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ADVANCES IN MOLECULAR STRUCTURE RESEARCH Volume3

9

1997

This Page Intentionally Left Blank

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


ix

List of Contributors Petr M. Zorky



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


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


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:


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


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


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


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


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


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


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) )


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


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


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


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


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


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-


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


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


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


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 0.5; 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


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.


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


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].


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


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


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


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


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


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 _ 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


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


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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50 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139.

140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158.

J. DEMAISON, G. WLODARCZAK, and H. D. RUDOLPH Jensen, E; Brodersen, S.; Guelachvili, G. J. Mol. Spectrosc. 1981, 88, 378. Btirger, H.; Schneider, W.; Somrner, M.; Thiel, W.; Willner, H. J. Chem. Phys. 1991, 95, 5660. BUrger, H.; Senzlober, M.; Sommer. M. J. Mol. Spectrosc. 1994, 164, 570. Breidung, J.; Hansen, T.; Thiel, W. J. Mol. Spectrosc. 1996, 179, 73. Morino, Y.; Tanimoto, M.; Saito, S.; Hirota, E.; Awata, R.; Tanaka, T. J. Mol. Spectrosc. 1983, 98, 331. Degli Esposti, C.; Cazzoli, G.; Favero, E G. J. Mol. Spectrosc. 1985, 109, 229. Lahaye, J. G.; Vandenhaute, R.; Fayt, A. J. Mol. Spectrosc. 1987, 123, 48. Puzzarini, C.; Tarroni, R.; Palmieri, E; Demaison, J.; Senent, M. L. J. Chem. Phys. 1996, 105, 3132. Degli Esposti, C.; Favero, E G.; Serenellini, S.; Cazzoli, G. J. Mol. Struct. 1982, 82, 221. Whiffen, D. H. Spectrochim. Acta 1978, 34A, 1165. Cazzoli, G.; Favero, E G.; Degli Esposti, C. Chem. Phys. Lett. 1977, 50, 336. Whiffen, D. H. Spectrochim. Acta 1978, 34A, 1173. Degli Esposti, C.; Tamassia, E; Puzzarini, C.; Tarroni, R.; Zelinger, Z. Mol. Phys. 1996, 88, 1603. Whiffen, D. H. Spectrochim. Acta 1978, 34A, 1183. Cazzoli, G.; Degli Esposti, C.; Favero, E G. J. Mol. Struct. 1978, 48, 1. Strey, G.; Mills, I. M. Mol. Phys. 1973, 26, 129. Creswell, R. A.; Robiette, A. G. Mol. Phys. 1978, 36, 869. Okayabashi, T.; Tanimoto, M. J. Chem. Phys. 1993, 99, 3268. Botschwina, E; Oswald, M.; Sebald, E J. Mol. Spectrosc. 1992, 155, 360. Botschwina, E; Oswald, M.; Fltigge, J.; Heyl, ,~,; Oswald, R. Chem. Phys. Lett. 1993, 209, 117; 215, 681. Botschwina, E; Seeger, S.; Horn, M.; Fltigge, J.; Oswald, M.; Mladenovic, M.; H/Sper, U.; Oswald, R.; Schick, E. Proceedings of the Meeting on Physical Chemistry of Molecules and Grains in Space (Mont Sainte-Odile, September 6-10 1993). In Conference Proceedings No. 312; Nenner, I., Ed. American Institute of Physics: New York, 1994, p. 321. Mladenovic, M.; Schmatz, S.; Botschwina, E J. Chem. Phys. 1994, 101, 5891. Botschwina, E; Seeger, S.; Mladenovic, M.; Schulz, B.; Horn, M.; Schmatz, S.; Fltigge, J.; Oswald, R. Int. Rev. Phys. Chem. 1995, 14, 169. Botschwina, E; Heyl, ,~,; Horn, M.; Fltigge, J. J. Mol. Spectrosc. 1994, 163, 127. Seeger, S.; Botschwina, E; Fltigge, J.; Reisenauer, H. E; Maier, G. J. Mol. Struct. (Theochem) 1994, 303, 213. Oswald, M.; Botschwina, E; Fliigge, J. J. Mol. Struct. 1994, 320, 227. Botschwina, E; Fltigge, J. Chem. Phys. Lett. 1991, 180, 589. Botschwina, E J. Chem. Phys. 1993, 99, 6217. Botschwina, P. Chem. Phys. Lett. 1994, 225, 480. McCarthy, M. C.; Gottlieb, C. A.; Thaddeus, E; Horn, M.; Botschwina, E J. Chem. Phys. 1995, 103, 7820. Gottlieb, C. A.; Killian, T. C.; Thaddeus, E; Botschwina, E; Fltigge, J.; Oswald, J. J. Chem. Phys. 1993, 98, 4478. Botschwina, E; Fltigge, J.; Seeger, S. J. Mol. Spectrosc. 1993, 157, 494. Botschwina, E; Horn, M.; Seeger, S.; Fltigge, J. Mol. Phys. 1992, 78, 191. Botschwina, E; Horn, M.; Seeger, S.; Fltigge, J. Chem. Phys. Lett. 1992, 200, 200. Botschwina, E; Fltigge, J.; Sebald, E J. Phys. Chem. 1995, 99, 9755. Botschwina, E; Reisenauer, H. E Chem. Phys. Lett. 1991, 183, 217. Botschwina, E; Tommek, M.; Sebald, E; Bogey, M.; Demuynck, C.; Destombes, J. L.; Waiters, A. J. Chem. Phys. 1991, 95, 7769. Carter, S.; Mills, I. M.; Handy, N. C. J. Chem. Phys. 1992, 97, 1606. Morino, Y.; Saito, S. J. Mol. Spectrosc. 1966, 19, 435. Thiel, W.; Scuseria, G.; Schaefer, H. E; Allen, W. D. J. Chem. Phys. 1988, 89, 4965.

