Pauling's Legacy (Theoretical and Computational Chemistry)

THEORETICAL A N D C O M P U T A T I O N A L CHEMISTRY Pauling’s Legacy Modem Modelling of the Chemical Bond THEORETI...

Author: Z.B. Maksi&cacute; | W.J. Orville-Thomas

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THEORETICAL A N D C O M P U T A T I O N A L CHEMISTRY

Pauling’s Legacy Modem Modelling of the Chemical Bond

THEORETICAL AND COMPUTATIONAL CHEMISTRY

SERIES EDITORS

Professor P. Politzer

Professor Z.B. Maksi6

Department of Chemistry University of New Orleans New Orleans, LA 70418, U.S.A.

Rudjer Bos"kovi~Institute P.O. Box 1016, 10001 Zagreb, Croatia

VOLUME 1 Quantitative Treatments of Solute/Solvent Interactions

P. Politzer and J.S. Murray (Editors) VOLUME 2 Modern Density Functional Theory: A Tool for Chemistry J.M. Seminario and P. Politzer (Editors)

VOLUME 3 Molecular Electrostatic Potentials: Concepts and Applications J.S. Murray and K. Sen (Editors)

VOLUME 4 Recent Developments and Applications of Modern Density Functional Theory J.M. Seminario (Editor)

VOLUME 5 Theoretical Organic Chemistry

C. Pdrkdnyi (Editor) VOLUME 6 Pauling's Legacy: Modern Modelling of the Chemical Bond Z.B. Maksic"and W.J. Orville-Thomas (Editors)

O

THEORETICAL AND C O M P U T A T I O N A L CHEMISTRY

Pauling's Legacy Modem Modelling of the Chemical Bond

Edited by Z.B. M a k s i ~

Rudjer Bo~kovid. Institute P.O. Box 1 0 1 6

Bijeni~ka 5 4 10001 Zagreb, Croatia W.J. Orville-Thomas

Caer Cae Melyn Aberystwyth Dyfed SY23 2HA, Wales,

UK

ELSEVIER 1999 Amsterdam - Lausanne - New York - Oxford - Shannon - S i n g a p o r e - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 1999 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail" [email protected]. You may also contact Rights & Permissions directly through Elsevier's home page (http://www.elsevier.nl), selecting first 'Customer Support', then "General Information', then 'Permissions Query Form'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 171 436 5931; fax: (+44) 171 436 3986. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Contact the publisher at the address indicated. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. Address permissions requests to: Elsevier Science Rights & Permissions Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

First edition 1999 Libr~y of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. ISBN: 0-444-82508-8 (~ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

PREFACE

Theory and experiment in chemistry today provide a wealth of data, but such data have no meaning unless they are correctly interpreted by sound and transparent physical models. Linus Pauling was second to none in the modelling of molecular properties, as we discuss later in the prologue to this book. Indeed, many of his models have served chemistry for decades, and that has been his lasting legacy for chemists all over the world. The aim of this book has been to put such simple models into the language of modern quantum chemistry, thus providing a deeper justification for many of Pauling's ideas and concepts. However, it should be stressed that many contributions to this book, written by some of the world's most prominent theoretical chemists, do not merely follow Pauling's footprints. By taking his example, they made bold leaps forward to overcome the limitations of the old models thus opening new scientific vistas. We are grateful for the effort, inspiration, care, and patience the authors have shown in the preparation of their contributions to this book. We trust that this spirit of "Pauling's legacy" will apeal to many chemists, both younger and older, in the areas ranging from chemical physics to physical organic chemistry. We thank Mr.B. Kova~evi5 for some technical help.

Z.B. Maksi5 W.J. Orville-Thomas

December 1998

This Page Intentionally Left Blank

vii

TABLE

OF CONTENTS

Prologue:

The Chemical Bond on the Eve of the 21st Century ....................... X I X Zvonimir B. Maksid and W. J. Orville - Thomas

Chapter 1. Theoretical Treatise on Molecular Structure and Geometry .............. 1 Jerzy Cioslowski 1. 2. 3. 4.

5. 6. 7. 8.

I n t r o d u c t i o n : T h e H i e r a r c h y of M o d e l s in C h e m i s t r y ......................... 1 M o l e c u l a r W a v e f u n c t i o n s ...................................................................... 3 D e c o u p l i n g of N u c l e a r and E l e c t r o n i c D e g r e e s of F r e e d o m ................. 4 The R e l e v a n c e of S p e c t r o s c o p i c States .................................................. 6 4.1. T i m e D e p e n d e n c e ........................................................................ 6 4.2. I n t e r a c t i o n s w i t h E x t e r n a l F i e l d s ............................................... 7 4.3. I n t e r m o l e c u l a r I n t e r a c t i o n s ....................................................... 8 D e s c r i p t i o n of M o l e c u l a r P h e n o m e n a with S p e c t r o s c o p i c and L o c a l i z e d States .................................................................................... 11 T h e C o n c e p t of M o l e c u l a r G e o m e t r y .................................................. 13 T h e C o n c e p t of M o l e c u l a r S t r u c t u r e ................................................... 15 C o n c l u d i n g R e m a r k s ............................................................................ 16

Chapter 2. Beyond the Born-Oppenheimer Approximation ................................. 21 D. B. Kinghorn and L. Adamowicz 1. I n t r o d u c t i o n .......................................................................................... 21 2. E q u i v a l e n t T r e a t m e n t of N u c l e i and E l e c t r o n s 2.1. E x p l i c i t S e p a r a t i o n of the C e n t e r - o f - m a s s M o t i o n ( M e t h o d I) ................................................................................... 22 2.2. E f f e c t i v e N o n - a d i a b a t i c M e t h o d ( M e t h o d II) ........................... 25 3. G r o u n d - s t a t e W a v e f u n c t i o n ................................................................. 29 4. V a r i a t i o n a l C a l c u l a t i o n s ....................................................................... 31 5. S a m p l e A p p l i c a t i o n s ............................................................................. 37 5.1. E x p l i c i t S e p a r a t i o n of the C e n t e r - o f - m a s s M o t i o n in V a r i a t i o n a l C a l c u l a t i o n s of E l e c t r o n Affinities of H-, D- and T-. ................................................................................... 37 5.2. C a l c u l a t i o n on H D ~ with E f f ect i ve N o n - a d i a b a t i c M e t h o d ....................................................................................... 39 6. G e n e r a l N - b o d y N o n - a d i a b a t i c W a v e f u n c t i o n .................................. 42 7. S u m m a r y ............................................................................................... 44

Chapter 3. The Mills-Nixon Effect: Fallacies, Facts and Chemical Relevance .............................................................................................. 47 Zvonimir B. Maksi~, Mirjana Eckert-Maksi~, Otilia M6 and Manuel Yd~ez 1. I n t r o d u c t i o n .......................................................................................... 47

viii 2. The M i l l s - N i x o n Effect: The first E x p e r i m e n t a l R e s u l t and T h e o r e t i c a l I n t e r p r e t a t i o n by S u t t o n and P a u l i n g .............................. 48 2.1. Definition of the M N - e f f e c t and some C o m m o n Fallacies ..................................................................................... 49 3. Structural C o n s e q u e n c e s of the M N - E f f e c t ......................................... 53 3.1. The R o l e of R e h y b r i d i z a t i o n ..................................................... 53 3.2. The Role of 7t-Delocalization .................................................... 57 3.3. P a r a d i g m a t i c I n d a n and T e t r a l i n Cases ................................... 59 3.4. The Ring Size Effect .................................................................. 61 3.5. The Effect of the D o u b l e Bond and L o n e Pair(s) ..................... 67 3.6. Amplification of the M i l l s - N i x o n Effect .................................... 72 3.7. E x t e n d e d ~ - S y s t e m s : [ N ] p h e n y l e n e s .......................................... 75 4. R e v e r s e d M i l l s - N i x o n Effect ................................................................. 79 5. C h e m i c a l C o n s e q u e n c e s of the M i l l s - N i x o n Effect .............................. 85 5.1. E l e c t r o p h i l i c S u b s t i t u t i o n R e a c t i v i t y ........................................ 85 5.2. M i s c e l l a n e o u s P h y s i c a l and C h e m i c a l Properties .................... 94 6. C o n c l u d i n g R e m a r k s ............................................................................ 96

Chapter 4. Predicting Structures of Compounds in the Solid State by the G l o b a l O p t i m i z a t i o n A p p r o a c h ................................................... 103 J.C. SchOn and M. Jansen 1. 2. 3. 4.

I n t r o d u c t i o n ........................................................................................ 103 T h e E n e r g y L a n d s c a p e ....................................................................... 105 T h e L i d - and the T r e s h o l d - a l g o r i t h m ............................................. 108 S t r u c t u r e P r e d i c t i o n at L o w T e m p e r a t u r e s ....................................... 110 4.1. G e n e r a l Aspects ....................................................................... 110 4.2. Specific O p t i m i z a t i o n A l g o r i t h m s ........................................... 111 4.3. Specific Empirical Potentials .................................................. 112 5. E x a m p l e s ............................................................................................. 114 6. C o n n e c t i o n s to Earlier Studies of the E n e r g y Surface of C o m p l e x S y s t e m s ................................................................................ 123

Chapter 5. P o l a r i z a b i l i t y a n d H y p e r p o l a r i z a b i f i t y of Atoms and Ions ............... 129 David M. Bishop 1. Historic T i m e s ..................................................................................... 129 2. T h e P a p e r ............................................................................................ 131 3. S u r v e y of P o l a r i z a b i l i t y and H y p e r p o l a r i z a b i l i t y C a l c u l a t i o n s ........ 134 3.1. Static Dipole Polarizabilities (~) ............................................. 135 3.1.1. The He Isoelectronic Series ....................................... 136 3.1.2. The Ne Isoelectronic Series ....................................... 138 3.1.3. The Ar Isoelectronic Series ....................................... 138 3.2. Static D i p o l e H y p e r p o l a r i z a b i l i t i e s ......................................... 138 3.2.1. The H A t o m ............................................................... 139 3.2.2. The He Isoelectronic Series ....................................... 139 3.2.3. The Ne Isoelectronic Series ....................................... 139 3.2.4. The Ar Isoelectronic Series ....................................... 140

ix

3.3. D y n a m i c Dipole Polarizabilities and H y p e r p o l a r i z a b i l i t i e s . . . 141 3.3.1. T h e H A t o m ............................................................... 142 3.3.2. The He Isoelectronic Series ....................................... 142 3.3.3. The Ne and Ar I s o e l e c t r o n i c Series .......................... 143 4. C o n c l u s i o n s and O t h e r Aspects ........................................................... 143

Chapter 6. Molecular Polarizabilities and Magnetizabilities .............................. 147

P~I Dahle, Keneth Ruud, Trygve Helgaker and Peter R. Taylor 1. I n t r o d u c t i o n ........................................................................................ 147 2. M o l e c u l a r Properties as E n e r g y D e r i v a t i v e s ..................................... 149 3. M o l e c u l a r Properties in the D i a g o n a l R e p r e s e n t a t i o n of the H a m i l t o n i a n ........................................................................................ 156 4. E x p l i c i t E x p r e s s i o n s for Electric and M a g n e t i c P r o p e r t i e s .............. 159 5. L o n d o n O r b i t a l s .................................................................................. 162 6. T h e C a l c u l a t i o n of M o l e c u l a r M a g n e t i z a b i l i t i e s : C o m p a r i s i o n with E x p e r i m e n t ........................................................... 170 7. P a s c a l ' s Rule and G-Ring C u r r e n t s ................................................... 172 8. A r o m a t i c M o l e c u l e s and ~ - B o n d C u r r e n t s ....................................... 178 9. T h e P o l a r i z a b i l i t y of N o r m a l - and C y c l o - alkanes .......................... 179 10. The P o l a r i z a b i l i t y of P o l y a r o m a t i c H y d r o c a r b o n s ........................... 183 11. C o n c l u s i o n s ......................................................................................... 184

Chapter 7. The Concept of Electronegativity of Atoms in Molecules ................. 189

Juergen Hinze 1. 2. 3. 4. 5. 6. 7.

