VIBRATIONALSPECTRAAND STRUCTURE Volume 21
OPTICALSPECTRAAND LATTICE DYNAMICS OF MOLECULARCRYSTALS
EDITORIAL BOARD
Dr. Lester Andrews University ofVirginia Charlottesville,Virginia USA
Dr. J. A. Koningstein Carleton University Ottawa, Ontario CANADA
Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA
Dr. George E. Leroi Michigan State University East Lansing, Michigan USA
Dr. A. R. H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA
Dr. S. S. Mitra University of Rhode Island Kingston, Rhode Island USA
Dr. William G. Fateley Kansas State University Manhattan, Kansas USA
Dr. A. Miiller Universit~it Bielefeld Bielefeld WEST GERMANY
Dr. H. Hs. G/inthard Eidg. Technische Hochschule Zurich SWITZE~
Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN
Dr. P. J. Hendra University of Southampton Southampton ENGLAND
Dr. Herbert L. Strauss University of California Berkeley, California USA
ViBRATiONAL SPECTRAAND STRUCTURE A SERIES
OF ADVANCES
JAMES R. DURIG (Series Editor) College of Science and Mathematics University of South Carolina Columbia, South Carolina
VOLUME
21
OPTICALSPECTRAAND LATTICE DYNAMICS OFMOLECULARCRYSTALS
G.N. Zhizhin and E. Mukhtarov Institute of Spectroscopy, Academy of Sciences of Russia, Troitzk, Moscow region, 142092 Russia
0
1995 ELSEVIER Amsterdam
- Lausanne - New York-
Oxford - Shannon - Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam The Netherlands
ISBN 0-444-82295-X 91995 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other.copyright questions, including photocopying outside of the U.S.A,. should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
PREFACE TO TI-IE SERIES It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fruitful avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the changes of a significant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular s ~ m e t r i e s ; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. Most of the recent reviews in the area of vibrational spectroscopy have appeared in other progress series and it was felt that a progress series in vibrational spectroscopy was needed. A flexible attitude will be maintained and the course of the series will be dictated by the workers in the field. The editor not only welcomes suggestions from the readers but eagerly solicits your advice and contributions. James R. Durig Kansas City, Missouri
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PREFACE TO VOLUME 21 The current volume in the series Vibrational Spectra and Structure is a single topic volume on the optical spectra and lattice dynamics of molecular crystals. The monograph is divided into two parts, with the first part on the vibrational spectra of organic crystals and their interpretation by means of atomatom potentials. The first part includes four chapters, with the first one on the dynamics of the molecular crystal lattices. Chapter II deals with the calculations of frequencies and normal vibrational forms in the approximation of the rigid molecule and Chapter III with the solution of dynamical problems with consideration of the intermolecular vibrations. Chapter IV provides information on the devices and methods of the experimental investigation of organic crystals by means of vibrational spectra. Therefore, Part I covers both the theoretical and experimental investigations of organic crystals. Part II of the monogram deals with the investigation of the structure, phase transitions and reorientational motion of molecules in organic crystals. This part has five chapters, with Chapter I dealing with the vibrational spectra and phase transitions in plastic crystals. In Chapter II, the reorientation motion of molecules and crystals without meso-phases is covered. In Chapter III, the stationery orientational disorder in trihalogen substituted methane crystals is analyzed. In Chapter IV, the vibrational spectra and phase transitions in crystals with conformationally unstable molecules is covered, and the final chapter in Part II deals with the application of vibrational spectra and the determination of structural investigations of phase transitions of crystals of phenantharene and pyrene. In addition, the appendices provide the parameters for the calculation of the lattice dynamics of molecular crystals, procedures for the calculation of frequency eigenvectors utilizing computers, and the frequencies and eigenvectors of lattice modes for several organic crystals. The Editor would like to thank the Editorial Board for suggesting the topic for this volume, and the two authors for their contribution and patience which was required when producing the monograph. The Editor would also like to thank his Administrative Associate, Gail Sullivan, and Editorial Assistant, Janice Long, for diligently typing all the articles in camera-ready copy form and quietly enduring some of the onerous tasks associated with the completion of the volume. He also thanks his wife, Marlene, for copy-editing and preparing the subject index. James R. Durig Kansas City, Missouri vii
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TABLE OF CONTENTS
PREFACE TO THE SERIES ........................................................................................ v PREFACE TO VOLUME 21 ....................................................................................... vii CONTENTS OF OTHER VOLUMES ....................................................................... xiii
PART 1 VIBRATIONAL
SPECTRA
OF ORGANIC
AND THEIR INTERPRETATION
CRYSTALS
BY MEANS OF
ATOM-ATOM POTENTIALS
D y n a m i c s of t h e M o l e c u l a r C r y s t a l L a t t i c e .............................................. 2 A. B. C. D.
II.
C a l c u l a t i o n of F r e q u e n c i e s a n d N o r m a l V i b r a t i o n a l F o r m s i n t h e A p p r o x i m a t i o n of R i g i d Molecules ........................................................... 71 A. B. C.
III.
Theory of Harmonic Vibrations of Molectdes in Crystals ........................ 2 Symmetrical Properties of Force Constants and Dynsmical Matrix ....................................................................................................... 23 Intermolecular Interactions and Atom-Atom Potentials ...................... 38 Typical Cases of Disorder in Molecular Crystals and Their Representation in Vibrational Spectra ......................................... 59
Calculation Technique for the Lattice Dynamics of Molecular Crystals by Means of AAP ....................................................................... 71 Choice of the AAP Parameters for the Calculation of the External Vibrational Frequencies of Molecular Crystals ...................... 81 Temperature Dependence of External Vibrational Frequencies. Model of Independent Anharmonic Oscillators .............................................................................................. 101
S o l u t i o n of t h e D y n a m i c a l P r o b l e m w i t h t h e C o n s i d e r a t i o n of t h e I n t e r m o l e c u l a r V i b r a t i o n ......................................................................... 122 A. B. C.
Display of the Intermolecular Interaction in Molecular Spectra of Crystals ................................................................................. 122 Interaction Between the Internal and External Vibrations ............................................................................................... 130 Calculation of the Low-Frequency Spectra of Crystals with the Conformationally Unstable Molecules. Half-rigid Molecular Approximation ...................................................................... 141
ix
x
V@
CONTENTS
Devices a n d Methods for t h e E x p e r i m e n t a l I n v e s t i g a t i o n of O r g a n i c Crystals by Means of the V i b r a t i o n a l S p e c t r a ..................... 151 A. B. C.
Measurement Technique for Molecular Crystals in the Far Infrared Region ...................................................................................... 151 The Technique of Sample Preparation and Methods of the Investigation of Raman Spectra of Organic Crystals ........................... 152 Modified Model of the Oriented Gas for the Calculation of the Relative Line Intensities of the Low-Frequency Ramsn Spectra of the Molecular Crystals ...................................................................... 163
R E F E R E N C E S - P A R T I ...................................................................................... 179
PART 2 INVESTIGATION OF THE STRUCTURE, PHASE TRANSITIONS AND REORIENTATION MOTION OF MOLECULES IN ORGANIC CRYSTALS
Vibrational Spectra and Phase Transitions in Plastic C r y s t a l s ......................................................................................................... 202
A.
B. C.
II.
R e o r i e n t a t i o n a l Motion of Molecules in C r y s t a l s W i t h o u t M e s o p h a s e s .................................................................................................. 245 A. B.
C.
IIl@
Investigationof the Phase Transition in Cyclohexane and Deuterocyclohexane Crystals with the Help of Vibrational Spectra .................................................................................................... 202 Anisotropy of the Rotational Reorientations of Molecules in the Crystals of Cyclopentane and Thiophene ....................................... 220 StatisticalModel of Orientational Phase Transitions in Plastic Organic Crystals with the Consideration of the Rotational Reorientation Anisotropy .................................................... 231
Premelting Effects in Naphthalene Crystal ......................................... 245 The R a m a n Spectra of Benzene Crystals. The Temperature Dependence of Spectra in the Vicinity of the Melting Point ............................................................................... 255 Conditions of the Display of the Reorientational Motion in Low-Frequency R a m a n Spectra of Crystals ..................................... 262
Stationary Orientational Disorder in Trihalogen S u b s t i t u t e d M e t h a n e Crystals ................................................................. 268
A. B.
Vibrational Spectra and Structure of Crystals of Bromoform and Iodoform ....................................................................... 269 Disordering of the Chloroform Crystal Structure in the Vicinity of the Phase Transition to the Liquid State ........................... 284
CONTENTS
IV@
Vibrational Spectra and Phase TrAnsition in Crystals w i t h Conformationally Unstable Molecules ......................................... 291 A. B.
V.
xi
Distortion of the Plane Conformation of Biphenyl Crystal Molecules in the Vicinity of the Melting Point ........................ 291 Low-Frequency Vibrations of Methyl Groups in Organic Crystals .................................................................................... 304
A p p l i c a t i o n of Vibrational Spectra and AAP for the D e t e r m i n a t i o n of Structures and the Investigation of P h a s e Transitions in Crystals of P h e n a n t h r e n e a n d P y r e n e ....................... 332 A. B.
Investigation of the Phase Transition in Phenanthrene Crystal by Means of Low-Frequency Raman Spectra .......................... 333 Calculation of the Molecular Packing, the Low-Frequency Vibrational Spectrum and Their Changes at the Phase Transition in Pyrene Crystal ................................................................. 354
R E F E R E N C E S -- P A R T II .................................................................................... 380
APPENDIX I AAP Parameters Used for the Calculation of the Lattice Dynamics of Molecular Crystals .................................................................................................... 393 A P P E N D I X II Compute~.zed Calculations of the Frequencies and Eigenvectors of the Normal Vibrations .................................................................................................. 396 A P P E N D I X III Frequencies and Eigenvectors for the Low-Frequency Normal Vibrations of Several Organic Crystals ..................................................................................... 404
S U B J E CT IN])EX ................................................................................................... 445
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CONTENTS OF OTHER VOLUMES
VOLUME 10 VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDowell INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Natie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O~Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes
VOLUME 11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF SUPPORTED HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Patricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED I N T E ~ R O M E T R Y , Part 1, D. E. Honigs, R. M. Hamms ker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS- CURRENT TRENDS AND F U T I ~ E DIRECTIONS, Elliot R. Bernstein
VOLUME 12 HIGH RESOLUTION INFRARED STUDIES OF SITE STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECUI~S, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi xiii
xiv
C O N T E N T S OF O T H E R V O L U M E S
S O M E P R O B L E M S O N T H E S T R U C T U R E OF M O L E C U L E S IN T H E ELECTRONIC EXCITED STATES AS STUDIED B Y R E S O N A N C E R A M A N SPECTROSCOPY, Akiko Y. Hirakawa and Massmichi Tsuboi VIBRATIONAL S P E C T R A A N D C O N F O R M A T I O N A L ANALYSIS OF SUBSTITUTED T H R E E M E M B E R E D RING C O M P O U N D S , Charles J. Wurrey, Jiu E. DeWitt, and VictorF. Kalasinsky VIBRATIONAL S P E C T R A OF SMALl. M A T R I X ISOLATED M O L E C U L E S , Richard L. Redington R A M A N DIFFERENCE SPECTROSCOPY, J. Laane
VOLUME 13 VIBRATIONAL S P E C T R A OF E L E C T R O N I C A L L Y EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, A N D M O L E C ~ IN CRYSTAI~, Roger Frech
PARAMETERS
R E C E N T A D V A N C E S IN M O D E L C A L C U L A T I O N S OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETRIES, L. Nemes APPLICATIONS OF D A V Y D O V SPLITTING F O R STUDIES OF C R Y S T A L PROPERTIES, G. N. Z~zhin and A. F. Goncharov R A M A N S P E C T R O S C O P Y O N M A T R I X ISOLATED SPECIES, H. J. Jodl
VOLUME 14 H I G H R E S O L U T I O N L A S E R S P E C T R O S C O P Y OF S M A L L M O L E C U L E S , Eizi Hirota ELECTRONIC S P E C T R A OF P O L Y A T O M I C FREE RADICALS, D. A. Ramsay AB INITIO CALCUI~TION OF FORCE FIELDS AND VIBRATIONAL SPECTRA,
G~za Fogarasi and Peter Pulay F O U R I E R T R A N S F O R M INFRARED SPECTROSCOPY, John E. Bertie N E W T R E N D S IN T H E T H E O R Y OF INTENSITIES IN I N F R A R E D SPECTRA, V. T. Aleksanyan and S. Kh. Samvelyan
VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIAI~, S. Nakashima, M. Hangyo, and A. Mitsuishi
C O N T E N T S OF OTHERVOLUMES
xv
VOLUME 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito
B A N D S H A P E S A N D D Y N A M I C S IN LIQUIDS, Walter G. Rothschild R A M A N S P E C T R O S C O P Y IN E N E R G Y CHEMISTRY, Ralph P. Cooney D Y N A M I C S OF L A Y E R CRYSTAI~, Pradip N. Ghosh THIOMETAIJ,ATO COMPLEXES: VIBRATIONAL SPECTRA A N D S T R U C T U R A L CHEMISTRY, Achim Miiller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken W H A T IS H A D A M A R D T R A N S F O R M SPECTROSCOPY?., R. M. Hammaker, J. A. Graham, D. C. Tilotta,and W. G. Fateley
VOLUME 16
S P E C T R A A N D S T R U C T U R E O F POLYPEPTIDES, Samuel Krimm S T R U C T U R E S O F ION-PAIR S O L V A T E S F R O M MATRIX-ISOLATION/SOLVATION SPECTROSCOPY, J. Paul Devlin L O W F R E Q U E N C Y VIBRATIONAL S P E C T R O S C O P Y OF M O L E C U L A R PLEXES, Erich Knozinger and Otto Schrems
COM-
T R A N S I E N T A N D TIME-RESOLVED R A M A N S P E C T R O S C O P Y OF SHORTLIVED I N T E R M E D I A T E SPECIES, Hiro-o Hamaguchi I N F R A R E D S P E C T R A OF CYCLIC D I M E R S OF CARBOXYLIC ACIDS: T H E M E C H A N I C S OF H - B O N D S A N D R E L A T E D PROBLEMS, Yves Marechal VIBRATIONAL S P E C T R O S C O P Y U N D E R H I G H PRESSURE, P. T. T. Wong
VOLUME 17A
SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Verma; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist,B. Darshan and P. I~ Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier S E M I C O N D U C T O R SUPERI~TTICES, M. V. Klein;A. Pinczuk and J. P. Valladares; A. P. Roy; I~ P. Jain and R. I~ Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. I~ Sood
xvi
CONTENTS OF OTHER VOLUMES
TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jha; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida
BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. St0ckburger,W. L. Peticolas;A. T. Tu and S. Zhen~, P. V. Huong and S. R. Plouvier;B. D. Bhattacharyya; E. Taillandier,J. Liquier,J.-P.Ridoux and M. Ghomi
VOLUME 17B STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATrERING, H. W. SchrStter and J. P. Boquillon; G. S. Agarwal; L. A. Rahn and R. L. Farrow; D. Robert; I~ A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, R. J. Carlson, M. T. Riebe, J. I~ Steehler, D. C. Nguyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and R. L. Cook
R A M A N S O U R C E S A N D R A M A N LASERS, S. Leach; G. C. Baldwin; N. G. Basov, b_ Z. Grasiuk and I.G. Zubarev; A. I.Sokolovskaya, G. L. Brekhovskikh and A. D. Kudryavtseva O T H E R APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi ~nd T. Ishioka;S. R. Ahmad; S. Singh and M. 1. S. Sastry;K. K~mogawa and T. Kitagawa; V. S. Gorelik;T. Kushida and S. Kinoshita;S. I~ Sharma; J. R. Durig, J. F. Sullivanand T. S. Little VOLUME 18
E N V I R O N M E N T A L APPLICATIONS OF GAS C H R O M A T O G R A P H Y / F O U R I E R T R A N S F O R M I N F R A R E D S P E C T R O S C O P Y (GC/FT-IR), Charles J. Wurrey and Donald F. Gurka DATA TREATMENT IN PHOTOACOUSTIC FT-IR SPECTROSCOPY, I~ H. Michaelian
R E C E N T D E V E L O P M E N T S IN D E P T H PROFILING F R O M S U R F A C E S U S I N G FT-IR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIES, Lester Andrews
VIBRATION A N D R O T A T I O N IN SILANE, G E R M A N E A N D S T A N N A N E A N D THEIR M O N O H A L O G E N DERIVATIVES, Hans Biirgerand Annette Rahner FAR I N F R A R E D S P E C T R A OF GASES, T. S. Littleand J. R. Durig
C O N T E N T S OF O T H E R V O L ~ S
xvii VOLUME 19
A D V A N C E S IN I N S T R U M E N T A T I O N F O R T H E OBSERVATION OF VIBRATIONAL OPTICAL ACTIVITY, M. Diem SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs D E T E R M I N A T I O N OF M E T A L IONS AS C O M P L E X E S I MICELLAR M E D I A BY UV-VIS S P E C T R O P H O T O M E T R Y A N D FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. RodriguezFernandez AB/N/T/O CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles and Ellak I. von Nagy-Felsobuki APPLICATION OF INFRARED A N D R A M A N S P E C T R O S C O P Y TO T H E S T U D Y OF S U R F A C E CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED S P E C T R O S C O P Y OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya. M. Kimerferd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED AND CONDUCTING POLYMERS, Issei Harada and Yukio Furukawa
VOLUME 20 APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE A N D F R E Q U E N C Y SHIFT STUDIES B Y R A M A N SPECTROSCOPY, B. P. Asthana and W. Kiefer M I C R O W A V E FOURIER T R A N S F O R M SPECTROSCOPY, AlfredBauder AB/N/T/O QUALITY OF S C M E H - M O CALCULATIONS OF C O M P L E X INORGANIC SYSTEMS, Edward A. Boudreaux C A I ~ U I ~ T E D A N D E X P E R I M E N T A L V I B R A T I O N ~ SPECTRA A N D F O R C E FIELDS OF ISOLATED PYRIMIDINE BASES, WillisB. Person and Krystyna Szczepaniak
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PART I V I B R A T I ON A L S P E C T R A OF ORGANIC CRYSTALS AND T H E I R I N T E R P R E T A T I O N BY MEANS OF ATOM-ATOM P O T E N T I A L S
2
ZHIZHIN AND MUKHTAROV
I.
DYNAMICS OF THE MOLECIKAR
A. T h e o r y o f H a r m o n i c
Vibrations
CRYSTAL LATTICE
of Molecules in Crystals
The interpretation of vibrational spectra using atom-atom potentials is based on a theory of the crystal lattice dynAmlcs; its application to the rushy-atom molecular crystals is supposed to use appro~mations, which in general can induce additional errors in calculation results. Therefore, considering the lattice dynamics of these crystals one needs to perform the exact (detailed) analysis of the used approximations. We have also paid great attention to expressing elements of the dynamic matrix by known values in the form convenient for practical applications. The theory of the dynamics of simple crystal lattice was developed long ago by Born and Kun Huang [1]. This theory is also considered in detail by Maradudin et al. [2]. It is convenient to begin the discussion of the lattice dynamics with the assumption that the crystal is infinite; it allows us to use the periodicity of the lattice for simplicity of our problem. Applying the above assumption, we have to deal with infinitely large values related to the whole crystal; they can be normalized by appropriate choice of boundary conditions. Let Xpa(~), X~ a(0)(~) beK a-components (a = x, y, z) of both the instant and equilibrium radius-vectors R(! ) and ~u)(~),'^ respectively, of an atom P (P = 1.... , n; n is the number of atoms in a molecule) of a molecule z (z = 1, ..., z; z is the number of molecules in the unit cell) of the unit cell with a number ! A "
(I.1)
where R ( l ) = ! 1~1 + ! 2~2 + t 3~3 is the translational vector of a lattice. R p(x) is a vector of the atom P in the coordinate system of the unit cell. The displacements of atoms from equilibrium states are defined by coordinates X
)[ = X
K
(I.2)
The total kinetic energy of lattice
T=
]~ ~ ~ 1
K
ap
mpXpa
(I.3)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
3
In the adiabatic approximation the potential energy V of the crystal depends only on nuclear (atomic) coordinates. Expanding V in a Taylor series of atomic displacement, we get ~V
V=Vo+X X X t
pa
K
lt" ~-'F,"ap,a'p"
~Xpa
~}Xp,a, (I.4)
The harmonic approximation is obtained by neglecting all powers which are higher than second. The derivatives in Eq. (I.4) are calculated for equilibrium configuration. We denote :
-
-
~V -
(I.5a)
~2V (I.5b)
Because of lattice spacing (periodicity) the coefficients (Eq. (I.5a)) are independent of the number of the unit cell l , and force constants (Eq. (I.5b)) depend only on the difference (! - l ') [ 1,2]
(I.6)
According to the definition (Eq. (I.5a and I.Sb))
(I.7)
4
ZI-IIZI-IIN AND MUKHTAROV
The inv ~ a n c e of the potential energy (Eq. (I.4)) with respect to the displacement of the whole crystal implies [2]
"V" ~. fpa,p'a' lc't' p"
=0 l'
(I.8)
The additional and very essential relationships between the force constants are followed from the invariance of V with respect to the symmetry operations of the crystal; they shall be considered later in Sec. I.B. In the equilibrium state the forces acting on any atom are equal to zero. Hence we find 3nz conditions of the equilibrium
po/,/:
f,
K
=0; a=1,2,3; p=l,...,n; K=l,...,z (I.9)
Besides, the assumption of crystal infinity imposes the requirement such that in crystals the initial microscopic torsions are equal to zero [1,4]. This gives the additional conditions that define the equilibrium form and sizes of the unit cell ~V ------ = 0; a , ~ = 1,2,3 (I.10) where COa~ are components of crystal deformation tensor. The number of equations (Eq. (I.10)) is defined by the crystal symmetry, and in a general case it is equal to six [1,4]. The fulfillment of the equilibrium conditions (Eq. 1.9) and (Eq. 1.10) is very important for practical calculations, since the use of approximated models for the potential V can reduce the violation of these conditions and the nonstability of the crystal lattice (s~. Sec. II.B). The stability conditions of molecular crystals are also considered in Refs. [4-7]. The Lagrange function of a crystal is L = T - V, where T and V are of the form of Eq. (I.3) and Eq. (I.4), respectively. Hence we get the equations of lattice atom motion
mp~pa I
l'K" a'p"
l'
(I.11)
Considering the lattice periodicity, the solution is sought as usual in the form of the plane waves
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
Xpa
(~)=
5
1 e-iOt+i~(t) ~m'p xpa (~:)
(I.12) Using Eq. (I.12) the system of equations (Eq. (I.11)) can be written in the form
=
z
o
a'P" pa,p a
~:
(I.13)
where elements of a dynamical matrix are equal to ~ fpa,p,a,/O ~ ' / e - i~J~(t) (I.14) Thus, using the lattice periodicity, i.e., the properties of force constants (Eq. (I.6)), we can reduce the problem of solving the infinite system of equations (Eq. (I.11)) to solving the system of 3nz linear equations (Eq. (I.13)). If the determinant
D p a , p ' a ' ( ~ ' ) - ~025aa'~pp'~knd = 0 (I.15) then the system is solvable. Equation (I.15) of degree 3nz in o)2 has 3nz roots for every value k, which will be denoted by ~(k), j = 1..... 3r~. According to Eq. (I.14) and Eq. (I.7) the dynamical matrix f)(k) is calculated for the equilibrium configuration of crystal (the conditions of Eqs. (I.9) and (I.10) are fulfilled), then ~ 2 > 0 and the frequencies r
are also real values. The relationship co = oj(k) is
known as the law of dispersion of j branch of the dispersional surface. At k = 0 three frequencies are equal to zero, and corresponding dispersional branches are called the acoustical ones (this follows from Eq. (I.8)). The violation of the conditions of Eqs. (I.9) and (I.10) can cause appearing imaginary frequencies ~ ( k ) which correspond to the failure of the crystal [1,2]. For every value r
there exists the vector ~
whose components satisfy
the equation system (Eq. (I.13)):
r176
a'p" Dpa'p'a'(k')ea'p'(K'lkj)
eaP(KI~) = E ~:K"
(I.16)
6
ZI-IIZI-IIN AND MUKHTAROV
The vector ~ L ] )
is determined by this system with an accuracy to the constant
factor. It is usually chosen such that the condition of the orthonormalization are satisfied:
For obtaining the final solution it is necessary to find possible values of vector k which are determined by boundary conditions. One usually uses either periodical boundary conditions (when it is assumed that the atoms of opposite faces of a finite crystals are moving in the same way) or cyclic boundary conditions (when the infinitely large crystal is divided into "microcrystals" with sizes L x L x L = N); besides one introduces the periodicity of atomic displacements with a period which is equal to the "microcrystar' size
..o
Considering Eq. (I.12) one gets the N possible values of vector k k =
b1+
b2 +
b3; hl, h2,h 3 = 1,...,L
where b a are the vectors of an inverse lattice.
According to the Lederman's
theorem the solution of the problem is independent of the choice of boundary conditions for N ~ oo [2]. If in the expressions for the kinetic (Eq. (I.3)) and potential (Eq. (I.4)) energy of a crystal one uses new coordinates Qj(k) with the help of transformation -o
(I.17) then the total vibrational energy can be represented in the form [1-3]
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
7
"*(fc)Qj(fc)_ + mj2 (k)Qj* (k)Qj(k)] H = 2I _~ [Qj
kj
(s
The coordinates Q(k) satisfy the equation of motion 2 {~j(k) + coj (fc)Qj(~) = 0
From this equation it follows that the new coordinates are the simple periodical function of time characterized only by one of frequencies ~(k); they are usually called the normal coordinates. Every norms! coordinate describes only one independent vibrational motions, which are called normal vibrations. For such a vibration all atoms have only one vibrational frequency, and their phase shifts are constant. From Eq. (I.17) it also follows that the amplitude of an atomic displacement is determined by a following factor, for a given normal vibration,
eI,ll i.e., the eigenvectors give a pattern of the atomic displacements, and are often called the vibration form. The form of vibration plays a very important role in the interpretation of vibrational spectra of crystals, since it allows us to associate the experimentally observed spectrum with the pecularities of the crystal structure and with the type of motion of each atom; then we can get the information concerning the change of a crystal state, w~en there are external influences on it. The determination of the form of norms! vibrations is one of the main problems of lattice dynamics. It should be noted that sometimes it is more convenient to use the expression for atomic displacements without the normalization factor 1 / ~ (see Sec. II.C). In this case the formula (Eq. (I.18))remains unchanged, if the energy related to the unit cell,i.e.,H/N is assumed to be the total energy H of a crystal. Thus, the solution of the dynnmical problem is reduced to the transformation of
coordinates (Eq. (1.17))which transforms the vibrational energy of a crystal to the form of Eq. (1.18). This means that the consideration of this problem by means of quantum mechanics leads to the same numerical result as in the case of classical mechanics. The distinctions appear if we take into account the terms of third and more powers in the expansion of a potential energy of atomic displacements (Eq. (1.4)),i.e.,this can be if you describe the anharmonic properties of a crystal [8] (see Sec. II.C).
8
ZHIZHIN AND MUKHTAROV
In the above considered dynomlcal problem we do not take into account any pecularities related to properties of molecular crystals such that the interaction forces between atoms of one molecule are much more than the forces of intermolecular interaction. Molecules in a crystm maintain practically their peculiarities. As a rule, their properties (including the vibrations) are inconsiderably perturbed by intermolecular forces. Moreover, the difference between the intermolecular and intramolecular forces is not only of the quantitative type, but rather of the qualitative one. Therefore, the different methods and models of their description are necessary. O n the other hand, the sharp difference between the intermolecular and intramolecular forces allows us to use some approximations considerably simplifying the solving of dynamical problem. In particular, in the first approximation one can regard the molecule as an absolutely rigid particle,i.e. one can consider only intermolecular (external) vibrations (one also uses expressions such as "the low frequency spectrum" [9], low frequencies, "lattice vibrations" [3,10,12] or "phonon" vibrations [11]) described by translational and angular (librational)displacements of molecules. The external normal vibrations are only determined by intermolecular forces, and therefore, they are rather sensitive to the crystal state. In addition, the problem of intramolecular (internal) vibrations can be independently solved. Considering the peculiaritiesof molecular crystals we can represent the total potential energy V in the form
v vm+.
z (I.19)
where V in is the potential energy of the atomic interaction inside the molecule, and U is the potential of intermolecular interaction forces. Accordingly, we get the following expressions for force constants (Eq. (1.5)) and equilibrium conditions (Eq. (I.9))
f
~:
+ bx
K
~U
=0 (I.20a)
VIBRATIONAL S P E C T R A OF O R G A N I C CRYSTALS
fpa,p'aK''( ~r'/ = OXpal~~
+
9
o2u <;) (I.20b)
In practical calculations it is convenient to describe the intramolecular potential energy V in by using the valent-force molecular field characterizing the energy as a function of bond length and of angular deformation between bonds, etc. It is successfully applied to the theory of isolated molecular vibrations [14,15]: 3n-6 (I.21)
where k~tg, is the force constant of a molecule, q~tt~) = 1, .... 3n-6 are natural intramolecular coordinates which can be expressed by atom displacements
(I.22) For a given configuration of a molecule the values of coefficients b~t,pa can be calculated by means of formulas that are well known in the theory of molecular vibrations [14,15]. Substituting Eqs. (I.21) and (I.22) into Eq. (I.20b) and considering the equilibrium conditions (Eq. (I.20a)), we get
fr~,p'a'
=
Z
k ~ , b ~ , ~ p b ~ t , p , a, -
+
0X
•
~b~t,a'p"
5Kx,
6tg
K:'
(I.23)
10
ZHIZHIN AND MUKHTAROV
The derivatives of atomic displacement coordinates can be calculated by means of the Crawford formula [15]. In a general case the values of the force constants of a crystal molecule k~ttt, differ from those of an isolated one, and making successive calculations of the crystal vibrations it is necessary to solve the inverse spectral problem considering the influence of the mtemolecular interaction forces [16]. The calculation technique according to Eq. (I.25) will be in detail considered in Sec. ]:II.A. For simplif~ng the problem one often neglects the first derivative of U in Eq. (I.23) and then the force constants will be of the form [16,17]: "pa,p'a' 8~,~l/,+k (ex) pa,p'a' ( ; fpa,p'a' ( ; ~/ =i~(in)
(in) a, kpa,p,
=
~'] (I.24)
~ ' k ~ , b~,ap bu',a'p'
~2U (I.25)
pa,pa
l'
are regarded as force constants describing the
intermolecular interaction force, and they are defined by solving the inverse spectral problem. At k = 0 this method practically coincides with the calculation technique for the case of the isolated molecule vibrations; and for this reason, this technique is called the matrix technique (FG-Wilson's matrix technique [16]). For the first time the matrix technique was used in [17] for calculating normal crystal vibrations of benzene and naphthalene. Since for the description of intermolecular interaction forces it is necessary to have a large number of force constants pa,p a'
l'
in the case of the real calculations one must additionally make
rather rude approximations whose consequences are partly compensated by fitting the force constants to the experimental data. As a matter of fact this technique is a semiphenomenological one. For this reason, at the present time the coefficients
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
pa,p'a"
l"
11
are calculated by means of atom-atom potentials when one uses
the matrix technique [18-22]. At the interpretation of vibrational spectra the theory of lattice dynamics of molecular crystals is inconvenient in the case of Cartesian rectangular coordinates of atomic displacements. First of all, in calculating, one deals with dynamical matrices of large sizes (for example, for comparatively simple crystal of naphthalene the size is equal to 108 x 108). Secondly, in this basis the amplitudes of separate atom displacements are described by the form of normal vibrations, and the interpretation of spectra in terms of external and internal vibrations requires the additional efforts. Therefore, for the mA~mA] consideration of molecular crystal peculiarities it is convenient to introduce the generalized coordinates from the very beginning. These coordinates must correspond to the internal and external displacements of molecules. The lattice dynamics in the generalized coordinate basis were firstly applied in Refs. [23-25] and considered, in general, in Refs. [3,10,26-29].
(in) K
Let Xpa ( l )
be the coordinates of atomic displacements in a deformed
_(ex)(K~
molecule; let ~pa I t )
be the coordinates of atomic displacements if the rigid
molecule is moving as a whole. The kinetic energy of a molecule (K. l ) t. (in) )2 +mp(• 2T= ~ mp~Xp~ pa
-(in) ~(ex)] ))2 +2mp~pa "pa
J
The first two terms are related to the kinetic energy of the internal and external motion of a molecule, and the last one is related to the Coriolis interaction between them. To describe the motion of a molecule by the internal and external coordinates, _(in) Z mp~pa
=0
pa
(I.26a)
[~(o) ~(in)] = o. J
p 9--pt-*p ,-p
(I.26b)
The first condition shows that the displacements X~a')~-must be set relative to the center of a molecular mass. The second condition is approximately fulfilled with an
12
ZHIZHIN AND ~ T A R O V
accuracy to terms of second order [30]. If the internal natural coordinates q which can be consequently chosen as the internal generalized coordinates are changed, the atom displacements are satisfied by Eqs. (I.26a) and (I.26b). It should be noted
that in some references [13,26] the terms of the second order are considered in Eq. (I.22); in general, this complicates the calculations. These terms which are nonlinear in the atomic displacement coordinates can be correctly used only in the basis of the Cartesian coordinates of atomic displacements where there is no division into the external and internal motions. To determine the generalized external coordinates we consider the local system of molecular coordinates whose origin is at the center of mass and whose axes coincide with the main axes U, V, and W of its inertia tensor. Then the atomic coordinates can be expressed as
~ p ( / ) = ~c.m.(/)+/~(/) ~p(~), (I.27)
()
where X c.m. are the coordinates of the molecular mass center,/~ K! -
o, ~e ~ e ~ <
~~e~
o, ~e m o . e ~ = m e ~
=e.
is the matrix
<:,,>,~o(:) ~
=e ~e
J
coordinates of an atom p in the local coordinate system.
For the atom displacements we have +
(L28) where ~ ( ~ = / ~ , ( ~ - ~(0)0r ~(0)(~)are the directing cosines for the equilibrium x,~J x,~J -. /K~ -./K~ -'-" m/K~ orientation of a molecule K, hxc.m.[tJ = X,/!J - X~" "t !J is the shift of the center of a molecular mass, zC~) "
"-
zC~)" - ~ p
" i s the intrainolecular
displacement of an atom, Zp are the atomic coordinates of nondeformed molecule in the system of its inertial axes.
~e.=
[..\
o, ~e.e~n~ order ~t~J z t ~ ) ~
<~"
.,ows ~at ~e
division of motions into internal and external ones is rather approximate; for this reason, it will not be considered in the future.
/'.j~
Let us choose the mass center displacement of a molecule tatS)
as the
translational generalized coordinates; it is convenient to set it in the local coordinate system
(I.29)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
13
To describe the angular displacements of a molecule in [31] it is assumed to use 9 f
\
elements of a matrix ~{~}. However, since/~(~) is the orthonormalized Inatrix, only three of them will be independent 3
u,~ = 1,2,3 p=l The application of dependent generalized coordinates leads to the noticeable complication of the problem, and for this reason, it is unfit for practical use. In a general case, the matrix of directing cosines can be represented as [31]
where .%(~)is some matrix ofthe rotation relative to its axes [32]
where
A1 =
[i 0 0/ cos0
sin0
-sin0
,
cos0 J
A2 =
fo OS i] \-sin0
1
0
0
cos
,
A3 =
f= -sin i] cos0
~-sin0
0
(I.30) As is known, the kinetic energy of the rotational motion of solids is equal to 3
2Trot = ~: J a ~2 a=l
(I.31)
where J a are the inertia moments of a molecule, ~ a are the angular velocities of its rotation relative to inertia axes. Let us express the angular velocities by angles 0a [30]
14
ZHIZHIN AND M U K H T A R O V f~l = (}1 cos02 c~
+ (}2 sin02
f~2 = - {}1cos02 sin 03 + (}2 cos02 ~3
= 01sin02 +03
Hence follows that, if the generalized external coordinates are used, it is necessary to consider not only the harmonic approximation of the potential energy, but also the harmonic approximation of the kinetic energy which depends on these coordinates in a complicated way.
The angular velocities assumed for an
equilibrium configuration, i.e. at 0a = 0, correspond to this approximation. In this
case Qa = (}a. It should be mentioned that the application of "die Eulerian angles given by rotational matrix A323 = A3(01)~k2(02)A3(03) is very inconvenient in this case, sinceat 0 a = 0
weget f~2=0.
Expanding the rotational matrix ~k123 by its Taylor series expansion up to the terms of the second order and performing simple manipulations, we finally get the relation between the Cartesian coordinates of atomic displacements and the
generalized coordinates ta(~), ea(~), ql~ ~P~ t =
I
~
+ ~ I ~p %
+
~=1 hPa'g qtt
p,6
/
'
(I.32)
where ..~p p(p) are the elements of the skew-symmetric matrix
l~(p)
I~
1
ZP3 -Zp2 = -Zp3 0 Zpl , Zp2 -Zpl 0
P ~ ) , T = P + 3(~- 1) are the elements of the matrix
(I.33)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAL~
P(P) --
I 0 0 -Zp2 Zp 1
-Zp3
0 0
0 -Zpl Zp 1 0
0 Zpl 0
0 0
0 0
15 0 0
-ZPl 1 -Zp2 -Zp3 Zp2 Zpl Zp2 0
(I.34)
and hpa,~ are the elements of a matrix which is calculated by the Crowford formula in the theory of molecular vibrations. It should be noted that solving the problem of the generalized coordinate choice, which seems to be simple at first, requires special attention because of approximations influencing the calculation results. For example, in [13,26] the
authors have used the coordinates of the infinitely small rotation relative to the equilibrium axes of a molecular inertia which were regarded as the projection on these axes of a molecular rotation relative to some arbitrary chosen direction. However, these coordinates are independent of the rotation angle in the approximation of the first order; in the expression (Eq. 1.32) the consideration of the terms of the second order leads to the wrong form of the matrix (Eq. 1.34) and, therefore, to a certain error at the calculation of normal vibrations. We denote the generalized coordinates for 7 = 1,...,6
for 7 = 1,2,3
for 7 = 4,5,6
for 7 = 7,...,3n
Let us express the kinetic and potential energy of crystal in terms of new coordinates
2T=
Z a
'Cl
Cl'
I
(I.35a)
16
ZHIZHIN AND MUKHTAROV
v=vo 3n
1 llt
~2u
.
(I.35b)
Equilibrium conditions ~V ~a
~U
= 0,
~t = 1 , . . . , 3 n - 6
(I.36a)
~U ex(K) = 0,
~qv
v= 1,...,6
t
(I.36b)
The first condition determines the equilibrium conformotion of the molecule in a crystal, and the second one determines the equilibrium position of its mass center and orientation. Let's express the force constants in Eq. (I.35b) by derivatives of the energy of intermolecular interaction forces with respect to coordinates of atomic displacements
= k~'8~'Stt' + 8~r
Z
~U
K:
!
,
K
+Z ap a'p'
(I.37)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
17
The further solving of the problem is in complete analogy with the above mentioned case of simple lattices and corresponds to the following transformation between the generalized coordinates and the normol ones
in K =Z {p~__l[TJ ep+6(Kl~: (I.38a)
qve x _-
1 (I,38b)
where
My = (M, M, M, IuIvIw ),
1
' / /~,=~ 8~, § - 89
:
and S -eigenvectors, ~t are the eigenvalues of the matrix T of the kinetic energy of a molecule [33]. Thus, the form of the vibrations describes now the degree of a participation of different internal and external motions of molecules in a given normal vibration. Besides, the application of a rigid molecule approximation is possible; in the expression in Eq. (I.37) it is sufficient to set equal to zero all force constants for Y > 6 or ~ > 6. In this case the solution of the dy~_amical problem gives 6 frequencies of normal vibrations whose forms show the mixing of different translational and vibrational motions of the molecules. However, as is easily seen from Eq. (I.38a), this method does not allow simplifying the problem by the consideration of internal vibrations due to the nondiagonalization of a kinetic energy matrix of a molecule T; it does not give any advantages in comparison with the more accurate method of the calculation using the Cartesian coordinate basis of the atomic displacements. For the sake of a further simplification of the problem, we assume that in most cases the properties of molecules are very little changed at the gas-crystal transition. Let's consider the approximation, when the geometric parameters and the valence-force field of a free molecule and of a molecule in crystal are assttmed to be equal.
In such a case one can choose the normal coordinates of an isolated
18
ZHIZHIN AND MUKHTAROV
molecule as the generalized intramolecular coordinates. In fact, the part of the total crystal energy related to the internal vibrations
H = T + V = ~1 ~
~ {ap4t qu . i n ( ~ ) .qtt' i n ( ~ ) + k ~ , q~in(~) q~in(~)}+ . . . . tttt"
(I.39)
coincides with the Hamiltonian function of a free molecule at aforementioned assumptions; it can be written as 1 H=~Z
3~6i=1[ ( ~ n ( ~ ) ) 2 +r
+
Kt
...
(I.40a)
ff we transform the coordinates
i=1
(I.40b)
where L~i, are the elements of eigenvectors, and (%i is the frequency of i th normal vibration of a free molecule which is described by a normal coordinate Qm(~) . For the first time the normal coordinates of a free molecule as the generalized coordinates were used in Ref. [34]. The general theory of the lattice dynamics applying these approximated basis coordinates was considered in Refs. [13,26,35]. The coordinates of atomic displacements are related to the normal coordinates Qin (~) by (see Sect. III.B)[14,15] Z a P - ~~,.,p
Y-" i=1
Lap,i Qi
(I.41)
where Lap~i is the form of normal vibrations of a free molecule; it describes the rectangular atomic displacements. In this case it is convenient to use the mass-weighted coordinates as the generalized external coordinates: Q:X(~) = ~f~ tv(~),
l)
-4i
-3 o _3 t '
v= 1,2,3
v = 4,5,6 (I.42)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
19
Let us introduce the general notation for coordinates y= 1,.... 6; Then we get the following expression for the total energy of a crystal 3n ( ~ ) + I N Z 3n (KK'~ H = - I Z Z (~V Z Z Z fw'~,g_t,jqv 2~ u 2 R" ~'K' ~'
(~)
(K') Qv' g' (I.43)
The solving of the dynnmical problem is usually performed by the diagonalization of two quadratic forms (Eq. 1.43) in the same time with the help of coordinate transformation Q~(~/= ~ e~+6/K:lkj/ e~(g)Qj( ~ )
Q~X( / ) = j~k ev (~:lh:j/ eil~(l)Qj (~:) (I.44) The frequencies and the eigenvectors of normal vibrations are determined by solving equations that are in analogy with Eq. (I.15) and (I.16); the dyvomlcal matrix is assumed to be of the form
(K K'~ e-if~[R(t)-R(t')] Dvr"(~:z') ~: = ~,fvz'~,g-g') l-t"
(I.45)
The force constants ~7Y'~g-g') in the coordinate basis QT t are expressed by the derivatives of the intermolecular interaction potential with respect to atomic displacement coordinates (Eq. 1.25) with the help of relationships (
~')
n 3 pp' [JfJ"
I gg' ) fJa'~ "'fJ' a"(K')
(I.46a)
20
ZHIZHIN AND MUKHTAROV
k(ex),-,pp,ppgl."
q M I - '~ pp' y-"[3~' Z
(K) /~(0)(Ic')I~,(P)[Ya' (I.46b)
+3,a,+3Lt_g,) -
1 Y--. Z ,4IaIa" pp" ~ '
{/~(0)(w:,)l~,(p')} + 1 n f~'a" VIaIa"]E
k(e-x),,,, PP,P P
/~(0)(w:)l~.(P)x
~)U
3
p(p)
-
fJfJ"bXafj(:) A(~'(W:) fj''a+3(a'-l)SK'z'B'g (I.46c)
K K']_
n 3 3 k(ex),,( "~A(0)-A(0~" x LP'~"'~t Z Z [3 ~ p[3,p[3 Vmp
1
e ,H+ Lt_t,)-
.....
pp' ~[3'
()Kx'
1
n 3 pp' ~ '
(I.46d)
}
3 t.(ex),(~j'.~(0) l~(p)
~a
VI:I:Ip"
(I.46e)
(K K'~
f6+~,6+~'Lt_ t, )
pp' ~ ' pp'
Lpp,p Lp,p,,~, ~mpmp,
+WO,uUl.~ ...2
~
t~
oK1r
t~
t
(I.46f)
where a = 1,2,3; ~ = 1 ..... (3n-6), Eqs. (I.32) and (I.41) being considered. The vibration form (Eq. 1.44) gives the obvious picture of the interactions of internal and external vibrations in a crystal.
This method gives also the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
21
opportunity to neglect any internal degree of freedom in the calculation. Then it is sufficient to cross the corresponding row and column in the dynamical m~trix. In a
general case this permits to divide the whole vibration spectrum into normal vibration groups with comparatively wide gaps between them and to solve the dynamical problem for each of them. The symmetrical properties of generalized coordinates Q are also their advantage:
they form the basis of irreducible
representation of the free molecule point group. The calculation technique will be in detail considered in Sect. III.B. If the molecule is conformationaUy unstable (for example, its separate parts can relativelyfree rotate around a single bond), the aforementioned method can not be applied, since at the gas molecule transition the molecule can noticeably be deformed. This case will be specially considered in Sect. IH.C. The results of solving the dynamical problem can be used not only for the interpretation of vibrational spectra, but
also for
the
calculation of
thermodynamical properties of molecular crystals [3,36,37], their elastic constants [35], etc. (the review is given in [38,39]). In particular, the Debye temperature of a crystal can be expressed by the root-mean-square frequency of external vibrations [3,12]
0D = k
~2
(I.47a)
where ~2
1 ~
1
v=l
(I.47b)
The rms values of atomic displacements along some direction given by a unit vector l can be also calculated. In the approximation of rigid molecules [40]
a,fJ= l "
(I.48)
where ~ is the radius-vector of an atom whose origin is in the molecule mass center.
22
ZHIZHIN AND MUKHTAROV
Formulas for tensors Ta~, Lal3 and Sa~ are given in Ref. [40]. For example, for librational tensor La~ we have
La~= 6Nzz []~]~ ~Rea+3(KIk'j/}IIme~+3(~[k'j/}N,~iai ~s176
(I.49)
where e(co)=h
"/-']
+ exp~-~
1
In this expression the real and imaginary parts of eigenvector of librational components are used. The diagonal components Taa and Laa correspond to the average values of squares of translational ~2 and librational ~2 displacements of the whole molecule.
In Ref. [40] the tensors T, L , S
are calculated for 9
hydrocarbon crystals. The root-mean-square displacements of molecules proved to be inconsiderable, and their values do not exceed 0,2 - 0,3/~ and 5~ respectively, i.e. the
harmonic approximation is sufficiently adequate
in the
case of
comparatively large molecules. For the evaluation of mean angular displacements of molecules, one uses the approximation such that the dispersion of external vibrational frequencies is neglected [3,41]
,(1o
ga K 0a = j=l K=I
(I.50)
where nlibr" is the number of frequencies of librational vibrations at k = 0. Equation (I.49) is applied only to the centrosymmetrical position of a molecule in a crystal. In conclusion, we consider the pecularities of dynomical problem solutions for nonideal crystal when the translational spacing is destroyed, and the relationships (Eq. 1.6) between the force constants cannot be, in general, used for simplicity of the problem.
The presence of a crystal surface is the most simple case of the
destruction of a crystal translational symmetry. In this case the problem can be
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
23
approximately solved by using the model of two-dimensional infinite crystal consisting of L layers [42]. Instead of a unit cell consisting of z molecules one uses the extended unit cell consisting of Lz molecules.
In the rigid molecule
apprommation the consideration of two-dimensional symmetry leads to solving the system of 6 equations that is analogous to the system (Eq. 1.13), which gives 6Lz frequencies and eigenvectors.
In this case the last ones demonstrate the
contribution of motions of different crystal layer molecules into the normal vibration.
The analysis of these motions permits to pick out and describe the
crystal surface vibrations. This method was applied to the crystal of naphthalene (the 001 surface) that consists of 12 layers. The presence of irregular oriented molecules [3] is also often occurred defect of molecular crystals. In this case some simplifications are impossible, and it is necessary to solve the initial system of 6Nz equations (rigid molecules) whose solutions will be, obviously, more or less localized normal vibrations described by spatially attenuated waves [43]. The practical calculations can be carried out by using a numerical method proposed by Dean [2,44]. The method is based on the negative eigenvalues theorem which permits to determine the number of eigenvalues of a very large matrix in a chosen spectral range (co1, m2). The calculation is in detail described in Ref. [45] for the case of molecular crystal where the calculation of the external vibrations of the orientationaUy disordered nitrochlorobenzene crystal is carried out. We note that the expressions obtained in Sect. I.A for the force constants are also applied to the nonideal crystals.
B. Symmetrical Properties of Force Constants and Dyn~mlcal Matrix The potential energy of a crystal and, consequently, the force constants are invariant to the symmetry transformation under which the interatomic distances are remained unchanged [46]. The invariance with respect to the translational lattice vector shift leads to the relationship (Eq. 1.6) of force constants which permits to simplify considerably the dynamical problem (as was shown in Sect. 1.1). For further simplification of this problem it is necessary to take into account the crystal symmetry with respect to symmetrical rotations and to partial lattice vector shills (the usual and spiral axes, symmetry planes, etc.). These symmetry elements are assumed to denote [47]
24
ZHIZHIN AND MUKHTAROV
(Z.5Z) where a is the symmetry element belonging to the point crystal group, ~(a) is the partial translation vector. According to Eq. (1.51) the coordinates of an atom p of a molecule (K, l ) are transformed by the symmetry element into the coordinates of the some atom of another molecule (K a ,la). The symmetry elements (Eq. 1.51) form the group (the factor-group) with an accuracy to the translation by a lattice vector; this group is isomorphic to the point group (class) of the crystal. Although the consideration of the crystal symmetry concerning the dynamics of molecular crystal lattice was discussed in m a n y articles, and the analysis was carried out either in rather general form [47-51] or in the form of some particular cases [10,13,52]. Let's consider in detail the lattice dynomics in a basis of generalized coordinates
o(;)[,e,(;) II (see Sect. I.A) given in a local system of mass center and inertia axes coordinates of a molecule (~, ! ), which is the most convenient for practical calculations. W e also suggest that in a crystal there is only one symmetrically independent molecule in the unit cell, i.e. the atomic coordinates of all molecules of the unit cell can be obtained by acting of some symmetry elements of a factor-group which will be denoted by
on coordinates of molecule atoms; the molecule is chosen as the original one (K = 1). The symmetry axes of a molecule will be also related by symmetry elements. Let's choose the directions of these axes such that they can be deduced from the original
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
25
molecule axes by symmetry elements az. Then the directing cosines matrix of a molecule K can be represented as /~(0)(K) =/k~/~(0)(1) = A~/~
(I.52)
where "~K is the matrix representation of an element a K of a point group in the rectangular coordinate system of a crystal, /~ is the directing cosine matrix of an original molecule. Consider the action of the symmetry operator on the generalized coordinates f
\
Q~[~}.
If in a crystal the molecules are in the general position, their number in
the unit cell is equal to the number of representative elements of the factor-group simply interchanging the generalized coordinates of various coordinate systems
(I.53b) where the indices (gK)mean the transformed radius-vector of the molecule mass center [51]
~(~,)=a-~l~(1)= a~l ~ ( 1 ) _ a ~ l ~(a~ ) -.
K u
(I.54) If the molecule has the symmetrical position in a crystal, the set of appropriate symmetry elements which will be denoted as ~i ={bii0} 9 i= 1,...,hpos. form the point group Gpos., being the subgroup of the crystal factor-group F. As before, let's denote the interchange symmetry elements a~ relating the molecule axes (K = 1) to other molecule axes and satisfying Eq. (I.53a). The other interchange symmetry elements o can be written in the form o = ~ a~
(I.55)
26
ZHIZHIN AND ~ T A R O V
This corresponds to the expansion of the factor-group in a series of conjugated classes of a subgroup Gpos. [46,53]. Considering that the internal coordinate p (the normal coordinates of a free molecule) are transformed by symmetry elements ~ in accordance with the irreducible representations of a free molecule symmetry group and taking only the non-degenerate internal vibrations for simplicity, we get
Q
R t"
~
in
~ex
(:)
~.!)
(I.56a)
in(l,);
= ~i"Q
{XK e x ( 1 ) Q v X ( ~ , )
t"
"Q
t"
>,, (,,) in K:"
(I.56c)
where o = ~ia,r
a,c,a,r = ~ i , a , r
(I.56d)
are established with the help of the table of factor-group element multiplication. I3i is the matrix representation of 6x6 d~mensions of molecule symmetry element ~i in the basis of the translational-librational coordinates of displacements.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
27
Since the infinitely small angular displacements are transformed as the components of an axial vector, we get [49,51 52]
(I.57) where t~i is the 3x3 matrix of the polar vector transformation, det ]3i -- 1 and -1 for the rotations describing the usual and mirror rotations, respectively. Consider the part of the vibrational potential energy of a crystal in the harmonic approximation related to two arbitrary chosen molecules (K,0) and (ICl,tl)
(I.58) where
ii--/"k:
Q(o)
~li
is the vector-row and [[QO-,IBH is the vector-column consisting of 3n r II ~\'lJg generalized coordinates Q~(~), f|KKlJ is the 3 n x 3n matrix consisting of force
constants ' L1 " Transform the generalized coordinates by some symmetry element a. Considering the invariance with respect to potential energy and relationships, Eqs. (I.55) and (I.57), we get
(I.59) Hence, we obtain
t l ) =~i"
fit." t'~ )'J" =
t'~-t"J'j"
(I.60)
where
=
Z(03n-6) (~) (det ~)13i
(I.61)
28
ZI-IIZHIN AND MUKHTAROV .-,,
,r..' o;',o<
K n
...,,
...f t~" x= 1R(K1 '
aK", aK~, ~i" etc., are determined by Eq. (I.56d). Expressions (I.56d), (I.59) - (I.61) establish, in general, the relationship between the force constants which follows from the crystal symmetry. Consider some consequences which are i m p o ~ t for the simplification of the dynamical problem at the calculations (a)
a =a~ 1" l~ - l"J"J"
(I.62)
This means that for constructing the dyvAmical matrix in the general case it is sufficient to find the force constants corresponding to the crystal molecule interaction only with the single original molecule K = 1.
Cv)
o=~Iic
KI ri - r')"i
R(l~ K - l"/= R-1R(K1 ~i Ct~)-~(r')
(I.63)
In particular, if the molecule has the centrosymmetrical position, then [3 = C~
~:1 11 = + fYr"
fYr'
-
1 1(1)
(I.64a)
K1 (I.64b)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
29
where the factor (-1) corresponds to coordinates Qj and Qj' with different parity with respect to the inversion (for example, librational a n d translational coordinates, internal normal coordinates of g- and u- modes of an isolated molecule and so on). Equations (I.64a) and (I.64b) show that in this case it is sufficient to calculate the force constants only for molecules whose radius-vectors are not related to the transformations, i.e. the number of necessary force constants is smaller by a factor 2. (c)
If the factor-group contains the element with symmetry a = ~-1
=a~:la-~l
which interchanges the coordinates Q(t) and Q ( ~ : ] , then we get
-
~:1
Lo)
(I.65)
By taking into account the properties of force constants, we consider the structure of the dyn~mlcal matrix consisting of z x z blocks D with 3n x 3n dimensions
E
(t-tl)
f o tl-t (I.66)
According to Refs. [47,48] the dynamical matrix is transformed by the s ~ m e t r y element of the factor-group 6 = {sl~(s)} in the following way
(I.67) Hence, using the property (Eq. 1.62) and denoting k 1 = {1~1 ~ we obtain
(I.68)
30
ZHIZHIN AND MUKHTAROV
Thus, the whole dynamical matrix can be constructed by submatrices of the first row ~(1w:~, ~:./ only with" another wave vector. If the molecule centrosymmetrical position (the property of Eq. (I.65)), then we get
has
the
(~K:I/ +D ..,('W:~l)= +I~, ('~:~:i'~
DW,~, kl )=
.,, L_kl
_
~, E1 ).
(I.69)
Finally, in the case of Eq. (I.65)we obtain
~,E1 )
:
/
O' E1
E1
9
(I.70)
As an example, we consider the crystal of glycine (the T-modification) whose space group is P31, z = 3(1) [54]. The factor-group C3: a 1 = E; a 2 = (C 1 1 001/3); a 3 = (C~ 1 002/3). In accordance with Eq. (I.68) the dynamical matrix can be written as
l~k,1)
Ikl)
If~l)
/
The further simplification of the problem is possible only for the wave vectors such that remains unchanged at action of some symmetry elements s: sk = k. The set of elements s form the point group which is called the group of a wave vector F~:. By applying the group theory one can find the unitary transformation which transforms the dynamical matrix to the block-diagonal form. In the general case the construction of such transformation is considered in [47-51] and represent the sufficiently complicated problem. -o
-o
At the description of optical vibrational spectra of the first order which are in detail discussed in the present work, one can limit oneself to the consideration of normal vibrations at k = 0, since the light wave vector in a crystal is very small (of the order of one millionth part of a phonon wave vector at the BriUouin zone boundary [4-7]). In this case it is possible to simplify the dynamical problem. For example, for the centrosymmetrical position of a molecule relationship (Eq. 1.69) will be of the form
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
-
{
r 0 if T and T" have identical parities = 0 if T and y' have different parities
31
(I.71)
This means, in particular, the complete division into the translational and librational vibrations. The whole dynamical matrix is divided into two diagonal blocks corresponding to normal vibrations which are even or odd ones with respect to the inversion; the first ones will be active only in the _l~m~n spectra; the second ones are active in the infrared absorption spectra (the exclusion rule [10,49]). We also note that for the coordinates of the same parity
(I.72) where the summation is carried out only over molecules whose radius-vectors are not associated by an inversion. Equation (I.70) implies for k - 0
b(K0
K1)
=
K1) l~(K0
i.e., the bl~ ~ a dynamical matrix D( ~:W:l )0 condition 0 = 0 -1 = {xK10~1
(1.73) are the symmetrical matrices under
In the general case the maximal number of diagonal blocks to which the dynamical matrix and their dimensions can be transformed is determined by means of a group theory according to formula [46,49,10]: n(i) = 1 ~ z(i)*(a ) Z(o~) g a
(I.74)
where n(i) is the number which shows that the given irreducible representation F (i) is included n times in the representation of a factor group; the representation F is constructed in the basis of generalized coordinates, ~(a) and Z(a) are the characters of representations 1"(i) and F, g is the order of the group. The characters of the representation F for the internal vibrations are calculated by using the formula (the Bhagavantam and Venkatarayudu method [46]) Zac(a) = _+1 +2 cos ~a Xtr(a) = (Ua'l)(+-I +2 cos Za) Zlib(a) = Ua(1 +_2 cos Xa)
(I.75)
32
ZI-IIZHIN AND ~ A R O V
where U a is the number of molecules of the unit cell being invariant to a transformation a, the signs (+) and (-) correspond to a simple and mirror rotation by an angle zo~ respectively. The classification of normal vibrations by irreducible representation of a factor-group is very important at the analysis of the Raman spectra and infrared absorption spectra, because it permits to predict the activity and polarization of the vibration lines [10,49]. At the group theoretical analysis of the optical vibrational spectra of molecular crystal the method of site symmetry is widely adopted; it was first suggested by Halford [55] and Hornig [56] and is in detail described in many papers [10,49,57-59]. The application of this method is based on the assumption such that the site symmetry group is the subgroup of a factor-group of a crystal and the subgroup of a symmetry group of an isolated molecule in the same time; there exists the certain relation between their irreducible representations which is defined by the correlation theorem [49]. In this case the group theoretical analysis can be carried out without any calculations according to the well known correlation tables [10,49]. In particular, if the crystal factor-group is known, the method of the site symmetry analysis gives an obvious interpretation of a line splitting of a given internal normal vibration of the gas, i.e., a crystal transition appears as the degeneracy is removed due to lowering the molecular symmetry (the Bete splitting) and to the dynamical interaction of translationaUy nonequivalent molecules (the Davydov splitting)[10,49]. It is possible to formulate the inverse problem for obtaining the information on the molecular structure from the data concerning the properties of a free molecule and the line fine structure in a crystal spectra. As is known from the systematic research in paper [10,60-62], the solving of this problem gives sufficient knowledge about the factor-group of a crystal,the quantity of molecules in the unit celland their site symmetry. The method which is similar to the site symmetry can be also used to construct the unitary transformation (the dynamical matrix is transformed to the quasi-diagonal one) which permits to go from solving the 3nz equation system to solving some independent systems of less dimension. Then it is necessary to find the unitary matrix relating the initial generalized coordinates to the symmetry coordinates which are transformed in accordance with irreducible representations of a factor-group. Since the molecules of different unit cells vibrate in the same phase at k = O, the transformation between a given molecule and translationally equivalent one can be regarded as a single transformation and, therefore, one can omit the index ..o
. ~ given the number ofthe unit celh QT{~}-->Q7(~:) coordinate, we get \--/
Thus, if Q~o s)- is the symmetry @
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
~(s) = ~
33
(I.76a)
(I.76b) where S is the unitary 3nz x 3nz matrix, Q is the vector-column, Q = ~Qj(K) . Comparing formula (I.76a) with (I.44) one can see that the rows of a matrix correspond approximately to the form of normal vibrations which defines only by a symmetry and does not depend on any specific interactions of molecules in a crystal. In some cases when only one normalized vibration corresponds to some irreducible representation, its form coincides with the symmetry coordinate. Thus, the group theoretical analysis permits to make some conclusions concerning the form of vibrations without the complete calculation of the lattice dynamics. Transformation (Eq. 1.76b) can be constructed by using projection operator [49-50]. Choosing the generalized coordinates QT(1) of the molecule (): = 1) as the starting ones, we get [40] Q~h,i) = A ~ T~h)'(o)oQ~,(1) o~f
(I.77)
where Q~.,i)lh are the symmetry coordinates corresponding to the irreducible representation ~ F (h) of the factor-group F and belong to ith line of a matrix ~(h) (o) of this representation. The values i = 1, ..., ! n correspond to degenerate symmetry coordinates. A is the normalization factor. In the general case the direct use of the projection operator is comparatively hard process. First of all, the summation in (I.77) is carried out over all elements of a factor-group. Secondly, for the determination of the complete set of symmetry coordinates it is necessary to carry out the calculations for all subscripts i and T, the part of coordinates may be either equal to zero or be linearly dependent [10,49], i.e., one must carry out the additional calculations. The most simple way of determining coordinates is in the case, when the molecule has the general position in a crystal, i.e., when oQ(1)= aKQ(1 ) = Q():) (~K)aKQ'(1)= A ~ K=I
K=I
T~h)*(~c)Q'()r (I.78)
34
ZHIZHIN AND ~ T A R O V
For example, in the case of the abovementioned crystal of 7-glycine we immediately get the following expressions using the table of irreducible representations [10,49] r = 3nA + 3nE
~(Ag) = ~3 [~(1) + ~(2) +~(3)]
Q(E,1) = ~ [ ~ ( 1 ) _ 1~(2)_ }~(3)]
Q(E,2) = ~22[~(2)_~(3)]
E
where I~ and 0 are the unit and zero 3n x 3n matrices, respectively.
6(lo
o
=
(lo)
i9
Consider the case, when Z chosen permutable symmetry elements form the subgroup Gper and commute with the symmetry element ~i of site symmetry group Gsite, i.e.,
akEGper, ~iEGsite, aK~i = ~iaK According to [54] this case includes more than 90% of the well known structures of molecular crystals and also includes all crystals of lowest syngoni with symmetry axes of the second order and the crystals with centrosymrnetrical positions of molecules and so on. Then the factor-group of a crystal can be represented as the direct product of its subgroups Gpe r and Gsite [46-53]
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
f = Gsite @ Gper
35 (I.79)
The matrix of the irreducible representation of any symmetry element o = ~iaK can be also represented as the direct product of appropriate matrices of some irreducible representation Fsite and Fper of groups Gsite and Gper [53] ~(h) (8) = ~(h) (~ia K) = ~(h 1) (~i) |
) (a~)
(I.80)
(h 1) and r(h2) The relationships between the representations I~), r aite per can be established by means of the correlation tables. Then we can transform the projection operator (I.77) by ~klng into account (I.80)
QT(h,i) =Q~h1h2ili2)=A
z
Z Z T~jh)'(~i'aK:Qy(1))= ~:=113i,
z T( h2 )* =AZ i2j 2 (~ K:=I
aK Z [3i,
T.(h 1)* 11j I (~i')~i'Qy(1)
Hence, we find Q~h'i)- Q~hlh2ili2) = A1
~ T!h'2)*12J2 (a~:)QT(hlil)(*:) t:=l
(I.81)
where Q~hlil)(K:)are the sitesymmetry coordinatesofa molecule Q(hlil)(~:) = A2 ~ T.(h.1)* Y 113i (~i)~i Q y (K) (I.82) One can also construct the coordinates of permutation symmetry z
Q(h2i2)=A1 ~
T!h.2)* ]232 (a~:)Q7 (t:)
(I.83)
Thus, in a given case the problem of the construction of symmetry coordinates is reduced to the independent determination of the site symmetry coordinates and the
ZHIZHIN AND MUKHTAROV
36
permutation symmetry ones. The complete set of coordirmtes is obtained with the replacement of the coordinates Qj(K) by QIhlil) (k:) in the expression (I.83). The use of correlation tables permits to present the appearance of zero and linearly dependent symmetry coordinates. Besides, the site symmetry coordinates for external vibrations are assumed to be well known, since they are cited in the tables of irreducible representation characters [10,49]. We explain the application of the suggested method; as an example, we regard the external vibrations of the acenaphthylene crystal, Pbam, Z = 4, the factor-group D2h [54]. We choose the following elements as the permutation symmetry elements which relate the original molecule axes to the axes of other molecules
~i = E,
a2 = C~,
(z3 = C~,
~4 = C~
These elements form the subgroup D 2. The sitesymmetry group C s includes [31 ffiE and [5'2= Cs- It is obvious that D2h = D 2 @ C s. Using the tables of irreducible representations of groups D 2 and C s, the relationship (I.83) and the correlation table [10],we find that
~(B1) =I[~(1)+ Q(2)- Q(3)- Q(4)]
D2 ~
D2h ~
A
A g ~ A Big
B
(B2) = 1[Q(1)- Q(2)+Q(3)- Q(4)]
/ \/B2g.\ /~ /~
B2
~(B3) =1[~(i)- Q(2)-Q(3)+Q(4)]
~
Cs
B3
~i"Blu~
' tu, tvOw
~" tw, 0u, 0v ;~
~B2u 7
B3u
(the inertia axis U of a molecule is perpendicular to its symmetry plane). The symmetry coordinates are now easily found
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAl,S
37
Q~Ag)= ~it [ u (1) + tu (2) + tu (3) + tu (4)] = ~[tv (1) + tv (2) + tv (3) + tv (4)]
Q~Ag)
10 = ~[ w(1)+0w(2l+Ow(3)*0w(4)]
Q(4Bg)
= ~[tu (1)+ tu ( 2 ) - t u ( 3 ) - t u (4)], etc.
By means of the appropriate orthonormalized matrix S the dynamical matrix is divided into 8 diagonal 3 x 3 blocks. In the abovementioned scheme it is sufficient to add the correlation between this scheme and the irreducible representation of symmetry group of a free molecule (we take into account the internal vibrations in the basis of normal coordinates of an isolated molecule). It is necessary to mention that the dynamical matrix can be factorized even at the transition to the symmetry coordinate (I.83) of this group, since the permutation elements form the group. The Z 3n x 3n motrices will be on the diagonal; in the case of acenaphthylene they will be of the form 15(sss) = 6(1,1) + 6(1,2) + 6(1,3) + 6(1,4)
f)(ASA)= I)(1,1)- I)(1,2)+ D(1,3)- D(I,4) f)(SAA)= I)(1,1)+ I)(1,2) - D(1,3)- f)(1,4) I) (AAS) = D(1,1)- I)(1,2) - I)(1,3)+ I)(1,4)
(I.84)
The dynamical m~trices for all factor-groups are reduced to the analogous form; the abovementioned conditions are valid for these matrices, and the group Gper does not contain the symmetry axes of the higher order than second, i.e., for the most molecular crystals. This property was first mentioned for some particular cases [3] in which the notations used in (I.84) were suggested. The reduction of the dyvamical matrix to the form which is analogous to (I.84) is sufficiently universal. It was applied to the program of the computer calculation of frequencies and of molecular crystal vibration form (~: = 0) by the method of atom-atom potentials. The eigenvectors of
38
ZI-I ZI-IIN AND MUKHTAROV
normM vibrations mentioned in this work are normalized to one molecule, since they are obtained by the diagonalization of symmetrical matrices of the form (I.84). The symmetry of blocks -~(x~') -of a dynamical matrix follows from the properties (I.70). The form of vibrations of the whole unit cell can be easily determined, if we transform the symmetry coordinates from (I.73) to the initial ones Q~K). The abovementioned relationships between the force constants were also considered to simplify the calculations. In the case of crystals containing some symmetrically independent molecules with the arbitrary positions and orientations the results obtained are remained completly unchanged, if the set of these molecules can be regarded as the one "macromolecule" under symmetrical transformations, when their site symmetries are identical. In some other cases which rarely r the simplification of the dynamical matrix is achieved with the help of the ordinary construction of symmetry coordinates for every system of homological molecular positions. Thus, the consideration of the crystal symmetry with the help of group theoretical methods permits to simplify essentially the dynamical problem and gives a certain quality information concerning the molecular motion type for various normal vibrations. For the final solution of the dynamical problem, i.e. for the quantitative description of the vibrational frequencies and forms, it is necessary to know the real interactions of molecules in crystals.
C. I n t e r m o l e c u l a r
Interactions
and Atom-Atom
Potentials
In the general case the theory of intermolecular interactions is a quantum mechanical problem and reduces to the solution of the Schrtklinger equation for a system of electrons and nuclei. The Hamiltonian of the system of two molecules is H = H A + H B + VAB
(I.85)
where H A and H B are the HAmiltonians of noninteracting molecules. VAB describes the interaction and is of the form [63-65]
VAB=-Z n i rni
Zm+ ZZm+E • 9rmj
mn r m
ij rlJ
(I.86)
where the subscripts n and j denote the nuclei and electrons of a molecule A, and m and i denote the nuclei and electrons of the molecule B; r is the distance between the particles, and Z is the nucleus charge.
VIBRATIONAL S P E C T R A OF O R G A N I C CRYSTAL~
39
One usually uses the adiabatic approximation which corresponds to the solution of the SchrSdinger equation with motionless nuclei. In this case the energy of electrons depends on the coordinates of nuclei as parameters, and it plays the role of the potential energy of a nuclear system, which is also called the intermolecular potential [63]. It is necessary to know this potential in order to study the behavior of an interacting molecular system. Discussing the intermolecular potential we can approximately consider three types of distances between the molecules, ff there is a sufficiently small overlap of electron shells [63]: 1. The case of small distances corresponds to the repulsion of molecules due to the overlap of their electronic shells (the exchange and Coulomb interactions). 2. In the case of average values of a distance with a van der Waals' minimum the situation is described by the balance between the repulsion and attraction forces. 3. In the case of large distances one can neglect the exchange effects. Then, we can apply the ordinary Rayleigh-SchrSdinger perturbation theory which permits the representation of the potential in the form of the sum of separate contributions. In the first approximation the perturbation theory describes the electrostatical interaction of electronic densities of molecules in the ground state which can be expanded into a series of multipoles. For example, in the case of a dipole-dipole interaction Edd= ~ 5 [ R2 (dAdB)-3(dA~.)(aBR,)] - R13 (1.87)
where d A and d B are the dipole moments of molecules A and B, respectively. The quadrupole interaction is changed by a distance of ~1~5 and so on. The terms of the second order describe the interactions between the constant multipoles of one molecule and the induced multipoles of another one (the induced interaction). For example, for the dipole-dipole induced interaction 1 [d 2 Eind - -R-6-[ A aB + d2 aA ]
(I.88)
where d and a are the dipole moments and the polarizabilities of the molecules, respectively. The direct electrostatic and induced interactions can be described in the frame of classical mechanics, and they are often called Keesom forces [63]. The other term of a potential appearing in the second order of the perturbation theory corresponds to the electrostatical interaction of two mutually induced electronic
40
ZHIZHIN AND MUKHTAROV
distributors; it is not the classical analogue, since it is determined by quantum mechanical fluctuations of the electronic density. Unlike Keesom forces, this interaction (the dispersive one) is universal, and its energy is not equal to zero even for spherically symmetrical systems. The dispersive energy can be expanded into a series of multipoles for large distances OO
Edisp
= -
~ RCnn n=6
(I.89)
where n is even and Cn > 0, i.e., the dispersive interactions always describe the attraction of molecules. Formula (I.89) was first obtained by London, and these interactions are sometimes called London forces [63]. The dominated term Cn/Rn in Eq. (I.89) corresponds to the dipole-dipole dispersive interaction [63]
|
Cn = "~ EIj OOioOOjo((.0~o+
Oio = E i - E o
jo/
A is the frequency of the transition from the ground state to the i th excited where OOio state, and fA is the oscillator strength of a transition 0 ~ i. In the case of average values of distances the overlap of electronic shells is rather small, but it is necessary to consider the exchange interaction. In this case, the division of the potential of intermolecular interaction forces into the separate parts is performed by means of exchange perturbation theory in which one uses the antisymmetrized products of molecular wave-functions as unperturbed wavefunctions [63-65]. In the case of short distances the perturbation theory is not applied, and the system of interacting molecules must be regarded as one molecule (the method of supermolecules [ 6 3 - 6 6 ] ) eint = EAB- E A - E B
(I.90)
where eint is the potential of the intermolecular interaction forces; EAB, E A, and E B are the total energies of electrons of a supermolecule and of separate isolated molecules. The calculations are carried out by the variational method. The classification and the theoretical methods of analysis of intermolecular interaction force potential are more completely discussed in Refs. [63-67]. In the general case the quantum mechanical calculation of this potential is a very complicated problem even for the simplest systems. This also concerns (to a
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
41
greater extent) the crystals, in which it is necessary to consider a large number of closely positioned molecules interacting with the closest neighboring molecules. The potential energy of intermolecular interactions in a crystal is usually represented in the form of a sum of effective pair potentials [3] , u =
u
tt'
, (K,t)
~l ~'l"
where U
(I.91)
is the potential energy of interaction between two molecules
ll"
(K,l) and (K',l').
The potential
energy of a crystal for the
equilibrium
configuration is often represented in the form of a lattice energy (corresponding to one molecule) which coincides with the energy of the crystal sublimation [3] with an accuracy to the zero vibrational energy at T = 0~ Considering the crystal symmetry gives
ULett
= "~-"~ = 2
r~t
(I.92)
Expression (I.91) is valid, if we assume that the charge densities of interacting molecule pairs are not perturbed by the presence of other molecules. The theoretical analysis and the direct calculations of trimers, of small molecules of
H20, CH 4 and N 2 [63-68,70], made by variational methods show that the nonadditivity appears mainly due to the exchange and induced interactions; its contribution into the lattice energy is equal to -10%. The nonadditive dispersive forces appear only in the third order of the perturbation theory, and their contribution is negligible [63]. The calculations of external vibrational frequencies of some molecular crystals carried out in [71] show that the consideration of nonadditive induction interactions has in practice no influence on frequency values. Thus, the representation of the intermolecular interaction potential of a crystal in the form of (I.91) is sufficiently verified. The nonadditivity is taken into account (to some extent) by the empirical fitting of some model potential parameters. However, the introduction of the potentialw "molecule-molecule" is insufficient to carry out, in practice, the calculations of crystal properties. For example, consider Kichara's potential, in which the molecules are approximated by a convex rotation body, and the interaction energy between them is calculated by the formula [63,72]
(I.93)
42
ZHIZHIN AND MUKHTAROV
where e and P0 are the parameters of a model and p is the distance between the nearest points on a molecular surface. The deficiencies of this model are obvious, since it is applied only to crystals of sufficiently symmetrical molecules. Besides, the application of the potential (I.93) to the lattice dynamics leads to awkward calculations. For the evaluation of Davydov's splitting one often uses a potential which describes the interaction between the oscillating dipole moments of a molecule during its vibrations [73,74]
___1
UAB = R5 bQ~n bQ~
in [eA'R][eB' R]- 2(~A~.)(~B~.) QAQBin
(I.94) where eA is the unit vector of a transition dipole. The applicabilityof this potential is limited by comparatively small molecules and by internal modes which are active and sufficientlyintensive in infrared absorption spectra [73]. Kitaigorodskii and his colleagues were the first ones who started the systematic research work concerning the properties of molecular crystals [75,76]. They have developed the theory of organic crystals on the basis of the atom-atom potentials (AAP). The method of A A P uses the additive model, in which the energy of intermolecular interactions of two molecules is represented in the form of a sum over the central interaction energies of atoms forming these molecules
pq
(I.95)
where p and q are the numbers of atoms of molecules (t:, ! ) and (K', ! '), ~pq(r) is the potential of an interaction of two atoms p and q at a distance rpq. The AAP method appeared as a result of the extension of the close packing approach [3,77] in which a certain form of a molecule was defined by van der Waars radii of molecular atoms. The crystal is considered as a close packing of these bodies; it is impossible to find a structure that will be more closelypacked than the real one. The atomic radii are determined from the experimental data. It was found that for each type of atom it is sufficientto have one universal radius. The derivation of possible space groups of molecular crystals is the verification of the close packing principle [3,77]. The above obtained rules of organic crystal chemistry have many proofs [3,77-79]. However, the crystal structure is not actually determined by geometrical factors; it is determined by the conditions of a free energy minimum. It is the AAP method that must associate the mechanical, structural and other crystal properties; the principle of close packing is its consequence. The atom-atom potentials correspond to the theory of the close packing in the form of rigid spheres (Fig. 1.1).
