Advances in Molecular Structure Research

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ADVANCES IN MOLECULAR STRUCTURE RESEARCH

Volume 2

91996

This Page Intentionally Left Blank

ADVANCES IN MOLECULAR STRUCTURE RESEARCH Editors" MAG DOLNA HARG ITTAI Structural Chemistry Research Group Hungarian Academy of Sciences Budapest, Hungary p

ISTVAN HARGITTAI Institute of General and Analytical Chemistry Budapest Tochnical University Budapest, Hungary ,

VOLUME 2

91996

@ Greenwich, Connecticut

JAI PRESS INC.

London, England

Copyright 91996 by JAi PRESSINC 55 Old Post Road, No. 2 Greenwich, Connecticut 06836 JAI PRESSLTD. The Courtyard 28 High Street Hampton Hill, Middlesex TWl 2 1PD 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: 0-7623-0025-6 Manufactured in the United States of America

CONTENTS LIST OF CONTRIBUTORS PREFACE

Magdolna Hargittai and Istv~n Hargittai

o~

VII

ix

CONFORMATIONAL PRINCIPLES OF CONGESTED ORGANIC MOLECULES" TRANSIS NOT ALWAYS MORE STABLE THAN GAUCHE

Eiji Osawa

TRANSITION METAL CLUSTERS" MOLECULAR VERSUS CRYSTAL STRUCTURE

Dario Braga and Fabrizia Grepioni

25

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

67

103

129

163

vi

CONTENTS

STRUCTURE DETERMINATION USING THE NMR "INADEQUATE" TECHNIQUE

Du Li and Noel L. Owen

191

ENUMERATION OF ISOMERS AND CONFORMERS: A COMPLETE MATHEMATICAL SOLUTION FOR CONJUGATED POLYENE HYDROCARBONS

Sven J. Cyvin, Jon Brunvoll, Bjlarg N. Cyvin, and Egil Brendsdal

INDEX

213 247

LIST OF CONTRIBUTORS Valerio Bertolasi

Department of Chemistry University of Ferrara Ferrara, Italy

Dario Braga

Department of Chemistry University of Bologna Bologna, Italy

Egil Brendsdal

Department of Physical Chemistry The University of Trondheim Trondheim, Norway

Jon Brunvoll

Department of Physical Chemistry The University of Trondheim Trondheim, Norway

Bjerg N. Cyvin

Department of Physical Chemistry The University of Trondheim Trondheim, Norway

Sven J. Cyvin

Department of Physical Chemistry The University of Trondheim Trondheim, Norway

Valeria Ferretti

Department of Chemistry University of Ferrara Ferrara, Italy

Gastone Gilli

Department of Chemistry University of Ferrara Ferrara, Italy

Paola Gilli

Department of Chemistry University of Ferrara Ferrara, Italy

Fabrizia Grepioni

Department of Chemistry University of Bologna Bologna, Italy

vii

viii

LIST OF CONTRIBUTORS

Clark R. Landis

Department of Chemistry University of Wisconsin Madison, Wisconsin

Kenneth R. Leopold

Department of Chemistry University of Minnesota Minneapolis, Minnesota

Du Li

Department of Chemistry and Biochemistry Brigham Young University Provo, Utah

Vladimir S. Mastryukov

Department of Chemistry The University of Moscow Moscow, Russia

Eiji Osawa

Department of Knowledge-Based Information Engineering Toyohashi University of Technology Toyohashi, Japan

Noel L. Owen

Department of Chemistry and Biochemistry Brigham Young University Provo, Utah

Stanley H. Simonsen

Department of Chemistry and Biochemistry The University of Texas at Austin Austin, Texas

m

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 will be critically evaluating a field or direction including their own achievements, and charting expected developments. The present volume is the second in this series. We would appreciate heating 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. Magdolna and Istv~ Hargittai Editors

ix

This Page Intentionally Left Blank

CONFORMATIONAL PRINCIPLES OF CONGESTED ORGANIC MOLECULES" TRANS IS NOT ALWAYS MORE STABLE THAN GAUCHE

Eiji Osawa

I. II. III. IV.

V.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Molecules Like to be Folded . . . . . . . . . . . . . . . . . . . . . . . . How Does Intramolecular Congestion Develop? . . . . . . . . . . . . . . . . . trans-Form is Not Always More Favorable than gauche . . . . . . . . . . . . . A. 2,3-Dimethylbutane and 1,1'-Dicycloalkyls . . . . . . . . . . . . . . . . . B. (t-Bu-C60)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ABSTRACT This short essay is concerned with the relatively narrow topic of predicting stable conformation in large and crowded molecules. Two new rules are presented: (1)

Advances in Molecular Structure Research Volume 2, pages 1-24 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

1 2 4 6 15 16 18 19 21 21

2

EIJI i~SAWA congestion does not occur unless unrelievable strain develops around GG' segments, and (2) the trans or extended form is not always a safeguard for the conformational preference. A magic intramolecular C...C distance of 3.9/~ between the end carbon atoms of 1,4-trans (T) and 1,5-GG sequences produces 200 cal/mol of stabilization per pair due to the van der Waals attraction. Warning is given to the unwarranted extension of conformational rules for small molecules to large molecules. A remarkable case of the gauche-conformer becoming the global energy minimum 10 kcal/mol below the trans-conformer is mentioned for a novel buckminsterfullerene-based dimer with tert-butyl substituents.

I. I N T R O D U C T I O N Chemical literature is abound with errors [1]. Sometimes, a seemingly secure experiment like X-ray crystallographic analysis may contain serious errors. For instance, I myself have twice found critically important bond lengths obtained from X-ray analysis in error [2]. Very rarely, the whole paper can be 100% wrong, creating embarrassments for both authors and editor [3]. As long as errors have been committed unintentionally, we cannot blame the authors or editor. Rather, errors may be regarded as the fertilizers of science. I wish to begin this chapter by citing an interesting case, which cannot exactly be called an error, but is difficult to detect. A conformation of the novel molecule l a has recently been depicted in a communication [4]. The molecule is a novel dimeric buckminsterfullerene with tert-butyl substituents, wherein the substituents are believed to exist near the pivot bond, namely in the para position in a six-membered ring relative to the pivot carbon atom. The authors of the paper [4] performed molecular modeling of this congested molecule using a commercial software package, hence it may be reasonably assumed that they imply an energyminimum by la. Many chemists will accept this conformation: the bulky, space-demanding substituents are positioned as far apart as possible and tert-butyl groups are rotated

la

Conformation of Congested Molecules

3

lb such that the largest possible separation is produced between the methyl hydrogens and the C60 cage to which they are pressed hard upon, even though Cq-Me bonds are eclipsed with the cage C-C bonds. According to our analysis [5], however, this conformation proved to be an energy maximum, and the global energy-minimum conformation turned out to be the unexpected, apparently crowded conformer, lb! Believe it or not, this particular example illustrates two key features which I believe are relevant to the conformational analysis of large and congested molecules. Let me explain further the purpose of this article. Chemists already know the basic rules for the conformation of small molecules [6]. Nevertheless, straightforward extension of these rules, such as "equatorial is more stable than axial on a cyclohexane ring" and "trans form of large molecules is more stable [ 7] than gauche to large molecules," may be dangerous [8]. As shown in the above example, we are inclined to apply these rules rather blindly. Because the first rule can be reduced to the second (an axial substituent produces two gauche interactions while an equatorial substituent produces two trans interactions, Scheme 1) [9], I will concentrate in this chapter on the validity of one of the cornerstones in conformational analysis, the preference oftrans over gauche, when applied to large and crowded molecules. axial

2 gauche interactions

equatorial

2 trans interactions Scheme 1.

4

EIJI i~SAWA

ii. LARGE MOLECULES LIKE TO BE FOLDED It is generally believed that the gauche form of, e.g. n-butane, is less stable than the trans form because there are close approaches of atoms in the former. Is it true then that the close approach of atoms always leads to crowding or destabilization? I begin this section by disclosing the fact that many conformations of large molecules which look crowded are not really crowded at all. In the course of the past few decades since molecular modeling began [ 10], we have learned that molecules have many ways of relieving internal strain. When atoms bump into each other, the molecule calls for various modes of nuclear movements in increasing order of energy expense--torsion or bond rotation first, then valence angle deformation, and finally bond stretching--to separate the close atoms apart. In most cases, a compromise is reached between the energy expense that is paid for deforming the structure and the energy gain that comes from the decreased nonbonded repulsions. A good example showing the smartness of molecules in avoiding overcrowding is podophyllotoxin 2. Its CPK model gives a very tight structure wherein the E ring has to be forcibly squeezed into a small space above C ring in order to maintain the axial disposition on the E ring. In reality, however, two ortho-hydrogens, H 2, and

OH >

O CH30~~~OCH3 cH3o

Clip

Conformation of Congested Molecules

5

H 6, of the E ring, appear as a singlet in its NMR spectrum at room temperature, which splits into a doublet only when cooled t o - 1 3 6 ~ This means that the E ring rotates fast at room temperature in the small space over the fused rings [11]. The rigid feeling that we perceive from a CPK model is an artifact caused by the lack of freedom of angle deformations and bond stretches, and also by the hard and smooth plastic surface. Actually all real molecules would feel flexible and soft like a sea slug if we could ever touch them. Tactics will be mentioned later in this chapter which smart molecules employ to avoid congestion. Not only are the molecules able to deform extensively to avoid congestion, but often times it seems that they rather like a crowded situation. Numerous examples [12,13] are known wherein folded conformations are preferred over extended conformations. In methotrexate 3, for example, hydrogen bonding of the O-H...N and N-H...O types, donor-acceptor interaction between stacked aromatic rings, and attractive van der Waals interaction in a number of atom pairs across the inner space,

H~~N.../N~

~H~CO~.

-W NH2

CH3

\

\

~ C

~

/

H H :

CH~,

O~ C ~

NH2

i CH3

6

EIJI GSAWA

combine to stabilize a folded conformation as shown. Indeed this conformation has been observed to dominate in THF/CH2CI 2 solution [12m]. In addition to the folded J-conformations [12], rings [14], and helices [15] are the prominent forms of molecules that we often encounter, especially in nature. Does nature have any special reason for preferring these over the extended forms? I would like to introduce here an obvious implication of molecular modeling functions as an ubiquitous source of attractive interactions that exist specifically in rings and helices. A consequence of atom-atom pair interaction theory is that an attractive force should operate in every nonbonded C...C pair separated by a van der Waals distance of 3.9 ]k [16]. At this distance, hydrogen atoms attached to these carbon atoms also tend to be in the regions of strongly attractive C...H and H...H interactions. This is a natural (albeit somewhat artificial) consequence of using conventional van der Waals potential functions of the Lennard-Jones type, which always produces an energy well at some distance between any combination of atom pairs. A pair of carbon atoms at the ends of gauche-gauche (GG) sequence are separated exactly by this distance, and, combined with C-..H and H...H attractive interactions involving the attached hydrogen atoms, produce a stabilization energy of-0.20 kcal/mol (per one GG sequence) according to MM3 force field [17]. No one would be convinced by such a conclusion based on empirical evidence from molecular mechanics, unless some ancillary supporting results are put forward. Fortunately, however, since the last few years it has become clear that we can replace some kinds of experiments with theoretical simulations. Consensus has now been reached in the chemical community that several types of data obtained from the high-level ab initio MO calculations using well-trained, large, and composite basis orbitals and including appropriate correction for the electron correlation effect are as good as experimental data [18]. Molecular properties of simple organic molecules that are reliably reproduced by these ab initio techniques include equilibrium structures and their relative stabilities [17]. Our MP4SDQ/631G*//HF/6-31G* calculations of n-pentane conformers published in 1991 [17] established that GG sequences really produce a similar value (--0.16 kcal/mol per pair) without using any artificial dispersion correction term [17,20]. In view of the facts that every comer [21] of a cyclic structure consists of a GG sequence and that helical structures are contiguous gauche sequences of the same sign ( .... GGG .... ), the attractive 1,5 atom-pair interaction appears to be working favorably for the frequent occurrence of cyclic and helical molecular structures.

II!. H O W DOES INTRAMOLECULAR CONGESTION DEVELOP? With so many kinds of intramolecular interaction available to stabilize compact conformations (fold, ring, and helix), when and how do molecules suffer from congestion [22]? Alternatively, one might ask how we can really produce conges-

Conformation of CongestedMolecules

7

tion. At this point, I ask the readers to allow me to deviate for the moment from my main theme and discuss congestion. I should first emphasize that naive strategy of simply packing bulky groups into a small intramolecular space to produce crowding has generally fallen short of its goal, despite great many attempts that must have consumed so much time and efforts of synthetic chemists in the past [23]. The reason for these failures is that molecules are not so stupid as to be loaded limitlessly with strain" they usually rearrange to less crowded structures, or simply refuse to react. We should stop the simplistic drive to produce intramolecular congestion, but instead seek those cases where enormous strain develops under seemingly uncrowded circumstances, or without letting molecules realize incipient crowding. I would like to set our strategy straight by defining molecular congestion as a

conformational situation wherein high internal strain cannot be effectively relieved by any internal freedom available to the molecule. According to this definition, destabilization by the eclipsing of bonds is not counted as a cause of congestion because, in most cases, eclipsing can be readily overcome by bond rotation as long as the bond is capable of rotating. We basically consider conformations containing only staggered bonds. On this basis, general causes of internal strain are G, GG' and GG'G sequences. Let us first analyze GG'. The sequence has been well known but rarely [24] systematically analyzed as the source of strain in conformational analysis. According to our analysis, this system is not a real threat in the open systems. The simplest example of an open GG' sequence is seen in n-pentane (Scheme 2), where both end methyl groups would approach very close if G and G' take the "standard" gauche dihedral angle of +60 ~. However, there are such large freedom of nuclear movements in n-pentane that strain can be well relieved by internal deformation. When GG'-n-pentane is subjected to energy-minimization by molecular mechanics under Cs constraints, the CH2-CH 2 bonds spontaneously rotate while opening up MeCH2-CH 2 and CH2-CH2--CH2 valence angles (see GG' in Scheme 2). When the Cs constraint is removed, then the conformation further undergoes extensive and peculiar skeletal deformation to give a strange structure having no element of symmetry, as shown by [ ~ ' and GIG --7] in Scheme 2 [25]. Dihedral and valence angles entered in Scheme 2 are obtained by high-level ab initio calculations [17,26], and almost identical values are obtained by molecular mechanics calculations as well [27-29]. Deviations of these angles in the optimized geometry of GG'-n-pentane from the "standard" values are quite large, and they contrast with those of G sequence in the optimized geometry of gauche-n-butane (4): the central CH2-CH 2 bond has been rotated by 6 ~ from the standard value to increase Me...Me distance and Me-CH 2 bonds by 4-7 ~ from a local C3v structure to increase the close 1,6-H...H distance [17]. Hence, compared to GG', G is easy to relax and will not contribute seriously to generating congestion according to our definition.

8

EIJi iSSAWA GG

"116"

"

78~

-78*

J \

T

-

"..116~ 115"~ 95~

/

V

-63*

63~

_95o

cD Scheme 2.

H.,~ ...... ~.-H H

H

( H

H

GG' strain is often referred to as "1,3-diaxial interaction" because GG' sequence is contained in the diaxial form of cis-1,3-dimethylcyclohexane 5. This molecule spontaneously transforms itself into the TT form (trans-trans, 5') by ring inversion, thus completely disposing of the GG' strain [6]. Such a transformation would be impossible, however, in fused cyclohexane systems like friedelin (6). Those portions of friedelin which contain 1,3-diaxial-dimethyl substituents should certainly be congested in accordance with my definition. Nonetheless, this large molecule Me

Me

5

5'

Conformation of Congested Molecules

9

o

Me M

Me

Me fl~

l~le

1~Ie

Me

Me Me ~ M e !

Me

must have a number of degrees of freedom for small deformations and, accordingly, the amount of total strain does not seem to be forbiddingly high to exclude its existence: indeed, friedelin is a natural product [30]. Let us turn to more congested systems where only very minor deformation is allowed, or any deformation is totally impossible around a multitude of GG' sequences, so that the unrelieved strain stays at a high level. In order to see how the congestion builds up, we start from the simplest possible hydrocarbon, methane, and replace its hydrogens one by one with tert-butyl groups. It is well known that the successive substitution succeeded up to the third stage to give di- [31] and tri-tert-butylmethane [32], but the fourth stage failed. Nonetheless, the unknown tetra-tert-butylmethane 7 can be geometry-optimized into several energy minima [33,34]. The global minimum has T symmetry wherein all Me-Cq bonds are rotated by about 15 ~ in the same sense [35]. In all of the computed minima, the molecule contains 12 pairs of GG' interactions. As they are distributed over a small spherical surface, there is no room to release strain except for very limited adjustments of bond angles and lengths. One aspect of this molecule which has so far been left M| e

Me

',Me

M e ~ ~ e _..._.._/ "'~~MMe Me"~'~Me Me

10

EIJI i~SAWA

Me,,,.".. ~,,.~M e,,,,Me Me l~d l~Ie ~de M

~ M e l~Ie ~ e Me Me.Me Me

tB~~_Me

Me

tB~" ~" ~tBu tBu

unnoticed is that it has 12 pairs of stabilizing GG interactions, which might have some relevance to its successful geometry optimization and gives some hope for its existence, although perhaps only as a transient species. Similarly, replacement of all hydrogens of ethane with tert-butyl groups gives another unknown molecule, hexa-t-butylethane (8), which must be tremendously strained, formally containing 12 GG', and 6 GG'G sequences. Although 8 is still an imaginary molecule, it is our first encounter with a GG'G sequence in this review. To my knowledge, no example of GG'G-containing conformer is known as yet. Even in a computer, GG'G conformation of n-hexane is not a stationary point, but spontaneously transforms itself into a different conformer during the geometry optimization [ 17,27]. This seems to be actually happening in reality. For example, the boat conformation of D and E rings in friedelin 6 [36] is apparently a consequence of avoiding GG'G sequence which would appear in the all-chair conformation (6'). As a matter of fact, the high congestion of 7 and 8 is evident by a glance at the structure. They belong to those old brute-force cases where the bulky tert-butyl groups are squeezed into a very small space. We should rather look for apparently unstrained but actually congested molecules. Let us start again from methane. Replacement of all four hydrogens with ethyl groups gives tetraethylmethane or 3,3'-diethylpentane 9, a known molecule. The conformation shown (D2d) may

MM~~~.~...~ e e Me Me Me

Conformation of Congested Molecules

11

appear stable: two molecules of n-pentane in its most stable TT conformation being jointed together at their centers. However, it does contain eight G interactions. The only available internal freedom, rotation of a Cq--CH2 bond by 120 ~ annihilates two G sequences but simultaneously creates one G and one GG' interaction. Hence any change from this conformation should increase energy. The reason for the surprisingly low strain in 9 (8.93 kcal/mol according to MM3) can be ascribed to the fact that this global minimum conformation contains four GG sequences [37]. Replacement of six hydrogens of ethane with ethyl groups provides hexaethylethane or 3,3,4,4-tetraethylhexane 10, which again may appear normal but is actually highly crowded. There are six GG' interactions on both ends and six G interactions across the central C--C bond. Despite an open look, ways of relieving strain are limited: rotation of a Cq-CH 2 bond by 120~ would eliminate two GG' interactions but produce one new GG' pair across the central bond. Conformation 10' proved to be the global energy-minimum by molecular mechanics calculation [38].

..............:

..............,

/

,

)

/

10

...........

10'

The crystal conformation of 11 has several conformational features common to 10 [39]: two GG' and four G in addition to eight GG'-like interactions involving ortho-Car carbons. The reason for 11 not taking an asymmetric conformation like 10' in the crystal must be the better packing of the all-trans chains compared to the gauche-containing chains.

~l||O|eO0

11

12

EIJI iSSAWA

Me~"~

Me

12a

Me M

e

~

~

Me

N~ "'Me 12b

The rigid conformation of 9 and 10 are caused by the fact that the neighboring conformations (those which can be reached by one-step bond rotation) are of high energy. An alternative possibility for a rigid conformation will be the one surrounded by a high-rotational barrier. On the other hand, it would not be surprising if a really congested, high-energy conformation becomes a barrier of conformational interconversion, even if it has a staggered arrangement of bonds. A remarkable e x a m p l e e x a c t l y m a t c h i n g this s i t u a t i o n is p r o v i d e d by N,N'-bi(cis-2,4-dimethylpiperidine) 12. When the rotational barrier of the N-N bond of this well-designed [40] molecule was first studied, it was intuitively assumed that a staggered conformation 12a had to be the global energy-minimum [41]. However, molecular mechanics calculations have shown that the conformation 12a is a saddle point that appears in the course of rotation about N-N bond, while the global energy-minimum conformation is gauche (12b) [42]. Inspection of the changes in steric energy during the pivot bond rotation reveals that a pair of GG' sequences, Me-Ct-N-N-CH2, which are generated as the bond is rotated towards the trans form 12a, are the cause of the unexpected high steric energy of this conformation. Due to the rigid chair piperidine tings [40] and the S2 symmetry, there is little freedom of deformation around the pivot bond in conformation 12a. It seems that the pair of GG' sequences are firmly locked into an inescapably difficult situation. In order to distinguish this novel conformational transition state from the conventional eclipsed ones, we named it a "staggered barrier". This case proved to be one of the early examples wherein molecular mechanics helped correct errors in experimental studies [42]. Nitrogen atoms in 12 may be replaced with carbon, hence ortho-substituted 1, l'dicycloalkyls are supposed to show staggered barriers [43]. As illustrated in Figure 1, introduction of one methyl group (13b) next to the pivot bond of dicyclohexyl

/

13a (o)

13b (e)

/

\

13d (z~)

13c (G)

C6'

H

C2'

A

(C)

D

A

(B)

,,,,,C],

~15 m

(C)

(B')

,~, ,,,,

D

o

E

m O .-~10 O') r

sl.... (9 (9

.o 5 t._ -J (9 n-

O

I

-5o

I

o

I

so

(016-1-1 '-6') (deg)

~

loo

!

1so

Figure 1. Relative steric energy vs. rotation angle of pivot bond in methylated dicyclohexyls (13a-d). Only the energy minima and saddle points are given. 13

14

EIJI OSAWA H

equatorial perpendicular

equatorial parallel

14 Scheme 3.

(13a) causes the trans or coplanar rotamer (C) becoming a small saddle point. Dimethyl dicyclohexyls (13e,d) produce distinct staggered barriers. All these occur by a common cause, namely unavoidable GG' interactions. Phenylcyclohexanes 14 (Scheme 3) acquires striking similarity to dicyclohexyls, when substituents are introduced at the ortho positions to the pivot bond: the equilibrium between equatorial-parallel and equatorial-perpendicular conformations shifts completely to the left (e-par) due to the strong GG'-like repulsive interactions involving substituents R and R' in the e-per conformation [43b]. Once again, prohibitively high dynamic congestion would develop if the pivot bond of 14 were forced to rotate. In other words, the rotation about the pivot bond will be made very difficult or forbidden if a small substituent or two are present in the strategic positions. The surprisingly high barrier ( 14.7 kcal/mol in CDC13) observed with cannabidiol 15 is due to the latent congestion in the e-per conformation [44]. There are other examples of arylcycloalkanes having stable e-par conformation protected by high rotational barrier about the pivot bond such as thalidomide (16) [37], a morpholinocyclohexene (17) [45], catechin (18) and its dimers [46], and 2'-methyl anabasine (19) [47,48].

CsHI1n

/

/ OH

15

o 16

Conformation of CongestedMolecules

15

SOzCH2CN

17

H

.""

~

OH

OH

OH OH

18

Me

19

IV.

TRANS.FORM IS NOT ALWAYS MORE FAVORABLE THA N GAUCHE

Having defined and briefly surveyed the nature of intramolecular crowding, we go back to the first-mentioned topic on the unexpected instability of trans forms in large molecules. As mentioned above, there is no a priori reason to believe that a compact conformation must always be less stable than an extended one. The lower energy of simple 1,4-trans forms (like trans-n-butane) compared to 1,4-gauche (like gauche-n-butane) arises at least partly from an incidental fact that the distance in the 1,4-gauche pair is such that the pair interaction falls in repulsive region. Beyond 1,4-pairs, relative stabilities among conformers should vary depending on individual situation. In the first place, I mention that the trans form is by no means strain-free, but contains several elements of strain. Let us take a zigzag conformation of n-hexane (20, C2h) as an example. This conformation contains eight pairs of 1,3-cis-H/H repulsive interactions along the backbone chain (double arrows). While certainly less pronounced than 1,3-cis-C/C repulsion (GG') mentioned above, these H/H interactions accumulate in the interior of chain and manifest themselves in small but significant increase in the internal C-C--C valence angles and C-C bond lengths

16

EIJI iDSAWA

V !

H H

-

H H

V _

H H

20 compared to those at the end of chain [49,50]. The numbers entered in 20 have been obtained by ab initio calculations [17,26,51]. One would naturally ask why a carbon chain of all zigzag form does not twist to get rid of these minute strain sources, thus giving rise to a helical structure with long periodicity. One easy answer to this question would be that the completely planar and linear all-trans chains are advantageous in close packing in crystals. As a matter of fact, n-alkanes always crystallize into planar C2h (even carbon atoms) or C2v (odd carbon) conformation. However, planar structures are also obtained for n-alkanes in vapor or solution phase by experiments as well as by theoretical calculations. To the best of my knowledge, a tenable interpretation has never been given to this question. I propose here a hypothesis which might well solve this fiddle: there is one more element of stabilization that has been overlooked and this stabilizing force keeps the all-trans form in planar conformation. The magic C...C distance of 3.9/k that produces 0.16 to 0.2 kcal/mol of van der Waals attraction mentioned above occurs not only in 1,5-GG sequence but also between the end atoms of 1,4-trans sequence [52]. This means that every 1,4-CH2/CH 2 and 1,4-CH2/CH 3 pair along the zigzag chain receives this attraction, although it may be smaller than when going through a vacuum because it works through those portions of space which are heavily populated with bonding electrons of high dielectric (21).

3.9/~ 21 A few examples are given below wherein the trans form is not necessarily more stable than the gauche form, as revealed by experiments and/or computation.

A. 2,3-Dimethylbutane and 1,1'-Dicycloalkyls A simple hydrocarbon, 2,3-dimethylbutane, is a classical example wherein the

trans form (22a) has almost equal energy as the gauche form (22b) [52]. The

Conformation of Congested Molecules

17

It Me.Me

H M

H

Mel " ~ ~Me

e

/ H

Me

(b) gauche

(a) t r a n s

22 explanation was given previously with emphasis on symmetry requirements (trans C2h, gauche C2). Little freedom exists in the former to relieve gauche-Me~Me interactions except for limited valence angle deformations, due to the presence of a mirror plane passing through H-Ct-Ct-H. On the other hand, axial symmetry in the latter allows the central bond to rotate so that the strain can be relieved by deforming towards the less crowded portion of the molecule. Dicyclohexyl (13a)can be regarded as a ring analogue ofdimethylbutane: Figure 1 already demonstrates that this molecule has comparable stabilities for trans (C) and gauche (A) forms. This prediction has been confirmed by MM3 calculations [53] and electron diffraction analysis [54] (Table 1). Smaller dicycloalkyls like dicyclobutyl and dicyclopentyl (22, n = 3, 4) shows the trans dominance, apparently

H

H

2.3

Table 1. Population of Gauche Forms in Dicycloalkyls (22) a n

gauche %

Method

Ref

2

52

EDb

53

Dicyclobutyl

3

72 40

MM3 ED

53 53

Dicyclopentyl Dicyclohexyl

4 5

6 10 47

MM3 MM3 ED

53 53 54

58

MM3

53

Dicyclopropyl

Notes: aFordetails of populating conformers, see references 53 and 54. Generally,junction is equatorial--equatorial and rings are in its most stable conformation. bElectron diffraction at 112 ~

18

EIJI DSAWA

because of the increase in 1,4-C/C distance across the pivot bond. (Dicyclopropyl (23, n = 2) is an exception [53]). Conformational energies in 22 and 23 depend basically upon 1,4-C/C interactions across the central C-C bond, hence these examples belong to the classical, shortrange chemistry. We return to the long-range interactions in the next example. B. (t'Bu'C6o)2 This molecule, touched upon at the beginning of this chapter, is unique in that its congestion is not caused by the unavoidable GG' or G sequences discussed above. First, we better simplify the situation by presenting an energy minimum (lc) for the trans form [55] of 1, wherein the eclipsed tert-butyl groups in l a have been rotated by 60 ~ to a staggered conformation. During this rotation, steric energy decreases by 16 kcal/mol. Analysis has revealed the following reasons for the novel preference of gauche (la) over trans (lc): 1. There is no repulsive interaction but rather distinct attractive stabilization (amounting to 1 kcal/mol) between the two tert-butyl groups in the optimized geometry of gauche form lb. Hence a generally held idea of repulsive interaction between close substituents does not apply to lb. 2. Nonbonded repulsion between a tert-butyl group and the C60 cage which it faces quickly develops as soon as the rotation about the pivot rotation enters into anti region (-90 ~ > dihedral angle > 90~ The situation can be readily understood by invoking a lever effect. A close look at the drawings le and l b will reveal that tert-butyl groups are squeezed into small space between the two C60 cages and produce such severe crowding that the pivot as well as the t-Bu--C60 bonds are greatly bent away from their normal directions. The swing motion (thick arrow) of

lc

Conformation of CongestedMolecules

19

trans

lc'

gauche

lb'

the C60 cage about the pivot bond with both its ends as the fulcrums generates contrasting effects, depending on whether the two tert-butyl groups are in anti or syn disposition relative to the pivot bond. In the trans form (lc'), swing of the left cage in order to decrease the critical nonbonded repulsion in the upper part (double-tipped arrow), increases a similar repulsive interaction in the lower part like a lever action and vice versa. By contrast, in the gauche form (lb'), swing of one cage simultaneously decreases the other repulsion. While lc corresponds to the worst point in the lever effect, l b represents a much less severe situation in the absence of the lever effect [56]. Going back to the extent of error in the study of la, we point out here that the computed energy difference between the global energy minimum l b and the energy maximum l a amounts to 26 kcal/mol.

V. PERSPECTIVES In retrospect, conformational analysis has been dominated by 1,4-interactions, which operate over relatively short atom-atom distances [57-59]. The seminal series of Dale's work which appeared in 1973 [60] on the then new concept in conformational analysis of cycloalkanes is based almost entirely on the gauche interaction (butane model). As a matter of fact, conformational analysis has so far been preoccupied with the idea of gauche interaction energy relative to the trans

20

EIJI OSAWA

GG'

GG

V Attractive

Repulsive

Scheme 4.

form as the standard strainless model. For a long time, counting the number of gauche sequences has served as practically the only way of estimating relative stabilities among possible conformations until molecular mechanics came to the fore. Larger systems involve longer range interactions than 1,4-type. I discussed above mainly 1,5-interactions, with only a few references to 1,6-interactions. Even though these have been expressed as the combinations of 1,4-interactions like GG' and GG'G, I have indicated that there should be phenomena peculiar to the longer range interactions, which could not have been foreseen or understood on the basis of the knowledge of small systems alone. I once again emphasize contrasting behavior of contiguous gauche sequences (Scheme 4). How far should we go in search of long-range 1,n-interactions (n > 5), in order to understand the conformational behaviors of large systems like biopolymers? It seems likely that there will be some limit on the number of essential interactions. We may have to find more, especially for those systems which contain heteroatoms. As illustrated in this chapter, the atom-atom interaction theory used in molecular mechanics is still useful in this area. Once we find all the essential rules, then we can switch to a more robust and faster theory than the meticulously elaborate but too-slow atom pair or LCAO-MO theories, so that we will be able to simulate actual biosystems efficiently [61]. Finally, I cannot finish this chapter without referring to the power of microscopic simulation. We have seen that the peculiar deformation of the free GG' sequence has been confirmed by "computer experiments", namely by the high-level ab initio calculations. This experiment eventually led to a new definition of intramolecular congestion and the mechanism of its genesis. A more positive application of computer experiments will be to analyze the results of simulation like the first principle molecular dynamics [62] in submolecular level to find out rules that control the conformation of large molecules. This should be the surest way to uncover the secrets of protein folding, molecular recognition, and self-assembly, conformational phenomena which dominate in nature but which we still cannot predict with confidence.

Conformation of Congested Molecules

21

ACKNOWLEDGMENTS I am grateful to my eldest son, Shuichi, who was my Ph.D. student at the time of writing this chapter, for discovering the generality of "staggered barriers", and for many helpful discussions.

REFERENCES AND NOTES 1. In this review, we do not talk about fraud but only of innocent errors. As to the former see, e.g. Hazen, R. M. The New Alchemists: Breaking Through the Barriers of High Pressure, Times Books: New York, 1994. 2. (a) Osawa, E." Onuki, Y.; Mislow, K. J. Am. Chem. Soc. 1981, 103, 7475. (b) Dougherty, D. A.; Choi, C. S.; Kaupp, G.; Buda, A. B.; Rudzinski, J. M." Osawa, E. J. Chem. Soc., Perkin Trans. 2 1986, 1063. 3. A recent example: Menger, E M.; Haim, A. Nature 1992, 359, 666. 4. Morton, J. R.; Preston, K. E; Krusic, P. J.; Hill, S. A.; Wasserman, E. J. Am. Chem. Soc. 1992,114, 5454. 5. Osawa, S.; Osawa, E.; Harada, M. J. Org. Chem., in press. 6. See for example, (a) Juristi, E. An Introduction to Stereochemistry and Conformational Analysis, Wiley: New York, 1991. (b) Anderson, J. E. In The Chemistry of Alkanes and Cycloalkanes; Patai, S." Rappoport, Z., Eds.; John Wiley & Sons: New York, 1992, Chapter 3. 7. Throughout this review, the word 'stable' refers to either All or AG depending on the context. 8. It is well known that 1,2-disubstituted ethanes with highly electronegative atoms or groups (F, OR, NO 2, CN, etc.) are more stable in gauche than in trans form: (a) ref. 6. (b) Tan, B. G.; Lam, Y. L." Huang, H. H.; Chia, L. H. C. J. Chem. Soc., Perkin Trans. 2 1990, 2031. (c) Lam, Y. L." Huang, H. H." Hambley, T. W. J. Chem. Soc., Perkin Trans. 2 19911,2039. (d) Chong, Y. S 9Chia, L. H. L.; Huang, H.-H. J. Chem. Soc., Perkin Trans. H 1985, 1567. (e) Koh, L.-L.; Lam, Y." Sim, K. Y.; Huang, H. H. Acta Cryst. 1993, B49, 116. (f) Lam, Y.-L.; Koh, L.-L.; Huang, H. H. J. Chem. Soc., Perkin Trans. 2 1993, 175. The "gauche effect" is caused by intramolecular charge interactions, but we do not discuss these complications. We concentrate on hydrocarbons. 9. With regard to violations of the equatorial preference in cyclohexanes, see (a) Osawa, E.; Varnali, T. In Conformational Studies of Six-Membered Rings, Juristi, E., Ed.; VCH Publishers: New York, Chapter 4. (b) Anderson, J. E.; Ijeh, A. I. J. Chem. Soc., Perkin Trans. 2 1994, 1965. 10. (a) Osawa, E.; Musso, H.Angew. Chem. Int. Ed. Engl. 1983, 22, 1. (b) Molecular Mechanics and Modeling; Davidson, E. R., Ed." a special issue in Chem. Revs. 1993, 93, 2337-2582. 11. Rithner, C. D.; Bushweller, C. H.; Gensler, W. J.; Hoogasian, S. J. Org. Chem. 1983, 48, 1491. 12. (a) Liberles, A.; Greenberg, A.; Eilers, J. E.J. Chem. Educ. 1973, 50, 676. (b) Tel, R. M.; Engberts, J. B. E N. Rec. Trav. Chim. Pays-Bas 1974, 93, 37. (c) Tel, R. M.; Engberts, J. B. E N. J. Chem. Soc., Perkin Trans. 2 1976, 483. (d) Ellefson, C. R." Swenton, L.; Bible, R. H.; Green, P. M. Tetrahedron 1976, 32, 1081. (e) Bleeker, I. P.; Engberts, J. B. E N. Rec. Trav. Chim. Pays-Bas 1977, 96, 58. (f) Tickle, I.; Hess, J." Vos, A.; Engberts, J. B. E N. J. Chem.. Soc., Perkin Trans. 2 1978, 460. (g) Visser, R. J. J.; Vos, A.; Engberts, J. B. F. N. J. Chem. Soc., Perkin Trans. 2 1978, 634. (h) Exner, O.; Engberts, B. E N. Coll. Czech. Chem. Commun. 1979, 44, 3378. (i) Murray, W. J.; Reed, K. W.; Roche, E. B. J. Theor. Biol. 19811,82, 559. (j) Creighton, T. E. J. Phys. Chem. 1985, 89, 2452. (k) Sanders, G. M.; van Dijk, M.; Koning, G. P." van Verdhuizen, A.; van der Plas, H. C. Rec. Trav. Chim. Pays-Bas 1985, 104, 243. (1) Simon, K.; Podanyi, B.; Ecsery, Z.; Toth, G. J. Chem. Soc., Perla'n Trans. 2 1986, 111. (m) Faupel, P." Buss, V. Angew. Chem. Int. Ed. Engl. 1988, 27, 422. (n) Grunewald, G. L.; Creese, M. W.; Weintraub, H. J. R. J. Comput. Chem. 1988, 9, 315. (o) Loncharich, R. J.; Seward, E." Ferguson, S. B.; Brown, E K.; Diederich, E" Houk, K. m

E

22

EIJI i~SAWA

N. J. Org. Chem. 1988, 53, 3479. (p) Stevens, E. S 9Sathyanarayana, B. K. J. Am. Chem. Soc. 1989, 111, 4149. (q) Nishio, M.; Hirota, M. Tetrahedron 1989, 45, 7201. (r) Ihn, K. J.; Tsuji, M.; Isoda, S.; Kawaguchi, A.; Katayama, K.; Tanaka, Y.; Sato, H. Macromolecules 1990, 23, 1781. (s) Shida, N.; Uyehara, T." Yamamoto, Y. J. Org. Chem. 1992, 57, 5049. (t) He, Y.-B.; Huang, Z.; Raynor, K.; Reisine, T.; Goodman, M. J. Am. Chem. Soc. 1993, 115, 8066. (u) Zhu, G.-D.; Van Haver, D." Jurriaans, J.; De Clercq, P. J. Tetrahedron 1994, 50, 7049. 13. Yonemitsu, O. J. Synth. Org. Chem., Jpn. 1994, 52, 946. 14. (a) Semlyen, J. A. Pure & AppL Chem. 1981, 53, 1797. (b) Semlyen, J. A. Chem. Brit. 1982, 18, 704. 15. Hargittai, I.; Hargittai, M. Symmetry, a Unifying Concept; Shelter Publ.: Bolinas, CA, 1994, Chapt. XII. 16. This distance is close to twice the van der Waals radius of carbon atom used in the MM3 force field [Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J. Am. Chem. Soc. 1989, 111, 8551]. 17. Tsuzuki, S 9Sch~ifer, L.; Goto, H.; Jemmis, E. D.; Hosoya, H.; Siam, K.; Tanabe, K.; Osawa, E. J. Am. Chem. Soc. 1991, 113, 4665. 18. (a) Boggs, J. E. Pure & Appl. Chem. 1988, 60, 175. (b) Kutzelnigg, W. J. Mol. Struct. (Theochem) 1989, 61, 11" 1988, 50, 1. See however, Mack, H.-G." Oberhammer, H. J. Mol. Struct. (Theochem) 1992, 258, 197. 19. Among the easiest are the gross electronic properties like dipole moments, while the hardest are the absolute energies like heat of formation. For the latter problem, see McGrath, M. P.; Radom, L. J. Chem. Phys. 1991, 94, 511. 20. Sch~ifer has performed extensive ab initio calculations at higher levels of theory to obtain still larger stabilization energy for the GG interaction (private communication). He has further indicated the possible role of the attractive GG sequence in the sensitivity of a molecule to conformational mobility [Cao, M." Schger, L. J. Mol. Struct. (Theochem) 1993, 284, 235]. 21. A simple ring structure consists of equal numbered of edges (all anti sequence) and corners (GG or G'G' sequence). For an extension of this definition, see Goto, H. Tetrahedron 1992, 48, 7131. 22. The terminology "congestion" has been used for intermolecular crowding between reagent and substrate: Wipke, W. T.; Gund, P. J. Am. Chem. Soc. 1976, 98, 8107. However, I restrict this term to the conventional internal crowding in the present article. 23. (a) Tidwell, T. T. Tetrahedron 1978, 34, 1855. (b) Bushweller, C. H." Hoogasian, S.; Anderson, W. G. Tetrahedron Lett. 1974, 547. (c) Riichardt, C.; Beckhaus, H.-D. Angew. Chem. Int. Ed. Engl. 19811, 19, 429. 24. (a) Anderson, J. E.; Pearson, H. J. Am. Chem. Soc. 1975, 97, 764. (b) Hoye, T. R.; Peck, D. R." Swanson, T. A. J. Am. Chem. Soc. 1984, 106, 2738. 25. The notation[C;] or ~-qmeans a gauche sequence containing the large absolute dihedral angle near 100~ 26. Some of the 6-31G*-optimized values given in Scheme 2 are taken from Wiberg, K. B." Murcko, M. A. J. Am. Chem. Soc. 1988, 110, 8029. 27. Goto, H.; Osawa, E.; Yamato, M. Tetrahedron 1993, 49, 387. 28. I had noticed this phenomenon soon after I started using molecular mechanics in the mid-1970s [Osawa, E. In Mathematics and Computational Concepts in Chemistry; Trinajstic, N. Ed.; Ellis Horwood: Chichester, 1986, Chapter 21 ], although I learned later that the earliest record of the C1 conformation of GG' n-pentane had already appeared in 1966 [Scott, R. A.; Scheraga, H. A. J. Chem. Phys. 1966, 44, 3054]. I long thought that this must be an artifact of molecular mechanics much as I felt for the GG attraction, but neither experimental method nor theoretical alternative to prove this point were available at that time. 29. The amount of energy released from the hypothetical Cs GG'-n-pentane structure when it is relaxed to the C 1 structure is not large, only 0.28 kcal/mol according to our ab initio calculation (ref. 17). However, in view of the fact that the optimized Cs structure already is deformed, the relaxation energy from the 'standard' to C I-GG' conformation must be quite large (the molecule works hard!). m

Conformation of CongestedMolecules

23

30. Aoyagi, R.; Yamada, S.; Tsuyuki, T.; Takahashi, T. Bull. Chem. Soc. Japan 1973, 46, 959. 31. Bushweller, C. H.; Anderson, W. G." Goldberg, M. J.; Gabriel, M. W.; Gilliom, L. R." Mislow, K. J. Org. Chem. 1980, 45, 3880. 32. (a) Lomas, J. S. Nouv. J. Chim. 1984, 8, 365. (b) Tiffon, B 9Lomas, J. S. Org. Magn. Reson. 1984, 22, 29. 33. Iroff, L. D.; Mislow, K. J. Am. Chem. Soc. 1978, 100, 2121. 34. Nachbar, R. B. Jr.; Johnson, C. A.; Mislow, K. J. Org. Chem. 1982, 47, 4829. 35. The same twist as 7 has been predicted by molecular mechanics and semiempirical calculations for tetra-tert-butyltetrahedrane ('lqT), which can be regarded as an analogue of 7 with somewhat relaxed congestion in the sense that the four ten-butyl groups are displaced from the center by 0.91 ,~ along the radial direction to the outside. However, X-ray crystallographic analysis of a single crystal of T I T revealed Td symmetry, namely there is no twisting about the Me-Cq bond in 'ITT [Imgartinger, H." Goldmann, A.; Jahn, R.; Nixdorf, M." Rodewald, H.; Maier, G.; Malsch, K.-D.; Emrich, R. Angew. Chem. Int. Ed. Engl. 1984, 23, 993]. There are a number of interesting problems behind this discrepancy (crystal packing effect, disorder, reliability of computations etc). 36. de Aquino Neto, F. R.; Sanders, J. K. M. J. Chem. Soc., Perkin Trans. 11983, 181. 37. Unpublished results. 38. Winiker, R.; Beckhaus, H.-D.- Riichardt, C. Chem. Ber. 1980, 113, 3456. 39. Littke, W.; Driick, U.Angew. Chem. Int. Ed. Engl. 1979, 18, 406. 40. Nitrogen inversion is suppressed by introducing cis-2,4-&methyl substituents. For an elegant work on the stereodynamics of open-chain tertiary amine involving nitrogen inversion and N-C rotation simultaneously, see (a) Bushweller, C. H.; Fleischman, S. H.; Grady, G. L.; McGoff, P.; Rithner, C. D.; Whalon, M. R." Brennan, J. G.; Marcantonio, R. P." Domingue, R. P. J. Am. Chem. Soc. 1982, 104, 6224, (b) Brown, J. H.; Bushweller, C. H. J. Am. Chem. Soc. 1992, 114, 8153. 41. Ogawa, K.; Takeuchi, T.; Suzuki, H.; Nomura, Y. J. Chem. Soc., Chem. Commun. 1981, 1015. 42. Jaime, C.; Osawa, E. J. Chem. Soc., Chem. Commun. 1983, 708. 43. (a) Jaime, C.; Osawa, E. J. Chem. Soc., Perkin Trans. H 1984, 995. (b) Jaime, C.; Osawa, E. J. Mol. Struct. 1985, 126, 363. 44. Kane, V.; Martin, A. R.; Jaime, C.; Osawa, E. Tetrahedron, 1984, 40, 2919. 45. Sammes, M. P.; Harlow, R. L." Simonsen, S. H. J. Chem. Soc., Perkin Trans. H 1976, 1126. 46. Vishwanadhan, V. N.; Mattice, W. L. J. Chem. Soc., Perkin Trans. H 1987, 739. 47. Prokai-Tatrai, K.; Zoltewicz, J. A.; Kem, W. R. Tetrahedron 1994, 50, 9909. 48. Further examples of reinforced stabilization of e-par conformation of arylcycloalkanes: (a) Earley, J. V." Cook, C.; Todaro, L. J. J. Am. Chem. Soc. 19911, 112, 3969. (b) Nasipuri, D.; Konar, S. K. J. Chem. Soc., Perkin Trans. 2 1979, 269. (c) Delgado, A.; Granados, R." Mauleon, D.; Soucheiron, I.; Feliz, M. Can. J. Chem. 1985, 63, 3186. (d) Colebrook, L. D. Can. J. Chem. 1991, 69, 1957. (e) de Marco, A.; Zetta, L." Consonni, R.; Ragazzi, M. Tetrahedron 1988, 44, 2329. (f) Rubiralta, M.; Jaime, C." Feliz, M.; Giralt, E. J. Org. Chem. 1990, 55, 2307. 49. For analogous CI...C1 parallel interaction see, Dempster, A. B." Price, K.; Sheppard, N. Spectrochim. Acta 1969, 25A, 1381. 50. Osawa, E. InReactivity in Molecular Crystals; Ohashi, Y., Ed.; Kodansha/VCH: Tokyo/Weinheim, 1993, pp. 1-9. 51. Similar trends have been obtained by molecular mechanics method as well. 52. Osawa, E." Collins, J.; Schleyer, P. v. R. Tetrahedron 1977, 33, 2667. 53. Mastryukov, V. S.; Chen, K." Yang, L. R.; Allinger, N. L. J. Mol. Struct. (Theochem) 1993, 280, 199. 54. Dorofeeva, O. V." Mastryukov, V. S.; Alemenningen, A.; Horn, A.; Klaeboe, P.; Yang, L.; Allinger, N. L. J. MoL Struct. 1991, 263, 281. 55. The term trans is used here not only in the sense of 1,4-relation but also in the sense that substituents are in anti-disposition across the pivot bond. The same applies to gauche.

24

EIJI/SSAWA

56. These are by no means the whole story about the conformational characteristics of 1. For example, lb and 1r are the positional isomers (difficult to see in the drawings). Details are described in ref. 5. 57. See any textbook of classical stereochemistry, e.g. Eliel, E. L. Stereochemistry of Carbon Compounds; McGraw-Hill Book: New York, 1962. 58. Mursakulov, I. G.; Guseinov, M. M.; Kasumov, N.K.; Zefirov, N. S.; Samoskin, V. V.; Chalenko, E. G. Tetrahedron 1982, 38, 2213. 59. See a special issue on rotational isomerism, J. Mol. Struct. 1985,126, 1-580, Kuchitsu, K.; Tasumi, M., Eds. 60. (a) Dale, J. Acta Chem. Scand. 1973, 27, 1115, 1130, 1149. (b) Dale, J. Stereochemistry and Conformational Analysis; Verlag Chemie: New York, 1979. 61. A referee reminds me of Einstein's pessimistic conclusion, "Chemistry is too complicated for chemists", in counterpoint to the optimism expressed above. 62. See, for example, a special issue on "Computers '94, Networks and Modeling," Science 1994, 265, 879-914.

TRANSITION METAL CLUSTERS:

MOLECULAR VERSUS CRYSTAL STRUCTURE

Dario Braga and Fabrizia Grepioni

I. II.

III.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. van der Waals Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Organometallic Crystal Isomerism . . . . . . . . . . . . . . . . . . . . . . C. Cation-Anion Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . D. Hydrogen Bondings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ABSTRACT The structures of representative examples of transition metal cluster complexes are discussed in terms of the interplay between intramolecular and intermolecular interactions. The structural features of some fundamental transition metal cluster species are reviewed with emphasis on the relationship between molecular characteristics (size, shape, bonding, dynamic behavior in solution, etc.) and crystal properties

Advances in Molecular Structure Research Volume 2, pages 25-65 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6 25

25 26 27 28 39 51 57 61 62 62

26

DARIO BRAGA and FABRIZIA GREPIONI

(intermolecular interlocking, cohesion, and solid state dynamics). Recent advances in the understanding of the effect of crystal packing forces on the molecular structure in the solid state are reported. These include: the role played by van der Waals interactions in crystals of neutral complexes and the existence of structural isomers of cluster species in the solid state; the relationship between ion organization in crystals of molecular salts formed by large cluster anions and organic cations and the presence of anisotropic arrangements of the ions in the crystals; and the patterns of hydrogen bonding intermolecular interactions and the involvement of carbon monoxide ligands in such interactions.

I. I N T R O D U C T I O N Transition metal clusters present an extraordinary structural variability due to the number of possible combinations and types of ligands, metal frameworks, and intramolecular bonding. The number of transition metal cluster complex species characterized to date is enormous and the field is still intensely explored. The subject of transition metal cluster structures has been extensively reviewed throughout the years and today extremely valuable books [1] and review articles [2] are available to satisfy all but the most curious of readers. For this very reason, it is not easy to add something new to the available literature on the topic. Even if attention is confined to the structural features of these molecules, a major problem is to circumscribe the subject matter. For this reason we have abandoned the initial idea of attempting a review of the structural chemistry of transition metal clusters. In the following we will try to offer to the reader alternative ways to look at the structural features of some of the most representative transition metal cluster complexes. Far too often the structural interest of the synthetic chemist (and of the crystallographer!) begins and ends with the presentation of a good picture and with the description of the ion or molecular structure in terms of structural parameters (bond lengths and angles, sometimes torsion angles or other slightly more esoteric geometrical parameters). The discussion of albeit extremely important features is carried out on objects extracted from their crystalline environment as if they were isolated (gas-phase type) molecules or ions. This is still the most common attitude in the field of metal cluster chemistry, where the characterization of structurally complex new species relies and depends on the (rapid) structural determination of the reaction products. X-ray diffraction still represents the most accessible and straightforward tool for unambiguous structural characterization. The contribution of crystallography to the growth of the chemistry of transition metal clusters parallels that of protein crystallography to molecular biology. However, the use of the diffraction experiment as a mere analytical tool leads to the "unfortunate" loss of a large body of structural information on the three-dimensional arrangement of the particles composing the crystalline material and, therefore, on the chemical and physical properties which depend on such arrangements. In this respect, the

27

Metal Clusters

recognition that the optimization of the intermolecular (or inter-ion) association in the crystal structure plays a vital role in the definition of the structural features of the component particles and that this role becomes fundamental when the molecule (or ion) is not rigid but flexible, has important implications on our way of looking at the structures of organometallic complexes and clusters. In this light many "old" problems of the pioneering days of structural cluster chemistry can offer "new" insights and a wealth of information. A second important aspect relates to the properties (both chemical and physical) of the solid state material formed by a collection of such molecules. It is widely recognized that the engineering of new materials with given chemical and physical properties would be of extreme practical interest [3]. The design of molecular aggregates to form crystalline materials must rely on a knowledge not only of the chemical and physical properties of the molecules or ions, but also on appreciation of the forces controlling intermolecular interactions, nucleation, and growth of the crystalline solid from the molecules or ions [4]. Crystals formed by organometallic molecules and clusters show characteristics that are intermediate between those of typical molecular organic solids and those of inorganic materials. In spite of the interest in these aspects, the crystal structure and solid state properties of transition metal clusters have been given very little attention in the field of organometallic chemistry. Whereas considerable progress has been made in the understanding of the interatomic bonding in metal clusters even of high nuclearity, systematic investigations of intermolecular interactions among cluster molecules and ions have only recently begun [5].

il.

DISCUSSION

Molecular aggregation and crystal stability depends on the complex interplay between different (and at times contrasting) interactions. These interactions can be grouped in two broad classes:

1. Internal interactions, both of the bonding and nonbonding type, arising from direct orbital overlap between atoms as well as from (steric) interatomic repulsions and attractions operating at molecular level. These interactions determine the minima and low energy regions of molecular potential energy hypersurfaces; 2. External interactions, i.e. van der Waals interactions, hydrogen bonds, and Coulombic interactions (both between ionic charges and between polar groups), which operate at the crystal level. These interactions are most sensitive to the number and type of atoms and to the molecular shape. Both types of interactions are simultaneously at work in the crystal (any crystal) and, although to a different extent, both contribute to determine the features of the solid state structure. The interplay of internal and external interactions becomes

28

DARIO BRAGA and FABRIZIA GREPIONI

especially intriguing when nonrigid molecules, or molecules that possess internal degrees of freedom, are involved. This is the case for many organometallic compounds where structural nonrigidity arises from two typical features of the bonding between the metal center(s) and the ligands, viz. the availability of almost isoenergetic, though geometrically different, bonding modes for the same ligand (CO, phosphines, arsines, NO +, CN-, etc.) and the delocalized nature of the bonding interactions between unsaturated x-systems (aromatic tings, alkenes, alkynes, etc.) and the metals [6]. These two distinctive features of the bonding in organometallic clusters suggest that with flexible organometallic complexes and clusters molecular (or ion) shape and crystal packing influence each other in a manner which is much more complex than for the majority of organic crystals [7a] or inorganic solids [7b]. Structural nonrigidity is also responsible for the dynamic behavior shown by molecules or ions in solution and/or in the solid state [8]. The characterization of the dynamic molecular structure will depend on the timescale of the technique adopted to investigate the process. The aim of this review is to discuss the interplay between molecular and crystal structure for transition metal cluster complexes. To this end we will examine some prototypical cluster molecules with a focus on the main types of intermolecular interactions at work in their crystals. The control exerted by the intermolecular assembly on the dynamic behavior in the solid state will also be briefly discussed. The following discussion will be organized in three main sections concentrated on van der Waals interactions, Coulombic interactions, and hydrogen bonds, respectively. It is important to stress, however, that this classification does not imply any precise borderline between the various types of interactions. Even when the clusters carry charges, for instance, the delocalization of the charge over the whole cluster body is so extensive so as to be almost ineffectual [9]. A second point to stress relates to the physical state of the material. It is important to appreciate that all the interactions acting among molecules in the solid state are the same as those acting among molecules in solution, maybe with the only (far from trivial) loss of the molecule-solvent interactions. The fixed positions of the particles in the solid state with respect to the rapid tumbling motion of the molecules in the liquid state imposes specific constraints to the patterns of interactions as well as on the type and energetic of the solid state dynamic processes. A very few among the species that are termed "highly fluxional" in solution do show dynamic behavior on the NMR timescale in the condensed state. A. van der Waals Solids

At the intermolecular level organic-type ligands (viz. molecular fragments formed by C, O, H, N, etc.) interact in the same way as organic molecules. This means that ligand-to-ligand interactions are ruled by optimization of van der Waals type interactions and hydrogen bonding. We have demonstrated in several cases that organic ligands control molecular packing arrangements as in the correspond-

Metal Clusters

29

ing organic crystals. For example, the relative orientation of the arene ribbons in solid naphthalene, anthracene, coronene, and benzene as well as in some mononuclear arene complexes such as (C6H6)2Cr is of the "herring-bone" type [10]. It is also true, however, that the interaction with one or more metal centers introduces additional constraints to the intermolecular assembly because of the relative orientation of the ligands belonging to the same molecule. In general the organization and assembly of mononuclear and polynuclear complexes in the solid state are governed by a balance between nondirectional close-packing forces and directional interactions between metal atoms [11] or charged groups. The situation is different with the carbonyl ligand, which is almost ubiquitous in transition metal cluster chemistry. It is crucial in the following discussion to recall that the CO molecule can bind to polynuclear metal complexes in terminal and various bridging fashions. Bridging carbonyls can span a metal-metal bond (la2bonding mode) or cap a triangulated metal face of an higher nuclearity cluster (kt3-bonding mode). These bonding modes are represented in Figure 1. In all these bonding modes, CO provides two electrons to the complex. CO is also known to interact with metal centers via the n system of C-O bond. End-on (or isocarbonyl) complexes are also well established [12]. The formation of these complexes requires strong Lewis acids such as alkali metal cations, other main group metals, or rare earth acceptors. The carbonyl ligand is a weak Lewis base because the good 6-donation from the more basic C-terminus is compensated by a very efficient back n-donation from metal to ligand [13,14], which leads to the formation of only a slightly negatively charged O-terminus. The study of strong hydrogen bonding in organometallic

T M

mb

Db M

M

Figure 1. Bonding modes of the CO ligand :terminal (T), doubly bridging (Db), triply bridging (Tb).

30

DARIO BRAGA and FABRIZIA GREPIONI

crystals has also provided evidence to support this conclusion, because the CO oxygen participates in hydrogen bonding networks only when stronger acceptor groups are absent [15]. The involvement of this ligand in hydrogen bonding increases as the basicity of CO increases on going from terminal to bridging bonding geometry. In high nuclearity clusters the bonding characteristics of CO (and other ligands) are coupled with the tendency towards formation of closed metal atom polyhedra which maximize the intermetal bonding. Hence, the external shape resulting from the coveting of the metal core with CO ligands is that of a "lumpy" object with the O atoms protruding from the surface. This molecular shape is characterized by the presence of bumps (the O atoms) and cavities (the space in between the CO ligands) which can be used for intermolecular locking and molecular crystal self-assembly. It has been pointed out that different molecular geometries can correspond to very similar molecular shapes. Thus, for example, both Fe3(CO)12 and Co4(CO)12 possess the similar icosahedral distribution of the peripheral atoms but quite different molecular structures, while Co2(CO)8 and Fe2(CO) 9 show very similar crystal structures in spite of the structural differences (see below). The very concept of a ligand envelope or outer ligand polyhedron, whichhas been used to interpret the dynamic behavior of many carbonyl clusters both in solution and the solid state [16], is essentially the recognition that a certain ligand distribution over the cluster surface is compatible with different orientations of the inner metal frame, i.e. with different molecular geometries. The understanding of the intermolecular assembly and of the way cluster molecules interlock and interact in the solid state requires an accurate knowledge of the intermolecular organization in the experimentally determined crystal structures and a method of categorizing and correlating. We have shown, previously, that a given crystal structure can be efficiently decoded by focusing on the number and distribution of the molecules surrounding the one chosen for reference [17]. These molecules are said to form an "enclosure shell" (ES). The ES-approach avoids the strictness of the lattice translational and point symmetry and affords a direct knowledge of the most relevant intermolecular interactions. As examples the ES present in crystalline Fe(CO)5 and Cr(CO) 6 are shown in Figure 2. This approach also facilitates the investigation of the effects that intermolecular interactions may have on the molecular structure observed in the solid state. We start from the known molecular structure and study how the observed crystal can be reconstructed. We proceed preparing a "one-dimensional" crystal by linking molecules to form a molecular row, then a "two-dimensional" one by coupling molecular rows, and, finally, a three-dimensional crystal by stacking molecular layers [18]. We will now proceed by discussing some selected cases. We analyze first the relationship between the crystal and molecular structures of two fundamental dinuclear metal carbonyl species, namely Co2(CO) 8 [19] (see Figure 3) and Fe2(CO) 9 (see Figure 4) [20]. These molecules have been the subject of much experimental and theoretical work, mainly because of the controversy on

Fe

fe

fe

f e Fe

Fe

Cr

Cr

\

Cr

Figure 2. The "enclosure shell" (ES) of molecules present in crystalline Fe(CO)5 (a) and Cr(CO) 6 (b). 31

32

DARIO BRAGA and FABRIZIA GREPIONI

~J

/// J

9

j

Figure 3. The solid state molecular structure of Co2(CO)8 showing the space-filling

outline.

the description of the bonding in Co2(CO) 8. The effective atomic number (e.a.n.) rule in this diamagnetic compound can be satisfied either by combining two metal orbitals pointing along the metal-metal vector ("straight bond" model)or two metal orbitals directed towards the empty bridging site ("bent bond" model) [21]. It has also been reported that Co2(CO) 8 shows dynamic behavior in the solid state [22], while Fe2(CO) 9 is static [23]. The dynamic behavior of Co2(CO) 8 has been explained on the basis of a ligand exchange process of the type occurring in solution [22a] or, alternatively, by assuming equilibration of terminal and bridging ligands via a librational motion of both the Co-Co axis and of the CO ligands [24]. More recently, a different interpretation has been proposed: a homolytic cleavage of the Co-Co bond in the solid state would generate "trapped" Co(CO) 4 radicals which would be able to reorientate and, eventually, recombine in the bridged structure leading to CO exchange on the NMR timescale [22b]. In spite of the fact that Co2(CO)8 possesses one bridging CO less than Fe2(CO) 9, the molecular organization in the crystals of the two species is almost identical. In both crystals the ES is formed by 12 first-neighbor molecules distributed in anticubooctahedral fashion. This results in the crystal of C02(CO) 8 being much less densely packed than that of Fe2(CO) 9. While the former species crystallizes in space

Metal Clusters

j

/"

33

//"

"~\

I' /_._ / ..... ~ \,

"

'\\ \

--\

'

Figure 4. The solid state molecular structure of Fe2(CO)9 showing the space-filling outline.

group P2Jm, with Z = 4 (the asymmetric unit contains two independent "half" molecules, each of which is related to the second "half" by a crystallographic mirror plane), Fe2(CO)9 crystallizes in the space group P63/m with Z = 2. The similarity between the two packing distributions can be appreciated from the layers shown in Figure 5. These sections are cut along the crystallographic mirror planes which comprise the bridging CO's. While the three bridging CO's in Fe2(CO) 9 are tightly interlocked with those of six surrounding molecules, the arrangement of the Co2(CO) 8 molecules results in the presence of continuous channels along the a-axis (van der Waals width in the range 2.4-4.5 A]. The channels are delimited along the b-axis by the umbrellas formed by the terminal tricarbonyl units. These channels are very likely filled with the electron density from the Co-atom orbitals pointing towards the "missing ligands". This electron density should be very diffuse and polarizable, giving rise to van der Waals type interactions that contribute to the crystal stability. Therefore Fe2(CO)9 and Co2(CO)8 have the same effective shape and crystallize in pseudo-isomorphous crystal structures. One may claim, however, that Co2(CO) 8 has not adopted a more efficient interlocking pattern because the unoccupied bridging sites cannot be efficiently used by neighboring molecules for intermolecular interlocking. In order to address

/

/

\

/

/

\

/

I f

("

Figure 5. Sections in the crystal packings of Co2(C0) 8 (a) and Fe2(CO)9 (b) along the metal-metal bonds and perpendicular to the planes defined by the bridging CO ligands; note how the layer of Co2(C0)8 molecules presents a channel along the a-axis in which CO-ligands are missing. 34

Metal Clusters

35

this question, alternative ways to pack Co2(CO)8 and Fe2(CO) 9 molecules in the solid state have been explored [25]. The calculated Structures can then be compared with the observed crystal structures. It has been possible to demonstrate that Co2(CO) 8 molecules can indeed be arranged in a way that uses the empty bridging sites. The hypothetical crystal structures generated are denser than the one experimentally observed and do not possess channels. This is consistent with the observation that, were it not for the stereoactivity of the electron density present in the channels, the shape of the molecule would be compatible with more densely packed crystal structures. Another example of the relationship between molecular and crystal structures is afforded by the binary carbonyl molecules Fe3(CO)I 2 and Ru3(CO)I 2. Both molecules occupy a special place in metal carbonyl cluster chemistry and have been the subject of many investigations. A number of papers on their structures in solution and in the solid state have been published. We only wish here to briefly summarize some key aspects which bear on the following discussion. The room temperature molecular structure of Fe3(CO)I 2 (see Figure 6) and the nature of the crystallographic disorder affecting its crystal were first established by Wei and Dahl [26a]. Cotton and Troup were subsequently able to resolve the light atom positions obtaining an image of the molecular structure characterized by the

? ~..,.~.\\ 9 ~,,

,>/

Figure 6. The molecular structure of Fe3(CO)12 as determined in the solid state at 1O0 K.

