). 18. K. Yamada, B. P. Winnewisser, and M. Winnewisser, J. Mol. Spectrosc., 56, 449 (1975). 19. B. P. Winnewisser and M. Winnewisser, J. Mol. Spectrosc., 56, 471 (1975). 20. P. R. Bunker, B. M. Landsberg, and B. P. Winnewisser, J . Mol. Spectrosc., 74, 9 (1979). 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

E. Fermi, Z . Physik, 71,250 (1931). F. C. De Lucia and W. Gordy, Phys..Rev., 187, 58 (1969). D. Marcuse, Proc. IRE, 49, 1706 (1961). A. K. Garrison and W. Gordy, Phys. Rev., 108,899 (1957). J. R. Rusk and W. Gordy, Phys. Rev., 127, 817 (1962). C. Huiszoon and A. Dymanus, Phys. Lett., 21, 164 (1966). F. A. Van Dijk and A. Dymanus, Chem. Phys. Lett., 2,235 (1968). F. A. Van Dijk and A. Dymanus, Chem. Phys. Lett., 5,387 (1970). W. E. Lamb, Phys. Reu., 134, A1429 (1964). C. C. Costain, Can. J. Phys., 47, 2431 (1969). R. S . Winton and W. Gordy, Phys. Lett., 32A, 219 (1970). G. Cazzoli, C. D. Esposti, and P. G. Favero, J. Phys. Chem., 84, 1756 (1980). T. Oka, Adv. Atom. Mol. Phys., 9, 127 (1973). R. M. Lees, J. Mol. Spectrosc., 69, 225 (1978). J. G. Baker, “Microwave-Microwave Double Resonance,” in Modern Aspects of Microwace Spectroscopy, G. W. Chantry, Ed., Academic, London, 1979, pp. 65-122.

36. R. L. Kuczkowski, D. R. Lide, and L. C. Krisher, J. Chem. Phys., 44,3131 (1966). 37. D. R. Lide and R. L. Kuczkowski, J. Chem. Phys., 46,4768 (1967). 38. C. Matsumura and D. R. Lide, J. Chem. Phys., 50,71 (1969). 39. E. F. Pearson and M. B. Trueblood, Astrophys. J., 179, L145 (1973). 40. E. F. Pearson and M. B. Trueblood, J. Chem. Phys., 58, 826 (1973). 41. D. R. Lide and C. Matsumura, J. Chem. Phys., 50,3080 (1969). 42. P. Kuijpers, T. Torring, and A. Dymanus, Chem. Phys., 15,457 (1976). 43. E. F. Pearson, B. P. Winnewisser, and M. B. Trueblood, 2.Naturforsch., 31a, 1259 (1976). 44. S . M. Freund, P. D. Godfrey, and W. Klemperer, 25th Symposium on Molecular Structure, The Ohio State University, 1970. Paper E8. 45. E. F. Pearson and R.V. McCormick, J. Chem. Phys., 58, 1619 (1973). 46. C. Kirby and H. W. Kroto, J. Mol. Spectrosc., 83, 130 (1980). 47. A. G. Smith, W. Gordy, J. W. Simmons, and W. V. Smith, Phys. Ret.., 75,260 (1949). 48. E. F. Pearson, R. A. Creswell, M. Winnewisser, and G. Winnewisser, Z. Naturforsch., 31a, 1394 (1976). 49. F. C. De Lucia, J. Mol. Spectrosc., 55, 271 (1975). 50. L. E. Snyder and D. Buhl, Asrrophys. J., 163, L47 (1971). 51. H. A. Gebbie, N. W. B. Stone, and F. D. Findly, Nature, 202,685 (1966). 52. D. R. Lide and A. G . Maki, Appl. Phys. Lett., 11, 52 (1967). 53. R. A. Creswell, E. F. Pearson, M. Winnewisser, and G. Winnewisser, Z . Naturforsch., 31a, 221 (1976). 54. R. J. Saykally, P. G. Szanto, T. G . Anderson, and R. C. Woods, Asrrophys. J., 204,143 (1976). 55. G . L. Blackman, R. D. Brown, P. D. Godfrey, and H. I. Gunn, Nature, 261, 395 (1976). 56. L. E. Snyder and D. Buhl, Bull. Am. Astron. SOC.,3, 388 (1971). 57. B. Zukerman, M. Morris, P. Palmer, and B. E. Turner, Astrophys. J., 173, L125 (1972). 58. J. Barsuhn, Astrophys. Lett., 12, 169 (1972).

REFERENCES

171

59. D. E. Milligan and M. E. Jacox, J. Chem. Phys., 39, 712 (1963). 60. N. W. Broten, T. Oka, L. W. Avery, J. M. MacLeod, and H. W. Kroto, Astrophys. J., 223, L105 (1978). 61. L. W. Avery, N. W. Broten, J. M. MacLeod, T. Oka, and H. W. Kroto, Asrrophgs. J., 205, L173 (1976). 62. H. W. Kroto, C. Kirby, D. R. M. Walton, L. W. Avery, N. W. Broten, J. M. MacLeod, and T. Oka, Asrrophjis. J., 219, L133 (1978). 63. A. J. Alexander, H. W. Kroto, and D. R. M. Walton, J . Mol. Specrrosc. 62, 175 (1976). 64. C. Kirby, H. W. Kroto, and D. R. M. Walton, J. Mol. Spectrosc., 83, 261 (1980). 65. E. B. Turner, Astrophys. J., 163, L38 (1971). 66. L. E. Snyder, “Molecules in Space,” M T P Internarional Review of Science, Physical Chemisrry, Vol. 3, Spectroscopy, D. A. Ramsey, Ed., Butterworths, London, 1972, pp. 193-240. 67. M. Winnewisser and J. J. Christiansen, Chem. Phys. Lett., 37, 270 (1976). 68. M. Winnewisser, E. W. Peau, K. Yamada, and J. J. Christiansen, Z. Naturforsch., %a, 819 (1981). 69. J. B. Simpson, J. G. Smith, and D. H. Whiffen, J. Mol. Spectrosc., 44,558 (1972). 70. G. Cazzoli, C. D. Esposti, and P. G . Favero, J. Mol. Struct., 4 8 , l (1978). 71. W. R. Thorson and I. Nakagawa, J. Chem. Phys., 33,994 (1960). 72. J. W. C. Johns, Can. J. Phys., 45, 2639 (1967). 73. K. Yamada and M. Winnewisser, Z . Naturforsch., 31a, 139 (1976). 74. M. Born and J. R. Oppenheimer, Ann. Physik, 84,157 (1927). 75. A. V. Burenin, E. N. Karyakin, A. F. Krupnov, and S . M. Shapin, J. Mol. Spectrosc., 78, 181 (1979). 76. T. R. Dyke, B. J. Howard, and W. Klemperer, J. Chem. Phys., 56,2442 (1972). 77. T.R. Dyke and J. S. Muenter, J. Chem. Phys., 60,2929 (1974). 78. J. W. Bevan, A. C. Legon, D. J. Millen, and S . C. Rogers, J. Chem. Soc. Commun., 341 (1975). 79. A. C. Legon, D. J. Millen, and S. C. Rogers, Chem. Phys. Lett., 41, 137 (1976). 80. A. C. Legon, D. J. Millen, and S . C. Rogers, Proc. R. SOC.(London)A370,213 (1980). 81. A. C. Legon, D. J. Millen, and S . C. Rogers, J . Mol. Spectrosc., 70, 209 (1978). 82. A. C. Legon, D. J. Millen, P. J. Mjoberg, and S. C. Rogers, Chem. Phys. Lett., 55, 157 (1978). 83. A. C. Legon, D. J. Millen, and P. J. Mjoberg, Chem. Phys. Lett., 47, 589 (1977). 84. L. W. Buxton, E. J. Campbell, and W. H. Flygare, Chem. Phys., 56, 399 (1981). 85. A. C. Legon, P. D. Soper, M. R. Keenan, T. K. Minton, T. J. Balle, and W. H. Flygare, J. Chem. Phys., 73, 583 (1980). 86. M. R. Keenan, T. K. Minton, A. C. Legon, T. J. Balle, and W. H. Flygare, Proc. Narl. Acad. Sci. USA, 77, 5583 (1980). 87. P. D. Soper, A. C. Legon, and W. H. Flygare, J. Chem. Phys., 74,2138 (1981). 88. A. C. Legon, P. D. Soper, and W. H. Flygare, J. Chem. Phys., 74,4944 (1981). 89. F. A. Baiocchi, T. A. Dixon, C. H. Joyner, and W. Klemperer, J. Chem. Phys., 74,6544 (1981). 90. C. H. Joyner, T. A. Dixon, F. A. Baiocchi, and W. Klemperer, J . Chem. Phys., 74,6550 (1981). 91. A. C. Legon, P. D. Soper, and W. H. Flygare, J . Chem. Phys., 74,4936 (1981). 92. J. S. Muenter, J. Chem. Phys., 56, 5409 (1972). 93. S . E. Novick, P. Davies, S. J. Harris, and W. Klemperer, J . Chem. Phj’s., 59, 2273 (1973). 94. S . J. Harris, S. E. Novick, and W. Klemperer, J. Chem. Phys., 60,3208 (1974). 95. S . J. Harris, S. E. Novick, and W. Klemperer, J . Chem. Phys., 61, 193 (1974). 96. S . E. Novick, S . J. Harris, K. C. Janda, and W. Klemperer, Can. J . Phj,~.,53, 2007 (1975).

172


97. S. E. Novick, K. C. Janda, S. L. Holmgren, M. Walden, and W. Klemperer, J . Chem. Phj:~., 65, 1114(1976). 98. F. A. Baiocchi, T. A. Dixon, C. H. Joyner, and W. Klemperer, J . Chem. Phys., 75,2041 (1981). 99. K. V. Chance, K. H. Bowen, J. S. Winn, and W. Klemperer, J . Chem. Phys., 70, 5157 (1979). , (1981). loo. T. A. Dixon, C. H. Joyner, F. A. Baiocchi, and W. Klemperer, J. Chem. P h j ~ .74,6539 71,2723 (1979). 101. T, J. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare, J. Chem. Phj)~., 102. T. E. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare, J. Chem. Phys., 72,922 (1980). 103. M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton, T. K. Minton, P. D. Soper, and W. H. Flygare, J. Chem. PAYS.,72, 3070 (1980). 104. M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle, and W. H. Flygare, J. Chem. P h j ~73, , 3523 (1980). 105. L. W. Buxton, E. J. Campbell, T. J. Balle, and W. H. Flygare, Chem. Phys., 54, 173 (1981). 106. M. R. Keenan, L. W. Buxton, E. J. Campbell, A. C. Legon, and W. H. Flygare, J. Chem. Phj)~., 74, 2133 (1981). 107. E. J. Campbell, L. W. Buxton, and W. H. Flygare, J. Chem. Phj’s., 74,813 (1981). 108. S. E. Novick, P. Davies, T. R. Dyke, and W. Klemperer, J. Am. Chem. SOC.,95, 8547 (1973). 109. G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand Company, New York, 1945, p. 173. 110. T. A. Dixon and R. C. Woods, Phys. Rev. Lett., 34,61 (1975). 111. K. B. Jefferts, P h j ) ~Rev. . Lett., 23, 1476 (1969). 112. D. Buhl and L. E. Snyder, Nature, 228,267 (1970). 113. W. Klemperer, Nature, 227, 1230 (1970). 114. E. Herbst and W. Klemperer, Astrophys. J., 185, 505 (1973). 115. L. M. Hobbs, Astrophys. J., 181, 795 (1973). 116. L. E. Snyder, J. M. Hollis, B. L. Ulich, F. J. Lovas, and D. Buhl, Bull. Am. Astron. SOC.,7, 497 (1975). 117. U. Wahlgren, B. Liu, P. K. Pearson, and H. F. Schaefer, Nature, 246, 4 (1973). 118. W. P. Kraemer and G. H. F. Diercksen, Astrophys. J., 205, L97 (1976). 119. R. C. Woods, T. A. Dixon, R. J. Saykally, and P. G. Szanto, Phys. Rev. Lett., 35, 1269 (1975). 120. M. Morris, B. Zukerman, B. E. Turner, and P. Palmer, Astrophys. J., 192, L27 (1974). 121. J. M. Hollis, L. E. Snyder, D. Buhl, and P. T. Giguere, Astrophys. J., 200, 584 (1975). 122. M. Bogey; C. Demuynck, and J. L. Destombes, Mol. Phys., 46,679 (1982). 123. B. E. Turner, Astrophys. J., 193, L83 (1974). 124. S. Green, J. A. Montgomery, and P. Thaddeus. Astrophys. J., 193, L89 (1974). 125. R. J. Saykally, T. A. Dixon, T. G. Anderson, P. G. Szanto, and R. 77. Woods, Astrophys. J., 205, LlOl (1976). 126. K. V. L. N. Sastry, P. Helminger, E. Herbst, and F. C. De Lucia, Chem. Phys. Lett., 84, 286 (1981). 127. P. Thaddeus, M. Gutlin, and R. A. Linke, Astrophys. J., 246, L41 (1981). 128. C. S. Gudeman, N. N. Haese, N. D. Piltch, and R. C. Woods, Astrophys. J., 246, L47 (1981). 129. C. S. Gudeman and R. C. Woods, Phys. Reb. Lett., 48, 1344 (1982). 130. R. C. Woods, C. S. Gudeman, R. L. Dickman, P. F. Goldsmith, G. R. Huguenin, W. M. Irvine, A. Hjalmarsoa, L.-A. Nyman, and H. Olofsson, Astrophys. J., 270, 583 (1983). 131. G. A. Blake, P. Helminger, E. Herbst, and F. C. De Lucia, Astrophys. J., 264, L69 (1983). 132. T. G. Anderson, C. S. Gudeman, T. A. Dixon, and R. C. Woods, J. Chem. Phys., 72, 1332 (1980).

REFERENCES

173

133. K. D. Tucker, M. L. Kutner, and P. Thaddeus, Astrophys. J., 193, L115 (1974). 134. L. M. Ziurys, R. J. Saykally, R. L. Plambeck, and N. R. Erickson, Astrophys. J., 254,94 (1982). 135. K. V. L. N. Sastry, P. Helminger, A. Charo, E. Herbst, and F. C. De Lucia, Astrophys. J., 251, L119 (1981). 136. M. Gutlin, S. Green, and P. Thaddeus, Astrophys. J., 224, L27 (1978). 137. S. Wilson and S . Green, Astrophys. J., 212, L87 (1977). 138. M. Gutlin and P. Thaddeus, Astrophys. J., 212, L81 (1977). 139. J. T. Hougen, J. Chem. Phys., 36,519 (1962). 140. S. Saito and T. Amano, J. Mol. Spectrosc., 34, 353 (1970).

Chapter

VI

S Y M M ETRIC-TOP MOLECULES MOLECULES IN THE GROUND VIBRATIONAL STATE

175

The Rigid Rotor Approximation The Distortable Rotor

175 177

2

MOLECULES IN EXCITED VIBRATIONAL STATES

182

3

MOLECULAR INVERSIONS

187

1

4

FORBIDDEN ROTATIONAL TRANSITIONS Molecules w i t h C, Symmetry Avoided-crossing Molecular Beam Resonance Method for Measurement of K-level Separations

5

6

194 196

Tetrahedral Spherical-top Molecules

202 205

LINE INTENSITIES

208

Optimum Regions for Detection of Symmetric-top Rotational Spectra

21 2

MOLECULAR STRUCTURES

21 3

Ground-state Structures Equilibrium Structures Hydrogen- bonded Symmetrie-top Complexes

21 3 21 6 221

A molecule in which two of the principal moments of inertia are equal is a symmetric-top rotor. This condition is generally met when the molecule has an axis of symmetry that is,trigonal or greater. The molecules PH3, CH3F, CH3CCH, CH3SiH3, and CF3SF5 are examples of symmetric-top molecules. A linear molecule can be treated as a special case of a symmetric top in which the angular momentum about the symmetry axis is zero. The rotational energies of the symmetric-top molecule were calculated first by Dennison [l] with matrix mechanical methods and later by Reiche and Rademacher [2] and by Kronig and Rabi [3] with the Schrodinger wave equation.

1

MOLECULES IN THE GROUND VIBRATIONAL STATE

The Rigid Rotor Approximation The classical mechanics of the symmetric top is discussed in Chapter 11, Section 1. Classically, the molecule rotates about the symmetry axis while this 175

176

SY M M ETR I C-TOP MOLECULES

axis in turn precesses about a fixed direction in space corresponding to the direction of the total angular moment P. This motion is illustrated by Fig. 6.1. In a symmetric top, one of the principal axes of inertia must lie along the molecular axis of symmetry. The principal moments of inertia which have their axes perpendicular to this axis are equal. If a, the axis of least moment of inertia (Ia< ZJ, lies along the symmetry axis, the molecule is a prolate symmetric top (CH,CCH, for example). If c, the axis of the greatest moment of inertia ( I a = I b < IC),lies along the symmetry axis, the molecule is an oblate symmetric top (BCI,, for example). Most of the symmetric-top molecules observed in the microwave region are prolate. With the a axis chosen along the symmetry axis ( I c= I b ) and with Pz= P,'+Pi P:, the Hamiltonian operator of (2.17) may be expressed as

+

Eigenvalues for the angular momentum operators Pz,P,, and Pz for the symmetric-top rotor (Chapter 11, Section 2) are ( J , K, MIPZJJ,K, M ) = h z J ( J + 1) ( J , K, MIPAJ, K, M ) = h K

where J=O, 1, 2, 3,. . . K=O, f 1 , f 2 , & 3,..., + J M=O, +1, +2, + 3 ,..., + J In the x, y , z system, z is chosen as the symmetry axis of the top. In the a, 6, c system used here for designation of the principal axes of inertia, z becomes a P

Fig. 6.1 Vector diagram of a symmetric rotor.

MOLECULES I N THE GROUND VIBRATIONAL STATE

177

for the prolate top, and z becomes c for the oblate top. In the field-free rotor, the rotational energies do not depend on M . It is evident that the Hamiltonian of (6.1) commutes with P2 and Pa and is therefore diagonal in the J, K representation. Its eigenvalues, which are the quantized rotational energies of the rigid prolate symmetric top, are therefore

With the designation A =h/(Sn2Za)and B=h/(87r2Zb),Ej,K can be written EJ,K=h[BJ(J

+ 1)+ ( A - B ) K 2 ]

(6.6)

Since Pa is a component of P, the values of K cannot exceed those of J in magnitude. Although K can have both negative and positive values, the + and - values do not lead to separate sets of energy levels because K appears only as a squared term in (6.6). Thus all K levels except those for K=O are doubly degenerate. This K degeneracy cannot be removed by either external or internal fields. In addition to the K degeneracy, there is a ( 2 J + l), M degeneracy in the field-free symmetric rotor as in the linear molecule. Unlike the K degeneracy, the M degeneracy can be lifted completely by the application of an external electric or magnetic field (see Chapters X and XI). In a true symmetric top, any permanent dipole moment must of necessity lie along the symmetry axis. All matrix elements of this dipole moment resolved along a space-fixed axis vanish except those between states corresponding to J - r J or Jf 1, K+ K. See Chapter 11, Section 6. The selection rules for the field-free rotor are therefore

AJ=O, + 1 AK=O (6.7) The rule corresponding to absorption of radiation is J+J+1 and K+ K. Application of these rules to (6.6)gives the formula for the absorption frequencies for the rigid symmetric top v =2B(J+ 1 )

(6.8)

which is exactly that for the linear molecule. However, as we shall see below, centrifugal stretching separates the lines corresponding to different IKI by small frequency differences which are usually sufficient to make these lines resolvable in the microwave region.

The Distortable Rotor Centrifugal stretching is treated as a perturbation on the eigenstates of the rigid rotor. If X r represents the Hamiltonian of the rigid rotor and %d represents that of the distortional energy, the rotational Hamiltonian is H"=Hr+%d

(6.9)

178

SYMMETRIC-TOP MOLECULES

In Chapter VI11 it is shown that the centrifugal distortional Hamiltonian has the form (6.10) where r , b, y, d represent the principal coordinate axes of the moments of inertia and where each must be summed over all three coordinate axes. However, in the first-order perturbation treatment, the terms with odd powers in any angular momentum which occur in the sum average to zero. Furthermore, by use of the angular momenta commutation rules and the fact that many of the 7’s are equal, the Hamiltonian can be further simplified. This is discussed in Chapter VIII. For the first-order treatment, which is adequate for the usual microwave measurements, the Hamiltonian can be written as

+ r:,,,(PIPZ” + P,’PI)+ r;yl:(P;PZ”+ PZ”P,Z)]

(6.11)

in which x, y , z represent the principal axes and z represents the symmetry axis of the top. The 5”s are defined in Table 8.4. To first order, the distortion energy is

E$”=(J, zqX$’’IJ,K )

(6.12)

With the angular momentum matrix elements given in Table 8.19, this expression can be evaluated readily. The resulting expression can be condensed because of the fact that for the symmetric top certain of the coefficients 5’ are equal, that is, 7kxXx= T : =~7 ~ u~v v Therefore, .~ the terms which contain contributions from and which are not simple functions of P 2 and K 2 cancel with those from (PIP; PSPI). The nonvanishing terms can be combined in an expression of the form

(e) (e) +

E$’)= -~[D.,J~(J+I ) ~ + D . , ~ J ( J1+ ) ~ 2 + ~ ~ ~ 4 (6.13) 1 in which the DJ,D J K ,and D K are the usual first-order centrifugal stretching constants of the symmetric-top molecule expressed in frequency units. The D’s obviously represent a combination of the 5’s. For molecules of (e3”symmetry, expressions of the centrifugal stretching constants in terms of 5’s are given in Table 8.1 1 and 8.12. Addition of E$’’ to the rigid rotor values of (6.6) gives the usual expression for the rotational energy of the nonrigid prolate symmetric-top molecule

+

=~ [ B ~ J ( J11+ ( A-~B ~ ) KD.,JZ(J+ ~ 112 -D . , ~J ( J + I ) K Z +

D ~ 41K (6.14)

For the oblate symmetric top, the expression in the parentheses of the second term becomes ( C - B ) instead of ( A - B ) . With the selection rules J+J+I,

179

MOLECULES IN THE GROUND VIBRATIONAL STATE

K-+ K, this equation gives the rotational frequencies as

This frequency equation holds for both the prolate and the oblate tops. Note that neither the second nor the last term in the energy expression of (6.14) influences the pure rotational frequencies. These terms involve only rotation about the symmetry axis. The last term of (6.15) splits the rotational transition into J 1 closely spaced components whose separation increases as K The next higher order correction for centrifugal stretching leads to the expression

’.

+

for the ground-state rotational frequencies of symmetric-top molecules. The first-order expression, (6.15),has been found entirely adequate for most measurements. Small, higher-order terms have been detected in a few cases [4, 51. Figure 6.2 indicates the appearance of the first few rotational transitions of a symmetric-top molecule. The K splitting is greatly exaggerated. Figure 6.3 shows the K lines of the 5=8-+9 transition of CF3H. Generally, pyramidal molecules such as NH, and PH, have DjK negative so that the lines of a given J transition occur at higher frequencies as K increases, whereas those for which the symmetrical group occurs on the end of a linear group, such as CH,CN and CH3CCH, have DjK positive and have the lines of increasing K falling at lower frequencies. Those such as CF3H (shown in Fig. 6.3) and CC13H, which have a heavy symmetrical group and the light H atom attached to a central carbon, have negative DjK like the pyramidal molecules, whereas those with the lighter H atoms forming the symmetrical group of CH3F, CH3C1, and so on, have DjK positive. The values of Bo, of D j , and of DjK for selected molecules of different types are compared in Table 6.1. For all types, D j is positive; therefore the D j term shifts the different J transitions to lower frequencies. The centrifugal stretching constants of symmetric-top molecules are related to the bond force constants and fundamental vibrational frequencies, but the

J = O+I

J = 1-2

J = 243

J = 3+4

J = 4-5

Fig. 6.2 Diagram indicating the general appearance of the rotational spectrum of a symmetric-top molecule. The lines of different K for the same J transition are separated by centrifugal distortion. This separation is extremely small compared with the separation of the different J transitions.

Fig. 6.3 Cathode ray display of the eight K lines of the J=8+9 transition of CF3H occurring at i =1.61 mm. The centrifugal distortion constant D ~ isK negative (- 18 kHz) so that the higher K components fall at higher frequencies to the right. The total frequency spread of the multiplet is 21 MHz. From C. A. Burrus and W.Gordy, J.Chern. Phys. 26,391 (1957).

0

0

I

2

3

4

LOG D

Fig. 6.4 Plots of logDJ and of logDJKversus log (4B2/v2)for the methyl halides where v represents the fundamental vibrational frequency of the parallel mode u3 for the DJ plot and the fundamental frequency of the perpendicular mode u6 for the DjK plot. From Orville-Thomas, Cox, and Gordy [7].

180

MOLECULES

IN THE GROUND VIBRATIONAL STATE

181

Table 6.1 Selected Rotational Constants of Some Symmetric-top Molecules in the Ground Vibrational State Molecule

Bo ( M H z )

DJ (kHz1

DJK(kHz1

PH3 ASH, '2'SbH3 NF3

133,480.15 112,470.59(3) 88,038.99(3) 10,681.02(1) 3,468.603(3) 25,536.147(2) 13,292.869(10) 9,568.19(5) 7,501.276(5) 9,198.897(6) 4,973.009(15) 8,545.869 4,828.687 3,914.796(4) 2,517.888(14) 1,516.040(1)

3950 2925(3) 1884(4) 14.53(7) 1.863(2) 59.87(20) 18.0 9.87 6.31 3.81(8) 1.48(10) 2.875 2.1 0.42(2) 0.9(2) 0.037(4)

-5180 -3718(4) - 2394(15) -22.69(10) - 3.015(6) 420.3(4) 198(3) 128(1) 98.7(1) 176.9(2) 63(2) 163.0 63 153.1(3) 641(13) 41.958(6)

~

3

5

~

CH3F CH335C1 CH379Br CH31 CH3CN SiH3CN CH3CCH SiH3CCH CH3CNO SiH3NC0 SiH3NCS

1

Ref: Y

b b c,d

e r,g g

9 9

h

i

i,k I

m n 0

"Helminger and Gordy [77]. *Helminger et al. [78]. 'A. M. Mirri and G. Cazzoli, J. Chem. Phys., 47, 1197 (1967). dotake et al. [117]. 'G. Cazzoli, P. G. Favero, and A. Dal Borgo, J. Mol. Spectrosc., 50, 82 (1974). IR. S. Winton and W. Gordy, Phys. Lett., A32,219 (1970). 'T.E. Sullivan and L. Frenkel, J. Mol. Spectrosc., 39, 185 (1971). hA.Bauer and S. Maes, J. Phys. (Paris),30, 169 (1968). 'Careless and Kroto [31]. 'C. A. Burrus and W. Gordy, J. Chem. Phys., 26,391 (1957). 'A. Bauer and J. Burie, Compt. Rend., 268B, 1569 (1969). 'Gerry and Sugden [30]. "H. K. Bodenseh and K. Morgenstern, Z. Naturforsch., 25a, 150 (1970). "M. C. L. Gerry, J . C. Thompson, and T. M. Sugden, Narure, 211, 846 (1966). "K. F. DosSel and A. G. Robiette, Z. Naturforsch., 32a, 462 (1977).

relationship is generally quite complicated because of the many vibrational degrees of freedom of the polyatomic molecules (see Chapter VIII). The methyl halides provide perhaps the simplest group of organic symmetric-top molecules for which attempts at relating the vibrational frequencies with the stretching constants have been made. From the infrared vibrational spectra Chang and Dennison [6] have calculated values of DJ = 18.4 kHz and DJK= 189 kHz for CH335C1which agree very well with the observed microwave values [7], 18.1 and 198 kHz, respectively. One can, to a useful approximation, treat the methyl halides as diatomic molecules in which the CH3 group as a unit vibrates against the halogen. To

182


this simplified model one can then apply (4.16) for diatomic molecules 4B3 Dj =T

(6.17)

mi

where m i represents the fundamental parallel vibrational frequency corresponding to the stretching of the C-Xbond. Figure 6.4 shows a plot [7] of log D j versus log (B3/m2)for the methyl halides with m itaken as the parallel frequency m3. The relationship is linear as expected from (6.17). In the same figure is given a plot of log DjK versus log (B3/m2)with mi taken as the frequency of the mode in which the atoms vibrate in planes perpendicular to the symmetry axis. This relationship is also found to be linear. As the mass of X is reduced to that of H, both the frequencies m3 and (36 go over into the deformation frequency m4 in methane [8]. Likewise, the parameters Dj and DjK go over into a single distortion parameter D for the spherical-top molecule. I t is therefore interesting to note that the two plots of Fig. 6.4 intersect at the point for methane if m iis taken as its deformation frequency m4 and if DJ=DJK = D represents the distortion constant of CH4 from optical spectroscopy.

2 MOLECULES IN EXCITED VIBRATIONAL STATES For symmetric-top molecules in excited vibrational states the effective rotational constants differ somewhat from those of the ground state. The effective values of the constants in excited states are expected to have, in the first approximation, variation with the vibrational quantum number similar to that for linear molecules. To the approximation usually required, (6.18)

A,=A,-C

a:

(6.19)

The higher-order expression B, is similar to that for the linear molecule, (5.26). In these relations Be and A , are the equilibrium values, v i is the vibrational quantum number for the ith mode, a: and a! are small anharmonic constants, and direpresents the degeneracy of the ith mode. Slight differences in the stretching constants from those of the ground vibrational state are also expected. These differences, which are usually negligible, may be represented by a formula similar to that of (4.12). Silver and Shaffer [9, 101 have shown that (6.18) and (6.19) hold for pyramidal molecules; Shaffer [ 111, for axially symmetric molecules. Nielsen [12] has given a general treatment of the subject. For molecules in nondegenerate vibrational modes, the rotational frequencies of either the prolate or the oblate symmetric top can be expressed by

v = ~ B , ( J + ~ ) - ~ D $ ’ ” ( J + ~1)K2 ) ~ - ~ D ~ ~ ( J(6.20) + where B, is given by (6.18), with di=1. For molecules in degenerate vibrational modes, di>1, the change in B, can also be expressed by (6.18), but (6.20) does

MOLECULES I N EXCITED VIBRATIONAL STATES

183

not adequately predict the rotational spectrum. In addition to the change in A , B, or C for the degenerate vibrational modes, there is a splitting of the rotational lines, which is similar to the I-type doubling for linear molecules. The displacement of rotational lines of symmetric-top molecules by vibrational excitation is illustrated in Fig. 6.5. This recording shows J=11+12 rotational transitions in the 21 GHz region of (CH3)3C-C =C35C1 in excited vibrational states up to v = 5 of the fundamental mode believed to be that corresponding to a bending mode between the tertiary C atom and the acetylene group [13]. The labeling on the figure indicates the main peaks of the rotational lines in the vibrational states u =O to v = 5 which have unresolved I-type doubling as described later, also unresolved nuclear quadrupole structure and K structure. From a fitting of the relative intensities of the lines with those expected from the Boltzmann relation (Chapter III), the fundamental frequency of the vibration was found to be 120 cm-’, approximately. For symmetric-top molecules such as CH3CCH which have a linear group along the symmetry axis, there are degenerate bending vibrational modes which give rise to I-type doubling of the rotational levels similar to that described for linear molecules in Chapter V. Also, degenerate vibrational modes involving perpendicular motions of the atoms which are off the axis can give rise to components of vibrational angular momentum along the symmetry axis. Thus degenerate modes leading to vibrational angular momentum occur in pyramidal XY, molecules and even in planar XY, molecules. Because of Coriolis interaction between vibration and rotation, a splitting of the levels occurs which is known as I-type doubling. There is a component of this vibrational angular momentum along the figure axis which adds to, or subtracts from, the pure rotational angular momentum about the symmetry axis. This component has the value (lii, where 5 is the Coriolis coupling constant and l=vd, vd-2,.

. ., - u d

(6.21)

where u d is the vibrational quantum number of the degenerate mode. For the most important case, v d = 1 and I = f 1. When the vibrating motions are perpendicular to the symmetry axis, ( = 1 and the vibrational component along z is I i i , just as in the linear molecules. In the symmetric top, however, the

Fig. 6.5 Recording of the J = 11-12 rotational transition of (CH3)3C-C=C35C1 in excited states up to t.=5 of a bending vibrational mode. The substructure of the lines caused by [-type doubling, centrifugal distortion, and nuclear coupling is not resolved. From Bodenseh, Gegenheimer, Mennicke, and Zeil [131.

184


component along the symmetry axis can, in general, be less than Ih, or Od lild 1. In any event, the quantum mechanics of the symmetric top requires that the overall angular momentum along the symmetry axis, that is, the sum of that caused by pure rotation and vibration, must be an integral multiple of h instead of that caused by vibration or rotation alone. Likewise, the square of the overall angular momentum including that arising from the degenerate vibrational modes is quantized with values J(J+ l ) h 2 . If P,, P,, and P, represent the overall angular momentum about the principal axis including that caused by pure rotation of the molecule and that caused by vibration, the pure rotational Hamiltonian can be expressed as (6.22) where p x , p , , and p z represent the components of the angular momentum which arises from the degenerate vibratory motions. Let us assume a prolate symmetric top with Z,=Z,=Ib, I z = I a , and px=py=O. The 2,.can then be expressed as

(6.23) In the last expression we neglect the term p,2/(2IO),which represents pure vibrational energy and which does not change with the rotational state. Since P2=h2J(J+1),

Pz= Kh,

and

p,=ilh

s = B , J ( J + I ) + ( A , - B,) K 2 - 2 I K 4 , h

(6.24) (6.25)

where

J=O, 1,2, 3, * . . K=O, f1, & 2, . . . , + J

(6.26)

and where 1 values are given by (6.21). To obtain the formula for the oblate symmetric top one simply replaces the A , by C,. A formula like that of (6.25) was originally derived by Teller [I41 and by Johnston and Dennison [15]. Since both K and I can take plus and minus values, (6.25) indicates a doublet splitting of the levels with each component retaining a double degeneracy. However, certain of the levels may be further split by interactions of higher order. When IKI =3n + 1 where n is an integer including zero, the term with KI positive represents two coinciding levels of species A whereas the term with KI negative represents a doubly degenerate level of species E. For I I(I = 3n + 2, the reverse is true. When 14= 3n but not including zero, the terms with KI positive or negative represent separately doubly degenerate levels of species E.


185

The double degeneracy of the levels of species E cannot be removed [16], but the coinciding levels of species A may be split by interactions of higher order. H. H. Nielsen [I71 has shown that the further splitting of the levels of species A can be appreciable for K = 1, but that it is negligible for other K values. More complete treatments of the interaction of degenerate vibrations with rotation in symmetric-top molecules have been given by Nielsen [17, 181, by Grenier-Beeson and Amat [19], and by others [20-221. For singly excited degenerate bending, with u d = 1, I = f 1, Nielsen’s formula for the rotational energy which includes stretching as well as higher-order interaction with the degenerate vibration is

+

EjK =h{B,J(J+ 1) ( A , - B,) K 2 - 2AuK/( - D j J z ( J + 1)’

+

- DjK J(J + X

1)K Z-DK K 4 2 [(20j+Dj,)J(J+ ~ ) + ( ~ D K + D ~ K ) K ~ ] K / (K, +P /)}( (6.27) J,

where for K=/= f 1

P = +iJ(J+ l)q

(6.28)

where q z 2 B Z / w , and o is the fundamental bending vibrational frequency. For K # / = f 1,

* [J ( J + 1)

+

T I)] [J(J 1)- ( K T 1)( K T 2)] q2 (6.29) 8 ( K T 1)[(1- O A - &I The upper signs are taken when K and / have the same sign; the lower signs, when K and 1 are of different sign. With the selection rules, P=

- K( K

J+ J+ 1,

K+ K,

(6.30)

l+I

the predicted frequencies are ~ V = 2 B U ( J + 1 ) - 4 D ~ ( J1)3-2&(J+ + 1)K2 4(2Dj +DjK)(J+ 1)K/( + AP(J, K, I )

(6.31)

AP= +q(J+ 1)

(6.32)

+

where for K = l = f 1, and ( J + l)[(J+

-(KT

u21

* 4( K T 1)[(1- ( ) A ,- B,]

q2

(6.33)

for K#l= f 1. Upper signs are taken when K and 1 have the same sign; the lower signs hold when K and I have different signs. The qualitative features predicted by Nielsen’s theory, including the rather wide separation of the K = l = f 1 lines by 2q(J+ I), were verified [23] by early measurements on CH3CCH. The theory has been applied to a number of other symmetric-top molecules [24], including SiH3NCS which, surprisingly, is a symmetric top, and to several different rotational transitions of CH3CN

186


[25]. Table 6.2 illustrates the extent of agreement of the calculated with the observed frequencies of the J = lo-, 11 lines of the 08 = 1 bending vibrational state of CH3CN. Although it was found that the positions of the various doublets for CH3CN were given adequately by Nielsen's theory, the separations of the individual doublets were not satisfactorily predicted. Venkateswarlu et al. [25] found that these separations could be closely predicted, as in Table 6.2, ifthe term 4&K5(J+1)where8 ~ 0 . 0 3 is 4 inserted into (6.31).Values of interaction constants were found to be (=0.878 and q=17.775 MHz. Measurements and analysis of CH3CN have been extended by Bauer [26] to include several rotational transitions of the second excited state, 08 =2. Nielsen's theoretical work on 1-type doubling has been extended and improved by others. Formulas for J+J+ 1 rotational frequencies including 1-type doubling are provided by Grenier-Beeson and Amat [19] for first excited states, ut = 1, of molecules having C3, symmetry. The formulation is extended by Tarrago [27] to second excited vibrations having 1=1, that is, ut=21 states.

Table 6.2 The Observed and Calculated Frequencies and the Assignments of the J=10+11 Lines Corresponding to the Molecules in the Excited State us = 1 of CH3CN" Transition

J 10+11

K i 1

I

Observed Frequency WHz)

Calculated Frequency (MH4

203,161.23 202,769.94 202,972.63 202,950.97 202,956.31 202,943.39 202,935.67 202,924.94 202,907.98 202,897.68 202,872.91 202,862.38 202,830.05 202,819.06 202,779.70 202,768.06 202,721.62 202,709.07 202,655.71 202,642.27

203,161.53 202,770.08 202,972.53 202,950.87 202,956.28 202,943.68 202,935.61 202,925.23 202,907.97 202,898.15 202,872.87 202,862.88 202,830.11 202,8 19.63 202,779,621 202,768.47 202,721.40 202,709.43 202,655.39 202,642.51

"From Venkateswarlu et al. [ 2 5 ] .


187

The formulas by Grenier-Beeson and Amat were applied by them [19] to the available data on F3CC -CH and found to provide consistent agreement. Others have successfully applied their theory: Otake et al. [28], to measurements on the degenerate u3 and u4 modes of NF,; Whittle et al. [29], to the degenerate u7 and ua modes of CF,CN; Gerry and Sugden [30], to the degenerate bending mode u l 0 of SiH,CCH, as examples. The formulation of Tarrago [27] has been applied with consistent fitting to rotational transitions of a number of C3" molecules in v=2' states: to CH,CN, by Bauer [26]; to SiH,CN, by Careless and Kroto [31]. The latter work extends the formulation to include ua=3 and 4 states of SiH3CN. Direct transitions between the I-type doublets in the J = 17 to 24 rotational levels of the u4 = 1 state of PF, have been observed and analyzed by Hirota [32]. The I-type doubling constant, q4 =29.49270(8) MHz, and the 2-1 interaction constant, r4 = 3.013(5) MHz, were obtained. Direct I-type doublet transitions have also been measured and analyzed in rotational levels of the 06'1 state ofCF3H and CF3D by Kawashima and Cox [33]. The values, 1461 =36.27917(13) MHz and Ir61 =0.97(2) MHz, were obtained. These papers provide instructive examples of the analysis of I-type doublet spectra in symmetric-top molecules. An example of Fermi resonance between different vibrational states is provided by the work of Morino and Hirose [34] in the measurement of shifts in the lines due to interaction of the u5 and u3 + 06 vibrational states of CH3I. Accidentally strong resonances between certain rotational levels of the u = 2 vibrational states of both CH31ZCNand CH313CN have been observed by Bauer and Maes [35].

3


According to quantum mechanics, pyramidal XY, symmetric-top molecules can execute inversion motion in which the X atom moves through the Y3 plane to achieve an identical but inverted pyramidal configuration. The inverted configuration is obtained mathematically by a change of sign of the coordinates of all the particles measured from the center of mass. Theoretically, such inversion is possible in any nonplanar molecule (Chapter 111, Section 4), but, practically, it is of significance for only a very few. The inversion potential curve of the NH, molecule is illustrated by Fig. 6.6. The potential energy of the molecule plotted as a function of the distance between the N atom and the H3 plane has double minima corresponding to the two equivalent equilibrium configurations. The wave functions of the molecule can be expressed as linear combinations of 4Land (PR of those of the molecule in the equivalent right and left configurations. These functions, 1 (6.34) (6.35)

188


v,= 2

*

v2= 0

)AU = 0.8 cm”

H, H-N H’

I

1 I I

I

-10

0

/H NH ; H

I

10

- x

Fig. 6.6

Potential curve of the NH, inversion.

have opposite symmetry. For an infinitely high barrier they represent degenerate states; but when the barrier is sufficiently low, as in NH,, the two states are separated by a measurable interaction energy. Selection rules (Chapter 111, Section 4) allow transitions +-- between the states giving rise to inversion spectra. The transition moment is given by (2.125). In addition to the inversion motion, the molecule has a symmetrical vibrational mode in which the X atom moves back and forth along the symmetry axis in opposition to the motions of the Y, atoms. In the first and second excited symmetrical modes this vibrational energy for NH3 is lower than the potential energy barrier between the two minima. In all other XY, molecules the potential hill between the vibrational and potential energy is much greater relative to the vibrational energy. Thus, in lower symmetrical vibrational modes the XY, molecule executes vibrational motions in the potential valley on either side of the potential barrier. Classically, it is not possible for the molecule to invert when in these vibrational states which lie below the potential maximum. Quantum mechanically, though, it is possible for the molecule to achieve inversion transition through “barrier tunneling.’’ In NH,, the N atom tunnels through the H, plane at approximately 23 GHz to give rise to the well-known inversion spectrum in the 1.3-cm wave region, the first spectrum to be observed in the microwave region [36-381. The corresponding inversion frequency of ND3 is approximately 1.7 GHz, and for NT3 it is 0.306 GHz. No other symmetric-top molecules have inversion frequencies that fall in the microwave region. The next highest inversion frequency [39,40], that of PH,, is predicted to be very small. So far, the inversion splitting of the rotational lines of PH3 has proved undetectable with the highest resolution available. Other related molecules such as ASH, are predicted to have such long inversion time, of the order of days [40], that no observable effects on microwave spectra are expected. However, related barrier-tunneling effects associated with internal rotation and torsional oscillation of groups within symmetric-top and asymmetric-top molecules are observable in microwave spectra. These effects are treated in Chapter XII.


189

Theoretically, one can obtain the inversion frequencies from a solution of the wave equation for the vibrating rotor, as is done for diatomic molecules with the assumption of a double minima potential of the appropriate form. This is a very difficult problem, however. In a first-order approximation one can neglect the interaction between vibration and rotation and can obtain separate solutions of the wave equation for the vibration and rotation and can then, with the perturbation theory, correct for the interaction. The simplest model that accounts for the inversion is one which treats the three hydrogens in a plane as a unit vibrating against the N atom, somewhat as a diatomic molecule. The reduced mass of the vibration is p=3mM/(3m+M). The wave equation for this simple model is (6.36) With a double minima potential function, the solution yields the vibrational levels split by inversion. Solutions have been obtained by Manning [41], also by Dennison and Uhlenbeck [42], whose solutions for inversion energies and frequencies are

AE inv. AE,

--

1

-z

where A=exp

(2z

or

lo

)’(

(6.37)

Vinv = V v i b

[2p(V(x)-

dx

(6.38)

in which

AE, = separation of the vibrational levels AEinv= inversion splitting of the vibrational levels Vinv= the inversion frequency vv,b = the fundamental vibrational frequency E, = the total vibrational energy x = x l when V=E, The quantity A measures the area under the potential curve between the potential minima and is not particularly sensitive to the exact shape of the curve. Dennison and Uhlenbeck used the W -K-B method and assumed the curves in the region of the minima to be parabolic. Manning assumed a potential function of the form X

X

2P

2P

I/ (in cm-’)=66,551 sech4 - - 109,619 sech’ -

(6.39)

where p=6.98 x IO-*/p”’. This formula was found to predict the frequencies quite well. I t allows, through the effects of p, the prediction of the isotopic shifts of the inversion frequency.

190


The effects of rotation on the inversion splitting have been taken into account in various semiempirical formulas, the nature of which is suggested by the centrifugal stretching terms of the noninverting symmetric-top rotor or by the nature of the function A of (6.38). The various constants in these expressions are then evaluated by a fitting of them to the accurately measured rotational fine structure of the inversion. The formula that fits most closely all the accurately measured fine structure is an exponential function suggested by Costain [43] from a consideration of the nature of the function of (6.37) as well as the nature of the centrifugal stretching terms in symmetric-top rotors. I t is evident that centrifugal distortion would alter slightly both I/ and E, of (6.38).Costain’s formula was extended to higher order by Schnabel et al. [44] and its various constants determined by a fitting of the formula to the most complete and accurate measurements on the NH, available to 1965. Their formula and values for the constants are given in the earlier edition of this book. Because the lowest rotational frequency of NH, and ND, falls in the submillimeter wave region, the many early microwave measurements on these symmetric molecules were limited to their inversion spectra, although combined rotation-inversion transitions were observed for the mixed species NHD2, N H 2 D which are asymmetric rotors. In 1957, observation of the J=O-+l rotational transition of ND,, which occurs at 2=0.97 mm, was reported [45], but the corresponding rotational transitions of 14NH3and 15NH3,which occur in the 0.52-mm wave region, were not observed until 1967 [46]. In 1974, the rotation-inversion spectrum of NT, in the v2=0 states was measured [47]. Figure 6.7 shows the energy level diagram of the first few rotation-inversion levels of NH, and ND3 with the observable transitions indicated. Note that for NH3, only one of the inversion levels occurs when K=O, whereas for ND,

K=2

K= I K=O

Fig. 6.7 Diagram of the lower rotation-inversion levels and transitions of NH, and ND, with the allowed transitions indicated.


191

both the + and - levels occur, but with differing statistical weights. This difference is due to the effects of nuclear spin statistics discussed in Chapter 111, Section 4 and conforms to the statistical weight formula given in Table 3.3. Figure 6.8 indicates how the measured centimeter wave inversion frequencies for the upper and lower levels were combined with the J=O-+I submillimeter wave rotation-inversion frequencies [48] to give accurate values for the rotational constants Bo for 14NH3 and "NH3. After correction of the rotational frequencies for inversion effects, they were used in (6.15) with infrared D j values for calculation of the Bo values listed on Fig. 6.8. Measured inversion frequencies of ND3 in the 2-GHz microwave region [49,501 have also been combined with the observed submillimeter wave rotation-inversion transitions for derivation of the rotational constants [48] Bo, D j , and DjK for the 14ND3 and 15ND3,entirely from microwave data. The inversion splitting of NT3, together with its rotational constants, has been obtained by Helminger et al. [47], entirely from measurements of millimeter and submillimeter wave rotation-inversion transitions. They used the frequency formula V=2Bo(J+ 1)-4Dj(J+ 1 ) 3 - 2 D ~ ~ ( J1)K2f + [(")o+

C1(Jf 1)2+C2K2] (6.40)

and adjusted the constants by the least-squares method to reproduce the observed frequencies as shown in Table 6.3. The resulting constants are given at the bottom of Table 6.3. The J=O+l frequency is corrected for 14N nuclear quadrupole splitting. The bracketed term in (6.40) represents the inversion 14

NH3

J K MHz

90~298,115.37MHZ

Bo = 297,388.12 M Hz

Fig. 6.8 Illustration of the derivation of Bo from the observed rotational frequencies of I4NH3 and 15NH3as measured. From Helminger et al. [48].

192


Table 6.3 Rotation-Inversion Frequencies and Derived Spectral Constants of 14NT3''

Transition

Inversion Component

Observed Frequency (MHz)

Calculated Frequency (MHz)

Dgerence (MH4

J=O+l, K=O

lower

2 10,814.885

2 10,815.023

-0.138

J=1+2, K = l J=1+2, K=O J=1+2, K = l

lower upper upper

42 1,891.743 422,482.040 422,500.800

42 1,891.608 422,482.014 422,500.871

0.135 0.026 -0.071

J=2+3, J=2+3, J=2+3, J=2+3, J=2+3,

lower lower lower upper upper

632,810.839 632,836.513 632,914.199 633,440.324 633,523.744

632,810.768 632,836.626 632,914.202 633,440.321 633,523.724

0.07 1 -0.113 -0.003 0.003 0.020

K=O K=l K=2 K=l K =2

Inversion Constants ( M H z ) ( v J O 4 0 5 . 8 9 kO.11

C1= -0.557$0.020 Cz =0.971 k0.038

Rotation Constants ( M H z )

Bo = 105,565.373k0.034 Djz2.5981 k0.0024 DjK = - 4.472 k0.006

"From Helminger et a]. r471

splitting; (vi)o is the inversion frequency: the C1and C2 terms correct the inversion frequency for centrifugal distortion effects. The effective and substitution structures for the ground state of ammonia, as derived from the Bo values of various isotopic species, are given in Section 6, Table 6.7. for the ground state are now known The inversion doubling frequencies accurately from microwave measurements for all three species, NH3, ND3, and NT3. The much larger inversion frequencies for the u2 = 1 excited state. have been measured with infrared vibration-rotation spectroscopy : for NH3 and N D 3 by Benedict and Plyler [51] and for NT3 by Rao et al. [52]. Figure 6.9 shows the relationship between loglo(vi/vo)and p1l2,where vi is the inversion frequency, vo is the fundamental vibrational frequency of the u2 mode, and p is the reduced mass of the particular isotopic species of the ammonia. The relationship in each state, u2 =O or 1, is approximately linear. The relationship revealed in Fig. 6.9 is suggested by the Dennison-Uhlenbeck formula, (6.37). Substitution of (6.38) into (6.37) shows that (6.41) and thus (6.42)


193

4

Y 3

P a-

-

0

-0" 2. 1.1

I

1.6

I

1.8

I

I

I

20

2.2

2.4

& Fig. 6.9 Correlation of the inversion splitting with the reduced molecular mass for the three isotopic species of ammonia in different u2 vibrational states. From Helminger, De Lucia, Gordy, Morgan, and Staats [47].

where, in Fig. 6.9, v i = v l n v and vo = v v l b . The fact that the experimental relationship for each vibrational state, u2 =O or v 2 = 1, is approximately linear indicates that the quantity expressed by the integral varies little with the change of isotopic species, that is, approximately ln(vijvo)-p1i2 for a given u2 vibrational state. The simple, one-dimensional, double-minimum potential model of Dennison and Uhlenbeck [42] and Manning [41], (6.37)-(6.39), has been extended and refined by Swalen and Ibers [53] and by Damburg and Propin [54]. A more comprehensive, vibration-inversion-rotation Hamiltonian that includes centrifugal distortion and Coriolis interactions in the ground and excited inversion states has been developed by PapouSek et al. [55]. The Hamiltonian is a logical extension of one developed by Hougen et al. [56] for triatomic molecules. Further development of this Hamiltonian has been carried out by Spirko et al. [57], who give procedures for a least-squares fitting of its solution to the vibration-inversion-rotation energy levels of NH3, ND3, and NT3. This later form of the ammonia Hamiltonian is related to the one for the inversion-rotation of triatomic molecules with large-amplitude bending motions, developed earlier by Hoy and Bunker [58]. The effective vibration-inversion-rotation Hamiltonian developed by Papouiek et al. [55, 571 has been used in successful analyses of extensive submillimeter wave measurements on the inversion and inversion-rotation spectrum of NH3 in the v 2 = 1 vibration state by Belov et al. [59]. A section of the submillimeter wave spectrum of 14NH3in the u2 = 1 vibrational state, obtained with the Krupnov spectrometer, is shown in Fig. 6.10. Scappini and Guarnieri

194

SYMMETRIC-TOP MOLECULES I

I

1

I

I

I

959

960

961

962

I

GHr Fig. 6.10 Part of the submillimeter microwave spectrum of 14NH3.The three strong lines result from inversion transitions, s(J, K ) + a ( J , K), of the excited u2 = 1 state. The weak line, denoted a(l$,O)+a(lO, 3), is a perturbation-allowed transition of the ground state, u2 =O. From S . Urban, V. Spirko, D. Papousek, J. Kauppinen, S. P. Belov, L. I. Gershtein, and A. F. Krupnov, J . Mol. Spectrosc., 88, 274 (1981).

[60] have measured the millimeter wave spectrum of ND, in the v i = l vibrational state and obtained the accurate value of 106,354.35(3) MHz for (vi)l in that state.

4

"FOR BI DDEN" ROTATIONAL TRANSITIONS

Because there is no permanent dipole component perpendicular to the symmetry axis, resonance radiation does not normally induce changes in the K quantum number of symmetric-top rotors; that is, the selection rule A K=O applies in first order. Thus, from normal rotational spectra, only one inertial constant, Zb or B, can be obtained for symmetric-top rotors, whereas three independent inertial constants can be obtained for most asymmetric rotors. Because of theoretical developments and considerable improvement in experimental techniques and spectrometer sensitivity, it has become possible to detect

“FORBIDDEN“ ROTATIONAL TRANSITIONS

195

in polar, symmetric-top rotors the “forbidden” rotational transitions of the ground vibrational state corresponding to changes in the K quantum number. Likewise, it has become possible to detect A J = 1, A K-0 rotational transitions of the ground vibrational states for certain nonpolar, spherical-top molecules. These advances, which came after publication of the earlier edition of this book (1970), have led to the precise microwave measurement of the spectral constant Co (or A . for prolate-top rotors) of several symmetric-top molecules and Bo values for some spherical tops such as CH4. For pyramidal XU3 molecules, such as PH3, that have only two structural parameters, these developments have made possible complete determination of the molecular structure of the ground state without isotopic substitution. Examples are given in Section 6, Table 6.8. The possibility of observing forbidden rotational transitions in the ground state of symmetric-top molecules was apparently reported first by Hansen [61]. In the same year, Oka [62] considered collision-induced A K= k 3 transitions of NH3 in mixtures with rare gases. Pure rotational spectra of spherical-top molecules in their ground vibrational states were theoretically predicted by Fox [63] and Watson [64] in 1971. Mizushima and Venkateswarlu [65] had predicted earlier that nonpolar, symmetric-top molecules having Td symmetry (CHI, CF4, SiH3, etc.) can have vibrationally induced dipole moments and possibly observable pure rotational spectra when in excited, degenerate vibrational states. Their theory was extended to other molecular types by Mills et al. [66]. Pure rotational spectra of nonpolar molecules in excited vibrational states would be very difficult to observe because of the relatively low populations of the excited states. So far as we know, none have been observed. The first detection of the forbidden AJ=O, AK= + 3 rotational transitions in symmetric-top molecules was achieved by Chu and Oka [67], who observed A J = O , A K = f 3 ( K = f l + f 2 ) transitionsinPH3, PD3,andAsH3.Thefirst observation of the forbidden rotational transitions in a nonpolar molecule of Tdsymmetry, CH4, was achieved by Rosenberg et al. [68] in far infrared spectroscopy. The first microwave detection of pure rotational of CH4 was accomplished by Holt et al. [69]. An excellent review of the earlier work on forbidden rotational transitional is given by Oka [70]. A general theory of ground-state, forbidden rotational transitions in polyatomic molecules, with predictions of the selection rules and transition probabilities, has been developed by Watson [64]. The theory for ground-state transitions of tetrahedral molecules like CH4 was developed independently by Fox [63,71] and was extended to other molecular types by Aliev and Mikhaylov [72]. Generally, the ground-state transition moment results from centrifugal distortions of the molecules. For those with C3“symmetry, the rotation of the molecule about the b axis produces a small distortion moment perpendicular to the symmetry axis which can give rise to observable A J = k 1, AK= & 3 transitions. In molecules having G symmetry and no permanent dipole moment (CH4, SiH4, etc.), rotation about a bond axis distorts the molecule so as to produce a weak dipole moment along that axis which can give rise to observ-

196


able A J = & 1, AK=O transitions. In addition to the ground-state, centrifugaldistortion mechanism, Watson [64] has shown that vibration-rotation interaction of the ground state with an excited, degenerate vibrational mode can also contribute to the probability of a forbidden transition. Watson [64] defined an effective dipole-moment operator which includes the induced distortion moments as well as any fixed dipole components. The operator for the effective dipole moment is expressed as a power series in the vibrational and rotational operators in a manner similar to that used in Chapter VIII, Section 2 for development of the Wilson-Howard Hamiltonian, (8.10). In the molecule-fixed axes, Watson's effective dipole operator is expressed by pa =psp)

+

e,8ypPp,

(6.43)

PY

where pg) signifies the components of the fixed dipole operator, and the last term signifies the components of the induced dipole operator in which P,, P, are molecule-fixed components of the total angular-momentum operator. The symmetrized, space-fixed components of the operator of (6.43) are

where @Fa are the direction cosines of the molecule-fixed axes with the spacefixed axes. In calculation of the intensities of the spectra lines, Watson used the matrix elements of the effective dipole operator pf of (6.44) in the representation that diagonalizes the Wilson-Howard rotational Hamiltonian, (8.10).

Molecules with C,, Symmetry Molecules having C3"symmetry comprise the largest class of symmetric tops investigated with microwave spectroscopy. Forbidden AK = f3 transitions have been observed in a number of the simpler ones. The A K = 3 selection rule for these not-strictly-forbidden transitions arises from the nature of the molecular wave function for the CJUsymmetry and an intramolecular interaction yi"d such as rotational distortion that mixes the K and K' = K f3 states. Exclusive of the A4 dependency, which we omit for simplicity, the undistorted symmetrictop rotational function may be expressed, (2.103), by @J K = e'+Ke'+K@

Jx

(6.45)

+

where K has both and - values and 4 is the angle about the symmetry axis. For a molecule of CJusymmetry, a rotation of 120" exchanges two pairs of identical nuclei. The two configurations are indistinguishable. Thus, IC/j&J*K is left unchanged for an operation that simply rotates the molecules through f 120" or f2n/3 about the symmetry axis. Under the C, operation , J ~ ~ = + * ~ " W K @J K

(6.46)

If J, the total angular quantum number exclusive of nuclear spin, is assumed to remain a good quantum number, the degree of admixture of the K and K'

"FORBIDDEN" ROTATIONAL TRANSITIONS

197

States is measured by (J, KIxdlJ, K ' ) , where i%?d is the Hamiltonian for the admixing interaction. With the restricted t,hJK=IJK) of (6.46), this can be expressed [70] (6.47) Only when K - K ' = 3n where n is an integer including zero or when the interaction ( J , KlXdlJ, K ' ) is zero, are the two sides of (6.47) equal. Thus, with a finite interaction, the selection rules AK = k 3, k 6, and so on, may occur in addition to the normal AK =O rule. The interaction xd, responsible for admixing of the IJ, K ) and IJ, K + 3 ) levels of the ground vibrational state of symmetric-top rotors having CJO symmetry, is the centrifugal distortion term (6.48) where T~~~~ is the centrifugal distortion constant and P, = P, k iP, are the rotational ladder operators. The IJ, K ) levels of the ground state can also intermix slightly with IJ, K k 2, 1 T 1) levels of excited degenerate vibrational states by means of the vibration-rotation interaction term of the form [70] (6.49) where v, is the frequency of the degenerate vibration, B is the rotational constant, u~:"'=(8ZX,/8Q,,), Qt,, Qt, are the pair of degenerate coordinates, and q+ and P , are the vibrational and rotational operators. Watson [64] shows that the effective dipole-moment operator of (6.44) may be expressed for molecules of D J hor CJOsymmetry by where (6.51) In (6.51)p f ) is the permanent dipole moment, T,,, is the centrifugal distortion constant, and B, and B, are rotational constants. The z coordinate is along the symmetry axis; hence, B, = B and B, = C for the oblate top and A for the prolate top. It should be noted that for molecules of D J hsymmetry, which have no permanent dipole moment, the last term of (6.51) vanishes and (fI?),,=e?. The line strength S is proportional to the squared transition matrix elements of pF. These squared matrix elements of the pF of (6.50) were derived by Watson [64] for molecules of D J hsymmetry, planar XY,-type. The line-strength factors of the allowed transitions (those for which the matrix elements are nonvanishing) are given in Table I of Watson's paper. These nonvanishing matrix elements prove the AK = k 3 selection rules for molecules of D J hand CJOsymmetry. By a unitary transformation of the X dof (6.48) and of the pF expressed by (6.50), he

198


derived the (OF) for molecules of C,, symmetry given by (6.51). Thus, by substitution of (e?),, from (6.51) for OF in the formulas of his Table I, the corresponding line strengths for symmetric-top molecules of C3c.symmetry are obtained. For example, the line strengths for the AJ=O, A K = 3 transitions of phosphine or arsine are

*

Q(J, K-J,

Kf3)=a(O?)i,(JT K ) ( J F K- l)(JT K-2) x ( J f K+ 1)(J+ K + 2 ) ( J i K+3)(2J+ l ) / J ( J + 1)

(6.52)

where (6.53)

and (6.54) is the induced dipole component resulting from the admixture of the excitf 1 degenerate vibrational modes u3 and u4 with the ground vibrational states. Note that Watson [64], also Oka [70], uses k for + and - values of K and defines K=Ikl, whereas we allow K to have + or - values defined by (2.62). Chu and Oka [67] used (6.52) in calculations of the intensities of forbidden (J, K ) = ( J , f l ) + ( J , T2) rotational transitions of PH, for J values ranging from 6 to 17 which they observed in the frequency range of 47.4 to 43.7 GHz. The intensities ranged in strength from 2.1 x lo-' cm-' for J = 6 to 5.5 x cm-' for J=lO and dropped to 1.1 x lo-' cm-' for J=17. Helms and Gordy [73] observed ( J , O)+(J, 3) transitions of PH3 for J values from 3 to 14 which range in frequency from 143.7 to 134.6 GHz. The intensity distribution of the observed lines, as calculated with (6.52), is shown in Fig. 6.11. The maximum intensity, 48.5 x lo-' cm-', occurs for J = 10. Intensities for the same transitions in PD3 are an order of magnitude lower than those for PH3. For ASH,, they are still lower [74], with the maximum intensity for the J-J, K = f 2 - T I transitions only 2 x cm-'. It is obvious that millimeter wave spectrometers of high sensitivity are required for observation of these forbidden spectra. The spectrometer used at Duke for measurement of the A K= f3 transition in phosphine [73] and arsine [74] employed a high-Q, Stark-modulated Fabry-Perot absorption cavity and had an estimated sensitivity of 2 x lo-'' cm-' at 144 GHz for a time constant of 1 sec. Figure 6.12 shows four 75Ashyperfine components of a AJ+O, AK= k 3 transition of ASH, recorded with this spectrometer. For C3"molecules such as phosphine and arsine having a relatively large y, and high degenerate vibrational frequencies, v 3 and v4, the intensities of the A K= k 3 transitions are due primarily to centrifugal distortion in the ground vibrational state. For example, Chu and Oka [67] calculated OF=1.6 x D and the distortion moment, last term of (6.53), to be 8.3 x l o e 5 D. Thus the

“FORBIDDEN” ROTATIONAL TRANSITIONS

PHJ

501

K.0 +3

199

TRANSITIONS

30 Y

I

I

I I ;

I

[ I 1 I

;

“

j

j

-L-L.L

15 16 17

143701.63MHz -1346,

.42MHz

Fig. 6.11 Calculated intensities of the J+J, K=0+3 rotational lines of PH3 in the ground vibrational state. The solid lines indicate observed transitions. From Helms and Gordy [73].

F

F

= 29/2

d

b-

=

31/2

1

10.84 MHz = -

Fig. 6.12 Single scan of the “forbidden” J=14, K = * 5 + k 2 rotational transition of ASH, showing F - t F components of the ”As hyperfine structure. From Helms and Gordy [74].

contribution of the vibration-rotation interaction to the line intensities is only (1.6)2/[(1.6)2+(8.3)2] =3.6%. Oka [70] describes the contribution from centrifugal distortion, which mixes the ( J , K ) and ( J , K & 3) rotational levels of the ground vibrational states, as “rotational intensity borrowing” from the normal A J = 1, A K=O transitions. The contribution of the vibration-rotation interaction, (6.49), which mixes the ground-state rotational levels with those of excited, degenerate vibrational states, he describes as “vibrational intensity borrowing.” In this description, the observed A K= 3 transitions in phosphine

200

SY M M ETRIC-TOP MOLECULES

Next Page

and arsine are due primarily to “rotational intensity borrowing” within the ground vibrational state. In the rigid-rotor approximation, Section 1, the I 4 levels of the symmetrictop molecule are all doubly degenerate except for K=O. In the distortable rotor, the 14 levels, except for 14=3, remain degenerate with E symmetry but are displaced by centrifugal distortion. The K=O and 14=3,6, . . . have symmetry A . In addition to displacement, the I 4 = 3 levels are split into two components with symmetry A , designated as A I and A 2 . These designations indicate the symmetric or antisymmetric classification of the coordinate wave function (exclusive of nuclear spin) under exchange of two identical atoms. The selection rules A H A and E-E conform to the A K= i3 rule. Within the transitions for the split K = 3 levels, Al-A2 are allowed. The signs + and - are used for designation of the symmetry of the wave function with respect to inversion of the coordinates (Chapter 111, Section 4). In addition to the K= f3 rule, the transitions must comply with the parity selection rules, sym-sym and antisym -antisym, for the overall wave function including nuclear spin, as described in Chapter 111, Section 4. The effects of nuclear-spin statistics on the intensities of the transitions for symmetric-top molecules having C g Usymmetry are given in Table 3.3. In measurements on PH,, PD,, and ASH,, the inversion levels were not separated, and hence the weights given at the bottom of Table 3.3 apply. In addition to the determination of the moments of inertia about the symmetry axis, measurements of the forbidden transitions in C3u symmetric-top molecules have provided considerable new information about the centrifugal distortion of the molecule in the ground vibrational state including the splitting of the K= 3 levels. Because the transitions can be measured over a wide range of J values (see Fig. 6.1 l), accurate evaluations of the centrifugal distortion constants can be made for relatively light symmetric tops such as PH, without need for measurement of very high frequencies extending into the submillimeter and infrared regions. The most extensive measurements of forbidden spectra to date have been made on PH,, PD3, and ASH,. Vibration-rotation transitions, including A(K-I)= f3, have been measured by Maki et al. [75] for PH, and by Olsen et al. [76] for ASH,. The first measurements of ground state, A K = f 3 , transitions are those by Chu and Oka [67] the J - J , K= 5-2- T1 transitions of PH,, PD,, and ASH,. Helms and Gordy [73, 741 extended these measurements to include: J-J, K=3-0 in PH,; J - J , K=3-0, and K = i 4 + + 1 in PD3; J-J, K = + 4 + f l , and K = f 5 - + 2 in ASH,. Each K transition was measured for several J values. Helms and Gordy combined the measured frequencies of all these forbidden transitions with those measured earlier [77, 781 for the normal A J = l , AK=O of these molecules in a computer analysis to obtain the values of their rotational constants, listed in Table 6.4. The molecular structures obtained from this analysis are given in Table 6.8. The analysis of the phosphine and arsine data consists of a least-squares computer fitting of the involved spectral parameters to the observed frequencies. The computer program was made to include all frequencies of the different types of transitions measured. It was designed to compute the differences in the

Previous Page

201


Table 6.4 Ground-state Spectral Constants4 of PH,, PD3, and ASH, Constant

PH3

PD3

ASH,

69,47l.lO(3) 58,974.37(5) 1.021(2) - 1.312(1) 1.023(2) 0.221(5)

112,470.597(30) 104,884.665(43) 2.9257(39) - 3.7164(4) 3.4126(21) 0.679(3) 1.990(47) - 0.239(9)

- 0.091(4)

9.41(22) - 1.729(67)

0.044(5)

0.701(11)

“These constants were calculated by Helms and Gordy [73, 741 from J-J, AK = 3 transitions measured by them and by Chu and Oka [67] and from normal A J = 1, AK =O transitions measured by Helminger et al. [77,78].

energy levels involved in these transitions with assumed values of the spectral parameters in the appropriate energy equations. Transition frequencies were thus automatically computed and compared with the measured frequencies to generate new estimates of the spectral constants. This iterative process was repeated by the computer until convergence was obtained. The degree of fitting of the composite set of frequencies determined the estimated errors (standard deviations) indicated in Table 6.4. The energy equations for the K = 3 levels involved in the observed K=0-3 transitions of PH, and PD3 include the term f h o J ( J + 1)[ J(J+ 1)- 2][J(J+ 1)-61, first derived by Nielsen and Dennison [79] in their calculation of the anomalous effects of the K = 3 lines in the ammonia inversion spectrum observed by early microwave spectroscopists. The ASH, energy equations include the nuclear hyperfine interactions described in Chapter IX. The energy expressions for all transitions included the pure rotational energy with the centrifugal distortion terms expressed in the form

+

+

E(J, K)/h = B o [ J ( J + 1)- K2] CoK2- DJJ2(J -DjKJ(J+ l)KZ-DKK4 HjJKJ2(J+ 1)’K2 +H,,J(J + 1 ) ~+4 L ~ ~ J +~1(1 J2 ~ 4 + H K K 6+ LJJKJ3(J+l),K2

+

(6.55)

Complete expressions for the energy terms for both K k 3 and K = 3 levels are given in the papers cited. See also Amat et al. [22]. Recently (1981) Belov et al. [SO] measured the PH, rotational spectrum in the region of 300 to 1070 GHz. In addition to the normal rotational transitions, AJ=l, AK =0, they measured the J-J, jK( =2+5 transitions which occur in the region of 310 to 334 GHz for J values ranging from 6 to 16. They

202


Table 6.5

Ground-state Spectral Constants‘ of 16031P19F3

Parameter

Value

A,, (MHz) Bo (MHz) DJ (kHz1 DJK(kHz1 DK (kHz1

4811.7579(18) 4594.2624(5) 1.0119(12) 1.2971(7) - 1.114(12)

Parameter

Hj HJK HKj

HK

Hz) Hz) Hz) Hz)

Value

- 6.7(10) 9.10(27) -648) -73(30)

“From Kagann et al. [82].

reanalyzed the previously measured millimeter wave frequencies together with their extended submillimeter measurements to obtain a new set of parameter values. These higher-frequency data made possible an improvement in the centrifugal distortion constants, particularly those of higher order. The symmetry selection rules permit direct transitions between the two components of the split K = 3 levels. The transition frequencies between the two components in PH3 have been measured with molecular beam resonance techniques for IK1=3, J = 3 to 9 by Davies et a]. [81]. These frequencies were included in the analyses of Belov et al. [go]. Belov et al. [59] have also used the Krupnov spectrometer to measure some AK = k 3 transitions in the rotation-inversion spectrum of NH3 in the excited u2 vibrational state as well as the u2 =O ground state. One of the ground-state lines, the J=lO, K=0+3 is shown in the center of Fig. 6.10. The theory used for analysis of these transitions, designated by them as “perturbation allowed,” is that developed by PapouSek et al. [59]. Ground-state, forbidden AJ=O, AK= f 3 transitions of OPF3, an X Y Z 3 type of prolate, symmetric-top molecule, have been observed by Kagann et al. [82]. A total of 152 lines of 16031P19F3 were measured in the frequency range of 9 to 18 GHz. These included six K transitions in the range from K=9+6 to 1 5 ~ 1 2each , for a wide range of high J values. For example, the K = 1l+8 transitions were measured for 31 J values ranging from 52 to 82. Thus the highorder distortion constants could be accurately obtained from the data analysis. As an aid to the assignment of the J values for the different K transitions, they included in their computer program a reanalysis of the normal J+J+ 1 transition (R-branch) measured earlier [83]. The various rotational and distortion constants obtained in the analysis are recorded in Table 6.5. The molecular structural parameters are listed in Table 6.9. Section 6.

Avoided-crossing, Molecular Beam Resonance Method for Measurement of K-level Separations Ozier and Meerts [84-871 have developed an effective method for measurement of the separation of K levels between which normal transitions are forbidden. The method is capable of measurement of symmetric-top energy-level separations corresponding to AK = k 1 and + 2 transitions as well as to the

"FORBIDDEN" ROTATIONAL TRANSITIONS

203

AK = 3 transitions discussed in Chapter 11, Section 3. In molecular-beam, electric resonance experiments a strong, uniform electric d field is applied to bring selected Stark components of the upper and lower K levels into collision. The Stark shift for the upper-level component must, of course, be negative and that for the lower be positive. In the vicinity of the crossing, or anticrossing, region, the wave functions of the upper and lower levels are strongly mixed. This admixing is similar to that described in Chapter X, Section 5 for the mixing by an imposed Stark field of the wave functions of the I-doublet states of a linear molecule in a degenerate bending vibrational state, or those of the inversion doublet states of NH3.It is also similar to the admixing of near-degenerate pairs of levels in slightly asymmetric rotors described in Chapter X, Section 3. As in other cases of near-degeneracy, the two colliding levels appear to repel, or to avoid, crossing. This effect is demonstrated by Fig. 6.13, where the M J = +_ 1 Stark components of J = 1, K = & 1 and J = 1, K =O states of CF3H are brought

Fig. 6.13 Schematic plot against the electric field of the energy levels in the J = 1 state of fluoroform. The nuclear magnetic quantum numbers and the hyperfine effects are not shown. The circle indicates the anticrossing region. For clarity, the curvature of the levels in this region and the quadratic Stark effect of the K = O state have been greatly exaggerated. From Meerts and Ozier [86].

204

S Y M M ETR I C-TOP MOLECULES

into proximity by an increasing Stark field. In the encircled “anticrossing” region, the IJ, K , MJ)=[1, f 1, T1) and Il,O, f 1) states are strongly mixed. As a result, M J components are reversed, giving rise to a detectable change in the strength of the molecular beam resonance signal. The admixing causes changes in the symmetry of the rotational wave functions and, in some instances, changes to ortho-para conversions in the nuclear spin states. If the effects of centrifugal distortion are neglected, the separation of the IJ, K =0) and IJ, K = f 1) states is (Bo- C,) for the oblate symmetric top CF3H, as indicated on the diagram of Fig. 6.13. From the diagram it is evident that the sum of the Stark energies required to bring the two MJ components to the critical anticrossing region is equal to (Bo- Co).The critical field 8,for the anticrossing signal can be accurately measured; from the known dipole moment p9, these Stark displacement energies can be calculated. Thus the value of (Bo- Co) is obtained. By inclusion of the centrifugal distortion terms and the measurement of additional transitions, the K-dependent distortion constants can be obtained. Obviously, the presence of nuclear hyperfine structure further complicates the solution, but it does not make the method unworkable. One then observes the critical anticrossing regions of Stark components of certain hyperfine levels of the two K states and proceeds to calculate the sum of the upper and lower level Stark displacement energies at the critical field 8,. One must also correct for the zero-field displacement of the hyperfine components employed. Hyperfine structure in symmetric-top molecules is treated in Chapter IX. The Stark effect of symmetric-top molecules with and without nuclear hyperfine structure is treated in Chapter X. For the case illustrated in Fig. 6.13, the Stark effect in the upper ( K =0) level is a second-order effect whereas that for the lower ( K = 1) level is a first-order effect. There is no nuclear quadrupole hyperfine structure for CF3H, but the nuclear magnetic hyperfine structure of the F and H caused a broadening of the signal of the order of 20 kHz. The avoided-crossing method is simplest and most accurate for symmetric tops for which (Ao-Bo) or (Bo-Co) is relatively small and the permanent dipole moment is large. The molecule OPF3, to which the method was first applied [84], fits these criteria well with (Ao-Bo)=217.495 MHz and po= 1.868 D. Because the crossing fields gCare not excessively high, the first-order Stark effect can be applied with good accuracy. The J-J first-order, Stark-component displacements for molecules in which nuclear splittings are negligible are given by (10.3), and the E(J, K ) rotational energies of a prolate top are given by (6.14). If 8,is the crossing field for two Stark components of levels differing by 3 in K which are undergoing avoided crossing, the energies corresponding to the transitions J-J, K+ K 3, as given by the first-order Stark effect, are

+

AEr(J+J, K+K

+ 3)= [(A0 - Bo)-Dj,J(J+

+

1)-DK(2KZ 6K +9)](2K

+3 ) (6.56)

where K is algebraically the smaller of the K values for the levels involved.

“FORB IDD E N” ROTATIONAL TRANS IT10 N S

205

Table 6.6 Rotational Constants A . and Co from Avoidedcrossing, MBER Measurements of “Forbidden” K Transitions Molecule

Bo’ ( M H z )

CF3H CF3 D CFjCHj

10,348.867(2)’ 9,921.126(2) 5,185.1387(24p

A o b (MHZ)

cob( M H z ) 5,673.46(10)d 5,673.21(10)‘

5,498.85(7)’

”From normal, J-J+ 1 rotational spectra. bObtained from ( A o - B o ) and (Bo- C,) measured with avoidedcrossing, MBER method and Bo values from normal spectroscopy. ‘From T. E. Sullivan and L. Frankel, J. Mol. Spectrosc.,39, 185 (1971). dFrom Meerts and Ozier [86]. ‘From Meerts and Ozier [85].

Derivations of similar first-order equations for AK = 1, or 2, will be obvious. If 8, for sufficient J, K levels are measured, the (Ao-Bo), DjK, and DK can be obtained from (6.56). However, DjK can be measured more accurately from the normal, A J = 1, AK =O transition. With Bo and DjK from the normal spectrum, the constants DK and A . (or Co for the oblate top) are easily evaluated from the 8,measurements. Meerts and Ozier [86] estimated the absolute accuracy of (Ao-Bo) or (Bo- Co), as determined mainly by the long-range stability and reproduction of the source voltage, to be 0.002%. For determination of the value of (Ao-Bo) for OPF3, Ozier and Meerts [84, 861 measured 8,for the K = T 1- + _ 2crossing with M J = J for levels of J = 2 to 6. They employed DjK and DK values from the distortion-moment spectra observed by Kagann et al. [82] and obtained (Ao-B0)=217.494(4) MHz, in good agreement with the value of 217.495(2) MHz derived from the distortion moment spectroscopy. The resulting A . value is 431 1.756(5)MHz. Values of A . or Co obtained for other molecules by this method are listed in Table 6.6. In the avoided-crossing experiments [85] on CH3CF3, internal rotational splittings were also observed for the levels corresponding to f 1 in the torsional quantum number 0,as well as for those corresponding to changes in K. From measurements of these splittings, the height of the barrier to internal rotation of the CH3 group relative to the CF3 was found to be V3=3.16(11) kcal/mole. The value of 3.480 kcal/mole (Table 12.7) was found from microwave spectroscopy. In addition, the moment of inertia of the C H 3 group relative to the symmetry axis, Z, =3.17(11) amu A’, was derived. This experiment demonstrates a new method of obtaining these constants.

Tetrahedral, Spherical-top Molecules Although spherical-top molecules, strictly speaking, are not symmetric tops, they have many properties in common with symmetric tops. They are the limiting case of a symmetric top as the moment of inertia about the symmetry axis

206


approaches Zb.Because they have no permanent dipole moment, spherical-top molecules have no normal rotational spectra. However, as predicted by Fox [63,71], Watson [64], and others [72], in the ground vibrational state they can have weak, but observable, pure rotational spectra resulting from distortion dipole moments. The moments, which are similar to the e? moment of (6.54), arise from the vibration-rotation interaction of the ground-state rotational levels with those of excited degenerate vibrational states. Terms like the last one of (6.53) make no contribution to the distortion moment because #=O. Consequently, the line strength results entirely from the “vibrational intensity borrowing” [70]. A distortion dipole moment is induced when the tetrahedral molecule rotates about a C3 axis; it is zero for rotation about an S4axis. In contrast, the centrifugal distortion energy is greatest when the rotation is about the S4 axis and least when it is about the C3 axis [70]. Thus the distortion moment spectrum results from rotation about a C3 axis and resembles that of a C3t:symmetric top. As for other molecules, the selection rules depend on the overall symmetry of the wave functions, including nuclear spin functions (Chapter 111, Section 4). The tetrahedral molecules, such as CH,, SiH,, and so on, belong to the point group & with symmetry A , E, F. There are two types of nondegenerate A wave functions, A l and A 2 ,one doubly degenerate species E, and two triply degenerate species, Fl and F2. The selection rules for the distortion moment spectra are

AM=O, f 1

AJ=O, f l , A1-A2,

E-E,

F1eF2

The nuclear spin functions have symmetries determined by the magnitude of the resultant spin: 7i= ZI+ Z2 + Z3 + Z, =2, 1,0, for spins off as in CH,. Those for F = 2 have symmetry A l ; for F=O, symmetry E ; and for F = 1, symmetry F2. The line-strength factor for rotational distortion spectra corresponding to the J+J+ 1 transitions of spherical tops as calculated by Watson [64] (neglecting effects of nuclear statistical weights), is

lk(J-J+ 1 ) = t (6:y)2J(J+2)(2J- 1)(2J+ 1)(2J+ 3)(2J+ 5 ) 35

(6.57)

D from the vibration For CH,, Watson calculated e:y=(2.6 or 1.2) x frequencies v 3 and v4 and vibrational intensities measured by Mills [88]. The distortion moment for the ground vibrational state of CH, has been measured by Ozier [89] with molecular beam resonance techniques. The value obtained D=(20)-”28:ygive~B:Y=2.41 x D, ingoodagreement forp,=5.38 x with the upper value estimated by Watson [64]. A rigid spherical-top molecule has only one value for the moment of inertia, and its pure rotational spectrum-if it could be observed-would be like that for a rigid linear molecule (Fig. 4.1) with rotational frequencies, v,=2B(J+ 1).


207

Centrifugal distortion is necessary, however, to produce the dipole moment making possible the detection of the lines. Thus the detected lines must include distortion effects in their frequencies. These effects can be taken into account by the inclusion of scalar distortion terms similar to those of a linear molecule. The frequency equation is v.,+.,+’ =2Bo(J+ l)-4Ds(5+ l ) 3 + H , ( J + 1)3[(5+2)3-53]

(6.58)

Rosenberg et al. [68] found that they could fit the lines they observed for 12CH4to (6.58) with the constants Bo=5.245(4) cm-’ and Ds=1.19(9) x IO-, cm-’, and H,=O. Ten J+J+ 1 transitions were observed in the region of 20 to 200 cm-’ for J = 7 to 16. Equation 6.57 was also used in the analysis of the observed J+J+ 1 frequency of SiH4 [90] and GeH, [91]. Despite the good fitting of the R-branch transitions achievable with (6.58), the scalar distortion constants thus obtained do not give a measure of the complete molecular distortion of the tetrahedral molecules, as it does for linear molecules. The distortion of the spherical top is more nearly like that of the symmetric top, for which the centrifugal distortion is not completely determined by the normal J+J+ 1, AK =O transition, but requires also the measurement of the AK # O transitions for determination of the centrifugal distortion constant DK.The spherical top is like the symmetric top in having, under centrifugal distortion, an observable Q-branch, AJ=O, spectrum. These Q-branch transitions are, however, more complicated than the “forbidden” Q-branch transitions of a symmetric top, described in Chapter 11, Section 3. They result not only from distortion dipole moments but also from centrifugal distortion effects on the inertial moments. For example, if the effects of distortion on the moments of inertia are omitted, the Q-branch, AJ=O, AK=3, frequency of the prolate symmetric top would be (6K + 9)(Ao-Bo). In the limiting case of a rigid spherical top, A . =Bo( = Co),these frequencies vanish. Centrifugal distortion caused by rotation about the b axis can, however, destroy the equality, causing A . (or Co)to differ slightly from Bo and thus producing a “forbidden” Q-branch spectrum analogous to that of a symmetric top. Because the inequality of the inertial moments caused by centrifugal distortion is not large, the Q-branch frequencies fall in the lower microwave or radiofrequency region. They have been measured directly with microwave spectroscopy by Ozier, Gerry, and their associates for CH4 [69, 92, 931, for SiH4 [94], and for GeH, [95]: they have been measured with microwave-infrared, double-resonance techniques by Curl et al. [96,97] for CH,, and by Kreiner and Oka [98] for SiH,. Because the A J = O (Q-branch) spectra depend entirely on differences in moments of inertia and on transition dipole moments caused by centrifugal distortion, the spectra are rather complex. As shown in the early theory by Hecht [99], and in later developments by Fox [71], the rotational energy of a tetrahedral molecule consists of a scalar term E, and a tensor term ET Em= E,

+

ET

(6.59)

208


The scalar term is like that for linear molecules, (5.13). The tensor term results entirely from distortion. Expressions for both E , and ET are given by Kirschner and Watson [IOO], Ozier [ l o l l , and others [94]. Values for the tensor distortion constants are obtained only from measurements of the Q-branch transitions, whereas the scalar energies as well as the scalar distortion constants are obtained from the A J + J + 1 transitions. However, it is evident from (6.59) that a precise evaluation of the rotational energies E,,,, of Bo, and the scalar distortion constants depends on a knowledge of the tensor distortion energies ET. Thus accurate measurements of both R-branch and Q-branch transitions, with a combined analysis of the frequencies of both, are required for precise values of the rotational constants. Such an analysis has been made for SiH, by Pierre et al. [I021 and improved with more accurate Q-branch frequencies by Ozier et al. [94]. Their final values for '%iH4 are: Bo=2.859065(10) cm-', D,=3.690(13) x cm-', and H,=2.14(48) x lo-' cm-'. Tabulation of the measured tensor distortion constants and Bo values for 12CH4, 28SiH4, and GeH, is given by Kagann et al. [95] (see their Table 7). For "CH,, Bo=5.241036(10) cm-' was obtained by Tarrago et al. [103].

5

LINE INTENSITIES

The absorption coefficients for microwave rotational lines of symmetricfrom top molecules can be found from (3.24), or the peak absorption ct, (3.27) by substitution of the appropriate values of F, and of the dipole moment matrix elements. The values of F m = FJ,K may be obtained from (3.66) by substitution of Qr from (3.68).The dipole moment matrix elements corresponding to the rotational absorption line transitions J + J + l , K + K are given by (2.124). When these expressions, along with the values of N from (3.25), are substituted in (3.24) with numerical values for the various constants, the resulting expressions for the absorption coefficient becomes a, =

11.58 x 1 0 - ' 2 F , i c p m m ~ 2 a g I g K B ~ v 2 T712 ( I - 24 T

1

Av (J+1)2-K2 x[(v-v0)2+(Av)21[ (J+1) - 48 .X 10-6[BJ(J+ 1)+ ( A - B)K '1 XeXP T

{

I

(6.60)

In this expression the frequencies v , Av and spectral constants A and B are expressed in MHz: ct,, in cm-'. Here and in later expressions it is assumed that the effects of centrifugal distortion cause the different K lines to be resolved, but these effects are otherwise neglected. Microwave spectroscopists are usually interested in the absorption coefficient amax at the center of the resonance corresponding to the frequency v = v o , for which the maximum absorption occurs. The lines of gases are most often

LINE INTENSITIES

209

observed under conditions such that the peak intensity is independent of the pressure (Chapter 111, see Section 2). Under these conditions we can set Av= 3 0 0 (Av),/T, ~ ~ (3.26).In the numerical calculations of E,,, it is more convenient to express the resonant frequency v o and the spectral constants A , B, or C in GHz units (1 GHz=lOOO MHz) rather than in MHz. Because of its smaller value, we shall leave (Av), in MHz units. With v0=2B(J+ l), the formula for the peak absorption can be expressed

[-

0.048(A - W K ' ] T

(6.62)

where

X=

6.11 X 10-4F,i,/120g~g~fi1'2 (Av), T5I2

(6.63)

and where vo, A and B are in GHz and ( A V ) ~is in MHz. When Jvo 4 T and ( A- B)K 4 as is true for relatively low J transitions, the simpler expression

'

(6.64) is often adequate. In these expressions F, =fraction of molecules in the particular vibrational state observed; see (3.61). i, = fractional concentration of the isotopic species observed T = temperature of observation in absolute scale p =molecular dipole moment in debye units v o =resonant frequency in GHz units ( A V )=line ~ breadth in MHz for p = 1 mm of Hg pressure when T = 300°K A =spectral constant in GHz units for rotation about symmetry axis. This constant becomes C for a prolate top B=spectral constant in GHz units for rotation about axis perpendicular to symmetry axis g K = l for K=O = 2 for K>O o =symmetry number described in Chapter 111, Section 4 gr = reduced nuclear statistical weight factor described in Chapter 111, Section 4

For molecules in symmetric ground electronic states having CJvsymmetry with no resolvable inversion splitting, such as CH3F or CH,CCH, which have

210


one set of identical off-axis nuclei with nuclear spin 1, 0=3 for K =0, 3, 6, . . .

(6.65)

for K not divisible by 3

(6.66)

and

For molecules of C3 symmetry in symmetric ground electronic states having two sets of three identical off-axis nuclei with spin Zl and Z2, such as CH3SiF3, 0=3 (211

+

1)2(212

+

for K =0, 3, 6, . . .

(6.67)

for K not divisible by 3

(6.68)

1)2

and (211

+

1)2(212

+

1)2

The use of these values for gr assumes that there is no resolvable inversion doubling. Such doubling has been found for ammonia only. From examination of the foregoing expressions it is evident that agI 21 when Z > f and for two sets of identical nuclei even when ZI= I2=f.For the latter case, with K divisible by 3, a g r = j z l ; for K not divisible by 3, agI= = 1. Thus, to the approximation normally desired in intensity calculations, one can set 091 = 1, or simply drop the factor ogr from the equations except for the important class of molecules with one set of three identical nuclei with Z=i. For these, agI=2 when K =0, 3,9, , . ., and 0gr=? when K is not divisible by 3. For these, the weights of the lines for the first group are twice those for the latter group. Note that when Z=O, gr=O for K not divisible by 3. Hence, only the lines for K =0, 3, 6, . . . occur. In computations of the absorption coefficients it is convenient to calculate those for the K =O lines first and multiply the coefficient for the K = O line by the K-dependent factor to obtain the amaxfor the other K components. Thus the formula can be expressed, from (6.61) as

(

&,,,(K =O)= X V ~1-

o'oro) ~

e-0.024Jv~/T

(6.69)

It is evident from comparison of (6.69) with (4.82) that the intensities of the K = O lines of the symmetric top vary with frequency in the same manner as do those for diatomic and linear polyatomic molecules. For J n o t too large, the amaxfor the K =O lines increases approximately with the cube of the frequency.

LINE INTENSITIES

211

The K lines for a given J transition decrease with increasing K if effects of g I and g K are neglected. As an illustration, we shall calculate the peak absorption coefficients for the microwave rotational lines of CH3CCH, which is a rather typical organic, symmetric-top molecule. For 12CH312C'2CH, the natural isotopic concentration i,=0.966, p=0.75 debye, Bo=8.545 GHz, A=153 GHz, a = 3 , g r = 4 for K =0,3,6,. . ., and gr=b for K = 1,2,4,5,7,8,. . .. Although the line breadth parameter is not accurately known and is not exactly independent of J and K, we assume for an approximate calculation that (Av)l = 10 MHz. We choose the temperature of dry ice, T=195"K, the one at which measurements on this molecule are commonly made. The molecule has five nondegenerate, symmetrical modes of vibration and five doubly degenerate modes. However, in calculating F, we need to consider only the two lowest modes. Other modes have negligible populations at T = 195°K. The C-CzC bending mode [I041 cola equals 336 cm-', and the C=C-H bending mode [I041 o9equals 643 cm- '. Both are doubly degenerate. With these frequencies substituted into (3.61) with u = O for the ground vibrational state,

~,=~=n

(l-,-hWkTdi

1'

(6.71)

i

F,=o(CH3CCH)=(1-e-1.44(336)/195)2(1-,-1.44(643)/195

12 . . . =0.83

(6.72)

With these various constants, X=9.68 x 10- lo. When centrifugal distortion effects are negligible, vo =2B(J+ 1) for the frequency of the K =O lines and the absorption coefficient for the K=O lines expressed in terms of J , the lower rotational quantum number for the transition, is ~t,,,(~=0)=4.83 x 1 0 - ~ ( ~ + 1 ) ~ [ 1 - 2 . 1x0I O - ~ ( J + I ) ],

- 2 . 1 o ~ ~ ~ - ~ ~ ( ~ + ~

(6.73) =0) as a function of J for all the significantly Figure 6.14 shows a plot of amax(K populated K=O lines of 12CH312C12CH.The coefficients for the lines with K f 0 can be obtained from these with the relation

(6.74) where g j =ffor K values divisible by 3 and gr =$ for K values not divisible by 3. In the inset of Fig. 6.14 we have plotted the K lines for the J= 10- 11 transition. At 195°K the strongest rotational lines of methyl acetylene occur in the wavelength region of 0.65 mm. The absorption coefficient of the K = O line of the J=26+27 transition which occurs in the optimum region is 0.02 cm-'. This indicates that a one-meter cell filled with the gas under the assumed pressure range would absorb (1 - F 3 ) = 8 6 % of the energy at the peak or resonant frequency. The strongest component of this transition is the K = 3 lines which has ama,=0.03 cm-'; it would absorb 95% of the energy with a cell one meter in length.

212

SYMMETRIC-TOP MOLECULES 0020 -

0.018 0.016

I

CH,CCH T = 195" K

-

0.014 -

E

2

0012

.-t

2 E

U

-

0.010 -

0006 0.008

0.004 0.002 -

7

700,

0

857

vo

in GHz

10+11

K-lines of J

-360

,

M

H

H

Fig. 6.14 Plots of the peak absorption coefficients of the K=O lines of different J transitions for the symmetric-top molecule CH3CCH. The inset shows the variation of intensity among the lines of different K for a particular J transition. The lines are assumed to be pressure broadened within the range in which the line width is directly proportional to pressure.

Optimum Regions for Detection of Symmetric-top Rotational Spectra One can predict the spectral region where the rotational absorption lines of symmetric-top molecules are strongest by equating the derivative of tx,,,,, with respect to Jequal to zero and by solving for Jopt, just as for linear molecules. Since we are interested only in locating the region where the strongest lines occur, we can simplify the problem by setting K =O. The problem thus reduces to that for the linear molecule, and we can employ (5.43) or (5.441, already derived in Chapter V, Section 4. By substitution of Jopt from (5.43) into (5.44) one obtains vopt (in GHz) = 2B

+ 11(BT)'I2

(6.75)

MOLECULAR STRUCTURES 20

-

213

T = 300’

- , cCH,HgI

-

-(CH,),CCCC13’ -

-

-

c73F

1

1

1

1

l

1

l

l

l

l

l

~

~

1

11

l1

1~

1

CH,F-

‘*

1

1

’

1

where B is in GHz and T is the absolute temperature on the Kelvin scale. The optimum wavelength region in millimeters is Aopt(inmm) =

300 2B+ ll(BT)’’’

(6.76)

For convenience we give in Figure 6.15 a plot of the wavelength in millimeters for optimum absorption as a function of B in GHz at a temperature of 300°K. The optimum regions for a few selected molecules are indicated on the chart. It is evident from this chart that the region of strongest absorption for practically all symmetric-top molecules falls between 0.4 and 2 mm wavelength. This is also true for diatomic or linear polyatomic molecules.

6


Ground -state Structures Ground-state structures for a very large number of symmetric-top molecules have been found from microwave spectroscopy. Listings of the structural parameters of those reported in the literature up to about 1969 may be found in Tables A.VIII.3 and A.VIII.4 of the 1970 edition of this book. A complete listing of the structures derived to a later date, 1982, may be found in the pertinent volumes of the Landolt-Bornstein tables [105]. Here, we give only a few examples of the many symmetric-top molecular structures found from microwave spectroscopy. Methods for their calculation from the observed spectral constants are described in Chapter XIII.

214


Probably the most complete information on ground-state structure is available for the group V hydrides: NH,, PH,, ASH,, and SbH,. These pyramidal molecules have only two independent structural parameters; hence only two independent rotational constants are required for their solution. Extensive measurements of rotational transitions of several isotopic species of ammonia and stibene in their ground vibrational states have made possible different kinds of solutions for the various structural parameters listed in Table 6.7. The effective ground-state structures, yo, are obtainable from only two isotopic forms whereas substitution structures, Y,, require three isotopic forms. Descriptions of both ro and r, structures with methods for their calculation are given in Chapter XIII. Because of the relatively large zero-point vibrations of the light hydrogen, a substitution of D for H is expected to cause a much greater degree of nonconformity of the structures than does isotopic substitution of the heavier central atom. Thus the effective structures for NH, and ND, and for SbH, and SbD3 listed in Table 6.7 provide an indication of the degree of nonconformity between the normal and the completely deuterated structures. Note that for both molecules the effective bond lengths in the deuterated species are slightly shorter than those in the corresponding hydrogen species. In Table 6.7 are also given substitution structures Y, for six different isotopic combinations of ammonia, which include tritiated as well as deuterated species. These structures Table 6.7 Ground-state Structures of Ammonia and of Stibene

Isotopic Combinations

Bond Length (A)

Bond Angle

Ref:

Effective Structures of Ammonia 1.0156 1.0143

14NH3,"NH3 14ND3,"ND3

107" 17' 107" 0 4

a (I

Substitution Structures of Ammonia l4NH3:l5NH3,l4ND3 15NH3:14NH3,15ND3 14ND3:15ND3,14NH3 15ND3:14ND3,"NH3 I4NH3:"NH,, 14NT3 14ND3:15ND3,14NT3

1.0138 1.0138 1.0136 1.0137 1.0132 1.0128

107" 14' 107" 14' 107" 04' 107" 04 107" 13' 107" 02'

a (1

0

a

b

b

Effective Structures of Stibene 'SbH3, lZ3SbH3 "'SbD3, IZ3SbD3 lZ

1.7102 1.7075

91" 42' 91" 42' ~

C

c ~~

"P. Helminger, F. C . De Lucia, and W. Gordy, J . Mol. Specrrosc., 39,94 (1971). bP. Helminger, F. C. De Lucia, W. Gordy, H. W. Morgan, and P. A. Staats,

Phys. Rev., A9, 12 (1974). 'Helminger et al. [78].


215

are remarkably consistent, and the bond lengths are slightly shorter than those for the ro structures. Note that the bond lengths decrease slightly as the combination used includes more of the heavier hydrogen species. The structures of phosphine and arsine, shown in Table 6.8, are among the few symmetric-top structures that have been derived without the aid of isotopic substitution. These derivations were made possible by observation of the moment of inertia about the symmetry axis from “forbidden” A K = + 3 transitions, as described in Section 4.With only two structural parameters, a complete structural solution is possible when Co (or A,, for a prolate top) as well as Bo is known. The effective ground-state structures derived from Co and Bo for PH3, PD3, and AsH3 are recorded in Table 6.8. As for ammonia, the bond length of the deuterated species is slightly shorter than that of the normal species. This slight shortening, 0.0024 8, for ro(PD), is expected from the lower vibrational energy of the deuterium. The effect of the lower vibrational energy on the bond angle is more difficult to assess, but the measured bond angle is slightly greater in PD3 than in PH3. Equilibrium structures have also been derived for PH3 and PD3. As expected, no measurable differences could be detected in the re structures of the PH3 and PD3. These re structures are given for comparison at the bottom of Table 6.8. Note that the equilibrium dimensions are slightly smaller than those of the effective structures.

Table 6.8 Structures of Phosphine and Arsine Derived from Bo and Co without Isotopic Substitution

Species

Bond Length (A)

Bond Angle (”)

Ref:

Effective Ground-state Structure

1.42OO 1.4176 1.52014

93.345 93.359 91.9758

Average Ground-state Structure

PH3 PD3 ASH,

1.4270(2) 1 .‘4227( 1) 1.5275(3)

93.229(5) 93.257(4) 91.93(5)

0,c

a,c

e

Equilibrium Structure

1.4116(6) 1.4116(6) 1.51 l(2) “Helms and Gordy [73]. bHelms and Gordy [74]. ‘Chu and Oka [67].

93.33(2) 93.33(2) 91.94(20)

0.c

a,c E

216


Table 6.9 Effective Ground-state Structures of OPF3, OPCI3, and OVC13 ro(0-V

0XY3

(A)

OPF3 OPC13 OVC13

1.437(4) 1.455(5) 1.595(5)

ro(OY)

(A)

L(YXY) Wg.1

1.5222(4) 1.989(2) 2.131(1)

101.14(10) 103.7(2) 111.8(2)

Re$ LI

b c

"Kagann et al. [SZ]. S . Li, M. M. Chen, and J. R. Durag, J. Mol. Struct., 14, 261 (1972). 'K. Karakida, K. Kuchitau, and M. Matsumura, Chem. Lett., 273 (1972).

The very interesting organometallic molecule C5H,BeCI (cyclopentabenzylberyllium) has been investigated with both microwave spectroscopy and electron diffraction. The microwave spectrum, observed by Bjsrseth et al. [lo61 conforms to that of a symmetric top having %," symmetry. Through isotopic substitution these observers were able to obtain a complete ro structure and a partial r, structure. The probable nature of the bonding of the Be to the cyclopental ring is described by Drew and Haaland [107]. Although the ground-state structures of the methyl, silyl, and germanyl halides were obtained in the early years of microwave spectroscopy, spectral measurements on the chemically unstable stannic halides, SnH, X,were not achieved until 1971. These measurements, by Wolf, Krisher, Gsell, et al. [108-1101, led to accurate values of the substitutional structures and nuclear quadrupole coupling, which provide information about the nature of the Sn-Hal bonds. Remeasurement of some of the germanyl halides [l l l-1131 has provided more accurate values of the ground-state structures of these halides. The complete ground-state structure ro of vanadyl chloride, OVCI,, has been obtained by Karakida et al. [1141 from their microwave measurements of its three isotopic forms. This structure is compared with that obtained by Li et al. [115] for the similar molecule 0 P C l 3 in Table 6.9. The newly prepared, reactive molecule CH3CP has been studied by Kroto et al. [116] with isotopic substitution. The r, structure of CH3CP was found to be: r,(CH)= 1.108(1), r,(CC)= 1.465(3),r,(CP)= 1.544(4), L,(HCC)= 110.30(9)o.

Equilibrium Structures The determination of the equilibrium constant Be =h/(87r21,) requires measurements on rotational lines for molecules in all the different fundamental vibrational modes. Symmetric-top molecules with n atoms have 3n - 6 fundamental vibrational modes, some of which are doubly degenerate. The

217


pyramidal AB3 molecules, the polar symmetric-top molecules with the fewest number of atoms, have six fundamental vibrational modes, two of which are doubly degenerate. Therefore, to obtain the equilibrium structure one must measure B, values for at least four different vibrational states even for this simplest symmetric top The doubly degenerate modes may also be complicated by I-type doubling, as described in Chapter VI, Section 2. Otake et al. [117] have measured two rotational transitions in all the fundamental vibrational states of two isotopic species of the molecule, 14NF3 and 15NF3.From the results they were able to derive the complete equilibrium structure of this molecule. The results are summarized in Table 6.10. From (6.18) for the ground state and the first excited state of the ith mode, v i = 1, all others zero, it is evident that c$ =Bo - B"

(6.77)

Of the fundamental modes, ul, u2, u3, and u4, the last two are degenerate. Thus the equilibrium value of B from (6.17) is

B, =Bo ++M;

>tab,+

.;

+x i

(6.78)

From the Be values of the two isotopic species the two equilibrium parameters can be found by methods described in Chapter XIII. As would be expected from the anharmonicity in the potential functions, the r,(NF) length is slightly less than the ro(NF) length. However, the equilibrium value of the bond angle

Table 6.10 Anharmonic Constants and the Equilibrium Structure of NF3 from Microwave Spectroscopy (from Otake et al. [117]) I4NF3 State YO v4

v2 v3 v1

2v4

B, ( M H z ) 10,681.02kO.01 10,676.54k0.01 10,642.3850.02 10,602.22f 0.03 10,724.47 k0.03 10,671.90-t0.05

15NF3 CY:

(MHz) -

4.48 k0.02 38.65 f0.03 78.81 k0.04 -43.45 k0.04 9.12 k0.06

B, ( M H 4 10,629.44f0.03 10,624.80f0.02 10,589.30k0.02 10,553.77k0.02 10,667.85k0.04 10,620.12k0.03

ci:

-

4.64 k0.05 40.14k 0.05 75.67 k0.05 - 38.41 k0.07 9.32k0.06

Be ( M H 4

Be W H z )

10761.91k0.2

10710.63k0.2 Molecular Structure

ro(NF)=1.371+0.0028, re(NF)=1.365f0.002 8,

(LFNF)o = 102"lO' & 2' ( L FNF), = 102'22' f2'

(MHz)

218


was found to be slightly less than the effective value. Both these structures are given at the bottom of Table 6.10. Duncan [l IS] has combined the A , values derived from infrared vibrationrotational spectra with Be values from microwave rotational spectra for calculation of the equilibrium structures of the methyl halides. Because the moment of inertia Z, about the symmetry axes is small, arising only from the inertial moments of the light, off-axis hydrogens, the A , values are sufficiently large for measurement with good accuracy from vibration-rotation spectra. The Zb values are an order of magnitude larger, and the small B, values are measured with significant accuracy only with microwave rotational spectroscopy. Since the A , values are not obtained by microwave spectroscopy, except in special cases of second-order transitions as described in Section 4, the combination of infrared and microwave data as employed by Duncan is a useful procedure. Next to the AB3 molecules, the methyl halides belong to the simplest class of symmetric-top molecules measurable with microwave spectroscopy. Even so, the complete determination of their equilibrium structures required an enormous amount of infrared and microwave data. Many spectroscopistscontributed to these data. The numerous papers, microwave and infrared, from which Duncan selected these data are referenced by him and will not be repeated here. Table 6.1 1 gives the a: (or a:) values that he used in calculation of the B, values of the chloride, bromide, and iodide. The corresponding infrared a: and A , constants may be found in his paper. The methyl halides have nine vibrational modes, three of which are degenerate, yielding six which are distinguishable. Thus, there are only six a; constants, as listed in Table 6.11. The three symmetric modes, vl, v2, v 3 (species AI), are nondegenerate whereas the perpendicular modes, v4, u5, 06 (species E), are each

Table 6.11 Equilibrium Constants B, a n d a! for Some Methyl Halides"

CH335C1 CD335C1 CH3A'Br

CD3A'Br CH31 CD31

13,407 10,925

1.8 5.2

121 179

9,635 7,753

9 6.6

63.9 60.6

7,564 6,077

5.1 4.05

41.1 30.9

-5.4 0

-47.4 -90

49.2 39

72.5 49.8

-4.2 0

- 16

34 24

54.3 37.8

-3.6 0

115 84

-27 -7.5 - 13

24 15

"From Duncan [118]. The u i values are those chosen by Duncan [118] from various sources, to which references may be found in his tabulations. Here the equilibrium constants given by Duncan in cm-' units are converted to MHz, and xp is used for his $. A corresponding set of A,, uf values obtained from infrared spectroscopy may be found in Duncan's paper. The CH3A'Br is the average of the "Br and "Br isotopic forms.


219

doubly degenerate. It is generally simpler to analyze the rotational spectra of the nondegenerate, parallel modes than the degenerate perpendicular ones, because of the I-type doubling in the latter. However, perpendicular modes usually have lower vibrational frequencies and hence higher excited-state populations which make their rotational lines easier to detect than those of excited parallel modes. In addition to 1-type doubling, the coincidence, or near coincidence, of certain excited states leading to Fermi resonance between fundamental or combination bands can complicate the analysis of the rotational structure of excited vibrational states and the evaluation of sli for symmetric-top molecules. Other complications in the evaluation of C L ~arise from effects of large Coriolis interactions between fundamental modes that are nearly coincident [118]. These effects were found to be especially large between the v 2 and u5 modes of CH3F and of CD3CI. The equilibrium structures for the methyl halides as obtained by Duncan [118] are given in Table 6.12. The structure for methyl fluoride was considered as least reliable of the four, primarily because of uncertainties in some of the cti values resulting from the strong overlapping of the u2 and u s fundamental bands already mentioned. Because of these uncertainties, the probable errors in the structural parameters were not estimated for the fluoride. With infrared, tunable-laser spectroscopy or with microwave-infrared double resonance it should be possible in the future to improve the accuracy of the A , values of methyl halides to the level of that already achieved for the B, values. Table 6.12 Equilibrium Structures of the Methyl Halides" Bond Distances (A) Halide

r,(CH)

r,(C-Hal)

Bond Angle L,(HCH)

CH3Fb CH3CI' CH,Br' CH31c

1.095 1.086(4) 1.086(3) 1.085(3)

1.382 1.778(2) 1.933(2) 2.133(2)

1lO"20' llO"40' k40 110"10+25' 111'1 7' 25'

"From Duncan [1 181. bDerived from the equilibrium constants A!, B f , &' for CH,F, CH,F, and CD3F, respectively. The values from A t , By, and are 1.0941,1.3821,and llO"26'.The limits of error were not determined for the fluoride. See discussion in text. 'These values are determined from two sets: those derived from the combinations (I) A!, B f , and those from (11) A t , BF, The sets are essentially indistinguishable for the bromide and iodide. For methyl chloride the sets are (I) 1.0871,1.778A, 110"47',and (11) 1.084A, 1.777A, llO"27': the chosen structure is weighted to favor set (I).

e

e.

e,

220


However, this improvement in accuracy may not bring about a corresponding improvement in accuracy of the structures for the following reasons. There are three independent structural parameters in the methyl halide molecules, and one must resort to use of isotopic substitution for a third observable. The degree of conformability of the normal and substituted structures will probably limit the accuracy of the structures to approximately that already obtained, regardless of the accuracy of the A , and Be values. The equilibrium structure of trifluorosilane (HSiF3) has been derived by Hoy et al. [119] from Be values of different isotopic species that were obtained principally from microwave spectroscopy. Because the molecule has three independent structural parameters, Be values for a minimum of three isotopic species are required for determination of its equilibrium structure. Six isotopic species, those listed in Table 6.13, were observed. Like the methyl halides, HSiF3 has six fundamental modes, three of which are doubly degenerate. The constants, ab,, ab,, a i , ab,, ab,, were measured for HZ8SiF3and Dz8SiF3;the ab,, which could not be measured, was theoretically deduced from the cubic anharmonic force field derived partly from vibrational force constants obtained from infrared spectroscopy. Also, some of the values of a! for the less abundant 29Siand 30Si species were derived from the cubic anharmonic force field. The Be values were then obtained from the measured Bo values with the expression 6

B,=Bo+

(6.79)

a!($) i= 1

Table 6.1 3 Rotational Constants and Equilibrium Structure of Trifluorosilane" Rotational Constants ( M H z ) Isotopic Species

Bo (obs.)

HZBSiF3 H2'SiF3 H3'SiF3 D2%iF3 D2'SiF3 D3'SiF3

7208.05 1195.75 7183.79 6890.10 6880.20 6870.58

Z~i(di/2) 21.97 21.89 21.82 22.97 22.90 22.80

Be (obs.)

Be (obs.-calc.)b

7230.02 7 2 17.64 7205.61 69 13.07 6903.10 6893.38

+0.02 - 0.00

-0.01 - 0.02 +0.01 +0.01

Equilibrium Structure r,(SiH) 1.4468(5)8,

r,(SiF) 1.5624(1)8,

L,(HSiF) 110.64(3)"

"From Hoy et al. [119]. "Difference between the observed values of B, and those calculated from the equilibrium structure.


221

The resulting Be values with the Bo and the summed c$ contributions for each isotopic species are given in Table 6.13. The equilibrium structure derived from these Bo values is shown at the bottom of Table 6.13. This paper also gives derived I-type doubling constants for the three degenerate modes and makes corrections for perturbations of Fermi resonance between the u2 and 2u3 vibrations.

Hydrogen-bonded Symmetric-top Complexes The first example of a hydrogen-bonded complex shown by microwave spectroscopy to have a symmetric-top structure was the hetero-dimer CH3CN. . HF. In 1980 Bevan et al. [120] measured pure rotational transitions of the complex and correctly assigned the transitions to a characteristic symmetric-top spectrum. Shortly thereafter, they observed [1211 a symmetrictop-like rotational spectrum attributable to the large complex (CH3)3CCN. . . HF. From analysis of the spectrum the C-C=N. . . H F fragment of the complex was shown to be linear. The hydrogen bridges in both these complexes are similar to that in the linear hetero-dimer HCN * . H F described in Chapter V, Section 8. The rotational constants and hydrogen-bridge lengths, ro(N. . . F) found for the cyanide complexes are listed in Table 6.14. Those for the linear HCN . . . H F complex are included for comparison. Although the bridge lengths are approximately the same for all three complexes, there is a definite trend to a shorter bridge length with increasing size of the R group. This trend, which

Table 6.14 Rotational Constants and Bridge Lengths for Some Hydrogen-bonded Complexes DJ

DJK

ro(N. . F) ro(Bz. . . Cl)"

or

Complex

Bo ( M H z )

(kHz)

(kHz)

3591.11 1853.37 1686.21 1832.42 876.25 862.10 1237.6836 1228.2440 1165.1542

5.2 0.82 1.o 1.o

61 50 62

1.22 1.19 1.08

13.35 14.57 10.93

2.796 2.759 2.768 2.153 2.725 2.725 3.594" 3.570" 3.591"

"Distance from center of gravity of benzene ring to the C1. bA.C. Legon, D. W. Millen, and S. C. Rogers, Proc. R. Soc. London, A370,213 (1980). 'Bevan et al. [120]. dGeorgiou et ai. [121]. 'Read et a]. [122].

ReJ:

222


indicates an increasing hydrogen-bond strength with R = H to (CH3),C, is borne out by hydrogen-bond stretching force constant k,(k,) =0.24 mdyne/A for R = H and 0.27 mdyne/W for R=(CH&C. A very interesting complex formed by HCI and benzene was shown from measurements on its rotational spectrum to have a c60 symmetric-top equilibrium structure [122]. The vibrationally averaged direction of the HCl is along the perpendicular symmetry axis of the benzene with its H directed toward the center of the ring. Evidently the positive H interacts with the ;rc-electron cloud to form the hydrogen-bonded dimer HCI . . . C6H6. From the rotational constants (Table 6.14), the averaged distance of the C1 above the plane of the ring is found to be 3.59 1.An approximate hydrogen bond-stretching constant, k,=0.080 mdyne/W, and bonding energy (well depth), E = 720 cmare obtained. Other information with discussion of the various results are given in the paper. From microwave measurements, HCN and HCI are found to form T-type hydrogen-bonded structures [123, 1241 with HCEC-H in which the HCN, or HCI, is along the perpendicular bisector of the acetylene so that the hydrogen forming the bond is nearest to the center of the negative lobes of the x bonds. Hydrogen cyanide forms a similar T-type complex with ethylene [125] in which the H of the HCN interacts with the x-bond lobe density of the H2C= CH2. Although these T-structures are asymmetric rotors, they have approximate (accidental) prolate, symmetric-top structures with B z C < A . The cyclopropane-HCN complex, found by Kukolich [126] from microwave spectral measurements, is also very close to an accidental prolate symmetric-top with B z C < A . The HCN was found to lie in the plane of the carbon ring and to be directed from outside along a perpendicular to the center of one of the C-C bonds with the H toward the bond. The distance between the center of this C-C bond and the carbon of the HCN, the hydrogen-bridge length, was found to be 3.474 A, the hydrogen bond stretching force constant to be k,=0.062 mdyne/A, and the bonding energy of the complex to be 860 cm- approximately. The “outward bending,” caused by the strain in the bonds of the saturated, three-membered, cyclopropane ring evidently gives rise to the interaction of the bond with the proton of the HCN. Hydrogen bonds are not known to occur through direct interaction with a normal o-bond pair. The most commonly occurring hydrogen bonds are formed by interaction of the hydrogen with a nonbonding electron pair like those of CH3CN. . . H F or the linear complexes described in Chapter V, Section 8. Hydrogen-bonded complexes, like those described, in which the bonding hydrogen interacts with n-electron lobes or with “strained” o bonds should be useful for locating reactive sites in molecules where reactions with other electrophilic agents are most likely to begin. The methods for derivation of various properties of the hydrogen-bonded complexes from microwave spectral measurements such as hydrogen-bond force constants and bond energies, bridge length, and dipole moment enhancement are like those described for linear hydrogen-bonded complexes in Chapter

’,

’,

REFERENCES

223

V, Sections 8 and 9. Also, the spectrometers used for detection and measurement of the symmetric-top complexes are similar to those described or referenced in those sections. For both types of structures the complex is produced in very cold beams by expansion of a pressurized mixture through a restricting nozzle into the evacuated observation cavity.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

D. M. Dennison, Phys. Rev., 28, 318 (1926); Rev. Mod. Phys., 3, 280 (1931). F. Reiche and H. Rademacher, Z . Physik, 39,444 (1927). R. de L. Kronig and I. I. Rabi, Phys. Rev., 29, 262 (1927). T. E. Sullivan and L. Frenkel, J . Mol. Spectrosc., 39, 185 (1971). J. G. Smith and I. Thompson, Mol. Phys., 32, 1247 (1976). Tsu-Shen Chang and D. M. Dennison, J . Chem. Phys., 21, 1293 (1953). W. J. Orville-Thomas, J. T. Cox, and W. Gordy, J . Chem. Phys., 22,1718 (1954). J. Wagner, Z . Physik. Chem., B40,36 (1938). S. Silver and W. H. Shaffer, J . Chem. Phys., 9, 599 (1941). W. H. Shaffer, J . Chem. Phys., 9,687 (1941). W. H. Shaffer, J . Chem. Phys., 10, 1 (1942). H. H. Nielsen, Phys. Rev., 60, 764 (1941). H. K. Bodenseh, R. Gegenheimer, J. Mennicke, and W. Zeil, Z.Naturjorsch., 22a, 523 (1967). E. Teller, Hand-u. Jahrb. G e m . Phys., 9, II,43 (1934). M. Johnston and D. M. Dennison, Phys. Rev., 48, 868 (1935). E. B. Wilson, J . Chem. Phys., 3, 818 (1935). H. H. Nielsen, Phys. Reo., 77, 130 (1950). H. H. Nielsen, Reo. Mod. Phys., 23, 90 (1951). M. L. Grenier-Beeson and G. Amat, J . Mol. Spectrosc., 8, 22 (1962). S. Maes and G . Amat, Can. J . Phys., 43, 321 (1964). S. Maes, Cah. Phys., 14, 125 (1960). G . Amat, H. H. Nielsen, and G. Tarrago, Rotation Vibration of Polyatomic Molecules, Dekker, New York, 1971. 23. R. Trambarulo and W. Gordy, J . Chem. Phys., 18,1613 (1950). 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

D. R. Jenkins, R. Kewley, and T. M. Sugden, Trans. Faraday SOC.,58, 1284 (1962). P. Venkateswarlu, J. G. Baker, and W. Gordy, J . Mol. Specfrosc., 6, 215 (1961). A. Bauer, J. Mol. Spectrosc., 40, 183 (1971). G . Tarrago, J . Mol. Spectrosc., 34, 23 (1970). M. Otake, F. Hirota, and Y. Morino, J . Mol. Spectrosc., 28, 325 (1968). M. J. Whittle, J. G. Baker, and G . Corbelli, J . Mol. Spectrosc., 40, 388 (1971). M. C. L. Gerry and T. M. Sugden, Trans. Faraday Soc., 61, 209 (1965). A. J. Careless and H. W. Kroto, J . Mol. Spectrosc., 57, 198 (1975). E. Hirota, J . Mol. Spectrosc., 37, 20 (1971); 38, 195 (1971). Y. Kawashima and A. P. Cox, J. Mol. Spectrosc., 61,435 (1976). Y. Morino and C. Hirose, J . Mol. Spectrosc., 22, 99 (1967). A. Bauer and S. Maes, J . Mol. Spectrosc., 40,207 (1971). C. E. Cleeton and N. H. Williams, Phys. Rev., 45, 234 (1934). B. Bleaney and R. P. Penrose, Nature, 157, 339 (1946).

224

SY M M ETR I C-TO P MOLECULES

38. W. E. Good, Phys. Rev., 70, 2 13 (1 946). 39. C. C. Costain and G. B. B. M. Sutherland, J . Phys. Chem., 56, 321 (1952). 40. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955, p. 307. 41. M. F. Manning, J . Chem. Phys., 3, 136 (1935). 42. D. M. Dennison and G . E. Uhlenbeck, Phys. Rev., 41, 313 (1932). 43. C. C. Costain, Phys. Rev., 82, 108 (1951). 44. E. Schnabel, T. Torring, and W. Wilke, 2. Physik., 188, 167 (1965). 45. G. E. Erlandsson and W. Gordy, Phys. Rev., 106,513 (1957). 46. P. Helminger and W. Gordy, Bull. Am. Phys. Soc., 12, 543 (1967). 47. P. Helminger, F. C. De Lucia, W. Gordy, H. W. Morgan, and P. A. Staats, Phys. Rev., A 9, 12 (1974). 48. P. Helminger, F. C. De Lucia, and W. Gordy, J . Mol. Spectrosc., 39, 94 (1971). 49. R. G. Nuckolls, L. J. Reuger, and H. Lyons, Phys. Rev., 89, 1101 (1953). 50. G . Hermann, J . Chem. Phys., 29, 875 (1958). 51. W. S. Benedict and E. K. Plyler, Can. J . Phys., 35, 1235 (1957). 52. K. N. Rao, W. W. Brim, J. M. Hoffman, L. H. Jones, and R. S. McDowell, J . Mol. Spectrosc., 7, 362 (1961). 53. J. D. Swalen and J. A. Ibers, J . Chem. Phys., 36, 1914 (1962). 54. R. J. Damburg and R. Kh. Propin, J . Phys. B : Atom. Mol. Phys., 5, 1861 (1972). D. Papougek, J. M. R. Stone, and V. Spirko, J . Mol. Spectrosc., 48,17 (1973). J. T. Hougan, R. R. Bunker, and J. W. C. Johns, J . Mol. Specfrosc.,34, 136 (1970). V. Spirko, J. M. R. Stone, and D. Papougek, J. Mol. Spectrosc., 60,151 (1980). A. R. Hoy and P. R. Bunker, J . Mol. Spectrosc., 52,439 (1974). S. P. Belov, L. I. Gershtein, A. F. Krupnov, A. V. Maslovskij, S. Urban, V. Spirko, and D Papouiek, J. Mol. Spectrosc., 84,288 (1980). 60. F. Scappini and A. Guarnieri, J . Mol. Spectrosc., 95,20 (1982). 61. H. M. Hanson, J . Mol. Spectrosc., 23, 287 (1967). 62. T. Oka, J . Chem. Phys., 47, 5410 (1967). 55. 56. 57. 58. 59.

63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

K. Fox, Phys. Rev. Lett., 27, 233 (1971). J. K. G. Watson, J . Mol. Spectrosc., 40, 536 (1971). M. Mizushima and P. Venkateswarlu, J . Chem. Phys., 21, 705 (1953). I. M. Mills, J. K. G. Watson, and W. L. Smith, Mol. Phys., 16, 329 (1969). F. Y . Chu and T. Oka, J . Chem. Phys., 60,4612 (1974). A. Rosenberg, I. Ozier, and A. K. Kudian, J . Chem. Phys., 57, 568 (1972). C. W. Holt, M. C. L. Gerry, and I. Ozier, Phys. Rev. Lett., 31, 1033 (1973). T. Oka, “‘Forbidden’ Rotational Transitions,” in Molecular Spectroscopy: Modern Research, Vol. 11, K. N. Rao, Ed. Academic, New York, 1976,pp. 229-253. K. Fox, Phys. Rev.,A6,907(1972). M. R. Aliev and V. M. Mikhaylov, J . Mol. Spectrosc., 49, 18 (1974). D. A. Helms and W. Gordy, J . Mol. Specfrosc., 66, 206 (1977). D. A. Helms and W. Gordy, J . Mol. Spectrosc., 69, 473 (1978).

75. A. H. Maki, R. L. Sams, and W. B. Olson, J . Chem. Phys., 58,4502 (1973). 76. W. B. Olson, A. G. Maki, and R. L. Sams, J . Mol. Spectrosc., 55, 252 (1975). 77. P. Helminger and W. Gordy, Phys. Rev., 188, 100 (1969). 78. P. Helminger, E. Beeson, and W. Gordy, Phys. Rev., A3, 122 (1971). 79. H. H. Nielsen and D. M. Dennison, Phys. Rev., 72, 1011 (1947).

REFERENCES

225

80. S. P. Belov, A. V. Burenin, L. I. Gershtein, A. F. Krupnov, V. N. Markov, A. V. Maslovsky. and S. M. Shapin, J . Mol. Spectrosc., 86, 184 (1981). 81 P. B. Davies, R. M. Neumann, S. C. Wofsky, and W. Klemperer, J . Chem. Phys., 55, 3564 (1971).

82. R. H. Kagann, I. Ozier, and M. C . L. Gerry, J . Mol. Spectrosc., 71, 281 (1978). 83. J. G . Smith, Mol. Phys., 32, 621 (1976). 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.

101.

102. 103. 104. 105.

106. 107. 108.

109. 110. 111. 112. 113. 114. 1 15. 116. 117. 118.

119. 120.

I. Ozier and W. L. Meerts, Phys. Rev. Lett., 40, 226 (1978). W. L. Meerts and I. Ozier, Phys. Rev. Lett., 41, 1109 (1978). W. L. Meerts and I. Ozier, J . Chem. Phys., 75, 596 (1981). I. Ozier and W. L. Meerts, Can. J . Phys., 59, 150 (1981). I. M. Mills, Mol. Phys. 1, 107 (1958); Spectrochim. Acta., 16, 35 (1960). I. Ozier, Phys. Rev. Lett., 27, 1329 (1971). A. Rosenberg and I. Ozier, Can. J . Phys., 52, 575 (1974). I. Ozier and A. Rosenberg, Can. J . Phys., 51, 1882 (1973). C. W. Holt, M. C. L. Gerry, and I. Ozier, Can. J . Phys., 53, 1791 (1975). I. Ozier, R. M. Lees, and M. C. L. Gerry, J . Chem. Phys., 65, 1795 (1976). I. Ozier, R. M. Lees, and M. C. L. Gerry, Can. J . Phys., 54, 1094 (1976). R. H. Kagann, I. Ozier, G. A. McRae, and M.C.L. Gerry, Can. J . Phys., 57, 593 (1979). R. F. Curl, T. Oka, and D. S. Smith, J . Mol. Spectrosc., 46, 518 (1973). R. F. Curl, J . Mol. Spectrosc., 48, 165 (1973). W. A. Kreiner and T. Oka, Can. J . Phys., 53, 2000 (1975). K. T. Hecht, J . Mol. Spectrosc., 5, 355 (1960); 5, 390 (1960). S. M. Kirschner and J. K. G. Watson, J . Mol. Spectrosc., 47, 347 (1973). I. Ozier, J. Mol. Spectrosc., 53, 336 (1974). G. Pierre, G. Guelachvili, and C. Amiot, J . Phys. Paris., 36, 487 (1975). G. Tarrago, M. Dang-Nhu, G. Poussigue, G. Guelachvili, and C. Amiot, J . Mol. Spectrosc., 57, 246 (1975). G. Herzberg, Infrared and Raman Spectra of Polyatornic Molecules., Van Nostrand, New York, 1945, p. 339. This collection of data initiated by B. Starck as Vol. II/4 is now extended to Vols. 11/6, 11/7, and 11/14,A complete reference is given in Chapter I, Ref. 65. A. Bjarseth, D. A. Drew, K. M. Marstokk, and H. Mallendal, J . Mol. Struct. 13,233 (1972). D. A. Drewand A. Haland, Chem. Commun., 1551 (1971). L. C. Krisher, R. A. Gsell, andJ. M. Bellama,J. Chem. Phys., 54,2287(1971). S . N. Wolf, L. C. Krisher, andR. A. Gsell,J. Chem. Phys., 54,4605 (1971). S. N. Wolf, L. C. Krisher, and R. A. Gsell, J . Chem. Phys., 55, 2106 (1971). J. R. Durig, A. B. Mohamad, P. L. Trowell, and Y. S. Li, J . Chem. Phys., 75, 2147 (1981). L. C. Krisher, J. A. Morrison, and W. A. Watson, J . Chem. Phys., 57,1357(1972). S. N. Wolf and L. C. Krisher, J . Chem. Phys., 56, 1040 (1972). K. Karakida, K. Kuchitsu, and C. Matamura, Chem. Lett., 293 (1972). Y. S. Li, M. M. Chen, and J. R. Durag, J . Mol. Struct., 14, 26 (1972). H. W. Kroto, J. F. Nixon, and N. P. C. Simmons, J . Mol. Spectrosc., 77, 270 (1979). M. Otake, C. Matsumura, and Y . Morino, J . Mol. Spectrosc., 28, 316 (1968). J. L. Duncan, J . Mol. Struct., 6, 477 (1970). A. R. Hoy, M. Bertram, and I. M. Mills, J . Mol. Spectrosc., 46, 429 (1973). J. W. Bevan, A. C. Legon, D. J. Millen, and S. C. Rogers, Proc. R . Soc. London A370, 239 (1980).

226

121. 122. 123. 124. 125. 126.

S Y M M ETRIC-TOP MOLECULES

A. S. Georgiou, A. C. Legon, and D. J. Millen, Proc. R. Soc., London A370,257 (1980) W. G. Read, E. J. Campbell, and G. Henderson, J. Chem. Phys., 78, 3501 (1983). P. D. Aldrich, S. G. Kukolich, and E. J. Campbell, J. Chem. Phys., 78, 3521 (1983). A. C. Legon, P. D. Aldrich, and W. H. Flygare, J . Chem. Phjs., 75,625 (1981). S. G. Kukolich, W. G. Read, and P. D. Aldrich, J . Chem. Phqs., 78, 3553 (1983). S. G. Kukolich, J . Chem. Phys. 78, 4832 (1983).