43
V I B R A T I O N A L S P E C T R A OF O R G A N I C C R Y S T A I ~
~(r), kcal/mole
a) 3
!
4
r,A
d
~(r), kcal/mole 0.2
0.1 b) 4
5
-0.1 r
kcal/mole
0.2
0.1
c)
r,h -0.1 FIG. 1.1. Atom-atom potential curves for H...H, H...C and C...C atom interactions, (a) the rigid sphere model [3]; (b) the Kitaigorodski AAP/2 model [3]; and (c) the Williams AAP/1 model [85].
#(2) =
oo0for r>r 0 for r
The various analyticalexpressionswhich are described in detailin [63,80] are used for atom-atom functions Spq(rpq). The modified Buckingham potential (6-exp) (the originalBuckingham potentialhas the term (1/r)Bexp(-ar))is the most widespread ~(r) =
----A+ Bexp(-ar) r6
(I.96)
44
ZHIZHIN A N D M U K H T A R O V
The Lennard-Jones potential is also well known (6-12)
r
=-~+rl
B 2 9
(I.97)
The first term in the expressions for these potentials imitates the dispersive London forces, the second one imitates the molecular repulsion at the overlap of their electronic shells. The potential in the form of (6-exp) is explained more by the theory, and its application leads, as a rule, to more precise results. The potential (6-12) is more convenient from the point of view of calculations (the term B/r 12 is chosen only from these considerations). The potentials (I.96) and (I.97) are often expressed by the equilibrium distance r 0, by the potential well depth e and by the value of the second derivative D O = r
For example, for the Buckingham
potential [3,80]
r
= 6 _ a r 0 - r--'~ +
Oe
/ (I.98a)
DO
6ca a r o - 7 r0 ar 0 - 6
~'~ "=" ~
(I.98b)
Formula (I.98) is convenient because the different parameters determine to some extent the different crystal properties: r 0 determines the unit cell parameters; e determines the sublimation energy, D O determines the vibrational frequencies and the elastic tensor. Regarding the parameter r 0 as the main one, Kitaigorodskii has suggested the one-parametrical universal potential for all atoms independent of the molecule, in which they are present [75,76]
0
+ 4 , 2 . 1 0 4 exp(-13,6 r ) ro
(I.99)
This potential is characterized by the same depth e of potential well e = -0,067 kcal/mol for any type of interacting atoms. Usually r0/2 is larger by approximately 15% than the van der Wards radius [3]. Kitaigorodskii et al. have compared the xray analysis data for some hydrocarbon crystals extrapolated to the data obtained at T = 0~ with structures calculated by a minimization of lattice energy u(a,~,T,a,b,c,x,y,z,~,XJl) as a function of the unit cell parameters and molecular coordinates. They have determined r 0 for the atoms of carbon and hydrogen [3,81]
VIBRATIONAL
SPECTRA OF ORGANIC
rocC = 3,8,~,; rortH= 2,8A;
CRYSTAI~
45
roo I = l(roc C + roHH)= 3,3 ,~
The curves obtained by Kitaigorodskii, and his colleagues are shown in Fig. 1.1. Afterwards, the universal potentials were determined more exactly by the variation of parameters e [ 8 2 - 8 4 ] . Another approach to the AAP par~meterization in the form (6-exp) was suggested by Williams [85]. The data concerning structure, sublimation energy and elastic constants of 9 crystals of aromatic hydrocarbons were used for fitting the parameters. Some parameters were fixed: the value aCX~ was determined from the data of graphite compressibility; aHt t and BHH were determined from the wave mechanical calculations of the dimer (H2)2, and OCH was calculated using the formula 1/2(aHH + aCC). The various properties of crystals were expressed by parameters Apq and Bpq in the form of linear equations. For example, the lattice energy AH0sub = Ulet t = AH H S(6 ) + BHH ~(exp) riri ~HH + "'" where
S(6)
1
HH = - ~
~:
~:
6
1
rd PHqH rpHqH
9
The structural data were used in the form of equilibrium conditions
~ULett = AHH ~a exp
~}S(6)
:~(exp) rln + BH H ~ H~}a H +... = 0 ~}a
The obtained redistributed system of 77 equations which are the functions of 5 unknowns A H H , A C H , ACC , BCC , B C H was solved by the least squares method. W h e n the lattice sums SHH and SCH were calculated, the interaction centers were not positioned on the nuclei of atoms of H; they were shifted to atoms of C over a distance of 1.047 ,/L Thus, one tries to take into account the electron density distribution of C-H bonds. The potential curves obtained by Williams are shown in Fig. 1.1;the parameters are given in Appendix (AAP/1). The parameters of A A P were obtained later by the identical method of fitting them to the properties of seven crystals of aliphatic hydrocarbons (AAP/4) and of fitting them to properties of both aliphatic and aromatic hydrocarbons (5 fitting parameters, 108 equations) at the same time [86].
ZHIZHIN AND MUKHTAROV
46
The atom-atom potentials with the Williams and Kitaigorodskii parameters and with parameters obtained later in some other works were most widely used for the calculation of molecular crystal structure. They proved the rather high accuracy for the calculated unit cell parameters (-1%); the orientation of molecules was 2-3 ~ etc. (the review of works is given in Ref. [84]). The first quantitative calculations of lattice dynamics were carried out independently [24,25,87]. The calculations were carried out by Kitaigorodskii's method of AAP/2 in the approximation of rigid molecules by numerical differentiation of crystal potential energy. The calculations of frequencies and the external vibrations of both naphthalene and anthracene crystals were carried out. For example, the following results were obtained for librational vibrations of naphthalene (in cm-1): Experimental. 125 109 74 71 51 46
I 126 134 94 7O 61 55
II 129 139 92 77 62 49
where column I contains the results of calculations carried out with the assumption of the libration only with respect to inertial axes of molecules, column II contains the results of the complete calculation which showed that the effective librational axis forms an angle of 30 ~ with respect to the inertial axis. In Refs. [24,87,88] the identical calculations were carried out for the Grtineisen parameters and the Debye temperature of some molecular crystals (Table 1.1). Considering both the simplified potential and the limited accuracy of the earlier calculations, we suggest that the agreement of the experiment with the theory is very good. ARerwards, for a large number of crystals with different sets o f AAP parameters the calculations were carried out more precisely in the approximation of rigid molecules [89]. The results of the calculations with Kitaigorodskii AAP/1 and Williams AAP/2 parameters are given in Fig. 1.2, where the calculated frequencies are plotted against the theoretical ones. As is seen, although there is a big difference between the potential curves (Fig. 1.1) for two sets of parameters, the experimental data are in good agreement with the theoretical ones to the some accuracy of approximately 10 cm -1. The application of the AAP method demonstrates some advantages of the AAP model, namely, the comparatively small number of parameters, the universal reproducibility of various properties of crystals, and the transferability of potentials
VIBRATIONAL S P E C T R A OF ORGANIC CRYSTAI~
47
160
120
/~,
o
80 o ~o
40 0
/
0
a) 9
,
160 ,--4 o
t9 >
~
120 ~ offo 80
o2~
~
40 b) 0
40
80
120
160
e- v(cm-1), experiment --> FIG. 1.2. Comparison between the calculated and experimental frequencies of external vibrations for some hydrocarbon crystals [89]: (a) the calculation by the Williams AAP/1 method and (b) the calculation by the Kitaigorodskii AAP/2 method.
TABLE I.l GriineisenParameters and Deby e Temperature of Some Molecular Crystals Crystal
ULett
AHsubl
Naphthalene Anthracene Biphenyl Benzene Durene Phenanthrene Pyrene
16.5 23.0 19.2 10.1 19.0 18.8 20.2
17 23 19 11 18 21 23
Ocalc, K 125 135 125 117 115 112 113
Oexp, K 131 134 120 118 120 117 123
ZI-IIZIIIN AND MUKHTAROV
48
of related molecule crystals [3,80,90]. Following the works of Kitaigorodskii and WiUiams, a large number of AAP parameters of different kinds were suggested, namely, the AAP (6-exp), (6-12), (6-9) and so on. Their values were determined by fitting them to the static properties of crystals (for more details see Refs. [80,84]). A wide variety of different parameters means the ombiguity of the empirical tmr~metrization, and it is the consequence of the correlation between parameters, the use of different fitting methods and different experimental data. In particular, in order to decrease the number of fitting parameters one uses the different combination rules for mixed atom-atom interactions. For e_xomple, in the case of hydrocarbons / - -
1
8CH = ~/eCC eHH
eCH =
2 e c c eHH eHH +ecc
2 a c c aHH aCH =
aHH + aCC
and so on. The most often applied rules are [85,86] = 1
ACH
=
~/Acc AHH
B~
= IBcc
B
(I.100)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
49
As was shown in [80,93], usually the application of various combination rules has a negligible influence on the calculation results, but it can essentially influence the A A P parameter values. By calculating the structural features of crystals, the limitations of the AAP model were found; it proved to be insufficiently precise to consider some fine differences in the crystal symmetry, which must be known for the calculation by the A A P method [84,90]. In particular, in the case of benzene the A A P method predicts the orthorombical structure at high pressures; however, one actually observes the transition into the monoclinic phase [94]. For this reason, efforts were made to improve the model by considering explicitlythe electrostaticalinteraction with the help of the additional term of atom-atom potentials [95]:
~(lXl ~ ) -- qp qq rpq
(I.101)
which describes the Coulomb interaction of point charges qq and qp on atoms. The appropriate AAP are usually denoted (6-exp-1), (6-12-1), etc. In some works for determining the charges on atoms one uses the approximate quantum mechanical methods [80,96]. However, the concept of a point charge is physically uncert~n, and the application of such charges can destroy the consistency of parametrization [97]. The most consistent consideration of the electrostatical interaction was suggested by Williams [98] in the frame of the AAP model. He has regarded the charges on atoms as the additional fitting parameter. It was considered also that the dipole momentum of C-H bonds was almost constant in different molecules; the condition (qH" qc = Ae) was imposed. Some sets of AAP (6-exp-1) parameters for H and C atoms [95,98] were obtained by least-squares fitting them to the sublimation energy and the structural data for 18 hydrocarbon crystals (see Appendix 1). The AAP parameters for atoms of C1 were obtained in the same way [99]. Later, Williams and his colleagues [100] proposed to determine the charge on atoms by fitting to the electrostatical potential (surrounding the molecule) which was obtained from ab initio calculations by the self-consistent field method for molecular orbitals. This method was applied to atoms of fluorine and oxygen for determining AAP (6-exp) parameters which were purified from electrostatics [100,101]. It should be mentioned that the consideration of the electrostatic interaction by means of multiple-multiple potentials is inadequate in the case of sufficiently complex molecules, since in such crystals the distance between molecules is less than the size of the molecules. In particular, in [102] it was shown that the consideration of the quadrupole-quadrupole interaction leads to the noticeable decrease of the accuracy of crystal structure calculations. According to some papers [3,35,102], the calculations of structural properties
50
ZHIZHIN AND MUKHTAROV
have practically no influence on the calculation results, if the electrostatic interaction was explicitly taken into account. According to some other papers, [94,98] they improve the accuracy of calculations. The only proof of the necessity of the interaction consideration is evidently the correct prediction of the monoclinic benzene phase at a high pressure [94]. The other attempt of improving the AAP model was the system of universal parameters proposed in [103]; it describes beth the properties of separate molecules (a dipole momentum, the conformation, and so on) and the interaction between them (the EPEN model). In such a model the interaction centers are positioned not on the atoms, but on the "electrons". The centers have the positive charge on atom nuclei and the negative charge outside the nucleus. The latter have the values -e and -2e (one or two electrons). The bond "electrons" are positioned along the line between two atoms, unpaired ones and the ~-electrons are positioned outside this line. The "electron" coordinates are regarded as the model fitting parameters. The interaction between the "electrons" is described by potential
~ qpqq/BepqrA /qPqq The rules (I.100) are applied to the cases of mixed interactions. The advantage of this model is that it is universal for the description of various properties of molecules and crystals. In particular, any special functions for the hydrogen bond are not required. The essential disadvantage of this method is the large number of interaction centers. For example, in the case of ethylene we have 13 centers instead of 6 for the ordinary AAP. As was shown by the calculations of some crystal structures [100], during the procedure of the calculation with the help of the EPEN model one needs more than ten times the computation time of the previous computation time,J and the calculation accuracy was either the same or even worse in comparison with the ordinary AAP model. The alternative efforts are well known; they are undertaken to decrease the number of the interaction centers.
In Ref. [104] one used 6 interaction centers
instead of 12 for benzene; they were l ~ z e d at C-H bonds. One obtained rather good agreement between the experiment for the structure and the external vibrational frequencies of this crystal. The most extreme case was considered [105], namely, when the molecule was simulated by a single point which was localized at a distance R from its mass center. The coordinates R = (x,y,z) were regarded as the fitting parameters. This model was applied to the calculation of phonon state density and to the calculation of temperature dependence of tetramine heat capacity.
One has obtained very good agreement with the experiment.
The
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
51
disadvantage of such models is that they are not universal and cannot be applied to related crystals (for exsmple, they cannot be applied to the internal vibration calculations). Thus, at the present time the AAP model is the most optimal one. A wide variety of these model parameters (excluding above mentioned reasons) can be associated with the small sensitivity of crystal statical properties with respect to the intermolecular interaction. For this reason, it is very interesting to try and use the dynamical crystal properties determined by the second derivative of the potentials for the empirical model parameterization. Besides, the vibrational crystal spectra give valuable, real information, and they can be obtained experimentally with good accuracy. Such efforts were undertaken first by Dows [34] and Shimanouchi et al. [106]; they fit the parameters of the simplified potential, Be -ar (only the interaction between the hydrogen atoms was considered), to the values of Davydov's splitting of crystal internal vibrations of ethylene and to the frequencies of crystal external vibrations of benzene. Afterwards, Pawley [107] carried out the m o r e precise calculations in the approximation of rigid molecules; he fit the APP (6-exp) parameters to the experimental dispersional curves of external vibrations of a deuteronaphthalene crystal. He also pointed out that it is necessary to take into account the extern_a_!-internal vibrational interaction which can influence the results the of fitting. This was done by Pawley et al. [92]. They used the parameters of the external vibrations of naphthalene-d 8 measured at four points in the Brillouin zone for the parameterization of AAP (6-exp). The correction considering the external-internal vibrational interaction was applied to the values for these frequencies. The data concerning the crystal structure were also used. However, one fails to obtain any noticeable improvement of the agreement with experiment (in comparison with the initial AAP/2 parameters of Kitaigorodokii). In [108] the authors have tried to fit the AAP (6-exp) parameters to the external vibrational frequencies in the same way (43 frequencies were considered). However, in this case the frequencies calculated with optimized parameters practically coincide with the frequencies calculated by means of the initial Williams AAP/3. The most careful parameterization of the AAP model with respect to the external vibrational frequencies of a naphthalene-d 8 crystal measured at a temperature of 5~ using five points in the Brillouin zone was performed in [109]. The experimental data concerning the parameters of the unit cell, the molecular orientation and the lattice energy obtained at a low temperature were also used. In addition, the quadrupole-quadrupole electrostatic interaction was also considered. The influence of the nonadditive induced interaction (at k = 0), which provides the correction-3-4 cm "1 for translational vibrational frequencies, was also studied. The AAP parameters obtained as a result of the fitting reproduced most precisely
52
ZHIZHIN AND MUKHTAROV
the dispersional curves and the whole vibrational spectrum of the naphthalene crystal [109,110]. However, the disagreement of some calculated frequencies with the experimental ones was not decreased. The authors of [109,110] think that it is the limit for this type of potential. The aforementioned examples of fitting the parameters to the properties of only one crystal is of limited significance, since one of the most i m p o ~ t properties of AAP (vamely, their transferability to some other crystals) is lost. For example, the AAP parameters obtained from the external vibrational frequencies of benzene describe the lattice dynamics of this crystal better than the Williams' AAP/3; however, the calculations with AAP/3 proved to be more precise for other crystals [111]. Besides, the use of the additional "molecule-molecule" potential for describing the electrostatical interaction requires the individual fitting of parameters for each crystal. This considerably decreases the heuristic worth of these calculations, and it is very essential for the objective interpretation of vibrational spectra. In our opinion the tendency to have the most precise coincidence of the calculation with the experiment in each case is confusing considering the physical uncertainty of AAP. It is more i m p o ~ t that the empirical potentials with minimal parameter number describe the properties of a large number of crystals with reasonable accuracy. To achieve such a result one must simultaneously use the dynamical properties of some crystals for determining the parameters. This was first done in [112] where the optical frequencies of externM vibrations of four crystals were used for determining the AAP (6-exp) parameters of chlorine atoms [CeCI6; 1,3,5 - C6H3C13; p - C6H4C12 (two phases)]. The procedure of fitting consists of the calculation of the Jacobian of the J-matrix, whose elements are the derivatives of the frequencies and other parameters with respect to potential parameters: ~Xi Jij = "Opj (1.102) where Pj are the AAP (6-exp) parameters
X i =
ULea = AH
is the sublimation heat
3QeX = 0
is the equilibrium condition
O~exp
is the external vibrational frequencies
One uses 35 frequencies, 15 equilibrium conditions and four values for the sublimation energy. The parameters A and B are varied for the C1--'C1, CI--'C and
VIBRATIONAL SPECTRA OF O R G A N I C CRYSTAI~
53
CI'"H interactions. The Williams AAP/1 were used as the parameters for the C---C, H--'H and H-'-C interactions. The fitting was performed by means of an iterative least-squares method:
(I.103) where "~/ is the weight matrix. A strong correlation between parometers was observed. It was also noted that if one preferred the equilibrium conditions, the fitting is improved, i.e. the addition of vibrational frequencies has no decisive influence. The AAP parameters obtained reproduce the frequencies with an error less than 4%, the orientation with an error of 1~, and the positions of their centers with an error of 0.01/~. Applying these AAP parameters, the calculations of the external vibrational frequencies of other chlorosubstituted benzene crystals have been made, and they have shown good transferability of parameters in these crystals [36,113]. An attempt [114] was made to improve the AAP model for the chlorosubstituted benzene crystals by using the potentials (6-exp-1). In performing the fit, the parameters of the unit cell were also varied. The external vibrational frequencies were not used directly, but they serve as a criterion of the fitting. A better agreement with the experiment was obtained. It was also found that the consideration of electrostatics] interaction had practically no influence on the frequency values. We note that the authors of [73,115] have arrived at the same conclusion by solving the dyrmmical problem for some other crystals. The above mentioned method was later used for obtaining the AAP parameters (6-exp) of Br atoms [116,117]. The problem of determining the AAP parameters from the external vibrational frequencies was investigated most carefully and precisely by Williams and Starr [118]. Excluding the statical properties of 18 hydrocarbon crystals (118 equations) [98] used before, 58 frequencies of the external vibrations of five crystals were added during the procedure of the least-squares method, i.e., 176 experimental conditions were used for determining 5 AAP (6-exp-1) parameters. The AAP parameters obtained by fitting only with the use of static properties were chosen as the starting ones. As a result, the AAP parameter values that were very dose to the starting ones were determined; the calculations with the latter practically led to the same frequencies as before fitting. Williams and Starr have come to the conclusion that the potentials adequately describing the crystal structure (i.e., satisfying the equilibrium conditions (I.93) and (I.94)) will also be good in describing the dynsmical properties of these crystals. The results of this work and of [112] raise doubts about the expediency of including the frequencies
54
ZHIZHIN AND MUKHTAROV
into the procedure of parameter fitting. Evidently, the empirical AAP in the form of either (6-exp) or (6-exp-1), which are optimized with respect to statical properties, reach their limits in describing the molecular crystal properties, and they cannot be improved by using the new additional experimental data. The disadvantage of the empirical parameterization of model potentials is that they are based on the approximate calculations of the crystal properties. In particular, some thermal effects (namely, the influence of zero vibrations on structural parameters, the influence of ~nharmonicity on external vibrational frequencies, the influence of nonadditivity of intermolecular interaction forces and so on) are not considered. The parameters obtained by fitting "absorb" these effects. This leads to the different values of AAP parameters obtained for the same crystal at different temperatures [109]. Besides, by fitting with consideration of macroscopic crystal properties, the different types of interactions are mixed, and the separate atom-atom potentials and their parameters lose their initial physical sense. All types of interactions are effectively taken into account in the value of the potential independently of their inclusion in the potential terms [90]. From this point of view, the justification of the AAP model can be, in general, the consideration of principles of the close packing of molecules which are the consequence of this model and the agreement between the calculated crystal properties and the experimental ones. Nevertheless, in spite of its disadvantages the AAP method is now the only model of the intermolecular interaction permitted to quantitatively describe the properties of composite systems such as organic compound crystals. Recently, the source of determining the model potential parometers, i.e. the direct nonempirical energy calculation of the nonbonded interaction of two molecules (the review and the details of ab initio calculations are contained in [63,66,80,119]), becomes available due to the development of the computational technique and the intermolecular interaction theory. Such an approach was first realized by Clementi [120] who had carried out the variational calculations of the potential surfaces of interactions between ~mlno acids and other biologically important molecules and the water molecule in the Hartree-Fock approximation by applying the supermolecular method. The results of these calculations were appro~m~ted by means of AAP (6-12-1) and applied to the investigation of a protein water solution. The serious disadvantage of the nonempirical calculations carried out by Clementi was the absence of dispersional interaction consideration
[80]. The calculations of this interaction were rather precisely performed by ab initio in [121] for 7 dimers of comparatively composite molecules of benzene, striazine, etc. One has used the Rayleigh-SchrSdinger theory (i.e., the exchange
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
55
interaction was not taken into account). The interaction operator was expanded in a series of multipoles (R = 7.9 10.6 ~):
AE(2) -_
C6 R6
C8 R8
C10 R 10"
(I.104)
The calculation results for 8400 randomly chosen configurations of dimers were used for fitting the parameters A for atoms of C, H and N. The statistical analysis shows that in most cases the discrepancy between the dispersional energies of the AAP is ab irdtio no more than 10%. The authors of [121] come to the conclusion that the dispersional interaction is sufficiently precise when described by the AAP method. The parameters Apq determined in this work showed the best agreement between the tetracyanethylene lattice dynamics calculation and the experimental results [122]. The identical calculations of electrostatical interactions performed in the first order of the perturbation theory showed that unlike the dispersional type, this interaction is i rm__dequately described by the model of point charges on atoms [123]. The paper by van der Avoird et al. [124] is of great interest, because when the ab initio calculations of potential curves for 8 ethylene dimer configurations were performed, all types of interactions were considered. The calculations were carried out by means of exchange perturbation theory with the use of sufficiently precise wave-functions of an ethylene molecule. The interaction energy of the first order was represented in the fore AE(1) = fAE (1) + AE(p~I)n ~ + AE (1) exch j mult where AE(1)exch and AE(p~I) n are the exchange and electrostatical energies of molecular repulsion at the overlap of their electron shells, respectively; AEmult (1) is the energy of the electrostatic interaction at large distances. The dispersional energy AE(2) was calculated in the second order of the perturbation theory without the consideration of the exchange. According to the evaluations by the authors of [124], the error of ab initio calculations for AE(1) was equal to -10%, and for AE(2) it was equal to -20%. The calculation results were used for fitting the AAP (6-exp-1) parameters where each term of AAP was fitted separately qPqqrpg --# AE(1)mult" Bpqe-~PqZPq "-> (AE(1) - AE(1)mult);
Apqr 6 --> AE(2 ). Pq
It was shown that the dispersional interaction is very well reproduced by the AAP functions A p q / r p6q 9 9t~s result verifies the conclusion made by the authors of -[121,123]. For the electrostaticinteraction the point charge model proved to be inadequate. Some fittingimprovement was achieved by the displacements of point
56
ZHIZHIN AND MUKHTAROV
charges from atoms to some position on a molecular plane which was determined in the fitting procedure. The repulsion interaction is reproduced considerably worse than the dispersional one; however, the discrepancy with ab initio calculations is not so large, as in the case of the electrostatic interaction. Thus, the separate term functions obtained by AAP have a certain physical sense, since they were fitted independently. The ethylene crystal structure calculated with "nonempirical" AAP and the externM vibrational frequencies led almost to the same agreement with the experimental as in the case of the empirical Williams AAP. In calculating the external vibrational Raman spectral line intensifies and anharmonic properties of this crystal the agreement with the experimental proved to be even better than for the empirical AAP [125]. The identical calculations were carried out by van der Avoird et al. for the dimer of (N2)2 [126]. The obtained "nonempirical" AAP (6-exp-1) very precisely reproduced the properties of the crystal a-N 2 [127,128]. One should mention the recent paper [129] in which the variational calculation of the energy of the nonvalent interaction of two benzene molecules was performed by the supermolectdar method. The dispersional energy was separately calculated by the approximate configuration interaction method. The calculation results were approximated by multiparametric AAP functions of the form A
B
C
D
E
0(r) = -r- + ~'~ + ~'~ + ~-~ + r l 2 9
(I.105)
Unfortunately, the AAP (I.105) were not applied to the calculation of any properties of the benzene crystal, which are the most sensitive to intermolecular interaction features. In Fig. 1.3 the comparison between the interaction energies for two configurations of the dimer C6H6...C6H 6 calculated by ab initio and by means of the empirical AAP is shown. It is seen that the latter reproduces very well the interaction potential for the configuration (b) which has the deepest van der Waals minimum, and it is inadequate to the configurations with the large molecular repulsion. Evidently, this provides the successful application of the empirical AAP, since the presence of neighboring molecules with a large repulsion in a crystal is scarcely probable. The empirical fitting of AAP is actually performed not for any orientations of molecules, but for separate, more convenient ones. The results of wave mechanical energy calculations of nonvalent molecular interactions carried out in the recent time [120-129] can be regarded to some extent as the theoretical justification of the AAP model. It is this fact that leads us to a further appearance of more improved universal and transferable models of the intermolecular interaction. The large number of papers concerning the molecular crystal lattice dynamics are dedicated to the use of vibrational spectra for the parameterization
V I B R A T I O N A L S P E C T R A OF O R G A N I C C R Y S T A L S
57
AE, kcal/mole 6
2 0 1
-2
2
2 6 AE, kcal/mole
~
10
R,~
R
y 2'
'
6
" i~o- R, i
FIG. 1.3. Dependence of the potential energy of the nonvalent interaction of two benzene molecules on their mass center distance [129]: (1) the ab initio dependence and (2) the calculation by means of the empirical AAP.
and the verification of different model potentials of the intermolecular interaction. The application of well known and very popular AAP models for the interpretation of spectral data was less considered. In our opinion, this can be explained by the absence of the systematic research which demonstrates thoroughly the possibilities of the calculation by the AAP method for obtaining the necessary information from these data. Consider first the application of spectral data and the calculations by the AAP method for the determine tion of crystal structure, when either its direct determination is complicated or it is impossible. As was mentioned above (see Sec. I.B), the group theoretical analysis of the tLqm~n and infrared spectra permits the establishment of the factor-group of the crystal and the number of molecules in the unit cell. On the other hand, by using these data and by the AAP method, one can
58
ZI-]TZT-IIN AND MUKHTAROV
calculate the possible packing of molecules in crystals, which correspond to the crystal energy minimum. By solving the dynamical problem for every possible packing and comparing the calculated frequencies with experimental ones, one can finally choose the crystal structure which is the most similar to the real one. This approach was first used for the improvement of the crystal structure of ethylene. In the first X-ray research [130] performed with low accuracy, one has established the space group Pnnm, z = 2(C2h) and d e t e ~ i n e d only the coordinates of carbon atoms. However, in [131] the other space group P21/n, z = 2(C 1) was suggested; it is in better agreement with results of polarization infrared measurements with monocrystals. Subsequently, Dows [34] has analyzed qualitatively the molecular packing in a monoclinic crystal. He shows the possibility of two structures which differ from each other by the rotation of the molecules around their CfC axes, and he tried to choose the most probable one by carrying out the simplified calculations of the Davydov splitting of the infrared absorption band by the AAP method. The possibilities of two molecular packings was then verified in [132,133] by the direct AAP calculation of the structure. The final choice of the ethylene crystal structure was made on the basis of the lattice dynamics calculation [132,133]. The Raman band intensities proved to be the most sensitive ones to the molecular packing; they were calculated in the approximation of an oriented gas and libration frequency. The ethylene crystal structure found by means of both the spectral and calculation investigations was completely verified later by direct X-ray and neutronographic measurements [134,135]. The analogous research was made in [136], while determining the crystal structure of 1,2-diiodobenzene. Unlike ethylene, the space group of this compound was ambiguously established. The only parameters of the unit cell that were known were obtained by measurements using the method of X-ray structural analysis, and the factor-group C2h, z = 4 (C 1) which was determined from spectral data. The crystal structure which was the closest to the real one was determined by the AAP calculation method of the optimal packing and external vibrational frequencies for all possible space groups C~h - C~h. From other studies it should be pointed out that there has been a determination of the low temperature phase of the 1,2,4,5-tetrachlorobenzene structure [36] and also, the determination of the two crystal phases of n-butane [137]. In the last case one uses only the spectral research data and the computational results by the AAP method. The form of vibrations allowed to describe each line of a spectrum as the representation of the translational, librational or internal vibration in a crystal is very important; one can say the same thing about the normal vibrational frequencies, which are given by solving the dynamical problem. As a rule, the form of vibration is less sensitive to the choice of the AAP model in comparison with the
VIBRATIONAL
SPECTRA OF ORGANIC CRYSTAI~
59
frequency values [89]. Consequently, it describes rather precisely the real situation concerning the molecular displacement at normal vibrations. This is verified by a coincidence between the external vibration eigenvalues (obtained by the AAP method and found independently of the nonelastic scattering intensities of neutrons [138,139]) and the _R~man band intensifies of naphthalene and anthracene crystals [1401. The calculation of frequencies and normal vibrational forms by means of AAP methods permits one to perform the correct interpretation of spectra from the point of view of vibrations of molecular chain or dimer complexes in the hydrogen bond crystals [10,141]. By means of such calculations it is possible now to assign the lines to the symmetry types in polycrystalline spectra and to also define the band frequencies allowed by the selection rules, but not present in the spectrum due to their weak intensities or a random degeneracy. The analysis of review data and our own calculation results show that the AAP method predicts the external vibrational frequencies with an accuracy to ~10 cm -1. In general, it is not sufficient for unambiguous assignment of the lines. However, the probability of the correct interpretation of spectra can considerably be increased, if one calculates the line intensitiesusing the normal vibrationalform. Thus, our analysis has shown that at the present time the most optimal model of the intermolecular interaction, which is necessary to calculate force ~. constants kex,~ ~ \ l l J z~, is the method of atom-atom potentials whose parameters are determined by sivdultaneous empirical fitting to structures and sublimation energies of some crystals. The consideration of the electrostatical interaction in the approximation of point charges on atoms considerably complicated the calculations due to low convergence of the appropriate Coulomb lattice sums [142]; it is not in agreement with wave mechanical calculations. Taking into acr.ount all reasons, we suppose that the AAP of the form (6-exp) are the preferable ones because they are simple and effective. These AAP were proposed for example by Kitaigorodskii and Williams in their earlier works [81,85,86]. The verification of the "nonempirical" AAP method proposed at present by means of lattice dynamics calculations is of interest.
D.
Typical Cases of Disorder in Molecular
Crystals and
Their Representation in Vibrational Spectra In molecular crystals there are many ways of distorting a regular threedimensional molecular packing: orientational,positional,conformational disorders existing in stationary and dyr~mical versions, lattice defects in the form of vacancies, admixtures, etc. [143]. The presence of any disorder in a crystal first leads to the symmetry distortion and, consequently, to the withdrawal of the
60
ZHIZHIN AND MUKHTAROV
spectral limitation, i.e. selection rules with respect to a wave-vector, exclusion rule, etc. The most complete representation of disorders in vibrational spectra can be obtained by investigating the crystals with simple defects of known nature. First of all, the finiteness of crystal sizes relates to such defects. The influence of a surface on lattice dynamics was first studied in the works by Lifshitz [144-146], where the existence of optical surface vibrations, which can appear in the form of additional lines in the Raman and infrared spectra, was predicted. In the general case the decrease of crystal size leads to the change of a lattice constant, heat capacity, the phonon free path [147,148] and, consequently, to the change of low frequency vibrational spectrum parameters. The influence on a Raman line width of grinding a naphthalene crystal to a powder was studied in [149]. It was found that only grinding to a very small particle size (10 -7 m) by acting with large energy flow waves leads to noticeable changes in the Raman spectr~m (namely, to line broadening, their shifts to the excitation line and to the appearance of an intensive continuous background). Thus, the influence of a finite crystal size proved to be negligible when performing the experiment at the ordinary conditions. The experimental investigation of the influence of defect substitution (or inhomogeneous doping substitution) on distorting the crystal transformational symmetry on its vibrational spectr~lm is discusssed in [150] for mixed naphthalene~-naphthol crystals. It was shown that in the region of reciprocal solubility of components the changes observed in the "externar' vibrational Raman spectra are comparatively small; this proved that from the authors point of view the mixture lattice is analogous to the crystal lattice of a dominating component. The noticeable broadening of "external" vibrational lines was observed, if the concentration of impurities was increased. The concentration of impurities of the order of several percent or less do not essentially influence line parameters in a vibrational spectrum in the "external" frequency region, and probably the crystal packing. This conclusion verified the experiments for studying the concentration dependence on the Davydov splitting [60]. However, even the presence of small concentrations of impurities in a crystal can considerably influence the phase transformation kinetics and the appearance of different pretransition effects. For this reason, the careful refining of samples is necessary during the research of these phenomena. The study of partially deuterated crystals is of great interest. The results are often used for the interpretation of the multiplet structure of internal vibrational lines [151-154]. These compounds also attracted our attention as the models for a complicated disordered system. On one hand, the partial deuteration distorts the molecular s ~ m e t r y , and on the other hand, it leads to an orientationally disordered structure, since every similar molecule in a crystal can have some possible orientations.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
61
The investigation of external vibrational spectra of "pure" ethylene crystals and of ethylene crystals formed by partially deuterated molecules showed that the spectra of these crystals can be interpreted by assuming some effective site symmetry, which is incompatible wit~ molecular symmetry and corresponds to the ordered ethylene crystal [154]. The analogous investigations of other materials show that the selection rules, in the optical spectrum of crystal external vibrations consisting of deuterated molecules, are the same as for "pure" crystals. These investigations show that the disorder of the molecular orientation in crystals (if it is not accomp~mied by the noticeable change of the intermolecular interaction energy) does not lead to the essential changes of the vibrational spectrum in the external vibrational region. The last conclusion is very important for the interpretation of external vibrational spectra of organic compound crystals, since the defects in the form of irregularly oriented molecules are a sufficiently popular type of distortion to their crystal structure [143]. There are the large number of orientationally disordered molecular crystals in which every molecule has one of some possible orientations; moreover, the energy barriers between these orientations are so large that they completely exclude the molecular jumps from one orientation to the other in the whole temperature range of a given crystal phase. These crystals are called the crystals with rigid disorder [3] or the statically disordered crystals [143]. One of the most widespread cases of this disorder is the formation of the centrosymmetrical crystals by molecules without symmetry centers. The pbromochlorobenzene, p-nitrochlorobenzene and azulene crystals and other ones are related to these structures. The analysis of molecular packing in these crystals [3] shows that, as a rule, the different mutual molecule positions in the unit cell are inconsiderably different in energies. The interesting case of the rigid disorder was observed in isomorphic crystals in the a-phase of bromoform and iodoform in which only the positions of the atoms of C and H are disordered, but the atoms of Br and I form the regular lattice. The polar space group P63, z = 2 with the same direction of all molecular dipole momenta corresponds to the ordered position of molecules in crystals in the a-phase of bromoform and iodoform. The nonpolar structure characterized by the effective space group P63/m, z = 2 [155] corresponds to the disordered positions of that molecule. In the last case the direction of molecular dipoles are parallel or antiparaUel to a crystal axis. Although the spectra of these compounds explicitly demonstrate all features of disordered structures (especially, in the low-frequency region), in a large number of studies their interpretation was performed on the basis of the P63 structure; this leads to essential contradictions in the assignments of external vibrational lines of
62
ZHIZHIN AND MUKHTAROV
the Raman spectra and of absorption spectra in the far infrared region [156-159]. It is evident that for the correct interpretation of these disordered crystal spectra it is necessary to proceed from some effective average symmetry as in the case of partially deuterated molecules. It is then necessary to perform a more careful investigation of spectra of external and internal vibrations of both the a-phase of bromoform and iodoform. One has not observed the disordered phase in chloroform (among halogen substituted methanes); however, one cannot exclude the possibility that it exists in a narrow temperature range in the vicinity of the melting point. In [161] the qualitative comparison between the low-frequency Raman spectra of a statically disordered bromochlorobenzene crystal and phonon densities of states g(~) = .Zgi(co) calculated by Dean's method with the use of AAP was performed. It wa~ established that the disorder in different crystal directions is differently displayed in different parts of the spectra. The appearance of additional lines is caused by local vibrations of certain clusters whose probabilities of appearance in a structure can be estimated by spectral intensities. As a rule, the Raman spectral lines of statically disordered crystals are considerably wider than the lines of ordered crystal spectra [156,157,161]. However, the line broadening is caused by both the phonon scattering at defects and the phonon-phonon scattering due to the vibrational anharmonicity. In [160] it was shown that the _l~man spectral line width is additive with respect to two mechanisms of phonon scattering. The separate contributions into the line width can be distinguished by studying its temperature dependence. In particular, the careful measurements of the L~man spectral line width (26 cm "1) of the statically disordered parabromochlorobenzene crystal showed that the width of this line is only of 0.6 cm "1 at 2~ i.e. the influence of statically orientationally disordered molecules on a Ramau spectral line width is very insignificant. However, the lines are considerably broadened in the low-frequency Raman spectra of statically disordered nitrochlorobenzene crystals [161]. This spectral difference is the evidence of a different character of the intermolecular interaction in these disordered crystals. In statically disordered structures the disorder does not depend on a temperature. In ordinary ordered crystals there are always the elements of more or less distortion of a structure. These defects (vacancies, dislocations) represent the thermodynamic equilibrium type of structure distortions, and their concentration does not depend on temperature. They are the indirect experimental data about the point defect existence in the form of irregular oriented molecules (for example, the photodimerization of molecules in the anthracene crystal in which the molecule in a regular lattice are packed in the unfavorable manner for this reaction [3]). In [162164] the model is considered in which the oriented defects can arise in the form of disordered molecule files beginning and ending on vacancies.
VIBRATIONAL
SPECTRA OF ORGANIC
CRYSTAI~
63
The presence of thermal defects must be displayed in the thermal dependence of vibrational spectra, in particular, in the region of external vibrations which are the most sensitive to the mutual molecular orientations. In [165-167] the external vibrational spectra of more than 20 molecular crystals were studied in a wide range of temperatures. The heating of crystals was accompamed by the tendency of forming the bands instead of lines which were grouped in pairs and relates to molecular librations with respect to the axes with one moment of inertia. These pairs merge simultaneously into broad bands and are fully smeared into the Rayleigh wing at the orientational melting. The observed temperature changes in spectra were accounted for by the appearance of an increased concentration of point defects with temperature in the form of irregularly oriented molecules. The authors of [168] have tried to explain the observed phenomena by the theory. They have obtained the expression for a contour of the external vibrational Raman spectral line. This expression was based on the one-dimensional model of a molecular crystal with point defects in the form of irregular oriented molecules and with the distance L between the defects which depends on a temperature and is defined by the condition of free energy minimum. The line width is determined by an opening of the Brillouin zone with scattering of wave vector IAKI~ 2 ~ / L ( T ) . The defect activation energy equal to 1.4 kcal/mol in a naphthalene crystal is determined by the comparison of the theoretical RamAn spectral line contours with the experimental ones. This value is considerably less than the value 21.7 kcal/mol which is obtained from the nuclear magnetic resonance data [169] in the vicinity of the melting point. One observes also the discrepancy between the theoretical and experimental dependencies of the line width. For this reason the observed thermal changes of the vibrational spectra should probably be assigned to the vibrational anharmonicity in most cases. By means of calculations in the AAP approximation, it was established [39,97] that the molecular reorientation barriers in crystals such as naphthalene, anthracene, phenanthrene, etc. are sufficiently large, and the sufficiently high vacancy concentration is necessary to form the point defects in the form of irregularly oriented molecules [97]. The molecules of the aforementioned crystals with disordered elements are in motions of the same type as in the ordered crystals.
However, if the potential
barriers dividing the different equilibrium molecular orientations in crystals are sufficiently small, the chaotic molecular reorientation, due to the thermM fluctuations, is possible in these crystals. The mean rate of reorientation is determined by the Arrenius equation Ua
1 v=-=Ae I;
kT .
64
ZHIZHIN AND MUKHTAROV
The presence of random orientations corresponds to the crystal disorder. The physical methods of the motion dynamics investigation (basically the line width measurements by the nuclear magnetic resonance method) show that the activation energy U a of reorientations is always more than kT; this excludes the completely free molecular rotation. The most real model of reoriented motion is the Frenkel model, according to which the time of allowed orientation in which the molecule performs the ordinary motion is considerably larger than the time of a molecular jump into another orientation. In the case of benzene and furane [170], for example, it was shown that the fraction of molecules in the jump state is negligible and has no influence on the crystal thermodynamics up to the melting point. In the case of symmetrical molecules all allowed orientations can be undistinguished; in this case there is no disorder from the thermodynamical point of view. The classical example is the benzene crystal [3]. At the range of 100 - 120~ the narrowing of the nuclear magnetic resonance signal occurs. This narrowing is the evidence of reoriented motion around the free molecular axis C6; however, the careful investigation of benzene crystal shows that there is no phase transition up to the melting point. Thus, the necessary condition for the phase transition of the order-disorder type is the thermodynamic distinguishing of the allowed orientation, i.e., it is necessary that the molecular state in any additional minimum was different; the basic state is defined by the crystal space group requirements. In other words, the n-fold degeneracy of a molecular state in a lattice mode must be released, the additional minima must be identical to the basic one, and the molecular symmetry group must not coincide with the symmetry group of allowed orientations. The "irregularly" oriented molecules localized at additional minima created, in the first place, the thermodynamic disorder and, in the second place, (being the source of phase failure), they caused the distortion of correlated vibrational motion of the rest "regularly" oriented crystal molecules [171]. In the infrared and l ~ m o n spectra this leads to broadening and a relative decreasing of phonon line intensities, and at the limit, when the orientation correlation is negligible, this leads to the complete smearing of the phonon spectrum of a disordered crystal modification. The assumption of the existence of rotational molecular mobility in crystals has appeared with the analysis of the thermophysical method data of the investigation of structure and compound properties: the measurements of heat capacity thermal behavior, the phase transition entropy and so on. Summing up these data, Timmermans has distinguished the special class, i.e., the plastic crystals, among a wide variety of organic compounds [43]. In a plastic phase the crystal has the high symmetry hexagonal unit cell (or the cubic one) whose symmetry is, as a rule, incompatible with the symmetry of its molecules.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
65
The main criterion of the assignment of crystals to the plastic ones is the small melting entropy (5 e.u.) [143]. This shows that in such compounds the orientation "melting" occurs before the "melting" of translational freedom degrees of molecules. The plastic crystal phase was called the rotationM crystal state by Kitaigorodskli [3]. The reorientation of molecules in the plastic phase are of a cooperative character. The comparison of the van der Waals sizes of a large number of molecules with the average distance between their centers obtained from the X-ray structural analysis data shows that, as a rule, the latter are smaller than the former, and that the molecule has no suffident amount of space for a comparatively free rotation. In the plastic phase there is the very high concentration of defects (one vacancy per 15 - 20 molecules). Undoubtedly, this makes the procedure of rotational reorientations easier. The Frenkel model is, consequently, quite adequate for describing the molecular motion in a plastic phase. The infrared and Raman spectra of plastic crystals are analogous to the liquid spectra [143]. If one formally applies the selection rules, the vibrational spectrum of the first order is prohibited, because the crystal symmetry is high. However, in the region of 100 cm -1 (in the vicinity of the Rayleigh line) one frequently observed 1 - 2 wide maxima [172-174] which increased in_intensities going to the phase transition temperature from above. At that temperature these maxima are sharply divided into a number of narrow lines corresponding to external vibrations of the ordered phase. In [175] the Rayleigh line wing appeared at the orientational melting of the crystal, and localizing near the nonshifted line (15 20 cm -1) is explained by the rotational diffusion of molecules. The rest of the part of the wing is described by librations in the equilibrium state. In this case the appearance of wide m~xim8 can roughly be explained by the violations of selection rules for the wave vector and by the appearance of all vibrations of the Brillouin zone in the spectrum [172,173,176], as it was in the case of statically disordered crystals. The appearance of internal vibrational lines in the infrared spectnm~ of 2,2-dinitropropanol in the plastic phase, which are prohibited in the molecular spectrum, is accounted for by the selection rule violation [173]. -
In [174] the authors have used the Brot-Lassier modified model applied previously for the description of internal vibrational line broadening for liquid molecules, in order to explain the absorption band in the far infrared spectra of thiophene and furane in the plastic phase. O n the basis of this model and the measurement of frequency and bandwidth of the far infrared spectrum it was shown that this absorption band is associated not with the jump reorientation of a molecular dipole, but with librational molecular vibrations in allowed orientations. One has determined the frequencies of librations (98 cm "1 for thiophene and 25 cm -1 for furane) and their relaxation times (6.8 and 6.1 psec, respectively).
66
ZH.IZHIN AND MUKHTAROV
It was found that the librational vibrations in thiophene and furane plastic phases were strongly damped, since the librational period is of the same order as the relaxation time of vibrations. O n the basis of Frenkers model [176] the general theory was developed. It relates the intensity of the depolarized Rayleigh line wing with the character of orientational order in plastic crystals. It was shown that in cubic and hexagonal plastic crystals the orientational order is characterized by two or three independent order parameters (the components of correlation orientation functions), respectively; they can, in principle, be determined by studying the dependence on a Rayleigh line wing intensity in the polarized emission. The quantitative data concerning the reorientational motion can also be obtained by means of the Fourier analysis of the Raman spectral line contour of an internal vibration [143]. The barriers (2.8 kcal/mol and 4.7 kcal/mol) and correlation times (1.1 psec and 1.1 psec) were thus determined by means of a Fourier arm lysis of the line contour of v (C-N) = 2250 cm"1 for succinonitrile and of v (CH 2) = 1450 cm -1 for cyclohexane [177]. It was noted that the reorientation barriers can actually be considerably larger than these values, since one has not taken into account the contribution of a vibrational relaxation into the line width, and one has used the approximation of independent molecular motions. To account for the reorientation molecular mechanism in a plastic phase one has used the quasielastic neutron scattering (the _anMogue of the Rayleigh one). Two models (the Frenkel model and the rotational diffusion model of the Brownian type) were suggested for the explanation of experimental data in a plastic phase of neopentane [143]. However, it was found that neither of these models can describe the experimental data. The dual nature of molecular motion in a plastic crystal was also mentioned in [143], whose authors have come to the conclusion that, most of the time, the molecules are not in the librational motion state, but in the reorientation one. At the present'time the question concerning the nature of molecular motion in a plastic phase has thus remained open. It is possible that the investigation of vibrational spectra in the vicinity of an anisotropic crystal - plastic crystal phase transition can clarify the situation, bemuse in this region one can observe the procedure of the transformation of the external vibrational spectrum into the Rayleigh line wing. With a decrease of the temperature the crystal has the transition of the order-disorder type from the plastic phase into the anisotropic one, in which the molecular orientations are fixed. Sometimes this transition occurs through some plastic phases and is accompanied by step-by-step "cooling" of molecular degrees of freedom. Conversely, the total reorientation disordering is achieved only in the high temperature phase. In the low temperature plastic modification the reoriented
VIBRATIONAL
SPECTRA OF ORGANIC
CRYSTAI~
67
T,K
3o4
3
o
3
3~176
_
~ _
II
,
FIG. 1.4. Temperature dependence of the low frequency Raman spectra of p-carborane polycrystal [178]: I, II - the plastic modification and I I I the ordered phase.
III
120
80
40 cm-1
motion has more or less explicit anisotropic character. The l ~ m ~ n lattice vibrational spectra of 1,12-dicarboclosodode-carborane(p-carborane) [178] are given in Fig. 1.4. This compound has two plastic modifications (crystals I and H) with phase transitiontemperatures at 240~ and 303~ respectively. The spectrum of low temperature modification with narrow, intense lines is typical for "rigid" crystal spectra; the Rayleigh wing, which has no structure in crystal I, is typical for the plastic phase spectrum with the totally disordered rotational reorientations; the presence of the second modification of "crystal" frequencies (50 cm -1 and 70 cm -1) (the low temperature one) in the spectrum proves the anisotropy of rotational reorientation molecular motion of para-carborane in this phase. The data of the X-ray structural analysis [179], of the calorimetry [180] and of the nuclear magnetic resonance [184] has proved that orientation molecular jumps around the axis which are close to the C-C axis are activated even in the lowtemperature anisotropic modification of 1,2-dichloroethane; this leads not to the f o ~ t i o n of an "isotropic"plastic crystal, but instead to the formation of a onedimensional one as the result of the phase transition.
68
ZI-IIZI-IIN AND MUKHTAROV
T, K ~ ~ ~ 2 3 T, K 5
J
J phase transition at 177~
J
l
7
i
100
'
cm-1
FIG. 1.5. _l~man spectra of the crystal lattice vibrations of 1,2-dichloroethane at different temperatures [182]. The temperature dependence of the 1,2-dichloroethane Raman spectra also demonstrates this fact in the region of external vibrations [182] (Fig. 1.5). The line at 122 cm -1 (at 4.2~ assigned to the librational vibrations around the axis which is close to the C-C axis [183], decreased in intensity and broadened more rapidly than other lines and is absent in the high-temperature phase spectnml. The similar behavior of the band at 75 an -1 (4.2 K) is also observed; it is assigned to librations with respect to the C-C axis in [123]. The phenomenon of one-dimensional disordering of rotationo_ 1 reorientations in normal paraphines is most widespread. For most of them one observes the phase transition into the hexagonal modification with the strong order of a long molecular axis position and with disorder of their azimuthal rotations in the vicinity of their melting point [3,184]. The special type of orientation disorder was found in molecular complexes with the charge transfer [182]. The peculiar feature of the compound structure is that alternatively succeeding donor and acceptor molecules are filled in quasihexagonal close packing. This leads to a situation such that the orientation disorder of one or two components is most probable at a molecular plane, since the molecular interaction in a file is much stronger that the interaction between molecules of neighboring files. For example, the naphthalene molecules (donor) in a 1:1 complex with 1,2,4,5-tetracyanobenzene molecules (acceptor) are disordered at a room temperature due to the existence of two possible orientations [185]; accord-
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
69
ing to nuclear magnetic resonance data [187] the barrier of 2.3 kcal/mol corresponds to jumps from one orientation to another by the angle of 36 ~ (which is smaller), and the barrier of 10.2 kcal/mol corresponds to the jump into the same orientation by an additional angle of 144 ~. The identical values for barriers are also obtained during calculation by the AAP method [186]. The presence of the order-disorder transition is found at 63~ in the investigation of the Raman spectra of this complex [187]. The disorder in molecular crystals can be achieved by disordering its separate fragments, i.e., as the result of methyl group reorientation. The reorientation of the CH 3 groups by means of the nuclear magnetic resonance method was studied only at a low temperature. The analysis of data has shown that all CH 3 groups are reoriented considerably faster than the transfer energy speed of molecular reorientation, i.e. the molecules can be regarded as unmovable ones. In [3] it was shown that the nuclear magnetic resonance data can be described by three models: by the disorder reorientations of the Brownian motion type, by the Frenkel model and by the tunnel effect. The internal rotation of CH 3 groups in molecular crystals leads to broadening of the CH-bond "normal" vibrational bands in infrared absorption bands. The CH 3 group reorientation barriers of some molecular crystals were found from the temperature dependence of the width of these absorption bands [188]. The rotational barriers of methyl groups in crystals were mainly determined by intermolecular forces. The CH bond vibration of the CHD 2 methyl group was used as a probe for the l ~ m a u spectrum analysis of the structure and the internal rotation in a- and ~phases of deuterated crystals of toluene C6H5CHD 2 and CeD5CHD 2. The fine structure of this vibrational line in the region of 2925 - 2950 cm -1 was observed. At low temperatures it was interpreted as a result of some reorientations of the CHD 2 group between potential mivima with different energies. It was also shown that even at very low temperatures the tunnelling effects are not displayed in the spectrum. By decreasing the temperature one observes the strong broadening and the intensity redistribution of multiplet components. These spectral changes cannot be described by the simple Frenkel model of the CHD 2 group reorientation. The useful information concerning the methyl group motion in a crystal can be obtained by the investigation of torsional vibrations of these groups in the lowfrequency region of the Raman spectra.
The assignment of these lines to these
types of internal vibrations is often contradictory and additional studies are needed. Another example of conformational disorder is the polyphenyl crystals. The molecules are not rigid in the simplest case of diphenyl; the phenyl groups are capable of rotating around the ordinary C-C bond. The c o n f o ~ t i o n of a crystal is in tension, and it is defined by a balanced equilibrium between the intra- and
70
ZI-IIZHIN AND lt~IKHTAROV
intermolecular forces. This is displayed in normal temperature increases by many tens of degrees of the diphenyl phase transition and of n-terphenyl ones at the pressure increase [190,191], and also in the change of the phase transition temperature of p-terphenyl at the exposure to the ultraviolet emission. At low temperatures the polyphenyl crystals have phase transitions accompanied by the rotation of phenyl groups at some angle, as a result of a balance failure between the intra- and intermolecular forces. The conformation of molecules is not planar in the low-temperature polyphenyl phases. In [193] the authors suggest that in diphenyl there is the doubling of the crystal unit cell. The torsional motion of phenyl rings causes the phase transition. In the case of the planar diphenyl molecule this vibration has the Au symmetry and is inactive in the Raman spectra. For the nonplanar molecules the torsional vibrations become active in the Raman spectra. For the phenyl ring torsional vibrations one must observe four components of factor-group splitting to which the frequencies 19, 24, 30, and 33 cm-1 (T = 4.2 K) are assigned; these results were obtained by the assumption that in the low temperature phase the unit cell contains four molecules. The study of polyphenyl crystals is of interest at high temperatures where, as a consequence of weakening the intermolecular interaction, the balance failure between the intra- and intermolecular forces must be displayed in the most prominent way. An interesting case of conformation disorder is observed in crystals with cyclobutane [194] and cyclopentane [195] molecules having the ring conformation in a tension; it is related to the "pseudorotation" of molecules in a crystal; then, the different ring conformations are not distinguished from the rotation of molecules as a whole by some angle; this leads to the additional broadening of the Raman spectral lines. The methods of vibrational spectroscopy are thus the effective way of studying the disorder in molecular crystals. However, the conclusions concerning the molecular motion in the anisotropic phase and the character of its structure change in a phase transition procedure (as a rule, obtained simply from the temperature evolution of spectra) are of sophisticated character, since the interpretation of vibrational transitions is either absent or is given on a basis of indirect data and, consequently, it is not free from errors. It should also be mentioned that most spectral investigations of phase transitions are reduced to the studying of peculiarities of systems before and after the phase transitions. The most important information can be received by studying the dynamics of the transformation procedure in the vicinity of a phase transition point. There have not been many investigations performed by different physical methods, including the vibrational spectroscopy methods. This is explained by serious difficulties in the experimental performance.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
H.
71
CALCUIATION OF FREQUENCIES AND NORMAL VIBRATIONAL FORMS IN THE APPROXIMATION OF RIGID MOLECULES
A. Calculation Technique for the Lattice Dynamics of Molecular
Crystals by Means of AAP
The lattice dynamics calculations by the AAP method are, in essence, nonempirical because the atom-atom potentials are transferable and universal. Nevertheless, the external vibrational frequencies calculated by the different authors using the same parameters differ from each other. For example, the differences between the symmetrical librational vibrational frequencies of naphthalene crystal calculated by the A A / 3 method are as follows (in cm -1) [1-3] Ref. [1] 114.3 80.9 52.1
Res [2] 104.9 75.7 51.5
Res [3] 105.8 78.9 52.3
with the differences reaching as much as 10 can -1. This is explained by both the application of different approximations and methods of dynamical matrix calculations and the different approach to the starting data choice (the molecular geometry, the crystal structure and so on). For this reason, we have obtained the detailed expressions for force constants k ,p, g, , and the choice of the optimal condition for the lattice dynamics calculation by the AAP method [4] was made. Consider the calculation of force constants (Eq. 1.25) by the AAP method. Taking into account the representation of the intermolecular interaction energy of a crystal in terms of effective paired potentials (Eq. 1.91), we get
k(ex ) (K: K'
~2U
(K,t)~(~',t') (II.I)
72
ZHIZHIN AND MUKHTAROV
ap,pp
l:
l"
K:T (II.2)
The constant (II.1) corresponds to the second derivative of the crystal potential energy with respect to the displacements of different atoms in the crystal, the expression (II.2) corresponds to the second derivative of the crystal potential energy with respect to displacements of single atoms in the isolated molecule (the "own" force constant [5]). To simplify the formulas we omit the indices (K,!) and denote the atoms of a molecule (K,!) with the subscript p and the atoms of a molecule (K',l') by the subscript q. Using the expression (I.95) of the intermolecular interaction energy of two molecules in terms of atom-atom potential functions, we get then the following expressions
k(ex> (K K') ~2~pq(rpq) (K,I)r (K',t') aP,[$q[l I' = 0Xap~Z~q '
=E Zq )r
ap,pp
~
(II.3)
~ p ; 8PP'
(II.4)
where rpq is the distance between atoms p and q in molecules (K,I)and (K',I'), and 8pp, is the Kroneker symbol. The derivatives are calculated for the equilibrium configurationof a crystal
,pq(rpq)
rpq
+ ~)Pq ~rpq Orpq
Ox~ ~Xpq = ~Pq ~)X~l~ p
Ox~k10Xap
(II.5a)
where d2~)pq (rpq) ! d, pq (rpq) I
%q
eq
~)~)q = dr2pq
eq (II.5b)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
73
The first term in (II.5) is often neglected (for example, during calculations by the FG matrix method); this can essentially influence the calculation results [6]. The distance between the atoms of molecules displaced from the equilibrium state can be expressed in the following form:
3 2 r2= Z (X~+xpa-x(O)qa -Xqa) a=l
(II.6)
where X pa (0) are the equilibrium atomic coordinates; and, by calculating the derivatives (II.5), we get the following expression for the force constants k(eX)ap,~t(: K')=l, ~(~-~-rpq ~q)Uqp-(a) n(~)9p- ~qrpq8a~ ' (K,l) ;e (~',t')
~
g" =-~ipp, ~_~ Z
~-~ k~;,~q( K (II.7)
where nqp(a)=(X(l~0) _ X(0))/qarpq is the a-component of the unit vector directed from the atom q to the atom p. In the case of AAP of the type (6-exp), for example, the derivatives ~pq and ~pq are of the form 6Apq ~pq=
- apq B pq exp(-apq rpq)
and =
_
42Apq ~ +
apq2Bpq exp(-apq rpq)
Putting the force constants from (II.6) and (II.7) in the expressions (I.46), we obtain the formulas for calculating the force constants on the basis of generalized coordinates and the dynamical matrix elements by means of AAP. The corresponding formulas are given in Appendix 2 in the matrix form which is convenient for computer calculations. The eigenvectors and eigenvalues were determined by the diagonalization of the dynamical matrix according to the Jacoby method using the standard algorithm.
ZHIZHIN AND MUKHTAROV
74
TABLE II.1 The Calculated Results of the External Vibrational Frequencies of the Napthalene Crystal in the Approximation of Rigid Molecules Rmax, ]k
v, c m "1
6
111.5
108.0
95.1
87.0
80.6
65.4
57.6
51.2
48.6
7
109.6
106.1
93.4
85.2
78.9
63.9
56.1
49.6
47.9
8.5
109.8
105.0
94.3
82.3
77.2
64.9
54.5
46.9
45.7
By calculating the dyv~mical matrix dements (I.45), it is necessary to limit the summing over the crystal lattice by some finite number of molecules surrounding the starting one (1.0). For decreasing the computation time of calculation by the AAP method it is convenient to restrict the lattice sums to a so called summing radius, Rmax, which shows the m~ximol atom-atom distance, rpq, considered in the calculations. To verify the convergence of the calculation results we have used the values of Rmax in the region 6 - 8.5 ~k, since in most studies the calculations were mainly carried out with Rmax = 4 - 6 ~k. The calculation results of the external vibrational frequencies of the naphthalene crystal in the approximation of rigid molecules are given in Table II.1 (the AAP/1 parameters). From the table it follows that, as a rule, the frequencies are decreased by 1-4 cm -1 by increasing the value of Rmax. T h e lattice energy is increased from 15.6 kcal/mol to 16.2 kcal/mol, i.e., it is more than 20%. The analogous calculations for the crystals of benzene show that by changing Rmax from 5 to 10 ~k the external vibrational frequencies are decreased only by 1-2 cm'l; at the same time the lattice energy is increased from 8.1 to 11.8 kcal/mol, i.e., it is more than 40%. The explanation is that the frequency values dependent on the second derivatives of the energy are mainly determined by the repulsion potential, which depends exponentially on the interatomic distance. It should be mentioned that at large Rmax the u n c e r t ~ n t y of the empirical potential curves is increased, since at their determination from the experimental data the parts of these curves which are close to the van der Waals miDimum are fitted (see Sect. I.C). By increasing Rmax the number of atom-atom contacts is also increased; this leads to appreciable timeconsuming computer calculations. Considering these factors for the calculation of the frequencies and vibrational forms, we have chosen the value of Rmax = 7 ~, which gives the frequency value error on the order I a n -1. For the estimation of a considerable part of the lattice energy which was not taken into account one can use the simple extrapolation method proposed in [7]. The calculation is performed up to Rmax- For interatomic distances which are more than Rma x the crystal is regarded as the continuous distribution of atoms of every type with the density
VIBRATIONAL SPECTRA OF ORGANIC
CRYSTALS
75
niz Pi= V where n i is the number of atoms of ith type in a molecule, V is the volume of the unit cell,and z is the number of molecules in the unit cell. The additional lattice energy can then be calculatedby the formula x 2x
z
Uadd= V ~~, nini' ~ ~ 0
f
.ii,(rii,)r 2, sinOdrii, dc~dO
0 Rm~
(II.8)
For example, for AAP (6-exp)
Uadd = "V- ~ nini' fi
~ {Xii'
(a2' R2max+ aii'Rmax + 2)e -eii'Rm~ -
Aii' 3R3max I
2~z
(11.9)
The second term in brackets gives the largest contribution to the additional energy. The calculationshows that even at Rma x = 5/~ the cryst~ latticeenergy calculated by means of this method differsfrom the precise calculationby no more than 1%. One of the main difficultiesin solving the dynamical problem by the A A P method is the choice of starting data concerning the crystal structure, which must satisfy the equilibrium conditions in (1.10) and (1.36) by defining the m i n i m u m of the crystal potential energy with respect to the unit cell parameters and molecular coordinates. If one uses the crystal structure determined experimentally from the calculation by the A A P method, these conditions will not be, in general, fulfilled, since the empirical potentials are only the approximate intermolecular interaction model. The homogeneous macroscopic tension and forces acting on molecules can lead to the instabilityof the crystal latticeand, consequently, to the appearance of imaginary frequencies from solving the dynamical problem. For this reason, during the successive procedure of solving this problem using the harmonic approximation,
it is necessary a priori to c~ctflate the crystal structure that satisfies the conditions in (I.10) and (I.36) for given empirical potentials. However, the frequencies calculated with such optimized structure will correspond to the crystal vibrations only at T = 0~ This scheme is not consequently useful for the interpretation of the vibrational spectra of polymorphic
76
ZHIZHIN AND U K H T A R O V
crystal modifications in which the phase transitions occurred. In the general case the equilibrium crystal structure is defined by the Gibbs free energy minimum at a given external condition, i.e., at a temperature T and at pressure P [5] G = ULatt - TS + PV
(II.10)
where S is the entropy and V is the crystal volume. The direct calculations of the structure of a number of molecular crystals carried out by the AAP method in [8-11] show that the changes of the entropy and crystal vibration energy determine mainly the values of unit cell parameters, and the influence of these factors on the orientation and coordinates of the molecular mass center is negligible. If the unit cell parameters are given, the molecular coordinates are defined only by the lattice potential energy minimum. The influence of thermal factors on the molecular coordinates is indirectly displayed as the displacement of the crystal potential energy minimum at its thermal broadening. At some nonzero temperature the vibrational spectrum of a crystal can be calculated by means of solving the harmonic dy~amlcal problem using the unit cell parameters given at the same temperature. This corresponds to the so called quasiharmonic approximation which describes the temperature dependence of external vibrational frequencies of the molecular crystal in an adequate manner as will be shown in Sect. II.C. The determination of the unit cell parameters by the direct mivlmization of free energy (II.10) is, however, related to very large amounts of computations [8-10]. For this reason their experiIhental values measured at some temperature [1,12,13] are usually used. The opinions of different authors concerning the equilibrium conditions (I.36) with respect to atomic displacements at fixed unit cell parameters are different. In most papers beginning with the Pawley paper [12] the preliminary minimization of the crystal potential energy with respect to the molecular displacements is regarded as a necessary step for the lattice dynamics calculation by the AAP method. In other papers, only experimental structural data are used in spite of the AAP parameter choice [14]. For the calculations by these two methods the approximate character of the empirical potential will be displayed differently in the external vibrational frequencies. In the first case it directly influences the frequency values, since the calculated molecular packing (unlike the real one) is used. In the second case, it leads to a situation such that the dynamical ~ t r i x
will no more be Hermitian, since the
generalized forces act on molecules in the equilibrium state. One can establish the most optimal method only by calculating the frequencies and by comparing them with the experimental data. It was the goal of our calculation of the external vibrational frequencies of a number of molecular crystals in the approximation of rigid molecules with the use
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
77
of both the experimental data and the preliminary structural optimization at fixed unit cell parameters. The dynamical matrix (k = 0) was, as a rule, antisymmetrical, if we use the experimental structure. For this reason, its -o
symmetrization by averaging nondiagonal elements was carried out during the calculation procedure. The lattice potential energy minimum as a function of six molecular coordinates was found by the Newton-Raffson iterative method according to the following scheme [15] q(n) ( n - l ) + Aq(n) v = qv
V= 1,...,6
(II.11) where U ' and lJ" are the column-vector and the matrix, respectively,
n'
~lv ~:lv' n
(II.12)
n is the iterative number, Aq -(n) =lAq(n)[I is the displacement of the equilibrium position of a starting molecule (1,0) at the n th iteration. The experimental molecular packing was chosen as the starting one. The derivatives U v and U~v, were calculated by the AAP method in the same way as the calculation of the dynamical matrix elements. Only the additional relation between the different atomic displacements in the molecule were considered; this relation follows from the crystal symmetry conservation (the appropriate formulas are given in Appendix 2). The calculation results from the AAP/1 and AAP/2 methods for the naphthalene crystal are given in Table II.2. In this crystal the molecules have the centrosymmetrical position; for this reason only the orientational molecular coordinates were varied during the structural optimization procedure. The unit cell parameters and the starting atomic coordinates measured at 300~ [16] were used. The external vibrational frequencies are changed by a value which is more than 30 cm -1, in spite of very insignificant change of the molecular equilibrium orientation with respect to their inertial axes U, V and W (I U > I v > I w)
AAP/I AAP/2
ev
~v
~v
0.77 ~ 0.52 ~
-0.13 ~ -0.27 ~
-1.96 ~ -3.12 ~
78
ZH ZHIN A N D M U K H T A R O V
TABLE II.2 The Calculated Results from the AAP/1 and AAP/2 Methods for the Naphthalene Crystal Exp. cm -1 Raman [17] IR [i8] 109 125 98
74 71 66 53 51 46
Optimized Structure AAp/2 ' AAP/1
135.7 124.2 78.9 87.5 72.8 51.7 42.4 58.9 45.9
Exp. Structure [16] A~/2 AAP/1 114.9 109.6 93.8 106.2
121.0 121.0 88.2 92.0 78.7 54.9 43.0 57.7 47.3
93.2
94.3
80.3 74.8 68.9 54.7 53.5 41.4
85.2
78.9 63.9 49.6 56.1 47.1
which is comparable with accuracy to the experimental determination of a structure. Our analogous calculations carried out for a number of molecular crystals have shown that the external vibrational frequencies calculated by means of experimental structures are, as a rule, in better agreement with the spectral data than those calculated by means of the optimized structure. In particular, the average discrepancies between the calculations (AAP/1 parameters) and the experiment are in the case of hydrocarbons which have the complete set of spectral data as follows (cm -1)
benzene naphthalene phenonthrene cyclohexane ethylene
Exp. Str.
Opt. Str.
3.7 6.0 5.9 7.9 7.5
6.1 9.0 5.7 10.4 8.0
Temp.; K 140 300 300 115 85
We have studied some crystals with the use of experimental structure, and only in the case of pyrene, tetroxocane and toluene do the violation of the equilirbium conditions (I.36) lead to the appearance of imaginary frequencies. The optimization of the pyrene crystal structure was performed with Rmax = 7 ~k (AAP/I) at fixed unit cell parameters; the experimental data were used as the starting structure [19]. It appears that the calculated packing differs very slightly from the experimental one. The pyrene molecules localized at general positions in a lattice
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
79
TABLE H.3 The Results of the Pyrene Lattice Dynamics Calculation for g-Modes. Exp. 1a
Exp. 2 b
Exp. 3 b
Exp. 4 c
Exp. 5 c
Exp. 6 d
95 95 77 (77) 67 56 46 41 (30) 30 --17
111 110 112 112 119 106 105 108 109 116 85 83 84 93 96 78 77 80 93 96 61 60 52 63 65 61 59 59 71 73 45 43 43 46 49 42 40 47 47 50 38 36 35 38 41 29 26 24 39 40 22 17 24 28 29 hnag !mag hnag 23 24 aThe Raman spectra, see Part IH, Sect. V.B. bThe calculation results with the experimental crystal structure (300~ at Rmax = 5.5 and 8.5/~, respectively. CThe calculation results with the optimized structure at Rmax = 8.5 and 5.5/~, respectively. dThe calculation results with the optimized structure, the optimized structure at Rmax = 5.5/~ being used as the starting one.
(the space group P21/C , z = 4) are shifted by the values -0,027; -0.014; and 0.068/~ and rotated by the Euler angles 0.00; -0.75; and -0.25 ~ However, the lattice dynamics calculation in the approximation of rigid molecules leads again to imaginary frequencies. The reason for this phenomenon is their positions in a false or local minimum of the potential energy to which the Newton-Raffson method is very sensitive. The formation of such false minimums can be caused by the restriction of lattice sums during the calculation of derivatives of the energy lattice. For this reason, the calculation of molecular optima_! packing and external vibrational frequencies were performed with the different values of Rmax = 5 - 10/~. It appears that the molecular packing for Rmax = 5.5/~ differs significantly from both the experimental (At = -0.225; -0.091; and 0.144 ,/k and 0 = -4.43; 0.71; and 5.06 ~ and the calculated structures for Rmax > 5.5/~. In this case all eigenvalues of the dynamical matrix have the positive values. The optimization of the structure at Rmax = 7/~ and 8/~ with the use of calculated molecular packing (at Rmax = 5.5/~) as a starting structure does not lead practically to a change in either the structure or the external vibrational frequencies. The results of the pyrene lattice dynamics calculation for g-modes are given in Table II.3 (the peculiarities of the lattice dynamics of this crystal are also considered in Sects. HI.B and P a r t 2, V.B).
80
ZHIZHIN AND MUKHTAROV
TABLE II.4 Influence of the C"'H Bond Length on the Calculated Values of the External Vibrational Frequencies of NaphthaJene dCH ffi 1.00 .~ AAP/2 AAP/1 99.7 84.7 86.7 64.9 59.3 57.2 45.4 43.5 36.6
97.7 86.2 85.9 70.7 56.6 62.5 43.3 46.0 39.8
dCH- 1.09/~ AAP/2 AAP/1 135.7 124.2 78.9 87.5 51.7 72.8 42.4 58.9 45.8
121.0 121.0 88.2 92.0 54.9 78.7 43.0 57.7 47.3
The solving of the dyImmical problem for tetroxocane and toluene crystals will be considered later in Sect. H.B and in Part 2 of Sect. IV.B. The abovementioned example of pyrene shows the close relationship between the description of the structure and dynamical properties of crystals by means of empirical potentials. Excluding the abovementioned difficulties of principles concerning the crystal structure choice, there are also difficulties associated with the uncertainties of hydrogen atomic coordinates. For example, the value of the C-H bond length dCH = 1.0 - 1.03/~, which is usually given by the X-ray structural analysis, is, as a rule, less than the value dCH = 1.09 ~ obtained from the neutron diffraction experiment. Such uncertainty in the localization of H'"H, H'"C, etc., atom-atom interaction centers noticeably influence the external vibrational frequencies as is seen from Table II.4 (naphthalene). By varying the C-H bond length in the limit of 1.00 - 1.10 A, one can determine its optim~! value for a given potential, at which the discrepancy between the calculated and experimental frequencies will be minimal. This actually means that the C-H bond length is regarded as the additional fitting parameter of the AAP model. These calculations were carried out for a naphthalene crystal and benzene crystal with the use of the experimental structures calculated at 300~ [16] and 140~ [20], respectively, with Rmax = 7 ~ and with the AAP/1 and AAP/2 parameters. The dependence of root-mean-square deviations for calculated and experimental frequencies on dCH is shown in Fig. II.1. The optimal value for the C-H bond length obtained thus proved to be equal to 1.050 ~ for AAP/2 and 1.054 for AAP/1. These values can be recommended for the lattice dynamics calculations of molecular crystals.
V I B R A T I O N A L S P E C T R A OF O R G A N I C CRYSTAI~
81
On the basis of our performed calculations we can come to the conclusion that the optimal starting data for the lattice dynamics calculation of molecular crystals in the "quasi-harmonic" approximation are the following: the summation radius which is no more than 7/~; the crystal structure determined by the X-ray analysis or by the neutron diffraction analysis at some temperature, excluding the C-H bond length which should be taken to be equal to 1.05 /~. The performance of preliminary calculations of the optimized molecular packing is basically necessary only in the cases when either the AAP model does not describe the equilibrium crystal structure in a sufficiently good manner or the structure data are absent.
B. Choice of the AAP Parameters for the Calculation of the External Vibrational Frequencies of Molecular Crystals In this section we consider the problem concerning the optimal choice of the AAP parameters for the interpretation of crystal external vibrational spectra. This problem is nontrivial, since at the present time one proposes a wide variety of sets of these parameters whose values differ essentially from each other (see Appendix 1). Their heuristic worth can be established only by the comparison between the calculated and experimental properties of crystals of a wide variety of compounds. The results of our calculations of frequencies and vibrational forms of 16 molecular crystals with the use of different AAP parameters are given in Appendix 3. The values of root-mean-square deviations ~ of external vibrational frequencies from their experimental values were calculated by the approximation of rigid molecules for a series of hydrocarbon crystals; these values are given in Table II.5. The Kitaigorodskii (AAP/2) and Williams (AAP/1, AAP/3, AAP/4) parameters and the parameters obtained by fitting them to the external vibrational frequencies of benzene crystal (AAP/8) were used. In the last case (as is seen from Table II.5) the use of the AAP/8 for the calculation of external vibrational frequencies of other crystals (naphthalene) leads to poor agreement with the experimental values. The fitting of AAP parameters to the properties of only one crystal violates their transferability properties. The remaining AAP/1 and AAP/4 parameters reproduced the experimental data with appro~mately the same accuracy (to -10 cm -1) independently of the use of a particular crystal for the fit of the AAP parometers (Table II.5, crystals Nos. 1 - 6 and Table II.5, crystals Nos. 7 - 10). This proved the good transferability of parameters of the Williams AAP and the Kitaigorodskii AAP between the hydrocarbon crystals. It should also be mentioned that solving the dyvamical problem for the cyclohexane crystals by means of AAP/3 and AAP/4, whose parameters were determined from the aliphatic hydrocarbons, reproduced the spectral data in a considerably better way than by using the calculation
82
ZI-IIZHIN AND MUKHTAROV
TABLE II.5 AAP Parameters for Some Aromatic Hydrocarbon Crystals No.
