Advances in Molecular Structure Research, Volume 3, First Edition (Advances in Molecular Structure Research)

ADVANCES IN MOLECULAR STRUCTURE RESEARCH Volume3

9

1997

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ADVANCES IN MOLECULAR STRUCTURE RESEARCH Editors: M A G D O L N A HARGITTAI Structural Chemistry Research Group Hungarian Academy of Sciences E6tv6s University Budapest, Hungary ISTV,'~N HARGITTAI Budapest Technical University and Hungarian Academy of Sciences Budapest, Hungary ,

,,

VOLUME3

*

1997

JAI PRESS INC.

Greenwich, Connecticut

London, England

Copyright 91997 by JAI PRESSINC. 55 Old Post Road, No. 2 Greenwich, Connecticut 06836 JAI PRESSLTD. 38 Tavistock Street Covent Garden London WC2E 7PB England All rights reserved. No part of this publication may be reproduced, stored on a retrieval system, or transmitted in any form, or by any means, electronic, mechanical, photocopying, filming, recording, or otherwise, without prior permission in writing from the publisher. ISBN: 0-7623-0208-9 Manufactured in the United States of America

CONTENTS

LIST OF CONTRIBUTORS PREFACE

Magdoina Hargittai and Istvgn Hargittai

vii

xi

DETERMINATION OF RELIABLE STRUCTURES FROM ROTATIONAL CONSTANTS

Jean Demaison, Georges Wlodarczak, and Heinz Dieter Rudolph

EQUILIBRIUM STRUCTURE AND POTENTIAL FUNCTION" A GOAL TO STRUCTURE DETERMINATION

Victor R Spiridonov

STRUCTURES AND CONFORMATIONS OF SOME COMPOUNDS CONTAINING C-C, C-N, C-O, N-O, AND O-O SINGLE BONDS" CRITICAL COMPARISON OF EXPERIMENT AND THEORY

Hans-Georg Mack and Heinz Oberhammer

53

83

ABSORPTION SPECTRAOF MATRIX-ISOLATED SMALL CARBON MOLECULES

Ivo Cermak, Geroid Monninger, and Wolfgang Kr~tschmer

SPECIFIC INTERMOLECULAR INTERACTIONS IN ORGANIC CRYSTALS" CONJUGATED HYDROGEN BONDS AND CONTACTS OF BENZENE RINGS

Petr M. Zorky and Olga N. Zorkaya

117

147

vi

Contents

ISOSTRUCTURALITY OF ORGANIC CRYSTALS: A TOOL TO ESTIMATE THE COMPLEMENTARITY OF HOMO- AND HETEROMOLECULAR ASSOCIATES

Alajos K~lm~n and L~szl6 P~rk~nyi

AROMATIC CHARACTER OF CARBOCYCLIC g-ELECTRON SYSTEMS DEDUCED FROM MOLECULAR GEOMETRY

Tadeusz Marek Krygowski and Micha! Cyrahski

COMPUTATIONAL STUDIES OF STRUCTURES AND PROPERTIES OF ENERGETIC DIFLUORAMINES

Peter Politzer and Pat Lane

CHEMICAL PROPERTIES AND STRUCTURES OF BINARY AND TERNARY SE-N AND TE-N SPECIES: APPLICATION OF X-RAY AND AB INITIO METHODS

Inis C. Tornieporth-Oetting and Thomas M. Klap6tke

SOME RELATIONSHIPS BETWEEN MOLECULAR STRUCTURE AND THERMOCHEMISTRY

Joel F. Liebman and Suzanne W. Slayden

INDEX

189

227

269

287

313 339

LIST OF CONTRIBUTORS

lvo Cermak

Max-Planck Institute for Kernphysik Heidelberg, Germany

Micha,l- Cyranski

Department of Chemistry University of Warsaw Warsaw, Poland

Jean Demaison

Laboratoire de Spectroscopie Hertzienne Universit~ de Lille Villeneuve d'Acq, France

Alajos K~lm~n

Central Research Institute of Chemistry Hungarian Academy of Sciences Budapest, Hungary

Thomas M. Klap6tke

Institute of Inorganic Chemistry University of Munich Munich, Germany

Wolfgang Kratschmer

Max-Planck Institute for Kernphysik Heidelberg, Germany

Tadeusz Marek Krygowski

Department of Chemistry University of Warsaw Warsaw, Poland

Pat Lane

Department of Chemistry University of New Orleans New Orleans, Louisiana

Joel F. Liebman

Department of Chemistry and Biochemistry University of Maryland Baltimore County Baltimore, Maryland vii

viii

LIST OF CONTRIBUTORS

Hans-Georg Mack

Institut ffir Physikalische und Theoretische Chemie Universit~it TLibingen T~Jbingen, Germany

Gerold Monninger

Max-Planck Institute fiJr Kernphysik Heidelberg, Germany

Heinz Oberhammer

Institut ffir Physikalische und Theoretische Chemie Universit~it TLibingen TCibingen, Germany

L,~szl6 P,#k,~nyi

Central Research Institute of Chemistry Hungarian Academy of Sciences Budapest, Hungary

Peter Politzer

Department of Chemistry University of New Orleans New Orleans, Louisiana

Heinz Dieter Rudolph

Department of Chemistry University of UIm UIm, Germany

Suzanne W. 51ayden

Department of Chemistry George Mason University Fairfax, Virginia

Victor P. 5piridonov

Department of Chemistry Moscow State University Moscow, Russia

Inis C. Tornieporth-Oetting

Department of Chemistry University of Glasgow Glasgow, Scotland

Georges Wlodarczak

Laboratoire de Spectroscopie Hertzienne Universit~ de Lille Villeneuve d'Ascq, France

OIga N. Zorkaya

Department of Chemistry Moscow State University Moscow, Russia

ix

List of Contributors Petr M. Zorky

Department of Chemistry Moscow State University Moscow, Russia

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PREFACE Progress in molecular structure research reflects progress in chemistry in many ways. Much of it is thus blended inseparably with the rest of chemistry. It appears to be prudent, however, to review the frontiers of this field from time to time. This may help the structural chemist to delineate the main thrusts of advances in this area of research. What is even more important though, these efforts may assist the rest of the chemists to learn about new possibilities in structural research. It is the purpose of the present series to report the progress in structural studies, both methodological and interpretational. We are aiming at making it a "user-oriented" series. Structural chemists of excellence evaluate critically a field or direction including their own achievements, and chart expected developments. The present volume is the third in this series. We would appreciate hearing from those, producing structural information and perfecting existing techniques or creating new ones, and from the users of structural information. This would help us gauge the reception of this series and shape future volumes. We would like to make a special acknowledgment with respect to Volume 3. Much of the editorial work has been done at the University of North Carolina at Wilmington where we held visiting positions during the academic year 1996/97 (MH as Visiting Scientist and IH as Distinguished Visiting Professor). We thank the Department of Chemistry of UNCW and our colleagues for excellent working conditions and for helpful interactions that were instrumental in the completion of this volume. Magdolna and Istvfin Hargittai Editors

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DETERMINATION OF RELIABLE STRUCTURES FROM ROTATIONAL CONSTANTS

Jean Demaison, Georges Wlodarczak, and Heinz Dieter Rudolph Abstract

Io II.

III.

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

D e t e r m i n a t i o n o f the E q u i l i b r i u m Rotational C o n s t a n t s . . . . . . . . . . . . . A. Centr ifugal C o r r e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4

B.

Electronic C o r r e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

C.

Vibrational C o r r e c t i o n

6

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structures f r o m O n l y the G r o u n d State Rotational C o n s t a n t s A.

IV.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

rz Structure

..........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 10

B.

rm Structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

C. D.

rc Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rPm Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 17

E.

rIe Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Least-Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

A.

19

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 1-51 Copyright 9 1997 by J A I Press Inc. AH rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

2

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

B. Assumptions of the Least-Squares Method . . . . . . . . . . . . . . . . . C. Collinearity or Ill-Conditioning . . . . . . . . . . . . . . . . . . . . . . . D. Corrective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Outlier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Sources of Additional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Ab Initio Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Liquid Crystal NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . D. Empirical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIo Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Fluorophosphaethyne, F C ~ P . . . . . . . . . . . . . . . . . . . . . . . . B. Formyl Cation, HCO + . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Phosgene, COC12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Chloroacetylene, HC~CCI . . . . . . . . . . . . . . . . . . . . . . . . . E. Methyl Chloride, CH3C1 . . . . . . . . . . . . . . . . . . . . . . . . . . F. Difluoroethyne, F C ~ C F . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 24 26 26 28 29 30 31 31 32 32 32 34 35 36 37 39 41 46 46

ABSTRACT The different methods of obtaining an accurate near-equilibrium molecular structure by fitting the rotational constants of a set of isotopomers are reviewed. The strong points of these methods and their limitations are discussed. One of the main and most often encountered problems is that of an ill-conditioned fit. This situation is more clearly revealed when the condition indexes and the variance decomposition-proportions are calculated. It can be overcome by including additional data from different sources (electron diffraction .... ). In particular, the usefulness of combining experimental data with the results of ab initio calculations is emphasized. A mixed fit of all available data allows us to check their compatibility and to improve the accuracy of the derived parameters. This method is illustrated by some typical examples.

i. I N T R O D U C T I O N The best description of the geometry of a molecule is the equilibrium (re) structure. An equilibrium internuclear distance is the distance between two hypothetically motionless nuclei and corresponds to the m i n i m u m of the potential energy function. A b initio calculations yield theoretical equilibrium geometries. The re structure is particularly important because of its well defined physical meaning, furthermore it is isotopically invariant (in the frame of the Born-Oppenheimer approximation), finally it allows one to make meaningful comparisons between the results of different methods and between the structures of different molecules. Although there

Determination of Reliable Structures from Rotational Constants

3

are many different ways to determine the structure of a molecule (ab initio calculations, electron diffraction, etc.), we will mainly concentrate in the present chapter on the determination of accurate molecular structure through rotation and vibration-rotation spectroscopy, or more precisely on the extraction of the molecular geometry from rotational constants. This aspect of the problem has been recently reviewed [1], therefore we shall mainly discuss, wherever possible by way of examples, the available methods to obtain reliable structures, i.e. structures with an uncertainty less than 0.2 degree for the angles and 0.002 A for the distances. Generally it is relatively easy to obtain the moments of inertia of a molecule in its vibrational ground state (a well known exception is the axial moment of inertia of a symmetric top), but as the nuclei undergo vibrational motions about their equilibrium positions, the experimental ground state moments of inertia (Io) are different from the equilibrium ones (Ie) and the difference depends in a complicated way on the force field (mainly quadratic and cubic). We will first discuss how to determine the equilibrium moments of inertia. This method suffers from many limitations, particularly it is difficult to use and is limited to small molecules. We will then describe different approximate methods devised to obtain a "nearequilibrium" structure from the ground state moments of inertia only. Almost all the methods use a least-squares fit to calculate the geometrical parameters from the moments of inertia. We will show that it can be one of the main weak points of all these methods and we will describe how it is possible to improve this situation. The effective structure (r o) and the substitution structure (r s) are not discussed here because they have been reviewed only recently [1, 2, 3, 4]. They are purely empirical structures and are sometimes only poor approximations of the equilibrium structure. Structure data of molecules have been collected in a series of volumes of group II of the new series of Landolt-Brrnstein. Volume 11/23 contains the most recent (up to 1994) experimentally determined structures [5], whereas volume 11/22 supplements these data by ab initio calculated structures [6]. These tabulations are frequently revised to bring them up to date.

II. DETERMINATION OF THE EQUILIBRIUM ROTATIONAL CONSTANTS Analysis of vibrotational spectra yields rotational constants Bg, g = x, y, z, whereas moments of inertia, Ig = K/Bg, are the experimental data normally used in a structure determination (K = h/8n 2 = 505379.07(61) MHz uA2). The determination of the experimental rotational constants will not be discussed here because the methods are classical and are described at length in general textbooks [2, 7] as well as in a recent review [3]. However, it is worth mentioning that important progress has been made towards obtaining accurate axial constants of symmetric tops [8, 9, 10]. Also, thanks to the high sensitivity of the microwave Fourier transform spectroscopy, it

4

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

is now much easier to determine the rotational constants of an isotopic species in natural abundance [11, 12]. The most important correction to the ground state rotational constants is the vibrational correction, but even when this has been taken into account, the constants obtained from fitting the rotational transitions will have small contributions from centrifugal distortion terms as well as from electron-rotation interaction effects for which corrections must be made.

A. Centrifugal Correction When a fit of the rotational spectrum of a nonplanar asymmetric top molecule was attempted, for the first time including also centrifugal distortion correction, it was found that the six quartic centrifugal distortion constants were fully correlated and hence not independently determinable. This problem was solved by Watson [13] who subjected the rotational Hamiltonian to a unitary transformation. The transformed rotational and distortion constants can be divided into three groups: 9 the constants of the first group are not affected by the transformation and are determinable, 9 the constants of the second group are only marginally affected by the transformation. To this group belong the rotational constantsmthey are determinable but must be corrected in order to obtain the "true" constants, and 9 the constants of the third group (centrifugal distortion constants) are significantly affected by the transformation--they cause the correlation problem, and at least one of them must be removed from the Hamiltonian. In conclusion, the rotational constants of an asymmetric top obtained from a fit using the Hamiltonian of Watson [13] are affected by a small centrifugal distortion contribution which depends on the choice of the reduction and of the representation. Watson has shown that the following linear combinations can be determined from the analysis of the spectra: B z = B ~A) + 2Aj = B ~s) + 2Dj + 6d 2

(la)

B x = B(xA) + 2As + AjK - 2fig- 25 K = B(xs) + 2Dj + DjK + 2d 1 + 4d 2 (lb) By = B~A) + 2Aj + AjK + 25j + 25 K = B~s) + 2Dj + D j K - 2d 1 + 4d 2 (lc) B~A) are the experimental constants in the A reduction, B~s) are the experimental constants in the S reduction, and Bg are the determinable constants (where g = x, y, z). However, these latter constants are still contaminated by the centrifugal distortion. As shown by Kivelson and Wilson [14], the true rigid rotor constants B~ are given by:

Determination of Reliable Structures from Rotational Constants

5

]'able ~. Centrifugal, Electronic, and Vibrational Contributions to the Moments of

Inertia (in u~ 2)

Corrections:

802 b

NO2 c

ONF d

Notes:

Centrifugal AI~d

Ia

0.000014.

Ib Ic

0.~ -0.001660

Electronic A I ~ ec

Vibrational A I ~ b

-0.002757

-0.003268 -0.002303

0.037978

-0.194612 -0.294790

I ea

8.350324(72)

48.783084(350) 57.132620(500)

Ia

0.0(0)O

-0.00231

0.02762

Ib Ic

0.00151 -0.00191

-0.04213 -0.04460

-0.23178 -0.29926

2.1319~(5)

Ia

0.00001

-0.00275

0.01551

Ib

0.00(~

-0.00257

-0.18780

42.47919(10)

Ic

-0.00155

-0.00169

-0.29035

47.80038(15)

38.59636(53) 40.726I_.Q(76) 5.32192(3)

aI a9 I a lelec + v~ The insignificant digits are underlined. = o + A I cd a +A -a Al~t. bRef. 27. CRef. 123. dRef. 124.

1 1 Bx = Bx + -~ (Xyyzz + X~yxy+ r'xzxz) + -~ Xyzyz

(2)

p

By and B"z are obtained by cyclic permutation of x, y, z. The x constants have units of MHz. The problem is that the z constants are experimentally determinable only for a planar molecule by means of the planarity relations of Dowling [15]. For a nonplanar molecule they can be calculated from the harmonic force field [16]. Compared to the other corrections, the centrifugal distortion correction is generally quite small even for very light molecules (see Table 1). Furthermore, it is different from zero only for asymmetric top molecules.

B. Electronic Correction Before using the rigid rotor constants for a structure determination, it may also be necessary to correct them for electronic contribution. This is due to the fact that the distribution of electrons contributes to the moments of inertia because an atom in a molecule is not a mass point and the center of mass of the electrons in an atom generally does not coincide with the position of the nucleus. The electronic contribution is related to the molecular g factor by the following relation [2, 17,

181, B~Gt

Bexp 9 1 + (m/Mp)gaa

(3)

where gas is expressed in units of the nuclear magneton, m is the electron mass, Mp the proton mass, and ct = a, b, c. The g-factor can be obtained experimentally from

6

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

the analysis of the Zeeman effect on the rotational spectrum [2, 17, 18]. The experimental values are tabulated in [19]. When no experimental value is available, it is now possible to calculate the g-factor ab initio [20]. The problem is that the magnetic vector potential depends upon the choice of the gauge origin. To solve that problem, the standard gauge-dependent approach [20] may be used. Another possibility is to use multiple origin gauges for localized orbitals (IGLO) [21, 22]. The electronic correction, although often much greater than the experimental uncertainty, is small compared to the vibrational correction (see Table 1). For a very light molecule, the electronic correction can be large, but it decreases rapidly as the corresponding rotational constant decreases. For instance, for LiD, CO, and OCS, it is (in MHz): 18.8, 8.48, and 0.095, respectively, with the corresponding rotational constants being (in MHz): 126905, 57899, and 6081 [2].

C. Vibrational Correction Due to the zero-point vibrations, the rotational constants of the vibrational ground state are different from those of the hypothetical vibrationless equilibrium configuration. In principle, it is possible to obtain the equilibrium constants if the rotational constants for all the vibrationally excited states can be determined. For the lower, and hence more populated vibrational states, they can be determined by rotational spectroscopy. For the higher vibrational states, the analysis of the rotational structure of the vibrational spectrum is often the best method.

From the Analysis of the Excited States The rotational constants in the vibrational ground state and excited states are related to those in the equilibrium configuration by the series expansion,

gvmXe-ECs X i

i+

+EE~[il.i i<j

+...

(4)

where X = A, B, C. The summations are over all the vibrational states, each characterized by a quantum number vi and a degeneracy di. For a nonlinear molecule with N atoms, there are 3N-6 modes of normal vibrations. The widespread presence of resonances (anharmonic or Coriolis), as well as the large number of vibrationrotation interaction constants (a, 7. . . . ) to be determined, explain why this method is limited to molecules containing a small number of atoms. The convergence of the Eq. 4 series has been studied for a few small molecules: SO2 [23], HCN [24, 25], OCS [26], etc. For most molecules it converges rapidly (when the resonances are properly taken into account, otx is about 2 orders of magnitude smaller than X, and y x 2 orders of magnitude smaller than otx), nonetheless the neglect of the 7q may lead to errors. In some cases the inclusion of third-order terms (eijk) has been found necessary [25, 26, 27].

Determination of Reliable Structures from RotationalConstants

7

Table 2. Comparison of the "Best" Equilibrium Structure (in ~) and the Structure Determined Using the Experimental (t-Constants and Neglecting Anharmonic Interactions Molecule

Parameter

"Best" re

re(a )

OCS a

r(C=O) r(C=S)

1.15617(14) 1.56140(14)

1.1545(2) 1.5630

0.00167(20) -0.00160

HCO + b

r(C-H) r(C-----O)

1.0919(9) 1.1055(3)

1.09725(4) 1.10474(2)

-0.0053(9) 0.0008(3)

HCPc

r(C-H) r(C=P)

1.0702(10) 1.5399(2)

1.0660(1) 1.54045(2)

0.0042(10) -0.0006(2)

FCP d

r(C-F) r(C--=P)

1.2761 (7) 1.5445(6)

1.284 1.538

-0.008 0.006

Notes:

Ar

aRef.125. bRef. 116. CRef. 126. dRef. 114.

There is still a more serious problem: Eq. 4 assumes that there is no anharmonic resonance between the vibrationaUy excited states (the Coriolis resonances complicate the problem too, but they are less insidious because it is easy to detect their existence). Although such resonances can be analyzed and unperturbed rotational constants can be derived, this is often a tedious task requiring the study of many overtone and combination levels [8]. This is the reason why a full set of vibrationrotation interaction constants has been experimentally determined for only a few small polyatomic molecules (mainly triatomic). The determination of an accurate equilibrium structure, only from the rotational constants, requires not only taking into account all the corrections (Eqs. 1, 2, 3, and 4) but also to carefully analyze all the possible resonances. Therefore, the collection of the necessary experimental data and the detection and analysis of possible anharmonic resonances is not an easy task, not even for simple triatomic molecules, and structures obtained from different research groups at different times may show large variations (see Table 2).

From the Experimental Anharmonic Force Field To overcome this difficulty, it was proposed quite early to calculate the vibrationrotation interaction constants from the force field [28]. This avoids the difficult problem of analyzing the anharmonic resonances. Furthermore, it allows us to obtain without difficulty the a-constants of rare isotopomers. Finally it is quite useful when a vibrational state cannot be experimentally analyzed as for example in the case of SiHF 3 [29] or HBS [30]. However, the number of cubic constants increases rapidly with the number of atoms and the loss of symmetry of the

8

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH Table 3. Number of Quadratic and Cubic Constants for Some Molecules Molecule

Quadratic

Cubic

1 3 4 4 6 6 10 12

1 3 6 6 10 14 22 38

Diatomic Linear XY2 Linear XYZ Bent XY2 Bent XYZ Pyramidal XY3 Planar X2YZ Pyramidal XY3Z

molecule (see Table 3). For this reason, this method is limited to small molecules. Nevertheless, it often gives more reliable results than the direct experimental determination of the vibration-rotation interaction constants (without a careful analysis of the anharmonic resonances)Bsee Table 4 and compare with Table 2. Of course, the accuracy of the structure depends on the quality of the force field. In the particular case of FCP (Table 4), the force field was determined from few experimental data; the vibrational correction could, nonetheless, be calculated with precision (note that the derived structure is not satisfactory because the problem is ill-conditioned as described in Section IV and illustrated in Section VI.A). Table 4. Molecule

Comparison of the "Best" re Structures (in ,g,) with that Obtained Using the a-Constants Calculated from the Force Field Parameter

"Best"

Ref.

Force Field

Ref.

FCN

r(C-F) r(C=N)

1.26405(74) 1.15680(81)

127

1.26504 1.15594

128

C1CN

r(C-CI) r(C=N)

1.6290(24) 1.1606(28)

129

1.62995 1.15923

130

BrCN

r(C-Br) r(C=N)

1.787490(4) 1.159523(6)

131

1.78611 1.15767

132

ICN

r(C-I) r(C=N)

1.99209(22) 1.16044(33)

133

1.99241 1.15965

132

HCP

r(C-H) r(C--=P)

1.0702(10) 1.5399(2)

126

1.0692(8) 1.5398(2)

134

FCP

r(C-F) r(C~P)

1.2761(7) 1.5445(6)

114

1.2717 1.5481

112

HNC

r(N-H) r(N=C)

0.996064(3) 1.168351(2)

136

0.9940(8) 1.1689(2)

135

Determination of Reliable Structures from Rotational Constants Table 5. Triatomic s

Equilibrium Structures Obtained by a Combination of Experimental and Ab Initio Data Tetratomic s

HCN [156], HNC [139] HCO§ 116] HCP [ 126] HOF [ 158] HNSi [155], HOSi + [137] [141], C 3 [140]

9

Pe ntat omic s

BH2C1[144], BH2F [144] HCNH § [142], H2CO [159] HC--=CCI [ 118], HC=CF [ 138] NCCN [ 145, 146] CNCN [145, 147] C3N [148] C30 [154], C3S [143]

S ix-A toms

C50 [153]

CH3F [49]

HNCCN+ [150] H2C--C--O [ 160] HC--------CCN[ 151 ] HC~----CNC [ 152] H 2 C = C = C [ 149]

From the Ab Initio Anharmonic Force Field Recently it has become possible to calculate the differences between equilibrium and ground state moments of inertia from a b initio potential energy functions either variationally (for triatomic and tetratomic molecules) or by means of conventional second-order perturbation theory in normal coordinate space. The perturbational approach has already been applied to many molecules with up to six atoms, mainly by P. Botschwina and his group. Table 5 gives a list of molecules whose equilibrium structure was determined by combining the experimental ground state constants and the a b initio e = Io - le. There are not many reliable experimental data to check the accuracy of this method. However, in the few cases where a comparison is possible, it produces results in excellent agreement with the purely experimental method (see Table 6). Furthermore, as shown in Section IV (and illustrated in Section VI), it is possible to assess the accuracy of the different data through a least-squares fit analysis.

Table 6. Comparison of the Mixed Ab Initio/Experimental Method with the Purely Experimental Method a'b Molecule

HCN HNC CO 2 N20

Notes:

Bond

CH CN HN NC CO NN NO

Experimental

1.06501 1.15324 0.9960 1.1684 1.15996 1.12729 1.18509

Mixed

1.0651 1.1534 0.9956 1.1686 1.16011 1.12702 1.18545

are structures in both cases [139]. bErrors of reasonable magnitude cannot be given because the imperfections of the input data other than those due to measurement were not fully assessed.

10

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

Order of Magnitude of the Vibrational Correction We have just seen that it is difficult to determine with accuracy the vibrational correction e = Io - le. On the other hand it is easy to estimate its order of magnitude. Demaison and Nemes [31] have found an empirical relation between the vibrational correction e and the ground state moment of inertia Io using a statistical treatment of existing data. They found that e varies as Ig with n = 0.611(10) for diatomic molecules and n = 1.247(5) for small polyatomic molecules. This relation is useful to estimate the order of magnitude of e. In particular cases it can also be used to calculate a structure. For instance, in methyl halides, CH3X (X = F, C1, Br, I, CN) ~;A varies roughly like I 1/2 (Ia is the axial moment). Thus it allows us to calculate near-equilibrium coordinates for the H atoms [32]. But this method is not always appropriate to predict the variation of ~ by isotopic substitution. This happens, e.g., when individual contributions of the normal vibrations to e have different signs and add up to an almost zero e. Then the variation of e from isotopomer to isotopomer may appear erratic, and it can no longer be simply estimated. Long linear chains, such as XYYZ, are an interesting special case because the positive contributions to the summation from the stretching modes are almost totally canceled by the negative contributions from the bending modes, resulting in le values very close to the Io values [33]. The consequence is that the ground state moments of inertia lead directly to a structure that should be near the equilibrium structure, at least if the problem is well conditioned (but see also Section VI.F). For instance, Tay et al. [33] calculated the ro structure for diacetylene and showed that it is nearly identical with the equilibrium structure except for the C - H bond length where the ro value is 0.0056/~ shorter than the re value, which is commonly found for a ro value [34].

!!!.

STRUCTURES F R O M O N L Y THE G R O U N D STATE ROTATIONAL

CONSTANTS

Whatever method is used, the determination of a "true" equilibrium structure is limited to small molecules (6 atoms or less). This is the reason why many methods have been devised to determine a structure without explicit knowledge of the vibration-rotation interaction constants. The most useful ones are described hereafter.

A. rz Structure The average (rz) distance is the distance between average nuclear positions in the ground vibrational state at 0 K [35]. Thus it has a clear physical meaning and permits the comparison of structures of different molecules. Furthermore, it differs from the equilibrium structure only because of the anharmonicity of the molecular vibrations. Finally, the most interesting point is that it is possible to extrapolate the

Determination of Reliable Structures from Rotational Constants

11

rz distance to the equilibrium value if we assume that the bond-stretching anharmonicity is the dominating term. In order to obtain the zero-point moments of inertia Izg (g = a, b, c), the ground state constants are corrected for the harmonic contributions to the ct's which are calculated by the classical expressions which may be found, e.g., in ref. 16. The rz structure is then obtained by doing a least-squares fit of the structural parameters to the Izg. It is well established that the rz structure is not isotopically invariant [36-38]. In order to obtain a reliable structure, primary isotopic differences in bond lengths (the isotopic substitution affects at least one atom of the considered bond) as well as secondary isotopic differences (the isotopic substitution does not affect any atom of the considered bond) have to be taken into account in the fit while isotopic differences in bond angles may be neglected. Although the differences in bond lengths are usually small, a very different structure may be obtained, if no allowance is made for them [39]. This is due to the problem of ill-conditioning, as discussed in Section IV. The isotopic changes 8rz in bond lengths are estimated from Kuchitsu's formula [39-40], 3 au 2 rz=re+ ~ -K

(5)

3 8r z = ~ aS(u 2) - 8K

(6)

hence,

where u 2 is the mean-square amplitude for the bond concerned, and K the meansquare perpendicular amplitude correction, both obtained from the harmonic force field, while a is the Morse anharmonicity parameter, the potential function for the stretching displacement Ar of a bond being 1

V ~ ~ k r [Ar 2 - aAr 3]

(7)

Equation 5 is an approximation (diatomic approximation) because it neglects all the anharmonic constants except the bond-stretching anharmonicity (kr is the harmonic-stretching force constant). This assumption was carefully checked in many different molecules and was found to be a reasonable one [41]. The a parameters are often assumed to be equal to those of the corresponding diatomic molecules, for which potential functions have been determined experimentally. Strictly speaking, a depends on the bond length [42] but its range of variation is small because the range of variation of the bond length is also small. Most of the parameters a for diatomics have been tabulated by Kuchitsu and Morino [42], but it is better, whenever possible, to estimate this effective anharmonic constant a for each molecule, either from isotopic differences in the average structure [43] or from differences in the average structure in vibrationally excited states [44]. Altematively, the constant a can be transferred from a similar molecule. This last method

12

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

was, for instance, used to calculate the rz- and re-structures of PH2F and PH2C1 using the value of a for PH3 [46] (the most recent molecular parameters for the PH radical [45] gives 1.674/~-1 to be compared with the tabulated value a(PH) = 1.659 ~-1, whereas an analysis of the PH3 molecule gives a = 1.993/~-1 [46]). Quite generally, the average bond angles are very close to the corresponding equilibrium values (see Table 7) where average and equilibrium angles of a few simple molecules are compared. For larger molecules significant differences are sometimes observed, but this is generally due to the problem of ill-conditioning as will be explained in the next section (Section IV). The extrapolation from r z to r e depends on the individual molecule. If an rz distance has been obtained without isotopic substitution (i.e. from the moments of inertia of a single isotopic species), the mass-dependence method of Nakata and Kuchitsu [47] may be used. The r e bond length is approximated by re obtained from the following equation, f m ~ + m i 1] r e ~ re = r z + z~.. ~ m ~ - m i +-~~~Srz(i)

Table 7. Comparison of Zero Point (Oz) and Equilibrium Molecule H20 H2S SO 2 C10 2 03 NO 2 F20 NF 2

SiF 2 C120 PH 3 PF 3 PH2F PH2C1 AsF 3 HECO C12CO CHaF

Angle HOH HSH OSO OC10 OOO ONO FOF FNF FSiF C1OC1 HPH FPF HPH HPF HPH HPC1 FAsF HCH C1CC1 FCH

Note: aln degrees.

0z 104.5 92.13 119.35 117.50 116.73 133.80 103.158(14) 103.18 100.88(2) 110.92 93.3 97.6 92.0 97.8 92.8 96.4 95.97 (28) 116.23(10) 111.85(5) 108.7(2)

Ref. 161 162 163 164 163 163 167 168 167 170 46 46 46 46 46 46 171 172 174 49

(8)

(Oe) Bond Angles a

Oe 104.48 92.11 119.330(3) 117.4033(27) 116.79 133.857(2) 103.07(5) 103.33 100.77(2) 110.88(l) 93.46 97.57 92.08 97.88 92.59 96.73 95.77 ( 12) 116.44 111.90(12) 108.8(3)

Ref 180 162 27 165 166 123 157 168 169 170 46 46 46 46 46 46 171 173 174 49

Determination of Reliable Structures from Rotational Constants

13

where 8rz(i) represents an isotopic difference for the rz distance when the ith atom af mass m i is substituted by one of its isotopic species m~. The sum is taken over all constituent atoms. This method is limited to a few simple molecules where the rz value may be obtained without isotopic substitutions (linear or bent x g 2, planar ar pyramidal XY3, spherical XY4, ZZ distance in planar XYZ2 . . . . ). In the more general case, Eq. 5 is used to calculate the re bond lengths. A weakness af this diatomic approximation is that it provides no correction for nonbonded atom pairs. If the cubic force field were known, it would be possible to calculate the correction r e - rz. In practice, this is feasible only for a few simple polyatomic molecules. However, it has been noticed that only a small number of "effective" cubic potential constants are of crucial importance for the structure and that those can be estimated from the available experimental data. This is the "triatomic approximation" of Kuchitsu [41]. For instance there are 60 cubic potential constants for 1,1-dichloroethylene, C12C=CH 2, whereas seven "effective" constants are enough to obtain an equilibrium structure [48]. This method has been tested on CH3F [49], H20, C120, C12C--O, and applied to Cl2C--CH 2 [41] and allene, H z C : C = C H e [50]. In short, the lack of precision of the equilibrium structure, when derived from the rz structure, is due to a number of different causes. Even if the problem is well-conditioned (see Section IV.C), the Morse anharmonicity parameters rarely are accurately known; also the diatomic approximation may be the cause of nonnegligible errors. Thus when the corrections (Eqs. 5 and 6) are large, the potential errors introduced are correspondingly large. An uncertainty equal to 30% of the correction r z - re (see Eq. 5) seems to be reasonable. B. rm Structure

Watson [51] has shown that the approximate equation, ie z im = 2i s _ io

(9)

is valid for any molecule, if it is justified to limit the expansion of the rovibrational contribution z of a singly substituted isotopomer to the first-order term: Oe

(10)

8* : 8 + ~ 5m i o~rl i

In Eq. 9 Is is the "substitution moment" of the parent molecule which is calculated from the substitution coordinates which must have been derived by the application of the full Kraitchman equations, where formally zero-valued squares of coordinates for substitutions on a principal plane or axis may happen to come out negative. Also these negative contributions must be included in the sums over coordinate squares.

14

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

A ro-type fit of the moments of inertia Im gives the rm structure which should be a good approximation of the re structure. But this method requires a large number of isotopic species and is not accurate except when the number of parameters is equal to the number of independent moments of inertia of a single isotopomer (i.e. for XY2 molecules). In fact, for OCS, OCSe, as well as for many other molecules [52], the difference between the r m structure and the r e structure is rather serious. For instance, in the particular case of OCSe, we have: Ar = re - rm = 0.0025/~ for the CSe bond and -0.0033 ]k for the CO bond [53]. It may be noted that these differences are similar to those found for OCS: Ar(CO) = -0.0025/~ and Ar(CS) = 0.0021 A. The origin of this discrepancy has been explained by Smith and Watson [52]. It is due to the use of finite changes in mass Ami, i.e. the second-order terms in the following expression [52], 1

,

~(Me)

Im=le+--M Z m ' ( m i - m i )

(11)

Om~ + ' " = / e + S

i

where m~ - m i represents the isotopic mass difference for the ith nucleus used for deriving the rs coordinate of the parent molecule. C.

rc Structure

Nakata et al. [54-56] have shown that it is possible to essentially cancel the influence of the second-order terms S (see Eq. 11) by taking an average of the rm structure of complementary sets. They first applied this method to C120 [54] and C12CO [55]. For instance in the particular case of OCSe [53], the two complementary sets correspond to the following isotopomers:

16012C785e, 18012C785e, 16013C785e, 16012C8~ for the first set 18013C8~

16013C8~ 18012C8~

18013C785efor the second set.

The S term (Eq. 11) for the parent species (16O12C78Se)may be written,

1 ~ ~(Me) , BE(Me) S = ~ [mo(m o - mo) Om2o + mc (m c - mc) Om 2 c

,

(12)

02(Me,)~

+ mse (m S e - mse) Om 2e

and the corresponding term for the 18Ol3CS~ parent species is:

S*

1 f

02(M*e*)

*

OZ(M*e*)

= M* l m~176 * - m~* ~ +Omo m ' c2 ( m c - m * c ) .

.

+ m Se (mse- m Se)

C3mc *-""--"7"-

(13) )

Determination of Reliable Structures from Rotational Constants

15

Since the approximations m i / M ~ mT/M* and 02(Me.)/Om 2 .~ O2(M*e.*)/Om~ 2 hold with sufficient accuracy, the relation S z -S* is obtained. A least-squares fit to the eight Im corresponding to the two complementary sets gives the re structure which is a better approximation to the re structure than the rm structure. However, the ro-type fit to the eight moments Im treats the most abundant isotopomer as the parent and the remaining seven isotopomers as three-monosubstituted and fourmultisubstituted forms, and thus introduces an amount of unbalance into the two complementary sets which makes the method still unsatisfactory. This can be explained by the fact that in Eq. 11 the cross-terms O2(Ms)/OmiSmj were neglected. It has been found [53, 57] that these terms are not at all negligible for the multisubstituted species. So, instead of using the experimental constants for the multisubstituted species, they should be calculated with the help of a method devised by Nakata and Kuchitsu [56] which, in the present case, accurately predicts the rotational constants of the multisubstituted species once the constants of all the monosubstituted species are known. For instance, the moment of inertia of a linear triatomic molecule may be written, I=

m l m 2 r a2 + m2m3 r2 + m l m 3 R 2 m I +m2+m

(14)

3

where m ~, m 2, and m 3 are the masses of the atoms, ra the distance between the atoms 1 and 2, rb the distance between the atoms 2 and 3, and R = ra + rb. A new parameter F is introduced: F=I

ml +m2+m3

mlm2m 3

?'a2 +

=~ m3

r2

R2

(15)

~ + ~ ml m2

If atom 1 is substituted, its new mass being m], the parameter F~ of the isotopomer is,

(r b + ~Srbl)2

(r a + 8ral) 2 E 1=

ra FI~ +

m3

+

m3

m 1

+

p m 1

R 2 2r a~l"al ~ + ~ + m2

(R + ~SR1)2 +

m2

2r b~)Fbl 2RSR 1 ~ ) m2 t

m3

(16a)

(16b)

m 1

where 8ral, ~Fbl and 8R~ = ~ral + ~Fbl are the changes in the interatomic distances due to the substitution of atom 1. A relation similar to Eq. 16b is obtained when atom 2 is substituted: r a2

r2

R2

F 2 "~ ~ + ~ +---r + m3 m I m 2

2raSra 2 m3

2rbSrb 2 +

ml

2RSR 2 +'"-'7-m 2

(17)

16

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

When atoms 1 and 2 are simultaneously substituted, the value of F for the corresponding isotopomer is: r a2 r 2 R2 2rafral2 2rbfrbl 2 2RfR12 El2 ~ ~ +-.--7. + - 7 + ~ + ~ + ~

m3

m2

ml

m3

m'1

(18)

m2'

If it is assumed that, ~ral 2 m, ~ral + ~ra2

(19a)

~rbl 2 m, 6 r b l + ~rb2

(19b)

~Rl2 ~ ~ral 2 + ~rbl 2

(19c)

we have:

Table 8. Comparison of re and rc Structures Molecule

Parameter

OCS

CO CS CO CSe NN NO CH r(CN) r(CH) r(NC) r(BH) r(BS) r(OH) Z(HOH) r(SH) Z(HSH) r(CIO) Z(C1OC1) r(OO) Z(OOO) r(SO) Z(OSO)

OCSe N20 HCN HNC HBS H20 H2S C120 03 SO 2

re

1.1562 1.5614 1.1533 1.7098 1.1273 1.1851 1.0650 1.1532 0.9961 1.1684 1.1698 1.5978 0.9578 104.48 1.3360 92.11 1.6959 110.88 1.2717 116.78 1.4308 119.33

Note: aDistancesin ./k and angles in degrees.

rc

1.1547 1.5623 1.1534 1.7098 1.1296 1.1826 1.0639 1.1534 0.9919 1.1694 1.1711 1.5971 0.9577 104.47 1.3360 92.20 1.6959 110.89 1.2719 116.79 1.4307 119.33

[56] a

r e - rc

0.0015 -0.0009 -0.0001 0.0000 -0.0023 0.0025 0.0011 -0.0002 0.0042 -0.0010 -0.0013 0.0007 0.0001 0.01 0.0000 -0.09 0.0000 -0.01 -0.0001 -0.01 0.0001 0.00

Determination of Reliable Structures from Rotational Constants

17

2 FI2 ~ F + Z (Fi - F) + 2 i=1

1

~

1 rb~rb2 +

ml

2

m2

RSR 1

(20)

The last two terms of Eq. 20 are usually small and may be neglected giving: 2

Fl2 ~ F + Z (Fi- F)

(21)

i=1

Correspondingly, the F of the trisubstituted molecule may be written: 3

FI23 ~ F + ~ (F i - F) i=1

(22)

With FI2 and FI23 known, the required inertial moments 112 and 1123 are obtained using (the first term of) Eq. 15. This method may be used for any triatomic (linear or bent) molecule as well as for a few more complicated molecules. In fact it is more general than Eq. 8 because it can be applied also to planar XYZ2, axially symmetric XYZ3, acetylene-like X2Y 2, ethylene-like X2Y 4, and ethane-like X2Y 6. Using this method, the cross-terms do not interfere and the derived structure does not depend any more on the isotopic species used and is in very good agreement with the re structure which is demonstrated for several molecules in Table 8. The rc method has recently been extended to linear tetraatomic molecules and applied to chloroacetylene [58], but the derived structure does not seem satisfactory due to ill-conditioning problems (see Section VI.D).

D. r*m Structure Following the work of Watson on the rm structure [51], Harmony and colleagues [59-65] have proposed an empirical procedure for obtaining near-equilibrium structures using only ground state data. This rPm structure goes back to Eq. 9 and is based on the assumption that the isotopic dependence of the ratio ISg (i)/Ig(i) is smaller than that of either moment and can hence be replaced by that of the parent Ig (1)/l~ The ground state moments of inertia I ~ of all isotopomers i are first scaled by a factor 2p - 1, IPm. (i) = (2pg - 1)I~(i)

where

g = a,b,c

ISg(1) pg = pg(1) = ~Ig(1) i = 1 = parent

(23)

(24)

and Ig(1), g = a,b,c are the "substitution moments" of the parent molecule (see Eq. 9).

18

J. DEMAiSON, G. WLODARCZAK, and H. D. RUDOLPH

The rfn method introduces e contributions which are linear in the inertial moments, eg = 2(1

pg)lg o

-

(25)

i.e., e is proportional to Io. In fact, since e is small compared to Io (a few percent), this relation is compatible with that proposed by Demaison and Nemes [31] (see Section II.C). As demonstrated by Harmony et al. [59], the approximate isotopic invariance of p is a good assumption. This could be checked experimentally in the case of OCSe because many isotopic species are available for this molecule [53]. A very small isotopic dependence could be pointed out. If the molecule contains hydrogen atoms, Harmony advocates correcting the data of the deuterated isotopic species to account for overscaling. This "Laurie" correction is equivalent to an elongation of the X - D bond by an amount 8r = 0.0028/~. For instance, for the a-axis, p o (Im,a)corr---- (IPrn,a)D + 2mo E (biSbi + Ci~)Ci)

(26)

i

and correspondingly for the b and c axes by cyclic permutation of a, b, and c, whereas a i, b i and ci are the cartesian coordinates of the D atoms and 5a i, 5b i, 5c i, the components of 8r. In cases where the deuteration leads to a non-negligible rotation of the principal axis system of the isotopomer, the approximation by Eq. 26 should be replaced by an exact calculation, which is easily possible, e.g. by a short computer routine, and requires no more information than is necessary for the application of Eq. 26. After the subsequent ro-type least-squares fit to the scaled moments as prescribed by Harmony's rules, the correction leads to a structure where the X - D bond is longer by approximately 0.0028 A and the respective X - H bond by 0.0056 A compared with the results obtained without this correction. Furthermore, this correction has been shown [66] to remain often restricted to the resulting C - H bond length, hardly affecting the rest of the bonds. The structural parameters of the molecule derived from the fit have been designated as the rfn structure. This method has already been applied to several molecules containing hydrogen atoms [32, 67, 61-64]. It appears to give rather accurate results, except for some X - H bonds (see Table 9). For HNC the discrepancy has been explained by the presence of a low-frequency, large-amplitude bending mode. But it is interesting to note that the largest positive residual, Ar = r e - rfn, is for the shortest bond length (HNC), whereas the largest negative residual is for the longest bond length (HCO§ The assumption that 8r = 0.0028 A is an invariant correction in Eq. 26 is actually only a rough one although it does rather well for most C - H bond lengths. If we calculate 8r using Eq. 6, we find 0.0078/~ for HNC and 0.0018 A for HCO § It would be worthwhile to try to improve the rfn method by the introduction of a bond-dependent 8r calculated with Eq. 6.

Determination of Reliable Structures from Rotational Constants Table 9. Molecule

HCN HCO§ HN~ HNC HC--=CH HC=CC1 H2CO CH2C12 H2C--CH2 H2C--C--O HC=---CCN CH3C1 CH3Br CH3CN CIt3C~_----CH CH3C~CI-I

19

Comparison of re and rPm C-H Bond Lengths re

rPm

Ar

1.065 1.092 1.034 0.996 1.063 1.061 1.101 1.084 1.081 1.076 1.062 1.084 1.082 1.087 1.089 1.062

1.067 1.096 1.035 0.992 1.063 1.060 1.101 1.085 1.080 1.074 1.063 1.084 1.082 1.089 1.089 1.059

-0.002 -0.003 -0.001 0.004 -0.001 0.000 -0.001 -0.001 0.001 0.002 -0.001 0.001 0.001 -0.002 0.000 0.003

Ref

62 62 62 62 62 62 62 62 62 160 33 32 32 32 67 67

E. r,~ Structure It is often a reasonable approximation to assume that the rovibrational contributions, eg (g = a,b,c), remain almost constant during an isotopic substitution, particularly for the larger molecules, because the range of variation of the moments of inertia of the different isotopic species is often quite small [4]. Then it is a good method to determine the eg by least-squares fits together with the structural parameters. The isotopic invariance of eg is also the basic assumption of the substitution structure, but as stated by Rudolph [4], for polyatomic molecules where one can obtain a number of isotopomers, the ri~ structure is preferable to a Kraitchman-type structure, where the Cartesian coordinates of each atom have been obtained by the separate application of Kraitchman's equations. This is particularly true when near-axis atoms are present. Furthermore it may be applied even to molecules which have one atom without an isotope. This method has recently been extensively reviewed [1] and applied to the determination of the structure of many moderately sized and larger molecules.

IV. LEAST-SQUARES METHOD A. Introduction All methods previously described use the least-squares method to determine the structural parameters. This method has been recently reviewed with emphasis on structure determination [1]. For details, the reader is referred to that review and the

20

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

appendix of this chapter; therefore, we will limit ourselves to describe some shortcomings of this method and how to circumvent them. In the following we use the notation of ref. 1 which is the same as that of the tutorial by Albritton et al. [68]. If we want to determine p parameters 13j(j = 1. . . . . p) from n experimental data Yi (i = 1 . . . . . n), the starting equations of the linear least-squares method are, after the model equations have been linearized in the neighborhood of the expected solution, y=XI3+~

(27)

where X is the Jacobian matrix and e the vector of residuals (which should not be confused with the vibrati6nal correction eg used elsewhere in this paper). In the following we shall assume that the errors are independent and that the experimental data are of equal precision (uncorrelated, unity-weighted case). In the more general case of unequal variances (data of different precision) or correlated errors, the problem has to be first transformed to the unity-weighted uncorrelated problem (see appendix). But the following discussion remains valid provided the transformed Jacobian, observations, and residuals are used. For part of the later discussion we need the diagonal elements hii of the "hat A matrix" H. The observed vector y is transformed into its least-squares estimate y by the H matrix, y^ = H y

(28)

H = X ( X r X) -1 X r

(29)

which is defined as,

where X T is the transpose of the matrix X. As H is symmetric and idempotent, the following inequalities hold for its diagonal terms hii (= H/i): 0 < hii

_< 1

(30)

B. Assumptionsof the Least-SquaresMethod The application of the least-squares method requires certain assumptions (Gauss-Markov conditions). The first one is that the errors have zero mean and that they are randomly (more precisely normally) distributed. If the observations to be fitted, the moments of inertia, were the highly accurate equilibrium moments of inertia (without systematic errors), it would be probably true. But, since generally only approximations of the equilibrium moments of inertia can be used, contaminated by systematic errors, this assumption does not hold. For instance, Figure 1 represents the residuals of the least-squares fit of the ground state moments of inertia of OCSe using the ri~ method. It is obvious that the errors are not at all random; on the contrary, they are correlated (in fact they are proportional to Io, justifying the

Determination of Reliable Structures from Rotational Constants

21

0.0020

916-12-82

0.0015

917-12-82

918-12-82

0.0010 916-13-82 0.0005 18-13-82 0.0000 1

- 9 126

124

128

9 130

132

134

136

917-12-76 -0.0005

916-12-74 -0.0010 918-12-74 -0.0015 916-13-74 -0.0020

918-13-76

Figure 1. Plot of/exp.- lcalc, versus Icalc" for the q~ structure of OCSe [53] (unit: uA2). In each of the five tilted-upward sequences of points, the atomic mass of selenium is increased while the masses of O and C are held constant. approximation of the rPm method). There have been a few rare attempts to take this correlation into account [69], but it is not easy because neither the theoretical form of the correlation matrix is known nor are there enough experimental data to use estimation procedures, as for instance in econometrics [70]. However, in the particular case of the rPm method, the correlation of the errors may be, at least partly, taken into account. The method was first devised by Epple et al. [66]. Indeed, the r~ method has apparently been mostly applied assuming that the scaling coefficients 9g are error-free and that the IPm are uncorrelated. In fact, the three I~ (1), g = a,b,c, are functions of all Ig (i) (but of no other variables); the same is true for the l~,g(i). The errors of the Ig (1) (Eq. 24) are almost invariably much larger than those of the I~ (1), in particular, when the molecule has atoms with small coordinates. Therefore, the errors of the pg and hence those of the scaling factors 2 p g - 1 (Eq. 23) are large enough to make the errors of the inertial moments l~,g (i) for all isotopomers i larger by orders of magnitude than those of the Ig (i). Even more important is the fact that the three scaling factor s which dominate the errorsofthel~,g (i), g = a,b,c, are assumed as isotopomer-independent. That means that the same factor, e.g. 2pa - 1, that of the parent, Eq. 24, multiplies the moments I ~ (i) of all isotopomers when the m o m e n t s IPrn,a(i) are calculated. Consequently, these moments are extremely highly correlated, separately for each g = a,b,c. In contrast, the correlations between moments of different g are only moderate and directly reflect those of the corresponding components Pg.

138

22

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

For the ro-type fit as prescribed by Harmony's work, the variance-covariance matrix of the data fitted [i.e. the errors and correlations of the moments IPm,g(i)] do affect also the numerical values of the internal coordinates obtained from the fit, not only their errors. In particular, when the very high correlations between the IPrn,g(i) with the same g but different i = 1. . . . . n are neglected, the numerical results may be distinctly different from those when these correlations are retained. The differences in the numerical structures when produced with or without the neglect of these correlations were conspicuous in the very stable cases of cyclopropylgermane [66] and of cyclopropylsilane [71]. In theory, the correlation should not bias the estimates, but this is true only in the limit of an infinite number of data. In the particular case of a structure determination, the number of experimental data is often not much greater than the number of parameters to be determined. Furthermore, as the mean of the errors is not zero, a bias is expected. Due to the nature of the system of equations, this bias can be large (see Section IV.C). Generally the weighted least-squares or the general least-squares method (correlated errors) is used. Usually the weight of an observation is taken as the reciprocal square of the experimental uncertainty. However, in the present application the inherent model limitations due to the lack of sufficiently well-known vibrational corrections e g generally cause much larger errors than the experimental errors of the inertial moments, although the latter may themselves differ greatly in magnitude, usually being larger for the less abundant species. This is disclosed, e.g., by a preliminary ri~-type fit with unit weights for all inertial moments and no correlations assumed. Even when the moments come from well-conducted spectral measA urements, the standard deviation ~u of this unity-weighted uncorrelated fit may, nonetheless, come out 1 to 2 orders of magnitudes larger than unity, which is the value expected for a perfect model and a realistic assessment of the observational errors (see remark on p. 39 of ref. 68). Epple et al. [66] have proposed a method to assign weights which is hopefully more appropriate in this situation. They assign to each inertial moment I~, (i) an additional "model-induced" error:

A(1 ~. go ( i )")j = ~A .

(i))2 3(I~ ~--,(g,()) Io i 2

for g =

a,b,c, all/

(31)

= a,b,c

The errors of the inertial moments could be increased up to the values given by Eq. 31 representing "apparent" (model-induced) errors, before the standard deviation of the fit would decrease to the expected value of approximately unity. In a crude form, this additional error even takes account of the different magnitude of the inertial moments for g = a, b, c. In practice, the square of Eq. 31 is added to the experimental variance of the respective moment on the diagonal of the covariance matrix of the observations. The procedure was used for the determination of the structure of 2-chloropropane [72], cyclopropylgermane [66], and several dicy-

Determination of Reliable Structures from Rotational Constants

23

anides [73] where the usual weighting scheme (by reciprocal squares of the experimental errors only) would have resulted in a molecular structure with very unbalanced errors due to very differently weighted moments. In all cases the additional "model-induced" variance was much larger than the experimental variance. Since the procedure does not change the covariances, the additional terms on the diagonal also lead to a substantial reduction of the correlations between the observations. From a theoretical standpoint, the method remains questionable because systematic (model) errors have been simulated by random errors. Essentially the same objection is to be made against the attempt to calculate weights by the iteratively reweighted least-squares method [74] which, in contrast to Eq. 31, makes no assumptions regarding the dependence of the weights on the magnitude of the inertial moments and may hence be better suited when the model errors go back to the inadequate approximation of the vibrational correction e g. Originally, the reweighted least-squares method was devised to detect outliers due to faulty measurements etc. This must, of course, be excluded in the present application. The only objective is to find suitable weights to compensate for the deficiencies of the model which, in principle, cannot be made perfect because the necessary inclusion of isotopomer-dependent vibrational corrections e,g, g = a, b, c, i of all isotopomers would introduce, with each new isotopomer, as many new variables as observations. If possible, the weight finding by the iteratively reweighted least-squares method should be applied separately to each of the three components sets g = a, b, c of the inertial moments I,r in particular when the errors of the three sets are very different. An initial set of residuals, /k

eg = I g (exp.) - I g (calc.)

(32)

(do not confuse with the vibrational correction e,g) is obtained by the application of, e.g., ordinary least-squares or more robust methods. Then weights are assigned to the experimental inertial moments which are functions of these residuals. Different weight functions are in use [74]. Some of them assign the weight zero to inertial moments whose residual, Eq. 32, is larger than a certain threshold value, which means that the respective inertial moment is dropped from the fit as a suspected "outlier." For other weight functions, the weights decrease more gradually with increasing residuals. Since for the problems treated here the number of input data is usually not much larger than the number of parameters to be determined, one can hardly afford the loss of data; therefore, the latter type of weight functions appears to be preferable. With the newly assigned weights a new (weighted) least-squares fit is performed which will result in new residuals and new weights. The treatment is iterated until consistence of the residuals is attained. The reweighted least-squares procedure was already used for the structure determination of a few molecules but there is not yet any systematic study of its performance. A number of problems remain, in particular the presence of autocorrelation of the

24

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

input data (non-zero covariances between the errors of inertial moments) requires further study. Another problem relevant to the weighting is the choice of the fitted moments of inertia for the structure determination of a planar molecule. As the principal axes a and b lie on the plane of the molecule, it seems to be better to fit only the moments of inertia Ia and Ibwsee for instance the example of phosgene, COC12, in Table 14. But, here again, there is not yet any systematic study of this problem, although it has been discussed in the particular case of the substitution structure [75]. In conclusion, most assumptions of the least-squares method are not fulfilled by the problem as originally given. The difficulties can be circumvented, at least partly, by taking proper account of the correlations of the observations (see appendix) and by a judicious manipulation of the variances of the observations. The conventional least-squares method is, no doubt, mainly used for its convenience. The more recent robust methods which are able to deal with non-normal errors might give more reliable results and hence deserve further attention. However, it should be noted that if outliers and extreme situations are absent, the "robust" results hardly differ from those of the more conventional method.

C. Collinearity or Ill-Conditioning It often happens that some parameters cannot be estimated with precision and that they are very sensitive to small perturbations in the data. This is due to "collinearity" or, more exactly, to the near-collinearity of the fit vector subspace spanned by p column vectors of the Jacobian matrix X in error space (the ndimensional space of the vectors of the observations and of their residuals, y and e, respectively). This collinearity increases the variances of the estimated parameters and is responsible for important round-off errors. This problem was pointed out early by Kuchitsu et al. [39]. There are diagnostics that determine whether a collinearity exists and that can identify the parameters affected. The correlation matrix is often employed for that purpose. But the absence of high correlations does not imply the absence of collinearity. Therefore many different procedures have been proposed. Lees [76] suggests to calculate the eigenvalues and eigenvectors of the square matrix xTx: for each vanishing eigenvalue, there is a linear dependency among the columns of X. So, a small eigenvalue is an indication of collinearity. Watson et al. [77] advocates the use of the diagonal elements of the inverse of the correlation matrix. This notion has later been generalized by Femenias [78] and then by Grabow et al. [79]. Belsley [80] has critically reviewed these procedures and has concluded that "none is fully successful in diagnosing the presence of collinearity and variable involvement or in assessing collinearity's potential harm?' To palliate the weaknesses of the existing diagnostics, he has introduced the condition indexes and has shown that they can be easily used to determine the strength and number of near-dependencies. First the columns of the Jacobian matrix X are scaled to have

25

Determination of Reliable Structures from Rotational Constants

unit length (each term of the vector column Xi is divided by the norm IIXill). Then the singular values (square roots of the eigenvalues of xTx) of the scaled X matrix are calculated: g l, g2 . . . . gp. The singular-value decomposition of X may be written [81], X = UDV r

(33)

where U is a n x p matrix with orthonormal columns, D is a p x p diagonal matrix whose elements are the singular values ~ti of X, and V is ap x p orthonormal matrix. The "scaled condition indexes" of the scaled matrix X are defined by: ~-Lmax

rlk = ~

k = 1. . . . p

(34)

gk The highest condition index is the condition number K(X). It is an error magnification factor and it is used to determine whether a matrix is ill-conditioned or not: if the data are known to d significant figures and if the condition number of X is 10 r, then a small change in the data in its least significant digit can affect the solution in the (d-r)th place. If we consider a perturbation 5y in y, then, 5y

(35)

A

A

A

where 13 is the least-squares solution and 8[3 the perturbation in [3 due to By. Thus a large ~: may be responsible for a large bias. The number of near-dependencies is equal to the number of high-scaled condition indexes. To determine which parameters are involved in the collinearities, Belsley defines the variance-decomposition ~roportions. The variance-covariance matrix A A 0([3) of the least-squares estimator 13 of 13 is (where ~ is the estimated standard deviation of the fit): =

(x

x)-i = awo-w 2

,

The variance-decomposition proportions are:

Belsley proposes the following rule of thumb: estimates are degraded when two or more variances have at least half of their magnitude associated with a scaled

26

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

condition index of 30 or more. Evidently the probability of encountering the problem of ill-conditioning rapidly increases with the number of parameters to be determined. However, it is already present for small molecules as will be illustrated in Section VI.

D. Corrective Action Once the problem of ill-conditioning has been diagnosed, it is important to reduce its influence. The first idea is to include the moments of inertia of additional isotopomers. But the number of isotopic substitutions in a molecule is limited and, furthermore, these new data will possess near dependencies similar to those of the original data. Thus it is preferable to use additional information derived in some other way (electron diffraction, ab initio calculations, etc.). The different (and most useful) possibilities are described in Section V. Unfortunately, it is not always possible to obtain new data which will substantially reduce the correlations. In such a case the introduction of appropriate "prior" information is the best solution. In the case of structure determination, the "mixed estimation" is often the most useful. Auxiliary information is added directly to the data matrix. It may be a set of reasonable values of parameters based on similar molecules [2, 80, 82]. When data of different origin are used, it is particularly important to check that the estimated weights are appropriate and that the data are compatible. Also, particular caution is indicated when one parameter appears to be determined by only one datum. This can be checked by an outlier analysis.

E. Outlier Analysis Several good books describe the different outlier diagnostics (see for instance, refs. 83-86). To check whether a particular datum is compatible with the other data, the following diagnostics may be used: 1. The "standardized" residuals defined as, A gi

A

Y i - Yi

0 ^ - ~[

1

(38)

n

~ ,,2 j=l

^

2.

where Ei is the ith residual and ~ the standard error of the fit. The "studentized" residuals defined as, A I~i

t i - O(~i) - ~/1-

A 1~i

hil

(39)

Determination of Reliable Structures from Rotational Constants A

3.

A

A

27

A

where I~(Ei) -- [40(E) ]ii= ~]1 -- hii is the estimated standard error of the ith residual. The "jackknifed" residuals defined as, A

A

o ~i t(i) = ti~(i ) - ~(i)ql hii

(40)

where ~(i) is the estimated standard deviation of the fit when the ith measurement is dropped. The last diagnostic is as easy to calculate as the first two ones and it is the most sensitive to errors. The variables t i and t(i) follow a t-distribution if the errors are Gaussian, and they have a near t-distribution under a wide range of circumstances. In least-squares fitting, it is desirable that none of the parameters is determined by a single experimental datum or a small subset of them because the disproportionate influence of only one or a few observations on the parameter would diminish the balancing effect of a regression analysis. However, such an isolated influential observation would certainly improve the estimate of the parameter, if the observation were known to be a legitimate datum. Therefore, an observation must not be discarded just because it is influential, although the method itself does not offer the means to conclude if the observation is good or bad. When, as in molecular structure fitting, the number of observations is not much larger than the number of parameters to be determined and when the nature of the problem, as in the rs-fit scheme, favors a certain separation into subsets of data, the occurrence of"influential observations" is not infrequent and cannot be avoided. The researcher must then know which data are of this type and he must convince himself that the corresponding measurements and assignments are correct beyond any doubt. The diagonal elements hii of the H matrix are useful to detect such influential data. One point is influential when its hii is near 1 (its residual is near zero from Eq. 28 and its studentized and jackknifed residuals are undefined). An observation with hii <_0.2 is safe, on the other hand, if hii > 0.5; the corresponding measurement has to be carefully checked. Another diagnostic which gives more specific information is, ~j

DFBETASj(i) = A

A

A

- ~j(i)

t~

^ ^ ~(13j) ~(i)

(41)

where 13,(i) is the estimate of the jth parameter when the ith data is omitted and A J A ~(~.) = [x/~)(a)].. the estimated standard error of ~i This test shows us how much J p ,JJ the estimation of the parameter 13jwould change if the ith measurement were to be deleted (for the practical calculation of these tests see refs. 83, 86). The critical or cut-off value for DFBETAS is 2/~-n. Another measure of the influence of the ith observation is Cook's squared distance which should be lower than 4/n:

28

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

CD2(i) =

1

hii

(42)

q 1 - h,, DFBETAS is well suited to detect an observation which substantially influences only one regression coefficient. When the influence of an observation spreads out over several coefficients, it is more readily detected by CD2(i) [86]. Finally DFFITS(i), which tells us how much the predicted value ~i would be affected if the ith observation were deleted, may also be useful (particularly when data of different origin are used in the fit),

^ Yi(/) ^ 9 (Y ^ YiDFFITS(i) = ~ ~(~i)

~Y(i)

h]i/2~i

(43)

~y(i)(1-hii )

^ ^ where (y(yi)= [41~)(~)]ii=~hli/2 is the estimated standard error of the predicted ^ Yi. Observations for which DFFITS exceeds 2qp/(n - p ) should be scrutinized. Other diagnostics are also useful; they can be found in refs. 83, 84, 85.

V. SOURCES OF ADDITIONAL DATA There are many techniques available from which information on the geometrical structure of an isolated molecule can be obtained: the most important are reviewed in ref. 87, some useful relations are described in ref. 34, and many more are reviewed in ref. 88. There are also many methods which are interesting in special cases, e.g.: 9 the centrifugal distortion constant R 6 is quite useful for determining the structure of C4v molecules [89], 9 a force field analysis may also be helpful to assess the reliability of a structure, as for instance in the case of CH3F [90], 9 the nuclear quadrupole coupling tensor furnishes important information on the structure, particularly in the case of complexes [91], and 9 analysis of the internal rotation splittings of the rotational spectra is also a source of structural data [92]. But we will only focus on a few general methods which produce good estimates of equilibrium bond lengths and angles:

9 ab initio methods, 9 electron diffraction and NMR spectroscopy, and 9 empirical relations between geometrical parameters and other types of parameters like the stretching force constants.

Determination of Reliable Structures from Rotational Constants

29

A. Ab lnitio Methods In the past, a great many studies were published that apply high level of theory and use large basis sets to produce ab initio geometrical parameters. Regarding the available methods, the state-of-the-art treatment which can be considered as a reference at the present time, is the CCSD(T) method. This method belongs to the coupled-cluster theories, which take account of the electron correlation in the most accurate way [93]. Unfortunately, this method is computationally very expensive: the computation time is proportional to the seventh power of the number of molecular orbitals n used in the calculation. This is the reason why only small atoms or molecules can be treated by this method. The other methods taking into account electron correlation and currently utilized are CCSD and MP2 methods, for which the computation times are proportional to n 6 and n 5, respectively. The QCISD method can be considered as a variant of the CCSD method. The other factor which has to be considered is the size of the basis set. For very accurate calculations two main families of basis sets are currently used: 9 the atomic natural orbitals (ANO) basis sets of Alml6f and Taylor [94], available for the first and second row atoms, and 9 the correlation consistent basis sets of Dunning [95], also available for the first and second row atoms, which seem to be favored for the optimization of molecular geometries. The cc-pVTZ (correlation consistent polarized valence triple zeta) and cc-pVQZ (correlation consistent polarized valence quadruple zeta) basis sets are the most commonly used. It was found necessary to incorporate at least f-type functions (d-type functions for the hydrogen atom) in the one-particle basis sets to obtain chemical accuracy [93]. Systematic calculations with correlation consistent basis sets at the CCSD(T) level were recently carried out on many small molecules (several examples are discussed in refs. 93 and 96). Even at these high levels of calculation, some systematic errors remain in the bond lengths, mostly due to the neglect of core-valence correlations [96]. The main conclusions from these studies are the following [96]: 9 CCSD(T) / cc-pVQZ calculations overestimate single bond lengths by about 0.001 ,~ and multiple bond lengths by 0.003 A (double and triple bonds show the same behavior), and 9 CCSD(T) / cc-pVTZ bond lengths are still longer by about 0.001 .~, for single bonds, 0.003 A for double bonds, and 0.004 ]k for triple bonds. For a smaller basis set (cc-pVDZ), single and multiple bonds are overestimated by an additional 0.01 and 0.02 A, respectively. These systematic errors (offsets) may be estimated with the help of molecules whose structure is accurately known and can then be used to correct the ab initio

30

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

structure of a new compound to produce what is generally called the best estimate structure. This method of empirically derived offsets may also be used at lower levels of theory, at least with treatments which include the electron correlation. In fact, it has already been used for a long time, but mostly to estimate rg or ro structures [97, 98]. We have recently derived new offsets to determine the equilibrium CH, CC, CN [99], CF [100], C--O [73], and CC1 [101] bond lengths. The calculations were made at the MP2 level with triple zeta basis sets including polarization. The corrected ab initio values are often used to check the consistency of experimental geometries. When discrepancies appear, they are often due to unreliable experimental structures. The accuracy depends on the kind of the bond, it may be given as 0.002 ,~ or better for C - H bonds, and 0.003 A for the other bonds. This approach is expected to be accurate only for molecules that are comparable with those for which the offsets were derived. The remaining residual error in CCSD(T) / cc-pVQZ calculations is mostly due to the neglect of core-valence correlations. When this effect is considered, the equilibrium bond distance in N2 is reduced by 0.002 ,~ [102], for H20 r(OH) is reduced by 0.00087 ,~, and for HF r(HF) is reduced by 0.00057/k [103], while in PH 3 the effect is still bigger: r(PH) is reduced by 0.005 ,~ [46]. The situation is much better for bond angles, because they are generally well estimated at a lower level of theory (MP2) provided the basis set is large enough [104]. For instance, the calculated values of the Z(CCH) bond angle in propyne, CH3C--=CH [67], at various levels of approximation (SCF/DZP, SCF/TZ2P, MP2/DZP, MP2/TZ2P, QCISD/DZP, QCISD/TZ2P) lie between 110.18 ~ and 110.61 ~ i.e. in a range of 0.5 ~ Without the SCF values the range is still much smaller. The same range of variation has also been noticed for the Z(CCH) bond angle in many other molecules [99]. Recent calculations on PHzF and PHeC1 also indicate that the calculated bond angles are not very sensitive to the choice of the correlation treatment (MP2, CCSD, or CCSD(T)) or to the basis set (cc-pVTZ, cc-pVQZ, cc-pV5Z), the range of variation being about 0.3 ~ [46]. At the highest level of theory we may trust that the ab initio value for a bond angle is within 0.2 ~ from the experimental equilibrium value.

B. Electron Diffraction Electron diffraction is a method complementary to rotational spectroscopy, but it provides rg bond lengths, which are quite different from equilibrium values [41, 105, 106]. The rg value can be converted into a r e value with the knowledge of the anharmonic force field, but this remains a difficult problem. On the other hand, the rg values can be easily converted to rz values provided the harmonic force field is known. Thus a combined fit of the rotational and electron diffraction data often gives an accurate average structure [41, 105].

Determination of Reliable Structures from Rotational Constants

31

Kuchitsu and Oyanagi [36] have shown that the difference, r g - re, remains almost constant for a given bond. In this case, as for the ab initio calculations, a simple offset correction should allow us to obtain the equilibrium bond length. For instance, for the CN bond, a highly rigid bond, the difference rg - re is estimated to be 0.005 ]k from the data on HCN and NCCN, for which both structures were determined. This offset value was then applied to methyl cyanide, CH3CN [32], acrylonitrile, H a C - C H C N [99], and carbonyl cyanide, OC(CN)2 [73] to obtain the equilibrium value of the CN bond in these molecules, and the results were in good agreement with experimental and ab initio estimates. It should be noticed that two molecules constitute a very small sample and that the accuracy of the rg distances is only a few thousandths of an angstrom. Thus the accuracy on the predicted re value is not high. In the general case, a constant offset is not expected. Indeed the relation between the re and rg bond lengths is (neglecting the small centrifugal distortion term) [105], 3

re= r g - ~ au

2

(44)

where u 2 is the mean square amplitude of vibration and a the Morse anharmonicity parameter (compare with Eq. 5). In general, mean amplitudes are expected to increase with increasing internuclear distances [107]. Likewise a should vary with the bond length (see Section III.A). But a good correlation between rg and re is expected and was indeed observed for the C - H bond [34].

C. Liquid Crystal NMR Spectroscopy Direct spin-spin coupling, which can be observed in nuclear magnetic resonance spectra of solutions in oriented liquid crystal solvents, may provide complementary information on ratios of average distances. This technique has been recently reviewed [108, 109].

D. Empirical Relations Since Badger's work [110], it is known that there is a relationship between a stretching force constant and the corresponding bond length. McKean [111] made an extensive study of the correlation between ro(C-H) bond lengths and the corresponding isolated stretching frequencies (the stretching frequency of the C - H bond when all H but one have been replaced by D). This correlation was extended to equilibrium C - H bond lengths in ref. 34. It can be considered as one of the best methods to determine C - H bond lengths: an 0.001 ]k increase in re(C-H) corresponds to a 15 cm -l decrease in vcH. This correlation may be generalized to other bonds but the accurate stretching force constant is generally no easier to determine than the bond length. A review of numerous kinds of relations between structural parameters and other parameters was published in a previous volume of this series [88]. For instance,

32

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

there is an empirical relationship between t h e / ( H C H ) angle and the corresponding C - H bond length. This relation has been used to check the accuracy of the structure of propyne, CH3C~=CH [67], malononitrile, H2C(CN)2, and 1,1'-dicyanoethene, H2C=C(CN)2 [73].

VI.

EXAMPLES

A. Fiuorophosphaethyne, FC~P The equilibrium structure of FC~=P was first determined using the ground state moments of inertia of the FI2Cp and F13Cp isotopic species, the vibrational correction being calculated from an experimental anharmonic force field [112]. Later it was redetermined using the experimental equilibrium moment of inertia for F12Cp isotopic species and an estimated equilibrium moment of inertia for FI3Cp species [113]. Both determinations are obviously incompatible (see Table 10). The force field was determined with very few experimental data, thus it could be argued that it is not reliable, but, in fact, the problem lies elsewhere. The carbon atom is near the center of mass; therefore the system of equations is ill-conditioned (condition number K = 450) and it is not possible to determine accurately both interatomic distances which are fully correlated. However, it is possible to improve the conditioning by using the "corrected" ab initio distances as additional data. The offsets used to correct the ab initio distances are estimated from the structurally similar molecules H C ~ P and FC~=N using the same method and the same basis set: CCSD(T)/6-311 +G(2d, p) [114]. The results of this fit are collected in Table 11; the condition number drops to 58 and the structure obtained with the mixed fit is almost identical with the corrected ab initio data.

B. Formyl Cation, HCO § The experimental equilibrium geometry of HCO § was determined by Woods" r(CH) = 1.09725(4) A and R(CO)= 1.10474(2)/~ [115]. Several high level ab initio studies indicate that the experimental equilibrium value of the C-H bond length is much too high, a more likely value being 1.092/~ (see ref. 116 for a complete list of references). One possible explanation for this discrepancy is that the rotational constants of HCO § and of the isotopomers HI3CO § and HC~80 § are not sufficient for the unequivocal determination of the geometry. Indeed, a least-squares fit using

Table 10. Comparison of Different Equilibrium Structures (A) of FC=P

Method

Force field Experimental Mixed

r( C-F)

r( C~--P)

Ref

1.272 1.284 1.276

1.548 1.538 1.544

112 113 114

Determination of Reliable Structures from Rotational Constants

33

Table 11. Data Used for the Least-Squares Determination of the re Structure of FC------P [ 114] Data

Exp.

e- c

Notes:

t(i)

hii

CD2

0.40 0.34 4.05 2.31

0.80f 0.20 0.56 0.43

0.55 0.03 1.22 0.63

unc. a

I(F12cp)C 95.83955 -0.0005 0.0045 I(FI3Cp)c 95.98510 0.0019 0.0091 re(C-F)d'e 1.277 0.0009 0.002 re(C~P) d'e 1.5455 0.0010 0.002

DB(CF) b DB(CP) b

0.06 -0.02 0.03 -0.04 -1.56 1.56 1.12 -1.13

aUncertainty used to weight the data. bDFBETAS. tin uA 2. din h. ~orrected ab initio data. fThe diagnostics are in boldface when significant.

the three experimental equilibrium moments of inertia gives a condition number K: = 104 and indicates that the two distances are fully correlated. This explains why the result is not reliable: small errors of the rotational constants are magnified by the large condition number. If we add a fourth experimental moment of inertia, that of DCO +, the condition number drops to 7 which is good, but the outlier diagnostics indicate that this fourth datum is not compatible with the other three and is an influential point (see Table 12). In fact, in DCO +, the state 0111 is nearly degenerate with 10~ and the resulting large anharmonic interaction leads to a much too small vibration-rotation interaction constant Ctl. As a result, adding the experimental Ie(DCO +) does not lead to better results although it improves the conditioning. One remedy of the problem is to estimate the equilibrium rotational constants by correcting the experimental ground state rotational constants with the a b initio Bo - Be as done by Puzzarini et al. [116] (see Section II.C). Another solution is to add as

T a b l e 12. Test

Threshold Value

t(i)

hii

9.92a 21

0.5 1

Outlier Diagnostics for I(DCO § DFFITS

2 379

CD2

1 659

DFBETAS r

1 -31

DFBETASR

1 24

Note: ap = 99%.

Table 13. Outlier Diagnostics for r(C-H) in HCO + Test

t(i)

hii

DFFITS

CD2

DFBETAS r

DFBETASR

Value

15

0.94

58

22

6.7

-6.6

34

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH Table 14. Comparison of Different Structures of Phosgenea r( C ' = O )

r o (a,b,c) b r o (a,b) c rIc (a,b,c) b ri~ (a, b) c rs re Notes:

r( C - C l )

1.166(10) 1.1794(17) 1.1871(22) 1.1825(8) 1.185(2) 1.1756(23)

/(ClCCl)

1.746(10) 1.7401 (8) 1.7347(13) 1.7371(6) 1.736(1) 1.7381 (19)

Ref

111.3(10) 111.93(8) 112.46(14) 112.17(5) 112.2(1) 111.79(24)

175 176 this work this work 55 55

aDistancesin ,~, angles in degrees. bFit of the moments of inertia, Ia, 1b, and I c. CFitof the moments of inertia, Ia and I b only.

fourth datum the ab initio equilibrium, r(CH) = 1.093 A, with an uncertainty of 0.003 A, which is rather pessimistic. In this last case the condition number is then only 13, and the two parameters are accurately determined: r(CH) = 1.0933(7)/~, and R(CO)= 1.1055(1) ,~. The ab initio r(CH) bond length is influential as shown in Table 13, but, as it is likely to be reliable, it does not harm the result.

C. Phosgene,COCI2 At first sight, COC12 is a good candidate for an accurate structure determination because it is formed from only four atoms; there is no H atom, no small coordinate either, and the rotational constants of nine isotopic species have been accurately determined [55]. As a consequence a substitution or ri~ structure should be a good approximation of the re structure. In fact, the result is disappointing as shown in Table 14 where the results of different investigations are gathered. These poor results may be understood if we calculate the condition number of the Jacobian. In all cases it is large and the variance-decomposition proportions indicate that all the parameters are highly correlated, thus they are not expected to be reliable (see Table 15) where the matrix of variance-decomposition proportions is reported for the particular case of the ri~(a, b, c) structure. The other structures Table 15. Matrix of Variance-Decomposition Proportions for the q,(a, b, c) Structure of Phosgene Proportions o f Variance b 1]a

r(C=O)

r(C-Cl)

L(CICCI)

e,a

e,b

e,c

81 395

0.017 0.983

0.010 0.971

0.019 0.977

0.002 0.775

0.889 0.046

0.810 O. 179

Notes:

aCondition index. bThe diagnostics are in boldface when significant.

35

Determination of Reliable Structures from Rotational Constants

Table 16. Comparison of Different Structures (A) of Chloroacetylene r(C-H)

ro rs

r(C-~--C)

1.0612(2) 1.0552 1.0615(2) 1.0604 1.063(2) 1.0605(5) 1.0616(18)

rlc

r~ rc re rc(mixed)

1.2038(5) 1.2038 1.2043(3) 1.2032 1.209(2) 1.2030(2) 1.2033(15)

r(C-Cl )

Ref

1.6371(3) 1.6366 1.6359(4) 1.6355 1.631(2) 1.6353(1) 1.6360(12)

117 117 117 117 58 118 this work

give similar results. In that particular case, the problem was solved by combining the microwave and electron diffraction data [55].

D. Chloroacetylene, HC_=CCI The rotational constants of many different isotopic species of chloroacetylene have been identified and the structure of this molecule has been determined by different authors using different methods [58, 117, 118] (see Table 16). The rc structure deviates from the re, r~ and rle-structures beyond the estimated errors limits, but the latter two structures might be not reliable with respect to the r(C-H) bond length (see Sections III.D and E). The re structure was calculated using the experimental ground state constants of 12 isotopomers and vibration-rotation constants calculated ab initio using the coupled electron pair approximation. The rc structure was calculated using the ground state rotational constants of complementary sets of 16 isotopomers, thus the rc structure could be more reliable, but, here again the condition number of the system of normal equations is large and the variance-decomposition proportions indicate that all the parameters are highly correlated, thus they are not expected to be reliable (see Table 17) where the matrix of variance-decomposition proportions is reported. Of course, in the case of the re structure, the ill-conditioning continues to exist, but it is no longer harmful because the data are highly accurate. In fact, a calculation

Table 17. Matrix of Variance-Decomposition Proportions for the rc Structure of HC=CCI a Proportions of Variance qb

160

r(C-H)

r(C~--C)

r(C-CI)

0.548

0.999

0.999

Notes: aForthe highestconditionindex. bConditionindex.

36

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

Table 18. Ab lnitio and Experimental re Structures (A) of HC-=CH, CICN, and HC----CCI r( C-H) HCCH

C1CN

r( C-~-~C)

calc. a b exp.

1.0637 1.0625

1.2024

offset c

-0.0012

-0.0074

1.2098

calc. a d exp.

1.6437 1.6290

offset c HCCC1

r(C-CI)

-0.0147

calc. a

1.0623

1.2094

1.6492

re e

1.0611

1.2020

1.6345

Notes: aCCSD(T)/cc-pVTZ. bRef. 177. Cre(eXp.) - r[CCSD(T)/cc-pVTZ]. dRef. 129. er[CCSD(T)/cc-pVTZ] + offset.

of the Ic moments of inertia with the parameters fixed at the re structure values gives a root mean square error which is not significantly higher than in the rc fit. It is possible to check that point by calculating the ab initio structure and by correcting the bond lengths with constant bond-specific offsets taken from similar molecules whose re structure is accurately known (see Table 18).These results confirm the validity of the re structure. Alternatively it is possible to make a mixed estimation using as input data the Ic of Nakata [58] (with an estimated standard deviation of 0.08 uA 2) and the corrected ab initio re of Table 18 (with an estimated standard deviation of 0.002 A). In that case, the fit is well-conditioned (K = 10), the correlations are much lower (-0.955 instead of-0.999), the different data seem to be compatible, and the results, which are given in the last line of Table 16, are in good agreement with the re structure of Horn [118]. The C - H bond length is not correlated with the other parameters, furthermore all the methods give compatible results which are in good agreement with the value found using the isolated stretching frequency v(C-H), r(C-H) = 1.0612 A [117].

E. Methyl Chloride, CH3CI The equilibrium structure of methyl chloride has already been determined by Jensen et al. [119]. But some experimental data were rather inaccurate; for instance the axial rotational constants of CH335C1 and CH337C1 are too different to be reliable. Furthermore the C - H bond length, re(C-H) = 1.0854 A, seems to be too large to be compatible with the isolated stretching frequency of the C - H bond as well as with the results of high level ab initio calculations [34]. Since that time, a

37

Determination of Reliable Structures from Rotational Constants

Table 19. Data Used for the Least-Squares Determination of the re Structure of CH3Cl [ 178] Data a

Exp.

la(CH335Cl) Ib(CH335C1) la(Ca337C I) Ib(CH337C1) Ia(CD335Cl) Ib(CD335CI) Ia(CD337CI) Ib(CD337C1) re(C-H ) re(C-C1) Ze(HCH) Notes:

3.1972429 37.692545 3.1986016 38.277959 6.390552 46.26345 6.390774 47.06163 1.0840 1.7760 110.410

t( i)

hii

--0.00218 0.00941 -0.00082 0.00401 -0.00337 0.0139 -0.00315 0.01447

Exp - Calc

0.012 0.048 0.1 0.5 0.012 0.048 0.012 0.048

unc. b

0.29 0.51 0.08 0.06 0.21 0.05 0.42 0.13

0.11 0.42 0.00 0.00 0.43 0.40 0.43 0.42

-0.0002 -0.0008 -0.098

0.002 0.004 0.5

1,46 2.93 2.21

@58c 0.20 0.00

aMomentsof inertia in uA2, distancesin A, anglesin degrees. bUncertaintyused to fit the data. Clnfluentialpoint.

great deal of work has been devoted to the determination of accurate molecular parameters for CH3CI (see Table 19). Nevertheless, the inaccuracy of the axial rotational constants of CH335C1 and CH337C1 remains and a fit of all available moments of inertia indicates that Ia(CH335C1) is not compatible with the other data (t(i) = 4.4) and that Ib(CH335C1) is an influential point (hii - 0.99). The result of the least-squares fit is hence not completely reliable. To solve that problem, we have calculated the ab initio structure at the CCSD(T) level using the cc-pVTZ basis set. It gives Z(HCH) = 110.46 ~ r(C-H) = 1.0859/~, and r(C-C1) = 1.7905 ~. The ab initio r(C-H) is known to be too long [93], and after a small offset correction, we obtain r(C-H) = 1.084 ~. The ab initio C-CI distance is also too long but there is not yet any reliable offset available. On the other hand, the structures of chlorine derivatives for which an accurate re is known have been recently calculated at the MP2 level using the 6-311 +G(2d, p) enabling us to derive an accurate offset [101]. The result for CH3C1 is (after offset correction)" r(C-C1) = 1.776 ~. These structural data have been incorporated into the least-squares analysis as additional observations using the uncertainty given in Table 19 for the weighting. The fit indicates that the system is well conditioned, K = 6, and that all the data are compatible (see Table 19). The final result is: re(C-H) = 1.0842(2)/~, re(C-C1) = 1.7768(2)/~, and Ze(HCH) = 110.21(3)/~.