Chapter

VI I

A S Y M M ETRIC-TOP MOLECULES

QUALITATIVE D ES C R I PTI 0 N

227

ENERGIES OF THE RIGID ASYMMETRIC ROTOR

232

Factorization of the Energy Matrix from Symmetry Properties Symmetry Classification of the Energy Levels Evaluation of the Energy Levels, Wave Functions, and Average Values

237 241

SLIGHTLY ASYMMETRIC ROTORS

250

SELECTION RULES AND INTENSITIES

254

IDENTIFICATION A N D ANALYSIS OF ROTATIONAL SPECTRA

264

244

Common Aids for Identification of Observed Transitions

264

Computer Assignment and Analysis Double Resonance as an Aid to Spectral Assignment

269 273

EFFECTS OF VIBRATION

281

Rotation-Vibration Hamiltonian Coriolis Perturbations

282 289

ASYMMETRIC ROTOR STRUCTURES

291

QUALITATIVE DESCRIPTION When none of the three principal moments of inertia of a molecule is zero and if no two are equal, considerable complexity is encountered in its pure rotational spectrum. The rotational frequencies can no longer be expressed in convenient equations, as can be done for linear or symmetric-top molecules. Only for certain low J values can the energy levels of the asymmetric rotor be expressed in closed form, even if centrifugal distortion effects are neglected. The increased complexity of the pure rotational spectrum over that of the symmetric-top rotor is illustrated in Fig. 7.1. The various methods that have been used to give the energy levels and wave functions for the asymmetric rotor are considerably more involved than are those for symmetric rotors. The general procedure is to assume that the wave functions can be expanded in terms of an orthogonal set of functions (a natural basis would be the symmetric-top functions) and to set up the secular equations for the unknown coefficients and energies. The resulting secular determinant can be broken down into a number 227

228

ASYMMETRIC-TOP MOLECULES

n

GHz Fig. 7.1 (a) A condensed-scale scan of the spectrum of 1,3-cycloheptadiene ( ~ ~ 0 . 8 which 5) has '%'-type transitions. Frequency markers are spaced 100 MHz apart and the sweep rate was about 5 MHz/sec. The zero field lines are down, Stark lobes are up. The separation between the *Q1band heads is approximately 3110 MHz. A=3419.424, B=3297.290, and C=1799.961 MHz. From Avirah et al. [48]. ( b ) Condensed-scale, R-band spectrum of 2-cyanothiophene. The sample pressure was about 30pm and the scan was about 3 MHz/sec. The Stark lobes are up and the zero field lines are down. Groups of closely spaced absorption lines ( J 4 + 1, K - l - + K - l ) are observed separated by approximately 3175 MHz(-B+C). From T. K. Avirah, T. B. Malloy Jr., and R. L. Cook, J . Mol. Struct., 29, 47 (1975).

of subdeterminants, the order of which increases with J. The solution of these subdeterminants yields the required energy levels and expansion coefficients. Details of these energy level calculations will be discussed in the next section. Despite this complexity, much information useful to the chemist can be obtained from the microwave spectrum of an asymmetric-top molecule, in many instances without long labor or advanced mathematical skill. This is a fortunate circumstance since most molecules are of the asymmetric-top class. In the first place, both qualitative and quantitative spectrochemical analysis can be accomplished by measurements of microwave rotational spectra without an interpretation of their patterns. Second, somewhere in the wide span of the now workable microwave region there are usually low J transitions whose

QUALITATIVE DESCRIPTION

229

frequency can be expressed with closed algebraic equations, from which the principal moments of inertia can be immediately evaluated. Furthermore, except for a few very light molecules such as H 2 0 , the centrifugal distortion effects on very low J transitions can usually be neglected. These lines cannot be identified from the simple pattern of the rotational spectrum as they can for the symmetric-top rotor, but the low J transitions can be readily identified by their Stark patterns from which the electric dipole moment can also be obtained (see Chapter X). In some cases, the transition can be identified by nuclear quadrupole hyperfine structure. Third, extensive tables are available from which energy levels for various degrees of asymmetry can be obtained to a useful degree of approximation for high J values. To apply the equations and selection rules developed in the following sections, one must first become familiar with the notation used to designate the levels. Here we shall outline briefly the qualitative characteristics of the asymmetric rotor energy levels. In an asymmetric rotor there is no internal component of the angular momentum that is a constant of the motion; that is P, no longer commutes with & and only J and M are “good” quantum numbers. As for the symmetric rotor, we can write Pz = ( h / 2 7 ~ ) ~ J (1) J +and PZ=hM/2n, where J and M are integers as previously defined. Pseudo-quantum numbers, customarily designated by subscripts on J, are employed in the designation of the levels. The double subscript system of King et al. [I] is perhaps the most descriptive and useful; it will be employed extensively. Their system is best understood by a comparison of the limiting prolate and oblate symmetric tops. In the conventional order, Za< & O . 9 are collected in Table 7.10. For a transition J , T-+J’, T’ caused by a component of the electric moment p g , a convenient quantity called the line strength is defined as (7.58)

where the sum extends over the three directions of the space-fixed system and over all values of M and M’. The latter sum over M and M‘ takes into account all possible transitions which in the absence of external fields will be degenerate. The line strength depends only on the inertial asymmetry parameter K and not on the individual rotational constants; thus it is easily tabulated. Extensive tables of line strengths for various values of K are available [23, 26-28]. The summation over F in the absence of external fields can be accomplished by multiplying the results for a given F by 3, that is,

&(J,

5;

I(J,T , M ~ Q ) F ~T’,JM’)12 ’,

J’T’)=3

(7.59)

MM‘

The dipole moment matrix element appearing in (3.24) may be expressed in terms of the line strength. W e have for a field-free rotor

(7.60) since p F

=egpgaFg and for a given transition only one term in the sum over g

260

o

3

-

l-

o

II

0

Y

II

3 "

II

0

Y

E ! 3

r"3

II

+

00

c!

--

N-

Y

II

N-

Y

I1

m^

Y

I/

m

Y

m A

II

0

-

I

-5I --

4-

N

r!

-N-

$2

3

r!

'10

e+

n r ! m

?!

e

II

II

A

-

Y

A

0

I3

U

4

-

v

m 4-

E!

e

N,

+ $2 m + *

I

2

Y

+

c!

& 4-

N

E!

II

x

II

Y

“From Lide [25]. b H e r e K s t a n d s f o r K - I a n d y = O i f J + K + K , i s e v e n a n d y = l i f J + K + K , isodd.Theparameterb=(B-C)/ ( A - C). The direction cosine matrix elements are evaluated from the relations: ( J , K , K,l@FqlJ’,K, K ; ) = [1 - (a2+b2 +a” +8’’)/2](5, K @ l I,:

S, K )+aa’(J, K -21@gglJ’, K

- 2)

+B/Y( J, K +2l@g,l J’, K + 2) and

( J , K , KII@FAJ‘,K + 1 , K ’ ~ ) = [ l - ( a 2 + 8 2 + ~ ’ 2 + / Y 2 ) / 2 ] ( J , K l @ : g l JK+l)+a’(J, ‘r Kl@$,IJ’, K-1)

+b(J,K+2(@,8,1J’,K+l)+aa’(J, K-2l@:,lJ’. +Bp(J,K + 21@8,1J’, K + 3)

K-1)

The (J, Kl@:,lJ’, K’) are the symmetric rotor direction cosine matrix elements and a,B, calculable from the above table, refer to the level J, K, K,, and a’,/3’ to the level J‘,K’, K ; . For ( J , Ol@:,JJ‘, 1) the expression given in Table 2.1 must, in this particular case, be multiplied by 2”’.

262

ASY M M ETRIC-TO P MOLECULES

remains from symmetry arguments. The results are Lg(J,

T ; J‘, T’)=(2J+

1 ((J,

1)

5, M I @ F g I J ’ , 9‘3

M’)I2

FM‘

=PJ+ u p , +(J‘,

(7.61)

f’,12/1.;

[(J’,T’,M’I@FgIJ,5, M)I2

=(25’+ 1) FM

=(2J’+ l)I(J’, T’IPL(J, 2)I2/PL,2

(7.62)

The direction cosine matrix elements can also be conveniently expressed in terms of the line strengths. For F = Z and any g, the line strength can be expressed in terms of the matrix elements I(J, T ( @ ~ ~ 7’)12 ~ J ’by , use of (7.57), Table 2.1, and carrying out the summation. Note that the sum over M‘ reduces to one term since QZg is diagonal in M [see (10.71)-(10.73)]. We find

l,(J, T; J’,T’)=$RI(J, TI@~,IJ’, ?)I2

(7.63)

where R is defined as 1/J, (2J+ l)/J(J+I), or 1/(J+1) for J ’ = J - 1, J, or J + 1, respectively. Since from (7.57)

one obtains from (7.63) and Table 2.1 J2 - M 2 IV, *, MI@zglJ- 1, T’,M)I2= 4 2 5 - 1)(2J+ 1) n,(J, 7 ;J- 1, T’)

I J, 7,M1@zglJ, T’, M )I

I(J,

7,M l @ z JJ+

1, f,M)I2=

=

M2 Ag(J,T ; J, 7’) J(J+ 1)(2J+ 1)

(J+ 1)2-M2 (J+1)(2J+ 1)(2J+3)

Lg(J,7 ;J+ 1, 5 ’ )

(7.65) (7.66) (7.67)

These expressions multiplied by represent the dipole matrix elements for particular Stark or Zeeman components with the field along Z analogous to (2.109)and (2.115) for a symmetric top

I(J,

5,

MlPzlJ‘,

T’, M)I2=PL,21(J,

f,

MI@Z,IJ’,

T‘,

MI2

(7.68)

It may also be readily shown that +.I

(7.69) since EM2 = j J ( J + 1)(2J+ 1). A number of useful sum rules exist because of the orthogonal properties of the direction cosine matrices [23]. Rudolph [29,30]

Next Page

SELECTION RULES A N D INTENSITIES

263

has also derived a number of additional relations involving the line strengths, reduced energy, and average of Pi. A knowledge of the line strength is important in accessing the intensity of a given line and how the intensity can be expected to change with K or variations in J or K . The larger \As1 in the transition the smaller the value of the correThe line strength is independent of the direction of the transition. sponding i. For a given K (7.70)

&(J, T ; J’,5’)=Ag(J’, 5 ’ ; J, T)

Also the line strength for an oblate asymmetric rotor transition JK - I K I + J i , - I K i can be obtained from the prolate asymmetric rotor transition J K I K JkiKf-l with the subscripts reversed by the means of the relation

Ag(J,5 ; J‘,5 ‘ ; K ) = I w g s ( J ,

-7;

J’,-T‘;

-K)

(7.71)

Here the explicit dependence on K is indicated, and the axes change g+g’ as follows: a-c, b-rb, and c-ra. For small asymmetry the direction cosine matrix elements may be evaluated with the aid of Table 7.10 and hence the line strengths [see (7.63)]. Explicit expressions for the line strengths of low J rotational transitions have been given by Gora [31]. The line strengths for the symmetric top limit, which are useful simple approximations to the line strengths for near asymmetric tops, follow from (7.63) and Table 7.10 with 6=0 and Table 2.1. The dipole matrix element factor appearing in (3.27) for the absorption coefficient is related to the line strength from (7.61) as follows: (7.72) If (7.72) and (3.25), (3.26), and (3.66) (the latter equation modified for an asymmetric rotor as indicated in Chapter 111, Section 3) are inserted into (3.27), the peak absorption coefficient for the transition J, T+J’, T’ may be written as

where it is assumed that hv, QkT The rotational constants A , B, C, the resonant ~ in megahertz, with ,ugin debye units frequency v,, and the line breadth ( A V )are and u,,, in cm-’. If the lowest vibrational frequency is much greater than kT (say 1000 cm-’ for room temperature), the vibrational partition function is nearly equal to unity, and, for the ground vibrational state, F , may hence be taken to be unity. Other things being equal (such as the Boltzmann factor), the strongest lines of a spectrum will be those which have the largest line strength. For asymmetric rotors with no symmetry o, g r = l . When equivalent nuclei may be exchanged, the rotational level populations are affected and hence the relative intensities. The effects of nuclear spin statistics and the evaluation of g r have been discussed in Chapter 111, Section 4.

264

ASY M M ETR IC-TOP MOLECULES

Previous Page

5 IDENTIFICATION AND ANALYSIS OF ROTATIONAL SPECTRA Before molecular information can be derived from the rotational spectrum, the major task of assignment of the rotational quantum numbers to the observed spectral transitions must be accomplished. Therefore some remarks need to be made on the assignment of the transitions, application of selection rules, and evaluation of the rotational constants.

Common Aids for Identification of Observed Transitions The assignment of spectra can be relatively easy for linear and symmetric-top molecules. Because of the greater complexity of the spectra of asymmetric-tops, assignment of their rotational transitions can be very difficult. In the investigation of any molecule it is possible to obtain an initial set of structural parameters from similar molecules, or other studies, for example, infrared, X-ray diffraction, or simply an intelligent guess, and thus the rotational constants with which the rigid-rotor spectrum can be tentatively predicted. Some idea of the dipole components pg, or of their relative magnitudes, is also required, since the strength of a particular transition will be proportional not only to the line strength but also to the square of the dipole component. Thus the transitions that dominate the spectrum are dependent on the size of p g . Bond dipole moments or dipole moments of similar molecules can be used for approximation of these components. Often the molecular symmetry can be used to argue for, or against, the presence of a particular component. Such calculations can usually indicate via the selection rules at least the type of transitions and the frequency region in which one can expect to find certain useful transitions, for example, particular low J transitions. Spectral patterns, the Stark effect (see Chapter X, Section 8), quadrupole hyperfine splittings (see Chapter IX, Section 4), nuclear spin statistics (see Chapter 111, Section 4),and the rigid rotor fit can all be helpful in establishing and confirming assignments. If the conditions permit, the Stark effect can be a useful technique. By observation of the number of components (Stark lobes) into which a given line splits in the presence of an externally applied electric field, the smaller of the two J’s involved in a transition may be determined. For Q-branch lines, the intensity of the Stark lobes increases with M J . The reverse is true for R-branch lines. If nuclear quadrupole hyperfine structure is present, it may be used as a guide, and transitions may be identified on the basis of their characteristic quadrupole splitting pattern, as well as the selfconsistency of the spectral fit. Often a characteristic spectral pattern is an aid to assignment. Both the line patterns and intensities provide useful information. Finally, the technique of microwave double resonance can be employed (see later in this section). One of the more useful patterns is obtained for molecules with “a”-type Rbranch transitions (“Rol)for a prolate asymmetric rotor and “,”-type for an oblate rotor (‘Rlo). Consider a prolate rotor. There are two transitions


265

(J+J+ 1, K-l-+K-l) for each value of K - l except for K - l =O. The structure hence consists of 2 J + 1 lines for a given J+J+ 1 transition. Figure 7.1 shows a condensed-scale spectrum of 2-cyanothiophene which is characteristic of a near-prolate asymmetric rotor with an “,”-type spectrum. (See also Fig. 7.5). The grouping of the lines J K - l K 1 - + ( J + l ) K - linto K I +clusters l for a given J+J+ 1 transition is clearly apparent. The intensity of the transitions within a given group fall off with increasing K - because of the Boltzmann factor. As J increases, the number of lines in each group increases, and the lines are spread out over a larger frequency range. The lower K - l-doublet transitions are more widely split by the asymmetry than the higher K - lines. Hence as K - increases, the splitting between a given K - doublet decreases. If the asymmetry is not too large, the pair of higher K - lines will coalesce. For very slightly asymmetric rotors, the frequencies of the K - = 1 and 2 lines can be readily used for evaluation of the rotational constants. These relations collected in Table 7.1 1 are also useful for more asymmetric rotors as rough approximations to the rotational constants. Another identifying characteristic for this kind of transition is the Stark effect. If the dominant contribution to the Stark effect arises from pafor a prolate top, the degenerate transitions at high K - will exhibit a very fast Stark effect and give rise to a first-order Stark pattern, that is, Stark components on both sides of the line which move away very rapidly with increasing electric field. This is due to the degenerate levels JK -IK1,JK -lK’, interacting via pa(see Chapter X). The K - =O line will show a characteristic second-order effect, whereas the lower K - lines with larger splitting will also show .a second-order pattern, although the Stark effect can be quite fast because of the near degeneracies of the interacting levels. The higher K - lines are hence more easily modulated and will often show up even at low modulation voltages. The Stark patterns of low K - lines can be useful even if not resolved. The Stark lobes for the higher frequency K - line are on the low-frequency side of the line; whereas, the Stark lobes for the low-frequency K - line will be found on the high-frequency side. This pattern, along with an approximate frequency separation of a given pair of low K - lines, can be very helpful in limiting assignment possibilities. With the development of broadbanded detectors and BWO tubes as microwave sources, rotational spectra can be conveniently recorded over wide frequency ranges. Under fast-sweep conditions, rotational spectra with significantly reduced resolution are obtained, and a whole microwave band (e.g., R-band 26 to 40 GHz) can be displayed on a recording similar in size to an IR spectral trace (see, e.g., Fig. 7.1). They are useful as an aid in the assignment of high resolution spectra. Such spectra were first reported by Scharpen [32]. Iltustrative applications to structure studies are given in Chapter XI1 (see also [33] and [34]). The ORol and ‘RIOtransitions discussed previously can often, under low resolution, give rise to a symmetric-top, bandlike spectrum. Useful band spectra arising from overlapping of high K lines have been obtained for molecules with 1x1 20.70 and paor,>0.50 D [35]. The frequency of the band center in terms

Table 7.11 Useful Relations for Characterization of Rotational Spectra of Asymmetric Rotors

Comments

Relation

I

Vh$Vf=2(B+C)(Jf1) J+J+1 =(B-C)(J+ 1) K - 1 = 1 + 1

Approximate location of the high ( v h ) and low ( v f )frequency components of the K - = 1 doublet of the ‘ROl series for a prolate rotor. Approximate splitting of the K - = 2 (B- C)2/4 J-+ J + 1 vh-v[= J ( J + 1XJ+2) K - =2+7 ~ - lines of the ‘Rol series for a prolate 2A - ( B C) rotor. For an oblate rotor (‘Rlo) interchange A and C here and above expression. Frequency of the band center for the v = ( B + C)(J + 1 ) J+J 1, K - 1 + K - 1 ORor transitions of a near prolate rotor. Av=(B+ C ) Band separation For transitions (‘Rlo) of a near oblate rotor replace C by A. Empirical relation [37] to correct low (B+C)/(Bo+Co)=l+(0.025~0.005)(1 +K) resolution observed (BsC) to groundstate quantity (Bo+ Co).K may be estimated from assumed structure. Approximate location (their mean) of v = ( A + B)(J + 1 ) (J+J 1, K i + K i + 1 ) the high K1 R-branch transitions of a (2C - A - B)(K + 1 i) highly asymmetric rotor. The separation between the different pairs of transitions is 2C. The K 1= J - r J + 1 transitions can give v =(25 l)C ( K = J-r J 1) rise to band spectra separated by 2C characterized by this relation. Location of the Q-branch transitions between K doublets where A WJKis the energy level splitting between a given K doublet. Q-branch series beginning ( J = K - 1 ) for a near-prolate rotor with “&’-type transitions. Separation between band heads is 2A-B-C. v = [ i ( A + B ) - C ] ( 2 K l - 1) K1+Kl - 1 Q-branch series beginning ( J = K l ) for a near-oblate rotor with “@’-typetransitions. Separation between band heads is A+&-2C. vh-vl

{

+

+

+ +

+

+ +

?$)

+

+

266


267

of the rotational constants is given in Table 7.11; a typical band spectrum [36] is shown in Fig. 7.7. Corrections to the band center to extract the ground state constant (Bo Co) given in Table 7.11, have been discussed by Farge and Bohn [371* The width of the band varies from about 50 to 200 MHz because of the spreading effect of the asymmetry; in fact, the half width of the band can be used as a qualitative measure of K [35]. This can be useful in comparing different conformers of a compound if the K ’ S differ by at least 0.05. The broadening of the band with application of a static field [38] can also be used to approximate the magnitude of pa. With large molecules there are often more than one configuration of the molecule that can give distinct spectra, and more than one band series will be found. The ratio of the integrated band intensities of two conformers of a molecule can be related to the ratio of the number of molecules, N, of each conformer [39]. For “a”-type R-branch transitions of a highly asymmetric toror (IKI < 0.3), the pair of transitions Jo,J+(J+ l ) o , J +and l J1,J+(J+ l ) l , J + lturn out to be coincident, or nearly coincident, in frequency, and adjacent pairs are separated by 2C. Because of this, and the fact they have large. line strengths, they tend to overlap and stand out in the spectrum. In addition, the Stark effect is particularly and J 1 ,levels; ~ fast when pc is nonzero due to the near degeneracy of the mirror-image Stark patterns can be obtained. The lines are then easily observable even at low modulation voltages. Such transitions have been found [40-441 to be particularly useful in the assignment of rotation spectra of highly asymmetric rotors. A similar situation holds for “8’-type transitions J1,,+(J+ l)o,J+ and Jo,J+(J+ l)l,J+1. The characteristics of these transitions are most easily explained by noting that for a given K at high Jmany of the higher K - energy

+

I

2

I

I

I

I

33

GHr

I

I

I

I

]

38

Fig. 7.7 Low-resolution O R o l spectrum of phenylisothiocyanate. The constant separation ( B + C) in the bands is apparent. From Higgins, et al. [ 3 6 ] .

268

ASY M M ETRIC-TOP MOLECULES

levels are prolate in character, whereas the lower K - levels are quite oblate in character [45] (see Table 7.11). These high K1-lines, when accompanied by vibrational satellite lines, can also give rise to band spectra under appropriate experimental conditions [46], but these band spectra are not as common as those described previously. Little application has been made of them so far. An analogous situation holds for “c”-type transitions of a highly asymmetric rotor from levels of high K - l , for example, JJ,o+(J+ l)J+l,o and J J , l - ( J + l)J+l,l. The separation between different adjacent pairs of such transitions is now 2A. Q-branch transitions which often give rise to recognizable spectral patterns can provide the initial key to an assignment (see, e.g., [45], [47]-[50]). The stronger subbranches have been listed in Table 7.9. Even when no special spectral pattern is present there may be certain prominent Q-branch transitions which can be used as an aid in the assignment [51]. For the bQ1-, series, a given K + K ’ transition consists of two transitions (except K=O) of the same J ; hence two series can be formed characterized by K and various values of J. These two series diverge from each other as J increases. Only one part of the series may be observed in a given frequency region. If K is large or a series with large (or K 1 ) is being studied, the two series can overlap and a series of strong lines relatively closely spaced can be obtained with each line consisting of two superimposed transitions. However, for a given K , the pair of superimposed transitions will split at sufficiently high J because of asymmetry effects, with their separation increasing with increasing J. The K =O+ 1 lines form a single series moving to higher or lower frequency depending on the type of transition involved. The details of the series structure depend on K , J, K and the type of transition. These details will be clear from calculated spectra, but the less the inertial asymmetry, the less the spliting and the more easily recognizable will be the series. If the splittings between lines of the series are not too large and if a high K series is studied, a broad cluster of lines can be obtained if recording is done under low resolution conditions, as illustrated in Fig. 7.1. Each cluster of lines or band is characterized by a particular value of K and various values of J. Expressions for the band origin and the separation between the band heads are given in Table 7.11. The Q-branch transitions between K doublets K + K form a series for each value of K (see, e.g., [52]). These are the subbranches “Qo- for a prolate rotor and ‘QIO for an oblate rotor. Expressions for the frequencies in the series are given in Table 7.11. The asymmetry splitting of the levels (K>O),and hence the frequency, decreases with increasing K . Often the type of transition to be expected is quite sensitive to a particular structural parameter(s). HzSz has been investigated by Winnewisser et al. [22]. By analogy with previous observations on H 2 0 2and FzS2,the molecule H2S2 would be expected to have a skew-chain structure (H-S-S-H) and hence a Cz axis of rotational symmetry. Furthermore, the dihedral angle (q), the angle defining the relative position of the two S-H bonds, might be expected to be

IDENTIFICATION AND ANALYSIS OF ROTATIONAL SPECTRA

269

close to that found in F2S2,that is, somewhere around 90”. Because of the C2 axis of symmetry, a variation in the line intensities is expected due to nuclear spin statistics (see Chapter 111, Section 4); preliminary calculation indicates that a dominant feature of the spectrum will be a series of closely spaced Qbranch-type transitions, namely, J-J, K - =O- 1. However, the asymmetry of the molecule and the type of transition which will be observed depend critically on the dihedral angle. Figure 7 . 6 ~gives the dependence of I,, II,, I, and (C- B) on the dihedral angle. It is apparent that and I, vary significantly as q is varied, although I, remains essentially constant. For values of q near 90” (or 270”) the molecule is a very slightly asymmetric rotor. At qo (90’5’) I b = I,, the molecule becomes an accidentally symmetric top, and the C2 symmetry axis changes from the b axis to the c axis. For values of q v;

Full P u m p P o w e r

D i s p o s i t to 11

L 01

Energy Level

Fig. 7.9 Frequency-modulated double resonance signals for various pumping conditions, and the three-level energy level arrangements in isoazole. The pump transition is indicated with a bold face arrow and the signal transition with a light arrow. Class I : pump transition lies outside the signal energy levels; Class 11: pump transition lies inside the signal energy levels. The positions of the weak double-photon transitions of Class 11 are indicated for clarity in the upper portion of the figure with an arrow. The sign displayed on the energy levels specifies whether the pump decreases or increases the population of the level. The zero-pump lines are pointing downward. Note for Class I, for example, with v , < v g , a derivative profile is obtained with a double-photon transition at higher frequency. For v p = v i , a symmetrical profile is obtained. Here and in other traces, the zero-pump line is directed downward opposite to the coherence doublet of Fig. 7.10. For isp> v: a derivative profile with the double-photon transition is at lower frequency. From Stiefvater [83].

and Y=lP,l

E,”

(7.87)

where E,” is the electric field amplitude of the pump radiation and lppl is the dipole moment matrix element for the pump transition. The choice of sign in


275

front of A v p in (7.84)and (7.85) depends on the relative order of the energy levels: plus sign for class I, negative sign for class 11. If the pump frequency is on resonance vp=v:, a symmetrical doublet is obtained that is centered about the v,” position and split by lyl

v,,=v,o-

IYI

2 Vs2=V,“f-

lYl 2

(7.88) (7.89)

This coherence splitting is illustrated in Fig. 7.10. On the other hand, when the pump radiation is above or below resonance, an asymmetrical doublet is obtained. One of the components, the “main line,” moves toward the unsplit position v,” and becomes stronger while the other component moves away from v,“ and becomes weaker, as shown in Figure 7.10. This component, which is called the “creeper,” corresponds to a double photon transition between, for example, the lower and upper state of the three-level system. For A v , B > y l

Fig. 7.10 The effect on an absorption line for the pump frequency on- and near resonance. (a)Absorption line pump radiation off. (6)Symmetrical doublet obtained when the pump radiation is on resonance =$). (c) Asymmetrical doublet obtained when the pump radiation is off resonance (v,f$). From Stiefvater [83]. (\ip

276

ASYMM ETRIC-TOP MOLECULES

we have

.v v,0

N

v," Avp

(7.90)

(7.91)

It is clear that the maximum signal absorption in the presence of the pump radiation does not occur at exactly v," but is shifted slightly. Likewise, the creeper is displaced from v," by essentially the amount that the pump frequency is off resonance but in opposite directions [taking the upper sign in (7.91)]. If, for example, v p> v;, then one signal peak for class I is shifted to higher frequency by a small amount; the other is shifted to lower frequency by essentially Av,. More accurate relations are available to higher order in IyI [69,82]. Various experimental configurations have been described for observation of microwave-microwave double resonance [74, 77, 83-92]. In a conventional Stark-modulated spectrometer, the usual spectrum is displayed and double resonances are detected by observation of changes in the line shape and intensity of a given line as the pump frequency is varied [74]. One of the more useful techniques makes use of double resonance modulation [77, 831 where only certain transitions that have a common energy level will be observed. Here one makes use of square-wave frequency modulation of the pump to modulate, via the double resonance effect, the absorption signal. A narrow-band amplifier tuned to the modulation frequency is employed along with the usual phasesensitive detector. The spectrometer is somewhat more complicated than a conventional Stark-modulated spectrometer because it requires two radiation sources. Both the signal and pump radiation propagate through the waveguide absorption cell. The lower frequency is usually chosen as the pump transition, since it is easy to prevent the modulated pump radiation from reaching the signal detector by use of a waveguide transition section with a cutoff frequency beyond that of the pump radiation. Often the pump frequency is selected, and the signal frequency is swept in the search for double resonant signals. A spectrometer may be designed to operate with either Stark modulation or double resonance modulation [83]. Figure 7.9 shows various energy level schemes for a three-level system that can give rise to a double resonance effect. Since the signal v, and the pump v p transitions share a common energy level, the signal transition will be affected by the high-power pump radiation when both are applied simultaneously to the sample. Since this double resonance effect allows the selective modulation of v, via vp, only the modulated v, reaches the detector. Thus, essentially only the transitions that share an energy level with the pump transition will be observed.


277

This virtually eliminates double resonance signals involving different rotational energy levels of a given molecule and from different molecules, for example, isotopic species. This is in contradistinction to Stark effect modulation where all possible absorptions are essentially modulated and hence dense spectra are possible. The appearance of the double resonance modulated signal for various pump frequencies is also illustrated in Fig. 7.9. It should be noted that the “zero-pump” line is slightly shifted from v,“ by an amount that depends on the size of pump power modulation. Under the usual conditions of the experiment, however, v,“ is obtained to better than f 0 . 2 MHz, and v,” should be better than f 0 . 5 MHz by adjustment of v p to give the symmetrical pattern [83] of Fig. 7.9. The attainment of the spectral patterns of Fig. 7.9 provides unambiguous evidence that the signal and pump transitions are coupled through a common energy level. This knowledge of the energy level connections, as pointed out previously, provides the basis for spectral assignments. A double resonance effect of reasonable intensity can be observed [93, 941 even when the pump frequency is away from the resonant frequency v,”. This detectability, can make it difficult to identify the desired transition. On the other hand, if the pump transitions are not too dense, this effect can be very useful. If more than one pump transition lies within say 50 MHz of another, the appropriately connected signal transitions will be simultaneously modulated and a scan of the signal region will reveal these transitions. Subsequent adjustment of v p to each resonance point allows measurement of the various v,” and v,”. Even when the pump frequency is far off resonance, a signal can be observed that arises from the so-called high-frequency Stark effect [95,96]. It is also possible to have a pump transition share an energy level with more than one signal transition. In isobutyryl fluoride [83], for example, if the pump frequency is (423+51~)~=24,249MHz, a signal can be observed at 25,103, 25,860, 31,416, and at 32,199 MHz corresponding, respectively, to the signal transitions (3 i 2 +423)~,(423 5 2 4 ) ~(4~1 3 5 1 4 ) ~(5~14-61 S ) S , and (5 14+625)~A further useful guide in the assignment of spectra, as pointed out by Woods et al. [77] and by Stiefvater [97], is the two different line profiles that are obtained, depending on the position of the common energy level with respect to the other two levels of the three-level system. In the Class I line shape category, the common energy level is the intermediate energy level; for Class 11, the common level is the highest or lowest energy level. For Class I, which is the most common, Fig. 7.9 shows that the double resonance lobes are stronger, the signal and lobes tend to cancel each other with a reduction of pump power in such a way that the lobes extinguish the “zero-pump” line, and the creeper moves to high (low) frequency if the pump radiation is tuned below (above) exact resonance by a few megahertz [upper sign in (7.91)]. On the other hand, for Class 11, the double resonance lobes are much weaker and virtually disappear when the pump power is reduced (leaving essentially the “zero-pump’’ line), and the creeper moves in the same direction as the pump frequency relative to the exact resonance position v,” [lower sign in (7.91)].

-

+

278


Clearly, a knowledge of the relative position of the common energy level further restricts the number of possible assignments that may be made for the pump and signal transitions, and hence provides additional information toward the assignment of an observed spectrum. Furthermore, these distinguishable line shapes are also useful in the assignment of isotopic species and vibrational excited states. For a particular double resonance signal of the ground state of the parent species, the line shape for the same three-level system for different isotopic species and vibrational states must be the same. A number of applications to the assignment of rotational spectra may be found in the previous references; see also [98]. The simplification possible by use of microwave-microwave double resonance is illustrated in Fig. 7.1 1 from the work of Pearson et al. [Sl]. The upper trace taken with Stark modulation shows a number of absorptions. The lower trace, a double resonance scan of the same region, shows only two features-an on-resonance line shape and an off-resonance profile.

n

Fig. 7.11 Illustration of the simplification possible with double resonance techniques. The upper trace is taken with the Stark modulation. The lower trace is taken with frequency modulated microwave-microwave double resonance, and only two signals are apparent. From Pearson et al. [91].


279

The power of the technique is illustrated by the assignment of 2,6-difluoropyridine by Stiefvater [99], who has assigned some 10 isotopic forms of the molecule, most of which were studied in their natural abundances. This can be difficult with use of Stark modulation only. Initial calculations indicated some key double resonance connections. For a detailed description of the process of the assignments, the reader should consult the original work [99]. A particularly convenient double resonance technique for identifying microwave transitions makes use of radiofrequency pump radiation. Transitions in this region arise from accidental near degeneraciesbetween levels of asymmetric tops and transitions between K-type doublets, as well as I-type doublets of linear molecules. Double resonance experiments with radio frequencies were first carried out by Autler and Townes [69]. More recently Wodarczyk and Wilson [82] have discussed the use of the technique for the assignment of complex asymmetric top spectra and have described details of the technique. The strong radiofrequency source is amplitude modulated with square-wave modulation. The modulated radiofrequency radiation can be introduced into the sample cell by its application across a conventional Stark waveguide cell as is done with a Stark square wave generator. Hence, both Stark spectroscopy and radiofrequency-microwave spectroscopy can be carried out with the same waveguide cell. Radiofrequency fields on the order of 3 to 15 V/cm are sufficient. Without paying special attention to impedance matching, the upper limit for the RF signal is about 100 MHz, and a RF power amplifier is usually required. As before, only those signal transitions will be observed which are modulated via the pump radiation; therefore, the pump and signal transitions must share a common energy level. Since the radiofrequency field is “off’ for half of the modulation cycle and “on” for the other half, the zero-field absorption signal and the double resonance signal can be displayed by use of phase-sensitive detection with opposite polarity. In the three-level systems of Fig. 7.9, when the energy levels of the pump transitions are close enough, the pump frequency can be in the R F region and R F pumping can be employed. Closely spaced levels are frequently encountered for slightly asymmetric tops. A typical energy level diagram for K-type doublets of a slightly asymmetric rotor is shown in Fig. 7.12. Two radiofrequency transitions (vpOl, vpOz) and two microwave transitions ( v : ~ ,vf2) are possible. Note that both signal transitions have energy levels in common with either pump transition. When v p is near vpOl or viz, the three-level situation applies. If the pump frequencies are close enough, say (vgz - vgl)? 15 MHz, the pump radiation over a range of pump frequencies will be near resonance to both transitions. For such a four-level system (Fig. 7.12) with v p near vpOl, or vjZ, a creeper and a main line appear about both zero-field signal frequencies and Figure 7.13 shows the line shape changes of two adjacent microwave double resonance signals arising from a pair of K-doublet transitions as the pump frequency is varied from slightly above vpOl to slightly below vgZ. The two creepers that face each other, clearly approach each other, merge at Cp, and again approach the zero-field positions as v p nears vEZ. This distinctive mirror-image pattern can be

280

ASYMM ETRIC-TOP MOLECULES

Fig. 7.12 A four-level double resonance system. A K-doublet energy level arrangement of a typical slightly asymmetric top is shown.

Q

TI

b

w

5

C

d

Fig. 7.13 Thedouble resonance signals ofskew-1-butene. T h e J = 3+4, K = 2 transitions are shown at different fixed pump frequencies, where v i , = 2.04 MHz and vpOt = 6.10 MHz. (a) v p = 3.00 MHz; (b) vp=4.05 MHz; (c) vp=4.50 MHz; (d) vp=6.00 MHz. From Wodarczyk and Wilson [82].

very useful in assignment of these transitions. The frequency at which the double resonance creepers from the adjacent double resonance signals of the four-level system merge is given by [82] (7.92)


281

This four-level effect can be used to measure vOpl, and vOp2 when direct measurement by tuning v p for a symmetrical line profile is not satisfactory (as with a usual three-level system). Here fp, v:~, v,“z are measured and (7.92) and the relation v:1

+

$2

= v,”,

+

v:2,

(7.93)

which follows from Fig. 7.12, are solved simultaneously for v , ” ~and v , ” ~ These . frequencies provide additional data for the determination of the rotational constants. In the radiofrequency region, higher-order terms affecting the intensities and positions of the double resonance components can be important although they are negligible in the microwave region. This is discussed elsewhere [82]. As before, the pump frequency does not have to be on resonance, and double resonance signals can be observed over a range of pump frequencies. Radiofrequency pumping is also particularly useful as an aid in the assignment of vibrational, excited-state transitions. Often the shift of the frequency of the pump transition with vibrational state is very small, and by pumping at the appropriate frequency for the ground state a whole series of excited-state transitions can be detected as the signal frequency is scanned [loo]. Other illustrative application of RFMDR may also be cited [101, 1021.

6


The vibrational motions of a molecule, which are ignored in the rigid-rotor approximation, affect the moments of inertia and result in a dependence of the moments of inertia on the vibrational state. The spectroscopic constants ascertained from an analysis of the rotational spectra are hence effective constants. The dependence of the effective rotational constants on the vibrational quantum numbers may be expressed by A” = A ,

-Ei

q u i ++)

(7.94) (7.95)

c, = c, -

+

af(ui +) i

(7.96)

where A,, etc., are the equilibrium constants and where the vibrational state u is specified by the quantum numbers (ul, u2, . . ., ui,. . . ) with vi the vibrational quantum number of the i th vibration, and where the sum is over the various vibrations. The vibration-rotation constants depend on both the potential constants and the Coriolis coupling constants. To account for various effects of vibration in the rotational spectrum, and to obtain explicit expressions for the interaction

282


constants such as the ai)s in terms of the fundamental molecular constants, a general rotation-vibration Hamiltonian is required. In this section, the appropriate Hamiltonian is described and a perturbation technique for obtaining the energies to the desired order is outlined. Nonrigidity effects arising from centrifugal distortion also shift the rigid rotor energy levels. These corrections are discussed further in Chapter VIII. Large amplitude motions are treated in Chapter XI1 and in Chapter VIII, Section 9.

Rotat i o n-Vi brat i o n Ham i Iton ia n If in addition to rotational motion vibrational motion is included, then an internal angular momentum 71, relative to the molecule-fixed axis system, will arise from the rotation-vibration interaction. If P, signifies the overall angular momentum components, then the rotational Hamiltonian may be generalized as (7.97) where pup (a, p=x, y , z ) is an element of the inverse moment of inertia tensor and is here a function of the vibrational coordinates, which may be taken as the 3N-6 normal coordinates Q k . The Pk is the vibrational momentum conjugate to Qk (Pk=- ih8/8Qk),and V a vibrational potential function. A rigorous derivation of this Hamiltonian for a nonlinear molecule starts with construction of the classical Hamiltonian, employing a center-of-mass, rotating, coordinate system and a set of coordinates that specify the positions of the atoms relative to each other in the rotating system. This is followed by transformation to a quantum mechanical Hamiltonian operator. The derivation of the general rotation-vibration Hamiltonian has been discussed by Wilson and Howard [103], Darling and Dennison [104], Nielsen [l05], and more recently further simplified by Watson [1061 to the form given in (7.97).This is a consequence of the commutator relation

1a [ E a , ~ a a I = o

(7.98)

A derivation of (7.97)starting with the Schrodinger equation has been given by Louck [107]. For further references discussing aspects of the rotation-vibration Hamiltonian, see Chapter VI11 and [lo81 through [I 131. The general Hamiltonian contains only terms quadratic in Pa, whereas we have seen in previous chapters that the effective rotational Hamiltonian contains higher terms in the angular momentum. The usual analysis procedure entails expanding papand V in terms of the vibrational coordinates followed by a perturbation treatment to obtain the energies correct to a given order. The zeroth-order vibrational Hamiltonian involves only the quadratic terms from V (7.99)


283

with &=47c2C20: and w k the harmonic vibrational frequency in cm-' units of the kth normal mode Q k .It is convenient to employ the dimensionless normal coordinates q k and momentum P k defined as follows (7.100) thus (7.101) These and other terms in the Hamiltonian (7.97) will be divided by hc to express all terms in consistent wavenumber units (cm- '). The eigenfunctions of X:,, are product functions of harmonic oscillator eigenfunctions

n

3N-6 yU=

(7.102)

*Uk(qk)

k= 1

where u represents the set of vibrational quantum numbers vibrational state and eigenvalues e i b

y u =E:yu

=E

wk(uk

uk

specifying the

-k $)$u

(7.103)

k

For a perturbation treatment, matrix elements of ( u k 1qklok (UklPkluk+

qk

and P k will be required

+ 1)= [ ( u k + 1)/2] 1'2, ( U k l q k l u k - 1)= [uk/2] l i Z

(7.104)

I ) = -i[(uk+

(7.105)

1)/2]'i2,

(UklPklUk-

1)=i[uk/2]1i2

Other matrix elements may be readily obtained from tabulations given elsewhere [Ill]. The general vibrational potential function is expressed as (7.106) where klmnare the cubic anharmonic potential constants. The next terms in the expansion would involve the quartic constants k,,,. The term U is a pseudopotential term of quantum mechanical origin [1061 (7.107) which depends only on the vibrational coordinates and represents a small mass-dependent correction to the vibrational potential energy. This term may be neglected except for very precise work. The effective vibrational angular momentum in (7.97) is defined by (in units of h ) (7.108)

284

ASY MM ETRIC-TOP MOLECULES

where (8’((8)=-(E) and (8)=0 if k = I ) are the Coriolis coupling constants, coupling Qkand QIvia rotation about cc (see Chapter XI11 for further discussion). The pas may be expanded about the equilibrium configuration as (7.109) which is assumed to converge rapidly (small oscillations approximation). Here (7.110) and the derivatives are given by [106, 1081 (7.111) (7.1

(7.1 3 ) Since the internal coordinates Ri are related to the normal coordinates by the transformation R i = Z LikQk,(7.113) can be written in the form Upa’

=

1i J$Lik

(7.114) (7.115)

The Lik are obtained from a solution of the vibrational eigenvalue secular equation (see Appendix H). Derivation of expressions for J $ are given in Chapter VIII. Insertion of (7.109) and (7.106)into (7.97) gives for the expanded rotationvibration Hamiltonian

(7.1 16)

Clearly the expansion can be carried to higher order if required. The various terms resulting in the expanded Hamiltonian may be classified with the notation Xnm, where n is the degree in the vibrational operators qn(qandlorp for x ~n ,=2)


285

and m is the degree in the rotational operators P",

*=

*02(K2)+

*20(K0)+

*21(K2)+

*0,2(K3)+

+

-@22(K4) *30(K)+

*31(~3)+

. . . (7.117)

The order of magnitude of the coefficients is specified in terms of the BornOppenheimer expansion parameter K =(me/mn)1/4 =& as suggested by Oka [1141 and further clarified by Pedersen [1151. This parameter measures the size of a term relative to the vibrational energy K " + ~ ~ - ~ EThe , . K-order is also indicated in (7.117). The rigid rotor rotational constants are thus smaller than E, by a factor K', and smaller than a typical electronic energy E , by K ~ This . classification allows one to choose the particular terms required to obtain the energies by means of a perturbation treatment to a particular K'. Terms such as &hornp of order K', which represent a purely vibrational correction, are often not required. Furthermore, since the Hamiltonian must be invariant to the symmetry operations of the particular molecular point group, many terms will be found to vanish in a specific application. For a term such as X12 -upp) qkpapp, a given upp)can be nonvanishing only if the product of the symmetry species of qk, Pa and P p is totally symmetric. Similarly, a Coriolis coupling constant Cfi) is nonvanishing only when qkplPabelongs to the totally symmetric species. An excellent discussion of the molecular symmetry group for both rigid and nonrigid molecules has been given by Bunker [1161(see also references cited therein). In addition, a number of sum rules exist for the inertial derivatives upp)(e.g., x k (up))'=4C) and the Coriolis coupling constants Cfi) which can be used to simplify the general interactions of (7.119)-(7.121). These relations are discussed elsewhere [117-1221, Similarly, isotopic relations between such parameters are very useful and have been reported [123]. We now consider some of the more important terms of (7.116). The zerothorder vibrational Hamiltonian X z 0of order K O has already been defined in (7.101). The zeroth-order or rigid-rotor rotational Hamiltonian is given by the leading term of pnp (7.118) where the rotational constant B,=h2/2hcG is in wavenumber units. Since the principal axes system is chosen for the equilibrium configuration, Gp(uf p) will vanish. In a similar way we find, for example,

(7.119)

286


r

1

=E Rkqk

(7.120)

k

(7.121) (7.122)

The definitions of the R-operators [124] are enclosed in brackets. The forms of Xnm in terms of the R’s are convenient for the perturbation treatment to be discussed. The dimensionless parameter CPp)is defined by (7.123) Recall that the operators q k , Pk, P a , ?I, are dimensionless and that the Xnm are in wavenumber units. The X z 1term represents a Coriolis interaction, X12and X Z 2represent correction terms to the rigid rotor which arise from the linear and quadratic dependence of p on the vibrational coordinates, respectively, and X 3 0represents an anharmonicity correction to the potential energy which can give rise to anharmonic resonance effects. For a perturbation calculation of the rotation-vibration energies, this Hamiltonian may be subjected to successive contact transformations chosen to effectively diagonalize the Hamiltonian in the vibrational quantum numbers, thereby giving an effective rotational Hamiltonian for each vibrational state. This procedure has been extensively discussed in the literature [log, 124-1261. Detailed expressions (although somewhat complicated) of the transformed Hamiltonian and appropriate matrix elements for various types of rotors have been compiled in a monograph by Amat et al. [lOS] to a high order of approximation. The Van Vleck transformation has also been applied [103, 1271. An alternate approach uses standard perturbation theory, to the appropriate order, with the harmonic oscillator functions as basis functions to effectively diagonalize X in v and thus provide the effective rotational Hamiltonian 2 for a given vibrational state. The K-order and the type of rotational operator dependence sought are used as a guide for choice of the appropriate terms to treat in the expanded Hamiltonian. A few examples are given here to illustrate the procedure. The first- and second-order perturbation correction are expressed by (VI 2 l v ) = ( o l

X”’v)

(7.124)

EFFECTS

OF VIBRATION

287

(7.125) where X ‘ represents the perturbing Hamiltonian and, in general, contains all the terms of (7.117) except the zeroth-order vibrational Hamiltonian 220. The transformed Hamiltonian_operator *nm also labeled by n and rn are given by the diagonal elements (ul Xplu) via the foregoing equations. This type of treatment requires that the spacing of the rotational energy levels be small compared to the vibrational spacings and that no near-degenerate vibrational levels are connected by the perturbation. This approach is analogous to the Van Vleck perturbation technique (Appendix C) under similar assumptions. The rotational operators in 2’are treated as noncommuting parameters independent of u. Third- and fourth-order perturbation formulas appropriate to this treatment, which take into account that the order of the *Lot elements in the perturbation formulae are important in view of the noncommuting character of the rotational operators, have been given by Georghiou [128]. In a particular rotation-vibration interaction, only those terms in X ‘ that contribute to the interaction need be retained. The leading term, for example, representing the effects of centrifugal distortion (considered further in Chapter VIII) involves purely rotational terms of P4. Thus the equilibrium quartic distortion Hamiltonian is specified by the transformed Hamiltonian Xo4of A second-order treatment of 2 n m and (~ Xr ne ) m gives r ( ~ sa )contribuorder tion of order IC‘+’and red-uces the power of the vibrational operators by two. Thus the desired form of 2 0 4 arises from a second-order treatment of XI2(rc3). A discussioc regarding the selection of the terms to be considered to obtain a particular Xnm has been given by Mills [129]. Setting X ’ = X12 in (7.125) we have, because of (7.104),

(7.126) which gives the correct rotational operator dependence; the distortion coefficient is defined by fapya

= -2

c

Okcfp’ct’a’

(7.127)

k

This may be expressed in terms of u f p ) from (7.123). When the vibrational and rotational energy differences are not widely separated, this perturbation treatment will be inappropriate. In this case, a more appropriate basis for the perturbation treatment are the product functions

288


of X 2 0and XO2, that is, YvYJz.The denominators in (7.126) will then also involve explicitly the rotational energy difference. An example of this type of effect is provided by HNCO [130] where the K - = 7 level of the ground state is very close in energy to the K - l = 6 level of the lowest bending vibrational state u5 = 1. The vibrational correction to the rotational constants aip)are coefficients of (uk ++)Pb which appears in the transformed Hamiltonian i%22(~4) since i . ( u k l q ~ + p ~ l ~ k ) = u k + + This type of term results [129] from a first-order treatment of X 2 2 ( ~and 4 ) ,second-order treatments of X 2 1 (x~X 22 ) 1(~ and 2) X 3 0 ( ~ x) X12(~3). The origin of these terms is clear from inspection of the individual X,,,, and the three contributions may be classified, respectively, as a harmonic, Coriolis, and anharmonic contribution. The resulting expression is [131]

+

+

aip)=r@)(har) af)(Cor) af)(anhar)

A first-order treatment of X 2 2X ( ’ = X22), for example, readily yields the first

term of (7.128). If it is desirable to obtain the definition of the equilibrium coefficients of the sextic distortion constants, which aresoefficientsof P6 of order K~ O, five terms [124,128] are found to contribute to 2 0 6 which require a laborious calculation. The appropriate terms arise from a third-order treatment of X 1 2 (x~X 31 ) 2(~3 x X22(~ and 4 ) a fourth-order treatment of X 3 0 ( ~ l x) X 1 2 (x~X31) 2 ( ~ 3 ) 2 0 2 ( ~ x~ 2) 0 2 ( ~ x X 1 2 ( ~ ~ ) , * 2 1 ( ~ ’ ) x X 2 1 ( ~ x~ % ) 1 2 , ( ~ ~x) * 1 2 2 ( ~ ~ ) , x X 1 2 ( x~X 31 ) 2 ( ~and 3 ) ,X 2 1 ( x~X 20) 2 ( ~ x2 X)1 2 ( x~X 31 ) 2 ( ~Each 3 ) . of

these coiitributions is of order of magnitude K ~ O The . presence of n products of Xnm indicates the use of nth-order perturbation theory. If X ‘ = 2 1 2 + X 2 2 is set in the appropriate third-order perturbation formula, which will give terms of the type (X12 x X12x X22)/(AEt,b)2, the contribution of the first-indicated term may be evaluated [l28]. Other product terms which might appear to contribute can be shown either to vanish or to have inappropriate rotational operator dependence. Also contributions of apparent degree greater than six which is diagonal in u, can be reduced to sixth degree, one power involving XO2, for each X O 2[128]. In fact, it is possible to drop either two powers of q or p , or one power of P for each AE,” in the denominator of a perturbation formula [129]. Hence, the terms to include are not always obvious and caution must be exercised that all the appropriate contributions have been included. For symmetric tops, complications arise from the presence of degenerate vibrations, but the analysis is similar to that just described. Discussions of vibration-rotation effects are given in Chapter VI, and further treatments [110, 129, 132-1341 and illustrative applications may be cited [135-1381. Coriolis interactions in asymmetric tops are treated in the next section.

~ )


289

Coriolis Perturbations Because of the lower rotational symmetry of asymmetric rotors there are no inherent degenerate modes of vibration, and hence there are no first-order Coriolis effects such as were discussed previously for symmetric rotors in a degenerate vibration state. However, higher-order Coriolis perturbations of the rotational energy level structure are possible and can be particularly significant when there is an accidental vibrational degeneracy. This condition has been discussed by Wilson [139], and the treatment may also be applied with appropriate modification to symmetric rotors having two nondegenerate nearby vibrational states. The Coriolis interaction is specified by 2’ given in (7.119), which we write here in the form (7.129) The n,(a = x , y , z), which are components of the vibrational angular momentum along the molecule fixed axes, are defined as (7.130) in terms of the dimensionless q k and p I .The previous perturbation treatment is not adequate when near-degenerate vibrational levels are connected by the perturbation term; rather, the energy matrix of the interacting levels is diagonalized directly. The Hamiltonian including coupling between rotation and vibration may be written as

where the first term is the rigid rotor Hamiltonian, yi“, is the vibrational Hamiltonian, and the second term represents the coupling between rotation and vibration. The energy matrix may be constructed with basis functions which are products of symmetric rotor functions and vibrational functions. The vibrational functions are solutions of the vibrational Hamiltonian and may, to a good approximation, be taken as harmonic oscillator functions. The energy matrix will have both diagonal and off-diagonal elements in u. The off-diagonal elements in v come from the operator n, in the coupling terms. For nondegenerate states the diagonal elements of na vanish, and hence the Coriolis interaction does not contribute elements diagonal in v. Part of the infinite energy matrix is illustrated in Figure 7.14.When the two vibrational states u and v‘ have nearly the same energy E , N E,, the rotation-vibration coupling can be large. If the other vibrational states have energies significantly different

290


Fig. 7.14 Illustration of part of the infinite matrix for X in the basis ri/Jh$L. The doubly shaded blocks contain the matrix elements connecting the two vibrational states of nearly the same energy. The unshaded blocks, which represent the connections between states of widely different energy, may be neglected. The 2 x 2 block submatrix may hence be factored from the remaining matrix. This submatrix is diagonal in J and has 2(2J + 1 ) rows and columns labeled by the values of K ( 0 , .t 1, . . . , . t J ) and u(u, d).

from E , and E,., the connections between these states and u or u‘ may be neglected, and only the elements connecting the nearly degenerate states u and u‘ need be retained. The submatrix composed of the diagonal and off-diagonal elements of the two interacting vibrational states u, ut can thus be separated from the rest of the matrix (see Fig. 7.14). Diagonalization of this 2 x 2 block submatrix gives the energy levels. Employing Table 7.2 we find that the nonvanishing matrix elements of the secular equation are: (7.132)

(J,K , u ~ JK, , u ) = K 2 + 0 , - c (J,K , u’IJ, K , d ) = K 2 +o,,-o

(7.133)

(J, K , v ~ JK+2, , u)=(J, K , dlJ, K + 2 , u’) =-fb{[J2-(Kf1)2][(J+1)2-(K+

1)2])”*

(7.134) (7.135)

(J,K , u ~ JK, , ut)= -(J, K , u’IJ, K , u)=iG,K (J,K , u ~ JK, + 1, u’)= -(J, K , d l J , K + 1, U ) =:(iG, T G , ) [ ( J T K ) ( J +K

+ l)]’”

(7.136)

where here

(7.137)

ASYMMETRIC ROTOR STRUCTURES

291

(7.138)

(7.139)

(7.140) In the above matrix elements of the secular equation it has been assumed that the effective moments of inertia are the same in the two vibrational states. The G, may be evaluated explicitly if sufficient information on the molecular force field is known, or they may be treated as empirical parameters. For G, to be nonvanishing, ll/vll/o.must have the same symmetry as II,under the molecular point group. With asymmetric rotors of symmetry r! V,, and C2,, no more than one of the G, will be nonvanishing for a given pair of states u, u‘. Furthermore, by use of a new set of basis functions SJKy 9, and SJK,t,bVj, where the rotational functions are the Wang functions, the matrix may be factored into four smaller submatrices [1391. For l k l to be of reasonable magnitude, the two states u and u‘ are required to differ in only two of their vibrational quantum numbers u k , and these two quantum numbers must change by one unit from u to u’. For the levels to have nearly the same energy, this essentially limits the state u’ to one in which u; =uk 1 and u; = uI- 1 for a pair of normal modes k and 1. Evaluating l k l with harmonic oscillator functions [See (7.104) and (7.105)] yields

+

(7.141) where a k is the frequency of the kth vibration. Some examples of Coriolis interactions studied in asymmetric rotors are SO2F2 [140], CH2(CN)2 [141], F2CO [142], H2CO [143], FNO2 [144], O 3 [145], OF2 [146], HNCS [147], HNCO [148], and (CH,), CCCH, [149]. Such studies give information on Coriolis coupling constants and on the separation of the interacting vibrational states.

7

ASYMMETRIC ROTOR STRUCTURES Most applications must be postponed until treatments of centrifugal distortion, hyperfine structure, internal rotation, Stark and Zeeman effects have

292

ASYM M ETRIC-TOP MOLECULES

been given and until methods for calculation of molecular parameters from spectral constants have been described. However, a few remarks will be made here regarding the use of rotational constants in the derivation of molecular structures. Illustrative values of spectral constants A, B, and C are given in Table 7.1. Analysis of the rotational lines of an asymmetric top gives the three principal moments of inertia from which, with known atomic masses, the internuclear distances and bond angles can be evaluated. However, except for the simplest asymmetric-top molecules there are more structural parameters to be determined than moments of inertia from one isotopic species; therefore a number of isotopic species must be investigated. The investigation of a sufficient number of isotopic species can be tedious, especially when the line intensities are not large enough for observation of the isotopic species in natural abundance and chemical synthesis of a number of isotopic species becomes necessary. Nevertheless, the structures of numerous molecules have been evaluated by this procedure. Various methods of calculating molecular structures of asymmetric rotors from rotational constants are described in Chapter XIII. In principle, very precise determinations of bond lengths and bond angles can be made from measurements of pure rotational spectra. Unfortunately, because the moments of inertia generally obtained are not equilibrium moments but effective ground-state moments of inertia, which are contaminated with zero-point vibrational effects, ambiguities are transmitted to the derived structural parameters. The methods of evaluation of molecular structures from pure rotational spectra and techniques for minimizing vibrational effects will be discussed in Chapter XIII. To obtain the equilibrium moments of inertia one must measure rotational transitions in excited vibrational states to enable the evaluation of the ai's in (7.94)-(7.96). Usually, some or all of the vibrational frequencies are greater than 500 cm-', and the rotational transitions in the excited vibrational states are weak because of the small population. The population F , in the excited vibrational state can be increased by heating the gas under observation. However, the line intensity cannot always be increased sufficiently because of the offsetting effect due to the inverse dependence of the intensity on the temperature [see (7.73)]. So far only a few equilibrium structures of asymmetric rotors have been evaluated (see Chapter XIII, Section 6). In a previous edition of this book [1501 we gave a reasonably complete listing of structures through about 1969. Since most molecules are asymmetric tops and since the output of molecular structures by microwave spectroscopists has been enormous during the last ten years, such a listing is no longer feasible. However, some illustrative structures of asymmetric tops are given in Chapter XIII. For more complete compilations the reader is referred to those referenced in Chapter XI11 and the Landolt-Bornstein tables [1513.

References 1. G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys., 1 1 , 2 7 (1943). 2. See, for example, H. H. Nielsen, Rev. Mod. Phys., 23, 90 (1951) and the references quoted therein.

REFERENCES

293

3. W. Gordy, W. V. Smith, and R. F. Trambarulo, Microwave Specrroscopy, Wiley, New York, 1953. 4. B. S. Ray, 2. Physik., 78, 74 (1932). 5. R. S. Mulliken, Phys. Rev., 59, 873 (1941). 6. S. C. Wang, Phys. Rev., 34,243 (1929). 7. J. H. Van Vleck, Phys. Rev., 33,467 (1929). 8. See, for example, A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers, Wiley, New York, 1960. 9. M. W. P. Strandberg, Microwave Spectroscopy, Wiley, New York, 1954. 10. J. D. Swalen and L. Pierce, J. Math. Phys., 2, 736 (1961). 11. R. M. Hainer, P. C. Cross, and G. W. King, J. Chem. Phys., 17, 826 (1949). 12. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. 13. H. C. Allen, Jr. and P. C. Cross, Molecular Vib-Rotors, Wiley, New York, 1963. 14. J. W. Blaker, M. Sidran, and A. Kaercher, J. Chem. Phys., 37, 684 (1962); M. Sidran, F. J. Nolan, and J. W. Blaker, J. Mol. Spectrosc., 11, 79 (1963); F. J. Nolan, M. Sidran, and J. W. Blaker, J. Chem. Phys., 41, 588 (1964). 15. R. H. Schwendeman, “A Table of ( P i ) and ( P t ) for Asymmetric Rotor Molecules,” Michigan State University, East Lansing, Michigan. 16. H. Dreizler and R. Peter, J. Chem. Phys., 39, 1132 (1963). See also H. Dreizler, R. Peter, and H. D. Rudolph, Z . Naturforsch., 21%2058 (1966). 17. D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys., 20, 1575 (1952). 18. J. K. Bragg and S . Golden, J. Chem. Phys., 75,735 (1949). 19. This may be easily seen as follows: the energy of an asymmetric rotor may be written in the form E = A ( P i ) + B ( P i ) + C ( P f ) ,where the average values are evaluated in the asymmetricrotor basis. This expression readily yields to first-order, (7.29)-(7.31). 20. R. H. Schwendeman, J . Chem. Phys., 27,986 (1957). R. H. Schwendeman, “A Table of Coefficients for the Energy Levels of a Near Symmetric Top,” Department of Chemistry, Harvard University, Cambridge, Massachusetts. 21. D. Kivelson, J. Chem. Phys., 21, 536 (1953). 22. G. Winnewisser, M. Winnewisser, and W. Gordy, J. Chem. Phys., 49, 3465 (1968). 23. P. C. Cross, R. M. Hainer, and G. W. King, J. Chem. Phys., 12,210 (1944). 24. R. H. Schwendeman, J. Mol. Spectrosc., 7,280 (1961). 25. D. R. Lide, Jr., J. Chem. Phys., 20, 1761 (1952). 26. R. H. Schwendeman and V. W. Laurie, Table of Line Strengths for Rotational Transitions of Asymmetric Rotor Molecules, Pergamon, London, 1958. 27. P. F. Wacker and M. R. Pratto, Microwave Spectral Tables, Vol. 11. Line Strengths of Asymmetric Rotors (Natl. Bur. Std. Monograph 70), 1964. 28. See [121, Appendix V. 29. H. D. Rudolph, Z.Phys., 211,419 (1968). 30. H. D. Rudolph, 2. Naturforsch., 23a, 1020 (1968). 31. E. K. Gora, J. Mol. Spectrosc., 2, 259 (1958). 32. L. H. Scharpen, Symposium on Molecular Structure and Spectroscopy, Paper 09, Ohio State University, Columbus, Ohio 1969. 33. L. H. Scharpen and V. W. Laurie, Anal. Chem., 44,378R (1972). 34. G. E. Jones and R. L. Cook, ‘CRC Crit. Rev. in Anal. Chem.,’ 3,455 (1974). 35. W. E. Steinmetz, J. Am. Chem. SOC.,96,685 (1974). 36. R. J. Higgins, L. L. Combs, T. B. Malloy, Jr., and R. L. Cook, J. Mol. Struct., 28, 121 (1975). 37. M. S. Farag and R. K. Bohn, J. Chem. Phys., 62,3946 (1975).

294

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

66.

67. 68. 69. 70. 71. 72. 73.


E. M. Bellott, Jr., and E. B. Wilson, Jr., Tetrahedron, 31, 2896 (1975). E. M. Bellott, Jr., and E. B. Wilson, Jr., J. Mol. Spectrosc., 66,41 (1977). W. J. Lafferty, J. Mol. Spectrosc., 38, 84 (1970). T. Ikeda, R. Kewley, and R. F. Curl, Jr., J. Mol. Spectrosc., 44,459 (1972). H. M. Pickett, J . Am. Chem. SOC.,95, 1770 (1973). R. W. Kitchin, T. K. Avirah, T. B. Malloy, Jr., and R. L. Cook, J. Mol. Struct., 24, 337 (1975). R. W. Kitchin, T. B. Malloy, Jr., and R. L. Cook, J. Mol. Spectrosc., 57, 179 (1975). D. R . Lide, Jr., “Microwave Spectroscopy” in Methods of Experimental Phjsics, Vol. 3, Molecular Physics, Part A, D. Williams, Ed., Chapter 2.1 Academic, New York, 1974. S. J. Borchert, J. Mol. Spectrosc., 57, 312 (1975). J. A. Wells and T. B. Malloy, Jr., J. Chem. Phys., 60, 3987 (1974). T. K. Avirah, T. B. Malloy, Jr., and R. L. Cook, J. Chem. Phys., 71,2194 (1979). T. Ikeda, K. V. L. N. Sastry, and R. F. Curl, Jr., J . Mol. Spectrosc., 56,411 (1975). R. A. Creswell and W. J. Lafferty, J. Mol. Spectrosc., 46,371 (1973). T. K. Avirah, T. B. Malloy, Jr., and R. L. Cook, J. Mol. Struct., 26, 267 (1975). M. Winnewisser, G. Winnewisser, T. Honda, and E. Hirota, Z. Naturforsch., Ma,1001 (1975). B. P. van Eijck, J. Mol. Spectrosc., 38, 149 (1971). A. B. Delfino and K. R. Ramaprasad, J. Mol. Struct., 25,293 (1975). A. Bouchy and G. Roussy, J. Mol. Spectrosc., 78,395 (1979). J. R. Durig, C. C. Tong, and Y. S. Li, J. Chem. Phys., 57,4425 (1972). S. Doraiswamy and S. D. Sharma, J. Mol. Spectrosc., 58,323 (1975). F. Kneubuhl, T. Gaumann, and H. Giinthard, J. Mol. Spectrosc., 3, 349 (1959). R. A. Beaudet, Ph.D. Dissertation, Harvard University, 1962. G. F. Pollnow and A. J. Hopfinger, J. Chem. Educ., 45,528 (1968). K. M. Marstokk and H. Msllendal, J. Mol. Struct., 4,470 (1969). J. G. Baker, “Microwave-Microwave Double Resonance” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, New York, 1979, pp. 65-121. H. Jones, “Infrared-Microwave Double Resonance Techniques” in Modern Aspects of Microwave Spectroscopy, G. W. Chantry, Ed., Academic, New York, 1979, pp. 123-216. J. I. Steinfeld and P. L. Houston, “Double Resonance Spectroscopy” in Laser and Coherence Spectroscopy, J. I. Steinfeld, Ed., Plenum Press, New York, 1978, pp. 1-123. K. Shimoda, “Infrared-Microwave Double Resonance,” pp. 29-44; T. Oka, “InfraredMicrowave (Radiofrequency) Two-Photon Spectroscopy,” pp. 413-43 1 ; J. C. McGurk, C. L. Norris, T. G. Schmalz, E. F. Pearson, and W. H. Flygare, “Infrared-Microwave Double Resonance Measurements of TI in Methyl Fluoride and Methyl Chloride,” pp. 541-554, in Laser Spectroscopy, R. G. Brewer and A. Mooradian, eds., Plenum Press, New York, 1974. K. Shimoda and T. Shimizer, “Non-Linear Spectroscopy of Molecules” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, Eds., Pergamon, Oxford, 1972, Vol. 2, pp. 43-139. T. Oka, Ad. At. Mol. Phys., 9, 127 (1973). T. Oka, “Forbidden Rotational Transitions” in Molecular Spectroscopy: Modern Research, K. N. Rao, Ed., Academic, 1976, Vol. 11, pp. 229-253. S. H. Autler and C. H. Townes, Phys. Rev., 100, 703 (1955). K. S. Shimoda, J. Phys. SOC. Japan 14,954 (1959). A. Battaglia, A. Gozzini, and E. Polacco, Nuouo Cimento 14, 1076 (1959). T. Yajima and K. Shimoda, J. Phys. SOC.Japan, 15, 1668 (1960). T. Yajima, J. Phys. SOC.Japan, 16, 1709 (1961).

REFERENCES

74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. loo. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114.

295

A. P. Cox, G. W. Flynn, and E. B. Wilson, Jr., J. Chem. Phys., 42, 3094 (1965). M. L. Unland, V. Weiss, and W. H. Flygare, J. Chem. Phys., 42,2138 (1965). M. L. Unland and W. H. Flygare, J. Chem. Phys., 45,2421 (1966). R. C. Woods, 111, A. M. Ronn, E. B. Wilson, Jr., Rev. Sci. Instrum., 37,927 (1966). P. G. Favero, F. Scappini, and A. M. Mirri, Boll. Sci. della Facolta’ di Chim. Indust. (Bologna), 24, 93 (1966). A. Javan, Phys. Rev., 107, 1579 (1957). A. DiGaicomo, Nuovo Cimento 14, 1082 (1959). T. Yajima, J. Phys. SOC.Japan, 16, 1594 (1961). F. J. Wodarczyk and E. B. Wilson, Jr., J. Mol. Spectrosc., 37, 445 (1971). 0. L. Stiefvater, Z. Naturforsch., 30a, 1742 (1975). H. Mader, H. Dreizler, and A. Guarnieri, Z. Natutjorsch., 30a, 693 (1975). H. D. Rudolph, H. Dreizler, and U. Andresen, Z. Naturforsch., 26a, 233 (1970). J. Ekkers, A. Bauder, and Hs. H. Gunthard, J. Phys. E. Sci. Instrum., 8,819 (1975). J. Ekkers, A. Bauder, and Hs. H. Gunthard, Rev. Sci. Instrum., 45, 311 (1974). M. C. Lee and W. F. White, Rev. Sci. Instrum., 43,638 (1972). G. K. Pandey and H. Dreizler, Z. Naturforsch., 31a, 357 (1976). T. Oka, Can. J. Phys., 47, 2343 (1969). R. Pearson, Jr., A. Choplin, V. Laurie, and J. Schwartz, J. Chem. Phys., 62, 2949 (1975). R. G. Ford, J. Chem. Phys., 65, 354 (1976). G. W. Flynn, J. Mol. Spectrosc., 43, 353 (1972). P. Glorieux, J. Legrand, B. Macke, and J. Messelyn, C. R. Acad. Sci. Paris 270B, 1412 (1970). P. Glorieux, J. Legrand, B. Macke, and J. Messelyn, J. Quant. Spectrosc. Radiat. Transfer, 12, 731 (1972). B. Macke and P. Glorieux, J. Mol. Spectrosc., 45, 302 (1973). 0. L. Stiefvater, Z. Naturforsch., 30a, 1756 (1975). See for example: 0. L. Stiefvater, J. Chem. Phys., 63,2580 (1975); 0.L. Stiefvater, H. Jones, and J. Sheridan, Spectrochim. Acta, 26a, 825 (1970); ref. 55. 0. L. Stiefvater, Z . Naturforsch., 30a, 1756 (1975). R. G. Ford, J. Mol. Spectrosc., 49, 117 (1974). R. D. Suenram, J. Mol. Struct., 33, 1 (1976). G. N. Mathur and M. D. Harmony, J. Mol. Spectrosc., 69, 37 (1978). E. B. Wilson and J. B. Howard, J. Chem. Phys., 4,260 (1936). B. T. Darling and D. M. Dennison, Phys. Rev., 57, 128 (1940). H. H. Nielsen, Rev. Mod. Phys., 23, 90 (1951). J. K. G. Watson, Mol. Phys., 15, 479 (1968). J. D. Louck, J. Mol. Spectrosc., 61, 107 (1976). G. Amat, H. H. Nielsen, and G. Tarrago, Rotation-Vibration of Polyatomic Molecules, Marcel Dekker, New York, 1971. J. E. Wollrab, Rotational Spectra and Molecular Structure, Academic, New York, 1967. H. W. Kroto, Molecular Rotation Spectra, Wiley, New York, 1975. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. H. M. Pickett, J. Chem. Phys., 56, 1715 (1972). A. Attanasio, A. Bauder, Hs. H. Gunthard, and H. J. Keller, Mol. Phys., 23, 35 (1971). T. Oka, J. Chem. Phys., 47, 5410 (1967).

296

ASY MM ETRIC-TOP MOLECULES

115. T. Pedersen, J. Mol. Spectrosc., 80,229 (1980). 116. P. R. Bunker, Molecular Symmetry and Spectroscopy, Academic, New York, 1979. 117. J. H. Meal and S. R. Polo, J. Chem. Phys., 24, 1119, 1126 (1956). 118. T. Oka and Y. Morino, J. Mol. Spectrosc., 6, 472 (1961); 11, 349 (1963). 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129.

130. 131.

132. 133. 134. 135. 136. 137. 138.

L. Nemes, J. Mol. Spectrosc., 28, 59 (1968). T. Oka, J. Mol. Spectrosc., 29, 84 (1969). J. K. G. Watson, J. Mol. Spectrosc., 39, 364 (1971). M. R. Aliev, J. Mol. Struct., 23,411 (1974). A. D. Bykov, Y. S. Makushkin, and 0. N. Ulenikov, J. Mol. Spectrosc., 85,462 (1981). M. R. Aliev and J. K. G. Watson, J. Mol. Spectrosc., 61, 29 (1976). A. Niroomand-Rad and P. M. Parker, J. Mol. Spectrosc., 85,40 (1981); 75,454 (1979). F. W. Birss, Mol. Phys., 30, 111 (1975). T. Pedersen, J. Mol. Spectrosc., 13, 360 (1978). C. Georghiou, Mol. Phys., 32, 1279 (1976). I. M. Mills, “Vibration-Rotation Structure in Asymmetric- and Symmetric-top Molecules,” in Molecular Spectroscopy: Modern Research, K. N. Rao and C. W. Mathews, Eds., Academic, New York, 1972. K. Yamada, J. Mol. Spectrosc., 81, 139 (1980). J. K. G. Watson, “Aspects ofQuartic and Sextic Centrifugal Effects on Rotational Energy Levels,” in Vibrational Spectra and Structure, Vol. 6, J. R. Durig, Ed., Marcel Dekker, New York, 1977. E. Hirota, J. Mol. Spectrosc., 43, 36 (1972). A. J. Careless and H. W. Kroto, J. Mol. Spectrosc., 57, 189 (1975). M. R. Aliev and J. K. G . Watson, J. Mol. Spectrosc., 75, 150 (1979). Y. Y. Kwan and E. A. Cohen, J. Mol. Spectrosc., 58, 54 (1975). P. N. Brier, S. R. Jones, J. G. Baker, and C. Georghiou, J. Mol. Spectrosc., 64,415 (1977). J. H. Carpenter, J. D. Muse, C. E. Small, and J. G. Smith, J. Mol. Spectrosc., 93, 286 (1982). M. Winnewisser, E. F. Pearson, J. Galica, and B. P. Winnewisser, J. Mol. Spectrosc., 91, 255 (1982).

139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150.

E. B. Wilson, Jr., J. Chem. Phys., 4, 313 (1936). D. R. Lide, Jr., D. E. Mann, and R. E. Fristrom, J. Chem. Phys., 26, 734 (1957). E. Hirota, J. Mol. Spectrosc., 7, 242 (1961). V. W. Laurie and D. T. Pence, J. Mol. Spectrosc., 10, 155 (1963). T. Oka, K. Takagi, and Y. Morino, J. Mol. Spectrosc., 14, 27 (1964). T. Tanaka and Y. Morino, J. Mol. Spectrosc., 5,436 (1969). T. Tanaka and Y. Morino; J. Mol. Spectrosc., 33, 538 (1970). Y. Morino and S. Saito, J . Mol. Spectrosc., 19,435 (1966). K. Yamada and M. Winnewisser, J. Mol. Spectrosc., 12,484 (1978). K. Yamada and M. Winnewisser, J. Mol. Spectrosc., 68, 307 (1977). J. Demaison, D. Schwoch, B. T. Tan, and H. D. Rudolph, J. Mol. Spectrosc., 68,97 (1977). W. Gordy and R. L. Cook, Microwaue Molecular Spectra, Wiley-Interscience, New York,

(1970). 151. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology. For a complete reference, see Chapter I, [65].

THE DISTORTABLE ROTOR ~

1

INTRODUCTION

297

HAMILTONIAN FOR THE DISTORTABLE ROTOR

301

Calculation of the

305

7

s

FIRST-ORDER PERTURBATION TREATMENT OF ROTATIONAL DISTORTION

31 3

Simplifications for Planar Asymmetric Rotors and Symmetric Tops

322

REDUCED H A M I LTO N IAN

324

Asymmetric Tap Reduction ( A ) Symmetric Top @eduction (S) Determinable Cofnbinations of Coefficients

329 333 335

Symmetric and Spherical Tops

337

EVALUATION OF THf DISTORTION CONSTANTS FROM OBSERVED SPECTRA

339

Symmetric and Spherical Tops Choice of Hamiltonian

357

PLANARITY RELATIONS FOR CONSTANTS OF THE REDUCED HAMILTONIAN

366


368

Centrifugal Defect

370

CENTRIFUGAL DISTORTION CONSTANTS A N D MOLECULAR FORCE FIELDS

372

Calculation of the Sextic Distortion Constants

382

ALTERNATE FORMULATIONS

382

Large Amplitude Motion-Rotation Hamiltonian The Pade Hamiltonian

382

361

383

INTRODUCTION

In the initial treatment of the asymmetric rotor we regarded the nuclear framework as rigid. This has proved to be a very useful approximation in the interpretation of rotational spectra. However, in reality, the nuclei are held together by finite restoring forces. Thus, bond distances and angles will vary because of the centrifugal force produced by rotation, which gives rise to a 297

298

THE DISTORTABLE ROTOR

centrifugal distortion. The rotating molecule will no longer be in its equilibrium configuration but in a distorted configuration. Rotation of CH,F, for example, about the symmetry axis would lead to a spreading of the hydrogens and a lengthening of the C-H bond, while end-over-end rotation would produce a closing of the methyl group umbrella and a stretching of the C-F bond. The centrifugal distortion of the molecular parameters of a few molecules for various rotational states is illustrated in Table 8.1. A general expression for the centrifugally induced changes in molecular geometry has been given [I]. Specific application has been made [2] to various isotopic forms of water and hydrogen sulfide which allows calculation of the geometric changes for any rotational level. The effects are usually large for light molecules because of the small moments of inertia. As a consequence of this distortion, the moments of inertia can no longer be considered constant and independent of the rotational state. Therefore, the rotational spectrum will not be simply that of a rigid rotor characterized by a set of equilibrium mdments of inertia. Even for low-lying levels with relatively small rotational energies, the precision of microwave measurements is such that the effects of centrifugal distortion can be observed.

Table 8.1 Centrifugal Distortion of Molecular Parameters" Equilibrium Conj?guration

Molecule

Change due to Centrijiugal Distortion

J=

10

30

-

OHb HClb

r(W)

CO*b

r(W)

NH3'

r(W)

r(W) tl (deg)

0.0109 0.0070

0.0922 0.0595

1.1615 O.oo00 J , K = 10,lO

0.0004

0.9706 1.2746

1.0124 106.67

B (deg)

0.00093 0.237 0.217

JI(-~,K~ 100.10 =

H20d

r(A) CI

0.9572

(deg) 104.52

0.00283 0.016

10,5

10,o

0.00246 -0.337 - 0.308

0.00297 -0.528 -0.483

103,7

1010,o

0.00352 0.781

0.00983

- 5.741

"From M. Toyama et al. [l]. bThe rotational state is indicated by the value of J. 'The rotational state is indicated by the value of J , K ; r denotes N-H distance; a, the H-N-H bond angle; and p, the angle between the N-H and the symmetry axis. dThe rotational state is indicated by the value of J K : r denotes 0-H bond angle. distance; and a, the H-0-H

bond bond bond

INTRODUCTION

299

The theory of centrifugal distortion is considerably more complex for asymmetric rotors, and the effects are in general larger for them than for linear or symmetric-top molecules. In asymmetric rotors numerous transitions between levels of large rotational energies can occur in the microwave region, especially for perpendicular-type transitions. On the other hand, for symmetric tops and linear molecules, transitions involving high J values usually are observed for molecules with large moments of inertia and consequently small rotational energies. In Table 8.2 we show for the asymmetric rotor NSF the shift of the observed frequencies because of centrifugal distortion for various rotational transitions. It is apparent from the table that the spectrum varies significantly from that of a rigid rotor. Although we must consider the influence of centrifugal distortion in order to accurately account for the positions of rotational transitions, its effects still represent only a small fraction of the rotational energy which is accounted for mainly by the rigid rotor term. Therefore in many cases it can be treated as a perturbation of the rigid rotor Hamiltonian. Besides introducing a complication in the rotational spectrum, the study of centrifugal distortion provides useful kinds of information. First of all, such a study provides very accurate ground-state spectroscopic constants. For small molecules these data can be profitably combined with infrared vibrationrotation measurements to yield even more accurate values for the rotational energy levels of the excited vibrational states. Such results are important in the study of the fundamental processes of complex molecular laser systems; see, for example, De Lucia et al. [3, 41. Second, a study of distortion effects allows

Table 8.2 Comparison of Observed and Rigid Rotor Frequencies for the Asymmetric Rotor NSF (MHz)"

Transition

Observed Frequency

Calculated Rigid Rotor Frequency

Distortion Sh@*

16,105.42 71,897.10 34,757.97 36,366.59 13,904.62 11,267.85 39,528.46 34,948.10 16,312.53 39,307.27 27,957.29

16,105.45 71,898.95 34,780.70 36,418.53 14,O14.98 11,267.46 39,281.63 35,440.36 15,373.17 4 4 4 5 1.29

- 0.03 - 1.85 - 22.73 -51.94 - 110.36 0.39 246.83 -492.26 939.36 -.2,144.02 18.44

27,938.85

"After Cook and Kirchhoff [163]. bobserved frequency minus rigid rotor frequency.

300


one to predict unmeasured transition frequencies, in many cases, with a high degree of confidence. The more varied the sample data set, the lower will be the calculated uncertainty limits in the predicted frequencies. This is particularly useful for the transitions of molecules of potential astrophysical interest. Radio astronomers have discovered to date some 50 molecules, and it can be expected that many more will be found as the sensitivity of radio telescopes increases. Contrary to previous expectations, which considered only diatomic molecules as likely to be found in interstellar space, most of the molecules found have been considerably more complex. The importance of laboratory measurements of accurate spectroscopic constants to aid in the detection of molecules in interstellar space through observation of rotational transitions is well documented in reviews on the subject by Winnewisser et al. [5] and Snyder [6]. Third, lines of most interest in a distortion analysis are also particularly useful in the application of microwave spectroscopy to chemical analysis. The location of absorption lines provides the basis for qualitative analysis. The transitions of most use for analytical identification purposes are naturally the most intense transitions. These strong transitions will usually be high J transitions, the ones assigned and employed in the analysis of distortion effects, but not the ones most likely used to evaluate just the rotational constants. However, if a distortion analysis is performed, high J lines can be assigned, and one can be sure that these lines belong to the molecule in question and not to a possible impurity. This is illustrated in Fig. 8.1, which shows a small portion of the R-band spectrum of methylene cyanide [7]. The spectrum extends some 800 MHz with the major frequency markers separated by 100 MHz. The three strongest lines correspond to transitions with J>30. The two lower J transitions are shifted around 300 MHz from their rigid rotor positions while the higher J transition is shifted around 1900 MHz. These transitions have been measured and assigned on the basis of the distortion analysis. It perhaps should be noted that the rigid rotor positions of these lines cannot even be indicated in Fig. 8.1 since the limited frequency range shown does not encompass these frequency positions. Such

38000MHZ . . . . . . m_*, ,

38400

38800

,

Fig. 8.1 A small portion of the microwave spectrum of methylene cyanide. The Stark modulation voltage is 800 V. The three stronger transitions are high J lines. The J = 4 7 - + 4 8 transition is shifted about 1900 MHz from its rigid-rotor position. From Cook et al. [ 7 ] .


301

strong lines would naturally be most useful for compilation in a microwave atlas. With the continued interest in microwave spectroscopy as an analytical tool, it might prove useful for molecules where the distortion effects have been studied, to specifically note some of the most intense lines of the spectrum. Finally, the distortion constants themselves provide direct information on the vibrational potential function. This has proved particularly useful in the case of small molecules. A knowledge of the molecular force constants provides fundamental information for the elucidation of interatomic forces and bonding. Examples of this application are reviewed in Section 8.

2


Before we can discuss the perturbation treatment of rotational distortion for a semirigid rotor, a knowledge of the quantum mechanical Hamiltonian is required. The general theory of centrifugal distortion of asymmetric rotors has been formulated by Wilson and Howard [8] and by Nielsen [9]. A simpler derivation, similar to that given earlier by Wilson [lo], is described here. In general, three coordinates are required to describe the translational motion of the center of mass of a molecule. In addition, three coordinates are needed for a nonlinear molecule (two for a linear molecule) to describe the rotational motion about the center of mass. Therefore, for an N-atom molecule a total of 3N-6 (or 3N-5 for a linear molecule) coordinates are necessary to describe the vibrations of the molecule. The molecule-fixed Cartesian coordinate system rotating with the molecule provides the reference framework for description of the vibrational motions. Various coordinates can be employed which measure the displacements of the atoms from their equilibrium positions, and since the potential energy is a function of the distortion of the molecule from its equilibrium configuration it can be expressed as a power series expansion in terms of the displacement coordinates. For our purposes it is convenient to set up the vibrational problem in terms of a set of 3N- 6 independent internal displacement coordinates. Consider a set of 3N-6 internal parameters (e.g., bond distances ri and angles a i ) required to describe the molecular configuration of a polyatomic molecule. Let { R i } be a set of 3N-6 internal displacement coordinates that describe the displacements of the internuclear distances and bond angles of the molecule from their equilibrium values (e.g., R1 =6rl = I l - r ? , R2 =6r2 = r 2 - r 5 , RJ=6cil =cil -gel,. . .). If harmonic forces are assumed, that is, only terms quadratic in the displacements are retained, the potential energy can be expressed as V =-1 C JjRiRj

2 i,j with fij the harmonic force constants and f i j = f j i . The nature of the internal coordinates Ri and R j determines whether the force constant f i j will represent a bond-stretching or angle-bending force constant, or an interaction force

302


constant, for example, a stretch-bend interaction constant. For small displacements this potential function is a good approximation to the actual potential of the molecule. Thus the classical Hamiltonian for a vibrating rotating polyatomic molecule may be written approximately as 1

1 ~xpPmPp+-C Gijpipj+ (8.2) a,p 2 i,j where u and /? take on the values x, y , or z of the molecule-fixed Cartesian coordinate system, where i and j enumerate the various internal coordinates, and where V is given by (8.1). The first term represents the rotational energy of the system with pupas an element of the inverse moment of inertia tensor and P, as the a component of the total angular momentum. At this point the pap coefficients must be considered to be functions of the vibrational coordinates, since the requirement that the molecule be rigid has been dropped. The last two terms made up of kinetic and potential energy terms represent the vibrational energy of the molecular system. Here p i is the momentum conjugate to the internal coordinate Ri, and Gij are the elements of the well-known Gmatrix that arises when internal coordinates are used in molecular vibration problems [l 11. The G-matrix elements are simply functions of the masses and molecular geometry. In the present discussion we are interested in the situation where the molecular system is not vibrating but only rotating. This rotational motion will result in a distortion of the molecule because of the stretching effects of the centrifugal forces and the nonrigidity of the molecular framework. One of Hamilton's equations of motion, pi = - a 2 / a R i , when applied to (8.2), with papdepending on the R/s, yields

x=j

since bothpi andpi are zero under the assumption of no vibration. This relation expresses the equilibrium condition between the restoring potential forces and the centrifugal forces due to rotation. The coefficient pcrpcan be approximated by a series expansion about the equilibrium value p & (i.e., the value when all displacement coordinates Ri are zero, which corresponds physically here to no rotation), as follows

with

This latter quantity, which is the partial derivative of the up component of the reciprocal moment of inertia tensor with respect to the internal coordinate R i , is to be evaluated at the equilibrium position. In the expansion, only linear


303

terms are retained since the displacements are assumed to be small. From this expansion in conjunction with the potential energy expression, (8.1), one immediately obtains from (8.3) 1 -1 p$PZPp+1fijRj=O,

i=1,2 , . . . , 3N-6 2 Z>P j This set of linear equations may be solved for R j giving

(8.6)

where Cf - ' ) j i is an element of the matrix inverse to the matrix of force constants hi.Inserting this in (8.4) allows one to express the instantaneous values of pap in terms of the components of the total angular momentum as follows

If (8.7) is inserted into (8.1), the potential energy becomes

XI

Note f i d f - l ) l j = d i j . When these expressions for pap and V are substituted in (8.2) one obtains (8.10) with

(8.1 1) This is the classical rotational Hamiltonian for a semirigid nonvibrating molecule; it represents the quantum mechanical Hamiltonian when the angular momentum components are taken as the corresponding angular momentum operators. The first term on the right in (8.10) represents the usual rigid rotor energy while the second term represents the energy of distortion. The T~~~~ are the distortion constants, and the distortion contribution will involve angular momentum about all three axes. The dependence of the distortion constants on the inverse elements of the molecular force constant matrix is physically reasonable, since, all things being equal, one would expect that weaker bonds with small force constants should show larger distortion effects than the strong bonds with large force constants. The equilibrium rotational and centrifugal distortion constants in 2 which are independent of the rotational coordinates are also seen to be independent of the vibrational coordinates. Within the approximation considered here, we have a rigid rotor with a correction term for centrifugal distortion. However,

304


in an actual molecule, vibrational motion is unavoidable, and the rotational and distortion constants must be considered as functions of the vibrational coordinates and thus dependent on the vibrational state in which the rotational spectrum is observed. In practice then, the principal moments of inertia obtained from the ground state spectrum are moments averaged over the ground vibrational state, that is, effective moments of inertia. The general vibration-rotation quantum mechanical Hamiltonian operator of a polyatomic molecule has been discussed by Wilson and Howard [8] and by Darling and Dennison [12]. Perturbation treatments to various order have been given for the evaluation of the vibration-rotation energy levels [8, 9, 13, 141. These general formulations admit vibration-rotation interaction terms in the Hamiltonian from which the vibrational dependence of the rotational constants and centrifugal distortion constants arises as well as other vibration-rotation interactions such as Coriolis coupling. We may also cite some of the more recent discussions [15-201 on the rotation-vibration Hamiltonian (see also Chapter VII). As a simple application of (8.10), let us consider the case of a semirigid diatomic rotor. The rotation of a diatomic molecule is mathematically equivalent to the rotation of a single particle of reduced mass m at a distance r from the axis of rotation and we have

(8.12) The distortion term of (8.10) with the aid of (8.11) reduces to

(8.13) wheref is the force constant appearing in the potential energy which is simply V = i f R 2 where R = d r = r - r , , re is the equilibrium bond distance, and r is the instantaneous value. Now the partial derivative, evaluated at equilibrium, of the reciprocal moment of inertia is (8.14)

In the case of polyatomic molecules the evaluation of such quantities is more complicated. If we now impose the quantum mechanical restriction that the angular momentum must be quantized in units of ii, that is, P 2 = k 2 J ( J + l), then (8.13) may be written in the familiar form -hDJJ2(J 1)2 with the distortion constant defined as

+

(8.15)

where (8.14) and the relations Be=h/(8n21e) and 0,=(1/27~)Cf/rn)”~have been utilized. This is the usual definition of Dj given for a diatomic molecule.


Calculation of the

305

7's

The distortion constants can either be evaluated from the rotational spectrum (this will be discussed in the next section) or if the force constants are known they may be obtained by means of (8.11) once the derivatives, which depend on the molecular geometry, are evaluated. For a general molecule the partial derivatives appearing in (8.11) are most easily determined by the method proposed by Kivelson and Wilson [21]. They first express the partial derivatives of the inverse inertia tensor components in terms of the partial derivatives of the components of the inertia tensor. In the matrix notation we have, since p = I - the equation p1 = E with E the unit matrix, and taking partial derivatives we obtain

',

av

a1

a R i

a R i

-I+p-=O

(8.16)

thus (8.17) The coordinate axes are now so chosen that the undistorted molecule will be in its principal axis system with origin at the center of mass. Then the inertia matrix is diagonal with equilibrium elements I&, I&, Ziz, and we find for the components of (8.17) evaluated at equilibrium

(8.18) since pza = l/&, etc. Here (8.19)

If (8.18) is inserted into (8.11), the distortion constants may be re-expressed as 7apya=

-$(CIipI&I&)-'

1

J$]eCf-')ij[J$]e

(8.20)

4 j

An alternate definition in terms of partial derivatives with respect to the normal coordinates has been given in Chapter VII, Section 6. Evaluation of the partial derivatives of components of the inertia matrix requires special consideration. The components of the inertia matrix are given by Iaa=C 1 rap

m(B? +Y?)

=~ p = , -

CI ml.&

(8.21)

(8.22)

where m1 and a1are, respectively, the mass and the wcoordinate of the Ith atom as measured in the body-fixed axis system. Here, and in the following equations,

306


p, and y are taken in cyclic order; that is, if a I = x l , then PI and y i stand for yl and z l , respectively. What is required are the variations of the components of the inertia tensor when the molecule is distorted from its equilibrium configuration. Now the variations in the 3N Cartesian coordinates, 6al, 6 p l , 6yi, cannot be made arbitrarily but must satisfy the six Eckart conditions [22, 231 which serve to define the rotating coordinate system so as to provide maximum separation of rotation and vibration. Physically, we may view these as restrictions on the distortions that insure that the center of mass of the molecule does not change and that no angular momentum is imparted to the molecule. With the Eckart conditions it can be shown that [21] a,

(8.23) (8.24)

Zor=ck

with mkai,and so on, and Zyy given by (8.21). These relations allow the displacements 6aI, and so on, to be entirely arbitrary while still satisfying the Eckart conditions. The expressions are to be evaluated at the equilibrium configuration of the molecule (Ri=O, i = 1 , . . , ,3N- 6) and are referred to the rotating principal axis system. The Cartesian coordinates a1are thus measured from the center of mass in the principal axis system. For the finding of a derivative, the principal axis system may be translated to a convenient position in the molecule. Small displacements are then given to the nuclei so that the set of increments 6aI will change the angle or internuclear distance that is involved in a given internal coordinate R i , in such a way that the internal coordinate will change by an increment 6Ri while leaving all other internal coordinates with their equilibrium values (see Fig. 8.2). The coordinates a1are related to those of the displaced coordinate system (x’, y’, z’) by a1=a; - E, where ii is the a component of the center of mass in the displaced system. When there are more internal displacement coordinates than vibrational degrees of freedom, that is, redundancies are present, additional considerations are necessary. These have been discussed by Gold et al. [24]. If the molecule has symmetry, calculation of the distortion constants can be simplified by use of a set of internal symmetry coordinates { S i >which are particular orthonormal linear combinations of the internal displacement coordinates. These symmetry coordinates are chosen so that, under the symmetry operations of the molecular point group, each coordinate may be classified as belonging to one of the symmetry species of the group. Equation 8.20 is still applicable when both the partial derivatives and the force constant matrix are expressed in terms of the symmetry coordinates. To obtain the J$, we choose increments 6al so as to produce an increment 6Si in a particular symmetry coordinate while the other symmetry coordinates are left unchanged. When symmetry coordinates are used, the matrix of force constants, and its inverse, will factor into submatrices

307


Fig. 8.2 Principal and translated axes for bent XY2-type molecule. Displacement of length (top figure). Displacement of valence bond angle (bottom figure).

rl

bond

corresponding to different symmetry species of the group; there will be no coupling between J$ and J$ if the ith andjth symmetry coordinates belong to different symmetry species [21]. In Table 8.3 (on pages 308-311) explicit expressions for the derivatives of some common types of molecules are given. Expressions for other types such as axially symmetry Z X 3 Y molecules [25], pyramidal X Y Z 2 molecules [26], and axially symmetric Z X 3 Y Wmolecules [27] have also been reported. General expressions for a number of commonly used internal displacement coordinates may be found in [21]. Corrections for nontetrahedral angles in the symmetric tops Z X 3 Y and X U 3 have been described [28]. Since calculation of the J$ elements is somewhat tedious, alternate methods of evaluation that are more convenient have been formulated. The Cartesian displacements in the principal inertial axis system can be conveniently expressed, as pointed out by Parent and Gerry [29], in terms of the p vectors (Pi=P!f’, P!?, pi?’, * * * P%) du, = pi?’ (8.25) with p$) the u component of the p vector for the Ith atom associated with the ith internal coordinate. These p vectors, introduced by Polo [30], can also be used to evaluate the inverse kinetic energy matrix G-’,which is useful in the solution of the vibrational problem. An alternate formulation which makes use of the Wilson s vectors has been discussed by a number of authors (see Aliev and Aleksanyan [31], Cyvin et al. [32-341, Pulay and Sawodny [35], and Klauss and Strey [36]. Let Ri denote an internal displacement coordinate and 6al=ul-af, the Cartesian displacement coordinate of the Ith atom (CI=X, y , or 2). These are 3

308


related by the well-known transformation (8.26) The B matrix elements are constructed from the Wilson s vectors and give the kinetic energy matrix G, (8.27)

a!),

with the three components l@’, 8:’ associated with atom 1 and coordinate Ri. The inverse transformation is denoted by

6crl

=E A p R i

(8.28)

i

The matrix A is not simple B-’ since B cannot be inverted because it is a rectangular matrix. However, Crawford and Fletcher [37] have shown that if the six Eckart conditions are added, the desired transformation is given by (8.29) This relation forms the basis for evaluating the derivatives of Zap satisfying the Eckart conditions. Taking the partial derivatives of the components of the inertia tensor, with respect to the internal coordinates, gives

(8.30) From (8.28) (8.31) and substitution in (8.30), with cognizance of (8.29), gives

(8.32) The definitions of Tit) and Ti? are apparent from the foregoing equations. Evaluation of these in the equilibrium configuration yields the desired quantities (8.33)

Table 8.3 The [J$lp Expressions for Some Particular Cases

Internal Coordinate

For Nonlinear X Y 2 Molecules" 6r

2myrs2

2mxmyrc2 M

1 +2mymi 's2

2myrsz

2mxmyrc2 M

-2myrsc 1 + 2 m y m i 's2

2myr2sc

- 2mxmyr2sc M

2myrsc

0

Symmetry Coordinate

s1

2 J z m yrs2

S2

2myr2sc

s 3

0

2 J?imxmyrc2 M

- 2mxmyr2sc M

0

0 2Jzmyrsc

0

1-t2mym1 '52

Definition of Symmetry Coordinatesfor XY2 A species B1 species 6rl -6r2

s3 =-

s1

JT

s 2

0 0

s3

0

S4

0

0

s5

0

0 IYY

309

Table 8.3

Continued

[J % ] e

[51'21e

Definition of Symmetry Coordinatesfor X Y Z 2 A l species

B , species

B2 species

S2=6r3

For Pyramidal X Y 3 Molecules' Sl s 2

myr2t(3my-mx) J ~ M

army [ M + ( m x - 3my)c2] M

Jz

s1x

-myr2t

a

S2Y

0

2J?myrs2

0

Jj

0 0

s2x

sly,

2

-myr't

0

0

Definition of Symmetry Coordinatesfor XY3 A l species

31 0

E species

MI,, 0


311

The Eckart conditions are satisfied automatically, and the G-matrix elements and the J$ can be calculated together with a set of s vectors evaluated in the principal inertial axis system of the equilibrium configuration. The equilibrium coordinates ctf, however, need not refer to the center of gravity of the molecule as their origin [33]. If desired, (8.20) may be written in an alternate form in terms of the T $ . The vibrational eigenvalue problem may be expressed as G FL = LA, where A is a diagonal matrix of eigenvalues ii=471'c20~and L relates the internal coordinates Rito the normal coordinates Qi, R=LQ. Since in matrix notation L F L = A and LG-'L=E, with E a unit matrix, it follows that G - ' F - ' G - ' - 2- ' A - I L - I . By use of (8.32), (8.20) may be rewritten as

-i(&lipI;yl&)-l

f,pi'g=

E T$,k41$'ekk,

(8.34)

k,k'

with (8.35) The elements of 6 k k ' are obtained from the usual normal-coordinate analysis of the molecular vibrations. (See Appendix H). Application of Table 8.3 may be illustrated by considering a symmetrical bent triatomic molecule, for example, SO2, for which there are four force constants and four independent 7's. From (8.20) and the [J$],'s of Table 8.3, with introduction of the factor h4/h, the explicit relations between the distortion constants (equilibrium values) and the force constants are found to be [38] --TaaaaR - 2(F- ')'

A2

+ tan'

d(F- ')" - 2 4 tan 6 ( F - ')I'

(8.36)

"After Kivelson and Wilson [21]. The xz plane is taken as the molecular plane with the x axis parallel to the C 2 axis, and M =2my mx, c =cos 0, s= sin 0, where 0 is one-half the Y - X - Y angle (a) and r is the Y - X equilibrium bond distance. Derivatives of I,, with respect to SI and Sz can be obtained from the table since I,, = I , , + I,, for a planar molecule.

+

bFrom M. G. K. Pillai and F. F. Cleveland, J. Mol. Spectrosc.,6,465 (1961). The symbols are defined as follows: mZ=mass of Z atom, m y = mass of Y atom, rl is the X-Z bond distance and r3 is the X-Y bond distance, c=cos(r/2), s=sin(x/2) with a the Z-X-Z angle, I is the distance between the center of mass and the X atom, measured along the X-Y bond, which is taken as the z axis. For symmetry coordinates: r l and r 2 are the X-Zi bond distances; r 3 , the X-Y bond distance; r, the Z-X-Z angle; pi, the Y-X-Zi angle; and 5, out-of-plane angle. Derivatives of I,, with respect to SI,S2,and S3 are easily obtained from the table since the molecule is planar. All other derivatives vanish. 'From P. N. Schatz, J . Chem. Phys., 29,481 (1958). Here the C3 axis has been chosen as the z axis. The symbols are defined as follows: c=cos 0, s=sin 0, with 0 the angle between the C3(z)axis and an X-Y bond (0>90"), mx=mass of X atom, my=mass of Y atom, M=total mass, r=X-Y bond distance, t=sin a, where a= Y-X-Y angle.

312


--T b b b b R - 2 ( - ~I ) I~+ cot2 e ( F -

I~~ + 2 J 5

B2

--T a b a b R - 2 M m i

AB

cot e(F -

I~

'(I + 2 m y m i 1sin2 6 ) - 2 ( ~ - 1 ) 3 3

(8.37)

(8.39)

Here h is in erg-sec, masses are in amu and

R=

r2 x IO-22 2h

with 8 as one-half the Y-X-Y angle, M=2my+mx, r as the X-Y bond distance expressed in angstrom units, and with the ?s and rotational constants in megahertz. The (F-l)ijare the elements of the matrix inverse to the potential constant matrix F (dynes/cm) written in terms of the symmetry coordinates given in Table 8.3 [see (8.199)]. It is clear that the observed T'S provide a basis for evaluation of the force constant matrix. Since Tabab depends only on ~ 3 ( B 1 ) , an even simpler expression may be written [39] in terms of the vibrational frequency 16cABC Tabab=-

(8.40)

___ 0:

with in Hz and A, B,C , m 3 in cm-', and c the speed of light in cm/sec. From these relations it is apparent that for triatomic planar molecules, where the C2" symmetry is unaltered by isotopic substitution, the distortion constants for the isotopic species can be calculated from the distortion constants of the parent species. In particular, for Y *-X*-Y * we have (8.41)

(8.42)

=rz)rs) [

(8.43)

+ 1 + (2m$/m$)sin2 8

1

1 (2my/mx)sin2 8

T,*bab

Tabab

(8.44)

where A, B, and C are the rotational constants; the asterisk refers to the isotopically substituted molecule. It can be noted that for the isotope X*Y2, where X is on the C2 symmetry axis (a), the corresponding moment of inertia is unchanged, and hence the corresponding T,,,,is unchanged. An even simpler isotopic relation follows from (8.40)for T a b a b . Isotopic relations for the T constants


313

of various type molecules have been given [40,41], as well as relations for the P6 constants of tetrahedral XY4 molecules [42]. From an examination of the extrema1 properties of T,,,,and Tapap, it has been shown that these constants are always negative [43]. Simple empirical [44] and theoretical [45] relationships between the distortion constants T,,,, and the rotational constants B, have been discussed to obtain order-ofmagnitude estimates of the distortion constants. Relations for the upper and lower limits of T,,, ~,,pp, and ~,p,p have been developed [43, 46, 471 that are determined from the values of the rotational constants and the lowest and highest frequency vibration. The relation for (- T , , , , ) ~ ~ ~ = 16B,3/oii, is particularly simple, and interchanging max-min gives the lower bound, (- T,,,,)~~,,. These relations may be employed to calculate an approximate value of a constant which can help in deciding whether such a constant should be included in the distortion Hamiltonian or whether the observed constant is consistent with the observed vibrational frequencies. Bounds for the spectroscopic constants D J , DjK, DK for symmetric tops [46,47] and A j , AjK, and so on, for asymmetric tops [47] have been discussed. From these results it may be concluded that D j is positive and DjK, DK may be positive or negative. For a planar symmetric top DjK and DK are negative and positive, respectively. The linear combinations (Aj + 26j), ( A j - 26j), and (Aj AjK + AK) are necessarily positive. For linear triatomic molecules, the experimental values of D j are very close to their upper limits and thus are not very useful as supplemental data for force constant evaluations. In this regard, a very simple formula for D j , applicable to a triatomic molecule with a very low stretching frequency, has been obtained [48]. This is particularly useful for linear van der Waals molecules. For nonlinear triatomic molecules, however, the experimental values are not close to their upper limits and hence are useful in force field evaluations, as has been found. The fact that the lower frequency vibrational modes are primarily responsible for the centrifugal distortion effects has been used to give information on the skeletal motions of some molecules of moderate size [49].

+

3 FIRST-ORDER PERTURBATION TREATMENT OF R OTATl 0 NA L D ISTO RT IO N Taking the quantum mechanical Hamiltonian for a semirigid rotor in the form of (8.10)and choosing the principal axes system for the rigid rotor problem, we may express the Hamiltonian as

A? = 2; + 2i 2;=A‘P;

+ B P : + C’PyZ

(8.45) (8.46) (8.47)

314


with the angular momentum in units of h, with a, p, y , 6 =x, y , or z, and with A’=h2/2Z,, and so on. As mentioned previously, a rigorous treatment of the problem indicates that the rotation and distortion constants in the foregoing Hamiltonian must be regarded as effective constants and thus dependent on the vibrational state. The effects of centrifugal distortion on the rotational spectra will depend on both the constants sapyd and the angular momentum operators. A simple closed-form expression for the effect of these operators, like that obtained for a symmetric rotor, cannot be obtained for an asymmetric rotor. However, the distortion effects for a large number of asymmetric rotors may be treated to a good approximation by first-order perturbation theory. Furthermore, the first-order energy expression is particularly convenient for analysis of the rotational spectrum for the distortion constants, since these constants enter linearly into the energy expression. A first-order treatment of centrifugal distortion has been discussed by Kivelson and Wilson [SO] and more recently by Watson [51]. The first-order perturbation treatment will involve averaging the perturbing operator 2;over the wave functions associated with %;, that is, the asymmetric rigid rotor wave functions. The firstorder distortion energy for a particular level J, is then Ed

= (Xi) =( J , 51 2il J, 0)

(8.48)

In the following discussion it will be seen that by consideration of only a firstorder approximation, or molecules with particular symmetry, the general distortion Hamiltonian, (8.47), may be greatly simplified. In Section 4, order-ofmagnitude arguments are used to provide a more general simplified Hamiltonian. As noted before, an important quantum mechanical property of angular momentum components is that they do not commute. When the total angular momentum is referred to axes fixed in the molecule (x, y, z), the following commutation rules are found to exist: P,P, - P,P, = - iP, P, P, - P, P, = - iP, (8.49) P,P, - P,P, = - iP, where Pa is in units of h. Except for the sign change on the right-hand side of the equations, these are similar to the set of commutation rules that apply for the components of the angular momentum resolved along the space-fixed (nonrotating) axes ( X , Y , 2).Because of the noncommuting character of the angular momentum components Pa, there is a total of 81 terms in the sum of (8.47).However, from (8.20) it is seen that many distortion constants are equivalent, and we have, in fact, %pya

=L p d y =Tpady = Tpayd = 5ysap =q d p a = %ypa =%yap

(8.50)

Furthermore, many of the terms will not contribute to first-order. The rigid asymmetric rotor wave functions are classified according to the Four-group V, and hence if the integral (8.48) is to be nonvanishing, the product of the


315

angular momentum components must be invariant to the symmetry operations of the group, that is, belong to the totally symmetric species A . The transformation properties of P,, P b , P, have already been given (see Chapter VII, Section 2), and we may summarize the results by saying that under the group V, P, transforms according to Bo, Pb according to Bb, and P, according to B,. The symmetry of a given angular momentum term will be given by the direct product of the symmetries of the individual members of the term. It follows, therefore, that all terms involving odd powers of any angular momentum component will not contribute to first-order because such terms will belong to one of the B symmetry species and their average values will vanish. Thus, for any asymmetric top only 21 constants (many of which are equal to each other) will contribute to a first-order approximation. Moreover, when the symmetry of the molecule is taken into account (&” and thus Xi must be invariant under the symmetry operations of the molecular point group), it is found that the distortion constants of these odd-power terms must vanish identically for molecules of orthorhombic symmetry [52, 531. Orthorhombic symmetry includes molecules belonging to either the Cz0,V,, or I/ point groups. The eight asymmetric rotor point groups are summarized in Table 8.4. Noting that Pa transforms as the corresponding rotation Ra of the point group, and carrying out the symmetry operations of the group on PaPpPyPa,Parker [53] found that the nonvanishing ~~~~a are as given in Table 8.4. All other terms are antisymmetric under one or more symmetry operations of the particular group. As a consequence, for molecules of orthorhombic symmetry there are only 21 nonvanishing coefficients in Xi to any approximation; since many of these constants are equal, one is left finally with only nine different quartic distortion constants : Taaaa, Taapp=Tppaa,

Tapap=~apga=~papa=~pzap

(a, P=x, .Y, or Z )

(8.51)

with a f p . Therefore, the various surviving P,PpP,Pa terms of Xi may be arranged into nine groups, each of which has the same T coefficient. A further reduction of the distortion Hamiltonian to six groups of terms is still possible by an appeal to the commutation rules. By means of the commutation rules, (8.49), the following relations may be derived [48] :

(PzPp+ Pp Pa)’

=~ ( P , Z P ~ Z

+ PpZP:) + 3P: - ~ P , Z- 2~pZ

(8.52)

here a f f i f y and with a, p, y to be taken in cyclic order. Note that since Papp do not commute, there will be four different P4 terms when the left-hand side is multiplied out. From these three relations, further simplification results, since one can eliminate the T ~ ~ ~ ~ ( P ~ P ~terms + P ~of PXi. ~ ) As ’ a result of this simplification, the coefficients Tabap are folded into those of (P;Pi + PjP,”),and terms in Pi, Pp’, and Py’ are introduced which can be absorbed into the rigid rotor part of the Hamiltonian Xi, thus giving a new definition to the rotational constants. This procedure leads to the following form of the Hamiltonian

x=X r + X*

(8.53)

Table 8.4 Asymmetric-Rotator Point Groups and Nonvanishing T Constantsa

Crystallographic Nomenclature

Group Symbol

Triclinic

Cl

Group Operations Other Than Identity Operation

Sets with Nonzero Distortion Constantsb

ci=s2

Monoclinic

CS=Clh

c2

C2h Orthorhombic

C2” V=D2 Kh=D2h Nonvanishing

~,g~a

Set

1 2 3

4 5

I1

6 7

8 9 10 11 12 13 14 15 16 17 18

19 20 21

”From Parker [53]. bCorresponding angular momentum operators of Set I may have nonvanishing matrix elements in a symmetric rotor basis of the type ( K I K ) , ( K I K k 2 ) ,(KlKk4); whereas Set I1 operators may have nonvanishing elements of the type ( K I K + 1) and ( K I K k 3 )only. 31 6


&, =AP:

+ BP: + CP;

31 7

(8.54) (8.55)

This Hamiltonian is applicable for molecules with orthorhombic symmetry or for any molecule to a first order of approximation. The relations between the old and new coefficients are given in Table 8.5. From the table we see that the corrected rotational constants A , B, and C depend on the distortion constants. This introduces another ambiguity into the definition of the moments of inertia in addition to that arising from the vibrational effects.If the distortion constants are not known, with the result that A’, B’, C‘ cannot be obtained, additional uncertainty is transmitted to the derived structural parameters although this effect is very small. The first-order distortion energy may now be written from (8.55) in the following form (8.56)

+

with ~ L , p p= and with (PjPpZ PpZP:) as the average value of the appropriate operators. To proceed further, one needs explicit expressions for the various average values appearing in the above equation. These are usually expressed in terms of P 2 , (Pg) ( n = 2 and 4), and the rigid rotor energy E,, since the latter quantities are easily determined from the diagonalization of the rigid rotor problem. Employing the exjressions for &, and P2, one may write P: =

[ 2,- ( A - C)P: - CP’]

( B - C) [&,- ( A - B)P: - BP’] Py”= (C - B)

Table 8.5 Definition of the Coefficients in the Distortion Hamiltonian

(8.57)

(8.58)

318


from which one may construct the desired average value equations noting that with P in units of h

(P4)= S ( J + 1)2 ( 2 ;=E; ) (X,.P2 P 2 X r )=2ErJ(J+ 1 ) ( HrPt P,”X r )=2Er (Pz’) (PZPz’ Pz‘P2) =2J(J+ l)(P,Z)

+ + +

(8.59)

The average values of these operators are evaluated in a rigid asymmetric rotor basis, the basis in which the matrix representation of 2, is diagonal with diagonal elements E, = ( X r )= A (P;) B(P:) C(P;). The appropriate average value equations of the quartic angular momentum terms are listed in Table 8.6. It may be pointed out that the average value equations can also be expressed as desired in terms of Ray’s E ( K )or Wang’s W(b,) reduced energy by simple definition of A , B, C in the appropriate manner, that is, A-1, B-K, C-+ - 1, E,+E(K) or A+l, B-t - b,, C-b,, Er-+ W(b,). Watson [51] has shown that, in addition to the five equations of (8.59), another relation may be obtained from the average over the commutation relation

+

i[ X r ,(P,P,P,

+ P,P,P,)]

+

=2(C- B)(P:P; + P p , ”+ 2 P 3 2(A - C)(P,ZP,” Pip; 2 P 3 2(B- A)(P:P,Z P,ZPz 2P;)

+ +

+ +

+ +

(8.60)

Here the following relations,

[ P i , (P,P,P,

+ P,P,P,)]

+

+

=2i(P,zP,2 Pp’Pi)- 2i(P,zP.: P.:Pi) 4i(P: - Pp’)

+

(8.61)

(with a, p, y =x, y, or z ; tl / p /y, and a, p, and y taken in cyclic order) derived from the angular momentum commutation rules, have been used to reduce the right-hand side of (8.60).Now the average value of the commutator < [2,O,] > vanishes for any operator 0, averaged over eigenfunctions of 2.Therefore, the average value of the commutator on the left-hand side of (8.60) vanishes, and one obtains a relation between the average values of the quartic and quadratic angular momentum terms in the rigid rotor basis. The vanishing of such commutators has also found application to internal rotation problems [54]. By insertion of the appropriate average value equations given in Table 8.6, the following expression is attained [51]

E; =(B- C)’(Pz’)+ ( A - C ) ( C - B ) ( P f )+ ( A - B)(B- c)(P;) +(B+ C)ErJ(J+ 1)- BCJ2(J+ 1)2- 2(AB+AC- 2BC) x J ( J + l)(P,”)- 3(A - B)(A- c)(P:) (8.62) + 2(2A- B- C)E,(P,”) This relation may now be used to eliminate E: from the average value equations of Table 8.6 so that any given average value will at most be a function of five different terms, P4,P2(P,”),(P:), P2Er,and E , ( P ; ) . As a result, the distortion


319

Table 8.6 The Average Values of the Distortion Operators in the Asymmetric Rigid Rotor Basis’

=(Ly

(e)B- C 1

(E)=(=)

2

[ C 2 P 4 + 2 C ( A - C)P2(Pt+ ( A - C ) 2 ( E ) - 2 C P 2 E , - 2 ( A - C)E,(Pt) +E?] [B2P4+2B(A-B)P2(Pi)+(A-B)2(P~)-22BP2E,

- 2(A - B)E,(PI) +E:]

(

+

(PX’PY’ PY’Pf)= - 2 B; c>2[BCP4+ (AB+ A C - 2BC)PZ(PZ)

+ ( A - BXA - C)(P:) - (B+ C)P2E,.- (2A - B- C)-E,(P:) + E?]

+

(PX’Pt PZ2PZ) = - 2

(-)

1

B-C

+

[CP2(Pf) ( A- C)(P,“)- E,(Pf )]

‘P2= J ( J + l), E , is the rigid rotor energy and (P:)is the average value of Pi in the rigid asymmetric rotor basis.

energy to first order will depend on only five combinations of the T & ~ . The terms in (P:), (P;), and ( P z ) introduced by the elimination of E: can be absorbed into E,, thus resulting in a redefinition of the effective rotational constants A , B, and C. This will not significantly affect the average values of the quartic terms, and the average values in (8.64)may be identified with the asymmetric rotor basis specified by (8.65). This produces, however, another small ambiguity in the definition of the moments of inertia, since the quantities A’, B, and C’ should define the effective moments of inertia, whereas those of (8.65)are determined in practice. By use then of (8.62)and Table 8.6 and by insertion of the average values in (8.56) followed by collection of terms and considerable manipulation, one arrives at the following first-order expression for the energy of a semirigid asymmetric rotor [50, 511 E =E , Ed=

+ Ed

(8.63)

- d j J 2 ( J + 1)’ - ~ J K J ( J +1)(P;) - d ~ ( P i ) -d,,E,J(J+ l)-d,KE*(PzZ)

(8.64)

Here E , is the energy of a rigid rotor with the following rotational constants d=A

+ 16&

%=C+

16&(A - B) B- C

(8.65)

320


The d coefficients of Watson [51] are defined in Table 8.7 in terms of the Ds, R’s, and dJ used by Kivelson and Wilson [50]. These latter constants are linear combinations of the 7’s and are also defined in Table 8.7. Equation (8.61) could equally well have been used for elimination of (P:‘) rather than E:. The calculation of (P:‘) could therefore be avoided, but the analogy with the symmetric top would be lost. The energy expression obtained by elimination of (P:) may be found elsewhere [ 5 5 ] . By application of (8.60) it is possible to express the average values of the quartic terms, (P:P$)( = (Pp’Pi)),in terms of the average values of the quadratic terms, ( P i ) . The detailed expressions are given elsewhere [%].Results for the dcoefficients of some asymmetric rotors are collected in Table 8.8. It has been found [56-581 in studies of a number of nonplanar molecules with the first-order expression, (8.63), that there is high correlation between dKand dEKand between dJKand dEK.Also, for K C -0.97, the normal equations become highly ill-conditioned (see Section 4).Equation 8.64 can be written in a more suitable form involving the A coefficients to be introduced later. In particular, elimination of E, from (8.64) by means of the relations Hr=i(g+‘%)J(J+ l)+[.d--f(B+%)]Pf+i(g- %)(P;-P;)

and

Table 8.7 Distortion Coefficients of the First-Order Energy Expression

dEK =

-8(R5 +20R,j)

B-C 2A-B-C 0= B-C

(8.66)

Table 8.8 Rotational a n d Centrifugal Distortion Constants (d Coefficients) of Some Asymmetric Rotors (MHz)"

Molecule (CH3)2S0 (CH3)ZSC CH~(OH)CHO~

CH,FCONH,' 1,3 Dioxaneb Pyridine 3-Fluorophenol' truns.CH,CHCH,NH'

CH3SCI HBF,' NZF4

d

93

'&

dJ

dJK

dK

dEJ(X lo6)

dbK( X lo6)

7,036.49 17,809.73 18,446.41 9,884.36 4,999.93 6,029.28 3,748.49 16,892.44 17,341.83 52,896.09 5,576.19

6,910.93 7,621.10 6,525.04 4,059.67 4,807.61 5,804.95 1,797.71 6,533.61 4,603.78 10,498.29 3,189.42

4,218.78 5,717.77 4,969.27 2,932.47 2,757.1 1 2,959.25 1,215.05 5,761.30 3,719.10 8,740.46 2,813.16

0.043 -0.03158 - 0.02714 -0.00206 0.00198 0.00367 0.00023 -0.00457 -0.01167 -0.0386 -0.01389

10.550 -0.1503 - 0.0732 - 0.0312 -0.2088 0.1738 -0.0110 0.020 -0.2087 -1.134 -0.2566

-4.172 0.058 0.0883 -0.0507 0.09118 -0.0868 -0.0150 0.020 -0.3692 -2.96 -0.2087

5.20 5.943 5.648 0.801 -0.1839 - 0.22 -0.20 1.228 3.468 4.9 1 4.7984

-

"Note constants dEJand dUKare dimensionless. bIII' representation. 'I' representation. dH. Dreizler, Z. Nuturforsch, 21% 1719 (1966). 'K. M. Marstokk and H. Msllendal, J. Mol. Struct., 5,205 (1970). /K. M. Marstokk and H. Mellendal, J. Mol. Struct., 22,287 (1974). gR.S. Lowe and R. Kewley, J. Mol. Spectrosc., 60,312 (1976). *R.R. Filgueira, A. C. Fantoni, and L. M. Boggia, J. Mol. Spectrosc.,78, 175 (1979). 'A. I. Jaman, R. N. Nandi, and D. K. Ghosh, J. Mol. Spectrosc., 86,269 (1981). 'C. F. Su and E. B. Beason, J. Chem. Phys., 59,759 (1973). 'A. Guamieri, Z. Nuturforsch., 25a, 18 (1970). 'T. Kasuya, W. J. Lafferty, and D. R. Lide, J. Chem. Phys., 48,1 (1968). "V. K. Kaushik and P. Venkateswarlu, Chem. Phys. Lett., 46,426 (1977). W

!2

-

1517 7.280 -3.40 8.41 42.21 -29.87 7.2 - 5.0 37.23 89.5 81.164

Rf$ d d e

f y

h 1

j k 1 rn

322


Table 8.9 Relations Between the d- and

A Distortion Coefficients"

"Here (x, y, z ) ~ ( bc, ,a), alternate choices made by reidentification of d,8, %.

yields Ed

= -A j J z ( J

+ 1)2- AjKJ(J + l)(P:)

-AK(P:) - 2 6 j J ( J + l)(PE-P:}-6,(PS(PE-P:)+(PE-P,2)PZ)

(8.68)

The relations between the d- and A coefficients are given in Table 8.9. In the present discussion we have used an I' representation (x-tb, y+c, z-ra) for .ri",. Alternate choices are readily obtained by reidentification of d , W,%.

Simplifications for Planar Asymmetric Rotors and Symmetric Tops For planar asymmetric rotor molecules, significant simplification results, as is shown by Dowling [59]. For a planar, prolate-type rotor, the unique symmetric-top axis z will lie in the plane of the molecule, whereas for an oblate top the limiting symmetric-top axis will be perpendicular to this plane. For purposes of discussion, assume that the molecule lies in the xz plane. Then if c( or p is y in (8.24), all the J$s will be zero since all the yl coordinates of the atoms are zero at equilibrium. Therefore it follows from (8.20) that T ~ T ~ ~ ~ ~Similarly, = O . (8.23) yields J$A+ J$;=Jf;' when all y I coordinates are set equal to zero. This relation in conjunction with (8.20) provides three relations among the T'S. The relations summarized in Table 8.10 are appropriate for both prolate, type I',and oblate, type III', cases. These relations can be used to reduce the number of independent T'S to only four, for example, T,,,~, T,,,,, T ~ and~ ox,,. ~ In ~this , particular case the first-order energy expression can be written in terms of four 7's. The coefficientsof the four T'S in the energy expression can be easily found by use of the average values of Table 8.6 and the relations of Table 8.10 in conjunction with (8.56). Evaluation of the four T'S from the

~

FIRST-ORDER PERTURBATION TREATMENT

OF ROTATIONAL DISTORTION

323

Table 8.10 Distortion Constant Relationships for Planar Asymmetric Rotor Molecules"

"The coordinate axes are to be assigned a-z, b-x, c-y for the prolate case and a-y, b-x, and cuz in the oblate case; in each case the molecule lies in the ab plane.

Table 8.11 Rotational Distortion Constants for Molecules of C3"Symmetry" Nonvanishing Txxxx = Tyyyy

T'S

Tzzzz

-2TxYxy - Txyzx = - Txxyrb

Txxzz = TYYzz

Txxyy = Txxxx

Txzxz = Tyzyz

Tyyyz =

Distortion Coefficients ti4

D J = - - Txxxx 4

ti4

D J K= - 205 - - { Txxrz 2

+2 T x z x z }

ti4

DK = -Dj -D J K- - T,,,, 4 ' y axis lies in a symmetry plane with I:, = R5=R6 = JJ =O. The symmetry axis is z. b D o not contribute in a first-order approximation.

spectrum and use of the relations allows direct evaluation of the seven individual 7's. The rotational constants to be used in the relations of Table 8.10 should be the equilibrium rotational constants, but in practice these usually must be replaced by the effective rotational constants. Also, if the harmonic approximation is not adequate, then the relations are not strictly correct. For a C,,(p#4) symmetric rotor, Rs,R6,and 6., are identically zero while the factors multiplying these constants remain finite in this limit. Furthermore, (PI")=K and (P:) =K and hence the first-order (8.64) will reduce to that given previously for a symmetric rotor. For molecules with C3, symmetry,

324


Table 8.12 Distortion Coefficients for Planar Symmetric Rotors”

“The z axis is taken as the symmetry axis that is perpendicular to the plane of the molecule.

symmetric rotors, direct calculation [60] or group theoretical considerations [21] lead to the nonvanishing T’S listed in Table 8.11, and the explicit definitions ofDJ,&, and DJKin terms of the T’S can be simplified [25]. These results are also included in Table 8.11. For planar symmetric rotors (oblate tops) additional relations exist between the T’S, with further simplification of the distortion coefficients [59]. These are given in Table 8.12. For this case it is evident that if any two distortion constants are determined, the remaining one may be immediately evaluated.

4

REDUCED HAMILTONIAN

From (8.64) it is apparent that no more than five linear combinations of the nine possible T’S can be obtained from an analysis of the rotational spectrum of an asymmetric rotor. It might appear from (8.56) and Table 8.6 that six linear combinations of 7’s could be determined from the observed spectrum; because of the linear relation (8.62),however, the resulting energy level equation would have an inherent indeterminacy and would thus not be satisfactory for evaluation of unique values of the distortion constants, although it would be applicable to calculation of the distortion effects with known constants. In (8.64) this indeterminacy has been removed by application of (8.62), leaving five determinable distortion coefficients. Such a problem does not arise for planar molecules when the planarity conditions (see the previous section) are applied, since this effectively removes the indeterminacy resulting from the linear relation between the average values. Most early analysis of centrifugal distortion employed the planar relations and thus avoided the problem. This indeterminacy was first reported by Dreizler, Dendl, and Rudolph [61, 621 in their analysis of dimethylsulfoxideand dimethylsulfide for six quartic distortion constants. They found the least-squares equations to be quite ill-conditioned, and determinate values for the constants could not be obtained. This is an example of a more general problem that can arise when one attempts to evaluate parameters in any Hamiltonian from the corresponding eigenvalues. The eigenvalues may depend on only certain linear combinations of the parameters. To properly fit the experimental data, one must employ a

REDUCED HAMILTONIAN

325

Hamiltonian that has any experimentally indeterminable parameters removed from it. This Hamiltonian, which may be obtained by a contact transformation, is termed a “reduced Hamiltonian.” The reduced Hamiltonian is to be used to fit experimental data. On the other hand, to interpret the quantities in terms of fundamental molecular parameters, the experimentally determined coefficients must be related to the constants in the original “first-principles Hamiltonian.” These relations are, of course, specified by the contact transformation which relates the two Hamiltonians. The nature of the problem can be readily appreciated by consideration of the rigid rotor Hamiltonian. For a general asymmetric top only three principal moments of inertia can be obtained from the eigenvalues of H r .For an arbitrary orientation of the molecule-fixed axis, however, the rotational Hamiltonian contains six terms corresponding to six elements of the inertia tensor; see (2.6). But, by application of a unitary transformation a particular axis orie_ntation can be selected for which the inertia tensor has a diagonal form; Hr then depends on only three independent constants, the three principal moments of inertia, and it is these parameters that can be evaluated from the observed spectrum. A unitary operator ( U - = U t ) can be conveniently written as

’

U= ,is

(8.69)

where S is Hermitian. The transformed Hamiltonian is given by

2=u-’Yeu

(8.70)

The result of apjlying the appropriate transformation is that the form and eigenvalues of Ye are unchanged from X, but the individual coefficients are altered. This fact, along with order of magnitude arguments, have been used by Watson in a series of papers [55, 63, 641 to obtain the reduced Hamiltonian. Since the eigenvalues must remain the same, only those linear combinations of coefficients which remain invariant to the transformation can be evaluated from the eigenvalues, and hence are determinable coefficients. The most general form of the rotational Hamiltonian, correct to terms in P6, is given by

From our previous where we denote the rotational constants by &=fi2/2Z,. discussion of the P4 terms ,we have shown that for orthorhombic symmetry there are nine independent (Table 8.4) and that reduction of the fourthpower angular momentum terms PaPaP,Pa of 2‘; by means of the commutation relations yields a Hamiltonian with six terms, (8.55). Similar simplifications are possible for the P6 terms. Initially, there are a total of 36 = 729 coefficients in the P6 sum. &’must, however, be invariant under all symmetry operations of the point group of the molecule, and a number of coefficients in the P6 sum must vanish. Application of the group operations of the asymmetric-top point groups

326


to the Hamiltonian reveals that for the orthorhombic point groups (C2",D2,DZh) a total of 183 coefficients are nonzero, for the monoclinic point groups (Clh, Cz, C 2 h ) 365 coefficients are nonzero, and for the triclinic point groups (Cl, Ci) all 729 coefficients are nonzero. Enumeration of the P6 distortion constants, T . ~ ~ S & for ~ , the orthorhombic point groups has been discussed by Chung and Parker [65]. They show that, in general (8.72)

Tapydeq = Tqedypcr

The 183 coefficients are hence reduced to 105 distinct ones as follows: 1. Three coefficients of the type T,,,,. 2. Fifty-four coefficients of the type ~ ~ ~ ~ ~ with ~ ~ the ( csubscripts ~ # p )occur, ring in any order, taking (8.72) into account.

3. Forty-eight coefficients of the type ~ ~ # ~ B # y)~, with ~ the ~ subscripts ~ ( occurring in any order, with (8.72) taken into account. Because of (8.72), coefficients such as T~~~~~~ and T~~~~~~ occurring in (2) are equal, and likewise coefficients such as and T~~~~~~ occurring in ( 3 ) are equal. Compilation of all 105 distinct coefficients may be found elsewhere [65]. Reduction of the sixth-power angular momentum terms of the rotational Hamiltonian appropriate to an orthorhomic molecule by means of the commutation relations has been discussed by Kneizys et al. [66]. This reduction results in various P6 coefficients being folded into other coefficients. It is found that the Hamiltonian can be written in the form [63]

B a P i + C TapPaZPi+C@)rraaPz a

ad

+C

a

oaap(P,"Pp"+p$p,")

aPP

+oxyz ( P : PY'PZ + P,zPY'P;)

(8.73)

where the coefficients are all real. (See also Table 8.13 for an alternate form.) The rotational constants B, are defined in Table 8.5, as Ba =(fi2/21u) + (fi2/4)(3~pypy - 2Tapap - 2Tayay),a # B # Y E (x,Y,or Z ) (8.74) Once a particular choice of representation has been made, the various B,, By, B, can be identified with the A , B, C in a particular order. Here an alternate notation has been adopted for the effective quartic distortion coefficients (8.75)

with Tap=Tpa.The 10 @upu coefficients are the effective sextic distortion coefficients. They are linear combinations of the various 105 P6 distortion constants mentioned previously, and explicit expressions are available [66]. The

t

REDUCED HAMILTONIAN

327

subscripts on the distortion coefficients are indicative of the angular momentum operator associated with the coefficient. An alternate and more convenient form of the general power series of (8.71) has been introduced by Watson [55], the so-called standard form, that is, m

(8.77) Any term in the general Hamiltonian, (8.71), can, by means of the commutation relations, be cast into this form at the expense of introducting terms of lower degree. There is one term for each combination of powers of P,, P,, P,. Equation 8.55 is hence in standard form, and the fact that T~~~~ cannot be separated from T , , ~ is~ taken care of automatically by writing the Hamiltonian in the standard form. (Note, e.g., T,, =hZZO.) If one exploits the requirement that the Hamiltonian is invariant to the operation of Hermitian conjugation and time reversal [55], only terms with even values of n = p + g + Y are allowed, with the corresponding coefficients hPqr real. The number of terms of total degree n = p g r is i ( n l)(n 2). Thus, for n =2,4, and 6, there are 6, 15, and 28 terms, respectively. Of these terms for a given n, there are i ( n + 2)(n + 4) terms with pgr =eee (e even) and in(n 2) terms each with pgr = e m , oeo, and ooe (o odd). Since the Hamiltonian must belong to the totally symmetric symmetry species of the molecular point group, one finds [55] for molecules of orthorhombic symmetry that p , q, and r are, in addition, each required to be even in (8.77).Thus, it is clear that for n =2,4, and 6, there are in the standard form &(n+ 2Nn 4) = 3,6, and 10 terms, respectively, in agreement with (8.73). It should be noted that the Hamiltonian (8.73) holds even for molecules of non-orthorhombic symmetry. For such molecules, additional terms such as ooe appear in the standard form, for example, h310,h l 1 2 , and these contribute in higher order. Elements of the unitary transformation S3 (S5 for Xi6))can, however, be chosen to eliminate such terms, and thereby to give a Hamiltonian of orthorhombic form with coefficients modified from their original values. The important changes occur in the higher-order P6 coefficients, and the modified definitions of the OapYare given elsewhere [64]. The orthorhombic form of the Hamiltonian can be further reduced by means of a unitary transformation as indicated previously. The transformation is conveniently represented as two successively applied transformations

+ +

+

+ +

+

(8.78) where terms to P6 are to be considered. The invariant properties discussed previously show that S has a form similar to (8.77) with p + q + r = n , and n odd, viz. ~ = ~ i S s ~ i S 5

s=

Spqr(

PpP;P:

+ P: P;Pxp)

(8.79)

Terms of the same degree are denoted by S,,. For molecules of orthorhombic symmetry, only the terms [55] s 3 =SI 1 l(P,P,P,

+ P,P,P,)

(8.80)

328


and S5 =$3 1 1(P: p y p z + p z p y p:) + s13 l(PXP,3PZ+ PZP,3PX) +sl13(pxpyp5

+p l p y p x )

(8.81)

are required. The spqr are real coefficients and are the parameters of the unitary transformation that are to be chosen to achieve the reduction. Other terms of S3 and Sscome into play when one wishes to remove nonorthorhombic terms in (8.77) as mentioned previously. For such molecules, the terms in J? that are not totally symmetric in the Four-group are removed by non-totally symmetric terms in S and they appear in terms of higher degree in the Hamiltonian. Now the order of magnitude of the terms in (8.77) may be taken approximately as

hPqr

(8.82)

~~(p+q+*)E,

where the rotational, vibrational, and electronic energy are related approximately by E,-ic2E,-ic4Ee(ic ~ 1 )The . rigid rotor Hamiltonian is thus of order ic4E,. Likewise, the quartic distortion Hamiltonian is of order ic8Ee,and the sextic distortion Hamiltonian is of order icl'Ee. The spqr satisfy the order-ofmagnitude relation Spq*

(8.83)

K Z ( ~ + q + r -1)

Thus, for example, s1 is of order of magnitude ic4, which is on the order of the ratio of a quartic distortion coefficient to a rotational constant (T/B).Expressing the exponentials in (8.70) as a power series expansion (see Appendix C) and arranging by order of magnitude, one finds

where (8.86)

*A6'=

J?A6)+i[J?r, S5]+i[Xi4), S3]-f[[Xr,

S3],

S3]

(8.87)

which is correct to order icl'Ee, where [ A , B] is the commutator AB-BA. Note, for example, that XA4)and [X,, S3]are both of order K*E,. Here s1 specifies the reduction of XA4)whereas , s3 1, s1 31, and s1 specify the reduction of *A6). Once a choice of the spqrparameters has been made to eliminate the maximum number of terms from X, the transformed Hamiltonian becomes a reduced Hamiltonian. In principle, an infinite number of reductions are possible, and a number have been discussed in the literature [ 5 5 , 63, 64, 67-71]. In the next section two specific reductions will be considered. The most general Hamiltonian for a symmetric-top and the procedure for obtaining the corresponding reduced Hamiltonian via (8.84),which depends on the molecular symmetry, are discussed elsewhere [64], though the particular results will be given here.