Crystal
Space Group
AAP/1
AAP/2
AAP/3
AAP/4 AAP/8
Naphthalene C10H8
P21/c z = 2(C i)
9.9
11.7
10.3
12.6
Benzene C6H e
P bca z = 4(C i)
4.4
5.3
4.7
6.0
Anthracene C14H10
P21/c z = 2(C i)
6.2
10.9
Phenanthrene C14H10
P21 z = 2(C I)
6.6
12.3
Chryzene C18H12
1 2/c z = 4(C i)
7.4
8.1
Biphenyl C12H10
P21/c z = 2(C i)
9.6
6.0
7
Ethylene C4H 2
P21/n z = 2(ci)
14.7
8
Cyclohexane C6HL2
C2/c z = 4(ci)
13.2
13.9
6.7
9
p--xylene C6H4(CH3) 2
P21/n z = 2(c~)
12.6
10.1
13.1
10
Toluene C6H5(CH 3)
P21/c z -- 8(C I)
10.1
6.7
2
4
10.1
6.9
(I) translational vibrations. (2) librational vibrations.
performed by means of parameters of AAP/1 and AAP/2 obtained from the properties of aromatic hydrocarbon crystals. In other words, the AAP are sensitive to a valent state of a hydrocarbon atom, and at the parameterization it is expedient to divide the hydrocarbons into the saturated and unsaturated ones. In the case of a crystal whose molecules contain not only the atoms of hydrogen and carbon, but also other atoms (N, S, O, C1, etc.), the problem of the AAP choice is considerably complicated, since the number of necessary parameters
VIBRATIONAL
SPECTRA
OF ORGANIC
CRYSTALS
83
TABLE II.6 Root-Mean-Square Deviations of the Calculated External Vibrational Frequencies from the Experimental Ones AAPII § + AAP/16 + AAPII7 (I, cm "1
8.6
12.6
AAPI2 + + AAPII6 20.1
+ AAPII6 8.2
AAPI3 + + AAPII7 13.9
sharply increases. For example, for the A A P of the form (6-exp) and molecules containing 2,3,4,...,N kinds of atoms the number of parameters is equal to 9,18,30 ...., 3N(N+1)/2, respectively. For this reason, the simpler methods of the A A P parameterization are necessary. For example, one can use the Williams or Kitaigorodskii parameters for H---H, C---H, and C'-'C interactions, and the parameters obtained separately from other crystals where the interaction of some type is determinative, for the interaction of other atoms X"'X. The fitting of two sets of A A P with respect to the mixed H...X interactions can be shown by means of the combination rules (see Sect. I.C). Consider the calculations of external vibrational frequencies of thiophthene (C6H4S 2) and chloroform (CHCI 3) crystals as examples. The thiophthene structure (space group Pbca, z = 4(Ci)) is determined at 300~ [22]. Eighteen external vibrations are active in the optical vibrational spectra (the interpretation of the lowfrequency R a m a n spectra of this crystal is considered in Sect. IV.C). With the calculation one uses the AAP/1, AAP/2, and AAP/3 for H'"H, C-"H, and C'"C interactions and the AAP/16 and AAP/17 obtained from the structural properties of orthorombical crystals S 8 [23,24] for S'"S interactions. The parameters of the A A P for mixed interactions of H'-'S and C-"S were calculated by means of the combination rules (Eq. 1.16). The root-mean-square deviations of exterrml vibrational frequencies obtained in the abovementioned way from the experimental data are given in Table H.6 (cm-1). It is seen that by using the (AAP/1 + AAP/16) and the (AAP/3 + AAP/16) one is able to achieve an accuracy compared with calculated results for hydrocarbon crystals (Table II.5). The chloroform crystal structure is determined at 88~ [25] (the space group Pnma, z = 4(Cs)). Eighteen vibrations are active in the spectrum (see Sect. III.B of Part 2). For the frequency calculations we have used the total sets of AAP/21 and AAP/22 parameters suggested in [26,27] for the description of the structure of chlorosubstituted hydrocarbon crystals and also the '~ydrocarbon" AAP/3 in combination with different sets of parameters of AAP/18, AAP/19, and AAP/20 for C1.'-C1 interactions obtained by fitting to the properties of chlorosubstituted benzene crystals in [28-30]. The values of root-mean-square deviations of
84
ZHIZHIN AND MUKHTAROV
TABLE II.7 The Values of Root-Mean-Square Deviations of Frequencies from Experimental Data
AAP/21 ~, cm -1
21.7
AAP/22 17.1
+ AAP/18 13.9
AAP/3 + AAP/19 6.9
+ AAP/20 3.2
frequencies from experimental data are given in Table H.7 from which it follows that the AAP/21 and AAP/22 are not fitted for the calculation of the crystal dyvomical properties. The combination of (AAP/3 + AAP/19) and (AAP/3 + AAP/20) in which the parameters of CI-'-CI interactions are obtained by fitting to the structure and external vibrational frequencies of the hexachlorobenzene crystal give the best agreement with the experimental results. The chloroform molecule has a sufficiently large dipole moment, and for this reason the external vibrational frequencies were also calculated by considering the electrostatical interaction in the approximation of the point charges on atoms whose values (in electron charge units) eH = 0 . 0 3 1 ; e C = 0 . 2 8 4 ; and eCl = -0.105 were taken from [31]. In spite of the significant contribution to the lattice energy (11.4%) the consideration of electrotstatical interaction does not practically influence the values of the external vibrational frequencies (the frequencies change is, on average, less than 1 cm-1). Thus, the abovementioned exomples show that the use of a combination set of AAP permits the description of the lattice dynamics of molecular crystals in a sufficiently precise manner. However, some violation of the consistency of AAP parameters which were obtained independently of crystal properties is possible. For this reason the paper of Williams et al. [32] is of interest; the authors have suggested the complete set of AAP parameters for oxohydrocarbon crystals. The parameters were fitted simultaneously to the properties of five crystals. The electrostatical interaction (of AAP in the form (6-exp-1)) was taken explicitly into account. The procedure of fitting was the same as in the case of hydrocarbons (see Sect. I.C). The calculations show the good transferability of obtained AAP/14 for describing the crystal structure [32]. To verify if the AAP/14 is universal we have carried out calculations of the frequencies and the external vibrational forms of trioxane and tetroxocane crystals whose structures were used by Williams during the procedure of obtaining the AAP/14 parameters. The tetraoxocane structure (C4H80 4) was determined at room temperature in [33] and described by space group C2/c, z = 4(C2). The solving of the dynamical problem with the use of experimental values of atomic coordinates led to the vibrational imaginary frequency of the type Bg (Table II.8). For this reason, the
VIBRATIONAL
SPECTRA
OF ORGANIC
CRYSTALS
85
TABLE II.8 The External Vibratior~l Frequencies for Tetroxocane Calculated with the Optimal Packing
0alculatioa With Optimizr Structure Exp. [36] Ag
61 38 88
Bg
77 48 32
Au
31
Bu
68 60
U -19.0 kcal/mol [37]
Calculation With Experimental Structure 65.5 48.7 95.6
electrostatical interaction was considered
electrostatical interaction was not considered
84.9 39.8 94.7
80.0 42.8 95.4
84.4 33.6 18.9
83.6 35.0 9.2
38.8
27.3
37.4
61.5 36.6
65.5 36.3
64.0 36.1
-16.6
-12.5
79.1 50.1 imaginary
optimization of structure was carried out. Since the tetraoxocane molecule has the
site symmetry C 2 in a crystal, only two coordinates of the molecule, ~Amely, the shiR Ay along the C 2 axis and the angle of A0 of the rotation around this axis w e r e varied during the procedure for the determination of the crystal potential energy minimum. It was found that the calculated packing of molecules differs from the experimental ones by the values Ay = 0.096 ,/k and A0 = 3.5~ The external vibrational frequencies calculated with the optimal pacing are given in Table II.8. The electrostatical interaction gives the essential contribution to both the lattice energy (24.5%) and the values of some frequencies (-10%). The crystal structure of trioxane (C3H60 3) was measured at temperatures of 103~ [35] and 293~ [34]. The space group R 3c, z = 2(C3). The external vibrational frequencies calculated at these temperatures are given in Table II.9. The contribution of electrostatical interaction into the lattice energy is -13% in this case, and it does not practically influence the frequencies. From Table II.8 and II.9 it is seen that the spectral data were badly reproduced by the AAP/14; the root-mean-square deviation of calculated frequencies from the experimental frequencies is equal to 14.3 (tetroxocane) and 19.4 cm -1 (trioxane). This is considerably more than that for the hydrocarbons (Table II.1). The AAP/14 parameters obtained by Williams are not thus fitted for describing the crystal dy~Amlcal properties. One of the reasons is that the consideration of the electrostatical interaction by means of the model of point charge on atoms is
86
ZHIZHIN ANDMUKHTAROV
TABLE II.9 External Vibrational Frequencies of Trioxane Calculated for Temperatures of 293~ and 103~ T = 293~
Exp. [15] E
85 62 ---
A2 A1
T = 103~
Calculation AAP/14 AAP/2
62
104.3 81.8 39.7
85.8 75.0 36.1
118.7 60.5
105.3 62.7
69.1
60.5
Exp. [15] 98 68 ---
68
Calculation AAP/14 AAP/2 118.8 97.4 49.7
98.0 82.9 42.5
131.9 75.9
114.5 67.5
79.3
72.8
inadequately reproduced in the ab initio results of dimer calculations (as was shown in Sect. I.C). This inadequacy influences the parameters of the short-range part of the AAP (6-exp) which determines mainly the external vibrational frequencies. The good agreement with experiment (a = 9.1 cm -1) obtained for trioxane during the calculation with the sufficiently rough universal AAP (6-exp) of Kitaigorodskii (I.99), which does not consider explicitly the electrostatical interaction (Table II.9), can verify this fact. In the general case the results of our calculations and the data from other authors (see Sect. I.C) show that the agreement between the experimental external vibrational frequencies and calculated ones by means of the AAP method within the limits of 10 cm-1 can be regarded as satisfying. The efforts to improve this agreement by fitting the AAP parameters to the external vibrational frequencies were not successful (as was mentioned in Sect. I.C). One of the reasons can be the strong correlation between the parameters A, B and a of A A (6-exp) which have no certain physical sense. For this reason we have tried to fit the parameters to the external vibrational frequencies of the AAP curves expressed by the parameters ro, e0 and D O which characterize the equilibrium state, the potential well depth and the second derivative of the potential curve at the equilibrium point (see Sect. I.C). The fitting was performed by the minimization of the function:
F(~) = ~
L.(exp))2 coi(~)-w i
I
by means of the iterativeleast-squaresmethod according to the followingscheme
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
87
~(~(n+l)) = ~ ( ~ ( n ) ) + ~(n) A~
9
~(n+l) = ~(n) + A~
(II.13)
where Pj = roll H, EoHH, DollH, roc H, eoCH, DoCH ... are the AAP parameters, ~ = (COl,...,O~6z.3) are the external vibrational frequencies, ~(n) is the matrix of the derivatives of frequencies with respect to the AAP parameters at the n th iterative
(J(n))ij -3 P~kOi j - (P)]n "
For the calculation of derivatives J~f) we have used the expression for frequencies in terms of eigenvectors Y and the elements of the dynamical matrix
2
= ~T
Dei
and we assume that 8Yi / bpj = 0, i.e. bcoi_ -
1 ~T ~D
e-i
The fitting was simultaneously performed to the external vibrational frequencies (k = 0) of naphthalene and benzene. The AAP/1 were used as the starting parameters
Pj= r 0, tk
e 0, kcal/mol
D 0, kcal/(mol ]k-2)
H---H
3.452
-0.0114
0.0634
C'--H
3.281
-0.0559
0.3130
C'--C
3.851
-0.0931
0.4556
~-I~IN
88
AND MUKHTAROV
TABLE II.10 NormM Matrix Calculated with the AAP/1 Parameters for a Benzene Crystal Pj'/Pi eoHH Doll H roll H eoCH DoCH roc H
eoHH
DoHH
10.8011 2.8103 0.2412 0.2181 0.0889 0.0072
roHH
2.8103 0.7341 0.0628 0.0569 0.0232 0.0191
eocH
0.2412 0.0628 0.0054 0.0049 0.0020 0.0016
DoCH
0.2181 0.0569 0.0049 0.0047 0.0019 0.0015
roCH
0.0889 0.0232 0.0020 0.0019 0.8 10 ~3
0.6 10 -3
0.0717 0.0020 0.0016 0.0015 0.6 10 .3 0.5 10 -3
However, during the firststep an unreal change of A A P parameters was obtained (1)oHH = 3.66 ^~
^~.(1)oHH= - 0 . 2 6
AD(1) oHH = L32
and so on.
This means that the convergence of the iterative least-squares method was absent. )~ks~j(is seen from Table II.10, the rows and columrts of the normal matrix (~(0) 0) calculated by the AAP/1 for a benzene crystal (in units 10 7 [v]2/([Pj][Pj,]), Iv] = cm -1 [Pj] =/~, kcal/mol, kcal ./k-2 tool-I) are almost linearly dependent (the elements of this matrix for C---C interactions are not given because their values are very small). The matrix which is the inverse of the normal m~trix in (II.13) is close to the singular matrix, and any insignificant i_n~_ccuracies in the calculations and also in the experimental data lead to the considerable uncontrolled changes of the fitted AAP parameters. The fitting of their parameters to the d y ~ m l c a l properties of crystals by means of the least-squares method leads, thus, to the necessity of solving for the badly justified equation system at each iterative independently of the way of the representation of AAP curves, i.e., the determination of AAP parameters from the external vibrational frequencies is the incorrectly formulated inverse problem. One can try to decrease the A A P parameter correlation by using other crystal properties (excluding the dy~amlcal ones) for fitting. Because of it, we have used the experimental values of the latticeenergy and the molecular packing in crystals of benzene, naphthalene and ethylene. The molecular packing was taken into account in the form of the equilibrium conditions, i.e.,the equality of generalized forces acting on molecules in the equilibrium state to zero. The fixed values (the "nonempiricar' AAP/13 ones) were used in order to prevent the nonlinear dependence of calculated crystal properties on the A A P parameters (6-exp) for apq aHH = 3.70
aCH = 3.43
acc = 3.16 ./k"1
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
89
and as the dynamical properties, one has used not separate frequencies ~i of external vibrations, but the values of ]~ o}~ = ~2 which are directly expressed by the trace of the dynamical matrix. Sinc~ in the studied crystals the molecules have the centrosymmetrical position, only three conditions of the equilibrium with respect to molecular orientations and two values of ~2 (for librational and translational vibrations, respectively) were considered for each of them. Thus, every crystal gives a system of 6 equations with respect to parameters Apq, Bpq ULet = AHH CLHH +BHH bHH + ACH aCH + ... = AHsub
DULet _____ k(a) + ACHaCH ^ (a) +''" = 0 ' a = 1 , 2 , 3 ~0a = AH H a ~ ) + BHH UHH
~br
= ~2[ ~ L.i. b r ) ]
= AHH a ( ~ r) + BHHb(~t~br) +
9OD
1
~ 2 = ~2 [r. .
= A HH a ( ~ + BHH b ~ H) +. .. (II.14)
1
where
aHH = - ~
1 ~ r6 , tzl PHPq PHPq
bHH= ~ ~ e Kt PHqH
-aHHrpHqH (II.15)
aHH' ( a )~(a) _ UHH'
etc., are expressed by the first derivatives of the lattice sums (II.15) with respect to the molecular orientations, and a(lib)HH,a~'-'~), etc., are expressed by the second derivatives of lattice sums with respect to the librational and translational displacements of molecules. The system of eighteen equations was, thus, used for the determination of six parameters AHH, BHH, ACH, BCH, ACC and BCC X(exp) = ~P where
(II.16)
90
ZHIZHIN AND MUKHTAROV
X!exp ) 1
-2 = AHsl,0,0,0,~21,o~11ib,
AHs2 ~0,0,0,~2tr
~'''~
9
^(~) is the 18 x 6 rectangular matrix consisting of the coefficients aHH, bHH, ~HH, etc. The solution of the system (II.16) by the least squares method leads to the normal equation system ~T~(exp) = (~T a)~
(II.17)
It was found that, as was in the abovementioned case, the matrix (~T~) is almost degenerate, i.e., the system of equations (II.17) is badly justified. For this reason, to solve this system one has used the regularization method suggested in [38], namely, instead of (II.17) one solves the system of normal equations in the following form ~T-~(exp) = (~T~)+ ~]~ (II.18)
where 1~ is the unit matrix. This system has the single solution for 13~ 0
P = [(~T~) + 13E]-I ~.Tx(exp) (II.19) m
The parameter ~ was chosen in such a way that the solution P~ depends slightly on the value ~. Then the numbers of unstable components of the solution ~ of the system (II.17) were determined by means of the determination of eigenvalues and eigenvectors of the matrix (NTN) (the smallest eigenvalue corresponds to the unstable part of the solution). One has fixed the value determined from (II.19) for this part of the solution, and it was put in the system of equations (II.17). The newly obtained system of equations which was of less dimension was solved by the ordinary method. This method proved to be sufficiently effective, and the following AAP parameters were obtained (A in kcal/~6/mol, B in kcal/mol): Aim
ACH
ACC
BHH
BCH
BCC
new AAP:
15
200
364
3360
5000
10840
AAP/13:
20
132
876
1500
6378
27116
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
91
The AAP/13 parameters are also given here; they were obtained with the use of the same values of apq in [39]. However, though these two sets of parameters differ considerably from each other, they give almost the same agreement between the calculated externM vibrational frequencies and the experimental ones. The abovementioned example demonstrates, thus, the ambiguity of the determination of the AAP parameters from the experimental data, i.e., the application of the rigorous mathematical methods of solving the inverse problem does not provide the most optimo] AAP parameterization. It can be said that the correct choice of the AAP parometers is defined to some considerable extent by the scientist's intuition and requires the tlme-consumlng calculations of different crystal properties of a wide variety of compounds. Finally, consider the possibility of using the AAP parameterization with respect to results from ab initio calculations of the dimer intermolecular interaction energy which has no disadvantages of empirical fitting (see Sect. I.C) for solving the dynomlcal problem. The most precise ab initio calculation of the intermolecular interaction energy of the dimer N2--'N2 was carried out by exchange perturbation theory (6 dimer configurations) [40] and by the supermolecular method (7 configuration) [41]. In [40] the results of these calculations were approximated by AAP (6-exp-1); the center interaction positions were fitted independently for every term of the AAP function A, kcal ~6/mol
B, kcal/mol
a, ~-I
hA, ~
hB ' .~
(a)
19
360.9
133509
3.949
0.471
0.547
(b)
21
402.7
64155
3.636
0.451
0.451
where (a): the "nonempiricar' AAP parameters; (b): the empirical parameters obtained by fitting to the statical and dynamical properties of crystal line nitrogen [42]; h A and h B are the distances between the center of molecule N 2 and the interaction centers of the potential functions A/r6 and B exp(-ar). The dependence of the intermolecular interaction potential energy on the distance between the molecular centers of N 2 obtained by the direct ab initio calculation and also by means of the "non-empiricar' and empirical APP's is given for some configuration of the dimer (N2)2 in Fig. II.1. Since the point charge model is inadequate, we have described the electrostatical interaction by means of the quadrupole-quadrupole interaction [5]
(II.20)
92
ZHIZHIN AND MUKHTAROV
/ C6H6
(AV2) 1/2, cm-1
12
/
////C6H;/
"\\
10
/ / /
/
\ ~
//
1/~ \
,/
///
/ / C10H8
/ \
/
\ ',,,
/
/
----- 1 .
..
.
.
.
.
"!
'
/
I / I -
"~.
~\\\~
I:00
CloH 8
/
/// / / /
,~,, \
/
1:04
'
1'.08
.
.
-
"
1.'12
dCH,/~
FIG. II.1. Dependence of the root-mean square deviation of calculated frequencies from the experimental values on C-H bond length for a naphthalene crystal (C10H8) and a benzene crystal (C6H6): (1) the parameter of AAP/2 and (2) the parameter of AAP/1. where fs = ($1,$2), fnl = (Sl,fi), fn2 = CS2,fi), $1 and $2 are the unit vectors directed along the axes of interacting molecules, fi is the unit vector in the direction of the radius-vector between the molecule centers; UQQ: in kcal/mol; and Q: in units of the charge of electron e and ~2. The value of the quadrupole moment Q = 0.495 e~ 2 was taken from [41]. In paper [41] the results of the ab initio calclations were approximated by the combination model potential consisting of the AAP of the form (6-9-12) and of the molecule-molecule potential of the type (6-9-12) and the quadrupole-quadrupole interaction UQQ: U = U AA P + UM. M + U Q Q
where
UAAp=
2 (a 6
~
-'V+
ij=l rij
a9
+
a12 /
~ij9- r i .2 ~
,
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
93
TABLE II.11 Calculated External Vibrational Frequencies for Nitrogen a t 30~ d a t a from [42]
exp. 1
nonemp,
nonemp.
[43]
AAP [40]
AAP [41]
exp. 2
Au Eu
46.8 54.0
50.4 55.8
49.8 56.3
---
50.2 53.8
Tu
48.4 79.4
49.9 72.8
49.4 74.0
48.8 70.0
49.9 73.3
Eg
32.3
36.1
38.6
31.5
25.1
Tg
36.3 59.7
42.2 52.8
45.2 56.8
31.5 35.8
29.9 37.4
empirical AAP
1the neutron scattering. 2the R a m a n and infrared absorption spectra.
UM-M
B6
B9
B12
= ~6- + ~-'~ + R ~m
rij is the distance between the nitrogen atoms, R is the distance between the molecule centers, Q = 0.508 e~ 2, a6 =-314.6
b 6 = -1327.2 kcal ~6/mol
a 9 = 9945.8
b 9 = 37861.3 kcal ~9/mol
a12 = 444496.5
bm = -113152.2 kcal ~12/mol
The calculated results of crystal external vibrational frequencies obtained at 30 K ~ (the low-temperature ordered phase a N 2 with space group Pa3, z = 4 (C 3 ) [42]) are given in Table II.11. The nitrogen molecule is linear, for this reason there are (5z3) external vibrations at k = 0, F = E u + 2T u + 2Tg + Eg + A u, and the vibrations A u and E u are inactive in the opticalvibrational spectra. -o
The most complete and precise experimental data concerning the low frequency vibrational spectra of this crystal are obtained by the measurement of nonelastic neutron scattering [43] (Table II.11). Our frequencies calculated by means of the "non-empiricar' AAP/19 and AAP/20 are in good agreement with these data (the root-mean square deviation is approximately equal to 5 cm'l). The frequencies calculated for two different intermolecular interaction models obtained by independent ab initio dimer
94
ZHIZHIN AND MUKHTAROV
calculations have very close values. This is the proof of the sufficiently high accuracy of wave mechanical intermolecular interaction energy calculations of dimers (N2)2. On the other hand, the empirical AAP's [21] show a large discrepancy with the neutron scattering experiment (Table II.11). The explanation is that the authors [42] have used the Raman spectra with a wrong interpretation (modes Eg and Tg, Table II.11) for determining the AAP parameters. This proves again that the empirical fitting of AAP parameters to the properties of only one crystal is of restricted significance and can lead to the wrong interpretation of the vibrational spectrum (see Sect. I.C ). The calculations which are a,m~logous to those for solid nitrogen were performed for crystal of CO 2 with the same space group Pa3, z = 4 [44] at 90~ We have used the results of ab initio calculations carried out by the CO2---CO2 dimer supermolecular method (9 configurations) [45]. The dimer intermolecular interaction energies were approximated by means of AAP of the form (6-exp) and quadrupole-quadrupole potential (II.20). The direct application of the least squares method has led to a badly justified equation system that shows the ambiguity of the AAP parameter determination from the dimer energies. For this reason, the abovementioned regularization method was used for fitting. Excluding the parRmeters A, B and a, the value of the molecular quadrupole moment (Qexp = 2.04 e~ 2 [46]) was varied; the interaction center coordinate of the oxygen atom was also varied along the axis of the linear CO2 molecule (i.e., it was suggested that the interaction center does not coincide with the oxygen atom). Two sets of parameters for AAP/a and AAP/b with the same approximate description as the ab initio calculations were obtained. The values of these parameters are given in Table H.12. The dependence of the dimer intermolecular interaction energy on the distance between molecular centers of atoms of O and C is given in Fig. H.2. The parameters of AAP of the form (6-12) ob~ined by fitting to the lattice energy and the unit cell parameter of crystal C02 and to the second virial gas coefficient are also given in the Table II.12. The negative values of some "nonempiricar' AAP parameters demonstrate the absence of a certain physical sense of some AAP curves.
The calculated results of the lattice dynamics of a CO2 crystal are given in Table II.13. A good agreement with the experimental values was obtained for all AAP's and for all lattice energies Ula~. The electrostatic interaction calculated in the quadrupole-quadrupole approximation gives a significant contribution to the lattice energy (-40%). The consideration of this interaction in the lattice dynamics leads to increasing the external vibrational frequencies, especially for the mode Au whose frequency is increased by 12 cm -1. The empirical AAP/c and the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
95
TABLE H.12 AAP Parameters for a CO2 Crystal
O"-O C---O
a,/k-1
C---C
AAP/a
AAP/b
3.6
3.1
3.5 3.4
2.95 2.8
Empirical AAP/c [47]
A, ~6
O'--O
kcal mo1-1
C---O C--'C
-4930 15414 -42130
-8614 15330 -67570
495.0 231.8 107.5
B, kcal tool -1
O"-O C"-O C---C
-43290 235400 -647500
-43290 115100 -228000
371005 137708 50167.2
h,~
0.9
0.9
1.16
Q, e/k2
1.60
1.20
2.04
Table II.13 The Calculated Results of the Lattice Dynamics of a CO 2 Crystal exp cm-1
Au
Eu Tu
Eg Tg
empirical AAP/c Q~0 Q=0
empirical AAP/14
nonempirical AAP AAP/a AAP/b
95 105 114 68 72 128
30 30 29 29 28 28
96.1 99.8 114.2 72.3 56.9 108.3
84.2 101.1 118.2 67.0 57.2 106.9
89.3 114.0 137.2 86.7 53.3 87.3
102.5 75.5 100.8 51.7 70.7 129.5
98.3 78.8 102.3 59.8 55.2 121.6
88
28
72.3
71.0
64.3
89.4
76.5
11
-6.88
-4.11
-6.05
-6.90
-6.92
-2.77
.
3.47
-2.71
-0.96
24.7
14.5
15.0
Ulatt -6.82 UQQ --o, cm -1
12.3
Ulattand UQQ: in kcal/mol
.
.
.
96
ZHIZHIN AND ~ T A R O V U, kcal/mole 0.50 0.25 \
0
3.8
R,h
4.2
3 \%C - -
-0.25
f
J -0.50-0.75 -1.00 -1.25 -1.50
U, kcal/mole
1 2
0.75 \ \
0.50 0.25
.
\
-0.50 -0.75
.
.
.
\
\.3.4 -0.25 -
.
\
3.8' ~ -- -.
R,/~
~.'2----
9
-1.00 -1.25 -1.50 -1.75 FIG. II.2. Dependence of the intermolecular interaction potential energy of the dimer (C02) 2 on the distance between the centers of molecules with ab initio calculations [45] (2), by means of the empirical AAP/c (1) and "nonempirical" AAP/a (3). "nonempiricar' AAP's reproduce the experimental frequencies approximately in the same way. The calculation with the AAP/14 parameters gives rather significant discrepancies with the experiment in the cases of trioxane and tetroxocane. It was interesting to study the transferability of the AAP parameters obtained from the wave mechanical calculations of dimers of small molecules for the description of crystal lattice dynamics of other related compounds which are more complex. In Table II.14 we have tabulated the external vibrational frequencies
VIBRATIONAL S P E C T R A O F O R G A N I C C R Y S T A L S
97
TABLE H.14 A Comparison of the External Vibrational Frequencies of Ethylene, Naphthalene and Benzene Crystals exp. a nonempirical b nonempirical r cm "1 AAP/1 AAP/13 [40] (6-9-12-1) AAP [51] Ethylene
Ag
177 97 90
164.0 97.7 83.1
177.6 90.5 71.2
Bg
167 114 73
156.5 111.9 49.8
202.5 129.3 23.7
Au
73 --
62.7 50.5
75.4 54.6
Bu
110
92.4
91.0
= 14.2
~ = 25.9
95.9 82.4 51.0
Naphthalene
Ag
109 74 51
105.4 77.9 51.2
Bg
125 71 46
102.0 71.5 42.9
92.7 74.8 40.6
Au
98 53
91.4 48.3
75.6 43.5
Bu
66
61.9
57.0
r = 9.1
a = 15.8
93 78 56
97.7 74.0 54.3
88.3 64.7 50.5
55.0 52.4 23.4
90 78
102.4 96.4 83.3
98.8 88.4 81.2
67.4 56.9 43.2
Big
128 100 56
132.5 94.1 62.1
123.5 94.7 63.3
79.4 64.0 24.3
B3g
128 84 61
129.5 91.5 59.9
124.8 80.8 43.3
76.6 55.1 37.0
Benzene
Ag
B2g
ZHIZHIN AND M U K H T A R O V
98
TABLE II.14 (continued) exp.a cm-Z
AAP/I
nonempirical b AAP/13 [40]
B2u
98 66
104.1 62.5
110.9 62.4
Blu
89 73
90.0 74.1
92.9 74.8
B3u
100 56
104.6 56.6 a = 4.4
110.1 59.0 ~ = 7.6
nonempirical c (6-9-12-1) AAP [51] 57.8 30.4 54.2 34.2 56.8 29.3 a =38.1
aEthylene [52], naphthalene [17,18], benzene [53,54] bFrom the ethylene dimer calculations CFrom the benzene dimer calculations
of the ethylene crystal,naphthalene crystal and benzene crystal calculated with the "nonempiricar' AAP/13 obtained from the ab initio calculations of the ethylene dimer intermolecular interaction energy [40], (see Sect. I.C). It is seen that the agreement with experiment is better in the case of the naphthalene crystal (in comparison with the case of ethylene crystal). The same result is obtained for the benzene crystal. This can be partially associated with the influence of the ethylene external vibrations anharmonicity (see Sect. H.C) which was not considered in the quasi-harmonic calculation. Nevertheless, the obtained results are the proof of the sufficiently good transferability of the AAP/13 parameters to the unsaturated hydrocarbons. An especially good agreement between the experimental and calculated results with the empirical AAP/1 was observed for the benzene crystal. It is expected that the accuracy of the lattice dyn~mlcs description for this crystal can be increased, if the dimer of benzene (C6H6) 2 is stillused as the basic system for the nonempirical parameterization of AAP. The wave mechanical calculation of the intermolecular interaction energy of this dimer (9 configuration) was performed by the supermolecular method in [51], and the calculated results were approximated by the A A P in the form of (6-9-12-1) (see Sect. I.C). However, the benzene external vibrational frequencies calculated by means of these A A P proved to be more than 1.5 times the experimental frequencies (Table II.14). The reason is, evidently, the use of the weighted factors w i = exp(-u.~/kT)at T = 300~ during the procedure of fittingthe A A P by using the least-squares method (Ui is the dimer intermolecular interaction potential energy of a given configuration). This does not permit one to approximate sufficientlyand precisely the repulsion part of the intermolecular potential, which gives the most contribution into the external vibrational frequencies. The dy~nmical properties of crystals are the very sensitive criterion of the A A P parameters obtained from the nonempirical calculations.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
99
Table II.15 The External Vibrational Frequencies of Cyclohexane-II as Calculated with the Intermolecular Interaction Model. exp. a Nonemuirical A b ~ h cm -1 AAP/4 Calc. I Calc. I/
As
110 88 72
102.3 85.8 61.0
106.3 89.0 65.6
104.5 87.1 63.9
Bg
120 97 63
124.8 97.7 59.1
126.4 100.3 65.8
125.7 99.1 63.0
An
101 65
112.9 67.2
115.7 66.3
98.3 61.5
Su
(65)
61.2
62.9
62.2
-11.4
-2.3
-10.4
-11.0
Ulatt
39
kcal;/mol
= 6.9
c~ = 4.9
aSee Part 1I, Sect. I.A. bCalculation I with AAP from [56], Calculation II for h A = h s = 0.92 ~ and A = 350 kcal ~6/mol. One can suggest that the transferability of A A P parameters obtained from the ab initiocalculations of small molecules to cryst~s of more complex compounds will be considerably better for saturated compounds with the distribution of molecular electron density localized at chemical bonds than for aromatic compounds with delocalized conjugated bonds. It was the aim of our calculation of the external vibrational frequencies of low-temperature ordered phase II cyclohexane crystal (T = 115~ the space group C2/c, z = 4(Ci)); the structure was determined [55] by means of A A P (6-exp) parameters obtained by fitting [56] to the dimer intermolecular interaction energies of methane, (CH4) 2, calculated by the supermolecule method (6 configurations). One has used the following A A P model: instead of two interaction centers on atoms of C and H, one has used only one center localized at the C-H bond, and its position was varied by a separate fittingof the repulsion potential part and the dispersion energy: A: ~k6 kcal/mol 153
B: kcal/mol
a,/~-1
h B, ~
h A, ~k
5533
3.677
0.92
0.17
where h A and h B are the distances between the interaction centers and hydrogen atoms for-A/r6 and B exp(-r). The external vibrational frequencies of cyclohexaneII calculated with this intermolecular interaction model are shown in Table II.15.
100
ZHIZHIN AND MUKHTAROV
It is seen that the calculation gives a considerably smaller value of Ulatt, although the frequencies are in very good agreement with the experimental values. It is evidently caused by the fact that in [56] one has used the rather approximated half-empirical method, based on data concerning the methane C-H bond polarizability, for the calculation of the dispersion energy. For this reason the calculations were carried out with the different values of the parameter A characterizing this interaction. However, it has been suggested that h A = h B - 0.92 ik. It was found that the choice of parameters h and A has a very small influence on the frequency values (excluding one translational vibration of A u -type). One has obtained very good agreement of both the frequencies and lattice energy with experiment when A = 350 kcal/~6/mol (Table H.15, the calculation If). The rootmean-square deviation of the frequency from the experimental data proved to be less than in the case of using the empirical AAP/4. Thus, the most optimal models of A A P are the following ones for the interpretation of molecular crystal vibrational spectra: (1) The AAP/1 and the AAP/4 of Williams and Kitaigorodskii in the case of hydrocarbon crystals. (2) In the case of organic compounds containing other types of atoms X, the combined A A P parameters consisted of 'hydrocarbon" A A P - a n d X"-X interaction parameters obtained independently from the crystal properties in which the X-atom interactions are determinative. It was also shown that the AAP/14 suggested by Williams for oxidated compounds are not valid for the lattice dynamics calculations. (3) The accuracy of the external vibrational frequency calculations by means of the empirical A A P is -10 cm -1. The discrepancy with the experimental values which are more than 10 cm -1 proved either the nonoptimal choice for the A A P parameters or the influence of factors which were not considered in solving the dyv~mical problem (the anharmonicity, the interactions of external and internal vibrations). (4) It was shown that the A A P parameterization with respect to macroscopic properties of crystals leads to the strong correlation between the A A P parameters independently of their representation form, i.e. in this case the number of actual independent parameters is less than the number of formal A A P model parameters. (5) The use of wave mechanical calculations for the intermolecular interaction energy has the most perspective for the further development of the intermolecular interaction models and their interpretation on the basis of the vibrational spectra of molecular crystals.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
101
C. Temperature Dependence of External Vibrational Frequencies -- Model of Independent Anharmonic Oscillators
In the consistent harmonic approximation, the vibrations of the crystal lattice are described by 6zN independent harmonic oscillators. The distance between the energy levels of every oscillator of this kind is the same, and it is characterized by one frequency ~ whose value is determined by the minimum of the crystal potential energy (see Chapter I). The vibrational spectra of the crystal in the frame of this approximation consists of infinitely nswow lines with frequencies which are independent of temperature. However, as follows from many papers [58-67], the vibrational spectra of molecular crystals differ by considerable temperature broadening and displacements of lines, which are especially noticeable in the low-frequency spectral region where the relative shift of lines is more than 10% in the temperature range from 0~ up to the melting point. The temperature dependencies of external vibrational frequencies of the naphthalene crystal (at k = 0) constructed according to the data concerning the low-frequency Raman and i ~ e d absorption spectra [58,65-70] are given in Fig. II.3 and Fig. II.4. The form of these dependencies is typical for molecular crystals and correlates with the thermal broadening of the unit cell volume (Fig. II.5). The external vibrational frequencies are decreased by 10-20 cm -1 at the change of temperature in the range from 0~ to 300~ Thus, the influence of thermal factors associated with the crystal vibrational anharmonicity must be taken into account for the correct interpretation of the vibrational spectra with the use of calculated frequency values. In particular, the necessity of studying the spectra at different temperatures arises from the investigation of polymorphic crystal modifications whose existence is restricted by different temperature ranges. In a general case the dependence of the external vibrational frequencies on the temperature is first caused by the intermolecular change as a result of the thermal expansion of a crystal lattice and, secondly, by purely anharmonic effects related indirectly to the anharmonicity of the crystal potential surface and by the interaction between the normal vibrations. It can be expressed by the following relationship [60,68,71,72]
V
(II.21)
102
ZHIZHIN AND MUKHTAROV v, cm -1
Ag
\
_
Bg
\
\
\ \ \
150
\
\ \
\ \ \ \
130
\
~.~-,.,,~ v4 \\
\ 110 " ~ . \ \
Vl
~
.
90
-'~
70
~
v2
~ 5~v..~. ~ -~ v6
5O
30
. . . . . ,
o
I;O 2oo
,
aoo
i'
o
I;O 2;o ;oo T , K
FIG. II.3. Temperature dependence of librational vibrational frequencies of the naphthalene crystal (1) experimental, (2) and (3) calculation from the "quasiharmonic" appro~mation with AAP/1 and AAP/II, respectively.
where ~ = ~(v,T) is the frequency of the jth vibration as a function of the unit cell volume and temperature,
TJ=-
/~/n~0j / ~tnv T
tnv]
")p
are the Grtineisen parameters
is the volume coefficient of t h e ~ !
expansion.
The first term in (Eq. II.21) corresponds to the change of frequency as the result of the thermal expansion, the second tern corresponds to the influence of the "pure" anharmonicity which is responsible for the temperature dependence of frequencies at the constant volume of the crystal. If one neglects the influence of
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
103
v, cm -1
120 -
I00
FIG. H.4. Temperature dependence of the translational vibrational frequencies of a naphthalene crystal.
"
\
80-
60-
400
100
200
3 0 T,K
V,h 3 o
360 -
FIG. II. 5. Temperature dependence of the unit cell volume of a naphthalene crystal (according to [80-82]).
o
350 "
340 v'
0
100
"
2()0
3()0 T,K
"pure" anharmonicity and considers that the Grtineisen parameters are slightly changed by the change in temperature [72,73], the relationship (Eq. II.21) implies the simple dependence of frequencies on the temperatures [68,72] oj(T) = mj(O)exp -Tj t n v ( o )
(II.22)
where ~(O) and v(O) are the frequency and the unit cell volume at 0~ The thermal expansion of the crystal, i.e. the dependence v(T), can also be expressed by the Grtineisen parameters in the frame of this approximation [74,75]. The thermal
104
Zt-IIZI-IIN AND MUKHTAROV
expansion of crystals is predicted by the considered model (which does not take into account the "pure" anharmonicity), but the vibrations of molecules in the vicinityof a new equilibrium position remain harmonic. The dependence of these vibrational frequencies on the temperature is determined only by the Griineisen parameters. Since these parameters describe the dependence of frequencies on the crystal deformation, they characterize the deviation from the precise harmonic model. For this reason the expression (II.22) corresponds to the so called quasi-harmonic approximation [74,75]. Since the Griineisen parameters and the thermal expansion of a crystal can be independently measured, the direct experimental verificationof this approximation can be performed by means of formula (II.22). Such verification was carried out for the naphthalene crystal in [62,68] and for some other molecular crystals in [60,62,72,76]. It was established that the temperature dependence of external vibrational frequencies is perfectly described by (II.22), and the quasiharmonic approximation is sufficientlyadequate. In the case of naphthalene it is indirectly verified by small values of mean amplitudes of molecular displacements which are equal, for example, to approximately 4 ~ for the angular displacements (see Chapter I). In the quasi-harmonic approximation, the temperature shift of frequencies can be comparatively simple to take into account by using the A A P method, which permits one to perform the direct calculation of the change of force constants at the thermal expansion of a crystal. In this case the calculation of the external vibrational frequencies at some temperature consists of solving the usual dyv~mical problem with the use of the fixed unit cellsizes at this temperature. W e have first used this approach [77-79] for the description of the thermodynamics of molecular crystals in the Debye approximation (the Debye temperature was expressed by the root-mean-square of the external vibrationalfrequencies). The dependencies of external vibrational frequencies of the naphthalene crystal calculated by means of the AAP/1 and AAP/3 (rigidmolecules) in the quasiharmonic approximation are shown in Fig. II.3 and Fig. II.4. The detail data concerning crystal structure in the range 4 - 296~ obtained by the neutron and Xray method in [80-82] were used. At every temperature the lattice energy was minimized with respect to the molecular orientations. The conditions of the calculation are described in detail in Sect. H.A. As is seen from Figs. II.3 and II.4, the calculated dependencies demonstrate the considerably sharper fall in the temperature decrease in comparison with the temperature; this is in good agreement with the results of the analogous calculations performed with AAP/2 in [83,84]. The large discrepancy between the calculated results with two sets of the AAP/1 and AAP/2 parameters attract our attention; they were obtained at low temperatures, and they were less noticeable at the high temperature. This is explained by the fact that the contribution to the values of molecular repulsion
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
105
Table II.16 The Values of the Relative Temperature Shifts of the Librational Vibrational Frequencies of Naphthalene in the Range of 0 to 300~ exp.
Calculation by Eq. II.23
Calculation with AApa
V1
0.11
0.20
0.19 (1) O.2O (2)
V2
0.18
0.26
0.22 (1) 0.29 (2)
V3
0.26
0.32
0.22 (1) 0.26 (2)
V4
0.14
0.20
0.22 (1) 0.26(2)
v5
0.13
0.26
0.21 (1) 0.27 (2)
v6
0.21
0.32
0.25 (1) 0.28 (2)
a (1) and (2): the calculation with AAP/1 and AAP/2, respectively.
frequencies, whose description by means of AAP/1 and AAP/2 differs considerably from each other (AAP/2 are more "rigid" then AAP/1, see Chapter I), increase at the decrease of the crystal volume (the decrease of temperature). The quasi-harmonic approximation and the AAP were also used for the calculation of the thermal dependencies of the external vibrational frequencies of paradibromobenzene [63], anthracene [13], biphenyl [13], and benzene [85] crystals. The discrepancy of the calculation with the experimental values, which is analogous to the discrepancy in the case of naphthalene (Figs. H.3 and H.4), is observed in all cases. This discrepancy cannot be explained by the interaction between the internal and external vibrations (at any rate, for naphthalene, see Sect. HI.B). Two reasons are'possible:
(1) the insufficient adequacy of atom-atom
potentials overestimating the contribution of the molecular repulsion potential at their approaching; (2) the influence of the "pure" anharmonicity which is not considered in the quasi-harmonic approximation. The values of the relative temperature shifts of the librational vibrational frequencies of naphthalene in the range of 0 - 300~ are given in Table II.16. They were calculated by the AAP and
106
ZHIZHIN AND MUKHTAROV
by (II.22) with the use of the Griineisen parameters [3] and the therm~l expansion of crystal in the quasi-harmonic approximation (Fig. II.5)
Arel =
c~ (0~ K)-~ (300~ o}j (0 ~K)
= 1-exp
-7j
In
v(300 ~K) ] . ~
(II.23)
It follows from this table that the AAP method satisfactorily describes the quasiharmonic frequency shift. This means that one of the main causes of discrepancy between the calculated and experimental temperature dependence of frequencies (especially,v 1,v 2 and v 5) is the influence of the "pure" anharmonicity. The anharmonic corrections to frequencies of some external vibrations can, thus, be sufficientlylarge; this can, in general, lead to the wrong interpretation of the low-frequency spectra of molecular crystals with the help of the quasi-harmonic calculation. The consideration of these corrections is also necessary, if one uses the spectral data for the parameterization of the model potentials, because the experimental determination of the contribution of the "pure" anharmonicity is a very complicated problem requiring the measurement of the temperature dependence of frequencies at a constant volume. At the present time it can be accomplished only for the crystal -N 2 [86]. One of the possible methods for the consideration of the molecular crystal external vibrational anharmonicity is the method of the self-consistent phonon approach in which the second derivatives of the crystal potential energy are calculated by means of AAP for the instant moiecular positions, and they are, afterwards, averaged over the ensemble [74,87]. The "average" (effective) dynamical matrix is used instead of the usual dyv~mical mat,fix calculated at the potential minimum. The application of this method is, however, conjugated with the repeated calculation of frequencies in the whole Brillouin zone, and this actually leads to the same results as was shown by calculations for a number of simple crystals [34-36]. For the correct calculation of the anharmonicity correction it is, thus, necesssary to take explicitlyinto account the deviation of the crystal potential from the harmonicity for solving the equation of motion. Consider, first the onedimensional anharmonic oscillatorwith the potential energy of the form
U(Q) = ~12
Q2 + a(3)Q3 +a(4)Q4
(II.24)
where co0 is the harmonic vibrational frequency and Q is the mass-weighted displacement coordinate (Q = m 89x). In the expansion (Eq. II.24) of the potential as a Taylor series of displacements, it is necessary to consider the terms of third and fourth orders at the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
107
same time, since their contribution to the harmonic frequency shifts is approximately the same [74,90]. The classic equation of motion is
Q + a i Q + ~ c ( ( ~+)~Qc ~( ( ~=) 0Q ~
(11.25)
The solving of this equation by the successive approximation method (the anharmonic terms in (Eq. II.25) are considered as the small perturbation of the harmonic oscillator) is given in the book by Landau and Lifshitz [911; this gives the following correction to the harmonic vibration frequency (11.26) where A&) and A&) are the shiRs in frequencies caused by the anharmonicity of the 3 d and 4th degrees; QO is the displacement amplitude; and d3) and d4) are the factors of anharmonicity
(11.27) The consideration of the anharmonicity of the third degree leads to the decrease in the frequency, since d3)is less than zero. The average value
(11.28) where n is the quantum number whose average value over the harmonic oscillator states is equal -1
ii=(exp$-l) (11.29)
Thus, unlike the harmonic approximation (see Chapter I), the classical and wave mechanical consideration of the anharmonic oscillator give the different temperature dependencies of frequency shifts coinciding only in the range of high temperatures when h a << kT .
108
MIZHIN AND MUKHTAFtOV
Studying the anharmonic properties of a crystal, one can expand its potential energy into a Taylor series of harmonic vibration normal coordinates 192,931
It permits one to use the symmetry properties of a crystal; the factors a(3) and a@) will be nonzero only in the case when the direct product of the appropriate irreducible representations of the wave vector group contains the completely symmetrical representation [921. The solving of the wave mechanical problem for atomic potentials by means of the perturbation theory is considered, for example, in [741. The necessity of the consideration of the kinematic anharmonicity related to the complicated dependence of the kinetic energy operator on the angular molecular displacement is additionally increased in the case of the molecular crystals. The wave mechanical problem of the anharmonic external vibrations of the molecular crystals is solved for the general form in [94,951. Without the consideration of anharmonicity the perturbation theory gives the expression for the line form of the j* optical phonon 1961
where ~ ( 0 is the ) harmonic frequency of the jth vibration at 2 = 0; +(a) = AcI#~) A c o , ( ~ ) (is ~ )the anharmonic shift of a frequency; rj(a)is the halfwidth of a line
+
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
109
-1
(11.32) The subscript p in the expressions of the form U(mj,(o))pmeans that the sums (integrals) are calculated in the sense of their principal values. The calculations by (Eq. II.32) are extremely complicated and time-consuming; they were mainly performed for the simple molecular crystals N2, C02, N H 3 , etc. 197-1011. The angular vibrational amplitudes of molecules in these crystals are rather large, and
110
ZHIZHIN AND MUKHTAROV
the anharmonicity consideration gives the essential corrections to external vibrational fkequencies. At the present time, only the calculation using Eq. II.32 for naphthalene (4°K) with the use of AAPl1 and the explicit consideration of quadrupolequadrupole interactions [%I is well known for the case of the crystals of more complex molecules. The main contribution to the frequency shift is caused by the anharmonicity of the 4th order that leads to systematic increasing of the frequencies. The anharmonic correction to the frequencies proved to be, in general, rather small, and they are, on the average, equal to 3.7 cm-1 and 0.6 cm-1 for librational and translational vibrations, respectively. The analogous calculation was carried out with M I 2 at 300°K in [102]; however, it was performed only for one vibration v2(A,J of a deuteronaphthalene crystal. The calculation of the anharmonic correction for only one frequency was performed during more than fourteen hours of computer time (computer of the CYBER 172 type). For this reason the methods permitting one to estimate the anharmonic corrections, to the external vibrational frequencies of sufficiently complex crystals consisting of molecules containing many atoms, by the more simple methods are necessary. The first effort of this kind was, evidently, the calculation of the corrections to the frequencies of librational vibrations of the n-hexane crystal, which were carried out by the different AAP [103]. The authors have used the simplified model in which they have used the interaction between three librational vibrations in the Brillouin zone center (the n-hexane crystal contains only one molecule in the unit cell). The kinetic anharmonicity which proved to give the negligible contribution to shifts of frequencies was also considered. The calculated corrections to frequencies were up to 40%, and they were very sensitive to the choice of the AAP parameters. The necessity for the calculation of the derivatives of the 3rd and 4th order of a crystal potential and the considerable complication of the problem by increasing the number of molecules in the unit cell make, however, this method sufficiently time-consuming and difficult, and it is not justified by performing approximations. The model of the independent harmonic oscillator is more attractive, because it is simple. It was successfully used for a description of the external vibration of the simple molecules N2 and 0 2 1104-1061. This model was also used for the deuteronaphthalene crystal (though it was slightly performed in the incorrect way, it will be described later) [102]. In the considered approximation one completely neglected the normal vibrational interaction, and the anharmonicity consideration is reduced to solving the abovementioned problem for the one-dimensional oscillator. At K = 0 the calculation method is the following one: (1) First, the ordinary dynamical problem is solved. "he values of quasi-harmonic frequencies oojand the appropriate eigenvectors are determined.
c:,"
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
111
(2) AU crystal molecules are shiRed together to the direction of eigenvectors; their motion is proportional to a given normal coordinate
(11.33) The "real" dependence of the crystal potential energy WQj> on the normal coordinate is directly calculated by the AAP method; i.e., the one-dimensional cross-section of the complete potential surface U(Q1, Q2, ...I is constructed. (3) Finally, the coefficients ay)and a?) were determined by fitting to the "real" potential according to the least-squares method for a polynomial of the fourth degree (11.34) The anharmonic corrections to the j* vibration were found by means of a(3)and I d4), according to the formulas (11.27) and (11.29). The following dimensions of J quantities were used rJl = amu A2, [MI = amu, 161 = ( m u )G,
All three parameters were varied during the procedure of fitting by means of the least-squares method 11051. However, in this case a@)and a(4)cannot be independently determined and t h i s leads to the loss of their physical sense. Besides, the parameter aj(2) is directly related to the harmonic frequency (a:) = and characterizes the potential form in the quasi-harmonic approximation. Because of it, we have varied two parameters, a(3)and ay),while j fitting only the anharmonic part of potential
cotj)
(11.35)
112
ZHIZHIN AND MUKHTAROV *
In order to limit the range of a coordinate Qj and to 1ly consider the form of a potential UCQj) in the region of the most probable molecular displacement, we have used the weighted factors during the procedure of a least-squares method
As was mentioned above, the equality of the factor d3)of (11.35) to zero can be J
predicted by means of group theory. The direct product rj 8 rj 8 rj contains the unit representation for only the completely symmetrical rj in the naphthalene case (the factor-group c2h) and also in all other cases, when the normal vibrations only related to one-dimensional irreducible representations rj of the factor-group. Consequently, the independent oscillator model gives the correction to the anharmonicity of the 3rd order only for the completely symmetrical vibrations, i.e. for the $- modes in the naphthalene case. For this reason, the nonzero anharmonic correction a(3)to the frequencies of the deuteronaphthalene crystal vibrations, B,, j obtained by the authors of paper [lo21 cause doubt about their method of calculation. Consider the calculation for the dependence of U(Qj) in detail. Since the translational symmetry is maintained during the vibrations of molecules at the Brillouin zone center, the potential energy of a crystal with molecules shifted h m the equilibrium state can be of the form
(11.36)
where K ~ K and ' f # f ' , N is the number of unit cells, and UAQj) is the energy of the interaction between the Icn molecule of the unit cell which was chosen as the origin of all other crystal molecules
In order to simplify the problem we have also used the factor-group symmetry properties of a crystal and of the n o d coordinate Let gK, K = 1, ..., z be the permutable symmetry elements of the crystal factor-group. They transform the Kth molecule of the unit cell into the same origin (K = 1). Then,
a,.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
113
The normal coordinates transform according to the irreducible representation of a factor-group. The potential energy of a crystal (II.36) can then be represented in the form
U(Qj) = I ~ I U I ( x ~ ) Q j )
where xKfJ) is the character of the jth irreducible representation of the symmetry element g~. The calculation of the potential energy of a crystal with molecules displaced in the direction of the eigenvector leads to the calculation of the interaction energy of one chosen molecule with all other crystal molecules. For example, in the case of naphthalene gl = C1, g2 = C2 or C s, and according to the table of characters [93], we get U(Qj) = U I(Qj)
Ag modes
U(Qj) = ~ [UI(Qj)+ UI(-Qj)]
Bg modes
U(~) = UI(~) ,U I(~) = UI(-~)
A u and B u modes
It is possible, in principle, to calculate the anharmonic corrections for every other point of the Brillouin zone. However, the violation of a translational symmetry considerably complicates the calculations for these vibrations, and the advantages of this model are lost. The aforementioned calculation technique for the approximation of independent anharmonic oscillators is illustrated by the exomple of naphthalene in Fig. II.6. The values of the anharmonic corrections to the external vibrational frequencies of this crystal obtained by means of AAP/1 and AAP/2 at 5~ and 296~ are given in Table II.17. It is seen that the corrections are insignificant at low temperature, and they do not exceed, in average, the value of 1 cm -1. Excluding the vibration v6(Bg) (the calculation with AAP/2), the anharmonic frequency shiI~ is also insignificant at 296~ and it is, on average, -4 cm 1. The large sensitivity of the calculation results to the AAP parameter choice attracts our attention; this means that the use of anharmonic crystal properties is a possibility for the improvement of the intermolecular interaction models. The external vibrational frequencies of naphthalene with the consideration of anharmonic corrections are given in Table II.18. It follows from this table that the application of independent anharmonic oscillators does not exclude the discrepancy between the experimental and calculated temperature dependencies of the frequencies (Figs. II.3 and II.4) (the calculation was carried out in the quasi-harmonic approximation), though it
114
ZHIZHIN AND MIYKHTAROV
v 1 (Ag)
v4 (Bg) /
AU, kcal]mole
/
AU, kcal/mole
\
! II
tl tl
! , -2
/'
i/
_l~_~
'. . . .
-1
1
10
Q
Q(amu) 1/2 A
2
-2
-1
0
AUan h, kcal/mole ,,~(3)~. Q3 4- a(4) Q4
,,J
_~_./'~,
,/,/
Q(amu) 1/2/~
1
2
AUanh, kcal/mole
I
.j
-1 FIG. 11.6. '~Real" dependencies of the crystal potential energy, AU, and their anhaz~onic component, AUanharm = AU -(1/2) Coo2Q2, on the normal coordinate Q for two librational vibrations v I (At) and v4(Bg) of the naphthalene crystal. The data were calculated by means of AAP/1.
improves the agreement between the spectral data for some vibrations at 296~ (especially for vibration v4(Bg)). The comparison, between the results of the "precise" calculation according to the formula shown in (II.32) and used in [96] and our data (Table II.17), shows that the independent oscillator approximation gives considerably lower values for A~(4) in spite of the precise estimation of the correction value of Aco(3) at the same time. Thus, the influence of the "pure" anharmonidty is mainly caused by the phonon interaction processes in the case of weakly anharmonic crystals such as naphthalene. In this case, the model of independent oscillators will be insufficiently effectively applied. However, the advantages of this model can be completely taken advantage of in the most interesting case of the strong anharmonicity, when the molecules are vibrating with large amplitudes, and the form of the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
115
T A B L E II.17 T h e V a l u e s of t h e A n h a r m o n i c C o r r e c t i o n s to E x t e r n a l V i b r a t i o n a l F r e q u e n c i e s of a N a p h t h a l e n e C r y s t a l O b t a i n e d b y M e a n s of AAP/1 a n d A A P / 2 a t 5 ~ a n d 2 9 6 ~ T = 5~ Ao)(3) C~-I AOo(4)c m ' i
vI
(1)*
-0.8
0.3
-2.3
(2)
-2.2 -1.0
2.2 2.2
-5.4
0.0 -0.2 -1.0
-0.7 0.4 7.0
0.0 -0.5
-0.9
-0.3 -0.6
-0.7 -0.6
(3) (1)
Ag
1.9 4.2
v2
(2) (3)
v3
(2)
-0.3 -0.5
(3)
-0.6
-0.3 0.1 4.8
(1)
---1.0
1.3 3.1 7.2
---
4.3 7.8
---1.8
-0.6 0.2 2.4
---
-0.8 0.9
0.7 2.4 4.9
---
3.7 12.4
(3)
---0.9
v7
(1) (2) (3)
---O.6
0.2 1.1 0.4
---
2.3 3.0
v8
(1) (2) (3)
---0.7
-0.3 0.1 1.9
---
-1.0 -0.1
(1)
---0.9
-0.2 0.1 1.6
---
0.0 0.2
(1)
v4
(2) (3)
v5
(2)
(1) Bg
T = 296~ A(0(3) cm-1 A0)(4) cm-1
(3)
v6
(1) (2)
0.9
Au
Bu
v9
(2) (3)
* (1) a n d (2) a r e t h e r e s u l t s of c a l c u l a t i o n s w i t h A A P / 1 a n d AAP/2, r e s p e c t i v e l y , (3) is t h e "precise" c a l c u l a t i o n a c c o r d i n g to f o r m u l a (II.32) [96].
116
ZHIZHIN AND MUK/-ITAROV
TABLE II.18 The External Vibrational Frequencies of Naphthalene With the Consideration of Anharmonic Corrections T = 5K
Vh~rm. vI
Ag
Bg
Bu
121
V~nh
Vexp"
T = 296K Vl.mrm V~nh
143.2 165.1
142.7 165.1
109
(2)
114.7 128.3
114.3 127.1
(1)
v2
89
(1) (2)
111.0 116.8
110.3 117.1
74
85.6 84.2
84.7 84.6
v3
69
(1) (2)
67.6 74.7
67.0 74.3
51
53.8 56.1
52.8 54.8
v4
141
(1) (2)
150.4 161.2
151.7 164.3
125
114.2 119.2
118.5 122.0
v5
83
(1) (2)
90.2 91.4
89.6 91.6
71
72.3 67.3
71.5 68.2
V6
57
(1) (2)
58.6 63.1
59.3 65.5
46
44.8 45.6
48.5 58.0
V7
108
(1) (2)
117.3 116.4
117.5 117.5
97
86.3 82.6
88.6 85.6
V8
57
(1) (2)
54.3 57.6
54.0 57.7
51
43.1 43.5
42.1 43.4
v9
78
(1) (2)
68.5 70.0
68.3 70.1
66
54.3 52.5
54.3 52.7
* (1) and (2) are the results of calculations with AAP/1 and AAP/2, respectively.
"precise" potential U(Qj) differs considerably from the quasi-harmonic one. The form of the potential U(Qj) for the crystal a-N 2 calculated in [105] is given in Fig. II.7. The molecules of this crystal perform the angular vibrations with the amplitude -26 ~ It is evident that the calculation of the derivatives, of the 3rd and 4 th order describing the change of the potential U(Qj) only at an infinitely small molecular displacement from the equilibrium point, cannot take into account the rather complicated dependence of the crystal potential energy U(Q) on the angular displacements. At the same time the effective anharmonic potential (II.34) used in the independent oscillator model corresponds to the potential form for the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
117
U, kcal/mole f f
i
-1.5 \ "
,'
/
\'
O, grad ....
o
4'0
8'o
lio
1;o
FIG. II.7. Dependence of the potential energy of the crystal a-N 2 on the angular displacements of molecule corresponding to the vibration Eg [105].
Av, cm -1 0.0
-2.0
-4.0
I !
10
I "1
20
....
--'W
30
4'0 T, K
FIG. II. 8. Temperature shift of the frequency of the librational vibration Eg of the crystal a-N 2 [105]: (1) the quasi-harmonic calculation; (2) the experiment and (3) the model of independent anharmonic oscillators.
118
ZHIZHIN AND MUKHTAROV
sufficiently large range of the normal coordinates Qj. The consideration of the anharmonicity of librational vibrations in the frame of this model leads to very good agreement with experimental values (Fig. II.8) [105]. Another example is crystalline ethylene, which is more typical of the organic crystals. The efforts to perform the quasi-harmonic calculation for the external vibrational frequency of this crystal by means of a large number of sets of AAP parameters were made in [107,108]. The poor agreement with the experimental results was obtained in all cases. The most probable reason is, evidently, the influence of the "pure" anh~rmonicity, since the ~mplitudes of molecular displacements have comparatively large values (Azu = 11~ Azv = 10~ and Axw = 15 ~ Ju > Jv > Jw)We have carried out the calculation of the anharmonic corrections to the ethylene external vibrational frequencies according to the independent oscillator model with the use of the anharmonic corrections to the ethylene external vibrational frequencies. These calculations were performed according to the independent oscillator model with the use of the crystal structure, which was measured by the X-ray structural analysis method at 85~ [109] and with two sets of parameters (the empirical AAP/1 method and the "nonempiricar' AAP/13 (6-exp-1) method) and which has taken into account the electrostatic interaction (see Sect. II.B). The calculation shows that the dependences of the "precise" potentials U(Qj) on the normal coordinates have noticeable deviations from the quasi-harmonic potential (Fig. II.9). As is seen from Table II.19, the anharmonic frequency shifts have sufficiently large values, especially, in the case of the vibration ve(Bg) whose frequency calculated in the quasi-harmonic approximation has a strongly lowered value. The calculation with "nonempiricar' AAP gives, however, the absolutely unreal value of the anharmonic correction to this vibration (Table ]:[.20). It can be related to the large sensitivity of the frequency ve(Bg) to the electrostatic interaction, which seems to be inadequately described by means of point charges on atoms [39]. Unlike the AAP/13, the calculation with the use of AAP/1 leads to good agreement with the experimental results (Table II.19), namely, the mean relative deviation from the experimental frequencies is decreased by the consideration of the anharmonicity in the range from 10.2% to 5.3%. It has proven the more real description of the ethylene molecular interaction in a crystal by means of the empirical AAP. Thus, our analysis of the influence of the temperature factors on the external vibrational frequencies of molecular crystals has shown that the quasi-harmonic shift of frequencies is satisfactorily described by the AAP, and the quasi-harmonic calculation by the AAP method can be used for spectral data interpretation. It is necessary to show that it is possible to obtain lowered values for the calculated frequencies in comparison with the experimental ones in the range of high temperatures. The simple and convenient calculation technique in the approximation of
119
VIBRATIONAL SPECTRA O F O R G A N I C CRYSTALS
Bg
AU
Ag
AU
[1/2 ~Oo2Q2 i
2
2
111 f/
//-
v4
Q .
.
.
.
,
~...
"'T
0
1
2
-1
3
0
1 AU
AU, kcal/mole
!
I
\
////
V2
L
-i
!
1
0
-
i
]/
v5
1
2
i
2
0
AU \
2
AU
2
\\
1
v3
6
Q -2
-1
0
1
2
3
Q-3
0
1
2
FIG. II.9. One-dimensional cross-sections of the potential energy with respect to the normal coordinates of librational vibrations of the ethylene crystal; the cross-sections were calculated by means of the AAP/12.
independent anharmonic oscillators has been suggested. It permits one to estimate the anharmonidty of separate normal vibrations without the difficult and timeconsuming calculations. It was shown that the application of this approximation is the most effective in the case of the strong anharmonic vibrations for which the
120
Z H I Z H I N AND M U K H T A R O V
TABLE II.19 Observed a n d Calculated H a r m o n i c a n d A n h a r m o n i c cm -1 of E t h y l e n e v exp. 1 cm -1 Ag
v1
177
v2
97
v3
90
v4
Bg
167
v5
114
V6
73
v7
Au
Bu
71
v8
--
v9
108
Vharm. 2 cm -1
Vanh. cm "1
Av(3)
Av(4)
(1)
164.0
159.0
-2.1
-2.9
(2)
177.6
170.4
-4.9
-2.2
(1)
97.7
96.9
-4.1
3.1
(2)
90.5
90.9
-0.7
0.3
(1)
83.1
92.7
-0.1
9.7
(2)
71.2
71.7
-16.7
17.2
(1)
156.5
150.3
--
-6.2
(2)
202.5
194.2
--
-8.3
(1)
111.9
113.0
--
1.1
(2)
129.3
127.4
--
1.9
(1)
49.8
85.4
--
35.7
(2)
23.7
(200)
--
(200)
(1)
62.7
65.1
--
2.4
(2)
75.4
76.6
--
0.9
(1)
50.5
51.3
--
0.8
(2)
54.6
55.1
--
0.5
(1)
92.4
97.0
--
4.6
(2)
91.0
95.0
--
4.0
1The R a m a n spectra [107]; t h e i n f r a r e d s p e c t r a [110]. 2(1) a n d (2) are t h e r e s u l t s of calculations w i t h t h e AAP/1 a n d AAP/13 (at 85~ respectively.
calculations
not
only give the
qualitative
information,
but
the
reasonable
q u a n t i t a t i v e e s t i m a t i o n of frequency shifts. The a f o r e m e n t i o n e d model will be u s e d (see P a r t 2) in our i n t e r p r e t a t i o n of the l ~ m a n spectra of cyclohexane a n d p h e n a n t h r e n e crystals. In
conclusion
we
note
that
the
technique
of a n h a r m o n i c
correction
calculations can be, in principle, applied to t h e s t u d y of the interactions b e t w e e n
VIBRATIONAL SPECTRA OF ORGANIC CRYSTAI~
121
the separate normal vibrations. It is necessary to find the effective anharmonic potential in the form of a polynomial of the 4 th degree approximating the twodimensional cross-section U(Qjr Qj2) of the crystal potential surface. However great difficulties arise. They relate to the problem of the physically correct independent evaluation of a large number (in the general case it is equal to 9) of polynomial factors according to the least-squares method.
122
ZHIZHIN AND M U K H T A R O V
HI.
SOLUTION OF THE DYNAMICAL PROBLEM W I T H 22-IE C O N S I D E R A T I O N O F T H E INTERMOLEC~ VIBRATION
A. Display of the Intermolecular Interaction in Molecular Spectra of Crystals
The intermolecular interaction forces are represented in the Rsmsn and infrared spectra of condensed materials (liquids, solutions and crystals) by the different phenomena, namely, shifts and band splitting, changes of a contour, polarization and an intensity [1,2]. In spite of the wide application of these phenomena for the investigation of properties of materials [2-5], the mechanism of intermolecular interaction representation in the vibrational spectra is not clear; and it is, therefore, difficult to perform the correct interpretation of experimental data. For this reason we have considered the possibility of applying the AAP model to the description of the displacement AC0d and the Davydov splitting Ams of the intrsmolecular vibrational frequencies at the change of the gas-crystal state [6]. By means of the lattice dynamics in the generalized coordinate basis (see Chapter I) the splitting Ams and the displacement of frequencies Acod can be directly expressed by the intermolecular interaction potential, if we neglect the interaction of different normal vibrations of a molecule. For example, in the case of two translationally nonequivalent molecules in the unit cell, the dynamical matrix (Eq. 1.45) for some internal coordinate Q will be the 2x2 matrix:
)
1
The eigenvalue of this matrix
~'1,2 = ~1,2 =
-+ D( 1~
(III.1)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
123
where, according to (Eq. 1.460
D(101)=fex(~ ~)+ X fex(~ 1)+CO~ l#0
(III.2) D(102) = X fex(1 2) t COOis the frequency of the normal vibration Q7 of an isolated molecule, the force f(ex)(~'- ~) are expressed by the derivatives of the intermolecular constants interaction potential with respect to the coordinate ~
v
f(ex) (~ ~)=
b2U
10. (III.3)
Assuming that ACOs and ACOd << r
we obtain from Eqs. (HI.l) and (III.2) [7]
1 ACOs= col-co2 =~00 ~'t f(ex)(01 2)
coI +c~ ACOd = ~ ~ -
_ COo_ ~
(III.4)
f(ex)(01 1)+ 2 + 0 ~t f(ex)(1 1). (IH.5)
Thus, the factor-group frequency splitting (111.4) is determined only by the vibrational interaction of the translationally nonequivalent molecules, unlike the external vibrations whose frequencies depend on interactions of all crystal molecules. In the expression (III.5) the first term describes the frequency shift caused by the static force field of a crystal and depends on the interaction between the starting molecule and all surrounding ones (the "site"shift [7]). The second term describes the frequency shiR as the result of the vibrational interaction of translationaUy equivalent molecules. From the equalities (III.4) and (III.5)it also
follows that the values Acos and Acod are basically determined by the intermolecular interaction potential, since the elements of intramolecular nornm] vibration eigenvalues in the formula for f(ex)(~ ~) have comparatively small sensitivities to the choice of the molecular force field (see Chapter I). The above mentioned characteristics of shifts and frequency splitting qualitatively keep their values in cases where there are close values of vibrational frequencies in a molecule, and it is necessary to consider their interaction in a crystal.
124
ZI-IIZHIN AND ~ A R O V
A convenient object for the study of the intermolecular interaction representation in the spectnnn is ethylene whose vibrational spectra of the gas [7,8] and of the crystal [7,9,10]are preciselymeasured. The crystallinestructure is well known at 30~ (P21/n, z = 2(C i)[11]). The generalized force fieldof a molecule is ab initio calculated [12]. The symmetry of an isolated molecule is D2h, and the symmetry of the molecule in a crystalis C i. The correlation table for the internal vibration is of the form:
D2h
3
Ag
1
B2g
1
Au
1
B l u ~
Ci
Cgh
Ag ~
A u ~
the Raman spectra
Bg
Au
the infrared spectra
In the Ramon and infrared absorption spectra of a crystal it is expected to observe splitting of lines of each of twelve normal molecular vibrations into two components of A and B types. For the frequency calculation by means of the AAP one has used the most precise method of solving the dynamical problem on the basis of the rectangular coordinates of atomic displacement (see Sect. I.A), i.e. all external and internal vibrations were considered in the same time. The following expression
which is part of the formula for the force constant (Eq. 1.23) was obtained by means of numerical differentiation. The dynomical matrix ]5 of the ethylene crystal considering its symmetry (see Sect. I.B) is reduced to the four matrices of 9x9 dimensions; the diagonalization of each of them gives the frequencies of internal and external vibrations of the given type of symmetry (Ag, Bg, Au, Bu). The same force field for the isolated molecule and for the molecule in a crystal and the same geometry of a molecule (dc__C = 1.339/~, dc= H = 1.085 ~,
VIBRATIONAL S P E C T R A OF ORGANIC CRYSTAI~
125
TABLE HI.1 Experimental and Calculated Values of the Factor-Group Splitting and Vibrational Frequencies for Ethylene
Vgas
Avsa
[8]
exp.
Blg Ag Ag Ag
v5 vI v2 v3
3102.5 3026.4 1630 1342.2
Big
v6
B2g B2u
b
c
Avua
b
[7]
c [7]
2.4 -7.1 19.5
1.4 -1.1 3.6 7.1
3.5 1.5 2.2 14.5
0.6
1220
4.3
-4.5
5.2
-0.7
v8 v9
939.9 3104.9
9.9 --
1.9 -0.4
-1.0
5.4 --
0.7 -0.2
B3u B3u Au
vii v4 v4
3021 1443.5 1023
-3.7 5.6
0.5 1.4 5.4
-0.2 -0.3 0.0
-2.3 4.9
0.2 0.7 2.8
-0.2
Blu
v7
949.3
7.9
-6.2
9.3
6.0
-2.8
6.4
B2u
rio
826.0
5.6
3.1
0.1
3.9
1.6
0.0
-0.6
0.I 5.9
-0.6
-0.3 0.0
aR~mon spectra [10], infrared spectra [9], absolute values of Avs are given bThe calculation results of this work (AAP/1); CThe calculation by the model of the dipole-dipole interaction [7].
appears that the calculated molecular orientation in a crystal is almost the same as the experimental one. The calculated external vibrational frequencies coincided completely with the values obtained by solving the dynamical problem in the rigid molecular approximation (Sect. II.B). This is natural, bemuse the regions of the external and internal vibrations are separated by more than 600 cm -1. In Table III.1 the values of the factor-group splitting of internal vibrations Avs = VAt- VBg (the R~man spectra) and Avu = VAu - VBu (the infrared absorption spectra) are shown. Since the polarization measurements are absent, the absolute values of Avs are given as the experimental ones. As is seen from Table HI.l, the calculation by means of AAP/1 satisfactorily describes the experimental data.