Determination of Reliable Structures from Rotational Constants 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180.

Carter, S.; Handy, N. C., Demaison, J. Mol. Phys. 1997, in press. East, A. L. L.; Allen, W. D.; Klippenstein, S. J. J. Chem. Phys. 1995, 102, 8506. Cook, R. L.; De Lucia, E C.; Helminger, P. J. MoL Spectrosc. 1974, 53, 62. Cook, R. L.; De Lucia, E C.; Helminger, P. J. Mol. Struct. 1975, 28, 237. Laurie, V. W.; Herschbach, D. R. J. Chem. Phys. 1962, 37, 1687. Clark, A. H. J. Mol. Struct. 1971, 7, 485. Miyazaki, K.; Tamoura, M.; Tanaka, K. J. Mol. Spectrosc. 1986, 116, 435. Colmont, J.-M.; Demaison, J.; Cosl6ou, J. J. Mol. Spectrosc. 1995, 171,453. Kirchhoff, W. H. J. Mol. Spectrosc. 1972, 41,333. Brown, R. D.; Burden, F. R.; Godfrey, P. D.; Gillard, I. R. J. Mol. Spectrosc. 1974, 52, 301. Shoji, H.; Tanaka, T.; Hirota, E. J. Mol. Spectrosc. 1973, 47, 268. Nakata, M.; Yamamoto. S.; Fukuyama, T.: Kuchitsu, K. J. Mol. Struct. 1983, 100, 143. Smith, J. G. Mol. Phys. 1978, 35, 461. Duncan, J. L. Mol. Phys. 1974, 28, 1177. Botschwina, P., private communication. Yamamoto, S.; Nakata, M.; Kuchitsu, K. J. Mol. Spectrosc. 1985, 112, 173. Robinson, G. W. J. Chem. Phys. 1953, 21, 1741. Carpenter, J. H.; Rimmer, D. E J. Chem. Soc. Faraday Trans. II 1978, 74, 466. Kostyk, E.; Welsch, H. L. Can. J. Phys. 1980, 58, 912. Bouddou, A.; Demaison, J., unpublished results. Botschwina, P.; Seeger, S. J. Mol. Struct. 1994, 320, 243. Hoy, A. R.; Bunker, P. R. J. Mol. Spectrosc. 1979, 74, 1.

51


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.

54 54

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

61 63

A. B.

63 64

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

60 60

54

VICTOR P. SPIRIDONOV

VI.

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


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,


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


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.


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


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-


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


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


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


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].


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


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)


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


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.


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


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)


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].


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


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


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


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


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


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.


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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.


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


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.


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


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


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


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) ~


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


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


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


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


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


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


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


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.


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


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


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


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


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


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.


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


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


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.


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


[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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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

Absorptions of Carbon Molecules

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.

120

IVO CERMAK, GEROLD MONNINGER, and WOLFGANG KRg,TSCHMER

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


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

122

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,


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

124

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.


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.


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.


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.


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


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 -.o o-.

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


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).


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


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


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).


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.


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.