I n t r o d u c t i o n ........................................................................................ 189 P a u l i n g ' s Definition of E l e c t r o n e g a t i v i t y ........................................... 190 M u l l i k e n ' s Definition of E l e c t r o n e g a t i v i t y ......................................... 193 Orbital E l e c t r o n e g a t i v i t y and Electrical P o t e n t i a l ............................ 195 Orbital E l e c t r o n e g a t i v i t y V a l u e s ........................................................ 199 E l e c t r o n e g a t i v i t y E q u a l i z a t i o n and C h a r g e D i s t r i b u t i o n .................. 202 M o l e c u l a r P r o p e r t i e s .......................................................................... 204 7.1. B o n d L e n g t h s ........................................................................... 205 7.2. B o n d E n e r g i e s .......................................................................... 208 8. C o n c l u s i o n ........................................................................................... 210

Chapter 8. On Hybrid Orbitals in M o m e n t u m Space ......................................... 213

B. James Clark, Hartmut L. Schmider and Vedene H. Smith, Jr. 1. I n t r o d u c t i o n ....................................................................................... 1213 2. F o u r i e r T r a n s f o r m s of P o s i t i o n - s p a c e H y b r i d s ................................. 214 3. H y b r i d s in M o m e n t u m Space ............................................................. 215 3.1. H y b r i d s of the spa-Type ........................................................... 215 3.2. H y b r i d s I n v o l v i n g d-Orbitals .................................................. 217 4. M o m e n t s of the H y b r i d Orbitals ........................................................ 226 5. C o n c l u s i o n ........................................................................................... 228

Chapter 9. Theory as a Viable Partner for Experiment- The Quest for Trivalent Silylium Ions in Solution .............................................. 231 Carl-Henrik Ottosson, Elfi Kraka and Dieter Cremer 1. I n t r o d u c t i o n ........................................................................................ 231 1.1. W h y to Investigate S i l y l i u m Ions in Solution ? ...................... 232 1.2. C o n n e c t i o n to P a u l i n g ' s W o r k and Scope of the Article ....... 234 2. The N M R / a b i n i t i o / I G L O M e t h o d ................................................... 235 3. The S i l y l i u m Ion P r o b l e m ................................................................... 242 3.t. Properties of S i l y l i u m Ions in the Gas Phase .......................... 243 4. S i l y l i u m and C a r b e n i u m Ions in Solution. I n t e r a c t i o n of S o l v e n t s and C o u n t e r i o n s ............................................................... 246 4.1. Definition of a Nearly Free Silylium Ion R3Si ~ in S o l u t i o n ................................................................................ 246 4.2. C a r b e n i u m Ions R 3 C + in Solution ........................................... 254 5. S o l v a t i o n of Neutral Silyl C o m p o u n d s R3SiX and S i l y l i u m Ions R3Si +.............................................................................. 256 5.1. N e u t r a l S i - c o m p o u n d s in S o l u t i o n ......................................... 257 5.2. Specific C o m p l e x a t i o n of R3Si + by Nucleophilic Solvent Molecules .............................................. 258 5.3. C o u n t e r i o n s used in R e s e a r c h on R3Si ~ a n d R 3 C ~- Ions in S o l u t i o n .............................................................. 260 6. Structure D e t e r m i n a t i o n of Silyl Cations in S o l u t i o n ....................... 262 6.1. S t r u c t u r e D e t e r m i n a t i o n by the N M R / a b i n i t i o / I G L O M e t h o d ................................................. 263 7. I n t r a m o l e c u l a r Solvation of S i l y l i u m Ions ......................................... 266 7.1. Strong I n t r a m o l e c u l a r S o l v a t i o n of Silyl Cations .................. 267 7.2. W e a k I n t r a m o l e c u l a r Solvation of S i l y l i u m Ions ................... 272 8. A p p r o a c h i n g a nearly Free S i l y l i u m Ion in S o l u t i o n ......................... 277 8.1. T r i a l k y l s i l y l i u m Ions in A r o m a t i c Solvents ............................ 277 8.2. Silyl Substituted S i l y l i u m Ions in Solution ............................. 281 8.3. D i a l k y l b o r y l Substituted S i l y l i u m Ions in S o l u t i o n ................ 284 9. The S o l u t i o n of the P r o b l e m : First G e n e r a t i o n of a Free S i l y l i u m Cation in C o n d e n s e d Phases ....................................... 287

Chapter 10. Bond Energies, Enthalpies of Formation, and Homologies: The Energetics of Aliphatic and Alicyclic Hydrocarbons and some of their Derivatives ................................................................... 303 Suzanne W. Slayden and Joel F. Liebman 1. T e t r a c o o r d i n a t i o n , T e t r a h e d r a l G e o m e t r y and H y b r i d i z a t i o n ..................................................................................... 303 2. The N u m b e r of C o m p o u n d s and the N e c e s s i t y for I n t e r c o n n e c t i o n s , H o m o l o g i e s and H o m o l o g o u s Series ................... 304 3. H o m o l o g o u s Series: The 1-Substituted Alkanes ............................... 304 4. H o m o l o g o u s Series: C y c l o a l k a n e s ..................................................... 310 5. H o m o l o g o u s Series: Saturated P o l y c y c l i c H y d r o c a r b o n s ................ 310 6. T e t r a h e d r a n e and [ 1.1. l ] P r o p e l a n e .................................................. 312

xi

6.1. T e t r a h e d r a n e ........................................................................... 313 6.2. [ 1 . 1 . 1 ] P r o p e l l a n e ...................................................................... 315

Chapter 11. Stabilization and Destabilization Energies of Distorted Amides ..... 321 Arthur Greenberg and David T. Moore 1. I n t r o d u c t i o n ....................................................................................... 321 1.1. 1.2. 1.3. 1.4. 1.5.

C h e m i c a l I m p l i c a t i o n s of S t r a i n e d A m i d e s and L a c t a m s ..... 321 B i o l o g i c a l I m p l i c a t i o n s ............................................................ 322 Effects of D i s t o r t i o n on A c i d / B a s e P r o p e r t i e s ........................ 323 D e f i n i n g D i s t o r t i o n of the A m i d e L i n k a g e ............................. 323 L a r g e r B r i d g e h e a d B i c y c l i c L a c t a m s : Are T h e y H y p e r s t a b l e ? ............................................................................ 324 2. B a c k g r o u n d ........................................................................................ 325 2.1. E n e r g e t i c s of D i s t o r t e d L a c t a m s ............................................. 325 2.2. B o n d i n g in L a c t a m s : Is there Still a R o l e for R e s o n a n c e ? .............................................................................. 326 2.3. P r o t o n Affinities of B r i d g e h e a d B i c y c l i c L a c t a m s : N vs. O ..................................................................................... 327 2.4. State of C a l c u l a t i o n a l Studies of D i s t o r t e d A m i d e L i n k a g e s ................................................................................... 328 3. C o m p u t a t i o n a l S t u d i e s ...................................................................... 328 3.1. M o l e c u l a r M e c h a n i c s .............................................................. 328 3.2. Ab initio C a l c u l a t i o n s ............................................................. 334 3.3. S e m i - e m p i r i c a l R e s u l t s ............................................................ 337 4. S u m m a r y ............................................................................................ 343

Chapter 12. Some Chemical and Structural Factors Related to the Metastabilities of Energetic Compounds ................................... 347 Peter Politzer and Jane S. Murray 1. I n t r o d u c t i o n ....................................................................................... 347 2. I m p a c t / S h o c k S e n s i t i v i t y and M o l e c u l a r St ruct ure: S o m e B a c k g r o u n d ............................................................................ 348 2.1. S t r u c t u r e - sensitivity R e l a t i o n s h i p ........................................ 348 2.2. S o m e Specific D e c o m p o s i t i o n P a t h w a y s ................................. 349 3. R e l a t i o n s h i p s B e t w e e n I m p a c t Sensitivities and M o l e c u l a r S u r f a c e E l e c t r o s t a t i c P o t e n t i a l s ...................................... 351 3.1. A n a l y s i s and C h a r a c t e r i z a t i o n of S u r f a c e P o t e n t i a l s ............ 351 3.2. U n s a t u r e t e d C - N i t r o D e r i v a t i v e s : N i t r o a r o m a t i c s and N i t r o h e t e r o c y c l e s ..................................................................... 352 3.3. I m p a c t S e n s i t i v i t y and S u r f a c e P o t e n t i a l I m b a l a n c e ............. 354 4. S u m m a r y ............................................................................................ 358

Chapter 13. Valence Bond Theory: A Re-examination of Concepts and Methodology ...................................................................................... 365 Roy Mc Weeny 1. I n t r o d u c t i o n ....................................................................................... 365

xii

2. T h e E l e c t r o n - p a i r B o n d : S o m e P r e l i m i n a r i e s .................................. 3. C l a s s i c a l VB T h e o r y : P e r f e c t - p a i r i n g and R e s o n a n c e ..................... 3.1. S y m m e t r y C o n s i d e r a t i o n s ....................................................... 3.2. C a l c u l a t i o n of the E n e r g y ........................................................ 4. T h e Rise and Fall of C l a s s i c a l VB T h e o r y ........................................ 5. M o d e r n VB T h e o r y ........................................................................... 5.1. VB T h e o r y w i t h O r t h o g o n a l O r b i t a l s ..................................... 5.2. T h e " N i g h t m a r e of the I n n e r S h e l l s " . ..................................... 5.3. VB T h e o r y with N o n - o r t h o g o n a l O r b i t a l s .............................. 5.4. C o n n e c t i o n w i t h O t h e r M e t h o d s ............................................. 6. S o m e I l l u s t r a t i v e A p p l i c a t i o n s .......................................................... 6.1. T h e W a t e r M o l e c u l e ................................................................ 6.2. M e t h y l l i t h i u m .......................................................................... 6.3. L i t h i u m F l u o r i d e ..................................................................... 6.4. B e n z e n e and Its Ions ................................................................ 7. C o n c l u s i o n ..........................................................................................

365 371 372 376 380 383 383 384 387 389 392 392 394 394 396 397

Chapter 14. Advances in Many-body Valence-bond Theory ............................... 403 Douglas J. Klein 1. 2. 3. 4.

I n t r o d u c t o r y S u r v e y .......................................................................... VB T h e o r y : Bases, M o d e l s , & R e s o n a n c e ......................................... M a n y - b o d y T h e o r y ............................................................................ M a n y - b o d y T e c h n i q u e s for VB M o d e l s ............................................ 4.1. C o n f i g u r a t i o n I n t e r a c t i o n ....................................................... 4.2. M a n y - b o d y P e r t u r b a t i o n T h e o r y ............................................ 4.3. C l u s t e r and M o m e n t M e t h o d s ................................................ 4.4. S p i n - w a v e s and G r e e n ' s F u n c t i o n s ......................................... 4.5. W a v e - f u n c t i o n C l u s t e r E x p a n s i o n .......................................... 4.6. M o n t e C a r l o C o m p u t a t i o n s ..................................................... 4.7. R e n o r m a l i z a t i o n - g r o u p T e c h n i q u e s ........................................ 4.8. M i s c e l l a n y ................................................................................ 5. O v e r v i e w and P r o s p e c t s ....................................................................