F. Difluoroethyne, FC~CF A high-resolution infrared study yielded rotational parameters of the ground and all singly excited states of the parent species [120]. The vibrational correction

38

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH Table 20. Interatomic Distances (]k)in FC----CF

Molecule

Method

FC_=CF

r( C~---C)

r(C-F)

Ref 121

ro

1.197(2)

1.276(1)

re

1.197(3)

1.277(2)

MP2/rz2P+f a

1.187

1.283

121 100, 120

CCS D(T)/cc-pV QZ a

1.1860(6)

1.2835( 4 )

122

HC~CF

re

1.1961 (2)

1.2765(2)

179

HC----CH

re

1.2024(1)

177

Note: aWithoffset correction.

e = Io - le was found very small as expected, 0.044% of Io (see Section II.C). Later, ground state constants were obtained for the isotopic species FI2c----13CF from ground state combination differences of the v3 and v2 + v3 bands [121]. From the two rotational constants, it is possible to determine the ro structure. On the other hand, to obtain the re structure, it is necessary to make an assumption on the variation of e with isotopic substitution. One may assume that e remains constant (rI~ structure); alternatively, one may assume that e' for the isotopic species may be obtained from the e of the parent species using the approximation,

with 0.5 _
Table 21. Least-Squares Determination of the Structure of FC_----CF Dataa

Exp.

Unc. b

Exp - Calc.

Ie(FCCF)

142.2393

0.006 c

-0.003

Ie(FCI3CF)

t( i) -84.2 e

hii 0.52

142.5932

0.006 c

0.003

36.2

0.52

r(C-F) d

1.283

0.003

0.0003

0.2

0.37

r(C=--C)d

1.186

0.004

-0.0004

0.2

0.58

Notes: aMomentsof inertia in U/~2, distances in/~,. bUncertainty used to weight the data, the experimental standard deviation of Ie(FCCF) is 0.0012 u~ 2. c10% ofe. dMP2 calculation with offset correction (see Table 20). eThe diagnostics are in boldface when significant.

Determination of Reliable Structures from Rotational Constants

39

the ab initio structures of these three molecules are more consistent than the experimental ones. Furthermore, quite recently the structure of FC-----CF has been recalculated at the CCSD(T)/cc-pVQZ level, confirming the results of the MP2 calculation [122]. A least-squares fit using as input data the two experimental moments of inertia and the two ab initio distances indicates that the system is ill conditioned (~: = 123) and that the two moments of inertia are likely to be incompatible, although the calculated le reproduce the experimental ones to within 3cr (see Table 21). If we exclude the 13C species from the fit, we still have t(i) = 5.0 for the parent species. On the other hand, if we exclude the parent species, all the data seem to be compatible. However, the number of experimental data is certainly not large enough to draw any safe conclusion. What can be said is that the problem (without the ab initio structure as additional data) is extremely ill-conditioned and that a very small change of the moments of inertia may have a big effect on the derived structure. It shows the usefulness of mixing data from different sources, which is much more effective than trying to improve the accuracy of the equilibrium moments of inertia. The structure derived by the mixed estimation is in very good agreement with the recent CCSD(T)/cc-pVQZ calculation [122] (see Table 20).

VII. CONCLUSION One of the main objectives of this review is to encourage the inclusion of additional structure data, in particular from ab initio calculations, when the structure of small molecules is to be determined from the fit to inertial moments evaluated from rotational or rovibrational spectra. To the computing chemist it may seem sufficient to correct the ab initio values by (preferably constant) offsets to obtain a reliable re structure. This is often true when the molecule is small and formed from only light atoms, when the basis set is large, and the electron correlation has been properly taken into account. However, a combined (mixed) fit of the experimental and the ab initio data allows us to check the compatibility of the data and to improve the accuracy of the derived parameters, thus giving us more confidence in the structure obtained. An interesting alternative to this method is to combine the ab initio corrections, e = Io - Ie, and the experimental ground state moments of inertia. This method does not require any empirical offset. Furthermore e is only a small correction, thus, in favorable cases (when the system is not ill-conditioned), it is not necessary to determine e with high accuracy. But these favorable cases are rather exceptional because ill-conditioning seems to be a property inherent to the structure determination. Furthermore, this method is also limited to small molecules with only light atoms. For the application of the available fitting procedures, insight into the possibilities and pitfalls of the least-squares method is required in any case. It is worth noting that small standard deviations are not sufficient proof of a reliable structure, in particular when the data set is small.

40

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

For medium-sized and larger molecules, the situation is less satisfactory. In principle, structure data from sources other than spectroscopy (e.g., from electron diffraction) can be easily incorporated, but sources with inherently higher accuracy are seldom available. Also, possible correlations between the additional data might then require attention. Two routes can be followed when inertial moments are fitted: the rs-fit method (derived from the substitution or rs-method), and/or the ro-derived rie-method. Both assume isotopomer-independent vibrational corrections, the former implicitly by fitting isotopic differences of moments, the latter explicitly. When in the set of available isotopomers all atomic positions have been individually substituted at least once, a third and probably more refined method can be applied to obtain the rPm -structure where the vibrational corrections are assumed proportional to the inertial moments, the proportionality factor for all isotopomers being derived from the parent's rs-structure. In favorable situations, where the number of isotopomers measured results in a sizable overdetermination, the results of these methods tend to be similar. While they are certainly a substantial improvement over the simple ro-structure, it can be safely said that they still fail to attain re-quality. Figure 2 collects the current means and methods to approximate the re-structure from available experimental data.

Figure 2. Flowchart to visualize the approximation of the molecular equilibrium structure by the methods discussed in this work.

Determination of Reliable Structures from Rotational Constants

41

APPENDIX Whenever the quantifies that are introduced into a least-squares fitting problem as observations have themselves been obtained as the result of a preceding leastsquares fitting process, the errors of these "observations" will, in general, be correlated. This is typically the case when the inertial moments of a set of isotopomers have been evaluated from the respective rotational spectra by a least-squares fitting of the transitions and are then used in another subsequent fit to obtain the structural parameters of the (parent) molecule. While it is a general assumption in spectroscopy that there is no correlation between consecutively measured spectra of different isotopomers (provided the instrument is well adjusted) and hence neither between the evaluated sets of inertial moments of different isotopomers, the three inertial moments within each set are often seriously correlated, depending on the type and number of transitions that were accessible and could be measured. Also, in both the rs-fit- and the r0rn -method, the quantities that enter structure fitting are not strictly the original inertial moments of a substitution set of isotopomers, but are, though still called inertial moments, rather derived by a process which necessarily leads to more or less correlation between all of them. Another typical feature of the present application is the inherent nonlinearity of the relation between the observations, i.e. the data set of inertial moments, and the regression coefficients, i.e. the structural parameters to be determined. The GaussNewton method of successive iterations allows nonetheless the application of the popular linear least-squares procedure. For this purpose the problem is linearized by an expansion in the neighborhood of the expected solution. For each iteration, the observations are the differences between the inertial moments of the original input and the approximations computed from the present structure, while the solutions produce corrective increments to this structure. The design matrix is the Jacobian of the expansion; it can and should be computed anew for each new iteration. The linear regression model equations in the "general least-squares" case for determining p variables 13from n correlated observations y are, Eq. 27, y = XI3 + e, while the covariance matrix of the correlated observations (and of the errors or residuals e) is | - O(e) - o ' 2 M . It is assumed that M is due to the known experimental errors and correlations of the observations. In Section IV it was shown that there may be situations where the experimental errors are small compared with unknown systematic errors. In the interest of a better balanced regression, the systematic errors may then have to be taken into account, however crudely, by cautiously simulating them by statistically distributed errors. At any rate, M is assumed to be a known matrix in the following discussion. In order^ to ^ obtain least-squares estimates of the variables 13and their covariance matrix 0([3) which are optimum in the Gauss-Markov sense, the positive definite n x n covariance matrix M must first be reduced by the congruent transformation p'r to the unit matrix, pTMp= 1; then the transformed covariance matrix I~)(~1) = ~21 is, as

42

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

required, that of uncorrelated unity-weighted errors, a property which is denoted by the subscript 1. By the transformation pT in the n-dimensional space of the observations and their errors, the vectors y and ~ and the n x p design matrix X are transformed accordingly: pTy = Yl, PTe = e l, and pTx = XI. As stated at the beginning of Section IV, this section was essentially limited to the discussion of a problem that is already in the unity-weighted form. For better readability, we have in Section IV not explicitly affixed subscripts (and have also left it to the context to distinguish between quantities for the scaled and unscaled unity-weighted problem). The case of "weighted least-squares" with observations of unequal precision but with zero correlations need not be treated as a separate case. The matrix M is in this case a diagonal matrix with the variances var(yi) of the observations as diagonal elements. When each observation Yi and all elements of the ith row of the design matrix X are divided by the square root of the corresponding variance [i.e., by cy(yi), the standard error of Yi], this simple scaling will immediately transfer the weighted problem to a problem that is unity-weighted from the outset. Of course, the solutions given below for the general least-squares case, Eqs. A7-A9, will formally apply also to the weighted least-squares, with M the prescribed diagonal matrix [68]. When least-squares diagnostics are to be discussed, it is useful to go one step further than unity-weighting and scale the design matrix X I by postmultiplication, XIS = Xls (s for scaled). The columns of Xls (the p fit vectors in error space) are thereby all given the same length, usually unity. The scaling matrix S is diagonal; the elements are the reciprocal column lengths of Xl. Rescaling [3 by S-113 = fls compensates for the scaling of Xl. Therefore, scaling is a transparent process with respect to error space, Yl = Xls[~s + el = Xll3 + el. The scaled matrix Xls is no longer sensitive to an arbitrary change of the dimension of any of the variables 13, but the "collinearity" of the problem (how well- or ill-conditioned it is) remains essentially unchanged. The problem has thus been "standardized" and the framework of the unity-weighted, uncorrelated, scaled transform of the problem can now serve as common grounds when different sets of observations or variants of the model are investigated with respect to their effects on the collinearity of the problem. The solution of the standardized problem is now obtained in its wellknown form, for the variables to be determined and their covariances, A

A

13s = [XTsXls]-lXlsYl = S - I [ x I x I ] - I X I T y l ,

A

0([3 s) = ~'2[XTsXls]-1

(A1)

and, after backscaling, for the variables of the original (correlated) problem, = S~s,

A

A

~( ) = SO(13s)S

(A2)

For the estimated observations of the unity-weighted problem and their covariances we have,

^ =Xls[XlsXls] T -! XlsYl T =Hlsyl =Xlt,-xTx Yl I 11-,-1xTlYl =Hlyl

Determination of Reliable Structures from Rotational Constants ^ ^ O(yl) = ff^ 2 Xls

[XTsXls]-i

XTs = ~y2Hls ^

=

~y2Xl[XlT Xl]

-1 X lTY 1

43

= ~^ 2 H 1

(A3)

where the fight-side equations clearly show the transparency of scaling with respect to error space. (By these equations, also the "hat matrix" H1 = His is defined: l~-- Yl (see Eqs. 28-30). Finally the estimated residuals and their covariances are obtained as: ^ (;1

=

^ T -1 T Yl - Yl = [1 - X l s [ X l s X l s ] X l s ] Y l = (1 - H l s ) Y 1

= [1 - XI[XTX,]-IXT]y 1 = (1 - H,)y I ^

A

O(e,) = ~2[1 - Xls[XlTsXlsl-lXTs ] = ~2(1 - His ) = ~2[1 - X, [ x T x 1 ] - I x 1 T] = ~y:(1 - H,).

(A4)

The estimated observations and residuals can also be readily backtransformed to those of the original (correlated) problem: ^ Y^ = 0pT)-I Yl, A

8 =

^ O^ ( y^) = 0 p T ) - l ^O(yl)P -1

A

A A

(pT)-I 81'

A A

O(~) = (PT)-Io(8)P-I

(A5)

(A6)

Without recourse to the "standardized" problem, the solutions of the original correlated problem, y = Xl3 + 8, | =| = o2M, are usually formulated directly [68] as: A

13- [XTM-IX]-IxTM-Iy,

yA = x[XTM-1X]-IxTM-ly, A

= y-

A

y =

~)(~) - ~2[XTM-IX] -l

(A7)

A A O(y) = ~2[XTM-IX]-Ix T

(A8)

A A [1 - x[xTl~l--1x]-lXml~l--1]y, O(13) =

~2[M-

x [ X T M - 1 X ] - I x T] ( A 9 ) A

In the above equations, we need not distinguish between ~ andS1 because the estimated variance of the fit ~2 is not changed by the transformation between the original and the unity-weighted problem:

~1 =

~T~l n -p

^ ________~g A = ~2 = gATppT_______A___~=g ~Tm-1. n -p

(AIO)

n -p

For the uncorrelated problem (for the correlated problem only after transformation), the singular value decomposition of the design matrix can be employed to simplify the solution and make it computationally more stable. Applied to the scaled, "standardized" n x n matrix Xls , the decomposition is Xls -- Uls Dis V Tls, cf. Eq. 33, where the elements of the p x p matrix Dis are the singular values

44

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

~tls(j), j = 1. . . . . p of Xls (non-zero ~tls for full rank of Xls), and where the n x p matrix Uls is column orthonormal, U~sUls = 1, and the p x p matrix Vls orthonormal, V~sVls = VisV~s = 1. (Corresponding equations hold for the unscaled Xl.) The greater numerical stability comes from the fact that the ubiquitous matrix [X~sXls] -l need not be computed by matrix inversion (except for the trivial one of D2s), but is given by [X ~sXls]-l = V IsD~2V its. Therefore, cf. Eq. 36, A 13s = V TsD ~ U lsYl,

A^ O(13s) = ~.2Vl sDlsVls -2 T

(All)

A

which can be readily backscaled to obtain 13and ~)(~) (see Eq. A2). The hat matrix is simplified to U lsU its = His = H1 = U IU ~, so we further have A

Yl = HlYl = U,U lrYl, A E 1 =

O(Yl)= ~Y2HI= ~y2uIUT

A

A

(Ale)

A

A

(A13)

(1 - Hl)Y l = (1 - U1U~)y 1, O(e~) = ~2(1 - H l) = ~2(1 - U1U ~)

All least-squares diagnostics discussed in Section IV can be immediately applied to the unity-weighted scaled (standardized) problem or any problem that can be modeled as such from the beginning (as in other fields of statistics). It should be noted that there is no known relation between the singular value decomposition of the matrix Xl and the matrix XlS = Xls (except when S is orthogonal), i.e., between any one of the three individual factor matrices UI, DI, and VI and their counterparts of the scaled problem, Uls, Dis, and Vls (a relation like U IU T = UlsU ~s is not excluded). For example, the two diagnostics, (scaled) condition indexes rlls J , j 1. . . . p, Eq. 34, and (scaled) variance decomposition proportions rCls,jj,, j, j' = 1,. ..,p, Eq. 37, used to disclose the number and severeness of possible near-collinearities among the p variables and to indicate which of them might hence be degraded (see Section IV), could not be retransformed from the scaled to the unscaled problem. These two diagnostics can and should be discussed for any problem only after it has been given the standard scaled form (if it is not originally in this form), because only then some simple questions can be simply answered; e.g., when a condition index rlls, j has to be considered as large (when rlls, j > 30, see Section IV) or when the magnitude of a variance decomposition proportion 7r,ls, jj, would require attention (when nu,jj' > 0.5 and rlls, j > 30, see Section IV). While the quantities rlls. j and rCls,jj, deal only with the p singular values and the variances-covariances of the p variables, which are sensitive to scaling, other diagnostic concepts are concerned with the n observations Yl.i and their influence on the problem, e.g., the standardized, studentized, and jackknifed residuals, Eqs. 38, 39, and 40, respectively, and the leverage, hl,ii--O~li/~Yli=Hl,ii [~)(~1)] ii ~or2, which represents the influence of the ith observation on its own estimate, as well as the diagnostics CD2, Eq. 42 and DFFITS, Eq. 43. Since the n-dimensional space of the observations (and errors) is transparent with respect to

Determination of Reliable Structures from Rotational Constants

45

scaling, only the subscript l is required on the constituent quantifies, but may be replaced by is if desired and convenient, e.g. for the leverage,

hl,ii-- Hl,ii-- [u1uT]ii = E u2l,ij j=l p

(A14)

----Hls,i i = [Uls U Ts]ii = ~ u 21s,ij

j=l where the terms of the sums are the leverage compAonents. (The diagnostic DFBETAS, Eq. 41, which references also the variables [3j, has been made insensitive to scaling by dividing the precursor, DFBETA, by the standard error of the variable [84].) Several of the diagnostics are based on the effect of deleting one observation from the problem, say Ylj, and removing the corresponding row xTlj from the design matrix X1 which is then denoted as Xl(i ). The diagnostics jackknifed residual, CD2, DFBETAS, and DFFITS are all of this type. Thanks to a "remarkable formula" [86], it is possible to express the matrix [X T (i)x l (/)]-I (still of dimension p x p), which is a constituent part of the row-deleted solution, in closed form, using only the matrix X 1 of the full problem and the row vector XTlj. Therefore, the quantities ~(i), ~ly(i), ~j(i) of the row-i-deleted solutions, i --- 1. . . . . n, required for these diagnostics, can all be computed without explicitly setting up and solving n new least-squares problems, each with n - 1 observations. In principle, the formula mentioned could be applied also when a row is deleted from a scaled matrix Xls; note, however, that the row-deleted matrix Xls(i) is then no longer scaled. Therefore, if rescaling of the row-deleted matrix is required, as for Belsley's "row-deleted scaled condition indexes" [84], the computationally expensive route of repeatedly solving n row-deleted and rescaled least-squares problems must be followed. Unfortunately, the same is true in the case of a general, i.e. correlated, leastsquares problem. Deleting a particular observation Yi (note: no subscript I in contrast to the previous paragraphs) means removing also the ith row and ith column from the c o r r e s p o n d i n g covariance matrix of the full problem, M = O ( y ) / (~2 (_~ M(i) = O[y(i)]/cr2(i), and there is no known useful relation between the two congruent transformations, of dimensions n x n and (n - l) x (n - 1), which reduce M and M(i) to respective unit matrices. A new congruent transformation is required to transform each new row-deleted correlated problem into a corresponding equally-weighted uncorrelated problem. Therefore, n row-deleted problems must be separately solved, an intense labor indeed, all the more since the final solution must usually be obtained iteratively. In conclusion, when dealing with a correlated least-squares problem, the diagnostics related to the variables and their covariances (viis, Xls) should be applied to the uncorrelated scaled ("standardized") transform of the original problem for the

46

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH

inspection of near-collinearity and possible harm done by it, because the p individual variables 13j do not lose their identity, neither by the congruent transformation p r of the observations and the design matrix nor by the scaling of the latter. In contrast, the n observations Yi of the original correlated problem do not retain their individual identity because they are transformed into n linear combinations Yl,i (except for weighted least-squares which hence need not be treated in the cumbersome way described below). If a diagnostic is to be based on the influence of an individual observation Yi on the problem, one must recur to the definition, e.g., for the leverage, ~)~)i (A15) i~yi -Ix[XTM-IX]-IxTM-11 ii--I( PT)-IX 1[XTX 1]-IxTpT 1 ii-[(PT)-IH1PT]ii which is less gainly than Eq. A 14, even when one has recourse to the transformed problem (i.e., to Hi), because the matrix in the last term ofEq. A15 is not symmetric and some of the relations which hold for the elements of the matrix I-I~ are no longer valid (e.g., Eq. 30). If a diagnostic requires the consecutive deletion of observations, one at a time, one has to put up with each time setting up and solving a new correlated problem, reduced by the oarticular observation; e.g., see Eq. 41, 9 A A . A A . A AI A A A A DFBETASj(t) [tT/tT(1)][pj-pj(l)]/o'(pj), where tr, 13j, and tr(13j) come from the solution of the full original correlated problem, y = XI3 + e, O(y) = c2M (backtransformed after it has been transformed to the uncorrelated problem and solved), whereas ~r(i)and ~3j(i) are the corresponding counterparts of the solution of the separate correlated problem, y(i) = X(i)I3 + e(i), | = ~2(i)M(i), which is reduced by the deletion of observation Yi, but otherwise correspondingly treated. =

ACKNOWLEDGMENTS The authors wish to thank Prof. J. E. Boggs, Dr. J.-M. Colmont, Dr. J. Coslrou, Dr. G. Graner, and Prof. K. Kuchitsu for a critical reading of the manuscript. They also thank Prof. H. BUrger, Prof. A. Fayt and Dr. Fleischer for helpful hints. H.D.R. is grateful to the Fonds der Chemischen Industrie, Frankfurt, for support.

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EQUILIBRIUM STRUCTURE AND POTENTIAL FUNCTION" A GOAL TO STRUCTURE DETERMINATION

Victor P. Spiridonov

I. II. III.

IV.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Deficiencies o f the Conventional Interpretational Scheme . . . . . . . . . Analysis o f Diffraction Data in Terms o f the Molecular Potential Function A. M o l e c u l a r Structure and Potential Energy Function . . . . . . . . . . . . . B. General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Problems and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . A. A n h a r m o n i c i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. C. D.

V.

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55 . . . 56 56 . 57 . 57 58 . 59

Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . N u m b e r o f Adjustable Parameters . . . . . . . . . . . . . . . . . . . . . . M o l e c u l a r Parameters Determination as an Example o f an Ill-Posed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffraction Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. B.

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C u m u l a n t - M o m e n t Representation o f the Intensity Equation . . . . . . . . C o m m e n t s on Perturbation Calculation . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 53-81 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9 53

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VII. VIII. IX. X. XI.

XII.

C. General Plan of Diffraction Analysis of Quasi-Rigid Molecular Systems . 65 Large-Amplitude Motion Analysis . . . . . . . . . . . . . . . . . . . . . . . . 65 A. General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 B. Adiabatic Separation of Large-Amplitude Motion . . . . . . . . . . . . . 66 C. Thermal Average Coordinate Distribution Function in the Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 67 D. Molecular Intensity in the Adiabatic Approximation . . . . . . . . . . . . 70 Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Quasi-Diatomic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 72 Badger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Combined Use of Electron Diffraction and Various Relevant Techniques . . . 75 A. Fundamental Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B. Complications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 C. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

ABSTRACT A method of analysis of molecules by electron diffraction in terms of the intramolecular potential function is presented. Some problems and limitations are discussed. Further developments in the field, particularly in a large-amplitude motion analysis are given. Special attention is paid to the choice of the model potential function and to some useful simplifications of the analysis scheme. Critical comparisons illustrating the approach are presented. Compelling evidence has been given that the combined use of electron diffraction and various relevant techniques on the basis of a common potential function is most powerful in efforts to solve equilibrium geometries and molecular dynamics.

I. I N T R O D U C T I O N In a routine gas-phase electron diffraction study the scattering intensity is measured from an ensemble of vibrating, randomly oriented molecules. Theoretically, an orientational averaging of the rigid scatterer is readily obtained but the vibrations and vibration-rotation interactions must still be considered. Generally, the latter were found to play a minor role. This leaves us practically with the intensities averaged over the vibrational motion. Thus it appears that when we make a diffraction measurement, we obtain primarily vibrationally averaged values of the distance and amplitude parameters of a molecule. This idea was practically realized in a conventional interpretational scheme of diffraction analysis [1, 2]. Within the scope of this scheme it is a current practice to determine the distance parameter

Equilibrium Structure and Potential Function

55

ra which is equal to the center of gravity of the P(r)/r function [P(r) is thermally averaged distance distribution function] or the parameter rg which is the position of the center of gravity of the P(r) function itself (for any details ref. 3 should be consulted). Indeed, hardly a practical diffraction analysis of fairly complicated molecular system has been published so far that was not based on this scheme. As a result, in the ra (or %) distance representation, a large body of structural information has been accumulated over the years concerning overall molecular shapes, bond distances, valence and dihedral angles, preferred orientations in conformers, etc. This is most exhaustively demonstrated in the MOGADOC database produced by the Center for Structure Documentation at the University of Ulm, Germany [4]. The most general conceptual considerations suggest, however, that the inherent capabilities of diffraction measurements certainly are not limited solely by determining the distance and amplitude parameters averaged over nuclear motion in thermal equilibrium. It is evident that potentially attainable end product which can be extracted from scattering intensities is dependent on a basic theoretical model adopted for the development of the interpretational scheme. Since the intramolecular potential function is the most fundamental molecular quantity governing both geometry and dynamics of a molecule diffraction analysis, a scheme duly formulated in terms of this function seems particularly appropriate for practical applications [5]. By pursuing such a scheme, some troublesome limitations of the current data treatment in terms of the vibrationally averaged distance and amplitude parameters can be, at least in principle, largely surmounted. A more detailed discussion of the most noteworthy aspects of this scheme and inherent problems will be given in the following sections. However, special mention should be made here of the main advantage of such an approach that particularly works toward its suitable development. It is that, indeed, electron diffraction starts speaking the same language as molecular spectroscopy and ab initio techniques. Once this has been done, the nontrivial problem of compatibility of data from all relevant techniques is automatically solved. Then, a self-consistent basis is provided for joint use of electron diffraction intensities, infrared, and microwave spectroscopy data together with high-quality theoretical calculations.

II.

SOME DEFICIENCIES OF THE CONVENTIONAL INTERPRETATIONAL SCHEME

In principle, the most annoying deficiency of the ra (or rg) distance representation is a result of its definition as the average characteristic of an ensemble of molecules at thermal equilibrium. For this reason the ra (or rg) distances lose the strict meaning of the true molecular constants of an individual molecule. An important consequence is that these distances for the nonbonded atom pairs generally tend to be shorter than would be expected in a motionless molecule with the same equilibrium angles (foreshortening, or shrinkage, effect; see e.g. ref. 3) and, therefore, do not rigorously represent any physically significant molecular geometry. Thus it appears

56

VICTOR P. SPIRIDONOV

entirely possible that even for relatively rigid systems serious distance inconsistency can be met in the interpretational scheme and final results cannot be obtained preserving the full accuracy afforded by the primary intensity data. This failure may lead to troublesome ambiguity in the analysis: whenever a distorted molecular structure is derived it is difficult to decide whether this is a consequence of vibrational motion or whether the equilibrium geometry itself is really distorted. Thus the initial goal to the reliable structure determination may not necessarily be approached. The aforementioned inconsistency can partly be alleviated by introducing into the ra (or %) distances vibrational corrections calculated from the harmonic force field ("shrinkage" corrections) yielding a geometrically more consistent ra structure of a molecule or r ~ structure corresponding to the ground vibrational state [6, 7]. The magnitudes of these corrections usually range from a few thousands of an angstrom unit for bonded distances to several hundred for nonbonded distances, depending on the individual case. However, in this development, the vibrational anharmonicity is totally ignored. Extensive experimental studies coupled with theoretical calculations (see, e.g. refs. 8-14; for a review see ref. 15) have shown that the anharmonic foreshortening effect on distances undoubtedly is significant and becomes particularly large (hundredths to tenths of an angstrom unit) when large-amplitude motion occurs either because the molecules are very hot or because they are intrinsically strongly flexible along some of the vibrational coordinate. This sort of a distance vibrational effect was termed "anharmonic shrinkage" to stress its similarity with the harmonic analogue. Because little is known about vibrational anharmonicity, no routine anharmonic distance corrections emerged so far that were both simple and reliable enough to be appropriate for incorporation into current diffraction analysis schemes. Thus, irrespective of the precision of the primary intensity measurements, the molecular structures derived may suffer more or less appreciable uncertainty caused by vibrational effects on internuclear distances, depending on the nature of the particular molecular vibrations.

III.

ANALYSIS OF DIFFRACTION DATA IN TERMS OF THE MOLECULAR POTENTIAL FUNCTION

As discussed in the Introduction, conceptually it is conceivable to formulate an alternative approach to diffraction analysis that deals directly with the molecular potential function. This section provides general formulation and considers most strong motivations for pursuing such an approach. Its problems and limitations and the ideas as to how these can be addressed will be outlined in the subsequent sections.

A. Molecular Structure and Potential Energy Function Within the framework of the Born-Oppenheimer approximation [16], the motions of the nuclei in a molecule can be fully understood if the potential function,

Equilibrium Structure and Potential Function

57

including anharmonicity, is known. However, it is an extremely formidable problem to characterize experimentally this function from the equilibrium to the dissociation limit even for a relatively simple molecular system. Fortunately, for an ordinary quasi-rigid molecule the nuclear displacements can be treated as almost infinitesimal, thus for such a system it is practically sufficient to confine the description of the vibrations to the region in the vicinity of the equilibrium nuclear positions. In this regard, following a rational suggestion by Morino [17] the concept "molecular structure" with regard to electron diffraction measurements can conveniently be defined as including the following sets of parameters: (1) equilibrium nuclear positions, and (2) quadratic and any higher order derivatives of the potential energy function taken at the minima positions (force constants). Accordingly, the distances between the equilibrium nuclear positions are the re distances. In practical applications, more sophisticated potential functions can be used. Thus the strict meaning of the particular re distance yielded by the analysis scheme will be determined by the chosen potential model.

B. General Formulation It undoubtedly is important that the above definition of a molecular structure includes both geometrical and force field parameters, thus giving some pointing on their close interplay. In conformity with this definition, the essence of the alternative treatment of diffraction data is that just the equilibrium geometrical parameters and force field constants are recognized as the quantities sought by the least-squares refinements in the course of the analysis. Thus a solution of the geometrical problem must be accompanied by that of vibrational motion for thermal equilibrium at a given temperature with an appropriate calculation of the molecular intensity function at each step of the parameters variation. The frequencies of vibration are simultaneously obtained in the analysis. However, when the large-amplitude motions are involved, more complicated potential functions and vibrational treatments must be used. This special case will be dealt with in a separate section below.

C. Motivation The strong motivation for pursuing such an analysis scheme comes from the fundamental significance of the potential function concept in a theory of molecules. The re representation of molecular geometry in anharmonic approximation is thus most rigorous and would ensure the adequate consistency of the distance parameters in the course of the analysis. Moreover, both geometrical and dynamical problems are treated simultaneously within the framework of a unified calculational scheme. It then seems fair to conclude that joint treatment of the two basic molecular problems is more adequate than either alone. Another compelling reason can be found in the studies of molecules characterized by large-amplitude motion. This motion (e.g. bending vibration, internal rotation, inversion, ring-puckering motion, etc.) is characterized by comparatively large

58

VICTOR P. SPIRIDONOV

nuclear excursions that are accompanied by relatively small changes in the potential energy. Accordingly, the large-amplitude motion can appropriately be described as a low-frequency oscillation of the nuclei which is governed by a potential with a single shallow minimum or, alternatively, with a set of minima separated by relatively low barriers surmountable within a time-scale specific for the particular experimental technique (see e.g. ref. 18). Of most significant geometrical implication of this motion is the fundamental conclusion by Thorson and Nakagawa [19] that, in the case of a shallow potential minimum, the idea of the equilibrium nuclear position with respect to the large-amplitude displacements loses its primary meaning and the usual structural classification of molecules in terms of the rigid nuclear frame becomes fully inappropriate. To advance a most realistic approach in this case, the authors appealed to the quasi-geometry concept (e.g. the "quasi-linear" term as applied to the XY2-type molecule describes a triatomic system geometry in the presence of large-amplitude bending motion, thus making an implicit pointing on a situation when both linear and bent geometries no longer are adequate structure representations and some intermediate case is at play). Despite the prominent place that conventional interpretational scheme still continues to hold, it follows from the foregoing considerations that the routine reduction of diffraction data based on a rigid frame molecular geometry representation may fail for molecules with large-amplitude motion. In contrast to this, an alternative approach to the analysis in terms of the molecular potential function remains fully valid in this case because the rigid frame should not necessarily be incorporated into the interpretational scheme. It leaves to be added that the presence of a shallow potential minimum is an important and significant feature of molecules of frequent occurrence. A wide application of the described scheme at Moscow University Electron Diffraction Laboratory (see e.g. recent review paper [20] and references cited therein) and elsewhere [21-27] conclusively demonstrated that sufficiently accurate diffraction measurements are not limited by determinations of the vibrationally averaged molecular quantities, and equilibrium geometries are certainly not beyond reach at the present state-of-the-art experimental methods. Under favorable conditions valuable information on the force constants and vibrational frequencies is also available. However, a most strongly supported and meaningful solution for equilibrium molecular geometry and dynamics results when diffraction observations are complemented by available evidence from alternative sources (see below).

IV. SOME PROBLEMS AND LIMITATIONS Considering the fact that diffraction analysis in terms of the molecular potential function is essentially a straightforward extension of the earlier techniques, many inherent problems appear just similar. The following discussion focuses on certain aspects that are most important in this approach also yielding some feeling for the new problems encountered.

Equilibrium Structure and Potential Function

59

A. Anharmonicity As discussed in the foregoing, anharmonicity of molecular vibrations is responsible for the systematic shifts in the internuclear distances measured by electron diffraction. These shifts, which are magnified for the larger internuclear separations, can make the uncertainty in the distances derived greater than that due to random and systematic errors in intensity measurements [15]. Thus this factor cannot be ignored in any accurate equilibrium distance determinations. The manifestation of the anharmonic effect may sometimes be large enough so that certain relevant parameters can be sufficiently reliably extracted from the evidence hidden within the scattering intensity itself. For example, for several selected diatomic (I2 [28, 29]) and simple polyatomic (CH4 [30], CF 4 [10], SiF 4 [10], SiF6 [10]) systems, fairly successful diffraction estimation of some anharmonic parameters was feasible. Since temperature apparently enhances anharmonic effects, high temperature [31] or hot-molecule [9-11] techniques of electron diffraction may be useful in this regard. However, it is not immediately obvious that such a favorable situation is generally the case, whereas it is also true that very few reliable evidence about anharmonicity is available from spectroscopy so far. Theoretical discussions have been developed with the result that some predictive theoretical models augmented by quantum calculations [9,13, 32] are able to yield a profitable source of valuable information about anharmonicity other than scattering intensities or spectroscopic transitions themselves. Moreover, many successful attempts have been made to introduce anharmonic Morse-like approximations for the stretching displacement of covalent bonds (see e.g. refs. 1, 2, 32-36) with the Morse a 3 parameter borrowed from the corresponding diatomic molecules for which sufficiently accurate potential functions have been determined by spectroscopy. Typical a 3 values are listed in a published table [37]. Notwithstanding that the assumption of the similarity of bond-stretching vibrations in polyatomic and diatomic systems has no rigorous theoretical basis, in one of the following sections evidence will be presented that a Morse-like model can be used advantageously in a fairly precise analysis of equilibrium geometry. However, the angle-bending anharmonic effect seems to be more troublesome. Nevertheless, there are some reasons for believing that the neglect of this effect for molecules, which can safely be envisioned as quasi-rigid, would not seriously interfere with the reliable determinations of the equilibrium valence angles at not too high temperatures. It does appear, however, that this would not remain valid for molecules with large-amplitude bending effect. It is entirely possible, then, that adiabatic separation of the bending motion with subsequent refinements of the appropriate potential function parameters directly from scattering intensifies would work satisfactorily in this case. A more detailed discussion of the plausibility of the above statements will be continued in the following sections.

60

VICTOR P. SPIRIDONOV

B. Curvilinear Coordinates In the conventional harmonic force field calculations it is customary to set up the potential energy function as a power series expansion in terms of the linearized internal coordinates, such as bond stretch and bend displacements [38]. However, it undoubtedly is significant in the precise experimental characterization of the equilibrium geometry and force field to formulate the intramolecular potential function in terms of the natural curvilinear internal coordinates (see e.g. refs. 39-42). The apparent rationale for such approach is provided by the firmly established observation that various motions have different force constants. For example, bond angle bending with bonded distances as rigid arms is best characterized by fairly circular trajectories of the terminal atoms, in general agreement with the understanding that bond-stretching force constants are much greater than those for bond angle bending. That is why any realistic potential functions must allow for the greater freedom of the angle-bending motion by making the terminal atoms move around arcs of circles. More strongly arced trajectories would then yield some extra contributions to the measured distance and force field parameters. To further clarify the implications of curvilinear motion for solution of molecular structure and dynamics an alternative consideration seems pertinent here. Since, by definition, curvilinear coordinates are nonlinearly related to normal (rectilinear) coordinates, which are practically used for the solution of a vibrational problem, the nonlinear coordinate transformation effect results in appearance of anharmonic potential constants with respect to normal coordinates even if the initial potential function formulated in terms of curvilinear internal coordinates is harmonic [39]. Thus the use of the latter coordinates is essentially an attempt to include anharmonic effects in the kinematic description of the vibrations by a more rigorous consideration of the true atomic trajectories. It is to be emphasized in this regard that the curvilinear motion effects not necessarily would lead to ambiguities into spectroscopic results quite as large as those in electron diffraction. However, experience gained so far in spectroscopic calculations seems to suggest that model force fields expressed in terms of the curvilinear coordinates generally tend to be simpler than those in terms of their rectilinear counterparts and the refinements converge more rapidly [43]. In principle, the curvilinear approach does not require introduction of additional parameters in the potential function. It seems also interesting to note that practically suitable calculational approaches relating the curvilinear coordinate force constants to the observed vibration-rotation spectrum of a molecule were developed not too far back [44].

C. Number of Adjustable Parameters The inherent problem of current diffraction studies is that they are characteristically underdetermined since, generally, the measured intensity function may not contain sufficient details to allow an unequivocal determination of all geomet-

Equilibrium Structure and Potential Function

61

rical and amplitude parameters (see e.g. ref. 3). Unfortunately, the number of force field parameters is even increased and becomes formidable for relatively complex systems. The situation worsens when certain parameters are severely correlated. Thus it may well be practically the case that the diffraction data at hand are not sufficient to obtain unambiguous solution for equilibrium structure and force field although, generally, a very useful and suggestive set of conclusions may still be available. However, it must not be assumed that this dramatic situation is specific solely for the electron diffraction technique. Suffice it to say that vibrational spectroscopy measurements are also frequently not self-sufficient, with very few exceptions, to characterize the complete force field of a molecule even by use of fairly extensive data. Under these circumstances, it will be most helpful to make best use of all available data from relevant experimental and theoretical sources as well as to reduce the number of adjustable parameters by introducing realistic potential models or to constrain some of them to fixed values transferred, for example, from related molecules. More detailed discussion of some of these possibilities will be given in the subsequent sections. D. Molecular Parameters Determination as an Example of an Ill-Posed Problem The foregoing discussion has outlined some problems which may be faced in the analysis of diffraction data in terms of the molecular potential function. It is well to point out, however, that a further troublesome offender exists which is characteristic to the analysis of experimental data in any area of science, not just in electron diffraction. It is the intrinsic mathematical incorrectness of a solution of the inverse problem which comes about for a reason that the necessary requirements for the correctness of a problem by Hadamard (solvability, uniqueness, stability of solution against small perturbations of input data) are practically never met. Such an unfavorable situation may lead to incomplete, or worse still, not unique final answers. The principal sources of this deficiency as applied to electron diffraction studies may be both laboratory-dependent experimental imperfections and shortcomings of the theoretical framework of today's scattering theory. Comprehensive description of these sources was given by Kuchitsu [45 ], Hargittai [3], and Bartell [15]. However, a general mathematical approach to deal with the incorrectness of a solution of ill-posed problems has not yet been explicitly discussed by practitioners in the field. Therefore, it seemed warranted to consider it here at some length. A theory and practical schemes for a stable and unique solution of ill-posed problems were developed with the aid of a basic concept of regularizing algorithm by Russian mathematicians with Prof. A.N. Tikhonov as a head (see e.g. ref. 46). A regularization theory was employed successfully to various fields of chemistry and physics (an excellent presentation of the background behind this theory and its numerous applications to inverse problems encountered in vibrational spectroscopy

62

VICTOR P. SPIRIDONOV

can be found in a recent monograph [47]). A brief sketch of the theory is given in the following. To make a reliable decision between multiple solutions of a given inverse problem two basic ideas were advanced: (1) limiting the set of possible solutions by some physically plausible a priory considerations, and (2) formulation of a certain selection principle which is expected to provide a stable approximate solution in the vicinity of a rigorous solution if the uncertainty of a primary observable is guaranteed to be sufficiently small. As easily recognized, an attempt to find a rigorous solution of a problem is thus abandoned and a guided search for the so-called normal pseudo-solution is formulated instead. Then, the Tikhonov functional is set up as follows [47],

M (a) (Z) = IlYcalc(2) - Yexpll2 + ~112- 20112

(1)

where Z is a solution vector sought, Z0 is some reference vector which for certain physical reasons is believed to be real, Yexp is the experimental observable, Ycalc (Z) is its calculated counterpart, ct > 0 is the regularization parameter the value of which must be appropriately coordinated with the estimated uncertainty of the primary observation. The first term on the right-hand side of Eq. 1 characterizes the usual goodness of fit, whereas the second one enforces some extraneous stabilizing constraint which is treated simultaneously. This term is seen to represent the extent to which Z deviates from Z0, the value of ct being a measure of penalty paid for this deviation. Thus the more accurate the constraint is the less likely the final results will be wrong. However, some critical point which may be raised in regard with such an approach is that an improperly chosen constraint condition can be misleading. The refinements of a solution vector are carried out by formulating the familiar extreme problem, i.e. that of finding a minimum: rain M (a) (Z) z

(2)

A final result of the refinements, Z (a), i.e. Z (a) = arg min M (a) (Z), can be obtained for any value of ct > 0. A critical problem in pursuing such a technique is to reliably decide on the value to be ascribed to ct. To approach this problem a generalized discrepancy principle [47] was formulated profitably by assuming that the estimate of the total uncertainty of observation ;5 > 0 is known. Then, the transcendental equation with regard to ct [47], prl(~) = IlYcalc(Z (~ - Yexpll- 5 = 0

(3)

is tO be solved. However, the nontrivial problem still remains of deciding what the illusory-free value should be given to 8 based on the technology of the hardware used in a particular laboratory. The inherent difficulty here is that 6 should include

Equilibrium Structure and Potential Function

63

both the random and systematic errors. While the former is assessed relatively easily the same is not true for the latter. This is because it is necessary to identify scrupulously all the principal factors contributing to systematic errors [3, 15, 45]. Toward this end in electron diffraction it seems preferable to estimate these errors from a study of some sample molecules. The appropriate conditions for such a molecule are [45]: (1) precise values of geometrical and vibrational parameters are known from spectroscopy, and (2) the molecular intensity is strong enough so that the parameters derived have sufficient accuracy. Of course, we must not be too dogmatic by pursuing a pure mathematical technique because the basic problem of a reasonable choice of a plausible constraint condition still remains. With regard to electron diffraction studies this dictatesthat some physical model of a molecule should be advanced based, for example, on a previous experience of related compounds or on a more or less likely guess of a geometrical and force field parameters. Apparently, the development of experimental technology and theoretical framework of scattering theory would contribute to the improvement of the situation. But, again, the most strongly supported constraints can be found in the case when scattering data are augmented by evidence from various alternative sources. A more detailed discussion of such a hybrid approach will be given in a separate section.

V. DIFFRACTION INTENSITY A. Cumulant-Moment Representation of the Intensity Equation The relationship between the potential energy function and intensities of diffracted electrons has long been established by perturbation [2, 32, 48-50] and variational [50-52] techniques. However, in practical work at the Moscow University Electron Diffraction Laboratory the cumulant representation of the molecular scattering function, sM(s), and perturbation calculation of thermal average normal coordinate moments transformed then into cumulants was found to be most appropriate [53, 54]. In this representation, the expression for sM(s) is of the form,

sM(s)=i~>j. . (rijic exp

(2k)! (r2k)c sin

t1(2k ) - -+- - - ~ (r2k+l)c

(4)

where (r}k) c and (r}k+l)~ are the cumulants defined with regard to the P(r)lrfunction with zeros as points of reference, gij(s) is the conventional scattering function for a given atom pair, and s is the standard scattering variable. In a practical calculation Eq. 4 can be reduced to [54]:

sM(s) = Y'~ gO(s) exp (r2.) sin s(ro.) c (r3.)c 9 . (rij)c --2 --~ t >j

(5)

64

VICTOR P. SPIRIDONOV

The cumulants entering Eq. 5 are readily expressed in terms of the more familiar moments (Am) (the subscripts ij are left out) [54]" (r)c = r e + (A r)c = r e + (A I") (r2)c = (A r2)c = (A r 2) - (A r) 2 (r3)c = (A r3)c = (A r 3) - 3(A r)(A r 2) + 2(A r) 3

(6)

One can immediately observe the close analogy between Eq. 5 and the current scattering theory expression (see e.g. Eqs. 1-63 in ref. 3) provided the following definitions are introduced: (r)c = r a

average internuclear distance

xl/2 r 2 2c = Im

effective mean vibrational amplitude

1 (r3) c =Z

frequency modulation constant

6

(7)

These quantifies are the three kinds of adjustable parameter to be refined in the current analysis schemes (for details ref. 3 should be consulted). B. Comments on Perturbation Calculation

The main advantage of the cumulant-moment method used in deriving Eq. 5 is that the cumulants entering this equation in a quite straightforward but not necessarily easy way can be calculated from the force constants by the perturbation algebra routine [54]. It is proven in general that cubic term approximation to the anharmonic component of the potential function of a quasi-rigid molecule is sufficient within the limits of today's average precision in yielding diffraction records (see e.g. refs. 15, 50, 55). In pursuing the perturbation technique caution is needed to provide against possible occurrence of resonances. The implications and significance of the Fermi resonance effect for diffraction analysis of sample molecules CO2 and CS2 was carefully studied in a classical paper by Morino and Iijima [56]. Treating anharmonicity and vibrational modes coupling by first-order perturbation theory these authors solved accurately for the vibrational states and calculated the Boltzmann averaged distance distribution function in normal coordinates transformed then in Cartesian displacements. It was found that when one is concerned with moderate temperatures the Fermi resonance does not alter the distance distribution function. It appears possible that this favorable conclusion can safely be extended to larger systems. However, it may not necessarily be correct for high vibrational excitation levels when anharmonic interactions are considerable and the number of vibrational states involved becomes formidable, particularly for medium- and large-size molecules. Thus a more sophisticated approach should be appealed in perturbation calculations of systems at elevated temperatures [54].

Equilibrium Structure and Potential Function

65

C. General Plan of Diffraction Analysis of Quasi-Rigid Molecular Systems1 It is important to realize that the difficulty and expense of a diffraction study of a molecule is critically dependent on the presence or absence of large-amplitude motion. This is the main reason why it is important to distinguish at the outset between small- and large-amplitude cases. First, we consider the former. Within the confines of a small-amplitude vibrational theory a calculation of diffraction intensities may be broken down into two main components: 1.

2.

The molecular potential function is set up as a power series expansion in terms of natural curvilinear internal coordinates and a nonlinear coordinate transformation is carried out into normal coordinates. At this stage the initial coordinate force constants are transformed into those of normal coordinate. The L tensor formalism originally due to Hoy et al. [44], which essentially is a straightforward generalization of a more familiar L matrix technique [38], seems particularly appropriate for the purposes of diffraction analysis [40]. Calculation of the thermal average normal coordinate moments by appropriate use of perturbation theory expressions [54] with subsequent transformation of these quantities into the cumulants of Eq. 6.

Accordingly, a practical calculation proceeds through several consecutive stages: 1.

2.

3.

4. 5.

Given the availability of the starting equilibrium geometry and force field in terms of the curvilinear internal coordinates. The harmonic vibrational problem is solved and the L matrix is determined. The L tensor elements follow directly. Once the L tensor is formulated the internal coordinate force constants are transformed to the normal coordinate force constants and the vibrational Hamiltonian of a molecule is set up as a normal coordinate series expansion through cubic terms. The average normal coordinate moments are calculated from the potential function coefficients entering the Hamiltonian of the previous item which are converted then into the expressions in internal coordinates. From the internal coordinate moments the cumulants are calculated and the classical centrifugal stretching correction is introduced into the first cumulant. Finally, the reduced molecular intensity function, Eq. 5, is calculated.

VI.

LARGE-AMPLITUDE M O T I O N ANALYSIS A. General Formulation

The scheme of analysis described in the preceding section was based entirely on the assumption that the molecule under study is quasi-rigid, i.e. has no largeamplitude motion. Typically with this assumption, the expansion of the molecular

66

VICTOR P. SPIRIDONOV

potential in a Taylor series in the vicinity of equilibrium converges sufficiently rapidly. In a large-amplitude case, however, a similar expansion has an unsatisfactory radius of convergence and becomes fully intractable. This underscores the need for an appeal to more sophisticated approaches. The basic assumption underlying conventional diffraction analyses for molecules exerting large-amplitude motion is that of full separation of this motion from skeleton vibrations supplemented by the use of classical treatment for thermal average coordinate distribution function (see e.g. ref. 3). At its best, this elementary dynamic model has proved realistic in many gas-phase electron diffraction studies. However, the condition that couplings with other modes are negligibly small may lack precision in quantitative evaluation of the relevant parameters in some cases [18]. The commonly cited further assumptions invoke the neglect of certain inherent factors. The most important among them appear the following [18]: (1) curvilinearity of the large-amplitude motion coordinate, (2) interaction between large-amplitude motion and a rotation of a molecule as a whole, and (3) variations of the skeleton geometry accompanying the large-amplitude motion. These deficiencies can be remedied, partly at least, by a more elaborate treatment of large-amplitude motion dynamics and by ab initio computed "local geometries" employed as useful constraints in diffraction analysis [57].

B. Adiabatic Separation of Large-Amplitude Motion In diffraction analysis of nonrigid systems an effective approach can be formulated by assuming that large-amplitude motion has a period of vibration much longer than that of small amplitude motion. Then the molecular vibration can be described as made up of rapid oscillations about some reference configuration changing more slowly and parametrically depending upon the large-amplitude motion coordinate. Under this approximation, which is seen to be essentially the adiabatic separation of small- and large-amplitude motions, the total Hamiltonian of a system can be represented as (see e.g. ref. 58), H = Hr( p, R, p) + Hnr(Jx, Jr, Jz, Jp' P)

(8)

where, Hr( p, R, p)= Tr(p, R, p) + Vr(R, p)

(9)

is the rigid subsystem Hamiltonian,

Hnr(Jx' Jy' Jz, Jp'

p) =

Tnr(Jx' Jy' Jz, Jp, P) + Vnr(P)

(1 O)

is its nonrigid counterpart, Tr (p, R, p) is the rigid subsystem kinetic energy operator, Vr (R, P) is the potential of this subsystem, and

Vnr(P ) = V0(P)

+

(H r (p, R, p) )

(11)

Equilibrium Structure and Potential Function

67

is the effective potential of the nonrigid subsystem, Vo(p) describes the largeamplitude motion potential. In the above equations R = (R I . . . . Rn) is a set of small-amplitude vibrational coordinates, p = (Pl . . . . Pn) is a set of momentum operators canonically conjugate to the coordinates, R, Jx, Jy, and Jz are the Cartesian components of the total angular momentum operator with respect to the molecule-fixed axis system, J~, is the momentum canonically conjugate to the large-amplitude coordinate P. A notation < . . . > denotes averaging over rigid motions (for systems in thermal equilibriumtemperature averaging).

C. Thermal Average Coordinate Distribution Function in the Adiabatic Approximation Within the framework of the adiabatic approximation the total thermal average coordinate distribution function P(R, p) can be represented by [58], P(R, p) = P r(R ,p ) + Pnr(P)

(12)

where Pr(R, p) is the rigid coordinate distribution function which is parametrically dependent on p, and P,,r(P) is its nonrigid counterpart. The Pr(R, r) function describes thermal distribution of the rigid coordinates in a hypothetical molecule with a fixed value for p and is usually referred to as the framework, or skeleton, vibrations. This function has long been evaluated quantum mechanically in a closed form in the harmonic approximation for the vibrational potential (see e.g. ref. 59). In contrast to this, no analytical quantum solution for P,,r(P) is available so far in a general case. For diffraction analysis purposes, however, it was possible to adapt a simple classical approximation which has proved to capture a significant portion of statistical physics involved since, usually, hco3~ < 1 (co~ is large-amplitude vibration frequency). Under classical approximation, for an arbitrary potential Vnr(P), P,,r(P) is given by [59],

Pnr(P) = ~e- nr(Jx" J'" J' Jp' P)/kTdJxdJydJzdJ p

(13)

where the operators Jx, Jy, Jz, and Jo are now considered as the classical momenta. For simplicity, the subscript nr will be left out in the following discussion. If the dependence of the Cartesian components of the principal moment of inertia of a molecule upon skeleton vibrational coordinates is neglected, a straightforward integration of Eq. 13 with respect to Jx, Jr, and Jz yields an approximate expression for the thermal average large-amplitude coordinate distribution function, P(P) = (Ix(P)ly(p)Iz(P)) 1/2 ~e- H(Jp"9)/k Tdj O

(14)

where Ix(p), ly(p), and Iz(p) are the Cartesian components of the principal moment of inertia of a molecule parametrically depending on p,

68

VICTOR P. SPIRIDONOV

H(Jp, p)= F(p)JZp + V(p)

(15)

is the classical Hamiltonian for the large-amplitude motion, F(p) is one-half the effective inverse moment of inertia for this motion,

hccoi(P) V(p) = V0(P) + ~ ~ coth

hco~i(p)

(16)

2k-----T--

i

is the effective large-amplitude motion potential. The sum on the right-hand side of Eq. 16 extends over all skeleton vibrations. Attention should be called to the following points: 1. The large-amplitude motion is governed by an effective potential function, Eq. 16, with an explicit parametrically contribution depending on p skeleton vibrations which, for the convenience of this particular case, can safely be assumed harmonic. Then, the frequencies 0)i(p) are readily calculated through a normal coordinate analysis performed at a set of fixed p values chosen with spacing such as to give an adequate representation of the continuous distribution function. It would seem most simple, in this calculation, to use a rigid skeleton geometry and harmonic force field model. This primitive model can be discarded, however, if the relaxation effect is estimated, roughly at least, by appropriate application of high-quality ab initio approaches [57]. 2. The p-dependence of the kinetic energy term in Eq. 15 apparently results from the curvilinearity of the large-amplitude motion coordinate. However, the assumption F(p) = const in a typical and readily visualized example of a hindered rotation was shown [60] to distort the V0(P) function spectroscopic determination, particularly the ratios of the higher order terms to the leading term in the Fourier expansion of V0(P). It is important to mention in this regard that the familiar classical coordinate distribution function with a constant preexponential factor,

P(r) = Ne -v(p)/kr

(17)

is not correct unless the coordinate P is truly rectilinear. As easily recognized, the algebraic and computational complexity of the largeamplitude curvilinear effects makes it impractical to formulate a theoretical evaluation of F(p) from rigorous dynamical considerations in a routine work. Accordingly, in ref. 60 an alternative approach for modeling F(p) from spectroscopic evidence was advanced. In the following development we adapt this idea to the purposes of diffraction analysis.

Equilibrium Structure and Potential Function

69

It was shown in ref. 60 that the formal replacement of F(p) by a constant is equivalent to assuming that the curvilinear coordinate p is replaced by the rectilinear coordinate 13through a F(p) specified transformation,

o~Fo J

dt

(18)

where, for the case of a hindered rotation, the condition that 13and P are equal at 0 and 2n radians should be added. One may well note that while the old coordinate P retains an obvious geometrical significance in the sense that it typically is a measure of the large-amplitude displacement of one part of a molecule relative to another, the new coordinate 13has essentially an operational meaning. For the case of a hindered rotation the following functional form was suggested for F(p) [60],

F(p) = a + B cos(p)

(19)

where A and B are empirically determined constants (B is supposed to be sufficiently small). It may well be assumed that similar simple relationships can be developed for F(p) in other large-amplitude cases of interest. Now, the rotation-vibration coordinate distribution function, Eq. 14, takes the following final form,

P(~) = (Ix(fS) . Iy(~) 9Iz(~)) 1/2 e-v(f~)/I,r

(20)

where V(13) is defined by Eq. 16 in which the curvilinear coordinate p is replaced by the rectilinear coordinate 13. The classical formula (Eq. 20) can be suitably modified to account for quantum effects by simple addition of a correction factor of (tanhZ)/Z, where Z = hccoo/kT, to the exponent [61]. The frequency o~0 can be evaluated by an appropriate numerical solution of a one-dimensional quantum problem specified by the effective potential: V(I3) = V0(13)+ ~

hcco i(f5) 2 "

(21)

However, the performance of the coordinate distribution function of the form of exp{ ~T~) (tanzh Z1 } is practically limited by a single-minimum potential [61]. For this reason it can be used profitably just in this case. When the double-well potential is expected to exist, a more effective approach to the quantum thermal average coordinate distribution function appears the numerical integration of the Bloch equation for the statistical

70

VICTOR P. SPIRIDONOV

density matrix using "short-time" propagators computed by a fast Fourier transform technique [62, 63].

D. Molecular Intensity in the Adiabatic Approximation Under the adiabatic approximation, the molecular intensity function for a nonrigid system with a single large-amplitude coordinate is given by [58],

(22)

sM(s) : ~ P(~3)sM(s;~3)d~3

where P(13) is the large-amplitude coordinate distribution function, Eq. 20, and

sM(s;~) is identical with that of Eq. 5 where the cumulants now become paramet-

rically dependent upon the rectilinear large-amplitude coordinate 13. The function sM(s;~5) is supposed to be calculated according to the scheme described previously for quasi-rigid systems. Then, Eq. 22 is evaluated numerically. The function sM(s) thus obtained is used for the usual least-squares fitting of observed scattering intensities. The described method is general enough and can suitably be extended to a molecule with multiple large-amplitude motions with due allowance for mixings between two or more large-amplitude modes. It should be borne in mind, however, that an accurate experimental or theoretical analysis of the structure and dynamics of a molecule having large-amplitude motions still remains nonroutine work despite all sound efforts.

VII.

POTENTIAL FUNCTION

In a search for plausible model potential functions with fewer number of adjustable parameters it is important to systematically reproduce the general pattern of the molecular potential. In this regard, the potential functions advanced in refs. 64 and 65 for linear and bent quasi-rigid molecular systems of the XY2-type are worth mentioning. They are of a general form, 2

v OrrtE1 exp a3 rl, l *E1 exp

2

Drr, I1- exp(-a3Arl)l. I 1 - exp(-a3Ar2)1+ Va

t (23)

where for linear molecules,

1

2 faare2 A~2exp(-a3Arl) exp(-a3Ar2)

For "- --

and for nonlinear molecules:

(24)

Equilibrium Structure and Potential Function

1 Va = ~frct(Arl + Ar2)

71

react

2 sin(cte/2) sin2 (Act/2) [

sin2(Act/2 )

sin(ct/2)

sin2(cte+ Act/2)

+ 2 faot re

/ 1+ L

TI

J

(25)

In the above equations re is the equilibrium X - Ydistance; cte is the equilibrium YXY angle; a3 is the Morse-like anharmonic parameter; Ar 1, Ar2, and Act are the instantaneous changes in the X - Y distance and the valence angle, respectively; and ct = cte + Act. The following simple correlations were shown to exist between the constants in Eq. 23 and those of the general valence force field (GVFF) [64, 65]:

f GVFF O rr =

rr 2a~

GVFF

f rr' ' o rr' = a~

' f aa - J f caet v r r 'frcL _- - df r co vt r e

favFF rrr

f G V F F _ _ l f G V F F COS(cte/2) ' a3 = 2..,.GI,'-'~" reJrr

(26)

The potential function, Eq. 23, was appropriately extended to include linear XYZ-type molecules [66]. This function and some of its modifications were shown [ 6 4 - 6 6 ] to be fairly satisfactory in predicting and interpreting the cubic (sometimes even quartic) anharmonic force fields in triatomic systems. Its main virtues are the following: 1. The complete harmonic GVFF is reproduced. 2. Only a single parameter a 3 is needed to account for anharmonic bond stretching effects. It is appropriate as a good approximation to assume this parameter to be equal to that of the corresponding diatomic molecule. 3. No additional bend-bend-bend constants, the most prominent contributors to the bending anharmonic effect, are needed to account for bending anharmonicity in nonlinear systems. If necessary, stretching-bending anharmonic interactions can also be suitably included in a simple way [64, 65]. It is hoped that the above considerations yield sufficient justification for exploration of the potential function, Eq. 23, in a greater detail to allow extension to more complex molecular systems. For purposes of illustration, extensive calculations have been performed recently [55] to check the practical performance of Eq. 23 in diffraction analysis using most precise CO2 and SO2 nonphotographic intensity data as an example. Some selection of the refinements is shown in Tables 1-3. A discussion will be given in a separate section below.

72

VICTOR P.SPIRIDONOV VIII. QUASI-DIATOMIC APPROXIMATION

A considerable simplification of a calculational scheme can be effected for moderately anharmonic quasi-rigid molecular systems provided the quasi-diatomic approximation is introduced. The implicit idea is to represent a part of the potential function associated with the bond-stretching vibration by the diatomic Morse oscillator function, whereas the harmonic component of the force field is fully included. Then, we can write for the anharmonic part of the first and third cumulants of the bonded distances [55], 3

(27)

(r)c = -~ a312

(r3)c =

l~ a314 I 3 - 2 -ff

I

(28)

where l2 is the harmonic mean square amplitude and l2 is its zero-point value. Equation 27 is similar to that first developed by Bartell in 1955 [67], whereas Eq. 28 for comparatively rigid bonds, when l ~ I0, transforms into an analogue of the familiar expression (see e.g. ref. 3): (29)

(r3)c = a314.

The most useful advantage of this approximation is that scattering intensity calculations can be carded out solely in terms of the harmonic force field and the diatomic

Table 1.

Equilibrium InternuclearDistance re(C-O)a in CO2b'c

T(K)

I

H

III

IV

V

298

1.1604

1.1603

1.1607

1.1605

1.1648

463

1.1610

1.1610

1.1615

1.1612

1.1656

502

1.1609

1.1609

1.1613

1.1611

1.1655

627

1.1607

1.1607

1.1614

1.1609

1.1656

731

1.1604

1.1603

1.1610

1.1605

1.1654

817

1.1601

1.1601

1.1611

1.1603

1.1653

828

1.1600

1.1599

1.1608

1.1601

1.1650

937

1.1605

1.1604

1.1615

1.1606

1.1660

Notes: aln angstroms. bExperimental spectroscopic value: re(C-O) = 1.1600 A [74]. c(I) Spectroscopic anharmonic force field in normal coordinates through terms of third order [74]. (II) Morse-like model potential function (see Eq. 23 in the text). A value of 2.39 ,g-1 [37] was assumed for a3(C-O ). Harmonic force constants were constrained at spectroscopic values [74]. (HI) Quasi-diatomic approximation. A value of 2.39 /~-t [37] was assumed for a3(C-O). Harmonic force constants were constrained at spectroscopic values [74]. (IV) Badger model. The following values were used for the force constants (in mdyrd~):frr= 14.8, fre = 1.11, fra = 0, fact = 0.52, frrr = --19.1 [73]. (V) Spectroscopic harmonic force field [74].

Equilibrium Structure and Potential Function

73

Table 2. Equilibrium Internuclear Distance re(S--O) a in 502 b'c

Notes:

T,K

I

H

I11

IV

V

288

1.4312

1.4312

1.4318

1.4310

1.4355

295

1.4306

1.4306

1.4312

1.4305

1.4339

485

1.4313

1.4313

1.4322

1.4312

1.4359

527

1.4311

1.4310

1.4320

1.4310

1.4359

547

1.4317

1.4316

1.4325

i.4316

1.4365

553

1.4318

1.4317

1.4326

1.4316

1.4366

623

1.4304

1.4303

1.4313

1.4303

1.4353 1.4350

682

1.4299

1.4298

1.4309

1.4297

728

1.4309

1.4307

1.4318

1.4307

1.4361

780

1.4317

1.4315

1.4327

1.4315

1.4371

828

1.4316

1.4313

1.4326

1.4314

1.4371

878

1.4311

1.4308

1.4322

1.4309

1.4368

979

1.4307

1.4304

1.4318

1.4305

1.4367

aln angstroms. bExperimental value from MW spectroscopy: re(S-O) = 1.43076(13) A [75]. c(I) Spectroscopic anharmonic force field in normal coordinates through terms of third order [75]. (II) Morse-like model potential function (see Eq. 23 in the text). A value of 2.103 A-I [37] was assumed for a3(S-O). Harmonic force constants were constrained at spectroscopic values [36]. (III) Quasi-diatomic approximation. A value of 2.103 A-I [37] was assumed for a3(S-O). Harmonic force constants were constrained at spectroscopic values [36]. (IV) Badger model. The following values were used for the force constants (in mdynL/~):fr r = 11.5,f,, = 0.58,fr a = 0,faa = 0.83, f~rr = --19.8 [75]. (V) Spectroscopic harmonic force field [36].

anharmonic parameter a 3 for the bonded distances. At the same time an obvious demerit of the approach is that no suitable prescription can be formulated for the nonbonded atom pairs corresponding to Eqs. 27 and 28. Some illustrative quasi-diatomic calculations of the geometrical parameters of the sample molecules CO2 and SO2 based on precise diffraction data were carried out in a recent study [55]. Some selection of the results obtained is shown in Tables 1-3. We defer discussion of them until a separate section below.

IX. BADGER MODEL Following the known Badger semiempirical formula for diatomic systems [68, 69], a new similar equation was formulated in refs. 70-72 as follows,

K(t)

AB =

CAs

(30)

(r:t' - d A ~

which was found to be valid for various excited electronic states of the molecule is the force constant for the t-th electronic state, r(et) is the

AB. In Eq. 30 K]~

74

VICTOR P. SPIRIDONOV Table 3. Equilibrium OSO Angle a of SO2b'c

T(K)

Notes:

I

H

m

uv

v

288

119.4

119.5

119.4

119.5

119.4

295

119.3

119.4

119.3

119.4

119.3

485

119.6

119.7

119.5

119.6

119.6

527

119.4

119.5

119.3

119.4

119.4

547

119.2

119.4

119.2

119.3

119.2

553

118.9

119.0

118.8

118.9

118.8

623

119.4

119.5

119.3

119.4

119.4

682

119.1

119.3

119.1

119.1

119.1

728

118.9

119.1

118.8

118.9

118.9

780

119.5

119.7

119.4

119.6

119.5

828

119.2

119.4

119.1

119.2

119.2

878

119.3

119.6

119.3

119.4

119.3

979

119.4

119.6

119.3

119.5

119.4

aln degrees. b Experimental value from MW spectroscopy: Z e OSO = 119018.8(0.7)' [75]. C(l)-(V) See the corresponding footnotes in Table 2.

equilibrium internuclear distance in this state, and CAB= CA + CB, dan = da + dB, CA, CB, da, dB are some atomic constants. In addition, the Badger atomic constants were significantly refined and new simple relationships were formulated fairly accurately correlating the various spectroscopic constants of diatomic molecules in the ground and excited electronic states. It was also shown that suitably modified Badger relations provide a simple predictive model for estimating the bond-stretching harmonic and anharmonic force constants in polyatomic systems. In addition, confirmative evidence was presented indicating that in triatomic molecules of XY2-type realistic evaluation of the bending force constant is feasible [72]. Thus the number of adjustable parameters in diffraction analysis of triatomic molecules can be reduced by constraining some of the force field constants at Badger values. This idea was checked against an appropriate analysis of precise diffraction data for the sample molecules CO2 and SO2 in ref. 73. In this analysis the force constants were fixed at Badger values and not varied. Some of the results obtained are shown in Tables 1-3. A discussion is given in the following section. X.

DISCUSSION

In this section we discuss in some detail the results of diffraction refinements of geometrical parameters of the sample molecules CO2 and SO2 in various approximations described in the previous sections and summarized in Tables 1-3. For

Equilibrium Structure and Potential Function

75

comparison, the results of a reference analysis based on most accurate spectroscopic force fields are also given. In the first place, the data exhibited in Tables 1-3 definitely confirm that highly precise equilibrium geometry is derivable from accurate diffraction data if complete and reliable spectroscopic anharmonic force field is used in the analysis (see columns 1 in these Tables). On the other hand, the results given in columns 2 of these Tables unambiguously demonstrate that the use of the potential function, Eq. 23, with a diatomic a 3 constant for the bonded distances does not lead to any apparent shift in the equilibrium distances as compared to the reference analysis. It is fair to conclude, then, that the Morse function undoubtedly yields sufficiently accurate representation of scattering intensities in the case of moderate anharmonicity. Another important observation is the relatively high apparent accuracy of the quasi-diatomic approximation (compare columns 3 and 1 in Tables 1-3). This lends additional support for the use of the diatomic a 3 constants for bonded distances in diffraction analysis of quasi-rigid polyatomic systems. Moreover, irrespective of the validity of some criticism regarding the form of the anharmonic component of the potential function in this approximation, the advantages of this simple calculation seem overriding. Generally, the values of the equilibrium parameters derived from diffraction analysis certainly are not too critically dependent upon the accuracy of the force field parameters used. Therefore the results obtained with the use of the Badger force constants (see column 4 in Tables 1-3) also seem promising although they reproduce the frequencies of vibration with only fairly moderate accuracy [73]. The data exhibited in column 5 of Tables 1 and 2 unequivocally confirm the statement in the previous sections that the use in the analysis only of the harmonic force field component results in a displacement of the bonded equilibrium distances exceeding the uncertainties currently ascribed to the distance parameters in routine diffraction studies. Damaging as this failure of the harmonic approximation appears at first glance, one apparently assuring aspect of the data presented in column 5 of Table 3 is that the molecules undergoing infinitesimal vibrational displacements would not necessarily suffer serious distortions of the shapes if bending anharmonicities or stretching-bending anharmonic interactions are not allowed for. Thus, just bonded distances appear the primary candidates for an appropriate treatment of anharmonic effects.

Xi. COMBINED USE OF ELECTRON DIFFRACTION AND VARIOUS RELEVANT TECHNIQUES A. Fundamental Advantages A joint analysis of electron diffraction and spectroscopic data was practically initiated in 1968 (see e.g. ref. 37) when the basic principles, main features, and some implications were first elaborated. Such combined work appeared as a natural

76

VICTOR P. SPIRIDONOV

extension of the studies of molecules by separate techniques. As a matter of fact, it was made possible mainly through the development of calculational approaches to account for vibrational effect on the geometrical parameters, which is needed because of the distinction in the physical backgrounds behind particular methods (see e.g. refs. 3, 37, 76-78). The apparent consequences of primary applications of combined procedures in 1968-1979 and of their more systematic use in the subsequent years was a substantial increase in the accuracy of the interatomic distances derived and in the ability to explore in more detail molecular properties such as vibrational force fields. New developments in the integrated approach confirming the original basic idea may result from the analysis of all available data from a molecule, including those from high-quality theoretical calculations, in terms of a common molecular potential function. The primary advantage of such an approach is that it enables the results based on data from various techniques to be coordinated in a most rigorous and systematic way within the framework of a calculationally unified scheme discarding the use of vibrational distance corrections. By testing the consistency of different views of the same molecular system would afford the best means of critical assessing reported uncertainties of the parameters evaluated in the individual studies and largely remedying the deficiencies latent within each particular technique in isolation. Another important merit is that a more justified treatment of highly correlated distance and force field parameters is provided. Simultaneously, estimates can be obtained for the parameters whether missing or deeply hidden in the observations by a separate technique. In favorable cases this may also serve as a powerful aid in the resolution of various geometrical and dynamical ambiguities. Thus the integrated approach is most suited to reveal fairly delicate features of a molecular structure and dynamics (e.g. relatively small distortions of a molecule or small structural differences in a series of molecules).

B. Complications Two inherent complications of the integrated approach involve appropriate weighting the various experimental data and possible bias in the parameters resulting from incorporation of theoretical results because of today's unfortunate dependence on the mode of calculation. The problem of how to weight experimental observations from different techniques has generated discussion for some time (see e.g. refs. 6 and 37). But, at least in principle, the appropriate answer is fairly simple: in conformity with the conventional error theory prescription the weights should be assigned to be inversely proportional to the squared standard errors of the observations or, better still, to the estimated total uncertainties. In this regard, particularly heartening is a published statement that the relative weights for the scattering intensities and microwave rotational constants can be altered over several

Equilibrium Structure and Potential Function

77

orders of magnitude without a significant change in the value of the refined parameters [76]. More troublesome seems the theoretical bias which may well exceed the experimental uncertainties. To properly avoid its possibly unfavorable consequences, an appropriate evaluation is needed of the measure to which the theoretical calculation mode used controls the analysis as well as of the extent to which this auxiliary evidence may be added into an analysis. In any event, it must be realized that the inclusion of theoretical results in the treatment of experimental observations is a very delicate affair.

C. Example As an illustration of a multimethod analysis, we present in Table 4 the results of a corefinement of all available spectroscopic and diffraction data on the C102

Table 4.

Experimental and Optimized Values of Spectroscopic Observables for ClO2a,b,c,d

Spectroscopic Observation

Experimental

Refined V a l u e

0)1

963.5(7)

966.0

0)2

451.7(5)

450.0

0)3 Ao Bo Co "Caaaa

1133.0(10) 52079.5(8) 9952.23(15) 8333.99(13) -7.8(10)

1133.4 52081.2 9952.46 8333.96 -7.37

--0.04(1) 0.5(1)

17bbbb 1;aabb 1;abab Xll

--0.054 0.407 --0.060 --4.7

--0.04(2) -4.2(1)

X22 X33 Xl2 Xl3

-0.15(7) -6.75(20) -4.5(2) -15.0(5)

-0.34 -6.23 0.08 -16.75

X23

-2.4(1)

-2.15

Geometrical and Force Field Parameters

Refined Value

r (Cl-O) Z OClO

1.4707(16) 117.48(8)

Fll (Al) Flz(Al)

6.817(82) -0.052(63)

F22(AI) F33(BI)

fr,

0.643(9) 7.255(22) -11.9(13)

frrrr

19.4(114)

Notes: aStructural(r in A, angle in degrees) and force field (in mdyn/]k) parameters. bSpectroscopic observables are given for the 35C102species. cVibrational frequencies are given in cm-I units, rotational (Ao, Bo, C0) and centrifugal distortion (Xaaaa,~:bb~, ~aabb,l:~t,ab)constants in MHz; spectroscopic anharmonic constants (X Ii, X22, X33,X12,X13'X23) are given in cm-" units. dFor the combined analysis the following potential function was assumed: V= Vna'n +frrr r;'( Ar3 + At3) +f,,',r re2(Ar4 + Ar4) formulated in terms of the curvilinear internal coordinates in which coordinates of symmetry were used for the harmonic component. Spectroscopic observables were taken from refs. 80, 81; diffraction intensities from r~f. 82.

78

VICTOR P. SPIRIDONOV

molecule [79]. The following spectroscopic observables were included in the analysis: vibrational frequencies, rotational constants, centrifugal distortion constants, and anharmonic constants. Because the original focus of the analysis was on structural parameters and on the harmonic force field, the simplified representation of the anharmonic component of the potential function was adopted. First of all, it is fair to observe in this Table 4 the high apparent accuracy of the refined values of structural parameters. This proves the validity and fitness of the combined analysis to establishing well-characterized equilibrium molecular geometries. A further conclusion is the availability of fairly reliable harmonic force field constants which are seen to sufficiently and accurately reproduce the measured frequencies of vibration. At the same time, the refined values for some spectroscopic anharmonic constants reveal fairly large discrepancies with measured counterparts, thus suggesting that the reported uncertainties of these measurements certainly are underestimated. Attention should also be called to a large value of the quartic anharmonic force constant standard deviation, thus this constant cannot be claimed to be reliably determined. A combined analysis actually has proved to be a very useful tool for gaining access to the internal consistency of experimental data used.

XIi. CONCLUDING REMARKS The molecular potential function offers the most rigorous conceptual basis on which any related techniques can be coordinated within the scope of a unified calculational scheme. Thus the integrated approach in terms of this function considerably enhances the value of each relevant technique and promises substantial future progress in the solution of equilibrium structures and molecular dynamics. However, it is hoped that the material presented in this chapter has left the impression with the reader that this area of research is still in an infant age. It should be noted that the above expressions can also be used for more complicated molecules other than those discussed as the examples in this review. Further developments along this line would afford deeper insights into molecular structures and dynamics augmented by greater accuracy. It is especially stimulating to have increasingly sophisticated techniques to simulate fairly complicated systems, as well as to continue to improve and expand the present collection of reliable molecular structures.

ACKNOWLEDGMENTS The author appreciates the critical reading of the manuscript by Dr. Yu.I. Tarasov from the Department of Chemistry of Moscow State University and his helpful comments and suggestions. Grateful thanks go also to my colleague Dr. V.I. Tyulin from the same Institution for fruitful consideration of the Badger model. The author expresses his sincere gratitude to Prof. A.G. Yagola from the Department of Physics of Moscow State University for his interest in this work and critical reading the part concerned with ill-posed problems. The author's

Equilibrium Structure and Potential Function

79

thanks are also due to Dr. V.S. Iorish from the Institute for High Temperatures of Russian Academy of Sciences for stimulative discussion of internal rotation and related problems. It is a particular pleasure to acknowledge assistance of Dr. L.I. Ermolayeva for sound help in preparation of the manuscript.

NOTES 1. Explicit expressions of all relevant quantities and calculation details can be found in refs. 44 and 54 which are also recommended for further in-depth consultation.

REFERENCES 1. Kuchitsu, K.; Bartell, L. S. J. Chem. Phys. 1961, 35, 1945. 2. Kuchitsu, K. Bull. Chem. Soc. Jpn. 1967, 40, 498, 505. 3. Hargittai, I. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 1. 4. MOGADOC Database, University of Ulm, Germany. 5. Spiridonov, V. P. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 265. 6. Kuchitsu, K.; Cyvin, S. J. In Molecular Structures and Vibrations; Cyvin, S. J., Ed. Elsevier: Amsterdam, 1972, p. 183. 7. Kuchitsu, K. In Accurate Molecular Structures. Their Determination and Importance; Domenicano, A.; Hargittai, I., Eds. International Union of Crystallography, Oxford University Press: London, 1992, p. 14. 8. Bartell, L. S. J. Mol. Struct. 1982, 84, 117. 9. Goates, S. R.; Bartell, L. S. J. Chem. Phys. 1982, 77, 1866, 1874. 10. Bartell, L. S.; Stanton, J. E J. Chem. Phys. 1984, 81, 3792. 11. Bartell, L. S.; Vance, W.; Goates, S. R. J. Chem. Phys. 1984, 80, 3923. 12. Bartell, L. S. J. Mol. Struct. 1984, 116, 279. 13. Bartell, L. S.; Barshad, Y. Z. J. Am. Chem. Soc. 1984, 106, 7700. 14. Stanton, J. E; Bartell, L. S. J. Phys. Chem. 1985, 89, 2544. 15. Bartell, L. S. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New York, 1988, Vol. A, p. 55. 16. Born, M.; Oppenheimer, J. R. Ann. Phys. 1927, 84, 457. 17. Morino, Y. Pure Appl. Chem. 1969, 18, 323. 18. Lowrey, A. H. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 367. 19. Thorson, W. L.; Nakagawa, I. J. Chem. Phys. 1960, 33, 994. 20. Spiridonov, V. P. J. Mol. Struct. 1995, 346, 131. 21. Gershikov, A. G.; Subbotina, N. Yu.; Girichev, G. V. Zh. Strukt. Khim. 1986, 27, 36 (in Russian). 22. Hargittai, M.; Subbotina, N. Yu.; Gershikov, A. G. J. Mol. Struct. 1991, 245, 147. 23. Hargittai, M.; Subbotina, N. Yu.; Kolonits, M.; Gershikov, A. G. J. Chem. Phys. 1991, 94, 7278. 24. Ischenko, A. A.; Ewbank, J. D.; Schafer, L. J. Phys. Chem. 1994, 98, 4287. 25. Vogt, N. Yu.; Haaland, A.; Martinsen, K.-G.; Vogt, J.J. Mol. Spectrosc. 1994, 163, 515. 26. Maggard, P.; Lobastov, V. A.; Schafer, L.; Ewbank, J. D.; Ischenko, A. A. J. Phys. Chem. 1995, 99, 1315. 27. Belova, I. N. Ph.D. Thesis, Ivanovo State Academy of Chemical Technology, 1995 (in Russian). 28. Ukaji, T.; Kuchitsu, K. Bull. Chem. Soc. Jpn. 1966, 39, 2153. 29. Spiridonov, V. P.; Gershikov, A. G.; Butayev, B. S. J. Mol. Struct. 1979, 51, 137. 30. Bartell, L. S.; Kuchitsu, K.; de Neui, R. J. J. Chem. Phys. 1961, 35, 1211.