It should be
mentioned that the measured large splitting value 19.5 cm -1 of the vibration v3(Ag) can be associated not with the Davydov splitting, but with the isotopic splitting due to the presence of atoms 13C [10]. The values calculated in [7] by means of a model of the interacting dipoles of the ethylene molecular transition (I.94) are also given in Table III.1. This model predicts the noticeable splitting only for the vibration v7(Blu) having the most
126
ZHIZHIN AND MUKHTAROV
TABLE Ill.2 Experimental and Calculated Values of the Frequency Displacement for C2H4 and C2D4 at the Gas-Crystal Transition C2H4
C2D4 ,,
AVexp
Areal9
[9,10] Blg
Ag Ag Ag Big B2g B2u B3u B3u Au Blu B2u
5 1 2 3 6 8 9 11 12 4 7 10
-35.4 -25.6 -5 -3.9 4 7.0 -15.7 -16.0 -4.9 14 -4 12.2
AVexp
AVcalc
[9,10] 2.0 3.2 2.9 4.7 4.0 8.1 1.7 2.5 5.1 15.2 4.6 8.8
-6 -15 -7 -5 (-11) 1 -16 -9.2 -6.3 19 2 6
1.2 2.0 1.4 4.6 2.3 3.9 1.0 1.7 3.4 11.2 3.3 6.1
q(C-H) q(C-H) q(C-C) (HCH), q(CfC) (CCH) CH 2 q(C-H) q(C-H) (HCH) CH 2 CH 2 (CCH)
intensive absorption band in the infrared spectn~n, and it cannot explain the frequency splitting of the other molecular vibrations. The values of the frequency displacement Avd = Vcryst - Vgas (Vcryst is the mean frequency of the Davydov doublet) at the gas-crystal transition are given in Table ILI.2, from which it follows that the calculation with AAP/1 predicts the shiit of all frequencies to the short wavelength region, but the experiment demonstrates the frequencies shift, for the most part, to the long wavelength region at this transition. There are also the calculated short wavelength shifts for all internal vibrational frequencies of the benzene crystal [13,14], the naphthalene crystal [15,16], the acenaphthene crystal [17] and so on. It is probable that this reflects the common phenomenon of the display of short range intermolecular interaction described by the AAP model without the consideration of the influence of other factors. One of these factors is, for example, the anharmonicity of the molecular force field. The problem concerning the anharmonic oscillator frequency displacement at the gas-condensed medium transition was first solved by Buckingham for a two-atom molecule [1,18]
~r
1 82U
2~Q2
3K3 ~U 1
(o0 aQ '
(IH.6)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
127
where U is the intermolecular interaction potential, Q is the vibrational coordinate, K3 is the anharmonicity constant, B is the rotation constant. The first term in UII.6) corresponds to the fresuency shift calculated by the AAP method, the second term corresponds to the additional frequency shift due to the change of the anharmonic oscillator equilibrium conditions. As is well known [191,the valent vibrations of molecular C-H bonds are the most anharmonic. It determines, probably, the largest wavelength frequency shift of these vibrations at the gascrystal transition (Table III.2). On the other hand, the influence of this Eactor is, evidently, small in the case of vibrations v6, Vg, v10 and v4 whose frequency shift6 are well reproduced by the calculation with the AAP. Consider the application of the AAP method for the interpretation of the intramolecular spectrum of the more complex phenanthrene crystal. Its structure is determined at 293°K [203; the space group is P21, z = ~ ( C I ) . "he phenanthrene molecule (C14H10) in a crystal is almost planar, i.e. the angle between the planes of the extreme benzene rings and the middle molecular plane is no more than 1". For this reason, we have used the model of the planar molecule with the symmetry C2". The appropriate correlation table is of the following form:
C1 planar 23 vibrations 22 nonplanar 11 vibrations 10
c2
A1
Bl\
A
Raman spectra B
Infrared absorption spectra
B2
Each of the 66 internal vibrations in a crystal is split into two components of the symmetry types A and B, and it is active in the Raman spectra and infrared absorption spectra of a crystal. The polarization measurements of these spectra for the monocrystal are made in Refs. 121-231. Some of the observed lines cannot be unambiguously assigned to fundamental vibrations, the Raman frequencies and the Davydov splitting frequencies [223. The application of the method of solving the dynamical problem which was used by us for the ethylene crystal is associated with very difficult and timeconsuming calculations, since it leads to the necessity of the construction of two dynamical matrices (for vibrations of the types A and B) of large dimension (the 72 x 72 matrix). For this reason, the method of the lattice dynamics calculation on the basis of generalized coordinates Q, was applied (see Sect. LA). As in the ethylene case, we have assumed that the planar conformation and the molecular force field
ZHIZHIN AND MUKHTAFtOV
128
does not change at the gas-crystal transition. First of all, we have solved the dynamical problem for the isolated molecule by the diagonalization of the dynamical matrix which was done separately for planar and nonplanar vibrations
--1
where d
2
--1
is the diagonal matrix with elements mp2 (q is the mass of an atom
k,. is the force constant matrix, (B)p,w=bp,crp is the matrix p), (6) = w' associating the MtUd coordinates q with the Cartesian coordinates of atomic displacements xap (1.22). This method of the molecular vibrational calculation is convenient because it directly gives the components of eigenvedors $p,p in the expression (1.46) for the crystal dynamical matrix elements. The force constants of a molecule for planar vibrations are taken &om 1223, and for nonplanar vibrations they are taken from 1241. The intermolecular spectrum was then divided by some frequency intervals which were separated by a distance of less than 40 cm-1. They contained &om t w o to ten internal vibrations. The calculation of the molecular vibrational frequencies for these i n t e ~ a l was s independently performed. Two sets of parameters of M I 1 and M I 2 were used. The calculations show that the normal vibrations of a molecule, whose frequencies differ &om each other by a value which is more than 5 cm-1,practically do not interact. In a crystal one has observed only the mixture of the nonplanar vibrations ~ 3 (B2) 1 and ~ 3 (A2), 2 v28(B2) and ~29(A2)and also of the C-H bond valent vibrations. As was shown in [223, the predictions of infiared absorption band intensities at different polarization conditions by means of the oriented gas model were experimentally verified. It also proved the weak interaction of normal molecular vibrations. The calculated results of the Davydov splitting hs= OA og of frequencies of a number of phenanthrene internal vibrations whose splitting was experimentally observed were given in Table III.3 (the calculated splitting of the remaining vibrations does not exceed 3 cm-l). Two nonplanar vibrations v 1 (B2) and v2 (A2) can strongly interact with the external crystal vibrations, since they have low fiequencies of 100 and 114 cm-l. The influence of intermolecular interaction on these vibrations will be considered in the following section. From Table III.3 it follows that the calculation by the AAP reproduces satisfactorily the spectral data. In particular, the sign of Avs is predicted in the right way, i.e. the relationship between the frequency values of the Davydov doublet components associated with the symmetry of the type A and B. The comparison between the calculated and experimental results permits one to also improve the interpretation of the vibrational spectrum suggested in [22].
-
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
129
TABLE III.3 Experimental and Calculated Davydov splitting values and the Frequencies of the Internal Vibrations of Phenanthrene *vexp AAPl1 AApf2 m12a Vexp [221 245 406 547 832 1164 1437 1608 2102 1142 1417 3024 3041 352
v4 v7 v13 AI
v23 v39 v50 v55 v65 v37 v47 BI v58 v61 A2 v6 v3 v8 VI 1 B2 v19 v22 v25 v28
228 427 497 735 815 865 947
[221
--
[211 6.8
7.5 7.5 -0.1 -1.5 -12.8 4.0 -0.2 10.2 -11.6 3.0 -10.3 -3.5
4 -9 -1 -1 -1 1 -1 1
-3 0 3
-0.2
0.9 -7.5 -4.2 -0.4 0.7 -1.2
-2.3 0.5
-1.8 -0.1
12
7.2
-1
-7 3
-0.1 -1.5 -15.9 4.1
2
-0.2
_-
8.7 -15.0 3.1 -6.1
-5
-6 5
--5 6
__
-4 -4 -3 -3 -4 -1
-4.3 0.8 -7.5 -2.8
-0.9 0.7
-2 4 0
-2
2 0
2 1 0
aThe calculation by the FG-matrix method. In this case one observes poor agreement with the experiment. It can be assoeated with the simplified force field of a molecule, with the small radius La= 4.2 A of summing up (we have used the value ha = 7 and also with the approximated character of the FG-matrix method used by the authors of [211 (see Sect. 1.A). Our method of solving the dynamical problem on the basis of generalized coordinates Q, thus, provides the more precise description of spedral data,and it is more simple than the FG-matrix method from the point of view of the calculation.
A)
Thus, the infiared absorption bands at 1142 (A) and 1148 (B) are assigned to the molecular vibrations of the types A1 and B1, respectively [221. These bands have approximately the same intensity, and we have assigned them to components of the doublet 1142 (A) - 1148 (B)cm-l of the normal vibration v37 (B1) whose calculated splitting value is from (-12) to (-15) cm-l. In the infrared absorption spectra one also observes the band at 1172 (B) cm-l which has no interpretation in [22]. We
130
ZHIZHIN AND MUKHTAROV
-
have assigned this band to the component of the Davydov doublet 1165 (A) 1172 (B)cm-1 of the molecular vibration v39 (All whose calculated value was equal to 13 - 16 cm-l. The good agreement with the spectral data was also obtained by the analogous calculation of the Davydov splitting of the benzene crystal [13,141 and the naphthalene crystal 1161. The values of phenanthrene calculated by the FG matrix method [211 were given in Table III.3. It is a pity that the experimental data concerning the vibrational spectrum of the phenanthrene molecule in the gas phase are absent. For this reason, our calculated values of the frequency ShiR Ava at the gas-crystal transition are not given here. We only note that the calculation by the AAP predicts the shift of all frequencies in the short wavelength range as in the case of ethylene. The value of this shift for nonplanar vibrations (11.4 cm-1, in average) is considerably larger than for planar ones (3.5 cm-1). Thus, the results of our investigations show that the factor-group splitting of the internal vibrational frequencies is adequately described by means of the AAP model. The calculations by this method can be very useful in the interpretation of the tine structure of spectral data. On the other hand, solving the dynamical problem by means of AAP in the harmonic approximation was insuflicient for the explanation of frequency shifts at the gas-crystal transition. Thus, it was associated with &he influence of the additional effects, namely, the internal vibrational anharmonicity,the molecular deformation, the light field effects etc. [l]. Nevertheless, the results of such calculations are of interest, since they allow us to determine the contribution of these effects to the frequency shifts of the different normal vibrations of a molecule.
B. Interaction Between the Internal and External Vibrations The calculations of lattice dynamics in the rigid molecular approximation based on the use of a small number of the universal AAP parameters and having, consequently, the undoubtfid advantages for the objective interpretation of spectral data are, in general, limited to crystals of comparatively simple compounds. In multi atom molecules there are, as a rule, one or two normal vibrations with low frequencies whose values are in the region of the crystal external vibrations (0 - 150 cm-1). In this case the strong interaction of the vibrations of two types and the formation of qualitatively new mixed normal modes in a crystal are possible. In particular, one can expect the appearance of some peculiarities and the dependence of frequencies and line intensities of such modes on temperature 1253. Consider first the mixture of normal vibrations in a crystal with one molecule in the unit cell and one low-frequency internal mode with frequency vo in the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
131
qualitative way (for example, the n-octane crystal, PI, z = I 1261, vo = 148 cm-l[27]). In this case there are only three external librational frequencies at K = 0, and the dynamical matrix, on the basis of generalized coordinates Q,, is of the simple form
D=
Dii D12 Di3 Di4
D12 D22 D23 D24
Di3 Di4 D23 D24 D33 D34 D34 DL:)+oi
We have noted by dashed lines the dynamical matrix block which relates to the external vibrations and coincides with the dynamical matrix in the rigid molecule approximation. D(0) determines the shift of the internal vibration frequency 00 at the gas-crystal transition (see Sect. IIIA). By means of the unitary transformation of external generalized coordinates $&(ex) this matrix can be represented in the following form
(111.7)
where R, are the frequencies of the external vibrations in the rigid molecule approximation, S is the appropriate matrix of eigenvectors. The nondiagonal elements D'24 describe the interaction between the internal vibration and the the general case all elements of the unitary transformation S external one. ~n diagonalizing the matrix (111.7) are nonzeros, i.e. the inclusion of the interaction of the internal and external modes leads also to some interactions between the "rigid normal vibrations. If we assume that D'14 = D'24 = 0, D'34 = A C 0. Then 01 = R1, 02 = R2 (qare the normal crystal vibrational frequencies). In order to find 03 and 04, it is necessary to determine the eigenvalues and eigenvectors of the following matrix
Hence, the mixing between the internal vibration Q4 and the external 6 ' 3 in the normal modes 0 3 and 0 4 can be evaluated by means of the relationships
132
ZHIMIN AND MUKHTAROV
(III.8)
where S3(ex) and &(ex) are the components of eigenvectors of vibrations 03 and 04 and S4(h) are the same components related to the external coordinate Q3; related to the internal coordinate 64.k o m the equalities (III.8) it follows that a3 = -1 and aq = 1 at (III.9)
i.e., the normal modes are represented by the complete mixed vibrations in which the internal and external displacements of a molecule are taking part to the same extent. The value of the fi-equency interval A at which the mixture of these vibrations become noticeable depends upon the interaction constant A. Thus, the applicability of the rigid molecule approximation in every case can be different and is determined by the intermolecular interaction peculiarities. Besides, from (III.9) it follows that how well this approximation fits must be estimated not by the internal vibration fiequencies of an isolated molecule, but by considering the shift of this frequency at the gas-crystal transition. T h e relationships (III.8) show that the vibrational form is sensitive to the internal vibrational hquency values ~0 of an isolated molecule, i.e., the correct quantitative calculation of eigenvedors of low-frequency crystal vibrations requires not only the AAP parameters, but sdiciently precise data concerning the force field of a molecule. In order to obtain a better agreement with the experimental values the frequency values 00 are often varied during the computation procedure [28,29]. This can lead, generally speaking, to the subjective error of the interpretation of the vibrational spectra. We have studied the interaction between the internal and external vibrations of the naphthalene crystal (P21/c,z = 2 ( q ) [30]), the phenanthrene crystal (P21, z = ~ ( C I[20]), ) and the pyrene crystal (P21/c, z = ~ ( C I[31]). ) Each of these molecules has two nonplanar vibrations at low frequencies: 175 (BI,) and 189 cm-1 (A,) (naphthalene), 100 (A2) and 125 (B2) cm-1 (phenanthrene) and 96 (B3,) and 170 (A,) cm-1 (pyrene) [19,24] which can interact with external vibrations (0 - 130 cm-1). The solution of the dynamical problem was carried out on the basis of the
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
133
generalized coordinates Qj (see Chapter I), i.e. we have suggested that the conformation of molecules and their force field are not changed at the gas-molecule transition. Only the above mentioned internal modes of a crystal were considered. The calculations for phenanthrene show that the consideration of the other internal vibrations with frequencies more than 200 an-1 practically does not influence the external vibrations. For calculations we have used two sets of parameters of the AAPl1 and the M I 2 and also the valent force field which is the same for all three molecules and was suggested in [MI for the nonplanar vibrations of aromatic hydrocarbon molecules. The calculated forms of internal low-l%equencyvibrations of the isolated molecules of naphthalene, phenanthrene, and pyrene are shown in Fig. III.1. The vibrations with the least frequencies 175(Blu), lOO(A21, and 96(B3J cm-1 are related to the vibrations of the type "of butterfly flight"; the others are related to the vibrations at which the twisting of a molecule along its long axis is performed. The correlation table for the considered crystals are of the form 1. Naphthalene D2h
Ci
3t Au Q2 (Blu) 2. Phenanthrene c2v
C1
C2h
<
Au BU
The infrared absorption spectra
c2
30
"he Raman spectra, The infrared spectra
3. Pyrene D2h
C1
CZh
The Raman spectra
The ifiared absorption spectra
ZHIZHIN AND MUKHTAROV
134
100 032) /+\+
+I
''' I
- /-\-/-
+/+\-/-
0.16 -
I ''10.30
0.16 -
10.13
'+0.33
0.34 -
I '-0.25
+ 0.35
I
+ 0.14
0.22 t
+'
'-/
0.05
0.40
/
'
f
\+/
1
\+
i\+4.-h+$o I I -1- 0.25
4-
1,, I0.21
0.40
-
+
\ 0 -0 /
'
-
+' 0.35
\ - -. /0.28 C
FIG. III.1. Low-frequency nonplanar vibration form of phenanthrene molecule, pyrene molecule and naphthalene molecule.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
135
TABLE III.4 Experimental Frequencies for the Internal Modes of Naphthalene, Phenonthrene and Pyrene, and Experimental and Calculated Frequency Shifts VOa VO AVS AVa CalC. exp.b M A MIZ exp.c MII MIZ exp.
-_
189.0
3
0.8
1.6
--
28.0
29.5
Blu
166
175.1
15
10.4
8.9
18.3
14.3
14.6
A2
(140)
125.0
-_
-5.2
-9.2
--
35.2
40.2
€32
(123)
100.2
16
9.2
1.2
17.8
18.5
--
1
0.0
1.3
28.0
26.9
170.0
--
2.0
2.1
----
10.0
11.5
10.3
--
37.5
34.2
96.0
_-
9.6
(124)
-10.9 -12.2
--
28.6
27.3
AU
AU
B3u
-6
aNaphthalene [28,341,phenanthrene 122,231,and pyrene [351;the frequencies observed in solutions are given in brackets. bNaphthalene 1281, pyrene and phenanthrene (see Part 11, Chapter V). CSeparately for the Davydov splitting components which are active in the-and infirared spectra.
where 9, t, Q are the coordinates of the librational, translational and internal vibrations. The interactions between all external vibrations and the low-frequency modes of a molecule can be in the crystals of phenanthrene and pyrene. In the case of naphthalene, only the mixtures of these modes with the external translational vibrations of the symmetry types A, and B, can occur at = 0. For t h i s reason, it is necessary to note that we have explained the strong interaction between the internal modes and the librational vibrations of naphthalene (observed in 1321 in the calculation by the AAP method) by means of the results of the earlier work [331 in which the internal vibration at 189 (B1,) cm-1 was wrongly assigned to the Bag type vibration (the lowest vibrational frequency of this type is actually equal to 400 cm-l [19,241). Our calculated results are given in Table 111.4 and 111.5;they are also shown in Fig. III.2. The frequencies of the internal modes 125 (A21 and 170 (A,) cm-1 obtained by means of the valent force field for phenanthrene and pyrene [24]were decreased by 114 and 147 an-1 for improving the agreement with experiment in our work (such change of frequencies does not influence the interpretation of spectra of these crystals). From Table III.4 it follows that the frequencies of the molecular internal modes have considerably short wavelength shifts (up to 40 cm-1) at the gas-crystal
ZHIZHIN AND MUKHTAROV
136
TABLE III.5 The InteractionBetween the Internal and External Vibrations in the Crystals of Naphthalene, Pyrene and Phenanthrene (& = 0). contribution of internal vibration,
v, cm-1
expo (a)
0)
(a)
(b)
3
4
5
6
1 2 1. Naphthalene 212 192 ~7 98 AU vg 49 BU 210 177 vg 65
216.6 194.8 85.6 39.7 217.4 184.4 54.1
2. Phenanthrened 142 125 106 84 60 47 (32) (142) 109 101 85 60 32
145.7 121.3 108.6 79.8 63.7 49.9 34.0 150.5 112.4 92.5 76.9 50.1 22.3
3. Pyrened 170 v1 v2
(127) 95 77
~3
56
Ag vq ~5
v6
(46) 30 17
175.1 138.4 105.0 91.6 57.7 45.9 36.7 18.5
--86.3 43.1
--
__
52.3
--
__
110.7 80.9 64.1 53.7 34.6
_-
__
97.2 77.9 50.3 22.8
--106.1 91.7 59.4 47.7 39.0 19.1
219.3 191.2 81.5 40.0 217.6 182.3 52.4
149.6 117.8 132.5 83.9 68.5 52.1 34.4 158.8 119.1 99.9 80.8 49.2 25.6
173.1 135.3 99.8 88.7 60.3 47.3 36.8 17.5
__ -82.6 43.5
_--
52.5
_-133.1 84.9 72.2 55.3 35.8
_-
113.1C 84.0 51.2 26.8
--
_100.3 89.0 62.6 48.5 39.5 17.8
% (AAP/l)
7
61(A), 93.1 6.2 0.4 0.2 99.4 0.6 0.0
61(A21 93.7 5.2 0.2 0.0 0.0 1.0 0.0 94.1 4.1 0.7 1.1 0.0 0.0 Q1(A),
98.6 0.4 0.4 0.1 0.0 0.5 0.0 0.0
8 Q2 (B1u) 6.2 93.1 0.0 0.6 0.6 99.4 0.0 6 2 (B2) 4.2 76.9 14.1 2.1 0.5 1.9 0.4 5.0 73.1 20.0 1.4 0.1 0.2 Q2 (B3u) 0.3 96.3 1.2 0.0 1.1 0.1 1.1 0.0
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
137
TABLE III.5 (continued) v, cm-1 exp. 1
2 169 126 v7 (95) v8 (77) vg 67 Bg v10 41 ~ 1 1 (30) v12
--
--
contribution of internal vibration, 56
AAp/'IAAp/2
(a) 3 175.1 128.8 108.8 90.9
68.5 44.0 33.0 21.4
(b) 4
__
-109.8 91.1 69.1 44.6 34.3 24.4
(a) 5 174.4 125.0 103.6 97.2 68.9 46.3 33.6 22.3
(b) 6
__
-104.7 87.7 69.6 47.1 34.9 24.4
(AAPf1) 7 98.0 0.8 0.7 0.1 0.0 0.1 0.2 0.1
8 0.8 96.9 0.8 0.1 0.7 0.1 0.0 0.6
__
159.3 -100.0 0.0 117.2 -_ 0.0 90.2 113.0 1095 109.5 0.0 8.8 81.3 74.6 75.0 0.1 0.0 -49.3 48.7 49.9 0.0 0.8 15 -_ 46.7 47.4 47.6 0.1 0.1 16 19.7 -19.1 19.8 0.0 0.1 17 _158 157.2 -100.0 0 .o BU -129 129.4 -0.0 98.6 105 112.0 105.8 105.8 0.0 0.7 v18 89 78.5 73.2 73.3 0.0 0.0 v19 _51.7 49.0 50.2 0.0 0.7 v20 -31.2 30.4 30.4 0.0 0.0 v21 aThe calculation with the consideration of internal vibrations. bThe calculations without the consideration of internal vibrations. This vibration is mixed to a large extent with the internal vibration of type B2 at the calculation with W f 2 . dThe experimental values (see part 11,Chapter V.B). 123 ~ 1 3 102 (71) Au v14
157.5 119.2 112.3 81.1 48.3 46.3 19.1 155.5 130.0 111.9 78.5 50.7 31.2
--
transition. For example, the frequency of 96 cm-lof the vibration of pyrene for an isolated molecule is less than the frequency 113 cm-l of the external vibration ~ 1 3 (A,) (Table III.5). It is shifted up to values in the range 119 - 138 cm-1in a crystal, and it becomes more than the external vibrational fyequency. Such behavior must lead to the decrease of the interaction between the internal and external modes in a crystal. In fact, the consideration of intermolecular vibrations does not practically influence either the frequency or the form of external vibrations of naphthalene and
138
ZHIZHIN AND MUKHTAROV
pyrene crystals. One has observed only the systematic decrease by an average of 1.5 cm-l for these vibrational frequencies in comparison with the calculated results in the rigid molecular approximation. In crystalline phenanthrene the calculation predids the interaction between the internal vibration at 100 cm-1 (B2) and the external high frequency vibrations vl and vg associated mainly with the molecular librations around the axis with the least inertial moment (in the rigid molecular approximation). In particular, this interaction is noticeable for the vibration V6 (B) which is completely mixed with the internal mode of 100 (B2) cm-1 in the calculation with the M I 2 parameters. In this case the calculated normal vibration of type B at 119.1 and 99.9 cm-l can not be unambiguously assigned either to external vibrations or to internal ones. In Fig. III.2 we have shown our dispersional dependences of normal modes of phenanthrene calculated by the AAP method. The low-frequency internal modes have the noticeable dispersion, and they are identical to the external (phonon) lattice vibrations in this sense. The dispersion of these modes in the Brillouin zone exceeds, as a rule, the values of their Davydov splitting (the so called "hidden" dispersion [281). We note that the interaction between the internal and external vibrations of the phenanthrene crystal decreases by increasing the wave vector, and it is practically absent at the boundary of the Brillouin zone. The polarization measurements of the temperature dependence of the lowfrequency Raman spectra were made on the phenanthrene monocrystals at 80 370°K in [37,38]. The polarized Raman spectra b(c*b)a in which the vibrations of type B are active are given in Fig. III.3. It is seen that the line at 115 cm-l is subject to the influence of the anomalous temperature dependence in the region from 120"to 280°K. (In our measurements we have obtained the value 109 cm-l at 300°K see Part I, Chapter W.B.) There is an intensity transfer from this line into the line at 101 cm-1. This phenomenon is, probably, related to the change of this vibrational form, since the calculation by the AAP predicts their mixed character. According to formula (III.8) the frequency interval and the interaction constant A of the internal and external modes are, evidently, changed by the change of temperature, and their relative contribution into the normal vibrations at 90.8(B) and 101(B) cm-1 of a crystal is also changed, respectively. In the Raman spectra of liquids the lines corresponding to the low-frequency nonplanar vibrations of the phenanthrene molecule have slightly noticeable intensity [21], i.e. the change of the polarizability of a molecule is small at such vibrations. For this reason one can suggest that at low temperatures of a crystal the internal vibration at 100 cm-1 (B2) is mixed with the external one at 97.2 cm-1(Vg) in the most strong manner, and the line at 109 cm-1 is the most intensive one. By increasing the temperature, the contribution of the external vibration of the normal mode at 109 cm-1 is decreased, and the intensity of the appropriate line is decreased. As is seen from the equality
139
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
v, cm-1
0.0
~
0.2 K = [OlO]
0.4
0.0
--f
K
0.2
0.4
= [OOl]
FIG. III.2. Dispersion dependences of the internal vibration frequencies (1) and the external vibration frequencies (2) of the phenanthrene crystal.
(III.8),the variation of parameters A and A can lead to sufficiently sharp changes to the vibrational form in the region of the resonance (A z 0). We note that the authors of [37,38]had no explanation of the phenomenon observed in phenanthrene. This can be accounted for by the fact that the authors have used the wmng interpretation of the vibrational spectra of the phenanthrene molecule [211 for the analysis of their spectral data 137,381. According to [Zll,there are four internal vibrations at 120, 92, 47 and 37 cm-1 which are several times lower than the
140
ZHIZHIN AND MUKHTAROV
cn1”
150
100
50
0
FIG. III.3. Temperature dependence of the low-frequency Raman spectrum of the phenanthrene monocrystal 137,383.
frequencies 242,227, 125 and 100 cm-1calculated in our work. However, there is considerably better agreement between our data and the calculated results for anthracene whose spectrum is analogous to the vibrational spectra of phenanthrene [221: 253,227,123,93 cm-l[241 and 321,235,127 and 96 cm-lC391. Thus, our analysis permits one to come to the conclusion that the rigid molecule approximation is sufficiently good for conformationally stable molecules such as the aromatic hydrocarbon. “his approximation can be adequate even in cases where the internal vibrational frequencies are less than the maximal external vibrational frequencies. Two mixed normal vibrations at 109 and 101 cm-1 are observed in the phenanthrene crystal. It was shown that in the vibrational spectra such mixed vibrations can be determined by means of the anomalous dependence of the appropriate line intensities on the temperature.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
141
C. Calculation of the Low-Frequency Spectra of Crystals with the Conformationally Unstable Molecules (Half-rigidMolecular Approximation) Molecules which have internal rotation around one or several bonds can considerably change their conformation at the transition from the free state to the crystalline one. The rigid molecular model can be inadequate, and the application of the ordinary consideration methods of intermolecular vibration (Sect. III.A and 1II.B) is very difficult, since they are based on the use of the results of the analysis of a force field and free molecular vibrations for the equilibrium conformation. The classic example is biphenyl C6H5-C6H5 whose molecule is planar in a crystal, and in the free state the phenyl groups are rotated by 42" around the C-C bond [40]. The planar conformation corresponds to the intermolecular energy maximum with respect to the internal rotation, the appropriate force constant is negative, and the molecular vibration in a crystal will be unstable without the consideration of the intermolecular forces. In the general case, the dynamical problem for similar crystals can be solved by means of the method of the calculation of the whole vibrational spectrum (Sect. IIIA) with the difference such that the force constants of a molecule can be determined by fitting them to the experimental vibrational frequencies with consideration of the intermolecular interaction influences. The consistent system of intra- and intermolecular potential parameters must provide the fulfillment of conditions of the atomic equilibrium in a crystal. However, the obtained force field of a molecule with the nonequilibrium conformation can have no transferability to other compounds, and the necessity of fitting large numbers of parameters arises every moment. The effort of solving this problem for cyclo- and n-alkane crystals was undertaken by Warshel and Lifson [27]. They have used the "consistent field" (28 parameters). The main feature of this field was the application of the same AAP (69-1) for the description of both the intermolecular and nonvalent intramolecular interactions. This permits one to automatically consider the change of the molecular force field by changing its conformation. Unfortunately, the authors of 1271 fail to reproduce the spectral data with the accuracy which is needed by scientists (the discrepancy is more than 200 cm-1). The satisfactory agreement was achieved only in the case of the low-frequency spectra in the region of the external vibrations (0 - 200 cm-1). This indicates that it is necessary to have more simple methods for the separate low-frequency vibrational calculations of conformationally unstable molecules which are based on the approximation of the separation of high and low frequencies. In this approximation which was called the half-rigid molecular
ZHIZHIN AND MUKHTAROV
142
approximation 1441 one has considered only a small number of the internal degrees of freedom describing the relative displacements of the rigid part of the molecule (for example). The half-rigid molecular approximation and the AAP were successfully used for the interpretation of the low-frequency spectra of the biphenyl f41-431 and durene crystals [441. In particular, it was found that the normal vibrations of biphenyl which are active in the infrared spectra cannot be divided into the external translational and internal torsional vibrations of the rigid phenyl groups. The results of these studies have shown the necessity for further development of the half-rigid molecular approximation requiring the introduction of the special generalized internal coordinates and the calculation schemes which are ditrerent, to some extent, from the common ones used in the theory of the molecular vibrations [451. Consider the application of the half-rigid molecules in the general case. Let n be the number of the considered internal degrees of freedom and xi be the appropriate generalized coordinates. According to the general formula (Chapter II), for the determination of the dynamical matrix elements by the AAP method it is necessary to find the relation between xi and the Cartesian displacements Zk of molecular atoms in their local coordinate system and to determine the kinetic energy matrix $2. We limit ourselves to the linear dependence of %k onxi.
(III.lOa) or in the matrix form:
A
is the 3N x n matrix (N is the number of atoms in a molecule) whose where elements are the vectors ak. Hence it follows:
(III.11) where mk is the mass of the atoms. The Lagrangian part of the crystal corresponding to the change of the internal coordinates (III.12) where Uh is the force constant matrix. The first derivative of the intermolecular energy with respect to xi, which is nonzero for the nonequilibrium molecular
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
143
conformation, is compensated for by the appropriate derivative with respect to the intermolecular interaction energy. Introducing the new internal coordinates xi(S)
(III.13) we can represent (III.12) in the form
(III.14) where
is the eigenvalue, and 8 is the matrix of the eigenvectors of the matrix 1 The expression (III.10) is, then, transformed to
(111.15) Thus, solving the dynamical problem with the use of basic coordinates xi(s) coincides formally with the method described in Sect. 1II.B. The difference is that the eigenvalues xi can be negative, and they should be considered as the varied force constants of a molecule in the basis xi(S). However, the coordinates form, as before, the basis of irreducible representations of the site group symmetry of a molecule in a crystal. This permits one to simplify the dynamical matrix structure at & = o . The special feature of the half-rigid molecular model is the finding of the relation between the Cartesian displacements of atoms xk, the generalized coordinates xi(s) (III.15) and the kinetic energy matrix 91 (or reduced mass coordinates xi). Consider the most widespread case where the conformational instability is related to the internal rotation around the ordinary bond. Let i be the n* bond which joins (n + 1) rigid parts of a molecule. The position of the ith bond in a space will be defined by the unit vector fi; and by the radius-vector Ri directed fiom the mass center of a molecule to the arbitrary point on a line through the ith bond. The generalized internal coordinates will be defined in the following form
(III.16) where Ah' and A@i" are angles of rotation from the equilibrium position of two parts of a molecule separated by the ith bond around this bond.
ZHIZHIN AND MUKHTAROV
144
The choice of the bond direction iii defines the sign of the change of this angle and the coordinates xi. The Cartesian atomic displacements of two molecular parts 2; and Z; in the motionless coordinate system coinciding with the main inertial axes and with the molecular mass center in the equilibrium condition are defined by
k = 1, ... ,Ni’
zk =[iii,ik-fii]
2; =
-[a, ,z;
(III.17)
-ai]
1 = N{ + 1, ... ,N
where ii and 2; are the radius-vectors of atoms k and 1 of two parts of a molecule; the brackets mean the vector product; Ni’,Ni” (Ni’+ N< = N) are the numbers of atoms in two parts of a molecule. For the determination of the atomic displacements in the coordinate system, which is moving with a molecule, it is necessary to subtract the displacements related to the molecular rotation by an angle A6 and to the displacement of its mass center by i. The latter can be found &om the condition such that the molecular angular momentum change and the molecular momentum change are equal to zero. We get the following expressions if we use the infinitely small displacements instead of derivatives 4, A;, h
. 7,
A‘
-*
= ri+ri,
M, = Mi +Mi
(III.18)
where Mi’, Ji’, Mi’’, J<, Mo; 3, are masses and tensors of inertia of two parts of a molecule and the whole molecule is in a system of its main inertial axes, are the radius-vectors of the mass centers of two parts of a respectively. and molecule. If the internal coordinates xi are changed, the atomic displacements are the following: 2; = Z k -[AG1
?;]-A,
k = 1, ... ,Ni’
2; = 2; -[AG1
?;]-A,
1 = N;’+ 1, ... , N
Using expression (III.18) for A6 and
we finally get:
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
145
(III.19)
where
-,,
M:'
_ n
-
B. = L [ i i i , R i -Ri].
Mo
By putting 2, fkom (III.19)into (III.11)one gets the complicated expressions for the elements of the kinetic energy matrix 'fx.For this reason, it is convenient 4 priori to calculate I ,from (III.19)in the general case. The values Ikiare directly used in the calculations by the AAP method. The calculations are s d c i e n t l y simple, if the coordinates and the atomic masses of a molecule are known. However, in some cases it is convenient to use the concrete expressions for the reduced masses in the interpretation of the spectrum with respect to the isotopic relationships. The problem is essentially simplified, if the lines coinciding with the bonds around which the rotation occurs are directed through the mass centers of the molecular parts. From (III.19)we find
(III.20)
ZHIZHIN AND MUKHTAROV
146
h m (IlI.20) in (III.ll),we get the sufliciently simple expressions for the Putting elements of the kinetic energy matrix +x :
(III.22)
In (III.22) ai' and jj' are the tensors of the moments of inertia of molecular parts having no common atoms. Consider the application of the half-rigid molecular method to the AAF' calculation of the low-frequency crystal vibrations in the example of p-xylene CH3C6H4-CH3 in a more detailed way. According to the X-ray s t r u c t d analysis data at 163°K (see Sect. IV.B of Part 11),the p-xylene crystal has the space group P 2 1 h z = 2(Ci). Introducing two internal generalized coordinates
where A $ M and ~ ~ A+1 are the rotation angles of the methyl group I and the rest of the molecule around the C-C axis. Let k be the force constant of the torsional displacement of the methyl group from the equilibrium position in a crystal. Considering the site symmetry and neglecting the interaction force constants we get the following expression instead of (III.12)
q = +cil'%2,(
)(. ) - 3
Tll
212
212
211
x1 x2
l(xlxz) (k
(III.23)
(III.24) where J Mis~the inertial moment of the methyl group at its rotation around the C-C bond. (The rotation axes of the two methyl groups are localized at one line coinciding with the axis U of the least inertial moment J, of the molecule, 61 = -ii2). "ransfomtion of (III.23) to the form (III.14) by means of the transition to give the new generalized coordinates (III.15)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
s=-( 1 1
Jz
1 1 1 -1
147
)
(III.25b)
Fi2),
Considering that ZL1) =we obtain the relationship between the Cartesian atomic displacements and ~ 0 0 r d i ~ t x(Au) es and z(1) = -
k
1 [ f i y ~ k l ) ] X ( A u = g p ) X(Au
K
ZHIZHIN AND MUKKTAROV
148
(III.26)
where ZL”, ip’,tkbne) are the atomic displacements of methyl groups and the benzene ring, respectively.
A
We have used the matrix with elements of (III.26) and the force constants (III.25b) in the basis ,(%) and x(*u) for the calculation of dynamic matrix elements of p-xylene crystal with the consideration of the internal torsional vibration of methyl groups. The values k were obtained from the expression of the internal rotation potential of the methyl group (the details are given in Sect. lV.B, Part 11):
k = 18 v6 cos 6 80
(III.27)
where v6 is the barrier of the internal rotation of the methyl group (v6 = 14 d m 0 1 [45]), 80 is the angle between the equilibrium orientations of methyl groups in a crystal and in the free state. One should expect thirteen normal vibrations (at = 0) in the low frequency
crystal spectrum of
r‘ = 4Ag+ 4Bg + 3Au + 2Bu.
Site Symmetry
Factor Group C2h
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
149
Table III.6 Normal Vibrational Frequencies for the P-Xylene Crystal Calculated by the M I 3 at Different Conditions rigid molecules d = 1.05 A
d = 1.05 A
95.8 73.3 43.0
107.9 81.9 47.6
105.1 82.8 54.7
98.3 78.9 55.2
111.1 89.8 70.8
108.4 91.1 71.4
A,
61.9 32.9
75.4 38.9
73.0 34.8
B,
52.4
56.4
54.5
Ag
B,
with consideration . . . of intermolecular rmmrmzedture d = 1.05 A d = 1.05 A d = 1.05 A motionless com lete k=O calAation benzenerinp 148.0 96.0 64.7 53.3 137.1 102.8 84.5 55.3 138.3 73.0 32.6 127.2 51.8
136.6 96.2 68.6 54.4 126.1 102.2 85.5 56.8 136.0 72.9 32.5 124.8 51.6
138.8 96.5 69.7 54.4 127.7 102.5 85.8 58.1 138.3 73.0 32.6 127.2 51.8
According to the previous correlation table the internal "vibration" x(k)interacts only with the librations, and x(*U) interacts with the molecular translations. The normal vibrational frequencies of the p-xylene crystal calculated by the AAp/3 at different conditions (163°K & = 0) are given in Table III.6. Since the X-ray data gave values of the C-H bond length that were too low (0.93 - 1.02 A), the changed coordinates H corresponding to d = 1.05 A (see Chapter II) were used for the calculation. This leads to a considerable increase of the external vibrational frequencies up to 15 cm-1 (Table III.6). The radius of summation was chosen as 7 A in the calculation by the AAP method. As a rule, the external vibrational frequencies calculated with the minimized structure differ only slightly from the frequencies obtained with the molecular orientation obtained experimentally. This provides proof of the correct description of the molecular packing by means of the AAP13. As will be shown later, the consideration of the torsional vibrations of the methyl groups leads to the significant change of the external vibrational frequencies (in comparison with the rigid molecule approximation)which is very noticeable in the case of the librational vibrations. Thus, the frequencies 82.8 (Ag), 71.4 (B,) cm-1 are decreased to values of 69.7 and 58.1 cm-1, respectively,
ZHIZHIN AND MUKHTAROV
150
in the consideration of the internal vibrations. Consequently, the consideration of the torsional vibrations of the methyl groups in the given case is very necessary for the adequate description of the low-fiequency spectrum. The calculation with the empirical constant (k # 0) estimated by (III.27) and the calculation with k = 0 = 0, = 0) lead practically to the same frequencies as shown in Table I11.6. Thus, the torsional vibrations of the methyl groups are defined only by the intermolecular forces, and in this sense they have much in common with the external vibrations of a molecule. In the case when one of the molecular parts has a moment of inertia which is considerably less than the moment of inertia of the whole molecule, one can significantly simplify the expression for the atomic displacements (III.191, (III.25), (III.26) assuming, for example, that JM$J~ = 0. However, from the results of the pxylene calculation (Table III.6) it follows that in spite of the fact that the moment of inertia of the methyl group is smaller by a factor 30 than the moment of inertia of the molecule, neglecting the displacements of the atoms of the benzene ring considerablyinfluences the normal vibrational frequencies (the difference is up to 10
(h
cm-1).
We have used the above mentioned half-rigid molecular approximation for the interpretation of the low-frequency Raman and infrared absorption spectra of the p xylene and toluene crystals by means of both the calculated results of the vibrational frequencies and forms and the isotopic relationships, Raman spectral intensities and the reorientation barriers of the methyl group (see Sect. IV.B of Part 11). In conclusion it should be mentioned that the half-rigid molecular approximation including a small number of empirical parameters is rather attractive for the description of the polymer low-frequency spectra, although one cannot, evidently, be limited only to the internal rotational coordinates in the case of macromolecules.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
151
IV. DEVICES AND METHODS FOR THE EXPERIMENTAL INVESTIGATION OF ORGANIC CRYSTALS BY MEANS OF THE VIBRATIONAL SPECTRA Two spectra (Raman and infrared) which complement one another permit one to obtain complete information concerning the vibrations of a crystal, since the selection rules for transitions between the vibrational levels are different in the cases of scattering and infrared absorption. There is no direct relationship between the scattering intensity and the absorption intensity even in the cases when the bands can appear in both types of spectra, and there is no symmetry. If some main mode has low intensity and cannot appear in the Raman spectra, it can be identified by means of the infrared spectrum, and vice versa. For t h i s reason, it is necessary to use both methods for the assignment of all vibrational frequencies. The translational vibrations of molecules in a crystal are displayed, as a rule, in the far infrared spectra. During the last fifteen to twenty years we have observed the rapid development of Raman and far infrared spectroscopy. Raman spectroscopy has been developed mainly due to the construction of lasers, which are ideal monochromatic sources, and also due to the construction of highly sensitive receivers, which have eliminated the di0iculties restricting its wide application. Far infrared spectroscopy is now widespread, because of the development of Fourier transform spectroscopy, whose revival was associated with the construction of compact computers and with the invention of the rapid Fourier-transform algorithm. The detailed information concerning Raman and far infrared spectroscopy can be found in some reviews [1,21. In the present chapter we will describe only our devices and methods which allowed us to improve the efficiency of our studies of organic crystals by means of analyzing the vibrational spectra and, first of dl,our studies in the neighborhood of the phase transition point 131.
A. Measurement Technique for Molecular Crystals in the Far Infrared Region In our experimental investigation we have used the Fourier-spectrometer FS720 (Beckman, USA - England) and the one-beam spectrometer FIS-21 (Hitachi, Japan). The former was mainly used to obtain the general information data, the latter for studying temperature dependencies in the vicinity of the phase transition.
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ZHmm AND MUKHTAROV
The metal vacuum cryostat was used for low-temperature measurements made using the FS-720, the block diagram is shown at Fig. IV-1. The temperature was measured by means of a copper-constantan thermocouple. It is comparatively easy to maintainthe temperature of samples witbin 22°K. During the study of molecular crystals by means of the FIS-21 we used the cryostat with the smooth adjustment of temperature in the range of 78 to 450°K [4] which was constructed in the laboratory of the Institute of Spectroscopy of the Russian Academy of Science (Fig. W-2). During the precise measurements when it was required to know the change of temperature with an accuracy of 0.1"K and to maintain the appropriate temperature with an accuracy of d0.O5"K,a special stabilization system was used. Its block diagram is given in Fig. IV-3. It was used also in our work with the cryostat and FS-720 (Fig. IV-1). The spectrometer FIS-21 was unfit for our investigation of molecular crystals because it could only be used in the case of large light sizes of examples (50x 50 mm2). It was found [41that this device was applied sufliciently well in combination with the device JPO-22 (LOMO, Russia). We could work without large temperature changes with the samples of sizes 4 x 11 (mm2), the light energy loss was 2 - 10%. The far infrared spectra in the temperature range 77 - 15°Kwere obtained by means of a helium cryostat (Hitachi). However, during the low-temperature experiments we used the cryostat constructed in the Physical Institute of the Russian Academy of Science in combination with the FS720. The temperature stabilization accuracy in our far infkared experiments was * 5°K in the region far fiom the phase transition temperature and 2 0.05"K in the vicinity of it. During the investigation of the temperature dependencies by means of the FS-720, the rate of temperature change of the sample was 20 Wh. In the case of applying FIS21 this rate was 1 K h for measurements in the vicinity of the phase transition. The spectra were registered with a resolution 1.6 - 3.2 a - 1during the work with both spectrometers. Liquid samples were placed in universal sealed quartz cuvettes (Fig. IV-4).
B. The Technique of Sample Preparation and Methods of the Investigation of the Raman Spectra of Organic Crystals The Raman spectroscopicmethod is essentially simpler and more informative
than the far infrared ones. The modem spectrometer pennits one to register the Raman spectra in the region 10 - 4000 a - 1 for most samples in the condensed
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
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43-
Ili-J -o-, I r
FIG. N-1. The "inertial"IR cryostat (1)cell with a sample; (2) copper block; (3) steel tube with thin walls; (4) housing; ( 5 ) flange; ( 6 ) rubber vacuum gasket; (7) ring flange; (8) heating element. 6
~
T
I
5-
2
$-I--
87
-
/
3 L
1
-
-
--I-
9
I' I
FIG. N-2.The diagram of the IR cryostat with the smooth temperature adjustment: (1) housing; (2) cooling bath, (3) gas-liquid chamber; (4) porous wad; (5) gas Carrying tube; (6) control valve; (7)sample holder; (8) heater; (9) copper constantan thermocouple.
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ZHIZHINAND M-UKHTAROV
FIG. IV-3. The block diagram of the installation for the precise adjustment and measurement of the sample temperature in the IR cryostat (1)semi-automatic potentiometer;(2) potentiometer; (3) power supply; (4)junction of the thermocouple at T=O"C; (5) junction of the thermocouple at the sample; (6) heating element; and (7) thermostat.
a
FIG. IV-4. "he crystalline quartz cell: (1) cell lid; (2) cell mounting; d thickness of the absorbing material.
-
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
155
h! FIG. IV-5.The scheme of installation of the vertical type for the zone melting of organic compounds: (1)glass container with the material; (2) Ni-Cr alloy heaters joined successively; (3) heat isolated infiltrations (asbestos, glass cloth); (4) glass tubes; ( 5 ) clock mechanism.
phase. The disadvantage of this method is that there is a strong luminescence background in the spectral region under study. It is necessary to ensure the careful purification of samples in order to avoid the influence of this background effect. In order to obtain complete qualitative information, we have performed the measurements on the oriented monocrystalline samples in the polarized emission. For this reason, we have constructed some special installations and developed the technique of compound purification, monocrystal growth, which were crystallized at both low and high temperatures, etc. (the low temperatures in the region Tmdt > Troom temperature; the high b m ~ e r a t u r e in s the region Tmelt < Troom temperature).
ZHIZHIN AND MUKEITAROV
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A
I
FIG. IV-6.The diagram of the installation for the monocrystal growth h m the melting by the Bridgman method (1)asbestos-cement lids with asbestos seals; (2) glass tubes; (3) two section heating; (4) asbestos lid with a hole; (5) clock mechanism; (6) glycerin, (7) cell with the compound; (8)silicone oil. The compound whose melting point temperature is higher than room temperature was carefully purified by means of zone melting 15-71(Fig. IV-5). The monocrystals whose melting point temperatures are higher than room temperature were grown in glass tubes with special capillaries by the Bridgman method [8,9]. This method is very convenient for the growth of monocrystals with sufficiently large sizes. This method is based on the directed crystallization of the compound from the melt with the condition of maintaining a certain temperature gradient. The installation is shown in Fig. IV-6. The purified compound was placed in the glass cell which was then pumped to a pressure of -10-3mm. The glass cell was then sealed. The cell was lowered into the special furnace a t a given speed provided by means of the clock mechanism. If the speed is chosen correctly (the speed of lowering the cell with the melt into the cooling region), the nucleus of a monocrystal is formed in the capillary which is at the end of the cell. This nucleus then fills the whole cell during its growth process. The capillary can have various forms which are chosen experimentally. The accuracy necessary for thermostating during crystal growth and the purification of compounds is provided by the electronic control of temperature. The grown monocrystal cooling must be performed over a period of several days in order to avoid its cracking.
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The quality of these monocrystals depends essentially on the thermal gradients which arise in the region of growth due to some faults in the construction of the heater. Because of these gradients, the monocrystal has some defects, and the crystal-melt boundary becomes curved instead of planar. For this reason the speed of the cell must be chosen such that this boundary remains planar. The calculations [7,9] have shown that in the case of the organic compound (of the naphthalene type) the speed of lowering the cell into the cooling region must be 5 0.5 mm/h and the axial temperature gradient in the region of growth must be g 0.5 grad./*. In our experiments this optimal speed was within 0.01 - 0.1 mmih. The grown specimen was verified by the X-ray method. The absence of double reflexes in the X-ray h u e pattern proved that the specimens were monocrystals. The growth and orientation of the monocrystals of compounds whose melting point is lower than room temperature are very difficult. As is known, there is only one paper concerning the polarized Raman spectra of a low melting point crystal such as benzene ITmdt = 278.5"KJ1101. We have constructed the special installation [ll] which permits one to grow the monocrystals of the low melting point organic compounds (Tmdt < 293°K) by the Stober method [12] and to study them by means of Raman spectra obtained with polarized emission (Fig. IV.7). This installation was made on the basis of the metal optical vacuum cryostat which was placed at the cell compartment of the spectrometer DFS-22. The liquid compound was placed in the special cell (1). The accuracy of thermostating was * 0.05"K. The temperature change rate was accomplished by increments of -0.l"K. The Grozen compound has a temperature which is below the melting point. The vertical temperature gradient is provided by means of constant heating by lamp emission through the side window of the cryostat. The temperature of the low part of the cell is such that the crystal-melt boundary was sunk until several polycrystalline nucleii remain in the capillary. This was provided by the heater (4). The growth of the nucleus of the monocrystal began when the lower part of the capillary was slowly cooled. After the nucleus was formed, the growth of the monocrystal was performed by slowly lowering the temperature (at a rate of 0.1 0.5 kh) of the cell holder in such a way that the surface of the growing monocrystal remains planar all the time. Control was performed by means of a microscope. In order to release the tension, the grown monocrystal was closely cooled up to the nitrogen temperature and was maintained in this state for several days.
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-< \ \ \
-la \ \
\
\ \
\
8
\ \
\ \ \
I;(
5
10
?-
%
'2 \ FIG. IV-7. The installation for the growth of the monocrystals of low melting point compounds: (1)cell; (2) glass window; (3) total reflection prism; (4) heater; (5) steel tube with thin walls; (6) springnut; (7) nitrogen bath, (8)cryostat housing; (9) optical windows. The arrows denote two versions of the propagation of the laser emission in a crystal during the procedure of polarized Raman spectra measurements.
We have grown monocrystals of organic compounds with a length of up to 10 mm and with a diameter of up to 7 mm. We have also developed the technique of growing oriented hexagonal monocrystals. It is well known that the capillary form which is at the end of the cell influences the orientation of monocrystal growth [12,131. In our experiment we have grown monocrystals with the hexagonal axis directed along and normal to it by means of varying the capillary form. The control of the orientation of the grown monocrystal was performed by means of analysis of the polarization of the linear polarized laser beam spreading through the crystal 1141. The Raman spectra were measured on a DFS-24 spectrometer 1151 (Fig. IV-8) using 4800 A and 5145 A excitation lines ftom an argon laser, a 4416 A excitation line from a He-Gdlaser and a 6328 A excitation line from a helium-neon laser.
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
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-.---
I-
FIG.N-8.The optical scheme of the monochromator DRS-24.The location of blackened screens with windows are denoted by the dashed lines. These windows are positioned corresponding precisely to the sizes of the elliptic crosssection of a beam.
The polarization measurements were of special interest to us (Fig. IV-9). It is well known that the intensities of Raman spectral lines in monocrystals are determined by the Raman scattering tensor components [16,171. These tensors of the f i s t order and the vibration types for the crystals are given in Table IV-1.
160
Q5
FIG.IV-9. The optical scheme of the installation for measuring the Raman spectra of monocrystals in the polarized light: (1)laser; (2) plate ll2; (3) rotational prism (4) Nicol prism; (5) focusing lens (f=200mm); (6) cryostaG (7) condensor; (8) plate 1/4;(9) slit of spectrometeq (10)polaroid; (11)pumping coupling for cryostat.
The components of the Raman scattering tensors are independent and have different polarizations, if the electric field vectors of the exciting and scattering lights in a crystal are directed along the principle axes of refraction index ellipsoid [16,171. The Port0 notation system is very convenient in the description of the experimental geometry. The effects which had an influence on the accuracy of performing the polarization experiments were described in detail in Refs. [19-221. The orientation of the principal axes of the refraction index ellipsoid of the crystal under study with respect to the propagation direction of the exciting and scattering lights and their electric field vectors was of special interest to us [23-271 (Fig. IV-10). The optical cryostat was made for performing the measurements of the Raman spectra at low temperatures(77 - 300'K) (including the polarized spectra of a monocrystalline specimen) 128-291(Figs. IV-11and IV-12).
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TABLE IV-1 First Order Tensors and Vibration Types for Crystals of Organic Compounds structure
Monoclinic
symmetry
Class
c2h
Raman Spectra Tensor and the Vibration Type
[: I:
(. ;
f,
Bg
("
Rhombic D2 c2v D2h
.1 Ld dl A Al(d
Bl(z) A2 Blg
if f1
[e B2(Y) BI(x) B2g
B3M
WY) B3g
FIG.IV-10. The block diagram of the installation for the determination of the principal axes of the refraction index ellipsoid in crystals by means of the linearly polarized laser emission: (1)laser; (2) total internal reflection prism; (3)
Nicol prism; (4) goniometer (5)crystal;(6) screen.
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ZHIZHIN AND MUKHTAROV
FIG. IV-11. The schematic view of a cryostat for studying the crystals by means of the polarized Raman scattering: (1) optical windows; (2) metal housing; (3)bellows; (4) glass nitrogen bath, (5) metal rods for fixing the platform to the limb;(6)foam plastic plug; (7) spring; (8)nuts for platform position control; (9) limb with grades; (10) limb slider with vernier; (11) microscrew; (12) vacuum-tight connection (13)platform for limb fixing; (14) hole for liquid nitrogen; (15) stainless steel tubes with thin walls (a, Imm); (16) vacuum coupling; (17)vacuum seal; (18) felt wad; (19)heater; (20) copper thermal screen; (21) crystal;(22) thermocouple.
The special installation was constructed for measurements of the Raman spectra of monocrystals in the range 290 - 500°K (Fig. IV-13). It permits one to study the successive transformations in the vicinity of the phase transition with a gradient of 0.1"Kand a thermostat accuracy o f f 0.01"K[30].
VIBRATIONAL, SPECTRA OF ORGANIC CRYSTALS
163
4
A
I
a
?
'
FIG. IV-12. The scheme of sample fixing in the cryostat for the lowtemperature investigations of the Raman spectra: (a) for liquid samples at room temperature and for decaying samples (the samples are positioned in a sealed thinwalled glass cell): (1) sample; (2) copper clamping plate; (3) indium gasket; (4) heater; (5) stainless steel tube with thin walls (a, Imm); (6) cryostat nitrogen bath, (7) copper sample holder; (b) for powders: (1)sample; (2) optical glass windows; (3) heater; (4) stainless steel tube with thin walls (0,Imm); ( 5 ) clamping nut; (6) cryostat nitrogen bath; (7) clamping nut; (8) washer.
C. Modified Model of the Oriented Gas for the Calculation of the Relative Line Intensities of the Low-Frequency Raman Spectra of the Molecular Crystals The practical application of the theory of Raman scattering was developed by Placek in a convenient form. In the first approximation, when the frequency of the exciting emission is far from the absorption band of a sample, the tensor of its dielectric susceptibility xpo depends on the nucleus coordinates. In the harmonic approximation it leads to the well known equation for integral intensities of the Stokes lines of Raman scattering 116,171 (non-degeneratemodes)
where v, and Q, are the frequency and coordinate of the normal vibration, respectively. P g ) are the elements of the Raman tensor for a mode n
164
ZHIZHIN AND MUKHTAFtOV
FIG. IV-13. The schematic view of the thermal compartment for the precise thermostat: (1) compartment base; (2) copper cell for thermostat; (3) cell with sample; (4) copper end of cell holder; (5)output optical windows; (6) optical windows for the control of a.laser emission polarization; (7)adjustment screws; (8) bellows; (9) input and output tubes for heating agent; (10) steel cylinder with thin walls; (11) limb with hex nut divisions; (12) thermocouple; (13) rotation head of a crystal holder; (14 and 15) steel tubes with thin walls; (16) vacuum-tight seal; (17) upper lid of compartment; and (18) copper cylinder.
IpJQn) is the integral intensity of a line in the case when the electric vector of the exciting emission is polarized along the fixed coordinate axis p, and the electric vector of a scattering light is polarized along the 0-axis. In the case of a polycrystal it is necessary to average (Iv-1) over all orientations
In the case of infrared absorption we have the analogous expression for the integral absorption factor 131,321:
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
165
where Mp is the component of the dipole moment of the unit cell. The semi-empirical methods for calculating the intensities by means of (N.l) and (N.4) are based on different assumptions concerning the relationships between the dielectric susceptibility and the dipole moment and the properties of the crystal's separate molecules. The region of practical application of similar calculations can be conditionally divided into two parts. The first part is related to solving the inverse problem, i.e. one must use the measured line intensities to obtain the quantitative information concerning the intermolecular interaction, the crystalline structure and the external vibrational form. The methods of the quantitative description of Raman scattering of molecular crystals concerning the external vibrations were considered in references [33-401. They are based on the representation of the dielectric susceptibility tensor in the following form:
where v is the volume of the unit cell; &(K) is the polarizability tensor of a free molecule in the crystal coordinate system; A G ( ~ )is the correction to B(K) describing the change of the molecular polarizability due to the crystalline environment. is the tensor relating to the macroscopic electric field Ri to the molecule: internal (local)field E(K) acting on the
The components of tensors &(K) in (N.5) depend only on the molecular orientations, and and A&(') depend both on the orientational and translational displacements of molecules. The local field factors d(K) can be independently calculated in the approximation of the point molecules if the crystal structure is known [40,411. The change of polarizability A&(K) due to crystal molecular vibrations is related to the deformation of their electronic cloud, and any half empirical description A&(') leads to the necessity for the introduction of a large number of empirical parameters. Thus, in reference [361 the authors have proposed a model (similar to the AAP method) in which the difference between the polarizability of a molecule in a crystal and its polarizability in a free state is
)'(A
ZHIZHIN AND MUKHTAROV
166
expressed in terms of the polarizability of atoms (empirical model parameters) and of their dependence on the interatomic distances. This model was used for the calculation of the intensities of Raman spectral lines and of the optical properties of halogen substituted crystals [34-361, including the determination of the forms of external normal vibrations by means of the measured absolute intensities of the Raman spectra of Bp-dichlorobenzene crystals 1373. For the description of inkired absorption spectra in the external vibrational region the dipole moment of the unit cell is usually represented in the form [42] =
c 2
( p ( K ) + @(K)
K= 1
+ &(K)
A h ) 2.
1 )
(IV.6)
where p(K)is the constant dipole moment of a free molecule, is the shortrange acting intermolecular interaction forces, &(K) A(K)ai is the dipole moment induced by the local field. In the case of a crystal with centrosymmetrical positions of molecules, only the translational vibrations are active in the infrared spectra, and the intensity of t h i s spectrum is determined by the two last terms in (IV.61, because p(‘) = 0 (the depend on translational molecular displacements 1421). All efforts in factors the calculation of the far infrared absorption spectra of such crystals have led to large discrepancies with experimental values, if only the induced local field of the dipole moment of the unit cell is considered 131-33,431. The authors have not suggested any models which take into account the change of the dipole moment @(K) induced by the shortrrange intermolecular interaction. It is necessary to note that, as shown in reference [32], neglecting the finite sizes of molecules usually permitted in the calculations of intensities is a sufficiently rough approximation, since the local field in the molecular center and its peripheral region can difFer considerably. Thus,a brief review of papers shows that the correct quantitative description of intensities of the low frequency vibrational spectrum of a molecular crystal is a sufficiently complicated problem. In order to solve t h i s problem it is necessary to overcome not only the theoretical, but also the considerable experimental difficulties associated with necessities of precise measurements of absolute absorption line intensities of Raman and far M a r e d spectra. For t h i s reason, we pay attention mainly to the second part of the application of the calculation of intensities concerning the qualitative interpretation of low frequency vibrational spectra. First of all, using the theory of groups one can predict zeroth values of separate elements of Raman scattering tensors (N.2) and of the derivatives of the dipole moment of the unit cell (IV.4) for certain symmetrical types of normal vibrations. There are detailed tables which describe the form of p(n) and a61/Wn for different irreducible representations of all factor-groups
dK)
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
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[16,17,411. If one uses these tables and performs the polarization measurements of the oriented monocrystalline spectra, one can unambiguously assign the Raman and infrared absorption lines to symmetry types. However, in the case of molecular crystals, the growth of monocrystals of a sufficiently large size is often related to considerable diEculties, and one usually deals with polycrystals. Only the use of the values of normal vibrations obtained by the AAP method is, generally speaking, insuf6cient for the unambiguous assignment of the low frequency spectra of polycrystals due to the limited accuracy of the calculations. For this reason, at any rate, it is necessary to qualitatively reproduce the relative intensity distribution in a spectrum which depends on the molecule disposition in the unit cell and normal vibrational form, excluding other factors. The simplest and sufficient universal model for solving this problem can be the approximation of the oriented ("freezed") gas in which the influence of intermolecular interaction on optical properties of molecules in a crystal is neglected, and the difference between the local and macroscopic field is not taken into account [44,451. Actually, the model of the oriented gas is the zeroth approximation for the calculation of the vibrational spectral intensities. In the region of the intramolecular vibration the model of the oriented gas is in good qualitative agreement with experiment in the analysis of relative intensities of the Davidov splitting component, and it is widely used in the spectral investigations [461. The application of this model to the spectra in the external vibrational region is considered in detail in references [38,44-461. In the approximation of the oriented gas the dielectric susceptibility (IV.5) and the dipole moment of the elementary unit (zV.6) are of the form
Hence, in particular, it follows that this approximation is unfit for the description of the infrared absorption spectra in the external vibrational region, since in the case of a crystal with a centrosymmetrical position of the molecules ii(x)= 0, and it predicts the zeroth intensity of these spectra. Consider the expression for the Raman scattering tensor in the approximation of the oriented gas. From (IV.2) and (IV.7a) it follows that:
ZHIZHIN AND MUKHTAROV
168
(i)
where qp are the displacement coordinates of a molecule K of the chosen unit cell given in the system of the main inertial axes; m is the number of the internal vibrations of a molecule. The tensor of the polarizability dK) of a molecule in the motionless coordinate system of a crystal is related to the molecular polarizability in the moving system of its main inertial axes by the relationship
where AK is the matrix of directing cosines of inertial axes of a molecule K in its equilibrium state; sC9u9v ew) is the Euler matrix of the rotational displacement of a molecule around its equilibrium inertial axes; & is the polarizability tensor of an isolated molecule in the system of coordinates of its inertial axes. Thus, in the model of the oriented gas only the rotational molecular displacements give the contribution to the intensities of lines, if the contribution of the internal displacements Q is neglected, i.e. if one assumes that the intensities of lines corresponding to the low frequency internal molecular vibrations in a crystal depend only on their mixture with librations. From (IV.9)we find that
(Iv.10) where u,v, w are the inertial axes of a molecule, and
-
as C, =-a %a
- + 6-a
9
%a
(Iv.11)
Since the inertial axes of a molecule are related by the interchange elements of the symmetry of a point group in a crystal, we get
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS 1
I
.
169
.
AK = G K A 1
(IV.12)
where GK is the matrix representation of the element of symmetry. Also, taking into accouzlf.also the relationship between the librational displacement coordinates and normal coordinates (see Chapter I), we finally get the expression for the Raman scattering tensor in the approximation of the oriented gas
(IV.13) where 1, are the moments of inertia of a molecule; L" are the elements of the eigenvectors of a normal vibration corresponding to the librational displacement of a molecule.
Ba = ileai ;
(IV.14)
If the interchange symmetry elements are the simple symmetrical operations Ci, Cs and C2 in (IV.12) it is more convenient to use the vibrational form L',"' for the relative molecular displacements (see Chapter I):
where S(,n) = k 1 is dependent on the vibration types S, A, SAA, etc. The direct differentiation of the Euler matrix in (IV.11) leads to the following expression for C, :
. c, =
c, = (IV.15)
ZHIZHIN AND MUKHTAROV
170 -2a,
]
(auu-aw) -ayw 2auv auw a, 0
where are the elements of the molecular polarizability tensor in the system of coordinates of its inertial axes. The calculations of the intensities of the external vibrational Raman spectral lines were performed [38,44-511 in the oriented gas approximation according to the aforementioned scheme. The contribution to the vibrational form of the libratiod displacement was calculated by the AAP method. Sufiiciently good agreement with the experimental values was obtained for the relative intensities of Raman spectral lines for a series of hydrocarbon crystals [38,44,47,491, of malonitrile 1501 and of St r i d e 1511. In Ref. 1451 the authors have studied the applicability of the model of the oriented gas to five crystals with strong intermolecular interactions. Considerable discrepancies between the calculated results and the measured absolute intensities of the Raman spectra was found (for the urea crystal it was one order of magnitude larger than the abovementioned one). However, the comparison of the relative intensities gives more or less good qualitative agreement with experiment. The detailed analysis of the oriented gas model was perforg~edin Ref. [14] in which 11 molecular crystals were discussed as examples, this model shows good agreement with experiment only for three of them. The authors of Ref. 1441 have come to the conclusion that the oriented gas approximation is very rough, and its applicabilityis rather limited. The efforts to improve the oriented gas model by the consideration of the internal local field does not eliminate the discrepancies with the experimental values 1453. Thus,the main disadvantage of the model is that it does not take into account the differences between the tensors of polarizability of an isolated molecule and the molecule in a crystal. In Ref. 1381 the authors have suggested the use of the effective polarizability of a molecule determined from the optical properties of a crystal. However, generally speaking, the polarizability is determined unambiguously 152,533 and the data from authors differs noticeably (Table Tv.2). We have proposed the method for the calculation of the Raman spectral intensity lines of crystal internal vibrations consisting of the introduction of the fitting half empirical parameter y which takes into account the influence of intermolecular interactions. This also makes it possible to calculate the relative intensities of Raman spectral lines even if the data concerning the optical properties of crystal molecules are absent. Suppose that the main axes of the effective polarizability tensors coincide with the axes of a molecular inertial moment. Then the tensor 6 will be of the diagonal form, and the elements of matrices (IV.15)
ea
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
171
TABLE IV.2 Polarizabilities for Naphthalene, Anthracene and Biphenyl for an Isolated Molecule and a Molecule in a Crystal Isolated Molecule (A13 1. Naphthalene
[541 20.2 18.8 10.7 y = 0.173
1531 24.4 18.2 9.6 y = 0.721
1551 29.2 15.5 12.5 y = 4.57
u
35.2 25.6 w 15.2 y = 0.923
43.2 20.6 17.0 y = 6.28
40.3 25.3 11.9 y = 1.043
1563 u 24.7 v 20.3 w 13.8 y = 0.677
35.2 16.7 15.8 y = 20.6
u v w
2. Anthracene
v
3. Biphenyl
Molecule in a Crystal (A13
puv =auu-aw
B,=a,-a, Pw=a,-a,
'
,
(lV.16)
Assume that I, > Iv > I,, and determine 7 by means of the formula (auUf -1:
(IV.17)
ZHIZHINANDMUKHTAROV
172
Taking into account that
B, +B,
+Buv
=0
(IV.18)
relations (IV.16) may be written in such a way:
8,
= Y 8,
(IV.19)
Since we deal with relative intensities, the calculated results are independent of multiplication of matrices (IV.15) by the same constant (Vk,). Consequently: 0 0 0
c,-[o
0 O Y O
cv=[
:
-1-y
0 -1-y
0 0
8
1,
.=['
0 1 0 0 0 01. 0
Thus, in our case the relative intensities depend only on one parameter y, characterizing relative optical anisotropy of the molecule in a crystal. Values of y for some molecules are shown in Table IV.2. The data for a n isolated molecule and for a molecule in crystal, determined from its optical properties, differ markedly. However, the same order of difference was obtained in different investigations. Therefore, for a better agreement with the experimental values it is advisable to vary the magnitude of y. Consider some examples of the application of the oriented gas model. The data which are necessary for the intensity calculations are given in Appendix 3. 1. Ethylene (C2H4)
The crystal structure was determined at 85°K in Ref. [58]. The space group is P21/n, z = 2(Ci). In the Raman spectra six librational normal vibrations (3% + 3Bg) are active. The calculation of the relative intensities of the Raman spectral lines was performed according to the oriented gas model with the use of a value y which is equal to 0.233 [57]. The distribution of intensities in the Raxnan spectra of a polycrystal was imitated by the Lorentz contour s u m in the comparison between the calculation and the experiment. From Fig. IV.14 and Table IV.3 it follows that the
173
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
50
70
90
110
130
150
170
-
-
50
cm-1
FIG. TV-14.
Experimental and calculated Raman spectra of ethylene
POlYCrYS~.
Vexp. cm-1
TABLE IV.3 Calculated and Experimental Raman Frequencies and Relative Intensities for Ethylene VdC. &el., %, 30°K Qn-1
exp. 1201
30°K Ag
Bg
CalC.
I, > I, > 1,
176 97 90
176.1 93.4 81.2
0.2 42.0 100.0
14.3 46.1 100.0
kl
167
161.0
0.6
1.5
114 73
107.3 69.2
20.0 19.0
17.0 25.6
Lu Lv
Lw
Lv
L,
174
ZHIZHIN AND MUKHTAROV
FIG.IV-15. Experimental [62] and calculated Raman spectra of anthracene monocrystals in the polarized light (in the oriented gas approximation). calculated results for the ethylene polycrystal are in good quantitative and qualitative agreement with the experimental data obtained at T = 30°Kin [47]. 2. Anthracene (C14H10) The space group of a crystal is P21/c, Z = 2(C;) 159-611. There are a large number of papers dedicated to the investigation of the Raman spectra of this crystal in the region of external vibrations. We chose the data of the polarization measurements and the Raman spectra of a polycrystal given in [62] for comparison with the calculated results. Six librational vibrations (3Ag + 3Bg)are active in the Raman spectra. The totally symmetrical vibrations are active at four conditions of the polarization of the incident and scattering emission aa, bb, c*c*, ac*, the vibrations of the type Bg are active at two conditions -- ab, bc. The calculated results obtained with the use of the isolated molecular polarizability (Table IV.2) given in Figs. IV.15 and IV.16 and Table IV.4 (y = 0.928) are in
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
140
120
100
80
175
60
40
FIG. N-16. Experimental [381 and calculated Raman spectra of anthracene polycrystals (inthe oriented gas approximation). TABLE IV.4 Calculated and Experimental Frequencies and Relative Intensities for Anthracene expr. calC. Relative Intensities, % cm-1
cm-1
[351 121 71 41
127.0 75.8 41.2
125 64 45
116.1 68.9 47.2
aa
bb c*c* ac* ab bc*
13 28 1
17 0 15
0 33 10
2 20 2
plycrystal 34.3 100.0 28.6
12 2 3 16 3 6
28.4 37.7 18.3
176
ZHIZHIN AND MUKHTAROV
qualitative agreement with the experiment. One gets considerably worae agreement with the experiment with the use of the effective molecular polarizability obtained from the optical properties of a crystal.
3. Thiophthene (C6H4S2) The space group of the crystal is Pbca, Z = 4(C2) 1631. Twelve librational normal vibrations are active in the Raman spectra. The low Grequency vibrational spectra of the thiophthene crystal were not studied before. The data concerning the polarizability tensor of the molecule are absent. This crystal can be considered as the example of the interpretation of the Raman spectra of a polycrystal by means of calculations of the external vibrational fkequencies and of the relative line intensities. The Raman spectra of a thiophthene polycrystal was measured on a DFS-24spectrometer using the 488 nm excitation lines &om an argon-ion laser at T = 293°K The calculation of the relative intensities of the Raman lines of a polycrystal does not lead to satisfadory agreement with the experiment, if one performs the variation of parameter y. This can be associated with the noncoincidence of the principal axes of the effective polarizability tensors and inertial moments of a molecule and, consequently, with the necessity of taking into account the nondiagonal elements of the polarizability tensors. Since the thiophthene molecule has a symmetry axis of the second order which is normal to its plane, one can suggest that the principal axes of the tensors of inertia and of polarizability differ &om each other only by the rotation of an angle around the axis w (Fig. IV.17); i.e. 6 is of the form
aUV
auv aw
a(0) ww
where aulql (0) ,avtvt, (0) am (0) are the basic values of the tensor of the polarizabdity of a molecule, S(4) is the Euler matrix. Using the definition of the parameter y (IV.17) in terms of the basic . values of the polarizability tensor, we get the following expressions for matrices c, (IV.15) which are necessary for the calculation of the relative intensities:
177
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
77K I
140
120
100
80
60
40
FIG. lV-17. Experimental (a,b) and calculated Raman spectra of the thiophthene polycrystal (c,d) calculated at y = -1,5,and for different orientations of the principal axes of the molecule'spolarizability tensor.
ZHIZHIN AND MUKHTAROV
178
c.=[
0
0
I
-cos $sin $
0 0 (y+sin2$ -cos$sin$ (y+sin2$) 0
,.
cv=
,.
c,
2cos$sin$ = (cos2$-sin2$) 0
-2cos$sin$
0 0
0
0
(cos2$-sin2$)
The angle is, actually, the additional fitting parameter, since the orientation of the polarizability tensor axes of a molecule in a crystal is a priori unknown. As seen in Fig. IV.17, the results of the calculations of the relative intensities depend essentially on the choice of an angle 9. If y = -1.5 and I$E 45", one gets the good qualitative agreement with the experiment (Fig. IV.17). This permits one to assign the lines of the Raman spectra of this crystal in a sufficiently unambiguous way (see Appendix III). Thus, our analysis of the oriented gas model has shown that its application for the qualitative description of the distribution of relative intensities of Raman spectra in the internal vibrational region is restricted to only a small number of crystals, basically, of hydrocarbon crystals with rigid molecules. It proves the conclusions of the authors of Ref. [44]. The oriented gas model modification, which is suggested in our work and includes the use of either one or two empirical parameters for the description of the effective polarizability tensor of a molecule in a crystalline state, permits one to extend, essentially, the field of application of this model, and is rather useful in the assignment of polycrystal Raman spectra (see
Part II).
VIBRATIONAL SPECTRA OF ORGANIC CRYSTALS
179
--
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54. C. L. Cheng, D. S. N. Murthy, and G. L. D. Ritchie, Aust. J. Chem., 25,1301 (1972). 55. M. F. Wuks, Optika i Spektros. (USSR), 20,361(1966).
56. R.J. W.Le Fevre and D. S. N. Murthy, Aust. J. Chem.,21,1903(1968). 57. G.W.Hills and W. J. Jones, J. Chem. SOC. Far.Trans., 271,812(1975). 58. G. J. H.Nes and A. Vos, Acta Cryst., 33B, 1653 (1977). 59. D. W.Cruikshank, Acta Cryst, 10,504(1957).
200
ZHIZHIN AND MUKHTAROV
60. R. Mason, Acta Cryst., 17,547(1964). 61. G.R. Charbonneau and Y. Delugeard, Acta Cryst., 33B, 1586 (1977). 62. M. Ito, M. Sudzuki, and T. Yokoyama,in "Excitom,Magnons and Phonons in Molecular crystal^", London: Cambridge Univ.Press, (19681,p. 1.