36

DARIO BRAGA and FABRiZIA GREPIONI

well known C 2 symmetric distribution of CO ligands with 10 terminal ligands and two asymmetric bridging carbonyls along the one edge of the triangle of iron atoms that is shorter than the other two [2.558(1) vs. 2.677(2) and 2.683(1) A at room temperature] [26b]. The approximate chemical twofold axis passes through the middle of the bridged Fe-Fe bond and the opposite Fe atom. Disorder in crystalline Fe3(CO)12 arises from the average of molecules in two centrosymmetric orientations. This is possible because the polyhedron described by the outer oxygen atoms is a (distorted) icosahedron which is invariant to inversion causing the molecules to be randomly distributed in two centrosymmetrically related orientations. The dynamic behavior in solution and the solid state, as well as the origin of the disorder have attracted much interest in the scientific community. Fe3(CO)12 is fluxional both on the IR [27] and the NMR [28] timescale in solution. The fluxionality process has been modeled in various ways [29]. Cross-polarization magic angle spinning (CPMAS) NMR experiments have also been carried out [30] showing that the solid state spectrum of Fe3(CO)12 is markedly temperaturedependent. The spectral features have been interpreted on the basis of a 60 ~ reorientational jumps of the Fe triangle within the CO-ligand shell. Fe3(CO)12 has also been studied by M/3ssbauer [31] and EXAFS [32] techniques, as well as by various theoretical approaches at different level of sophistication [33]. Recently, we have, in collaboration with others [34], reinvestigated the molecular and crystal structure of Fe3(CO)12 on diffraction data measured between 100 and 320 K. It has been found that, although the crystal retains the centrosymmetric P21/n structure, the temperature greatly affects the molecular structure. The bridging ligands become progressively more symmetric as the temperature is decreased. The symmetrization of the bridging ligands is accompanied by a decrease of the bridged Fe-Fe bond length [from 2.554(1 )/~ at 320 K to 2.540(2)/~ at 100 K]. The other two Fe-Fe bonds do not change appreciably with respect to the room temperature structure. There is also good agreement between these data and those obtained by Cotton and Troup [26b]. The anisotropic displacement parameters of the iron atoms at the various temperatures indicate that the Fe triangle undergoes a rigid-body libration about the molecular twofold axis passing through the bridged bond and the unique Fe atom. The preferential orientation persists down to 100 K. This could be either due to the persistence, even at 100 K, of the librational motion of the Fe triangle or to a component of static disorder which overlaps the dynamic disorder revealed by the temperature dependence illustrated above. Recent molecular mechanics calculations [35] have shown that both rigid-body librational motion about the molecular twofold axis and threefold reorientational jumps of the Fe triangle are possible in the crystalline state. The former process is that of lowest energy whereas the energy barrier to reorientational jumps is ca. 12 kcal.mol -l. This result can afford a rationale to both the observation of a preferential orientation of the Fe-atom anisotropic displacement parameters which is main-

Metal Clusters

37

tained down to 100 K, and to the solid state NMR evidence for a complete equilibration of the 13C nuclei of the terminal and bridging ligands. It is interesting to mention at this stage that OsFe2(CO)12 [36] is almost isostructural with Fe3(CO)12, although its crystal shows a much smaller degree of disorder with only 8% of the molecules in alternative "centrosymmetric" orientation. In contrast to the situation for Fe3(CO)I 2, the species Ru3(CO)I2 [as well as Os3(CO)12]does not show any extensive solid state dynamic behavior. Ru3(CO)I2 possesses an "all terminal" structure with each ruthenium atom carrying four CO ligands (see Figure 7) [37]. The overall molecular symmetry is close to D3h.The most important packing motif in terms of intermoleculax cohesion in crystalline Ru3(CO)I 2 is based on the insertion of one axial CO of one molecule into the tetracarbonyl unit generated by two axial and two radial CO's on a next neighboring molecule (a "key-keyhole" interaction). The same feature is also present in the isomorphous crystals of the isostructural species Os3(CO)12[38]. This intermoleculax interaction is responsible for the deviation of the molecular symmetry from idealized D3h symmetry. The Ru-Ru bond involved in the interlocking interaction is slightly longer than the other two [2.8595(4) A versus 2.8521(4) and 2.8518(4) ~]. This effect is also observed in crystalline Os3(CO)12 [2.8824(5) /k versus

()

()

Figure 7. The molecular structure of Ru3(CO)12 in the solid state.

38

DARIO BRAGA and FABRIZIA GREPIONI

Figure 8. The molecular structure of Co4(C0) 12 in the solid state.

2.8752(5) and 2.8737(5) ~]). Once a row of Ru3(CO)12 molecules is formed, a two-dimensional substructure can be generated by placing other rows on both sides of the central one. The outermost packing motif above and below the surface of the layer consists of parallel rows of Ru(CO)3 and Ru(CO) 4 units. New layers of molecules will interact in "Velcro-like" fashion generating a stacking of molecular layers and the full three-dimensional structure [18]. The crystal structure ofCo4(CO)l 2 is also disordered [39]. The molecule has three carbonyl groups in edge-bridging positions around a tetrahedral face with the remaining carbonyls being terminally bound. The peripheral polyhedron described by the O atoms is again approximately icosahedral, and the molecular symmetry is very close to C3v (see Figure 8). The CPMAS NMR spectral features were initially interpreted assuming reorientation of the Co 4 core within the ligand envelope [40]. The high field CMPAS spectrum of Co4(CO)12 at room temperature has been reinterpreted on the basis of either a "static", poorly resolved, spectrum, or by invoking a rotational motion of the Co 4 core exclusively around the crystallographic twofold axis that relates the two disordered orientations of the cluster [40b]. The anisotropic displacement parameters of the cobalt atoms indicate a preferential oscillation of the metal frame around the tetrahedron threefold axes.

Metal Clusters

39

B. Organometallic Crystal Isomerism Several examples are known of flexible cluster molecules that crystallize in different isomeric forms. These isomers interconvert in solution via CO exchange or similar fluxional processes observable by NMR spectroscopy. In these cases the energy difference between the isomers is sufficiently small to the point that less stable isomers can be isolated in crystalline forms because crystal packing interactions afford the necessary additional stabilization. The possibility of isolating sufficiently stable and long-living intermediates of fluxional processes has not yet been investigated in a systematic manner. Nonetheless, there are several examples of this type already available in the literature. Only in few cases the different stability of the isomers has been demonstrably related to differences in crystal cohesion and stability. We will use the term crystal isomerism to indicate isomeric forms in different crystal structures. This approach recalls the conformational polymorphism well characterized in organic crystals [41]. Ir6(CO)l 6 is one of the most interesting cases for which crystal isomerism is observed [42]. Two structural forms have been characterized in the solid state: one with four face-bridging CO ligands, and another with four edge-bridging ligands (see Figure 9). Only a small movement of these ligands is required to interconvert the two isomers. It has been argued [43], on the basis of molecular volume and packing coefficient calculations, that the edge-bridged isomer is more efficiently packed in the lattice than the face-bridged one, i.e. a more efficient packing is attained at the expense of a more symmetric distribution of the ligand-cluster bonding interactions. Ru6C(CO)I 7 is known in two different crystal forms [44], one of which contains an asymmetric unit comprised of one and a half crystallographically independent molecules. The resulting symmetry generated three full molecular structures. The three molecular structures differ essentially in the rotameric conformation of the tricarbonyl units above and below the equatorial plane containing the bridging ligands, and in the pattern of terminal, bridging, and semibridging CO's around the molecular equator (see Figure 10). The presence of three different rotameric conformations indicates that the apical (CO) 3 units lie on a rather fiat potential energy surface without well-defined conformational minima. The conformational choice is therefore controlled primarily at intermolecular level. The observation of different isomeric forms for Ir6(CO)16 and Ru6C(CO)I7 is in agreement with recent molecular mechanics calculations on octahedral carbonyl clusters which show that the ligand distribution can be easily controlled by optimization of the intermolecular interactions in the solid [45]. The fundamental packing motif in both crystal forms of Ru6C(CO)I 7 is characterized by rows of molecules linked via a "key-keyhole" interaction of the kind discussed above for Ru3(CO)I 2 (see Figure 11). The interlocking is attained by inserting the unique bridging ligand in the middle of the cavity generated by four

3:

v~

c~

v

0

..t

0

3

0

,..-I

3

0

0 m.

.-,I

--,I r-

0

O

~

l

-% 9

%

b

(continued) Figure 10. Solid state molecular structures of the three conformers of Ru6C(CO)17 (a), (b), and (r note how the three molecules differ chiefly in the rotameric conformation of the tricarbonyl units above and below the octahedron equatorial plane containing the bridging CO ligands. 41

Figure 10. (Continued)

/

f---~

)

Figure 11. Rows of molecules linked via "key-keyhole" interactions in crystalline

Ru6C(CO)17.This is a typical packing feature which is often observed in crystals of

derivatives of Ru6C(CO)I 7 (see Figure 17).

42

Metal Clusters

43

terminal CO's linked to the opposite Ru-Ru edge. This motif is also observed in crystals of mono-arene derivatives of Ru6C(CO)I 7 (see below). The hydrido-borido cluster HRu6B(CO)I 7 [46] is also known in two crystalline forms (see Figure 12) one of which is isomorphous with one of the crystalline forms of Ru6C(CO)lT. The two molecules of HRu6B(CO)I 7 differ not only in the orientation of the tricarbonyl units above and below the equatorial plane [as in the case of Ru6C(CO)I7] but also in the location of the H(hydride) atom over the cluster surface. As in the previous cases, the difference in internal energy between the two isomers must be small and well within that in packing energy between the two respective crystals. Two different structural units have been observed within the same crystal in the case of RusS(CO)Is [47]. Both molecules carry a quadruply bridging S ligand and differ chiefly in the mode of bonding of the basal CO ligands (see Figure 13). One ofthese ligands is doubly bridging in one molecule, while it occupies an asymmetric triply bridging position in the second molecule. Interestingly, the different CObonding mode appears to have a significant effect on metal-metal bond lengths within the two metal cores, in keeping with previous observations on the relative "softness" of intermetallic bonding in cluster systems [48]. Another example of solid state isomerism is offered by the neutral compound Ira(CO)9(P3-trithiane) [trithiane = 1,3,5-(SCH2)3] [49], which is present with two main structural forms in solution: the "all-terminal" form with no bridging CO ligands and the "bridged" form with three edge-bridging CO ligands (see Figure 14) [50]. Packing potential energy calculations [49] have shown that the most cohesive packing is associated with the bridged form Ir4(CO)6(~-CO)3(~3trithiane), which also possesses a smaller volume and a higher molecular density than the unbridged Ir4(CO)9(~-trithiane ). This is in agreement with the positive activation volume obtained by variable-pressure NMR for the bridged-unbridged interconversion process [50]. Interestingly, a network of C-H..O hydrogen bonding interactions is present in these crystals. In the bridged isomer only the bridging CO ligands are involved in short hydrogen bonds (see also below). These interactions affect the molecular geometry to a great extent and are apparently responsible for the asymmetry of the CO bridges, i.e. for the deviation of the molecular geometry from idealized C3vsymmetry [49]. Another related example is that offered by the ionic species [Ir4(CO)II(SCN)]-, and [Ira(CO)8(kt-CO)3(SCN)]- which will be discussed below in the section devoted to cluster crystalline salts [51]. Another family of compounds well suited for the study of the relationship between molecular and crystal structure is that of the carbonyl-arene clusters. In these complexes the availability of various modes of bonding for the arene ligands over the cluster surface generates additional degrees of structural freedom which are reflected in the number of structural isomers that can be isolated and structurally characterized [52]. For example, isomeric pairs are known for the pentanuclear species Ru5C(CO)I2(I.13-TI2:q2:TI2-C6H6) and RusC(CO)I 2(TI6-C6H6)(see Figure 15) [53], and

.~

~ rib ,-'," e0 "~

i'D

0

v

oO

: : r "r"

3

~

0

~g.~-

0

~

~"

~

_--: ..,

~

I'D ~

N.o

.<

-rR

5-q-o

~. ~.

~

~

o~K

0

.~

0

rD

::r 3

o ~0 "1

~.

.

.

.

i

S

Figure 13. Molecular structure of the two isomers of RusS(CO)I 5 (a) and (b), the two molecules are present in the same crystal and differ in the bonding mode of one bridging CO ligand. 45

C

.

Figure 14. Molecular structure of the two isomeric forms of ir4(CO)9(F3-1,3,5trithiane) in the solid state (a) and (b). 46

~v

t~

9

I

v

~n

O

~D

m.

c~-- T

~3

t~

tjY ~

b

9

Figure 16. The solid state molecular structures of the two isomeric bis-benzene derivatives Ru6C(CO)11(116-C6H6)(l13-q2-'q2.112_C686)(ill) and Ru6C(CO)11(116-C6H6)2 (b). 48

Metal Clusters

49

for the hexanuclear species Ru6C(CO)II(TI6-C6H6)(~3-TI2"I]2.'I]2-C6H6) and RtI6C(CO)II(1"I6-C6H6)2 (see Figure 16) [54]. These isomers differ chiefly for the bonding modes of the benzene ligands. For these species chemical and spectroscopic evidence provide a clear sequence of stability of the different isomers in solution. The bonding in these complexes has also been investigated using extended Htickel calculations [55]. The relative stability of the facial (~t3-112:l]2:l"I2-C6H6 bonding) or apical isomer (116-C6H6bonding) has been related to the chemically characterized interconversion process occurring in solution. The calculations have led to the conclusion that the apical isomers are the more stable, although the local benzene-ruthenium interaction is stronger in the facial isomers. The molecular organization in the respective crystal structures, as well as the relative cohesion of the solid materials, has also been investigated by empirical packing potential energy calculations. The relationship between stability of the individual arene cluster molecules and that of the same molecules in the solid state has been addressed in terms of the relative crystal cohesion. Hydrogen bonds of the C-H..O--C type have been detected in crystals of the apical isomers. In crystalline transRu6C(CO)II(q6-C6H6)2 , molecular piles are formed by molecules joined by direct benzene-benzene interactions; a similar packing motif is also present in crystalline Ru6C(CO)II(1]6-C6H6)(kt3-q2:I"I2:TI2-C6H6) [55]. As mentioned above, the "keykeyhole" interaction present in crystalline Ru6C(CO)I 7 is preserved upon substitution of three CO ligands for a benzene ligand, irrespective of the mode of coordination of the latter. This is shown in Figure 17. The structural flexibility of carbonyl-arene cluster systems can also be exploited to construct heteromolecular crystals in which molecules of similar shape and size but different in chemical composition can be cocrystallized. Contrary to organics [56], the cocrystallization of organometallic complexes and clusters is not a

f

......N/

Fiflure 17. The "key-keyhole" interaction in crystalline Ru6C(CO)11(116-C6H6)(~t3 -

1"1:112:q2-C6H6) and

trans-Ru6C(CO)11(116-C6H6)2(compare with Figure 11).

C

,

/

Figure lB. The solid state molecular structures of Ru6C(CO)14(q6-C6H4Me2) (a) and trans-Ru6C(CO)]l(Ti6-C6H4Me2)2(b); these two molecules have been observed to form

both homo-molecular crystals of each type and an hetero-molecular co-crystal in which they are present simultaneously. 50

Metal Clusters

51 .,

( (

Figure 19. A space-filling representation of the co-crystal formed by Ru6C(CO)14(q6C6HaMe2) and trans-Ru6C(CO)ll(rl6-C6HaMe2)2 (CO-ligands are omitted for sake of clarity); note the AA-BB-AAstacking sequence of molecular layers.

common phenomenon. One example is afforded by the cocrystallization of two arene clusters, Ru6C(CO)14(TI6-C6H4Me2) and trans-Ru6C(CO)llOl6-C6HaMe2)2 , which retain in the cocrystal the same packing features shown by the two molecules in their separate crystals [57]. The solid state molecular structures are shown in Figure 18. The cocrystal has been described as being composed of homomolecular crystals of Ru6C(CO) 14(TI6-C6HaMe2)and of homomolecular crystals of Ru6C(CO) 11(TI6-C6HaMe2)2, both extending for only two layers. A space-filling representation of the cocrystal is given in Figure 19.

C. Cation-Anion Interactions After discussing the relationship between crystal and molecular structure in van der Waals type solids, we can now proceed to discuss systems in which the presence of a fixed charge on the component particles could (at least in principle) give rise to more directional intermolecular interactions and specific packing patterns. Transition metal cluster anions form molecular salts with a variety of organic cations. The effect of the counterion choice on the crystallization process and on the properties of the crystalline material has only recently begun to be systemati-

52

DARIO BRAGA and FABRIZIA GREPIONI

cally investigated [9,58]. Crystals formed by transition metal cluster anions and organic counterions have characteristics which are intermediate between those of molecular crystals and those of typical ionic salts. For this reason we chose the term molecular salt to designate materials in which the component particles are ions of opposite sign which interact essentially in van der Waals fashion [9]. In an initial attempt to study the effect of the counterion choice on the crystal structure organization, we have searched the Cambridge Structural Database [60] for crystals containing octahedral cluster anions [9]. The relationship between shape, size, and charge of the component ions and the formation of preferential aggregates (piles, "snakes", layers) in the crystal architecture has been analyzed. It has been shown that, with large organic cations, the packing pattern is essentially that of molecular cocrystals, while small cations drive towards monodimensional aggregation of the anions, thus demonstrating that the volume of the counterion can be chosen to attempt generation ofpredefined one-dimensional or two-dimensional networks. Similar results are obtained if the analysis is extended to high nuclearity systems. For example, it has been shown that in crystalline [Os10C(CO)24][(Ph3P)2N] the cluster anions form piles surrounded by a cation belt [61]. The same cluster pile

Figure 20. The solid state molecular structures of the dicarbido cluster anions [Rh12C2(CO)24]2-.

Metal Clusters

53

constitutes the fundamental packing motif in the crystal of the neutral dihydride H2OSloC(CO)24 [62]. Preferential aggregation of the anions in monodimensional arrays is also observed in the family of carbonyl cluster anions [M6X(CO)Is] n- [M =Co, Rh; X = N, n = 1; X =C, n = 2] [61]. The two closely related families of dicarbido cluster anions [Rh12C2(CO)24]2and [Rh12C2(CO)23]n- (n = 3, 4) and [Co13C2(CO)24] n- (n = 3, 4) and the pair of isomers [Rhll(CO)23] 3- have also been examined [63]. The dianion [Rh12C2(CO)24]2- differs from the trianion [Rh12C2(CO)23]3- and the tetraanion [Rhl2C 2(CO)23]4- only for the presence of one additional CO ligand (see Figure 20). These two latter complexes are isostructural but possess different anionic charges. Furthermore, the dianion and the tetraanion are diamagnetic while the trianion is paramagnetic. Similarly, [Co13C2(CO)24 ]3- and [Co13C2(CO)24]4- are isostructural but not isoelectronic (see Figure 21). The Rhl2 and Co B anions have been crystallized with a variety of counterions such as (Ph3P)2N§ NMe~, NPr~, and NMe3(CH2Ph) +. Packing coefficients for all these crystals fall in the range 0.57-0.64 (mean 0.61), i.e. strictly comparable to packing coefficients calculated for neutral binary carbonyls. The least densely packed are the Rhll trianions [0.57]. In these crystals, as well

Figure 21. The solid state molecular structures of the dicarbido anions [Co13C2(C0)24] n- (n = 3, 4).

54

DARIO BRAGA and FABRIZIA GREPIONI

as in [Co13C2(CO)24]4-, the solvent of crystallization is accommodated in the structure interstices, thus contributing to crystal cohesion. Anisotropic arrangements of the anions are observed in these crystal structures when the contribution of the cations to the overall molecular volume is small. Let us discuss briefly the pair of isomers [Rhll(CO)23][NMe413.Me2CO and [Rhll(CO)za][NMe413.C6HsMe [64]. The two cluster anions differ in the number and distribution of the bridging CO ligands as well as in the pattern of metal-metal bond lengths. As pointed out in the original paper [64] the CO-ligand distributions in the two species can be considered as two frozen steps of the fluxional process that occurs in solution where both isomers have identical IR spectra and are still fluxional on the timescale of 13C-NMR spectroscopy at -90 ~ Interestingly, the separate crystallization of these fluxional isomers depends simply on the choice of crystallization solvent. While the two molecular ions differ in geometry, the respective crystals are similar (in spite of the differences in the space-group symmetry). The two crystals are quasi-isomorphous with the anions forming piles throughout the crystals. The cluster anions have nearly the same relative orientation along the piles while the CO ligands show two different distributions. Crystals containing large or very large anions and cations have characteristics that resemble more closely those of molecular crystals, where the component particles can be discriminated on the basis of the interatomic separations, than those of typical ionic salts, where the ions are surrounded by counterions and held together by Coulombic forces. With high nuclearity cluster anions and large organic counterions the actual ionic charge is diffused over the whole cation and anion bodies and has no recognizable effect on the organization of the molecular ions in the crystal structure. As a consequence, anions (and cations) can approach each other as if basically neutral. These materials behave very much like molecular cocrystals (see previous section), whose components are packed in the lattice according to the same rules which govern the packing of neutral molecules in single component systems. The association of the component anions and cations is essentially governed by their relative volumes. When the anion is large with respect to the cation(s), anisotropic arrangements are generated in the crystal. It is worth recalling that high nuclearity cluster anions tend to undergo a different type of reaction chemistry compared to lower nuclearity systems. Monoions and dianions with nuclearity greater than ten behave very much like neutral species and do not easily react with cations because the charge is not sufficiently localized. The degree of flexibility of the counterion is another factor that plays a relevant role in the packing choice. Organic-type cations containing long aliphatic arms (such as NEt~, NPr~, and NBu~, etc.) are highly adaptable and can vary volume and shape by folding or unfolding around the anions. When the even charge distribution over the CO ligands is "perturbed" by the substitution of a polar heteroligand for one or more CO's, there appears to be a tendency to segregate the polar part of the molecule in a cage of counterions. This is usually achieved in two alternative ways" (1) either by forming "dimers" of anions

Metal Clusters

55

with the heteroligands roughly pointing towards each other; or (2) by forming anion piles of the type discussed above with the heteroligands protruding laterally and interacting with the counterions. These behaviors had been noticed first in the crystalline salts of [Ira(CO)II(SCN)][N(CH3)2(CH2Ph) 2] and of [Ir4(l.t2CO)3(CO)8(SCN)] [(Ph3P)zN], and observed again in the systematic study of carbonyl cluster anions containing halide ligands [65]. For example in crystalline [Ir6(CO)laBr] [PhaP] the bromine atoms are segregated from the surroundings by a belt of [PPh4]§ cations (see Figure 22). Cation-dependent molecular structures have also been observed for the dianion [Fe4(CO)13]2- [66]. This cluster possesses a slightly distorted all-terminal structure in its PPN § salt [66a], while in its [Fe(py)6] 2§ (py = pyridine) salt the anion possesses asymmetric bridging CO's [66b]. Together with the structures of the neutral analogue HFeCo3(CO)9{P(OMe)3}3 [66c], the three complexes were regarded [66a] as belonging to the same CO-ligand interconversion pathway. A similar structural correlation between solution behavior and solid state geometry is shown by the octahedral carbido-carbonyl cluster anions [Co6C(CO)13]2- [67a], [Rh6C(CO)13]2- [67b], and [Co2Rh4C(CO)I3] 2 [67c]. The CO-ligand distribution in the latter mixed-metal species is intermediate between those possessed by the

#

/ff

Figure 22. Space-filling diagram of the packing arrangement in crystalline [Ir6(CO)14Br] [PPh4]2, note how the bromide ligands oftwo neighboring molecules are segregated in a cation cage.

b

(continued) Figure 23. The CO ligand distribution in [C06C(CO) 13] 2- (a), [Rh6C(CO)I3] 2- (b), and

[Co2Rh4C(C0)13]2 (c); note how the ligand distribution in the latter mixed-metal species is intermediate between those possessed by the two homometallic clusters (darker atoms represent the cobals atoms). 56

Metal Clusters

57

Figure 23. (Continued) two homometallic clusters as shown in Figure 23. Analogously, the octahedral dianion [Ru6C(CO)I6]2- shows different distribution of the bridging CO ligands in its AsPI~ and NMe~ salts [68a, b]. Solid state isomerism has also been detected by Mtissbauer spectroscopy of the PPN § NEt~ and NMe~ salts of the anion [FesN(CO)14]- [68c]. In all these examples the structural differences are due to the different distribution of the CO ligands and, therefore, can be regarded as "snap-shot" pictures of the fluxional process affecting the CO ligands in solution. CO-ligand migration usually occurs via interconversion from terminal to bridging mode of bonding. This process has been investigated within the structural correlation method [69] by comparison of CO-ligand geometries in different crystal environments [70].

D. Hydrogen Bondings Hydrogen bonding networks play a distinct role in the three dimensional organization of molecules in the solid state. In the case of transition metal clusters [71] the presence of strong hydrogen bond donors and acceptor groups (such as carbonylic -COOH, and oxydrilic -OH groups) on the metal-bound ligands has essentially the same effect on the intermolecular organization as in the corresponding organic solids where the ligands exist as free molecules [72].

58

DARIO BRAGA and FABRIZIA GREPIONI

Derivatives carrying -COOH groups preferentially form ring systems joining two molecules via interaction of the OH donor with the C--O acceptor of the neighbor; the O-H---O bond is usually linear and the ring is planar. These types of packing motifs are present in crystals of transition metal cluster species such as H3Os3(CO)9(kt-CCOOH) [73a] and HRu3(CO)I0(Ia-S-CH2-COOH) [73b] (see Figure 24) indicating that ring formation is not sterically hindered by the approach of these cluster systems. In general, when -COOH and CO groups are present in the same cluster complex, the CO ligand is not involved in the hydrogen bonding network, whereas with the weaker hydroxyl group, which is a t~-donor, there is competition for hydrogen bond formation between -OH itself and other acceptors, such as the CO ligand. This is in keeping with the general observation that less acidic protons are used for hydrogen bonding only after all the more acidic protons have been used [71]. The

f

b

C

;,

?3

Figure 24. Hydrogen bonding patterns in crystals of H3Osg(CO)9(la-CCOOH) (a) and HRu3(CO)lo(II-S-CH2-COOH) (b). Note the formation of characteristic eight-membered rings between the -COOH groups of molecular pairs.

Metal Clusters

59

capacity of the oxygen atoms of CO ligands to form intermolecular hydrogen bonds is, though weak, is significant. In fact, the good t~-donation capacity of CO from the C-terminus is compensated by a very efficient x back-donation from metal to ligand, which leads to the formation of a slightly negatively charged oxygen atom [74]. The weak Lewis basicity of CO is, for instance, responsible for the formation of the "end-on" (or isocarbonyl) complexes [75] with strong Lewis acids such as alkali metal cations or other main group metals or rare earth acceptors. Complexes with transition metal acceptors are also well established, mainly with early transition metals in high oxidation states which are relatively strong Lewis acids. CO ligands have also been observed to participate in strong bonds with water molecules in alkali metal solvate salts of transition metal cluster anions. Networks of C-H---O bonds, where the O atom belongs to terminal or bridging CO ligands, are present in crystalline cluster complexes. These networks are common in cluster systems because of the simultaneous presence of a large number of CO ligands as well as of ligands carrying acidic C-H groups [71,77]. As the basicity of CO ligands increases on going from the terminal to the bridging bonding geometry, the strength of the hydrogen bonding also increases (see below).

r.,,

,.

/

(

Figure 25. Intermolecular C-H---O interactions between the two independent molecular units in crystalline HFe4(TI2-CH)(CO)12.

60

DARIO BRAGA and FABRIZIA GREPIONi

A particularly well-established case is represented by the pair of tetranuclear methylidene clusters HFe4(rl2-CH)(CO)12 [76] and HFe4(rl2-CH)(CO)II(PPh3) in which also "agostic" Fe--H interactions are observed [77]. The existence of C-H---O interactions, involving one terminal CO ligand and the methylidene H atoms (see Figure 25), in crystalline HFea(rl2-CH)(CO)12 was first discussed by Muetterties et al. [76]. Due to the presence of two independent molecular units these interactions differ in H---O separation (2.58 and 2.48/~, respectively). In crystalline HFea(rl2-CH)(CO)II(PPh3) [77] C-H-.-O interactions are not possible because the methylidene H atom is completely embedded within the ligand shell and screened from the surroundings by one phenyl group, whereas in both systems short intermolecular contacts between the H(hydride) atoms and the CO ligands are present (see Figure 26). C-H---O(carbonyl) intermolecular interactions between H atoms belonging to organic ligands and CO ligands have been detected in crystals as the dienyl species HRu3(CO)9(I.t3:(Y:rl2:rl2-C6H7) [78] and the benzene clusters RusC(CO)12(TI6-C6H6) and RusC(CO)I2(~3:rl2:rl2:rl2-C6H6). The participation of these "weak" hydrogen bonds in the cohesion of organic solids has been reviewed by Desiraju [79]. Recently, the importance of this type of interaction in solid organometallic materials, including transition metal clusters, has been the subject of a statistical survey [80]. Finally, it is worth mentioning that in charged organometallic species hydrogen bonding to the metal centers has been observed. Examples of this type are the

f % Figure 26. Short intermolecular contacts between the H(hydride) atoms and the CO ligands in crystalline HFe4(r12-CH)(CO)ll(PPh3).

Metal Clusters

61

Me3NH + and Et3NH + salts of the monoanion [Co(CO)4]-where an hydrogen atom bridges the ammonium N atom and Co [81]. Another important example in the cluster area is offered by the ipr2NH~ salt of [HFe3(CO)II]- where both the unique bridging CO and one terminal ligand are directly H-bonded by surrounding counterions in the crystal structure [82].

III. CONCLUSIONS AND OUTLOOK The preparative and structural chemistry of transition metal clusters has seen an enormous expansion over the last three decades. The combined efforts of a large number of scientists all over the world has transformed what were regarded as mere chemical curiosities into classes of new compounds. This thrust has led to the creation of an entirely independent field of research, and new discoveries and papers reporting on transition metal cluster systems are nowadays counted by the thousands. Although the field is still actively explored, the focus has, in recent times, changed from the synthesis of new molecules (of increasing nuclearity and structural complexity) to that of chemical systems to be employed as substrates of chemical reactivity (the "surface-cluster" analogy) or chemical materials with their own characteristic properties. For instance, there is accumulation of evidence that transition metal clusters have properties that are intermediate between those of dispersed and bulk metals [83]. These properties depend on the complex interplay between the properties of the individual molecular entity and those of the collection of molecules present in the solid material. This contribution has had its focus on the intermolecular interactions acting in crystalline transition metal clusters. We have discussed the molecular and crystal structures of a number of "old" and "new" compounds in terms of the interplay between molecular characteristics (size, shape, charge, type and number of ligands, fluxionality and isomerism, etc.) and crystal characteristics (Coulombic, van der Waals, and hydrogen bonding interactions, crystal aggregation, cohesion, and stability). In transition metal clusters the presence of relatively "soft" metal skeletons combine with ligand(s) migration over the cluster surface to yield extremely flexible molecular systems. In this condition, the collective properties in the solid state differ from those of the single molecule or ion in the gas phase or in solution. The possibility of combining organic and inorganic components makes cluster systems a bridge between organic solid state chemistry and inorganic materials. The way in which cluster molecules or ions interact and interlock to generate extended molecular arrays and crystals can be of importance in the area of surface and solid state chemistry. The relatively small difference in energy between structural isomers allows isolation and characterization in different crystalline environments and, hence, the study of the effect of intermolecular interactions on the molecular structure.

62

DARIO BRAGA and FABRiZIA GREPIONI

It is our opinion that transition metal cluster chemistry has reached the stage of maturity of other branches of chemistry and can now open up to new challenges in less explored areas such as that of material chemistry and physics.

ACKNOWLEDGMENTS We acknowledge the Ministero per 1' Universita' e la Ricerca Scientifica e Tecnologica (MURST) for financial support. A number of colleagues have contributed throughout the years, most of them are co-authors of the papers quoted in this chapter. In particular we wish to express our gratitude to Professor Brian F. G. Johnson (Edinburgh) and Professor Jack Lewis (Cambridge) for continuous help and encouragement. We wish also to thank Professor Vincenzo G. Albano (Bologna) for having introduced us to the field of transition metal cluster structural chemistry. It is a pleasure to acknowledge Professor A. Guy Orpen (Bristol) for helpful comments and constructive criticisms.

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68. (a) Johnson, B. E G.; Lewis, J.; Sankey, S. W.; Wong, K.; McParflin, M.; Nelson, W. J. H. J. Organomet. Chem. 1980, 191, C3. (b) Ansell, G. B.; Bradley, J. S. Acta Crystallogr. 1980, B36, 726. (c) Hourihane, R.; Spalding, T. R.; Ferguson, G.; Deeney, T.; ZaneUo, P. J. Chem. Soc. Dalton Trans. 1993, 43. 69. (a) Btirgi, H. B.; Dunitz, J. D. Acc. Chem. Res. 1983, 16, 153. (b) Structure Correlation; Biirgi, H. B.; Dunitz, J. D., Eds.; VCH: Weinheim, 1994. 70. Crabtree, R. H.; Lavin, M. Inorg. Chem. 1986, 25, 805. 71. Braga, D.; Grepioni, F.; Sabatino, P; Desiraju, G. R. Organometallics 1994, 13, 3532. 72. (a) Leiserowitz, L. Acta Cryst. Sect. B 1976, B32, 775. (b) Leiserowitz, L. ; Schmidt, G. M. J. Chem. Soc. A 1969, 2372. (c) Etter, M. C. Acc. Chem. Res. 1990, 23, 120. (d) Etter, M. C.; MacDonald, J. C.; Bernstein, J. Acta Crystaliogr. Sect. B 1990, B46, 256. 73. (a) Krause, J.; Deng-Yang Jan, D.-Y.; Shore, D.J. Am. Chem. Soc. 1987, 109, 4416. (b) Jeannin, S.; Jeannin, Y.; Lavigne, G. Inorg. Chem. 1978, 17, 2103. 74. Chang, T.-H.; Zink, J. I. J. Am. Chem. Soc. 1987, 109, 692. 75. Horwitz, C. P.; Shriver, D. F. Adv. Organomet. Chem. 1984, 23, 218. 76. (a) Beno, M. A.; Williams, J. M.; Tachikawa, M.; Muetterties, E. L. J. Am. Chem. Soc. 1980, 102, 4542. (b) Beno, M. A.; Williams, J. M.; Tachikawa, M.; Muetterfies, E. L. J. Am. Chem. Soc. 1981, 103, 1485. 77. Wadepohl, H.; Braga, D.; Grepioni, F. Organometallics 1995, 14, 24. 78. (a) Braga, D.; Grepioni, F.; Parisini, E.; Johnson, B. F. G.; Martin, C. M.; Nairn, J. G. M.; Lewis, J.; Martinelli, M. J. Chem. Soc. Dalton Trans. 1993, 1891. 79. Desiraju, G. R. Acc. Chem. Res. 1991, 24, 290. 80. Braga, D.; Grepioni, E; Peddireddi, V. R.; Kumar, B.; Desiraju, G. R. J. Am. Chem. Soc. 1995, 117, 3156. 81. Calderazzo, F.; Facchinetti, G.; Marchetti, F.; Zanazzi, P. F. J. Chem. Soc., Chem. Commun. 1981, 181. 82. Chen, C. K.; Cheng, C. H.; Hsen, T. H. Organometallics 1987, 6, 868. 83. (a) Kharas, K. C. C.; Dahl, L. F. Adv. Chem. Phys. 1988, 70, 1. (b) Johnson, D. C.; Benfield, R. E.; Edwards, P. P., Nelson, W. J. H.; Vargas, M. D. Nature 1985, 314, 231. (c) Teo, B. K.; DiSalvo, E J.; Waszczak, J. V.; Longoni, G.; Ceriotti, A. Inorg. Chem. 1986, 25, 2265.

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A NOVEL APPROACH TO HYDROGEN BONDING THEORY

Paola Gilli, Valeria Ferretti, Valerio Bertolasi, and Gastone Gilli

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance-Assisted Hydrogen Bonding (RAHB) . . . . . . . . . . . . . . . . A. Structural Data Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Choice of a Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . C. The Overall Structural Correlation . . . . . . . . . . . . . . . . . . . . . . D. Interpretation of the Results: The RAHB Model . . . . . . . . . . . . . . . III. A Comprehensive Theory for Homonuclear Hydrogen Bonding . . . . . . . . . A. Interpretation of the Results: Covalent Nature of the Strong H-Bond . . . . B. Conclusions: Homo- and Heteronuclear H-bond . . . . . . . . . . . . . . IV. Selected Applications of Hydrogen Bonding Theory . . . . . . . . . . . . . . . A. Effect of Dissymmetry on Hydrogen Bond Strength . . . . . . . . . . . . B. Changing the Length of RAHB ~-Conjugated Chain . . . . . . . . . . . . C. On the Origin of the Keto-Enol Tautomerism . . . . . . . . . . . . . . . . D. RAHB Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. II.

Advances in Molecular Structure Research Volume 2, pages 67-102 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6 67

68 68 69 71 72 73 75 78 85 88 90 90 93 96 97 100

68

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLI

ABSTRACT Though the first explicit paper on H-bonding (Latimer and Rodebush, 1920) is more than seventy years old and in spite of the thousands of papers published in this field over the years, no definitive answer has ever been given to this simple chemical question: "which molecules (or molecular moieties or chemical groups) are most likely to form the strongest H-bonds?". In an attempt to give the long-awaited answer to such a question, a method has been developed which consists in making a provisional tabula rasa of all previous hypotheses, assembling a complete database that collects structural data for all strong (short) H-bonds presently known, and trying to find out the factors that determine the H-bond strength by simple induction from experimental evidence. Such a research project has been made possible by the systematic use of the Cambridge Structural Database and has been completed for the homonuclear O-H--O bond.

!.

INTRODUCTION

Though a few hints have appeared in the previous literature, it is generally admitted that the first definitive paper on H-bonding is that published in 1920 by Latimer and Rodebush [1], where, however, they acknowledged M.L. Huggins for having had the original idea. Three years later [2] G.N. Lewis, chief of the laboratory where the three scientists were working, recognized that the H-bond was "the most important addition to my theory of valence." From then on, thousands of papers have been published on the subject, the most interesting for the history of chemistry being the review written in 1971 by Huggins [3] to summarize his continuous interest in H-bonding over a period of 50 years. A number of relevant books have also been written [4], the most comprehensive and most recent ones being, respectively, the series edited by Schuster, Zundel, and Sandorfy in 1976 [5a] and the book published by Jeffrey and Saenger in 1991 [5b]. The H-bond story is, however, far from being concluded because, in spite of the remarkable achievements in H-bond theory registered in the last 70 years, we can still state that "the chemical factors determining its strength remain widely unknown if exception is made for the role played by the electronegativities of the hydrogen bond donor and acceptor atoms" [6]. The case of O-H--O bond, to which the present chapter is mainly devoted, is perfectly suited to illustrate such a paradoxical lack of knowledge of the chemical forces involved in H-bond formation. Homonuclear and nearly linear O-H--O bonds display an extremely wide interval of d(O--O) contact distances ranging continuously from 2.36 to ca. 3.69/~ which is obtained by summing up the O-H distance (= 1 .A.)and the O and H van der Waals radii. They are formally classified as very strong (2.37-2.50/~), strong (2.50-2.65/~), medium (2.65-2.80/~), and weak (> 2.80/~), but no certain rule is known that can predict which chemical species are able to form the strongest (or shortest) H-bonds.

Hydrogen BondingTheory

69

The prevailing opinion on the matter until 1991 is well described in Jeffrey and Saenger's book [5b], for which there are three classes of very strong H-bonds, the first due to severe intramolecular strain and the two others associated with protonated oxyanions -O-H----O- or solvated protons --O--H+--O = . It is clear that this position, admitting such an incidental event as steric hindrance, can hardly lead to a general chemical theory on the nature of the H-bond. Moreover, the fact that the strongest bonds occur in two out of three cases as a charge-assisted phenomenon has supported a kind of widespread belief that all H-bonds, irrespective of their strength, are essentially electrostatic interactions, though the idea that very strong hydrogen bonds may have covalent character has been taken into account by some authors [7] and, most recently, by Kollman and Allen [8] and Noble and Kotzeborn [9], particularly in relation with the F-H--F- bond. On the other hand, the electrostatic model was also supported by the results of theoretical studies indicating that it was essentially correct for a number of weak and medium range H-bonds [10]. A first improvement towards a more general chemical theory of the H-bond was accomplished by showing that the few cases of very strong intramolecular H-bonds observable in neutral systems were not imputable to steric strain, but to an interplay between H-bond strengthening and delocalization of a re-conjugated system connecting the two oxygens, which was called resonance-assisted hydrogen bonding (RAHB) [11]. Further investigations [12] confirmed this hypothesis for a variety of other resonant intra- and intermolecular O-H--O bonds and, on these grounds, a new comprehensive treatment able to identify all chemical classes allowing strong H-bond formation was advanced [6]. The present chapter is devoted to a review of these most recent models of homonuclear O-H--O bonds and to give a quick overview of their further generalization to more complex systems, paying particular attention to the biologically important heteronuclear N-H--O bond.

II. RESONANCE-ASSISTED HYDROGEN BONDING (RAHB) Until recently [11], no systematic investigation on the possible interplay between H-bond strengthening and n-delocalization of conjugated double bond systems had been accomplished, though some authors have hinted at its possible existence from time to time [3,4d,5b,13]. The first explicit mention of such effect has been made, at least at our knowledge, by Huggins in his 1971 review [3] where, discussing the adenine-thymine association, he remarks that the "important stabilization of these [hydrogen] bonds by electron resonance.., is rarely mentioned." We became interested into the problem in 1984 through the more or less fortuitous crystal structure determination of CGS8216 [14], a high-affinity ligand of the benzodiazepine receptor. The crystal packing (Figure 1) consisted of infinite chains of molecules linked, head-to-tail, by an abnormally short N-H--O bond having an N--O distance of 2.694(3)/~, while the pattern of bond distances appeared to be heavily perturbed in the H N - C = C - C = O enaminone (1-enamin-3-one) fragment

70

P. GILLI, V. FERRETTI,V. BERTOLASI,and G. GILLI

.,.-.~:.:~,o"

:~ro:.~-.z..~:~.....:~y3"--...~:t'' ~\~._...~\ ~ ~

....

Figure 1. The crystal packing of CGS8216 [ 14]. The shading indicates the chain of resonance-assisted hydrogen bonds connecting the molecules inside the crystal. Distances in ,~. (Reproduced by permission from Gilli et al., 1993 [ 12c]).

because of an unusually large contribution of the H N + - - C - C = C - O - polar canonical form. The authors remarked that such n-delocalization was to "be ascribed to the fact that C : O and N-H are implicated in infinite chains of CO--HN hydrogen bonds; this causes a partial obliteration of the charges and a consequent increased contribution of the polar canonical form." Starting from the idea of a synergism between hydrogen bond shortening and ~-delocalization, a search on the inter- and intramolecular N-H--O bonds (la,b) present in the Cambridge Structural Database (CSD) [15] was undertaken. Though promising in some respects, the investigation

~N/H

.... 0

could not be conclusive because of the insufficient number of accurate crystal structures available at that time. It was then decided to study the very similar O--C-C--C-O--H 2-en-3-ol-l-one (enolone or 13-diketone enol) fragment, a system for which, because of its constant interest for inorganic, organic, and physical chemistry, a wealth of structural and physicochemical data had been collected and recently reviewed in remarkable detail by Emsley [16]. Also enolones were found to form both intra- (Ha) and intermolecular (llb) H-bonds, and the possibility of a definite interdependence between H-bond strengthening and increased contribution

Hydrogen BondingTheory

71

0/H.... 0 d~I IId` R'~~R3 l~z lla ~/H....O llIa

. . , ,~. Ho ~"~

,,,.

lib

~/H ....e IIIb

of the polar canonical form ][lib to the ground state of the fragment was supported by the following facts: 1. O--O distances were often unusually short, up to 2.40-2.42 /~ for the intramolecular (lla) and 2.46-2.48 ]k for the intermolecular (Hb) case, at variance with other nonresonant O-H--O bonds (e.g. 2.75/~ in water and 2.77 [7] A in polyalcohols and saccharides [ 17,18]). 2. Such d(O--O) shortening was generally associated with an increased delocalization of the O--C--C--C-OH x-conjugated system, a lengthening of d(O-H) from 0.97 up to 1.20-1.25/~, a lowering of the IR v(OH) stretching frequency from 3600 to 2500 cm -1, and a downfield shift of the enolic proton 1H NMR from the usual value of 5-8 to 17 ppm [16,19,20]. To show that these somewhat diverse experimental data were all pointing to a specific type of O-H--O bond, called resonance-assisted hydrogen bond (RAHB) was, however, a much more complex affair requiting systematic investigations on large sets of homogeneous data. Since IR and NMR information was rather scattered and not ordered in proper databases, the structural data collected in the CSD [15] were used and analyzed by crystal structure correlation methods [21]. The final aim was to obtain evidence of a systematic dependence of the H-bond strength on n-delocalization in all strong (i.e. having d(O--O) < 2.65 A) intra- or intermolecular O-H--O bonds occurring in or between neutral molecular moieties.

A. Structural Data Retrieval In our first paper [11] only 20 X-ray or neutron structures of 13-diketone enols were analyzed, most of them forming the intramolecular H-bond lla. Though sufficient for establishing the RAHB model, the original set was rather poor and has been successively enriched by the crystal structure determinations of nine 1,3-diaryl-l,3-propandione enols displaying intramolecular H-bonds (lla) [12b, 12e], and seven 1,3-cyclopentanedione or 1,3-cyclohexanedione enols forming infinite chains of intermolecular hydrogen bonds (Hb) [22]. At the same time a systematic investigation of 22 crystals containing enolone (H) or enediolone (2-en-2,3-diol-1-one; IV) fragments connected by intermolecular hydrogen bonds

72

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLI

o•o•,. o

IV

H

(lib) was accomplished [12c]. All these compounds were found to share the common geometry displayed in II where the enolone frame, as well as O-H bonds and carbonyl lone-pairs, are essentially coplanar and conforming to the trigonal geometry of the sp 2 hybridization [23]. The following part of this paragraph makes use of this complete set of data (54 unique fragments), which are analyzed by the same methods used in the original paper [11]. This procedure is aimed to show that the results of our first analysis are in agreement with (or cannot be falsified by) all structural data so far collected on [~-diketone, 13-ketoester, and [~-ketoamide enols forming both inter- and intramolecular H-bonds. It has to be noted that CSD [15] includes several other crystal structures containing the enolone fragmentBnot all, however, suited to our present purpose. Some of them lack in precision, and, generally, only X-ray or neutron structures having R < 0.07, average standard deviations a(C-C) < 0.008, no disorder in the group of interest, and refined enolic protons have been considered. The few cases where the proton was lying on a crystallographic element of symmetry were, likewise, rejected because of suspected disorder. Moreover, nresonance is not the only factor affecting the H-bond strength, which turns out to be lowered by further H-bonds accepted by the two O--O--H oxygens. All these cases, excluded from the present analysis, have been discussed in detail in a previous review [12a].

B. Choice of a Coordinate System In order to confirm a relationship between H-bond strengthening and n-system delocalization starting from crystal data, it is necessary to quantify these two factors from a geometrical point of view. Both d(O--O) and d(O--H) can be taken as natural indicators of H-bond strength, though the first is to be preferred because of the low precision of X-ray determined proton positions. A simple way for describing the n-system delocalization consists in using symmetry coordinates for the in-plane antisymmetric vibration of the O = C - C = C - O H group that is ql = d l - d 4 and q2 = d 3 - d2 (for symbols see lla and Va). The physical meaning of ql and q2 is better understood by taking the sum Q = q l + q2. Positive, zero, and negative Q values now correspond to the keto--enolic (KE = Va, a'), totally n-delocalized (TnD = Vb, b'), and enol-ketonic (EK = Vc, c') forms, respectively. Maximum and minimum Q values for the completely localized forms are calculated as + 0.32 ,~ by assuming [11] standard pure single and double bond distances: d 1 = 1.37, d 2 = 1.33, d 3 = 1.48, and d 4 = 1.20/~. Alternatively, the fragment geometry can be

Hydrogen Bonding Theory

73

KE

TxD

EK

0 ''H ....0

o'/H"'O

O'""'H~o

0.320

0

I"

I

1.0

0.5

Q

-0.320

~

0.0

I

described by a coupling parameter ~,, according to which the geometrical state of the fragment is a mixture of the KE and EK forms according to ~,(KE) + (1 - X)(EK). It assumes values of ~ = 1 for KE, 0.0 for EK, and 0.5 for TroD. It is easy to s h o w that ~ = ( I + Q / Q o ) / 2 with Q = d 1 - d 4 + d 3 - d 2 and Qo = (d I - d 4 + d 3 - d2)st, where st stays for standard pure single and double bond distances.

C. The Overall Structural Correlation Having so chosen d(O--O) and d(H--O) as geometrical indicators for H-bond strength and qx and q2 (or Q = ql + q2) for n-system conjugation, it is now interesting to find out how they are related. Accordingly, Figure 2 displays the [d(O--O), EK

T~D 2.70

d(O.--O) 2.60 _

tl) 2.50 _

2.40

1.75

1.50

1.25

1.00

I

I

I

I

9:,,t

+..

Pc.."

',

,.

o'~..~

',

o O0o,+ o.O ~oldm"

- ~ 0

oo

+ 2t

l

9

. I O o~

dill---Of (~) cliO-H) ~-----'~m~q~'-~m,.-~_ I .

Figure 2.

I

.+.

q, c;~l "b'-O.~5 0.17

p~

o

9

%

Oc

"~0

q2 1'~1

0.15

Four-dimensional scatterpiot and its projections for the 54 enolones forming intramolecular (open circles) and intermolecular (full circles) H-bonds. The variables ql, q2, d(O--O), d(O-H) and d(H--O) are defined in the text. The stars indicate the geometries of the enolone fragment not perturbed by H-bonds. (KE = ketoenol, EK = enolketo, TxD = totally x-delocalized forms; cfr. Va-c and Va'-c').

74

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLI

d(O-H), q l, q2] four-dimensional scatter plot for the sample of enolones described above. The d(O--O) versus d(O-H) and d(H--O) plot on the left illustrates a well established relationship in H-bond theory: that is that the shortening of the O--O contact distance causes both a shortening of d(H--O) and a lengthening of d(O-H), the two distances becoming nearly equal for d(O--O) = 2.40/~. There are less points in this part of the plot because some X-ray determined O-H distances were too inaccurate to be taken into account, thus the plot can be considered only as a qualitative indication of the d(O-H), d(H--O), and d(O--O) interdependence (a much more reliable correlation based on accurate neutron data is reported in the

0.30

[i',i!i',i!i',i!i!i!i',i',iii',i',i',i',i',ili',i!!i',i!i',i',i',-i!iiiiiiil ~~

-

0.25 0.20

Q

-o.g

o

oO.~ e

-

~e

-o.B

(i) 0.15

0.10 0.05

0.00

0.7

11',

,

~ o (~) o

: -

o

-

o oo O

0.6

0.5

. . .2.4.0 . I . . . . I , , - , - - , , , I , " , I t ' "2.60 1'2.50

' 89

. . .2.80 . I,,,

,-

d(O--O) (~,) 0.32 I

~.o 0 / H .... 0

0 '

Q

I

o.5 0"

.H.,.

.,~ 0

-0.32

(A)

'1

o.o 0 .... H'~O

Figure 3. Scatter plot of the x-delocalization parameters (Q and Z,) versus d(O--O) distances for the set of 54 enolones forming intramolecular (open circles) and intermolecular (full circles) H-bonds. The upper right rectangle indicates the range of normal nonresonant O-H--O bonds. The changes of Q and k along the pathway from ketoenolic (KE) to enolketonic (EK) through totally x-delocalized (TxD) form is shown at the foot of the drawing.

Hydrogen Bonding Theory

75

next section). The three-dimensional [d(O--O), ql, q2] scatter plot is displayed in the fight side box of Figure 2. Points representing each structure are plotted twice because of the symmetry of the fragment. The two stars indicate the geometries of the KE and EK forms not perturbed by H-bonding, determined as an average of eight structures containing the O--C-C----C-OR enolone alkylether fragment; they correspond to a barely delocalized r~-conjugated system describable as an 87:13 mixture of the canonical forms Ilia and rob. The shortening of d(O--O) is seen to be associated with a progressive decrease of (ql, q2) until the central point (0,0), corresponding to the totally n-delocalized form (T~D). All points essentially lie on the diagonal plane of the plot, thus showing that ql and q2 are linearly dependent and that the d(O--O) versus (ql, q2) correlation can be replaced without loss of generality by that of d(O--O) versus Q = ql + q2" This correlation is displayed in more detail in Figure 3, which illustrates the scatterplot of the r~-delocalization parameters, Q and ~, versus the O--O distances for all compounds studied. For comparison, the range ofd(O--O) values [17] observed for nonresonant (~ = 1.0 or 0.0) H-bonds have been added (elongated rectangle shown on the upper right corner).

D. Interpretation of the Results: The RAHB Model The correlations of Figures 2 and 3 confirm the interdependence between rc-delocalization (Q or X) and O--O contact distance in H-bonded enolones, an effect which has been called RAHB [11] and is qualitatively interpretable by the model illustrated in Figure 4. Let us firstly imagine that the double C--C bond is substituted by a single C--C bond. In this case the nonresonant intramolecular O-H--O bond will have its most common d(O--O) length of 2.77 + 0.07 ]k (see Figure 3). Re-establishing the double C--C bond, the resonance Illa ~ m b will

do.. 0 HBond

Figure 4. A graphical representation of the RAHB model. (Reproduced by permission from Gilli et al., 1989 [ 11]).

76

P. GILLI, V. FERRETTI, V. BERTOLASi, and G. GILLI

produce a certain delocalization of the n-conjugated system and the establishing of positive and negative partial charges on the enolic and ketonic oxygens, respectively. These have the correct signs for strengthening the O-H--O bond with consequent shortening of d(O--O) and lengthening of d(O-H). Since the movement of the proton to the right (Figure 4) is equivalent to moving a negative charge to the left, the global effect will be the annihilation of the partial charges initially set up by resonance, which will allow a further delocalization of the n-system and a further strengthening of the H-bond. Iteration of such an imaginary process will inevitably lead to the complete delocalization of the n-conjugated system and to a very short O--O distance of some 2.40 A, unless stopped at an intermediate level by repulsion forces or by the imperfect symmetry of an unsymmetrically substituted enolone fragment. This simple model suggests that RAHB is a synergism of n-delocalization and H-bond strengthening which can lead to very short O-H--O bonds by a mechanism of generation and annihilation of charges. This may suggest that the H-bond is becoming increasingly covalent in nature, a point which will be discussed in detail in the next section. Notwithstanding, the ionic model of Figure 4 is capable of semiempirical parametrization, and this can be done by minimizing the energy of the system, partitioned according to, E = EHB+ ERES + EBp + EvdW,

(I)

a scheme having the advantage that empirical equations for the evaluation of all terms are known (for a detailed treatment see the original paper [II]). The bond energy, EHB,can be calculated by the Lippincott and Schroeder equation [24], the resonance energy, Ep.ES,by the HOSE method proposed by Krygowski [25], and the energy of bond polarization required to create the opposite fractional charges on the oxygens, EBp , from the curves of atomic ionization energy versus electron affinity given by Hinze and Jaff'e for the main group elements [26], while the van der Waals energy, EvdW, is easily evaluated by normal molecular mechanics methods. The final potential energy surface (PES), calculated for acetylacetone (R t = R3 = CH3 and R2 = H in IIa), is shown in Figure 5 as a function of two independent variables: (I) the Pauling bond number of the O-H bond, n(O-H) [ I _>n _>0 while 0.97 _d(O-H)_< oo], and (2) the usual coupling parameter X, (and its related quantity Q). The map is centrosymmetric because of the fragment symmetry, and shows a low-energy diagonal valley corresponding to the pathway of proton transfer that transforms KE (upper left corner) into EK (lower right corner). The experimental structures belonging to the set of 54 enolones for which proton positions were determined with reasonable accuracy are indicated by open and full circles for intra- and intermolecular H-bonds, respectively; they are seen to be arranged along the minimum energy pathway. The open lozenges mark the positions of the gas electron diffraction structure of acetylacetone [27], the molecule for

Hydrogen Bonding Theory

77 |

0"/H.... O

..H~| 0

01

0.197 1

1.0L_

1.06 .8[

1.62 d (O~H)~{ ~ ) ~ ~ .'~ 0

1.18 1.25 1.34 n COcH)

9 . . . . . . . . . .

.4

I .....

I ......

~ 0.32

I

16

0.75.

0.5

0,25

16

o

0.0_ G/H.. e 0 O "'0

2

4

n CO~-H)

i PROTON

.6

0.32 8

.

I

x O""H\o Z -

TRANSFER

Figure 5. Potential energy surface (PES) in kJ mo1-1 for acetylacetone as a function of Q or k (the x-delocalization parameters) and n(O-H) Pauling's bond number (the H-bond strength parameter). The most accurate experimental geometries in the set of 54 enolones considered are indicated by open (intramolecular H-bonds) and full (intermolecular H-bonds) circles; the shaded rectangles are clusters of 6 open and 17 full circles. The stars mark the enolone geometry in the absence of H-bond and the open lozenges the g.e.d, structure of acetylacetone [27]. (Adapted by permission from Gilli et al., 1989 [11]). which the PES has been evaluated. The agreement is remarkable, and simple measurements on the energy map indicate that for acetylacetone EHB and the proton transfer barrier amount to 14.9 and 8.3 kcal mo1-1, respectively. A three-dimensional representation of the same PES is reported in Figure 6.

78

P. GILLI, V. FERRETTI,V. BERTOLASI, and G. GILLI

O

o. O

,z

J

\

Figure 6.

Three-dimensional representation of the PES of Figure 5. (Reproduced by permission of Oxford University Press from Gilli, 1992 in Fundamentals of Crystallography, C. Giacovazzo, Ed. [12c~).

Iii.

A COMPREHENSIVE THEORY FOR H O M O N U C L E A R HYDROGEN BONDING

The RAHB model has succeeded in rationalizing a class of short O-H--O bonds occurring in neutral molecules. There are, however, other classes of strong or very strong homonuclear H-bonds that occur in electrically charged systems (i.e.-O-H---O- and =O--H+--O = ) [5,28,29], and it may be wondered whether it is possible to develop a comprehensive model able to predict all chemical situations giving rise to abnormally strong H-bonds. To this end, we have undertaken [6] a systematic study on short H-bonds with d(O--O) < 2.69 ]k to be found in organic (CSD [15]) or inorganic (Inorganic Crystal Structure Database, ICSD [30]) databases. Since the number of structures found was more than adequate, we could be very selective about the quality of the data. The final selection can be considered essentially exhaustive as far as neutron data of short H-bonds are concerned, and X-ray data of good quality were mainly used to cover chemical classes for which neutron information was inadequate. Moreover, all cases where the two O-H--O oxygens were perturbed by further solvation were excluded and, for intermolecular bonds, only homomolecular associations of identical molecules were taken into account (e.g., cases of water bridges connecting two molecules were not considered).