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31. Spiridonov, V. E; Zasorin, E. Z. In Characterization of High-Temperature Vapors and Gases; NBS Special Publication No. 561; Hastie, J. W., Ed. National Bureau of Standards: Washington, DC, 1979. 32. Kuchitsu, K.; Bartell, L. S. J. Chem. Phys. 1962, 36, 2460, 2470. 33. Bartell, L. S. J. Chem. Phys. 1955, 23, 1219. 34. Bartell, L. S. J. Chem. Phys. 1979, 70, 4581. 35. Bartell, L. S. J. Mol. Struct. 1981, 63, 253. 36. Kuchitsu, K.; Morino, Y. Bull. Chem. Soc. Jpn. 1965, 38, 805, 814. 37. Kuchitsu, K.; Nakata, M.; Yamamoto, S. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 227. 38. Wilson, E. B., Jr.; Decius, J. C.; Cross, E Molecular Vibrations. Mc Graw-Hill: New-York, 1955. 39. Bartell, L. S. J. Chem. Phys. 1963, 38, 1827. 40. Bartell, L. S.; Fitzwater, S. J. Chem. Phys. 1977, 64, 4168. 41. Gershikov, A. G.; Spiridonov, V. P.; Zasorin, E. Z. J. Mol. Struct. 1983, 99, 1. 42. Gershikov, A. G.; Spiridonov, V. P. J. Mol. Struct. 1981, 75, 291. 43. Suzuki, I. Appl. Spectrosc. Rev. 1975, 9, 249. 44. Hoy, A. R.; Mills, I. M.; Strey, G. J. Mol. Phys. 1972, 24, 1265. 45. Kuchitsu, K. In Molecular Structure and Vibrations; Cyvin, S. J., Ed. Elsevier: Amsterdam, 1972, p. 148. 46. Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G. Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publishers: Dordrecht, the Netherlands, 1995. 47. Kochikov, I. V.; Kuramshina, G. M.; Pentin, Yu. A.; Yagola, A. G. Inverse Problems in Vibrational Spectroscopy. Moscow University: Moscow, 1993 (in Russian). 48. Reitan, A. Acta Chem. Scand. 1958, 12, 785. 49. Toyama, M.; Oka, T.; Morino, Y. J. Mol. Spectrosc. 1964, 13, 193. 50. Fink, M.; Kohl, D. A. In StereochemicalApplications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 139. 51. Hilderbrandt, R. L.; Kohl, D. A. J. Mol. Struct. Theochem. 1981, 85, 25. 52. Hilderbrandt, R. L.; Kohl, D. A. J. Mol. Struct. Theochem. 1981, 85, 325. 53. Spiridonov, V. P.; Ischenko, A. A.; Ivashkevich, L. S. J. Mol. Struct. 1981, 72, 153. 54. Ischenko, A. A.; Spiridonov, V. P.; Tarasov, Yu. I.; Stuchebrynkhov, A. A. J. Mol. Struct. 1988, 172, 1955. 55. Tarasov, Yu. I.; Spiridonov, V. P. J. Mol. Struct. 1996, 376, 207. 56. Morino, Y.; Iijima, T. Bull. Chem. Soc. Jpn. 1963, 36, 412. 57. Schafer, L.; Ewbank, J. D.; Siam, K.; Chin, N.-S.; Sellers, H. L. In Stereochemical Applications of Gas Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A,p. 301. 58. Spiridonov, V. P.; Gershikov, A. G.; Lyutsarev, V. S. J. Mol. Struct. 1990, 221, 57. 59. Landau, L. D.; Lifshitz, E. M. Statistical Physics. Nauka: Moscow, 1964 (in Russian). 60. Ewig, C. S.; Harris, D. O. J. Chem. Phys. 1970, 52, 6288. 61. Spiridonov, V. P.; Butayev, B. S.; Nasarenko, A. Ya.; But, O. N.; Gershikov, A. G. Chem. Phys. Lett. 1984, 103, 363. 62. Hellsing, B.; Nitzan, A.; Metiu, H. Chem. Phys. Len. 1986, 123, 523. 63. Ermakov, K. V.; Butayev, B. S.; Spiridonov, V. P. Chem. Phys. Lett. 1987, 138, 153. 64. Butayev, B. S.; Saakjan, A. S.; Spiridonov, V. P. Chem. Phys. Lett. 1987, 138, 133. 65. Ermakov, K. V.; Butayev, B. S.; Spiridonov, V. P. J. Mol. Struct. 1990, 240, 295. 66. Butayev, B. S.; Lyutsarev, V. S.; Saakjan, A. S.; Spiridonov, V. P. J. Mol. Struct. 1990, 221, 149. 67. Bartell, L. S. J. Chem. Phys. 1955, 23, 1219. 68. Badger, R. M. J. Chem. Phys. 1934, 2, 128. 69. Badger, R. M. J. Chem. Phys. 1935, 3, 710.

Equilibrium Structure and Potential Function

81

70. Tyulin, V. I. Vibrational and Rotational Spectra of Polyatomic Molecules; Moscow University: Moscow, 1987 (in Russian). 71. Tyulin, V. I. Izv. Vusov, Ser. Khim. i Khim. Tekhnol. 1991, 34, 34 (in Russian). 72. Tyulin, V. I.; Erokhin, E. V.; Matveyev, V. K. Vestn. Mosk. Univ. Ser. Khim. 1997, in press (in Russian). 73. Tarasov, Yu. I.; Tyulin, V. I.; Spiridonov, V. P. Vestn. Mosk. Univ. Ser. Khim. 1997, in press (in Russian). 74. Suzuki, I. J. Mol. Spectrosc. 1968, 25, 479. 75. Saito, S. J. MoL Spectrosc. 1969, 30, 1. 76. Geise, H. J.; Pyckhout, W. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 321. 77. Rankin, D. W. H. In Stereochemical Applications of Gas-Phase Electron Diffraction; Hargittai, I.; Hargittai, M., Eds. VCH: New-York, 1988, Vol. A, p. 451. 78. Hargittai, M.; Hargittai, I. Int. J. Quantum Chem. 1992, 44, 1057. 79. Spiridonov, V. P.; Gershikov, A. G.J. MoL Struct. 1986, 140, 173. 80. Richardson, A. W.; Redding, R. W.; Brandt, J. C. D. J. Mol. Spectrosc. 1969, 29, 93. 81. Pulai, M. G. K.; Curl, R. E, Jr. J. Chem. Phys. 1962, 37, 2921. 82. Clark, A. H.; Beagley, B. J. Chem. Soc. A, 1970, 46.

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STRUCTURES AND CONFORMATIONS OF SOME COMPOU N DS CONTAINING C-C, C-N, C-O, N-O, AND O-O SINGLE BONDS" CRITICAL COMPARISON OF EXPERIMENT AND

THEORY

Hans-Georg Mack and Heinz Oberhammer

I~ II.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Compounds with C-C Bonds: Halogenated Malonic Acid Derivatives CIC(O)-CH2C(O)CI, FC(O)-CH2-C(O)F, and FC(O)-CF2-C(O)F . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Compounds with C-N Bonds: Carbonylisocyanates of the Type XC(O)NCO (X = CH 3, F, El) . . . . . . . . . . . . . . . . . . . . . . . . C. Compounds with C--O Bonds: Fluoroformylhypofluorite, FC(O)OF, and Bis(fluorooxy)difluoromethane, CF2(OF)2 . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 83-115 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

83

84 84 87

87 93 96

84

HANS-GEORG MACK and HEINZ OBERHAMMER

D. Compounds with N-O Bonds: Halogen Nitrates, XONO2 (X = F, C1, Br), and O-Nitrosobis(trifluoromethyl)hydroxylamine, (CF3)ENONO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Compounds with O-O Bonds: CHaOOCH3, FC(O)OOC(O)F, CF3OOOCF3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 107 111

112

ABSTRACT Experimental and theoretical structures and conformations of some compounds which contain C-C, C-N, C-O, N-O, and O-O single bonds are discussed. Experimental results are based on gas electron diffraction and vibrational spectroscopy. The theoretical structural and conformational properties were obtained by various ab initio and density functional methods. The following compounds were considered: halogenated malonic acid derivatives, carbonylisocyanates, fluoroformylhypofluorite, bis(fluoroxy) difluoromethane, halogen nitrates, O-nitrosobis(trifluoromethyl)hydroxylamine, ((CF3)2NONO), peroxides, and bis(trifluoromethyl)trioxide. The main interest is the comparison between experimental and theoretical conformational properties. The selected examples demonstrate that predicted relative stabilities of different conformers may depend strongly on the computational method.

I. I N T R O D U C T I O N The geometric structure of a molecule is one of the most important properties for characterization of a compound and several methods for structure determinations have been used for many decades. The most widely applied experimental technique is X-ray diffraction for single crystals which started about 75 years ago, ca. 25 years after the discovery of X-rays by Roentgen in 1895. Extensive automatization of collection and analysis of diffraction data during the last two decades made this method very effective, and routine structure analyses can be performed in a few days or even faster. Besides this method for the study of single crystals, techniques for the determination of molecular structures of gaseous compounds are available, which either use gas electron diffraction (GED) intensities or rotational constants from microwave (MW) or high-resolution infrared (IR) spectroscopy. It has been demonstrated that a joint analysis of GED intensities and rotational constants of a single isotopic species leads to the most accurate structural parameters for small compounds which contain about 5-15 atoms. These gas-phase techniques, however, are in general limited to small or medium-sized molecules with up to about 30 atoms and require a much higher effort than X-ray diffraction. A complete structure determination by GED takes months and by MW or IR a year or longer. On the other hand, such results have the advantage of representing the structure and

Structures and Conformations

85

conformational properties of the free molecule, i.e. unperturbed by intermolecular interactions which are present in the solid phase. With the introduction of the gradient technique by P. Pulay [1] and with the rapid increase of computer capacities in the past decades, the theoretical determination of molecular structures with quantum chemical methods has become very effective and widely used. In most cases ab initio approaches with Gaussian-type split valence basis sets are applied, either in the Hartree-Fock (HF) or Mr (MP) approximation. The latter includes electron correlation effects. Many results reported in the literature are based on HF/3-21G, HF/6-31G*, or MP2/6-31G* calculations. For small molecules with up to about eight "heavy" atoms, configuration interaction (CI) or coupled cluster (CC) methods can be used. In recent years also density functional theory (DFT) methods in the local (LDFT) or nonlocal (NLDFT) approximation are applied for free molecules. These methods, which include electron correlation effects, have the advantage of considerably lower computational effort, especially for larger molecules. On the other hand, only few data exist so far which compare results of such calculations to experimental structures and conformational properties. Various program systems for ab initio or DFT calculations such as GAUSSIAN, SPARTAN, GAMESS, or DGAUSS are available and very easy to use. The combination of experimental and theoretical methods for the determination of structures of free molecules has become very important in the last decade. Most experimental studies (GED or MW) reported in the literature now include theoretical calculations. Such combinations are performed for two reasons: (1) to support the experimental analysis, and (2) to check how well the applied computational method reproduces the experiment for a certain molecule. In many cases some structural features of a molecule, such as differences between very closely spaced interatomic distances or angles are badly determined in a GED experiment. It is generally assumed that such differences between bonds or angles of the same type are predicted reliably by theoretical methods and, therefore, can be used as constraints in the experimental analysis. This combination has been called "molecular orbital constrained electron diffraction" (MOCED) [2]. In conformational studies, theoretical calculations can be very helpful to determine how many and which structures correspond to minima on the energy hyperface and, thus, have to be considered in the experimental analysis. Furthermore, in the case of mixtures of two or more conformers, only mean values for bond lengths and bond angles can be determined in a GED experiment and the differences between the individual parameters in the various conformers can be obtained only from theoretical calculations. All quantum chemical calculations use approximations and the quality of the result for a certain molecular property cannot be judged a priori, but only by comparison with experimental values. Since calculations are normally performed for an isolated molecule, geometric structures and conformational properties of gaseous molecules are much better suited for such a comparison than solid-state

86

HANS-GEORG MACK and HEINZ OBERHAMMER

data. This is especially true in the case of conformational properties, which may be affected very strongly by packing effects in the crystal. Most compounds, which exist as a mixture of two or more conformers in the gas phase, possess only a single conformation in the solid state. For a strict comparison between experimental and theoretical geometric parameters, their systematic differences have to be taken into account. Quantum chemical methods predict equilibrium structures which correspond to the minima on the energy hyperface, whereas vibrationally averaged geometric parameters are derived by experimental methods. In the case of GED results, equilibrium bond lengths (re) can be estimated from the experimental rg values, using the diatomic approximation: 3

re= rg--~ a3 l

2

With anharmonicity parameters 1.5 < a 3 < 2.5/~-1 [3] and vibrational amplitudes 0.04 < 1 < 0.05 A [4], re distances are estimated to be 0.004 to 0.009 ,~ shorter than experimental r values. This difference is larger for distances to hydrogen (ca. 0.014-0.018/~%or O-H, N-H, or C - H bonds). The ro bond lengths obtained from rotational constants lie in general between re and rg values and no simple expression for r o - r e can be given. Bond angles are assumed to be affected very little by bending vibrations, as long as those are reasonably harmonic and calculated bond angles can be compared directly with experimental values. Vibrationally averaged dihedral angles, however, can differ by up to 10 ~ or more from the equilibrium value if a large amplitude torsional vibration occurs in an anharmonic potential. For hydrogen peroxide, the vibrationally averaged dihedral angle (6o(HOOH) = 120.0(5) ~ from rotational constants [5]) and the experimental equilibrium value (re(HOOH) = 111.5 ~ from IR data [6] or l~L8 ~ from a nonrigid analysis of rotational constants [7]) differ by ca. 8 ~ Thus, ab initio calculations which reproduce the 5o value perfectly are by far not as good as claimed by the authors [8, 9]. It was shown that larger basis sets are necessary to reproduce the experimental ~e value [10]. In conformational analyses, GED experiments determine the differences between free enthalpies, AG~ or between enthalpies, AH ~ For a comparison of AH~ with theoretical A E values, which represent the energy difference between the minima in the energy hyperface, corrections for different zero point vibrational energies and the temperature dependence, A H ~ - AH~., have to be taken into account. Furthermore, the entropy term AS~ has to be considered for a comparison of AG ~ and AE. All these terms can be derived from ab initio results by applying simple statistical thermodynamics. For reasonably rigid molecules and equal statistical weights of the conformers, AG~ and AE differ by less than 0.2 kcal mo1-1. In many experiments these differences are smaller than the experimental uncertainties.

Structures and Conformations

87

In general, theoretical calculations reproduce experimental bond lengths to within + 0.03 A and bond angles to within +3 ~ independent of the calculational level. The agreement may be considerably worse for dihedral angles. The theoretical predictions for conformational properties, i.e. the relative energies of various conformers, depend very often strongly on the computational method, which makes reliable predictions for medium-sized compounds very difficult. The obvious reason for this failure is the high sensitivity of the conformational composition on the relative energies of the various structures. A variation of this energy by + 1.0 kcal mol -~, which is only a minute fraction of the total electronic energy, causes a large variation in the conformational composition. In this chapter we will discuss experimental results for geometric structures and conformational properties of some compounds and their comparison with various theoretical calculations. We have selected compounds which can possess different conformations due to rotation around C-C, C-N, C-O, N-O, or O-O single bonds.

!1. EXAMPLES A. Compounds with C-C Bonds: Halogenated Malonic Acid Derivatives CIC(O)-CH2C(O)CI [11], FC(O)-CH2-C(O)F [12], and FC(O)-CF2-C(O)F [ 12] fI-Dicarbonyl compounds of the type XC(O)-CH2-C(O)X' have been of considerable interest in chemistry for many years because of possible keto-enol tautomerism. A comprehensive review of their chemical and physical properties is given by Emsley [13]. In almost all 13-dicarbonyls the hydrogen bonded U-cis enol form (Scheme 1) is found to be the predominant tautomer. Structures with asymmetric (Cs symmetry) or symmetric hydrogen bonds (C2v symmetry) are discussed in the literature. Assuming planarity of the carbon-oxygen framework of the keto tautomer, three conformations are conceivable: U-cis, W-trans, and S-trans (Scheme 1). A considerable number of structural investigations of 13-dicarbonyls in the crystal and in the gas phase was reported in the literature. According to MW spectroscopy the parent compound, malondialdehyde (X = X' = H), possesses a planar U-cis enol structure with an asymmetric intramolecular O-H.--O hydrogen bond (Cs symmetry) [14]. This result was reproduced correctly by various quantum chemical methods [15-18]. GED analyses for acetylacetone (X = X' = CH3) [19], trifluoro(X = CH3, X' = CF3) [20], and hexafluoroacetylacetone (X = X' = CF3) [21, 22] also result in enol forms. In the case of CH3C(O)-CH2-C(O)CH3 an asymmetric intramolecular hydrogen bond was determined, whereas the GED intensities of the CF 3 substituted derivatives were interpreted in terms of a symmetric O-..H...O bond. Similarly, all fl-dicarbonyls studied in the solid state are enol tautomers, except for the 1,3-diphenyl derivatives, PhC(O)-CHR-C(O)Ph, with R = methyl or 4methoxyphenyl, which possess non-planar diketo structures [23].

88

HANS-GEORG MACK and HEINZ OBERHAMMER H

x

/c2

H

I

~Cl

x'

\C2 ~ C 3

%3 /

II

I

x

2,,

I

Ol

02

"%.,.

--..H ~

enol, U - cis, Cs

H

x\cl

II

Ol

H

%3

,,,,.X'

II

02

keto, U-cis, C2v

Ol

~

enol, U-cis, C2v

H

9,, ...,." /C2"

H

/X"

%,

I

x

H

\ ......" /C2" //02

%3 I

x'

keto, W-trans, C2v

H

H

\ ......" /C2"

Ol

I

x

II

02

keto, S-trans, Cs

Scheme 1.

IR and NMR spectra of the halogenated malonic acid derivatives C1C(O)-CH2C(O)C1 (1) and FC(O)-CHz-C(O)F (2) demonstrate that these compounds exist in the diketo form. For FC(O)-CFz-C(O)F (3) no enol form is possible. The analysis of the three compounds was started with ab initio calculations in order to determine the possible minima in the conformational space due to rotation around the two C-C bonds. Calculations were performed only at the HF level (HF/3-21G and HF/6-31G*(*)), since it is generally assumed [24] that the shape of the potential function for rotation around a C-C single bond is reproduced correctly at this level of theory. In the first step, structures with C2 symmetry were considered, i.e. both dihedral angles ~1(C 1C2C302) and 82(C3C2C 101) (see Scheme 1 for atom numbering) are equal and have the same sign. Geometries were optimized with fixed dihedral angles in steps of 30 ~ Furthermore, geometry optimizations were performed for structures with CI symmetry. Hereby, 62 was fixed at 0 ~ and 61 was varied in steps of 30 ~. In the final step, both dihedral angles were optimized for all stable structures. The calculated potential curves for 1 are presented in Figures 1 and 2. Both computational procedures result in three stable structures (Figure 3): U-cis (C2v symmetry), W-gauche (C2), and S-gauche (C1). The W-gauche and S-gauche forms represent non-planar structures with both C--O double bonds or one C = O

Structures and Conformations H

\/

Cl~

89

H

s

H

/Cl

01~C

~r ~

3

o,

H

\/

o,

ir~'-i -~~ 3

ct

c,

/ 3 l- 18 I

/

f

I0

/

6 -

_J~

/

~

,

0

i

60

.,,,

/

,

AEt

,

4

~E ~

~

"2

I-

'

o,=oo 120

0

180

Figure 1. Calculated relative energies for various conformations of malonyl dichloride possessing**C2symmetry (c31(C1 C2C302) = a2(C3C2C1 O1 )). HF/3-21G (full line) and HF/6-31G (broken line). H

\/

H

a~c/c2. " 01

H

\/

/cl

c1.~,(/c2~.///02

0z

01

i----60

/

f

._.. t 6 AE~ 0

/

j

2

-t 61

8

Cl

/-

0

H

120

0

~

IBO

Figure 2. Calculated relative energies for various conformations of malonyl dichloride possessing C1 symmetry. HF/3-21G (full line)and HF/6-31G** (broken line).

90

HANS-GEORG MACK and HEINZ OBERHAMMER

U-cis '

W-gauche

S-gauche

Figure 3. Predicted stable conformations for malonyl dichloride. double bond, respectively, gauche with respect to the opposite C - C bond and 5(CCCO) > 100 ~ The calculated relative energies of these three conformers depend on the basis sets. Calculations with the HF/3-21G method predict the U-cis form (51 = 82 = 0 ~ to be lowest in energy. The S-gauche (81 = 105 ~ 52 = - 6 ~ and the W-gauche structures (81 = 52 = 122 ~ are higher in energy by 1.6 and 2.8 kcal mol -l , respectively. In the HF/6-31 G* approximation the S-gauche form (51 = 102 ~ 52 = - 4 ~ represents the global minimum and the W-gauche (81 = 82 = 123 ~ and U-cis conformers are less stable by 1.3 and 1.5 kcal mol -l, respectively. The enol tautomer of 1 (Cs symmetry) is predicted to be higher in energy by 3.5 (HF/3-21G) or 6.6 kcal mol -l (HF/6-31G*) than the most stable diketo form. Similarly, three conformational minima exist for 2 and 3 (see Table 1). According to the HF/3-21G method the global minimum of both compounds corresponds to a W-gauche structure and the highest energy conformer either to U-cis (2) or U-gauche (3) geometry (30 ~ < 81,2 < 60~ The HF/6-31G* approximation results in different relative stabilities for the stable structures. Now, for both compounds the S-gauche rotamer is predicted to be lowest in energy. The W-gauche forms are slightly less stable and the U-gauche structures are highest in energy. In the case of 2 the structure of the high energy form changes with the basis set from U-cis to U-gauche (Table 1). Depending on the computational method, the enol form of 2 is higher in energy than the most stable keto tautomers by 1.8 (HF/3-21G) or 8.9 kcal mol -l (HF/6-31G*). In the analysis of the experimental GED intensities these various theoretically predicted conformations were considered as possible structures. Comparison of calculated radial distribution functions (RDF) with the experimental curve for 1 (Figure 4) demonstrates, that the S-gauche structure predominates in this compound. The agreement between experimental and calculated RDFs improves if a contribution of 30(15)% W-gauche conformer is added. The U-cis structure can be excluded, because the C1...C1 distance expected at ca. 5.5 A for this form is not observed in the experimental RDE From the experimental composition we obtain AG ~ = G~ - G~ = 0.5(5) kcal tool -l and AH ~ = 0.1(5) kcal

Structures and Conformations

91

Calculated and Experimental Energies AE and A H ~ relative to the

Table 1.

S-gauche Structure for CIC(O)-CH2-C(O)CI (1), FC(O)-CH2-C(O)F (2), and FC(O)-CF2-C(O)F (3) a

CIC(O)-CH2-C(O)CI (1)

FC(O)-CH2-C(O)F (2)

FC(O)-CF2-C(O)F (3)

Method

U-cis b

HF/3-2 IG HF/6-31G** GED

-1.6 +1.5

HF/3-21G HF/6-31G GED

+0.4

HF/3-21G HF/6-31G GED

-

-

-

-

-

-

U-gauche

W-gauchb + 1.2

0.0

+ 1.3 +0.1(5)

0.0 0.0

-0.6

0.0 0.0 0.0

+1.9

+0.3 +0.9 (6)

+1.7 +1.9

-1.0 +0.4

-

S-gauche b

0.0 0.0 0.0

-0.9 (4)

-

Notes: aAE and AH~ in kcal mol-i. bBold values correspond to the global minimum.

A W-gauche S-gauche

I

/ \

Exp. I

I

/I

92 9 ?

I \~

~oo~oooo

/\\~

I

o

I

I

!

I

I

0

1

2

3

4

/\

oo

I

5

I

6

R/A Figure 4. Calculated and experimental RDFs for malonyl dichloride and difference curve for mixture. Important interatomic distances of the main conformer (S-gauche) are indicated by vertical bars.

92

HANS-GEORG MACK and HEINZ OBERHAMMER

mol -~ if only the different multiplicities of the two conformers are taken into account in the entropy term. Comparison with the ab initio calculations demonstrates that the HF/6-31G** method predicts the correct main conformer, but overestimates the energy difference between S-gauche and W-gauche forms by about 1 kcal mo1-1 (see Table 1). The HF/3-21G method predicts a wrong sequence for the relative stabilities of the three conformations. In the case of 2, again the S-gauche conformer represents the most stable structure in the GED analysis. A small contribution (10(10)%) of the W-gauche form is possible. From this composition a AH~ value of 0.9(6) kcal mo1-1 is derived. The experimental result is reproduced correctly by the HF/6-31G** calculations, but not by the HF/3-21G method which predicts the W-gauche conformer to be lower in energy. For FC(O)-CF2-C(O)F (3) the relative stability is reversed according to the GED experiment, i.e. the W-gauche conformer predominates and a contribution of 30(15)% of the S-gauche form is present in the gas phase. This composition corresponds to AH~ = H~ - H~ = -0.9 kcal mo1-1. For this molecule the conformational properties are predicted correctly by the small basis set, whereas calculations with the larger basis set lead to a preference of the S-gauche form. All these experimentally studied 13-dicarbonyls exist in the gas phase as mixtures of two nonplanar conformers. These conformations are characterized by eclipsed orientations of the C--O double bonds relative to vicinal single bonds. In S-gauche structures one C--O bond eclipses a methylene C-H or C - F bond, the other C--O bond eclipses the opposite C-C bond. In the W-gauche forms both C--O bonds are eclipsed with respect to C - H or C - F bonds of the methylene group. In the case of C1C(O)-CH2-C(O)C1 (1) our calculations do not describe the experimental composition adequately. For FC(O)-CH2-C(O)F (2) the larger basis set (HF/631G**) reproduces the experiment and for FC(O)-CF2-C(O)F (3) the smaller basis set (HF/3-21 G) gives a correct result. In the GED analyses the geometric parameters of the main conformers only could be determined. For the minor conformers, the differences between bond lengths and bond angles were set to the HF/6-31G *(*) values and the dihedral angles were constrained to the calculated values. For 1 the HF/6-31G** method reproduces bond lengths of the S-gauche conformer to within + 0.03/~ and the largest deviation for bond angles occurs for C - C - C where the calculated value is ca. 4 ~ larger than the experimental angle. The dihedral angle 51(C1C2C303) = 95(6) ~ is predicted closely by the larger basis set (~il = 102.0~ The HF/3-21G method predicts the C-C1 bonds by about 0.11 A too long, as expected for a basis set without d functions on chlorine. In the case of 2 and 3 both theoretical methods predict bond lengths better than + 0.03 ,~ and angles better than + 2 ~ The calculated dihedral angle in the S-gauche form of 1, ~51(C1C2C303), is too large by ca. 12~ with the HF/3-21G method and is in agreement with the experimental value of 112.0(20) ~ with the larger basis set. For the W-gaucheform of 3 the calculated dihedral angles are larger by ca. 11 o (HF/3-21 G) and by ca. 6 ~ than the experimental result of 120.0(16) ~

Structures and Conformations

93

B. Compounds with C-N Bonds: Carbonylisocyanates of the Type XC(O)NCO (X = CH3, F, CI) Compounds with 1,3-conjugated double bonds can, in general, adopt three different conformations: planar s-trans (dihedral angle ~5= 180~ gauche, or planar s-cis (5 = 0 ~ (Scheme 2). The shape of the potential function for internal rotation around the central single bondwi.e, the type of conformations, their relative stabilities, and the barriers between stable formswdepends on several effects: (1) conjugation between the double bonds; (2) interactions between double bonds A = X and B--Y; (3) interactions between double bonds and single bonds, if the central atoms are carbon atoms; (4) interactions between double bonds and lone pairs, if the central atoms are nitrogen atoms; and (5) interactions between terminal atoms or groups. Whereas conjugation favors planar structures, the various steric interactions lead to planar trans or to gauche forms. Since it is difficult to estimate the relative magnitude of these effects on the basis of the limited experimental data available, the prediction of stable conformations and of their relative energies is highly ambiguous. In 1,3-butadiene, H2C=CH-CH--CH2, two stable conformations exist: the more stable one having a planar trans structure [25-27], and the high energy rotamer (Mar~ = 2.1 - 3.1 kcal mol -l) possessing a gauche form [26-31]. IR matrix studies, however, suggest that the high energy conformer is planar cis [32-34]. Similar conformational properties are observed for the isoelectronic 2,3-diaza-l,3-butadiene (formaldazine), H2C--N-N-CH2, where the gauche form is 1.2(5) kcal mo1-1 less stable than the trans conformer [35]. In glyoxal, O--CH-CH--O, the ground state structure is again planar trans [36], but the high energy form possesses a planar cis structure (M-/~= 3.2(6) kcal mo1-1) [37, 38]. Acrolein, H 2 C = C H - C H = O , where a CH2 group and an oxygen atom are at the terminal positions, has the same conformational properties as glyoxal [39], but the enthalpy difference (AH~ = 1.7(6) kcal mol -l [40]) is smaller. Various ab initio calculations [26, 30, 41] reproduce the conformational properties of 1,3-butadiene correctly and predict the high energy form to possess a gauche structure. Similarly, calculations for glyoxal [41, 42] and acrolein [41, 43, 44]

\

/B X~A

\

~Y

/B =---"

\ trans

:

\

....":~Y

A

/B --

"-

~Y

~A

"~x

"%x

gauche

cis

Scheme 2.

94

HANS-GEORG MACK and HEINZ OBERHAMMER

reproduce the experimental results. In the case of formaldazine inclusion of electron correlation effects in the MP2 approximation is required in order to obtain a second stable conformer [41], whereas HF/6-31G* calculations predict only one minimum with trans structure [45]. Substitution of the hydrogen atoms by halogens or other groups may have drastic effects on the conformational properties of these systems. In perchloro- and perfluoro- 1,3-butadiene gauche conformations with dihedral angles around 80 ~ and 50 ~ respectively, are the preferred structures and no planar trans structures were observed [46, 47]. The result for the fluorinated compound was reproduced by ab initio calculations [48]. In the formaldazine system, fluorination has no effect on the conformational behavior [45], whereas in the chlorine- [49] or bromine-substituted [50] derivatives only gauche forms are found. In the case of glyoxal, chlorination does not change the ground state conformation (planar trans) but, according to the interpretation of vibrational data, the high energy form changes from planar cis in the parent species to gauche in oxalyldichloride [51]. In this context we were interested in the effect of various substituents on the conformational properties of carbonylisocyanates XC(O)NCO (X = CH 3, F, C1). Only planar trans and cis structures are expected for these systems (Scheme 3). According to MW spectroscopy [52], GED [53], and vibrational spectroscopy [54] for acetylisocyanate, CH3C(O)NCO, only the cis form is observed in the gas phase. On the basis of the vibrational analysis, the contribution of the trans structure is estimated to be less than 3%, i. e. AG ~ = G~ - G~ is larger than 2 kcal mol -l. This experimental result is reproduced correctly by ab initio calculations, which predict AE = 3.8 kcal mol -l (HF/6-31G*) or AE = 2.4 kcal mo1-1 (MP2/631G*) [53]. Acetylisocyanate is the only 1,3-conjugated system whose structure was determined in the gas phase for which only the cis form is found. GED of fluorocarbonyl isocyanate, FC(O)NCO, results in a mixture of 75(12)% cis and 25(12)% trans forms which corresponds to AG ~ = 0.7(3) kcal mo1-1 [55]. A similar result is obtained from IR spectra (AG ~ = 0.4(2) kcal mo1-1) [55]. Full structure optimizations with the HF/6-31 G* and MP2/6-31 G* methods predict energy differ-

x

/9 /C

N

//o

o _.

,,--

0

// c

/ C~N x

trans

cis Scheme 3.

95

Structures and Conformations

ences of AE = 1.7 kcal mol -l, which are too large by about 1 kcal mo1-1. The experimental conformational properties are reproduced correctly by MP2 and MP4 single point calculations with the HF/6-31G* optimized structures (AE = 0.6 kcal mol-l). Similarly, DFT calculations (AE = 0.7 and 0.4 kcal mol -l for local and nonlocal approximation, respectively) are in agreement with the experiment. Chlorocarbonylisocyanate, C1C(O)NCO, was also studied by GED and vibrational spectroscopy [56]. The calculated RDFs for the two conformers (Figure 5) differ markedly for r > 3 A. The peaks near 4.5 and 5 ,~ in the trans and cis curves correspond to the longest intramolecular O..O and C1-.O distances, respectively. Analysis of the experimental scattering intensities demonstrates that for this compound the main conformer possesses a trans structure with a contribution of 25(8)% cis form. If the necessary corrections are applied to the free enthalpies (AG ~ = G~ - G~ = -0.7(3) kcal mo1-1) an experimental AE value of-0.6(3) kcal mol -~ is derived. From the splitting of the C--O vibration in the IR gas spectrum a composition of 79(5)% trans and 21 (5)% cis is obtained if the ab initio values for the C = O vibrational intensities (squared dipole moment derivatives) are taken into account [56]. This corresponds to AE = -0.7(2) kcal mol -~, in agreement with the GED result. In combination with these experimental studies, various ab initio calculations at the HF and MP level, using 4-31G*, 6-31G*, and D95" basis sets, 1

C[kC2

//

N ~'cl -

-

02

cis

trans

Exp.

/1\\

/~\~

0000o

.

.

.

.

.

.

.

zozozo

.

\

/\ \

I

000

0

.

I

I

I

i

I

!

0

1

2

3

4

5

,J

6

R/A Figure 5. Calculated and experimental RDFs for chlorocarbonylisocyanate and difference curve for mixture. Interatomic distances of the main conformer (trans) are indicated by vertical bars.

96

HANS-GEORG MACK and HEINZ OBERHAMMER

Table 2.

Selected Values for Theoretical Energy Differences AE = E(trans)- E(cis) for CIC(O)NCO a zkE

Ref

(I) (II)

HF/4-31G(d) HF/6-31G(d)

Method

+1.3 +1.6

56 56

(HI) (IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (xIII) (XIV)

MP2/4-31G(d) MP4SDTQ/D95(d)//MP2/4-31G(d) MP2/6-31G(d) MP4/6-31G(d) HF/6-311G(d)//MP2/6-31G(d) MP2/6-311G(d)//MP2/6-31G(d) MP2/6-311+G(2d)//MP2/6-31G(d) HF/6-31G(2d)//HF/6-31G(d) MP2/6-31G(2d)//HF/6-31G(d) MP2/6-31G(2d)//MP2/6-31G(d) LDFTb NLDFTc GED Experiment

+0.6 +0.4 +0.6 -4.9 +7.4 - 1.7 -2.5 +0.8 -0.7 -0.7 -0.6 -0.5 -0.6(3)

56 56 58 58 58 58 58 59 59 59 60 60 56

Notes:

aln kcal mol -!. bLocal density functional theory [61]. CNonlocal density functional theory, exchange functional of Becke [62] and correlation functional of Perdew [631.

were performed. Disappointingly, all these calculations predict the preference of the cis form (AE between +0.4 and +1.6 kcal mol-~), which is in disagreement with the experiment. This discrepancy between experiment and theory prompted us to ask the question: "How reliable are ab initio calculations?" [57]. This question challenged two other research groups [58, 59], which reported several additional ab initio calculations for this molecule. A selection of the ca. 40 different calculations is given in Table 2. The theoretical AE values vary from -4.9 (VI) to +7.4 kcal mol -l (VII). MP2 single point calculations with two sets of d functions (XI and XII) reproduce the experimental value perfectly, independent of the geometry used. Further increase of the basis set, however, makes the agreement (AE = -2.5 kcal mo1-1 in IX) worse than that obtained with the standard MP2 method (AE = +0.6 kcal mol -l in V). Density functional methods (XIII and XIV) lead to full agreement with the experiment [60].

C. Compounds with C-O Bonds: Fluoroformylhypofluorite, FC(O)OF, and Bis(fluorooxy)difluoromethane, CF2(OF)2 Flu o ro f o rm ylh ypo flu o rite

The structural and conformational properties of formic acid, HC(O)OH, have been studied extensively by GED [64, 65] and MW spectroscopy [66-68]. Two

Structures and Conformations

0

\\ /

/ C

0

97

F

_

0

\\

-"

F

/

C

\ F

F

trans

0

cis Scheme 4.

conformers were observed with the O - H bond trans or c/s to the C - H bond, and the trans form was determined to be more stable by 3.903(86) kcal mol -l [69]. The perfluorinated derivative, fluoroformylhypofluorite FC(O)OF, was first synthesized by Cauble and Cady [70] in 1967. The IR (gas) spectrum indicates the presence of a mixture of two planar conformers (Scheme 4) [71]. The experimental RDF, derived from GED intensities, is shown in Figure 6 [71]. The calculated curves for pure cis and trans conformers differ mainly in the range around 2.6 A. Comparison with the experimental RDF demonstrates that the major component is the trans structure. A detailed analysis of the molecular intensities results in a small contribution (18 + 12%) of the cis conformer, which corresponds to a free enthalpy difference of AG ~ = G~ - G~ = 0.9(4) kcal mol -l. The enthalpy difference was determined more accurately from IR matrix spectra of equilibrium mixtures obtained from beam sources in the temperature range -45 to +140~ These experiments resulted in AH ~ = 1.17(7) kcal mo1-1. The entropy difference was estimated from rotational constants for the GED structures and from vibrational wavenumbers to be AS~ 0.35 cal mol -l K -1. With these data a composition of 14% cis and 86% trans forms at 298 K is obtained. These figures are in good agreement with the directly measured equilibrium from NMR (gas) experiments (13% cis and 87% trans) [72]. Comparison of FC(O)OF with HC(O)OH shows that fluorination has a considerable effect on the conformational properties. In both compounds the trans form is preferred but the energy difference between cis and trans decreases from 3.903(86) kcal mo1-1 in HC(O)OH to 1.17(7) kcal mol -l in FC(O)OE The energy difference AE between these two conformers was calculated with different ab initio and density functional methods (Table 3). The geometric structures of both forms were fully optimized. In order to allow for a strict comparison between the experimental AH ~ value and the theoretical AE values, the corrections for different zero point vibrational energies and the temperature dependence of AH ~ were calculated with the MP2/6-31G* procedure. According to these calculations AH ~ and AE differ by 0.13 kcal mo1-1, such that AH ~ has to be decreased by this difference. All three ab initio calculations predict the trans form to be the more

98

HANS-GEORG MACK and HEINZ OBERHAMMER

Figure 6. Calculated and experimental RDFs for fluoroformylhypofluorite and difference curve for mixture. Interatomic distances of the main conformer (trans) are indicated by vertical bars.

Table 3.

Experimental (Matrix Spectra) and Theoretical Energy Differences between cis and trans FC(O)OF IR(matrix) HF/3-21G HF/6-31G* MP2/6-31G*

AE = E ( c i s ) - E(trans) (kcal mol -l)

1.04(7) a

1.73

Notes: aCorrected experimental AE (see text). bSee footnote a in Table 2. CSee footnote b in Table 2.

1.95

1.62

LDFT b -0.50

NLDFT c -0.80

Structures and Conformations

99

stable one, in agreement with the experiment, but the energy difference is overestimated. The density functional calculations, however, result in a slight preference of the cis form, which is in disagreement with the experiment. The HF/3-21G and MP2/6-31G* methods reproduce the experimental bond distances very well, whereas all bond lengths are predicted too short by the HF/6-31G* approximation. The largest discrepancy occurs for the O-F bond distance (1.418(5) A vs. 1.357 A). LDb-'r results also in good agreement with the experimental bond lengths, somewhat better than the NLDFT method. Bond angles depend little on the computational procedure and are close to the experimental values. Recently, FC(O)OF became of interest as a precursor in the synthesis of difluoro dioxirane, CF202 [73], the only dioxirane which can be isolated as a pure substance and which is stable in the gas phase at room temperature. The geometric structure of this CF202 isomer was determined by high-resolution IR spectroscopy [74] and by joint analysis of GED intensities and rotational constants [75]. The three-membered CO2 ring in this compound possesses an extremely long O-O bond of 1.578(1) A. The experimental structure is reproduced perfectly by MP2/6311G(2d) calculations. Again, the HF/6-31G* approximation predicts all bonds too short, especially the O-O distance (1.578(1) vs. 1.487 A).

Bis(fluorooxy)difluoromethane The conformational properties of compounds of the type XO-CY2-OX (X, Y = H or Me) have attracted considerable interest both from experimentalists [76, 77] and theoreticians [78-81]. Depending on the dihedral angles around the C-O bonds these compounds can adopt four different conformations: (ap, ap), (sc, ap), (+sc, +sc) and (+sc, -sc) (Scheme 5). Extensive ab initio calculations for the model compound CH2(OH)2 result in a clear preference of the (+sc, +sc) conformation. Structures (+sc, -sc) and (sc, ap) are predicted to be higher in energy by 3-5 kcal mo1-1. The (ap, ap) form does not correspond to a stable structure and is calculated to be higher in energy by 8-11 kcal mo1-1. The strong preference of the (+sc, +sc) conformation can be rationalized by the generalized anomeric effect [82]. HyperY

\ ,,,..--Y X\o/C\o/X

Y

\ ,,,,.--Y

o/ t

C\o/X

o

I

X

( ap, ap )

Y

X

( sc, ap ) Scheme 5.

\,,,... Y

/c

\o X

( +sc, +sc )

y

o 1

\/

/c

X

Y

\o 1 X

( +sc,-sc )

100

HANS-GEORG MACK and HEINZ OBERHAMMER

conjugative orbital interactions between the oxygen lone pairs, lp(O), and the antibonding 6*-orbital of the opposite C-O bonds, lp(O) ~ 6"(C-O), favor the synclinal orientations of the O-X bonds. On the other hand, such an orbital interaction between the oxygen lone pairs and the C-Y bonds, lp(O) ~ 6*(C-Y), would stabilize the (ap, ap) structure. Apparently, in CH2(OH)2, the lp(O) --> 6*(C-O) interaction is much stronger because the 6*(C-O)-orbital is lower in energy than the a*(C-H)-orbital. Gas-phase structural studies of dimethoxymethane, CH2(OMe)2, and 2,2-dimethoxypropane, Me2C(OMe)2, confirm the above theoretical predictions [76, 77]. Experimental studies of partially fluorinated dimethylethers, CH2F-O-CH3 [83] and CF3-O-CH 3 [84], demonstrated that orbital interaction between oxygen lone pairs and C - F bonds, lp(O) ~ cr*(C-F), have a strong effect on the structural and conformational properties of these compounds. In CH2F-O-CH3 only the synclinal conformation is observed (C-F bond synclinal with respect to the O-CH 3 bond and anti to one of the sp 3 hybridized oxygen lone pairs). Thus, it was of interest to determine the conformational properties of CF2(OF)2 where the orbital interactions, lp(O) ~ a*(C-O) and lp(O) ~ a*(C-F), compete with each other. According to the GED analysis [85] a mixture of 65(15)% (+sc, +sc; (72 symmetry) and 35(15)% (sc, ap; C1 symmetry) is present in the gas phase of CF2(OF)2. The refined dihedral angles 8(OCOF) are 62.8(10) ~ in the structure with C2 symmetry. This conformational composition corresponds to AG~ = G~ ap) - G~ +sc) = 0.4(3) kcal mol -l. Taking into account the different multiplicities of the two conformers and neglecting further entropy differences, a AH~ value of 0.8(3) kcal mol -l can be derived. This enthalpy difference is much smaller than the theoretical energy differences obtained for the parent compound CH2(OH)2 (3-5 kcal mol-l), and it demonstrates that the orbital interactions, lp(O) --> ~*(C-O) and lp(O) --> c*(C-F), are of similar magnitude in CF2(OF)2. Table 4 compares the experimental AH~ value with the relative energies predicted by different theoretical methods. The results of the HF/6-31G* and MP2/6-31G*

Table 4. Experimental Enthalpy Differences and Theoretical Energy Differences for Various Conformers of CF2(OF)2a GED b

(+sc, +sc)

0.0

(+sc, ap)

0.8(3)

HF/3-21G

HF/6-31G*

MP2/6-31G*

LDFT c

NLDFT d

0.0

0.0

0.0

0.0

0.0

-0.002

0.72

0.71

0.15

-0.27

(+sc, -sc)

D

2.08

2.79

2.84

3.47

3.14

(ap, ap)

m

0.22

1.86

2.55

4.70

4.29

Notes:

aln kcal mol -I. bAH~ from GED experiment. CSee footnote a in Table 2. dNonlocal density functional theory, exchange functional of Becke [62], correlation functional of Lee et al.

[8o3.

Structures and Conformations

101

calculations for the energy difference between the (+sc, +sc) and (sc, ap) structures are in perfect agreement with the experiment. The low level ab initio procedure (HF/3-21G) and the density functional approaches underestimate this energy difference or even predict the Cl form to be more stable. In general, all relative energies of the four possible conformers of CF2(OF)2 are smaller than those obtained for the parent compound CHz(OH)2.

D. Compounds with N-O Bonds: Halogen Nitrates, XONO2 (X = F, CI, Br), and O-Nitrosobis(trifluoromethyl)hydroxylamine, (CF3)2NONO

Halogen Nitrates Covalent halogen nitrates are attracting increasing interest by chemists and atmospheric scientists because of their presence in the atmosphere and their participation in the halogen and NO cycles for ozone depletion in the stratosphere [87]. The geometric structures of halogen nitrates have a long and controversial history. In 1937 Pauling and Brockway derived from GED intensities of FONO2 a nonplanar structure with the O-F bond perpendicular to the NO3 plane [88]. About 25 years later IR spectra of FONO2 and C1ONO2 were interpreted in terms of planar structures [89, 90]. At this time planar structures have also been established for nitric acid, HONO2 [91], and methyl nitrate, CH3ONO2 [92]. Two independent Raman studies of liquid C1ONO2 concluded from polarization data that this nitrate possesses a nonplanar structure with Cl symmetry [93, 94]. On the other hand, similar measurements have been interpreted by other authors in terms of a planar configuration [95]. In 1976 Suenram et al. derived unequivocally from the MW spectrum a planar structure for CIONO2 [96]. The rotational constants of two isotopomers (37C1and 35C1),however, did not allow a determination of the geometric parameters. At about the same time the presence of two rotamers, planar and nonplanar, was deduced from the far infrared and from the low-resolution microwave spectrum [97]. The GED intensities for FONO2 [98], C1ONO2 [99], and BrONO2 [99] clearly demonstrate that in these nitrates only one conformer is present which possesses a planar structure (Cs symmetry). This is shown by the RDF for BrONO2 (Figure 7) where the positions of the O--Br distances (Oc..Br and Oc.Br) are compatible only with a planar configuration. Whereas the geometric parameters of the NO2 group (N=O and O--N--O) are very similar in all covalent nitrates (XONO2), the N-O single bond increases considerably with increasing electronegativity of X. The following bond lengths have been reported: 1.406(3) A for HONO2 [100], 1.402(5) ,~ for CH3ONO 2 [101], 1.456(5)/~ for BrONO2, 1.499(3) ,~ for C1ONO2, and 1.507(4) for FONO2. Various ab initio calculations for C1ONO2 confirm the planarity of this molecule, but the values obtained for the N-O and O-C1 bond lengths depend strongly on the computational method (Table 5): especially the N-O bond increases by almost 0.2

102

HANS-GEORG MACK and HEINZ OBERHAMMER

Figure 7. Experimental RDF and difference curve for bromine nitrate. Interatomic distances are indicated by vertical bars. /~ upon inclusion of electron correlation. A similar dependence on the computational procedure is observed for the N - O and O - F bond lengths in FONO2 (Table 6). In order to allow for a direct comparison with the theoretical results, re values for the N - O and O - F bond lengths were derived from the experimental ra values using the diatomic approximation [3]. The predicted N - O bond lengths vary from 1.385 ~ (II) to 1.656 ,~ (XII) and O - F bond lengths from 1.350/~ (II) to 1.476 ~, (XII). A surprisingly close agreement with the experimental structure is obtained with the most simple ab initio method (I), but the predicted vibrational frequencies deviate from the experimental values by up to + 300 cm -1. The two widely applied standard methods, (II) and (III), yield N - O distances which are 0.12 ~ too short or 0.05/~ too long and O - F distances which deviate by -0.05 ]k or +0.03 ]k from the

Table 5. Experimental and Theoretical N-O and O-CI Bond Lengths in CIONO2 a Exp. b

HF/4-31G c

HF/3-21G* d

HF/6-31G* e

MP2/6-31G* e

N-O

1.496(3)

1.443

1.464

1.372

1.547

O-C 1

1.665(2)

1.765

1.684

1.666

1.701

Notes:

aln angstroms. bra values from joint GED / MW analysis. CRef. 102. dRef. 99. eRef. 103.

103

Structures and Conformations

Table 6. Experimental and Theoretical N-O and O-F Bond Lengths in FONO2 a Method

(I) (II) (HI)

(IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) (XIV) Notes:

N-O

O-F

Experiment (re values )b

1.500(6)

1.402(7)

HF/3-21G HF/6- 31 G* MP2/6-31 G* MP3/6-31 G* MP4SDTQ/6-31G* CCD/6-31G* CCS D/6- 311G(2d ) CISD/6-31 G* QCISD(T)/6-31 G* LDFT/TZP c LDFT/TZP (S) d NLDFT/TZP (S,B) d NLDFT/TZP (S,P) d NLDFT/TZP (S,B,P) d

1.494 1.385 1.551 1.458 1.609 1.467 1.467 1.427 1.518 1.500 1.518 1.656 1.582 1.600

1.421 1.350 1.429 1.409 1.447 1.413 1.403 1.382 1.440 1.403 1.415 1.476 1.434 1.446

aln angstroms. bEstimated from rg values [98]. CSeefootnote a in Table 2. ds = Stoll correction [104]; B = nonlocal exchange functional of Becke [62]; P = nonlocal correlation functional of Perdew [63].

experimental value. Higher order perturbation theories (IV) and (V) do not converge smoothly, but show strong fluctuations for the two bond lengths. Larger basis sets (6-311G* or 6-31G(2d)) in methods (II) to (V) have only small effects on the geometry. The two coupled cluster methods, (VI) and (VII), result in nearly identical bond lengths. The calculated N - O distance is too short by ca. 0.03 ,~ and the O - F distance agrees very well with the experiment. The low-level CI method (VIII) predicts the N - O bond too short by 0.07 A. The higher level calculation (IX) reproduces the N - O bond length very well but results in an O - F bond which is too long by 0.04 A. LDFT calculations in its basic version (X) reproduce the experimental bond lengths excellently. The Stoll correction (XI) has only small effects. The application of nonlocal gradient corrections (XII)-(XIV), however, leads to lengthening of the N - O bond by 0.07-0.16 ,~ and of the O - F bond by 0.03-0.07 ,~ and makes the agreement with the experimental data drastically worse. It is unsatisfactory that higher level calculations with ab initio or DFT methods do not necessarily lead to closer agreement with the experiment.

O-Nitrosobis(trifluoromethyl)hydroxylamine, (CF3)2NONO The conformational properties of compounds which possess one or more atoms with lone electron pairs depend on the interactions between these lone pairs and

104

HANS-GEORG MACK and HEINZ OBERHAMMER

Scheme 6.

between lone pairs and ~*-orbitals of vicinal bonds (anomeric effect). (CF3)2NONO (NONO) possesses three adjacent atoms with stereochemically active lone pairs. Assuming planarity of the NONO skeleton, as suggested by the structures of hydroxylamines [105] and nitrites [106], four conformations are conceivable: trans-anti, trans-syn, cis-anti, and cis-syn (Scheme 6). Trans or cis refers to the orientation of the O-N bond relative to the CNC bisector, and anti-syn describes the position of the N = O double bond relative to the N - O single bond. Trans or cis and anti or syn also applies to the relative orientations of the lone pairs in the N - O - N chain. Since very little is known about the relative magnitudes of the different interactions between atoms, bonds, and lone pairs, a reliable prediction of the relative stabilities of these conformations is impossible. The region of the N--O vibration in the IR(gas) spectrum [107] (see Figure 8) with two bands at 1820 and 1800 cm -1 indicates the presence of two conformers. The intensity ratio of these two bands is 4:1. The stronger band at higher wave number possesses a PQR contour with a strong Q-branch typical for a c-type vibration. Only in the trans-syn conformer the N = O bond lies exactly parallel to the c-axis. As indicated by ab initio calculations (see below), the cis-anti and cis-syn forms can be excluded, and the weak band must correspond to the

105

Structures and Conformations

X Z ..-1 O Z N

i

1750

i

I

I

I

1800 Cm

i

I

I

l 1850

-1

Figure 8. Experimental IR(gas) spectrum of the N=O stretching region in (CF3)2NONO.

trans-anti form. In this structure the N = O bond is between the a- and c-axis and the expected hybrid is consistent with the contour of this band. If equal absorbances (squared dipole moment derivatives) for the N = O vibrations in the two conformations are assumed, a contribution of 80(10)% trans-syn is obtained from the relative intensities. This corresponds to AG ~ = G~ - G~ = 0.8(3) kcal mo1-1. The GED intensities are consistent only with the trans-syn form being the predominant component [107]. However, the RDF is not very sensitive towards the ratio of the conformers and the composition derived from the IR spectrum was used in the final analysis. Most bonds in this molecule have rather similar lengths and all bond angles are close to tetrahedral. This causes high correlations in the least-squares analysis and leads to large uncertainties in the geometric parameters of the NONO group. Furthermore, the structure of this compound was determined

106

HANS-GEORG MACK and HEINZ OBERHAMMER

by low temperature X-ray diffraction [107]. In the crystal only the trans-syn conformation is present. Ab initio calculations at the HF/3-21G, HF/6-31G*, and MP2/6-31 G* levels of theory were performed [107]. The cis-syn conformer possesses unreasonably short contacts between the nitrite oxygen and the fluorine atoms (1.8 ,~ for standard bond lengths and bond angles) and, therefore, is not expected to correspond to a stable structure. According to the HF/3-21G calculations the cis-anti form does not represent a minimum. Geometry optimization leads to an inversion at the central nitrogen and results finally in a trans-anti structure. The HF/6-31G* method predicts an energy minimum for the cis-anti form, which lies, however, 5.6 kcal mol -l higher than the trans-anti conformer. No MP2 calculations were performed for this structure and only the trans-anti and trans-syn conformations were considered in the further theoretical and experimental analyses. Experimental and theoretical results (relative energies of trans-anti and transsyn conformers and geometric parameters of the NONO group of the trans-syn form) are collected in Table 7. Two results of this study are remarkable, i.e. the unexpected conformational properties of this compound and the unusually long O - N single bond. According to the experimental studies NONO exists in the gas phase as a mixture of trans-syn and trans-anti conformers with the former structure energetically favored. In the solid state only the trans-syn conformation is present. In this form the lone pairs at the amine N atom and at the central O atom are trans to each other, whereas the lone pairs at the central oxygen and at the nitrite nitrogen are cis to each other. This result is unexpected, since it leads to a very short contact between the terminal oxygen and the amine nitrogen of 2.52 A in the gas phase and 2.48 ]k in the crystal. This distance is much shorter than the sum of the van der Waals radii (2.90/~). Ab initio calculations result in contradicting predictions for the relative stabilities of the two conformers. Whereas the HF/3-21G and MP2/6-31 G* methods predict the correct main conformer (trans-syn) but overes-

Table ;r. Geometric parametersa of the N - O - N = O Group in (CF3)2NONO (trans-syn) and Relative Stabilities of trans-anti and trans-syn Conformers From Experiment and Ab Initio Calculations GED

N-O O-N

1.410(15) 1.572(21)

N=O N-O-N O-N=O AG~ ~ Notes:

1.156(8) 107.6(19) 115.7(43) 0.8(3)

aln angstroms and degrees. bAG~= G~ - G~

X-ray

HF/3-21G

HF/6-31G*

MP2/6-31G*

1.376(3) 1.669(3)

1.417 1.527

1.371 1.401

1.382 1.717

1.120(4) 106.9(2) 110.9(2) m

1.150 111.2 113.0 1.8

1.136 114.4 115.0 -0.6

1.156 103.1 110.1 3.3

from IR(gas) spectrum, in kcal mol-I.

Structures and Conformations

107

timate the energy difference, the HF/6-31 G* calculations lead to a predominance of the trans-anti structure. The O - N bond in gaseous NONO (1.572(21) ,~) is much longer than the O - N single bonds in nitrites, R O - N = O : 1.392(10) A in cis-HON--O and 1.398(5) A in cis-MeO-N=O [106]. In the crystal this bond (1.669(3) ,~) is still longer by ca. 0.1 ,~. This large difference between gaseous and solid state may be rationalized by the possible decomposition into two stable radicals: (CF3)2NONO ~ - (CF3)2NO-+ .NO The ab initio results demonstrate that this O-N bond length depends drastically on the computational method: 1.527 ,~ with HF/3-21G, 1.401 A with HF/6-31G*, and 1.717 A with MP2/6-31G*. The experimental gas phase value is intermediate.

E. Compounds with O-O Bonds: CH3OOCH3, FC(O)OOC(O)F, CF3OOOCF3 The structural feature of principle interest in peroxides ROOR is the dihedral angle, 5(ROOR). For noncyclic peroxides dihedral angles from less than 90 ~ (81.03(1) ~ in CIOOC1 [108] and 88.1(4) ~ in FOOF [109]) to nearly 180~ (166(3) ~ in ButOOBu / [110]) have been reported. This large range of dihedral angles indicates that this parameter is a delicate balance between two opposing effects: (1) interaction between the oxygen lone pairs and between a lone pair and the opposite ~*(O-R)-bond (anomeric effect), which favor a dihedral angle of about 90 ~ and (2) repulsion between the substituents, which tends to increase this angle.

Dimeth yiperoxide For CH3OOCH 3, experimental as well as theoretical studies produce rather controversial results for the dihedral angle. Two photoelectron spectroscopy (PES) investigations interpret the splitting of the ionization potentials of the oxygen lone pairs in terms of an exactly planar (5 = 180 ~ C2h symmetry [111]) or near planar (5 = 170 ~ [112]) trans configuration. However, analysis of IR and Raman spectra [113] and a normal coordinate analysis based on these data [114] indicate a nonplanar structure with C2 symmetry. Microwave spectra have been recorded in four different laboratories [115-118], which demonstrate a nonzero dipole moment and, thus, exclude a planar trans configuration. So far, these very complicated and dense spectra have not been assigned. Semiempirical and ab initio calculations predict dihedral angles ranging from 96.5 ~ (MINDO/2 [119]) to 180 ~ (CNDO/2 [111], GVB-CI [120], and MP2/6-31 G* [121, 122]). The MINDO/3 method ( 110.7 ~ [123]) and ab initio calculations with STO-2G (101.7 ~ [124]) and 4-21G* (115.5 ~ [125]) basis sets lead to intermediate values. The analysis of the GED intensities definitely result in a nonplanar skew structure [126]. For a rigid molecular model an effective dihedral angle of 135(5) ~ is obtained. A dynamic model with a double minimum potential,

108

HANS-GEORG MACK and HEINZ OBERHAMMER

Table 8. Experimental and Theoretical Skeletal Geometric Parametersa (A, deg) and Barriers Vtb for the trans Configuration of CH3OOCH3 HF/3-21G HF/6-31G** MP2/6-31G* CISD/6- 31G* MP4/6-31G* //HF/3-21G MP4/6- 31G** //HF/6- 31G** MP4/6-31G* //MP2/6- 31G* Exp. (GED) Exp.(FIR) Notes:

0-0

o-c

ooc

5e(cooc)

v,

1.464 1.399 1.478 1.450 1.462

1.445 1.397 1.419 1.417 1.448

104.3 106.0 103.2 104.2 105.1

180 180 180 180 116

m 0.22

1.393

1.399

107.3

122

0.13

1.471

1.421

104.3

121

0.10

1.457(12) 1.449(1)

1.420(7) 1.420c

105.2(5) 103.92(4)

119(4) 120.01(3)

0.25(+25/-10) 0.223(3)

---

aln angstromsand degrees. bin kcal mol-I. CNotrefined.

v= v~[ (~1~) 2-

1

]2

(~ = 180 - B) leads to an equilibrium dihedral angle of Be = 119(4) ~ and to a barrier Vt = 0.25 (+0.25/-0.10) kcal mol -l for the planar trans structure. This result was confirmed by the analysis of the high-resolution far infrared spectra, where more than 300 transitions were fitted with Be = 120.01(3) ~ and Vt = 0.223(3) kcal mol -l [127]. Results of various ab initio calculations [121] are compared to the experimental data in Table 8. Three widely used standard ab initio methods (HF/3-21G, HF/631G**, MP2/6-31G*) and a CI calculation (CISD/6-31G*) predict planar trans configurations with C2h symmetry (Se = 180~ in contrast to the experiments. Only recently, MP2/6-31G* results, which have been shown several years before to be wrong [121], were published again [122]. The experimental dihedral angle can be reproduced with MP4 single-point calculations using geometries optimized at the HF/3-21G, HF/6-31G**, or MP2/6-31 G* levels of theory. The predicted barriers for the trans structure are between 0.10 and 0.22 kcal mol -l. B is (flu o ro carbon yl)pe ro x ide

In this oxygen-rich FC(O)OOC(O)F three conformers are possible, depending on the orientation of the two carbonyl groups, regardless of the dihedral angle B(COOC): syn-syn, syn-anti, and anti-anti (Scheme 7). The C = O vibrational bands in the IR(gas) and IR(matrix) spectra indicate that only one conformer is present in the gas phase. The band shape of the symmetric vibration suggests a

109

5tructures and Conformations 0

/ 0 ~

\ C~F /0

0

-- 0

F--C

F~C 0 syn- syn

/

\

C ~0

/ /O m O

0 --C

0 syn- anti

C ---0

\ F

anti- anti

Scheme 7.

syn-syn structure [128]. Considering the small energy difference between skew and planar trans configurations in CH3OOCH3 (0.223(3) kcal mol -l [127]), it could be

expected that conjugation between the C = O n-bonds and oxygen lone pairs stabilizes the planar trans structure of this compound. The GED intensities [129] are reproduced only with a nonplanar syn-syn conformation, in agreement with the interpretation of the IR spectra, and contributions of other conformers larger than 10% can be excluded. The experimental dihedral angle (5(COOC) = 83.5(14) ~ is surprisingly small. Planarity of the FC(O)OO moieties was assumed in the GED analysis. Ab initio calculations in the HF approximation confirm, that the syn-syn structure is the most stable conformation. The HF/6-31 G* method predicts the syn-anti and anti-anti forms to be higher in energy by 3.2 and 6.4 kcal mol -l, respectively. The trans barrier in the syn-syn conformation is calculated to be 0.6 (HF/3-21G) and 2.5 kcal mol -l (HF/6-31G*). Both approximations reproduce the experimental dihedral angle reasonably well (87.3 ~ with HF/3-21 G, 89.5 ~ with HF/6-31G*). The small basis set results in an O - O bond length (1.445/~) which is slightly longer than the experimental value (1.419(9)/~), whereas the large basis set leads to an O - O distance which is too short (1.369 ~).

Bis(tritIuorometh yl)trioxide The only straight-chain trioxides ROOOR, which are stable at room temperature, are those with R = perfluoroalkyl groups. The simplest derivative in this class of compounds is CF3OOOCF 3. Analysis of the radial distribution function (Figure 9) yields a skew structure with trans orientation of the two CF 3 groups (6"2 symmetry) [130]. This result agrees with the interpretation of Raman spectra [131], but is in contrast to the interpretation of IR spectra which indicate C~ symmetry [132]. Very similar geometric parameters were determined for the solid state [130] (Table 9). In particular, the OOOC dihedral angle, which is most easily affected by intermolecular interactions in the crystal, possesses identical values in both phases 05(OOOC) = 96.0(38) ~ vs. 95.9(8)~ Surprisingly, the experimental structure of this trioxide is reproduced very well with the low-level HF/3-21G method. A second

110

HANS-GEORG MACK and HEINZ OBERHAMMER

Figure 9. Experimental RDF and difference curve for bis(trifluoromethyl)trioxide. Interatomic distances are indicated by vertical bars.

Table 9. Experimentally and Theoretically Determined Structural Parameters of CF3OOOCF3 Gas a

Crystal b

HF/3-21G

C-F

1.326(3)

1.315(2)

O-C

1.378(12)

1.389(2)

1.393

O-O

1.452(5)

1.437(2)

1.453

1.327

O-O-O

106.7(20)

106.4(1)

104.4

O-O-C

105.8(6)

106.5( 1)

107.4

F-C-F

109.6

108.3(4)

109.3(7)

ot (CF3) c

5.4(7)

5.0(2)

4.0

O-O-O-C

96.0(38)

95.9(8)

97.6

Notes:

aElectron diffraction; ra distances (/~) and Z a angles [deg]; in parentheses 30 values. bX-ray diffraction; mean parameters; in parentheses o values. CTilt angle between the C 3 axis of the CF 3 group and O-C bond direction.

Structures and Conformations

111

stable conformer with cis orientation of the CF3 groups (Cs symmetry) is predicted to be higher in energy by 3.3 kcal mo1-1.

i!1. C O N C L U S I O N The main interest of this review is a comparison between experimental and theoretical conformational properties of compounds containing C-C, C-N, C-O, N-O, and O-O single bonds. The selected examples demonstrate that predicted relative energies of different conformers may depend strongly on the computational method. A calculation is considered to be good if experimental energy differences are reproduced within their error limits. In the case of the malonic acid derivatives, two HF methods (HF/3-21G and HF/6-31G*(*)) were applied. For C1C(O)-CH2-C(O)C1 none of the two methods, for FC(O)-CH2-C(O)F) only the HF/6-31G** approximation, and for FC(O)CF2-C(O)F only the HF/3-21G calculations give good agreement with the experimental conformational data. For the halogenated carbonyl isocyanates the standard ab initio methods (HF/3-21G, HF/6-31G*, and MP2/6-31G*) do not predict the experimental energy differences correctly, whereas DFT calculations work very good. It was demonstrated, however, that ab initio approaches can be found, which reproduce the experiment (e.g. MP2/6-31G*//HF/6-31G* for FC(O)NCO or MP2/6-31G(2d)//HF/6-31 G* for C1C(O)-NCO). The cis-trans energy difference in FC(O)OF is reproduced reasonably well with HF/3-21G and MP2/6-31G* calculations, but DFT methods lead to wrong results in this case. DFT approaches also result in bad agreement for the conformational properties of bis(fluoroxy)difluoromethane, whereas HF/6-31 G* and MP2/6-31G* calculations agree perfectly with the experiment. On the other hand, standard ab initio methods do not reproduce the conformational characteristics of (CF3)2NONO and of dimethyl peroxide, whereas the HF approximation yields good results for FC(O)O-OC(O)F and CF3OOOCF 3. In addition to the conformational properties, geometric parameters such as bond lengths may also depend strongly on the theoretical approach. This is especially true for fluorine nitrate, FONO2, and (CF3)2NONO. For both compounds only the simplest ab initio method (HF/3-21 G) gives reasonable agreement with the experimental N-O/O-N bond lengths. In the case of FONO2 local DFT calculations reproduce the geometric structure perfectly, whereas nonlocal methods make the agreement drastically worse. In all cases, except for CIC(O)NCO and FONO2, only a very limited number of computational procedures has been tested. For molecules, where the applied methods do not reproduce the experimental data, it would certainly be possible to find a suitable theoretical approach which leads to agreement with the experiment. Without knowledge of the experimental results, however, it is not possible to select this suited method a priori for each individual molecule. This general problem of computational chemistry has been addressed by M. J. S. Dewar several years ago

112

HANS-GEORG MACK and HEINZ OBERHAMMER

[133]: "Any valid use of ab initio procedures in chemistry has therefore to be on a purely empirical basis, limited to situations where specific tests have shown the procedure in question to give satisfactory results."

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Structures and Conformations 37. 38. 39. 40. 41. 42. 43. 44.

45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

75. 76. 77. 78.

113

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ABSORPTION SPECTRA OF MATRIX-ISOLATED SMALL CARBON MOLECULES

ivo Cermak, Geroid Monninger, and Wolfgang Kr~tschmer

I. II. III. IV. V.

VI. VII. VIII.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Isolation o f Carbon Molecules . . . . . . . . . . . . . . . . . . . . . Carbon Molecules in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon Molecular Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Stretching Modes in Linear Molecules . . . . . . . . . . . . . . . . . . . B. Electronic Spectra o f Linear Species . . . . . . . . . . . . . . . . . . . . C. Bending Modes o f Linear Molecules . . . . . . . . . . . . . . . . . . . . D. Cyclic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assignments o f IR Bands in Matrices . . . . . . . . . . . . . . . . . . . . . . Recent Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. IR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 117-146 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9 117

118 118 119 121 122 124 124 125 128 128 129 134 137 137

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IX.

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~TSCHMER B. UV-vis Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Correlations of UV-vis and IR Absorptions . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 143 144 144

144

ABSTRACT An overview is given about the IR and UV-vis absorption spectra of pure carbon molecules (Cn; n > 2). Most of the discussed results concern carbon molecules trapped in argon-ice matrices, but reference is also given to studies of other researchers employing solid neon matrices or using gas-phase laser spectroscopy. In matrices, the growth of carbon vapor species (C1, C2, and C3) into larger molecules can be initiated and controlled by thermal annealing. The IR-spectroscopic methods are presented and discussed by which the produced molecules can be identified. The methods are based on line correlations, comparison with data obtained from quantum chemical calculations, and on isotopomeric studies. In the IR domain, species up to linear C 13 could be identified. An evaluation of the available IR and UV-vis data suggests that upon matrix annealing the carbon molecules grow to linear chains containing 15 and more atoms. In addition, there is evidence for the formation of nonlinear, probably cyclic species.

I. I N T R O D U C T I O N It is well known that carbon can assume various kinds of bonds not only with other atoms but also with itself. This ability contributes to the intriguing complexity of carbon compounds in the organic world. Only recently it has been realized that carbon alone, without the involvement of any other element, can form fascinating molecules as well: e.g. the closed-cage fullerenes [1]. Our present knowledge of carbon molecules is distributed very unevenly. In 1989, Weltner and van Zee [2] in their comprehensive review article on carbon molecules stated that the knowledge of Cn molecules is almost monotonically decreasing with n. Species up to C3, which are abundant in carbon vapor are comparatively well studied. Clustering of carbon vapor leads to the formation of C60 [1, 3]. This discovery set free an avalanche of research on fullerene species. Now we face the curious situation that C60, C70, and some other fullerenes are much better known than most of the smaller carbon species. "Small" molecules in our context means species containing between 2 and about 20 atoms. Such species are important in many fields, as e.g. in combustion chemistry, in astrophysics, and as fullerene-precursors. This article reports the advances in studying these molecules by optical absorption spectroscopy. To understand fullerene formation, ion c h r o m a t o g r a p h i c studies on massselected b e a m s of carbon molecules were carried out by several researchers. These experiments were particularly successful in elucidating the structures of

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119

various ionic carbon clusters C n [4, 5]. It tumed out that the small clusters are linear chains and that for n beyond about 10, monocyclic rings occur. These features were already predicted by early theory [6, 7]. At larger n, more complicated molecular shapes were detected, e.g. bi-, tri-, and multicyclic rings [8]. The occurrence of graphene flakes was also reported [9]. In the molecular size range between about 10 and 30 atoms, the cluster ions show a large number of isomeric structures. However, most of these isomers seem to belong to energetically unfavorable configurations and relax to simpler chains or rings [10]. Compared to the ionic species, the structures of neutral clusters are not so well known. Over the last years, it was debated whether linear or cyclic species are the ground state configurations of small clusters. It thus appears that isomerism occurs in neutral species as well. The structure of the Ca molecule may be taken as example: Coulomb-explosion experiments suggest a cyclic (i.e. rhombic) C4 structure [11], while cluster and matrix spectroscopy showed the linear form of Ca [12, 13]. Apparently, carbon clusters can assume various structures depending on their size and on the conditions under which they form. The quenching of carbon vapor is an especially interesting process. In a rare-gas atmosphere, quenching of carbon vapor leads to formation of fullerenes and other fascinating carbon clusters. Trapped within a cryogenic matrix, carbon vapor tends to form larger clusters as well. As will be shown, the matrix technique provides the unique opportunity to control and to follow the growth of carbon molecules into larger species. For the detection of carbon species, e.g. in chemistry or astronomy, the spectroscopic fingerprints of these molecules have to be known. Matrix-isolation studies can perform pioneering work in this field. Based on matrix data, high-resolution gas-phase laser spectroscopy can be carried out yielding data which are not accessible by the matrix technique. Recently, Saykally and co-workers successfully measured the IR absorptions of jet-cooled carbon clusters by high-resolution laser-diode spectroscopy (for an overview, see ref. 14). The efforts culminated in the detection of the linear C 13molecule, which could be identified from the analysis of a single rotation-vibration band [15].

II. MATRIX ISOLATION OF CARBON MOLECULES The early studies in this field were carried out by William Weltner and co-workers in the 1960s [16-19]. These researchers matrix-isolated carbon vapor in solid argon and neon and measured the UV-vis and IR absorption spectra. Carbon vapor at 3000 K roughly consists of atomic carbon (20%), C2 (10%), C3 (70%), and trace amounts (less than 1%) of larger species. The early work focused on C3, the main species of carbon vapor. It was also observed that upon thermal annealing of the matrix, a molecular growth process takes place. Within the matrix, the initial population of trapped vapor molecules grows to larger species Cn (n > 3) according to the scheme Cn + Cm ~ Cn+m.

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Matrix-isolation has advantages and shortcomings. A rather attractive feature is the ease by which reactive species can be accumulated to sufficient concentrations to perform conventional spectroscopy. An overview of the electronic and vibrational absorptions of the trapped species can be readily obtained. Furthermore, at the low temperatures applied, transitions occur exclusively from the molecular ground states, leading to a considerable simplification of the spectra. The major shortcoming of this method is the loss of useful spectral information. The interaction between matrix and host molecules shifts and broadens the levels, and does not allow free molecular rotation. Depending on the matrix environment, the levels may also split into different "site" components. Although in cryogenic rare-gas ices the degree of influence of distortions is limited, matrix effects have to be observed very carefully. The identification of the absorbing molecular species is a difficult task, especially if the matrix contains a large number of different species. Trapping carbon molecules in a matrix may lead, especially after matrix annealing, to a large variety of different species. Such samples contain, what in our laboratory slang is called a "carbon soup". Using models for diffusion and chemical kinetics, several authors tried to calculate the molecular growth development starting from the initial vapor composition [20, 21]. These calculations provide some help and guideline, but are too uncertain to predict the abundance of larger sized molecules. To facilitate the molecular identification work, a better definition of the trapped species is helpful. This can be achieved e.g. by the photolysis of hydrocarbons, which have a suitable backbone of carbon atoms (e.g. diacetylene C4H2 or CaD 2 is used for preparing C4; the hydrogens are removed by exposure to UV light). Usually, a mixture of the precursor and the matrix gas is deposited under UV irradiation and the pure carbon species is obtained along with some by-products, which can be discerned in additional photolysis experiments with isotopically modified hydrocarbons [12]. An attractive method was recently applied by Maier and co-workers. These researchers successfully produced mass-selected beams of carbon molecules (and of other species) for matrix isolation [22]. Such beams can be obtained from ion sources in which the species of interest (usually anions) becomes mass-analyzed, then neutralized, and finally trapped into the matrix. This technique led to considerable progress in the assignment of IR- and UV-vis transitions of linear carbon molecules [23-26]. However, the carbon anions prefer the linear conformation and thus no data on cyclic species could be obtained. The use of carbon vapor as seed material in argon matrices has, despite the mentioned limitations, the advantage that a large set of molecular structures can be readily obtained in the matrices. It appears that various carbon molecules ranging from linear C3 up to Cl5 or even larger can be prepared and also nonlinear, probably cyclic species can be studied. The concentrations of these species are usually sufficient also for the more demanding requirements of IR spectroscopy. To overcome the uncertainties in molecular identification, supporting evidences are required. These are provided by the already mentioned matrix spectroscopy of

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121

mass-selected species and by the gas-phase spectroscopy of carbon clusters [14]. In the IR domain, specific identification methods are applied. These are based on line-growth correlations and comparisons with the data obtained by quantum chemical calculations. After an appropriate linear scaling of the frequencies, the IR absorptions of some carbon molecules can be fairly accurately predicted (for an early work on Ca, see e.g. ref. 27, and for larger species, see the more recent results by Martin and co-workers [28, 29]). Further important supporting evidence is provided by the IR spectra of the 12C-13C isotopically substituted molecules: Under total isotopic substitution, pure carbon molecules show a typical shift of their IR absorption lines. Partially substituted species yield IR absorption spectra which reflect the symmetry and size of the molecules. More details on identification of carbon molecules can be found in later chapters.

!11. CARBON MOLECULES IN SPACE Astrophysics is the discipline in which the spectra of carbon molecules played an early role. C2 and C3 occur in the coma of comets as a photolytic decay product of hydrocarbon precursor molecules. The cometary C3 band at 405 nm was discovered more than 100 years ago, but the molecular carrier remained unidentified until the work of Douglas [30]. Another astrophysical environment with interesting spectroscopic properties is the interstellar medium. In a highly diluted state, this medium hosts dust grains and molecules which produce characteristic absorptions. Most of the features in the UV-vis part of the spectrum could not yet be identified or satisfactorily explained. Starting at about 440 nm and extending into the red portion of the spectrum, a set of more than 200 absorption lines is especially intriguing: the diffuse interstellar bands, or DIBs for short [31, 32]. Very likely, most of the DIBs originate from molecular carders in which the cosmic abundant elements H, C, O, and N are involved. The detection of interstellar polyine species HC2n§ by radio astronomy suggested that linear chain molecules may be responsible for the DIBs. The distribution of the DIB-lines may be related to the known feature of linear molecules since the wavelength positions of their UV-vis absorptions increases with the chain length (see below). Douglas suggested pure carbon chain molecules as DIB carriers and estimated that the species C5 to C7 may be suitable candidates

[33].

In our early matrix-isolation work we found that some carbon species--which however at that time could not be properly identifiedmin fact have strong absorptions at about the DIB positions [34]. Again by matrix spectroscopy but based on a much larger set of data on various individual molecules, Maier and co-workers recently concluded that highly unsaturated hydrocarbon species like CnH m (with n = 14, 16 and m < 3) may be responsible for some of the DIBs [35]. Because of matrix shifts, a spectroscopic proof of this suggestion is difficult. Gas-phase data would be required to settle the intriguing problem of DIB-carrier identification. That such spectroscopy is feasible has been shown by the already quoted work of

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IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,A,TSCHMER

the Saykally group. Last but not least it should be mentioned that the IR absorptions of gaseous carbon species, e.g. C3, have been detected in the atmospheres of carbon-rich stars [36].

IV. EXPERIMENTAL SETUP Our experimental setup for studying absorption spectra of carbon molecules is shown schematically in Figure I. The apparatus consists of a carbon evaporator, a quadrupole mass spectrometer, and a matrix isolation chamber in which the cryogenic matrices are prepared. The chamber contains an inlet opening for the carbon sample molecules, a matrix-gas inlet, a cold substrate onto which the matrix is deposited, and windows for the source beam of a spectrometer to take the absorption spectra of the matrix. For thermal reasons, the substrate is a Rh-coated sapphire mirror contacted directly with the cold finger of the cryostat. Spectra of the coated substrate are taken in reflection. The required temperature (for argon about 10-20 K) is attained by a commercial two-stage Joule-Thompson refrigerator which uses helium as working gas. A thermocouple-controlled electrical heater is employed to reach and maintain elevated sample temperatures. The carbon molecules are produced by an evaporator in which graphite rods are heated by a high electrical current. In a repetitive mode, the rods are pushed together gently so that only the carbon in vicinity of the contact area evaporates. This method avoids excessive heating of the apparatus. We also apply a continuous evaporation mode, in which the flux of emitted carbon vapor is low but time-constant and thus more homogeneous matrices can be prepared. For producing isotopomeric species, commercially available 13C isotopically enriched carbon powder is sintered into rods and evaporated in the same fashion. We also stuffed highly enriched ~3C powder into hollowed out natural graphite rods, but found that the isotopomeric compositions of the produced molecules is not homogeneous. The composition of the carbon vapor and of the rest-gas in the apparatus is checked by a quadrupole mass spectrometer flanged between the evaporator and the matrix chamber. The matrix gas is usually applied in large excess, typically about 1000 times the amount of the sample. In case of the argon matrices, the deposition occurs between 10 and 20 K, and for thermal annealing, matrix temperatures are raised subsequently up to 40 K where argon sublimation starts. Spectra are taken at low temperatures before and after the annealing process and also at elevated annealing temperatures. Especially for UV studies, the lightscattering by the matrix has to be kept small and the matrix must be very carefully deposited. When the carbon evaporator is running, water vapor, CO, and CO2 may develop and contaminate the matrix. These molecules are known to form compounds or complexes with the rather reactive carbon species, leading to unwanted additional absorption lines and other distortions of the IR spectra (see, e.g. ref. 37 for H~O complexes with C3). Pressures of less than 10-8 are attained in the whole set-up,

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123

Figure 1. Schematical representation of the apparatus used for the matrix isolation of carbon vapor. Carbon vapor molecules are produced by evaporating graphite rods under vacuum (top). The molecules pass the ion source of the mass spectrometer, and are cocondensed with an excess of argon onto a cold (10-20 K) substrate (bottom). For further details, see text. the evaporator is carefully baked out, and a differential pumping stage between the evaporator and the cold substrate is applied to reduce the deposition of these species. Spectroscopy in the UV-vis and the IR could be performed on the same matrix sample; however we mainly concentrated our efforts in the IR domain. For the IR, we use a FT (Fourier-transform) spectrometer equipped with a liquid nitrogen

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IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KRATSCHMER

cooled In-Sb detector, a liquid helium cooled Cu-Ge detector, and a Ge bolometer which all together cover the spectral range 10,000-20 cm -1. Throughout our IR work we used resolutions between 0.1 and 0.05 cm -l, which is less than the molecular line width in the matrix. For the UV-vis range, a diode-array spectrometer was employed (200-900 nm) with a resolution between 2 and 0.1 nm.

V. CARBON MOLECULAR SPECTRA Small carbon molecules can be distinguished by their structure, i.e. divided into a linear and a cyclic family. Within such a family, there are common features which should also show up in the spectra. The spectra may also suggest to subdivide the families further, e.g. the family of linears into species with an even and odd number of atoms. The following chapter presents some ideas what the spectra of such families should look like. Naturally, emphasis will be on linear species since this family so far has been exclusively studied.

A. Stretching Modes in Linear Molecules For better understanding the spectra, some simple theoretical considerations may be appropriate. Halford [38] derived a formula for calculating the stretching mode eigenfrequencies coi of a linear chain consisting of n equal masses M. Each mass is assumed to be coupled to its nearest neighbor by an elastic spring. The spring constant D is the same for all springs. The formula reads:

O)i

=

2 q D / M sin ~n (i = 1, 2 , . . . , n - 1).

The eigenvectors which describe the atomic displacements for even i have ungerade symmetry and thus belong to IR-active vibrations. The case of odd i describes Raman active stretchings. Chains with more than four atoms have more than one IR-active vibration. While the stretching frequencies for long chains may become rather low, they cannot exceed the high frequency limit of 2~D/M. Halford's formula assumes uniform bonds and thus should model a cumulenic structure of the carbon chain, i.e. :C=(C--C)m=C:. The dangling bonds at both ends of the molecule indicate the reactivity, which is substantial even at cryo-temperatures and which gives rise to the already mentioned molecular growth within the matrices. Pozubenkov gave a formula which assumes two different kinds of springs arranged in an alternating manner [39]. This would concern even n molecules which might have an acetylenic bond arrangement, i.e. " C ( = C - C ) m ~ C ". The available IR data strongly suggest the cumulenic type of bond in both, in even as well as in odd n-chains. This conclusion may be drawn from Figure 2 which compares the known stretching frequencies with the predictions calculated by the Halford formula. In view of the simplicity of the approach, the agreement is surprisingly good.

Absorptions of Carbon Molecules

125

Figure 2. The frequencies of the IR-active stretching modes of a linear chain calculated from the elastic spring model of Halford [38] (solid lines). For comparison the known IR-band positions of linear carbon molecules are shown (broken lines).

For practical application however, the deviations between observed and calculated frequencies are intolerable. In the range between 2000 and 1500 cm -1, where most of the stronger IR lines occur, more accurate quantum-chemical calculations are required to get an idea of which molecule is responsible for a particular absorption. Figure 3 compares the known IR absorption frequencies with data using (a) an AM 1-Hamiltonian, and (b) a high level ab initio calculation. The results are scaled in a linear fashion to fit the data. Even the high level calculations spread about + 30 cm -1 around the data.

B. Electronic Spectra of Linear Species For describing the electronic transitions in linear Cn molecules, simple methods like e.g. the free-electron model or the Htickel method work surprisingly well. Both approaches explain the basic features (for comparison and connections with other

126

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~TSCHMER

Figure 3. The known IR-line positions of linear carbon molecules compared with the results obtained from quantum chemical semiempirical calculations using an AM1 hamiltonian (top) [57] and a B3LYP/cc-pVDZ ab initio method (bottom). The latter data are taken from ref. 28. The solid line is the best linear fit to the data, the dotted curves show the prediction interval.