403 405 407 409 409 410 411 411 412 413 413 413 414

Chapter 15. Ab Initio Valence Bond Description of Diatomic Dications ............ 423 Harold Basch, Pinchas Aped, Shmaryahu Haz and Moshe Goldberg 1. I n t r o d u c t i o n ....................................................................................... 424 2. He2 + .................................................................................................. 426 3. 022+ ..................................................................................................... 428 4. N F 2+.................................................................................................... 433 5. S u m m a r y ........................................................................................... 438

Chapter 16. One-electron and Three-electron Chemical Bonding, and Increased-Valence Structures ........................................................... 449 Richard D. Harcourt 1. I n t r o d u c t i o n ....................................................................................... 449

xiii

2. The O n e - e l e c t r o n B o n d ..................................................................... 450 3. The O n e - e l e c t r o n B o n d and N o n - p a i r e d Spatial O r b i t a l S t r u c t u r e s ............................................................................. 452 4. A T h e o r e m ......................................................................................... 454 5. The T h r e e - e l e c t r o n Bond, or T h r e e - e l e c t r o n H a l f - b o n d ................. 454 5.1. P a r a m a g n e t i c Electron Rich M o l e c u l e s and M o l e c u l a r Ions that I n v o l v e A t o m s of M a i n - g r o u p E l e m e n t s ................. 456 5.2. H y p o l i g a t e d Transition Metal C o m p l e x e s , such as High-spin (S=2) [Fe(H20)6] a+ .................................................. 456 5.3. F+-type C o l o u r Centers ............................................................ 457 5.4. n - T y p e S e m i c o n d u c t o r s ........................................................... 458 5.5. C o n d u c t i o n in Alkali Metals in the Solid State ...................... 458 6. Instability of T h r e e - e l e c t r o n Bonds .................................................. 458 7. The T h r e e - e l e c t r o n B o n d with F o u r or M o r e A O s .......................... 460 8. T h r e e - e l e c t r o n B o n d s and I n c r e a s e d - v a l e n c e Structures for F o u r - e l e c t r o n T h r e e - c e n t r e B o n d i n g ................................................ 462 9. I n c r e a s e d - v a l e n c e Structures and M u l l i k e n - D o n o r acceptor C o m p l e x e s ........................................................................... 464 10. I n c r e a s e d - v a l e n c e Structures and SN2 Reactions ............................. 465 11. T h r e e - E l e c t r o n B o n d s and F i v e - e l e c t r o n T h r e e - c e n t r e B o n d i n g .............................................................................................. 466 12. T h r e e - E l e c t r o n B o n d s and I n c r e a s e d - v a l e n c e Structures for E x t e n d e d S i x - e l e c t r o n F o u r - c e n t r e B o n d i n g .............................. 469 13. T h r e e - e l e c t r o n B o n d s and I n c r e a s e d - v a l e n c e Structures for Cyclic Six-electron Four-centre B o n d i n g .................................. 473 14. T h r e e - e l e c t r o n B o n d s and C o v a l e n t - i o n i c R e s o n a n c e ..................... 475 15. Conclusions ........................................................................................ 477

Chapter 17. Valence Bond Description of ~-Electron Systems ............................ 481 Joseph Paldus and X. Li 1. 2. 3. 4.

I n t r o d u c t i o n ....................................................................................... 481 P P P - t y p e H a m i l t o n i a n s ..................................................................... 483 PPP-VB M o d e l .................................................................................. 486 A p p l i c a t i o n s ....................................................................................... 488 4.1. C o r r e l a t e d G r o u n d States, Basic T r a n s f e r a b i l i t y and The R o l e of Ionic Structures ............................................ 488 4.2. Spin P r o p e r t i e s ........................................................................ 490 4.3. Electron Delocalization, R e s o n a n c e and B o n d L e n g t h A l t e r n a t i o n .............................................................................. 492 4.4. Excited States ........................................................................... 493 4.5. VB C o r r e c t e d C o u p l e d C l u s t e r M e t h o d ................................. 493 4.6. Ionization Potentials and Electron Affinities .......................... 494 5. Conclusions ........................................................................................ 495

xiv

Chapter 18. The Spin-coupled Description of Aromatic, Antiaromatic and Nonaromatic Systems ........................................................................ 503 David L. Cooper, Joseph Gerratt and Mario Raimondi 1. 2. 3. 4. 5. 6.

I n t r o d u c t i o n ....................................................................................... S p i n - c o u p l e d W a v e f u n c t i o n s ............................................................ B e n z e n e .............................................................................................. C y c l o b u t a d i e n e .................................................................................. C y c l o o c t a t e t r a e n e ............................................................................... C o n c l u s i o n s ........................................................................................

503 505 507 511 514 514

Chapter 19. Aromaticity and Its Chemical Manifestations ................................. 519 Keneth B. Wiberg 1. H i s t o r i c a l P r e l u d e .............................................................................. 519 2. 3. 4. 5.

V a l e n c e B o n d vs. M o l e c u l a r O r b i t a l T h e o r y .................................... M a n i f e s t a t i o n s o f " A r o m a t i c " S t a b i l i z a t i o n ..................................... S i g m a C o n t r i b u t i o n to the G e o m e t r y o f B e n z e n e ............................ M a g n e t i c P r o p e r t i e s .......................................................................... 6. O r i g i n of the S t a b i l i z a t i o n o f B e n z e n e .............................................. 7. H e t e r o c y c l i c A r o m a t i c S y s t e m s ........................................................ 8. S u m m a r y ...........................................................................................

521 523 527 529 532 532 533

Chapter 20. Hypercoordinate Bonding to Main Group Elements: The Spin-coupled Point of V i e w ............................................................... 537 David L. Cooper, Joseph Gerratt and Mario Raimondi 1. I n t r o d u c t i o n ....................................................................................... 537 2. d - O r b i t a l P a r t i c i p a t i o n V e r s u s D e m o c r a c y ..................................... 538 3. H y p e r c o o r d i n a t e B o n d i n g to F i r s t - r o w A t o m s ................................ 543 3.1. 1.3-Dipoles ................................................................................ 543 3.2. O x o h a l i d e s of H y p e r c o o r d i n a t e N i t r o g e n and P h o s p h o r o u s 547 4. F u r t h e r E x a m p l e s .............................................................................. 548 4.1. O x o f l u o r i d e s of H y p e r c o o r d i n a t e S u l f u r ................................ 548 4.2. C h l o r i n e F l u o r i d e s and C h l o r i n e O x i d e F l u o r i d e s ................. 550 4.3. F l u o r o p h o s p h o r a n e s ................................................................ 550 4.4. Y X X Y D i h a l i d e s and D i h y d r i d e s of D i o x y g e n and Disulfur..551 5. C o n c l u s i o n s ........................................................................................ 551

Chapter 21. The Electronic Structure of Transition Metal Compounds ............. 555 Gernot Frenking, C. Boehme and U. Pidun 1. I n t r o d u c t i o n ....................................................................................... 555 2. C o m p u t a t i o n a l D e t a i l s ...................................................................... 556 3. R e s u l t s and D i s c u s s i o n ...................................................................... 558 3.1. C h e m i c a l B o n d i n g in [(CO)sW-A1CI(NH3)2] a n d [(CO)sW-A1C1] ........................................................................ 558 3.2. T h e Series [(CO)sW-XCI(NH3)2] (X = B,A1,Ga,In,T1) ........... 562 3.3. T h e series [ ( C O ) s W - Y ] (Y = [SiC12(NH3)], [A1CI(NH3)2], [Mg(NH3)3], [Na(NH3)3]) ......................................................... 565

XV

4. S u m m a r y and C o n c l u s i o n s ............................................................... 568

Chapter 22. Fundamental Features of Hydrogen Bonds ..................................... 571 Steve Scheiner 1. I n t r o d u c t i o n ....................................................................................... 571 2. 3. 4. 5.

H y d r o g e n B o n d i n g S t r e n g t h ............................................................. C o n t r i b u t i o n of E l e c t r o s t a t i c s ........................................................... R e l a t i o n s B e t w e e n V a r i o u s P r o p e r t i e s ............................................. C o o p e r a t i v i t y ..................................................................................... 5.1. G e o m e t r i e s ............................................................................... 5.2. E n e r g e t i c s ................................................................................. 5.3. V i b r a t i o n a l S p e c t r a ................................................................. 5.4. E n e r g y C o m p o n e n t s ................................................................ 6. S u m m a r y ...........................................................................................

572 574 578 581 582 584 585 586 589

Chapter 23. Molecular Similarity and Host-guest Interactions ........................... 593 Paul G. Mezey 1. 2. 3. 4.

I n t r o d u c t i o n ....................................................................................... 594 F r o m F u n c t i o n a l G r o u p s to E x t e n d e d M o l e c u l a r R e g i o n s .............. 596 E l e m e n t s of E l e c t r o n D e n s i t y S h a p e A n a l y s i s .................................. 600 E l e c t r o n D e n s i t y A n a l y s i s o f I s o l a t e d and I n t e r a c t i n g R e a c t i v e R e g i o n s of M o l e c u l e s .......................................................... 602 5. S h a p e S i m i l a r i t y M e a s u r e s in the S t u d y of H o s t - g u e s t I n t e r a c t i o n s ........................................................................................ 607 6. S u m m a r y ........................................................................................... 609

Chapter 24. Chemical Bonding in Proteins and Other Macromolecules ............ 613 Paul G. Mezey 1. I n t r o d u c t i o n ....................................................................................... 614 2. M a c r o m o l e c u l a r Q u a n t u m C h e m i s t r y B a s e d on A d d i t i v e F u z z y D e n s i t y F r a g m e n t a t i o n ( A F D F ) .............................. 616 3. " L o w D e n s i t y G l u e " ( L D G ) B o n d i n g in P r o t e i n s ............................. 624

Chapter 25. Models for Understanding and Predicting Protein Structure ......... 637 Dale F. Mierke 1. I n t r o d u c t i o n ....................................................................................... 2. M e t h o d s ............................................................................................. 2.1. H o m o l o g y M o d e l l i n g ............................................................... 2.2. S e c o n d a r y S t r u c t u r e P r e d i c t i o n .............................................. 2.3. P r i m a r y to T e r t i a r y P r e d i c t i o n ............................................... 2.4. E n e r g e t i c F o r c e F i e l d s ............................................................. 2.5. R e d u c e d A t o m R e p r e s e n t a t i o n ................................................ 2.6. R e d u c e d C o n f o r m a t i o n a l S p a c e / L a t t i c e M o d e l s .................... 3. C o n c l u s i o n s ........................................................................................

637 640 641 643 644 645 646 649 650

xvi

Chapter 26. Possible Sources of Error in the Computer Simulation of Protein Structures and Interactions ............................................. 655 J.M. Garcia de la Vega, J.M.R. Parker and Serafin Fraga 1. 2. 3. 4.

I n t r o d u c t i o n ....................................................................................... 655 Deficiencies of Potential E n e r g y Functions ...................................... 656 C o n f o r m a t i o n a l C h a r a c t e r i z a t i o n .................................................... 658 C o n c l u s i o n s ........................................................................................ 661

Chapter 27. The Nature of Van der Waals Bond ................................................. 665 Grzegorz Chalasinski, Malgorzata M. Szczesniak and Slawomir M. Cybulski 1. 2. 3. 4.