63. A. I. Kitaigorodskii, "Organic Cristallochemistry", MOSCOW, USSR Acad. of Science, (19551,p. 558.
PART I1
INVESTIGATION OF THE STRUCTURE,PHASE TRANSITIONS AND REORIENTATION MOTION OF MOLECULES IN ORGANIC CRYSTALS
201
202
ZHIZHIN AND MUKHTAROV
I. VIBRATIONAL SPECTRA AND PHASE TRANSITIONS IN PLASTIC CRYSTALS
A. Investigation of the Phase Transition in Cyclohexane and Deuterocyclohexane Crystals with the Help of Vibrational Spectra
In recent years crystalline cyclohexane was studied in many papers 11-20]. Such interest was caused by the polymorphism and high-temperature plastic modification of the crystal I which is also referred to as "rotationally-crystalline" one [141. The s d melting entropy inherent to the plastic phase is the proof of the 'liberation" of the translational degrees of freedom and of the translational order destruction during the melting process, since the orientational order is ruined within the limits of the plastic phase due to thermally activated molecular reorientation [2]. The plastic crystal is transformed to the anisotropic monoclinic one (the crystal 11) with a fixed molecular orientation. If the temperature is lowered (186.1"K) 1161 the isothermic phase transition evidently occurs over the intermediate (metastable) crystalline modification which exists within a temperature interval of less than one degree 13,131. For a long time (since 1958) only incomplete X-ray data about the structure of the cyclohexane low-temperature crystalline modification (crystal II) have existed. They were obtained from Ref. [17]. "he unit cell was assumed to be monoclinic with the space group C i h (Z = 8) on the basis of this data. This structure seemed to be dubious according to the infrared [41 and Raman spectral 151 data in the region of the intermolecular vibrations. The low symmetry of the unit cell might cause the essential change of selection rules which cannot be observed in the infrared and Raman spectra of anisotropic crystals. The correct choice of the space group of the plastic crystal according to spectral data and the validity of the criticism of the anisotropic crystal symmetry were verified in a careful X-ray analysis [ 181. 6 The cyclohexane crystal II is related to the space group C 2/c (CZh) with four molecules in the unit cell, and the symmetry of the molecule is D3d. When it is in the crystalline state, its symmetry becomes Ci (Fig. 1.1). The plastic crystal I belongs to the space group F m3m; its unit cell has four molecules. If the mass centers of the molecules have ordered positions (the regular cubic lattice), their
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
203
FIG. 1.1. Crystallographic cell fragment of the cyclohexane crystal 11. 0 is the carbon atom, @ is the hydrogen atom; u, v, w are the orientations of the molecule's main inertial axes with respect to the crystallographic axes (a, b, c and c*) [181. orientations are chaotic and change with time. This conclusion is proved also by the N M R line width data 1191. In order to explain the reconstruction of crystal structure in the vicinity of the phase transition (the anisotropic crystal I1 - the plastic crystal I) we have investigated the temperature dependence of cyclohexane far infrared and Raman spectra in a wide temperature region including the closest vicinity of the phase transition. We have also calculated the external vibration spectra of the lowtemperature crystalline modification (the crystal 11). Starting with the structure of the crystal I1 we can expect to have nine optically active crystal vibrations; three of them, the translational vibrations (Mu+ B,) are active only in the infrared absorption spectra, and six of them, the librational vibrations (3Ag + 3Bg)are active only in the Raman spectra. In Fig. 1.2 and 1.3 we have shown the far infrared absorption spectra and Raman spectra of solid cyclohexane and its deuteroanalog, respectively, at different temperatures. In the region (20 - 90 cm-1) of the Fourier infrared spectra of the
ZHIZHIN AND MUKHTAROV
204
0 0.5 1 1.5
3 w, B
20
60
cm-1
FIG. 1.2. Temperature dependence of the cyclohexane FIR spectra: 1-15°K; 2-85°K; 3-113"K; 4-150°K; 5-180°K; 6-205°K; 7-258°K. The thickness of the layer for curve I is 0.7 mm, and for curves 2-7 it is 2 mm. The optical density scale is given for curve 1. Curves 6 and 7 correspond to the plastic modification. anisotropic modification we have observed an absorption band with a maximum at 65 at T = 15°K. Its existence is proved by results of studies [8,113 in which the additional absorption band (101 cm-1) was observed at T = 65°K [lll. Both bands can be unambiguously assigned to the crystal lattice vibrations, since the internal vibration of cyclohexane has its lowest frequency at 240 cm-l[201. The Raman spectra of cyclohexane was investigated by B. P. Nevsorov and A. B. Sechkarev [91. The Raman spectrum of the crystal I1 was obtained at T = 4.2"K by Ch. E. Sterin and B. W. Mavrin [23]. The results of these investigations and our data are shown in Table 1.1. From this table it is seen that all the Raman spectral data are in good agreement, if the thermal frequency shift (Fig. 1.4) is taken into account and if the frequencies at 30, 35 and 48 cm-1 are excluded, because they were erroneously assigned to frequencies of the Raman cyclohexane spectra by the authors in Ref. [91. Thus, in the far infrared absorption spectra of the cyclohexane crystal II we observed two bands instead of three as predicted by selection rules. In the Raman spectrum we observed all six bands assigned to the external librational vibrations. One usually uses the polarization measurements for the interpretation of vibrational spectra of organic compounds, however, in the case of cyclohexane it is impossible to obtain a sufficiently large monocrystalline sample with the necessary
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
1.1 120
vl
v5 v2
A,
B, A, I
90
205
v3 v6
A,% L
A
I
-
1
60
FIG. 1.3 Temperature dependence of phonon Raman spectra of cyclohexane (C6Hu): 1,2 - 115°K; 3 - 140°K 4 - 160°K; 5 - 170°K; 6 - 180°K and deuterocyclohexane (C6Du): 1- 115°K; 2 - 140°K 3 - 160°K 4 - 180°K. The curve 1 (C6Hu) is obtained by means of the filter eliminating the plasma line (105 cm-l).
orientations and to perform the measurements with polarized light because of the phase transition at low temperatures. For this reason, all experiments were performed with polycrystalline samples, and the assignments of lines to the symmetry types and their interpretations were performed on the basis of lattice dynamics' calculations and additional spectroscopic data, i.e. the Raman spectra of completely deuterized cyclohexane (the use of isotopic relations). The spectrum was calculated using the approximation of rigid molecules which is suf6ciently justified, since the lowest frequency (240 cm-1) of the cyclohexane intramolecular vibrations is twice as large as the highest frequency of the external vibrations. The AAP method was not used before for the description of cyclohexane crystal properties; for this reason, the M e r e n t sets of parameters AAP/l- M I 4 were applied. The cyclohexane molecule contains a large number of hydrogen atoms forming the short-range contact H-.H in a crystal (Fig. 1.1). The
TABLE 1.1 Experimental and Calculated Frequencies for Cyclohexane and Deuterocyclohexane at Different Temperatures Calc. (AAP/4, CH = 1.07 A, 115K), cm-1 Experiment, cm-1 C6H12 c6D12 C6H12 C6H12 1&
Ag *g
Bu
4.2Ka
77Kb
115Kc
120 94 77
115 90 74
110 85 69
97 75 61
138 102 66
128 99 64
120e 92 62
83 54
65
_-
61
115Kd
--
__
anharmonic corrections
Irel.%
harm.
102.3 85.8 61.0
-0.1 -1.5 -1.2
0.6 1.6 -0.1
100.0 51.5 23.0
89.4 75.9 54.3
1.144 1.130 1.130
88L,,+lOL, 72&+28L, 24L+lOL,,+65L,
124.8 97.7 59.1
_____
4.3 2.0 1.1
2.8 31.0 71.1
100.8 87.0 52.0
1.137 1.123 1.137
95L+5L,, 22L,,+78L, 5&+73L,,+22L,
115K
harm.
1.13 1.13 1.13
__
1.11 1.15
_-
61.6
aRaman [22,231, FIR 1233. bRaman 193, FIR 1111. CRaman 1221. Qaman 1223. eExtrapolationto 115K of frequencies 138 cm-1 at 4.2K and 218 cm-1 at 77K [9,23].
V H b
vibration forms. %
105.6 62.8
100 T,* 100 T,
57.6
100 Tb
s
6,
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
207
d
8 120
100
80
60
FIG. 1.4. Temperature dependence of the cyclohexane phonon spectral frequencies using the anisotropic plastic modification. (It is obtained by the use of data from Table 1.1). inaccuracy of the coordinate determination of these atoms which is inherent to X-ray analysis can substantially influence the calculation results. For this reason, the length of all C-H bonds were established as equal and were varied during the procedure of the calculation of normal vibrations within the limits 1.00 - 1.10 The influence of the calculation conditions on the external vibrational frequencies (cm-1)at 115°Kis shown in Table 1.2. The best agreement with experiment was obtained for M I 4 and dCH = 1.07 A. The comparison of the experiment with the calculation is presented in Table 1.1 in which the frequency assignments to the vibration types are given.
A.
ZHIZKIN AND MUKHTAROV
208
TABLE 1.2 Calculated External Vibrational Frequencies ( ~ m -at ~ )115°K and M e r e n t C-H Distances for Cvclohexane
AApI3
AAp/IAAp/2
1.09 b;
1.07b;
1.05 b;
1.05 b;
102.3 85.8 61.0
106.8 89.7 63.0
100.1 83.9 60.5
114.4 95.8 69.0
111.0 92.8 59.0
118.9 93.1 56.0
124.8 97.7 59.1
131.0 102.3 62.3
123.0 96.5 59.3
140.3 110.3 68.5
145.3 111.8 55.1
104.0 62.1
108.4 64.6
112.9 67.2
117.5 69.9
109.5 66.3
124.4 74.9
118.5 75.5
54.9
58.1
61.1
65.3
61.3
70.3
69.9
X-ray analysis
1.03b;
1.05 b;
1.07 b;
Ag
87.5 78.0 60.5
93.6 78.6 57.1
97.9 82.1 59.0
Bg
102.0 82.0 55.9
113.2 88.7 53.1
Au
94.6 57.5
B,,
57.6
The interpretation of the low-frequency vibrational spectra of cyclohexane II (Table 1.1) is based on the calculation in the quasi-harmonic approximation whose application is obvious, if we take into account the mobilities of molecules in this Crystal.
However, the anharmonic corrections to the librational vibrational frequencies calculated by means of the independent oscillator model (see Part I, 1II.B) appear to be very small and have no influence on the interpretation of the spectra (Table 1.1).
Dependence of Far Inbred Spectra on Temperature During the process of the investigation of the thermal dependence of the far infrared spectra in a wide temperature region we have basically paid attention to the band at 65 cm-1(at 15"K),since the second band in the far infrared spectra (101 cm-1 at 65°K) is broadened rapidly; its intensity is decreased, and it becomes unobservable at T = 130°K 1111. If the temperature of crystal 11 is increased, the band at 65 un-1 and the other lines of the vibrational spectra are shifted to the low-frequency region (Fig. 1.4). The absorption band of cyclohexane has a frequency at 55 cm-l before the transition. It disappears after the phase transition, and the spectrum of the hightemperature (plastic) modification has the form of an extended continuum (Fig. 1.2). 1.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
209
The phase transition in cyclohexane is of the order-disorder type. Because of the high rotational ability of the molecules, their orientations in a plastic modification are almost completely disordered, but in the low-temperature (anisotropic) phase they are essentially ordered. On the basis of this fad one can suggest [231 that in the vicinity of the transition one can expect a change of the intensity in the infrared absorption spectrum (or in the Raman spectrum) which is similar to the behavior of the order parameter in the Landau theory 126-311. The possibility of this relation was theoretically predicted [32]. From the analysis of the normal vibrational symmetry changes which define the changes in the infrared and Raman spectra, the authors of this work have shown that in the closest vicinity of the transition the following relation is valid in the first approximation:
I-+
(1.1)
where I is the intensity of "rigid modes" appearing in the infrared and Raman spectra of the asymmetrical phase at the transition. The relation (1.1)is valid for a wide variety of structural phase transitions including the transitions of the 6rst order (which are near to the transitions of the second order) on the condition that one can indicate the order parameter which characterizes the distortions of the symmetrical phase structure. For this reason, the behavior of the band at 55 cm-1of the crystalline vibration was studied in the vicinity closest to the phase transition. If the sample is cooled to T = 182.4OoK, the cyclohexane spectrum has the form which is inherent to the plastic modification spectrum (Fig. Ma, curve 1). At T = 182.40"K the jump-like of the absorption increase occurred; besides, one has observed the thermal emission of the sample. As a result, its temperature was increased by 0.4" at T = 182.4"K and the cyclohexane spectrum remained the same as the plastic phase spectrum, but it was shifted by 0.5 on the optical density scale. The spectrum ( c w e 3) recorded a second time at the same temperature had a sharp absorption band 55 cm-l. The following records performed step-by-step with an interval of 0.2 show the gradual increase of the intensity of this band. If the sample is heated (Fig. I.5b) one observes a similar picture, the difference being that the different points correspond to the transitions at the decrease and increase of temperature. Thus, if the plastic phase is cooled, the boundary of its stability relating to the temperature To is 182.8"K. The boundary of the anisotropic modification stability is 186.2", respectively, if the sample is heated up to Th. Hence, the value of the thermal hysteresis Th - To is equal to 3.4"K. The frequency of the band at 55 cm-1 remains practically unchanged in the vicinity closest to the phase transition. The presence of thermal hysteresis and the thermal emission of the sample at the transition justifies the conclusion that the phase transition under investigation in cyclohexane is a transition of the first order.
MIZHIN AND MUKHTAROV
210
0
b
._
a
O(1) -
The Landau theory gives the following expression for first order transitions in the case of the temperature dependence of the parameter in the vicinity of the transition C261:
)B 1
q = +[C(l+,/l-(T-T,)/AC
where C = -b/2c, A = -W2%, %, b, c are the coefficients of the expansion in a series of the fiee energy with respect to the order parameter degree; To is the stability boundary of the plastic phase during the cooling process; AC is equal to (Th - To) at the stability boundary of the low-temperature phase. If we take into account the relation I 92 [321 we obtain the following expression for the intensity in the vicinity of the transition:
-
This relationship allows us to verify the applicability of the Landau theory to the description of the cyclohexane phase transition and the validity of the relationship I q2.
-
STRUCTLJRE AND PHASE TRANSITION OF ORGANIC CRYSTALS
182
184
211
186 T , K
FIG. 1.6. Intensity of the band at 55 cm-1 of C6Hu in the vicinity of the phase transition vs. temperature. The solid line corresponds to the dependence of dl+d-C); the dashed line corresponds to the dependence of
41-.
The temperature dependence of the integral intensity of the band at 55 cm-1 in the vicinity of the phase transition was processed by means of the least squares refinement method. This procedure was repeated until the values A = 7.421 and C = 0.454 were obtained. In Fig. 1.6 the experimental values of the intensity of the band at 55 cm-1in the vicinity of the phase transition are represented by points; the calculated dependence I = c(~+,/~-(T-T,)/Ac)
with determined coefficients A and C is represented by a solid line; the calculated dependence I = C(1-J) corresponding to the free energy maximum [this solution exists only in the interval (To, Th)l is represented by a dashed line. As is seen from this figure, the dependence I = C(I+,/~-(T-T,)/AC)
212
ZHIZHIN AND MUKEITAFtOV
describes sufficiently well the behavior of the intensity in the vicinity of the phase transition; the product A = 0.37 reproduces almost precisely the value of the temperature hysteresis Th- To = 3.4"K If the coe5cients A = - W2a, and C = - W2c are known, one can obtain the starting values %, b and c using the following approach: the transition entropy for the phase transition of the h t order is determined by the expression according to the Landau theory [331
Here q2(TC)= 3/2C = 0.68 is the value of the order parameter in the transition point T, = To + (316) W/a,c) = 185.35 K The experimental value of the transition entropy in cyclohexane AS is equal to 8.655 kcaYmol[15]. Simple calculations with these data lead to the values of a,= 0.39, b = -0.52, and c = 0.64. Thus,on the basis of the spectroscopicinvestigation of the phase transition in cyclohexbe one can state that this transition is a transition of the first order. Furthermore, the same temperature dependence as predicted by the Landau theory for the square of the order parameter was obtained by experiment for the band intensity (55 cm-1) in the vicinity of the transition. Consequently, the intensity of spectral bands is the experimental value like the frequency of the soR mode (in the case of the phase transition of the second order); the temperature dependence of which can give the information concerning the order parameter behavior in the vicinity of the transition. Thus, the relation between the intensity and the order parameter was experimentally verified; it was theoretically established by Petzelt and D v o d [321.
2.
Temperature Dependence of Raman Spectra,Estimation of the Barriers of Molecular Reorientations in a Crystal
From the N M R data 1191 it is known that in cyclohexane at a temperature above 150°K one observes the considerable narrowing of the resonance line and the drop of the second momentum value (M,)(Fig. 1.7). The type of motion which is the reason for this drop is usually obtained from the comparison between the experimental values of M2(at m e r e n t temperatures) and the calculated ones for a crystal with the rigid lattice and for the case when this supposed type of motion takes place. The narrowing of the NMR signal in cyclohexane can be explained, if one assumes that at temperatures above 150°K the activation of rotational molecular rotations around the C3axis (of a free molecule) o m s [191.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
100
180
100
180
213
260
T, K
FIG.1.7. The NMR line width and second momentum of solid cyclohexane vs. temperature 1191.
The activation barrier of these reorientations is evidently sufficiently large (more than 2 kcdmol); it cannot be determined from the temperature dependence of the width of the infrared absorption lines corresponding to the intramolecular vibrations [2]. It was interesting to find the appearance of this phenomenon in the Raman phonon spectrum of the crystal II which is directly related to molecular librations [35]. We have obtained sufIiciently complete information concerning the correctly interpreted spectrum and the data about the vibrational eigenvectors (see Table 1.1 and Fig. 1.3). The temperature dependence of the Raman phonon spectra of cyclohexane and deuterocyclohexane in the region of 115 - 180°K is shown in Fig. 1.3. The interpretation of the Raman lines obtained from the calculations and verified by means of isotopic relations is also given there. The experimental frequencies of cyclohexane and deuterocyclohexane and the calculated vibrational forms are presented in Table 1.1. From the temperature dependence one can see that the lines v3 and v5 of cyclohexane and the analogous lines of the deuterocyclohexane spectrum are broadened more rapidly than other lines; their intensities decrease in the vicinity of the phase transition. They become practically unobsenrable at 160 - 170"K,i.e. at a distance 20-30"from the transition point and before it. From the calculation of the eigenvectors it follows that the lines v3 ($) and v5 (B,) are related to vibrations with the essential contributions of rotations around axis W (C3in a iiee molecule) which is near to the crystallographic axis C* (Fig. 1.1 and Table 1.1). Thus, we can assume that the barrier to rotational reorientations with respect to this axis is lower than this barrier with respect to axes u and v. For this reason, the intensity decreases, and the broadening of the appropriate phonon lines occurs at a long distance from the transition point.
ZHIZHINANDMUKHTAROV
214
In a plastic modification the barriers of the cyclohexane rotational reorientations were estimated from the temperature dependence of the width of the infrared absorption bands corresponding to intramolecular vibrations 121 and from the data about the inelastic scattering of slow neutrons 1363. According to recent evaluations the value of the barrier changed inconsiderably and is, on the average, equal to 2.0 f 0.3 k d m o l in the whole range between the melting point and the phase transition to the anisotropic modification (see Table 1.3). It is clear that as a result of the phase transition (when the volume is decreased and the unit cell symmetry is lowered) one can expect both the change of the reorientation motion character and its barrier increase. Nevertheless, the value of the barrier which is equal to 11 1k d m o l and obtained 1191 from the measurement of the time versus temperature dependence of the spin-lattice relaxation TI is obviously too high. Uo and Fedin 1371 have developed a simple method for obtaining the approximated estimation of the barrier &om the temperature dependence of the NMR line width. They have suggested that the potential energy of the reoriented molecule can be written in the form
where U, is the orientation barrier height, n is its multiplicity, I$ is the angle of the molecular rotation. It is assumed that the average activation energy can be identi6ed with the full height U, of the barrier, and the reorientation frequency is, as usual, proportional to exp(-U/RT). If T, is the temperature of the starting moment of the N M R line narrowing, and A is its "surplus" width (the line width for the crystal with a rigid lattice; at T, it is equal to the reorientational frequency), then the potential barrier U, is approximately determined by the following expression [371
(1.3) where I is the molecular moment of inertia. The calculation [371 shows that the righthand side of (1.3)is changed by no more than 10% if one proceeds 6.om one compound to another. A more simple expression is valid for this accuracy U, z 0.037 T, .
(1.4)
The main advantage of the estimation (1.3)is that the heights of the barriers obtained on the basis of this estimation are in good agreement with the barrier heights obtained from the measurements of the spin-lattice relaxation time 1373.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
215
However, in the case of cyclohexane the barrier estimation made by means of the rough formula (1.4) gives a value of approximately 5.7 k d m o l (To = 155°K [19]), i.e. almost twice as small as the value in 1373. The use of (1.2) does not appreciably change this value providing 5.5 - 5.7 kcal/mol for A = lo4 lo5 Hz 1193. The fad that the orientation barrier in the crystal 11 of cyclohexane is enlarged [191 is verified by means of the data of recent research 1381 in which the value of the barrier was estimated by means of the data of the measurement of the time versus temperature dependence of the spin-lattice relaxation Tlp in the rotational coordinate system. In these experiments one discovers an additional phenomena which was not noticed by the authors of Ref. 1383. As is known [39], the value Tlp is minimal, if the fkquency of the molecular rotation is close to the resonance frequency up. The contribution of the rotational jumps to the relaxation of the nuclear magnetization is proportional to the correlation time T (time of jumps) of the molecular rotation for 9 7 << 1 and >> 1. In any case one can estimate the activation inversely proportional to Ifor energy U, of the relaxation process from the inclination of the linear dependence ln(Tlp) on the inverse temperature (to the leR side and to the right side from the minimal value). One should remember that I exp(U,,/RT). If the process of the spin-lattice relaxation is not defined only by one correlation time (it is also defined by a number of different correlation times), then each of them has its minimum. If these times are close to each other, their minima become one. The inclinations of the straight lines ln(Tlp) on both sides of this common minimum w i l l be different; it testifies that at least two reorientation motions with different activation energies give contributions into the nuclear magnetization relaxation. This situation was realized in the case of NH4Cl 1401. In the case of cyclohexane one observes an identical situation; however, this fact was not even discussed in [38]. The time dependence of the spin-lattice relaxation on the inverse temperature obtained in this work is given in Fig. 1.8. It is clearly seen that there is an inclination symmetry of the function ln(Tlp) on the leR and right side of the minimum at T = 173°C. Thus, one can make the conclusion that, apart from the reorientations around the C3 axis in the crystal 11, there also exists the rotational reorientations having the larger orientation barrier. According to estimations [38] the value of the reorientation barrier around the C3 axis (straight line on right side of minimum) is equal to 7 k d m o l ; it is less than the value in Ref. [191 by almost 40%. It is difficult to precisely estimate the value of the second barrier by means of the experimental data. There is a rather large scattering of the experimental points in the vicinity of the phase transition. Thus, the values T1p (the two last ones before the transition point) differ
-
-
MIZHIN AND MUKHTAROV
216
T,K 200
180
170
160
0.01 I-
5.0
6.0
5.5
6.5
103m
FIG. 1.8. Temperature dependence of the spin-lattice relaxation time TI, in the cyclohexane crystal II [38]. considerably at T = -184.4"Kand T = -185°K (see Fig. 1.8). It is evident that the point at T = 185°K should be excluded from the consideration. If one takes it into account, the value of the second barrier is very high, i.e. 42 kcal/mol. In the opposite case (the dashed line in Fig. 1.3) its value is probably more acceptable (approximately19 kdmol). The barriers to the rotational reorientations around the inertial axes of cyclohexanein the crystal II were estimated by means of the AAP method according to the following scheme [40,41].We have calculated the energy of the interaction between the chosen molecule and the surrounding molecules of the first coordination sphere. We have used the same parameters of the potential curves as used with the calculation of the phonon spectra (see Table 1.2). Since even in the vicinity of the phase transition the frequency of the orientational jumps is 106 Hz [191 (this frequency is some orders of magnitude smaller than those of the librational vibrations of cyclohexane (-1012 Hz)), it is clear that the reorientation is a very rare phenomenon. For this reason, it is assumed that the chosen molecule rotates around its inertial axes, and the surrounding molecules can be considered immobile and localized in accordance with their structural data.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
217
The dependence of the potential energy of the chosen molecule on the rotational angle around the appropriate axes is given in Fig. 1.9. The barriers of 4.5 k d m o l create obstacles to the rotations around the axis w, while the calculated values of the barriers with respect to the axes u and v seem to be considerably larger, i.e., -19 and -30 k d m o l , respectively (see Table 1.3). It should be mentioned that these large values of the barriers give only the general picture of the orders of the real value. This takes place, because the parameters of the atom-atom potential curves are usually chosen to reproduce the interaction energy in the van der Waals potential minimum. In the vicinity of the energy minimum the interaction energy increases considerably, the qualitative calculation accuracy is lost; the calculation usually gives only the upper limit of the barrier values [41].Besides, the real barriers can be less for the reason such that it is more probable (according to the data of Table 1.1)that the cyclohexane molecule is reoriented in a more complex way than the simple orientational jumps around the appropriate axes. For example, in the case of pyrene the analogous calculations [41]give a value of 27.9 k d m o l for the rotational barriers of a molecule in the plane. However, the calculation considering small motions of molecules around another two axes lower this value by 1/3part. Thus, the calculation of barriers by means of the AAP also verities the possibility of the anisotropic orientational molecular motion in the crystal II of cyclohexane. The various values of barriers of these reorientations correlate, in general, with the estimation according to formulae (1.3)and (1.4)and with the experimental data [381. However, since both the estimations and the experimental data [19,38]differ, it makes no sense to choose the model giving a better agreement with the experiment. To clarify the situation concerning the freedom of the reorientational motion having a large barrier we have studied the Raman spectra of cyclohexane and deuterocyclohexane in the vicinity of the phase transition (Fig. 1.10). The spectrum of the crystal II (curve 1) has the form inherent with the anisotropic crystalline modification up to T = 184°K. Heating to 184.20"Kdoes not influence this spectrum (curve 2), but if the sample is heated at this temperature for approximately 30 minutes, its spectrum smears, the intensities of the lines are decreased, and the lines are broadened; the intensity of the Rayleigh scattering wing increases ( m e 3). Further heating causes the appearance of the structureless spectrum of the plastic modification (curve 4). The cooling of the sample by 0.10"K and keeping it at this temperature for approximately 1.5 hours recreates the spectrum of the anisotropic phase (curve 5). A similar picture was also observed for deuterocyclohexane.
ZHIZHIN AND MUKHTAROV
218
-6
w-axis
I-
L
0
I
I
160
1
I
q0
320
FIG.1.9. Calculation of orientational barriers (kcdmol) in cyclohexane 11by the AAP method. From these experiments it follows that the lines v1 ($1 (the rotation around the axis u), v2 ($), (the rotation around the axis v), v6 (Bg)(the rotation around the axis w) are maintained in the spectrum up to temperatures differing &om the transition point by no more than 0.2"K These reorientations are essentially hindered by the molecular packing in a crystal (Fig. 1.1 and 1.9). They are completely "released only at the transition into the plastic phase. In this case all Raman spectral lines are transformed into the structureless Rayleigh scattering
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
219
TABLE 1.3 Experimental and Calculated Barriers to the Rotational Reorientationsof Cyclohexane Crystal I and II with Respect to Axes u, v and w 0.4 * 0.7 In691.ed t21 Crystal I 1.7 * 2.3 Neutron Scattering [361 AXiS
V
U
crystal 11
19
19
30
W
11
I'MR, Ti 1191
7 5.5 - 5.7 4.5
NMR,T1p C381 Estimation according to Fedin C371 Calculation.AAP
5
5
/
I "1
90
70
50
90
v2
70
I
v6
50
FIG. 1.10. Raman spectrum of cyclohexane in the vicinity of the phase transition. C@u (T"K):1-1842-4 - 184.2;5-184.1;C6Du (YK): 1-184.5;2,3 184.6;4-184.7;5-184.6.
220
ZHIZHIN AND MURHTAROV
wing. The data obtained allows us to state that at the phase transition to the plastic crystal only reorientations of molecules with respect to axes u and v are essentially "released, and the reorientations around w are activated long before the transition in the crystal II. Thus,since we have known the forms of the crystalline vibrations which are active in the Raman spectra and studied the temperature dependence of phonon spectra approaching the phase transition from the low temperature side, we can obtain additional data (to the NMR data) concerning the reorientation molecular anisotropy of cyclohexane and deuterocyclohexane in the crystal IL: it was &st shown that the molecular reorientations around the axis which is near the C3axis are essentially activated up to the phase transition of the plastic crystal. In recent investigations there have been many examples of the stage-by-stage "melting" of the degrees of freedom in crystals which were accompanied, as a rule, by the phase transition. Thus, the transformation between the solid crystalline phase and the nematic liquid crystalline phase is caused by the "melting" of the translational degrees of freedom, but the transition between the anisotropic phase flow-temperature one) and the plastic phase (high-temperature one) is the consequence of the "melting"of the rotational degrees of freedom. In the latter case, it is assumed that all rotational degrees of freedom are simultaneously "melted at the same point. However, the aforementioned results of the spectral investigations of cyclohexane and deuterocyclohexanehave unambiguously shown that some types of rotations can be activated before others in the anisotropic phase because of their structural features (for example, because of their forms which are like disks).
B. Anisotropy of the Rotational Reorientations of Molecules in the Crystals of Cyclopentane and Thiophene Cyclopentane Solid cyclopentane is interesting because it has two plastic crystalline modifications: crystal I and crystal II. Crystal I exists in the following range of temperatures: 179.6 - 138.1"K; crystal 11 exists in the interval 138.1 - 122.4"K; crystal III (the anisotropic phase) forms below 122.4"K [421. The infrared and Raman spectra of cyclohexane in the intramolecular region have been investigated many times [43-471. Deuterocyclopentanehas been studied in Ref. [44-451, and the monodeuteroanalog has been studied in 146-471. In Ref. [44-451, from the comparison of the infrared and Raman multiplets of the Davydov splitting in the intramolecular vibrational region, the authors have suggested that crystal 111(the 3 anisotropic modification)belongs to the space group CZh of the site symmetry group C,, assuming that there are four molecules in the unit cell. 1.
22 1
S T R U C W AND PHASE TRANSITION OF ORGANIC CRYSTALS
TABLE 1.4 Frequencies of Cyclopentane-dg,-dlo, and -dl for Crystal 111 and Crystal I1 Forms FIR Slowneutron [451 [451 M71 [451 absorption scattering[B] -dlo -dl -4 -4 OurData spectrum
crystal m (85K)
__-
--
105 98 82 73
115 (sh) 107 87 69 58 46
crystal II (130K)
90
_-
-_
116 (sh) 108 89 71 58 (sh) 48
116 110 91 88 74 71 59 50
117 (sh) 110.5 91.5 88.5 (sh) 74 71 (sh) 60 (sh) 50
97 84
73 65 58
85
--
60
60
5
According to the group theoretical analysis of the structure CZh7one can expect twenty-one optically active crystalline vibrations. The vibrations ( 5 4+ 4B,) are active only in the infrared absorption spectra, and the vibrations (6% + 6Bg) 3 are active only in the Raman spectra. In the case of the structure c,h there are nine vibrations. The vibrations + B,) are active in the infrared absorption spectra and the vibrations (3% + 3B.J are active in the Raman spectra. In Ref. [451 the authors observed eight lines in the Raman spectra of the cyclopentane crystal In at T = 85°K. This proved, evidently, the first suggestion. However, the Raman spectrum of deuterocyclopentane had only four lines at the same temperature 1451;the spectrum of the monodeuteroanalog 1471 contains only six lines (Table 1.1). In the preliminary experiments with cyclopentane 1471 one observed only six lines in the Raman spectrum of the crystal 111 (Table 1.4). As is known, the far infrared absorption spectra was not studied until now. 5 3 Thus,it was difficult to choose the appropriate structure (either C2h or CZh) on the basis of these incomplete and contradictory data. We have studied the temperature dependence of the far infrared absorption spectra of the cyclopentane crystals I - 111 and repeated the experiment to obtain the Raman spectrum of the sample which was additionally purified 121,491.The results of these measurements are given at Fig. 1.1 and in Table 1.4.
222
ZHIZHINANDMUKHTAROV
The total number of the observed maxima in the far infrared and Raman spectra of the crystal ID is proof of the suggestion 144,453 concerning the space 5 group C ,, if the unit cell contains four molecules. Probably, because of the low intensity of some vibrations, one observed a lesser number of these maxima in the spectra. In particular, for example, the absorption band at 97 cm.1 in the spectrum of the cyclopentane crystal III can be registered only by means of repeated measurements at T = 85°K.Generally speaking, it is necessary to emphasize that the bands in the far infrared absorption spectra of solid cyclopentane have low intensities (the dipole moment of the molecule is very small), and the recording of the spectra is provided by thick absorbinglayers. If the temperature is increased to the point of the phase transition (122.4OK) into the plastic modification (crystal II), the intensities of the bands of the far infrared and Raman spectra are decreased, the bands are broadened and shifted to the low-frequency side; the dependence of this process on the temperature is practically linear. The rate of the shift is approximately equal to 0.12 cm-1PK for the far infrared high-fkequency region and 0.06 cm-1PK for the far infrared lowfrequency region. The Raman spectra of the cyclopentane crystal I and the cyclopentanecrystal 11 have the form of the broad strudureless wing of the Rayleigh scattering line. This is typical for the plastic modification spectra. Unlike this,in the far idtared spectra of the crystal 11(Fig. 1.11,curve 9)one can observe a weak structure in the form of two smeared absorption maxima (60 and 85 cm-1) in the background of the very broad diffuse band. These maxima are absent in the spectra of the crystal I and the liquid. In Ref. 1471 it was mentioned that in the intramolecular vibrational region both the infrared and Raman spectra of plastic modifications of cyclopentane C-& and -dl) are so identical that one cannot unambiguously discuss the phase transition "crystal II - crystal I" only by these means. The authors of Refs. 143,441 have come to the analogous conclusions about the unsubstituted cyclopentane. The NMR data [501 are also insuf€icient to discuss the retardation rate and the cyclopentane molecular rotation in the crystals I and II. By studying the curve of the dependence of the NMR line width on the temperature (Fig. I.121,one can see the low-temperature transition III-II. The transition II-I is not so clearly observed, and the melting transition is unobserved. The narrowing of the NMR signal in the crystal 11and its gradual narrowing in the crystal I to the value which is practically the same as the line width in a liquid shows that the rotational mobility of cyclopentane molecules in these modificationsis high. In the case of the crystal I it is comparable with the identical mobility in a liquid. However, the question concerning the freedom of the rotation during the transition 111-11is s ti l l open. According to the opinion of the authors in Ref. 1501,it
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTAL3
223
FIG. 1.11. Raman spectra (1-3) and FIR absorption spectra (4-11) of cyclopentane at different temperatures. The values T ( K ) are given to the right side of figure.
I
.,..
N
0
1
100
140
180
Tr MP
100
140
180
T. K FIG. 1.12. NMR line width and second momentum (Mz) of polycrystalline cyclopentane vs. temperature 1501.
224
ZHIZHIN AND MURHTAROV
OCCUI+I around all axes.
Probably, the considerablejump of entropy at the transition 111-11 (9.5 e.u.) [42] can also be explained by the hct that the rotation of the molecules in the crystal II occurs either around all or almost all axes. One of the main reasons for smearing the plastic phase phonon spectrum is that the molecular motions in a crystal are not coordinated because of the isotropic disordered reorientations. However, this reorientational motion cannot lead to the complete smearing of the spectrum,if an anisotropy of some kind is inherent to the features of this motion. If the molecular crystal has two plastic modifications it is natural to assume that the reorientations with the smaller barrier are mainly activated during the h t phase transition (in the low temperature region), and the reorientational motion of molecules is more or less anisotropic. Then in the phonon spectrum (if it is not prohibited by selection rules) one can observe broad "crystalline"frequencies with small intensities in the far infrared absorption spectra, as they originate in the cyclopentane crystal or in the Raman spectrum; for example, this was observed in the paracarborane spectrum [511. The reorientations with a large barrier are activated as a result of the second phase transition. The reorientational motions become essentially isotropic. The molecules obtaining the effective symmetry of the sphere become translationally equivalent, and the phonon spectrum has the form of the extended continuum. The neutron scattering experiment [36]in cyclopentane (Fig. 1.13)is certain proof of t h i s statement. The spectrum of plastic cyclopentane I is completely structureless, while a weak structure is observed in the spectrum of plastic cyclopentane II; its maxima practically coincide with the maxima of the far inkired spectrum (Fig. 1.13,Table 1.4). In spite of the reorientational motion in the cyclopentane crystal 11, its phonon spectrum does not smear. Probably, this means that some anisotropy of rotational reorientations remain in this modification. Thus, one can id en^ the plastic crystalline modification (the crystal II) and the high-temperature plastic modification (the crystal I) with isotropic rotational reorientations by means of the far i n f k e d absorption spectra. One can choose unambiguously one of the possible spatial structures by means of the far infrared absorption and Raman spectra of the anisotropic crystal m.
Thiophene From the thermophysical measurement [521 it is known that crystalline thiophene has five modifications: phase I is above 1745°K; phase 11 is in the interval 170.5' - 174.5"K;phase III is in the interval 136.8 - 170.5"K; phase IV is in the interval 111.3 - 136.8"K; and phase V is at temperatures lower than 111.3"K (Fig. 1.14). The mechanism of phase transitions and the structure of 2.
STRUCTURE! AND PHASE TRANSITION OF ORGANIC CRYSTALS
225
x of neutron, A 1
2
3
4
5
-1-1
100
20
5 2
0
-2
energy, MeV
FIG. 1.13. Neutron scattering spectra vs. transit time in solid and liquid cyclopentane (the scattering angle is equal to 80.5")1363.
150
100 0 0 .
000 0 0 0
50
100 150 T,K FIG. 1.14. Temperature dependence of heat capacity in crystalline thiophene 1521. 000: rapid cooling (3"Wmin)in the range of 190-77°K;00 u:the slow cooling (0.1"Wmin) in the range of 190-162°K; 0.0: the heating of the preliminary recrystallized sample.
ZHIZHIN AND MuKH.TARov
226
A A.
n
A 1083
1413
2
FIG. 1.15. Fragments of Raman intramolecular spectrum of four crystalline phases of thiophene: 1-85.0(phase IV); 2-223.0 (phase III); 3-163.0(phase II); r223.0 (phase I); 5-243.0"K(liquid).
crystalline phases are not unambiguously established. The X-ray analysis of thiophene I has shown that the orientations of molecules are disordered in this phase [53]. The investigation of the molecule mobility performed by the NMR method has demonstrated the dynamic character of disorder 1393. In Ref. 1541 the authors mentioned that the disorder is evidently partial and is related only to the rotations of molecules in its planes. A rather large difference between values of orientational barriers for the thiophene molecule reorientation in a plane and n o d to it has also testified to it 1551. These data allow us to suppose that the phase transitions in thiophene are accompanied by the orientational order change and the change of the orientational mobility of molecules in a crystal. Consequently, the polymorphism in thiophene
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
227
can be studied by means of Raman spectra in the region of intermolecular
vibrational fiequencies which are sensitive to the changes of molecular orientations. In this work we have for the 6rst time studied the Raman spectra in the region of external vibrational frequencies of thiophene crystalline phase; we have investigated also the nearest vicinity of the phase transition pointa 156,573. The intramolecular vibrational spectra of four crystalline moditications of thiophene were studied in detail and correctly interpreted in Ref. [54,58-611. Our intramolecular Raman spectra whose parts are given in Fig. 1.15 are in good agreement with data of other works. The Raman spectra of intermolecular vibrations of four thiophene crystalline phases in the region 20 - 200 cm-1 are shown in Fig. 1-16. The experimental frequencies are represented in Table 1.5. From the experimental data it is seen that the spectrum of crystalline thiophene IV and the spectra of other phases differ essentially. The lines of this spectra are most prominent. It testifies that the structure of the crystal IV is more ordered in comparison with the other phases. The parts of the intramolecular Raman spectra verified this fact (Fig. 1.15). It is interesting that the most intensive line of thiophene IV (54 cm-1) is anomalously changed with the temperature in comparison with the other spectral lines if the temperature approaches the point of the phase transition IV - III. The most intensive line is broadened more rapidly than other lines. Its intensity is decreased, and it practically disappears (Fig. 1.17). The parameters of other lines do not demonstrate essential changes. The Raman spectrum of e x t e d vibrations of thiophene III has the form of two broad maxima with explicit structure which disappear at the phase transition between thiophene 111 and thiophene II. The Raman spectra of thiophene II and thiophene I are identical. In our Raman spectroscopic experiments the thiophene modification observed in Ref. 1521 in the temperature range 170.5 - 174°K was not discovered. However, some changes in the spectrum of the crystal I were observed. This spectrum consists of two broad diffuse maxima with frequencies 52 and 116 cm-1at 175°K. If the temperature is increased in the interval of the phase I, one observes the successive "disappearance" of thc maximum 116 cm-1in the vicinity of the melting point, and one can see the "disappearance" of the maximum 52 cm-1 (Fig. 1.18). Thus, the thiophene crystal is completely orientationally disordered in the vicinity of the melting point, since in the low-frequency region one has observed the structureless wing of the Rayleigh line which is characteristic for the disordered system. The data concerning the crystalline phase structure of thiophene does not permit one to perform the calculation of the spectrum and its interpretation. However, we can make some conclusions on the assignment of some lines of Raman spectrum on the basis of our experimental data. As it was repeatedly mentioned,
ZHIZHIN AND MUKHTAROV
228
6
5
4
3
2
1 0
80
160
vcm-1
FIG. 1.16. Raman external vibration spectra of four crystalline phases of thiophene: 1-85.0(phase IV); 2-128.0(phase III); 3-158.0(phase II); r-213.0(phase I);5-233.5(phase I in completely disordered state); 6-243.0"K(liquid).
Table L5 T,WV~ m - 1
~1
v2
v3
v4
85.0 (phase IV) 128.0 (phase III) 158.0 (phase 11) 213.0 (phase I)
38.0
54.0
72.0
78.0
108.0
118.0
126.0
37.0
53.0
71.0
78.0
---
118.0
---
---
_--
62.0
---
__-
116.0
---
___
__-
52.0
---
__-
116.0
---
v5
v6
v7
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
I
229
!
FIG. 1.17. Temperature behavior of the line 54 cm-1 of the intermolecular vibration spectrum of thiophene in the vicinity of the phase transition "thiophene IV-thiophene 111": 1-85.0 (phase IV);2-112.0; 3-112.2; 4-112.4; 5-112.6; 6-113.0; 7113.5"K(phase III).
the activation of molecular reorientations around some axis leads to the broadening of spectral lines, their intensity decreases, and they smear into the Rayleigh line wing (it relates to lines of normal librational vibrations around this axis). According to NMR data [391 in the phase IV of thiophene at 93°K (i.e. long before the transformation to the phase 111) one has observed the noticeable orientational mobility of molecules. Assuming that in thiophene the reorientations around the axis normal to the plane of a molecule are most probable, one can suggest that it is the most intensive line at 54 cm-1that can be assigned to the vibrations with the partition of molecular rotations around this axis. This line, 54 cm-1, disappears in the nearest vicinity of the transition between the thiophene IV and thiophene III.
MIZHIN AND MUKHTAROV
230
7
6
5 4
3 2
1 0
80
160
v,cm-l
FIG. 1.18. Temperature dependence of Raman external vibration spectra of the tbiophene crystal I 1-215.0;2-227.0; 3-230.0;4-231.5; 5- 233.3; 6-233.5; 7-235.0 (liquid). Thus, the anomal temperature dependence of the 54
cm-1 line shows that the
activation of rotational molecular reorientations are anisotropic in the narrow temperature range in the vicinity of the phase transition IV - III. Successive transitions in thiophene (111- IT - I) are, evidently, related to the W e r increase of the anisotropic orientational mobility of molecules. It is proven by means of data in Ref. 1621 on the basis of the X-ray analysis that, in thiophene I, the orientational jumps around the axis perpendicular to the molecular plane are
STRUCTURE:AND PHASE TRANSITION OF ORGANIC CRYSTALS
231
possible for twenty equivalent orientations. In the case of thiophene 11 they are possible only for ten orientations. It is interesting to compare the obtained results for thiophene with the results of investigations of the furane crystal where the molecule is nearly identical to thiophene. Unlike thiophene, furane has only two crystalline phases. Furane II is ordered (whether P41212 or P43212, z = 4) [631, and furane I has the disordered crystalline lattice which is like the lattice of thiophene I. Their unit cells are characterized by the space group whether Cmca or Abaz. They contain four molecules [53,62,63,64]. The N M R data [391, the dielectric measurement data 1651 and the X-ray data help to establish that in furane I the molecules performed the orientational jumps around the axis perpendicular to the molecular plane. These jumps are allowed for four equivalent orientations [63]. The Raman external vibrational spectrum of furane I [64] which consists of two broad lines with frequencies at 50 and 120 cm-1is nearly identical to the spectrum of thiophene I. However, in spite of the identity of the high temperature phase of thiophene and furane, in the case of the last one there were not intermediate modifications observed in thiophene (crystals 11and III). Probably, it can be explained by the fact that, in the low temperature phase of furane, the intermolecular hydrogen bonds [661 create obstacles to the molecular reorientations. Their noticeable activation becomes possible only in the case of the high-temperature phase.
C. Statistical Model of Orientational Phase Transitions in Plastic Organic Crystals with the Consideration of the Rotational Reorientation Anisotropy To describe the molecular crystal dynamics having phase transition into the plastic state some authors have proposed several statistical models [67-691, which describe satisfactorily the main features of the behavior of such systems. These models are the natural extensions of the site melting theory (the Lennard-JonesDevonshire theory) to the case of the orientational disorder processes. They are of two types: one the simplest type when the molecules perform the orientational transitions into one of the two allowed orientations [67,681 or where there are several identical orientations, but the potential barriers are equal, and the reorientational motion is isotropic [69]. However, the experimental data show that the polymorphic transformations can be more complex in these crystals. Thus, the investigations of solid cyclohexane by the N M R method and our investigations of the Raman spectra with the dynamical calculations by the AAP method allows us to state that the activation of the reorientational molecular motion around the axis close to the Cs-axis of the free
ZHIZHIN AND MLTKHTAROV
232
molecule begins long before the phase transition to plastic modXcation. These reorientations (unlike the other ones) are activated practically completely before the transition. Thiophene has demonstrated the identical phenomenon in the vicinity of the transition "crystal IV - crystal III". The reorientations of molecules begin in the low temperature phase in the cases of 1,2-dichloroethane and cyclohexane. They lead to the phase transition when the one dimensional plastic crystal is formed. We know several compounds having two plastic modifications (cyclopentane, dl-camphor, para-carborane, etc.). This gives the justification to assume that for the more complete description of the plastic molecular crystal thermodynamics it is necessary to take into account the anisotropy of the rotational reorientations, i.e. to consider the presence of orientational barriers with different values. For t h i s reason, the statistical model proposed in 1691 is extended to the case of reorientations with two different potential barriers.
--
Building of the model the thermodynamical characteristics of phase transitions. The site disorder of a crystal (as in the Lennard-JonesDevonshire model) is taken into account by the introduction of two interpenetrating sublattices: a-basic and &hypothetical sublattices of the interstitial sites. At the nodes there are molecules occupying them &om the nodes of the basic lattice. In the general case, every node of the type a is surrounded by z nearest neighboring & nodes and by z' nodes of the type a and vice versa. To take into account the orientational disorder we suppose that in every one of the sites of a and p the molecules can have any orientation of two different sets having D1 and D2 thermodynamically distinguishable allowed orientations, apart from the orientation defined by the crystal space group. We also assume that there is not another short-range order of localizing the molecules in different sites and orientations, besides such one which is defined by the long-range order, i.e. the Bragg-Williams approximation is used [701. At sufficiently low temperatures, the largest number of molecules of a crystal are in sites of the basic lattice and oriented according to the space group requirements, i.e. if S is the number of these molecules in unit parts, then S tends to the unit for T=O. If the temperature is raised, the reorientational motion is activated, and certain parts of molecules S1acquire D, orientations of first set and another part Sz has D2orientations of second set. Evidently, the equality is always valid
The degree of the site ordering Q is defined by the relationship Q = NaN, where N = Na + NB,. N is the total number of molecules in a crystal; N a and Np are the number of molecules in appropriate lattices.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
233
The total energy E of a crystal is assumed to be equal to the s u m of energies of pair interactions of nearest neighbors of different kinds. Then it is not difficult to obtain the following expression
E = zWNQ(1 - Q) + Z'NR, (WiEl+ W2E2) where W is the energy of the interaction (repulsion) of the pair of neighboring aand fl- molecules; W, is the interaction energy of neighboring molecular pairs in sublattices, when each of them either keeps the starting orientation or has any orientation of the first set; W, is the analogous value for the case when each of the neighbors can have any orientation of the second set. We introduce the following notations
% = 1 - 2Q + 282 El = D1S1 (S+ (D1 - 1) S1/2) E2 = D2S2 (1 - (D2 + 1) S42) The interaction between the site disordering process and the orientational disordering processes are taken into account by the artificial method as in Refs. [67,68,69]: the pair a+ of nearest neighbors do not contribute additionally in the energy, even if their orientations differ. Define Y (S, S, S2, Q);Y is the number of possibilities for localizing the crystal molecules at sites a and and for all orientations:
Thus, we have the following expression for the configurational contribution into the statistical s u m R
The s u m in this expression is performed in all possible orientation sets and molecular positions in a crystal. Applying the standard way of the replacement of the total s u m by the maximal term (for N +00 1, using the Stirling formula for N! and differentiating Inn in S, Sl,S2,Q , we obtain the following system of equations which allows us to seek
234
ZHIZHINANDMUKHTAROV
the equilibrium values S, S1,4, and Q (together with the relationship (I.l)), mnaimizing the distribution function:
Here the following notations are introduced (the letter v with the index is used only to denote the model parameters in this section): x = zWkT and
The relationships v1 and v2 are the measures of energy barriers that prevent the orientational and site molecular disordering, speaking more precisely, they are the measures of relative increasing of the potential energy during the change of the molecular orientations in comparison with the increase of the energy at site jumps. Ifvl = v2, the system of equations (1.5) - (1.8) is reduced to the appropriate system given in [691; it is quite justifiable, since the model described in 1691 is the particular case of the model under consideration. The dependence of values S and Q on the parameter X is given in Fig. 1.19. In the low temperature region (large values X) the crystal is practically ordered S and Q are close to 1. If the temperature is increased the value S is gradually decreased, and at a certain temperature it drops by jumping to the value which is close to 1/(1 + Dl); it becomes equal to Sl,i.e. almost complete "release" of reorientations with the smaller barrier occurring. Other regions of drastic changes of the value S correspond to the activation of reorientations with a large barrier at high temperature. The degree of site ordering Q differs little from 1 all this time, and at further increasing temperature it drops almost by jump to the value Q = 1/2 verifying the site disordering. The smaller the value vl, the lower the temperature of the first jump; and ifvl is constant, the smaller the difference between v2 and v1, the closer the temperature (the temperature range) of the second jump to the temperature of the first jump. Thus, the possibility of existence of plastic crystalline modifications with activated reorientations (with smaller barrier) and with "frozen" ones (with large barrier) follows from the dependence of the values S and Q on the parameter x. However, the most complete information concerning the behavior of the system is obtained from the equation of state.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
235
FIG. 1.19. Dependence of degrees of the orientation and positional disorder on the parameter x = zW/kT: 1-S;4-Q; D1=3, D2=20; Vl=O.O6;V2=O.25; 2-S; 3-6; D1=2; D2=20; ~1=0.1; V2=0.45
Assuming that the thermodynamic functions of a crystal (f+ee energy, enthropy, etc.) can be divided into two parts, we can use the Lennard-JohnesDevonshire cell model for the description of the ordered state. One of the parts relates to the ordered state, and another one describes the disorder effects. It is assumed that the crystal has the face centered cubic structure, and the interaction between the molecules is described by the pair potential in the Lennard-Johnes form
Here E is the potential minimum; ro is the distance at which the attraction is equal to the repulsion. In Ref. [711 the method of calculation of W e r e n t thermodynamic values, including the isothermic compressibility and the enthropy, is given on the basis of the abovementioned model. The thermodynamic values with dependence on the reduced temperature kT/& and volume V N , were calculated 1711 with consideration of three nearest neighbors; V, is the ensemble volume consisting of N molecules packed in the face centered lattice in which the distance between the nearest neighbors minimizes the interaction potential (r = 2v6 r,).
ZHIZHIN AND MLTKHTAROV
236
To take into account the disorder effects it is necessary to make additional assumptions concerning the dependence of values W, Wland W, on a volume. Assuming as in Ref. [67-691, that the main contributions are made by the repulsion forces, we can write 4
w=wo(+).
w1 - w O (V0)1 lV ’ w2 -- w0
(2 V7 g
where Wo,Wol, WO2 are constants. Hence, it follows that the dimensionless parameters v1 and v2 do not depend on a temperature and a volume and are defined only by the nature of a compound. In calculating the phase transition isotherms we have also used the empirical relationship
WO = 0.977 obtained in Ref. [671. &
Thus,
in the case of compressibility we have the following expression for the term caused by configuration effects (p is the pressure, R is the gas constant)
The equation (1.9) is the isothermic compressibility W&T as the function of the reduced volume VN, and the temperature kTk and also the parameters v1,v2, Dl and D2. To obtain the complete isotherm it is necessary to add the contribution corresponding to the ordered state. The latter together with other thermodynamic functions was calculated by the same method as in Ref. [71]. The values corresponding to the phase transitions were sought according to the Maxwell rule for the zeroth pressure. In studying the thermodynamical characteristics of phase transitions we have paid the most attention to estimation of the conditions of the existence of the orientational transitions. The calculated results show that there is some region of parameter values v1 (vl 7 0.5) in which either one or two transitions caused by the reorientation activation and the phase transition describing the positional disorder (melting) of a crystal exist. The number and the character of the orientational transitions, apart from the values of the relative barriers v1 and v2 essentially depend on the number of allowed orientations D1and D2. The calculated dependences of the phase transition temperatures and of melting on the value of v2 at fixed values of other parameters are shown in Fig. 1.20. In the region where Dl is equal to 2 and D2 < 6 for all allowed parameter values v1 and v2 (it does not matter how they differ in values) one has observed only one orientational transition which is the transition of the first order in the frame of
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
I 20'
237
I 02
0,4
"2
FIG. 1.20. Dependence of phase transition temperatures and melting on the large orientational barrier: D1=2, vl=O.1. Values of D2 are indicated by numbers; the regions of the second order phase transitions are indicated by dashed lines.
I
1.05
I
I
1.09
I
VIVO
FIG. 1.21. Isotherms of the second orientational transition vs. value of D1 (the values are indicated by numbers). vl=O.l; v2=0.275; Dz=20.
238
ZHIZHINAND MUKHTAROV
the describing model. Its isotherm has the form which is similar to the curve 2 of Fig. 1.5. Consequently, we can conclude that only the difference between the barriers cannot be the single reason for the existence of the successive orientational transitions. IfD2 is equal to 6, there is some region of values v2 for which the activation of rotational reorientations associated with the overcoming of a larger barrier causes one more transition (being the transition of the second order in the frame of the model) in the vicinity of the melting point. Its isotherm has the form which is similar to the curve 5 in Fig. 1.21; the point of inflection on the isotherm corresponds to the moment of the transition. The picture becomes more complicated at D2 = 8. The region of the transitions of the second order is shifted to the side of lower values of v2, and the appropriate region of phase transitions of the first kind appeared in the vicinity of the melting point. At further increasing D2 the region of second order transitions is shifted fiwther (in temperature and in value vp) to the region of transitions related to the overcoming of smaller barriers. Finally, it is completely displaced by the corresponding phenomena of the first order. It is interesting to note that the dependence of the orientational phase transition of higher temperatures on the value of D, has the inverse character in comparison with D,. It is a transition of the first order for all D, < 4,a transition of the second order at D1 = 5, and this transition vanished completely for Dl > 5 (see Fig. 1.21). The regions of changing of the parameter v2 can be divided into three parts (see Fig. 1.20) for sufficiently large values D2 (D2 $: 20). I f v 2 is lower than some limiting value vz(, the activation of rotational reorientations with the smaller and larger barriers occurs during the process of the single transition. The calculation of the dependence of the orientational order rate on the temperature shows that the smaller the difference between v2 and vz(, a larger number of molecules participates in the reorientational motion with smaller barriers at the approach to the transition point, i.e. the more prominent is the anisotropy of rotational reorientations in the low temperature modification. The reorientations with the larger barrier are substantially activated only in the process of the phase transition. There is also the region of two phase transitions with the successive activation of reorientations with Werent barriers. There is again only one orientational transition, if v2 is more than the second limiting value vn; the reorientations with the smaller barrier are practically completely activated, and the reorientations with the larger barrier are "frozen" and activated only during the melting of a crystal. As was mentioned, this situation can qualitatively correspond to the case of the one-dimensional plastic crystal, for example, to the crystal of 1,2dichloroethane [72,731or to some straight-chain par&ns [14,741.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
239
With increasing the parameter v1 the region of two successive orientational transitions diminishes sufEciently rapidly, and for every D1 and D2 there is such limiting value above which there is only one orientational transition. In this case the increase of the larger barrier of v2 leads to the approach of temperatures of this orientational transition and of the melting transition until they coincide and make one common transition at which the orientational and positional disorder occur simultaneously. The typical dependences of relative volume and entropy changes of transition and melting on the parameter v2 are given at Fig. 1.22 and 1.23.
The estimation of the phase transition enthropy in cyclohexane. The discussion of conditions of the model applicability. To estimate the transition entropy in cyclohexane by means of the model, it is necessary to b t determine the values of parameters Dl,D2,vl, v2. The presence of rotational reorientations of molecules (around C3-axis) in the crystal I1 which have a smaller barrier in comparison with barriers of another reorientation shows that in the frame of the model it is natural to assume that Dl + 1 = 3, i.e. Dl = 2. To estimate the value of D2one can use the results of Ref. 1181in which the authors have shown that the assumption of isotropic orientational molecule jumps between 24 difFerent orientations in crystal I is in better agreement with the X-ray data than the assumption of the continuous (retarded) rotation of a cyclohexane molecule in this phase. Thus, in the frame of this model it is necessary to assume that D1+ D, + 1= 24;hence, it follows that D2= 21. The values of v1 and v2 are estimated more diflticultly. As in the models of the Pople-Karasz [67,681 and Amzel-Becka 1691 these parameters are not related directly with the experiment, and they cannot be calculated theoretically at the modem state of the knowledge concerning the intermolecular force nature. The relationship between the rotational reorientation barriers and the self-diffiion barriers gives the general picture of them 175,761. The barrier value is in the limits 4.5-7 k d m o l for the C3-axis (see Table 1.3 in LA). The value of 11k d m o l obtained in Ref. 1191must be excluded as explicitly exceeding one. The value of the second (larger) barrier is about 19 k d m o l according to the NMR data [38], our calculation for axis u and v gives the value of 19 and 30 k d m o l , correspondingly. However, these values should probably be reduced by a third (13 and 20 k d m o l ) , as in the case of pyrene. Unlike the reorientations, the diffusion process in crystals requires the presence of vacancies. The energy of forming the Shottky simple defect is equal approximately to the energy of the lattice of a crystal, but the energy which is necessary for transferring the molecule into a site between nodes is approximately three times as much according to the calculations for the solid inert gas 1391. If we
ZHIZHIN AND MUKHTAROV
240
0.1
0 0.2
0.4
v2
FIG. 1.22. Dependence of the relative change of the volume for the orientational phase transitions and melting on the value of the large barrier v s
/
I
0.2
I
I
1
0.4
v2
FIG. 1.23. Dependence of the enthropies of the orientationar. transitions and melting on the value of v2.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
241
TABLE 1.6 Phase Transition Entropy and Melting Entropy for Cyclohexane V1
0.15 0.15 0.14 0.16 0.175 0.18 0.19 0.21 0.21
Experiment
v2
0.4125 0.3 0.315 0.32 0.28 0.306 0.304 0.3255 0.315
Tph.pmelt
&melt/R
@ph.t/R
0.951 0.717 0.751 0.758 0.667 0.728 0.717 0.762 0.74
0.62 1.28 1.20 1.17 1.39 1.25 1.25 1.13 1.19
4.44 3.87 3.78 3.99 4.00 4.04 4.11 4.24 4.19
0.665
1.15
4.35
assume that it is valid for cyclohexane, then we have the following estimations for v1 and v2: v1 = 0.14 - 0.21 and v2 = 0.4 - 0.6 (the lattice energy of C a 1 2 is equal to 11kcal/mol[771). In fact, the mechanism of diffusion in cyclohexane is more complex than the formation and migration of a simple vacancy [78]. It leads to the decrease of its activation energy and to the essential increase of parameters v1 and v2 such that in the frame of the model the reorientation activation occurs only at the melting point (the reorientations with the large barriers). For this reason, the aforementioned estimations were taken as the starting ones, and their corrections were performed by the coincidence of the calculated melting entropy value with the experimental data and by comparison of the phase transition temperature with the melting temperature (i.e. by their ratio as in Ref. [69]). The calculation results are given in Table 1.6 (D1= 2; D2 = 21). As is seen &om Table 1.6, the region of possible values of the parameter v1 remains unchanged, and the region of the values of the parameter v2 is shiRed a little to the side of smaller values in comparison with the preliminary estimation. According to the aforementioned reasons we have not tried to attain the maximal coincidence with the experimental values T,h.flrnelt and Asmelt; however, in general, we can conclude that in the case of the reasonable chosen parameters the proposed model reproduced sufficiently the values of phase transition entropy and melting entropy and the ratio of the appropriate temperatures for cyclohexane. Thus,from the aforementioned, it is seen that the consideration of rotational reorientation anisotropy allows us to extend considerably the region of phenomena under study. We have all the justifications to assume that the given model can be
ZHI2;HIN AND MUKHTAROV
242
applicable not only to plastic crystals, but also to the molecular ones which do not have mesophases as the description of the orientational disorder processes in molecular crystals which do not give mesophases up to the melting point. The question concerning the comparison of the quantitative predictions of the model with the experiment is raised because a number of simplifications and approximations are used in the construction of a model. Thus, one does not consider the fhct that the crystalline lattice symmetry is changed at the phase transition of the order-disorder type in the plastic crystals. Instead of this, one assumes that the system symmetry before the transition remains the same as after the transition (the face centered cubic lattice). The cooperative effects which are, in general, often essential 1551 are ignored for the sake of simplicity. Furthermore, the real force field of a crystal is approximated by the simplified potential of the intermolecular interadion in the form of Lennard-Jones, and in considering the values of w, wl, and w2 one supposes additionally that they are caused only by repulsion forces. A number of the serious simplifications is associated with the difficulties arising in calculating the statistical sum. Consider this problem in detail. The calculation of the statistical s u m is the central problem of statistical mechanics. In our case of systems with transitions of the order-disorder type, it is necessary to take into accoullt the distribution of molecules in positions and orientations of Merent types. If the generalized parameter of the long-range order is denoted by P, we have the following expression for the statistical sum Z(P)
(1.10) where we have the following notations for the crystal in kth configurational state at a given value P is the potential energy obtained at the condition such that all molecules are at rest; E, is the energy of the vth vibrational state. In the harmonic approximation this energy is expressed by the well known formula
where ni are the quantum numbers of n o d vibrational modes, o$ are their frequencies. We have the following expression for the vibrational statistical sum of a crystal in a given configurational state
n
Zk(P)= I: exp(-Et/kT) = Zok 3N (l-exp[-h"k/kT]) -1 V
i=l
STRUCTLlRE AND PHASE TRANSITION OF ORGANIC CRYSTALS
243
where Z,k is the contribution of zeroth vibrations of a crystal. If the expression (1.10) is now rewritten in the form
(1.11)
then the necessity of the analysis of the dependence of normal vibrational frequencies on the configurational crystal state is perfectly clear. However, it is very dillicult to perform this analysis in practice. The problem of the interaction between the order-disorder processes and lattice vibrations was solved by the use of the Einstein model 1791 and the Born von Karman model [801 with the description of vibrational modes for the statistics of the disordered systems of the binary allowed type. Dean 1811 has made a successful effort to calculate the distribution of frequencies for the disordered solid solutions by means of the computer simulation. The phonon state density calculated by this method in Ref. [821is compared to the Raman spectra of the lattice vibrations of the statistically disordered crystal of nnitrochlorobenzene. In the case of the systems in which the order-disorder phenomena have a prominent dynamic character, as in plastic crystals, the problem of the investigation of vibrations for different configurational states is more complicated. For this reason, one introduced the assumption such that the problem related to the calculation of the vibrational and configurational contributions in the statistical sums can be divided. In practice this division can be realized by introduction of the configurational statistical s u m
(I.12) and assumption such that the thermodynamical crystal functions are the s u m of values calculated by means of Z, and values obtained from vibrational contributions. It is clear that this approach will be successful in a certain case, when the vibrational statistical s u m will be considerably less sensitive to the system order degree than the configurational one; and only in this case one can hope that the analysis based on this division leads to the qualitatively correct description of the order-disorder phenomena. Let us formulate the conditions under which this division is more or less correct. First of all, the crystalline field must be so strong that the thermal motion cannot essentially destroy the picture of localized molecular orientations. Furthermore, the depth and curvature of potential wells of angular dependence of
244
MIZHIN AND MUKHTAROV
the intermolecular interaction energy must not change noticeably at the phase transitions. Finally, the phase transition must have mainly the orientational character and be performed in a well defined jump way. The second of these conditions is often not fulfilled in plastic molecular crystals. As a result, the additional contribution from the "noncodigurational" disorder processes appeared in the transition entropy; it is necessary to distinguish the increase of fteedom of the vibrational motion at the phase transition. As in the Pople-Karasz and Amzel-Becka models, in the present model this contribution is not taken into account; however, the problem of the division of the transition entropy on the configurational and nonconfigurational terms was not the goal of thi s work due to its complicated character. This problem is the theme or the subject for an individual or separate investigation. For this reason, we have not tried to obtain the complete coincidence between the calculated values and the experimental ones at estimating the transition entropy of cyclohexane by means of this model (see Table 1.6). The precise calculation of the configurational statistical sum (1.11) still remains an extraordinary complicated problem, since it is necessary to know the number of configurational states with a given energy at a fixed value of the order degree P. The combination problem which must be solved is so complicated that to solve it one must use the various approximation methods, in particular, the BraggWilliams approximation [70]. This approximation relates to the number of widely used approximation methods which are called the "molecular" or "mean" field theories. The main assumption of the method concludes with neglecting all correlations of the short-range order and assuming the correlations of the longrange order. The heuristic meaning of this approximation is evident; however, the estimation of the error which arises is difficult. In the direct vicinity of the transition point the Bragg-Williams approximation gives, probably, the same results as the Landau theory of phase transitions. Thus, it is evident that if the approach is so simplified, one cannot require the quantitative agreement of numerical model predictions with the experimental data for the concrete crystals, and it is possible to obtain only the qualitative description of their properties. The abovementioned statements are applied completely to the models of the Pople-Karasz and Amzel-Becka. All three models are essentially based on the assumption that the main role in phase transitions in plastic crystals belongs to the orientational disorder. However, our constructed model considers the orientational jumps with Werent barriers, unlike the abovementioned ones. It permits one to describe the following experimentally observed phenomena ftom the single point of view: the anisotropy of rotational reorientations in the vicinity of the phase transitions, the one-dimensional plastic crystal and two successive transitions into the rotationally crystalline state [831.
S T R U C m AND PHASE TRANSITION OF ORGANIC CRYSTALS
245
II. REORIENTATIONAL MOTION OF MOLECULES IN
CRYSTALS WITHOUT MESOPHASES
We have discussed in the previous chapter the statistical model of orientational phase transitions in plastic crystals considering the anisotropy of rotational reorientations. This model predicts the possibility of the existence of the anisotropic rotational reorientational motion of molecules in crystals without the plastic modification at the phase transition "anisotropic crystal - isotropic liquid. To verify the stage-by-stage approach in the "melting" of rotational degrees of &eedom of molecules in a crystal and the possibility of its extention to the crystals without mesophases, we have studied the vibrational spectra of naphthalene and benzene crystals in a wide range of temperatures including the close vicinity of the phase transition into the liquid state.
k Remelting Effects in Naphthalene Crystal Naphthalene crystallizes at monoclinic syngony with a P21/a space group, and two molecules in the unit cell are in position Ci [l] (Fig.II.1). According to the group theory analysis in the Raman spectrum of this crystal six librational frequencies (3% + 3B,) are active, while in the far infrared spectrum three translational frequencies (2A, + B,) are active. Low frequency spectra of a naphthalene crystal were studied many times [2-101. All the frequencies allowed by the selection rules have been detected. The experimental results and our calculations are summarized in Table II.1. Calculations of the frequencies and normal coordinates were made by means of the AAPI1 parameters (the summation radius 6A) and with the use of the structure determined by a n X-ray analysis at 296°K [HI."he frequencies were calculated with consideration of anharmonicity corrections (see Part I, 1I.C).
ZHIZHIN AND MUKHTAROV
246
0
2
4
6
8 10
A
FIG. II.1. Unit cell of naphthalene crystal. Orientation of librational axes W', V, W') relative the the principal inertia axes (U,V, W)of naphthalene molecule in a crystal [l] (I, < I, <.),I
TABLE II.1 Experimental and Calculated Lattice Modes for Naphthalene Calculation (cm-1) Experiment (cm-1) 4K [51 121 89 68
77K I21 120 88 67
B,
141 83 57
141 83 56
130 76 50
125 71 46
118.5 71.5 48.5
Au
108 57
106 (44)
102 (41)
98 53
88.6 42.1
Ag
195K [21 293K [2,6] 113 109 80 74 59 51
296K 114.3 84.7 52.8
vibration forms, % 9OL, 90Lv lOOL,
lOOL, 80 + 20 L, 80 L, + 20 L, 65 TA + 35 T,* 65 T,* + 35 T,
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
0
100
cm-1
0
100
247
cm-1
FIG. II.2. The Raman spectra of naphthalene crystal: experimental [4]and calculated ones i n the approximation of the oriented gas approximation. The Raman spectra of naphthalene observed experimentally are given in Figs. II.2 and 11.3. The calculated results for Raman line relative intensities in the oriented gas approximation with the use of isolated molecular polarizability are given in Figs. 11.2 and 11.3 and in Table 11.2 (see Table lV.2,Part I, IV.C). In the Raman spectrum, the totally symmetrical vibrations are active under four conditions of the polarization of the incident and scattered emission aa, bb, c*c*, ac*; the vibrations Bg - under two conditions (ab, bc*). The calculated results are in good agreement with the experimental data. The spectral lines are grouped in pairs of $- and B, species. As the temperature increases, appropriate pairs merge into broad bands. The use of effective polarizabilities of molecules obtained fiom the optical properties of crystals provides much poorer agreement with the experiment. The lines vl($) and v4(Bg) which correspond to the librational molecular motion relative to the U' axis, close to the axis U of the smallest moment of inertia, are much broader than the other, and their bandwidths depend exponentially on the temperature (Fig. II.4). According to the opinion of the authors of Ref. [lo], the temperature dependence of bandwidths are caused by the reorientations around the axis with the smallest moment of inertia in the naphthalene crystal. The authors of 181 have suggested that the broadening of the lines ~ ~ ( $ and 1 v4(Bg) is caused by strong anharmonicity of vibrations; this is verified by the discrepancy between the temperature evolution of frequencies calculated in the quasi-harmonic approximation compared to the experimental one, which is particularly large for lines v1 and v4 (see Figs. 11.3,11.4).
248
cm-'
140
120
100
60
80
40
20
FIG.II.3. Raman spectra of polycrystalline naphthalene: experimental [21 and calculated in the oriented gas approximation.
exp. an-1
TABLE II.2 Calculated Relative Intensities for Naphthalene .. % Relative . calC. n I an-1 aa bb c*c* ac* ab bc* polycrystal
43
109 74 51
110.4 83.0 53.7
BF.
121 71 46
105.6 77.9 44.1
34 2 0
29 10 7
0 20 5
0 34 5
--
--
64.1 100 20.7
22 1 1
1 12 1
46.9 25.4 4.5
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
249
FIG. II.4. Dependence of the band widths of some Raman spectral lines of intermolecularvibrations of naphthalene crystal on the temperature [8].
24
1 v kcaVmol
12
0
-12
-24
FIG. 11.5. Dependence of the potential energy of crystalline naphthalene on the molecular rotation angle ($1 around the W axis (solid line) and around the Vaxis (dashed line) at room temperature.
250
ZHIZHIN AND MUKHTAROV
The dependence of the lattice energy calculated at room temperature on the angle of rotation of chosen molecules around their main axes of inertia with fixed positions of the rest of the molecules is shown in Fig. 11.5 [1,12]. From this picture it is seen that the smallest reorientation barrier arises at the rotation of a molecule around the axis W which is the axis of the largest moment of inertia. T w o other barriers arise with the rotation of molecules around axes U and V which are considerably larger (not shown in Fig. 11.5) than axis W. Attention is drawn to the barrier of reorientations around the axis W. The barrier is relatively large (about 16 k d m o l ) . The curve has some sufEciently deep minima. The conditions for the reorientational motion are complicated. It was shown by the X-ray structural studies of the complex naphthalenetetracyanobenzene that at 250°K in this crystal the naphthalene molecules have been randomly rotated in their plane inside an angle limit of 36" [13]. For this temperature the NMR signal begins to narrow indicating the reorientational motion [131. Calculation of the reorientational barrier around the axis perpendicular to the molecular plane made in Ref. 1131 for a naphthalene-tetracyanobenzenecomplex gives a curve very close to ours of pure naphthalene (Fig. II.5). This permits one to suppose that for high temperatures in pure naphthalene a static disorder is realized: naphthalene molecules are statically disordered around the W axis inside an angle limit of about 60" with preferential conservation of the orientations relative to other axes. The increase in the number of the irregularly oriented molecules results in the violation of the selection rules for the wave vector -- the opening of the Brillouin zone and broadening of the lines. The preferential broadening of v1 and v4 lines in this case may be explained by a strong dependence of the dispersion branches corresponding to these vibrations on the wave vector (Fig. II.61, and high sensitivity of the librational vibrations around the U axis for such type of disorder arising from strong anharmonicity of these vibrations. The anomalous behavior of lines v1 and v4 in comparison with the other lines of the spectrum is especially displayed in the premelting region where one may expect to have the redistribution and a considerable decrease of the orientational barriers to the values sufficient for the activation of the reorientational molecular motion. In particular, the data in Ref. [14] where the temperature dependence of the volume of the unit cell of the naphthalene crystal was calculated by the methods of the molecular dynamics predict the anomalously large increase of the volume in the vicinity of the melting point (Fig. 11.7). "he temperature dependence of the Raman spectra of a naphthalene crystal was investigated by us in the premelting region with the temperature step 0.1"K and the thermostatic accuracy off 0.01"K 1151.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
100
251
u
75
I
I
50
I I _
I
0
0,lf:
02;
0,3;
0,4if
0,5:
K
FIG. 11.6. Cross-section of dispersional surfaces a(g) of naphthalene.
100 300 T,K FIG. II.7. Experimental (El and calculated (M, W, K) temperature dependences of the unit cell volume of naphthalene crystal [14].
ZHIZHIN AND MUKHTAROV
252
I 200 200
v cm-1
FIG.11.8. Temperature dependence of external vibration Raman spectra of naphthalene single crystal in the polarized radiation in the premelting region. a(ba)c 1-352.6;2-353.65;4-353.7;5-353.75;6-353.8;7-354.0K (melt). a(ca)b 1-348.0;2-352.7;3-353.1;4-353.4;5-353.6;6-353.9;7-354.0K (melt). a(bb)c 1-350.0;2-353.6;3-353.65;4-353.7;5-353.75;6-353.9;7-354.0K (melt).
As the lines in the external vibrational spectrum of t h i s
crystal overlap at
temperatures above room temperature (Fig. II.3), single crystal specimens and polarized radiation were used for the experiments. The temperature dependences of these spectra in the closest vicinity of the melting point are shown in Fig. II.8. One can see that the lines v1 and v4 decrease in intensity much faster than the other Raman lines. They disappear completely near the melting point. The disappearance of these lines happens in the temperature interval -0.02"K. Beyond 0.25"K up to the melting point these lines are practically unobservable, whereas the other lines of the external vibrational spectrum are clearly observed (librations around the axes close to the axes of the largest and average moment of inertia of a
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
253
140 v, cm-1
100
60
345
349
353
FIG. 11.9. Temperature dependence of b a n spectra frequencies of naphthalene single crystal external vibrations in the vicinity of the melting point. molecule). As the temperature decreases from 353.8 to 353.6"K the v1 and v4 lines appear in the spectnun again. One has not observed the anomalous behavior of frequencies: in the premelting region the frequencies of all lines depend linearly on temperature up to their complete smearing into the Rayleigh line wing (Fig. II.9). The experiment was repeated with two single crystal specimens in four crystal orientations. In every case, in the closest vicinity of the melting point one has observed the predominant smearing of the v l and v4 lines into the Rayleigh line wing; the other lines remain unchanged. Considering that the anomalous behavior of the v1 and v4 lines m a y be caused by their intensive depolarization as the temperature of the single crystal approaches the melting point, we have carried out a n additional study of the polycrystalline specimen in the premelting region (see Fig. 1I.lOa). In the polycrystalline sample the lines v1 and v4 in the premelting region are even more
254
ZHIZHINAND MUKHTAROV
a
I
0
I
b
I
100
v, cm-1
FIG. 11.10. Temperature dependence of Raman spectra of external vibrations of naphthalene (a) and solid solution of naphthalene (95%) + @naphthol (5%) (b) in the premelting region. 1-353.0;2-353.0;3-353.3;4-353.4;5-353.5;6-353.6;7-353.7;8-353.9;9-354.0; a) 10-355.2K (melt). b) 1-352.0;2-353.1;3-354.3;4-353.6;5-353.7;6-353.8;7-353.9;8-355.2K(melt). broadened and not resolved. Nevertheless, it is clearly seen that here also the lines smear into the Rayleigh line wing much faster than the other librational lines of the spectrum. This fact is also clearly demonstrated by Raman spectra of the naphthalene@naphthol mixture (Fig. ILlOb). The selected concentration of a mixture of naphthol is 5%. It is well known that, when an impurity is introduced, the
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
255
temperature interval of the premelting effects in the crystal increases. We have chosen naphthalene-pnaphthol as the mixture because the crystalline structure of this mixture is well studied in wide ranges of temperatures and pressures 116,171. For the selected concentration of the admixture (5%) the solid solution lattice is similar to the structure of a pure naphthalene crystal [171. In spite of the fhct that the increase in the temperature interval of the effect in this case w a not greatly increased (apparently, no more than 0.5"K),one can see distinctly from the spectra, as compared to the other spedral lines, the preferential disappearance of the lines which refer to the molecular librations around the U axis. Thus,the phenomenon of rotational reorientation anisotropy was observed in naphthalene crystal in the course of precise measurements of the temperature dependence of the low frequency Raman spectra in the narrow premelting region. The calculations of frequencies and eigenvectors show that the orientational disorder happens around the U axis, close to the axis U of the smallest moment of inertia of a molecule. The orientational disorder of molecules around two other axes (V,W takes place only at the transition from the crystal to the liquid state. The results of Ref. [18] also indicate that the certain pretransition occurs in the Vicinity of the melting point of naphthalene crystal. The authors of Ref. [18] have observed the maxima at the curves of melting and solidification obtained by the DTA method. It is interesting to compare the results of the Raman spectra with the NMR data [19]. In this work the authors have found that the temperature dependence of the spin-lattice relaxation time of TIis not anomalous in the whole interval of the solid phase. Unlike this, the dependence of TI,,(the time of the spin-lattice relaxation in the rotational coordinate system) has slight minimum in the vicinity of the melting point (Fig. II.11). It is the proof of the appearance of a certain molecular mobility. However, the authors cannot determine the exact type of motion that caused the anomaly.