Absorptions of Carbon Molecules

127

LCAO theories, see e.g. refs. 40-42). Originally designed for hydrocarbon chains with conjugated bonds, these models have to be slightly adapted to be applicable for pure carbon chains. In such a molecule, each carbon atom contributes six electrons, of which the two electrons in the K-shell are so strongly attached to the nucleus that they do not play a role in the molecular bonds. There are thus 4n electrons available for a molecule of n atoms, and 2n + 2 electrons provide the g-electrons which maintain the atomic framework of the chain molecule. This number includes the two "dangling bond" electrons at either end of the molecule. The g-electrons are assumed to be tightly bound and do not participate in the optical transitions of concern. The remaining 2n - 2 electrons move free and independently in n-orbitals. If the molecular axis points along the z-axis of a carthesian coordinate system, the lobes of the two equivalent molecular x-orbitals are oriented in the x and y direction, respectively. These orbitals are derived from the Px and py orbitals of the individual carbon atoms in the molecule. In total, the twofold degenerate molecular x-orbitals can accommodate four electrons. Com' pletely filled orbitals will occur when n -1 is an even number. Therefore, the molecules with an odd number of atoms will have a closed shell. The even species will exhibit half-filled orbitals. From Hund's rules follows that the former have singlet and the latter triplet ground states. With the exception of C2, which has a singlet ground state, this picture was confirmed experimentally [2]. To determine the approximate location of the energy levels of the molecule, the Htickel method may be applied. For a linear chain of n atoms the formula for the energy of a n-level is E = -or -213 cos [kn/(n + 1)], where k = 1, 2 . . . . . n numbers the (twofold-degenerated) re-levels, while a and 13are the HiJckel parameters. The odd n species exhibit completely filled levels up to k = (n -1)/2. The transition to the next higher level is dipole-allowed (E u ~ Eg) and yields a strong absorption band. Inserting the numbers into the formulas one can easily find that the transition e n e r g y is AE= 213 sin[rc/(n + 1)], w h i c h for larger n a m o u n t s to about AE = 2[Sn/(n + 1). The wavelength of the E. -~ ]~g band thus increases proportionally with the number n of atoms in the linear chain. The spectrum of a sample containing a range of odd n carbon molecules thus should show a pattern of absorption features equidistant in wavelength. Counting the absorption features from the blue to the red means counting the number of (odd) atoms in the absorbing chain molecule. For even n carbon molecules a similar trend between wavelength of absorption and molecular size should be expected. Since such species have half-filled rc-orbitals, their Eg ~ Eu transitions may result in more complex spectra. For carbon species, the linear dependence between n and the wavelength of the main absorption has recently been verified by Maier and co-workers who studied the spectra obtained from mass-selected species [24, 25] (see also Figure 11). Data which are of concern to our work are shown in Figure 6, depicting the spectrum of a heavily annealed matrix. Such samples contain a large variety of carbon species. Clearly, the most intense features show a regular spacing of about 40-50 nm. This is precisely the pattern expected for linear molecules. It will turn out that the main

128

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KRA,TSCHMER

absorptions (Figure 6) belong to odd n species. Even n molecules are also present in this sample but their absorptions are less striking (see below).

C. Bending Modes of Linear Molecules The bending modes of carbon species seem to be all low in energy. The far-IR range is difficult to study by matrix isolation and conventional spectroscopy since the required large amounts of sample molecules are not easy to prepare. Therefore, experimental data are sparse. The only known bending frequencies are 62 cm -1 for C3 [43], and about 172 cm -1 for Ca [44]. The substantially lower bending frequency of C3 (as compared to Ca) may originate from a particular orbital feature, namely the ungerade symmetry of its highest occupied n-orbital [45]. In this case, bending about the central atom is facilitated by an attractive overlap of the n-lobes of the wavefunction. The larger species having the same orbital feature are C7, Cll, etc. These molecules should also be very flexible around the central atom and thus should have very low-energy bending modes. Furthermore, a look at the wavefunctions of the species C3, C7, C11, etc. shows that these can steadily merge if both ends of the chain are connected into a cycle. All these properties may imply a very low-energy barrier for the conversion of linear C7, C 11, etc. into cyclic species [14].

D. Cyclic Molecules Plane, cyclic molecules should have a high enough symmetry to exhibit twofold degenerated IR-active modes [46, 47]. Very likely, distortions (e.g. deformation by matrix interaction) will remove this degeneracy giving rise to two adjacent IR bands. Such line pairs may serve as indicators for cyclic species. As far as the strengths of the IR absorptions are concerned, the more compact arrangement of atoms in a cycle suggests smaller dipole moments and thus also smaller IR intensities compared to linear cluster of the same size. The weakness of IR lines may be one of the reasons why cyclic structures have not yet been identified by absorption spectroscopy. The electronic transitions of cyclic carbon species are not known. In the Htickel or free-electron model, cyclic molecules are regarded as chains which are bent into themselves. Thus, like linear molecules, cyclic species should exhibit strong electronic transitions. Unlike the linear structure, the two n-systems in a plane cycle will not be degenerated. The in-plane and the out-of-plane n-system will exhibit its own energy levels and only in large tings with low curvature will both systems tend to merge energetically. The UV-vis spectra associated with the two different n-systems may be helpful in searching for cyclic species in the UV-vis domain. Cyclic, i.e. rhombic Ca, may be a special case different from the other cyclic species. According to theory, this species should be bicyclic: Two triangles are attached to each other via a carbon double bond stretching over the diagonal of the rhombus (see e.g. ref. 48). It certainly is a challenge for future research to prepare this and other cyclic species for absorption spectroscopy.

Absorptions of Carbon Molecules

VI.

129

EARLY RESEARCH

The carbon vapor species C 3 can be easily studied by matrix isolation. C 3 is the cartier of the cometary 405 nm band, which is a transition between the leg ground state and the ~I-luelectronically excited state. Matrix interaction shifts the band to 410.2 nm in argon and to 405.7 nm in neon matrices. Figure 4 shows the spectrum of trapped carbon vapor in an argon matrix in which the 410 nm C3 transition is dominating. The fine structure within the band originates from different trapping sites of the C3 molecule in the solid argon. The weaker features at the blue side of the 410 nm absorption are caused by transitions into bending vibrations of the excited lI-lu state. The weak features between 275 and 300 nm belong to a different electronic transition in C3. The band at 238 nm is a strong Mullikan band of C2 [49], and the feature adjacent to the red and centered at about 247 nm belongs to a so-far unknown larger carbon species. A possible cartier will be discussed below.

q_ v-,

I

o!._ o

I

I

I

I

I

I

I

I

I

I

o

AI- o OI cq._ O o vI~ u-, o,.i I/I c:~.Q i

m._ o

o

Qo._ o

i I

200

I

225

I

250

i

275

I

300

I

325

I

350

i

375

I

400

, ,,' i

425

I

450

Wavelength [nm]

Figure 4. The UV-vis absorption spectrum of carbon vapor trapped in solid argon at about 15 K. The strong band at 410 nm and the weaker features between 280 and 300 nm originate from C3. The absorption at 238 nm is the Mullikan band of C2. The carrier of the structured absorption centered at 247 nm is not known, but might be cyclic C6.

130

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,g,TSCHMER

Thermal matrix annealing leads to molecular growth within the matrix. The carbon molecules apparently migrate through the argon ice to an extent which is controlled by the applied annealing temperature. In encounters with other carbon molecules, chemical reactions occur and the initially abundant smaller species grow

Figure 5. Matrix spectra taken in the UV-vis range after the matrix was thermally annealed--i.e, warmed up--to the indicated temperatures. The decay of the C3 line and the rise of broad, structured absorption features shows that molecular growth processes take place within the matrix.

Absorptions of Carbon Molecules

131

to larger chains. After keeping the matrix for a short time at an elevated temperature level, the reactions cease and the spectrum becomes stationary. The growth process can clearly be recognized in the UV-vis and IR spectra. Figure 5 shows the UV-vis spectra of a matrix sample under increasing degrees of annealing. The 410-nm C3 line (and the less pronounced line of C2 at 238 nm) decays, while other broad absorption features emerge. Especially in the UV, these new bands show strong vibrational substructures. Apparently, atomic carbon C~ (which has no absorptions in the studied UV-vis range), and the molecules C2 and C3, are consumed by feeding the growth of larger species. Upon warming the argon matrices to about 35 K the new features grow further, while the C3 band continues to decrease. In each annealing step, the newly emerging bands appear further to the red. Features in the UV and blue appear in the early and red in the later states of matrix annealing. This indicates that the former absorptions belong to smaller and the latter to larger molecules.

Figure 6. The UV-vis spectrum of a matrix sample as in Figure 5, but after severe thermal annealing. In this process, large molecules are formed by chemical reactions between the species C, C2, and C3 which are abundant in carbon vapor. The C2 and C3 lines have completely disappeared and a variety of broad features has emerged. Each broad feature seems to belong to an individual linear molecule Cn with n > 3.

132

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~,TSCHMER

The spectrum of a highly annealed matrix sample is shown in Figure 6. The C 3 band has disappeared completely. From the different growth characteristics described above, one can distinguish a pattern of absorption features which have their centers at about 247, 311,348, 394, 447, 492, 529, 586, and 643 nm wavelength [50]. In view of very recent results it appears that each of these absorption features belongs to a different carbon molecule. In the early days, the situation was less clear since some of these bands, e.g. those at 311 and 348 nm, appeared to grow correlated. The wavelength spacing between the features is roughly regular, as can be noticed in Figure 6. Furthermore, the bands are rather strong. We reported these observations in our early work [50, 51] in which we suggested that the absorptions belong to the Z ~ E transitions of linear species. If one would have identified the carriers of a few of these features, the other could be assigned by simple feature counting. Even though the basic idea turned out to be correct, we were mistaken in the band assignments. Figure 7 shows a similar thermal annealing process, but recorded in the IR range. The spectra are taken from our early work done by Nachtigall [51, 52]. In the initial spectrum, the most intense line occurs at 2039 cm -l. Weltner and co-workers identified this absorption as the stretching vibration of linear C3 [17]. Upon annealing, the C3 absorption line decreases. Other lines grow or appear. Most striking is the strong increase of the feature at 1998 cm -l which becomes dominating in the later steps of matrix annealing. Figure 7 also shows that carbon vapor containing matrices which are freshly prepared and deposited at low temperatures already exhibit complex spectra. The additional IR lines originate from larger species Cn (n > 3). This finding indicates that molecular growth to some extent also occurs at low temperatures during matrix deposition, i.e. without thermal annealing. Matrix samples prepared with higher (lower) carbon concentration, show a higher (lower) amount of heavier species after deposition. Even at very low carbon concentrations the spectral signatures of larger species are distinctly present (Figure 4 shows the UV-vis spectrum of such a case). It appears that during the deposition of the carbon-argon mixture, the impinging carbon molecules can easily diffuse through the matrix surface, i.e. the interface between the growing argon ice layer and the stream of incoming argon gas. Under these conditions the probability of chemical reactions and molecular growth is enhanced. It is interesting to note that this kind of growth seems to lead to a population of molecules which is slightly different in composition from that obtained by thermal matrix annealing [53]. Early researchers tried hard to assign the various IR absorptions to molecules, but most of the assignments were incorrect and had to be revised later. Nevertheless, several important results emerged: Complete isotopic 12C-13C replacement experiments showed that most of the weaker and all of the stronger observed IR features originate from vibrations of pure carbon molecules [19]. For the 12Cn species these bands are located at about 2164 (C5), 2128 (C7), 2039 (C3), 1998 (C9), 1953 (C6), 1894 (C7), 1543 (Ca), 1447 (C5), and 1197 cm -1 (C6). For convenience, the

Absorptions of Carbon Molecules

133

C molecules in Argon matrix

05 0.4 0.3 0.2

~'

0.1-

o =

7" 0oz

0

I-"

@.. rY

0 u~ a~

0.2 0.1

0.3' (12 0.! 0.0

lio'o . . . . . . .

i~'oi) . . . . . . .

i6~ ........

iio'o . . . . . . .

id~ . . . . . . i~do "

WAVENUMBER (cm-11

Figure 7. The molecular growth process seen in the IR part of the spectrum (resolution 2 cm -1). The line at about 2039 cm -1 originates from C3 and is the IR-active stretching vibration v3. This line decreases under annealing. The growing lines belong to larger carbon species. Striking is the growth of the line at about 1998 cm -1 which comes from the 1,6 vibration of linear C9. The figure is from ref. 52.

now-established carriers are given in parenthesis. (The traditional v, numbering of the frequencies can be found in Table 1.) With the exception of the C3 band at 2039 cm -1, all the assignments were not known at that time. To identify the carders of the other IR absorptions, a search for intensity correlations between various IR bands was undertaken. First results were rather

134

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~TSCHMER

discouraging. Even the bands which (as was found out later) belong to the same molecule and thus ought to go together did not show striking correlations. Apparently, insufficient spectral resolution and matrix impurities severely degraded the data. However, one very strict correlation was found between the lines at about 1804 and 1844 cm -l [51]. Our new data confirm this correlation. These two bands probably belong to a yet unidentified species which may be cyclic; quantum chemical calculations suggest cyclic C8 or C ll as possible candidates [54, 55].

VIi. ASSIGNMENTS OF IR BANDS IN MATRICES In case the structure and composition of the trapped carbon species are not known beforehand, these have to be deduced from the spectra. One can realize immediately that the rather distinct IR absorptions are much better suited for such task than the broad and structured UV-vis features. Furthermore, the molecular vibrations are easier to interpret than electronic transitions. The method applied for identification consists of three steps, namely (a) the search for IR line correlations, (b) comparison of the spectral data with quantum chemical predictions, and (c) the study of the spectra of isotopomeric species. The identification is regarded as definitive if the data of (a)-(c) yield a consistent picture. Otherwise, the assignment is regarded as tentative. Group theory can predict the number of IR active modes; however it does not predict their frequency and strengths. Whereas the vibration frequency can be roughly estimated by a suitable "spring model" the latter have to be calculated by quantum chemistry. Figure 8 shows one of our recent IR spectra in which the so-far established IR modes are indicated. All of the major bands belong to IR active vibrations of linear species. Notice the very large variation in intensity, e.g. of the C9 lines. Of the four IR active stretching fundamentals (vs-Vs), the second vibration 1,'6at about 1998 cm -~ is by far the most intense line. The v8 stretching mode should be located at around 800 cm -t. It must be very weak, and has not yet been discerned. One notices that the IR line intensities all fall off very strongly with decreasing wavenumber. Thus, not all the stretching modes of larger species are easily accessible for intensity correlations. Correlations between lines can also be found by looking for peculiarities in line shapes, e.g. the occurrence of line splittings. Within a certain temperature regime of annealing, the absorption lines of the stretching vibrations of C5 (and also other species, like C8 and probably also C1 l) exhibit a splitting into several components. The splitting only occurs during the first warming and disappears irreversibly at still higher temperatures. Figure 9 shows this "site-effect" splitting--as we call it--for the v 3 line of C5 at about 2164 cm -~ [56, 57]. Similar splittings occur in all IR lines of C5, including its high frequency combination vibrations v 3 + v2 [57]. We believe that C5, C8, and C ll experience an exceptional strong interaction with the matrix. The commensurability between the size of the matrix atoms (argon) and that of the trapped species may provide an explanation: linear C5, C8, and C ll can

Absorptions of Carbon Molecules 1.6

,

.........

.........

C

- 9 1.2 -

I .........

I .........

I .........

I .........

I .........

I .........

I ........

,v

C6

I

c7

1.0 :

c7

.,~ 0.8 - c , .~

I .........

13 (~',9 [

1.4

i..

I .........

135

0.6

0

:

:

C9

c4

C13"9

~0.4 <

.

C9

C6

0.2 0.0

2200

2100

2000

1900

1800

1700

1600

1500

1400

1300

1200

1100

Wavenumber [cm I]

Figure 8. A recent high resolution (0.05 crn-1) IR spectrum of a thermally annealed argon matrix containing carbon vapor molecules and species already grown to larger sizes. Most of the intense absorptions originate from linear molecules, and the established identifications are shown at the respective lines. Further details can be found in Table 1.

closely replace a row of 1, 2, and 3 argon atoms, respectively. When the matrix atoms rearrange thermally, these carbon molecules may become strongly deformed. In any case, the splitting effect very much facilitates the search for correlated IR lines. For an identification of the cartier, the existence of line correlations supported by an agreement with theoretical predictions is a necessary but not a sufficient condition. Confirmation provides the analysis of the isotopomeric line pattern which is associated with each IR band. Isotopomers substantially increase the number of spectral lines. In case samples are prepared by a homogeneous mixture of comparable amounts of 12C and 13C, all isotopomeric lines show up in the IR spectra. For example, the linear species C3 shows 6, C5 shows 20, and C9 exhibits 272 different lines (for more details, see ref. 58). These lines are located in a relatively narrow frequency window between the isotopically pure 12C and J3C components at wavenumber co and, to a good approximation, o~q12/13. For larger species like C9 it is impossible to disentangle all the lines. For lighter species the conditions may be more favorable, and the IR lines of the molecules C3, C5, and C6 have been identified in this way [17, 59, 60]. We usually work with either natural graphite (about 1% 13C) or only moderately enriched carbon (10% 13C). Under

136

iVO CERMAK, GEROLD MONNINGER, and WOLFGANG KRATSCHMER ' ' ' t ' ' ' l ' ' ' l " ' ' ' l ' ' ' l ' ' ' ' l ' ' ' " '

1.2

1.1

1.0

38K

0.9

"~' 0.8 ,.0.7

~

0.6

.~ 0.s

3,

o

<

,~

0.4

0.3

-1

0.2

/ ~

0.1

0.0

9

2166

'

'

18K '

'

2164

'

'

'

'

2162

9

'

'

9

2160

'

~

'

'

2158

'

'

'

'

2156

'

'

'

'

2154

'

'

2152

Wavenumber [cm -1]

Figure 9. The

stretching absorption line of linear C5 during thermal annealing. For clarity, the spectra are shifted vertically. Note the two additional lines (site peaks) which emerge and disappear with increasing annealing temperature. The IR lines associated with stretching modes of certain other linear species like C8, and probably Cll also show such peculiar splitting into different components. The effect may be related to an especially strong interaction of trapped molecule and matrix. 1)3

these conditions, the intensities of the isotopomeric lines are reduced such that only molecules containing a few ~3C atoms produce detectable absorptions. The frequencies of isotopomeric lines can be either directly calculated by quantum chemical methods or by an appropriate "spring model" which describes the molecular vibrations. A simplification can be introduced by regarding the 13C isotope mass excess as a small perturbation. Sufficiently accurate eigenfrequencies are obtained from the eigenvectors by applying perturbation theory [61]. Considering the number and relative intensities of the isotopomeric lines, one can estimate the number of atoms of the absorbing molecule and get an idea about

Absorptions of Carbon Molecules

137

its possible structure. Assuming the molecule Cn is formed by 12C and 13C in a homogeneous manner, the abundance ratio between all monoisotopic variants 13C12C,,_l to that of the pure 12C,, species is just n times the lac/12C ratio. Provided that the strength of the absorption per molecule does not depend on its isotopic composition, the intensifies of the IR lines are proportional to the respective molecular abundances. The size n of the molecule can be readily obtained from the intensity ratio of all monoisotopomeric lines added together and the pure 12C,~line. For the two extreme cases of either linear or cyclic structure, the intensity pattern of isotopomeric lines differs greatly. Let R be the 13C/12C abundance ratio and I0 the intensity of the main line of the isotopic pure species. For the sake of simplicity, let us consider only the case where one 13C atom in the molecule is substituted. Because the linear carbon species have a center of symmetry (they belong to the point group Dooh), molecules with even n have n/2 isotopomeric lines of equal intensity 2.R.I o. Those with odd n have 1 + (n - 1)/2 lines, from which (n - 1)/2 lines have an intensity 2-R-10, and one line has half the intensity. This absorption belongs to the isotopomer where the 13C atom is located in the center (see e.g. the IR lines of C7 isotopomers in Figure 10). In case of a perfect cyclic carbon molecule C,, (point group Dnh), just one isotopomeric line of intensity n.R.I o would show up. The line pattern simply reflects the number of equivalent positions of atoms within the molecule. Cyclic species of lower symmetry, or nonlinear bent-chain molecules, may show intensity patterns intermediate between these extremes. In practice, intensity patterns are not easy to evaluate. Line maskings, background absorptions, and intensity anomalies limit the amount of useful and clear-cut information. We thus regard the data based on the numbers and positions of isotopomeric lines as more important and decisive for molecular identification, and use these in the first place. The conclusions drawn from line intensities we consider as supporting evidence.

VIII.

RECENT RESEARCH

From the beginning of the 1990s up to the present day substantial progress in the assignment and identification of carbon molecule absorptions has been reported. Major breakthroughs were achieved through laser spectroscopy of jet-cooled carbon clusters in the gas phase and through the matrix spectroscopy of massselected carbon species. We give an overview of the recent status of band assignments in the IR and UV-vis domain. The available IR data--which also contain our own results--are compiled in Table 1. Finally, we want to show the implications the new results have on the research of matrix-isolated carbon vapor.

A. IR Spectra Graham and co-workers found that linear C 4 is the carrier of the 1543.4 cm -1 IR line in argon matrices [12]. This is the v3 mode predicted by theory [27]. These

138

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~,TSCHMER

researchers prepared C 4 by UV photolysis of a mixture of argon and C4H 2 or C4H 4. The products of the photolysis were deposited in the matrix and analyzed. The spectra of the C4 species prepared in this way are identical to that of C4 obtained from carbon vapor deposition in matrices. Gas-phase cluster spectroscopy further confirmed the linear structure of this species [13]. The photolysis technique was also employed to detect the v5 bending mode of C4 at 172.4 cm -1 [44]. So far, no cyclic C4 isomer could be detected in matrix spectra. Vala and co-workers identified linear C5 as the cartier of the line at 2163.9 cm -I in the spectra of carbon vapor in argon matrices [59]. This assignment to the v 3 stretching mode of C5 was the first onewbesides the early identification of C3mwhich passed the test of time. The identification was achieved by evaporating 13C-enriched carbon and by counting the isotopomeric IR lines. The other stretching mode, u4, was found in later research at 1446.5 cm -~ [62]. In this work, the matrix preparation of C5 occurred in two ways: by photolysis of methyl-butadiene C5H8 and by carbon evaporation. At that time there was some confusion as to whether or not the line at 1543.4 cm -l is the second stretching mode of C5. The study in ref. 62 as well as theoretical work by Martin and co-workers [63] made clear that the 1543.4 cm -l carrier is Ca rather than C5. Gas-phase spectra obtained in the laboratory [64] and from observations of carbon-rich stars confirmed the linear structure of C5. Also noteworthy is the already mentioned "site-effect" splitting which we discovered [56, 57] and which is shown for the v3 vibration in Figure 9. For the C6 molecule one may intuitively assume a hexagonal structure like that of benzene. Calculations, however, show that the cyclic form of C6 should not have the perfect D6h symmetry but should rather look like an equilateral triangle with the sides bent outward, i.e. exhibit D3h symmetry [65]. In any case, argon-matrix spectroscopy clearly shows that the strong line at 1952.5 cm -l and the weaker band at 1197.2 cm -l belongs to C6 in a linear conformation [60, 66]. Also the pattern of isotopomeric lines shows that C6 is linear and the above frequencies are the two stretching vibrations v4 and 1,'5. Gas-phase spectra confirm this conclusion [67]. Two stretching frequencies v4 and v5 of linear C7 were identified by gas-phase cluster spectroscopy [68, 69]. These lines correspond to the long-known absorptions at 2127.7 cm -l and 1894.2 cm -l in solid argon, which could not be unambiguously identified by matrix absorption spectroscopy. Figure 10 shows the spectrum of isotopomeric lines around the v 5 feature of C7 which we obtained by evaporating regular graphite and enriched carbon. One recognizes the pattern expected for an odd linear species. One of the isotopomeric lines is hidden behind the main absorption of 12C7, making the identification of the line carrier difficult. The third stretching mode 1,'6of C7 should occur at around 1000 cm -1, but has not yet been identified. The gas-phase data show that C7 is extraordinary floppy, which is in accord with its filled ungerade x-orbital [70]. Data on C8 are sparse. Maier and co-workers deposited a mass-selected C8 beam in neon matrices and found two absorptions which were assigned to the 1,'5and V6 vibrations of linear C8 [24, 71]. In close vicinity, namely at 2071.6 cm -l and

Absorptions of Carbon Molecules

139

2.0 1.8 1.6 10% J3C

,---,1.4

~ ~

B

C

1.2

1.0

t___.__a

o

.-, 9 0.8 o

<

0.6

1%

0.4

13C

0.2

B

0.0 1895

1890

1885

E 1880

C 1875

1870

Wavenumber [cm l ]

Figure 10. The pattern of isotopomeric lines of the v5 stretching vibration of linear C7. Natural (1% 13C) and enriched carbon (10% 13C) was used as source material. For clarity, the two spectra are shifted vertically. The intensity distribution of the lines is characteristic for a linear species. The main line (A) belongs to the pure 12C7 isotopomer, while the lines (B)-(D) come from species, in which each molecule contains just one 13C atom. Line (E) originates from i s o m e r 12C3-13C-12C3 in which the central atom is substituted. Thus (E) has about half the intensity of (B) and (O. An evaluation shows that (B) originates from the arrangement 1 3 C - 1 2 C 6 , and (C) is coming from 12C-13C-12C5. The line (D) is hidden by the main absorption and corresponds to the isotoporner 12C2_13C_12C4" 1710.5 cm -1, we observed lines which exhibited a splitting effect--very similar to the case of C5. For this reason, we already suspected C8 as a cartier. The observed lines are located close to the positions suggested by theory (AM1 [57], and ab initio [28] calculations). The absorptions of C8 are usually low in intensity even after strong matrix annealing. This is in striking contrast to the rather intense line at 1997.8 cm -1 which earlier researchers erroneously assigned to C8 but which in fact belongs to C9.

140

lVO CERMAK, GEROLD MONNINGER, and WOLFGANG KRATSCHMER

Table 1, The IR-Absorptions of Linear Carbon Molecules in Matrices of Solid

Argon and in the Gas Phasea

IR Absorption [cm -1] Cluster C3

In Ar

Gas Phase

3245.2 2038.9

2040.0 63.4

C4

1699.8 1543.4

C5

1548.9

Confirmation of the Matrix Data Mode

Calculation Correlation Isotopes

Other

References

v3 + v I

+

+

76

v3

+

+

17, 36

v2

43

v5 + v3

+

v3

+

77 +

44,13

172.4

I,'5

2939.1

V3 + V2

+

v3

+

+

+

v4

+

+

+

62

1559.9

v4

+

+

+

60, 67

2163.9

2169.4

1446.5

44 site peaks

57 59, 64

C6

1952.5

v5

+

+

+

66

C7

2127.7

2138.3

v4

+

+

+

68

1894.2

1898.4

v5

+

+

+

C8

2071.6

v5

+

+

site peaks

24, 57

171.0.5

v6

+

+

site peaks

71, 57

v5

+

+

71, 57

v6

+

+

71, 72

1197.2

C9

2078.0 1997.8

Clo CII

C12 C13 Note:

2014.3

69

1600.8

v7

+

+

2074.6

v6

AM 1

+

1915.5

v7

AM1

+

1945.9

v7

AM1

+

site peaks

57

1856.5

v8

AM1

+

site peaks

57

1360

v9

AM1

site peaks

57

2013.0"

v8

mass

24

1819.5"

v9

selection

24

1809.0

v9?

+

61 24, 57 57

15

aThe data are taken from the literature and from our own research. The identification methods are indicated. In case no argon matrix data are available, neon matrix measurements are taken from the literature and marked by an *

Absorptions of Carbon Molecules

141

The very intense rise of the 1997.8 cm -l absorption band upon thermal annealing and its dominating strength thereafter has puzzled researchers from early on. Matrix spectroscopy failed to identify the mysterious carrier of this line. The extended width and the fuzzy and complex isotopomeric line pattern renders an interpretation difficult. However, gas-phase cluster work and mass-selected beam deposition studies unambiguously identified the carrier as linear C9 (mode v6) [71, 72]. With the latter method, also a weaker feature at 2078.0 cm -1 could be detected and this apparently is the v5 stretching mode with the highest frequency [71]. Intensity correlations suggested that a third line located at 1600.8 cm -l also belongs to linear C9 (vibration vT) [61]. Compared to the other bands, this line shows a much clearer isotopomeric line pattern and the carrier could be identified as C9. The low energy difference between the u5 and v6 modes and the predicted vicinity of other Raman active modes leads to a distortion of these high-frequency stretching modes of C9 [61, 73] and makes the interpretation of the isotopomeric absorption spectra extremely difficult. Very recently, spectral data for the molecules Cl0 to C12 became available. Tentative assignments were given by Maier and co-workers for C10 and C12 in neon [24]. Our argon matrix work also suggests assignments for Cl0. From line splittings we suspect that we also have identified linear C11 in our spectra. These recent and tentative data are included in Table 1. The gas-phase detection of linear C13 by Saykally and co-workers has already been mentioned [15]. The identification of the molecule was achieved from analysis of a vibration-rotation band at 1809.0 cm -~. From theory it is quite unexpected that such a large cluster exists in linear form rather than in the form of a monocyclic ring. In argon matrices, the only close-by line with reasonable intensity is at 1817.7 cm -1. One would be tempted to identify this line with an absorption by C13; however, the rather small and positive (!) matrix shift renders such an assignment unlikely. The line shifts of all known carbon species are negative and increase with the molecular size [70]. The absorption lines of linear C 13in solid argon thus remain to be detected.

B. UV-vis Spectra Neon-matrix-isolation experiments performed by Maier and co-workers on mass-selected species show that there is a clear linear correlation between the wavelength of the E ~ E band and the molecular size. These findings, compiled into a diagram are shown in Figure 11. The strong Eu ~ Eg absorptions of the odd n molecules move from 250 nm for C7 to 420 nm for C15, and the wavelength increment between subsequent odd features is 40-45 nm [25]. The corresponding E g ~ E, band spectra of the even n species are distinctly different in appearance. These features are much more dispersed in wavelength and move from 380 nm for C4 to 980 nm for Cl4. The wavelength increment between subsequent even species amounts to about 100-130 nm [24, 26].

142

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~TSCHMER

1000

evenn

/

odd n 80O E

600

-

0

o~,,~

~400 ~

Eu-'~Eg

200

0

5

10 n

15

20

Figure 11. The band positions of the s -~ T_.gtransitions in even and T_.g--~ T_.u in odd

n linear Cn molecules in neon matrices. The data are taken from studies of Maier and co-workers (refs. 23-26). For odd-n-species, the band maxima, and for even-n-species the band origins were taken. Note the linear dependence on the molecular size.

To translate the above neon data into the corresponding wavelengths in argon matrices, an additional matrix shift has to be invoked. The shift goes into the red and seems to be cluster-size dependent, ranging from 5 nm for C3 to about 25 nm for C15. On the basis of these facts, the features shown in Figure 6 might all be assigned to odd species: C9 (311 nm), Cll (348 nm), C13 (394 nm), and C15 (447 nm). The absorptions of the even species are very hard to discern. Three weak narrow lines at 456, 470, and 520 nm (see Figure 5) belong to linear C6 and are the most distinct features of all even n molecular absorptions. The other intense bands at 492, 529, 586, and 643 nm (see Figure 6) fit well into the pattern of the odd species. However, it remains to be seen whether these bands belong to still longer chains. A feature count would suggest that the relatively intense band at 586 nm belongs to linear C21! Provided this assignment could be confirmed, it would imply

Absorptions of Carbon Molecules

143

that the range of linear carbon molecules extends far more than expected. However, the rather special cryogenic matrix environment may stabilize such structures which in free state would rearrange into cycles or would decay.

C. Correlations of UV-vis and IR Absorptions In IR and UV-vis, the matrix spectra of linear carbon species up to C 9 are now sufficiently well known to look for absorption features which do not belong to this family. For this kind of research, correlation studies between UV-vis and IR are valuable. Kurtz and Huffman performed such work for the first time in order to characterize the carriers of the strong UV-vis bands in argon [53]. These authors found an intensity correlation between the 1997.8 cm -1 line and the 311 nm feature. This finding, confirmed by a similar study of Vala and co-workers [73], passed the test of time and it is now firmly established that linear C9 is the carrier of both bands. There are also other correlations reported by Vala and co-workers which concern weaker UV-vis absorptions of linear C5 and C3 with their respective IR bands [73]. Unexpected in the light of recent results is the reported correlation between the already discussed 586-nm band and a relatively intense IR feature at 1695 cm -1. The 586-nm band may originate either from a cyclic cluster Cn with n > 6--as assumed by Vala and co-workersmor from a large linear molecule C21mas the feature count suggests. One of the most intense and striking UV-vis absorption bands which emerges in matrix-isolated carbon vapor is centered at 247 nm. As noticed already by the early researchers, this band appears almost inevitably along with C3 (see Figure 4) and grows strongly during the first annealing steps [50, 53]. In neon matrices, this feature peaks at 236 nm [75] and is not coincident with the 253-nm absorption of linear C7 [25]. C7 as a carrier is further ruled out by the absence of an intensity correlation between the 247-nm band and the known IR lines of linear C7. However, some loose correlation with the UV-vis and IR absorptions of linear C6 seems to exist [51, 53], indicating that linear C6 and the 247-nm carrier may be related, e.g. produced in similar processes. Since Maier and co-workers could definitely exclude linear C6 as the 247-nm band carrier [23], one may speculate that cyclic C6 is a reasonable alternative. Cyclic C6 might be easily formed by the conjunction of two C3, which are known to be very flexible. The lack of any striking correlations indicates that the IR lines of the 247-nm carrier are weak or hidden behind other absorptions (as suggested in ref. 74). It certainly would be rewarding to resume the correlation studies by including also rather weak IR lines. The quenching of carbon vapor in a hot helium or in other rare-gas atmospheres leads to formation of fullerene clusters. In our matrix work, we never could detect any trace of C60, neither in the spectra nor in the recovered soot. Thus it seems that the clustering in the cold matrix follows reaction pathways different from that in a hot and gaseous environment. Very likely, the molecular rearrangement and restruc-

144

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KR,~TSCHMER

turing processes which are essential for fullerene formation can readily occur in a hot quenching gas but not in a cryogenic matrix.

IX. OUTLOOK The task of studying carbon molecules in their anticipated linear, cyclic, or other isomeric structures requires that sufficiently different preparation methods have to be applied. As far as linear and cyclic isomers are concerned, entropy favors the formation of linear carbon species at high temperatures. Carbon vapor molecules certainly are formed in a "hot" process and thus are mainly linear. "Cold" carbon molecule production processes may allow the cyclic conformation to dominate over the linear form. Such production methods may start from clusters--e.g, from fullerene molecules--and use these as raw-materials. Fullerenes sublime easily and it may be possible to break them into interesting molecular pieces. Heavily fragmented fullerenes should at least give an interesting source of C2 molecules [78]. For future matrix work, not only do other preparation techniques have to be applied, but also a better definition of the trapped species should be attempted. The use of mass-selected molecular beams is one possibility. However, the ionic species used in the conventional mass selection already predetermines the structure of the subsequently obtained neutrals. Furthermore, the relatively large amounts of sample required for IR spectroscopy are difficult to collect. It may also be rewarding to apply more selective spectroscopic methods. For example, there is the possibility to optically excite or chemically activate a certain carbon species and to observe the response or decay of this species by the changes in the spectra. We are presently pursuing this kind of research. Last but not least, matrix work helps to elucidate the chemistry among pure carbon molecules at low temperatures. Starting the molecular growth with, e.g., a mixture of pure 12Cn and 13Cn species, the IR spectroscopy of the obtained isotopically substituted products may yield additional information about the reaction kinetics in the cryogenic matrices.

ACKNOWLEDGMENTS We thank John E Maier (Universit~t Basel) and his co-workers for sending us their data prior to publication, and for many helpful discussions. The engagement of W. Schulze (FritzHaber-Institut Berlin) in preparing and measuring neon matrices is kindly acknowledged. We thank Mrs. Cermak for helping in the data evaluation. We are grateful to the Deutsche Forschungsgemeinschaft for generous financial support.

REFERENCES 1. Kroto,H. W.; Heath, J. R.; O'Brien, S. C.; Curl, R. E; Smalley,R. E. Nature 1985, 318, 162.

Absorptions of Carbon Molecules 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

145

Weltner, W. Jr.; Zee, van R. J. Chem. Rev. 1989, 89, 1713. Kr~itschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990, 347, 354. Bowers, M. T.; Kemper, P. R.; Helden, von G.; Koppen, van P. A. M. Science 1993, 260, 1446. Jarrold, M. E; Bower, J. E. J. Chem. Phys. 1992, 96, 9180. Pitzer, K. S.; Clementi, E. J. Am. Chem. Soc. 1959, 81, 4477. Hoffmann, R. Tetrahedron 1966, 22, 521. Helden, von G.; Hsu, M. T.; Gotts, N.; Bowers, M. T. J. Phys. Chem. 1993, 97, 8182. Shelimov, K. B.; Clemmer, D. E.; Hunter, J. M.; Jarrold, M. E The Chemical Physics of Fullerenes 10 (and 5) Years Later, Andreoni, W., Ed. Kluwer Academic Publishers: Dordrecht/Boston/London, 1996, p. 71. Handschuh, H.; Gantef/Sr, G.; Kessler, B.; Bechtold, P. S.; Eberhardt, W. Phys. Rev. Lett. 1995, 74, 7, 1095. Algranati, M.; Feldman, H.; Kella, D.; Malkin, E.; Miklazky, E.; Naaman, R.; Vager, Z.; Zajfman, J. J. Chem. Phys. 1989, 90, 8, 4617. Shen, L. N.; Graham, W. M.J. Chem. Phys. 1989, 91, 8, 5115. Heath, J. R.; Saykally, R. J. J. Chem. Phys. 1991, 94, 4, 3271. Heath, J. R.; Saykally, R. J. On Cluster and Clustering; Reynolds P.J., Ed. North-Holland: Amsterdam, London, New York, Tokyo, 1993, p. 7. Giesen, T. E; Orden, van A.; Hwang, H. J.; Fellers, R. S.; Provencal, R. A.; Saykally, R. J. Science 1994, 265, 756. Weltner, W. Jr.; Walsh, P. N.; Angell, C. L. J. Chem. Phys. 1964, 40, 5, 1299. Weltner, W. Jr.; McLeod, D. Jr. J. Chem. Phys. 1964, 40, 5, 1305. Weltner, W. Jr.; McLeod, D. Jr. J. Chem. Phys. 1966, 45, 8, 3096. Thompson, K. R.; DeKock R. L.; Weltner W. Jr. J. Am. Chem. Soc. 1971, 93:19, 4688. Sorg, N. Diploma work, University of Heidelberg, 1984. Szczepanski, J.; Pellow, R.; Vala, M. Z Naturforsch. 1992, 47a, 595. Maier J. P. Mass Spectrom. Rev. 1992, 11, 119. Forney, D.; Fulara J.; Freivogel, P.; Jacobi M.; Lessen D.; Maier, J. P. J. Chem. Phys. 1995, 103, 1, 48. Freivogel, P.; Fulara, J.; Jakobi, M.; Forney, D.; Maier, J. P. J. Chem. Phys. 1995, 103, 1, 54. Forney, D.; Freivogel, P.; Grutter, M.; Maier, J. P. J. Chem. Phys. 1996, 104, 13, 4954. Freivogel, P.; Grutter, M.; Forney, D.; Maier, J. P. Chem. Phys. Lett. 1996, 249, 191. Michalska D.; Chojnacki, H.; Hess, B. A. Jr.; Schaad, L. J. Chem. Phys. Lett. 1987, 141,376. Martin, J. M. L.; E1-Yazal, J.; Francois, J. P. Chem. Phys. Lett. 1995, 242, 570. Martin, J. M. L.; El-Yazal, J.; Francois, J. P. Chem. Phys. Lett. 1996, 252, 9. Douglas, A. E. Astrophys.J. 1951, 114, 466. Herbig, G. H. Annu. Rev. Astrophys. 1995, 33, 19. Jenniskens, P.; D6sert, E X. Astron. Astrophys. Suppl. 1994, 106, 39. Douglas, A. E. Nature 1977, 269, 130. Kr~itschmer, W. Astrophysics and Space Science 1986, 128, 93. Freivogel, P.; Fulara, J.; Maier, J. P. Astrophys. J. 1994, 431, L151. Hinkle, K. H.; Keady, J. J.; Bernath, P. E Science 1988, 241, 1319. Szczepanski, J.; Ekern, S.; Vala, M. J. Chem. Phys. 1995, 99, 8002. Halford, J. O. J. Chem. Phys. 1951, 19, 11. Pozubenkov, A. E Soviet Physics-Doklady 1966, 10, 7, 600. Ruedenberg, K.; Scherr, Ch. W. J. Chem. Phys. 1953, 21, 9, 1565. Platt, J. R. Encyclopedia ofPhysics; Fltigge, Ed. Springer: Berlin, 1961, Vol. 37, Chap. 2, p. 173. Heilbronner, E.; Bock, H. Das HMO Modell und seine Anwendung--Grundlagen und Handhabung. Chemie GmbH: Weinheim/Bergstr, 1968. Schmuttenmaer, C. A.; Cohen, R. C.; Pugliano, N.; Heath, J. R.; Cooksy, A. L.; Busarow, K. L.; Saykally, R. J. Science 1990, 249, 897.

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44. Withey, E A.; Shen, L. N.; Graham, R. M. J. Chem. Phys. 1991, 95, 2, 820. 45. Fan, Q.; Pfeiffer, G. V. Chem. Phys. Lett. 1989, 162, 6, 472. 46. Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules. D. van Nostrand Company, Inc.: Princeton, 1962. 47. Raghavachari, K.; Binkley, J. S. J. Chem. Phys. 1987, 87, 2191. 48. Magers, D. H.; Harrison, R. J.; Bartlett, R. J. J. Chem. Phys. 1986, 84, 6, 3284. 49. Milligan, D. E.; Jacox, M. E.; Abouaf-Marguin, L. J. Chem. Phys. 1967, 46, 12, 4562. 50. Kr~itschmer, W.; Sorg, N.; Huffman, D. R. Surface Science, 1985, 156, 814. 51. Kr~itschmer, W.; Nachtigall, K. Polycyclic Aromatic Hydrocarbons and Astrophysics; Leger, A. et al., Eds. D.Reidel Publishing: Dodrecht, Boston, Lancaster, Tokyo, 1987, p. 75. 52. Nachtigall, K. Diploma-Work, University Heidelberg, 1987. 53. Kurtz, J.; Huffman, D. R. J. Chem. Phys. 1990, 92, 30. 54. Martin, J. M. L.; Taylor, P. R. J. Phys. Chem. 1996, 100, 6047. 55. Martin, J. M. L.; Francois, J. P.; Gijbels, R.; Alml/Sf J. Chem. Phys. Lett. 1991, 187, 4, 367. 56. Monninger, G. Ph. D. Thesis, University Heidelberg, 1995. 57. Cermak, I.; Monninger, G.; Kr~itschmer, W., in preparation. 58. Pellow, R.; Vala, M. Z. Phys. D.--Atoms, Molecules and Clusters 1990, 15, 171. 59. Vala, M.; Chandrasekhar, T. M.; Szczepanski, J.; Van Zee, R.; Weltner W. Jr. J. Chem. Phys. 1989, 90, I, 595. 60. Vala, M.; Chandrasekhar, T. M.; Szczepanski, J.; Pellow, R. Science 1990, 27, 19. 61. Kranze, R. H.; Withey, P. A.; Rittby, C. M. L.; Graham, R. M. J. Chem. Phys. 1995, 103, 16, 6841. 62. Kranze, R. H.; Graham, R. M. J. Chem. Phys. 1992, 96, 4, 2517. 63. Martin, J. M. L.; Francois, J. P.; Gijbels, R. J. Chem. Phys. 1989, 90, 6, 3403. 64. Heath, J. R.; Cooksy, A. L.; Gruebele, M. H. W.; Schmuttenmaer, C. A.; Saykally, R. J. Science 1989, 244, 564. 65. Raghavachari, K.; Whiteside, R. A.; Pople, J. A. J. Phys. Chem. 1986, 85, 11, 6623. 66. Kranze, R. H.; Graham, W. R. M. J. Chem. Phys. 1993, 98, 71. 67. Hwang, H. J.; Orden, van A.; Tanaka, K.; Kuo, E. W.; Heath, J. R.; Saykally, R. J. Molecular Physics 1993, 79, 4, 769. 68. Heath, J. R.; Sheeks, A. L.; Coosky, A. I.; Saykally, R. J. Science 1990, 249, 895. 69. Heath, J. R.; Orden, van A.; Kuo, E.; Saykally, R. J. Chem. Phys. Lett. 1991, 182, 1, 17. 70. Heath, J. R.; Saykally, R. J. J. Chem. Phys. 1991, 94, 3, 1724. 71. Forney, D.; Grutter, M.; Freivogel, P.; Maier, J. P. Proceeding SASP; Maier, J.P.; Quack, M., Eds. 1996, p. 106. 72. Heath, J. R.; Saykally, R. J. J. Chem. Phys. 1990, 93, 11, 8392. 73. Orden, van A.; Hwang, H. J.; Kuo, E. W.; Saykally, R. J. J. Chem. Phys. 1993, 98, 9, 6678. 74. Szczepanski, J.; Vala, M. J. Phys. Chem. 1991, 95, 2792. 75. Schulze, W.; Kr~itschmer, W.; unpublished data, 1986. 76. Szczepanski, J., Vala, M. J. Phys. Chem. 1993, 99, 7371. 77. Shen, L. N.; Withey, P. A.; Graham, R. M. J. Chem. Phys. 1991, 94, 4, 2395. 78. Gruen, D. M.; Zuiker, C. D.; Krauss, A. R. SPIE, Fullerenes and Photonics H 1995, 2530, 2.

SPECIFIC INTERMOLECULAR INTERACTIONS IN ORGANIC CRYSTALSCONJUGATED HYDROGEN BONDS AND CONTACTS OF BENZENE RINGS

Petr M. Zorky and Olga N. Zorkaya

Io II. III.

IV.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugated Hydrogen Bonds Illustrated by Crystalline Derivatives of Hydroxy- and Dihydroxybenzene . . . . . . . . . . . . . . . . Classification o f Contacts and Aggregates of Benzene Rings . . . . . . . . . A. Types of BzC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Types o f BzC Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . C. BzC in Crystalline Derivatives of Benzene, Phenol, and Dihydroxybenzenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 147-188 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

147

148 148 154 163 164 170 172 187 188 188

148

PETR M. ZORKY and OLGA N. ZORKAYA

ABSTRACT Two types of specific intermolecular interactions playing an important role in the formation of organic crystal structures are described. Those are conjugated hydrogen bonds (CHB) and some special (particularly advantageous) arrangements of benzene rings among different benzene contacts (BzC). The objects of the analysis were mainly the benzene derivatives that contain hydroxyl groups and alkyl and alkoxyl radicals. The analysis has revealed the typical systems of CHB: spiral chains and cyclic systems; their topological and geometric characteristics are considered. The role of these systems in the formation of the crystal structure is discussed. A classification of "effective" contacts of benzene tings has been worked out. Ideal and shifted sandwiches, some dihedral and orthogonal arrangements, are regarded as the basis types of BzC. The possible distortion of the basic types are described. The benzene ring agglomerates generated by contact of various types are listed; the most important among them are zigzag chains, parquet layers, chains and layers of ~ orthogonal BzC, and stacks. A system of parameters for a quantitative description of the variety of BzC is proposed. The results of the estimation of energy of these contacts using atom-atom potentials are presented. A close examination of specific intermolecular contacts in derivatives of benzene, phenol, and dihydroxybenzenes was carried out. It is shown that in some cases the combination of systems of CHB and BzC leads to some extremely interesting intricate molecular arrangements, which by no means fit into the concept of close molecular packing. An important conclusion is that the symmetry of benzene rings often manifests itself in molecular subsystems though it is virtually always absent in the crystal structure as a whole.

!. I N T R O D U C T I O N It is noteworthy that the term "packing" which not so long ago used to be very common in descriptions of molecular arrangements in crystals, has gradually become more and more rare. Indeed, one can rightfully speak about molecular packing only when the filling of space by voluminous bodies representing individual molecules is actually considered. As a rule, such consideration did not take place. However, the concept of the close packing, established by Kitaigorodskii's books, has occupied such a stable place in organic crystal chemistry that sometimes it is automatically implied without good reasons even when there is no need for it. Most often it would suffice to speak about a spatial arrangement of molecules. In recent years the interest in intermolecular interactions and in the structures generated by them (it is not only crystals, liquid crystals, and liquids, but also other supramolecular formations, e.g. clusters) has continually increased partly due to the development of supramolecular chemistry. In this new stratum of scientific publications we no longer come across the habitual word "packing" as an ordinary term.

Formerly something of the kind was taking place in inorganic crystal chemistry. Originally the detection of the densest sphere packings appeared to be very

Specific Intermolecular Interactions

149

promising. However, it has been soon found out that the vast majority of inorganic crystals have more complicated structures, which in no way can be modeled by regularly packed spheres. The concept of close packing was found to be not very efficient as applied to organic crystals, too. The point is that compact packing is, at best, a necessary but not sufficient criterion of the optimum structure of a molecular crystal. Already in early works [1-3] we were able to demonstrate, using some chelate compounds of nickel and copper as examples, that there is a multitude of arrangements of space-filling molecules with approximately the same coefficient of packing density, and that by no means all of them are realized as optimal. The minimization of energy calculated using atom-atom potentials also does not, as a rule, lead to an unambiguous prediction of a molecular arrangement. The multidimensional surface of potential energy calculated by this method has many minima that are close in depth. For this reason alone the selection of the preferable structure becomes very complicated (even not to mention the necessity of taking into account thermal motion and other factors). The vast experimental material on molecular arrangements, e.g. the data contained in the Cambridge Structural Database (CSD), shows that an important and often decisive role in the formation of a crystal structure is played by specific intermolecular interactions. In condensed systems, i.e. in systems that are formed by closely spaced molecules, such interactions manifest themselves in specific intermolecular contacts. The best known of them are hydrogen bonds. Many structures clearly show the influence of contacts: halogen-halogen, sulfur-sulfur, metal-oxygen, and others. The study of specific intermolecular contacts leads to a new approach to the structure of the organic crystal [4]. It is viewed not as a close packing of three-dimensional bodies but as a spatial arrangement of molecules stitched by specific intermolecular contacts. While considering molecular arrangements, one deals with molecule-molecule contacts and with atom-atom contacts, and both these types of contacts are called "intermolecular contacts." We shall also use this term on the assumption that in each specific case one can understand the exact meaning from the context. The rigorous definition of the specific intermolecular contacts (as well as of most of fundamental chemical notions [5]) appears to be unrealizable. It is unlikely that any simple energy criterion can play a decisive role here. Eventually the energy of the phase being formed is somewhat more important than the energy of individual contacts; even without considering cooperative effects (i.e. non-additivity) one must take into account the whole set of realized atom-atom interactions, not just some selected contacts even if they are particularly advantageous. Besides, detailed analysis of a wide range of structural data shows that the optimum phase state of a substance realized under given conditions only slightly differs in energy from several other hypothetical phases, which are almost equally advantageous. At the same time, no reliable and precise method of energy calculation is available at present for complicated structures.

150

PETR M. ZORKY and OLGA N. ZORKAYA

Probably the best way of establishing the specificity of an intermolecular contact is to detect frequent recurrence of a given local structure (in similar and in completely different situations). Of course the energy justification will ultimately become necessary, but with the current state of our calculation resources such a justification can succeed only if the answer is known in advance. Not infrequently specific intermolecular contacts are conjugated; i.e. they are immediately adjacent to one another, and this gives rise to aggregates of such contacts. Moreover, specific intermolecular contacts (and their aggregates) cause the formation of molecular agglomerates, the study of which appears to be exceptionally promising since fragments of such agglomerates must survive in liquid phases (in solutions and melts of organic substances). The following circumstance, which is very important methodologically, should be taken into account. Nowadays symmetrical and topological characteristics of a structure as a whole or of its large substructures, among which are the aforementioned agglomerates and aggregates [6, 7], prove to be more helpful than local geometrical and energy characteristics of individual bonds or of other fragments of the structure when studying atomic and molecular systems and their properties. Such a consideration gives the most reliable, even if indirect, information on the structuring role of fragments of a substance, in particular of specific intermolecular contacts. The specific intermolecular contacts of two types are discussed here: conjugated hydrogen bonds (CHB) and specific contacts of benzene tings. Such interactions have been mentioned by Desiraju [8]. However, a closer analysis of the problem seems to be useful. The list of the considered structures is given in Tables 1-3. From CSD we have picked all homomolecular crystalline substances with the formula C6+kH6+2kOre, where k = 0, 1. . . . . 18; m = 0, 1, 2, and which are: 1. hydrocarbons containing a six-membered aromatic ring and alkyl radicals with no more than five C atoms (m = 0); 2. similar compounds, in which substituents contain alkoxyl groups, again with no more than five C atoms, besides (or instead of) alkyl radicals (m = 1, 2); 3. phenol derivatives which, besides the hydroxyl group, contain the above substituents (m = 1, 2); and 4. the derivatives of dihydroxybenzenes with alkyl substituents (m = 2). The substances belonging to types 1 and 2 have been used to study benzene contacts (BzC); they are listed in Table 3. The substances belonging to types 3 and 4 (Tables 1 and 2) have been used for the investigation of both conjugated H-bonds and B zC.

Table 1. Conjugated H-Bonds and Contacts of Benzene Rings in Crystalline Derivatives of Phenol a

Substance

Refcode CSD

Structural C l a s s

System of Conjugated H-Bonds

6

C6HsOH

PHENOL03

P21, Z = 6(13)

[[pctx)31] ]

7

7

C6HsOH (meta) (CH3)C6H4OH (para)

MCRSOL CRESOL01

P21[c, Z= 24(16) P21]C, Z = 8(12)

[Pc(y)31] [4/m]

7 7 8 8 8 8

(CH3)C6H4OH (ortho) (CH30)C6H4OH (para) (CH3)2C6H3OH (2,3) (CH3)2C6H3OH (2,5) (CH3)2C6H3OH (2,6) (CHa)2C6HaOH (3,4)

OCRSOL MOPHLC DIMPHE10 DMPHOL 11 DMEPOLI0 DPHNOL 10

P3 I, Z = 9(13) P21/c, Z = 4(1) P212121, Z= 4(1) P2 l, Z = 2(1) P21/c,Z=4(1) P], Z = 6(13)

Pc(z)3 l Pc(y)21 Pc(z)21 Pc(y)2 i Pc(y)21 [[6/m]](])

(iso-C3HT)C6H40H(para) (CH3)(iso-C3H7)C6H30H(3,4) (CH30)(CH3)4C6HOH (4,2,3,5,6) (iso-C3H7)2C6H30H(2,6) (CH3)(tert-C4H9)2C6H2OH (4,2,6) (CH3)(tert-C4H9)2C6H2OH (4,2,6) (CH30)(tert-C4H9)2C6H2OH (4,2,6)

IPRPOL MIPHOL MOPHLA GAPTOG MBPHOL01 MBPHOL10 MOPHLD

P4 l, Z = 4(1) P4 l, Z = 4(1) P211c, Z = 4(1) C2/c, Z = 16(12) C2/c, Z = 8(1) P212121, Z= 4(1) e21/C, Z= 4(1)

Pc(z)41 Pc(z)41 Pc(y)cl [4/m]

9 10 11 12 15 15 15

Note: anmnumber of atoms C.

Effective Contacts of Benzene Rings In H-aggl.

BetweenH-aggl.

BzC aggr.

Ob(2q)l, Ob(2q)2

m

S(m2) S/xy S(ml) S(ml)

{Ob(21) }, [[PL]], TW Ob(2q) {Ob(21 ) } O(2)/h, S(ml)/h, Complex S/xy, OB(2q) framework O(4)/s, O(4), O(1)/s Spiral Pr ) Ob(2q) {S(m 2) }, PL -{S/xy } -{S(mt) } --

S(m2), S/xy, O(4)/s D

S/xy

Ob(2q) Ob(2q) --

{S(ml)

}

Complex framework {Ob(21) } {Ob(21) } {S/xy}

Table 2. Conjugated H-Bonds and Contacts of Benzene Rings in Crystalline Derivatives of Dihydroxybenzenes a

Substance

Refcode CSD Structural Class

C6H4(OH) 2 (para)

HYQUIN02

R3, Z = 54(13)

C6H4(OH) 2 (para)

HYQUIN05

C6H4(OH)2 (para) ~l C6H4(OH)2 (meta)

HYQUIN

System of Conjugated HBonds

Effective Contacts of Benzene Rings In H-aggl. O(2)/h b

R3 Z = 9(1)

[6/m1(3) & Pctz)31(15) [6/m] (3)

P211c, Z = 4(-f2)

Pc(y)21

S(m 2)

RESORA13

Pna21, Z = 4(1)

Ob(2q)

C6H4(OH) 2 (ortho)

CATCOL

P211c, Z = 4(1)

8 10 10

(CH3)2C6H4(OH) 2 (2,5,1,4) (CH3)4C6(OH) 2 (2,3,5,6,1,4)

Pca21,Z = 4(1)

11 14

(tert-CaH9)C6H3(CH2OH)(OH) (4,2,1)

DOGWOL DOGWUR GASJOZ DINWlG 10

P21/c, Z = 2(1) 141/a, Z = 16(1) 141/a,Z= 16(1)

[[Pctz)41]] (Pc(z)21) [[Pc(z)41]] (Pc(z)21) Pc(y)21 Pcty)21 Pc(z)41 c Pc(z)41 c

(tert-C4H9)2C6H2(OH)2 (3,6,2,1)

COXRUC

141/a,Z= 16(1)

(iso-C3H7)C6H3(CH2OH)(OH) (4,2,1)

~

m

m S(m 2) S/xy ~ ~ ~

Between H-aggl. --

BzC aggr.

O(2)/h

{O(2)]11} & composite tubes Framework

Ob(2q)

{S(m 2) },PL

m Ob(2q) ~ m ~ m S(ml)

{Ob(21)} {Ob(21) } {S(m2) } {S/xy }

S(ml)

Notes: an--number of atoms C. t'l'here are four kinds of the stereotype; some of them have angular deformations. CThe chains of conjugated H-bonds have a composite structure, such a chain includes a sequence of H-bonds formed by CH2OH groups, and adjacent H-bonds formed by OH groups.

Table 3. Contacts of Benzene Rings in Crystals of Alkyl and Alkoxy Derivatives of Benzene a Effective Contacts of Benzene Rings Substance

t,.rl

6 6 7 8 8 10 10 12 12 12 12 12 13

C6H6 C6H6 CH3C6H 5 (CH3)2C6H4 (para) (CH30)2C6H4 (para) (CH3)4C6H 2 (1,2,4,5) (C2HsO)2C6H 4 (para)

(iso-C3H7)2C6H4 (para) (C2H5)3C6H 3 (1,3,5) (CH3)(iso-CsH l I)C6H4 (1,4) (CH3)6C6 (CHaO)E(CH3)4C 6 (1,4,2,3,5,6) (CHa)(n-C6HI3)C6H 4 (1,4)

Notes: an--number of atom

Refcode CSD BENZEN BENZEN03 TOLUEN ZZZITY01 MOXBEN DURENE01 DEXBEN DIPRBZ FIPDUD MEPRBZ ~ MXTMBZ MHXBEN

C.

bThe Ob(2q) contacts in toluene are close to the Obi(mi)/h type. CUnusual ribbons formed by Ob(2q) contacts of different kinds.

Structural Class m

Pbca, Z = 4(1) P21/c, Z = 2(1) P21/c, Z = 8(12) P21/c, Z = 2(1) Pbca, Z = 4(1 ) P21/c, Z = 2( 1) m

F2 l, Z = 2(1 ) P2jlc, Z = 2(1 ) Prima, Z = 4(m) P21/c, Z = 4(1) PI, Z = 1(1) P1, Z = 1(1) P1, Z = 2(1)

Types of BzC

BzC-Aggregates

O( 1)/s, Ob(2q) l, Ob(2q) 2 Ob(2q) Ob(2q)l b, Ob(2q)2 b S(m I) Ob(2q) S(m 2) Ob(2q) 1, Ob(2q) 2 m

{S(ml)} PL

Ob(2q) S(m 2) S/xy

{Ob(21)} (S(m2)} {S/xy} m

OBzL, PL l, PL 2 PL {Ob(2q)l, Ob(2q)2 }c

{S(m2)} [PL]

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PETR M. ZORKY and OLGA N. ZORKAYA

!1. CONJUGATED HYDROGEN BONDS ILLUSTRATED BY CRYSTALLINE DERIVATIVES OF HYDROXY- AND DIHYDROXYBENZENE Conjugated hydrogen bonds (CHB) are rather common in diverse organic (but not only organic) substances. They form an uninterrupted sequence ...OH...OH...OH .... with O atoms being bound to some other atoms. Classic examples are provided by derivatives of hydroxybenzene (phenol) and of three isomeric dihydroxybenzenes" p-dihydroxybenzene (hydroquinone), m-dihydroxybenzene (resorcinol), and o-dihydroxybenzene (catechol), which were first considered in an earlier work [9]. In the present work we have restricted ourselves to alkyl and alkoxyl derivatives of the above compounds, and have carried out a systematic study of CHB systems using the Cambridge Structural Database (CSD). An important feature of this analysis lies in the fact that structuring functions of H-bonds and contacts of benzene rings (BzC) have been considered simultaneously. (It should be noted here that halogen contacts have an even greater influence than BzC on the formation of a structure, and we plan to review the structures of halogen-bearing derivatives of phenol and hydroxybenzenes in a near-future communication.) As is seen from Tables 1 and 2, most of considered crystals contain spiral systems of CHB with either crystallographic or approximate axes of symmetry n l, where n = 2, 3, 4. The corresponding group of symmetry is written as Pcnl. (Note that in our studies we commonly use groups of symmetry of three-dimensional chains and layers. Their symbols are similar to those of space groups; to specify these symbols the indices c and ~ are utilized. See, for example, ref. 10 where the concept of "structural class" used in this work is also covered.) Index c is usually followed by a parenthetical designation of the axis, along which the chain is directed. Symbols of the groups that describe the pseudosymmetry of a CHB system are bracketed. This pseudosymmetry is sometimes realized with considerable deviations; in such cases the symbol of the group is twice bracketed, and has rather a topological, not geometric, meaning. Sometimes molecules are united by CHB into finite agglomerates. In these cases the symmetry of the CHB system is expressed by a point group. If there is a considerable number of aliphatic carbon atoms, steric hindrances arise which hamper the formation of H-bonds. That is the reason why we only consider substances with small alkyl and alkoxyl radicals. Tables 1 and 2 show that chains with screw axis 21, i.e. those described by group Pc21, are the most frequent. As an example, Figure 1 shows the chains that are present in the crystal structure of p-methoxyphenol and the arrangement of these chains in the crystal. Some other substances from Table 1 contain quite similar chains. However, the period of the chains varies from 4.47 ,~ (in 2,6-dimethylphenol) to 5.29/~ (in p-methoxyphenol). There are three examples of Pc21 chains in Table 2 ()'-form of p-hydroquinone, period 5.20/~; 2,5-dimethylhydroquinone, 4.56/~; 2,3,5,6-tetramethylhydroquinone, 4.73/~). The presence of two hydroxyl groups in a molecule leads to the formation of layers where molecules are bound

Specific Intermolecular Interactions

15 5

Figure 1. Crystal structure of para-methoxyphenol (MOPHLC). Chains of conjugated H-bonds with screw axes 21 are present in this structure. (a) View along Y; (b) molecular chain projected along Z.

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PETR M. ZORKY and OLGA N. ZORKAYA

Figure 2. Crystal structure of 2,5-dimethylhydroquinone (DOGWOL) viewed along Y. There are layers where molecules are bound by spirals of conjugated H-bonds.

by CHB spirals, as one can see for instance in 2,5-dimethylhydroquinone (Figure 2). An example of a CHB chain with screw axis 31 is provided by o-cresol (Figure 3). These trigonal crystals have three varieties of symmetrically nonequivalent chains Pc31; yet the period of the chains is the same in all three cases, and the chains are very similar in structure. The crystals of m-cresol and phenol also have chains Pc31. However, these crystals belong to the monoclinic system, and here axis 31 is an axis of pseudosymmetry. This pseudosymmetry is rather exact in m-cresol; in phenol the actual structure of the chain substantially deviates from symmetry Pc31. The periods of the chains Pc31 shown in Table 1 are 6.05 A in phenol, 6.20 ~ in m-cresol, and 5.94 in o-cresol. o-Cresol and 13- and ~t-hydroquinones (Figures 3-5) constitute an exceptionally interesting sequence of structures. All have chains with symmetry Pc31, though in hydroquinones these chains are formed not by CHB but by orthogonal benzene

Specific lntermolecular Interactions

157

Figure 3. Crystal structure of ortho-cresol (CRESOL01). A subsystem of molecules that is infinite in one dimension lies along axis Z. This subsystem has a complicated structure, which looks like a stem of a plant with its shells. There are other similar fragments in other crystals that are also called stems. (a) Projection along Z, dashed lines represent the spirals of conjugated H-bonds here and below orthogonal benzene contacts are designated by arrows; (b) spiral system of molecules called stern I.

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PETR M. ZORKY and OLGA N. ZORKAYA

Figure 4. Crystal structure of 13-hydroquinone (HYQUIN05). (a) Projection along Z; (b) spiral stem 2; (c) composite stern 3, formed by stems 2 with the construction and stacking of hexamers.

contacts (and by nonconjugated H-bonds too). In this case, distinguishing such chains is a matter of convention since they overlap: each molecule is shared among two adjoining chains (excluding molecules III in ot-hydroquinone). Besides, the structure ofhydroquinones is cemented by finite systems of CHB, namely six-membered tings. In addition, in its ~t-form one can see an elegant system of CHB, i.e. a spiral containing 15 H-bonds per period (see Figure 5a). It should be noted, that the

Specific Intermolecular Interactions

159

Figure 5. Crystal structure of c~-hydroquinone (HYQUIN02). (a) Projection along Z, the spiral formed by conjugated H-bonds is marked; (b) spiral stem 4.

structures of o-cresol and [3- and ~t-hydroquinones can be realized only as the combination of CHB and BzC, and we shall revert to this subject after presenting a method of describing BzC. CHB systems with screw axes 41 (with the period 8.33 and 8.89 A) are found in the structure of 4-isopropylphenol (Figure 6) and in the similar structure of 3-methyl-4-isopropylphenol (Table 1). Analogous chains of CHB have been observed in crystals of dihydroxybenzenes (Table 2). There are also spirals which are topologically similar to CHB chains with 4~ axes, but are significantly distorted; they retain only symmetry Pc21. Such CHB chains occur in m-dihydroxybenzene (also called resorcinol) (Figure 7) and o-dihydroxybenzene (catechol) (Figure 8).

160

PETR M. ZORKY and OLGA N. ZORKAYA

Figure 6. Crystal structure of para-iso-propylphenol (IPRPOL) projected along Z. There are chains of conjugated H-bonds with screw axes 41.

The peculiarity of these structures is that CHB spirals bound molecules to form a three-dimensional agglomerate or molecular layers. The chain agglomerates which are present in 2-hydroxymethyl-4-isopropylphenol and its analog with t-Call9 have a more complicated structure than usual Pc41 chains (Figure 9). Each molecule is attached twice to the CHB system; one of two OH groups is included in spiral Pc4~, the second one is adjacent to this spiral. Finally, finite aggregates of CHB from Tables 1 and 2 should be mentioned. In p-cresol and in 2,6-di-isopropylphenol cyclic CHB systems are formed with approximate symmetry 4/m (the exact crystallographic symmetry of these systems is ]-), and in 3,4-dimethylphenol and in two polymorphs of hydroquinone (a and 13) there occur cycles with pseudosymmetry 6/m (the exact symmetry is 3).

Specific Intermolecular Interactions

161

Figure 7. Crystal structure of meta-dihydroxybenzene (resorcinol) (RESORA13) viewed along Z. Spirals of conjugated H-bonds bind molecules to form a three-dimensional agglomerate.

Thus in all examined crystalline substances there is a steady and pronounced trend towards the formation of CHB. However, as noted above, the enlargement of alkyl and alkoxyl substituents and the increase in their number results in steric hindrances, which rule out the possibility of formation of such CHB systems. In the examined compounds the maximum number of aliphatic C atoms in a molecule (n - 6) that does not exclude the possibility of formation of CHB is six. In two polymorphic modifications of 4-methyl-2,6-di-tert-butylphenol, ordinary hydrogen bonds OH...O are absent (weak bonds CH3...O apparently play some structuring role in the orthorhombic modification). In 4-methoxy-2,6-di-tert-butylphenol, OH

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PETR M. ZORKY and OLGA N. ZORKAYA

Figure 8. Crystal structure of ortho-dihydroxybenzene (catechol) (CATCOL) viewed along Y. In this case spirals of conjugated H-bonds bond molecules to form molecular layers.

groups form H-bonds with the O atoms of methoxyl groups, but these H-bonds are not (and cannot be) conjugated. In 3,6-di-tert-butylpyrocatechol too, only the formation of localized (not conjugated) H-bonds of the OH...OH type appears to be possible; the H-atom of the second of these groups cannot find an acceptor to form an H-bond since it lies opposite to the benzene ring of a neighboring molecule. Structures of this kind rarely come into the view of researchers and appear odd. Yet they refute the often postulated assertion that the functional groups which in principle can form H-bonds always form them. It can be seen that even the fairly narrow range of the crystalline substances discussed above (which have been selected to meet rather rigid criteria) shows the considerable diversity of CHB aggregates. Is it possible to predict or at least to interpret a posteriori the type of the formed CHB system in each specific case? This problem, far from being easy, does not seem to be unsolvable. It is clear that the choice of one or another CHB aggregation is determined by steric factors which operate, first, within the molecular agglomerate formed on the basis of CHB, and second, in the superposition ("packing") of such agglomerates. For the substances being examined one of the most important factors is the optimal way of formation of the contacts of benzene tings (BzC). The diversity of BzC (those possible in principle and actually observed) is discussed below.

Specific Intermolecular Interactions

163

Figure 9. Crystal structure of 4-iso-propyl-2-hydroxymethylphenoi (GASJOZ) viewed along Z. There are chain agglomerates which are more complicated than usual chains with screw axes 41 . Each molecule is twice attached to the system of conjugated H-bonds.

III. CLASSIFICATION OF CONTACTS AND AGGREGATES OF BENZENE RINGS Chemical compounds containing benzene rings (more comprehensively, aromatic six-membered tings) are very common in laboratory practice, both among natural and artificial substances. In some cases such a ring forms the frame of a molecule, to which various substituents are added. In other cases a benzene ring itself in the form of a phenyl radical (or in the form of some other radical C6H6_k) plays the role of a substituent adding to some framework. Another typical form of existence of a benzene ring in an organic molecule is the involvement of such a ring in a system of conjugated aromatic tings (e.g. benzene tings form parts of naphthalene, anthracene, and similar condensed nuclei). Being very common and rather voluminous substituents or being a part of a condensed system, benzene tings very often take part in the formation of intermo-

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PETR M. ZORKY and OLGA N. ZORKAYA

lecular contacts in crystals and other condensed phases. As this takes place, some typical and frequently occurring arrangements often arise. There are grounds to believe that such arrangements are particularly advantageous. The systematic analysis of effective contacts of benzene rings (BzC) required the classification scheme which is described below. We will begin by listing possible ways of mutual arrangement of two benzene rings (in the literature this pair of rings is often called "dimer" but we prefer the term "contact"), and after that we will consider finite and infinite structures which contain some set of BzC. Finally, on this basis the structure of several dozens crystalline substances containing benzene tings are observed.

A. Types of BzC Some characteristic appearances of fragments of atomic and molecular arrangements are often called stereotypes in our studies. It is convenient to distinguish the stereotypes of BzC shown in Figures 10 and 11. When describing them (and other BzC) we shall designate two contacting benzene rings as A and B. These stereotypes are as follows: 1. a sandwich with rings A and B identically oriented and situated exactly over the other; such a pair of benzene molecules has symmetry 6/mmm; the contact will be designated as S; 2. a sandwich with ring B rotated through 30 ~ with reference to ring A; symmetry is 122m; designation is Si; 3. a mirror-symmetrical-shifted contact derived from contact S through a shift of ring B along a major diagonal of ring A; symmetry is m; designation is S(ml); 4. a mirror-symmetrical-shifted contact derived from contact S through a shift of ring B along a perpendicular to a major diagonal of ring A; symmetry is m; designation is S(m2); 5. mirror-symmetrical-shifted contacts similarly derived from contact Si; symmetry is m; designations are Si(ml) and Si(m2); 6. mirror-symmetrical oblique contacts similar to contacts 3-5, but differing from them in that ring B is rotated about the perpendicular to the direction of the shift; the planes of rings A and B make angle e; designations are Ob(ml), Ob(m2), Obi(ml), Obi(m2); and 7. T-shaped (orthogonal) contacts; symmetry is mm2; there are four varieties of such contacts designated O(1)-O(4); the conformation of these BzC (which are also designated as OBzC) is sufficiently clear from Figure 11. It has been described in ref. 11. The above stereotypes of BzC are listed in Table 4 along with their geometric and energy numerical characteristics explained below. Because of high symmetry

Specific Intermolecular Interactions

165

S(m2)

S(ml)

>-.o o-.<

o---<

Si(ml)

Si

)

o.--<

Ob(ml)

Ob(m2)

Obi(ml)

Figure 10. Stereotypesof sandwich, shifted and oblique benzene contacts.

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PETR M. ZORKY and OLGA N. ZORKAYA

O(1)

0(2)

0(3)

O(1)/h

0(1)/,4,

0(4)

ro

o(1)/s

o(~)/s~

Figure 11. Stereotypes of ideal and deformed orthogonal benzene contacts. of the benzene ring identical arrangements of benzene tings also arise at other values of the geometric characteristics (we skip the detailed consideration of this question for brevity). Every so often there occur situations when while varying stereotypes one arrives at the same BzC, the identity of which is established by renaming rings A and B. Thus among other things one can deduce that in fact stereotypes Si(m2)

Specific Intermolecular Interactions

167

Table 4. Geometric and Energy Characteristics a of Some Special BzC Values of Parameters Corresponding to the Minimum of Energy

Stereotypes

x

y

z

tp

0

~

U

S

0

0

dI

0

0

0

-0.835

d I = 3.76

Si

0

0

d2

0

0

30

-0.843

d 2 = 3.75

S ( m l)

x

0

z

0

0

0

-1.293

x - 3.19, z -- 3.35

S ( m 2)

0

y

z

0

0

0

-1.261

y = 3.20, z = 3.37

Si(m l)

x

0

z

0

0

30

-1.279

x - 3.21, z = 3.35

Ob(ml)

x

0

z

0

eI

0

-1.309

x = 3.89, z = 2.62, e I

O b ( m 2)

0

y

z

-e 2

0

0

-1.273

y = 4.11, z = 2.42, e2 = 31.1

O b i ( m l)

x

0

z

0

I~3

30

-1.311

x = 3.91, z = 2.59, I~3 -- 26.3

O(1)

0

0

Dl

90

0

0

-1.229

D l -- 5.07

0(2)

0

0

DE

90

90

0

-1.232

D 2 - 5.07

0(3)

0

0

D3

90

0

90

-1.179

D 3 = 5.17

0(4)

0

0

D4

0

90

0

-1.180

D 4 = 5.17

O(1)/s

x

0

z

90

0

0

-1.237 b

x = 0.75, z = 5.03

O(2)/h

0

y

z

90

90

0

-1.248 b

y = 0.76, z = 5.02

Notes:

-

-

25.5

aDistances in ]k, angles in degrees, energy in kcal/mol. bWhen varying angle u (without a loss of mirror symmetry) the minimization of energy gives for O( l)/s: x = 0.69, y = 0, z = 5.03, go = 90, 0 = 0, ~ = 4.6, U = -1.249, and for O(2)/h: x = 0, y = 0.76, z = 5.02, go = 90, 0 = 90, ~ = 3.5, U = -1.258.

and Obi(m2) reproduce stereotypes Si(ml) and Obi(ml); they are named above for completeness, but they are omitted in Figure 10 and in Table 4. To exactly describe the conformation of BzC it is convenient to use the following coordinate system and set of parameters: the origin of coordinates is in the center of one of the rings; axis X passes along a major diagonal of the six-membered ring; axis Y lies in the plane of the ring perpendicular to axis X; x, y, z--the cobrdinates of the center of the second molecule in the coordinate system of the first (reference) molecule (Xl, YI, Zl); and tp, 0, ~gmEulerian angles characterizing the orientation of the coordinate axes of the second molecule (X2, Y2, Z2) with respect to the reference coordinate system. Here angle tp corresponds to the rotation of the second molecule about axis X1, angle 0 corresponds to the rotation about the new axis Y l(tP) (which appears after the rotation through angle tp), and angle ~g corresponds to the rotation about the new axis Zl(tp, 0) = Z2 (which appears after the rotations through angles ~pand 0). Two other characteristics of BzC are also useful: angle e between the planes of tings A and B and distance D between the centers of the tings. Deviations of some real contacts from ideal stereotypes reveal themselves in deviations of the above numerical characteristics from the values given in Table 4. We use the following designations for the qualitative characterization of deformed BzC (examples drawn from a number of deformed OBzC are shown in Figure 11): s is a shift of ring B along a major diagonal of ring A; h is a shift of ring B in a perpendicular direction; and tp, 0, V in the symbol of a deformed contact indicate a deviation of the respective angle from its ideal value.

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PETR M. ZORKY and OLGA N. ZORKAYA

Of course lines of demarcation between a BzC that is close to some ideal stereotype, a significantly deformed BzC of this type, and another relative arrangement of benzene tings is rather a matter of convention. Thus we regard BzC as deformed if the deviation of at least one of its characteristics from the ideal values (Table 4) is more then 0.5/~ or 5 ~ (but less than 1.5/~ or 15~ However, these limits are not assumed to be very strict. For the sake of clarity it should be noted that the difference between the similar contacts classed as S(ml) and S/s (or S(m2) and S/h) lies in the value of the shift, which is small for contacts S/s and S/h (<1.5/~) and is substantially larger for contacts S(ml) and S(m2), and just this circumstance leads to qualitatively distinct relative arrangements of the tings. Yet another important type of contacts of benzene tings is worthy of special attention. It is the arrangement of the rings such that one of them can be transformed into the other by a 180 ~ turn and a shift along the axis of the turn, i.e. the tings are related by hypersymmetry operation 2q (see e.g. ref. 10). In particular cases the shift is zero (q = 0), or it is a half of the translation along the corresponding axis (q = 1) and the axis 2q becomes a rotation axis 2 or a screw axis 21, respectively. Since a benzene ring has an inversion center, the presence of operation 2q leads to operation mp, which is a reflection across a plane perpendicular to axis 2q in combination with a corresponding shift. (An introduction of some substituents into the benzene ring can break centrosymmetricity and result in a contact in which only axis 2q or only plane mp is present.) Strictly speaking the contacts of this type should not be regarded as specific, for their conformation has a large number of degrees of freedom (three translational and two rotational) and the interaction energy of the tings may vary over a wide range. However, such contacts often lead to advantageous ways of aggregation (agglomeration) and therefore occur rather frequently. Going to energy characteristics of the considered contacts, we should point out that there is no definite correlation between them and the frequency of realization of these contacts in crystal structures. The fact is that the advantage of a structure is determined by the totality of intermolecular interactions in the crystalline substances, and the contribution of the considered contacts to this total energy may be large or may be insignificant. Nevertheless, in the systematic analysis of all kinds of BzC, the data on the energy of interaction of benzene rings as a function of their mutual arrangement are instructive. The benzene dimer was the object of numerous quantum-chemical calculations (including ab initio), the results of which were far from unambiguous. For instance, the conformation of the dimer that corresponds to the 0(3) type was found to be the preferable in one study [12], while in another [13] the "parallel displaced structure" (the sandwich with the second molecule shifted and rotated: x = 0, y = 1.63/~, z = 3.49/~, 9 = 0, 0 = 0, ~ = 30 ~ D = 3.85/~, e = 0) is more advantageous. Thorough calculations carried out on the basis of atom-atom approximation [14] showed that the dimer with mirror symmetry (x = 3.91/~, y = 0, z = 2.59/~, tp = 0, 0 = 26 ~ = 30 ~ D = 4.69/~, e = 26 ~ has the optimum conformation. However, the results

Specific Intermolecular Interactions

169

of energy calculations for the benzene dimer do not allow conclusive judgements about the geometrical parameters of optimum contacts of benzene tings in molecular agglomerates or in the spatial crystal structure. Such judgements require optimization of the infinite systems (chains, layers, crystal) under consideration. The works of Dzyabchenko [15, 16] and Gibson and Sheraga [17], in which the minimum-energy crystal structures of benzene have been calculated using atomatom potentials in good agreement with X-ray structural data, are of interest in this regard. From what has been said it is evident that quantum-chemical calculations ab initio, being very labor-consuming, do not give a sufficiently clear picture even for the benzene "dimer". Hence there are even less reasons to hope that this approach can be successfully applied to more complex molecular systems of benzene tings (chains, layers, three-dimensional structures). At the same time the approximation based on atom-atom potentials gives quite reasonable results. Therefore we have chosen the latter approach for estimating the energy of BzC. The energy calculations were realized by means of Dzyabchenko's program [18] using the parameters of the atom-atom potential and effective atomic charges (qH = Iqcl = 0.153 e) taken from ref. 14. For each of the above-listed ideal stereotypes the minimization of the energy (U) of interaction between two benzene molecules was carried out. The obtained values of U along with the values of the parameters that determine the optimal conformation of the contact are given in Table 4. These values show that stereotypes S and Si are not sufficiently advantageous; indeed, such contacts do not ordinarily occur in crystals. However, distinguishing them is convenient for purposes of classification and allows presenting the stereotypes in the form of a logical sequence. The plot presented in Figure 12a shows how the value of U changes with the shift of ring B along axis X. This shift, which is accompanied by a turn q~at sufficiently large values of x, leads from type S to type S(ml), then to type Ob(ml). Quite similar dependencies occur in transitions S-S(m2)-Ob(m2), Si-Si(mi)-Ob(ml) and Si-Si(m2)-Ob(m2). It is remarkable that there is a wide range of geometric parameters at which contacts Ob(m) and S(m) have very close (and rather low) values of energy U. The contact of type Obi(ml) with parameters x, z, and e3 given in Table 4 corresponds to a global minimum of function U(x, y, z, q~, 0, ~); the numerical values of the minimum agree with those of ref. 14. For the six-dimensional potential surface U, or, more precisely, for its symmetrically independent part this minimum is the only one. Other minima occur only in sections of function U (at fixed values of some variables), i.e. they are conditional. The following circumstance is noteworthy and important. The symmetries of all the distinguished BzC are not less than m. Indeed, in all cases the loss of plane m increases energy U (decreases IUI). In other words, the neighborhood of a point with coordinates Xl, Yi, zl, tpl, 01, ~l, which corresponds to a pair of benzene molecules

170

PETR M. ZORKY and OLGA N. ZORKAYA

u

Kcal tool

-1.10

-1.20

- 1.30

D I

I

I

I

I

I

I

I

1'

,.~ ~o ,.~ ,.o ,.,

I

I

I

I

~-

,0 ,, x(~,)

I..

u

Figure 12. Dependence of potential energy of the benzene contact on its geometric parameters. (a) Energy of contacts S(ml) (solid line) and Ob(ml) (dashed line); (b) energy of the deformed contact O(1 ). with symmetry m (or higher), does not have points that would correspond to asymmetrical pairs of molecules for which U < UI. Stereotypes O(1)-O(4), also designated OBzC, correspond to T-shaped contacts with symmetry mm2. In the cases of 0(3) and 0(4) the loss of one of the two planes of symmetry, i.e. a shift of ring B designated as s or h (the transition to deformed contacts O/s or O/h), results in less advantageous energy characteristics. Yet a shift s leads to a decrease of the energy of contact O(1) as it is shown in Figure 12b. The point of tangency of curves 1 and 2 corresponds to contact O(1) with D1 = 5.07 A. Curve 1, which lies in plane A, characterizes the variation of energy U with shift h, while curve 2, which lies in a perpendicular plane B, with shift s. Curve 3 (plane C, where s = 0.75/~) corresponds to the change of U with varying D and has a minimum at D1 = 5.03/~. Curve 4 (plane D) is a result of further minimization with varying angular characteristics of the contact. Similarly a shift h leads to a decrease of the energy of contact 0(2). This circumstance appears to be one of the contributory factors for the formation of contacts O(1)/s (it is the most frequently encountered type of orthogonal BzC in crystals). The energy of the above-mentioned contacts Ob(2q) may vary over a wide range. The contacts of this type occurring in the crystals of orthorhombic benzene appear to be close to the most advantageous ones found among organic crystals (-0.79 and -0.59 kcal/mol).