Hydrogen Bonding Theory

79

Table 1 summarizes the chemical classes where strong and very strong O-H--O bonds are encountered, together with the number of structures found [distinguishing neutron (N) and X-ray (X) data] and their ranges of d(O--O) and k values. The coupling parameter ~, is the same as for enolones but calculated only from the pair of C - O distances: for a generic X - O - H - - O - - Y bond it is obtained as ~, = (1 + q/qo)/2, where q = d(X-O) - d(Y--O), and qo is the same quantity calculated for a system not perturbed by hydrogen bonding. Class D has been added to the table for comparison and includes non-resonant and non-charged H-bonds for which ~, can be assumed to be 1 (or 0). The last column of Table 1 reports the codes of the most representative compounds for each chemical class, whose structural formulas are shown in Charts 1, 2 and 3. All short H-bonds found can be grouped into three main classes:

Table 1. Summary of the Chemical Classes Where Strong or Very Strong O-H--O Bonds Can Be Observed (Classes A, B or C)a

Class

HB b

Sample c

d(O--O) d Range

~e Range

Chemical Schemes f

Ala. Carboxylic acid-carboxylates

i

9N, 4X

2.44-2.49

0.50-0.68

AI-1, A1-2

Alb. Carboxylic acid-carboxylates

I

7N

2.39-2.42

0.53--0.60

A1-3, A1-4

A2a. Metal oximes

I

6N, 3X

2.39-2.48

0.56--0.61

A2-1

A2b. Metal glyoximes

I

5X

2.44-2.69

0.54-0.78

A2-2

A3. Alcohol-alcoholates

i

IN, 1X

2.39-2.43

0.5 g

A3-1

A4. Water-hydroxyl A5. Inorganic acid salts

i i

2X 9N

2.41-2.44 2.36-2.43

0.5 g n.c. h

A4-1 A5-1, A5-2

B. O--H+--O

i

3N, 4X

2.36-2.43

0.5 g

B.1-B.4

Cla. ~-Diketone enols Clb. ~-Diketone enols

I i

IN, 10X 2N, 14X

2.43-2.55 2.46-2.65

0.51-0.72 0.56--0.76

CI-1 C1-2

C2. ~-ketoester or ketoamide enols

i

1N, 9X

2.55-2.69

0.64--0.88

C2-1

C3. ~-Diketone enols

I

2X

2.42-2.44

0.52-0.53

C3-1

C4. ~-Diketone enols

I

4X

2.43-2.51

0.51-0.53

C4-1

C5a. Carboxylic acids (chains) C5b. Carboxylic acids (dimers)

i i

IN, 6X 4N, 6X

2.62-2.70 2.62-2.67

0.74-0.84 0.68-0.83

C5-1 C5-2

--

2.77-.k0.07

1.0

D. Alcohols and saccharides

any

Notes: aClass D is reported for comparison (see text). Adapted from Ref. 6 to which the reader is addressed for a complete list of references. bHB = i(inter), I(intra). CSample = number of neutron (N) and X-ray (X) structures taken into account. dd(O--O)range in ,~,. ex = coupling parameter defined in the text. fChemical Schemes - codes of the structural formulas given as examples in Charts l, 2, and 3. gAssumed for symmetry. h~, not computable.

m~

ILl

Lu

O

A

ua

O

~

,o~

,-,

Z

co

~

,o

~ IO ILl

A

O O zI.LI -"Z~

ILl Z 0 Z

0.--

o-~

Zn

0 I-.

!

".~

v

un

v

"O

!

O

.1_

9

I

7-

O

"O

$

r

c~ 00

O

O..

E

r0 X

E

0

,4

Chart 1.

(Continued)

81

Chart 2. Some examples of positive charge-assisted O--H+--O H-bonds (Class B).

82

Chart 3. Some examples of resonance-assisted O-H--O H-bonds (Class C).

83

84

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLi

Class A: - O - H - - - O - o r (-)CAHB;

negative charge-assistedhydrogen bonding,

Class B" ---O--H+--O= or positive charge-assisted hydrogen bonding, (+)CAHB; Class C" -O-H--O-- where the two oxygens are interconnected by a system of n-conjugated double bonds, or resonance-assisted hydrogen bonding, RAHB.

While only the H-bonds belonging to these three classes can be classified as short or very short, those belonging to class D, which represent the great majority of all the other possible O-H--O interactions, only give rather long d(O--O) distances (i.e. longer than 2.70 A). The linear relationship between d(O--O) and X already observed for enolones (now classified in class C) is found to hold for all the compounds studied, as illustrated in Figure 7 which reports the X versus d(O--O) .

1.0

::::::::::::::::::::::::::::::::::::::::::::

0.9

~A"k..k 0.8

X

,i A

[]

0.7

II

[] I

,i

A

Ii

i

~bn 130 1 I

0.6

~=Oi~ 0.5

.

ill

+ r i 1 I'i

I I Ill

2.40

[]

I i i it

" Ili

2.50

i II

i I I I1

III

2.60

a(o--o)

J I i f'i

I I i J iil"l

2.70

li

il

iri

2.80

i

(i)

Figure 7. Scatter plot of the coupling parameter ~ versus d(O--O) for compounds classified in Table 1. Class symbols: A1 = open circles; A2 = open squares; A3 and A4 = larger open circles; B = crosses; C1 - full squares; C2 = stars; C3 and C4 = full circles; C5 = full triangles; D - shaded upper right rectangle. (Reproduced by permission from Gilli et ai., 1994 [6]).

Hydrogen Bonding Theory 1.25

.....

85

g

o

r o _ _- 0 . 9 6 ,

1.20

Rm~n..... -2.40 A

r~

Rmln=2.40/~

1.15 o,~ v

0

1. I 0

I

1,05

1.00

=1.0 ,&

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

0.95

O,cJo

T ~ ~

1.10

1'i~()

~ ~'i'

i'"~",

1.,30

~'~ 'I

~ ~ ,

1.40

, i"'~'~

1.50

~ ~'I

1.60

'~ ~ ~ ' I

d(H--0)

1.70

,

~ ~ i,

1.80

i,

1'.9s

,

' 2.00 '''

r"~

,

I

2.1(~

(A)

Figure 8. Scatter plot of d(O-H) versus d(H--O) for selected neutron structures containing nearly linear H-bonds (O-H-O > 165~ Continuous and dashed lines are interpolations calculated by Eqs. 4 and 5 with the parameters given in the upper right corner. Class symbols: A1, A2, B, Cl, and C5 as in Figure 7; A5 = inorganic acid salts = diagonal crosses; alcohols, saccharides, nonresonant acids and amino acids = open lozenges; ice Ih = open triangles. (Reproduced by permission from Gilli et al., 1994 [6]).

scatter plot shown in Figure 3 for classes C and D with the addition of the new classes A and B (data from Table 1). Because of the intercorrelation between d(O--O) and X, very short O-H--O bonds are inevitably associated with X = 0.5, thus a situation for which X1-O and O = X 2 distances become practically identical. A parallel behavior is displayed by the O-H and H--O distances as shown by the d(O-H) versus d(H--O) and d(O-H) versus d(O--O) scatter plots of Figures 8 and 9. These have been obtained from the most accurate neutron data presently available and clearly show that the two O-H distances become identical when d(O--O) tends to 2.40 A.

A. Interpretation of the Results: Covalent Nature of the Strong H-Bond The progressive equalization of both O-X and O-H distances occurring when the H-bond becomes increasingly shorter seems to indicate that very short O-H--O bonds are to be considered as totally delocalized O.-.H---:O three-center-fourelectron covalent bonds. This is expressed in VB terms by saying that they are linear combinations,

86

P. GILLI, V. FERRETTI,V. BERTOLASi, and G. GILLi 1.25

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

'"i

r~

1.20

o

r ~ =0.925, Rmin=2.40 A

o

1.15

m.

1.10

I O

Rm~,=2.40 .A

1.05

i•

.~,.

-.

o

d(O-H)=l.0 A

1.00 0.95 -

o.o:i

i

I 1 :I I i

2.40

t

i'

1 lt

zs0

'

'Lk;

i

2.70.

' "

.

'2.~d . . .

2.90

'3.(~0

Figure 9. Scatter plot of d(O-H) versus d(O--O) for selected neutron structures containing nearly linear H-bonds (O-H--O >_165~ Continuous and dashed lines and symbols have the same meaning as in Figure 8. (Reproduced by permission from Gilli et al., 1994 [6]). qJ = alqJl + anq~"

(2)

of two resonance forms, -O-H---O-- e+ -O----H-O += (/)

(3)

(I/)

which are isoenergetic and mix with identical a coefficients. The derivation of this simple rule makes it possible to predict which chemical substances are able to form strong H-bonds: only chemical situations for which the resonance forms I and H may become energetically (and then chemically) equivalent can lead to strong or very strong H-bonds. How such chemical equivalence can be achieved for the O-H--O bond and, in general, for any homonuclear H-bond is illustrated in Figure 10; there are only three ways: 9 by adding an electron and giving rise to the-O-H--O-- bond; 9 by removing an electron and producing the situation for which a proton is captured by two oxygens, that is ---O--H+--O=; 9 by connecting the two oxygens by a chain of conjugated double bonds. It is evident that these three classes do correspond to those already identified by the systematic analysis of the experimental structural data, that is, in the

Hydrogen Bonding Theory

87 --O--H . . . . O =

I

II

- -

+e E

A

s

O N

---O- H. . . . O - - I

~

OW---H. . . .

N

---O . . . . H--O--

| Oy H'~O

cE

II

~ 0

.

.

.

O"-" I

H~O . "--

.

II

cJ o-'H'o

o f H'-o

II Figure 10. / h e three ways for making equivalent the two resonance forms i and II of Eq. (3): by adding (A) or removing (B) an electron, and by connecting the two oxygens by a chain of conjugated double bonds (C). (Reproduced by permission from Gilli et al., 1994 [6]).

order, (-)CAHB, (+)CAHB, and RAHB. All previous considerations point, therefore, to the idea that strong homonuclear H-bonds are essentially covalent in nature and can be described by the three-center-four-electron formalism and this hypothesis is widely confirmed for the O-H--O bond by the scatterplot of Figure 7, which clearly shows that in a generic XI-O-H--O----X 2 bond the two X - O distances must become equal in the strongest H-bonds. Further evidence comes from a more accurate analysis of the scatterplots of Figures 8 and 9. Symmetrical three-center-four-electron b o n d s - X - - - Y - - - : X - " (such as O-H--O) are known [31a-c] to follow the bond number conservation rule n I + n 2 = 1, where n I and n 2 are the bond numbers of the X - Y and Y--X bonds, respectively, and n is defined according to Pauling's formula [31d] Ad = - c lOgl0n. It is straightforward to show that for a linear-O.-.H---:O= system these relations lead to, r* = r ~ - c log10 (1 - 10 R = r + r*

-(r-r~

(4) (5)

where r = d(O-H), r*= d(H--O), R = d(O--O), r ~ = d(O-H) in the absence of an H-bond (usually 0.96/~), and,

88

P. GILLI, V. FERRETTI, V. BERTOLASi, and G. GILLi

c = [Rmin/2- r~

(6)

where Rmin is the shortest d(O--O) value observable (usually 2.40/~). This implies that only two known parameters, r ~ and Rmin, control the form of the functions r* versus r and R versus r. It is important to note that the exclusion of bent hydrogen bonds does not actually decrease the number of cases to be taken into account because of the well-known tendency displayed by the O-H--O system to be the more linear the shorter the d(O--O) distance is. It has been shown [32] that this condition becomes stringent for strong H-bonds: e.g., d(O--O) values in the range 2.40-2.62 ,~ do not allow O-H--O angles outside the 163-180 ~ interval

[32c]. Figures 8 and 9 show the interpolation of the experimental points by Eqs. 4-6 using the values of R = 2.4 and r ~ = 0.96 ~ (dashed curves) or r ~ = 0.925/~ (continuous curves). The best fit is obtained with the value of 0.925 A in spite of the fact that microwaves and gas electron diffraction experiments give larger values of r ~ of the order of 0.96-0.97 ,~. This apparent discrepancy can be interpreted by saying that the dashed curves represent the r versus r* (or R) relationships which would be observed if the H-bond were totally covalent, and that another attractive contribution, probably electrostatic in nature, is able to further shorten the interaction and to shift the covalent dashed curves into the position of the continuous ones. Assuming, for the sake of simplicity, that the homonuclear O-H--O bond energy, EHB, can be factorized according to, EHB = ECOv + EEL + EREP

(7)

(Eco v and EEL being the covalent and electrostatic attraction energies, and EREP the interatomic repulsion energy), the two dashed curves represent, for any r, the contribution of Eco v to the H-bond geometry, while the horizontal displacements from the dashed to the continuous curves are a measure of the further contribution of EEL. Such horizontal displacements are rather large for long H-bonds which, accordingly, are to be considered mostly electrostatic in nature, but rapidly decrease for shorter bonds and vanish when R = d(O--O) becomes smaller than ca. 2.45-2.50 ,~ (see Figure 9), the point beyond which the H-bond becomes essentially covalent. Beyond this point the increasing degree of covalency causes a remarkable lengthening of the O-H bond, and all the shortest H-bonds (2.40 < R < 2.44/~) display more or less perfectly centered protons corresponding to the symmetrical O--H--O geometry.

B. Conclusions: Homo- and Heteronuclear H-bond The previous section can be summarized as follows: While the O--O distance is shortened from ca. 2.80 to 2.40 ,~, the H-bond is transformed from a weak and unsymmetrical O-H--O electrostatic interaction to a strong, covalent, and symmetrical O--H--O bond. By generalization, such behavior could be considered a

Hydrogen BondingTheory

89

common feature of all homonuclear H-bonds (O-H--O, N-H--N, F-H--F), at variance with heteronuclear ones (e.g. N-H--O) which are assumed to give weaker bonds of mostly electrostatic nature in view of their intrinsic lack of symmetry. In agreement with the previous paragraph, it is generally recognized that heteronuclear H-bonds are weaker than homonuclear ones, and, in this context, Jeffrey and Saenger have remarked in their recent book [5b] that "unlike the O-H--O bonds, there are no examples of strong N-H--O hydrogen bonds." The reason for this is now identified in the hindered covalency occurring in the N-H--O bond (or, in general, in any X-H--Y heteronuclear bond). Taking as an example the RAHB in enaminones (I), the condition for establishing a really strong N-H--O bond would require the nearly 1"1 mixing of the two isoenergetic forms, O=C-C=C-N-H

(19

H-O-C=C-C=N

(113

but this is made impossible by the greater affinity of the proton for the N than for the O atom, which stabilizes I' with respect to I f by some 19.4 kcal mo1-1, according to our preliminary 4-31G* ab initio calculations. The mixing is then imperfect, with the result that the H-bond in pure enaminones (la and Ib), though assisted by resonance, is weaker than in the homologous enolones lla and lib. This hindering of covalency as an effect of dissymmetry is convincingly confirmed by the results of neutron diffraction studies [33] on the N-H--O system, which shows that the N-H bond can be only slightly stretched by H-bond formation (from 1.01 to 1.06 /k) in spite of the strong electronegativities of the donor and acceptor atoms involved. A detailed study of the heteronuclear H-bond is beyond the scope of the present chapter. It may be remarked, however, that strong resonant X-H--Y bonds can be actually obtained by introducing in the enaminone fragment proper substituents able to equalize the proton affinities of the X and Y atoms. This has recently been discussed in a series of papers on the N-H--O and O-H--N bonds to which the interested reader is addressed to for additional information [34]. It is much more simple to show [6] that other homonuclear X-H--X bonds can display features not dissimilar from those of the O-H--O bond. The N-H--N system can produce very short RAHBs in compounds such as enaminoimine Vl (d(N--N) = 2.65 ]k [35]) and cis-formazan VII (d(N--N) = 2.57 ]k [36]), as well as positive charge-assisted H-bonds in the so-called proton sponges, which display very short N--N distances (d(N--N) = 2.54-2.65 A in VIII and 2.53-2.60 ]k in IX [37])

~N'H.~~N / ~N"H....N/ ~,,I

II

N~ i

o

,,,~.,N...H...N/~ ~..,F~

IIN

~VII

......

VIII

~LX

90

P. GILLi, V. FERRETTi, V. BERTOLASi, and G. GILLI

together with elongated N-H bonds and even centered protons. Likewise, difluoride anions are well known to form homonuclear F-H--F- bonds of negative chargeassisted type which are extremely short (d(F--F) = 2.26-2.28 ,~) and have essentially centered proton positions [38].

IV. SELECTED APPLICATIONS OF HYDROGEN BONDING THEORY A. Effect of Dissymmetry on Hydrogen Bond Strength The effect of dissymmetry on the strength of H-bonds has been already discussed in the previous section as far as the differences between homonuclear and heteronuclear H-bonds are concerned. However, dissymmetry effects can also be observed in homonuclear bonds because of uneven substitutions in the molecules involved. Two well documented examples are described below.

Effect of Dissymmetric Substitution on Intramolecular O-H--O RAHB It is well known from solution chemistry that ~-ketoesters (Xb) and 13-ketoamides (Xc) undergo enolization to a lesser extent than 13-diketones (Xa). It might be supposed that this is due to the different strengths of the intramolecular H-bonds which stabilize the enol form (X) in the two classes of compounds.

~ .--H%0

O,.--H~o

Xa

Xb

~

Xc

Accordingly, a preliminary investigation has been undertaken on a sample of enolones retrieved from the CSD [15] (56 13-diketone, 5 ~-ketoamide, and 37 ~-ketoester enols). The results are summarized in Table 2 and in the histograms of Table 2. Average Values of the Parameters Characterizing the Intramolecular

H-Bond in 13-diketone, 13-ketoester and I~-ketoamide Enols d(O--O)(,~)

13-diketone enols 13-ketoesters and 13-ketoamide enols

~

EHB (kcal mol-l) b

na

Range

Average

Range

Average

Range

Average

57 37

2.37-2.59

2.46(4)

0.49-0.77

0.60(7)

16.1-5.1

9.6(2.0)

2.50-2.64

2.57(3)

0.74-0.90

0.82(4)

7.9-3.7

5.4(8)

5

Notes: an = sample dimension. bEHB= H-bond energy calculated according to Lippincott and Schroeder [24] for an average O-H--O angle of 149 ~.

Hydrogen Bonding Theory

91 -16

16II-ll i !--11

14-

',~,,

12. 71,

[o-~

10.

I:::1 IP" ~ .IL

I"

,._!

......

i~

'~176 IE] Eli ',ooq

I'" .... I. . . . . .

[] ,~1 u [] [] u,

,

i',',',','.', . 0.50

d(O--O)

.

I.....

,,~~ - ~ o o ~,,

,

'nl

(i)

'14 '12

',o5,,

d.: :.~:L

" " " " "I D D D D D I

F .....

I.I

I.~

',~

In', ID O I I. . . . I , ~ 5 [] o 5 ~ A . . . . ~ o D O []Ol

O-

b

--1 ~-~ ,oon, i I'~t~l I i"t~i

tn[]n, ,o [] oi

"," it~,t~lu,n,~ 0.65

0.80

-0 0.95

2,

Figure 11. Histograms of O--O contact distances (a) and coupling parameters ~, (b) for a series of intramolecularly H-bonded 13-diketone (full points) and 13-ketoester and 13-ketoamide (open squares) enols.

Figure 11. Amides and esters are statistically indistinguishable and have been grouped together. The distributions ofd(O--O) (Figure 11a) and coupling parameter (Figure 1lb) are clearly bimodal, showing that the H-bond in esters (and amides) is longer and their system of conjugated double bonds less delocalized than in ketones, therefore confirming that the dissymmetry due to the substituents produces a weakening of the H-bond, in agreement with the model proposed. In Table 2 an estimate of H-bond energies (EHB) is also attempted. Energies have been calculated by the equation proposed by Lippincott and Schroeder [24] giving EHB = f(R, O0, where R = d(O--O) and o~ = O-H--O angle; all energies were evaluated by using the average value of e~ for the global sample [ = 149(5)~ Though the results must be taken with some caution because of uncertainties in the method of calculation used, they clearly show that energies of resonant H-bonds can be several times larger than those for nonresonant bonds [e.g. En8 for linear O-H--O bond in ice (d(O--O) = 2.74 A) is calculated to be not greater than 3.7 kcal mol-1].

Effect of Dissymmetry on O-H--O CAHB Meot-Ner (Mautner) [39] has measured by pulsed-electron-beam mass spectroscopy a large number of dissociation enthalpies (AH oo ) and entropies (AS oD ) concerning the gas-phase dissociation of charged molecular complexes of the types O-H--O- and O--H+--O occurring between identical or different molecular moieties. O-H--O-complexes are obtained by associating alcohols, carboxylic acids, or water with their anions (alcoholates, carboxylates, or OH-) and both homomolecular (A-H--A-) and heteromolecular (A-H--B-) associations have been studied. The

92

P. GILLI, V. FERRETTI,V. BERTOLASI, and G. GILLI

O--H+--O case corresponds to two equal or different molecules capturing a proton, and the associations of water, ketones, amides, alcohols, and ethers with a water molecule have been investigated, as well as the mutual homo- or heteromolecular associations of ethers, ketones, and dimethylsulfoxide. The dissociation enthalpies, AH ~D, can be considered a direct measure of the energy of the H-bond connecting the two partners, and the data collected make evident that the largest AHg values are always associated with homomolecular pairs. For instance, the dissociation enthalpy of CH3COO----HOOCCH 3 is 29.3 kcal mo1-1, while those of CH3COO----HOC2H 5 and CH3COO---HOH are 20.7 and 16.0 kcal mol-1; analogously, the enthalpy of H20--H§ is 31.6 and that of (CH3)20--H+--OH2 24.0 kcal mol -l. These findings unquestionably support our previous hypothesis that very short homonuclear H-bonds are essentially three-center-four-electron covalent interactions. Moreover, they suggest that the H-bond is stronger the more similar are the proton affinities of the two partners of the association equilibrium. As a matter of fact, Meot-Ner (Mautner) [39] was able to show that the dissociation enthalpies of several O-H--O- and O--H§ bonds decrease linearly with the increasing difference (APA) of the gas-phase proton affinities (PA) [40] of the linked molecules. Figure 12 reproduces the correlations obtained for O-H--O- (open circles and regression line A) and O--H+--O (crosses and regression line B); the two regression lines have equations: 35.0

+

30.0

0

-

25.0

0 o

20.0

15.0 -

10.0

. . . .

' . . . . 20.0 ' " o,o' . . . . lo.o

'6'o'.6 ' i o t ~ ' " '~ , o . . . . so.o ' "

A PA (kcal mol-')

Figure 12. Correlation between gas phase dissociation enthalpies (AH~) and proton affinity differences (APA) of the associated molecules. Open circles and correlation line A = O-H--O- bonds; crosses and correlation line B = O--H+--O bonds. (Data from Meot-Ner (Mautner) et al. [39]).

HydrogenBondingTheory

93

O-H--O-(line A): A/-Po(_-_+0.9)=28.1(_-+0.4)-0.29(_-_+0.01) APA (n = 13, r-- 0.981, units = kcal mo1-1) O--H+--O (line B): A/-Po(_-_+0.8)= 30.4(_+0.4)- 0.30(_-_+0.01) APA (n - 20, r = 0.979, units = kcal mol-1). From a chemical point of view, set B corresponds to complexes A--H§ of different electron donors with the same water molecule; the intercept of 30.4 kcal mo1-1 therefore represents the H-bond energy of the homomolecular HzO--H +-OH 2 association. Very interestingly, the average dissociation energy of the A--H+--A homomolecular complexes (where A are the donors previously associated with water) of 30.5(+1.8) kcal mo1-1 is practically identical to this intercept, which shows that all symmetrical O--H§ bonds give a same maximum energy of some 30-31 kcal mo1-1 independent of the nature of the attached molecules. Similar considerations hold for the O-H--O- case. Set A refers to A-H--B- complexes (where AH and BH are carboxylic acids, alcohols, or water). The regression intercept of 28.1 (_-_+0.4)kcal mo1-1 represents the H-bond energy in the homomolecular A-H--Asituation which is always the same irrespective of the nature of A. The AH~ value obtained at APA = 0 is slightly different for the O--H+--O and O-H--O- cases (30.4 and 28.1 kcal mo1-1, respectively). The difference might be imputed to the greater electronegativity of a positively charged oxygen with respect to a negatively charged one, in agreement with the general fact that electronegativity is the essential factor inferred to control H-bond formation. On the contrary, the slopes of both regressions are statistically indistinguishable, suggesting a common dependence of AH~ on PA for all O-H--O bonds.

B. Changing the Length of RAHB n-Conjugated Chain RAHB has so far been discussed mainly in relation to inter- and intramolecular H-bonds in [~-diketone enols containing the O=C-C--C--OH heteroconjugated system, though other heterodienes can ensure conjugationmin particular carboxylic acids, O--C-OH, 8-diketone e n o l s , O - - C - C - - C - C = C - O H , and ~-diketone enols, O = C - C - - C - - C = C - - C - - C - O H . The comparative study of such compounds [6,12a] can help us to assess specific effects due to the length of the resonant chain. The intervals of d(O--O) and ~ values observable in these compounds are given in Table 1 and some chemical formulas depicted in Chart 3. Carboxylic acids are known to form both resonant chains (C5-1) and dimers (C5-2), but there are only few cases of compounds which can be classified as ~5- or ~-diketone enols, all forming intramolecular H-bonds (see C3-1 and C4-1, respectively). The only examples of 8-diketones are three rubazoic acid derivatives XI, which are characterized by very short O--O distances (2.42-2.45 A) and almost total delocalization of the HO-C---C-X=C--C--O heterodienic system (~, = 0.52-0.53). Though direct

94

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILL! _ . . , H . . . . #_ , X: N, C H ~

\

V

_ '~\1'

, fH,,,,.,,.

/

xI

evidence for ~-diketones is not available, some vicinal diacetylcyclopentadienes (XII) can be shown to behave like them. Apparently these molecules are to be classified as y-diketone enols, but simple inspection of bond distances shows that they instead are conjugated ~-diketone enols where the heterodiene is bridged by bond 5, which however does not participate in the conjugation. The structures of five derivatives of XII are known, and all show very short (2.43 < d(O--O) < 2.51 /~) and practically linear O-H--O bonds associated with an almost total r~-delocalization (0.51 < X < 0.53). Figure 13 summarizes the distribution of d(O--O) distances for O-H--O bonds formed by chains of different lengths (data from Table 1). The order of increasing distances is: 15, ~-diketones (C3-C4), 13-diketones (intra) (Cla), I]-diketones (inter) (C1 b), 13-ketoesters (and ~-ketoamides) (C2), carboxylic acids (C5), and nonresonant bonds (D). The case of ~-ketoesters and ~-ketoamides has already been discussed in the previous section. The comparison among the other classes is complicated by the fact that the O-H--O angle is not constant. As an example, the intramolecular bonds in ~-diketones appear to be stronger than the intermolecular ones, but this is clearly an artefact due to the different value of the O-H--O angle (- 150 ~ and 170-180 ~ respectively) which produces an abnormally short value of d(O--O) in the intramolecular case. In these conditions, d(O--O) distances cannot I!i!i~i!iii!i!i~:i!i!i]

C5&C4: 6 - and (-diketones

I!iii!ii!iiii!iiiii!iii!iiiil

Clo: ,8-diketones (inf.ra)

Ii:iii!iiiiiiii!!iiiiiii!i!iiii!iiii!i!il ~iiii!ii~!i!i!!iii!!iiiiii!!iii~iiiii~;ili]

Clb: ,8-diketones (inter) C2: ,8-kef.oesters and amides

Iii~ii!iiii!ii!.]

C5: carboxylic acids D: non-resonant

2.40

2.6o

2.~o

z{o

2.go

d(O--O) (i)

~_.~o

3.do

Figure 13. Summary of the d(O--O) distances observed in the inter- and intramolecu-

lar O-H--O bonds formed by resonant systems of different length. The d(O--O) range for nonresonant bonds is added for comparison (Data from Table 1 corrected according to the histogram of Figure 11 ).

Hydrogen Bonding Theory

95

Table 3. The Shortest O-H--O Bonds Observed in Noncharged Resonant Systems and Their Energies Estimated by the Lippincott and Schroeder Formula as a Function of d(O--O) and of the O-H--O Anglea d(O--O) (,~) O-H--O (o) EHs(kcal mol -l) ~-diketone enols

2.431

171

18.7

/5-diketone enols

2.425

177

20.3

Ref. [42] [43]

13-diketone enols (intra)

=2.39

=150

14.8

m

13-diketone enols (inter) carboxylic acids (chains and dimers)

2.465 =2.62

176 =180

15.5 6.7 b

[44]

nonresonant

=2.70

= 180

4.7 c

m

Notes: aEstimated nonresonant limit given for comparison. bExperimental thermodynamic values in the range 7-8 kcal mol-l [45]. CExperimental thermodynamic values in the range 3.2--4.0 kcal mo1-1 [45d].

be used anymore as reliable indicators of H-bond strengths, and it seems more reasonable to compare the values of the H-bond energies, EHB, calculated as a function of the (O--O) distance and O-H--O angle by the equations given by Lippincott and Schroeder [24] for both linear and bent O-H--O bonds. Table 3 reports the results of such calculations for the shortest H-bonds for which there is reasonably reliable crystallographic documentation. EHB values are seen to increase in the order: nonresonant < carboxylic acids < l~-diketone enols (intra and inter) < 8- and ~-diketones. Though many EHB values cannot be tested against experimental values for lack of thermodynamical data on [3, ~5, and ~-diketones, ab initio calculations carried out on malondialdehyde [41a] and acetylacetone [41b] by the use of extended basis sets at the Hartree-Fock and second-order M611er-Plesset level indicate EHB values of 11.6-12.4 and 12.0-12.5 kcal mo1-1, respectively. These values are comparable with those calculated in Table 3 for 13-diketone enols. This seems to indicate that the Lippincott and Schroeder force field can provide reasonable estimates of absolute H-bond energies, which should be even more reliable on a relative scale. In this case, the order of energies found in Table 3 can be then interpreted according to what could be called a RAHB polarization rule: the strength of resonant H-bolufs tends to increase with the polarizability of the n-conjugated chain, that is with the number of sp 2 atoms (n) connecting the H-bond donor and acceptor. This increase, illustrated in Figure 14, is not constant; it is smaller when going from nonresonant systems (for which n is formally assumed to be zero) to carboxylic acids which, accordingly, form rather weak RAHBs; the greatest increments occur in passing from carboxylic acids (n = 1) to [3-diketone enols (n = 3) and, then, to 8-diketone enols (n = 5), while a further increment up to n = 7 (~-diketone enols) does not produce any further H-bond strengthening. The simplest interpretation can be given in terms of the stated RAHB polarization rule, though differences in steric repulsion occurring in the different

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLi

96

30.0 I

o

20.0 ,,--..,

(3 0 V

I:Q -1-ILl

. :

~ i

.

c

"

0

t-

10.0

0.0

T_ o

' ....

6 ....

1""'"'"g2

....

.... 7""g

72,

Figure 14. Energy ofthe strongest H-bond observed for each class of resonant systems as a function of the number of C(sp2) atoms present in the resonant chain (n = 0: non resonant systems; n = 1: carboxylic acids; n = 3,4,5: 13, 8 and ~,-diketone enols, respectively). H-bond energies EHB are evaluated from the crystal geometries by means of the Lippincott and Schroeder [24] force field. Data are from Table 3.

classes could also play some role, which is so far difficult to assess. The intrinsic weakness of RAHB in carboxylic acids due to their poor rc-delocalizability suggests that a similar effect can be expected in the other short resonant systems and, in particular, in the biologically relevant amide group [46]. C. On the Origin of the Keto-Enol Tautomerism

13-diketones are the best known case and the example traditionally illustrated in organic chemistry textbooks of keto-enol tautomerism, a phenomenon for which a chemical equilibrium is established between two constitutional isomers whose formulas differ for the localization of a proton. The origin of tautomerism has been object of much debate. The process of enolization, - C H 2 - C O - ~ CH--C(OH)is, in itself, energetically unfavored, as can be easily confirmed by a glance at any compilation of bond energies. The reason is that it moves the double bond from a strongly heteropolar (C--O" D O= 191 kcal mol -l) to a homopolar (C--C: D O- 144 kcal mo1-1) pair of atoms. Similarly, our ab initio calculations on the model system CH3-CHO --> CH2=CHOH carried out at the 3-21G* level have shown that the enol is less stable by 8.46 kcal mol-l. For [~-diketones electronic factors have often been invoked, in the sense that the enol form should be stabilized by the resonance I H a <-->IIIb.

However, recent studies [12a] have suggested that electronic factors alone are not sufficient to stabilize the enol. A tentative energy scale for the different isomers

HydrogenBondingTheory

97

H~

/

O,,,.H....O

:

12.0

/

k

:

6.9

kcal/mol

i

5.0

I

/

i 0

/

of malondialdehyde, the prototype of [3-diketones, has been evaluated by Emsley [16] by compiling a large collection of different data and is illustrated in Xllla-d. The open cis-enol form Xllla is some 5.1 kcal mo1-1 higher in energy than the keto form Xlllb, while the intramolecularly H-bonded enol form XllId is 6.9 kcal mo1-1 lower because of a strong RAHB of some 12 kcal mo1-1 (the same value also obtained by the most accurate ab initio calculations [41]). It is evident that a normal nonresonant H-bond (whose maximum energy is ---4.7 kcal mol-1; see Table 3) would not be strong enough to make the ketoenol more stable than the [~-diketone and only the specific features of the resonant H-bond make this possible. It can therefore be stated that: the larger H-bond energy of RAHB is the primary cause of the enolization process, or, in other words, no enolization can occur without the concomitant formation of a resonant H-bond. Some compounds, typically 1,3-cyclohexanediones and 1,3-cyclopentanediones, cannot form an intramolecular H-bond because of their specific constitution. In this case the enolization can only occur by formation of infinite resonant chains XIIIc, a kind of pattern which can easily be formed in crystals. Though, according to Emsley [16], form XIIIc is less stable by some 5 kcal mo1-1 than Xllla (at least for malondialdehyde), the RAHB energy is still sufficient to enolize the diketone and, in fact, practically all ~-diketones are found to assume the enolic form in crystals by forming intramolecular H-bonds Xllld when sterically possible, or, otherwise, infinite RAHB chains of the XIIIc type. D.

RAHB Generalization

As defined in Section II, RAHB is a mechanism of synergistic interplay between hydrogen bond strengthening and resonance reinforcement. It has firstly been discovered for [3-diketone enols and extended by the arguments of Section IV.B to include resonant chains of different lengths. In principle it may be extended to any heteroconjugated system such as XIV, A--R(-R--R)N-DH ~ A--R(--R-R)N=+DH XIVa XIVb

(N = 0,1,2,..)

A--Rn-DH (n = 2N + 1) XV where A and D are the H-bond acceptor and donor atoms (mainly O and N but also S), R is a C or N atom, N a positive integer including zero, and n = 2N + 1 the total

98

P. GILLI, V. FERRETTI, V. BERTOLASI, and G. GILLI

o/H--.O

~N/H--.O

~N/H--.N/

o/H--.N/

XVl

XVII

XVIH

XIX

number of C/N atoms in the resonant spacer R n connecting A and D, as shown in the graphical symbol of the RAHB unit given in XV. Such a unit has different ways of self-aggregation, the most frequently observed being infinite chains, closed intramolecular tings (only when n > 3), and dimers, though larger tings including four or six units have sometimes been observed. From a chemical point of view, a relevant number of molecular fragments are found to correspond to schemes XIV and XV: the first-order term (n = 1) can depict carboxylic acids O--CR-OH, amides O--CR-NHR, and amidines NR--C-NHR; the third-order term (n = 3) may be represented, besides the usual enolones (XVI), by enaminones (XVII), enaminoimines (XVIII), or enolimines (XIX), while it has already been remarked that higher order terms correspond to 6-diketone (n = 5) and ~-diketone (n = 7) enols. Though the formation of RAHBs is easily generalized in its chemical and topological aspects by means of schemes XIV and XV, a rationalization of the H-bond energies would require a much more detailed analysis because the efficiency of RAHB has been already shown to depend also on other factors, such as the homoor heteronuclearity of the bond (Section III) and the length of the interleaving resonant chain (Section IV.B). A complete analysis of all these different factors is far beyond the scope of the present treatment, and we want to conclude by illustrating a few aspects of this problem which may be of interest for their remarkable biochemical and biological relevance.

H-.Nf

6.y

'

,

,

~ O'yN~ XX

~N.

XXI

-~

..N~,.."-.:'" j ]

XXlII

T

/ ~ . ~ .,#O,.

XXII

N

I

c

"'H,,N/ ,,~,.,,

iilr,..--.....::. --.,0,

"I , ~N~N..-.:....JIIL. '

H

\

HydrogenBondingTheory

99

Vl

In XX-XXllI are shown some typical RAHB associations of first order terms (n = 1), such as dimers of carboxylic acids (XX) and amides (XXI), an amide-amidine complex (XXH), and a chain of amide fragments (XXIII). The complex XXII assumes a particular importance because it is present in both thymine-adenine (XXIV) and cytosine-guanine (XXV) H-bonded pairs determining the double-helix architecture of DNA; in the second pair cytosine and guanine are also connected by another much wider cycle of conjugated double bonds closing a third resonant N-H--O bond. Similar considerations hold for the H-bonds determining the secondary structure of both fibrous and globular proteins. Both o~-helices (XXVI) and [3-pleated sheets (XXVII) contain long chains of r~-heteroconjugated systems

VII

I !

I

100

P. GILLi, V. FERRETTI, V. BERTOLASI, and G. GILLi

connected by H-bonds of the amide type shown in XXIII, which differ only in their mutual orientation. The residues of the tx-helix are connected by three such chains which are parallel and isooriented, contributing therefore to the large dipole moment of the helix itself. In 13-pleated sheets (XXVII) the chains are parallel but antidromic, giving rise to an essentially apolar structure. The role played by resonant H-bonding in biological structures is certainly a complex problem which has attracted the attention of some authors from time to time [3,5,11,47] but is far from a convincing solution. The hypothesis that nature itself may have taken advantage of the greater energy of RAHB to keep control of molecular associations whose stability is essential to life [12c] seems to be, however, promising and deserving of further investigation in the near future.

REFERENCES 1. Latimer, W. M.; Rodebush, W. H. J. Am. Chem. Soc. 1920, 42, 1431. 2. Lewis, G. N. Valence and Structure of Atoms and Molecules; The Chemical Catalog: New York, 1923, p. 109. 3. Huggins, M. L.Angew. Chem. Int. Ed. 1971, 10, 147. 4. (a) Hadzi, D., Ed. Hydrogen Bonding; Pergamon Press: Oxford, England, 1959. (b) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; Freeman: San Francisco, 1960. (c) Hamilton, W. C.; Ibers, J. A. Hydrogen Bonding in Solids; Benjamin: San Francisco, 1968. (d) Vinogradov, S. N.; Linnell, R. A. Hydrogen Bonding; Van Nostrand-Reinhold: New York, 1971. (e) Joesten, M. D.; Schaad, L. J. Hydrogen Bonding; Marcel Dekker: New York, 1974. (f) Gutmann, V. The DonorAcceptor Approach to Molecular Interactions; Plenum Press: New York, 1978. 5. (a) Schuster, P; Zundel, G.; Sandorfy, C., Eds. The Hydrogen Bond. Recent Developments in Theory and Experiments; North-Holland: Amsterdam, NL, 1976. (b) Jeffrey, G. A.; Saenger, W. Hydrogen Bonding in Biological Structures; Springer: Berlin, 1991. 6. Gilli, P.; Bertolasi, V.; Ferretti, V.; Gilli, G. J. Am. Chem. Soc. 1994, 116, 909. 7. (a)Coulson, C. A. Valence; Oxford University Press: Oxford, 1952. (b)Coulson, C. A.; Danielsson, U. Ark. Fis. 1954, 8, 239. (c) Tsubomura, H. Bull. Chem. Soc. Jap. 1954, 27, 445. (d) Pimentel, G. C. J. Chem. Phys. 1951, 19, 1946. 8. Kollman, P. A.; Allen, L. C. J. Am. Chem. Soc. 1970, 92, 6101. 9. Noble, P. N.; Kortzeborn, R. N. J. Chem. Phys. 1970, 52, 5375. 10. (a) Umehyama, H.; Morokuma, K. J. Am. Chem. Soc. 1977, 99, 1316. (b) Morokuma, K. Acc. Chem. Res. 1977, 10, 294. (c) Singh, U. C.; Kollman, P. A. J. Chem. Phys. 1985, 83, 4033. 11. Gilli, G.; Bellucci, E; Ferretti, V.; Bertolasi, V. J. Am. Chem. Soc. 1989, 111, 1023. 12. (a) Gilli, G.; Bertolasi, V. In The Chemistry of Enols; Rappoport, Z., Ed.; J. Wiley & Sons: New York, 1990, Chapter 13. (b) Bertolasi, V.; Gilli, P.; Ferretti, V.; Gilli, G. J. Am. Chem. Soc. 1991, 113, 4917. (c) Gilli, G.; Bertolasi, V.; Gilli, P.; Ferretti, V. Acta Crystallogr. Sect. B 1993, 49, 564. (d) Gilli, G. In Fundamentals in Crystallography; Giacovazzo, C., Ed.; Oxford University Press: Oxford, 1992, Chapter 7. (e) Gilli, P.; Ferretti, V.; Bertolasi, V.; Gilli, G.Acta Crystallogr. Sect. C 1992, 48, 1798. 13. Haddon, R. C. J. Am. Chem. Soc. 1980, 102, 1807. 14. Ferretti, V.; Bertolasi, V.; Gilli, G.; Borea, P. A.Acta Crystaliogr. Sect. C 1985, 41, 107. 15. Allen, E H.; Bellard, S.; Brice, M. D.; Cartwright, B. A.; Doubleday, A.; Higgs, H.; Hummelink, T.; Hummelink-Peters, B. G.; Kennard, O.; Motherwell, W. D. S.; Rodgers, J. R.; Watson, D. G. Acta Crystallogr. Sect. B 1979, 35, 2331. 16. Emsley, J. Struct. Bonding 1984, 57, 147.

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101

17. Kroon, J.; Kanters, J. A.; Van Duijneveldt-Van de Rijdt, J. C. G. M.; Van Duijneveldt, E B.; Vliegenthart, J. A. J. Mol. Struct. 1975, 24, 109. 18. Ceccarelli, C.; Jeffrey, G. A.; Taylor, R. J. Mol. Struct. 1981, 70, 255. 19. Floris, B. In The Chemistry ofEnols; Rappoport, Z., Ed.; John Wiley & Sons: New York, 1990, Chapter 4. 20. Novak, A. Struct. Bonding 1974, 18, 177. 21. (a) Biirgi, H.-B. Angew. Chem Int. Ed. Eng. 1975, 14, 460. (b) Biirgi, H.-B.; Dunitz, J. D. Acc. Chem. Res. 1983,16, 153. (c) Dunitz, J. D.X-ray Analysis and the Structure of Organic Molecules; Cornell University Press: Ithaca, NY, 1979. (d) Biirgi, H.-B.; Dunitz, J. D., Eds. Structure Correlation; VCH: Weinheim (Germany), 1994. (e) Ferretti, V.; Gilli, P.; Bertolasi, V.; Gilli, G. Crystallography Rev. 1995, in press. 22. Bertolasi, V.; Gilli, P.; Ferretti, V.; Gilli, G. Paper in preparation. 23. (a) Murray-Rust, P.; Glusker, J. P. J. Am. Chem. Soc. 1984, 106, 1018. (b) Glusker, J. P.; Lewis, M.; Rossi, M. Crystal Structure Analysis for Chemists and Biologists; VCH: New York, 1994. 24. (a) Lippincott, E. R.; Schroeder, R. J. Chem. Phys. 1955, 23, 1099. (b) Schroeder, R.; Lippincott, E. R. J. Chem. Phys. 1957, 61, 921. 25. Krygowski, T. M.; Anulewicz, R.; Kruszewski, J. Acta Crystallogr. Sect. B 1983, 39, 732. 26. (a) Hinze, J.; Jaff'e, H. H. J. Am. Chem. Soc. 1962, 84, 540. (b) Hinze, J.; Jaff'e, H. H. J. Phys. Chem. 1963, 67, 1501. (c) Hinze, J.; Whitehead, M. A.; Jaffa, H. H. J. Am. Chem. Soc. 1963, 85, 148. 27. Iijima, K; Ohnagi, A.; Shibata, S.J. Mol. Struct. 1987, 156, 111. 28. Speakrnan, J. C. Struct. Bonding, 1972, 12, 141. 29. Emsley, J. Chem. Soc. Rev. 1980, 9, 91. 30. Bergerhoff, G.; Hundt, R.; Sievers, R.; Brown, I. D. J. Chem. Inf. Comput. Sci. 1983, 23, 66. 31. (a) Johnston, H. S.; Parr, C. J. Am. Chem. Soc. 1963, 85, 2544. (b) Biirgi, H.-B. Inorg. Chem. 1973, 12, 2321. (c) Biirgi, H.-B.Angew. Chem. Int. Ed. Eng. 1975,14, 460. (d) Pauling, L. J. Am. Chem. Soc. 1947, 69, 542. 32. (a) Chidamberam, R.; Sikka, S. K. Chem. Phys. Lett. 1968, 2, 162. (b) Steiner, Th.; Saenger, W. Acta Crystallogr. Sect. B 1992, 48, 819. (c) Steiner, Th.; Saenger, W. Acta Crystallogr. Sect. B 1994, 50, 348. 33. (a) Ref. 5, Chapter 5. (b) Olovsson, I.; Ji3nsson, P.-G. in Ref. 4, Chapter 8, p. 412. 34. (a) Bertolasi, V.; Ferretti, V.; Gilli, P.; Gilli, G.; Issa, Y. M.; Sherif, O. E. J. Chem. Soc. Perkin Trans. 2 1993, 2223. (b) Bertolasi, V.; Nanni, L.; Gilli, P.; Ferretti, .V.;Gilli, G.; Issa, Y. M.; Sherif, O.E.New J. Chem. 1994,18,251. (c) Bertolasi, V.; Gilli, P.; Ferretti, V.; Gilli, G.Acta Crystallogr. Sect. B 1994, 50, 617. 35. Furmanova, N. G.; Kompan, O. E.; Strutchkov, Yu. T.; Michailov, I. E.; Olekhnovich, L. P.; Minkin, V. I. Zh. Strukt. Khim. 1980, 21, 98. 36. Cunningham, C. W.; Burns, G. R.; McKee, V. J. Chem. Soc. Perkin Trans. 2 1989, 1429. 37. (a) Staab, H.A.; Saupe, T.Angew. Chem. Int. Ed. Engl. 1988, 27, 865. (b) Alder, R. W. Tetrahedron 1990, 46, 683. 38. (a) McGraw, B. L.; Ibers, J. A. J. Chem. Phys. 1963, 39, 2677. (b) Ibers, J. A.J. Chem. Phys. 1964, 40, 402. 39. (a) Meot-Ner (Mautner), M. J. Am. Chem. Soc. 1984, 106, 1257. (b) Meot-Ner (Mautner), M.; Sieck, L. W. J. Am. Chem. Soc. 1986, 108, 7525. (c) Meot-Ner (Mautner), M. In Molecular Structure and Energetics, Liebman, J. F.; Greenberg, A., Eds.; VCH: Weinheim (Germany), 1987, Vol. IV, Chapter 3. 40. Lias, S. G.; Liebman, J. E; Levin, R. D. J. Phys. Chem. Ref. Data 1984, 13, 695. 41. (a) Frisch, M. J.; Schneider, A. C.; Schaeger, H. E III; Binkley, J. S. J. Chem. Phys. 1985, 82, 4194. (b) Dannenberg, J. J.; Rios, R. J. Phys. Chem. 1994, 98, 6714. 42. Bruce, M. I.; Walton, J. K.; Williams, M. L.; Hall, S. R.; Skelton, B. W.; White, A. H. J. Chem. Soc. Dalton Trans. 1982, 2209.

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43. Driick, U.; Littke, W. Acta Crystallogr. Sect. B 1980, 36, 3002. 44. Bideau, J. P.; Bravic, G.; Filhol, A.Acta Crystallogr. Sect. B 1977, 33, 3847. 45. (a) Calis-van Ginkel, C. H. D.; Calis, G. H. M.; Timmermans, C. W. M.; de Kruif, C. G.; Oonk, H. A. J. J. Chem. Thermodynamics 1978, 10, 1083. (b) Chao, J.; Zwolinski, B. J. J. Phys. Chem. Ref. Data 1978, 7, 363. (c) Murata, S.; Sakiyama, M.; Seki, J. J. Chem. Thermodynamics 1982, 14, 723. (d) Curtiss, L. A.; Blander, M. Chem. Rev. 1988, 88, 827. 46. (a) Gilli, P. Doctoral Thesis, 1994. (b) Bertolasi, V.; Gilli, P.; Ferretti, V.; Gilli, G.Acta Co, stallogr. Sect. B 1995, 51, in press. 47. Saenger, W. Principles of Nucleic Acids Structure; Springer: New York, 1983, Chapter 6.

PARTIALLY BONDED MOLECULES A N D THEIR TRANSITION TO THE C RYSTALLIN E STATE

Kenneth R. Leopold

I. II.

III.

IV.

V.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unusually Large Gas-Solid Structure Differences . . . . . . . . . . . . . . . A. Some Representative Examples . . . . . . . . . . . . . . . . . . . . . . B. General Remarks About Gas-Solid Structure Differences . . . . . . . . . Intermediate Bonding in the Gas Phase . . . . . . . . . . . . . . . . . . . . . A. Empirical Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Some Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . Crystallization of Partially Bonded Molecules and the Origins of Large Gas-Solid Structure Differences . . . . . . . . . . . . . . . . . . . . . A. Qualitative Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantitative Predictions of Gas-Solid Structure Differences . . . . . . . C. How "Molecular" is the Effect? . . . . . . . . . . . . . . . . . . . . . . Applications to Chemical Dynamics: Tests and Extensions of the Structure Correlation Method for Mapping Reaction Paths . . . . . . . . . . A. Structure Correlations in the Gas Phase . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 2, pages 103-127 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

103

104 104 107 107 110 111 111 113 115 115 116 118 120 120

104

KENNETH R. LEOPOLD

B. Extension of the Correlation Method . . . . . . . . . . . . . . . . . . . . VI. Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 124 125 125

ABSTRACT The nature of molecules which lie in the intermediate regime between van der Waals complexes and chemically bound species is discussed. Representative examples are given, and the systems are shown to exhibit dramatic differences in structure between the gas phase and solid state. The nature of the bonding in the gas phase species is described as are the origins of the large gas-solid structure differences. Systematic variations in structure among members of series of related systems in the gas phase suggest both tests and extensions of the well known structure correlation method for determining reaction paths.

I. I N T R O D U C T I O N An accurate knowledge of molecular structure lies at the foundations of our understanding of virtually all chemical phenomena, and it is truly fortunate that with a remarkably simple set of principles the structures of a wide variety of molecules can readily be estimated. With the simple concept of orbitals, for example, bond angles can often be approximated with essentially no calculation at all, while rough interatomic distances are easily obtained by simple addition once sets of covalent, ionic, or van der Waals radii have been established [1]. It is implicit in the use of such procedures that one regards a molecule as a microscopic building block whose internal geometry is insensitive to the environment which surrounds it. Indeed, one normally expects to be able to make reasonable predictions of molecular structure and to further have the results be consistent between the gas phase and solid state. For most common systems this is a very reasonable idea. Certainly, there is ample experimental support for the use of simple rules predicting molecular structure, at least within the recognized limits of their applicability [1]. Moreover, for most substances which form molecular crystals, a comparison of gas phase and solid state structural data do reveal only small differences in molecular geometry, amounting typically to a few hundredths of an angstrom or a few tenths of a degree for bond lengths and bond angles, respectively [2]. In effect, there is normally a kind of separation between intra- and intermolecular structure whose origins may be considered to lie in the widely differing energy scales associated with weak intermolecular interactions and chemical bonds. Indeed the use of "bond radii" (e.g., covalent, van der Waals, etc.) irrespective of the state of aggregation is, in some sense, predicated on this same separation.

Partially Bonded Molecules

105

When intramolecular bond energies lie between those of truly chemical and weak intermolecular interactions, however, such simplifications may not be realistic. This is so because lattice energies in this regime can be comparable with molecular binding energies, and thus the intermolecular forces in a crystal might be expected to introduce non-negligible and system-dependent perturbations on the nature of intramolecular bonds. The separation between inter- and intermolecular structure would lose its usual clarity in this case, and molecules could defy the traditional notion of intramolecular bonding as separable from the state of aggregation. A necessary consequence, of course, is that bond lengths and bond angles could no longer be readily predicted by simple sum and valency rules. The observation of such effects, however, requires the identification of systems which lie precariously on the edge between bonding and nonbonding. Much work is now available about the nature of closed-shell--closed-shell interactions, with numerous weakly bound (van der Waals) complexes having been characterized in the gas phase by microwave and high resolution infrared techniques [3]. Needless to say, the chemical bond has been studied for decades [1]. Yet it is rather curious that the overwhelming majority of bonds with which chemists are familiar lie at or near one of these two limits, and surely this is why the concepts of"van der Waals" and "covalent" interactions figure so prominently in traditional chemical thinking about structure and bonding. Yet there is no fundamental reason why intermediate bonds should not exist as well (and indeed they do) though they have been generally more elusive than either their van der Waals or fully formed counterparts. In this chapter we explore some of the unique features of "quasi-bound" species. While systems with only partially formed bonds have been known for years, mostly in the condensed phase, it is only recently that high resolution gas phase spectroscopy has taken a more active role in their study. Indeed, with an increasing volume of gas phase and solid state data now becoming available for small, theoretically tractable systems, the study of partially bound species has created some new and interesting perspectives on structure, bonding, and dynamics. It is the goal of this chapter to illuminate these perspectives via a few particularly illustrative examples. Three important topics form the cornerstones of the discussion: 1. the nature of"quasi-bound" species in the gas phase; 2. unusually large intramolecular gas-solid structure differences; and 3. the experimental determination of reaction paths from gas phase structure correlations. The interrelation between these is schematically illustrated in Figure 1. At the corners of the triangle lie the topics themselves, while along the edges lie the broader areas which their interconnection impacts in some new way. For example, we have already indicated that partial bonding in the gas phase is closely connected with the observation of large differences in structure between the gas phase and

106

KENNETH R. LEOPOLD PARTIAL BONDING

MOLECULAR STRUCTUR

CHEMICAL AMICS

GAS-SOLID

REACTION

STRUCTURE

PATHS

DIFFERENCES

"COOPERATIVE CHEMISTRY"

Figure 1. A schematic representation of the relationship between topics discussed in this chapter. solid state, and this impacts significantly on the concept of "molecular structure" as a relatively immutable property of a molecule. Also, it will become apparent that a system's display of"partial bonding" in the gas phase often signals the possibility of a continuum in the degree of bonding across a series of related compounds. By borrowing established ideas from the crystallographic literature, this permits the experimental determination of generalized reaction paths for simple gas phase processes and thereby offers some new perspectives on the study of chemical dynamics. Finally, with the connection to chemical dynamics established, the observation of large gas-solid structure differences is cast in a new light. In particular, the large changes in structure and bonding which take place upon crystallization of partially bound species become viewed as a kind of"cooperative chemistry" in which it is the formation of a crystal as a whole which drives the progress of a chemical reaction at the single-molecule level. The organization of this chapter is as follows" In Section II, we present some empirical data on gas-solid structure differences, exemplifying the types of systems to be considered and the magnitudes of the effects which are observed. In Section III, we give a similarly phenomenological description of "partial" bonding in the gas phase, followed by a summary of some particularly interesting and pertinent conclusions from ab initio theory. Section IV links partial bonding in the gas phase to the occurrence of large gas-solid structure differences, and describes some recent theoretical work on the origins of this unusual effect. In Section V, we draw a parallel with well-known work on crystallographic structure correlations and the determination of reaction paths, with emphasis on some of the newer aspects

Partially Bonded Molecules

107

introduced by gas phase spectroscopy. Finally, in Section VI, we conclude with a few comments about the future of studies of partially bound systems.

UNUSUALLY LARGE GAS-SOLID STRUCTURE DIFFERENCES

I!.

A. Some Representative Examples Table 1 presents bond lengths and bond angles for a number of systems which exhibit large changes in structure between the gas phase and in the solid state. Figure 2 gives a pictorial view of the changes accompanying crystallization for a number of cases. A classic example is the silatranes [19] which have a tri-bridged structure in which a silicon atom is held near a nitrogen by three-O-CH2-CH 2- linkages. The nitrogen lone pair faces the silicon. The structures of the fluoro and methyl

Table 1. Summary of Some Gas-Solid Structure Differences Compound

Solid

Difference

c~(O-Si-F) (deg)

2.324(14) a 98.7(3) a

2.042( 1)b 93.8(6) b

0.282(15) 4.9(9)

R(N-Si) (/~) o~(O-Si-C) (deg)

2.453(47) c 101.0(15) c

2.175(4) d 97.2(5) d

0.278(51) 3.8(20)

H3N-BH 3

R(B-N) (/~) c~(N-B-H) (deg)

1.6576(16) e 104.69(11) e

1.564(6) f

0.0936(76)

CH3CN-BF 3

R(B-N) (/k) o~(N-B-F) (deg)

2.011 (7) g 95.6(6) g

1.630(4) h 105.6(6) n

0.381 (11) -10.0(12)

HCN-BF 3

R(B-N) (/~) oc(N-B-F) (deg)

2.473(29) i 91.5(15) i

1.638(2) j 105.6(3) j

0.835(31) -14.1(18)

(CH3)3N--SO 2

R(N-S) (/~) c~(O-S-O) (deg) ]~(N-S-C 2) (deg)

2.260(30) k 116.9(20) k 101.5(20) k

2.046(4) ka 113.7(3) k'l 106.4(2) ka

0.214(34) 3.2(23) -4.9(22)

H3N-SO 3

R(N-S) (/~)

1.912 m 1.957(23) n 97.8 m 97.6(4) n

1.7714(3) ~

0.186(23)

102.46(2) ~

-4.9(4)

~ 1

Parameter

~~-0

O~Si~o

~

I

R(N-Si) (/~,)

Gas

c~

o~(O-S-N)

Notes: aRef. 4; bRef. 5; CRef.6; dRef. 7; eRef. 8; fRef. 9; gRef. I0; hRef. 11; tRef. 12; JRef. 13; kRef. 14; IRef. 15; mTheoretical Valueof Ref. 16; nRef. 17; ~ 18.

108

KENNETH R. LEOPOLD AR(N-Si) = 0.282(15)/~ aa(O-Si-F) = 4.9(9) ~

FJ (a)

C2

/

CH 3

.-----CH

AR(N-S) = 0.214(34) A Aa(O-S-O) = 3.2(23) ~ A/3(N-S-Ca) = -4.9(22) ~

CH3

(b)

F N F

C----H

AR(B-N) = 0.835(31)/~ Aa(N-B-F) = -14.1(18) ~

F

(c) Figure 2. Some representative gas-solid structure differences. The arrows indicate the direction of change of the most important structural properties of the molecule upon formation of the crystal. The numbers give the magnitude of the change (gas phase values minus solid state values). derivatives are given in Table 1. The key features of interest here are the S-N distance and the coordination number and geometry about the silicon. In the absence of a silicon-nitrogen bond, the coordination number is four and the geometry is essentially tetrahedral, while bonding to the nitrogen raises the coordination number to five and places the silicon at the center of an approximate trigonal bipyramid. It is a particularly striking feature of the data in Table 1 that for both methyl- and fluorosilatrane, the formation of the crystal reduces the Si-N distance by 0.28/~ and decreases the O-Si-R angle by 4-5~ These distances and angles are also seen to differ considerably between the two compounds, but this is a characteristic feature of the silatranes and is readily understandable as a substituent effect which causes a variation in the electrophilicity of the silicon. The change of bond length for the same molecule in different phases, however, is clearly

Partially Bonded Molecules

109

a more complex phenomenon which must derive from the cumulative effect of some unknown number o f longer range interactions. Although such interactions individually may be weak, their net effect evidently imposes a significant perturbation on the energetic balance between bonding at the silicon and relief of strain in the O-CH2-CH 2- bridges. The relatively complex tri-bridged structure and the existence of ring strain are not necessary in order to realize large gas-solid structure differences, however. Indeed some remarkably simple systems can show similar effects, as is illustrated by the amine and nitrile adducts of BH 3 and BF 3, also included in Table 1. One of the earliest examples is H3N-BH 3, for which comparison of the gas phase [8] and solid state [9] structures reveals a shortening of the B-N bond by 0.094 .~ upon crystallization. Although this effect is smaller than that in the silatranes, it is still quite significant on the scale normally characteristic of gas-solid structure differences. Perhaps the most dramatic effects are seen in the complexes of BF 3 with HCN and CH3CN. In CH3CN-BF 3 and HCN-BF 3, changes in the B-N bond lengths upon crystallization are 0.38 and 0.84/~, respectively, with concomitant changes in the NBF angle of 10 ~ and 14 ~ respectively [10-13]! It is of some interest to compare these changes with those noted above for the fluoro- and methylsilatranes, for apart from the larger magnitude of the effect in the BF 3 adducts, there are significant qualitative differences as well. First, it is seen that for the BF 3 systems, the changes in B-N bond length upon crystallization differ by over a factor of two for the HCN and CH3CN adducts (0.84/~ and 0.38/~ for the HCN and CH3CN adducts, respectively). This is in sharp contrast with the silatranes, where the Si-N distance decreases uniformly by 0.28 .~ for both the fluoro and methyl derivatives. It is interesting to note, also, that for the B F 3 complexes, the B-N distances and N - B - F angles are virtually identical in the crystals, and essentially all of the gas-solid structure differences arise from the differences in gas phase structures. In the silatranes, both the gas phase and crystal phase bond lengths differ for the two compounds. The sulfur systems given in Table 1 provide a few additional examples. SO3-NH 3 is the zwitterionic form of sulfamic acid, and recent ab initio calculations indicate that SO3-NH ~ lies only about 0.5 kcal/mole higher in energy than HSO3-NH 2 in the gas phase [16]. In the solid, however, the zwitterion is more stable, and indeed the substance crystallizes in the SO3-NH ~ form [18]. The nitrogen-sulfur distance in the crystal is 1.771 ~,, and the NSO angle is 102.5 ~ Ab initio calculations, on the other hand, predict a gas phase N-S bond length of 1.912 A (for the staggered configuration), and a NSO angle of 97.8 ~ [16,20]. Microwave data are now available for the zwitterionic isomer in the gas phase [ 17] and are in good agreement with these results: The N-S bond length and NSO angle are 1.957 ~ and 97.6 ~ respectively, indicating about a 0.19 ,~ contraction of the N-S bond and a 4.9 ~ widening of the NSO angle upon crystallization. Effects of similar magnitude are also obtained for the adduct (CH3)3N-SO 2 [14,15]. Here, the nitrogen-sulfur bond

110

KENNETH R. LEOPOLD

is shortened by 0.21 ]k upon crystallization, and the O - S - O and angles are changed by 3 ~ to 5 o.