B. The Raman Spectra of Benzene Crystals. The Temperature Dependence of Spectra in the Vicinity of the Melting Point No phase transitions occur in crystalline benzene at normal pressure; its 1201 at 140°K and 218°K as well as by X-ray diffraction [21] at 270°K (the melting point being 278°K). The space group of benzene crystal is Pbca, Z = 4 (Ci) (Fig. 11.12). structure was defined by neutron *action
ZHIZHIN AND MUKHTAFtOV
256 357.1
I
0 102
I
B
I I I
."!a
I
UY
'B 0
-8
10
8.l
345.0
333.3
322.6
/
I lyJ
312.5
T,K
T1
I
10-2
I
/
I I
I
10-3
I
I
T1p
'L4JP I 10-4 2.8
2.9
3.0
3.1
FIG. II.11. Temperature dependence of spin-lattice relaxation times TI and T1p in naphthalene crystal [19].
-;t
%
pH'
H2
V
FIG. 11.12. CrystA structure of benzene [201.
l c I
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
257
t
100
4
180
6
260
T,K
8
FIG. 11.13. (a) Temperature dependence of the N M R line bandwidth of benzene crystal [221 and 6) temperature dependence of spin-lattice relaxation time Tu >in benzene crystal [271.
Beginning with the temperature llO"K, i.e. long before the phase transition to the liquid state, one has observed the drastic narrowing of the resonant NMR line (Fig. II.l3a), which was explained by the activation of the molecular reorientations around their C6 axis [22]. The value of the second moment is decreased by jump from 9.73 Gs2 (90°K) to 1.6 Gs2 (120°K). The values of the reorientational barrier of benzene molecules around their C6 axis measured by the NMR method in Ref. 122-241 are equal to 3.7, 3.5 and 4.2 kcdmol, respectively.
ZHIZHIN AND MUKHTAROV
258
~
U,lrcavmol
3-
2-
1 4
140
220
180
TX
U kcaVmol -6 .
b
/
-10
1
30
1
L
150
I
I
270
I
I
0"
FIG.II.14. (a) Dependence of benzene crystal lattice energy on the molecule rotation angle around the W-axis at 138°K and 6)dependence of the value of the benzene molecule reorientation barrier with respect to the W-axis on the temperature (0- our data; A - the data of 1241).
The calculated barriers which are close to the experimental values (2.7,3.2 and 4.9 k d m o l ) were obtained in Ref. [1,24,251 using the AAP method and the crystalline structure measured at 140°K [201. The barriers were determined by the change in the lattice energy during the rotation of the chosen molecule around the c6 axis and the fixed positions of the neighboring molecules (the rigid lattice approximation). We have performed the analogous calculation with the AAP/1 parameters (Fig. II.14a). The barrier value obtained is 3.2 kcdmol. The hydrogen atom repulsion, which is very sensitive to changes in the distance between the molecules and consequently to the thermal expansion of the crystal, contributes mostly to the barriers. In particular, the calculation for benzene has shown that the barrier value at 270°K is almost twice as low as that at 140°K (Fig. II.14b).
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
259
The rotational reorientations of benzene in the plane of a molecule are observed (also considerably) in the temperature dependence of the spin-lattice relaxation and dipole times and Tm, respectively) [26,27]. Besides, the sharp increase of Tm at temperatures above 230°K (Fig. II.13b) proved that in the vicinity of the melting point there is the diffusion of molecules in the crystal. The activation energy of this motion is 21.1 - 22.5 kcaVmol[27]. Thus, the benzene crystal is an example of the structure with m e r e n t types of molecular disorder. However, in spite of the large amount of studies concerning the mobility of molecules in benzene crystal and its deutero analog (the detailed review in [12]), the detailed investigation of its s t r u m disorder by the spectra in the external vibrational region was not performed. The spectroscopy studies performed recently [3,28,29] were dedicated mainly to the interpretation of the vibrational spectrum, but not to its temperature evolution. It was interesting to observe the display of benzene crystalline structural disorder in the Raman spectra in the external vibrational regions. For this reason, we have made the measurements of the Raman spectra of polycrystalline benzene in a wide range of temperatures including the vicinity of the phase transition [El. In the low frequency Raman spectrum of benzene, 12 external librational normal modes are active:
in the Fourier infrared absorption spectra of benzene 9 translational normal modes are active:
Generalizing the results of studies [3,15,28-311 on the polarization measurements of the Raman spectra of the benzene single crystal, the experimental data and calculation of eigenvectors and frequencies performed in this work in the rigid molecular approximation with the AAPI1, we have assigned the lines to the symmetry types and performed the interpretation of frequencies according to the form of the vibrations. The results of these investigations are summarized in Table II.3. The complete calculations of the external vibrational spectra of the benzene crystal are given in Appendix III. As is seen from Table II.3, the normal vibrations correspond mainly to the librations of molecules around the axes which are close to their principal axes of inertia.
260
ZHIZHIN AND MUKHTAROV
TABLE II.3 Librational Modes of Crystalline Benzene Experiment (cm-1) Calculation (cm-1) 4.2K 85K 273K [311 1301 [our data1
-_
140K [291
140K
92 79 57
95.8 77.9 46.0
vibrationforms 0.30 L, - 0.36 Ly - 0.34Lw 0.50 & - 0.49 & 0.19 L, - 0.16 & - 0.65 L,
Ag
100 86 64
95 80 60
136 107 69
130 105 65
110
Big
-_
__
128 100 (57)
138.8 94.7 51.0
0.75 L, - 0.23 & 0.20 L, - 0.75 L, 0.95LW
__
__
-_
--
(65)
90 79
101.8 91.5 84.0
0.96& 0.16 & - 0.94 L, 0.8 L, - 0.16 L,
128 84* 61
135.4 86.7 65.3
0.81 & - 0.19 & 0.25 L, - 0.68 L, 0.12 & - 0.55 & - 0.33 L,
Bzg
(100) (86) (136)
B3g
__
(69)
(95) (80)
65 39
--
(130)
(110)
-_
-_
(65)
(39)
*The line was observed at high pressures [321.
In the spectrum there are six pairs of lines arising from librations around one of the inertia axes of a molecule (W, U, V). At 112°K they merge into three pairs of lines; now every line of a pair relates to the librations around the Merent inertia axes of a molecule. At 223°K they merge into three lines at 44,67,108 cm-l. The Raman spectra of the external vibrations of benzene and its deuteroanalog are shown in Fig. II.15 at different temperatures. From the temperature dependence it is seen that all spedral lines are simultaneously broadened, decreased in intensity with increasing temperature and completely smeared into the Rayleigh line wing at the melting temperature of benzene crystal. The line at 57 cm-1with comparatively large intensity relating to the normal vibration of v3 with the predominant participation of molecular librations around the W axis is observed in the spectrum up to the melting point (Fig. II.15). According to the abovementioned analysis, this line transforms to the line at 44 cm-1at 223°K and gives to it the largest contribution.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
26 1
b
0
60
120 v m - l 0
60
120 vcm-l
FIG. 11.15. Temperature dependence of Raman spectra of external vibrations in polycrystalline samples. (a) benzene: 1-112; 2-223; 3-273; 4-277.8; 5-277.9; 6278; 7-279 K (liquid). (b) benzene: 1-112; 2-173; 3-223; 4-273; 5-274; 6-274.9; 7278; 8-278.3; 9-279 K (liquid).
Thus, the Raman spectra is sensitive to the reorientational motion in benzene (which influences the NMR lines) as broadening and decreasing intensities of all spectral lines even at 112°K It is evident that this type of motion of molecules in a crystal of benzene also causes the translational disorder; the absence of benzene plastic phase test* to it. The translational molecule diffusion in a crystal in the vicinity of the melting point is codrmed by the NMR data.
MIZHINAND MUKHTAROV
262 112°K
323°K
w
U
V
C. Conditions of the Display of the Reorientational Motion in Low Frequency Raman Spectra of Crystals The molecule reorientation process is usually described only by the correlation time zc which characterizes the average life time of a molecule in the state of small vibrations [32,33]. The number of molecules having enough energy for overcoming the potential barrier Vop is proportional to exp(-V0#T). For this reason the temperature dependence for T~ can be expressed in the form
In a general case, the spectral distribution of the scattering light intensities is defined by the Fourier-transform of the time autocorrelation function 1341:
where QPO is the element of the polarizability tensor of a crystal, coo is the frequency of the radiation incident on a crystal. If one considers the interruption of the
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
263
molecular vibration by its random reorientations as the relaxation process, one can represent C d d in the following form [MI: Cp&)
- ei(oj + o0he g ~ o
(11.3)
As a result, we obtain the Raman line in the form of the Lorentz contour with a width Aoj =
2n
2,
27t = Kexp(-V,,,./kT)
(11.4)
Thus, the qualitative analysis shows that the Raman line broadening caused by the reorientations of molecules depends exponentially on the orientational barrier and temperature (it is seen in experiments). It also follows that with increasing temperature the Raman lines of the anisotropic phase referred to the librations of molecules around their axis (the reorientational motion is activated relative to it) will be broadened faster than others and decreased in intensity. Then it is necessary that the molecules will be optically anisotropic with respect to the reorientation axis, and the random jumps modulate directly the scattering radiation. In fact, if in a crystal the polarizability tensor has the form:
(11.5) where a1 I and a1 are the polarizabilities along the reorientation axis and normal to it, respectively, then in the stationary coordinate system of a crystal the polarizability tensor can not be dependent on the angle of a molecular rotation around this axis: (11.6) where i($,,,vlqL) is the Euler matrix. The correlation function (11.2) will be defined only by the time dependence of angles of the molecular rotations around axes perpendicular to reorientation axes. For this reason, it was interesting to understand why the reorientational motion, which o m s rather often in the case of the high-symmetrical molecule and does not lead to the fluctuation of polarizability of a crystal, is displayed in the low frequency Raman spectra in some cases in cyclohexane and is not displayed in other cases of benzene.
!ZHIZHIN AND MUKHTAROV
264
TABLE II.4 Relative Intensities of Raman Lattice Modes of Benzene Exp.
Calc.
92
79 57
95.8 77.9 46.0
128 100 (57)
138.8 94.7 51.0
Sym. cm-1
Ag
Big
__ B2g
90 79 128
B3g
84 61
cm-1
101.8 91.5 a4 135.4 86.7 65.3
m u m INTENSITIES, 96 aa bb
cc
ac bc ab
Assignments
18.6
12 59 2 38 1 12 16 1
POlYcryst.
6
100.0 17.9
1
1.5 0.8 3.6
0 2
17.1 4.6 4.7
9 2 2
10 0 1
19.9 0.4 2.4
The calculated results (140°K the AAP/l) of relative intensities of benzene Raman lines in the approximation of the oriented gas (auU= u, C &w, I,, = Iv c 1,) are given in Table II.4 and in Fig. II.16. It was assumed that the polarizability tensor of a molecule in a crystal has the form (11.5), i.e., the polarizability with respect to axes U and V in the molecule plane is equal. This model reproduces very well the real Raman spectra of both the mono- and polycrystals (Fig. II.16). Thus, the rotational motion of benzene molecule around the W-axis(the C6 axis of a 6.ee molecule) gives the zeroth contribution in the Raman spectra intensity. Therefore, the display of random molecular reorientations in the form of anomalous behavior of Raman lines corresponding to librations of molecules around this axis has small probability. The form of the cyclohexane molecule with the D3d symmetry remains practically unchanged in a crystal. For this reason, in calculating the relative Raman line intensities it is natural to suggest that the molecules of cyclohexane in the crystal 11 are optically isotropic with respect to their rotations around the Waxis (it is analogous t o the case of benzene).
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
265
b)
1 I
I
0
60
120
an-1
FIG.II.16. (a) Raman spectra of the benzene single crystal with polarized radiation at 140°K [29] (the calculation with AAP/l, s v = O ; s u = (b) ~ Raman ); spectra of benzene polycrystals: 1 - calculated spectrum (140"K, AAPA); 2 experiment with C6H6 (112°K); 3 - experiment with Cf& (112°K).
However, the Raman spectra calculated by means of this model does not agree even qualitatively with the experiment (Fig.II.17b). In particular, the calculation predicts considerable intensity of one of the highest frequency lines v4 a t 115°K which was not observed in the Raman spectra at this temperature and is displayed only at very low temperatures. Thus,unlike benzene, the oriented gas approximation here is unsatisfactory. In calculating the intensities we have used the effective polarizability tensor of a molecule in a crystal (Part I, Sect. IV.C),which takes into account empirically the influence of the internal local field and the intermolecular interaction. However, unlike the Sect. IV.C,it is necessary to keep in mind that in a general case the main axes of the cyclohexane molecular polarizability tensors do not coincide with its inertial axes U and V whose choice of direction in the plane perpendicular to C3 axis of a molecule is arbitrary, i.e. the polarizability tensors of the cyclohexane molecule must have the nonzeroth diagonal elements in its coordinate system. We have used the tensor of the effective polarizability in the following form for the calculation of the relative intensities.
ZHIZHINAND MUKHTAROV
266
&
C812
2 1
an-1
120
AP
100
80
60
on-1
100
80
60
40
FIG. II.17. Experimental (the spectrum 2 was obtained with the filter for excluding the plasma line 105 cm-1) and calculated Raman spectra (a - qV= 0,9 (qu - awW),b - qv= 0)of cyclohexane-du and - h u at 115°K.
(11.7) The derivatives of the tensors (11.7)with respect to the angles of a molecular rotation around its axes can be expressed by one varied parameter
STRUCTURE AND PHASE "SITION
sym. Ag
B,
OF ORGANIC CRYSTALS
267
TABLE II.5 Experimental and Calculated Librational Modes of Cyclohexane. calc., a - 1 exp., cm-1 relative cal2 C6D 12 CSHn C @ n intensities, % assignments 110 85 69
97 75 61
102.3 85.8 61.0
89.4 75.9 54.3
100 51.5 23.0
Lv
120 92 62
--
124.8 97.7 59.1
109.8 87.0 52.0
2.8 31.0 71.1
Lu
83 54
[
= -2auv 0 0
0 2auv
0
0
0
o] +
+ i)
LU LwLl
Lw
Lv
(11.8)
where k is a varied parameter and
The calculated results for the value of k = 0.9 which gives the best agreement with the experiment are given in Fig. II.17a and in Table II.5. The agreement between the calculated and the experimental data must be considered sufficiently good, because the approximations of this model were rather rough. Consequently, unlike benzene, the cyclohexane molecule in the crystal II form becomes optically anisotropic with respect to its C3 axis, and in the Raman spectra we have seen (see Sect. LA) the direct display of the reorientational molecular motion around the axis. Thus, the reorientational motion of the crystal molecules will be displayed in the low fkequency Raman spectra only in such a case when the molecule is optically isotropic relative to the axis around which the reorientational motion is activated either because of its unsymmetrical structure or because of the deformation of its electron cloud caused by its surroundings.
268
ZHIZHIN AND MUKHTAROV
III. STATIONARY ORIENTATIONAL DISORDER IN TRIHALOGEN SUBSTITUTED METHANE CRYSTALS
The approach of the stage-by-stage "melting" of the molecular degrees of freedom in a crystal is the convenient basis for the interpretation of phenomena associated with the processes of increasing the dynamical disorder in molecular crystals with increasing the temperature. However, there are several examples of forming the high temperature intermediate phases with the elements of the stationary disorder. The transition from the orientationally ordered state into the isotropic liquid OCCUTS in such crystals through the intermediate state whose feature is not the activated, but the stationary ("rigid) orientational disorder. In crystals with the rigid disorder the molecule has several orientations divided by so large a n orientational barrier that the reorientations are practically excluded here. The crystals of bromoform (CHBq), iodoform (CHI3) and chloroform (CHC13) are convenient substances for studying the orientationally disordered molecular solids; in these crystals the atoms of carbon and hydrogen can be statistically distributed with equal probability below and above the halogen atoms without the noticeable change in the lattice energy, because of the considerably large atomic radii of halogen atoms in comparison with the radii of atoms of H and C [l]. From the neutron diffraction and X-ray analysis data the crystalline phase with such structure is observed in a comparatively small range of temperatures in the vicinity of the melting point of bromoform crystal. However, it is necessary to mention that the data of the diffraction measurements may be equally successfully interpreted also by the dynamical orientational disordering which is inherent to plastic crystals (as an example). At lower temperatures the bromoform crystal has the phase transition into the ordered phase [21. In earlier investigations of the iodoform crystal structure [3,41, the authors could not establish the structure because of the difficulties during the X-ray determination of coordinates of H and C atoms in the presence of a large amount of iodine atoms. According to these data the crystal belongs to the polar space group P63, z = 2 ((331, according to which the dipole molecules are equally oriented in the direction of C3 axis. However, the absence of the pyroelectrical properties 151 and the presence of such a morphological feature as the symmetry plane perpendicular to the C3 axis are in explicit contradiction with the suggested ordered structure. The data obtained show also the absence of the phase transitions in iodoform [3-93.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
269
The ordered structure Pnma, z = 4 (C,) [lo] is unambiguously determined for chloroform crystal; it differs noticeably from the structures of bromoform and iodoform crystals. One has not observed the phase transitions [ll-141. Thus,the crystalline phase structure of trihalogen substituted methanes and their degree order are considerably merent. The remons for these difFerences are not established completely. It is possible that the essential reason is due to the halogen type. In the present chapter we discuss the result of our investigation of crystals of trihalogen substituted methane (bromoform, chloroform, iodoform) in a wide range of temperatures including the close vicinity of the melting point [151; this investigation is performed to study the low temperature phase structural features, determining the structure and degree of order of molecules in the high temperature modifications.
k Vibrational Spectra and Structure of Crystals of Bromoform and Iodoform From the different investigation data [2,9,10,16,17] we know that at 270°K the bromoform crystal has the reversible phase transition a fi(a is the high temperature phase, the melting temperature Tmelt = 281°K); and at temperatures above 193°K there is the irreversible transition (y + either from the y-phase which is obtained by fast cooling either the liquid or from the a-phase below the boiling point of the liquid nitrogen into the gphase [2,9]. From the X-ray analysis 1163 and neutron diffraction data 1101 of a-phase its structure can be described only approximately by means of space group P63, z = Z(C3) or P3, z = 2 (C3) corresponding to ordered structures with equally and oppositely oriented molecular dipoles along the C3 axis. In a general case, the structure of a-phase is characterized by the pseudosymmetry P63/m, z = 2 (c3h) and by the average cell which maintains two pairs of superimposed molecules (Fig. III.l). The unit cell parameters of Ref. [lo] (Table III.1) show the large similarity of all three bromoform phase structures; the transitions between them are characterized mainly by the change of the molecular orientational disordering and crystal symmetry. Several hypotheses for forming the disorder in the a-phase are possible. First of all, it is the consequence of the dynamical disordering, i.e. the activation of molecular reorientations around their axis perpendicular to the C3 axis during the time which is smaller than the characteristic times of the experimental methods. The possibility of such disordering is confirmed by the dielectric spectroscopic data El81 indicating the noticeable thermal mobility of molecular dipoles in the a-phase of bromoform. In this case the formation of the -p and B-phases can be considered as
ZHIZHIN AND MUKEITAROV
270
6 FIG. III.l. Crystal structure of a-phase of bromoform 1163.
TABLE III.l Unit Cell Parameters of Bromoform Unit Cell Parameters c b a,deg B,deg
Phase
ab
bA
1. a,273"K P63, Pi,P6gm
6.333
6.333
7.226
90.0
90.0
120.0
2. fl,220"K pi,Z=2
6.287
6.276
7.448
81.3
81.3
119.7
3. 7, 14°K P3,z=2
6.253
6.253
6.886
90.0
90.0
120.0
7,deg
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
271
the ordering and "frozen" of dipoles. In all three phases the activation of molecular reorientation around their symmetrical axis is also possible; this activation does not lead to the orientational disorder [l]. Secondly, the disorder in the a-phase can be rigid (stationary); and it is formed when the crystal is growing fi-om the melt as is shown in Ref. [17]. In this case the a-phase can consist of the discrete order "islands" (domains) (P63and P3) which are shown by means of the conditional one-dimensional model:
In a given case the transition a + y can be regarded as the transition between the many domain and the one domain structure P3 the more stable analogous structure of P i (the transition a + p). The vibrational spectra are very sensitive to the reorientational motion of molecules and, in consequence of the short range feature of intermolecular forces, to the change of short order in the molecular position. Besides, fi-om the data of the external vibrational spectra and their interpretation by means of the AAP method one can verify the determination (in Ref. [lo]) of the structures of three bromoform phases obtained fi-om the neutron diffraction experiments on the polycrystalline samples, since this method does not guarantee, in general, the unambiguous determination of the structure. In earlier studies [2,8,9,14,19-231the spectral investigations of bromoform were performed mainly for the determination of the unknown factor groups of three phases; for this reason, in our work the temperature dependence of Raman spectra was studied in detail. In the case of an isolated molecule (the Csv symmetry) six intramolecular vibrations are active in the vibrational spectra [61 v(A1) = 3020 cm-l
v(A1) = 539 cm-1
v(A1) = 222 cm-1
v@) = 1145 cm-1
v(E) = 655 cm-1
v@) = 152 cm-1
ZHIZHIN AND MSJKHTAROV
272
In the ordered crystalline y and &phases in accOrdance with the correlation tables: y,
sp.gr. P i
one can expect the splitting of the doubly degenerate intramolecular vibrations only in the @phase because of the removal of the degeneracy by decreasing the symmetry of a molecule due to the crystalline field (the site splitting, or the beta splitting). This splitting was observed in our Raman spectra of pbromoform (for the line v5 (E)),Fig. III.2. By decreasing the temperature one observed the gradual broadening and merging of the doublet; after the transition into the a-phase one observes only one line which has asymmetrical contour indicating the presence of the beta splitting also in this phase. Since the splitting of the intramolecular frequencies is impossible for suggested space groups P63 and P s and for the pseudosymmetry of the phase P63/m, the asymmetrical form of the observed line v5 is the direct indicator of the stationary (rigid) disordering of this phase structure. One can also suggest that the v5 line asymmetry is associated with domains of two types in the a-phase. The experimental and calculated Raman spectra of the external vibrations of y and &bromoform are shown in Fig. III.3 (the calculation by the AAP method). The calculations were performed with the different known sets of the AAP (6-exp) parameters; the M I 2 2 were chosen; they were proposed in Ref. 1241 for the calculation of bromosubstituted methane crystal structures. The authors have also chosen the M i 2 3 and the M I 2 4 obtained in Ref. [251 by fitting the external vibrational frequencies of bromosubstituted benzene crystals (Appendix 1). Using the data about the unit cell parameters (Table III.l) and minimizing the crystal lattice energy with respect to the Euler angles of the molecular orientation and their mass center coordinates we have also calculated the equilibrium structures of the ordered 1” and &phases and the suggested structures P63 and P? for the a-
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
a
b
I
660
273
650
550
I
I
570
FIG. III.2. Splitting of lines v5 (Elin the Raman spectra of bromoform (a) and iodoform (b) crystrals in dependence of temperature. (a) 1-112.0;2-203.0; 3250.0;4-260.0(@phase);5-275.0K (a-phase);(b) 1-85.0;2-300.0K
phase. It seems that the calculated structures differ comparatively little &om the structures determined by the neutron diffraction performed on the powder samples in Ref. [lo].The mass center coordinates differ by the shift of 0.09 - 0.12 A,and the molecular orientations differ by the rotation angle 2.5". The calculated frequencies and eigenvectors of the y- and &phases are given in Appendix III. The assignments of the Raman spectrum lines of these phases were made in terms of both the frequencies and the relative intensities calculated in the oriented gas approximation. In the case of fkbromoform the intensities were calculated by the fitting of the relative optical anisotropy of a molecule. In the y-phase the polarizability tensor form of a molecule is fixed by the positional C3 symmetry (auU = % # c+J. Here the calculation predicts the zeroth intensity of the line at 52 cm-1 which is assigned in terms of the eigenvectors to the translational-librational vibration with respect to the W-axis(the C3 axis of a molecule). At the same time the maximal intensity of this line was experimentally observed. It proves the restricted applicability of the oriented gas model and the necessity of considering the change of the bromo atom polarizability at the translational vibrations of a
ZHIZHIN AND MLTKHTAROV
274
25
0
26
60
60
76
100
75
100 cm-1
FIG.III.3. Experimental and calculated Raman spectra of polycrystals of and y-phases of bromoform.
molecule. Considering this we have obtained very good agreement with the experiment, i.e. the mean square root deviations of the calculated frequencies kom the experimental ones are 6.0, 5.7 and 9.7 cm-1 for the AAPl23, M I 2 4 and AAP122, respectively. Thus, the data obtained from the Raman external vibrational spectra confirm the reality of the y and &phases determined in Ref. [lo]. The differences between the spectra of these two phases are defined mainly by the
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
I
0
L
I
I
25
50
76
275
FIG. III.4. Raman spectra of intermolecular vibrations of the three crystalline phases of bromoform. 1-85K (y-phase); 2-85.0;3-203.0;4-260.0 K @phase); 5-275.0K (a-phase).
change of the crystal symmetry and degeneracy of molecule vibrations with respect to the axes U and V which are perpendicular to the C3-axis (the lines 30,33,38,47 cm-1- p-phase, 85"K,and the lines 38,43 cm-1- y-phase, 14°K). The comparatively broad line at 65 cm-1 having small intensity refers predominantly to the translational molecular vibration along the W axis in both phases, and the line at 52 cm-1refers to the molecular librations around this axis. T h e temperature dependence of the Raman spectra of the &phase in the vicinity of the point of the phase transition is shown in Fig. 111.4. One has not
ZHIZHIN AND MUKHTAROV
276
TABLE III.2 Reorientation Molecular Barriers of Bromoform Barriers of reorientations
U 1 ,
(kcaVmo1)
1. 2. 3. 4.
a-phase, 2736 P63 a-phase, 273K P bphase, 2206 P 7-phase, 146 P
19.8 20.0 20.3 21.5
U 28.6 30.6 53.4 43.9
V
33.9 37.8 100.0 49.0
W
30.7 31.3 30.7 41.4
observed some noticeable anomalous dependences of lines on the temperature which testify to the stage-by-stage melting of molecular degrees of freedom and are inherent to the phase transitions into the dynamically disordered state. The absence of a rather considerable orientational mobility of molecules in the y- and phases is confirmed also by large values of reorientation molecular barriers (Table III.21, calculated with the AAP/24in the rigid lattice approximation. The calculated results for the hypothetical structures P63,Pi of the a-phase are given also in Table III.2. If the barrier of the reorientation around the C3 axis is the smallest one in the case of the y- and pphases, then in the case of the a-phase the barrier of the molecular reorientation by 180" around the U axis perpendicular to the C3 axis is the minimal one (the directions of the axes are given in Appendix
III). However, the value of this barrier remains comparatively large, and the reorientations of molecules with the change of their dipole directions on the opposite ones are discovered only by such a "slow" method as the dielectric spectroscopy [18]. It should be mentioned that there are practically the same values of the lattice energies of all three phases (Table III.2);it is the consequence of the very smaU differences between their structures. The Raman spectra of the polycrystal are shown in Fig. III.4. The Raman spectra of the single crystal in the polarized radiation (the z axis is parallel to the C3 axis of the crystal) are shown in Fig. III.5. One has observed four lines at 22 0 , 3 6 0 , 4 3 CyZ)and 60 cm-1(in brackets we have indicated the conditions of the polarization under which the Raman spectra are observed). There is a large similarity between the Raman spectra of the a-phase of bromoform and the iodoform crystal (Fig. III.5). The structure of the iodoform crystal is defined by means of the X-ray analysis in earlier studies [3-51in which the authors have proposed the space group P63, Z = 2 (C3). In these papers the disordering of molecules can not be discovered, since only the coordinates of iodine atoms were determined with insufficient accuracy. However, the Raman spectral data indicate
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
I
20
I
40
I
277
I
60
v,an-l
FIG. III.5. Raman spectra of poly- (1) and monocrystals of a-phase of bromoform (a), T-275°K and of iodoform (b), T=293"K [2] in polarized radiation.
convincingly the disordering of the iodoform structure which is very similar to the structure of the a-phase of bromoform. The form of the intramolecular vibration line vg(E) indicate t h i s fact also (Fig. 111.2); the splitting of t h i s line must not be observed for the ordered structure P63,z = 2 (C3). Thus, the spectral data indicate the rigid stationary character of the orientational disorder in the crystals of a-bromoform and iodoform. The interpretation of the Raman spectrum of the a-bromoform crystal by means of the
ZHIZHINANDMUKHTAROV
278
hypothetical structure P63 and Ps and the lattice dynamics calculation by the AAP method is shown in Table III.3. It is seen that the effort to reach the agreement between the experimental and calculated external vibration fiequencies leads to the contradiction between the observed and predicted polarized Raman lines and vice versa. Besides, in the Raman spectra of the isomorphic iodoform crystal one observes contidently at least five lines at 38,32,25 ([Sl), 60,15 cm-1 (Figs. III.4 and III.5); this number exceeds the number of vibrations which are active in the Raman spectra for groups P63 and P i with two molecules in the unit cell. Thus,the direct use of the grouptheoretical methods is complicated for the interpretation of the disordered bromoform and iodoform crystals. The analysis shows that the spectral data concerning the a-bromoform and iodoform can be satisfactorily interpreted by means of some hypothetical structure with the pseudosymmetry P6dm obtained by the diffraction methods, not with z = 2 but with z = 4 (C3) (Table III.4). Probably, two possible dipole orientations cause the doubling of the unit cell sizes. If the number of these orientations was more than two, the effective sizes of the cells will be, probably, enlarged a t appropriate times. In a general case, in calculating the frequencies and interpretating the abromoform disordered spectrum the vibrational modes should be considered as almost localized and proportional to the product of the spatial damping factor exp (r/A) and the plane wave factor exp (ik)."he value of wave vector i; is no longer a good quantum number, and in principle, there are no selection rules in wave number & = 0. Shuker and Gamon suggested [26] that the space-time correlation is proportional to exp(&,?)exp(-r / A) for the j* mode. The Fourier-transform of this expression is proportional to the intensity of the Raman j* mode. As a result, they have obtained that the nonresonance Stokes component with the frequency shift o is described by the following expression
(III.1) where n(a,T) = [exp(holkT)-l]-l is the function of the Bose-Einstein distribution; c:")(W) is the bond constant for every vibrational mode ((T is the set of indices describing the polarization of the incident and scattered radiation); g$o) is the phonon state density of the j* mode. The main result of this work is that the Raman spectrum of the disordered solid must reflect the main features of phonon state densities.
STRUCTUREAND PHASE TRANSJTION OF ORGANIC CRYSTALS
279
TABLE III.3 Assignment of Lattice Modes of a-Bromoform experiment (cm-1) polarization symmetry
__ cyz) 0 0
-cyz) 0 0
calculation (cm-1) polarization MI24 symmetry V Space group P63 A, 0 48 El, cyz) 27 E2, cyy) 39 E2, (YY) 12 Sub group Pi 49 $., (yY) 43 Ag? 0 31 Eg, 0, cyz) Eg, 0, (yz) 22 &, (IR) 47 E,, (IR) 28
V
(60) 43 36 22 (60) 43 36 22
-_ _-
MI23
MI22
V
V
47 28 38 10
51 29 49 11
48 42 30 21 45 27
61 54 36 25 55 35
TABLE m.4 Spectral Lattice Data for a-Bromoform and Iodoform Assignment bromoform iodoform P63,2=2 P63/m, Z=4
Raman Spectra
V,Cml
v,cm-l
(60)
60
--
%>0
43 (rn)
38 CYZ)
El
El,,
36 cyy)
32 cw)
A
$9
E2
E2g9
25 0
E2
E2g, 0
15
_-
El,, CYZ)
24
El
El,
36
A
AU
_-
22 0
-FIR
absorption spectra [71
__
[6,71
this work
cyz)
280
Z H m I N AND MUKHTAROV
In review [27] there are many examples of the successful application of the expression (Eq. m.1) for the interpretation of the Raman spectra of amorphous semiconductors in which the short-range order in atomic arrangements differs considerably fkom the structure of the appropriate ordered crystal. Here it was assumed that the dispersion of the constant bonds cy'(a) is either absent or is a slowly changing function of the frequency w, and the experimental spectrum was compared directly with the common density of states = gj(O), calculated on the assumption of the ordered semiconductor structures. It is interesting to verify the possibilities of such a n approach for the interpretation of the Raman spectra of disordered molecular crystals, in particular, of a-bromoform and iodoform. The calculation of the phonon state density of the aphase of bromoform was performed by the use of the M I 2 4 and hypothetical ordered structures P63 and P3. In a given case it was justified, because (as was mentioned before) the change of the molecular orientation along the C3 axis for the opposite one does not essentially influence the interaction forces between the molecules. Densities of states of two ordered structures P63 and Ps which correspond probably to two types of short-range order in a-bromoform describe correctly the features of the Raman spectra of the real disordered crystal. The calculation was performed according to the method in Ref. [281 by scanning of the wave vector in the symmetrically independent part of the Brillouin zone with the step, 0.1 Wa, where a are the parameters of the unit cell. The calculated densities of states are shown in Fig. ITI.6. The total density of states describing approximately the vibrational a-bromoform spectrum which maintains the equal number of "islands" with short-range order of the P63 and P3 types is also given in Fig. III.6.The reduced total density of states is given in Fig. 111-7 J
The "broadened densities of states gl(w) and gZ(o) are also given there. They were obtained by the convolution with the Lorentz function
where L(w,r) is the Lorentz function, r is the halfwidth of this function which is equal to 3 cm-1. From Fig. III.3 it follows that the calculated spectra are shiRed with respect to the experimental Raman spectra to the low-frequency side by about 15 cm-1. It can be explained by several reasons:
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
28 1
Raman spectrum, polycrystalline sample
I
10
20
30
40
50
60
7c
10
20
30
40
50
60
7(
10
20
30
40
50
60
70
10
20
30
40
50
60
70
FIG.111.6. Calculated densities of phonon the AAP method).
cm-1
states of a-bromoform crystal (by
282
ZHIZHIN AND MUKHTAROV
Raman spectrum,polycrystal,d-phase
FIG.III.7. Total reduced density of states - 1; broadened reduced densities of
- 2, 3, 4: (P63+P8,8 and P63, respectively) of a-phase of bromoform crystal (calculated by the M I 2 4 method). states
(a) the short-range order in molecular arrangements in the real crystal can differ a little from the ideal structures P63 and P s used in the calculation; (b> the
anharmonic corrections which can give the noticeable contribution to the values of frequencies were not considered;
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
283
(c) one does not consider the electrostatical interaction which leads usually to the increase of frequencies, as was mentioned before. However, these factors essentially cannot influence the general form of the calculated spectrum which is in good qualitative agreement with the Raman spectra and consists of one broad line at 21 cm-1 and has the shoulders at 6,14,31,46 un-1 (in the Raman spectra - one broad line at 38 cm-1 and shoulders at 15 (iodoform), 22,43,60 cm-1, respectively). This is seen in Fig. III.7. The calculated results show the noticeable difference of the relative intensity of the shoulder at 31 cm-1between the spectra of two structures, P63 (Fig. III.7 (4)) and Pj (Fig. III.7 (3)); it can be used for studying the short-range order and its change with temperature. In particular, it permits one, in principle, to study the change in the relationship between the number of "islands" with molecules of the same direction along the C3 axis and the opposite one. Since the a-phase of bromoform exists in the comparatively narrow range of temperatures from 270 281"K, it is more convenient to perform such investigations at the isomorphic iodoform crystal in which there are no phase transitions up to the boiling temperature of the liquid nitrogen, as is shown in our investigations. As is seen in Fig. III.8, the relative intensity of the Raman spectral shoulder at 32 cm-1 (iodoform)is considerably decreased with decreasing temperature. According to the calculated data for a-bromoform, it can be explained by gradual ordering of the iodoform structure at the expense of the increase of the domain part with molecules having the same orientations along the C3 axis at low temperatures. Thus, the obtained data show that the method of the interpretation of disordered solids proposed in Refs. [26,271 is applicable not only to the amorphous semiconductors, but also to the molecular crystals with the rigid orientational disordering to which the a-bromoform and iodoform relate, as was shown by the spedral and calculated data. The application of this method allows us to extract the valuable information concerning the change of the short-range order in these crystals from the temperature dependence of the external vibrational Raman spectra. In particular, it was established that with decreasing the temperature from 300 to 85°K the domain part with the same orientation of molecular dipoles is gradually increased in the iodoform crystal. On the other hand, in the a-phase of the bromoform crystal such change in structure leads to the phase transition of phase (pi) whose structure is characterized by almost oppositely directed dipoles of molecules.
ZHIZHIN AND MUKHTAROV
284
I a)
0.3
'2
t
I,<
,
1
,'",
280
1
0
>
25
60
76
v, cm-1
FIG. III.8. (a) Temperature dependence of Raman iodoform polycrystal spectra; (b) Temperature dependence of the relative intensity of Raman spectra at 32 cm-1.
B. Disordering of the Chloroform Crystal Structure in the Vicinity of the Phase Transition to the Liquid State In a series of iodoform-bromoform crystals one observes the dependence of the temperature region of the disordered phase on the halogen type. This region for iodoform is hundreds of degrees and is spread, at least, from 77°K to the melting
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
285
n
I
A
v
I'
3.F
I
I
I
FIG. III.9. Projections (cb) and (ab) of chloroform crystal structure [ll]; U,V,W are the inertia axes of a molecule.
point of a crystal (392°K). It is considerably smaller for bromoform and is less than 10" (Tmelt = 281°K). If such law exists, one can suggest the forming of such a disorder phase in the narrower temperature interval than in bromoform in the vicinity of the melting point (210"K),if one transfers to chloroform. The chloroform crystal structure is determined at 180"K.by the X-ray analysis [ll]. The structure is ordered, the space group Pnma, there are four molecules in the unit cell. As is seen fiom Fig. III.9, the C-H bonds of all molecules of chloroform are oriented along the direction which coincides practically with the
ZHIZHIN AND MUIMTAROV
286
\
25
50
75
100
v, cm-1
FIG. lII.10. Temperature dependence of external vibration Raman spectra of chloroform crystal: 1-112.0; 2-180.0; 3-209.0; 4-209.5; 5-209.8; 6-210.O"K (melt).
crystallographic axis a. Consequently, the disordering like observed in the a-phase of bromoform and iodoform (two opposite directions of the C-Hbonds and two possible dipole orientations, respectively)must lead to the one-dimensional disorder along the a axis in the case of chloroform. This anisotropic disordering can be displayed in the temperature dependence of the chloroform crystal vibrational spectra, as was abovementioned. In the Raman spectra of external vibrations of chloroform crystal there are twelve vibrations (3% + 3B1, + 3B2, + 3B3,), and in the FIR absorption spectra six vibrations (2B1, + 2B2, + 2B3,) are active according to the data of the grouptheoretical analysis. In Fig. III.10 there are the external vibrational Raman spectra
STRUC"URJ3AND PHASE TRANSITION OF ORGANIC CRYSTALS
287
of polycrystalline chloroform obtained in a wide range of temperatures. Eight lines were observed in a polycrystal spectra at 112°K Their existence is confirmed also in Ref. [13], in which two more lines with very s m a l l intensity were found in the Raman spectra of a sample at 20°K We have performed the calculations of frequencies, eigenvectors and line intensities by the AAP method in the "rigid molecular approximation for assigning the experimentally observed fiequencies. This approximation is justified here, since the lowest fkequency of the intramolecular vibrations of chloroform is 259 cm-1 [29]; hence, it is two and one-half times as large as the highest frequency of the external vibrations. We have used the combined sets of parameters (AAP13+ AAPI19) and ( M I 3 + AAP/20) in which the parameters of Cl-*CI interactions were obtained by fitting the structure and external vibration frequencies of the hexachlorobenzene crystal [30]. The consideration of the electrostatical interaction does not influence the values of the calculated frequencies (see Part 1, Section 11.2). The calculation of the line intensities was performed by the method described in Part I, Section IV.3. The normal modes in chloroform crystal are the mixed librational-translational vibrations, since the molecules are arranged in the position C, (Fig. III.9). Since the contribution of the translational vibrations to the Raman line intensities was not considered in the oriented gas model, chloroform can be the example for verifying the applicability of this model to the crystals of this kind. The chloroform molecule has the symmetry C3", i.e. a,f g, = G,, (the axis of inertia is shown in Fig. III.9). For this reason, instead of y it is more convenient to use the parameter y' in the form:
where y' = 0 corresponds to the isolated molecule. The calculation of the relative intensities of the polycrystal Raman lines with this value of y' (i.e. on the assumption that the polarizability of a molecule of chloroform remains the same at the transition to the crystalline state) has led to considerable discrepancy with the experiment (Fig. 111.11). For this reason, the parameter f was varied at the calculation. The angle Q between the main axes of the polarizability tensor and inertia moment in the UW plane of a molecule was simultaneously changed (analogous to the thiophene case). The orientation of the third main axis of these tensors was fixed by the site symmetry (the axis V of a molecule) (Fig. KlI.9). The satisfactory agreement with the experimental Raman spectra of a polycrystal was obtained at values y' = -3 and Q = 45" (Fig. III.ll); it allows us to assign the
ZHIZHIN AND MUKHTAROV
288
\
nr
I
I
m
7c
100
FIG. III.ll. Experimental and calculated (at 180°K) Raman spectra of chloroform polycrystal.
observed lines to the symmetry types unambiguously (Table ID,Appendix III). The predicted, very intensive line at 21.1 an-1(Bsg) was not observed in the polycrystal Raman spectra at temperatures above lOO"K, because it is close to the Rayleigh line; it is observed only at very low temperatures (20°K)1131. The analysis of the temperature dependence of the Raman spectra shows that up to the melting point it has no anomalous changes: the lines gradually broadened, their intensities are decreased (Fig. III.10, spectrum 2 - 209°K). However, in the narrow premelting region (Fig. III.10, spectrum 3 - 209.5"K; spectrum 4 - 209.8"K) the lines v3 (38 cm-1) and v2 (48 cm-1) have predominant broadening in comparison with the other lines of the Raman spectra; their intensities are decreased, and they are smeared into the structureless Rayleigh line wing.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
289
Table III.5 Lattice Modes of Chloroform Calculation (cm-1) Experiment (cm-1) 110°K
180°K
185°K
80 48 38
72 43 34
75 42 29
-_ --
v6
87 76 55
--
79 71 41
v7
87
_-
93
V1
%
v2
v3 Blg
B2g
B3g
v.4 v5
v8
--
V9
63
v10 v11
96 69 *22.5
VB
--
__
88 62
--
Vibration form, % 30 T, 87 T, 47 T,
+ 67 L,
+ 28 T, + 25 & 42 Tb + 27 L, + 31 L , 55 Tb + 36 L, 37 L, + 60 L, 73 T, + 26 L,
62
95 T, 23 T, + 73 L,
93 63 21
93 L, 27 Tb + 73 L , 67 Tb + 26 L ,
64
T is the translation with respect to axes a, b and c* L is the librations around the inertia axes U, V, W of a molecule * is 20°K Ref. [131. From the calculation of the eigenvectors it follows that the lines v2(A& and v3(Ag) refer to the vibrations with the predominant participation of translations along the crystallographic axes a (Fig. III.9, Table III.5) in whose direction the disordering occurs (as we have suggested). The dependence of the lattice energy calculated at 185°K on the rotation angle of a chosen molecule around its main inertial axes at fixed positions of the rest of the molecules is shown in Fig. III.12. The calculated values of the reorientation barrier are as follows: U, = 25 - 43; U, = 20 - 32, U, > 100 k d m o l (these values were calculated kom these curves by means of varied sets of the AAP parameters). The relatively small values of the barrier W (the calculation in this approximation gives the upper limit of reorientational barrier values 1311) show that the molecules of a crystal irregularly oriented in the direction of the a axis can occur (by means of their rotation around the W axis by 180"). Thus, the experimental and calculated data obtained in this work show that in a series of crystals of trihalogen substituted methane there is the dependence of their structure and properties on the halogen type and temperature. We have
290
ZHIZHIN AND MUKHTAROV
U,W m o l t34
W
FIG. III.12. Dependence of the lattice energy of chloroform crystal (at T=185"K)on the rotation angle of a molecule around its main inertia axes.
isomorphic phases in a series of chloroform, bromoform, iodoform. In chloroform this phase exists within the limits of 1°K and is broken by the melting processes. The character of crystal structure disordering is very similar and is caused by two possible dipole orientations. If one considers the chain of the phase transitions in these crystals, one can suggest that the ordered phase of iodoform exists at the temperatures below 77°K The investigations performed show that the use of vibrational spectra and the AAP is effective for studying the stationary orientational disorder in organic crystals.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
291
IV. VIBRATIONAL SPECTRA AND PHASE TRANSITION IN CRYSTALS WITH CONFORMATIONALLY UNSTABLE MOLECULES
The melting of the substances and the transformations in the solid phase are often accompanied by the change in the molecular conformation which in a crystal state under tension are due to the influence of intermolecular forces. The behavior of the conformationally unstable systems is most interesting in the critical temperature region in the vicinity of the phase transition point where the most essential distortion of the conformation equilibrium takes place because of the change of the balance between intra- and intermolecular forces. The best representatives of such systems are the polyphenyl crystals and the crystals of organic substances which contain the methyl groups.
A. Distortion of the Plane Conformation of Biphenyl Crystal Molecules in the Vicinity of the Melting Point The simplest of polyphenyls is biphenyl; its molecules in the gas and liquid are nonplanar; they have the symmetry D2;the planes of the phenyl groups are rotated around the C-C bond by angles 42" and 32", respectively [l-51. In the high temperature crystalline phase (40 - 343.3"K)the molecule is planar, and it has its symmetry center (according to the X-ray measurements), and consequently, is under tension. At T = 40°K the balance between intra- and intermolecular interaction is distorted, and the crystal undergoes the phase transition of second order, accompanied by the rotation of phenyl groups by 10" and displayed in the form of the soft mode corresponding to the intramolecular torsional vibration of the phenyl group around the C-C bond 181. The low-temperature polymorphism of biphenyl crystal and other polyphenyl substances has been thoroughly studied [9-131. It was interesting to investigate the behavior of such systems in the region of high temperatures where the distortion of the balance between the intra- and intermolecular interactions occurs, and consequently, the change in the molecular conformation (the rotation of phenyl groups around the C-C bond) takes place as a result of the thermal expansion and the weakening of intermolecular forces. These changes may be observed and successfully studied by temperature changes in Raman spectra [14-161.
292
ZHIZHIN AND MUKHTAROV
FIG. IV-1. Dependence of the intensity of the line 409 cm-1 of biphenyl crystal Raman spectrum on the temperature in the premelthg region: 1-293.0;2340.0; 3-342.5;4-342.6;5-342.7;6-342.8;7-342.9;8-343.0;9-343.1;10-343.2;11343.3OK (melt).
The exclusion rule was observed for the high-temperature phase, if one compares the biphenyl Raman and infrared absorption spectra [1,3];it is additional proof of the existence of the molecular symmetry center. The transition of crystalline biphenyl into the melt accompanied by the distortion of the plane molecular conformation due to the rotation of these phenyl groups around the C-C bond leads to the loss of the inversion center of the last one and to the release of the
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
293
1612 I
7
6
5
FIG.IV.2. Temperatue behavior of the intensity of the line 1596 cm-1of the biphenyl crystal Raman spectnun in the premelting region: 1-293.0; 2-340.0; 3342.6; 4-342.8;5-343.0; 6-343.2;7-343.3"K(melt).
exclusion rule. In the Raman spectrum of the melt the new lines appear: 135,177, ='348°K); these lines are prohibited for the plane 409, 849, 870, 1250 cm-1 ('I conformation of a molecule [171. These lines are sufficiently intensive in the spectrum of liquid and are absent in the spectrum of the high-temperature phase of a crystal. The temperature behavior of intensities of these lines must serve as the convenient indicator for predicting the distortion of the biphenyl molecular plane geometry in the vicinity of the melting point of a crystal.
ZHIZHIN AND MUKHTAROV
294
342.5
343.0
FIG. IV.3. Dependence of torsional angle 8 between the phenyl groups in biphenyl crystal at the temperature at the crystal-liquid phase transition.
We have studied in detail the temperature dependence of the integral intensity of an isolated line a t 409 cm-1which corresponds to the vibration of the & type for the plane conformation of the biphenyl molecule and is nonactive in the Raman spectrum of the high-temperature phase. From Fig. IV.l it is seen that the intensity of this line in a crystal is equal to zero at room temperature, but it increases essentially in the vicinity of the melting point. Thus, even in the crystalline phase of biphenyl there is the distortion of the plane conformation of a molecule in the premelting point. This conclusion confirms also the decrease of the intensity of the line at 1596 cm-1 corresponding to the benzene ring vibration and sensitive to the change of conjugation (Fig. IV.2). The data obtained permits one to estimate the dependence of the angle between the phenyl groups on the temperature in the vicinity of the melting point. As is shown in Refs. [4,51, the intensity of the Raman lines corresponding to benzene ring vibrations in the conjugated systems of biphenyl type can be described by the expression I I, cos48, where I,, is the intensity of the line in a spectrum of a plane molecule; 8 is the angle between the planes of phenyl rings. The line of plane deformational vibrations of benzene ring at 1000 cm-1 was used for the
-
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
295
b(ab)cBg
Maah Ag
0
~(42)
loo
v, cm-1
FIG. N . 4 . Dependence of the external vibration Raman spectra of biphenyl w:1-337.5; 2-341.2; 3-342.6; 4-342.9; 5-343.0; 6-343.1; 7-343.2; 8-343.3"K(melt).
crystal on temperature in the premelting region.
calculation of intensities (as an internal standard). The change of conjugation does not influence the line at 1000 cm-1 (AIg, y (CCC)), and for this reason, does not change a t the gas-liquid transition [4,5]. The temperature dependence of the angle 8 is given in Fig. IV.3. It is seen that in the investigated range of temperatures the angle between the benzene ring planes is increased practically linearly with increasing the temperature, and its drastic increase occurs only in the closest vicinity of the melting point (in the interval 3 0.15"K). This jump corresponds to the phase transition of a crystal into
ZHIZHIN AND MUKHTAROV
296
v, cm-1
I
90
I 0
'
-'Ag
I
80
50
40
I
I ,
I I
337
I
339
I
341
343
m
.lmelt T,K
FIG. N.5. Dependence of displacements of some frequencies of external vibrational Raman spectrum of biphenyl at the temperature in the vicinity of the melting point.
the liquid state. Our value of the torsional angle (n 31") for a liquid biphenyl molecule is in good agreement with the values of other studies. The abovementioned changes in intramolecular vibrational spectra in the vicinity of the melting region are accompanied by noticeable changes in the external vibrational spectra of biphenyl (Fig. N.4). The lines at 53 cm-1 ($) and 42 cm-l (Bg)(T = 293°K) are broadened, and their intensities are essentially decreased, when the temperature of a sample approaches the melting point. With the further increase of the temperature the line at 53 cm-1 "disappears" practically and is smeared into the Rayleigh line wing. The temperature behavior of frequencies does not demonstrate any anomalies (Fig. IV.5). In spite of the numerous experimental investigations of the external vibrational Raman spectrum of biphenyl crystal [11,17,18-201, the complete and unambiguous interpretation of the phonon spectrum is absent. According to
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
297
/
FIG. IV.6. The unit cell of biphenyl crystal [6].
group theoretical analysis, six librational vibrations (3% + 3B,) must be active in the Raman low-frequency spectrum of biphenyl crystal, and five mixed translational-torsional vibrations (3A, + 2B,) must be active in the infrared absorption low-frequency spectrum of biphenyl (the space group PBllc, z = 2 (Ci)) (Fig. IV.6)[6,71. The authors of [213 have observed only three considerably depolarized lines at 4 2 , 5 7 , 8 8 cm-1at room temperature in the polarized radiation (three lines in the Raman spectra of a single crystal); and at T = 77°K there were only five lines at 43, 65, 71, 107 and 108 cm-I in the Raman spectra of a polycrystal. The authors of this study have suggested that the line at 43 cm-I is double and is not resolved even at low temperatures. The missing line of the low-frequency Raman spectra of biphenyl was not discovered in all subsequent experiments of other authors. To ascertain the interpretation of low-frequency Raman spectra we have performed the calculations .of frequencies and eigenvectors of the external vibrations, line intensities. We have also studied the spectra of oriented monocrystalline biphenyl samples measured in the polarized radiation at low temperatures with the use of a cryostat which permits one to adjust the sample during the procedure of temperature meausrements (see Part 1, N.B). The calculations of frequencies and eigenvectors were performed in the rigid molecular approximation with the M I 1 parameters, the radius of sllmming 6A,
ZHIZHIN AND MUKHTAROV
298
TABLE IV.l Raman Data for Biphenyl Relative Intensities*, %
calc.
exp. cm-l
an-1
aa
bb
c*c*
ac*
Ag
88 53 42
83.5 63.5 50.2
22 4 26
28 4 17
0 15 7
1 35 16
B,
88 54 42
84.5 68.3 47.4
ab
bc*
polycrystal 51.9 91.5 81.2
3 3 5 8 6 4 4
12.8 24.9 100.0
*With respect to the mostintensive Raman spectrum line of a polycrystal.
CH = 1.05 A and with the use of the structures determined by the X-ray analysis at 110°K and 293°K [6,7,22,23]. The use of the rigid molecular approximation is justified in the case of biphenyl because the centrosymmetric librational vibrations which are active in the Raman spectra do not interact with the low-frequency noncentrosymmetric torsional vibration at 70 cm-1 of biphenyl molecule (the appropriate nondiagonal elements of the dynamical matrix are equal to zero). It is continued also by the isotopic relationships calculated in accordance with the eigenvedor. With the calculation of line intensities, the applicability of the oriented gas model to biphenyl crystal is not obvious, since the biphenyl molecular conformation is noticeably changed at the transition from the free state to the crystalline one, as was mentioned above. From Table IV.l (see Part 1, I V . 0 it follows that the polarizability of an isolated molecule and the effective polarizability in a crystal are Werent. The intensities of the Raman spectral lines calculated by means of data listed in this table describe the experimental spectrum poorly (Fig. IV.7 a,b). For this reason, we have calculated the intensities. The sufficiently good qualitative agreement with the Raman spectrum (T = 293°K) of a polycrystal (Table lV.l, Fig. I v . 7 ~and ) data of the polarization change (Fig. IV.8) was obtained at y = 1.5. According to the calculations, the missing sixth line of the spectrum should be expected in the high-frequency region. The line with weak intensity is assigned to the B, type. In fact, we have actually seen all six frequencies of the external vibrational Raman spectrum of biphenyl crystal (43, 65, 70, 100, 105, 118 cm-l) at the orientations (aa), (ab) and (bc) (Fig. lV.9). The sixth line (very weak) at 100 cm-l (B,) is seen only at very good adjustment of a single crystal in the cryostat as the shoulder of the intensive line at 105 cm-1.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
20
.
100
299
cm-1
FIG. IV-7. The Raman spectrum of biphenyl polycrystal: experimentally measured a t 293°K [24] and calculated with the various values of y parameters. (a) y = 0.677; (b)y = 20.6; (c) y = 1.5.
The experimental and calculated values of frequencies and eigenvectors of external vibrations of the high-temperature biphenyl phase are summarized in Table IV.2. The comparison of the calculation with the experiment given in the table for low-frequency vibrations (active in infrared spectrum) is conditional, since the torsional vibrations of molecules which interact strongly with the external translational vibrations are displayed in this spectrum. From Table IV.2 we can see that the lines at 42 (B,) and 53 ($1 an-1 undergo the anomalous temperature dependence in the premelting region; they
ZHIZHIN AND MUKHTAROV
300
lL:
0
120 cm-1
0
120 an-1
FIG.IV.8. Experimental Raman spectra of biphenyl crystal [21] and spectra calculated in the oriented gas approximation.
correspond to most low-frequency librational vibrations of molecules around the axis with the largest and mean inertial moment and the average one. One should expect the strong interaction between the intramolecular torsional vibration of phenyl groups and the low-frequency lattice modes because the frequency of this vibration in a crystal is of the same order and is equal to 50-70 cm-l [18-20,241. Since this interaction is possible only in such cases when the molecule of biphenyl has lost its symmetry center due to its plane conformational distortion, the anomalous behavior of lines at 42 (B,) and 52 (A& cm-1is the additional factor which proves the conformational disorder of biphenyl crystal in the premelting region. Our experimental data and the data of other authors allows us to suggest the following schematic model of the temperature behavior of such a conformationally unstable crystal as biphenyl. The dependence of the energies on the torsional angle and also the dependence of the energy at different temperatures are given in Fig. lV.10. The calculations with the AAP predict that the contribution Ul,tt(B) is predominant at low temperatures (there is the sharp minimum at 8 = 0); they predict the plane conformation of a molecule in a crystal (curve 1,Fig. IV.lOb). The weakening of intermolecular interaction occurs with increasing the temperature as
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
0
40
80
120
160
301
v,cm-l
FIG. lV-9. The external vibrational Raman spectra of biphenyl monocrystal in polarized radiation at 77°K. The line 100 cm-1is indicated by an arrow and it was not observed before.
a result of the thermal expansion of the lattice, the contribution into the torsional potential lattice energy is enlarged also. The dependence of U1,t.(0) has the form of a curve with two minima and the maximum at 0 = 0 (curve 2, Fig. Iv.10b). In this case the plane conformation is now unstable, and the appearance of torsional
ZHIZHIN AND MUKKTAROV
302
TABLE IV.2 Frequencies and Eigenvectors of Biphenyl experiment (cm-1)
calculation (cm-1)
Isotopic Relations***
CuH10 CuH10 CuH10 CuH10 CuH10CBH10 vibration forms Exp. 77K* 8OK* 293K* llOK llOK 293K (llOK)**
B,
4
B,
105 70 65
97 67 62
88 53 (42)
111.5 79.2 56.7
101.8 75.0 53.9
83.5 63.8
118 100 43
108 41
88 54 42
108.4 92.9 47.7
103
99
91
--
--
_-
_-
39
--
38
83 72
79 69
(74) 68
dmi
50.2
97 L, 71L,+28L, 28L,+ 71L,
1.08 1.05 1.05
1.097 1.058 1.05
100.6 86.8 45.2
84.5 68.3 47.4
31 L, + 66 L, 58 L, + 35 L, 88 L, + 12 L,
1.09
1.097 1.050 1.058
100.8
97.7
82.4
99 T,
--
--
--
72.1
69.8
56.6
--
--
--
76.8
74.4
67.6
-1.05
99 T,
* FIR spectra at 90 K 1181, at 293 K [171 ** I, > I, > I, *** Isotopical relations were calculated according to vibration forms. vibrations of the phenyl group with large amplitude is possible. This picture is in good agreement with the X-ray analysis data at 110°K and 293°K [6,73 which predict the plane conformation on one hand and the anomalously large amplitudes of librations around the long axis of a molecule on the other hand.
R$(O)2
110°K
293°K
R, R,
45.7 3.45 2.40
109.17 11.47 8.39
RUU
However, in the Raman spectra one has not observed any anomalous changes with temperature both in the intramolecular region and in the lattice vibrational region. It can be explained by the fact that the potential barrier dividing the nonplanar conformations of biphenyl molecules is small in comparison with kT, and the
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
303
b) FIG. N.lO. Schematic drawing of the dependence of the biphenyl molecule potential energy on the torsional angle: (a) for separate contribution to the lattice energy; (b) for the total lattice energy at different temperatures (TI < T2 < T3).
molecules are not "kept" in the certain conformation. Only in the premelting region (1-2" before the melting point) does the barrier (the maximum at 0 = 0) increase so (curve 3, Fig. IV.10b) that it becomes comparable with kT; the molecules are fixed in one of nonplanar conformations; it is displayed in the Raman spectra in the region of intramolecular vibrations and anomalous changes in intensities of most low-eequency lines in the external vibrational region 1251.
ZHIZHIN AND MUKHTAROV
304
B. Low-FrequencyVibrations of Methyl Groups in Organic Crystals The crystals of organic molecules containing methyl groups between the molecular crystals are of special interest. In a f%eemolecular group, CH3 has the torsional vibrations with large amplitudes and almost flee rotation with respect to the core of a molecule. For this reason, the quantum-mechanical theory of the torsional motion of CH3 groups is usually used for the interpretation of vibrational spectra of these substances (the details are in Refs. [26,271). According to it, every methyl group is regarded as the symmetric one-dimensional rotator. The potential energy is the periodic function of the rotational angle of the CH3 group around the C-C bond V($)= V($ + 27th) (n-fold potential), and it can be expanded in a Fourier series. It appears that it is sufficient to restrict ourselves by the first term of this expansion in many cases
where V, is the reorientation barrier. The eigenvalues of torsional vibrations are found by solving the Schrodinger equation which is reduced to the well-known Mathieu equation 1263:
are the eigenvalues; v, a are the quantum number and where 2x = n$ + n; b, sublevel of the torsional vibration, s is the dimensionless parameter. The energy levels E , and the banier V, are related to b,, and S by formulae: E&= , zn 1 2 Fb, (IV.3a)
where F(cm-1)= W8x2a; T is the reduced moment of inertia (see Part I, lII.C). By determining the frequency of the transition (v + v') from the experiment and using the tables of values b,, for different s [26,281, one can determine the value of the barrier of the internal rotation V,. Since the frequency of these vibrations is in the region of low-frequencyRaman spectra and infrared absorption spectra, vibrational
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
305
and microwave spectroscopy have wide application and are the most reliable methods of determining the internal rotation barriers of methyl groups. The motion of these groups in a crystal becomes more complicated due to the intermolecular interaction influence. However, the numerous data show that the relatively large mobility of CH3 groups also remain in the crystalline state. The potential barriers have, as a rule, the same order of value as in the case of free molecules. In connection with this fact several authors have undertaken the efforts to use the Mathieu equation method for interpretation of low-frequency vibrational spectra of crystals [29-321. The selection of lines related to the torsional transitions of the methyl group was performed by means of the isotopic relationships, when the CH3 group is substituted by CD3. The frequencies of these vibrations are in the interval at 200 - 100 cm-1, as a d e . The reorientation barrier of methyl groups in the hexamethylbenzene lowtemperature phase 1.4 k d m o l [29] obtained in such a way is in good agreement with the NMR data (1.9 kcdmol) [331. We have performed the analogous investigation of the Raman spectra of acetonitrile crystal (CH3CN). The dependence of these spectra on temperature is shown in Fig. IV.ll. At 229°K there is the phase transition [341 which is displayed by the noticeable change of the spectrum, as is seen from Fig. IV.ll. Eight lines were observed in the a-phase at 85°K: 155, 145, 131, 120, 116, 108, 86, 72 m-l; four lines were observed in phase at 218°K: 110, 83, 74, 50 m-1. Incomplete data about the structure (the space group of a-phase P21/c, z = 4 [351) do not permit one to perform the calculation and detailed interpretation of the spectrum. However, the lines at 145 and 155 cm-1 in the a-phase can be unambiguously assigned to torsional vibrations of methyl groups, since even at low temperatures they are considerably broadened in comparison with the other lines and practically completely smeared into the Rayleigh line wing long before the phase transition. Assuming that the frequency of the transition 0 -j 1 at 85°K is equal to 155 cm-1 and using the Mathieu equation method for the threefold potential, we have obtained the reorientation barrier of CH3 groups V3 = 1.6 k d m o l ; this value is very close to 2 kcdmol measured by the NMR method [36]. The torsional potential and the levels of the rotator are shown in Fig. IV.12. The line at 145 cm-1 is referred, probably, to the hot transition 2 t 1 (Fig. IV.12) whose calculated frequency is 137 cm-1. However, in some cases (for example, in the case of oxylene), the discrepancy between the N M R data and other methods is very large. The inadequacy of the application of the independent rotator approximation in a crystal, when the reorientation barriers are sufficiently small and the torsional vibrational frequency is less than 100 cm-l, is the most probable reason for this fact. The considerable
ZHIZHIN AND MUKHTAROV
306
50
100
160
v. cm-1
FIG. IV.ll. Temperature changes in the low-frequency Raman spectrum of crystalline acetonitrile at a-p transition: 1-112.0; 2-160.0; 3-200.0; 4-216.8 (Uphase); 5-218.0 @phase); 6-230.0"K(melt).
mixing of torsional motions of methyl groups with external vibrations in a crystal can occur in this case. The p-xylene and toluene crystals can serve as examples. The methyl groups have almost kee rotations h isolated molecules. According to the microwave measurements [37] the potential of this rotation for toluene is described well by the expression
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
307
v=4 v=3
v=2
vrl
v=o
FIG. IV.12. Levels of rotator energy and torsional potential of methyl group in a-acetonitrile crystal.
The value of the barrier v6 is only 0.014 kcdmol. Consequently, the vibrations of CH3 groups in a crystal are defined, mainly, by intermolecular forces. To study the motion of methyl groups in p-xylene and toluene crystals we have performed the calculation and the interpretation of their low-frequency spectra [38]. 1. Para-xylene (CH3 - C6H4 - CH3)
The structure of p-xylene crystal (Tmelt = 286°K) remained unknown for a long time. One has established by studying the Raman and infrared absorption spectra that the molecules in a crystal have the centrosymmetric position [39,401; the possible space group P2l/m, z = 2(Ci) [401. These data are in good agreement with the X-ray analysis data about p-xylene which were obtained comparatively not
ZHIZHIN AND MUKHTAROV
308
so long ago at the Leiden University (Netherlands)and were kindly given to us by Dr. C. Gorder. The structure was determined at 163°K; it has the space group P21/c, z = Z(Ci). The molecular packing in the unit cell is shown in Fig. IV.13. The reorientation barrier of the methyl group (0.35 kcal/mol) was determined by the dependence of the heat capacity C, of pxylene crystal on the temperature in 1411. The measurements of relaxation times T1 and Tlp by the NMR method have shown that the reorientations of CH3 groups are activated even at 50°K 142,431 (Fig. IV.14a). The following values of barriers were obtained: 0.58 0'1, 1433); 0.43 Crl, 1421) and 0.50 0'1, 1421) k d m o l . In the infkared spectra the rotation of methyl groups in the pxylene crystal is displayed in the predominant broadening of absorption bands in the region of 2975-2950 cm-1 [39,441 corresponding to C-H vibrations of the CH3 group with the transitional dipole moment directed perpendicular to the rotation axis of this group. The analogous changes were observed by us at 170 - 200°K in the Raman spectrum (Fig. IV.15). In 1393 the Ramari and infrared absorption spectra were investigated in the rauge of 0 - 200 un-1 where one should expect the direct display of torsional vibrations of methyl groups. The polycrystals of various deuterated compounds C&I4(CH3)2, C ~ H ~ ( C Dand ~ ) ZC$4(CH3)2 were studied for the interpretation of these spectra. According to the selection rules (Part 1,III.C) eight Raman lines and five infrared absorption lines are allowed; the authors have observed six Raman lines and three infixwed absorption lines at 93°K; their frequencies are as follows:
IR,cm-1
Raman, cm-1 C&4(CH3)2:
130
108
94
88
69
64
152
95
79
C&&(CD3)2:
121
111
99
80
69
60
120
95
79
The authors have come to the conclusion that the observed Raman lines do not refer to the vibrations of the CH3 groups, since the isotopic ratios of frequencies are considerably smaller than the expected value of 1.41. However, this conclusion does not agree with our results: the line at 135 cm-1 is referred to the torsional vibrations of methyl group. Even at low temperatures (112°K) this line is several times broader and weaker than the other lines; it is broadened with the increase of the temperature and becomes practically unobservable long before the melting point (Fig. IV.16) 1451.
FIG.IV.13. Packing of molecules and their conformation in p-cylene crystal (1 - experimental, 2 - calculated).
310
I
I I
1 0
20
80
60
160
100
240
140
T,K
T,K
FIG. lV-14.Temperature dependence of spin-lattice relaxation times T1 and TI, in crystals of p-xylene (a)and of a-toluene modification (b) [42,481.
The far infrared absorption spectrum band at 152 cm-1 with the isotopic relation 1.27 [39] can be referred to as torsional vibrations of methyl groups. However, the use of this frequency in the independent oscillator model gives considerably larger values of this barrier. Thus, these data indicate that there is a strong interaction between the vibrations of CH3 groups and the strong interaction
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
3 100
3000
2900
311
v, cm-1
FIG.JY.15. Temperature changes in p-xylene Raman spectrum in the region of fkequencies of valent C-Hvibrational: 1-112.0;2-280.0;3-288.0"K(melt).
of these groups with the external vibrations of a crystal. The anomalous increase of frequencies of several lines in the Raman spectra of p-xylene with the deuteration of the methyl group proves it. The lattice dynamics calculations for the p-xylene crystal and its deuterated species were performed by us in the semirigid molecular approximation (part 1, lII.C). We have used three sets of parameters AAP(6-exp): AAP/l, 2,3 (Appendix 1). The potential energy of a crystal was taken in the following form:
v = v, + v,
(Iv.5)
ZHIZHIN AND MLTRHTAROV
312
60
100
136
150
v,cm-l
FIG. IV.16. Temperature dependence of low-frequency Raman spectra of p=xylene crystal. 1-112.0; 2-215.0; 3-258.0; 4-273.0; 5-286.0; 6-286.2; 7-2865°K (melt).
where V,, is the intramolecular torsional potential. Since the mutual intluence of methyl groups of pxylene molecule is very small (the distance between the H atoms of methyl groups is more than 6.5 A), the potential (Eq. N.4)obtained in 1371 for the toluene molecule was chosen as V,*. The structure of a crystal determined by C. Gorder at 163°K was used for the calculation. Since large errors are possible during the procedure of determining the hydrogen atom coordinates by the X-ray analysis method, we have ascertained the orientation of CH3 groups in a crystal by the potential energy minimization (Eq.
STRUCTURE AND PHASE TRANSlTION OF ORGANIC CRYSTALS
3 13
IV.5). Since p-xylene molecules in a crystal are centrosymmetric, only four coordinates were varied, i.e. three Euler angles and one torsional angle of the methyl group. It appears that the orientation of the CH3 group relative to the molecular core calculated by the AAp/1,2,3 dif€era from the experimental one by angles 1.6; 4.0 and 2.9" (Fig. N.13). The rotational angle of the whole molecule around its experimental orientation is also small and is equal to 1.8; 2.4 and 1.6". The lattice energy calculated by the AAP/1,2,3 is equal to 16.38; 18.35 and 16.44 k d m o l . The rest of the conditions of p-xylene lattice dynamics calculations were discussed in detail in Part 1, II1.C). The Raman spectra of polycrystals of C&(CH3)2 and C$€4(CD& calculated for 7 = -0.2 (11> I3 > 12) with various AAP and measured at 93°K in 1393 are shown in Fig. IV.17. The calculation (T = 163°K) reproduces quite well the features of pxylene Raman spectrum. The application of the M I 2 gives worse agreement with the experiment than the application of the AAp/1 and 3. The interpretation of the Raman spectrum of t h i s crystal is performed unambiguously; the results are given in Table IV.3. The broad line at 130 cm-1is referred to the doublet ag - b, which corresponds to the torsional vibrations of methyl groups (the contribution of these vibrations is 60 - 70%)considerably mixed with the librations of molecules around their axis V of the smallest inertial moment. The noticeable factor-group splitting 138(a& - 128(b,) cm-l predicted by the calculation indicates also the strong interaction between the torsional vibrations of CH3 groups of translationally nonequivalent molecules. Thus, the independent rotator model is unfit for the description of the CH3 group motion and for the determination of their reorientation barriers in p-xylene crystal. In the case of p-xylene -(CD3)2 the agreement with the experiment is worse (Fig. N.17). However, the calculated results permit one to conclude that there are changes in the Raman spectra with the deuteration of methyl groups. First of all, the frequencies v1 and v5 (Table N.3) are decreased by 40 - 70 cm-1, and the vibrations of -CD3 groups are completely mixed with the librations of molecules. The eigenvectors of all % and bb modes is noticeably changed (Appendix III). Secondly, the fi-equenciesv1 and v5 become lower than the frequencies (vs, v2, vr); it leads to the "repulsion" of the latter to the high-frequency side. It is in agreement with the anomalous shift of the line at 108 cm-1 experimentally observed as the kequency 121 cm-1(Fig. IV.17). At the same time the anomalous isotopic shift of the line at 88 un-1(vg) is not reproduced. We have associated t h i s fact with the following argument: the line at 130 cm-1 (vl, v5) is actually shifted to the frequency region which is not higher (as the calculation predicts), but lower than the frequency of the v3 line at the substitution of CH3 by CD3; it leads to the "repulsion" of the latter to the high-
a)
150
100
100
+
2 50
-
150
100
50
150
100,
,!
50
50
150
FIG. IV.17. Raman spectra of p-xylene -(CH& and -(CD)3I2;experimental one (93°K [39]) and calculated with AAP/3 (a), AAP/l (b)and AAP/2 (c) at 163°K.