B. Typesof BzC Aggregates Consecutive repetition of a series of identical BzC in the volume of a crystal can be regarded as their conjugation and leads to the formation of diverse aggregates

Specific Intermolecular Interactions

171

(Figure 13). These aggregates form part of molecular agglomerates (or another subsystem of the crystal structure). Since the benzene ring is often just a fragment of a much more complex molecule, the agglomerates can differ significantly from the aggregates. The enumeration of possible molecular agglomerates would be too cumbersome. We shall speak only about particular agglomerates that are present in the considered structures. At the same time a variety of possible BzC aggregates can be foreseen. Here we shall list some infinite and finite BzC aggregates. In both cases we shall restrict ourselves to the analysis of homogeneous aggregates, which consist of identical or similar BzC. Infinite sequences of contacts of a definite type are designated by braces, e.g. {S(ml)}. Contacts S and Si form straight columns {S} and {Si}; contacts S(m), Si(m), Ob(m), and Obi(m) give rise to corresponding oblique columns (or packs). Very important and frequent are zigzag chains {Ob(21)}. They arise from Ob(2q) contacts. When forming a chain, these contacts generate a 2~ axis which can be either crystallographic (exact) or approximate. The chains {O} formed by OBzC can hardly be called columns (they are not like piles), but these chains are topologically

{S} .

_

{Si}

_

.

-

{S(m)}

.

{Ob(21)}

{0}

_

-

{Si(m)}

{Ob(m)}

PL

/

/

/

/

/

[

I

{Obi(m)}

OBzL

/.//

/N/N/

Figure 13. Chain and layer aggregates of benzene contacts.

172

PETR M. ZORKY and OLGA N. ZORKAYA

close to columns {Si} and are characterized by the same symmetry groups and structural classes. A special type of the chains arises when contacts of types S(m), Si(m), O/s, and O/h are infinitely repeated with a change of the direction of the shift in accordance with the hexagonal symmetry of the benzene ring (by exhaustion of equivalent directions). For example, if the direction of the shift in consecutive BzC of chain {S(ml) } changes by 60 ~ there arises a chain with the 61 axis. Similar spiral systems can be formed in sequences {O/s} or {O/h}, too. However, such arrangements are of low probability and, as far as is known, do not occur in organic crystals. Instead, the chains {O(21) } and {O(31) } (with ideal or deformed OBzC) can be considered more important. Yet they are not present in Figure 13 because the {O(21)} chains can be regarded as a special case of the {Ob(21) } chains and the {O(31) } chains are difficult to draw using the chosen style. But we shall present some examples of such chains below. The construction of a sequence that includes OBzC of different types appears to be more close to reality. Thus in the crystals of o-cresol, the structure of which is discussed in detail below, there are spirals in which contacts O(4)/h, O(4), and O(1)/s are consequently repeated. The special role of OBzC and of contacts Ob(2q) lies in the fact that they can serve as bases for the formation of layer aggregates (Figure 13). Of course layers can be made up of columns or other chains formed by BzC, but in such layers there is only one optimal contact per molecule; much less advantageous contacts are realized between chains. In layers formed by contacts Ob(2q) or OBzC there are two optimal contacts per each molecule. A two-dimensional aggregate of contacts Ob(2q) formed by benzene tings gives rise to aparquet layer(PL). The layer formed by OBzC can be viewed as a particular case of the parquet layer. However, it has some important features, which gives grounds to distinguish "orthogonal benzene layers" (OBzL). Ideally, this layer is tetragonal and has symmetry Pl 4 bm. Obviously four variants of such layers are possible in principle (for such is the number of OBzC stereotypes). Besides, the diversity of the structures of OBzL is increased by various deformations of BzC. Finally, it should be noted that sometimes finite aggregates of BzC occur in crystals: for example, six-membered cycles or simply pairs of benzene tings forming some BzC.

C. BzC in Crystalline Derivatives of Benzene, Phenol, and Dihydroxybenzenes Benzene Derivatives First we turn to the benzene derivatives that do not contain hydroxyl groups (Table 3). Benzene, which is known in the forms of common orthogonal (orthorhombic) and monoclinic high-pressure modifications, naturally has the maximum number of BzC. In orthogonal benzene, parallel to each of the coordinate planes

?

X

0

v

m

~ ,.., 0

~'~z

-~

~ ~ z

o _ I:lo i-r'l

r~

-<9 N ~

tl)

0

~

0

I:~ --"~'. 0 0

x~.~

a,, ~--..- m

t"~ 0~

~s.

.,,,.

~-. -u ,...-

r-" --"~'.

-~o

o"o

,.-,. 8 ~-"" %" ~-,~. ~

o.~

S%

,.

0

x

,,,, ....

"X x

th

.,,,..

rh c::

0

rh

rb

174

PETR M. ZORKY and OLGA N. ZORKAYA

Y

r

0

f Figure 14. (Continued) 2~

2I

there are molecular layers with symmetry Pt(xz)ll -E, Pt(vz)l c 1 and 21 Pt(xY) $11, respectively. The crystal structure can be considered as formed by antiparallel stacking of any of these layers. Figure 14 shows not only the structure of the layers but also the way of stacking these layers.

I

Figure 15. Perspective view of the L(YZ)layer in orthorhombic benzene. It can be seen that, viewed along the axis of the rectangular projection, the orthogonal benzene layer is stepped.

Specific Intermolecular Interactions

175

Besides ordinary projections (P) along X, Y, Z axes rectangularprojections (RP) along axes X', Y', and Z' have been constructed. The method of their representation is based on the existence of 21 axes parallel to the drawing plane. This method includes a x rotation about the 21 axis as a result of which all tings become perpendicular to the drawing plane. Thus the character of BzC becomes clearly seen. The layer plane forms an angle 1: with the plane of the rectangular projection (this angle is Xl or x2, or "c3, depending on the coordinate axis around which the crystal structure is rotated).

Figure 16. Space-filling representation of orthogonal benzene layers in benzene (a), 2-aminophenol (b), 4-chloro-2-aminophenol (c, d) (two sides of the layer). Rectangular projections are shown.

1 76

PETR M. ZORKY and OLGA N. ZORKAYA

Figure 14a,b show that intermolecular contacts, the conformation of which is close to the O(1)/s type, are present in layers L(XZ). These contacts are particularly pronounced in the RP(Y') projection, the angle T3 being equal to 18.9 ~ For OBzC in orthorhombic benzene: x = 1.23 A, y = 0.12 A, z = 4.93 A, q~= 94.8 ~ 0 = -2.1~ = -2.3 ~ D = 5.08 A, and e = 85.2 ~ The drawing of this contact as well as some other characteristic BzC is given below. From Figure 14(c-f) one can see that L(YZ) and L(XY) are parquet layers formed by Ob(2q) contacts; it is clearly illustrated by the RP(X') and RP(Z') projections (17 2 and 1; 1 are equal to 45.8 ~ and 70.6 ~ respectively). The perspective three-dimensional representation of the L(XZ) layer is shown in Figure 15. The Y' axis passes very close to the diagonal of the ring (the angle between these directions is 1.9~ The figure gives a very clear picture of this "stepped" layer and the OBzC conformation occurring in it.

Figure 17. Crystal structure of para-xylenel (77ZITY01) viewed along Y. There are columns formed by shifted sandwiches S(ml ).

Specific Intermolecular Interactions

177

It is noteworthy [11] that, as shown in Figure 16, very similar OBzL exist in crystalline 2-aminophenol and 4-chloro-2-aminophenol (their refcodes in CSD are AMPHOM02 and AMCPHO, respectively). The aggregate of O(1)/s contacts proved to be so stable that even H-bonds and C1...C1 contacts don't destroy it. Just the same OBzL can be seen in crystals 1,2-diaminobenzene (refcode BAGFIY). A very common ("naphthalene-like") crystal structure with classic parquet layers is found in monoclinic benzene. Topologically similar are the structures ofp-xylene (Figure 17) and durene; however they do not contain parquet layers, p-Xylene provides a prominent example of contacts S(ml), which form shifted columns along axis Y. Shift s is -3.0/~, i.e. the energy of the contact is close to the minimum shown in Figure 12a. Durene has a closely similar structure differing only in that here shifted packs are formed by contacts S(m2). Drawings of BzC in these substances (and OBzC in benzene) are present in Figure 18.

a

b ~)

()

d

e

Figure 18. Some typical benzene contacts; (a)-(d) projected onto plane A, (e)-projected along axis 2q, (a) Shifted orthogonal contact O(1)/s in benzene; (b) shifted sandwich contact S(ml) in para-xylene; (c) shifted sandwich contact S(m 2) in durene; (d) asymmetrically shifted sandwich contact S(xy) in 1,4-dimethoxy-2,3,5,6-tetramethylbenzene; (e) contact Ob(2q) in para-diethoxybenzene.

178

PETR M. ZORKY and OLGA N. ZORKAYA

Toluene has a highly nontrivial crystal structure with two symmetrically independent molecules (more precisely, the molecules occupy two orbits) (Figure 19). BzC are concentrated in the layers passing parallel to plane (104). In such a layer molecules B form chains {Ob(21) } along axis Y. On two sides these chains contact with molecules A, which form BzC AIB l, A2B2, and so on. Eventually very unusual

Figure 19. Crystal structure of toluene (TOLUEN)" Interesting and unusual ribbons are present in these crystals. (a) View along u (b) projection along u (ribbons formed by Ob(2q) contacts are framed); (c) a ribbon projected along Z.

Specific Intermolecular Interactions

179

and interesting ribbons are formed as BzC aggregates. The arrangement of such ribbons is shown in Figure 19b. In crystalline p-dimethoxybenzene contacts of type Ob(2q) form parquet layers which are superimposed antiparallel and are connected by bonds CH...O. Layer of type PL are present in the crystal ofp-diethoxybenzene (Figure 20). Here, however, a deviation from Kitaigorodskii's centrosymmetricity rule is observed (molecules with intrinsic symmetry 1 do not occupy inversion centers of the crystal), the reason of which is unclear. Contacts A~BI and B IA2 are nonequivalent. Each of them generates a chain {Ob(21) } along axis Y. The conjugation of such chains gives rise to a layer parallel to plane YZ; this layer may be classified as "pseudoparquet" (in Table 3 it is designated as [PL]). Chains {Ob(21)} are also present in 1-methyl-4isoamilbenzene, where they are superimposed antiparallel (without forming BzC) in a layer parallel to plane YZ. Crystalline hexamethylbenzene (I) and 1,4-dimethoxy-2,3,5,6-tetramethylbenzene (II) have a highly interesting structure. Crystals I were among the first crystals studied using X-ray structural analysis [19, 20]. However, afterwards for some reason or other they did not attract the attention of researchers, and coordinates of atoms in this substance are absent in CSD. To construct structure I (Figure 21) we used the data of ref. 20. These crystals have pseudohexagonal layers with approximate symmetry Pt6/m. Here we have a rather rare case of planar molecules lying m

Figure 20. Parquet layer in the crystal of para-diethoxybenzene (DEXBEN). A deviation from Kitaigorodskii's centrosymmetric rule is observed here: molecules with intrinsic centers of symmetry do not occupy inversion centers in the crystal.

Figure 21. Crystal structure of hexamethylbenzene and 1,4-dimethoxy-2,3,5,6-tetramethylbenzene (MXTMBZ) (methyl groups are shown as spheres). In hexamethylbenzene local mirror planes arise in the form of shifted sandwiches S(m2); the crystal is a packing of IS(m2)} chains. In 1,4-dimethoxy-2,3,5,6-tetramethylbenzene such local symmetry is absent and the crystal appears as a set of chains {S/xy]. (a, c) Projections onto the plane of rings; (b, d) the same projections in the space-filling representation. 180

Specific Intermolecular Interactions

181

in the plane of the layer. When stacking in crystal, these layers lose their symmetry except the centers of inversion. At the same time local mirror-planes arise in the form of contacts S(m2). They are easily seen in Figure 21 a. The infinite set of such contacts is a shifted pack {S(m2)} lying along axis Z. The symmetry of this chain (without considering H atoms) is Pc(y)21m.The crystal as a whole may be regarded as a packing of such chains oriented parallel. It is instructive to see what changes occur in the crystal structure if two CH 3 groups are replaced by OCH 3 groups as it takes place in II, where the molecule is nonplanar and reduces its pseudosymmetry

Figure 22. Crystal structure of phenol (PHENOL03). In crystalline phenol there are chains formed by conjugated H-bonds with very approximate axes 31. The packing of such spiral chains is stabilized by compactly organized "trihedral wells." (a) View along Y; (b) a pseudoparquet layer; (c) two adjacent trihedral wells; (d) another trihedral well.

182

PETR M. ZORKY and OLGA N. ZORKAYA

to 2/m. Crystals II (Figure 2 lc,d) also have approximately planar layers, which are parallel to plane (101) and contain all atoms except C atoms of methoxyl groups (without considering hydrogen). However, these layers are far from hexagonal; they may be approximately characterized by pseudosymmetry Pt2/m. Joining of the layers occurs with the substantial participation of methoxyl groups. Benzene tings form sandwich contacts S/xy with considerable shifts both along axes X and Y (in the coordinate system of one of two benzene tings), and the crystal appears as a set of chains {S/xy}. Naturally the probability of formation of BzC decreases with increasing number and dimensions of substituents adjoining the benzene ring. As seen from Table 3, BzC are absent in the crystals of some compounds with 12 and 13 carbon atoms in the molecule.

Phenol and Dihydroxybenzene Derivatives We now return to the above-discussed crystalline derivatives of phenol and dihydroxybenzenes (Table 1 and 2) in which systems of conjugated H-bonds (CHB) occur. The presence of specific intermolecular interactions that are stronger than those in BzC determines the secondary role of the latter. Consequently it is natural to distinguish BzC, first, occurring within H-agglomerates, and second, connecting individual H-agglomerates. In crystalline phenol (Figure 22a) effective BzC are not found within the molecular agglomerate [[Pc(x)31]]; in other words, such an agglomerate is rather loose. This is compensated for by the formation of a complicated nontrivial system of BzC between H-bonded spirals. A clearly defined chain {Ob(21)} arises in the contact ofpiles A1 and A]. A similar chain is formed by piles A] and A2. The above piles comprise a corrugated molecular layer (Figure 22b), which is close in

Figure 23. Crystal structure of meta-cresol (MCRSOL) viewed along Y (atoms H are omitted). Here there are analogs of the "wells" that exist in phenol, but methyl groups substantially reduce intermolecular interaction when they penetrate the system of benzene contacts.

Specific Intermolecular Interactions

183

character to the parquet layer. The structure is further stabilized by "triple" BzC, which occur in compactly organized "trihedral wells" (TW) (Figure 22c, d). Such wells, in which cooperative interactions of a great number of benzene tings presumably take place, are a peculiar feature of phenol. The crystals of m-cresol (Figure 23) have a similar structure, but on closer examination this similarity proves to be superficial. As in phenol, in m-cresol there too are chains {Ob(21) }; however, in the latter substance there are no pseudoparquet layers. There also exist analogs of the above-described "wells" but methyl groups substantially reduce intermolecular interaction when they penetrate the system of B zC. Crystalline p-cresol (Figure 24) also exhibits a complicated system of multiple BzC. Here attention is primarily drawn to orthogonal BzC of the O(2)/h type, which connect tetramers in chains directed along axis X. This contact (together with others belonging to OBzC) is shown in Figure 25. Such contacts arise between molecules A~ and B2 (as well as between molecules B~ and A2) of neighboring tetramers. Besides, in the piles of tetramers stacking in the direction of axis X there are effective BzC of the S(ml)/h and S/xy types between equivalent molecules AI and

,,I

~..

9

Figure 24. Crystal structure of para-cresol (CRESOL01) projected along X. In this crystal there are orthogonal benzene contacts of the O(2)/h type between molecules, i.e. A~ and B2 or B~ and A2; such contacts unite tetramers into chains directed along axis X.

184

PETR M. ZORKY and OLGA N. ZORKAYA

p-cresol

13-hydroquinone

o-cresol

a-hydroquinone

Kl

Kl

K2

K2

K3

K3

K4

%

Figure 25. Orthogonal benzene contacts projected onto plane A. A 2, and B l and B 2, correspondingly. Yet another notable feature of this structure is the formation of zigzag chains of the {Oh(21)} type (in Figure 24 these chains, which also lie along axis X, a marked by horizontal bars). Finally, antiparallel rings A 3 and A~ of two close tetramers form BzC of type S/xy. It is rather difficult to describe all observed BzC as a whole. We shall restrict ourselves to calling this totality the complex framework. The structures of o-cresol, [3-hydroquinone, and a-hydroquinone turn out to be extremely interesting and amazing when considered as the result of combined

Specific lntermolecular Interactions

185

action of H-bonds and BzC. These structures also give excellent examples of the structuring role of OBzC. In o-cresol there are three symmetrically independent molecules I, II, and III (Figure 3). This fact predetermines the nonequivalence of BzC (KI, K2, K3). However, all these BzC are orthogonal (Figure 25) and form an aggregate, namely a spiral Pc3 l, Z = 9(13). Such a spiral looks like a stem. The core of this stem (stem 1) is a helix of conjugated H-bonds formed by molecules II. The shell consists of benzene tings linked by consecutive (conjugated) OBzC. The connection of these complicated spirals occurs through the helical systems of conjugated H-bonds, formed by molecules I and III. In o-cresol each molecule has two H-bonds (as a donor and as an acceptor) and two OBzC (also as a donor and as an acceptor). The subsystem of molecules linked by H-bonds (or by OBzC) is a chain. At the same time, the molecular agglomerate formed by both H-bonds and OBzC is a complicated framework containing all molecules. Like o-cresol, crystalline 13-hydroquinone (Figure 4a) contains spiral chains, which, if each chain is regarded as isolated, belong to class Pc31, Z = 3(1). However, these chains have a different structure. Again, the chain ...-O0-Oi-O2... looks like a stem (stem 2, Figure 4b), but the core of this stem is more voluminous and includes molecules Ok which are consecutively linked by OBzC of O(2)/h type (Figure 25). Qualitatively this can be easily seen in Figure 26a. In OBzC OkOk§ molecule Ok is an acceptor, and molecule Ok§ is a donor (that is molecule Ok acts as ring A, and molecule Ok+l as ring B). The outer shell of the chain is weak-pronounced; it is formed by H-bonds, which unite the molecules into two rows" ...-00-02-04-... and ...-Oi-O3-O5-... Each of the rows can be transformed into another by operation 3~. Since the crystal belongs to structural class R3, Z = 9(]-), all molecules are equal; each of them lies at an inversion center and forms four OBzC: two as a donor, and two as an acceptor. Accordingly, each of the above chains overlaps with three neighboring chains (which is why these chains are not true molecular agglomerates in the ordinary sense). At the same time, there are conjugated OBzC which unite the molecules into "hexamers" (Figure 26b) with symmetry D3d, i.e. 3m (again, these hexamers are not genuine, for they overlap in the crystal). The hexamers are stacked along the Z axis; the arising pile is additionally stabilized by six adjacent spiral chains. Thus a complicated stem 3 shown in Figure 4c is formed; this stem is hollow, and that is why 13-hydroquinone can produce clathrates. Stem 3 overlaps with six similar stems. The totality of OBzC forms a framework, which is additionally stabilized by six-membered tings of conjugated H-bonds. The crystal structure of ct-hydroquinone (Figure 5a) reproduces some features of o-cresol and 13-hydroquinone; but it appears to be still more complicated. Since the substance belongs to class R3, Z = 54(13), there are three varieties of molecules (I, II, III) which occupy three orbits. Once again, in this structure we see spirals belonging to structural class Pc31, Z = 3(1). They are very like the spiral stems that occur in the 13-form (stems 213);

186

PETR M. ZORKY and OLGA N. ZORKAYA

Figure 26. Space-filling representation of the orthogonal benzene contacts in crystalline 13-hydroquinone. (a) Model of two overlapping stems; (b) model of a hexarner.

their core consists of consecutive OBzC, and the outer shell is sewn with H-bonds. In this crystal, however, there are two types of such spirals: (1) stems 2otl formed by molecules IIIwthey may be regarded as independent molecular agglomerates because they do not share molecules with other stems, though they are linked with them by H-bonds; and (2) overlapping stems 2ot2 formed by molecules I and II. Stems 2(z2 and the way of their connection in the crystal require a more detailed description. In such a spiral two molecules I and one molecule II are consecutively linked by OBzC; molecules I are shared by two spirals. The spirals coupled in such a way form a chain hexamer. This is really a molecular agglomerate since it does not overlap with the neighboring similar stems; its structural class is Pc3, Z = 12(12). The hexamer is, on the whole, the same as stem 3 in the 13-form. The spiral containing 15 H-bonds per period sews molecules into stem 4 (Figure 5b). In ot-hydroquinone stem 4 overlaps with three neighboring similar stems; the arising structure is stabilized by the above hexamers. In general, this construction is like the structure of 13-form; however, stem 4 is much more complicated then stem 2 that generates class R3, Z = 54(13) instead of the R3, Z = 9(]-) class. There are four unequal OBzC in (~-hydroquinone (Kl, K2, K3, K4) (Figure 25). Their parameters and projections show that these OBzC are of the O(2)/s, O(2)/s0, O(2)/sga0~, and O(2)/s0 types, respectively. Each molecule takes part in four H-bonds (twice as a donor and twice as an acceptor) and in four OBzC (in the same way). The totality of H-bonds unites the molecules into a framework; the totality of OBzC forms stems 2 and 3. Of course, the joint system of H-bonds and OBzC is a spatial framework which is very intricate in this case. The structure of ct-hydroquinone as that of its 13-form is suitable for clathration. Naturally the contribution of BzC to intermolecular interaction and to the formation of the crystal structure diminishes with increasing number and dimensions of substituents. In the other crystalline substances named in Table 1 and 2, aggregates of BzC take very simple forms or do not exist.

Specific Intermolecular Interactions

187

In p-methoxyphenol, 2,3-dimethylphenol, 2,5-dimethylphenol, 2,6-dimethylphenol, and 4-methoxy-2,3,5,6-tetramethylphenol BzC formed by parallel benzene rings that are shifted relative to each other give chains of translationally equivalent molecules. In the case ofp-methoxyphenol (Figure 1) there are additional contacts of the Ob(2q) type, resulting in the formation of parquet layers parallel to plane YZ. In 3,4-dimethylphenol, there arise OBzC of O(4)/s type; they form a complex framework together with shifted sandwich contacts S(m2) and S/xy. Very simple aggregates of BzC are found in 2,5-dimethylhydroquinone and 2,3,5,6-tetramethylhydroquinone; once again, it is the chains formed by translationally equivalent (hence parallel) benzene tings shifted relative to each other. In p-isopropylphenol, 3-methyl-4-isopropylphenol, resorcinol, and catechol the contacts of the Ob(2q) type form zigzag chains {Ob(21) }. As Tables 1 and 2 show, within H-agglomerates the BzC are, as a rule, shifted sandwiches; they most often belong to stereotypes S(ml) or S(m2), but in some cases the symmetry plane of the contact disappears and there arise a deformed contact S/xy. The BzC that unite H-agglomerates often belong to orthogonal BzC or to stereotype Ob(2q). Sometimes a combination of different BzC leads to complex frameworks of such contacts as it takes place in p-cresol and 3,4-dimethylphenol.

IV. CONCLUSION The presented material shows that both of the discussed types of specific intermolecular contacts (conjugated hydrogen bonds and some typical variants of arrangement of adjacent benzene tings) frequently occur in crystalline derivatives of benzene, phenol, and dihydroxybenzenes. Undoubtedly, they are particularly advantageous and play an important structuring role. Only in the presence of large and/or numerous alkyl or alkoxyl substituents do steric factors arise hindering the formation of these intermolecular contacts. A noteworthy and important circumstance lies in the fact that in the vast majority of cases the resulting benzene contacts have either a plane of symmetry (47% of cases) or a hypersymmetry axis 2q (40% cases). The benzene contacts that are devoid of any of these traits occur rarely (13% of cases). This means that in the formation of one or another of local molecular subsystems observed in the crystal the intrinsic symmetry of benzene tings shows up very often (while it is virtually always absent in the crystal structure as a whole); in other cases the possibility of formation of advantageous zigzag chains or of parquet layers is realized as usual. In our previous paper [21] an influence of the own symmetry of molecules on the crystal structure has been considered. Now we can say that even the symmetry of some moieties of molecules, e.g. benzene tings, often manifests itself in molecular arrangements. Specific intermolecular interactions cause the formation of definite aggregates of specific contacts and of corresponding molecular agglomerates. In some cases

188

PETR M. ZORKY and OLGA N. ZORKAYA

this leads to nontrivial intricate molecular arrangements. Particularly interesting diverse crystal structures arise as a result of a combination of these conjugated hydrogen bonds and specific benzene contacts. The approach we have taken (the description of aggregates of conjugated specific contacts and molecular agglomerates united by such contacts from the standpoint of their symmetry and topology) in some cases provides a way of interpreting the observed structures or, at least, of presenting them in the form of a well-systematized survivable set.

ACKNOWLEDGMENTS The authors are grateful to A.V. Kamchatldn for appreciable help in preparing this manuscript. The work was supported by the Russian Foundation for Fundamental Research, Grant 96-03-32455.

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

Zorky, P. M. DokL Akad. Nauk SSSR 1%1, 38, 2, 355-357. Zorky, P. M.; Porai-Koshits, M. A.; Psalidas, V. S. Zh. Strukt. Khim. 1966, 7, 4, 577-582. Zorky, E M.; Porai-Koshits, M. A.; Sveshnikova, S. N. Zh. Strukt. Khim. 1966, 7, 5, 752-758. Zorky, P. M. Russ. J. Phys. Chem. 1994, 68, 870-876. Zorky, E M. Ross. Khim. Zhurn. (Mendeleev Chem. J.) 1996, 40, 3, 5-25. Zorky, P. M.; Lanshina, L. V.; Korableva, E. Yu. J. Struct. Chem. 1994, 35, 256-259. Zorky, E M. J. Struct. Chem. 1994, 35, 364-366. Desiraju, G. R. Crystal Engineering: The Design of Organic Solids. Elsevier: Amsterdam, 1989. Zorky, E M.; Zasurskaya, L. A. In Problemy Kristallokhimii; Porai-Koshits, M.A., Ed. Nauka: Moscow, 1986, p. 7-31. Zorky, E M. J. Mol. Struct. 1996, 374, 9-28. Zorky, E M.; Zorkaya, O. N.; Lanshina, L. V. J. Struct. Chem. 1995, 36, 704-717. Hobza, E; Selzle, H. L.; Schlag, E. W. J. Chem. Phys. 1990, 93, 5893-5897. Hobza, P.; Selzle, H. L.; Schlag, E. W. J. Phys. Chem. 1993, 97, 3937-3938. Williams, D. E. Acta Cryst. 1980,A36, 715-723. Dzyabchenko, A. V. Zh. Strukt. Khim. 1984, 25, 3, 85-90. Dzyabchenko, A. V. KristaUografiya 1989, 34, 226-229. Gibson, K. D.; Scheraga, H. A. J. Phys. Chem. 1995, 99, 3765-3773. Dzyabchenko, A. V.; Belsky, V. K.; Zorky, P. M. KristaUografiya 1979, 24, 221-226. Lonsdale, K. Proc. Roy. Soc. London 1929, A 123, 494. Brockway, L. O.; Robertson, J. M. J. Chem. Soc. 1939, 1324. Zorky, P. M.; Potekhin, K. A.; Dashevskaya, E. E. Acta Chim. Hung.--Models in Chemistry 1993, 130, 221-233.

ISOSTRUCTURALITY OF ORGANIC CRYSTALS"

A TOOL TO ESTIMATE THE

COMPLEMENTARITY OF HOMOAND HETEROMOLECULAR ASSOCIATES

Alajos K~lm,~n and L,~szl6 P~rk,~nyi

I~ II.

III.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Main Part Isostructuralism . . . . . . . . . . . . . . . . . . . . . . . . . Isostructurality o f Homomolecular Crystals . . . . . . . . . . . . . . . . . . A. Numerical Descriptors o f Isostructurality . . . . . . . . . . . . . . . . . B. Conditions and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Correlation between Isostructurality and Co-Crystallization . . . . . . . D. Isostructurality in General . . . . . . . . . . . . . . . . . . . . . . . . . E. The Relaxed Forms o f Isostructurality . . . . . . . . . . . . . . . . . . . F. Forms of Isostructurality in the Ph3E-E'Me3 and Ph3E-E'Ph3 S e r i e s . . .

Advances in Molecular Structure Research Volume 3, pages 189-226 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

189

190 190 191 191 195 196 196 201 203 204 208 211

190

ALAJOS KALMAN and LASZLO P,~RKANYi

IV. Isostructurality of Supramolecules . . . . . . . . . . . . . . . . . . . . . . . . 214 A. Adducts with Slight Difference in the Guest (or Host) Molecules . . . . . 214 B. Clathrates Formed by Basically Different Guest Molecules . . . . . . . . 216 C. Homostructural Adducts Formed by Different Host and Guest Molecules. 218 D. Clathrates of2,2'-Bis(3,4,5-trimethoxyphenyl)-l,l-bibenzimidazole . . . 221 V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

ABSTRACT The pioneering work by Kitaigorodskii in the 1950s resulted in several conclusions concerning the isomorphism and close packing of crystals, though the lack of adequate experimental data did not permit one to address or clarify some important aspects. To elucidate these questions left unanswered by Kitaigorodskii, examples are shown which represent different forms of packing similarity. Main-part isostructurality, conditions and limits of the isostructurality of homomolecular crystals, and the numerical descriptors of isostructurality are demonstrated for a series of cardiotonic steroids. Iso- and homostructural relationships and "morphotropic steps" (Kitaigorodskii) are shown in the series of the group 14 homologues of Ph4E (E = C, Si, Ge, Sn, Pb) and Ph3E-E'Me3, (E, E ' = Si, Ge, Sn). Isostructurality of supramolecules is discussed through cases of adducts with slight differences in the guest (or host) molecules, clathrates formed by basically different guests, homostructural adducts formed by different hosts and guests, and clathrates of 2,2'-bis(3,4,5trimethoxyphenyl)- 1,1 ,-bibenzimidazole.

!. I N T R O D U C T I O N From a topological point of view, the close packing of each crystal is governed by the complementary surfaces [1] of "host" and "guest" molecules. The simplest model of this concept is a triclinic crystal (Z = 1) with space group P1 in which every "guest" molecule is surrounded by 26 isometric "host" molecules. In general, the self-complementary surfaces of the homochiral molecules reach their optimum in those space groups which possess the best combination of rotational symmetries (e.g. space group P212121). If self-complementarity is extended (i.e. the bumps of the bulky molecules fit perfectly into the hollows formed by themselves [2]) t h e n - - a s revealed by the discovery [3, 4] of the recurrent isostructurality of numerous cardiotonic steroids (cardenolides and analogous bufadienolides)-these associations tolerate atomic replacements, even substitutions, or a change in chirality (epimerization) without visible decrease of the already established close packing. The preserved, or just slightly modified, molecular harmony indicates a certain ability by which the ca. 30% empty space (voids or small cavities) in the crystal lattices is exploited without any morphotropic rearrangements [2]. Consequently, it has been shown [5, 6] that the study of the forms and degree of

Isostructurality of Organic Crystals

191

isostructurality is a suitable tool for better understanding both homo- and heteromolecular close packing. Present work attempts to review the conditions of isostructurality exhibited by zero- and first-order supramolecules. The latter in general are termed as binary adducts. Whenever we encounter the phenomenon of isostructurality we must bear in mind that it is inversely related to the widely observed phenomenon termed as polymorphism. It is hardly necessary to point out that polymorphism is of great industrial, in particular pharmaceutical, importance [7]; e.g. the tableting behavior of powders, physically stable dosage forms, and chemical stability are equally dependent on it. The differences between the various polymorphs (at least dimorphs) of a compound manifest themselves as differences in solubility, rate of dissolution, and vapor pressure. Just as different chemical compounds can have different polymorphs, solvates (many of them are in fact inclusion compounds [8]) of different compounds can exhibit polymorphism as well. Our attention was first drawn to the relationship between polymorphism and the less common isostructurality by the study ofpara-disubstituted benzylideneanilines [9] which are mutually cross-linked by these contrasting phenomena.

!1. HISTORICAL BACKGROUND A. Isomorphism The first step towards the knowledge of the very important phenomenon of isomorphism frequently encountered in nature was made by Rom6 de l'Isle in 1772. He noticed that by placing potassium alum crystals in saturated ammonium alum solution the crystals start growing and become coated with an ammonium alum layer. After this and other early observationsme.g. Wollaston (1809) recognized that calcite, magnesite, and siderite (Ca, Mg, FeCO3) all crystallize as rhombohedra with slightly different faces--in 1819 Mitscherlich found that certain pairs of salts (e.g. KH2PO 4, KH2AsO 4, NHaH2PO4, and NHaH2AsO4) developed the same crystalline form and such salts had similar chemical formulae; one kind of atom in one compound being replaced by another kind in the "related" salt. These pairs were called isomorphous. Soon it became clear that isomorphism plays an important role in chemistry. For example, analyzing such a pair of salts immediately gave the relative atomic weights of these atoms. Thus Berzelius (the teacher of Mitscherlich) established the atomic weight of selenium in 1828 following the discovery of the isomorphism of Na2SO4, Ag2SO4, Na2SeO4, and Ag2SeO4. The characterization of the phenomenon was developed further by von Groth (1874), N~ay-Szab6 (1969) [10], and others. They concluded that the basic condition for isomorphism of compounds is that crystals are closely similar in shape while one of the crystals continues growing in the solution of the other. In other words, lattice-type crystals should be identical and the formation of a solid solution (mixed crystals) is guaranteed. To define isomorphism precisely, however, has remained impossible.

192

ALAJOS K,~LM,~Nand LASZL(9 P,~RKANYI

This can be attributed to the fact that the term refers only to the external similarity of crystalline substances. Some authors (e.g. Wells [11] and Bloss [12]) therefore have introduced the terms isostructural and isostructurality (or isotypism), whereas others, like Zoltai and Stout in the book, Mineralogy [13], gave the following definitions: 9 Isomorphism: solid solution series in which the crystal structure is the same throughout the series. 9 Isostructural: relationship among minerals of different chemical composition and identical crystal structures. 9 Isotypism: relationship among minerals constructed by different stacking of the same structural unit, but containing different cations in certain structural sites. This controversial situation forced the IUCr Subcommittee on Inorganic Structure Types [14] to recommend the use of the terms isotypism (and homeotypism), but exclusively for inorganic compounds. As far as organic compounds are concerned, prior to our activity in the field [5] Kitaigorodskii was the first who in his fundamental work [2] summarized the contemporary knowledge of the phenomenon by using the inaccurate and thus misleading term, isomorphism. He simply assumed that "ions are of precise spherical symmetry in the crystals, so that changing one ion for another merely involves replacing one structural element (a sphere) by another of the same shape. The situation is different with organic crystals where molecules can replace one another: thus strict geometrical similarity (i.e. proportionate change in all external measurements) never occurs" (page 222 in ref. 2). His observations on isomorphism can be summarized in four conclusions: 1. " . . . molecules will be termed approximately isomorphous if their structures are obtained on replacing atom by atom, such cases being common in organic chemistry" (page 223 in ref. 2). 2. Approximate isomorphism of the C1-Br-I series may result in the "isomorphism" of their crystals if the replacement does not diminish the already existing "high" packing coefficient. Otherwise in a "morphotropic step" a new packing will be developed resulting in a new crystal lattice. 3. "In some cases, however, it is meaningful to speak of a species of conditional isomorphism and thereby imply the rough similarity that occurs between two large molecules which differ only in small radicals [e.g. methylcoronene and nitrocoronene, or hydroxynaphthalene and aminonaphthalene]. We shall not seek to define what the molecular size must be so that conditional isomorphism will result in crystal isomorphism, for insufficient data are as yet available" (p. 225 in ref. 2). 4. Series of homologous molecules (series of condensed and non-condensed aromatic compounds, each of which contains one benzene ring more than

Isostructurality of Organic Crystals

193

the preceding one, e.g. biphenyl, terphenyl, quarterphenyl, etc.) "usually adopt exceptionally similar ways of packing: not only is the cell symmetry the same, but so are cross sections normal to the directions in which the molecules extend. This can termed as ~homologous isomorphism>> we have given examples earlier" (p. 228 in ref. 2). These conclusions are still basically valid. The lack of appropriate refinement of the atomic positions in the 1950s, however, made Kitaigorodskii share the misbelief that inorganic crystals are formed by the close packing of ions of very precise spherical symmetry and in such array one spherical element can just be replaced by another. At that time crystal structures like Na2SeO4 [15] isomorphous with thrnardite (an orthorhombic polymorph of Na2SO4) [10] were still frequently depicted in such a classical form; i.e. each ion is shown as a perfect sphere (Figure 1). Without experimental data the boundaries of approximate molecular isomorphism could not be estimated either, while the term conditional isomorphism did not specify what sort of small radicals would generate substantially different hydrogen bond networks which forbid any packing similarity. To elucidate these

Figure 1. Packing diagram of Na2SeO4 as represented by spherical ions. The dark small balls represent Se6+ ions, the large balls are for O 2+ ions, while the Na + cations are shown as grey balls of medium size.

k~

~ ~

X~ ~

O X T ~

~~ C~ m~

o~

O ~o

Q_ ~D

~D Q_

--,

~o

9

~2

C~

~~

~

o

X

~

~o

~8

3~

2~

c~

Z

r---

N

r~

Z

f~

9

rm.

Isostructurality of Organic Crystals

195

questions left unanswered by Kitaigorodskii, examples will be shown which represent different forms of packing similarity. B. Main Part Isostructuralism

The term quasi-isostructural was first applied to two related cardenolide crystals, digirezigenin and digitoxigenin, when we published the X-ray analysis of digirezigenin [16] formed from digitoxigenin [17] via 14,15,[3-ring-closure. Through continuing X-ray studies of analogous bufadienolides isolated from Ch'an Su (the dried venom of the Chinese toad), a second pair formed by arenobufagin and gamabufotalin was also found quasi-isostructural [18]. Simultaneously, we observed that the 5-OH group of cinobufotalin [19] does not alter the crystal lattice found for cinobufagin [20] either, while a search in the literature revealed that both 3-epi-digitoxigenin [21] and (21R)-methyldigitoxigenin [22] are also quasi-isostructural with digitoxigenin (Figure 2). These observations led to the first characterization of this recurrent phenomenon shown by numerous cardenolides and bufadienolides [3]. The structure determinations of (21S)-methyldigitoxigenin (also isostructural with digitoxigenin) and uzarigenin (configurational isomer of digitoxigenin about C5) also helped to shed more light on their persistent structural similarities. Bearing in mind the classification of inorganic substances published in ref. 14 which discourages the use of the terms "isomorphous" and "isomorphism", we felt that the structural similarities exhibited by organic crystals also cannot be termed as isomorphous. Instead, the term isostructural (recommended by Wells and Bloss, vide supra) is adequate since it refers directly to the chemical conditions of structural similarity developed in a similar morphological (isomorphous) environment. To emphasize the principal difference from the conventional definition of "isostructuralism" given basically for inorganic substances by Bloss [12]Eeach related group of compounds can be characterized by a general formula in which the quality of the constituent atoms is changed while their relative quantity remains constantm in ref. 3 the adjective quasi (isostructural) was replaced by the term main-part. Namely, cardenolides (y-lactones) and bufadienolides (8-1actones) can be described by the formula Ck+rHl§ Om+twhere k, l, and m denote the number of common atoms in a pair (or group) of the compounds related by the phenomenon in question, whereas r, s, and t (0, +1, +2) express the differences in their chemical composition. This term equally refers to the approximate isomorphism (more precisely isometricity) of molecules which is prerequisite of any packing similarity. Since small molecules like urea and thiourea cannot be isometric, they cannot develop similar packing motives either [2]. This is the reason why close packing similarity among organic crystals is less common than in inorganic systems. Namely, in inorganic crystals anions and cations varying in shape from spheres to different polyhedra (trigonal pyramids, tetrahedra, octahedra, etc.) frequently de-

196

ALAJOS IG~LMANand LASZLO P,~RKANYI

velop similar packing arrays which can be regarded configurationally and even chemically as isotypes [14]. For example, N~ay-Szab6 [10] listed 23 crystal structures (SrMoO4, NaLa(MoO4)2, KCe(WO4)2, etc.) which are isotypes of the tetragonal scheelite (CaWO4). The accepted definition of isotypism permits the addition to this list of crystals such as NalO4 [23] and KIO4 [24]. This can be attributed to the fact that the hierarchy of terms recommended [14] for the characterization of the inorganic crystals (isopointal, configurationally isotypic, crystalchemically isotypic, and homeotypic structures) is built up along the identical Wyckoff positions (both in letters and multiplicity), which for most organic substances are limited only to general positions. Further studies on the main-part isostructuralism of cardiotonic and other related steroids helped to understand how far the isometricity of these relatively rigid molecules can be altered without changing their identical packing arrays. Consequently, the phenomenon termed "isostructurality" indicates that beyond unit cell and space group similarities both molecular isometricity and self-complementarity are substantial.

III. ISOSTRUCTURALITY OF HOMOMOLECULAR CRYSTALS

A. Numerical Descriptors of lsostructurality The rigid 14-isoaethiocholane skeleton common to 14 steroid structures (Table 1 and Figure 3) proved to be a genuine model to estimate the degree of isostructurality and its dependence on the molecular isometricity by the use of three descriptors. They have been inferred from the unit cell parameters and the absolute atomic coordinates of the related molecules assumed in the unit cells possessing identical (or at least fairly similar) space group symmetry. These descriptors should reflect the crystallographic consequences caused either by tolerable conformational (even configurational) differences or by changes in the chemical compositions of the isometric pairs expressed by the indices r, s, and t in the general formula of the steroids given above. Although molecular isometricity is prerequisite to isostructurality, in itself does not guarantee the similar packing motif for any related organic crystals. Consequently, the morphological descriptors (the similar brutto formulae accommodated in similar unit cells etc.) can always be recognized first, which is then followed by the estimation of the degree of isostructurality. Of course, the latter already requires the knowledge of the atomic coordinates. Finally the isometricity index can also be calculated with Eq. 3 by using the properly minimized parameters inferred from the least-squares fit of the superimposed molecules. It is evident that the structural differences influence the volume of the asymmetric unit (V'), the cell parameters, and the fractional coordinates of the common atoms of CkHIOm. The first can be estimated from thepacking coefficient increments A(pc),

M,.?__~o

0

i H.

H-,.I HO

OH

.o.

I .

0

0

o~

0

f

6 O~

HO

H ~,

o

H.o

I OH

I:1

w

0

0

0

13

OH 14

0

0

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/I

HO

H

1

C)

HI?

H

o ~ ~o

H

8

H

O

~

10

~ 0%__0

0

o

0

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H

2

",.J

HO,~7~J

! H

Io.

_ 7

MeOH 9

,.

HO

OH 9

:oZ

I

.L J~ " H

12

Figure 3. Structural diagrams of cardenolides and bufadienolides: (1)digitoxigenin, (2)digirezigenin, (3) 3-epi-digitoxigenin, (4)(21R)-, (5) (21S)-methyldigitoxigenin, (6)uzarigenin, (7)sarmentogenin, (8) 19-nordigitoxigenin, (9)scillarenin, (10)bufalin, (11) gamabufotalin, (12) arenobufagin, (13) cinobufagin, (14) cinobufotalin.

198

ALAJOS K,~LMAN and Lg,SZLO Pg,RKt~NYI

Table 1.

Lattice Parameters (Space Group P212121)and Densities for Cardenolides and Bufadienolidesa

Digitoxigenin (1) Digirezigenin (2) 3-epi-Digitoxigenin (3) (21R)-Methyldigitoxigenin (4) (21S)-Methyldigitoxigenin (5) Scillarenin (9) Bufalin (10) Gamabutotalin (11) Arenobufagin (12) Cinobufagin (13) Cinobufotalin (14)

a (,~)

b (i~)

c (1~)

Dx (Mg m -3)

7.250(2) 7.288(2) 7.279(3) 7.249(1) 7.193(1) 10.387(2) 10.726(1) 7.850(1) 7.826(1) 7.663(2) 7.631(1)

15.015(4) 14.686(3) 14.685(9) 15.109(1) 15.208(3) 12.075(4) 12.381(2) 14.766(1) 14.864(2) 15.900(5) 15.727(5)

18.464(8) 18.480(3) 18.541(10) 19.268(3) 19.277(4) 16.431(6) 15.717(1) 17.836(1) 17.841(2) 19.291(5) 19.557(2)

1.238 1.251 1.255 1.223 1.224 1.239 1.230 1.293 1.333 1.249 1.297

aEstimatedstandarddeviationsin parentheses.

A(pc) =

A(rst) - A V' V

(1)

f

where AV' is the change in volume of the asymmetric unit while A(rst) is the net difference between the volume of the newly added and/or deleted atoms, by including the volume alteration resulting from the adjoining carbon atoms. A low A(rst) suggests a high degree of isostructurality. Of course, to avoid any marked change in the packing, A(pc) must be kept as low as possible (ca. +2.0%) in such a way that the lattice parameters change slightly with the increment of some cell edges being balanced by a decrease of others. This can be expressed by a second descriptor termed as the unit cell similarity index. Beyond the similar axial ratios and interaxial angles, the internal motion of the lattice parameters can be estimated by Eq. 2 where a, b, c and a', b', c' are the orthogonalized lattice parameters of the related crystals. H

~

a+b+c -1 a' + b' + c'

~0

(2)

In case of great similarity between two unit cells FI is practically zero. Of course, both A(pc) and FI refer only to alterations in the cell dimensions, therefore the internal degree of isostructurality should be quantified in some other way. The isostructurality index serves this purpose, 1/2

li(n)=I 1-(ZARzI~n )

]xlO0

(3)

Isostructurality of Organic Crystals

199

where n is the number of the distance differences (Aei) between the absolute coordinates of the identical non-hydrogen atoms within the same section of the asymmetric units of the related (A and B) structures. It is worth noting that the AR i values should not be refined (i.e. their positional differences caused by rotational and translational operations must not be corrected). Namely, a full, or partial least-squares fit of the positions occupied by the identical (ia and in) atoms results in different possible superpositions of the molecules. When refined AR~ values are used in Eq. 3 then the new index I i (n*), termed as molecular isometricity index, is a direct measure of the degree of the approximate isomorphism [2] of molecules A and B. As the hydrogen atomic positions are generally less precisely determined than those of the heavy atoms (at least by X-ray diffraction) they are excluded from both kinds of li (n) calculations. For the cardenolides and bufadienolides discussed, the /~l~ i distances were calculated between the common non-hydrogen atoms (n = k + m) within the same section of the asymmetric units of the related structures and applied in Eq. 3. From the above definitions, we observe the following (see Figure 3, Table 1): 1. The highest degree of isostructurality (95%), in accordance with the low FI = 0.002 and A(pc) = -0.1 values, is shown by 11 and 12. This is also confirmed by the excellent agreement between I i (19) and I i (29). The second best agreement is shown by cinobufagin 13 and cinobufotalin 14. Even the 16fl-acetoxy group occupies a similar position as is shown by the value Ii(28)-Ii(32)=-1%. 2. The lowest degree of isostructurality is exhibited by the scillarenin 9 [25] and bufalin 10 [26] pair which can be attributed to the diminished isometricity of these molecules, i.e. to the relevant difference (see above) between the conformation of ring A and its surroundings (half chair versus chair). This can be estimated from the difference between//(19) and the average of 1/(19) for pairs 11/12 and 13/14 (A/i(19)= 43.5%). In contrast, the almost identical position (conformation) of the ~5-1actone tings is illustrated by the small value of I i (21) - I i (28) = 2%. 3. The similar//(19) values (90%) obtained for the pairs 1/4, 1/5, and 4/5 suggest that neither the presence of the 21-methyl group nor a change in the chirality at C21 alter considerably the positions of the 19 skeletal atoms. Similarly, they do not influence the crystallographic positions of the skeletal oxygens 03 and O 14 either. The average of the//(20) and I i (21) values is still 89.5(5)%. 4. The 21-methyl group and its position, however, account for the degree of rotation about C17-C20 within the first potential-energy valley at q~ = 670-97 ~ as calculated by H6hne and Pfeiffer [27]. This is shown by the Aq~ values (-8 ~ 17~ and 25 ~ for the pairs 1/4, 1/5 and 4/5" t h e / / ( 2 7 ) - / / ( 2 1 ) values (5, 10, and 16%) are proportional to them (A/i(27) = 4% corresponds to ca. Aq~= 10~ of internal rotation of ring E). As suggested by the low

200

5.

6.

ALAJOS KALM,~N and Lg,SZLO P,~,RKANYI

//(27) -//(28) = 3% for the isomeric pair 4/5, the internal rotation of ring E brings the 21-methyl group into a common, favorable position in the voids of the isostructural lattices. The//(19) (average) - / / ( 1 9 ) (1/2) - 5% difference is attributable to the change in the conformation of ring D pertaining to 2 (the envelope shape 14E found in (1) is shifted to 17E). This is in accordance with the formation of the 14,1513-epoxy ring which is indicated directly b y / / ( 2 0 ) - / / ( 2 1 ) = 6%. Despite these alterations the position of the l-lactone ring is undisturbed (//(21) -//(27) = 0% versus Aq) = 0 ~ Finally, for the diastereomeric pair 1/3, in addition to the effect of epimerization around C3, indicated by li(19)-//(20) = 29%, and the rotation (Atp) = 10 o about C17-C20 (Ii(21)-Ii(27)= 5%) Ii(26)=49%), the second hydrogen bond accounts directly for a significant decrease in isostructurality as estimated by//(19) (1/2) - li(19)(average) = 12%. This points to the significant contribution of even one hydrogen bond to the lattice packing. Of course, the common 14-isoaethiocholane skeleton is indicated by the high degree of isometricity between 1/3,//(26*) = 88% (the corresponding isostructurality index is only//(26) = 49%). I, (n) (%) 100 f 90 80

70 x

60

1,A

o6v

2

9 3,A

7 08n

4 +5

9

50

40

C2

30

1

10

Figure 4.

15

I

20

I

25

.

I

30

35

l(n) (%) versus n curves for nine isostructural steroid pairs. Each starts from n = 10 and runs until the last common atom of the isostructural pairs. After n = 21 or 22, the atoms of the y- and 8-1actone rings are treated as compact units. In curve 2, there is an additional 1613-acethoxy group. (I) arenobufagin-gamabufotalin; (2) cinobufotalin-cinobufagin; (3)(21S)-methyldigitoxigenin-digitoxigenin; (4)(21 R)- and (21S)-methyldigitoxigenins; (5) digitoxigenin-digirezigenin; (6) 5ot-androstane-3 13,1713-diol monohydrate-5-androstene-3bg,1713-diol monohydrate; (7) 3-epidigitoxigenin-digitoxigenin showing the dramatic change in the function when 0 3 is taken into account in Eq. 3; (8) scillarenin-bufalin; (9) 5or- and 5ig-epimers of androstane-3ot, 1713-diol.

Isostructurality of Organic Crystals

201

li(n) can be calculated not only for the total set of common non-hydrogen atoms [5] but also as a function of n. In this form, n starts from a well-chosen "isostructural core" and goes to the last of the common pair of atoms taking into account each relevant group in stepwise mode (Figure 4). Since four 1 ~ i ValUeS determine the relationship between two structures in three-dimension, the isostructural core should be defined at least by the four shortest ' ~ i ValUeS. For the relatively large steroid molecules (cf. Table 6 in ref. 5) the isostructural core was defined deliberately by the 10 shortest Z ~ i values pertaining to the steroid skeleton. From these 10 up to the total 19 skeletal atoms, the atomic sequence is determined by the monotonously increasing AR i values. In the next step the adjoining functions (e.g. y- or ~5-1actonetings) are taken into account. In general, li (n) calculations may apply to a different grouping of atoms. In this case, however, li (n) function depends on the arbitrary sequence of the Z ~ i values especially if there are substantially different A R i values. This ambivalency of the li(n*) and li(n) calculation, however, enables us to explore marked differences in molecular isometricity and their degree of isostructurality in the crystalline state. B. Conditions and Limits Beyond a minimum degree of isometricity, isostructurality is also dependent on numerous other intra- and intermolecular properties. First they are summarized for the above-mentioned homomolecular and homochiral steroids. Table 1 shows that the cardenolides 1-5 are isostructural, whereas the closely related uzerigenin 6 (monoclinic, space group P21) is not. The common feature of the isostructural cardenolides (2-5), arranged around 1 in Figure 3, is the similar conformation of the 3,-lactone ring (with the ring oxygen on the left-hand side). The non-isostructural 6 and 8 (19-nordigitoxigenin) [28] exhibit the opposite lactone ring conformation equally (with the ring oxygen on the fight-hand side). Of the three bufadienolide pairs (9/10, 11/12, and 13/14) the first bears the 8-1actone ring in the opposite position to that assumed in the other two, but the orientation of the lactone-ring is similar to that found in 6 and 8. A review of the 14 steroid structures shown in Figure 3 suggests that rotation of the lactone ring about C 17-C20 bond in either of the two energetically preferred ranges [27] is one of the principal factors which controls the isometricity between two (or more) closely related steroid structures. (The equal probability of both conformers is supported by the disordered crystal structure of sarmentogenin (7) in which both conformers appear in equal proportion [3]). A comparison of the q~ torsion angles shows that in 8 the 7-1actone ring is rotated by ca. 180 ~ from the range of 650-93 ~ (preferred by 1 and 2-5) to a second energy minimum around -100 ~ [tps = -97(1)~ which is also revealed by the potential-energy calculations of H0hne and Pfeiffer [27]. This conformational change apparently alters "the head-to-tail" molecular packing found in 1 and hinders the isostructurality. On the other hand, 9 (scillarenin: 313,14-dihydroxy-1413-bufa-4,20,22-trienolide), which is

202

ALAJOS K,g,LM,~,N and L,~,SZL6 P,g,RKANY!

formed from helleborogenone (14-hydroxy-3-oxo- 14[3-bufa- 1,4,20,22-tetraenolide) via partial A-ring saturation [25] becomes isostructural with 10 (bufalin) in such a way that its 6-1actone ring, at least in the crystalline state, undergoes a rotation of 171 o from the first to the second potential-energy valley. Despite the difference between the conformation of ring A (half-chair versus chair), the similar 6-1actone ring position (q~9 = 93(1) and qho = 87(1) ~ is sufficient to establish isometricity and isostructurality between them (cf. Figure 4. in ref. 6). The lack of isostructurality between 6 and 1 arises from the differences in orientation of the ?-lactone ring and from the configurational isomerization (epimerization) about C5, which results in a rigid trans-A/B junction. The different A/B junction (trans versus cis) also excludes the isometricity of 6 and 8 although the orientation of the lactone tings is similar (q)6 = -93(1)~ versus q~s = -97(1)~ As shown by the superposition of the steroid skeletons 6, 9, and 10 (Figure 5), the flexible A4-half-chair shape of ring A 9 displaces 03 from the position observed in 10 by 1.33/~, but retains, at least in part, the half-moon shape of the 14-isoaethiocholane skeleton. Presumably this is the reason why 9 remains isostructural with 10. In contrast, the rigid trans-A/B junction in 6 alters the 14-isoaethiocholane skeleton, displacing 03 still further (by 1.75 A). This new 03 position does not allow the formation of the head-to-tail hydrogen bond developed by I and 10. Within the group of five isostructural digitoxigenin derivatives there are two isostructural pairs related by epimerization. The chemical formulae of 1 and 3 (C23H3404)and 4 and 5 (C24H3604),respectively, is the same and differ only by epimerization either about C3 of ring A or C21 of the ?-lactone ring. A further condition of isostructurality is that the changes in chemical composition expressed by the indices r, s, and t should not alter the existing hydrogen-bond network. For example, in spite of the 14,1513-epoxy ring closure 2 (digirezigenin) remains isostructural with 1. This can be attributed to the long intermolecular O14--.O 1 distance observed in 1 (and 2, 4, and 5). In other words, the formation of a

Figure 5. A superposition of the steroid skeletons of 6 (full circles) 9 (dotted circles) and 10 (open circles) showing the differences in the conformations of ring A.

Isostructurality of Organic Crystals

203

14,15[3-epoxy ring in 2 does not hamper the formation of hydrogen bonds. This is not the case, however, when cinobufagin is formed from bufotalin [19]. C. Correlation between Isostructurality and Co-Crystallization Co-crystallization is the most classical concomitant of isomorphism. It appears that the co-crystallization of the steroid pairs investigated is also dependent on the degree of their isostructurality. The components related by high li(n) (1:1 mixtures of gamabufotalin-arenobufagin li(19) =//(28) = 95% and cinobufagin-cinobufotalin//(19) = 94,//(28) = 89%) co-crystallize readily forming a disordered solid solution, while those which possess low Ii(n ) values (bufalin-scillarenin (Ii(19) = 51,//(28) = 48%) could be co-crystallized only after several fruitless attempts from n-butyl acetate in the form of an 1:1 adduct. In contrast to these solid solutions (Table 2) the asymmetric unit of the monoclinic adduct of digitoxigenin (1) and digirezigenin (2) contains two molecules (A and B). Therefore, the four symmetry related positions of P212121 split into 2 x 2 positions in the monoclinic unit cell retaining space group P1121 with ~' = 90.35 ~ Consequently, the adduct remains quasi-isostructural with both component crystals. Molecule A is characterized by the 14,1513-oxirane ring, which, however, is distorted in comparison with that in

]'able 2. Crystal Data of Isostructural Steroid Pairs together with their Co-Crystallized Mixtures OH

HOtm,..

OH

HOt, e,,

H 15

16

Space group

Reference

a (~)

b (/{)

c (/{)

1"1 mixture of 11/12 1"1 mixture of 13/14 1:2 mixture of 1/2

7.844(4) 7.645(1) 7.290(3)

14.807(5) 15.789(8) 14.817(4)

P21212 l P212121 Pll21

5-Androstene-313,1713-diol mono-

6.250(1)

12.143(3)

17.849(6) 19.479(2) 18.520(7) ,/=90.35 ~ 23.044(2)

6.444( 1)

12.150(1)

23.024(1)

P212121

23.234(4) 12.159(4) 10.347(5) 7.191(4) 13=114.1(1) ~ 10.960(2) 7.157(1) 11.875(2) 13= 114.70(2) ~

P212121 P1211

4 37

P1211

38

P212121

hydrate

5 ct-Androstane- 313,1713diol monohydrate 1:1 mixture

5~t-Androstane-3~t, 1713diol 15 513-Androstane-3~t, 1713-diol 16

6.352(1) 12.327(5)

204

ALAJOS KALMANand LASZLOP,~RKANYI

digirezigenin and cinobufotalin [19]. This is in agreement with the final result of refinement of this molecular position where a 10% disordered O atom was found. In contrast, molecule B resembles digitoxigenin with a rather short nonbonded distance of C15'..-O14' = 1.94(1) /~, indicating the presence of mismatching digirezigenin molecules. No spurious electron density peaks were found around this molecular position. The puckering parameters [29] for ring D found in these symmetry-independent A and B molecules revealed substantial differences to both digitoxigenin and digirezigenin (cf. Figure 3 in ref. 4). The observed ring puckerings are the averaged superpositions of different amounts of digitoxigenin [laEenvelope] and digirezigenin [ 17E_envelope]). From these observations the following conclusions were drawn: in position A approximately 20% of digirezigenin is substituted by mismatching digitoxigenin molecules, whereas in position B digitoxigenin (53%) somewhat exceeds digirezigenin (47%), meaning that the adduct crystallized from a solution of equivalent amount of the components contains 64(6)% digirezigenin and 36(6)% digitoxigenin, thus forming a partially ordered crystal lattice. The two symmetry-independent molecular sites are linked separately by infinite head-to-tail hydrogen-bonded chains similarly to that shown by digirezigenin [16].

D. isostructurality in General The complex analysis of the 14 cardiotonic steroids built on the isometric 14-aethiocholane skeleton led to the following conclusions: isostructurality is a connection between the three dimensional arrays of a molecule and their somewhat "modified" derivative(s). These modifications can be either epimerization (isomerization), or substitution or replacement(s) of atoms with others without any substantial change of the molecular isometricity. A number of questions arise at this point. To what extent are these rules valid for other related crystal structures (racemates, and heteromolecular associates)? To what extent does the size of the molecules and their substituents permit isostructurality? How does the chemical character of the substituent and the site of the substitution permit or hinder the formation of similar molecular arrays in solid state?

The Size of Molecules Even without accurate experimental data at the time, Kitaigorodskii [2] felt that small related molecules such as urea and thiourea could not develop isostructural lattices. Namely, if A(rst) in Eq. 1 with respect to V' is large, then the shape of a small molecule will be substantially altered by such a replacement as O ~ S. For example, the 5-halo-substituted indol-3-ylacetic (IAA) acid (Figure 6) containing 14 non-hydrogen and 8 hydrogen atoms (CIoH8NO2X) displays quite a high isostructurality index of Ii(14) = 87% for the F ---) C1 replacement, but with Br it becomes too large to remain isometric with the small 5-F-IAA and 5-CI-IAA molecules [30]. In contrast, with 22 non-hydrogen and 24 hydrogen atoms, the C1,

Isostructurality of Organic Crystals

205

CH2C00H

X

'>

5 ~

7

1

4-,5-F-IAA 4-,5-,7-C1-1AA 5-Br-IAA

X=F X= CI X= Br

Figure 6. 5-Halo-substituted indol-3-ylacetic acids [30]. Br, and I analogues of 17-halo-3-methoxy- 16,17-secoestra- 1,3,5-triene- 16-nitrile [31] with 22 non-hydrogen atoms exhibit perfect isostructurality (Table 3). Similarly, the 23 non-hydrogen atoms of the 2'-halo- l'2'3'4'-tetrahydrospiro[ 1,3-dioxolane-2,1'-naphtalene]-4,5-dicarboxylates [32] retain a high degree of isomet_ricity (Figure 7) and also guarantee their isostructurality. The limited isostructurality of IAA derivatives (R = F, C1 versus Br) compared to those of the two C1 --->Br --->I series is partly due to the small number of hydrogen atoms (8 versus 24 and 17, respectively), contributing also to the molecular volumes. This suggests that with their increasing volumes, molecules increasingly tolerate substitution(s) or replacement(s) without losing their isostructural relationships. For example, the native enzyme, xylose isomerase (from Arthrobacter B3728) remains isostructural (space group: P3121) with its D254E/256E double mutant after the replacement of 254-Asp and 256-Asp units with the larger glutamic acids [33]. In special cases the replacement of even two or more heavy atoms is also tolerated by the existing molecular self-complementarity. For example, the replacement of the halogen atoms of the 5-chloro-7-iodo-8-quinolinol molecule [34] with two bromines does not alter its isostructurality. This is attributable to the

Table 3. Descriptors 1-I, A(pc) and ll(n) for the Isostructural Pairs formed by the Three C16H1706x (X = CI, Br, I) Derivatives [31]a Pair of Compounds

1-I

CI/Br

0.006

0.005

Br/I

0.012

0.015

CI/I

0.018

0.018

Note:

A(pc)

1/(4)

//(22)

//(23)

99%

96%

95%

98

94

93

97

90

88

aln Ii (n) n = 4, 22 and 23 are the number of atomic pairs which have been taken into account; n = 23 means that finally halogen atoms are also taken into account.

206

ALAJOS KALMAN and LASZLO P,~RKANYI

C)c16

06

~

Cllsr

Figure 7. A superposition of three naphtalene derivatives [32] differing in halogen atoms: CI, Br, and I. The H atoms are omitted for clarity. fact that the atomic volume (and mass) goes up at one site, while at the other it goes down, resulting in a slight decrease (AV'= 13/~3) of the asymmetric unit volume accompanied by 0.1 Mg m -3 increase of the crystal density.

The Size and Character of Substituent(s) and the Site of the Substitution There are only a few substituent pairs whose members can replace each other. Apart from the CI --->Br ---> I series, CH 3 can almost equally be replaced with H, C1, and occasionally an ethyl moiety [5]. The hydrophilic moieties such as -OH and --O are replaceable only in special circumstances. The = O atom in arenobufagin that replaces two hydrogen atoms in the isometric gamabufotalin [18] is prevented from participating in hydrogen bonding. Cinobufagin and cinobufotalin [19] are isostructural since the -OH group is embedded in the A/B ring junction. It forms only an intramolecular hydrogen bond. Their isostructurality can be attributed to the special site of the H --->OH replacement. The types of isostructural pairs that occur among para-disubstituted benzylideneaniline derivatives (R-C6H4 CH--N-C6H4-R ') support this conclusion [9]. Although conformational disorder hinders precise refinement of these structures, it is certain that the R and R' positions are not equivalent (the lattice parameters of the related compounds are summarized in Table 2 of ref. 5). The replacement of either substituent of C1-C6H4-CH=N C6H4-CI (C1C1) and Br-C6Hn-CH=N-C6Hn-Br (BrBr) by the opposite function is ambivalent. While C1Br remains isostructural with C1C1, its contrasubstitutional analogue of BrC1 is isostructural with BrBr (cf. Figure 2 in ref. 5). This asymmetrical packing relationship of CIBr and BrC1 with the homosubstituted pair indicates

05O2 0502 N3

0 ",4

", c ' k ~

Figure 8. The crystal structures of 1-methyl-5-nitro-2-phenylimidazole and 2-(p-aminophenyl)-1-methyl-5-nitro-imidazole, the latter with the hydrogen bonds generated by the entering-NH2 moieties.

208

ALAJOS KALMAN and LASZLO P,g,RKANYI

that the substituent on the benzene ring plays the most important role in determining the crystal structure. Sometimes the size and character of the substituent (-NH2) and the site of the substitution lead to unpredictable pairs of isostructurality. For example, the crystals of 1-methyl-5-nitro-2-phenylimidazole (Figure 8) and its 2-p-amino derivative [35] exhibit a high degree of isostructurality 1/(15) = 90%), although in the latter there are two novel hydrogen bonds formed by the entering -NH2 groups with the -NO2 moieties. The complementary site and distribution of the entering active group(s) is shown by the low increase of the asymmetric unit volume (AV'= 13/~3). The absence and presence of the double O...H-N-H.-.O synthon requires further investigations, however. At any rate, it is worth noting that there is a strong O 14-H-..O 1 hydrogen bond only in one (3) of the five isostructural cardenolides (1-5). This molecular synthon is practically missing from the others.

E. The Relaxed Forms of Isostructurality Finally, how far does diminished molecular isometricity guarantee the isostructurality of related crystals? For example, 5ct- and 513-epimers of androstane-3,17diol (15, 16) differ in shape but their crystals still exhibit a rather similar packing array. They had been suspected to be "isomorphous" by Norton et al. [36]. The structure determinations [37, 38] revealed that in both crystal packings close contacts are formed between the molecules translated by the 21 operator at 1/2, y, 1/2) via head-to-tail OH...O hydrogen bonds arranged helically by a second 21 operator at (0, y, 1/2). However, close inspection of these molecular packings revealed [4] that the different A/B junctions (cis for the 513-epimer and trans for the 5a-epimer) permit only a low degree of isostructurality (1/(21)= 41%). The 513epimer is bent along the molecular axis, towards the 21 operator at (1/2, y, 1/2). This results in a slight increase in the unit-cell volume (AV = 9/~3). By partitioning li (n) for the rigid part of the skeleton it can be seen that the C and D tings exhibit a higher degree of isostructurality (li (16) = 76%). (For atoms C7-C9 and C 11-C 18 the remaining deviation can be attributed to a mandatory balance of the A ring displacement, thus maintaining a similar packing mode). From these results it follows that in special circumstances a similar packing motif in these systems can be maintained even with considerable changes in the molecular shape, i.e. with diminished isometricity. The alteration of the molecular isometricity within a group of related structures is demonstrated by a series of group 14 homologues of tetraphenylmethane [39], silane [40], germane [41], tin [42], and lead [43]. Here the atomic replacement happens inside the PhaE molecule which hardly influences the electronic surface of the molecules. But the size of the molecules alters monotonously with the increasing difference in the lengths of the C-E bonds, limiting the isometricity of the PhnE tetrahedra. While each pair formed by two neighbors in this series is practically isometric, the remote derivatives like Ph4C and PhaPb, due to the

lsostructurality of Organic Crystals

209

substantial difference in the atomic radii of the core atoms, are not and consequently their crystals exhibit low degrees of isostructurality. It can be concluded that whenever the strict rules of isostructurality are relaxed (e.g. molecular shape and/or size are different, etc.) to some extent (i.e. Ii(n ) < 30% or so), the related pairs are considered as homostructural, which naturally implies a greater variety of packing arrays. Expediently, such molecules (similar shape, but different absolute size) are considered as homometric. The simultaneous iso- and homostructurality of these homologues are visibly indicated by the motions of the unit cell parameters of the C --->Pb series. It appears as if the unit cell of PhnC were squeezed along the c-axis in a stepwise mode through Ph4Si and PhnGe towards PhaSn and PhaPb (Figure 9). With a visible distortion of the PhnC tetrahedron towards the bisphenoids of Ph4Sn and Ph4Pb sitting on axis 4, the unit cell vectors a versus c, in accordance with the unit cell similarity indices (FI), are subject to inversely related changes as summarized in Table 4. At any rate, the governing role of self-complementary surfaces is well demonstrated by the ortho-methyl derivative of tetraphenyltin [42] which remains isostructural with the parent compound in their common space group P-421c. Even the m- and p-tolyl derivatives crystallize in closely related space groups 141/a and 14. There are several known cases in which homostructurality is pronounced [5]. The butterfly-like chiral molecules of diphenyldiacyloxyspirosulfurane 17 [44], for example, and its binaphthyl analogue 18 [45] (Figure 10) possess C2 molecular symmetry on the short twofold axes of the orthorhombic unit cell (space group Fdd2). Despite the larger naphthyl tings, the packing array of 18 is quite similar to that of 17. Of course, li(n) calculation is limited to a few common central atoms, and the large naphthyl groups account for the elongated (by ca. 25%) a-axis (H = 0.077). Spirosulfuranes 17 and 18 exemplify well Kitaigorodskii's homologous isomorphism described by the lattice parameters of biphenyl, p-terphenyl, and p-quarterphenyl [2]. Recently a par excellence case of homostructurality has been found among the isomeric pair of 3-cyano- and 4-cyano-cinnamic acids where the packing arrays formed by slightly different dimers are quite similar. Their solidstate reactivity is, however, different [46]. The proper use of this term particularly m

Table 4. Lattice Parameters and Density of the Isostructural Tetraphenyl Derivatives of Group 14 Elements Compound Ph4C Ph4Si Ph4Ge PhnSn Ph4Pb

a (t~)

c (t~3)

V (,~)

dx (Mgm-3)

E-C (,~)

10.896(2) 11.450(2) 11.656(11) 12.058(1) 12.092(3)

7.280(1) 7.063(1) 6.928(7) 6.581(1) 6.589(2)

864.3 926.0 941.3 956.8 963.4

1.22 1.21 1.31 1.48 1.75

1.50 1.87 1.96 2.14 2.19

ALAJOS KALM,~N and L,~,SZL6Pg,RK,~NYI

210

C iO~e Sn, Pb

Figure 9. The comparison of the tetragonal unit cells of Ph4E (E = C, Si, Ge, Sn, Pb), showing the change in the rate of the a/c axes.

"~" u30~,~ C~

0

~..0

0

84

s0

0 17

(X= S, Se)

18

Figure 10. Projections of the full unit cells of two diaryldiacycloxyspirosulfuranes (space group Fdd2) perpendicular to the shortest axes c.

Isostructurality of Organic Crystals

211

seems to be useful in the case of supramolecules where the guest or even the host molecules are somewhat different within the related pairs.

F. Forms of Isostructurality in the Ph3E-E'Me3 and Ph3E-E'Ph3 Series

Homo- and Isostructurality within the Ph~E-E'Me3 Series If one of the phenyl groups of the above-discussed Ph4E compounds (apart from Ph4C) is replaced by a EMe3 moiety a novel class of disilanes, digermanes, etc. and their isomers Ph3E-E'Me3, also related by isostructurality, is formed. Ph3Si-SiMe3 [47] and Ph3Ge-GeMe3 [48] give high isostructurality index (//(23) = 94%) in the common space group P3 (Figure 11). In the first approximation Ph3Sn-SnMe3 [49] is also isostructural with the former ones. Nevertheless, due to the sharp increase in the atomic radii between Ge and Sn (Ar = 0.19/~), it is only homostructural with the Si/Ge derivatives: the substantially elongated Sn-Sn "dumb-bell" (2.782(1)/~) is stretched on the threefold axis with enlarged (2.138(5) and 2.152(4)/~) Sn-C distances. The isomeric derivatives of Ph3Ge-SiMe3 [50] and Ph3Si-GeMe3 [51] also retain the C3 molecular symmetry in space group P3. The high isostructurality index of Ph3Ge-SiMe3 with Ph3Si-SiMe3 (//(23) = 95%) and a slightly lower (//(23) = 81%) with Ph3Si-GeMe3 indicates that the Si-Ge dumb-bell does not

Figure 11. Close packing of the Me3E-E'Ph3 derivatives (E, E' = Si and/or Ge) in the trigonal unit cell having space group P3.

212

ALAJOS KALMAN and LASZLO P,g,RKANYI

alter visibly the isometricity of these molecular pairs (for comparison: Si-Si = 2.355(1), Ge-Si = 2.384(1), Si-Ge = 2.394(1), and Ge-Ge = 2.418(1)/~).

Morphotropic Steps within the PhzE-E'Mez Series When either of the Ge atoms of Ph3Ge-GeMe3 is replaced by the larger Sn atom, the new isomers are no longer isostructural with the parent compounds. With the elongated distances of Ge-Sn (2.652(2) A) and Sn-Ge (2.599(3) A), Ph3GeSnMe3 and Ph3Sn-GeMe3 remain isostructural [52] in their new orthorhombic (pseudohexagonal) unit cell with space group Pna21 (Figure 12). Such rearrangements, termed as morphotropic steps, were expected by Kitaigorodskii [2] whenever the atomic replacement substantially alters the existing packing coefficients. In these pseudohexagonal unit cells, the bumps of the molecules stacked with the similar orientation along the polar c-axis fit perfectly into the hollows of the adjacent columns generated by the glide planes, thereby forming new efficient close packing. The morphotropic step does not substantially alter the close packing of the isomers. A joint projection of the trigonal and orthorhombic close packing perpendicular to the c-axis (the difference between the lattice parameters is not significant) reveals their strong relationship (Figure 13). The volume of the new orthorhombic unit cell is doubled relative to the trigonal cell. Three trigonal cells form a hexagon with a

Figure 12. Close packing of the Me3Ge-SnPh3 and Me3Sn-GePh3 isomers in the orthorhombic unit cell (space group Pna21).

Isostructurality of Organic Crystals

213

Figure 13. Joint projection of the related trigonal (shaded area) and orthorhombic unit cells of Si, Ge and Sn derivatives (Me3E-E'Ph3) perpendicular to the c axes. The intersections of the glide planes (a) (y = +1/4) and (n) (x = +1/4) define the central lines of the trigonal unit cells.

21-axis of the orthorhombic cell in its center. The six Ph3E-E'Me 3 molecules in these three trigonal cells are situated with altering polarity on the bc planes of the orthorhombic cell at x = 0, 1/2, and 1, respectively. They altogether contribute four molecular units to the latter. After the Ge --->Sn replacement they are shifted parallel to axis a to their new positions (1/2 - x) in opposite directions for the two halves of the new unit cell bisected at y = 1/2 and assume similar orientation with respect to the polar axis. Presumably, this rearrangement with the relaxed C3 molecular symmetry provides a good fit of the bumps of the molecules into the hollows of their neighbors, as predicted by Kitaigorodskii by the use of close packed layers in plane groups. The necessity of the morphotropic step can be demonstrated by the following lattice transformation. As the Sn-Ge vector of Ph3Sn-GeMe3 forms a 0.7 ~ angle with the c-axis it is a facile transformation to conform this molecule with a P3 structure, e.g. with Ph3Sn-SnMe3. The transformation involves bringing the Sn-Ge vector exactly parallel with and rotate about the c-axis, retaining only one phenyl and one methyl group. The new c-axis in P3 is fixed by the Sn-Ge bond distance, while the a = b axis is varied until the molecular geometry becomes identical with that of the Pna2~ structure. An intermolecular distance calculation reveals that Ph3Sn-GeMe3 in the transformed P3 unit cell (a = 11.739, c = 8.871/~) is entrapped by six short H...H intermolecular distances (2.148/~ (2x), 2.206 A (2x), and 2.449 /~ (2x)), whereas in its original orthorhombic cell (Pna21, a = 20.267 A, b = 12.388 A, c = 8.044 g,) it has only two longer (2.472 ]k) H...H close contacts. Therefore, m

214

ALAJOS KALMAN and LASZLO P~,RKANYI

to avoid contacts falling in the repulsive region of the nonbonded H-..H potentialenergy function, Ph3Sn-GeMe3 favors crystallizing in the orthorhombic unit cell. From this it follows that a second morphotropic rearrangement occurs again when the retained Ge core is also replaced by a Sn atom in these orthorhombic crystal structures. Then the symmetric Sn-Sn dumb-bell reaches again the optimum of molecular complementarity by sitting on the 3 axis in the trigonal (P3) unit cell.

Homostructurafity within the PhaE-E'Ph3Series Except for the Ph3S"-CeMe3 pair, the PhnE and Ph3E-E'Me3 series molecules reach their bump-in-hol(ow packing with relatively high packing coefficient of about 0.73 by giving their own molecular symmetry (either $4 or C3) to the crystal lattice. Interestingly, this rule is relaxed in the crystals of the bulkier hexaphenyl molecules. They do not (possibly cannot) develop the optimum packing around their $6 molecular symmetry. Only hexaphenyldigermane has a metastable hexagonal (space group P6322) form [53]. The stable form crystallizes in a triclinic unit cell (space group P1) built up by achiral bipropellers. When one of the Ge atoms is replaced by Pb then the asymmetric Ge-Pb dumb-bells beating six phenyl groups exhibit orientational disorder in a triclinic unit cell isomorphous with that of Ph3Ge-GePh3. Since it now has an inversion center (space group P1) it can be considered homostructural with the parent compound. The replacement of the remaining Ge atom of Ph3Pb-GePh3 by Sn [54] is followed by a morphotropic rearrangement. In the new monoclinic unit cell (space group P21/c) there are two symmetry-independent molecules providing the optimum close packing. This conclusion is supported by the fact that this positionally disordered Ph3Sn-PbPh3 is homostructural with both Ph3Sn-SnPh3 and Ph3Pb-PbPh3 [55, 56]. The presence of more than one molecule in the asymmetric units of Ph3E-E'Ph3 (E = Sn, Sn, Pb; E' = Sn, Pb, Pb), combined by positional disorder of the asymmetric dumb-bells, gives rise to the optimum of the molecular self-complementarity. Ge-

n

m

IV.

ISOSTRUCTURALITY OF SUPRAMOLECULES

A. Adducts with Slight Difference in the Guest (or Host) Molecules Since we have reported [4] on the main-part isostructuralism of the binary 5-androstene-3f3,17-diol monohydrate with that of the saturated 5ot-androstane313,1713-diol monohydrate showing high index of Ii(21) = 78% and forming readily an isostructural solid solution, discovered cases of isostructurality shown by heteromolecular associations have increased rapidly. In these "simple" clathrates either the host (A) or the guest (B) molecules exhibit slight differences. The Me2SO clathrates [57] of two isomeric host molecules, thieno[3,2-b]thiophene and thieno[2,3-b]thiophene differing only in the position of one of the S atoms are almost perfectly isostructural in triclinic unit cells. The complementarities of the isomers is shown by the packing coefficient increment A(pc) = 0.5%.

isostructurality of Organic Crystals

215

Rutherford and co-workers reported [58] on the isostructural clathrates of adamantane with thiourea and selenourea with FI = 0.02 for trigonal unit cells (space group R3-c). It is worth noting that the crystals of the host molecules, which can be regarded as the homologues of the group-16 elements, are not isostructural. Nevertheless, they form infinite tunnels on the surface of which there is a dense hydrogen-bond network. Each heteroatom (S and Se) receives four hydrogen bonds from the four N-H groups. In the center of these tunnels there are the entrapped adamantane molecules sitting on the 3 axes. Perchloro(2,2'-biphenylene)diphenyl-silane and -germane form isostructural adducts [59] with two molecules of benzene in triclinic unit cell (space group P1). 1,1'-binaphthyl-2,2'-dicarboxylic acid forms isostuctural (1:2) inclusion complexes [60] in the monoclinic space group C2/c with ethanol and propanol (differing in one CH2 moiety) In the voids of the clathrates both relatively small guest molecules exhibit positional disorder, indicating the predominance of the A...A [61] interactions. The host lattices exhibit a high degree isostructurality (Ii(13)= 88.%) showing similar self-complementarity of the bulky scissor-like [8] host molecules. The relatively similar positions of the disordered alcohol molecules is shown by the isostructurality index (//(16) = 70%) calculated with the participation of the alcohol atoms. Perchloro-9-spirocyclohexa-2',5'-diene forms isostructural inclusion compounds with benzene and cyclohexadiene [62]. w

m

Figure 14. Molecular compounds formed by androsta-1,4-diene-3,17-dione with o~-naphthol and 13-naphthol. The oxygen atoms forming the OH...O hydrogen bonds are marked by full circles.

216

ALAJOS Kg,LMAN and Lg,SZLO Pg,RKANYI

Not only clathrates, but molecular compounds can also exhibit isostructurality. B6cskey and co-workers reported [63] on isostructural molecular compounds formed by androsta-1,4-diene-3,17-dione with a-naphthol and 13-naphthol (Figure 14). In these crystal lattices the partner molecules are bound by hydrogen bonds (A...B interaction [61]). The steroid molecules assume a favorable position in the orthorhombic unit cell (space group P21212~) from which it can form hydrogen bonds either with the a-OH or 13-OH moiety of the donor naphthol ring that is sitting almost perfectly in the same site in both crystals. This is shown by the very high degree of isostructurality: FI = 0.005,//(31) = 92%, Ii(31") = 93%. In addition to these binary systems, a pair of ternary adducts, namely nitrate and perchlorate salts of the double betain (L = cis-pMe2N+CsHnN)C2(CO0-)2) monohydrate, has recently been found to be isostructural [64] with quite a high index of //(28) = 88%. The disordered NO3- and C104-anions are sitting on mirror planes (x, y, 1/4) in space group Pbcm (No. 57). These mirror planes also bisect the double betain cations, while the water molecules are localized on the twofold axes (x, 1/4,0).