I n t r o d u c t i o n ......................................................................................... 66 5 F u n d a m e n t a l I n t e r a c t i o n E n e r g y C o m p o n e n t s ............................... 666 Ab Initio A p p r o a c h to I n t e r m o l e c u l a r F o r c e s ................................. 667 4.1. E x c h a n g e R e p u l s i o n versus M o l e c u l a r Shape ........................ 670 4.2. D i s p e r s i o n as the I n t e r m o n o m e r C o r r e l a t i o n Effect .............. 673 4.3. I n d u c t i o n , C h a r g e - t r a n s f e r and S C F D e f o r m a t i o n ............... 675 4.4. E x a m p l e 1. Ar-CO2: Dispersion B o u n d C o m p l e x .................. 676 4.5. E x a m p l e 2. W a t e r D i m e r : I n t r o d u c i n g E l e c t r o s t a t i c s ........... 679 4.6. G e n e r a l C o n s i d e r a t i o n s ........................................................... 682 5. M o d e l l i n g of PES and its C o m p o n e n t s ............................................. 682 5.1. A r - C O 2 ..................................................................................... 683 5.2. W a t e r D i m e r ............................................................................ 684 6. T r i m e r s and N o n a d d i t i v e Effects ...................................................... 687 6.1. A r 2 - C h r o m o p h o r e Clusters: E x c h a n g e and D i s p e r s i o n N o n a d d i t i v i t y ........................................................................... 688 6.2. W a t e r T r i m e r : I n d u c t i o n N o n a d d i t i v i t y ................................ 695 7. S u m m a r y ........................................................................................... 696

Chapter 28. The Nature of the Chemical Bond in Metals, Alloys, and Intermetallic Compounds According to Linus Pauling ................... 701 Zelek S. Herman 1. 2. 3. 4.

I n t r o d u c t i o n ....................................................................................... 701 Q u a n t u m M e c h a n i c s and the N a t u r e of M e t a l s ............................... 703 T h e M e t a l l i c Orbital .......................................................................... 705 The D e t a i l e d A n a l y s i s of the Statistical T h e o r y of U n s y n c h r o n i z e d R e s o n a n c e of C o v a l e n t Bonds ............................... 710 5. C a l c u l a t i o n of the N u m b e r of M e t a l l i c Orbitals per A t o m from the Statistical T h e o r y of the U n s y n c h r o n i z e d R e s o n a n c e of C o v a l e n t Bonds ........................................................... 715 6. T h e C r y s t a l S t r u c t u r e s of the M e t a l s and the M a x i m u m Values of the metallic V a l e n c e .......................................................... 718 7. The C o m p i l a t i o n of M e t a l l i c S i n g l e - b o n d Radii and Radii for L i g a n c y 12 .................................................................................... 722

xvii

8. The Structure and P r o p e r t i e s of E l e m e n t a l Boron. Is it a M e t a l ? ............................................................................................. 724 9. The Nature of the M e t a l - M e t a l B o n d in Alloys, I n t e r m e t a l l i c C o m p o u n d s , and on the Surfaces of Alloys ...................................... 726 10. S u p e r c o n d u c t i v i t y I n t e r p r e t e d in T e r m s of the U n s y n c h r o n i z e d - r e s o n a t i n g - c o v a l e n t - b o n d T h e o r y of Metals ........ 732 11. Conclusions ........................................................................................ 738

Epilogue:

Linus Pauling, Quintessential Chemist ............................................ 749 Dudley Herschbach

Index

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

755


xix PROLOGUE T h e C h e m i c a l B o n d o n t h e E v e of t h e 21st C e n t u r y

Zvonimir B. Maksi5 and W.J. Orville-Thomas Linus Pauling is rated as the most prominent American scientist and the greatest chemist of this century. To some people he was Moses who has led chemists to the promised land, whereas others imagined him as the mythical Prometheus who brought quantum mechanical fire to classical chemistry. One thing is beyond doubt - nobody made so many important discoveries in so many different branches of chemistry and related disciplines as Linus Pauling. As Dudley Herschbach put it at the end of this book, he was the quintessential chemist. The younger generation considered him, like his grandson Alexander Kamb, as a force of nature. The latter is also the title of an excellent biography skilfully written by Thomas Hager [1]. This book is dedicated to Pauling and his work focusing on the chemical bond. It is, therefore, appropriate to begin with Pauling's own words: "The concept of the chemical bond is the most valuable concept in chemistry. Its development over the past 150 years has been one of the greatest triumphs of the human intellect. I doubt that there is a chemist in the world who does not use it in his or her thinking. Much of modern science and technology has developed because of the existance of this concept" [2]. This is perfectly true: the chemical bond is one of the three most important cornerstones of classical chemistry, together with the notion of atoms in chemical environments and the idea of molecular structure and geometry. The latter reflects a multitude of properties of molecules stored in their structural parameters, size, shape and symmetry. These classical pillars received a proper interpretation and physical meaning from quantum mechanics with one notable exception - molecular structure - which still poses a problem not rigorously solved as yet from first principles. Many researchers have contributed to this remarkable progress over many decades, and one could characterize the development of quantum chemistry as a permanent crawling revolution in molecular sciences particularly taking into account recent advances in computational chemistry. Linus Pauling was, however, the pioneer and champion of quantum chemistry in the pre-computer era. It took a genius and a vivid imagination to tackle intricate and perplexing chemical problems by using a slide rule and to make tremendous leaps in understanding chemical bonding, which has substantionally contributed to dramatic growth in the life sciences that we have witnessed in recent years. By using his astonishing ability to reduce the complex to the simple, Pauling shed light on the architecture of molecules and crystals. He explained the directional properties of covalent bonds in an elegant way by introducing polarized local hybrid (chemical) orbitals and inaugurated the concept of resonance within the classical valence bond (VB) theory, which in turn is undergoing a remarkable renaissance. Pauling was the first to establish a quantitative electronegativity scale thus enabling a simple description of charge distributions in molecules and providing a rationalization of the ionic component of chemical bonding. Combining resonance with the electrostatic interactions, Pauling discovered the important role of hydrogen bonding in determining weak intra- and intermolecular interactions. These interactions proved crucial in understanding essential features of molecules

XX

of life to mention only proteins. His work on the nature of the peptide bond and on the structural patterns of proteins in terms of alpha helices and beta-pleated sheets are milestones in the development of biochemistry and molecular biology. Instead of listing all discoveries and work which stimulated others to unravel the secrets of Nature - to single out only the Crick-Watson model of DNA as an enlightening example - we shall succintly say that he erected a more lasting scientific monument than those made of brass or stone. Pauling was a grand master of modelling in science. His models were very simple, reflecting the quintessence of a phenomenon or property under scrutiny and satisfying the Occam razor principle at the same time. They provide a qualitative understanding of the fundamental principles of chemical structure, bonding and reactivity, thus serving as a guide in the research process. These models are close to chemical intuition by building bridges between a rich chemical experience on one side and rigorous quantum mechanics on the other [3]. It should be stressed that Pauling's models did not only have a heuristic value, but also provided important semiquantitative information on a variety of molecular properties in the pre-computer age. They involve elementary, back of the envelope, calculations, illustrating in the highest sense of the word van't Hoff's statement that imagination and shrewed guess work are powerful instruments in acquiring scientific knowledge. The success of the Pauling's approach is best described by the Figure below, where the accuracy of theoretical models in reproducing a particular property of a very large compound or system of chemical interest is schematically plotted against the rigour of the applied theoretical procedures:

ca. true value

~

~

/

f

Pauling Ph.D. point point

Figure

~

rigour approaching full theory postdoctoral result

xxi This curve possesses several characteristic maxima and minima asymptotically approaching the exact value for the full theory. The first maximum corresponds to the Pauling point, where a simple and transparent physical model gives insight and reasonable agreement with experiment by focusing on the dominant effect(s) only. The quantitative description requires a much more sophisticated theoretical approach and meticulous calculations. It should be emphasized, however, that simple conceptual models have led to great discoveries in the molecular sciences, which cracked some very important codes of nature, more frequently than the exact theories and detailed calculations. The latter usually came in the a posteriori stage to confirm that a bold hypothesis was correct. Not all of Pauling's models and concepts were new and original. For instance, the electronegativity idea dates back to Avogadro and Berzelius in the beginning of the 19th century [4]. However, he gave to many of them a deeper meaning and showed their chemical relevance by utilizing his encyclopedic knowledge. His views on chemical bonding were summarized in a superb landmark book "The Nature of the Chemical Bond" [5], which inspired generations of chemists. It is frequently cited as one of the most influential scientific books of our century. This is not surprising because well established models provide in general a scientific vocabulary and lend themselves to classification purposes. They give a pervasive physical insight and extract the key features of very complex phenomena, thus revealing their essence and simplicity. It should be stressed that reliable models possess a grain of truth, which is not always realised. They are true within the limits of the approximations involved and within a carefully determined range of applicability - no more but, at the same time, no less. Metaphorically speaking, models extend the range of our senses and make it possible to "see" mentally what cannot be seen [6]. In the meantime, breakthroughs in tackling molecular many-body problems by computational quantum chemistry based on new theoretical schemes, novel numerical methods and the dramatically fast development of computer technology made possible quantitative description of versatile chemical bonding phenomena comparable to that offered by experiments. It is timely to give a modern, present-day, theoretical description to many of the apparently successful Pauling models and seminal ideas and to present a refined interpretation of many subtle effects, which were not amenable to theoretical analysis earlier. Coverage of the recent advances in modelling of chemical bonding is therefore the main task of this book written by some of the most prominent experts in the field. Chapters on the molecular structure, geometries of fused aromatics and their electrophilic reactivity, bond energy, electronegativity, hybridization, aromaticity, contemporary VB methods, hydrogen bonding and the structure of the proteins and other large biological compounds reflect much of the leading current thinking. They are prepared by carefully avoiding dangers of the Scylla of intricacy and the Charybdis of oversimplification and by putting a considerable emphasis on the interpretation of theoretical results. It can be stated safely that most of Pauling's models have stood the test of time and found rigorous justification. However, it should be pointed out strongly that many authors, by building on Pauling's ideas and by standing on his shoulders, overcame the limitations of the old, crude and sometimes fully empirical models by making bold steps forward thus expanding the frontiers of molecular sciences. Although such a book is never complete, it is our belief and hope that it will contribute to better understanding of the ubiquitous chemical bond and become an indispensable textbook for post-

xxii graduate/doctoral students in physical and advanced physical organic chemistry. It is important to point out in this connection that quantum chemistry - the Holy Grail of molecular sciences - will have an ever-increasing role in the 21st century particularly in establishing strong links between chemistry and molecular biology and thus featuring as a unifying methodology. Finally, we would like to use this opportunity to thank all authors for their scholarly written and intellectually stimulating chapters, which made this book possible. REFERENCES 1. 2. 3.

4. 5. 6.

T. Hager, Force of Nature - The Life of Linus Pauling, Simon & Schuster, New York, NY, 1995. L. Pauling, The Nature of the Chemical Bond - 1992, J. Chem. Ed., 69 (1992) 519. Z.B. Maksi(~, On the Significance of Theoretical Models of Chemical Bonding- Prologue to the Special Subject Issue on Conceptual Quantum Chemistry: Models and Applications, Part 1, Croat. Chem. Acta 57 (1984) No. 5. W.B. Jensen, Electronegativity from Avogadro to Pauling, J. Chem. Ed., 73 (1996) 11. L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals, Third Ed., Cornell University Press, 1960. Z.B. MaksiS, Modelling - A Search for Simplicity, in Theoretical Models of Chemical Bonding, Vol. 1, Z.B. MaksiS, Ed., Springer Verlag, Berlin- Heidelberg, 1990, p. 13.