N-S---C2(SO2)

B. General Remarks About Gas-Solid Structure Differences A few general points about these gas-solid structure differences are important to note. First, to reiterate, they are enormous by any measure, and indeed their magnitude quells any suspicion that they are simply artifacts arising from differences in measurement techniques. Hargittai and Hargittai have compared a number of common methods for determining of molecular structure (e.g., microwave spectroscopy, X-ray diffraction, etc.) with specific attention to apparent discrepancies arising from the differing molecular quantities to which the techniques are sensitive [2a]. For example, X-ray crystallography measures the centroid of the electron charge distribution, while microwave spectroscopy determines vibrationally averaged inverse moments of inertia, and one therefore expects slightly different interatomic distances to result from these two methods. Such differences, however, are not normally large enough to preclude meaningful interpretation of ordinary gas-solid structure differences, and thus certainly do not interfere with the comparisons made here. Second, the question may be raised as to whether the gas phase and solid state structures arise from two distinct minima in the molecular potential energy surface, with the different structures corresponding to systems trapped in one or the other of the wells. While it is difficult to rigorously exclude such a scenario on the basis of failure to observe two isomers in the same phase, there is strong evidence to suggest that such a double-well viewpoint is not correct. First, ab initio calculations on these systems have not revealed multiple minima along the radial bond coordinate [16,21-25]. Second, for CH3CN-BF3, extensive microwave searches revealed only a long bond-length isomer in the gas phase, though a shorter bond-length form (which would presumably be more stable) would almost certainly have been produced under the experimental conditions [10]. Perhaps most importantly, however, is the observation of a wide and nearly continuous variation in the bond lengths and bond angles within a series of related complexes. This feature of these systems is somewhat evident from Table 1, and will be more thoroughly demonstrated in Section Iii. The idea is that because the systems are inherently quite similar, it is more reasonable to regard each structure as corresponding to a single minimum of variable location, than to assume that each system is trapped in a different one of many local minima along the bond coordinate. This reasoning follows closely arguments surrounding the study of crystallographic structure correlations [26] which interpret correlated variations in bond distances and angles as depictions of reaction paths for simple chemical processes. Indeed, we will adopt and more thoroughly discuss this viewpoint in Section V. But the point is important to raise here as well, since it underlies our basic view that there

Partially Bonded Molecules

111

is a single minimum in these systems whose location may be adjusted by either chemical and environmental factors. A final point about large gas-solid structure differences is that while large changes in bond lengths upon crystallization are also commonly observed in such species as metal dimers or ion pairs, such changes are of a fundamentally different nature. Metallic and ionic crystals are not collections of dimeric units. Therefore, the comparison between gas phase and crystal distances is not strictly analogous to that in molecular crystals where an individual molecule in the gas phase retains its integrity as a distinct unit in the solid. This is not to say that metal dimers and ion pairs are uninteresting problems, of course, and indeed a series of very elegant experiments by Legon and co-workers have explored the properties of the gas phase complexes R3N-HX, which may be thought of as the gas phase precursors to the ammonium halides [27]. But our focus here, at least initially, is on molecular crystals, for which the observed gas-solid structure differences are truly measures of the effect of environment on a particular, identifiable intramolecular bond.

III.

INTERMEDIATE B O N D I N G IN THE GAS PHASE

From the examples in Table 1, it is clear that the quantitative aspects of large gas-solid structure differences are system-dependent. The obvious questions are, "What, if anything, do these systems have in common?", and "What is the nature of the isolated species that gives rise to this unusual behavior upon crystallization?" Useful insight into both of these questions can be found at a number of different levels, the simplest of which resorts to basic empirical observations about bond lengths and molecular geometry. Modem ab initio theory, of course, can in principle provide the most rigorous answers. Between the two lie a variety of high resolution spectroscopic measurements which provide an experimental approach to the physical characterization of these systems. In this section we concentrate first on the simple empirical observations, and then briefly summarize some of the pertinent conclusions of related ab initio studies. A further discussion of the physical properties of these species will be deferred to Section V.

A. Empirical Observations From the simplest standpoint, it is useful to examine structures of individual compounds in the broader context of series of similar systems to which they are related. For example, Table 2 presents some relevant data for the BF 3 adducts with nitrogen donors. Ar-BF 3 is included as a truly weakly bound reference. First, we see that the observed bond length in Ar-BF 3, together with the standard van der Waals radius of argon [1] give an effective van der Waals radius of boron equal to 1.41 /~. Hence, we expect a weak bond length of 2.91 /~ for a boron-nitrogen interaction, which is quite close to that observed in N2-BF 3. Moreover, we observe a dramatic decrease in the B-N distance across the series, accompanied by steady

112

KENNETH R. LEOPOLD Table 2. Physical Data for Some Gas Phase BF3 Complexes

Donor

R(BN) (it)

Ar

--

a(NBF) (deg) 90.5(5) a

ks (mdynefft) 0.029 a

eqQ(llB) (MHz) AE (kcat/mol) 2.70(5) a

N2 NCCN

2.875(20) a 2.647(3) b

90.5(5) a 93 c

HCN

2.473(29) e

91.5(15) e

0.103 e

2.813(16) e

CH3CN

2.011 (7) i

95.6(6) i

0.094 i

2.377(9) i

NH 3

1.68 g'k

4.14 m

1.21 (2) k'n

N(CH3) 3

1.636(4) ~

(k,l) 106.3(2) ~

__ ~

3.6 ~ 4.6 f 7.2 g,h 5.7 d 9.1 g'h 12.0(8) j 19.2 f 22 g,h 32.9 g 31.0(11) i

Notes: aRef. 28; bRef. 29; CLow resolution microwave data of Ref. 29 are consistent with an NBF angle spanning 90 ~to 100 ~ Table quotes the ab initio value from Ref. 22a. ;dRef. 22a; eRef. 12; fRef. 22b; gRef. 21 a; hlncludes thermal corrections; iRef. 10; iExperimental value, Ref. 30; kRef. 31; IA tetrahedral geometry at the boron was consistent with experimental data, but the precise angle was not determined; mRef. 32; n"A" internal rotor state value; ~ 33.

increase in the N - B - F angle as the Lewis basicity of the nitrogen donor increases. Muetterties suggests 0.70/~ and 0.88 ]k for the single bond covalent radii of nitrogen and boron, respectively [34], and thus, a fully formed single bond between boron and nitrogen should be about 1.58 ]k in length. Clearly, the observed distance of 1.64 ~, in the N(CH3) 3 adduct [33] is very close to this value. Thus, it is apparent that the gas phase species span nearly the entire range between van der Waals and covalent bonding, at least as defined by their bond lengths. The monotonic progression in NBF angles from 90 ~ in the N 2 complex (undistorted BF 3) to 106 ~ in the N(CH3) 3 adduct (a near tetrahedral boron) is further consistent with this conclusion. Thus, the HCN and CHaCN adducts, which have been shown above to exhibit large gas-solid structure differences, are seen here to lie somewhere between the van der Waals and covalent limits of the series. Similar observations can be made for the sulfur-nitrogen systems, and Table 3 presents the corresponding physical data. Clearly, a wide range in N-S distances and interactions is represented. The "van der Waals radius" of sulfur depends significantly on oxidation state [37], but using the known Ar-S distance in the complex Ar-SO 2 (3.66/~ [38]) together with the standard van der Waals radii of argon and nitrogen (1.92 and 1.5 A, respectively) one estimates an N-S van der Waals distance for HCN-SO 2 of about 3.24/~. The observed distance falls slightly short of, but is still reasonably close to this value. On the other hand, the sum of the single bond covalent radii for nitrogen and sulfur is 1.74/~, which is almost exactly equal to that in the crystalline form of H3N-SO 3 [39]. Not too surprisingly, HCN-SO 2 with its long N-S bond distance exists only in a supersonic jet, while

Partially Bonded Molecules

113

Table 3. Physical Properties of Some Nitrogen-Sulfur Speciesa-d Adduct

Phase

R(N-S) (,~)

k s (mdyne/,~)

HCN-SO 2 H3N-SO 2

gas gas

2.98 a'e 2.79 b'f

(CH3)H2N-SO 2

gas

2.62 b'f

(CH3)2HN-SO 2 ( CH3)3N-SO 2

gas gas

2.34(3) a'g~ 2.26(3 )a,i,j

0.211 a,g 0.301 i 0.8a,l,n

(CH3)3N-SO 2 H3N-SO 3

crystal gas

2.046(4) c'i'k 1.957 (23)a'l'm

H3N-SO 3

crystal

1.7714(3) c'd'p

AE (kcal/mol)

0.026 a'e 7.0 b'f 9.4 b'f 10.3 b'f 13.4 b'f 19.1 b,o 32.6 b'~

Notes: aMicrowave spectroscopy; bAb initio calculation; CX-raydiffraction; dNeutron diffraction; eRef. 35; fRef. 25; gRef. 36; nAb initio value from Ref. 25 is 2.46/~,; iRef. 14; JAb initio value from Ref. 25 is 2.36 A; kRef. 15; IRef. 17; mAb initio value from Ref. 16 is 1.972/k; nValue obtained from analysis of centrifugal distortion, subject to the usual assumption that the intermolecular bond is the primarily responsible for the effect--given the strength of this bond, significant error may be incurred by this approximation; ~ 16; PRef. 18c.

H3N-SO 3, as noted above, is the crystalline form of sulfamic acid. Once again, a wide and continuous variation in bond lengths is observed, and the species which exhibit large gas-solid structure differences lie midrange in the series. Finally, we note that for the silatranes, the true "van der Waals" limit (at least as defined by bond lengths) is hard to identify since the appropriate van der Waals radius of silicon is difficult to assign. Pauling, however, gives the single bond, covalent radii for silicon and nitrogen as 1.17 and 0.70/~ [1], respectively. Thus, a fully formed single bond between silicon and nitrogen ought to be about 1.87 ,~ in length, which is considerably shorter than either of the bond lengths given in Table 1. Here, however, full realization of the limiting van der Waals and chemical bond lengths in a series of silatranes is neither expected, nor required since clearly ring strain must play a significant role. Perhaps more to the point in this case is that among the silatranes as a class, Si-N bond lengths are found to vary by up to 0.87 /~, [19] as the substituent on the silicon is varied, and the X-Si-O angles vary by as much as 15 ~. Although these numbers derive from crystal phase data (which is considerably more abundant for silatranes than is gas phase data), the quite large range again suggests that a broad spectrum of interactions is possible in these systems.

B. Some Theoretical Results In light of these comparisons, it would appear that the complexes which exhibit large differences in structure between the gas phase and solid states have bonds which are neither distinctly van der Waals nor distinctly chemical. But what, then, is the nature of the interaction? Is there charge transfer or electron pair sharing, or does the interaction simply arise from particularly strong intermolecular electro-

114

KENNETH R. LEOPOLD

static forces? If the latter, what is the closest approach possible before chemical interactions become important? If the former, how much charge is transferred? Such questions, of course, can be addressed by ab initio methods. Jurgens and AlmlOf, for example, have compared the binding energies in the complexes NCCN-BF 3 and CH3CN-BF 3 (3.6 and 5.7 kcal/mole, respectively) and have argued that the difference can be rationalized by simple electrostatics [22a]. Using their calculated structure of the CH3CN complex, for which the BF 3 is distorted by 8 ~ out of plane, the dipole-dipole energy between CH3CN and the slightly pyramidal BF 3 is about 2 kcal/mole. This is very close to the excess binding energy of CH3CN-BF 3 over that of NCCN-BF 3, whose bond length is more nearly characteristic of a van der Waals interaction, and for which little distortion of the BF 3 is calculated to occur. Bond orders were not determined, but the results suggest a large electrostatic contribution to the interaction in the CH3CN complex. This is interesting since the significant out-of-plane deformation of the BF 3 has at least a superficial appearance of partial rehybridization at the boron, yet little or no "chemical" interaction (in the sense of electron pair sharing) has taken place. Indeed the binding energy of the CH3CN complex is rather small compared with that of a conventional chemical bond. More recently, Jonas et al. have published a very comprehensive theoretical paper on the structure and bonding in a total of 18 Lewis acid-base complexes of BH 3, BF 3, BCI 3, A1C13, and SO 2 [21a]. Among the many conclusions they reach is that there is apparently no correlation between the degree of charge transfer in Lewis acid-base complexes and the strength of the dative bond! For example, comparing the complexes OC-BH 3, H3N-BH 3, and (CH3)3N-BH3 , the amount of charge transferred to the boron is 0.44, 0.35, and 0.34e, respectively, while the respective binding energies are 26.4, 33.7, and 43.6 kcal/mole. The greater degree of charge transfer in the CO complex is noted as being consistent with a shorter bond, which arises from the sp hybridization of the donor, as opposed to the sp 3 hybridization in the amines. Yet, the OC-BH 3 complex has the highest degree of covalent character of the three (as revealed by the energy density), and the excess binding energy of the amine complexes arises from the net effect of electrostatic and deformation terms [2 la,23]. The BF 3 complexes with the same donors, on the other hand, show the CO complex to arise entirely from electrostatic interactions while the amine complexes show more covalent character than those in the corresponding BH 3 adducts. Yet the binding energies of H3N-BH 3 and (CH3)3N-BH 3 are greater than those of H3N-BF 3 and (CH3)3N-BF3 ! The inevitable conclusion of the work is that "neither the degree of covalency nor the electrostatic interactions alone determine the strength of the donor-acceptor bond." Interestingly, the most strongly bound of the complexes studied, (CH3)3N-A1C13, has essentially no covalent character at all, with the interaction completely dominated by electrostatic contributions. It is clear that a dative bond is not as simple as Lewis dot structures would imply.

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Theoretical work on the sulfur-nitrogen systems presents a similar picture in which neither ionic nor covalent interactions are uniformly dominant. Wiberg and co-workers, for example, have recently presented detailed analyses of the bonding and energetics of both the SO 2 and SO 3 complexes listed in Tables 1 and 3. In a systematic theoretical study of the sequentially substituted methylamine-SO 2 adducts [H3N-SO 2 through (CH3)3N-SO2], the donor-acceptor interaction was found to be strongly ionic [25], with significant degrees of charge transfer from the donor (amine) to the acceptor (SO2). As expected, the charge transfer increases with increasing methylation of the amine and parallels the trend in binding energies and N-S distances. Yet, the amount of electron pair sharing in these systems is negligible, i.e., the N-S bonds have essentially no covalent character. This picture of the bonding was also deduced by Jonas et al. [21a] for the trimethylamine complex. In the H3N-SO 3 adduct, on the other hand, Wong, Wiberg, and Frisch [16] again find a strongly ionic donor-acceptor interaction. However, electron pair sharing is also quite important in this complex, and indeed, the calculated covalent bond order in the gas phase system is 0.46. Thus, to summarize, there is evidence that systems which exhibit large gas-solid structure differences each belong to a class of complexes in which a large range in bonding is represented. Though the evidence is perhaps circumstantial at this point, a connection between large gas-solid structure differences and a large accessible range in bonding is a very reasonable one (see below). Moreover, ab initio theory confirms binding energies in these systems which are larger than those for van der Waals interactions and smaller than those of ordinary chemical bonds. The nature of the bonding is complex, with ionic, covalent, and intermolecular electrostatic deformation terms assuming varying degrees of prominence. As noted above, further insight into the nature of all of these adducts can be gained via examination of other physical properties not yet considered. Some of these have already been listed in Tables 2 and 3, but their discussion is reserved for Section V.

IV. CRYSTALLIZATION OF PARTIALLY BONDED MOLECULES A N D THE ORIGINS OF LARGE GAS-SOLID STRUCTURE DIFFERENCES A. Qualitative Ideas The possibility of tuning the degree of bonding so widely in the gas phase BF 3 systems comes about because BF 3, though stable on its own, is electron-deficient and therefore has a built-in propensity to form a new bond. The extent to which this takes place can be tuned across the series of gas phase complexes by adjusting the basicity of the electron pair donor. In the sulfur and silatrane systems, new bonds can be added to pre-existing stable structures because hypervalency is possible, and thus similar effects can be realized without the electron deficiency prerequisite. In either scenario, a large "dynamic range" is accessible for the bonding, which is

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logically a necessary condition for large differences in bond lengths and angles between phases. Thus, we fully expect partial bonding in the gas phase and large gas-solid structure differences to be closely related phenomena, and it is natural to regard the anomalous differences in structure between phases as the manifestation of yet another mechanism for tuning the bonding. But what precisely is the mechanism? One idea which has been put forth in several forms focuses on the increase in dipole moment accompanying bond formation. For HCN-BF 3 and CH3CN-BF 3, for example, a strong increase in dipole moment is expected as the bond is formed and the initially planar BF 3 distorts toward a tetrahedral geometry. Calculations of the dipole moment of HCN-BF 3 as a function of B-N distance support this expectation [22b], indicating a 3.6-D enhancement in dipole moment of the fully formed (short bond length) complex relative to that of the isolated adduct. Kuczkowski et al. have obtained similar results for (CH3)3N-SO 2 [14]. Thus, one might expect a cooperative effect in these systems in which the formation of bonds increases the stabilization of the crystal through enhancement of the long-range dipolar interactions. Indeed, a semiquantitative demonstration of this idea has been possible in the case of (CH3)3N-SO 2 [14] (see Section IV.B). It is worth noting here that a dipolar enhancement mechanism is expected to operate only when the lattice energy is comparable to, and therefore competitive with, the bond energy. Interestingly, Jonas et al. have demonstrated a strong correlation between the calculated binding energies of donor-acceptor complexes, and the differences between the ab initio (i.e., gas phase) and solid state bond lengths [21a]. Not unexpectedly, the most dramatic effects are seen when the binding energies are the least.

B. Quantitative Predictions of Gas-Solid Structure Differences As noted above, in the case of (CH3)3N-SO 2 the dipolar enhancement mechanism has been shown by Kuczkowski et al. [14] to give very reasonable quantitative agreement with the observed change in the N-S bond length. In particular, using ab initio calculations and the known X-ray crystal structure, it was shown that the combination of a relatively fiat potential minimum and a sharply rising dipole moment function could allow the total dipolar interaction energy in the solid to exceed the energetic cost of contracting the N-S bond. Though the authors commented that the calculations were only approximate, the agreement nicely showed the feasibility of the mechanism, at least in that case. Certainly the dipolar interactions in the crystal form only a part of the total lattice energy. It is well known, for example that higher order multipoles are crucial to the realistic description of charge distributions for use at close range [40], and one might conceivably envision calculations which sum not only dipole-dipole interactions, but also the energetic contributions from higher order (or even distributed) multipoles as well. However such a calculation is likely to become quite cumber-

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some and, moreover, the convergence problem for finite lattice sums is well known [41]. An approach which circumvents these problems is the use of self-consistent reaction field (SCRF) theory [42], which has been quite successful in its quantitative prediction of large medium effects on intramolecular bonding in a number of cases [16,43,44]. In this approach, an individual molecule is placed in an environment which is treated as a dielectric medium and which polarizes to produce an electric field in a spherical cavity around the "solute" molecule. The molecule further polarizes in response, and the medium readjusts until self-consistency is obtained. In short, the electric field produced by the solute-induced polarization of the medium acts as a perturbation in the calculation of the solute's molecular and electronic structure. Using this technique, Wong et al. [16] have examined the structure of H3N-SO 3 and have predicted a gas phase N-S bond length of 1.912/~ (for the staggered configuration) which contracts to 1.817/~ in a dielectric medium of e = 40. The N - S - O angle correspondingly increases from 97.8 ~ to 101.0 ~ As noted earlier, the experimental solid state structure determined from X-ray crystallography and neutron diffraction experiments give a N-S distance of 1.771/~ and NSO angle of 102.5 o [18], and new microwave work [17] is in good agreement with the gas phase predictions (see Table 1). The calculated gas phase bond length is somewhat short, however, and that in the dielectric medium is somewhat long, so that the errors add, giving predicted values for the gas-solid structure differences which are about a factor of two smaller than those actually observed [0.095/k vs. 0.186 ,~ for R(NS) and 3.2 ~ vs. 4.9 ~ for ct(NSO)]. Nonetheless, the calculation correctly predicts both the existence and magnitude of a strong medium effect on the bonding. The dipole moment is indeed predicted to increase substantially as the bond forms, with a 6.61-D moment obtained for the gas phase, staggered form of+NH3-SO3 increasing to a value of 9.69-D in a dielectric medium of e = 40. The dissociation energy increases from 19.1 kcal/mole in the gas phase to 32.6 kcal/mole in the dielectric environment. A particularly dramatic success of SCRF theory has recently been demonstrated in the prediction of the large gas-solid structure difference observed in HCN-BF 3 [43]. Figure 3 shows the B-N bond distance and the dipole moment of the complex calculated as a function of the dielectric constant of the medium. The striking feature of this curve is the abrupt onset and rapid completion of the bond formation as the dielectric constant is increased. Both the structure and the dipole moment are seen to "saturate" at a value of e equal only to 10. The predicted gas-solid structure differences are 0.746/~, for the B-N distance and 11.8 ~ for the NBF angle, both of which are in very reasonable agreement with the experimental observations. The SCRF approach has also successfully accounted for the difference in gas phase and solid state structures in H3N-BH 3 [44].

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C. How "Molecular" is the Effect? With the success of the SCRF model, it is interesting to comment on how a continuum picture of the crystal is apparently a suitable representation of the environment surrounding a molecule despite the obvious discrete order of the lattice. The situation provides an interesting contrast with solution-phase, where solvent polarization is free to develop in an arbitrary direction and to a variable extent, since the mechanism primarily involves molecular reorientation. In the crystal, polarization of the medium is constrained to occur at fixed lattice points and in fixed directions, since the changes in dipole moment are overwhelmingly largest along the bond axes of the molecules. Also, it is a unique feature of the situation that the formation of the lattice drives the formation of the bond, and thus the polarization produced is intimately connected with the creation of the environment itself. The presumption in applying the SCRF theory is that the crystal has already been formed, in which case most, if not all, of the polarization has already taken place. Nonetheless, the calculation might ultimately succeed anyway because the real and calculated electric fields are both large enough to produce the maximum possible change in a "solute" molecule's

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structure. It would be interesting to know whether a small number of nearest neighbors can account for most of the structural changes which are observed, or whether the effect requires some large number of longer range interactions to become fully developed. Several results are pertinent to this question. First, and perhaps most directly, Jonas el al. have calculated the structures of the dimer and tetramer of H3N-BH 3, and the dimer of H3N-BF 3 [2 la]. For (H3N-BH3) 2, the shortening of the B-N bond length relative to that ofH3N-BH 3 is 0.025/~, while in (H3N-BH3)4, the shortening is 0.058/~. Since the total gas-solid structure difference for this species is 0.093 A, the calculations reveal that 60% of the total bond shortening has taken place with only four molecules aggregated. It should be noted, however, that this calculation does not represent a true simulation of the crystal since the optimized orientation of monomers in the small clusters does not match that in the fully formed lattice. Nonetheless, the development of such a substantial effect even at the very small cluster level suggests the importance of highly localized interactions in the crystal. Significant shortening of the B-N bonds was obtained for the dimer of H3N-BF 3 as well. It is also interesting to take a brief look at this same issue from a thermodynamic standpoint and to compare the lattice energy of the crystal with the calculated dipole-dipole interaction energy for a pair of adjacent monomers in the crystal. Here, HCN-BF 3 provides an interesting test case. The crystal structure of HCNB F 3 is such that each monomer is paired with an antiparallel partner at a (nitrogen to nitrogen) distance of about 3.8 ,~ [13]. Thus, using the calculated dipole moment of 8.2 D [22b] for the fully formed adduct, the interaction energy for two antialigned dipoles in the crystal is roughly 9 kcal/mol. This is a tantalizing result, since vapor pressure measurements above the crystal [45] give a value of AH~ = 22 kcal/mol for the reaction HCNBF3(c) ~ HCN(g) + BF3(g). The lattice energy can be estimated from this number by a thermodynamic cycle which uses the ab initio binding energy and radial potential function to account for differences in gas phase and solid state structures of the complex. The result is an enthalpy change of about 24 kcal/mole for the reaction HCN-BF3(c) ~ HCN-BF3(g,s), where (g,s) refers to a hypothetical gas phase adduct with the (short) crystal phase bond length. This reaction represents disruption of the lattice without the concomitant structural rearrangement and/or dissociation of the gas phase products, and hence is the correct number to compare with the dipole--dipole interaction energy. The fact that the total molar lattice energy appears less than a factor of three higher than the molar dipole-dipole interaction energy for nearest neighbors again suggests the dominance of a small number of interactions. It would seem that the question of a molecular versus continuum picture of the crystalline environment would be an interesting one for further exploration.

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V. APPLICATIONS TO CHEMICAL DYNAMICS: TESTS AND EXTENSIONS OF THE STRUCTURE CORRELATION METHOD FOR MAPPING REACTION PATHS A. Structure Correlations in the Gas Phase In the previous sections, we have discussed wide variations in structure among closely related systems. While much of the attention has been focussed on gas phase species, it is important to note that such variations are certainly not new to workers in the solid state. Indeed, the crystallographic literature abounds with examples of bond lengths and bond angles in related systems spanning those characteristic of weak, nonbonded interactions through those suggestive of incipient and ultimately fully developed chemical bonds [46]. In effect, each structure provides a view of a particular bond at a different stage of its formation. A classic body of work, largely pioneered by Btirgi and Dunitz and co-workers, has developed and exploited this interpretation, using X-ray crystal structures for series of related compounds to map the transformations which take place during certain simple chemical reactions [26]. Correlated variations in key structural parameters, when observed, are interpreted as "reaction paths", and the results have generally shown good qualitative agreement with expectations based on independent chemical evidence. The use of X-ray structure correlations has thus been widely regarded as a "static means" of observing "chemical dynamics". That is to say, one interprets "static" structures of individual compounds as snapshots along generalized reaction paths. What is new in the study of quasi-bonded molecules is the observation of a continuum of bonding in the gas phase. This is graphically illustrated in Figure 4 for BF 3 and BH 3 adducts using the complexes in Table 2, and additional data from the literature. The "X"'s represent gas phase data and the circles correspond to crystal structures. We have already noted that these structures represent a continuum of interactions between the van der Waals and chemical limits, and what the plot highlights is that the gas phase points span a significant portion of the transition from one to the other. In the spirit of established ideas from crystallographic structure correlations [26], these bond lengths and bond angles are now interpreted together as a reaction path for the formation of the B-N dative bond, i.e., as a glimpse of the response of the initially planar BX 3 to the approach of the nitrogen donor. Assuming that such an interpretation is correct, one can note that by the time the nitrogen has reached half way between the van der Waals and covalent bond distances, the out-of-plane distortion of the BX 3 is only about 5 ~ Stated differently, most of the out-of-plane deformation of the BX 3 which occurs when a B-N bond is formed takes place during the second half (or second 0.5/~) of the approach. The ability to apply the structure correlation method to relatively simple systems in the gas phase has a number of interesting ramifications, both in terms of testing its assumptions and extending its utility. First, we note that the gas and crystal phase data in Figure 4 splice together cleanly to form what appears to be a smooth curve.

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This is important since it is an explicitly stated assumption in the interpretation of structure correlations [26] that the crystalline environment does not significantly perturb the system from the presumed reaction path. The lack of any apparent discontinuity in Figure 4 would tend to support this assumption, though certainly a more optimum distribution of points would be one in which there is additional intermingling of gas phase and solid state data. Nonetheless, it appears that while the effect of the crystalline environment is to introduce a tremendous perturbation on at least some of these systems (as evidenced by the large structural changes that it renders), the perturbation is one which drives the system along the reaction path, but does not displace it significantly from that path. Another implicit assumption of the structure correlation method which can be tested with these data is that the series of structures for different compounds does indeed provide a reasonable map of the path taken by any one system. That such a test is possible arises from the combination of the relative simplicity of the adducts and the absence of a perturbing environment, which together make ab initio computations particularly well suited for the problem. Hankinson et al. have calculated the reaction paths for the reactions HCN + B F 3 --->HCN-BF 3 and H3N

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BF 3 ---) H3N-BF 3 and have compared plots of the optimized NBF angle as a function of B-N distance with the experimental data of Figure 4 [22b]. While the interpretation of the results requires some care, it appears that there is reasonable agreement between the calculated and experimental reaction curves, though the theoretical paths differ somewhat for the two systems. Further details will be given elsewhere [22b]. This "dynamical" picture presents an interesting view of the crystallization process in these compounds. It is easily verified from Table 1 that in all cases, crystal phase bond lengths are shorter than those in the gas phase, and that, where appropriate, solid state bond angles are larger. Indeed, one might surmise that the basic tendency in all cases is to form bonds upon crystallization, though the extent to which this occurs varies among systems. For the BF 3 adducts, for example, the formation of the solid drives the dative bond virtually to completion. This effect might be thought of as a kind of"cooperative chemistry" in which it is the formation of the lattice which drives individual systems along their reaction paths. In the sulfur-nitrogen and silatrane systems, the chemistry is similarly driven, but not always to completion. What causes it to stop short of a fully formed bond is probably obvious in the silatranes, but perhaps less so in some of the sulfur-nitrogen adducts. +

B. Extension of the Correlation Method The strong focus on molecular structure is a natural one, but structure provides only one measure of a molecule's status as a weakly bound or chemically bonded species. Energetics, bond rigidity, and electronic structure, for example, are also expected to vary between the van der Waals and chemical limits, and fortunately these are properties which are readily accessible from gas phase spectroscopy and/or ab initio computation. Thus, it is of interest to reconsider the nature of the partially bonded systems discussed above in these terms. Moreover, in view of the relationship between structure correlations and reaction paths, it is natural to speculate that any correlated variation in a molecular physical property across a series of related compounds can be similarly interpreted as a glimpse of the physical changes which take place as any one system proceeds along its reaction path. This would represent a generalized extension of the Bfirgi and Dunitz ideas. Rigorously speaking, it is in the realm of conjecture, though it follows quite naturally from the large body of crystallographic work. Moreover, the quantitative validity of such a viewpoint could presumably be tested by direct ab initio calculation, and such would be an interesting study. In any case, trends in physical properties across a series of systems can certainly be safely taken at face value, with the more adventuresome dynamical view at least raised as plausible speculation. Some examples are found for the BF 3 complexes in Table 2 where ks and eqQ(llB) are the stretching force constant for the intermolecular bond and the nuclear quadrupole coupling constant of boron, respectively [47]. AE is the binding energy derived from ab initio theory, with some thermochemical data included as well. It

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is a remarkable feature of these data that the force constants and binding energies vary quite slowly among the systems with the longer bond lengths, despite the large range of B-N distances and NBF angles represented. For example, for the HCN and CH3CN adducts, the force constants are virtually identical and the binding energies differ by only 20%, though the B-N bond lengths differ by 0.46/~! In contrast, the NH 3 and N(CH3) 3 adducts, which have the shortest bonds in the series, differ in B-N distance by only about 0.04/~, yet the binding energies differ by nearly a factor of two. The absolute magnitudes of these numbers are at least as interesting as their comparison. For example, while it is perhaps not too surprising to find that the force constants of the HCN and CH3CN complexes are about a factor of three larger than that of Ar-BF 3, we note that they are still characteristic of truly weakly bound systems. This is readily seen by comparison with (HF) 2, for example, which is a strongly hydrogen-bonded system, with an intermolecular stretching force constant of 0.14 mdyne/A [48]. The binding energies of the BF 3 adducts, on the other hand, are somewhat larger than those for a typical van der Waals bond, though they most certainly do not compare with those of ordinary chemical bonds. The nuclear quadrupole coupling constants of boron, eqQ(llB), provide yet another view of these systems. In chemically bound species, these constants are sensitive to electronic structure via their dependence on the molecular electric field gradient at the nucleus [49]. In weakly bound complexes, the monomer charge distributions are largely unperturbed, but the quadrupole coupling constants are always reduced from their free-molecule values due to large amplitude vibrational averaging [50]. It follows that in systems for which the bonding is intermediate, both electronic and vibrational effects need to be considered as potential contributors to the observed quadrupole coupling constant. It is seen in Table 2 that for the HCN complex, eqQ(llB) is quite similar to that observed in Ar-BF 3, while for the CH3CN complex, there is a small but marked change in the direction toward that of the more strongly bound H3N-BF 3. Detailed arguments have been presented elsewhere [10] which conclude that the reduction in eqQ(llB) for the CH3CN complex relative to the HCN complex is probably largely electronic in origin. Thus, HCN-BF 3 and CH3CN-BF 3, while indistinguishable in terms of their force constants, show small but significant differences in electronic structure. This is in accord with the differences in angular distortion of the BF 3 in the two complexes. The value of eqQ(llB) for the NH 3 adduct, on the other hand, is seen to be quite distinct from the rest, and this, too, is consistent with the other properties of the system (i.e., the force constant and binding energy). All in all, a dynamical interpretation of the binding energies, force constants, and quadrupole coupling data suggest the abrupt onset of a strong interaction somewhere between B-N bond distances of about 2.0 and 1.7/~. In the context of examining such data, it is again interesting to compare the BF 3 systems with the sulfur-nitrogen adducts, and the appropriate information has been given in Table 3. We have already noted the wide range in sulfur-nitrogen distances

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represented by this series of complexes. Now, in contrast with the BF 3 systems, we see a steady rise in the binding energies and force constants across the series. For example, the N-S bond lengths in dimethylamine-SO 2 and trimethylamine-SO 2 differ by only 0.08(6)/~, but the bond stretching force constant of the latter is larger by 50%, and the binding energy is greater by 30%. Whether or not this difference between the BF 3 and SO 2 systems can be simply related to the differences in the nature of the interactions noted in Section III.B is an interesting question. Finally, one other important class of systems deserves mention in the context of examining mechanical and electronic aspects of partial bonding, and this is the amine-hydrogen halide systems studied by Legon and co-workers [27]. As noted in Section II.B, these systems are somewhat different as they form ionic crystals upon transferring a proton, and therefore the issue of gas-solid structure differences takes on a different flavor from that of the discussion in Sections II and IV. However, the question of how much proton transfer takes place in the gas phase complex is certainly a relevant one, and indeed a broad spectrum of degrees of proton transfer are observed in these systems. The complexes all have a hydrogen bond to the nitrogen lone pair, but since the proton is hard to locate, their nature is revealed in most cases by nuclear quadrupole coupling constants and intermolecular stretching force constants. In the series of NH 3 complexes with HF, HCI, and HBr, the evidence presents a picture of essentially weakly bound complexes, with little or no transfer of the proton. Methylation of the ammonia, however, increases the degree of ionic character, in accord with chemical intuition. In the series (CH3)3NHX (X = C1, Br, I), the percent ionic character in the gas phase increases from 62% in the HCI complex to 93% in the HI complex! A very interesting and complete review of these and related systems is given elsewhere [27].

VI. FUTURE The study of partially bound systems connects a rich variety of topics which span gas phase and solid state chemistry, as well as traditional and nontraditional views of chemical bonding. The opportunities for new interactions between theory and experiment are also particularly strong here, and it is indeed fortunate that now more than ever, both experimental and theoretical tools exist which can provide a precise characterization of species with only partially formed bonds. The possibility of building a bridge between van der Waals and chemical interactions is a particularly exciting one. It not only provides an opportunity to link two separate but fundamentally important types of interactions, but in doing so provides a new area in which experiments and theory can touch a common base in the study of chemical dynamics. The observation of large gas-solid differences offers new connections between spectroscopy and crystallography in a way that impacts on conventional ideas about molecular structure. So what might the future hold? Although we have considered a relatively limited number of systems here, the expandable octet likely renders many of the phenom-

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ena discussed quite general, and much chemistry across the periodic table remains to be explored. For example, we have considered the changes in bonding which take place when substituents are varied, but not when the donor or acceptor atoms themselves are systematically varied within a column of the periodic table. Such experiments might, for example, offer an opportunity to examine the interplay between optimum orbital overlap and the increased tendency toward hypervalency in heavier systems [51]. In any case, we have already seen that the dative bond is not a single, simple interaction, but rather is a complex combination of interactions whose particular mixture is highly system-dependent. The study of a broader sample of main group compounds, both experimentally and theoretically, should provide a more complete picture of these bonds and of the general role of intermediate interactions in chemistry. It will also be particularly interesting to continue to explore the connection between gas phase high resolution spectroscopy and condensed phase chemistry. This, of course, applies to gas-solid structure differences, but is not limited to it. For example, one might ask how far a disordered environment can promote the formation of a bond, compared with the highly ordered environment of a crystal. Such a question makes solution phase studies quite interesting and, although molecular structure may be difficult to determine under such conditions, other properties (e.g., NMR chemical shifts, dipole moments, vibrational frequencies, etc.) may prove enlightening. The chemical shifts of H3N-BH 3 have already been a topic of much discussion [44], and the vibrational frequencies of the SO2-amine series have been calculated [25]. Moreover, in view of the SCRF results, in which structure is saturated in moderately dielectric media, solvents and/or solvent mixtures with low and varying dielectric constants may offer the most critical tests of the SCRF theory [52]. It is hoped that this chapter, together with the work contained in the references, will help promote further work in this area.

ACKNOWLEDGMENTS It is a pleasure to acknowledge the students and co-workers who have contributed to the development of ideas presented in this chapter. I am particularly indebted to William Burns, Edward Campbell, Manjula Canagaratna, Mike Dvorak, Randy Ford, Heather Goodfriend, Alex Grushow, Jim Phillips, and Scott Reeve for their efforts at the University of Minnesota. I am also grateful to my colleagues, Professors Jan AlmlOf and Doyle Britton, for many helpful discussions and much valuable assistance, and to Drs. Rick Suenram and Frank Lovas at NIST for allowing us the initial opportunity to become involved in this area. Dr. Jiao and Prof. von R. Schleyer kindly provided a copy of Figure 3.

REFERENCES AND NOTES 1. Pauling, L. C. The Nature of the Chemical Bond, 3rd Edition; Cornell UniversityPress: Ithaca, NY, 1960.

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2. See, for example, (a) Hargittai, I.; Hargittai, M. In Molecular Structure and Energetics, Liebman, J. E; Greenberg, A., Eds.; VCH Publishers: Deerfield Beach, 1987, Vol. 2, p. 1; (b) Hargittai, M.; Hargittai, I. Phys. Chem. Miner. 1987, 14, 413, and references therein. 3. See, for example Leopold, K. R.; Fraser, G. T.; Novick, S. E.; Klemperer, W. Chem. Rev. 1994, 94, 1807, and references therein. 4. Forg~ics, G.; Kolonits, M.; Hargittai, I. Struct. Chem. 1990, 1,245. 5. P~k~yi, L.; Hencsei, P.; Bih~itsi, L.; MUller, T. J. Organomet. Chem. 1984, 269, 1. 6. Shen, Q.; Hiiderbrandt, R. L. J. Mol. Struct. 1980, 64, 257. 7. P~k,~nyi, L.; Bih~itsi, L.; Henscei, P. Cryst. Struct. Commun. 1978, 7, 435. 8. Thorne, L. R.; Suenram, R. D.; Lovas, E J. J. Chem. Phys. 1983, 78, 167. 9. Biihl, M.; Steinke, T.; Schleyer, P. v. R.; Boese, R.Angew. Chem. Int. Ed. Engl. 1991, 30, 1160. 10. Dvorak, M. A.; Ford, R. S.; Suenram, R. D.; Lovas, F. J.; Leopold, K. R. J. Am. Chem. Soc. 1992, 114, 108. 11. (a) Hoard, J. L.; Owen, T. B.; Buzzell, A.; Salmon, O. N. Acta. Crystallogr. 1950, 3, 130. (b) Swanson, B.; Shriver, D. E; Ibers, J. A. lnorg. Chem. 1969, 8, 2182. 12. Reeve, S. W.; Bums, W. A.; Lovas, F. J.; Suenram, R. D.; Leopold, K. R. J. Phys. Chem. 1993, 97, 10630. 13. Bums, W. A.; Leopold, K. R. J. Am. Chem. Soc. 1993, 115, 11622. 14. Oh, J. J.; LaBarge, M. S.; Matos, J.; Kampf, J. W.; Hillig, II, K. W.; Kuczkowski, R. L. J. Am. Chem. Soc. 1991, 113, 4732. 15. van Der Helm, D.; Childs, J. D.; Christian, S. D. Chem. Commun. 1969 p. 887. 16. Wong, M. W; Wiberg, K. B.; Frisch, M. J. J. Am. Chem. Soc. 1992, 114, 523. 17. Canagaratna, M.; Goodfriend, H.; Phillips, J. A.; Leopold, K. R., manuscript in preparation. 18. (a) Kanda, E A.; King, A. J.J. Am. Chem. Soc. 1951, 73, 2315. (b) Sass, R. L.Acta Crystall. 1960, 13, 320. (c) Bats, J. W.; Coppens, P.; Koetzle, T. F. Acta Crystallogr. 1977, B23, 37. 19. For an excellent review of the chemical and physical properties of these systems see Voronkov, M. G.; Dyakov, V. M.; Kirpichenko, S. V. J. Organomet. Chem. 1982, 233, 1. 20. The calculated bond length is increased to 1.950/~ in the eclipsed conformation. 21. (a) Jonas, V.; Frenking, G.; Reetz, M. T. J. Am. Chem. Soc. 1994,116, 8741. (b) Jonas, V.; Frenking, G. J. Chem. Soc. Chem. Commun. 1994, p. 1489. 22. (a) Jurgens, R.; Almltf, J. Chem. Phys. Len. 1991, 176, 263. (b) Hankinson, D.; Almltf, J.; Leopold, K. R., submitted. 23. Glendening, E. D.; Streitwieser, A. J. Chem. Phys. 1994, 100, 2900. 24. Binkley J. S.; Thorne, L. R. J. Chem. Phys. 1983, 79, 2932. 25. Wong, M. W.; Wiberg, K. B. J. Am. Chem. Soc. 1992, 114, 7527. 26. See, for example, Biirgi, H. G.; Dunitz, J. D. Acc. Chem. Res. 1983, 16, 153. 27. Legon, A. C. Chem. Soc. Rev. 1993, 22, 153. 28. Janda, K. C.; Bernstein, L. S.; Steed, J. M.; Novick, S. E.; Klemperer, W. J. Am. Chem. Soc. 1978, 100, 8074. 29. Leopold, K. R.; Fraser, G. T.; Klemperer, W. J. Am. Chem. Soc. 1984, 106, 897. 30. Gal, J.-F.; Maria, P. C. Proc. Phys. Org. Chem. 19911, 17, 159. 31. Legon, A. C.; Warner, H. E. J. Chem. Soc. Chem. Comm. 1991, p. 1397. The boron atom in this complex lies almost exactly on the center of mass, making a microwave spectroscopic determination of the B-N distance quite difficult. The authors' best estimate of 1.59 ,~, has been disputed by theoretical calculations of Refs. 21b and 22b, which give a slightly higher value of 1.68 ,~,. In light of the unfortunate location of the boron with respect to the center of mass, and the success of similar ab initio calculations in related systems, the latter value is probably more nearly correct. 32. Taylor, R. C. In Boron-Nitrogen Chemistry, Advances in Chemistry Series; Gould, R. F., Ed.; American Chemical Society: Washington, DC, 1964, Vol. 42, p. 59. 33. (a) Cassoux, P.; Kuczkowski, R. L.; Serafini, A. hzorg. Chem. 1977, 16, 3005. (b) Bryan, P. S; Kuczkowski, R. L. Inorg. Chem. 1971, 10, 200.

Partially Bonded Molecules 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

48. 49. 50. 51.

52.

127

Muetterties, E. L. The Chemistry of Boron and its Compounds, Wiley: New York, 1967. Goodwin, E. J.; Legon, A. C. J. Chem. Phys. 1986, 85, 6828. Oh, J. J.; Hillig, II, K. W.; Kuczkowski, R. L. J. Phys. Chem. 1991, 95, 7211. Bowen, K. H.; Leopold, K. R.; Chance, K. V.; Klemperer, W. J. Chem. Phys. 1980, 73, 137. DeLeon, R. L.; Yokozek, A.; Muenter, J. S. J. Chem. Phys. 1980, 73, 2044. We note, however, that the N-S bond in this case has a significant ionic component (see Ref. 16), and therefore the use of covalent radii may not be entirely appropriate. See, for example, Dykstra, C. E. Chem. Rev. 1993, 93, 2339, and references therein. For an interesting pedagogical account, see Burrows, E. L.; Kettle, S. E A. J. Chem. Ed. 1975, 52, 58. (a) Tapia, O.; Goscinski, O. Mol. Phys. 1975, 29, 1653. (b) Wong, M. W.; Frisch, M. J.; Wiberg, K. B. J. Am. Chem. Soc. 1990, 113, 4776. Jiao, H.; Schleyer, P. von R. J. Am. Chem. Soc. 1994, 116, 7429. Buhl, M.; Steinke, T.; Schleyer, E von R.; Boese, R. Angew. Chem. Int. Ed. Engl. 1991, 30, 1160. Pohland, V. E.; Harlos, W. Z. Anorg. Allg. Chem. 1932, 207, 242. Bent, H. A. Chem. Rev. 1968, 68, 587. We note that the stretching force constants describe the curvature of the bottom of the intermolecular potential well. Thus, this particular property of a system would not under any circumstances be interpreted as snapshots along a reaction path involving the intermolecular bond coordinate since an arbitrary point on such a path would not correspond to a potential minimum. Howard, B. J.; Dyke, T. R.; Klemperer, W. J. Chem. Phys. 1984, 81, 5417. Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy; Dover: New York, 1975. See, for example, T. R. Dyke in Topics in Current Chemistry; E Schuster, Ed.; Springer-Verlag: Berlin, 1983. We note a few gas phase studies of adducts of PH 3 and some substituted phosphines: (a) Kuczkowski, R. L.; Lide, D. R. J. Chem. Phys. 1967, 46, 357. (b) Kasten, W.; Dreizler, H.; Kuzckowski, R. L. Z. Naturforsch 1985, 40a, 920. (c) Odom, J. D.; Kalasinsky, V. E; Durig, J. R. Inorg. Chem. 1975, 14, 2837. (d) Bryan, E S.; Kuczkowski, R. L. Inorg. Chem. 1972, 11,553. We note that dipole moment of CH3CN-BF 3 has been determined to be 5.8 D in benzene (e = 2). Laubengayer, A. W.; Sears, D. S. J. Am. Chem. Soc. 1945, 67, 164. An SCRF calculation of this complex would be most interesting.

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VALENCE BOND CONCEPTS, MOLECU LAR MECHAN ICS COMPUTATIONS, AN D MOLECU LAR

SHAPES* Clark R. Landis

Io II. III. IV.

V. VI.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid Orbital Strengths and Molecular Shapes . . . . . . . . . . . . . . . . Molecular Mechanics Methods . . . . . . . . . . . . . . . . . . . . . . . . . Valence Bond Concepts and Molecular Mechanics Computations . . . . . . . A. Nonhypervalent Main Group Molecules . . . . . . . . . . . . . . . . . . B. Hypervalent Main Group . . . . . . . . . . . . . . . . . . . . . . . . . . C. Transition Metal Hydrides . . . . . . . . . . . . . . . . . . . . . . . . . Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary ..................................... Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

tThis chapter is dedicated in memory of Linus Pauling.

Advances in Molecular Structure Research Volume 2, pages 129-161 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

129

130 130 131 136 138 140 147 152 156 159 160 160

130

CLARK R. LANDIS

ABSTRACT The principles of valence bond theory, first enumerated over 60 years ago by Linus Pauling, form the basis of an amazingly versatile and robust theory for the rationalization of molecular geometries. In this contribution, I show how these principles guide the construction of new molecular mechanics potential energy functions and enlighten our understanding of the geometries of simple transition metal hydrides. The cornerstone of this work is the hybrid orbital strength function. This function emphasizes the role of hybrid orbital orthogonality in determining molecular shapes. From the hybrid orbital strength functions, potential energy functions suitable for a molecular mechanics description of angular distortions can be derived. Molecular mechanics computations using this valence bond basis are shown to reproduce many geometric and vibrational data with an accuracy similar to other popular molecular mechanics programs, but require far fewer empirical parameters. Extension of this model to hypervalent molecules requires the incorporation of another old valence bond idea, resonance, to simulate the effects of delocalization. The model that we have constructed leads to accurate geometries, vibrational frequencies, and fluxional pathways for hypervalent molecules. A relatively new area of exploration concerns the application of valence bond ideas to describing the bonding in transition metal hydrides. I show that a very simple set of valence bond rules permits the complicated and often unexpected geometries of transition metal hydrides to be understood. Furthermore, these ideas translate readily into an accurate molecular mechanics force field for transition metal hydrides. I.

INTRODUCTION

A recurring theme in the progress of chemical thought from the mid-nineteenth century to the 1930s concerns the interplay of theory and molecular shapes. Consider Kekulr's molecular structure theory, Lewis' rules for fulfilling atomic valencies in molecules, and the Heitler-London-Slater-Pauling quantum theory of the covalent bond. From these developments one can trace clearly the origins of valency, the concept of stereochemistry, and the theoretical constructs of hybridization and resonance. Both the concept and the experimental observation of molecular shapes played a key role in these developments. For example, the rationalization of optical isomers inspired the conception of tetrahedral carbon centers. Similarly, awareness of the geometries at carbon centers stimulated Pauling [1,2] and Slater [3,4] to create the construct of hybridization which reconciled quantum mechanics with both Lewis' electron pair bonds and the tetrahedral carbon. The power of the idea of three-dimensional molecular shapes is that the observed properties of stereoisomers can be rationalized. The importance of the observation of molecular shapes is that it guides the application of quantum principles to molecules. Thus we see that our understanding of molecular properties

VB Concepts, MM Computations, and Molecular Shapes

131

and the nature of chemical bonds is enhanced, and even driven, by the conceptualization and observation of the shapes of molecules. With the onset of the computer age in the 1950s, quantitative computations emerged as new tools for exploring molecular shapes and energetics. The driving forces were twofold: (1) to use computational results to verify the underlying correctness of quantum theoretical models of chemical bonding, and (2) to harness computers as practical tools for predicting, and even designing, molecules with specific properties. Molecular mechanics (MM) methods [5] emerged primarily in response to the latter of these two forces. The fundamental idea of MM is to use classical mechanics to represent the forces controlling the three-dimensional structures of molecules. In essence, most MM computations treat molecules as collections of balls (atoms) and springs (bonds). The advantage of this model is that the functions describing the forces in a molecule are simple, so that computations can be performed on very large molecules. However, the disadvantage is that such simple force functions will be poor imitations of the actual interatomic forces for geometries far from the equilibrium geometry. Given the strong influence of molecular shapes on Pauling's conception of valence bond theory and given that a main purpose of the application of MM is the computation of molecular structures, it seems logical that valence bond concepts might form the basis of a useful MM computational method. Indeed, this line of thought is supported at a more fundamental level by a closer examination of the MM method. MM computations use molecular topologies, i.e. graphs of localized bonds, to define the kinds of forces acting between atoms in molecules. Thus, the perspective of the MM computation intrinsically is that of valence bond theory: atoms are joined together by localized bonds. A theory built from the description of molecules in terms of localized bonds is the natural basis for the derivation of MM functions. It is the purpose of this article to summarize the ways in which I and my coworkers have used the concepts of valence bond theory to formulate a MM description of molecular shapes. As we will show, this approach not only leads to an accurate and general MM computational method, but indirectly constitutes a demanding, quantitative test of valence bond concepts.

Ii.

HYBRID ORBITAL STRENGTHS A N D MOLECULAR SHAPES

Pauling's classic 1931 paper [2], "The Nature of the Chemical Bond. Application of Results Obtained from the Quantum Mechanics and from a Theory of Paramagnetic Susceptibility to the Structure of Molecules", clearly enumerated the following principles of the theory that we now call valence bond (VB) theory: 1. The electron-pair bond is formed through the interaction of an unpaired electron on each of the two atoms.

132

CLARK R. LANDIS

2.

The spins of the electrons are opposed when the bond is formed, so that they cannot contribute to the paramagnetic susceptibility of the substance. 3. Two electrons which form a shared pair cannot take part in forming additional pairs. 4. The main resonance terms for a single electron-pair bond are those involving only one eigenfunction from each atom. 5. Of two eigenfunctions with the same dependence on r, the one with the larger value in the bond direction will give rise to the stronger bond, and for a given eigenfunction the bond will tend to be formed in the direction with the largest value of the eigenfunction. 6. Of two eigenfunctions with the same dependence on 0 and r the one with the smaller mean value of r, that is, the one corresponding to the lower energy level for the atom will give rise to the stronger bond. The primary problem concerns how to construct these bond-forming eigenfunctions. Pauling supplied a solution: hybridization. Thus, four bonds can be formed at a carbon by formally exciting the carbon from its ground state 2s22p 2 configuration to a 2s12p 3 state. Each of the unpaired electrons can then interact with the unpaired electron on an H. The enduring power of the concept of hybridization lies in the strong correlation of molecular structures with simple hybridization schemes. According to rule 5, the best bond eigenfunctions (hybrid orbitals) that can be created in forming four bonds from the 2sl2p 3 state of C are obtained by maximizing the bond eigenfunctions. As the radial distributions of the 2s and 2p orbitals of carbon are similar, the best bond eigenfunction is obtained by maximizing the angular part of the function. This occurs for the sp 3 hybrid, for which the value of the angular eigenfunction is 2.0, shown below (Eq. 1). Pauling demonstrated that three more equivalent and orthogonal eigenfunctions could be formed and that these hybrid orbitals are directed to the comers of a regular tetrahedron. Similarly, Pauling showed that three equivalent sp 2 hybrids could be formed that point to the vertices of an equilateral triangle and that two equivalent sp ~ orbitals that point in opposite directions along a line can be found. Thus, the tetrahedral shape of CH 4, the trigonal planar shape of B H 3, and the linear shape of Bell 2 are understood as consequences of the maximization of bond eigenfunctions. 1

best sp m hybrid = -~s + - - T p z

(1)

In the 1970s, Pauling and associates [2,6-8] attacked the problem of hybridization in a less precise but more general way. This change in strategy was driven by a desire to rationalize the more complex geometries of transition metal complexes. Again we see experimental observations of molecular shapes forcing the advent of new theoretical models. Using the principle of maximization of angular hybrid

VB Concepts, MM Computations, and Molecular Shapes

133

orbital functions, Pauling found that the strongest pairs of equivalent, orthogonal hybrid orbitals correspond to hybridizations sp 3, sp3d 5, and sp3d5f 7. In Pauling's terminology, the term strength refers to the maximum value of the angular part of the wavefunction. The angle ({t) dependencies of the strength functions (S) are given in Eqs. 2-4. Maxima in the strength functions occur with S = 2.0 at 109.47 ~ for sp 3 hybridization; S = 3 at 73.15 ~ and 133.6 ~ for sp3d 5 hybridization; and S = 4.0 at 54.88 ~ 100.43 ~ and 145.37 ~ for sp3d5f 7 hybridization. Note that these strengths apply only to a pair of equivalent orbitals. Pauling proposed that the strengths of a collection of equivalent hybrid orbitals could be approximated by the pair-defect-sum [8] (Eq. 5). The advantage of this method is that it enables rapid computation of accurate hybrid orbital bond strengths. A major disadvantage is that it is limited to the maximum-value basis sets (sp 3, sp3d 5, and sp3dSf7). Ssp(oQ = (0.5 + 1.5cos2(R/2)) 1/2 + (1.5 - 1.5cos2(0~/2)) 1/2

(2)

sspd(oQ = (3 - 6cos2(0~/2)) + 7.5cos4(0~/2)) 1/2 + (1.5 + 6C0S2(0~/2) - 7.5cos4(0~/2)) 1/2

(3)

sspaf(ot) = (3 + 15cos2(o~/2) - 45cos4(o~/2) + 35cos6(~/2)) 1/2 + (5 - 15cos2(0~/2) + 45cos4(0~/2) - 35cos6(~/2)) 1/2

allangles sappr~ smax- Z [smax-S(o~)]

(4)

(5)

We have generalized Pauling's expressions for hybrid orbital strengths to any arbitrary combination of s, p, and d hybridization [9]. Thus for any pair of equivalent spmdn hybrid orbitals, the hybrid orbital strength at the interorbital angle, ix, is given by, q 1 - ~(1 - A2) S(c~) = Smax 1 2

(6)

where the nonorthogonality, or overlap, integral, A, at the interhybrid orbital angle, ~, for two hybrids with spmdn hybridization is given by:

A = 1 + m1 + n ( ls + mcos ot,pz + ~( 3cos2o~- 1)dz2)

smax =.~/

I l+m+n

(I + ~ +

5~-nn)

(7)

(8)

Although the form of Eq. 6 appears quite different from those of Eqs. 2 and 3, the strength formulas are identical when appropriate values of m and n are substituted.

134

CLARK R. LANDIS

In addition, these new generalized functions accurately reproduce more rigorously derived functions for non-maximum-strength hybrids such as sp 1 and sp 2. It is unfortunate that Pauling chose the term "strength" to describe the hybrid orbital angular functions. Although the overlap of two hybrids, hence the bond energy, does depend on the magnitudes of the angular parts of the hybrid orbitals, the radial electron density distribution also is important. Thus, one finds experimentally that C-H bond energies follow the trend sp 1 > sp 2 > sp 3, counter to the trend of Pauling's strength functions: 1.932(sp l) < 1.991(sp 2) < 2.00(sp3). Numerous lines of investigation suggest that localized electron pair bonds are in fact good descriptions of molecular electron density distributions. In my opinion, the clearest quantitation of the agreement of accurate electronic structures computations with a simple localized bond picture comes from Weinhold's natural bond orbital (NBO) analysis [10-12]. Using the NBO method, Weinhold and others find that the simple Lewis picture, for which electron pairs bonds and lone pairs are accommodated in localized hybrid orbitals, commonly accounts for >99.95% of the one-electron density matrix. For example, the Lewis structures for B H 3, CH 4, NH 3, and H20 account for 99.98, 99.98, 99.99, and 99.99% of the total electron densities, respectively. Exceptions do arise, not surprisingly, when hypervalent molecules such as XeF 2 are considered. Thus a single Lewis configuration accounts for just 99.54% of the total density (N.B., although this number is close to 100%, the density left unaccounted in the Lewis description rises by a factor of more than 25 in going from CH 4 to XeF2). The prediction of molecular structures using valence bond concepts requires a method for the estimation of the preferred hybridization of the individual bonds. For example, the H-A-H bond angles of AH 3 molecules vary from 106.7 ~ to 93.8 ~ to 91.7 ~for A = N, P, and As, respectively. The simplest valence bond rationalization is that the intrinsic preferences for p-character of the lone pairs relative to the A-H bond varies systematically in the series. That is, the p-character of the lone pairs decreases going down the column of the periodic table (alternatively, one can view the s-character as increasing). This trend is in keeping with the increased average radius of the valence p-orbitals relative to the valence s-orbitals in moving down a column of the p-block of the periodic table. Bent [13] has devised a rule that provides a qualitative rationalization of hybridization preferences: more electronegative ligands prefer greater p-character. Thus, F has the greatest preference for p-character and a lone pair (no ligand) has the least preference for p-character. This rule is useful as a gross rationalization but there are many known exceptions. We have formulated a quantitative procedure for assigning hybridizations that reproduces empirical structures; in effect this procedure is a quantitative, empirical formulation of Bent's ideas. Our hybridization algorithm [9] begins with a Lewis structure and assigns gross hybridizations according to the rules: (1) use as few p-orbitals as possible, and (2) each ~-bond occupies a p-orbital of the constituent atoms. Thus, the gross hybridi-

VB Concepts, MM Computations, and Molecular Shapes

135

zation of the central atoms of BH 3 and CH20 are sp 2 and the gross hybridizations of CH 4, NH 3, SO2C12, and AsH 3 are sp 3. The distribution of p-character (Pi) among the individual hybrids making bonds to the ith ligand is accomplished by a weighting algorithm, %Pi

np* wt i = ligands, lone pairs, radicals

E i

(9)

wt i

where the weights (wti) are empirical parameters, %Pi = %P character of the bond to the ith ligand, np= number of p-orbitals used in gross hybridization, and hybridization of ligand i is" i = sp %pi/(1-%pi)

(10)

Let us examine the case of propene. The carbon weighting parameters for bonds to H and C are 1.05 and 1.0, respectively. The gross hybridizations a r e sp 3 for the methyl group and sp 2 for the vinylic carbons. Application of the hybridization algorithm yields sp 2"2l and sp 1"9~for the vinylic C-H and C-C bonds respectively. Thus we anticipate that the H-C-H bond angle of the terminal vinyl group will be less than 120 ~ (i.e. more p-character than sp2). This prediction agrees with experiment (
136

CLARK R. LANDIS

+CIF2). Please note that although these resonance structures permit a normal VB description to be maintained, they do not readily rationalize the shapes of hypervalent molecules. The shapes of transition metal bond configurations also can be rationalized in terms of cPspm hybridizations [14], such a s d2sp 3 hybridization for an octahedral metal complex. Once again, however, these models do not agree with the electronic configurations derived from high level ab initio computations. Thus, a current challenge in the application of valence bond concepts to the description of molecular shapes is a simple and consistent treatment of hypervalent and transition metal-containing molecules.

i11. MOLECULAR MECHANICS METHODS From the force field work of Bjerrum [18] early in this century, to Westheimer's [19] seminal applications of MM methods in the middle of the century, to the development of Allinger's MM programs [20-24] over the last three decades, researchers have sought classical mechanical descriptions of the structures and energetics of molecules. The simplicity of the MM method allures computational chemists in two ways. First, the computations are fast, enabling large molecules (even proteins) to be modeled by computation. Second, the mechanical expression of interatomic forces parallels the way in which most chemists think of molecules" collections of particles that interact through bonds and through space. In keeping with chemical intuition, MM computations partition the energy of all interatomic interactions into bonded and nonbonded (or through space) interactions. Whether a particular pair of atoms are considered to interact through bonds or through space is determined from the molecular topology. First we consider the bonded interactions. Atoms that are bonded directly to one another (i.e. 1-2 interactions) have an interaction energy that is dependent on the bond length, the interaction energies atoms joined via bonds to a common atom (i.e. 1-3) depend on the angle made by the three atoms of the molecular graph, and topologies of four atoms (1-4 interactions) have energies which depend on the torsion angle of the graph. Atoms whose connectivities are longer range (i.e. 1->4) are considered to interact through space. Thus the energies of these interactions depend on the distance between the pair of interacting atoms. In some instances, such as planar, sp 2 carbon centers, an improper torsion coordinate is defined to describe the through bond interaction energies associated with deformation from planarity. Simple functions commonly are chosen to describe the potential energies of molecules. The total energy (Eq. 13) is a summation of the bonded (Eq. 11) and nonbonded (Eq. 12) terms. The simplest MM force fields (such as CHARMM25) are purely diagonal and employ very simple functions. Thus bond stretch, bond

VB Concepts, MM Computations, and Molecular Shapes

137

Ebond terms "- Z Ebonds + E Eangles 1,2 interactions 1,3 interactions -t- E Epr~ ertOrsiOns -I- ~ Eimpropertorsions proper 1,4 improper 1,4 interactions interactions Enonbond terms ---- Z

Evan der Waals +

Z

Eelectrostatic

1 ,>4

1,>4

interactions

interactions

Etotal = Ebond terms + Enonbond terms

(11)

(12)

(13)

angle bend, and out-of-plane bend terms are harmonic functions in these appropriate internal coordinates: Ebond = k r( r -- r0) 2

(14)

where r 0 is the strainless bond length in/~ units, and k r is the force constant in units of kcal/mol.fk2; Eangle = k0(0 - 00) 2

(15)

where 0 o is the equilibrium bond angle and k0 is the force constant; and Eimproper torsion - kco(0)- 0)0)2

(16)

where 0)0 is the equilibrium torsion angle and ko~ is the force constant. The proper torsion energy function is based on the simple Fourier series, Epropertorsion = E k,(1 + c o s ( n r all expansion terms

(17)

where kr is one-half of the barrier height, n is the periodicity, and 8 is the phase shift. The nonbond energy terms comprise simple Lennard-Jones 6-12 potentials for the van der Waals interactions, Evan der Waals = 4e[ ((I/r) 12 -- ((~/r) 6]

(18)

where e is the Lennard-Jones well depth (= "~reiej or the geometric mean of the well depths of the individual atoms), t~ is the sum of the two Lennard-Jones, radii, and r is the internuclear distance, and Coulomb's law for the electrostatic interaction, Eelectrostatic _

qiqi er

where r is the internuclear distance and q's are the partial atomic charges.