REFERENCES 1. Pratt Brock, C.; Dunitz, J. D. Chem. Mater. 1994, 6, 1118-1127. 2. Kitaigorodskii, A. J. Organic Chemical Crystallography. Consultants Bureau: New York, 1961,pp. 222-231. 3. K~ilm~in,A.; Argay, Gy.; Scharfenberg-Pfeiffer, D.; Hthne, E., Rib~ir, B. Acta Crystallogr., Sect. B. 1991, B47, 68-77. 4. K~ilmtin,A.; Argay. Gy.; Vladimirov, S.; ~ivanov-Stakic, D.; Rib~, B. Acta Crystallogr., Sect. C. 1992, B48, 812-819. 5. K~ilm(m,A.; P~k~inyi, L.; Argay, Gy. Acta Crystallogr., Sect. B. 1993, B49, 1039-1049. 6. K~ilm~in,A.; Pfirk~inyi,L.; Argay, Gy. Acta Chim. Hung. 1993, 130, 279-298. 7. Haleblian, J. K. J. Pharm. Sci. 1975, 64, 1269-1288. 8. Weber, E.; Czugler, M. Topics in Current Chemistry. Springer Verlag: Berlin, Heidelberg, 1988, Vol. 149, pp. 45-135. 9. Bar, I.; Bernstein, J. Tetrahedron 1987, 43, 1299-1305. 10. N~iray-Szab6, I. Inorganic Crystal Chemistry. Akadtmiai Kiad6: Budapest, 1969. 11. Wells, A. E Structural Inorganic Chemistry. Clarendon Press: Oxford, 1962, pp. 182-186. 12. Bloss, E D. Crystallography and Crystal Chemistry. Holt, Rinehart and Winston: New York, 1971, pp. 249-250. 13. Zoltai, T.; Stout, J. H. Mineralogy: Concepts and Principles. Burgess Publishing: Minneapolis, 1985, p. 490. 14. Lima-de-Faria, J.; Heller, E.; Liebau, E; Makovicky E.; Partht, E. Acta Crystallogr., Sect. A. 1990, A46, 1-11. 15. Nfiray-Szab6, I.; Argay, Gy. Acta Chim. Hung. 1963, 39, 85-92. 16. K~ilm~in,A.; Argay. Gy.; Rib~, B." Vladimirov, S.; ~ivanov-Stakid, D. Croat. Chem. Acta 1984, 57, 519-528. 17. Karle, I. L.; Karle, J. Acta Crystallogr., Sect. B. 1969, B25, 434-442. 18. Argay, Gy.; K~ilm~in,A.; Rib(u', B.; Vladimirov, S.; ~ivanov-Stakie, D. Acta Crystallogr., Sect. C. 1987, C43, 922-926. 19. K~ilmtin,A.; FtilOp, V." Argay, Gy.; Rib~, B." Lazar, D.; ~ivanov-Staki~, D.; Vladimirov, S. Acta Crystallogr., Sect. C. 1988, C44, 1634-1638. 20. Declercq, J.-P.; Germain, G.; King, G. S. D. Abstr. 4th Eur. Crystallogr. Meet. Oxford, 1977, pp. 279-280. 21. Messerschmidt, A.; Htihne, E.; Megges, R. Cryst. Struct. Commun. 1981, 10, 149-156. 22. Prasad, L.; Gabe, E. J. Acta Crystallogr., Sect. C. 1983, C39, 273-275.


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Weber, E.; Cs/Sregh, I.; Stensland B.; Czugler, M. J. Am. Chem. Soc. 1984, 106, 3297-3306. Herbstein, E H. Acta Chim. Hung. 1993, 130, 377-387. Gall, J. H.; MacNicol, D. D.; Mallinson, P. R.; Welsh, P. A. Tetrahedron Lett. 1985, 26, 4005-4008. Bticskei, Zs.; Simon, K.; Ambrus, G.; IlkiSy,1~.Acta Crystallogr., Sect. C. 1995, C51, 1319-1322. Wu, D-D.; Mak, T. C. W. J. Chem. Cryst. 1994, 24, 689-694. Caira, M. R.; Mohamed, R. Supramolec. Chem. 1993, 2, 201-207. Giuseppetti, G.; Tadini, C.; Bettinetti, G. P.; Giordano, E Cryst. Struct. Commun. 1977, 6, 263-274. Caira, M. R. J. Chem. Cryst. 1994, 24, 695-701. Ung, A. T.; Bishop, R.; Craig, D. C.; Dance, I. G.; Scudder, M. L. Chem. Mater 1994, 6, 1269-1281. Speier, G.; P~k(myi L. J. Org. Chem. 1986, 51, 218-221. Czugler, M.; Bombicz, P., to be published. Jones, W.; Theocharis, C. R.; Thomas, J. M.; Desiraju, G. R. J. Chem. Soc., Chem. Commun. 1983, 1443-1444. Reddy, D. S.; Craig, D. C.; Desiraju, G. R. J. Am. Chem. Soc. 1996, 118, 4090-4093, (and references therein).

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

?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

PETER POLITZER and PAT LANE

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.

Computational Studies of Energetic Difluoramines

285

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.


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


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)


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


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


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].


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].


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


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].


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].


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


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].


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

,

18o*(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+.


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


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.


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.


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


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


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


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


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


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


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


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


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.


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


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


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


~(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


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


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


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.


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


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.


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.

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