Z.B. Maksid and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modern Modelling of the Chemical Bond Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

THEORETICAL TREATISE ON MOLECULARSTRUCTURE AND GEOMETRY Jerzy Cios lowski Department of Chemistry and Supercomputer Computations Research I n s t i t u t e , ,

Florida St a t e University, Yallahassee, F l o r i d a 32306-3006, USA

1. INTRODUCTION: THE HIERARCHY OF MODELS IN CHEMISTRY The primary o b j e c t i v e of modern science f a i t h f u l l y describe the r e a l i t y . subject

to perpetual

tency.

When a p a r t i c u l a r

reasons.

is

to construct

theories

that

Theories are nothing but models that are

experimental

s c r u t i n y and checks of i n t e r n a l

theory is abandoned,

it

consis-

is for one of several

Some t h e o r i e s , such as those of f l o g i s t o n and c a l o r i c , are simply

proven wrong.

Others,

such as the g e o c e n t r i c theory of the unive rs e ,

superseded by simpler d e s c r i p t i o n s

of r e a l i t y .

Finally,

are

some formalisms

(such as the Newtonian mechanics) are found to possess only a limited val i d i t y or to c o n s t i t u t e special cases of more general ( u n i f i e d ) t h e o r i e s . In this r e s p e c t , chemistry does not d i f f e r from other sciences.

Contem-

porary chemical research is organized around a hierarchy of models that aid its practitioners phenomena.

in t h e i r everyday quest for the understanding of n a t u r a l

The building blocks of the language of chemistry, including the

representations

of molecules

in terms of s t r u c t u r a l

the very bottom of this h i e r a r c h y . as

reaction

etc.

[2],

types and mechanisms,

come next.

formulae

[1],

occupy

Various phenomenological models, such thermodynamics and chemical

Quantum chemistry, which at present

kinetics,

is the supreme

theory of e l e c t r o n i c s t r u c t u r e s of atoms and molecules, and thus of the ent i r e realm of chemical phenomena, resides at the very top.

This research was supported by the National Science Foundation under the grant CHE-9632706.

E-mail address: [email protected], web page:

ht tp: //www. scr i. fsu. edu/~j erzy.

2

Being first formulated in the second half of the nineteenth century, the concept of molecular structure has evolved from a working hypothesis to the major tenet of chemistry by the time of the advent of modern quantum mechanics. At last, the new theory provided the means for predicting and explaining properties of atoms and molecules. However, with its description of matter that was (and still is) alien to many chemists conditioned by the experiences in the macroscopic world, the new theory stood little chance of displacing the existing models of chemical species and their transformations. Consequently, peculiar hybrid formalisms that invoke conventional chemical notions dressed up in the language of quantum mechanics have soon emerged. These formalisms, which are collectively known as the electronic structure theory, are in use t o this day. Modern electronic structure theory employs two levels of simplification. The use of various mathematical approximations is dictated by the limitations of computer hardware and the need for keeping the cost of quantumchemical calculations within reasonable limits. In contrast, the avoidance of quantum-mechanical treatment of nuclei i s deeply rooted in the aforementioned conceptual prejudices. While the severity of mathematical approximations is on a constant decrease thanks to the ever-increasing speed and availability of computers (note the gradual disappearance of semiempirical calculations from the chemical literature!), the validity of views that regard molecules as quasi-rigid assemblies of nuclei held together by electron clouds is rarely questioned by the majority of researchers. Twenty years have passed since the publication of the original paper by Woolley [3] in which the incompatibility of the molecular structure concept with the rigorous quantum-mechanical description of isolated molecules has been eloquently brought to the attention of chemists. The ensuing flurry of research publications clarified several misconceptions but did little to familiarize the broader scientific audience with this important issue. Regretfully, few quantum or computational chemists are aware of these papers, which are nowadays seldom discussed or quoted. This short treatise is intended to provide the reader with a concise summary of the current theoretical status of the molecular structure and geometry concepts. A fully quantum-mechanical treatment of molecules is employed where necessary. The relevance of stationary states of isolated molecules is discussed and the notion of molecular geometry is contrasted with that of molecular structure.

3 2 . MOLECULAR WAVEFUNCl'IONS A nonrelativistic description of molecules is provided by the well-known

Hamiltonian, A

h

A

A

H = T n + Te where

t V

,

A Tn = (-1/2)1 mI- 1 V,2

,

(2)

I A

Te = (-1/2)1 Vi2

,

(3)

1

and

In Eqs.(2-4), {m,} and {Z,} are the vectors of nuclear masses and charges, respectively; whereas R = {RI} and r = {ri} stand for the positions of nuclei and electrons. The atomic units are used throughout the text. A s the consequence of its translational invariance, H possesses infinitely many eigenstates {Y (r,R)} and its spectrum is continuous. The PN entire set {Y (r,R)} can be readily reconstructed from a finite manifold PN of eigenstates that correspond to zero linear momentum p,

(5) PN(r,R) = exp (ip-RCM ) YON (r-RCM,R-RCM) , is the position vector of the center of mass. In the following, where both r and R always refer to vectors relative to RcM, and YON (r-RCM,R-RCM) (the spectroscopic s t a t e s [4]) are denoted simply by YN(r,R). One should note that the removal of three degrees of freedom from (r,R) makes the spectroscopic states normalizable. Y

RCM

A

The invariance of H under other transformations of the coordinate system [5] imposes certain symmetries upon the elements of the set {YN(r,R)}. In particular, rotational invariance implies that these states must be eigenfunctions of the total angular momentum operators, "2 J YN(r,R) = J(J+l) YN(r,R) , AJz YN(r,R) = M YN(r,R) . (6)

A

In addition, thanks to the space-inversion symmetry of H, spectroscopic states must possess definite parities, YN(-r,-R)

=

Il YN(r,R)

,

Il

E

{-l,l}

.

Consequently, spectroscopic states can be labeled {YNJMn(r,R)}.

(7)

^

Although H is n o n r e l a t i v i s t i c , in the r e a l i z a b l e e i g e n s t a t e s

relativistic

e f f e c t s manifest

{~NJM~(r,R)} through permutational

themselves symmetry.

Spectroscopic s t a t e s must be t o t a l l y antisymmetric under the permutation of the labels of any two e l e c t r o n s . with respect

In a d d i t i o n ,

total antisymmetry/symmetry

to the permutations of the labels of fermionic/bosonic nuclei

must be exhibited.

For this reason, the set {~NJMH(r,R)} is not determined

by {mi,Zi} alone but by {mI,ZI,SI}, where {S I} is the set of nuclear spins. Let ~q(r,R) be an a r b i t r a r y t r i a l wavefunction expressed in terms of r and R r e l a t i v e to RCM. The preceding c o n s i d e r a t i o n s lead to the conclusion that the symmetry-adapted ~q(r,R) must be of the form ~JM~ (r'R) = An Ae f WjM(fl) [~(Ur,IJR) + I q~q(-Ur,-IJR)] dfl A

,

(8)

^

where An is the nuclear antisymmetrizer,

symmetrizer/antisymmetrizer,

WjM(~) is

the appropriate

weight

Ae is the e l e c t r o n i c function of

the Euler

angles ~ ~ (@,0,~) [6], and U ~ O(fl) is given by I cos~osOcosr 0 --

r

cos ycos Os i n t i s i n~costp

-cosTsinO

-sin}'cosOcostp-cosysintp

-sin),cosOsintp+cosTcostp

sin),sinO

s i nOcosr

s inOs inr

.

(9)

cosO

3. DECOUPLINGOF NUCLEAR AND ELECTRONIC DEGRF~ OF FREEDOM The e l e c t r o n i c Hamiltonian, A

He^ - "Fe + V

,

(10)

is parameterized by the nuclear coordinates R.

Its eigenstates

{~N(rIR)}

and the corresponding eigenenergies {eN(R)} transform as follows upon rotations and space inversion of R: VtN(UrIUR) = WN(rIR)

,

eN(IJR) - eN(R)

,

(11)

~N(-rl-a) - WN(rlR)

,

es(-R ) - eN(a )

.

(12)

It ensues from the property (11) that i t is s u f f i c i e n t

to

define {1/tN(rla ) }

and {eN(a)} only within the domain of internal nuclear coordinates a. replacement of R by R = {RI}, where RI = {xi,Yi,Zi},

which r e s u l t s

The

in the

removal of three degrees of freedom (two for linear molecules), corresponds to adopting a r o t a t i n g

("body-fixed")

fixed ("space-fixed")

one.

system are p o s s i b l e ,

the most natural

coordinate

Various d e f i n i t i o n s

system in place of

the

of the former coordinate

involving the requirement

that

the

5

tensor of inertia,

B

2: mIaIBI

, a,B E { X , Y , Z ) , (13) I is diagonal. The ground eigenstate yo(rlR) can be employed as a continuous basis set for the trial wavefunction %(r,R),

afi

=

$(rN

=

I IV&IR)

.

dR

Symmetry adaptation [Eq.(8)] of such %(r,R)

JMn(

?R)

produces

=

fin II wjM(~) [ V ~ W ~ I %(URIR) R)

+

n

yo(Url-R) %(-URIR)I

d~ d~

(15)

where Eq.(12) has been used and proper electronic antisymmetry of yo(rIR) has been assumed. The simplest choice for the function %(R~R) is provided by the Born-Oppenheimer ("clamped nuclei") approximation [7]

%(R[R)

=

6(R-R) ~ ( R - R ~ ) ,

(16)

where R N i s the position of one of the minima of E ~ ( R ) . It is important to note that such a choice of %(R~R) leads to %JMn(r,R) that cannot be an eigenstate of the molecular Hamiltonian with finite nuclear masses. This problem is rectified in the adiabatic approximation [8], in which

q.J(RIR)=

W - R )

?

(17)

and thus

where R is such that U(R) R coincides with R. Both the Born-Oppenheimer and adiabatic approximat ions decouple the nuclear degrees of freedom from the electronic ones. This coupling is restored in more sophisticated representat ions of %(R~R) that employ the generator coordinate method (GCM) [6,9]. Such representations, which are capable of correctly describing the instantaneous following of electronic motions by nuclei, provide partial justification for the approximate separation of electronic, vibrational, and rotational degrees of freedom that is commonly invoked in theories of molecular spectra [6]. However, the simple representation (18) suffices for the purpose of discussing the concepts of molecular geometry and structure.

4. THE RELEVANCE OF SPECTROSCOPIC STATES Since, as mentioned in Section 2, the spectroscopic s t a t e s

{~NjMH(r,R)}

are f u l l y determined by the set {mI,ZI,SI}, isomeric systems with i d e n t i c a l isotopic compositions share

the same molecular Hamiltonian [10],

implying

that {~NJMH(r,R)} represent s t a t i o n a r y s t a t e s of i s o l a t e d systems of nuclei and e l e c t r o n s r a t h e r than i d e n t i f i a b l e molecules. to question species~

the relevance of such s t a t e s

Hence, it is l e g i t i m a t e

to the d e s c r i p t i o n

of chemical

A

Although H has a continuous

spectrum,

its

subspectrum {ENJM~} that

a s s o c i a t e d with {~NJM~(r,R)} is mostly d i s c r e t e . {ENJMI} is

characterized

by several

energy

is

The d i s c r e t e p o r t i o n of

scales.