(19)

138

CLARK R. LANDIS

The treatment of inorganic molecules strains the framework of MM methods

[26]. Much of the difficulty originates in the higher coordination numbers and more complex shapes of many inorganic complexes. For example, the square planar complexes XeF 4 and PtCI42- have coordination numbers identical to methane, but the description of the square planar molecular shape is complicated by the presence of both 90 ~ and 180 ~ equilibrium F - M - F bond angles. If the MM computation employs the harmonic function of Eq. 15, the method must uniquely label the F's such that both 90 ~ and 180 ~ equilibrium angles are used appropriately. If the MM computation is to model large angular distortions that would lead to fluxionality (such as interchange of two adjacent ligands), then the F labels must change during the fluxional process. As the number of ligands is increased, so do the number of equilibrium bond angles and the difficulties associated with atom labeling. Other difficulties associated with transition metal complexes include the presence of strong electronic influences, such as the Jahn-Teller effect, the softness of the angular deformation potentials for some complexes, and the highly delocalized bonding modes of organotransition metal complexes (e.g. cyclopentadienyl complexes) that may make it difficult to decide where and how many bonds should be used. To date there has been no generally satisfactory treatment of these problems. MM computations in their common forms are not well suited to the description of reactions. The problems are that chemical reactions involve breaking and making bonds. As the MM coordinates are based on the bonded topology, a chemical reaction is a major perturbation in the coordinates upon which the potential energy computation is based. The potential energy terms also are a source of difficulty in modeling reaction trajectories. For example the energy of a harmonic bond stretch term continues to rise ad infinitum as the bond is stretched. However real bond dissociation potential energies lie closer to the shape of a Morse potential. Similar arguments can be made concerning other potential energy functions. The essence of the problem is that MM computations generally do not yield accurate descriptions of the potential energy surface of a molecule except in the region of the equilibrium geometry. Thus, most modeling has been limited to a different approach, the transition state modeling method [27,28]. In this model a new set of parameters is created for the transition state and it is treated as a minimum on the potential energy surface.

IV. VALENCE BOND CONCEPTS A N D MOLECULAR MECHANICS COMPUTATIONS Our recent work on MM methods [9,29,30,31] has focused on the application of localized bonding theory to the molecular mechanics description of molecular shapes. We call the resulting force field VALBOND. The basic premise of the VALBOND [9] approach is that the energies associated with deformations of molecular shapes are reflected in the overlaps, or nonorthogonalities, of directed hybrid orbitals located on the same atom. For example, the equilibrium bond angle

VB Concepts, MM Computations, and Molecular Shapes

139

80

Water O

60

o

valbond

=

harmonic

x

E

m

t~ O

ab initio

40 O~ t_

19 r UJ

20

0

40

.

,

.

60

,

80

.

~ = C "

100

Angle

,

120

.

,

140

.

,

160

9

180

(degrees)

Figure 1. Comparison of ab initio(HF/6-311G**), harmonic, and VALBOND potential energy surface for bending the H-O-H bond angle at constant bond length.

of 104 ~ for water corresponds to sp 4"4 hybridization of the O-H bonds. That is, two sp 4"4 hybrids are orthogonal at the equilibrium bond angle of H20. If we v i e w sp 4"4 as the intrinsically preferred hybridization of the O-H bonds in water, then deformation of the bond angle from the equilibrium value will lead to a "defect", the quantity S max- Si(etij ) where etij is the angle made by the two O-H bonds, due to the nonorthogonality of the O-H hybrids. As a result the total energy of the molecule increases. It turns out that the computed increases in the total energy are mirrored nearly perfectly by the loss in strength of the preferred orbital hybridization (Figure 1) as computed by Eq. 20. In this equation, k i is a scaling parameter that converts the "defect" into energy units. This parameter may vary depending on the nature of the bond, but we generally find that a single parameter works well for most bonds. We conclude that the notion that the molecular deformation energies correlate with loss of hybrid orbital strengths is both qualitatively and quantitatively correct. all ligands all ligands

Eangle = E i

E

ki(S max-

Si(eij))

(20)

j(~)

The idea that molecular energies quantitatively are related to hybrid orbital overlaps is not new. To the best of my knowledge, the first example in which changes in hybrid orbital overlaps formed the basis of a valence force field was the

140

CLARK R. LANDIS

orbital valence force field (OVFF) of Heath and Linnett [32]. These authors state, "it is assumed that the bond-forming orbitals of an atom X are set at definite angles to each other, and that the most stable bond is produced when one of these orbitals overlaps the bond-forming orbital of the second atom Y to the maximum possible extent". Other examples of empirical potential energy functions with similar conceptual bases are the bond energy bond order (BEBO) method [33] of Johnston and Park and the iterative maximum overlap (IMO) method [34] of Maksic. VALBOND differs from these other methods primarily in its use of the generalized strength functions. The VALBOND method generalizes to central atoms containing more than two ligands by summation of pair defects [9]. Thus, for a carbon with four different substituents, each with a different preferred hybridization, the total defect of Eq. 20 involves a summation of twelve pair defects, or three for each ligand. The interested reader should refer to the original paper describing VALBOND for the details of the computations. The computation of pair defects requires assignments of hybridization to all of the bonds. This is accomplished with the Bent's rule algorithm given in Eq. 9. In the following sections we examine the application of the VALBOND force field across much of the periodic table. (N. B., the VALBOND force field only uses new angular terms; until recently all other terms for bond stretch, torsions, etc. use the CHARMM formulas.) Currently we are working on new terms for both proper and improper torsions that are derived from valence bond concepts.

A. NonhypervalentMain Group Molecules VALBOND has been parametrized for nonhypervalent molecules containing elements from the p-block of the periodic table [9]. Two parameters are required for each central atom-ligand atom pair: a scaling parameter (k in Eq. 20) and a weighting parameter (wt in Eq. 9) that relates to the preference of the ligand for p-orbital character. These parameters have been determined empirically, i.e. by finding values that reproduce experimental geometries and vibrational frequencies well. A partial listing of VALBOND parameters are given in Table 1. The values of the VALBOND parameters exhibit interesting trends. In general we find that the scaling factor (k) does not vary much from one ligand to another with the exception of H as a ligand. One could simply use genetic values of 150 kcal/mol for H ligand and 220 kcal/mol for all other ligands and achieve reproduction of vibrational frequencies and molecular geometries at a level of quality that is similar to those of other popular MM force fields. That bending frequencies are reproduced with genetic values of the scaling parameter most likely reflects the response of the bending frequencies to the bond hybridization. As the p-character of the hybridization increases so does the curvature of the potential energy surface. Thus, changes in vibrational frequencies that accompany changes in the ligand type mostly are accommodated in VALBOND computations via the change in bond

VB Concepts, MM Computations, and Molecular Shapes

141

Table I. Weighting Factors (wt) and VALBOND Constants (k, kcal.mo1-1) for Central Atom and Ligand Combinations Ligands Central Atom C

N

0

Si

P

S

H

B

wt

1.05

0.958

k

163

220

wt

1.04

0.878

k

145

C

N

0

F

Cl

Br

1.0

1.0

0.875

0.99 a

0.98

0.95

212

220

220

220

220

220

0.851

1.0

0.815

1.114

1.058

1.0b

220

220

190

300

175

220

220

wt

1.016

0.825

0.922

1.151

1.0

1.181

0.871

1.0b

k

163

220

220

220

220

195

220

220

wt

1.152

0.97 a

1.114

1.005

1.01

1.251

1.18 a

0.95

k

163

220

220

220

220

220

220

220

wt

1.152

0.898 a

0.97

0.397

0.342

0.442

0.437

0.402

k

122

220

220

220

220

200

220

220

wt

1.177

0.993

0.929

1.155 a

0.7

0.65 a

0.871

1.0b

k

146

220

220

220

220

220

220

220

Notes: aWeightingfactor determined by extrapolation. bDefault weighting factors.

hybridizations. That H is unique among all ligands is no surprise given the limited (essentially pure s) flexibility of H to hybridize. Originally we had anticipated that the weighting factors (wt) should exhibit trends in agreement with Bent's rule. That is, the weighting factors systematically would increase as the electronegativity of the ligand relative to the central atom increased. Several factors come to play in determining the weighting factors. First, they are empirical values that are determined from experimental data. Therefore, all forces influencing molecular shapes, including 1,3 ligand-ligand electrostatic and van der Waals interactions, are folded into the value of wt. Also, for the sake of simplicity, we employ the same values of wt for a ligand regardless of whether the bond is a single, double, or triple bond (recall that the gross hybridizations do change with the bond order). For an ester, the C--O bond has the same preference forp-character as the C-O bond. It would be reasonable to expect that the preference of an oxygen for p-character will be less when the bond is short (e.g., a double bond) than when it is long (e.g., single bond). Overall, the trends that one might expect the weighting parameters to exhibit are obscured by other factors that influence the molecular geometries. Bond angles are described well by the VALBOND force field. A short compilation of inorganic and organic examples are given in Tables 2 and 3. As shown by these structures, VALBOND reproduces geometries quite well despite the simplic-

142

CLARK R. LANDIS

Table 2. Some Computed and Observed Bond Angles (degrees)

for Organic Molecules Molecule

Angle

VALBOND

Experimental

1CHz

H-C-H

103.0

102.4 a

3CH2

H-C-H

131.7

136b

CH 3

H-C-H

120

120c

1CF2

F-C-F

103.8

104.8 d

1CHF

H-C-F

103.6

101.8 e

propane

C-C--C H-C(2o)-H

112.0 107.2

112.0 (0.2) f 107.8 (0.2)

butane

C-C-C

112.0

113.3 (0.4) g

isobutane

C-C-C

110.7

110.8 (0.2) h

C2-C1-CS C1-C2-C3 C2-C3-C4

103.7 104.9 106.4

103.0 i 104.2 105.9

C 1--C2-C3 C2--C1-C6 C1--C7-C4

102.4 109.0 91.8

102.7 i 109.0 93.4

ethylene

C=C-H

120.9

121.4 (0.6) j

cyclohexadiene

C--C=C C-C--C

123.2 113.6

122.7 k

C1--C7-C4 C1-C2=C3

93.4 106.7

95.31 107.7 (MM3) m

O-C-H(trans)

112.8 105.4

107.2 n

H-C-H 1,4-dioxane

C-C--O C-O-C

112.5 111.7

109.2 (0.5) ~ 112.6(0.5)

acetaldehyde

C-C-O C-C-H

123.9 115.4

124.1 (0.3) 0

cyclopentane(C s) 1

5 ~ 4

\ 3

2

norbornane 6

4

3

norbomene 6

4

113.3

3

methanol

108.5

115.3 (0.3) (continued)

VB Concepts, ~VIMComputations, and Molecular Shapes

143

Table 2. (Continued) Molecule

Angle

VALBOND

Experimental

acetone

C--C-C C--C=O

116.0 122.0

116.0 (0.2) q 122.0(0.2)

methyl formate

O-C=O

125.9

125.9 (1.0) r

C-O-C

111.9

114.8 (1.0)

C--C-N

110.8

109.8 (0.5) s

C-N--C

113.8

112.6 (0.5)

109.8

107.2 (1.0) t

N-C-C

117.5

115.1 (1.6) u

piperazine

nitromethane acetamide

Notes: aHerzberg, G.; Johns. J. W. C. Proc. Ro): Chem. Soc. Ser. A 1966, 296, 107. bHerzberg, G.; Johns, J. W. C. J. Chem. Phys. 1971, 54, 2276. CHerzberg, G. Proc. Roy. Soc. Ser. A 1961, 262, 291. aKirchhoff, W. H.; Lide, D. R.; Powell, F. X. J. Mol. Spect. 1973, 47, 491. eMerer, A. J.; Travis, D. N. Can. J. Phys. 1966, 44, 525. fWiberg, K.; Boyd, R. H. J. Am. Chem. Soc. 1972, 94, 8426. gHeenan, R. K.; Bartell, L. S. J. Chem. Phys. 1983, 78, 1270. hHilderbrandt, R. L.; Wieser, J. D. J. Mol. Struct. 1973, 15, 27. iAllinger, N. L.; Geise, H. J.; Pyckhout, W.; Paquette, L. A.; Gallucci, J. C. J. Am. Chem. Soc. 1989,111, 1106. JBartell, L. S.; Roth, E. A.; Hollowell, C. D.; Kuchitsu. K.; Young, J. E. J. Chem. Phys. 1965, 42, 2683. kDallinga, G.; Toneman, L. H. J. Mol. Struct. 1967, 1,117. IChiang, R.; Chiang, R.; Lu, K. C.; Sung, E.-M.; Harmony, M. D.J. Mol. Struct. 1977, 41, 67. mAllinger, N. L.; Li, E; Yan, L. J. Comp. Chem. 1990,11,848. nLees, R. M.; Baker, J. G. J. Chem. Phys. 1968, 48, 5299. ~ M.; Hassel, O. Acta Chem. Scand. 1963, 17, 1181. Vlijima, T.; Kimura, M. Bull. Chem. Soc. Japan 1969, 42, 2159. qIijima, T. Bull. Chem. Soc. Japan 1972, 45, 3526. rCurl, R.E J. Chem. Phys. 1959, 45, 3526. Syokoseki, A.; Kuchitsu, K. Bull. Chem. Soc. Japan 1971, 44, 2352. tCox, A. P.; Waxing, S. J. Chem. Soc., Farad. Trans. H 1972, 68, 1060. UKitano, M.; Kuchitsu, K. Bull. Chem. Soc. Japan 1973, 46, 3048.

Table 3. Some Computed and Observed Bond Angles

(degrees) for Nonhypervalent Main Group Inorganic Molecules

Molecule

Angle

VALBOND

Experimental

BHF 2

F-B-F

118.1

118.3 (1.0) a

BHCI 2

CI-B-CI

119.6

119.7 (3.0) b

BFENH 2

F-B-F H-N-H

119.9 115.0

117.9 (1.7) c 116.9 (0.3)

NH 3

H-N-H

106.8

106.7 d

NC13

C1-N-CI

106.5

107.1 (0.5) e

NHCI 2

H-N-CI CI-N-C1

106.7 106.1

102 f 106

NHEC1

H-N-H H-N-CI

107.1 106.3

107 (2) g 103.7 (0.3) (continued)

144

CLARK R. LANDIS

Table 3. ( C o n t i n u e d ) Molecule

Angle

VALBOND

Experimental

HN 3

N-N-N

180

180 h

NO 2

O-N--O

149.2

134.1 (0.1) i

NCIO

CI-N-O

120.0

113.3 (1.0)j

PH 3

H-P-H

93.8

PCI 3

CI-P--CI

100.1

CH3PH 2

C-P-H H-P-H

97.1 91.3

96.5 m 93.4

(CH3)2PH

C-P-C C-P-H

98.5 95.6

99.7 (0.5) n 97.0 (0.7)

(CH3)3P

C-P-H

95.6

98.9 (0.2) ~

03

O-O--O

116.8

116.78 (0.03) p

(CH3)20

C-O-O

111.6

111.7 (0.4) q

(CH3)2S

C-S-C

99.3

98.9 (0.2) r

(CH3)2Se

C-Se-C

96.2

96.2 (0.3) s

(SiH3)20

Si--O--Si

142.7

144.1 (0.9) t

(SiH3)2S

Si-S-Si

99.6

97.4 (0.5) u

(SiH3)2Se

Si-Se-Si

98.0

96.6 (0.7) v

Si404C8H24

Si-O-Si O-Si-O

143.4 112.4

144.8 (1.2) w 110(1.8)

C-Si---C

107.3

111.2 (1.8)

<Sn--Sn-Sn> C-Sn-C

112.4

112.5 x

105.5

106.7

(CH3)2Si~O. 0z \Si(CH3) 2

I

(CH3)2Si N

93.345 (0.003) k 100.1 (0.5) l

I

i,,0 O--Si(CH3) 2

Sn6(Ph2) 6

/ P h 2 S n-....SnPh.~.~SnPh2 Ph2s/~SnPh2~s2nP~h2

VB Concepts, MM Computations, and Molecular Shapes

145

Table 3. (Continued) Molecule

Angle

Ga2Pyr2Br 4

Br--Ga-Br Ga-Ga-Br

pyridm~

VALBOND

107.0 115.1

Experimental

105.8 y 116.3

Br ~---Br

BrT/ Br

pyridine

As3(CH3)6In3(CH3)6

~m

C-In-C

101.9

C-As-C

124.7

99 z 126

In

/

\

Notes: aKasya, T." Lafferty, W. J.; Lide, D. R. J. Chem. Phys. 1968, 48, 1. bLynds, L.; Bass, C. D. J. Chem. Phys. 1964, 40, 1590. CLovas, F. J.; Johnson, D. R. J. Chem. Phys. 1973, 59, 2347. '/Benedict, W. S.; Plyler. E. K. Can. J. Phys. 1957, 35, 1235. eB0rgi, H. B.; Stedman, D.; Bartell, L. S. J. Mol. Struct. 1971, 10, 31. fMoore, G. E.; Badger, R. M. J. Am. Chem. Soc. 1952, 74, 6076. gCazzdi, G.; Lister, D. G.; Favero, P. G. J. Mol. Spect. 1972, 42, 286. hForman, R. A.; Lide, D. R. J. Chem. Phys. 1963, 39, 113. iBird, G. R.;

Baird, J. C.; Jache, A. W.; Hodgeson, J. A.; Curl, R. F.; Kunkle, A. G.; Branford, J. W.; Rastrup-Andersen, J.; Rosenthal, J. J. Chem. Phys. 1964, 40, 3378. JMillen, D. J.; Pannell, J. J. Chem. Soc. 1961, 1322. kMaki, A. G.; Sams, R. L.; Olson, W. B. J. Chem. Phys. 1973, 58, 4502. IKislink, P.; Townes, C. H. J. Chem. Phys. 1956,18, 1109. mBartell, L. S.J. Chem. Phys. 1960, 32, 832. nNelson, R. J. Chem. Phys. 1963, 39, 2382L. ~ P. S.; Kuszkowski, R. L. J. Chem. Phys. 1971, 55, 3049L. PTanaka, T.; Merino, Y. J. Mol. Spect. 1970, 33, 538. qBlukis, U.; Kasai, P. H.; Myers, R. J. J. Chem. Phys. 1963, 38, 2753. rpierce, L., Hayashi, M. J. Chem. Phys. 1961, 35, 479. SBeecher, J. G. J. Mol. Spect. 1966, 21,414. tAlmenningen, A.; Bastiansen, O.; Ewing, V.; Hedberg, K.; Traetteberg, M. Acta Chem. Scand. 1963,17, 2455. UAlmenningen, A.; Hedbert, K.; Seip, R. Acta Chem. Scand. 1963, 17, 2264. VAlmenningen, A.; Fernhoft, L.; Seip, H. M. Acta Chem. Scand. 1968, 22, 51. wOberhammer, H.; Zeil, W.; Fogarasi, G. J. Mol. Struct. 1973, 18, 309. XHaas, A.; Kutsch, H. J.; Krfiger, C. Chem. Ber. 1987, 120, 1045. YSmall, R. W. H.; Worrall, I. J. Acta Co'stallogr., Sect. B: Struct. Sci. 1982, B38, 86. ZCowley, A. H.; Jones, R. A.; Kidd, K. B.; Nunn. C. M. J. Organomet. Chem. 1988, 341, C 1.

ity of the method. Indeed geometries of simple molecules are reproduced with similar rms deviations in bond angles as is observed for much more heavily parametrized methods such as MM2 and MM3 [21-24,35,36]. For a simple comparison, complete parameterization of the p-block of the periodic table using the VALBOND scheme involves about 1400 potentially unique values comprising both weighting and scaling parameters. For MM2, this total number of parameters

146

CLARK R. LANDIS

would be 26 times larger ifMM2 used just one atom type per element. Since MM2 uses many different atom types for some elements the extent of parameterization is even greater. Simple Lewis structures in combination with the VALBOND method yield good descriptions of radicals and electron deficient molecules. Generally we find that singly occupied radical orbitals, such as the radical orbital of .CH 3, have a strong preference for p-orbital character, whereas lone pairs have a strong preference for s-character. Thus, if we permit ourselves to represent the singlet and triplet states of CH 2 as, :~ C

/"

\

and

H

i

.~CRH I H

then the different H - C - H bond angles (103 ~ and 136 ~ respectively) are seen to arise from the different distributions of p-character among the two systems. The methyl radical, although it has gross sp 3 hybridization, yields a planar molecule due to the high p-orbital preference of the radical orbital. Another interesting example in which hybridization-based ideas correlate well with observations concerns the Si--O-Si linkage of siloxanes, an important component of zeolite structure. According to Bent's rule we would anticipate that Si would

16-

o\

14-

~si/

12-

8

-

6

-

4

-

2

-

0

/ si"

l

120

130

140 Si-O-Si

150 Bond Angle

160

170

180

(degrees)

Figure 2. Frequencies of occurrence (expressed as percent probabilities) of various Si-O-Si bond angle values for hexamethyldisiioxane taken from a 300K molecular dynamics simulation using the VALBOND force field.

VB Concepts, MM Computations, and Molecular Shapes

147

have a low preference for p-character when bonding to O because of the low electronegativity of Si with respect to O. Therefore we expect the large Si-O-Si bond angle associated with bond orbitals of high s-character; the observed bond angle for disiloxane is 144 ~ Furthermore, the shape of the bending potential energy surface should be very shallow due to the low curvature of hybrid orbitals with large s-character. We expect then that the energy for linearization of the Si-O-Si linkage should be quite low. The value of 1.3 kcal/mol computed with VALBOND for the linearization of the Si--O-Si angle agrees well with experimental estimations (ca. 0.3 kcal/mol) and ab initio computations (ca. 1 kcal/mol). As a consequence of the shallow bending potential, the distribution of Si-O-Si bond angles for a 300K molecular dynamics simulation of hexamethyldisiloxane is broad and highly anharmonic (see Figure 2), a feature that must be considered when evaluating, e.g., the gas phase electron diffraction data for disiloxanes. In summary, the hybridization effects incorporated into the VALBOND force field lead to a simple MM description of molecular shapes. In addition to the features emphasized above, we have examined vibration frequencies and inversion barriers (for AX 3 molecules). The consistent, quantitative agreement of experimental and computed data provides strong testament to the utility of valence bond concepts.

B. HypervalentMain Group Localized bond descriptions of hypervalent molecules are troublesome because the formation of 2-center 2-electron bonds from the central atom to each ligand requires use of more than just the valence s and p orbitals. The term hypervalency has two distinct but related meanings [37]. It may be used to indicate either that the central atom has more ligands than can be accommodated by Lewis' octet rule (e.g. PFs) or that the central atom has more bonds than its normal valency (e.g. O--PF3). We will concern ourselves with the former meaning, only. Pauling proposed dnsp m hybridizations as a way of expanding the central atom valency while maintaining consistency with observed shapes. For example, the trigonal bipyramidal geometry of PF 5 can be rationalized on the basis of dsp 3 hybridization and the octahedral structure of XeF 6 correlates with d2sp 3 hybridization. Kimball has provided a more complete set of shape-hybridization pairings [15]. Despite the success of dnsp m hybridizations in rationalizing molecular shapes of hypervalent molecules, it seems quite clear now that such hybridizations invoke far more d-orbital character than is consistent with high level ab initio computations (vide supra). Rather than focus on dnsp m hybridization in extending VALBOND to hypervalent structures we have concentrated on the angular dependence of the resonance stabilization of ionic configurations. Thus, the hypervalent or 3-center 4-electron bonding interaction in XeF 2 is broken into a resonance of the two configurations: F-Xe § F- +--)F- +Xe-E The problem is how to model the geometry dependence of

148

CLARK R. LANDIS

the resonance interaction ? Again we use the NBO method [ 10-12] of Weinhold and co-workers as a tool for decomposing the results of molecular orbital computations into localized bond concepts. A perspective [11] of hypervalent 3-center 4-electron bonding interactions arising from NBO analyses is to view them as donor-acceptor interactions (noncovalent delocalization effects) in which the lone pair of the Fligand is donated into the o* antibonding orbital of the +Xe-F fragment. The stabilization energies associated with this donor-acceptor interaction can be estimated in terms of a second-order perturbation analysis of the interacting NBO's, V

~+ :

AE=-2(lplh~

F"

(21)

e~, - e/p

where lp and o* are the donor and acceptor NBO's, respectively, F is the Fock operator, and eo. and etp are the energies of the acceptor and donor NBO's. Figure 3 illustrates the angular dependence of this stabilization energy. It is clear that the resonance stabilization of XeF 2 maximizes at 180 ~ and minimizes in the vicinity of 90 ~ as expected for a donor-acceptor interaction in which the acceptor is a high p-character antibond orbital. (The failure of the delocalization interaction to minimize at exactly 90 ~ and the large increase in delocalization energies at small angles is attributed to bending of the covalent bond in the +Xe-F fragment away from the 500

-

,-,

490

,I

E

480-

O

- -210.55

-210.6

'~"

-210.65

=

-210.7

= w

m

ca

470 m

>. "I.-

460 4 5 0 -

uJ

440-

"

430-

O '~, N .m O O

-

U,. "1"

-210.75

420-

m m

410-

O

-210.8

400-

390

IZ:

~"

-

0

380 0

0

;

:

0 O~

0 0

: 0 I-

: 0 0~I

:

I

I

0 03

0 ~

0 It)

I

;

0 QD

0 r,,.

-210.85 0 o0

F-Xe-F Bond Angle (degrees)

Figure 3. Plots of the XeF2 RHF energy (right absicca) and the F- +Xe-F donoracceptor delocalization energy (left absicca) as a function of the F-Xe-F bond angle at a constant Xe-F bond length of 1.97 ,~.

VB Concepts, MM Computarons, and Molecular Shapes

149

internuclear axis as the F-Xe-F bond angles approaches unrealistically small values). Our algorithm [30] for the angular dependence of the 3-center 4-electron bonding interaction follows the prescription of the donor-acceptor interaction. We focus on the orbital overlaps where the 6" acceptor orbital is simply the antiphase form of the Xe bond-forming hybrid. A simple function that expresses the loss of stabilization as the F-Xe-F bond angle changes from 180 ~ is, E(~) = BOF • ka[ 1 - A(o~+ ~)2]

(22)

where ot is the bond angle, ka is the VALBOND parameter, BOF is a bond order factor (equal to 0.25 in this example), and A is the overlap integral. The phase shifting of the hybrid overlap is accomplished by adding 180 ~ to the bond angle (o0 and the overlap integral (A) is computed according to Eq. 7. The bond-order factor is present to account for the lower formal bond-order of 3-center 4-electron bonding interactions (each partial 2-center bond has an order of 0.5). For cases such as XeF 2 there is just one bond angle; each "bond" in the angle has an order of 0.5 leading to a BOF - 0.5 • 0.5 = 0.25. XeF 2 is perhaps the simplest hypervalent molecule. More complex hypervalent molecules have more ligands and more resonance configurations. The simplest way to bookkeep electrons in more complex hypervalent molecules is to count the total number of electrons associated with the central atom from the best Lewis structure. By this formula, XeF 2 has 10 electrons at Xe, XeF 4 has 12 electrons, and XeF 6 has 14 electrons. Similarly, CIF3 has 10 electrons at C1, SF4 has 10 electrons at S, and SF 6 has 12 electrons at S. For every pair of electrons at the central atom beyond the octet of electrons, there must be one 3-center 4-electron bonding interaction. Therefore, molecules with one 3-center 4-electron bond include XeF 2, SF4, and CIF3. Those with two 3-center 4-electron interactions include XeF 4 and SF 6 and XeF 6 has three "hypervalent bonds", or 3-center 4-electron interactions. From a qualitative viewpoint, we can rationalize the shapes of these hypervalent molecules simply by considering: (1) the high p-character of the bonds, and (2) the preference of 3-center 4-electron interactions for linear bond angles. Hypervalent main group molecules should consist primarily of fight angles from the former consideration and 180 ~ from the latter. For example, C1F3 adopts the T-shape because there is one "hypervalent bond" that is linear and one 2-center 2-electron bond that has high p-character and thus prefers 90 ~ bond angles. The same reasoning leads to a square planar geometry for XeF 4 which has two 3-center 4-electron bonds, each preferring to be linear, and high p-character which forces the two hypervalent bonds at 90 ~ with respect to one another. Although it is trivial to assign the 3-center 4-electron bonding interaction of, e.g., C1F3 to the linear bond in the T-geometry, the challenge in applying the resonating valence bond picture to MM computations is to devise a method that works at any geometry. A good MM potential energy surface should be free of cusps and

150

CLARK R. LANDIS

O1 F1

el

tt,,,,~"~F2

F

~"~

02 "F2

F1 ",,,s/r

,~,,F3

F3

F3

F2

A

B

C

Scheme 1. discontinuities and should permit the molecule to undergo its natural fluxional motions. In order to satisfy these constraints, we have adopted a geometry-dependent, linear combination of MM configurations framework for hypervalent molecules. Consider the structure and dynamics of CIF3. The electronic structure of C1F3 involves one 3-center 4-electron bonding interaction and one 2-center 2-electron covalent bond. Three limiting arrangements of the bonding interactions are shown below; we call each of these a MM configuration. (N.B. that, in our terminology, a VB resonance structure and a MM configuration are slightly different in that each MM configuration comprises the two F- +CI-F resonance structures of the 3-center 4-electron interaction.) In the diagram shown below, the 3-center 4-electron interaction is represented by the symbol F .... CI ....E As the geometry of the molecule changes, so does the contribution made by each of the MM configurations (A, B, and C) to the total MM energy as shown in Eq. 23. Eto t = c A X E A + CB X E n + c c x E c

(23)

The contributions, or fractional populations, of the individual MM configurations is based on a simple angular formula, hype l-I COS20i

Cj

i=1 = res hype

(24)

l-I COS20i

j=l i=1 where res = number of resonance configurations and hype = number of hypervalent angles. In the limit of the perfect T-geometry only one MM configuration is populated. At Ca-symmetric geometries each configuration is populated equally. The VALBOND picture of CIF3 makes some surprising predictions concerning the nature of the CIF3 axial-equatorial interchange process. One can imagine four possible exchange pathways: those that pass through D3h (trigonal planar), C3~ (trigonal pyramidal), C2v (planar, Y-shape), and Cs (pyramidal, Y-shape) geometries. Computations with VALBOND yield lower energies for planar geometries

VB Concepts, MM Computations, and Molecular Shapes

151

but show a substantial preference (ca. 5 kcal/mol) for the C2v geometry rather than the more symmetric D3hgeometry. To a good approximation, one can characterize the axial-equatorial exchange pathway as maintaining the near 90 ~ angle between one of the axial ligands and the equatorial ligand while moving the other axial ligand from a cis to trans relationship with the equatorial bond. Also surprising is the magnitude of the exchange barrier. VALBOND computations using generic parameters predict a barrier of nearly 46 kcal/mol, nearly an order of magnitude greater than the Berry pseudorotation barrier in PFs! Perhaps most surprisingly, we find that the VALB OND results mirror those of good quality ab initio computations. Thus, at the MP2 level using effective core potential basis sets we compute an axial--equatorial exchange barrier of ca. 37 kcal/mol with a C2v transition state geometry (the O3hgeometry lies 3 kcal/mol higher in energy). The geometries of the VALBOND and ab initio transition states are quite similar; the F-C1-F bond angles computed by the two methods are 91 o, 134.5 o, 134.5 o and 100 ~ 130 ~ 130 ~ respectively. These results are particularly interesting because other qualitative theories, such as VSEPR [38,39], are not designed to analyze transition states. If one attempts to extend the principles of VSEPR to transition state geometries, it is difficult to rationalize such a low-symmetry transition state and such a high barrier. Apparently, quantitative computations based on a localized bond perspective illuminate our understanding of the forces controlling shapes in unique ways. Initial results [30] for the application of VALBOND to other hypervalent structures are promising. For the simple hypervalent fluorides (AF n) the VALBOND scheme consistently generates the correct gross shape (e.g., SF 6 and XeF 6 are octahedral, XeF 4 minimizes at the square planar shape, PF 5 adopts the trigonal bipyramidal geometry, SF4 has the seesaw geometry, and BrF 5 is square pyramidal; see Figure 4). The axial--equatorial exchange pathway in PF 5 has a low barrier (3 kcal/mol) and proceeds through a square pyramidal transition state. However, some of the finer details of the geometries systematically are discrepant with experimental structures. The most glaring discrepancy is the tendency of the F's to slightly splay away from one another, whereas the experimental geometries commonly distort from the idealized geometry by drawing the F's closer together. For example, the experimental structure of CIF3 is not a perfect T-shape. The two axial F's make angles with the equatorial F that are slightly less than 90~ i.e., the shape is an arrowhead. In contrast, the VALBOND structure of C1F3 distorts in the opposite direction by a degree or two to yield a slightly Y-shaped geometry. Similarly, we find the VALBOND structures distort from the idealized geometry in the opposite sense of the experimental distortions for BrF 5 and SF4. Apparently, the failure of the VALBOND method to model these fine distortions arises from the lack of explicit lone pairs in the computations (vide infra). Of course one of the tremendous successes of VSEPR is the ease with which these distortions are rationalized. In VSEPR theory, the primary forces controlling the movement of the F's closer together are the large volumes of lone pair domains and the small volume of bond pairs to highly electronegative ligands, such as E

152

CLARK R. LANDIS F ,,,,,91 ~

/\

~

F

92~

18~176176176

900(90 ~)

F

)"q ~2~;P--- F F "F~120o(120o)

Br~-F F ~ F 91~

F

Figure 4. VALBONDcomputed and experimental ( ) geometries of some hypervalent molecules.

For heteroleptic hypervalent molecules of the main group, it is commonly found that more electronegative ligands occupy the axial sites. Such a finding is not unexpected from a valence bond description which emphasizes stabilization of the 3-center 4-electron bond due to resonance of ionic configurations. We have begun to model this important effect. Essentially the approach is to vary the various weighting coefficients in Eq. 23 according to the electronegativities of the ligands constituting the 3-center 4-electron interaction. Our preliminary work [30] indicates that site preference energies and geometric distortions in heteroleptic hypervalent molecules are modeled well by this approach.

C. Transition Metal Hydrides Perhaps the most difficult geometries to rationalize, at any simple level of theory, are those of transition metal complexes. The high coordination numbers, the strong influence of the d-electron configuration and spin states on structure, the high ionic character of many of the bonds, the presence of Jahn-Teller distortions, and the highly delocalized binding of unsaturates such as cyclopentadienyl groups all sum to yield complex electronic and molecular structures. Given the multitude of effects at play, a broad examination of very simple transition metal complexes is warranted. We have initiated such an examination (as have others, notably Prof. Siegbahn [40] at Stockholm) beginning with the homoleptic hydrides and alkyls of the transition metals. The geometries of simple hydrides and alkyls of transition metals are surprisingly complex. Here our information comes primarily from the computational work of Albright [41-43], Schaefer [44], Eisenstein [42], and Siegbahn [40], and the experimental work of Haaland [45,46] and Girolami [47]. Let us use structures of WH 6 and WMe 6 as illustrations. The gas phase electron diffraction data for WMe 6

VB Concepts, MM Computations, and Molecular Shapes

153

led Haaland and co-workers [45] to conclude that the structure is not octahedral. Rather the structure is either trigonal prismatic (D3h) o r a distorted trigonal prismatic (C3v). Morse and Girolami [47] determined that the structure of the isoelectronic [Li(tmeda)2]Zr(CH3) 6 is best described as a distorted trigonal prism. Computational predictions of the structures of WH 6 and WMe 6 using ab initio methods agree with experiment. Thus, Albright and co-workers [43] find that the trigonal prism is the global minimum for WMe 6. The structure of the methyl complex results in part from intermethyl steric interactions that are absent in the case of WH 6. Extensive searching of the WH 6 conformational space yields at least four true minima, two with C3v symmetry and two with C5v symmetry. Interestingly, Schaefer [44] estimates that the energy of octahedral WH 6 is ca. 140 kcal/mol less stable than the minimum energy structures. From a thermodynamic viewpoint, the enthalpic penalty for homolytic cleavage of a W-H bond in WH 6 is less than that for distorting to an O h geometry! The strong preference of WH 6 for nonoctahedral structures has been rationalized as a second-order Jahn-Teller distortion and as a method of maximizing s and d orbital participation in the formation of covalent W-H bonds. Our application of the VALBOND force field to the structures of simple metal hydrides and alkyls forms a compelling case for the validity of localized bond concepts in describing the shapes of simple transition metal complexes [29]. Our primary tool for probing the electronic structure of metal hydrides has been NBO analysis of good ab initio density matrices. We have devised a set of simple rules guides the descriptions of localized, covalent bonds in simple metal hydrides and alkyls. These are: 1. Use only s and d orbitals on the metal in forming hybrid orbitals. 2. In forming n bonds use sd n-1 hybrids (i.e. maximize the s-character of the bond hybrids). 3. Place lone pairs on the metal in pure d-orbital. 4. If the metal electron count exceeds 12 electrons, form linear H - M - H 3-center 4-electron bonds by resonance of the Lewis structures H-M :H ~ H: M-H. According to these rules, WH 6 requires sd 5 hybridization leading to six W-H electron pair bonds; ZrH 4 has sd 3 hybridization and Tall 5 has sd 4 hybridization. The 12 valence electrons of TcH5 are accommodated in one pure d lone pair and five bond-forming sd 4 hybrids. The dihydride, PtH 2, has four pure d lone pairs and two sd ~ hybrids that make the Pt-H bonds. Molecules with more than 12 valence electrons are hypervalent. Thus, PtH42- (16 electrons) is described as four lone pairs and two 3-center 4-electron bonding interactions. Each 3-center 4-electron interaction of PtH42- involves resonance of two Lewis structures, each having a Pt-H covalent bond with sd hybridization and an unbound hydride (:H) ligand. Molecular shapes of the simple metal hydrides correlate beautifully with the bond hybridizations. According to the VALBOND scheme, the predicted structures are

154

CLARK R. LANDIS

Table 4. VALBOND Predictions and Ab Initio Structures for Nonhypervalent

Metal Hydrides

Bond Hybrid Hybrids Angles

sd I

VALBONDMinima

Ab Initio Results

90 ~ YH2 + at M P 2 geometry NBO: sdl'6p 03

sd 2

c~ 90 ~

~~

S

93 ~

-90 ~

ZrH3 + at M P 2 geometry NBO: sdl97p 0"13

C4_z] sd 3

71& 109 ~

RuH 4 at M P 2 geometry N B O : sd2"9~ 0"10

c~

sd 4

%

1

66 ~ &

'

1~~/59o

75 ~ NbH 5 at M P 2 geom. TcH5 + at M P 2 geom. NBO: sd3"68p ~ N B O : sd399p ~176

114 ~

C L ' ~ I I , , , C2z]

,,@ sd 5

63 o& 117 ~

116 ~ 114 ~ M o l l 6 at M P 2 geom. TcH6 + at M P 2 geom. NBO: sd4"9~ ~176 N B O : sdS"~ ~

those which minimize (Eq. 20). For sd 5 hybridization minimization of Eq. 20 occurs at geometries having H-M-H bond angles of 63 ~ and/or 117 ~ Four geometries which achieve these combinations are exactly the minima (two C3v and two C5,,) found independently by Schaefer and Albright! Some of our results for nonhypervalent metal hydrides are summarized in Table 4.

VB Concepts, MM Computations, and Molecular Shapes

155

WMe6 Inversion Coordinate

AE=3.0 kcal/mol

Scheme 2.

MM computations can model the steric effects that might cause WMe 6 to distort from the minima preferred by WH 6. VALBOND computations [29] on WMe 6 suggest that the minimum energy structure is distorted approximately midway between the D3htrigonal prism and the lowest energy C3v structure adopted by W H 6. Dynamics simulations reveal a facile inversion coordinate that interconverts two C3v structures via a O3htrigonal prismatic transition state (Scheme 2). The inversion barrier is computed to be ca. 3 kcal/mol. Preliminary results indicate that the internuclear distance distributions obtained in dynamics simulations are consistent with those determined by electron diffraction. According to the VALBOND scheme, many transition metal complexes should be considered hypervalent because most transition metal complexes formally have more than 12 valence electrons. The 18-electron rule is the most fundamental rule for predicting the coordination numbers and stabilities of organotransition metal complexes. Although the VALBOND scheme does not allow one to understand why special stability is associated with 18-electron complexes, it does allow one to rationalize the geometries of hypervalent hydrides. For PtH42- mentioned above, the crystallographically determined square planar geometry is expected due to the presence of two 3-center 4-electron bonds by analogy with XeF 4. Recall that for XeF 4 the linear bond angles arise from the resonance stabilization of the 3-center 4-electron bonds, whereas the 90 ~ bond angles are typical of ligands that prefer hybrids with high p-character. For PtH42- the 180 ~ bond angles are rationalized similarly and the 90 ~ bond angles are the natural consequence of sd 1 hybridization. This strong analogy between hypervalent main group shapes and hypervalent transition metal shapes is general. For example, molecules that are hypervalent by

156

CLARK R. LANDIS

one electron pair include the pair PdH 3- and C1F3, which are T-shaped, and RhH 4and SF 4, which both adopt seesaw geometries. Both Fell 4- and XeF 6 exceed 12-electron counts by three electron pairs and each adopt a structure that is close to octahedral. A classic polyhydride with no main group analog for which VALBOND rationalizes the molecular shape is ReH 2- . This complex essentially is an sdS-hybridized ReH~ fragment with three H- ligands stabilized through 3-center 4-electron bonding interactions. The structure is constructed from a C3v shape for the ReH~ to which three linear hypervalent bonds are added. Minimization of the total strain leads to the observed trigonal prism with capping of the three rectangular faces. Preliminary indications suggest that the VALBOND approach can be extended to ligands other than hydrides and alkyls [31]. For example, simple phosphine complexes, such as Rh(PR3) ~, have geometries, such as the T-shape, that can be rationalized using the rules listed above. However, at the current level of development the VALBOND rules are insufficient for all transition metal complexes. Several factors must be considered: (1) many metal-ligand bonding interactions are highly ionic; the VALBOND hybridization schemes are poor approximations to the electronic structures in these instances; (2) often metal-ligand interactions involve significant re-components with the ligands acting as either donors (e.g. F) or acceptors (e.g. CO)---the hybridization prescriptions of VALBOND do not account for multiple bonds at this stage; (3) the hybridization rule that precludes valence p-orbital participation is inaccurate for some early transition metal atomsmfor example, the p-orbitals of Y lie low enough in energy and the s2d I configuration is sufficiently stabilized relative to the sd 2 configuration that YH 3 has considerable p-character in its bond orbitals and adopts a trigonal planar geometry; (4) in some instances two Lewis structures, such as molecular H 2 versus two M-H bonds, lie so close in energy that the structure might best be considered a mixture of the two (e.g., the MP2 optimized geometry of PdH 2 has a bond angle of just 70 ~ perhaps reflecting a mixing of the molecular hydrogen and dihydride limiting geometries). Nonetheless, the current work on simple metal hydrides forms an excellent point of departure for considering the structures of more complicated metal-ligand bonding scenarios.

V. FUTURE DIRECTIONS Although our application of valence bond (VB) concepts to molecular mechanics (MM) computations leads to some improvements of MM methods and provides some unique insights into the origins of molecular shapes, it also raises many questions and new opportunities. I will briefly consider four areas of active research: (1) new force field terms derived from a VB perspective, (2) the role of lone pairs in determining molecular shapes, (3) reconciling valence bond concepts with structures of nonhydridic metal complexes, and (4) modeling chemical reactions with MM techniques.

VB Concepts, MM Computations, and Molecular Shapes

157

Scheme 3.

Although we have employed VB concepts primarily to the derivation of new angular terms in MM computations, a localized bond perspective should apply to the derivation of torsion terms (both proper and improper) as well. For example, consider the torsional motions of ethylene. These motions can be decomposed into two types: pyramidatization of the sp 2 hybridized carbons and twisting of the C = C double bond (Scheme 3). We have found [48] that the energy penalty associated with the pyramidalization motion can be modeled by considering the nonorthogonalities of the p-orbital of the n-bond with the ca. sp2-hybridized orbitals of the framework (see below). For the twisting of the double bond, the energy changes are modeled well by considering the loss of overlap between the two p-orbitals making the n-bond. For the VALBOND methods summarized above, the influence of lone pairs is indirect only. That is, the positions of lone pairs are not computed and do not affect the energies. Lone pairs do exert an influence, however, in the determination of bond hybridizations. Our simple algorithm has separate weighting factors for lone pairs and singly occupied orbitals. In general, lone pairs have high intrinsic preferences for s-character; this preference leads to substantial p-character in the bonds to atoms. Thus, ammonia is pyramidal. Lone pairs have been excluded from explicit representation in MM computations primarily for simplicity and because lone pairs generally are not located in structure determinations. In principle, the nonorthogonalities of lone pair hybrids with each other and with bond hybrids are as important in determining molecular structure as the nonorthogonalities of bond hybrids. The inclusion of lone pairs brings the emphases of VALBOND closer to those of VSEPR theory. Lone pairs are particularly important in rationalizing the distortions to small bond angles of hypervalent molecules. For example, the small distortion of C1F3 from

158

CLARK R. LANDIS

the T-shape to the arrowhead geometry is consistent with the nonorthogonalities of the lone pairs forcing the F's closer together. The advantage of incorporating VB methods into a MM computation is that this notion can be tested quantitatively. Indeed, we find that explicit inclusion of two lone pairs into our VALBOND model of CIF 3 does indeed force the Feq-Cl-Fax angles from values greater than 90 ~ to smaller values (89~ Similar results are found for SF4. It is interesting to explore the interconnections between VALBOND with explicit lone pairs, VSEPR [38,39], and Bader's analyses of electron density distributions [49]. One could assert that the effect of hybridization and minimization of nonorthogonalities is analogous to partitioning electron density surfaces and minimizing repulsions between them. For this analogy to work we must assert that hybridization provides a reasonable approximation of electron density distributions. Bader finds [49] angles between regions of bonded and nonbonded charge density which are consistent with those based on simple hybridization ideas for molecules in which bond bending is not too severe. A more profound distinction between the hybridization model and the VSEPR/charge concentration model arises for the structures of simple metal hydrides. The structures of minima on the potential energy surface ofWH 6 and many other simple metal hydrides are fully consistent with hybridization ideas. VSEPR theory apparently is incompatible with the structures of all minima on the WH 6 surface (unless, of course, one uses s-d hybridization to obtain the regions of charge concentration). If one considers the valence shell of WH 6 to consist of six equivalent bonding pair of electrons, VSEPR principles suggest an octahedral structure. However, such results do not invalidate VSEPR ideas; VSEPR has limited applications to transition metals and does not claim wide applicability [39]. The case of the Y-shaped transition state for axial--equatorial ligand exchange in CIF 3 is another example in which the VB concepts prove to be more enlightening. Therefore, although the VSEPR/charge concentration model and the VB model share many features they are distinguishable. In some instances it appears that hybridization models provide more powerful prescriptions for predicting and quantitating the structures of molecules. The structures of nonhydridic transition metal complexes place formidable challenges on simple computational models, such as VALBOND. For example, although VALBOND accounts for the nonoctahedral structure of WH 6, the structure of WF 6 is octahedral. Both the high ionicity and re-donor interactions of WF 6 presumably contribute to the change in structures. However, the problems of identifying simple rules and potential energy expressions that deal with these effects are considerable. Added to these problems are those associated with Jahn-Teller distortions, x-acceptor interactions (e.g. CO), and the bonding of highly delocalized ligands (e.g. cyclopentadienyl groups). Our approach is to carefully examine the structures of simple molecules containing these more complex ligands using ab initio methods and NBO analyses. One of the most exciting areas for application of MM methods is the modeling of reaction transition states. Superficially, it would appear that MM computations

VB Concepts, MM Computations, and Molecular Shapes

159

are ill-suited to modeling processes in which bonds are made or broken and atom types change because the framework of MM computations consists primarily of the bond topologies and the atom typing information. However, Rapp6 and coworkers [50] recently have shown that the VALBOND scheme enables an accurate and smooth description of the bond-breaking process. Consider the homolytic cleavage of a C-H bond in methane. To model this potential energy surface with MM computations one must smoothly transform the shape of the carbon fragment from tetrahedral to trigonal planar as the bond is stretched. This can be accomplished by (a) using a Morse or Rydberg potential for the bond that is broken, and (b) changing the hybridization weighting factor of the dissociating H ligand from that of H to that of radical as the C-H bond order changes from 1 to 0. Thus, the hybridization becomes distance-dependent. Rapp6 and co-workers find that this approach yields a model of the three-dimensional (where the dimensions are the C-H distance, the H--C-H bond angles of the nondissociating bonds with threefold symmetry imposed, and the potential energy) potential energy surface that varies from that computed by CASSCF/CI calculations with an rms deviation of ca. 2 kcal/mol. This treatment of bond-breaking (or its reverse, bond-making) has been combined by Rapp6 with a modification of Jensen's seam-searching technique [28] to create a method that models the transition states of simple concerted reactions (such as Diels-Alder reactions and Cope rearrangements) with impressive accuracy.

VI. SUMMARY Let us emphasize that the conceptual framework of VALBOND does not invoke any fundamentally new ideas about bonding and molecular geometries. Rather, the context of developing molecular mechanics (MM) methods that span the periodic table presents the opportunity to refine and adapt the simple, but robust, concepts of valence bond (VB) theory into quantitative potential energy expressions. For example, our development of generalized hybrid orbital strength functions, a quantitative hybridization algorithm, a geometry-dependent multiconfiguration model of hypervalency, and simple rules for hybridization in metal complexes all are refinements of previous ideas that were prodded by our desire to improve MM methods. Although we are at an early stage of developing and testing these expressions, I believe that the results summarized in this chapter constitute a vigorous demonstration of the power of VB descriptions applied to the MM description of molecular shapes. An alternate view of the work summarized in this chapter emphasizes the validation of VB concepts through MM computations. That is, MM methods can be considered the computational expression of our simplest bonding models. Qualitative VB theory, along with qualitative MO theory and VSEPR theory, is a widely used model of bonding and molecular structure. Most chemists think in terms of localized bonds, hybridizations, and resonance because these concepts are

160

CLARK R. LANDIS

simple and deeply rooted in our educational system. I have attempted to demonstrate that the V A L B O N D M M scheme enlightens our understanding of p h e n o m e n a such as the fluxional pathways and energetics of C1F 3 and the shapes of simple metal hydrides more effectively than other simple models, such as qualitative M O methods and V S E P R theory. Perhaps it is not too surprising that VB concepts are so well adapted for describing molecular shapes: the VB concept of hybridization arose in order to rationalize molecular geometries. What is truly remarkable about these VB concepts, first developed by Linus Pauling over six decades ago without the aid of digital computers or large structural databases, is how they have not only endured but continue to illuminate our understanding of molecular structures.

ACKNOWLEDGMENTS This work was supported in part by a grant from the National Science Foundation and by a studentship from Molecular Simulations, Inc. The bulk of this work has been carried out in collaboration with three graduate students, Timothy K. Firman, Thomas Cleveland, and Daniel M. Root. This work benefited from many discussions with colleagues. I gratefully acknowledge Prof. Frank Weinhold for his frequent doses of wisdom and encouragement, Prof. Chuck Casey for his critical analyses, and Prof. Tony Rapp6 for his selfless advice and sharing of ideas. This manuscript benefited from many thoughtful suggestions by the reviewer.

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

Pauling, L. Proc. Natl. Acad. Sci U.S.A. 1928, 14, 359. Pauling, L. J. Am. Chem. Soc. 1931, 58, 1367. Slater, J. C. Phys. Rev. 1931, 37, 481. Siater, J. C. Phys. Rev. 1931, 38, 1109. Burkert, U.; Allinger, N. L. Molecular Mechanics; American Chemical Society: Washington DC, 1982, pp. 339. Pauling, L. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 4200. Pauling, L. Proc. Natl. Acad. Sci. U.S.A. 1976, 73, 274. Pauling, L.; Herman, Z. S.; Kamb, B. J. Proc. Natl. Acad. Sci. U.S.A. 1982, 1982, 1361. Root, D. M.; Landis, C. R.; Cleveland, T. J. Am. Chem. Soc. 1993, 115, 4201. Reed, A. E.; Weinhold, F. J. Chem. Phys. 1985, 83, 1736. Reed, A. E.; Curtiss, L. A.; Weinhold, E Chem. Rev. 1988, 88, 899. Carpenter, J. E.; Weinhold, E J. Am. Chem. Soc. 1988, 110, 368. Bent, H. Chem. Rev. 1961, 61,275. Pauling, L. The Nature of the Chemical Botut; Cornell University: Ithaca, 1960. Kimball, G. E. J. Chem. Phys. 1940, 8, 188. Magnusson, E. J. Am. Chem. Soc. 1990, 112, 7940. Coulson, C. A.J. Chem. Soc. 1964, 1442. Bjerrum, N. VerhandL Deut. Physik. Ges. 1914, 16, 737. Westheimer, F. H. Steric Effects in Organic Chemistry; John Wiley: New York, 1956. Allinger, N. L.; Sprague, J. L. J. Am. Chem. Soc. 1973, 95, 3893. Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J. Am. Chem. Soc. 1989,111,8551. Allinger, N. L.; Lii, J.-H. J. Am. Chem. Soc. 1989, 111, 8566.

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23. Allinger, N. L.; Liu, J.-H. J. Am. Chem. Soc. 1989, 111, 8576. 24. Allinger, N. L.; Li, E; Yan, L. J. Comput. Chem. 1990, 11,848. 25. Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M. J. Comput. Chem. 1983, 4, 187. 26. Allured, V. S.; Kelly, C.; Landis, C. R. J. Am. Chem. Soc. 1991, 113, 1. 27. Eksterowics, J. E.; Houk, K. N. Chem. Rev. 1993, 93, 2439. 28. Jensen, E J. Comput. Chem. 1994, 15, 1199. 29. Landis, C. R.; Cleveland, T.; Firman, T. K.J. Am. Chem. Soc. 1995, 117, 1859. 30. Cleveland, T.; Landis, C. R., unpublished results. 31. Firman, T. K.; Landis, C. R, unpublished results. 32. Heath, D. E; Linnett, J. W. Trans. Faraday Soc. 1948, 44, 873. 33. Johnston, H. S.; Park, C. J. Am. Chem. Soc. 1963, 85, 2544. 34. Kovacevic, K.; Maksic, Z. B. J. Org. Chem. 1974, 39, 539. 35. Allinger, N. L. J. Am. Chem. Soc. 1977, 99, 8127. 36. Allinger, N. L.; Yuh, Y. H. "MM2," Quantum Chemistry Program Exchange, 1980. 37. Musher, J. Angew. Chem. Int. Ed. Engl. 1969, 8, 54. 38. Gillespie, R. J. Molecular Geometry; Van Nostrand Reinhold: London, 1972. 39. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular Geometry; Allyn and Bacon: Boston, 1991. 40. Siegbahn, P. E. M.; Blomberg, M. R. A.; Svensson, M. J. Am. Chem. Soc. 1993, 115, 4191. 41. Albright, T. A.; Tang, H. Angew. Chem. Int. Ed. Engl. 1992, 31, 1462. 42. Kang, S. K.; Albright, T. A.; Eisenstein, O. Inorg. Chem. 1989, 28, 1611. 43. Kang, S. K.; Tang, H.; Albright, T. A.J. Am. Chem. Soc. 1993, 115, 1971. 44. Shen, M.; Schaefer, H. E; Partridge, H. J. Chem. Phys. 1992, 98, 508. 45. Haaland, A.; Hammel, A.; Rypdal, K.; Volden, H. V. J. Am. Chem. Soc. 1990, 112, 4547. 46. Pulham, C.; Haaland, A.; Hammel, A.; Rypdal, K.; Verne, H. P.; Volden, H. V. Angew. Chem. Int. Ed. Engl. 1992, 31, 1464. 47. Morse, P. M.; Girolami, G. S.J. Am. Chem. Soc. 1989, 1989, 4114. 48. Root, D. M.; Landis, C. R., unpublished results. 49. Bader, R. W. Atoms in Molecules: A Quantum Theory; Clarendon: Oxford, 1990. 50. Rapp6, A. K., personal communication.

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EMPIRICAL CORRELATIONS IN STRUCTURAL CHEMISTRY

Vladimir S. Mastryukov and Stanley H. Simonsen

I. II.

III.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Bond Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Bond Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 165 165 173 174 178 185 185

ABSTRACT Seventy four empirical relationships including in an explicit form one or two of the structural parameters (bond length, bond angle, torsion angle) are collected and classified. Additional material (27 cases) refers to those relationships which were reported in the literature only in the graphical form. Three empirical equations published earlier have been updated using new data (Figures 1,5, and 8). Furthermore, seven new relationships (Figures 2-4) are reported for the first time. The geometrical

Advances in Molecular Structure Research Volume 2, pages 163-189 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

163

164

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN

parameters are correlated mostly in a linear way, although other types of correlations also occur. Structural geometrical parameters are found to correlate to the following basic characteristics: 9 spectra (frequency of vibration, force constant, amplitude of vibration, d-d transition); 9 energy (energy of dissociation, strain energy, ionization potential); 9 miscellaneous (spin-spin coupling constant, electronegativity, Taft constant).

I.

INTRODUCTION

There are many empirical relationships observed in physics and chemistry. Some of these have proven to be very helpful and important either in classifying experimental data or in stimulating the theoreticians to find an explanation. Probably one of the best examples of this kind is an empirical formula relating to the spectrum of hydrogen suggested by a Swiss teacher, Balmer [1], and later confirmed by Bohr [2] on the basis of his theory for the hydrogen atom. Another outstanding classical example is the Mendeleev periodic system of elements. As Maksic notes: "It is an empirical finding which is purely symbolic and abstract. The subunits are atoms but explicit dependence and periodicity of their properties is difficult to express in a mathematical form. In spite of its qualitative nature, the periodic system had a predictive power, and anticipated existence and properties of several missing elements which were subsequently discovered. It is common knowledge that the periodicity of atomic properties and the Mendeleev system itself can be interpreted by the shell model of the quantum electronic structure theory of atoms" [3]. (A more quantitative application was presented in a recent paper by Allen et al. [4] which describes the use of Van Arkel-Ketelaar triangles [5, 6] to classify binary compounds according to their bonding: metallic, ionic, or covalent. By representing each element with a number, its configuration energy in electron volts, the bonding described in the periodic table is quantified.) Empirical relationships and models play an important role in phenomenological chemistry. They help to classify similar compounds into families exhibiting characteristic properties. The purpose of this review is to give an overview of some empirical relations used in modem structural chemistry. Because the term "structural chemistry" seems to be rather diffuse, only those relationships which contain at least one of the three independent structural (geometrical) parameters are included. These parameters are: internuclear distance (bond length), r; bond angle (valence angle), or; and dihedral (torsion) angle, tp. (More information about these parameters is given in refs. 7-9.) The structural parameters referred to above can be measured for free molecules using electron diffraction and microwave spectroscopy, or for those in crystals by X-ray and neutron diffraction methods.

165

Empirical Correlations

Unfortunately, many authors did not treat their data by curve-fitting methods to obtain analytical expressions, which greatly facilitate their subsequent application, but present it only in graphical form. Many examples can be found in books by Hargittai [10] and Hefferlin [11]. However, because such information may be useful or important, graphical examples are collected in descriptive form. The reader will find a group of equations devoted to each of the parameters, r, t~, and tp, respectively. This order also refers to an appearance of particular relations. For example, if relations of the type: r=f(~)

or

t~=f(r)

are considered, they appear only in the section devoted to bond lengths, although the cross-references are given in the appropriate text. To avoid a lengthy presentation, the analytical forms are given with very short descriptions; details may be found in the original publication. A limited number of figures which show typical trends is included. A certain inconsistency in symbols used for designation of the structural parameters in the empirical relationships arises from the wish to reproduce without modification the analytical forms cited in the original references. Obviously, the relations presented are not a complete list of those reported in the literature. It is impossible to conduct a systematic search because empirical correlations are not usually mentioned in titles, abstracts, or keywords; they form a part of the discussion in the original paper. For this reason, this compilation is principally based upon our own experience in the field of structural chemistry, and, therefore, may be very subjective.

I!.