B. Clathrates Formed by Basically Different Guest Molecules Caira and Mohamed [65] reported on three solvates of 5-methoxysulfadiazine formed with (a) dioxane, (b) tetrahydrofuran, and (c) chloroform. These clathrates are based on the common isostructural sulfamide host framework with the solvent molecules occupying the framework cavities. As shown by the stereoscopic views of these binary adducts depicted in Figure 15, the guest molecules occupy the same relative position in the infinite channels formed by the host molecules along the c-axis of the common space group P21/c. The great similarity in the cavity occupied by the guest molecules is well presented by Figure 6 in ref. 65. The adducts by pairs exhibit three relatively high isostructurality indices accompanied by low FI values (Table 5). The highest isostructurality index is shown by the dioxane/chloroform pair which suggests that the morphologically and electronically different guest molecules hardly influence the self-complementarity of the host molecules governed by A..-A interactions [61]. It is dubious as to whether the lowest isostructurality index shown Table 5. Isostructurality Ii (19) and Unit Cell Similarity Indices H with Volume

Differences A V'(,g,3) for the Asymmetric Units of the Clathrates of 5-Methoxy-sulfadiazine (5) with Dioxane (C4H802), Tetrahydrofurane (C4HaO) and Chloroform (CHCI3)

Guest Molecules

1i (19)%

FI

AV'

C4HsO2/CHCI3 C4HsO/CHCI3 C4HsO2/C4H80

90 86 83

0.011 0.0004 0.011

9.2 2.8 12.1

Isostructurality of Organic Crystals

217

,~ .~. r9

f

G

7

.I

Figure 15. Stereoscopic view of clathrates of 5-methoxysulphadiazine with (a) dioxane, (b) chlorofom, and (c) tetrahydrofurane showing the channels built up from the host molecules giving room for the guest unit approximately in the same positions (CHCI3 molecules exhibit positional disorder).

218

ALAJOS KALMAN and LASZLOP,~RKANYI

by the dioxane and tetrahydrofuran clathrates are attributable to their substantially different conformations, or not. When any conspicuous difference between these pairs is studied, we must bear in mind that the differences in their asymmetric unit volume is rather small even for the dioxane/tetrahydrofuran pair (less than 18 A3). At any rate, from the point of view of the guest molecules they have to be considered homostructural. The close relationship between isostructurality and polymorphism is shown by the three polymorphs of this biologically active sulfonamide which were also characterized by X-ray crystallography [66, 67].

C. Homostructural Adducts Formed by Different Host and Guest Molecules A large group of binary adducts formed by closely related host and guest molecules has been synthesized and characterized by X-ray structure determinations [68]. An Australian team has co-crystallized three helical tubuland diols 19, 20, and 21 (Figure 16) with phenol and five phenol derivatives: p-chlorophenol, hydroquinone, p-methoxyphenol, p-hydroxythiophenol, and phloroglucinol (1,3,5trihydroxybenzene). Since phloroglucinol substantially differs from the other guest molecules its adduct with host 21 is orthorhombic with the polar space group Fdd2, while the others crystallize in space group P21/c with similar unit cell parameters (cf. Table 1 in ref. 68). Only hydroquinone possessing a molecular center of symmetry crystallizes with a 2:1 ratio of the host and guest molecules in a smaller unit cell (V = 1370 ]k3) with respect to the mean of the other five of 1792 A.3 The other five adducts form a group of homostructural crystals denoted as: A:19 (p-chlorophenol); B:20 (p-chlorophenol); C:20 (p-methoxyphenol) D:20 (p-hydroxythiophenol); and E:21 (phenol) of which A, B, and D are depicted in Figures 17-19. The similar complementarity of the helical tubuland diols can be estimated by their superposition, but to quantify their degree of isostructurality is difficult. Nevertheless, in the canals formed by diols 19, 20, and 21 (cf. Figure 1 in ref. 68) the guest molecules assume approximately the same positions, even if they are disordered (C). In these channels besides the A...A interactions [61] the effect of the B...B contacts has also to be taken into account. The highest degree of isostructurality could be expected between structures B and D (H = 0.005) where the same dio120 is co-crystallized with p-chlorophenol and p-hydroxythiophenol. Indeed, they show very high isostructurality index for the guest molecules (//(8) = 93%) and somewhat lower for the host molecules (Ii(15) - 67%). Naturally, their common isostructurality index is still high (//(23) = 73%). In contrast, for the A/E pair (H - 0.023) where both hosts (19 versus 21) and guests (p-chlorophenol versus phenol) are different only the guest molecules show high isostmcturality (//(7) 86%). The fit of the host molecules has not been satisfactory so far. Similarly, the proper fit of host (19 and 20) skeletons for the A/B pair should be solved. At any

Isostructurality of Organic Crystals

219

Figure 16. The non-hydrogen atom skeleton of the three helical tubuland diols: 2,6-dimethylbicyclo[3.3.1 ]nonane-exo-2,exo-6-diol (19); 2,7-dimethyltricyclo[4.3.1.1 ]undecane-syn-2,syn-7-diol (20); 2,8-dimethyltricyclo[5.3.1.1 ]dodecane-syn-2,syn-8diol (21). rate, the isostructurality of the guest molecules is high (//(8) = 84%). The corresponding isometricity index is Ii(8") = 99%. The B/C pair has the same host 20 with the lowest isometricity of the guest molecules. Their//(8) is only 48% disturbed by the positional disorder of the p-methoxyphenol molecules. In spite of this, the self-complementarity of the host molecules are similar, consequently their isostructurality is high (//(15) = 74%). The common isostructurality index of the host and guest molecules is still//(22) = 63%. The highest unit cell similarity index (1-I = 0.04) is shown by the adducts A and C is due to the substantial differences between both host and guest molecules,

220

ALAJOS KfiLM,~N and L,~SZL6 P,g,RKANYi

Figure 17. Perspective view of the unit cell of the inclusion compound of 19 with p-chlorophenol (A).

Figure 18. Perspective view of the unit cell of the inclusion compound of 20 with p-chlorophenol (B).

Figure 19. Perspective view of the unit cell of the inclusion compound of 20 with p-hydroxythiophenoi (D).

Isostructurality of Organic Crystals

221

respectively. Without the appropriate calculation of isostructurality and isometricity indices for the different host molecules only their homostructurality can be visualized (Figures 17-19). The proper fit of these host molecules depends on the formation of the distance differences (z~i) between the coordinates of the positionally similar non-hydrogen atoms within the same section of the asymmetric units (cf. Eq. 3). It seems to be advisable if these calculations for supramolecules were extended to the whole unit cell, instead of one asymmetric unit.

D. Clathrates of 2,2'-Bis(3,4,5-trimethoxyphenyl)-l,l-bibenzimidazole The crystal structure of the bulky 2,2'-bis(3,4,5-trimethoxyphenyl)-l,1bibenzimidazole molecule (Figure 20) had been determined in 1986 [69]. Using the concept of the selection ofpowerful host molecules summarized in [8], Czugler and Bombicz [70] have recently made an attempt to co-crystalize this bulky molecule possessing C2 molecular symmetry in space group C2/c with achiral formic, acetic, propionic, butyric, and the bulky trimethylacetic acids (their lattice parameters are listed in Table 6). They revealed that the smallest (formic) acid forms

Figure 20. The crystal structure of 2,2'-bis(3,4,5-trimethoxyphenyl)-1,1-bibenzimi-

dazole.

222

ALAJOS K,~LMAN and L/~SZLO P,~,RK,/~NYI

Table 6. Unit Cell Dimensions of the Clathrates Formed by 1,1 '-Bibenzimidazole

with Different Aliphatic Carboxylic Acids

Type of Guest

No guest Formic acid Acetic acid Propion acid Propionic+butyric acid (1:1) Butyric acid Trimethyl-acetic acid

Space Group

a (,~)

b (,~)

C2/c C2/c P3121 P3221 P3121 C2/c C2/c

17.911 18.691 10.431 10.526 10.543 17.968 17.674

10.717 10.351 10.431 10.526 10.543 10.431 10.567

c (,~)

15.219 17.955 28.498 29.265 29.459 21.765 22.925

Angle (o)

96.33 111.15 120.00 120.00 120.00 105.19 111.18

an adduct with the host molecule which crystallizes in the monoclinic space group C 2 / c and remains homostructural (FI = 0.04) with the parent compound. Although their monoclinic angles differ (96.33(1)~ for the host and 111.15(2) ~ for the adduct crystal), the isostructurality index for this adduct and the host molecule is high: Ii(21) = 76% (without the three terminal methyl groups li(19) = 78%, especially if we take into account that the asymmetric unit of newly formed clathrate incorporates an additional HCOOH molecule. The high self-complementarity of the host molecules is also indicated by the moderate increase of the asymmetric unit volume (AV'= 42 A 3, i.e. 14 A3 for each entering heavy atom). In contrast, the 1:2 adducts with acetic and propionic acids via spontaneous resolvation form isostructural conglomerates in the enantiomorphic hexagonal space group(s) P3121/P3221. In these supramolecules the host unit preserves its 6"2 molecular symmetry which is now perpendicular to the chiral 31- or 32-axes. With 1-I = 0.02 the host molecules have a very high isostructurality index of

iS

9

IS S

Figure 21. The general packing pattern of 2,2'-bis(3,4,5-trimethoxyphenyl)-l,1bibenzimidazole inclusion complexes. The large open circles indicate the voids occupied by aliphatic carboxylic acids in the monoclinic unit cell (space group C2/c) projected onto the c-axis.

Isostructurality of Organic Crystals

223

Ii(21) = 96% and they cannot be distinguished from each other. The positionally disordered guest molecules, however, occupy visibly different positions in the large voids of the crystal lattices. Accordingly, the formation of these clathrates are basically governed by A...A interactions [61]. With butyric and trimethyl acetic acids the host molecule forms adducts which crystallize again in monoclinic unit cells with space group C2/c and are isostructural again with the clathrate formed by formic acid. A general pattern of their packing similarities is depicted in Figure 21.

V. CONCLUDING REMARKS Prior to the publication of ref. 5 only sporadic references have been made to the phenomenon termed simply as. isomorphism even if characteristic properties of the related crystals (e.g. chloro-methyl interchangeability, etc.) have been described [71]. Since the early work of Kitaigorodskii [2], the phenomenon has been widely regarded so trivial that no one has attempted to characterize it per se. With the increasing interest in the phenomena which have been shown by organic molecules in the solid state isostructurality has proven more frequent than one would have thought it earlier. In particular, the heteromolecular associates are capable of showing great variety of isostructurality even with the interchangeability of both host and guest molecules [68]. Double morphotropic steps may also occur within the series of the clathrates as listed in Table 6. In summary, a high degree of isostructurality can be expected for homomolecular associates whenever the related molecules are of great degree of isometricity, and the supramolecular synthons [72] are not only interchangeable but located at the same site of these molecules. Such crystals are of similar unit-cell parameters and the same space group symmetry. Even partial isometricity of the molecules may give rise to rather similar packing motifs [37, 38] (termed as homostructural) if the supramolecular synthons, based on strong hydrogen bonds, are located at the same site of the molecules. Crystals of molecules like PhnC [39] and PhaPb [43], having similar shape but different size are also considered homostructural. When the related molecules possess more than one supramolecular synthon, then the rank (type and position) of these synthons will permit or exclude the formation of isostructurality. The increasing molecular symmetry of e.g. $4 of the series PhaC ~ PhaPb seems to govern the position-dependent effect of the supramolecular synthons. For example the methyl substituent of PhaSn [41] in an ortho-position retains isostructurality with the parent compound (space group P-421c), while the meta- and para-tolyl derivatives crystallize in different, but closely related space groups: 141/a and 14, respectively [42]. Mutatis mutandis, in tetrakis(4-bromophenyl)methane the paraposition of the bromine atom gives rise to a similar tetragonal unit cell with space group 14. This tetragonal crystal with its supramolecular Br4 synthon of tetrahedral symmetry is "nearly isostructural" [72] with the heteromolecular complex of

224

ALAJOS K,g,LMAN and LASZLO P,g,RKANYI

tetraphenylmethane [39] and tetrabromomethane. This homostructural pair exemplifies the interchangeability of supramolecular synthons (Br4) with molecular synthons (CBr4) applied purportedly in crystal engineering.

ACKNOWLEDGMENTS We wish to express our sincere thanks to our colleagues Drs. Gyula Argay, Petra Bombicz and M~ity~isCzugler (Budapest) for their invaluable help in data collections, and brainstorming discussions. Thanks are also due to Professors Mino Caira (Cape Town, SA), Roger Bishop (Kensington, Australia) and John Rutherford (Umtata, Transkei) for their cooperation and useful information concerning their own works. We are very grateful to Mrs. GyiSrgyi Csfikvfiri for her assistance in preparing the manuscript. This work has been sponsored by the Hungarian Research Fund, Grant No. OTKA T014539.

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

AROMATIC CHARACTER OF CARBOCYCLIC x-ELECTRON SYSTEMS DEDUCED FROM MOLECULAR GEOMETRY

Tadeusz Marek Krygowski and Micha[ Cyraffski*

Io II. III. IV. V.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavioristic versus Structural Definitions o f Aromaticity . . . . . . . . . . . Indices B a s e d on M o l e c u l a r G e o m e t r y . . . . . . . . . . . . . . . . . . . . . Relations b e t w e e n Indices o f Aromaticity Based on Molecular G e o m e t r y . . . R e v i e w o f M o l e c u l a r Problems . . . . . . . . . . . . . . . . . . . . . . . . . A. B e n z e n o i d H y d r o c a r b o n s . . . . . . . . . . . . . . . . . . . . . . . . . . B. Polysubstituted B e n z e n e Derivatives . . . . . . . . . . . . . . . . . . . . . C. Aromaticity and the H a m m e t t - L i k e Substituent Effects . . . . . . . . . . D. A r o m a t i c i t y and the Steric Substituent Effects . . . . . . . . . . . . . . . E. A n g u l a r G r o u p I n d u c e d B o n d Alternation ( A G I B A ) . . . . . . . . . . . F. A r o m a t i c i t y o f the Ring in the Strain-Affected B e n z e n e Derivatives . . .

*Stipendiarius of the Foundation for Polish Science Advances in Molecular Structure Research Volume 3, pages 227-268 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved.

227

228 228 229 231 234 239 239 244 246 248 251 251

228

VI.

TADEUSZ MAREK KRYGOWSKI and MICHAL CYRANSKI G. Intermolecular Interactions Affecting Aromaticity . . . . . . . . . . . . . H. Nonbenzenoid Hydrocarbons . . . . . . . . . . . . . . . . . . . . . . . . I. Aromaticity of Molecules in the Excited State . . . . . . . . . . . . . . . J. Fullerenes and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

256 258 264 264 265 266 266

ABSTRACT After a short historical introduction a review of definitions of aromaticity is presented indicating their structural and behavioristic nature. The most effective indices of aromaticity based upon molecular geometry comprise Julg's Aj, Bird's 15 and I6, bond alternation index BAC, and HOMA index, with its separation into the energetic and geometric terms. This shows how the molecular geometry (bond lengths) may provide an information on energetic and geometric contributions to the aromatic character, and that these two terms are mutually independent. It is also shown that benzene tings in the benzenoid hydrocarbons differ dramatically in aromatic character, depending strongly on the topological embedding of the given ring. The observed changes are either due to the energetic term or the geometric term, or to both of them. Much less dramatic are the substituent effects on aromaticity; nevertheless in p-substituted benzene derivatives geometric aromaticity is related to the Hammett substituent constants. The same is also found for steric substituent effects in exocyclically monoand disubstituted derivatives of benzylic cation: the change of charge delocalized from the exocyclic part of the cation onto the ring is associated with a decrease of its aromatic character expressed by geometric indices. The geometric term in changes of aromaticity as a result of the substituent effects of nonsymmetric substituents is also mostly related to changes via geometric factor. The geometric factor is also most important in the case of changes of aromatic character due to the intermolecular interactions. In the case of non-alternant systems, either geometric or energetic factors decide about the overall aromaticity of these systems. Aromaticity of molecules in the excited state (S1) is also discussed.

!. HISTORICAL SUMMARY Friedrich August Kekul6 von Stradonitz [1] first used the term "aromatic" to distinguish chemical compounds containing a benzene ring. A year later Erlenmeyer [2] used this term for compounds which have similar chemical properties to the derivatives of benzene. The formally unsaturated ring was accepted as a necessary component of aromatic molecules. Derivation of cyclooctatetraene derivatives by Willstaetter [3] demonstrated that this idea was false. The situation became more clear after fundamental work by Htickel [4] who introduced his famous 4n + 2 rule, which may be now summarized as follows: a cyclic n-electron system which contains 4n + 2 electrons exhibits enhanced stability in comparison

Carbocyclic n-Electron Systems

229

to those with 4n re-electrons. This rule represents a unified view on the phenomenon of aromaticity: the structural requirements (cyclicity, defined re-electron structure) are associated with a physicochemical property (energetic stability). The historical period of the development of understanding the nature of aromaticity or aromatic character ends in beginning of the 1970s when the Jerusalem Symposium on Quantum Chemistry and Biochemistry was devoted to problems of "aromaticity, pseudo-aromaticity, and antiaromaticity" [5]. The main conclusion was not decisive [6], namely t h a t . . . "classification and theory are not ends in themselves. If they generate new experimental work, new compounds, new processes, new methods--they are good; if they are sterile--they are bad" Undoubtedly, numerous results in the field of research of aromaticity in the past 25 years have showed that the first alternative won; even if we do not agree precisely what aromaticity means, attempts at understanding chemical and physicochemical facts associated with this notion have nevertheless made a considerable impact on the development of organic chemistry and related fields of research.

!1.

BEHAVIORISTIC VERSUS STRUCTURAL DEFINITIONS OF AROMATICITY

At the beginning of the 1960s [7-8] a view was established that a cyclic r~-electron system in its ground state is aromatic if x-electrons are delocalized which is documented by the following features: 1. 2. 3.

such systems are more stable than their acyclic analogues (by the "resonance energy"); C-C bonds exhibit a trend to have intermediate lengths between those typical for the single and double bonds (so-called aromatic bond lengths); and in an external magnetic field a x-electron ring current is induced and in consequence characteristic chemical shifts in proton NMR are observed (aromatic chemical shifts).

Sometimes organic chemists insist on using the reactivity criterion--aromatic systems are more inclined to undergo substitution reactions than addition [9]mand a few theoretical indices of aromaticity were based just on this property [10-11]. Nevertheless, most of the above presented properties of x-electron systems (1-3) have been applied for defining the quantitative measures of aromatic character [12-16], often called indices of aromaticity. Most often these indices were treated in the past as equivalent. This almost idyllic situation broke down following the important papers by Katritzky et al. [17] and Jug and K/Sster [18]. They applied principal component analysis to the set of aromaticity indices for a group of hetero-x-electron systems and showed that variability of these data needed at least two or three orthogonal vectors. The samples of molecules for both these studies were rather inhomogeneous, i.e. consisting of aromatic, weakly aromatic and even non- and antiaromatic

230

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKI

compounds, mostly hetero-rc-electron systems. Perhaps this was the reason why only about 80% of the total variance was explained by two or three [17, 18] principal components. Further studies were carried out with more homogenous systems, like benzenoid hydrocarbons in which individual tings were subject to estimation of their aromatic character [19], or para-disubstituted derivatives of benzene [20], or finally exocyclically substituted derivatives of fulvene and heptafulvene [21]. All these studies supported the conclusion of Katritzky et al. [17] and Jug and Ktister [18] that aromaticity is a Multidimensional Phenomenon. Sometimes a problem arises, e.g. which of the dimensions are the most representative or most useful? In a series of papers, Schleyer together with co-workers presented the view that the most representative are indices expressed by so-called NICS (nuclei-independent chemical shifts) based on diamagnetic susceptibility [22]. The arguments for this view are based upon the statement that diamagnetic susceptibility is entirely dependent on x-electron structure, and that aromatic character is a property of the x-electron structure, whereas the other properties are not. This may be opposed by arguments which are based on series of papers by Shaik, Hiberty, and co-workers [23-24] as well as Jug and Ktster [25]. They studied the relation between the role of re- and ~-electrons in determining the regular hexagonal shape of the benzene ring, and concluded that [24] "the re-system is forced by the ~-frame to adopt a regular hexagonal geometry?' Thus, in this view, rc-electrons, even if they are responsible for the aromatic character of benzene, are "forced" to adopt such a structure by the ~-skeleton. A recent precisely determined geometry of chrysene [26-27] showed that the aromaticity of chrysene increases if it is involved in an EDA (electron donoracceptor) complex with TCNQ or fluoranil: acceptors in these complexes pump out electrons from the chrysene molecule, causing an increase of its aromatic character. This is the first experimental argument for the idea of Shaik and Hiberty. The problem is whether we should look at the phenomenon of aromaticity from the point of view of one representative property, or try to see its complexity as indicating more distinct properties, depending on the kind of situation. In this case certain problems arise: How many dimensions should be taken into account? What is their physical basis? In many cases, aromaticity indices are accessible for only a small number of molecular systems for which decisive conclusions could be reached. Additionally, many indices of aromaticity are based on models which apply more or less approximate methods of quantum chemistry (for recent review cf. ref. 16). However we should be aware that, apart from the uncertainty due to the models used in estimating aromatic character, there is also some uncertainty due to the approximations in the quantum chemical method which is applied. Therefore, in addition to this fruitful source of information on aromaticity, experimental data on geometries, for example, should be taken into account. Due to the very fruitful development of X-ray diffraction techniques for determining molecular geometries, and large

Carbocyclic n-Electron Systems

231

databases in which this information is collected [28], it seemed reasonable to us to show how important this source of information about aromaticity of molecules or even of their fragments may be. It is worth mentioning that by applying the structural data--essentially bond lengthsmto obtain variously defined indices of aromaticity unifies to some extent the structural and behavioristic definitions of aromatic character. This aspect will be discussed in more detail later.

I!1. INDICES BASED ON MOLECULAR GEOMETRY Julg and Franqois [29] were the first to apply molecular geometry to define a quantitative measure of aromaticity, i.e. the index of aromaticity. They defined it as a function of the variance of the perimeter bond lengths in the molecule,

n

A j = I - 225 ( l ~rn

2 - -~1

(1)

where n is a number of C - C bonds in the system; R is the mean bond length, and the summation runs over all C - C bonds r = 1 . . . n; 225 is a normalization factor which gives A = 1.00 for a system with all bonds of the same length; and A = 0.00 for the Kekul6 structure of benzene with alternation of double and single bonds. The procedure of Julg et al. [29] has an obvious limitation: it can be used only for carbocyclic n-electron systems. It was then surpassed by a model called HOMA (harmonic oscillator model of aromaticity [30]), H O M A = 1 -- not Z [Ropt - gi]2

(2)

in which the average bond length was replaced by a so-called optimal bond length Rop t which was defined by,

R(s) + w.R(a) R~ =

(3)

1+ w

where R(d) and R(s) stand for lengths of pure double and single bonds, respectively; w is the ratio of their stretching force constants, w = k(x=~.; and XY may be any kind k(Xof bond for which we know precise double and single ~ond lengths. The ratio of force constants was assumed to be w = 2, in rough agreement with ratios for particular kinds of bonds [31]. The final formula for HOMA in its actual form is [32], HOMA = 1 -

{ot(CC)Z [R(CC)opt- Ri] 2 + a ( c x ) Z [R(CX)opt

+a(CY)Z [R(CY)opt- Ri]2 + ot(YY ) Z

-

Ri] 2

[R(yY )opt- ni ]2 }/n (4)

232

TADEUSZ MAREK KRYGOWSKI and MICHAt CYRANSKI

Table I. Structural Parameters of the HOMA Index CC CN CO CP CS NN NO

R(s)

R(d)

Rop,

1.467 1.465 1.367 1.814 1.807 1.420 1.415

1.349 1.269 1.217 1.640 1.611 1.254 1.164

1.388 1.334 1.265 1.698 1.677 1.309 1.248

257.7 93.52 157.38 118.91 94.09 130.33 57.21

where ot(XY) is an empirical constant which accounts for the ability of the specific bond R(XY) to undergo compression or expansion, as well as taking into account different ranges of bond length variability depending on the nature of the bond. Table 1 presents HOMA parameters for various bonds. Thus the HOMA index can be used for all hetero-n-electron cyclics for which there are parameters listed in Table 1. Additionally it has an advantage over the older Julg index that in the case of rings with low alternation of bond lengths, i.e. with "partly" equalized bond lengths but with a large value of the mean value (e.g. so-called "empty" rings in Clar classification [33]), HOMA correctly predicts a low aromatic character (in opposition to Aj). This aspect will be discussed in detail later. Another attempt at extending the use of molecular geometry in determining aromatic character comes from Pozharskii [34] and in a similar way by Bird [35]. The last treatment is known as an index of aromaticity--15 (for five-member rings) and 16 (for six-member rings)--which was applied to many carbo- and heterocyclics [36]. The main idea of this index is that the aromatic character depends on the variance of bond orders, N, which are calculated from bond lengths by use of the Gordy [37] formula, a N = --~ - b

(5)

where a and b are empirical constants. Then the index of aromaticity I is defined as,

(6)

,: lOOl, where, 100

V=N-~av

~

~ [N-Nav]2 n

(6a)

Carbocyclic x-Electron Systems

233

with Nav being the arithmetic mean of bond order; and Vk a constant depending on kind of system--equal to 33.5 for six-membered rings and 35 for five-membered ringsmexpressed as a percentage. Benzene and all cyclic x-electron systems with equal bond orders (independent of their values) have 100% aromatic character. In other words this index suffers from the same disadvantage as that of Julg: empty rings in benzenoid hydrocarbons will have a relatively high aromatic character. Very recently it has been found that the HOMA index may be decomposed into two terms, of which one accounts for the energetic contribution to the aromatic character and the other term describing the geometric contribution [27]. The separated form of HOMA reads, HOMA = 1 - EN - GEO

(7)

GEO = not Z (Rav-

(8)

where,

Ri)2

EN = (x(eop t - R a v ) 2

for Rav > Ropt

(9a)

EN = -ot(Ropt - Rav)

for Rav< Ropt

(9b)

and,

The above formulation (Eqs. 9a and 9b) is important for rings in which the mean bond length, Rav, is shorter than Ropt. Then EN becomes negative and HOMA may attain a value greater than unity. It should be mentioned that geometric indices of aromaticity need not necessarily be based on the concept of variance. If we take into account that, in rings with alternating bond lengths, the sum of consecutive squared differences directly measures the degree of bond length altemation, then we arrive at the definition of the aromaticity index called BAC (bond alternation coefficient) [19-21], which is defined as, BAG :

~/Z[Rr- Rr+1]2

(10)

r

or in the normalized form: BAC n :

1 - 3.46~/~--'[R

r - R r + 1] 2

(11)

r

For a system with all bonds of equal lengths BAC = 0 or BACn = 1. Undoubtedly this index is purely geometric in nature.

234

TADEUSZ MAREK KRYGOWSKI and MICHAf_ CYRANSKI

IV.

RELATIONS BETWEEN INDICES OF A R O M A T I C I T Y BASED O N M O L E C U L A R GEOMETRY

There are two kind of problems in using different indices of aromaticity. One is associated with a difference in physical or chemical grounds of models used as a base for defining a given index. The other is the analysis of similarities and dissimilarities of aromatic character for various groups of molecules or their fragments, when using different indices of aromaticity. One of the most fundamental features of nonaromatic compounds is alternation of their bond lengths. Hence the proposal of Julg and Franqois [29] to use as a measure of aromaticity the value of the variance in the ring (or perimeter) bond lengths (Eq. 1), a safe and very advantageous statistical method describing dispersion of the data around the mean value. The same idea of using the variance of bond orders was applied by Bird in the construction of his 15 or 16 (Eq. 6) index of aromaticity. Application of the Gordy [37] bond orders was necessary to extend the

1.447~~47

1.447

(b)

.

(c)

. 1.422

~

Fe

1.432

1.437

1.437

P<:(CH3)3 C~Si(CH~3

~

1.420

1.420

1.420

Figure 1. Plot of (a) the fullerene C60 and (b) a ferrocene complex with bond lengths and the ring numbering scheme; (c) Plot of (E)-I-t-Butyl-2-(carbonyl-OlS-cyclopentadienyl)-nitrosylrhenium)-2-tr i methyl si Ioxy-( 0)- 1-phosph aethen e.

Carbocyclic x-Electron Systems

23 5

Table 2. Indices of Aromaticity for Pentagon Ring of Fullerene C60 for Two Rings of Ferrocene Molecule for Rhenium and Calcium Complex Compound

Fullerene C6o (pentagon ring) Ferrocene (1) (2) Rhenium complex Calcium complex

HOMA

EN

GEO

BA Cn

15

Aj

0.103

0.897

0.000

1.00

100

1.00

0.691 0.434 0.736 0.986

0.308 0.564 0.264 0.001

0.000 0.001 0.000 0.013

0.993 0.975 1.00 0.929

99.6 98.1 100 94.5

1.00 0.999 1.00 0.994

main Julg idea to systems with heteroatoms. Both these treatments suffer from the same shortcoming: they do not take into account that equalization of bond lengths is not the only factor determining aromatic character. When we look at the pentagonal rings in the fullerene C60, we find that it has all bonds of the same length equal to 1.447 ~ [38] (Figure 1a) and hence 15 = 100, but H O M A is only 0.103. Of course the low aromaticity monitored by HOMA is due to the energetic factor, EN = 0.897 since GEO = 0.000. The same effect is found for the cyclopentadienyl ring in a complicated compound like (E)-l-t-Butyl-2(carb~ny~-(q~-cyc~pentadieny~)-nitr~sy~-rhenium)-2-t~methy~si~xy-~)~-ph~sphaethene [39] (Figure 1c). All C - C bonds in the ring are of length 1.420 A and hence 15 = 100, whereas HOMA = 0.736, and dearomatization is due to the term EN = 0.264 since the other term GEO = 0. A similar picture is often met in derivatives of cyclopentadienyl and ocenes. Ferrocene [40] may be a good illustration, as shown in Figure lb. Table 2 summarizes the results showing that the purely geometric indices like 15, Aj and GEO fail in these cases, since the aromatic character of them is only or mostly due to the energetic factor.

(a)

1.392

1.395

/

1.394

(b)

/

,.,,o \

/

1.393

Figure 2. Plot of perylene molecule (a) with bond lengths (b) the numbering scheme of rings.

236

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKI

In all three cases the rings have THE SAME 15 and Julg's A j values: 100% and 1.00, respectively. Undoubtedly the five-member carbocyclics in these three molecules are different as far as their aromatic character is concerned. The pentagon ring in C60 is part of the weakly aromatic fullerene [41], and is less aromatic than the hexagons which exhibit a considerable alternation of bond lengths. This conclusion is in line with the calculated ring currents and diamagnetic susceptibility measurements [42-43]. On the other hand, the cyclopentadienyl anion is considered as aromatic, or at least almost aromatic [44-45]. HOMA values for cyclopentadienyl in the rhenium complex and ferrocene are much higher and the dearomatization is due only to the energetic term. In some cases, however, cyclopentadienyl ring exhibits high aromatic character in terms of both kinds of indices: 15 and HOMA. This is met in the case of calcium salt of cyclopentadiene [46] for which HOMA = 0.986 and 15 = 94.5. Table 2 presents relevant values of indices of aromaticity discussed in this review. A similar but not so dramatic situation is met in the condensed benzenoid hydrocarbons with Clar's [33] empty tings. Consider e.g. perylene [26] (Figure 2). The 16 value for the external tings and for the internal one are 79.2 (mean value) and 83.0, respectively. The equivalent values of Aj are 0.931 and 0.957, respectively. Undoubtedly the central ring is weakly (if at all) aromatic, whereas the external ones are quite aromatic. Again the 16 index predicts almost the same aromaticity for these two very chemically different tings. Table 3 summarizes these results. The reason for these strange results is the fact that indices based on variance of bond lengths or bond orders do not take into account changes of bond energy. The same is true for BAC. It is well known that shorter bonds have greater energetic content. Empirically it may be shown, at least for C-C bonds that the dependence of bond energy, E(n), on the bond lengths, R(n) [19, 26], is: (12)

E(n) = 87.99.exp{ 2.255[ 1.533 - R(n)] }

Equation 12 was tested on heats of formation (from atoms) of eight wellmeasured benzenoid hydrocarbons, and the agreement found was very encouraging

Table 3. Indices of Aromaticity for Individual Rings of Perylenea'b Ring

HOMA

EN

GEO

BA C n

16

Aj

1

0.816

0.045

0.139

0.742

80.3

0.938

2

0.798

0.036

0.165

0.701

78.4

0.926

3

0.133

0.765

0.101

0.709

83.0

0.957

4

0.799

0.038

0.162

0.721

78.6

0.928

5

0.812

0.038

0.150

0.743

79.6

0.933

Notes:

aSmalldifferences in values of indices of aromaticity for rings 1,2,4,5 of perylene are due to the fact that perylene molecule in the crystal cell does not occupy the special position. t'l'he numbering scheme according to the Figure 2.

Carbocyclic re-Electron Systems

237

Table 4. Bond Energy of Individual Rings of Perylene

and Triphenylene Molecules

Type of Ring(s)

Perylene

Triphenylene

Central

648.1

653.4

External

711.7 714.0 713.5 713.4

721.9 718.9 716.8

[19]. When we apply this formula to the geometry of perylene [26], the difference in C - C bond energy of the central ring and the external ones is about 60 kcal/mole, as shown in Table 4. Evidently the central ring must have much less resonance energy in comparison to those which are external. In other words, the central ring is energetically much less aromatic than the external tings. Similar results are found for triphenylene, for which the precise geometry is known [47]. Table 4 presents relevant data for the central and external tings for perylene and triphenylene. The bond energy (BE) is a simple sum of energies of particular bonds of the ring. It is worth presenting the BE value for benzene of 717.4 or 724.6 Kcal/mole [19], depending on the minor differences in the experimental bond lengths taken into calculation. Evidently the external tings have comparable energetic content (i.e. resonance energy) to benzene itself, whereas the central tings are considerably less stable. It is noteworthy that the HOMA index as shown in Eq. 7 is composed of two terms, one of which (GEO) is proportional just to the variance of bond lengths, i.e. closely related quantity to that of Julg (Eq. 1) and that of Bird (Eq. 6), provided that in the latter, the bond lengths are replaced by bond orders (Eq. 5). However, the other term (EN) depends directly on the difference between the mean C - C bond length in the ring and the Rop t, i.e. a single value. The shorter the mean bond, the stronger it is, i.e. it has more energy content. Figure 3 illustrates the dependence of mean C - C bond lengths on BE calculated by use of Eq. 12. Indeed it may be concluded that the term EN in Eq. 7 represents the contribution to aromatic character of changes in resonance energy; hence the name of the term EN. Thus it may now be asked how these two terms in Eq. 7 are related to the above-mentioned indices of aromaticity, which were claimed to be geometric in nature (i.e. based directly on the alternation of bond lengths as BAC, or based on the variance of bond lengths or bond orders). Table 5 presents correlation coefficients for regressions of EN on BE (Eq. 12) and GEO on 16 or BAC for a few samples of n-electron systems. Finally we plotted EN against GEO to test their mutual correlation. As shown by correlation coefficients there is no mutual dependence between the energetic and geometric contributions to the aromaticity index HOMA. The only exceptions are cyclopentadienyl complexes with Rh. This is an effect of strong statistical bias by

238

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKi 73o [

720

710

BE 700

690

1

139

9

,

,

,

I

i,i

139.5

i

i

I

140

. . . .

I

. . . .

140.5

R

1

. . . .

141

I

,

141.5

,

,

9

I

142

(x O.Ol)

Figure 3. Linear d e p e n d e n c e of C-C bond energy (in kcal) of the ring of T C N Q complexes and salts calculated by use of Eq. 12 on their mean bond length values, R, (in ,g,). Correlation coefficient r = --0.989 for n = 90.

a few points which determine the least-square linear correlation. Undoubtedly, if the sample is not normally distributed, correlation analysis should not be used for studies of mutual dependence of two-dimensional random variables [48]. In this particular case 4 of 48 data points determine this relatively high correlation coefficient as shown in Figure 4. Table 5. Correlation Coefficients for Linear Relationships between EN, GEO, and

Other Indices of Aromaticity (BAC, BE, and I6) for Four Samples of Structurally Different Topological Patterns

Sample

n

BAC/GEO 16/GEO

BE/EN

EN/GEO

Benzene rings in TCNQ-EDA complexes and salts Benzene rings in benzenoid hydrocarbons Benzene rings in p-disubstituted benzene derivatives Cyclopentadienyl rings in complexes with Rh

90

-0.991

-0.992

-0.977

0.390

169

-0.929

-0.983

-0.948

0.048

116

-0.935

-0.995

-0.906

0.092

48

-0.949

--

-0.899

0.780

Carbocyclic x-Electron Systems

239

4.9

3.9

2.9

EN 1.9

0.9

-0.1 ~t

,

,

o

l

o.s

,

,

I

,

~_

I

t

...,

o8 GEO 09

I

1.2

,

,

I

1.5

Figure 4. Scatter plot of EN vs. GEO for 48 five-membered rings from complexes of cyclopentadienyl with Rh. V.

REVIEW OF MOLECULAR

PROBLEMS

Since there are several aromatic compounds, it seems appropriate to present the rules of selection which have helped us in the presentation of structural aspects of aromaticity. First, we employ only the most precise experimental data, which in the CSD are denoted by AS = 1 and only sometimes AS = 2 [28]. It means that the mean esd of bond lengths for systems taken into consideration is not greater than 0.01 A. In some cases, for which experimental data are not accessible at all, we have used for illustration some computed bond lengths, but applying the most accurate ab initio techniques feasible. Second, the presented compounds are chosen in a way to show some regularities, or third to show some typical or characteristic situations. A.

Benzenoid Hydrocarbons

Benzenoid hydrocarbons represent a convenient group of compounds for which there is a sufficient amount of reliable experimental geometries and for which the separation of energetic and geometric terms may be carded out. Table 6 presents HOMA, EN, and GEO terms for 171 tings of 27 benzenoid hydrocarbons [27, 49]. Only unsubstituted compounds have been taken into consideration.

240

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKI

Table 6. Aromatic Character of Individual Benzene Rings in Benzenoid Hydrocarbons a

o~o.,21 I o=o.121 I

I. BENZENE

III.A N T H R A C E N E

il NAPIITALENE

IV.PHENANTHRENE Complex with TCNQ E==0.00~6 ~.

D=0.378 [G= 0

V. CHRYSENE

VI. CHRYSENE

13=0.3261 G~.323 I

i o;o.l:~

~I. B E N Z O P H E N A N T H R E N E

.VIII TRIPHENYLENE

IX. PYRENE(neutron diffraction)

X. PYRENE(X'-raydiffraction)

XI. PERYLENE

XII. PENTAHELICENE

Table 6.

Continued

~=o.3o5 10=o.o51 I (3=0.305 I

PICENE

XIII. DIBENZOANTHRACENE

XlV.

I G=0"34~G--01520 I I E=0"01~ I c~:o.14

XV. TETRABENZONAPHTALENE

~

0

XVI.HEXAHELICENE

G--o.o9

13o\ G=0.133 )'---"'6=o.31o

It=o.~/ \

~

--" E--0.17S G=0-25(

XVII. HEPTAHELICENE

XVIII. PYRENOPYRENE

G~~.256 [

XX. TETRABENZOPENTACENE

X~.CORONENE

XXII. OVALENE

XX1.TETRABENZOPERYLENE 241

242

TADEUSZ MAREK KRYGOWSKI and MICHAf_ CYRANSK!

Table 6. Continued

=0.1|5

o-4.325 ~

~,&ii~ ~

XXIII. ANTHRABENZONAPtlTOPENTACENE

XXIV. DIPliENANTHROPICENE

XXV.QUATERRYLENE

)~XVI.TRIBENZOPHENANTHRAPENTAPIlENE

XXVII. DIBENZONAPtlTOPYRANTHRENE

Note:

aE, G, and H stand for EN, GEO, and HOMA, respectively.

A few observations can be made while looking at the above mentioned data gathered in Table 6. First of all, a great diversity in aromatic character for individual phenyl tings is observed depending on the topological embedding [19], or less precisely the closest molecular environment. Another important finding is that the dearomatization of the ring may be due to two different reasons: either it is due to bond length alternation, or to the elongation

Carbocyclic g-Electron Systems

243

of the mean bond length in the ring. The first is monitored by the GEO term, and the other by the EN term. In some, neither mechanism is dominant. The HOMA value for the central rings in perylene (XI) and triphenylene (VIII) are as low as 0.133 and 0.077. In both cases the GEO terms are rather low: 0.101 and 0.239, respectively. In contrast to this, the EN terms are much larger at 0.765 and 0.683, respectively. The same may be also observed for the central ring of coronene (XIX). Evidently the dearomatization of these tings is due to the large EN terms, i.e. these tings have lower resonance energy. An opposite situation is found in the central ring in phenanthrene (IV) for which the HOMA value is 0.400. However, the GEO term in this case is relatively large (0.419) compared to the EN term (0.181). Evidently, the dearomatization of the central ring in phenanthrene is due mostly to the alternation of the bond lengths. In general, the angularly condensed tings are of this type, as may be seen also from the HOMA, EN, and GEO values for chrysene (V), benzophenanthrene (VII), or helicenes (XVI, XXIV). A statistical treatment (factor analysis [50]) of the aromatic character HOMA and its geometric and energetic contributions is helpful in deciding which of these components is decisive in dearomatization of particular benzene tings. Figure 5 is

1

.H(,,)

i

G(III)

,

E(VIII) "~'J

0.6 r

9

" H(IX) E(XI)

.

G(IX)

~ G(X) 9 E(V)

0.2 I!" Rec.I

9E(I)

.E(III)

8E(II)

9H(VIII)

9G(VI)

9

G(XI)

-0.2

-0.6

-1

r

i

l~ 0

1

H(I)

9H(III)

9

H(V)

H(VI)

. G(Vlll) " E(IX) lY H(X) H~.I)

2 3 4 Number of rings

5

Figure 5. Plot of resources of the first factor against the number of rings fused to the ring in question. Numbers in parenthesis stand for numbers of groups to which belongs the given ring (see Table 7).

244

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKI

a plot of resources of the first factor (which described 65.7-91.9% of the total variance) in H(i), G(i), and E(i) against the number of rings fused to the ring in question. Due to the form of relations between HOMA, EN, and GEO, at least one component has to be negative. The most important information is which of these two terms, GEO or EN, is a counterpart of HOMA. This counterpart is the most important in determining the HOMA value. The analysis of the results leads to the conclusion that the aromatic character of benzene rings with three or fewer fused tings is due mostly to geometric contributions, whereas in other cases energetic contribution is decisive [49]. Another interesting observation is that the central tings in chrysene (V) are less aromatic than if they are in chrysene (VI) involved in the EDA complex with TCNQ or fluoranil [26-27]. The partial loss of n-electrons of value -0.35 e in the EDA complex with TCNQ (for the method of estimation see ref. 51) due to the intermolecular charge transfer causes an increase of aromaticity observed in the decrease of bond alternation. This finding seems to be the first evidence based on experimental data which supports the Shaik and Hiberty [23, 24] idea of the decisive role of the a-electron skeleton in determining the averaging of the bond lengths in aromatic compounds. Pumping out n-electrons from the chrysene moiety results in an increase of its aromatic character, particularly expressed in the central rings. These changes are due to the decrease of alternation of bond length, shown by a decrease of GEO term, whereas EN terms remain practically unchanged.

B. Polysubstituted Benzene Derivatives Another large sample which is worth investigating are substituted benzene derivatives. Since the number of polysubstituted benzene derivatives is very large, data for precisely determined geometries of 2045 species have been statistically analyzed [52] using HOMA, EN, and GEO terms calculated from experimental bond lengths (retrieved from CSD [28]). The most striking feature of this group is that the diversity of the aromatic character of benzene rings in this group of n-electron systems is much smaller than in the case of benzenoid hydrocarbons. Table 7 shows the comparison of HOMA, EN, and GEO values for phenyl tings substituted in various ways and these indices estimated for analogous closest topological embedding of phenyl tings in the benzenoid hydrocarbons. The abovementioned observation is now well illustrated: differences between mean HOMA values for the first group (tings in substituted benzene) and second group (tings in benzenoid hydrocarbons) are sometimes dramatic in magnitude. For example between Class 8 and class III or V are differences of 0.3-0.4 units of HOMA. For class 12 of the first type of group there are five topological analogues in the second type of group (classes VII-XII). The differentiation between these analogous classes is huge: from the mean value of HOMA = 0.22 for class X to the value 0.73 for class XII.

Carbocyclic rc-Electron Systems

245

Table 7. HOMA, EN, and GEO Values for Substituted Phenyl Rings Type of Ring

Class

Sample Size HOMA

EN

GEO

Group Class HOMA

(T1

,26

0.994 _000503 0.0,35

(~2

379

0.973 000003 0.0297 ~

~~

~4

0.99~

000~90.01~6

~~

~6

0.990

0.00~0.0,4~

~1~

0.866

0.00~6,0.,,~

6

312

0.975 -0.00035 0.0247

~.

7

22

0.995 -0.00016 0.00587

~

8

108

0.937

.s

~

0.00623 0.0572 ( ~ ~

EN

GEO

i

0.8620.009280.128

~

08~,00~610.,41

HI 0.503 0.134 0.329 V 0.639 0.155 0.192

~

9

154

0.964

0.0014 0.0335

~ } ~ 10

119

0.966

0.0014 0.0295 ~

IV 0.818 0.124 0.0574

87

0.946

0.0098 0.0390 ( ~

VI 0.645 0.146 0.164

~

11

~ ~

12

161

0.967

VIII 0.794 0.157 0.0531

0.0078 0.0218 ( ~

VII 0.314 0.484 0.202

(~

IX 0.271 0.549 0.192

~

X 0.219 0.662 0.0722

~

XI 0.459 0.436 0.0845

~

XII 0.728 0.212 0.0599

246

TADEUSZ MAREK KRYGOWSKI and MICHAL CYRANSKI

The main conclusion from these results is that topological requirements are much stronger than chemical ones! In benzenoid hydrocarbons, the closest atoms neighboring the rt-electron structure of the phenylic ring are either carbon or hydrogen atoms. In both cases effects of electronegativity are minor or absent, although this is a main driving force in the substituent effects. The above presented statement may be a good ground for the understanding why the so-called graph-topological approaches [53-55] applied to benzenoid hydrocarbons or even to nonalternant hydrocarbons may be so successful in prediction of their aromaticity. In the case of polysubstituted benzene derivatives, variation in electronegativity of the closest neighbors (i.e. substituents) is much greater. But in spite of this, the observed variation of aromaticity is much smaller. In all cases of substituted benzene derivatives, dearomatization due to substitution is mostly due to the geometric term. The dominant contributions determining the changes in overall aromatic character are also those coming from variation of the GEO terms.

C. Aromaticity and the Hammett-Like Substituent Effects The induction of the quinoidal structure in the molecules of para-disubstituted benzene derivatives with substituents strongly differing in their electronegativity have been reported in the first papers on their structure determination. A typical example is p-nitroaniline, studied as early as in 1961 [56], for which canonical structures I, II, and VI may be drawn with an indication of enhanced contribution of VI, i.e. the quinoidal one (Scheme 1). This view has been stressed even stronger in the latest report on this compound [57]. Application of the HOSE model [58], which allows estimation of canonical structure weights from the molecular geometry, made these efforts much more quantitative in nature and has been used many times in the last decade. It indicates that small but concerted changes in bond lengths in the ring of substituted species lead to the substantial changes in weights of canonical structures which, in turn, correlate well with the Hammett substituent constants. In this way, weights of canonical structures of 14 para-substituted nitrobenzene have shown good or very good correlations [59], as do canonical structures of benzene and pyrylium tings in 2,6-diphenyl-4-(4-substituted-phenyl)-

X

X

X+

X+

(I)

(II)

(III)

(IV)

X+

~

Scheme 1. Considered canonical structures.

X+

{~/I)

Carbocyclic g-Electron Systems e/oiL11} */, f llt-V | "/, IVll

~--

-

]

-2.0

247 '

~

"II

-- 1.0

Q

9

0 ~ I~p I

Figure 6. The dependence of % (I,I!) (empty circles) % (Ill-V) (full circles) and % (VI) (crosses) calculated for the traditional resonance theory scheme of substituent effect

on cr+ for electron-donating and Op for electron-accepting substituents.

pyrylium perchlorates [60]. Figure 6 shows how various weights of canonical structures depend on substituent constants. We should note that slopes for weights of I + II and III + IV + V are opposite, but similar in their absolute values, whereas slope for the regression of weight of VI canonical structure is about 4 times less, i.e. this effect is 4 times less sensitive to the substituent effect. Undoubtedly the induction of the quinoid structure, VI, and partial quinoid structure, V, in the ring of para-disubstituted benzene derivatives are well-monitored via HOSE estimated weights of canonical structures [59]. Consequently, since geometric patterns (bond lengths) of these two canonical structures contribute mostly to the variance of bond lengths in the ring, they tell a lot about the aromatic character of the ring. The HOMA and other type of indices are slightly less sensitive to changes in geometry, and hence less results are reported in this field. Nevertheless, it was shown that the geometrical indices work much better in the correlation with Hammett's substituent constants than the energetic ones [20] in describing the variation of the aromatic character with the nature of the substituents. Figure 7 clearly shows these kinds of relationships. For para-substituted anilines two typically geometric indices (BAC and/6) correlate well (the correlation coefficients are -0.908 and -0.924, respectively) with ap provided (~p > 0. This condition says that, only in the case when the counter substituent in aniline is electron-accepting,

248

TADEUSZ MAREK KRYGOWSKI and MICHAL CYRANSKI (x ool) 100

9

o

."

.'""

9 9

135

92

BAC

BE

U 9 ~

O

9

72O

9

9

125

9\ ~

,,s

9

oo

" O! 0

"

71c -01

-03

Ol O'

~t

I

-01 . . . . .

09

05

f

t

-03 -

"

~

" 05

"

vn 9

I

9

Z

9

18 O2

.0'~~ " g ~

-

0',

@

....

0's " -0;

Figure 7. Scatterplots of BACn, BE, and 16, respectively, vs. Hammett's substituent constants (for aniline derivatives). is the deformation of the ring meaningfully interpreted as the formation of the quinoid structure. In consequence, the geometric indices monitor it. As seen from the graph of BE versus o, the correlation is bad (correlation coefficient equal 0.676). Application of the separation model, i.e. Eqs. 7-9, leads to similar results (Figure 8). All relevant data are gathered in Table 8. It may be concluded that the substituent effect in para-disubstituted benzene derivatives on the aromatic character of the ring is due mostly to the geometric contribution to aromaticity, and hence the most appropriate indices for these compounds seem to be geometric indices like BAC and Bird's 16.

D. Aromaticity and the Steric Substituent Effects Very recently it has been shown that the steric hindrance in exocyclically substituted benzylic cations causes a dramatic change in the aromatic character of the ring. Application of HOMA, EN, and GEO and BAC indices to the ab initio

Carbocyclic x-Electron Systems

249

(x O.Ol)

(X 1E-3) 36

loo

HOI~ 96

9

o

o.+

o.,

o.,,

',

oo

~ o

0

0.2

0.4

0.0

0.8

!

o

9

o

.

.

.

0.0

.

i

0.0

.

.

.

t

t

0.08

O.Oe GEO 0.04

0.02 $

0

0.2

"

0.4

o

Figure 8. Correlation HOMA, EN, and GEO, respectively, vs. Hammett's substituent constants (for aniline derivatives). 6-31G calculated molecular geometry of 18 (Scheme 2) 7-monosubstituted and 7,7-disubstituted derivatives of benzyl cation has led to the conclusion that variation in the aromatic character is due chiefly to the geometric term, GEO, or equivalently correlated well with BAC [61]. Moreover, it was found that the increase of aromatic character is associated with an increase of the C1-C7 bond length (Figure 9). This

Table 8. Correlation Coefficients for Scatter Plots of Aromaticity Indices vs. Hammet's c~for 13 para-Substituted Derivatives of Aniline a

ot

HOMA

EN

GEO

BAC

BE

-.7346

-.4894

.8723

-.9079

.4155

-.9243

.0042

.0896

.0001

.0(O

.1579

.0000

Notes: an stands for significance level of the regression line.

16

250

TADEUSZ MAREK KRYGOWSKI and MICHAL CYRANSKI R2

R1

C1

Cs ~ . C

\oy

3

Scheme 2. Atom numbering of substituted derivatives of benzyl cations.

finding is in line with the following interpretation: For the bulky substituent at C7, the plane of the substituent is perpendicular to the plane of the ring and, consequently, there is no opportunity for electron delocalization. The ring is not too much perturbed from its geometry (and aromatic character) of the unsubstituted benzene. When the substituents become less bulky (or even not at all) ~,then the interplanar

1.05

J

0.95

,-

0.85

0.75

o.6s 0.55

0 y..//

/~

0.45 . . . . . . . . . . . . . . . . . . . . 1.340 1.360 1.380 1.400 1.420 1.440 1.460 1.480

1.500

ClC7 Figure 9. The BAC values plotted against R(CI-C7). Correlation coefficient r= 0.974.

Carbocyclic x-Electron Systems

251

(~R2

CR2

CR~

CR2

CR2

(i)

Oi)

(ul)

or)

(v)

Scheme 3. Considered canonical structures of benzyl cations.

angle between two planes decreases and the n-structure of the ring interacting with the positively charged exocyclic group becomes more and more positively charged. Consequently a partially quinoid structure III becomes more favored, and decrease of aromatic character is observed (Scheme 3). Finally one may observe the relationship between the charge at C7 and the values of aromaticity indices (BAC or HOMA as shown in Figure 10).

E. Angular Group Induced Bond Alternation (AGIBA) Very recently it was reported that applying a HOSE model [58] to the 6-31G* geometry of anisole methoxy group in anisole induces a slight imbalance of the Kekul6 structure weights [62]: 52.6:47.4. This finding was then confirmed experimentally. Low-temperature X-ray diffraction measurements of 1,3,5-trimethoxy benzene [63] and 2,4,6-trimethoxy-s-triazine [64] gave reliable geometry; then imbalance of the Kekul6 structure was calculated as 56.4:43.6 and 57.7:42.4, respectively. Similar effects were observed in the case of diazobenzene derivatives [65]. The mean geometry of the ring in 21 well-determined derivatives of diazobenzene (substituent varying only at N2) led to the imbalance of Kekul6 structures as 54.2:45.8. The main conclusion of these studies [62-65] is that angular substituents XY with the single X - Y bond induce more double cis-C1C2 bonds in the ring, whereas if XY has a double bond, X'-Y, then the C 1C2 bond becomes less double than the C6C1 bond. These effects are well monitored by use of the HOSE calculated weights of canonical structures, but they also are observed, in a less significant way, when aromaticity indices are applied. Tables 9-10 present the collection of data [66] which show that the dearomatization of the benzene ring due to angular substituents is mostly due to the geometric term.

F. Aromaticity of the Ring in the Strain-Affected Benzene Derivatives It is well known that the bond alternation in substituted benzene is observed whenever the aromatic ring is fused to one or several small rings. Usually it is interpreted in terms of the so-called Mills-Nixon effect [67] and may be described

252

TADEUSZ MAREK KRYGOWSKI 1.05

.

.

.

.

.

.

.

and M I C H A L

.

.

.o

0

0.95

I

CYRANSKI

~

,'"

,. -.

..o / ,

o~ ..~~ .o..'" ~ 0 .j~ ..~ ...~ ~ ..'"i ~176176 ~176149 ..~

0.85

.~176176

HOM,~.75

........

/'

:....."oo /

...6()

0.95 /

0.55

9

..

:'"'"

...~ ~149 ~149149 --

-0.3

0

-0.2

~

-0.1

0

0.1

0.2

0.3

0.4

q(c7)

1.05

~O

0.95

/

0.85

o

0.75

0.65

0.45 -0.3

..

/

-0.2

/,.,,.

-0.1

0

0.1

0.2

0.3

0.4

qC7

Figure 10o The H O M A and BAC values plotted against q(C7). Correlation coefficient r = 0.88 and r = 0.864, respectively (significance level in both cases ~ = 0.0038%).

Carbocyclic re-Electron Systems

253

Table 9. Aromaticity Indices and the Kekul6 Structure Weights for Various Conformers of 1,3,5-Trimethoxybenzene Estimated from 6-31 G* Calculations and from Experimental Geometry

Z(~ Me~

45

90

135

180

all 90

Exp.

o

O~O" I x Me Me~

0

H = 0.956 H = 0.973 H = 0.975 H = 0.980 H = 0.977 H = 1.002 H = 0.989 MeE=0.000 E=0.000 E=0.000 E=0.000 E=0.000 E=-0.002E=0.007 G=0.044 G=0.027 G=0.025 G=0.020 G=0.023 G=0.000 G=0.004

o

o ~ ~ 1 ~ o ..Me I Me Me~O

70.2%

65.8%

64.9%

63.4%

62.0%

50.0%

56.4%

29.8%

34.2%

35.1%

36.6%

38.0%

50.0%

43.6%

Canonical structure K I

0~0~ Me I Me structureK2 Canonical

by an imbalance of the two Kekul6 structures. Obviously this kind of distortion of the bond lengths in the ring should be associated with the loss of aromatic character. In the last decade there have appeared many papers dealing with these kinds of problems and many compounds were studied experimentally and theoretically. As a consequence, precise geometry of many of them is known. It seems to be of interest to look at the separation of energetic and geometric contribution to the overall aromatic character of them. Table 11 presents the relevant data. Experimental geometry of molecules I, HI, V, and VII leads to a very slight decrease of the aromatic character of the phenylic ring The HOMA values are in the range of 0.91-0.97, and the loss of aromaticity is on both sides: in the geometric and energetic terms of HOMA. Values obtained from theoretically calculated geometries lead to a similar picture (molecules II, VI, and VIII). A very considerable change is observed for unsaturated analogues. Comparing aromaticity indices for 1,2-dihydrobenzocyclobutene (I) and 1,2-benzocyclobutene (IV) we observe a substantial change: aromaticity of (IV) dropped down to 0.56 and the loss of aromatic character is mostly due to the geometric effect, i.e. the huge increase of bond alternation. Experimental geometries of two complicated benzene derivatives, (IX) and (X), led to the same result as observed for (IV): a substantial loss of aromaticity which

254

TADEUSZ MAREK KRYGOWSKI and MICHAL CYRANSKI Table 10. Aromaticity Indices and the Kekul~ Structure Weights of Averaged

Geometry of the Ring of 21 Diazobenzene Derivatives and for Calculated Geometry of 1,3,5-Tridiazobenzene with Varying CNN Angle a tp (o)

128

124

123.869

120

116

112

Average (exp.)

R I

~N

i)

H = 0 . 9 9 4 H=0.971

O

E=0.000

H=0.971

H=0.929 H = 0 . 8 6 6 H = 0 . 7 7 9

E=-0.001E=-0.001E=-0.000E=-0.000E=0.000

H=I.011 E=-0.015

G = 0 . 0 0 6 G = 0 . 0 2 9 G = 0 . 0 2 9 G=0.072

G=0.134 G=0.220

G=0.004

57.63%

66.53%

66.54%

74.72%

81.95%

88.21%

54.20%

42.37%

33.47%

33.46%

25.28%

18.05%

11.79%

45.80%

R I N.~ N

I

(3

Canonical structure

KI

R I

Canonical structure

IC2

Note: aOnesubstituent shown.

is mostly due to the geometric term. Very illustrative is a comparison of imbalance of the Kekul6 structure weights for these molecules. In Table 11 there are given Kekul6 structure weights for a canonical structure of the ring presented in the structural scheme. All calculated geometries were obtained within 3-21 G basis set and the resulting Kekul6 canonical weights are in qualitative agreement with those estimated from experimental geometry. When 6-31 G* MP2 geometries were used for VI [70] and VIII [70] the obtained Kekul6 weights and aromaticity indices are in a qualitative agreement with the experimental ones. For the first class of compounds, with fused saturated rings the K value varies in the range of 92.3-50%, whereas for the other class represented by IV is as large as 95.2%. Evidently the loss in aromaticity is associated with a huge alternation of bond lengths. The observed changes in bond alternation leading to the observed dramatic changes in aromatic character due to the strain have recently been discussed by Baldridge and Siegel [73] in terms of two factors:

Carbocyclic re-Electron Systems

255

Table 11. The Aromaticity Indices and the Kekul~ Structure Weights of

Strain-Affected Benzene Derivatives

I [68]

~

~

56.5%

II [69]

50.0%

IV [691

~

~

63.0%

Expelirnentlll

III [681

~-~~A,, Exi)erimental

95.2%

IICH

/<>"-,-=-,/
V [70]

63.1% Exp~mtmtJ

-<>t

VII

c~

#' 68.9%

t ol

VI [70]

c~ ~ c ~

VIII

r o]

i

~

73.5%

Calculated

I

81.9%

I

CH,~ C H ,

Expelirnental

IX [711

~,

C~li~uliled

89.3%

X [721

CH/;""O"~CH' CH 2

~,o\/---X Experimental CH~

92.3% 9.1"12

i ~/

-

CHz_CH i CH2_CH 2 Exliedmiltil

1. the classical Mills-Nixon effect [67], and 2. the n-electron effect. The latter is rationalized in terms of Htickel rule that cycles with 4n + 2 n-electrons are more stable than isolated double bonds, which in turn are more stable than Htickel cycles of 4n n-electrons. It was originally illustrated by the geometry of triphenylene (I) and starphenylene (II) (Figure 11). In both molecules (I) and (II) the three endo-bonds of the central ring are part of an annelated n-system and the three exo-bonds are not. In the triphenylene the annelation ring is a 4n + 2 n-electron system and the endo-bonds are shorter than the exo-bonds, 1.411 and 1.470/~, respectively. Contrary to that is the situation in starphenylene whereby the annelation ring is a 4n n-electron system and the exo-bonds are shorter than the endo-bonds, 1.336 and 1.494 A, respectively.

256

TADEUSZ MAREK KRYGOWSKI and MICHAt CYRANSKi

9

1.470 (exo) I

endo)

1.494 (endo)

1.336 (exo) II

Figure 11. Structures of triphenylene (I) and starphenylene (11).

G. Intermolecular Interactions Affecting Aromaticity In the case of molecules or molecular ions with a mobile n-electron system, considerable variation may be observed in geometry, depending on intermolecular interactions. This effect is often observed in crystallography when two or more independent molecules exist in the unit cell or if a formally symmetric molecule (or part of it) becomes unsymmetrical [74-75]. In these cases they differ in their closest environment and also in geometry (more often and more clearly in conformation). The molecular environment in the crystalline phase may be well-defined (interatomic distances may be measured with high precision). Analysis of changes in geometrical patterns of the molecule embedded in the crystal lattice, and the associated aromatic character changes in relation to the intermolecular interactions (monitored by interatomic distances), is a very good way for understanding how flexible a molecule's aromaticity is. This problem has been undertaken first for the case of the p-nitroso-phenolate anion. Talberg [76] showed that the geometry of this anion varies considerably in magnesium and sodium salts. Further study of these systems [77] showed that this effect is due to the different hydratation of nitroso- and oxo-terminating groups in the anion in these two cases. The lithium salt [78] represents the intermediate change of geometry. Figure 12 presents molecular geometry, interatomic contacts for hydratation of anions and HOMA, and EN and GEO values. It is worth mentioning that the most important contribution to the variation of aromaticity of the phenylic ring inp-nitrosophenolate anion is due to the geometric factor, which in turn is due to variation of the quinoid structure weights in these cases [78]. In conclusion, it may be said that molecules or molecular ions with

Carbocyclic x-Electron Systems

257 +or

:

+or

H

+a

:H ,

H':

+(~

I01

+or

H........IoI ........H

Na +

H ....
"'"

: ......+or H "+Or H

+Or

"

H

Li +

",_..-

," o,"

I01

M g 2+

+or

"'...

,,

,

"

,,,:9'......+or.....H H

_.N,

+Or

~

..+Or " H

+or

............ i::::H

Figure 12. The HOMA, EN, and GEO values for p-nitrosophenolate anion in three salts: sodium, magnesium, and lithium, respectively.

mobile and easily polarizable n-electron structure are potential systems in which the intermolecular interactions may considerably affect the aromaticity of these systems. Moreover, the changes in aromatic character are mostly due to the geometric factors. Interestingly, p-nitrophenolate anions are not susceptible to these kind of effects: the aromatic character of the phenylic ring in these anions is relatively high (HOMA = 0.844, EN = 0.035, and GEO = 0.121), and does not vary too much in various solid-state environments. This is due to the fact that the nitro group has its own mesomeric effect (Scheme 4) and hence is less susceptible for mesomeric interactions with the ring and via through-conjugation with the oxo group than nitroso [79]. Consequently the rt-electron system is less mobile and hence less sensitive to intermolecular interactions. The above-described effects have a good illustration in Katritzky et al. studies [80] where it was shown that classical, i.e. geometrical, aromaticity of some carbocycles such as azulene and most heterocycles increases with the polarity of the medium.

i

R

i

R

)l)

R+

Scheme 4. Model of mesomeric effect of nitro group.

258

TADEUSZMAREKKRYGOWSKIand MICHALCYRANSKI H. Nonbenzenoid Hydrocarbons

The last class of hydrocarbons are alternant (but not benzenoid) and nonalternant carbocyclics. Table 11 presents HOMA, EN, and GEO terms for a few typical monocyclic nonaltemant hydrocarbons. They illustrate exceedingly well the famous Hiickel 4n + 2 rule. Fulvene itself (unsubstituted) is too unstable to accurately determine its geometry by X-ray diffraction, but 11 of its derivatives substituted exocyclically by not too strongly interacting substitents have given the final result similar to that presented in Table 11 as structure I. HOMA is close to zero, and the GEO term is about 3 times larger than the EN term. When strongly electron-donating substituents are present at the exocyclic carbon atom, e.g. the N,N-diamine group (structure II of the table), then the situation changes dramatically: HOMA becomes large, EN term very low, GEO still large, but much smaller than in the case of I. The situation is reversed in the case ofheptafulvene and its derivatives. Heptafulvene, III, is sufficiently stable, and X-ray geometry provided us with HOMA, EN, and GEO terms. Its aromatic character is very low (HOMA = 0.257) but dearomatization is due almost entirely to the geometric term: the strong double bond

Table 11.

~

Indicesof Aromaticityfor Fulvene,Heptafulvene,and Two CylopentadienylSalts[21, 27] ......

NMe2

~~== CH-Ph-p-NMe2 NMe2 II

~ ~

EI-0"-227~ 1=0.769/

/CHO

C \

CHO III

IV

V

VI

Carbocyclic re-Electron Systems

259

[~CHD

=

CHA

~

~

_-

=

[~CHD@

CHA|

Scheme 5. Illustration of the HCickel rule.

alternation is typical for both, fulvene (I) and heptafulvene (IV). If the electron-accepting substituents are attached at exocyclic carbon atoms, the formyl groups (III), then the picture changes in a striking way: HOMA becomes large (0.769) and the GEO terms reduce by a factor of about 3. Of course, these two systems are beautiful illustrations of the Hiickel rule, and the role of substituent is in these cases decisive. Of notable interest is the case of the cyclopentadienyl anion which may be involved in various salts and complex (coordination) compounds. Thorough statis-

1.1

9I

"

"

"1

"

""

"

r

1

'

'

''"

"

I

'

"

"

/--

0.9

HOMA 0.7

0.5

0.3

,

1.5

,

,

,

J 1.7

.....

I ...... 1.9

~ 2.1

.......

l 2.3

DELTA

Figure 13. Scatterplot for the relation between the average of HOMA and A.

260

TADEUSZ MAREK KRYGOWSKI and MICHA{_ CYRANSKI

Figure 14. Frequency histogram distribution of HOMA values for chromium (a) and molybdenum derivatives (b), respectively.

Carbocyclic n-Electron Systems

261

tical analysis of 4079 derivatives of cyclopentadienyl and cyclopentadiene bound to all elements from lithium to thallium (except carbon) showed a great diversity in aromaticity of the ring in these compounds [44]. In typical ionic compounds, such as salts with cations of lithium and berylium groups of the Periodic Table, the ring is aromatic with a mean value of HOMA between 0.9 and 0.6. An interesting regularity was found for cyclopentadienyl compounds with s- and p-block elements. When the mean values of HOMA for subsamples of a given element, say X, are plotted against the difference between the mean C - X interatomic distance R(C-X) and the ionic radius of X, the scatter plot presented in Figure 13 is obtained. The interpretation is clear--the longer the C - X distance, the less ionic interactions and the less uniform interactions between the element X and the cyclopentadienyl moiety, which in turn exhibits lower aromatic character. In typical coordination compounds high values of HOMA are also typical as shown in frequency histograms for compounds with Cr for 145 entries (Figure 14a) or Mo for 481 entries (Figure 14b). In the case of coordination compounds, another regularity is observed. If the average HOMA values for the ring in the cyclopentadienyl compounds with a given element are plotted against the measure of dispersion of the HOMA values in this subsample, the plot as in Figure 15 is found. Since the sets of HOMA values in subsamples are not normally distributed as a measure of dispersion, the interquartile range (IR) was applied [48, 81].

0.93 - i . . , ; - ' ~ , . . . , . . , i . . . . . . . i . . . . . i . . . . , - l ~ . ~ . . . ,

I

,ND

0 83

~ .Ti MO ~ .W

~

,,~In

AVERAGE HOMA

0.73

. Fe

.Co

Ru

|

*" Ni

1

.S

0.63 "P

.Re

.Rh ~

0.53

01

015

02

~ $i

025

IR HOMA

03

035

04

Figure 15. Scatter plot of average of HOMA vs. interquartile range (IR) of HOMA.

262

TADEUSZ MAREK KRYGOWSKI and MICHAt_ CYRANSKI

Figure 16. Structure of tris(cyclopentadienyl)nitrosylmolybdenium and aromaticity indices HOMA for cyclopentadienyl rings. It is obvious from Figure 15 that the more dispersed values of HOMA index, i.e. the greater diversity of the aromatic character of the ring in cyclopentadienyl and/or cyclopentadiene compounds with a given element, the less is its averaged aromatic character. This finding is nicely illustrated by Figure 16 which presents the structure of tris(cyclopentadienyl)nitrosylmolybdenium [82] in which three tings differ in their kind of interactions with the Mo central ion. One of the tings is "almost covalently bound" and hence the interatomic distances between Mo and carbon atoms of the ring are strongly dispersed. The variance of five R (Mo-C) interatomic distances is large and amounts to 0.58 ~2. The HOMA value for this ring is extremely low, equal to 0.023. Two other tings differ considerably from the former one; they are interacting with the Mo central ion in a more uniform way, which is documented by much lower values of variances for R(Mo-C) with interatomic distances being 0.025 and 0.016/~2. In consequence, HOMA values for these two rings are high: 0.913 and 0.717. An interesting conclusion based on these results may be summarized in the picture given in Figure 17 in which the averaged HOMA values for subsamples specified by a given element are plotted against the averaged interatomic distance R(C-X). The rough conclusion may be said that the high values of HOMA, i.e. the aromatic character of the ring in cyclopentadienyl and/or cyclopentadiene compounds with various elements, are mostly met for systems in which R(C-X) interatomic distance is short.

Carbocyclic x-Electron Systems

263

0.93 ,Nb 0.83

~ Or 9

AVERAGE HOMA 0.73

Mn 9

.Fe ,Co Ni 9

,TI "Mo "W

.Ru

.S ,P

0.63 .Re oRn

0.531

olr 2

2.2

-,Si 24 2.6 2.8 AVERAGE RCX

3

3.2

Figure 17. Scatterplot of averaged HOMA vs. averaged R(C-X).

S0

S

E=0-028 [

I E=0.028

So

1

I E=0.484

S 1

Figure 18. Indices of aromaticity for azulene and naphtalene molecules in So and $I

states.

264

TADEUSZ MAREK KRYGOWSKI and MICHA/_ CYRANSKI

I. Aromaticity of Molecules in the Excited State Recently Tyutyulkov [83] has shown that the singlet-state geometry (calculated) of azulene and naphthalene differs significantly when the ground and the first excited states of these molecules are compared. Figure 18 shows the resulting aromaticity indices for these two molecules in So and S ~. It is immediately seen that changes in aromatic character on coming from the ground to the first excited singlets are dramatic. Naphthalene almost completely loses its aromatic character, HOMA drops down from 0.75 to 0.19 with a great contribution of the energetic factor in this dearomatization, and a rather slight increase of the geometric term occurs. The picture for azulene is quite different. One ring, the seven-member one, increases its aromatic character in S l, and the increase is due to the decrease of the energetic factor. Opposite to that, the five-member ring decreases its aromatic character, but this decrease is due to both an increase of the energetic and geometric terms. During the excitation the charge is transferred from the seven-member ring to the five-member ring, thus the seven-member ring becomes poorer in x-electrons and the five-member ring richer. In this direction is observed the decrease of aromatic character, which could be a supporting finding for the Hiberty-Shaik [23-24] concept of the dominant role of the o-skeleton in the balance with x-systems in determining aromaticity.

J. Fullerenes and Their Derivatives Finally, a new, interesting, and important group of nonaltemate carbocycles should be mentionedBfullerenes. Their stability has been the subject of many

/ ~- i_.o12_/

,, E=0.136 ~ ,

/~, G=0.94y H---0.823 "~=0.292

/ \ ~

E-o.933 ~ ,

0.511

\ E-o.l~2

~.-1.2~5 /

Figure 19. Aromatic character of particular rings in (a) fullerene C60 [38], (b) C6o-[(Me3Si)3Si]2*CS2 [98], (c) fullerene C70"6S8 [99], and (d) C84 derivative [100], respectively (star denotes place of substituent addition).

Carbocyclic x-Electron Systems

265

H=-0.089

(c)

15sI ~=o.946 /

G:

HO

392[ (3--0.114/

E=0.200

\ o0

\

.....

o/?S

\ ~-'~

\u=o.o~

/

IH=-o.o'~, lE=0.222]

E=0.224

o o o,

/

0

G=0 157

Figure 19. Continued studies in the past [84-92] and hence their aromatic character has often been discussed [42-43, 93-97]. Recently it has been shown quantitatively that pentagon and hexagon tings in C60, C70, and in their derivatives differ significantly in aromatic character [41]. Figures 19a-d shows HOMA, EN and GEO values for a few of them. It is apparent from these data that HOMA values for pentagons are always low and dearomatization is due to the energetic term. Hexagon tings are usually more aromatic and contributions to HOMA are from both kinds of terms. A dramatic change in the aromatic character appears in tings containing the substituted carbon atom that is caused by change of hybridization of this atom from sp 2 to sp 3, and in consequence lengthening of the bonds with that atom. From these cases results that apart of these changes, the other tings were not touch by substitution.

VI.

CONCLUSIONS

It seems that molecular geometry carries the information which may be applied in studies of the aromatic character of carbocyclic compounds. A large body of structural data stored in the Cambridge Structural Database [28] strongly encourages this view, and the relative easy access to new structural data due to routine X-ray diffraction measurements is a convincing argument for applying molecular geometry in the determination aromatic character. To summarize this review I suggest the reader consider the following by Roald Hoffmann in a forward to a Vilkov, Mastryukov, and Sadova monograph [101]:

266

TADEUSZ MAREK KRYGOWSKi and MICHAL CYRANSKI

There is no more basic enterprise in chemistry than the determination of the geometrical structure of a molecule. Such a determination, when it is well done ends all speculation as to the structure and provides us with the starting point for understanding of every physical, chemical, and biological property of the molecule.

ACKNOWLEDGMENTS The BST/24/96 grant provided financial support for this work. We are grateful to Dr S.T. Howard (Cardiff) for careful reading of the manuscript and providing useful comments. We are also very grateful to Prof. Joel Liebman for his thoughtful and inspiring comments.

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COMPUTATIONAL STUDIES OF STRUCTURES AND PROPERTIES OF ENERGETIC DIFLUORAMINES

Peter Politzer and Pat Lane

Io II. III.

IV.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

270 270

C o m p a r i s o n o f the Nitro and Difluoramino Groups . . . . . . . . . . . . . . Survey o f Computational Studies . . . . . . . . . . . . . . . . . . . . . . . . A. T h e r m o d y n a m i c Stabilities . . . . . . . . . . . . . . . . . . . . . . . . . B. Heats o f Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S u m m a r y and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272 273 273 273 273 280 282 283 283

Advances in Molecular Structure Research Volume 3, pages 269-285 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

269

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270

PETER POLITZER and PAT LANE

ABSTRACT In designing energetic molecules (e.g. explosives and propellants), it can be advantageous to include some difluoramine (-NF2) groups along with the usual nitro (-NO2). The former can have the desirable effects of increasing the density and the number of moles of gaseous products, although there may also be a decrease in the amount of energy released. The overall result is likely to be improved energetic performance. Although we find, through density functional calculations, that C-NF2 and N-NF2 bonds are slightly stronger than C-NO2 and N-NO2, the stabilities of difluoramines are a matter of some concern. Molecules of the type RaRbN-NO2 may have a tendency to lose F-. We have shown, however, that this can be prevented by making Ra and Rb sufficiently electron-attracting. With molecules of the type RaRbC(H)NF2, the danger is the intramolecular elimination of HF; the activation barrier to this considerably less than either C-NO2 or C-NF2 bond energies, making it a definite threat to stability. This can be avoided, however, by structuring the molecule so that there are no hydrogens on carbons bearing -NF2. It is also advisable to not have -NO2 and -NF2 on the same carbon, since this leads to a destabilizing weakening of both the C-NO2 and C-NF2 bonds. Despite these cautionary notes, however, the results of our computational studies should stimulate interest in carefully designed mixed nitro/ difluoramine compounds.

I. B A C K G R O U N D A characteristic of many organic energetic compounds (i.e. explosives and propellants) is the presence of the nitro group, -NO2 [1, 2]. It is typically linked to a carbon (nitroalkanes, nitroaromatics, nitroheterocycles), a nitrogen (nitramines), or an oxygen (nitrate esters). The nitro group has the desirable features that it increases the density of the compound and promotes the formation of very stable gaseous decomposition products (N2, CO, CO2, and H20), resulting in a greater energy release and pressure buildup per unit volume of material. An alternative that has been of interest for some time [2, 3] is to replace some -NO2 groups by -NF2, the difluoramino group. This would have at least two potential advantages: (1) the compound's density is likely to increase, due to the greater mass of fluorine compared to oxygen; and (2) depending upon the overall stoichiometry, there may be an increase in the number of moles of gaseous products. This is because -NF2 makes more efficient use of the hydrogens present in the molecule, by forming HF(g), which requires only one hydrogen, in contrast to the H20(g) produced by -NO2. Thus, while it is important to have enough oxygen to oxidize all of the carbon to CO or CO2, it is desirable to convert as many of the hydrogens as possible to HF rather than H20. On the other hand, a disadvantage of replacing -NO2 by -NF2 is that the heat of formation is decreased [4-6], because the reference compound F2 is much less stable than 02. This may result in a lesser release of energy upon decomposition of the compound.

Computational Studies of Energetic Difluoramines

2 71

These various points can be illustrated by considering 1,3,3-trinitroazetidine, 1, which is known [7], and its difluoramine analogue 2, which has not yet been synthesized. It will be assumed that their ultimate decomposition products are as shown in Eqs. 1 and 2 [4]: O2N ~ N

F2N__ N

x

/x CH 2

\ / \

~,~

2N 2 + 2CO + CO 2 + 2H20

(1)

2N 2 + 3CO + H20 + 2HF

(2)

NO 2

CH 2 NO 2

~

2

It is seen that 2 produces more moles of gases than does 1; on a gram basis, which is what is important for energetic performance [8], 1 yields 0.036 moles of gaseous products per gram of starting material, while 2 results in 0.040 moles/gram. The masses of 1 and 2 are 192 and 198 g/mol, respectively, while their molecular volumes (obtained by taking the 0.001 au contour of the electronic density to be the molecular surface [9]) are 184 and 182/~3, respectively; thus 2 has a greater molecular mass but smaller volume than does 1, and can be anticipated to have a higher density. Finally, our calculated solid-phase heats of formation [4, 6] are 9 kcal/mol for 1 and -6 kcal/mol for 2. (The former is in excellent agreement with the experimental value for 1, 8.7 kcal/mol [10]). Using these calculated data for 1 and 2 in conjunction with experimental heats of formation for the decomposition products [11], the heats of reaction for Eqs. 1 and 2, at 298 K, are -271 kcal/mol (- 1413 cal/g) and -261 kcal/mol (- 1321 cal/g), respectively. The decomposition of the difluoramine compound, 2, is accordingly expected to release less energy than that of 1. The net effect upon energetic performance of these three factors--density, moles of gaseous products, and energy output--has been estimated for 1 and 2 using codes obtained from the Naval Air Warfare Center (China Lake, CA). These predict that 2 would have only slightly better detonation properties than 1 (roughly 2% [12]), but would have a significantly higher (6% [13]) specific impulse; the latter is a measure of the thrust developed by a propellant upon combustion [14, 15]. Despite the improvement in energetic properties that can result from the judicious introduction of-NF2 groups, there has not been, over the past 30 years, a sustained and extensive synthesis effort aimed at preparing nitro/difluoramine compounds [1]. One reason for this is the hazardous natures of the fluorinating agents [1, 16-21] which have included HNF> N2F4, F> and HE A second concern is instability of the difluoramine products toward loss of HE e.g. Eq. 3, or loss of F-, e.g. Eq. 4 [4,

16-18, 21, 22].

272

PETER POLITZER and PAT LANE

---C--NF 2

~-

mC=NF

+

I-IF

(3)

--N~'~NF 2

~-

--N=NF

+ F

(4)

I

I

I

I

4

These potential sources of instability shall be examined in the discussion to follow. Our general purpose shall be to summarize and review the results of several computational studies of nitro/difluoramine systems, most of which have not yet been synthesized. Our focus shall be upon their structures, stabilities, and energetic properties. Most of our calculations have involved density functional (DF) techniques due to the sizes of the molecules and the need to take electronic correlation into account; the latter factor is particularly significant for molecules such as these, that contain several of the "electron-rich" N, O, and/or F atoms [23-27], which have high outer-shell electron densities [28].