Energy levels

i d e n t i c a l N, J, and M but d i f f e r e n t p a r i t i e s H are c l o s e l y spaced.

with

For ex-

ample, the s p l i t t i n g AE = IEooo,+I-Eo00,_ll of ca. 4 [#Hartree] (which is equivalent to h/&E = 40 [ps] and AE/k = 1 [K]) in NH3 is considered q u i t e large [11];

the value of h/AE for AsH3 that

being probably more typical identical [#Iartree]

N but

[12].

different

angular

is estimated at ca. 3000 [h]

The spacing between energy levels with momenta J

is of

the order

of

10-100

for small diatomics and decreases very quickly with increasing

molecular size.

Finally,

the spacing between energy l e v e l s with d i f f e r e n t

N can be as small as 1 [mHartree] and as large as 200 [mHartree]. Whether or not two e i g e n s t a t e s of a given Hamiltonian can be i n d i v i d u a l ly observed depends on t h e i r energy d i f f e r e n c e . ious f a c t o r s

In the following, the var-

that determine the o b s e r v a b i l i t y of spectroscopic s t a t e s

(and

thus t h e i r relevance) are discussed.

4.1. Time dependence Time scales of many experiments are short enough to make the issue of the s t a t i o n a r y c h a r a c t e r of the observed s t a t e s

totally

irrelevant.

For

example, consider two spectroscopic s t a t e s ~+ and ~_, ~+ - ~NJM,+I

and

~ _ - ~NJM,- 1

whose energies d i f f e r by AE.

(19)

'

Being n o n s t a t i o n a r y ,

the m i x e d - p a r i t y s t a t e s

~R and qJL' ~R = 2-1/2 (~+ + ~_) are subject that

,

(20)

to quantum beats with the period of T -

h/AE (one should note

the r a d i a t i o n a l

and

q~L = 2-1/2 (~+ _ ~_)

decay can be s a f e l y neglected h e r e ) .

Consequently,

when a mixed-parity state YR is prepared, i t may evolve very little in the course of the experiment, provided that AE is sufficiently small. Considerations of this nature give rise to the notion of the so-called feasible symmetry operations [ 5 ] . From a chemist's point of view, space inversion is a feasible symmetry operation for NH3, as the lifetime of the mixedparity states of this molecule is only ca. 40 [ps] (see above). On the other hand, AsH3 may stay in one of its mixed-parity states for several months. Thus, the description of AsH3 in terms of the mixed-parity states is as legitimate as that in terms of spectroscopic states with definite pari ties. Similar observations pertain to the phenomenon of optical activity [13,14]. The fact that spectroscopic states cannot exhibit such activity is known as the Hund paradox [13]. However, once prepared, the enantiomers that are described by the wavefunctions IR and YL can persist for very long times thanks to the extreme smallness of the energy splittings AE in these cases (AE as small as 10- 50 [au], which corresponds to h/AE of the order of [years] [15]) 4.2. Interactions with external fields The spectroscopic states are eigenfunctions of the unperturbed molecular Hamiltonian. The behavior of these states and their energies upon weak external perturbations is governed by perturbation theory. However, when matrix elements of the perturbation operator between the spectroscopic states are much greater than the corresponding energy spli ttings, {YNJMn(r,R)} have to be replaced by functions that form a basis in which the perturbation operator is diagonal. Such basis functions, which are linear combinations of the spectroscopic states, possess reduced symmetries. An example involving a molecule interacting with a homogeneous electric field of strength & helps clarify the above statement [16]. Due to their definite parities, spectroscopic states cannot possess permanent dipole moments. When the field is weak enough, spectroscopic states retain their identities to the first degree of approximation, and the changes in their energies are quadratic in & - the second-order Stark effect is observed (the fact that the perturbation series is an asymptotic expansion is ignored here for the sake of simplicity). However, once the magnitude of interaction with the electric field exceeds the spacing between the adjacent

energy levels of the unperturbed Hamiltonian, localized

states

is more appropriate

the use of broken-symmetry

than that of {~NJMH(r,R)} [17].

The

l o c a l i z e d s t a t e s , which describe the molecule under c o n s i d e r a t i o n with i t s dipole moment oriented along the d i r e c t i o n of e l e c t r i c f i e l d have energies that change l i n e a r l y with s - a phenomenon known as the f i r s t - o r d e r Stark effect.

In systems such as AsH3 or CH3F, the s p l i t t i n g

between energy

levels of d i f f e r e n t p a r i t y is so small that the f i r s t - o r d e r Stark e f f e c t is always observed in p r a c t i c e . symmetric-top rotors"

In the conventional treatment of these " r i g i d

[11],

this phenomenon is a t t r i b u t e d

of a permanent dipole moment.

to the presence

However, in light of the present d i s c u s s i o n ,

this dipole moment is not a property of the molecule alone but is acquired through i n t e r a c t i o n with the external f i e l d . In the "nonrigid symmetric-top r o t o r s " Stark

effect

is

observed

under

(such as NH3), the second-order

normal

circumstances.

Indeed,

field

strengths of the order of 1 600 000 [V/m] are required to bring the i n t e r action

into the f i r s t - o r d e r

regime in this case [18].

In c o n t r a s t ,

very

weak i n t e r a c t i o n s s u f f i c e to make the mixed-parity s t a t e s ~R and ~L appropriate neutral

for the d e s c r i p t i o n of o p t i c a l l y a c t i v e systems. currents

have been proposed as

molecular Hamiltonian enantiomers [14,19].

[Eq.(1)] At p r e s e n t ,

that

is

the

interaction

responsible

for

this hypothesis is s t i l l

Parity-violating missing from the the e x i s t e n c e

of

awaiting experi-

mental v e r i f i c a t i o n . External p e r t u r b a t i o n s can mix not only spectroscopic s t a t e s of d i f f e r ent p a r i t i e s but also those of d i f f e r e n t

tota l angular momenta.

If the in-

t e r a c t i o n s are s u f f i c i e n t l y strong, almost complete l o c a l i z a t i o n of nuclei may be achieved, However, it

y i e l d i n g molecular

states

with well defined geometries.

is important to emphasize that no n o n r e l a t i v i s t i c p e r t u r b a t i o n

is capable of breaking the nuclear antisymmetry/symmetry of the molecular wavefunction. 4.3.

Intermolecular

The fact

that

interactions

s t a t e s with d e f i n i t e p a r i t i e s

are almost never observed

experimentally cannot be explained alone by the p e r s i s t e n c e of mixed-parity states.

Obviously, symmetry-breaking phenomena are in operation whether or

not external

perturbations

are p r e s e n t .

For species

in condensed media,

symmetry breaking is brought about by intermolecular i n t e r a c t i o n s . extreme case of s o l i d s ,

these i n t e r a c t i o n s are so strong that it

In the is proper

9

to describe molecules with the localized states discussed in Section 4.2 of this treatise. However, one should realize that even very weak intermolecular interactions suffice to effect symmetry breaking in cases where energy levels of the molecular Hamil tonian are closely spaced. These interactions do not have to break symmetry themselves as long as they introduce nonlinearities in the Hamiltonian 112,201. Consider a system that can exist in two spectroscopic states Y+ and B [Eq.(19)]. Quantum evolution of this system under the following nonlinear Hamiltonian, A

h

HI = H +

A

A

a

v

(211

,

where measures the strength of the intermolecular interaction, affords a useful model of symmetry-breaking phenomena 1211. Hamiltonians of this type describe interactions of molecules with solvents within the dielectric continuum models 1221. Since the states B+ and B are eigenfunctions of the unperturbed molecular Hamiltonian, A

A

,

HB+=E+B+

H B = E- B -

(22)

,

A

the matrix elements of H for the mixed-parity states YR and YL [Eq.(20)] are given by A

A

=

+E

)

=

Eo

. c Y f ~(Ef-Ei)-l ~ ~ Y i > exp

(-Ei/kT)

i f#i

x { 1 - exp

[(Ei-Ef)/kT]}

,

(41

that involves the second-order perturbation theory for the spectroscopic state Y..1 In Eq.(41), the subscripts i and f are abbreviations for the respective sets of quantum numbers, i = NJMn and f = N'J'M'I1'. For a pair of states that share the same N, the energy difference Ef - Ei is much smaller than kT at room temperature (see Section 4). Consequently, the contribution from these pairs of states to x can be accurately approximated by

12

x1

A

=

A

(2/kT) [I exp (-Ei/kT)]-' 1 1 - exp (-Ei/kT) i i f>i

=

(I/kT) [! exp (-Ei/kT)]-' C (1/3kT) i = a . (43) In other words, the conventional Langevin-Debye formula is recovered but with the expectation value of the square of the dipole moment operator in place of the square of the permanent dipole moment. Second, consider an experiment in which molecules are deflected by inhomogeneous electric field. According to the conventional description, the deflection is caused by the interaction between the field and the molecules represented by material points with permanent dipole moments. On the other hand, a rigorous quantum-mechanical treatment of this phenomenon cal Is for the consideration of the coupling between the overall motion of the centerof-mass and the perturbed spectroscopic states [24]. This coupling arises as a consequence of the fact that the total momenta of molecules are not conserved in the course of such an experiment. Molecules for which a temperature-dependent dielectric susceptibility i s observed in gas phase are commonly called polar. Polar molecules have microwave spectra with transitions corresponding to AJ=+l and are deflected by inhomogeneous electric fields. In the conventional approach, these phenomena are attributed to the presence of permanent dipole moments in such molecules. In contrast, the notion of permanent dipole moments (which are zero for spectroscopic states) plays no role at all in the fully quantummechanical treatment out 1 ined above. The temperature-dependent component of x arises from the existence of low-lying spectroscopic states for which =

2

1

A

the transition matrix elements do not vanish [Eq.(42)].

Transi-

13 tions between these s t a t e s are responsible for the observed microwave spectra.

Coupling to these s t a t e s also explains the d e f l e c t i o n by inhomogene-

ous e l e c t r i c f i e l d s . The conventional d e s c r i p t i o n of molecules, which is obviously much more intuitive

and s t r a i g h t f o r w a r d

often adequate.

than i t s quantum-mechanical counterpart,

is

the manifestations of quantum e f f e c t s

are

Nevertheless,

e a s i l y d e t e c t a b l e experimentally.

For example, species such as HC~CD, HD,

or CH3D, which are c l e a r l y nonpolar by the conventional d e f i n i t i o n , do possess temperature-dependent X and observable microwave s p e c t r a , flect

in inhomogeneous e l e c t r i c

upon the conventional approach,

fields

[11,16].

In f a c t ,

and do de-

if one i n s i s t s

these observations can be c o n s i s t e n t l y ac-

counted for by assuming the presence of small (of the order of 0.01 [D]) permanent dipole moments in these molecules.

However, a rigorous quantum-

mechanical treatment of such cases is c l e a r l y p r e f e r a b l e .

6. THE CONCEPT OF MOLECULARGEOMETRY When nuclei

in molecules are t r e a t e d c l a s s i c a l l y ,

ular geometry emerges in a natural way.

the concept of molec-

To be more p r e c i s e ,

the e q u i l i b -

rium geometry is defined as the set of i n t e r n a l coordinates R for which the ground-state eigenvalue eo(R ) of the e l e c t r o n i c Hamiltonian a t t a i n s a local minimum. Different minima in eO(R) correspond to equilibrium geometries of isomeric species with i d e n t i c a l compositions.