REDUCED HAMILTONIAN

329

Asymmetric Top Reduction ( A ) For simplicity we consider only the reduction of *A4) in demonstrating the will be simply quoted. Consult procedure. The results for the reduction of 2j6) [63] for further details. Evaluation of the commutator in (8.86) has been given in (8.60), and with a change in notation we have

+

+

i[ X v ,S,] =2s11 I{(& - B,)(PZPy” Py”P,’ 2 P f ) +(Bz-By)(P,2P,2+P,2P,z+2P,’) P:Pf +2Py”)} +(B,-B,)(PtPf

+

(8.88)

and hence, from (8.73) and (8.84), (8.89) Z>P

01

This has the same form as 2 ;however, the coefficients are modified as follows:

(8.90) and

(8.91) It will be convenient to recast (8.73) into a form involving the constants D j , D J K ,DK, dj, R5 , and R6 defined in Table 8.7, which is more suitable for numerical calculations. With considerable manipulations one finds [64]

*=(Bx- 4R6)P: + (By- 4&j)Pz+ (B,+ 6Rs)P: -D J P4 -D J K P ’ P ~-D K P-~6jP2(P: +RS{P;(P: +PZ_)+(P: +PZ_)P:} &(P$ P?)

+

+

+P t ) (8.92)

+

with P* =(Px& iP,) and P z =PE Py”+ P;. A more compact notation [72] for this Hamiltonian is found in Table 8.13. The relations between the coefficients of the cylindrisal tensor form and that of (8.73) are also given in Table 8.13. The form of X of (8.92) will also be the same, with the coefficients identified by a tilde, however; with use of (8.90), (8.91), and Table 8.7, they are defined as

(8.93)

330


Table 8.2 j Relations Between the Cylindrical and Cartesian Coefficients

''P* =(PxkiPJ, P 2 = P: + Pj + P:. Note the subscripts 21,2m,and 212 refer, respectively, to the powers of P, P,,and P+. bFrom Watson [64].

with the effective rotational constants (8.94) The most advantageous reduction, initially introduced by Watson, involves chosing s1 so that R6 =O. In this case, the ( K / K & 4)matrix elements of 2 in a symmetric roto: basis are eliminated. *A4) then contains only five terms, and the matrix of Xh4)has the same tridiagonal form as that for a rigid rotor. The condition i?, =O requires (8.95)

(8.96) where the determinable constantsZJ and so on, are given an alternate notation.

REDUCED HAMILTONIAN

331

Note that A K is invariant to a permutation of axes, whereas Aj, A ~ K6 ,5 , and 6~ change on permuting axes. Hence these latter constants depend on the axis representation used in the analysis. From (8.92) with P: + P!. =2(Pf - P,”),the reduced Hamiltonian for the A reduction including the P6 terms is, therefore, *(A)

+ +

=

xr+ XA4’ + xp

X r= B‘p’P: B‘yA’Py2 =f (B’!’ B’;”)P2

(8.97)

+ BIA’P,2 + -i.(B’p+ B’yA’)]Pz” [&A’

(8.98)

++(By’- B p ) ( P : - P;)

%A4)

%A6)

- 26j P 2 (p ; - p,”) (8.99) - 6 K [Pi(P: - p,”)+ (Pf - P,”)Pt] =@J P 6+ @JKP4Pt + @KJP2P;+ @KP,6+ 2$ JP4(Pf- P,”)

+

= A jP4 - AjKP2Pj - AKP;

+

$ ~ K P ’ [ P ~ ( P ; -P,”) ( P ; -P,”)P,Z] +$K[P~(Pf-P,”)+(P:--,”)P~]

(8.100)

where Aj, A j K , A K , J J , and aK are the quartic distortion coefficients defined in , K JO , K ,$ J , $ J K , terms of various notations in Table 8.14, and where OJ,O J K O and $ K are the sextic distortion coefficients. Note Pi and.(Pf - P,”)do not commute. The notation H,, HjK, HKj, HK, hJ, hJK,hK has been used in the past for the P6 constants, and many constants have been reported with this notation using the A reduction. However, as has been suggested [64], we have used the

Table 8.14 Quartic Distortion Coefficients of the Reduced Distortion Hamiltonian”

332


+.,

alternate notation O J , @ j K , @ K j , Q K , 4 J K , and 4K in order to retain the H,, . . . ,H K notation for the symmetric-top reduced Hamiltonian of the next section, since the diagonal P6 constants for the A reduction do not correlate appropriately with the symmetric-top constants. The P6 spectroscopic constants are defined in Table 8.15 in terms of the OaPy.The effective rotational constants are defined as

(8.103) where B,, B y , and B, are given in (8.118) and are readily derivable from the effective rotational constants using the A-constants obtained from the spectrum. This Hamiltonian has been found to be applicable to a large range of asymmetric tops. It is apparent that the effective Hamiltonian has been reduced to one that contains only determinable constants and is appropriate for fitting purposes. The reduced Hamiltonian in general contains (n + 1) distortion coefficients for each degree n in the angular momentum. There are (n/2 1) purely diagonal angular momentum operators, and the remaining ones are offdiagonal, involving (P:- Pi). For light molecules higher-order effects are important, and numerous distortion coefficients are required to adequately fit the spectrum. The reduced

+

Table 8.15 Sextic Distortion Coefficients of the Reduced Distortion Hamiltonian in Terms of the Cylindrical Tensor Coefficientsa @J=@600

+ 2@204

@JK=@~ZO- 12@204+2@024+ @ K J = @ 2 4 0 +?@420-

~ 6 ~ @ 0 0 ~ + 8 ~ 0 2 2 ~ 0 0 4 / ~ 0 0 z

~O@ZO~-?@JK

@~=@060-3~420f28@204f:@)~~

4 J = @ 4 0 2 +@006 4 J K = @ 2 2 2 +40@204-

10@006 ~

~

~

~

~

2

0

~

~

~

~

~

0

2

~

~

~

0

0

4

“See Table 8.13 for definition of the coefficients in the cylindrical tensor form of z? in terms of of (8.73). From Watson [64].

~

~

0

0

REDUCED HAMILTONIAN

333

The L, 1 are octic centrifugal distortion constants, and the P , p are the corresponding dectic constants. These coefficients must be regarded at this point as empirical constants in the sense that they have not as yet been theoretically related to the vibrational potential constants. On the other hand, the quartic and sextic distortion constants have been related to the potential constants, in particular to the quadratic and cubic force constants, respectively.

Symmetric Top Reduction ( S ) For very slightly asymmetric tops, the previous reduction can present problems. It is clear from the definition of s1 that if B,- By,s1 gets very large and the order-of-magnitude arguments employed previously no longer apply. Specifically, the breakdown depends on the ratio of a P4distortion constant to ( B x - By). Different representations can be chosen to minimize this effect by making ( B x - B y )larger and R6 as small as possible. However, it is preferable to employ an alternate reduction that reduces in a welkdefined way to the symmetric top limit. In this case it is convenient to take R5 = O which requires 8111 =

2R5 2Bz - B, - By

(8.106)

This reduction has been proposed by Winnewisser [69] and Van Eijck [70]. It has been demonstrated by Van Eijck [70], who discussed si4’,and by Typke [71] who extended the results to &‘h6), that a more satisfactory analysis can be obtained for slightly asymmetric tops with this reduction. Watson [64] also was instrumental in the introduction of this form of the reduction. The coefficients in the reduced form of (8.92) are now

334


-_ -

_

(8.107)

2Bz-Bx-By Bx - By

(8.108)

6K

40

where 0=

for an I* and 111’ representation, 0 = - l/b, and - l/bo, respectively. The notation adopted to retain the similarity of the reduced Hamiltonian to that of a symmetric top is *

-

*

*

D j = D j , D J K Z D J KD, ~ = D K , d l = - h j , d 2 = R t j

(8.109)

The reduced Hamiltonian for the S reduction is therefore

+ 4 4 ’ + 2f$6’

9 s ’ = 2fr

(8.110)

where the P6 terms have also been included. Note, however, that the constants D j , DjK, and DK used in the foregoing Hamiltonian are not the same constants as those defined in Table 8.7. The effective rotational constants, with cognizance of (8.90), are

(8.114)

REDUCED HAMILTONIAN

335

(8.115)

= B2-2Dj-6d2

(8.116)

where B,, By,B, are defined in (8.118). The coefficients in 2‘’) and X(’) can be related to each other (see Table 8.16), and the coefficients of the two reduced Hamiltonians provide the same information. It is apparent that this Hamiltonian can be readily generalized to include higher-order constants. If we extend the Hamiltonian to P8 terms, the following terms must be added

+

+

+

+

L J P 8 L J JKP6Pt L JKP4P: L K K JP2P,6 LKP,8 +IIP6(P: +p2-)+12p4(p4+ +p4_)+l3p2(P6+ +p6_)+I4(p8+ +P?)

(8.117)

Determinable Combinations of Coefficients From (8.90) and (8.91) s1 can be eliminated from the nine equations to give eight linear combinations of quadratic and quartic coefficients. These coefficients are the determinable combinations of coefficients since the energies cannot depend on s1 The experimentally determinable coefficients can be conveniently taken as B, =B, - 2 Tyz By= By- 2 T,, B,=B,-2TXy

(8.118)

Table 8.16 Relations Between the Distortion Coefficients in the A and S Reduced Hamiltonians”

‘u =(2B, - B, - By)/(&- By).From Watson [ 1401.

336


(8.119) In terms of the T notation, the determinable coefficients are

(8.120) L x X ,

~Lzz,

&yy,

+

+c c x y y ~2 =Bx~;yzz+ B y ~ L z + z Bz~Lxyy T1 = 7;yzz

TLXZZ

(8.121)

The Tl (or T ~ and ) T2 (or T ~ have ) the same value for any permutation of the axes. The constant T2 (or z 2 ) is often scaled so that it has same units as the other constants, that is, T2 T2+or S

72

T ~ + -

S

(8.122)

with S = B x + B y + B,. Likewise the P6 determinable coefficients can be taken as [64] @xxx, @ Y Y Y , @m @1

=3

C

@aap+@xyz

afP

=(Bx - BzPxxy + (Bx - By P x x z -2(Txx - Txy)(Txx- Z z ) Tyz)(Tyy- T x ) 0 4 = (Bz - By P z z x + (Bz - Bx)@zzy- 2 ( L - Z x )(Zz- Z y )

0 2

@3 = ~ ~ y - ~ x ~ @ y y , + ~ ~ y - ~ z ~ @ y y x - ~ ~ ~ y y -

(8.123)

These determinable combinations of constants are hence directly related to the constants of the first-principles Hamiltonian and can thus be directly related to the molecular force field. The foregoing determinable combinations may be expressed in terms of the constants evaluated from the rotational spectrum via the Hamiltonians of (8.97) and (8.110). These relations are given in Tables 8.17 and 8.18. Once an axis representation is chosen, those relations yield the constants T,,, T b b , . . . , referred to the principal inertial axis system. In reporting the rotation and distortion constants, the effective rotational constants d ,a,V and A, B, C should be listed. The type of reduction ( A or S ) used and the representation I", II", and so on, that is employed should be indicated in the table of constants. In the cases where large numbers of constants are required in the fit, the addition of an A or S superscript to the constants would avoid the need for two sets of symbols for the higher-order constants, for example, P ' constants ( ~ 5 ' ) or ~ 5 ~ ) ) .

REDUCED HAMILTONIAN

337

Table 8.17 Determinable Combinations in Terms of the Spectroscopic Constants of the A Reduction‘

“Rg=&(TJs+ T,,-2Tx,). From Watson [64].

Table 8.1 8 Determinable Combinations in Terms of the Spectroscopic Constants of the S Reduction’

“Rs=&(TX,+T,,-2Tx,). From Watson [64].

Symmetric and Spherical Tops For a symmetric rotor B’!’=B’:’, and the reduced Hamiltonian has the form

2 = B, P2 + (B, - B,)P: -k H j P 6

-D J P ~ DJKP2Pz -DKP t

+ H J K P ~ P-k: HKJP’P; -k H K P ,+~

2split

(8.124)

338


The D’s have their usual meaning for a symmetric top. The 2 s p i i t term arises from the off-diagonal terms in X(’) and leads to splitting of the K-degenerate levels of a symmetric top. The form of the reduced splitting term depends on the molecular point group. In particular, correct to P6 terms =(d2+h2P2)(P4++Pf)+h3(P: +P!)

2split

(8.125)

where, for ap-fold axis of rotation [61] d2=h2=h3=0, d2=h2=0,

h3=0,

i f p = 5 or a 7

(8.126)

i f p = 3 or 6

(8.127)

ifp=4

(8.128)

The constants d2, h2, and h3 are particularly important in the splitting of the K = L 2 and K = L 3 levels, respectively. For a C S v molecule [72]

(8.129) The Hamiltonian for a spherical rotor has been discussed by a number of authors [73-801. It may be conveniently written in terms of spherical tensor operators Rl of rank 1(1=0,2,4,6,. . .) [78]

+

Y?=BP2 -DsP4+D4TR4 +HsP6 H4,P2sZ4+ Htj~f&,

(8.130)

where the D’s and H’s are, respectively, the effective quartic and sextic distortion constants. The subscript T indicates a coefficient of a tensor operator, 1f0, and S indicates coefficients of the scalar terms, 1=0. For a Td molecule [72] 1 (45QxXx 36@,,, 105

Hs =-

H

+

-1 (6@,,,, - 12@,,,, 6T-231

+ 2@,,,)

+ 4@,,,,)

(8.131)

The energy levels are conveniently labeled by J,M J , C, t , where J, M J have their usual meaning, C specifies the irreducible symmetry representation of the level, and t distinguishes different levels of the same representation. Precise measurements [81] have indicated the need for higher-order terms in the Hamiltonian. The P8 effects add the following terms %(8’=LSP8 +L4TP4n4+L6TP2R6 +L8TRs

(8.132)

339

EVALUATION OF DISTORTION CONSTANTS FROM OBSERVED SPECTRA

It has been shown that this can be written in an alternate form [SO]

+

x p ( 8 ) = L S P 8L 4 T P 4 n 4 + L 6 T P 2 f i 6 + L 4 4 T n :

(8.133)

which replaces the evaluation of the matrix of 0 8 with that for ai. The latter matrix is readily constructed from that for R4. The coefficients of lower degree are also modified slightly by this formulation. In a similar way one obtains the following additional terms to higher-order for the reduced Hamiltonian to twelfth degree [SO] 0 ) = pspl 0

+

P4TP6R4

+

P6TP4n6

+P 4 6 T ( a 4 n 6 +n6R4)

+

P44TP2Ri

(8.134)

s % ( ~ ~ ' = R+sRP4~~~P ~ n 4 + R 6 ~ P ~ a 6 + R 4 4 ~ P ~ n : + R 4 6 T P 2 ( R 4 R 6 + n6a4) + R 4 4 4 T n : (8.135) where the P's and R's are additional distortion constants. A discussion of the evaluation of the energy levels is found in Section 5 (see also Chapter VI). The tensor operators in the foregoing expressions are constructed from the angular momentum components P, so as to be totally symmetric under the operations of the molecular point group, for example, [77] !24=6P4- 10(P$+P:+P:)-2P2

(8.136)

and

0, = 1 5P6 +T(Pz+ Py" + P,")- y (P$ + P; + P4)(3P2- 7) - 9 P 4 + 1 9P2 (8.137)

5 EVALUATION OF THE DISTORTION CONSTANTS FROM OBSERVED SPECTRA For molecules with very small moments of inertia (e.g., H 2 0 , H2S) or levels with high rotational quantum numbers, a first-order treatment of centrifugal distortion may not be adequate. A detailed study of the distortion effects in O F 2 by Pierce et al. [82] has shown that at high Jvalues, 20< JG40, secondand higher-order distortion effects, though small compared to the first-order effects, are nevertheless significant. For these cases, as with the rigid rotor problem, the complete energy matrix can be set up and diagonalized directly to give the total energy of the molecule, including distortion corrections. Earlier distortion studies employed the Hamiltonian of (8.53) and were carried out on planar molecules; the planarity relations were used to reduce the number T b b b b , To&, and T o b o b . The of T constants from six to four, for example, T,,, indeterminacy mentioned previously was thus removed. For the case of molecules with orthorhombic symmetry, (8.53), the Hamiltonian matrix in a symmetric rotor basis is diagonal in J a n d M , with nonvanishing matrix elements in K of the type ( K I K ) , (KIKF2), and (KIKf4). For molecules with non~ are ~intro~ orthorhombic symmetry, additional distortion constants (e.g., T duced which also contribute in higher order [50, 64, 72, 831. In the reduced

~

340


Hamiltonian, these constants do not appear explicitly but rather modify the definitions of the higher-order reduced constants. The matrix elements for the distortion Hamiltonian (8.55) may be obtained, as before, by matrix multiplication and use of the matrix elements given previously. Explicit expressions are given in Table 8.19. The matrix will still factor into four submatrices upon application of the Wang symmetrizing transformation [84]. Portions of the submatrices in terms of the original matrix elements are:

Table 8.19 Matrix Elements of P:Pi in a Symmetric Rotor Representation”

“The phase choice used here is that of G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys., 11, 27 (1943). For simplicity we do not display the J and M labeling, it being understood the matrix elements are diagonal in these quantum numbers.


341

For E-, the first row and column must be deleted as indicated by the dashed lines. In these derivations, use has been made of the relations

E K ,K = E - K , - K ;

E K , K+ i = EK+ i , x = E

- K , -K

- i = E -K -i , -K

(8.139)

with 1=2 or 4. Table 7.5 is still applicable for identification of the explicit energy levels contained in each submatrix. Each matrix element of (8.138) will be made up of contributions from X r and X,,. Diagonalization of the complete energy matrix gives the rotational energy, including distortion corrections. For sufficiently low values of J , the subdeterminants of (8.138) may be solved directly to give expressions for the energy levels, including the effects of centrifugal distortion. Oka and Morino [85] and Chung and Parker [19] have reported expressions for the rotational energies of particular low J levels which are in the form of linear or quadratic expressions. In Table 8.20 we give explicit expressions for the energy levels that can be expressed in linear forms. By use of this table, the frequency for certain low J transitions can be easily evaluated if the 7's are known. Sum rules involving the rotational and distortion constants have also been obtained [86-901. These are derived by use of the invariance of the trace of a matrix to a similarity transformation. Thus, for each J , the trace of a submatrix of, for example, (8.138) equals the sum of the corresponding energy levels.

Table 8.20 Linear Expressions for the Rotational Energy Levels Including Centrifugal Distortion Effects" JK-I~KI

E

00,o 10,l

11,l

11,o

21,2

21,l

22,l

3 2 , ~

"From Oka and Morino [ 8 5 ] . Note that for planar molecules

T . ~ ~ ~ = ? ~ ~ ~ ~ = O .

342


Because the latter information is often unavailable, however, the sum rules are hence of limited value. Sum rules for the reduced Hamiltonian are also available [64]. For higher values of J , the order of the secular determinant requires numerical diagonalization techniques. To set up the energy matrix, (8.138), the numerical values of the 7’s are required. If the force constants and structural parameters are available, the distortion constants may be calculated directly, and the corresponding energy matrix diagonalized to give the effects of centrifugal distortion. The centrifugal distortion constants are, however, best evaluated from the experimental data of the rotational spectrum. An approximate set of rotation and distortion constants may be obtained by use of the first-order expression (8.56) for a planar molecule; all seven individual 5’s may be obtained from the analysis (if the relations of Table 8.10 are assumed). These constants along with the rotational constants are used for setting up the complete energy matrix, (8.138), and the higher effects may be taken into account by use of the rigid-rotor basis distortion analysis to be discussed later. Distortion constants of some earlier studies, as well as more recent ones employing the planar relations in the analysis, are listed in Table 8.21. Although the form of %,, (8.55), is applicable to planar molecules with use of the planarity conditions, it is not suitable for the general case. The maximum number of quartic distortion coefficients that can be obtained from the experimental data is five, whereas %d of (8.55) contains six coefficients. Furthermore, as we shall see, use of the planarity relations usually introduces significant model errors. To remove this problem, the Hamiltonian was transformed by means of a unitary operator to a form that contains only five independent terms in the angular momenta. The coefficients of each term are combinations of the original coefficients. This reduced Hamiltonian, which has been discussed in Section 4, has been given correct to PI0 terms in the Hamiltonian of (8.97)(8.105)

+ %d

=Xr

(8.140) (8.141)

This form is particularly suited to the empirical determination of the centrifugal distortion constants if higher-order effects are to be considered in nonplanar molecules or in planar molecules where the planarity conditions are not employed. The energy matrix, like the rigid rotor energy matrix, has only ( K I K ) and (KIK f2) matrix elements in a symmetric rotor basis, and the Wang transformation gives four separate submatrices E+, E-, O’, 0- for each J , the form of which is specified by (7.22). Furthermore, the continued-fraction technique, which is particularly efficient, may be used for diagonalization of the matrix (see Appendix B). Thus, if a program applicable to calculation of the rigid rotor energy levels is available, it can be readily applied to the analysis of distortion effects by modification of the matrix elements. In particular, the matrix elements of (7.22) contain terms arising from the various distortion constants

Next Page

343


Table 8.21 Rotational Distortion Constants for Some Planar Asymmetric Rotors'

so2 0 3

OF2

ClO2 Cl20

S(CN)2 NF2 NOF NSF NOCl NOzCl H"BF2

CINCO CHO-COOH "BH20H

ArCO,

- 10.1557 - 23.2588 -6.6154 - 7.8422 - 5.3889 -0.5838 - 7.75 - 15.5869 - 8.8343 - 17.322 - 0.0427 - 9.04 -59.138 - 0.0397 -25.64 - 0.0160b

- 0.0356 - 0.0762 - 0.0950 - 0.0433 - 0.009 1 -0.0144 - 0.08 1 - 0.0996 - 0.0564 - 0.0292 -0.0173 - 0.0563 - 0.0107 - 0.0033 - 0.362 -0.0983

0.4644 0.4538 0.3996 0.4748 0.1146 0.0833 0.297 0.4186 0.4480 0.2555 0.0038 0.409 0.6923 0.0018 0.54 Ob

- 0.0485 - 0.3009 -0.1625 -0.0416 -0.0188 - 0.0023 -0.126 - 0.2896 - 0.0896 -0.1101 - 0.0253 - 0.099 -0.0101 - 0.0368 - 1.676 - 0.907

c

d e

J 9

h I

j

k 1 m n 0

P 4 r

"The planarity conditions have been invoked in these analyses. *Fixed. 'Kivelson [38]. dPierce [153]. 'Pierce et al. [82]. IM. G. K. Pillai and R. F. Curl, Jr., J. Chem. Phys., 37, 2921 (1962). 8Herberich et al. [138]. hL. Pierce, R. Nelson, and C. Thomas, J. Chem. Phys., 43, 3423 (1965). 'R. D. Brown, F. R. Burden, P. D. Godfrey, and I. R. Gillard, J. Mol. Spectrosc., 52,301 (1974). 'Cook [157]. 'Cook and Kirchhoff [163]. 'G. Cauoli, R. Cervellati, and A. M. Mirri, J. Mol. Spectrosc., 56,422 (1975). "R. R. Filgueira, P. Forti, and G. Corbelli, J. Mol. Spectrosc., 57, 97 (1975). "Robiette and Gerry [178]. "W. H. Hocking and M. C. L. Gerry, J. Mol. Spectrosc., 42, 541 (1972). 'K. M. Marstokk and H. Msllendal, J. Mol. Struct., 15, 137 (1973). 'Y. Kawashima, H. Takeo, and C. Matsumura, J. Chem. Phys., 74,5430 (1981). 'J. M. Steed, T. A. Dixon, and W. Klemperer, J. Chem. Phys., 70, 4095 (1979).

The matrix elements required for X,, in a symmetric rotor basis are summarized in Table 8.22. Those for the rigid rotor part, yi", have been given in Chapter VII. The terms independent of ( P j - P j ) are purely diagonal in the symmetric top basis, while those involving (P: - P:) are off-diagonal with AK=+2. For evaluating the centrifugal distortion constants from the rotational spectrum, two procedures are particularly convenient-a rigid rotor basis [91, 921 and a semirigid rotor basis [93] distortion analysis. The former procedure makes use of the rigid rotor basis, that is, the basis in which X r is diagonal, to derive a first-order energy expression from which the

344


Previous Page

Table 8.22 Matrix Elements for the Angular Momentum Operators in a Symmetric Rotor Basis'

"Phase choice that of G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys., 11, 27 (1943). Angular momentum is in units of h.

rotation-distortion constants can be obtained with least-squares techniques. If the distortion effects are treated as a perturbation on the rigid rotor, the first-order distortion energy may be found by evaluating (8.97) in the rigid asymmetric rotor basis. If the Wang representation of X r is employed, (7.35) with (x-rb, y+c, z-a, that is, Ir), and if it is recognized that P 2 and %(b,) are diagonal in the asymmetric rotor basis, the following average value equations for the angular momentum operators may be derived from (7.35)

(PZ- P:) =.[ W(b,) - (Pi)]

(8.143)

(Pi(PZ-P,z)+(P~-Pyz)Pt)=20[W(bp)(P,z)-( E ) ]

(8.144)

(P:(P,2-P,2)+(P: -P,')~)=2o[W(b,)(P,4)-P,6)]

(8.145)

The first-order expression for the semirigid prolate asymmetric rotor for the (8.146) (8.147)

(8.148)

(8.149)

where 0 = - l/b, =( 2 6 - L@ - %')/(.% - %), W(b,) is the Wang reduced energy, and (P:) is the average of P: in the rigid asymmetric rotor basis. The latter quantities are obtained in conjunction with the diagonalization of the rigid rotor energy matrix. The d (=@I), .%(= @/'), %( = @!)) are the effective rotational constants


345

If these results are generalized to include P8 and PI0 effects, the following terms are added to the first-order energy expression [92]

One may apply these equations to an oblate top (x+b, y+a, z+c; i.e., 111') by interchanging d and $and setting a = - l/bo in the foregoing energy expression. For a 111" representation, a = l/bo. The effective rotational constants for the 111' representation are now defined by

346


In order to obtain the rotational and distortion constants, a least-squares analysis of the observed frequencies is carried out based on (8.146).In particular, the difference between the observed frequency and the calculated rigid rotor frequency is taken as the distortion effect, and the distortion constants are fitted to these distortion shifts ( v o ~ v,) by means of the least-squares technique. Usually, linear variations of the rigid rotor energy with respect to the three rigid rotor parameters are also considered, that is,

(8.154) The three variables 6d,6 9 , 6% are thus added to the analysis. For (8.147), variations in (9 %)/2,(9- %)/2,and d - (9+ %)/2would be more appropriate (see Chapter VII, Section 5). The observed frequency for a transition can thus be expressed in the form

+

vat, = V ,

+

A(ni)Ti

(8.155)

where v, is the rigid rotor frequency, ( n i ) are the appropriate operators averaged in the rigid rotor basis, A(Q) is the average value difference between the upper and lower energy level of the transition, and the IT;. represent the distortion constants and corrections to the rotational constants which are to be evaluated by use of the least-squares criterion. If higher-order distortion effects, such as P6 distortion terms, have to be considered in the analysis, then a first-order treatment will be be adequate, and higher-order perturbation effects must be considered. When the higherorder effects are not extremely large, a convenient iterative procedure may be used which was first employed by Pierce et al. [82] and subsequently extended to light asymmetric tops by Helminger, et al. [91]. This procedure takes into account higher-order perturbation effects while allowing the actual analysis for the distortion constants to be linear. The technique entails first analyzing the spectrum by means of (8.146) to obtain an approximate set of rotational and distortion constants. These constants are then employed to construct the complete energy matrix of (8.140).The energy for a given Jvalue may be written, correct to P6 terms, for example, as

~ = ++(%)J(J+ a 1)- A,J~(J+

+

+

112 H J ~ ( J +113 [d-+(B

+ $11w

(8.156)

where W is an energy eigenvalue of the reduced energy submatrix W, which is composed of K-dependent matrix elements of H,, %A4), and HA6),in which the original distortion constants have been divided by the factor [ d -+(9 This scaling is convenient for the diagonalization problem. Diagonalization of W gives the energy levels of the complete Hamiltonian; by taking energy differences, the transition frequencies are obtained which include the corrections due to centrifugal distortion. These frequencies are “exact” in that the complete energy matrix, (8.142), has been considered. The difference in the transition frequency found from (8.156) via diagonalization of the complete energy matrix

+ %)I.


347

and the frequency computed from the first-order energy expression, (8.146), gives the second- and higher-order centrifugal distortion effect to the given transition (the so-called “second-order” effects). The observed frequencies are then corrected for these effects and reanalyzed with first-order theory. ‘“second-ordei‘

‘corr.obs

-

- ‘exact ‘obs

- ‘first-order

- ‘“second-order“

(8.157) (8.158)

The observed frequencies, when corrected for these higher-order effects, essentially represent a set of first-order observed frequencies, that is, those which would be observed if there were no higher-order distortion effects present. If the corrected-observed frequencies, (8.158), are then reanalyzed with the firstorder theory, a slightly different set of rotation and distortion constants is obtained. The whole procedure is then repeated with these new constants, and another set of exact frequencies is thus obtained, which are again compared with first-order frequencies (computed with the new constants), to give the second- and higher-order corrections and subsequently a new set of correctedobserved frequencies. This iterative procedure may be continued until the higher effects are stable. In this way accurate distortion constants can be obtained while the higher-order distortion effects are still taken into account. This procedure may be applied in similar problems where a first-order treatment is not sufficient and higher-order effects necessitate diagonalization of a matrix in which the constants to be evaluated appear explicitly. Usually only a few iterations are required to stabilize the higher-order effects. This mode of analysis is very convenient, since only the average values of the operator, Pi in the rigid rotor basis are required, and is applicable to all heavy molecules; it has also been applied successfully to the light molecules HDS [91] and H D O [92]. However, when the “second-order” effects become very large, the convergence of this procedure becomes slow. It is then convenient to employ an alternative procedure which involves linearization of the problem from the start and is very similar to that used in obtaining the rotational constants from a rigid rotor spectrum. We follow the discussion of Cook et al. [93] and for convenience, denote the Hamiltonian of (8.140) as (8.159) where T correspond to the various rotation and distortion constants of (8.140) which we are interested in evaluating, and the ni represent the corresponding quantum mechanical operators. If (8.159) is evaluated in the semirigid rotor basis, that is, the basis in which 2 is diagonal, we have (8.160)

The (Hi) are averages such as ( P : - P : ) , (P:), and so on. If the 7; are the correct set of constants, then (8.160) represents the true energy of the level.

348


In general, the energy may be expanded in a Taylor series about the approximate parameters ( T iO), and to first order we have (8.161) This expression, which is linear in the parameter corrections ST,, may be used as the basis for a least-squares analysis in which the corrections to the constants are calculated. The derivatives are given by (8.162) where the average value on the right is calculated in the semirigid rotor basis employing the TP. These require the matrix elements of Table 8.22 and the eigenvectors for each energy level that should be obtained as part of the diagonalization procedure. The procedure for the analysis in the semirigid rotor basis is as follows. Approximate values of the spectral constants TO are used for the construction of the energy matrix of (8.140), and subsequent diagonalization of this matrix From (8.161) we can write for an observed yields the E(T?) and the (ni)o. transition (8.163) where vCalc=AE(TP). Here A indicates the difference between the energy (or average value) of the upper and lower level of the transition. Equation 8.163 and the least-squares criterion are used in an analysis of the observed transitions for the calculation of the 6rs.These corrections yield an improved approximation to the constants given by x=TP+STi

(8.164)

This procedure is then repeated with the improved approximations to the rotation and distortion constants. This numerical iteration procedure is continued until the 6T/s are negligibly small. Both the rigid rotor basis distortion analysis and the semirigid rotor basis distortion analysis will produce the same parameters. In general, when the “second-order’’ effects are large, the rigid rotor basis tends to be inconvenient because of the number of iterations that are necessary, whereas the semirigid rotor basis has been found to converge very rapidly [93]. Reasonably good approximations to the constants can usually be obtained from a rigid rotor basis first-order analysis. As additional higher-order distortion constants are subsequently added to the analysis, these constants may be initially started at zero with success. In fact, with an initial choice of zero for all the distortion constants, no difficulty is usually experienced in the convergence of the analysis. In all the calculations described, double precision (16-digit) arithmetic should be employed.

EVALUATION

OF DISTORTION CONSTANTS FROM OBSERVED SPECTRA

349

Examples of distortion constants obtained from rotational spectra with use of the A reduction are given in Table 8.23. The recent advances in submillimeter wave rotational spectroscopy and the accompanying theoretical advances in centrifugal distortion theory have provided significant impetus to the study of light asymmetric rotors. In the past, such molecules have eluded an accurate microwave study because of the occurrence of many, if not most, of their rotational transitions at frequencies inaccessible to conventional microwave spectrometers. Reviews of millimeter and submillimeter spectroscopy have been given by Winnewisser et al. [94], by De Lucia [95], and by Krupnov and Burenin [96]. In addition, inadequacies in the previous formulations of centrifugal distortion theory have handicapped the empirical determination of the distortion constants from observed spectra. Present theoretical formulations provide an adequate solution, but require on the order of 20 parameters to properly characterize the rotational spectrum of this class of molecules. Results of a typical analysis requiring a complete set of P4 and P6 terms, six P8 terms, and one Plo term are given in Table 8.24 for D2S [93]. In addition, studies of HDS [91], H2S [97], H 2 1 6 0 [98], H 2 1 7 0 [99, 1011, H 2 1 8 0 [loo, 1011, D2160 [102, 1031, D2173180[104], HDO [92, 1031, T 2 0 [105, 1061, HTO [105, 1071, DTO [107], H2Se [108], NH2D and N D 2 H [109, 1101 have been reported. Table 8.25 summarizes some of the results for this class of molecules. For the symmetric-top reduction, the analysis procedure discussed previously can be applied; however the energy matrix of Y P )= yi",

+ HL4' + 2 j 6 '

(8.165)

has matrix elements of the type (KIK),( K I K + 2 ) , (KIKf4), (KIKf6), and the four submatrices given in (8.138) contain more elements, as indicated in (8.166).

(8.166)

w

cn

0

Table 8.23

Rotational and Centrifugal Distortion Constants of Some Asymmetric Rotors ( A Reduction)(MHz)”

Molecule

d

D

AJ(103)

AJK(102)

Rf$

AK(lOO)

~

HOF

so2 s20 SCL, F2S0 DN, HN3-t HN03 HC=CCOOH CH,(CN), HCOOH H,C=CHCN CH, =CHF H2C0 trans.-HCOSH cis-HCOSH COF2 COCL, CH,DF H,C =C =CHCN cis-CH ,CH,COF

585,631.20 60,778.52 41,915.44 14,613.60 8,614.80 344,746.59 610,996.2 13,011.03 12,110.02 20,882.75 77,512.23 49,850.70 64,584.69 281,970.52 62,036.09 62,927.71 11,813.54 7,918.79 119,675.05 25,981.00 10,042.54

26,760.31 10,318.07 5,059.10 2,920.87 8,356.95 11,350.98 12,034.15 12,099.86 4,146.94 2,942.30 12,055.10 4,971.21 10,636.88 38,836.04 6,125.30 6,134.26 11,753.06 3,474.96 24,043.44 2,689.29 3,762.21

25,510.19 8,799.70 4,507.16 2,430.69 4,952.94 10,964.75 11,781.45 6,260.64 3,084.49 2,616.72 14416.15 4,513.83 9,118.03 34,002.20 5,569.64 5,584.75 5,880.90 2,412.21 22,959.37 2,474.8 1 2,832.69

94.14 6.5875 1.895 1.3249 4.586 4.281 4.673 14.038 0.5376 1.8554 9.989 2.2448 8.4352 75.2979 3.4287 3.6786 6.130 0.9712 49.371 1.28 0.776

241.46 - 11.724 - 3.192 - 1.4604 - 0.2475 44.45 1 79.1186 -2.0178 2.132 - 6.7922 - 8.625 - 8.5442 - 7.593 129.038 -4.3167 - 4.7962 - 0.3073 -0.1533 34.268 - 8.55 0.3474

82.520 2.5899 1.197 0.1380 0.01001 92.242 230.0 0.00742 -0.0077 15 0.8622 1.7023 2.7183 1.3294 19.4125 1.2507 1.3057 0.0133 0.01041 1.3774 1.94 0.0042

43.2 17.008 3,453 3.397 15.50 1.864 0.888 11.828 1.605 4.8926 19.492 4.5716 17.6591 104.4571 4.3473 4.63 12 25.775 3.481 23.29 3.10 1.832

1 12.0

b

2.5259 1.223 0.3927 0.2225 36.51 0 - 2.0565 1.219 0.6750 4.260 2.4575 3.5654 102.5757 1.6516 3.7653 0.4293 0.1793 6.87 0.6 0.1774

c

d c

f 9

h

j

k I m

” 0

P

P 4

5

t

”

? A K fixed and 6, set to zero. "I' representation employed. Constant times 10' is tabulated. bPearson and Kim [120]. 'Carpenter [115]. dCook et al. [155]. 'Davis and Gerry [152]. /Lucas and Smith [170]. gJ. Bendtsen and M. Winnewisser, Chem. Phys., 40,359 (1979). 'J. Bendtsen and M. Winnewisser, Chem. Phys. Leri., 33, 141 (1975). 'Cazzoli and De Lucia [3]. jR. W. Davis and M. C. L. Gerry, J. Mol. Specrrosc., 59,407 (1976). 'Cook et al. [7]. 'Willemot et al. [142]. "M. C. L. Gerry and G. Winnewisser, J. Mol. Specrrosc., 48, 1 (1973). "Gerry [174]. OR. Cornet and G. Winnewisser, J. Mol. Spectrosc., 80,438 (1980). 'W. H. Hocking, G. Winnewisser, Z. Narurjorsch., 31a, 422 (1976). qCarpenter [115]. 'J. H. Carpenter and D. F. Rimmer, J.C.S. Faraday 11,74,466 (1978). "W. W. Clark and F. C. De Lucia, J. Mol. Strucr., 32,29 (1976). tBouchy et al. [58]. "F. Scappini and H. Dreizler, 2. Narurforsch., 36a, 1327 (1981).

352


Table 8.24 Rotational and Distortion Constants of D2S(MHz)”

d = 164,571.118 k 0.045 9= 135,380.313k0.045

%=73,244.068 kO.071 AjK = - 41.7800 k 0.0066 65= - 1.95725k0.00068 AK=29.2170 k0.0113 6,=47.2516k0.0037 @~,=-3.206 X 10-2&0.020X @~=3.783 x 10-3k0.041 x @ ~ ~ = 6 . 3X7 10-2+0.044X 3 lo-’ @ K = -3.620 x 10-’k0.075 x $ J =-9.84 X 10-4+0.16 x 4jKz2.443 X 10-’&0.019 X lo-’ 4K=8.804 x 10-’+0.065 x lo-’ LJJK=2.747 x i o - 5 ~ o . i oxi io-’ LjK=-8.85 ~ l O - ~ k 1 . 0 7 ~ 1 0 - ~ 1J=7.72x10-7~1.02x ~ O - ~ x 10-5 ~ ~ ~ . , = x710-5f2.1 .8 l J K = - 1.750 x 10-5f0.131 x l K = -5.76 x 10-4f0.25 x pK=1.95 x 10-6f0.32 x

A j = 13.0763k0.0026

“The errors quoted are for 95% confidence limits. There is a high positive correla; a high tion (lpI>O.90) between ( @ J , Aj), (4J,6~1,( L J x ,@ K ) , and ( I J , L J J K )and negative correlation between ( L K K j , @ K ) , ( L K K J , LJK),( [ J , 4 ~and ) (IK, 4 K ) . A-reduction and 111’ representation employed in analysis. From Cook et al. [93].

It is not usually necessary to consider these higher-order distortion effects to obtain satisfactory distortion constants if high J transitions are excluded from the analysis, because only for high J (except for light molecules such as H 2 0 with large rotational energies even at low J ) do these higher-order effects become appreciably large. Therefore, a first-order treatment that is particularly simple can be profitably employed. It may be remarked that even for high J certain bQ1- transitions exhibit very small centrifugal distortion effects. The procedure for finding the values of J that correspond to these transitions is discussed elsewhere [1111. Since the matrices no longer have tridiagonal form, the continued fraction diagonalization procedure is no longer directly applicable, and a slower diagonalization procedure is required, for example, the Jacobi method [1121 which employs a series of rotation matrices to bring the energy matrix to diagonal form. Alternately, the Givens [112] method can be used which first reduces the matrix to tridiagonal form and then proceeds with a continued fraction procedure to complete the diagonalization. The off-diagonal terms (KIK f 4)arise from the constants d2 and h2, whereas the (KIKf6) elements arise from the h3 term in the Hamiltonian. If these terms are not too large, they may be neglected; the submatrices then have the same form as that for the rigid rotor problem. The neglected terms could then be treated by perturbation theory

EL1’=d2(P$ + P f ) + h 2 J ( J + l ) ( P $ +P?)+hj(P6+ +P6)

(8.167)

The approximation should be quite good for the P6 terms, but may not be satisfactory for the d2 term.

Table 8.25 Rotational, Quartic, and Sextic Distortion Constants of Some Light Molecules (MHz)"

H,Oh

H DOh

TzOb

H2SC

835,840.29 435,351.72 278,138.70 37.594 - 172.91 973.29 15.210 41.05 15.66 -4.21 - 5.10 3.733 78.0 - 2.52 10.97

701,931.50 272,912.60 192,055.25 10.838 34.208 377.078 3.647 63.087 1.128 7.344 - 2.740 1.4651 6.55 3.096 5.549

338,810.92 145,665.42 100,259.42 4.1456 - 22.039 144.138 1.6098 5.441 0.557 - 0.245 - 0.273 0.2036 2.68 0.0277 0.3637

310,182.24 270,884.05 141,705.88 49.851 - 159.696 111.851 - 6.0191 262.17 28.1 - 22.83 4.59 - 0.277 - 58.4 24.3 28.70

292,35 I .30 147,861.80 96,704. I2 2.6134 28.6933 - 11.297 0.8554 19.4078

f

9

h

d

W

Ul

w

"A-reduction. "1' representation. '111' representation. dDe Lucia et al. 1981. 'De Lucia et al. 1921. /De Lucia et al. [106]. gHelmingeret al. 1971. hHelmingeret al. [91]. 'Helminger and De Lucia [lo81

HD S ~

HZnoS,'

244,095.5 3 232,566.1 3 116,874.62 34.822 - 118.293 88.965 - 2.0788 463.32

-

-

1.326 0.2027 0.0130 1.07 0.53 0.283

~

-

117.2 259.4 -

-

58.44 22.34 85.1 1

354


The important new kinds of matrix elements required to set up the energy matrix are given in Table 8.26, and the remaining ones may be found in Table 8.22. Indications are that for a given set of data, the correlations between the constants are somewhat less for the S-reduced Hamiltonian than for the A reduced Hamiltonian, but the quality of the fit for the same number of constants is the same [64]. Most of the results reported have been obtained with the Areduced Hamiltonian. Examples employing the S reduction, which is particularly useful for slightly asymmetric rotors, are given in Table 8.27. To date, only a few molecules have been analyzed by means of the S-reduced Hamiltonian. A limited number of molecules have also been analyzed by the K reduction [68, 1131. In this reduction, the value of s1 is constrained to zero, and a sixth parameter ‘53 is fixed by means of the planarity relations. Details of this reduction may be found elsewhere [68]. Different representations affect the size of s l l l and the distribution of the angular momentum operators Pa, P b , P, appearing in the distortion Hamiltonian. The importance of the choice of axis representation (choice of a, b, c to x, y , ) has been studied by Typke [1141 and Carpenter [1151for the A reduction. If only the quartic distortion effects are considered, it is found that the standard deviation of the fit and the value of the distortion constants depend on the representation. However, when the sextic constants are added to the analysis, these effects are essentially eliminated. Different representations result in a different distribution of terms in the Hamiltonian between quartic and sextic terms, since s1 is different, as indicated by (8.86) and (8.87). Truncation of the Hamiltonian (S or A ) can produce a significant dependence on the representation because a substantial quartic effect in the standard Hamiltonian can wind up in the sextic terms of the reduced Hamiltonian. The addition of higherorder constants ensures there will be no higher-order contributions absorbed in the lower-order constants. On the other hand, for molecules where a large number of higher-order terms are required, a representation that minimizes the size of s l l l and hence improves the convergence of the Hamiltonian would be more appropriate [116]. Since AK should be invariant to the choice of representation, it has been useful as a check on the representation chosen and whether sufficient terms have been retained in the Hamiltonian. The T,,, G b , Table 8.26 Matrix Elements of (P:”+ P?) in a Symmetric Rotor Basis”

“Phase choice that of G. W. King, R. M. Hainer, and P. C . Cross, J . Chem. Phys., 11, 27 (1943). Angular momentum in units of h. P* = P , k i P , , n , m = l , 2 ,....

Table 8.27 Rotational and Centrifugal Distortion Constants of Some Asymmetric Rotors (S Reduction) (MHz)"

H2C=C=Sb HNCS HNCO H2CNNb H,N-NC(O+F H,N-NC(O-F NH,D CD2Sb

286,655 1,357,250.0 918,504.4 272,979.1 282,757 282,616 290,125.38 146,399.87

5,659.48 5,883.46 11,071.01 11,305.43 10,757.28 10,765.77 192,194.18 14,904.28

5,544.51 5,845.61 10,91 0.57 10,845.22 10,525.34 10,525.25 140,795.26 13,495.86

1.0857 1.1939 3.4863 4.1462 5.2720 2.2669 15,820.7 12.579d

23.5 16.8269 102.537 5 1,570.0 93.1 70 6,065.6 39.682 22.0 42.265 214 42.092 214 - 2394.5 10.954 29.61 5.208 -

25.46 13.781 - 72.995 - 192,550. - 176.75 175.33 4.182 x loh - 1409.

"I' representation employed except for NH,D. Constant times I@ is tabulated. bDKfixed. ' O + , 0- inversion states. dFixed by planarity relation. 'M.Winnewisser, E. Schafer, 2. Nuturjorsch., 35% 483 (1980). 'K. Yamada, M. Winnewisser, G . Winnewisser, L. B. Szalanski, and M. C. L. Gerry, J. Mol. Spectrosc., 78, 189 (1979). 9K. Yamada, J. Mol. Spectrosc., 79, 323 (1980). hE. Schafer and M. Winnewisser, J. Mol. Spectrosc., 97, 154 (1983). 'E. Schdfer and M. Winnewisser, Ber. Bunsenges, Phys. Chem., 86,780 (1982). 'Cohen and Pickett [1lo]. 'A. P. Cox, S. D. Hubbard, and H. Kato, J. Mol. Spectrosc., 93,196 (1982).

-

-5.21 - 4.59 - 36.5 - 57.27 - 34.34 - 34.34 1.324 x lo5 - 287.2

r 9

h I

J

k

356


T,,, TI, TZ(or Taaaa, Tbbbb, T ~ t2) , referred to the principal inertial axis should also be invariant to the choice of representation or the reduction chosen for the Hamiltonian. It should also be noted that the signs of the spectroscopic constants can change from one representation to another and this must be considered in comparing constants (P" and P6, etc.) evaluated with different representations. It is apparent from Table 8.14, for example, that the quartic constants d,(III")= - GJ(III') and GK(III")= - GK(III') change sign, whereas Aj, AjK, and AK are invariant to this change in representation. Likewise, the P6 constants 4,, 4 J K , and 4Kare reversed in sign in going from a 111' to a 111"representation. Relations between the quartic distortion constants for various representations and reductions have been given by Yamada and Winnewisser [117]. Some of the pertinent results are summarized in Table 8.28. Studies by G. Winnewisser et al. [69, 118, 1191 of HSSH ( K = -0.9996) and DSSD (K = - 0.9999993), which are very slightly asymmetric tops, revealed that the asymmetry splitting of the K levels can be smaller than that arising from centrifugal distortion effects; usually the reverse is true. The K doubling arises from the inertial asymmetry operator (P: + P?) and the centrifugaldistortion operator (P: + P?). The coefficient of (P: P?) is dependent on

+

Table 8.28 Relations Between the Spectroscopic Constants in Different Axis Representations" Transformations between representations: D,=QT'QiD,, (i,j=I', II', and 111')

0

1

D=

-?

0

0

1

0

;

i.

0

:

:

-4 o--

3

-7

A-C 2(A - B)

1

0 A-C

-2

B- C

-2(A - B )

I'+III' 1 0 0

0

1

0

A-B

lr+IIr 1 -3 0

1

:

0 0

-4

0

-; -4

-: 0 -

:

0

0 0 0 B- C

A-B

0

0

0 -

-4 1

A-B 2(A - C )

0 0 0 0

-1

3 0

-4 0

B- C -__ A-C

For Q;'Qlll, replace last row of Qfi'Q! by (0, 0, ( A - C ) / { Z ( B - C ) } , 0, ( A - B ) / ( B - C ) ] For Q;'Q,,, replacelast row of QlltQl by [ O , O ( A - B ) / { 2 ( B - C ) } , O , - ( A - C ) / ( B - C ) ] For QfilQlll,replace last row ofQfitQl by [ O , O - ( B - C ) / { Z ( A - C ) } , 0, - ( A - B ) / ( A - C ) ] "The vector D of spectroscopic constants depends on the choice of axis representation. I'(xyz=bca), IIr(xyz=cab), and III'(xyz zabc). From Yamada and Winnewisser [117].


357

the rotational constants; the coefficient of (P: + P!.) depends on the distortion constants and hence on the molecular force field. For the K = 2 levels the distortion splitting contribution is so large that the order of the levels ( J 2 , j - > J 2 , ~ is reversed from that expected for a rigid rotor. This could produce problems in labeling the energy levels properly [1201. This large centrifugal-distortion splitting of the K levels is illustrated in Fig. 8.3 along with that expected from the asymmetry splitting only.

Symmetric and Spherical Tops The diagonal terms of the symmetric-top Hamiltonian, (8.124), give the usual energy level expression to sixth degree

+ +

+

-DjK J(J+ 1)K2-DK K E J K=& J(J 1) (& - &)K -DjJ 2 ( J + H J ~ ( J +1 ) 3 + ~ ~ ~ ~~ )2~(K~ ~++ H ~ J1( J) + ~ 4 + ~ ~(8.168) ~ 6 is important, off-diagonal matrix elements are also For cases where *split present in the energy matrix. Additional symmetry factoring of the energy matrix, (8.166), in the Wang basis is possible [64]. For p = 3 or 6, the reduced splitting Hamiltonian is *s,,i,

=h3(P6+

+ P!)

(8.169)

and matrix elements of the type (KIK & 6) are introduced. A first-order treatment is given by the diagonal matrix elements of (8.166). Thus, the usual expression, (8.168), applies except for the K = & 3 levels. Matrix elements of (8.169) directly connect the pair of originally degenerate K = f3 states, and the E - 3 , 3 element gives rise to a splitting of the IK1=3 level

AE=

fE-3,3=

Th3J(J+l)[J(J+l)-2][J(J+1)-6]

(8.170)

The K = 3 splitting constant 2h3 depends, for example, on T~~~~ and the cubic potential constants. A detailed expression in terms of molecular parameters is available [72]. The coefficient has also been denoted by the symbol A and A3 in the literature. This type of splitting has been observed [121-1251. This splitting, for example, in the ground state of PH3 has been studied in the radiofrequency region by Davies et al. [123]. The observed splitting as a function of J was found to fit (8.170). The energy level diagram for the J = 7 , K = 3 state is given in Fig. 8.4. The observed K-doubling amounts to 12,969.0 kHz. Additional hyperfine splitting is also shown in the figure. The splitting for higher K =6, 9,. . . is caused by higher-order interactions and is usually very small. The next first-order effect would arise from the higherorder distortion term ( P i 2+Pk2) in the reduced Hamiltonian which leads to a splitting of the K = 6 level. In general, the splitting for K = 3n(n = 1,2,. . .) arising from such terms may be written to first-order as

A v ~ ,=A3,, ,

(J+3n)! (J- 3n)!

~

(8.171)

36 3.33

363.34

372.36

371.37

184.1

419.3

1841’26.5

,

+4.1I I

363#

3 6 3 ~ -37z,~, 372,36

184.1

419.3 GHz

198doe.7 , MHz -93.44 MHz 353,$%w 3539L 362.35

’

Fig. 8.3 Energy level diagram for HSSH. (a)K doubling due to inertial asymmetry. (b)K doubling due to asymmetry and centrifugal distortion splitting. The pattern of the P z ( 3 7 ) doublet is reversed in going from (a)to (b).The calculated intensity pattern of the P z ( 3 7 )doublet may be compared with the observed transition P2(37). The effects of nuclear statistics are also illustrated for the Pz(37) and P2(36)doublets, where the intensity pattern is weak, strong; strong, weak. From G. Winnewisser [1191.

358

EVALUATION OF DISTORTION CONSTANTS

J.7, K . 3

FROM OBSERVED SPECTRA ENERGY

F

6979.41 6932.49 i6 8 7 3 . 8 5 6aoi.51

6 7

6115.67 6070. 15 6 0 1 2 . 15 5939.39

5 6 7

359

a

9

a

1869.0 (K - d o u b I i ng)

f~

€

-5989.59 -6036.51 -6095.15 -6167.49

6 7

-6853.33 -6898.85 -6956.05 -7029.61

5 6 7 8

a

9

Fig. 8.4 Energy level pattern for J = 7, K = 3 state for phosphine. From Davis et al. [1231.

Small K = 6 splittings have been observed for NH, [122]. On the other hand, large splittings have been observed for K = 3 , 6, and 9 in excited vibrational states of PH, by Bernard and Oka [125]. To account for these splittings, the perturbation treatment described above was found inadequate. Here the dominant term in the reduced constant A3 arises from T,,,,. Instead of the transformed or reduced Hamiltonian, the original general Hamiltonian for a symmetric top was used in the analyses. The terms off-diagonal in K by k 3 arise from the three second-order T’S of Table 8.11. Because of (8.50) (See also Table 8.4), there are 16 angular momentum operators associated with these T ’ S , all of which (because of the relations in Table 8.11) may be associated with the same T coefficient usually taken as T,,,, (x in symmetry plane). These angular momentum operators may be reduced by the commutation relations (8.49)and written in the compact form given in (6.48). To account for the observed splittings, the distortion term (6.48) was added directly to the usual symmetric rotor Hamiltonian, and the resulting secular equation was solved directly. The off-diagonal

360


elements of the T,,,, distortion terms, (KI K 3), effectively explain the observed K=3n splittings and provide T,,,, values of reasonable magnitude [125]. This operator of (6.48) is also important in allowing observation of “forbidden” rotational transitions as described in Chapter VI. Forp=4

={dz+hZJ(J+ l)}(P$ + P t )

*split

(8.172)

and matrix elements of the type (KI K k 4) are introduced. Thus dz and h2 contribute to first-order to K = _+2doubling. The energy expression (8.168) applies with the additional term AE=fE-2,2= rt{d2+hzJ(J+l)}J(J+l)[J(J+1)-2]

(8.173)

for 1KI =2. This K-type doubling effect has been observed for the first time in BrFs (symmetry C4“)by Bradley et al. [126]. The usual selection rules A J = 1, AK =0, indicate the K = rt 2 lines would fit a formula of the type

+

V

=V o & 4dzJ(J+ 1)(J+ 2)

(8.174)

where vo contains the contributions from the usual symmetric-top frequency expression and effects of the quadrupole interaction from Br. The splitting constant d2 is the more important coefficient in (8.173) and is equivalent to R6. The observed spectra are complicated by nuclear quadrupole structure and contain many individual and overlapped lines. When the quadrupole coupling constants from earlier work are used, the appearance of the spectrum depends on only DJKand R6. Figure 8.5 shows theoretical spectra for various values of these constants. The best agreement with the appearance of the observed spectrum is obtained with IR61 =0.07 kHz corresponding to a splitting of 2.8 MHz for I KI =2. Direct measurement of the 1K1=2 splitting for a lower J transition (J=8+9) has also been obtained [127]. Both the quadrupole structure and the 1 KI =2 splitting are resolved, and the value of the splitting constant obtained from fitting the spectral line contours is confirmed. The IKI =2 splittings in SF&l [128], CIFS [129], and IFS [130] have also been analyzed. Additional discussions are available [131, 1321. For spherical tops, the calculation of the energy levels has been considered by a number of authors [73-801. The eigenvalues f ( J , K ) of the fourth-rank tensor f14 have been discussed by Hecht [73]. Kirschner and Watson [77] have treated the sixth-rank tensor f l 6 by first-order perturbation theory employing the eigenfunctions of the operator f14 g v , K ) = ( J , K I ~ ~ ~ K) IJ,

(8.175)

These quantities are independent of the distortion constants and are readily tabulated; tables for the f’(J, K ) and g(J, K ) for J G 2 0 are available [77]. To sixth degree, the rotational energy is given approximately by

E(J, K ) = B J ( J + I)-DJ~(J+ 1 ) 2 + ~ ~ ~ 113 3 ( ~ + + [D4T+H4TJ(J+ l)]f(J, K ) + H 6 T g ( J , K ,

(8.176)


361

I

DJK= -0 5 kHz

b

Re= 0

-12 -8

-4

0

4

8

12

16 MHz

Fig. 8.5 Calculated spectra for the 5=16+17 transition of 79BrF5for the values of DJKand

R6 shown. The bottom pattern is similar to the observed spectrum. From Bradley et al. [126].

Perturbation treatment of 0 8 has been discussed by Ozier [78]. This adds the following terms to the energy expression J??~’(J, K)=LsJ4(J+

+L4,J2(J+ 1 ) 2 f ( J ,K)+&,J(J+

l)g(J, K )

(8.177) Tables to include the effect of the octic terms (h,g) have also been given ( J < 20) [78]. Here the K = C t )label is used to distinguish the energy levels for each J. Matrix elements for the Hamiltonian are available [64, 761 for numerical diagonalization if higher-order effects are to be considered. An idea of the order-of-magnitude of the constants may be given by quoting the results obtained for SiH4:D4,=74,751.4; H4,= -6.044; H6,=2.598; L4,=46.5 x L 6 ~ = - 3 7 . 9 - 1 0 - ~ ; a n d L g , = - 7 6 . 6 X 1 0 - ~ H Z [133].

Choice of Hamiltonian Assignment procedures and applications of statistical techniques for model testing have been discussed by Kirchhoff [68] and others [7, 91-93, 102, 134, 1351. A detailed treatment of least-squares fitting of spectroscopic data has been given by Albritton et al. [136]. The distortion analyses techniques discussed here give rise to a set of observational equations, such as (8.1551, appropriate to a linear least-squares analysis.

362


The least-squares criterion leads to the usual normal equations for estimates of the parameters, ?(j=l, 2, . , . , p ) employing the n observed transition frequencies, where p < n . The so-called square matrix of normal equations is designated here by A Usually, unit weight (K = 1) will be associated with the microwave measurements unless there is a known difference in the measurement accuracy of the transitions in the fit. However, when combining, for example, microwave and infrared data, different weights will be required. The weights, in general, are taken as inversely proportional to the variances, u/;=l/o?. A typical set of weights could be constructed [136] for the IR data with oIR2: cm-', and for the microwave data subset, o M % cm-I. The weighting factors can be conveniently introduced by multiplying each observational equation by ( W)'l2 and treating the system as unweighted where r/c: is the weight assigned to the ith transition. An important measure of the quality of the least-squares fit and precision of the data is provided by the standard deviation o,which is the square root of the variance 02, defined as the sum of squares of the deviations between observed and calculated frequencies divided by the number of degrees of freedom (n-p). It is possible to establish a confidence interval for o, (ol < o < 0 2 ) since o2 follows a chi-square (x2) distribution [1351. If, for example, the data set is divided into two groups (one with predominately high J transitions), and analyzed, two 95% confidence intervals can be obtained from the two standard deviations of the fits. If these do not overlap, then some evidence exists for the need of higher-order distortion terms. An example of the use of confidence intervals on o to indicate the need for higherorder constants in SO2 has been given [68]. An estimate of the standard error in the ith parameter is given by (8.178) with A- ' the inverse of the normal equation matrix and o the standard deviation of the fit. Usually, the uncertainties in the parameters are quoted as 7;f20i, which corresponds to essentially a 95% confidence level; that is, one can be 95% confident that the true values lie within the interval. The factor of 2 is satisfactory [136] as long as (n-p)N>30. For ease in constructing other confidence limits, one might report the number of degrees of freedom (n - p ) explicitly. The uncertainty in the derived parameters gives rise to an uncertainty in the calculated frequency. The standard deviation of the calculated frequency ov is given by I,

D

(8.179) where oi is the standard error in the ith spectral constant and Ci the corresponding coefficient in the frequency expression. The p i j are the correlation coefficients [see (8.180)]. The ov are useful for detection of bad transition data and as a guide to selection of transitions for measurement.


363

Table 8.29 shows results of an analysis of the data of HDS used to confirm the assignment of the lines. The transitions listed in the table have been omitted from the data set and the frequency of each line calculated, along with its corresponding standard deviation. It is apparent that the underlined transition is not satisfactorily predicted when compared to the expected a". When this transition was remeasured, it agreed with that calculated well within the predicted uncertainty. On the other hand, when this transition was included in the analysis, it was not apparent that the transition was mismeasured. This often happens with least-squares analysis where the error is often distributed over other transitions or particular constants. Another useful statistical test for location of a bad transition is the standardized residuals [68] t(Av) = Av/a(Av), where Av = Vobs - vcaic and ~ ( A v is ) the standard deviation of Av. The t(Av) measures the number of standard deviations by which Av differs from the expected value of zero; a large value indicates the transition is suspect. The values of this ratio should follow a student's t,,-p distribution. Details of application of this test and expressions for ~ ( A v may ) be found elsewhere [68, 1371. The correlation coefficients p i j between the ith and j t h constants may be evaluated from the inverse normal equation matrix, that is,

(8.180)

(-1 Qij$1)

pij=A,'/Jzmm

These are of interest in themselves since they give information on the linear dependence of the constants [I381 and associated errors [136]. The diagonal elements are always 1. A value of p i j = + 1 indicates a perfect positive correlation or interdependence, while a pij=O indicates no correlation. Note that the p i j can be calculated for a given Hamiltonian and choice of transition data and does not involve the actual measurements. The variation of p i j with the data set can thus be studied, and transitions which might lead to a decrease in a given correlation can be determined. When the spectroscopic constants are highly correlated, a number of significant figures beyond that indicated by the standard error are required to re-

Table 8.29 confirmation of HDS Assignment

Transition 11,0-~1,1

2 1,1-21,2 5 3.2-44, I 62,4-61,5 73,4-72,5

Observed Frequency (MHZ)

Calculated Frequency (MH4

Calculated Error" (MHz)

5 1,073.27 153,179.16 112,689.59b 598,805.17 628,018.53

51,073.36 153,179.32 112,684.90 598,804.92 628,016.90

2 0.08 kO.18 _2.32 _ f0.74 f3.80

'95% confidence limits. *Remeasurement 112.685.45.

364


produce the calculated quantities. The number of significant figures required can be determined by trial and error, if additional digits in each spectral constant are kept until the rounded numbers actually reproduce the observations. Three or four digits beyond the one standard error digit should be a reasonable rule of thumb. More specific criteria have also been reported [137,139]. If the confidence interval 17;.k220iincludes zero, then to a 95% confidence limit, 17;. is not significant and can be omitted from the fit. A more sophisticated statistical criterion of whether an additional constant is needed is given by the F test. This is essentially a measure of the improvement in the sum of squares of the deviations, S, obtained by introduction of another fitting parameter. In particular, F = ( n - p ) (SP- 1 - SP)

SP

(8.181)

with n the number of transition frequencies and S , the sum of squares of the deviations for the p-parameter fit. For example, consider a p versus a p- 1 parameter P6 fit. If comparison of the calculated F with the tabulated F l , n - p value (5% critical value) indicates that F < Fl,n-p,the fit withp constants does not significantly improve the fit over that withp - 1 constants, and the constant with 95% confidence may be omitted. An example of use of the F-test to decide on the significance of certain terms in Xi6)has been given for CH2(CN)2[7]. In evaluating the distortion constants from the rotational spectrum one usually selects a set of low J lines, with small distortion effects, to obtain the three rigid rotor parameters. Next, additional lower J transitions are assigned which have sufficient distortion effects to provide credible predictions. Some difficulty may be experienced in this first stage, if an incorrectly measured or misassigned transition is present in the data set. Once enough data are available for a reasonably reliable analysis, the frequency predictions and their standard deviations become meaningful. Those predicted transitions with small estimated uncertainties can then be readily found. Attempts to assign transitions with large uncertainties (e.g., > 100 MHz) will usually be unsuccessful. These new transitions may then be added to the distortion analysis to improve the spectroscopic constants and thus predict other higher J transitions with greater reliability. This technique is repeated as one observes and adds to the analysis higher and higher J transitions, until an adequate data set is obtained that yields accurate spectroscopic constants and predicts the remainder of the spectrum to a given J value with uncertainties near the measurement error. One, of course, must be careful about employing predictions for high J transitions or certain subbranches, if such data are not in the data set. The determination of accurate distortion constants requires a number of rotational transitions with distortion effects of reasonable size. Also, it is important to select transitions from as many different subbranches as possible. (A subbranch is designated by the changes in J, K - l , and K1, Le., AJ, A K - l , and A K 1 . ) To guard against possible misassignments or incorrect measurements, the following procedure can be followed. Separate distortion analyses are carried


365

out by use of the final Hamiltonian with one line (or a small arbitrary group of lines) omitted from the data set. The spectral constants, line-frequency predictions, and the standard deviation cV in each line-frequency prediction are evaluated for each analysis. By use of an analysis that does not include the line itself, each line is compared with its predicted frequency and required to fall within, or very close to, the range expected considering 2avand the experimental uncertainty. A useful measure for this comparison is also given by the values of ~ ( A v )discussed , previously. No attempt at this stage should be made to improve the predictions by including more constants in the Hamiltonian. The standard deviation of the least-squares fit should also be virtually the same for the various analyses and, in addition, the various analyses should produce essentially the same set of spectral constants. As additional transitions are measured and used in the analysis, higherorder, for example, P6 constants may have to be included to adequately fit the experimental observations. This will become apparent from a significant increase in the standard deviation of the least-squares fit as higher J transitions are added, with the assumption that the measurements and assignments of the higher J transitions are correct. However the observed data may not be sensitive to all the P6 constants. The P6 constants selected to describe the data may be obtained as follows. The data are analyzed with a full complement of sextic terms, and the most statistically undetermined constant (standard deviation greater than the constant) is removed and the analysis repeated. It will usually be clear that the constant omitted did not improve the fit. For the six-parameter fit, another constant may now be statistically undetermined; this constant can be removed and the analysis repeated with one less fitting parameter. This process is continued until all the remaining constants are judged to be statistically determined. The selection process is complicated if there are high correlations between the constants. Because of the possible high correlation among some of the constants, it is not always clear in which order the terms should be eliminated. In such situations, inspection of the results of the several alternatives will usually indicate the proper choice. In certain cases, the uncertainty in the constant may be such that it is not clear whether the constant should be retained or rejected. In fact, as judged from the standard deviation of the least-squares, the constant could possibly be omitted without significantly affecting the fit. For such borderline cases a more sophisticated statistical criterion, the F-test, might be used to test for the significance of a particular constant as indicated previously. Instead of constraining one or more P6 constants at zero, as judged from the previous discussion, all the constants could, of course, be included in the analysis. However, their inclusion would add nothing to the fit, and there is little reason to expect that their inclusion would improve the prediction of unmeasured lines, particularly those which might be sensitive to these constants. Since a sextic planar relation is now available, it would be preferable to use the planarity condition to eliminate a particular constant rather than to fix the constant to zero. If the data are still not sensitive to the remaining constants,

366


then the above procedure can be used to remove additional constants from the fit. Such a statistical treatment essentially allows the available data to decide which of the distortion terms are important in the characterization of the spectrum. For light asymmetric rotors such as HzO, DzO, HzS, D2S, and so on, the distortion analysis is further complicated [91-93, 1021. For these light molecules, the series converges very slowly, which necessitates the inclusion of a large number of terms in the Hamiltonian to adequately characterize the observed spectra. This slow convergence also makes difficult the accurate prediction of transitions too far outside the particular data set used. Even the relatively low J lines have significant effects due to the higher-order distortion coefficients. Of critical importance to the proper analysis of these molecules is the selection of distortion terms to be retained in the Hamiltonian. This selection is complicated by the rather slow convergence of the power series in the angular momentum as well as by the high statistical correlation among many of the terms. The analysis procedure is similar to that described previously and is considered in more detail elsewhere [92, 931. One error that should be guarded against is the truncation of 2 too early when the data contain higher-order effects. That the importance of a term is not simply characterized by its degree, but depends on both the coefficient and the corresponding operator, has been pointed out by Steenbeckeliers and Bellet [102]. For small K-l, both diagonal and off-diagonal terms are of similar importance, whereas for large K - 1, the diagonal terms are the more important. In general, for both light and heavy molecules, the final selection of terms (choice of model) should produce a Hamiltonian that [92,93]

1. Fits the data to within experimental uncertainty. 2. Adequately predicts lines removed from the fit, to within expected theoretical and experimental error. 3. Has the minimum number of terms consistent with (1) and (2). 4. Contains constants that are statistically meaningful. 5. Is consistent with a converging power series. With regard to (5), this implies, for example, that should be better determined than P K .It is furthermore important that the Hamiltonian, as has been found, results in essentially the same values of rotation and lower-order distortion constants independent of the details of either the data set or the higher-order terms retained in the Hamiltonian.

6 PLANARITY RELATIONS FOR CONSTANTS OF THE REDUCED HAMILTONIAN For a planar molecule, the number of coefficients in the reduced Hamiltonian can be reduced. The planarity relations discussed previously can be used to

PLANARITY RELATIONS FOR CONSTANTS OF REDUCED HAMILTONIAN

367

write the following planar relations

C2 2A2

Tnncc =-T n n n n -

== C2

Tbbcc

C2A2

A2

cccc

(8.182)

Tcccc

(8.183)

2 p T b b b b + -2 T c2

B2C2

Tbbbb - 2 A 4 Taaaa

+ B2

+

Recall that Only Tabnb #o and hence Tbnbb = Tnrrbb 2 T n b a b , T i n c c = Tnacc, and '&bcc = Tbbcc. These relations and the planarity condition A B = ( A + B ) C can be used to give a single relation between the five determinable coefficients. In particular 7 2 = CT 1

+( A +B

) T ~ ~ ~ ~

(8.184)

or in terms of the T ' s

+

T2= CT, + ( A b)Tcc

(8.185)

These relations are strictly correct o d y in the small oscillations approximation. In principle, equilibrium rotational constants and distortion constants should be used in these relations, although one is usually forced to use effective constants which contain vibrational contributions. If the planarity relations are to be used to eliminate one of the P4 coefficients in the reduced Hamiltonian leaving only four quartic coefficients, it is convenient to express the planarity relation in terms of the parameters of the A- or S-reduced Hamiltonians [ 1 4 0 ] . If the previous equations are employed, the relations are given in Table 8.30 for two particular representations I'(x-+b, y+c, z-w) and III'(x-w, y+b, z+c), appropriate for a prolate and oblate top, respectively. A relation for the sextic coefficients has also been derived by Watson [ 1 4 0 ] 3 ( A +B)@ccc+2{ -(B-C)@cca-(A-C)@ccb -2(T,c-T,c)(T,c-Tbc)}+4T~c=0

(8.186)

The expression in braces corresponds to Qf3, 02,or O4 for I', II', or 111' representations, respectively. (Note for II':x+a, y+b, z--rc). This relation leads to an expression between the parameters of the A - or S-reduced Hamiltonian also given in Table 8.30. Application of this planarity constraint can be used to reduce by one the number of sextic coefficients in the reduced Hamiltonian, leaving only six terms. For a planar symmetric top, the molecule is an oblate top A = B = 2 C and dl =0, T,,=TbC=2T,,. Using these conditions for the S reduction in a 111' representation gives the two relations for a symmetric top, also included in Table 8.30. Planarity relations for the P6 constants of X Y X and X Y Z asymmetric tops have also been discussed by Niroomand-Rad et al. [ 1 4 1 ] . Here, four planarity relations among the 1 0 original sextic coefficients of ( 8 . 7 3 ) are given. These could be employed to eliminate four of the ten sextic constants and thus reduce the general "first principles" Hamiltonian so that it would contain only determinable coefficients and thus remove the inherent indeterminacy in Y?0(6).

368


Table 8.30 Quartic and Sextic Planarity Relations for the Spectroscopic Constants of the Reduced Hamiltonian'

"Here, A , B, C are the equilibrium rotational constants. From Watson [140].

Unless the data are such that least-squares equations are ill-conditioned, the quartic planarity constraint should not be applied (see next section). On the other hand, the sextic constraint can, in many cases, be profitably employed to improve the determination of the constants when the uncertainties in the sextic constants are large.

7


The derived distortion coefficients obtained from an analysis of the observed ground state spectrum are effective values containing contributions from the vibrational averaging effects. The vibrational dependence can be expressed, for example, as (8.187)

(8.188)

369


and so on, and the dependence in the T constants might be expressed by (8.189)

where the superscript e indicates equilibrium values, [ u ] represents the set of vibrational quantum number, us, specifying the vibrational state, with the sum over the various vibrations of degeneracy d,. For asymmetric rotors d , = l . Very general definitions of ptfl are available [19]. Just as with the rotational constants, in order to obtain the distortion coefficients associated with the equilibrium vibrationless state, measurements must be made in all ( 3 N - 6 ) vibrational states of the molecule. Because of intensity considerations, such an analysis has been made in only a limited number of cases. Examples of distortion constants obtained in the ground and excited vibrational states are available [142]. The difference in these values gives one some idea of the size of the vibrational effects and hence the possible error introduced into force field calculations when effective rather than equilibrium values are employed in the analysis. Differences between the equilibrium and ground state distortion constants on the order of 5 % or less are common. Observed variations in the distortion constants with vibrational state which are particularly large, can often be explained in terms of a Coriolis interaction [143] or a large amplitude motion. A detailed study of trimethylene oxide (TMO)with Czvsymmetry by Creswell and Mills [144] shows for the distortion constants an anomalous zig-zag dependence on the ring puckering vibrational quantum number. Similar dependence has been exhibited by the rotational constants [145] (see Chapter XII). The distortion constants were determined for the ground state and in the excited puckering vibrational states through up = 5. The vibrational dependence is illustrated in Fig. 8.6. All states could be analyzed with the same effective Hamiltonian so the vibrational dependence in the quartic constants is not affected by the choice of model. The constant dJ is essentially independent of the puckering vibration, while AK, - AjK, and - d K show qualitatively similar patterns in the vibrational dependence. The experimental uncertainty in A j is believed to be too large to draw significant conclusions. The low-frequency anharmonic ring puckering vibration of T M O can to a good approximation be separated from the remaining vibrational modes which are essentially harmonic in nature. The contribution of this mode to T~~~~ may hence be separated from the contributions of the other modes which are independent of u p . The contribution from the puckering mode which depends on up may be evaluated from the known ring puckering potential constants (see Chapter XII, Section 14). From the symmetry of the puckering coordinate Q p , one finds the only nonvanishing ugp)is @). Thus only T,,,, will carry a vibrational dependence which implies only AjK, AK, and AK (111' representation) are affected. Explicit calculation shows that the vibrational dependence for d j , AjK, AK, and aK given in Fig. 8.6 is accounted for. Similar results have been obtained for 1-pyrazoline [146]; however, for u p = O and 1, because of the close separation of these vibrational levels

370


0

I

3

2

4

5

V Fig. 8.6 Observed values of the five quartic distortion constants of trimethylene oxide (TMO) as a function of the quantum number in the puckering vibration. Uncertainty in AJ is shown as an error bar equal to the standard error in each case; the uncertainty in the remaining four distortion constants is too small to show on these graphs. From Creswell and Mills [144].

(7.61 cm-'), it was necessary for a vibration-rotation coupling term to be included in the effective 2 and for these two states to be analyzed together.

Centrifugal Defect A number of examples are available of the failure of the planarity relations when applied to effective ground state constants rather than equilibrium quantities. This can be well demonstrated if an unconstrained fit of a planar molecule is carried out to five quartic constants that are then used to test how well (8.184) is obeyed. This has been demonstrated by Steenbeckeliers [134] for SO2 and for a number of molecules by Kirchhoff [68]. In terms of the scaled value of 5 2 , one can define a centrifugal defect analogous to the inertial defect 6cd=Tcccc-?i-

C

-(T>-Tl)

A+B

(8.190)

with T > = r 2 / ( A+ B+ C). Values of 6 c d calculated from the observed constants are collected in Table 8.31. If the relations were exactly obeyed, 6,d=O. The

EFFECTS

OF VIBRATION

371

Table 8.31 Values of the Centrifugal Defect for Several Planar Molecules“

so2 F20

- 0.248 10.009 - 0.56 & 0.06

NSF SiF2 NOF COFI CF2 SF2 NHzCHO HNCS HCICO SCl2 KCN CH 2 =CHNC

0.323k0.015 - 1.1910.37 -0.08 f0.18 010.12 - 0.22 k0.07 - 0.42 k0.14 -0.12f0.07 4.363 f0.003 - 0.017 0.016 - 0.045 k0.032 - 0.86 0.02 - 0.133 k0.001 -

b b b b b b C

d c

J 9

h

“Uncertainties represent one standard deviation. *Kirchhoff [68]. ‘Kirchhoff et al. [137]. dKirchhoff et al. [154]. ‘W. H. Kirchhoff and D. R. Johnson, J. Mol. Spectrosc., 45, 159 (1973). ’K. Yamada, M. Winnewisser, G. Winnewisser, L. B. Szalanski, and M. C. L. Gerry, J. Mol. Spectrosc., 19,295 (1980). gM. Suzuki, K. Yamada, and M. Takami, J. Mol. Spectrosc., 88, 207 (1981). hDavis and Gerry [152]. ‘T. Torring, J. P. Bekooy, W. L. Meerts, J. Hoeft, E. Tiemann, and A. Dymanus, J. Chem. Phys., 73,4875 (1980). ’K. Yamada and M. Winnewisser, Z . Naturforsch., 30a, 672 (1975).

results indicate that planarity condition should not be used unless the data are clearly insufficient to evaluate all five determinable coefficients. Examples of the quartic defect employing the planarity relations for the parameters of the reduced Hamiltonian are given in Table 8.32 for isotopes of water. The effects of vibration for water are readily apparent from the rather large nonzero values obtained. It is further apparent that there is a systematic decrease in 6 with increasing mass or effectively decreasing amplitude of vibration. A study of the sextic planarity relations for several triatomic molecules has also been reported [140]. From Table 8.32 it is apparent that the sextic defect is significant for H 2 0 and for most of the isotopic forms of water. For H2S and D2S the sextic defect is only barely significant. For O3 it has been found that there is good cancellation of terms and that the sextic defect is insignificant.

372


Table 8.32 Terms in Planarity Relations for the Reduced Hamiltonian for Isotopes of Water'sb Isotopes

Quartic Planarity Relation (GHz)' 4CAj - ( B - C)AJK - 2(2A + B+ C)6J 2(B- C ) ~ K Sum

41.83 8.326 -2.766 27.18 -72.56 -13.632 12.91 10.202 9.36 2.130 50.15 k0.009

5.397 3.328 -9.499 1.517 0.743 k0.003

3.137 -2.316 -4.820 4.900 0.901 k0.007

1.663 2.477 1.001 0.821 -4.316 - 2.974 1.366 0.494 0.348 0.184 + 0.O3 i0.002

Sextic Planarity Relation [102(MHz)*]

Refs. to Data

261 66 -459 -79 57 82 -72 It22 [98]

13.0 -59.4 -29.5 50.1 4.7 8.2 -12.9 k1.6 [92]

15.2 5.6 -28.6 -3.9 3.5 5.4 -2.8 i0.5 [IO21

4.7 - 18.8 -7.2 16.4 1.1 2.0 -1.8 i0.9 [lo71

4.3 - 2.2 - 8.0 3.1 1.1 1.9 0.2 0.9 [io71

+

3.35 1.11 - 6.03 0.25 0.69 1.26 f0.63 k 0.34 [IO61

"The uncertainties of the sums are calculated from the uncertainties of the individual terms without correlation. All constants refer to the zero-point vibrational state. From Watson [140]. *The relations for a I'A Hamiltonian are employed.

For SO2 a rather large sextic defect is obtained, which is probably due to truncation of the Hamiltonian at the sextic terms in analysis of the data.

8 CENTRIFUGAL DISTORTION CONSTANTS A N D MOLECULAR FORCE FIELDS A considerable amount of effort has been expended in the calculation of force constants from the fundamental vibrational frequencies. The fundamental vibrational frequencies v of a molecule are related to the 3 N - 6 (or 3 N - 5 for linear case) roots 1 of the secular determinant [1I] /FG-LEI=O

(8.191)

where E is a unit matrix, F the force-constant matrix with elements fij, and A =4n2v2. However, in most cases the number of quadratic force constants exceeds the number of normal vibrations. This difficulty is avoided by either the use of isotopic information or the assumption that the interaction force constants may be neglected. Unfortunately, the latter assumption is not always

CENTRIFUGAL DISTORTION CONSTANTS AND MOLECULAR FIELDS

373

justifiable. Furthermore, in many cases it is found that the isotopic vibrational frequency data, especially when hydrogen is not substituted, are not sufficient to uniquely determine the force field, and additional data are necessary [147]. The experimentally determined distortion constants provide one such valuable source of force-constant information. These data can be used to supplement the vibrational data and thus uniquely fix the force field or enable the determination of additional force constants. In special cases, for diatomic or bent symmetric triatomic molecules, they can be used directly in calculation of the force constants. In addition to the distortion constants, the inertial defect [148] and Coriolis coupling coefficients [1491, as well as mean-square amplitudes [150, 1511, are also useful sources of information, since they depend on the quadratic potential constants. The force constant information available from distortion constants is, of course, limited by the number of independent combinations of 5’s that can be derived from the rotational spectrum. As we have seen, in the general case, no more than five independent linear combinations can be determined. For planar asymmetric molecules, only four distortion constants are independent; for a symmetric rotor, although no more than three independent linear combinations exist, only two can be determined from pure rotational spectra. All of the earlier work to obtain information about molecular force constants, utilizing distortion data of asymmetric rotors determined from microwave spectra, was centered around planar molecules. Several calculations utilizing only distortion information have been carried out for symmetrical bent triatomic molecules, for example, SO2, for which there are four force constants and four independent 5’s. The inverse relationships to those of (8.36)-(8.39) are [138]: 1

(F-l)ll=- -Rsin28cos28 2 ( F - 1 ) 2 2 = - R sin2 8 cos2 8

cot’

(-+-

8Taa~a

+

Taaaa

Tbbbb

A2

B2

tan2 8 T b b b B2

+-)

2Taabb

AB

2Taobb --

(8.192) (8.193)

AB

(8.195) Or alternately in terms of

(F-l)ll=-

2

T~~~~rather

than

Taabb

[152]: B tan2 8- A B3

Tbbbb

+ 3Tcccc AB

1

(8.196)

374


[

(y

( ~ - ~ ) ~ ~ = - ~ s i n ~( A e +c @ o sT+-~ e );T

(F-’)12=

-

2 ]

--

R s i n 2 0 c o s 2 0 B(tanO-cot8)-2AcotQ A3 2 4

[

+ A(tan0-cotB38 ) + 2 B t a n 8

Tbbbb

-

T

~

~

~

~

(8.197)

Taaaa

AB(tan 8 - cot 0)

c4

Tcccc

1

(8.198)

The expression for (F-1)33is still given by (8.195). Here, as before, R=

r2 x 2h

angle, M=2my+rnx, r as the with h in erg-sec, 8 as one-half the Y --X-Y X-Y bond distance expressed in angstrom units, and with the T’S and rotational constants in megahertz. The F matrix has the following form (elements in d y nes/cm)

0

0

5-5,

(8.199)

These relations are sufficient to determine the complete quadratic-valence force field 2 I/ =f,(6r:

+ 6r:) +f,6u2 + 2fr,(6r +6r2)Sct+ 2 f d r

6r2

1

(8.200)

from the distortion constants obtained by an analysis of the rotational spectrum. The force constants can, of course, then be used for calculation of the vibrational frequencies of the molecule. In certain cases, force constants determined in this manner have yielded vibrational frequencies which were in fairly good agreement (-4%) with observed values [82, 138, 1531. Such calculations, as well as yielding force constants, provide an independent confirmation of the assignment of the vibrational frequency. If the planarity relations are employed and the spectrum analyzed for 4 T’S, the model errors are averaged in a complicated way. The use of these constants to evaluate a force field for XY2 can, of course, be expected to introduce an unspecified error in the derived constants, although the results have, in fact, been quite useful. Alternately, if the five quartic constants have been evaluated, then the planarity relations can be used to extract, for example, T a a a a , T b b b b , T a a b b , and Tab&, and subsequently the force constants from (8.192)-(8,195). However, the evaluation of T a o b b and is overdetermined and different values can be obtained depending on which combination of constants is employed because the planarity relations are not satisfied exactly. This ambiguity has been pointed

CENTRIFUGAL DISTORTION CONSTANTS A N D MOLECULAR FIELDS

375

out by Kirchhoff [68, 1541 and by Cook et al. [93, 155, 1561, and a detailed consideration has been given by Yamada and Winnewisser [117]. In Table 8.33 the results for OF2 are given for three sets of derived 7’s. In particular, T o & and Tobob are evaluated from T,,,,,T b b b b , and ( T ~ T,~ ) , ( z l , T,,,,), or ( 7 2 , T,,,,). The value will have the same value for the latter two combinations but not Tab&. of This gives three sets of force constants. Note that F 3 3 is evaluated solely from 7 , b a b . The spread of the force constants and calculated vibrational frequencies thus obtained, gives some idea of the effects of using effective rather than equilibrium parameters in the planarity relations. The use of effective rotational constants can also amplify the model errors. It is clear that the vibrational effects can be significant. It would be better to minimize the use of the planarity relations. Since T,,,,,T b b b b , T,,,, can be obtained directly from the analysis of the rotational spectrum without the use of the planar relations it is more appropriate to use these constants to extract information on the force constants F11, F12, F z z (Al vibrations) [155, 1561, for example, from (8.196)-(8.198). The correct calculation procedure for obtaining a value of T o b o b from the observed spectroscopic constants that minimizes the vibrational effects remains uncertain. The most fruitful approach for obtaining information about the molecular potential function is the combination of both microwave and infrared data. A technique for combining vibrational and rotational distortion data for symmetric bent triatomic molecules has been discussed by Kivelson [38] and Pierce et al. [82,153]. The criterion applied is that the force constants reproduce the observed vibrational frequencies exactly and fit as consistently as possible the frequency shifts in the rotational spectrum which are due to centrifugal distorTable 8.33 Force Field Calculations from Microwave Data for OFz” Force Constants (rndyneJ1)

I I1 111

403.4k2.4 385.3k2.7 385.3k2.7

- 158.2k0.9 4.65k0.04 0.202k0.004 - 142.5k2.2 4.45k0.04 0.206+0.004 - 140.0k2.4 4.45k0.04 0.206k0.004

0.7051 k0.0009 2.969k0.016 0.7161 k0.0014 3.297k0.050 0.7161 k0.0014 3.354k0.057

Infrared Spectrum Predictions (cm- ‘)e Wl

1 11 111

916.5 5 3.2 897.9 k 3.0 897.9 k 3.0

0 2

455.9k0.3 458.3 k0.5 458.3k0.5

0 3

807.5 k 2.2 851.0k6.5 858.3 f7.3

“The uncertainties represent one standard deviation as estimated by the least-squares fit. From Kirchhoff [68]. bSet I: Calculated from T,,, rbbbb, T ~ r2; , Set 11: Calculated from r,,., Tbbbb, T,,,,, and r l ; Set 111: Calculated from T,,,,, rbbbb, T ~ and~ T ~ . ~ , ‘Observed: 928,461, and 831.

376


tion. Extension of this method to bent unsymmetrical triatomic molecules has been discussed by Cook [157]. For larger molecules the number of force constants increases rapidly. For an unsymmetrical molecule, the number of harmonic and anharmonic potential constants is $n(n + 1 ) and &(n + 2), respectively with n = 3 N - 6 (or 3 N - 5). For molecules of higher symmetry there are fewer constants and the number of independent constants can be obtained [158] by group-theoretical arguments and the representations of r v i b , r,",b and r$b.The most general and convenient approach for combining distortion data, and other spectroscopic quantities dependent on the force field, with the vibrational isotopic frequency data makes use of a least-squares analysis. The analysis procedure is very similar to that discussed previously for evaluation of spectroscopic constants from rotational spectra. Linear equations are constructed describing the changes in the observed quantities due to small changes in a particular force constant. From an initial trial force field, the least-squares criterion is employed to evaluate the changes in the force constants required to obtain agreement between observed and calculated quantities. The vibrational problem is usually linearized by means of a Taylor series expansion

where & = h 2 c 2 w : , with c the speed of light, and wk a fundamental vibrational frequency in cm-' units. The L i j are elements of the L-matrix, where % k = LikLjkFij.The 0: and L t quantities are calculated from the approximate set of force constants Fo by solution of the vibrational eigenvalue problem (see Appendix H). The foregoing expression and the following equations are used as the basis for a least-squares analysis in which the corrections, 6 F i j , to the constants, F;, are calculated. The quantities 02 and L.: are calculated after each iteration with the improved force constants, F $ + 6 F i j . The iterations are continued until all the 6 F i j become sufficiently small. It is assumed in applying such a procedure that the initial Fo represents a good approximation to the force field and that the data are reasonably sensitive to the force constants so that convergence of successive iterations is assured. Likewise, for the distortion constants (8.202)

with (8.203)


377

where the derivations of the distortion constants with respect to the F i l , follow directly from (8.20). By taking the partial derivations of the matrix equation F- F = E,we find (8.204) which gives (8.205) Note F;'=Fj'. These relations provide the basis for combining the vibrational frequency data and distortion constant data. The relations between the 7's and the observed spectroscopic constants have been given previously. In addition to the distortion constants, the Coriolis constants lg)= xi,jLilCly)LG1depend on the force constants (through L) and measure the coupling of the two vibrations Qkand Ql via rotation about the a axis. The Ciy) elements depend on the masses and geometry of the molecules. The derivatives (a(g)/aFij)have been given by Mills [159] and others [160]. For planar molecules, the inertial defect for the vibrational state [ u ] , A["]= x s A s ( ~ , + ~ ) + A c e +A,,,, nt defined in (13.134) is a function of the harmonic vibrational frequencies and the Coriolis constants. The differences between the inertial defect in successive vibrational states are particularly useful because they are virtually independent of electronic and centrifugal distortion contributions. The derivatives (aA,/dFij) are derivable from (13.135). Mean square amplitudes [151] also depend on the force field and are obtained in gas-phase electron diffraction structure studies. The sensitivity of all these quantities to the force field can vary from molecule to molecule, and usually as much information as possible is required to uniquely specify the force constants. Some illustrative examples of force fields determined from both microwave (MW) distortion data and infrared (IR) data for some simple molecules are given in Table 8.34. In the combination of IR and MW data a weighted least-squares analysis is usually employed. However, weighting factors are complicated by the use of ground state rather than equilibrium values. Simply specifying the weights W=l/o? in terms of the estimated experimental uncertainties does not take zero-point vibrational effects into consideration. The weighting scheme should allow for the effects of anharmonicity in the vibrational frequencies, which are usually larger for the highest frequencies. Furthermore, the distortion data should be weighted with the realization that the effective ground state constants may be contaminated with vibrational effects of 2 to 5%. Water and hydrogen sulfide are two of the molecules for which the anharmonic corrections to the observed vibrational frequencies are known. In these rotors the corrections are particularly large. Hence, it is possible to judge directly the merits of the potential function obtained by combination of observed IR and MW data. Least-squares calculations employing only IR data,

Table 8.34 Potential Constants Determined by Combination of Infrared and Microwave Date (mdynelk) Molecule

so2 0 3

OF2 CL20Q

CL02 NO2 GeFz Se02 SiF2

CF2 SCL2 NF2"

NOF~ NOCLb NOBrb NSF' NSCL'

f; 10.006 5.70 3.950 2.88 7.018 1 1.043 4.08 6.91 5.03 6.19 2.27 5.14

JI 0.024 1.52

0.806 0.31 -0.170 2.140 0.26 0.03 0.31 1.56 0.25 0.18

frl

fr2

15.08 15.29 14.3 10.703 10.031

2.09 1.30 1.42 2.872 1.465

fzl*

0.193 1.28 0.724 0.423 0.651 1.109 0.316 0.488 0.439 1.373 0.262 1.103 &Id2

1.08 0.56 0.43 0.411 0.244

f;Z,

0.189 0.332 0.137 0.17 0.006 0.481 -0.01 0.009 0.138 0.545 0.065 0.335 frlzld

Od

0.09 -0.19 Od Od

0.17 0.07 0.02 0.014 - 0.038

1.85 1.45 - 1.5

' '

Od

f

Od

u

"Determined from microwave distortion data only *For the N O X molecules rl = N - 0 , r 2 =N-X, and d=(rlrJ' '. 'Here r l =S-N, r2 = S-X. dAssumed value. 'Kivelson [38]. 'Pierce [1531. @Pierceet al. [82]. hM. G. K. Pillai and R. F. Curl, Jr., J. Chem. Phys., 37, 2921 (1962). 'Herberich et ai. [138]. 'G. R. Bird, J. C. Baird, A. W. Jache, J. A. Hodgeson, R. F. Curl, Jr., A. C. Kunkle, J. W Branstbrd, J. Rastrup-Anderson, and J. Rosenthal, J. Chem. Phys., 40, 3378 (1964). kH. Takeo, R. F. Curl, and P. W. Wilson, J. Mol. SpecrroSc., 38,464 (1971). 'H. Takeo, E. Hirota, and Y. Morino, J. Mol. Spectrosc., 34,370 (1970). "H. Shoji, T. Tanaka, and E. Hirota, J. Mol. Spectrosc., 47, 268 (1973). "Kirchhoff et al. [137]. "Davis and Gerry [1521. 'R. D. Brown, F. R. Burden, P. D. Godfrey, and I. R. Gillard, J. Mol. Spectrosc.,52, 301 (1974). 'Took [157]. 'Mirri and Mazzariol [147]; and G. Cazzoli, R. Cirvellali, and A. M. Mirri, J . Mol. Spectrosc., 56, 422 (1975). "Mirri and Mazzoriol [147]. 'Cook and Kirchhoff [163]. "T. Beppu, E. Hirota, and Y. Morino, J. Mol. Specrrosc., 36,386 (1970).

378

CENTRIFUGAL DISTORTION CONSTANTS AND MOLECULAR FIELDS

379

MW data ( T ~ Tbbbb, ~ ~ T ~~ ,~and ~ ~IR) and , MW data have been reported [156, 1611, and various weighting schemes have been used. A comparison of the force fields for H 2 0 obtained in various ways is summarized in Table 8.35. For the IR and MW entry, F 3 3 comes from the o3data only, because of the limitations in calculating an accurate value of Tobob, as discussed previously. Even for these light molecules, the vibrational frequencies (wl, w 2 ) predicted by use of the microwave data only are quite satisfactory. The largest discrepancy is about 5%. Clearly, one advantage of the IR and MW force field for the ground state is that it fits both spectral regions, and the interaction constant is positive and close to the value found for the harmonic force field. The valence bond force constants obtained from the symmetrized force constants are also compared in Table 8.35. These results indicate that if corrections for the effects of vibration are to be ignored, the “best” or most representative force field is obtained from a combination of both IR and MW data, even when the vibrational frequency data are rather extensive. Furthermore, even though the anharmonicity effects are particularly large, the vibration-rotation data, when included, lead to a meaningful potential function. There is, in general, more than one possible solution to the force field that will reproduce the vibrational frequency data. For an n x n secular equation, there are n ! sets of force constants corresponding to the n ! possible set of frequency assignments, that is, the correlation of a vibrational frequency with a Table 8.35 Force Constant Calculation for

Symmetry Force Constants

Fll(mdyne/A) F 2 ~ ( dyne/A) m F 1 2 ( mdyne/A) F33(mdyne/A)

MW(w,)* 8.355 0.761 0.345 8.555

H20“sf

MW(wo)* MW‘ 7.396 0.771 -0.517 7.838

7.481 0.675 0.443

IR(wo) & MWd 7.653 0.700 0.536 7.838‘

Valence bondforce constants (m dyne/A)

IR(w,)

W@O) IR(w0) & M W

-

8.455 7.617 7.746

0.761 0.771 0.700

-0,100

-0.221 -0.093

0.244 -0.366 0.379

. vibrational frequencies corrected for “Weighting l/m and l / ~ IR(oic): effects of anharmonicity. IR(oo): observed uncorrected vibrational frequencies. b H 2 0 ,DzO, HDO data. ‘ H 2 0 , D 2 0 , TZOdata. dHzO,D 2 0 data. ‘From IR(o,) calculation. From [156].

380


certain force constant or internal displacement coordinate. This results because the force constants are derived from the solution of a set of polynomial equations. If the frequency assignments are in doubt or not largely dependent on a given normal mode, other data such as isotopic frequency shifts, distortion constants, and so on, are needed to resolve the ambiguity. A general method of arriving at the various possible solutions has been discussed by Toman and Pliva [162]. An example of the problem is provided by the nonlinear NSF molecule [1631 where the centrifugal-distortion information is employed to decide between the possible assignments. Even with a rather varied data set, the data may not be sufficient to discriminate between two or more physically realistic force fields. For example, for NF,, two harmonic force fields [164] can be found for the E-species (2 x 2) which reproduces not only the vibrational frequencies but also the distortion constants, Coriolis coupling constants, and nitrogen isotopic frequency shifts. This ambiguity associated with a 2 x 2 degenerate species has been considered in a general way by Hoy et al. [165]. In favorable cases, one of the solutions can be rejected on physical grounds. In principle, for XY, molecules resolution of the ambiguity can be obtained by use of vibrational frequencies of an unsymmetrical substituted species XYzY *. However, this is not possible for trifluorides. Fortunately, the mean square amplitudes do provide a means for discriminating between the two force fields. Some force-constant investigations have already been mentioned in the previous tables. Over the last 10 years a considerable number of studies have employed MW data to aid in the determination of molecular force fields and one cannot cite all of them here. Recent reviews on force constant calculations have been given by Duncan [166] and by Mills [167]. Harmonic force field studies have been reported for C12C0 [168], N O X [169], (X=F, C1, Br), S 2 0 [155], SOF2 [170], HCOOH [171], HCOSH [172], HONO [173], H2= CHF [174], FC102 [29], CH,X(X=F, C1, Br, I) [175], BF, [165], PFJ [165], NCl, [176], PCl, [176], SbC1, [177], HBFz [178], N 0 2 C l [179], HzCO [180], and FzCO [181], to mention a few. In general, limitations on the accuracy of the potential constants arise from neglect of certain interaction constants and from use of moments of inertia averaged over the ground vibrational state rather than equilibrium moments of inertia, as well as from the fact that the distortion constants and the vibrational frequencies contain contributions from anharmonic terms in the potential energy which are usually neglected. Some feeling for the significance of these latter effects on the derived constants has been obtained. However, the evaluation of distortion constants for the equilibrium state for a few asymmetric tops will allow a better assessment of the errors introduced when present calculation procedures are employed. In addition to work on the harmonic part of the potential function, work has been done on small molecules to extract information on the anharmonic part of the potential function. Evaluation of the rotation-vibration interaction constants from the rotational spectrum provides information on the cubic force


381

Table 8.36 Comparison of Calculated and Observed Sextic Centrifugal Distortion Constants H2S

Constants @J

4J @JK

4JK @KJ @K

4K

SO2

Calc."

Obs.b

25.2 5.7 - 218.6 -201.8 447.3 -254.0 -2767.3

28.1 f 1.1 5.8 f0.1 -228.3 f2.7 -242.815.2 459.0 116.0 -276.5f14.1 -2870.3f39

Calc."

Obs.'

0.159

0.148$0.014

- 0.080

- 0.074 k0.007

-0.863 0.349 1.271 -0.568 - 0.282

- 0.818 10.073 0.332 f 0.029

1.220kO.100 -0.553+0.041 - 0.275 5 0.020

NH3

CakdSe

0bs.f

1.944

2.38 f0.34

- 6.693

- 8.78& 0.9 1

7.786

10.5f1.2

- 2.962

-

0.064

0.1203 CH4

H H4T

- 12.0

137.1

- 16.984k0.023

H6 T

2.9

11.034 k0.019

"Calculated values from Georghiou [201]. Units: kHz. *III' axis representation. From Helminger et al. [97]. '11' axis representation. From Carpenter [ 1151. dCalculated values from Aliev and Watson [72]. 'Units: lO-'cm-'. IJ. M. Dowling, J. Mol. Spectrmc., 27, 527 (1968) and E. Schnabel, T. Torring, and W. Wilke, Z. Physik, 188, 167 (1965). gunits: Hz. hC.W. Holt, M. C. L. Gerry, and 1. Ozier, Can. J. Phys., 53, 1791 (1975).

constants F i j k .An accurate harmonic force field is needed in such calculations. A useful review has been given by Morino [182]. Relations involving the rotation-vibration information and the anharmonic potential constants has been given by Mills [1831. Anharmonic potential constants have been reported for HCN [184], HNC [185], HCP [186], ClCN [187], OCS [188], SOz [189], O 3 [190], OFz [191], SeOz [192], SFz [193], SiFz [194], GeFz [195], and

382


H 2 0 [196], to list a few. A summary of cubic force fields obtained from microwave data may be found in a review on anharmonic force fields by Pliva [197].

Calculation of the Sextic Distortion Constants The recent theoretical elucidation of the explicit relations between the sextic distortion constants and the cubic potential constants (8.206) (qk= y l i 2 Q k dimensionless normal coordinate) now makes possible the use of these constants to provide information on the anharmonic potential constants. Derivation of these relations requires application of fourth-order perturbation (see Chapter VII) and presents a particularly grueling computation. Theoretical treatments have been given by Aliev [198] for axial symmetric tops, by Sumberg and Parker [199] for nonlinear triatomic molecules; by Chung and Parker [200], Georghiou [201], and Aliev and Watson [72] for a general polyatomic molecule. The latter two formulations are particularly convenient for practical calculations, and relatively compact expressions of them have been formulated. Comparison between observed and calculated sextic coefficients are given in Table 8.36 for HIS, SOz, NH,, and CH4. The rotational parameters and the cubic potential constants employed in the calculations may be found in the references cited. The results, though quite satisfactory, could undoubtedly be improved by adjustment of the force constants, particularly for CH4. However, it should be realized that the observed sextic constants have undetermined zero-point vibrational contributions and that some discrepancy is to be expected, as with the quartic constants. Similiar calculations have been reported for O 3 [200].

9

ALTERNATE FORMULATIONS

In the usual treatment, based on the small oscillations approximation, the vibrational potential function and elements of the inertia tensor are expanded in terms of the vibrational coordinates, and a perturbation treatment is applied to give an effective rotational Hamiltonian for a given vibrational state (see Chapter VII). However, the Hamiltonian, represented by a Taylor series expansion in components of the angular momentum, converges very slowly for certain molecules. These extreme distortion effects can arise because of the strong dependence of a moment inertia on a bending coordinate such as in water, or the presence of a large amplitude coordinate such as the inversion coordinate in ammonia and related molecules. In such cases, the previous treatment may not be the most appropriate and alternate formulations have been described.

Large Amplitude Motion-Rotation Hamiltonian The general rotation-vibration Hamiltonian allowing for a large amplitude bending motion in triatomic molecules has been derived by Hougen et al. [18].

ALTERNATE FORM U LATl ONS

383

Further developments and applications of this Hamiltonian have been given by Bunker and Stone [202] (rigid-bender Hamiltonian), Hoy and Bunker [203] (nonrigid bender Hamiltonian), and Bunker and Landsberg [204] (semirigid bender Hamiltonian). Extension to tetraatomic molecules has been discussed by Sarka [205] and Stone [206]. Treatments for a large amplitude inversion motion have also been given [207-2091. To calculate the energy levels in the nonrigid bender formulation requires for H 2 0 a knowledge of the structure and potential constants to quartic terms21 parameters [203]. The particular advantage of this approach is not in a better fit to the rotational data but in the direct way information on the potential energy surface is obtained. In HCNO, with a large amplitude H-bending motion, the vibrational and rotational energy levels have been fit with the semirigid bender formulation by use of geometrical parameters and a simple quadraticquartic bending potential function [210]. The results indicate a linear equilibrium structure but a bent form in the ground vibrational state and a zero-point HCN bending amplitude of +34". A similar treatment has been applied to carbon suboxide [211].

The Pade Hamiltonian An alternate treatment to overcome the slow convergence of the angular momentum power series is to express the energy as a Pad6 approximant. The success of Pad6 approximants to approximate the value of a function is well known [212, 2131, and its application has been growing in various areas of physics although little use of the method has been made to problems in spectroscopy [214]. The usual Pad6 approximant to a function F ( x ) is the ratio of two polynomials Pn(x)/Qm(x) of degree n and m. The first application of this technique was made by Young and Young E2151 to fit the inversion spectrum of NH,. The Pad6 approximant with 19 terms, for example, gave a much better fit than the usual power series in J(J+ 1) and K with 21 terms. Based on a representation of the effective Hamiltonian in the form of a Pad6 operator, Belov et al. [216] have obtained an improved fit of the rotational spectrum of PH3 in the ground vibrational state. The Pad6 Hamiltonian has the form %Pad&

f[ (&)+(A)

=

H

H]

(8.207)

where the operators H and h both have the form of the usual Hamiltonian operator and in general do not commute. Both H a n d h contain n constants and hence Ytopadt. consists of 2n constants. The constants associated with h are dimensionless. It is convenient [216] to use basis functions which are eigenfunctions of h

(RlhlR)=&

(8.208)

where R denotes the collection of quantum numbers required to specify the particular eigenfunction and energy level. The eigenfunction aR is actually a

384


linear combination of appropriate basis functions. The energy matrix of the Pad6 Hamiltonian then has the form (8.209)

For a symmetric top

H = BP2 4-(c- B)P: -D j P 4 -D J K P ~ P- D : K p:

’ ‘ ‘

(8.210)

and

+

h = b P 2 ( c - b)P: - d j P 4 - djKP2Pt - dKpt + ’ ‘ ’

(8.211)

By expansion of the PadC operator, the Pad6 constants of (8.210) and (8.211) may be related to the spectroscopic constants of the usual Hamiltonian. For example [216],

+ Bb D X =DjK + (C - B)b + B(c - b ) DSp =DK + ( C - B)(c- b) DSp = D j

(8.212) (8.213) (8.214)

These spectroscopic constants may be interpreted with existing theories. This analysis technique should be useful for molecules with large rotational constants to provide a better fit of the observed rotational frequencies than is possible with the usual Hamiltonian containing 2n constants.

References 1.

2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14.

M. Toyama, T. Oka, and U. Morino, J. Mol. Spectrosc., 13,193 (1964). R. M. Garvey, J . Mol. Spectrosc., 65,330 (1977). G. Cazzoliand F. C. De Lucia, J . Mol. Spectrosc.,76,131 (1979). F. C. De Lucia and P. Helminger, J . Chem. Phys., 67,4262 (1977). G . Winnewisser, E. Churchwell, and C. M. Walmsley, “Astrophysics of Interstellar Molecules,” in Modern Aspects of Microwave Spectroscopy, G . W. Chantry, Ed., Academic, New York, 1979. L. E. Snyder, “Molecules in Space,” in M T P International Review of Science, Vol. 3, Spectroscopy, D. A. Ramsey, Ed., Butterworth, London, 1972. R. L. Cook, R. T. Walden, and G . E. Jones, J . Mol. Spectrosc., 53,370 (1974). E. B. Wilson, Jr., and J. B. Howard, J . Chem. Phys., 4,260(1936). H. H. Nielsen, Rev. Mod. Phys., 23, 90 (1951). E. B. Wilson, Jr.,J. Chem. Phys., 4,526(1936). E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. B. T. Darlingand D. M. Dennison, Phys. Rev.,57,128(1940). M. Goldsmith, G. Amat, and H. H. Nielsen, J. Chem. Phys., 24, 1178 (1956); 27, 838 (1957). G. Amat and H. H. Nielsen, J . Chem. Phys., 27, 845 (1957); 29, 665 (1958); 36, 1859 (1962). See also M. L. Grenier-Besson, G. Amat, and H. H. Nielsen, J . Chem. Phys., 36, 3454(1962); G. Amat, H. H. Nielsen, and G . Tarrago, Rotation-Vibration of Polyatomic Molecules, Dekker,NewYork, 1971.

REFERENCES

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

385

J. K. G . Watson, Mol. Phys., 15, 479 (1968). L. S. RothmanandS. A. Clough,J. Chem. Phys.,54,3246(1971). A. M. Walsh-Bakke,J. Mol. Spectrosc.,40,1(1971). J. T. Hougen, P. R. Bunker,and J. W. C. Johns,J. Mol. Spectrosc.,34,136(1970). K. T. Chung and P. M. Parker, J. Chem. Phys., 38,8 (1963); M. Y. Chan, L. Wilardjo, and P. M. Parker, J . Mol. Spectrosc..40,473(1971). J. D. Louck,J. Mol. Spectrosc., 61,107(1976). D. KivelsonandE. B. Wilson, Jr.,J. Chem. Phys., 21,1229(1953). C. Eckart, Phys. Rev., 47,552(1935). S. M. Ferigle and A. Weber, Am. J . Phys., 21,102 (1953). R. Gold, J. M. Dowling, and A. G. Meister, J . Mol. Spectrosc., 2,9 (1958). J. M. Dowling, R. Gold, and A. G. Meister, J. Mol. Spectrosc., 1, 265 (1957); 2, 411 (1958). G. Thyagarajan, S. Sundaram, and F. F. Cleveland, J . Mol. Spectrosc., 5,307 (1960). S . Sundaram and F. F. Cleveland, J . Chem. Phys., 32,166 (1960). I. McNaught, J . Mol. Spectrosc., 91,492 (1982). C. R. Parent and M. C. L. Gerry, J . Mol. Spectrosc., 49, 343 (1974). S . R. Polo, J . Chem. Phys., 24,1133 (1956). M. R. Aliev and Aleksanyan, Dokl. Akad. Nauk SSSR,Ser. Khim. 169, 1229 (1966). See also [40] and [41]. S . J. Cyvin and G. Hagen, Chem. Phys. Lett., 1,645 (1968). S. J. Cyvin, B. N. Cyvin, a n d G . Hagen, 2.Naturforsch.,23A, 1649(1968). G. 0.Ssrensen, G. Hagen, and S . J. Cyvin, J . Mol. Spectrosc., 35,489 (1970). P. Pulay and W. Sawodny, J . Mol. Spectrosc., 26,150 (1968). K. Klauss and G. Strey, Z. Naturforsch., 23A, 1308 (1968). B. L. Crawford, Jr.andW. H. Fletcher,J. Chem. Phys., 19,141 (1951). D. Kivelson,J. Chem. Phys., 22,904(1954). I. Gamo, J . Mol. Spectrosc., 30,216 (1969). A. P. Aleksandrov, M. R. Aliev, andV. T. Aleksanyan, Opt. Spectrosc., 29,568(1970). A. P. Aleksandrov and M. R. Aliev, J . Mol. Spectrosc., 47,1(1973). J. K. G. Watson, J . Mol. Spectrosc., 74,483 (1979). A. Muller, N. Mohan, and A. Alix, J . Chem. Phys., 59,6112 (1973). J. Demaison, J. Mol. Struct., 31, 233 (1976). L. Nemes,J. Mol. Srruct.,46,67(1978). M. R. Aliev, Opt. Spectrosc., 31,568 (1971); 33,858 (1972). M. R. Alievand J. K. G. Watson, J . Mol. Spectrosc., 74,282(1979). S . E. Novick,J. Mol. Spectrosc.,68,77(1977). R. L. Cook, J . Mol. Struct.,26,126(1975). D. KivelsonandE. B. Wilson, Jr.,J. Chem. Phys.,20,1575 (1952). J. K.G. Watson,J. Chem. Phys.,45,1360(1966);48,181(1968). E. B. Wilson, Jr.,J. Chem. Phys.,5,617(1937). P. M. Parker,J. Chem. Phys.,37,1596(1962). R. M. Lees, J . Mol. Spectrosc.,33,124(1970). J. K. G. Watson, J. Chem. Phys., 46,1935 (1967). K. M. Marstokkand H. Msllenda1,J. Mol. Struct.,8,234(1971). V. Typke, Z. Naturforsch., 26A, 1775 (1971). A. Bouchy, J. Demaison, G. Roussy, and J. Barriol, J . Mol. Struct., 18,211 (1973). J. M. Dowling, J. Mol. Spectrosc., 6, 550 (1961).

386


60. 61. 62. 63. 64.

Z. I. Slawsky and D. M. Dennison, J . Chem. Phys., 7,509 (1939). H. Dreizler, G. Dendl, Z . Naturforsch., 20A, 30 (1965). H . Dreizler and H. D. Rudolph, Z . Nuturforsch., ZOA, 749 (1965). J . K . G. Watson, J. Chem. Phys., 48,4517 (1968). J . K. G. Watson, “Aspects of Quartic and Sextic Centrifugal Effects on Rotational Energy Levels,” In Vibrational Spectra and Structure, J . R. Durig, Ed., Vol. 6 , Marcel Dekker, New York (1977).

65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

K . T. Chung and P. M. Parker, J . Chem. Phys., 43,3865,3869 (1965). F . X . Kneizys, J. N. Freedman, and S.A. Clough, J . Chem. Phys., 44,2552 (1966). K . K . Yallabandiand P. M. Parker,J. Chem. Phys., 49,410(1968).

75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94.

95. 96.

97. 98.

W. H. Kirchhoff, J . Mol. Spectrosc., 41,333 (1972). G. Winnewisser, J. Chem. Phys., 56,2944 (1972). B. P. Van Eijck, J. Mol. Spectrosc., 53,246 (1974). V .Typke, J . Mol. Spectrosc., 63,170 (1976). M . R. Alievand J. K. G . Watson, J . Mol. Spectrosc., 61,29 (1976). K. T. Hecht, J . Mol. Spectrosc, 5,355, 390 (1960). J . Moret-Bailly, J. Mol. Spectrosc., 15, 344 (1965). J. Moret-Bailly, L. Gauthier, and J. Montagutelli, J. Mol. Spectrosc., 15, 355 (1965). A. J. Dorney and J. K. G . Watson, J . Mol. Spectrosc.,42,135 (1972). S. M. Kirschnerand J. K. G .Watson, J . Mol. Spectrosc., 47,347 (1973). I. Ozier, J . Mol. Spectrosc.,53,336 (1974). F. Michelot, J. Moret-Badly, and K. Fox, J . Chem. Phys., 60,2606 (1974). J . K. G. Watson, J . Mol. Spectrosc., 55,498 (1975). C. W. Holt, M. C. L. Gerry, and I. Ozier, Phys. Rev. Lett., 31,1033 (1973). L. Pierce, N. Di Cianni, and R. H. Jackson, J . Chem. Phys., 38,730 (1963). D. W. PosenerandM. W. P. Strandberg, Phys. Rev.,95,374(1954). R. E. Hillgerand M. W. P. Strandberg, Phys. Rev., 83,575 (1951). T. OkaandY. Morino,J. Mol. Spectrosc.,8,300(1962). W. S.Benedictand E. K . Plyer, J . Res. Nar. Bur. Stand.,#, 246, (1951). H . C. Allen, Jr. and W. B. Olson, J . Chem. Phys., 37,212 (1962). P. E. Fraleyand K. Narahari Rao,J. Mol. Spectrosc., 19,133 (1966). L. H. Ford, K. K. Yallabandi, and P. M. Parker, J . Mol. Spectrosc., 30,241 (1969). R. H. Hilland T. H. Edwards,J. Mol. Spectrosc., 11,433 (1963). P. Helminger, R. L. Cook, and F. C. De Lucia, J . Mol. Spectrosc., 40,125 (1971). F . C. De Lucia, R:L.Cook, P. Helminger, and W . Gordy, J . Chem. Phys., 55,5334 (1971). R. L. Cook, F. C. De Lucia, and P. Helminger, J . Mol. Spectrosc., 41,123 (1972). G. Winnewisser, M. Winnewisser, and B. P. Winnewisser, “Millimeter Wave Spectroscopy,” in M T P International Review of Science, Vol. 3, Spectroscopy, D. A. Ramsay, Ed., Butterworths, London, 1972. F. C. De Lucia, ‘‘Millimeter- and Submillimeter-Wave Spectroscopy,” in Molecular Spectroscopy: Modern Research, Vol. 11, K. N. Rao, Ed., Academic, New York, 1972. A. F. Krupnov and A. V. Burenin, “New Methods in Subillimeter Microwave Spectroscopy,” in Molecular Spectroscopy: Modern Research, Vol. 11, K . N. Rao, Ed., Academic, New York, 1972. P. Helminger, R. L. Cook, and F. C. De Lucia, J . Chem. Phys., 56,4581 (1972). F. C. De Lucia, P. Helminger, R. L. Cook, and W. Gordy, Phys. Rev., A5487 (1972).

REFERENCES

99. 100. 101. 102. 103.

104. 105. 106. 107. 108. 109. 110. 111. 112. 113.

114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134.

387

F. C. De Lucia and P. Helminger, J . Mol. Spectrosc., 56,138 (1975). F. C. DeLucia, P. Helminger, R. L. Cook, and W. Gordy, Phys. Rev., A6,1324(1972). P. Helminger and F. C. DeLucia, J . Mol. Spectrosc., 70,263 (1978). G.Steenbeckeliersand J. Bellet,J. Mol. Spectrosc.,45, lO(1973).

J. Bellet and G. Steenbeckeliers, Compt. Rend., 2718, 1208 (1971). See also G. Steenbeckeliers and J. Bellet, Compt. Rend., 273B, 471 (1971) and W. Lafferty, J. Bellet, and G. Steenbeckeliers, Compt. Rend., 273B, 388 (1971). J. Bellet, W.J. Lafferty, and G.Steenbeckeliers,J. Mol. Spectrosc., 47,388 (1973). J . Bellet, G. Steenbeckeliers, and P. Stouffs, Compr. Rend., 275B, 501 (1972). F. C. DeLucia, P. Helminger, W. Gordy, H. W. Morgan, and P. A. Staats, Phys. Rev., AS, 2785(1973). P. Helminger, F. C. DeLucia, and W. Gordy, P. A. Staats, and H. W. Morgan, Phys. Rev., A10,1072(1974). P. Helmingerand F. C. DeLucia, J . Mol. Spectrosc., 58,375 (1975). F. C. DeLuciaandP. Helminger, J. Mol. Specfrosc.,54,200(1975). E. A. Cohenand H. M. Pickett, J . Mol. Spectrosc., 93,83 (1982). A. Bouchy and G. Roussy, J . Mol. Struct., 84,1(1982). See for example, M. Newman, “Matrix Computations,” in A Survey ofNumerical Analysis, J. Todd, Ed., McGraw-Hill, New York, 1962. See also, e.g., W. H. Kirchhoff and D. R. Johnson, J. Mol. Spectrosc., 45, 159 (1973); D. R. Johnson, F. J. Lovas, and W. H. Kirchhoff, J. Chem. Phys. Ref. Data, 1, 1011 (1972); Y. Beers, G. P. Klein, W. H. Kirchhoff, and D. R. Johnson, J. Mol. Spectrosc., 44,553 (1972). V. Typke, Z. Naturforsch., 26A, 175 (1971). J. H. Carpenter, J . Mol. Spectrosc., 46,348 (1973). L. L. Strow, J . Mol. Spectrosc.,97,9 (1983). K. Yamadaand M. Winnewisser, Z. Naturjorsch.,31A, 131 (1976). G. Winnewisser and P. Helminger, J . Chem. Phys., 56,2967 ;2954 (1972). G. Winnewisser,J. Chem. Phys.,57,1803(1972). E. F. Pearson and H. Kim, J. Chem. Phys., 57,4230 (1972). C. C. Costain, Phys. Rev.,82,108(1951). E. Schnabel, T. Torring, and W. Wilke, Z . Physik, 188,167 (1965). P. B. Davies, R. M. Neumann, S . C. Wofsy, and W. Klemperer, J . Chem. Phys., 55, 3564 (1971). W. B. Olson, A. G. Makiand R. L. Sams,J. Mol. Spectrosc., 55,252 (1975). P. Bernardand T. Oka,J. Mol. Spectrosc., 75,181 (1979). R. H. Bradley, P. N. Brier, and M. J. Whittle,J. Mol. Spectrosc.,44,536(1972). S . R. Jones, P. N. Brier, D. M. Brookbanks, and J. G. Baker, J . Mol. Spectrosc., 47, 351 (1973). R. Jurek, J. Chanussot, and J. Bellet, Compt. Rend., 277B, 53 (1973); J. Chanussot, R. Jurek, and P. Suzean, Compt. Rend., 276B, 729 (1973). R. Jurek, P. Suzean, J. Chanussot, and J. P. Champion, J. Physique, 35, 533 (1974); P. Goulet, R. Jurek, and J. Chanussot, J . Physique, 37,495 (1976). B. BalikciandP. N. Brier,J. Mol. Spectrosc.,85,109(1981);89,254(1981). G. Amat and L. Henry, J . Phys. Radium, 21,728 (1960). P. Kupecek,J. Phys.,25,831(1964). I. Ozier, R. M. Lees, and M. C. L. Gerry, Cand. J . Phys., 54,1094 (1 976). G. Steenbeckeliers, Ann. SOC.Sci. Brux.,82,331(1968).

388


135. W. C. Hamilton, Statistics in Physical Science, Ronald, New York, 1964. 136. D. L. Albritton, A. L. Schmeltekopf, and R. N. Zare, “An Introduction to the Least-Squares Fitting of Spectroscopic Data, in Molecular Spectroscopy: Modern Research, Vol. 2,K. N. Rao, Ed., Academic, New York, 1976. 137. W. H. Kirchhoff and D. R. Lide, Jr., and F. X. Powell, J . Mol. Spectrosc., 47,491 (1973). 138. G. E. Herberich, R. H. Jackson, and D. J. Millen, J. Chem. SOC.,(A),336 (1966). 139. J. K. G. Watson,J. Mol. Spectrosc.,66,500(1977). 140. J. K. G. Watson, J . Mol. Spectrosc., 65,123 (1977). 141. A. Niroomand-Rad, M. Y. Chan, and P. M. Parker, J . Mol. Spectrosc., 69,450 (1978). 142. See, e.g., J. G. Smith, Mol. Phys., 35,461 (1978);K. M.Marstokk and H. Msllendal, J . Mol. Struct., 32, 191 (1976);K. M. Marstokk and H. Msllendal, J . Mol. Struct., 40,1 (1977);A. Dubrulle and J. L. Destombes, J . Mol. Struct., 13,461 (1972);E.Willemot, D. Dangoisse, and J. Bellet,J. Mol. Spectrosc., 73,96(1978). 143. J. Demaison, D. Schwoch, B. Tan, and H. D. Rudolph, J . Mol. Spectrosc., 59,226 (1976). 144. R. A. Creswell and 1. M. Mills, J . Mol. Spectrosc., 52,392 (1974). 145. S.I. Chan, T. R. Borgers, J. W. Russell, H. L. Strauss, and W. D. Gwinn, J . Chem. Phys., 44, 1103(1966). 146. L.Halonen, E.Friz, A. G. Robiette, and I. M. Mills, J . Mol. Spectrosc., 79,432 (1980). 147. See, for example, J. L. Duncan and I. M. Mills, Spectrochim. Acta., 20,523 (1964);A. M. Mirriand E. Mazzariol, Spectrochim. Acta., 22,785(1966). 148. T.OkaandY. Morino,J. Mol. Spectrosc., 6,472(1961). 149. J.H.MealandS.R.Polo,J.Chem.Phys.,24,1119,1126(1956). 150. S . J. Cyvin, Molecular Structures and Vibrations, Elsevier, Amsterdam, 1972. 151. S. J. Cyvin, “Molecular Vibrations and Mean Square Amplitudes,” Elsevier, Amsterdam, 1968. 152. R. W. Davis and M. C. L. Gerry, J . Mol. Spectrosc., 65,455(1977). 153. L. Pierce, J . Chem. Phys., 24,139 (1 956). 154. W. H. Kirchhoff, D. R. Johnson, and F. X. Powell, J . Mol. Spectrosc., 48,157 (1973). 155. R.L. Cook, G. Winnewisser, and D. C. Lindsey, J . Mol. Spectrosc., 46,276(1973). 156. R. L. Cook, F. C. DeLucia, and P. Helminger, J. Mol. Spectrosc., 53,62 (1974). 157. R. L. Cook, J. Chem. Phys., 42,2927(1965). 158. J. K. G. Watson, J . Mol. Spectrosc., 41,229 (1972). 159. I. M. Mills, J . Mol. Spectrosc., 5,334 (1960);17,164 (1965). 160. See also G . Strey, J . Mol. Spectrosc., 17,265(1965)and CH. V.S.R. Rao, J . Mol. Spectrosc., 41,105 (1972). 161. R. L. Cook, F. C. DeLucia, and P. Helminger, J . Mol. Spectrosc., 28,237 (1975). 162. S. Toman and J. Pliva, J . Mol. Spectrosc., 21,362 (1966). 163. R. L. Cook and W. H. Kirchhoff, J . Chem. Phys., 47,4521 (1967). 164. A. Allan, J. L. Duncan, J. H. Holloway, and D. C. McKean, J . Mol. Spectrosc., 31,368(1969). 165 A. R. Hoy, J. M. R. Stone, and J. K. G. Watson, J . Mol. Spectrosc., 42,393 (1972). 166. J. D. Duncan, “Force Constant Calculations in Molecules,” in Molecular Spectroscopy, Vol. 3, The Chemical Society of London (1973). 167. I. M. Mills, “Harmonic Force Field Calculations,” in Critical Evaluation of Chemical and Physical Structural Information, D. R. Lide and J. A. Paul, Eds., National Academy of Science, Washington, 1974. 168. J. H. Carpenter and D. F. Rimmer, J . C . S . Faraday I I , 74,466(1978).

REFERENCES

389

169. L. H. Jones, L. B. Asprey, and R. R. Ryan, J. Chem. Phys., 47,3371 (1967); L. H. Jones, R. R. Ryan, and L. B. Asprey, J. Chem. Phys., 49,581 (1968); J. Laane, L. H. Jones, R. R. Ryan, and L. B. Asprey, J. Mol. Spectrosc., 30,489 (1969). 170. N. J. D. Lucas and J. G. Smith, J. Mol. Spectrosc., 43, 327 (1972). 171. R. W. Davis, A. G. Robiette, M. C. L. Gerry, E. Bjarnov, and G. Winnewisser, J. Mol. Spectrosc., 81, 93 (1980). 172. B. P. Winnewisser and W. H. Hocking, J. Phys. Chem., 84, 1771 (1980). 173. D. J. Finnigan, A. P. Cox, A. H. Brittain, and J. G . Smith, J . C. S . Faraday II,68,548 (1972). 174. M. C . L. Gerry, J . Mol. Spectrosc., 45, 71 (1973). 175. J. L. Duncan, D. C. McKean, and G. K. Speirs, Mol. Phys., 24,353 (1972). 176. G. Cazzoli, J. Mol. Spectrosc., 53, 37 (1974). 177. G. Cazzoli and W. Caminati, J . Mol. Spectrosc., 62, 1 (1976). 178. A. G. Robiette and M. C . L.Gerry, J. Mol. Spectrosc., 80,403 (1980). 179. R. R. Filgueira, P. Forti, and G. Corbelli, J. Mol. Spectrosc.,57,97 (1975). 180. J. L. Duncan and P. D. Mallinson, Chem. Phys. Lett., 23, 597 (1973). 181. P. D. Mallinson, D. C. McKean, J. H. Holloway, and I. A. Oxton, Spectrochim. Acta, 31A, 143 (1975). 182. Y. Morino, Pure Appl. Chem., 18, 323 (1969). 183. I. M. Mills, “Vibration-Rotation Structure in Asymmetric- and Symmetric-Top Molecules,” in Molecular Spectroscopy: Modern Research, K. N. Rao and C. W. Mathews, Eds., Academic, New York, 1972. 184. T. Nakagawa and Y. Morino, Bull. Chem. Soc. Japan, 42,2212 (1969). 185. R. A. Creswell and A. G. Robiette, Mol. Phys., 36, 869 (1978). 186. G. Strey and I. M. Mills, Mol. Phys., 26, 129 (1973). 187. C. B. Murchison and J. Overend, Spectrochim. Acta. 27A, 1801 (1971). 188. Y. Morino and T. Nakagawa, J . Mol. Spectrosc., 26,496 (1968). 189. Y. Morino, Y. Kikuchi, S. Saito, and E. Hirota, J. Mol. Spectrosc., 13, 95 (1964). 190. T. Tanaka and Y. Morino, J. Mol. Spectrosc., 33, 538 (1970). 191. Y. Morino and S . Saito, J. Mol. Spectrosc., 19, 435 (1966). 192. H. Takeo, E. Hirota, and Y. Morino, J. Mol. Spectrosc., 41, 420 (1972); 34, 370 (1970). 193. Y. Endo, S. Saito, E. Hirota, and T. Chikaraishi, J. Mol. Spectrosc., 77, 222 (1979). 194. H. Shoji, T. Tanaka, and E. Hirota, J. Mol. Spectrosc., 47, 268 (1973). 195. H. Takeo, R. F. Curl, Jr., and P. W. Wilson, J. Mol. Spectrosc., 38,464 (1971). 196. D. F. Smith and J. Overend, Spectrochim. Acta., 28A, 471 (1972; A. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys., 24, 1265 (1972). 197. J. Pliva, “Anharmonic Force Fields,” in Critical Evaluation of Chemical and Physical Structural Information, D. R. Lide and M. A. Paul, Eds., National Academy of Science, Washington, 1974. 198. M. R. Aliev, J. Mol. Spectrosc., 52, 171 (1974). 199. D. A. Sumberg and P. M. Parker, J. Mol. Spectrosc., 48,459 (1973). 200. M. Y. Chan and P. M. Parker, J. Mol. Spectrosc., 65,190 (1977). See also A. Niroomand-Rad and P. M. Parker, J. Mol. Spectrosc., 75, 177 (1979). 201. C. Georghiou, Mol. Phys., 32, 1279 (1976). 202. P. R. Bunker and J. M. R. Stone, J. Mol. Spectrosc., 41, 310 (1972). 203. A. R. Hoy and P. R. Bunker, J . Mol. Spectrosc., 52,439 (1974); 74. 1 (1979). 204. P. R. Bunker and B. M. Landsberg, J. Mol. Spectrosc., 67, 374 (1977). 205. K. Sarka, J. Mol. Spectrosc., 38, 545 (1971).