100
50
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
315
Gequency side ( h m 88 to 99 cm-l). For this reason, we have assigned the line at 80 cm-1 with the d intensity observed in the Raman spectra of C&(CD3)2 crystal at 93°K to the line shifted from 130 cm-l as a result of the deuteration of methyl groups (Fig. IV.17). The aforementioned discrepancy with the experiment for the aforementioned crystal concerning the succession of lines can lead to the distortion of the calculated eigenvedors; it is expressed by the discrepancy between the experimental and calculated intensities of the Raman lines of this crystal. In the case of normal vibrations which are active in the inhued absorption spectrum the vi&ations of methyl groups practically do not interact with translational vibrations. The character of their motion remains with the deuteration. The comparison of calculated and experimental values of frequencies (Table IV.3) permits one to refer unambiguously the absorption bands at 152 and 120 cm-l of C6H4(CH3)2 and C6H4(CD3)2 crystals to the torsional a,-vibrations of methyl groups. However, the factor-group splitting is 10 un-1, according to the calculation, i.e. the vibrations of methyl groups of different molecules interact considerably. Our complete interpretation of p-xylene vibrational spectra is shown in Table IV.3. Considering that the spectra were measured at 93°K in Ref. [391, and the calculation was performed in the quasi-harmonic approximation at 163"K, and the AAP parameters used were obtained without the p-xylene data, the agreement of calculated frequencies with the experimental ones is satisfactory. The application of the M I 3 parameters gives better results than the application of the other parameters. 2. Toluene (C&-CH3)
Unlike p-xylene crystal, toluene can have three forms: glass-like and two crystalline (metastable bphase ("melt = 154OK) and stable a-phase, Tmelt = 178°K) [47,48]. By heating the samples the glass-like and metastable phases are transformed irreversibly to the a-phase at 136°K [481. Let us consider in detail the properties of the a-phase of toluene; many studies have been dedicated to this problem. A rather small value of the reorientation barrier of methyl groups in this phase (0.2 kcdmol) was obtained fkom the temperature dependence of C, [41]. According to the NMR data [481, the reorientations of CH3 groups are activated even at 20 - 40°K (Fig. IV.14b). The torsional barrier determined by the ordinary method in [48] is equal to 0.49 kcdmol. The more complicated model considering the quantization of the rotational motion and also the interaction between the spin and rotational degrees of freedom and phonon vibrations of a crystal was used [49]
TABLE IV.3 Low-frequency Raman Data of p-Xylene
1
Ag
exp. [391 2
calc.*
calc.*
3
4
130
163.8 135.0 138.2
108
103.9 96.6 96.5
88
79.4 64.7 69.7
92.6 81.2 82.8
99
78.7 63.5 68.4
87.4 76.8 78.7
81
72.9 59.2 64.1
87.0 76.2 77.4
64
62.4 54.4 54.0
62.9 54.7 55.4
60
58.3 50.8 49.4
58.5 50.9 51.5
60.2 52.5 52.7
61.2 53.2 53.8
130
151.6 121.7 127.7
80
87.5 79.8 78.1
108
113.3 104.5 102.5
120.4 111.5 108.4
111
118.0 104.1 102.7
94
97.3 97.8 85.8
105.1 102.8 91.1
99
100.9 96.2 88.1
C
a
Bg
V6 b C
a
Bg
v7 b C
163.3 133.9 137.6 113.1 105.1 102.3
a
v5 b
130
103.0 97.5 95.6
C
Bg
10
105
a
v4 b
calc.*
9
109.7 101.3 98.0
C
Ag
dc.*
122.7 106.7 105.1
a
v3 b
7
121
C
Ag
6
exp. [391 8
116.9 108.3 105.3
a
v2 b
Calc. *
91.6 83.2 84.3
C
Ag
calc.*
80
a
v1 b
exp. [39] 5
__ 130
150.8 120.3 126.8
111.1 102.6 99.9
105
112.0 102.4 101.7
118.4 110.0 106.8
99.5 96.2 86.4
87
94.3 96.1 82.5
99.8 99.1 86.2
assignments** 11
a Bg
"8
66.9 53.5 58.1
152
164.5 134.5 138.3
95
83.8 91.5 73.0
84.0 91.6 73.0
95
81.5 88.8 71.0
81.7 89.1 71.0
__
82.3 89.8 71.6
82.5 89.9 71.6
35.7 37.6 32.6
37.7 40.4 34.8
_-
34.7 36.3 31.6
36.7 39.3 33.8
--
35.1 36.9 32.0
37.0 39.6 34.1
(120)
107.7 87.0 91.1
79
54.0 52.5 49.8
C
a Au
v9 b C
a Au v10 b C
82.4 74.7 71.4
a
79 C
* Ag and A,
55.9 55.4 51.8
69
117.0 120
151.1 119.6 127.2 v13
77.7 67.0 67.3
69
58.2 59.9 54.5
66.0 51.2 56.6
164.5 96.2 98.4
are the torsion shifb of two methyl groups of a molecule
** T, L are the translational and librational shifts of molecules
65
-_
61.3 49.1 53.6
78.0 69.5 67.7
134.5 138.3
150.9 119.5 127.1 56.6 58.3 53.0
_-
54.9 54.4 50.8
57.2 58.8 53.5
318
ZHIZHIN AND MUKHTAROV
for the interpretation of the NMR data. The value of the barrier 0.1 - 0.2 k d m o l is in good agreement with the thermodynamical data 1411. The investigations of low-frequency vibrational spectra of a-toluene were reported in papers [32,40,45,501. The Raman and FT-i&ared absorption spectra of polycrystals of a-CgH5CH3 and a-CgH5CD3 were obtained at 20 - 163°K in 1501. The authors have come to the conclusion that the torsional vibrations of methyl groups are not observed in these spectra, since the isotopic relations do not exceed 1.032. The new line at 24.5 cm-1 whose intensity increases anomalously at heating the crystal up to 80°K(the authors of Ref. [501 have suggested that it is associated with the phase transition in this crystal) is displayed in the Raman spectrum of aC6H5CD3 crystal at 40°K 1501. The changes in the C-H bond vibration region were observed in the Raman spectrum of C6H5CHD2 crystal at these temperatures [51]. The effort to interpret the low-frequency Raman spectnun of the a-toluene polycrystal by means of the isotopic relations was undertaken also [32]. However, the displacement of lines was also only 1- 4 cm-1(the measurements of the Raman
(IV.6) Mathieu equation method (Eq.IV.2); it was done to confirm the assignments 1323. Unlike the latter, the more complicated torsional potential was used which imitates the influence of the intermolecular interaction. The parameters V2, V, and V u were fit in such a way that the most intensive, calculated lines have the frequencies 77 and 105 cm-1. It was assumed that the methyl group polarizability depends on the torsional angle (a= a, sin 20). The following values were obtained V2 = 0.08; v6 = 0.28, V u = -0.12 k d m o l ; the reorientation barrier of methyl group was equal to 0.3 kcal/mol; these data were very close to the experimental ones. The disadvantage of this work is the use of the potential (IV.6) only for two values of fiequencies in the fitting procedure for three parameters. The calculation of normal vibrations of a-toluene crystals (CgH5CH3, C6H&D3 and C6D5CD3) was performed by us in the half-rigid molecular approximation (part 1, III.C). We have used the X-ray data obtained for a-toluene at 160°K in 1521. The crystal has the space group P21/c, z = 8 (C1). The packing of molecules in the unit cell is shown in Fig. IV.18. The structure of a-toluene is considerably more complicated than the structure of pxylene. Two symmetrically independent molecules forming two sublattices in a crystal (Fig. IV.18) are contained in the unit cell. In particular, it thus follows that two different values of
STRUCTURE AND PHASE TRANSITIONOF ORGANIC CRYSTALS
3 19
the reorientation barriers of methyl groups related to different sublattices are possible. The normal vibrations at = 0 are divided by following irreducible representations of the factor-group c2h with the consideration of the methyl groups
All
known experimental data about the low-frequency vibrational spectra of a-
toluene are summarized in Table IV.4. According to the most complete lowtemperature data [501, only seventeen lines and h n infrared bands are displayed in the spectra; 28 Raman lines and 25 infrared bands were allowed by the selection rules. We have used the same conditions as in the Case of pxylene for the calculation of the dynamical problem. Two sets of parameters were used (the AAP/2 and AAP/3). The frequencies and eigenvectors were found by the diagonalization of blocks A and S of the dynamical matrix (Part 1, Chapter I) having 14 x 14 dimensions (which is twice as large due to the presence of two independent molecules. The eigenvectors describe the contribution to the n o d vibration of spectra of C6H&H3 and C&CD, crystals were performed at 130°K 1323). Nevertheless, the authors of Ref. [321 have assigned the comparatively narrow lines with small intensities (105 and 77 un-l) to torsional vibrations of CH3 groups, because they disappear with the deuteration of molecules. It was suggested that the intensity of these vibrational lines must decrease twice at the substitution of CH3 by CD3. The calculation of the methyl group vibrations in the independent rotator approximation was performed by the method which is similar to the degrees of freedom of two molecules instead of one molecule as before; two molecules belong to the different sublattices of a crystal. The calculation with the use of the experimental structure of a crystal [52] leads to imaginary frequencies. For th i s reason, we have performed the optimization of the structure and ascertained the orientations of methyl groups for every set of AAP parameters. The lattice energy was minimized by varying fourteen coordinates of two independent molecules, including the torsional angles of methyl groups. As a result, the lattice energy was increased by 0.5 k d m o l (in absolute value) and becomes equal to 13.78 kcdmol for AAP/2 and 12.73 k d m o l for AAP/3. The equilibrium orientations of methyl groups calculated with two AAP are practically the same, but they differ considerably from the X-ray analysis data (Fig. IV.18). An especially large difference (55")was obtained for the orientation of CH3 groups in the sublattice 2. It appears also that, unlike the X-ray analysis, the
w
h3
0
‘H H
\i
H
- experiment
FIG. IV.18. Packing of molecules and their conformationin a-toluenecrystal (two sublattices are indicated by numbers 1 and 2) [52].
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
321
TABLE IV.4 Low-Frequency Vibrational Spectra of a-Toluene I&aredAbso tion SpeCtra(cm? 1i
Raman spectra ( a - 1 ) 133K [321
85K M51
_-
__
28 32
--
44 52 59 65
-_ 77 83 94 99 103 106 115
--
-48 55 63 68 70
-93 99 108 118
-135
20K 1501
20K "501
19.5 27.5 31 35.5 (39) 44.5 49 56 64.5 70.5 75 81.5 93.5 100 106 119.5 125 135
20
--
30 35
_-
133K [321 18 23 28 31
20K [501
--
--
44 46.5
43
46
53.5 62 69 74
56 64
--
-_
9-
_-
__
--
88
--
96 105 114
95 101
120 133
114
---
64 69.5 76.5 81.5 94 102
-_
115 123 130
20K 1501
_66 73
_110.5 118 124
calculation predicts practically the same conformation of molecules in two sublattices (Fig. IV.18). The structure calculated with M I 3 (AAP/2) differs &om the experimental one also by the shift of the molecule mass center 0.1 A (0.2 A) and by the rotation angle 5" (7"). Thus, the M I 3 describes the crystal structure of atoluene better than M 1 2 . The results of the calculation of frequencies and eigenvectors of a-toluene crystal (T = 160"K, M I 3 1 are summarized in Tables IV.5 and IV.6; the detailed description is given in Appendix III; they are as follows:
2;HIZHINAND MUKHTAROV
322
1. The calculation with the consideration of the methyl group vibrations leads to decreasing frequencies (in comparison with the rigid molecular approximation)by no more than 8 cm-l(3 cm-1, in average).
2. Only five normal vibrations (from eight) are predominantly torsional vibrations of methyl groups (80 - 90%). 3. Every normal vibration corresponds to the motion of CH3 groups belonging to only one sublattice of a crystal (whether 1 or 2). The vibrational frequencies of CH3 groups in the sublattice 1 (AAP/3) 133.2($) - 132.9(Bg)- 134.1(&) 133.8(BU)~ m - 1 are close to the calculated results obtained for p-xylene. Only one normal vibration at 99 cm-1 ( v ~ )can be referred to vibrations of methyl groups in sublattice 2. 4. The contribution of the methyl group displacements to other normal vibrations is less than 60%;moreover, the torsional vibrations interact considerably with the librations of molecules around their W axis which coincides with the rotational axes of methyl groups. 5. In a crystal of a-toluene, C6H5CD3, the vibrations of CD3 groups are considerably mixed with the librational translational motions of molecules. Only one normal vibration, 71.4 cm-1, can be assigned to torsional
-
We have calculated the relative intensities of Raman lines in the oriented gas approximation for assignment of Raman spectra of C6H5CH3 and C6H5CD3 polycrystals measured at 20°K 1501. Since the data for the tensor of molecular polarizability are absent, we have performed the calculations with various values of y (Part 1, IV.C). The experimental and calculated Raman spectra of a polycrystal for y = 0 (I3 > I2 > 11) are shown in Fig. IV.19. The agreement with the experiment is worse than in the case of p-xylene. Nevertheless, the calculation is in qualitative agreement with the'experiment concerning the number of intensive lines. The calculation predicts also the presence of a line at 12.9 cm-1 whose intensity is more than four times as large as the intensity of other lines. In our opinion, this line was not observed because of its closeness to the Rayleigh line. Our interpretation of Raman spectra of polycrystals C&jC& and C6H5CD3 is given in Table IV.5 and in Fig. IV.19. We have taken into account the isotopic relations (Table IV.6) and considered that the calculated frequencies (at 163°K) must be smaller than the experimental ones measured at 20°K in 1501. As in the case of p-xylene, the M I 3 parameters give better agreement with the experiment than the M I 2 parameters. For this reason, we have not given here the calculated results with AApI2 parameters. According to our assignments, the purely torsional
STRUCTURE AND PHASE TRANSITCONOF ORGANIC CRYSTALS
323
vibrations at 133.2 (A& - 132.9 (Bg)cm-1 are not observed in the Raman spectra because of their small intensities and strong broadening by the methyl group reorientations. With deuterating these groups their vibrational frequencies are shifted to 102.8 (A& - 101.9 (Bg)cm-1, and the motion of CD3 groups are mixed with the external vibrations. This leads to the noticeable change in line intensities in the range of 130 - 100 cm-1; it is in qualitative agreement with the changes in the experimental Raman spectra (Fig. IV.19). Thus, none of the lines observed in the low frequency Raman spectrum of atoluene crystal can be assigned to the purely torsional vibrations of methyl groups. Consequently, the interpretation of this spectrum in [32] is incorrect. The line at 24.5 cm-1 whose intensity increases anomalously with changing the temperature from 40" to 80°K ( U - C ~ H ~ C 1501 D ~ ) corresponds to the n o d vibration v26 of molecules of the sublattice 2 (Fig. IV.18, Table IV.5). As in the case of phenanthrene, the observed phenomenon can be also explained by the redistribution of the intensities between the internal and external vibrations which are considerably mixed. The assignments of the infrared absorption spectrum of a-toluene made by us on the basis of the calculated frequencies and isotopic relations are given in Table N.5. The assignments are less unambiguous here in comparison with the Raman spectra. The absorption band with the fkequency of 115 cm-l at 18°K [50] (100 cm-1 at 165°K [501) is assigned by us to an &-vibration (Table IV.5) of the CH3 groups in the sublattice 1. However, the calculation with this frequency by means of the Mathieu equation method gives the value of the reorientation barrier 0.73 kcdmol, which is several times larger than the experimental one. Thus, the calculation of the dynamics lattice by the AAP method has shown that the low-frequency vibrational spectra of the p-xylene crystal and a-toluene crystal can not be used for a reasonable evaluation of the reorientation barriers of their methyl groups by means of the independent rotator model. However, the values of these barriers can be directly estimated by means of the AAP, if one calculates, for example, the dependence of the lattice energy on the angle of the rotation of one CH3 group around the molecular core with fixed positions of surrounding molecules (the rigid lattice approximation). The calculated results are given in Fig. N . 2 0 and in Table N.7.The calculation of the barriers was performed independently for each of two symmetrically nonequivalent molecules in the case of a-toluene. The calculated barriers are in good agreement with the experimental data. The torsional potentials V(4) are described sufficiently well by the threefold potential of the form (IV.21), unlike the assumptions of the authors, Ref. 1323. It appears also that the reorientation barriers of methyl groups in two sublattices of a-toluene crystal are almost the same.
ZHIZHINAND MuKHTARov
324
TABLE IV.5 Calculated and Observed Frequencies for a-Toluene C6H5CH3 expKO]* calc. 20K
a-1
V1
C6D5CD3 exp. [321 130K calc.
a-1
a-1
m-1
133.2
114
102.8
99.5
a-1
=-l
v2
135 (115)
117.8
133
110.3
103.6
v3
119.5 (106)
99.4
105
86.3
84.0
_-
88.1
-_
79.2
77.7
v4
Ag
--
C6H5CD3 exp. 1503 20K calc.
v5
93.5 (83)
80.0
88
75.5
73.6
v6
75 (70)
75.0
__
65.4
63.7
v7
70.5 (65)
65.2
69
62.9
60.2
VS
64.5 (59)
56.4
62
54.7
52.5
V9
56 (51)
49.4
53.5
46.8
44.9
49
44.2
46.5
43.0
41.4
v11
44.5 (44)
37.1
44
36.5
35.3
vr2
35.5 (32)
32.4
35
31.8
30.7
v13
31
25.4
30
24.1
14
--
12.9
__
24.9 12.8
12.1
15
__
132.9
114
101.9
99.3
16
125
116.7
120
108.1
101.0
106 (102)
97.5
96
87.1
84.6
v10
v17
(-1
(-1
assignments**
12yx2
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
TABLE W.5 (continued) C6H5CH3 exp[501*
20K
=-l
1
2
calc.
a-1
3
C6H5CD3 exp. 1503 20K calc.
m-1
4
--l
5
C6D5CD3
exp. 1321 130K =-l
6
calc.
=-l
7
18
86.6
78.9
76.5
v19
80.3
74.3
71.7
B, v20
73.1
68.6
65.5
v21
71.4
64.2
62.7
v22
53.4
51.6
49.9
v23
48.6
45.4
44.3
v24
37.8
36.9
35.6
v23
33.3
32.8
31.5
v26
29.9
29.4
28.5
v21
22.0
21.5
20.4
v28
13.3
13.2
12.7
v29
124.1
104.5
101.7
v30
99.0
71.4
71.1
v31
107.1
103.1
96.9
Au v32
89.4
84.4
80.4
v33
83.7
78.2
76.3
v34
65.5
64.4
61.5
v35
63.3
60.4
57.9
v36
49.9
47.9
46.6
v37
45.6
44.4
42.0
v38
41.4
40.5
assignments** 8
325
ZHIZHIN AND MURHTAROV
326
TABLE IV.5 (continued) C6H5CH3
exp1501* 20K
a-1
1
2
C6H5CD3
calc. a-1
3
exp. 1501 20K
a-1
calc. a-1
4
5
30.8 31
c6D5cD3
exp. [321 130K calc. a-1
6
m-1
assignments**
7
30.2
29.2
8
30.5
_--
26.6 9.3
__
25.9 9.2
25.0 8.8
133.8
103.4
100.6
102
98.0
---
72.6
70.9
123
103.9
118
99.5
93.8
94
90.0
--
85.7
81.3
81.5
79.9
--
75.3
72.9
76.5
72.4
72
68.8
66.1
55.1
53.7
55.8
54.5
52.8
42.0
40.5
38.8
36.4
34.9
57.7
--
64
46
43.5 38.3
31
26.2
30.5
25.9
24.9
--
17.7
--
17.5
16.7
* Frequencies measured at 130°K in Ref. 50 are given in parenthesis.
**XI,L1, T1 and x 2 , 0 , P are the displacements of methyl groups, librations and translations of molecules in the sublattices 1and 2, respectively.
Using the Mathieu equation method and the calculated values of the barriers V3, we have found the energy levels and frequencies of the transitions VO-, 1 of the rotator (see Table IV.7).
STRUCTURE AND PHASE TRANSJTlON OF ORGANIC CRYSTALS
Bg
__
327
1.015 1.137 1.063 1.022 1.040 1.047 1.054 1.011 1.014 1.03
1.068 1.152 1.060 1.037 1.031 1.056 1.028 1.016 1.019 1.020
16 v17 v26 v27
1.042 1.104 1.12 0.98
1.080 1.119 1.017 1.023
v31 v35 v36
1.048 1.053
1.039 1.048 1.042
1.105 1.093 1.071
1.027 1.020
1.086 1.055
1.027
1.064
Au
1.116
_-
1.016 1.054
-1.023 1.032 1.00
-_ 1.288 1.065 1.05
1.137 1.183 1.083 1.074 1.100 1.068 1.051 1.055 1.054 1.156 1.153 1.049 1.078
1.058 v37 v39
1.016
v47
1.042 1.063
1.044 1.053
1.108 1.095
v50 v51
1.058
1.037 1.052
1.083 1.097
v52
1.016
1.012
1.052
v44
4
*u
aRef. [24]. bRef [7].
ZHIZHIN AND MUKHTAROV
328
I
v, cm-1
v, cm-1
,
,
I
,
,
,
,
,
(
150
150
60
FIG. IV.19. Raman spectra of a-toluene polycrystals (-CH3 and -CD3): experimental ones (a) and (b) [50] and calculated with AAPI3 (c) and (d) at 160°K.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
329
In the case of the a-toluene crystal this model predicts the retarded rotation of methyl groups, since the energy of the first excited level of the rotator is larger than the value of the reorientation barrier V3. However, the frequencies of the harmonic vibrations of CH3 groups calculated with AAP/3 in the independent torsional oscillator approximation (we have considered only the second derivative of the crystal potential energy with respect to the torsional displacements of CH3 group of one molecule) have the values which are close to the values of vo+1 (Table IV.7). The results justify, to some extent, the use of the quasi-harmonic approximation for the calculation of frequencies and forms of vibrations of p-xylene and a-toluene crystals. As can be seen from Tables IV.3, IV.5, and IV.7, the frequencies of the CH3 group vibrations calculated for independent oscillators and for the case of the complete solution of the dynamical problem differ by more than a factor of two. Thus, the consideration of the vibrational interactions is absolutely necessary for the construction of an adequate model of the methyl group motion in the crystals of p-xylene and toluene. It is also necessary in some other cases when the reorientation barriers are determined mainly by the intermolecular forces and have the values on the order 0.5 k d m o l and below it. The application of the isotopic relations only is insufficient for the assignments of lines of CH3 group torsional vibrations in the low frequency spectra of these crystals. The application of the AAP method allows us to ascertain the crystal structure of p-xylene and a-toluene, to calculate the reorientational barriers of methyl groups, to predict a very intensive line close to the excitation line in the Raman spectrum of a-toluene and to make certain conclusions about the reconstruction of the structure of this crystal in the range of temperatures from 40 80°K. For the first time we have used the AAP method for the interpretation of the complicated low-frequency vibrational spectrum such as the spectrum of a-toluene crystal. We have shown also that the display of normal vibrations corresponding to the motion of molecules in one of the sublattices of a crystal is possible in spectra of these crystals. In conclusion we must point out that the calculations indicate good transferability of the AAP/3 parameters to the p-xylene and a-toluene crystals; the AAP method was not previously applied to their description.
ZHI2;HINAND MUKHTAROV
330
7
80
160
240
320
400
480
-12.5
-12.6
a)
-12.7
-12.8
FIG. IV.20. Dependence of the energy of interaction between the chosen molecule and other molecules of a crystal on the rotation angle of its methyl group calculated with AAPI3 for a-toluene (a) ((1) and (2) are symmetrically independent molecules) and for p-xylene (b).
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
TABLE IV.7 Calculated and Experimental Methyl Barriers U,, k d m o l "o-, 1 experiment
dC.*
pxylene
0.35 1411; 0.43 1421 0.50 [421; 0.58 1431
a-tohene**
0.20 [41]; 0.49 [48]
0.45 - a 0.41 - b 0.30 - c 0.16(2)
0.10 - 0.2 [49]
0.230) 0.17(1) 0.14(2)
*
Ib
'
331
bm.
m-1
m-1
68.4
73.7
56.3(2)
49.4(2)
59.0(1)
55.9(1)
a,b,c - AAp/1,2,3 **(1)and (2) are the numbers of the sublattice of a-toluene crystal.
332
ZHIZHIN AND MUKHTAROV
V. APPLICATION OF VIBRATIONAL SPECTRA AND AAP FOR THE DETERMINATION OF STRUCTURES AND THE INVESTIGATION OF PHASE TRANSITIONS IN CRYSTALS OF PHENANTHRENE AND PYRENE In a number of cases the application of basic methods for the determination of a molecular crystal structure (the X-ray analysis and the neutron =action method) is either impossible or gives incomplete data. First of all, it relates to crystals that have undergone polymorphic transformations which make it difficult to grow monocrystals of good quality. In particular, there is only a small number of papers concerning the investigation of organic compounds forming crystals at low temperatures and also low-temperature crystal modifications. The problem of the determination of molecular crystal structures can be solved in principle by the AAP method with the use of data on the molecular geometry [l]. The fact is that one of the minima of the lattice potential energy expressed in terms of AAP as a function of unit cell parameters and coordinates of a molecule U (a, b, c, a,p, y, T,, Tb, Tc*, (I, (It,, correspond to the structure, which is very close to the real molecular packing in a crystal [I]. However, the problem becomes very complicated and time-consuming because of the necessity of varying a large number of parameters (no less than twelve in a general case) and space groups. A larger problem is that there is no single method of seeking the global minimum of many variable functions, although the effective methods of seeking the separate local minima are well known [2,3]. Excluding time-consuming calculation problems, there are some other principal diaculties. First of all, the lattice energy is calculated by the AAP method to a n accuracy of about 1 kcdmol and, consequently, the choice of the molecular packing with respect to the local minimum depth, which differs by less than 1k d m o l , becomes ambiguous. Secondly, the contribution of enthropy i n a free energy is not considered, i.e. the calculation of energies lead only to some hypothetical structure of a crystal at T=O"K. Thus, it is necessary to obtain additional information concerning the structure. It is very important to know the size of the unit cell. The thermal expansion takes into account to some extent the enthropy factor (see Part 1, Sect. II.C) and performing the calculation of the packing with the fixed known parameters of the unit cell first decreases the number of varied parameters, and it permits, secondly, one to study the dependence of the structure on temperature and, consequently, the structural aspects of phase transitions. At the present time there are comparatively simple diffraction methods for the determination of the unit cell
STRUCTURE AND PHASE TRANSITION OF ORGANIC C R Y S T m
333
parameters, even in such cases where the total determination of the structure is impossible [4]. The important information about the structure can be obtained by means of molecular spectroscopic methods. For example, at the present time the following methods are widely used the spectroscopic methods of determination of the crystal factor-group by means of the multiple structure of the intramolecular infrared absorption bands and the Raman spectrum one8 151. The spectra in the external vibration region are of particular interest. The external vibrations are most sensitive to the intermolecular interaction and, consequently, to the mutual positions of the molecules. On one hand, the external vibrational fkequencies can serve as a sufficiently strong criterion of the reality of the calculated molecular packing, and on the other hand, they are of interest, together with the line width and their intensities, because the different normal vibrations can have unequal sensitivity to different reconstructions of the crystal structure due to their mutual positions and the molecular geometry 161. We have chosen the phase transitions in phenanthrene and pyrene crystals to study the vibrational spectra and to calculate the molecular packing and lattice dynamics.
A. Investigation of the Phase Transition of Phenanthrene Crystal by Means of Low-Frequency Raman Spectra The unusual phase transition of phenanthrene at 70°K has been observed [71 by performing thermal physical measurements. The anomalous changes of various physical properties at t h i s transition (the review of papers is given in [81) occur continuously in the temperature region above 10°C; the structure of phenanthrene crystal was defined completely by the X-ray analysis and by the neutron *action method 19,101 only at the room temperature (Phase I). The structure is characterized by the space group P21 with two molecules per unit cell and it is shown in Fig. IV.5. At high temperatures (Phase 11)only the parameters of the unit cell measured by the method of the X-ray disaction on polycrystal [111 are known:
A
T,K
a, A
b,
299 293 293 353
8.4744 8.472 8.46 8.54
6.1720 6.166 6.16 6.25
c,A 9.4805 9.467 9.48 9.55
P,degree
Ref.
98.01 98.01 97.8 99.0
r 121 r101 r111 1111
MIZHIN AND MUKHTAROV
334
98.8 8.61 8.48
98.4 20
40
60
80
T,”C
98.0
L
20
40
60
80
T,”C
FIG.V-1. Packing of phenanthrene molecules in a crystal (space group P21, 2=2) [lo] and dependence of the unit cell parameters on temperature 1111.
It was discovered that these parameters were anomalously changed in the region of the phase transition (Fig. V-1). Studying the changes in the diffraction pattern, the authors have come to the conclusion that, in the phenanthrene crystal, the phase transition occurs at high order without changing the space group and with rather inconsiderable reconstruction of the structure at temperatures 45 72°C. This transition is associated usually with the changes in the planar molecular configuration of phenanthrene in consequence of the strong
-
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
335
intramolecular repulsion of the hydrogen atoms [8,10,11]. However, investigations of the temperature dependence of the infrared absorption spectra [51 and Raman spectra [13] in the region of the internal vibrations which, as a rule, are sensitive to these temperature changes do not give any evidence for the molecular deformation. The authors have observed only insignificant shifts and the decrease of the Davydov splittings of the bands at 498 and 850 cm-1 in the infrared absorption spectra a t the transition I + II 151. The Raman spectra in the region of the external vibrations at the transition I + 11exhibit a noticeable shift of several lines to the low-frequency region 1131 as shown in Ref. [13,141. This can not be explained by changes in phenanthrene molecule conformation. The analogous shift was observed for the band at 140 cm-1 in the FT infrared spectrum of phenanthrene 151. It was interesting to understand, whether the phase transition in the phenanthrene crystal and, in particular, the shift of low-frequency lines in the infrared absorption and Raman spectra, is relevant to the change in the molecular orientation. The possibility of such reconstruction of the structure results from the very loose molecular packing in this crystal whose dense packing fador 0.69 is the smallest in a series of polycyclic aromatic hydrocarbons. For this reason, we have calculated the changes in the molecular packing and in the external vibrational frequencies of the phenanthrene crystal at the phase transition, using the temperature dependence of the unit cell parameters measured in ill]. Since there are discrepancies in the data concerning both the number of observed lines in the low-frequencyRaman spectrum and their interpretation 113-161,the Raman spectra (0 - 150 cm-1) of a monocrystal in a polarized radiation and their dependence on temperature in the region 20 - 100°C (Tmelt = 104°C) [17] were investigated for comparison of the calculated results with the experimental data. In the low-frequencyRaman spectrum all nine external vibrations (5A + 4B) are active according to the selection rules. The Raman spectra of a monocrystal at 20°C obtained by us at Merent polarization conditions are shown in Fig. V.2. The frequencies of thirteen repeatedly observed lines are marked. Our data are in good agreement with the results of analogous measurements in 1161, excluding the lines 47 and 85 cm-1 which were not observed in Ref. [161 (Table V.1). The authors of 1161 and 1181, where the calculation of the external vibrational frequencies in the rigid molecular approximation was performed by the AAP method, have referred all low-frequency spectral lines to the external vibrations of the crystal. However, in the 'isolated phenanthrene molecule there are some low-frequency nonplanar vibrations displayed in the infrared spectrum of the solution at 124 cm-1 [13,15] and having frequencies 100 (A2) and 125 (B2) cm-1 (according to the calculation in Ref. [19]) which can be seen in the spectrum of the crystal in the external vibrational region. In particular, at low temperatures (-195°C) in the Raman
ZHIZHIN AND IvIUKHTAROV
336
B
32
60
A
I
I
B a(ba)c*
32
106
A
a(bb)c*
60 I
FIG. V-2. Raman spectra of phenanthrene single crystal at 300°K in polarized radiation.
195°C)in the Raman spectrum [15] the large number of additional weak lines are discovered; they were assigned to the nonplanar internal vibrations on the basis of the calculation of the total vibrational spectrum of a crystal by the AAP method and FG-matrices. The faults of this calculation was discussed by us in Chapter III Part 1). It should be also mentioned that the molecular crystals with the small packing factor have, as a rule, rather stable metastable modifications [l] whose formations can lead to displaying the new lines in the spectrum. "he Raman spectra of phenanthrene polycrystals at 20°C with M e r e n t thermal treatment of samples are given in Fig. IV.3. Phenanthrene was heated in the sealed copper cell up to 200 - 400"C,afterwards the melt was rapidly cooled by merging into liquid nitrogen. Depending on the starting temperature difference and cooling rate, this leads either to the complete transition of a sample to the metastable state (Fig. V.3, spectrum 3) or to the partial transition (Fig. V.3, spectrum 2) into this state which can exist during several days at room temperature.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
337
TABLE V.l Observed and Calculated Frequencies of Phenanthrene Single Crystal at 20°C Our Data (cm-1)
A
CalC.
CalC.
MI2
M I 1
Assignment
143 127 108
142 125 106
149.6 132.5 117.9
145.7 121.3 108.6
v1 (Q(A2)) ~2 (Q(B2)) v3(LW)
89 62 62
85 60 60 47
83.9 68.5 68.5 52.1
79.8 63.7 63.7 49.9
32
34.4
34.0
142
158.8
150.5
~8 (Q(A2))
60
109 101 84 60
119.9 99.9 80.8 49.2
112.4 92.5 76.9 50.1
vg (Q(B21, L ,) vio(L,) vll&,) vu (T,, L,)
31
32
25.6
22.3
exp. 1161cm-1
142 123 108 L, 97 90 L 62 61 47 41 L, 32
B
exp. cm-1
exp. 1151cm-1
140 127 115 98 L, 84 60 Lu 36 T, 32 L, 21
33
109 (99)
v4(Tu) v5(LV) ~5 &,I
v6(Tw) v7 (L,)
~ 1 (L 3,)
Our proposed interpretation of the low-fkequency Raman spectrum of the phenanthrene single crystal at 20°C is given in Table V.l and is in good agreement with the calculated frequencies, eigenvectors and relative intensities of the Raman lines (Fig. V.2 and V.3, Appendix III). The calculation was performed by the AAP method with the experimental structure determined by the neutron diffraction method at 20°C in 1103 and with consideration of the two internal nonplanar vibrations 100 (A21 and 113.5 cm-1 (B2) (see Part 1, Sec. III.B). The parameters M I 1 and AAP12 were used, and the average discrepancy between the theory and experiment was 6.4 and 8.8 cm-1, respectively. For this reason, all further calculations were performed with the
ZHIZHIN AND MUKHTAROV
338 32
20
20
100
100
v, cm-1
v, cm-1
FIG.V-3. Raman spectra of phenanthrene polycrystal at 20°C (1); (2) and (3) are obtained by merging the melt into liquid nitrogen and subsequent heating up to 2OOC; (4) is the spectrum calculated with the packing II for the high-temperature phase.
AAPI1 parameters. The relative intensities of the Raman lines were calculated in the oriented gas approximation using an effective polarizability tensor which was found by means of the dielectric susceptibility of the phenanthrene crystal obtained in Ref. [201.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
u
v
339
w
The distribution of intensities in the Raman spectrum of the crystal (Fig. V.3) was simulated by the s u m of the Lorentzian contours with line widths of 5 cm-1. According to the calculated eigenvectors (Appendix III), the external vibrations can be unambiguously referred to either translational or librational molecular vibrations with respect to certain inertia axis. The calculation of the spectxal intensities predicts the highest intensity for the Raman lines which correspond to vibrations with predominant librations of the molecule around its inertia axes, as it should be expected. The interaction between the internal and external vibrations is small (Part 1, Sect. III.B). The consideration of this interaction leads to a decrease of the external vibrational frequencies by only 2.4 cm-1. Thus, in the Raman spectrum of a phenanthrene single crystal at 20°C we have discovered and interpreted all nine lines related to the external vibrations which, in good approximation, can be regarded as predominantly translational or librational vibrations of rigid molecules. The line at 109 cm-1was referred to as an internal vibration by us, unlike the authors of Ref. 1163. The temperature dependence of the frequencies of the most intensive Raman lines of the phenanthrene single crystal is shown in Fig. V.4. As in Ref. 113,141, the anomalous shift of these lines was observed in the region of the phase transition temperatures (50 - 75°C). The analysis of the remaining lines was considerably complicated because of their broadening and the depolarization of the spectrum at temperatures close to the melting point. The investigation of the crystal structure in Phase II was performed by the minimization of the lattice energy with the Gxed unit cell parameters measured at 80°C in Ref. 1111. The authors have used the molecular geometry determined by the neutron -action method at 20°C [lo]. The crystal structure was defined by the orientation of one chosen molecule described by three Euler angles and by the position of its mass center described by two independent coordinates in the d c plane (Fig. V.l) for a given molecule conformation, unit cell parameters, and space group P21, z = 2. The c 0 0 r d i ~ t eof~ the other molecules are derived by the elements of the crystal symmetry. The equilibrium values of five of these coordinates were found by the steepest descent method, then they were ascertained by the Newton-Rafson method [2]. The summing radius is 6 A and the inconsidered part of the lattice energy was evaluated by means of the method discussed in Sect. II.A, Part 1. The test calculations at 20°C have shown that the calculated structure differs insignificantly from the experimental one: the mass center coordinates by -0.003;
340
ZHIZHIN AND MUKHTAROV
FIG. V.4a. Temperature dependences of the vibrational frequencies of phenanthrene crystal (A-modes):(1) experimental ones (unbroken points - FIR [51); (2) calculated in the quasi-harmonic approximation; and (3) calculated with the consideration of the anharmonicity.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
341
30 -
25
-
20
-
I
, t -
20
40
60
80
100
To,C
FIG. V.4b. Temperature dependences of the vibrational frequencies of phenanthrene crystal @-modes): (1) experimental ones; (2) calculated in the quasiharmonic approximation; and (3) calculated with the consideration of the anharmonicity.
ZHIZHIN AND MUKHTAROV
342 0.000; 0.000
A,and their orientation with respect to axes u, v, w by the Euler angles
0.22;0.71;0.00". This discrepancy determines the accuracy of the prediction of the molecular packing in the crystal; however, the predicted accuracy of its relative changes may be considerably higher. To seek the minima of the lattice energy at 80°C the authors have taken the experimental structure at 20°C [lo]as the starting one; they have taken also four other structures which differ fiom the experimental one by the mass center shift + AT and by the Euler angles (rotation around the axes parallel to a, b, and c*) (Fig. V.1): ATab 0.2 0.1 0.0 0.0
2 3 4 5
ATbb 0.0 0.0 0.0 0.0
ATpA 0.2 -0.3 0.0 0.0
Ad,deg. 20 20 13 0
Ab,deg. A$,*,deg. 0 20 0 30 10 0 -31 0
The calculation using the first four starting structures has led to a fast convergence to the molecular packing in the crystal structure corresponding to the minimum of the lattice energy at 20°C and Mered from the last structure by the orientation of the molecule with respect to its axes u, v, and w with angular shifts of 0.14;-0.82; -0.22"and by the mass center shifts along axes a, b, c* of -0.018;0.000; 0.000 (the packing I). The calculation with the starting structure 5 has led to the other local minimum which differs from the global minimum at 20°C by the molecular rotations around its axes u, v, w with Euler angles of -4.83;-31.82;and 24.55" and by the mass center shifts along the axes a, b, c* of -0.087, 0.000; and 0.178 A (the packing II). The directional cosines of the molecular inertia axes, the mass center coordinates and the lattice energy have the following values at the various minima of the potential energy:
A
V
W
-0.168
0.440 0.881 -0.176
0.230 0.079 0.970
U
V
W
U
a 0.868
A20" c =
b -0.467 C
-
0.869 0.444 0.217 -0.470 0.879 0.084 -0.154 -0.175 0.973
-
2.059
1789
0.011
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
343
-
u
V
W
0.819 -0.416 0.395
0.364 0.909 0.204
-0.444 -0.024 0.896
1954
1789
0.189
The energy of the lattice ( k d m o l ) is: 20"C, exp. 22.1 1211
20°C 23.55
80°C (I) 22.94
80°C (11) 22.09
Thus, the calculation predicts the existence of two possible molecular packings in phase 11 whose lattice energies differ only by 0.85 kcal/mol. It appears that the calculation of the frequencies using the packing I which differs insignificantly h m the experimental one at 20°C reproduces better the Raman spectral data (the average discrepancy is 7.4 cm-1) than the calculation using the packing II (the average discrepancy is 23.7 cm-1). "he relative intensities of the Raman spectrum lines calculated using packing I practically does not differ from the calgdation at 20°C; this is in good agreement with the absence of some noticeable changes in the intensity distribution in the low-frequency Raman spectrum at the phase transition. At the same time the calculation using packing 11predicts the drastic changes in intensities of the Raman spectral lines at this transition (Fig. V.3). Thus, from the comparison with the spectral data it follows that packing I better represents the real structure of phase I1 which differs rather insignificantly from the structure of phase I and has deeper minimum of the potential energy. It is possible that packing 11describes the structure of the metastable phase observed in this study (Fig. V.5). This is proved also by the similarity of the intensity distributions in the low-frequency Raman spectrum of this phase and by the calculations using packing 11(Fig. V.3, spectrum 2). The positions of the calculated Raman lines shifting to higher frequencies can be explained by the fact that the unit cell parameters of the metastable phase can have values which differ from the values used in our calculations. In order to investigate the peculiarities of the structural reconstruction in the closest vicinity of the phase transition we have performed the calculation of the temperature dependences of the molecular packing in the region 20 - 90°C. The minimization of the lattice energy was done by the Newton-Rafson method with the fixed unit cell parameters taken with a step of 5°C starting from 20°C. The temperature dependence of these parameters obtained in [113 (Fig. V.1) was used.
344
FIG.V.5. Packing of the phenanthrene molecules in a
crystal
calculated at
80°C.
The calculated dependence of the Euler angles on the temperatures is shown in Fig. V.6; these angles describe the changes in the equilibrium orientation of phenanthrene molecule with respect to its inertial axes at 20°C. This dependence has an anomalous character and correlates with the change of the monoclinic angle 6 (Fig. V.l) at the phase transition. The dependence of the mass center coordinates on the temperature has no peculiarities and is determined completely by the homogeneous deformation of the crystal lattice during the thermal expansion. At 80°C the structure calculated in such a way coincided completely with the packing I obtained earlier. The orientation of a molecule with respect to its axis V is the most sensitive; it has a maximal change of 0.5" in the region 55 - 75°C. This change begins long before the phase transition. The appropriate lines v5 and v~ relating to the external vibrations with the essential partition of the librations of molecules around this axis have drastic shift in the region 60 - 75°C (Fig. V.4).
STRUC-
AND PHASE TRANSITION OF ORGANIC CRYSTALS
345
0.8 -
0.6 -
0.4
-
0.2 -
0 -
20
40
60
80
100
T,"C
FIG. V.6. Calculated temperature dependence of the equilibrium orientation of the phenanthrene molecule with respect to its inertia axes U,V,W (I, > I, > I,,,).
The question arises whether these insignificant structural changes are sufficient for the quantitative description of the effects observed in the vibrational optical spectra. For this reason, we have undertaken the effort to calculate the temperature dependence of the total vibrational spectrum of phenanthrene crystal in the quasi-harmonic approximation in the phase transition region. The solution of this problem was obtained by the method described in Sect. III.B (Part 1) for every equilibrium molecular packing calculated at temperatures 20 - 90°C. The calculation of the intramolecular vibrations in the region of 3100 - 200 cm-1predicts the systematic decrease of the Davydov splitting values by 0.5 cm-1(in average) and by 2.5 cm-1(maximal one) for nonplanar vibrations at the transition I - II. The average shift of the Davydov doublets to the low-frequency side is equal to 1.3 cm-1 (according to the calculated data) and is maximal for internal vibrations with lowest frequencies. The investigations of the temperature depeddence of the infrared absorption spectra of phenanthrene crystal in the polarized radiation [51 have shown that the Davydov splitting values (4cm-1 at 20°C) and the shift of the band at 495 cm-1corresponding to the nonplanar vibration B2 of a molecule are the largest at the transition I - II. However, the calculation gives a considerably lowered value of the Davydov splitting for this vibration (0.9cm-1); it is changed by
MIZHINAND MURHTAROV
346
Av, cm-1
1
1.0 20
40
60
80
T,"C
FIG. V.7. Calculated temperature dependence of "gas-crystal" shift of the internal vibrational frequencies 476 (B2) (l), 517 (A2) (21, 469 (B1) (3) cm-1 of the phenanthrene molecule.
0.1 an-1at the phase transition, and this change may be associated with the large contribution of the interaction of the dipole-transition-dipole-transitiontype into Av; this interaction is not taken into account by the AAP model. On the other hand, the predicted maximal frequency shift of this vibration is in a good agreement with the experiment (Fig. V.7). The results of the calculation of the temperature dependence of the lowfrequency spectrum performed in the quasi-harmonic approximation are given in Fig. V.4. Analogous changes with the temperature were obtained also for vibrational frequencies in other points of the Brillouin zone. "he dispersional curves at 20 and 80°C, calculated with the consideration of two low-frequency internal vibrations in the direction [OlOI, are shown in Fig. V.8. As evident from Fig. V.4, the calculation suiliciently reproduces the temperature dependence of frequencies v3, v5 and vu; in particular, it predicts the most drastic shift (in agreement with the aforementioned) at the phase transition of the frequency v5 related to the libration of molecules around the axis V. However, a considerable discrepancy from the experiment was observed for frequencies v7, v10 and ~ 1 3 . It can be connected to the fact that in the instability region of the crystalline lattice the drastic change of the equilibrium molecular orientation
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
20°C
i6a
347
80°C
140
7 7
120
100
80
60
40
20
0
0.1 0.2 0.3 0.4 0.5
0
0.1 0.2 0.3 0.4 0.5
FIG. V.8. Dispersional curves in the direction 010 of the Brillouin zone calculated with the consideration of two low-frequency internal vibrations of phenanthrene molecule.
must be accompanied by a considerable increasement of its librational amplitude. Thus, in the closest vicinity of the phase transition one should expect a strong anharmonism of vibrations v7, v10 and ~ 1 which 3 leads to the inadequacy of the quasi-harmonic approximation in t h i s temperature range. The results of the calculation of the anharmonic corrections (a(3), a(4)) to the external vibrations at 20°C and 80°C in the independent oscillator approximation are given in Tables V.2 and V.3 and Fig. V.9. It appears that the main contribution is due to the fourth order anharmonicity which leads to the systematic increase
ZHIZHIN AND MUKHTAROV
348
TABLE V.2 Unharmonic Corrections to the External Vibrations of P h e n a n h n e a(3)
vham. cm-1 20°C ~~
vham cm-1 80°C
aC4)
>kcal/mol
dkcavmol
(mall)-= A-3 20°C 80°C
~~~
A
~
v3 v4 v5 v6
v7
B
(mau)" A4 20°C 80°C
v10 v11 vl2 13
110.7 80.9 64.1 53.7 34.6
98.6 72.6 57.9 49.0 30.7
5.01 -0.25 -1.23 -0.03 0.07
3.03 -0.49 -0.78 -0.02 0.02
10.00 2.41 1.68 0.31 0.06
3.94 2.35 1.70 0.38 0.10
97.2 77.9 50.3 22.8
84.8 70.1 45.5 21.3
--
___-_
10.02 4.28 0.31 0.55
9.73 3.20 0.28 0.51
__
-_-
of the external vibrational frequencies (Table V.3). The anharmonic corrections * 1 (especially for B-modes) are larger at 80°C than at 20"C, as a rule. The contribution of the anharmonicity is given to the vibration with the lowest frequency, ~ 1 3and , this leads to the increasement of its frequency by more than two times. This fact shows the inapplicability of the independent oscillators model for this vibration, which should be treated by the perturbation theory. The temperature dependence of the other external vibrational fkequencies calculated with the consideration of the anharmonism is given in Fig. V.4. The agreement between theory and experiment is improved for vibrations v3, v10 and v ~ This . is especially noticeable for the frequency v10 for which almost complete agreement between theory and experiment is achieved. On the other hand, the consideration of the anharmonism leads to the smoothing of the anomalous shifts of the frequencies v5 and v7 at the phase transition and enlarges their discrepancy with the experiment in phase II. Thus, it is probable that the anharmonic processes of phonon interactions begin to play a more important role at the transition to phase II. The three phonon decay process with two channels can be one of the probable processes, considering
TABLE V.3 Observed and Calculated External Vibrational Frequencies of Phenanthrene Crystal and Calculated Unharmonicity Shifts v, 20°C, cm-1 v, 80°C, cm-1 anh, 2O0C, c m l anharm, 8o0C,cm-1 packing I1 packing I vexp
"harm
Vanharm
108.6 79.8 63.7 49.9 34.0
110.7 82.2 65.5 51.1 34.7
96
VT
106 84 60 47 32
v10
101 85
~ 1 2
60 32
92.5 76.9 50.1 22.3
98.2 81.6 51.3 46.2
95
v11
~3 v4
A
v5 v6
B
v13
vexp
-53
__
27
__
53
27
Vharm
Vharm
Vanharm
Av(3)
Av(4)
Ad3)
Av(4)
127.4 102.8 86.2 38.9 18.8
98.6 73.3 58.8 46.1 30.4
99.9 76.9 63.1 48.1 32.5
-1.72 -0.02 -1.51 -0.00 -0.11
3.86 2.37 3.31 1.02 0.77
-1.30 -0.16 -1.18 -0.00 -0.01
2.57 3.81 5.45 1.99 2.07
120.4 106.4 52.1 27.9
83.3 70.6 46.0 21.2
93.2 76.4 47.9 53.9
---
__ __
5.69 4.68 1.24 23.86
9.94 5.76 1.86 32.68
350
110.7(a)
1.
Au
':\
64.1(A)
3
\
I
7-
2
4
Q
t'A
1
2
1 -10
-8
-6 -4
34.6(A) "
.'
0
-1
6
2
4
6
-Q
Q - 4 - 2 0
'
-1
I
-8 -6 -4 -2 0
-1
-- - 2
2
4
6
8
Q
FIG. V.9. Anharmonic component of the potential energy of the intermolecular interaction for various normal vibrations of rigid phenanthene moleculesin a crystal. 1-20°C;2-80°C.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
that 2 x v7 E
V and ~
2 x v13 E
351
V ~
~7 (32
4
v5 (60 cm-1)
~7 (32 m-1)
it must lead to decreasing frequencies v7, v5 and ~ 1 in 3 phase II a mr d in g to Sect. II.c (Part1). The anharmonic interaction of these phonons can be, in principle, studied quantitatively by the AAP method, if one constructs two-dimensional cross-sections U(Q5, 6 7 ) and U(Q5, Qu) of the crystal potential energy as functions of the normal C o o r d i ~ t e sand approximates them by polynomials, as it was done by us in the case of the one-dimensional cross sections in the independent oscillator model. However, in t h i s case it is necessary to fit too large of a number of parameters, and the problem of their unshiRed evaluation by the LSR method arises. The cross sections of the potential energy during a consistent symmetrical rotation of the molecules by angles A$a, A$b(A$e* = 0) around axes through their center of mass and parallel to axes a, b of a crystal are shown in Fig. V.10. The equipotential curves &om the bottom of the potential well with the step 0.3 k d m o l are shown there. Since the crystal axis c* practically coincides with the molecular axis W (Fig. V.11, and the vibrations refer predominantly to the molecular librations around the inertia axes U and V, Fig. V.10 characterizes simultaneously the two-dimensional cross section U(Q5,Q7) of the potential surface with respect to the normal coordinates of these vibrations. This cross section has a n irregular form (the "ravine" form) whose bottom becomes almost planar at the transition to the high temperature phase (Fig. V.11). Fig. V.11 shows that a tendency of forming a potential with two minima and inconsiderable barrier between them (3 l k d m o l ) in phase II is observed. The probability of t h i s process is not excluded considering the restricted accuracy of the lattice energy calculated by the AAP method. However, in any case the calculation predicts the librational motion of molecules around some axis perpendicular to the W axis (the axis of the smallest moment of inertia) with a larger amplitude (-25") in phase 11 and the strong interaction between the normal vibrations v5 (L,) and v7 (LJ. This leads to their softening at the transition I-II and to the broadening of the appropriate Raman lines which is especially noticeable for the frequency v7 (Fig. V.12). The other two-dimensional cross-sections, WU and WV, have comparatively regular form which does not change at the phase transition (Fig. V.lSa,b). This explains
ZHIZHIN AND MUKHTAROV
352
6
1 kh ca^
10
-a
8
-4
FIG.V.10. Cross-section of the potential surface (kcdmol) of phenanthrene crystal as the function of the Euler angles Aqa, Aob, (A$,*=O).
STRUCTURE AND PHASE TRANSITIONOF ORGANIC CRYSTALS
353
I u, kcaVmol -8
-4 I
,
0 ,
,
4 1
\
8 1
12
1
,
,
16 ,
A h grad
AA -18
-
, 20°C /
-20
.
-22 .
-24
FIG. V.ll. One-dimensional cross-section of the potential energy (A&* = 0) along the "ravine"bottom A-A (Fig. V.10).
the good agreement of the calculations with the experiment for the temperature dependence of fi-equenciesv3 (L), and v10 (L,). Thus, in accordance with the calculated and spectral data, the phase transition into the high-temperature phase of phenanthrene before the melting is related to the considerable change of the form of the crystal potential surface and to the activation of the consistent rotational motion of molecules with large amplitudes around their axes U and V. In conclusion it should be mentioned that the form of the lattice energy minimum must be very sensitive to the presence of admixtures in the crystal; it is probable that t h i s leads to smearing of the phase transition observed at normal conditions. In fact, as was shown in Ref. [221, the phase transition in the superpure phenanthrene crystal undergoes the special chemical and physical purification occurring very drastically at 72°C and is accompanied by the mechanical destruction of the sample. The introduction of anthracene admixture essentially simplifies the phase transition process.
ZHIZHIN AND MUKHTAROV
354
a)
k-----,
20
40
60
80
t,”C
32 cm-1
FIG.V.12. Temperature dependence of the width of the Raman line vg(A) and of the contour of a line v,(B).
B. Calculation of the Molecular Packing, the Low-Frequency Vibrational Spectrum and Their Changes at the Phase Transition in Pyrene Crystal The pyrene crystalline structure has been determined by the X-ray analysis method [231 and by the neutron a c t i o n method 1241 at room temperature. The
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
355
-@ it 8 0
-8
-4
0
4
crystal belongs to the space group P2 lk with four molecules in the unit cell. There
is the rare case of the packing in which the molecule with the symmetry mmm in free state has the general position in a crystal. Pyrene is characterized also by the unusual dimer structure: the pair of molecules connected through the symmetry center (Fig. V.14)plays the role of one
356
ZHIZHIN AND MUKHTAROV
FIG.V.14. The packing of molecules in a pyrene crystal (P21,Z = 4 123,241). Directions of the molecular inertial axes and the character of their shifts at the phase tranistion I - II (calculation) are shown. '*crystalline'*molecule. The distance between the molecular planes is 3.53 A. ~n Ref. [25] the authors have observed the excimer fluorescence of pyrene, i.e. the radiation of light during the radiational damping of the electronic excitation is localized at molecules with strong interaction. Pyrene is the classic object for studying this phenomena in the crystalline state. According to the thermophysical data 1261 the crystal of pyrene has a phase transition in the region 100 - 120°K. The phase transition is irreversible according to the birefringence study 1271 of a thin monocrystalline 61m (26 p); it occurs in the interval betwwen 117 - 129°K and has a temperature hysteresis which is associated with the change in the domain structure. In some cases, when one domain spreads all over the whole crystal, one does not observe the anomalies in the transition region. The authors of 1271 have suggested that the phase transition of a high order takes place in pyrene. There is little information about the low-temperature phase (phase II) and about the structural aspects of the phase transition. The X-ray analysis is complicated due to the great destruction of the single crystal at its cooling below the phase transition temperature, as was shown in Ref. 128,291. By analyzing the changes in the diffraction picture, the authors of 128,291 concluded that the space
SI'RUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
357
group does not change at the phase transition, and the crystal structure changes insignificantly. The parameters of the unit cell were determined by the X-ray analysis method (a) and by the joint electron microscopy and electron difhctometry methods (b):
(a) (b)
TS
4
293 98 98
13.649 12.31 12.4
bJ 9.256 10.03 10.2
4 8.470 8.27 8.02
Btdem 100.28 96.69 93.86
[23] [29] [29]
Thus, the situation for the pyrene case is considerably analogous to the situation in the aforementioned case of crystalline phenanthrene. It is usually assumed that in phase II the dimer structure is maintained 128-301. It is confirmed by the absence of changes in the excimer fluorescence at the transition [30]. It is interesting to note that in [281the crystalline modification of pyrene in which there is no excimer fluorescence was obtained by growing the crystal from the solution in ether. In spite of a large number of efforts the phase transition in this modification was not discovered. It is very interesting to study the phase transition in pyrene by means of the vibrational spectra, which are very sensitive to rather inconsiderable changes in the molecular packing, as was shown by us in the case of the crystal of phenanthrene. The low-ftequency Raman spectrum of the polycrystalline solid (0- 130 cm-1) was measured by varying the pressure and temperature within the limits of 0 10 kbar (293°K)and 12 - 340°K (0 kbar), respectively [311. At 110°K the authors have observed a drastic change in the Raman spectra which they have interpreted 191 in terms of the disappearance of old lines and the appearance of the new ones. The transition was fast and reversible, the hysteresis was not discovered. The analogous change in the Raman spectnun was observed at a pressure of 4 kbar and T = 293°K. The dependence of the line frequencies on the pressure measured in [31]is shown in Fig. V.15. The observed changes in the Raman spectrum can be regarded also as the result of the drastic line shift at the phase transition, i.e. the line in the spectrum of phase 11corresponds to the line in the spectrum of the phase I. This is proven by the set of data showing that the reconstruction is small, and the similarity between the Raman spectra is large (Fig. V.16). As is seen ftom Fig. V.15, the frequency shift is Merent in various regions of the spectrum; it can be used for obtaining detailed information concerning the phase transition features and the structure of phase II. Some necessary information concerning the interpretation of the spectra in terms of the symmetry types and vibration forms is needed tirst.
-
ZHJZHIN AND MUKHTAROV
358
v, crn-1
130 120
110
100 90 80
70 60 60 40
30
r”
J//
/@--
/A
20
10 1
1
2
I
4
6
8
1
P,kbar
FIG. V.15. Dependence of the Raman line frequencies of a pyrene crystal on pressure 1311.
In the Raman spectrum there are twelve active normal vibrations (6% + 6Bg),and nine normal vibrations (5Au + 4Bu) are active in the infrared absorption = 0. At mom temperature these spectrum according to the selection rules at spectra of a single crystal are studied using the polarized radiation [321; the calculation in the rigid molecular approximation was performed by the AAP method in [18,33,34]. All twelve lines in the Raman spectrum observed at frequencies below 150 cm-1 were referred to by the authors as the external vibrations in a crystal. However, as was shown in Chapter 111(Part l), in the vibrational spectrum of a free molecule there is the low frequency vibration at 96 cm-1, which may be displayed in the spectrum in the region of the external vibrations. Furthermore,
Sl'RUCTLTRE AND PHASE TRANSITION OF ORGANIC CRYSTALS
359
17
56 ag
T = 293 K
a)
73
FIG. V.16. Raman spectrum of a polycrystalline pyrene solid (a)phase I; (b) phsae It. Vertical lines: the relative intensities of the lines calculated in the oriented gas approximation.
the additional line at 17 cm-1was observed in the polycrystalline Raman spectrum 1311. We have observed the analogous line in the Raman spectrum of polycrystalline solid at 16 cm-1 (Fig. V.16a). To make the assignments of the Raman spectrum more precise we have performed the calculation of the frequencies
ZHIZHIN AND MUKHTAROV
360
TABIX V.4 Interpretation of the Low-FrequencyRaman Spectrum of Pyrene at Room Temperature Our Assignments (cm-1) Vexp. cm-1
%
B,
Vem. 1101
vcfilc.
170
_-
170 127
175 138
internal vibrations
127 95 77 56 (46) 30
95 77 56 (46) 30 17*
105 92 58 46 37 19
external vibrations
169
--
169 126
175 129
internal vibrations
126 95 67 56 41 30
95 (77) 67 41 (30)
109 91 69 45 34 24
external vibrations
__
*We have observed t h i s line also in [321. and eigenvedors of a pyrene crystal at = 0 by considering two low-frequency intramolecular vibrations and also the relative intensities of the Raman spectral lines in the oriented gas approximation. The optimized structure was used; it dif€ers &om the experimental one by the molecular mass center shift of 0.23 A and by the angle of rotation of 5" (the details are given in Chapt. II, Part 1 , ) . Since the polarizability of a molecule is not known, the calculation of the relative intensities was performed by varying the optical anisotropy fador y. The best agreement with the experimental results was obtained for y = -1; it corresponds to the same polarizability of a molecule with respect to the axes V and W (Fig. V.14). The calculated results are given in Fig. V.16a; the distribution of the intensities was simulated by the sum of the Lorentzian contours with a width of 5 cm-1. Our interpretation of the low frequency Raman spectrum of pyrene at mom temperature is given in Table V.4. All calculations performed by the AAP method and discussed here were made with the Williams AAPh parameters. As shown in Table N.4, the consideration of the intramolecular vibrations and the analysis of the relative intensities lead to the essential reconsideration of the assignments of the pyrene Raman spectral lines. In particular, the lines at 127 ($1 and 126 (B& cm-1 were assigned to the intramolecular vibrations. All results of the calculations of the frequencies and vibrational forms are given in Appendix III.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
361
It is necessary to express the eigenvectors in terms of basic coordinates which describe the peculiar packing of the pyrene molecule in order to obtain a more obvious interpretation of the spectrum and to establish its connection with the crystal structure. In particular, the calculated interaction energy for two dimer molecules (14.8 k d m o l ) seemed to be three times as large as the energy of the interaction of other pairs of molecules. It permits one to use the coordinates describing translational (T)and librational (v) shifk of dimers (as a whole) and relative shifts (x) of molecules inside the dimer as a basis. z(1), $(I) and f(2), are the initial coordinates of shifts of two inversion bound molecules in local systems of their inertial coordinates. The problem is to transform the eigenvectors calculated in this basis to the eigenvectors in the basis - of dimer coordinates T, W and 'ii. Since the consideration of intramolecular vibrations leads to a rather insignificant change in the frequencies and external eigenvectors, we are able to solve the problem in the rigid molecule approximation. In t h i s case, six degrees of fkeedom of the dimer (the total number is 12)correspond to its motion as a whole, and six dimer degrees of fkeedom correspond to intradimer shiRs of a molecule. The factor-group c2h can be represented as c2h = &OC1, where Ci is the symmetry point group of a dimer in a crystal, and C2 is the commutation group (see Chapter I, Part 1). Hence, one follows the dimer symmetry coordinates:
where $g and q p are six-dimensional column-vector of the appropriate centrosymmetric and noncentrosymmetric coordinates of shih of the dimer I. It is obvious that
ZHIZHIN AND MUKHTAROV
362
where F are the librational shifts of dimer I as a whole, TI are the translational ones; f g and ZAu are the three-dimensional column-vector of the intradimer molecular shifts. The appropriate correlation diagram has the form:
Position of Molecule in a Crystal
Factor-Group of a Crystal
Position of a Dimer in a Crystal
(3t, 30)A
Thus, three generalized intradimer coordinates must be symmetric with respect to the inversion, and three such coordinates must be antisymmetric. To simplify the problem it is necessary to introduce the antisymmetric intradimer coordinates as the relative, infinitely small rotations of molecules around their inertial axes in various directions
The symmetric coordinates may be introduced by several methods, for example, as a change of the distance between the mass center of two dimer molecules, and as two relative shifts in the perpendicular direction. However, two last coordinates are chosen ambiguously, and their description by vibrational forms is not so obvious. For this reason, we have chosen coordinates describing the relative translational shifts of two molecules in the direction of their inertial axes. Here it is necessary to extract the molecular shifts during the rotation of the dimer as a whole from the relative shifts in the laboratory coordinate system. Considering that the inertial axes of the dimer molecules are parallel, we have obtained
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
363
where square brackets indicate the vector product of the axial vector and the radius vector R of the mass center of the molecule I with coordinates given in the inertial axes system of this molecule. This expression may be written in the matrix form
67-3) where
0 -Rw
Rw
-%
0
Ry
-%
R, 0
To find the connection between the coordinates of dimer shifts (the dimer as a whole) with the starting coordinates, we use the expressions for the total momentum and angular momentum of a dimer
(x,
where primes indicate that the shifts of separate molecules $1, their inertia tensor .i:, and the radius-vector of the mass center of the molecule I, are given in the system of the mass center and inertial axes of the dimer; i is the diagonal matrix of the main inertia moments of the dimer; and m is the mass of the molecule. Proceeding &om the derivatives to the infinitely small shifts and to the coordinates of shifts of separate molecules in their coordinate systems, we obtain
where is the matrix of directing cosines between the inertia axes of a dimer and the molecule I. To create an analogy with the theory of molecule vibrations one can represent these expressions in a matrix form (see Eqs. (V.11, 67.31,(5.4b))
67.5)
ZHIZHIN AND MuKH!llARov
364 where
E I
-R
E
i-lit
-&lit
i-l&
E
0
-R -E
B=
1
where E and 0 are the unit and zeroth matrices with 3x3 dimensions, respectively. The inverse matrix of reduced masses of new generalized coordinates is
where
Hence, follows the matrix
The eigenvedors, i h ,in the mass-weighted dimer coordinate basis can be calculated now with the use of the k n o w n eigenvectors, i, given in ordinary coordinates by means of the formula:
The calculated results at 293°K (Phase I) and 98°K (Phase II) are given in Tables V.5 and V.6. The frequencies and eigenvectors given in brackets were
exp. cm-1
TABLE V.5 Observed and Calculated Frequencies and Eigenvectors of a-pyrene crystal at 293 K calc. Au Au cm-1 XU x,Au x, Tudim Tvdim Twdim Assignment
Raman Active Modes 95
106.1
0.563
-0.359
0.334
(80.0)
0'481
(0.395)
(0.487)
(0.779)
Ag
XU
9XWdim
91.7
-0.753
-0.552
0.108
-0.059
0.288
0.241
xuA g A g
56
59.4
-0.152
0.160
0.362
0.591
-0.581
0.365
XUdim? %?dim
(0.776)
(-0.567)
(0.278)
-0.630
0.353
-0.461
(-0.493)
(0.828)
(-0.268)
46
47.7
0.276
-0.345
0.103
(45.1)
1
0.347
77
(56.2)
$.
0.298
2 X "
Xudim, Xwdim
xvA g, xAwg, w
30 71
39.0 19.1
-0.123 -0.036
0.640 -0.173
0.440 0.734
-0.393 -0.094
0.356 0.642
0.304 -0.078
2
3
4
5
6
7
8
9
95
109.8
0.500
-0.284
0.451
0.394
0.288
0.476
&Ag > XVdim
(0.560)
(0.381)
(0.735)
0.051
0.101
0.148
0.257
&&,hAg
(82.2) 77
91.1
-0.771
-0.548
X,Ag ,Xvdim
10
0
aa aa
TABLE V.5 (continued) 1
2
3
4
5
6
7
8
9
10
67
69.1
-0.365
0.551
0.584
0.430
-0.154
-0.101
&,A g) XAwg9Xudjrn
41
44.6
-0.017
-0.042
0.242
-0.149
0.765
-0.576
&dim! Xwdim
(-0.568)
(0.823)
(0.007)
-0.510
0.365
o.596
(-0.602)
(0.422)
(0.678)
(45.9)
%! 30
34.3
-0.138
0.473
0.034
(40.4)
--
24.4
Xwdirn9 & Ag
0.058
-0.274
0.627
-0.607
-0.380
-0.081
XwAg Xudirn
-0.019
0.715
0.687
0.021
0.005
0.132
X,
9
Infrared Active Modes
102
AU
113.0
71
81.3
0.228
-0.656
0.709
-0.074
0.001
-0.098
_-
49.3
0.963
0.171
-0.145
0.136
-0.014
-0.060
__
46.7
0.002
-0.174
0.010
0.572
-0.006
0.803
(0.547)
(-0.002)
(0.836)
(49.0)
Au
Au 9
Au
xW ,x, xuAu
~
w
Au
Twdjm9 Tudim
z
1
2
3
4
5
6
7
8
9
10
19.7
-0.142
0.016
0.066
0.803
-0.095
-0.658
Tudjm, Twdim
(21.8)
BU
(0.833) (-0.096) (-0.545) Au
Au
105
112.0
0.020
0.651
0.754
-0.007
-0.086
0.004
xw ,xv
89
78.5
0.106
0.702
-0.642
-0.023
-0.288
0.015
X,
-_
51.7
0.189
0.253
-0.116
0.076
0.937
-0.048
Tvdjm
(54.8)
--
31.2
0.976
-0.139
0.077
(0.081)
(0.995) (-0.051)
-0.012
-0.042
NOTE: u, v, w and q j m , Vdjm, Wdjm correspond to inertial axes of molecules and dimers. Iwdim;experimental data were taken from Refs. 131,321.
0.008 2,
Ag
Au ,xW
Au
xu
> 2, > zw; Iudjm > Ivdim>
TABLE V.6 Observed and Calculated Frequencies and Eigenvectors of a Pyrene Crystal at 98 K Raman Active Modes (98K, Phase 11)
108
126.0
-0.432
-0.541
0.355
(95.4)
Oa213
(0.304)
(0.928)
(0.216)
99.3
0.883
-0.295
0.115
-0.077
0.340
0.037
73
77.7
0.070
0.385
0.284
0.490
0.244
-0.683
(0.413)
(0.076)
(-0.908)
0.408
-0.053
44 37
55.1 42.7
0.151
0.491
0.291
0.078
-0.490
-0.283
0.726
-0.377
-0.250
0.778
(0.859) -0.146
(50.7)
1
0.623
91
(71.6)
Ag
0.197
Oe6”
Wvdim9Xv AgAg
xuAg Wudim, Wvdim
Ag
WWdim’, Xv
?
WUdim
Ag
-0.002
Wudim,, Xv
(-0.365) -0.540
(0.360) -0.111
Xw Ag,Vvdim
29
30.3
0.014
2
3
4
5
6
7
8
9
10
122.4
-0.425
-0.505
0.350
0.245
0.552
0’276
AgAg WVdim’h
0.837
-0.479
0.002
(0.334) -0.076
(0.881) 0.131
(0.335) 0.220
108
(88.7) 91
100.9
?x,
x,A g,x, Ag
5
a
77
81.2
0.318
0.523
0.702
0.313
0.163
-0.059
50
59.6
-0.090
-0.069
0.206
0.214
-0.684
0.657
(0.178) (-0.408)
(57.8) 34
39.1
-0.055
0.323
-0.743
0.263
0*5l1
(0.239) -0.339
(0.239) -0.424
-0.088
-0.366
0.568
(-0.926) -0.488
--
30.3
WVdim’ WWdjm
(0.895)
0.098
(46.3)
xw A g,Wv A g
WUdimt WWdjm
xw Ag
Wudjm’ Wwdim
Infrared Active Modes
~
AU
Au
Au
Au
Au
--
129.7
0.072
0.126
0.005
-0.130
0.750
0.633
__
xV ,xw
83.4
-0.114
-0.021
0.006
-0.208
-0.639
0.731
xw , x ,
--
68.3
0.762
0.672
0.003
0.063
-0.171
-0.001
__
(70.7)
(0.726) (0.687)
--
49.8
0.921
-0.022
0.206
-0.266
0.193
Tudim, Twdim,
(0.004)
0.034
AU
xu
w
4
TABLE V.6 (continued)
0
Inf'rared Active Modes
__
BU
27.6
-0.625
0.699
0.100
(31.5)
(-0.684)
(0.722)
(0.106)
--
131.1
-0.006
0.007
--
98.5
0.007
--
56.0
0.069
(56.9)
--
30.0
0.016
-0.149
-0.008
-0.154
0.751
0.636
-0.008
0.103
0.110
-0.618
0.771
-0.078
0.967
-0.214
0.091
-0.026
TVdim
0.958
0.212
0.008
xuAu
(0.070) (-0.080) 0.014
Twdim,Tudim
-0.297
-0.015
Au
Au
xv ,xw Au Ag xw ,xv
(0.994) 0.190
NOTE: u,v, w and Udjm, Vdjm, Wdjm correspond to inertial axes of molecules and dimers. z, > z, >; ,7 experimental data were taken from Refs. [31,32].
Iudim> Ivdim > Iwdim
B
8
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
371
calculated in the rigid h e r approximation, i.e. two molecules bound by the inversion were considered as one rigid molecule. The assignments of the frequencies for the low-frequency Raman spectrum of the polycrystal measured at 100°K (phase II) in Ref. 1311 (Table V.6) were performed by us with the use of calculated frequencies and the relative intensities (y = -1) of the Raman lines (Fig. V.16b); the frequencies were calculated with the consideration of intramolecular vibrations. All results of the calculations for phase II are given in Appendix IXI. Since the structure of the low-temperature phase is not known, one must first seek the optimal packing of molecules in this modification. The lattice energy minimization was performed by varying all six independent coordinates of the starting molecule with fixed parameters of the unit cell determined from the X-ray analysis at 98°K in Ref. [29] and with the assumed space group P2+, z = 4. The sought minimum was obtained by the steepest descent method at the first stage, then the molecule coordinates were verified by the Newton-Rafson method. Three structures were chosen as the starting ones: one of them was the experimental obtained at 293°K [23], and the others have the molecules rotated around the U axis by the angle k30" (Fig. V.14). In all cases the fast convergency to the single structure was achieved; its mass center coordinates and the matrices of directing cosines of the inertial axes of the molecule have the following values: T=293"K (I,
I'
a -0.704 0.700 c * 0.124
=b
I,
T = 98°K
),1
-
"
W
0.529 0.632 -0.567
0.475 0.333 0.815
hc= 1.342; - 1215; 0.30781
U
a -0.771 A = b 0.612 C * 0.178
V
W
0.426 0.703 -0.570
0.474 0.364 0.802
kc= 1.221; -1.348;
0.235A
The calculated value of the lattice energy in phase II(26.96 kcdmol) at 98°K appears to be more than the value at room temperature by 0.73 kcdmol. The h e r character of the structure of the phase II remains completely. This is proven by the data of the excimer fluorescence. The interaction energy of two molecules is almost not changed (14.65 and 14.81 kcal/mol) in phases II and I, respectively.
372
ZHIZHIN AND MUKHTAROV
Some decrease in this energy is caused by the increase of the dimer molecule repulsion, which is due to the decrease of the distance between their planes by 0.07 A; it is in agreement with the fluorescence shift to the long wavelength region [301. The change of the structure at the transition I (293°K) - II(98"K) can be expressed in terms of the h e r rotation by the Euler angles (1.93; -1.03 and 7.04") around the axes coinciding with the axes U, V, and W of a molecule and directing through the maas center of a dimer, and in terms of the relative shift of molecules in a dimer in these directions by 0.07; -0.08 and 0.16 A. From Tables V.5 and V.6 it follows that the noncentrosymmetric vibrations which are active in the infrared spectrum are practically completely divided by intra- and interdimer vibrations in both phases. The vibrations which are active in the Raman spectrum are considerably mixed. The doublet 77 ($1 - 77 (B,) cm-l (293°K according to the calculation 91.6 90.9 cm-1)in phase I and 91 ($1 - 91 (B,) cm-1 (lOO"K,calculated data: 99.3 - 100.9 cm-1)in phase 11can be referred to as the intradimer vibration corresponding to the change of the distance between the The lines at 56 (A&, 46 ($1, 41 (B,), and 30 parallel planes of molecules (X?). (B,) cm-1 in phase I and the lines at 73 ($1, 50 (Bg),44 ($1, and 34 (B,) cm-1 in phase 11can be referred unambiguously to the librations of dimers as a whole. The eigenvectors change insignificantly at the phase transition, i.e., every normal vibration of one phase can be compared with the analogous vibration of another phase. Our assignments in terms of the symmetry types and eigenvectors of experimental dependencies of the Raman spectrum on the temperature and pressure ([31], Fig. V.15), which are suggested on this basis, are shown in Figs. V.17 and V.18. Thus, the changes in the Raman spectrum observed at the phase transition I + 11in [31] correspond to the drastic shift of all lines to the side of high frequencies. Two lines with frequencies 17 (4)and 56 (4) cm-1 (at room temperature) have the largest shift; they are twice as large as the shift of other lines. To study the reconstruction character of a pyrene crystal structure in the transition point, we have performed the calculation of the temperature dependence (300 - 90°K) of the molecular packing and normal vibrational frequencies (with the consideration of two internal vibrations) in phase I in the quasi-harmonic approximation. The optimal molecular packing was calculated by the lattice energy minimization by the Neuton-Rafson method with the fixed parameters of the unit cells given at temperatures &om 300°K with the step being 20°K The temperature dependence of unit cell parameters was found by means of the tensor, a,of thermal expansion measured in the range of temperatures 130 - 300°K by X-ray *action methods in [35]: all = 67 x 10-6, a22 = 101 x 10-6,a= = 82 x 10-6K-1and 4 = 34"; here % are the main values of the tensor a; I$is the angle characterizing the main axis of the tensor, I, with respect to the direction a in the unit cell. It was also
-
S T R U C m AND PHASE TRANSITION OF ORGANIC CRYSTALS
200
400 300 v, cm-1
100
I
--
I
,
373
I
/
A,
14C
.
130
/-:
140
130 / I
--
/
120
120
I I
I
I
I
/
110
110 I
I--
100
100
90
90
80
80
70
70
/
60
50
..
40
,
,/‘
I
,
60
50
, I
L-.
40
e
30
30
20
20
10
10
0 400 300 200
100
0 400 300 200
100
0
T,K
Fig. V.17. Calculated (1) and experimental (2) dependencies of the Raman spectral line frequencies of a pyrene crystal on the temperature.
ZHIMIN AND MUKHTAROV
v, Cm-1 ~
140
. 130 120
110 100
'90 80
'
70
50
40
0
2.25
4.5
6.75
0
2.25
4.5
6.75
P, kbar
FIG. V.18. Calculated (1) and experimental (2) dependencies of the Raman spectral line frequencies of a pyrene crystal on the pressure.