Needless to say, this naive

p i c t u r e is i n e v i t a b l y lost in a f u l l y quantum-mechanical treatment. Molecular geometries measured with condensed-phase

techniques

such as

X-ray d i f f r a c t i o n or NMR cannot be regarded as inherent p r o p e r t i e s of isolated species.

Similarly,

as the "determination" of molecular geometries

from microwave spectra involves c o l l a t i o n of data p e r t a i n i n g to many spectroscopic s t a t e s of species d i f f e r i n g in isotopic compositions, such geomet r i e s are merely c o l l e c t i o n s of f i t t i n g parameters that cannot be viewed as quantum-mechanical observables. All the information about p o s i t i o n s of nuclei that can be obtained for a given spectroscopic s t a t e ~NJMI(r,R) is contained in the corresponding nuclear p r o b a b i l i t y d e n s i t y PNJMI(R),

PNJMI(R) = f ~NJMI(r,R) ~NJMI(r,R) dr

.

(44)

14

It is obvious that the point and permutational symmetries of YNJMn(r,R) are reflected in PNJMn(R). The presence of these symmetries precludes the retrieval of the "classical-type" molecular structures [18] (geometrical parameters describing quasi-rigid assemblies of atoms that undergo vibrations and rotations) from the wavefunctions of spectroscopic states. This point is nicely illustrated by an example involving the simple adiabatic approximation [Eq.(18)] to YNJMn(r,R). In molecules that are not fluxional, %(R) is narrowly peaked around R that corresponds to the equilibrium geometry. It is often mistakenly believed that this fact implies localization of nuclei and thus the validity of the classical concept of molecular structure. However, a closer inspection of Eqs.(l8) and (44) reveals that this is certainly not true. First of all, PNJMn(R) derived from such %(R) is not peaked around any particular R because of the angular averaging brought about by the integration over the Euler angles Q. Second, thanks to the presence of the nuclear symmetrizer/antisymmetrizer, PNJMn(R) is fully symmetrical with respect to interchange of coordinates of any two nuclei of the same type. Consider, for example, a molecule with the composition AB2 [ 4 ] . Clearly, only one value of the internuclear separation NJMn,

NJMn

leading to a permanent dipole

42 moment.

There is a simple reason for the asymmetry of the electronic distribution in

H D +. For deuterium, the reduced mass and binding energy are slightly larger, and the corresponding wave function smaller, than for hydrogen. This leads to the contribution of the ionic structure H+D - being slightly larger than that of H - D +, and in affect to a net moment H+~D -~. The values of < r2HD >, < r2He > and < r~e > for the wave functions of different lengths are presented in Table 6, indicate that the electron shift towards the deuterium nucleus is correctly predicted in this approach. 6. G E N E R A L

N-BODY NON-ADIABATIC

WAVE FUNCTION

As we have suggested recently [68] the technique involving separation of the CM motion and representation of the wave function in terms of explicitly correlated gaussians is not only limited to non-adiabatic systems with coulombic interactions, but can also also extended to study assembles of particles interacting with different types of two- and multi-body potentials. In particular, with this approach one can calculate the vibrationrotation structure of molecules and clusters. In all these cases the wave function will be expanded as symmetry projected linear combinations of the explicitly correlated Ck of eqn.(29) multiplied by an angular term, Y~M"

LMr = "Pr ~ Y~MCk. k

(89)

Here 7~r is an appropriate permutational symmetry projection operator for the desired state, F, and YLkM is a product of coupled solid harmonics labeled by the total angular momentum quantum numbers L and M. Permutational symmetry is handled using projection methods in the same manner as described for the potential expansion in the previous section. Again, the reader is referred to the references for details[9,10,12]. Y~M is a vector coupled product of solid harmonics[69] given by the Clebsch-Gordon expansion, n

Y{M =

~ (LM; k I l l m l . . , l~m,~) 1-~ Ytj,~j. {lj, mj} J m l + . . . + m,~ = M

(90)

The solid harmonics are given by [ 2 / + 1 (l + m)' ( l - m)! Ytm (rj) -- [ 47r "

2zp+'~ (p + m)! p! (1 - m - 2p)!

(91)

The Yt,~ (rj) are single particle angular momentum eigen-functions in relative coordinates which transform the same as spherical harmonics, i.e., have the same eigen-values. Since the Ck are angular momentum eigen-functions with zero total angular momentum, the product with Y~M can be used, in principle, to obtain any desired angular momentum eigen-state. Note the k dependence of Y~M; this is included since there are many ways to couple the individual angular momentum lj to achieve the desired total angular momentum L and it may be necessary to include several sets of the lj in order to obtain a realistic description of the wave function. Varga and Suzuki[66] have recently proposed

43 representing the angular dependence of the wave function using a single solid harmonic whose argument contains additional variational parameters, u = (Ul, u 2 , " - , u~): n

~LMr -- VrYLM(V) ~ CkCk, with v k

~ u~r~.

(92)

i=1

There appears to be several advantages in doing this and we are investigating the possibility of using this approach in our full N-body implementation. The strict separation of the angular and "radial" variables is eqns.(89) and (92) allows separate consideration of the vibrational states with different total angular momentum quantum number, L. The magnitude of the Coriolis coupling for the particular L-state will determine whether the most general form, eqn.(89), or more simplified form, eqn.(89), of the total wave function should be used. There have been several highly accurate non-adiabatic variational calculations on atomic and exotic few particle systems using simple correlated gaussians [10,12,18,75,66,67]. By simple we mean they only contain the exponential part of the Ck, (no rij pre-multipliers). However, attempts at non-adiabatic molecular calculations have been plagued by problems with linear dependence in the basis during energy optimizations. This problem occurs in calculations on atomic systems also, but to a much lesser extent. We anticipate that we understand this phenomena and that the basis including pre-multiplying powers of rij will eliminate or at least drastically reduce the linear dependence problems. Our reasoning is as follows: In systems with more than one heavy particle there will be large particle density away from the origin in relative coordinates. That is, the wave function will have peaks shifted away from the origin. There are three ways to account for this behavior in the wave function using correlated gaussians: 1) use correlated gaussians with shifted centers, i.e., exp[- (r - s)'A (r - s)]; 2) Use near linearly dependent combinations of simple correlated gaussians with large matched + linear coefficients, or 3) Use pre-multiplying powers of rij. The first option is unacceptable since it results in a wave function which no longer represents a pure angular momentum state. The second option is what we believe causes the linear dependence and numerical instability which we are trying to avoid. The third option is what we are proposing. The linear dependence that we have observed in our calculation using the simple correlated gaussians looks, in some sense, like an attempt by the optimization to include in the wave function derivatives of the basis functions with respect to the non-linear parameters. The near linear dependent terms resemble numerical derivatives. Removal of these near linear dependent terms has an adverse affect on the wave functions, as manifested by poor energy results, but leaving them in leads to numerical instabilities which hinder optimization or cause complete collapse of the eigen-solutions. Now, derivatives of simple gaussians with respect to non-linear parameters, elements of the matrices Ak, bring down pre-multiplying (even) powers of r~j. Thus, explicitly including pre-multiplying powers of rij in the basis functions should add the needed flexibility to the basis in a numerically stable way. Also, we expect the rate of convergence to be improved by these pre-multiplying ri~ terms in the same way that they effect convergence in the Hyllerass basis. The Ck are similar to the Hyllerass basis functions with the Slater-type exponentials replaced by fully correlated gaussian type exponentials.

44

The above conclusion is supported by our recent prototype calculations for the vibrational structure of the/-/2 molecule, which was done with the above-described methodology [68]. However, instead of including all four particles (two electron and two protons) in the calculations, we only considered the nuclei interacting with the potential obtained by analytical gaussian fit to the B-O energy values of Kolos and Wolniewicz [70]. The most interesting feature of the results was a dominating contribution from higher powers of the H - H internuclear distance to all vibrational levels. For example, when only one Ck was used in the expansion, the lowest ground state vibrational energy was obtained with the r = r17e-43496r2. This result suggests, that constructing non-adiabatic wave functions for molecules, one needs to use Ck's with high powers of rij t o describe relative motion of nuclei. The superposition of these types of functions with functions describing the relative motion of the correlated electrons in the attractive field of nuclei, which we obtain in B-O calculations using correlated gaussians [71-75], will be used as the starting wave function in the fully non-adiabatic molecular calculations. 7. S U M M A R Y In this review we focused on a practical approach allowing quantum-mechanical description of the dynamics of the collective motion of nuclei and electrons in molecular systems. This approach allows investigation of the chemical bonding as a dynamical phenomenon which includes electrostatic, induction, charge-transfer and dispersion effects due to both electrons and nuclei. In this the procedure differs from the conventional approach, which treats the chemical bonding as an electron phenomenon. It is clear that more work is needed to develop a computational procedure, which can be used to study dynamics of the chemical bonding in the way described in this review. Application of explicitly correlated gaussians in one of the most promising approaches, which can lead to practical applications. REFERENCES

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45 13. G.W.F. Drake, Atomic, Molecular, and Optical Physics Handbook, American Institute of Physics, Woodbury, New York, 1996. 14. P.M. Kozlowski and L. Adamowicz, J.Chem.Phys., 95 (1991) 6681. 15. P.M. Kozlowski and L. Adamowicz, J.Comput.Chem., 13 (1992) 602. 16. P.M. Kozlowski and L. Adamowicz, J.Chem.Phys., 96 (1992) 9013. 17. P.M. Kozlowski and L. Adamowicz, J.Chem.Phys. 97 (1992) 5063. 18. P.M. Kozlowski and L. Adamowicz, Phys.Rev. A 48 (1993) 1903. 19. P.M. Kozlowski and L. Adamowicz, Chem.Rev. 93 (1993) 2007. 20. P.M. Kozlowski and L. Adamowicz, Int.J.Quantum Chem. 55 (1995) 245. 21. P.M. Kozlowski and L. Adamowicz, Int.J. Quantum Chem. 55 (1995) 367. 22. P.M. Kozlowski and L. Adamowicz, J.Phys.Chem. 100 (1996) 6266. 23. S. Gartenhaus and C. Schwartz, Phys.Rev.A 108 (1957) 482. 24. S.F. Boys, Proc.R.Soc. London Ser. A258 (1960) 402. 25. K. Singer, Proc.R.Soc. London Ser. A258 (1960) 412. 26. H.F. King, J.Chem.Phys. 46 (1967) 705. 27. W.A. Lester, Jr. and M. Krauss, J.Chem.Phys. 41 (1964) 1407; 42 (1965) 2990. 28. J.V.L. Longstaff and K. Singer, Proc.R.Soc. London, Ser. A, 258 (1960) 421. 29. J.V.L. Longstaff and K. Singer, Theoret.Chim.Acta 2 (1964) 265. 30. J.V.L. Longstaff and K. Singer, J.Chem.Phys. 42 (1965) 801. 31. N.C. Handy, Mol.Phys. 26 (1973) 169. 32. L. Salmon and R.D. Poshusta, J.Chem.Phys. 59 (1973) 3497. 33. K. Szalewicz and B. Jeziorski, Mol. Phys. 38 (1979) 191. 34. B. Jeziorski and K. Szalewicz, Phys.Rev. 19 (1979) 2360. 35. W. Kolos, H. J. Monkhorst and K. Szalewicz, J.Chem.Phys. 77 (1982) 1323; 77 (1982) 1335. 36. K.C. Pan and H.F. King, J.Chem.Phys. 56 (1972) 4667. 37. L. Adamowicz and A.J. Sadlej, J.Chem.Phys. 67 (1977) 4298. 38. L. Adamowicz and A.J. Sadlej, Chem.Phys.Lett. 48 (1977) 305. 39. L. Adamowicz, Int.J.Quantum Chem. 13 (1978) 265. 40. L. Adamowicz, Acta Phys.Pol. A 53 (1978) 471. 41. L. Adamowicz and A.J. Sadlej, J.Chem.Phys. 69 (1978) 3992. 42. L. Adamowicz and A.J. Sadlej, Acta Phys.Pol. A 54 (1978) 73. 43. L. Adamowicz and A.J. Sadlej, Chem.Phys.Lett. 53 (1978) 377. 44. K. Szalewicz, B. Jeziorski, H.J. Monkhorst and J.G. Zabolitzky, J.Chem.Phys. 78 (1983) 1420. 45. K. Szalewicz, B. Jeziorski, H.J. Monkhorst and J.G. Zabolitzky, J.Chem.Phys. 79 (1983) 5543. 46. B. Jeziorski, H.J. Monkhorst, K. Szalewicz and J.G. Zabolitzky, J.Chem.Phys. 81 (1984) 368. 47. K. Szalewicz, J.G. Zabolitzky, B. Jeziorski and H.J. Monkhorst, J.Chem.Phys. 81 (1984) 2723. 48. K.B. Wenzel, J.G. Zabolitzky, K. Szalewicz, B. Jeziorski and H.J. Monkhorst, J.Chem.Phys. 85 (1986) 3964. 49. D. Feldmann, Phys.Lett. 53 A (1975) 82. 50. K.E. McCulloh and J.A. Walker, Chem.Phys.Lett. 25 (1974) 439.