390 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216.


J. M. R. Stone, J . Mol. Spectrosc., 54, 1 (1975). D. Papousek, J. M. R. Stone, and V. Spirko, J . Mol. Spectrosc., 48, 17 (1973). V. Spirko, J. M. R. Stone, and D. PapouSek, J . Mol. Spectrosc., 60,159 (1976). V. Danielis, D. PapouSek, V. Spirko, and M. Horak, J . Mol. Spectrosc., 54, 339 (1975). P. R. Bunker, B. M. Landsberg, and B. P. Winnewisser, J. Mol. Spectrosc. 7 4 9 (1979). P. R. Bunker, J . Mol. Spectrosc., 80,422 (1980). G . A. Baker, Jr., Essentials of Pad&Approximants, Academic, New York, 1975. E. B. Saff and R. S . Varga, Eds., Pad; and Rational Approximation, Academic, New York, 1977. M. Mizushima, The Theory of Rotating Diatomic Molecules, Wiley, New York, 1975. L. D. G. Young and A. T. Young, J . Quant. Spectrosc. Radiat. Transfer, 20, 533 (1978). S . P. Belov, A. V. Burenin, 0. L. Polyansky, and S. M. Shapin, J . Mol. Spectrosc., 90, 579 (1981).

Chapter Ix

NUCLEAR HYPERFINE STRUCTURE I N MOLECULAR ROTATIONAL SPECTRA 1

INTRODUCTION

392

2

CLASSICAL HAMILTONIAN FOR NUCLEAR QUADRUPOLE COUPLING

392

3

NUCLEAR QUADRUPOLE INTERACTIONS IN FIXED MOLECULES OF SOLIDS

395

4

QUADRUPOLE INTERACTIONS BY A SINGLE COUPLING NUCLEUS IN A ROTATl N G M 0 LEC U LE

399

General Theory of the Interaction Diatomic and Linear Polyatomic Molecules Symmetric-top Molecules Asymmetric Rotors

399 407 41 2 41 3

QUADRUPOLE COUPLING BY MORE THAN ONE NUCLEUS IN A ROTATING MOLECULE

423

Coupling by Two Nuclei Coupling by Three or More Nuclei

423 425

6

SECOND-ORDER EFFECTS IN NUCLEAR QUADRUPOLE INTERACTIONS

426

7

MAGNETIC HYPERFINE STRUCTURE OF MOLECULES IN SINGLET

5

Z STATES 429

Theory of Magnetic Interaction Diatomic and Linear Polyatomic Molecules Symmetric-top Molecules Asymmetric Rotors

a

429 432 433 435

PLURAL NUCLEAR COUPLING-QUADRUPOLE

AND MAGNETIC

436

Hybrid Coupling by Two Nuclei Strong Quadrupoie and Weak

9

Magnetic Coupling

436

Couplings of Comparable Strength Coupiing by identical Nuclei S p i n-S pi n I nteractions Observed Spectra in Asymmetric Rotors-Maser

437 440 441 441

Resolutions

MOLECULES WITH UNBALANCED ELECTRONIC ANGULAR MOMENTUM

444

10


444

11

SELECTION RULES AND RELATIVE INTENSITIES OF HYPERFINE COMPONENTS

445 391

392

1

NUCLEAR HYPERFINE STRUCTURE IN MOLECULAR ROTATIONAL SPECTRA

INTRODUCTION

Nuclear hyperfine structure in molecular rotational spectra may arise from either magnetic or electric interactions of the molecular fields with the nuclear moments, or from a combination of the two. The most important of these interactions is that of the molecular field gradient with the electric quadrupole moments of the nuclei. Such an interaction is not possible, however, for isotopes with nuclear spins of 0 or since such nuclei are spherically symmetric and hence have no quadrupole moments. The nuclear quadrupole hyperfine structure provides a measure of the molecular field gradients from which much information about the electronic structure and chemical bonds can be obtained. Nuclear quadrupole interaction in molecules was first detected in molecular beam experiments on diatomic molecules in the radiofrequency region [l, 21. The basic theory worked out by Casimer3 was extended to diatomic molecules by Nordsieck [4] and by Feld and Lamb [5]. The first detection of quadrupole hyperfine structure in the microwave region was that of the ammonia inversion spectrum by Good [6]. The theory for the interaction was extended to symmetric tops by Coles and Good [7] and by Van Vleck [8], to asymmetric rotors by Bragg and Golden [9, 101. Practically all stable molecules, particularly organic molecules, have singlet C electronic ground states. All their electrons are paired, either in the atomic subshells or in the molecular orbital valence shells, so that in the first-order approximation their electronic magnetism whether from electronic spin or from orbital motion is canceled. When the molecules are not rotating, the molecular magnetism is canceled even in the higher orders of approximation. The end-over-end rotation of the molecule, however, generates weak magnetic fields which can interact with the nuclear magnetic moments to produce a slight magnetic splitting or displacement of the lines. Although the complexity of this slight magnetic field generated by rotation usually prevents measurements sufficiently accurate to give reliable information about chemical bonding like that given by nuclear quadrupole interactions, one must evaluate these small magnetic displacements when they are evident if the evaluation of the nuclear quadrupole coupling and rotational constants is to be the most precise. A general theory for nuclear magnetic interactions of rotating molecules in X singlet states has been worked out by Gunther-Mohr et al. [ll]. For convenience in the prediction of hyperfine structure we have provided a table of nuclear moments (Appendix E).

2 CLASSICAL HAMILTONIAN FOR NUCLEAR QUADRUPOLE COUPLING The nuclear quadrupole interaction results from a nonspherical distribution of nuclear charge which gives rise to a nuclear quadrupole moment and a nonspherical distribution of electronic charge about the nucleus which gives rise to an electric field gradient at the nucleus. If either the nuclear charge or the

CLASSICAL HAMILTONIAN FOR NUCLEAR QUADRUPOLE COUPLING

393

electronic charge about the nucleus is spherically symmetric, no such interaction is observed. The interaction puts a twisting torque on the nucleus, tending to align its spin moment in the direction of the field gradient. As a result of this torque, the spin axis will precess about the direction of the resultant field gradient, giving rise to precessional frequencies and nuclear quadrupole spectra. In solids, the field gradients are fixed in direction, and pure nuclear quadrupole spectra analogous to nuclear magnetic resonance are observable. In gases, the field gradients at the nucleus depend on the rotational state of the molecule; the nuclear quadrupole interaction differs for each rotational state and leads to a hyperfine structure of the rotational levels. Let us first consider the classical interaction energy of a nuclear charge with a static potential V arising from extranuclear charges. Suppose V ( X Y Z ) to be the potential at a point with coordinates X, Y, Z in a Cartesian system fixed in space and having its origin at the center of the nucleus, as indicated in Fig. 9.1. The value of V at the origin is V,. To a first approximation, the electrical energy is ZeV,, the usual energy that holds the electrons in their orbits. Because of the finite size of the nucleus, however, the electrical energy is correctly expressed by

where p,,=p(X, Y, Z ) represents the density of the nuclear charge in the elemental volume d ~ , = d X d Y d Zand where the integration is taken over the

Z

Fig. 9.1

Reference system for expressing nuclear interaction with extra-nuclear charges.

394

NUCLEAR HYPERFINE STRUCTURE I N MOLECULAR ROTATIONAL SPECTRA

nuclear volume. The charge density p n can be taken as uniform throughout the volume. In evaluation of the integral it is convenient to express the potential as a Taylor's expansion. In this expansion,

where X n = A X , Y,=AY, and Z n = A Z represent the nuclear coordinates at the point of the elemental volume drn in the nucleus. Substitution of I/ from (9.2) into (9.1) shows the first term in the expansion

to be the usual monopole interaction which is independent of nuclear orientation. The second or dipole term,

vanishes because p ( X , Y , 2)is a symmetric function, and the integral with + and - values of the coordinates cancels. This is in agreement with the fact that no nuclear electric dipole moment has ever been observed experimentally. The term of next higher order represents the quadrupole interaction,

Let us now choose a body-fixed system x, y, z with z along the spin axis I. As a consequence of rotational symmetry about the spin axis; x,y, z are principal axes; the integrals of the cross terms vanish; and

Consequently, (9.5) can be expressed as

NUCLEAR QUADRUPOLE INTERACTIONS IN FIXED MOLECULES

395

Because the charge giving rise to the field gradient can be considered to be zero over the nuclear volume, Laplace's equation V2 I/ =O holds, and

(!g)o+($)o=-rg) 0

With this relation, the classical interaction can be expressed as (9.9) The quantity Q* =;

1

J pn(3zf

- rf) dr,

(9.10)

is defined as the intrinsic nuclear quadrupole moment. Hence (9.11) In this classical expression z is along the spin axis. Quantum mechanically, however, the spin I and Q* are not observable quantities since only I2 and its components along a fixed direction in space have eigenvalues. However, the value of Q* can be obtained from the observable quantity Q with the relationship, (9.20), derived later. Q* is a measure of the deviation of the nuclear shape from spherical symmetry. A positive Q* indicates that the nucleus is elongated along the spin axis, is prolate; a negative Q* indicates that the nucleus is flattened along this axis, is oblate. For a spherical nucleus Q* vanishes.

3

NUCLEAR QUADRUPOLE INTERACTIONS I N FIXED MOLECULES OF SOLIDS

Let us now transform the field gradient of (9.11) to a space-fixed system X , Y, Z, with its origin at the center of the nucleus and chosen so that X , Y, and Z are principal axes of the field gradient. In the principal system the cross terms vanish, and the transformation is expressed by

(9.12) The last form follows from the fact that I is along z and Z, Zy,and Iz are components of I; hence the direction cosines are

ax --cos ( X , z)=--,I X . . . _

aZ

I I/

(9.13)

396


By treating the spin components as quantum mechanical operators and substituting for I’ its eigenvalue ](I+ l), one can obtain a spin Hamiltonian operator by substitution of (9.12) into (9.11).Thus

When the field gradient is axially symmetric about Z ,

(gqo=&qo=2 (E)o 1

a2v

(9.15)

where use is made of Laplace’s equation V 2 V = 0 . With (9.15) and Z2 =I;+ Z’, + If, (9.14) becomes, for the axially symmetric coupling, (9.16) and (EQ)MI=

eQ*(d’ V/dZ’)), [3M:- I(I+ l)] 8Z(Z+ 1)

(9.17)

where M I is the eigenvalue of Zz in units of A. Except for the use of the intrinsic quadrupole moment Q*, (9.14) and (9.16) are the usual Hamiltonians for pure quadrupole resonance in the solid state. A simple, if not rigorous, derivation of the relationship between Q* and the observable Q can be found by comparison of (9.17) with the classical energy, (9.11), derived for axially symmetric coupling about 2, the spin axis. Conventionally, Q is defined as the effective component for the most complete alignment along 2, that is, the component for M I = I. From (9.17), eQ*(a’ V/822)0(2Z- 1) (9.18) 8(Z+ 1) Since the effective quadrupole moment for coupling with (d2V/aZ2)is Q, the coupling energy found by analogy with (9.11) is (EQ)MI =I

=

EQ’j

(””)

0

eQ

(9.19)

Equation of this energy to (EQ)MI= I from (9.18) yields 2(I+ 1) Q* =(21-1)Q

(9.20)

Substitution of this value into (9.14) gives the solid state Hamiltonian operator in the form (9.21)

NUCLEAR QUADRUPOLE INTERACTIONS IN FIXED MOLECULES

397

where the customary designations for the principal values of the coupling constants have been made: xxx

eQqxx,

xyy

=e Q q y y ,

xzz

eQqzz

(9.22)

where (9.23) With axial symmetry about Z , it is evident from (9.17) that xzz

E ‘-41(21-

l)

(9.24)

[3M?- Z(Z+ I)]

where M*=Z,Z-I, 1-2 , . . . , -1

(9.25)

In the general case when there is no axial symmetry, the secular equation must be set up and solved as described in Chapter 11, Section 4, or perturbation methods must be used. The required matrix elements of the squared operators of (9.21)are

( 4 MI11 l; 1,M I )=M21 ( 4 MIIEII, MI)=(&MIIIz.II,M1)=3[W+ 1)-M2] (1,MrI 1.i 1, MI f2)= - (1,MII1 I; 1,Mr 2) l)-MI(MIf

= -$[Z(Z+

l)]l’2

x [Z(Z+ l)-(MIfl)(MI$_2)]”2

(9.26)

These matrix elements have the same form as those for P:, Py’, and Pz in the symmetric-top representation if I is identified with J and M I with K . However, I and M I may have half-integral as well as integral values, whereas J and K have only integral values. Note that (9.21) for has the same form as the Hamiltonian for the rigid asymmetric rotor if the couplings are related to the moments of inertia by xxx

21(21- 1)

+- 1

21, ’

XY Y

+ --1

21(21- 1)

21, ’

1 xzz +21(21- 1) 21,

(9.27)

With these relations the various solutions already obtained for the rigid asymmetric rotor can be used for the finding of quadrupole coupling energies for integral spin values. This relationship was first pointed out by Bersohn [12]. Because the values of I are small, exceeding 3 for only a very few isotopes, the problem of finding the quadrupole coupling energies in solid state asymmetric field gradients is simpler than that of calculating the energies of the asymmetric rotor for which populated states with high values are possible. Solutions for various spin values are given by Cohen and Reif [131and by Das and Hahn [141. Frequently, the field gradient is almost axially symmetric, if not completely so, about a bond to the coupling atom. Such cases are most conveniently

398


treated by perturbation theory in a manner similar to that for treatment of the slightly asymmetric rotor. The most nearly symmetric axis is defined as the Z axis, and the asymmetry parameter is defined by

v=

xxx - X Y Y

(9.28)

Xzz

+ +

Because Laplace’s equation holds, xXx xYy xZz=0, there are only two independent coupling parameters. These are chosen as xzz and q. The Hamiltonian can then be expressed so that the off-diagonal terms contain q as a coefficient. The off-diagonal terms can then be evaluated with perturbation theory. The asymmetry parameter q has been related by Das and Hahn [I41 to the asymmetry parameter b of the slightly asymmetric rotor so that the energy formula for the slightly asymmetric rotor (Chapter VII) can be used to give the quadrupole energies when q < 1. Transitions between the quadrupole hypefine levels in solids can be observed through coupling of the magnetic dipole moment of the nucleus to the radiation field. The selection rule for pure quadrupole absorption spectra is lAMIl = 1. For an axially symmetric field, the frequencies from (9.24) are (9.29) where M I is the larger of the two quantum numbers involved. Figure 9.2 shows an energy level diagram indicating the transitions and pure quadrupole spectrum for a nucleus with spin Z=; which is in an axially symmetric field.

V-

Fig. 9.2 Energy level diagram and predicted pure quadrupole resonance spectrum for a nucleus with spin I =.:

399

QUADRUPOLE INTERACTIONS BY A SINGLE COUPLING NUCLEUS

Pure quadrupole resonance frequencies of solids, first detected by Dehmelt and Kriiger [15], generally occur at radiofrequencies below those of the microwave region. For this reason we shall not discuss them further. Comprehensive treatments are given by Cohen and Reif [13], and by Das and Hahn [14]. The coupling values obtained are closely related to those measured by microwave spectroscopy of gases, and in Chapter XIV we shall discuss some of the solid-state results. Pure quadrupole resonance in solids is a simpler phenomenon than quadrupole coupling in gases, where the coupling gradient is a function of the rotational state. This is particularly true when there is more than one coupling nucleus in the molecule. In solids, the interaction of one nucleus does not perturb the field gradient at the other coupling nuclei, and the theory derived for a single coupling nucleus applies when there are other coupling nuclei in the same molecule. In contrast, plural coupling greatly complicates the problem in rotating molecules.

4 QUADRUPOLE INTERACTIONS BY A SINGLE COUPLING NUCLEUS IN A ROTATING MOLECULE

General Theory of the Interaction The quadrupole coupling of a nucleus in a rotating molecule of a gas is more complicated than that in frozen molecules of a solid because the field gradients depend upon the rotation. For an external-field-free molecule, the nuclear spin I is coupled to the molecular rotational angular momentum J to form a resultant F. In the vector model, J and I can be considered as precessing about F, as indicated in Fig. 9.3. The total angular momentum of the molecule with nuclear coupling is thus represented by F rather than J, which designates the total angular momentum exclusive of nuclear spin. However, in the first-order treatment, which is adequate for most coupling cases, J 2 is still a constant of the motion. The good quantum numbers are thus F , M F ,J, and I. The new angular momentum quantum numbers are F = J + Z, J+ I - 1, JS 1 - 2 , . . ., /J- I1

(9.30)

MF=F, F-1, F - 2 ,..., - F

(9.31)

and where J and I have the values assigned previously in Chapter 11. In units of h the eigenvalues of the square of the total angular momentum and its components along an axis in space are (F, MFIF’IF, MF)=F(F+ 1)

(9.32)

(F, MFlFZlF,

(9.33)

MF)=MF

Let us consider again the general expansion of (9.5) for the classical quadrupole interaction. By subtraction and addition of &V2I/ .f p,R; d7, to the right

400

NUCLEAR HYPERFINE STRUCTURE IN MOLECULAR ROTATIONAL SPECTRA F

Fig. 9.3 Vector diagram of coupling of a nuclear spin I with molecular rotational momentum J.

side, this expression can be put into the form:

(9.34) For reasons already given, Laplace's equation V2 V = O holds over the nuclear volume; hence the last term is vanishingly small and can be omitted. The remaining expression can be written as the scalar product (double-dot product) of two symmetric dyadics (9.35) where

vij=

-VEij and where the dyadic V E is the gradient of the electric field


401

of the extra nuclear charges (9.36) and Q is the quadrupole moment dyadic Q=

s

Pfl[3R,R,- OR:] dr,

(9.37)

Here R, represents the vector locating points of the nuclear volume in the space-fixed X, I: Z reference system. U=exex+eyey+ezez is the unit dyadic and e x , e y , and eZ are unit vectors along X, I: and Z. We now develop the quantum mechanical operator for the quadrupole interaction energy. The potential V at the nucleus arises from all the various extra nuclear charges of the molecule, but we need consider only the extra nuclear electrons since the charges of other nuclei are well screened. We can therefore express the potential as

(L)

V = Ck e Re

k

(9.38)

where (1/Re)krepresents the distance from the kth electron to the nucleus and where the summation is taken over all electrons contributiong to L! Let us substitute this value of V into the tensor K j . We consider only one component because the transformations of other components of the tensor are similar. Since

Thus it is evident that

(9.40) where ex is the angle between Re and the X axis fixed in space (see Fig. 9.1). The average is taken over the electronic orbitals and over the vibrational and rotational states of the molecule. However, ( ( l / R z ) k )is independent of the rotation and in a particular vibrational state may be treated as a constant. We can then write (9.41) Because the orbitals of all the electrons are fixed in the molecule and rotate with the molecular frame about the axis of J, the angular dependence of their averaged sum will be the same as that of J, and their resultant field gradient

402


will resolve along space-fixed axes in the same proportions as the components of J ; the rapid rotation about J will effectively average out the components of the field gradient perpendicular to J and make J an axis of symmetry for the gradient. From (9.41) one thus obtains (9.42) where Cis a proportionality constant to be evaluated, a constant which depends on the electronic distribution in the whole molecule but primarily in the atom of the particular coupling nucleus. In units of h, the eigenvalues of J 2 are J ( J + 11, and hence (9.42) can be written (VXX)OP

C [3J:- J(J+ l ) ] J(J+ 1 )

=-

(9.43)

The expressions for Vyyand Vzz are similar. Because the component operators do not commute, however, the analogous operators for the cross-product terms must be symmetrized. For example, (9.44) and the conjugate operator is (J!XY)OP

=-

c

(3 J ~ J ~ ; J ~ J ~ )

J(J+ 1 )

(9.45)

It is customary to evaluate the constant C by defining the coupling constant as that observed for the maximum projection of J along a space-fixed axis, that is, for the state MJ = J. Hence from the 22 component ‘J

‘J=(J,

JIVZZIJ ,

J

)

[3J2-J(J+l)]

=

J(J+ 1 )

p

(9.46)

and J+ 1 e=----

25-1 ”

(9.47)

Hence (Vzz)op=

‘5

J(2J- 1 )

[ 3 J i - J(J+ I ) ]

(9.48)

The generalized expression for the elements of the tensor operator of the field gradient is consequently

where i, j = X ,

E: or 2 and dij= 1 when i = j

and 0 when i #j.


403

Since the nuclear quadrupole tensor Qij, (9,37), has rotational symmetry Accordingly, about the spin axis, it can be treated in a manner similar to the nuclear quadrupole operator can be expressed as

xj.

(9.50) where C , is the proportionality constant evaluated by definition of Q as the value for maximum resolution along a space-fixed axis, that is, when M I = I. Thus eQ-(I, llQzzlZ, Z)=C,(Z, ll3Z;-Z(Z+I)lZ,

(9.51)

Z)=C,[3Z2-Z(Z+1)]

Therefore eQ c, =-4211)

(9.52)

and

Substitution of the operators of (9.49) and (9.53) into (9.35) yields the Hamiltonian operator for the quadrupole interaction Ye- 1

eqJQ (3 J(2J- l)Z(2Z- 1) i,j=x,y,z

zizji

Ijzi

+

Ji Jj JjJi

-6ijJ2 (9.54)

If the indicated multiplication is performed and the results expressed in terms of vector products, this Hamiltonian can be put into a more convenient form. Because the components of I commute with those of J , the term z i z j J i J j = c (ZiJi) i,j

i

=(ZxJx

(ZjJj) j

+ ZyJy + ZzJz)(ZxJx + Z y Jy + Zz J z ) = (1.5)’

(9.55)

The sum of all such terms is $(I. J)’. Because dij=0 when i f j , the terms

and (9.56)

404


Because the components of J and of I do not commute among themselves, that is, J i J j f J j J i and Z i Z j f Z j Z i when i f j , the commutation rules given in Chapter I1 must be applied for evaluation of the remaining terms which are of the form ZiZjJjJi.The transformations, which are rather lengthy, are given by Ramsey [ 161. The results are

1 IiZjJjJi=(I*J)’+ id

1.J

(9.57)

and the total of such terms is $[(I. J)2 + ( I . J)]. Collection of all these terms, with slight simplification, yields the quadrupole Hamiltonian in the compact form, 2 eQqJ [3(I. J)’+:I. Q-2J(2J- l)Z(2Z- 1)

J-12J2]

(9.58)

This Hamiltonian, originally derived by Casimer [3], is applicable to a coupling nucleus in any type of molecule, or to a free atom if J is treated as the total electronic angular momentum exclusive of nuclear spin. The quantity qJ depends on the particular type of molecule, whether linear, symmetric rotor, or asymmetric rotor. The advantage of the general Hamiltonian of (9.58) is that its eigenvalues for the terms in the bracket can be obtained easily for the field-free rotor in which J and I are coupled to form a resultant F, as indicated in Fig. 9.3. Since F is the vector sum of J and I,

F ’ = ( J + I)’=J2+21. J + J 2

(9.59)

and 1. J = i ( F 2 - J 2 -

12)

(9.60)

The eigenvalues of 1.J are therefore

( F , J , ZII’J)IF, J, Z)=t[F(F+l)-J(J+l)-Z(Z+l)]=9C

(9.61)

( F , J, Zi(I. J)’)F, J, Z)=iC2

(9.62)

and

Substitution of these values with J 2I 2=J(J+ 1)Z(Z+ 1) into (9.58) yields eQ9J E [?C(C+ 1)- J(J+ I)Z(Z+ I)] Q-2J(2J- l)Z(2Z- l )

(9.63)

where C=F(F+1)-J(J+l)-I(Z+l)

(9.64)

Let us now evaluate the quantity qJ.To do this we must first transform the field gradient to a reference system fixed in the molecule. Although later transformations will be made to special axes chosen to coincide with the bond axis to the coupling nucleus, the principal inertial axes a, b, c at this stage in the


405

development have the advantage of applying to molecules of all classes. The cross terms in the field gradient, such as (a2V/aaab), vanish when averaged Over the rotational state, as will be explained later. As defined previously,

where Z is fixed in space and where i represents any internal rotational quantum numbers such as K . In the inertial axes a, b, c,

)(ac )

+ 2 5 ( ab abac E

E

(9.66)

The quantities (9.67) are the direction cosines of the principal axes with the space-fixed 2 axis. The quantities (9.68) are the field gradients with reference to the principal inertial axes. They depend on the electronic state and to a slight extent on the vibrational state of the molecule, but to a high order of approximation they are independent of the molecular rotational state and can in the present evaluation be treated as constants. Thus

a2V

+q b b @ i b +q c c @ i c +2 q a b @ Z a @ Z b +2qac@Za@Zc +2qbc@Zb@Zc

y = q a a @ i a

az

(9.69)

To obtain qJ we must average the terms on the right side of (9.69) over the rotational wave function y y J = J The . factors qaa, q a b , . . . , are constant and can be removed from the integral. When averaged over the rotational state, the off-diagonal terms for an asymmetric rotor vanish, for example, (9.70) This results from the symmetry properties of the momenta1 ellipsoid (see Chapter 111). Either Y r must be symmetric (+) or antisymmetric ( - ) with respect to a

406


rotation of n degrees about a principal axis. In either case, however, the product ‘Pry,* will be positive for the same rotational function. It is evident that for a rotation of n about the a axis, mZawill be unchanged while mZbwill change sign only. Similar relationships apply for b and c. Thus the integrals of (9.70) must change in sign for at least two of the operations Ci, Ci, C.; However, these definite integrals cannot change sign for such operations as change the system into an indistinguishable one. They are therefore identically zero. The offdiagonal terms, however, are not zero in the second-order approximation. For linear and symmetric-top molecules the principal inertial axes are principal axes of the quadrupole coupling tensor, and hence the cross terms vanish automatically since qab, . . . vanish. Expressed in the principal inertial axes, qJ becomes qJ=qaa(J, i, MJ=Jl@&lJ, i, MJ=J)+qbb(J, i, MJ=Jl@,%blJ,i, M J = J )

+ qc,(J, i, M J =Jl@)%l J, i, M J = J )

(9.71)

where qaa,qbb, and qccare the molecular field gradients at the coupling nucleus with reference to the inertial axes. For brevity, the coupling constant eQq is often designated by X. With this designation the general expression for the quadrupole coupling energy is obtained by substitution of (9.71) into (9.63)

E,=

g = a,b,c

Xgg(J,i, M J= Jl@idJ, i, M J = J )

1

i C ( C + l)-J(J+ 1)1(1+ 1)

[

2J(2J- 1)1(21- 1)

(9.72) where Xaa=eQqaa,

Xbb=eQqbb,

XCC=eQqcc

(9.73)

Because of the relationship 2 1 (9.74) (J,ilJ,ZIJ, i)+(J+1)(2J+ 3) 2J+3 which exists between the diagonal matrix elements of @ig when M J = J and the diagonal matrix elements of J ; in the J , i, M J representation of the unperturbed rotor, (9.72) can be put in the alternative form

(J,i, M j = JI@iglJ, i, M J = J)=

The term ( 2 J t 3)-’ of (9.74) does not enter in this expression since we have used Laplace’s relation, (9.76). This expression of the quadrupole coupling energies, which was first obtained by Bragg and Golden [lo], is particularly useful in the calculation of the nuclear quadrupole perturbation in the asymmetric rotor. Equation 9.74 can be readily proved for the symmetric-top wave functions and hence for the asymmetric-top functions by comparison of the matrix elements derived from Table 2.1 (J,K , M=JI@;,IJ, K , M J = J ) with those for the symmetric top (J, KIJ,Z(J,K ) , which are given in Chapter 11.


407

Because Laplace’s equation holds, Xaa

+ Xbb f x c c = o

(9.76)

and there are only two independent coupling constants in the most general case. These are usually expressed in terms of the coupling constant with reference to one of the axes and an asymmetric parameter v]. If the reference axis is chosen as the c axis, the two coupling constants would be xCcand

(9.77) The reference axis is usually chosen as the one for which the coupling is the most nearly symmetric, that is, for which v] is smallest. The matrix elements of the squared direction cosines differ for the different classes of molecules. For diatomic or linear polyatomic molecules and for symmetric-top molecules they can be evaluated explicitly, as will be done below. For asymmetric rotors a closed-form evaluation can be made only for low J states, but values can be obtained by numerical techniques (especially with computers) for higher J values. It should be appreciated that the values of qJ as expressed in the preceding formulas are not exact, but are first-order perturbation values obtained when the field gradient is averaged over the wave functions of the unperturbed rotor. However, the off-diagonal elements that are omitted in the average are entirely negligible except for the coupling by the few nuclei that have large quadrupole moments. For them, the second-order corrections described in Section 6 are adequate.

Diatomic and Linear Polyatomic Molecules In a linear molecule, diatomic or polyatomic, the molecular field gradient is symmetric about the bond axis. Let us designate a molecule-fixed coordinate system x, y , z with z along the bond axis. Obviously, this system coincides with the principal axes of inertia with a+z. From the symmetry and from Laplace’s equation qxx

(9.78)

=q y y = -114 2 2

Because there is only one coupling axis it is customary to drop the subscript and to set qz2=q. Substitution of these relations into (9.71) gives 45 =q(J, MJ

= J1@&

-f[@.k+ @iy]l

J,

MJ = J ) (9.79)

The matrix elements of Oz2 for the linear molecule correspond to those of the symmetric top with K=O. Although the diagonal matrix elements of 0 for a linear molecule are zero, those of Q2 are not. From the matrix product rule, ( J , M J = Jl@i21J , M J = J ) = c ( J , JI@z21 J’, J)(J’, JI@zzIJ , J)=J’

1 2J+3

(9.80)

408


where the elements of the product are evaluated from Table 2.1 with K =O and M J = J. Substitution of (9.80) into (9.79) yields qJ q3=- 2J+3

(9.81)

The first-order quadrupole coupling energies for a single coupling nucleus in a linear molecule are found from combination of (9.63) and (9.81) or (9.72) and (9.80) to be

E,= -eQqY(J, I, F ) = -xY(J, Z, F )

(9.82)

where Y(J, Z, F)=

$C(C+ 1)- Z(Z+ 1 ) J ( J + 1) 2(2J- 1)(2J+ 3)1(2Z- 1)

(9.83)

For convenience in the calculation of hyperfine structure, numerical tabulations of Y(J, 1, F ) and of the relative intensities of the hyperfine components for J+J+ 1 transitions are given in Appendix I. Selection rules for hyperfine transitions in rotational absorption spectra are J+J+l,

F+F,

F+F+l,

I-tl

The rotational frequencies perturbed by quadrupole coupling are V=

E,(J+l)-E,(J) h

+E,(J+1,

Z,F’)-EQ(J, Z,F ) h

(9.84)

where F ‘ = F , Ff 1. It is customary to express the coupling constants in frequency units so that v=vo-eQq[Y(J+l,

Z, F ’ ) - Y(J, Z,F)]

(9.85)

where vo is the “unperturbed’ rotational frequency that would be observed if there were no quadrupole coupling. By measurement of only two frequencies of the hyperfine multiplet, ZeQq and vo may be obtained. For best accuracy one should measure the most widely spaced components provided that they are sufficiently strong for precise measurement. As a test of the accuracy, other components are usually measured. It is evident that wherever the rotational transitions have resolved hyperfine structure one must measure eQq in order to obtain vo and hence the rotational constant Bo described in Chapters IV and V. Figure 9.4 illustrates the calculation of the frequencies of the hyperfine components caused by l Z 7 I in the J = 7 + 8 rotational transition of lZ7ICN. The spin of ”’1 is 3, and the total number of components for all transitions for which J > Z is 15. However, it will be noted that only the six F+F+ 1 components are of significant strength. Figure 9.5 shows these six F+F 1 components as they appear on the cathode ray scope. The splitting caused by 14N (I= 1) is not resolvable for this J transition. For transitions where J > Z, the variation in the hyperfine pattern is rather uniform with increasing J. The F+F- 1 components become weaker and

x

+

J

F

1112 21 I 2

a

1312 1912 1512 1712

912 1912

7 1112 1712 1312 1512

Fig. 9.4 Energy level diagram and calculated nuclear quadrupole hyperfine structure for the 5 = 7 4 rotational transition of a linear molecule with a single coupling nucleus having a spin of$. All F+F- 1 components are less than one-thousandth the strength of the strongest component and hence are omitted. Here e Q q is assumed to be -2400 MHz, as it is for ICN in Fig. 9.5.

Fig. 9.5 The F + F + 1 components of the J=7+8 transition of ICN observed at 5.87 mm wavelength (compare with the rheoretical components of Fig. 9.4). The observed components are due to 12’1 (spin 4,eQq= -2400 MHz). The substructure caused by 14N (spin 1, e Q q = -3.8 MHz) is not resolved. 409

410


eventually undetectable; the F+F + 1 components remain strong but converge in frequency and eventually become unresolvable as J continues to increase. Thus for high J values only a single line is observable except when the coupling is very large. When J > 1, there are 21+ 1 components for F + F + 1,2Zcomponents for F+F, and 21- 1 components for F+F- 1. However, for the J=O+1 transition, there are only three hyperfine components whatever the value of I. These variations are illustrated in Fig. 9.6 for a coupling nucleus with I=+. Although there are four F+F+1 transitions expected for Z=$ when J > 1, two pairs of the four lines coincide so that only two F+F+ 1 lines are observable. Table 9.1 illustrates some of the nuclear quadrupole coupling constants of linear molecules that have been measured from analysis of rotational hyperfine structure. Note the rather large differences in the I4N coupling of the related molecules FCN, CICN, and BrCN. Also note the large difference between the

Spin I = p

J=O-I

1 ,

I

+ J =2+3

* 5.4-5

-0.40 -0.20

0

0.20

040

vu

eQq

Fig. 9.6 Theoretical patterns of the first five rotational transitions of a linear molecule resulting from quadrupole interactions by a single nucleus with spin I =.:

Table 9.1 Nuclear Quadrupole Coupling in Some Diatomic and Linear Polyatomic Molecules in the Ground Vibrational State Molecule

D coupling DBr DCN DCP DCCF

eQq ( M H z )

Ref:

l4

0.1469(14) 0.1944(25) 0.233(40) 0.212(10)

cs

b

SiS

C

GeS

d

4.5(4) - 3.71(3)

- 5.2031(5)

4

- 4.7091(15) - 2.67(5)

b

f

ocs

12.83(3) 10.90(20) 6.96(30) -29.07(1)

m

n n 0

0.245(5) - 4.28(5) 4.48(10) - 1.32(7)

HCI ClCn HCCCl

-67.80(10) - 83.2752(4) - 79.67

P

4

,

19Br coupling

h I

j

''0 coupling

co ocs

Refi

CI coupling e

N coupling

PN HCN FCN HCNO HCCCN

eQq ( M H z )

33Scoupling a

' 'B coupling BF HBS

Molecule

k 1

DBr BrCN HCCBr

530.65(7) 685.6(4) 648.10(2)

P 4 5

'" I coupling HI ICN HCCI

- 1828.42(20)

P

- 2418.8(5)

f

- 2250.6(55)

U

"B. P. Van Eijck, J. Mol. Spectrosc., 53, 246 (1974). *F.C. De Lucia and W. Gordy, Phys. Rev., 187, 58 (1969). 'S. L. Harford, W. C. Allen, C. L. Allen, C . L. Harris, E. F. Pearson, and W. H. Flygare, Chem. Phys. Lett., 18, 153 (1973). *V. W. Weiss and W. H. Flygare, J. Chem. Phys., 4 5 8 (1966). 'F. J. Lovas and D. R. Johnson, J. Chem. Phys., 55,41 (1971). 'E. F. Pearson, C. L. Norris, and W. H. Flygare, J. Chem. Phys., 60,1761 (1974). uJ. Raymonda and W. Klemperer, J. Chem. Phys., 55, 232 (1971). hJ. Sheridan, J. K. Tyler, E. E. Aynsley, R. E. Dodd, and R. Little, Nature, 185, 96 (1960). 'W. Hiittner, H. K. Bodenseh, P. Nowicki, and K. Morgenstern, J. Mol. Spectrosc., 71, 246 (1978). 'J. K. Tyler and J. Sheridan, Trans. Faraday Soc., 59,2661 (1963). 'W. H. Flygare and V. W. Weiss, J. Chem. Phys., 45,2785 (1966). IS. Geschwind, R. Gunther-Mohr, and J. Silvey, Phys. Rev., 85,474 (1952). "M. Bogey, C. Demuynck, and J. L. Destombes, Chem. Phys. Lett., 81, 256 (1981). "J. Hoeft, F. J. Lovas, E. Tiemann, and T. Torring, J. Chem. Phys., 53, 2736 (1970). "C. H. Townes and S. Geschwind, Phys. Rev., 74,626 (1948). PF. C. De Lucia, P. Helminger, and W. Gordy, Phys. Rev., A3, 1849 (1971). 4J. M. L. J. Reinartz, W. L. Meerts, A. Dymanus, Chem. Phys., 45, 387 (1980). 'H.Jones, M. Takami, and J. Sheridan, Z . Naturforsch., 33a, 156 (1978). 'A. P. Porter and P. D. Godfrey, J. Mol. Spectrosc., 68,492 (1977). 'T. Oka, H. Hirakawa and A. Miyahara, J. Phys. SOC.Japan, 12, 39 (1957). "E. Schafer and J. J. Christiansen, J. Mol. Struct., 97, 101 (1983). 41 1

41 2


coupling of the middle nitrogen and that of the end nitrogen in NNO, as well as that between the 33Scoupling in CS and in OCS. It is evident that the nuclear quadrupole coupling depends upon the nature of the chemical bonding to the coupling atom. This relationship is treated in Chapter XIV. Symmetric-top Molecules The most commonly studied symmetric-top hyperfine structure arises from a single coupling nucleus on the symmetry axis. A single coupling isotope, or two such isotopes which are off the axis, would destroy the symmetry and convert the molecule into an asymmetric rotor. It is possible, however, to have three off-axis coupling isotopes, for example, 35Cl in 35C13CH,in a symmetric top. At this point we consider the simpler case of a single coupling nucleus that is on the symmetry axis. The symmetry axis will, of course, be a principal axis of inertia, either a (prolate top) or c (oblate top). To include both cases we designate this axis as z and the other two body-fixed axes as x and y . With (9.78), which also holds for symmetric tops, (9.71) becomes (9.86) From the matrix elements given in Table 2.1

(J,K, Mj=JIO&IJ, K, M j = J ) = ( J , K, JIO,zzIJ,K , J ) 2 +(J, K, JIOzzl J + 1, K, J)’+(J, K , JIOzzI J - 1, K , J)’ -- K 2 (J+ 1)2

+

(J+1)2-K2 (J+1)’(2J+ 3)

(9.87)

Substitution of these values into (9.86) gives

[

qj=q_3K2 __- 11 2 J + 3 J(J+l)

(9.88)

The resulting formula for the quadrupole energy of the symmetric top is, from (9.631,

E Q = e Q q3K2 [ m ) -1

]

Y ( J ,Z , F )

(9.89)

where Y ( J , Z,F) is the function of Eq. (9.83), which is numerically tabulated in Appendix I, and (9.90) is the coupling constant with reference to the molecular axis of symmetry. SelectionrulesareAJ= & I ; AK=O;AF=O+l;AZ=O. It is evident from examination of (9.89) that each K component of a J+ J+ 1 rotational transition of a symmetric top will have its own hyperfine multiplet.

413


For K =O the equation reduces to that for the linear molecule, and the pattern for this component alone will be similar to those illustrated in Figs. 9.4-9.6. The splitting of other K components differs by a scale factor only and hence individually resembles the patterns for the linear molecule. When, as is often true, the separation of the K components by centrifugal stretching is less than the quadrupole splitting, the multiplets for the different K components are scrambled so that they produce a rather complicated spectrum, such as is illustrated in Fig. 9.7. Comparison of these various patterns was used for proof that the nuclear spin of radioactive 1291is 3. The slight disagreement between the observed and calculated patterns for Z=$ are due to second-order effects, described in Section 6, which were not included in the calculations. Since the spin of the stable isotope "'I is$, the hyperfine structure of the J = 2 + 3 transition of normal methyl iodide is like that for the theoretical pattern for 5 I=z. Selected values of quadrupole couplings in symmetric-top molecules as obtained from microwave spectroscopy are given in Table 9.2.

Asymmetric Rotors The general expression for nuclear quadrupole interaction, (9.72), applies to all classes of molecules. By multiplication and division of this equation by J/(2J+ 3) it can be expressed as

In the asymmetric-top representation,

The function Y ( J , I, F ) is expressed by (9.83) and is tabulated in Appendix I. Except for low J values, the matrix elements of the squared direction cosines in (9.92) cannot be expressed in closed form; hence qJ cannot be expressed in closed form. Because the line intensities depend on the squared matrix elements, the problem of calculation of qJ can, however, be reduced to that already encountered in the calculation of line intensities. For example, from the matrix product rule (JK-IKI,

J)@ialJK-lK1,

J)=

1 I(JK-~K~,

M J = J I @ Z ~ I J ' K ,MJ=J)I' -~K~,

JK'-,K;

(9.93) If the dipole moment is along the principal axis a, the line strength of the transitions ( J K- I K I , MJ = J)+(JkS-,K;,M J = J ) is proportional to the squared matrix elements of this expression. By thus relating the diagonal matrix elements of

Observed spectrum CH3”’1 J=2+3

iI

I

I

Lu

I IIL

45,000

4 5 1 00

Calculated 5

I=p

Ca Icu lated 3

I=

7

I

I

I

1111

I

I

I

Fig. 9.7 Observed nuclear quadrupole hyperfine pattern for the J = 2 + 3 rotational transition of CH, 1291 (top figure) compared with the patterns predicted with first-order theorj for various nuclear spin values assumed for the iodine (lower figures). The close similarity of the two top patterns proved that the spin of 1291 is -$.From R. Livingston, 0. R. Gilliam, and W. Gordy, Phps. Rev., 76, 149 (1949).

41 4

Table 9.2 Nuclear Quadrupole Coupling in Some Symmetric-top Molecules in the Ground Vibrational State

Molecule

eQq (MfW

T I coupling

D coupling CH3D CF3D CD3CN

0.192 0.17 l(2) 0.168(4)

a

b C

' 'B coupling BH3CO CH3BS

1.60(10) - 3.714(20)

d

- 4.0842(3) - 7.07(10) - 4.2244( 15) - 0.4885(10) -4.77(3)

f

e

I4N coupling NH3 NF3 CH3CN CH3NC SiH3CN

B C

-93.0(1) 32.5

+

- 108(3) - 74.77(10) - 40.0 - 46.95(2) - 41.6(3)

i

m n 0

P

"Br coupling CH3Br SiH3Br GeH3Br SnH3Br

577.1 5( 10) 334.970(6) 384(2) 350(6)

m 4 i

5

h I

' Ge coupling GeH3F GeH,CCH

NC13 CH3CI SiH3CI GeH3C1 SnH3CI

i k

" I coupling CH31 SiH31 GeH31 SnH31

- 1940.41(7) - 1240(30) - 1381(4) - 1273(8)

f U i

li

"S.C. Wofsy, J. S. Muenter, and W. Klemperer, J. Chem. Phys., 53,4005 (1970). *S.G. Kukolich, A. C. Nelson, and D. J. Ruben, J. Mol. Spectrosc., 40,33 (1971). ' S . G. Kukolich, D. J. Ruben, J. H. S . Wang, and J. R. Williams, J. Chem. Phys., 58,3155 (1973). *A. C. Venkatachar, R. C. Taylor, and R. L. Kuczkowski, J. Mol. Struct., 38,17 (1977). 'C. Kirby and H. W. Kroto, J. Mol. Spectrosc., 83, 1 (1980). IGunther-Mohr et al. [41]. #J. Sheridan and W. Gordy, Phys. Rev., 79,513 (1950). hS. G. Kukolich, Chem. Phys. Lett., 10, 52 (1971). 'A. J. Careless and H. W. Kroto, J. Mol. Spectrosc., 57, 198 (1975). 'L. C. Krisher, J. A. Morrison, and W. A. Watson, J. Chem. Phys., 57, 1357 (1972). 'E. C. Thomas and V. C. Laurie, J. Chem. Phys., 44,2602 (1966). 'G. Cazzoli, P. G. Favero, and A. Dal Borgo, J. Mol. Spectrosc., 50, 82 (1974). "J. Kraitchman and B. P. Dailey, J. Chem. Phys., 22, 1477 (1954). "A. H. Sharbaugh, Phys. Rev., 74, 1870 (1948). OS. Geshwind, R. Gunther-Mohr, and C. H. Townes, Phys. Rev., 81,288 (1951). pL. C. Krisher, R. A. Gsell, and J. M. Bellama, J. Chem. Phys., 54, 2287 (1971). 4K. F. Dossel and D. H. Sutter, Z. Nuturforsch., 32a, 1444 (1977). 'S. N. Wolf and L. C. Krisher, J. Chem. Phys., 56, 1040 (1971). ' S . N. Wolf, L. C. Krisher, and R. A. Gsell, J. Chem. Phys., 54, 4605 (1971). 'A. Dubrulle, J. Burie, D. Boucher, F. Herlemont, and J. Demaison, J. Mol. Spectrosc., 88, 394 (1981). "A. H. Sharbaugh, G. A. Heath, L. F. Thomas, and J. Sheridan, Nature, 171, 87 (1953). "S. N. Wolf, L. C. Krisher, and R. A. Gsell, J. Chem. Phys., 55,2106 (1971).

41 5

41 6


mig to those of the transition dipole moments, Bragg [9] has derived the expression

+qcc&(J, - I K l ; JK'4;)I

(9.94)

in which the A's are the line strength factors for the transitions JK -lKl-+JKI-IK tabulated by Cross et al. [17]. Although these tables include only values for line strengths of molecules having an asymmetry parameter K of T1, T0.5, 0, + O S , and + l , these can be interpolated to give reasonably approximate values of qJ for other degrees of asymmetry. More extensive tabulations are referenced in Chapter VII. A second, and perhaps more useful, expression of qJ in terms of the reduced energies and Ray's asymmetry parameter K has been derived by Bragg and Golden [lo]. This derivation is based upon the relationship between the matrix elements of the direction cosines and the angular momentum expressed by (9.74), which allows qJ to be expressed by (9.95) where the asymmetric-top representation is used for the average values

(Ji)=(J, K - 1 , KiIJilJ,

K-1,

Ki)

(9.96)

As discussed in Chapter VII,

J(J+ ~ ) + E ( K ) - ( K +1) -

(9.97) (9.98) (9.99)

which lead to an expression for the quadrupole energy in the form

in which E(K) is the reduced energy of the asymmetric rotor described in Chapter VI1 and Ray's parameter is K=

2B-A-C A-C

(9.101)


417

where A , B, and C are the rotational constants with regard to the respective a, b, and c axes. The reduced energies for various values of K have been tabulated (see Chapter VII). For low J values, for which nuclear quadrupole coupling is best resolved, explicit expressions for E ( K )have been obtained. Selection rules &=O, 1 and AZ=O apply with those given for J K - l K l in Chapter VII. We illustrate the application of (9.100) with a simple example, the l o l + l l o transition of H232S.For the l o l level:

and For the l l olevel:

and For both levels:

The resulting energy levels, allowed transitions, and predicted lines are indicated in Fig. 9.8. Comparison of theoretical with observed frequencies [181 leads to the values

xaa= - 32 MHz,

Xbb =

- 8 MHz,

xcc=40 MHz,

V O = 168,322.63 MHz

and hence the asymmetry parameter of the quadrupole coupling is Xaa

q =---

- Xbb = -0.60

xcc

For slightly asymmetric rotors, (9.100)can be expressed in a more convenient form by the use of expansions for W(b)given in Chapter VII. The resulting expression for the nearly prolate or oblate symmetric top is

+q(C1 +2C2b+3C3bZ+4Cqb3)]Y(J,

Z, F )

(9.106)

where the b’s are the inertial parameters defined in Chapter VII, Section 3 and where the C’s are parameters given in Table 7.8. For a prolate rotor b=b,,

418


Ei#F

F

JK-~K,

1/ 2

'lo

-0.05ha

5/2

101

+O.20ha

312

Ea

+0.20&

-0.05~~ -0.25~~

II

I

l

l

I

II II

I

l

l

I

l

l

I I

I

I

I

-0.25ha '

1

/

2

Fig. 9.8 Energy level diagram and predicted nuclear quadrupole hyperfine structure caused by 33S(Z=$)in the l o l - + l l orotational transition of H233S.

x=xaa and ~ = ( x ~ ~ - x ~whereas ~ ) / x ~for~ an , oblate rotor b=b,, x=xcc, and q =(Xaa-Xbb)/xcc. Although the diagonal elements zoo,X b b , and xcCof the coupling with reference to the principal inertial axes are the coupling constants directly observable from the rotational hyperfine structure, they are not necessarily the most convenient for interpretation of the quadrupole coupling in terms of the properties of chemical bonds. While the cross terms x a b , . . ., are averaged out by the relation [see (9.70)], the principal inertial axes are not generally principal axes of the nuclear coupling in the tensor, even though they may be in special cases, such as linear molecules. Let us therefore transform zoo,X b b , and xcc to the principal axes of the quadrupole coupling tensor which we shall designate by x, y , z. Since eQ is constant, we need only to transform q.

with similar expressions for q b b and qcc.Cross terms such as (a2V / a x d y ) are zero because x, y , z are principal axes of the field gradient. Now dX

-=cos aa

ox,, .

*

Next Page QUADRUPOLE INTERACTIONS BY A SINGLE COUPLING NUCLEUS

419

and xx=eQqxx,. . .

(9.108)

x,, =xx cos2 ex,+ x, cos2 e,, + xr cos2 e,,

(9.109)

Therefore

+ cos’ + x z cos’ xcc=1, cos’ e,, + x, cos2 e,, + x, cos2 e,,

xbb =xx

cos2 e x b

xy

eyb

ezb

(9.110) (9.111)

The difficulty in the solution for the principal elements is apparent from these equations. In the general case where the angles as well as the magnitudes x x , x,, and xz are unknown, there are too few equations linking these unknowns with observable quantities. To diagonalize the tensor in the usual way we must also know the off-diagonal elements x a b , x,,, and so on. However, these offdiagonal elements are not directly observable in the first-order quadrupole hyperfine structure usually measured. For nuclei with large quadrupole coupling it is sometimes possible to obtain very approximate values of off-diagonal elements from measurement of the second-order displacements in the hyperfine component (Section 6), but only diagonal elements in the inertial system are usually measured. Devious methods for use of these diagonal elements to give information about the principal elements are described later. In some molecules the directions of one or more of the principal axes in the coupling can be ascertained from a consideration of the symmetry of the electronic structure around the coupling nucleus. Then, with the molecular structure and the inertial axes a, b, and c known from analysis of the rotational spectra, the angles 6 can be found; hence the coupling constants x,, x,, and xz can be obtained from the measured quantities xaa,x b b , and x,, by solution of (9.109)(9.111). In some cases the coupling has axial symmetry about a bond axis or other axis. If z is chosen as such an axis of symmetry, xx=xy = -fx,. There is then only one independent coupling constant, and (9,109)-(9.111) can be put in the form Xaa=fXz(3 COS’ e,=- 1) (9.112) Xbb=iXz(3 cos2 e z b - 1)

(9.113)

ezc- 1)

(9.114)

Xcc =+XZ(~ COS’

where xz represents the coupling constant with reference to the symmetry axis. In a molecule having a plane of symmetry, the axis perpendicular to the plane will be a principal axis of inertia and also a principal axis of the field gradient or coupling. We assume that this is the c axis of the inertial moment and that it coincides with the principal axis y of the field gradient. Then (9.109)(9.1 11) reduce to the form (9.115) (9.116)

Previous Page Table 9.3 Nuclear Quadrupole Coupling in Some Asymmetric-top Molecules in the G r o u n d Vibrational State Coupling Constants ( M H z ) Inertial A x e s

Bond Axis System xxx

xaa

Molecule

Xbb

XYY

xcc

XI2

Orientation of x, y , z Axes

Ref:

D Coupling 0.2765(9) - 0.11lO(15)

HOD

-0.1658(11) CH 3 0 D

-0.1658 -0.1477 0.3132 x,,=0.303(12) l4

- 1.51

S(CN12

CH3C

0.30 1.21

4

0

\ CN

-4.34(7) 2.03(31) 2.31(31)

- 4.88(4) 1.43(3) 3.45(2)

Pyridine

z along OD bond z along OD bond

N Coupling 2.27 1.21 - 3.48

b

C

2.06 2.31 -4.37 1.43(3) 3.45(2) - 4.88(4)

LI

d

See Fig. 14.4 for coordinate directions

"0 Coupling

xaa= - 1.90(12) v = - 12.0(8)

H2CO HZC-CHI

9

12.6(1)

\ /

x I to plane h

- 5.2(1)

- 7.4(1)

0

z

II to cc

35CI Coupling

- 41.07(1) CH2C12

1.24(2) 39.83(1)

CH2FCl CH3CH2CI

0

4 CH3C-CI 420

- 76.92(2)

37.09(2) 39.83(1) 31.6(10) 38.8(10) - 70.5(10) xaa-= - 49.20(10) zsr= -68.80(15) v =0.4479(34) v =0.035(3) - 58.0(10) - 59.2 37.6 36.4 21.6 21.6

x along C-CI bond y in CI-C-CI plane

I tox z along C-CI bond y I to molecular plane z along C-CI bond = 260

,

J

k

e,,

x along C-CI bond z I to C-0-CI plane eax=6-28'

1

Table 9.3 Continued 35CI Coupling (continued) xrz= -78.7(10) t7=0.12(1) - 56.64(40) 36.72(22) xZ,= -71.40(51) i9.92(22) =o.o29(ii)

CH2=CC12 CH2-CHCI

\ / CH2

z along C-CI b o n d

m

x Ito molecular plane

"

z along C-CI b o n d ea,=2i.90

79BrCoupling Br2C0

CH,-CHBr

3 16.9(21) - 83.5(17) -233.4(11) 472.1 - 220.7 -251.4

- 229.4 - 397.1 626.5 -251.4 - 307.5 558.9

C6H,Br

x..

CH3CH2I

- 1478.1l(7) 564.46(8)

893(10) 878(50)

91 3.65(7)

- 1771(10)

= 567(4)

xrr =567(4) 11= - 0.049(20)

e.,

= 18.450

z along C-Br b o n d

4

2'1 Coupling z along C-I bond

1,5

X a b = 896.4(5)

CH,=CHI

- 1656(10) 770(10) - 654(10) X o b = - 765

886 99 1 - 1877

z along C-I b o n d x Ito molecular plane

'3"

"H.A. Fry and S. G. Kukolich, J. Chem. Phys., 76,4387 (1982). bK.H. Casleton and S. G. Kukolich, Chem. Phys. Lett., 22, 331 (1973). 'L. Pierce, R. Nelson, and C. Thomas, J. Chem. Phys., 43, 3423 (1965). dF. Scappini and H. Dreizler, Z. Naturforsch., 3111, 840 (1976). 'G. 0. Sorensen, J. Mol. Spectrosc., 22,325 (1967). 'G. 0. Sorensen, L. Mahler, N. Rastrup-Andersen, J. Mol. Struct., 20, 119 (1974). gR. Cornet, B. M. Landsberg, and G. Winnewisser, J. Mol. Spectrosc.,82,253 (1980). hR. A. Creswell and R. H. Schwendeman, Chem. Phys. Lett., 27, 521 (1974). 'W. H. Flygare and W. D. Gwinn, J. Chem. Phys., 36,787 (1962). IN. Muller, J. Am. Chem. SOC.,75, 860 (1953). 'R. H. Schwendeman and G. D. Jacobs, J. Chem. Phys., 36,1245 (1962). 'K. M. Sinnott, J. Chem. Phys., 34,851 (1961). "S. Sekino and T. Nishikawa, J. Phys. SOC.Japan, 12,43 (1957). "R. H. Schwendeman, G. D. Jacobs, and T. M. Krigas, J. Chem. Phys., 40, 1022 (1964). "J. H. Carpenter, J. G. Smith, I. Thompson, and D. H. Whiffen, Chem. SOC.Faraday Trans., 2,73, 384 (1977). pJ. Maroor and M. de Hemptinne, Bull. Acad. R. Belg., 58,956 (1972). 4E.Rosenthal and B. A. Dailey, J. Chem. Phys., 43,2093 (1965). 'T. Kasuya and T. Oka, J. Phys. SOC.Japan, 15,296 (1960). 'D. Boucher, A. Dubrulle, and J. Damaison, J. Mol. Spectrosc., 84,375 (1980). 'H. W. Morgan and J. H. Goldstein, J. Chem. Phys., 22, 1427 (1954). "C. D. Cornwell and R. L. Poynter, J. Chem. Phys., 22, 1257 (1954).

421

422

NUCLEAR HYPERFINE STRUCTURE IN MOLECULAR ROTATIONAL SPECTRA xcc

(9.117)

=x y

where Oza is the angle between the a and z axes. From the first two of these equations, X X

=

x,,sin’ O z a - x b b cos’ OZa sin’ O,,-COS~ O,,

(9.118)

I,, cos’ O,, - x b b sin’ O,, cos’ OzO- sin’ O,,

(9.119)

xz =

When the direction of one principal axis of the field gradient is known, even if the molecule does not have a symmetry plane, the coupling equations can be reduced by rotation of the coordinates a, b, and c to forms which have only

one unknown angle. Nevertheless, there will remain four unknown quantities and only three independent equations. An important way of providing the additional equations needed in solutions for the principal values of the quadrupole coupling tensor is isotopic substitution, which shifts the axes a, b, and c to new ones a‘, b‘, and c‘ that produce different values of the measurable couplings x,,,,, X b ’ b ‘ , and xcSc,. If the isotopic substitution is made in some nucleus other than the coupling one, there will be no change in the principal values xX, xy, and x,. If the principal quadrupole axes system is transformed (by a rotation of coordinates) to either the parent or isotopic principal inertial axis system, the coupling tensor will have off-diagonal elements. If the coupling nucleus is substituted, no new unknown couplings will be introduced when the ratios p of the quadrupole moments of the substituted isotopes are known. As an illustration, consider an isotopic substitution of an atom which is in the plane of symmetry, where (9.115)-(9.117) apply. This substitution will change neither the coupling axes nor the inertial axis c which is perpendicular to the symmetry plane, but will alter the direction of the inertial axes a and b relative to the coupling axes x and z and hence xafa.,X b ’ b ’ , and o,,,. We indicate the new values by primes and set x : = p x X , x:=pxz when the substitution is in the coupling nucleus. Thus

Xb’b‘

Since x,,,, and

Xb’b’

sin’ O,,,+px, cos’ O,,,

(9.120)

=pxx cos’ Oza, +pxz sin’ O,,,

(9.121)

are measurable quantities and since the quadrupole ratio

p is presumed to be known, only one new unknown quantity, O,,,, is introduced,

whereas two new equations are provided. Thus the four equations (9.115), (9.116), (9.120),and (9.121)have only four unknowns and are therefore solvable. Also, A0 =O,,, - O,, can be obtained from the change in the principal moments of inertia AZ, and AZb which was caused by the isotopic substitution. Diagonal elements of nuclear quadrupole coupling constants with reference to the inertial axes a, b, and c have been measured for numerous asymmetrictop molecules. Selected ones are given in Table 9.3. A complete tabulation of

QUADRUPOLE COUPLING BY MORE THAN ONE NUCLEUS

423

measurements of these elements up to 1982 may be found in the LandoltBornstein tables [19]. For certain of these, principal elements of the coupling tensors have also been found.

5 QUADRUPOLE COUPLING BY MORE THAN ONE NUCLEUS IN A ROTATING MOLECULE If there are two or more nuclei with quadrupole coupling in a molecule, its rotational hyperfine structure is generally quite complicated. The reason for this complexity is easy to see. The coupling of each nucleus perturbs the rotational axes and alters the averaged field gradient which interacts with all the other coupling nuclei. One cannot use the wave function Yr of the rigid rotor to average the quantity (a2V/dZ2),as was done for a single coupling nucleus, but must use the functions for the perturbed state to find the correct average of the field gradient along Z . Since the complexity in the quadrupole couplings by plural nuclei arises through their perturbations of Y r ,it is obvious that no such complexity arises for solid state resonances. In frozen molecules the quadrupole interaction of one nucleus is virtually independent of that of all others. Thus when there are two or more coupling nuclei in the same molecule, the coupling constants x are most simply obtained through measurement of pure quadrupole resonance in the solid state. However, the coupling constants of gaseous molecules and of frozen molecules are not exactly the same, and it is often an advantage to have measurements of both. When the coupling of one nucleus is large as compared with that of a second, the problem of a dual nuclear coupling in a rotating molecule is not very formidable, as we shall explain later. Also, when an experimentalist is better at chemistry than at mathematics, he or she may often circumvent the complexity in the gaseous state problem by substitution of noncoupling isotopes for all except one of the nuclei in a given molecule. Fortunately, nature has already done this in many isotopic species of common molecules.

Coupling by Two Nuclei The theory of quadrupole interaction by two nuclei in a rotating molecule has been worked out by Bardeen and Townes [20], who give formulas for analysis of the hyperfine structure for certain spin combinations. The simplest case occurs when the coupling by one nucleus is large as compared with that of the other. When 2, &10312, the spin I, will in the firstorder approximation couple to J to form a resultant F, about which they both precess. The spin of the second nucleus I2 will then couple with F, to form a resultant F about which they both precess. This coupling case is indicated by the vector diagram of Fig. 9.9. For this vector model to hold, the precession of J about F1 must be sufficiently rapid as compared with the precession of I2 that the components of the gradient at nucleus 2 which are normal to F1 average out, leaving an effective component only along F1. In this model

424


1

Fig. 9.9 Vector diagram of the nuclear quadrupole interaction by two nuclei in a molecule with the coupling of I, assumed to be stronger than that of I,.

F, F,, J, 11, and I, are constants of the motion. The associated quantum numbers are F1

and

=J+

I1,

F = F1 + I,,

J + I , - 1, J + I1 -2, . . ., 1 J - Ill

F1

+

12

- 1, F 1 + 21 - 2, . . .) I F1 - I21

(9.122)

Suppose that XQlrepresents the Hamiltonian operator for the interaction energy of I I and J, and that XQ2represents the operator for the interaction of I, and J. The coupling scheme F, = J + I l and F = F , + I2 generates a set of functions YF1,Ffor the combined system XQl + These are eigenfunctions of Yipo, ; if the effect of XQ2is negligible, states specified by a given F1 and different values of F will be degenerate. This degeneracy will, however, be removed if XQ2is not negligible. Likewise, the coupling scheme F, =J + I, and F=F, + I I generates a set of functions YF2,F for the combined system

QUADRUPOLE COUPLING BY MORE THAN ONE NUCLEUS

425

which are eigenfunctions of XQ,.These two sets of basis functions are related yF2,F=C Fi

C(F2, F1)yF1,F

or

yF1,F=c F2

C(F29 F1)yF2,F

(9.123)

where the transformation coefficients are chosen to be real. With the assumed coupling scheme of Fig. 9.9, the eigenvalues of XQlare (F1lXQl,lF1)=EQl(F,), the same as those already obtained in Section 4 for the coupling by a single nucleus. The allowed values of XQ2treated as a perturbation on XQlare

(F11XQ21F1)=1 [C(F2, F1)12(F2\xQ21F2)=x [C(F2, F1)I2EQ2(F2)

(9.124)

The combined quadrupole coupling energy values to first-order are EQ=EQ,(Fl)-k 1[c(F2, F1)I2EQ2(F2)

(9.125)

Fz

The values of E Q , ( F 1 and ) of EQ2(F2)are obtained by substitution of F1 and F 2 , respectively, for F in the formulas for a single nucleus (Section 4). The selection rules for Fi are AF1=0, & 1 and the selection rules for F are AF =0, 1. The weighting Coefficients C(F2, F1) for a few spin combinations (Zl=Z, Z 2 = 1 and 3)are given by Bardeen and Townes [20] and reproduced by Townes and Schawlow [21]. Transformqtion coefficients relating the two schemes of coupling three angular momenta find extensive use in quantum mechanical problems. In terms of the so-called 6j symbol, which have been tabulated for spin values up to 8 (discussed nex!), the foregoing coefficients are expressed as C(F2, F , ) = ( -

l ) J + F + 1 1 + 1 2 [(2F1+

+

1)(2Fz

1)]”2

{;: ;E;}

(9.126)

To obtain the energy values for the intermediate case where the couplings of the two nuclei are of comparable magnitude and for accidental degeneracies in the strong-weak coupling described previously, one must set up and solve the secular equation. Treatment of these more involved cases is given in the original paper by Bardeen and Townes [20]. The theory has been extended and applied to asymmetric rotors by Myers and Gwinn [22], also by Robinson and Cornwell [23].

Coupling by Three or More Nuclei Quadrupole coupling by three nuclei in a molecule can produce a very complex hyperfine structure of rotational spectra. Matrix elements of the have been worked out by Bersohn combined Hamiltonian XQ,+ &’Q2 [24]. Using his results, Mizushima and Ito [25] have calculated the expected theoretical pattern for the J=O+1 transition for spin values of 1, 3,2, and 3. In only a few instances of symmetric-top molecules having three identical symmetrically placed coupling nuclei, including CHC13 and CFC13, have attempts been made at application of the theory to observed hyperfine patterns

426


[26-281. In these efforts some inconsistencies have been encountered with transitions involving K = O because of failure to take into account off-diagonal elements caused by contributions from K = + 1 and K = - 1 states in the averaging of ( a 2 V / d Z 2 )This . difficulty was overcome by Wolf et al. [28] who made use of a procedure developed by Svidzinskii [29] from group theory for calculation of the hyperfine interactions in symmetrical molecules. A description of the rather complex theory involved in the calculations is given by Wolf et al. [28]. They also provide specific formulas and tables to aid in the calculation of the hyperfine structure for the case of three identical nuclei with spin I=: in symmetric-top molecules. They compared their theoretical calculation with the observed hyperfine patterns for the J = 1 + 2 and 2-3 transitions for CH3'C13 and CFC13 and reported good agreement. The derived bond axial coupling constants of 35Cl, -77.9k0.05 MHz in CHCI3 and -79.9k0.6 MHz in CFC13, are close to the solid-state values, - 76.98 and - 79.63 MHz, observed by Livingston [30]. The case of three identical nuclei with spin I= 1 has been treated theoretically by Hadley [31] and observed experimentally for ND3 by Hermann [32]. The general case for any number of coupling nuclei has been treated by Thaddeus et al. [33], who expressed the matrix elements of XQin terms of the 6j Wigner coefficients [34, 351, which are tabulated [36]. This treatment is similar to that used for magnetic structures of plurally coupling nuclei described in Section 8 and can be applied when the plural coupling is both magnetic and quadrupole. The matrix elements of S'QLfor the Lth coupling nucleus are expressed in the representation (J,i, ZL, F1 * . + F N - 1, F ) chosen as described in Section 8. The diagonal elements are obtained from the assumed vector model of the coupled system; the off-diagonal elements, from the tabulated 6j coefficients [36] with (9.185) of Section 8, from Thaddeus et al. [33]. A general discussion employing irreducible tensor methods is given in Chapter XV.

6 SECOND-ORDER EFFECTS IN NUCLEAR QUADRUPOLE I NTERACTI 0 NS The formulas for EQ in the previous sections represent the diagonal elements of X Qin the representation J, i, F , I, and M F , where i represents any inner quantum numbers such as K. However, the matrix of X Qis not exactly diagonal in this representation because the rotational part of the molecular wave function indicated by J, i is perturbed by the nuclear interactions, and hence J is not a perfectly good quantum number. Thus

EQ=(J, i, 1, F , MFIHQIJ,i, 1, F , M F )

(9.127)

earlier evaluated in Section 4, is a calculation of first-order perturbation only. The off-diagonal elements are negligible, however, except for coupling by a few nuclei, the most notable of which is "'1, which in some molecules give large couplings. Even then the off-diagonal elements are small, and adequate

SECOND-ORDER EFFECTS IN NUCLEAR QUADRUPOLE INTERACTIONS

427

correction can be made by inclusion of the second-order term in the perturbation theory, except perhaps in a few instances of near degeneracy in the levels of asymmetric rotors. The second-order term is (9.128) where J’,i’fJ, i. In this expression i represents the inner quantum number K for the symmetric top and the pseudo-quantum numbers T or K - lK1 for the asymmetric top. For the linear molecule, i=O. For the asymmetric rotors, the off-diagonal elements, like the diagonal elements, cannot be expressed in closed formulas except for every low Jvalues. They can be evaluated numerically by high-speed computers when the degree of asymmetry of the molecule is known. Bardeen and Townes [37] have evaluated the off-diagonal elements of XQ for symmetric tops, and Bragg [9] has given a general expression for the matrix elements of an asymmetric top. The nonvanishing off-diagonal matrix elements of HQin the JiIFM, representation are of the type (J,ilJk 1, i’) and (J,ilJf2, i‘). In particular

x[F(F+l)-I(I+l)-J(J+2)] x [(I+ J+ F 2)(1- J+ F ) ( J - I+ F + 1)(J+ I - F + 1)]1’2

+

[(I+ J + F + 2)(I + J + F + 3)(I - J+ F - 1)(I- J + F ) x (J- I+ F + 1)( J - I+ F + 2)(I+ J- F + 1)(I+J- F + 2)] x

’”

(9.129)

(9.130)

Since the matrix is Hermitian, the matrix elements for J’ = J- 1 and J’=J- 2 may also be obtained from these expressions. For symmetric tops the matrix elements of the electric field gradient appearing in the foregoing equations can be readily evaluated from Table 2.1 by expressing the electric field gradient along the space-fixed Z axis in terms of the field gradients along the principal axes of inertia; see (9.69). Since for a symmetric top with a single coupling nucleus the principal inertial axes coincide with the principal axes of the coupling tensor and qxx=qyy= -i q z z= -34,we have (9.131) The matrix elements will hence be diagonal in the internal quantum number K .

428


Table 9.4 Illustration of Second-Order Effects in Nuclear Quadrupole Hyperfine Structure‘ Theoretical Displacement f r o m the F K = O Line

=I-;,

F Transitions

First Order

Second Order

( A V ) (~M H z )

( A V ) (~M H z )

( A V )+ ~ (AV)~

Observed Displacement (Av)obs

For K = O

+

74.59 - 174.06 - 450.35 +406.14 129.85 +33.15 - 276.29 - 372.99

+

- 0.21 -0.37 +2.19 0.40 2.95 -0.39 3.35

+ +

+

0

+

74.32 - 174.43 - 448.16 406.54 132.70 32.76 - 272.94 - 372.99

+ + +

+

74.33 - 174.47 - 448.04 406.47 132.72 32.73 - 273.04 - 373.04

+ + +

0 For K = l b

t-f t-t t-: :-t

i-i 2-3

4-4 t--S 4-3

+281.73 163.02 +24.85 - 127.08 - 265.25 - 313.60 -62.18 - 110.53 75.99

+

+

+ 2.76 + 6.00 + 3.20 + 3.12 +0.92 + 2.22 $2.17 + 3.47 +OS7

+ 284.49 + 169.02 + 28.05 - 123.36 - 264.33 -311.38 - 60.01 - 107.06 76.56

+

+ 168.96 +

28.09 - 123.49 - 264.28 -311.28 -60.15 - 107.12 -k 76.65

“The observed and calculated displacements are from the 4-3, K =O line in the J = 1+ 2 transition of methyl iodide (CH31).The splitting arises from 12’1 with eQq = - 1934 MHz. Data taken from W. Gordy, J. W. Simmons, and A. G. Smith, Phys. Rev., 74, 243 (1948). *The K = 1 lines have an additional correction of -0.38 MHz for effects of centrifugal stretching.

The (JIJ+ 1) matrix element is evaluated as follows (J, K , J l$lJ+2,

K , J ) = q (J, K , J l v l J + 1, K , J) K , Jl@%lJ+ 1, K , J ) - 3qK[(J+ 1)’ - K 2]112 (J+1)(J+2)[2J+ 3]’l2

Likewise

(9.132)

MAGNETIC HYPERFINE STRUCTURE OF SINGLET STATE MOLECULES

429

These expressions along with (9.129) and (9.130)yield the off-diagonal elements of HQ for symmetric tops. They apply also to linear molecules when K is set equal to zero. It is evident that for linear molecules 2fQ has only off-diagonal elements of the type ( J / J + 2 ) . The rigid rotor approximation of EJ,K is adequate for evaluation of the energy difference of the denominator of (9.128). By substitution of the nonvanishing matrix elements and the energy differences E j + l , K - E j , K =2B(J+ l), EJ+Z,K-EJ,K=2B(2J+ 3) into (9.128) followed by summation, the secondorder correction energies are found. Because these calculations are rather tedious, numerical evaluations are given in Appendix J for most of the spins and levels for which they are needed. In Table 9.4 the second-order corrections are shown for the l Z 7 I nuclear quadrupole hyperfine structure of the J = 1+2 transition in CH31, for which the coupling eQq=Xz = 1934 MHz. Although these corrections are not large, they are well beyond the experimental error. For CH379Br,xz=577 MHz; the second-order corrections are still measurable, but very much smaller than those for CH31. For asymmetric rotors off-diagonal elements in T will appear. These must be evaluated numerically for each individual case. Nonvanishing contributions to ( J ,718' 7//dZ21J',T') will appear whenever the direct product of the symmetries of J,, J:,,@zg,and @zg, belong to the representation A of the Four-group (see Table 10.1 for the symmetries of the direction cosines and their products).

7 MAGNETIC HYPERFINE STRUCTURE OF MOLECULES IN SINGLET STATES Theory of Magnetic Interaction The classical Hamiltonian for the interaction of a magnetic dipole with a magnetic field is XM=-p-H (9.134) The dipole moments we are considering here are the nuclear spin magnetic moments, and the field H is that generated by the molecular rotation. Let us first consider the magnetic interaction with a single nucleus having spin I. Its magnetic moment will be Pr=SrPrI (9.135) where PI is the nuclear magneton and gr is the dimensionless gyromagnetic ratio or g factor for the particular nucleus. Because the interaction of with the molecular field is weak, J can be considered a good quantum number, and the vector model of Fig. 9.3 applies. The molecular rotation is so rapid compared with the precession of I that the component fields normal to J are effectively averaged out, and pI interacts only with the averaged field which is in the direction of J. Thus we can set (9.136)

430


Substitution of (9.135) and (9.136) in (9.134) yields the first-order expression (9.137) When the principal magnetic field axes x,y , z are also principal inertial axes, as is true for linear and symmetric-top molecules and for certain asymmetrictop molecules which have a plane of symmetry, we can set

= H,

-+ IJI Jx

H y -+ H, Jy

I4

Jz -

1-4

= [J(J+ l)] - 1/2(hxxJf

+ hyyJ,2+h,,J,2)

(9.138)

This last step follows from the assumption that Hx=h,,Jx, and so on. Since the magnetic field is generated by the molecular rotation, it is reasonable to assume that the field components are proportional to the components of angular momentum. This assumption is verified by the treatment in Chapter XI, Section 4. In the general case where the principal axes of magnetic susceptibilities do not coincide with the principal inertial axes, off-diagonal elements will be generated, and

Hx=hxxJx+hxyJy+hx,J, H y = hyyJy+ hyxJx+ hyzJz H , =h,, J, +hzx Jx + h z y J y

(9.139)

Substitution of these values into (9.138)yields

HJ = [ J ( J + I)] - ‘I2

1

hgg,JgJg,

(9.140)

gg’=x,y,z

where x, y , z are the principal axes of inertia. For reasons similar to those given for cancellation of the field gradients when they are averaged over the rotational state (Section 4), the off-diagonal elements in HJ will drop out, leaving ( H J ) =[J(J

+ 1)]-”2[hxx(J?)

+h y y ( J ; )

f

h z z (JZ’)]

(9.141)

where the average is taken over the rotational state. Therefore the Hamiltonian operator can be expressed as ;Ce,= Cj,i I . J

(9.142)

where (9.143) and

c,, = -SIPIhg,

(9.144)


431

The term C,, represents coupling constants that are independent of the rotational state, whereas CJ,iis constant only for the particular rotational state J, i over which HJ is averaged. Here i represents any inner quantum numbers such as K of the symmetric top or pseudo-quantum numbers T or K - , K 1 in the asymmetric rotor. The I -J term can be expressed (Section 4) as

I . J = ~ ( Fz 12 - 5 2 )

(9.145)

where F is the vector sum of I and J, as indicated in Fig. 9.3. Since J is a good quantum number in the assumed conditions, as are Z and F also, the evaluation can be reduced to E ~ = cJ,i ~[F(F+l)-Z(Z+l)-J(J+I)]

(9.146)

where 1 CJ,i=(J,i I C , , J ~ + C y y J ~ + C r z J ~ il )J , J(J+ 1)

F=J+Z,J+Z-l,

(9.147)

. . . , IJ-ZI

and where the C,,’s are the diagonal elements of the nuclear magnetic coupling tensor. When the principal magnetic axes coincide with the principal inertial axes, these Cg,’s represent principal elements of the nuclear coupling tensor. This is true for linear and symmetric-top molecules, also for certain asymmetric rotors which have a plane of symmetry. Although the parameter CJ,idepends on the rotational state, it is a constant for a particular rotational state. In spectral analysis it is convenient to measure first the CJ,I)sor their differences for connecting rotational states and from these values to derive the constants Cgg.For an R-branch J + J + 1 transition, the frequencies of the strong F+F+ 1 components are displaced by the magnetic interaction thus J-J+

(Av)F-F+

1

1=

(cJ’,i‘cJ,i)[ F ( F + 1)2h

J(J+ 1)- I(Z+ I)]

+* h

( F - J) (9.148)

For a Q-branch transition J - J , the F-+F components are the stronger. Their frequencies are ( A v )J F -+FJ=

cJ,if-

2h

[ F ( F + 1)- J(J+ 1)- I(I+ l)]

cJ,i

(9.149)

where the primed values are for the upper state. These magnetic displacements are to be added to any quadrupole displacement by the same nucleus for the same transitions. To simplify the derivation, we have assumed that the molecular magnetic field generated by the rotation is proportional to the angular momentum. A Hamiltonian of the same form is obtained if more rigorous theoretical treatments are applied. The magnetic field arises from two sources, the more obvious

432


of which is the rotation of formal charges with the molecular frame. The second, not so obvious, source is the induction by the rotation of a slight admixture of excited electronic states with angular momentum L into the ground electronic state of the molecule. A molecule in a 'X state when at rest can have no electronic angular momentum because of the strict quantization of the total angular momentum. When the molecule is rotating, the total angular momentum is still strictly quantized, but the total can now represent an admixture of end-over-end rotation with a component of orbital angular momentum. The interaction between the electronic and end-over-end rotation is a second-order effect which results in nuclear coupling constants proportional to E,,l(OlL,ln)12/ (E,,- E o ) where L , represents the components of the electronic orbital moment along the principal coupling axes and where the summation is taken over all excited electronic states having orbital angular momentum. The quantity E,, - Eo represents the difference in energy between the excited electronic states n and the ground state. The nuclear coupling has the same origin as does the rotational g factor and is derived in a similar manner. See Chapter XI, Section 4. Attempts to relate the molecular magnetic coupling constants of 'C state molecules to the electronic structure of the molecule have been made by Townes et al. [38]. Although their treatment gives an insight into the nature of the coupling, it does not provide a means for obtaining reliable information about the chemical bond because of the very small values of the observed coupling combined with the complex nature of the coupling mechanism.

Diatomic and Linear Polyatomic Molecules Linear molecules in the ground vibrational state and in 'C electronic states have no resultant angular momentum about the molecular axis, chosen as the z inertial axis. Because of symmetry, C,, = Cy,= Cr. With these conditions, the matrix of (9.147) transforms to

+

1 J ( J 1) CI(JJ21J)=C1-CJ=J(J+ i ) v C 1 J ( J + 1)

(9.150)

and from (9.146) the magnetic hyperfine energies are found to be

CI

EM=T [F(F+l)-Z(I+l)- J ( J + l ) ]

(9.351)

where C1 is the nuclear magnetic coupling constant. Selection rules are the same as those for quadrupole hyperfine structure. For a rotational absorption transition J+J+l,

F+Ffl

and F+F,

I+I

(9.152)

Equation 9.151 gives the frequency of the J+ J + 1 transition to be vF-,F+~=vo-

):(

-

):(

( J + l ) +- ( F + l )

(9.153)


433

(9.154) (9.155) where vo is the frequency of the line unperturbed by the magnetic interaction. As will be seen from an examination of the intensity tables in Appendix I, only the F+F+ 1 lines have significant intensity except for very low J values. Therefore, except for JS 3, the noticeable hyperfine components are the F+F+ 1 lines. When J > I, the values of F range in integral values from J - I to J+Z. Thus there are 21+1, F + F + l components with equal spacing of Cr/h. Usually the value of Cr/h for molecules in ‘Z states is only of the order of 10 kHz and hence cannot be resolved in most experiments because of Doppler broadening (Chapter 111, Section 2). When, however, the F+F 1 components are already separated by nuclear quadrupole coupling, as is usually true for spins greater than f, the extra displacement caused by the magnetic hyperfine interaction is often measurable. Note that the outside components are the most displaced in frequency, one up and the other down. The small splitting of the magnetic hyperfine structure can often be resolved with molecular beam masers or molecular beam resonance techniques which avoid both collision broadening and Doppler broadening. Some of the magnetic coupling constants given in Table 9.5 were measured by molecular beam resonance techniques. From the observed nuclear coupling constants Cr, it is possible to calculate the molecular magnetic field at the respective coupling nucleus for the different rotational states. Since gI=/.ir (nm)/Z, the field values from (9.141) and (9.144) can be expressed as

+

or (9.157)

As an illustration, let us calculate the field at the nitrogen nucleus in HCN, for which Cr(14N) is 10 kHz (see Table 9.5). For I4N, Z=1 and ~1=0.407. Substitution of these values into (9.157) shows that ( H J )=32.5[J(J+ l)]l” gauss Thus for the rotational state J = 1, the magnetic field at the N nucleus is approximately 46 gauss; for the J = 10 rotational state it is about 340 gauss. Symmetric-top Molecules In the symmetric-top molecule two of the coupling coefficients of (9.147) are equal because of axial symmetry. We choose z as the symmetry axis and set

434


Table 9.5 Illustrative Values of Nuclear Magnetic Coupling Constants in Linear Molecules Molecule

Nucleus

C I (kHz1

Ref:

3 5 c 1

0.026( 11) 0.148(9) 0.165(6) 0.349(10) 2.2(6) 32.9(1) 10.4(5) - 78.2(5) 10.4(3) - 0.6(3) 2.5(8) - 4.0( 15)

0

DCI DBr DI HI

79Br 1271 1271

LiF

7Li

LiF

19F

PN PN HCN DCN

4N 31P

4N D

I4N

ClCN

ocs

170

a (1

a

b

b C E

d d b b

"F. C. De Lucia, P. Helminger, and W . Gordy, Phys. Ret;., A3, 1849 (1971). bWhite [45]. 'J. Raymonda and W. Klemperer, J. Chem. Phys., 55, 232 (1971). dF.C. De Lucia and W . Gordy, Phys. Ret;., 187,58 (1969).

C,, = C y ,= C N and C,, = C K . In the symmetric-top representation the matrix elements of (9.147) are then

=---

1

J(J+ 1)

=

( J , K IC,( J 2 - J:)

c, + ( C , - C,) J (KJ +2 1) ~

+ C KJiI J , K ) (9.158)

Substitution of these values into (9.146) yields the formula for the nuclear magnetic splitting of the symmetric-top rotational levels by a single coupling nucleus

where C K is the principal value of the nuclear coupling along the symmetry axis and where C , is the principal value normal to this axis. Selection rules for F are the same as those for linear molecules, (9.152). Only the F+F+ 1 components of the rotational absorption transition J + J + 1, K-+K have significant strength except for very low J values. From


435

(9.159) the magnetic displacements of these transitions are found to be

Z(Z+ I ) - J ( J + l)]

(9.160) Small deviations in the positions of the hyperfine components of the NH3 inversion spectrum from those predicted from nuclear quadrupole theory were first measured by Simmons and Gordy [39] and threated theoretically by Henderson [40]. The observed displacement can be fitted to a formula [40] like (9.159)with CN= 11.4 kHz and C K = 13.6 kHz. From a relationship like that of (9.157) CK (kHz) ZK ( H K )(gauss)=1.311 (9.161) PI (nm) it is found that the magnetic field at the nitrogen nucleus in the direction of the symmetry axis is approximately 44 gauss when K = l . For large K values it is a few hundred gauss. The NH3 hyperfine structure is further complicated by magnetic coupling by the hydrogen nuclei (see Section 8). x [ J ( J - F ) + F ( F + 1)-

Asymmetric Rotors The magnetic coupling by a single nucleus in an asymmetric rotor is given by (9.147), but the constants C,,, Cy,,, and C,, with reference to the principal inertial axes are not necessarily the principal elements of the magnetic coupling in the molecular system: they are simply the averaged values with reference to the inertial axes a, b, c. We designate these constants as C, cb, and C,. The CJ,iof (9.143)can then be expressed as 1 cJx - I X I=J(J+) ( J , K - 1 , K,ICaJ,Z+CbJb2+CcJC2IJ, K-1, K1) (9.162) where the average is over the wave function for the rigid asymmetric rotor. It is obvious that the quantity in vertical bars has the same form as the Hamiltonian of the rigid asymmetric rotor (see Chapter VII). Hence the energies already found for the rigid rotor can be used for its evaluation. The values of ( J ; ) , already used in the evaluation of quadrupole coupling energies, are given by (9.97)-(9.99). Substitution of these expressions into (9.162) and combination with (9.146) yields the first-order values E -

J(J+~)+E(K)-(K+~)+2CbaE0

dK

+C,

[

J(J+~)-E(K)+(K-~)--x [ F ( F + 1)-

for the magnetic coupling energies.

Z(Z+ 1)- J(J+ l)] (9.163)

436


8

PLURAL NUCLEAR COUPLING-QUADRUPOLE AND MAGNETIC

Effects of plural nuclear magnetic coupling were noted in microwave spectra when distortions of certain NH, inversion lines by the hydrogen nuclei were observed with a conventional gaseous spectrometer by Gunther-Mohr et al. [41]. The invention of the molecular beam maser by Townes and associates, which first operated on the NH, inversion spectrum, made possible further resolution of this closely spaced hyperfine structure as well as the resolution of composite hyperfine structure in other molecules. The many lines arising from the four coupled nuclei in NH, were partially resolved by Gordon [42] with the first molecular beam spectrometer, and later, by Shimoda and Kondo [43]. An exceptionally high resolution of this hyperfine structure has since been achieved by Kukolich [44] with a two-cavity molecular beam maser. Stimulated by these initial observations on NH,, Gunther-Mohr et al. [ l l ] developed a comprehensive theory of plural nuclear coupling in NH3 which has been extended and adapted to molecules of different types by others. Reviews of the early treatments are given by Townes and Schawlow [21] and by White [45]. Theory for the combined nuclear quadrupole and magnetic hyperfine interactions by plural nuclei has been treated by Posener [46], and a convenient formulation with adaptation to a symmetric rotor is provided by Thaddeus et al. [33]. We give here only an outline of the essential elements. A more detailed discussion is given in Chapter XV.

Hybrid Coupling by Two Nuclei: Strong Quadrupole and Weak Magnetic Coupling When there is coupling by two nuclei, one of which has both a quadrupole moment and a magnetic moment and the other only a magnetic moment, the quadrupole coupling is usually much the stronger for molecules in C singlet states. The outstanding exception is the combination of D coupling with H coupling which must be treated by the more general methods discussed later. The more typical case of strong nuclear quadrupole coupling combined with weak magnetic coupling can be treated by use of the vector model, as here described. The quadrupolar nucleus with spin I, forms a resultant F1 = J + I1

(9.164)

and the magnetic nucleus with spin I2 couples to F1 to form the resultant

F=F,

+ 12

(9.165)

as in the vector model of Fig. 9.9. The quadrupole interaction of the first nucleus is calculated in the usual manner with the neglect of the weak magnetic coupling of Z2. This energy can be expressed as (9.166)


437

AND MAGNETIC

where 45 depends on the nuclear type and where Y ( J , Zl,F , ) is defined by (9.83). Nucleus one will also have an 1. J interaction caused by its magnetic moment which can be expressed from (9.146) by (9.167) In the assumed vector model, J precesses so rapidly about Fl that only the components of ( H J )of (9.137) that are parallel to F1effectively interact with I2 which precesses much more slowly about F. The Hamiltonian for the magnetic interaction is therefore (9.168) under the conditions that F1, Z2, and F are all good quantum numbers. Hence

EM,=[+]

[F(F+1)-Z2(Z2+1)-F,(F1+1)]

(9.169)

where

(9.170) with cos OJ,F1 evaluated from the vector model and the law of the cosine

The total hyperfine energy is (9.172) The hyperfine quantum numbers are F1= J

+ ZI , J+

F =F1+

Z2,

z1

F1+

- 1, . . . , 1 J- I1 I I 2 - 1,

. . . , IF1 - Z2J

(9.173) (9.174)

The selection rules for F1 and F are the same as those for the F of a single coupling nucleus. If F12 Z2 each level of F1 which is due to nucleus one is split into 2Z2+ 1 components by the interaction of nucleus two. For F1< Z2 the level splits into 2F1+ 1 components. Because nuclei with spin Z=f have magnetic moments only, with no quadrupole displacements, their splitting can seldom be resolved except in molecules with unbalanced electronic angular momentum (Section 9).

Couplings of Comparable Strength When two or more of the nuclei have couplings of comparable strength, either quadrupole or magnetic, the vector model treatment given in the previous

438


section breaks down. Because the off-diagonal elements of the coupling matrix are large, one must solve a secular equation to find the correct level splitting. A general method for finding the required matrix elements for N-coupling nuclei, which may be a combination of quadrupole and magnetic coupling, has been described by Thaddeus et al. [33]. Their treatment will be described briefly with the omission of the spin-spin interaction, which is quite complicated and usually negligibly small. The Hamiltonian for the I - J interaction of any one of the coupling nuclei is like that for a single coupling nucleus just described. Thus the Hamiltonian for the combined I . J interactions can be expressed as

(CJ,i)LIL'J

* M = c

(9.175)

L

where IL is the spin of the Lth coupling nucleus, where the summation is taken over all coupling nuclei, and where (9.176)

In this form of the Hamiltonian J is assumed to be a good quantum number. The problem is to express the matrix elements of .Ye, in a suitable representation so that its quantized values can be found either by perturbation theory or from solution of the secular equation. It is desirable to express the matrix of .J?,~ in the representation in which off-diagonal elements are smallest. Since all the magnetic perturbations are small, J can be assumed to be a good quantum number. The matrix AfMis therefore diagonal in J a s well as in all the individual spin numbers Zl . IN. It is diagonal also in F , the number representing the total angular momentum, which is always a constant of the motion. Thus .ri"M commutes with the operators J, I,, . . . , IN, F but not necessarily with the component operators F1, F2, . . ., FN defined next. The representation which makes the off-diagonal elements smallest when the degree of coupling is substantially different, is [33] 9

111 '

' *

IN,

J, F1

' ' '

FN-1,

F)

(9.177)

where the F , . . . F N - l indices are chosen by the following coupling scheme. Of the distinguishable nuclei the strongest coupling one with spin I1 is assumed to be coupled first to J to form a resultant F1;then the nucleus having coupling next in strength with spin indicated by I, is assumed to be coupled with F, to form the resultant F2, and so on

Fl=J+I1,

F,=F,+I*,

...,

F=Fly-l+IN

(9.178)

The evaluation of the interaction for each nucleus can be separated. The total energy in first-order is the sum

E,M=E1+E2+ . . . +EN

(9.179)

PLURAL NUCLEAR COUPLING-QUADRUPOLE A N D MAGNETIC

439

where EL is the diagonal matrix element of the interaction of the particular nucleus in the representation of (9.177). In this representation the matrix of the Hamiltonian for the Lth nucleus diagonal in J and i is

The diagonal elements of the matrix of IL*J can be obtained from the assumed vector model of the coupled systems. The off-diagonal elements can be calculated [33] most easily from the Wigner 6j coefficients tabulated by Rotenberg et al. [36] with the formula [33-351

where L- 1

(Fi-

r = ( L - 1)+ i= 1

+ zi+Fi)+(F~-+ ZL+ FL)

(9.182)

The derivation of (9.181) and (9.184) is given in Chapter XV, Section 3. The quantities in the large braces are the 6j coefficients. The operator IL.J is diagonal in all the Fi not lying in the range F1 . * * F L - 1, and does not depend on the quantum numbers F L + . . . F. In these coefficients Fo=J=Fb and FN=F. When L = 2 the product of the 6j symbols immediately on either side of the leaders in (9.181) reduces to one symbol. When L = 1, only the symbol on the far right of the chain remains, and the product [(2F 1)(2F1 I)] * * . [(2Fi- + 1)(2FL- + l)] reduces to unity. For two coupling nuclei the offdiagonal elements of (12.J)are of the type (F1+lIF1). Some of the N-coupling nuclei may also have quadrupole interactions. The composite Hamiltonian of the quadrupole interaction can likewise be represented as the sum of the interactions of the separate nuclei, as described in Section 4,

+

= ;

(eQqJ)L [ ~ ( I L * J ) ’ + ~ ( I L . J125’1 )+ 2425- l)ZL(2ZL - 1)

+

(9.183)

In terms of the 6j Wigner coefficients the matrix elements of the quadrupolar Hamiltonian for the Lth coupling nucleus in the representation just described

440


(2J+ 1)(2J+2)(2J+3) [(2F; 8J(2J- 1)

{

x [(2Fi-

1

+ 1)(2F1+ l)]

* *

.

+ 1) ( ~ F L+- I)] (2IL+ 1)(21,+2)(2IL+3) 8 I L IL (~ - 1) 1

I

li2

(9.184) where L- 1

(9.185) As before, when L = 1, only the 6j symbol on the far right of the chain remains, and the product [(2F; 1)(2F1 l)] . . . [(2Fi-, 1)(2FL-1 l)] reduces to unity; for L = 2 the product reduces to [(2F; 1)(2F1 l)], and only two 6 j symbols remain. See also Chapter XV, Section 3.

+

+

+

+

+

+

Coupling by Identical Nuclei When there are identical nuclei in the molecule, linear combinations of their identical spin functions must be chosen to give independent functions. In this case, however, the plural hyperfine coupling is sometimes simplified. Consider the coupling by two identical nuclei in a molecule such as H 2 1 6 0 or C H 2 0 . Because the over-all symmetry of the wave function must be antisymmetric for an operator which exchanges these identical nuclei (see Chapter 111, Section 4), only spin functions that are antisymmetric can combine with the even rotational states (if the electronic and vibrational functions are assumed to be even), and only even nuclear spin functions can combine with the odd rotational levels. For Il = I2 =fthe combinations (3,f)and (-f, -f) are symmetric since an exchange of the two nuclei produces an indistinguishable combination. For the degenerate combinations (-f,+) and (3,-f), the independent functions are the linear combinations (9.186) (9.187) The first of these is symmetric; the second, antisymmetric. Thus there are three symmetric combinations. These are equivalent to an effective spin I' = 1 with M I values of 1, 0, and - 1. Therefore the hyperfine structure of the odd rotational levels corresponds to that of a single nucleus with spin I=1,


AND MAGNETIC

441

and the calculation is the same as that for a single nucleus with I = 1. There is only one spin combination that is antisymmetric with effective spin I'=O, and thus the even rotational levels are singlets and cannot be split by the nuclear interaction. For methods of finding the symmetry of rotational levels in asymmetric rotors, see Chapter 111, Section 4. The hyperfine interaction by three identical nuclei is likewise reduced by the symmetry of the rotational wave function, but its calculation is not so simple as that for two nuclei, described previously. These effects are treated for the three identical coupling H nuclei in NH3 by Gunther-Mohr et al. [I I].

S p in-S p i n Interact ions The direct dipole-dipole or spin-spin interaction energy between the different nuclei is of the order of only a few kilohertz and is usually not resolvable even with the beam masers. For this reason we shall not treat it here. The Hamiltonian of this interaction is derived by Gunther-Mohr et al. [111, and a generalized method for finding its matrix elements which employ the tabulated 6 j Wigner coefficients is given by Thaddeus et al. [33]. A brief discussion is given in Chapter XV, Section 5.

Observed Spectra in Asymmetric Rotors- Maser Resolutions Nuclear magnetic hyperfine structure has been measured in the rotational spectra of only a few, relatively simple asymmetric-top molecules. Most of the coupling nuclei are protons or protons and deuterons. The splittings expected from (1.J) interactions of the various nuclei are of the order of 10 kHz or less. Such splittings are usually obscured by Doppler broadening in the conventional microwave spectrometer used for the study of gases, but further displacements of this order might be observed when the 21+1 degeneracy caused by a given nucleus is already lifted by quadrupole splitting. Since the Doppler broadening decreases with frequency, the best possibilities for measurement of the shifts for gaseous molecules in thermal equilibrium are on low frequency transitions. Posener [47] as well as Treacy and Beers [48] have measured the D and H couplings in HDO with sensitive gaseous absorption spectrometers designed to operate on low frequency transitions. Similarly, Flygare [49] has measured these constants in HDCO. However, the instrument best suited for observation of magnetic hyperfine structure is the molecular beam maser which avoids both Doppler and pressure broadening. Because of difficulties in experimental techniques and low sensitivity, it can, unfortunately, be used only for highly selected molecules and transitions. One of the simplest organic molecules for which magnetic hyperfine structure has been studied with the beam maser is "CH2160. In this molecule there is no nucleus having a quadrupole moment, and the hyperfine structure arises solely from the magnetic moments of the two protons. The spins of the two protons combine to form a resultant spin T = 1 or 0, which gives a coupling like that of a single nucleus with I = 1, or with no coupling Z=O, according to

442


the symmetry of the rotational levels. The splitting of the rotational levels which combine with the symmetric spin functions can be calculated by use of (9.163) with I = 1. The rotational levels are split into a triplet corresponding to F = J + I, J, J- 1. For Q-branch transitions J-J, there are three strong lines that correspond to the F+F transition. Resolution of this hyperfine structure was first achieved by Okaya [50], who detected two of the strong F+ F components in the 414+413transition. Later Takuma et al. [51] observed the three expected F+F components of the 312+313 transition. More recently, Thaddeus et al. [33] resolved beautifully all three of the F+F components of the 2,,+2,, transition, and, in addition, detected one of the weaker F+ F - 1 components. This spectrum, which is shown in Fig. 9.10 provides an illustration of the resolution which can be achieved with the molecular beam maser. The total spread of the spectrum is about 40 kHz. The observed proton magnetic coupling constant for the 211 level CH(211) is only 0.65 kHz, and CH(211)-CH(212)= 2.26 kHz. When a deuterium atom, I= 1, is substituted for one of the hydrogens, the symmetry is destroyed, and the splitting results from the quadrupole coupling of the D atom, for which eQg=170 kHz, as well as the additional, slight magnetic displacement by D and the magnetic splitting by the remaining H. Figure 9.11 shows the spectrum for the same 211+212 transition of CHDO, which, when compared with Fig. 9.10, illustrates the marked changes produced by the D substitution.

40

I FREQUENCY (kHz)

Fig. 9.10 Magnetic hyperfine structure of the 211-+212transition of CH20caused by the two protons. Bars represent calculated components. The curve simulates the observed maser pattern. From Thaddeus et al. [33].

Fig. 9.11 Composite quadrupole and magnetic hyperfine structure in the 2, + 2 1 2 transition of CHDO caused by the proton and deuteron. Bars represent calculated components. The curve simulates the observed maser pattern. From Thaddeus et al. [33].

Table 9.6 Hyperfine Splitting Parameters of D and H for Particular Rotational States, as Measured with the Molecular Beam Maser

Molecule

HDO HDS NH2D

Rotational State

220 22 1 220 221 313 303

"Thaddeus et al. [33]. bThaddeus et al. [52].

( ~ ~ J Q ) D

(kHz)

79.3 k0.3 79.6f0.3 42.9 f 0.4 43.3 f 0.4 -62.5 f 1.0 -73.95 1.0

(CH)J (kHz)

(CD)J &HZ)

-43.4750.11

-2.33 k0.02

- 43.63 50.13

- 2.20 k0.02

- 25.03 f 0.13

- 27.45 50.13 - 13.650.5 - 12.3k0.5

- 0.47 50.02 - 0.22 f 0.02

-2.750.3 - 2.7 f 0.3

ReJ: (I

7' 0 (I

b b

444


Similar maser observations of composite hyperfine structures have been made by Thaddeus et al. [33, 521 for HDO, HDS, and NH2D. The observed H and D coupling constants are given in Table 9.6.

9 MOLECULES WITH UNBALANCED ELECTRONIC ANGULAR MOMENTUM Molecules with unbalanced electronic spin or orbital momentum generally have large magnetic nuclear coupling and widely spaced magnetic hyperfine structure. However, stable molecules of this kind are not prevalent and are mostly small, inorganic molecules such as 0 2 ,NO, and NO2. Transitions between A doublet states of the free radicals O H and O D have been observed in the microwave regions (see Chapter IV), and their H or D hyperfine structure has been analyzed [53]. Magnetic hyperfine structure of the rotational transitions of N O in the 'llli2state [54,55] and in the 2113i2state [56] has also been studied. Theoretical treatments are given by Frosch and Foley [57], by Dousmanis [58], by Mizushima [59], and by Lin and Mizushima [60]. The hyperfine structure of chlorine in C102 has been analyzed by Curl et al. [61] and that of 14N in 14N1602by Bird et al. [62]. Foster et al. [63] observed and analyzed some 150 components of the 808+717 and 919+ 10o,lorotational transitions of the paramagnetic molecule 17014N160.This very complex hyperfine structure is due to two nuclei, 14N and " 0 , which have both quadrupole and magnetic couplings. However, the magnetic component is the predominant one for these high J values. Magnetic hyperfine structure for such molecules depends upon the nature and degree of the coupling between the various angular momenta and spin vectors. Because of its complexity and limited applications in rotational spectra we shall not treat the subject here. The original literature cited on these various types of molecules provides the necessary theory and illustrations for those who are concerned with analysis of hyperfine structure in paramagnetic molecules.

10


Although the analysis of nuclear quadrupole transitions of molecules in excited vibrational states is generally similar to that for the ground vibrational state, the spacing of the hyperfine structure and the nuclear coupling constants depend significantly on the vibrational state. Most of the microwave measurements of hyperfine structures in excited states have been made on the heavier diatomic molecules. Examples of the observed quadrupole couplings in them are given in Table 9.7. The coupling constant eQqv for the parallel vibrational states may be fitted to the power series expansion eQqv =eQqe + e Q q h +f)+ e Q q d u + +I2

+.

1

*

(9.188)

where u is the vibrational quantum number and e&,, eQql, and so on, are constants. For example, the observed eQqv values of 1271 in 69Ga1271listed in

445

SELECTION RULES AND RELATIVE INTENSITIES

Table 9.7 Nuclear Quadrupole Coupling of Selected Diatomic Molecules in Excited Vibrational States Coupling in M H z Coupling

iMolecule

Atom

BiF RbBr

'09Bi

IBr GaCl

85Rb "Br '"I 79Br 69Ga 35c1

GaI

69Ga 1,271

v=O

- 1148.2(1) - 47.1(2)

v=l

v=2

Ref:

v=4

- 1139.7(3) - 46.7(2) - 46.6(3)

- 1144.0(2)

3.5(3) 4.1(3) 5.0(3) -2753.5(1) -2752.9(3) -2752.4(3) 696.8(2) 697.7(2) 698.5 -92.1(2) -91.4(2) -90.8(2) - 13.3(1) - 13.5(1) - 13.7(1) -81.1(8) -80.8(1) -80.0(2) - 370.6(1) - 373.3(2) - 375.9(2)

(I

b b C C

-90.0(2) - 13.9(2) -79.8(2) - 378.3(2)

* e e

"P. Kuijpers and A. Dymanus, Chem. Phys., 24,97 (1977). *E. Tiemann, B. Holzer, and J. Hoeft, 2. Nuturforsch., 32a: 123 (1977). 'E. Tiemann and T. Moller, Z . Nuturforsch., Ma, 986 (1975). dE. Tiemann, M. Grashoff, and J. Hoeft, Z. Nuturforsch., 27a, 753 (1972). 'K. P. R. Nair and J. Hoeft, J. Mol. Spectrosc., 85, 301 (1981).

Table 9.7 can be fitted to (9.188) with eQqe= - 369.4(1) MHz, eQql= -2.5(2) MHz, and eQqII =0. An examination of the ccuplings listed in Table 9.7 will show that the magnitude of the coupling cons-ant increases with the vibrational quantum number u for the more electronegative atom of the bond (i.e., the atom forming the negative pole of a polar bond) and decreases with u for the less electronegative atom (Le., the one forming the positive pole of the polar bond). This trend is evident for other single-bonded, diatomic molecules listed in the Landolt-Bornstein tables [19]. In contrast, the coupling of the more electronegative 14N atom in the triply bonded PN molecule decreases with increase in the vibrational quantum number, as it does in the triply bonded, nonpolar N2. The 79Br coupling in the nonpolar Br2 decreases with increase of u, whereas in the polar IBr the 79Br coupling increases with u. Thus the nuclear quadrupole coupling may increase or decrease with an increase of the vibrational state, depending on the nature of the bonding to the coupling atom. The relationship of quadrupole coupling and chemical bonding is discussed in Chapter XIV.

11 SELECTION RULES A N D RELATIVE INTENSITIES HYPERFINE COMPONENTS

OF

Selection rules for hyperfine structure, magnetic or quadrupole, depend on the nonvanishing matrix elements of the molecular dipole moment along space-fixed axes. The relative intensities are proportional to the square of these

446


matrix elements expressed in the appropriate wave functions describing the states. When there is only one coupling nucleus and J forms with I the resultant F, the proper representation is J , i, I, F, M F , and the line intensities Z are proportional to

Z-

(9.189)

l(J’, i’, Z‘,F’, M$lj.t/J, i, I, F, MF)12 MFM’F

The wave functions representing the hyperfine states can be expanded in terms of the uncoupled basis functions IJ, i, M J , I, M I ) of the molecule as follows $ ( J , i, I, F, M F ) = C(JZF; ~ MJMIMF)$(J, i, Mj)&Z,

MI)

(9.190)

MJ

where the expansion coefficients are the Clebsch-Gordon coefficients and where $(J, i, M J ) and $(Z, MI) are the unperturbed rotational and nuclear spin functions, respectively. The molecular dipole moment is independent of the nuclear spin functions $(Z, M I ) which can be factored out of (9.189)and normalized to unity. Equation 9.189 then becomes IC(J’ZF’; M;MIMk)121C(JZF; MJMIMF)~’

1-

MFM~MJM~

x l(J’,

i‘, M;I plJ, i, MJ)12

(9.191)

The squared matrix elements l(J’, i’, M ; ( plJ, i, MJ12are those already given in Chapter 11, Section 6 for the rigid rotor. Although the foregoing discussion brings out the basic procedure for evaluating the intensities, (9.191) is not in a form very convenient for calculation. Evaluation of (9.189) is most easily accomplished by the use of the calculus of tensor operators which is discussed further in Chapter XV, Section 6. It is found [64] that the relative intensities of the components of a hyperfine multiplet of a particular rotational transition are given by For J - 1-J F-1-F:

B(J+ F

+ I + l)(J+F + Z)(J+ F - Z)(J+ F - Z- 1) F

-

B(J + F + I + I)( J + F - Z)(J- F + Z)(J- F - I - 1)(2F+ 1) F(F 1)

+

(9.192)

(9.193)

F+l-F:

+

B(J- F + Z)(J- F I - 1)(J- F - I - 1)(J- F - Z- 2) F+l

(9.194)

SELECTION RULES A N D RELATIVE INTENSITIES

447

For J-, J F-l-tF:

- A ( J + F + I + l ) ( J + F - I ) ( J - F + I + 1)(J- F - I ) F

(9.195)

F-F:

A[J(J+1)+F(F+1)-Z(I+1)I2(2F+1) F(F 1 )

+

(9.196)

F+l-,F:

- A ( J + F + I + 2 ) ( J + F - I+

1)(J- F + I ) ( J - F - I - 1 ) F+l

(9.197)

where A and B are constants that depend on the strength of the unsplit rotational line. For convenience, the relative intensities calculated from these expressions are given in Appendix I. They are the same as those tabulated for atoms in various books on atomic spectra where F, J, I are replaced, respectively, by J, L, S. When there is plural nuclear coupling, the foregoing formula no longer applies. The relative intensities must be found by evaluation of the squared matrix elements of the dipole moment in the representation of the coupled system. Understandably, the evaluations are more involved than those for a single coupling nucleus. For the case of two coupling nuclei, in which the coupling of Il is much stronger than that of I,, the relative intensities of the components arising from Il can be obtained from the foregoing formulas, and the relative intensities of the additional hyperfine structure on the Il components due to 1, may likewise be computed from (9.192)-(9.197) by replacement of J with F 1 and I with I,. The general case for N-coupled nuclei is treated by Thaddeus et al. E331 and is discussed further in Chapter XV, Section 6. The relative intensities evaluated in the representation specified by (9.178) may be written Z(a’, F‘+a, F ) - IC(a’, F‘; aF)I2

(9.198)

where C(U’,F‘; ~F)=(-1)”{[(2F;+1)(2F1+1)].‘ ’ [(2FL+1)(2FN+l)]}”’

and N

(Zi+F:-,+Fi)+N

U=

(9.200)

i= 1

Here CI stands for Fo, F 1 , . . . FN-1 and J‘=Fb, J = F o ; F=FN. Numerical values of the 6 j symbols may be obtained from tables [ 3 6 ] . Simple formulas

448


for some special cases are given by Edmonds [34]. If the intermediate Fi are not good quantum numbers, the intensities given by (9.198) may not be adequate. The matrix of C must then be transformed by means of a similarity transformation to the basis which diagonalizes the hyperfine Hamiltonian, Le.,

C’= ‘TicTf

(9.201)

where Ti diagonalizes the energy matrix of the initial state of the transition and Tf diagonalizes the energy matrix of the final state. The squares of the elements of C’yield the relative intensities.

References 1. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev., 57,677 (1940). 2. N. F. Ramsey, Molecular Beams, Oxford Univ. Press, London, 1955. 3. H. B. G. Casimer, On the Interaction between Atomic Nuclei and Electronics, Teyler’s Tweede Genootschap, E. F. Bohn, Haarlem, 1936. 4. A. Nordsieck, Phys. Rev., 58, 310 (1940). 5 . B. T. Feld and W. E. Lamb, Phys. Rev., 67, 15 (1945). 6. W. E. Good, Phys. Rev., 70, 213 (1946). 7. D. K. Coles and W. E . Good, Phys. Rev., 70,979 (1946). 8. J. H. Van Vleck, Phys. Reo., 71,468 (1947). 9. J. K. Bragg, Phys. Rev., 74, 533 (1948). 10. J. K. B r a g and S. Golden, Phys. Rev., 75,735 (1949). 11. G. R. Gunther-Mohr, C. H. Townes, and J. H. Van Vleck, Phys. Rev., 94, 1191 (1954). 12. R. Bersohn, J. Chem. Phys., 20, 1505 (1952). 13. M. H. Cohen and F. Reif, “Nuclear Quadrupole Effects in Solids,” in Solid State Physics, Vol. 5 , F. Seitzand D. Turnbull, Eds. Academic, New York, 1957, pp. 321438. 14. T. P. Das and E. L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Academic, New York, 1958. 15. H. G. Dehmelt and H. Kriiger, Naturwiss., 37, 111 (1950). 16. N. F. Ramsey, Nuclear Moments, Wiley, New York, 1953. 17. P. C. Cross, R. M. Hainer, and G. W. King, J. Chem. Phys., 12, 210 (1944). 18. C. A. Burrus and W. Gordy, Phys. Rev., 92,274 (1953). 19. Landolt-Bornstein, Numerical Data and Functional Relations in Science and Technology. For a complete reference, see Chapter 1, [65]. 20. J. Bardeen and C. H. Townes, Phys. Rev., 73,97 (1948). 21. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. 22. R. J. Myersand W. D. Gwinn, J . Chem. Phys., 20,1420 (1952). 23. G. W. Robinson and C. D. Cornwell, J . Chem. Phys., 21, 1436 (1953). 24. R. Bersohn, J . Chem. Phys., 18, 1124 (1950). 25. M. Mizushima and T. Ito, J . Chem. Phys., 19,739 (1951). 26. P. N. Wolfe, J. Chem. Phys., 25,976 (1956). 27. M. W. Long, Q. Williams, and T. L. Weatherly, J. Chem. Phys., 33, 508 (1960). 28. A. A. Wolf, Q. Williams, and T. L. Weatherly, J . Chem. Phys., 47,5101 (1967). 29. K. K. Svidzinskii, “Theory ofthe Hyperfine Structure in the Rotational Spectra of Molecules,” in Soviet Maser Research, D. V. Skobel’tsyn, Ed., Consultants Bureau, New York, 1964. (Translated from Proc. P . N . Lebedev Phys. Inst., 21,88 (1963).

REFERENCES

30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

64.

449

R. Livingston, Phys. Rev., 82,289 (1951). G. F. Hadley, J . Chem. Phys., 26, 1482 (1957). G. Herrmann, J . Chem. Phys., 29, 875 (1958).

P. Thaddeus, L. C. Krisher, and J. H. N. Loubser, J . Chem. Phys., 40, 257 (1964). A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed., Princeton Univ. Press, Princeton, 1960. B. R. Judd, Operator Techniques in Atomic Spectroscopy, McGraw-Hill, New York, 1963. M. Rotenberg, R. Bivens, N. Metropolis, and J. K. Wooten, The 3 j a n d i j Symbols, The Technology Press, Massachusetts Inst. Tech., Cambridge, 1959. J. Bardeen and C. H. Townes, Phys. Rev., 73,647, 1204 (1948). C. H. Townes, G. C. Dousmanis, R. L. White, and R. F. Schwarz, Discussion Faraday Soc., 19, 56 (1955). J. W. Simmons and W. Gordy, Phys. Rev., 73,713 (1948). R. S . Henderson, Phys. Rev., 74, 107 (1948). G. R. Gunther-Mohr, R. L. White, A. L. Schawlow, W. E. Good, and D. K. Coles, Phys. Rev., 94, 1184 (1954). J. P. Gordon, Phys. Rev., 99, 1253 (1955). K. Shimoda and K. Kondo, J . Phys. SOC.Japan., 15, 1125 (1960); 20, 437 (1965). S . G. Kukolich, Phys. Rev., 156, 83 (1967). R. L. White, Rev. Mod. Phys., 27, 276 (1955). D. W. Posener, Australian J . Phys., 11, 1 (1958). D. W. Posener, Australian J . Phys., 10, 276 (1957); 13, 168 (1960). E. B. Treacy and Y. Beers, J . Chem. Phys., 36, 1473 (1962). W. H. Flygare, J . Chem. Phys., 41, 206 (1964). A. Okaya, J . Phys. SOC.Japan., 11, 258 (1956). H. Takuma, T. Shimizu, and K. Shimoda, J . Phys. SOC.Japan., 14, 1595 (1959). P. Thaddeus, L. C. Krisher, and P. Cahill, J . Chem. Phys., 41, 1542 (1964). G. C. Dousmanis, T. M. Sanders, and C. H. Townes, Phys. Rev., 100, 1735 (1955). C. A. Burrus and W. Gordy, Phys. Rev., 92, 1437 (1953). J. J. Gallagher and C. M. Johnson, Phys. Rev., 103, 1727 (1956). P. G. Favero, A. M. Mirri, and W. Gordy, Phys. Rev., 114, 1534 (1959). R. A. Frosch and H. M. Foley, Phys. Rev., 88, 1337 (1952). G. C. Dousmanis, Phys. Rev., 97,967 (1955). M. Mizushima, Phys. Rev., 94, 569 (1954). C. C. Lin and M. Mizushima, Phys. Rev., 100, 1726 (1955). R. F. Curl, J. L. Kinsey, J. G. Baker, J. C. Baird, G. R. Bird, R. F. Heidelberg, T. M. Sugden, D. R. Jenkins, and C. N. Kenney, Phys. Rev., 121,1119 (1961). G. R. Bird, J. C. Baird, A. W. Jache, J. A. Hodgeson, R. F. Curl, A. C. Kundle, J. W. Bransford, J. Rastrup-Anderson, and J. Rosenthal, J . Chem. Phys., 40,3378 (1964). P. D. Foster, J. A. Hodgeson, and R. F. Curl, Jr., J . Chem. Phys., 45, 3760 (1966). E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge Univ. Press, Cambridge, England, 1959.

Chapter

x

EFFECTS OF APPLIED ELECTRIC FIELDS

1

LINEAR A N D SYMMETRIC-TOP MOLECULES WITHOUT NUCLEAR COUPLING

451

2

LINEAR A N D SYMMETRIC-TOP MOLECULES WITH NUCLEAR QUADRUPOLE COUPLING

458

The Weak-field Case The Strong-field Case Single or Plural Nuclear Coupling The Intermediate-field Case

459 462 465

ASYMMETRIC-TOP MOLECULES WITHOUT NUCLEAR COUPLING

468

The Nondegenerate Case The Degenerate Case

469 474

4

ASYMMETRIC-TOP MOLECULES WITH NUCLEAR QUADRUPOLE COUPLING

478

5

MOLECULES IN EXCITED VIBRATIONAL STATES-INVERSION A N D / DOUBLETS

481

3

DOUBLETS

6

ANISOTROPIC POLARIZABILITIES FROM THE STARK EFFECT

484

7

RELATIVE INTENSITIES

489

8

IDENTIFICATION A N D MODULATION OF ROTATIONAL LINES WITH THE STARK EFFECT

491

9

DIPOLE MOMENTS FROM THE STARK EFFECT

493

The Stark effect is a particularly useful auxiliary in microwave spectroscopy. From it, the most accurate evaluation of electric dipole moments of gaseous molecules can be made. It is also useful in the identification of pure rotational lines, particularly of asymmetric-top molecules, and it is widely employed as an aid to the detection of spectral lines [13.

1 LINEAR AND SYMMETRIC-TOP MOLECULES WITHOUT NUCLEAR COUPLING Classically, the interaction energy of a dipole moment p in a field 8 is - p. 8. In the Stark effect of rotational spectra, 8 is an electric field fixed in space, and 451

452


p is an electric dipole moment fixed in the molecule. Here d is assumed to be constant in magnitude and to have the fixed direction Z in space, and p is assumed to be constant in the molecule-fixed reference system chosen as the principal inertial axes x, y , z. With these conditions, the Stark effect Hamiltonian operator can then be expressed as

(10.1)

where aZsare the direction cosines of the x, y , z axes with reference to the space-fixed 2 axis. A small term caused by anisotropic polarizability of the molecule which is neglected in (10.1) is treated in Section 6. Linear and symmetric-top molecules have a dipole moment component only along the symmetry axis z. For them, p L , = p , p x = p L y = O ;the Hamiltonian then becomes simply %PI = - p m z z (10.2) where p and I are constants and where OZzis the direction cosine of the axis of molecular symmetry with reference to the direction of the applied field. Although the magnitude of the field is periodically changed in a Stark modulation spectrometer, a square-wave modulation of low frequency is generally employed so that I can still be considered constant in value during the time interval when the Stark components are displaced. In comparison to the effects of the permanent dipole moment which must be present for detection of rotational spectra, those of the much small polarization moments induced through electronic displacements by the applied field are entirely negligible. We can consider p as having a constant value in the molecule-fixed reference system even though centrifugal distortion causes it to vary slightly with vibrational state. In the present treatment we shall neglect these small effects and consider only Q, as varying with rotation. For rotational lines observed in the microwave region with the field values usually applicable, the Stark energies can be evaluated with significant accuracy from perturbation theory. The first-order energy is simply the average of 2 8 over the unperturbed rotational state. Since the linear molecule can be treated as a special case of the symmetric top with K =0, we shall express the average first in the unperturbed symmetric-top wave functions. In this representation the first-order Stark energies are p8 -(J, K , MJIH8\(J,K , MJ) = - p8(J, K , M J I @ Z z I J , K , MJ)

(10.3) where the direction cosine matrix elements are evaluated from Table 2.1. Note that M J is designated by M in Table 2.1. When K =0, it is seen from (10.3) that Ef’=O. Thus there is no first-order Stark effect for a linear molecule nor for the K =O levels of a symmetric-top

MOLECULES WITHOUT NUCLEAR COUPLING

453

molecule. We can also prove this by averaging QZz over the wave function I j / J , M , of the linear molecule. The first-order energy can be obtained from the vector model of Fig. 10.1 which we shall describe because of the insight it gives into the Stark effect. In this model, K represents the direction of the symmetry axis and hence the direction of p. In the normal motion of a symmetric top, K, which is a component of J, precesses about the direction of J while J precesses about the direction of the applied field, as indicated in the diagram. When the rotational energy is large as compared with the Stark energy, the precession of K about J is so rapid compared with that of J about d that the components of p normal to J are averaged out, leaving only the component p J along J which effectively interacts with the field. Therefore the first-order Stark energy can be expressed as

E (6’ )- - 8pJcosOZ,J

(10.4)

where (10.5)

and (10.6)

Substitution of these values into (10.4) gives the expression already obtained, (10.3). It is seen that the vanishing of the first-order effect when K=O can be attributed to the fact that cos Oz, J=O and hence that the molecular axis z is normal to J. The second-order term of the Stark perturbation for the symmetric-top molecule is

Fig. 10.1 Vector model of a symmetric-top molecule in an electric field.

454


The direction cosine matrix elements are found from Table 2.1 and the energq differences Ej,K-Ej+i,K=

Ej,K

-2hB(J+ 1)

(1 0.8)

- E j - 1 , K =2hBJ

(1 0.9)

are evaluated from (6.6). Substitution of the values into (10.7) shows that the second-order Stark energy of the symmetric-top molecule is

The Stark energy for the linear molecule, obtained by setting K = O in (10.10) is

E

(2)

~'8' [J(J+ 1)- 3Mj]

' -- 2hB J(J+ 1)(2J- 1)(2J+ 3)

(10.11 )

Although the J=O level cannot be split by the field since for it MJ=O, it has a second-order Stark displacement which is given by (1 0.12)

as can be seen by substitution of J=O, K=O, and MJ=O into (10.10). The M J selection rules for rotational lines are

MJ+MJ

and

MJ-MJkl

(10.13)

The first of these transitions AMj =0, sometimes designated as n components, are observed when the Stark field is parallel to the electric vector of the microwave radiation. The AMj = & 1 transitions, designated as 0 components, are observed when the dc field is perpendicular to the electric vector of the radiation. The most commonly observed are the AMj=O components. The first-order frequency displacement of the K / O lines of these components are seen from (10.3) to be (10.14)

To this must be added the small second-order corrections when the splitting becomes large (when p 8 is large). For linear molecules or for symmetric-top molecules with K =0, the displacement of the AMj =O components caused by the second-order effect are

8 p'8' Av"'(AM~=O,J = O - + l ) = - 15 h2vo

(IO. 15)

and Av"'(AM~=O,J+J+ 1, J f O )

+

3M3(8J2 16J+ 5 ) - 4 J ( J + 1)'(J+2) J(J+ 2)(2J- 1)(2J+ 1)(2J+ 3)(2J+ 5)

1

(IO. 16)


455

In these equations vo represents the frequency of the unsplit rotational line observed when no field is applied; Av represents the displacements caused by the applied field 8. If the factor 2/h2 in (10.16) is replaced by 0.5069 and the factor 8/15h2 in (10.15) is replaced by 0.1352, the equations give Av in MHz when vo is in MHz units, ,u is in debye units, and 8 is in volts per centimeter. It is evident from the foregoing equations that the magnitude of the Stark splitting of the rotational levels decreases with increase in J. The largest displacement occurs for M J = J . For high J values the maximum second-order displacement varies inversely with J z approximately. It also varies inversely with B so that the second-order Stark effect becomes increasingly difficult to observe as the submillimeter wave region is approached. If one chooses a light molecule so that high-frequency transitions occur at low J values, the large B value then makes low the second-order Stark sensitivity. Note, however, that the first-order effect, (10.3), which is very much more sensitive than the secondorder effect, is independent of B. The maximum first-order Av, which occurs for K = J, MJ = J, varies approximately as 1/J;but, because of the great sensitivity of the first-order effect, the splittings of symmetric-top lines for K / O can be observed with ease for very high J values of transitions occurring in the submillimeter wave region. Figure 10.2 illustrates the first-order Stark effect

J=5-6

E=0

-

Fig. 10.2 Illustration of the first-order Stark effect of a symmetric-top molecule. The J = 5 + 6 transition of CH,CCH. From P. A. Steiner, Ph.D. dissertation, Duke University, 1964.

456


of a symmetric-top molecule observed in the 2.9-mm wave region with a field of 110 v/cm. Figure 10.3 illustrates the second-order effect for a linear molecule as observed in the centimeter wave region with a field up to 1070 v/cm. Figure 10.4 shows an energy level diagram of the Stark splitting of the 5=1+2 transition of a symmetric-top molecule with the n and o absorption transitions indicated by solid arrows. Such a diagram cannot be drawn to scale because the separations of the rotational levels are generally of the order of 100 times greater than the Stark splitting; and for the same field value, the first-order splitting of the K = 1 lines is of the order of 10 to 100 times that of the second-order splitting of the K =O lines. Nevertheless, the diagram gives a qualitative indication of the expected Stark effect. The K =0, AMj=O transitions correspond to the observed components shown for a linear molecule in Fig. 10.3. Because the field orientations required for observations of the n and o components are different, these components are generally not observed at the same time. Nearly all measurements in the past have been made on AMj=O components, which require the dc electric field ‘to be parallel to the electric vector of the microwave radiation. These are the components observed with a rectangular waveguide Stark cell having for the Stark electrode a metal strip held by dielectric supports in the center of the waveguide. The development of the parallel-plate, millimeter-wave Stark cell made possible the observation of the AMJ= & 1 components, which require that the dc electric field be imposed at right angles to the electric vector of the microwave radiation.

Zero field

750 v/cm

1070 v/cm Fig. 10.3 Illustration of the second-order Stark effect in the splitting of the J = 1 -+2 rotational line of OCS. From T. W. Dakin, W. E. Good, and D. K. Coles, Phys. Ret.., 70, 560 (1946).


457

AM=OI AM=-IIAM=+IM~

-2

-I

J K

0

I T - - - - - - - - - -

+I

+2

1st order

Fig. 10.4 Diagram (not drawn to scale) of the Stark effect of the J = 1 4 2 transitions of a symmetric-top molecule.

Perturbation theory to second order is sufficient to account for the Stark splitting within the accuracy of most measurements which are made with an imposed field of a few thousand volts/centimeter. When there is a first-order effect, one needs to apply only a few hundred volts/centimeter to obtain a very wide separation of the Stark components (- 100 MHz or more), and first-order plus second-order terms are completely adequate. When there is no first-order effect, one often finds it advantageous to apply several thousand volts/centimeter to obtain sufficiently large displacements for precise measurement of Av. With voltage of the order of 5000v/cm, Muenter and Laurie [2] detected deviations of as much as 1% from the second-order theory in K=O transitions of some symmetric-top molecules. For linear molecules or symmetric tops with K=O, all odd perturbation terms are zero. Hence the next term of significance is of fourth order. To correct for the fourth-order perturbation in the precise measurement of a dipole moment it is fortunately not necessary to employ explicitly the complicated fourth-order formula. The displacement of a particular Stark component for which there is no odd-order effect can be expressed by the simple equation

AV=ap2b2+ bp4g4+ . * .

(IO.17)

where a and b are constants. The constant a can be obtained from the secondorder formula, (10.1l), or for the AMJ =0 component more simply from (10.16).

458


.604LII

1

1

I

1

1

1

50

1

I

I

I

100

1

1

1

1

I

150

I

'

1 I 200

8' (statvolts/cm)*

Fig. 10.5 Plot demonstrating fourth-order Stark effect in the J of methyl acetylene. From Muenter and Laurie [2].

= 1 +2,

K =0, A M J =O transition

If one expresses (10.17)in the form

(IO.18) it is seen (bp4/u)is the slope and p z is the intercept of a straight line obtained

by a plotting of ( A v / a b 2 )versus 6'. Figure 10.5 shows such a plot for the J = 1-2, K = O transition of CH,CCH by Muenter and Laurie. From projection of the straight line one obtains pz as the intercept. With this value one can then obtain b from the slope if this is desirable. Effects of anisotropic polarization, detectable in some molecules at very high field values, are treated in Section 6. The relative intensities of the Stark components of molecules without hyperfine structure depend only on J and MJ and hence are the same for all classes of molecules. The relative intensity formulas given in Section 7 are simply the MJ-dependent terms of the squared direction-cosine matrix elements given in Chapter 11.

2 LINEAR A N D SYMMETRIC-TOP MOLECULES WITH NUCLEAR QUADRU POLE CO U PLI NG The theory of the Stark effect in the rotational spectra of linear molecules with nuclear quadrupole coupling was first treated by Fano [3]; that of symmetric-top molecules, by Low and Townes [4]. Buckingham and Stephens [5] developed this theory further to take into account such cross-product terms as X6 X Qand &""gQ which have measurable effects for molecules having large nuclear quadrupole coupling. Coester [ 6 ] has treated the combined StarkZeeman effect for symmetric-top molecules with nuclear quadrupole coupling. The complete Hamiltonian for a rotating molecule having nuclear quadrupole coupling with an imposed electric field can be expressed as

H=

xr+&Q+ X

6

(10.19)

MOLECULES WITH NUCLEAR QUADRUPOLE COUPLING

459

where 2,.is the Hamiltonian operator for the pure rotational energy described in Chapter 11, XQis that for the nuclear quadrupole interaction described in Chapter IX, and X t is that just described for the Stark interaction of molecules without hyperfine structure. In finding the characteristic energies it is convenient to consider three cases separately. First is the weak-field case, in which St < X Q; second, the intermediate-field case, in which 2 8 X Q; and third, the strong-field case, in which X t 9 X Q .The weak- and strong-field cases are simpler than the intermediate one, and the greater number of measurements of Stark effect and dipole moments have been made with applied fields such that one or the other of these simple cases holds. When the nuclear coupling is relatively strong, as that for Br or I in organic molecules, the weak-field case can be used; when the coupling is relatively weak, as that for B or N, the strongfield case can be easily achieved. Perturbation theory allows the derivation of a closed formula for weak- and strong-field cases; for the intermediate field-cases a secular equation must be solved. Plural nuclear coupling will be treated in the strong-field case only, for which the effect is particularly simple.