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
375
0.18
0.14
0.10
0.06 4
3 a
0.02 0
-0.02
-0.06
I -0.10
I
'\
\\
I
I FIG.V.19. Calculated temperature dependence of the relative shift of the pyrene dimer molecule equilibrium position in the direction of axes U, V, and W of the molecule 11. assumed that the monoclinic angle p does not depend on the temperature. It appears that with decreasing the temperature of phase I the equilibrium orientation of dimers remains practically constant up to T and changes by the jump of the Euler angles: 1.97;-1.033;and 7.06" around directions U, V and W. The temperature changes of the mass center coordinates of the dimer molecule 11,given in coordinates of the mass center and inertial axes of the molecule I (Fig. V.14)and describing the deformation of a dimer, are shown in Fig. V.19. The most relative shift of 0.12 A belongs to the molecules in the direction of their long axis W. The distance between the molecular planes practically does not change (the U axis is perpendicular to the planes of molecules). Thus, the change of the s k c t u r e at the phase transition shown schematically in Fig. V.14 can be regarded, in general, as the opposite sliding of the molecular planes along the a x i s W accompanied by the
ZHIZHIN AND MUKHTAROV
376
dimer rotation around the axis (dimer as a whole), w ‘ch is close to the C axis of the unit cell. The generalized intradimer coordinate which corresponds to the abovementioned deformation of a h e r , gives the main contribution to the abovementioned normal vibration with the frequency of 17 cm-1 undergoing the most drastic change at the phase transition, which is accompanied by the noticeable shift of the line 56 cm-1($1 relating to the vibration of a dimer as a whole (Table V.5). It should be mentioned that the extrapolation of the temperature dependence of the structure of phase I below 110°K (shown by the dashed line in Fig. V.19) leads to the drastic decrease of the distance between the planes of molecules. The repulsion of two dimer molecules increases sharply, and the conservation of such molecular packing below 110°K becomes energetically disadvantageous. The calculated temperature dependence of the frequencies of the normal vibrations in phase I is shown in Fig. V.17. Excluding the abovementioned lines at 17 and 56 cm-1, the calculation does not reproduce the line shifts both in value and sign at the phase transition. The rate of the changing of the frequencies with the temperature does not reproduce either one, especially for high-frequency vibrations for which the calculation gives the frequency values considerably higher at low temperatures. It can be associated both with the use of inaccurate values of the unit cell parameters and with the anharmonicity whose influence may be considerably large, as was shown in Chapter 111, Part 1 (naphthalene as an example). It can be clarified by comparison of the temperature dependence of frequencies calculated by the quasi-harmonic approximation with the frequency dependence on a pressure measured in 1311, since in both cases the frequency shifts are caused only by the change of the unit cell volume. In fact, according to the data of Sect. II.C, Part 1, the following relationship is valid in the quasi-harmonic approximation
9 Xd,
is the compressability of a crystal. To perform such a comparison one needs to know the dependence of the unit cell volume on the pressure, i.e. the compressability K whose data are absent. The value of K can be found from (Table V.7), if the value of
‘T
(atnvi)€l
is taken from the calculation, and the value of
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
377
TABLE V.7 Calculated and Experimental External Vibrational Frequencies of a Pyrene Crystal at Various Temperatures and Pressures Phase I ( a l > Phase II ( a - 1 ) 293°K 0 kbar
%
B,
293°K 4 kbar exp.
110°K 0 kbar exp.
calC. 126.9 111.2 70.7 55.3 44.9 24.4
117 100 78
131.8 110.8 85.7 55.7 42.0 26.5
117 100 87 49
exp.
calC.
95 77
105.0 91.6 57.7 45.9 36.7 18.5
113 95 66
104 84 61
38 20
36 17
108.8 90.9 68.5 44.0 33.0 21.4
113 95 82 49
104 84 70 46
56 (46) 30 17 95 77 67 41
--
--
-_
--
_-
_-
--_
293°K 4 kbar exp.
__
41 33
--
--
1OO"K, 0 kbar exp.
calc.
108 91 73 44 37 29
125.8 99.2 76.5 53.1 41 29.5
108 91 77 50 34
121.6 100.7 80.6 59.5 37.8 25.1
--
9.2 12.4 20.5 5.8 9.2 A is the root mean square deviation of calculated frequencies from the experimental ones. A
(y)T
is taken &om the experimental data [31]. Using the data of Table V.7 in
which there are the values of the calculated and experimental external vibrational &equencies of a pyrene crystal at various temperatures and pressures we get k - (4L3 k 8.9)deg/ kbar -
P-
For $ = 250 x 10-6K-1 [35] we have found the average value (in the interval 0-4 kbar) of the compressability, I
*
K T=--p+293"K $ The evaluation shows that at 4 kbar, i.e. in the vicinity of the phase transition I - II, T = (127.8 * 35.6)K; this value is close enough to the transition temperatures Tph.tr.
ZHIZHIN AND MUKHTKROV
378
250
300
350
temperature, K
FIG.V.20. Temperature dependence of the line width and second momentum of NMR 1363. = 110°K. Assuming that at p = 4 kbar, T = 293°K and p = 0 kbar, T = 110°K the unit cell volume is the same, and it is linearly dependent on the pressure in the interval 0-4 kbar, we have performed the comparison between the calculated frequency dependence on temperature and the measured data 1311 at 293°K and increased pressures (Fig. V.18). It appears that in this case the calculation is in better agreement with the experiment than in the case of the data h m temperature measurements (Fig. V.17). As is seen fkom Table V.7, the root-mean-square deviation of the calculated and experimental frequencies A (A = 5.8 m-l)in phase 11 at 0 kbar, T = 293°K is more than 1.5 times smaller than at 0 kbar, 293°K at which the structure is determined experimentally; this confirms the reality of our molecular packing in phase 11and shows the strong anharmonicity of vibrations in phase I. The latter may be associated with the anomalous temperature dependence of the N M R line width and relaxation time TI in the region of 255 - 285°K discovered in 1361 (Fig. V.20). This anomaly is associated with the activation of random reorientations of molecules around the axis perpendicular to the r plane, with the energy of activation equal to 13.6 k d m o l . As follows &om Fig. V.19 (the curve U),while increasing the temperature from 210"K,the distance between the molecule planes begins to grow; it is accompanied by sufiiciently drastic shifts of the h e r molecules in the direction of the W axis (Fig. V.14). In the vibrational spectrum the calculation predicts the change of the slope of the temperature dependence of the frequencies at this temperature, which is especially noticeable for low-frequency vibrations. These data show the possibility of the existence of the smeared phase transitions in a pyrene crystal at temperatures above 200°K which are associated
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
379
with the dimer deformation. The confirmation of this is the insignificant anomaly of the heat capacity at 210.8"K, which was observed in the procedure of the thermophysical measurements [361. Thus, we have performed the calculation of the molecular packing, frequencies and eigenvectors, and the relative intensities of the Raman lines at different temperatures by the AAP method; the results of these calculations permit one to assign the data regarding the low-frequency Raman spectrum of two phases of the pyrene crystal and its dependence on the temperature and pressure. The reconstruction of the crystal structure at the phase transition agreed with the calculated and observed data and consists of the dimer deformation by means of the shift of parallel planes of molecules by 0.12 A along their axis of smallest inertia moment and of the dimer rotation around the axis close to the C d s of the unit cell by an angle of -7". The existence of the smeared phase transition of the high order at T = 210°K associated with the analogous dimer deformation was predicted. It was shown that the low-frequency vibrations of the pyrene crystal are considerably anharmonic in the region 20 - 110°K1371. In conclusion it should be mentioned that our suggested method of the calculation of the eigenvectors allows us to describe more obviously the crystal normal vibrations, since it gives direct information concerning the relative motion of total groups consisting of two or more molecules, excluding the relative amplitudes of separate molecular shifts. This can be rather useful for the interpretation of low frequency molecular spectra of crystals consisting not only of such dimer formations (as in the case of the pyrene crystal or the crystals with hydrogen bonds), but also of crystals with the ordinary packing of molecules.
ACKNOWLEDGEMENT The authors wish to acknowledge our colleagues, Drs. Yu. N. Krasjukov, V. N. Rogovoj and N. V. Sldorov, who made considerable contributions to this book.
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5. G. N. Zhizhin, B. N. Mavrin, and V. F. Shabanov, "Optical Vibrational Spectra of Crystals", USSR, Moscow, Nauka, (19841,236pages. 6. E.I. Mukhtarov, in: "Condense Media Spectroscopy", USSR,Kiev, Naukova Dumka, (19881,p. 88. 7. S.Matsumoto, Bull. Chem. Soc. Jpn., 39,1811 (1966). 8. D. H.Spielberg, R. A. Arndt, and A. C. Damask, J. Chem. Phys., 54,2597 (1971). 9. I. Trotter, Acta Cryst, 16,605(1963).
STRUCTURE AND PHASE TRANSITION OF ORGANIC CRYSTALS
391
10. I. Kay, Y. Okaya,and D. E. Cox, Acta Cryst, B27,26 (1971). 11. S.Matsumoto and T. F'ukuda, Bull. Chem. Soc. Jpn., 40,747(1967).
12. E.T. Peters, A. F. Armington, and B. Rubin, J. AppL Phys., 37, no. 1, 226 (1966). 13. L. Colombo, Chem. Phys. Lett., 48,166(1977). 14. L. Colombo, I. Blazevic, and G. Baranovic, Fizika, 9,11(1977). 15. I. Godec and L. Colombo, J. Chem. Phys., 65,4693(1976). 16. A. Bree, F. G. Salven, and V.V.B. Vilkos, J. MoL Spectrosc., 44,292(1972). 17. G.N. Zhizhin, Ju. N. Krasjukov, E. I. Mukhtarov and N. V. Sidorov, h e p rin t No. 6, Institute of Spectroscopy, USSR Acad. of Sci., (1987),35pages. 18. G. Fillipini, C. M. Gramaccioli, and M. Simonetta, J. Chem. Phys., 59,5088 (1973). 19. G. F'illipini, M. Simonetta, and C. M. Gramaccioli, MoL Phys., 91, 445 (1984). 20. P.S.Bounds and R.V. Munn, Chem. Phys., 24,no. 3,343(1977). 21. C. G. De Kruif, J. Chem. Thermodyn., 12,no. 3,243(1980). 22. B. I. McArdle, I. N. Sherwood, and A. C. Damask, J. Cryst. Growth, 22,193 (1974). 23. A. Camermann and J. Trotter, Acta Crystallogr., 18,636(1965). 24. A. C. Hazell, F. K. Larsen, and M. S. Lehmann, Acta Crystallogr., B28, 2977 (1972). 25. T. Forster and K. Kasper, Z. Electrochem., 59,976(1955). 26. H.C. Wolf,SoL State Phys., 9,no. 13,23(1959). 27. A. Mutsui, K. Tomioka, and T. Tomioka, Solid State Comm., 25,237(1978). 28. W.Jones and N. D. Cohen, Mol. Cryst. Liq. Cryst, 41 (Letters), 103 (1977). 29. W.Jones, S.Ramdas, and I. M. Thomas, Chem. Phys. Lett, 54,no. 3, 490 (1978). 30. D.Fisher, G. Naundorf, and W. Kipffer, Z. Naturforsck,28u,973 (1973).
392
ZHIZHIN AND MUKHTAFtOV
31. R. Zallen, C. H. GrifEths, M.L. Slade, M. Hayek, and 0. Brafman, Chem. Phys. Lett., 39,no. 1,85(1976). 32. A. Bree, R. A. Lydd, T. N. Misra, and V.V.B. Vilkos, Spectrochim. Acta, 27A,2315 (1971). 33. G. Fillipini, C. M. Gramaccioli, M. Simonetta, and G. B. S d i i t t i , Chem. Phys., 8, 136 (1975). 34. L. Starr and D. E. Williams,Acta Cryst, B13,771(1977). 35. Ju. P.Ivanov, USSR, Moscow, Dissertation, 1984,24pages. 36. C.A. Fyfe, D.F.R. Gilson, and K. H. Thompson, Chem. Phys. Lett, 5, no. 4, 215 (1970). 37. G. N.Zhizhin, Ju. N. Krasjukov, E. I. Mukhtarov and N. V. Sidorov, Institute of Spectroscopy, USSR Acad. of Sci., preprint no. 16,(1987),31 pages.
393
APPENDIX I
AAP Parameters Used for the Calculation of the Lattice Dynamics of Molecular Crystals
AAP parameters used for the calculation of the lattice dynamics of molecular crystals
have unit of kcal/mol; ru has a unit of A;A u has a unit of A6 kcdmol; B u has a unit of k d m o l ; a12 has unit A-1; q has a unit of electron charge.
N
Interaction type
A 36 139 535 57 154 358 27.3 125 568 32.3 128 505 40 449 32.5 576 33.3 137.6 528.5 44.1
B
a
9
4000 9411 74460 42000 42000 42000 2659 8766 83630 2630 11000 61900 2864 71360 2787 87449 11993 37026 72525 34308
3.74
--
3.67 3.60 4.86
--
4.12 3.58 3.74
--
3.67 3.60 3.74 3.67 3.60 3.74 3.60 3.74 3.60 4.73 4.08 3.60 3.79
0.162 0.153
_-
__
n0te1
APPENDIX
394
Interaction
N
type
A
B
C-H
150.0 443.2 30 578 30.7 83.9 546.0 51 369 -26.5 330 10 20 132 876 268.2 128 2346 1430 1005 -631 3650 1744 2300 29 118 42 1 322 1055 2980 1468 1847.4 3628 40 1 421 7830
97730 65821 2343 97757 4252 5733 45544 31651 31735 1264 15475 41800 1500 6378 27116 54907.9 42000 235000 220800 33300 44160 263200 212432 426000 4900 18600 71600 45600 16700 4580 243300 220686 270600 20400 118000 64300
c-c 9
H-H
c-.c 10
Ha-H C-H
c-c 11
H-H
c-c 12
H*-H C-H
13
H*-H C-H
c-*c
c.0-c 14 15 16 17 18
19 20 21
0.-0 0-0
s-s
s-s H-Cl c-*c1 c1-Cl Cl*--Cl Cl-.*Cl H-H C-H
c-c
22 23 24 25
He-C1 C..Cl Cl-Cl Cl-Cl Cl-Cl Br-Br H-Br C.-Br B r -Br
a 3.67 3.61 3.74 3.60 3.928 3.567 3.916 4.78 3.33 3.74 3.67 3.60 3.70 3.43 3.16 3.96 4.25 3.49 3.62 3.623 3.653 3.51 3.51 3.56 4.29 3.94 3.68 3.07 2.94 2.26 3.66 3.51 3.28 3.55 3.57 2.78
9
n0te1
0.159
[7l,s,d,k [91,s1,d1
--
[lOI,sl,dl,k
--
111 51,d1
0.264
APPENDIX
N 26
27 28 29
30
395
Interaction type He-Br C-Br Br-Br F-F
1-1
c-c C-N N-N N-N
A 555 730 4580 217 8373 568 374 762 259
B 18050 18500 149000 71200 372900 83630 11340 105600 42000
note1
Q
3.44 3.37 3.14 4.25 3.03 3.46 3.46 3.46 3.78
--
[221,s,d,k(2)
---
[26],~',k(2) [25],d',k(2) [231,s,d,k(3)
--
[241,s1,k(2)
1The following notations are adopted AAP parameters obtained from data for the structure of several crystals
These parameters obtained from data for the structure of one crystal At fitting the AAP parameters the lattice vibrational frequencies of several crystals were used Frequencies of one crystal were used At fitting the AAP parameters the electrostatic interaction was considered in the form of a quadrupole-quadrupoleinteraction Results of quantum-mechanical calculations were used for the AAP parametrization For mixed X...Y interactions the AAP parameters are calculated by means of the combination relationships
The combined AAP; the number of a set AAPN is indicated in brackets; this set of parameters is used in combination with this number
APPENDIX
396
APPENDIX I1 Computerized Calculations of the Frequencies and Eigenvectors of the Normal Vibrations The expressions for force constants in the basis of generalized coordinates Qy given in Chapters I and 11of Part I can be represented by the following unit matrix form which is convenient for computer calculations
(AII.2)
-
(the sign means the matrix transposition); &K) is the matrix representation of the point symmetry element which transforms the axes of molecule (1,O) to the axes of molecule (KJ) in the rectangular crystal coordinate system; W is the matrix of the directing cosines of a molecule (1,O);
APPENDIX
397
Lpa,p is the form of a n intramolecular normal vibration of a fkee molecule in the
basis of Cartesian atom displacements; mp is the mass of an atom p; ooyis the internal vibrational frequency y of a fkee molecule; (o,y = 0 for y I6); is the ath coordinate of a n atom p of a molecule in the coordinate system of its mass center and main inertia axes,
Gu
is the 0th column of the matrix: 0 Rp - - $2
$1
-$3
0
[
&=
{
0
i 0 -Zp1 0 izpl 0 0
$2
i 0
0 0
i 0 i 0
0 0
zp3 zp3 i Z p l z p 2
-$I -$2] 0
1 for y and y'=4,5,6 0 in other cases
My={
M,M,M,I1,12,13 at y = l ,...6 1 fory>6
where M is the mass of a molecule and ,I are its main inertia axes;
( I is~the atom-atom potential; rw is the distance between the atoms p and q of
molecules (1,O)and ( I C , ~ )Epq ; is the column-vector corresponding to the unit vector directed fiom the atom p to the atom q. The data for the distances rw and the coordinates of the directional vectors of npq between the different atoms are necessary for the calculations using formulas (AII.1)and (AII.2). These values can be found, if the coordinates of all atoms in the crystal are known. The structural data are usually represented in the form of atomic coordinates of only one symmetrically independent molecule (1,O) given in
APPENDIX
398
must perform several coordinate transformations extending the starting molecule (1,O) to the whole crystal before the calculation procedure by the formula (AII.1) and (AI1.2).
where x,y,z are the rectangular system; al,a2,ag are the axes of the crystallographic coordinate system; Cl,C2,C3 are the angles between them. The transition Eom the skew-angular coordinate system to the rectangular one is accomplished by the matrix B:
where
b1=cos C3,
ba = sin C3,
b31= cos Cz,
b32 = (COS C1- cos C3 - cos C@in C3 b,
-
Yp
-
Kp
cosc2 cosc, ; sin1 3 &in 2 c3-cos2c~-cos2c~+2cosc~ = 7
- coordinates of atom p in the rectangular system, - coordinates of atom p in crystallographic system.
The coordinates of the atoms of a molecule (1,O) in the system of its mass center are:
399
APPENDIX
-
- , Yp = Yp
- Yc.m.
where
The tensor, I, of the molecule's inertial moments is:
The main axes of inertia Iu and the directions of the main axes of inertia (the matrix of directing cosines A) can be obtained by the diagonalization of the symmetric matrix i by the Jacoby method
Thecoordinates
Xp (1)
of the atoms in a molecule U,O) in the system of its w
s
center and inertia axes are:
where
The coordinates (AII.3) are the starting ones for the calculation of the force constants,i.e. the solving of the dynamical problem is performed in the rectangular coordinate system of the crystal which coincides with the coordinate system of the inertial axes of the starting molecule (1,O). In this system the coordinates of an atom in the molecule ( ~ , lwhose ), interaction with the starting molecule is considered, can be calculated by the following formula
APPENDIX
400
where E is the unitary m a w , are the crystallographic "coordinates" of the permutation element of symmetry, %, which mean the coordinates of the point of crossing of the coordinate plane and the rotation axis; they mean also the c o o r d i ~ t eof~ the point of crossing the crystallographic axis and the symmetry plane; they mean the crystallographic coordinates of the symmetry centre; qK1is the vector of the translational transfer of a molecule, including the translation along the lattice vector and the partial translation for the screw axes and slipping planes. is defined by the The form of the matrices &K) and the vectors OK and crystal space group. The appropriate information can be given by means of two 61es @ and 6 with 3x(3z+N) and 3x2 dimensions (N is the total number of considered molecules): 1
.._
- .
-
.
-
F = [E i Ti1.....T h l :..... :G*(z) i Tzl.....T%I
where the first row consists of zeros. As an example, consider the starting data which are necessary for the calculation of the lattice dynamics of naphthalene crystal with AAP in the rigid molecule approximation:
:; 3
P = 42000 42000 42000 4[:
- AAPparameters
a l =8.235 a2 =6.003 a3= 8.658A ~ 1 = 9 0 ~2=122.91 c3=90°
I
the unit cell parameters
I = -1;I is the centrosymmetricposition of a molecule (I = 1in other cases); n H = 8, nc are the numbers of atoms of each kind; IZ = 2, IZ is the
r q = 10; n H and
number of the translationally nonequivalent molecules in the unit cell; INT(1) = 9, INT(2) = 6; INTis the number of considered molecules (ql);
;]
0 112
it=[;
APPENDIX
401
1 0 0 1 0 0 1 1 0 0-1 0 0 0 0 0 0-1-1
F =
0 1 0 0 1 0 1 - 1 1 1 0 1 0 21- 1 2 21 31 21 31 0 0 1 0 0 1 0 0 1-1 0 0-1 0 0 1 1 1 1
The elements of the 6les 6' and F are given in unite of the unit cell parameters. The centroeymmetric position of the molecules was considered also at the construction of the file F (only such translationally equivalent molecules whose mass centers are not binded by the inversion were chosen, see Part 1,Sect. 1.B); the coordinates of atoms are as follows (in units of the unit cell parameters): dal 0.1375 0.1888 0.1490 0.0345 -0.0345 -0.1490 -0.1888 -0.1375 0.0856 0.0116 0.1148 0.0749 0.0472 -0.0472 -0.0749 -0.1148 -0.0116 -0.0856
Ylaz 0.0657 0.3176 0.4056 0.2999 -0.2999 -0.4056 -0.3176 -0.0657 0.0186 0.1869 0.1588 0.2471 0.1025 -0.1025 -0.2471 -0.1588 -0.1869 -0.0186
da3
0.4663 0.2752 -0.0233 -0.3394 0.3394 0.0233 -0.2752 -0.4663 0.3251 -0.2541 0.2200 -0.0784 0.0351 -0.0351 0.0784 -0.2200 0.2541 -0.3251
The coordinates of the other atoms in a molecule can be easily found, if its site symmetry is considered. The dynamical matrix is real and symmetrical at = 0. Its eigenvalues and eigenvedors were found by the Jacobian method diagonalization. An algorithm in the form of a subroutine was used JACOB (D,S, Z, ALPH1,N) where D is the initial symmetric matrix, S is the orthogonal matrix of eigenvedors, Z is the one-dimensional matrix of eigenvalues of the matrix D, ALPHl is the
APPENDIX
402
parameter defining the accuracy of the reduction of the matrix to a diagonal form, N is the dimension of the matrices D, S and Z. D and S are the complex matrices at g#Oin a general case. The equation is in the following form:
The real and imaginary parts give two combined systems of equations:
These systems can be written in the form of one matrix equation:
The real matrix D with a dimension which is twice as large as the dimension of the starting complex matrix D is symmetric. This follows from the Hermitian nature of the dynamical matrix (see Part I, Chapter I). For this reason, one can use the ordinary Jacobian method for the diagonalization. In this case we can obtain two complex conjugate eigenvectors of the starting system for every eigenvalue (AI1.4). It is convenient to use the units of k d m o l A and m.a.u. for the calculation of the eigenvalues. The frequencies in wave number units can be calculated by the formula:
v(cm-')=208.56
m k ikcal/(mol amu
The abovementioned calculation method can also be used almost without any changes for the optimization of the crystal structure by means of the NewtonRafson method (see Part 1, Chapter II). It is necessary to consider only an additional relation between the atomic displacements of different molecules which follows from the requirement of the crystal symmetry conservation at the minimization of the lattice energy
403
APPENDIX
The expression for the second derivatives
is obtained h m (AII.21, if one makes the substitution
and assumes that
= 1 for all ‘y.
APPENDIX
404
APPENDIX III Frequencies and Eigenvectors for the Low-Frequency Normal Vibrations of Several Organic Crystals
TABLE A-1 Benzene (C~HG), Pabc, z = 4(C9
b -0.437 0.894 0.102
C
-
0.682 0.256 0.658
1,=88.2 a.u.m.A2 I, =88.2 a.u.m.A2 I,= 176.3 a.u.m.A2
v exp.a, cm-1
I
v calc.b, cm-1 I1
Vibration forms (140K)C
I1
140K 3
218K 4
140K 5
Lv
Lw
6
7
8
73 53
98 74 54
87 71 45
96 78 46
0.548 0.714 0.439
-0.600 0.699 0.390
0.584 0.050 0.810
119 102 53
133 94 62
123 104 48
139 95 51
0.865 0.447 0.230
-0.484 0.864 0.141
-0.135 -0.233 0.963
102 96 83
93 84 75
102 92 84
-0.219 -0.414 0.884
0.975 -0.057 0.215
-0.038 0.909 0.416
140K 1
195K 2
$.
93 78 56
Big
128 100 56
__
Ta 9
Tb
10
TC 11
v exp.8, cm-1
B3g
AU
140K 1
195K 2
128 84 62
119
n.a.
I 140K
TABLE A-1 (continued) v calc.b, cm-1 11 I1 218K
140K 5
3
4
____
130 92 60
119 79 65
135 87 65
n.a.
103 70 57
87 58 50
100 67 57
Vibration forms (140K)C
Lv
Lw
6
7
8
0.891 0.283 0.355
-0.453 0.493 0.743
-0.036 0.823 -0.567
Ta 9
Tb
10
Tc 11
0.808 -0.398 0.435
0.335 0.917 0.217
0.488 0.030 -0.874
0.724 -0.690
0.690 0.724
Blu
89 73
---
90 74
74 62
87 72
B2u
100 56
--
104 63
88 48
103 58
0.404 0.915
B3u
98 66
__
105 57
83 46
97 53
0.978 -0.207
__-
aRaman 195K 111, 140K 121, IR 140K 133. bI and I1 are results of calculations with AAP/2 and AAP/1 crystal structure at 140 and 218K 143. CL are librational shifts;T are translational shifts.
0.915 -0.404 0.207 0.978
TABLE A-2 Naphthalene (CloHs), P21/a, b
= 2(Ci)
1,=564.4 a.u.m.A2 I, =404.3 a.u.m.A2 I, = 161.1 a.u.m.A2
v exp.a
v ca1c.b
Vibration forms (140K)
L
Lv
Lw
I
I1
$.
109 74 52
115 80 54
110 85 56
-0.108 0.185 0.979
0.569 0.817 -0.092
0.815 -0.546 0.192
B,
125 71 46
94 75 41
106 79 47
0.039 0.405 0.914
0.080 0.910 -0.407
0.996 0.089 -0.003
AU
98 53
93 55
93 50
BU
66
69
64
300K
2
ahman 151, LR 161 b1 and I1 are calculations with AAP/2 and AAP/l crystal structure at 300K [7].
T*
Tb
0.812 -0.584
Tc
0.584 0.812 1.000
b
Iu=1348.8 a.u.m.A2 1,=1117.8 a.u.m.A2 I,=231.0 a.u.m.A2
v exp.a
v calc.b
Vibration forms (95K)
~~
11 290K
11 95K
92K
290K
I 95K
133 83
123 71 41
153 101 45
127 76 41
134 97 43
123
B,
140 72 57
__
128 81 57
116 69 47
133 79 52
Au
127
_-
120
125 49
99 38
116 45
Bll
68
61
74
61
67
Ag
50
46
--
Lv
Lw
-0.195 -0.267 0.944
0.268 0.911 0.313
0.944 -0.313 0.106
-0.082 0.064 0.998
0.122 0.991 -0.063
0.993 -0.122 0.016
~
aRaman, IR [81. bI and I1 are calculations with AAP/2 and AAP/l; crystal structure at 95 and 290K [91.
Ta
Tb
0.848 -0.531
Tc
0.531 0.848 1.000
TABLE A-4 Phenenthrene C14H10, P21, z = 2(Ci) 1,=1222.7 a.u.m.A2 I, =910.3 a.u.rn.A2 I, = 312.5 a.u.rn.A2
v exp.a 300K 142 125 106 84 60 47 (32)
v ca1c.b
Vibration formsc
I
I1
150 118 133 84 69 52 34
147 122 110 82 66 51 36
LU -0.010 -0.012 0.068 0.369 -0.293 -0.028 0.879
Lv 0.014 -0.087 0.094 -0.016 0.941 -0.011 0.312
LW
TU
T"
0.055 -0.382 0.910 0.029 -0.088 0.010 -0.116
0.068 -0.125 -0.106 0.785 0.106 -0.186 -0.293
0.046 -0.073 -0.058 0.395 0.053 -0.185 -0.151
TW
Q B ~
QA2
-0.109 0.068 0.012 0.263 0.045 0.949 -0.067
-0.204 0.877 0.376 0.143 0.068 -0.138 -0.062
0.968 0.228 0.039 -0.012 -0.002 0.099 0.018
152 0.022 -0.056 (142) 159 0.002 -0.026 0.048 0.004 109 119 113 0.017 -0.028 -0.475 0.015 -0.028 -0.003 B 101 100 94 0.012 0.308 0.830 0.044 -0.082 -0.007 85 81 -0.150 0.843 80 -0.282 0.189 -0.355 -0.032 60 49 52 0.183 0.434 -0.067 -0.413 0.773 0.070 32 26 25 0.971 0.047 -0.033 0.107 -0.200 -0.018 aRaman, 300K, our data. bI and I1 are calculations with AAP/2 and AAF'/l; crystal structure at 300K [lo]. c & ~ and 2 Qm are internal normal coordinates of the nonplanar vibrations at 100 (Bz) and 114 (A2) cm-1.
0.230 0.855 0.447 -0.117 0.000 -0.044
0.970 -0.202 -0.085 0.103 -0.028 -0.004
A
TABLE A-6 Chryzene (ClIH11), P21/c, z = 4(Ci)
b
1,=2296.8 a.u.m.A2 1,=1901.1 a.u.m.A2 I,=395.8 a.u.m.A2
v exp.a
300K
v ca1c.b
I
Vibration forms
I1
Lu
L
Lw
Ag
121 84 57
108 75 56
-0.030 0.887 0.461
-0.321 0.446 0.836
0.947 -0.123 0.298
B,
102 83 68
95 76 60
0.096 0.389 0.916
0.085 0.914 -0.397
0.992 -0.116 -0.055
AU
89 34
115 28
96 27
BU
57
53
46
Tfl
Tb
0.984 -0.177
Tc
0.177 0.984 1.000
aIR, 300K [lll. bI and I1 are results of calculations with AAP/2 and AAP/l; crystal structure at 300 K [12].
TABLE A-6 Pyrene (C16H10), P21/a, z = 4(Ci)
A=
I a c*l
b u 0.529 0.632 -0.567 v -0.704 0.700 0.124 w 0.475 0.333 0.815
v exp.a 300K 1
I,=896.6 a.u.m.A2 I,=1390.0a.u.m.A2 I, = 493.3a.11.m.A~
v ca1c.b
I 2
Vibration formsc
I1 3
Lu
Lv
LW
TU
T"
Tw
QB~U
QAU
4
5
6
7
8
9
10
11
$.
170 127 95 77 56 (46) 30 17
173 135 100 88 60 47 36 18
175 138 105 92 58 46 37 19
0.026 0.009 0.610 0.009 0.776 -0.026 0.088 0.130
-0.059 0.024 -0.096 0.184 0.149 0.878 0.151 -0.371
0.067 -0.097 0.652 0.542 -0.491 0.044 -0.142 -0.068
0.034 -0.130 0.125 -0.109 -0.128 -0.260 0.828 -0.430
-0.035 -0.065 0.403 -0.807 -0.287 0.293 -0.118 0.023
-0.033 -0.043 -0.028 0.093 -0.160 0.261 0.489 0.809
-0.055 0.981 0.107 -0.131 -0.103 -0.025 0.104 -0.020
0.993 0.065 -0.063 -0.022 0.014 0.067 0.009 0.020
B,
169 126 (95) (77) 67 46 (30)
174 125 104 87 69 46 34 22
175 129 109 91 69 44 33 21
0.032 0.120 0.681 0.015 0.509 0.441 -0.248 -0.073
-0.038 -0.024 -0.211 -0.003 -0.399 0.879 0.123 0.084
0.068 -0.001 0.581 0.519 -0.577 -0.149 0.166 0.078
0.060 0.001 0.146 -0.212 0.159 0.023 0.874 -0.376
-0.007 0.027 0.341 -0.827 -0.391 -0.096 -0.119 0.149
-0.017 0.090 0.024 0.008 0.252 -0.012 0.341 0.901
-0.092 0.984 -0.092 0.023 -0.085 -0.031 0.008 -0.078
0,990 0.088 -0.085 -0.026 -0.008 0.025 -0.046 0.033
~~
--
TABLE A-6 (continued) v exp.a
I
II
Lu
Lv
2
3
4
5
Lw 6
TU 7
T" 8
28
159 117 110 75 49 47 19
158 119 112 81 48 46 19
0.036 -0.222 0.678 -0.655 0.146 -0.199 0.023
-0.009 0.079 -0.048 0.224 0.953 -0.128 -0.126
-0.016 -0.204 0.656 0.709 -0.138 0.031 0.062
0.006 -0.021 -0.001 -0.050 0.148 0.373 0.661
0.005 0.018 -0.060 0.084 -0.143 -0.637 -0.272
-0.022 0.003 0.128 -0.081 0.019 0.631 0.634
-0.003 0.950 0.296 0.016 0.090 -0.033 0.031
0.999 0.007 -0.010 0.035 0.001 0.022 -0.018
158 129 105 67 63 35
157 129 106 73 49 30
156 130 112 79 51 31
0.003 -0.067 0.648 0.702 0.252 -0.140
0.009 -0.009 0.020 0.106 0.192 0.975
-0.020 -0.003 0.751 -0.643 -0.111 0.077
0.007 -0.049 -0.061 -0.181 0.593 -0.970
0.008 -0.054 -0.067 -0.200 0.656 -0.107
0.004 -0.026 -0.032 -0.095 0.312 -0.051
0.005 0.993 0.084 -0.015 0.084 -0.008
0.999 -0.005 0.013 -0.013 -0.015 -0.005
300K 1
-AU
123 102 89 61
__
*U
$ h3
Vibration formsc
v calc.b
~~
Tw 9
Q~3u
10
~
a h a n 300K [13,14], IR 300K 1141, IR 223L 1151. bI and I1 are calculations with AAP/2 and AAP/l; crystal structure at 300K 1163 CQBSuand Q A are ~ internal normal coordinates of the nonplanar molecular vibrations with frequencies 96 and 147 cm-1.
QAU 11
a
A=
b
C*
u 10.736 0.677 0.0001 I, =1506.0a.u.m.A2 v -0.677 0.736 0.000 Iv=3025.5 a.u.m.A2. w 0.000 0.000 1.000 I,=1506.0a.u.m.A2
I
I
v exp.*
v ca1c.b
I
$-
Bu
Vibration forms
I1
Lll
L,
Lw
90 56 49
88 54 51
0.307 0.914 -0.265
0.170 0.221 0.960
0.936 -0.340 -0.088
75 52 33
82 58 36
0.373 0.881 -0.291
0.154 0.251 0.956
0.915 -0.401 -0.043
62 47
51 40
41
39
aData are absent. bI and I1 are calculations with AAP/2 and AAP/l; crystal structure at 300K [17].
Ta
Tb
0.774 -0.631
Tc*
0.631 0.774 1.000
TABLE A-8 Ethylene (C2H4), P24n, z = 2(Ci)
I, =3.0a.u.m.A2 I,=18.9a.u.m.~2 I, = 16.0 a.u.m.A2 v exp.8
v ca1c.b
Vibration forms (C2H4)
I
11
I1
C2H4
C2D4
CzH4
CzD4
C2H4
LLl
Ag
177 97 90
135 93 75
178 95 76
176 93 81
121 82 73
0.979 -0.190 0.074
0.201 0.959 0.202
0.003 -0.213 0.977
B,
167 114 73
123 95 60
164 104 64
161 107 69
116 94 53
0.998 -0.015 -0.062
0.054 -0.303 0.951
0.331 0.953 0.301
AU
73
__
68
__
58 41
62 48
58 45
BU
108
102
78
88
81
aRaman 35K [181, IR 20K 1193. bI and I1 are calculations with AAP/l and M / 3 ; crystal structure a t 85K [201.
Lv
Lw
T*
Tb
0.879 -0.477
Tc*
0.477 0.879 1.000
b 0.559 0.801 -0.216
c*
-
0.308 0.041 0.951
I,=113.7a.u.m.A2 I,=115.9a.u.m.A2 I,=201.3a.u.m.A2 v ca1c.c
v exp.a
Ag
B,
LU
Lv
Lw
102 86 61
0.215 0.849 -0.483
0.939 -0.044 0.341
-0.268 0.527 0.807
52
125 98 59
0.967 -0.056 0.250
-0.246 0.474 0.845
-0.071 0.879 0.472
106 63
113 67
58
62
C6H12
C6D12
C6H12
110 85 69
97 75 61
89 76 54
--
110 87
1202 92 62
A,
98 61
BU
--
83 54
--
--_
Vibration forms (C6H12)
CGD12
a h m a n 115K [21,221, IR 115 [21,221. bRaman 115K 1231. CCalculations with AAP/4; crystal structure a t 115K [241.
T*
Tb
0.168 0.986
Tc*
0.986 -0.168 1.000
TABLE A-10 Biphenyl (C12H10), P2l/c, z = 2(Ci)
I, =1083.4 a.u.m.A2 1,=910.7 a.u.m.A2 I,=172.8 a.u.m.A2
v exp.*
v ca1c.b
Vibration forms (293K)
~~
I 293K
llOK
I1 llOK
105 70 65
88 53 42
119 82 57
115 79 57
118 100 42
89 54 42
103 92 49
110 94 48
110K
Ag B,
I 293K 86 59
I1 293K
L"
L
49
84 64 50
0.340 0.758 -0.556
-0.036 0.601 0.798
0.940 -0.252 0.232
81 57 46
85 68 47
0.282 0.126 0.951
0.778
-0.662 0.826 0.057
0.550
-0.304
Lw
aRaman [25,261. bI and I1 are calculations with AAp/2 and AAP/l; crystal structure a t llOK 1271, a t 293K [281. Translational modes are not given, because they are mixed considerably with torsional vibrations of phenyl groups.
b I,=439.7
a.u.m.A2
I,=91.5 a.u.m.A2 I, = 353.4 a.u.m.A2 v exp.a
v ca1c.b
Vibration formsc
I
I1
I11
LU
L
Lvf
130 108 88 64
164 104 79 62
135 97 65 54
138 97 70 54
0.020 0.208 0.356 0.911
0.475 -0.048 0.818 -0.320
-0.263 0.926 0.109 -0.248
0.840 0.312 -0.438 0.081
130 108 94 69
152 113 97 67
122 105 98 54
128 103 86 58
-0.030 0.201 0.861 0.466
0.510 0.139 0.392 -0.763
-0.363 0.908 -0.143 -0.153
0.779 0.339 -0.290 0.440
Au
152 95
-_
165 84 36
135 92 38
138 73 33
-0.044 -0.379 0.564
-0.048 -0.394 0.616
-0.064 0.837 0.543
0.996 0.018 0.089
BU
(152) 79
151 56
120 55
127 52
-0.107 0.725
0.100 -0.673
-0.002 0.011
0.989 0.146
83K
Ag
Bg
TU
aRaman 126,291. bI, 11,I11 are calculations with AAP/l, AAP/2, AAFV3; structure at 163°K (data of S.Gorder). %,,xu are coordinates of the torsional shitts of the methyl groups.
T"
Tw
xg
xu
a b 0.500 -0.732 -0.463 v 0.548 0.681 -0.485 0.742 W 0.670 -0.012 c* U
A=
v exp.a
Bg
v calc.8
Vibration formsa
I
I1
I11
LU
Lv
LW
121 80 99 60
123 92 79 58
107 83 64 49
105 84 68 51
0.012 0.255 0.239 0.937
-0.649 0.269 0.671 -0.236
0.545 0.790 0.121 -0.253
-0.531 0.488 -0.691
111 99 80 69
118 101 88 66
104 96 80 51
103 88 78 57
0.050 0.462 -0.723 0.510
-0.529 0.628 -0.036 -0.569
0.777 0.537 0.292 -0.149
-0.336 0.321 0.624 0.628
117 82 35
96 89 36
98 71 32
-0.059 -0.381 0.561
-0.065 -0.395 0.613
-0,101 0.835 0.540
0.991 0.036 0.129
108 54
87 53
91 50
-0.174 0.711
0.162 -0.662
-0.003 0.011
0.971 0.238
83K
4%
1
I,=510.4 a.u.m.A2 I, =97.0 a.u.m.A2 1,=423.9 a.u.m.A2
Wee notation to the Table A-2.
TU
T"
TW
xg
XU
0.050
A=
I
TABLE A-13 Toluene (C6H6(CH3)), a-phase, P21/c, z = 8(C1)*
b u 0.8811 0.465 0.&5] v -0.584 0.683 0.439 w -0.038 -0.563 0.825
11
b 0.i37 0.577 0%6] A= v 0.832 -0.483 -0.273 -0.139 -0.660 0.739 ~
v exp.b
v calc.c
I 20K 1
-A,
1, = 282.5 a.u.m.A2 I,=197.5 a.u.m.A2 I,=87.8 a.u.m.A2
Vibration formsd
130K 2
163K 3
I1 163K 4
--
164
138
135
115
154
120
106
122
--
--
93
LU 5
LW
6
7
TU 8
T"
TW
x
9
10
11
6)0.131
0.004 0.263
-0.022 0.023
0.030 0.033
0.001 0.045
0.091 -0.046
-0.002 0.945
ab0.139 0)-0.063
0.008 0.326
0.580 -0.074
-0.056 -0.033
-0.008 -0.093
0.001 -0.005
0.716 -0.047
99.4
a) 0.068 O ) 0.030
-0.066 0.687
0.164 -0.264
-0.139 -0.026
0.142 -0.068
-0.104 0.096
-0.499 -0.163
88
a) 0.079 01-0.393
0.345 -0.333
0.442 0.266
-0.362 -0.154
0.105 -0.282
0.016 0.007
-0.262 0.171
118
a) 0.078
Lv
R
TABLE A-13 (continued) v exp.b
I
I1
163K 3
163K 4
94
83
90
80
a) 0.088 (T) 0.284
75
77
79
75
71
65
71
65
LU
Lv
LW
TU
T"
TW
x
5
6
7
8
9
10
11
0.710 0.038
0.169 -0.013
0.055 0.440
-0.142 0.349
0.085 0.143
-0.057 -0.090
a)-0.153 01-0.054
0.374 0.104
-0.576 -0.080
-0.597 -0.153
0.043 -0.037
0.017 -0.059
0.314 0.000
a) 0.085 0.567
0.136 -0.255
0.191 -0.377
-0.010 -0.514
0.181 0.027
-0.050 -0.327
-0.034 0.004
a) 0.421 0)-0.155
0.098 0.282
-0.072 0.415
0.168 -0.164
0.366 0.253
0.252 -0.431
0.069 -0.160
(T)
a) 0.730 0.178
-0.178 -0.147
-0.043 -0.089
-0.258 0.384
0.070 -0.304
-0.127 0.034
0.209 -0.019
a) 0.085 01-0.378
0.191 -0.061
-0.080 -0.370
0.291 -0.006
0.518 0.161
-0.461 0.235
0.094 0.108
a) 0.069 0 ) 0.142
0.253 0.141
-0.151 0.026
0.375 -0.215
0.113 -0.601
0.353 0.422
0.025 -0.060
a) 0.235 (T) 0.085
0.130 0.168
-0.049 0.353
0.096 -0.377
-0.446 0.006
-0.629 0.134
0.039 -0.011
(T)
65
59
68
56
56
52
60
49
49
_-
53
44
45
44
39
37
35
32
35
0
Vibration formsd
130K 2
20K 1
$.
v C81C.C
32
$ ;4 ~
31
28
27
25
--
20
13
--
163
133
a) 0.104
o) 0.161
-0.235 -0.113
0.073 0.188
-0.267 -0.261
0.198 0.456
0.205 0.648
0.082 0.005
a)-0.371 0)0.408
-0.027 0.014
0.002 0.477
-0.046 0.247
0.507 -0.176
-0.341 -0.029
-0.003 0.008
a)-0.078 -0.009 0.235
0.051 0.031
-0.034 0.016
-0.012 0.011
-0.097 -0.053
0.020 0.952
0)0.120
125
--
154
117
ab0.145 0)0.077
0.014 -0.211
0.568 0.090
-0.037 0.098
-0.003 0.091
0.013 0.063
0.758 -0.018
106
103
116
98
a)-0.029 0)-0.018
0.064 0.612
-0.190 -0.348
0.369 -0.228
-0.219 -0.186
0.095 -0.002
0.413 -0.112
100
99
93
87
a) 0.134 o) 0.248
0.535 0.268
0.515 -0.331
-0.284 -0.076
0.026 0.056
0.051 -0.034
-0.292 -0.098
82
--
92
80
a)-0.014 0.551 obO.190 -0.503
-0.338 -0.260
-0.049 -0.315
0.051 0.021
0.214 0.011
0.186 0.201
82
73
68
71
a) 0.177 -0.188 0.442 01-0.169 -0.252 -0.193 ab0.020 0.413 0.168 0)-0.498 0.135 0.565
0.672 -0.233 0.106 -0.025
0.056 -0.032 -0.152 -0.329
0.120 0.032 -0.117 -0.238
-0.259 0.127 -0.048 -0.014
61
52
0.178 0.355
-0.160 -0.254
0.081 -0.524
-0.125 -0.034
(75) (75)
__ __ __
a)-0.553 a) 0.231
0.051 0.001 -0.200 -0.230
TABLE A-13 (continued)
v calc.c
v exp.b
B,
I
II
20K 1
130K 2
163K 3
163K 4
_-
__
48
49
LU
L,
LW
TU
T"
TW
x
5
6
7
8
9
10
11
O)
a) 0.582 0.085
0.088 -0.030
-0.097 -0.110
0.103 0.423
0.462 -0.356
-0.091 -0.218
0.194 0.035
__
__
44
38
a)-0.072 O ) 0.395
0.209 -0.024
-0.109 0.199
0.335 -0.305
0.234 0.326
-0.562 -0.220
-0.072 -0.103
__
--
40
33
a)-0.440 0b0.042
0.230 0.097
-0.020 -0.005
0.201 0.254
0.435 -0.172
-0.079 0.635
-0.094 -0.012
28
25
34
30
ab0.186 0)-0.109
-0.085 0.241
0.015 0.170
-0.022 -0.137
0.591 0.249
0.555 -0.245
0.021 -0.025
20
18
22
22
a) 0.158
01 0.593
0.145 -0.062
-0.082 0.451
0.187 -0.039
-0,191 -0.182
0.492 0.201
-0.031 0.013
a) 0.140 01-0.156
0.254 0.085
-0.088 0.007
0.293 0.544
-0.231 0.650
0.132 0.018
-0.011 0.023
a) 0.064
01 0.122
-0.078 0.216
0.176 0.037
-0.006 0.017
0.050 0.022
0.021 -0.056
0.049 0.942
a) 0.156 O) 0.010
0.066 0.385
-0.097 -0.243
0.114 -0.144
-0.146 0.109
0.049 0.057
-0.826 0.013
-B,
ih3%
Vibration formsd
(123)
102
__
12
13
_-
164
134
--
141
98
Bu
123
11
115
104
94
--
101
90
a)-0.027 a) 0.010
0.082 0.385
-0.609 -0.243
0.221 -0.144
-0.087 0.109
0.080 0.057
0.463 0.013
a) 0.024
a) 0.109
0.491 0.281
0.669 -0.274
-0.095 -0.169
-0.032 -0.008
0.152 -0.058
0.229 -0.164
82
--
84
80
a)-0.O 18 0b0.330
0.506 -0.586
-0.157 -0.099
0.266 -0.314
0.009 -0.045
0.097 0.004
-0.087 0.261
77
72
73
72
a)-0.005 0)-0.501
0.076 0.265
0.151 0.647
0.230 0.070
-0.067 -0.352
-0.054 -0.199
-0.026 -0.041
67
58
a)-0.496 a) 0.078
0.423 0.090
-0.097 0.116
-0.271 0.203
-0.335 -0.056
-0.307 0.472
-0.028 0.035
56
56
a) 0.188 a) 0.241
0.323 -0.096
-0.118 -0.176
0.029 0.575
0.195 -0.345
-0.284 -0.420
-0.045 0.030
50
42
a) 0.106 a) 0.333
0.409 0.130
-0.262 0.491
-0.205 -0.087
0.192 0.376
0.339 -0.183
-0.086 -0.035
41
38
a) 0.800 O)-O.l92
0.137 0.057
-0.014 0.044
-0.187 0.064
-0.120 0.112
-0.215 0.389
0.160 0.099
a)-0.173 0)-0.346
0.090 0.233
-0.013 -0.126
0.071 0.030
0.682 0.389
-0.369 0.092
-0.065 -0.034
64
46
31
26
26
Ip
N
w
TABLE A-13 (continued) v exp.b
A,
v d c. C
I
11
LU
L,
LW
TU
T"
TW
X
5
6
7
8
9
10
11
-0.027 -0.060
0.048 0.267
0.518 -0.310
0.174 -0.155
-0.378 0.270
0.020 -0.041
0.133
0.074 0.233
-0.016 0.029
0.003 0.029
-0.059 0.057
-0.026 -0.053
-0.048 0.938
IS)
a)-0.144 0.392
-0.017 0.289
-0.041 -0.121
-0.027 0.025
0.084 -0.010
-0.017 0.111
0.926 -0.011
130K 2
163K 3
163K 4
-_
--
15
18
a) 0.087 IS) 0.528
(130)
--
165
134
a) 0.067 IS)
--
141
$2
Vibration formsd
20K 1
115
IP
99
130
122
107
a) 0.009 0)-0.011
-0.079 0.661
0.624 -0.202
-0.208 -0.022
0.070 -0.116
-0.082 0.062
-0.227 -0.044
94
101
89
a) 0.020 0)-0.204
0.435 -0.391
0.643 0.253
0.052 0.074
0.003 -0.129
0.153 0.068
0.199 0.215
82
80
84
a) 0.024 IS) 0.450
0.459 0.156
0.046 -0.146
0.342 0.412
-0.136 0.432
0.086 0.072
-0.066 -0.173
66
a) 0.185 IS)-0.443
-0.236 0.192
-0.130 0.309
0.043 0.622
-0.044 0.066
0.172 0.377
-0.036 -0.009
63
a) 0.193 IS) 0.426
-0.563 -0.351
0.246 -0.321
0.140 0.149
0.097 -0.102
0.174 0.256
0.044 0.138
77 70
50
a) 0.618 0)-0.039
0.246 0.098
-0.130 -0.004
0.204 -0.363
0.473 -0.017
-0.037 0.360
-0.006 -0.043
46
a) 0.358 a) 0.215
-0.311 0.071
0.220 0.590
0.066 -0.123
-0.169 0.338
-0.253 -0.267
0.156 -0.072
44
41
a)-0.396 0)-0.304
-0.182 -0.106
0.144 0.084
0.229 -0.107
0.337 0.615
-0.260 0.189
-0.063 0.125
33
31
a) 0.267 01-0.228
-0.053 0.009
0.030 -0.235
-0.177 0.061
0.253 0.348
0.553 -0.536
0.067 0.009
28
27
a)-0.345 0 ) 0.111
-0.124 0.246
0.002 0.332
0.514 -0.253
0.133 -0.184
0.543 -0.051
-0.071 -0.018
a) 0.213 01-0.393
-0.074 0.049
0.061 -0.373
0.559 -0.148
-0.624 -0.078
-0.096 -0.140
0.069 0.024
59 46
{ {
55
14
9
BToluene crystal contains two symmetricallyindependent molecules in the unit cell, the directing cosines are given for each independent molecule (a) and (a). &man, IR at 20K [30], Raman, IR a t 130K [30,26]. CI, I1 are calculations with AAp/2 and AAP/3; crystal structure a t 163K [31]. dThe upper and lower lines are related to the coordinates of shifts of two independent molecules; x is the coordinate of torsional shift of the methyl group of a molecule.
6 g
TABLE A-14 Toluene-d3 (CH(CD$), a-phase, P21/c, z 3: 8(Cl)a U
A1= v W
a b C 0.812 0.465 0.354 -0.583 0.683 0.440 -0.037 -0.564 0.824
b
v exp.8
A,
I, = 306.7 a.u.m.%i2 I, = 221.5 a.u.m.A2 I, = 90.8 a.u.m.A2
v calc.8
Vibration formsa
20K 1
I
I1
Lu
L
LW
Tu
T"
TW
x
2
3
4
5
6
7
8
9
10
133
134
110
ab0.075 01-0.021
0.007 0.509
0.685 -0.118
-0.122 -0.020
0.031 -0.105
0.000 0.017
0.448 0.152
114
123
103
a) 0.127
-0.052 0.445
-0.306 -0.106
0.042 0.098
0.010 0.149
0.078 -0.027
-0.223 0.714
a)-0.017 -0.203 01-0.073 0.481
-0.173 -0.281
-0.429 -0.114
0.155 -0.132
-0.178 0.098
-0.382 -0.429
a)-0.081 -0.147 0.237
-0.083 -0.226
0.499 0.340
-0.179 0.350
-0.037 0.106
0.187
0 ) 0.334
ab0.123 -0.775 0)-0.046 -0.073
-0.120 -0.061
0.259 -0.386
0.063 -0.211
-0.058 -0.186
0 ) 0.280
105
111
86
__
86
79
88
82
76
--
78
65
a)-0.294 0.310 0 ) 0.295 -0.071
-0.390 -0.292
-0.333 -0.303
-0.023 0.081
0.018 -0.197
0.489 -0.019
69
67
63
a)-0.243 -0.092 0b0.520 0.243
-0.402 0.347
-0.135 0.329
-0.196 -0.037
0.104 0.189
0.329 0.039
62
64
55
a) 0.250 0.119 0)-0.130 0.317
-0.082 0.371
0.150 -0.250
0.354 0.303
0.261 -0.470
0.073 -0.232
54
55
47
a) 0.653 -0.218 a) 0.294 -0.086
-0.088 0.077
-0.320 0.336
-0.081 -0.283
0.055 -0.139
0.288 -0.108
47
52
43
a) 0.292 0.123 0)-0.343 -0.097
-0.144 -0.337
0.161 0.134
0.528 0.087
-0.452 0.150
0.258 0.122
(43)
38
37
a) 0.082 a) 0.135
0.270 0.150
-0.174 0.009
0.362 -0.201
0.162 -0.600
0.333 0.411
0.064 -0.087
35
34
32
a) 0.240 0 ) 0.072
0.148 0.180
-0.066 0.337
0.098 -0.388
-0.429 0.002
-0.633 0.129
0.063 -0.004
30
26
25
a) 0.122 -0.240 6)0.161 -0.116
0.052 0.183
-0.248 -0.258
0.195 0.460
0.201 0.648
0.120 0.011
__
20
13
a)-0.384 -0.030 6)0.420 0.011
0.004 0.478
-0.047 0.248
0.490 -0.152
-0.346 -0.027
-0.004 -0.004
TABLE A-14 (continued) v Calc.8
v exp.a
20K
B,
120
I 133
LU
Lv
L,
Tu
a)-0.109 0.095
0.018 -0.283
0.742 0.152
-0.125 0.151
0.049 0.135
-0.011 0.059
0.509 -0.054
O)
a)-0.089 0.230
0.043 0.508
0.169 -0.133
0.009 -0.012
-0.075 -0.020
-0.093 -0.066
0.118 0.774
II 108
Q)
114
120
Ei
Vibration form@
102
T"
Tw
X
96
108
87
a) 0.056 01-0.067
0.256 0.331
0.157 -0.436
0.422 -0.295
-0.184 -0.167
0.199 0.029
0.317 -0.368
99
87
79
a)-0.113 0)-0.110
-0.628 0.306
-0.052 0.399
0.388 0.204
-0.144 -0.142
-0.162 -0.008
0.215 -0.138
82
84
74
ab0.114 01-0.250
-0.092 -0.563
-0.201 -0.069
0.460 -0.242
0.027 0.029
0.137 0.058
0.247 0.446
(75)
80
69
a)-0.067 01-0.450
0.550 0.063
0.030 0.531
-0.017 -0.061
-0.143 -0.304
-0.106 -0.262
0.026 0.078
__
63
64
a)-0.367 0.107
0.099 0.048
-0.538 -0.030
-0.404 0.070
-0.157 0.054
0.054 0.050
0.568 -0.085
O)
--
56
52
a)-0.361 0.063 O) 0.258 -0.212
0.012 -0.248
0.197 0.457
-0.020 -0.319
0.066 -0.565
-0.128 -0.054
__
47
45
a) 0.610
-0.181
0.034
0.489
-0.087
0.357
0.091
%
1
5;1
a) 0.011
--
A"
0.007
-0.052
0.337
-0.300
-0.019
0.065
43
37
a)-0.101 0.231 a) 0.406 -0.009
-0.125 0.166
0.322 -0.274
0.280 0.324
-0.561 -0.164
0.062 -0.147
40
33
a)-0.478 01-0.077
0.219 0.121
0.019 0.000
0.166 0.244
0.422 -0.167
0.009 0.621
-0.156 -0.012
28
34
29
abO.154 -0.105 O)-0.120 0.250
0.010 0.159
-0.031 -0.164
0.560 0.258
0.553 -0.323
0.043 -0.029
20
21
22
a) 0.159 0.141 a) 0.608 -0.067
-0.073 0.452
0.168 -0.053
-0.173 -0.167
0.491 0.196
-0.041 -0.026
12
13
a) 0.148 01-0.154
0.270 0.088
-0.089 0.017
0.282 0.543
-0.228 0.647
0.138 0.020
-0.014 0.032
130
105
a) 0.097 a) 0.216
0.004 0.686
0.040 -0.200
-0.083 0.060
-0.028 0.064
-0.086 -0.008
-0.032 0.652
117
103
ab0.060 -0.067 0b0.144 0.325
0.764 -0.132
-0.187 -0.034
0.119 -0.195
-0.021 0.121
0.154 -0.362
96
84
a)-0.066 -0.635 ab0.099 0.243
-0.431 -0.179
-0.260 -0.293
0.101 -0.159
-0.179 -0.066
-0.131 -0.252
78
78
a)-0.073 a) 0.340
0.083 0.307
-0.223 -0.275
0.255 0.359
-0.047 0.386
0.023 0.144
0.138 -0.517
TABLE A-14 (continued)
v CalC.8
v exp.8
A"
8 0
Vibration formsa
20K 1
11 3
Lu
Lv
LW
Tu
T"
TW
2
4
5
6
7
8
9
__
104
71
a)-0.210 01-0.088
-0.015 -0.036
-0.205 -0.010
-0.025 -0.072
0.123 -0.112
0.021 0.111
0.912 0.161
__
72
64
a)-0.147 O ) 0.55
0.141 -0.246
0.155 -0.414
0.029 -0.532
0.059 0.032
-0.128 -0.297
0.036 0.000
66
69
60
a) 0.241
-0.563
a) 0.313 -0.331
0.169 -0.250
0.141 0.276
0,083 -0.141
0.230 0.333
0.015 0.201
I
X 10
57
48
a) 0.527 01-0.083
0.262 0.103
-0.129 -0.050
0.212 0.364
0.514 -0.012
-0.045 0.411
-0.043 -0.045
52
44
a) 0.147 O ) 0.257
-0.314 0.073
0.164 0.560
0.091 -0.150
-0.124 0.305
-0.247 -0.208
0.252 -0.087
43
41
a)-0.374 -0.231 01-0.316 -0.133
0.179 -0.064
0.198 -0.063
0.287 0.635
-0.250 0.132
-0.090 0.190
33
30
a) 0.290
-0.050 0.025
0.010 -0.215
-0.164 0.043
0.260 0.315
0.576 -0.531
0.100 -0.002
a)-0.371 -0.136 0 ) 0.120 0.248
0.021 0.336
0.513 -0.255
0.130 -0.173
0.526 -0.042
-0.107 -0.013
44
01-0.216 28
26
118
72
14
9
a) 0.217 -0.077 0)-0.400 0.054
0.047 -0.364
0.560 -0.156
-0.517 -0.096
-0.088 -0.134
0.094 0.027
128
103
a)-0.063 -0.039 a) 0.231 0.340
0.480 -0.035
-0.040 0.032
0.065 0.053
0.049 -0.080
0.131 0.747
111
100
a)-0.044 -0.081 a) 0.010 -0.541
0.613 0.335
-0.242 0.216
0.130 0.188
-0.086 0.025
0.086 -0.195
96
86
a) 0.076 0.629 0)-0.061 -0.069
0.469 -0.327
0.207 -0.291
-0.077 -0.064
0.174 -0.029
0.104 -0.288
81
75
ab0.115 0.254 a)-0.191 -0.659
-0.259 -0.082
0.163 -0.239
0.113 0.076
0.083 -0.010
0.155 0.495
104
73
a)-0.172 0.056 0)-0.233 0.203
-0.091 0.407
0.148 0.082
0.049 -0.203
-0.013 -0.230
0.751 -0.095
69
69
a) 0.095 0.121 01-0.459 0.066
0.157 0.507
0.242 -0.034
-0.083 -0.292
-0.032 -0.111
-0.533 0.175
63
55
ab0.481 a) 0.055
0.436 0.066
-0.077 0.105
-0.231 0.193
-0.340 -0.072
-0.320 0.483
-0.029 0.080
55
55
a) 0.172 0.318 a) 0.239 -0.138
-0.090 -0.194
0.034 0.570
0.203 -0.341
-0.287 0.417
-0.080 0.040
TABLE A-14 (continued)
v calc.a
v exp.a
20K 1
Vibration formsa
I
II
LU
Lv
L,
TU
2
3
4
5
6
7
48
41
a) 0.005
0.421 0.122
-0.233 0.454
0 ) 0.368
BU
x
T" 8
Tw 9
-0.201 -0.079
0.200 0.368
0.348 -0.209
-0.153 -0.046
10
44 39
36
a) 0.798 0.185 0)-0.160 0.062
-0.076 0.105
-0.197 0.065
-0.069 0.150
-0.174 0.331
0.230 0.141
31
26
25
a)-0.174 0b0.360
0.093 0.247
0.002 -0.136
0.073 0.026
0.669 0.380
-0.365 0.095
-0,094 -0.051
__
15
18
a) 0.092 -0.051 0 ) 0.539 -0.063
0.023 0.256
0.509 -0.310
0.187 -0.136
-0.376 0.273
0.028 -0.053
Wee notations to Table A-13.
TABLE A-16 Thiophtene ( C ~ H ~ S Z Pbca, ), z = 4(Ci)
A=
1
1 0.t40
b -0.333 0.&31 0.368 0.924 0.105 w -0.678 0.188 0.714 u v
,I = 522.0 a.u.m.A2 I, = 381.5 a,u.m.A2
,I = 140.5 a.u.m.A2
v exp.*
300K 1
$.
99 53
_-
117
v ca1c.b
Vibration forms
I
I1
I11
2
3
4
5
6
7
78 52 25
91 51 32
85 50 28
0.055 0.982 -0.184
0.361 0.152 0.929
0.931 -0.117 -0.345
81 51 44
108 57 35
100 53 32
0.310 0.951 0.012
0.166 -0.066 0.984
0.936 -0.303 -0.178
91 50 28
94 62 51
87 55 48
-0.305 0.502 0.810
0.666 -0.496 0.558
0.681 0.709 -0.183
84
101
94
0.200
0.260
0.945
8
9
10
B3g
85 (38)
70 37
83 38
74 37
93 34 27
85 38 29
87 36 27
0.951 0.036 -0.298
98 41
90 36
0.141 0.990 0.894 0.453
B2u
77 51
80 45
86 51
82 47
B3u
86 51
79 41
90 57
82 51
-0.604 0.772
0.792 0.552
-0.090 -0.316
IP
0 Ip
-0.057 0.560 0.950
0.145 0.828 0.102 0.990 -0.141
-0.453 0.894 0.000 1.000
aRaman and Infrared at 300K, our data. bI, 11, and 111: Calculation with -12 + AAP116, AAP13 + M I 1 6 and AAp/3+AAPl17;crystal structure is given at 293K C321.
1.000 0.000
TABLE A-16 Chloroform (CHC13), Pnma, z = 4(CJ U
A= v W
a b C 0.969 0.000 0.246 0.000 1.000 0.000 -0.246 0.000 0.969
I, = 291.9 a.u.m.A2 I, = 152.9 a.u.m.A2 I, = 150.6 a.u.m.A2
v ca1c.b
v exp.a llOK 1
180K 2
I
Vibration forms
3
I1 4
I11 5
52 37 24
69 38 27
75 42 29
65 56 34
71 64 37
79 71 41
75 57 52
85 56
55
93 64 62
78 52 19
83 57 18
93 63 21
6
7
8
0.820 -0.277 -0.501 0.523 0.599 0.606
10
0.042 0.844 -0.535 -0.552 -0.304 0.777
-0.507 -0.090 0.857 0.963 0.028 0.269
9
0.571 0.460 0.680 0.650 -0.741 0.172
0.794 -0.435 0.424 0.118 0.852 -0.510
11
0.335 0.896 0.292 -0.243 0.523 0.817
P
AU
68 47 32
prohibited
69 52 38
78 58 41
-0.194 0.981 0.O 18
-0.253 -0.068 0.965 -0.516 0.857
B2u
B3u
---
_--
57 52
77 52
86 57
_-
--
69 50
74 75
81 65
__
__
0.991 -0.043
0.948 0.182 0.261 0.830 0.500
0.043 0.991 0.515 0.857
~~
*Raman 1331. bI, 11, I11 are calculations with AAFV18, AAp/19, AAP/20;crystal structure at 185K [34].
w
Q)
0.127 -0.211 0.000 0.900
-0.211 0.127
0.831 -0.499
TABLE A-17 Bromoform (CHBrS), P-phase, Pi, z = 2(C1) b
C
I, = 408.2 a.u.m.A2 I, = 408.2 a.u.m.A2 I, = 804.4 a.u.m.A2
v exp.a
%!
Au
v ca1c.b
Vibration forms (220K)
203K 1
84K 2
220K 3
14K 4
Lu
L,
Lw
Tw
6
7
Tll 8
T"
5
9
10
(60) 48 39 32 29 27
(67) 52 47 38 33 30
55
46 35 30 21 21
67 54 43 37 26 37
0.191 0.247 -0.220 0.650 0.256 0.605
-0.179 0.101 0.782 0.346 0.408 -0.244
0.620 0.717 0.136 -0.221 -0.078 -0.168
0.006 -0.070 0.380 -0.572 0.193 0.697
0.088 -0.045 -0.382 -0.251 0.851 -0.239
0.734 -0.639 0.176 0.137 -0.005 -0.052
50 38 31
58 47 38
-0.224 0.232 0.947
-0.063 0.966 -0.252
0.973 0.116 0.202
aRaman spectra, our data. bCalculation with AAP/26; crystal structure is determined in Ref. [38].
TABLE A-18 Bromoform (CHBra), y-phase, PB, z = 2(C3)
b
v exp.a 85K
I,, = 409.0 a.u.m.A2 I, = 409.0 a.u.m.A2 I, = 806.2 a.u,m.A2 v calc., 14Kb
I
I1
81
Vibration forms
LU
L
Lw
TU
T"
A,
74 52
65
65 54
E,
43
49
41
0.682 0.585
-0.585 0.682
-0.369 -0.238
0.238 -0.369
E,
38
34
29
0.421 -0.129
0.129 0.421
0.815 -0.376
0.376 0.815
A"
64
54
E"
50
41
0.456 0.890
0.890 -0.456
1.000
0.000
0.999 -0.031
0.031 0.999
Tw
0.000 0.000
0.000 0.000
*Raman, our data. b 1 Calculation with AAP/24,11: Calculation with AAP/26; crystal structure is determined at T = 14 K in Ref. 1383.
TABLE A-19 1,3,5,7-Tetraoxcane (C4H804), CWc, z = 4(c2)
b
I, = 199.9 a.u.m.A2 1, = 352.7 a.u.m.A2 I, = 185.1 a.u.m.A2
v exp.a
v ca1c.b
300K
I
61 38
85 40
88 77 48 32
95 84 34 19
31
27
68 60
66 36
Vibration forms LU
L,
Lv?
TU
0.981 -0.197 0.134 0.701 0.655 0.249
*Raman, infrared 1353. bI, calculation with AAP/14; crystal structure a t 300K [39].
Tv?
0.197 0.981 -0.105 0.708 -0.685 -0.137
-0.037 -0.081 -0.270 0.959
0.985 -0.023 -0.172 -0.013 1.000
0.000
0.956 0.093
T"
-0.093 0.956
0.000 0.000
0.000 0.000
TABLE A-20 1,3,5-Trioxane (C3H6031, R3c, z = 2(C3)
b C u [0.;40 -0.298 -0.5421 0.000 0.835 -0.462 0.453 0.550 0.702
1, = 96.6 a.u.m.A2 I, = 96.6 a.u.m.A2 I, = 173.6 a.u.m.A2
I
v exp.a
E
A2
v ca1c.b
Vibration forms (103K)c
I
II
300K
lOOK
103K
300K
I11 103K
86
98
119
86
93
0.313 0.870
0.870 -0.313
0.247 -0.289
0.289 0.247
62
68
97
75
83
-0.522 0.612
0.612 0.522
0.576 0.145
-0.145 0.576
50
36
42
0.296 -0.347
-0.347 -0.296
0.755 -0.477
-0.477 -0.755
132 76
105 63
115 68
0.874 -0.486
0.486 0.874
79
61
73
1.000
0.000
62
68
Lu
L,
8Rama.n [351. bI, 11: Calculation with AAP/14 and AAP/15; crystal structure at [361 and a t 103K [371. Fhvo forms of degenerate vibrations correspond to E-modes.
Lw
Tu
T"
Tw
441
APPENDIX
REFZRBNCES TO APPENDIX 1 1. D. E. Williams, J. Chem. Phys., 45,3770 (1966).
2. A. I. Kitaigorodski, MoL Cryst, USSR, Moscow, Nauka, 1971,424 pages. 3. D. E. Williams,J. Chem. Phys., 47,4680 (1967).
4. D. E. Williams, Comp. Chem., I, 173 (1971).
5. D. E. Williams, Acta Cryst., 30,71(1974). 6. G. Taddei, M. Bonadeo, M. P. Mamocci, and S. Califano, J. Chem. Pays., 58 (3), 966 (1973). 7. T. L. Starr and D. E. Williams, Acta Cryst, A33,771(1977). 8. G. Taddei, R. Righini, and P. Manzelli, Acta Cryst,A33 (41,626 (1977).
9. R. Righini, S. Califano, and S. H. Walmsley, Chem. Phys., 50 (l),113(1980). 10. G. A. Mackemie, G. S. Pawley, and 0. W. Dietrich, J. Phys. C: SOL State Phys., 10,2723 (1977). 11. B. M. Powell, G. Dolling, and H. Bonadeo, J. Chem. Phys., 69 (121, 5288 (1978).
12. T. Wasiutynsky, A. van der Avaird, and R. M. Berns, J. Chem. Phys., 69 (121, 5288 (1978). 13. S. R. Cox, L. Y.Hsu, and D. E. Williams, Acta Cryst., 37,293 (1981).
14. V. V. Nauchitel and K. V. Mirskaja, Kristallografiya (USSR),6,507 (1961). 15. E. Giglio, L. Mazzarelli, and A. M. Liquori, Nuovo Cimento, 25B,57 (1968). 16. H. Bonadeo and E. D'Alessio, Chem. Phys. Lett, 19 (11,117 (1973). 17. J. B. Bates and W. R. Busing, J. Chem. Phys., 60 (61,2414 (1974). 18. P. A. Reinolds and J. K. Kiems, in "Symp. on Neutron Inelastic Scattering", (F. M. Makkof, ed.), IAEA-SM-155llB2,p. 195 (1972).
19. K. Mirski and M. D. Cohen, Chem. Phys., 28,193 (1978). 20. A. Cavezzotti and M. Simonetta, Acta Cryst, 31 (l), 645 (1975). 21. L. Y. Hsu and D. E. Williams, Acta Cryst, 36,277 (1980).
APPENDIX
442 22. E. Burgos and H. Banadeo, Chem. Phys. Lett., 49 (91,475(1977).
23. T.Luty,A. van der Avoid, and A. Mienejewski, Chem. Phys. Lett., 61 (1),10 (1979). 24. K, V. Mirskqja and V. V. Nauchitel, Kristallografiya (USSR), 17,73(1972).
25. G.Faerman and H. Bonadeo, Chem. Phys. Lett., 69 (1),91(1980). 26. L. F.Konshina, K. V. M i r s k ~ aV. , M. Kozhin, and I. E. Kozlova, prikl Teor. Fiz (USSR),vip. 6,Alma-Ata, 1974.
REFERENCES TO APPENDIX 3 1. W. D.Ellensen and M. Nicol, J. Chem. Phys., 61,1380(1974). 2. M.It0 and T. Shigeoka,Spectrochim.Acta, 22,1029 (1966). 3. Y.A. Sataty and A. Ron, Chem. Phys. Lett., 23,500(1973). 4.
G. Bacon, N. Curry, and S.Wilson, Proc. R Soa, 279A,98 (1964).
5. M.Suzuki and K Yokogama, Spectrochim. Acta, 24A,1091 (1968).
6. I. Harada and T. Shimannuchi, J. Chem. Phys., 44,2016(1966). 7. D.W. Gruikshrank,Acta Cryst., 10,504(1957). 8. Y.A.Vovelle and A. Ron, C. R Acad Sci., 1279,125 (1974).
9. R.Mason, Acta Cryst., 17,547(1964). 10. I. Kay, Y.Okaya, and D. E.Cox, Acta Cryst., B27,26 (1971). 11. C. Corradini and G. Avitabile, Eur. Polym. J., 4,385(1968). 12. D.M.Burns and I. Iball, k. R Soc., 257,497(1960). 13. A. Bree,R.A. Kydd, T. N. Misra, andV. V. B. Vilkos, Spectrochim. Acta, 27A, 2315 (1971). 14. R.Zallen, C. H. GrifEts, M. L. Slade, M. Hayak, and 0. Brafman, Chem. Phys. Lett., 39,85(1976). 15. F. Brehat, B. Wincke, and A. Hadni, Proc. 12th Eur. Congr. MoL Spectrosa, Strasburg, (19751,Chapter II, p. 225.
APPENDIX
443
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APPENDIX
SUBJECT INDEX
Coulomb interaction, 49 cyclohexane, 78,82,120,202,219, 239,241,267,415 cyclopentane, 220
A
AAP method, 71 acetonitrile, 306 anharmonic oscillators, 101
anthracene, 47,62,82,171,174,408 atom potentials, 2 atom-atom potentials, 38 atomic displacements, 18
D Davydov splitting, 32,42,60,345 Debye temperature, 21,104 kp-dichlorobenzene, 166 1,2-dichloroethane, 232 l,Z-diiodobenzene, 58 disorder in molecular crystals, 59 disordered structures, 62 durene, 47 dynamical matrix, 37
B bandwidths, 247 benzene, 47,56,78,82,88,97,405 biphenyl, 47,82,141,171,291,294, 296,302,416 Born von Karman model, 243 Bragg-Williams approximation, 232, 244 Brillouin zone, 30,51 bromoform, 62,268,269,276,279 Brot-Lassier model, 65
E ethylene, 50,78,82, 97,172,414 Euler angles, 342 external vibrations, 130
C
F 1,3,5-C&$13,52 C&hj, 52 camphor, 232 carborane, 224 center of mass, 12 chloroform, 268,284 chryzene, 82,410 coz crystal, 95 computerized calculations, 396 conformationally unstable molecules, 291 coronene, 413
factor group, 24 far infrared region, 151 far infrared spectra, 208 FG-matrix method, 129 force constants,19 Frenkel model, 64,66,69 furane, 231
G gas-crystal transition, 17
445
446
SUBJECT INDEX
Griineisen parameters, 104
oriented gas, 167 oxylene, 305
H P hexamethylbenzene,305
I internal vibrations, 26 iodoform, 62,268,269,279
L Landau theory, 210 lattice dynamics, 393 Lennard-Jones-Devonshiremodel, 232,242
London forces, 44 Lorentz contour, 263 Lorentz function, 280 low fkequency spectra, 262,333,354 low frequency vibrations, 304,404
P-C&c12,52 p-carborane, 67 p-terphenyl, 70 p-xylene, 82,306,307,329,417,418 permutation symmetry, 36 phase transitions, 202,222 phenanthrene, 47,78,82,120,133, 332-335,349,353,409
plastic crystals, 202 plastic phase, 65 polarizability tensor, 170 polymorphism, 202 Pople-Karasz models, 244 Port0 notation, 160 premelting effects, 245 pyrene, 47,79,133,332,354,355, 358,365,379,411
Q M Mathieu equation, 326 Maxwell rule, 236 methane, 100 methyl barriers, 331 methyl groups, 304 molecular crystals, 81
N
quasi-harmonic approximation, 247, 340
R Rayleigh line, 322 Rayleigh wing, 253,260 Rayleigh-Schrodingertheory, 54 reorientational barrier, 250,305,315 reonentational motion, 245
naphthalene, 46,47,63, 74, 78,80,82, 97,133,171,407
naphthalene-dg, 51 Newton-Rafson method, 343,371 nitrochlorobenzene, 23 nitrogen, 93 0
order-disorderphenomena, 243 organic crystals, 152,304,404
S
Stober method, 157 subgroup, 25
T 1,2,4,5-tetrachlorobenzene,58 1,2,4,5-tetracyanobenzene,68 tetraoxocane, 85 thiophene, 220,224
SUBJECT INDEX thiophthene, 176 toluene, 69,82,306,315,324,329,419 torsional potentials,323 translational symmetry, 22,113 translational vector, 24 trioxane, 85
V van der Waal's radii, 42
W Wilson's matrix, 10
447
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