I!. COMPARISON OF THE NITRO A N D D I F L U O R A M I N O GROUPS The nitro and the difluoramino groups each consist of nitrogen bonded to two highly electronegative atoms. It is accordingly not surprising that the groups themselves have been estimated to have rather high electronegativities: 3.45 for -NO2 [29] and 3.26 for -NF2 [30], on essentially the Pauling scale. A more detailed picture can be obtained from their Hammett and Taft substituent constants [31, 32]. For-NO2, these are [31]: CYm -" 0.71, C~p = 0.81, ~I 0.67, and ~R = 0.15. For -NF2, experimentally-based values are not available" however, using correlations developed earlier, we have predicted [33, 34]: Cym= 0.54, C~p= 0.49, ~I = 0.53, and CYR= -0.04. These data show that the primary effect of both -NO2 and - N F 2 is strong inductive electron withdrawal; resonance interactions with the remainder of the molecule are relatively minor factors in both cases, as has already been noted in the past [35-37]. Despite these similarities, and notwithstanding the analogies that have sometimes been drawn between the energetic effects of these two substituents [1, 38], there is a major structural and electronic difference between them. The -NF2 group is essentially a doubly substituted amino group; the bonds are in a pyramidal arrangement (5) [4, 6, 39] and there is a localized lone pair above the nitrogen. An indication of the latter is the presence of a strongly negative electrostatic potential; in the case of CH3NF 2, for example, it reaches a minimum o f - 6 2 kcal/mol (HF/STO6G//DF/B3PW91/6-31G**). In contrast, the-NO2 group is planar and features a -

-

Computational Studies of Energetic Difluoramines /

F

F

+2

--N"o

~

273

--N

"0-

considerable degree of internal electron delocalization (6); the nitrogen is formally positive, and indeed nitro derivatives do not show negative regions above the nitrogens [40-44]. Thus there are very significant qualitative differences between these two groups, both electronically and structurally.

III. SURVEY OF COMPUTATIONAL STUDIES A. Thermodynamic Stabilities One of the motivations for our computational analyses has been to ascertain the possible thermodynamic stabilities of specific molecules or molecular ions by determining if their optimized geometries correspond to energy minima. This can be confirmed by verifying that the structure has no imaginary vibrational frequencies [45]. We have established in this manner the thermodynamic stabilities, at the computational levels indicated, of each of the molecules or molecular ions in Table 1.

B. Heats of Formation A second objective in these studies has been to determine the heats of formation of various difluoramine compounds, since these are a measure of energy content. Initially we were able to compute only gas-phase heats of formation (DF/BP86/631G**) [46], but the recent development of a procedure for predicting heats of sublimation [47] now permits us to obtain solid-phase values as well. Our results are in Table 2. As mentioned already, the effect of replacing -NO2 by -NF2 is to decrease the heat of formation. This is shown by the data given earlier for compounds 1 and 2, and can be seen again by comparing the data in Table 2 with the values for the nitro analogues, e.g. (H3C)2N-NO2 (calc. AHe(298) = -2.7 kcal/mol [4], exp. AHf(298) = -1.1 kcal/mol [48]) and O2N-C~C-NO2 (calc. AHf(298) = 89 kcal/mol [5]).

C. Structures We have computed optimized structures for all of the molecules listed in Tables 1 and 2. Many of these have been published, as indicated; the remainder are available upon request. A particularly interesting structural feature has been observed in molecules of the type RaRDN-NF2 [4, 49]. This can be seen from the data in Table 3, all of which were computed at the DF/BP86/6-31G** level. The N - F and N - N bond lengths in

274

PETER POLITZER and PAT LANE Table 1. Difluoramines and Related Systems for which True Energy Minima have been Verified

Molecule or Ion

Computational Method

ON-C--=C-NF2 O2N-C~-C-NF2 FN2O+ N(NF)~ N(NF)3 FaN+-N--F [HaN-NF2] + NO2 N N /~.._.~ O2N

Reference

HF/6-31 G* HF/6-31 G* DF/BLYP/6-31 +(3** DF/BLYP/6- 31 +G** DF/BLYP/6-31 § DF/BLYP/6-31G** DF/BLYP/6- 31G**

a a b b b c c

DF/BP86/6-31G**

d

DF/BPW91/6-31G**

e

DF/BPW91/6-31G**

e

DF/BPW91/6-31G**

e

DF/BP86/6-31G**

c

DF/B P86/6- 31G** DF/BLYP/6-31G** DF/B 3LYP/6-31G** DF/B3PW91/6-31G** DF/B LYP/6-31G** DF/B 3LYP/6-31G** DF/B3PW91/6-31G** DF/BLYP/6-31G** DF/BLYP/6- 31G** DF/BLYP/6-31G** DF/BLYP/6-31G** DF/BLYP/6-31G** DF/B LYP/6-31G** DF/B 3LYP/6-31+(3* DF/B 3PW91/6-31 +G* DF/B3P86/6-31 +G*

f g

F

(H3C)2N-NF2

/N F2N--N

V

N--NO 2

F2N-- N ~

NO2 NO 2

O2N_N~

NF2 NO 2

(NC)2N-NF2 HaC-NF 2 NF2-CH2-NO 2 H2C=NF O2N-HC=NF (H3C)3C-NF2 O2N-(H3C)2C-NF2 H-(H3C)2C-NF2 (H3C)-CH-NF 2 NF 2

g h h h h h h i

Computational Studies of Energetic Difluoramines

275

Table 1. Continued Molecule or Ion

Computational Method

Reference

NF 3

DF/B 3LYP/6- 31 +G* DF/B3PW91/6-3 I+G* DF/B 3P86/6- 31 +G*

i

HNF 2

DF/B 3LYP/6- 31 +G** DF/B 3PW91/6-31 +G** DF/B3P86/6-31 +G** DF/B 3LYP/6-31 +G*

i

C1NF 2 HO-NF 2 FNO

Notes:

i

DF/B 3LYP/6-31 +G** DF/B3LYP/6-3 l+G* DF/B3PW91/6-3 I+G* DF/B 3P86/6- 31 +G*

i i

apolitzer, E; Lane, E; Sjoberg, P.; Grice, M. E.; Shechter, H. Struct. Chem. 1995, 6, 217. bGrice, M.E.; Politzer, E Tech. Rep. 79, Office of Naval Research, Contract No. N00014-91-J-4057, Arlington, VA, March 16, 1995. CGrice, M. E.; Politzer, E, unpublished work. dpolitzer, E; Murray, J. S.; Grice, M. E. Tech. Rep. 90, Office of Naval Research, Contract No. N00014-951-0028, Arlington, VA, March 7, 1996. ePolitzer, E; Lane, E; Grice, M. E.; Concha, M. C.; Redfern, E C. J. Mol. Struct. (Theochem) 1995, 338, 249. fPolitzer, E; Grice, M. E. J. Chem. Res. (S) 1995, 296. gPolitzer, E; Lane, E J. Mol. Struct. (Theochem.), in press. hPolitzer, E; Lane, E; Grice, M. E. J. Mol. Struct. (Theochem.) 1996, 365, 89. ipolitzer, E; Lane, E; Wiener, J. M. M., unpublished work.

Table 2. Calculated Heats of Formation of Difluoramines and Related Compounds AHf ( 298K; kcal/mol ) Compound

Gaseous

(H3C)2N-NF 2 ON-C~C-NF2 O2N-C~=C-NF2 O2N--N

/~

V /~

NBNF2

Reference

-20 95 74

a b b

29

a

16

a

7

c

/NO2

F2N--NX~NO2

O2NmN~

Solid

NF2 NO 2

276

PETER POLITZER and PAT LANE

Table 2. Calculated Heats of Formation of Difluoramines and Related Compounds AHf (298K; kcal/mol) Compound

/

Gaseous

Solid

Reference

NO2

N"~N O2N

77

56

d

73

43

e

141

114

e

11

-21

f

5

-39

f

-24

-67

f

F

F2N

/ F2N F2N.

/ F2N

F 2 N ~ NF2

02N~

NvN~

o2 2

NO2

N--NO 2 O2N-- NF2N~NF 2

N--NO2 O2NmNF2N~NF 2

Notes:

apolitzer, P.; Lane, P.; Grice, M. E.; Concha, M. C.; Redfem, P.C. J. Mol. Struct. (Theochem) 1995, 338, 249. bpolitzer, E; Lane, E; Sjoberg, E; Grice. M. E.; Shechter, H. Struct. Chem. 1995, 6, 217. CPolitzer, E; Murray, J. S.; Grice, M. E. In Decomposition, Combustion, and Detonation Chemistry of Energetic Materials; Brill, T. B.; Russell, T. E; Tao, W. C.; Wardle, R. B., Eds. Materials Research Society: Pittsburgh, PA, 1996, Vol. 418, p. 55. dpolitzer, E; Murray, J. S.; Grice, M. E. Tech. Rep. 90, Office of Naval Research, Contract No. N00014-951-0028, Arlington, VA, March 7, 1996. ePolitzer, E; Lane, E; Grice, M. E. Tech. Rep. 9 l, Office of Naval Research, Contract No. N00014-95-10028, Arlington, VA, June 4, 1996. rPolitzer, E; Grice, M. E.; Lane, E Tech. Rep. 92, Office of Naval Research, Contract No. N00014-95-10028, Arlington, VA, June 18, 1996.

Computational Studies of Energetic Difluoramines

277

Table 3. Optimized DF/BP86/6-31G** Geometries of Some RaRbN-NF 2 Molecules a Distance

Molecule (H3C)2N-NF2

(,~)

N-N: 1.358 Ca, Cb-N: 1.469, 1.474 N-F a, Fb: 1.431, 1.542

Ca, Cb-N(NO2): 1.485, 1.487 O 2 N - - N v N _ _ N F 2 C a, Cb-N(NF2): 1.507, 1.498 N-NO2:1.423 N-NF2:1.368 N-O: 1.237, 1.238 N-F a, Fb: 1.418, 1.536

Ca, Cb-C: 1.536, 1.534 /NO2 C-(NO2) a, (NO2)b: 1.540, 1.534 FEN--N \ / / ~ Ca, Cb-N(NF2): 1.501, 1.511 v NO2 N-N: 1.366 N-O: 1.233-1.236 N-F a, Fb: 1.417, 1.538

Angle (degrees) N - N - C a, Cb: 109, 118 N - N - F a, Fb: 105, 108 C-N-C: 116 F-N-F: 98 Ca-N-N-F a, Fb: 171, 85 Cb-N-N-F a, Fb: 54, 49 C-N(NO2)-C: 92 C-N(NF2)-C: 91 N-C-N: 89 C-N-NO2:117 Ca, Cb-N-NF2:119, 114 N-N-O: 116 N - N - F a, Fb: 103, 107 O-N-O: 128 F-N-F: 99 C-N-C-N: 1 N-C a, Cb-N-NO2: 120, 121 N-Ca, Cb-N-NF2:119, 124 C-N-N-O: 38, 145 Ca-N-N-F a, Fb: 63, 42 Cb-N-N-F a, Fb: 168, 64 C-C-C: 90 Ca-C-(NO2)a, (NO2)b: 114, 117 Cb-C-(NOE)a, (NOE)b: 116, 116 C-C-N: 88 NO2-C-NO2:105 C-N-O: 114-118 C-N-C: 92 Ca, Cb-N-NF2:114, 120 N - N - F a, Fb: 103, 107 O-N-O: 128 F-N-F: 99 C-C-C-N: 12 C-C-N-C: 12 N-C-C-(NO2) a, (NOE)b: 105, 131 C-C a, Cb-N-NF2: 136, 132 Ca-N-N-F a, Fb: 169, 65 Cb-N-N-F a, Fb: 61, 43

Note: aThese data are taken from ref. 4.

Table 3 will be compared to those found in H3C-NF 2 and (H3C)2N-NH 2 by the same computational procedure; the values for the latter two molecules are assumed to be reasonably typical for these bonds, and will be used as reference points. In each molecule in Table 3, one N - F distance is in the 1.417-1.431/~ range, which is similar to what is obtained for both N - F bonds in HCa-NF2, 1.436/~.

278

PETER POLITZER and PAT LANE

However the other N - F bond in the RaRbN-NF2 molecules is considerably longer, by more than 0.10/~ in each case. The N - N bonds, on the other hand, are much shorter than the 1.492/~ computed for (H3C)2N-NH2. We have suggested that these observations can be explained in terms of a delocalization of electronic charge such as is shown in structures 7 [4, 49], which can be regarded as an example of the anomeric effect [45, 50-52]. This delocalization is expected to be most effective when the lone pair on N a is concentrated in the same plane as are the Na-Nb and one of the Nb-F bonds; the latter should then be lengthened and the former shortened, as is indeed observed in Table 3.

~

..r

~i+..... r m

J

N b ~F

,,j~a

Nb,,.F~i-

We have examined these electronic and structural factors in greater detail in the case of (H3C)2N-NO2 by investigating the effects of rotation around the N - N bond. We reoptimized the geometry, at the DF/BP86/6-31G** level, at 60 ~ intervals relative to the eclipsed conformation. The results are in Figure 1, which also includes the energy of each conformer relative to the ground state; the latter, which is not shown, corresponds to an angle of rotation of 69.9 ~. In both the 60 ~ and the 120 ~ conformers the lone pair on N a is coplanar with the N - N and one of the N - F bonds. The latter is consequently much longer than the other N - F bond, which has a normal length. Furthermore, the N - N distance is considerably shorter in each case than its reference value of 1.49/~. In the 0 ~ and 180 ~ conformers, on the other hand, the N a lone pair is concentrated in a plane that bisects the angle between the N - F bonds; accordingly it affects them equally. The N - F lengths are now the same, and are intermediate between their normal and their highly elongated values. Similarly, the N - N distances are not shortened as much as in the 60 ~ and the 120 ~ conformers. Thus, there is still some delocalization of the N a lone pair in these conformers, but not to the same extent as when it is coplanar with one of the N - F bonds. Not surprisingly, the eclipsed conformation has the highest energy. It might have been anticipated that the 180 ~ conformer, in which the lone pairs and the substituents are maximally separated, would be the most stable. The fact that the 60 ~ and the 120 ~ conformers both have lower energies presumably reflects their greater degrees of charge delocalization and consequent stabilization. The data in Figure 1 fully support the charge delocalization interpretation, as depicted in structures 7, of the anomalous N - F and N - N bond lengths given in Table 3. Thus, even though each of these molecules does correspond to an energy minimum, the elongated N - F bonds reinforce concern regarding the possible

Computational Studies of Energetic Difluoramines

~

H3C

1.421

a

279

Nb ~ 1.493

1.49

H3C

F

angle of rotation: 0~ (eclipsed conformation) relative energy: 13.1 kcal/mole

,yF

1.362 H3C / j a H3C

angle of rotation:60~ relative energy: 0.4 kcal/mole !.376 H3C

~

x"b ~

F

Na

H3C angle of rotation: 120~ relative energy: 7.9 kcal/mole F 1.423 H3C ~ N

a

b ~

F

-N

H3C

angleof rotation: 180~ relative energy: 8.9 kcal/mole Figure 1. Computed (DF/BP86/6-31G**) properties of conformers of (H3C)2N-NF2 produced by rotation around N-N bond. The energies are given relative to the ground state, which corresponds to a rotation of 69.9 ~ instabilities of RaRbN-NF2 molecules toward the loss of fluoride ion [22], as depicted in Eq. 4. We have shown, however, that this problem can be avoided or at least diminished by having electron-attracting groups on N a [49], which would oppose the charge delocalization shown in 4 and in 7. For example, a DF/BP86/631G** geometry optimization of (NC)2N-NF2 produced N - F bond lengths of 1.410 and 1.420 A, while the N-N was 1.460 ~ [49]. All of these are quite normal.

280

PETER POLITZER and PAT LANE

Thus the strongly electron-withdrawing cyano group prevents N - F bond elongation and presumably stabilizes the >N-NF2 grouping toward the loss of F-. This should provide encouragement regarding the feasibility of synthesizing compounds of the type RaRbN-NF 2, provided that Ra and Rb are electron-attracting.

D. Energetics Loss of HF from RaRbC(H)NF2 Molecules It has already been mentioned that a possible problem with difluoramines of the type RaRbC(H)NF2 is instability with respect to loss of HF [16-18, 21]"

~C--NF 2

I

~

--=C=NF

I

+

(3)

HF

This process normally requires a catalyst, such as a base. Taking H3C-NF2 as our model, we have confirmed (at the DF/BLYP/6-31G** level) that the elimination of HF is indeed thermodynamically favorable [53]. For the reaction, H3C-NF 2 -+ H2C--NF + HF

(5)

we found AE (0 K) = -20.1 kcal/mol and AG (298 K) = -28.9 kcal/mol (Table 4). We have investigated two possible mechanisms for HF elimination [53]; the first proceeds through a four-centered transition state, Eq. 6, while the other begins with the homolytic cleavage of an N - F bond, Eq. 7. As can be seen in Table 4, the energy barrier for the formation of the transition state in Eq. 6 is AE* (0 K) = 38.0 kcal/mol, AG* (298 K) = 37.9 kcal/mol, whereas the energy requirements for N - F bond cleavage are AE (0 K) = 69.3 kcal/mol, AG (298 K) = 60.6 kcal/mol. A recent computational study of NF3 by various density functional procedures suggests that the DF/BLYP/6-31G** may overestimate the N - F bond energy by a few kcal/mol, and that AE (0 K) is probably about 65 kcal/mol [54]. Nevertheless it is clear that the formation of the four-centered transition state, Eq. 6, is energetically favored over initial N - F bond rupture, Eq. 7, as the pathway for the elimination of HF from H3C-NF2. Since the activation barrier for Eq. 6 is relatively high, about 38 kcal/mol, some degree of kinetic stability can be anticipated, even though the loss of HF is thermodynamically favored (Table 4).

H3 C--NF 2

~,~

n--~ ', ; H2C---NF

~

H2C = N F

+

HF

(6)

Computational Studies of Energetic Difluoramines

281

Table 4. Energetics (DF/BLYP/6-31G**) Related to HF Elimination a Energies (kcal/mole) Process H3C--NF2 ~

H 2 C ~ N F + HF

H3C--NF2 H3C--NF2

~ ~

~

H2C-'-NF + HF

298 K

AE = -20.1

AG = - 2 8 . 9

AE ~ = 38.0

AG* = 37.9

AE = 69.3

AG = 60.6

AE = -19.4

AG = -28.2

AE ~t = 35.2

AGat= 35.7

AE = 65.7

AG = 56.4

H3C--NF + F

02 H2C--NF2

>

?o~ H2C----NF2

- F lift-!

' H2CmNF

0K

?02 HC=NF + HF

No~ ~

No~

HC--NF I !

~

HC--NF + HF

i~--~

1•O2

H2C--NF2 ~

?02 H2C--NF + F

Note: aThesedata taken from ref. 53.

H3C-NF 2 ~

H3C-NF + F

(7)

We have also examined the effects of introducing a nitro group to give the H E C ( N O E ) N F 2 [ 5 3 ] ; there is a recent report that such compounds have been synthesized [55]. Table 4 shows that the presence of the strongly electron-attracting nitro group somewhat facilitates both the formation of the transition state and N - F bond-breaking, although the overall AE (0 K) and AG (298 K) are very little affected. We have shown, by means of an isodesmic reaction that there is a weakly destabilizing interaction between -NO2 and - N F 2 when substituted on the same carbon [53]. On the other hand, methyl groups have the opposite effect; we found the N - F bond dissociation energies of (H3C)aC-NF 2 and (HaC)EC(NO2)NF 2 to be 71.2 and 66.8 kcal/mol at 0 K, respectively [53].

gem-nitro/difluoratrfine

C-NF2 and N-NF2 Bond Strengths as Compared to C-N02 and N-N02 There is considerable evidence that the strengths of the C - N O 2 and N - N O 2 bonds are an important factor in determining the stabilities of energetic compounds, and their sensitivities to external stimuli such as shock and impact [56-68]. It is accordingly relevant to the consideration o f - N F 2 as a substituent in energetic molecules to ascertain the strengths of C-NF2 and N-NF2 bonds. We have used the DF/B3P86/6-31+G** procedure to compute AH (298 K) for the bond-breaking processes shown in Eqs. 8 and 9:

282

PETER POLITZER and PAT LANE H 3 C - N O 2 --~ H3C + N O 2

(8)

H 3 C - N F 2 --~ H3C + N F 2

(9)

We find the C-NO 2 dissociation energy, AH (298 K) for Eq. 8, to be 59.8 kcal/mol, very close to the experimental 60.8 kcal/mol [11]. Our calculated C-NF2 energy, AH (298 K) for Eq. 9, is slightly higher, 63.3 kcal/mol. Thus the substitution of -NF2 on carbons should not have a destabilizing effect, provided that the molecular environment is such that the elimination of HF, Eq. 3, cannot occur. However the presence of both -NO2 and -NF2 on the same carbon (gem-nitro/difluoramine) does pose a problem; we have found that this weakens both the C-NO2 and the C-NF2 bonds [39], lowering their dissociation energies by roughly 10 kcal/mol and presumably destabilizing the molecule. It has indeed been reported that the gemnitro/difluoramine compounds that have been synthesized are highly sensitive [55]. Proceeding to N-NO2 and N-NF2 bonds, for N-NO2 dissociation in (H3C)2NNO2, Eq. 10, we find AE (0 K) = 43.8 kcal/mol at the BP86/6-31G** level [4]. This is reasonably consistent with the experimental AH (298 K) = 44 kcal/mol [11]. For the analogous N-NF2 bond-breaking, Eq. 11, we find AE (0 K) = 47.3 kcal/mol. (10)

(H3C)2N-NO 2 -~ (H3C)2N + NO 2

(11)

(H3C)2N-NF 2 --) (H3C)2N + N F 2

The same conclusion that the N-NF2 bond is somewhat stronger than the N-NO2 bond was reached for 1-nitro-3-difluoramino-l,3-diazacyclobutane [4]. For the bond dissociations in Eqs. 12 and 13, AE (0 K) is 38.7 kcal/mol for Eq. 12 and 42.7 kcal/mol for Eq. 13. Thus the substitution of-NF2 on nitrogens should not be destabilizing, compared to -NO2 on nitrogen, provided that there are sufficiently strongly electron-attracting groups present to prevent loss of F-, Eq. 4. /N F2N--NvN--NO2

F2N--N

/N

V

N--NO 2

~

/N F2N--NvN

~

N

/N

V

+ NO2

N--NO 2 + NF2

(12)

(13)

IV. SUMMARY AND CONCLUSIONS The judicious introduction of the difluoramino group into energetic compounds, whether on carbon or on nitrogen, may have the positive effects of increasing the density and the number of moles of gaseous products, and the negative effect of decreasing the energy released upon decomposition. The net result is likely to be an improvement in energetic performance.

Computational Studies of Energetic Difluoramines

283

However the stabilities of difluoramine systems are a matter for concern, although not a reason to avoid them altogether. We have found that the - N F 2 group binds somewhat more strongly to both carbon and nitrogen than does the -NO2, so this is not a source of instability. The problem, in molecules of the type RaRbN-NF2, is a possible tendency toward the loss of F - , as depicted in Eq. 4. We have shown that this can be prevented by making R a and Rb sufficiently electron-attracting. W h e n - N F 2 is on carbon, the danger is the loss of H E as shown in Eqs. 3, 5, and 6. The activation barrier for this is about 38 kcal/mol, considerably less than the energy requirement for either C - N O 2 or C - N F 2 bond rupture. This must accordingly be viewed as an undesirable destabilizing and sensitizing factor. The obvious way to avoid this is of course to structure the molecule so that there are no hydrogens on the carbon beating the -NF2; examples are given in Table 2. It is also suggested that - N O 2 and -NF2 groups not be present on the same carbon, since both the C - N O 2 and the C - N F 2 bonds are then much weaker. Overall, we feel that the results and discussion that have been presented should stimulate interest in carefully designed mixed nitro/difluoramine compounds.

ACKNOWLEDGMENTS We greatly appreciate the assistance of Dr. Jane S. Murray, and the financial support of the Office of Naval Research, through contract N00014-97-1-0066 and Program Officer Dr. Richard S. Miller.

REFERENCES 1. Urbanski, T. Chemistry and Technology of Explosives. Pergamon: New York, 1984, Vol. 4. 2. K6hler, J.; Meyer, R. Explosives, 4th ed. VCH Publishers: New York, 1993. 3. Gould, R. E, Ed. Advanced Propellant Chemistry. Advances in Chemistry Series, No. 54. American Chemical Society: Washington, DC, 1966. 4. Politzer,P.; Lane, P.; Grice, M. E.; Concha, M. C.; Redfern, P. C. J. Mol. Struct. (Theochem.) 1995, 338, 249. 5. Politzer, P.; Lane, P.; Sjoberg, P.; Grice, M. E.; Shechter, H. Struct. Chem. 1995, 6, 217. 6. Politzer,P.; Murray, J. S.; Grice, M. E. In Decomposition, Combustion, and Detonation Chemistry of Energetic Materials; Brill, T. B.; Russell, T. P.; Tao, W. C.; Wardle, R. B.; Eds. Materials Research Society: Pittsburgh, PA, 1996, Vol. 418, pp. 55-66. 7. Archibald, T. G.; Gilardi, R.; Baum, K.; George, C. J. Org. Chem. 1990, 55, 2920. 8. Kamlet, M. J.; Jacobs, S. J. J. Chem. Phys. 1968, 48, 23. 9. Bader, R. E W.; Carroll, M. T.; Cheeseman, J. R.; Chang, C. J. Am. Chem. Soc. 1987, 109, 7968. 10. Archibald, T. G.; Garver, L. C.; Malik, A. A.; Bonsu, F. O.; Tzeng, D. D.; Preston, S. B.; Baum, K. "Report No. ONR-2-10," Office of Naval Research, Arlington, VA, 1988. 11. Lias, S. G.; Bartmess, J. E.; Liebman, J. E; Holmes, J. L.; Levin, R. D.; Mallard, W. G. J. Phys. Chem. Ref Data 1988, 17, 1. 12. Desibhatla, V., private communication. 13. Grice, M. E., private communication. 14. Politzer, P.; Murray, J. S.; Grice, M. E.; Sjoberg, P. In Chemistry of Energetic Materials; Olah, G. A.; Squire, D. R., Eds. Academic: New York, 1991, Chap. 4.

284

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15. Holzmann, R. T. InAdvancedPropellant Chemistry. Advances in Chemistry Series, No. 54; Gould, R. E, Ed. American Chemical Society: Washington, DC, 1966, Chap. 1. 16. Smith, P. A. S. The Chemistry of Open-Chain Organic Nitrogen Compounds. W. A. Benjamin: New York, 1965, Vol. I. 17. Logothetis, A. L.; Sausen, G. N. J. Org. Chem. 1966, 31, 3689. 18. Petry, R. C.; Freeman, J. P. J. Org. Chem. 1967, 32, 4034. 19. Grakauskas, V.; Baum, K. J. Org. Chem. 1969, 34, 2840. 20. Baum, K. J. Org. Chem. 1969, 34, 2049. 21. Grakauskas, V.; Baum, K.J. Org. Chem. 1970, 35, 1545. 22. Craig, A. D.; Ward, G. A.; Wright, C. M.; Chien, J. C. W. In Advanced Propellant Chemistry. Advances in Chemistry Series, No. 54; Gould, R. E, Ed. American Chemical Society: Washington, DC, 1966, Chap. 15. 23. DeFrees, D. J.; Levi, B. A.; Pollack, S. K.; Hehre, W. J.; Binkley, J. S.; Pople, J. A. J. Amer. Chem. Soc. 1979, 101, 4085. 24. Clabo, D. A.; Schaefer III, H. E Int. J. Quantum Chem. 1987, 31,429. 25. Coffin, J. M.; Pulay, P. J. Phys. Chem. 1991, 95, 118. 26. Seminario, J. M.; Concha, M. C.; Politzer, P. J. Comput. Chem. 1992, 13, 177. 27. Phillips, D. H.; Quelch, G. E. J. Phys. Chem. 1996, 100, 11270. 28. Politzer, P.; Murray, J. S.; Grice, M. E. In Chemical Hardness; Structure and Bonding No. 80; Sen, K. D., Ed. Springer-Verlag: Berlin, 1993, p. 101. 29. Wilmshurst, J. K. J. Chem. Phys. 1957, 27, 1129. 30. Ettinger, R. J. Phys. Chem. 1963, 67, 1558. 31. Exner, O. Correlation Analysis of Chemical Data. Plenum: New York, 1988. 32. Hansch, C.; Leo, A.; Taft, R. W. Chem. Rev. 1991, 91, 165. 33. Sjoberg, P.; Murray, J. S.; Brinck, T.; Politzer, P. Can. J. Chem. 1990, 68, 1440. 34. Haeberlein, M.; Murray, J. S.; Brinck, T.; Politzer, P. Can. J. Chem. 1992, 70, 2209. 35. Baum, K. J. Org. Chem. 1970, 35, 1203. 36. Lipkowitz, K. B. J. Am. Chem. Soc. 1982, 104, 2647. 37. Politzer, P.; Lane, P.; Jayasuriya, K.; Domelsmith, L. N. J. Am. Chem. Soc. 1987, 109, 1899. 38. Leroy, G.; Sana, M.; Wilante, C.; Peeters, D.; Bourasseau, S. J. Mol. Struct. (Theochem.) 1989, 187, 251. 39. Politzer, P.; Lane, P. Journal of Molecular Structure (Theochem.) 1966, 388, 51. 40. Politzer, P.; Domelsmith, L. N.; Sjoberg, P.; Alster, J. Chem. Phys. Lett. 1982, 92, 366. 41. Politzer, P.; Abrahmsen, L.; Sjoberg, P. J. Am. Chem. Soc. 1984, 106, 855. 42. Politzer, P.; Laurence, P. R.; Abrahmsen, L.; Zilles, B. A.; Sjoberg, P. Chem. Phys. Lett. 1984,111, 75. 43. Politzer, P.; Murray, J. S. In Organic Energetic Compounds; Marinkas, P. L., Ed. Nova Science: Commack, NY, 1996, Chap. 1. 44. Murray, J. S.; Lane, P.; Politzer, P. Mol. Phys. 1995, 85, 1. 45. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory. Wiley-Interscience: New York, 1986. 46. Habibollahzadeh, D.; Grice, M. E.; Concha, M. C.; Murray, J. S.; Politzer, P. J. Comput. Chem. 1995, 16, 654. 47. DeSalvo, M.; Miller, E.; Murray, J. S.; Politzer, P., unpublished work. 48. Pedley, J. B.; Naylor, R. D.; Kirby, S. P. Thermochemical Data of Organic Compounds, 2nd ed. Chapman and Hall: London, 1986. 49. Politzer, P.; Grice, M. E. J. Chem. Res. 1995, 296. 50. Szarek, W. A.; Horton, D., Eds. Anomeric Effect, Origins and Consequences; ACS Symp. Ser. 87; American Chemical Society: Washington, 1979. 51. Deslongchamps, P. Stereoelectronic Effects in Organic Chemistry. Pergamon: Oxford, 1983.

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52. Carey, E A.; Sundberg, R. J. Advanced Organic Chemistry, Part A; 3rd ed. Plenum: New York, 1990. 53. Politzer, P.; Lane, P.; Grice, M. E. J. Mol. Struct. (Theochem.), 1996, 365, 89. 54. Politzer, P.; Lane, P.; Wiener, J. M. M., unpublished work. 55. Litvinov, B. V.; Fainzil'berg, A. A.; Pepekin, V. I.; Smirnov, S. P.; Loboiko, B. G.; Shevelev, S. A.; Nazin, G. M. Doklady Chem. 1994, 336, 86. 56. Delpuech, A.; Cherville, J. Propellants Explos. 1978, 3, 169. 57. Sharma, J.; Owens, E J. Chem. Phys. Lett. 1979, 61,280. 58. Kamlet, M. J.; Adolph, H. G. Proc. 7th Symp. (lnternat.) Detonations. Office of Naval Research, Arlington, VA, 1981. 59. Sharma, J.; Garrett, W. L.; Owens, E J.; Vogel, V. L. J. Phys. Chem. 1982, 86, 1657. 60. Owens, E J. J. Mol. Struct. (Theochem.) 1985, 121,213. 61. Gonzalez, A. C.; Larson, C. W.; McMillen, D. E; Golden, D. M. J. Phys. Chem. 1985, 89, 4809. 62. Brill, T. B.; Oyumi, Y. J. Phys. Chem. 1986, 90, 2679. 63. Tsang, W.; Robaugh, D.; Mallard, W. G. J. Phys. Chem. 1986, 90, 5968. 64. Murray, J. S.; Lane, P.; Politzer, P.; Bolduc, P. R. Chem. Phys. Lett. 1990, 168, 135. 65. Murray, J. S.; Politzer, P. In Chemistry and Physics of Energetic Materials; Bulusu, S. N., Ed. Kluwer: Dordrecht, The Netherlands, 1990, Chap. 8. 66. Politzer, P.; Murray, J. S.; Lane, P.; Sjoberg, P.; Adolph, H. G. Chem. Phys. Lett. 1991, 181, 78. 67. Politzer, P.; Murray, J. S. Mol. Phys. 1995, 86, 251. 68. Politzer, P.; Murray, J. S. J. Mol. Struct. 1996, 376, 419.

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CH EMICAL PROPERTi ES AN D STRUCTURES OF BINARY AND TERNARY SE-N AND TE-N SPECIES"

APPLICATION OF X-RAY AND AB INITIO METHODS

lnis C. Tornieporth-Oetting and Thomas M. Klap6tke

Io II.

III.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparative Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Preparation o f Se4N4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Preparation o f Binary S e N Cations . . . . . . . . . . . . . . . . . . . . C. Preparation o f Ternary S e l e n i u m - N i t r o g e n - H a l i d e s . . . . . . . . . . . . D. Preparation o f a Ternary T e l l u r i u n ~ N i t r o g e n - H a l i d e . . . . . . . . . . . Structural Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Aromatic 6x Cation [Se3N2] 2+ and the ~ * - ~ * Bound 7re Cation [Se3N2] + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Polymorphism: Se4N4 and Se2NCI3 . . . . . . . . . . . . . . . . . . . . C. Solid-State Structures o f [E2NCI4] + (E = Se, n = 1; E = Te, n - 2)

Advances in Molecular Structure Research Volume 3, pages 287-311 Copyright 9 1997 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0208-9

287

288 288 289 290 290 291 292 292

....

293 296 299

288

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

Structure of[(SeCl2)2N]+: A Comparison of X-ray andAb Initio Studies Structure of[(SeC1)2N]+: An Unusual Structurally Very Flexible Ion . . Valence Bond Description for the Molecules [Se3N2]2§ [Se3N2] +, SenNa, and the Hypothetical (SeN)x . . . . . . . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. E. F.

. 300 . 303 307 308 309 309 309

ABSTRACT The chemical properties and structures of binary and ternary Se-N and Te-N species are discussed. The aim is to focus on the preparation and structural aspects of Se-N and Te-N compounds. The molecular structures are discussed on the basis of results from X-ray diffraction studies and are compared with results originating from high-level quantum-mechanical computations in order to elucidate and understand intrinsic bond properties of individual species. This chapter is not exhaustive in scope, but rather focuses on the most recent 8 years of work in this still developing area. In particular, this chapter emphasizes the preparation and structural properties (experimental X-ray data and ab initio computed data) of the following compounds: two modifications of SenNa, namely (x-SenNa (monoclinic) and ~-SenN4 (monoclinic); [Se3N2]~[AsF6]~ and [Se3N2]2+[AsF6]~; two modifications of Se3NC12, namely ct-Se3NCl2 (orthorhombic) and 13-Se3NCI2 (monoclinic); three isomers of the cation [(SeC12)2N]§ in the salt [(SeCI2)2N]+[AsF6]-; the cation [TeaN2CI8]2+ in the salt [TeaN2CI8]2+[AsF6]~; and three isomers of the cation [(SeCI)2N] § in the salt [(SeCI)2N]+[X] - (with X- = GaCI~, FeCI~, SbCI~).

I. I N T R O D U C T I O N During the last 5 to 8 years, significant advances have been made in the area of selenium-nitrogen chemistry, as indicated by the number of recent reviews covering various aspects of the subject [1-8]. The impetus for such studies has, in part, been derived from the possibility that the polymer (SeN)x may exhibit even more unusual properties than those of the metallic superconductor (SN)x [7]. Since the only known binary S e - N compound is the highly explosive cage molecule SenNa [9, 10] ternary selenium-nitrogen-chlorine compounds are of great interest as building blocks in preparative S e - N chemistry. Several S e - N chlorides that are potential sources of the SeNSe unit have been recently synthesized for the first time: Se2NC13, [11], [(SeCI)2N] § [11-14], [(SeC12)2N] § [15, 16], and [Se3N2C1] §

[17, 18]. The only well-characterized binary tellurium-nitrogen species is the azide cation Te(N3) ~ [19, 20]. In addition, two neutral binary tellurium-nitrogen compounds have been reported in the literature: TenN4 and Te3N4 [21, 22]. Whereas the c o m p o u n d of the composition Te3N4 was p r e p a r e d f r o m the reaction o f

289

Se-N and Te-N Compounds

K2[Te(NH)3] with an excess of [NH4][NO3] in liquid ammonia [23], the existence of TenN4 is still doubtful. The spectral data, structures, and especially the chemistry of all neutral tellurium nitrides are still unknown and largely unpredictable. The chemistry of ternary tellurium-nitrogen-halogen compounds is also very limited. Well established is the neutral compound FsTeNC12 [24] and the explosive azides, C13TeN3 and C12Te(N3)2 [25]. It was only recently when the first ternary telluriumnitrogen-halogen salt, Te4N2C18(AsF6)2-2 SO2, was reported [26]. This compound is a potential precursor to simple Te-N compounds and may well turn out to substantially facilitate tellurium-nitrogen chemistry.

II.

PREPARATIVE ASPECTS

In an emerging area like Se-N and Te-N chemistry, the early stages of development are primarily concerned with defining the descriptive aspects of the subject. Thus, the emphasis has been on the synthesis (this section) and structures (following section) of novel compounds. A significant advantage in Se-N and Te-N chemistry (compared to S-N) is the availability of a convenient NMR probe (775e, I = 1/2, 7.7%; 125Te, I = 1/2, 7%). Whereas the number of reported 125Te NMR chemical shifts of binary Te-N and ternary Te-N-C1 compounds is still very small, there has been extensive work on selenium compounds which has recently been reviewed [27]. Table 1. Colors of Binary and Some Ternary Se-N and Te-N Species Species ct-Se4N4 I3-Se4N4 [Se3N2]+ (AsF6- salt) [Se3N2]2+ (AsF6- salt) ct-Se3NCI2 13-Se3NC12 [(SeC12)2N]+ (AsF6- salt) [(SeC1)2N]+ (SbC16- salt) [Te(N3)3]+ (SbF6- salt) [TeaN2C18]2+ (AsF6- salt) FsTeNC12 C13TeN3 C12Te(N3)2

Color and Appearance yellowish orange to red solid red-brown crystal needles red-brown SO 2 solution; dark brown solid; crystals are black with dark green appearance in reflected light (yellow)-brown SO 2 solution; yellow brown solid; yellow crystals red, when crystals viewed against the light; shiny, metallic green when crystals viewed against a background golden microcrystalline solid with metallic luster chrome-orange (crystals and in SO 2 solution) deep orange solution; orange powdery solid; orange red crystals yellowish solution; colorless crystals pale yellow solution; white solid orange-yellow liquid colorless benzene solution; colorless crystals colorless benzene solution; colorless crystals

Ref. 1 10 36 36 11, 37 37 15 12 19 26, 38 24 25 25

290

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

Many of the binary (and ternary) selenium-nitrogen-(halogen) and telluriumnitrogen-(halogen) species are intensely colored. A compilation of the colors which may help the preparative chemist to initially identify reaction products is given below (Table 1).

A. Preparation of Se4N4 The only known binary Se-N compound is the highly explosive cage molecule SenNa. Despite this hazard, several applications of SenNa in the synthesis of other Se-N compounds have been reported [18, 28, 29, 30]. There are two well-established methods for the synthesis of SenNa. The first involves the reaction of (CH3CH20)2SeO with gaseous ammonia in benzene [31, 32]. The disadvantages of this method are low yields and time-consuming synthesis of (CH3CH20)2SeO. The second method uses the reaction of SeXn (X = Br, C1) or SeO2 with liquid ammonia at 70-80 ~ which produces SenNa in up to 75% yield but requires high-pressure apparatus [28, 33, 34]. Quite recently, two new practical routes for the preparation of small amounts of SenNa have been reported [35]. These methods involve: (1) the reaction of (Me3Si)2NLi (Eq. 1), and (2) the reaction of [(Me3Si)aN]2Se (Eq. 2). 12 (Me3Si)2NLi + 2 Se2CI2 + 8 SeC14 ~ 3 SenN4 + 24 Me3SiC1 + 12 LiC1 (1) 2 [(Me3Si)2N]2Se + 2 SeCl 4 --~ SenN4 + 8 Me3SiC1

(2)

Se4N4 in the metastable 13-modification has been prepared by the reaction of selenium dioxide with the phosphane imine Me3SiNPMe3 in acetonitrile, forming red-brown crystal needles (Eq. 3) [10]. 3 SeO 2 + 4 Me3SiNPMe 3 ~ 1/2 ~-Se4N4 + Se + 2 (Me3Si)20 + 4 0 P M e 3 (3)

B. Preparation of Binary Se-N Cations The first and so far only well-characterized examples of binary selenium-nitrogen cations are the five-membered Se-N heterocycles [Se3N2]+ (7n system) and [Se3N2]2+ (6n system) which can be made by direct oxidation of Se4N a with arsenic pentafluoride in liquid sulfur dioxide (Eqs. 4, 5) [28, 36]. From 775e NMR spectroscopic data there is evidence that in SO2 solution the dication [Se3N2]2§ is in rapid equilibrium with SeN § and Se2N§ (Eq. 6); however, the equilibrium is shifted very far to the left [36]. Both cations, SeN § and Se2N§ are still unknown in the condensed phase; however, Se2N§ was recently detected in the gas phase [13] and is also a likely intermediate formed during the reaction of [(SeC1)2N]§ with SnC12 as a chloride acceptor [13]. 3 SenN4 + 6 AsF 5 --~ 2[Se3N2]~ [AsF6]2 + 2 N 2 + 2 AsF 3

(4)

Se-N and Te-N Compounds

291

3 Se4N4 + 12 AsF 5 ~ 4[Se3N2] 2+ [AsF6] 2 + 2 N 2 + 4 AsF 3

(5)

[Se3N2] 2+ ~ SeN + + Se2N+

(6)

In analogy to the chemistry of S4N4 and the preparation of ($3N2C1)2 the corresponding reaction of SenN4 and SeC12 in dichloromethane yields (Se3N2C1)2 as a dimeric compound with two [Se3N2]+ heterocyles (Eq. 7) [18]. The same compound may also be obtained in essentially quantitative yield by a reaction according to Eq. 8 [18]. SenNa + Se2C12 _.~ [Se3N2]~[CI]2

(7)

2 [(Me3Si)2N]2Se + SezC12 + 2 SeC14 ~ [Se3N2]~[C1]2 + 8 Me3SiC1

(8)

C. Preparation of Ternary Selenium-Nitrogen-Halides Several Se-N chlorides have been recently synthesized for the first time. The attempted preparation of the SeN + cation (Eq. 9) produced instead the [(SeClz)zN] + cation (Eq. 10) [15]. [SeC13]+[AsF6]- + N(SiMe3) 3 ~ [SeN]+[AsF6]- + 3 Me3SiC1

(9)

6 [SeC13]+[AsF6]- + 5 N(SiMe3) 3 3[(SeC1)zN]+[AsF6]- + 9 Me3SiF + 3 AsF 3 + N 2 + 6 Me3SiC1

(10)

In a very related reaction the first known neutral selenium chloride nitride was obtained [11]. This compound was first prepared by Dehnicke et al. by the action of tris(trimethylsilyl)amine on a suspension of selenium tetrachloride in boiling dichloromethane (Eq. 11) [11]. Diseleniumtrichloride nitride, Se2NC13, has proven to be a very reactive and suitable synthetic reagent which, for example, by treatment with suitable Lewis acids (e.g. GaC13, FeC13, SbC15) can easily be transferred into the corresponding cation [(SeC1)2N] + [11-14]. Originally, Se2NC13 was recrystallized from dichloromethane, and the crystals when viewed against light appeared red but viewed against a background appeared shiny, metallic green [11]. This crystalline form of diseleniumtrichloride nitride was characterized by X-ray diffraction analysis and was shown to be orthorhombic. 2 SeC14 + N(SiMe3) 3 ~ Se2NC13 + 3 CISiMe 3 + C12

(11)

Quite recently, there was a report about a new convenient one-pot preparation to give monoclinic SezNC13 (Eq. 12) [37]. This route does not require a recrystallization procedure and produces Se2NC13 as a microcrystalline material with a golden,

292

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

metallic luster. Therefore, Se2NCI3 was shown by X-ray analysis to exist in two polymorphic forms, orthorhombic and monoclinic [37]. 3 SeC14 + Se2[N(SiMe3)2] 2 ~ 2 Se2NC13 + 4 Me3SiC1 + SeCI 2

(12)

D. Preparation of a Ternary Tellurium-Nitrogen-Halide Until today, only one cationic ternary tellurium-nitrogen-halide has been reported. The white thermally stable and moisture-sensitive compound, TenN2C18(AsF6)2.2 SO2, was prepared in a two-step synthesis from TeC14, N(SiMe3)3, and AsF5 [26, 38].

III.

S T R U C T U R A L ASPECTS

The electronic as well as the structural properties of any molecule containing heavy atoms such as selenium or tellurium can in principle be determined by quantummechanical calculations. Computational chemistry is now an established part of the structural and preparative main-group chemist's armory. It can be used as an analytical tool in the same sense that an NMR spectrometer (e.g. 14/15NNMR, 77Se NMR, 125TeNMR) or X-ray diffractometer can be used to rationalize the structure of a molecule. An important feature of ab initio computations is that they always yield the equilibrium geometry for an isolated molecule (i.e. an isolated molecule in the gas phase with no interactions to other molecules). It is, however, well known that molecules in the gas phase retain very nearly the same structure when assembled in a molecular (not an ionic!) crystal. Although small gas-solid structure differences are common (for covalent molecules forming molecular solids; e.g., SeaNC13), their magnitudes are typically hundredths of an angstrom or less for lengths and tenths of a degree for angles, reflecting the relatively minor perturbation that a crystalline environment imposes on any particular molecule (see ref. 52). In extreme cases the structure of a substance may, however, be quite different in the solid state and in the gas phase. This is especially true when strong interionic interactions favor structures for either the cation or the anion which do not represent the global minimum structure for the isolated species (e.g. the cation (SeC12)2N+: the isolated cation has a C2v structure as the global minimum; in the crystal the cation possesses Cs structure, see below). The computational determination of molecular geometry (for isolated molecules, see above) has developed considerably in recent years as the capabilities of computers and the theoretical understanding of molecular structure have increased. There are still severe limitations to the size of the molecule for which ab initio computations are feasible, but they have no other limitations and most species discussed in this chapter are "small enough" to be calculated at high level of theory including electron correlation. However, even if electron correlation is taken

293

5e-N and Te-N Compounds

completely into consideration (full CI; note this corresponds to the exact solution of the Schr/Sdinger equation and is of no real practical interest--cpu time!), the computed energy does not meet the true energy value for atoms or molecules. This remaining difference between computed (full CI) and true energy is due to relativistic effects. As the core or total relativistic energies go like Z 4 (the valence contribution roughly goes like Z 2) they become relevant especially for the very heavy elements but also for those following copper, i.e. selenium; relativistic effects are comparable to the usual shell-structure effects. Due to the relativistic mass increase, the effective Bohr radius will decrease for inner-shell electrons with large average speed. The most widely used approach to include relativistic effects into quantummechanical computations is the pseudo-potential method. In this method for very large (i.e. heavy) nuclei, electrons near the nucleus are treated in an approximate way, via effective core potentials (ECPs, frozen inner shells). The corresponding nodes in an atomic valence wave function are omitted by considering instead the eigenvalue problem for a nodeless one-component pseudo-wave function. The pseudo-potential or effective core potential (ECP) corresponds to all interactions between the valence and the core electrons. Whereas relativistic SCF calculations for molecules containing one or more selenium atoms are extremely time-consuming, the ECP method is fast and can describe molecular properties almost as accurately as all-electron calculations (see ref. 50). All calculations discussed in this chapter and dealing with selenium containing molecules are based on the ECP method. A. The Aromatic 6n Cation [SeaN2] 2+ and the n*-n* Bound 7n Cation [Se3N2] + The identities of both binary selenium-nitrogen cations were confirmed by the determination of their X-ray crystal structures [28, 36]. The compound N

N

~

,~ S e

s~ Se

tl Se

N

Figure 1. The structure of the crystallographically unique centrosymmetric [Se3N2]~ dimer in [Se3N2]~[AsF612128,36].

294

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

Se

Figure 2. Nature of the SOMO of [Se3N2]~ [28, 36].

[Se3N2]~[AsF6] ~ contains the indefinitely stable 7n radical [Se3N2]+ which dimerizes in the solid state to give the cation [Se3N2]~ (Figure 1). The 7n radicals in the dimers are weakly linked in a centrosymmetric trans arrangement ( n , - n * interaction) through the diselenium portions of each of the tings by overlap of the selenium portions of the two singly occupied molecular orbitals (SOMO) (Figure 2). The Se-Se bond (2.334(3) ,~) in the 6n [Se3N2]2+ dication (Figure 3) is significantly shorter than that in the 7n system [SeaN2] + (2.398(3)/~). This change in the Se-Se bond distances which accompanies the oxidation of 7n [Se3N2]+ to 6n [SeaN2] 2+ is therefore consistent with the nature of the SOMO of 7n [SeaN2]+ (cf. Figure 2). The antibonding interactions between the two selenium atoms cause the bonds to shorten on removal of the single electron. The species [Se3N2]2+ and [Se3N2]+ are formally 6n and 7n systems, respectively. The n-bond orders estimated from the bond distances indicate the presence of thermodynamically stable delocalized 4pn(Se)-2pn(N) bonds consistent with a simple MO derived from that of the sulfur analogue [$3N2]2+ [39]. Using a simple extension of the Hiickel rules, the planar 6n cation [Se3N2]2+ can be regarded as formally aromatic (Figure 4a) [40, 41]. The most important VB representations for the [Se3N2]2+ cation are shown in Figure 4b.

$e

N

Figure 3. The structure of the 6n cation [Se3N2]2+ in [Se3N2]2+[AsF6]~ [28, 36].

Se-N and Te-N Compounds

295

Figure 4. (a) Qualitative ~-MO diagram for [Se3N2] 2+ (based on a CNDO calculation) [40, 41]. (b) VB representation for [Se3N2]2+

However, the calculated Se-Se bond orders in [Se3N2]2+ (1.02) and [Se3N2]+ (0.83) are very much lower than expected on the basis of the MO treatment which indicates a substantial 4p~-4p~ contribution from the totally symmetric bonding MO in the Se-Se region. 1 Long Se-Se bonds are also observed in [SeNSNSe] 2+ (bond order 1.10) and [SeNSNSe]~ (0.96), although they are significantly shorter than those in [Se3N2]n+ (n = 1 or 2; cf. Figures 1 and 3). Similar and even more pronounced weakening effects are present in [SNSNS]~ (d,S-S = 2.147 ]k, bond order 0.71) [43] 1and [SNSNS] 2+ (d,S-S = 2.093 ~, bond order 0.84) [39] 1in which the S-S bonds are 0.107 and 0.053 .&, respectively, longer than the normal S-S single bond (2.04-2.08 A) [40], whereas there is significant x-bonding within the S-N bonds. The long chalcogen-chalcogen bond may result in part from some lengthening of the bond due to repulsion of the two adjacent positively charged chalcogen atoms as illustrated by the long Se+-Se + bond (2.382/~, bond order 0.86) and S+-S + bond (2.124 ]k, 0.76) in [C6HI2Se2]2+ and [C6H12S2]2+, respectively (Figure 5) [44].

S

o

Figure 5. Se-Se and S-S bond lengths in [C6H12Se2]2+ and [C6H1252] 2+ [44].

296

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAP(~TKE

B. Polymorphism:Se4N4 and Se2NCI3 Tetraselenium Tetranitride Although SenN4 and the sulfur analogue SaN4 are not isostructural, both compounds have the same cage-like molecular structure (Figure 6). SenNa exists in at least two modifications. The metastable monoclinic 13-modification (P21/n) is isostructural with SaN4, whereas the more stable modification is the also monoclinic a-form (C2/c) [9]. Just as the a-form, ~-SenN4 forms cage molecules without symmetry and intramolecular Se-Se contacts of 2.732 and 2.740 A. There are strong Se--.N interactions between the SenN4 molecules which are stronger than in the case of SnN4; this might be the reason that no solvent which dissolves SenNa has been found [10].

Diselenium Trichloride Nitride In contrast to the material obtained from a reaction according to Eq. 11, the reaction according to Eq. 12 always led to microcrystalline Se2NC13 which had a golden color and a metallic luster. Subsequently, the X-ray structure analysis revealed Se2NC13 to be polymorphic and to exist in two modifications. Whereas the a-form was shown earlier to crystallize in an orthorhombic space group [11], in recent work there was established that there is also a monoclinic form of 13-Se2NC13 (Table 2) [37]. Both structures are made up of discrete Se2NC13 molecules the structural parameters of which are very similar (Figure 7, Table 3). However, in the monoclinic form there are short intermolecular Se-..Se contacts of 3.82 ,~, which is substantially shorter than the sum of the van der Waals radii of 4.00/~ [37] (Figure 8). This may well explain the golden color and the metallic luster of monoclinic 13-Se2NC13 [37]. In contrast, the shortest intermolecular Se..-Se distances in orthorhombic a-Se2NCI3 are 4.10/~ [11]. In the solid state, the monoclinic form of 13-Se2NCI3 is stable under an inert gas atmosphere below -5 ~ only. At higher

Figure 6.

Structure of a Se4N4 molecule in solid 13-Se4N4 [ 10].

5e-N and Te-N Compounds

297

Table 2. Crystallographic Data for Orthorhombic ~t- and Monoclinic 13-Se2NCI3 a

Mr space group a //~ b //~ c //~ 13/ o V //~3 Z

Orthorhombic t:t-Se2NCl3

Monoclinic ~- Se 2NCI3

278.28

278.28

P bca

P211a

12.290(6) 8.046(4) 24.336(12)

7.605(3) 8.7643(20) 8.966(3) 93.23(3) 596.6(3) 4

2406.5 16

Note: aFromrefs. 11 and 37.

temperatures I3-Se2NC13 d e c o m p o s e s to give Se2C12 and other jet to be identified d e c o m p o s i t i o n products. The metallic luster of monoclinic 13-Se2NC13 naturally led to an investigation of its electrical conductivity. However, due to the great air sensitivity and especially due to the instability at ambient temperature and even at 0 ~ these m e a s u r e m e n t s were not easy to carry out and were p e r f o r m e d under nitrogen in a dry box with

Table 3. Selected Bond Lengths and Angles a for Orthorhombic t~- and Monoclinic [3-Se2NCI3 b,c Orthorhombic a-Se2NCI 3 Monoclinic ~-Se2NCl 3

Se 1-N 1 Se2-N 1 Se 1-CII Se2-C12 Se 1-C13 Se2-C13 Sel-CI3-Se2 SeI-N1-Se2 C11-Se I-N 1 C12-Se2-N 1 CI3-Se 1-N 1 CI3-Se2-N 1 Se...Se Notes:

1.742(5) 1.733(5) 2.269(2) 2.198(2) 2.519(2) 2.676(2) 70.7(1) 119.9(3) 91.4(2) 93.0(2) 87.1 (2) 82.4(2) 4.10

aln angstromsand degrees. bSeeFigure7 for atom labels. CFromrefs. I 1 and 37.

1.79(3) 1.73(3) 2.166(9) 2.196(9) 2.759(8) 2.682(9) 67.75(21) 119.4(17) 95.9(9) 93.4(11) 84.6(9) 88.1 (11) 3.82

298

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

Figure 7. Molecular structure of 13-Se2NCI3in the crystal [37]. precooled samples. Two measurements on single crystals were carried out at 0 ~ and at 20 ~ (+5 ~ It could be established that monoclinic 13-Se2NC13 does not show significant electrical conductivity. The measured electrical resistivity was > 106 ohm cm (at 0 and 20 ~ The relative high value may well be explained by partial decomposition and formation of elemental selenium (cf. electrical resistivity of elemental gray selenium at 25 ~ 101~ohm cm, very dependent on purity) [40].

Figure 8. Packing diagram for I3-Se2NCl3 in the monoclinic system [37].

Se-N and Te-N Compounds

299

Figure 9. Structure of the [(SeCI2)2N] + cation in solid [(SeCI2)2N]+[AsF6]-.CH3CN (a different view of the same cation is shown in Figure 12) [ 15].

C.

Solid-State Structures of [E2NCI4]+n (E = Se, n = 1; E = Te, n = 2)

So far, only two examples of the class of [E2NC14]n+ type compounds have been reported: 1.

2.

the selenium cation [(SeCI2)2N] + that exists in the salt [(SeC12)2N]+[AsF6] CH3CN 9 with n = 1 (Figure 9) [15] (for a more detailed discussion see Section III.D), and the tellurium cation [Te2NC14]~ which exists in the salt [Te2NCI4]~[AsF6]2-2 SO 2 formally with n = 2 (Figure 10) [26, 38].

The tellurium salt contains the cation [(CI3Te)NTe(C1)N(TeCI3)Te(CI)] 2+, which may be regarded as the dimer of [CI3Te-N=TeC1] +, cf. the n-bonded monomeric [(SeCI2)2N] +, reflecting the situation for related R I N = E - N R 2 compounds (E = S, monomer and R l, R 2 = tBu, SiMe3; E = Te, dimer and R l = tBu and R 2 = PPh2NSiMe3) [46, 47] that appear to be stabilized by bulky groups. The centrosymmetric [Te4N2C18]2+ cation contains a planar four-membered Te2N2 ring with Te-N bond lengths of 2.04(9) and 1.98(4)/~, similar to that expected (2.04/~) for

Figure 10. Structure of the [Te4N2CI8] 2+ cation in solid [Te4N2CI8]2+[AsF6]2.CH3CN [26].

300

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPtDTKE Cl

Cl

I

Te CI--Te--N

I

Cl

CI N--Te--CI

Te

I

CI

Cl (~

Cl ~

i

Te

CI--Te--N CI e

CI O N--Te--CI

Te

i

I

CI

~

Figure 11. Valence bond structures for the [Te4N2CI8] 2+ cation [26]. a Te(IV)-N bond and similar to distances in other related Ze2N 2 rings (cfl Figure 10) [47, 48]. The two exocyclic Te-N bond distances (2.05(4) ~) are not significantly different from those in the Te2N2 ring, i.e. each tellurium atom is joined to three nitrogen atoms by single bonds. The 10 tellurium, nitrogen, Cl(1), and C1(3) atoms are roughly coplanar. The tellurium-chlorine bond lengths (Te(1)-Cl(2), 2.27(2) ,~, Te(2)-Cl(4), 2.20(2) ]k, perpendicular to the plane) are similar to those in [TeCla]+[AsF6] - (av. 2.264 ]k) [49], i.e. are of bond order one. The longer tellurium-chlorine distances (Te(1)-Cl(1), 2.51 (2) ,~, Te(1)-Cl(3), 2.50(2) /k) within the plane correspond to a bond order of about 0.5, implying that the bonding within the cation may be described approximately by the valence structures shown in Figure 11 and related resonance structures [26, 38].

D. Structure of [(SeCI2)2N]+" A Comparison of X-ray and Ab Initio Studies The solid-state structure of the [(SeCI2)2N] + cation was determined by low-temperature X-ray diffraction (Figure 12, Table 4) [15, 16]. The geometry of the cation can be approximated by Cs symmetry with two significantly different Se-C1 bond lengths (2.169(4) and 2.141(4) A, Table 4). The Se-N distances of 1.741(11) and 1.760(11) A are essentially identical (within 3a). The RHF calculations located three isomers of [(SeClz)zN] + in the geometric configuration space which was searched; these isomers were fully optimized at HF level (Figure 13). Their energies at different theoretical levels are summarized in

Figure 12. X-ray structure of the cation [(SeCI2)2N] + in solid [(SeCI2)2N]+[AsF6] CH3CN 9 (cf. Figure 9) [15"].

Se--N and Te-N Compounds

Figure 13,

301

H F - o p t i m i z e d structures for [(SeCI2)2N] + [ 16].

Experimental a and Computed b Structural Parameters c for Different Isomers of [(SeCI2)2N] § and Absolute (-au) and relative energies d'e

Table 4,

X-ray d(Sel-N1) d(Se2-N1) d(Cll-Sel) d(Cl2-Se2) d(Cl3-Sel) d(Cl4-Se2) Z(SelN1Se2) Z(N1SelCll) Z(N1Se2C12) Z(N 1Se 1C13) Z (N 1Se 2C14) / ( C I 1Se 1C13) Z(C12Se2CI4) HF, -au (in kcal/mol) MP2, -au (in kcal/mol)

Notes:

1.741(11) 1.760(11) 2.169(4) 2.169(4) 2.141(4) 2.141(4) 117.6(2) 102.9(4) 103.6(4) 93.3(4) 92.8( 4 ) 98.2(2) 98.5(2)

d-HF, Cs) 1.800 1.800 2.163 2.163 2.139 2.139 115.5 103.5 103.5 95.5 95.5 99.6 99.6

d-MP2, Cs

d-HF,C2v

u-HF, C2v

1.823 1.823 2.221 2.221 2.168 2.168 114.5 104.6 104.6 93.2 93.2 100.1 100.1

1.796 1.796 2.150 2.150 2.150 2.150 114.6 99.6 99.6 99.6 99.6 99.6 99.6

1.760 1.760 2.166 2.166 2.166 2.166 135.7 108.2 108.2 108.2 108.2 98.3 98.3

132.29690 (7.5)

131.37031 (5.6) 132.29690 (7.5)

131.37924 (0.0) 132.30888 (0.0)

131.37106 (5.1)

aX-ray. bN: 6-3 I+G* basis set; CI: [5s5pld]/(3s3pld) (DZ+P), 10 electron core potential; Se: [5s5p]/(3s3p) (DZ+P, extended with one set of d functions, dexp = 0.334), 28 electron core potential. CBond lengths in angstroms, angles in degrees. din kcal mol-t. eFrom ref. 16.

302

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE C

-'ON

/ it I

! o E

"1

t._ t.J

0

100

rotational

~'~

3C0

angle / *

Figure 14. Rotation of the N atom from the (a) d-HF, C2v (0 ~ via the (b) d-HE Cs (24 ~ into the (c) u-HF, C2v position (180 ~ and rotational barrier.

Table 4. In the following section we use the following notation: "d-HE C2v" for the [(SeC12)2N] + cation with the nitrogen down; "u-HE C2v"for the nitrogen up; "d-HF, Cs" designs the experimentally observed Cs isomer with nitrogen down. For the isolated [(SeC12)2N]+ cation, the nitrogen up-C2v structure is most favored both at the HF and the correlated MP2 levels (Table 4). Moreover, the experimentally observed d-Cs isomer (local minimum) had considerable higher energy. Unfavorable crystal lattice effects in the hypothetical u-C2v cation in the solid state

J/I

/Se ~

~

j N ~

~

Figure 15. Lewis representation of the cation [(SeCI2)2N] + according to NBO analysis [16].

Se-N and Te-N Compounds

5el N

303

~

~

C1!

C13

Figure 16. Negative LP(N) ---~c~*(Se-CI)hyperconjugation in [(SeCI2)2N]§ [16, 50]. (AsF~ salt) due to repulsion of the negatively charged nitrogen in the cation and the AsF~ anion, may be responsible (for a more extensive discussion see ref. 16). The rotational barrier of the process moving the nitrogen atom from the d-HF, Cev (0 ~ via the experimentally observed d-HF, Cs (24 ~ HF level; 10.7 ~ at MP2 level) into the mostly favored u-HF, C2v position (180 ~ was also computed (Figure 14) [16, 50]. According to strictly localized natural bond orbital analysis (NBO) 2 the central nitrogen atom in d-Cs [(SeC12)2N]§ possesses two lone pairs of electrons (LP: one sp-hybrid and one p-orbital). Therefore, Figure 15 shows the best Lewis representation for the cation [(SeC12)2N]§ The relatively short Se-N distance (1.741-1.760/~; cf. sum of covalent radii of N and Se, 1.870/~ [16]; cf. also Table 4) can best be attributed to LP(N)---~r~*(Se-C1) negative hyperconjugation (Figure 16) [16, 50]. This also explains nicely the two different sets of Se-C1 bond distances in [(SeC12)2N]§ [16, 50].

E. Structure of [(SeCI)2N]§ An Unusual Structurally Very Flexible Ion The cation [(SeC1)2N]+ exists in the solid state stabilized by various anions, i.e. [(SeC1)2N]§ - [11], [(SeC1)2N]§ - [13, 14], and [(SeC1)2N]§ [SbC16][12 ]. Whereas the cation [(SeC1)2N]§ in one of the compounds [(SeC1)2N]§ [11] or [(SeC1)2N]§ - [13, 14] has an u-shaped C2v structure in the solid state (Figure 17), it has an s-shape Cs structure in the crystal when the counterion is SbCI~ [12, 50]. This structural diversity can be qualitatively rationalized assuming the existence of several (at least two) minima (that are close in energy) on the potential energy hypersurface of [(SeC1)2N]§ and slightly different cation-..anion interactions in the crystal and other lattice effects can favor either of the energetically similar isomers. The internal structural parameters of the various isomers of the cation [(SeCI)2N] § are summarized in Table 5. It is noteworthy that the two

304

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

Table 5.

Structural Parameters for the Two Isomers of the Cation [(SeCI)2N] § a

Isomer

d(Se 1-N)//~, d(Se2-N)//~ d(Cl 1-Se 1)//~ d(Cl2-Se2)/]~ Z(Se 1-N-Se2)/~ Z(C11-Sel-N)/~

Z(CI2-Se2-N)/~ Z(CI_Se_N_Se)/O e Notes:

[GaCI4]- Salt b

[FeCI4]- Salt c

[SbCI6]- Salt d

u-Isomer

u-lsome r

s-Isomer

1.695(4) 1.694(4) 2.151(2) 2.143(2) 146.6(3) 108.3(1) 108.3(1) 0.0

1.699(9) 1.695(9) 2.154(3) 2.157(2) 146.9(5) 108.2(3) 108.0(3) 0.0

1.644(12) 1.753(22) 2.156(7) 2.136(7) 127.0(13) 102.2(8) 99.5(7) 180.0

aDeterminedby X-raydiffraction. bFromref. 11. CFromref. 14. dFromref. 12. edihedralangle(C1l-Se I-N-Se2-CI2).

S e - N and Se-C1 bond distances in the s-shaped isomer are just slightly different (2 cs) [12]. In order to understand this structural diversity more fully, high-level a b i n i t i o computations were carried out. The RHF calculations located three isomers of [(SeC1)2N] § in the geometric configuration space which was searched (Figure 17). These isomers were fully optimized at HF level and at correlated MP2 level (Table

(•CI

CI

~$

Cll

1 N Se2

Figure 17. MP2 optimized structures for the u-, w- and s-isomers of [(SeCI)2N] + [ 12].

5e-N and Te-N Compounds

305

Table 6. Computed Structural Parameters for the s- and u-Isomers of [(SeCI)2N] § a H f f ~ s-Isomer d(Se 1-N)/~

M P 2 b s-Isomer

d(Se2-N)//k d(Cl 1-Se 1)//~ d(C12-Se2)/~, Z(Sel-N-Se2) / ~ Z(C11-Se I-N) / ~ Z(CI2-Se2-N) / ~

1.717 1.736 2.162 2.144 132.4 103.8 98.7

1.759 1.787 2.190 2.159 124.0 105.7 97.5

Z(C1SeNSe)/O c

180.0

180.0

Notes:

H F b u-Isomer

MP2 b u-Isomer

1.708 1.708 2.156 2.156 151.0 107.9 107.9

1.747 1.747 2.184 2.184 142.5 111.4 111.4

0.0

0.0

aFrom refs. 12 and 50. bN: 6-3 I+G* basis set; CI: [5s5pld]/(3s3pld) (DZ+P), 10 electron c o r e potential; Se: [5s5p]/(3s3p) (DZ+P, extended with one set of d functions, dexp -- 0.334), 28 electron c o r e potential. Cdihedral angle (CI l - S e I - N Se2-C12).

6). In good agreement with the experimental results the total energies for both observed isomers (s- and u-isomer) are very similar. Therefore, only marginal differences in the cation-..anion interactions can favor either of these species. The experimentally not-observed w-isomer was computed to be 9.2 kcal mol -~ less stable than the u-isomer and 8.2 kcal mol -~ less stable than the s-isomer (MP2). Since the two experimentally established isomers (u and s) which are very close in energy can interconvert by a 180 ~ rotation along one of the S e - N axes (i.e. by variation of the dihedral C12Se2NSe 1 angle from ) to 180 ~ Figure 18), the energy required for this isomerization was computed ab initio. It was shown that both

"1

D

z~-

"

,

/

4

transition s t a t e

I

0 G

1=

u

~.) l... ,,t

5

(~

,

0

45

dihedral

i,

i

90

angle

,

135

,

18<3

(C1-Se-N-Se) /*

Figure 18. Energy profile for the interconversion of the u- (0 ~ to the s-isomer (180 ~ of [(SeCI)2N] + (HF/6-31 +(3*) [12, 50].

306

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAPOTKE

t

I

C12

CI1

Figure 19. Negative p-LP(N)->o*(Se-CI) hyperconjugation in the s-isomer of [(SeCI)2N]+ [12, 50]. isomers are separated by a barrier of 19 kcal mol -], which should allow interconversion in solution when there are no cation.--anion interactions [12, 50]. In agreement with this prediction it was observed that the experimental Raman spectrum of [(SeCI)2N] § [12] [SbC16]- changed dramatically after dissolving of the sample in liquid sulfur dioxide. Whereas the Raman spectrum of the crystalline solid compares with the computed frequencies of the s-isomer, as expected (at MP2 level the u-isomer is more stable than the s-isomer by 1 kcal mol -l [12, 50]) the solution spectrum showed the slightly more stable u-isomer (as an "isolated" cation) to be present [12]. Finally, it might be interesting to compare the experimental structures with the results of the natural bond analysis. 2 As indicated by NBO analysis, the S e - N - S e unit in all three isomers (u, s, w) of the [(SeCI)2N] § cation has strongly polarized Se-N bonds. In all cases the NBO charges on nitrogen are close to -1, whereas for each selenium they are nearly +1 [12]. In agreement with NBO analysis the best Lewis representation for these cations would clearly be a species containing four o-bonds and one n-bond that is delocalized within the Se-N-Se unit [12]. The different Se-N and Se-CI bond distances in the s-isomer are best explained by a fairly strong interaction of the p lone pair (p-LP) on nitrogen with one of the unoccupied, antibonding o*-orbitals of the Se-C1 bond (in our notation Se l-C11): p-(LP)--~o*(Se l-C11), 10 kcal mo1-1 (Figure 19). Obviously, the interaction with the other Se-CI bond (Se2-C12) is rather weak for geometrical reasons:

..

9Se

..

Se. -

A

-.S e m S e

6

.0

:

+]~

. .S e ~ - - b

:

C

Figure 20. VB representations for the 6~ dication [Se3N2]2+.

Se-N and Te-N Compounds

307

b N(+~ i

:,,,

;

9S e - - - - ; - - - S e : / /

9

N(+~ i .Se

;

~

Se:

S e ~ S ~

D

E

Figure 21. VB representations for the 7re dimer {[Se3N2]+}2. p-LP)---~a*(Se2-C 12), 3 kcal mol -l (Figure 19) [12]. This not only explains the two different N-Se distances but also accounts for the differences in the Se-C1 bond lengths [12].

F. A Valence Bond Description for the Molecules [Se3N2]2§ [Se3N2]+, Se4N4, and the Hypothetical (SeN)x The species [Se3N2]2+ and [Se3N2]+ are formally 6x and 7re systems, respectively

[40, 41]. The most important VB representations (A, B, C) for the 6re [Se3N2]2+ cation are given in Figure 20. The Sen linkage of the {[Se3N2]+} dimer can best be described using the increased valence structures D and E as indicated in Figure 21. The dimer has D2h symmetry for the Sen linkage; overlap considerations for it suggest that "increased-valence" structure D has a larger weight than has E. The intermoiety Se-Se bonds have a length of 3.12/~ and are therefore ca. 0.8 A longer than a "normal" Se-Se single bond (2.34/~) [53]. The structure of the cage-type SeaN4 can also be rationalized in terms of VB considerations (structures F, G; Figure 22). It was suggested that one set of nitrogen lone-pair electrons of the standard Lewis structure F could delocalize appreciably

.

+

"L i-i

+ :

o

:

N.

..N

I.. +i ..I

.N 9

~ ..

F

! i i"

So: +

Se:

I-. .-I Se

N..

G

Figure 22. VB representations for the 12-electron 8-center bonding in Se4N4.

308

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAP(DTKE

9,

N~Se.

N~Se

,,

x._./'. N------,Se. 9 9

.~

9

+I/2

N------So

-t/2

o,

o"

+I/~

Figure 23. VB bond description for (SeN)x using polymerized Pauling "3-electron bonds" and polymerized 6-electron 4-center bonding units. into the antibonding Se-Se ~*-orbital, thereby reducing the Se-Se bond order well below the value of unity that pertains for F. If these electrons are delocalized into the adjacent Se-N bonding orbitals, "increased-valence" structure G is obtained with Se-Se or-bond numbers less than unity. This structure indicates that the Se-Se and Se-N bonds should be respectively longer and shorter than single bonds, and this they are found to be. The measured Se-Se and Se-N bond lengths are 2.76 and 1.78 ,~,, respectively, and the estimate of an Se-N single-bond length is 1.87 A (sum of covalent radii). A 12-electron 8-center "increased-valence" bonding unit is present in "increased-valence" structure G; it involves the eight nitrogen n-electrons and the four Se-Se ~-electrons of F [53; cf. also 54, 55]. The still hypothetical (SeN)x can be regarded either as a polymer of SeaN 4 or, more appropriately, as a polymer of the also as yet unknown Se2N2. The impetus for such studies has, in part, been derived from the possibility that the polymer (SeN)x (for a classical VB representation see H, Figure 23) may exhibit even more unusual properties than those of the metallic superconductor (SN)x, [7]. A comparison with the bond situation in the well-known sulfur nitride (SN)x which can be described by polymerized Pauling "3-electron bonds" and polymerized 6-electron 4-center bonding units (structure I; Figure 23) indicates that the electron delocalization should be favored by a high electronegativity of the pnicogen and a low electronegativity of the chalcogen [53]. Therefore, it can be expected that (SeN)x as well as (TeN)x may possess even higher conductivity than (SN)x does. IV.

CONCLUSIONS

The recent discoveries of readily handled binary Se-N and ternary Se-N-C1 and Te-N-C1 reagents will facilitate the development of selenium-nitrogen and tellurium-nitrogen chemistry. Significant differences between the structures of Se-N compounds and their sulfur counterparts have been observed and it is likely that Te-N compounds will be again structurally very different from their selenium analogues. The recently established first example of a ternary Te-N-C1 cation, i.e. [TeaN2C18]2+ [26], already indicates this trend. The construction of molecular

309

Se-N and Te-N Compounds

conductors based on S e - N species with strong intermolecular interactions in the solid state is a distinct possibility (cf. the structure of monoclinic Se2NC13) [37]. The synthesis of S e - N polymers related to the well-known metallic conductor (SN)x is also an interesting challenge [6]. The electronic as well as the structural properties of any molecule containing heavy atoms such as selenium or tellurium can now be investigated by ab initio quantum chemistry [50]. In particular, pseudopotential methods facilitate ab initio calculations on compounds involving heavier elements. The major relativistic effects can be included conveniently by fitting the atomic pseudopotential parameters to relativistic or quasirelativistic all-electron data.

ACKNOWLEDGMENTS The authors should like to thank Prof. Magdolna Hargittai for suggesting the problem to us. We are indebted to and thank Professor Richard D. Harcourt for many helpful and stimulating discussions. Our part of the work reported in this chapter has been supported by the North Atlantic Treaty Organization (CRG 920034, 1992-1997), the Deutsche Forschungsgemeinschaft, and the University of Glasgow.

NOTES 1The bond orders b have been estimated from the experimentally determined bond lengths d using the empirical relationship suggested by L. E Dahl et al. [40, 42] in which lg b - (d I - d)/0.71 (with d in/~ and d I = bond length for an E-E single bond), dl(S) = 2.08 ~, dl(Se) = 2.35/~, dl(N) = 1.44/~. 2NBO analysis: In the quantum mechanical computation (subjecting the HF density matrix as represented in the localized NBOs to a second-order perturbative analysis) the energy was computed according to,

Eqxp. ( 2=)-2

((p I hF I (p*)2

E~,- E~

with hF being the Fock operator [50, 51].

REFERENCES 1. Klap6tke, T. M. In The Chemistry of Inorganic Ring Systems; Steudel, R., Ed. Elsevier: Amsterdam, 1992, p. 409. 2. Cordes, A. W.; Haddon, R. C.; Oakley, R. T. In The Chemistry oflnorganic Ring Systems, Steudel, R., Ed. Elsevier: Amsterdam, 1992, p. 295. 3. Kelly, P. E; Slawin, A. M. Z.; Williams, D. J.; Woollins, J. D. Chem. Soc. Rev. 1992, 246. 4. Woollins, J. D. In The Chemistry oflnorganic Ring Systems; Steudel, R., Ed. Elsevier: Amsterdam, 1992, p. 349. 5. Bj6rgvinsson, M.; Roesky, H. W. Polyhedron 1992, 10, 2353. 6. Chivers, T.; Doxsee, D. D. Comments Inorg. Chem. 1993, 15, 109. 7. Chivers, T. Main Group Chem. News 1993, 1, 6. 8. Broschag, M.; Klaptitke, T. M. Phosphorous, Sulfur, and Silicon 1994, 93-94, 181. 9. Bhrnighausen, H.; v. Volkmann, T.; Jander, J.Acta Cryst. 1966, 21,751.

310

INIS C. TORNIEPORTH-OETTING and THOMAS M. KLAP(~TKE

10. Folkerts, H.; Neumtiller, B.; Dehnicke, K. Z. Anorg. Allg. Chem. 1994, 620, 1011. 11. Wollert, R.; Htillwarth, A.; Frenking, G.; Fenske, D.; Goesmann, H.; Dehnicke, K. Angew. Chem. 1992, 104, 1216; Angew. Chem. Int. Ed. EngL 1992, 31, 1251. 12. Broschag, M.; Klap/Stke, T. M.; Schulz, A.; White, P. S. lnorg. Chem. 1993, 32, 5734. 13. Borisenko, K. B.; Broschag, M.; Hargittai, I.; KlapOtke, T. M.; Schr~ler, D.; Schulz, A.; Schwarz, H.; Tornieporth-Oetting, I. C.; White, P. S. J. Chem. Soc., Dalton Trans. 1994, 2705. 14. Broschag, M.; KlapiStke, T. M.; Schulz, A.; White, P. S. Chem. Ber. 1994, 127, 2177. 15. Broschag, M.; Klap6tke, T. M.; Tornieporth-Oetting, I. C.; White, P. S. J. Chem. Soc., Chem. Commun. 1992, 1390. 16. Schulz, A.; Buzek, P.; Schleyer, P. v. R.; Broschag, M.; Tornieporth-Oetting, I. C.; Klap6tke, T. M.; White, P. S. Chem. Ber. 1995, 128, 35. 17. Wollert, R.; Neumtiller, B.; Dehnicke, K. Z. Anorg. Allg. Chem. 1992, 616, 191. 18. Siivari, J.; Chivers, T.; Laitinen, R. S. Inorg. Chem. 1993, 32, 4391. 19. Johnson, J. P.; MacLean, G. K.; Passmore, J.; White, P. S. Can. J. Chem. 1989, 67, 1687. 20. Tornieporth-Oetting, I. C.; Klal~tke, T. M. Angew. Chem. 1995, 107, 559; Angew. Chem. Int. Ed. Engl. 1995, 34, 511. 21. a) Garcia-Fernandez, H. Bull. Soc. Chim. France 1973, 1210; b) Garcia-Fernandez, H.; Pascal, M. P. C. R. Acad. Sc. Paris 1964, 258, 2579. 22. Strecker, W.; Mahr, C. Z. Anorg. Allg. Chem. 1934, 221, 199. 23. a) Schmitz-Du Mont, O.; Ross, B. Angew. Chem. 1967, 79, 1061; Angew. Chem. Int. Ed. Engl. 1967, 6, 1071; b)Schmitz-Du Mont, O.; Ross, B.; Klieber, H.Angew. Chem. 1967, 79, 869; Angew. Chem. Int. Ed. Engl. 1967, 6, 875. 24. Hartl, H.; Huppmann, P.; Lentz, D.; Seppelt, K. Inorg. Chem. 1983, 22, 2183. 25. Wiberg, N.; Schwenk, G.; Schmid, K.-H. Chem. Ber. 1972, 105, 1209. 26. Passmore, J.; Schatte, G.; Cameron, T. S. J. Chem. Soc., Chem. Commun. 1995, 2311. 27. Klap6tke, T. M., Broschag, M. Compilation of Reported 77Se NMR Chemical Shifts. Wiley, Chichester, New York, 1996. 28. Awere, E. G.; Passmore, J.; White, P. S. J. Chem. Soc., Dalton Trans. 1993, 299. 29. Kelly, P. E; Slawin, A. M. Z.; Wdliams, D. J.; Woollins, J. D. J. Chem. Soc., Chem. Commun. 1989, 408. 30. a) Adel, J.; Dehnicke, K. Chimia 1988, 42, 413; b) Adel, J.; EI-Kholi, A.; Willing, W.; Mtiller, U.; Dehnicke, K. Chimia 1988, 42, 70; c) Adel, J.; Ergezinger, C.; Figge, R.; Dehnicke, K. Z. Naturforsch. 1988, 43b, 639. 31. Strecker, W.; Schwarzkopf, X. Z. Anorg. Allg. Chem. 1935, 221, 193. 32. Ginn, V.; Kelly, P. E; Woollins, J. D.J. Chem. Soc., Dalton Trans. 1992, 2129. 33. Jander, J.; Doetsch, V. Chem. Ber. 1960, 93, 561. 34. Gowik, P. K.; Klaptitke, T. M. Spectrochim. Acta 1990, 46 A, 1371. 35. Siivari, J.; Chivers, T.; Laitinen, R. S. Inorg. Chem. 1993, 32, 1519. 36. Awere, E. G.; Passmore, J.; White, P. S.; Klal~tke, T. M. J. Chem. Soc., Chem. Commun. 1989, 1415. 37. Broschag, M.; Klap~tke, T. M.; Rien~icker, C. M.; Tornieporth-Oetting, I. C.; White, P. S. Heteroatom Chem. 1996, 7, 195. 38. Passmore, J.; Schatte, G.; Cameron, T. S. 1995 Intl. Chem. Congr. of Pacific Basin Societies, Honolulu, Hawaii, December 17-22, 1995, Abstract Inorg. 345. 39. Brooks, W. V. E; Cameron, T. S.; Grein, E; Parsons, S.; Passmore, J.; Schriver, M. J. Chem. Soc., Chem. Commun. 1991, 1079. 40. Klap6tke, T. M.; Tornieporth-Oetting, I.C. Nichtmetallchemie. VCH: Weinheim, 1995. 41. (a) Banister, A. J. Nature Phys. Sci. 1972, 237, 92; (b) Woollins, J. D. Non-Metal Rings, Cages and Clusters. Wdey: Chichester, New York, 1988, p. 96; (c) Gleiter, R. Angew. Chem. 1981, 93, 442; Angew. Chem. Int. Ed. Engl. 1981, 20, 444.