DATA S U M M A R Y

A. Bond Lengths Chemists were and are today always interested in estimating the length of chemical bonds and distances between nonbonded atoms. For this reason many systems of atomic radii have been suggested. Among them the most popular systems are: atomic radii (covalent radii), van der Waals radii, and 1,3-radii (see

[8,12-14]). However, it was noted a long time ago that for some chemical bonds the length was shorter than that calculated by summing the covalent radii. In order to take into account this shortening, Schomaker and Stevenson [15] in 1941 suggested the helpful relationship, rAB = r A + r B - CI~A-- XBI where Z = the electronegativity; c = constant. This rule was modified by Blom and Haaland [ 16] in 1985 who used new structural information (114 experimental data) which appeared after the original paper,

166

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN rAB = r A + r B - C I ~ A -- ~ B In

where c = 8.5 pm; n = 1.4. More recently, O'Keefe and Brese [17] showed that bond lengths could be estimated for diverse bond types (ionic, covalent, and metallic) by,

rirj(~r i + ~rj) 2 Rij= ri + rj -

ric i + rjcj

where r is an atomic size parameter (might be considered as the "covalent single bond radius") and c is a parameter which may, or may not, be related to electronegativity. The authors tabulated the best values of r and c for 92 elements fitted by least squares using 883 bond lengths from crystal and molecular structures. Rij can be corrected for variation of bond length with valence by use of the empirical correction,

d o = R i j - b In ~Jij where dij is the corrected value of the bond length; b is a constant = 0.37/k; and aJij is the valence of a bond between two atoms i andj and is defined so that the sum of all valences from a given atom i is:

2'Dij-- Vi. J Therefore, as can be seen from above, there are many ways to assess the bond lengths of common chemical bonds. van der Waals' radii are frequently used in interpretation of crystallographic data when two molecules in a unit cell approach each other. These radii can also be applied when a single molecule adopts such a shape that parts of the molecule come into close contact. The usefulness of 1,3-radii (sometimes called "intramolecular van der Waals' radii"), which were suggested by Bartell [18,19] and Glidewell [20], was demonstrated by Hargittai [21] in discussing the difference between two models found for tetrafluoro-1,3-dithietane by different authors who studied this molecule:

The 1,3 F...F distance in the molecule was reported to be 2.154 + 0.004 and 2.090 /~ (no uncertainty reported), respectively. The fluorine 1,3 radius, as postulated by Bartell [18], is 1.08/~ which implies that the F-..F distance should be 2.16 ]k. This is consistent with the value found for the first model, whereas the 2.09 ,~ seems to

Empirical Correlations

167

Table 1. Empirical Relations Including Internuclear Distances, r Empirical Relation r(C--C) = 1.299 + 0.040n

Equation

References

(1)

[22]

(2)

[23]

(3)

[24]

(4)

[24]

(5)

[25]

(6)

[24]

(7)

[26]

(8)

[27]

(9)

[28]

(10)

[28]

(11)

[28]

(12)

[29a, b]

(13)

[30]

(14)

[31]

(15)

[321

(16)

[33]

Carbon-carbon bonds in aliphatic hydrocarbons (Stoicheff's rule), n = number of bonds adjacent to the C - C bond in question, n = 2-6. rg(C-C) = 1.285 + 0.0533n - 0.0020n 2 An improved version of the previous equation. r(C-C) = 1.079 + 0.058n Carbon-carbon bonds in closo- and nido-carboranes, n = 6,8,9,10. r(C-C) = 1.510 + 0.0020n Exopolyhedral bonds in carboranes. r(C-B) = 1.249 + 0.77n c B - C bonds in the most symmetrical closocarboranes, n c is the coordination number of carbon (n c = 4,5,6). r ( B - B ) = 1.780 + 0.006m B - B bonds in carboranes are classified by types [k, l] where k and I indicate the number of carbon atoms linked to every boron atom; m = k + l = 0,1,2,3,4. E s = -5.74 + 1.56Ar Strain energy (E s) in cycloalkanes is a function of the difference between the standard value. 1.533/~, and the C - C bond in question. A r = 11.533 - ri 103.

K = c(r e - do) -3 Badger's rule. K is the force constant; c and d are constants.

(O= Ho(re - dij) 2/3 tO= frequency (Eqs. 9-11 can be deduced from Badger's rule, Eq. 8). ZeOe = No(r e - dij) 1/2 (Xe(%) = anharmonicity. D e = laij/(r e - dij) 5/2 D e = dissociation energy. is ro(XH) = a - b VXH is Vxa_l = isolated XH stretching frequency. X = C: a = 1.3982: b = 0.0001023. X = Si: a = 1.87229; b = 0.001798. u(CC) = 0.013837 + 0 . 0 2 3 3 9 8 r - 0.000147 2 u = amplitude of vibration. 1.217 < r < 5.618 ,~, (71 points). u(CC) - --0.071855 + 0 . 1 2 4 1 6 2 r - 0.028974 2 1.2086 < r < 1.549 ,g, (57 points). u(CH) = 0.050135 + 0 . 0 2 7 3 6 8 r - 0.001805r 2 1.080 < r < 4.617/~, (74 points). < O - - S - - O = - 1 . 4 7 7 r + 331.7 [r = r ( S = O ) ] . In a large series of XSOzY molecules with increasing electronegativity of X and Y, the S = O bond shortens and the Z O = S = O opens.

(continued)

168

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN

Table 1. (Continued) Empirical Relation = 120.9 - 17.4r(N---)Si)

Equation

References

(17)

[34]

(18)

[351

(19)

[36a]

(20)

[37]

(21)

[38]

(22)

[39]

(23)

[40]

(24)

[41]

(25)

[42]

(26)

[43]

(27)

[10]

(28)

[101

(29)

[44]

(30)

[I0]

(31)

[101

= LN---)Si-Requatorial; 16 compounds of this type.

Ar = -0.0027Aoc Distortion of the benzene ring in 1,3,5-X3C6H 3 derivatives in comparison with the standard values, r = 1.399 ,~ and oc = 120 ~ o~ = / C C C (ipso). r(eC) = 1.6711- 0.00234~ 1,3,5-X3C6H 3 derivatives. X = CH 3, Br, CI, F, NO 2.

Er = Constant E = energy of dissociation. Diatomic hydrides.

Er 2 - a + br; Diatomic and polyatomic molecules.

K r / E =a K = force constant. 148 Diatomic molecules of 18 different families.

E = K(a + br + cr 2) Diatomic molecules, a, b, and c are constants. Z X - M - X = 101.3 + 0.082r x - 0.088r N + ~y X3MY molecules, r x and r M = covalent radii of atoms X and M; )~y = electronegativity of Y. r(CS) = l l . l n 2 + 52.311 + 221.2 n = bond order. r(CS) =-33.4n~ + 185.5 n~ = ~ bond order. log v ( S = O ) = 4.301 log r ( S = O ) + 3.3770 Sulfoxides. v = - 3 6 . 9 5 0 r + 6549.4 Sulfoxides. l o g f ( S e = O ) = -4.551 log r ( S e = O ) + 12.887 f = force constant. E~ = - 0 . 9 1 2 8 r ( S - - O ) + 135.54 XSO2Y molecules. s

= group electronegativity.

I~X = 0.043699r2(S=O) -13.2865r(S=O) + 1012.00 XSO2Y molecules.

(continued)

Empirical Correlations

169

Table 1. (Continued) Empirical Relation r(CC) = 1.5734- 0.3022 cos 2 c~

Equation

References

(32)

[45]

(33)

[46,47]

(34)

[48]

(35)

[49,501

(36)

[51,52]

(37)

[51,52]

(38)

[51,521

(39)

[51,52]

(40)

[53]

(41)

[54]

(42)

[55]

(43)

[56]

(44)

[57]

(45)

[58]

ct = ZC-C-C (13 points). r(CH) = 1.27035 - 0.00146ct c~ = Z H - C - H (cycloalkanes, 9 points). r(NN) = 1.37 - 0.006466x + 0 . ~ 6 2 1 5 x

2

x = out-of-plane amino bend (o) in nitroamines (60 X-ray data).

g = qa/(b + N) N = bond order, R = bond length, a and b = constants characteristic of the bonded atoms. ASi = 0 . 6 6 1 r - 124 Silatranes. r is the Si-N distance (in pm) and ASi is the distance from the plane of the 3 pseudoequatorial oxygens (n = 32 data points; corr. coef. = 0.970). AN = --0.218r + 84 AN is the distance from the plane of the 3 carbons bonded to N (n = 30, corr. coef. = -0.674). d(Si-C) = -0.260d(Si-N) + 246 C refers to the carbon of the R group bonded to Si (n = 24. corr. coef. = -0.731). d(Si-N) = - 3 7 . 5 s

+ 253

a] = inductive constant. d(M-M) = b + c cos o~ M - M is the metal-metal multiple bond distance in dinuclear compounds of the types M2X 8, MeX8L, and M2XsL 2 (M = Cr, Mo, W, Tc, Re, Os, Rh). b corresponds to the expected M-M distance when t~ = 90~ c gives the measure of the M - M bond length to pyramidalization, and ct is the M - M - X bond angle. A = -0.441d + 1.123 A is the distance of C 3 from the C3--C14-O plane, and d is the N4--C3 distance in strychnos alkaloids (4 compounds). r = 1.647 - 0.1841n K r = ( C = O ) bond length and K is the corresponding force constant for 24 transition metal carbonyls.

kr = 180 XxZy - 160 k = the force constant and X = the Pauling electronegativity for 1 1 diatomic alkali halides, XY. kr is in kcal mol- /~- (corr. coef. = 0.9948). r e (C-H) = 1.1133(33)- 0.0001890(174)j(13CH) j(13CH) = the spin-spin coupling constant (19 compounds; Corr. coef = 0.935).

kr = c~ (ro)5; N = bkr; N = bc/(ro) 5 r o = M - M bond distance between group VIB metals; kr = force constant of M - M bond; N = bond order; b and c = constants specific to the bimetal bond. Mo: b = 0.9537 and c = 137.4. Cr: b = 1.633 and c = 45.02.

1 70

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN

be too low. Thus, one result m a y be preferred o v e r the other just on s t e r e o c h e m i c a l grounds. T h e empirical relations listed in Table 1 include internuclear distances, r. In the majority of cases r refers to the distance b e t w e e n atoms c o n n e c t e d by a c h e m i c a l bond; however, in the cases o f Eqs. 13, 15, 17, and 3 6 - 3 9 , r refers to n o n b o n d e d distances. T h e r e are only a few cases, like the B a d g e r rule, w h e r e the particular

Table 2, Graphical Correlations

1

10 11 12

13 14 15 16

17

18

Correlation

References

"Ring centroid-metal-ring centroid" angles as a function of metal radii in decamethylmetallocenes in both the solid and gas phases. Relation between open-chain, cyclic, bicyclic polyhedral compounds. The average C-C distances are given for three families of tetrahedrane, prismane, and cubane (gas phase data). Bond lengths C-I and C--C as a function of coordination number of carbon. The extrapolation of data known from stereochemistry of organic compounds is shown to be inadequate for carboranes with coordination number six. Periodicity of internuclear distances A-H in diatomic hydrides (distances are a function of periodic number ZA) Mutual dependence of A-CI and A-H distances in the corresponding diatomic molecules. Plot of observed deuterium-isolated C-H stretching frequencies versus ab initio calculated C-H bond lengths (linear relation). Calculated (6-31G) C-H bond lengths versus experimental isolated frequencies. Variation of bond lengths P--S and P=O in X3P--S and X3P--O molecules (X = Me, Br, CI, F) versus electronegativity of ligands. Correlation of Si--O bond lengths, and Si-O--C bond angles for acyclic Si-O--C fragments (X-ray crystallographic data). Similar correlation for C-O/Si-O-C is also given, augmented by ab initio calculations. Bond order-bond distance relationship, C-F bonds Plot of N-F force constants versus N-F bond lengths for NE NF2, NF 3, NFO, and F3NO. Ln-O and O-O distances vs. atomic number of Ln in dipivaloylmethane complexes of lanthanides, Ln(dpm)3 (Ln = Sm, Eu, Gd, Tb, Dy and Ho)" dpm = (CH3)3CCOCHCOC(CH3). (Gas phase data.) JBH coupling constant versus B-L distances in BH3L complexes. (L = H-, OH, CN-, CI-, NH 3, and PHF2). N-N bond lengths in sesquibicylic hydrazines are correlated with O~av pyramidality. N-F force constants versus bond lengths in the series MenNF3_n with n = 0-2. Correlation between ~H and r(O.. O) for 108 compounds with a short hydrogen bond (O-H-O or O-D--O) based on X-ray and neutron diffraction data. ~n = proton stretching vibrational frequency. 2.40 < r(O- 9O) < 2.69 ,~ Correlation of nuclear magnetic deshielding constant, & with compression of interstitial atoms in transition-metal clusters, i5(13C) versus r(C), carbides; ~(ISN) versus r(N), nitrides. Linear relation between the C-NH 2 bond length in 6 aromatic amines and the deviation from planarity (pucker) of the amino group (X-ray data).

[59] [60]

[60]

[61] [61] [62] [63] [64]

[65]

[66a1 [66b1 [67]

[68] [69] [70] [71]

[72]

[73]

Empirical Correlations

171

equation is applied to many chemical bonds (see, e.g. Eqs. 20-24, 40, 43, and 45). In the vast majority of cases the relations found pertain only to some particular bond, i.e., C--C, C-H, C-Si, C-B, B-B, N-N, N-+Si, C-S, S--O, or Se=O. There are examples where r is related to certain parameters, but it sometimes acts as a variable. In addition, there are 18 cases where correlation was demonstrated by the authors only in graphical form (Table 2). These include the following distances: C--C, C-H, C-I, A-H and M-CI, P = X (X = O,S), C--O, Si-O, C-F, N-F, N-N, C-N, O-.-O, Ln-O. Although, as mentioned earlier, internuclear distances and bond angles in a given molecule are usually independent, they seem to be correlated in some series of compounds. This relationship is demonstrated by Eqs. 18 and 19 which refer to bond lengths C-C and bond angles CCxC in the following compounds: X

xj t, x Figure 1 shows an updated version ofEq. 19 and also includes electron diffraction data reported for 1,3,5-(CN)3C6H 3 (model b) [36a] and reinvestigated data of

1.402-

'

o1

I

'

'

'

'

I

'

y - 1.690 5 + 0.0024433x

\

1.400

o

'

'

'

I

'

J

|

!

R = 0.95893

2

1.398 ,,< 1.396

o d 1.394

3

1.392 4 ~

1.390 60-

1.388 1.386 118

119

120

121 ZC-C-C, o

122

123

124

Figure 1. Empirical correlation between the C-C bond length in the benzene ring of

1,3,5-X3C6H 3 and the ipso C-Cx-C bond angle. The data points are numbered as follows: (1) X = CH3, (2) CN, (3) Br, (4) CI, (5) NO2, (6) F. (The equation given in the figure includes new data for X = F [36b] and is an updated version of Eq. 19 mentioned in Section II.A. R is the correlation coefficient.)

172

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN Table 3. Analytical Expressionswhich Include Bond Angles

Expression

Equation References

Z X - M - X = 92 + 8~;M + 37(x + ~ Molecules X3MY; )C = electronegativity. / H - C - H = 126.1-0.175 ZC-C--C

(46)

[74]

(47)

[75,76]

51 ~ < ZC--C-C < 117 ~ in cyclic and alicyclic hydrocarbons (CCH2C fragments). It was later applied to compounds having CCX2C, CAX2C fragments (X = F, CI, Br, CH3; A = Si, Ge, Sn). ZC--C--C = 0.836 + 18.282n

(48)

[77]

1J13c_n = 256.05 - 2.09170~ + 0.008202t~ 2 Ij13c_H = coupling constant; o~ = bond angle C - C - C in cyclic hydrocarbons.

(49)

[78]

1J13c_a = 251.715 - 1.963t~ + 0.0073t~ 2

(50)

[26]

Independent version of above equation. ct = 1.816~; + 116.146

(51)

[79]

E s = 111.2-0.895t~1; E s = 100.6- 0.894ct 2 cxI = Z C - C = C , and o~2 = / C - C - C in cycloalkenes; E s = strain energy. v ( C = O ) = 1278 + 6 8 K - 2.2o~ v ( C = O ) = frequency of vibration in unconjugated ketones; K = force constant; tx = ZCCC in C - ( C = O ) - C .

(52)

[26]

(53)

[80]

tXl5 = 0.569~24 + 150.0; o~24= -0.5778~24 + 151.9

(54)

[81]

(55)

[48]

(56)

[76,82,831

(57)

[76]

(58)

[76]

(59)

[84]

A n g l e b e t w e e n two b r i d g i n g c a r b o n atoms in s y m m e t r i c a l bicyclo[m.m. 1]alkanes. n = m + 3 = ring size.

o~ = ipso C - C - C angle in C6HsX (X = H, I, Br, CI, F); Z = electronegativity of X.

F1 2.~

3

13

4 ~

I2

Variation of the axial angle oq5 and the equatorial angle o~24 versus the dihedral angle ~24 as structural distortion for five-coordinate Si(IV). Z C - N - C = 121.312 + 0.376x - 0.02x 2 x is the out-of-plane bend in nitroamines (60 X-ray data).

IP = IP o - nOkzxc Saturated heterocycles of formulas X(CH2) n where X = NH, O, or S, and n = 2,3,4,5. IP = molecular ionization potential; IP o = energy of the electron around the heteroatom if a hypothetical zero degree angle could exist without a dramatic change in orbital contribution. Linear regressions fit of experimental and theoretical data. v(C--O) = 1600.9 - 6.2054ct - 0.04486 ct2 tx is the C - C - C bond angle in cyclic ketones. v ( C = C ) = 1824.82- 1.5289t~ o~ is the C-C--C bond angle in methylenecycloalkanes. ZN-P-Nendo = 0.215~ip + 110.65 is the 31p NMR chemical shift in cyclophosphazenic structures investigated by X-ray crystallography.

(continued)

173

Empirical Correlations Table 3. (Continued) Expression

Equation References

ZkEo_d = 144.250- 9784 Isolated CuC12- chromophore. AEd_d = maximum d-d transition (crn-1); O = average of the two largest trans C1-Cu-CI angles (a measure of the distortion of the CuC1]- chromophore from Ta symmetry). (8 points). Note: the equation given above corrects a typographical error in original paper. AEd_.d = -34110 + 506.480 - 1.345202 Room temperature band maxima. (24 points). See discussion concerning use of limited data to establish a linear relationship. Also, Figures 6 and 7.

(60)

[85]

(61)

[86]

1,3,5-F3C6H 3 [36b]. S o m e other e x a m p l e s o f similar correlations, r/m, can be found in Eqs. 16, 32, 33, and 40.

B. Bond Angles B o n d angles are usually associated with a theory of directional valence of atoms, and together with dihedral angles (see Section II.C below) are mostly responsible for the shape of the molecules. S o m e idea about possible b o n d angles c o m e s f r o m the previous section where 1,3-radii were m e n t i o n e d [18-20]. B o n d angles can s o m e t i m e s be rationalized in terms of the packing of spherical atoms. For instance the t r i g o n a l - p l a n a r h e a v y - a t o m skeleton of N(SiH3) 3 m a y be v i e w e d as conseq u e n c e of a Si...Si contact. In a pyramidal a r r a n g e m e n t which is characteristic for the vast majority o f amines, NX3, the Si...Si distance w o u l d be prohibitively short.

Table 4. Graphical Correlations Involving Bond Angles Correlation 1

Bond angles at sulfur in analogous sulfones (502X2), sulfoxides (SOX2),and sulfides (SX2) (gas phase data). The ipso angle and the mean C-C bond length of the ring in paradisubstituted 1,4-C6H4X2 benzene ring derivatives. X = E C=--N, H, CH 3, SiMe 3 (gas phase data). Variation of ipso angle versus electronegativity of ligand in many benzene derivatives (crystal data). Variation of ipso angle versus electronegativity of ligand in C6H5X derivatives (X = H, I, CI, F); structural parameters derived from direct couplings measured by NMR in liquid crystal solvents. Correlation between the separation of CH2 stretching frequency and the H-C-H bond angle in molecules containing the HzC= group. Relationship between the proton affinity in zeolites and the bond angles around the oxygen atoms.

References [87]

[87] [88] [89] [90] [91]

174

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN

Table 3 list 16 analytical expressions (Eqs. 46--61) which include bond angles. Additionally, there were several other cases mentioned in Section II.A, Eqs. 16-19, 24, 32-34, and 40. Also, six cases of graphical relations involving the bond angles are given in Table 4. Additionally, in Section II.A similar cases may be found in correlations 9, 14, and 18 in Table 2. Recently some empirical relationships involving bond angles were reviewed by Mastryukov and Zefirov [76]. However, the most general way to assess the bond angles is via application of the VSEPR model rules which in spite of being very empirical in nature have recently received a strong support from quantum mechanical calculations [8].

C. Dihedral Angles There is no other part of structural chemistry which is more related to the study of dihedral angles than conformational analysis. Many regularities for dihedral angles holding for particular families of molecules were established by conformational studies carried out during the last decades [92-95]. Table 5 lists eight expressions (Eqs. 62-69) that involve dihedral angles. See also Eq. 54 of Section ll.B. Three additional examples of graphical relations are also included (Table 6). Figures 2 and 3 illustrate the relationship between the dihedral angles and the average interior ring angles of the central ring of thianthrenes and phenoxathiins. Most of the following compounds were studied by one of us using X-ray diffraction methods (the correlations have not been published, thus the equations were not included in Sections II.B or II.C):

//'-,.o ~ The molecules have a folded C2v configuration; the fold angle is defined as the dihedral angle between the planes of the two outer aromatic tings (1 and 2), not including the heteroatoms. As the 1,4-dithiin and 1,4-oxothiin rings flatten, the interior C--C-X and C-X-C angles increase slightly to accommodate the change. In Figure 2 the average C--C-S and C-S--C angles are plotted versus the dihedral angle using data from the crystal structures of: thianthrene and 1-azathianthrene [102]; 2,7- dimethylthianthrene [103], 4-nitro-2-azathianthrene (2 mols/asymmetric unit) [104]; 1,4-diazathianthrene (2 mols/asymmetric unit) [105]; perfluorothianthrene [106], 2,7-dichlorothianthrene [ 107]; 1,4,6,9-tetraazathianthrene [108]; 1,6-dinitro-3,8-bis(trifluoromethyl)thianthrene [109]; (rl6-2-methylthian threne)-(rlS-cyclopentadienyl)iron(II) hexafluorophosphate [110]; (rlS-cyclopen tadienyl)(rl6-thianthrene) iron(H)hexafluorophosphate (2 mols/asymmetric unit) [111]; benz[b]-l,4-diazathianthrene [112]; dibenzo-[b,i]-l,4,6,9-tetraazathianthrene [113]; and 2,3,7,8-tetramethoxythianthrene (3 mols/asymmetric unit) [114].

Empirical Correlations Table 5.

175 Empirical R e l a t i o n s I n c l u d i n g D i h e d r a l A n g l e s

Expression l Jl3c_n = -176.0 + 3.123~ - dihedral angle in bicyclo[n.l.0]alkanes, n = 1-4. ~ f =-5.75 + 7.39n ~ f = dihedral angle in fused four-membered ring of bicyclo[n.2.0]alkanes. n - 1, 2, 4. ~4~ = 69.5 -8.75n ~4~ = dihedral angle in bridged four-member ring of bicyclo[n.l.1]alkanes. n = 1-3. ~5 = 71.83 - 8.02n q)5 = dihedral angle in five-membered ring of bicyclo[n.2.1 ]alkanes. n = 1-3. (~6 -- 7 5 . 2 - 6.4n ~6 = dihedral angle in six-membered ring of bicyclo[n.3.1 ]alkanes, n = 1-3. Vmax =-106A + 16260 Vmax = highest energy d-d transition (cm -1) of CuC12- chromophore; A = dihedral angle. J{In = A + B cosO + C cos20 J{IH is a vicinal proton spin-spin coupling constant, O is the dihedral angle H--C-C'-H'. J = A cosO + B cos20 + C cos30 + D cos220 + W(E cosOE Agi cos~j + FY~Agicos2~i+ G A~i) + H{o~1 +032)/2- 110} + l(rc_C - 1.5) + K Z Ag~ cos 2Vj + Lr -4 + M

Equation

References

(62)

[26]

(63)

[96]

(64)

[96]

(65)

[96]

(66)

[96]

(67)

[97]

(68)

[98]

(69)

[99]

is the group electronegativity; ~ is the dihedral angle H--C-CH; 0)l and or2 are the C = C - H bond angles; rc_c is the C-C bond length; W is the dihedral angle between a coupling proton and the 13-substituent; and r is the distance between a coupled proton and a non-bonded atom within the same molecule in close proximity to this proton.

Table 6.

Graphical Correlations Involving Dihedral Angles

Correlation 1

2

Relationship between Av(OH) values for intramolecular hydrogen bond formation OH--M and the torsion angle for energetically favorable conformations of a-metallocenylcarbinols of the iron subgroup (M = Fe, Ru, Os). (13 points). Relations among different torsion angles, e.g. 13- ~' and 13- ~5showing conformations of various oligo- and polynucleotides (established and proposed). Angle names are arranged in the following order: - C 4, - C 3, - 03, - p - 05 , - C 5, -

3

C 4, -

References [100]

[101]

C 3,

Linear relation between the C-NH 2 bond length in 6 aromatic amines and the dihedral angle between the ring and the NH 2 plane. (X-ray data).

[73]

176

VLADIMIR

130

....

, ....

S. M A S T R Y U K O V

, ....

, ....

and STANLEY H. SIMONSEN

, ....

125

I

--

t

, ....

,'

'''

.----8

_.,....

j~

--~O8o~

._-o---

120 o

~r~ 1 1 5
o C-C-S

"o9 o 110

r-, C - S - C

----

-y

-~

105.39 + 0.11916x

..... y = 84.658

+ 0.12102X

R= 0.9814 R= 0.95348

133

..., -'- ""

105

_ []

t~tr

100

95 .... 120

-

I ~ o

_[] n

......

E l - - -

' .... 130

' ..... 140

' .... 150

' .... 160

' .... 170

' .... 180

190

Dihedral Angle, ~

Figure 2.

Linear relationship

between

the dihedral

angle and interior angles of the

central ring of thianthrenes.

Each molecule of those compounds containing multiple molecules/asymmetric unit was plotted separately. The regression lines are nearly parallel indicating that the C--C-S and C-S-C angles deform equally as the 1,4-dithiino ring flattens, increasing by 0.12 ~ for each 1~ increase in dihedral angle. Figure 3 is a plot of the average C-C-S, C-S-C, C-C-O, and C-O-C angles versus the dihedral angle: phenoxathiin [115]; 1-nitrophenoxathiin (2 mols/asymmetric unit); 9-nitro- 1-azaphenoxathiin (2 mols/asymmetric unit) [116]; 3-azaphenoxathiin [117]; 7-chloro- 1-azaphenoxathiin; 8-chloro-l-azaphenoxathiin [118]; 6,7,9-trimethyl-4-azaphenoxathiin [119]; 1,3-diazaphenoxathiin [120]; 1-N,N-dimethyl-amino-2,3-diazaphenoxathiin [121]; 1,4,9-triazaphenoxathiin [122]; 1c h l o r o - 2 , 3 , 9 - t r i a z a p h e n o x a t h i i n (2 m o l s / a s y m m e t r i c unit) [123]; (rlS-cyclopentadienyl)(rl6-phenoxathiin)iron(II) hexafluorophosphate [124]; (1]5cyclopentadienyl)(rl 6-1-azaphenoxathiin)iron(II) hexafluorophosphate [125]; and benzo[b]-l,4- diazaphenoxathiin (2 mols/asymmetric unit) [102]. The change in the C--C-S angle is very similar to that of the thianthrenes although the magnitude of the change is a little less. The slope of the C-S--C line is about one-half that of the C--C-S line, indicating a smaller change in the C-S-C angle with flattening as compared to the thianthrenes. The slope of the C--C-O line is similar to that of the C-C-S lines of the thianthrenes and phenoxathiins, but the C-O-C line has a steeper slope, indicating that the C-O--C angle is more sensitive to flattening of the

Empirical Correlations 130

....

i ....

125

i ....

i ....

i ....

i ....

i ....

i .... O

. . . . . . . . 0

~

-

"

.....,

120

o

1 77

~

~

~

o

o i g . ~ -

~.. ~ X -

"R-"-

X_

-

ci_o ~

X'-~ X

"" X

~ )

"

-

x--'-x

(D D'}

<= 115

...... y = 107.85 + 0 . 0 9 5 9 2 x R= 0 . 8 6 0 6 6

o C-C-S

c o

---y

o C-S-C

m 110

.

C-C-O

..... y = 9 0 . 0 4 4 + 0 . 0 5 8 6 4 7 x R= 0 . 6 4 1 1 6

x C-O-C

105

- 1 0 6 . 3 9 + 0 . 0 9 9 5 0 5 x R= 0 . 8 5 9 4 8

. . . . y = 9 5 . 6 2 4 + 0 . 1 5 1 1 5 x R= 0 . 7 8 7 0 5

o ~ ~- . . . .

100

~ n _.El ~ ~- ocu,--~

n

O

[]

95 145

150

155

160

165

170

175

180

185

Di hedral Angle, ~

Figure 3.

Linear r e l a t i o n s h i p b e t w e e n the di hedral angle and interior angles of the

central ring of p h e n o x a t h i i n s .

oxathiino ring. The C-C-S and C-S--C angles in phenoxathiins increase by 0.10 ~ and 0.06 ~ respectively; the C-C-O and C-O-C angles increase by 0.10 ~ and 0.15 o, respectively, for each 1o increase in folding angle. Caldwell et al. [126] observed in a series of similar compounds a linear relationship between 13C-NMR chemical shifts and the dihedral angle of five phenoxathiins (phenoxathiin, 7-chloro-l-azaphenoxathiin, 8-chloro-l-azaphenoxathiin, 1-nitrophenoxathiin and phenoxathiin-10-oxide). This relationship may have some basis in theory, as stated in a footnote (17) in their paper: "In effect, plotting 13C-NMR chemical shifts of CI0 a (the carbon atom in the aza-ring that is closest to the sulfur atom) versus the dihedral angle is analogous to correlating electron densities, which have been shown to be reflected linearly in aromatic carbon chemical shifts (Olah, G. A.; Matescu, G. D. J. Amer. Chem. Soc. 1970, 92, 1430) with dihedral angles. Further, since sulfur-containing tricycles have been shown to exhibit changes in dihedral angle as a result of changes in electron density (Advances in Heterocyclic Chemistry; Katritzky, A. R.; Bolton, A. J., Eds.; Academic Press: New York; Vols. 8 and 9), it is not particularly surprising that such changes would conversely be reflected in 13C-NMR chemical shifts." The above data has been updated by including nine additional compounds and a redetermination of the parent compound, phenoxathiin [115], as shown in Figure 4. The new compounds are: 1-nitrophenoxathiin [116,127]; 9-nitro-l-azaphe-

1 78

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN 155 150

' "'

' '''

'"

I'''

' ' ' ' ' '

I ''

' ....

''1

w'''

....

' ' ' ' ' 1 ' ' ' ~

/

/

''''

.

"

/

8,, /

O

'-- 130 E

/

125 120

I ' ' ' '

5a

t~ 140 E a~ t- 135

t'~ r..-

' ' ' ' ' ' ' '

/

:.E 145

O9

c

' I'

- - - y = 12.783 + 0.72668x R= 0.96735

/

/

/

,o

o

o

/ x lb

3a ,9 o"/P-/ ~a "18"

3b 9 9 4b

115 ......... ' ......... ' ......... ' ......... ' ......... 4,......... , ........ 130 140 150 160 170 180 190 200 Dihedral Angle, ~

Figure 4. Linear relationship between ]3C-NMR chemical shift and dihedral angle of phenoxathiins. Point l b (X) is from the original structure determination of phenoxathiin, and l a is the redetermined value. The pairs (3a,3b), (4a,4b), and (5a,5b) represent compounds which contain two molecules per asymmetric unit, each molecule having a different dihedral angle.

noxathiin [116,128]; 1-azaphenoxathiin- 1-oxide [129,130]; 2-aza-phenoxathiin-2oxide [131], 1,3-diazaphenoxathiin [120]; benzo[b]-l,4-diazaphenoxathiin [102]; 1,4,9-triazaphenoxathiin [122]; 7-nitro- 1-azaphenoxathiin- 10,10-dioxide [132]; and 6,7,9-trimethyl-4-azaphenoxathiin [119,133]. The linearity of the plot in Figure 4 is an indication that the conformation in the solid state is almost the same as in solution. Point 1b is from the original structure determination of phenoxathiin, and l a is the redetermined value; the deviation of 1b from the line is an indication that the dihedral angle was in error and emphasizes the importance of empirical correlations. Point 2 is 6,7,9-trimethyl-4-azaphenoxathiin, in which the molecule in the crystal lies in the mirror plane and is thus required by symmetry to be planar. This is a case where the geometry in the solid state is probably different than in solution: packing forces are more important in influencing the dihedral angle than the electron density of the or-carbon (Cl0a). The pairs (3a,3b), (4a,4b), and (5a,5b) represent compounds which contain two molecules per asymmetric unit, each molecule having a different dihedral angle. The conformation of the points on the line probably have conformations similar to the solution conformations. The outliers, 1b, 2, 3b, 4b, and 5b, are not included in the regression analysis.

Iil.

DISCUSSION

There are many ways to classify the equations displayed in Sections II.A-II.C. Consider first the mathematical expressions used, which can be placed into three

Empirical Correlations

1 79

categories: (1) linear, (2) quadratic, and (3) miscellaneous. The total number of equations belonging to each particular group is as follows" 43, 14, and 17, respectively. From this it is quite clear that the linear relationships are by far the most numerous. Linearity is the simplest kind of relationship, implying some additive effects at work. Another feature of linear dependence is that when two such relations have a common variable, i.e., Yl = k l x and Y2 = k2x, a simple transformation yields: Y2 = k z y l / k 1 = (k2/kl)y 1. This means that if Yl and Y2 are linearly related to the same argument they are mutually related linearly. This can be illustrated using Equations 19 (Section II.A) and 51 (Section II.B). They may be rewritten as follows: r = a 1 + blot and ct = a 2 + b2g showing the linear relationship between internuclear distances, r, and bond angles, ct, and between bond angles and electronegativity, ~, respectively. C o m b i n i n g these two relations gives: r - a 3 +b3~, w h e r e a 3 = a 1 + a2b 1 and b 3 = b~b 2. In this case the linearity of the relations involved helps to establish a correlation not previously observed: a linear relationship of internuclear distances versus electronegativities. Ideally, the compounds involved in generating such transformations should be identical, or at least, as in the example cited above, very similar. Further, the ranges of the variables for which the empirical correlations were established should be the same, or nearly so. McKean [134] derived a number of linear relations which include frequency, v, as a variable related to different molecular characteristics: 9 9 9 9 9 9 9

internuclear distance, r bond angle, HCH coupling constant, JCH dissociation energy, D2~ ionization potential, IP force constant,f/j Taft constant, Eta*

Using these kinds of transformations, this list may be further augmented because some of the variables, such as r, tx, JCH, IP, andfu in turn are linearly related to other characteristics (see Sections II.A and II.B). An example of a linear relationship is shown in Figure 5. This is an updated version of Eq. 7 in Section II.A. It relates the strain energy in cycloalkanes, E s, and Ar, the absolute value of the difference between the "strainless" value for the C-C bond length, 1.533/~,, suggested by Bartell [137] and the current value of C - C of every cycloalkane molecule studied: (CH2) n, n = 3-10. Since the publication of the original paper [26] in 1982, cyclononane has been studied [138] and cyclooctane has been reinvestigated [139]. These new data were fitted by least squares to give the new correlation shown in Figure 5. There are also some special cases. For example, Eq. 24 (Section II.A) is a linear relation except for two variables, r x and r M. Equations 32 and 40 (Section II.A) can

180

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN 30

~

,

,

,

---y

,

!

.

.

.

.

i

,

,

,

,

I

'

'

,

= -4.3461 + 1487.7x R= 0.90928

,

25

'

"6 20

/

,

'

'

o

n = 3

/

/

/

/ /

/

0

w

i

n=4 o

.,15

n=9 / n=8

10

/ n~

/

/

i , i

0.005()

9"

~" ~

/

o

n=7

~

/

/

n=5 0

.

I

.

o olo

.

IArl,

.

|

I

o o~;

.

.

l

I

.

o o~o

,

.

.

o025

A

Figure 5.

Linear relationship between strain energy, Es, and the difference, Ar = I1.53 3 - r(C-C) I, the latter being an estimate of the "geometric" strain. The numbering of the points indicates the number of carbon atoms in the cycloalkane, ( C H 2 ) n. This is an updated version of Eq. 7 in Section II.A; see discussion.

be linearized by setting COS2~= X or coso~ = x, respectively. In these cases the authors preferred the correlations r/cos2~ or r/cosct instead of the simple form r/ct. The correlation, in spite of being empirical, may have some theoretical background which influences the reasonable choice of a variable. A word of caution should be added in dealing with the linear relationships: an apparent linearity may be simply a consequence of a relatively narrow range of variables and it may disappear when more data outside of the previous range become available. The danger of using limited sets of data is illustrated nicely in Figures 6 and 7. The average trans-angles of the CuC12- chromophores are plotted versus the d-d transition energies using data compiled by Halverson et al. [86]. Figure 6 uses the 2B 2 --~ 2A 1 transition versus 19, whereas Figure 7 uses the maximum transition energy versus 19. In each figure, the curve calculated by the angular overlap model [86,135,136] is represented by the small, filled squares: Energy (2B 2 ----> 2A1) = 3easin 4 o t - 4%(cos 2 o ~ - 0.5sin2(x) - 2e~e~sin22o~ + 16das(COS2et - 0.5sin2(x) - 13.3edosin 4 O~cos2 00, where (x = (9/2. e<~and err represent the ~- and n-bonding effects of the ligands (5100 and 820 crn-1, respectively); ea~ takes into account the lowering of the dz2 orbital caused by configuration interaction (1450 cm-1); and eds represents the possible effects of mixing of metal d and p orbitals (550 crn-1).

Empirical Correlations 1 89 1 0 4

' ' ' ' ' ' ' ' ' 1 ' ' '

181 '''

'''

I'

----y

....

' ' ' ' 1 ' ' ' ' ~

= -9607

.

....

+ 144.53x

' ' ' ' ' 1 ' ' ' ' ' ' ' ' ' 1

R= 0.98613

1 . 6 104

O

m

m,,,,,"~

m

/

- 9 1.4104

J

' ' ' ~

m o

f

f /

ui < 1.2 10 r

0 m

/

"C 9 o

1 . 0 104

O/ II

~g

o

,,7

9

~

0

8.0103

6.0 10 3 ......... 120

' ......... 130

' ......... 140 Average

' ......... 150

' ......... 160

trans CI-Cu-CI

' ......... 170

' ......... 180

190

Angle, ~

Figure 6. Linear relationship between average trans-CI--Cu--CIangle and AE of the 2B2 ~ 2A1 transition. The filled squares are the values calculated using the angular overlap model [86,135,136]; the dashed line is the least-squares fit.

In cases where there were some deviations from a linear relationship, sometimes the data can be fitted to a quadratic curve. An example of a quadratic relation is shown in Figure 8. It relates the spin-spin coupling constants, 1J13 H and the cC--C-C bond angle in cycloalkanes in a similar way as Eq. 50 of Section ll.B. However, it uses new data for cyclononane [138] and cyclooctane [139]; a new equation was obtained by a least-squares fit. In the miscellaneous category, (3), data are fitted to cubic, parabolic, log, cos, or cos 2 functions. In the last two cases, the choice of cos and cos 2 was suggested by the theory of hybridization. As an example of this category, Figure 9 shows a hyperbolic dependence between dissociation energy, E, and internuclear distances in diatomic hydrides [37] (Eq. 20, Section II.A). Another way of classifying the equations collected in Sections II.A-II.C could be in the nature of the characteristics correlated: 9 spectra (frequency of vibration, force constant, amplitude of vibration, d-d transition) 9 energy (energy of dissociation, strain energy, ionization potential) 9 miscellaneous (spin-spin coupling constant, electronegativity, Taft constant)

182

V L A D I M I R S. M A S T R Y U K O V and STANLEY H. SIMONSEN 1.8104 . . . . . . . . .

, .........

, .........

, .........

, ..........

Y = -34110.9 + 506.485X -1.34516X 2 R = 0.980 1.6 104

1

. ........

1.4 104 - - --'i

%-

E O

td <3

O

1.2 10 4 J

J

f

1

O ~" ~

1.0 104 /

~

8.0 103

/

6.0103 ......... 120

~.o. ~

Y "8*

...;.

, ......... 130

, ......... 140

, ......... 150

, ......... 160

, ......... 170

180

Average trans CI-Cu-CI Angle,"

Figure 7. Quadratic relationship between average trans-CI--Cu-CI angle and AE of the band m a x i m u m . The filled squares are the values calculated using the angular overlap model; the dashed line is the least-squares fit. See Eq. 61.

Still another method of classification could be in the relationship with respect to atomic or molecular properties: 9 atomic (coordination number, electronegativity, atomic radius) 9 molecular (energy of dissociation, strain energy, bond order, ring size in cyclic compounds, group electronegativity, ionization potential, frequency of vibration) The usefulness of atomic characteristics such as electronegativity and atomic radius (together with valence-electron number) has been recently recognized and applied to quasicrystals, ferroelectrics, and superconductors to develop computerized search strategies for the prediction of new materials [140]. In Section II.A we mentioned some empirical relations which exist between formally independent geometrical parameters--bond lengths and bond angles. New relationships can be found in a recently published paper [141]; additionally a review paper [142] contains a complete bibliography and describes ab initio calculations which have been carried out to justify these correlations.

170

.........

, .........

, .........

\

, .........

, .........

\

\

\

150

\

1"

(3

\

erj 1--

\

140

\

n--4

o

130

o .

120

, .........

Y = 255.36-2.0491X+0.0078994X 2 R = 0.996

% n--3

160

, .........

..................................................................... 50 60 70 80 90 100

n--5 ~ n=7,8,9 n=6 T M , 4 ~ e ) n=10o 110

120

C-C-C Angle, Deg.

Figure

8. A parabolic relationship between the spin-spin coupling constants, lj13 ,` . and the C - C - C bond angle in cycloalkanes. The numbering of the points indi(:-ates the number of carbon atoms in the cycloalkane, (CH2) n. This is an updated version of Eq. 5 in Section II.B; see discussion. 140

. . . . . . . . . , .

120

. . . . . . . . , .

. . . . . . . . , . . . . . .

. . . , . . t . . . . , ~ .

\ 9 \

\

I00

\

\

\

80

\

" 60 W .

O ~

40

-.,...

20 0

I

0.500

l

1

*

I

I

i

i

,

I

,

1.00

I

.

9

,

,

,

*

I

I

,

1.50

1

,

l

,

,

!

r, ~,

,

l

I

I

2.00

I

l

l

9

l~

'

'

9

I

I

2.50

i

,

I

l

I

*

,

I

3.00

Figure 9. A hyperbolic relationship between the dissociation energy, E, and internuclear distances, r, for diatomic hydrides, A - H (A = O, S, Si, Au, AI, Bi, Cs). For more details see Eq. 20 of Section II.A. 183

184

VLADIMIR S. MASTRYUKOV and STANLEY H. SIMONSEN

Among relationships including the torsional angle and collected in Section II.C, the most important is the Karplus relation (Eq. 68). It was formulated as an empirical equation, although it was based on some theoretical considerations developed earlier by the same author (see [98] and references therein). Ab initio calculations have been used to show the existence of a Karplus-like dependence of 3Jnn and the dihedral angle in ethane [143], and recently, a Karplus-type dependence was found for some Cd-substituted metalloproteins [144]. Chemists have always been interested in finding correlations between structure and reactivity, and a great deal of literature exists, mostly on the qualitative level. In Section II.B the relation between proton affinity in zeolites and the bond angles (correlation 6, Table 4, [91]) was noted. There is also a linear relationship between C-O bond lengths and reactivity in a series of compounds derived from 2-phenoxytetrahydropyran [145] and a quantitative structure-stability relationship for potassium ion complexation by crown ethers [146]. Many other examples can be found in contributions by Btirgi [147] and Ferretti et al. [148], as well as in a recent book, Structure Correlation [149]. Several extensive data bases containing structural parameters are now available: 1. Cambridge Structural Database, Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge, B2 1E2, U.K. (The 1994 file contains --119,000 organic and metalloorganic compounds, and is updated several times a year). 2. Inorganic Crystal Structure Database, FIZ Karlsruhe, D-76344 EggensteinLeopoldshafen, FRG; or Scientific Information Service, Inc., 7 Woodland Ave., Larchmont, NY 10538, U.S. (Currently the data base contains 35,000 entries and will be updated with approximately 1400 new entries per year.) 3. Metals Data File, National Research Council of Canada, Ottawa, Canada. (Intermetallic compounds). 4. Protein Data Bank, Brookhaven National Laboratories, U.S. These files are very useful for systematic investigations involving similar compounds or for preparing compilations. Allen [150] has given an overview of the information content, software for information retrieval, search strategy, and numerical analysis of the data for the Cambridge Structural Database. Examples of some applications are given by: Btirgi and Dunitz [151] (dynamic aspects of molecular structure), and by Allen et al. (compilation of lengths of bonds in which a number of elements are involved [152a], and in a systematic analysis of structural data in studies of bonding theories, conformational analysis, reaction pathways, and hydrogen bonding [152b]). A similar, but less extensive data base exists to retrieve information about structural data in the gas phase [153]. Structural information found in these files can also be used to verify parameters predicted by empirical relations and to provide data used to formulate new relationships. In conclusion, the search for empirical correlations is not always based solely upon experimental results; sometimes experimental and calculated data are com-

Empirical Correlations

185

bined [146,154]. A special case occurs when "empirical" relationships are based only upon calculated values, as illustrated in the paper of Schleyer et al. [155]. Calculated values of the rotational barrier in ethane-like compounds were plotted versus calculated bond lengths of the central atoms in an attempt to discover: (1) whether a relationship existed, (2) the origin of the single-bond rotation barriers, and (3) the effect on these barriers when heavier group 14 atoms are involved. As mentioned above, our compilation is by no means comprehensive, but we hope it is representative of correlations existing in modern structural chemistry. We also hope that it will encourage the experimentalists either to add new data to previous relationships or to find new relationships. For those who are interested in formulating theoretical models there is a wealth of opportunity to explain empirical correlations.

ACKNOWLEDGMENTS V S M is grateful to Prof. J. E. Boggs for his hospitality and support during his stay in Austin in 1991/92 and 1993/95 and to Professor I. Hargittai for his encouragement and help at an early stage of this project. The authors acknowledge the support of the Robert A. Welch Foundation.

REFERENCES 1. Balmer, J. J. Wied. Ann. 1885, 25, 80. 2. Bohr, N. The Theory of Spectra and Atomic Constitution; Cambridge Press: 1922. 3. Maksic, Z. B. In Theoretical Models of Chemical Bonding. Part I. Atomic Hypotheses and the Concept of Molecular Structure Maksic, Z. B., Ed.; Springer-Verlag: Berlin, 1990, p XVI. 4. Allen, L. C.; Capitani, J. F.; Kolks, G. A.; Sproul, G. D. J. Mol. Struct. 1993, 300, 647. 5. van Arkel, A. E. Molecules and Crystals in Inorganic Chemistry; Interscience: New York, 1941. 6. Ketelaar, J. A. A. Chemical Constitution, An Introduction to the Theory of the Chemical Bond, 2nd ed.; Elsevier: New York, 1958. 7. Vilkov, L. V.; Mastryukov, V. S.; Sadova, N. I. Determination of the Geometrical Structure of Free Molecules; Mir: Moscow, 1983. 8. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular Geometry; Allyn and Bacon: Boston, 1991; Prentice Hall International: London, 1991. 9. Ladd, M. F. C.; Palmer, R. A. Structure Determination by X-Ray Crystallography, 2nd ed,; Plenum Press: New York, 1985. 10. Hargittai, I. The Structure of Volatile Sulfur Compounds; Reidel: Dordrecht, 1985. 11. Hefferlin, R. Periodic Systems and Their Relation to the Systematic Analysis of Molecular Data; The Edwin Mellon Press: Lewiston, NY, 1989. 12. Pauling, L. The Nature ofthe Chemical Bond, 2nd ed.; Cornell University Press: Ithaca, NY, 1948. 13. Slater, J. C. J. Chem. Phys. 1964, 41, 3199. 14. Wells, A. E Structural htorganic Chemistry, Fifth ed.; Claredon Press: Oxford, 1986. 15. Schomaker, V.; Stevenson, D. P. J. Amer. Chem. Soc. 1941, 63, 37. 16. Blom, R.; Haaland, A.J. Mol. Struct. 1985, 128, 21. 17. O'Keefe, M.; Brese, N. E. J. Amer. Chem. Soc. 1991, 113, 3226. 18. Bartell, L. S. J. Chem. Phys. 1960, 32, 827. 19. Bartell, L. S. J. Chem. Educ. 1968, 45, 754.

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58. O'Neill, E M. M. Ph.D. Thesis, University of the Witwatersrand, Johannesburg, South Africa, 1992. 59. Williams, R. A,; Hanusa, T. P.; Huffman, J. C. Organomettalics 1990, 9, 1128. 60. Mastryukov, V. S. Vestn. Mosk. Univ., Ser. 2, Khimiya 1988, 29, 539. 61. Mastryukov, V. S. Zh. Strukt. Khim. 1972, 13, 379; J. Struct. Chem. (USSR) 1972, 13, 359. 62. Snyder, R. G.; Aljibury, A. L.; Strauss, H. L.; Casal, H. L.; Gough, K. M.; Murphy, W. E J. Chem. Phys. 1984, 81, 5352. 63. Aljiburi, A. L.; Snyder, R. G.; Strauss, H. L.; Raghavachari, K.; J. Chem. Phys. 1986, 84, 6872; ibid. 1988, 88, 7257. 64. Jacob, E. J.; Samdal, S. J. Am. Chem. Soc. 1977, 99, 5656. 65. Shambayati, S.; Blake, J. E; Wierschke, S. G.; Jorgensen, W. L.; Schreiber, S. L. J. Am. Chem. Soc. 1990, 112, 697. 66. In Fluorine Contahling Molecules. Structure, React&it),, Synthesis and Applications; Liebman, J. E,Greenberg, A. Eds.; VCH: New York, 1988: (a) Skancke, A. Chapter 3, p. 43. (b) Peters, N. J. S.; Allen, L. C. Chapter 10, p. 199. 67. Shibata, S.; Iijima, K.; lnuzuka, T.; Kimura, S.; Sato, T. J. Mol. Struct. 1986, 144, 351. 68. Ouassas, A.; El Mouhtadi, M; Frange, B. J. Mol. Struct. 1987, 152, 255. 69. Nelsen, S. E; Wang, Y.; Powell, D. R.; Mayashi, R. K. J. Am. Chem. Soc. 1993, 115, 5246. 70. Christen, D.; Gupta, O. D.; Kadel, J.; Kirchmeier, R. L.; Mack, H. G.; Oberhammer, H.; Shreeve, J. M. J. Am. Chem. Soc. 1991, 113, 9131. 71. Sokolov, N. D.; Vener, M. V.; Savelev, V. A. J. Mol. Struct. 1990, 222, 365. 72. Mason, J. J. Am. Chem. Soc. 1991, 113, 24. 73. Chao, M.; Schempp, E. Acta Cryst. 1977, B33, 1557. 74. Konaka, S.; Kimura, M. Bull. Chem. Soc. Jap. 1973, 46, 413. 75. Mastryukov, V. S.; Osina, E. L. J. MoL Struct. 1977, 36, 127. 76. Mastryukov, V. S.; Zefirov, N. S. J. Org. Chem. ( USSR) 1993, 28, 1973. 77. Mastryukov, V. S.; Osina, E. L.; Dorofeeva, O. V.; Popik, M. V.; Vilkov, L. V.; Belikova, N. A. J. Mol. Struct. 1979, 52, 211. 78. Baum, M. W.; Guenzi, A.; Johnson, C. A.; Mislow, K. Tetrahedron Lett. 1982, 23, 31. 79. Brunvoll, J.; Samdal, S,; Thomassen, H.; Vilkov, L. V.; Volden, H. V. Acta Chem. Scand. 1990, 44, 23. 80. Halford, J. O. J. Chem. Phys. 1956, 24, 830. 81. Holmes, R. R. Chem. Rev. 1990, 90, 17. 82. Montero, L. A.; Medina, S. A.; Paneque, A.; Alvarez-Idaboy, J. R.; Mastryukov, V. S. J. MoL Struct. 1991, 249, 305. 83. Pupyshev, V. I.; Mastryukov, V. S. Zh. Fis. Khimii 1992, 66, 1922. 84. Labarre, M.-C.; Labarre, J.-E J. Mol. Struct. 1993, 300, 593. 85. Harlow, R. L.; Wells, W. J.; Watt, G. W.; Simonsen, S. H. Inorg. Chem. 1975, 14, 1768. 86. Halverson, K. E.; Patterson, C.; Willett, R. D. Acta Cryst. 19911,B46, 508. 87. Hargittai, I. Pure and Applied Chem. 1989, 61, 651. 88. Domenicano, A.; Vaciago, A.; Coulson, C. A.Acta Cryst. 1975, B31, 1630. 89. Ugolini, R.; Diehl, P. J. MoL Struct. 1990, 216, 325. 90. Duncan, J. L. Spectrochim. Acta 1970, 26A, 429. 91. Redondo, A.; Hay, P. J.J. Phys. Chem. 1993, 97, 11754. 92. Eliel, E. L.; Allinger, N. L.; Angyal, S. J.; Morrison, G. A. Conformational Analysis; WileyInterscience: New York, 1965. 93. Internal Rotation in Molecules; Orville-Thomas, W. J. Ed.; Wiley: London, 1974. 94. Burkert, U.; Allinger, N. L. Molecular Mechanics, American Chemical Society: Washington, DC, 1982. 95. Stereochemical Applications of Gas-Phase Electron Diffraction, Part B; Hargittai, I.; Hargittai, M., Eds.; VCH: New York, 1988.

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96. Mastryukov, V. S. J. Mol. Struct. 1983, 96, 361. 97. Battaglia, L. P.; Bonamartini, Corradi, A.; Marcotrigiano, G.; Menabue, L.; Pellacani, G. C. Inorg. Chem. 1979, 18, 146. 98. Karplus, M. J. Am. Chem. Soc. 1963, 85, 2870. 99. lmai, K; Osawa, E.; Magn. Reson. Chem. 1990, 28, 668. 100. Shubnina, Ye. S.; Epstein, L. M. J. Mol. Struct. 1992, 265, 367. 101. Nishmura, Y.; Tsuboi, M.; Sato, T.; Aoki, K. J. Mol. Struct. 1986, 146, 123. 102. Larson, S. B.; Simonsen, S. H.; Martin, G. E.; Smith, K.; Puig-Torres, S.Acta Cryst. 1984, C40, 103. 103. Weakley,T. J. R. Cryst. Struct. Comm. 1982, 11,681. 104. Lam, W. W.; Martin, G. E.; Lynch, V. ~ "..; Simonsen, S. H.; Lindsay, C. M.; Smith, K. J. Heterocyclic Chem. 1986, 23, 785. 105. Larson, S. B.; Simonsen, S. H.; Lam, W. W.; Martin, G. E.; Lindsay, C. M.; Smith, K.Acta Cryst. 1985, C41, 1784. 106. Rainville, D.; Zingaro, R. A.; Meyers, E. A. Cryst. Struct. Comm. 1980, 9, 909. 107. Thomas, S. N.; Lynch, V. M.; Simonsen, S. H.; Ternay, A. L. Jr., unpublished data. 108. Lynch, V. M.; Simonsen, S. H.; Davis, B. E.; Martin, G. E.; Musmar, M. J.; Lam, W. W.; Smith, K.Acta Cryst. 1994, C50, 1470. 109. Dahm, D. J.; May, F. L.; D'Amico, J. J.; Tung, C. C.; Fuhrhop, R. W. Cryst. Struct. Comm. 1978, 7, 745. 110. Simonsen, S. H.; Lynch, V. M.; Sutherland, R. G.; Pi6rko, A. J. Organometal. Chem. 1985, 290, 387. 111. Abboud, K. A.; Lynch, V. M., Simonsen, S. H., Pi6rko, A., Sutherland, R. G. Acta Cryst. 1990, C46, 1018. 112. Smith, K.; Lindsay, C. M.; Matthews, I.; Wahlam, W.; Musmar, M. J.; Martin, G. E.; Hoffschwelle, A. E; Lynch, V. M.; Simonsen, S. H. Sulfur Lett. 1992, 15(2), 69. 113. Pignedoli, A.; Peyronel, G.; Antonlini, L.J. Co'st. Mol. Struct. 1977, 7, 173. 114. Hinrichs, W.; Riedel, H.-J.; Klar, G.J. Chem. Res. 1982, (S)334, (M)3501. 115. Kimura, M.; Simonsen, S. H.; Martin, G. E., unpublished data. 116. Hossain, M. B.; Dwiggins, C. A.; van der Helm, D.; Gupta, P. K. S.; Turley, J. C.; Martin, G. E. Acta Cryst. 1982, C38, 881. 117. Caldwell, S. R.; Martin, G. E.; Simonsen, S. H.; Inners, R. R.; Willcott, M. R., II. J. Heterocyclic Chem. 1981, 18, 47. 118. Martin, G. E.; Korp, J. D.; Turley, J. C,; Bernal, l.J. Heterocyclic Chem. 1978, 15, 721. 119. Lynch, V. M.; Simonsen, S. H.; Martin, G. E.; Puig-Torres, S.; Smith, K.Acta Cryst. 1984, C40, 1483. 120. Puig-Torres, S.; Martin, G. E.; Larson, S. B.; Simonsen, S. H. J. Heterocyclic Chem. 1984, 21, 995. 121. Womack, C. H.; Turley, J. C.; Martin, G. E.; Kimura, M.; Simonsen, S. H.J. Heterocyclic Chem. 1981, 18, 1173. 122. Larson, S. B.; Simonsen, S. H.; Martin, G. E.; Smith, K.Acta Cryst. 1985, C41, 1781. 123. Lynch, V. M.; Simonsen, S. H.; Martin, G. E., unpublished data. 124. Lynch, V. M.; Thomas, S. N.; Simonsen, S. H.; Pi6rko, A.; Sutherland, R. G. Acta Cryst. 1986, C42, 1144. 125. Sutherland, R. G.; Pi6rko, A.; Lee, C. C.; Simonsen, S. H.; Lynch, V. M. J. Heterocyclic Chem. 1988, 25, 1911. 126. Caldwell, S. R.; Turley, J. C.; Martin, G. E. J. Heterocyclic Chem. 1980, 17, 1145. 127. Turley, J. C.; Martin, G. E.J. Heterocyclic Chem. 1981, 18, 431. 128. Martin, G. E.; Turley, J. C. J. Heterocyclic Chem. 1978, 15, 609. 129. Martin, G. E. J. Heterocyclic Chem. 1980, 17, 429. 130. Kimura, M.; Simonsen, S. H.; Martin, G. E., unpublished data.

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131. Caldwell, S. R.; Turley, J. C.; Martin, G. E.; Dwiggins, C. A.; Hossain, M. B.; Mendenhall, J. V,; van der Helm, D. J. Heterocyclic Chem. 1984, 21,449. 132. Mehta, M. U.; Seginy, M. O.; Martin, G. E.; Kimura, M.; Simonsen, S. H., unpublished data. 133. Puig-Torres, S.; Martin, G. E.; Smith, K.; Cacvioli, P.; Reiss, J. A. J. Heterocyclic Chem. 1982, 19, 879. 134. McKean, D. C. Croat. Chim. Acta 1988, 61,447. 135. Smith, D. W. Coord. Chem. Rev. 1976, 21, 93. 136. McDonald, R. G.; Riley, M. J.; Hitchman, M. A. Inorg. Chem. 1988, 27, 894. 137. Bartell, L. S. J. Am. Chem. Soc. 1959, 81, 3497. 138. Dorofeeva, O. V.; Mastryukov, V. S.; Allinger, N. L.; Almenningen, A. J. Phys. Chem. 1990, 94, 8044. 139. Dorofeeva, O. V.; Mastryukov, V. S.; Allinger, N. L.; Almenningen, A. J. Phys. Chem. 1985, 89, 252. 140. Rabe, K. M.; Phillips, J. C.; Villars, P.; Brown, I. D. Phys. Rev. 1992, 45, 7650. 141. Mastryukov, V. S.; Palafox, M. A.; Boggs, J. E.J. MoL Struct. 1994, 304, 261. 142. Mastryukov, V. S.; Schaefer, H. E III; Boggs, J. E. Acc. Chem. Res. 1994, 27, 242. 143. Edison, A. S.; Markley, J. L.; Weinhold, E J. Phys. Chem. 1993, 97, 11657. 144. Zerbe, O.; Pountney, D. L.; Philipsborn, W. V.; Va~ak, M. J. Am. Chem. Soc. 1994, 116, 377. 145. Jones, P. G.; Kirby, A. J. J.C.S. Chem. Comm. 1978, 288. 146. Hay, B. P.; Rustad, J. R.; Hostetetler, C. J. J. Am. Chem. Soc. 1993, 115, 11158. 147. Biirgi, H.-B. Angew. Chem Int. Ed. Engl. 1975, 14, 460; Ferretti, V.; Bertolasi, V.; Gilli, P; Gilli, G. J. Phys. Chem. 1993, 97, 13568. 148. Ferretti, V.; Dubler-Steudle, K. C.; BiJrgi, H.-B. In Accurate Molecular Structures; Domenicano, A.; Hargittai, I. Eds.; Oxford University Press: Oxford 1992, Chap. 14. 149. Structure Correlation; BUrgi, H.-B.; Dunitz, J. D. Eds.; VCH: Weinheim-New York, 1994. 150. Ferretti, V.; Dubler-Steudle, K. C.; Allen, E H. In Accurate Molecular Structures; Domenicano, A.; Hargittai, I. Eds.; Oxford University Press: Oxford 1992, Chap. 15. 151. BiJrgi, H.-B.; Dunitz, J. D.Acc. Chem. Res. 1983, 16, 153. 152. (a) Allen E H.; Kennard, O.; Watson, D. G.; Brammer, Orpen, A. G.; Taylor, R. J. Chem. Soc. Perkin Trans. II, 1987, S1. (b) Allen, E H.; Kennard, O.; Taylor, R. Acc. Chem. Res. 1983, 16, 146. 153. Vogt, J. J. Mol. Spectrosc. 1992, 155, 413; Struct. Chem. 1992, 3, 147. 154. Batista de Carvalho, L. A. E.; Teixeira-Dias, J. J. C.; Fausto, R. Struct. Chem. 19911, 1,533. 155. Schleyer, P. v. R.; Kaupp, M.; Hampel, E; Bremer, M.; Mislow, K. J. Am. Chem. Soc. 1992, 114, 6791.

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STRUCTURE DETERMINATION USING THE NMR "INADEQUATE" TECHNIQUE

Du Li and Noel L. Owen

I. II. III. IV. V. VI.

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Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Inadequate E x p e r i m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experimental D e v e l o p m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications o f I N A D E Q U A T E . . . . . . . . . . . . . . . . . . . . . . . . .

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Small Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Natural Products and other Large Molecules . . . . . . . . . . . . . . . Other Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

204 208 208

Future D e v e l o p m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Advances in Molecular Structure Research Volume 2, pages 191-211 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

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DU LI and NOEL L. OWEN

ABSTRACT The INADEQUATE NMR technique is discussed in relation to its potential and usefulness in determining the carbon skeletal structures of organic molecules. A brief introduction to the principles of the method is followed by an account of some of the experimental and computational developments and improvements associated with this technique. The chapter also includes a summary of recent applications of the INADEQUATE method to study the structure of a variety of different compounds.