-

The Weak-field Case When &t < HQ, the Stark interaction can be treated as a perturbation on the hyperfine state. The first-order Stark energy of a symmetric-top is then the average of the &,g over the wave function $(J, K , I, F,M F ) of the nuclearcoupled molecule unperturbed by the Stark field, that is, the wave function of the operator X r + H0. This wave function can be expanded in terms of the function for the molecule without nuclear coupling: $(J, K , 1, F, M F ) = C(JIF; ~ MJMIMF)$(J,K , MJ)$(],M I ) (10.20) MI

where the C’s are known as Clebsch-Gordon coefficients, values of which are given by Condon and Shortley [7]. (See also Rose [8].) These coefficients arise when, as here, the problem of combining two commuting angular momenta to form a resultant is considered, in particular F = J + I. These coefficients are independent of K and vanish unless MF=MJ + M I and unless F is one of the following values (see Chapter XV for further discussion). F=J+

z, J + z-

1,.

. . , IJ-

ZI

The first-order Stark term is

EV’=(J, K , 1, F , M~lXoglJ, K , 4 F, M F )

IC(JIF;MJMIMF)I’(J,K , MJI@Z~IJ, K , M J ) (10.21)

= -Pg MJ

Since the direction cosine operator Ozz is independent of the nuclear spin function $(Z, M I ) , these spin functions are factored out and normalized to unity ($I$)= 1. The quantity (J,K , MJlQZ,(J,K , M J ) has been evaluated as

460


K M J / [ J ( J +I)] in (10.3). Therefore

(10.22) The quantity in the summation is the average of MJ over the function +(J, K , Z,F , M F )which is simply the component of J along the space-fixed axis in the 1. J-coupled vector model of Fig. 10.6. This value can easily be found from the vector model. For example, the component of J along F is I J / cos (F, J), and the averaged component along Z is

( M J )=I JI COS (F, J) COS (F, Z)

(10.23)

From application of the law of the cosine with the vector model COS

+

F ( F + l)+J(J+1)- Z(Z+ 1) (F, J )= F2 J 2 - I2 2(FI JJI 2[F(F+ 1)]1/2[J(J+ 1)]1/2

(10.24)

also COS

MF (F, Z)=-= IF1

MF [F(F+

(10.25)

F ( F + I ) + J ( J + 1)- Z(Z+ 1) 2F(F 1)

(10.26)

and hence (MJ)=MF

+

Substitution of this value for ( M J )for the summation into (10.22) shows the first-order Stark energy to be

/ibKMF[F(F+ l ) + J ( J + 1)- Z(Z+ p6 -2J(J+ 1)F(F+1)

l)]

(10.27)

Fig. 10.6 Vector model of the weak-field Stark effect of a symmetric-top molecule with nuclear coupling.

MOLECULES

WITH NUCLEAR QUADRUPOLE COUPLING

461

One can derive (10.27) entirely from the vector model of Fig. 10.6 by first resolving the component p, which is wholly along the symmetry axis K, from K to J. Thus, PJ=P

K

[J(J+ 1)11’2

(10.28)

and (10.30) The Stark energy is just - d times the average dipole moment in the direction of the field. Therefore

-

E&= -p&=

PdKMF [ J ( J + l)]”’[F(F

+ l)]’’’ cos (F, J)

(10.31)

Substitution of the value of cos (F, J) from (10.24) into (10.31) yields (10.27). It should be noted that the first-order Stark effect vanishes when K =0, as it does for molecules without nuclear coupling. This condition also applies for linear molecules. When MF=F = J+ Z, the first-order formula reduces to the simple form, when MF=F = J + Z

(10.32)

which is equivalent to (10.3) for molecules without quadrupole coupling when M j = J. The second-order term in the weak-field case is

I(J,

K , 1, F, M F I X ~ ~KJ,’1, , F’, M F ) ~ ~ (10.33) F‘,J‘ EJ$K , I , F , M F - E J ~K , ,I , F ! ,MF Note that the term F =F’, J = J’is to be excluded from the sum. This expression can be separated into two sums. By substitution of the &‘&from(10.2) it becomes EF’=

E(’) 6 -p212

I

c I(J,

F’ZF

K , 1, F , MFI%zIJ, K , 1, F‘, M F ) ~ ’ EQ(F)- EQ(F’)

where the approximation EJ,K,F- FJ’,K,F’ zEJ,K- Ej,,K has been made. The first of these expressions is the larger because of the smaller energy difference in the denominator. To evaluate the terms on the right, one expresses the functions of IJ, K , Z,F, M F )in terms of those of the field-free rotor with the nucleus decoupled, as in the expansion of (10.20). When evaluated, the first term on the right contains the factor K and hence vanishes when K =O. The second term gives the second-order correction for symmetric-top levels when K =0, or for

462


linear molecules. It has the values [3,4] =O)= -

p2d2[3Mi-F(F+ 1)][3D(D- 1)-4F(F+ l ) J ( J + I)] hB2J(J+ 1)(2J- 1)(2J+ 3)2F(F + 1)(2F- 1)(2F 3)

+

(10.35)

in which D=F(F+l)-I(I+l)+J(J+1)

(10.36)

As before, the case where M F = F = J + I is particularly simple since the Stark effect is independent of the effects of nuclear quadrupole coupling. As noted previously, because of the smaller energy difference in its denominator, the first term on the right of (10.34) is larger than the second term, but it is absent unless the much larger, first-order effect of (10.27) is present. Since the field values must be sufficiently small to insure that the Stark splitting is small compared with the quadrupole splitting for the weak field case to apply, the secondorder terms are not of much value except when K =O. Hence we shall not reproduce here the rather complicated second-order term which vanishes when K =O. An explicit formula for the term is given by Coester [6]. In addition to the above terms, Buckingham and Stephens [SI show that for strong nuclear quadrupole coupling, which is necessary for practical application of the weakfield case, cross terms of the form ( J , K , 1, F ,

2 f ‘ #1f

MFI x p b IJ’,K , I, F , MF)(J’,K , I, F , MFI *Q I J, K , 1, F , M F1 E(+-E(J’)

are important. With this term included, the Stark displacement of a particular component in the weak-field case can be expressed [5] as (10.37)

where a, 6, and c depend only on the quantum numbers for the transition. The MF selection rules are MF+MF

and

MF-’MF+~

(10.38)

The AMF=O transitions are observed when the dc Stark field is parallel to the radiofrequency electric field, while AMF = k 1 transitions are observed when these fields are perpendicular.

The Strong-field Case: Single or Plural Nuclear Coupling The simplest and probably the most useful case is that in which the applied field d is sufficiently strong to break down the nuclear coupling between I and J but not large enough to perturb significantly the rotational state. This case occurs when 2 f Q < 2 8 < 2fr.It is represented by the vector model of Fig. 10.7 in which both J and I precess about the direction of the field. This model is similar to that of the familiar Paschen-Back effect observed with strong magnetic fields in atomic spectra.


463

Fig. 10.7 Vector model of the strong-field Stark effect of a molecule with nuclear coupling.

Since J and I are decoupled, the quantum number F is destroyed as a good quantum number and the J, I, F scheme is no longer appropriate. The representation appropriate to the strong-field case is J , K , Z, M J , M I , and in this representation the required matrix elements of P,., ZQ, and 2 8 are easily found. The rotational Hamiltonian 2,.is independent. of Z and M I , and the values are those already obtained in Chapter VI. Since the matrix elements of P g are likewise independent of the nuclear spin functions IZ,M I ) in this decoupled case, the values of the Stark energy are the same as those for molecules without nuclear coupling described in Section 1. Hence we have only to evaluate XQ. Under the assumed conditions, only the 2 component of the molecular field component, or Vzz =(B2V/BZ2),interacts with the nuclear quadrupole moment, and only with the Z component of eQzz. The precession of J about Z is so rapid relative to that of I that the X and Y components are averaged out. The quadrupole Hamiltonian can be expressed as ~Q=~QZZVZZ

(10.39)

With the value of (Qzzbpobtained from (9.53) with i=j=Z and with Vzz given by (9.48), it is seen that eQ9J [3Ji- J ( J + 1)][3Zi- Z(Z+ l)] (xQ)zz=4J(2J- 1)Z(2Z- 1)

(10.40)

This Hamiltonian is diagonal in the specified representation with Jz and Zz having the eigenvalues M J and M I , respectively. The quantity qJ, which is independent of M J and M I , has already been evaluated in Chapter IX, Section 7. For the symmetric top, its values are given to first order by (9.88). Thus for the symmetric top in the strong-field case the quadrupole energies are eQq E Q-4(2J- 1)(2J+ 3)Z(2Z- l ) x [3M3 - J ( J + 1)][3Mj - Z(Z+ I)]

(10.41)

464


which for the linear molecule becomes

EQ= -

eQq[3M; - J(J+ 1)][3M? - Z(Z+ l)] 4(2J - 1)(2J 3)Z(2Z- 1)

+

(10.42)

The nuclear quadrupole interaction in the strong-field case is very similar to the interaction of a quadrupole moment with an axially symmetric field in a solid. If qz2 of (9.24) for the solid state is replaced by the value of V,, from (9.48), it is seen that the same value of EQ is obtained as is given by (10.41). The strong-field Stark effect has the great advantage that coupling by more than one nucleus causes no theoretical complication, as it does for field-free molecules or for the weak-field or intermediate-field Stark effect. The value for the E, of each coupling nucleus is calculated from (10.41), and the results are added: EQ=EQ,(11 1+ EQ,( 1 2 ) + * . * The presence of magnetic hyperfine structure also causes no particular complication in the strong-field case, for which all nuclear coupling is broken down. For molecules in ‘Cstates the magnetic coupling is weak and easily decoupled. The strong-field energies are given by Em=(J, K , M J ,M I I P m l J , K , M J , M I ) = C J , K M J M I (10.43) where CJ,xdepends only on the quantum numbers J and K and is given by (9.158). The magnetic energy for the different nuclei are additive, like those for the quadrupole energy. This simplification is possible because the precession of J is so dominated by the Stark field that coupling by a given nucleus does not significantly alter the value of the electric field gradient or the magnetic field value at the other nuclei. This is similar to the situation in solids where the nuclear coupling does not alter the field gradient because the directions of the field gradients are “frozen in.” Hence the quadrupole couplings by the different nuclei of the solid state are simply additive, as in the strong-field case. In addition to the usual selection rules for M J ,we have AMri=O. The resultant energy for the strong-field case is the sum E = E,+ E r ’ + E f ’ +

Emi i

(10.44)

i

where the values of Ey’ and EY’ are those given in Section 1 and where EQi and Emiare those given above. Although the strong-field case can generally be achieved when eQq is not large, there is one important exception for K=O which occurs for integral spin values. When M J = k 1 and also M I = k 1, there is an admixture of the MJ = k 1 and the M I = T 1 states which causes a breakdown in the M J = i1 degeneracy and hence a failure of the strong-field formulas. To obtain the energy for this case one must solve the secular equation


465

for the intermediate-field case described in the next section. The abbreviation IMJ,M I ) is used for IJ, Z,M J ,M I ) . It is apparent from (10.45) that there are off-diagonal matrix elements that directly connect M J = 1 and M j = - 1. For Et(lMJI= l ) , the values from (10.10) can be used. The matrix elements of XQ are obtained from (10.53). Solution of this equation is like that for (10.55).

+

The Intermediate-field Case When the applied field is such as to make X 8 of comparable magnitude to XQ,one must treat X Q +X J as a common perturbation on the pure rotational states instead of treating H8 as a perturbation on the functions of X,+ XQ,as was done for weak fields. This can be achieved by solution of a secular equation which is obtained from %$=E*

(10.46)

where X=Xr+2fQ+&t

(10.47)

To find E, it is convenient to express $ in the eigenfunctions of the operator Hr of the unperturbed rotor

$

=E Ci$i(J, K , 1, M J ,MI)

(10.48)

Subsitution of these functions into (10.46), multiplication by t,bj, followed by integration and transformation, yield Ci[('/'jI%r+

2 8 +

x~I$i)-Edij]=O

(10.49)

i

Since Hr is diagonal in all the quantum numbers J,K , Z, M J ,M I ,and since X Q is diagonal in J, K , Z, (10.49) can be expressed as

1 Ci[(J,K , 1, M J ,MII XtlJ', K , 1, MJ,M I ) i

+(J,

K,1, M J ,MrlXQIJ, K , Mi,Mi)+(EJ,K-E)dij]=O

(10.50)

where EJ,K is the unperturbed rotational energy. Because Ht has off-diagonal elements only in J,the matrix can be conveniently reduced by a procedure that begins with evaluation of Htfor the rotor without nuclear coupling, as was made in Section 1. Let the resulting values be E t (MJ 1=Eh' '(MJ)+ Eh2WJ

(10.51)

are given by (10.3)and (10.10).The secular equation where Eh')(MJ)and Eh2)(MJ) can then be expressed as

I(J,

K , 1, M J ,MIIXQIJ,K,4 Mi,Mi)+[EJ,K+E~(MJ)-E~M,M;~M,M;I= (10.52)

This simplification is possible because the second-order perturbation in J by the Stark effect can be neglected in the evaluation of the matrix elements of

466


where

P=

eqQ 4(2J- 1)(2J+ 3)Z(2Z- 1)

and where the 6’s are all zero except when the indicated subscript values are equal. Because XQis diagonal in J, K , I, an independent set of equations is obtained for each J, K , I combination. Furthermore, .Ye, is diagonal in M = M J + M j , and hence the matrix can be factored into submatrices with common values of M J + M I . In writing out the matrix it is convenient to group together the common values M = M J + M I . For the maximum values M J = J, M I = I, and M = I+ J , the secular equation is of first degree and gives

E =E j , K + E 8 ( M j =J)+ ( M j = J, MI = I1 2 ~ l M=j J, M j = I )

(10.54)

when M = I+ J - 1, the secular equation is the quadratic

In these equations we have used the abbreviation IMJ,M I )for I J , K , I, M J , M I ) . The equation of next higher order, that for M = M J + M I = J + I - 2, is a cubic; the next, for M = J + I - 3, is a quartic. High-speed computers can be used for solution of these equations of higher power when it is desirable. However, one can often employ transitions involving only the energy values obtained from equations of first and second power for accurate evaluation of dipole moments. The relevant selection rules for M are AM=O, f 1 ; M J and M I are no longer good quantum numbers. Explicit values of E for symmetric rotors ( K # O ) for which M = M J + MI= J + I are derived from (10.54).


467

pLbK p 2 6 2 [ ( J + 1)2- K Z ] E J ,K , M j = J . M , = I =EJ,K- __ J+ 1 2hB(J+ 1)3(2J+3) - J(J+ l)] +eQq[3K2 4(J+ 1)(2J+ 3)

(10.56)

For linear molecules this equation reduces to E J , M=~J , M , = I=EJ-

p w

eQqJ

(10.57)

2hB(J+ 1)(2J+ 3) -

With a Stark spectrometer which permits measurement of the CJ component M + M ' ( = M + 1) of the J - J ' ( = J + I ) transition, one can measure frequency shifts determined entirely by (10.56)or (10.57). From these shifts one can obtain p and eQq. The quadratic equation (10.55) is readily solvable and yields energies of the levels for M = M J + MI = J + I - 1 which are E + = E j , K ++(U

+ fl

+U

+ b ) i + [ ( U - fl + U - b)2+ 4 ~ ~ ] ' ' ~ (10.58)

where

-_ -

}

p I K ( J - 1) (/A&)' (5'- K 2 ) 4J[(J+ 1)2- K2] J(J+ 1) +2hB{J3(2J+ 1) - (J+ 1)3(2J+ 1)(2J+3)

(10.60)

(10.62) (10.63) P=

eQq 4(2J- 1)(2J+ 3)Z(2Z- 1)

(10.64)

The levels corresponding to E + and E - in (10.58) represent a mixture of MJ = J and M J = J- 1 states. The r-type transitions (AM=O) are possible from, or to, either the plus or minus levels to other levels having the same M values, provided that the selection rules for J and K are not violated. A commonly observed transition for measurement of dipole moments is a n-type one from the M = M J + M I = J + I (the energy of which can be calculated with 10.56) to the next highest rotational level, represented by J' = J+ 1, M' = J' I- 1 = J+ I. Since M = M ' , this is a AM=O transition which is observed with the dc electric field parallel to the electric vector of the microwave radiation. However, a doublet

+

468


will be observed that corresponds to transitions to the plus and minus substates having the same M value. The relative intensities of the doublet will depend on the relative weights of the M$ = J‘ and M i = J’- 1 components in the admixed states. These weights can be found by substitution of the E* values into (10.50) followed by solution for the C’ values. The relative intensities will be proportional to the C’ values of the two admixed functions. In deriving the secular equation for the intermediate-field values Low and Townes [4] used the representation J , Z,F , M in which X Qis diagonal and X J is therefore not diagonal. This representation is a bit complicated since one must then express the basis function in terms of the set IJ, K , Z,M J ,M I )in which the matrix elements of 2 8 are known. This transformation can be achieved by use of (10.20) with the Clebsch-Gordon [7, 81 coefficients.

3 ASYMMETRIC-TOP MOLECULES WITHOUT NUCLEAR COUPLl NG The theory of the Stark effect for asymmetric rotors has been developed by Penney [lo] and by Golden and Wilson [ll]. The Stark effect for asymmetrictop molecules is calculated with perturbation theory in a manner similar to that for linear or symmetric rotors. However, the evaluation of the direction cosine matrix elements becomes more involved since the wave functions for asymmetric rotors are considerably more complicated. With symmetric or linear molecules the dipole moment was associated with one of the axes, whereas for asymmetric rotors we have now the possibility of permanent molecular dipole components along each of the three principal axes of inertia. The Stark effect Hamiltonian operator will thus have the form given in (10.1).The application of an electric field will, via the interaction of the dipole moment and the field, perturb the rotational motion and thus the rotational energy levels. In particular, the levels will no longer be (2J+l)-fold degenerate in the space orientation quantum number M J . Each energy level of the asymmetric rotor consequently splits up into a number of sublevels corresponding to various values of lMJI. In a perturbation treatment of 2 8 , the asymmetric rotor functions IJ, 7,M J ) for the unperturbed or field-free rotor are used to give the matrix elements of the direction cosines. The relevant direction cosine matrix elements in an asymmetric rotor basis may be expressed as (J,7,M J I @ Z g I J ’ ,

T’, M J ) = ( J I @ Z g I J ’ ) ( J ,

TI@ZglJ‘,

T ’ ) ( J , MJl@ZglJ’,M J )

(10.65)

The J and J , M J dependent factors are the same as for a symmetric rotor basis and are found in Table 2.1. The direction cosine matrix elements will be nonvanishing only if the product of the symmetries of I//JrM,, Q Z g , and $ J ; , M , belongs to species A . The symmetries of the direction cosines (see Chapter VII, Section 4) and their products, under the rotation group V ( x , y , z), are summarized in Table 10.1.The correlation of the species of V(x,y , z ) with those of V(a,b, c) may be readily found by use of Table 7.5. With the aid of Table 10.1 the possible


469

The Direction Cosine Symmetries in V ( x ,y , z ) Table 10.1

Direction Cosine

Symmetry

nonvanishing matrix elements of YYt may be found. These are listed in Table 10.2. It is apparent that the perturbation connects the various diagonal blocks of .yi",. From (10.71)-(10.73), which give the direction cosine matrix elements in terms of the line strengths, we see that the connections are of the type (JIJ) and (JIJk 1). The complete energy matrix with 2 8 included can, of course, be constructed and subsequently diagonalized to give the energy levels [121. Usually, however, perturbation techniques are sufficiently accurate. The diagonal elements of 2 8 , in the basis which diagonalizes .Ye,, are the first-order perturbation corrections. However, since X t has (from Table 10.2)no diagonal matrix elements, there should be no first-order effects. Usually the Stark effect for asymmetric rotors will be of second-order, that is, proportional to 8'. It frequently happens, however, that a pair of interacting levels for the unperturbed rotor are degenerate or nearly degenerate. Under these conditions a first-order effect can arise. This situation will be deferred until consideration is given to the case where the unperturbed energy levels are nondegenerate, that is, widely separated.

The Nondegenerate Case From Table 2.1 it is seen that the Ozgmatrix elements are diagonal in M,; hence, the total rotational energy with a Stark field perturbation can be expressed for a specific value of M J as: (10.66)

Table 10.2 Structure of the X 8 Matrix

470


where EJq is the rotational energy of the unperturbed rotor, and the secondorder Stark energy is given by the conventional nondegenerate perturbation expression as

The summation over J’, 5‘ includes all states that interact with the state J,, that is to say, all states which are connected by a nonvanishing direction cosine matrix element. The level Jz,however, is to be excluded as indicated by the prime on the summation. There are no cross products of dipole components in (10.67) because of symmetry restrictions (see Table 10.1).The total Stark energ) of a level characterized by J , 7,MJ is the sum of the contributions arising from each dipole component. As mentioned previously, the only nonvanishing direction cosine matrix elements are those for which J’=J- 1, J, and J+ 1. If (10.65)is used to separate out the J and MJ factors, the energy shift arising from the gth component of the permanent dipole is [1I]

I(J, +4 ( J +(J+l)’-MJZ 1)’(2J+ 1)(2J+ 3) c T‘

T1@ZSIJ+

1, T’)I2

E:,- E!+

1,.

1

(10.68)

This expression in conjunction with (10.66)yields to second order the rotational energies in the presence of an applied electric field. The Stark energies are seen to depend on Mf, and hence the (2J+l)-fold degeneracy in M J is partially removed, a given rotational level being split into ( J + 1) distinct sublevels. To calculate the Stark shifts one must know the dipole moment, the direction cosine matrix elements, and the energy level differences. In many instances, some of the energy differences will not correspond to observed frequencies and will have to be evaluated by known rotational constants. Conversely, when the energy levels and Stark shifts are known, the dipole moment may be calculated. The evaluation of I(J, T I O ~ ~ T’)[’ ~ J ’ ,in an asymmetric rotor basis is rather tedious for any except the lowest J values. Separating the M j dependence, Golden and Wilson [ll] write (10.68) in the equivalent form, (10.69)

where M. = ( A- C ) / ( A+ C) and the ( A + C)/2 dependence of the Eo’s has been factored out. They have tabulated these reduced Stark coefficients A j , andBJr for various values of M. and K up to J = 2 . By means of these tabulated quantities, the second-order Stark energies may be calculated with a minimum of effort.


471

For higher values of J, the calculations can be further simplified if it is noted that the matrix elements of interest are related to the line strengths, (7.58), for which extensive tables are available. In the absence of external fields, X , I:and 2 are equivalent, and the line strengths may be written as ng(J,

T;

J’,T‘)=3

1 I(J, M

T,

MJl@ZgIJ’, T‘, MJ)12

r

1

(10.70) In writing this expression we have taken cognizance of the fact that Ozg has nonvanishing elements only for M;=MJ. The term in brackets may be readily evaluated. Using Table 2.1 and carrying out the sum over M J , noting that i J E M I = - J Mj=(+)J(J+ 1)(2J+ l), one obtains:

Ag(J, 7 ; J+ 1, T‘)=

(10.71) (10.72) (10.73)

Thus the required matrix elements I(J, 7J@zglJ’,T’)\ may be evaluated from the foregoing equation and the tabulated line strengths [13]. Interpolation of the line strength tables may be used when necessary. As J increases, the number of possible interacting levels increases rapidly, especially when there is more than one dipole component. However, in many cases certain of the matrix elements or line strengths connecting two levels will be vanishingly small and can be neglected. Usually the electric field applied is such that A M j = O transitions are observed. A completely resolved spectral line with A J = & 1 will have (J+ 1) Stark components, while for AJ=O there will be J components, where J is the smaller of the two Ss involved in the transition. These cases are illustrated in Fig. 10.8. The loss of one component for AJ=O transition is due to the vanishing intensity of the M J =O component which is apparent from the intensity expressions given later. According to the magnitude and sign of the Stark coefficients for the two levels involved in the transition, the Stark components may be on one or both sides of the zero-field line. When the Stark coefficient differences, that is, AA and AB, have the same sign, the largest displacement will occur for the Stark component associated with the largest possible M J value. The evaluation of the lMJl associated with a particular Stark component can be made if necessary from measurements on the frequency displacements [113. Let the frequency shift for a given MJ component be expressed as AvM, = d 2 ( A ’ + B ’ M i )

(10.74)

472

EFFECTS OF APPLIED ELECTRIC FIELDS I

I>

L

I

E = 4.9 esu/cm

000-

Stark Component M=O

(I,

Fig. 10.8 ( A ) Recording of the 2 0 2 + 2 1 1 transition of SO2 at 53,529 MHz. (8)Recording of the Ooo+ 1 transition of SO2 at 69,576 MHz. Obtained with a Stark-modulation spectrometer, employing phase-sensitive detection. After Crable [151.

The MJ value for successive components will differ by unity and the following relations for three consecutive components may be written

-A v M J

T b’B‘(2Mj 1)

(10.75)

A V M ~ ~ ~ - A V TMb ~2 B~ ’ ~( 2=M j ? 3

(10.76)

AvMJ

1=

Taking the ratio of the component separations gives r = AVMJ-AvMJ f 1 --~ M fJ1 ~ MfJ3 A v M ~f 1- A v h l ~f 2

(10.77)

which in turn yields (10.78) The values of M J can also be related to the relative intensities of the components [l 13; however, frequency measurements can be made with greater accuracy, and hence the lMJl can be most satisfactorily estimated from the frequency displacements. With very large electric fields, higher-order Stark corrections may need to be considered. For nondegenerate levels the odd-order corrections vanish [l I],


473

and hence the next correction will be proportional to g4.If fourth-order Stark corrections are important, the procedure for handling the data discussed in Section 1 can be employed which, although explicitly including fourth-order effects, does not require a knowledge of the fourth-order Stark coefficients. The second-order Stark coefficients in (10.18) are obtained from (10.68). This method may be generalized for the case where the Stark effect depends on more than one dipole component pe, but additional intercepts from different transitions (e.g., different MJ values) will be required in the solution for the dipole components. As an illustration of the application of (10.68), consider the evaluation of the dipole moment of SO2 from the Stark splitting of the Oo0+ 1 transition for which only one Stark component is observed, viz., MJ=O(see Fig. 10.8), and no near-degeneracies occur. The symmetry of SO2 is such that the dipole moment will be entirely along the twofold axis of symmetry which is the b axis. Therefore, the levels which perturb the Ooo level by means of the dipole component pb must have symmetry Bb. Thus only the 1 level perturbs the Ooo level. O n the other hand, the 1 level interacts via the Stark perturbation with the Ooo, 202,and 220 levels. The spectroscopic constants of SO2 are [14]: A =60,778.79 MHz, B=10,318.10 MHz, and C=8,799.96 MHz. Keeping only three decimal places we find K = -0.942. The reduced energies of the various levels may be obtained from the expressions of Tables 7.6 and 7.7 or existing reduced energy tables. The total rigid-rotor energy of the various levels is found to be (MHz): Eooo=O El,, = A

+ C=69,578.75

E202=3(A

+ C)+- ( A-2

(- 5.8273)=57,288.13

= 3(A

+ c)+---( A-2

(2.0593)=262,256.25

The appropriate matrix elements of the direction cosines are obtainable from the tabulated line strengths and (10.71)-(10.73). The necessary quantities may be summarized as follows:

000-111 111-202 111-220

1.oooo 0.5191 1.4809

4.0000 4.1528 11.8472

These are for a K of -0.95 from [13] which is sufficiently accurate for our purpose. In practice, however, an interpolation would be carried out to give the values corresponding to the observed K . With these results, the Stark energy for the Ooo and 111 levels may be readily calculated from (10.68). The

474


separation Av of the Stark component (MJ=0) from the zero-field rotational transition is given by the following expression AV= [EA2'] i l l - [ E $ 2 ' ] 0 0 0=(0.50344)2&F2[9.3973- ( - 4.79071 x = p i d2(0.50344)2(14.1880x Here the conversion factor 0.50344 (MHz) (debye volt/cm)- is included SO that the Stark splittings may be expressed in megahertz with pg in debye units, and d in v/cm. For d=2005.1 v/cm, the observed Stark splitting is found to be Av=36.2 MHz [15], and one obtains from the above expression p = p b = 1.58 D for the dipole moment. The dipole moment obtained with interpolation and measurement of different splittings is 1.59 fO.O1 D [16].

The Degenerate Case In asymmetric rotors degeneracies frequently occur; both approximate symmetric-rotor degeneracy and various types of accidental degeneracies are possible. Table 10.3 indicates for 5 < 3 which pairs of levels can become accidentally degenerate. When degeneracies are present, (10.68) obviously fails, since it will contain terms with vanishing denominators. Thus, if rotational degeneracies or near-degeneracies are encountered, a simple second-order perturbation treatment is not applicable. Van Vleck [17], however, has developed a convenient perturbation technique which is very effective for dealing

Table 10.3 Possible Degeneracies in the Asymmetric Rotor' J L I K ,

000

101

111

110

202

212

211

221

22,

"From Golden and Wilson [ll]. Entries indicate which component of the dipole moment becomes important in the accidental degeneracy. The asterisk denotes no coupling term. Blanks indicate no accidental degeneracy possible.


475

with the problem of degeneracies or near-degeneracies. It consists of application of a transformation to the energy matrix, that is, the matrix of X r + 2 “ ~ evaluated in an asymmetric-rotor basis. The transformation is so constructed that the elements connecting degenerate blocks after the transformation are reduced to second order and can hence be neglected for results correct up to fourth order in the energy. This perturbation technique is discussed in more detail in Appendix C and in the section on internal rotation. The net result of the transformation is that one obtains a number of small submatrices, each associated with a group of degenerate or nearly degenerate levels. The diagonal elements of these submatrices contain correction terms from the Van Vleck transformation and are given by (10.66) and (10.68), except that in the case of degeneracies or near-degeneracies the summation is only over those levels for which E;,,, is not near E& The perturbed energies are then found from solution of the usual secular determinant. Usually the degeneracy occurs only between a pair of levels; the secular equation then has the form (10.79) where 5 is the appropriate off-diagonal matrix element. The two possible solutions are

where for E j , M J > E j , r , M J , the plus sign is associated with the level J,Mj and the minus sign is associated with the level &MJ, since two such interacting states repel one another. The off-diagonal elements are made up of the Stark coupling term for the levels and correction terms from the transformation

St =(.TI, , MJI XtlJ’,T’, MJ)+Correction terms

(10.81)

These off-diagonal elements depend on the two levels under consideration, and Golden and Wilson [111 have given expressions for various types of neardegeneracies. We shall discuss only the case of near-degeneracies between asymmetry doublet states, which is often encountered even in rather asymmetric rotors. These levels will appear in pairs having different symmetry, except for the K=O level which is nondegenerate in K . For a near-prolate rotor the pairs of K levels will have either symmetries A , B, or symmetries Bb, B,, whereas for a near-oblate rotor the symmetries will be B,, Bb or A , B,. For two members of an , interact in the presence of an electric asymmetry doublet, J , and .I,,which field by means of a dipole component pg, the off-diagonal matrix element connecting the two states will have the simple form

where g is a or c depending on the symmetry of the two near-degenerate K levels. The secular determinant, (10.79), with the off-diagonal elements given

476


by (10.82), is just what is expected from conventional perturbation theory for two such nearby levels, except that the diagonal elements here contain, in addition, the second-order Stark contributions of the remaining nondegenerate levels. Each of the two levels, J, and J7,, splits into ( J + 1) distinct sublevels given by (10.80) and (10.82), the + M J and - M J levels coinciding. If IEJ~M~-EJ,.M O to exhibit a first-order Stark effect that is dependent on the barrier height. This is discussed more fully in Chapter XII. It suffices to say here that because of the presence in the Hamiltonian of a linear term in the angular momentum, the + M J degeneracy can be removed and a first-order Stark effect obtained. This splitting between the + M J and the - M J Stark components is sensitive to the barrier height and thus allows a means of its evaluation. The degree of planarity of molecules and the nature of the potential function which governs the out-of-plane bending vibration are questions in which the Stark effect can yield valuable information. A discussion of this may be found elsewhere [19, 201.

478


Previous Page

4 ASYMMETRIC-TOP MOLECULES WITH NUCLEAR QUADRUPOLE COUPLING As with linear and symmetric-top molecules, the presence of nuclei with quadrupole coupling complicates the Stark effect of asymmetric rotors. Plural nuclear coupling will be treated for the strong-field case only. The appropriate Hamiltonian is given by (10.19)with YYr, now the rotational Hamiltonian for an asymmetric rotor. For the weak-field case where the Stark energy is much smaller than the quadrupole energy, the functions appropriate for the field-free Hamiltonian YYr + X Qare the IJ, T , I, F , M F )basis functions for which J2, Iz,F’, and MF are all diagonal. A particular level is characterized by the set of good quantum numbers J, I, F, MF and the pseudo-quantum number T . In the weakfield case when an external electric field is applied, the degeneracy of the spatial orientations of F will be removed. Each F level is split into a number of different Stark levels ( F + 1 distinct levels for a second-order Stark effect) corresponding to the values of M F . The basis functions required for the perturbation treatment of 2 8 may be constructed from product functions of the orthonormal asymmetric-rotor ) the orthonormal nuclear spin functions + ( I M I ) wave functions $ ( J T M J and as follows

The expansion coefficients, C’s, are the Clebsch-Gordon coefficients [7, 81 noted previously. These coefficients are independent of 5. For asymmetric rotors without degenerate or near-degenerate rotational energy levels, the second-order Stark perturbation energies of the hyperfine levels are given by (10.89) with

Usually the separation of the rotational energy levels coupled by the Stark perturbation are large compared to the hyperfine splitting of these levels and E j , , I , F - E j ; , , I , F , z E J , - E j ; , . With this approximation and the substitution of (10.88)into (10.90) one obtains, upon integration [4,21, 221, (10.91) Note that

1C*(J’IF’;MjMrMF)C(J‘IF‘;MJMIMF)=1 C’

Expressing [Ef)]J,,M,

as in (10.69) gives

(10.92)

ASYMMETRIC TOPS WITH NUCLEAR QUADRUPOLE COUPLING

EM,

479

cM,

since IC1 = 1 and IC12MJ” = (MJ”).This latter quantity is simply the quantum mechanical average of MJ”in the J, Z, F , M F representation and when evaluated can be expressed as [3] [3M;-F(F+

(W=

1)][3D(D- 1)-4F(F+ l)J(J+l)] J(J+ 1) 6 F ( F + 1)(2F- 1)(2F+3) 3

+-

where D = F ( F + l)+J(J+1)- Z(Z+l). The Stark coefficients AJr and Bj, are those discussed previously where quadrupole coupling was absent. When Z=O, it is easily seen that (MJ”)= M $ = M j and hence, (10.93) reduces to the Starkeffect expression for asymmetric rotors without quadrupole hyperfine structure. When a nucleus with a quadrupole moment is present, (10.93) indicates how each hyperfine level characterized by J, T, F is further split by the Stark effect into ( F + 1) sublevels corresponding to the values of M F , each MF level being doubly degenerate except for MF=O. A given hyperfine line of a rotational transition will thus exhibit a number of Stark components. In addition to the selection rules for the unperturbed rotational transition we have AF=O, k 1 ; and for the usual experimental set up have AMF =O. When the hyperfine splitting as compared with the rotational splitting can$ Ejr- Ej;,, then the expression not be ignored, that is, when Ej,,,,F- Ej:, becomes more complicated. The evaluation of the direction cosine matrix elements in the IJr,I, F , M F )basis has been carried out by Mizushima [22], using Racah’s [23] method. The Stark energy for the weak-field case is now given by

480


in which the A i s are the line strengths and the f coefficients are given in Table 10.4. For the strong-field case where the Stark energies are much larger than the quadrupole coupling energies, F is no longer a good quantum number. This case may be obtained if the quadrupole coupling constants are small, or if the electric fields are sufficiently large. The IJ, 7,M J , I, M I ) functions are suitable basis functions for treating the strong-field case. In this representation J2, 12, MJ, and MI are diagonal. A particular level in the strong-field case will be characterized by the set of quantum numbers J , I, M J ,M I , and the label z. The Stark levels are those determined previously for asymmetric rotors without quadrupole coupling effects. If only diagonal elements need be considered, that is, if 2 8 is large enough that the effect of the off-diagonal elements of X Qcan be ignored, the quadrupole perturbation energies of the Stark levels [see (10.53) and (10.98)] are given by

Eg’=(J,

7,

=eQqJ

1, MJ, M I I ~ Q I JT,, 1, M J , Mr)

[3MJ’- J(J+ 1)][3Mj - Z(Z+ l)] 4J(2J- 1)1(2Z- 1)

(10.96)

where qJ is given by (9.88). The foregoing expression indicates how each Stark level characterized by J, z, M J is split into a number of hyperfine levels identified by different values of M I . Note that the + M I and the - M I hyperfine levels have the same energy. A given Stark line of a rotational transition will thus have an accompanying quadrupole hyperfine structure. The additional selection rule AMI=O is applicable. If two or more coupling nuclei are present, their respective quadrupole energies may be simply added because of the nuclear decoupling in the strong-field case.

Table 10.4 f Coefficients for the Stark Energy in the Weak-field Case“ fi(JIFM,)= f,(JIFM,)= f,(JIFM,)=

f,(JIFM,)= f5

(JIFMF1=

M:(J+ I + F + 2 ) ( I + F - J ) ( I + J + 1 - F ) ( F + J + 1 - I) 4F’(F+ 1 ) 2 ( 2 J + 3 ) ( 2 J +l)(J+1 )

( J + I+ F

+ 3)(J+ I+ F + 2XJ-

I+ F + 2)(J- I+ F + 1)((F+1j2 -

4 ( F + 1)’(2F+3)(2F+ 1)(2J+ 3)(2J+ l ) ( J + 1)

( I + F - J ) ( I + F - J - 1)(I+J - F + 2 ) ( I + J - F + 1)(F’- M;) 4 F 2 ( 2 F - 1)(2F+ 1)(2J+3)(2J+ 1 ) ( J + 1) (I+ J+F+2)(I+ J-FXZ+F-

J + 1 X J + F - I+ l){(F+1j’ - M:}

4 J ( J + 1)(2J+ 1)(F+1)’(2F+ 3)(2F+ 1)

M:{J(J+ ~ ) + F ( F +1)-1(1+1)}~ 4 J ( J + 1)(2J+ l)F’(F+ 1)’

“From Mizushima [22].


481

For the intermediate-field case, whether the weak-field or the strong-field quantization scheme is chosen, both diagonal and off-diagonal matrix elements are important, and it is necessary to solve secular equations to obtain the desired energy levels. Mizushima [22] has given the following secular equation for the intermediate-field case in the IJ, Z,M J , M I )basis

I(J,

T, 1, MJ,

MIIxQIJ,

T, 1, Mi,M;)+(EJ,,I=O,M,-E)BM,,M;BM,,Mjl=O(10.97)

Where E j , , r = O , M , is obtained from (10.66). The matrix elements of SQare given by (10.53)where P=(+)eQqJ[*J(2J- 1)1(21I 1) 1

(10.98)

with qJ given by (9.95). The secular equation can be factored into blocks for each value of M = M J + M I , since matrix elements of xQbetween different M's are zero, as noted previously. For the usual experimental arrangement the selection rule for M is AM=O. In the transition from a weak-field case to a strong-field case, M remains a good quantum number, becoming MF in the weak-field limit and M J M I in the strong-field limit. The strong-field Stark effect for molecules with two quadrupole nuclei of spin: has been discussed by Howe and Flygare [24]. They note that the J=O+1, MJ=O component will, in the strong-field limit, appear as a symmetric triplet with the center component the most intense and independent of the quadrupole coupling. This obviously simplifies the determination of the dipole moment. Eagle et al. [25] have discussed the Stark effect when 2 8 , X Qand rotational near-degeneracy are comparable. Off-diagonalelements of X Qand Y f 8 coupling near-degenerate rotational levels must be considered since they can make significant contributions to the energy. Approximations are discussed, and the secular equation applicable to planar near-prolate rotors such as NOCl is given in the IJ, t,M J , Z,M I )basis.

+

5

MOLECULES IN EXCITED VIBRATIONAL STATESINVERSION DOUBLETS AND / DOUBLETS

Although the theory described in other sections for the Stark effect in rotational spectra of ground vibrational states is also applicable to molecules in excited, nondegenerate vibrational modes, it is not applicable to I-type doublets of degenerate bending modes or to vibrational modes with inversion doubling. The Stark effect of I-type doublets and inversion doublets can be treated with the same theory. The treatment given here is very similar to that in Section 3, where the Stark effect of near-degenerate levels in asymmetric rotors is considered. The two near-degenerate levels of these doublets are assumed to be close in energy as compared with the energy separation from the doublets of all other levels of the molecule with which intermixing induced by the applied field is possible, that is, with which the off-diagonal elements of &'t are not zero.

482


This approximation holds for inversion doublets of symmetric tops like N H and for the I-type doublets commonly observed in linear molecules. In this approximation the problem can be treated as a two-level system with eigenfunctions $(f') and $bo) corresponding to the doublet energies E\'' and E',"'. The Hamiltonian operator of the perturbed system is

x=X(')+ xt

(10.99)

Since there are only two states, the wave functions of the perturbed can be expressed as the linear combination of the functions $1') and $$'I. these functions are eigenfunctions of %(O) but not of X g , there are diagonal elements of X(') and no diagonal elements of 2 8 . Hence the equation is

system Since no offsecular

(10.100)

where (11X(')11)=E$'),(21iW(0)12)=E(?O),and (11X&j2)=(212?8/1)= d(lIp12). Solution of the equation yields the perturbed energies For the symmetric-top inversion levels, the matrix elements are

and the Stark-perturbed levels can be obtained by substitution of these elements into (10.101).The average of the two unperturbed levels (E\')+ E$0))/2=EJ,Kis simply the rotational energy the molecule would have in the absence of inversion doubling; it is given by (6.14). The energy difference E$')-E$')=hv inv where vLnVis the inversion frequency for the particular rotational and vibrational state. The energies in the presence of a field d can thus be expressed as (10.103) When the inversion splitting is zero or negligible, as it is for all symmetric tops except ammonia, the formula becomes that of the ordinary first-order effect for a symmetric-top. In Chapter V, Section 2 it is pointed out that the linear molecule in an excited bending mode has angular momentum Ih about the figure axis. Its motion simulates that of a symmetric top with I-. K. As was first shown by Penney [lo], the off-diagonal elements of p connecting the I-type doublet states are like those of the symmetric top with 1 replacing K . For the I doublets, therefore (10.104) The Stark-perturbed I-doublet levels where E\')> E$') are then


483

and

I t is evident from these equations that when the Stark perturbation is large as compared with the I-doublet separation, the Stark displacement is a firstorder effect, like that of the symmetric top. When the Stark splitting is small as compared with the doublet separation, a quadratic Stark effect is observed. This can be seen if (Elo)- E$o))2is factored from the radical and if the term under the radical is expanded by the binomial theorem. When (E$')- Ei'))' 9 [(2pblMJ)/J(J+ the resulting second-order formulas can be expressed as

(10.107) and (10.108) The Stark components of the higher doublet level (1) are shifted upward, and those of the lower level (2) are shifted downward. the first-order In the region where [(2pblMJ)/J(J+ 1)129 (E\')effect holds, and the M j + M j Stark components of the rotational transitions J+J+ 1 have the frequencies

(IO. 109) (IO. 110) where vLo) =(vio)+ v(2'))/2 is the rotational frequency that would be obtained if the 1 doubling were not present. This approximation, which neglects the 1 doubling (Eio)-E$o)), shows that the linear molecule can have a first-order Stark effect when in excited bending vibrational modes. Although the doublet energies El and E 2 for the same J and MJ are pushed apart, the M j - M j components of the rotational frequencies v 1 and v2 are pushed together by the Stark field. Thus for a particular field value the Stark components of v1 and v2 for the same M J values will coincide. When this coincidence occurs in the region where the first-order effect holds, the field value for the intersection is (10.111) where (Av)") = vio)- ~(2') is the separation of the frequencies of the rotational doublets in the absence of a field. Figure 10.10 from Strandberg et al. [26] shows the Stark frequencies as a function of 6 for the lMJl = I and 2 Stark components of the J = 2 + 3 transition

484


24,160.00 58.00 56.00 54.00

2 5 c

s

'-

52.00 50.00

c

$ 48.00

?!

U

46.00 44.00 42.00 40.00 38.00

0 E , in voltslcrn

Fig. 10.10 Stark effect of /-type doublets. Plots of the lMJl =1 and 2, n components versus field strength for the J=2+3 transition of OCSe in the u2 = 1, I = 1 vibrational state. From Strandberg et al. [ 2 6 ] .

of OCSe in the u2 = 1, I= 1 bending vibrational state. The components for the same lMJl cross at the field values given by (10.111).This crossing point provides a convenient method for evaluation of p since the only frequency difference which must be measured is the field-free (Av)'') which is greater than the Stark splitting except at very high fields.

6 ANISOTROPIC POLARIZABILITIES FROM THE STARK EFFECT In the preceding discussions we have not considered the Stark effects of induced dipole moments caused by polarizability of the molecule. This omission is justified because the small effects of polarizability are seldom detected and indeed for most molecules are probably not measurable with meaningful accuracy. However, with favorably selected molecules and with accurately


485

ground parallel plate Stark cells which will permit measurements up to 50,000 v/cm it is possible to measure polarizability anisotropy with useful accuracy. As early as 1957, Marshall and Weber [27] detected polarizability effects in their measurements of the Stark effect of OCS, from which they derived a rough value of polarizability anisotropy. They made measurements with fields up to 24,000 v/cm on Stark components of the J = 1 - + 2 transitions. More recently, in 1967, Scharpen et al. [28] extended the measurements to higher J transitions and to higher fields (up to 50,000 v/cm) and thus achieved better discrimination between the effects of the permanent and induced moments and thereby obtained a more reliable measurement of the polarizability anisotropy in the molecule. They obtained for OCS the value (a,,- a,) = (5.34 & 0.06) x cm3 where all and a, are the polarizabilities parallel and perpendicular to the molecular axis. A reevaluation has led to correction [29] of this value to 4.63 x cm3. Mizushima [30] has predicted polarizability effects for several diatomic molecules and for HCN. Most of these molecules have nuclear quadrupole coupling which complicates the measurement and evaluation because of the changing degrees of coupling as the field is increased. Mizushima derived the formulas for the weak-field case, but these are not very useful because of the high fields that must be applied to produce measurable polarizability effects. Here we shall consider only molecules without nuclear coupling. The Hamiltonian for the interaction of a field with an induced moment pa is - pa 8,but here, in contrast to the dipole interaction considered earlier, pa is itself a function of the field strength. Because the polarizability is anisotropic, the induced moment is also a function of the orientation of the molecule in the field. Generally, the polarizability is a tensor quantity, and the principal axes of the polarizability do not necessarily coincide with the principal axes of inertia. However, in linear or symmetric-top molecules considered here, and in many asymmetric tops, these axes are coincident. If g =x, y , z are the principal axes of the polarizability tensor in the molecule, the classical energy which is due to the polarization by the field is 9

--_

a-

11

g=x,y,z

a'es&'g2

(10.112)

where clggrepresents the principal values of the polarizability and 8,represents the components of the field with reference to the molecule-fixed axes. Since &' is imposed along the space-fixed axis chosen as Z,

(IO.113) 8,= Cmzg where QZg represents the direction cosine of the axis g with 2. Therefore, the polarizability Hamiltonian can be expressed as

+

+

yi", = - -p2(~,,@;, "yy@;y a&&)

(10.114)

For symmetric-top or linear molecules, the polarizability axes are the same as the inertial axes, and only the diagonal elements of the polarizability tensor,

486


a, are nonzero. Although there are no permanent components of the dipole moment normal to the symmetry axis, there are components of the induced moment perpendicular to the molecular axis even in linear molecules. Nevertheless, the polarizability is axially symmetric about the figure axis z, and u e can set cixx=a,,=a, and azz=ctll. With these conditions, with the omission of M,&’ which does not depend on the orientation, and with the relation = 1, (10.114) transforms to

1

,yi“,= -+(all -MI)&’@&=

(10.11 5 )

-+Ma2@,BZ

where for simplicity we designate (10,116)

ci=cill-ML

The omitted term shifts all levels by the same amount and hence has no effect on the observed spectra. The first term in the contribution of the polarizabilitj to the Stark energy is the average of yi“, over the unperturbed wave function IJ, K , MJ).Actually, this represents a second-order contribution to the total Stark effect. Since QZzis diagonal in K and M J , EJ,x,M,(CO=(J,K , MJI yi“.lJ, K , M J ) = -+ci&’(J, K , MjI@izIJ,K , M j ) = -$a&’

1(J,K , MjI@zZIJ’,K , M J ) ~

(10.117)

J’

The direction cosine matrix elements, which may be obtained from Table 2.1, are the same as those used in calculation of the second-order term of the permanent moment. In practice it is unlikely that the polarizability term will be observable for levels with K / O when the large first-order contribution from the permanent moment is present. Hence we shall write the specific formula only for symmetric-top levels of K =0, which applies for linear molecules also. When the direction cosine matrix elements for K=O are substituted into (10,117)it is seen that E j , ~ , ( c i ) =- 4 ~ 8 ’

]

J~ - MJ” (J+1)2- MJ” (2J+ 1)(2J+3)+(2J+ 1)(2J- 1)

(10.118)

This term is added to that caused by the permanent dipole moment, given by (lO.ll), for a total Stark perturbation to second order: EJ,M , =EJ,M,(PI

+ EJ,M,

(10.119)

The displacement of a particular Stark component can be expressed as

AV= (a1p’

+u ~ E ) & ’=ab’

(10.120)

where a l and a2 depend on the quantum numbers but are constant for the particular Stark component. These constants are calculable: a l from (10.11) and a2 from (10.118). By measurement on two or more different Stark components with different values of a l and a’, both p and ci =(xII - x , ) can be obtained. Notice that a l decreases more rapidly with increasing J than does a2.


487

Thus for measurement of the small term in cx, it is advantageous to use relatively high J values with very high field values to achieve splitting which are accurately measurable. When carried to higher order, the Stark shift of a particular component can be expressed as

+

Av = a B 2 b d 4

(10.12 1)

a=alp2+a2cc

(10.122)

b=blg2+b2gp2+b3p4

(10.123)

where

and If (10.121) is divided by tZ2, it is seen that a plot of values of Av/d2 versus b2 gives a straight line with b as the slope and a as the intercept. Thus a values, from which p and c( are obtained, can be evaluated from (10.121) without specific knowledge of the coefficients b l ,b2,and b3.However, Scharpen et al. [28] give the higher-order formulas required for calculation of these constants if they are needed. Only the b3p4 term is likely to have measurable effects. It is interesting to compare the anisotropic polarizability obtained by the Stark effect for OCS with the values of 2.2 x cm3 for C 0 2 and 9.6 x cm3 for CS2 evaluated by other methods [31]. The average of these two values cm3 is not far from that of 4.63 x cm3 obtained for OCS [29]. 5.9 x Probably many other polarizability anisotropies will be measured by the microwave Stark effect as experimental techniques are further improved. The electric polarizability anisotropy of the nonpolar oxygen molecule has been measured through observation of the splitting of one of its millimeterwave, fine structure lines by a strong external electric field [32]. The observed splitting of the J=O+ 1, N = 1+ 1 line that occurs at 118 GHz is shown in Fig. 10.11. Since O2 has no permanent dipole moment, the splitting is due entirely to anisotropic polarization induced by the applied field. From the Hamiltonian of (10.115), the anisotropic polarization energy for O2 can be obtained in the form [32] 2J+ 1

where the first quantity in the summation is a squared 3 j symbol explained in Chapter XV. This expression leads to the splitting for the J=O+ 1, N = 1+ 1 oxygen line of 1 (IO. 125) (Av),, = o =- (511, - u,)@ 15h (10.126)

488

EFFECTS OF APPLIED ELECTRIC FIELDS AY-152kHz A

I I

A Y = 322 kHz

Fig. 10.11 The Stark shift of the 118GHz line of oxygen ( A M J = kl). From Gustafson and Gordy [32].

Equation 10.126, when applied to the observed splittings (see Fig. lO.ll), gives the anisotropic polarizability of O 2 as

A comparable magnitude of 1.09 cm3 x has been estimated from laser depolarization scattering and bulk polarizability measurements [33, 341. Perturbation corrections to the Stark energy arising from polarization for asymmetric rotors have been discussed by Golden and Wilson [ll]. Although the off-diagonal elements of the polarizability tensor do not vanish, in general, for asymmetric rotors, only the diagonal elements of the tensor will contribute in a perturbation treatment correct to second-order in the field. The polarization contribution is obtained if ifa, (10.114) is averaged over the functions IJ, T , M J )

(10.127) This may be expressed in terms of the line strengths by use of (10.71)-(10.73) or more conveniently in terms of the average values of the angular momenta, that is, (J:), ( J i ) , and (J:) (see Chapter VII, Section 2).


7

489


The relative intensities are important for the interpretation of the observed spectra. When the MJ degeneracy is lifted by an external field, the intensities of the Stark components will depend upon M j . Since the MJ dependence of the direction cosine matrix elements is the same for asymmetric rotors as for symmetric rotors, the MJ dependence of the intensities for a particular line will be proportional to I(J, MjIQ)pSIJ', M;)12. Thus the relative intensities of the M j + M j Stark components are from Table 2.1 seen to be (10.128)

Q

IM, =T (JfM j P

+ I)(Jf M j + 2)

-

A J= + 1

ZM,

=% (J+Mj)(Jf M j + 1)

AJ=O

ZM,

- Mj)(JT M j + 1) =TQ (J+

AJ=-1

(10.130)

>AMj=fl

(10.131) (10.132)

J = 5-6 AM.0 First-order Stark effect

J = 5-6 AM-fl First-order Stark effect

iL

J-5+6 AM.0 Second-order Stark effect

J.5-6 AMIf 1 Second-order Stark effect

I

I

J.4-4 AM-0 Second-order Stark effect

J=4+4 AM=* I Second-order Stark effect

,

yo Fig. 10.12 Some typical first- and second-order Stark patterns.

490

IDENTIFICATION A N D MODULATION OF ROTATIONAL LINES

491

causes a mixing of the zero-field wave functions with the degree of mixing being dependent on the field strength. Even though these effects may be small, an originally forbidden transition, for example, J A J l > 1, can, when an electric field is applied, have a nonvanishing transition probability, and observation of such forbidden transitions becomes possible [ll, 211. The presence of neardegeneracies can cause rather large perturbations, as in the case of an 1-type doublet in linear rotors, and of a K-type doublet in asymmetric rotors. The relative intensities, when there is quadrupole hyperfine structure can be readily found if the limiting weak- or strong-field approximation is valid. For strong-field case the JZMjM, representation is appropriate, and the relative intensities of the various Stark components are the same as for molecules without hyperfine structure. For the usual AMJ = O transitions the intensities are hence given by (10.128) and (10.129).However, a given Stark line is further split into I + 1 hyperfine lines, all of which have the same intensity except for M I=0, which is half as strong because of the loss of the twofold degeneracy, viz., + M I . For the weak-field case the JZFMF representation is appropriate. In this representation the possible projections of the total angular momentum F along the field direction are specified by the values of M F .The intensities are hence obtained by replacing J by F and M F in the preceding equations. For the MF-MF transitions, the Stark component intensities are

z,,=RM; I,+,, =S[(F+ 1)2- M i ]

AF=O AMF=O AF= _+ 1

(10.133) (10.134)

where R and S are parameters independent of the quantum number M F and where F is the smaller of the two values involved in the particular hyperfine transition. The intensities for the intermediate-field case can be approximated by an interpolation between the strong- and weak-field limits. The exact intensities are proportional to the squares of the dipole-moment matrix elements evaluated with the intermediate-field wave functions. Evaluation of the intensities hence requires the solution of the secular equation and evaluation of the wave function for each level. The intermediate-field wave functions are linear combinations of the original basis functions chosen to construct the intermediate-field secular equation. They are defined once the expansion coefficients, the C/s, are specified [see (10.48)].

8

IDENTIFICATION A N D MODULATION OF ROTATIONAL LINES WITH THE STARK EFFECT

A knowledge of the intensities, spacing, and number of Stark components is frequently helpful in the assignment of rotational transitions, especially for asymmetric rotors. For the usual AMj = O transitions the relative intensities are useful for distinction between the AJ=O type transitions and the A J = k 1 type. In the former case, the components with the largest values of MJ will have

492


the greater intensity, while in the latter case just the reverse is true. The measured spacing between successive M J components gives information regarding the value of the quantum number M J corresponding to a particular component of a Stark pattern as well as to the smaller J involved in the transition. This latter information is obtained by determination of the maximum value of jMJl which must be equal to the smaller J. For relatively low J lines, where it is usual11 possible to obtain a completely resolved Stark pattern, a direct count of the number of M J components leads to the determination of the smaller of the two Ss involved in a transition. Note that for a second-order Stark effect there will be a maximum of (J+1) Stark components when A J = i.1 and J components when AJ=O, while for a first-order Stark effect there will be ( 2 J + 1 ) components when A J = f 1 and 2 J components when A J = O . Once the value of J a n d the type of transition are known, the assignment of the 5 and 7’ involved in a transition may often be found by a trial-and-error fitting of observed transition frequencies to calculated rigid rotor frequencies. A valuable application of the Stark effect is its use for modulation of rotational absorption lines [35] in the Stark modulation spectrometers introduced by Hughes and Wilson [I]. The modulation is accomplished by insertion of an electrode in the waveguide absorption cell and application of a low radiofrequency (e.g., 100 kHz) square wave voltage. The electrode consists of a flat strip of brass placed parallel to the broad side of the rectangular waveguide and mounted on a grooved dielectric such as Teflon. This arrangement results in a modulation of the microwave energy in the vicinity of an absorption line. If a narrow bandwidth detector is tuned to the modulating frequency, a significant increase in sensitivity is obtained. If one side of the square wave is clamped to zero voltage, the undisplaced absorption line is observed during the zero field part of the cycle; during the “field-on” portion, a series of absorption lines which are displaced because of the Stark effect will appear. When phasesensitive detection is used, the zero-field line and its Stark components will be displayed with opposite sense; see, for example, Fig. 10.8. For the millimeter wave region, source-frequency modulation is more useful because of the high losses of power in conventional Stark cells. If Stark modulation is desired, a parallel plate cell can best be employed. Rather thorough treatments of the Stark effect for the varying fields often used in Stark modulation spectrometers are given by Autler and Townes [36] and in the text by Townes and Schawlow [37]. A method for approximate measurement of dipole moments which depends on Stark modulation is described by Lindfors and Cornwell [38]. The method consists in measurement of the peak-to-peak intensity of an absorption line for various field strengths (modulation amplitude). I t requires the observation of an absorption line with a first- or second-derivative line shape over the range of modulation amplitudes. This is usually possible at low modulation amplitudes when one uses a spectrometer employing zero-based square wave Stark modulation and phase-sensitive detection. In general, a range of field strengths must be chosen such that the Stark shifts are smaller than the linewidth. Although

-


493

this method, “the rate of growth of line intensity,” is not as accurate as the Stark splitting method for determination of the dipole moment, it has the advantage of not requiring resolution of individual Stark components and hence can be applied to high J transitions for which the Stark pattern can be modulated but not completely resolved. The necessary working equations and a discussion of experimental procedures are given in the original paper [38].

9


Although the Stark effect is a valuable aid in the detection and identification of rotational lines, its most important chemical application is in the measurement of electric dipole moments. In contrast to that of the methods based on measurement of the dielectric constant, its accuracy is not influenced by impurities in the sample. With the Stark effect, the dipole moment is measured for a particular isotopic species and for a particular vibrational state and rotational transition. Hence it is possible under favorable conditions to measure the small differences which are due to isotopic substitution or change in vibrational or rotational state. In the earlier period of microwave spectroscopy, up to 1957, the Stark field was applied to an electrode in the center of a rectangular waveguide. The uncertainty in the effective spacing of the electrode in such ace11 and the tendency for potential breakdowns at high voltage limited the accuracy of these measurements to a percent or so. Improvement by an order of magnitude in the accuracy of dipole moment measurement was made possible by the introduction of the parallel plate Stark cell [27,39,40] with optically ground quartz spacers. Such a cell is highly practical in the shorter millimeter wave region where the radiation is easily focused with horns and lenses of convenient size and where the rotational absorption lines are so strong that a cell of short length can be used. With such a cell it is possible to apply a higher voltage without breakdown than is possible with the conventional waveguide Stark cell. Furthermore, the parallel-plate cell can easily be designed to transmit modes which allow measurement [40,41] of the 0,or AM., = f 1, components. In measurements of Stark splitting it is convenient to use as secondary standards the dipole moments or Stark splitting values of molecules which have been calibrated previously by careful, high-precision measurement. This practice eliminates the necessity for making precise voltage measurements with a potentiometer and precise electrode spacing with an interferometer every time a dipole moment is measured. Dipole moments of the linear molecules OCS and HCN, also the symmetric-top molecules CH3F, CD3F, CH3CN, and CD3CN, have been measured to high precision with parallel-plate cells. These are absolute measurements in the sense that the plate separation, the potential, and the frequency were measured directly with precision instruments without the use of a secondary molecular standard. These moments can be used as secondary standards for calibration of Stark spectrometers. Errors in the OCS values mentioned later are, hopefully, now cleared up. Those of carbonyl

494


sulfide and methyl fluoride are more satisfactory to use because they offer no complications from nuclear quadrupole hyperfine structure. Chemically, OCS is simple to prepare and easy to handle. Its microwave spectrum is simple. All three nuclei of the most abundant isotopic species 1 6 0 1 2 C 3 2have S zero spins, and hence the rotational spectrum of this species has no hyperfine structure. The Stark pattern of the J=1-2 transition is the simple, strong doublet shown in Fig. 10.3. Unfortunately, there is a rather large discrepancy in the value of the dipole moment of this molecule originally measured from the M j - M j Stark components of the J = 1-2 transition with a parallel-plate cell by Marshall and Weber [27] in 1957 and the recent measurement (1968) with molecular beam electric resonance on the J = l , M j - M j k 1 transition, also with a parallel-plate cell, by Muenter [29]. The value obtained Marshall and Weber is 0.7124 D with a stated error limit of f0.0002, whereas the value obtained by Muenter is 0.71521 f0.00020 D. Although the two values have the same estimated error limit, they differ by a factor of more than ten times this limit. The reason for this discrepancy is not clear, but we accept the new Muenter value because it, when used as a secondary standard for measurement of the CH3F moment, gives a result that is consistent with the absolute value of the CH3F moment measured in our own laboratory, as described later. It is of interest that the measurement of the OCS moment which was made with a waveguide cell and widely used as a secondary standard until replaced by the measurement of Marshall and Weber is 0.709 f0.003 D, a value lower by 0.0062 D (twice the estimated error limits) than the most recent OCS value. Therefore it appears that the many dipole moment values reported in the literature up to 1968 for which OCS was used as a secondary standard have absolute errors greater than the error limits quoted for their standard. Because of the decreasing sensitivity of the Stark effect with increasing J, the OCS moment is not satisfactory for calibration of Stark spectrometers operating in the shorter millimeter range. For that region of the spectrum, the dipole moments of HCN, CH3F, or CD3F provide accurate secondary standards. The lines of HCN are the strongest, but the most abundant isotopic species of HCN is complicated by I4N quadrupole coupling. Absolute measurements of the moments of CH3F and CD3F have been made [41,42] with both K =O and K = 1 lines and with TC and c components of the J = 1-+2 transition, which falls at 102 GHz for CH3F and at 81.7 GHz for CD3F. A summary of the results is given in Table 10.5. There is no measurable difference in the values obtained for K=O and K = l transitions although there is a difference in the moments of the two isotopic species. Figure 10.4 shows the energy level diagram (not drawn to scale) and indicates the different transitions employed for these measurements. In some symmetric-top molecules with K =O at high fields, Muenter and Laurie [2] found measurable contributions of fourth-order Stark terms. Though the fourth-order perturbation term is calculable, the resulting expression is complicated and inconvenient to apply. They therefore used the method described in Section l which bypassed the calculations of the fourth-order co-


495

Table 10.5 Summary of Dipole Moments of CH3F and CD3FMeasured with 0 and TK Components of the J = 1-+2 Transition" Dipole Moment p (0) Transition K=l K=l K=O K=O

AM=O AM=fl AM=O AM=+l Average

CH3F

CD3F

1.858 +0.0010 1.857+0.0011 1.857 & 0.0010 1.857+0.0011 1.8572fO.0010

1.868 -1.0.0010 1.86810.0010 1.868 & 0.0013 1.868iO.0011 1.8682 +0.0010

"From Steiner and Gordy [42].

efficient but allowed the fourth-order effect to be included in the evaluation of the dipole moment. Using a short waveguide Stark cell calibrated with the HCN moment, Burrus [43,44] observed the Stark effect on a number of molecules in the wavelength range from 2.5 to 0.93 mm and derived the dipole moments of the simple molecules DCl, DBr, DI, CO, and PH3, which have no microwave transitions in the lower microwave frequencies. The moments of the deuterium halides which have quadrupole coupling by the halogen were measured with the weakfield formula, (10.35).The weak-field Stark displacement of each component is a quadratic function of the field strength, Av = k ( J , I, F , MF)p2g2

(10.135)

where k is constant for a particular component and is calculable from (10.35). Thus if one plots Av as a function of g2,a straight line is expected where the weak-field case holds. Such plots of the observed data by Burrus for D8'Br are shown in Fig. 10.13. It is seen that the straight-line function holds well up to about 8000 v/cm. The slope of these plots is k p 2 from which the dipole moment is obtained with the k's calculated from the weak-field formula. Burrus similarly observed that the weak-field case holds for D 3 T l to voltages of approximately 4000 v/cm, for which eQq is only 67 MHz. Because CO has a very small moment it was necessary to use fields up to 24,000 v/cm to produce accurately measurable shifts. With such fields, the polarizability corrections (Section 6) become important. Mizushima [30] has calculated this effect which, he indicates, would add about 0.002 D to the moment as given by Burrus [43]. With this correction the Burrus value of the permanent C O dipole moment would be 0.114 D. The dipole moment of CH3CN of Table 10.10 was measured with the strongfield Stark effect which is readily achieved with the first-order effect on the K f 0 lines having the relatively weak quadrupole coupling eQq z 4 MHz. Shulman et al. [45] used the intermediate-field theory to derive the dipole moments of the methyl halides from their Stark measurements. The value that

496


Fig. 10.13 Displacement of t h e n Stark components as a function of 8 ' for hyperfine components of the J=O+l transition of D8'Br. From Burrus [44].

they obtained from CH31 has been recalculated to higher order by Buckingham and Stephens [ 5 ] , who reduced the value from 1.647 to 1.618 D. Likewise, SiH3Cl, SiH3Br, and GeH3CI were treated with intermediate-field theory by Mays and Dailey [46]. Since the effective bond length r , is generally longer for excited bondstretching vibrational states, one might expect that, other things being equal, there would be an accompanying increase in the dipole moment for these states. In LiCl, where the bonding is simple and largely ionic, this is found to be true; but in CS, for which the bonding is largely covalent with a rc-bond component, the dipole moment actually decreases with vibration (see Table 10.6). Unfortunately, not many examples are available for comparisons. I t has not been feasible to measure such light molecules as HBr and CO in excited vibrational states. As a general rule, the dipole moments of diatomic molecules tend to increase with the bond-stretching vibrational quantum numbers. This may be expected from the greater effective bond lengths of the higher vibrational states. The trend is followed by the highly ionic molecules LiCl and KCI, shown in Table 10.6, but not by CS, for which the bonding is more complex. The dipole moments of linear polyatomic molecules tend to be slightly lower for excited bending states than for the ground state. This trend is illustrated by a few triatomic molecules in Table 10.7. Qualitatively, this effect may be attributed to the effective shortening of the overall molecular lengths of the bending vibrations. Since there is also some redistribution of the molecular charges resulting from the vibrations, no such simple rule can be expected to hold for all molecules. Deuterium substitution would be expected to lower the dipole moment of a particular bond because of the lowered zero-point vibrational energy com-


497

Table 10.6 Illustrations of the Effect of Bond-stretching Vibration on Dipole Moment

Dipole Moment (0)

Molecule

u=O

c=l

12c32S 6Li CI

1.966 7.1 195 10.269,,

1.944 7.2069 10.3290

39K35C1

u=2

u=3

c=4

Ref: (1

7.2964 10.3894

7.3865 10.4502

b

10.51l 4

c

"G. Winnewisser and R. L. Cook, J. Mol. Specrrosc., 28, 266 (1968). Values are here corrected to conform to new value of OCS moment used as a standard. bD.R. Lide, Jr., P. Cahill, and L. P. Gold, J. Chem. Phys., 40, 156 (1964). 'R. van Wachem and A. Dymanus, J. Chem. Phys., 46,3749 (1967).

Table 10.7

Illustrations of Effect of Bending Vibration

on Dipole M o m e n t Dipole Moment p (0) Molecule

HCN OCS OCSe

J-+J+1

v=O

u2=1

u2=3

0-4 1

2.985" 0.71521' 0.754'

2.957b 0.700' 0.728"

0.730'

1-42 2-43

"8. N. Bhattacharya and W. Gordy [40]. bR.G. Shulman and C. H. Townes, Phys. Ret.., 77,421 (1948). 'Muenter [29]. d R . G. Shulman and C. H. Townes, Phys. Rea., 77,500 (1950). 'Strandberg et al. [26].

bined with a slightly anharmonic, Morse-type potential function. However, the electronic structure of the molecule is also perturbed by this change in vibrational energy so that it is not possible to predict in a simple way the effect of isotopic substitution on a dipole moment. For example, deuterium substitution decreases the moment in CH3CCH from 0.7809 to 0.7668 D for the same transition in CH3CCD, but increases the moment from 0.390D in HCP to 0.397 D in DCP. The dipole moment is also raised by deuterium substitution in methyl fluoride (see Table 10.5), but is lowered in methyl silane and methyl germane. The effects of centrifugal distortion on the dipole moments of HNCO, DNCO and HN3 have been observed by White and Cook [47] from measurements of the rotational Stark spectra in the ground vibrational state. Analysis of the Stark effect for the J=3+4 transitions of these molecules yields the dipole moment component along the axis of smallest moment of inertia. For

498


these very slightly asymmetric prolate rotors, with A rotational constants between half a million and a million megahertz, the dipole moment is observed to decrease with increasing K , the changes being surprisingly large. Recent measurements of the &( = 1.35 D) component of HNCO by Hocking et al. [48] further clarified the variation of pa with K and the results are collected in Table 10.8. A simple model [47] can explain the observed K dependence of which is essentially quadratic. K can be approximately interpreted as t h e magnitude of the spin about the NCO molecular chain. Thus as K increases, the HNC angle decreases due to centrifugal distortion and the projection of the N-H bond moment on the a axis decreases. The calculated [48] change in the HNC angle and corresponding changes in pa are also shown in Table 10.8. Although only the absolute magnitudes of the dipole components along the three principal axes of an asymmetric-top molecule are determined from the rotational Stark effect, the direction of these components can usually be decided by an appeal to electronegativities, bond moments, or other chemical information. Therefore, the orientation of the dipole moment vector in the molecule can be determined. However, when a clear-cut decision as to the direction of the dipole components cannot be made, the effect of isotopic substitution can sometimes be used to give the relative direction of the components. Numerous determinations of the dipole moment direction employing isotopic substitution have been reported in the literature. We give here only two examples to demonstrate the method. Consider, for example, formyl fluoride (HCOF) studied by Favero and Baker [49] for which pa=OS8 ? 0.02 and p b = 1.91 i0.03 (debye units). The location of the principal axes is shown in Fig. 10.14. I t is most probable that the oxygen and fluorine atoms because of their greater electronegativity are negative with respect to the hydrogen atom and p b is oriented as

Table 10.8 Illustration of the Effect of Centrifugal Distortion on the Dipole Moment of lsocyanic acid ( H N C O )

Transition J=3+4 K=O K=l K=2 K=3

Apf)

A@(')

1.577

-

-

-

1.572 1.540 1.500

0.005 0.037 0.077

0.32 1.30 2.92

0.036

&)

Apt)

0.009 0.081

"Dipole component along the axis of smallest moment of inertia in debye units. From White and Cook [47], and Hocking et al. [48]. *Observed decrease in po relative to K =O. 'Calculated decrease in HNC angle (6 in degrees) and in pLyrelative to K = O from Hocking et al. [48].


499

Fig. 10.14 Principal inertial axes of HCOF. The direction of pb may be selected from chemical arguments but not that of pn However, for pa to increase on deuteration, which results in a rotation of the b axis towards the D atom, the direction must be as shown. The arrowhead indicates the negative end of the dipole moment components. After Favero and Baker [49].

shown. The arrowheads indicate the negative end of the dipole moment. Substitution of deuterium for the hydrogen atom will result in a rotation (- 1") of the principal axes of inertia, the b axis moving towards the D atom. For DCOF the dipole moment components are found to be pa=0.61 f0.02 and pb= 1.96f0.03. In order for pa to increase, its orientation must be as shown in Fig. 10.14 if it is assumed that the increase arises from the rotation of the axes and that the orientation of the molecular dipole relative to the molecule is essentially unchanged. A method for finding the sign of the dipole moment from the Zeeman effect is described in Section 6 of Chapter XI. For formic acid, HCOOH, which has been discussed by Kim et al. [50], the direction of pa (see Fig. 10.15)can be similarly selected, but not that of pb because of its small size. The measured dipole components for HCOOH are (in debye units): pa=1.391 f0.005, pb=0.26&0.04, and p=1.415+0.01 and for HCOOD they are pa=1.377&0.005, Pb=0.22-+_0.02,and p=1.39+0.01. From the experimental value of p i (HCOOH)=0.07, the angle 6 between the total dipole moment and the a inertial axis is found to be + 10.8" or - 10.8",depending 1.415 f .01 debye

\3,?.44 * , 2o

\0-H Fig. 10.15

The direction of the dipole moment in HCOOH. After Kim et al. [50]

500


on the direction of p b . Deuteration of the acid hydrogen will produce a rotation of the inertial axes which can be calculated from the known structure given by Kwei and Curl [51]. If the positive direction of 8 is taken as clockwise in Fig. 10.15, then the inertial axes are rotated by - 1.8" upon deuteration. Therefore, 8 in HCOOD should be either + 12.6" or -9.0'. From p i for HCOOD one finds 8 = f9.0"; and if we assume that the direction of the dipole moment relative to the molecule remains basically the same upon deuteration, then it follows that 8 is -10.8" in HCOOH. Therefore p b in HCOOH is directed counterclockwise from the p, direction. These results are summarized in Fig. 10.15. Table 10.9 Dipole Moments of Some Diatomic and Linear Polyatomic Molecules in the Ground Vibrational Staten Molecule

DCI DBr DI LiCl

-cos

-cs+ CSe

- ocs +

HCN

P (0) 1.12(4) 0.83(2) 0.445(20) 7.119(8) 0.1 12(5) 0.10980(3) 1.966(5) 1.99(4) 0.7152(2) 2.985(4)

ReJ b b b C

d

e

f B h

i

Molecule

HCP DCP FCP NNO HCNO HCCD HCCF DCCF HCCCI FCCCI

P (0)

Ref:

0.390(5) 0.397(5)

j

0.279(1) 0.166(2) 3.06(13) 0.01001(15) 0.7207 0.7329 0.44(1) 0.25(10)

k

1

1

m

" 0

P m

4

"All measurements are made from the J=O+1 transition except those of OCS(1+2), FCP(2+3), and the acetylene derivatives. Signs of certain of the moments, as indicated were determined by the Zeeman method as described in Chapter XI, Section 6. *Burrus [44]. 'D. R. Lide, P. Cahill, and L. P. Gold, J. Chem. Phys., 40, 156 (1964). dBurrus [43]. 'From molecular beam resonance, by J. S. Muenter, J . Mol. Specfrosc., 55,490 (1975). 'G. Winnewisser and R. L. Cook, J. Mol. Spectrosc., 28, 266 (1968). #J. McGurk, H. L. Tigelear, S. L. Rock, C. L. Norris, and W. H. Flygare, J . Chem. Phys., 58, 1420 (1973). hMuenter [29]. 'Bhattacharya and Gordy [40]. 'J. K. Tyler, J . Chem. Phys., 40, 1170 (1964). 'W. H. Kroto, J. F. Nixon, and N. C. P. Simmons, J . Mol. Spectrosc., 82, 185 (1980). 'Shulman et al. [45]. "M. Winnewisser and H. K. Bodenseh, Z . Naturjorsch., 22a, 1724 (1967). " K. Matsumara, T. Tanaka, Y. Endo, S. Saita, and E. Hirota, J . Chem. Phys., 84, 1793 (1980). "T. Tanaka, C. Yamada, and E. Hirota, J . Mol. Specfrosc.,63, 335 (1978). PK.Matsumura, K. Tanaka, C. Yamada, and T. Tanaka, J . Mol. Spectrosc., 80, 209 (1980). qA. Guarnieri and M. Andolfatto, Z . Naturjorsch., Ma, 899 (1981).


501

The total dipole moment lies between the C-H and C=O bonds, making an angle of 42.4" L- 2" with the C=O bond, A method for finding the orientation of the resultant dipole moment from the observed dipole components of the isomers for molecules exhibiting rotational isomerism has been described by Botskor [52]. Illustrative values of dipole moments of some diatomic and linear polyatomic molecules are given in Table 10.9. Similar listings for symmetric tops are given in Table 10.10 and for asymmetric tops in Table 10.11. Interpretations of dipole moments in terms of chemical bond properties may be found in Chapter XIV, Section 16. Dipole moments of hundreds of molecules-diatomic, linear, polyatomic, symmetric and asymmetric tops-have now been measured with the Stark Table 10.10 Dipole Moments of Symmetric-top Molecules from the Stark Effect in Microwave Rotational Spectra"

Molecule

p (0)

BH3CO BH3NH2 CH3F CH,CI CH3Br CH31 CH3CN CH3CP CF3H CF3C1

1.795(10) 5.216(17) 1.8572(10) 1.869(10) 1.797(15) 1.618 3.913(2) 1.499(1) 1.6526 0.5OO( 10)

Re$ b E

d e e

f d

B h l

Molecule SiH3F SiH3CI SiH3Br GeH3CI SiH3CH3 GeH 3CH SnH3CH3 CH3CCH SiH3CCH GeH +2CH

p (0) 1.298(4) 1.303(10)

Ref: j

k

1.318(8)

1

2.124(20) 0.7380 0.6450 0.68(3) 0.7835 0.317(3) 0.136(2)

k d d rn

d n 0

"All values are for the ground vibrational state. *M. W. P. Strandberg, C. S . Pearsall, and M. T. Weiss, J. Chem. Phys., 17, 429 (1949). 'L. R. Thorne, R. D. Suenram, and F. J. Lovas, J. Chem. Phys.,78,167 (1983); see also R. D. Suenram and L. R. Thorne, Chem. Phys. Lett.,78,157 (1981). dSteiner and Gordy [42]. 'Shulman et al. [45]. fBuckingham and Stephens [5]. gH.W. Kroto, F. J. Nixon, and N. P. C. Simmons, J. Mol. Spectrosc., 77, 270 (1979). hMuenter and Laurie [2]. Values are here corrected to conform to new value for OCS moment used as a standard. 'R.C. Johnson, T. L. Weatherly, and Q. Williams, J. Chem. Phys., 35,2261 (1 96 1). jC. Georghiou, J. G. Baker, and S . R. Jones, J. Mol. Spectrosc., 63,89 (1976). 'J. M. Mays and B. P. Dailey, J. Chem. Phys., 20, 1695 (1952). 'K. F. Dossel and D. H. Sutter, Z.Nuturforsch., 32a, 1444 (1977). "D. R. Lide, J. Chem. Phys., 19, 1605 (1951). "J. S. Muenter and V. C. Laurie, J. Chem. Phys., 39,1181 (1963).See footnote h. "E. C. Thomas and V. C. Laurie, J. Chem. Phys., 44,2602 (1966).

502


Table 10.1 1 Dipole Moments of S o m e Asymmetric Rotors from the Stark Effect of Microwave Spectraa Magnitude of Dipole Moment (debye unirs) Molecule

HN3 NSCl

H IC0 HCOOH H3NH2

CH30Cl CD30Cl

Pa

pb

0.837b 0.57(3)

1.48(5) 1.77(2)

1.39l(5) 0.304 1.373(4) 1.409(4)

0.26(4)

(CH312SO

H3CH2CN CHzCHCN CH3CHC0 CH3CHCHZO CH2CH2CH2S

CH2CH2CO

3.85(2) 3.68 1.755(10) 0.95(1) 1.85(3) 3.67(3)

Pc

1.247 1.146(7) 1.106(8) 3.94(4) 0.123(2) 1.25 0.35(3) 1.67(2)

0.40(1)

0.56(1) 0.00(3)

1.99(2)

P

1.70(5) 1.87(2) 2.339 1.415(10) 1.238 1.788 1.791(8) 3.96(4) 4.05(3) 3.89(8) 1.79(2) 2.00(2) 1.85(3) 4.17(3)

Ref. c

d

9 h

1

J

k 1

m n 0

L O A "All values are for the ground vibrational state and, with the exception ofCDAOC1, are for the most abundant isotopic species. bK. G. White and R. L. Cook, J. Chem. Phys., 46, 143 (1967). 'J. Bendtsen and M. Winnewisser, Chem. Phys. Lett., 33, 141 (1975). dA. Guarnieri, Z. Naturjorsch., 26a, 1246 (1971). 'J. N. Shoolery and A. H. Sharbaugh, Phys. Rev., 82,95 (1951). ,H. Kim, R. Keller, and W. D. Gwinn, J. Chem. Phys., 37,2748 (1962). gD. R. Lide, Jr., J. Chem. Phys., 27, 343 (1957). hM. Suzuki and A. Guamieri, Z . Naturjorsch., 31a, 1242 (1976). 'H. Dreizler and G. Dendl, Z . Narurjorsch., 19a, 512 (1964). 'H. M. Heise, H. Lutz, and H. Dreizler, Z . Naturforsch., 29a, 1345 (1974). 'W. S. Wilcox, J. H. Goldstein, and J. W. Simmons, J. Chem. Phys., 22, 516 (1954). 'B. Bak, J. J. Christiansen, K. Kunstmann, L. Nygaard, and J. Rastrup-Andersen, J. Chem. Phys., 45, 883 (1966). "J. D. Swalen and D. R. Herschbach, J. Chem. Phys., 27, 100 (1957). "M. S. White and E. L. Beeson, Jr., J. Chem. Phys., 43, 1839 (1965). "D. W. Boone, C. 0. Britt, and J. E. Boggs, J. Chem. Phys., 43, 1190 (1965).

effect in microwave rotational spectra, many of them to five significant figures. It is feasible to record only a small fraction of them in Tables 10.9 through 10.11 . Comprehensive tables of measured dipole moments are available in the LandoltBornstein tables [53]. These tabulations are frequently revised to bring them up to date.

References 1. R. H. Hughes and E. B. Wilson, Phys. Rev., 71,562 (1947). 2 . J. S. Muenter and V. W. Laurie, J . Chem. Phys., 45, 855 (1966).

REFERENCES

3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

503

u. Fano, J . Res. Natl. Bur. Stand., 40, 215 (1948). W. Low and C. H. Townes, Phys. Rev. 76, 1295 (1949). A. D. Buckingham and P. J. Stephens, Mol. Phys., 7,481 (1963164). F. Coester, Phys. Rev., 77, 454 (1950). E. U. Condon and G . H. Shortley, The T2eory of’Atomic Spectra, Macmillan, New York, 1935, pp. 76-77. M. E. Rose, Elementary Theory ojdngular Momentum, Wiley, New York, 1957. J. B. M. Kellogg, I. I . Rabi, N. F. Ramsey, Jr., and J. R. Zacharias, Phys. Rev., 57,677 (1940). W. G . Penney, Phil. Mag., 11,602(1931). S. Golden and E. B. Wilson, Jr., J . Chem. Phys., 16, 669 (1948). See, for example, B. G. West and M. Mizushima, J . Chem. Phys., 38, 251 (1963); C. C. Lin, Bull. Am. Phys. Soc., 1, 13 (1956). P. E. Wacher and M. R. Pratto, Microwave Spectral Tables, Vol. 11. Line Strengths of Asymmetric Rotors (Natl. Bur. Stand. Monograph 70), 1964. See Chapter VI1 for further references. D. Kivelson, J . Chem. Phys., 22, 904 (1954). G. F. Crable, Ph.D. dissertation, Duke University, 1951. G . F. Crable and W. V. Smith, J . Chem. Phys., 19, 502 (1951). J. H. Van Vleck, Phys. Rev., 33, 467 (1929). See also 0. M. Jordahl. Phys. Rev., 45, 87 (1934); E. C. Kemble, Fundamental Principles of Quantum Mechanics, Dover, New York, 1958. G. H. Kwei and D. R. Herschbach, J . Chem. Phys., 32, 1270 (1960). S. I. Chan, J. Zinn, J. Fernandez, and W. D. Gwinn, J. Chem. Phys., 33, 1643 (1960). S. S. Butcher and C. C. Costain, J . Mol. Spectrosc., 15,40 (1965). C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. M. Mizushima, J . Chem. Phys., 21,539 (1953). G. Racah, Phys. Rev., 62, 438 (1942). See also [SI. J. A. Howe and W. H. Flygare, J . Chem. Phys., 36,650 (1962). D. F. Eagle, T. L. Weatherly, and Q.Williams, J . Chem. Phys., 44,847 (1966). M. W. P. Strandberg, T. Wentink, Jr., and A. G . Hill, Phys. Rev., 75, 827 (1949). S. A. Marshall and J. Weber, Phys. Rev., 105, 1502 (1957). L. H. Scharpen, J. S. Muenter, and V. Laurie, J . Chem. Phys., 46,2431 (1967). J. S. Muenter, J . Chem. Phys., 48, 4544 (1968). M. Mizushima, in Advances in Molecular Spectroscopy, Vol. 3 (Proc. IVth International Meeting on Molecular Spectroscopy), A. Magnini, Ed., Macmillan, New York, 1962, pp. 1 167-1 182. J. 0. Hirschfclder, C. F. Curtiss, and R. B. Bird, Molecular Theory ofGasesand Liquid, Wiley, New York, 1954. p. 950. S. Gustafson and W. Gordy, Phys. Lett., 49A, 161 (1974). N. J. Bridge and A. D. Buckingham, Proc. R. Soc. (London), A295, 334 (1966). R. L. Rowell and G . M. Aval, J . Chem. Phys., 54, 1960 (1971). K. B. McAfee, Jr., R. H. Hughes, and E. B. Wilson, Jr., Rev. Sci. Instrum.. 20, 821 (1949). S. H. Autler and C. H. Townes, Phys. Rev., 100, 703 (1955). Townes and Schawlow, Reference 21, pp. 273-283. K. L. Lindfors and C. D. Cornwell, J . Chem. Phys., 42, 149 (1965). B. N. Bhattacharya, W. Gordy, and 0. Fujii, Bull. Am. Phys. Soc., 2, 213 (1957). B. N. Bhattacharya and W. Gordy, Phys. Rev., 119, 144 (1960). D. M. Larkin and W. Gordy, J . Chem. Phys., 38,2329 (1963). P. A. Steiner and W. Gordy, 1. Mol. Spectrosc., 21, 291 (1966). C. A. Burrus, J . Chem. Phys., 28,427 (1958).

504 44. 45.

46. 47. 48. 49. 50. 51. 52. 53.


C. A. Burrus, J . Chem. Phys., 31, 1270 (1959). R. G . Shulman, B. P. Dailey, and C. H. Townes, Phys. Rev., 78, 145 (1950). J. M. Mays and B. P. Dailey, J . Chem. Phys., 20, 1695 (1952). K. J. White and R. L. Cook, J . Chem. Phys., 46, 143 (1967). W. H. Hocking, M. C. L. Gerry, and G . Winnewisser, Can. J . Phys., 53, 1869 (1975). P. Favero and J. G . Baker, Nuovo Cimento, 17, 734 (1960). H. Kim, R. Keller, and W. D. Gwinn, J . Chem. Phys., 37,2748 (1962). G. H. Kwei and R. F. Curl, Jr., J . Chem. Phys., 32, 1592 (1960). I. Botskor, 2. Naturforsch., 35a, 748 (1980). Landolt-Bornstein, Numerical Data and Functional Relations in Science and Technolog),. For a complete reference, see Chapter I, [65].

Chapter

XI

EFFECTS OF APPLIED MAGNETIC FIELDS 1

ZEEMAN EFFECT IN MOLECULES WITHOUT NUCLEAR COUPLING

506

Interaction Energies Linear Molecules

506 509 51 2 51 5

Symmetric-top Molecules Asymmetric Rotors

2 ZEEMAN EFFECT OF NUCLEAR HYPERFINE STRUCTURE

51 8

Weak-field Case Strong-field Case

51 8 521

3

THEORY OF MAGNETISM IN CLOSED-SHELL MOLECULES ( ' E STATES)

523

4

ANISOTROPIC MAG N ET1C SUSCEPTI BI LIT1ES

528

Measurement of Molecular Anisotropies Calculation of Molecular Susceptibilities from Local Atomic Components

528

5

536

SECOND MOMENTS AND MOLECULAR QUADRUPOLE MOMENTS FROM THE ZEEMAN EFFECT

538

Molecular Quadrupole Moments Second Moments of the Electron Charge Distributions

538 542

6

SIGN OF THE ELECTRIC DIPOLE MOMENT FROM THE ZEEMAN EFFECT

544

7

ELECTRONIC EFFECTS ON INERTIAL CONSTANTS RELATION TO g FACTOR

546

8


549

Molecules with Orbital Angular Momentum ( n or A States) Molecules with Electronic Spin but Not Orbital Angular Momentum ('C or 3 C States)

550

9

Hyperfine Structure in Molecules with Electronic Angular Momentum

551 551

GAS-PHASE ELECTRON PARAMAGNETIC RESONANCE SPECTRA

552

Molecules with Orbital Angular Momentum Molecules in Multiplet E States

553 5 60

10 STARK EFFECT OF GAS-PHASE ELECTRON PARAMAGNETIC RESONANCE

561

505

506

EFFECTS OF APPLIED MAGNETIC FIELDS

Most stable molecules have C singlet ground electronic states and hence when not rotating are nonmagnetic in the sense that they have no permanent magnetic dipole moment exclusive of nuclear moments. However, magnetic moments are generated by molecular rotation which gives rise to Zeeman splittings [l-31, usually resolvable with fields of several kilogauss. Magnetic moments are also induced by the applied field because of the magnetic susceptibility of the molecules, but these induced moments are very much smaller than those caused by rotation, and only the anisotropic components can affect the observed spectra [4]. The nuclear magnetic moments influence the rotational Zeeman effect only when the moments are coupled to the molecular axes so that an observable hyperfine structure results. First we shall consider molecules in 'C states without nuclear coupling.

1 ZEEMAN EFFECT IN MOLECULES WITHOUT NUCLEAR COUPLING Interaction Energies The classical Hamiltonian for the interaction of a magnetic dipole with a field H is %H=-pL'H

(11.1)

In the molecular rotational Zeeman effect the magnetic moment is fixed in the molecular frame, and H is fixed in space. The component of the molecular magnetic moment along the total angular momentum vector J (or along F when there is nuclear coupling) interacts with H, causing J to precess about the direction of H, as indicated by the vector diagram of Fig. 11.1. With the H

Fig. 11.1 Vector model of the first-order Zeeman effect.


507

weak magnetic moments of molecules in C states, the precession of J about H is very much slower than the molecular rotation, and only the components of p along J effectively interact with H. Those normal to H are averaged out by the rotation. Even in molecules with unbalanced electronic moments this approximation holds for ordinary field strengths. If (p J ) represents the averaged value of the molecular magnetic moment along J, the Zeeman Hamiltonian can be expressed as X.q= - (

J*H ~

j

-= )

1JI

(11.2)

- (pJ)Hcos (J, H)

From the vector diagram of Fig. 11.1 it is seen that COS

JZ

(J, H)=-=

JJI

MJ [J(J+l)]l’z

(11.3)

Therefore (1 1.2) can be expressed as (11.4) Although (11.4) is a rather general first-order expression which applies to molecules of different types, symmetric tops or asymmetric rotors, the quantity ( p J ) varies with molecular type and rotational state. Therefore, if we are to find the specific value of EH in (11.4), we must find the corresponding operator for p J and average it over the molecular wave function. When the principal magnetic axes are the same as the inertial axes x, y , z, the components of p generated along the axes are proportional to the respective components of rotational momentum, as verified in Section 3. They can be expressed as Px=PIgxx

Jx,

Py=PIgyy

Jy.,

Pz=PIgzz

Jz

(11.5)

where P r g x x ,and so on, are proportionality constants. The nuclear magneton PI is introduced so that g will represent the ratio of p in nuclear magneton units to the angular momentum J in units of h, the customary designation used in the definition of the dimensionless gyromagnetic ratio of the nuclei. It gives the g factors in numbers of convenient magnitude for molecules in C states. The magnitude of p J is found by resolution of the components of (1 1.5) along J :

508


When the principal magnetic axes are not the same as the inertial axes, off-diagonal elements in the magnetic moment are generated so that P x =P l ( S X X

Jx

PLY = P l k I Y X J x PLL = p l ( g z x J x

+g x y J y +g x z J z ) + S Y Y JY + g y z J z )

(11.7)

+ g z y J y + g z z Jz)

Resolution of these components along J gives PI

P J = [ 4 J + 1)-p2

c

g,g'=x,y,z

S S S ' J S JS.

(11.8)

When, however, p J is averaged over the rotational wave functions, the offdiagonal elements will vanish, leaving only the diagonal elements of the g tensor. Thus, (11.9) Because ( p J ) is independent of M J , it is possible to define an effective g factor for the particular rotational state I J, 7)by (11.10) and hence (PJ) = g J , r P I [ J ( J +

1)]liz

(11.11)

where 5 becomes K for symmetric tops, is zero for linear molecules, and represets the pseudo-quantum numbers K - - K1 for the asymmetric rotor. Substitution of this value of ( p J ) from (11.11) into (11.4) gives the Zeeman energies as EH(PJ)= - g J , T P I H M J (11.12) in which (11.13) where g x x , gYY,and gzzare the diagonal elements of the g tensor with reference to the inertial axes and where the averages are taken over the wave functions of the unperturbed rotors. The quantity gJ,T is called the rotational g factor or the spectroscopic splitting factor. As shown in Section 3, its origin in 'C molecules is due partly to the rotation of formal charges and partly to a slight admixture of electronic states having orbital angular momentum with the C singlet ground electronic state. We did not include in (11.1)a small term arising from the moments induced in the molecule by the applied field. At very high fields this term can have detectable effects on the Zeeman displacements when the magnetic susceptibility of the molecules is anisotropic. These effects are discussed in Section 4.


509

Strictly speaking, we should include in (11.1) the Hamiltonian for the nuclear magnetic interaction with the applied field (11.14) where the sum is taken over all nuclei having nonzero spins. O n the other hand, when the nuclear coupling is negligible, as for nuclei with no quadrupole interaction, the perturbation of the molecular rotation can be neglected, and the allowed values of X,, are EH(PI)=

-PrH

1giiMri

(11.15)

i

Because of the selection rule AMIi=O, there is no observable effect of E H ( p I ) on the observed microwave spectra.

Linear Molecules The remaining problem in the evaluation of the Zeeman splitting is the specialization of (11.13) to the particular molecular type under consideration. For linear molecules J,=O, g,,=O, and from symmetry gxx=grr=g. Since J,’ + J,? = J’ and, in units of h, ( J ’ ) = J ( J + l), (11.16) Therefore, in the first-order treatment for linear molecules gJ is a constant, independent of the rotational state. The associated magnetic moment ( p J ) is perpendicular to the molecular axis and gives rise to a first-order Zeeman effect. The level splitting is given by (11.12). In contrast, there is no first-order Stark effect for linear molecules because the electric dipole moment is parallel to the molecular axis. The M J selection rules for the Zeeman effect are the same as those for the Stark effect Mj-Mj

and

Mj+Mj&l

(11.17)

For transitions which are induced by electric dipole coupling with the radiation, the magnetic field must be parallel to the electric vector of the radiation for observation of the M j + M j transitions, called n components; for observation of the M j + M j k 1 transitions, called CT components, it must be imposed perpendicular to the electric vector of the radiation. With the foregoing selection rules and with (1 1.12), the frequencies of the Zeeman components of the J + J + 1 rotational transition are v ( x )=v g

(11.18) (11.19)

510


where vo is the frequency of the line when no field is applied. Since from (11.16) g J is a constant, these first-order effects of a linear molecule are independent of J. One therefore expects the same Zeeman pattern for all the lines of an unsplit n component and for a doublet 0 component, as shown in Fig. 11.2. Together they form the normal Zeeman triplet. Since it does not depend on J , the c splitting can be observed for very high J transitions. Only the undisplaced rotational line, the n component, is observable when the magnetic field is parallel to the microwave E vector. In the higher-order approximations, g J is not exactly the same for all values of J. The frequency expressions for this situation may be obtained from (1 1.23) and (11.24) by replacement of gJ,K and g J + l , K with g J and g J + 1 , respectively. When g J # g J + 1, there are (2J+ 1) n components corresponding to the possible values of MJ.Although g J and g J + 1 are not likely to differ sufficiently to make this structure resolvable, they may differ enough to cause a detectable broadening of the line in some molecules. Likewise, a difference in g J and g J + would cause each component of the d doublet to be split into 2 J + 1 components, but such a splitting has not been resolved. Nevertheless, differences in g J and g J might cause a distortion of this line shape, as indicated by the theoretical pattern of Fig. 11.3. The difference between g J and g J + 1 assumed in these patterns is much greater than that likely to be encountered for linear molecules. Note that the center of the sub-multiplet structure does not occur at the peak absorption and that the asymmetry reverses when g J + changes from greater than yJ to smaller than g J . If 2Av represents the frequency difference between the centers of the d doublet multiplets, g J + 1 can be found from (11.20)

AM=O 7-

I I

I I I AM=T1 0-

I

I

I

AM=+I 0-

I

I I

I I I Fig. 11.2 Theoretical Zeeman components for a J + J ’ transition when y J = y J


Fig. 11.3 The A M

=

1 Zeeman pattern

(g components) for

a J-.J

51 1

+ 1 transition when g J fgJ+

1.

The most complete data available are for OCS. The A M = f 1 components of the Zeeman splitting have been observed for several OCS transitions, and the simple doublet pattern expected from (11.19)is found for each. That g J does not vary significantly with the rotational state is demonstrated more quantitatively in Table 11.1, where the effective g J values observed for different transitions are shown. Not surprisingly, the greatest deviation is observed for the J=O+l transition. This doublet splitting was first observed by Jen [ 5 ] , who found the g value from J = 1+2 transition to be 0.029 & 0.006. By use of circularly polarized radiation, Eshbach and Strandberg [3] later established the sign of g J as negative and found for the J = 1+2 transition the average g J = - 0.0251 f0.002. The magnitude of their value appears too small in comparison to other values, and its deviation is well outside the estimated limits of error of the highly accurate value later measured by Flygare et al. which is given in Table 11.1. The g J factor of CO is an order of magnitude larger than that of OCS. It has been measured from the J=O+1 transition by three different groups of workers [6-81. All the values are in good agreement. Rosenblum et al. [7], who made measurements on different isotopic species, obtained measurably different g J values for them. Burrus [8] extended the measurements on 1 2 C ' 6 0 to J = 1-2 transition and found g J to be independent of J within the accuracy of measurement. The g J measured for the J=O+1 transition is strictly that for

512


Table 11.1 Molecular Rotational g Factors for Selected Diatomic and Linear Molecules"

Molecule

J State 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 9 ' 5 9 - 0.269 l(5) - 0.2623(5) -0.2570(5) - 0.2562(5) - 0.2466(5) -0.2442(5) - 0.272(2) - 0.2431(16) 0.181(15) 0.184(15) 0.096(10) - 0.02871(4) -0.02813(4) -0.0195(2) - 0.0962 -0.0917(7) - 0.430( 10) 0.07606(10) - 0.0077(2) - 0.00630(14)

+ + +

+

ReJ: b b b b b

b c,d

e

I

I I qh h 1

j

k

e I m n

"The g factors are consistent with the magnetic moment expressed in nuclear magneton units. bRosenblum et al. [7]. 'Gustafson and Gordy [31]. dH. F. Bates, J. J. Gallagher, and V. E. Derr, J. Appl. Phys., 39,3218 (1968). 'McGurk et al. [40]. 'Burrus [SI. OHuttner et al. [17]. hFlygare et al. [20]. 'Shoemaker and Flygare [30]. 'B. N. Bhattacharya, Ph.D. dissertation, Duke University, 1958. 'Gustafson and Gordy [41]. 'Flygare et a]. [22]. "'Shoemaker and Flygare [18]. "Allen and Flygare [36].

the J = 1 state since MJ equals zero in the lower state. The gJ measured for the J = 1+2 transition is essentially that for the J = 2 state, as indicated by (11.20).

Symmetric-top Molecules For the symmetric top, as for the linear molecule, z represents the symmetry axis, and we can set gxx=gYY=gr, g r z = g K , and 5 = K . Since J,'+ J;= Jz- J 2 2,


513

the average is ( J , K, MJ1J,2+J$J, K , MJ)=J(J+1)-K2. Equation 11.10 specialized to the symmetric top becomes (11.21) and from (11.12) we get EJ,K,MJ=-gJ,KflIHMJ

(1 1.22)

Equation 11.21 can also be derived from the vector model of the symmetric top in an applied field, Fig. 10.1. The selection rules governing MJ are the same as those for a linear molecule, (11.17), and for a rotational transition J-J+l and K-K. Since K does not change, g K is the same for both upper and lower transitions. Furthermore, the assumption that gs(J,K ) = g s ( J + 1, K)=gl seems justified from the data on linear molecules, Table 11.1. From (11.22) the Zeeman frequencies are found to be (11.23) (11.24) where v o is the rotational frequency when no field is applied. When gr are constants

gK

and

(11.25) Because gK-91 is not negligible for many molecules, the 71 splitting is observable for low J values. It vanishes for K =O and for a given K decreases approximately as l / J 3 , but for K = J it decreases only as 1/J, approximately. We illustrate the application of the foregoing theory with CH3CCH. The upper curve of Fig. 11.4 shows the zero field and the components of the J = 3-4 transition. The different K lines are separated by centrifugal distortion. The bottom curve shows the same transition with a field of 10 kG imposed at right angles to the electric vector of the microwave radiation polarized in such a way that only the Mj+Mj+ 1 components are observable. Note that the K =O line is unsplit and not even measurably broadened by the field. When K=O, the splitting is due entirely to gr, as may be readily seen from (11.21). The expected splitting is like that for a linear molecule with gJ = g Thus the failure to observe splitting or broadening of the K=O line proves that g s is small. For the other components we can set g s z0 and obtain gKK2 gJ,K =J(J+1)

ggJ + 1,K =

gKK2 ( J + 1)(J+2)

The splitting is best resolved for the K = 3 line for which 9 3 3 = - 39K

4

and

9 4 3 = -9 9 K

20

(11.26)

514


Fig. 11.4 Zeeman effect of a symmetric-top molecule. The 5=3+4 transition (K=O, 1, 2, 3 components from right to left) of CH3CCH with the field arranged for observation of the AMJ = 1 components. The upper figure is for zero field; the lower, for a field of 10 kgauss. From Cox and Gordy [ 6 ] .

The splittings of the two energy levels are given by E33=- ~ ~ K P I H M Jand 4h

E43=-

99K P I HMJ 20h

in frequency units. The energy level diagram (not drawn to scale) and the indicated M j + M j k 1 transitions are shown in Fig. 11.5. The spacing of the upper multiplet level is only 3 that of the lower multiplet. The relative intensities of the M j - t M j - 1 components with increasing frequencies are 2, 6, 12, 20, 42, and 56 as calculated from (10.127). Those of the M j + M j + 1 components are the same, but in reverse order. The superimposed M j + M j - 1 and M j + M j 1 components obtained when g43=($)g33 are shown in Fig. 11.6. The four observed Zeeman components of the K = 3 line shown in Fig. 11.4 correspond to the four strong lines of the theoretical pattern. The others are too weak to be observable. From the separation of the outer of the four observed lines, g K is found [6] to be 0.298 k0.006. The corresponding gJ,K factor for the K = 2 line is only $ that for the K = 3 line, and the spread of the unresolved multiplet is about one-half that of the K = 3 line. The g factor for the K = l line, only 6 that of the K = 3 line, is too small for any resolution of the splitting. Shoemaker and Flygare [ 9 ] have remeasured the Zeeman effect of methyl acetylene with a magnetic field of 30,oOO G and obtained the more accurate value gll = +0.312 f0.002. They were also able to measure the small g L for CH3CCH. These and g factors for some other symmetric-top molecules are listed in Table 11.2. The g value caused by rotation of the methyl group in different symmetric tops appears to have the approximately constant value of 0.310.

+

ZEEMAN EFFECT I N MOLECULES WITHOUT NUCLEAR COUPLING

MJ

515

J

-4

-3

-2 -1

0 4 +1

t2 +3

+4

-2

Fig. 11.5 Energy level diagram of the Zeeman splitting of the J = 3 + 4 transition when g4=(:)g3 with indicated AMJ = 1 transition. This diagram corresponds to that for the K = 3 line shown for CH3CCH in Fig. 11.4 and to the theoretical spectrum shown in Fig. 11.6.

Asymmetric Rotors The first-order Zeeman perturbation energies of asymmetric-top molecules are given by (11.12) with the gJ,r given by (11.13). The problem of obtaining the average (J,”) over the wave functions of the asymmetric rotor is the same as that already encountered in the evaluation of the rigid rotor energies. The values given in terms of the reduced energies, (9.97)-(9.99), can be employed.

516


Fig. 11.6 The theoretical pattern of the J=3-+4, A M J =

1 Zeeman components when g4=($Jg3.

Table 11.2 Molecular Rotational g Factors for Some Symmetric-top Molecules Molecule 15NH3 PH3 PF3 CHjF CH335C1 CH3"Br CH31 SiHj8'Br CHjCN CH3NC CH3CCH SiH3NCS

gL

0.563(2)

- 0.033(1) -0.0659(3) - 0.062(5)

- 0.01653(28) -0.00569(30)

- 0.00677(40) - 0.02185( 13)

-0.0338(8) - 0.0546(15)

O.O0350(15)

- 0.01521(10)

911= g K

Ref:

0.500(2) 0.017(1) -0.0815(20) [0.31(3)]" [0.305(20)] 0.294(16) 0.310(16) -0.3185(5) 10.31(3)1 CO.31(3)1 0.312(2) -0.315(5)

"Values in brackets are assumed from methyl group contribution of 0.31. See Flygare and Benson [57]. bS. G. Kukolich and W. H. Flygare, Mol. Phys., 17,127 (1969). 'Kukolich and Flygare [28]. dStone et al. [23]. 'Cox and Gordy [6]. 'Shoemaker, Kukolich, and Flygare, reported by Flygare and Benson [57]. WanderHart and Flygare [21]. hDosel and Sutter [51]. 'Pochan et al. [32]. 'Shoemaker and Flygare [24]. kDossel and Sutter [52].

With these expressions we have


517

where a, b, and c are the principal axes of inertia, E ( K )is the reduced rotational energy, K =(2B-A - C ) / ( A- C ) is the asymmetry parameter, and A , B, and C are the usual spectral constants (see Chapter VII). In analysis of the spectra and evaluation of goo,gbb, and gcc,it is convenient to use first the frequency equations v*(MJ*

+ PrH (gJ,T- gJ',T')MJ h

MJ)=vO

(1 1.28)

~

to find the gJ,r values for different states from the observed Zeeman patterns and then to employ these values with (11.27) to solve for goo, gbb, and gee. Obviously, g J , T must be measured for three different states for evaluation of these elements. M'

MAGNETIC FIELD IN GAUSS

M

10857.1 110 M-+M'

111

-+

-1

-1

I I1 1009

I

1

I

--

-21-1 I I: :I

1: I:

-1

1511 492 14691 8594 1_: 15501 1:1 526 15081

I'

- y 1 1 - 2 -11 I1 2 1545110151473/

3

-

10856.0

10694.3

- - 10694.3

- 1 0

1

1-21-1

Io

2

3 I 1 12 460 13825 552 I542 I537 15101489 - 0754*0

; :1

I

I

Fig. 11.7 Theoretical Q-branch (AJ =0) Zeeman pattern with observed splittings for H,CO. From Hiittner et al. [17].

518

EFFECTS

OF APPLIED MAGNETIC FIELDS

Because J does not appear explicitly in the energy formula (11.12), the foregoing frequency formulas are applicable to J+J transitions as well as the J - J F 1 transitions, all of which may occur in asymmetric rotors. Since the intensity formulas are different, (10.125)-(10.129), the appearance of the Zeeman patterns for the Q, P, and R branches are generally different. Figure 1 1.7 illustrates the types of patterns which occur for the AJ=O, Q-branch transition. The small differences in the observed component spacings of a particular set are due to effects of anisotropic susceptibility, described in Section 4. Table 11.5, in Section 4, gives examples of principal g values of asymmetric rotors measured with the microwave Zeeman effect. For these molecules that have a plane of symmetry normal to the molecular plane, the magnetic axes, x, y , and z, coincide with the inertial axes, and the diagonal elements yuu, gbb, gcccan be taken as principal values of the g tensor.

2 ZEEMAN EFFECT OF NUCLEAR HYPERFINE STRUCTURE Weak-field Case When the nuclear coupling energy is much stronger than the Zeeman interaction, the rotational angular momentum J will form with the various coupled nuclear spin vectors a resultant angular momentum F which will precess about the direction of the field, as indicated in Fig. 10.6. In this weak-field case only the averaged values of the magnetic moment along F will effectively interact with H since the components of pJ or of pI normal to F will be averaged out by the precession about F. The Zeeman interaction energy will then be

By comparison with (11.11) gF is so defined that (PF)=gFPr[F(F+ 1)]''2

(11.31)

EH(MF= - gF PIHMF

(11.32)

Hence where MF=F,F-l,F-2,

..., -F

(11.33)

where F is the hyperfine quantum number defined in Chapter IX. The selection rules for M F are the same as those for M J ,that is, AMF =0, f 1. The frequencies of the Zeeman component can be expressed as (11.34)

ZEEMAN EFFECT OF NUCLEAR HYPERFINE STRUCTURE

519

where vo represents the frequency of the F-F' component when no field is applied and where MF and gF are for the lower rotational level. When the magnetic field is parallel to the electric vector of the microwave radiation, only the TC components corresponding to AMF=O are observable. When the field is perpendicular to this vector, the AMP = k 1 or 0 components are observable. The relative intensity formulas are the same as those for the Stark effect (10.130)and (10.131) and are like those for molecules without hyperfine structure if J is replaced by F and M J by MF. From analysis of the observed Zeeman patterns with (11.34) values of gF for the different hyperfine states can be obtained. From them it is possible to derive the nuclear g factor and the diagonal elements of the molecular g factor and pJ with reference to the inertial axes. To do this we must resolve both along F. When I and J are coupled, the nuclear magnetic moment which is along I contributes to the Zeeman splitting of the lines, whereas it does not when the coupling is negligible. In fact, the contribution of to the Zeeman splitting of the hyperfine components is of the same order of magnitude as that of pJ. We now express gF in terms of gJ and gl. First let us consider a molecule with a single coupling nucleus. The derivation is independent of the type of coupling (magnetic or quadrupole), but only quadrupole coupling is sufficiently strong to prevent decoupling of J and I for fields sufficiently strong for resolution of the Zeeman patterns. For the weak-field case the vector model of Fig. 10.6 applies. The angles between J and F and between I and F are constant, and cos (I, F ) with the relation ( p F )=gJPI[ J(J+ l)]'/' cos (J,F)+gjPj[Z(Z+ we find gF

=gJ a J

+gIaI

(11.35)

where

[ J ( J + 1)11/2 cos(J, F ) J-[F(F+ 1)]'12

a -

and

CI

-

[Z(I+

1)]"2

'-[F(F+ 1)]'12

cos (I, F )

From the law of the cosine applied to Fig. 10.6,

+ 1 ) + J(J+ 1)- Z(Z+ 1) 2F(F + 1) F(F + 1 ) + Z(Z+ 1)- J ( J + 1) = 2F(F + 1)

aJ= XI

F(F

(11.36) (11.37)

where F=J+Z, J+Z-1,. . ., IJ-11

(11.38)

Equation 11.32 for the Zeeman energies can now be expressed in the more explicit form EH(MF)= -(gJ~J+g1~I)PIHMF

(11.39)

520


For a linear molecule, 95'91 is approximately constant, independent of the rotational state (Section 1). For symmetric-top molecules, the value of g J - g J , K is given by Eq. 11.21; for asymmetric rotors, g J = g J , r is given by (11.27). With the g I measured from the Zeeman effect one can obtain values for the nuclear magnetic moment from PI

=srPrI

(1 1.40)

The value of p I for radioactive 1311was first measured with this method [lo], also that of j3S [ll]. However, the Zeeman effect cannot compete with the more accurate nuclear resonance method, which has already been used for measurement ofpI for essentially all known isotopes in their ground states. It is common practice to use these known nuclear moments (tabulated in Appendix E) to simplify the determination of g J from (11.39). When there is a second nucleus with coupling weaker than the first, one first finds (11.41)

gF1 = g J a J + g I l % l

~ , F1 the resultant of J and 11,as described above: by resolution of pJ and p ~along one then resolves pFl and P I , along F, which is the resultant of F, and I, (see vector diagram of Fig. 9.9). The result is gF=gFlaF1 +gIi'12

(11.42)

F(F 1) F1(F1+ 1)- I2( I, + 1) 2F(F 1)

+ +

(11.43)

+ 1)+ Iz( + 1)- F (F1+ 1) 2F(F + 1)

(11.44)

where aF1

=

EI,

=

F(F

+

I2

1

The Zeeman splitting of the hyperfine level designated by F is given by (11.32) with the g F of (11.42). Note that g F is a function of g J , S I , , and g 1 2 . Definition of the quantum numbers is given by (9.122) and (9.123). When there is a third nucleus with still weaker coupling, derivation of g F is accomplished by a further compounding of the vectors in the same manner. Jen [Z] first observed the Zeeman effect of rotational hyperfine structure in the 0-1 transition of CH3Cl. Burrus [8] has observed the effect in hyperfine components of DBr and DI. Figure 11.8 shows the and n components which he observed for the F=$+3 transitions of D13'I. The measured values of g J are listed in Table 11.1. Note that the sign as well as the magnitude of g J is given. The known sign of g I makes possible the determination of the sign of the molecular g factor from application of (11.35). For molecules without nuclear coupling, determination of the sign of g J requires the use of circularly polarized radiation [3].

ZEEMAN EFFECT OF NUCLEAR HYPERFINE STRUCTURE

521

Fig. 11.8 Zeeman pattern observed for the J=O+l, F=:+Z transitiun of DI at 195 GHz. The upper trace shows the z components (AMF=O). The lower trace shows the o components ( A M F = 1). From Burrus [SI.

Strong-field Case When the strength of the magnetic field is such as to make the Zeeman energy large as compared with the nuclear hyperfine energy, the nuclear coupling is broken down, and J and I precess separately about the field direction, as indicated in Fig. 10.7. The resultant F is not formed, and F and M F are no longer good quantum numbers. Instead, the quantum numbers which describe the state are J, M J , Z, and M I . The strong-field case is often called the PaschenBack effect from analogy with the breaking down of the spin-orbit coupling in atoms by strong magnetic fields or the Back-Goudsmit effect, which applies to the similar breaking of the nuclear and the electron coupling in atomic spectra. It is very similar to the strong-field Stark case described in this section. The Zeeman energy for the decoupled, strong-field case per se is the same as that for molecule without nuclear coupling, Section 1. However, the nuclear coupling energies are not canceled but simply altered by the applied field, and they must still be taken into account. The nuclear magnetic coupling energy in strong fields is very simple:

Ed”(MJMI)=CJ,,MJMI

(11.45)

where CJ,Tis the coupling constant for the particular rotational state J, 7 derived for various molecular types in Chapter IX, Section 7. The nuclear quadrupole coupling energy for the strong-field case, which is the average (J,M J ,I, MII XQIJ, M J ,Z, M I ) ,already evaluated in Chapter X, Section 2, is

Eb“(MJM,)=

eQqJ [3M3 - J ( J + 1)][3M! - I(Z+ l)] 4J(2J- 1)1(2Z- 1)

(11.46)

where qJ for different molecular types is given in Chapter IX. If there is more than one coupling nucleus, the EP for each one is calculated with the same

522


formula, (11.46), and the results are added. The total first-order perturbation energy of the rotating molecule in the strong-field case is

(11.37)

where the sums are taken over all nuclei with coupling moments. The strongfield Zeeman effect has the same advantage as the strong-field Stark effect in simplifying the hyperfine structure caused by more than one coupling nucleus. The selection rules are MJ+MJ

MI+MI

n components

(1 1.38)

MJ+MJf 1

MI+MI

0

components

(1 1.49)

Because g r , is the same for all states and because the nuclear spin does not change orientation; AMli =0, the second term on the right, has no influence on the spectrum and can be dropped. For molecules in ' 2 states the term in C J , ,is usually negligible. Since the first term corresponding to EH(Mj) is larger than Ep(MJMr),it is convenient to calculate first the splitting caused by this term and then to superimpose the hyperfine splitting on each of the Zeeman states. The relative intensities of the gross components of the Zeeman pattern are the same as those for molecules without hyperfine structure. Each gross Zeeman component is split into a hyperfine substructure. Because of the selection rules AMI=O, there is no reorientation of the nuclei during a transition, and the nuclear orientation states are equally populated or very nearly so. Thus the hyperfine components corresponding to different values of M I are all of equal intensity. However, for nuclear quadrupole interaction the levels corresponding to + M I and - M I are degenerate. For this reason the components for M I not zero are twice as strong as those for M I =O. When there is a single nucleus with quadrupole coupling having Z=$, each Zeeman component will be split into a doublet of equal intensity corresponding to M r = ki and M I = f?.For Z=1, the substructure would still be a doublet, but the components would have intensity ratios of 1 to 2 corresponding to Mr=O and M I = f 1. If the molecule has two nuclei with quadrupole coupling, it is convenient to calculate first the hyperfine pattern which is due to the stronger coupling nucleus and then to calculate the substructure of each of its components caused by the coupling of the second nucleus. Each calculation is independent of the other. Suppose, for example, that the stronger coupling nucleus has Z=$. Each Zeeman component would have a calculated triplet substructure of equal intensity caused by this nucleus. If the second coupling nucleus has I = 1, each component of this triplet is further split into doublets with intensity ratios of 1:2. Figure 11.9 illustrates the strong-field spectrum [12] for N 2 0 in which there are two 14N nuclei with weak but unequal quadrupole coupling. The

THEORY OF MAGNETISM IN CLOSED-SHELL MOLECULES

Calculated

523

I

I

Experi menta 1 I

I

I

I

I

I

1

Calculated

I

I

1

Experimental I

nuclear magnetic coupling is negligible. The gross triplet is due to the 71 and c components arising from the term - g J p I H M J ,the substructure to the nuclear quadrupole coupling. The coupling by one 14N splits each Zeeman component into a doublet with intensity ratios 1 :2; the second 14N further splits each of these into doublets with intensity ratios 1:2, thus giving each Zeeman component a four-line substructure. The splitting by the end nitrogen is the greater. Because of its infrequent application we shall not treat the intermediatefield case in which the Zeeman and the nuclear coupling energies are of comparable magnitude. The treatment is similar to that described for the intermediate-field Stark effect in Chapter X, Section 2. Usually the external field may be adjusted so that either the weak- or strong-field case applies.

3 THEORY OF MAGNETISM IN CLOSED-SHELL MOLECULES ( I C STATES) The most common molecules have closed electronic shells ('E states) and have no magnetic moments when field-free and not rotating. Molecular rotation, however, generates a small magnetic moment which is proportional to the rotational angular momentum, as is assumed in Section 1. Also, an externally imposed magnetic field may generate a magnetic moment caused by the

524


molecular susceptibility which is proportional to the strength of the magnetic field, as is assumed in Section 11.4. The magnetic susceptibility is a familiar quantity which is treated thoroughly in Van Vleck's classic work on magnetic and electric susceptibilities [131. The susceptibility per molecule consists of a diamagnetic as well as a paramagnetic contribution, the principal elements of which can be expressed as

In these expressions (0)represents the ground electronic wave function of the molecule; and In), that of the excited state; L, represents the electronic orbital angular momentum operator; and En- Eo, the difference in energy between the states In) and the ground state. The subscripts signify the coordinates of the ith electron, and the summation is taken over all electrons in the molecule. The constants e, m, and c represent the charge and mass of the electron and the velocity of light, respectively. The molar susceptibility may be obtained by multiplication of the susceptibility per molecule by Avogadro's number N . The quantity measured in the usual bulk susceptibility experiments is the average

x = 3 ( x x x +x y y +

xzz)

(11.53)

The first term on the right of Eqs. (11.50)-(11.52) arises from the Larmor precession of the electronic charge about the applied field and represents the diamagnetic component. The last term is the paramagnetic component that arises from an induced admixture of excited paramagnetic states with the nonmagnetic ground states. Although these excited states have orbital angular momentum, this momentum is counterbalanced by the opposite circulation of charges giving the diamagnetic term so that the total, overall angular momentum remains zero. The electronic spins remain paired and thus make no contribution to the magnetism. Condon [14] originally treated rotational magnetic moments as caused by the rotation of charges fixed in the molecule. However, the early measurements of rotational magnetism by the molecular beam resonance method showed that the concept of fixed charges rotating with the molecular frame was inadequate. Wick [15] considered the separate contribution of the electron and nuclei and showed that the measured moment of H 2 could be explained if the electrons were considered as having angular momentum apart from their rotation with the molecular frame. Jen [5] extended this theory to polyatomic molecules and assumed that the inner-shell electrons rotate with the nuclei and that only the valence electrons have angular momentum relative to the molecular frame.


525

With the Wick theory one can consider the rotation of the charges as generating a magnetic field which then induces a magnetic moment in the electronic cloud because of its magnetic susceptibility. In the Hamiltonian for the rotating molecule Eshbach and Strandberg [3] approached the problem in a more basic manner by considering separately the angular momentum due to the nuclear system from that due to the electrons. Their treatment, which does not explicitly employ the concept of magnetic susceptibility, is followed here. Rotating molecules in 'C states can have a slight component of orbital electronic angular momentum which is due to a rotation-induced admixture of electronic states having orbital angular momentum with the ground electronic state. Because of the rigorous requirement that the total angular momentum must be a constant of the motion, no such components are possible in the nonrotating molecule, for which the electronic orbital angular momentum must either be exactly zero, as for 'C states, or must have the magnitude h[L(L 1)11/' where L is an integer, as for ll states. In the rotating molecule let us represent the total angular momentum by

+

J=N+L

(11.54)

where N represents the angular momentum caused by rotation of the nuclei alone and L, that due to the electrons. Only J, the total angular momentum, must be precisely quantized. The rotational Hamiltonian for the nuclear system plus the Hamiltonian for the unperturbed electronic energies is

x=-c-+x--c 1 N,' 1 2

Ig

"-2

(Jg g

- LgI2 yl", +

Ig

where g =x, y , z represents the principal inertial axes and where I, represents the principal moments of inertia due to the nuclear masses alone. We have assumed the nuclear frame to be rigid, that is, have neglected the effects of vibration and centrifugal distortion. Since L, is very small, the third term in (11.55) can be neglected,.and the second term can be treated as a perturbation. Therefore H=HO++' (11.56) where (11.57) and (11.58) Here the first term in 2' is like the Hamiltonian of the rigid rotor considered in earlier chapters except that Ig represents the moment of inertia resulting

526


from the nuclear masses alone, whereas in the earlier chapters the electronic masses were lumped with the nuclear masses in determination of Zg. See the discussion in Section 7. The electronic Hamiltonian is a function of the kinetic and potential energy of the electrons; however, the exact form of is not required for our discussion. The angular momentum of the electrons will have an associated magnetic moment e L 2mc

p L = - __

(11.59)

Averaged over the unperturbed ground electronic state (pure C state), this will be zero. e 2mc

(oli(Llo)= - -(OlLlO)=O

(11.60)

However, the term 2'perturbs the ground electronic state when the molecule is rotating. To obtain the electronic contribution to the rotational magnetic moment one may correct the electronic basis functions for the perturbation term. According to perturbation theory the first-order perturbed wave function is (11.61) where $bo) signifies the wave function of the unperturbed state and $Lo) signifies those of other states for which the matrix elements of 2' are not zero. The E , are the unperturbed electronic energies. Since the unperturbed electronic functions are independent of the Jgoperators, the first-order perturbed electronic function is found from (11.58) and (11.61) to be (11.62) along with Is assumed to be constant. The averaged component (p)e,ectronic the g axis caused by the electronic angular momentum is

where higher-order terms are omitted. The first term on the right is zero, from (11.60). The second term is a tensor quantity, the diagonal elements of which give the observable quantities along the principal axes of inertia.


527

These are (11.64) where P (=eh/2mc) is the Bohr magneton, PI is the nuclear magneton, and Jg is now in units of h. The electronic contribution is seen to arise purely from an admixture with paramagnetic states. There is no diagmagnetic contribution to the rotational magnetic moment from the precessional motions of the electronic cloud. The contribution to pg by the rotating nuclei is easy to calculate because it need not be averaged over the electronic functions. It is simply the classical expression for the magnetic moment produced by the fixed nuclear charges Zke rotating about the principal axes with angular velocity u g = hJg/Ig.The component about the principal axis x is, for example, Jx

(Px)nuclear=MpPI

I x

2 Zk(yk2+Zk2)=/3Ig:.xJx

(11.65)

k

where M , is the proton mass, PI is the nuclear magneton efi/2Mpc,zk is the atomic number, and Yk and zk are the coordinates of the kth nucleus in the principal inertial axes, and the summation is taken over all nuclei in the molecule. The y and z components are similar. The foregoing equation defines the diagonal elements of the g"tensor; g", like g", is, in general, a tensor quantity, but in first-order only the diagonal elements enter, as observed previously. When the inertial axes and the magnetic axes are the same, as in symmetric-top molecules, the diagonal elements of g represent the principal values. The total rotational moment along the principal inertial axis g is Pg

=(Pg)nuciear

+ (Pg)electronic

(11.66)

where the values on the right are obtained from (11.64) and (11.65). It is seen that pg is proportional to J g , as assumed in Section 1. The diagonal elements of the molecular rotational g factor in the principal inertial axes are readily obtainable from these values of pg and from (11.5) which defines ggg=g;g+g;g. These elements are (11.67) (11.68) (11.69) The first term on the right in these expressions is easily calculable from the moments of inertia and the molecular structures which are found from analysis of the rotational spectra. Since I,M p z k ( y i + z i ) , and so on, these quantities are very closely equal to unity. The last, the paramagnetic term, can then be

2

528


measured from the principal elements of the g factors. These values of the paramagnetic terms can then be substituted into (11.50)-(11.52). When the corresponding elements of paramagnetic susceptibility are measured, the summed averages of all the electrons (OlyZ +zflO) over the electronic ground state function can be obtained. From the sum of these expressions (Olr?lO) can be evaluated. For this, it is convenient to eliminate the paramagnetic term between these expressions and to solve for the summed averages. Thus

xi

ci

(11.70)

(11.71)

For a linear molecule, gxx=gyy=g and (11.67)-(11.69) reduce to the simpler expression (11.73) where z k is the distance of the k t h nucleus from the center of gravity. For a linear molecule the summed average from (11.70) to (11.72) is (11.74) where x=~(xl1+2xI), z k is the distance of the kth nucleus from the center of gravity, I is the moment of inertia due to the nuclear masses alone, and r?=xf+y?+z?.

4

AN I SOTR 0 PI C MAG N ET1C S USC EPTl 6 I LIT1ES

Measurement of Molecular Anisotropies In addition to the molecular magnetic moments generated by rotation, which were considered previously, there are much smaller magnetic moments introduced in the molecule by the applied Zeeman field. When the magnetic susceptibility of the molecule is completely isotropic the induced moment is independent of the rotational state, and there is no detectable effect on the spectra. An anisotropic component can, in sufficiently strong fields, produce a measurable effect on the Zeeman pattern. Essentially all molecules, except the spherically symmetric ones which generally have no observable rotational spectra, do have anisotropic susceptibility. The effects of anisotropic susceptibility on the Zeeman effect are very similar to those of anisotropic polarizability on the Stark effect considered in

ANISOTROPIC MAGNETIC SUSCEPTIBILITIES

529

Chapter X, Section 6. It is convenient to express the induced moments with reference to the inertial axes. Although these axes are not necessarily the principal axes of magnetic susceptibility, the off-diagonal elements will average out in the first-order treatment so that we need to consider only the diagonal elements. The induced value of the moment along the principal inertial axis x is Yx(X)= f ( X x x H x

+x x y H y + + x x y @ z y +x x z @ z z ) X X Z H Z )

(11.75)

=fH(Xxx@zx

with H taken along the space-fixed 2 axis. The interaction of this component with the field will be

-j.tx’H= - p x H @ z x (11.76) = -t H ’ ( x x x @ i x + x x y @ z x @ z y + x x z @ z x @ z z ) The interactions with p y and pz are similar, and the total interaction is (11.77) where xsgj are elements of the magnetic-susceptibility tensor referred to the principal inertial axis system. The first-order energy is the average of this Hamiltonian over the wave function of the rotor. In this average the offdiagonal terms do not contribute; only the diagonal terms are involved. Thus EH(X)= - t H 2 [ x x x ( @ ; X > +XYY(@iY> + x Z Z ( @ 2 z > l (11.78) in a symmetric-top or linear molecule the principal inertial axes are also ~ ~ = where x ~ ,z is the symmetry axis. principal axes of x, and ~ ~ ~ = x xzz=~II, Therefore we write EH(x)= - 4 H 2 [ X . ( @ i x + @ i y > + X I I ( ~ i z > l

(11.79)

1 -aiz and with neglect of the term - f x I H 2 which does With @ix+@i,= not depend on the rotational quantum numbers, we obtain E H ( X ) = -fH’(X,, where for a symmetric rotor basis (@$z>=(J,

-xJ@L>

(11.80)

K , [email protected]/J,K , MJ)=C I(J, K , M J / @ Z z l J ’ , K , MJ)12

(11.81)

J’

The required cosine matrix elements are given in Table 2.1. Substitution of these matrix elements gives the perturbation energy for the symmetric top, EJ,K,M,(X)=

--sex,, -X.)H’

{

[(J+ 1)’- K ’][(J+ 1)’-M5”] (J+ 1)’(2J+ 1)(2J+ 3)

K ’M; + P(J+ 1)2

+

1

( J 2- K ’ ) ( J 2- MJ’) J2(4J’ - 1)

(11.82)

530


The formula for a linear molecule is obtained when K=O is set in (11.82), which then becomes

One must add E&) to E H ( p )to obtain the total perturbation of a magnetic field. The total energy of the rotating molecule in a magnetic field is EH=E:+EH(pL)+EH(X)

(1 1.84)

where E: is the energy of the unperturbed rotor. With the magnetic fields commonly available in the laboratory, one is not justified in extending the calculations to higher order. Here E H ( p )represents the first-order magnetic interaction derived in Section 1. The anisotropic susceptibility energies for asymmetric-top molecule5 are obtained from the average of the squared direction cosines of (11.78) over the functions of the asymmetric rotor. These averages can be expressed in terms of the line strengths. However, a more usable formula is obtained by expression of the average values of the squared direction cosines in terms of the average values of the squared angular momentum components. The resulting formula can be expressed in the form [16]

where X=$(Xaa+Xbb+Xcc)

(1 1.86)

is the isotropic component and where a, b, and c represent the principal axes of inertia. Values of ( J : ) in terms of the reduced rotational energies and their derivatives are given by (9.97)-(9.99). See also Chapter VII. The total energy of the asymmetric rotor in a strong magnetic field, exclusive of nuclear interactions, may be expressed by (11.84) with E H ( p )from (11.12) and EH(x)from (11.85). Thus

where the first term on the right, E,( J , 7, M J ) ,is the field-free rotational energy which may be found as described in Chapter IX. A term, - f H 2 , is omitted because it displaces all levels equally and has no influence on the spectra. The Zeeman splitting of the rotational lines caused by the second term on the right is described in Section 1. Further shifts of these components resulting from the last term are measured as a function of the field H. From these measurements are found the magnitudes of the three components, gaa, g b b , and g",


531

along the principal inertial axes, a, 6, and c, of the asymmetric rotor, together with the anisotropies. 3(Xaa

- 1)=2Xaa - Xbb - xcc

(11.88) (11.89)

(11.90) These expressions are easily derived by addition of xaa,X b b , or xCcto both sides of (11.86) and rearrangement of the terms. The third 3(xcc-x) is seen to be the negative sum of the first two. Because of the relationship of (11.86), there are only two independent molecular anisotropies. These are usually expressed by the right-hand sides of (11.88) and (11.89). When x can be reliably calculated or derived from bulk susceptibility measurements, the diagonal elements, xan, X b b , and xcc,can be obtained from (11.88)-(11.90). As indicated in Section 3, the molecular susceptibilities defined by (11.50)(11.52) have diamagnetic as well as paramagnetic components. xss =

xis + xgq

(11.91)

The diamagnetic components correspond to the first terms on the right in (11.50)-(11.52), and in the asymmetric rotor, inertial axes, a, b, and c, are electrons

X daa

-

4mc’

E I

(olbf + cflo)

( 11.92)

(11.93) (11.94)

If these values are substituted in (11.70)-(11.72) with the coordinate change x+a, y+b, z+c, solutions for xaa-xta, and so on, yield the paramagnetic components (11.95) (11.96) (11.97) where we have substituted Ia=h/8n2A, and so on. In these expressions Zk is the atomic number of the k th nucleus, and the summation is taken over all nuclei in the molecule. To express the susceptibility components in mole units, the terms on the right are multiplied by Avagadro’s number, N .