5e-N and Te-N Compounds

311

42. (a) Campana, C. E; Yip-Kwai Lo, E; Dahl, L. E Inorg. Chem. 1979, 18, 3060; b) Klap6tke, T.; Passmore, J. Acc. Chem. Res. 1989, 22, 234; (c) Burford, N.; Passmore, J.; Sanders, J. C. P. In From Atoms to Polymers; Liebman, J. E; Greenberg, A., Ed. VCH: New York, 1989, p. 53. 43. (a) Banister, A. J.; Clarke, H. G.; Rayment, I.; Shearer, H. M. M. Inorg. Nucl. Chem. Lett. 1974, 10, 647; (b) Gillespie, R. J.; Kent, J. P.; Sawyer, J. E Inorg. Chem. 1981, 20, 3784. 44. Iwasaki, E; Morimoto, M.; Yasui, M.; Akaishi, R.; Fujihara, H.; Furukawa, N. Acta Cryst., Sect. C 1991, 47, 1463. 45. (a) B~irnighausen, H.; Volkmann, T. v.; Jander, J. Angew. Chem. 1965, 77, 96; Angew. Chem. Int. Ed. Engl. 1965, 4, 72; (b) B~'nighausen, H.; Volkmann, T. v.; Jander, J.Acta Cryst. 1961,14, 1079; (c) B~'nighausen, H.; Volkmann, T. v.; Jander, J. Acta Cryst. 1962, 15, 615. 46. (a) Sharpless, K. B.; Hori, T.; Truesdale, L. K.; Dietrich, C. O. J. Am. Chem. Soc. 1976, 98, 269; (b) Fockenberg, E; Haas, A. Z. Naturforsch., Part B 1986, 41,413; (c) Herberhold, M.; Jellen, M. Z. Naturforsch., Part B 1986, 41, 144. 47. Chivers, T.; Gao, X.; Parvez, M. J. Chem. Soc., Chem. Commun. 1994, 2149. 48. (a) Chivers, T.; Gao, X.; Parvez, M. J. Am. Chem. Soc. 1995,117, 2359; (b) Haas, A.; Kasprowski, J.; Pryka, M. Chem. Ber. 1992, 125, 789. 49. Christian, B. H.; Collins, M. J.; Gillespie, R. J.; Sawyer, J. E Inorg. Chem. 1986, 25, 777. 50. Klap6tke, T. M.; Schulz. A. Quantenmechanische Methoden in der Hauptgruppenchemie. Spektrum, Heidelberg, 1996. 51. (a) Reed, A.; Schleyer, P. v. R. J. Am. Chem. Soc. 1987, 109, 7362; (b) Reed, A.; Schleyer, P. v. R. Inorg. Chem. 1988, 27, 3969. 52. Hargittai, M.; Hargittai, I. Phys. Chem. Miner. 1987, 14, 413, and references therein. 53. Harcourt, R. D. In Lecture Notes in Chemistry, Berthier, G.; Dewar, M. J. S.; Fischer, H.; Fukui, K.; Hall, G. G.; Hartmann, H.; Jaff6, H. H.; Jortner, J.; Kutzelnigg, W.; Ruedenberg, K.; Scrocco, E., Eds. Springer: Berlin, Heidelberg, New York, 1982. 54. Harcourt, R. D. J. Inorg. Nucl. Chem. 1977, 39, 237. 54. Harcourt, R. D.; Htigel, H. M. J. Inorg. Nucl. Chem. 1981, 43, 239.

This Page Intentionally Left Blank

SOME RELATIONSHIPS BETWEEN MOLECULAR STRUCTURE AND THERMOCHEMISTRY

Joel F. Liebman and Suzanne W. Siayden

Io II. III. IV. V. VI. VII. VIII. IX.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Congested Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexes o f Metals with Carbon Monoxide and rt-Systems Strong Hydrogen Bonds in Keto-Enols . . . . . . . . . . . . . . . . . . . . . Lewis Acid/Lewis Base Complexes . . . . . . . . . . . . . . . . . . . . . . . Hypervalent Species: Sulfur Fluorides . . . . . . . . . . . . . . . . . . . . . Rules and Regularities: Hydrocarbons and Their Derivatives Spiro Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polyenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 3, pages 313-337 Copyright 9 1997 by JAI Press Inc. All fights of reproduction in any form reserved. ISBN: 0-7623-0208-9

313

.........

.........

314 314 314 318 321 324 326 328 331 333 333 333

314

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

ABSTRACT Relationships of molecular structure and thermochemistry are exemplified by species explicitly chosen to relate to the earlier Volume 2 in this book series. In particular, we discuss congested hydrocarbons, metal complexes, hydrogen-bonded keto-enols, Lewis acid/Lewis base complexes, sulfur fluorides, regularities in organic homologous series, spiro compounds, and polyenes.

I.

INTRODUCTION

Molecular structure and thermochemistry are inextricably connected. Especially if the more general term "energetics" is employed, the structure/energy relationship is seen to provide much of the conceptual and pedagogical basis of chemistry. This interdependence also motivated the publication of this series of books in which the current chapter appears, as well as the journal, Structural Chemistry, the book series "Molecular Structure and Energetics", and "SEARCH" (Structure, Energetics, and Reactivity in Chemistry) that the editors of this series and one of this chapter's authors have participated in. Accordingly, given these long-standing and powerful interests, the demand to write a chapter of reasonablemi.e, readable and writeablemlength required including only a small number of possible topics. But what topics to choose, and why? Rather than reflecting our own current research interests or reviewing the spectacular chemical findings of the last few years, we decided to write a collection of short vignettes, each of which is based on an earlier chapter in Volume 2 of this book series [1]. The intent is to demonstrate how thermochemical data, principles, and procedures can be applied to arbitrarily selected examples already of interest to readers of this series. As such, our chapter gives an abbreviated, not exhaustive, sense of the thought processes involved in exploring a thermochemical point of view. Although thermochemistry pertains to chemical and physical energy changes such as enthalpies of formation, reaction, and vaporization/fusion/sublimation; heat capacities; and entropies, we will generally restrict our consideration to enthalpies of formation [2], enthalpies of reaction and of phase change [3], and of bond dissociation. In addition, our preference and prejudice is for phenomena in the gas phase. This decision results in the molecule per se being considered without complications from lattice or solvation effects. We note that this is also the implicit phase for results from the majority of computational theory, and so findings from experiment, theory, and concept are more easily and directly compared.

Ii.

CONGESTED MOLECULES

We thus commence with Osawa's chapter [4] dealing with congested molecules. As discussed in this review, the development of conformational analysis has been

Molecular Structure and Thermochemistry

315

dominated by the concept of short-range 1,4 and 1,5 interactions. Although most calculations of conformational energies are not determined directly from thermochemical measurements, conformational equilibria studies make use of derived AH, AS, and AG values. Writing formal reactions and obtaining enthalpies of formation from our preferred compendium of values [2], we demonstrate some basic thermochemical calculations and call attention to a few questionable enthalpy of formation values in the literature. Figure 1 shows isomerization reactions (1-7) of dialkylcyclohexanes from the less stable a,e diastereomer to the more stable e,e diastereomer and from the 1,2to the 1,3- and 1,4- structural isomers. Table 1 lists the resulting enthalpies of isomerization for R, R' = CH3, CH 3, and CH 3, CH3CH 2. These are the only alkyl groups for which there are enough experimental enthalpy of formation data; even so we lack enthalpies of formation for the gaseous ethylmethylcyclohexanes and for trans-1-ethyl-3-methylcyclohexane altogether [5]. We wish to calculate the enthalpy associated with the gauche interactions, G(*, X, Y'), where * defines the phase and X and Y are the two sterically interacting groups. In the present cases, the interactions include those between a substituent alkyl group and the ring methylene group (axial groups have two such interactions and equatorial groups have none) and between the substituent alkyl groups themselves (in the 1,2-substituted cyclohexanes). The enthalpy of formation, an indicator of molecular stability, is comparatively less negative (more positive) for the isomeric compound which exhibits greater steric repulsion or strain. Thus, the enthalpy of an isomerization reaction which relieves the gauche steric interaction is exothermic. The enthalpies of isomerization for the four reactions in the gas phase (4-7) which relieve the two methyl groups' gauche interaction are thermochemically indistinguishable and so we assign G(g, CH3, CH3) = 4.55 + 0.13 kJ mo1-1. The same interaction in gaseous n-butane [6] is ca. 3.7 kJ mo1-1. The G(1, CH3,CH 3)

, ~

]_..~~~R

3T

~

~7

5

,••'•R

4 ~~.

'T

~ R

R'

Figure 1.

R

21 6

316

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

Table 1. Enthalpies of Isomerization of Disubstituted Cyclohexanes a R, R' = C H 3 , C H 3

Notes:

R, R' = C H 3, C H 3 C H 2

Reaction

l~isom , Gas b

l~nisom , L i q u i d b

l~lisom , L i q u i d c

1

-7.8

-6.4

-4.0

2

-8.1

-7.2

3

-7.9

-6.8

-7.5

4

4.7

-4.7

-6.9

5

-4.6

-4.2

-6.2

6

-4.4

-3.9

7

-4.5

-3.8

-2.7

aln kJ mol -I. bUncertainties are in the range 2.5-2.6 kJ mol -I. CUncertainties are in the range 1.3-1.5 kJ mol -j.

values are slightly smaller for these same reactions in the liquid phase, most noticeably for the isomerization of the a,e cis-l,2-dimethylcyclohexane. This observation is consistent with the diminution of the gauche effect in liquid n-butane [6] to 2.3 kJ mo1-1. That both of these G values are larger than for n-butane, and thus indicate a more severe repulsion, is attributable to the difficulty of the relatively rigid cyclohexane ring in relieving the interaction by conformational deformations. The three reactions (1-3) which relieve the gauche interactions between the exocyclic axial methyl group and the endocyclic methylene groups are again nearly identical in the gaseous phase, G(g, CH 3, CH2) = 4.0 + 2.0 kJ mo1-1 per gauche interaction, and show some decrease in the liquid phase. The G(1, CH 3, CH 2) is slightly smaller, and thus less repulsive, than G(1, CH 3, CH3). Turning our attention to the enthalpies of isomerization for the ethyl-, methylsubstituted cyclohexanes, we notice apparent inconsistencies. While some irregularities are expected for the liquid phase based on the dimethylcyclohexanes precedent, the enthalpy values for reactions 1 and 7, both of which involve a,e cis-l-ethyl-2-methylcyclohexane, seem too small by about the same amount. A lower interaction enthalpy implies either that the gauche interaction between the axial and equatorial substituent groups [7] or between the axial alkyl group and the ring methylenes is less repulsive than in the correspondingly substituted dimethyl compound, or that some stabilizing attraction reduces the gauche interaction when an ethyl replaces a methyl group [8]. Why the gauche interaction should be so different for an axial/equatorial substituent pair (7) and an equatorial/equatorial substituent pair (4, 5) is unclear [9]. The difference in entropy between cis- and trans-l,2-dimethylcyclohexane was suggested [10] to be the result of lesser steric interference in the trans-isomer in which the chair distorts to move the methyl groups away from each other. If this steric effect also affects the enthalpy difference

Molecular Structure and Thermochemistry

8a

317

8b

and becomes more severe upon replacing a methyl with an ethyl group, the result would be to increase the G(1, CH3CH2, CH3) value relative to G(1, CH3, CH3) for reaction 1, contrary to the actual results. Apart from the reactions questioned above, the G(1, X, CH3) and G(1, X, CH2) values for X = CH3CH 2 are larger than for X = CH3, the former value appreciably so [11]. From a thermochemical perspective, we might suspect the reliability of the measured enthalpy of formation of cis-1-ethyl2-methylcyclohexane [12]. Discussion concerning the 1,2-dialkyl substituted cyclohexanes leads us next to consider the fused bicyclic decalins in which exocylic methyl groups are replaced by ring methylene groups. There are three gauche interactions in cis-decalin (Sa) which are absent in trans-decalin (Sb). The cis- to trans-isomerization, calculated from enthalpy of formation data, is exothermic by -12.9 kJ mo1-1 in the gas phase and by -11.2 kJ mo1-1 in the liquid state. G(g, CH2, CH2) is thus 4.3 kJ mo1-1 and G(1, CH2, CH2) is 3.7 kJ mol -l. That the values are intermediate between the corresponding values for G(*, CH 3, CH3) and G(*, CH 3, CH2) may result from a combination of decreasing steric size (from CH 3 to CH2) and increasing conformational rigidity of the bicyclic ring system. Substituting a methyl group for one of the ring junction hydrogens produces 9-methyl-cis- and -trans-decalin, for which there are only liquid-phase enthalpy of combustion data [13]. The exothermic cisto-trans-isomerization enthalpy of-5.9 + 2.7 kJ mo1-1 represents a net loss of one gauche interaction (methyl substitution increases the gauche interactions in the trans-isomer to four and in the cis-isomer to five). This relatively large G(1, CH3, CH2) value is unexpected. The authors of the combustion study note that each of the compounds was contaminated with --2% of its isomer, but no mention is made of any correction made to account for the impurity. An isomerization enthalpy of -2.3 + 1.2 kJ mo1-1 was determined by temperature-dependent equilibration [14]. These authors admit they were unable to completely separate the equilibrated isomers nor was a wide temperature range achieved and thus the accuracy of the study was compromised. Taking into account the different magnitudes of the uncertainties associated with the above isomerization enthalpy measurements, the weighted mean of these two determinations is -2.9 + 1.1 kJ mo1-1, closer to the value calculated above for G(1, CH3, CH2) of 3.4 kJ mol -l. A reexamination of this isomerization enthalpy, either directly or indirectly, is desirable in that the 9methyldecalin substructure, as found in cholesterol and other steroids, is widely distributed in nature. Osawa also discusses tert-butylated methanes and ethanes. Before introducing the large tert-butyl groups, let us consider the thermochemical effects of introducing

318

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

smaller alkyl groups in the series of hydrocarbons defined by CH 4, RCH 3, R2CH 2, R3CH, and R4C. The differences in the gas-phase enthalpies of formation [2] for the smallest alkyl group, R = CH 3, i.e. the enthalpies of sequential methylation, are -9.4,-20.9,-28.5, and-33.9 kJ mol -~. Thus, the sequentially methylated methanes are increasingly stabilized, and methylation of methane encourages further methylation. For the next and larger alkyl group, R = C2H 5, the enthalpies of sequential ethylation are -30.3, -42.2, -42.7, and -42.7 IO mo1-1. After the first step, within experimental error, ethylation neither encourages nor discourages further ethylation. As the simplest electronic reasoning (i.e. equating the substituent inductive effects of methyl and ethyl groups) would have suggested increased exothermicity for ethylation as for methylation, it appears that there are some steric effects, genetically recognized as strain, that do not increasingly favor more highly ethylated species. There being a lack of data for R = CH(CH3)2, now we are ready to consider the consequences of sequential tert-butylation. We find for R = tert-butyl, enthalpies of sequential tert-butylation [15] of-93.7, -73.5, and +58 kJ mol -~. It appears that tert-butylation discourages further tert-butylation to the point that the third tert-butylation is strongly endothermic. The tri-tert-butylmethane is unequivocally strained. We eagerly await the synthesis and thermochemical characterization of tetra-tert-butylmethane to see how strained it really is.

Iii. COMPLEXES OF METALS WITH CARBON M O N O X I D E AND x-SYSTEMS Turning now to Braga and Grepioni's chapter [16], we briefly consider intermolecular interactions of compounds on the organic/organometallic/inorganic boundaries. We limit our attention solely to enthalpies of sublimation of some metal carbonyls, and bis(hydrocarbon)metals containing cyclopentadienyl and benzene ligands. In Table 2 are listed the enthalpies of sublimation [17] for: the isovalent, isoelectronic hexacarbonyls of chromium, molybdenum, and tungsten; the trimetallic dodecacarbonyls of iron, ruthenium, and osmium; the tetrametallic dodecacarbonyls of cobalt, rhodium, and iridium; and the bimetallic decacarbonyls of manganese and rhenium. All of the above metal carbonyls obey the transition metal counterpart to the octet rule for main group elements--the 18-electron rule that asserts the structure and stoichiometry of metal complexes is generally consonant with 18 valence electrons associated with the metal [18]. We also include the sublimation enthalpies for the likewise rule-obeying, but otherwise stoichiometrically and structurally disparate, Fe(CO)5, Co2(CO) 8, Fe2(CO)9, and R h 6 ( C O ) I 6. Ignoring doubts [3] regarding regularities in sublimation enthalpies, we wonder if there is a simple rule of the type: ~subMx(CO)y

=

ax + by + c

(9a)

120

110

100 O

C 0

=

90

80

E

9

o,-

L~

~

70

0 0. .D

.c

60

Transition metal series:

C nl

9Row 1

50

9Row 2

9Row 3

40

30

t

I

t

t

O

I

6

8

10

12

14

16

Number of (CO) Groups

Figure 2. Enthalpies of sublimation of metal carbonyls.

320

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

l'able 2. Enthalpies of Sublimation a of Metal Carbonyl Compounds and Constants

from the Linear Regression Analysis b of Equation 9b

Mx(CO)y

M

x

y

AHsub

Row 1

Fe

1

5

40.2 + 0.8

Cr c

1

6

72.0 + 4.2

Co

2

8

65.2 + 3.3 75.3 _+ 21.0

Fe

2

9

Mn

2

10

80.3 + 4.2

Fe

3

12

96.0 + 21.0

Co

4

12

96.2 + 4.2

AHsubMx(CO)y + 1.39 = 8.06 + 0.23 * y - 0.07 + 2.26

Row 2

Mo

1

6

Ru

3

12

(100.0 + 20.0)

73.8 + 1

Rh

4

12

(100.0 + 20.0)

Rh

6

16

117.2 + 20.0

AHsubMx(CO)y + 0.11 = 4.35 + 0.02 9 y + 47.69 + 0.19

Row 3

W

1

6

76.4 + 1.3

Re

2

10

93.3 + 3.2

Os

3

12

104.6 + 20.0

Ir

4

12

104.6 + 20.0

AHsubMx(CO)y + 1.16 = 4.28 + 0.24 9 y + 50.73 + 2.45

Notes:

aln kJ mol-I; estimated enthalpies of sublimation are shown in parentheses. bin the least-squares analyses of Eq. 9b, the individual enthalpies were weighted inversely as the squares of the uncertainty intervals. In all cases, r 2 > 0.9999. The standard errors in the regression equations were generated from the unweighted enthalpies. CThis compound was not included in a regression analysis.

Upon further scrutinizing the data, it is apparent that all the enthalpies of sublimation cannot be accommodated by a single linear equation but instead they must be categorized in some way, perhaps either by the column or row position of the metal in the periodic table. Multiple linear regression analysis shows, with one exception, that the enthalpies of sublimation of metal carbonyls in each transition series row are fitted by Eq. 9a (correlation coefficient > 0.996). The exception is Cr(CO)6 which is an obvious outlier from the linearity apparently established by the other six carbonyl compounds containing metal atoms in the same row of the periodic table (see Figure 2). The a constants in Eq. 9a are very small, or negative, compared with the b constants, and thus we can simplify Eq. 9a and write Eq. 9b: AHsubMx(CO)y = b'y + c'

(9b)

A least-squares analysis of the enthalpies of sublimation of compounds containing metal atoms in the same row generates the regression constants in Table 2. Although

Molecular Structure and Thermochemistry

321

the enthalpy of vaporization of the volatile Ni(CO)4 is known [17], its enthalpy of fusion and/or sublimation is not. Extrapolation using the constants in Table 2 predicts its enthalpy of sublimation to be 32.2 ++ 1.4 kJ mol -~. However, if the measured enthalpy of sublimation of Cr(CO)6 is accurate, it may be that regularity of the metal carbonyl sublimation enthalpies is associated not with the row but with the column occupied by the metal atom. Considering the hexacarbonyls, we observe about a 2.2 kJ mol -~ increase in sublimation enthalpy with increasing atomic number for the chromium family; considering the dodecacarbonyls, we observe about a 4.2 kJ mol -l increase with increasing atomic number for the iron and cobalt families. (The enthalpies of sublimation for the carbonyls of both Ru and Rh were estimated as interpolated values in their respective families. These estimated values are consistent with both row- and column-associated regularity.) We thus expect the decacarbonyls to exhibit about a 3 kJ mol -~ increase within a family. However, the increase in the sublimation enthalpy for the decacarbonyls of the two members of the manganese family spans 13 kJ mol -~ , or about a 6.5 kJ mo1-1 increase with increasing atomic number. Until more data are available, we are unable to satisfactorily correlate metal carbonyl structure with enthalpies of sublimation. Cyclopentadienyl ligands are much more accommodating to violations of the above 18-electron rule than are CO ligands. As such, we consider all of the bis(cyclopentadienyl)metals as belonging to one category. For the first row transition metals--vanadium, chromium, manganese, iron, cobalt, and nickel--the enthalpies of sublimation are 58.6, 62.8, 75.7, 73.6, 70.3, and 72.4 kJ mo1-1" the relationship between the metal and enthalpy of sublimation is puzzling [19]. Benzene ligands generally return us to the more precise electron-counting demands of the 18-electron rule. To a first approximation, the enthalpy of sublimation of a bis(benzene)metal should be ca. 12.2 kJ mo1-1 higher than its bis(cyclopentadienyl) counterpart, the difference being estimated as twice the contribution to enthalpies of sublimation by two aromatic carbons [20]. The only metals for which a comparison can now be made are vanadium and chromium with differences of 70.0 - 58.6 = 11.4 and 78.2 - 62.8 = 15.4 kJ mo1-1, respectively. The agreement is quite good but we hesitate to generalize from our two examples [21].

IV. STRONG HYDROGEN BONDS IN KETO-ENOLS The chapter by Gilli, Ferretti, Bertolasi, and Gilli [22] reviews strong hydrogen bonds as found in ions and neutrals, and in particular, the role of resonance and electron delocalization in making these particular hydrogen bonds stronger than one might otherwise have deduced by analogy to those found in other more localized compounds. Keto-enols (unsaturated hydroxyketones) figure prominently in these authors' discussion. As befits our earlier enunciated prejudices and preferences, we will discuss results in the gas phase. An archetypical example is the enol form of acetylacetone(9), known from experiment to be some 10.0 kJ mo1-1

322

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

0

a~,

"0

II

0

I

0

II C

!1

C H3C/ \CH2/ \CH3

H3C/ C \ CH//C\ CH3 9

10

/CH2CH3 O

O

O

II HO / H

I

II

C C 1-13C/ \CH ~" \CH3

C C H3C/ NCH2/ \CH3

11

12

more stable than the diketo tautomer (10), a value scarcely changed for the condensed phase [23]. Gilli and co-workers assert the importance of intramolecular hydrogen bonding as a determinant for the relative stability of tautomers. For example, gaseous cyclohexane-1,3-dione, where intramolecular stabilization cannot occur, fails to be enolic, whereas the solid is composed of extensively intermolecularly hydrogen-bonded aggregated enols. We now ask two simple questions: what is the stabilization due to conjugative interaction in acetyl acetone enol compared to that in acetylacetone O-ethyl ether (11); and what is the stabilization due to hydrogen bonding in acetyl acetone enol compared to that in the saturated 4-keto-2-pentanol (12)? In principle, the energetics of the following isodesmic reaction should prove useful:

O ~CH\ H3C/ \CH CH3

H~O +

/CH2~ j]C\ CH3 CH CH3

0 H~O H3c/C\cH~ \ C H 3 9

cH/CH2~cH//CH\cH3 15

However, we know of no reliable experimentally measured enthalpy of formation for either 3-penten-2-one (13) or 2-penten-2-ol (14), nor for most of the other enones and enols that would be the natural counterparts of most of the keto-enols that are of relevance to us now. An alternative is to remove both the ketone and

Molecular Structure and Thermochemistry o

323

H\

0 16

17

hydroxy groups concurrently and replace them by H2 and H, respectively [24]. Thus we consider the enthalpy of formation difference between acetylacetone enol and (E)-2-pentene (15). This difference is numerically equal to -348.7 kJ mo1-1, a value which when considered alone without counterpoint and comparison is seemingly without conceptual meaning. What about other keto-enols? Thermochemically relevant data are sparse. The enthalpies of formation of the related tropolone (16) and tropilidene (17) differ by -336.3 kJ mol -l. Per ketone, 3,4-dihydroxycyclobutene- 1,2-dione and cyclobutene differ by ca. -320 kJ mol-~--this is a difficult number to appraise if for no other reason than there are two "keto" and "oi" groups and but one "en" group that are interacting. The logic of Gilli et al. is consonant with stabilization decreasing in the order acetylacetone, tropolone, and the dihydroxy-cyclobutenedione with their increasingly nonlinear hydrogen bonds. Recognizing that the keto-enols are vinylogous carboxylic acids, we ask now about carboxylic acids themselves. The corresponding difference is naturally that between RCOOH and its counterpart RCH3, for which the derived values for R = CH 3 and C6H5 are -349.0 and -344.5 kJ mo1-1, respectively [25]. These values are not that different from each other or the other hydroxyketones discussed above. Yet we are bothered somewhat. While resonance stabilization is well-established for carboxylic acids, intramolecular hydrogen bonding is not expected to be particularly important for these compounds: it appears quite unreasonable that carboxylic acids and the enols of 13-diketones should have nearly identical intramolecular hydrogen-bonding derived stabilization. It is then remembered that gas-phase monocarboxylic acids are dimeric and, correcting for the ca. 30 kJ mol -~ hydrogen-bond strength (per RCOOH molecule) in the dimer, the above COOH/CH 3 differences are reduced to ca. 315 kJ mol -l. These new results are more plausible. There were many other interesting compounds in this chapter. However, we are thwarted from making thermochemical comparisons because we lack enthalpies of formation of most of these species and also of their related, although more "normal" species. For example, enaminoketones are of relevance here and there are enthalpies of formation for some key compounds in the recent calorimetric literature. However, the necessary data for the simple enamines are absent. We have the enthalpies of formation for solid-phase 1,5-diphenyl- (18a) and 1,3,5-triphenylformazan (18b) of 457.9 and 543.1 kJ mo1-1, respectively. The ca. 85 kJ mo1-1 difference for solid-phase C-phenylation is not reflected in the ca. 54 kJ mol -l difference for the

324

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

Ph~N HNN/Ph II I N..~N

/ N

N

II

H

I

N.,,..pN

R 19 lga R = H lgb R=Ph

9

Ph-HC---CH~Ph

(20)

10

(2D 5-phenylation of both 1H-tetrazole (19) and its 1-phenyl derivative, or in the ca. 60 kJ mol -I difference per phenyl as found in benzene [26], biphenyl, and 1,3,5triphenylbenzene. Does this mean that at least one of the formazan enthalpies of formation is in error? Even if we would wish to ignore the intramolecular N-H-..N hydrogen bond that is intrinsic to formazans, we lack the enthalpy of formation of suitable azo compounds, hydrazones, and amidines with which to make the desired thermochemical estimates. Alternatively, it may reflect the relative absence of aromaticity in the pseudocyclic formazan. After all, solid phase mono (or)- and diphenylation (ot,(x') of stilbene (20) and 9,10-diphenylation of anthracene (21) is accompanied by ca. 80 kJ mol -l per phenyl changes in enthalpies of formation.

V. LEWIS ACID/LEWIS BASE COMPLEXES We now turn to Leopold's chapter [27] on "partially bonded molecules", most generally Lewis acid/Lewis base complexes that show major geometry changes between the gaseous and condensed phase. While numerous interesting molecules were discussed therein, we limit our attention solely to species containing trivalent boron, BX 3, as the Lewis acid [28]. The reader may recall the general order of Lewis acidity: B(CH3) 3 < BF 3 < BCI 3 < BBr3 and the textbook explanation in terms of the conflict of o-withdrawal and rr-backbonding by the groups on the boron [29]. A simple probe of this order is the enthalpy of solvation of the various BX 3 species in some electron-donating solvent. That is, we consider the enthalpy of solvation of the formal reaction, L (1) + BX 3 (g) ---) L 9 BX 3 (soln) instead of the enthalpy of solution of the formal reaction,

(10)

Molecular Structure and Thermochemistry L (1) + BX 3 (std state) ~ L 9 BX 3 (soln)

325

(11)

in order that the complexation by the Lewis base solvent does not have to overcome the intermolecular association energy of a condensed-phase BX3 Lewis acid in its standard state. For L = nitrobenzene, reaction 10 is found to be exothermic by 25.5, 36.2, 59.8, and 86.4 kJ mol -l for X = CH 3, F, CI, and Br, respectively, in accord with the order above. Substituent effects of substituted nitrobenzenes on the enthalpies of solvation would prove interesting; however the data are lacking and so we consider other Lewis bases for which relevant data are available. Consider the dialkyl sulfides in reaction 12: R2S (1) + BH 3 (g) --~ R2S 9 BH 3 (1)

(12)

For R = CH 3 and C2H5, the reaction is exothermic by ca. 122 and 152 kJ mo1-1, respectively. Since diethylsulfide is thus found to be a better Lewis base than dimethylsulfide, we deduce that the steric effects are probably small and the larger, more electron-donating ethyl group can better disperse the newly formed formal charge. Let us now consider the enthalpies of vaporization of the dialkylsulfide borane complexes formed in reaction 12. For R = CH 3 and C2H5, the experimental values, 42.2 and 43.1 El mol -l, respectively, are essentially independent of the alkyl group. This result is very surprising. Consider the highly polar dialkyl sulfoxides, R2S--O, which are isoelectronic with the sulfide boranes. The difference between the two vaporization enthalpies [30] for diethyl and dimethylsulfoxide, 62.4 and 52.9 kJ tool -l, respectively, is 9.5 kJ tool -l. The alkanes which are approximately isosteric with the sulfide borane complexes, (CH3)2CH-CH 3 and (CH3CH2)2CH-CH3, have a vaporization enthalpy difference of 30.3 - 19.3 = 11.0 kJ mo1-1. And the essentially nonpolar sulfides in their uncomplexed form have a vaporization enthalpy difference of 35.8 - 27.9 = 7.9 kJ mol -l. Indeed, all these values are close to the 9.4 kJ mol -~ predicted for any pair of compounds that have a common functional group and differ by a total of two carbons [31], whether it be C2HsCI and C4H9C1 or (n-C3HT)20 and (n-CaH9)20 for which the differences are 8.8 and 8.6 kJ mol -l, respectively. A related surprise is that the enthalpies of sublimation of the highly symmetric (CH3)3NeB(CH3)3, H3NeB(CH3) 3, and (CH3)3NeBH3 are very similar: 58.6, 57.7, and 57.6 kJ mol -l. In contrast, the enthalpies of sublimation of the alkanes which are isoelectronic and isosteric to these amine boranes formed by replacement of the BN linkage by a C-C bond are: 43.4, 33.2, and (of course, identical to the second) 33.2 kJ mo1-1. As observed for the sulfide boranes, the near constancy of phasechange enthalpies do not parallel the enthalpies of complexation. For these amineboranes, the relevant gas-phase complexation enthalpies are -71.5, -56.4, and -160.9 kJ mol -l. We expected the last species to have the most exothermic

326

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

complexation because intuitivelywshould we ignore steric effects--the Lewis acidity of BH3 is expected to be greater than B(CH3)3 and the Lewis basicity of (CHa)aN to be greater than NH 3.

VI.

HYPERVALENT SPECIES: SULFUR FLUORIDES

We now tum to Landis's chapter [32] on molecular shapes as understood using valence bond concepts and molecular mechanics. Let us consider just one class of molecules discussed therein: hypervalent main group compounds [33]. Although in this class of compounds geometry and energy are interrelated, they are unequivocally different aspects of a molecule. Admittedly, we will not attempt to contrast Landis' predictions with extant literature results: we lack a copy of his computer code and the literature is surprisingly sparse. Taking results solely from a highquality quantum chemical study in the recent literature on sulfur fluorides [34], we find that the recommended bond dissociation energies for the series of S F , - F vary with n: n = 5,444 + 6; n = 4, 159 + 7; n = 3,398 + 7; n = 2, 227 + 6; n = 1,374 + 6; and n = 0, 345 + 6 kJ mo1-1. For the related cations, the recommended bond dissociation energies for the series of SF~ - F also vary with n: n = 4, 371 + 6; n = 3, 56 + 6; n = 2, 400 + 6; n = 1,372 + 6; and n = 0, 367 + 8 kJ mol -~. There is a rough "flipflopping" with odd and even n: this corresponds to relatively stable species having an even number of electrons for both neutrals and cations alike. We immediately note that hypervalence unto itself does not dictate strong or weak bonds. Furthermore, in that bond energies are directly relatable to atomization energies and thereby to enthalpies of formation, it is seen that hypervalence unto itself does not dictate low or high thermochemical stability. More precisely, in the simplest picture, S has two unpaired valence electrons among its total of six, and awaits two more electrons to fill its octet to form a classically stable species. Assuming the S - F bonds in SF and in SF 2 are classical two-center/two-electron bonds, simple bond additivity logic suggests that these two bonds would be expected to have nearly equal dissociation energies, or equivalently to be equally strong. In fact, they are found to be nearly equal. It may be argued that bond order in SF is closer to 1.5 than 1 because this diatomic molecule is valence isoelectronic to both $2 and F ~ . So, why isn't the SF bond in the diatomic molecule stronger than in the triatomic SF2 .9 One molecular orbital derived answer is that this supplemental bond order or stabilization is maximized when the two atoms are the same [35]. It may be additionally argued that the SF bond in SF 2 is strengthened via the anomeric effect, i.e. two double bond/no bond resonance structures of the type FS § F- accompany the classical, covalent structure F - S - F and increasing fluorination provides increasing stabilization such as found in increasingly fluorinated methanes. Both of these arguments may well be valid: experience with the energetics of classical S - F bonds is almost nonexistent and it would be interesting as well as informative if the above two effects were almost equal as well.

Molecular Structure and Thermochemistry

327

This logic suggests that ionization of SF would result in a stronger bond because its ion would have bond order equaling 2. We might also expect ionization of SF2 to result in a weaker bond because FS+F lacks the ionic stabilization of the neutral. Yet, the S - F bond lengths in SF + and SF~ are not that different from each other or from those of the neutral fluorides from whence they are made by electron loss. SF 3 may be understood in terms of its formal synthesis from a fluorine atom approaching a closed shell, octet-obeying SF2 molecule. We would deduce that there are two normal S-F bonds and one S-F bond that is considerably weaker. The first two are two-center/two-electron interactions, while the latter is a two-center/three-electron interaction. This latter type of bond can be recast in terms of a resonance hybrid of >S: F. and >S+F. Indeed, the SF2-F bond in SF 3 is considerably weaker than the SF-F bond in SF2 itself. We would also have predicted that on ionization a "regular" two-center/two-electron bond is formed and so SF~ would have three equivalent, and quite normally strong two-center/two-electron S - F bonds. Equivalence of the bonds is in fact found for SF~. While changes in bond lengths and bond strengths need not parallel, encouragingly all three two-center/two-electron bonds in the above neutral sulfur fluorides have nearly the same length, and those in the derived ions nearly a constant ca. 0.07/~ shorter. By contrast, the unique two-center/three-electron bond in neutral SF3 is some 0.07 ]k longer than any other S - F bond found in these neutrals. SF4 is expected to have two types of S-F bonds: two which are two-center/twoelectron and relatively short, and the other a relatively long three-center/four-electron bond. The calculations confirm this with a 0.08/~ difference between the two types of bonds. That there are now two equivalent and ionic resonance structures that compose the hypervalent F - S - F linkage, F_SF~2F- ~ F-+SF2_F

(13)

makes it surprising that the bond distance difference between the two types of S-F bonds is so large, but nonetheless suggests a strong S-F bond in SF4 which indeed is corroborated by the calculations. Ionization or loss of an electron would be expected to seriously weaken the S-F bond. After all, electrostatic attraction is largely lost and the sticking of F- to SF~3 is expected to be less than that to SF2. Accordingly, a very weak S ~ . F complex is expected. The energy prediction is confirmed: the weakest S - F bond for the entire study is found in SF~. But with what geometry? A species with three significantly different types of S-F bonds, Fa-S(-Ff0~--- F~r, is not unreasonable, nor is one with an essentially undistorted SF~ ion cojoined by a dangling, appended E Neither is found. Instead, an essentially normal structure is found, which like the neutral has two types of S - F bonds. These bonds in the ions are "normally" shortened by comparable amounts to the above, 0.09 and 0.06/~. Why are our predicted geometries incorrect and how high in energy are these alternative structures?

328

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

SF4 would not be expected to stick particularly well to F-. Owing to the electron-withdrawing power of the four F's already bonded on the sulfur, the SFn-F bond energy is expected to be less than that of the already weak SF2-F bond. Indeed, this is found with the structure in which a sulfur with four equivalent fluorines is also bonded to another fluorine some 0.04/~ further away than the others. By contrast, SF~4 is expected to strongly bond to F. and indeed, the new S - F bond in SF~5 has been shown to be of comparable strength to that of the nonhypervalent two-center/two-electron bonds in SF +, S ~ , and SF~3. This result is compatible with SF~4 being described in terms of neutral SF4 with its two two-center/two-electron S - F bonds and one three-center/four-electron F - S - F unit, and a rather localized (mostly sulfur) orbital. In the neutral, this last orbital is doubly occupied and is recognized as a lone pair from which there is loss of an electron upon ionization. On formation of SF~5, the new S-F bond is of the two-center/two-electron type and so is really rather strong. Interestingly, the two types of S - F bonds in SF~5 are short and differ in length by less than 0.03/~. We close this discussion with the observation that the totally hypervalent SF 6, i.e. lacking all two-center/two-electron S - F bonds, has strong S-F bonds. Indeed, it has the strongest S-F bonds of any of the species in the current study. It is now acknowledged that experimentally derived data has been particularly difficult to obtain and unreliable for all of the neutral and cationic sulfur-fluorine species in this discussion, and the well-known, highly stable SF6 has been no exception or exemption from this general behavior. This is troublesome in that SF 6 is undoubtedly among the best characterized hypervalent species, and a quick examination of Chemical Abstracts or of any inorganic chemistry textbook shows it is certainly better chronicled than its more "normal" two-center/two-electron bonded counterpart, SF2.

VII. RULES A N D REGULARITIES: HYDROCARBONS A N D THEIR DERIVATIVES We now turn to Mastryukov and Simonsen's chapter [36] on empirical correlations in which are collected rules and regularities by which numerous data may be correlated and numerous phenomena unfolded. Let us present some empirical correlations that we have found useful in a thermochemical context. Regularities in the thermochemical properties of a variety of homologous series have often been demonstrated by their linear dependence on the number of carbon atoms in the hydrocarbyl substituent. Thus, Eq. 14 expresses the standard molar enthalpies of formation of a given homologous series as a linear function of the number of carbon atoms, no, in the compound [37]. Similarly, the enthalpies of vaporization for each series are also linear functions of the number of carbon atoms [38]. A least-squares analysis of the measured enthalpy data for a small selection of organic homologous series produces the numerical values appearing in Table 3.

329

Molecular Structure and Thermochemistry

Table 3. Constants from the Linear Regression Analysis a of Equation 14 (g) for

Several Homologous Seriesb

Homologous Series

~ (g)

C .~c

f3 (g)

Standard Error in

artS(g)

n-RH

C4-C12, C16, C18

-20.63 _+0.05

-43.20 _+0.53

0.71

n-RCH--CH 2 n-RC~CH

C4-CIo, Ct2, Cn6 C6-C~o

-20.54 _+0.06 -20.03 _+0.39

81.55 _+0.60 242.58 _+3.17

0.69 1.23

n-ROH

C4-CIo, Cl2, Cn6

-20.14 :!: 0.04

-194.73 _+0.40

0.46

n-RCHO

C2-C 4, C 7

-19.45 _+0.06

-127.20 _+0.27

0.23

n-R(CH3)C=O

C4-C6, C 9, Ct2

-20.59 +_0.11

-156.19 + 0.84

0.70

n-RSH

C4-C 7, Cno

-20.46 +_0.14

-7.03 + 0.95

0.65

n-RCI

C 4, C 5, C 8, C12, Cl8

-20.83 _+0.05

-71.34 _+0.59

0.63

n-RBr

C4-C 8, C~2, C~6

-20.23 _+0.08

-26.78 _+0.70

0.79

Notes:

aln the least-squares analyses of Eq. 14, the individual enthalpies were weighted inversely as the squares of the experimental uncertainty intervals. In all cases, r2> 0.998. The standard errors in the regression equations were generated from the unweighted enthalpies. bin kJ mol-I. Cnc is the total number of carbon atoms in the compound.

Anf (l or g) = ct-(nc) + 13

(14)

The regression equations are useful for calculating the enthalpies of interest for experimentally unmeasured members of the series and for evaluating the reliability of suspect measurements. Quite often we have identified enthalpies of formation and of vaporization which are most probably incorrect because they are outliers from otherwise linearly related enthalpies. As yet, there is no demonstrated correlation between the slope or intercept values derived from these equations and any physical properties of the functionalized series. Such equations cannot be used to estimate the enthalpy of formation of the methyl derivative in a homologous series: examination of graphical plots of the enthalpies of formation in the gaseous or liquid phase versus the number of carbon atoms clearly shows that the enthalpy of formation of most methyl-substituted compounds deviate from the otherwise apparently linear relationships. This methyl effect is well-known. Montgomery and Rossini [39] calculated the deviations from the "universal" slope [40] of the experimental enthalpies of formation of the methyl derivatives in various organic functional group homologous series. They then derived an empirical quadratic equation which correlated these methyl deviations with the Pauling electronegativity of the attached element in the functional group. We re-evaluated the relationship between the methyl deviations and electronegativity by incorporating more recently measured enthalpies of formation [41]. The methyl deviation, 5(CH3-Z), was defined as,

330

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

~(CH3-Z ) = A/-/f (CH3Zexp) - z ~ f (CH3Zcal)

(15)

When the 5(CH3-Z ) values were plotted versus the Pauling electronegativity of the element in the functional group, instead of a higher order correlation, two independent linear relationships were apparent: one containing the second-row elements Li, Be, B, C, N, and O; and the other, nearly parallel, containing the third row elements A1, S, and CI. Using other electronegativity scales did not improve the correlation. This example serves to emphasize the necessity of re-examining any empirical correlation when new data become available. Because the enthalpies of formation of each homologous series correlate with the number of carbon atoms according to Eq. 14, the enthalpies of formation of any one series must correlate with the enthalpies of formation of any other series with like nc, as in Eq. 16: ?

(16)

P

AHf(RZn) - O[,R,R, 9AHf(R Z.,) + I~R,R,

This equation has been remarkably useful in correlating and analyzing the abundant data for the alkyl halides in particular [42]. Two series may be related as reactant and product in a formal reaction such as Eq. 17 where, for example, a nucleophilic substitution reaction converts one alkyl halide into another Eq. 17a or hydrogenation of an alkene produces an alkane Eq. 17b.: (17a)

R - X + X ' - --+ R - X ' + X -

\ /

C--C

/ \

+ H2

~

I I

H--C--C--H

I I

(17b)

Because the enthalpy of reaction for any one reactant and product pair may be expressed as in Eq. 18, (where the reagents are halide ion or diatomic hydrogen in the examples above), A/-/rx n = A/"/f [product] - {A/"/f [reactant] + Z ~ f

[reagents] }

(18)

it is tempting to recast it as Eq. 19 in the linear form: A/-/f (product, g) = m . A/-/f (reactant, g) + "A/-/rxn"

(19)

The enthalpies of formation of reagent(s) in the series of reactions are constant and do not appear in Eq. 19. It would seem, therefore, that the enthalpy of reaction is constant throughout the series. However, only in the case where the slopes from Eq. 14, Otproductand ~reactant, are identical (and thus m = 1 in Eq. 19) is the enthalpy of reaction constant and equal to the y-intercept ("AHrxn") of Eq. 19 (and thus to the difference in intercepts ~product and ~reactant from Eq. 14). If the two slopes are

Molecular Structure and Thermochemistry

331

significantly different, i.e. as the lines generated from Eq. 14 become noticeably nonparallel, the enthalpy of reaction is not constant but instead exhibits a regular increase or decrease. We showed this to be true for several different reactions involving double-bonded compounds [43] where we defined, ~A/'/rx n "- A/-/rx n (experimental mean) = "z~I~rxn" (y-intercept from Eq. 19)

(20)

and plotted ~Anrx n v e r s u s the slope, m, from Eq. 19. For hydrogenation reactions of various alkene homologous series, when m was between 0.999 and 1.001 (very close to 1), ~A/-/rx n w a s clustered around _+0.2 kJ mol -l and the experimental Anrx n w e r e fairly constant. For m less than 0.98 or greater than about 1.001, ~A//rx n showed a steep, nearly linear increase and the experimental Anrx n changed monotonically.

VIII. SPIRO COMPOUNDS We now turn to Li's and Owen's chapter [44] on the INADEQUATE NMR method. Our knowledge is "inadequate" to make use of the structural determination methodology per se in this chapter to relate it to our own methodologies. Instead, we single out one species discussed by these authors, "bistramide A"(22) and in particular note its dioxaspiro[5,5]undecane substructure. What is this spiro linkage "worth"? To the best of our knowledge, the thermochemistry of unsubstituted compounds with a single spiro linkage is limited to the hydrocarbons spiropentane, spiro[4,4]nonane, spiro[4,5]decane, spiro[5,5]undecane, and spiro[5,6]undecane. Their respective enthalpies of formation are: 157.7 + 0.8, -143.8 + 1.3, -200.0 + 2.1, -244.5 + 2.8, and -253.0 + 1.3 kJ mol -l for the liquids, and 185.2 + 0.8, N. D. (no data), - 145.1 + 2.1, - 188.3 + 2.8, and N. D. for the gases. Let us compare these spiro compounds with the corresponding uncoupled monocyclic compounds via the formal reaction: (CH2)mC(CH2) n + C(CH3) 4 ~

(CH2)mC(CH3) 2 + ( C H 2 ) n C ( C H 3 ) 2

(21)

Unfortunately there are no enthalpy of formation data for 1,1-dimethylcycloheptane needed for the [5,6] species, and some desired gas phase enthalpies of formation for the remaining spiro and monocyclic species have not been measured. We thus use only the liquid-phase data and find the following reaction exothermicities: 34.1, 10.0, 0.5, and 2.7 kJ mol -l. The near-thermoneutrality of the reactions involving spirodecane and spiroundecane suggest they are "normal" species, i.e. the spiro linkage does not add to the strain. Does it therefore follow that their hetero derivatives are normal as well? Thermochemical data for hetero derivatives of spiro compounds are all but nonexistent. However, the enthalpy of formation of 2,4,8,10-tetraoxaspiro[5,5]undecane (23) has been measured and may be a useful comparison. By analogy with

332

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN

OH

~ 0

0

OH

(22)

its carbocyclic analog in Eq. 21, we consider the formal reaction whereby the monocyclic species formed is 5,5-dimethyl-1,3-dioxane (24): OAO

OAO + C(CH3)4 ~

2

(22)

OvO (23)

(24)

From archival data [45] we find this reaction to be exothermic by ca. 48 and 43 kJ mol -l for the liquid and gaseous species, respectively. This unexpectedly large exothermicity can arise from: (a) the relative instability of the spiro compound; (b) the relative stability of the dimethyldioxane; or (c) experimental error. The experimentally measured enthalpy of formation of 24 is reproduced by assuming the following reaction is thermoneutral: spiro[5,5]undecane + 2 (1,3-dioxane) ~ 24 + 2 (cyclohexane)

(23)

This suggests that interpretation (a) is unlikely. The experimentally measured enthalpy of formation of 24 is, however, different by ca. 20 kJ mol -l (ca. one-half the exothermicity of reaction 22) from that which would have been predicted by assuming the following reaction is thermoneutral: 1,3-dioxane + 1,1-dimethylcyclohexane ~ 24 + cyclohexane

(24)

The dimethyldioxane is apparently more stable than we would have thought and it appears that explanation (c) is not needed. An explanation of why the dimethyldioxane is stabilized and the tetraoxaspiro compound is not stabilized eludes us. What

Molecular Structure and Thermochemistry

333

remains needed, however, are some more examples of hetero derivatives of spiro compounds from which to derive general rules of their stability.

IX. POLYENES We close our chapter with a discussion of the last chapter in Volume 2 of this series by Cyvin, Brunvoll, Cyvin, and Brendsdal [46] on the enumeration of isomers and conformers. Most of the authors' study deals with conjugated polyenes and catacondensed polynuclear aromatic hydrocarbons. As with so many other classes of compounds, the available thermochemical data is disappointingly sparse. A recent review of this literature on dienes and polyenes in general [47] suggested that the first three members of the unsubstituted, conjugated polyene series CH2--CH2, CH2--CH-CH--CH2, ( E ) - C H 2 - - C H - C H = C H CH--CH2 are thermochemically homologous in that the differences in gaseous enthalpies of formation, ca. 57 kJ mol -t between the first and the second species, is nearly identical to that between the second and the third. However, that is where the data ends, even with estimation techniques for conjugated polyenes [48]. The nearly identical difference is found for the (E)-methyl derivatives of the first two species: CH2--CH-CH3 and CH2--CH-CH--CH-CH3, and so is suggestive of a "universal" vinylene increment, i.e. an essentially constant enthalpy of formation of a - C H = C H - unit. However, such enthusiasm is squelched upon noting that the enthalpies of formation of the enols [49] CH2--CH-OH and ( E ) - C H 2 - C H C H = C H - O H differ by but 40 kJ mo1-1. Is it this 40 or the earlier enunciated 57 kJ mol -l more "typical" of the enthalpy of formation differences of increasingly long, conjugated polyenes and their derivatives. We close this chapter by imploring for more data. There is something superbly frustrating about having to acknowledge how easily we are thwarted in our efforts to find regularities between structure and energetics.

ACKNOWLEDGMENTS One of us (J. E L.) wishes to thank the Chemical Science and Technology Laboratory of the U. S. National Institute of Standards and Technology for partial support of his research.

REFERENCES A N D NOTES 1. Hargittai, M.; Hargittai, I. (Eds.).Advances in Molecular Structure Research, Vol. 2; JAI Press: Greenwich, CT, 1996. 2. Pedley,J. B.; Naylor, R. D.; Kirby, S. P. Thermochemical Data of Organic Compounds, 2nd ed. Chapman & Hall: New York, 1986.This is our chosen archival source of enthalpy of formation data for organic compounds, and so any unreferenced datum may be assumed to be from this source. Relatedly, unreferenced enthalpies of formation of organometallic compounds may be assumed to be from (a) Pilcher, G.; Skinner,H. A. In: The Chemistry of the Metal-Carbon Bond; Hartley, E R.; Patai, S., Eds. Wiley: Chichester, 1982; and of inorganic compounds from (b)

334

3.

4. 5.

6.

7.

8. 9.

10. 11.

12.

13. 14. 15.

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. J. Phys. Chem. Ref. Data 1982, 11 (Supplement 2). Enthalpies of vaporization may be estimated according to the analysis of: Chickos, J. S.; Hyman, A. S.; Ladon, L. H.; Liebman, J. E J. Org. Chem. 1981, 46, 4294 for hydrocarbons and Chickos, J. S.; Hesse, D. G.; Liebman, J. F.; Panshin, S. Y. J. Org. Chem. 1988, 54, 3424 for their simple derivatives. No such conclusion is generally applicable for the enthalpies of fusion or of sublimation. Fortunately, phase change enthalpies involving numerous organic solids may conveniently be found in the following sources: (a) Chickos, J. S. In: Molecular Structure and Energetics (Physical Measurements); Liebman, J. E; Greenberg, A., Eds. VCH Publishers: Deerfield Beach, 1987, Vol. 2; (b) Acree, W. E. Thermochim. Acta 1991, 189, 37; 1993, 219, 97; (c) Domalski, E. S 9Hearing, E. D. J. Phys. Chem. Ref. Data 1996, 25, 1. Osawa, E. In: Ref. 1, Conformational Principles of Congested Organic Molecules: Trans is Not Always More Stable than gauche, pp. 1-24. We can estimate a liquid enthalpy of formation for this compound. The liquid enthalpies of formation for the a,e 1,3- and 1,4-dimethylcyclohexanes differ by only 0.1 kJ mol -l, while those for the e,e 1,3- and 1,4-dimethylcyclohexanes and for the e,e 1-ethyl-3-methyl- and 1-ethyl-4methylcyclohexanes differ by 0.5 and 0.7 kJ mol -l, respectively. Thus, the liquid enthalpy of formation for trans-l-ethyl-3-methylcyclohexane (a,e) may be virtually identical to that of cis- 1-ethyl-4-methylcyclohexane (a,e), -238.9 + 1.1 kJ mol -l. (a) Wiberg, K. B.; Murcko, M. A. J. Amer. Chem. Soc. 1988, 110, 8029; (b) Anderson, J. E. In The Chemistry of Alkanes and Cycloalkanes; Patai, S." Rappoport, Z., Eds. Wiley: Chichester, 1992. That there is less of the anti conformer present in the liquid state shows the intrinsic preference for that conformer is lessened when in close contact with other molecules. This may result from a better packing of the more compact gauche conformer in the liquid phase. The calculated enthalpy difference between an equatorial and an axial ethyl group is 1.71 kcal mol -l (1) (7.15 kJ tool-l), essentially the same as for the methyl group difference of 1.6-1.8 kcal mol -l. See Eliel, E. L.; Allinger, N. L.; Angyal, S. J.; Morrison, G. A. Conformational Analysis. Wiley: New York, 1965. An attractive 1,5-interaction of the type described by Osawa is too small (ca. 0.67 kJ mol -l) to account for the discrepancy. The equilibrium concentration of the trans- 1,2-dimethylcyclohexane a,a conformer is ca. 1% (ref. 7) and contributes negligibly to the enthalpy of formation of the e, ela, a mixture. The amount of the trans-1-ethyl-2-methylcyclohexane conformer would not be expected to exceed 1% either. Eliel, E. L. Stereochemistry of Carbon Compounds. McGraw-Hill: New York, 1962. Temperature-dependent equilibration studies were performed on the cis- and trans-isomers of 1,3and of 1,4-diethylcyclohexane. Although the gas chromatographic resolution of the isomer mixtures at equilibrium was not as high as desired, the derived AHisom(1)values for reaction 3 were within experimental error of the corresponding reaction shown in Table 1 for 1-ethyl-4-methylcyclohexane. Allinger, N. L." Hu, S-E. J. Am. Chem. Soc. 1962, 84, 370. Good, W. D. J. Chem. Thermodynamics 1970, 2, 399. The substituted cyclohexanes used in this study were donated to the author. There is no documentation concerning synthesis, purification, or physical properties. Dauben, W. G.; Rohr, O.; Labbauff, A.; Rossini, E D. J. Phys. Chem. 1960, 64, 283. Allinger, N. L." Coke, J. L. J. Org. Chem. 1961, 26, 2096. The enthalpy of formation of gaseous tri-t-butylmethane (-184 kJ mo1-1) was obtained by summing the enthalpy of formation of the solid: Flamm-Ter Meer, M. A.; Beckhaus, H.-D.; Rtichardt, C. Thermochimica Acta 1986, 107, 331, and an averaged value of the more recently measured enthalpy of sublimation from: Chickos, J. S.; Hesse, D. G." Hosseini, S." Liebman, J. E; Mendenhall, G. D.; Verevkin, S. P.; Rakus, K.; Beckhaus, H.-D.; Rtichardt, C. J. Chem. Thermodyn. 1995, 27, 693.

Molecular Structure and Thermochemistry

335

16. Braga, D.; Grepioni, E In: Ref. 1, Transition Metal Clusters: Molecular versus Crystal Structure, pp. 25-66. 17. All enthalpies of sublimation in this section are taken from Pilcher and Skinner, op. cit. (ref. 2) where their values are either from experiment or well-reasoned estimates. 18. The 18-electron rule for transition metals may be understood as the metal making use of (doubly occupying) a single s orbital, three p-orbitals and five d-orbitals, much as the octet rule for nonmetals may be understood in terms of the use of but the s- and p-orbitals. See, for example: Cotton, E A.; Wilkinson, G.Advanced Inorganic Chemistry; 5 th ed. John Wiley & Sons: New York, 1988, pp. 37, 1021, 1041; and Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry: Principles of Structure and Reactivity, 4 th ed. Harper Collins: New York, 1993, pp. 624-630. 19. We note now that the low temperature form (< 159 ~ of manganocene ((CsHs)zMn) lacks discrete molecules, the high temperature solid (> 159 ~ and the gaseous form mimic the archetypical, molecular, ferrocene. All of these forms are best described as ionic. By contrast, the gaseous 1, l'-dimethyl derivative exists as a 2:1 ratio of ionic to covalent species and the decamethylated species is only covalent. (See: Cotton and Wilkinson, op. cit., p. 82.) From the above we would have thought that the sublimation energy of manganocene should be larger than ferrocene because of ionic stabilization of the lattice. 20. We used a literature procedure that had been designed for the study of essentially planar polynuclear aromatic hydrocarbons: Stein, S. E.; Golden, D. M.; Benson, S. W. J. Phys. Chem. 1977, 81, 314. 21. By contrast, the enthalpy of sublimation of bis(naphthalene)chromium is suggested to be 105.0 kJ mol -l, a value corresponding to a contribution of but 3.5 kJ per mol-(aromatic carbon) beyond that of its bisbenzene analog. 22. Gilli, P.; Ferretti, V.; Bertolasi, V.; Gilli, G. In: Ref. 1, A Novel Approach to Hydrogen Bonding Theory, pp. 67-102. 23. Hacking, J. M.; Pilcher, G. J. Chem. Thermodyn. 1979, 11, 1015. 24. It would be better to replace the carbonyl oxygen with an exo-methylene group and replace the hydroxyl group with a methyl. After all, these are valence isoelectronic and presumably at least roughly isosteric. It is now acknowledged that the requisite thermochemical data are absent here as well. Nonetheless, to document the potential future usefulness of this approach, see the related comparison of CONH 2 and C(CHz)CH 3 in: Abboud, J. M.; Jim6nez, E; Roux, M. V.; Turri6n, C.; Lopez-Mardomingo, C.; Podosenin, A.; Rogers, D. W.; Liebman, J. E J. Phys. Org. Chem. 1995, 8,15. 25. The comparison of COOH and CH 3 has been made before in a thermochemical context: Colomina, M.; Turri6n, C.; Jimgnez, P.; Roux, M. V.; Liebman, J. E Struct. Chem. 1993, 5, 141. 26. The necessary enthalpy of formation of solid benzene was obtained by summing the enthalpy of formation of the liquid and the enthalpy of fusion which corresponds to the value at the melting point and is unmodified by any heat capacity correction. 27. Leopold, K. R. In: Ref. 1, Partially Bonded Molecules and the Transition to the Crystalline State, pp. 103-128. 28. Unless otherwise stated, all enthalpies of formation and reaction in this section are taken from the data in Wagman et al., op. cit., ref. 2. 29. This is also related to, but not identical to, the question of the resonance energies of the various BX 3 molecules: cf. Liebman, J. E Struct. Chem. 1990, 1,395. 30. Numerical values of enthalpies of formation and of vaporization of these and the other, solely organic, sulfur compounds in this paragraph are taken from Pedley et al., op. cit., ref. 2. 31. This, too, is a corollary of the analysis of Chickos et al., op. cit., in ref. 3. There it was asserted that the enthalpy of vaporization of a monosubstituted, but otherwise arbitrary, organic compound equals 4.7ffc + 1.3nQ + 3.0 + b where ffc, nQ, and b are respectively the number of non-quaternary carbons, the number of quaternary carbons, and a substituent-dependent parameter. Accordingly,

336

32. 33.

34. 35.

36. 37.

38. 39. 40.

41. 42.

43. 44. 45. 46.

47. 48.

JOEL F. LIEBMAN and SUZANNE W. SLAYDEN when nc differs byl 2 for a given class of compounds, the enthalpy of vaporization differs by 4.7 * 2 or 9.4 kJ m o l - . Landis, C. R. In: Ref. 1, Valence Bond Concepts, Molecular Mechanics Computations, and Molecular Shapes, pp. 129-164. These species have main group elements with formally an excess of the optimal eight valence electrons as noted in ref. 18. The term, hypervalence, is customarily not applied to transition metal species such as (CsHs)2Ni that have more than 18 valence electrons around the central Ni. Irikura, K. K. J. Chem. Phys. 1995, 102, 5357. This is more properly a level splitting and orbital occupancy argument: the n-orbitals in the + * homonuclear S 2 and F 2 have considerable bonding character and the corresponding n -orbitals have considerable antibonding character. By contrast, the related orbitals in the heteronuclear SF show much less bonding or antibonding character. Mastryukov, V. S.; Simonsen, S. H. In: Ref. 1, Empirical Correlations in Structural Chemistry, pp. 153-190. (a) Eq. 14 is a modified form of the more general relation, AHf[Y-(CH2)m-H] = A +Bm + 5, first proposed for homologous hydrocarbon series by: Prosen, E. J.; Johnson, W. H.; Rossini, E D. J. Res. Natl. Bur. Stand. 1946, 37, 51. A is a constant associated with a specific end ~roup Y; B is a constant for all normal alkyl series independent of the end group (-20.6 kJ mol-'), and 5 is the deviation from linearity for a given member of the series. (b) Liebman, J. E; Crawford, K. S.; Slayden, S. W. In: The Chemistry of Functional Groups, Supplement S: The Chemistry of Sulphur-Containing Functional Groups; Patai, S.; Rappoport, Z., Eds. Wiley: Chichester, 1993. (c) Liebman, J. E; Campbell, M. S.; Slayden, S. W. In: The Chemistry of Functional Groups; Supplement F2: The Chemistry of Amino, Nitroso, Nitro, and Related Compounds; Patai, S., Ed. Wiley: Chichester, 1996. It is because of this relationship that the authors of ref. 3 are able to derive an estimation protocol for enthalpies of vaporization. Montgomery, R. L.; Rossini, E D. J. Chem. Thermodynamics 1978, 10, 471. (a) The "universal" slope, or methylene increment, is an idealized value, identical to the constant a obtained from Eq. 14 for the n-alkanes by which each member of a homologous series should differ from the member of the next larger carbon number. In reality, the methylene increment for each functionalized series is different for the C2-C16 members typically measured. See also ref. 37. (b) Prosen, E. J.; Johnson, W. H.; Rossini, E D. J. Res. Natl. Bur. Stand. 1946, 37, 51. (c) Sellers, P.; Stridh, G.; Sunner, S. J. Chem. Eng. Data 1978, 23, 250. Liebman, J. E; Martinho Simfes, J. A.; Slayden, S. W. Struct. Chem. 1995, 6, 65. Slayden, S. W.; Liebman, J. E; Mallard, W. G. In: The Chemistry of Functional Groups, Supplement D: The Chemistry of Halides, Pseudo-halides, and Azides; Patai, S.; Rappoport, Z., Eds. Wiley: Chichester, 1995. Slayden, S. W.; Liebman, J. E In: The Chemistry of the Double Bonded Functional Groups; Patai, S., Ed. Wiley: Chichester, 1997. Li, D.; Owen, N. L. In: Ref. 1, Structure Determination Using the NMR 'Inadequate'Technique, pp. 191-212. The enthalpy of formation of the liquid is from: Fletcher, S. E.; Mortimer, C. T.; Springall, H. D. J. Chem. Soc. 1959, 580. Cyvin, S. J.; Brunvoll, J.; Cyvin, B. N.; Brendsdal, E. In: Ref. 1, Enumeration of Isomers and Conformers: A Complete Mathematical Solution for Conjugated Polyene Hydrocarbons, pp. 213-246. Liebman, J. E In: The Chemistry of Functional Groups: Dienes and Polyenes; Rappoport, Z., Ed. Wiley: Chichester, 1997. This is not completely true. Using assumptions about phenyl and vinyl substituents, and enthalpies of sublimation, the discussion in ref. 47 on 1,6-diphenyl- 1,3,5-hexatriene allows for discussion on the totally conjugated decapentaene. However, in that it was noted that the enthalpy of formation

Molecular Structure and Thermochemistry

33 7

of this diphenyl compound is discrepant by some 30-40 kJ mol -l discouraged any analysis of this type in either the earlier paper or the current one. 49. TuruEek, E; Havlas, Z. J. Org. Chem. 1986, 51,4061.

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INDEX

Absorption spectra of matrixisolated small carbon molecules, 117-146 carbon molecular spectra, 124-128 bending modes of linear molecules, 128 cyclic molecules, 128 electronic spectra of linear species, 125-128 Halford's formula, 124-125 Hi~ckel method, 127, 128 stretching modes in linear molecules, 124-125 experimental setup, 122-124 introduction, 118-119 complexity, 118 fullerenes, 118 "small," meaning of, 118 IR-bands, assignment of in matrices, 134-137 isotopomeric line patterns, 135137, 139 "site-effect" splitting, 134, 136, 138 matrix isolation of carbon molecules, 119-121 advantages and disadvantages, 120 carbon vapor as seed material in argon matrices, 120, 123, 129-130

outlook, 144 research, early, 129-134 research, recent, 137-144 correlations of UV-vis and IR absorptions, 143-144 cyclic C6, 143 IR spectra, 137-141 UV-vis spectra, 141-143 schematic representation of apparatus, 123 space, carbon molecules in, 121122 in astrophysics, 121 diffuse interstellar bands (DIBs), 121 Anharmonicity, 56, 59 Aromatic character of carbocyclic 7r-electron systems deduced from molecular geometry, 227-268 behavioristic versus structural definitions of, 229-231 aromatic bond lengths, 229 aromatic chemical shifts, 229 chrysene, 230 indices of aromaticity, 229-233 as multidimensional phenomenon, 230 conclusions, 265-266 historical summary, 228-229

339

340

indices based on molecular geometry, 231-233 HOMA, 231-232, 235, 236, 237 Kekul6 structure of benzene, 231 molecular problems, review of, 239-265 angular group induced bond alternation (AGIBA), 251 aromaticity of ring in strainaffected benzene derivatives, 251-256 benzenoid hydrocarbons, 239244 fullerenes and their derivatives, 264-265 graph-topological approaches, success of, 246 and Hammett-like substituent effects, 246-248 HOSE model, 246-247, 251 Hiickel rule, 255, 258-263 intermolecular interactions, 256257 Kekul6 structure, 251,253, 254 Mills-Nixon effect, 251-256 molecules in excited state, aromaticity of, 264 nonbenzenoid hydrocarbons, 258-263 p-nitroaniline, 246 and p-nitrophenolate anions, 257 polysubstituted benzene derivatives, 244-246 and steric substituent effects, 248 relations between indices of aromaticity based on molecular geometry, 234-239 alternation of bond length, 234 condensed benzenoid hydrocarbons, 236-237

INDEX

EN, 237-239 ferrocene, 234, 235 fullerene, 235, 236 GEO, 237-239 Astrophysics, carbon molecules in, 121 diffuse interstellar bands (DIBs), 121 Benzene, 172-182 (see also "Crystals...") rings (BzC), 148, 163-87 (see also "Crystals...") Born-Oppenheimer approximation, 2, 56-57 Cambridge Structural Database (CSD), 149, 154, 265 Carbocyclic w-electron systems, aromatic character of, 227-268 (see also "Aromatic...") Carbon molecules, absorption spectra of, 117-146 (see also "Absorption spectra...") CCSD(T) method, 29-30 Center for Structure Documentation, University of Ulm, Germany, 55 Centrifugal distortion constants, 4 (see also "Rotational constants...") Compounds containing C-C, C-N, C-O, N-O, and O-O single bonds, structures and conformations of, 83-115 with C-C bonds, 87-92 with C-N bonds, 93-96 with C-O bonds, 96-101 bis(fluoroxy)difluoromethane, 99-101 fluoroformylhypofluorite, 96-99 conclusion, I 1l-112

Index

introduction, 84-87 density functional theory (DFT) methods, 85 DGAUSS, 85 GAMESS, 85 gas electron diffraction (GED), 84 GAUSSIAN, 85 gradient technique, 85 Hartree-Fock approximation, 85 microwave (MW), rotational constants from, 84-85 molecular orbital constrained electron diffraction (MOCED), 85 Moller-Plesset approximation, 85 SPARTAN, 85 theoretical calculations, 85-87 X-ray diffraction, 84 with N-O bonds, 101-107 halogen nitrates, 101-103 O-nitrosobis (trifluoromethyl)hydroxylamine, [(CF3)2NONO], 103-107 with O-O bonds, 107-111 bis(fluorocarbonyl)peroxide, 108-109 bis(trifluoromethyl)trioxide, 109-111 dimethylperoxide, 107-108 Conjugated hydrogen bonds, 148, 154-163 (see also "Crystals...") Coriolis resonances, 6-7 Crystals, organic, isostructurality of, 189-226 abstract, 190 concluding remarks, 223-224 historical background, 191-196 digirezigenin, 195, 197, 203-204 digitoxigenin, 195, 197, 203-204

341

homeotypism, 192 isomorphism, 191-195 isostructuralism, definition of, 195 isotypism, 192, 196 main part isostructuralism, 195196 quasi-isostructural, 195 Wyckoff positions, 196 of homomolecular crystals, 196-214 conditions and limits, 201-203 homostructurality, 209-211 isostructurality and cocrystallization, correlation between, 203-204 isostructurality, forms of within Ph3E-E'Me3 series, 211-214 isostructurality, relaxed forms of, 208-211 isostructurality in general, 204208 isostructurality index, 198-199 molecular isometricity index, 199 molecular size, 204-206 morphotropic steps, 212-214 numerical descriptors of, 196201 packing coefficient increments, 196-198 substituents, size and character of, 206-208 substitution, site of, 206-208 unit cell similarity index, 198 introduction, 190-191 binary adducts, 191 complementary surfaces of molecules, 190 polymorphism, 191 of supramolecules, 214-223 adducts with slight difference in guest or host molecules, 214-216

342

clathrates of 2,2'-bis (3,4,5trimethoxyphenyl)- 1,1bibenzimidazole, 221-223 clathrates formed by basically different guest molecules, 216-218 homostructural adducts formed by different host and guest molecules, 218-221 Crystals, organic, specific intermolecular interactions in, 147188 abstract, 148 benzene rings (BzC), classification of contacts and aggregates of, 148, 163-187 aggregates, types of, 170-172 atom-atom potentials, 169 in benzene derivatives, 172-182 in crystalline derivatives, 172187 in dihydroxybenzene derivatives, 182-187 dimer, 168-169 energy characteristics, 168-169 Kitaigorodskii's centrosymmetricity rule, 179 meta-cresol (MCRSOL), 182183 o-cresol, 185 para-cresol (CRESOL01), 183 parquet layer (PL), 172, 177, 179 in phenol derivatives, 181, 182187 "pseudoparquet" rule, 179 shift, 168 stereotypes, 164-168 symmetries not less than m, 169-170 toluene, 178 "trihedral wells," 181, 183 types, 163-170

INDEX

conclusion, 187-188 conjugated hydrogen bonds (CHB) illustrated by crystalline derivatives of hydroxy- and dihydroxybenzene, 148, 154-163 catechol, 154, 159-160 halogen contacts, 154 hydroquinone, 154 phenol, 154 resorcinol, 154, 159-161 introduction, 148-153 agglomerates, formation of, 150 benzene rings, specific contacts of, 148, 150-153 conjugated hydrogen bonds (CHB), 148, 149, 150-152 "packing," 148-149, 162 spatial arrangement, 148, 149 Density functional theory (DFT) methods, 85 (see also "Compounds...'3 DGAUSS, 85 Energetic difluoramines, computational studies of structures and properties of, 269-285 background, 270-272 difluoramino group, advantages of, 270 difluoramino group, disadvantages of, 271 instability, 271,283 nitro group, 270 computational studies, survey of, 273-282 energetics, 280-282 heats of formation, 271,275-276 structures, 273-280 thermodynamic stabilities, 273

Index

nitro and difluoramino groups, comparison of, 272-273 summary and conclusions, 282283 Equilibrium structure, 2-3 (see also "Rotational constants...") Fermi resonance effect, 64 Fourier transform spectroscopy, 3-4, 123 Fullerenes, 118, 235, 236 (see also "Absorption spectra...") and derivatives, 264-265 GAMESS, 85 Gas electron diffraction (GED), 8485 (see also "Compounds...") Gas-phase cluster spectroscopy, 138, 140-141 Gauss-Newton method of successive iterations, 41 (see also "Rotational constants...") GAUSSIAN, 85 Halogen nitrates, 101-103 Hartree-Fock (HF) approximation, 85 HOMA (harmonic oscillator model of aromaticity), 231-232, 235, 236, 237 Homeotypism, 192 Hi~ckel rule, 127, 128, 228, 255, 258263, 294 Isomorphism, 191-195, 223 (see also "Crystals...") co-crystallization, 203-204 lsostructurality of organic crystals, 189-226 (see also "Crystals...") Isotypism, 192, 196

343

Jerusalem Symposium on Quantum Chemistry and Biochemistry, 229 Julg index, 232, 233, 235, 237 Kitaigorodskii's centrosymmetric rule, 179, 190, 193-223 (see also "Crystals...") Lewis acid/Lewis base complexes, 303, 306, 324-326 Mills-Nixon effect, 251-256 MOCED (molecular orbital constrained electron diffraction), 85 (see also "Compounds...") MOGADOC database, 55 Molecular Structure and Energetics book series, 314 Molecular structure and thermochemistry, relationships between, 313-336 congested molecules, 314-318 gauche interactions, 315-316 hydrocarbons and their derivatives, 328-331 hydrogen bonds, strong, in ketoenols, 321-324 enaminoketones, 323 hypervalent species: sulfur fluorides, 326-328 introduction, 314 gas phase, phenomena in, 314 thermochemistry, meaning of, 314 keto-enols, 321-324 Lewis acid/Lewis base complexes, 303, 306, 324-326 metals with carbon monoxide and r-systems, complexes of, 318-321

344

polyenes, 333 spiro compounds, 331-333 Moller-Plesset approximation, 85 Moscow University Electron Diffraction Laboratory, 58, 63 Naval Air Warfare Center, 271 "Packing," molecular, 148-149, 162, 190, 196 (see also "Crystals...") Parquet layer (PL), 172, 177, 179 (see also "Crystals...'3 Pauling scale, 272 Photoelectron spectroscopy (PES), 107 Polymorphism, 191 (see also "Crystals...") Potential function, equilibrium and, 53-81 Badger model, 73-74, 75 concluding remarks, 78 conventional interpretational scheme, deficiencies of, 5556 anharmonic shrinkage, 56 diffraction data, analysis of in terms of molecular potential function, 56-58 Born-Oppenheimer approximation, 56-57 formulation, general, 57 large-amplitude motion, 57-58 molecular structure and potential energy function, 56-57 motivation for, 57-58 diffraction intensity, 63-65 cumulant-moment representation of intensity equation, 63-64 perturbation calculation, 64 quasi-rigid molecular systems, diffraction analysis of, 65

INDEX

discussion, 74-75 electron diffraction and various techniques, combined use of, 75-78 advantages, fundamental, 75-76 complications, 76-77 example, 77-78 introduction, 54-55 intramolecular, 55 MOGADOC database, 55 large-amplitude motion analysis, 65-70 adiabatic separation of, 66-67 formulation, general, 65-66 framework vibrations, 67 molecular intensity in adiabatic approximation, 70 thermal average coordinate distribution function in adiabatic approximation, 67-70 potential function, 70-71 problems and limitations, 56-63 adjustable parameters, number of, 60-61 anharmonicity, 59 curvilinear coordinates, 60 and molecular parameters determination, 61-63 quasi-diatomic approximation, 7273, 74, 75 Resorcinol, 154, 159-161 Rotational constants, determination of reliable structures from, 1-51 appendix, 41-46 Gauss-Newton method of successive iterations, 41 weighted least-squares, 42 conclusion, 39-40 data, additional, sources of, 28-32 ab initio methods, 28, 29-30, 39 best estimate structure, 30

Index

CCSD(T) method, 29-30 electron diffraction, 28, 30-31 empirical relations, 28, 31-32 NMR spectroscopy, 28, 31 QCISD method, 29 equilibrium rotational constants, determination of, 3-10 centrifugal correction, 4-5 centrifugal distortion constants, 4 electronic correction, 5-6 Hamiltonian, 4 vibrational correction, 5, 6-10 (see also "Vibrational correction") Zeeman effect, 6 examples, 32-39 chloroacetylene (HC--CCI), 3536 DCO § 33 difluoroethyne (FC--=CF), 37-39 fluorophosphaethyne (FC--P), 32-33 formyl cation (HCO+), 32-33 methyl chloride (CH3C1), 36-37 phosgene, 34-35 flowchart, 40 ground state constants, structures from, 10-19 Kraitchman-type equations, 13, 19 Kuchitsu, triatomic approximation of, 11, 13 re structure, 14-17 re structure, 14-17, 19 ri,, 19 rm structure, 13-14 rm ~ structure, 17-19, 21 rz structure, 10-13 least-squares method, 19-28 assumptions of, 20-24 collinearity or ill-conditioning, 24-26, 42

345

condition indexes, 24-26 corrective action, 26 Gauss-Markov conditions, 20 introduction, 19-20 jackknifed residuals, 27, 44, 45 "model-induced" variance, 22, 23 outlier analysis, 26-28 standardized residuals, 26, 44, 45-46 studentized residuals, 26-27, 44 weighted, 22-24 introduction, 2-3 Born-Oppenheimer approximation, 2 equilibrium structure, 2-3 inertia, equilibrium moments of, 3 reliable structures, how to obtain, 3 Se-N and Te-N species, binary and ternary, chemical properties and structures of, 287-311 conclusions, 308-309 introduction, 288-289 preparative aspects, 289-292 (see also "...synthesis") structures, 292-308 aromatic 6rr cation [Se3N2] 2+ and rr*-rr* bound 7rr cation [Se3N2]+, 293-295 computational chemistry, importance of, 292 diselenium trichloride nitride, 296-299 effective core potential (ECP), 293 [E2NCI4].+, solid state structures of, 299-300 Pauling "3-electron bonds", 308 polymorphism: Se4N4 and Se2NC13, 296-299

346

pseudo-potential method, 293 Raman spectrum, 306 [(SeC1)2NI§ structure of, 303307 [(SeCla)aN]§ structure of, 300303 [TeaN2CI8]2+cation, 300 tetraselenium tetranitride, 296 valence bond description for molecules [Se3N212§ [Se3N2]+, Se4N4, and hypothetical (SeN)x, 307-308 synthesis, 289-292 of binary Se-N cations, 290-291 colors, 289 NMR probe, availability of, 289 of SenN4, 290 of ternary selenium-nitrogenhalides, 291-292 of ternary tellurium-nitrogenhalide, 292 SEARCH, 314 SPARTAN, 85 Specific intermolecular interactions in organic crystals, 147-188 (see also "Crystals...")

INDEX

Spiro compounds, 331-333 (see also "Molecular structure...") Supramolecular chemistry, 148-149 Te-N species, binary and ternary, chemical properties and structures of, 287-311 (see also "Se-N...") Thermochemistry and molecular structure, relationships between, 313-336 (see also "Molecular structure...") Toluene, 178 Vibrational correction, 5, 6-10 anharmonic force field, ab initio, 9 anharmonic force field, experimental, 7-9 Coriolis resonances, 6-7 excited states, analysis of, 6-7 in long linear chains, 10 magnitude, order of, 10 X-ray diffraction, 84 (see also "Compounds...") Zeeman effect, 6

Advances in Molecular Structure Research Edited by Magdolna Hargittai, Structural Chemistry Research Group, Hungarian Academy of Sciences, Budapest, Hungary and Istvdn Hargittai, Institute of General and Analytical Chemistry, Budapest Technical University, Budapest, Hungary Volume 1, 1995, 352 pp. ISBN 1-55938-799-8

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CONTENTS: Introduction to the Series: An Editor's Foreword, Albert Padwa. Preface, Magdolna Hargittai and Istv,~n Hargittai. Measuring Symmetry in Structural Chemistry, Hagit Zabrodsky and David Anvir. Some Perspectives in Molecular Structure Research: An Introduction, Istv,~n Hargattai and Magdolna Hargattai. Accurate Molecular Structure from Microwave Rotational Spectroscopy, Hans Dieter Rudolph. GasPhase NMR Studies of Conformational Processes, Nancy S. True and Cristina Suarez. Fourier Transform Spectroscopy of Radicals, Henry W. Rohrs, Gregory J. Frost, G. Barney Ellison, Erik C. Richard, and Veronica Vaida. The Interplay between X-Ray Crystallography and AB Initio Calculations, Roland Boese, Thomas Haumann and Peter Stellberg. Computational and Spectroscopic Studies on Hydrated Molecules, Alfred H. Lowrey and Robert W. Williams. Experimental Electron Densities of Molecular Crystals and Calculation of Electrostatic Properties from High Resolution X-Ray Diffraction, Claude Lecomte. Order in Space: Packing of Atoms and Molecules, Laura E. Depero.

Volume 2, 1996, 255 pp. ISBN 0-7623-0025-6

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CONTENTS: Preface, Magdolna Hargittai and Istvan HargittaL Conformational Principles of Congested Organic Molecules: Trans is Not Always More Stable Than Gauche, Eiji Osawa. Transition Metal Clusters: Molecular versus Crystal Structure, Dario Braga and Fabrizia Grepioni. A Novel Approach to Hydrogen Bonding Theory, Paola Gilli, Valeria Ferretti, Valerio Bertolasi and Gastone Gilli. Partially Bonded Molecules and Their Transition to the Crystalline State, Kenneth R. Leopold. Valence Bond Concepts, Molecular Mechanics Computations, and Molecular Shapes, Clark R. Landis. Empirical Correlations in Structural Chemistry, Vladimir S. Mastryukov and Stanley H. Simonsen. Structure Determination Using the NMR "Inadequate" Technique, Du Li and Noel L. Owen. Enumeration of Isomers and Conformers: A Complete Mathematical Solution for Conjugated Polyene Hydrocarbons, Sven J. Cyvin, Jon Brunvoll, Bjerg Cyvin, and Egil

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Advances in Molecular Electronic Structure Theory Edited by T h o m H. Dunning, Jr., Molecular Science Research Center, Pacific Northwest Laboratory, Richland, Washington Volume 1, Calculation and Characterization of Molecular Potential Energy Surfaces 1990, 275 pp. $109.50 ISBN 0-89232-956-4 CONTENTS: Introduction to the Series: An Editor's Foreword, Albert Padwa. Introduction, Thom H. Dunning, Jr. Analytical Representation and Vibrational-Rotational Analysis of Ab Initio Potential Energy and Property Surfaces, Walter C. Ermler and Hsiu Chinhsieh. Calculation of Potential Energy Surfaces, Lawrence B. Harding. The Analytical Representation of Potential Surfaces for Chemical Reactions, G.C. Schatz. Characterization of Molecular Potential Energy Surfaces: Critical Points, Reaction Paths, and Reaction Valleys, Elfi Kraka and Thom H. Dunning, Jr.. Long-Range and Weak Interaction Surfaces, Clifford E. Dykstra. The Von Neumann-Wigner and Jahn-Teller Theorems and Their Consequences, Regina F. Frey and Ernest R. Davidson.

Volume 2, 1994, 209 pp. ISBN 0-89232-957-2

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REVIEW: " This choice makes this an excellent and very readable volume for those chemists who want a broad familiarization of the topics without becoming mired in specific details>"

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CONTENTS: Introduction, Thom H. Dunning, Jr. Electronic Structure Theory and Atomistic Computer Simulations of Materials, Richard P. Messmer. Calculation of the Electronic Structure of Transition Metals in Ionic Crystals, Nicholas W. Winter, David K. Temple, Victor Luana and Russell M. Pitzer. Ab Initio Studies of Molecular Models of Zeolitic Catalysts, Joachim Sauer. Ab Inito Methods in Geochemistry and Mineralogy, Anthony C. Hess and Paul F. McMillan.