I. I N T R O D U C T I O N In the arsenal of structural techniques currently available to the organic chemist, few will dispute the preeminent role that nuclear magnetic resonance spectroscopy (NMR) now holds. Its development from an interesting phenomenon discovered by physicists in 1945 [1] to its now dominant structure determination role for the chemist is really quite remarkable. The advent of large superconducting magnets and the development of Fourier transform techniques, have enabled scientists to produce a proliferation of new radio frequency pulse sequences capable of identifying specific intramolecular connectivities. Many of these have acronyms that are now found in the common parlance of organic chemistry (e.g. COSY, DEPT, etc.). NMR spectroscopy is concerned with spectral transitions between nuclear spin states, and most chemists utilize nuclei with spin quantum number 1/2 (i.e. 1H, 13C, I9F, 15N, 3113,etc.), although NMR spectra can be obtained from any nucleus with a nonzero spin quantum number. Isolated nuclei that are commonly used have two possible spin states + 1/2 or-l/2, and an NMR spectral line or transition is the result of excitation of a nucleus from the lower spin state to the upper one. Spectra can be obtained for samples in all three states of matter, although the vast majority of structural work is conducted in solution with compounds in deuterated solvents. The basic parameters that NMR spectroscopy provides include the chemical shift of a magnetic nucleus, and the spin-spin interaction or coupling that occurs between magnetic nuclei which are connected by one, two, three, and sometimes more, bond lengths. In a highly symmetrical molecule such as cyclohexane, the six carbon atoms are in identical magnetic environments (as opposed to the hydrogen atoms that are grouped into axial and equatorial), and so the proton-decoupled 13C NMR spectrum comprises one signal at a frequency (for a particular field strength) which is characteristic of a ~3C nucleus in that precise molecular environment. On the other hand, the carbon atoms in a substituted six-membered ring such as menthol (Figure 1) all resonate at different frequencies when the compound is placed in a strong, homogeneous, magnetic field, because they are all in different magnetic environments. The concept of spin-spin coupling is equally well known, and the patterns for proton coupling for simple molecular systems can easily be predicted. Such patterns are routinely used to unravel proton-proton interactions. When viewing ~3C spec-

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tra, where the signal intensities are much weaker, due to the approximately 1% natural abundance of this nucleus, all protons are usually decoupled using a complex train of pulses. The result of this is that carbon spectra usually comprise single line patterns, and are a great deal simpler than proton spectra. An associated disadvantage of the low natural abundance, however, is that normally no Jcc coupling data is available from the NMR spectra. Similarly, the low natural abundance of the 13C magnetic isotope eliminates the necessity of decoupling carbon nuclei when protons are being viewed. A fairly wide range of more sophisticated experiments (both 1D and 2D) are now available to enable one for example, to correlate protons to protons and protons to carbons. For the natural product chemist who uses NMR and other techniques to help elucidate the structure of a new compound, it is a relatively easy matter to obtain information about specific or isolated fragments of the molecule. For example, infrared (IR) spectroscopy can identify the presence of functional groups such as carbonyls, carboxylate, or alcohols, whereas mass spectrometry (MS) can often provide an accurate value for the molecular mass. In many instances, this latter piece of data will also give the most likely molecular formula. From a quantitative 13C NMR spectrum, it is usually possible to determine the total number of carbon atoms in the molecule, and the number of CH, CH 2, CH 3 groups, and quaternary carbons can be gained from an NMR DEPT experiment. Despite these, and other items of data, fitting all the "jig-saw"pieces together to generate the total molecular structure of a complex molecule is often a very formidable task. In addition, it is often very difficult (and sometimes impossible) to distinguish between two possible isomeric structures using standard NMR methods.

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DU LI and NOEL L. OWEN

What the INADEQUATE method (Incredible Natural Abundance Double Quantum Transfer Experiment) can provide is a means of establishing unequivocally, and often in one experiment, the precise sequence of carbon atoms in the molecular skeleton of a compound. It does this by detecting the spin-spin coupling between adjacent 13C nuclei. If one carbon atom in a molecule can be assigned, for example from its chemical shift, then in principle, all other connected carbons can be subsequently assigned by means of this experiment. The main reason why this technique is a relative latecomer to the NMR spectroscopist's armory of weapons is the very low abundance of adjacent 13C nuclei in natural abundance and the resulting lack of sensitivity associated with the experiment. Since the natural abundance of 13C is 1.1%, the chances of having two adjacent 13C nuclei is approximately 0.01%, or 1 in 10,000. However, the immense potential of the INADEQUATE experiment has been known for over a decade, but its apparent drawbacks and disadvantages have tended to discourage many from utilizing the technique. Of course, the insensitivity problem largely disappears if large amounts of sample are available or if enrichment of 13C is feasible [2], but these alternatives are most often not applicable. The negative features normally associated with this method include its insensitivity, and as a result: (1) the requirement of large amounts of material, or (2) the requirement of large amounts of spectrometer time. Indeed, statements such as the following can be read in many NMR texts: ..this experiment is virtually the ultimate weapon so far as tracing the skeletons of organic molecules is concerned. Unfortunately it is also just about the least sensitive experiment available. [3] The method is very elegant, and the spectracan be interpreted in a straightforwardmanner, but its applications are limited. Since its sensitivityis low, large amounts of sample and measuring time are needed. [4] ..since INADEQUATE needs samples that contain about 100 times more substance than the samples necessaryfor... [5] What we hope to achieve in this chapter is to show that the improvements and the modifications made recently to the INADEQUATE method have succeeded in making this a very important new tool for structure determination, and to review some recent applications of the technique. Indeed, the INADEQUATE pulse sequence is currently offered as a standard package by all major instrument manufacturers.

II. THE INADEQUATE EXPERIMENT The INADEQUATE technique in its one-dimensional form was first proposed by Bax, Freeman, and Kempsell in 1980 [6], and this was followed quickly by a two-dimensional version [7]. The underlying rationale for the experiment is the

INADEQUATE

195

S

,,, o , I P " o

Figure 2. The energy level diagram and possible transitions for an AB or an AX spin system (s - single, d - double, z - zero quantum transitions). discrimination between isolated 13C nuclei and directly coupled 13C nuclei on the basis of the presence or absence of the homonuclear spin-spin coupling Jcc-Any potential complications from an isotopomer with three adjacent 13C nuclei can be conveniently neglected since in general the possibility of this occurrence is no better than 1 molecule in 106. Although a general rule in NMR spectroscopy is that the magnetic quantum number m may change only by one unit, by using several excitation pulses this rule can be circumvented. The pulse sequence used in the INADEQUATE experiment excites a double quantum coherence between coupled ]3C nuclei (i.e. t~, ct --) 13, 13; Figure 2), and eliminates the single quantum coherences through judicious phase-cycling schemes. Isolated 13C spins cannot generate double quantum coherences. Whereas simple NMR pulse sequences can be adequately and conveniently described by vector diagrams, the INADEQUATE experiment cannot be properly described without the use of density matrix methods. However, in common with many chemical techniques, utilization of the experiment to elucidate aspects of molecular structure does not require the practitioner to be conversant with density matrices! It is however important to understand the basis of the method and to appreciate the parameters that control the acquisition of meaningful data. Multiple quantum coherences cannot be directly observed in pulsed FT NMR spectroscopy, but can be observed by the application of a radio frequency pulse to convert double quantum into single quantum coherences which indirectly measure the double quantum density matrix elements. An excellent comprehensive review (with 130 references)of the INADEQUATE experiment was written in 1987 by Buddrus and Bauer [8], wherein they outline the theoretical aspects and include all the published applications on structure determination to that date. In addition, the technique is described in considerable

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DULI and NOEL L. OWEN

detail in other review articles [9-12] and in several texts which deal with two-dimensional NMR spectroscopy [13-15]. Consequently, other than reviewing some of the basic principles, this chapter will not deal with the physics of the experiment, but will concentrate instead on the newer developments and the more recent applications, together with the future potential of this technique.

!11. THE ONE-DIMENSIONAL EXPERIMENT The basic pulse sequence for the 1D INADEQUATE experiment is shown in Figure 3. In the simplest version of the experiment, two FID's (free induction decays) of the four-step pulse sequence are required. The sequences are identical except that their final 90 ~ pulses and the detector have different phases. Coaddition of these FID's results in elimination of the isolated 13C signals and in detection of the double quantum coherences. The crucial variable in this sequence is the parameter, x. By setting a: to be 1/4Jcc and using a series of phases for the final or "read" pulse and detector, the unwanted single quantum coherences are removed. Often a value of about 5.5 ms (Joe = 45 Hz) for a: works adequately for both aromatic and aliphatic compounds. The delay parameter A is usually set at about 10 lus (in the 2D experiment this fixed delay is replaced by an incremented value (t 1) which defines the double quantum domain). In practice, more complex phase cycling is needed to completely remove the single quantum coherences and to overcome phase imperfections, and up to 1024 steps are often used (any multiple of 16 cycles may be utilized). In the 1D experiment the double quantum transitions appear as weak doublets centered around the single quantum frequency; one doublet is seen for a carbon connected to one other ~3C nucleus; a set of two different doublets if the central nucleus is connected to two other carbons, etc. (Figure 4). However, if the 13C/13C coupling constants are very similar, the double quantum signals can sometimes overlap each other, and real problems in peak assignment can occur. These difficulties can largely be overcome through using the 2D procedure. 90Ox

180oy

90~

9000 A

Dl

DI = I - 3 TI

"c = I/(4 Jcc)

Figure 3. The basic pulse sequence for the 1D INADEQUATE experiment [the delay D1 is usually of the order of 1-3 times the largest T1 value].

INADEQUATE

197

'

Jc,.,Cc

=--

Figure 4. The 1 D INADEQUATE spectrum of a carbon (A) coupled to two neighboring carbons (B) and (C).

IV. THE TWO-DIMENSIONAL EXPERIMENT As mentioned above, the only significant change in the pulse sequence for the 2D version as compared with 1D INADEQUATE, is the fact that the delay (A) between the final 90 ~ pulse of the excitation sequence and the conversion pulse is replaced by an incremental time parameter t I (evolution time). By varying t I in such a way \

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Figure 5. The 2D INADEQUATE spectral pattern for a CH-CH2-CH 3 fragment showing coupling between adjacent carbons. (Reprinted with permission from Anal. Chem. 1992, 64, 3133. Copyright 1992 American Chemical Society).

198

DULI and NOEL L. OWEN

that l i t 1 covers all possible double quantum frequencies of the spectrum, the Jcc connectivities can be presented in two dimensions as functions of t I and t 2 (detection time). Fourier transformation results in coherences between coupled 13C nuclei appearing in the double quantum dimension (F 1 or y axis) at the algebraic sum of the chemical shifts of the two nuclei. On the other axis, (F 2 or x axis) is recorded the resonant frequencies of the individual doublet components for each carbon nucleus. Figure 5 illustrates the pattern one would expect for a three carbon fragment (CH-CH2-CH3). For each pair of adjacent (i.e. bonded) carbon atoms a four-line pattern (two doublets, restricted to two small regions) occurs, the center position of which coincides with the average value of the two chemical shifts. Each doublet is centered at the chemical shift frequency of one of the bonded carbon atoms, and the components are found at a frequency va +_Jcc/2, where va is the resonant frequency of carbon "a" and Jcc is the carbon-carbon scalar coupling constant. Since the spectral lines are derived from double quantum coherences, they represent antiphase doublets. In Figure 5 there are only two connectivities, and consequently only two pairs of antiphase doublets. No signals arise from the methyl/methine nuclear interaction, because the coupling is too small to be detected

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with an input Joe parameter corresponding to adjacent carbons. In this instance the trivial problem of sequencing the carbons could be done if, for example, the methyl carbon could be assigned from its chemical shift. Then the INADEQUATE pattern proves that this nucleus is connected to the methylene, which in turn is shown to be linked to the methine group. All sets of doublet pairs in a spectrum must have their centers-of-gravity on the diagonal line of slope 2. Returning briefly to the example of menthol, Figure 6 shows the connectivities established for that compound using the 2D INADEQUATE method. Although, strictly, the acronym INADEQUATE is not applicable to protons, which tend to be in plentiful abundance in nature, the technique of measuring double quantum coherences has been employed to study proton-proton interactions in both AX and AMX first-order spin systems [ 16]. For example, it was successfully used by Boyd and others to help resolve NMR spectral assignments in a mixture of amino acids in hen egg white lysozome [17].

V. EXPERIMENTAL DEVELOPMENTS In view of the enormous potential of the INADEQUATE experiment, much effort has been expended in the last decade or so to improve the sensitivity of the method and to overcome some of the disadvantages associated with the technique. Since signals due to double quantum coherences appear on the double quantum frequency axis as the sum of the coupled resonant frequencies, there are large segments of the data matrix that are void of all signals. This represents a considerable waste of computer memory. Partial rectification of this problem was brought about by using one-half of the frequency necessary to cover the whole range of double quantum coherences, and allowing the spectrum to be "folded" in the single quantum frequency direction. Originally, the lack of quadrature detection meant that the sign of the double quantum frequency could not be determined, but several schemes are now available that utilize quadrature detection [18]. Attempts to improve the sensitivity of the experiment include using combined pulse sequences such as INEPT-INADEQUATE [19] or DEPT-INADEQUATE [20]. In these cases, polarization transfer from protons to 13C nuclei can increase the intensity of the signals slightly, although unfortunately the improvement least affects quaternary carbons, which often are of great interest in the INADEQUATE experiment. Use of an indirect detection method (INSIPID-INADEQUATE) [21] also carries potential of increased sensitivity, but the experiment is not easy to carry out in a routine fashion. Kover et al. [22] have proposed a selective filtering method to improve the sensitivity of the 1D INADEQUATE experiment where only a limited number of connectivities are of interest. Lambert and Buddrus [23] report a method of improving the sensitivity and the signal/noise level (by a factor of 2) of 2D INADEQUATE by considering symmetry and isotopic shifts. Nakai and McDowell [24] describe a refocused INADEQUATE experiment which gives higher resolution for determining coupling constants, and Podkorytov [25] pro-

200

DU LI and NOEL L. OWEN

poses a new seven-pulse sequence for enhanced sensitivity. The inclusion of 2D INADEQUATE data in a computerized program that analyses molecular structure, by Funatsu and others [26], illustrates another way of usefully utilizing 13C-13C connectivities in structure determination. In their contribution to improving the sensitivity and applicability of the INADEQUATE technique, Torres and co-workers [27] address the problem that when a typical value for the coupling constant is entered as the Jcc parameter, sometimes some of the bonds are not detected. One reason for this is that 13C/13Ccoupling constants for different types of C-C bonds vary considerably in magnitude, and consequently the experiment may have to be repeated with a different input coupling parameter in order to detect the missing bond connectivities. Their proposed pulse sequences include both offset and J compensation, with the result that the experiment has a higher sensitivity, and is less sensitive to the input Jet value. A significant development in this area is the computerized approach of Dunkel and co-workers [28,29] whereby, using the program CCBond, INADEQUATE signals can be extracted from background noise where the rms S/N ratio is as low as 2.5 (i.e. the signals lie within the peak-to-peak noise level). This represents a factor of up to 10 improvement over what can be done using the normal visual/graphical analysis. The use of large bytes of computer memory is avoided by searching only those areas where coupling patterns could exist consistent with a pair of resonances in the 1D 13C spectrum (i.e. the rectangle "window" areas in Figure 5). The area to be searched for a Jcc coupling pattern typically comprises about 0.03% of the total data set. For a molecule with n carbon atoms it turns out that the time required to search all potential bond regions is proportional to n 2. Increased sensitivity is gained by examining, by means of a least-squares fitting process, a parameter response surface incorporating the full hypercomplex space (i.e. use is made of the fact that the doublets represent antiphase pairs). With this method, Dunkel and others were able to detect, with a confidence level better than 99%, 21 out of a possible 30 bonds in a 20 micromol sample of cholesterol (i.e. 8 mg) [30]. Seven other couplings corresponding to bonds were detected at a slightly lower confidence level, leaving only two bonds undetected for this sample. Of the two missing bonds, one was a double bond with a significantly different coupling constant (73 as compared with the value of 45 Hz, which was used as the Joe parameter in this work), and the other, a bond between two accidentally degenerate carbons having the same chemical shift. Two carbons forming an A 2 spin system cannot be detected by the INADEQUATE technique, and for an accidental A 2 system, unless a different solvent or a shift reagent is used to separate them, this represents one limitation of the method. However, successful analysis of a 20 micromol sample brings this powerful method into serious consideration for structure determination of relatively small amounts of sample. The experiment is still very time-intensive when compared with other NMR techniques because of the need to accumulate many transients. In addition, the delay time between pulses should be set 1-3 times as long as the longest 13C T 1

INADEQUATE

201

(spin-lattice relaxation time) value of the molecule in order to avoid signal saturation, and quaternary carbon atoms often have long T1 values. One possible method of reducing this pulse delay time is to add a small quantity of a relaxation reagent to the sample [e.g. Cr(acetylacetonate)3]. It has been shown in certain cases that a few milligrams of added shift reagent can decrease Tl values of small organic molecules by a factor of 10, without having a very significant effect on the line widths (i.e. without affecting T2) [31]. This can result in considerable time reduction when conducting an INADEQUATE experiment.

VI. APPLICATIONS OF INADEQUATE Although there are now many different types of problems to which the INADEQUATE experiment has been applied, it is probably in the area of natural products, where the amount of pure material is limited and the molecular complexity is fairly formidable, that the technique has potentially the most to offer. However, a brief review of other applications is included in this section.

A. Small Molecules Where quantity of material is not a limiting factor, then the INADEQUATE experiment can be applied to determine the structure of a compound without too much concern regarding excessive spectrometer time. For example, 20 and 24 resonances, respectively, of abietic and maleopimaric acids were unambiguously assigned by Kruk and others [32]; Kolehmainen and co-workers have used the INADEQUATE method together with other techniques to determine the absolute structure of a dimer ofpinocarvone i33], and Swapna and colleagues used both 1D and 2D INADEQUATE to help determine the correct isomeric structure of a photoadduct derived from 4,4-dimethylcyclohex-2-en- 1-one and acrylonitrile [34]. The method has also been successfully utilized in establishing carbon--carbon connectivities in several antiviral nucleosides [35], and in the unambiguous assignment of all nine possible dithieno[b,d]pyridines [36]. Other compounds studied using 1D or 2D INADEQUATE techniques include" substituted derivatives of adamantanes [37]; benzisoxazoles, some of which had antibacterial activity [38]; naphthalenes [39,40]; polycyclic ketones [41]; and a substituted phosphine [42]. Schraml and co-workers, looking at dicyclohexylphenylphosphine, have shown that when one-bond 13C connectivities cannot completely solve a structure, then a long-range INADEQUATE experiment may provide the requisite data [43].

B. Mixtures The conventional NMR method used for analyzing the component compounds in a mixture is through chemical shift recognition. For example, the chemical shifts of all resonance lines in a 13C spectrum could be read into a computer, and these shifts would then be matched against an appropriate data base of NMR 13C signals.

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Figure7. 13C NMR spectrum of a mixture of quinoline and 3-acetyl pyridine. The marked signals correspond closely to those of pyridine.

203

INADEQUATE

A good computer fit between the experimental spectrum and resonance lines of a certain compound in the data base would lead to the conclusion that a particular substance was present in the mixture. However, there are potential difficulties in such analyses, whether they be carried out by a computer or manually. A mixture of compounds can have lines that coincide (or nearly coincide) with those of a substance that is not actually present, and this can lead to misassignments. Although very little work has been done in this area of mixtures with the INADEQUATE experiment, it has the potential to become a powerful analytical tool. For example, Figure 7 shows the 13C NMR spectrum of a mixture of quinoline and 3-acetyl pyridine. The chemical shift of the three marked resonances are almost the same as those of pyridine (123.5, 135.7, and 149.8 ppm), and the latter would certainly be a candidate component for this mixture. The 2D INADEQUATE spectrum of this mixture is shown in Figure 8 [31]. Most carbon--carbon bond connectivities are clearly shown in this spectrum; the exceptions include two carbon signals, "a" and "g" that lie outside the frequency range shown in the figure, and carbons "3" and "4" where the close proximity of these two nuclei makes it difficult to clearly observe their connectivity on the scale used in Figure 8. The dashed lines illustrate the observed connectivities and hence the carbon skeleton for one of the compo-

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204

DULI and NOEL L. OWEN

nents. The carbon atoms that appeared as though they could have come from pyridine do not connect together, thus ruling out the presence of that substance. This is certainly a potential area of development for the INADEQUATE method.

C. Synthetic Polymers Several research groups have successfully applied the 2D INADEQUATE method to help solve the structures of different polymers. 13C spectra have been assigned and detailed connectivities found between methyl and methine carbons using this technique for copolymers of ethylene-propene [44,45] and for atactic polypropylene [46]. Asakura et al. have studied regio-irregular polypropylene [47] and poly(1-butene) [48], and the presence of specific sequencing units was confirmed in these compounds using the INADEQUATE experiment. The same investigators have also applied the method to examine the tacticity of poly-l-pentene and poly-l-hexene [49]. In their study of cyclopolymerization in a new allyl-ether-substituted acrylate polymer, Thompson and colleagues used the INADEQUATE method to determine the ring size of the new copolymer [50]. They conclusively confirmed that cyclization produced five-membered rings and the ratio of trans to cis ring configuration in the polymer backbone was determined. An epoxy monomer (4-vinylcyclohexene oxide) was shown to be composed of four stereochemical isomers using INADEQUATE by Udagawa and others [51]. Application of the INADEQUATE experiment to the study of poly(vinyl chloride) by Nakayama et al. [52] helped to solve a discrepancy that they had discovered between the relative ratio of methylene tetrad peaks as found from NMR spectra compared with that predicted from Bernoullian propagation statistics. They were able to reassign some of the 13C peaks and resolve the apparent contradiction in relative peak areas.

D. Natural Products and other Large Molecules To date, there are over 20 examples of fairly complex molecules whose structures have been determined with the help of INADEQUATE data or almost exclusively through application of the INADEQUATE technique. Most of the samples contained 13C in natural abundance but involved fairly large quantities of material (i.e. several hundred mg). However, several recent studies (also using natural abundance) have been carried out with considerably less than 100 mg.

Terpenes The structure of a new bis-nor-diterpene (C18H2803) extracted from a Brazilian plant was established by Pinto and colleagues [53], with the help of 13C-13C coupling information (in natural abundance), using both 1D and 2D INADEQUATE. A diterpene hemiacetal (C24H3807) was shown to have the sacculatane skeleton by Tori [54], employing a wide range of NMR experiments including INADEQUATE. They used 110 mg of the material for their analysis. The same

INADEQUATE

205

research group also used similar methods to determine the structure of the oxidation products of dammarane diterpenes [54]. The correct sequence of carbon atoms, as well as their unambiguous assignments was established for three sesquiterpenes using 2D INADEQUATE by Faure et al. [55], who also comment on the usefulness of this technique in routine structure applications for this type of natural product. Patra et al. have used INADEQUATE to reassign 13C signals and establish the correct structure for the triterpene friedelin [56]. In their work on odolactone, Pradhan and colleagues used 2D INADEQUATE data to revise the structure of this compound, and their results have implications for the structure of several other similar compounds [57]. A complete 13C NMR assignment for the sesquiterpene lactone, 11,13-dihydroparthenolide was determined by Parodi and colleagues [58].

Antibiotics In their study of erythronolide-B, Neszmelyi et al. used 1D and 2D INADEQUATE to determine one-bond 13C-13Ccoupling constants and to revise incorrect 13C assignments in the compound [59]. A similar study of a derivative of the antibiotic erythromycin A (azithromycin; 500 mg) by Barber [60] also illustrates the great power and potential of this technique, especially when concentrated samples are available.

Steroids Freeman and colleagues used 2D INADEQUATE to determine whether a (delta 4,6)-diene-3-keto steroid photodimerized in a head-to-tail or in a head-to-head manner simply by establishing the carbon connectivities in the compound [61]. Complete carbon connectivities were also established in 1982 using both 1D and 2D methods for two anthrasteroids by Jacquesy and colleagues [62], and Neszmelyi et al. have determined 13C-13Ccouplings in two other steroids [63]. More recently, Lee used the INADEQUATE technique to establish the 13C assignments in cholestane [64].

Saponins In their analysis of the structure of a new saponin from the Samoan medicinal plant, Alphitonia zizyphoides, Perera and co-workers [65], faced with only 60 mg of material of molecular weight in excess of 1000 daltons, used the computerized INADEQUATE development of Dunkel et al. [29], and they identified 35 of a possible 53 C-C bond connectivities with a statistical confidence level of 99.9%. Another six more bonds were confirmed at the slightly lower level of 99.5 %. Figure 9 illustrates the structure of the saponin, with the bonds determined using the INADEQUATE experiment shown in bold. The numbers shown at each carbon position represent the measured relaxation time (T1) for each nucleus, and it is probably significant that most of those carbons not observed using this technique all have relatively short relaxation times. The structure of another smaller saponin (C15H1609),derived from a Chinese medicinal herb, again with a relatively small

206

DULI and NOEL L. OWEN .31 o,

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Figure 9. The molecular structure of the natural product PT-2, showing (in bold) the bond connectivities established by the INADEQUATE experiment (the numbers correspond to the T] relaxation times for the carbon atoms). quantity (30 mg), was established by Zheng et al. using the INADEQUATE method coupled with computerized analysis [66]. In this case, the problem of using more established techniques to determine the structure was exacerbated by the very large numbers of quaternary carbons and oxygen atoms present in the molecule (Figure 10).

Peptides The total structure of a cytotoxic cyclic peptide ( C 4 9 H 7 7 N O I 3 S ) derived from marine organisms found in the seas around Fiji was established with the help of carbon connectivities deduced from the 2D INADEQUATE experiment on 313 mg

0

/H

O

OH

Figure 1t9. The molecular structure of a Chinese medicinal compound. Bonds shown

in bold were detected by means of the INADEQUATE method; those shown with dashed lines were not detected. (Hydrogens attached to ring carbons are not shown.)

INADEQUATE

207

0

HO

Figure 11. The skeletal structure of the peptide bistramide A showing the CC bonds (in bold) detected using the INADEQUATE method.

of material [67]. The NMR assignment of another complex peptide, bistramide A (C40H68N208) derived from the same marine environment was reinvestigated using INADEQUATE coupled with the computer program CCBond, and all 35 of the carbon-carbon single bond connectivities were found using only 50 mg of material [30, 68] (Figure 11).

Alkaloids The 13Cassignments for harman, a 13-carboline type plant alkaloid, were reinvestigated and the structure confirmed using carbon-carbon bond couplings by Welti, and errors of assignment for this molecule in the literature were corrected [69]. 2D INADEQUATE spectra of an indole alkaloid extract from a Panamanian plant also helped Harper and colleagues in establishing its structure and spectral assignments using only 70 mg of material [70].

Vitamins The connectivity of the carbon skeleton was established and the 13CNMR signals unambiguously assigned for vitamin D 3 using 2D INADEQUATE data by Kruk et al. [71]. The authors used a shift reagent added to a solution of the sample in order to reduce the recycling time and so shorten the spectrometer time.

An intriguing application of the INADEQUATE experiment was conducted by Guittet and co-workers, whereby they studied the 13C connectivities in soluble lignin fractions derived from poplar wood [72]. They helped to overcome the

208

DU Li and NOEL L. OWEN

inherent insensitivity of the method by growing their poplars in a 13C-enriched atmosphere, and thus were able to achieve an 11% enrichment. They were able to confirm the presence of many of the major subunits of lignin (e.g. syringyl, guaiacyl, etc.) and correct some of the 13C assignments in previous literature. The same research workers have continued their 2D INADEQUATE work into unraveling the structure of wood [73].

Coordinated Metal Complexes The presence of several very similar subunits in a compound such as heptamethyl cobyrinate makes unambiguous assignment of all the NMR signals very difficult. Benn and Mynott, in their study of this corrinoid complex, demonstrated the power and potential of the INADEQUATE method in assigning all the 13C signals for this molecule [74].

E. Other Nuclei The preponderance of carbon chains in nature makes it inevitable that the INADEQUATE experiment will be applied primarily to detect carbon-carbon connectivities; however considerable research using this method has been conducted using different magnetic nuclei. Connectivities between 2H-2H and 6Li-6Li, both of which have a spin quantum number of 1, have been observed [75-77]. The presence of 29Si in compounds containing silicon frameworks has also been utilized to great advantage in structure elucidation for various silane compounds; both 29Si-29Si and one-, two- and three-bond 29Si-13C coupling interactions have been measured [78-82]. Similar work has also been done using 183W in tungsten complexes [83].

F. Solid State In light of its potential usefulness, application of the INADEQUATE NMR technique to the solid state was inevitable, and data has been obtained for 13C, 29Si, ll9Sn, and 125Tenuclei in various compounds [84-86]. The biggest challenge in this medium is that line widths for CP-MAS spectra tend to be fairly broad and the smaller coupling between nuclei cannot be observed. It would appear that improved resolution (e.g. faster spinners for homonuclear coupling) will be needed before the method can be of general applicability.

VII.

FUTURE DEVELOPMENTS

This brief review of the literature on the application of the INADEQUATE method over the past decade or so clearly shows that it has been successfully utilized to help resolve structural features of a range of very different compounds. Most of those studied were available in reasonably large quantities, and thus the investigators were able to circumvent the inherent insensitivity of the experiment. However,

INADEQUATE

209

the newer experimental and software developments referred to in this chapter give rise to an optimistic projection that the technique is now becoming increasingly applicable to much smaller quantities of material. Although the very nature of isotopic ratios inevitably condemns the INADEQUATE experiment (carried out on 13C in natural abundance) to be much less sensitive than many other N M R techniques, the inescapable fact that NMR spectrometers will continue to improve their performance and capabilities implies that this method will become more useful and applicable. Although we have concentrated entirely on the INADEQUATE method in this chapter, it should be noted that there are several other (often more sensitive) NMR pulse experiments which, for specific instances, can sometimes provide equivalent structural information. For details of these, the reader is directed to the reviews [8,9] or to the NMR texts that are available. In addition, there is the related and so far, underutilized, experiment of 2D zero quantum coherence (see Figure 2), which offers both some advantages and some disadvantages in relation to the INADEQUATE method [87-89].

ACKNOWLEDGMENTS The authors are grateful to Brigham Young University and the Chemistry Department for encouragement and continual support, and to Professor Robin Harris (Durham University, U.K.) for his comments and advice. We would like to acknowledge the help and collaboration of Professor David Grant and his research colleagues at the University of Utah. One of us (NLO) is grateful to Durham University and its Chemistry Department for their hospitality.

REFERENCES 1. (a) Bloch, E; Hansen, W. W.; Packard, M. Phys. Rev. 1946, 69, 127. (b) Purcell, E. M.; Torrey, H. C.; Pound, R. V. Phys. Rev. 1946, 69, 37. 2. Oh, B. H.; Westler,W. M.; Darba, E; Markley,J. L. Science 1988, 240, 908. 3. Derome, A. E. In Modern NMR Techniques for Chemistry Research; Pergamon Press: Oxford, 1987, p. 234. 4. Schraml,J.; Bellama, J. M. In Two Dimensional NMR Spectroscopy; Wiley: New York. 1988, p. 89. 5. Kintzinger,J. E In Modern NMR Techniques and their Application in Chemistry; Marcel Dekker: New York, 1990, p. 408. 6. Bax, A.; Freeman, R.; Kempsell, S. E J. Am. Chem. Soc. 1980, 102, 4849. 7. Bax, A.; Freeman,R.; Frenkiel, T. A.J. Am. Chem. Soc. 1981, 103, 2102. 8. Buddrus, J.; Bauer, H. Angew. Chem. Int. Ed. Engl. 1987, 26, 625. 9. Benn, R.; Gunther, H. Angew. Chem. Int. Ed. Engl. 1983, 22, 350. 10. Freeman,R. J. Mol. Struct. 1988, 173, 17. 11. Hull, W. E. In Two Dimensional NMR Spectroscopy: Applications for Chemists and Biochemists; Croasman, W. R.; Carlson, R. M. K., Eds.; VCH: New York, 1987, p. 191. 12. Kessler,H.; Gehrke, M.; Griesinger,Ch. Angew. Chem. Int. Ed. Engl. 1988, 27, 490. 13. Schraml,J.; Bellama,J. M. Two Dimensional NMR Spectroscopy; Wiley: New York, 1988.

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14. Chandrakumar, N.; Subramanian, S. Modern Techniques in High Resolution FT-NMR; SpringerVerlag: New York, 1987. 15. Martin, G. E.; Zektzer, A. S. Two Dimensional NMR Methods for Establishing Molecular Connecth, ity; VCH: New York, 1988. 16. Mareci, T. H.; Freeman, R. J. Magn. Reson. 1983, 51,531. 17. Boyd, J.; Dobson, C. M.; Redfield, C. J. Magn. Reson. 1983, 55, 170. 18. (a) Bax, A.; Freeman, R.; Frenkiel, T. A.; Levitt, M. H.J. Magn. Reson. 1981, 43, 478. (b) Mareci, T. H.; Freeman, R. J. Magn. Reson. 1982, 48, 158. 19. Sorensen, O. W.; Freeman, R.; Frenkiel, T. A.; Mareci, T. H.; Schuck, R. J. Magn. Reson. 1982, 46, 180. 20. Sparks, S. W.; Ellis, P. D. J. Magn. Reson. 1985, 62, 1. 21. Keller, P. J.; Vogele, K. E. J. Magn. Reson. 1986, 68, 389. 22. Kover, K. E.; Uhrin, D.; Liptaj, T.; Batta, G. Magn. Reson. in Chem. 1992, 30, 68. 23. Lambert, J.; Buddrus, J. J. Magn. Reson. 1993, 101,307. 24. Nakai, T.; McDowell, C. A. J. Magn. Reson. 1993, 104, 146. 25. Podkorytov, I. S. J. Magn. Reson. 1990, 89, 129. 26. Funatsu, K.; Susuta, Y.; Sasaki, S. J. Chem. Inf. Comput. Sci. 1989, 29, 6. 27. Torres, A. M.; Nakashima, T. T.; McClung, R. E. D.; Muhandiram, D. R. J. Magn. Reson. 1992, 99, 99. 28. Dunkel, R.; Mayne, C. L.; Curtis, J.; Pugmire, R. J.; Grant, D. M. J. Magn. Reson. 1990, 90, 290. 29. Dunkel, R.; Mayne, C. L.; Pugmire, R. J.; Grant, D. M. Anal. Chem. 1992, 64, 3133. 30. Dunkel, R.; Mayne, C. L.; Foster, M. P.; Ireland, C. M.; Li, D.; Owen, N. L.; Pugrnire, R. J.; Grant, D. M.AnaL Chem. 1992, 64, 3150. 31. Li, D. Ph.D. Thesis, Brigham Young University, 1993. 32. Kruk, C.; De Vries, N. K.; Van der Velden, G. Magn. Reson. in Chem. 1990, 28, 443. 33. Kolehmainen, E.; Rissanen, K.; Laihia, K.; Malkavaara, P.; Korvola, J.; Kauppinen, R. J. Chem. Soc., Perkin Trans. 2 1993, 437. 34. Swapna, G. V. T.; Lakshmi, A. B.; Roa, J. M.; Kunwar, A. C. Tetrahedron 1989, 45, 1777. 35. Buchanan, G. W.; Bourque, K. Magn. Resort. in Chem. 1989, 27, 200. 36. Gronowitz, S.; Hoernfeldt, A. B.; Yang, Y.; Edlund, U.; Eliasson, B.; Johnels, D. Magn. Reson. in Chem. 1990, 28, 33. 37. Abdel-Sayed, A. N.; Bauer, L. Tetrahedron Lett. 1985, 26, 2841. 38. Prameela, B.; Rajanarendar, E.; Shoolery, J. N.; Murthy, A. K. Indian J. Chem. 1985, 24B, 1255. 39. Berger, S. Organic Magn. Reson. 1984, 22, 47. 40. Becker, D.; Gottlieb, L.; Loewenthal, H. J. E. Tetrahedron Lett. 1986, 27, 3775. 41. Lambert, J.; Kuhn, H. J.; Buddrus, J. Angew. Chem. 1989, 101,797. 42. Benn, R. Org. Magn. Reson. 1983, 21, 60. 43. Schraml, J.; Blechta, V.; Krahe, E. Collect. Czech. Chem. Commun. 1992, 57, 2005. 44. Hikichi, K.; Hiraoki, T.; lkura, M.; Higuchi, K.; Eguchi, K.; Ohuchi, M. Polym. J. 1987,19, 1317. 45. Hayashi, T.; Inoue, Y.; Chujo, R.; Asakura, T. Polym. J. 1988, 20, 895. 46. Miyatake, T.; Kawai, Y.; Seki, Y.; Kakugo, M.; Hikichi, K. Polym. J. 1989, 21, 809. 47. Asakura, T.; Nakayama, N.; Demura, M.; Asano, A. Macromolecules 1992, 25, 4876. 48. Asakura, T.; Nakayama, N. Polymer 1992, 33, 650. 49. Asakura, T.; Hirano, K.; Demura, M.; Kato, K. MakromoL Chem. 1991, 12, 215. 50. Thompson, R. D.; Jarrett, W. L.; Mathias, L. J. Macromolecules 1992, 25, 6455. 51. Udagawa, A,; Yamamoto, Y.; Inoue, Y.; Chujo, R. Polym. J. 1990, 22, 719. 52. Nakayama, N.; Aoki, A.; Hayashi, T. Macromolecules 1994, 27, 63. 53. Pinto, A. C.; Garcez, W. S.; Hull, W. E.; Neszmelyi, A.; Lukacs, G.J. Chem. Soc., Chem. Commun. 1983, 464. 54. Toil, M. In Studies in Natural Product Chemistry; Rahman, A., Ed.; Elsevier: Amsterdam, 1988, Vol. 2, p. 81.

INADEQUATE 55. 56. 57. 58. 59. 60. 61. 62.

211

Faure, R.; Gaydou, E. M.; Rakotonirainy, O. J. Chem. Soc. Perkin Trans. 2 1987, 341. Patra, A.; Chaudhuri, S. K.; Ruegger, H. J. Indian Chem. Soc. 1990, 67, 394. Pradhan, B. P.; Hassan, A.; Schoolery, J. N. Indian J. Chem. 1990, 29B, 797. Parodi, F. J.; Nauman, M. A.; Fischer, N. H. Spectrosc. Lett. 1987, 20, 445. Neszmelyi, A.; Hull, W. E.; Lukacs, G. Tetrahedron Lett. 1982, 23, 5071. Barber, J. Magn. Reson. in Chem. 1991, 29, 194. Freeman, R.; Frenkiel, T.; Rubin, M. B. J. Am. Chem. Soc. 1982, 104, 5545. Jacquesy, R.; Narbonne, C.; Hull, W. E.; Neszmelyi, A.; Lukacs, G.J. Chem. Soc., Chem. Commun. 1982, 409. 63. Neszmelyi, A.; Hull, W. E.; Lukacs, G.; Voelter, W. Z. Naturforsch. 1986, 41B, 1178. 64. Lee, S. G. BulL Korean Chem. Soc. 1993, 14, 156. 65. Perera, P.; Andersson, R.; Bohlin, L.; Andersson, C.; Li, D.; Owen, N. L.; Dunkel, R.; Mayne, C. L.; Pugmire, R. J.; Grant, D. M.; Cox, P. A. Magn. Reson. in Chem. 1993, 31,472. 66. Li, Z.; Li, D.; Owen, N. L.; Len, Z.; Cao, Z.; Yi, Y. Mag. Reson. in Chem. 1996, in press. 67. Zabriskie, T. M.; Mayne, C. L.; Ireland, C. M. J. Am. Chem. Soc. 1988, 110, 7919. 68. Foster, M. P.; Mayne, C. L.; Dunkel, R.; Pugmire, R. J.; Grant, D. M.; Kornprobst, J-M.; Verbist, J-F.; Biard, J-E; Ireland, C. M.J. Am. Chem. Soc. 1992, 114, 1110. 69. Welti, D. H. Magn. Reson. in Chem. 1985, 23, 872. 70. Harper, J.; Cates, R. E.; Owen, N. L.; Li, D.; Wood, S. G.; Dunkel, R.; Grant, D. M. J. Chem. Soc., Perkin Trans. 1995, in press. 71. Kruk, C.; Jans, A. W. H.; Lugtenburg, J. Magn. Reson. in Chem. 1985, 23, 267. 72. Guittet, E.; LaUemand, J. Y.; Lapierre, C.; Monties, B. Tetrahedron Lett. 1985, 26, 2671. 73. Lapierre, C.; Monties, B.; Guittet, E.; Lallemand, J. Y. Holzforschung 1987, 41, 51. 74. Benn, R.; Mynott, R. Angew. Chem. Int. Ed. EngL 1985, 24, 333. 75. Mons, H. E.; Bergander, K.; Guenther, H. Helv. Chim. Acta 1993, 76, 1216. 76. Eppers, O.; Guenther, H.; Klein, K. D.; Maercker, A. Magn. Reson. in Chem. 1991, 29, 1065. 77. Eppers, O,; Fox, T.; Guenther, H. Helv. Chim. Acta 1992, 75, 883. 78. Past, J.; Puskar, J.; Alla, M.; Lippmaa, E.; Schraml, J. Magn. Reson. in Chem. 1985, 23, 1076. 79. Maxka, J.; Adams, B. R.; West, R. J. Am. Chem. Soc. 1989, 111, 3447. 80. Hengge, E.; Schrank, E J. Organomet. Chem. 1989, 362, 11. 81. Schraml, J.; Past, J.; Puskar, J.; Pehk, T.; Lippmaa, E.; Brezny, R. Collect. Czech. Chem. Commun. 1987, 52, 1985. 82. Kuroda, M.; Kabe, Y.; Hashimoto, M.; Masomune, S.Angew. Chem. hit. Ed. Engl. 1988, 27, 1727. 83. Jon'is, T. L.; Kozik, M.; Casan-Pastor, N.; Domaille, P. J.; Finke, R. G.; Miller, W. K.; Baker, L. C. W. J. Am. Chem. Soc. 1987, 109, 7402. 84. Early, T. A.; John, B. K.; Johnson, L. E J. Magn. Reson. 1987, 75, 134. 85. Benn, R.; Grondey, H.; Brevard, C.; Pagelot, A. J. Chem. Soc., Chem. Commun. 1988, 102. 86. Gay, I. D.; Jones, C. H. W.; Sharma, R. D. J. Magn. Reson. 1991, 91, 186. 87. Muller, L. J. Magn. Resort. 1984, 59, 326. 88. Muller, L.; Pardi, A. J. Am. Chem. Soc. 1985, 107, 3484. 89. Zektzer, A. S.; Martin, G. M. J. Nat. Prod. 1987, 50, 455.

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ENUMERATION OF ISOMERS AND CONFORMERS"

A COMPLETE MATHEMATICAL SOLUTION FOR CONJUGATED POLYENE HYDROCARBONS

Sven J. Cyvin, Jon Brunvoll, Bjorg N. Cyvin, and Egil Brendsdal

Io II.

III.

IV.

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

214

Isomers, Conformers, and Graphs . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

215

A.

Isomers and Conformers . . . . . . . . . . . . . . . . . . . . . . . . . .

215

B.

Graphs

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

217

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

218

Alkanes A.

Constitutional Isomers

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

218

B.

Stereoisomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218

C. Staggered Conformers Alkanoids and Catafusenes

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

220 223

A.

Introduction and Def'mitions . . . . . . . . . . . . . . . . . . . . . . . .

223

B.

Unbranched Catafusenes and Unbranched Alkanoids . . . . . . . . . . .

224

C.

Branched Catafusenes

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

224

D.

Branched Alkanoids

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

226

Advances in Molecular Structure Research Volume 2, pages 213-245 Copyright 9 1996 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-7623-0025-6

213

214

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

V. Conjugated Polyene Hydrocarbons Represented by Polyenoids . . . . . . . . 227 A. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 B. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 C. Enumeration of Unbranched Polyenoids: Elementary Mathematical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 D. Enumeration of Polyenoids: More Advanced Mathematical Solution . . . 233 VI. Nonoverlapping Polyenoids: Computer Programming . . . . . . . . . . . . . 239 VII. Overlapping Polyenoids: Combinatorial Constructions . . . . . . . . . . . . . 240 VIII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

I. I N T R O D U C T I O N A (connected) graph may be defined as a set of vertices and a set of edges, where each edge connects two vertices. This is not the stringent mathematical definition, but is sufficient for our purpose. Some representations of graphs have a striking resemblance to current representations of chemical structures, and it is not surprising that the term chemical graph [1] has been introduced in mathematical chemistry. Applications of graph theory in chemistry have long traditions and have actually been discovered and rediscovered repeatedly [2-4]. In particular, the enumeration of chemical graphs is a broad area [1-10] beginning 120 years ago when Cayley [11] identified certain chemical isomers with "trees" [12], a graph-theoretical concept. The first enumeration of constitutional isomers of alkanes by this celebrated mathematician [13] seems to have inspired several chemists, although the reactions were not always as positive as the following citation [14]' "Wir Chemiker dtirfen es im Gegentheil mit Dank anerkennen, dass ein so ausgezeichneter Mathematiker sich unserer Wissenschaft angenommen hat und wir hoffen, dass er auf dem eingeschlagenen Wege fortfahren und auch kiinftighin seine Th/itigkeit der Behandlung chemischer Fragen zuwenden mOge." The present chapter deals with enumerations of selected chemical graphs. This selection is limited to three classes" C,,H2,,+2 alkanes, CnH(n+6)/2catacondensed polycyclic conjugated hydrocarbons with exclusively benzenoid rings (catafusenes), and finally the CnHn+2 conjugated polyenes. All these classes consist of hydrocarbons which can be represented by special graphs, called trees. In spite of this seemingly limited scope, the present treatment illustrates many fundamental features. In particular, three entirely different approaches to the enumerations are demonstrated: (1) combinatorial constructions, a paper-and-pencil method (without use of computers); (2) exact numerical solutions by computer programming; and (3) algebraic solutions, leading to explicit combinatorial formulas on one hand, and generating functions on the other. The treatment of alkanes starts by revisiting briefly the traditional enumeration of constitutional isomers and its extension to stereoisomers. However, in the realm of structural chemistry, the considerations of staggered alkane conformers which

Isomers and Conformers: Polyenes

215

follow are supposed to be even more interesting. There exists a beautiful one-to-one correspondence between the unbranched staggered conformers of alkanes and the isomers of unbranched catafusenes, for which the enumeration problem has been solved completely. The corresponding problem for branched catafusenes is not simple, but has also been solved by sophisticated graph-theoretical methods. A general solution for the numbers of branched staggered alkane conformers is still an open problem, but it is suggested that it might be solved by the same methods. In fact, some preliminary results towards this end have been obtained, but they are still far from the goal. Instead, an original solution of the enumeration problem for conjugated polyenes (polyenoids) is reported. This is a more manageable problem and well suited for illustrating the sophisticated methods.

il.

ISOMERS, CONFORMERS, AND GRAPHS A. Isomers and Conformers

A short recapitulation of some simple concepts about isomers [15] is needed. Two chemical isomers have the same composition, i.e. the same empirical formula. However, there are such diverse hydrocarbons as, for example, butatriene and cyclobutadiene, which have the same formula, viz. C4H4, but not the same bondings: the carbon skeleton is a linear chain in the former case, a ring in the latter. The two Call 4 molecules under consideration are said to be different constitutional isomers. A constitutional (or structural) isomer is determined by its atoms and the bondings between them. Figure 1 (top row) shows three isomers of C2H2C12. Here 1,2-dichloroethene and 1,1-dichloroethene are undoubtedly different constitutional isomers. However, there also are two configurational isomers of 1,2-dichloroethene, distinguished as cis and trans on chemical grounds because of the lack of internal rotation around the CC double bond. The situation is different in the apparently analogous case of buta-l,4-diene (bottom row of Figure 1) where methylene groups are (formally) substituted for the chlorines. Now the central bond is (formally) a single bond, which allows internal rotation [16]. The two forms, which both are planar, are referred to as conformers (shortly for conformational isomers), distinguished as syn and anti (formerly cis and trans, respectively, and still often designated so even in the most recent literature). Special conformers are sometimes called rotamers (or rotational isomers). Figure 2 (top row) shows the well-known staggered (in contrast to eclipsed) conformers of 1,2-dichloroethane [16] in the Newman projection [15]" anti and gauche. The gauche form displays chirality; the two conformers (see Figure 2) are mirror images which cannot be superimposed on each other. Such a pair of structures is called enantiomers. The conformers of n-butane (bottom row of Figure 2) are quite analogous to those of 1,2-dichloroethane.

216

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL H

Ct

\ /

C=C

H

/ \

Ct

ds-l,2-Dichloroethene

H

\

// CH2

C --C

/ %

H CH2

syn-Buta-l,4--diene

H

C[

\

C=C

/

/ \

Ct

Ct

H

Ct

tran~l,2-Dichloroethene

H

/

C=C

/ \

H

H

1,1-Dichloroethene

//CH2

\

\

*CH2

\

C=CH2

C --C ,? \ CH2 H

/ *CH 2

anti-Buta-l,4---diene

Trimethylenemethane

Figure 1. Configurational isomers of C2H2CI2 dichloroethene and C4H6 conjugated polyenes. Conformers of buta-l,4-diene.

Stereoisomers occur when the spatial arrangements around each atom are taken into account. The classical examples are molecules which possess a carbon atom with four different atoms or groups of atoms attached to it. Such an atom is called chiral (or asymmetric), and it gives rise to chirality and enantiomers. Figure 3 shows an example: 2,3-dimethylpentane. Sometimes the stereoisomers are counted without counting all the distinct conformers. Under this scope there are just two stereoisomers of 2,3-dimethylpentane.

Ct

Ct

Ct

Ct

H

H

._..__._._~,.~_~m~m=.-=~----*,,.~

gauche--1,2-Dichloroethane

antC-'l,2-Dichloroethane

H

CH3 ,

CH3

CH3

H

anti-n-Butane

~

CH3

,

H" "'-'~ "H H gauche-n-Butane

Figure 2. Staggered conformers of C2H4CI2 1,2-dichloroethane and of C4Hlo nbutane.

Isomers and Conformers: Polyenes

217

H

CH3~~CH3 CH3" ~

H

" CHzCH a

Figure 3. 2,3-Dimethylpentane,

C7H16 ,

CH2CH31 H- C-CH(CH3)2 I CH3 which exhibits stereoisomerism.

B. Graphs Only some of the most elementary properties of chemical graphs [1,17] and relevant definitions are needed in this chapter. We shall only be dealing with connected graphs (without multiple lines or loops); simply, vertices connected by edges. Two vertices which are connected by an edge, are said to be adjacent. Two edges which have a vertex in common, are said to be incident. The degree of a vertex is the number of (neighboring) vertices which are adjacent to it. Two graphs are isomorphic (equivalent, identical) if there exists a one-to-one correspondence between their vertices which preserves adjacency. In other words, two isomorphic graphs have the same connectivity, a topological property. A tree is a connected graph with no cycles (closed formations ofedges). Ageneral relation for trees in graph theory is,

m=n-1

(1)

where m is the number of edges and n the number of vertices. In the representation of hydrocarbon graphs the hydrogens are suppressed, as also often done in chemical structural formulas. Thus the vertices and edges of hydrocarbon graphs represent carbon atoms and carbon-carbon bonds, respectively, and it is tacitly assumed that all carbons are saturated by hydrogens if they are not already saturated by neighboring carbon atoms. Vertices of degree one, two, three, and four in a hydrocarbon graph correspond to primary, secondary, tertiary, and quaternary carbon atoms, respectively. It is clear that graphs are perfectly suited for representations of constitutional isomers. Figure 4 shows three versions of a graph which represents 2,2,4trimethylpentane, a constitutional isomer of C8H18 octanes. The counting of constitutional isomers is the same as the counting of nonisomorphic graphs. In other chemical applications it may be of interest to impose certain geometrical restrictions on the representation of a graph. Important examples are trees with angular restrictions: only certain angles are allowed for all pairs of incident edges. Geometrical planarity is another restriction. In order to distinguish this kind of virtually geometrical objects from graphs in the mathematical sense they are sometimes called systems. Several examples are detailed in the subsequent sections.

218

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

]

]

Figure 4. 2,2,4-Trimethylpentane, C8H18, as a chemical graph. The version at the extreme right emphasizes the longest chain of carbons. ili.

ALKANES

A. Constitutional Isomers Every student in organic chemistry is likely to confront the traditional enumeration of alkane isomers: one methane, one ethane, one propane, two butanes, three pentanes, five hexanes, etc., and perhaps even 366,319 C20H42 isomers. This is the enumeration of constitutional alkane isomers; only the bondings between carbon atoms matter or, in the language of graph theory, the connectivity between vertices. Thus, for instance, there is only one constitutional isomer of each normal alkane: n-butane, n-pentane, n-hexane, etc. Figure 5 shows the 9 and 18 constitutional isomers of heptanes and octanes, respectively. Numbers up to n = 15, where n indicates the carbon content, are found in Table 1 with references to original works

[11,13,18,19]. The route to a correct enumeration of the alkane isomers under consideration, apart from the lowest members, has been long and troublesome. It is an interesting story, but has been told so many times in more or less detail [1,4-6,9,10,20] that only a very brief survey will be given. The pioneering contribution by Cayley [11-13] is already mentioned above. Further attempts to solve the problem were made by others during that period [18,21,22], but no satisfactory solution was achieved before the work of Henze and Blair [19] considerably later. These authors devised effective recurrence relations, which also were applied by Perry [23] in order to extend the range of n values. Hereafter, it seems that all further significant progress in the enumerations under consideration was made by means of computers

[10,24-27]. Rouvray [28] considered the ordering of constitutional alkane isomers according to lengths of carbon chains, using just octanes (as in Figure 5, n = 8) for illustration. Ordering according to the longest chain was treated by Balaban et al. [29], who depicted the constitutional alkane isomers for n < 7; see also Figure 5 where the same ordering is employed.

B. Stereoisomers In the counting of alkane constitutional isomers and stereoisomers (see Table 1) the first difference between the two sets of numbers appears at n = 7. The reason is

Isomers and Conformers: Polyenes n=7 . . . . . . .

219

(7) \

\ /-

......

\ ~/ -

-

-- _-(6)

9 . j

@

-~

_- ~_ (5)

(~)

n=8 9

-_

/

:

:

:

:

:

: (8) (7)

-

-- - - / (8)

(5)

Figure 5. The 9 and 18 constitutional isomers of alkanes with n = 7 and 8, respectively. Chiral carbons are marked by asterisks, quaternary carbons by white dots. Numerals in parentheses indicate the lengths of the longest chains in terms of the numbers of carbons.

traced back to the occurrence of two configurations with one chiral carbon atom each (see Figure 5). These configurations give rise to enantiomeric pairs. In consequence, there are seven isomers at n = 7 listed as achiral, while the two remaining configurations are counted double and characterized as chiral. For n = 8 (see again Figure 5 and Table 1), four configurations with one chiral atom each occur, giving rise to eight chiral stereoisomers. In addition, one finds a configuration with two chiral carbons; it gives rise to two chiral and one achiral stereoisomer. Robinson et al. [30] is the main reference for the numbers of alkane stereoisomers for n < 14, as shown in Table 1. The totals therein reproduce the earlier numbers given by Blair and Henze [31]. It is noted that the terms "achiral" and "chiral" [30]

220

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL Table 1. Numbers of Isomers of Aikanes Stereoisomers Constitutional Isomers

A chiral

Chiral

Total

1

1a

1b

2

1a

1b

1c 1c

3

1a

1b

1c

4

2a

2b

2c

5

3a

3b

3c

6

50

5b

7

9d

7b

5c

4b

1 lC 24 c

8

18 d

14 b

l0 b

9

35 d

21 b

34 b

55 c

10

750

40 b

96 b

136 c

11

159 d

61 b

284 b

345 c

12

355 e

118 b

782 b

900 c 2412 c

13

802 e

186 b

2226 b

14

1858 f

365 b

6198 b

6563 c

15

4347 f

t

t

18127 g

Notes: aCayley (1874) [11]; wrong number therein (for n = 6) is omitted. The smallest numbers were o f course known before this publication; we have not attempted to trace the details.

bRobinson et al. (1976) [30]; misprint corrected (4226 ~ 2226). CBlair and Henze (1932) [31]; wrong number therein (for n = 15) is omitted. dCayley (1875) [ 13]; wrong numbers therein are omitted. eHerrmann (1880) [ 18]. fHenze and Blair (1931) [19]. gRead (1976) [6]. tNot given.

used in Table 1 are not synonymous with "nonstereoisomeric" and "stereoisomeric," respectively, as used elsewhere [31]. This point and further complications with stereoisomerism, focusing on the 136 stereoisomers at n = 10, are explained by Whyte and Clugston [32]. Klein [33] has studied sterically "forbidden" constitutional isomers and stereoisomers of alkanes; in the framework of his definitions the enumeration of forbidden isomers starts with a unique system at n = 23. Davies and Freyd [27] have enumerated both constitutional isomers and stereoisomers, and they have considered steric problems with surface hydrogens taken into account.

C. Staggered Conformers Introductory Remarks The staggered conformers (conformational isomers) are supposed to be the kind of alkane isomers of most interest in structural chemistry. Whereas the previous

Isomers and Conformers: Polyenes

221

work on the enumeration of constitutional alkane isomers is very extensive (see above) and on stereoisomers not so extensive, but still substantial, very little work has been done on the enumeration of alkane conformers. In fact, one single paper stands as a pioneering work in this area: Funck [34] achieved a complete mathematical solution for the staggered conformers of unbranched alkanes (for precise definitions, see below). To wit, Funck's work, which did not refer explicitly to graph theory, was ahead of its time and passed unnoticed among graph theoreticians and mathematical chemists who later performed similar enumerations. In fact, Balaban [35] rediscovered Funck's formulas independently after 18 years. With regard to the staggered conformers of branched alkanes, only some rudimentary results are available so far [35,36].

Basic Concepts It is well known that the carbon skeleton of any staggered alkane conformer (in an idealized configuration) can be embedded in the d i a m o n d l a t t i c e m a three-dimensional lattice (or network) of equidistant edges where the angle between any pair of incident edges is tetrahedral. In this lattice, also the carbon skeletons of the

Symmetry C2h / Oooh

Combinatorial Formula

Generating Function

1

"~[1 -- (--1)m]

x

z.

1 -

x2 2

C2v

89 1 + (-l)m]

x 1 -

x2 x

Cs + Ci

1 L(m-Z)/2J 1 (m>2) "~3 --~ _

x5

Ci

G

13L(m-l)/2J

(1 - x2)(1 - 3x2)

~3

1

( 1 - x2)(1 - 3xa) 4 ( 3 m-2 + 1) - 3 (m-3)/2 (m = 3,5,7

Cl

4

Cs (1-2)(1-3x 2)

t

),

0 form = 1, 13 m-2)/2 - 112 (m = 2,4,6 )

x

4

(1 - x)( 1 - 3x)( 1 - 3 2 )

l(3m-2 + 1)- 3(m-3)/2 (m = 3,5,7,..), Total

1 form= 1, 13 m'-2)/2- 1 ]2 (m = 2,4,6,...)

x(1 - 3 x - 2x 2 + 8x 3 - 3x 4)

( 1 - x)(1 - 3x)(1 - 3x2)

Chart 1. Formulas for the numbers of unbranched alkanoids (staggered conformers of n-alkanes).

222

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

Table 2. Numbers of Unbranched Alkanoids (Staggered Conformers of n-Alkanes)* m

Formula

1 2

C2H 6 C3H 8

C2h / D~oht 1 0

C2v .

C2 .

1

Us .

m

Ci

. --

--

--

Total 1 1

3

C4Hlo

1

0

1

--

2

4 5

C5H12 C6H14

0 1

l 0

1 4

1 0

-1

4 10

6

C7H16

0

1

4

4

0

25

7

C8HI8

1

0

13

0

4

70

8

C9H20

0

1

13

13

0

196

9

CloH22

1

0

40

0

13

574

10

CIIH24

0

1

40

40

0

1681

11

C12H26

1

0

121

0

40

5002

12

C13H28

0

1

121

121

0

14884

13

C14H30

1

0

364

0

121

44530

14

C15H32

0

1

364

364

0

133225

15 16 17

C 16H34 C 17H36 C18H38

1 0 1

0 1 0

1093 1093 3280

0 1093 0

364 0 1093

399310 1196836 3589414

18

C19H40

0

1

3280

3280

0

10764961

19

C20H42

1

0

9841

0

3280

32291602

20

C2 iH44

0

1

9841

9841

0

96864964

Notes:

*Fromthe formulas of Funck [34]. tD**h only for m = 1.

"chair" configuration of cyclohexane [37] and of adamantane [38] are recognized. On the contrary, the "boat" configuration of cyclohexane cannot be embedded in the diamond lattice. Here we give a few references to relatively recent works in mathematical chemistry where the diamond lattice is invoked [39-42]. Some of these works [40,42] and references cited therein include several carbon lattices or networks which, in addition to the diamond lattice, have been proposed. An alkanoid system (or simply, alkanoid) is defined as a connected set of edges with no cycle (a tree), which can be embedded in a diamond lattice. An alkanoid is nonoverlapping if no pair of vertices in it overlap; otherwise it is overlapping. An alkanoid (system) is clearly a model of the carbon skeleton of a staggered alkane conformer and represents as such the CnH2n+2 conformer. The vertices in an alkanoid may have the degrees one, two, three and four. An alkanoid (or the corresponding alkane) is said to be an unbranched alkanoid if all its vertices are of degree one or two; it is a branched alkanoid if it has at least one vertex of degree three or four.

Isomers and Conformers: Polyenes

223

Enumeration of Unbranched Alkanoids The above definition assures that the number of nonisomorphic alkanoids is identical to the number of distinct staggered alkane conformers. It is noted in particular that enantiomeric pairs are not counted double. As far as the unbranched systems are concerned, the nonoverlapping and overlapping alkanoids taken together correspond exactly to the conformers considered by Funck [34], which he called spectral isomers. The remarkable formulas of Funck [34] for the numbers of unbranched alkanoids belonging to different symmetry groups and in total are given in Chart 1. They were rendered into a slightly modernized form, where the "floor" function is employed: kx_] is the largest integer smaller than or equal to x. Numerical values are listed in Table 2. Herein a column for the C 1 symmetry group was omitted for the sake of brevity; the missing numbers are easily retrieved by subtractions from the totals. In Chart 1 and Table 2, the number of edges (m) is used as the leading parameter. Generating functions represent a powerful tool in different enumeration problems [6,8,26,30,43,44]; see also the below treatment of polyenoids. The appropriate generating functions in the problem at hand were derived as an alternative to Funck's formulas and are included in Chart 1.