532


Because of their symmetry, the principal axes of magnetic susceptibility of linear and symmetric-top molecules coincide with their principal inertial axes, indicated simply as the parallel and perpendicular axes in (11.82) and (11.83) and in the tabulated values for these quantities. See Tables 11.3 and 11.4. The axes, a, b, and c, in which gee or xes are expressed for the asymmetric rotors, as in (11.87), are principal axes of inertia and do not necessarily coincide with the principal magnetic axes. In general, the derived magnetic quantities yss and xgg are diagonal elements of magnetic tensors that may have nonzero, off-diagonal elements. However, any such off-diagonal elements of i: are averaged out by the molecular rotation and hence are not detected in these symmetry, such as those listed in experiments. In planar molecules having qZL. Table 11.5, the principal magnetic axes designated as x, y , z, coincide with the principal inertial axes a, b, c; for these, the principal values of the magnetic quantities are obtained directly from the analysis of the spectra.

Table 11.3 Anisotropy in Magnetic Susceptibility a n d Molecular Q u a d r u pole M o m e n t of Some Linear Molecules

Molecule

co cs CSe Si0 AIF

CuF

ocs OCSe

HC15N HCP NNO FCCH 35C1CCH

Susceptibility Anisotropy,

Quadrupole Moment,

XI - XI1 (10- 6erg/G2 mole)

Q II

8.2(9) 24(3) 27.8(14) 11.1(9) 5.2(5) 6.5(7) 9.27(10) 10.06(18) 7.6(8) 8.4(9) 10.15(15) 5.19(15) 9.3(5)

-2.0(10) 0.8(14) -2.6(16) -4.6(11) - 5.9(5) -6.1(8) -0.88(15) - 0.32(24) 3.5(9) 4.4(9) - 3.65(25) 3.96(14) 8.8(4)

’Gustafson and Gordy [31]. bMcGurk et al. [40]. ‘Hornerjager and Tischer [38]. dHornerjager and Tischer [42]. ‘Hornerjager and Tischer [45]. /Flygare et al. [20]. 9hoemaker and Flygare [30]. hGustafson and Gordy [41]. ‘Flygare et al. [22]. ’Shoemaker and Flygare [18]. ‘Allen and Flygare [ 3 6 ] .

esu.cm2)

ReJ: u

a,b

b C

d

e

r 9 h

b L

j k

533


Table 11.4 Anisotropy in Magnetic Susceptibility and Molecular Quadrupole Moment of Some Symmetric-top Molecules

Molecule

Susceptibility Anisotropy, X I - XI1 ( l o 6 ergfG2*mole)

Molecular Quadrupole Moment, Qa esu-cm’)

- 0.3 7(4) 2.7(8) 1.3(5)

- 2.32(7) - 2.1 (10) 24(3) 1.2(8) 3.4(8) 5.4(9) - 1.8(12) - 2.7( 16) 4.82(23) 9.9(8) - 0.1(4) 16.6(37)

NH3 PH3 PF3 CH3CI CH,Br CH31 CHCN CH3NC CHjCCH CH3CCCCH SiH3Br SiH3NCS

8.0(5) 8.5(4) 11.0(5) 10.2(10) 13.5(17) 7.70(14) 13.1(2) 2.7(3) 28.0(30)

Ref: u

b C

d d d e

e

/ 9

h I

“Kukolich [29]. *Kukolich and Flygare [28]. ‘Stone et al. [23]. dVanderHart and Flygare [21]. ‘Pochan et al. [32]. ’Shoemaker and Flygare [24]. %hoemaker and Flygare, reported by Sutter and Flygare [58]. hDossel and Sutter [51]. ’Dose1 and Sutter [52].

Taft and Dailey [4] have measured the anisotropic susceptibility for OCS from the J = 1 +2, M J + M J transition with a magnitude field of 23 kG. The M p M J or IT transitions are observed with the field imposed parallel to the microwave electric vector. With this arrangement no shift or broadening of the rotational lines is caused by the rotational magnetic moment when gJ =gJ+ 1. For linear molecules the g value for the different rotational states is expected to be the same except for very slight effects caused by centrifugal distortion. In OCS the possible difference in the g value observed for different rotational states can give rise to a splitting of the n components of only a few hertz, even at the 23 kG fields employed by Taft and Dailey. However, it is seen from Eq. (11.82)that the magnetic states have a slight J dependence caused by the anisotropic susceptibility. As a result, the J = 1+2, MJ=O+O component is component. Taft and Dailey separated from the J=1+2, M J = + 1 + + 1 observed that the I + & 1 component was shifted from the position of the zero-field line by 132 kHz. Using this observation they obtained the anisotropic susceptibility from (11.83)as xL -xil =8.35 x erg/(G2*mole).A remeasurement by Flygare et al. with a field of 30 kG gives the more accurate value listed in Table 11.3.

+


534

In an asymmetric rotor the molecular g factor differs according to the rotational state, and therefore the AMJ=O transitions are split, even in the absence of anisotropic susceptibility. Without anisotropic effects, however, all the Zeeman components of a given multiplet (whether n or 0)are equallj spaced. Addition of the quadratic term E&) of (11.85) cause a difference in the spacing which allows measurement of the (x,,-x). Hiittner et al. [17] were the first to measure the components of the magnetic susceptibilities of an asymmetric rotor, H,CO, from the differences in the spacing of the Zeeman components, as shown in Fig. 11.7. Later values of the magnetic constants of H 2 C 0 obtained with magnetic fields of 30 kG are listed in Table 11.5. Following the experiments just discussed, there have been a large number of measurements of anisotropies in magnetic susceptibilities, some of which are cited here [18-561. Description of the method and review of the essential results to 1970 are given by Flygare and Benson [57] and to 1976, by Sutter and Flygare [58]. Selected experimental values are given in Tables 11.3, 11.4, and

Table 22.5 Zeeman Parameters and Molecular Quadrupole Moments for Selected Asymmetric-top Molecules" Y

gxx

SYY

X

I-

Molecule

gzz

2xxx2Xw-

xyy-

xzz

xxx -x z z

erglG2*rnole)

Qxx

QYY Q Z Z

RCif.

H

\

0.71 8(7) 0.657( 1)

0

/

- 0.199(48) 0.464(24)

- 0.1 3(3)

2.63(2)

bd

- 2.50(2)

0.645(6)

H H

\

c=o

-2.9017(8) -0.2243(1) - 0.0994(1)

c=c=o

/

25.5(5)

- 0.1(3)

- 3.9(3)

0.2(2) - 0.1(5)

- 0.4182(9) -0.0356(13) -0.0238(6)

- 5.0 -0.2(6)

-0.7(3) 3.8(4)

- 0.042 1( 5 ) - 0.0466(4) - 0.01 19(4)

- 2.3(6) 7.7(5)

d

H H

\ /

E there is no possibility of passage from one potential well to another because the system does not have enough energy to surmount the barrier. Quantum mechanically, however, the molecule may pass from one configuration to another by tunneling through the barrier since

576

INTERNAL MOTIONS

A

A

?6

E

+-5

-

3

+ -

V3 = 0 Free Rotor

E

Potential Barrier

V3 =

00

Harmonic Oscillator

Fig. 12.4 Correlation between the energy levels of free internal rotation specified by the quantum numbers m, and those for harmonic torsional oscillation which are labeled by the quantum numbers v. For barriers between the two extremes some of the degeneracy of the two limiting cases is removed. The degeneracy of the torsional sublevels is indicated by A and E, the E levels being doubly degenerate whereas the A levels are nondegenerate.

the wave functions extend through the classically forbidden regions. The tunneling rate depends on both height and width of the barrier. The effect of tunneling through the potential barrier is to split the triply degenerate torsional level v into two levels, a nondegenerate level designated as an A level and a doubly degenerate level designated as an E level. The connections between the harmonic oscillator limit and the free rotor limit are shown schematically in Fig. 12.4. That each triply degenerate level of the harmonic oscillator limit is split into two levels is apparent. For the free-rotor states, with m a multiple of three, the fm degeneracy is removed. For the other values of rn this degeneracy remains. The symmetry of the torsional energy levels (the A and E designations) will be considered later. In Fig. 12.3 the internal rotational energy levels are shown superimposed on the potential curve. Note that the torsional sublevel spacing increases as the torsional energy increases, that is, as v increases. The greater the sublevel splitting, the faster the rate of tunneling. Near the bottom of the well the energy levels approximate those of a harmonic oscillator. When the torsional energy E is greater than the barrier height V3, the energy levels go over to those of a free rotor. The qualitative aspects of the torsional energy levels have been presented, but quantitative discussion for intermediate values of the barrier height requires consideration of the solutions of the original torsional wave equation, (12.6). A knowledge of the torsional eigenfunctions and eigenvalues is useful for the evaluation of various perturbation sums that arise in the perturbation treatment of internal rotation, which will be described subsequently. Because of the choice of the potential function, (12.6) can be transformed to

TORSIONAL ENERGY LEVELS

577

Mathieu’s equation by the substitutions:

3a + 71 =2x V 3 =($)Fs

V ( M=) M ( x ) E = (2)Fb

(12.15)

giving

d2M(x) dx2

+ (b- s COS’

x)M(x)=O

(12.16)

where b is an eigenvalue, S. is a parameter termed the reduced barrier height, and M ( x ) a Mathieu function. The eigenfunctions V(a)of interest must satisfy the boundary condition V(a)= V(a 271) or for the Mathieu functions, M ( x )= M ( x + 371). The Mathieu functions which meet this boundary condition are those with period 71 in x (period 2n/3 in a) and period 371 in x (period 271 in a). The eigenvalues and eigenfunctions of the torsional wave equation can hence be obtained from existing tables [15-191 on periodic Mathieu functions with the aid of (12.15). The solutions with period 71 are associated with nondegenerate eigenvalues while those of period 37t are associated with doubly degenerate eigenvalues. For distinction between the eigenvalues (Euuor bob)and the wave functions, the labels u and cr are introduced. The series of degenerate and nondegenerate solutions obtained are ordered in terms of increasing size. The lowest degenerate and nondegenerate eigenvalues are then labeled by u =O. The next lowest degenerate and nondegenerate eigenvalues are labeled by the index u = 1 and so on. For a high barrier, the degenerate and nondegenerate states corresponding to a given u have nearly the same energy, whereas those of different u are more widely separated. The degenerate and nondegenerate energy levels associated with a given u are regarded as the torsional sublevels. As V3+co, the different energies of the torsional sublevels coalesce, and u becomes the quantum number for the limiting harmonic oscillator state. The usefulness of the label u is thus readily apparent; it is called the principal torsional quantum number. The index cr =0, +_ 1, which gives the symmetry or periodicity of the torsional wave functions, serves to distinguish the torsional sublevels. The cr =O levels are the nondegenerate A levels, and the 0 = L 1 are the degenerate E levels. Appropriate solution of the torsional wave equation, therefore, leads to a set of torsional levels u, each of which is split into two sublevels with one nondegenerate and the other twofold degenerate. A periodic solution of (12.6) or (12.16) may be represented by a Fourier expansion. A solution of the torsional equation, for example, can be written in the following convenient form

+

00

Vu&)=

C 4 ei(3k+u)or k= 0)

(12.17)

m

where cr is an integer. For the appropriate periodic solutions of a threefold barrier cr takes on the three values - 1, 0, and + 1. As mentioned previously, the index cr gives the symmetry of the torsional wave functions. The torsional

578

INTERNAL MOTIONS

Table 12.1 Character Table for the Group C! Symmetry Species

E

c3

c:

A

1

1

1

1

E

E*

1

E*

&

E

i::

"Here ~=exp(i2n/3) where E* designates the complex conjugate and &E* = 1, (E)' = E*, (&*)2 = E .

functions may be conveniently classified under the rotational subgroup C3 of the torsional Hamiltonian. The character table of this group is given in Table 12.1. For classification according to the symmetry species of the group, it is necessary to know how the functions, (12.17), transform under the symmetry operations C3 and C;.The operations C3 and C: on UV&) will affect only the angle a, transforming it in the following manner-C3: a+a +2n/3; C:: a+a+ 4n/3 = a - 2n/3. The identity operation, of course, leaves the function unchanged; the effects of the remaining operations on Vu&) are c3 :

U",(a)+&' U"&4

c;: U",@)+ (E*)" vu, (4 From the foregoing relations and the character table, it is evident that the U,, with o=O belong to the nondegenerate species A and give rise to the nondegenerate eigenvalues. The functions with a= 1 belong to one part of the doubly degenerate E species, viz., El ; those with a = - 1 belong to the other part, E 2 , and they give rise to the pairs of degenerate eigenvalues. Some insight into the nature of the A(a = 1) and E(a = & 1)internal torsional states may be obtained from the following considerations. The internal motion for the A states [ U(a)periodic in 2n/3] resembles a back-and-fourth oscillation localized in the potential wells. For these states the &VI degeneracy of the freerotor states is removed (see Fig. 12.4). The internal motion of the E states [ U(a)periodic in 2n] has some of the character of free rotation passing from one potential well to another by tunneling, the + V I degeneracy of the free rotor which remains for the E states being related to the two possible directions of internal rotation. In general, for a molecule with N equivalent internal configurations the appropriate periodic solutions [17, 181

+

Uua(a)=ei~a

2

A:u )eiNkrr

(12.18)

k= -m

are obtained by choice of N integer values of a, such that - N/2 ea< N/2

TORSIONAL ENERGY LEVELS

579

with a # - N/2 for N even. Except for a=O or a = N/2 ( N even) which give nondegenerate solutions, the remaining values of a give pairs of degenerate solutions corresponding to +a. The number of distinct eigenvalues for any ti are ( N + 2)/2 for N even and ( N + 1)/2 for N odd. In the limit VN+m the sublevels coalesce, leading to an N-fold degeneracy. For finite barriers the N-fold degeneracy is lifted by the effect of tunneling. Explicit solutions of (12.6) or (12.16)can be obtained by numerical iterative techniques similar to those employed in the determination of the eigenvalues of a rigid rotor (see Appendix B). Substitution of the expansion equation (12.17) into (12.6) and subsequent multiplication by and integration from 0 to 2n gives Ak-1 + ( i - k f k ) A k + A k + I

=o

(12.19)

where

and

k=O, k l , k 2 , + 3 ,... Here the ti labeling of the Fourier expansion coefficients has for the present been omitted, and the substitution of cos 3cc =(ei3a + e - i 3 a)/2 and the fact that ei(k-k’)a dcc =0, unless k = k with k and integer, have been used for derivation of Eq. (12.19). If this infinite set of linear homogeneous equations is to have a nontrivial solution for the A’s, the determinant of the coefficients must vanish. The infinite order determinantal equation has the form :

Jt*

=O

(12.20)

There will be three separate determinants, one for each value of a. However, it is apparent that the determinants corresponding to +a have the same eigenvalues and thus constitute the degenerate E species solutions with U”,- l(a)= V,*,(c(). The eigenvalues E,, and thus the torsional energies associated with a particular value of a, may be conveniently obtained by solution of the appropriate determinant by the method of continued fractions [8, 17-20]. Once the eigenvalues are known, the Fourier coefficients of the expansion

580

INTERNAL MOTIONS

can be determined by means of the recursion relations between the A’s, (12.19). Solving these relations and making use of the normalization requirement, l;‘ U*(a)U(a)da= 1, or 271 A: = 1, one obtains the torsional eigenfunctions. Both the eigenvalues and the expansion coefficients depend on the reduced barrier height s. Since the values of k are not bounded, a truncated Fourier expansion which leads to a determinant of finite size must be used if the problem is to be mathematically tractable. The number of terms to be retained in the Fourier expansion is dictated by the accuracy desired in the eigenvalues and eigenfunctions. In the limit of free rotation only one of the Fourier coefficients will be nonvanishing and

1

(12.21) while the eigenvalues are given by the diagonal elements of (12.20):

E,, = F(3k

+

a)Z

(1 2.22)

Since (3kt-a) can have any integral value, a comparison of (12.22) with (12.10) reveals that m =a, modulo 3 (see Fig. 12.4).In the high barrier limit the energies are those of a harmonic oscillator and are given by (12.14)as Eu=3(V3F)”2(u+f) The energies are independent of a and hence are threefold degenerate. Each torsional function is a linear combination of three harmonic oscillator functions, each centered about one of the potential minima [8]

(1 2.23) (12.24) 1 U ~ , - ~ ( U ) = -[ ~ : 1 ) + & * ~ : 2 ) + 4 3 ) 1

(12.25)

where E =exp(i2~/3) and where HZ”, Hi2), and HL3)are harmonic oscillator functions centered, respectively, about 0, 2x13, and 47113. Fortunately, the detailed calculations outlined here are usually not necessary because the required perturbation sums, which can be obtained from a knowledge of the torsional eigenfunctions and eigenvalues, have been tabulated for various values of s, and interpolation of these tables is sufficiently accurate for barrier determinations. Up to this point we have been concerned with the torsional energy levels for a molecule undergoing hindered internal rotation. However, besides these levels, there is a set of energy levels arising from the overall rotation of the entire molecule which will be associated with each of the torsional sublevels. The interaction between overall and internal rotation provides the major

HAMILTONIAN FOR INTERNAL AND OVERALL ROTATION

581

mechanism (for asymmetric rotors) by which the splittings of the internal torsional levels are transmitted to the rotational spectrum. Since it turns out that the coupling effect differs for the two torsional sublevels, a given rotational transition associated with the A and E sublevels of a particular torsional state v will appear as a doublet rather than a single line. This doublet separation is a sensitive function of the barrier height, and it is from these splittings that the barrier height may be determined. The ensuing discussion will be concerned with the derivation of the Hamiltonian that describes a molecule undergoing both internal and overall rotation. From this Hamiltonian, a quantitative discussion of the effects of internal rotation on the rotational spectrum can be made.

4


The formulation of the internal rotation problem follows the usual procedure which first requires choice of a molecular model from which the classical kinetic and potential energies are developed and subsequently the classical Hamiltonian. The transition to the quantum-mechanical Hamiltonian is then usually accomplished in a straightforward way. The model chosen for the problem of internal rotation is comprised of a rigid symmetric top (e.g., a CH, group) attached to a rigid frame, which may or may not be asymmetric. The two rigid parts rotate about the bond connecting them. The symmetric top (CH,) is regarded as rotating with respect to the other part which is taken as the framework. The other modes of internal vibration are ignored. This separation of the internal rotation from the remaining ( 3 N - 6 - 1)modes of vibration has proved to be a satisfactory approximation for most of the molecules investigated. The separation is expected to be valid if the torsional frequency is not close to any of the other vibrational frequencies. The entire molecule made up of symmetric top and framework is also rotating and translating in space. The translational motion may be readily separated, and there are then four degrees of freedom-three for overall rotation, which may be described by the three Eulerian angles 8,4, x of the framework; and one for hindered rotation of the two groups, which is described by the relative angle a between the two groups. Two methods of approach have been used to handle the problem of internal rotation. These methods differ in choice of coordinate system, and each leads to a somewhat different mathematical formalism. The method introduced by Wilson [21] and Crawford [ 2 2 ] uses the principal axes of the whole molecule as the coordinate system. The axis of the internal rotation coincides with the symmetry axis of the top, but this axis may not be coincident with any one of the principal axes. The terms in the Hamiltonian which describe the interaction between internal and overall rotation are treated by perturbation theory. The boundary condition of invariance under a+a 2n applied to the total wave function requires that U(a)= U(cr+ 2n), as applied previously. Physically, this

+

582

INTERNAL MOTIONS

implies that when the original configuration of the system is restored, the wave function describing the system must take on its initial value, that is, be singlevalued. This leads to torsional eigenfunctions corresponding to periodic Mathieu functions which have been tabulated. This method is commonly referred to as the principal-axis method (PAM). An alternate method developed by Nielsen [23] and Dennison et al. [7, 24-26] uses a coordinate system in which the symmetry axis of the top is chosen as one of the coordinate axes. The other two axes are fixed with respect to the framework, and their orientation may be judiciously chosen. One or more coordinate transformations are employed to eliminate or minimize the rotationtorsion coupling; the interactions are then more readily treated by perturbation theory. Some complications, however, arise because the coordinate system chosen is, in general, not a principal axis system and the products of inertia do not vanish. Also the boundary condition of invariance now requires nonperiodic Mathieu functions which have not been tabulated. This method is usually referred to as the internal-axis method (IAM). In many cases either method is equally applicable, but for slightly asymmetric molecules with light frames, such as methyl alcohol, the internal axis method is to be preferred. The review article by Lin and Swalen [8] discusses these two methods of approach in detail. We shall limit consideration to the principal axis method by which a large class of molecules can be handled. One advantage of the principal axis method is that the necessary perturbation sums have been extensively tabulated; hence analysis of the spectra can be carried out by use of these tables and those developed for rigid rotor molecules. Only the theory for molecules with one internal rotor which is a symmetric top will be treated in detail. Two-top molecules are discussed briefly in Section 10. For the model of a rigid symmetric top attached to a rigid asymmetric framework (an asymmetric molecule), the principal-axis method uses a coordinate system x, y, z rigidly attached to the framework, the origin of which is located at the center of mass of the molecule and the orientation of which coincides with that of the principal axes of the entire molecule. The model is

i

Y

\

I

z

Fig. 12.5 Model used to treat overall and internal rotation.


583

illustrated in Fig. 12.5. The products of inertia therefore vanish since the coordinate axes are principal axes. Because of the cylindrical symmetry of the top, the principal axes of the molecule are not altered by the rotation of the top relative to the frame, and the moments of inertia of the entire molecule are constants independent of the angle of internal rotation. The moments of inertia, however, would not be independent of r if the internal rotor were not a symmetric top, for example, CH2DCH0. One of the first molecules whose microwave spectrum was analyzed by means of the PAM is acetaldehyde, studied by Kilb et al. [27]. With the system of coordinates described above, the kinetic energy for a general type of asymmetric molecule (no symmetry apart from that of the CH3 group) can be written as [22]:

where I, are the principal moments of inertia of the entire molecule, Z, is the moment of inertia of the top about its symmetry axis, cog are components along the principal axes of the angular velocity of the framework, Oi is the angular velocity of the top relative to the framework, and A, are direction cosines between the symmetry axis of the top and the principal axes of the entire molecule. The first term in the kinetic energy expression represents the energy of overall rotation of the molecule. The second term is the kinetic energy of internal rotation of the top, and the last term represents the coupling between internal and external rotation. To obtain the Hamiltonian we re-express the kinetic energy as a function of the angular momenta. The angular momenta are defined classically by the relations : aT P --=zgwg+%gza~ (12.27) g-aw,

(12.28) From these relations it is evident that the components P, of the total angular momentum of the molecule contain a contribution from the internal rotation of the top, whereasp, the total angular momentum of the internal rotor, contains contributions from both internal (Z&) and overall rotation. Employing (12.27) and the expression for the kinetic energy, (12.26), we may write 2T -

1 P2 =r Z,i2

(1 2.29)

1,

The coefficient rZ, is interpreted as the reduced moment of inertia for internal rotation of the two rigid parts of the molecule with r a reducing factor defined as (1 2.30)

584

INTERNAL MOTIONS

For the case of two coaxial symmetric tops, rZ,= Iz(Z,--Zz)/Zz = ZzZp/Zz, in which Z, is the total moment of inertia of the two tops about the symmetry axis. This is equivalent to the reduced moment expression given previously in (12.7). From (12.27) and (12.28) it is seen that p - 9’=rZzd

(12.31)

with

9’=C p,P,

(12.32)

9

The quantity p - 9 represents the relative angular momentum of the top and frame. By substituting (12.31) into (12.29), we obtain the kinetic energy in the following convenient form given by Herschbach [28] 1 P i 1 (p- 9 ) 2 T=-C-+-----2 I, 2 rZz

(12.33)

,

If the internal motion is frozen, that is, d =0, then the relative angular momentum vanishes, and (12.33)or (12.29) yields the usual expression for the kinetic energy of a rigid rotor in terms of the total angular momentum components. In addition, the angular momenta would no longer include contributions from internal rotation. Addition of the potential energy to the above kinetic energy expression gives the classical Hamiltonian. The coefficients appearing in (12.33) are constants; therefore the classical kinetic energy becomes a quantum mechanical operator if P, and p are simply considered as operators. If the angular momentum is measured in units of h, the Hamiltonian operator is expressed as

H = X r+ F ( p - 9 ) 2 + V(N)

(12.34)

in which the potential energy restricting internal rotation is usually assumed to have the form given in (12.5). The 2,.is the usual rigid rotor Hamiltonian and F ( =fh2/rZa) is the reduced rotational constant for internal rotation. The operator 8,a reduced angular momentum, is a linear combination of the P , as defined by (12.32). The commutation rules for the angular momenta P, are the standard ones for components of the total angular momentum along the molecule-fixed axis [see (2.58)]. In addition, there is the commutation relation: PP,-P,P=O

(g=x,y,

4

(12.35)

This follows from the fact that as a quantum mechanical operator [29] (12.36) whereas the corresponding operators for P , involve the Eulerian angles 8,4, but not a. Moreover, it follows from (12.35) that the operators p and 9 commute with each other.

HIGH-BARRIER APPROXIMATION

5

585

H IG H - BAR R I E R A PP R 0x1MATI0N

The Hamiltonian, (12.34),can be divided into three parts as follows X = X R + X T + X T R

(12.37)

with

X R =X , . + F B 2 XTR = - 2F p p

(12.38)

( 1 2.40)

where we shall write =A,P:

+ B,P; + C,P;

(12.41)

in which A,=h2/21,, and so on. The terms X Rand X T are, respectively, the rotational and torsional parts, and X T R represents the coupling between angular momenta of internal and overall rotation. The internal rotation or torsional Hamiltonian X T leads to Mathieu’s differential equation and hence to eigenvalues and eigenfunctions discussed in Section 3. The term X R is simply a quadratic form in the total angular momentum. The term F B2in X Rintroduces terms of the type P,P,. and Pl. The latter terms can be absorbed into the usual rigid rotor expression (X,.)by a modification of the definition of the -rigidrotor rotational constants. As long as the coupling term X T R is ignored, the Hamiltonian is separable into torsional and rotational parts, and the internal rotation does not directly affect the rotational spectrum. This follows since X Tis a function only of CY and is hence independent of the rotational quantum numbers, whereas X Ris independent of the internal coordinate u and therefore of the torsional quantum numbers. Inclusior, of - 2 F P p prevents the Hamiltonian from being separable, since this term depends on both sets of quantum numbers. For molecules with sufficiently large barriers, so that the separation of torsional levels of different u is large compared with the rotational levels, this coupling term can be conveniently treated as a perturbation. For evaluation of the energy levels, the matrix formulation is employed in which the matrix of X is constructed in a suitable representation. If the unperturbed Hamiltonian is taken as X R X T ,a convenient basis would be the product functions $ R $ T , with t+hRa rotational eigenfunction of X Rwith energy E R and Ij/T(=Uua)a torsional eigenfunction of X T with energy ET(=Euu). Here the set of rotational quantum numbers is represented by R and the set of torsional quantum numbers by T In this basis both X Rand X T are diagonal, but X T R contributes off-diagonal elements. In particular, for the functions U,, the operator p of X T R is diagonal in o but has both diagonal and offdiagonal elements in the principal torsional quantum number u [30]. The infinite-order Hamiltonian matrix can be grouped into blocks corresponding to different u. Within each block are the matrix elements of X , all of which are diagonal in u and c,but not necessarily in R.Associated with each block corresponding to a given u there are different submatrices corresponding to the dif-

+

586

INTERNAL MOTIONS

Van Vleck I

Tronsforrnation

Fig. 12.6 A portion of the energy matrix before and after the Van Vleck transformation. The blocks along the diagonal contain the matrix elements of internal and overall rotation, which are diagonal in the torsional quantum number T, in particular, in u and u. The unshaded area contains the matrix elements connecting the rotational levels of the various torsional states u. After the transformation these elements are reduced to second order, 1,', and corrections are introduced within each diagonal block. If the nondiagonal terms in u are now neglected, the transformed diagonal blocks can be diagonalized separately to give the rotational energy levels for each torsional state correct to second order in the separation of rotation and torsion.

ferent torsional sublevels. Blocks of different v are connected by matrix elements of H T R , the nondiagonal elements in v coming from the factor p . A portion of the energy matrix is illustrated in Fig. 12.6. To proceed with the solution further, we must attain some simplification of the matrix. To achieve approximate v diagonalization of this matrix we use a Van Vleck transformation [31, 321. The effect of this transformation will be to reduce the off-diagonal matrix elements in v from H T R so that they may be neglected. With neglect of the reduced off-diagonal elements, the transformed matrix may be factored into smaller rotational submatrices, one for each torsional state (see Fig. 12.6). We are then left with the problem of diagonalizing smaller submatrices. The transformation has the effect of modifying the elements diagonal in v by folding the elements off-diagonal in v into the v blocks. To second order, the matrix elements of 2 within a T block, from Appendix C , will be replaced after the Van Vleck transformation by: 1 (R,TJ2rIR'T)+Z (R, TI X I R " , T")(R"T"l XIR'T)

1'

R", T"

1

+

EjTE $ , T - lE ; f f T , ), (12.42) where E $ T = E R + E T and 8'is the perturbation giving rise to nondiagonal elements in T. The term T" = T is to be excluded from the sum over R" and T". The perturbation term X' is to be taken as 2 T R ( = - 2 F Pp). Since the perturbation is diagonal in c and hence no degenerate or near-degenerate torsional states are connected, it is a good approximation, for reasonably high barriers, to neglect the energy spacing between rotational levels compared to that between torsional states, Le., E j T E T - E T " = Eo,- E,,,,. This will allow us to express the energies of a given torsional state in terms of an effective Hamiltonian involving only operators of the total angular momentum.

HIGH-BARRIER APPROXIMATION

587

If the rotational energy differences are neglected, the denominators of (12.42) will no longer involve R,and the summation in R" and T" can be separated. Since each matrix element in the numerator can be factored as (Rl PIR").(Tip1T"), the R" summation is simply matrix multiplication giving a factor (RI P'IR'). Explicitly, we obtain for the matrix elements of a given ucr block:

where A,,! = E , , - E,.,. This is just the RR'th matrix element of the following effective rotational Hamiltonian operator for the torsional state ucr :

+

pub = .yi", F [W,bo'+ W,'b'P + WV(a2'9'1

(12.44)

where F=fh2/rI,, 9 = cpgPgand where (12.45)

(12.46)

WCbl'= -2(U, cr(plU, a )

(12.47) with

and (12.49)

This effective rotational Hamiltonian is correct to second order. The W coefficients introduced by Herschbach [28] depend on the barrier, in particular on the ratio V3/F. The Hamiltonian of (12.44) can also be written in the equivalent form [33, 81:

Z,, = A,,P.?

+ B,,Pj + C,,Pt + F Wi!'

pgPg 9

+fFW,',Z'

c PgPg~(PgPg~ +P,,P,)+E,U

(g=x, y , 2)

(12.50)

93 9' gig'

where (12.51) and

+ + +

A,, =A, W,';'Fp.? B,, =By W,',2'Fp,Z C,, = C, W,',Z'Fp;

(12.52)

ul W W

Table 12.2

Perturbation Coefficients for a Threefold Barrierozb Degenerate Sublevel ( E symmetry)

Nondegenerate Sublevel ( A symmetry) S

n=O

n=2

n=4

n=O

n=l

n=2

n=3

n=4

Ground Torsional Level ( v =0)

8 10 12 16 20 28 36 40 48 56 60 68 72 80 88 92 100

5.5936 6.4056 7.1226 8.3688 9.4499 11.3091 12.9096 13.6418 15.0029 16.2539 16.8457 17.9726 18.5111 19.5449 20.5280 21.0028 2 1.9222

0.280631 0.18071 6 0.1 17586 0.052068 0.024473 0.006261 0.001856 0.001055 0.0003643 0.0001354 O.oooO845 O.ooOo343 o.oooO2220 0.00000963 0.00000433 0.00000294 0.00000140

-0.0799

-0.05725 -0.03948 -0.01841 -0.0088 19 - 0.00228 1 - 0.000678 -0.000386 -0.0001331 -0.0000495 -O.oooO309 -0.oooO1252 -0.00000812 -0.00000352 -0.00000158 -0.00000107 - 0.00000051

5.7972 6.5335 7.2047 8.4047 9.4667 11.3134 12.9108 13.6425 15.0031 16.2540 16.8457 17.9726 18.5111 19.5449 20.5280 21.0028 21.9222

-0.2617257 -0.1602753 -0.1014166 - 0.0437764 -0.0203828 -0.0051860 -0.0015359 -0.0008723 -0.0003012 -0.0001120 -0.oooO6992 -0.oooO2835 -0.oooO1838 -0.00000796 -0.00000358 -0.0243 - 0.00000115

-0.137544 -0.089791 - 0.058664 - 0.026025

-0.012236 -0.003130 -O.OOO928 -0.0005272 -0.0001819 - O.oooO6769 0.00004227 -O.oooO1714 - o.oooO1111 -0.00000482 -0.00000217 -0.00000147 - 0.00000069

0.21 3762 0.124274 0.077241 0.032531 0.015012 0.003797 0.001123 0.0006379 0.0002202 O.oooO819 O.oooO5 112 O.oooO2074 O.oooO1345 0.00000582 0.00000262 0.00000178 0.00000084

0.0308 0.02676 0.019312 0.009 178 0.004408 0.001141 -0.0003391 0.0001928 O.oooO666 O.oooO2469 O.oooO1545 0.00000626 0.00000406 0.00000176 0.00000079 0.00000054 0.00000026

16 20 28 36 40 48 56 60 68 72 80 88 92 100

24.1805 27.2238 32.6645 37.4427 39.6401 43.7320 47.4959 49.2765 52.6667 54.2864 57.3951 60.3509 61.7780 64,5416

- 1.27334

-0.654112 -0.200728 - 0.06941 1 - 0.042117 - 0.0162968 - 0.0066663 -0.0043405 -0.0018985 -0.0012734 -0.0005876 -0.0002796 -0.0001948 -0.0000964

First Excited Torsional Level (v = 1) 0.7870 23.4277 0.7977206 26.8077 0.4695574 0.31265 32.5300 0.1594403 0.07947 0.02603 37.3955 0.0566696 0.015628 39.6114 0.0345699 0.005989 43.7209 0.0134408 47.4914 0.0055072 0.002442 0.00358727 0.0015888 49.2735 52.6654 0.00156961 0.0006942 54.2855 0.00105298 0.0004655 0.00048597 0.0002148 57.3947 60.3507 0.00023119 0.0001022 61.7779 0.00016112 O.oooO712 64.5415 O.oooO7974 O.oooO352

0.590647 0.319985 0.100170 0.034699 0.02 1057 0.0081484 0.0033332 0.002 1702 0.0009493 0.0006367 0.0002938 0.0001398 O.oooO974 0.0000482

-0.42680

-0.294937 -0.1 11855 - 0.040897 -0.0250835 - 0.0097997 -0.0040219 -0.0026207 -0.001 1471 -0.0007698 -0.0003553 -0.001690 - 0.0001178 -0.oooO583

-0.242 1 - 0.13408 - 0.03905

-0.012993 -0.007809

0.002995 -0.0012210 - 0.0007944 - 0.0003472 - 0.0002327 -0.0001074 -0.oooO511 - O.oooO356 -0.oooO17 -

Second Excited Torsional Level (v=2)

36 40 48 56 60 68 72 80 88 92 100 fn

03 (D

58.4614 62.3641 69.4772 75.9067 78.9255 84.6483 87.3744 92.5971 97.5549 99.9466 104.5751

0.890641 0.619322 0.291871 0.137167 0.09449 0.045504 0.031829 0.0158271 0.0080372 0.0057707 0.0030175

-

0.2370

-0.18380

-0.09759 -0.04822 -0.03365 -0.016435 - 0.011540 -0.005763 -0.002932 -0.002107 - 0.001102

59.1178 62.8093 69.6812 76.0014 78.9905 84.6795 87.3962 92.6080 97.5604 99.9505 104.5772

-0.8593042 -0.5670974 -0.2522742 -0.1156628 -0.0791721 -0,0378571 - 0.02643006 -0.01311438 -0.00665303 -0.00477549 - 0.00249627

"From Herschbach [ZS]. bThe coefficients, W$), are tabulated as functions of the dimensionless parameter s =$(V;/F).

0.428291 -0.304883 -0.145546 -0.068549 -0.047238 -0.022751 -0.0159144 -0.0079136 -0.0040186 -0.0028885 -0.0015088

-

0.721421 0.456080 0.192538 0.086201 0.058634 0.027842 0.0194013 0.0096063 0.0048686 0.0034936 0.0018255

0.06245 0.07625 0.04752 0.02400 0.016791 0.008215 0.005769 0.002882 0.001466 0.001053 0.000552

590

INTERNAL MOTIONS

Note that the rotational constants of the rigid molecule A,, . . . , are modified by the internal rotation terms arising from W,',2)P2. The perturbation treatment of X T Rcan be carried to higher order by application of successive Van Vleck transformations. The effective rotational Hamiltonian for a given torsional state obtained from an nth order perturbation calculation can be expressed simply as [28] (12.53) The first term is the rigid asymmetric rotor Hamiltonian, and the last term, which is a power series in 9, represents the effect of internal motion on the rotational motion. The W$) are the usual nth order perturbation sums, except for n = 2 which contains an added contribution arising from the F P 2 term in the original Hamiltonian [see (12.47)]. The dimensionless perturbation coefficients Wv(a")depend on the reduced barrier height s, and these coefficients (except n=O) decrease in size with increasing barrier, becoming proportional to the torsional sublevel splittings ( E u E - E u A )and hence vanishing at the limit of infinite barrier height [28]. At this limit, Xobreduces to the rigid rotor Hamiltonian plus the energy of a torsional harmonic oscillator, and a usual rigid rotor spectrum would be obtained. The zeroth order term Wl'p) is just the torsional energy contribution from ZT, and does not affect the rotational spectrum. The first-order term W,b')is nonzero only for the degenerate E sublevels. The second-order term differs both in sign and magnitude for the A and E sublevels of a given torsional state v. The following properties of the perturbation coefficients may be listed [28]: WJl)=O

for n odd

(12.54)

W::) =( - 1)" Wi,? 1

(12.55)

W(")uo - - 2 Wv';) for n > 0 and even

(12.56)

-(2~/3)~ --WL(1+') + 2)] W&") [(n + l)(n

for n>O

(12.57)

The first two relations are true in general, whereas the latter two relations hold for relatively high barriers. The perturbation coefficients have been tabulated for various values of n and v and for a wide range of the parameter s [18, 19,281. The recent tabulation by Hayashi and Pierce [19] lists the perturbation coefficients for v=O, 1, 2 and n=O, 1,. , . , 6 , d for s values from 4 to 200 with the interval in s being 2 for s < 100 and 4 for s> 100. These tables have also been reproduced in the book by Wallrab [121]. A short tabulation of the perturbation coefficients for the v=O, 1, and 2 torsional levels are given in Table 12.2. A discussion of the method for evaluating these perturbation coefficients, which avoids the laborious evaluation of the perturbation sums, may be found

H IG H - BARR1ER ASY M M ETR IC ROTORS

591

elsewhere [28, 341. This is the so-called “bootstrap” procedure which is based on the fact the W,k“)are not dependent on the molecular asymmetry. In the derivation of the effective rotational Hamiltonian the rotational energy differences were neglected in comparison with the torsional energy differences. This approximation is adequate for molecules with high barriers but may not be so for those with low and intermediate barriers or small moments of inertia. The error caused by ignoring rotational spacing can be corrected by expansion of the denominators in a Taylor series. This has been discussed by Herschbach [28] and by Stelman [35] and its consideration requires introducing another perturbation coefficient Wit) into the effective rotational Hamiltonian. These denominator corrections have no appreciable effect on the determination of the barrier height for high barrier cases [see, e.g., (12.61)] because the contributions to the rotational constants are essentially the same for both torsional sublevels. Nevertheless, they can contribute several megahertz to the rotational constants.

6

HIGH-BARRIER ASYMMETRIC ROTORS We have seen, by treating the coupling of internal and overall rotation by the degenerate perturbation theory of Van Vleck, that an effective rotational Hamiltonian may be obtained for each torsional level. For a particular torsional state u there will be a set of rotational energy levels associated with each of the two torsional sublevels o=O or A and o= k 1 or E(a= 1 or - 1 lead to the same energy levels). The two sets of rotational energy levels are obtained by of (12.53).The number of solution of the effective Hamiltonians Xu*and correction terms to X r that must be retained in (12.53)for analysis of the effects of internal rotation varies significantly from molecule to molecule and is dependent on the barrier height and the asymmetry. Also the complexity of the operator B depends on the molecular geometry. For molecules with a plane of symmetry containing the internal rotor axis, for example, CH,CHO, some simplification results since the direction cosine Ay vanishes ( y axis perpendicular to the symmetry plane). For two planes of symmetry, for example, CH3BF2, both i., and i yvanish, and B reduces to pzP,. Even when the molecule has only one plane of symmetry, the internal rotor axis may lie so close to one of the principal axes in the plane that the direction cosine referring to the other principal axis in the plane will be negligibly small and B will again depend only on the component P,. It may be mentioned that it is usually convenient to choose the axis of quantization, the z axis, to be the axis making the smallest angle with the top axis, even if this is not the best choice for evaluation of the asymmetric rigid rotor energy levels. Hence in dealing with specific molecules the symbols A,,, B,,, and C,, [e.g., in (12.50)] may have to be permuted to satisfy the usual convention A 2 B> C.

+

Energy Levels and Spectra of the Pseudorigid Rotor When barriers are relatively high (s>30) [36], terms up to n = 2 in (12.53) are often sufficient, and the Hamiltonian is then given by (12.50). The cross

592

INTERNAL MOTIONS

terms of the form (P,P,, + P,,P,) present in this expression can be eliminated as usual by a rotation of the coordinate axes to obtain a new set of principal axes in which these terms vanish. If, for example, only one such cross term were present, a simple 2 x 2 rotation in the gg’ plane may be chosen to remove the term. The orientation of the new coordinate system, referred to as the “effective principal-axis system” will vary with the torsional state since the coefficients of the cross terms contain WJ:). This transformation alters the rotational constants and the coefficients of the linear terms [8,27]. Often the modifications produced by such a rotation of coordinate axes is small, and negligible error is introduced by just ignoring the cross terms. The linear terms in P,, present only for the degenerate E levels since W J i ) = O ,can be neglected for a barrier greater than 1000 cal/mole and an asymmetry of IKI B,, > C,, and the xz plane is taken as the plane of symmetry of the molecule. Employing the structural parameters for acetaldehyde (Table 10 of [27]) we find for the moments of inertia (amu A') I,( Zb)=49.723 1,(1,)=55.545

Zz(Ia)= 8.945 Z, =3.123

If we take the symmetry axis of the methyl group as coaxial with the carboncarbon bond the direction cosines are ).==COS

23"1'=0.9204

E,,=cos

66"59'=0.3910

with i,=O, since the y(c) axis is perpendicular to the plane containing the methyl group axis. For the internal rotation parameters we have p,=0.02456 r=0.6946

pz=0.32134 F=233.0 GHz

We may apply (12.63)-(12.67) to the splitting of the O o o + l o l transition, and since i,=O we, have AC=O. For the Ooo level E(ic)=O, BE/BA=aE/aB=O, and we have

AE (000 ) =0

606

Next Page

INTERNAL MOTIONS

while for the l o llevel E ( K ) = K - 1, BE/BA=O, BE/BB=l. This gives AE(101)=AB= BOA - BOE The expression for the splitting (Av =vA-vE)

of the transition is therefore

Av=AB=Fp;[WJ:'Wdi'] = 140.54[ WJ9- WJ;'] From the observed ground torsional state splitting of 3.0 MHz one obtains the difference in the barrier-dependent perturbation coefficients

A WJ2' = WJi)- WJi' =0.02135 By making use of the following quantities: S

WJ?)

WJ;'

A WJ2)

22 23 24

0.01710 0.01436 0.01210

-0.008550 -0.007179 -0.006046

0.025650 0.021539 0.018146

a linear interpolation yields s =23.06, which gives sF v, =-4.6602 (23.06)(233.0) = 1153 cal/mole 4.6602

for the barrier height. The conversion factor 4.6602 gives V, in cal/mole when F is expressed in GHz.

7

HIGH-BARRIER SYMMETRIC ROTORS

The method for determining barrier heights described for asymmetric rotors, which utilizes splittings of rotational transitions arising from the direct effects of internal rotation, is not applicable for symmetric rotors. To demonstrate this, let us look at the effective rotational Hamiltonian for a symmetric rotor. For a symmetric top with an internal rotor, illustrated in Fig. 12.12, B , = A , and 9 reduces to: 9 = p P , , where p=Za/Zz. The Hamiltonian of (12.53) can thus be written as Z",=A,(P; + P : ) + C x P t + F

W,',"'(pP,)"

(12.8 1)

n

where F =h2Zz/[2Za(Zz- ZJ]. The W,C,.)are the barrier-dependent perturbation coefficients defined previously. The rigid symmetric rotor functions yield a diagonal matrix for this effectiverotational Hamiltonian. The rotational energies for the torsional state 00 are

E J K "=~A x J ( J + 1)+ ( C , - A,)K

+F

Wi:)(pK)" n

(12.82)

Previous Page

Fig. 12.12

LOW- BAR R IER APPROXI MATION

607

Model for a symmetric rotor with internal rotation.

The first two terms represent the rotational energy of a rigid symmetric rotor, and the last term represents the corrections to the rotational energy due to the effects of internal rotation. The o =O (A-type)levels retain the usual & K degeneracy for a symmetric rotor; however, for r ~ = + 1 (E-type levels the K degeneracy is removed by the coupling of internal and overall rotation. The r~ = + 1 or - 1 leads to the same energy levels; the rotational levels are doubly degenerate with respect to o but nondegenerate with respect to K. The three pairs of doubly degenerate rotational levels illustrated in Fig. 12.7 are o=o, + K

0=+1, +K, a = + l , -K,

~ = - l ,- K ~ = - l ,+ K

+

The selection rules for rotational transitions in the torsional state u are A J = 1, AK=O and Ao=O. The three possible absorption transitions are of the type

J,o=O, + K - + J + l , o = O , + K J,o=+l, +K+J+l,o=+l, f K J , a = + l , T K + J + l , o = + l , TK in which the upper or lower signs are taken together. Because of the selection rules it is apparent from (12.82) that the rotational transitions will not be affected by the contributions from internal rotation; hence the three possible transitions are coincident, and no splitting occurs (see Fig. 12.7).Other methods must, therefore, be used for the detection of the effects of internal rotation in the rotational spectra of symmetric top molecules. One technique, of course, is to introduce an asymmetry by isotopic substitution [37,72]. Coriolis interaction in a degenerate vibrational mode also allows observation of internal rotation effects [73, 741. In an excited degenerate vibrational state when both the Coriolis and internal rotation effects are taken into account, the interna1,rotation effect is found to depend on J and hence to enter into the transition frequency and thus to become directly observable. Other methods which can be utilized are discussed in Section 9.

8

LOW-BARRIER APPROXIMATION

In the previous sections a perturbation treatment of internal rotation was discussed in which the top-frame coupling term was treated as a perturbation.

608

INTERNAL MOTIONS

a

0'

30"

60"

90'

Fig. 12.13 As the methyl group with threefold symmetry is rotated with respect to the frame (NO,) of twofold symmetry, indistinguishable configurations are obtained at intervals of 2n/6. Hence the barrier has sixfold symmetry.

This method is best suited for molecules with high barriers. As the barrier is lowered or as KZa/Zz becomes large, the higher-order perturbation terms become more important and have to be included, as discussed previously. If the higher-order corrections for the coupling become excessively large, the internal axis formulation, where the coupling term is initially minimized by a coordinate transformation, is more convenient especially if the asymmetry of the rotor is small. However, for molecules with very low barriers (s"'

+re

(13.128)

which is not equal to re even when the vibrations are harmonic. Also, since t(, depends on the reduced mass via Be and we, ro can be expected to vary with isotopic substitution. In Table 13.7 the bond lengths of some diatomic molecules are compared. As the effective rotational constant is less than the equilibrium

EVALUATION OF EFFECTIVE STRUCTURES

681

rotational constant, the ro distance is larger than re. The difference is largest for bonds involving light hydrogen atoms where ro is about 0.01 8, larger than re. Furthermore, the ro values are different for different isotopic species, the variation being particularly large when light atoms such as hydrogen are substituted. Here a big effect on amplitude of vibration results, and the vibrational average over the zero-point vibration is significantly different. For substitution of heavier atoms little effect is observed. O n the other hand, the re values are intensitive to isotopic substitution. Since the re refers to the equilibrium position where the nuclei would be at rest, this type of effect does not enter the calculation of re from Be. Variations, like those for ro, are also found for the average bond distance, ( r ) , which is also given in Table 13.7. This measure of the bond distance will be discussed more fully in Section 9. For polyatomic molecules ro is no longer simply the (r-2)-1/2 because of Coriolis constributions. Also, isotopic data and the assumption that the effective structural parameters are not affected by mass changes are required for evaluation of the structure. Therefore, when more than a minimum number of isotopic species are available, different ro structures can be obtained because the molecular vibrations and hence the effective parameters are modified by isotopic substitution. For the linear molecule HCN, four independent moment of inertia equations may be obtained from the data on HI2CI4N, H13C14N, DI2Cl4N, and D13C14N. These equations may be combined in six different ways to give values of the two bond lengths. The results are given in Table 13.8 along with the ro structures of OCS. It is apparent that different ro structures are obtained when different isotopic data are used in the calculation. Variations of about 0.01 8, are observed. Obviously the neglect of the zero-point vibrational effects by assumption that ro parameters are the same for different isotopic species has seriously limited the accuracy of the derived structural parameters. With more complicated molecules, least-squares fitting techniques are convenient and have been employed. The problem is linearized in the usual way (13.129) where Zi corresponds to the i th experimental moment of inertia, fi represents the ith moment of inertia calculated from the initial assumed structure, and pj is the j t h structural parameter. The derivatives aZi/dpj = AZJApj are evaluated by calculating the change in I p with a small change i n p j while keeping all other parameters constant. The foregoing expression forms the basis of a leastsquares analysis in which the corrections Apj to the initial structure are obtained. The process is repeated until the corrections are small enough. A discussion of some computational problems is available [33]. Ambiguities caused by zeropoint vibrations are, however, still present. A particular problem arises if the normal least-squares equations are illconditioned. One must then resort to obtaining extra data, or must add restrictive conditions, for example, hold a particular bond length constant or evaluate only certain linear combinations of the parameters using techniques for the

682

DERIVATION OF MOLECULAR STRUCTURES

Table 13.8 and OCS

Comparison of ro Structures of HCN

Isotopic Species Used

1 6 0 12c32s 1 6 0 12 ~ 3 4 s

1 6 0 1 2c32s 16013C32s

,

16012C34S, 16013C34s 1 6 0 12c32s, 1 8 0 12c32s

Average Range

C-H

C-N

1.0674 1.0623 1.0619 1.0625 1.0624 1.0658 1.0637 0.0055

1.1557 1.1567 1.1568 1.1566 1.1563 1.1555 1.1563 0.001 3

c-0

c-s

1.1647 1.1629 1.1625 1.1552 1.1613 0.0095

1.5576 1.5591 1.5594 1.5653 1.5604 0.0077

"From J. W. Simmons, W. E. Anderson, and W. Gordy, Phys. Rev., 77, 77 (1950); 86, 1055 (1952). *From C . H. Townes, A. N. Holden and F. R. Merritt, Phys. Rec., 74, 1113 (1948).

solution of singular systems [34, 35,361. When the limits of uncertainties of the structural parameters can be reasonably estimated, the technique of Section 11 can be applied. In some cases, the best that can be done with the available data is to assume some structural parameters from similar molecules and evaluate only certain parameters which ideally do not depend significantly on the assumed parameters (this can be tested). Such structural parameters are useful but less accurate. The effects of zero-point vibrations are readily manifested when an atom is substituted which lies on a principal axis (e.g., the sulfur atom of SO2).If the atom lies on, say, the x axis, then in the rigid rotor approximation the corresponding change in the moment of inertia AZx must be zero. In practice, however, small changes are observed because of the effects of vibration. These effects present serious problems to the structural determination of molecules with near-axis atoms or atoms near the center of mass. For such atoms, isotopic


683

Table 13.9 Comparison of ro Structures Obtained from Different Pairs of Moments of Inertia SOz"

Calculated From

(I:, I:) (1%

'$1

(I:, 1,) Range

CIZ0*

r

e

r

e

1.4322 1.4336 1.4351 0.0029

119'32' 119'36' 119'8' 0'28'

1.6986 1.6994 1.7031 0.0045

111"6' 111'8' llO"39' o"29'

"From Morino et al. [ 2 7 ] . *From Herberich et al. [61]. 'Equivalent to I t , I : + A . dEquivalent to I:, I," + A ,

substitution produces only small changes in the moments of inertia, and these changes may be masked by the zero-point vibration effects. In the well known example of N 2 0 , the observed moment of inertia for 15N14N160is slightly larger than for 15N15N160. , as a consequence, the application of Kraitchman's equation yields an imaginary coordinate for the central nitrogen atom in 15N'4N160, an impossible result for a rigid molecule. Even for a simple, bent, triatomic molecule where two moments of inertia of one molecular species are sufficient to give the complete structure, ambiguities result because of the finite inertial defect A, If a molecule is planar in its equilibrium configuration, then

z;i z:- zg=o

(13.130)

where c is the axis perpendicular to the plane. However, the effective moments defined by (13.116) have somewhat different vibrational effects associated with the different axes and

C-Z:-Z:=A

(13.13 1)

where A is called the inertial defect. Although this quantity is very small compared to the moments of inertia, it nevertheless produces ambiguities in structural calculations. In the case of a bent X Y2 molecule, three different structures are obtainable from the three choices of data available, that is, the three pairs: ,:Z Zt; I:, I:; I t , I:. This is illustrated in Table 13.9 for SO2 and C102. Such effects are particularly large for the light molecules H 2 0 , D 2 0 and T 2 0 illustrated in Table 13.10. It will be observed that the range in the bond distance decreases as the mass of the vibrating atom increases. On the other hand, it is not clear whether the tendency is for the bond distance to decrease or increase with isotopic substitution. The length, for example, as calculated from Zf, :Z increases from H 2 0 to T 2 0 , whereas it decreases when calculated from the other two combinations. The trend for the average values also included on the

684


Table 13.10 Effective Structures of H 2 0 , D 2 0 , and T20" Effective Structuresb

(I:, I:) (I:, I:)c (I:, I:)d Range Average

105.1 106.3 102.9 3.4 104.8'

0.9560 0.9688 0.9703 0.0143 0.9650 8,

0.9567 0.9652 0.9673 0.0106 0.9631 8,

104.9 105.7 103.3 2.4 104.6'

0.9570 0.9637 0.9662 0.0092 0.9623 8,

104.9

105.5 103.5

2.0 104.6'

"Data from: (H,O) F. C. De Lucia, P. Helminger, R. L. Cook, and W. Gordy, Phis. Rec., A5, 487 (1972); (D,O) J. Bellet and G. Steenbeckeliers, Compt. Rend., 271B, 1208 (1970); (T,O) F. C. De Lucia, P. Helminger, W. Gordy, H. W. Morgan, and P. A. Staats, Phis. Rev., A8, 2785 (1973). bObtained from different pairs of moments of inertia. 'Equivalent to I,", I : + A . dEquivalent to I f , I," + A .

table are, however, consistent with the lower zero-point vibrational energy of the heavier isotopic species.

Inertial Defect The problem of calculation of inertial defects has been studied in some detail. The vibration-rotation parameters E: in (13.1 16) are separable into harmonic and anharmonic contributions &:=E:

(har)+E: (anhar)

(13.132)

(The Coriolis contributions are included in the harmonic part.) Darling and Dennison [37] have shown that the contribution of the anharmonic part of E: to the inertial defect vanishes exactly for planar molecules and hence A is given by Avib=C

(uS+ds/2)[~S (har)-E: (har)-E: (har)]

(13.133)

5

Therefore, A v i b correct to terms linear in the vibrational quantum numbers does not depend on the anharmonic potential constants and may be calculated from a knowledge of the harmonic force constants, masses, and geometry of the molecule. Oka and Morino [38] have given a general expression for the inertial defect of a planar molecule. In addition to the contributions from vibrational motion, there are small contributions from centrifugal distortion and electronrotation interaction, that is

A = Avib+ A cent + A elec

( 1 3.134)

Next Page


685

where

(13.135) (13.136) and (13.137) In these equations [g), and so on, are the Coriolis coupling constants; t runs only over the out-of-plane vibrations; me and mp are the mass of the electron and proton; gaa, and so on, are the rotational magnetic moment g tensor elements. For A in amu A2, the vibrational frequencies ofare expressed in cm-'; the centrifugal distortion constant h4T,bab and rotational constants A , B, and C are expressed in MHz; moments of inertia Z,, and so on, in amu A'; and h/n2c= 134.901. The Coriolis coupling constants are expressed by ("=L -

'@(t)-

(13.138)

1

p=

The r" (a=x, y , z ) matrices are skew-symmetric, that is - r". The C" matrices depend only on the masses and molecular geometry. Their evaluation has been discussed by Meal and Polo [39]. The matrix L which connects the normal coordinates and the internal coordinates, R=LQ, can be determined from a normal coordinate analysis. If the Coriolis coupling constant lg,! is to be nonvanishing, the direct product of the symmetry species of the vibrational coordinates Qsand Qs.must contain the species of the rotation R, [40]. For example, a nonlinear X Y , molecule, which belongs to the point group C Z vhas , two vibrational modes (ol, 02) of symmetry A l and one (03: antisymmetric stretch) of symmetry B1.The three rotations R,,R,,and R, under the CZvgroup belong to the species B2, B1,and A 2 , respectively. Since we have A l x B1=B1, A l x A l = A l for the direct products, it follows that the possible nonvanishing Coriolis constants are [(lys, [$,I, and [Ly3, ( y is perpendicular to the molecular plane). The vibrational contribution to A is the most important, although it has been found that for molecules with out-of-plane electrons, for example, the n electrons in the C = O bond of H 2 C 0 , the electronic contribution is not negligible. In the case of ozone the electronic contribution is particularly large, amounting to about 10% of the total inertial defect. The calculation of the electronic contribution to the moments of inertia requires a knowledge of the g tensor of the rotational magnetic moment. Experimental values of the g tensor have been given in Chapter XI. Good agreement between calculated and observed inertial defects has been obtained for a number of simple planar molecules [41], as illustrated in Table 13.11. Note that the inertial defect does not show a large

[LYJ,

686

Previous Page


Table 13.11 Inertial Defect for Some Three- and Four-Atom Molecules in the Ground Vibrational State (amu A') Calculated Molecule" H2O D2O H2S H2Se D2Se 0 3 '8002

so2 34s02 H2CO H DCO D2CO F2CO ClZCO ClNOz CIF3 BrF3

Avib

A cent

0.0460 0.0627 0.063 1 0.0736 0.1034 0.1107 0.1148 0.1376 0.1395 0.0597 0.071 1 0.0810 0.1554 0.2628 0.2154 0.202 0.220

0.0008 0.0008 0.0007 0.0008 0.0008 0.001 1 0.001 1 0.0004 0.0004 0.0016 0.0014 0.0014 0.0004 0.0005 0.0012

0.0000 O.oo00 0.0001 Ob Ob

-0.0104 -0.0104 - 0.0037 - 0.0037 - 0.0052 - 0.0052 - 0.0052 - 0.1N52~ - 0.0052b -0.0104b

Ob

Ob

Ob

Ob

Obserced

A

A

0.0467 0.0635 0.0639 0.0744 0.1042 0.1014 0.1055 0.1343 0.1362 0.0561 0.0673 0.0772 0.1506 0.2581 0.2062 0.202 0.220

0.0486 0.0648 0.0660 0.0595 0.1045 0.1017 0.1046 0.1348 0.1365 0.0574 0.0679 0.0777 0.1556 0.251 0.2079 0.125 0.260

"From Oka and Morino [41]. bAssumed.

change with isotopic substitution. This can be useful in distinguishing between an assignment of isotopic species as compared to an excited vibrational state where a larger change can occur. Both in-plane and out-of-plane vibrations contribute to the inertial defect and Avibmay be divided into in-plane A(i) and out-of-plane A(o) contributions. Usually the in-plane vibrations make a positive contribution, and the out-ofplane vibrations make a negative contribution to the inertial defect [42]. The in-plane contribution is ordinarily larger, and the total inertial defect is positive, as is observed for the tetratomic molecules in Table 13.11. If one or more of the out-of-plane vibrations is low, much of the in-plane contribution can be canceled with a very small inertial defect resulting, as observed in planar ring molecules. A negative value for the inertial defect is possible if A(o) is greater than A@), (see later discussion). The values for some planar rings are listed in Table 13.12. A convenient empirical approximation for calculation of the inertial defect of a planar unsymmetrical XYWZmolecule has been given [41] which requires taking the geometric mean of the inertial defects for the corresponding symmetric molecules: Axr =(AxxAYY)1'2

(13.139)


687

Table 13.12 Inertial Defects for Some Planar Ring Molecules

Molecule Furan" Thiopheneb Pyrrole' Pyridined 2-Chloropyridine'

Molecule

A (amu A')

Fluorobenzenef Chlorobenzeneg m-Dichlorobenzeneh Benzoni trile' Phosphabenzene'

0.033 0.053 0.098 0.084 0.052

A (amu A') 0.046

0.065 0.076 0.039

0.055

"B. Bak, D. Christensen, W. V. Dixon, L. Hansen-Nygaard, J. Rastrup-Andersen, and M. Schottlander, J. Mol. Spectrosc., 9, 124 (1962). *B. Bak, D. Christensen, J. Rastrup-Andersen, and E. Tannenbaum, J. Chem. Phys., 25, 892 (1956). 'B. Bak, D. Christensen, L. Hansen, and J. Rastrup-Andersen, J. Chem. Phys., 24, 720 (1956). dF. Mata, M. J. Quintana, and G. 0. SBrensen, J. Mol. Struct., 42, 1 (1977). 'R. T. Walden and R. L. Cook, J. Mol. Spectrosc., 52, 244 (1974). 'L. Hygaard, I. Bojesin, T. Pedersen, and J. Rastrup-Anderson, J. Mol. Struc., 2, 209 (1968). %. Roussy and F. Michel, J. Mol. Struct., 30, 399 (1976). hM. Onda, 0. Ohashi, and I. Yamaguchi, J. Mol. Struct., 31,203 (1976). ' J . Casado, L. Nygaard, and G. 0. SBrensen, J. Mol. Struct., 8, 211 (1971). 'R. L. Kuczkowski and A. J. Ashe, 111, J. Mol. Spectrosc., 42,457 (1972).

Thus the inertial defect for HFCO is just (AH,C0.AFzCO)112. Except for simple molecules the evaluation of the inertial defect, which requires a normal coordinate analysis, is rather tedious. Herschbach and Laurie [42] have considered simple approximations for calculating A v i b which depend on one or two modes of vibration and give results within 10 to 20% of the experimental values. A particularly simple approximation attributes the major part of the inertial defect to the lowest in-plane vibration, A"ib

4K =w

(13.140)

with w the frequency of the vibration in cm-' and K = h / 8 x 2 = 16.863 amu

W 2 cm-'.

Comparison of calculated and observed inertial defects provides the most conclusive evidence for the planarity of molecules. It has become apparent that the observation of a small inertial defect cannot of itself be taken as a decisive test of planarity. For example, in formamide (NH2CHO)the observation of a small positive inertial defect was originally interpreted as evidence of planarity [42]. However, a subsequent detailed study of various isotopic species by Costain and Dowling [44] revealed rather anomalous behavior in the inertial defects. Specifically, isotopic substitution on the N H 2 group resulted in a decrease in the inertial defect, and for some species there was a negative inertial defect. This behavior suggests that possibly formamide is nonplanar, and all

688

DERIVATION

OF MOLECULAR STRUCTURES

the evidence is, in fact, consistent with the H2N-C group forming a shallow pyramid. Observed inertial defects also provide information dependent on the harmonic force constants, especially A s for excited vibrational states, while calculated inertial defects are useful in the assignment of rotational transitions in vibrationally excited states. Furthermore, evaluation of A allows calculation of one of the moments of inertia if the other two are known. For a nonplanar molecule, the planar moment equation, (13.25), yields 2P: =

+Zi-

= 2P;

- A,

(13.141)

where, as before, A? represents the inertial defect often termed a pseudoinertial defect. For a nonplanar molecule, the effective planar moment can be large because of the presence of out-of-plane atoms. In a planar molecule P;( rn$) for the out-of-plane direction vanishes, and (13.131) is obtained. Although the anharmonic contribution to the inertial defect vanishes for a planar molecule, A, for a nonplanar molecule contains both harmonic and anharmonic vibrational contributions

=I

A? =A, (har)+ A? (anhar)

(13.142)

For molecules with a plane of symmetry and with only out-of-plane atoms in symmetrically equivalent pairs, certain generalizations can be made [42]. In this case the vibrations may be classified as symmetric or antisymmetric with respect to the plane of symmetry. The only contribution then to A, (anhar) arises from the symmetric vibrations of the out-of-plane atoms, whereas for A, (har) contributions arise from the symmetric and antisymmetric vibrations of the out-of-plane atoms and from antisymmetric vibrations of in-plane atoms. Only the symmetric vibrations of the in-plane atoms do not contribute to A, (har). In the structural analysis of nonplanar molecules complications arise because of A?. These will be considered further in the following sections. Since Ac is expected to be small compared to P,, the quantity Z:- ZE- Z:, where c is the out-of-plane direction, will be negative, as follows from (13.141). If, however, the out-of-plane coordinates are very small, Le., the molecule is only slightly nonplanar, then Zi- Z:- ZE may be positive. For amines and amides that are nonplanar, the quantity I," - I,"- Zp is small and negative, and deuteration of NH2 usually gives a large decrease in this quantity. On the other hand, for planar amines and amides Zp - I," - Zp is small and positive, and changes on deuteration, though negative, are much smaller [45]. Even for a planar molecule, if a very low-frequency out-of-plane vibration is present, the inertial defect can be larger than usual and negative. The inertial defects for some molecules of this type are listed in Table 13.13, along with the variations of the inertial defect in excited torsional vibrational states. For benzoyl fluoride [46], the variation with the torsional state can be expressed in amu A' by AzO.286- 1.222(~,+4)

(13.143)

689


Table 13.13 Inertial Defects for Some Molecules in the G r o u n d and Excited Torsional States

Molecule

C~HSNO

u=o

-0,146 -0.4811 C6H5CHO -0.128 C~HSCOF -0.325 F C ~ H ~ C H T C H Z -0.775

C6H5N02

v=l

-0.968 -1.8626 -0.966 -1.528 -2.327

Inertial Defect A (amu A’) v=2 2;=3 v=4

-1.801 -3.1858 -3.040 -2.765

-3.510

-2.589 -4.4704 -2.747 -3.963 -4.606

-3.376 -

w,O

82 49

-

80

-5.186

56 43

-

Ref:

e f

autin cm-’, u t =-67.45/SA, where 6A is taken as the change in the inertial defect between the first excited and ground state. bHanyu et al. [48]. ‘H$g et al. [49]. dKabar et al. [47]. ‘Kabar [46]. lRalowski et al. [50].

where - 1.222 amu 8, is the average change of A between successive torsional states. This variation will not necessarily be linear unless the vibration is harmonic in nature. This is a good approximation for benzoyl fluoride. Extrapolation of the inertial defect to the vibrationless torsional state (u,= -3) gives +0.286 amu 8,’ for the residual inertial defect. This value has the correct sign and magnitude for a planar molecule in the ground vibrational state with small amplitude vibrations. The -COF group is thus not bent or twisted out of the plane. If the torsional contribution is subtracted, similar results are obtained for benzaldehyde [47] (0.308 amu A’), nitrosobenzene [48] (0.264 amu A’), ni‘trobenzene [49] (0.249 amu 8,’) and p-fluorostyrene [50] ( ~ 0 . 1amu A2). Other examples may be cited [51, 52, 531. As pointed out for nitrosobenzene by Hanyu et al. [48], the variation of the inertial defect can supply information about the torsional vibration. The difference in the inertial defect between two successive singly excited vibrational states (us)is, from (13.135),

(13.144) where 6,, = 1 if s refers to one of the out-of-plane vibrations t, otherwise 6,, =O. Two out-of-plane modes do not interact, 5:)=0, ci=a, b, and c, while for two interacting in-plane modes, (2)=[:!I =0, [gt #O, and for an in-plane mode interacting with an out-of-plane mode, #O, 51;) #O, [g?= O [42]. The torsional mode thus interacts with an in-plane mode, and the [kt term drops out of (13.144); 6 6 between two adjacent excited states of the out-of-plane torsional mode is given by

690


If the torsional mode is much lower in frequency than any of the other modes, it couples with o: I:>I~

(1 3.193)

708


A detailed analysis by Herschbach and Laurie [85] has shown that similar considerations also hold for polyatomic molecules. By means of a perturbation treatment to terms linear in (~+f), it is found that the moments of inertia of the average configuration of the atoms are given by (13.194) or (13.195) where the sum is over all vibrations with degeneracy d,. As for the diatomic molecule, the moments of inertia for the average configuration can be derived from the effective moments without a knowledge of the anharmonic potential constants; if the vibrations are harmonic, then I,* = Zz,and the average structure will be identical to the equilibrium structure. The anharmonicity of the potential, which is always present to various degrees for real molecules, has the effect of displacing the average configuration of the vibrating nuclei from the equilibrium configuration. Convenient procedures for evaluation of (har) have been given [85] as well as explicit expressions for some simple molecules [86,90]. In general [85]

cm-'. Other forms are possible since

with K = 16.863 amu

(13.197)

where a:, = (al,,/aQ,)* and A!: = (a2l,,/dQs")*.In particular, for a linear X Y, molecule the ground state moment of inertia I$ is given by [86]

I$=Ip+K

(

3 -1 - -+1

(13.198)

-

0 1

w3

w2

0 2 + 043

,

For a linear X Y Z molecule [86] I,*=IE-K

[-+-+1

1

1

0 1

0 2

0 3

-4cf2

($+--)-4G31 0 1+ 0 2

(-&+-)I

1

w2+w3

(13.199)

and the Coriolis coupling constants are found from the equations [91] (!2

+( ? 3 = l

L1G3+M2=(rLF11+2rxYryZF13+~ZyZF33)/Ib

(13.200) (13.201)

where FI1,F33, F I 3are the quadratic potential constants for the X - Y bond stretch, the Y - 2 bond stretch, and the interaction, respectively.

EVALUATION OF AVERAGE STRUCTURES

709

For a bent X Y , molecule with b the symmetry axis and with c the out-ofplane axis [86] I t = I,”+3K

sin’ x cos’ x I & / I c ) ( 7 +0 2

(13.202)

0 3

(13.203)

(13.204) where x is a parameter determined from ((23 O) (13.205) The Coriolis coupling constants which refer to the out-of-plane axis (c axis) are related to the force constants and vibrational frequencies by the relations

(13.206) (13.207) where As=47c2cz0~with the F and G matrices expressed in terms of the usual symmetry coordinates for a bent X Y , molecule. Explicit expressions for the average moments of tetrahedral X Y , molecules and linear X 2 Yz molecules have also been given in [86]. In the above expressions the harmonic vibrational frequencies 0, are to be expressed in cm-’ with K=h/87c2=16.863 amu A’ cm- and As =5.8893 x 10- ’0;. By employment of these expressions the average moments of inertia I,* can be evaluated; they in turn can give information on the average positions of the atoms in the molecule. Laurie and Herschbach [86] have calculated average structures for several simple molecules and pointed out a number of interesting features. In Table 13.7 along with the effective and equilibrium bond lengths the average bond lengths of some diatomic molecules have been given. As expected, ( r ) > ro > r e . For the hydrides the ( r ) distance is about 0.02 larger than re. In the diatomics made up of heavy atoms the difference is smaller. The replacement of H by D, which decreases the amplitude of vibration, causes a shortening in ( I ) of 0.003 to 0.005 A. Isotopic s’ubstitution of the heavier atoms gives a much smaller shortening, of about 0.0001 A; when the heavy atom is bonded to hydrogen, the shortening is negligibly small since the hydrogen atom does essentially all the vibrating. The average parameters of some bent X Y , molecules are illustrated in Table 13.20. It is again observed that replacement of H and D shortens the bond length

710

DERIVATION

OF


Table 13.20 Average Structures of Bent X Y , Molecules in the G r o u n d Vibrational State

0.9724 8, 0.9687 0.9671 1.3518 1.3414 1.4754 1.4711 32s02d 1.4350 3 4 ~ 0 2 d 1.4350 3 5 c ~ 0 2 e 1.4756 3 7 ~ ~ ~ z 1.4755 e 1.2794 03d NOz* 1.2000 1.4124 OF/ 1.3528 NF2' SFzh 1.5921 sa,' 2.0153

HzOa DZO" TzO' H,Sb DzSb H Se' D2SeC

104"3 0 104"21' 104"16' 92%' 92"7' 90"52' 90"53' 119"21' 119'21' 117'30 117'30 116'44' 133"48' 103"lO' 103"ll' 98.20" 102.73"

"Cook et al. [94]. *Cook et ai. [93]. 'Oka and Morino [MI. dLaurie and Herschbach [86]. 'A. H. Clark, J. Mol. Struct., 7,485 (1971). fL. Pierce, N.DiCianni, and R. H. Jackson, J. Chem. Phys., 38,730 (1963). BR. D. Brown, F. R. Burden, P. D. Godfrey and I. R. Gillard, J. Mol. Spectrosc., 52, 301 (1974). hW. H.Kirchhoff, D. R. Johnson, and F. X. Powell, J. Mol. Spectrosc., 48,157 (1973). 'R.W.Davis and M. C. L. Gerry, J. Mol. Spectrosc., 65, 455 (1977).

by 0.004 A in H 2 0 and H2Se. Replacement of H by T in H 2 0 gives a shortening of 0.005 A. Substitution of a heavy atom has a negligible effect. Since for the average structural moments of inertia the inertial defect vanishes [92], different pairs of moments of inertia will give consistent structures, in contrast to the results obtained when effective moments are used. Another interesting isotopic variation found [ 8 6 ] is the decrease in I: (also I:) for SO2 with replacement of 32Sby a heavier isotope. In particular, I t for 3 2 S 0 2 , 3 3 S 0 2 ,and 3 4 S 0 2is, respectively, 49.0934,49.0931, and 49.0921 amu A2,Here the very small decrease in the average S - 0 bond causes a decrease in I t . For a rigid molecule like

71 1


SOz where S is on the b axis, I : would be unaffected. Similar effects have been noted in COz when "C is replaced by 13C [86]. Average structures in various vibrational excited states have also been evaluated for SOz [26] and O F z [27]. These are summarized in Table 13.21. It is apparent that the average structures show the characteristics of the excited vibrational states. When the symmetric (ul) or antisymmetric ( u 3 ) stretching mode is singly excited, the bond lengths increase relative to the ground state, while the bond angles change very little. When the deformation mode ( u 2 ) is excited, the bond angle clearly increases, and there are small changes in the bond lengths. On the other hand, the effective structures of SO2 do not clearly indicate the variation of the bond angle with vibrational state [26]. It is possible to obtain an estimate of the equilibrium structure from the average structure data. For a diatomic molecule, the average structure is related to the equilibrium structure as follows, with a. =02/4Be, (13.208) where p is the reduced mass, and where a. (cm-') and a l (dimensionless) are, respectively, the harmonic and cubic potential constants. From the average bond distance and the reduced mass of at least two isotopes, (13.208) can be used to calculate re. For an Y X z molecule, if we assume the Y-X bonds may be treated separately as one-dimensional oscillators, we may apply the above relation. This approximation has been used to derive the equilibrium structure of HzSe [88] and H2S [93] and H 2 0 [94]. The equilibrium structure obtained for H z O with use of the data in Table 13.20 is re =0.9587 A, 8, = 103.89'. In the derivation of Be, the same reduced mass dependence has been assumed for Table 13.21 Average Structures in Excited Vibrational States' State Moleculeb

so2

OF2

UIVZU3

(r )

000 100 010 001 020

1.4349 1.4388 1.4346 1.4396 1.4343 1.4124 1.4183 1.4139 1.4196 1.4155

000 100 010 001 020

a(r)b 0.0039 -0.0003 0.0047 -0.0006 0.0059 0.0015 0.0072 0.0031

(6 ) 119'21' 119'24' 119'32' 119"ll' 119'44' 103"lO 103'2' 103"34' 103'4' 103'58'

Wb 3' 11' -1 0 23'

- 8' 24' - 6' 48'

"From Morino et al. [26, 271. *Difference between average parameter in the vibrationally excited state and in the ground state.

712


(e) as for ( r ) . The extrapolation t o p - 1/2_t0is, however, rather long. This result for water may be compared to the more accurate infrared result [95] of re= 0.9575 8, and 104.51". For molecules like CH3COX with a single pair of out-of-plane hydrogen atoms, the average out-of-plane distance of the H atoms may be calculated if correction is made for the harmonic part of the inertial defect Ay, that is [42] Z,"+ I t - Z , O = ~ W Z ~ ( ~ (har) ~)~-A,

(13.209)

with c(y) the out-of-plane direction. The anharmonic part of the inertial defect simply shifts (y) fromy,. If A, (har) is essentially the same for the CH, and CD, species, then Ay (har) may be evaluated from data on CH3COX and CD3COX if the small variation is neglected and if an estimate is made of the difference between (YH) and ( y ~ ) .Alternately, A, (har) may be estimated from the vibrational frequencies [42]. When corrections are made for Ay (har), the ( y H ) obtained no longer shows the decrease in the methyl group size observed for the effective distance (see Table 13.12).It is found [42] that (yH) is 0.896,0.891, and 0.896 8, in CH3CH0, CH3COF, and CH3COCI, respectively [obtained by estimation of A, (har) from vibrational data]. These average values of the outof-plane coordinate compare favorably with that found in methane, namely, 0.896 8, [86]. The consideration of the average structure of a linear X Y Z molecule such as HCN reveals a fundamental problem in the calculation of average structures as well as of effective and substitution structures. Here, at least two molecular species are required for determination of the structural parameters: but, as we have seen, the average parameters clearly vary with isotopic substitution. However, the C-H bond distance is not expected to change significantly with a change in mass ofcarbon, since this will have little effect on the C-H vibration. Also the isotopic variation of C E N with 13C substitution is expected to be small. Hence, the smallest error is introduced in the structural calculation if either the pair Hl2Cl4N,H13C14N or D12C14N, DI3Cl4N is used. When the structure is evaluated in this way, it is found [86] that (rcH) is larger than (rm) by 0.003 8, and that the ( ~ c N is ) almost the same in HCN and DCN (decrease of 0.004 8,). For H 2 C 0 [90], the derivation of an accurate average structure required an estimate of the variation in the C =O bond with I 8 0 substitution. The difference (0.000123 8,) was estimated by means of an approximate onedimensional potential function with an assumed anharmonic force constant. More recent calculations on H 2 C 0 have been made by Duncan [96], who used data on seven isotopic species (Hi2C160,Hi3C160, H i 2 C ' * 0, D2 l 2C l 60 > D213C160,D2l2Cl8O,HD12C160).The data are sufficient for the evaluation of the isotopic effects for all three structural parameters. Corrections were made for the electronic contributions to the moments of inertia. It was assumed that 13C or l80substitution causes negligible change in the CH(CD) bond lengths or HCH(DCD) bond angles. The average structure and the major isotopic effects are given in Table 13.22. The data for HDCO were not employed in the calculation of the structure; however, the final structure and shrinkage effects were used

713


Table 13.22

Average Structure of Formaldehyde" rZ(CH) r,(CD) r,(CO) a,(HCH) z,(DCD)

1.1171 kO.0010 8, 1.1130$0.0010 8, 1.2072$ 0.0005 8, 116"14$6'[~,(HC0)121"53'$3'] 116'19' 6'[~,(DC0)12lo50'& 3'1

Isotopic Mass Effects on Structural Parameters [H-D] 0.0041 $0.0005 8, Gr,(CH) Sr,(CO) [12C-13C] 0.00003 f0.00005 8, [160-'80] 0.00010$0.00005 8, 6r,(CO) [H2-D2] 0.0005 k0.0002 8, [0.00025 8, per D substituion] 6r,(CO) - 5'+2[2.5' per D substitution] Gz,(HCH) [H2-D2] "From Duncan [96].

to calculate the average moments of inertia for this molecule. The agreement with those derived from the harmonic force field is as good as, or better than, +0.0006 amu A'. In average structure evaluations, the isotopic changes 6 ( r ) in bond lengths with isotopic substitution are often estimated from the approximate relationships given by Kuchitsu et al. [97, 981 (r)=re

+-3a2 u2 - K

6 ( r ) = -3a 6u2-6K 2

(13.2 10) (13.211)

where u2 is the zero-point, mean-square amplitude of the particular bond in question and K , the corresponding mean-square perpendicular amplitude correction. These quantities may be evaluated from the harmonic force field [99]. Here a is the Morse anharmonicity parameter which may be estimated from similar diatomic molecules [loo]. The isotopic effects on bond angles d(0) are small and are usually ignored. Equation 13.210can be used for estimation of the re structure by application of the appropriate correction to the ( r ) structure. These corrections, however, are rather large. In summary, for even relatively simple molecules, where sufficient harmonic force constants are available for evaluation of I:, if isotopic data must be combined in calculation of the structure, it is necessary to either ignore the isotopic effects or attempt to estimate the isotopic variation in ( r ) . This, of course, introduces some uncertainty in the average structural parameters although in favorable cases only small uncertainty. Such isotopic effects also introduce ambiguities in effective and substitution structures, since in the calculation of these structures it is tacitly assumed that the bond lengths are unaffected by isotopic substitution. Because of the isotopic shrinkage of bonds, the substitution

714


method gives coordinates which are too small. (The bond length, however, may be either too short or too long.) In the case of HCN, for example, the “substitution” C-H bond length, which is evaluated by neglect of the shortening caused by deuterium substitution, is found to be some 0.003 8, less than the equilibrium bond length [60]. It has been demonstrated [86] that an isotopic change of 0.0001 8, in a bond length can give an error of 0.002 8, in the substitution coordinate. When an atom is near a principal axis, this effect can be greatly amplified. Unfortunately, the accurate calculation of isotopic changes requires a knowledge of the anharmonic potential constants which are available for only a few polyatomic molecules. The isotopic changes, however, may prove to be characteristic of a bond and hence transferable from simple molecules to more complicated ones. In attempting to estimate the uncertainty in a structure, such as a substitution structure, consideration should be given to the effect of bond shortening and bond angle changes upon isotopic substitution, for example, by consideration of the effect of deuterium substitution where r ( X D) =r(XH)- 6 and 6 is varied from 0.002 to 0.005 8,. If X is substituted, 6 =O. For two heavy atoms, 6 is varied between O.ooOo5 to 0.0001 8,. One procedure [33] would involve calculation of the moments of inertia for all isotopic species with the structure obtained from the experimental moments of inertia. For each isotopic species, the bond distances associated with one or more isotopic substitutions are adjusted by selection of a particular value of 6, and the moments of inertia are recalculated. The difference between the moments of inertia with and without the shrinkage effects is then used for correction of the experimental moments of inertia. Finally, the structure is evaluated from the corrected moments of inertia in the usual way, thus giving an estimate of the uncertainties. In the absence of equilibrium structures, average structures provide a welldefined structural parameter. Ambiguities introduced by inertial defects of planar molecules are removed, and it is possible to relate ( r ) to the electron diffraction rg so that a meaningful comparison between the two can be made. This is in contrast to the effective and substitution parameters. O n the other hand, the evaluation of average structures requires a knowledge of the harmonic force constants which are usually not available except for relatively simple molecules.

10

EVALUATION OF MASS-DEPENDENCE STRUCTURES

A method of structural calculation has been introduced by Watson [57] which represents an extension of the substitution method but requires a more extensive set of isotopic data. It uses the I o values of a number of isotopic species to determine, to first-order, the mass dependence of the vibrational contributions and thus to obtain an estimate of the equilibrium structure. This method is called the “mass-dependence’’method and gives the so-called rm structure. In the following derivation one exploits the fact that various molecular parameters are homogeneous functions of the masses of degree n, that is, F ( p m l , p m z , . . . p m i , . . . ) = p n F ( m l ,m 2 , . . . mi... )

(13.212)

EVALUATION OF MASS-DEPENDENCE STRUCTURES

71 5

for any value of p. Simultaneous isotopic substitution of all the atoms of a I of molecule mi+pmi, for example, shows that Z:, a:, ca, afs,?,gyd, G I ,and C ~are degree 1, -3, f,f, -2, 0 and - 1, respectively [57]. Furthermore, the position of the center of mass and the orientation of the principal inertial axis are unchanged. A more detailed list of molecular parameters and their degree in the masses is given elsewhere [57]. Only in certain cases such as for ozone, 1 6 0 3 + can this isotopic substitution actually be realized, and in these cases the isotopic relations in the foregoing quantities are particularly simple. It is clear that the vibrational contributions to I," are of a different degree in the masses than I:. For a linear molecule we may write from (13.116)

&I;+&

(1 3.213)

with (1 3.214) With isotopic substitution of atom i, we have A I8 = A I;+AE

(1 3.215)

Introducing the relations A I" [ze(i)I2 =b

(1 3.216)

Pi

A P [zS(i)l2=b

( 1 3.21 7)

Pi

with p i=MAmi/(M+ Ami) and expanding AEas a Taylor series in Ami gives [57] (13.218) This may be simplified by multiplying by mi and summing over all atoms i. Thus

where I b = c mi[z,(i)]2. The first sum may be evaluated from Euler's theorem on homogeneous functions (1 3.220) where E is a homogeneous function of degree n =). Employing this relation in (13.219) and defining the mass-dependence moment of inertia Z r as

II: =2 I;- I:

(1 3.221)

gives (13.222)

716


Here Ami is the isotopic mass difference for the i th atom employed in deriving the rs coordinates of the particular parent species, and M (total mass) and m , refer to the parent species. We see that if linear and higher-order terms in Am) are ignored, IF represents an approximation to the equilibrium moment of inertia I : . To the same approximation

and the substitution coordinates defined by

(13.224) are well-defined properties of the parent molecule. It is clear, however, that the substitution structure of different parent molecules is mass dependent because a&/amiis of degree in the masses, though the differences are usually small. To determine an r,,, structure one employs various isotopic species to evaluate the substitution coordinate of each atom from (13.217)for a given parent. These coordinates are then employed for evaluation of I s and hence of I" from (13.221). Once a sufficient number of I" for different parent isotopic molecules have been evaluated, the moment-of-inertia equations may be solved for the rm structure. The validity of this procedure depends on the relation of (13.224) where the vibrational contribution to [zS(i)l2is given by &/ami. Therefore very small or imaginary coordinates, which can arise when &/am, is large compared to [ze(i)]' and negative, should be retained in calculation of I s . This yields a physically significant substitution moment of inertia regardless of the reliability of the coordinates. Furthermore, first moment equations should not be employed for calculation of z, coordinates. Ideally, to ensure the correctness of (13.224), the substitution coordinates, (13.218), should be extrapolated to zero values of

-4

Ami.

For polyatomic molecules it can be shown [57] in a similar manner that firstorder estimates of the equilibrium moments of inertia are 1:=21:-

I,"

z ~ = Cmi[cc,(i)12

cc=a,

b, and c

(13.225) (13.226)

i

where each nonequivalent atom must be substituted for determination of the I f of a parent molecule. The values of I: for a sufficient number of parent molecules are then solved for the rm structure. Double-substitution coordinates from substitution of a symmetrically equivalent set of atoms can be employed if care is used to maintain the internal consistency of the method. It has been shown that for bent symmetrical triatomic molecules, (13.225) and (13.226) apply with the coordinates of B located employing the substitution A B 2 + A B ; .

DIAGNOSTIC LEAST SQUARES

717

They apply for a C3”molecule ABC3 where ABC; data are employed. For CO the rm values are good estimates of r e , but the spread obtained for the various isotopes is appreciable, though only half the spread of the ro values [57]. Using the parent I 2 C l 6 0and the isotopic species 13C and 14C, one finds that the squares of the substitution coordinates are slightly different. Extrapolating them to Amc =O yields an improved substitution coordinate in which the higher-order effects in (1 3.218) are effectively eliminated. The same procedure may be carried out by use of ” 0 and data for the oxygen substitution coordinate. These data provide an improved I” value and an rm bond length that differs from re by only 0.00019 A. A critical test [ l o l l of the rm method has been carried out on OCS. It was found that the rm structure differs significantly from the re structure and that the errors are comparable in the rs and rm methods. On the other hand, if a corrected set of substitution coordinates is employed for OCS by extrapolation of the coordinates for, say, two different values of Ami for each atom i of each parent molecule to Ami=O, the results are in good agreement with the equilibrium structure. This procedure is clearly of limited applicability and reflects the breakdown of the first-order approximation employed previously. The rm structure of SO2, which is in good agreement with the re structure, is given in Table 13.5. A detailed study of the rm structure of 0C12 and C0C12 has been carried out by Nakata et al. [102,103]. If averages are taken of the rm parameters of appropriate isotopic combinations (termed “complimentary sets”), the higher-order effects of (13.222)may be reduced, since these effects will have opposite signs [57,102]. Parent species which have completely different isotopic make-up so that the signs of Ami for any ith atom are opposite, constitute a complimentary set. However, only one meaningful rm structural parameter can be obtained from the moment of inertia data about a given axis (e.g., two for C0Cl2) [103]. Unfortunately, the rm method is of limited applicability. Data of very high precision from a large number of isotopic species are required. The first order approximation I ” N P is not sufficient particularly for light atoms such as hydrogen atoms. The method cannot be applied to molecules containing atoms such as fluorine, which cannot be substituted; and even for moderate size molecules, a complete set of reliable rm parameters cannot be evaluated.

11


In calculations of structure, the number of parameters frequently exceeds the number of rotational constants. If one wishes to find a “physically reasonable” set of parameters that reproduce the rotational constants found from the experimental data, it is possible to vary the parameters one at a time until such reasonable parameters are found, but this procedure becomes unrealistic with a large number of parameters. On the other hand, some parameters or some linear combinations of the parameters may not be well determined by the experimental data; yet clearly some linear combinations of the parameters are determined by the data. Curl [35] has, devised a systematic method, referred

71 8


to as “diagnostic least squares”, which is applicable to such situations through use of adjustable parameters weighted by the inverse of their estimated uncertainties. A set of parameters is found that reproduces the experimental data with a minimum distortion in a least-squares sense from the initial set. An outline of this procedure is presented in the following discussion. The technique can also be applied to a combination of data from different sources. An electron diffraction structure, for example, might be adjusted to include the microwave data. In addition to the rotational constants, the dipole moment components can be profitably included when these data are sensitive to the orientation of one part of the molecule relative to another. Another application could involve a set of molecules which constitute a closely related series when one wishes to see if a reasonable set of structural parameters could be obtained that would correlate the microwave data of these molecules. With the assumption that similar structural parameters can be transferred, there may be, in principle, more data than parameters. However, if it is found that the data are not sensitive to all the structural parameters, a diagnostic least-squares analysis can be carried out to obtain a set of structural parameters which adequately fit the spectral data of these molecules. Clearly, the technique can be applied to problems other than calculations of molecular structure. We consider a set n observables Zi = libj), such as moments of inertia, which are considered functions of a set of rn parameterspj. To evaluate the parameters from the observations, the problem is linearized by a Taylor series expansion about the initial set of parameters pg

or m

Apj,

i=l, ...,n

(13.228)

j= 1

with Apj=pj -pP. To proceed where ordinary least squares will not apply, one defines a new set of parameters which have equal uncertainties. Structural parameters are predicted from analogous molecules along with an estimated uncertainty for each parameter. The latter may be done by consideration of the range of values obtained for this parameter in similar molecules. If the range pj” to p y is considered a normal distribution with 90% confidence interval, the best estimate of pj’ is given by

J

- PL +pu L 2

(13.229)

-

with estimated standard deviation [35] Isj =-PY-Pj”

3.290 A new set of weighted parameters is then defined by

(13.230)


719

(13.231) and (13.228) in terms of these parameters can be written in matrix form as JAP = AI

(13.232)

where AP is a column matrix of the m parameter changes A P j , AI is a column matrix with II elements composed of the difference between the observed and and T is the n x m Jacobian calculated experimental quantities AZj =Ifbs- lica', matrix (or design matrix) with elements defined by (13.233) The derivatives may be evaluated numerically. A small increment is given to the initial value of thejth parameter and the change in the observable Ziis computed to give the derivative. If equivalent parameters are present, the derivatives should reflect this fact. The least-squares normal equations are obtained by multiplication of both sides of (13.232) by 3, where the tilde denotes the transpose of the matrix. Thus, jJAP =j A I

(13.234)

where 3J is a m x m symmetric matrix (the normal matrix). In ordinary least squares where j J is nonsingular, the parameter shifts are given by AP =(5J) -

3A I

(13.235)

If the number of adjustable parameters, P j , exceeds the number of experimentally determined quantities, Zi, or if some linear combinagons of the parameters are not well determined by the experimental data, then JJ may be singular or nearly so. In this case, the matrix inversion leading to (13.235)is not possible. However, some linear combinations of the parameters are determined by the data. The inversion can be effectively obtained by an alternate procedure. The matrix of coefficients of the normal equations may be diagonalized by an orthogonal transformation, T .?.jJT=A

(13.236)

As a result of the parameter scaling, (13.231), the size of a particular eigenvalue i jreflects how well determined is that linear combination of P j described by the corresponding eigenvalue. Equation 13.234 can then be rewritten as

?jJT?AP= ? j A I

(13.237)

or

AI

(13.238)

AP=TA-'?jAI

(13.239)

A?AP= and inverting gives

720


The shifts in the original parameters are then computed from (13.240)

with p j = p p + Apj and where i runs up to the number of experimentally determined quantities, I runs up to the number of adjustable parameters, k runs over only those values for which & > C 2 , with o2=the variance of the experimentally determined quantities. If the indexk runs over all the possible values (Le., all the 1.k > 0 2 ) ,then keeping in mind that one may invert j J by diagonalizing, inverting the eigenvalues, and then performing the inverse of the transformation indicated in (13.236), it is seen that (13.240)and (13.235) are equivalent (with oj= 1). The difference comes when one or more Ak < 0 2 ,for then that eigenvector is not used in computation of the parameter shifts. Once the parameter shifts have been computed, the model calculation is repeated to see if the improved parameters do indeed reproduce the experimental data. This is necessary because a linear least-squares approximation is used and must be tested for convergence. The parameters obtained will be increasingly unrelated to the data as the eigenvalue decreases. The linear combinations which are better determined by the observations than by the a priori estimates will have the corresponding & > 0 2 . O n the other hand, the a priori estimates are expected to be more reliable for jlk 2 already given by (14.93). We have not included H F in the plot of Fig. 14.7 because of the uncertainty in the evaluation of the rather large pi,,.+ The electronegativity difference of I x F - x ~ I = 1.8 indicates an almost complete ionicity of the bond, 90%.However, the ionic radius of F- is 1.33 A, and yet the measured internuclear distance of H F is only 0.92 A. This means that if the bond were completely ionic, the proton would actually be buried deep in the negative charge cloud of the F- ion. If we assume that the proton is at the edge of a semi-ionic radius equal to the observed interatomic distance of 0.92 A, (14.166) would indicate a polarization moment of pid = 1.68. With this assumption, the ionic character is found from the observed [77] dipole moment 1.8195 D to be i,=0.82. This value is not unreasonable, but the structure of H F is evidently more complex than this analysis indicates. The value of 1.4 D for its moment predicted from molecular orbital theory [73] is not in good agreement with the accurately observed value.

Polyatom ic M01 ecu les The electric dipole moment of a molecule is determined by the charge distribution over the entire molecule, unlike quadrupole coupling which is determined primarily by the charge distribution in the immediate vicinity of the particular coupling nucleus. For this reason dipole moments of polyatomic molecules are more difficult to interpret quantitatively than are quadrupole couplings. Molecular orbital theory has not yet advanced to the point where molecular dipole moments can be reliably predicted for polyatomic molecules. However, from comparisons of the dipole moments of related polyatomic molecules it has been possible to assign approximate moment values to certain bonds in polyatomic molecules [78-801. One can then use these empirical bond moments to predict roughly the overall moments of certain molecules. However, bond moments are not vectorially additive because of the changes in bond character from molecule to molecule. For estimation of the primary bond moments in polyatomic molecules the relationship between ionic character and electronegativity as derived from quadrupole couplings (Fig. 14.1)and confirmed by dipole moments of diatomic

800

QUADRUPOLE COUPLING AND THE CHEMICAL BOND

molecules (Fig. 14.7) should be of much help when used with the proper group electronegativity values (Section 15). Likewise, the approximation methods described earlier in this section for estimation of other components ps,p h , and p i d for diatomic molecules should be useful in the prediction and understanding of the bond moments in polyatomic molecules. That the hybridization moment can be of much importance in polyatomic molecules is evidenced by the very small moment [Sl] (0.234 D) observed for NF3. Because of the large difference in electronegativity of N and F, a large primary N-F bond moment of 3.2 D is expected. The orbital overlap component is negligible because of the nearly equal covalent radii of N and F, but the induced component of each bond 1.0 D is significant because of the large primary moment. There will also be an N hybridization component directed along each bond, but this will be included in the resultant N hybridization moment to be estimated. Exclusive of this hybridization component, the component along each N-F bond will be approximately 3.2-1.0=2.2 D. Together the three bonds will give rise to an axial component of 3(2.2) cos 64' z2.9 D, with the positive pole on the N. The magnitude of the N hybridization moment ph=/.& -2.9 is estimated to be -2.7 D or 3.1 D, depending on the direction of the observed moment, which is unknown. This ph results from three bonding orbitals having 17% s character, and electron populations of 0.5 each, together with a counterhybridized sp orbital having 49% s character and two unpaired electrons. The overall p h is equivalent to that of an 1.5 electron population in the 49% hybridized orbital. The estimated ph is reasonable. A similarly hybridized orbital of C with nh=1.5 would give rise to ph z3.3 D (see Fig. 14.6). Although the N orbitals of NH, are comparably hybridized, the ph of this molecule is appreciably less than that of NF3 because of the more complete filling of the sp shell on the N. Because of the many uncertainties involved, we shall not attempt further interpretation of dipole moments in polyatomic molecules. In many organic molecules II character in the bonding increases the complexity of the moment. Selected values of moments measured with microwave spectroscopy are given in Tables 10.9-10.11.

-

-

References 1. 2. 3. 4. 5.

6. 7. 8. 9.

A. Nordsieck, Phys. Rev., 58, 310 (1940). C. H. Townes and B. P. Dailey, J . Chem. Phys., 17,782 (1949). C. H. Townes and B. P. Dailey, J . Chem. Phys., 20, 35 (1952). C. H. Townes and B. P. Dailey, J . Chem. Phys., 23, 118 (1955). W. Gordy, J . Chem. Phys., 19, 792 (1951). W. Gordy, J . Chem. Phys., 22, 1470 (1954). W. Gordy, Discussions Faraday SOC.,19,9 (1955). R. Bersohn, J . Chem. Phys., 22, 2078 (1954). J. H. Goldstein, J . Chem. Phys., 24, 507 (1956).

REFERENCES

801

IO. W. Gordy, W. V. Smith, and R. F. Trambarulo, Microwave Spectroscopy, Wiley, New York, 1954. 11. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, Mc Graw-Hill, New York, 1955. 12. T. P. Das and E. L. Hahn, Solid State Physics, Suppl. 1, Academic, New York, 1958. 13. P. Grivet and A. Basssompierre, Mem. R. Acad. Belg., 33, 217 (1961). 14. C. T. O'Konski, in Determination of Organic Structures by Physical Methods, Vol. 2, F. C. Nacnod and W. D. Phillips, Eds., Academic, New York, 1962, pp. 661-726. 15. T. M. Sugden and C. N. Kenney, Microwave Spectroscopy of Gases, Van Nostrand, London, 1965. 16. C. A. Coulson, Valence, Oxford Univ. Press, Oxford, 1953. 17. B. Pullman and A. Pullman, Quantum Biochemistry, Wiley, New York, 1963. 18. J. H. Goldstein, J . Chem. Phys. Rev., 84, 244 (1951); 86, 316 (1952); 95, 736 (1954). 19. R. M. Sternheimer, Phys. Rev., 84,244 (1951); 86, 316 (1952); 95, 736 (1954). 20. R. M. Sternheimer and H. M. Foley, Phys. Rev., 92, 1460 (1953); 102, 731 (1956). 21. H. W. White, Introduction to Atomic Spectra, McGraw-Hill, New York, 1934. 22. H. Kopfermann, Nuclear Moments (trans. by E. E. Schneider), Academic, New York, 1958. 23. J. C. Slater, Phys. Rev., 36, 57 (1930). 24. R. G. Barnes and W. V. Smith, Phys. Rev., 93,95 (1954). 25. H. B. G. Casmir, On the Interaction between Atomic Nuclei and Electrons, Teyler's Tweede Genootschop, E. F. Bohn, Haarlem, 1936. 26. G. Koster, Phys. Rev., 86, 148 (1952). 27. E. Tiemann, J. Mol. Struct., 97, 331 (1983). 28. H. M. Foley, R. M. Sternheimer, and D. Tycko, Phys. Rev., 93, 734 (1954). 29. R. Livingston, J . Chem. Phys., 19, 1434 (1951). 30. H. G. Dehmelt, Z . Physik, 130,480 (1951). 31. H. G. Dehmelt, Naturwiss., 37, 398 (1950). 32. N. Bettin, H. Knockel, and E. Tiemann, Chem. Phys. Lett., 80, 386 (1981). 33. A. Yokozeki and J. S . Muenter, J . Chem. Phys., 72,3796 (1980). 34. H. G. Robinson, H. G. Dehmelt, and W. Gordy, J . Chem. Phys., 22, 511 (1954). 35. R. S. Mulliken, J . Am. Chem. SOC.,72,4493 (1950). 36. A. D. Buckingham, Trans. Faraday SOC.,58, 1277 (1962). 37. L. Pauling, The Nature of the Chemical Bond, 2nd ed. Cornell Univ. Press, Ithaca, New York, 1948. 38. R. S. Mulliken, J . Chem. Phys., 46, 497 (1949). 39. L. Pierce, R. Nelson, and C. Thomas, J . Chem. Phys., 43, 3423 (1956). 40. H. G. Dehmelt, Discussions Faraday Soc., 19, 263 (1955). 41. M. H. Cohen and F. Reif, in Solid State Physics, Vol. 5, F. Seitz and D. Turnball, Eds., Academic, New York, 1957, pp. 321438. 42. H. 0. Hooper and P. J. Bray, J . Chem. Phys., 33, 334 (1960). 43. R. S. Mulliken, C. A. Reike, and W. G . Brown, J . Am. Chem. SOC.,63, 41 (1941). 44. P. A. Casabella and P. J. Bray, J . Chem. Phys., 28, 1182 (1958); 29, 1105 (1958). 45. E. Schempp and P. J. Bray, J . Chem. Phys., 46, 1186 (1967). 46. R. A. Marino, L. Guibe, and P. J. Bray, J . Chem. Phys., 49, 5104 (1968). 47. J. Sheridan and W. Gordy, Phys. Rev., 79, 513 (1950). 48. Y. Kato, U. Farukane, and H. Takeyama, Bull. Chem. SOC.Japan, 32, 527 (1959).

802 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.

QUADRUPOLE COUPLING AND THE CHEMICAL BOND

E. A. C. Lucken, Trans. Faraday SOC.,57,729 (1961). D. DeSantis, A. Lurio, T. A. Miller, and R. S. Freund, J . Chem. Phys., 58, 4625 (1973). F. W. Terman and T. A. Scott, Bull. Am. Phys. SOC.,3, (1958). H. Meyer and T. A. Scott, Phys. Chem. Solids, 11, 215 (1959). L. Nygaard, J. T. Nielsen, J. Kirchheiner, G. Maltesen, J. Rastrup-Andersen, and G. 0 . Sorensen, private communication. G . A. Sorensen, J . Mol. Spectrosc., 22, 325 (1967). W. Arnold, H. Dreizler, and H. D. Rudolph, Z. Naturforsch., 23a, 301 (1968). H. D. Rudolph, H. Dreizler, and H. Seiler, Z. Naturforsch., 22a, 1738 (1967). C. A. Burrus and W. Gordy, Phys. Rev., 92,274 (1953). H. G. Robinson, H. G. Dehmelt, and W. Gordy, Phys. Rev., 89, 1305 (1953). J. E. Griffths and K. B. McAfee, Proc. Chem. SOC.,1961,456. C . H. Townes, J. M. Mays, and B. P. Dailey, Phys. Rev., 76, 700 (1949). V. W. Laurie, J . Chem. Phys., 30,1210 (1959). E. F. Pearson, C. L. Norris, and W. H. Flygare, J . Chem. Phys., 60, 1761 (1974). C. Kirby and H. W. Kroto, J . Mol. Spectrosc., 83, 1 (1980). W. Gordy, H. Ring, and A. B. Burg, Phys. Rev., 74, 1191 (1948); 78, 512 (1950). H. G. Dehmelt, Z. Physik, 134,642 (1953). P. R. Wells, in Progr. Phys. Org. Chem., Vol. 6, A. Streitwieser, Jr., and R. W. Taft, Eds.. Wiley, New York, 1968, pp. 11 1-145. J. Hinze, M. A. Whitehead, and H. H. Jaffe, J . A m . Chem. SOC.,85, 148 (1963). W. Gordy, J. Chem. Phys., 14, 305 (1946). F. A. Matsen, J . Chem. Phys., 34,337 (1961). L. Wharton, L. Gold, and W. Klemperer, J . Chem. Phys., 37, 2149 (1962). P. Cade and W. Huo, J . Chem. Phys., 45, 1063 (1966). C. F. Bender and E. R. Davidson, J. Chem. Phys., 49,4222 (1968). B. J. Ransil, Rev. Mod. Phys., 32,239 (1960). L. C. Allen and A. M. Karo, Rev. Mod. Phys., 32, 275 (1960). R. S . Mulliken, J . Chim. Phys., 46,497 (1949). N. B. Hannay and C. P. Smyth, J. Am. Chem. SOC.,68, 17 (1946). R. Weiss, Phys. Rev., 131, 659 (1963). R. J. W. Le Fevre, Dipole Moments, Their Measurement and Application in Chemistry. Chemical Publishing Co., New York, 19?8. J. W. Smith, Electric Dipole Moments, Butterworths, London, 1955. C. J. F. Bottcher, Theory of Electric Polarisation, Elesvier, Amsterdam, 1952. S . N. Ghosh, R. Trambarulo, and W. Gordy, J . Chem. Phys., 21, 303 (1953).

Chapter

xv

IRREDUCIBLE TENSOR METHODS FOR CALCULATION OF COMPLEX SPECTRA

INTRODUCTION

803

SINGLE COUPLING NUCLEUS: REDUCED MATRIX ELEMENTS

81 2

PLURAL, UNEQUAL COUPLING

81 8

PLURAL, EQUAL, OR NEARLY EQUAL, COUPLING

82 3

N U CLEAR S PI N-S PI N I NTE R ACTIO N S

829

R E L A 1IVE I NTE N S IT1ES

831

ELECTRONIC SPIN INTERACTIONS

835

STARK EFFECT

838

I NTRO D U CTI 0 N [n this section we wish to demonstrate how the irreducible tensor method can be used to extract information on molecular interactions. Direct calculation techniques discussed previously in which wave function expansions are employed to change to a more appropriate basis give rise to particularly complicated expressions. With irreducible tensor methods, such complications are taken care of in the derivation of the relatively simple theorems associated with the theory. Irreducible tensor methods thus provide a highly efficient and powerful way to analyze hyperfine structure arising from plural nuclear coupling, as well as other molecular interactions. Since more applications of these methods are appearing, some familiarity with these techniques is required for an adequate appreciation of the literature on hyperfine interactions, as well as other areas of microwave spectroscopy. The theory of spherical or irreducible tensor operators has been discussed in detail by Rose [13, Edmonds [2], Fano and Racah [3], and Judd [4]. In this section we follow much of the expository discussion given by Cook and De Lucia [ 5 ] . Important theories will be simply quoted, and we will concentrate on their application. The reader may 'consult the previous references for derivations of these theorems. Discussions and specific applications of irreducible tensor methods to the 803

804

IRREDUCIBLE TENSOR METHODS FOR COMPLEX SPECTRA

analysis of complex spectra have been given by Curl and Kinsey [6], Thaddeus et al. [7], and others [8-441, to cite but a few (see also references cited therein). These include discussions applicable to symmetric tops [15, 19, 24, 26, 28, 29, 38, 40, 421 and molecules with unbalanced electronic angular momentum [6, 12, 20, 32, 35, 36, 391. Techniques for inclusion of the effects of identical particles in a particularly straightforward way have been given by Cederberg [17]. The theory of hyperfine interactions in molecules with internal rotation has been discussed by Heuvel and Dymanus [21], and Ribeaud et al. [43]; that of nuclear hyperfine interactions in spherical tops, by Michelot et al. [34]. By definition, the quantities Tbk)(q=-k, - k + 1,. . , , + k ) constitute an irreducible tensor of rank k (an integer) if they transform like the spherical harmonies Ybk)(O,4) under a rotation. The transformation properties of an irreducible tensor Ttk)under a rotation of the coordinate system is given by [I]

=E Tbk’D&@y)

(15.1)

Th!“’

4

where Tf) are the components defined with respect to the old axes, T:?) are the components with respect to the new or rotated coordinate system, and D$ is the (2% 1) dimensional matrix representation of the rotation operator. The D$ depend on the Euler angles (CI, p, y) that specify the relative orientation of the two frames. By inverting ( 1 5 . 1 ) we may express Tik)in terms of the components, T$). Multiplying by D;,’,(k)and summing over q’ give

+

(15.2) where the unitary character of the D(k)matrix and the relation D,;,‘(k) =D*(,W44 = ( - 1)4 -4’D‘k) -q, -4’

(15.3)

have been employed. This relation will be used to express the tensor components Tik) in terms of the components Tk(k)with respect to the rotating molecule-fixed principal inertial axis system. ) components The scalar product of two irreducible tensors Ttk)and U k with Tf) and V:) is defined as (15.4) The rule for the product of two irreducible tensor operators is specified by T:)=[Ttli) T(lz)];)

=(21+1)’12

E

(-l)-h+lz-q

4142

(15.5)

+

which yields irreducible tensors ofrank I, where lmay be equal to I I 1 2 . . . Ill - lzl. Note, in the multiplying of spherical tensors, the ranks are added vectorially

INTROD UCTI0N

805

not algebraically. The notation T('1)x T(i2)makes clear the origin of the irreducible tensor T t ) .The symbol above in brackets is a 3j symbol and is related to the Clebsch-Gordon coefficients. The 3j symbol has the property that the value is left unchanged for an even permutation of columns and is multiplied by ( - 1)'1+'2+'for an odd permutation of columns or for a sign change of all the elements of the second row. Simple algebraic formulas for some special cases are available [2] as well as tabulated numerical values [45]. They may be readily evaluated from their general definition by means of a computer algorithm. They are nonzero only if the elements of the second row sum to zero, that is, q1 +q2-q=O, and if 11, 12, I satisfy the triangular condition, that is, a triangle can be formed with the length of each side specified by a quantum number. A very useful orthogonality relation for the 3j symbols is

To apply the methods of irreducible tensors, the interaction must be written in the form of spherical tensors. In some cases, the interaction can be formulated directly in the form of spherical tensors such as the magnetic and quadrupole interactions [5, 81. In other cases, it may be convenient to formulate the interaction in Cartesian form. The general hyperfine Hamiltonian in Cartesian tensor form can be expressed as N

~ ~ r s = C ( ~ V i : Q i + I i . C i . J ) + C(Ii-Dij.Ij) i

(15.7)

i j+i

The sum is over all coupling nuclei, and the terms represent, respectively, the electric quadrupole (product of two dyadics), magnetic dipole and spin-spin interactions. This, however, presents no serious limitation to the application of irreducible tensor methods, since the elements of a Cartesian tensor can be cast into forms which transform as spherical tensor operators. The spherical unit vectors e f ) are defined as [2] eh1)=e,,

e$\ = T(l/JZ)(e,+ie,)

(15.8)

The components of a first-rank irreducible tensor T f ) in terms of the Cartesian of a vector are given by components (T,, Ty,

x)

T y ) =e f ) . T = ebl).(exTx+ e,Ty + e, T,)

(15.9)

which yields T(1)0 -z,

T "+ 1' -- - +(l/dmxkiTy)

( 15.10)

These linear combinations transform as a first-rank spherical tensor. Any interaction written in terms of two vectors can hence be readily formulated in spherical tensor notation. The electric dipole interaction with an external field would be written from (15.4) as

806


with fl)the electrical dipole moment tensor and with the components of the tensors chosen as in (15.10). A second-rank Cartesian tensor may, in general, be decomposed into three irreducible tensors: of rank 0, 1, 2. To define the irreducible tensor components in terms of the Cartesian components we first obtain the appropriate unit vectors. The spherical basis vectors may be constructed from (15.5). In particular, the spherical unit vectors obtained by combining eh',) and eh:) are given by

Employing a table of 3j symbols, one finds

To define the irreducible tensor components in terms of the Cartesian components we express T as a dyadic in terms of its Cartesian components

and with the aid of (15.8) and the relation where 1=0, 1 , 2 and q = -1, -1+1,. . ., +I, we find

It is apparent that if the Cartesian tensor is symmetric and traceless, only the second-rank irreducible tensor is nonvanishing. These relations thus allow an interaction involving first- and second-rank Cartesian tensors to be transformed to irreducible tensor form, where irreducible tensor methods may then be employed.

INTRODUCTION

807

Consider, for example, the case of the spin-rotation interaction. We may write for the interaction in Cartesian coordinates .X‘M = I. C * J

(15.18)

where all elements are referred to space-fixed axes and C is a second-rank Cartesian tensor written as a dyadic. To formulate this interaction so that the irreducible tensor techniques may be employed, we decompose C as in (15.12) and write the interaction as a sum of scalar products L

xM= 1 Nl[C(Ox

1“)

J(’)](’).

(15.19)

I=O

where the elements of I(1)and J(l)are defined by (15.10) and those of C“)by (15.17). The irreducible products [C‘O) x J(’)](’’, and so on, are obtained from (15.5). The constants N f are appropriate normalization factors such that when (15.19) is written out explicitly it is equivalent to (15.18). The normalization factors are found to be No = - 1/3’12, N1= - 1, and N 2 = - 5112/3’1z.An alternate formulation is also possible in which I(l)and J(’)are combined to give irreducible tensors of rank 0, 1, and 2 and the scalar products of these with the corresponding irreducible tensors of C are formed. The matrix element equations for irreducible tensor operators, which will be employed in the derivations of the following sections, are briefly summarized below. These theorems are taken from Edmonds [2]. For the basis functions Isjm) that are eigenfunctions of jz and j,, the Wigner-Eckart theorem states (Tj”m’lThk)(T,j, m ) = ( - l)j’-m’

(15.20) is defined as the reduced matrix element, or double-bar where (r’j’llT(k)II~j) matrix element, of T(k),it is, however, merely a number. The quantum number T , in general, may be considered a set of quantum numbers required to more clearly specify the state. This factorization theorem shows that the entire projection quantum number dependence of the matrix element is contained in the 3j symbol. A knowledge of the matrix element of any one of the components, T r ) ,allows the evaluation of the reduced matrix element. If the tensor operators T(k)and U kact, ) respectively, on subsystems 1 and 2 of a system with a subsystem 1 characterized by the angular momentum j, and subsystem 2 by the angular momentum jz, then the matrix elements of the scalar product of these two tensor operators in the coupled basis J =j, + j, are given by

where the symbol in braces is the 6j symbol, and and 5 2 represent additional pertinent quantum numbers characterizing subsystems 1 and 2. Numerical

808


values of the 6j symbol have been tabulated [45], and expressions for certain simple cases are available [2]. The value of a 6j symbol is unaffected by a permutation of the columns or by an interchange of the upper and lower elements in any two columns. They may be conveniently computed from a general formula due to Racah (see Ref. 2, p. 99). Whereas the 3j symbols are involved in the coupling of two angular momenta, the 6j symbol is associated with the coupling of three angular momenta. In particular, they are proportional to the expansion coefficients relating the representation of the coupling scheme j, + j, = j', j' + j, =J and the representation of the coupling scheme j, + j, = j', j'+j,=J [see (9.126)]. They are nonzero only if the three numbers of each row and each column satisfy the triangular condition. In applying (15.21) it is important to recognize that T(k)and U k must ) be commuting tensor operators. The relation between the reduced matrix element of a single operator in the coupled and uncoupled representation is given next. For the tensor operator T(k)acting only on the space of subsystem 1, that is, T(k)commutes with j2, the nonvanishing matrix elements are (TijiTd2J'llT(k)llT1jlTd2J) = (- 1)"

+ j z +J + k

[(2J+ 1)(2J'+1)]'''

For U k )operating only on subsystem 2, that is, Uk)commutes with j l , the nonvanishing matrix elements are (T1j17i.h

J' I I

u ( k ) I I T 1 j 1 ~ 2 j J)=(2

1)' '1 + j z + J ' + k [(2J+ 1)(25'+ 1)11/'

Let T(kl)and U(k2) be tensor operators acting on subsystem 1 and 2, respectively. The reduced matrix element of the product T ( k lx) U(kZ)of these two irreducible operators in the coupled representation, in terms of the reduced matrix elements of the individual operators in the uncoupled representation, is given by (T'T;j;T;j;J'IIITckl) x U ( k 2 ) ] ( k ) 1 ( ~ ~ 1 j l ~[(2J+ 2 j 2 J )1)(2J'+ = 1)(2k+ l)]'/'

r;

kl

x c j ; j 2 k2 "'

J'

J1 J

( ~ ' ~ ~ j ~ l ~ ~ ( ~ ~ )U~( k"2') l' l~~ ~, 2j j~2 )(15.24) ) " ' ' ~ ~

k

The symbol in braces, known as a 9j symbol, can be written in terms of the 3j or 6j symbols, and can arise when the coupling of four angular momenta are considered. The sum over the additional classifier T" is usually not required. On the other hand, for a tensor product which involves operators acting on the same subspace, jm, the reduced matrix elements are related by

INTRODUCTION

809

Such reduced matrix elements can arise, as, for example, in the application of the Wigner-Eckart theorem or (15.21). The summations in the foregoing equation, where T” represents other quantum numbers needed to characterize the state, are usually not required. Derivations of such relationships as given above, applicable to both the rotation group and the point groups, are discussed by Cederberg [17]. The matrix element of a scalar product is a special case of the more general relation of (15.24) with k l = k 2 , k = 0 . The 9j symbol with two parameters repeated and a single zero element is found to be proportional to a 6j symbol. A short list of expressions for some 3j and 6j symbols which are of particular use here is given in Table 15.1. An interesting application of the Wigner-Eckart theorem is in the evaluation of the matrix elements of the angular momentum. In particular, we have

where Jb’)= J,; Jy{ = T 1/2’I2(Jx& i J y ) are components referred to the spacefixed coordinate system. In this section the space-fixed coordinates will be designated as (x,y , z), and the body-fixed coordinates will be designated as Table 15.1 Expressions for Certain 3 j and 6 j Symbols

J

J

=(- 1)J-M

(J-M)(J+M+1)*2 ( 2 J + 2 ) ( 2 J + 1)(2J)

1

1/2

2[3M2 - J ( J + l)] [ ( 2 J + 3)(2J+ 2)(2J+ 1)(25)(2J- 1)]’’2

t s ;I

=(-l)s+l

{;

=(- 1)” [(2b - 1)2b(2b 1)(2b 2)(2b 3). (2c - 1)2C(2C 1)(2c 2)(2c+ 3)11/2

s a?

where s=a

2[b(b+l)+c(c+l)-a(a+l)] [2b(2b+ 1)(2b+2)2c(2c+ 1)(2c+2)]”* 2[3X(X- 1)-4b(b+ l)C(C+ l)]

+

+

+

+ 6 + c, X =b(b+ 1 ) + c(c+ l)-a(a + 1)

+

+

810


(x’, y’, z’). To evaluate the reduced matrix element we choose the simplest case

for the LHS of the foregoing equation (J’,

I Jb”I

IJ,IJ, MJ)=M d J d M , M,,

J , MJ) =(J’,

(15.27)

in units of h. Evaluation of the 3j symbol (q=O) from Table 15.1 gives MJ [J(J+ 1)(2J+ l ) ] l ’ Z and hence

( J l p l l J ) = [J(J+ 1)(2J+1)]1’2

(15.28)

Thus, in general

+ +

Note - M i q MJ =O. Replacing the 3j symbol by its equivalent yields the well-known matrix elements ( J , MJIJb”IJ, MJ)=MJ

(15.30)

Note in the foregoing expression that the upper (or lower) sign applies throughout. Matrix elements for Jx and J, may be readily found from these since

1

(I))

( J ( 3- J + 1

and

J,=-$J!!\+J‘:l) i

(15.32)

This yields (in units of h )

(J,M + 1IJxIJ, M ) = + [ ( J - M ) ( J + M + l)]l’Z

(15.33)

(J,M - lIJ,IJ, M=I [ ( J + M ) ( J - M +

(15.34)

2

l)]l’2

Likewise, the matrix elements of the angular momentum J$’) referred to the body-fixed axis in the symmetric rotor basis ( J K M J )may be obtained by use of (15.1) (15.35) hence,

( K ’ J ’ M ;J~! : ) ~ K J M= ~ )C ( K ’ J ‘ M ;J;’)JK’J’M[;) ~ (K’J’M[;~D:;),~ KJMj) qM$

(15.36) since from (15.29) Ji’)is diagonal in K and J. The symmetric rotor functions are related to the D matrix as follows [5]

INTRODUCTION

81 1

IC/J K M , (ab?)=( - l ) M J- K [(2J+ 1 ) / 8 ~ ’ ]”’0- M,, - K ‘ J ’ ( ~ b y ) (15.37) The matrix elements of Ob”,‘ in the symmetric rotor basis may now be obtained from the above identification and are given by [46] (K’J’M>ID$$K J M j ) =( - l ) M J- K [ ( 2J + 1)(2J‘ + l ) ]‘jZ

( 15.38) where K is the projection quantum number of the rotational angular momentum J in the molecule-fixed frame. Therefore, (K’J’M; 1 J:: ’1 K J M j ) =( - 1 )’[(2J + 1)(2J’ + 1)2J’( J’ + 1 )] ‘I’ (15.39) with t = J ’ - M ; + M J - K + l and k = l and two of the 3j symbols have been rearranged. Using the orthogonality property of the 3j symbols, (15.6), we obtain (K’JMjIJ$’’lKJMj)=(- 1)J+’-K[(2J+l ) J ( J + l)]”’ X ( J

K

-4‘

( 15.40)

-K’

Evaluation of this expression gives the matrix elements of the angular moment components in the molecule-fixed axis system for the symmetric rotor basis

( KJ M I ~J

~

K J M j )=K 1 ( K T ~ J M J ( J $ ’ , ’ I K J M ~T) = [ ( J f K ) ( J T K + 1)]’/’

Jz

(15.41) (15.42)

Note that the raising and lowering character of the operators is reversed in the two frames, as found in Chapter 11. The components J,, and J,. are defined in terms of J$’) as in (15.32),and this yields for the components referred to the molecule-fixed axis system ( K - 1, J, MjIJ,,IK, J, M J ) = f [ ( J + K ) ( J - K

+ l)]’’’

( K + 1 , J, M j I J J K , J , M J ) = ~ [ ( J - K ) ( J + K + ~ ) ] ” ’ 2

( 15.43)

(15.44)

The phase choice in this section is consistent with both J , and J,. as real and positive which is consistent with Condon and Shortley [47], but is opposite to the choice used throughout other portions of this text. In application of the techniques discussed in this chapter it is important to use a consistent phase choice throughout.

812


2 SINGLE COUPLING NUCLEUS: REDUCED MATRIX ELEMENTS By way of introducing the definitions of the reduced matrix elements for the quadrupole and magnetic interaction, we consider first the case of one coupling nucleus. This case has already been discussed in Chapter IX, Section 4; however, we formulate the problem here in terms of irreducible tensors. The quadrupole and magnetic dipole interactions may be developed in terms of irreducible tensor operators [ 5 , 81. Since the Cartesian tensor for the quadrupole interaction is symmetric and traceless, only second-rank irreducible tensors are needed to characterize this interaction.

2hfs- - V ( 2 ) . Q ( 2 ) +

m(’).p(l)

(15.45)

where V2)is the electric field gradient tensor, Q ( 2 ) the nuclear quadrupole tensor, m(’) the magnetic field tensor, and p(’) the nuclear magnetic dipole tensor. The basis is taken as IrJZFMF)where z has the usual meaning for an asymmetric rotor and equals K for a symmetric top. From (15.21) for the matrix elements of a scalar product, we find for the quadrupole interaction

(15.46)

All terms off-diagonal in F vanish, and the matrix elements are diagonal and independent of the projection quantum numbers M F . In order for the preceding relation to be useful, the reduced matrix element must be evaluated in terms of the usual spectroscopic constants. This is easily accomplished by application of the Wigner-Eckart theorem. The quantum mechanical observable, Q, termed the nuclear quadrupole moment, is defined via the relation (1,M I = ZlQb2)1Z, M I = Z)=i(eQ)

(15.47)

where I Z, M I )are the nuclear spin basis functions and where M I = 1corresponds to the nuclear state of maximum alignment along z. Application of the WignerEckart theorem, (15.20) to (15.47) gives therefore (15.48) Expressing the 3j symbol in terms of its algebraic equivalent from Table 15.1, we find for the reduced matrix element of the nuclear quadrupole tensor (1l l Q ( z ) l l ~

1=eQfU 1

(15.49)

where f(Z)= [(2Z+ 1)(21+ 2)(2Z+ 3)/8Z(2Z- 1)]1’2

(15.50)


813

The reduced matrix element of the field gradient tensor V2)is evaluated in a similar manner. The electric field gradient coupling constant 45.5 is defined as (J’ 2 J ) q J , J = ( T ’ , J’, M;= JI(d2V/dZ2)01T,J, M J = J )

(15.51)

and (T’,

J , M i = JIV~”(T,J, M J = J)=iq,,J

( 1 5.52)

where ~ T J M Jare ) the rigid rotor basis functions (e.g., the symmetric rotor or asymmetric rotor functions) with M J the projection quantum number in the space-fixed frame. It should be noted that in general 45.3 depends on the quantum number T. The Wigner-Eckart theorem yields

Thus where for J‘ 2 J

=[ (

f (J’)

2 J 0

“)I- ’

( 15.55)

-J

or explicitly

1 1’

(2J+ 1)(25+2)(2J+3)

li2

8J(2J- 1 ) (25+2)(2J+3)(2J+4)

1/2

f(J‘)=[

485 f (I)= - r J ( 2J 1 )(2 J 2)] 48(J- 1 )

+

+

l2

1

(2J+3)(2J+4)(2J+5)

f w=[

48 [2J(2J+ 1)(2J- 1)]’/’ f (J’)= 48

‘j2

J’ = J, J’=J+l, J’=J-1, J’ = J + 2, J’=J-2

(15.56)

where for completeness we have also included the case where J’ < J for which M ; = M J = J in the definition of 9J.J. In application to specific molecules the quantity qJ,J,which is defined with respect to axes fixed in space, is conveniently re-expressed in terms of the principal inertial axis system (x’, y’, z’) fixed in the molecule, since as previously pointed out, the field gradients with respect to the molecule-fixed axis system are constants independent of the rotational state. We have from the definition of 45.5 and (15.2)

814


~ . J , J = ~ ( T ‘ JI/b2’lTJJ) ‘JI -2 ( - ~ ) - ~ ’ v ~ ! ~ ) ( T ’ J ’ J I D ~ ~ ) - ~ , ~ T J(1J5.57) )

1 q‘

In the symmetric-top basis we may write from (15.38) qJ,J=( - 1 ) J - K

-9‘

X(J

[(2J+ 1)(2J’+ 1)]”2 J‘)2V;!2)

J‘)(J

J

0

K

-J

-K’

q‘

+

=( - 1)J-K-4‘[(2J+1)(2J‘ 1)]”2[2f(J‘)]-

x( K J

q’

-K’ J’)2V2?’

(15.58)

where q’= K ‘ - K , J ’ 2 J. Note that the latter expression for qJ,.Jalso applies to the case J’< J with f ( J ’ ) as given in (15.56). Usually, the interaction energy is defined as V ( 2 ) * Q ( 2in ) spherical tensor notation and as i V : Q in the Cartesian notation. For both expressions to be equivalent, a factor of 6-’’2 must be included in the definition of the components of Q(2) and V2).The irreducible field gradient tensor components V:?) are thus related to the Cartesian field gradient tensor elements as follows 2Vy’ = 2 V y i = T(Jz/Js)( VXTZ, iv,.,,) 2V’i2j=6- ”2(Vx,x,-VYty,f2iVxty,)

*

( 1 5.59)

In the usual spectroscopic notation, the molecular constants eQVxCx,, and so on, are the quadrupole coupling constants denoted by xxCx.,and so on. For a symmetric top with the coupling atom on the symmetry axis, only one of the V:?) is nonvanishing, viz., 2V$’)= = ( ~ 3 ~ V / a z=q, ’ ~ )and the coupling constant for the rotational state J, K is qJJ=(-

1)J-K(2J+I ) (

J K

2 0

J -K

)(

J 2 J O

’)

-J

q

(15.60)

which is the usual definition of qJJwhere only diagonal elements in J are considered, see (9.88). The above is also applicable to a linear rotor where K =0, and this gives (9.81). For an asymmetric rotor if only diagonal elements in J are to be considered, it is more convenient to return to the definition of qJJ and evaluate it in the asymmetric rotor basis as was done in Chapter IX, Section 4. Otherwise, the symmetric top basis may be conveniently employed, and (15.58) applies. Alternately, (15.57) may be evaluated in the asymmetric rotor basis [see also (9.69)]. Usually the interaction energy is sufficiently small that the rotational quantum numbers may be considered good quantum numbers, and only

815


matrix elements diagonal in J need be considered. This so-called first-order effect is by far the most important although for quadrupole coupling with nuclei of large Q second-order effects can be important, as discussed in Chapter IX, Section 6. In what follows, only the matrix elements diagonal in J will be considered for the magnetic coupling, while for quadrupole coupling general expressions will be given. For the magnetic spin-rotation interaction we find from (15.21)

We now consider the evaluation of the reduced matrix element. Conventionally the observable,

( 4 Mr=IlpZlL M r = I ) is defined as the magnetic moment, g I P I I, and since pz =pd') we have ( I , MI=Il/&"lZ, M I = I ) = p I = g I P I I

(15.62)

where P I is the nuclear magneton and is the appropriate nuclear g factor. For the molecular magnetic field tensor we may employ the following decomposition theorem for first rank tensors [l]

which relates the matrix elements of Ti') to those of the projection of T(') along the angular momentum J(').Applying this to mb') gives (r JJI rnb'

)ITJJ)=(7JJI 4,' )(J('). m''))I r J J ) / J (J + 1)

The latter factor may be interpreted as the average magnetic field along J ( H J )=(tJJIJ(".m(')lrJJ)/[J(J+l)]'"

(15.65)

and we have ( T J M J=

JJ~~~)ITJMJ

=J)= { J / [ J ( J + 1)I '") ( H J )

(15.66)

I t is recognized that the left-hand side of this equation is the average field in the z direction, and the above is a relation that is readily derived from the vector model (see Chapter IX, Section 7). The reduced matrix elements can now be found by application of the Wigner-Eckart theorem to (15.62) and (15.66) which yields

'

( I IIP' )II I 1= S I

PI 4 1 )

(15.67)

where h ( Z ) = [ I ( I + 1)(21+ 1)]1'2

(15.68)

816


and similarly, where h ( J ) = [ J ( J + 1)(2J+ 1)]"2

(15.70)

and the spectroscopic constant is defined by (15.71) Compare this with (9.137) and (9.142).The C J , ,depends on the rotational state J , T and is best expressed in terms of the elements of the magnetic coupling tensor with respect to the principal inertial axis system. The derivation now follows closely that discussed in Chapter IX, Section 7. Referring the elements of J(')and m(') to the molecule-fixed x'y'z' frame, we can write the scalar product in terms of the Cartesian components of the tensors J(1).

m ( ' ) = x (- 1)4'Ji'J')m'i'), 4 4'

= Jim; I(g =XI,

y', z')

(15.72)

9

For molecules in singlet electronic states the magnetic field arises from the rotation of the molecule and the field components along the molecule-fixed axes may be taken as proportional to the angular velocity, that is, m$=(grPr)-'

1 B,g,w$, 9'

=(grPr)-'x

c;g,J$,

(15.73)

9'

since w$= J;/Zg.Combining this with (15.72) and (15.65) we have (noting that, in general, only the diagonal elements of C contribute to the spectrum in first order) (15.74) CJ,T=[J(J+~)]-' c$g(JLz)

1 9

where J i are the angular momentum components with respect to the moleculefixed principal inertial axis system and where the average is taken over the rotational state J,T. The C;,, are the coupling constants with respect to the principal axes and are constants independent of the rotational state. This relation agrees with that given previously in (9.143).The quantity CggZg/gI=Bgg is invariant under isotopic substitution. It is of interest to consider the evaluation of the matrix elements o f X , when defined by (15.18). If for simplicity we consider one coupling nuclei, the matrix elements (diagonal in J)of the terms in (15.19)are from (15.21): I

(T'JZFI[C(') x J(')](')*I ( ' ) ~ T J I F ) =( 1)J + I + F il

J

'

11

x (o'JII[C(')x J(')](')I~TJ)(Z~~Z(')~~Z) (15.75)


817

The reduced matrix element (Z~~Z(1)~~Z)=[Z(Z+l)(2Z+l)]1'2=h(Z) is readily obtained from the Wigner-Eckart theorem. The reduced matrix element of the tensor product which involves operators acting on the same subspace is given by (15.25) 1

(15.76)

J As before

Collecting terms we have finally

This may be compared to (15.85). Here (15.79) The (T'JllC(i)llTJ)are the spectroscopic constants to be evaluated. For a symmetric rotor basis ( T = K ) we have

(15.80) where q' = K' - K and C$I) are the components with respect to the moleculefixed axes and are defined as in (15.17). These are the molecular constants independent of the rotational state. Equation 15.80 is obtained by use of (15.20) to rewrite the reduced matrix element in terms of an ordinary matrix element, followed by transformation of the C ( * )to principal axes by (15.1), and finally evaluation of the matrix elements in the J K M basis by (15.38). For a symmetric top with coupling atom on the symmetry axis (z') the off-diagonal elements C x f y fand , so on, vanish and C,,,, = Thus the nonvanishing elements are NoCb'O)=&y

+ 2C,!,!)

N2C$Z'= - [(10)"2/3](C,.,, - C,,)

(15.81) (15.82)

and hence in (15.80) q'=O, K ' = K . Evaluating (15.80) we find CJ= C,,xr

+ (cz,~, - Cx'x')[K2/J(J+I)]

(15.83)

which is equivalent to (9.158). If the previous definition of the reduced matrix elements (15.49) and (15.54)

818

IRREDUCIBLE TENSOR METHODS

FOR COMPLEX SPECTRA

are substituted into (15.46), the matrix element for the electric quadrupole interaction is

where t = J+ I + F, and where (eQ4J.J) is the quadrupole coupling constant for the rotational state JT.The quantities f(J’)and f ( Z )are specified by (15.56) and (15.50), respectively. For the magnetic interaction (15.61), (15.67), and (15.69) yield

where r = J + I f F , and h ( J ) and h(Z)are specified by (15.70)and (15.68),respectively. Equations (15.84) and (15.85) yield the usual results for the energy levels ( J = S ) with replacement of the 6j symbol by its algebraic equivalent. See Table 15.1. In general, once a particular value of J , I and F is specified, the 6 j symbol may be evaluated and hence the corresponding matrix element. This form is particularly convenient because of the ease with which the 6j and 3j symbols may be evaluated by machine computation.

3


We now consider explicitly the derivation of the matrix elements for the case of plural nuclear coupling. For N nuclei with spins Zl, Z2,. . . ZL, . . . ZN and different coupling energies, it is convenient to employ the coupling scheme J + 1, =F1 F1+12=F2

FL-1+ IL=FL FaV-l+IN=FN~F

(15.86)

where J is the rotational angular momentum and where the nuclear spins are coupled in order of decreasing coupling energy. The basis functions for this coupling scheme are denoted by ITJZ1FI.. . ZLFL.. . ZNFNMF) where T represents any additional quantum numbers, besides the rotational momentum quantum numbers J and M J , that are necessary to specify the unperturbed rotational state. To be specific, we consider the case of three coupling nuclei. The Hamiltonian is 3

YiDhss=

1 V(i).Q(i)+ i= 1

3

m(i).cl(i)

(15.87)

i= 1

Hereafter the explicit rank designation will be omitted. As indicated by the


819

foregoing equation, the matrix elements of%,,,-, will be composed of a sum of contributions from nucleus 1, 2, and 3. We wish to show first that the electric or magnetic interaction for the Lth nucleus does not depend on the quantum numbers ZL+ l F ~ + . . . F N for the coupling scheme of (15.86). For convenience we consider the case of two nuclei. For nucleus 1 we have the following matrix element to evaluate:

(T’J’I1Fi12F’Mklx(1)l7JI1F1Z2FMF) where%( 1) designates, for example, the electric quadrupole or magnetic dipole interaction for nucleus 1. Applying the Wigner-Eckart theorem we have (T’J‘ZlF;

12F’M;I %(1)lTJI1 F1 IZFkfF)=bhfpM~, 6FFf(2F+ 1)- 1’2 X ( T f J ’ 1 1 F ; I 2 F ’ I ( &(1)11TJI1F112F)

(15.88)

where the 3j symbol has been evaluated noting that X(1)is a scalar and hence k=q=O. The reduced matrix element may be reduced by use of (15.22) since X(1) does not operate on Z2 : F ; Z, FI p?( l ) l l T J Z 1 F1 I2 F ) =6~,,;(2F+ 1)[(2F+ 1)(2F1 1)I-l” (t’JfZiF;IIX(l)II7JZiFi) (15.89) (T’J’Ii Fi MF; I x(1) JIi I Fi MF ,) = 6 F F i 6MF MFgl (217 where we again employ the Wigner-Eckart theorem. Therefore we may write (T’J’Z1

+

(T’J’Zi

+

F ; 1 2 F’M; I x(l)lrJI1F1I2 FMF)= 6 M p M i d F F ’ d F , F ; ~ M ~ , M P , x ( T ’ J Z ~F ; Mk, I x(l)lsJI1F1 MF,)

(15.90)

which shows the matrix elements of Y(1) depend only on the quantum numbers rJI1F1.In general, the matrix elements of X ( L )are diagonal in all quantum numbers FLZL+ . . . FN and depend only on the quantum numbers T J I 1F . . . F L . Thus for nucleus 1 we have already discussed the matrix elements; from (15.84) with the appropriate change in notation we have

where t l = J +Zl+F1, and where (eQqJ?J),refers to nucleus 1. The quantities f ( J ’ ) and f ( Z l ) are specified by (15.56) and (15.50), respectively. For the quadrupole coupling of nucleus 2, (15.21) yields, with J = FZ, j 2 = Z2, jl z F l , and T~ =sJZ1, ( T ’ r Z 1 F ; I 2 F 2 IV(2). Q(2)17JZ1F 1 Z 2 F 2 )

The reduced matrix element of the field gradient of nucleus 2 may be reduced to the desired form by application of (15.22) since V(2) commutes with I,. This gives

820


The contribution that the rank (k)makes to the phase factor in (15.22) or (15.23) can obviously be omitted when the rank is even. The reduced matrix element on the right-hand side of this equation is given by (15.54). Therefore, taking cognizance of (15.49), (15.54), and (15.93), we may write for (15.92) for nucleus 2

(T’J’Z~ F ; Z ~ F ~ ~ V ( ~ ) . Q ( ~Z )2 F~z T ) JZ~F~

(15.94)

+ + +

+

where t 2 = J’ Zl Z2 2F F2. If there were only two coupling nuclei involved, one would stop here, and F 2 would be replaced by F . For the quadrupole coupling of nucleus 3, (15.21) yields

(T’YZI F ;I2 F ; 13 F3 I V(3). Q(3)IT J ZF~ 1ZzFz 13 F3)

(15.95) Successive application of (15.22) yields for the reduced matrix elements (7’11 sF ; 1 2 F;I

x [(2F2

I 7/(3)(I TJI1

+ 1)(2F; + l ) ] l / Z

F z ) =( -

{fi i:

+I2

fF2

~ } ( T ~ J ’ Z ~ F ~ ~ ~ V ( ~ )(1, 5.96) ~~JZ~

and

Combination of (15.95)-(15.97) and the definitions of the reduced matrix elements gives for the matrix elements (T’J’I1 F Z2 F; Z3 F3 IV(3)* Q(3))7511 F 1 1 2 F z Z3 F3) =( - 1)‘3(eQq~,~)3f(J’)f(Z3)[(2Fl1)(2F; 1)(2F2+ 1)(2F;

+

+ +

+

+

+ 1)]1’2

where t3 =J’+ Il Z2 Z3 +F1+ F ; +2F2 F3. A similar procedure for the magnetic dipole interaction operator yields the


821

following matrix elements for the three coupling nuclei,

+

+

xh(J)h(I2)[(2Fi 1)(2F; I)]”’

l{ ;

‘J;

:I{:2

2

;:}

(15.100)

(sJZIF;I z F l 1 3 F 3 ( m ( 3 ) . ~ ( 3 ) / s J 112F213F3)=(-1)Y3(C~,r)3 1F1 x h(J)h(Z3)[(2Fl+ 1)(2F\ 1)(2F2 1)(2F; I)]”’

+

+

+

+

+

+

+ +

+

where r l = J + ZI F1, r2 = J + Il I , +2F1 + F z 1, r 3 = J + Zl Zz Z3 +F1 F ; +2F2+ F3, and where h ( J ) and h(Zi) are specified by (15.70) and (15.68), respectively. If two coupling nuclei are present, (15.99) and (15.100) apply, with F z replaced by F. The procedure just outlined may obviously be extended to any number of nuclei. The matrix elements of the electric quadrupole interaction for the Lth nucleus may be written in the form [5] (t’FAF;. . . Ft-lFLIV(L).Q(L)l~FOF1.. .FL-~FL)

(15.102) where the phase factor is determined by L- 1

t=

1 (F[- 1 + Zi+ Fi)+(FL- + ZL+ FL) 1

i= 1

and Fo = J, Fb = S,and FN= F. As before, f ( J ’ ) is given by (15.56) where the expression to be used depends on the value of J’. Likewise, f ( I L ) is given by (15.50) where the spin of the Lth nucleus is to be substituted. The matrix elements of the magnetic dipole interaction may be written as [5] (rFbF; . . . F L - l F L l m ( L ) . ~ ( L ) l s F o F. l. . F L - I F L )

where the phase factor is determined by

822

IRREDUCIBLE TENSOR METHODS FOR COMPLEX SPECTRA L+ 1

~=(L-I)+

(FI-1+Zi+Fi)+(FL-1+zL+FL) i= 1

and F o =FA = J , and F N= F . The factors h ( J ) and h(ZL)are given, respectively, by (15.70) and (15.68). For both (15.102) and (15.103) in the special case of L = l , the product n[I1l2{ } is set equal to one. The diagonal elements yield the hyperfine energy levels to a first-order approximation. Exact energy levels are obtained by construction of the hyperfine energy matrix, which may be factored into blocks of different F , and subsequent diagonalization. For asymmetric rotors if the symmetric rotor basis is employed, the complete energy matrix +.Ye,, must be considered. This amounts to inclusion of the matrix elements of the rotational Hamiltonian in the symmetric-top basis along with those given for the hyperfine Hamiltonian in the coupled basis. A consistent phase choice for the matrix elements should be employed. In this regard, the rigid rotor Hamiltonian can be conveniently written as

xr=@,, + By.)J(J+ 1)+ [B,. - +(B,f+ B y ) ]{ Jb'

- b(J;"1+ P I ) }

(15.104)

with b = ( B y ,- Bx.)/(2B,.- B,. - By.). The study of DCECCI by Tack and Kukolich [41] provides a simple example of two nuclei with unequal couplings J + LJ = F1, F, ID= F. All the hyperfine components of the J=O+1 transition are shown in Fig. 15.1. The

+

1

1

1

-40

1

I

-30 -20

1

-20

1

1

1

0

I

r

-10

0

1

1

20

I

10

I

40kHZ

I

20

30

4 0 kHZ

-5

0

5 kHZ

Fig. 15.1 Molecular beam maser resolution of all hyperfine components of the 1-0 transitions of 3sC1C=CD. The components are labeled for the J = l level since all hyperfine matrix elements vanish for J =O to first order. ( a ) Recorder tracing of F , =+components. Frequencies in kHz relative to 10,358,017 kHz. (b) F 1=+ components. Frequencies relative to 10,377,921 kHz. ( c ) F ,=$ components. Frequencies relative to 10,393,892 kHz. From Tack and Kukolich [41].

823


spectrum was analyzed with a hyperfine Hamiltonian consisting of the deuterium quadrupole interaction and chlorine quadrupole and 35Cl spinrotation interaction. Second-order quadrupole corrections were calculated for chlorine by perturbation methods (see, e.g., Appendix J) and added to the first-order energies from matrix diagonalization. The coupling constants obtained are: eQq(35C1)=79,739.5(10)kHz, eQq(D)=202.5(15) kHz, and C,( 'CI) = 1.3(1) kHz.

4


When the coupling energies are the same or nearly the same, the coupling scheme

(15.105) is usually employed. This scheme is appropriate in cases such as NaCl where the nuclei have nearly identical quadrupole coupling [13]. It is also particularly convenient when the coupling nuclei are identical and may be exchanged by symmetry operations since some factoring of the energy matrix is then possible. The basis functions for the foregoing coupling scheme are denoted by lTJIl1292 . . . I L ~ L. .. 1 ~ 9 , F h f ~ ) It is convenient for this representation to start with the general Nnuclei problem. From (15.21) with T ~ = T , ~ ~5 2= =J1,1 1 2 9 2 , J 2 = 9 Nand J = F , the matrix element for the electric quadrupole interaction of the Lth nucleus is (7'1'11 129;

. . . 1 ~ 9. ;. . Z N ~ ~ F I V ( L ) . Q ( L ) ~1292 T J Z .~. .' 1 ~ 9. . I~N.~ N F )

(15.106) The matrix element for the magnetic dipole interaction of the Lth nucleus can also be written as (TJ11129;. . . 1 ~ 9 ;. .. INShF(m(L).p(L)ITJI11292... 1 ~ 9. .~ 1 ~.9 , F )

-(-

1)J+f h + F

{:

7 ij

9hlI~(L)II111292...9N)

( ~ J I I ~ ( ~ ) l ( ~ J ) ( ~ l ~ Z ~ ; .

(15.107) The reduced matrix element of the field tensors which appears in both (15.106)

824


and (15.107)has been given previously, and the problem reduces to the evaluation of the reduced matrix element of the nuclear quadrupole tensor operator and the reduced matrix element of the magnetic dipole tensor operator. Evaluation of these quantities depends on the number of coupling nuclei in the molecule. We consider explicitly the case of three coupling nuclei. The reduced matrix element of the nuclear quadrupole tensor operator of nucleus 1, (111 2 9 ; Z ~ ~ ~ I I Q (1~2 )3 I2 1I 3~9I3 ) is brought to the desired form by successive application of (15.22), and the matrix elements are found to be given by (r’J’Zl Z 2 9 ; Z39;FIV(1). Q(I)ITJZ~ Z2f2 Z 3 9 3 F )

+

+

+

+

=(- l)“(eQq~,~)lf(J’)f(Z1)[(2s2 1)(29; 1)(293 1)(29t; 1)I1l2

+ +

with t l = J +Zl Z2 Z3 +9, +Si+ f 3 +9; +F. Similarly, for nucleus 2, successive application of (15.22) and (15.23) bring the reduced matrix element of the quadrupole tensor operator in (15.106) to the desired form. This procedure gives as a final result Z3s;FIV(2)* Q ( ~ ) ( T1 2J9Z2 ~ Z393F) =( - l)”(eQq~,~)2f(J’)f(Z~)[(29~ 1)(29; 1)(2Y3 1)(29j 1)]’12

(T’rZIZ29;

+

+

+

+

+ +

with t 2 = J +Zl Z2 Z3 + 2 9 ; +Y3 +Si+ F . In the case of nucleus 3, (15.23) is used because 92commutes with Q(3). This yields specifically (I11 2 9 2 1

3 9 ; I IQ(3)II I1 1 2 9 2 Z 3 9 3 ) = ( - 1)”

+ +

+

+

I’

+

”;

with t 3 = J+ Z3 92 2 9 ; F. Note that the matrix elements are diagonal in S 2 ,as follows from (15.110),in contrast to the results for nucleus 1 and 2. The matrix elements for the magnetic dipole interactions are summarized below

825


+ 1)]1'2 {: + 2 9 ; + F + 1. with r3 = Jf Z3 +9, +

9J'

x [(W3 1)(29;

'}{

9 3

9l3 3

.a' 13

91 z}

(15.114)

Generalizing the preceding discussion to the case of N coupling nuclei ( N > l), we have for the matrix elements of the quadrupole interaction for the Lth coupling nucleus [5]

(t'JZ1Z 2 9 2

. . . Z L 9 L . . . ZN9EJFIV(L)-Q(L)(rJIl Z2.a2 . . . ZLYL.. . Z N 9 N F )

=(- l)'(eQq,,J)tf(J')f(ZL)

(15.115) where the phase factor is determined by

+n

N- 1

r=J+F+IN+YL-I

(Zi+9i+l

+9;+1)+29;

i=L

and for the magnetic interaction [SI (rJzl Z2Y2. . . Z L 9 L . . . ZN9kFlm(L)-p(L)lrJZ11292 ...ZL9L. ..INSNF)

(15.116)

826


where r=J+ F+

IN+9L-

1

+

N- 1

i=L

(1 +

si+1 +9:+ I)+

( 2 9 i - k 1)

In the foregoing expressions 9,=0, 9,=Si=Il, 9N=9. For the product factor in (15.115) and (15.116), if the upper product limit is less than the initial value of i, this factor is to be set equal to one. Likewise the summation term in the expressions for t and r is to be set equal to zero if the above condition holds. When L = l , the factorsinlargesquare bracketsin(15.115)and(15.116)areto be set equal to one; the last term ( 2 9 i )in the expression for t and the last r ( 2 9 ; + 1) in that for r is to be set equal to zero. Note that the matrix elements for a given L (except L = 1 , 2 ) are diagonal in all 4’s below in the coupling sequence [see (15.105)]. Usually only diagonal elements in J need be considered. Each state may be specified by a given value of J and F . Off-diagonal elements can occur for the intermediate quantum numbers Si, and these intermediate quantum numbers can be used to further classify the levels. The complexity introduced with plural nuclear quadrupole coupling may be illustrated by cyanogen azide (N,CN) with four coupling nuclei, studied by Blackman et al. [22]. The computed and experimental line shapes are shown in Fig. 15.2 for the 111-’212 transition. This transition has 864 hyperfine components. It is difficult to unambiguously assign coupling constants to particular nitrogen atoms in the azide groups. Solution of this problem, as well as considerable simplification on the spectrum, can be obtained, however, by study of 15Nisotopically substituted species of cyanogen azide. Some of the simplification possible when there are identical nuclei in the molecule may be illustrated by specialization of the general expression (15.115) to two coupling nuclei. In particular, for the quadrupole interaction we find (?./’I1

Z29’FlV(l)*Q(l ) l T J Z 1 I 2 9 F ) =( - 1)”(eQqJfJ)1

with t 2 = J + Z l + 1 2 + 2 9 ’ + F . Here, 1 1 + 1 2 = f , f + J = F . A study, for example, of (15.117) and (15.118) readily yields an interesting and well-known result when Il = I 2 and (eQqJ,J)l= ( & J , J ) ~ . Adding the contributions of nucleus 1 and 2, one finds that the elements of the quadrupole coupling matrix may be written as a factor times [( - 1)”’+ (- l)”]. Hence, it follows that matrix elements of the type (9f 119) vanish, while those of the type (919) and ( 9 + 2 1 9 ) are nonvanishing, and the matrices for a given F thus separate into two submatrices of even and odd 9.This is virtually true for


-24

-*O

lb

827

10

2o Nm Fig. 15.2 Comparison of experimental and computed line shape for the 1 -'212 transition of cyanogen azide. The quadrupole coupling constants were obtained by comparing the experimental spectral traces with computer simulated traces. The coupling constants found are: x.,(1)=4.82, X b b ( l ) = -0.70; Xya(2)= -0.85, ~bb(2)=0.70; xoa(3)= -0.75, xbb(3)'1.55; x,,(4)= -2.27,Xbb(4)=1.15, where the atom designations indicate the nitrogen position [N(3)-N(2)-N(l)-CN(4)]. From Blackman et al. [22].

NaCl [13] with nonequivalent coupling nuclei and is the case for molecules such as C0Cl2, D2S, where two identical quadrupolar nuclei are involved which occupy identical molecular sites. For a given J,the submatrices are conveniently grouped into matrices for a given F with elements within a submatrix labeled by values of 9. For the case of C0Cl2(ZI= Zz=$)the possible values of the quantum numbers are: 9 = 0 , F = J ; 9 = 1 , F = J , J+1; 9 = 2 , F = J , J 2 1 , J + 2 ; 4 = 3 , F = J, J& 1, J& 2, J 2 3. There are thus seven submatrices (without factoring) corresponding to the various values of F. Symmetry considerations provide further restrictions, and only certain values of 9 are allowed for a given rotational state JK , K I . For C0Cl2, where the rotation C; interchanges identical chlorine nuclei (Fermions), the total wave function must be antisymmetric with respect to this symmetry operation. A total of (2Z1 1)(2Z2 1)= 16 spin functions can be constructed. To divide the possible values of 9 into symmetric or antisymmetric spin functions, we note that the lowest 9 state is symmetric when Zl and Z2 are integral or antisymmetric when Zl and Z2 are half-integral. The remaining symmetries alternate with increasing values of 9.Thus, for ZI= I 2 =$, the 9 = 0 (1 spin function) and 9 = 2 (5 spin functions) give antisymmetric functions, while 9 = 1 (3 spin functions) and 9 = 3 (7 spin functions) give symmetric functions. Consider a totally symmetric electronic-vibrational state. For symmetric rotational levels A(ee),&(oo), only antisymmetric nuclear states (9=0,2) exist, while for the antisymmetric rotational levels &(eo), BJoe),only symmetric nuclear states (9= 1,3) exist. Further discussions for the cases Z=$ and $ are available [48,49]. As a further specific example we may consider DzS, where the hyperfine structure of the llo-+loltransition has been studied [16]. The symmetry axis is the b axis, and the llo(Bc)and lol(Ba)levels are antisymmetric with respect to exchange of the two identical deuterium nuclei (Il = Z2 = 1, Bosons). Thus

+

+

828


since 9 =O (symmetric), 1 (antisymmetric) and 2 (symmetric), 4 can take only the value 1 for these rotational levels in order for the total wave function to be symmetric. Specializing (15.117) and (15.118) to the case J = l , Zl= 1 2 = Y = 1 with (eQqJ)l =(eQqJ)2= eQqJ, one obtains, for the electric quadrupole energy ( 15.119)

with F=O, 1, 2. Similar simplified expressions may be obtained for the spinrotation and spin-spin interactions [16] (15.120) 1 1

(15.121)

It may be noted that the expressions for the electric quadrupole and magnetic dipole energies are equivalent to that for a single spin 1 nucleus except for the sign of the quadrupole energy. The calculated l o l + l o ltransition [28] of D 2 0 , which is like that for D2S, is compared with the observed pattern in Fig. 15.3. The hyperfine interaction by three identical nuclei in symmetric tops is also reduced by the symmetry of the rotational wave function, but its calculation is more involved than that for two nuclei, described previously. The reader is directed to the references cited previously. The two coupling schemes considered in (15.86) and (15.105) are equivalent in that they lead to the same hyperfine structure, and one coupling scheme or coupling order does not offer any great advantage over the other (excluding symmetry considerations) if the energy matrix is diagonalized. The choice of one coupling scheme over the other rests on the first-order approximation being significantly better in one scheme than in the other so as not to require matrix diagonalization. To this end, other coupling schemes, which are mixtures of the two considered, may prove useful. For example, if two nuclei (spins Z2 and I,) have comparable couplings which are smaller than that of a third nucleus (spin 11),the coupling scheme J + I 1=F1, Z2 + Z 3 = 9 , 9 + F 1 = F may prove more convenient. This is particularly true if I 2 and Z3 arise from identical particles. Symmetry restrictions will then limit the possible values of 4 associated with a given rotational level and these restrictions can more easily be introduced into the matrix elements. The matrix elements of%,,,, in this representation may be readily evaluated using the procedures already discussed; that for nucleus (1) is, for example, given by (15.84)and (15.85).For nucleus 2 and 3 we would have, for example,

NUCLEAR SPIN-SPIN INTERACTIONS

829

A

Fig. 15.3 Theoretical hyperfine pattern (lower figure) of the 1 l o l transition of D 2 0 compared with the observed pattern (upper figure). The structure is due to the magnetic and quadrupole coupling by the two identical D nuclei. It was calculated by irreducible tensor methods and observed with a molecular beam maser. The transition falls at the submillimeter wave frequency of 317 GHz. From Garvey and De Lucia [28].

with t=2F1+9'+F+J'+Z1 and L = 2 or 3. The reduction of the reduced matrix element follows from (15.22) for L = 2 and (15.23) for L=3. Symmetry restrictions for identical particles can be conveniently introduced at this stage in evaluation of the 6j symbol arising from this reduction.

5

NUCLEAR SPIN-SPIN INTERACTIONS

We may consider nuclear spin-spin interaction by reference to a specific example, D13C14N,studied by Garvey and De Lucia [23]. In addition to the quadrupole and spin-rotation interactions of nitrogen and deuterium, and the spin-rotation interaction of 3C, there is a spin-spin interaction involving the nuclear spins of nitrogen and carbon, and nitrogen and deuterium. The


830

direct spin-spin interaction may be written in Cartesian form as *spin-spin

. + 4.* DCN* IN =13*D32.12+ 13.D31'11 = 4.* DCD ID

(15.123)

The spin-spin interaction between deuterium and nitrogen is negligibly small due to the l/r3 dependence of the coupling constant and the large D-N separation. Since the largest interaction involves nitrogen, the coupling scheme

J + I1 =F1 F1+ I2=F2 F2+13=F3=F is appropriate, where the subscripts 1, 2, 3 refer to the nitrogen, deuterium and carbon atoms, respectively. The elements of the Cartesian coupling tensor are defined by (1 5.124) where I L and I K are the nuclear spins and p L and p K the corresponding magnetic dipole moments of nuclei L and K . Here r defines the relative position vector of nucleus K with respect to L. Rewriting the interaction in spherical tensor form, noting that D is a traceless and symmetric Cartesian tensor, we have = -J5[D'j2J x I\')](')*I'j1)-G[D\2]

&spin-spin

x l\l)](l)*l'jl) (15.125)

where D(2)is a second-rank spherical tensor and the normalization factor has been chosen such that Db2)=Dzz.Application of (15.21) to the first term of (15.125) gives (JI1 F ; 1 2 F; 13F(- 4 [ D i 2 J x

x&JZIF;

I\')IJZ1F1 I z F 2 13 F )

(1 5.126)

12F;II[D'j2J x I\l)](l)lIJZIF11 2 F 2 )

The latter tensor product is composed of tensor operators which operate on different subspaces; therefore application of (15.24) yields (J11F;Z2F;I/[D\2J x I\1']"~J~JI~F~Z~F~)=[(2F~+1)(2F~+1)3]1'2 x

{

F;

F1

z2

12

F;

F2

2 1 1

}

(JZ1

F ; IlD'j2JII JZ1 F 1 )(Z2 II z(21)II 1 2 )

(1 5.127)

Since D(2)commutes with I1 we apply (15.22) ( J I F ; l I D ' j ~ l l J I I F l ) = ( -1 ) J + l , + F 1 + 2 [(2F1+ 1)(2F;

+ (1 5.128)


831

( 1 5.1 29) Similarly, application of (15.21)to the second term of *spin-spin and subsequent use of (15.22) and (15.24) gives finally for the matrix elements (J11F ; Z2 F ; Z3 F I - G[DL2) x I\’)](’)* IL1)lJ11 F1 Z2 F2 13 F ) -(- ~ ) F I + I ~ + F + F ; + I ~ + F[Z1~5 ( 2 F z 1)(2F;+ 1)(2F1+ 1)(2F; l)]’”

+

+

The reduced matrix element of the rotation-dependent operator is found from the Wigner-Eckart theorem (J(1D‘2’11 J ) = 2 f ( J ) ( J J I D f ) J[ J )

(15.131)

and (JJ/Db2)1JJ)can be taken as the spectroscopic constant to be determined. More often the small effects of the spin-spin interaction can be accurately accounted for by calculation of Df)from the molecular geometry. Likewise we have (1,II

I‘d )II 1,)=h( 1 IU

( 15.1 32)

with h(ZJ defined in (15.68).

6

RELATIVE I NTENSlTl ES

In the interpretation of rotational hyperfine spectra the relative intensities of the transitions are required. This may also be readily formulated in terms of irreducible tensor methods. The line strength of the transition s‘J’a‘F‘+zJaF is defined by [2]

where p r ) are the components of the irreducible electric dipole moment tensor operator [pL1)=p,, p(i\ = T ( l / 2 1 / 2 ) ( p , fip,,)] and where T J specifies the pure rotational state. To be able to discuss plural nuclear coupling we let a stand for the set of intermediate momenta in the coupling scheme. From the WignerEckart theorem

832


The last term in this equation can be rewritten as

F‘ -Mk

1 F)lz=c 9 MF 4

(

M,M;

F‘

1

F ) ( F’ -& 4 MF -Mk

4 MF

( 15.136)

with the aid of the symmetry properties of the 3j symbol. The sum over M F and Mk is equal to i. This follows from the orthogonality property of the 3j symbols in (15.6). Thus the sum over the three components of q gives unity, and we have finally for (15.133)

S(t’J’a’F’+ JuF) = J(t’J’c”’I1~~(’)llrJcrF)I

(15.137)

Evaluation of the reduced matrix element depends on the coupling scheme. For a single coupling nucleus ct stands for I which does not change during a transition, and (15.22) gives

(T’J‘ZF’~~~-L(’)~~TJIF)=( - l)J’+’+F+l [(2F+ 1)(2F’+1)]1’2 X (7’

J’ 1 I/L( ‘’1 I T J )

(15.138)

The square of the preceding reduced matrix element gives the strength of the hyperfine transition according to (15.137) S(TtJ’IF’+tJIF)=(2F+1)(2F’+1)

{i7

: Y / ( T ’ J ‘ ~ ~ ~ ( ~ ) ~ ~(15.139) TJ)I~

The reduced matrix element ( ~ ’ J ’ ] l p ( ~ ) I which l t J ) represents the intensity of the unsplit rotational transition r’J’+tJ may be omitted in evaluation of the relative intensities of the hyperfine transitions. If it is desired, this reduced matrix element may be evaluated by first expressing p f ) by means of (15.1), in terms of the dipole moment components p::) along the molecule-fixed axes. The matrix elements of pb’) in the symmetric rotor basis may then be obtained by application of (15.38) and finally the reduced matrix element can be found is from the Wigner-Eckart theorem. For asymmetric rotors I(T’J‘I(~(~)I(TJ)I related to the line strength Ag introduced in Chapter VII. We may define a relative intensity of a particular hyperfine line by first


833

summing over F and F’ and rearranging the 6j symbol S(r‘J’IF’-+rJIF) F,F‘

The sum over the term in brackets is (2J’+ l)-’ by the orthogonality property of the 6j symbols. Likewise, the sum over the initial F’ states from lJ’-ll to J‘ I [for J’ > I and J‘ < I number of F‘ values are (21+ 1) and (2J’ + l), respectively] is (25’+ 1)(21+ 1). Hence

+

The relative line strength ofa hyperfine lineis the ratio of(15.139) to (15.141)[ll] s( J‘IF‘+J I F )=

S(J’t’lF’+ J T I F ) S ( J r ‘ I F ’ +JslF) FF’

-

(15.142)

(21-t-1)

where the notation for s takes into account that it is independent of r, 7‘. Furthermore, it is clear that

2 s(J’IF’+JIF)=l

(15.143)

FF’

As discussed by Rudolph [ll], these results can be used to give the unsplit line, corrected for the effects of quadrupole coupling, without actual assignment of the hyperfine structure. The transition frequency of a given hyperfine line J’z’lF’-t J r l F is (to first-order) v =v,+EQ(J?IF)- EQ(J’r’1F’)

(15.144)

where v , is the transition frequency J’z’+Jt if no hyperfine structure were present, If we multiply each hyperfine component by its relative intensity and sum over all components, we have V’S=V, S+ {E~(J~IF)-E,(J’~’IF’)}s (15.145) FF‘

E

FF‘

FF‘

By substituting from (15.84)for EQ, one can show from orthogonality relations of the 6 j symbol that the sums on the right-hand side vanish identically, yielding [ll]

v r(J‘T‘+Jr ) =

1 V( J’ r ’1F’

+J r l F )s(

J’ I F ‘ +J I F )

(15.146)

FF‘

Therefore, the unsplit frequency is given to first-order by the intensity-weighted average of the multiplet frequencies. This allows, in principle, the evaluation of v, without a detailed analysis of the hfs patterns and assignment of individual F’-+F components. The v,)s can then, of course, be used to give the rotational


834

constants. Partially resolved or coinciding components are properly accounted for by choice of the intensity weighted mean or center-of-gravity of the multiplet. Of course there are problems in application where weak components with large frequency displacements are not observed. The relative intensities for the two representations discussed previously require evaluation of the reduced matrix element of (15.137). For the case of unequal coupling since Ii commutes with p('), successive application of (15.22) can be used to bring the reduced matrix element 112..

. lNFhllpu")l(7J11F112. INFN) I

to the desired form (7' J'

II p' '1 1 T J)

For N nuclei the relative intensities for unequal coupling are given by [5]

S ( J ' F ; .. . F h + J F 1 . .

. FN)=IC(J'F;. . . F;V+JFi.. . FN)JZ

(15.147)

where

(15.148) and N

(Ii+F;- +Fi)+N with F,=J, Fb=J' and F N = F

t= i= 1

For evaluation of the relative intensities of the hyperfine transitions in the equal or nearly equal coupling scheme, the reduced matrix element (T'J'111292..

. l L 9 L . . ~ N ~ N " I I ~ " ' 1 ~ 7. JI L~Y 1L .~. .~INYNF) ~~. s

must first be evaluated. Since the 9 ; s commute with p('), (15.22)gives (7'J'I1F'llpL(''115J.. .9NF)"(-

JN+F+l

[(2F+ 1)(2F'+ 1)]1!2

The resulting relative intensities for equal coupling are [5]

S ( J ' . . . 9 N F ' - t J . . . YNF)=IC(J'. . . $NF'+J.. . 4NF)12

(15.150)

where

(15.151) and t =J'

+ YN + F + 1 with Y N=9.If the intermediate Fi (or Yi)are not good


835

quantum numbers, the first-order intensities given by (15.148) and (15.151)may not be adequate. The matrix of C must then be transformed by means of a similarity transformation to the basis which diagonalizes the hypefine Hamiltonian, that is, C' ='fiCT, (15.152) where Ti diagonalizes the energy matrix of the initial state of the transition and T, diagonalizes the energy matrix of the final state. The squares of the elements of C' yield the relative intensities.

7


The oxygen molecule, which has a 3Cground state, is a classic example of a diatomic molecule having both spin-rotation and electronic spin-spin interactions. Its microwave spectrum, analyzed in some detail in Chapter IV, Section 2, is most simply treated by irreducible tensor methods [SO, 511. Exclusive of centrifugal distortion and nuclear coupling isotopes, the 3C oxygen Hamiltonian (Chapter IV, Section 2) may be written as

X = B N 2 +yN.S +343SZ, -S2)

(15.153)

where the first term is the rotational part P,., the second term is the spinrotation part Xsr, and the last term is the spin-spin interaction [50, 511 Xss. Here Sz, is the body-fixed component of the electron spin along the internuclear axis, N is the rotational angular momentum, and S is the total electron spin operator. Evaluation of the matrix elements of this Hamiltonian allows the energy levels to be evaluated by direct diagonalization of the energy matrix. The appropriate basis functions are IS, N, J, M J ) where the total angular momentum J = S + N. The operators J2, J,, N2 and S2 are diagonal in this basis. It is readily seen that the matrix elements for Xr and X s rare given by (S, N, J, MJI HRlS, N, J, M d = B N ( N + 1)

N , J, M;IXs,lS,

N , J,M j ) = ; y [ J ( J + 1)- N ( N + 1)-S(S+ I)]

(15.154) (15.155)

noting N.S=i[J2 - N2 -S2]. The matrix elements for the spin-spin term may be conveniently evaluated by application of irreducible tensor methods by writing this term in terms of the where aiZ. is space-fixed components of S, that is,Xss=3A(c @iz,@jz,Si,Sj-S2), the direction cosine between the body-fixed z' axis and the space-fixed i axis ( i = x , y , z). In Cartesian tensor form Xss =AS* T .S

=A

1sisj

7;.j

(15.156)

i,j

where 7;.j=3(3@iz~@jz,-&j).This may be written in spherical tensor form as 2

1

Xss= Ni[S'"x S ( l ) ] ( W ~ N ) ;1 = ( - 1)' I=O

(15.157)

836

IRREDUCIBLE TENSOR METHODS

FOR COMPLEX SPECTRA

with S") the first rank spherical tensor form of the first rank Cartesian tensor S defined via (15.10),and C")is the Ith rank (l=O, 1, 2) spherical tensor form of T defined via (15.17). In this particular case, C")=O except for 1=2. Therefore &LL[S(1)xs'"]'2',

C'Z'

( 15.158)

From (15.21) we have

x (SII[S(')x S'1)]'2)11S)(N'lIC'2)llN)(15.159)

Application of (15.25) gives for the reduced matrix element of the electron spin

{s

(SJ1[S'"xS(')]'2)1IS)=~

}

1 1 2

(15.160) s s [(~lIs~1~11~)12

Here S= 1, and the 6 j symbol is equal to 4.From the Wigner-Eckart theorem, it follows from the matrix element (S, Ms=S(S&')lS,Ms=S)=S that

+

+

(SIIS")I(S)= [S(S 1)(S 2)] 1'2

=Jz

(15.161)

Similarly, application of the Wigner-Eckart theorem gives

where C&2)=(2/6"2)(30~z, - 1). Evaluation of the direction cosine elements for a linear molecule gives

(15.163) The matrix elements of the Hamiltonian are hence given by the sum of (15.154),(15.155),and (15.159)with cognizance of (15.160),(15.161),and (15.163). The Hamiltonian is diagonal in J,M J ,and S, and since S = 1 for each value of J , we have N = J - 1, J and J+ 1 with J = l , 2,. . . . For J=O, N = l . The energy matrix may hence be factored into a set of 3 x 3 blocks except for J=O which is 1 x 1. To take into account distortion effects, the constants B, y, and imay be expanded in terms of N . For the diagonal matrix elements, the constants are expanded as in (4.57). For the off-diagonal term ( N ' = J + l J N = J - l ) , 3.= ,lo +,11(J2 + J + 1). Elaboration of this theory and calculation of transition strengths are given in the dissertation of Steinbach [51]. The complexity of the spectra of polyatomic molecules with unpaired electron spins is illustrated in Fig. 15.4 for C F 3 which has a doublet electronic ground state with three identical coupling nuclei-the first such molecule analyzed by

Fig. 15.4 Observed spectrum of the N = 6 t 5 transition of CF,. The spectrum was synthesized from hatches of records, each covering a region 10 MHz wide and representing the accummulation of 400 scans at a repetition rate of 5 Hz. The spectrum consists of K components split into many lines by the spin-rotation and the hyperfine interactions. From Endo et al. [39]. 00 W 4

838


microwave spectroscopy. The hyperfine Hamiltonian is significantly complicated by the presence of the electronic spin momentum. The coupling scheme employed is J = N +S and F = J + I, with I, = I, + I, I,. A detailed analysis by Endo et ai. [39] allowed evaluation of the spin-rotation constants, the Fermi interaction parameter, and the diagonal elements of the dipole-dipole hyperfine interaction tensor. Elements of the latter tensor split the K = 1 level in first order. A discussion of the Hamiltonian may be found elsewhere [39, 35, 201. Varied applications of irreducible tensor methods to diatomic molecules with electronic spin momentum are given in the book by Mizushima [44].

+

8

STARK EFFECT

As a final example we consider the Stark effect for a linear or symmetric top molecule with a single quadrupole coupling nucleus. For a coupling nucleus with large nuclear spin I , a direct diagonalization of the energy matrix may be more appropriate than a perturbation treatment. The matrix elements of the Stark effect Hamiltonian will hence be considered. With the electric field d along the space-fixed axis z, the Hamiltonian has the form Aft = - 6 p f )

(15.164)

The basis functions are IKJIFMF), and from the Wigner-Eckart theorem we have

x (K’J’ZF’IIp6’)((KJIF) (15.165)

Further reduction of the matrix is accomplished with the theorem of (15.22)

(K’J’IF’11 pp1 IK J I F ) =( -

+I

+

+

[(2F

+ 1)(2F’+ l)]’”

Application of the Wigner-Eckart theorem again gives

Projection of ph’) on the molecule-fixed axis with (15.2) yields

For a linear or symmetric-top molecule only, p $ ’ ) = p is nonzero. The matrix element of D(’)may be evaluated from (15.38); thus K’=K throughout, and

STARK EFFECT

839

(KJ‘Mjjp‘d’lKJM,)=p(1)MJ-K[(2J+ 1)(23’+ l)]”’

Therefore

x(K J

0

J)

-K

(15.170)

(KJllpb1’lIKJ)=pK(2J+l)”’/[J(J+l)]”’, for J ’ = J + l , For J ’ = J , (KJ + 1llpb1)11KJ)=p[(J+1)’- K2]’12/(J+ 1)”’ and for J ’ = J - 1, (KJ- l)//pb’)IlKJ)=-p(J’ - K2)’’2/J’12. For a linear molecule, K=O, which implies that all diagonal elements vanish. The matrix elements for Xps are thus given by (15.165) with the auxiliary equation (15.106) and the values of (J’1lpL’)IlJ).To these may be added the matrix elements for the rigid rotor, the magnetic spin-rotation interaction, and the nuclear quadrupole interaction. The zero-field frequencies are found by setting 6=0, and the Stark shifts are found by setting 6 to the desired field value and diagonalizing the energy matrix for each value of M F . Since the energy matrix of the total Hamiltonian is no longer diagonal in F with the addition ofXps, a truncated matrix of appropriate dimensions must be chosen. If there is more than one coupling nucleus, additional quantum numbers will appear in the reduced matrix element of (15.165). Equation (15.22) may be repeatedly used until the reduced matrix element appears in its most reduced form. This will introduce products of 6 j symbols. It may be noted that the general form of (15.167) written in terms of pg)(q=O, f 1) yields the matrix elements of the dipole moment tensor in terms of the constant components along the body-fixed axis with the aid of (15.2). The matrix elements of D(’)are the matrix elements of the first rank spherical components of the usual direction cosines, 0.

Ref ere nces 1. M. E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957. 2. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton Univ. Press, Princeton, N.J., 1960. 3. U. Fano and G. Racah, Irreducible Tensorial Sets, Academic, New York, 1959. 4. B. R. Judd, Operator Techniques in Atomic Spectroscopy, McGraw-Hill, New York, 1963. 5. R. L. Cook and F. C. De Lucia, Am. J . Phys., 39, 1433 (1971). 6. R. F. Curl, Jr., and J. L. Kinsey, J. Chem. Phys., 35, 17 (1961). 7. P. Thaddeus, L. C. Krishner, and J. H. N. Loubser, J . Chem. Phys., 40,257 (1964). 8. N. F. Ramsey, Molecular Beams, Clarendon Press, Oxford, England, 1956. 9. W. H. Flygare and W. D. Gwinn, J . Chem. Phys., 36, 787 (1962). 10. H. P. Benz, A. Bauder, and Hs. H. Gunthard, J . Mol. Spectrosc., 21, 156 (1966). 1 1 , H. D. Rudolph, 2. Naturforsch., 23a, 540 (1968). 12. A. Carrington, B. J. Howard, D. H. Levy, and J. C. Robertson. Mol. Phys., 15, 187 (1968).

840

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

46. 47. 48. 49. 50. 51.


J. W. Cederberg and C. E. Miller, J . Chem. Phys., 50, 3547 (1969). G. L. Blackman, R. D. Brown, and F. R. Burden, J . Mol. Spectrosc., 36, 528 (1970). S . G. Kukolich, and S. C. Wofsy, J . Chem. Phys., 52, 5477 (1970). F. C. De Lucia and J. W. Cederberg, J . Mol. Spectrosc., 40, 52 (1971). J. W. Cederberg, Am. J . Phys., 40, 159 (1972). E. Herbst and W. Steinmetz, J . Chem. Phys., 56, 5342 (1972). J. T. Hougen, J . Chem. Phys., 57,4207 (1972). 1. C. Bowater, J. M. Brown, and A. Carrington, Proc. R. SOC.London, Ser. A333,265 (1973). J. E. M. Heuvel and A. Dymanus, J . Mol. Spectrosc., 47, 363 (1973). G. L. Blackman, K. Bolton, R. D. Brown, F. R. Burden, and A. Mishra, J . Mol. Spectrosc., 47, 457 (1973). R. M. Garvey and F. C. De Lucia, J . Mol. Spectrosc., 50, 38 (1974). R. M. Garvey, F. C. De Lucia, and J. W. Cederberg, Mol. Phys., 31, 265 (1976). J. T. Murray, W. A. Little, Q.Williams, and T. L. Weatherly, J . Chem. Phys., 65, 985 (1976). H. K. Bodenseh, W. Huttner, and P. Nowicki, Z . Naturforsch., 31a, 1638 (1976). A. Dymanus, “Beam Maser Spectroscopy,” in M T P International Reviews of Science, Physical Chemistry Series 2, Vol. 3 Spectroscopy, D. A. Ramsay, Ed., Butterworths and University Park Press, 1976. R. M. Garvey and F. C. De Lucia, Can. J . Phys., 55, 11 15 (1977). S . G. Kukolich, G. Lind, M. Barfield, L. Faehl, and J. L. Marshall, J . Am. Chem. Soc., 100, 7155 (1978). W. Hiittner, H. K. Bodenseh, P. Nowicki, and K. Morgenstern, J . Mol. Spectrosc., 71, 246 (1978). K. P. R. Nair, J. Hoeft, and E. Tiemann, Chem. Phys. Lett., 58, 153 (1978). J. M. Brown and T. J. Sears, J . Mol. Spectrosc., 75, 11 1 (1979). K. P. R. Nair, J. Hoeft, and E. Tiemann, J . Mol. Spectrosc., 78, 506 (1979). F. Michelot, B. Bobin, and J. Moret-Bailey, J . Mol. Spectrosc., 76, 374 (1979). J. T. Hougen, J . Mol. Spectrosc., 81, 73 (1980). Y. Endo, S . Saito, and E. Hirota, J . Chem. Phys., 74, 1568 (1981). H. A. Fry and S. G. Kukolich, J . Chem. Phys., 76, 4387 (1982). A. M. Murray and S . G. Kukolich, J . Chem. Phys., 77, 4312 (1982). Y. Endo, C. Yamada, S . Saito, and E. Hirota, J . Chem. Phys., 77, 3376 (1982). S . G. Kukolich and C. D. Cogley, J . Chem. Phys., 76, 1685 (1982); 77,581 (1982). L. M. Tack and S. G. Kukolich, J . Mol. Spectrosc., 94, 95 (1982). C. D. Cogley, L. M. Tack, and S . G. Kukolich, J . Chem. Phys., 76,5669 (1982). M. Ribeaud, A. Bauder, and Hs. H. Gunthard, J . Mol. Spectrosc., 42,441 (1972). M. Mizushima, The Theory of Rotating Diatomic Molecules, Wiley, New York, 1975. Numerical values have been tabulated by M. Rotenberg, R. Bivens, N. Metropolis, and J . K. Wooten, The 3jand 6j Symbols MIT Press, Cambridge, Mass., 1959. Equation 15.38 may be obtained by application of (4.62) of Ref. 1 or (4.6.2) of Ref. 2 and the definition of the symmetric rotor functions. E. U. Condon and G. H. Shortley, Theory of Atomic Spectra, Cambridge University Press. Cambridge, England, 1959. G. W. Robinson, and C. D. Cornwell, J . Chem. Phys., 21, 1436 (1953). J. S . Rigden and L. Nygaard, J . Chem. Phys., 44, 4603 (1966). W. Steinbach and W. Gordy, Phys. Rev., AS, 1753 (1973). W. Steinbach, Ph.D. dissertation, Duke Univ., 1974.

Appendix

A

NOTES O N MATRIX MECHANICAL METHODS

To facilitate the understanding of certain parts of this book by those not well versed in matrix mechanical methods, we give here a concise description of elements essential to interpretation of microwave spectra.

PROPERTIES OF SPECIAL MATRICES We start by considering the properties of some special matrices. There are two types of matrices that are of particular importance in quantum mechanics. They are Hermitian matrices and unitary matrices. Before defining these it is useful to define the transpose and Hzrmitian adjoint matrix. The transpose of A denoted by A is obtained from A by an interchange of rows and columns thus * Anm=Amn

(-4.1)

The transpose of a column matrix is a row matrix. The Hermitian adjoint A t of a matrix A we obtain by transposing A and taking the complex conjugate of each element, that is,

At =(A)*

(A4

or in terms of the matrix elements A Lm =(inm)* =AZn

(44.3)

where the asterisk signifies complex conjugation. A matrix A is said to be Hermitian if it is equal to its Hermitian adjoint, that is, A= A ~

('4.4)

or for the elements Anm=AZn

(A.5)

The operators of quantum mechanics representing observables are Hermitian operators since such operators yield real eigenvalues and expectation values. The matrix corresponding to a Hermitian operator is likewise a Hermitian matrix. A Hermitian matrix whose elements are real is a symmetric matrix 841

842

APPENDIX A

for which

-.

(A4

A=A

A matrix A is unitary if its inverse is equal to its Hermitian adjoint A-1

('4.7)

=AT

hence

AA-~=AA'=E where E is a unit matrix (E,,=6,,) matrix

also commonly designated I. A real unitary ",

('4.9)

A-I=A

is an orthogonal matrix. Transformations between different orthonormal basis systems are accomplished by unitary matrices. Unitary transformations have the property that they preserve the value of a scalar product, which is naturally required since the observable quantities are obtained from scalar products and must hence be independent of the choice of basis system. A unitary transformation is a generalization to complex space of an orthogonal transformation in real space.

MATRIX REPRESENTATION AND OPERATORS

OF WAVE

FUNCTIONS

In wave mechanics, the dynamical variables or observables are represented by linear operators. Each linear operator, A , has an eigenvalue equation associated with it.

A h =an44

(A.lO)

where 4, is the eigenfunction and a, the corresponding eigenvalue of the operator A . The most important operator is the energy or Hamiltonian operator &. The solution of its eigenvalue equation, the Schrodinger equation, yields the energy levels (eigenvalues) and energy eigenstates (eigenfunctions) of the quantum system. In matrix mechanics, the quantum mechanical operators are represented by matrices, and the problem of solving the eigenvalue equation (a differential equation) is replaced by the problem of diagonalizing a matrix. Consider a set of linearly independent functions 41,&, . . ., &, . . ., such as the set of eigenfunctions generated by the solution of an eigenvalue equation. Throughout our discussion we employ a single index to distinguish the various members of the set (4,) of basis functions. If more than one quantum number is required in order to specify the basis functions, then the index n must be regarded as standing for the complete set of quantum numbers, viz., n = i, j , k, . . .. We shall assume that the basis functions are orthogonal and normalized, Le.,

(4nl4rn)=

J' 4n*+rndr=6nrn

(A.11)

MATRIX REPRESENTATION OF WAVE FUNCTIONS AND OPERATORS

where the Kronecker delta is defined by 0 ifnfm L l = 1 ifn=rn

843

(A. 12)

Frequently the notation ($,l&,) is further abbreviated to (nlrn). The basis functions are said to form a complete set in that any arbitrary wave function may be expanded in terms of them. An arbitrary function $ can hence be expressed as

*=Icn4n

(A.13)

n

where the C’s are the expansion coefficients or coordinates of $ referred to the basis and are defined by

I&}

Cn =(4nl*)

(A.14)

The function $ is uniquely specified by enumeration of the coefficients Cn (in general complex). These may be written as a column matrix

(A.15)

In the language of matrix mechanics we refer to this as the matrix representation of $ referred to the basis (&}. The basis system is, of course, arbitrary, and a different basis may be chosen. Then, however, the list of elements in the column matrix will be altered, and a different matrix representation will be obtained. The scalar product of two functions x and $ denoted by (XI+) is given by the product of a row matrix and a column matrix

(XI$)=

11B,*4,* n

C n 4 , d z = I B,*C,=[BTB,*. . . B,* * * . n

n

1

(A.16) C,

Note that the first factor in the scalar product is the Hermitian adjoint of the column matrix representing x. If the two functions are orthogonal, the above scalar product will vanish. The matrix representation of the operator A in the basis ( 4 n }is defined by the equation A,,=(+~IAIQ,)=

J

(A.17)

The array of numbers (called matrix elements) calculated by the above relation

844

APPENDIX A

may be written as a square matrix A, that is,

rAll

A12

A13

. .1 (A.18)

The operator matrix depends on the basis chosen and will have different forms in different basis systems. If the operator represents a measurable quantity, the matrix operator is Hermitian and its elements satisfy (A.5). The matrix of the sum of two operators C ( = A + B ) is just the sum of the matrices representing the operators A and B in the basis (&}. Hence, the elements of C are defined by Cnm=Anm+ Bnm (A.19) The matrix of the product operator C ( = A B ) is obtained from the matrix of A and B by the usual rule for multiplication of two matrices. The matrix elements of the matrix operator C are hence given by (A.20) Note that all elements are defined with respect to the same basis. Similarly, the matrix for the product of any number of operators is obtained by multiplication of the corresponding matrices. In general, AB # BA since matrix multiplication is noncommutative. Any matrix A does, however, commute with its inverse and with a unit matrix E; any two diagonal matrices will also commute. If basis functions are chosen which are simultaneous eigenfunctions of two operators A and B, the corresponding matrices will be diagonal [see (A.33)] and hence commute. This is analogous to the statement that if a function is an eigenfunction of two operators, then the operators commute. We are now in a position to rewrite any equation of wave mechanics in matrix form. In matrix mechanics, for example, operation on a function $ with an operator A is represented by the product of a square matrix A times the column matrix representing $. The result is a new column matrix (or function). With this in mind and taking cognizance of the definition of a scalar product, we may write A12 A13 A22

]I] (A.21)

This gives the average or expectation value of an observable represented by the operator A, for the quantum state $. Evaluation of the average value of an observable thus requires a knowledge of the matrix elements of the operator in a given basis and the expansion coefficients of the state $ in that basis.

CHANGE OF BASIS

845

CHANGE OF BASIS Many times it is advantageous to change from one basis to another. Two salient questions then arise: first, how the coordinates of a wave function, that is, its matrix representation, are related in the two basis systems and second, how the matrix of an operator is related in the two basis systems. Let the connection between the original orthonormal basis functions (q5,,} and the new set of basis functions {$;} be written

Smn4,

n=l, 2,. . .

(A.22)

m

with

Smn=(4mI49

(A.23)

The coefficients Smndefine a square matrix S, a transformation matrix, which is a unitary matrix. This insures, for example, that if the original basis functions are orthonormal, then the transformed basis remains orthonormal. If the functions 4, and 4; are regarded as elements of two column matrices, (A.22)may be written in matrix form as

(A.24)

s

where is the transpose matrix of S. The relation between the expansion coefficients of a given wave function $ in the two different basis systems is found as follows

where cm=

C SmnCA

(A.26)

n

which gives the old coefficients or coordinates in terms of the new ones. The transformation may be equivalently written as

=S

=S t

(A.27)

846

APPENDIX A

Note that if the elements ofS are real, then both the basis functions and coordinates transform in the same manner. The basis functions { &,,} generate a representation of the operator A through the relation A n m =(4nIA

while in terms of the basis functions

{&I,

I4m)

(A.28)

the operator matrix is defined by

A hrn=(4hlA146)

(A.29)

The relation between the matrix A and A' is found by use of (A.22)in the foregoing equation

(A.30) In matrix form A' =S 'AS

(A.31)

This gives the matrix representation of the operator referred to the new basis in terms of the matrix representation of the operator in the old basis and of the unitary transformation S connecting the two basis systems. The matrix A is said to be transformed from the 4nrepresentation to the 4; representation by a similarity transformation. If the transformation has the property that it converts A to diagonal form, then it may be referred to as a principal axis transformation. If A is Hermitian, then because of the unitary character of S, A' is also Hermitian [l]. Furthermore, the form of an equation is unaltered [2] by a transformation such as S, and the eigenvalues of a matrix are invariant to a change of basis [3]. If the transformation matrix is real, as is often the case, then (A.31)reads A' =sAS

(A.32)

D IAG 0 NAL IZATl 0 N 0 F MATR I C ES : E I G ENVA L U E S AN D El G EN FU N CTI 0 NS In treating spectra we are especially interested in the eigenfunctions and eigenvalues of Hermitian operators. If the eigenfunctions I J ~. ,. ., ( j n , . . . of a Hermitian operator are known and these functions are used as basis functions for construction of the matrix of this operator, then the matrix will

DIAGONALIZATION OF MATRICES: EIGENVALUES AND EIGENFUNCTIONS

847

be diagonal. This follows from the orthonormality of the eigenfunctions ($rnlAI$n)=an($ml$n)

=andmn

(A.33)

The elements a, of the diagonal matrix are the eigenvalues of the operator A . The diagonal representation of the operator matrix is hence of special interest. Even if the eigenfunctions are initially unknown, it is possible to find a transformation which will diagonalize the operator matrix by means of a similarity transformation. Consider the operator matrix A with respect to some orthonormal basis q51, ( 6 2 , . . ., &, . . .. Let the basis functions $1, $ 2 , . . ., $,, . . ., the eigenfunctions of A , be related to the arbitrary basis functions by (A.34) The A matrix will be transformed from the 4 representation to the $ representation by

S +AS and for this basis the transformed matrix must be diagonal with elements d m=andmn , so that

S+AS=d

(A.35)

If we premultiply by S, since SSt =StS=E, we have

AS=Sd

(A.36)

and for the matrix elements

or m

( A k m - 8kman)Smn =o

(A.38)

Hence, if the transformation diagonalizes the matrix, we are led to a set of simultaneous linear equations defining the Sm, in terms of the a, and the A k m . For a given value of n, (A.38)gives a set of linear equations corresponding to the various values of k ; these have nontrivial solutions for the Sln,S2,, . . . ifand only if-the determinant of the matrix of coefficients vanishes IAkm

- d k m a n l =o

(A.39)

The roots of this equations (the secular determinant or secular equation) in

a, provide the eigenvalues a l , a2, . . . ; by substitution of each of the eigenvalues

into the set of (A.38) the corresponding S matrix may be evaluated. The nth eigenvalue, for example, yields the nth column of S that is Sln,S 2 , , . . . ; these are the expansion coefficients of the eigenfunction $, in the basis {4n}. The

848

APPENDIX A

columns ofS are also referred to as the eigenvectors of the matrix A. In evaluating the S ,, use is made of the normalization condition, that is,

1k snks:k =

(A.40)

In summary, the diagonalization of the matrix representation of the operator which is tantamount to the solution of the secular equation gives the eigenvalues. The determination of the transformation matrix S which diagonalizes the operator matrix yields the eigenfunctions. In the text, the secular equation is arrived at from a slightly different point of view, viz., by a rewriting of the eigenvalue equation in matrix form. Of particular importance is the diagonalization of the Hamiltonian matrix. The diagonalization of H can be simplified by a knowledge of the operators which commute with the Hamiltonian. If an operator A commutes with 2 then

[ 8, A ] = 2 A - A H=O

(A.41)

The matrix representation of this equation based on functions which are eigenfunctions of A , so that A is diagonal, is H n m ( A m m - A n n ) =O

(A.42)

It therefore follows that 2,, vanishes if the states n and m belong to different eigenvalues of A(A,,#A,,). The 8 matrix will hence factor into submatrices with each submatrix involving states with the same eigenvalues for A . In many cases, more then one operator will commute with 2.The greatest simplification will result if basis functions are chosen which are simultaneous eigenfunctions of the complete set of commuting operators. The operators which may be considered are not only the commuting observables such as the square of the angular momentum P 2 but also what we might classify as symmetry operators, operators which leave 2 invariant and hence commute with 2. Consideration therefore of the symmetry of the Hamiltonian also facilitates solution of the diagonalization problem.

References 1. If A is Hermitian we may write A=S'A'S: taking the Hermitian adjoint we have (A')'= (S'A'S)' =S'AS = A showing that the Hermiticity is invariant. 2. In the matrix equation %=A$ if we multiply by St then x'=S'X=S'A$=S'ASS'J=A'JI'

showing the transformed equation has the same form. 3. The eigenvalues are solutions of the secular equation IA-aEJ =O. Since 1ABI =IAl IBI and StS=E, we may write / A - & = IA-uEl IS'S( +'I

/ A - u E ~IS1 =/S'AS-& = 1A - u E ~

Hence the eigenvalues do not depend on the choice of basis.

Appendix

B

CALCULATION OF THE EIGENVALUES A N D EIGENVECTORS OF A HERMITIAN TRIDIAGONAL MATRIX BY THE CONTINUED FRACTION METHOD

When the matrix representation of the Hamiltonian operator has a tridiagonal form

-HI

1

H2 1

L

H12

H2 2

H2 3

Hk,k-l

Hk,k

Hn.n-1

the secular equation may be written in continued fraction form. The discussion given by Swalen and Pierce' on the continued fraction method will be followed here. Let S be a unitary transfo.rmation which diagonalizes H, that is,

StHS=A

(B.2)

where A is a diagonal matrix. Denoting by s a column of S, we have for a particular eigenstate the system of homogeneous linear equations

(H - AE)s =0

(B.3)

where 1, is an element of A and E is a unit matrix. To develop the continued fraction about the k t h diagonal element of H, let the eigenvector s be written as 849

850

APPENDIX B

The column matrix is an unnormalized eigenvector of H with the kth element set equal to unity, and N k is the appropriate normalization factor defined explicitly by the equation

IN k l -

=1

+

[ek*+agk

+ a + e?- l e k

a

-a ]

(B.5)

+

Here and in following equations when the subscript k c1 occurs, the running index a takes on the values 1,2, . . ., n - k ; whenever the subscript k - c( appears, a takes on the values 1, 2,. . ., k - 1. The simultaneous equations of relation (B.3) may be written in terms of the elements of the relative eigenvector, the k th equation is H k , k - lek

-1

+( H k k - l ) +H k , k + lek

+ 1 =o

(B.6)

Employing the (k + a)th and (k -a)th equations we obtain the following recursion relation for the elements of the relative eigenvector ek*a --

-Hkfa,kfaT

-

Ok*aT 1

1

Hk*a,kfa-L+Hkfa,k*af

l(ck*af llckfa)

(B.7)

Defining the recursion relation R k fa

=A-H k * a , k f a - hk2 f a *

l/Rk+ai 1

(B.8)

we may write the above recursion relation for the g’s in compact form Ok+a --H k f a , k f a T 1 ekfaT1

(B.lO)

&+a

The components defined by (B.7) or (B.lO) constitute an eigenvector only if (B.6) is satisfied. Note that the limits on c1 specified above arise because the quantity R k f aor hkia is zero when ( k - a ) < l or ( k + x ) > n . If the expressions for c k - 1 and g k + l are inserted in the kth simultaneous equation, (B.6), one obtains the continued fraction form of the secular equation (B.ll) The above expression represents the kth development (the leading term is Hkk) of the continued fraction form of the secular equation. There are n such developments corresponding to the n possible values of k.

CALCULATION OF EIGENVALUES AND EIGENVECTORS

851

To evaluate the eigenvalues of H, we may use a first-order iterative procedure. If interest lies in the kth eigenvalue and if the off-diagonal elements are small, then Hkk will most nearly approximate the kth eigenvalue. Substitution of this initial approximation to L on the right-hand side of (B.ll) gives an improved approximation to the eigenvalue. Note that the quantities Rk+ and R k - are functions of A. The improved approximation may now be inserted in (B.ll) to give a better approximation, and this process may be continued until the desired accuracy is obtained. A much more efficient procedure is to use the Newton-Raphson technique, a second-order iterative procedure. Let the functionfk(A) be defined as (B.12) The roots of fk(2) are the eigenvalues of H. There are n functions f(1)which may be defined; the above equation gives the kth development of f ( A ) . If fk(L)is expanded about the ith approximation A(') to an eigenvalue, then

fk(A)=&(A(')) +f;(A('))(n - 2')) where higher-order terms are neglected. From this, we see that for should be chosen as zero the next approximation

(B.13)

fk(A) to be (B.14)

in which

(B.15)

As before, with an initial approximation to the eigenvalue, the function fk(A) and its derivative are evaluated, and (B.14) is employed to give an improved approximation. The iterations may be continued until the change in the eigenvalue is small enough to be ignored. This second-order iterative procedure yields the eigenvalues much more rapidly. Since the sum of the diagonal elements of H is equal to the sum of the eigenvalues of H, that is, Hkk=c k

l k

(B.16)

k

one of the eigenvalues may be computed without solution of the continued fraction equation. The derivative fk(1) is also related to the normalization constant Nk. By insertion of (B.10) into (B.5) it can be shown that

852

APPENDIX

B

(B.17)

for any A. If 4, is, in particular, an eigenvalue of H, that is, (B.11) is satisfied, then f#) is related to the normalization constant of the eigenvector (B.18)

For convergence that is rapid and to the desired root, both the choice of the initial approximation to the eigenvalue and the choice of development are important. For A’ an eigenvalue of H, the best development may be defined as the kth if f;(A’)

Spin

Nuclear Magnetic Nuclear Electric Dipole Moment p (in units of Quadrupole Moment Q nuclear [in units of barns cm’)] magnetons)

~

40 42 43 44 46 48 45 46 47 48 49 50 50 51 50 52 53 54 55 54 56 57 58

39.962589 41.958625 42.958780 43.955491 45.95369 47.95253 44.955919 45.952632 46.951769 47.947950 48.947870 49.944786 49.9471 64 50.943961 49.946055 51.940513 52.940653 53.938882 54.938050 53.939617 55.934936 56.935398 57.933282

96.97 0.64 0.145 2.06 0.0033 0.185 100

7.93 7.28 73.94 5.51 5.34 0.24 99.76 4.31 83.76 9.55 2.38 100 5.82 9 1.66 2.19 0.33

-1.317

4.7564

-0.22

- 0.7884 - 1.lo40 -1.1040

3.3470 5.148

-0.4744 3.468

0.0905

0.27

-

0.03 0.4

27 28

Cobalt Nickel

co

29

Copper

cu

Ni

59 58 60 61 62 64 63 65

58.933189 57.935342 59.930787 60.931056 61.928342 63.927958 62.929592 64.927786

100 67.88 26.23 1.19 3.66 1.08 69.09 30.91

;

4.649

0.4

2.226 2.385

-0.18 -0.17

py=

1.0806 f0.0003

30

31

32

33 34

Zinc

Gallium

Germanium

Arsenic Selenium

Zn

Ga

Ge

AS Se

64 66 67 68 70 69 71

70 72 73 74 76 75 74 76 77 78

63.929145 65.926052 66.92715 67.924857 69.925334 68.925574 70.924706

69.924252 71.922082 72.923463 73.921 181 75.921405 74.921 596 73.922476 75.9 19207 76.91991 1 77.917314

48.89 27.81 4.1 1 18.57 0.62 60.4 39.6

20.52 27.43 7.76 36.54 7.76 100

0.87 9.02 7.58 23.52

3

1 3

1

0.8757

0.17

2.016 2.562

0.19 0.12

-0.8792

1.439

0.534

[$I=

1S867 +O.o004

-

0.22

0.29

Isotopic Abundances. Masses. and Moments

Atomic Number Element

Mass Symbol Number ~

35

Bromine

Br

Mass ( a 4

Relative Abundance

e/,>

Spin

Nuclear Magnetic Dipole Moment p (in units of nuclear magnetons)

Nuclear Electric Quadrupole Moment Q [in units of barns cm’)]

~

80 82 79 81

79.916527 81.916707 78.918329 80.916292

49.82 9.19 50.537 49.463

0 (0)

4

2.106 2.270

T

0.31 0.26 (%).=1.19707 f0.00003

36

37

Krypton

Rubidium

Kr

Rb

78 80 82 83 84 86 85 87

77.920403 78.916380 81.913282 82.914131 85.91 1503 85.910616 84.91 1800 86.909187

0.354 2.27 11.56 11.55 56.90 17.37 72.15 27.85

(0) 1

-0.970

0.23

(0) (0)

;

1.3527 2.7506

1

[Zr

0.28 0.14

= 2.0669

38

39

Strontium

Yttrium

Sr

Y

84 86 87 88 89

83.91 3430 85.909285 86.908892 87.905641 88.905872

0.56 9.86 7.02 82.56 100

+0.0005

(0) 1

-

1.093

(0) 1

-0.1373

0.36

40

41 42

44

45 46

Zirconium

Niobium Molybdenum

Ruthenium

Rhodium Palladium

Zr

Nb Mo

Ru

Rh Pd

90 91 92 94 95 96 93 92 94 95 96 97 98 100

89.904700 90.905642 91.905031 93.906313 94.905839 95.908286 92.906382 91.906810 93.905090 94.905839 95.904674 96.906022 97.905409 99.907475

51.46 1 1.23 17.11 17.40 15.72 2.80 100 15.84 9.04 15.72 16.53 9.46 23.78 9.63

96 98 99 100 101 102 104 103 102 104 105 106 108 110

95.907598 97.905289 98.905936 99.904218 100.905577 101.904348 103.905430 102.905511 101.90561 103.90401 104.90506 105.903479 107.903891 109.90516

5.51 1.87 12.72 12.62 17.07 31.61 18.58 100 0.96 10.97 22.23 27.33 26.7 1 11.81

3

-

1.303

3 6.167

-0.9135 - 0.9327

-

0.63

-0.69

I

1

- 0.0883

5

-0.6015

1

0.22

Isotopic


47 48

Silver

Cadmium

Mass Symbol Number

Ag Cd

49

Indium

In

50

Tin

Sn

107 109 106 108 110 111 112 113 114 116 113 115 112 114 115 116 117 118 119 120 122 124

Abundances. Masses, and Moments

Mass (am4

106.905094 108.904756 105.906463 107.904187 109.903012 110.904188 I 1 1.902763 112.904409 113.903360 115.904762 112,904089 114.903871 111.90484 113.902773 114.903346 115.901745 116.902958 117.901606 118.903313 119.902198 121.903441 123.905272

Relative Abundance (%)

51.35 48.65 1.215 0.875 12.39 12.75 24.07 12.26 28.86 7.58 4.28 95.72 0.96 0.66 0.35 14.30 7.61 24.03 8.58 32.85 4.72 5.94

Spin

4

f

Nuclear Magnetic Nuclear Electric Dipole Moment p Quadrupole Moment Q (in units of nuclear [in units ofbarns magnetons) cm’)]

-0.1 135 -0.1305

(0)

f f

- 0.5950

(0) - 0.6224

(0) (0)

E E

4 (0) 1

5.523 5.534

-0.918 - I .om

(0) 1

(0)

-

1.046

0.82 0.83

51

Antimony

Sb

121 123

120.903816 122.904213

57.25 42.75

5 1

3.359 2.547

- 0.29 -0.37

*0.000010 52

Tellurium

Te

53 54

Iodine Xenon

I Xe

55 56

Cesium Barium

Cs Ba

120 122 123 124 125 126 128 130 127 124 126 128 129 130 131 132 134 136 133 130 132 134 135 136 137

119.90402 121.903066 122.904277 123.902842 124.904418 125.903322 127.904476 129.906238 126.904470 123.9061 125.904288 127.903540 128.904784 129.903509 130.905085 131.904161 133.905397 135.907221 132,90536 129.90625 131.9051 133.90461 134.9056 135.90430 136.90550

0.089 2.46 0.87 4.61 6.99 18.71 31.79 34.48 100 0.096 0.090 1.919 26.44 4.08 21.18 26.89 10.44 8.87 100 0.101 0.097 2.42 6.59 7.81 11.32

-0.79

-0.12

2.579

- 0.003

0.8372

0.18

0.9366

0.28

Isotopic Abundances, Masses, and Moments ~


Relative Abundance

Mass Symbol Number ~

(%I

Spin


Nuclear Electric Quadrupole Moment Q [in units of barns cm’)]

~

~

138

137.90500

71.66

(gy=

(0)

1.543+0.003

57

Lanthanum

La

58

Cerium

Ce

59

Praseodymium Neodymium

Pr Nd

60

138 139

136 138 140 142 141 142 143 I44 145 146 148 150

137.90691 138.90614

135.9071 137.90583 139.90539 141.90914 140.9070 141.90766 142.90978 143.9 1004 144.91254 145.9 1309 147.91687 149.92092

0.089 99.91 1

0.193 0.250 88.48 11.07 100 27.1 1 12.17 23.85 8.30 17.22 5.73 5.62

5 7

1

5

1

3.707 2.778

4.5

($y=

kO.8 0.22

T3.5

- 0.06

I

7

-1.1

-0.6

7

-0.71

- 0.3

1

62

Samarium

Sm

63

Europium

Eu

64

Gadolinium

Gd

159 156 158 160 161 162 163 164

143.91199 146.91487 I 47.9 1479 148.91718 149.91728 151.91976 153.92228 150.91984 152.92124 151.91979 153.92093 154.92266 155.92218 156.92403 157.92418 159.92712 158.92535 155.9239 157.92445 159.92520 160.92695 161.92680 162.92876 163.92920

3.09 14.97 1 1.24 13.83 7.44 26.72 22.71 47.82 52.18 0.200 2.15 14.73 20.47 15.68 24.87 21.90 100 0.0524 0.0902 2.294 18.88 25.53 24.97 28.18

165 162 164 166

164.93042 161.92874 163.92929 165.93031

100 0.136 1.56 33.41

144 147 148 149 I50 152 I54 151 153 152 154 155 156 157 I58 160

65 66

67 68 W 6)

W

Terbium Dysprosium

Holmium Erbium

Tb Dy

Ho Er

7

0.90

1

-

7

-0.75

1

3.464 1.530

-0.27 -0.36

< f0.7 < f0.7

0.95 2.42

I .3 1.5

3 7

f1.7

1

5

k0.42

k1.1

z

5

+OS8

k1.3

7

4.1

1

3.0

Isotopic Abundances, Masses, and Moments

a, 4 0


69 70

Thulium Ytterbium

Mass Symbol Number

Tm Yb

71

Lutetium

Lu

72

Hafnium

Hf

Mass

a>

(mu)

167 168 170 169 168 170 171 172 173 174 176 175 176

166.93206 167.93238 169.93556 168.93425 167.9342 169.93502 170.93643 171.93636 172.93806 173.93874 175.94268 174.94064 175.94266

174 176 177 I78 1 79

173.94036 175.94157 176.94340 177.94388 178.94603 179.9468

180

Relative Abunrlance

22.94 27.07 14.88 100

0.135 3.03 14.31 21.82 16.13 31.84 12.73 97.41 2.59

0.18 5.20 18.50 27.14 13.75 35.24

Spin

Nuclear Magnetic Nuclear Electric Dipole Moment p (in units of Quadrupole Moment Q nuclear [in units of barns magnetons) cm’)]

-0.56

2.8

-0.229

0.4930 - 0.679

3.0

2.23 3.18

5.6 8.0

0.61

3

-0.47

3

(,=.I 177Hf

0

=0.’$9+0.02

73

Tantalum

Ta

74

Tungsten

w

75

Rhenium

Re

180 181 180 182 183 184 186 185 187

179.94754 180.94801 179.94700 181.94830 182.95032 183.95103 185.95444 184.95306 186.95583

0.0123 99.9877 0.135 26.41 14.40 30.64 28.41 37.07 62.93

I

2.36

4.2

(0) 1

0.117

(0) (0) z

4

3.172 3.204

2.6 2.6

(z:]= 1.056+0.005

76

77 78

79 80

Osmium

Iridium Platinum

Gold Mercury

Os

Ir

Pt

Au Hg

184 186 187 188 189 190 192 191 193 190 192 194 195 196 198 197 196 198 199

183.95275 185.95387 186,95583 187.95608 188.95830 189.95863 191.96145 190.96064 192.96301 189.95995 191.96115 193.96273 194.96481 195.96497 197.96790 196.96654 195.96582 197.966756 198.968279

0.01 8 1.59 1.64 13.3 16.1 26.4 41 .0 37.3 62.7 0.0127 0.78 32.9 33% 25.3 7.21 100

0.146 10.02 16.84

0.067

3 4

4

0.6566

0.8

0.18 0.18

1.3 1.2

0.6060 3

z

0.14486

0.5027

0.58

Isotopic Abundances, Masses, and Moments


Mass Symbol Number

81

Thallium

TI

82

Lead

Lb

83 90 92

Bismuth Thorium Uranium

Relative Abwtdance

Bi Th U

“Quadrupolemoment ratio.

200 20 1 202 204 203 205 204 206 207 208 209 232 234 235 238

E) 199.968327 200.970308 201.970642 203.973495 202.972353 204.974442 203.973044 205.974468 206.975903 207.976650 208.980394 232.03812 234.04090 235.04392 238.05077

23.13 13.22 29.80 6.85 29.50 70.50 1.48 23.6 22.6 52.3 100 100 0.0056 0.7205 99.2739

Spin


Nuclear Electric Quadrupole Moment Q [inunits of barns (10-24cm2)]

0.5567

0.45

-

1.6115 1.6274

0.5895 4.080

-0.35

- 0.34

4.1

Appendix

F

BOND RADII

Covalent Radii

Atom H B C

(AT

Single Bond Radius

P S

0.32b 0.81 0.772 0.74 0.74 0.72 1.17 1.10 1.04

c1

0.99

Ge As Se Br Sn Sb Te

1.22 1.21 1.17 1.14 1.40 1.41 1.37 1.33

N 0 F Si

I

Van Der Waals Radii

Double Bond Radius

Triple Bond Radius

0.71 0.667 0.62 0.62 0.60 1.07 1.00 0.94 0.89 1.12 1.11 1.07 1.04 1.30 1.31 1.27 1.23

0.64 0.603 0.55 0.55

1.00 0.93 0.87

(A) H 0 S

1.2 1.40 1.85 2.00 2.20

N P

1.5

As Sb CH, C6H6

2.0 Se Br 2.2 1 Te 2.0 1.70 =half-thickness of benzene molecule

1.9

F

c1

1.35 1.80 1.95 2.15

“From L. Pauling, Nature of the Chemical Bond, 3rd ed., Cornell Univ. Press, Ithaca, N.Y., 1960. The bond length rAB between (continued)

873

874

BOND RADII

two atoms A and B is given by T A B =r A

+ r B - B l x -~ X B /

where r A and rB are the covalent radii of atoms A and B : x A and XB are the corresponding electronegativities. A table of electronegativities is given in Appendix G. The empirical constant fi is assigned the following values: P=O.OS A, for bonds involving one (or two) first-row atom(s). fi=0.06 A, for bonds between Si, P, or S and a more electronegative atom not in the first row. fi =0.04 A, for bonds between Ge, As, or Se and a more electronegative atom not in the first row. fi =0.02 A, for bonds between Sn, Sb, or Te and a more electronegative atom not in the first row. p = O , for bonds between C and elements of Groups V, VI, and VI1 not in the first row. fi=O.O6 A, for bonds to hydrogen.* *From W. Gordy, W. V. Smith, and R. T. Trambarulo, Microwave Spectroscopy, Wiley, New York, 1953.

Appendix

G

ELECTRONEGATIVITIES OF THE ELEMENTS

Electronegativities as Derived from Atomic Number

1 2 3 4 5 6 7

8 9 10 11 12 13 14 15 16 17

18 19 20 21 22

Element H He Li Be B C N 0 F Ne Na

-

Mg

Si

P S

c1 sc Ti

24

Cr

25

Mn

VIV

VV Cr" Cr"' Cr" Mn" Mn'" MnVil

2.15 -

1.o 1.5 2.0 2.5 3.0 3.5 4.0

0.95 1.5 2.0 2.5 3.0 3.5 3.95 0.9 1.2 1.5 1.8 2.1 2.5 3.0

-

-

-

A K Ca

0.8

0.82 1.03 1.3 1.6 1.4 1.7 1.9

-

1.3 1.5 2.2 1.3 1.5 2.6

--

Selected Value'

2.1 -

0.9 1.2 1.5 1.8 2.1 2.5 3 .O

0.90 1.16 1.48 1.82 2.19 2.58 3.00

AI

V

Bond Energiesb

2.17 0.96 1.38 1.91 2.52 3.01 3.47 3.94

""I

23

Atomic Radius and Nuclear Screening"

1.o 1.3 1.6 1.35 1.65 1.8 1.5 1.6 -2.1 1.4 1.5 2.3

-

0.80 1.o 1.3 1.6 1.4 1.7 1.9 1.4 1.6 2.2 1.4 1.5 2.5

875

Electronegativities as Derived from Atomic Number

Element

26

Fe

27 28 29

co Ni

30 31 32 33 34 35 36 37 38 39 40 41 42

cu

Ma Mo" MoV

'

44 45 46 47 48 49 50

Ru Rh Pd

4%

51

Sb

52 53 54. 55 56 57 58

59 60 61

876

Cd In Sn

Tcv TcV' '

In' Sn" Sn" Sb"' Sbv

Te

1.7

1.48 1.77 2.04 2.35 2.68 -

1.65 1.8 1.7 1.7 1.8 2.0 1.5 1.6 1.7 2.0 2.3 (2.4) 2.8 -

0.79 0.98 1.21 1.48 1.7 1.6 2.1 1.9 2.3

1.36 1.61 1.82 2.08 2.36 -

1 Xe cs

Ba La Ce Pr Nd 11

Selected Value'

cuiI

Ge As Se Br Kr Rb Sr Y Zr Nb

Tc

Bond Energiesb

CUI

Zn Zn" Ga

43


Ce"'

0.78 0.93 1.2 1.2 1.2 1.2 1.2

1.8 1.7 1.8

1.8 2.0 1.5 1.5 1.8 2.0 2.4 2.8 -

0.8 1.o

0.8

1.o 1.2 1.5 1.7 1.6 2.1 1.9 2.3 2.0 2.1 2.0 1.8 1.5 1.5 1.7 1.8 1.8

1.2(1.3) 1.4 (1.6) 1.6 1.6 -2.1

--

2.05 2.1 2.0 1.8 1.5 1.6 1.65 (1.7) 1.8 1.8 (1.8) 2.1 2.1 2.6 (2.5) 0.7 0.85 (0.9) 1.1 1.05 1.1

--

2.1 2.55 0.75 0.9 1.1 1.1 1.1 1.2 1.2

Electronegativities as Derivedfrom Atomic Number 62 63 64 65 66 67 68 69 70 71 72 73

Element

Sm Gd

Tb DY

Er

Tm Yb Lu Hf

w

75

Re Ir Pt Au

80

Hg

81

TI

82

Pb

83 84

Bi

WlV W"'

ReV

1.8

ReV"

2.2

--

85

At

90

Rn Fa Ra Ac Th

91

Pa

92

u

93 94 95

NP Pu

Am

1.3 1.4

Selected Value'

------

TI' TI"' Pb" Pb"

1.8 1.9 1.5 1.9 1.6 1.8

1.34 1.56 1.8 2.0 2.2 0.76 0.92 1.1

1.o

Th" Th" Pal1' Pa"

1.4 1.3 1.7

UJV U"'

1.5 1.9 1.1 1.3 1.3

1.5 1.9 1.6 1.8 1.8 2.0 2.2 -

1.8 2.0 2.4 -

--0.8 .o 0.7 1

1.1

-

1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.2 1.4 1.7 1.6 2.0 1.8 2.2 2.0 2.1 2.1 2.3 1.8

1.6 -2.1

Hg' Hg"

1.2 1.1

1.3

2.1 2.1 2.1 2.3

Po

86 87 89

Ta"' Tav

1.2 1.1 1.2 1.4 1.3 1.7 1.6 2.0

os

76 77 78 79

88

1.2 1.2

Ho

74

Bond Energiesb

1.2 1.1 1.2 1.2 1.2

EU

Ta


1.4 1.3

--

0.7 0.9 1.1 1.o 1.4 1.3 1.7 1.4 1.9 1.1 1.3 1.3

877

Electronegativities as Derivedfrom Atomic Number

96 91 98

Element

Cm Bk Cf


1.3 1.3 1.3

Bond Energiesb

Selected Value'

--

1.3 1.3 1.3

"From method derived by W. Gordy, Phys. Rev., 69,604 (1946). bPauling's scale [ J . Am. Chem. SOC.,54, 3570 (1932)l extended by Haissinsky [ J . Phys. Radium, 7, 7 (1946)l. 'Values selected by Gordy and Thomas from those derived by four different methods [ J . Chem. Phys., 24, 439 (1956)l.

878

Appendix

H

COMPUTATIONAL PROCEDURE FOR THE VIBRATIONAL EIGENVALUE PROBLEM

Relation between internal (S)and normal ( Q )coordinates

S=LQ Vibrational, kinetic, and potential energy matrices -,

2T=SG-lS, 2V=SFS Vibrational secular determinant (GF nonsymmetric)

GFL = LA -+ 1 GF - EA I =0

*

A diagonal matrix of eigenvalues &, L eigenvector matrix, E a unit matrix. Diagonalize symmetric kinetic energy matrix and normalize orthogonal eigenvector matrix t8GL8=AG, LG=L,$A&’2[(LG)ij=(L$)ij(Ahi2)j] AG diagonal matrix of eigenvalues AG of G. Nonorthogonal transformation LGmatrix thus defined so

-

L G L G =G

*

Transformation LGoperates on F to give a new secular determinant

L&FL~LG ‘L =LA

FL, =L ~ A +I F - EA^ =o F=tGFLG and LF=LGIL

*

Diagonalization of the symmetric transformed F-matrix gives the frequencies and the orthogonal eigenvector matrix

L,FL,

*

=A

The orthogonal eigenvector matrix gives the original nonorthogonal transformation matrix

L=LGLF 879

880

*

APPENDIX

H

Computational checks

LLG,~FL=A The eigenvalue problem in the form FL=G-'LA or IF-G-'A/=O can be solved by the same sequence of steps. F and G matrices in terms of symmetry coordinates may, of course, be used.

Appendix

I

ENERGIES A N D RELATIVE INTENSITIES OF NUCLEAR QUADRUPOLE HYPERFINE STRUCTURE

The function Y ( J , I , F ) and the relative intensities are tabulated for the most common nuclear spins Z=t, 1, 5, 3,g and 4. The Y ( J , I , F ) is defined as Y ( J ,I , F ) =

+

+

$C(C 1)- Z(Z l)J(J+1) 2(2J+ 3)(2J- 1)1(2Z- 1)

where C = F ( F + 1 ) - I ( l + l)-J(J+1)

and F takes on the values

J+Z, J+Z-1,. . ., IJ-Zl The quadrupole energies are given by

Here the appropriate qJ for a linear, symmetric, or asymmetric rotor must be used (see Chapter IX). The relative intensities [see e.g., (15.142)] have been normalized such that the sum of the intensities of the various hyperfine components of a transition is 100. The intensities for the J + 1+ J transitions are obtained by reversal of the arrows in the entries for J + J + 1. Thus the entries for F+F+ 1 correspond to F + 1-F and those for F - F - 1 to F - 1-F.

881

Relative Intensities

o t 1

;

t 2 . :

3 3 4

3

.: ;

; 5 y P

6

9

$ 7

y 13

T

8

7 15

T

9

9 17

T

10 11 12

q 9 y

T

21

0

+?

0 0 0 0 0 0 0 0

23

T

13 25

T

14

9 27 T

15 29

T

882

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

66.667 60.000 33.333 57.143 40.000 55.556 42.857 54.545

33.333 6.667

0.0

0 0 0

0.0 0.0 11.111

0.0

0.0

0 0 0 0

1.010

0

0.0

44.444

0.0

0

1.235

53.846 45.454 53.333 46.154 52.941 46.667 52.632 47.059 52.381 47.368 52.174 47.619 52.000 47.826 51 3 5 2 48 .OOO 5 1.724 48.148 51.613 48.276 51.515 48.387

0.699

0

0.0

0.0

0.826

0.0

0 0 0

0.392

0

0.0

0 0

0.0 0.444 0.0

2.857

0.0 1S87

0.513

0.310

0.0 0.25 1

0.0 0.207

0.0 0.174

0.0 0.148

0 .o 0.128

0.0 0.111

0.0 0.098

0.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4.000

0.0 2.041

0.0 0.592

0.346

0.0 0.277

0.0 0.227

0.0 0.189

0.0 0.160

0.0 0.137

0.0 0.119

0.0 0.104

0.0 55.556 22.222 56.000 36.000 55.102 40.816 54.321 43.210 53.719 44.628 53.254 45.562 52.889 46.222 52.595 46.713 52.355 47.091 52.154 47.392 5 1.985 47.637 51.840 47.840 51.715 48.01 1 5 1.605 48.157 51 .509 48.283

Relative Intensities J+ J+ 1 J

F

Y(J,I , F )

F-tFfl

F+F

J+ J F-F-1

F++F+l

F+F

I= 1 0 1

1 2 1

0 2

3

4

5

6

7

8

3 2 1 4 3 2 5 4 3 6 5 4 7 6 5 8 7 6 9

8 9

7 10 9

8 10

11

12

13

11 10 9 12 11 10 13 12 11 14 13 12

0.0 0.050000 - 0.250000 0.500000 0.071429 - 0.250000 0.250000 0.038888 -0.250000 0.200000 0.090909 -0.250000 0.178571 0.0961 54 -0.250000 0.166667 0.100000 - 0.250000 0.159091 0.102941 - 0.250000 0.1 5 3846 0.105263 - 0.250000 0.150000 0.1071 43 -0.250000 0.147059 0.108696 -0.250000 0.144737 0.1 10000 -0.250000 0.142857 0.1 11 11 1 -0.250000 0.141 304 0.112069 -0.250000 0.140000

55.556 46.667 25.000 11.111 42.857 29.630 20.000 40.741 31.250 23.810 39.394 32.000 25.926 38.462 32.407 27.273 37.778 32.653 28.05 37.255 32.813 28.889 36.842 32.922 29.412 36.508 33.000 29.825 36.232 33.058 30.159 36.000 33.102 30.435 35.803 33.136 30.667 35.632 33.163 30.864

33.333 8.333 8.333

0.0 3.704 3.704

0.0 2.083 2.083

0.0 1.333 1.333

0.0 0.926 0.926

0.0

11.111 0.556

0.0 0.0

0.0 0.0

13.889 11.111

0.106

0.0

0.0 0.0

5.185 5 .OOO

0.033

0.0

0.0 0.0

2.679 2.646

0.01 3

0.0

0.0 0.0

1.630 1.620

0.006

0.0

0.0 0.0

1.094 1.091

0.003

0.0

0.0 0.0

0.785 0.783

0.521 0.521

0.002

0.0

0.0 0.0

0.590 0.590

0.412 0.412

0.001

0.0

0.0 0.0

0.460 0.460

0.001

0.0

0.0 0.0

0.368 0.368

0.001

0.0

0.0 0.0 0.000 0.0 0.0 0.000 0.0 0.0 0.000 0.0 0.0

0.302 0.302

0.680 0.680

0.0 0 .o 0.0 0.333 0.333

0.0 0.275 0.275

0.0 0.231 0.231

0.0 0.197 0.197

0.0 0.170 0.170

0.0

0.0 0.252 0.252

0.0 0.213 0.213

0.0 0.183 0.183

0.0 41.667 8.333

0.0 41.481 23.148 15.000 40.179 28.009 21.164 39.1 11 30.083 24.306 38.300 31.148 26.182 37.677 31.765 27.422 37.187 32.153 28.299 36.795 32.414 28.952 36.474 32.597 29.456 36.206 32.730 29.857 35.980 32.830 30.183 35.787 32.907 30.454 35.620 32.968 30.682

883


J+ J + 1 J

F

Y(J,I,F)

F+F+l

F+F

J-+ J F-F-I

F-F+l

F-+F

I = 1 (continued) 14

15

15 14 13 16 15 14

0.1 12903 -0.250000 0.138889 0.113636 -0.250000 0.137931

0.0 0.050000 - 0.200000 0.250000 0.07 1429 - 0.1 78571

0.0

5

I

5 ' : 11

T 9

P

7

7

6

45 13 ?

884

0.250000 0.083333 - 0.166667 - 0.050000 0.200000 0.090909 -0.159091 - 0.07 1429 0.178571 0.096154 - 0.1 53846 -0,083333 0.166667 0.100000 - 0.150000 - 0.090909 0.1 59091 0.102941 - 0.147059 -0.096154 0.1 53846 0.105263 - 0.1 44737 - 0.100000

35.484 33.185 31.035 35.354 33.203 31.183

50.000 40.000 2 1 .ooo 8.333 35.714 24.490 16.000 10.000 33.333 .25.463 19.133 14.286 31.818 25.785 20.741 16.667 30.769 25.888 21.694 18.182 30.000 25.905 22.316 19.231 29.412 25.887 22.750 20.000 28.947 25.854 23.068

0.148 0.148

0.0 0.130 0.130

0 .o

33.333 9.000 10.667 8.333 4.082 5.224 4.000

0.0 2.315 3.023 2.296

0.0 1.488 1.959 1.481

0.0 1.036 1.369 1.033

0.0 0.762 1.010 0.761

0.0 0.584 0.775 0.583

0.0 0.462 0.613 0.461

0.000 0.0 0.0 0.000 0.0 0.0

16.667

1 .ooo

1.667

0.0 0.204 0.286

0.0 0.0 0.066 0.085

0.0 0.158 0.158

0.0 0.139 0.139

0.0 0.0 15.000 13.889

0.0 5.714 7.000 5

.ooo

0.0

0.0 0.0

2.976 3.827 2.857

0.028 0.034

0.0 1.818

0.0 0.0

2.377 1.786

0.013 0.016

0.0

0.0 0.0

1.224 1.612 1.212

0.007

0.0

0.008 0.0 0.0

0.879 1.162 0.874

0.004 0.005

0.0 0.0 0.003 0.003

0.0

0.0 0.662 0.877 0.659

0.0 0.516 0.685

35.474 33.017 30.876 35.345 33.056 31.044

0.0 35.000 4.444 2.778 34.286 17.286 8 .OOO 5 32.738 21.769 14.745 11.429 31.515 23.583 1 8.060 14.881 30.594 24.437 19.904 lti.970 29.890 24.882 21.041 18.357 29.338 25.128 21.797 19.341 28.896 25.270 22.330

.ooo


J+ J + 1

J

F

Y(J, I , F )

F-+F+1

F+F

J+ J F+F-l

F++F+l

F+F

I =+(continued)

9

f f Y 17 T

Y 10

23

T

12' 19 T

17

T

11

25

T

Y

f 19 T

12

21

T

25

2

23

T

13

f 29 T

27

T

25

T

23

T

14

31

T

29

T

27

T

25

T

15

33

T

31

Y

v

0.150000 0.107143 - 0.142857 - 0.102941 0.147059 -0.108696 - 0.141 304 - 0.105263 0.144737 0.110000 -0.140000 - 0.107143 0.142857 0.1 11 11 1 -0.138889 -0.108696 0.141 304 0.1 12069 - 0.137931 - 0.1 10000 0.140000 0.1 12903 -0.137097 -0.1 111 11 0.138889 0,113636 -0.136364 - 0.1 12069 0.137931

20.588 28.571 25.816 23.310 21.053 28.261 25.778 23.500 21.429 28.000 25.740 23.653 21.739 27.778 25.704 23.778 22.000 27.586 25.671 23.883 22.222 27.419 25.640 23.971 22.414 27.273 25.61 1 24.047 22.581

0.0

0.0

0.515

0.374 0.497 0.374

0.002 0.002

0.0

0.0 0.309 0.412 0.309

0.0 0.260 0.346 0.260

0.0 0.222 0.295 0.222

0 .o

0.191 0.254 0.191

0 .o

0.166 0.222 0.166

0.0 0.146 0.195 0.146

0.0

0.0 0.0 0.001 0.001

0.0 0.0 0.001

0.001 0.0 0.0 0.001 0.001

0.0 0.0 0.000 0.001

0.0 0.0 0.000 0.000 0.0 0.0 0.000 0.000

0.0 0.0

0.414 0.549 0.41 3

0.0 0.339 0.450 0.338 0.0 0.283 0.376 0.282

0.0 0.239 0.318 0.239

0.0 0.205 0.273 0.205

0.0 0.178 0.237 0.178

0.0 0.156 0.207 0.156

20.074 28.534 25.353 22.722 20.640 28.233 25.401 23.021 21.090 27.978 25.428 23.255 21.457 27.761 25.442 23.442 21.761 27.573 25.447 23.596 22.017 27.408 25.447 23.723 22.236 27.264 25.443 23.830 22.425

I=$

0

:

1

;

2

4 t

H T

t t

0.0 0.05oooO -0.16oooO 0.140000 0.071429 - 0.121429 -0.071429 0.071429

44.444 33.333 17.143 6.222 28.571 19.400 12.245 6.857

33.333 9.524 12.190 9.333 4.409 6.450 6.612 5.418

22.222 1.587 4.000 6.667 0.353 0.816 1.143 1.058

0.0 0.0 15.873 15.556

0.0 6.173 8.571

8.000

0.0 28.571 1.905 6.667 27.161 11.922 3.429 9.148

885


J+ J + 1 J

F

Y(J, I , F,

F+F+l

F+F

J+ J F-*F-1

F-+F+l

F+F

I = $ (continued)

i3

0.200000 0.083333

+

e

13

T

Y H

-0.100000 -0.100000 - 0.006667 0.11m 0.200000 0.090909 -0.086364 - 0.107792 - 0.037662 0.071 429 0.178571 0.096154 0.076923 -0.110256 -0.053846

-

i

0.050000 0.050000

1

0.166667 0.1 00000 0.070000 0.1 10909 -0.063636 0.036364 0.159091 0.102941 -0,064706 -0.110860 -0.070136 0.026923 0.152846 0.105263 -0.060526 - 0.110526 - 0.074737 0.020000 0.150000 0.1071 43 -0.057143

i 6

y 15

T

13

T

9

P 2

7

9 17

T 15

T

13 -211

-r

8

e

% 19

T

17

T

15

T

13 T 11

T

9

9 21

T

886

-

2.963 25.926 19.697 14.550 10.393 7.143 4.762 24.242 19.580 15.598 12.256 9.524 7.407 23.077 19.373 16.138 13.350 10.999 10.999 9.091 22.222 19.160 16.440 14.051 11.988 10.256 21.569 18.963 16.619 14.529 12.692 11.111 21.053 18.788 16.729 14.873 13.217 11.765 20.635 18.634

3.704 2.525 3.848 4.157 3.628 2.381

0.0 1.632 2.532 2.785 2.463 1S 8 7

0.0 1.140 1,785 1.981 1.758 1.122 1.122

0.0 0.840 1.324 1.475 1.312 0.833

0.0 0.645 1.020 1.140 1.014 0.641

0.0 0.511 0.809 0.906 0.806

0.0

5.185

0.120 0.265 0.340 0.265

0.0

0.0 0.0 0.052 0.110 0.135 0.096

0.0 0.0 0.026 0.054 0.064 0.043

0.0 0.0 0.0 0.014 0.029 0.034 0.022

0.0 0.0 0.009 0.017 0.020 0.013

0.0 0.0 0.005

3.247 4.850 5.102 4.286 2.646

0.0 1.994 3.064 3.333 2.910 1.852

0.0 1.347 2.098 2.314 2.043 2.043 1.299

0.0 0.970 1.522 1.691 1.499 0.950

0.0 0.731 1.153 1.286 1.142 0.722

0.0

0.508 0.0

0.008 0.0 0.0

0.571 0.903 1.009 0.897 0.566

0.414 0.658

0.004 0.007

0.458

0.01 1 0.012

0.0

1.481 25.325 15.713 9.095 4.898 2.593 2.116 23.932 17.164 12.121 8.571 6.349 5.556 22.896 17.767 13.770 10.795 8.780 8.780 7.192 22.107 18.021 14.736 12.196 10.372 9.307 21.491 18.166 15.339 13.128 11.470 10.390 20.998 18.134 15.735 13.780 12.263 11.199 20.595 18.115

Relative In tensit ies

J+ J - t 1 J

F

Y(J,I,F)

F-tF+l

F+F

J-* J F-F-1

F-F+I

F-*F

I =+(continued 19

T I7

T 15

T

9

10

? 23

T

21

T 19

T

17

T

15

T

1

y 25

T 23

T

21

T

19

T

17

T

12

9

27 T

25

T 23 T 21 -2 19

T

13

y

29

T 27 T

25

T

23

T 21

T

14

9 31

T

29

T 27

T

25

T

23

T

15

9 33

T

31

T 29 T 21 T 25

T

- 0.1 10084 -0.078151 0.014706 0.147059 0.108696 - 0.054348 -0.109611 - 0.080778 0.010526 0.144737 0.1 10000 -0.052000 - 0.109 143 -0.082857 0.007143 0.142857 0.1 1 1111 - 0.050000 - 0.108696 - 0.084541 0.004348 0.141304 0.1 12069 - 0.048276 -0,108276 - 0.085931 0.002000 0.140000 0.1 12903 -0.046774 -0.107885 - 0.087097 0.0 0.138889 0.113636 - 0.045455 - 0.107524 - 0.088088 - 0.001724 0.137931

16.799 15.129 13.622 12.281 20.290 18.498 16.844 15.326 13.944 12.698 20.000 18.379 16.873 15.482 14.205 13.044 19.753 18.273 16.891 15.608 14.421 13.333 19.540 18.179 16.903 15.711 14.603 13.580 19.355 18.095 16.909 15.797 14.758 13.793 19.192 18.019 16.912 15.870 14.892 13.979

0.737 0.656 0.413 0.0 0.343 0.545 0.61 1 0.543 0.342 0.0 0.288 0.459 0.514 0.458 0.280.0 0.246 0.391 0.439 0.391 0.245 0 .o 0.212 0.338 0.379 0.337 0.212 0 .o 0.185 0.294 0.331 0.294 0.184 0.0 0.162 0.259 0.291 0.259 0.162 0.0

0.008 0.005 0.0 0.0 0.003 0.005 0.005 0.003 0.0 0.10 0.002 0.003 0.004 0.002 0.0 0.0 0.001 0.003 0.003 0.002 0.0 0.0 0.001 0.002 0.002 0.001 0.0 0.0 0.001 0.001 0.002 0.001 0.0 0.0 0.001 0.001 0.001 0.001 0.0 0.0

0.726 0.813 0.722 0.455 0.0 0.375 0.596 0.668 0.594 0.373 0.0 0.313 0.498 0.559 0.497 0.312 0.0 0.265 0.422 0.474 0.421 0.265 0.0 0.228 0.363 0.407 0.362 0.227 0.0 0.197 0.315 0.353 0.314 0.197 0.0 0.173 0.276 0.310 0.275 0.173

16.005 14.255 12.858 11.826 20.260 18.076 16.196 14.611 13.318 12.325 19.977 18.029 16.335 14.887 13.684 12.731 19.735 17.979 16.437 15.105 13.981 13.069 19.526 17.928 16.514 15.280 14.226 13.353 19.343 17.879 16.573 15.424 14.431 13.596 19.182 17.831 16.619 15.544 14.606 13.806

887

~


J+ J

J-r J f 1

J

F

0 1

2

3

;

13

-2-

11

T

e

-i

; ;

4

4

15

T

13

T

11

T 9

t

1

x

L

5

17

T

15

-

113 1

11

T 9

L

6

; 19

T

17

T

15

T

13

T

888

0.0 0.050000 - 0.142857 0.107143 0.071429 - 0.096939 - 0.081633 0.025510 0.15306 1 0.083333 -0.071429 -0.097619 - 0.047619 0.035714 0.1 19048 0.178571 0.090909 - 0.055195 - 0.097403 - 0.071892 - 0.009276 0.065399 0.132653 0.178571 0.096154 -0.043956 - 0.094322 -0.082418 -0.032053 0.036630 0.107143 0.166667 0.100000 -0.035714 -0.090909 -0.087662 -0.045455 0.018831 0.090909 0.159091

F-+F+l

F+F

F-rF-1

41.667 30.000 15.278 5.357 25 .OOO 16.883 10.476 5.612 2.143 22.222 16.827 12.311 8.598 5.612 3.274 1.488 20.455 16.485 13.054 10.124 7.660 5.268 4.000 2.778 19.231 16.120 13.374 10.974 8.900 7.139 5.682 4.545 18.333 15.789 13.510 11.484 9.700 8.152 6.838 5.769

33.333 9.722 12.698 9.643 4.545 6.926 7.483 6.531 4.286 2.618 4.196 4.885 4.838 4.209 3.175 2.083 1.697 2.785 3.338 3.428 3.127 2.494 1.556 0 .o 1.188 1.974 2.400 2.504 2.318 1.865 1.136 0.0 0.877 1.470 1.801 1.893 1.764 1.421 0.855 0.0

25 .OOO I .944 5.357 10.000 0.455 1.190 2.041 2.857 3.571 0.160 0.406 0.661 0.850 0.893 0.694 0.0 0.071 0.175 0.275 0.337 0.325 0.212 0.0 0.0 0.036 0.087 0.1 34 0.159 0.146 0.087 0.0 0.0 0.020 0.048 0.073 0.085 0.075 0.043 0.0 0.0

F-F+1

0.0 0.0 16.204 16.071 0.0 6.364 9.167 8.929 6.000 0.0 3.365 5.276 5.952 5.612 4.464 2.679 0.0 2.074 3.365 3.977 4.012 3.571 2.750 1.620 0.0 1.404 2.318 2.797 2.893 2.652 2.1 10 1.273 0.0 1.012 1.689 2.060 2.156 1.998 1.603 0.962

F+F

0.0 25.463 1.058 8.929 23.636 9.470 1.905 0.071 4.000 21.635 12.787 6.629 2.721 0.638 0.0 0.893 20.148 14.005 9.324 5.899 3.527 2.012 1.185 1.157 19.051 14.460 10.794 7.947 5.820 4.329 3.435 3.273 18.219 14.607 11.635 9.246 7.385 6.015 5.128 4.808

Relative intensities

J+ J + 1 J

7

F

21

T

19

T

17

T

15

-T 13

T

11

T

4 8

4

23

T

21

T

19

T

17

T

15

T

13

T

11

T

9

e

25

T

?? 21 T

19

T

f 15

T

13

T

11

T

10

27

T 25

T 22

T

21

T

19

T

17

T

15

T

13

T

11

29

T

21

T

25

T

23

T

21

T

19

T

Y

15

T

Y(J,I , F,

F-+F+1

0.102941 -0.029412 -0.087750 -0.090498 - 0.054137 0.006787 0.079670 0.153846 0.105263 -0.024436 - 0.084962 - 0.092105 -0.060150 -0.001880 0.071429 0.150000 0.107143 - 0.020408 -0.082533 -0.093037 - 0.064526 - 0.008403 0.065126 0.147059 0.108696 - 0.01708 1 - 0.080418 - 0.093576 - 0.067833 -0.013485 0.060150 0.144737 0.1 10000 - 0.014286 - 0.078571 -0.093877 -0.070408 -0.017551 0.056122 0.142857

17.647 15.502 13.558 11308 10.245 8.866 7.670 6.667 17.105 15.256 13.563 12.024 10.634 9.391 8.296 7.353 16.667 15.043 13.546 12.173 10.922 9.791 8.781 7.895 16.304 14.859 13.518 12.279 11.142 10.104 9.167 8.333 16.000 14.698 13.484 12.357 11.314 10.356 9.482 8.696

F+F

0.674 1.135 1.398 1.476 1.379 1.111 0.663 0 .o 0.534 0.903 1.115 1.181 1.105 0.889 0.528 0.0 0.433 0.734 0.910 0.965 0.903 0.727 0.430 0.0 0.359 0.609 0.756 0.802 0.752 0.604 0.357 0.0 0.302 0.513 0.637 0.678 0.635 0.510 0.300 0.0

J-, J F-F-1

0.012 0.029 0.043 0.049 0.042 0.023 0.0 0.0 0.008 0.018 0.027 0.030 0.026 0.014 0.0 0.0 0.005 0.012 0.018 0.020 0.017 0.009 0.0 0.0 0.004 0.008 0.012 0.013 0.01 1 0.006 0.0 0.0 0.003 0.006 0.009 0.009 0.008 0.004 0.0 0.0

F-F+I

0.0 0.764 1.283 1.576 1.659 1.545 1.242 0.741 0.0 0.597 1.007 1.242 1.312 1.225 0.985 0.585 0.0 0.479 0.810 1.003 1.062 0.993 0.798 0.472 0.0 0.393 0.666 0.826 0.876 0.820 0.659 0.389 0.0 0.382 0.557 0.692 0.735 0.688 0.553 0.326

F+F

17.569 14.620 12.141 10.099 8.463 7.213 6.351 5.926 17.050 14.573 12.458 10.682 9.227 8.084 7.254 6.768 16.626 14.500 12.661 11.094 9.787 8.735 7.940 7.422 16.274 14.417 12.794 11.393 10.208 9.235 8.476 7.944 15.976 14.332 12.882 11.617 10.534 9.629 8.905 8.370 a89


J+ J + 1 J

12

F

31

T

29

T

27

T

25

T

23

T

21 -T 19

T

17

T

13

33

T

31

1

29

T

21

T

? 23

T

21

T

19

T

14

35

T

33

T

31

T

29

T

27

T

25

T

23

T

"1' 15

37

T

35

T

33

T

Y 29

T 27

T

25

T

23

T

0

4

1 y 4 4 2 890

9

Y(J, I , F )

F-+F+I

0.11 1 1 1 1 -0.011905 -0.076950 -0.094030 - 0.072464 - 0.020876 0.052795 0.141304 0.112069 -0.009852 - 0.075517 -0.094089 -0.074138 -0.023645 0.050000 0.140000 0.112903 - 0.008065 - 0.074245 - 0.094086 - 0.075525 - 0.025986 0.047619 0.138889 0.113636 - 0.006494 - 0.073108 - 0.094044 - 0.076690 - 0.027989 0.045566 0.137931

15.741 14.558 13.449 12.414 11.451 10.561 9.744 9.000 15.517 14.433 13.413 12.457 1 1.563 10.732 9.964 9.259 15.323 14.323 13.379 12.490 11.656 10.876 10.152 9.483 15.152 14.224 13.345 12.515 11.733 10.999 10.314 9.677

0.0

0.050000 -0.1333333 0.091667 0.071429

40.000 28.000 14.182 4.889 22.857

F+F

0.257 0.438 0.545 0.579 0.543 0.436 0.256

J-, J

F+F-1

0.002 0.004 0.006 0.007 0.006 0.003 0.0

0.0

0.0

0.222 0.378 0.471 0.501 0.470 0.377 0.221

0.001 0.003 0.005 0.005 0.004 0.002 0.0

0 .o

0.0

0.194 0.330 0.411 0.438 0.410 0.329 0.193

0.001 0.003 0.004 0.004 0.003 0.002

0.0

0.170 0.290 0.362 0.385 0.361 0.290 0.170 0.0

33.333 9.818 12.929 9.778 4.615

0.0 0.0

0.001 0.002 0.003 0.003 0.002 0.001 0.0 0.0

26.667 2.182 6.222 12.000 0.527

F*F+l

0.0

0.278 0.473 0.588 0.625 0.585 0.470 0.276 0.0

0.238 0.406 0.505 0.537 0.504 0.404 0.237 0.0

0.207 0.353 0.439 0.467 0.438 0.351 0.206 0.0

0.181 0.309 0.385 0.410 0.384 0.308 0.181

0.0 0.0

16.364 16.296 0.0

F-F

15.722 14.249 12.940 11.788 10.790 9.945 9.254 8.724 15.502 14.170 12.977 11.920 10.996 10.203 9.544 9.022 15.310 14.096 13.001 12.205 11.164 10.418 9.787 9.277 15.141 14.026 13.016 12.109 11.303 10.598 9.995 9.497

0.0

23.636 0.673 10.370 21.538

Relative In tensities

J+ J + 1 F+F+l

F+F

J+ J F+F-1

F*F+1

F+F

I =$ (continued) 11

T

E

4 4 3

$5

13

T

11

T 9

f

4

; 4

y 15

T 13

T

11

T

Q 4 43 1 f 5

9 17

T

15

T

-0.083333 - 0.083333 0.005952 0.130952 0.83333 - 0.055556 -0.091667 -0.061111 0.005556 0.083333 0.152778 0.090909 - 0.037879 -0.086580 -0.079545 - 0.037879 0.020563 0.081169 0.132576 0.166667 0.0961 54 -0.025641 -0.080128 - 0.085470 -0,057692 -0.010684 0.043803 0.096154 0.138889 0.166667 0.100000 -0.016667 - 0.074242 - 0.087121 - 0.068 182 - 0.028788 0.021212 0.073485 0.121212 0.159091

15.385 9.455 4.952 1.796 20.000 15.111 10.989 7.576 4.815 2.653 1.048 18.182 14.631 11.539 8.876 6.612 4.714 3.152 1.891 0.889 16.923 14.170 11.724 9.566 7.682 6.052 4.662 3.497 2.545 1.818 16.000 13.769 11.757 9.955 8.352 6.938 5.706 4.650 3.768 3.077

7.161 7.879 6.966 4.490 2.667 4.376 5.245 5.387 4.913 3.918 2.444 1.733 2.921 3.625 3.906 3.826 3.448 2.836 2.069 1.333 1.215 2.078 2.626 2.892 2.910 2.712 2.331 1.790 1.09 1

0.0 0.898 1.551 1.980 2.206 2.249 2.125 1.849 1.429 0.848

0.0

1.455 2.667 4.082 5.714 0.190 0.513 0.909 1.323 1.701 2.000 2.222 0.086 0.226 0.392 0.551 0.673 0.727 0.679 0.485

0.0 0.044 0.115 0.196 0.269 0.318 0.326 0.280 0.168

0.0 0.0 0.025 0.065 0.109 0.147 0.169 0.168 0.137 0.075

0.0 0.0

6.462 9.455 9.333 6.286

0.0 3.429 5.495 6.364 6.190 5.102 3.143

0.0 2.118 3.526 4.308 4.545 4.321 3.714 2.800 1.630

0.0 1.435 2.439 3.055 3.329 3.306 3.030 2.545 1.891 1.091

0.0 1.036 1.781 2.262 2.505 2.536 2.378 2.051 1.570 0.923

8.084 1.212 0.381 5.714 19.429 11.077 5.285 1.732 0.136 0.327 2.571 17.882 12.134 7.722 4.480 2.245 0.854 0.152 0.015 0.593 16.746 12.490 9.052 6.344 4.275 2.755 1.697 1.018 0.655 0.727 15.887 12.567 9.802 7.540 5.728 4.317 3.263 2.533 2.122 2.154 891

Relative In tensit ies

J-t J + 1 J

F

Y(J, I , F )

F-tF+l

F-tF

J-t J

F-tF-1

F*F+l

F-tF

I =$ (continued) 0.102941 -0.009804 -0.069193 - 0.087104 - 0.074284 -0.040347 0.006222 0.058069 0.108974 0.153846 0.105263 -0.004386 - 0.064912 - 0.086403 -0.078070 - 0.048246 -0.004386 0.046930 0.100000 0.150000 0.107143 0.0 -0.061275 - 0.085434 -0.080532 -0.053922 - 0.012255 0.038515 0.093137 0.147059 0.108696 0.003623 -0.058162 -0.084382 - 0.0821 89 -0.0581 62 -0.018307 0.031941 0.087719 0.144737

15.294 13.427 11.726 10.183 8.792 7.549 6.448 5.487 4.667 4.000 14.737 13.137 11.666 10.320 9.095 7.986 6.993 6.117 5.348 4.706 14.286 12.889 11.596 10.403 9.309 8.310 7.406 6.595 5.879 5.263 13.913 12.676 11.523 10.454 9.466 8.557 7.727 6.975 6.303 5.714

0.691 1.200 1.543 1.731 1.776 1.689 1.477 1.141 0.667 0.0 0.547 0.956 1.234 1.390 1.433 1.368 1.198 0.924 0.535 0 .o 0.444 0.778 1.008 1.140 1.178 1.126 0.987 0.760 0.437 0.0 0.368 0.646 0.839 0.950 0.984 0.942 0.826 0.635 0.364 0.0

0.015 0.039 0.065 0.087 0.098 0.095 0.075 0.039 0.0 0.0 0.010 0.025 0.041 0.054 0.061 0.058 0.044 0.023 0.0 0.0 0.007 0.017 0.027 0.036 0.040 0.037 0.028 0.014 0.0 0.0 0.005 0.012 0.019 0.025 0.027 0.025 0.019 0.009 0.0 0.0

0.0 0.783 1.356 1.737 1.941 1.984 1.879 1.636 1.259 0.735 0.0 0.612 1.066 1.373 1.543 1.586 1.510 1.320 1.016 0.588 0.0 0.491 0.859 1.111 1.253 1.293 1.234 1.081 0.831 0.478 0.0 0.403 0.707 0.916 1.037 1.072 1,025 0.898 0.690 0.396

15.217 12.528 10.240 8.322 6.741 5.470 4.485 3.771 3.339 3.265 14.682 12.440 10.503 8.849 7.459 6.316 5.406 4.723 4.278 4.118 14.246 12.334 10.662 9.215 7.980 6.947 6.106 5.456 5.006 4.785 13.883 12.224 10.758 9.476 8.368 7.427 6.648 6.030 5.581 5.319

Relative intensities

J-t J + 1 J

F

Y(J, I , F)

F+F+l

F+F

J+ J F+F-1

F-F-tI

F+F

I = Q (continued) 11

31

T

29

T

21

T

25

T

23

T

21

T

19

T

17

T

15

T

13

T

12

33

T

31

T

29

T

27

T

25

T

23

T

21

T

19

T

Y 15 T

13

35

T

33

T

? 29

T

Y25 T

23

T

21

T

19

T

17

T

14

37

T

35

T

33

T

31

T

29

T

27

T

25

T

23

T

21

T

19

T

0.110000 0.006667 -0,055476 -0.083333 -0.083333 -0.061429 - 0.023095 0.026667 0.083333 0.142857 0.111111 0.009259 -0.053140 -0.082327 - 0.084138 - 0.064010 -0.026973 0.022343 0.079710 0.141 304 0.112069 0.01 1494 -0.011092 -0.081379 -0.084713 - 0.066092 - 0.030172 0.018736 0.076667 0.140000 0.1 12903 0.013441 -0.049283 - 0.080496 -0.085125 -0.067802 -0.032855 0.015681 0.074074 0.138889

13.600 12.490 11.452 10.483 9.583 8.749 7.983 7.282 6.650 6.087 13.333 12.328 11.384 10.499 9.672 8.903 8.190 7.535 6.938 6.400 13.103 12.186 11.320 10.506 9.741 9.027 8.362 7.746 7.181 6.667 12.903 12.059 11.261 10.506 9.796 9.129 8.505 7.925 7.388 6.897

0.310 0.545 0.709 0.804 0.833 0.798 0.700 0.358 0.307 0.0 0.264 0.465 0.606 0.689 0.714 0.685 0.601 0.461 0.262 0.0 0.228 0.402 0.524 0.596 0.619 0.594 0.521 0.399 0.227 0.0 0.199 0.351 0.458 0.521 0.541 0.519 0.455 0.349 0.198 0 .o

0.003 0.008 0.014 0.017 0.019 0.017 0.013 0.006 0.0 0.0 0.002 0.006 0.010 0.013 0.014 0.012 0.009 0.004 0.0 0.0 0.002 0.005 0.007 0.009 0.010 0.009 0.007 0.003 0.0 0.0 0.001 0.004 0.006 0.007 0.008 0.007 0.005 0.002 0.0 0.0

0.0 0.337 0.592 0.769 0.871 0.902 0.864 0.757 0.581 0.332 0.0 0.285 0.502 0.654 0.742 0.769 0.737 0.646 0.496 0.282 0.0 0.245 0.432 0.562 0.639 0.663 0.636 0.557 0.427 0.243 0.0 0.212 0.375 0.489 0.556 0.578 0.554 0.486 0.372 0.21 1

13.576 12.115 10.814 9.665 8.661 7.799 7.075 6.488 6.043 5.755 13.315 12.012 10.844 9.805 8.889 8.094 7.417 6.858 6.422 6.118 13.088 11.916 10.858 9.910 9.068 8.330 7.696 7.164 6.737 6.424 12.891 11326 10.860 9.989 9.21 1 8.524 7.926 7.419 7.003 6.685

893


J+ J

J-1 J+ 1 J

F

Y(J, I , F)

F-tF+l

F+F

F-F-1

F*F+I

F+F

I =Q(continued) 15

9 37

T

9

”:

3 9 9 25

T

23

T

21

T

894

0.1 13636 0.015152 -0.047675 -0.079676 -0.085423 -0.069227 -0.035136 0.013062 0.071839 0.137931

12.727 11.946 11.205 10.503 9.839 9.214 8.627 8.078 7.568 7.097

0.175 0.309 0.403 0.459 0.477 0.458 0.401 0.307 0.174 0.0

0.001 0.003 0.004 0.006 0.006 0.005 0.004 0.002 0.0 0.0

0.0 0.186 0.329 0.429 0.488 0.507 0.487 0.427 0.326 0.185

12.717 11.743 10.855 10.050 9.327 8.683 8.119 7.634 7.230 6.912

Appendix

J

NUCLEAR QUADRUPOLE SECOND-ORDER CORRECTION ENERGIES FOR LINEAR OR SYMMETRIC-TOP MOLECULES To obtain ELZ’,the entries given are to be multiplied by the factor

J

F

K=O

K= 1

K=2

K=3

K=4

K=5

K=6

- 10.4167

-6.0000 -2.2500 0.0000 - 4.0999 -2.1866 10.4167 0.0000 -3.0864 -1.9290 6.0000 2.2500 -2.4652 -1.6904 4.0998 2.1866 -2.0482 - 1.4935 3.0864 1.9290 -1.7500 - 1.3333 2.4652 1.6904

-9.4688

- 10.8750 - 11.7188 - 5.6487

2.4561 5.2082 11.7188 -3.8520 0.2170 3.4466 5.6668 -2.8917 -0.5944 2.8205 3.5660 -2.3084 -0.8679 2.3715 2.6111 -1.9198 -0.9447 2.0288 2.0749

- 7.2885

-3.8875 - 10.4170

0.0000 -5.3818 3.4721 -2.6042 10.4170 -3.9090 1.8570 -0.4613 6.4915 -2.9800 0.7160 0.4443 4.2779 -2.3779 0.0971 0.8175 3.0805

-5.3758 1.717119 - 7.3244 0.0000 -4.7287 -4.0384 3.1549 -0.8835 -4.0782 -5.2500 7.3240 0.0000 - 3.7373 -4.0366 2.3783 2.6536 -2.0424 -4.0004 5.7901 5.2500 -2.9701 -3.4295 1.4220 2.4120 -0.8767 -2.5661 4.2634 4.8839

- 3.1171

-0.5096 - 3.9066 0.0000 - 3.4365

-2.4684 2.2022 -0.3188 - 3.5684 - 3.0065 3.9066 0.0000 895

J

F

K=O

K= I

K=2

K=3

K=4

K=5

K=6

I=$ - 5.8333

-2.8930 - 2.0829 - 0.5400 - 1.8039 - 1.8039 4.9478 -0.1968 0.0000 - 1.2767 - 1.4773 2.0002 1.8072 0.5400 0.0000 - 0.9753 - 1.2289 0.9756 1.5081 0.8856 0.1968 -0.7835 - 1.0446 0.5225 1.1840 0.8928 0.2756 -0.6519 -0.9052 0.2900 0.9468 0.8283 0.2958

- 6.1475 - 3.6386

- 8.2350 -3.1402 2.6858 0.0657 5.6991 -0.8333 - 1.9108 0.3006 0.7866 1.7500 1.7341 0.8333 - 1.3193 - 0.4042 0.6127 1.2013 1.2000 0.7918 - 0.9894 - 0.6066 0.3992 0.9640 0.9656 0.6230 - 0.7844 - 0.6482 0.2466 0.8020 0.8321 0.5081

- 4.8439

-0.8261 - 3.9850

-5.1118 - 2.0832 - 3.2421 3.1402 - 1.6667 0.7640 3.0208 2.0833 -2.1601 1.4534 -0.2797 0.3300 1.7349 2.0910 - 1.5290 0.5031 0.0704 0.3562 1.0935 1.4861 - 1.1456 0.0404 0.1258 0.3961 0.8202 1.0693

- 3.5571

- 0.4417 - 1.7969 - 3.5926 - 2.6368

0.0000

- 2.9227

- 2.6497

2.4952 -1.1141 - 0.9597 1.2654 2.6368 -2.1682 1.6706 -0.3410 -0.4822 1.0039 2.3272 -1.6258 0.9144 0.0453 -0.1836 0.7229 1.7527

-0.3603 -0.9101 - 2.4238 - 2.2500 0.0000 -2.5167 1.8734 -0.6301 - 1.2888 0.2435 2.2500 -2.0431 1.5623 - 0.2204 -0.7938 0.4239 2.1803

-2.0281 -0.3198 -0.5106 - 1.6970 - 1.8228 0.0000 -2.1421 1.4089 -0.3361 -1.2316 - 0.2394 1.8228

- 1.5948 - 0.2855

-0.3097 -1.2380 - 1.4762 0.0000

“Numerical calculations were made by J. W. Simmons and W. E. Anderson. From W. Gordy, W. V. Smith, and R. F. Trambarulo, Microwave Spectroscopy,Wiley, New York, 1953.

896

AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that the author’s work is referred to although his name is not mentioned in the text. Numbers in italics show the pages on which the complete references are listed.

Links

Index Terms A Adachi, M.

627 (155)

644

Albritton, D. L.

106 (68)

122

Aldrich, P. D.

222 (123)

226

Aleksandrov, A. P.

313 (40)

385

Alexander, A. J.

146 (63)

171

Aliev, M. R.

195 (72)

361 (135)

388

206 (72)

224

285 (122)

286 (124)

288 (124)

296

307 (31)

313 (40, 461)

329 (72)

338 (72)

357 (72)

381 (72)

382 (72, 198)

Alix, A.

313 (43)

385

Allan, A.

380 (164)

388

Allen, C. L.

411

Allen, H. C., Jr.

246 (13)

251

341 (87)

386

Allen, L. C.

789 (74)

802

Allen, W. C.

411

512 (36)

546 (36)

565

Almy, G. M.

110 (83)

122

Altman, R. S.

155

Amano, T.

106 (70)

Amat, G.

385

(13)

389

293

532 (36)

534 (36)

107 (75)

116 (96, 99)

122

169 (140)

173

559 (108)

560 (94)

564 (108, 111)

567

133 (8, 13)

135 (15)

169

185 (19, 22)

187 (19)

201 (22)

223

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links 282 (108)

284 (108)

286 (108)

295

304 (13)

360 (131)

384

387

Amiot, C.

208 (102)

225

Andersen, U.

534 (39)

565

598 (52)

618 (52)

623 (52)

641

Anderson, P. W.

50 (6)

69

Anderson, T. G.

96 (43)

121

145 (54)

165 (125)

166 (132)

170

172

Anderson, W. E.

682

896

Andolfatto, M. K.

500

Andresen, U.

276 (85)

295

Arnold, W.

773 (55)

802

Asprey, L. B.

380 (169)

389

Attanasio, A.

282 (113)

295

Autler, S. H.

273 (69)

277 (69)

492 (36)

503

Aval, G. M.

488 (34)

503

Avery, L. W.

146 (60)

147 (60)

171

Avirah, T. K.

228 (48)

228

267 (43)

268 (48, 51)

294

629 (168)

644

690 (54)

159 (89)

160 (98, 100)

279 (69)

294

722 Aynsley, E. E.

411

Azrak, R. G.

635 (186)

645

Babcock, H. D.

106 (69)

122

Baiocchi, F. A.

154 (89)

157 (89)

161 (100)

171

Baird, D. H.

619 (99)

643

Baird, J. C.

444 (61)

449

Bak, B.

231

502

B

534 (37)

565


Index Terms

Links 627 (153)

644

691 (55)

698 (63)

4 (32)

7 (32, 62)

8

143 (35)

170

186 (25)

187 (29)

223

273 (62)

288 (136)

294

296

360 (127)

383 (212)

387

390

444 (61)

449

498 (49)

501

504

602 (62)

642

Balikci, B.

360 (130)

387

Ball, J. A.

116 (98)

123

Balle, T. J.

5 (41)

9

155 (85)

159 (86)

160 (101)

161 (102)

448

722 Baker, J. G.

156 (86)

171 Bardeen, J.

423 (20)

427 (37)

Barfield, M.

804 (29)

840

Barnes, C. E.

560 (96)

567

Barnes, R. G.

733 (24)

734 (24)

Baron, P. A.

639 (215)

646

Barrett, A. H.

5 (49)

9

115 (90)

Barriol, J.

320 (58)

351 (58)

385

Barsuhn, J.

145 (58)

170

Bartell, L. S.

706 (89)

723

Bassompierre, A.

727 (13)

801

Bates, H. F.

535

Battaglia, A.

273 (71)

Bauder, A.