IV. A L K A N O I D S A N D CATAFUSENES A. Introduction and Definitions Balaban [35,36] pointed out an interesting one-to-one correspondence between the unbranched alkanoids and the unbranched catafusenes. An alkanoid (system) is defined above. The systems, called catafusenes [45-47], form a subclass of simply connected polyhexes [1,7,47,48]. A polyhex is a connected system of congruent hexagons such that any two hexagons either share exactly one edge or are disjointed. These systems correspond to polycyclic conjugated hydrocarbons with exclusively six-membered (benzenoid) rings. A simply connected system should not contain any hole. The catafusenes are catacondensed (in contrast to pericondensed), which means by definition that they have no internal vertex, corresponding to an internal carbon atom. An internal vertex is a vertex shared by three hexagons. In conclusion: a catafusene is a catacondensed, simply connected polyhex. At this stage an elucidating example seems to be warranted. It is furnished by Figure 6. In catafusenes, the overlapping systems are allowed (as in alkanoids). A catafusene with overlapping has been called catahelicene [47]. It is often convenient to represent a catafusene as a dualist [1,45,46,49,50], where a vertex represents one hexagon, and a dualist edge is drawn between two vertices whenever the corresponding two hexagons share (exactly) one hexagon edge. The dualist is included in the example of Figure 6.

224

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

Figure 6. A catafusene (/eft) representing a C26H16 polycyclic conjugated hydrocarbon, and its dualist (right).

A dualist D of a catafusene is clearly a tree, where the angle between any two incident edges is either 120 ~ or 180 ~ The vertices of D may have the degrees one, two, and three. If all the vertices of D have the degrees one and two, then D corresponds to an unbranched catafusene; if D has at least one vertex of degree three, then D corresponds to a branched catafusene. The symbol h is used for the number of hexagons in a catafusene, equal to the number of vertices in its dualist.

B. Unbranched Catafusenes and Unbranched Alkanoids Algebraic formulas for the numbers of unbranched catafusenes were derived by Balaban and Harary. [49] Balaban [35,36] pointed out their correlation with the corresponding formulas for unbranched alkanoids (cf. Chart 1). This correlation is clearly seen when the formulas are written in a slightly modernized form, as in Cyvin et al. [46]; then one only has to identify h in the catafusenes with m in the alkanoids. In consequence, also the numbers of unbranched alkanoids (cf. Table 2) apply to the unbranched catafusenes when the symmetry groups are properly correlated. These features are explained in Chart 2.

C. Branched Catafusenes In a pioneering work of fundamental importance, Harary and Read [43] derived the generating function H(x) for the numbers of nonisomorphic catafusenes. Their work has been quoted many times, not at least by Balaban [7,9,35,36,40,45,51,52], who characterized H(x) as a complicated function, in contrast to the combinatorial formulas for unbranched catafusenes [49]. In some of the q u o t a t i o n s [9,35,36,51,52] he associated H(x) with branched catafusenes, which is not strictly correct. We have deduced the generating function Y(x) for branched catafusenes, which is displayed in Chart 3 for the first time. It is admittedly complicated, even more than H(x). In Chart 3, the generating function Z(x) for the numbers of

m

C2h/Oooh*

C2,, - -

C2

Cs

- -

- -

--l 0 4

Total

CI

1

1

1

2

0

1

3

1

0

4

0

1

5

1

0

6

0

1

-1 l 4 4

7

1

0

13

0

8

0

1

13

h

D2h/D6h t

C2v

1

1

--

2

1

--

3

1

4

1

1

5

1

4

6

1

4

7

1

13

8

1

13

1

ci

1

2 1

4

1

4

10

16

25

52

70

13

0 4 0

169

196

C2h

Us

Total

--

--

--

--

1

-1 1 4 4 13

--

2

1

1

4

4

10

6

25

52

70

169

196

*D,,,,h only for m = 1. tD6j ~only for h = 1. Unbranched alkanoids

s

,

m=4: '

,,

. . ....... . .i -~'" .:,, "

:

"v"

~

,

,

- - .... ""

,

i

tx

- ~

,

',--..~. " C

"\ "~c""~--,r-' ' J"\"'

,

,,

,

.... , ",.c ,

l

Unbranched

catafusenes

h = 4:

%v

%h

G

Chart 2. Numbers of n o n i s o m o r p h i c unbranched alkanoids with m edges (top table) and unbranched catafusenes with h hexagons (bottom table). The one-to-one correspondence between unbranched alkanoids and unbranched catafusenes is illustrated for m = h = 4 (incidentally four systems in both cases). 225

226

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL Z(x) =

~ h=l

Z h x h = x(1 - 3 x - 2x 2 + 8x 3 - 3 4 ) (1 - X)(1 - 3x)(l - 3X2)

= X + X2 + 2x 3 + 4X4 + 10X 5 + 25 6 + 70x 7 + 196X 8 + 574 9 + 1681x l~ + ... h=5:

._._Z-"

S oo

r(x) = ~

rhxh = (1 - z~)(1 - 12~2 + 29x4 - 4x5 - lz~6)

/r--4

2x2(1 - x ) ( 1 - 3x)(1 - 3x 2)

+ ~ 1 [(1 _ x)3/2( 1 - 5x)3/2- 3(3 + 5x)(1 _x2)1/2( 1 - 5x2)l /2 24x2 - 4(1 - x3) l/2(1

- 5x3) 1/2]

= x 4 + 2x 5 + 12x 6 + 53x 7 + 250x 8 + l l 1 5 x 9 + 5012x l~ + ... h=6:

dr

~

g-

-

./--

_

,/'-

__

Chart 3. Generating function Z(x) for the Zh nonisomorphic unbranched catafusenes, and Y(x) for the Yh branched catafusenes. The underlined terms in the expansions, viz. 10x s in Z(x) and 12x 6 in Y(x), pertain to the illustrations [46,49], where the appropriate 10 unbranched and 12 branched systems are depicted in the form of dualists.

n o n i s o m o r p h i c u n b r a n c h e d catafusenes is included. T h e H a r a r y - R e a d function [43] is n(x) = Z(x) + Y(x).

D. Branched Aikanoids T h e o n e - t o - o n e correlation between unbranched catafusenes and u n b r a n c h e d alkanoids [35,36] is treated above; see especially Chart 1 where Z(x) is r e c o g n i z e d a m o n g the generating functions. Thus, for instance, there are 10 u n b r a n c h e d alkanoids with m = 5, correlated with the depicted dualists of u n b r a n c h e d catafusenes in Chart 3. Such a correlation does not exist for the branched systems. T h e

Isomers and Conformers: Polyenes

227

smallest numbers of branched alkanoids are known [36]: 1, 3, 14, and 68 for m = 3, 4, 5, and 6, respectively. They are seen to have (almost) nothing to do with the Yh coefficients as given in Chart 3. A mathematical solution for the numbers of branched alkanoids (or alkanoids in total) is not known. It is tempting to ask whether a solution in terms of a generating function can be produced following the same methods as in Harary and Read [43]. We believe that this should be possible, but great complications are to be anticipated. A more manageable case is treated in the remainder of this chapter: the enumeration problem of polyenoids, for which a complete mathematical solution is derived for the first time.

V. CONJUGATED POLYENE HYDROCARBONS

REPRESENTED BY POLYENOIDS A. Introductory Remarks

The homologous series of C,,Hn+2 conjugated polyenes, starting with C2H4 ethene, is of great interest in organic and structural chemistry. A few relevant and relatively recent works in mathematical chemistry are cited here [53-56]. The members for n = 4 are shown in the bottom row of Figure 1 as the three isomers of interest. Some features should be pointed out in connection with this example. First, the C,,H,,+2 "isomers," which are taken into account, are in general a mixture of constitutional and configurational isomers, as well as conformers (conformational isomers). Second, the relevant (chemically unstable) radicals are included.

B. Basic Concepts A hexagonal lattice (also called graphite lattice [53]) is currently employed in connection with polyhexes or benzenoids [57-59]. It consists of congruent regular hexagons in a plane. The angle between any pair of incident edges is 120 ~ A polyenoid system (or simply, polyenoid) is a connected set of edges with no cycle (a tree), which can be embedded in a hexagonal lattice. A polyenoid is nonoverlapping if no pair of vertices in it overlaps; otherwise it is overlapping. The class of nonoverlapping polyenoids coincides with the class of systems considered by Kirby [55]. The members of this class are models of the carbon skeletons of conjugated polyenes and represent C,,H,,+2 molecules and radicals (see above). The vertices in a polyenoid may have the degrees one, two and three (as in a catafusene). Apolyenoid is an unbranched polyenoid if all its vertices are of degrees one or two; it is a branched polyenoid if it has at least one vertex of degree three. Examples are furnished by Figure 7. The number of vertices in a polyenoid is n, while its number of edges shall be identified by the symbol m.

228

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

Figure 7. Polyenoids (two unbranched and one branched) embedded in the hexagonal lattice. They correspond to the C4H6conjugated polyenes shown in the bottom row of Figure 1.

Symmetry plays an important role in the present work. A nonoverlapping polyenoid with m > 1 can obviously belong to one of the six symmetry groups D3h, C3h, D2h, C2h, C2v or Cs. The symmetry group D~h is unique for m = 1 and reflects the symmetry of the skeleton of ethene. We shall also assign the mentioned six symmetry groups to the overlapping polyenoids, disregarding the distortions which would be needed in order to realize the corresponding chemical compounds. Thus, for instance, all the coiled polyenoids, which can be embedded in one hexagon, are attributed to C2v. An illustration for m = 8 (C9HI1) is given in Figure 8. Two types of C2v systems are distinguished and identified by the symbols C2~(a) and C2~(b). Apolyenoid belongs to C2~(a) when its (unique) twofold symmetry axis (C 2) passes through a vertex; a polyenoid belongs to C2v(b) when C 2 bisects perpendicularly an edge. The coiled system in Figure 8, for instance, is of the type C2v(a), as in any coiled CnHn+2 polyenoid where n = 3, 5, 7 . . . . . The coiled C,,H,,§ polyenoids for n = 4, 6, 8 . . . . belong to C2v(b). The three systems in Figure 7 belong to (from the left): C2v(b), C2h, D3h.

Figure 8. The polyenoid corresponding to the coiled C9Hll conjugated polyene; in the right-hand drawing the C2v symmetry is accentuated.

Isomers and Conformers: Polyenes

229

C. Enumeration of Unbranched Polyenoids: Elementary Mathematical Solutions Preliminaries The smallest systems of polyenoids can be generated and enumerated by systematic drawings. This is the case for systems of any class, and in fact the way any enumeration usually starts. The six unbranched polyenoids with m = 5, for instance, are found in some of the works cited above [54,56]. In Figure 9 the smallest polyenoids are depicted, both unbranched and branched. For m > 1, the unbranched polyenoids are identical with the dualists of special unbranched catafusenes, called fibonacenes [46,60]. The combinatorial formulas for the numbers of nonisomorphic fibonacenes have been derived [60] and are reviewed elsewhere [46]. The corresponding derivation, adapted to unbranched polyenoids, is treated below. Furthermore, the appropriate generating functions are developed for the first time. The following exposition is worked out as an elementary course in enumerations within the two selected techniques. For the sake of clarity, it is emphasized that in the below algebraic solutions the nonoverlapping and overlapping systems among unbranched polyenoids are counted together.

Combinatorial Formulas An enumeration method where the symmetry is exploited is called "stupid sheep counting" [61]. Here "stupid" refers to the counting, not to the sheep. This name was coined under reference to Redelmeier [62], who says: "There is a well-known way to count cattle in a herd: count the number of legs and divide by four." In the application of stupid sheep counting to unbranched polyenoids, the "crude total" (corresponding to the number of sheep legs) is, Jm =

2m-1

(2)

The background of this simple formula is: start with one edge (m = 1) and add one edge at a time in each of the two allowed directions to an end of it, then add one edge in each allowed direction to the last added edge of the m = 2 systems, and continue in the same way. The number of systems is doubled in every round. The process generates successively all the unbranched polyenoids, but the systems are generated more than once--actually a number of times depending on symmetry. Specifically, the C2h and C2v systems are all generated twice, while each C~ system is generated four times. Finally, the m = 1 system, which is the only one of symmetry D.~h, is generated once as the initial system. Other symmetries are not possible for unbranched polyenoids. In summary, one obtains, Jm - Dm "t" 2 C m + 2 M m -t- 4 A m

(3)

m=l

/n,=2

/

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

l[

m=3

~ /

/

;\

\

/

,

~,-4

0

m=5

/

"~__

C,,-

/

A__ ,__,,-

v

\

/ Figure 9. All nonisomorphic polyenoids with m < 5.

/

\

Isomers and Conformers: Polyenes

231

where Din, Cm, Mm, and A m indicate the numbers of systems with symmetries

D.oh, C2h, C2v, and C s, respectively. The I m total number of nonisomorphic systems (corresponding to the number of unbranched polyene "isomers") is of course:

I m = D m+ Cm + Mm + Am

(4)

On eliminating A m from Eqs. 3 and 4, one obtains:

1 Im=-4(Jm + 3Din + 2Cm + 2gm)

(5)

Notice the division by four as in the sheep counting. In order to determine I m, we need D m, C m, and M m, in addition to "In which is given by Eq. 2. Here D m is trivial" D 1 = 1, while D m = 0 for m > 1. In order to determine C m and M m, it should be observed that from each system of symmetry C2h or C2v with m edges, two systems with the same symmetry and m + 2 edges can be generated; one edge is added appropriately to each of the ends. In this way, only nonisomorphic systems are generated, and no systems of the C2h and C2v symmetries will be missed if m is increased successively. In summary, the recurrence relations,

Cm+2-- 2Cm, gm+2 = 2M m

(6)

are valid. However, the initial conditions for the two symmetry groups are different. One has C e = 0, C~ = 1, but M 2 = Mz = 1. Now the combinatorial (explicit) formulas for C m and M m are readily obtained with the results listed in Chart 4 for Ceh and Cev, respectively. Consequently, also the total (In) is accessible as explained above, and finally A m for the C~ symmetry can be deduced from the same formula apparatus. Numerical values from the above analysis are entered in Table 3.

Generating Functions The basic properties of the kind of generating functions which are employed in the present work, are explained in Chart 3. Take, for instance, Z(x). Here the expansion of Z(x) around x = 0 converges for Ixl < 1. Otherwise it is of no interest to assign specific values to the variable x. It is the very expansion which we concentrate upon; it reproduces the numbers Z h as coefficients for the appropriate powers of x. Such an expansion is sometimes called counting series. The counting series in the following is a geometrical series: oo

S(x) = (1 - 2x)-1 = ~ 2 i x / = 1 + 2x + 4x 2 + 8X3 + 16X4 + . . . i=0

Herefrom one obtains readily:

(7)

Symmetry Do.h

Combinatorial Formula

Generating Function

1 for m = 1

x

C2h

[,~(m-3)/2 . ~-tm = 3,5,7 .... ) 10 for m = 1,2,4,6 ....

3 x 1 - 2x 2

Czv

[ 0 for m = 1

Cs

[ 0 for m = 1,2

x2(1 + x)

2L(m-2)/zj (m > 2)

1 - 2x 2

2m-3 _ 2L(m-3)/2J (m > 3)

x4 (1 - 2x)(1 - 2x 2)

2 m-3 + 2L(,n-3)/2J (m > 3)

Total

x(1 - x -

( I - 2x)(~-2xz)

1 f o r m = 1,2

Formulasfor the numbers of unbranched polyenoids.

Chart 4.

Table 3. m

2x 2 + x 3)

Formula

Number of Unbranched Polyenoids* C2h/D.oht

C2v --

Cs

Total

1

C2H 4

1

2

C3H 5

0

1

--

1

3

C4H 6

1

1

--

2

4

C5H 7

0

2

1

5

C6H 8

2

2

2

6

6

C7H 9

0

4

6

10

7

CsHI0

4

4

12

20

8

C9HII

0

8

28

36

9

CIoH12

8

8

56

72

10

CllH13

0

16

120

136

11

C12H14

16

16

240

272

12

C13HI5

0

32

496

528

13

CI4HI6

32

32

992

1056

14

ClsHl7

0

64

2016

2080

15

C16Hl8

64

64

4032

4160

16

C17H19

0

128

8128

8256

17

CIsH2o

128

128

16256

16512

18

C19H2]

0

256

32640

32896

19

C20H22

256

256

65280

65792

20

C21H23

0

512

130816

131328

Notes: *Fromformulas of Balaban [60]; see also the text. *D**h only for m = 1.

232

--

1

3

Isomers and Conformers: Polyenes

233

oo

J(x) = xS(x) "- Z 2m-lxm =

x + 2x 2 +

4x 3 + 8x4 + 16x5 + . . .

(8)

m=l

This is the generating function for the crude total in Eq. 2: oo

J(x) = Z Jmxm -" x ( 1

- 2 x ) -1

(9)

m=l

The generating function for D m is trivial" D (x )= x. The generating functions for C m and M m, viz. C(x) and M(x), respectively, are obtained, as well as J(x), by means of the auxiliary function S(x) of Eq. 7. First, one has, S(x2) = (1 - 2x2)-1 = 1 + 2 ~ + 4x4 + 8x6 + 16x8 + . . .

(10)

and consequently: C(x) = x 3 S(x2), M(x) = x2(1 + x) S(x2)

(11)

The explicit forms of these functions are found in Chart 4. Relations analogous to those of Eqs. 3 and 4 hold also for the appropriate generating functions. Therefore, one has in analogy to Eq. 5: l(x) = l[j(x) + 3D(x) + 2C(x) + 2M(x)]

(12)

Herefrom the generating function l(x) for the total numbers of unbranched polyenoids was deduced as given in Chart 4, and finally the function for C s was determined.

D. Enumeration of Polyenoids: More Advanced Mathematical Solution

Introductory Remarks The enumeration problem was solved for polyenoids, and the solution is presented below in terms of generating functions. The mathematical methods which were employed basically follow Harary and Read [43] in their enumeration of catafusenes. The Redfield-P61ya theorem [8] is implied, but has not been invoked explicitly. Chemical graphs representing planted, rooted, and free (unrooted) trees are introduced. In contrast to Harary and Read [43], all kinds of symmetry were allowed already in the enumeration of rooted trees. Nevertheless, the method of Otter [63] for passing from rooted to unrooted trees could be adapted. Thus the treatment became more surveyable, and the somewhat cumbersome considerations of the mirror symmetry for unrooted trees was avoided. In the below exposition the developments are outlined without going into details.

234

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

m-I Um+l= 2Urn+ E UiUm-i(m

> 1), U 1 = 1, U 2 = 2

i=1 oo

U(x)= E Umxm=--~[1- 2x- (I - 4x) 1/2] m=l

= x + 2x 2 + 5x 3 + 14x 4 + 4 2 x 5 + 132x 6 + 4 2 9 x 7 + 1430x 8 + 4 8 6 2 x 9 + 16796x l~ + ...

\/ eo

n(x) = ~

~ =

= ~33 1 -

x2 -

2x3)

m=l

1 [(1 -x)(1 -4x) 1/2 + 3(1 +x)(1 -4x2) 1/2 + 2(1 -4x3) 1/2] 12x 3 = x + x 2 + 6x 3 + 1 6 r 4 + 52x 5 + 170x 6 + 5 7 9 x 7 + 1996x 8 + 7 0 2 1 x 9 + 2 4 8 9 2 x 1~ + ...

(continued)

ChartS. From the top: formulas for the Um numbers of planted polyenoids, including the generating function U(x); generating function A(x) for the Am polyenoids rooted at a vertex (atom-rooted); generating function ~x) for the Bm polyenoids rooted at an edge (bond-rooted). The U4 = 14 planted, A4 = 16 atom-rooted and B4 = 12 bond-rooted polyenoids are depicted.

Isomers and Conformers: Polyenes

235

m=l

=x +x 2 + 5x3 + 12x4 +45.5

+ 143x6 + 511x7 + 1768xs + 6330x9 + 22610xl~+ ...

/ - ._/-'x

\ .../

Y Chart 5. (Continued)

For the sake of clarity, it is emphasized that the unbranched and branched polyenoids taken together are counted in the following. Furthermore, both nonoverlapping and overlapping systems are included.

Planted and Rooted Polyenoids In a r o o t e d polyenoid, at least one vertex is distinguished as not being equivalent with anyone of the other vertices. If this unique vertex has the degree one, we speak about a p l a n t e d polyenoid. As a starting point, the U m numbers of unsymmetrical planted polyenoids are to be determined. Among the depictions in Chart 5, an arrowhead points at the unique vertex in the examples of these systems. If the arrowhead itself is interpreted as an edge, one gets the symmetry right: all these systems belong to C s and are referred to as "unsymmetrical" because they only have a trivial plane of symmetry since they are planar. However, this "arrowhead edge" should not be counted among the m edges of each system. Chart 5 includes a recurrence relation for U m and its initial conditions. Furthermore, the derived U(x) generating function is displayed along with the counting series up to m = 10. The next task is to enumerate the polyenoids rooted at a vertex, here referred to as a t o m - r o o t e d polyenoids. In Chart 5 the unique vertex in such systems is marked by a black dot larger than the dots which indicate the other vertices. The number of nonisomorphic atom-rooted polyenoids with m edges each is identified by the symbol A m. The number A m can be interpreted chemically as the number of nonisomorphic polyenes with m bonds, where exactly one carbon atom is labeled, e.g. as 13C while all the other carbons are 12C. The deduction of the appropriate generating function, A(x), is somewhat complicated, but the methods of Zhang et

236

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

al. [64] for catafusenes rooted at a polygon core can be adapted to the actual case. In terms of the function U(x), which is given in Chart 5, it was found: 1

1

.,q(x) = ~- [U(x) + U2(x) + - ( 1 + x)U(x2)] +

1

[U3(x) + 2U(x3)]

x

(]3)

The explicit form of ~ x ) is found in Chart 5. Also the polyenoids rooted at an edge, viz. the Bm bond-rooted polyenoids need to be enumerated. The unique edge of these systems is drawn heavy in the pertinent depictions of Chart 5. A chemical interpretation of such a system is a polyene where exactly two adjacent carbons are labeled. Again the approach of Zhang et al. [64] with modifications is applicable for the relevant generating function, B(x). It was arrived at the following result in terms of U(x):

B(x) = -~x + xU(x) + -~xU2(x) +

u(x 2) + xug(x) + -~xU4(x)

(14)

The explicit form is found in Chart 5.

Polyenoids of Specific Symmetries The numbers of (free, unrooted) polyenoids belonging to the different symmetry groups are of interest. The relevant generating functions, which were deduced, are collected in Chart 6. The functions for the symmetrical systems (all but Cs) were determined by exploiting the method of stupid sheep counting (see above) to a high extent. The function for Cs was determined by subtractions from the total. But now we are anticipating the development; the l(x) function for the numbers of nonisomorphic free polyenoids has to be determined first. For this purpose, we shall find that the systems of symmetries Do.h, D2h, and C2h are especially important. These systems possess one central edge each. Hence their numbers are identical with those of the bond-rooted polyenoids of the same symmetries and relatively easy to deduce. It was found, D(X) "- X[ l + U(X4)]

(15)

for the systems of Doohand D2h symmetries taken together, and,

I C(x) = ~I - U(x2)_ ~-x[l + U(x4)]

(16)

for C2h. The generating function for the numbers of C2v systems, say M(x), which is given explicitly in Chart 6, splits into Ma(X) + ~ ( x ) , where the terms pertain to the symmetry types C2v(a) and C2v(b), respectively. There is a one-to-one correspondence between the C2h and C2~(b) polyenoids so that Mb(X) = C(x). A pair of such corresponding systems are the syrdanti-conformers, of which the smallest pair representing the buta-1,4-dienes is found in Figure 7 (cf. also the bottom row of Figure 1).

237

Isomers and Conformers: Polyenes

Symmetry

Generating Function

O~h

x

D3h

A_[ 2x 3 1 - (

1 - 4x6)1 /2]

Gh

-~[(1

46) 1/2

D2h

- ~ [ 1 - 2x4 - (1 - 4x4)] 1/2]

C2h

4-~[(1 4.4)1/2 (1 4x2)1/2 2x2]

C2v

1 2x3[(1-4x6) 1/2 + ( 1 - 4x4)1/2 -(1 +x)(1 -4x2) 1/2 - 1 + x - 2.x2 - 2 x 3]

G

- ~ 1 + x)+ 24--~[(1 -4x) 3/2 + 3(3 + 2x)(l -4x2) 1/2

-

(1

_

-

4x3) 1/2 2x 3] _

2x -

_

-

_

+ 2(1 - 4x3)1/2 - 6(1 - 4x4)1/2 - 6(1 -4x6) 1/2]

Total

~ 3 1 + x)(1 - 2x 2) + 24-~[(1 -4x) 3 / 2 - 3(3 + 2x)(1 - 4x2)1/2 - 4(1 -4x3) 1/2]

Chart 6. Generating functions for the numbers of polyenoids.

Free

Poiyenoids

The next step is to pass from the rooted to unrooted (free) trees. The method of Otter [63], which was explained and applied by Harary and Read [43], is also applicable to the present problem. The sets of symmetrically equivalent vertices and edges in the free polyenoids are to be counted under the different symmetry types. Denote the numbers of these sets by n* and m*, respectively, and refer to the sets themselves as symmetrically equivalent sets; in mathematics they are usually called equivalence classes. In the case of Cs, all the vertices and edges form symmetrically equivalent sets each by themselves; hence n* = n, m* = m, and therefore,

n*-m*=n-m=

i

(17)

where Eq, 1 has been employed. As another example, consider a polyenoid of s y m m e t r y C3h. S u c h a system has one central vertex, while the other n - 1 vertices are distributed into symmetrically equivalent sets with three vertices in each set. All the m edges, on the other hand, are simply distributed by three edges in each symmetrically equivalent set. Hence,

238

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

n*=l+~-(n-1),

(18)

m*=3 m

and consequently: n*-m

,

=3(n-m)+~-=l

(19)

2

In a similar way, one finds that n* - m* = 1 altogether in the cases D3h, C3h, Czv(a), and Cs. A polyenoid of symmetry C2h exhibits different properties. Such a system has a central edge, while the other m - 1 edges are distributed into symmetrically equivalent sets with two edges in each set. Simultaneously, the n vertices are simply distributed by two vertices in each symmetrically equivalent set. Hence, 1 n*=~n, m*=

1 1 + ~-(m-

1

1)=~(m

+

(20)

1)

and consequently: (21)

, , 1 1 n - m = Zx(n -- m) - -~-= 0

Table 4. m

Formula

Numbersof Polyenoids

D3h

C3h

D2h/Dooh*

C2h

C2v w

1

C2H 4

--

--

1

w

2

C3H 5

--

--

0

w

3

C4H 6

1

D

0

1

Total 1

1

1

1

3

4

CsH 7

0

--

0

0

2

4

5

C6H 8

0

w

1

2

4

12

6

CTH9

0

1

0

0

5

27

7

CsHI0

0

0

0

7

14

82

8

C9H11

0

0

0

0

14

228

9 10

CloHl2 CllHl3

1 0

2 0

2 0

20 0

39 42

733 2282

11

C12H14

0

0

0

66

132

7528

12

Cl3Hl5

0

7

0

0

132

24834

13

Cl4H16

0

0

5

212

424

83898

14

ClsHl7

0

0

0

0

429

285357

15

C16H18

2

20

0

715

1428

983244

16

C17H19

0

0

0

0

1430

3412420

17

CIsH2o

0

0

14

2424

4848

11944614

18

C19H21

0

66

0

0

4862

42080170

19

CzoH22

0

0

0

8398

16796 149197152

20

C21H23

0

0

0

0

16796 531883768

Note: *D.h only for m = 1.

Isomers and Conformers: Polyenes

239

The same relation, viz. n* - m* = 0, is altogether found in the cases D=h, D2h, C2h, and C2v(b). Consider a free polyenoid P. It is clear that P is counted n* times among the Am atom-rooted polyenoids and m* times among the B m bond-rooted polyenoids. Therefore A m - B n catches up every free polyenoid of the symmetry types D3h, C3h, C2v(a), and C s exactly once, while those of O.oh, D2h, C2h, and C2v(b) are missed. Notice that the number of C2h and C2v(b) systems with a given m is the same. In conclusion, when passing to the language of generating functions, one obtains for the total number of nonisomorphic free polyenoids: l(x) = A ( x ) - B(x) + D(x) + 2C(x)

(22)

This function was worked out in terms of U(x) by means of Eqs. 13-16, and finally in the explicit form as given in Chart 6.

Numerical Values In Table 4 the total numbers of nonisomorphic polyenoids are listed. The distributions into symmetry groups are included, but a column for the Cs symmetry was omitted for the sake of brevity. These missing numbers are easily retrieved by subtractions from the totals.

VI.

NONOVERLAPPING POLYENOIDS: COMPUTER PROGRAMMING

The prospects to synthesize polyenes which correspond to the overlapping polyenoids are bad because of severe steric hindrances. Therefore it is obviously of chemical interest to sort out the nonoverlapping polyenoids. These systems are exactly those which were considered by Kirby [55]. However, the steric hindrances are still not eliminated completely when the hydrogen atoms are taken into account. For the numbers of nonoverlapping polyenoids, no mathematical solution is available and is not likely to be found. Therefore we resorted to computer programming, like Kirby [55], in order to produce these numbers, but using quite different methods which allowed an extension of his data. The results up to m = 14, which were achieved, are collected in Table 5. In this table again (as in Table 4), the numbers pertaining to C s symmetry are suppressed. All the polyenoids depicted in Figure 9 are nonoverlapping. The nonoverlapping polyenoids were simulated by a certain class of benzenoids via their dualists because of the large experience which has been accumulated on computerized enumerations of benzenoids or polyhexes [10,46,48,50,65-67]. In the present computations, the very useful DAST (dualist angle-restricted spanning tree) code [50,67] was employed. The symmetries of the generated systems were recognized by means of an algorithm, of which the principles are described elsewhere [66].

240

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL Table 5. Numbers of Nonoverlapping Polyenoids D3h

C3h

D2h/O.oh*

C2h

C2v

1

C2H 4

--

--

1

~

~

2

C3H 5

~

~

0

--

3

C4H 6

1

--

0

1

4

CsH 7

0

~

0

0

2

4a

5

C6H 8

0

m

1

2

4

12 a

6

C7H 9

0

1

0

0

4

26 a

7

C8H10

0

0

0

7

12

77 a

8

C9HII

0

0

0

0

10

204 a

9

CIoH12

1

2

2

20

29

10

CIIHI3

0

0

0

0

27

m

Formula

Total 1a

1

1a

1

3a

624 a 1817

11

C12H14

0

0

0

63

88

5585

12

Cl3Hl5

0

7

0

0

76

17007

13

C14H16

0

0

3

191

247

52803

14

C15H17

0

0

0

0

217

164001

Notes: *D.a~only for m = 1. aKirby (1992) [55]; the smallest numbers were of course known before this publication.

VII. OVERLAPPING POLYENOIDS: COMBINATORIAL CONSTRUCTIONS

The numbers of overlapping polyenoids up to m = 14 are now available on subtracting the numbers of Table 5 from the corresponding numbers of Table 4. The results are shown in Table 6. Thus it is seen, for instance, that the smallest overlapping polyenoid occurs at m = 6. The form of this C2v(a) system is depicted in Fig. 10, together with the forms of the five overlapping polyenoids at m = 7. Also the number of these systems and their symmetries are consistent with the data of Table 6. The first of these m = 7 systems (from the left in Figure 10) belongs to C2v(b); the last one to C2v(a). Table 6 has no entries for the symmetry groups O3h and C3h. Overlapping polyenoids of these symmetries do exist, however, but the smallest of them occur just beyond the range of the listing. In Figure 11 these m = 15 systems are shown. In Figures 9-11, the forms of the smallest systems of certain categories are displayed. They were generated just by systematic drawings and demonstrate the method of combinatorial constructions in its simplest form. A systematic approach according to this method, supported by graph-theoretical proofs, has been developed and applied to helicenes [68-69]. This approach can straightforwardly be adapted for the present purpose. The following definition is analogous to the definition of irreducible helicenes [68]. An overlapping polyenoid is irreducible if no smaller overlapping polyenoid can be produced from it by deleting one edge; otherwise it is reducible.

Isomers and Conformers: Polyenes

241

Table 6. Numbers of Overlapping Polyenoids m

Formula

6 7 8 9 10 11 12 13 14

D2h

C7H 9

--

CsHIo

--

C9Hll

--

CIoHI2 CllH13

-~

C12H14

~

Cl3Hl5

~ 2 0

C 14H 16

ClsHl7

C2h

C2v

Cs

Total

---~ -3 0 21 0

1 2 4 10 15 44 56 177 212

-3 20 99 450 1896 7771 30895 121144

1 5 24 109 465 1943 7872 31095 121356

It is clear that an irreducible overlapping polyenoid must be unbranched and possess exactly one pair of overlapping vertices. The system is associated with a cycle which is broken, but would be restored if the two overlapping vertices coalesce. All reducible overlapping polyenoids can clearly be generated by adding edges to the irreducible polyenoids. The smallest irreducible overlapping polyenoid is the m = 6 system found in Figure 10. Denote it by 0 6. The five overlapping polyenoids with m = 7 (see Figure 7) are all reducible and generated by adding one edge at a time in different positions to 0 6. Symmetry must be taken into account in order to avoid isomorphic systems. In the following, we shall demonstrate the systematic generation of the m = 8 overlapping polyenoids according to the method of combinatorial constructions. The next smallest irreducible overlapping polyenoids are three nonisomorphic systems with m = 10. Therefore the m = 8 overlapping polyenoids are additions of edges to 0 6 only; more specifically, they are obtained by adding one edge at a time

C) m=7

m=6

-C)

Figure 10. All nonisomorphic overlapping polyenoids with m < 7.

242

S.J. CYVIN, J. BRUNVOLL, B.N. CYVIN, and E. BRENDSDAL

( _

v

. .

%h

%h

Figure 11. The smallest trigonal overlapping polyenoids; symmetries D3h (one system) and C3h (two systems).

to the five overlapping polyenoids with m = 7. Denote by 07 the coiled polyenoid with m = 7 (found among the forms in Figure 10), and by 08 the coiled polyenoid with m = 8 (see Figure 8). Case 1. Addition of two edges to 06 SO that 07 is not formed as a subgraph (or fragment). In Figure 12 (top row) one edge added to 06 is supposed to be fixed, while the possibilities for adding the second edge are indicated by numerals. Precaution was taken so as to avoid isomorphic systems, taking symmetry into account. The sum of the numerals, viz. 19, indicates the number of nonisomorphic systems under Case 1. Case 2. Addition of one edge to 07 so that 08 is not formed. The 4 possibilities are mapped in the bottom part of Fig. 12. Case 3. The system 08.

1 1

:

1

2

2\ 1

1

Figure 12. Systematic generation of the overlapping polyenoids under Case 1 (top row) and Case 2 (bottom drawing); see the text.

243

Isomers and Conformers: Polyenes

In conclusion, there are found to be 24 overlapping polyenoids with m = 8, in perfect agreement with the appropriate entry in Table 6. Also the prediction of four C2v systems with m = 8 is verified: three of them are found under Case 1, in addition to 08 (Case 3). The method of combinatorial constructions was also used to generate the 10C2v + 9 9 C s = 109 overlapping polyenoids with m = 9, again in agreement with Table 6. For the sake of brevity, we omit the details.

VIII.

CONCLUSION

This chapter deals with enumerations of chemical graphs or systems, reflecting the traditional enumerations of isomers and conformers in chemistry. A particular interest of this topic for structural chemists is exemplified by a comprehensive survey of alkane isomer/conformer enumerations, including new developments for the staggered conformers. Challenging unsolved problems in this area are pointed out.

Several methods of enumeration, specified as (1), (2) and (3) in the Introduction, are treated in detail. All of them are demonstrated in the application to polyenoids, which correspond to conjugated polyenes. An algebraic treatment involving generating functions is characterized as elementary for the unbranched polyenoids and more advanced for the polyenoids in total (unbranched + branched). Results for the numbers of nonoverlapping polyenoids from computer programming are reported. Next, the numbers of overlapping polyenoids are determined by taking differences between the totals from the deduced generating functions, and the computer-generated numbers for nonoverlapping systems. In this way, relatively small numbers emerge as differences between larger numbers, e.g. 82 - 77 = 5,228 - 204 = 24. Some of these smaller numbers are verified by the method of combinatorial constructions. This positive test of internal consistency between numbers from the algebraic solution, computerized enumeration, and a paper-and-pencil method (combinatorial constructions) increase our confidence in the correctness of the present analyses.

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Isomers and Conformers: Polyenes

245

49. Balaban, A. T.; Harary, E Tetrahedron 1968, 24, 2505-2516. 50. Nikolie, S.; Trinajsti~, N.; Knop, J. V.; Mtiller, W. R.; Szymanski, K. J. Math. Chem. 1990, 4, 357-375. 51. Balaban, A. T. Rev. Roum. Chim. 1977, 22, 45-47. 52. Balaban, A. T. Pure Appl. Chem. 1982, 54, 1075-1096. 53. Balaban, A. T. Rev. Roum. Chim. 1976, 21, 1045-1047. 54. Randi~, M. Int. J. Quant. Chem., Quant. Biol. Symp. 1988, 15, 201-208. 55. Kirby, E. C. J. Math. Chem. 1992, 11,187-197. 56. Pogliani, L. J. Chem. Inf. Comput. Sci. 1994, 34, 801-804. 57. Gutman, I. Bull. Soc. Chim. Beograd 1982, 47, 453-471. 58. Cyvin, S. J.; Gutman, I. Comput. Math. Applic. 1986, 12B, 859-876; reprinted in: Symmetry Unifying Human Understanding; Hargittai, I., Ed.; Pergamon: New York, 1986. 59. Gutman, I.; Cyvin, S. J. Introduction to the Theory of Benzenoid Hydrocarbons; Springer: Berlin, 1989. 60. Balaban, A. T. Commun. Math. Chem. 1989, 24, 29-38. 61. Cyvin, S.J.; Cyvin, B. N.; Brunvoll, J. J. Mol. Struct. (Theochem) 1993, 300, 9-22. 62. Redelmeier, D. H. Discrete Math. 1981, 36, 191-203. 63. Otter, R. Annals Math. 1948, 49, 583-599. 64. Zhang, E J.; Guo, X. E; Cyvin, S. J.; Cyvin, B. N. Chem. Phys. Lett. 1992, 190, 104-108. 65. Brunvoll, J.; Cyvin, S. J.; Cyvin, B. N. J. Comput. Chem. 1987, 8, 189-197. 66. Brunvoll, J.; Cyvin, B. N.; Cyvin, S. J. J. Chem. Inf. Comput. Sci. 1987, 27, 171-177. 67. Miiller, W. R.; Szymanski, K.; Knop, J. V.; Nikoli6, S.; Trinajsti~, N. J. Comput. Chem. 1990, 11, 223-235. 68. Guo, X. F.; Zhang, E J.; Cyvin, S. J.; Cyvin, B. N. Polycyclic Aromat. Compd. 1993, 3, 261-272. 69. Zhang, E J.; Guo, X. E; Cyvin, S. J.; Cyvin, B. N. Polyc3,clic Aromat. Compd. 1994, 4, 115-124.

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INDEX

Ab initio MO calculations, 6

Bis(pyridine N-oxide)tetrachloroaurate(III), and hydrogen bonding, 82 Bis(trifluoroacetate), and hydrogen bonding, 80 Bond angles, 173 in cyclic hydrocarbons, 172 in cyclic ketones, 172 in sulfur-containing compounds, 173 in zeolites, 173 of alicyclic hydrocarbons, 172 of bicyclo alkanes, 172 of thianthrenes, 174 phenoxathiins, 174 Bond lengths, determinations of, 165 Bond number conservation rule, 87 Bond order, calculated by VALBOND, 149 Bond radii, 104 Borane complexes, binding energy of, 114 (t-Bu-C60) 2, 18-19 and congestion, 18 and lever effect, 19 and nonbonded repulsion, 18 n-Butane, conformations of, 4 hexa-t-Butylethane, 10 tetra-tert-Butylmethane, 9

Alkanes constitutional isomers, 218 stereoisomerism, 218-220 n-Alkanes, crystal structures of, 16 Alkanoids, 223-227 branched, 222, 226-227 description of, 222 unbranched, 225 Anisotropic displacement parameters, 36 Ar-BF 3, nuclear quadrupole studies of, 123 Badger's rule, 167 Bent's rule, 141 Berry pseudorotation barrier, 151 BF 3 complexes, 111, 112 N,N'- B i ( c is- 2, 4-dimeth y l p iperdi ne ), 12 cz-Bis( 1,2-ben zoqu inone dioximato)Pd(ll), and hydrogen bonding, 80 Bis(2-amino-2-methyl-3-butanone oximato)Ni(II) chloride, and hydrogen bonding, 80 B is(N,N-dimethyl acetamide)tetrachloroaurate(III), and hydrogen bonding, 82 247

248

(-)CAHB (see Negative charge-assisted hydrogen bonding) (+)CAHB (see Positive charge-assisted hydrogen bonding) Cambridge Structural Database, 52, 70 Cannabidiol, conformational studies of, 14 Carbonyl-arene clusters, 43 Carboxyl, hydrogen bonding with carbonyl groups, 58 Carboxylic acids, and hydrogen bonding, 83 Catafusenes, 223-227 branched, 224-226 description of, 223 unbranched, 224, 226 Catahelicene, 223 Catechin, conformational studies of, 14 Cation-anion interactions, 51-57 CCBond, use of in NMR spectroscopy, 200 2-Center 2-electron bonds, 147 3-Center 4-electron bonds, 147 (CH3)2HN-SO 2, physical properties, 113 (CH3)3N-A1C13, binding energies of, 114 (CH3)3N-BH 3, binding energies of, 114 (CH3)3N-SO2 and dipole moment, 116 gas-solid structure differences, 107 physical properties, 113 CH3CN-BF3 gas-solid structure differences, 107 nuclear quadrupole studies of, 123 CH3H2N-SO 2, physical properties, 113

INDEX

Charge-assisted hydrogen bonding, 91 and dissymmetry, 91-93 Chemical dynamics, 106, 124 and mapping reaction paths, 120-124 Chemical graphs, 217 Chemical literature, and errors, 2 Chirality, 215 ElF 3 Lewis structure of, 149, 150 VALBOND structure of, 150 Clusters, of transition metals, 25-65 [Co13C2(CO)24], structure of, 53 Co2(CO)8, 30 and bridging ligand, 32 and crystal packing, 34 dynamic behavior in the solid state, 32 structure of, 31, 32, 33 [Co2Rh4C(CO)I3] 2-, 57 Co4(CO)12 crystal disorder of, 38 structure of, 38 [Co6C(CO)13]2-, structure of, 56 Combinatorial formulas, 229 Compact conformations, 6 J-Conformation, 6 Conformation analysis, 3 Conformation, folded, 5 Conformational polymorphism, 39 Conformers, 220-223 enumeration of, 213-245 Congested organic molecules, 1-24 conformational principles of, 1-24 Contact distances, in hydrogen bonding, 91 Cooperative chemistry, 106, 122 Counting series, 231 Coupling parameter (~,), 79 plot, 84 Covalent bond descriptions, in transition metal hydrides, 153

Index

Crystal conformation, 11 Crystal formation effect of carbonyl ligand, 29, 30 intermolecular assembly, 29, 30 Crystal isomerism, definition of, 39 Crystal stability, 27 Crystallization, of partially bonded molecules, 103-127 Crystals, and CO ligands, 54 Crystals, containing large anions, 54 Dative bond, 120 DEPT NMR, 193 DEPT-INADEQUATE NMR, 199 Diamond lattice, 221 1,3-Diaxial interaction, 7 1,1'- Dicy cloaklyls conformational studies of, 12, 16-18 Dicyclobutyl, population of gauche form, 17 Dicyclohexyl, conformational studies of, 17 Dicyclohexyl, population of gauche form, 17 Dicyclopentyl, population of gauche form, 17 Dicyclopropyl, population of gauche form, 17 3,3'-Diethylpentane, 10 Dihedral angles and NMR studies, 178 empirical relationships, 175 of thianthrenes, 176 Diketone enols, and hydrogen bonding, 83 Dimeric buckminsterfullerene conformation of, 2 global energy-minimum conformation of, 3 Dimethylamine-SO 2, and force constant, 124

249

2,3-Dimethylbutane, conformational studies of, 16-18

cis-l,3-Dimethylcyclohexane, 7, 8 Dipolar enhancement mechanism, 116 Disordered environment, 125 DNA, 99 Donor-acceptor interaction, 5 Double quantum coherence, 195 Dualist, 223,224 Dynamic behavior in the solid state, 32 Electron density distributions, 158 Electron diffraction data, of trisubstituted benzenes, 171 Electronic structure computations, 134 Empirical correlations, in structural chemistry, 163-189 Empirical relations in benzene tings, 168 in carboranes, 167 in cycloalkanes, 167, 169 in nitroamines, 169 in silatranes, 169 in sulfoxides, 168 of internuclear distances, 167-169 Enclosure shell (ES), 30 Enumeration, of isomers and conformers, 213-245 Equivalence classes, 237 ES (see Enclosure shell) External interactions, in the crystal, 27 Fez(CO)9, 30 structure of, 31, 33 Fe3(CO)12 anisotropic displacement parameters of, 36 crystalline disorder in, 36 dynamic behavior in solution, 36 NMR studies of, 36 structure of, 35

250

Fibonacenes, 229 First principle molecular dynamics, 20 Fluxional pathways, 130 Fock operator, 148 Folded conformation, 5 Free induction decay, 196 Friedelin, 8, 9 Gas phase, and intermediate bonding, 111-115 Gas-solid structure differences, 105, 106 and BF3 complexes, 109 and potential energy surface, 110 and sulfur systems, 109 general remarks, 110-111 large, 107-111 limitations, 111 magnitude of, 110 trends, 110 gauche Interactions, 3 gauche-gauche Sequence, 6 Generating functions, 231 1,5-GG sequence, 16 GG' interactions, 9, 11 GG' sequence, 7, 20 GG' strain, 7 Global energy-minimum conformation, 3 Graph theory, 217 Graphical correlations, of various molecular structures, 170 Graphs, chemical, 217 H3N-BF 3 calculated structures, 119 binding energy of, 114 calculated structures, 119 gas-solid structure differences, 107 prediction of bond length, 117 H3N-SO 2 bonding in, 115

INDEX

physical properties, 113 physical properties, 113 H3N-SO 3 gas-solid structure differences, 107 prediction of bond length, 117 H3Os3(CO)9(~t-CCOOH), hydrogen bonding in, 58 HCN-BF 3 calculated structure, 119 dipole moment of, 116 gas-solid structure differences, 107 nuclear quadrupole studies of, 123 HCN-SO 2, physical properties, 113 Hetermolecular crystals, 49 Heteronuclear hydrogen bonding, description, 88, 89 Hexaethylethane, 11 Hexagonal lattice, description of, 227 HFe4(rl2-CH)(CO)12, and hydrogen bonding, 60 HFe4012-CH)(CO)12(PPh3) hydrogen bonding in, 60 structure of, 60 Homonuclear hydrogen bonding, 78-90 and charged systems, 78 description, 87, 88, 89 HRu3(CO)9(~t3:O:TI2:TI2-C6H7), H-carbonyl interactions in, 60 HRu3(CO)I0(~-S-CH2-COOH), hydrogen bonding in, 58 HRu6B(CO)17, 43 Hybrid orbital, 130 Hybrid orbital strengths, and molecular shapes, 131-136 Hybridization, 132 Hydrogen bonding, 57-61 and bond lengths, 94, 95 and chemical equivalence, 86 and contact distances, 91 and covalency, 92 and dissociation energies, 92 and dissymmetry, 90

Index

and noncharged resonant systems, 95 and three dimensional organization of molecules, 57 covalent nature of, 85-90 delocalized nature of, 85 description of, 58 intramolecular, 90 plot of nearly linear bonding, 86 Hydrogen bonding theory, 67-102 applications of, 90-100 Hydrogen bonds strong and very strong, 79 types, 69 Hypervalent molecules, 130, 135 Imidazolium maleate, and hydrogen bonding, 80 INADEQUATE NMR, 191-211 and pulse sequence, 196 applications of, 201-208 description of, 194-196 menthol study, 198 one-dimensional, 196-197 solid state applications, 208 studies of alkaloids, 207 studies of antibiotics, 205 studies of coordinated metal complexes, 208 studies of lignin, 207, 208 studies of mixtures, 201-204 studies of natural products and large molecules, 204 studies of peptides, 206-207 studies of saponins, 205-206 studies of small molecules, 201 studies of steroids, 205 studies of synthetic polymers, 204 studies of terpenes, 204 studies of vitamins, 207 two-dimensional, 197-199 INEPT-INADEQUATE NMR, 199 INSIPID-INADEQUATE NMR, 199

251

1,4-Interactions, and conformational analysis, 19

1,5-Interactions, and conformational analysis, 20

1,6-Interactions, and conformational analysis, 20 Intermediate bonding, 111-115 and ab initio studies, 114 and BF 3 complexes, 111, 112 and bonding interactions, 115 and charge transfer, 114 and high resolution spectroscopic measurements, 111 physical properties of some species, 113 theoretical results, 113-115 Intermediate bonds, 105 Intermolecular assembly, 30 Intermolecular forces, 105 Intermolecular interactions, and intramolecular bond energies, 105 Internal interactions, in the crystal, 27 Intramolecular bond energies, and intermolecular interactions, 105 Intramolecular congestion, 6-15 [Ir4(kt2-CO)3(CO)8(SfN)] [Ph3P)2N], 55 [Ir4(CO)11(SCN)][N(CH3)2(CH2Ph)2], 55 Ira(CO)9(l.t3-trithiane, 43 structure of, 46 [Ir6(CO)laBr][PPh4] 2, structure of, 55 Ir6(CO)16 and crystal isomerism, 39 structure, 40 Isomers configurational, 215 constitutional, 215, 218 description of, 215 enumeration of, 213-245 Karplus equation, 184 Keto-enol tautomerism, 96

252

Key-keyhole interaction, 49 Large molecules and crowding, 4 and folding, 4-6 and overcrowding of substituents, 4 Lattice energies, 105 Lennard-Jones potentials, 137 Lewis basicity, 59 Lewis structures, 145 Ligand bonding modes, 28 Linear relationships, 180 classifications, 182, 183 Localized bond model, 135 Localized bonds, 131 Mendeleev periodic system, 164 Menthol, NMR study of, 193 Methotrexate, conformation of, 5 T-Methyl anabasine, conformational studies of, 14 Methylated dicyclohexyls and gauche interactions, 14 and staggered barriers, 12-14 steric energy graphs, 13 Microscopic simulation, 20 MM3 calculations, 6, 17 Molecular aggregation, 27 Molecular characteristics, 179 Molecular cocrystals, definition, 52 Molecular congestion, definition, 7 Molecular crystals, 104 Molecular geometries, 130 Molecular mechanics, 130 and inorganic molecules, 138 computations, 129-161 methods, 136-138 Molecular piles, 49 Molecular salt, definition, 52 Molecular shapes, 129-161 Molecular structure prediction, 134 Morpholinocyclohexene, conformational studies of, 14

INDEX

MP4SDQ/6-31G*//HF//6-31 G* calculations, 6 Na3H(SO4)2, and hydrogen bonding, 81 Na3H3bis[tris(glycolato)aluminate(III)], 81 NCCN-BF 3, binding energy of, 114 Negative charge-assisted hydrogen bonding ((-)CAHB), 84 Networks, of hydrogen bonded molecules, 59 NH3-BF3, nuclear quadrupole studies of, 123 Nitrogen and sulfur complexes, 113, 115 to replace carbon atoms for conformational analysis, 12 Nonbonded repulsion, 18 Noncharged resonant systems, and hydrogen bonding, 95 Nuclear quadrupole coupling constants, 124 Organometallic crystal isomerism, 39-51 [Os 10C(CO)24][(Ph3P)2N], crystal structure, 52 OsFe2(CO)12, as isostructural with Fe3(CO)I2, 37 Packing coefficients, 53 Partial bonding, 106 Partially bonded molecules and gas-solid structure differences, 115-119 and lattice energy, 116 crystallization of, 103-127, 115-119 predictions of gas-solid structure differences, 116-118 n-Pentane, conformational analysis of, 7

Index

PF5, 147 Phenylcyclohexane, conformational studies of, 14 Podophyllotoxin and overcrowding of, 4 CPK model of, 4 NMR analysis of, 5 Polyene hydrocarbons, 227-239 enumeration of isomers and conformers, 213-245 Polyenoid system and symmetry, 228 description, 227 Polyenoids and combinatorial constructions, 240-243 and symmetry, 236 branched, 227 enumeration of, 229-239 free, 237 nonoverlapping, 239-240 overlapping, 240-243 planted, 235,236 rooted, 235,236 unbranched, 227 Polyhex, 223 Positive charge-assisted hydrogen bonding ((+)CAHB), 84 PtCI42-, molecular mechanics predictions, 138 Pyridine-2,3-dicarboxylic acid, and hydrogen bonding, 80 Quasi-bonded molecules, 120 and B-N bond length, 120, 121 and force constants, 122,123 and gas phase bonding, 120 and nuclear quadrupole coupling constant, 122, 123 and reaction path, 121,122 Quasi-bound species, 105

253 RAHB (see Resonance-assisted hydrogen bonding) Reaction paths, 105,106 Redfield-P61ya theorem, 233 Regression lines, 176 Resonance, 130 Resonance-assisted hydrogen bonding (RAHB), 69-78 and chain length, 94 and DNA, 99 and H-bond energies, 98 and heteroconjugated systems, 93-96 and interatomic distances, 74 and secondary protein structure, 99 and tautomerism, 96 associations, 99 choice of a coordinate system, 72 description, 76 diketone enol studies, 71 enolone studies, 70 generalization, 97 historical, 69 model, 75 polarization rule, 95 potential energy surface, 76, 77 structural correlations, 73-75 Rh(III)-DMSO complexes, and hydrogen bonding, 82 [Rhll (CO)23][NMe4]3.C6HsMe, 54 [Rhll (CO)23][NMe4]3.Me2CO, 54 [Rh12C2(CO)24]2-, 53 [Rh12C2(CO)24]2-, crystal structure, 52 [Rh6C(CO)13]2-, structure of, 56 Ru3(CO)I2 crystal packing of, 38 molecular symmetry of, 37 structure of, 35 Ru5C(CO)I20q6-C6H6), 43 structure of, 47 RusC(CO)120q6-C6H6), H-carbonyl interactions in, 60

254

Ru5C(CO)12(I.I.3:1"I2:I"I2:l"I2-C6H6), 43 H-carbonyl interactions in, 60 structure of, 47 RusS(CO)Is, 43 structure of, 45

Ru6C(CO)11(1"16-C6H4Me2)2) space filling representation of, 51 structure of, 50

Ru6C(CO)11(q6-C6H6)(~t3-1]2:1]2:q2_ C6H6),49 key-keyhole interaction of, 49 structure of, 48 Ru6C(CO)I 1(116-C6H6)2, 49 key-keyhole interaction of, 49 structure of, 48

Ru6C(CO)14016-C6H4Me2) space-filling representation of, 51 structure of, 50

Ru6C(CO)17 crystal forms of, 39 crystal packing diagrams, 42 fundamental packing motif of, 39 structure, 41, 44 Saddle point, 12 Self-consistent reaction field theory, 117, 118 and medium effects, 117 prediction of B-N bond length, 117 prediction of H3N-SO 3 bond length, 117 SF4, Lewis structure of, 149 SF6, Lewis structure of, 149 Silatrane systems, and crystallization, 115 Silatranes, gas-solid structure differences, 107 Silicon-nitrogen bond distances, 109 Siloxanes and Bent's rule, 146, 147 VALBOND studies of, 146, 147 Sn(II)HPO 4, and hydrogen bonding, 81

INDEX

Stereoisomers, 216, 218 Strength functions, 133 Structural chemistry, empirical correlations in, 163-189 Structural parameters of molecules, 184

Structure correlations, crystallographic, 106 Structure determination, using INADEQUATE NMR, 191-211 Stupid sheep counting, 229 Sulfamic acid, 109 Sulfur, and BF 3 complexes, 112 Sulfur, and nitrogen complexes, 113, 115 Sulfur-nitrogen bond, 123, 124 Tautomerism, 96 and electronic factors influencing, 97 3,3,4,4-Tetraethylhexane, 11 Tetraethylmethane, 10 Tetrakis(ethylenediamine-N,N')-kt 2hydroxo-aqua-O,O')-dinitro-diCo(Ill) triperchlorate, and hydrogen bonding, 81 trans Form and strain, 15 versus gauche form, 15-19 trans Interactions, 3 Transition metal clusters, 25-65 and crystals, 27 and structural variability, 26 Transition metal hydrides, 130 calculations on, 152-156 covalent bond descriptions, 153 geometries of, 152 predictions of bond lengths and angles, 154 VALBOND calculations on, 153 Trimethylamine-SO 2, and force constant, 124

Index

Trisubstituted benzenes, bond length correlations of, 171 VALBOND, 138 and bond order, 149 and hypervalent fluorides, 151 and hypervalent molecules, 147 and nonhypervalent molecules, 140-147 and Pt complexes, 155 and transition metal ligands, 156 and use of Lewis structures, 145 and VSEPR, 158 approach, 139, 140 calculation of inversion barriers, 147 calculation of vibrational frequencies, 147 hydrocarbon studies, 142, 143 inorganic molecular studies, 143-145 parameters, 140, 141 siloxane studies, 146 transition metal hydride studies on, 152-156 weighting factors for different elements, 141 Valence bond concepts, 129-161, 138-156 Valence bond theory, 130

255

principles, 131-132 van der Waals attraction, 16 van der Waals complexes, 105 van der Waals interactions, in crystals, 28 van der Waals potential functions, 6 van der Waals radii, 104 van der Waals solids, 28-38 Van Arkel-Ketelaar triangles, 164 Vanadium-triflate complexes, and hydrogen bonding, 82 WH 6, 153 WMe 6 and VALBOND calculations, 155 structure of, 155 XeF bond energy, versus bond angle, 148 XeF 2, 147, 148 as simplest hypervalent molecule, 149 XeF 4 Lewis structure of, 149 molecular mechanics predictions, 138 XeF 6, 147 Lewis structure of, 149

ZrH4,

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Advances in Molecular Modeling Edited by Dennis Liotta, Department of Chemistry, Emory University "... as a result of the revolution in computer technology, both the hardware and the software required to do many types of molecular modeling have become readily accessible to most experimental chemists. Because the field of molecular modeling is so diverse and is evolving so rapidly, we felt from the outset that it would be impossible to deal adequately with all its different facets in a single volume. Thus, we opted for a continuing series containing articles which are of a fundamental nature and emphasize the interplay between computational and experimental results."

From the Preface to Volume 1

REVIEWS: 'q'he first volume of Advances in Molecular Modeling bodes well for an exciting and provocative series in the

future."

- - Journal of the American Chemical Society " . . . . provide a useful overview of several important aspects of molecular modeling. Their didiactic approach would make them particularly valuable for readings in a molecular moiling course."

Journal of the American Chemical Society Volume 1, 1988, 213 pp. ISBN 0-89232-871-1

$97.50

CONTENTS: Introduction to the Series: An Editor's Foreword, Albert Padwa. Preface, Dennis Liotta. Theoretical Interpretations of Chemical Reactivity, Gi//es K/opman and Orest T. Macina. Theory and Experiment in the Analysis of Reaction Mechanisms, Barry K. Carpenter. Barriers to Rotation Adjacent to Double Bonds, Kenneth B. Wiberg. Proximity Effects

on Organic Reactivity: Development of Force Fields from Quantum Chemical Calculations and Applications to the Study of Organic Reaction Rates, Andrea E. Dorigo and K.N. Houk. Organic Reactivity and Geometric Disposition, F.M. Menger. Volume 2, 1990, 165 pp. ISBN 0-89232-949-1

$97.50

J A 1:

P R E S S

J A 1 P R E S S

CONTENTS: Preface, Dennis Liotta. The Molecular Orbital Modeling of Free Radical and Diels-Alder Reactions, J.J. Dannenberg. MMX, an Enhanced Version of MM2, Joseph J. Gajewski, Kevin E. Gilbert and John McKelvey. Empirical Derivation of Molecular Mechanics Parameter Sets: Application to Lactams, Kathleen A. Durkin, Michael J. Sherrod and Dennis Liotta. Application of Molecluar Mechanics to the Study of Drug-Membrane Interactions: The Role of Molecular Conformation in the Passive Membrane Permeability of Zidovudine (AZT), George R. Painter, John P. Shockcor and C. Webster Andrews. Volume 3, 1995, 224 pp. ISBN 1-55938-326-7

$97.50

'qhe field of molecular modeling continues to evolve rapidly and to grow in its diversity. Consistent with this growth, Volume Three of this series presents a potpourri of articles dealing with several different aspects of this field. Some of the studies report on the use of modeling techniques to gain insight into the origins of enantioselectivity and diastereoselectivity in a number of important chemical reactions. Others deal with modeling methodology and the problems associated with obtaining accurate structural information about a ubiquitous class of compounds, carbohydrates. Finally, the creative merging of experimental and theoretical methods demonstrates the power modeling techniques for elucidating accurate, three dimensional models of important biomacromolecules, such as the polymerase domain of HIV reverse transcriptase. Although vastly different in scope, each of these studies shares a common theoretical foundation and each has the simple goal of providing insight which is complementary to that which can be obtained from experimental methods. I hope the reader will find all of these contributions to be interesting, informative and thought-provoking." m From the Preface CONTENTS: Preface, Dennis Liotta. Origins of the Enantioselectivity Observed in Oxazaborolidine-Catalyzed Reductions of Ketones, Dennis C. Liotta and Deborah K. Jones. Semi-Empirical Molecular Orbital Methods, D. Quentin McDonald. Construction of a Three-Dimensional Model of the Polymerase Domain of HIV Type 1 Reverse Transcriptase, George R. Painter, C. Webster Andrews, David W. Barry and Phillip A. Furman. Episulfonium Ions May Not Be the Stereodeterminants in Glycosylations of 2-Thioalkyl Pryrnosides, Deborah Jones and Dennis Liotta. Stereospecific Synthesis of 1B-Methylcarbapenem Intermediates: Understanding the Selectivity Observed, Deborah K. Jones and Dennis Liotta. Molecular Modeling of Carbohydrates, Ross Boswell, Edward E. Coxon and James M. Coxon. The Molecular Modeling Potential Energy Surface, D. Ross Boswell, Edward E. Coxon and James M. Coxon. Index.