133 (7)

169

181

186 (26)

187 (26, 35)

223

276 (86)

282 (113)

295

598 (48)

600 (58)

603 (48)

618 (48, 51)

622 (124, 126)

603

122

294

604 (71)

623 (51, 129, 140) 626

801

624 (48) 641

624

689 (52)

626 (126) 701 (68)


Index Terms

Links 722

804 (10, 43) 839

Bauman, L. E.

628 (164)

633 (176)

638 (164)

644

Beason, E. B.

321

Beaudet, R. A.

231

272 (59)

294

604 (69)

605 (70)

642

671 (17)

704 (75)

721

723

Bedard, F. D.

115 (95)

123

Beers, Y.

354 (132)

387

441 (48)

449

560 (99)

567

181 (77)

200 (77)

224

502

Beeson, E. L.

214 (78)

Bell, M. B.

146

Bell, R. P.

636 (204)

645

672 (18)

721

Bellama, J. M.

216 (108)

225

415

770

Bellet, J.

349 (102)

351 (142)

360 (128)

361 (102)

366 (102)

369 (142)

387

679

Bellott, E. M., Jr.

267 (38)

294

635 (183)

645

Belov, S. P.

2 (14)

8

193 (59)

194

201 (80)

202 (59, 80) 224

383 (216)

389 Bender, C. F.

789 (72)

802

Bendtsen, J.

351

502

Benedict, W. S.

192 (51)

224

712 (95)

724

516 (57)

534 (33, 35, 57)

535 (33)

537 (61)

565

695 (62)

723

Benz, H. P.

804 (10)

839

Berendths, B. T.

51

Beringer, R.

552 (74)

553 (74)

554 (77)

556 (76)

557 (76)

561 (74)

566

Benson, R. C.

341 (86)

386

536 (60)


Index Terms

Links

Bernard, P.

357 (125)

359 (125)

387

Bersohn, R.

397 (12)

425 (24)

448

730 (8)

767 (8)

800

Bertram, M.

220 (119)

225

Bettin, N.

745 (32)

801

Bevan, J. W.

153 (78)

171

221 (120)

225

Bhattacharya, B. N.

493 (39)

497 (40)

500 (40)

503

619 (99)

643

54 (9)

69

449

727 (8)

535 Bird, G. R.

444 (61)

449

Bird, R. B.

487 (31)

503

Birnbaum, G.

50 (9)

51

Birss, F. W.

286 (126)

296

Bivens, R.

426 (36)

439 (36)

447 (36)

805 (45)

838 (45)

840

Bjarnov, E.

380 (171)

389

Bjørseth, A.

216 (106)

225

Blackman, G. L.

145 (55)

170

804 (14, 22)

826 (22)

840 Blake, G. A.

164 (131)

172

Blaker, J. W.

246 (14)

293

Blanch, G.

577 (16)

640

Bleaney, B.

188 (37)

223

Blickensderfer, R.

534 (26)

565

Blukis, U.

231

Bobin, B.

804 (34)

840

Bodenseh, H. K.

135 (14)

169

181

183 (13)

223

411

500

706 (80)

723

804 (26, 30) 840

96 (44)

98 (44)

121

133 (7)

164 (122)

169

172

411

Bogey, M.

Boggia, L. M.

321


Index Terms Boggs, J. E.

Links 51

502

689 (48)

722

Bohn, R. K.

267 (36)

293

Boland, M. R.

636 (198)

645

Bolton, K.

635 (193)

645

620 (103)

643

634 (178)

645

804 (22)

826 (22)

636 (203)

645

622 (126)

626 (126)

501 (52)

504

600 (43)

840 Boone, D. W.

502

Borchert, S. J.

268 (46)

294

Borgers, T. R.

369 (145)

388

Born, M.

150 (74)

171

Born, W.

80 (7)

120

Bossert, W.

598 (49)

604 (71)

626

641

3 (28)

8

632 (174)

645

Böttcher, C. J. F.

799 (80)

802

Boucher, D.

415

421

597 (43)

641

766

770

269 (55)

294

320 (58)

349 (111)

351 (58)

385

387

620 (108)

643

689 (53)

722

560 (91)

567

804 (20)

838 (20)

Botskor, I.

Bouchy, A.

Bowater, I. C.

840 Bowen, H. J. M.

653 (6)

721

Bowen, K. H.

160 (99)

172

Bradley, R. H.

360 (126)

387

Bragg, J. K.

248 (18)

293

392 (9)

406 (10)

413 (9)

416 (10)

427 (9)

448

Bransford, J. W.

444 (62)

449

Bray, P. J.

767 (42)

772 (44)

Breivogel, F. W.

797

801


Index Terms

Links

Breslow, R.

695 (62)

723

Bridge, N. J.

488 (33)

503

Brier, P. N.

288 (136)

296

361 (126)

387

Brim, W. W.

192 (52)

224

Britt, C. O.

51

502

620 (103)

643

689 (48)

722

Brittain, A. H.

380 (173)

389

Broida, H. P.

105 (66)

122

Brookbanks, D. M.

360 (127)

387

Brossel, J.

3 (21)

8

Broten, N. W.

146 (60)

147 (60)

171

Brown, J. M.

115 (89)

122

560 (91, 96)

567

804 (20, 32) 838 (20)

840

145 (55)

170

679

710

804 (14, 22)

826 (22)

840

Brown, W. G.

768 (43)

801

Buck, L.

649 (4)

721

Buckingham, A. D.

458 (5)

462 (5)

488 (33)

496 (5)

501 (5)

503

539 (62, 65)

542 (65)

566

753 (36)

801

Budenstein, P. P.

2 (10)

8

Buhl, D.

87 (34)

121

144 (50)

147 (50)

163 (112)

164 (116, 121)

170

172

135 (20)

170

193 (56, 58)

224

285 (116)

296

304 (18)

382 (18)

385

389

Brown, R. D.

Bunker, P. R.

383 (202, 210)

Burden, F. R.

360 (126, 130)

145 (56)

620 (112)

643

677 (25)

722

679

710

804 (14, 22)

826 (22)

840 This page has been reformatted by Knovel to provide easier navigation.

Index Terms Burenin, A. V.

Links 2 (12)

8

152 (75)

171

201 (80)

225

349 (96)

383 (216)

386

390

Burg, A. B.

781 (64)

802

Burie, J.

415

770

Burkhard, D. G.

571 (7)

582 (7)

623 (132)

639

1 (3)

2 (10)

7

109 (81)

115 (81)

118 (103)

122

137

180

417 (18)

444 (54)

448

495 (43)

496 (44)

500 (43)

503

511 (8)

520 (8)

565

778 (57)

477 (20)

503

638 (210)

154 (84)

159 (84)

160 (103)

161 (103)

162 (103)

171

285 (123)

296

Cade, P.

789 (71)

802

Cahill. P.

84 (16)

121

497

500

Callomon, J. H.

649 (4)

721

Caminati, W.

380 (177)

389

636 (199)

645

5 (41)

643 Burrus, C. A.

802 Butcher, S. S.

231 645

Buxton, L. W.

Bykov, A. D.

C

Campbell, E. J.

Careless, A. J.

Carpenter, J. H.

443 (52)

449

624

635 (194)

9

154 (84)

155

159 (84)

160 (101)

161 (103)

162 (102)

171

221 (122)

222 (122)

226

181 (31)

187 (31)

223

288 (133)

296

415

770

288 (137)

296

351 (115)

351


Index Terms

Links 354 (115)

380 (168)

381 (115)

387

421

766

770

Carreira, L. A.

628 (163)

638 (163)

644

Carrier, L. A.

628 (164)

638 (164)

644

Carrington, A.

5 (44)

6 (53)

9

559 (87)

559

560 (91, 96)

561 (85, 105)

564 (106)

567

804 (12,

735 (25)

558 (85,

87)

20) 838 (20)

839

Casabella, P. A.

772 (44)

801

Casimir, H. B. G.

392 (3)

404 (3)

448

801 Casleton, K. H.

421

766

770

Castle, J. G., Jr.

552 (74)

553 (74)

556 (76)

561 (74)

143 (32)

148 (70)

160 (101)

170

181

299 (3)

351 (3)

380 (176)

384

389

415

678 (30,

823 (13)

827 (13,

566 Cazzoli, G.

32)

Cederberg, J. S.

722

770

804 (13, 16, 24)

809 (17)

16) 828 (16)

840

Cervellati, R.

635 (194)

645

678 (32)

722

Chan, M. Y.

304 (19)

341 (19)

369 (19)

382 (200)

385

389

369 (145)

388

477 (19)

503

636 (201)

638 (201)

645

Chan, Y.

367 (141)

388

Chance, K. V.

160 (99)

172

Chan, S. I.


Index Terms

Links

Chang, Tsu-Shen

181 (6)

223

Chantry, G. W.

5 (39)

7 (39)

Chanussot, J.

360 (128)

387

Chao, S.

629 (168)

644

Charo, A.

166 (135)

173

Charpentier, L.

618 (96)

643

Chen, M. M.

216 (115)

216

Cheung, A. C.

5 (50)

9

Chikaraishi, T.

381 (193)

389

679

Childs, W. J.

737

Choplin, A.

276 (91)

278 (91)

295

Christensen, D.

231

698 (63)

723

Christiansen, J. J.

147 (67)

151 (67)

171

231

411

502

627 (153)

644

Christy, A.

109 (79)

114 (79)

122

Chu, F. Y.

195 (67)

198 (67)

201 (67)

215 (67)

326 (65)

341 (19)

369 (19)

6 (52)

9

115 (92)

122

300 (5)

384

Chutjian, A.

666 (12)

721

Clark, A. H.

710

Clark, W. W.

81 (10)

82 (13)

84 (10)

89 (10)

107 (74)

116 (74)

120

351

Cleeton, C. E.

188 (36)

223

Cleveland, F. F.

307 (26)

311

385

Clough, S. A.

304 (16)

326 (66)

385

Clouser, P. L.

85 (31)

121

Codding, E. G.

624

Coester, F.

458 (6)

9

560 (101)

567

225

224 Chung, K. T.

304 (19) 385

Churchwell, E.

462 (6)

466 (6)

503


Index Terms

Links

Cogley, C. D.

804 (40, 42)

840

Cohen, E. A.

288 (135)

296

349 (110)

355 (110)

399 (13)

448

767 (41)

448

387 Cohen, E. R.

857

Cohen, M. H.

397 (13) 801

Cohen, V. W.

859

Cohn, E. R.

51

Coles, D. K.

392 (7)

415 (41)

436 (41)

456

506 (1)

565

Combs, L. L.

267 (36)

293

Condon, E. U.

446 (64)

449

459 (7)

468 (7)

478 (7)

503

524 (14)

565

811 (47)

840

Cook, J. M.

560 (103)

567

Cook, R. L.

7 (69)

10

100 (59)

107 (71)

122

228 (48)

228

265 (34)

267 (36), (43)

268 (48, 51) 292 (150)

293

294

296

300 (7)

313 (49)

344 (91)

345 (92)

343 (91, 147, 163) 346 (91) 347 (91)

349 (97)

351 (7, 155)

352 (93)

353 (91, 97)

361 (7, 91)

364 (7)

366 (91)

375 (93, 155)

376 (157)

378 (157, 163) 379 (156,

380 (155, 163)

381 (97)

384

388

497

498 (47)

500

502

504

551 (72)

566

628 (166)

629 (166, 168)

630 (166)

633 (176)

644

649 (2)

690 (54)

710 (93)

721

724

803 (5)

805 (3)

161)


Index Terms

Links 810 (5)

812 (5)

821 (5)

825 (5)

834 (5)

839

Cool, T. A.

3 (22)

8

Cooper, T. A.

145

Corbelli, G.

187 (29)

223

380 (179)

389

Comet, R.

351

421

766

770

Cornwell, C. D.

421

425 (23)

448

492 (38)

503

766

770

827 (48)

141 (30)

170

188 (39)

190 (43)

224

357 (121)

387

477 (20)

503

617 (85)

621 (85)

635 (181)

638 (210)

642

645

688 (44)

691 (58)

694 (58)

698 (58)

722

730 (16)

793 (16)

793

796 (13)

75

840 Costain, C. C.

Coulson, C. A.

801 Cowan, M.

2 (6)

7

Cox, A. I.

678 (29)

722

Cox, A. P.

187 (33)

223

273 (74)

276 (74)

295

380 (173)

389

616 (81)

617 (81)

627 (149)

642

644

180 (7)

511 (6)

514 (6)

516 (6)

550 (70)

565

Coxon, J. A.

115 (86)

122

Crable, G. F.

472 (15)

474 (16)

503

Crawford, B. L.

581 (22)

583 (22)

640

Crawford, B. L., Jr.

308 (37)

385

Creswell, R. A.

90 (29)

121

144 (48)

145 (48, 53)

146

170

421

268 (50)

294

369 (144)

381 (185)

388

Cox, J. T.


Index Terms

Links 624

689 (51)

722

766

229 (1)

232 (1)

770 Cross, L. C.

653 (6)

721

Cross, P. C.

31 (5)

35

234 (1)

Curl, R. F., Jr.

236 (1)

242 (1)

246 (1, 11)

259 (23)

262 (23)

282 (111)

292

295

302 (11)

340

372 (11)

384

416 (17)

448

849 (1)

852

207 (96, 97)

225

231

267 (41)

268 (49)

294

381 (195)

389

444 (61)

449

500 (51)

504

560 (103)

567

618 (97)

622 (125)

626 (125)

633 (177)

635 (185, 188)

643

645

679

682 (35)

717 (35)

720 (35)

722

804 (6)

839 Currie, G. N.

561 (105)

567

Curry, J.

87 (36)

121

Curtiss, C. F.

487 (31)

503

Cyvin, B. N.

307 (33)

311 (33)

Cyvin, J. J.

713 (99)

724

Cyvin, S. J.

307 (32)

311 (33)

373 (150)

377 (151)

385

388

706 (89)

723

534 (39, 44, 48)

541 (49)

565

231

421

495 (45)

496 (46)

500 (45)

501

504

506 (4)

511 (4)

533 (4)

540 (4)

565

624 (145)

644

727 (2)

741 (2)

766

770

772 (2)

779 (60)

Czieslik, W.

385

D Dailey, B. P.


Index Terms

Links 800

802

Dakin, T. W.

456

Dal Borgo, A.

181

415

770

Damaison, J.

421

766

770

Damburg, R. J.

193 (54)

224

Dang-Nhu, M.

208 (103)

225

Dangoisse, D.

351 (142)

369 (142)

Danielis, V.

383 (209)

389

Daniels, J. M.

561 (104)

567

Darling, B. R.

282 (104)

295

Darling, B. T.

304 (12)

384

674 (19)

684 (37)

448

727 (12)

767 (12)

388

721 Das, T. P.

397 (14) 801

Davidson, E. R.

789 (72)

802

Davies, P. B.

160 (93)

161 (93, 108)

225

357 (123)

359 (123)

387

351 (152)

351

371 (153)

372 (153)

380 (171)

388

282 (111)

295

302 (11)

372 (11)

Davis, R. W.

Decius, J. C.

171 202 (81)

384 de Hemptinne, M.

421

766

770

Dehmelt, H. G.

399 (15)

448

737 (58)

737

745 (30, 34)

746 (30)

767 (40)

770

782 (65)

801

4 (23)

7

de Leeus, F. H.

797

Delfino, A. B.

269 (54)

294

Delpuech, J. J.

689 (53)

722

De Lucia, F. C.

2 (8)

3 (17, 23)

75

81 (10)

83 (32)

84 (10, 28)


Index Terms

Links 85 (32)

86 (28)

89 (10)

96 (42)

100 (59)

107 (74)

116 (74)

120

131 (3)

141 (22)

143 (22)

144 (49)

145 (22)

164 (126, 131)

164

165 (126, 131)

166 (135)

169

172

190 (47)

191 (47)

192 (47)

214

224

299 (3, 4)

343 (91)

344 (91)

345 (92)

346 (91)

347 (91)

349 (95, 97, 106)

351 (3)

351 352 (93) 366 (91)

353 (91, 97, 106, 108) 375 (93, 156)

361 (91)

379 (156, 161)

381 (97)

384

386

434

444 (63)

449

560 (101)

567

702 (71)

710 (93)

723

803 (5)

804 (16, 23, 28)

805 (3)

810 (5)

821 (5)

825 (5)

827 (16)

812 (5) 828 (16, 28)

829 (23, 28)

834 (5)

839

291 (149)

296

313 (44)

329 (58)

351 (58)

369 (143)

385

388

415

597 (43)

600 (43)

625

641

706 (79)

723

770

Demtroder, W.

4 (31)

8

Demuynck, C.

96 (44)

98 (44)

121

164 (122)

172

411

Dendl, G.

324 (61)

386

502

Dennison, D. M.

29 (2)

32 (2)

35

65 (15)

69

175 (1)

181 (6)

184 (15)

189 (42)

193 (42)

201 (79)

223

282 (104)

295

304 (12)

324 (60)

384

386

571 (7)

Demaison, J.

582 (7, 24)


Index Terms

Links 603 (66)

618 (92)

639

674 (19)

684 (37)

721

797

642

Depannemaecker, J. C.

679

Derr, V. E.

82

535

DeSantis, D.

772 (50)

802

Destombes, J. L.

96 (44)

98 (44)

121

164 (122)

172

351 (142)

369 (142)

388

339 (82)

343 (82)

346 (82)

374 (82)

378 (82)

386

710

Dickman, R. L.

165 (130)

172

Diercksen, G. H. F.

164 (118)

172

DiGaicomo, A.

273 (80)

295

Dijkerman, H. A.

619 (101)

643

Disch, R. L.

539 (65)

542 (65)

Dixon, R. N.

110 (84)

122

Dixon, T. A.

5 (46)

9

96 (41, 43)

121

154 (89)

157 (89)

159 (89)

160 (98, 100)

161 (100)

163 (110)

164 (119)

165 (125)

166 (132)

171

Dixon, W. B.

231

698 (63)

723

Dobyns, V.

595 (39)

597 (39)

641

Dodd, R. E.

411

Donohue, J.

653 (6)

721

Dorain, P. B.

561 (104)

567

Doraiswamy, S.

272 (57)

294

Dorney, A. J.

338 (76)

360 (76)

386

Dossel, K.-F.

181

415

501

533 (51)

534 (49, 51) 546 (51)

625

770

109 (80)

115 (80)

411 Di Cianni, N.

Dousmanis, G. C.

566

122

97 (43, 48)

516 (51) 566

432 (38)


Index Terms

Links 444 (53, 58)

449

540 (67)

552 (79)

554 (79)

566

306 (24)

307 (25)

322 (59)

385

688 (44)

722

247 (16)

276 (84, 89) 293

295

321

324 (61)

351

386

421

502

534 (37, 39)

565

572 (9)

597 (43)

598 (46, 52)

600 (43)

603

615 (78)

618 (46, 52, 95)

643

621 (118, 120) 622 (122)

623 (52, 136)

624 (144)

625 (123)

625

639

641

667 (15)

721

766

770

773 (55)

802

Drew, D. A.

216 (106)

225

Drigas, T. M.

766

770

Dubrulle, A.

351 (142)

369 (142)

388

415

421

597 (43)

600 (43)

641

766

770

Duckett, J. A.

706 (83)

710 (84)

723

DuMond, J. W. M.

857

Duncan, J. L.

218 (118)

225

299 (166)

378 (147)

380 (164, 166, 175, 180)

388

603 (67)

642

712 (96)

724

Dunham, J. L.

73 (4)

78 (4)

120

Dunmur, D. A.

539 (65)

542 (65)

566

Durig, J. R.

216 (111, 115)

216

225

271 (56)

272 (56)

294

598 (53)

621 (54)

624

626 (53)

626

637

641

666 (14)

721

Dyer, P. N.

558 (87)

559 (87)

567

Dyke, T. R.

153 (76)

161 (108)

171

Dowling, J. M.

Dreizler, H.

544 (67)

324 (25, 59)

617 (83)

373 (147)


Index Terms Dymanus, A.

Links 84 (26)

86 (26)

121

133 (6)

141 (26)

143 (42)

169

411

445

497

535

600 (57)

619 (101)

642

797

804 (21, 27)

840

E Eagle, D. F.

481 (25)

503

Eck, T. G.

737

Eckart, C.

306 (22)

385

Eckhardt, W.

1 (4)

7

Edmonds, A. R.

426 (34)

439 (34)

448 (34)

449

803 (2)

805 (2)

807 (2)

831 (2)

276 (86)

295

121

381 (193)

839 Edwards, H. D.

51

Edwards, T. H.

341 (90)

386

Ekkers, J.

5 (40)

9

604 (71)

642

El-Sayed, M. A.

5 (45)

9

Ellder, J.

115 (94)

123

Endo, Y.

97 (49)

116 (101)

389

560 (95, 102)

567

603

(67) 624

642

679

804 (36,

520 (3, 11)

525 (3)

39) 837 (39)

840

Engerholm, G. G.

639 (213)

646

Englebrecht, L.

617 (83)

642

Erickson, N. R.

166 (134)

173

Erlandsson, G. E.

190 (45)

224

Esbitt, A. S.

619 (100)

643

Eshbach, J. R.

506 (3)

511 (3)


Index Terms

Links 546 (3)

565

143 (32)

148 (70)

170

645

678 (30)

722

105 (66)

106 (67)

115 (87, 89)

560 (97)

567

602 (60, 64)

603 (60, 68) 618 (60)

Fabris, A. R.

559 (89)

567

Faehl, L.

804 (29)

840

Fano, U.

458 (3)

503

Fantoni, A. C.

321

Farag, M. S.

267 (36)

293

Farukane, U.

772 (48)

801

Fateley, W. G.

574 (14)

601 (14)

640

Favero, P. G.

109 (82)

122

143 (32)

148 (70)

170

181

231

273 (78)

295

415

444 (56)

449

498 (49)

504

678 (30)

722

35

Esposti, C. D.

Evenson, K. M. Ewig, C. S.

635 (194)

122

642

F

803 (3)

839

770 Feenberg, E.

17 (1)

31 (1)

Feld, B. T.

392 (5)

448

Feldman, P. A.

146

Ferigle, S. M.

306 (23)

385

Fermi, E.

136 (21)

170

Fernandez, J.

477 (19)

503

636 (201)

638 (201)

645 Filgueira, R. R.

321

380 (179)

389

Findlay, F. D.

4 (33)

8

144 (51)

Finnigan, D. J.

380 (173)

389

Fitzky, H. G.

84 (17)

121

Fleming, J. W.

5 (42)

7 (42)

170

9


Index Terms

Links

Fletcher, W. H.

308 (37)

Fliege, E.

625

Flygare, W. H.

5 (40)

385

9

154 (84)

155 (84,

156 (86)

159 (84, 86)

91) 155 160 (101, 104)

161 (102, 104)

162 (102, 104)

171

222 (124)

226

273 (65, 75)

294

295

411

421

441 (49)

449

481 (24)

500

503

512 (17, 20, 30, 36, 40)

514 (9)

516 (21, 23, 28, 32, 57)

517 (17)

530 (16)

516

532 (18, 20, 30, 36, 40)

533 (21, 23, 28, 32, 58) 534 (17, 23, 32, 40, 57)

535 (19, 25, 33)

536 (58, 60)

537 (58, 61) 538 (61)

541 (58)

545 (17, 20, 24)

565

624

695 (62)

723

766

770

780 (62)

802

804 (9)

839

Flynn, G. W.

273 (74)

276 (74)

277 (93)

295

Foley, H. M.

97 (47)

121

444 (57)

449

554 (81)

566

731 (20)

735 (20)

736 (20)

742 (20, 28) 801

Ford, L. H.

341 (89)

386

Ford, R. G.

276 (92)

281 (100)

Forti, P.

380 (179)

389

Foster, P. D.

444 (63)

449

534 (19, 24)

535 (19, 24) 545 (20)

565

195 (63, 71)

206 (63, 71) 207 (71)

224

338 (79)

360 (79)

341 (88)

386

Fox, K.

Fraley, P. E.

540 (17, 66) 546 (17, 36, 40)

295

624

512 (20)

532 (20)

386


Index Terms

Links

Freedman, J. N.

326 (66)

386

Frenkel, L.

4 (35)

8

205

223

Freund, R. S.

772 (50)

802

Freund, S. M.

144 (44)

170

Fristrom, R. E.

291 (140)

296

Friz, E.

369 (146)

388

Fröhlich, H.

50 (5)

69

Frosch, R. A.

97 (47)

121

554 (81)

566

421

179 (4)

181

558 (86)

567

444 (57)

449

766

770

804 (37)

493 (40)

497 (40)

500 (40)

503

713 (98)

717 (102)

724

632

636 (200)

645

Gailar, N.

712 (95)

724

Galica, J.

288 (138)

296

Gallagher, J. J.

82

111 (85)

115 (95)

116 (85)

122

444 (55)

449

535

Fry, H. A.

840 Fujii, O. Fukuyama, T. Fuller, G. H. Fuller, M. J.

859

G

797 Gallaher, K. L.

704 (73)

723

Gamo, I.

312 (39)

385

Garrison, A. K.

84 (20)

121

141 (24)

170

Garvey, R. M.

3 (24)

8

298 (2)

384

804 (23, 28)

828 (28)

829 (23, 28)

840

Gaumann, T.

272 (58)

294

Gauthier, L.

338 (75)

360 (75)

386

Gaylord, A. S.

628 (162)

638 (162)

644

Gebbie, H. A.

4 (33)

8

144 (51)

170


Index Terms

Links

Gegenheimer, R.

183 (13)

223

Genzel, L.

1 (4)

7

Georghiou, C.

287 (128)

288 (136)

389

501

146

Gerry, M. C. L.

296

381 (201)

181 (30)

181

187 (30)

195 (69)

202 (82)

205 (82)

208 (94)

216 (82)

221 (121)

223

226

307 (29)

338 (81)

343 (178)

351 (152, 174)

351

361 (133)

371 (152)

385

498 (48)

700 (67)

380 (29, 171, 174, 178)

207 (69, 92)

504 635 (184)

645

688 (45)

706 (84)

710 (84)

722

2 (14)

8

193 (59)

201 (80)

202 (59, 80) 224

Geschwind, S.

411

415

Ghosh, D. K.

321

Ghosh, S. N.

800 (81)

802

Giguere, P. T.

164 (121)

172

Gilliam, O. R.

51

414

Gershstein, L. I.

194

770

520 (10)

565

170

411

679 Gillies, C. W.

704 (73)

723

Givens, W.

852 (2)

852

Glorieux, P.

277 (94)

295

Godfrey, P. D.

144 (44)

145 (55)

679

710

Gold, L.

789 (70)

802

979

Gold. L. P.

82

84 (16)

121

497

500 Gold. R.

306 (24)

307 (25)

324 (25)

385

Golden, S.

248 (18)

293

392 (10)

406 (10)


Index Terms

Links 416 (10)

448

468 (11)

470 (11)

488 (11)

491 (11)

503

Goldsmith, M.

304 (13)

384

Goldsmith, P. F.

165 (130)

172

Goldstein, J. H.

421

502

727 (9)

766

770

800

Golub, G. H.

682 (36)

722

Good, W. E.

188 (38)

224

392 (6)

415 (41)

436 (41)

448

456

506 (1)

730 (18)

565 Goodman, L. S.

737

Gora, E. K.

263 (31)

293

Gordon, J. P.

3 (20)

8

436 (42)

449

Gordy, W.

1 (1, 3)

2 (5, 11)

3 (15, 23)

4 (23)

7 (57)

7

47 (2)

50 (8)

51

54 (8)

67 (18)

69

72 (1)

75

81 (10)

82 (1)

84 (1, 10, 20, 28)

85 (1, 21, 30, 33)

86

(22, 28, 33) 87 (37)

89 (10)

99 (57)

102 (61)

104 (57, 61)

105 (57, 61, 65)

106 (61)

109 (81)

115 (81)

120

137

141 (22, 24, 31)

142 (31)

143 (22)

144 (47)

145 (22)

170

180 (7)

180

181 (7, 77)

181

182 (7)

185 (23)

186 (25)

190 (45)

191 (47)

192 (48)

198 (73)

200 (77)

201 (73, 77)

214 (78)

214

215 (73)

223

232 (3)

256 (22)

257 (22)

268 (22)

292 (159)

293

296

107 (71, 73)

349 (98, 106)


Index Terms

Links 353 (98, 106)

386

411

414

417 (18)

428

434

435 (39)

444 (54, 56)

448

487 (32)

493 (39)

494 (41)

495 (42)

497 (40)

500 (40)

501 (42)

503

511 (6)

512 (31, 41)

514 (6)

516 (6)

520 (10)

532 (31, 41)

534 (31, 41)

540 (68)

541 (69)

548 (68)

550 (70)

551 (70)

565

623 (141)

635 (187)

644

649 (2)

677 (23)

682

702 (71)

721

737 (58)

737

740 (10)

745 (34)

750 (7)

754 (6)

761 (7)

770

772 (47)

778 (57)

781 (64)

783 (7)

785 (5)

787 (7)

788 (68)

795 (7)

800 (81)

800

835 (50)

840

874

878

896

Gottlieb, C. A.

116 (98)

123

Gottlieb, E. W.

116 (98)

123

Gozzini, G.

273 (71)

294

Grashoff, M.

445

Green, S.

6 (56)

9

168 (136)

172

185 (19)

187 (19)

Grenier-Beeson, M. L.

727 (5, 10)

165 (122)

223

304 (14)

384 Griffths, J. E.

779 (59)

802

Grivet, P.

727 (13)

801

Groner, P.

598 (53)

621 (54)

624

626 (53)

641 Gsell, R. A.

216 (108)

225

415

770

Guarnieri, A.

193 (60)

224

231

276 (84)

295

321

500

502


Index Terms

Links 534 (43, 46, 53)

566

96 (43)

121

166 (132)

172

Guelachvili, G.

208 (102)

225

Guélin, M.

165 (127)

168 (136)

Guggenheim, E. A.

672 (18)

721

Guibe, L.

772 (46)

801

Gunn, H. I.

145 (55)

170

Günthard, Hs. H.

272 (58)

276 (86)

282 (113)

294

295

598 (48)

600 (58)

603 (48)

604 (71)

618 (48, 51, 93)

623 (51, 139)

624 (48)

626 (126)

641

701 (68)

723

804 (10, 43)

839

392 (11)

411

415 (41)

415

436 (11, 41)

441 (11)

448

770

487 (32)

503

534 (31, 41)

565

369 (145)

388

421

425 (22)

448

477 (19)

499 (50)

502

608 (75)

615 (79)

616 (76)

628 (162)

636 (201)

638 (162, 201, 206, 211)

639 (213)

642

644

770

804 (9)

839

426 (31)

449

164 (28)

165 (128)

172

307 (32)

311 (33)

385

397 (14)

448

727 (12)

767 (12)

31 (5)

35

229 (1)

232 (1)

234 (1)

236 (1)

242 (1)

246 (1)

Gudeman, C. S.

Gunther-Mohr, G. R.

Gustafson, S.

Gwinn, W. D.

164 (28)

165 (128)

169 (136, 138) 172

622 (124, 126)

512 (31, 41)

766

532 (31, 41)

H Hadley, G. F. Haese, N. N. Hagen, G. Hahn, E. L.

801 Hainer, R. M.


Index Terms

Links 259 (23)

262 (23)

292

340

416 (17)

448

849 (1)

852

Haissinsky

878

Haland, A.

216 (107)

Hallowell, C. D.

797

Halonen, L.

369 (146)

388

Hamer, E.

534 (37)

565

Hamilton, W. C.

361 (135)

388

Hamphill, D. C.

633 (177)

645

Handelman, E. T.

615 (79)

642

Hannay, N. B.

795 (76)

802

Hanson, H. M.

195 (61)

224

Hanyu, Y.

620 (104)

643

689 (48)

722

Hardy, W. A.

137

Harford, S. L.

411

Harmony, M. D.

281 (102)

295

636 (197)

645

649 (3)

691 (56)

721

Harrington, H. W.

638 (206)

645

Harris, C. L.

411

Harris, C. O.

602 (64)

642

Harris, D. O.

623 (133)

632 (133)

643

645

Harris, S. J.

160 (93)

161 (93, 95) 162 (96)

Harrop, W. J.

106 (68)

122

Hasegawa, A.

624

Hayashi, M.

577 (19)

579 (19)

590 (19)

624

626

627 (146, 154)

632

640

644

Heath, G. A.

415

770

Hebb, M. H.

99 (51)

121

Hebert, A. J.

797

225

638 (206)

639 (213)

171

621 (19)


Index Terms Hecht, K. T.

Links 207 (99)

225

338 (73)

360 (73)

386

582 (26)

618 (92)

623 (141)

640

642

644

674 (19)

721 Hedberg, K.

706 (89)

723

Heidelberg, R. F.

444 (61)

449

Heise, H. M.

502

Helminger, P.

2 (8)

7

47 (2)

69

75

96 (42)

121

131 (3)

164 (126, 131)

166 (135)

169

172

181 (77)

190 (46)

191 (47)

192 (47)

200 (77)

214 (78)

214

224

299 (4)

343 (91)

344 (91)

345 (92)

346 (91)

347 (91)

349 (97, 106)

352 (93)

353 (91, 97, 106, 108)

361 (91)

366 (91)

375 (93, 156)

356 (118) 379 (155, 161)

381 (97)

384

386

411

434

702 (71)

710 (93)

84 (25)

90 (25)

121

198 (73)

201 (73)

215 (73)

224

Henderson, G.

221 (122)

226

Henderson, R. S.

435 (40)

449

Henry, A. F.

552 (77)

554 (77, 80) 556 (77)

566

Henry, J. C.

5 (49)

9

115 (90)

122

Henry, L.

360 (131)

387

Herberich, G. E.

343 (138)

363 (138)

373 (138)

378 (138)

388

616

683 (61)

693 (61)

85 (32)

96 (42)

121

723 Helms, D. A.

723 Herbst, E.

83 (32) 164 (114, 126, 131)

164

165 (126, 131)

166 (135)


Index Terms

Links 172

558 (86)

567

804 (18)

840 Herlemont, F.

415

770

Herrmann, G.

191 (50)

224

426 (32)

449

Herschbach, D. R.

477 (18)

502

574 (11)

577 (18)

584 (28)

587 (28, 33) 589 (28)

591 (28, 34)

593 (32)

594 (11, 32)

596 (28)

599 (11)

601 (11, 59) 602 (18, 28)

612 (28)

618 (28)

639

680 (86)

686 (42)

706 (85)

708 (85)

709 (86)

714 (86)

722

7 (68)

10

57 (14)

69

87 (36)

94 (40)

106 (69)

121

162 (109)

172

211 (104)

225

Herzberg, L.

87 (36)

121

Heuvel, J. E. M.

600 (57)

642

804 (21)

840

Hickernell, F.

633 (175)

645

Higgins, R. J.

267 (36)

293

Hill, A. G.

483 (26)

497 (26)

503

Hill, E. L.

108 (76)

111 (76)

122

Hill, R. H.

341 (90)

386

Hill, R. M.

99 (54)

122

551 (70)

566

Hillger, R. E.

340 (84)

386

Hills, G. W.

560 (103)

567

Hinze, J.

784 (67)

802

Hirakawa, H.

411

Hirose, C.

187 (34)

Hirota, E.

97 (49)

Herzberg, G.

590 (18, 28)

712 (42, 86)

539 (64)

542 (64)

520 (11)

565

223

701 (69)

723

106 (70)

107 (75)

116 (96,

99, 101) 121

187 (28, 32) 223

231


Index Terms

Links 268 (52)

288 (132)

291 (141)

296

381 (189, 192)

559 (108)

560 (95, 102)

294 389

564 (108, 111)

567 596 (42)

603 (67)

607 (74)

620 (74)

621 (119)

623 (133)

624

628 (156)

630 (169)

633 (156)

641

649 (4)

677 (26)

678 (26, 31)

679

711 (26)

721

804 (36, 39)

837 (39)

840

Hirschfelder, J. O.

487 (31)

503

Hjalmarson, A.

165 (130)

172

Hobbs, L. M.

164 (115)

172

Hocker, L. O.

4 (35)

8

Hocking, W. H.

90 (29)

121

351

380 (172)

389

498 (48)

504

632

Hodgeson, J. A.

444 (62)

449

Hoeft, J.

81 (11)

84 (19)

87 (38)

89 (11)

89

120

411

445

804 (31, 33)

840

Hoffman, J. M.

192 (52)

224

Høg, J. H.

620 (105)

643

689 (49)

722

Holden, A. N.

51

137

682

Hollis, J. M.

164 (116, 121)

172

Holloway, J. H.

380 (164, 181)

388

Holmgren, S. L.

160 (97)

172

Holt, C. W.

195 (69)

207 (69, 92) 224

627 (133, 152) 635 (190)

338 (81)

386 Hølzer, B.

445

Honda, T.

268 (52)

Honerjäger, R.

89

294 532 (38, 42, 45)

534 (38, 42, 45)

565


Index Terms

Links

Honig, A.

83 (15)

85 (15)

Hooper, H. O.

767 (42)

801

Hopfinger, A. J.

272 (60)

294

Horak, M.

383 (209)

389

Horsfall, R. B.

110 (83)

122

Hougen, J. T.

169 (139)

173

193 (56)

224

304 (18)

382 (18)

385

560 (98)

567

804 (19, 35) 838 (35)

Houston, P. L.

273 (64)

294

Howard, B. J.

153 (76)

171

Howard, C. J.

560 (97)

567

Howard, J. B.

282 (103)

Howard, R. R.

121

840

804 (12)

839

286 (103)

295

301 (8)

304 (8)

384

586 (32)

641

674 (19)

721

48 (3)

69

539 (63)

542 (63)

224

380 (165)

538 (55)

566 Howe, J. A.

481 (24)

503

Hoy, A. R.

193 (58)

220 (119)

383 (203)

388

Hrubesh, L. W.

7 (70)

10

Hsu, S. L.

624

Hüber, K. P.

7 (68)

10

Hiibner, D.

534 (55)

535 (56)

536 (55)

540 (56)

542 (56)

566

502

Hudson, S. D.

626

Hughes, R. H.

451 (1)

492 (1, 35)

Huguenin, G. R.

165 (130)

172

Huis zoon, C.

141 (26)

170

Hunt, R. H.

623 (141)

644

Huo, W.

789 (71)

802

Hüttner, W.

411

512 (17, 20) 516 (23)

517 (17)


Index Terms

Links 530 (16)

532 (20)

533 (23)

535 (25)

540 (17)

565

804 (26, 30) 840

Ibers, J. A.

193 (53)

224

Ibushi, N.

626

Ikeda, T.

267 (41)

534 (17, 23, 25)

545 (17, 20)

546 (17)

I

268 (49)

294

618 (97)

165 (130)

172

643 Imachi, M.

627 (154)

644

Ingram, D. J. E.

7 (59)

9

Irvine, W. M.

115 (94)

123

Irwin, J. C.

623 (132)

643

Ito, T.

425 (25)

448

Ivash, E. V.

582 (25)

640

J Jaccarino, V.

737

Jache, A. W.

444 (62)

449

Jackson, R. H.

339 (82)

343 (82, 138)

346 (82)

374 (82, 138)

375 (82)

363 (138) 373 (138) 378 (82, 138) 386

683 (61)

693 (61)

770

710

723 Jacobs, G. D.

421

766

Jacox, M. E.

145 (59)

171

Jaeschke, A.

615 (78)

642

Jaffe, H. H.

784 (67)

802

Jahn, H. A.

685 (40)

722

Jalilian, M. R.

626

Jaman, A. I.

321

Janda, K. C.

160 (96)

171


Index Terms

Links

Javan, A.

4 (35)

8

273 (79)

295

Jefferts, K. B.

87 (35)

97 (45)

121

163 (111)

506 (2)

511 (5)

520 (2, 11)

523 (12)

524 (5)

546 (5)

565

Jenkins, D. G.

653 (6)

721

Jenkins, D. R.

185 (24)

223

444 (61)

449

Johns, J. W. C.

145

150 (72)

171

193 (56)

224

304 (18)

382 (18)

385

111 (85)

115 (95)

116 (85)

122

444 (55)

449

3 (19)

8

116 (99)

123

146

149

354 (113)

371 (154)

375 (154)

387

411

559 (108)

564 (108)

567

710

172 Jen, C. K.

Johnson, C. M.

Johnson, D. R.

Johnson, R. C.

501

Johnson, R. D.

608 (75)

642

Johnson, R. H.

52 (11)

69

Johnston, M.

184 (15)

223

Jones, G. E.

2 (7)

7 (69)

7

10

72 (2)

120

265 (34)

293

300 (7)

351 (7)

361 (7)

364 (7)

3 (28)

4 (30)

7 (30)

8

273 (63)

294

411

635 (189)

384 Jones, H.

645 Jones, L. H.

192 (52)

224

380 (169)

389

Jones, S. R.

288 (136)

296

360 (127)

387

501 Joosen, P.

602 (65)

642

Jordahl, O. M.

586 (31)

640


Index Terms Joyner, C. H.

Judd, B. R.

Links 154 (89)

157 (89)

161 (100)

171

426 (35)

439 (35)

159 (89)

160 (98, 100)

449

803 (4)

690 (46)

722

839 Jurek, R.

360 (128)

387

Kabar, R. K.

688 (46)

689 (46)

Kaercher, A.

246 (14)

293

Kagann, R. H.

202 (82)

205 (82)

207 (95)

216 (82)

123

620 (106)

632

225

K

225 Kakar, R. K.

116 (97) 643

Kalasinsky, K. S.

624

Kalasinsky, V. F.

624

Karakida, K.

216 (114)

216

Karlsson, H.

618 (97)

643

Karo, A. M.

789 (74)

802

Karplus, R.

51 (10)

54 (12)

Karyakin, E. N.

152 (75)

171

Kasai, P. H.

231

621 (114)

Kastler, A.

3 (21)

8

Kasuya, T.

321

421

Kato, Y.

772 (48)

801

Kauppinen, J.

194

Kaushik, V. K.

321

Kawashima, Y.

187 (33)

Keenan, M. R.

69

643

766

770

223

678 (29)

722

5 (41)

9

155 (85)

156 (86)

159 (86)

160 (101, 106)

161 (102, 106)

162 (102)

Keller, C.

689 (52)

722

Keller, H. J.

282 (113)

295

171


Index Terms Keller, R. A.

Links 499 (50)

502

504

639 (213)

646 Kellogg, J. M. B.

392 (1)

448

466 (9)

503

Kemble, E. C.

475 (17)

503

586 (31)

640

Kemp, J. D.

570 (1)

639

Kennard, O.

653 (6)

721

Kenney, C. N.

7 (60)

9

65 (17)

70

444 (61)

449

727 (15)

801

Kerr, C. M. L.

115 (89)

122

Kewley, R.

185 (24)

223

250

267 (41)

294

321

633 (177)

645

389

677 (26)

711 (26)

231

577 (17)

578 (17)

583 (27)

592 (27)

594 (37)

600 (27)

605 (27)

607 (37)

640

Kilpatrick, J. E.

639 (212)

646

Kim, H.

351 (120)

357 (120)

387

499 (50)

502

504

639 (213)

646

King, G. C.

31 (5)

35

King, G. W.

112

246 (1)

292

340

416 (17)

448

849 (1)

852

Kieffer, L. J.

737

Kikuchi, Y.

381 (189) 722

Kilb, R. W.

King, J. G.

737

King, W. C.

2 (5)

7

Kinsey, J. L.

444 (61)

449

804 (6)

839

Kirby, C.

144 (46)

146 (64)

147 (62)

170

415

700 (65)

723

770

781 (63)

802

Kirchheiner, J.

773 (53)

802

Kirchhoff, W. H.

328 (68)

343 (163)

354 (68, 113)

361 (68)


Index Terms

Links 363 (37, 68) 375 (68, 154)

364 (37)

370 (68)

371 (68, 137, 154)

378 (137, 163)

380 (163)

386

Kirschner, S. M.

617 (87)

642

710

208 (100)

225

338 (77)

360 (77)

386 Kirtman, B.

602 (60)

603 (60, 68) 617 (89)

618 (60, 89)

642 Kitchin, R. W.

267 (43)

294

628 (166)

644

Kivelson, D.

247 (17)

251 (21)

293

305 (21)

311 (21, 38)

314 (50)

319 (50)

33 (50)

324 (21)

343 (38)

375 (38)

378 (38)

385

473 (14)

503

617 (88)

618 (88, 90)

642

Kiwada, K.

626

Kizer, K. L.

624

666 (14)

Klauss, K.

307 (36)

385

Klein, G. P.

354 (113)

387

Klemperer, W.

82

87 (39)

121

144 (44)

153 (76)

154 (89)

155

156 (89)

156

157 (89)

159 (89)

160 (93)

721

161 (93, 95, 99, 108)

162 (96, 100)

163 (113)

164 (114)

170

202 (81)

357 (123)

359 (123)

387

411

415

434

558 (86)

567

770

789 (70)

802

979

Kneizys, F. X.

326 (66)

386

Kneubuhl, F.

272 (58)

294

Knöckel, H.

745 (32)

801

Knopp, J. V.

623 (127, 130)

643

225


Index Terms

Links

Kohima, T.

620 (106)

643

Kohler, F.

3 (29)

8

Kohler, J. S.

582 (24)

640

Kojima, T.

231

624

Kondo, K.

436 (43)

449

Kopfermann, H.

732 (22)

Korolkov, V. S.

737

Koster, G. F.

689 (47)

722

733 (22)

735 (22)

801

735 (26)

737

801

Kraemer, W. P.

164 (118)

172

Kraitchman, J.

231; 415

655 (8)

Kramers, H. A.

99 (50)

121

Kreiner, W. A.

207 (98)

225

616

Krisher, L. C.

143 (36)

170

216 (108, 112)

225

231

415

436 (33)

438 (33)

443 (33, 52)

447 (33)

449

594 (38)

597 (38)

600 (55)

641

700 (66)

723

721

770

426 (33)

770 Krishner, L. C.

804 (7)

839

Kronig, R. de L.

29 (4)

35

175 (3)

223

Kroto, H. W.

7 (63)

9

133 (9)

144 (46)

145 (9)

145

146 (60, 63)

147 (60)

169

181 (31)

187 (31)

216 (116)

223

225

295

415

500

571 (4)

639

700 (65)

723

770

781 (63)

802

Krüger, H.

399 (15)

448

Krupnov, A. F.

2 (12)

7 (13)

8

152 (75)

171

193 (59)

194

201 (80)

202 (59, 80)

224

349 (96)

386

282 (110)

288 (110, 133)


Index Terms Kuchitsu, K.

Links 216 (114)

216

225

388

649 (4)

706 (89)

717 (102)

721

723

143 (36)

170

415

649 (3)

704 (73, 76)

721

723

770

Kudian, A. K.

195 (68)

207 (68)

224

Kuhler, M.

618 (96)

643

Kuijpers, P.

84 (26)

86 (26)

121

143 (42)

169

445

Kuiper, T. B. H.

116 (97)

123

Kukolich, S. G.

222 (123, 125)

226

415

421

436 (44)

449

516 (28)

516

533 (28)

535

565

766

Kuczkouski, R. L.

770

804 (15, 29, 37, 40)

Kundle, A. C.

444 (62)

449

Kunstmann, K.

231

502

Kupecek, P.

360 (132)

387

Kurland, R. J.

687 (43)

722

Kusch, P.

737

Kutner, M. L.

166 (133)

173

Kwan, Y. Y.

288 (135)

Kwei, G. H.

477 (18)

Kwok, S.

146

373 (150) 713 (97, 100)

133 (6)

840

627 (153)

644

296

603 (66)

642

500 (51)

503

L Laane, J.

380 (169)

389

Lada, C. J.

116 (98)

123

Lafferty, W. J.

267 (40)

268 (50)

294

321

349 (104)

387

628 (161)

644

649 (3)

721

Lainé, D. C.

3 (25)

8

Lamb, W. E.

141 (29)

170

392 (5)

448


Index Terms Landolt-Börnstein

Links 7 (65)

9

292 (151)

296

423 (19)

445 (19)

448

502 (53)

504

623 (143)

644

752 (65)

135 (20)

170

383 (204, 210) 389

421

571 (4)

639

723

766

770

Larkin, D. M.

493 (41)

494 (41)

503

Larsen, N. W.

623 (135, 137)

643

Laurie, V.

276 (91)

278 (91)

295

Laurie, V. W.

259 (26)

265 (33)

291 (142)

293

296

415

457 (2)

458 (2)

485 (28)

487 (28)

494 (2)

501 (2)

501

607 (72)

617 (86)

628 (158)

638 (209)

642

644

649 (3)

674 (20)

680 (86)

686 (42)

703 (20)

706 (85, 89)

708 (85)

709 (86)

714 (86)

721

770

756 (65) Landsberg, B. M.

706 (82)

712 (42, 86) 779 (61)

802 Le Fevre, R. J. W.

799 (78)

802

Leacock, R. A.

623 (141)

644

Leavell, S.

635 (188)

645

LeCroix, C. K.

622 (125)

626 (125)

Lee, M. C.

276 (88)

295

Lees, R. M.

143 (34)

170

318 (54)

361 (133)

385

387

602 (61)

603 (61)

642

682 (34)

720 (34)

722

Legell, H.

598 (47)

641

Legon, A. C.

7 (66)

10

153 (78)

154 (80)

155 (80, 85, 91)

155

156 (80, 86)

157 (80)

159 (80, 86)

160 (106)

171

221 (120)

643


Index Terms

Links 221

222 (124)

225

635 (184)

645 Legrand, J.

277 (94)

295

Levine, I. N.

627 (147)

644

Levy, D. H.

5 (43)

9

558 (87)

559 (87)

559

561 (105)

564 (106)

567

804 (12)

839

Lew, H.

737

Li, Y. S.

216 (111, 115)

216

225

271 (56)

272 (56)

294

598 (53)

624

626 (53)

626

637

641

666 (14)

721

4 (34)

8

84 (16)

107 (72)

121

143 (36, 41)

170

231

259 (25)

261 (25)

268 (45)

291 (140)

293

296

321

363 (137)

371 (137)

378 (137)

388

497

500

501

502

563 (107)

567

571 (3)

584 (29)

608 (29)

613 (29)

615 (29)

617 (29, 86)

639

642

231

444 (60)

449

572 (8)

579 (8)

582 (8)

583 (27)

584 (29)

587 (8)

592 (8, 27)

594 (8)

599 (8)

600 (27)

603 (8)

605 (27)

608 (8,

609 (29)

610 (8, 29)

611 (29)

612 (8)

613 (8, 29)

614 (8)

615 (8, 29)

617 (29)

623 (129)

627 (129)

639

643

Lind, G.

804 (29)

840

Lindfors, K. L.

492 (38)

503

Lide, D. R.

Lin, C. C.

144 (41, 52)

29)


Index Terms

Links

Lindsey, D. C.

351 (155)

375 (155)

380 (155)

388

Linke, R. A.

165 (127)

172

Linzer, M.

556 (82)

567

Lister, D. G.

572 (10)

620 (109)

624

635 (191)

639

643

645

Little, G. B.

688 (45)

722

Little, R.

411

Little, W. A.

804 (25)

840

Liu, B.

164 (117)

172

Livingston, R.

414

426 (30)

449

520 (10)

565

745 (29)

770 (29)

801

Ljunggren, S. O.

689 (50)

722

Lo, M.-K.

512 (17, 20)

517 (17)

532 (20)

535 (19)

540 (17)

545 (17, 20)

546 (17)

620 (113)

534 (17, 19)

565 Long, M. W.

426 (27)

448

Longuet-Higgins, H. C.

539 (65)

542 (65)

566

644

643 Lord, R. C.

628 (163)

638 (163)

Lorentz, H. A.

48 (4)

69

Loubser, J. H. N.

426 (33)

436 (33)

438 (33)

449

804 (7)

839

Louck, J. D.

282 (107)

295

304 (20)

385

Lovas, F. J.

81 (11)

82 (14)

89 (11)

89

120

164 (116)

172

354 (113)

387

411

501

649 (3)

468 (4)

478 (4)

503

389

559 (89)

447 (33)

721 Low, W.

458 (4)

Lowe, R. S.

321

Loyd, R. C.

691 (56)

722

Lucas, N. J. D.

351 (170)

380 (170)


Index Terms

Links 567

Lucken, E. A. C.

772 (49)

802

Lum, D. K.

633 (176)

645

Luntz, A. C.

638 (206, 208, 211)

639 (213)

Lurio, A.

737

772 (50)

802

Lutz, H.

502

622 (122)

625 (122)

625

779 (59)

802

645

643 Lyons, H.

191 (49)

224

McAfee, K. B.

492 (35)

503

McCormick, R. V.

144 (45)

170

McDonald, C. C.

556 (84)

561 (84)

567

Macdonald, J. N.

7 (66)

10

572 (10)

628 (165)

639

644

McDowell, R. S.

192 (52)

224

McGurk, J. C.

273 (65)

294

500

512 (40)

532 (40)

534 (40)

546 (40)

565

Macke, B.

277 (94)

295

McKean, D. C.

380 (164, 174, 181)

388

McKinney, P. M.

622 (125)

626 (125)

643

McKnight, J. S.

105 (65)

122

MacLeod, J. M.

146 (60)

147 (60)

McNaught, I.

307 (28)

385

McRae, G. A.

207 (95)

225

Mäder, H.

276 (84)

295

598 (52)

618 (52)

623 (52)

641

Maes, S.

181

185 (20)

187 (35)

Mahler, L.

421

766

770

Mahler, R. J.

105 (66)

122

Makhanek, A. G.

737

Maki, A. G.

4 (34)

M

8

171

144 (52)

603

223

146


Index Terms

Links 149

170

200 (75)

224

649 (3)

357 (124)

387

721

Makushkin, Y. S.

285 (123)

296

Mallinson, P. D.

380 (180)

389

Malloy, T. B., Jr.

228 (48)

228

267 (36, 43)

268 (47, 51)

293

294

628 (163, 167)

629 (168)

633 (176)

638 (163)

644

690 (54)

Maltesen, G.

773 (53)

802

Mandel, M.

83 (15)

85 (15)

Mann, D. E.

291 (140)

296

Manning, M. F.

189 (41)

193 (41)

224

Manns, K.

89

Manson, E. L.

84 (10, 28)

86 (28)

87 (37)

722

121

89 (10)

120 Marcuse, D.

141 (23)

170

Margenau, H.

554 (80)

566

Mariella, R. P.

558 (86)

567

Marino, R. A.

772 (46)

801

Markov, V. N.

201 (80)

225

Maroor, J.

421

766

Marshall, J. L.

804 (29)

840

Marshall, M. D.

155

Marshall, S. A.

485 (27)

493 (27)

503

Marstokk, K.-M.

216 (106)

225

272 (61)

294

320 (56)

321

351 (142)

369 (142)

385

388

635 (195)

645

Martin, D. H.

7 (62)

9

Martin, M. C.

704 (74)

723

Maslovskij, A. V.

2 (14)

8

193 (59)

201 (80)

202 (59, 80)

224

770


Index Terms

Links

Masson, M. A.

689 (53)

722

Mata, F.

704 (74)

723

Matamura, C.

216 (114)

225

Mathier, E.

598 (50)

623 (137)

641

643

Mathur, S. N.

281 (102)

295

691 (56)

722

Matsumara, K.

500

Matsumura, C.

143 (38, 41)

144 (41)

170

181 (117)

217 (117)

225

621 (119)

643

717 (102)

724

504

779 (60)

Matsumura, M.

216

Mattauch, J. H. E.

859

Matthews, H. E.

146

Mays, J. M.

496 (46)

501

802 Mazur, U.

704 (76)

723

Mazzariol, E.

373 (147)

378 (147)

388

Meakin, P.

623 (133)

632 (133)

643

Meal, J. H.

285 (117)

296

373 (149)

685 (39)

709 (39)

722

Meeks, M. L.

5 (49)

9

115 (90)

122

Meerts, W. L.

202 (84)

203 (86)

204 (84)

205 (85)

225

411

618 (110)

620 (110)

621 (110)

643

Meier, J.

622 (124)

643

Meise, H. M.

603

Meister, A. G.

306 (24)

307 (25)

324 (25)

385

Mennicke, J.

183 (13)

223

Merritt, F. R.

51

137

Messelyn, J.

277 (94)

295

Messer, J. K.

2 (9)

8

Metropolis, N.

426 (36)

439 (36)

388

682

447 (36)

449


Index Terms

Links 805 (45)

838 (45)

840

Meyer, H.

773 (52)

802

Meyer, R.

598 (50)

603

618 (93)

641

Michelot, F.

338 (79)

360 (79)

386

804 (34)

840 Mikhaylov, V. M.

195 (72)

206 (72)

224

Millen, D. J.

7 (66)

10

153 (78)

154 (80)

155 (80)

159 (80)

171

221 (120)

221

225

343 (138)

363 (138)

373 (138)

378 (138)

388

627 (148)

635 (182, 184)

644

683 (61)

693 (61)

840

723 Miller, C. E.

804 (13)

823 (13)

827 (13)

Miller, F. A.

574 (14)

601 (14)

640

Miller, S. L.

99 (53)

121

Miller, T. A.

559 (88)

561 (105)

772 (50)

802

Milligan, D. E.

145 (59)

171

Mills, I. M.

195 (66)

564 (106)

567

206 (88)

220 (119)

224

287 (129)

296

369 (144, 146) 370 (144)

373 (147)

377 (159)

378 (147)

380 (167)

381 (183, 185)

388

674 (19)

706 (83)

721

723

Minton, T. K.

155 (85)

156 (86)

159 (86)

171

Mirri, A. M.

109 (82)

122

181

231, 273

295

373 (147)

378 (147)

388

444 (56)

449

624

635 (194)

645

678 (32)

722

Mishra, A.

804 (22)

826 (22)

840

Mitchell, A. D.

653 (6)

721

(78)


Index Terms

Links

Miyahara, A.

411

Mizushima, M.

7 (67)

10

105 (66)

106 (62, 67) 115 (88)

122

195 (65)

224

383 (214)

389

425 (25)

444 (59)

448

469 (12)

478 (22)

485 (30)

487 (30)

495 (30)

503

550 (70)

566

804 (44)

838 (44)

840

Mizushima, S.

571 (5)

639

Mjöberg, P. J.

154 (82)

155 (83)

171

689 (50)

99 (54, 58)

104 (62)

722 Mochel, A. R.

620 (103)

643

Mohamad, A. B.

216 (111)

225

Mohan, N.

313 (43)

385

Møllendal, H.

216 (106)

225

272 (61)

294

320 (56)

321

351 (142)

369 (142)

385

388

635 (195)

645

Moller, K. D.

623 (136)

643

Möller, T.

445

Molloy, T. B., Jr.

628 (166)

644

Moloney, M. J.

600 (55)

642

Montagutelli, J.

338 (75)

360 (75)

386

Montgomery, J. A., Jr.

6 (56)

9

165 (122)

Moret-Bailly, J.

338 (74, 79)

360 (74, 79)

804 (34)

840

190 (47)

191 (47)

224

349 (106)

353 (106)

387

421

702 (71)

723

766

770

Morgenstern, K.

181

411

804 (30)

840

Morino, Y.

3 (18)

8

107 (75)

116 (96, 99)

122

149

Morgan, H. W.

172 386

181 (117)

187 (28, 34)


Index Terms

Links 217 (117)

223

225

231

285 (118)

291 (143)

296

298 (1)

341 (85)

373 (148)

381 (182, 184, 188)

384

386

388

559 (108)

564 (108, 110)

567

621 (119)

628 (156)

633 (156)

643

677 (26)

679

683 (27)

684 (38)

686

706 (88)

711 (26, 88)

713 (98, 100)

722

Morris, M.

145 (57)

164 (120)

170

172

Morrison, J. A.

216 (112)

225

415

770

Morse, P. M.

73 (3)

120

Morton, R. J.

627 (148)

644

Muenter, J. S.

153 (77)

158 (92)

171

415

457 (2)

458 (2)

485 (28)

487 (28)

494 (2, 29)

497 (29)

500 (29)

501 (2)

500

745 (33)

770

801

Müller, A.

313 (43)

385

Muller, N.

421

766

770

Mulliken, R. S.

108 (77)

109 (79)

114 (79)

122

130 (2)

169

239 (5)

242 (5)

293

747 (35)

755 (38)

768 (43)

790 (75)

791 (35)

798 (35)

801

Mun, I. K.

633 (175)

645

Murchison, C. B.

381 (187)

389

Murray, A. M.

804 (38)

840

Murray, J. T.

804 (25)

840

Murty, A. N.

635 (185)

645

Muse, J. D.

288 (137)

296

Myers, R. J.

231

425 (22)

448

608 (75)

616 (76)

621 (114)

622 (125)

626 (125)

685 (41)


Index Terms

Links 642

N Nair, K. P. R.

445

625

804 (31, 33)

840

Nakagawa, I.

149 (71)

171

Nakagawa, J.

626

627 (155)

632

644

Nakagawa, T.

149

381 (184, 188)

Nakata, M.

717 (102)

724

Nandi, R. N.

321

636 (198)

7 (64)

9

Naylor, R. E., Jr.

608 (77)

612 (77)

615 (77)

642

Nelson, A. C.

415

Nelson, J. F.

624

Nelson, R.

231

421

627 (150)

644

765 (39)

766

770

801

285 (119)

296

313 (45)

385

706 (79)

723

75 (9)

80 (9)

120

511 (7)

545 (7)

548 (7)

565

676 (22)

357 (123)

359 (123)

389

645

National Bureau of Standards

Nemes, L.

Nethercot, A. H.

722 Neuenschwander, M.

689 (52)

722

Neumann, R. M.

202 (81)

225

387 Newman, M.

352 (112)

387

Nielsen, A. H.

127 (1)

169

674 (19)

721

Nielsen, H. H.

131 (4)

132 (4)

133 (8, 13)

169

182 (12)

185 (17, 22)

223

232 (2)

282 (105, 108) 284 (108)

286 (108)

292

295

301 (9)

304 (9, 13)

384

582 (23)

640

674 (19)

721

201 (22, 79)


Index Terms

Links

Nielsen, J. T.

773 (53)

802

Niroomand-Rad, A.

286 (125)

296

367 (141)

382 (200)

169

388 Nishikawa, T.

766

770

Nixon, J. F.

133 (9)

145 (9)

145

Nixon, J. R.

216 (116)

225

500

Nolan, F. J.

246 (14)

293

Nolting, H.-P.

89

Nordsieck, A.

392 (4)

448

726 (1)

800

Norris, C. L.

273 (65)

294

411

500

512 (40)

532 (40)

534 (40)

536 (61)

546 (40)

565

780 (62)

802

598 (51)

618 (51)

623 (51)

641

701 (68)

723

160 (93)

161 (93, 95, 108)

162 (96)

171

313 (48)

385

Nowicki, P.

411

804 (26, 30) 840

Nuckolls, R. G.

191 (49)

224

Nygaard, L.

231

502

620 (105)

623 (135)

627 (153)

643

666 (13)

689 (49)

698 (63)

704 (13)

721

773 (53)

802

827 (49)

840

165 (130)

172

727 (14)

740 (14)

767 (14)

772 (14)

574 (13)

627 (151)

640

644

852 (2)

852

695 (62)

723

Nösberger, P.

Novick, S. E.

Nyman, L.-Å.

315 (48)

O O’Konski, C. T.

801 O’Reilly, J. M. Oak Ridge National Laboratory Oda, M.


Index Terms

Links

Oelfke, W. C.

623 (141)

644

Ogata, T.

616 (81)

617 (80)

Ohashi, O.

145

Ohl, R. S.

2 (10)

8

Oka, T.

3 (26)

8

143 (33)

147 (60)

170

195 (62, 67, 70)

197 (70)

198 (67, 70) 199 (70)

201 (67)

206 (70)

207 (96, 98) 215 (67)

224

273 (65, 67)

276 (90)

285 (114, 118, 120)

291 (143)

294

295

298 (1)

341 (85)

357 (125)

359 (125)

373 (148)

384

386

388

411

421

684 (38)

685 (41)

686

706 (87)

711 (88)

712 (90)

722

766

770

Okaya, A.

442 (50)

449

Olofsson, H.

165 (130)

172

Olson, W. B.

200 (75)

224

341 (87)

357 (124)

642

146 (60)

386 Oppenheimer, J. R.

150 (74)

171

Oppenheimer, R.

80 (7)

120

Orville-Thomas, W. J.

180 (7)

223

571 (6)

639

Otake, M.

181 (117)

187 (28)

217 (117)

223

225 Overend, J.

381 (187)

382 (196)

389

Owen, N. L.

572 (10)

624

628 (160)

639

644

Oxton, I. A.

380 (181)

389

Oyamada, M.

626

627 (154)

Ozier, I.

195 (68)

202 (82, 84) 203 (86)

205 (82, 85)

206 (89)

635 (193)

644 204 (84)

207 (68, 90)


Index Terms

Links 208 (94, 101)

216 (82)

224

338 (78, 81)

360 (78)

361 (78, 133)

620 (110)

621 (110)

643

Pake, G. E.

17 (1)

31 (1)

35

Palke, W. E.

602 (60)

603 (60, 68) 618 (60)

642

Palmer, P.

145 (57)

164 (120)

170

172

Pandey, G. K.

276 (89)

295

534 (50)

566

603

625

667 (15)

721

2 (14)

8

193 (55, 57, 59)

202 (59)

224

383 (207)

Parent, G. R.

307 (29)

380 (29)

385

Parker, P. M.

286 (125)

296

304 (19)

315 (53)

326 (65)

328 (67)

341 (19, 89)

367 (141)

369 (19)

382 (199)

385

388

Pascal, P.

536 (59)

566

Pasinski, J. P.

671 (17)

721

Pauling, L.

38 (1)

69

73 (5)

120

754 (37)

761 (37)

768 (37)

770 (37)

795 (37)

801

873

878

84 (22)

90 (29)

386

618 (110)

P

Papousek, D.

194 389

Pearsall, C. S.

501

Pearson, E. F.

72 (1)

82 (1)

120

143 (39, 43) 144 (39, 43, 45, 48)

145 (48, 53)

170

273 (65)

288 (138)

294

296

351 (120)

357 (120)

387

411

540 (68)

541 (69)

544 (68)

566

677 (23)

722

780 (62)

802

Pearson, P. K.

164 (117)

172

Pearson, R., Jr.

3 (19)

8

149

276 (91)

278 (91)

295


Index Terms

Links

Peau, E. W.

147 (68)

171

Pedersen, T.

285 (115)

286 (127)

296

120

623 (135, 137)

643 Pekeris, C. L.

73 (6)

77 (6)

Pence, D. T.

291 (142)

296

Penfield, H.

116 (98)

123

Penney, W. G.

468 (10)

481 (10)

Penrose, R. P.

188 (37)

223

Penzias, A. A.

87 (35)

97 (45)

Peter, R.

247 (16)

293

Peters, C. W.

623 (141)

644

Pickett, H. M.

267 (42)

282 (112)

294

349 (110)

355 (110)

387

635 (192)

645

231

246 (10)

293

339 (82)

343 (82, 153)

346 (82)

374 (82, 153)

378 (153)

378 (82)

386

421

574 (12, 13) 594 (12, 37)

597 (38)

607 (37)

621 (116)

627 (146, 150)

639

643

698 (64)

700 (64, 66)

710

723

765 (39)

766

770

801

849 (1)

164 (28)

165 (128)

639 (212)

639

Pierce, L.

503

121

388

852 Pierre, G.

208 (102)

Pillai, M. G. K.

311

Piltch, N. D.

96 (43)

225

121

172 Pinkstone, J.

560 (96)

567

Pitzer, K. S.

570 (1)

571 (3)

646 Plambeck, R. L.

166 (134)

173

Pliva, J.

380 (162)

382 (197)

388


Index Terms Plyler, E. K.

Pochan, J. M.

Links 192 (51)

224

712 (95)

724

516 (23, 32)

532 (32)

341 (86)

386

533 (23)

534 (23, 32)

565 Polacco, E.

273 (71)

294

Pollnow, G. F.

272 (60)

294

Polo, S. R.

285 (117)

296

307 (30)

373 (149)

385

388

685 (39)

709 (39)

441 (47)

722 Polyansky, O. L.

383 (216)

Porter, A. P.

411

Posener, D. W.

339 (83)

386

436 (46)

449

849 (1)

852

Pote, C. S.

649 (4)

721

Poussigue, G.

208 (103)

225

Powell, F. X.

107 (72)

116 (99)

122

371 (154)

375 (154)

388

559 (108)

563 (107)

564 (107)

567

710

421

704 (75)

723

766

293

471 (13)

473 (13)

Poynter, R. L.

389

770 Prakash, V.

51

Pratto, M. R.

259 (27) 503

Pringle, W. C.

638 (205)

645

Propin, R. Kh.

193 (54)

224

Pulay, P.

307 (35)

385

Pullman, A.

730 (17)

801

Pullman, B.

730 (17)

801

Putley, E. H.

7 (62)

9

618 (94)

620 (106)

Q Quade, C. R.

623 (127, 138)


Index Terms

Links 627 (129)

632

642

689 (47)

29 (4)

35

175 (3)

223

392 (1)

448

466 (9)

503

Racah, G.

479 (23)

503

803 (3)

839

Rademacher, H.

29 (3)

35

175 (2)

223

Radford, H. E.

115 (87, 89)

122

552 (78)

554 (78)

556 (78, 82)

557 (78, 82) 558 (82)

560 (97)

722

R Rabi, I. I.

566 Ralowski, W. M.

689 (50)

722

Ralston, A.

246 (8)

293

Ramadier. J.

135 (15)

169

Ramaprasad. K. R.

269 (54)

294

624

Ramsey, D. A.

6 (53)

9

649 (3)

721

Ramsey, N. F.

392 (1)

404 (16)

448

466 (9)

503

804 (8)

812 (8)

839

Rank, D. M.

5 (50)

9

Ransil, B. J.

789 (73)

799 (73)

Rao, CH. V. S. R.

377 (160)

388

Rao, D. R.

4 (35)

8

Rao, K. N.

192 (52)

224

341 (88)

386

Rastrup-Andersen, J.

231

444 (62)

449

502

627 (153)

644

698 (63)

723

773 (53)

802

Rastrup-Andersen, N.

421

766

770

Rawson, E. B.

552 (77)

554 (77)

556 (77)

566

Ray, B. S.

233 (4)

293

Raymonda, J.

87 (39)

121

411

434

Read, W. G.

155

221 (122)

222 (122, 125) 226

Reed, P. R.

616

802


Index Terms

Links

Reiche, F.

29 (3)

35

175 (2)

223

Reif, F.

397 (13)

399 (13)

448

767 (41)

641

801 Reike, C. A.

768 (43)

801

Reinartz, J. M. L. J.

411

Renwanz, E. F.

89

Reuger, L. J.

191 (49)

224

Rhodes, I.

577 (16)

640

Ribeaud, M.

598 (50)

600 (58)

623 (140)

644

804 (43)

840

Rigden, J. S.

231

827 (49)

840

Rimmer, D. F.

351

Rinehart, E. A.

616

620 (106)

643

689 (47)

722 Rinehart, P. B.

616

Ring, H.

781 (64)

802

Riveros, J. M.

627 (149)

632 (170)

Robertson, J. C.

804 (12)

839

Robiette, A. G.

181 381 (185)

343 (178)

644

369 (146)

380 (171, 178)

388

706 (83)

710 (84)

723 Robinson, G. W.

425 (23)

448

827 (48)

840

Robinson, H. G.

737 (58)

737

745 (34)

770

500

512 (40)

532 (40)

534 (40)

546 (40)

565

Rodriguez-Kuiper, E. N.

116 (97)

123

Rogers, S. C.

153 (78)

154 (80)

155 (80)

159 (80)

171

221 (120)

221

225

Rohart, F.

617 (82)

642

Rohwer, F.

534 (53)

566

801 Rock, S. L.


Index Terms

Links

Ronn, A. M.

273 (77)

276 (77)

295

Rose, M. E.

459 (8)

468 (8)

478 (8)

803 (1)

815 (1)

839

Rosenberg, A.

195 (68)

207 (68, 90) 224

Rosenblum, B.

75 (9)

80 (9)

120

511 (7)

545 (7)

548 (7)

565

676 (22)

770

503

722 Rosenthal, E.

421

766

Rosenthal, J.

444 (62)

449

Rotenberg, M.

426 (36)

439 (36)

447 (36)

805 (45)

838 (45)

840

Rothman, L. S.

304 (16)

385

Roussy, G.

5 (39)

7 (39)

9

269 (55)

294

320 (58)

349 (111)

351 (58)

385

387

620 (108)

643

689 (53)

722

Rowell, R. L.

488 (34)

503

Ruben, D. J.

415

770

Rudolph, H. D.

3 (29)

8

231

247 (16)

262 (29, 30)

276 (85)

291 (149)

293

295

324 (62)

369 (143)

386

388

595 (40)

603

615 (78)

616

625

628 (157)

641

644

664 (11)

700 (11)

704 (72)

721

723

773 (55)

802

804 (11)

833 (11)

839

Ruitenberg, G.

620 (102)

643

Rusk, J. R.

51

84 (21)

121

141 (25)

636 (203)

645

449

170 Russell, J. W.

369 (145)

388

Ryan, R. R.

380 (169)

389


Index Terms Rydbeck, O. E. H.

Links 115 (94)

123

Saff, E. B.

383 (213)

389

Sage, M. L.

621 (117)

643

Saito, S.

3 (18)

8

S

97 (49)

116 (96, 99, 101)

121

123

169

(140) 173

291 (146)

381 (189, 191, 193) 560 (93, 100, 102)

Sams, R. L.

296 389

564 (108, 111)

559 (108) 567

603 (67)

624

642

677 (26)

679

683 (27)

711 (26)

804 (36, 39)

837 (39)

840

149

200 (75)

224

357 (124)

109 (80)

115 (80)

122

444 (53)

449

552 (79)

554 (79)

566

722

387 Sanders, T. M.

Sarachman, T. N.

231

Sarka, K.

383 (205)

389

Sastry, K. V. L. N.

83 (32)

85 (32)

96 (42)

107 (71)

121

164 (126)

164

165 (126)

166 (135)

172

231

250

268 (49)

294

534 (43)

551 (72)

566 Satten, R. A.

737

Sawodny, W.

307 (35)

385

Saykally, R. J.

5 (47)

9

145 (54)

164 (119)

165 (125)

166 (134)

170

172

193 (60)

224

273 (78)

295

351

421

534 (47)

566

598 (52)

618 (52)

623 (52)

641

Scappini, F.


Index Terms

Links 766

770

Schaefer, H. F.

164 (117)

172

Schäfer, F.

411

Schaffer, W. H.

674 (19)

721

Scharpen, L. H.

265 (32)

293

485 (28)

503

633 (175)

638 (207, 209) 645

487 (28)

Schatz, P. N.

311

Schawlow, A. L.

1 (2)

7

50 (7)

54 (13)

65 (7)

69

188 (40)

224

246 (12)

251 (12)

293

415 (41)

425 (21)

436 (21, 41) 448

478 (22)

491 (21)

492 (37)

503

727 (11)

740 (11)

741

770

772 (11)

801 Schempp, E.

772 (45)

801

Schimizer, T.

273 (66)

294

Schlapp, B.

99 (52)

121

Schmalz, T. G.

273 (65)

294

536 (61)

566

Schmeltekopf, A. L.

106 (68)

122

363 (137)

371 (137)

378 (137)

388

224

357 (122)

359 (122)

Schmidt, R. E.

632

Schnabel, E.

190 (44) 387

Schottlander, M.

231

Schwartz, J.

276 (91)

278 (91)

295

Schwarz, R. F.

432 (38)

449

540 (67)

544 (67)

247 (15)

251 (20)

259 (24, 26)

293

421

624

632 (172)

632

645

649 (3)

654 (7)

681 (33)

701 (33)

704 (33)

706 (78)

714 (33)

566 Schwendeman, R. H.


Index Terms

Links 721

766

Schwinger, J.

51 (10)

69

Schwoch, D.

291 (149)

296

Scott, T. A.

773 (51)

802

Sears, T. J.

560 (96)

Seiler, H.

231

770

369 (143)

388

567

804 (32)

840

615 (78)

642

773 (56)

802 Seip, H. M.

713 (99)

724

Sekino, S.

766

770

Serenellini, S.

678 (30)

722

Serratrice, G.

689 (53)

722

Shaffer, W. H.

131 (4)

132 (4)

169

182 (9)

152 (75)

171

201 (80)

225

383 (216)

389

Sharbaugh, A. H.

415

502

Sharma, S. D.

272 (57)

294

Shenk, B.

81 (12)

89 (12)

121

Sheridan, J.

7 (66)

10

137

145

411

415

624

626

628 (165)

635 (193)

644

770

772 (47)

801

Shiki, Y.

624

626

Shimizu, T.

3 (26)

8

442 (51)

449

Shimoda, K.

3 (27)

8

273 (65, 72)

294

436 (43)

442 (51)

Shimoda, K. S.

273 (70)

294

Shoemaker, R. L.

512 (18, 20, 30)

514 (9)

532 (18, 20, 30)

533 (23, 32)

223 Shapin, S. M.

534 (18, 20, 23, 30, 32, 35) 545 (20, 24)

770

449

516 (23, 32)

535 (20)

540 (66)

565


Index Terms

Links

Shoji, H.

381 (194)

389

679

Shoolery, J. N.

502

Shortley, G. H.

446 (64)

449

459 (7)

468 (7)

478 (7)

503

811 (47)

840

133 (10)

169

495 (45)

496 (46)

500 (45)

504

Sidran, M.

246 (14)

293

Silver, S.

182 (9)

223

Silvey, G.

137

Silvey, J.

411

Simmons, J. W.

144 (47)

170

428

435 (39)

449

502

682

896

133 (9)

145 (9)

169

216 (116)

225

500

Simpson, J. B.

148 (69)

171

Sinnott, K. M.

600 (56)

642

766

770

Skaarup, S.

691 (55)

722

Slater, J. C.

733 (23)

785 (23)

Slawsky, Z. I.

324 (60)

386

Small, C. E.

288 (137)

296

Smith, A. G.

144 (47)

170

Smith, D. F.

382 (196)

389

Smith, D. L.

616 (81)

617 (81)

Smith, D. S.

207 (96)

225

Smith, J. G.

148 (69)

171

179 (5)

202 (83)

223

225

288 (137)

296

351 (142, 170)

369 (142)

380 (170, 173) 388

421

678 (28)

717 (101)

724

766

770

Smith, J. W.

799 (79)

802

Smith, L. G.

571 (2)

639

Shulman, R. G.

Simmons, N. P. C.

801

428

642

722


Index Terms

Links

Smith, W. L.

195 (66)

224

Smith, W. V.

1 (1)

7

48 (3)

50 (8)

54 (8)

69

144 (47)

170

232 (3)

293

474 (16)

503

539 (63)

542 (63)

566

727 (10)

733 (24)

734 (24)

740 (10)

801

874

896

Smyth, C. P.

795 (76)

802

Snyder, L. E.

87 (34)

115 (91)

121

144 (50)

145 (56)

147 (50, 66)

170

172

Somerville, W. B.

6 (51)

9

Soper, P. D.

155 (85, 87, 91)

155

163 (112)

164 (116, 121)

300 (6)

384

156 (86)

159 (86)

171 Sørensen, G. L.

773 (53)

802

Sørensen, G. O.

307 (34)

385

421

620 (105)

643

689 (49)

704 (74)

722

766

770

Speirs, G. K.

380 (175)

389

Špirko, V.

2 (14)

8

193 (55, 57, 59)

202 (59)

224

383 (207)

389

Spitzer, R.

639 (212)

646

Srivastava, G. P.

635 (181)

645

Srivastava, S. L.

51

Staats, P. A.

190 (47)

191 (47)

224

349 (106)

353 (106)

387

702 (71)

723

Starck, B.

213 (105)

225

649 (4)

721

Steenbeckeliers, G.

349 (102)

361 (102, 134)

194

366 (102)

370 (134) 387 Steinbach, W.

2 (11)

8

99 (57)

100 (57, 60)


Index Terms

Links 102 (61)

103 (60)

104 (57, 61)

106 (61)

122

835 (50)

836 (51)

493 (42)

495 (42)

501 (42)

293

633 (175)

496 (5)

501 (5)

840 Steiner, P. A.

455 503

Steinfeld, J. I.

273 (64)

294

Steinmetz, W.

804 (18)

840

Steinmetz, W. E.

265 (35)

267 (35)

645 Stelman, D.

591 (35)

641

Stephens, P. J.

458 (5)

462 (5)

503 Sternheimer, R. M.

731 (19)

735 (19)

736 (20)

742 (20, 28)

801 Stiefvater, O. L.

274 (83)

277 (97)

278 (98)

295

632 (171)

645

Stitch, M. L.

83 (15)

85 (15)

121

Stølevik, R.

713 (99)

724

Stolze, M.

534 (55)

535 (56)

536 (55)

540 (56)

542 (56)

566

534 (56)

535 (56)

540 (56)

Stolze, W. H.

279 (99)

538 (55)

542 (56)

566 Stone, J. M. R.

145

193 (55, 57) 224

383 (202, 206)

388

Stone, N. W. B.

4 (33)

8

Stone, R. G.

516 (23, 32)

532 (32)

144 (51) 533 (23)

380 (165)

170 534 (23, 32)

565 Stouffs, P.

349 (105)

387

Strandberg, M. W. P.

7 (58)

9

52 (11)

69

99 (55)

122

246 (9)

293

339 (83)

340 (84)

386

483 (26)


Index Terms

Links 497 (26)

501

503

506 (3)

511 (3)

520 (3)

525 (3)

546 (3)

551 (73)

561 (73)

565

849 (1)

388

636 (203)

645

377 (160)

381 (186)

385

295

501

623 (131)

7 (60)

9

65 (17)

70

181 (30)

181

185 (24)

187 (30)

223

444 (61)

449

727 (15)

179 (4)

181

642

852 Strauch, R.

797

Strauss, H. L.

369 (145)

Street, K.

797

Strey, G.

307 (36) 388

Stroke, H. H.

737

Strow, L. L.

354 (116)

Su, C. F.

321

Suenram, R. D.

281 (101)

387

643 Sugden, T. M.

801 Sugie, M.

717 (102)

724

Sullivan, T. E.

4 (35)

8

205

223

Sumberg, D. A.

382 (199)

389

Sundaram, S.

307 (26)

385

Sung, E.

636 (197)

645

Sutherland, G. B. B. M.

188 (39)

224

Sutter, D.

617 (83)

618 (96)

622 (122)

Sutter, D. H.

415

501

516 (51)

533 (51, 58)

534 (27, 37, 44, 48, 54, 58)

535 (27, 56)

536 (54, 58) 537 (58)

538 (55)

540 (56)

541 (48, 58) 542 (56)

546 (51)

565

625

770


Index Terms

Links

Sutton, L. E.

653 (6)

721

Suzean, P.

360 (128)

387

Suzuki, M.

502

534 (46)

Svidzinskii, K. K.

426 (29)

448

Swalen, J. D.

193 (53)

224

246 (10)

293

502

572 (8)

574 (11)

579 (8)

582 (8)

587 (8, 32)

592 (8)

593 (32)

594 (8, 1, 32)

599 (8, 11)

601 (11)

603 (8)

608 (8)

610 (8)

612 (8)

617 (85)

621 (85)

639

641

849 (1)

566

852 Szalanski, L. B.

624

700 (67)

723

Szanto, P. G.

5 (47)

9

96 (43)

121

145 (54)

164 (119)

165 (125)

170

172

T Tack, L. M.

804 (41)

822 (41)

840

Taft, H.

506 (4)

511 (4)

533 (4)

540 (4)

296

624

712 (90)

8

116 (101)

123

389

679

717 (102)

369 (143)

388

565 Takagi, K.

291 (143) 723

Takami, M.

3 (27) 411

Takeo, H.

381 (192, 195) 724

Takeyama, H.

772 (48)

801

Takuma, H.

442 (51)

449

Tan, B. T.

291 (149)

296

616

625

Tanaka, K.

500

Tanaka, T.

291 (144)

296

381 (190, 194) 389


Index Terms

Links 500

679

Tang, C. L.

4 (38)

8

Tannenbaum, E.

231

608 (75)

616 (76)

642

Tarrago, G.

133 (8)

169

185 (22)

186 (27)

201 (22)

208 (103)

223

225

282 (108)

284 (108)

286 (108)

295

304 (14)

384

Taylor, R. C.

415

770

Teller, E.

184 (14)

223

Terman, F. W.

773 (51)

802

Thaddeus, P.

6 (54, 56)

9

168 (136)

169 (136, 138)

436 (33)

438 (33)

443 (33, 52)

447 (33)

449

804 (7)

839

765 (39)

766

Thiele, W.

859

Thomas, C. H.

421

624

770

801

Thomas, E. C.

415

501

Thomas, L. F.

415

770

Thompson, I.

179 (5)

223

165 (124, 127) 166 (133) 172

426 (33)

770

421

766

770 Thompson, J. C.

181

Thorne, L. R.

501

Thornton, D. D.

5 (50)

9

Thorson, W. R.

149 (71)

171

Thyagarajan, G.

307 (26)

385

Tieglear, H. L.

512 (40)

532 (40)

534 (40)

546 (40)

81 (11)

82 (14)

87 (38)

89 (11)

89

120

411

445

741 (27)

745 (27, 32)

565 Tiemann, E.

801


Index Terms

Links 804 (31, 33)

840

Tigelear, H. L.

500

Timms, P. L.

616 (81)

617 (81)

642

Tinkham, M.

99 (55)

122

551 (73)

561 (73)

81 (11)

89 (11)

89

99 (56)

120

122

532 (38, 42, 45)

534 (38, 42, 45)

565

Tobiason, F. L.

706 (78)

723

Tolles, W. M.

615 (79)

642

Tolman, C. A.

639 (213)

646

Toman, S.

380 (162)

388

Tong, C. C.

271 (56)

272 (56)

294

Tong, C. K.

637

Törring, T.

81 (11)

84 (18, 26)

86 (26)

87 (38)

89 (18)

120

133 (6, 12)

135 (12)

137 (12)

143 (42)

169

190 (44)

224

357 (122)

359 (122)

387

1 (2)

3 (20)

5 (50)

7

50 (7)

51

54 (13)

65 (7)

69

75 (9)

80 (9)

83 (15)

85 (15)

99 (53)

109 (80)

115 (80)

120

133 (10)

137

169

188 (40)

224

246 (12)

251 (12)

267 (69)

273 (69)

279 (69)

293

392 (11)

411

415

423 (20)

425 (21)

427 (37)

432 (38)

436 (11, 21)

441 (11)

444 (53)

448

458 (4)

468 (4)

478 (4, 21)

491 (21)

492 (36)

495 (45)

500 (45)

503

511 (7)

566 Tischer, R.

411 Townes, C. H.


Index Terms

Links 540 (67)

544 (67)

545 (7)

548 (7)

552 (79)

554 (79)

565

676 (22)

682

722

727 (2, 11)

740 (11)

741 (2, 11)

741

770

779 (60)

800

Toyama, M.

298 (1)

384

Trambarulo, R.

185 (23)

223

800 (81)

802

Trambarulo, R. F.

1 (1)

7

50 (8)

54 (8)

69

232 (3)

293

727 (10)

740 (10)

801

874

896

Treacy, E. B.

441 (48)

449

Trinkaus, A.

625

Trowell, P. L.

216 (111)

225

True, N. S.

634 (178)

645

Trueblood, M. B.

143 (39, 43)

170

Tsuchiya, S.

624

Tucker, K. D.

166 (133)

173

Turner, B. E.

6 (54)

9

115 (93)

123

145 (57)

147 (65)

164 (120)

165 (123)

170

171

172

Tycko, D.

742 (28)

801

Tyler, J. K.

137

145

411

635 (191)

645

797

320 (57)

328 (71)

333 (71)

354 (114)

385

625

701 (70)

723

Uehara, H.

564 (110)

567

Uhlenbeck, G. E.

189 (42)

193 (42)

Ulenikov, O. N.

285 (123)

296

Ulich, B. L.

164 (116)

172

Unland, M. L.

273 (75)

295

Typke, V.

772 (2, 11)

500

U 224


Index Terms Urban, S.

Links 2 (14)

8

193 (59)

202 (59)

534 (21)

565

224

V VanderHart, D.

512 (21)

532 (21)

Van Dijk, F. A.

141 (27)

170

van Eijck, B. P.

269 (53)

294

328 (70)

333 (70)

386

444 (63)

449

602 (65)

632 (173)

642

645

705 (77)

50 (5)

69

80 (8)

108 (76)

109 (78)

111 (76)

115 (78)

119 (8)

120

122

239 (7)

293

392 (8, 11)

436 (11)

441 (11)

448

475 (17)

503

524 (13)

544 (13)

565

586 (31)

640

676 (21)

723 Van Vleck, J. H.

721 van Wachem, R.

497

797

Varga, R. S.

383 (213)

389

Varma, R.

7 (70)

10

Veazey, S. E.

85 (30)

121

Venkatachar, A. C.

415

770

Venkateswarlu, P.

186 (25)

195 (65)

624

223

635 (187)

645

Verdier, P. H.

617 (98)

619 (98)

643

Verhoeven, J.

535

Volltrauer, H. N.

632 (172)

632

645

Wacher, P. E.

471 (13)

473 (13)

503

Wacker, P. F.

259 (27)

293

Wagner, J.

182 (8)

223

Wagner, R. S.

624 (145)

644

321

W


Index Terms

Links

Wahlgren, U.

164 (117)

172

Walden, M.

160 (97)

172

Walden, R. T.

300 (7)

351 (7)

361 (7)

364 (7)

6 (52)

9

115 (92)

122

300 (5)

384

Walsh-Bakke, A. M.

304 (17)

385

Walton, D. R. M.

146 (63)

147 (62)

Wang, J. H. S.

415

534 (26, 33)

384 Walksley, C. M.

171 565

770 Wang, S. C.

239 (6)

251 (6)

293

Wapstra, A. H.

859

Warren, I. D.

635 (196)

645

Watson, J. K. G.

195 (64, 66)

196 (64)

206 (63)

208 (100)

216 (112)

224

282 (106)

285 (121)

286 (106, 124)

288 (124, 131, 134)

295

304 (15)

313 (42, 47) 314 (51)

318 (51)

320 (51, 55)

325 (55, 63) 326 (63)

327 (55, 64)

328 (55, 63)

329 (63, 72) 330 (64)

335 (140)

336 (64)

338 (72, 76, 80)

342 (64)

250 (6)

357 (64, 72)

339 (64, 72, 80)

360 (76, 80)

361 (64, 76)

364 (139)

367 (140)

371 (140)

376 (158)

380 (165)

381 (72)

385

388

667 (16)

677 (24)

678 (57)

691 (57)

714 (57)

717 (57, 101)

Watson, W. A.

415

770

Wayne, F. D.

115 (89)

122

Weatherly, T. L.

426 (27)

448

481 (25)

503

804 (25)

840

Weber, A.

306 (23)

385

Weber, J.

485 (27)

493 (27)

721

724

501

503


Index Terms

Links

Weinreb, S.

5 (49)

9

115 (90)

122

Weiss, M. T.

501

Weiss, R.

799 (77)

802

Weiss, V. W.

273 (75)

295

Weisskopf, V. F.

50 (5)

69

Welch, W. J.

5 (50)

9

Wells, J. A.

268 (47)

294

628 (167)

Wells, J. S.

105 (66)

115 (87)

122

Wells, P. R.

784 (66)

789 (66)

802

Welsh, W. M.

104 (64)

106 (64)

122

Welti, D.

623 (137)

643

Wendling, P.

615 (78)

642

Wentink, T., Jr.

483 (26)

497 (26)

503

West, B. G.

104 (63)

106 (63)

122

469 (12)

797

802

653 (6)

411

644

503 Westerkamp, J. F.

133 (11)

169

Wharton, L.

82

789 (70)

Wheatley, P. J.

653 (6)

721

Whiffen, D. H.

148 (69)

171

421

721

766

770

White, H. W.

732 (21)

733 (21)

735 (21)

White, K.

477

White, K. G.

498 (47)

502

504

White, M. S.

502

White, R. L.

415 (41)

432 (38)

434 (45)

449

540 (67)

544 (67)

566

White, W. F.

276 (88)

295

Whitehead, M. A.

784 (67)

802

Whittle, M. J.

187 (29)

223

360 (126)

387

Wick, G. C.

524 (15)

565

Wiese, J.

534 (48, 54)

536 (54)

541 (48)

566

801

436 (41, 45)


Index Terms

Links

Wilardjo, L.

304 (19)

341 (19)

369 (19)

385

Wilcox, W. S.

502

Wilf, H. W.

246 (8)

293

Wilke, W.

190 (44)

224

357 (122)

359 (122)

369 (142)

388

387 Willemot, E.

351 (142)

Williams, G.

624

Williams, J. R.

415

770

Williams, N. H.

188 (36)

223

Williams, Q.

426 (27)

448

481 (25)

503

804 (25)

840

Willis, R. E.

82 (13)

121

Wilson, E. B., Jr.

38 (1)

65 (16)

69

73 (5)

120

185 (16)

187 (16)

223

247 (17)

267 (38)

273 (74, 77, 82)

276 (74, 77, 82)

277 (77)

279 (82)

282 (103, 111)

283 (111)

286 (103)

289 (139)

293

295

301 (8, 10)

302 (11)

304 (8)

305 (21)

311 (21)

314 (50)

315 (52)

319 (50)

324 (21)

339 (50)

372 (11)

384

451 (1)

468 (11)

470 (11)

488 (11)

491 (11)

492 (1, 35)

502

503

581 (21)

583 (27)

584 (29)

586 (32)

592 (27)

600 (27)

604 (70)

605 (27, 70) 608 (29, 77)

609 (29)

612 (77)

613 (29)

616 (77)

617 (29, 98)

619 (98, 100)

501

615 (29, 77)

621 (115)

628 (159) 632 (170, 171)

632

635 (183, 186, 196) 687 (43)

721

633 (159) 636 (186, 200)

640

674 (19)

722


Index Terms Wilson, P. W.

Links 87 (35)

97 (45)

121

381 (195)

389 Wilson, S.

168 (137)

173

Winn, J. S.

160 (99)

172

Winnewisser, B. P.

3 (16)

8

84 (24)

121

135 (16)

136 (20)

143 (43)

169

288 (138)

296

349 (94)

380 (172)

383 (210)

386

389

632

706 (81)

723

3 (16)

6 (52)

8

67 (18)

70

84 (25)

90 (25, 29)

115 (92)

121

144 (48)

145 (48, 53)

145

149

170

256 (22)

257 (22)

268 (22, 52)

293

300 (5)

328 (69)

333 (69)

349 (94)

351 (155)

351

Winnewisser, G.

356 (69, 118) 358 (119) 375 (155) 380 (155, 171) 384

386

421

497

498 (48)

500

504

623 (142)

644

700 (67)

723

766

3 (16)

8

67 (18)

70

84 (23)

90 (25)

107 (71, 73)

121

135 (14, 16)

136 (14)

144 (48)

145 (48, 53)

146

147 (67)

150 (73)

151 (67, 73)

152 (67)

169

250

257 (22)

268 (22, 52) 288 (138)

291 (147)

293

296

349 (94)

351

356 (117)

375 (117)

386

500

502

551 (72)

566

623 (142)

644

700 (67)

706 (80)

723

135 (17)

170

770 Winnewisser, M.

Winther, F.

256 (22)


Index Terms Winton, R. S.

Links 141 (31)

170

181

623 (142)

644 Wodarczyk, F. J.

273 (82)

276 (82)

279 (82)

295

Wofsy, S. C.

202 (81)

225

357 (123)

359 (123)

387

415

770

804 (15)

415

770

840 Wolf, A. A.

426 (28)

448

Wolf, S. N.

216 (109, 113)

225

Wolfe, P. N.

426 (26)

448

Wollrab, J. E.

7 (61)

9

282 (109)

295

590 (121)

616

622 (121)

643

5 (46)

9

96 (41, 43)

97 (43, 48)

121

145 (54)

164 (119, 128)

165 (125, 128)

166 (132)

170

172

273 (77)

276 (77)

295

597 (44)

641

426 (36)

439 (36)

447 (36)

805 (45)

838 (45)

840

85 (33)

87 (37)

121

Yajima, T.

273 (72, 81)

294

295

Yallabandi, K. K.

328 (67)

341 (89)

386

Yamada, C.

500

804 (39)

837 (39)

840

Yamada, K.

135 (18)

147 (68)

150 (73)

170

288 (130)

291 (147)

296

356 (117)

375 (117)

387

700 (67)

723

Yokozeki, A.

745 (33)

801

Young, A. T.

383 (215)

389

Young, L. D. G.

383 (215)

389

392 (1)

448

466 (9)

503

Woods, R. C.

Wooten, J. K. Wyse, F. C.

163 (110)

449

Y

Z Zacharias, J. R.


Index Terms Zare, R. N.

Links 106 (68)

122

363 (137)

371 (137)

378 (137)

388

Zeiger, H. J.

3 (20)

8

Zeil, W.

183 (13)

223

Zijderveld, G. R. D.

619 (101)

643

Zimmerer, R. W.

104 (62)

106 (62)

122

Zinn, J.

477 (19)

503

636 (201)

638 (201)

145 (57)

645 Ziurys, L. M.

166 (134)

173

Zukerman, B.

115 (93)

116 (97)

123

164 (120)

170

172


SUBJECT INDEX Index Terms

Links

A Absorption, optimum region for:

122

133

diatomic molecules

120

linear molecules

139

140

symmetric-top molecules

212

213

133

133

198

223

253

254

Absorption cells: attenuation constant cavity double resonance modulation free space high temperature

43 5

160

276

279

3

84

82

84

parallel plate

493

Stark modulation

492

493

Absorption coefficient: asymmetric-top molecules

263

diatomic molecules

117

general formula, with Lorentzian line shape

42

linear molecules

138

measurement of

43


118

208

Angular momenta: average values of, in asymmetric rotor basis

247

248

271 classical time rate of change commutation rules

11 13

15

16

17

19

21

282

314

318

584


Index Terms

Links

molecule -fixed axes

21

space-fixed axes

16

eigenfunctions

27

eigenvalues

17

17

matrix elements, see Matrix elements, angular momentum operators

15

16

23

raising and lowering

17

18

21

spin

19

20

19

811

237

315

136

187

phase convention transformation properties under V Anharmonicity, Fermi resonance Anharmonicity constant, ωexe

219

286

74

relation to spectroscopic constants

77

79

table of

83

86

88

90

258

264

273

491

492

686

273

491

Assignment of spectra

Associated Legendre polynomials Asymmetric rotor

28 227

analysis of spectra for rotational constants

269

assignment of spectra

258

264

492

686

254

257

258

288

297

286

289

branches notation centrifugal distortion

257 287

339

361 Coriolis interaction

285

energy levels, rigid rotor

232

algebraic expressions example calculation notation of

244

245

26

27

248

229

230

232

249


Index Terms

Links

sum rules

341

342

energy matrix

240

241

intensities

256

peak absorption coefficient internal motions

851

263 569

See also Internal rotation matrix elements

24

235

236

250

816

817

827

413

429

436

814

827

828

66

440

441

600

827

828

290 nuclear magnetic coupling

435 828

nuclear quadrupole coupling

nuclear spin statistics

representations

236

selection rules

254

slightly asymmetric rotors

250

417

418

474

Stark effect

468

488

603

604

structures

291

292

678

679

683

684

696

702

703

710

711

713

61

233

717 vibration, correction to rotational constant effect on energy levels wave functions

example calculation symmetry classification Zeeman effect

281

288

281 29

60

237

247

248

249

60

61

241

515

530

534

535

617


Index Terms Asymmetric rotor group V

Links 238

Asymmetry parameter: in asymmetric rotors, inertial

229

250

251

in quadrupole coupling

398

407

739

764

735

785

Atomic masses

859

Atomic number, effective

732

733

Atomic term energy

732

733

5

160

Atom-molecule, complexes force constants

162

structure

161

α-type transitions

256

Avogadro’s constant

857

Avoided-crossing technique for, internal rotation barriers

205

K-level separations

202

Back-Goudsmit effect

521

Band spectra

265

620

621

633

634

B

frequency relations

266

Barriers, hindered rotation, table of

603

616

624

Barrier tunneling

188

575

576

Basis functions

842

843

Bohr magneton

857

Bohr radius

857

Boltzmann distribution law Boltzmann’s constant

39

632

619

857

Bond length: average árñ

649

evaluation of

706

relation to r0 and re

707

effective r0

648

706

679


Index Terms evaluation of equilibrium re evaluation of mass dependence rm evaluation of substitution rs

Links 679 648 672 649

648 691

relation to r0 and re

693

Bose-Einstein statistics b-type transitions

714

714

evaluation of

Born -Oppenheimer expansion parameter

674

150

691

692

151

285

328

297

339

62 256

C Casmir’s function, see Y(J,I,F) Center of mass

653

Centrifugal distortion

297

asymmetric rotors

287

288

361

617

derivation of P4 Hamiltonian for

287

288

first-order

313

344

higher-order

339

diatomic molecules

74

301

75

78

79

95

99

110

340

342

349

352

matrix elements

340

344

354

Hamiltonian, general

325

304 with electronic angular momentum

94 111

energy matrix, form of

orthorhombic form number of terms Padé

326

327

327 383

384


Index Terms reduced, derivation of

Links 324

asymmetric-top reduction

329

symmetric-top reduction

333

standard form number of terms

327 327

linear molecules

127

131

132

spherical tops

207

208

338

339

360

361

177

185

191

197

200

201

322

337

338

357

approximation for methyl halides

181

182

bent triatomic, expression for

311

312

dectic

333

339

defect

370

determinable combinations

335

339

symmetric rotors

Centrifugal distortion constants:

diatomic, expression for

77

evaluation from spectra

339

choice of Hamiltonian

361

effect of representation

354

relations between representation

356

rigid rotor basis method

343

semirigid rotor basis method

347

304

356

348

extremal properties

313

isotopic relations

312

linear triatomic, expression for

127

octic

333

335

338

planarity relations

321

323

366

quartic

319

320

322

313

330


Index Terms relation to ταβγδ, asymmetric rotor

Links 333

334

338

320

322

330

331

334

337

352

337 symmetric rotor

323

relation between A and S reduction

335

sextic

331

324

332

338 calculation of

288

enumeration of

326

382

sum rules

341

342

table of, asymmetric rotors

321

343

350

353

355

381

75

76

83

88

90

98

106

108

142

148

155

161

164

168

201

202

305

311

diatomic molecules

linear molecules

spherical tops

381

symmetric rotors

181

192

221

381

vibration effects

368

Centrifugal distortion constants ταβγδ : calculation of

305

definition of

287

303

312 nonvanishing of, asymmetric tops

316

in first-order

314

315

symmetric tops

323

324

planarity relations

322

367

potential constants, from

372

Character table:


Index Terms

Links

C3 group

578

V group

238

Chemical analysis

7

Chi-square distribution

362

Clebsch-Gordon coefficients

446

301

459

468

279

280

478

805 See also Three-j symbol Coherence splitting, creeper

274

Commutation rules, see Angular momenta Commutator

328

Complexes: atom-molecule

5

160

hydrogen bonded

5

153

221

13

20

23

Constants of rotational motion

229 Continued fraction, evaluation of energy levels of asymmetric rotor

849

Conversion factors

857

Coordinates, structural, see Kraitchman’s equations Coriolis coupling constants: calculation of

685

expression for bent XY2

709

expression for linear XYZ

708

rule for nonvanishing of

285

sum rules

285

Coriolis force

131

685

Coriolis interaction: asymmetric rotors

289

linear molecules

128

symmetric rotors

183

219

Correlation coefficients

362

363

607


24

Index Terms

Links

Correlation diagram: asymmetric rotor

230

Mathieu eigenvalues

576

Π states

112

2

Covalent bond radii, table of Crystal harmonic generator c-type transitions

873 1

2

256

D Degrees of freedom Detectors Diagnostic least squares, for structural parameters

301 2 717

Diatomic molecules

71

bond lengths

76

83

86

90

106

680

74

75

78

79

94

95

99

110

111

304

centrifugal distortion

88

Dunham’s solution

78

with electronic angular momentum

87

90

444

550

551

835

836

838

71

78

79

87

90

107

103

104

energy levels

intensities peak absorption coefficient

117

117

Λ-type doubling

107

nuclear magnetic coupling

432

444


407

426


63

64

selection rules

34 95

444

445

72

74

92

102

112

113


Index Terms Stark effect

Links 451

484

vibration, effect on energy levels

72

78

wave functions

28

Zeeman effect

509

520

521

529

530

532

550

551

Dipole moment matrix elements, asymmetric rotor 259

262

263

Dimers, see Hydrogen bonding l-doublets linear

133

482

33

Λ -doublets

113

562

inversion doublets

33

482

symmetric rotor

31

33

498

544

497

498

isotopic substitution

494

496

vibration

496

497

from intensity measurements

492

493

from Stark splitting measurements

493

35

Dipole moments: direction of effects of, centrifugal distortion

example calculation

497

473

474

495

497

154

156

induced

195

206

207

interpretation of

788 794

795

tables of enhanced

bond moment

789

498

500

799

800 hybridization moment

791

induced moment

794

798

ionic character from

754

755

overlap moment

790

791

795


Index Terms primary moment Direction cosine, symmetry under V

Links 790 255

469

259

468

Direction cosine matrix elements: asymmetric rotor linear

33

symmetric rotor

31

32

Dissociation energy

74

77

Doppler broadening

45

Doppler frequency shift

45

Double resonance

3

assignment aid

276

four-level system

280

line profiles

279

280

information from

3

4

three-level system line profiles Dunham constants

4

471

143

273

273

274 274

277

78

79

relation to potential constants

79

table of

83

86

88

90

Dunham correction, to rotational constant

79

80

676

677

Dunham’s solution, diatomic

78

306

308

311

74

79

108

128

129

131

182

281

880

E Eckart conditions Effective rotational constant

672 Eigenvalue equation

842

Eigenvalues, methods of finding

24

846

879

Einstein coefficients

33

38

39

Electric field gradient in atoms from atomic beam measurements

731 735


Index Terms

Links

configuration interaction correction

736

from doublet find structure

733

effects of formal charge

733

effects of screening

732

733

relativistic correction

735

736

Sternheimer polarization correction

735

736

table of qnl0

734

737

approximations and assumptions

730

731

calculation of

727

738

formal charge corrections

741

742

hybridization effects

742

Electric field gradient in molecules:

interpretation of quadrupole coupling constants

745

755

See also Nuclear quadrupole coupling unbalanced p-electrons

738

units

738

Electronegativities

values of

753

762

795

875

761

778

763

783

783

785

875 Electronic angular momentum, coupling cases molecules in, Π or ∆ states

91 107

169

2

Σ states

94

166

836

3

Σ states

98

835

836

molecules with, hyperfine structure

444

notation

87

90

91

tables of derived constants

98

106

164

168

557

564

165


Index Terms Zeeman effect

Links 549

Electronic corrections to: inertial constants inertial defect Electron paramagnetic resonance of gases

80

546

675

684 552

molecular constants derived from

557

Stark effect

561

564

Electronic spin -spin interaction

99

835

Energy ellipsoid

14

15

Energy level index for asymmetric rotors, K -1, K1 , τ 229

230

232

76

83

86

88

90

106

143

149

217

219

220

678

679

680

219

286

Equilibrium structure, tables of

Euler’s angles

29

Euler’s equations of motion for free rotation

13

Euler’s theorem on homogeneous functions

715

F Fermi-Dirac statistics

62

Fermi resonance

136

Fine structure constant

857

Forbidden transitions, internal rotation

599

604

195

205

spherical tops line strengths

206

selection rules

206

Stark effect

489

symmetric tops

194

interaction term

197

line strengths

198

selection rules

197

187

491

562

200

Force constants, see Potential constants This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Four group, see Asymmetric rotor group Frequencies of transitions: l-doublet

133

Λ -doublet

113

135

rotational: in diatomic molecules

72

Hund’s case (a)

93

3

Σ states

102

in linear molecules

127

degenerate bending mode in symmetric rotors degenerate bending mode rotation-inversion

74

78

128

131 177

179

182

185 191

of some rotational transitions, in asymmetric rotors 266 Frequency, of inversion

188

F-test

364

Fundamental constants, table of

857

194

G g-factor, electron spin g s

549

molecular gJ asymmetric rotor

516

general definition of

508

linear molecules

509

symmetric rotor

513

550

molecules with electronic angular momentum 550

551

molecules with nuclear coupling

520

1

origin in Σ molecules

519 523

molecular g tensor elements ggg definition

527

effects of isotopic substitution

544

545

from microwave spectra

512

516

nuclear spin gI

429

520

534

535


Index Terms

Links

G-matrix, see Vibrational energy Gyromagnetic ratio

429

507

303

313

331

333

H Hamiltonian: centrifugal distortion: asymmetric rotor linear molecule

127

spherical top

338

339

symmetric top

178

337

Coriolis coupling

285

289

electric polarizability

485

harmonic oscillator

282

283

hyperfine, general

805

818

internal rotation

584

585

606

609

torsional

317

325

338

587

590

167

585

K-level splitting in symmetric tops

338

magnetic susceptibility

529

molecules with electronic angular momentum, Hund’s coupling case (a)

92

2

94

95

97

99

100

835

nuclear magnetic coupling (I-J)

430

438

805

818


396

404

439

805

333

337

Σ states

3

Σ states

818 solid state

396

Padé

383

reduced, centrifugal distortion

331

rigid rotor:


Index Terms

Links

asymmetric-top

232

233

reduced

233

250

linear molecule

72

126

spherical-top

23

symmetric-top

23

ring puckering spin-rotation interaction

250

176

638 94

100

spin-spin interactions

100

805

830

835

Stark effect

452

vibration-rotation

130

184

282

284

3

5

141

143

144

750

758

289 expanded form

284

general (nonlinear)

282

Zeeman effect nuclear magnetic

507 509

Harmonic oscillator, see Hamiltonian Hermitian, definition of, matrix operator High resolution spectroscopy

234

841

19 2 441

High temperature spectroscopy

82

Homogeneous function

714

Euler’s theorem

715

Hybridization: from nuclear quadrupole coupling table of

755 756

relation to bond angle

777

relation to dipole moment

749

794

795 rule

749

750


Index Terms

Links

Hydrogen bonded complexes: asymmetric-tops

157

222

bridge lengths

155

161

enhanced dipole moment

154

156

force constants

158

159

linear

153

symmetric tops

221

Hydrogen bonding in rotational isomers

635

Hyperconjugation from quadrupole coupling

767

table of Hund’s coupling cases case (a) case (b) Hypergeometric series

221

636

770 91 91 93 29

I Inertial defect effect on structure calculations

683

703

710

712

683

684

691

703

705

712

expression for

684

information on, planarity

687

688

torsional vibrations

689

690

substitution

691

tables of

686

Intensity of absorption, asymmetric -top

687

691

689

703

256

diatomic molecule

117

forbidden transitions

198

general theory

37

integrated

43

linear molecules

138

nuclear hyperfine structure

445

206

831

881


Index Terms irreducible tensor methods peak

Links 831 43

117

138

209

518

519

522

263 Stark effect with quadrupole coupling

489 491

symmetric-top molecule

208

Zeeman effect

514 549

Intensity measurements, relative

619

Internal axis method

582

Internal rotation

570

barriers, effects of vibration

620

600

602

603

from avoided crossing method

205

620

621

from relative intensities

618

from rotational fine structure

591

612

example calculation

605

606

from satellite frequency pattern

617

618

tables of

603

616

classical kinetic energy

583

Coriolis effects

607

denominator corrections

591

direct diagonalization methods

598

effective principal axis system

592

forbidden transitions

599

free internal rotation

609

Hamiltonian, derivation of

582

624

604

high-barrier approximation: derivation of effective rotational Hamiltonian585 higher-order effects

593

methyl group equilibrium orientation

624

590

591

625


Index Terms

Links

methyl group tilt

625

molecules with low sixfold barriers

612

distortion effects

617

Zeeman effect

617

molecules with twofold barriers

620

622

623

nuclear statistical weights

600

perturbation coefficients

587

621

622

table of

588

589

potential function

573

602

V6 effects

602

evaluation of

600

608

631

607

pseudocentrifugal distortion effects

596

597

pseudorigid rotor

591

quadrupole and magnetic coupling

600

selection rules

592

599

600

610

612

615

spectra in, ground torsional state

598

excited torsional states

600

601

Stark effect

603

604

615

symmetric rotor

606

607

609

610

617 torsional energy levels

574

two-top molecules

621

622

vibration-rotation interactions

600

617

618

5

6

96

115

116

145

163

300

189

194

436

441

Interstellar space molecules

Inversion frequency effect of isotopic substitution hyperfine structure

187 188 193 435


Index Terms

Links

potential, illustration of selection rules Stark effect Ionic character, definition of

188 34

188

481

482

749

754

759

760

762

790

796

797

754

755

795

797

798

748

772

799 from dipole moments table of values from quadrupole coupling table of values tables of values

Ionic character-electronegativity relation

752 752

756

760

770

797

798

753

798

799

763

Ions, see Radicals and ions Irreducible tensor, definition of

804

from Cartesian tensors

805

matrix element theorems

807

transformation properties

804

Irreducible tensor methods

803

electronic spin-spin interaction

835

nuclear quadrupole and magnetic interaction

812

relative intensities

831

nuclear spin-spin interaction

829

rotational angular momentum, matrix elements

809

Stark effect

838

Isotopic abundances, table of Isotopic relations distortion constants Dunham’s constants Isotopic shrinkage of bonds

815

839

859 81

285

312

313

312

313

81 713

714


Index Terms

Links

J Jacobi diagonalization procedure

246

K

196

K-level splitting

229

230

232

251

266

357

357

avoided crossing from

202

expressions for

201 360

forbidden transitions from Kraitchman’s equations

196 656

choice of

663

multiple isotopic substitution

665

single isotopic substitution

657

704

L Laplace’s equation, in quadrupole coupling

395

400

407

Large amplitude motions

149

187

382

383

3

4

219

299

270

271

361

681

682

701

702

717

407

426

706 See also Internal rotation; Ring, puckering Lasers Least squares analysis

Legendre equation Linear molecules

28 125


127

energy levels

126

Fermi resonance

136

intensities

138

peak absorption coefficient

138

ι -type doublet spectra

133

matrix elements

126


136

432


136

137


Index Terms

Links 814

quasi-linear selection rules

149

706

34

127

130

131

149

133 Stark effect

451

structures

143

tables of

vibration, effect on energy levels level notation

481

143

145

146

165

682

694

520

529

530

263

471

128 130

wave function

28

Zeeman effect

509 533

Line shape, Doppler illustration of, Doppler-broadened

45 47

first and second derivative

53

pressure-broadened

49

instrumental distortion

50

Lorentzian

48

Van Vleck-Weisskopf

50

Line strength

46

50

51

259

262

595

831

definition of

259

831

properties of

263

relation to matrix elements of, angular momentum 263

595

dipole moment

259

262

direction cosine

262

471

262

263

sum rules Line width collision with walls

263

44 52


Index Terms

Links

Doppler

45

modulation and detection effects

53

in molecular beams

52

natural

44

power effects

50

pressure broadening

43

table of

54

45

47

51

L-matrix, see Vibrational energy Lórentzian line shape function l-type doubling, linear molecules

48 129

constants, table of

135

spectra

133

Stark effect

481


183

spectra Λ -type doubling

185 107

expressions for

110

in electron paramagnetic resonance

552

spectra

113

L-uncoupling correction to rotational constant

137

552

107 80

546

675

M Magnetic moment: electron spin

549

molecular

506

origin

523

nuclear spin

429

orbital angular momentum

549

Magnetic susceptibility anisotropy in, table of

549

520

528 532


Index Terms

Links

bulk

524

calculation of

536

table of atomic susceptibilities

537

Magnetic -susceptibility tensor

524

Magneton, Bohr

857

nuclear

857

Manning potential Maser resolution and spectra from Mathieu’s equation eigenfunctions and eigenvalues Matrix, diagonalization of

529

531

3

141

436

3

441

822

352

846

189

829

577 577 246

continued fraction method

849

Givens method

352

Jacobi method

246

352

18

234

340

344

354

397

399

595

235

236

250

31

33

113

133

259

262

263

482

32

259

468

595

610

613

Matrix elements: angular momentum

809 relation to direction cosine matrix elements 406 asymmetric rotor

24 290

dipole mo ment

562 direction cosine

31 471

internal rotation

594 614

linear molecule

126

127


Index Terms nuclear magnetic coupling (I·J)


Links 404

439

817

818

821

825

404

427

428

440

818

821

825

reduced, see Reduced matrix elements spherical-top

23

symmetric-top

23

vibrational coordinates

283

vibrational momentum

283

Matrix mechanical methods Microwave region, extent of Modulation

672

841 1 53

54

276

279

492 double resonance

276

Stark

492

Molecular quadrupole moments

538

from line width

542

table of

532

tensor elements

539

relation to elements of g and x from Zeeman effect

279

541

539 540

Molecular structures, see Structure Moment of inertia average configuration expressions for

649

653

708 708

calculation of

649

effective

673

electronic corrections

546

expressions for

654

mass-dependence

715

of methyl group, evaluation of

703

709

655

712


Index Terms

Links

relation to structural parameters

656

substitution

695

Moment of inert ia tensor

715

716

649

derivatives of

284

inverse

282

derivatives of

284

expansion of

284

transformation property

650

Momentum sphere

14

Morse potential illustration of

74

306

305

74

N 44

45

44

45

Near-axis atoms, locatio n of

664

698

Normal equation matrix

362

Natural lifetime in state Natural line width

Nuclear hyperfine structure, effects of identical nuclei 440 table of relative intensities Nuclear magnetic coupling, asymmetric tops general theory irreducible tensor methods

441

804

881 435 429 815

831

linear and diatomic molecules

432

magnetic field, origin of

431

432

magnetic field at coupling nucleus

433

435

97

98

551

552

more than one coupling nucleus

436

818

831

in paramagnetic resonance of gases

554

with quadrupole coupling

436

818

831

selection rules

432

434

437

molecules with electronic angular momentum in Zeeman effect

826

444

445


Index Terms

Links

in Stark effect

464

symmetric tops

433

in Zeeman effect

518

441

Nuclear magnetic coupling constants, CJ i , expressions for

430

432

816

817

information from

432

table of

434

Nuclear magnetic moments, table of

859

Nuclear masses, table of

859

434

435

425

429

443

Nuclear quadrupole coupling: asymmetric tops

413

416

439 example calculation

417

derivation of Hamiltonian

399

general theory

392

399

812

818

831


136

137

407

with magnetic coupling

436

818

831

in paramagnetic resonance of gases

555

second-order corrections

426

815

823

895

896

398

408

412

417

425

432

434

437

440

818

irreducible tensor methods

table of selection rules

418

423

445 in solids

395

in Stark effect

458

478

symmetric tops

412

425

two or more nuclei

423

439

831


Index Terms in Zeeman effect

Links 518

Nuclear quadrupole coupling constants, atomic p-electrons, table of

737

determination of principal elements

418

419

effects of vibration

444

445

gJi , expressions for

405 416

relation to electronic structure

664

406

408

412

813

814

726

in centrally bonded atoms

776

group electronegativities

783

hybridization

755

hyperconjugation

422

777

767

ionic character

748

in nitrogen ring compounds

772

π character

758

772

775

776

780

411

415

420

421

443

445

395

396

401

403

783 pure covalent bonds tables of

Nuclear quadrupole moments

745

812 from atomic beam measurements

736

737

tables of

737

859

Nuclear spins, determination of, from microwave spectra 413 table of Nuclear spin -spin interaction Nuclear statistical weights

table of

859 441

805

829

61

210

440

600

827

828

64

65

441


Index Terms O Oblate symmetric rotor Operator matrix change of basis diagonalization of Orthorhombic symmetry Overlap integral

Links 176

229

843

844

845

846

246

352

230

236

846

849

138

211

315 747

789

Padé Hamiltonian

383

384

Partition function

54

118

Paschen -Back effect

462

521

P-branch

254

Perturbation theory, contact transformation

286

P

ordinary

286

855

Van -Vleck transformation

286

287

474

475

477

586

590

595

613

853

Phase convention

19

811

Phase-sensitive detection

53

54

492

758

775

776

780

763

766

770

521

522

π character from quadrupole coupling

783 from asymmetry of coupling

764

due to hyperconjugation

767

involvement of d orbitals

762

from magnitude of coupling

758

tables of

760

π feedback

762

π transitions: in Stark effect

454

in Zeeman effect

509

519

Planar moments of inertia

655

656

Planck’s constant

857


Index Terms

Links

Planck’s radiation law

39

Polarizability, electric

484

Population of energy states fraction of molecules, general definition

54 55

in rotational states

56

in vibrational states

55

118

3

40

603

616

624

83

381

382

379

inversion of Potential barriers, internal rotation, table of

138

211

Potential constants: anharmonic from centrifugal distortion constants

372

sources of information

373

tables of

375

378

multiple solution problem

379

380

relation to Dunham’s constants, diatomic molecule table of Potential function

79 83 74

78

283

301

622

631

139

211

374 illustration of Potential function for hindered rotation

74

675

573

608

effect of V6 term

602

illustration of

573

Potential function for ring puckering Power absorption fractional, calculation of

631

638 43 119

120

Power attenuation constant of absorption cell, αc

43

Power saturation

50

Pressure broadening

43

47

Principal axes of inertia

12

649

location from symmetry

651

Principal axes of quadrupole coupling constant This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

tensor, determination of

418

419

422

666

667

664

Principal axis coordinates, evaluation of by isotopic substitution

656

in asymmetric -tops

660

in linear molecules

658

sign of coordinates

670

671

in symmetric-tops

658

665

Principal axis method

581

582

12

649

expressions for

654

655

Products of inertia

649

653

Prolate symmetric rotor

176

229

Pseudorotation

639

Principal moments of inertia

Pumping

3

666

230

236

96

106

273

Q Q-branch

254

Quadrupolar center, effective

542

Quasi-linear molecules

149

criterion

706

150

R 38

Radiation density

5

Radicals and ions

6

163 Ray’s asymmetry parameter Ray’s operator

K

H ( K)

229 233

R-branch

254

Reduced angular momentum, P, internal rotation

584

591

Reduced barrier height

577

857

Reduced distortion Hamiltonian, see Centrifugal distortion Reduced energy, asymmetric rotor coefficients in perturbation expansion of W(bp)

232

250

252


Index Terms

Links

relation between Ray’s E(K) and Wang’s W(b)

253

submatrices

240

241

matrix elements of

235

236

250

symmetry classification of

241

table of Reduced mass

244 73

for inversion

189

192

193

for isotopic substitution

657

665

668

Reduced matrix element, for irreducible operators

807

Reduced moment of inertia, internal rotation

574

583

584

57

64

210

Reduced nuclear statistical weight, g, Reduced rotational constant F, internal rotation

584

Reduced Stark coefficients

470

ρ-type triplet ρ-vectors

98 307

Ring: conformation

627

puckering

636

Rotating coordinate system conditions on

12

688

232

306

Rotational constants: conversion factor dependence on vibrational quantum numbers

857 74

79

128

129

131

182

281

288

672 tables of, asymmetric-tops diatomic molecules

231

678

75

76

83

86

88

90

98

106

140

142

143

677 linear molecules

132


Index Terms

symmetric tops

Links 145

146

148

155

161

164

168

693

181

192

201

202

205

217

218

220

221 Rotational isomerism table of isomers

501

627

632

Rotational kinetic energy, classical

12

14

Rotational spectrum, assignment of

258

264

273

491

72

126

176

25

846

847

492 Rotation of coordinate axes, transformation matrix 650 Rotators, classification of

71 227

Rydberg constant

857

S s character, in terms of bond angle Schrodinger equation Second moments, electron charge distribution Secular equation, derivation of

777 24

25

542 24

Selection rules: asymmetric-top

254

basis for

33

diatomic molecules

34

72

74

92

95

102

112

113

607

electron paramagnetic resonance

555

internal rotation

592

599

600

610

612

615

inversion spectra

34

188

linear molecules

34

127

130

131

133 l-type doubling

133


Index Terms

Links

Λ -type doubling

113

nuclear hyperfine structure

398

408

412

417

425

432

434

437

464

466

188

445 pure quadrupole resonance

398

Stark effect

454

462

471

479

34

177

185

196

197

200

513

518

519

symmetric-tops

from symmetry Zeeman effect

58 509 522

σ transitions: in Stark effect

454

in Zeeman effect

509

519

521

522

425

426

447

448

Six-j symbol

807 symmetry

808

tables of

809

Slater’s rules

733

Slightly asymmetric rotors

250

417

418

474

Slightly nonplanar molecules

477

636

687

688

23

195

205

338

339

360

361

381

19

20

108

109

Spin -rotation interaction constant

94

100

Spin -spin interaction constant

99

831

Spin uncoupling

107

111

Standard deviation of least-squares fit

362

Spherical tensors, see Irreducible tensors Spherical top

Spin operators Spin -orbit coupling constant.

111


Index Terms

Links

Standard error in least-squares parameter

362

Stark effect

451

anisotropic polarizabilities from

484

assignment aid

491

492

asymmetric-tops

468

488

dipole moments from

493

tables of

156

495

497

498

500 forbidden transitions

489

491

562

604

fourth-order

457

458

472

473

487

494

495

high frequency

277

intermediate field case with one quadrupolar nucleus 465

481

internal rotation

477

603

604

615

inversion doublets

481

482

irreducible tensor methods

838

839

K-type doublets

475

l-type doublets

482

linear and symmetric-top molecules

451

481

Mj , evaluation of

471

472

476

492

with magnetic hyperfine structure

464

modulation with

492

paramagnetic resonance of gases

561

patterns

455

456

472

477

490 relative intensities

489

491

secondary standards for measurement of Stark splittings

493

494

selection rules

454

462

464

471

479

462

480

strong-field case, with one quadrupolar nucleus

466


Index Terms plural nuclear coupling weak-field case with one quadrupolar nucleus Stark effect constant

Links 464

480

459

478

481

857

Statistical weights, see Nuclear statistical weights Structural parameters, number of

667

Structure, evaluation of

647

average structure

706

from diagnostic least squares

717

effective structure

679

equilibrium structure

672

from least squares

681

mass-dependence structure

714

substitution structure

691

example calculation

693

with small coordinates

664

698

in asymmetric tops

291

292

tables of

678 702

682

701

702

679

683

684

703

710

711

Structures:

713 in complexes

153

221

tables of

155

161

221

76

83

86

90

106

680

in linear molecules

143

163

tables of

143

145

146

165

682

694

in diatomic molecules tables of

77

in symmetric tops

213

tables of

214

88

149

219


Index Terms vibrational effects on

estimation of

Links 214

215

680

702

712

717

704

714

Student’s t distribution

363

Sum rules

262

263

342

851

28

804

Surface spherical harmonics s-vectors

307

Symmetric rotor

175


341

185

191

197

200

201

322

337

338

357

609

610

175

forbidden transition

194

intensities

208

internal rotation

285

177

energy levels

peak absorption coefficient

694

209 606

607

617 inversion K-level splitting, from avoided crossing from forbidden transitions

34

187

202

620

621

194

l-type doubling

183

matrix elements

21

31

35


433

441

817


412

425

814


64

210

441

selection rules

34

177

185

196

197

200

Stark effect

451

481

482

structures

213

234

188

484


Index Terms tables of

Links 214

219

182

219

607

wave functions

29

810

811

Zeeman effect

512

520

528

529

57

58

63

66

209

210

vibration, effect on energy levels

533 Symmetry number σ

T Term symbols

91

Thermal equilibrium, Boltzmann’s law

39

Three -j symbol

804

symmetry

805

tables of

809

Torsional angle

572

Torsional energy levels

574

Torsional frequency

575

from intensity measurements Torsional satellite lines

805

809

619

620

618 600

601

Transformation: contact Van Vleck

286 286

287

474

475

477

586

590

595

613

853

to diagonalize asymmetric rotor energy matrix

245

of mo ment of inertia tensor

650

between parent and isotopic form

664

Transformation coefficients, in coupling three angular momenta two angular momenta

425

808

808

U Unbalanced p electrons Up

740


Index Terms

Links

Uncertainty principle

44

Unitary matrix

842

Unitary transformation

325

846

V Valence orbitals, hybridized: sp 2 sp 3

774 776

Van Vleck transformation

286

287

853

in internal rotation

586

590

595

in Stark effect

474

475

477

Van Vleck-Weisskopf line shape function

613

50

Van der Waals complexes, see Atom-molecule complexes Van der Waals radii, bond

873

V6 Barrier term

573

602

Vector model: Hund’s case (a)

92

Hund’s case (b)

94

nuclear coupling

400

nuclear quadrupole coupling, two nuclei

424

95

strong-field Stark effect, molecule with nuclear coupling

463

symmetric-top

176

Stark effect

453

weak-field Stark effect, symmetric-top with nuclear coupling

460

Zeeman effect

506

Vibration, effects on energy levels: asymmetric-top diatomic molecules

281 72

linear molecules

128

symmetric-tops

182

78

219

607


Index Terms

Links

Vibrational energy, coordinates

282

301

307

308

302

307

308

311

372

374

376

377

879

880

Hamiltonian

282

283

secular equation

372

879

solution of

879

880

Vibrational frequency, from intensity measurement 183

618

G matrix, etc.

relation to spectroscopic constants

77

Vibrational modes: resulting on dimer formation

157

triatomic linear molecule

129

triatomic nonlinear molecule

677

Vibrational potential function, see Potential function Vibration -rotation constants, expression for, asymmetric -tops 288 Vibration -rotation Hamiltonian

282

expansion of

284

order of magnitude classification

285

perturbation solution

286

Vibration -rotation perturbation parameter Vibration -torsion -rotation interactions

285

328

600

617

250

251

618

W Wang asymmetry parameter Wang functions transformation properties under V Wang operator: H (b) Wang transformation Wave function: asymmetric rotor

example calculation

239 242 250 239

240

29

60

237

247

248

249

61

233


Index Terms

Links

internal rotation

585

inversion

187

linear molecule

609

610

459

478

578

580

28

matrix representation

843

for nuclear hyperfine states

446

nuclear spin

63

torsional function

577

Wigner-Eckart theorem

807

Wigner 6j symbol, see Six-j symbol Wobble -stretching correction to rotational constant 80

675

Y 408

Y(J,I,F)

881

table of

Z Zeeman effect: asymmetric tops

515

530

534

535

520

529

530

518

519

522

617 determination of, molecular quadrupole moments

538

second moments electron distribution

542

sign of dipole moment

544

electronic corrections to inertial constants, relation to g factor

546

evaluation of anisotropic magnetic susceptibility

528

general theory

506


509 533

1

magnetism in Σ states, origin of

523

molecules with electronic angular momentum

549

molecules with nuclear coupling

518

relative intensities

514


Index Terms

Links

selection rules

509

513

518

519

520

528

529

516

532

541

214

215

680

694

702

712

717

522 symmetric tops

512 533

tables of derived constants

512 545

transition between Zeeman components, electron paramagnetic resonance Stark effect

552 561

Zero-point vibrations: effect on structures


Microwave Molecular Spectra

Microwave Molecular Spectra (Techniques of Organic Chemistry)

Spectra

Vibrational Spectra and Structure : Optical Spectra and Lattice Dynamics of Molecular Crystals

Vibrational Spectra and Structure : Molecular Approach to Solids (Vibrational Spectra and Structure)

Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra

Inverse spectra

Molecular Vibrations. The Theory of Infrared and Raman Vibrational Spectra

Introduction To Atomic Spectra

Purity, Spectra and Localisation

Organic Structures from Spectra

microwave applications

Microwave Measurements

Microwave Projects

Microwave Applications

Microwave Technology

Microwave Engineering

Microwave Projects

Microwave Component Mechanics (Artech House Microwave Library)

Modern Microwave Circuits (Artech House Microwave Library)

Microwave Component Mechanics (Artech House Microwave Library)

MICROWAVE HEATING

Microwave Knowhow

Microwave Engineering

Microwave Photonics

Spectra of Graphs

Raman Spectra from Acetone

Microwave Component Mechanics (Artech House Microwave Library)

Vonda N. McIntyre-Spectra

Spectra of Graphs

Vibrational Spectra and Structure

Microwave Molecular Spectra