45 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

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W.A. Chupka, P.M. Dehmer, and W.T. Jivery, J.Chem.Phys. 63 (1975) 3929. L.R. Scherk, Can.J.Phys. 57 (1979) 558. K.R. Lykke, K.K. Murray, and W.C. Lineberger, Phys.Rev. A 43 (1991) 6104. C.L. Pekeris, Phys.Rev. 126 (1962) 1470. K. Frankowski and C.L. Pekeris, Phys.Rev. 146 (1966) 46. G.W.F. Drake, Nucl.Instr.Meth.Phys.Res. B 31 (1988) 7. J.D. Baker, D.E. Freund, R.N. Hill, and J.D.M. III, Phys.Rev. A 41 (1990) 1247. A.J. Thakkar and T. Koga, Phys.Rev. A 50 (1994) 854. J. Ackermann, Phys.Rev. A 52 (1995) 1968. B.W. Petley, Phys.Scr. T40 (1992) 5. G. Audi and A.H. Wapstra, Nucl.Phys. A 565 (1993) 1. H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- And Two-Electron Atoms (Plenum, New York, 1977). S.G. Nash, SIAM J. Numer. Anal. 21 (1984) 770. netlib can be accessed by ftp at [email protected] and [email protected] or by World Wide Web access at http://www.netlib.org. D.B. Kinghorn, Explicitly Correlated Gaussian Basis Functions: Derivation and lmplementation of Matrix Elements and Gradient Formulas Using Matrix Differential Calculus, Ph.D. dissertation, Washington State University, 1995. K.Varga and Y.Suzuki, Phys.Rev. A 52 (1995) 2885. K. Varga and Y. Suzuki, Phys.Rev. C (1996), in press. D.B. Kinghorn and L. Adamowicz, J.Chem.Phys. in press. L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics. Theory and Application, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA, 1981. W. Kolos and L. Wolniewicz, J.Chem.Phys. 43 (1965) 2429. E. Schwegler, P.M. Kozlowski and L. Adamowicz, J.Comp.Chem. 14 (1993) 566. Z. Zhang, P.M. Kozlowski and L. Adamowicz, J.Comp.Chem. 15 (1994) 54. Z. Zhang and L. Adamowicz, J.Comp.Chem. 15 (1994) 893. Z. Zhang and L. Adamowicz, Int.J.Quantum Chem. 54 (1995) 281. D.W. Gilmore, P.M. Kozlowski, D.B. Kinghorn and L. Adamowicz, Int.J.Quantum Chem., accepted for publication.

Z.B. Maksi6 and W.J. Orville-Thomas (Editors)

47

Pauling's Legacy: Modern Modelling of the Chemical Bond

Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

T h e M i l l s - N i x o n Effect: Fallacies, F a c t s a n d C h e m i c a l R e l e v a n c e Zvonimir B. Maksid a , Mirjana Eckert-Maksid b, Otilia M5 c and Manuel Ys

c

~Quantum Organic Chemistry Group, Division of Organic Chemistry and Biochemistry, Rudjer Bo~kovid Institute, P.O.B 1016, 10001 Zagreb, Croatia and Faculty of Science and Mathematics, The University of Zagreb, Marulidev trg 19, 10000 Zagreb, Croatia bphysical Organic Chemistry Laboratory, Division of Organic Chemistry and Biochemistry, Rudjer Bo~kovid Institute, P.O.B 1016, 10001 Zagreb, Croatia CDepartamento de Qu{mica C-9, Universidad Aut6noma de Madrid, Cantoblanco, 28049-Madrid, Spain

1. I n t r o d u c t i o n The Mills-Nixon (MN) effect has a long, respectable but controversial history because it has been praised, questioned and disputed over decades culminating sometimes in heated arguments and debates. Although the MN effect is only a very small episode, it is the case interesting for the history of science and perhaps for the science on science. It possesses all ingredients neccessary for a thrilling saga about a winding path of the research process and its stepwise progress. This story begins with an interesting finding accompanied by a bold working hypothesis, followed subsequently by some evidence which supported the original experimental findings. These results were ensued by attempts to invalidate or to dismantle the pioneering discovery by focusing only on the wrong initial premise, by a "burial" of the original work and simultaneous "rediscovery" of the same effect albeit in a somewhat disguised form. The entire process was blended with misconceptions, charming naivety of some researches and/or disregard of other contributions to the subject lying next to fraud and swindle. Fortunately, there are also some unpretentious contributions representing perhaps small but signifficant improvements of the existing knowledge, which raise a hope that things will be settled as time elapses. It is the aim of this chapter to review the main achievements some seventy years after the original paper of Mills and Nixon in 1930 [1] concentrating on the structure, properties and reactivity of aromatic compounds annelated to strained rings. Last but not least, some common fallacies are discussed in order to solve this conundrum. It is beyond the size and scope of this chapter to provide a comprehensive coverage of the field. We apologise, therefore, for unintentional omission of some papers, which would otherwise have been mentioned.

48 2. The M i l l s - N i x o n Effect" The First E x p e r i m e n t a l Result and T h e o r e t i c a l I n t e r p r e t a t i o n by S u t t o n and Pauling Mills and Nixon found that an annelated small ring had directional capability in the electrophilic substitution reactions taking place at the benzene fragment [1]. More specifically, they established that the free/3'-position in fl-hydroxyindan is more susceptible to electrophilic substitution than the c~-site (Fig. 1) as revealed by bromine or diazo group substitutions. The opposite should be the case in/3-hydroxytetralin.

H2)n

(a)

(b)

(c)

Figure 1. (a) Schematic representation of/~-hydroxybenzocycloalkenes. (b) Predominant Kekul~ electron pairing scheme in/3-hydroxyindan according to the MN hypothesis. (c) Preconceived dominating Kekul~ structure in/~-hydroxytetralin.

This intriguing regioselectivity induced by the fused carbocycle triggered a cascade of papers addressing the question of the electrophilic reactivity of annelated benzenes [2-7], which have continued to flow until nowadays [8-14]. An interpretation of this effect was put forward by Mills and Nixon first. It was based on the tetrahedral distribution of the carbon atom valencies. Consider bond angles around the carbon junction atom in indan as depicted in Fig. 2:

.125.3

o

Figure 2. Hypothesis about the "Kekul6-resonance structure fixation" in indan by Mills and Nixon

49

The angle in the five-membered carbocycle is close to 109.5 ~ leaving for the two remaining valence angles in the molecular plane 125.3 ~ each. The double bond is then given in the bent bond representation by overlapping of two pairs of the sp 3 hybrid orbitals. This simple model implies that the joint CC bond is described by sp3-sp 3 hybrids - a supposition which was proved to be essentially correct by later quantum mechanical calculations performed on many small ring fused aromatics. If this is the case, then the Kekul6 structure of the benzene fragment in/3-hydroxyindan shown in Fig. 2 should be preferred leading to its freezing and to a concomitant dramatic alternation of CC bond distances. However, Mills and Nixon erroneously concluded, employing this simple pictorial concept, that/3-hydroxyindan existed in a form of two isomers corresponding to the two possible Kekul6 spin-coupling schemes of the benzene fragment and that the one presented in Fig. 2 predominates in their mixture [1]. This interpretation is obviously wrong and outdated, but it is perfectly clear to almost everyone that this kind of rationalization of the experimentally observed data should be distinguished from the MN-effect itself. Generally speaking, a genuine phenomenon should not be completely identified with a model developed for its description - not to mention a possibility that the employed simplified model could be oversimplified, inadequate or completely wrong. The first correct interpretation of the selectivity in the electrophilic reactions of fused aromatics was given by Sutton and Pauling by the valence bond (VB) method in its most elementary form [15]. The total molecular wavefunction was described in terms of two VB components corresponding to Kekul6 structures of benzene. It appeared that the one related to the situation depicted in Fig. 2 possessed a larger coefficient and more weight as was intuitively expected. In performing actual calculations, some approximations about the distribution of the strain energy induced by the five membered ring have been invoked. It was concluded, owing to the substantial energy associated with resonance between two VB structures, that these changes caused by strain should be relatively small. Nevertheless, they were large enough to account for all experimental data available at that time thus providing the first quantum mechanical explanation of the MN-effect. 2.1. D e f i n i t i o n of t h e M N - E f f e c t a n d s o m e C o m m o n Fallacies It is useful to have at hand an operational definition of the MN-effect in order to avoid unnecessary semantic difficulties. We propose the following conceptually simple and intuitively appealing definition: The Mills-Nixon effect is a perturbation of the aromatic

moiety exerted by fusion of one (or several) nonaromatic angularly strained molecule(s). This perturbation is reflected in the characteristic partial bond localization leading to modifications of a number of physical and chemical properties of the aromatic moiety. Thus the notion of the Mills-Nixon effect is free of any predetermined underlying mechanism pertaining to the exerted perturbation. This is important because the mechanisms and manifestations of the effect may be different in various molecular systems. More precisely, the MN-effect is generally a result of an interplay of several types of intramolecular interactions. Their contributions vary from one family of compounds to another as a rule. It should be also emphasized that by angularly strained fragments, annelated to an aromatic nucleus, molecular systems strained by the ring closure requirements were tacitly understood. They are not necessarily small and/or monocyclic. Finally, a useful rule of thumb can be applied in identifying the Mills-Nixon systems. It is given by a diagnostic tool

50 provided by the length of the aromatic CC bonds placed ortho to the bond of coalescence of two rings. If they are shorter than in a free aromatic molecule, then the MN distortion takes place. It is important to realize that there are reversed MN-systems too, which require extension of the proposed definition (vide infra). It follows that the MN-effect is reflected in changes of diverse properties of the free aromatic compound upon annelation. Concomitantly, it is obvious that various experimental techniques and appropriate theoretical models are required to provide a comprehensive description of the observed effect. Unfortunately, there are numerous misconceptions of this phenomenon in the literature, which introduce a lot of confusion. We give here a short catalogue of the main fallacies and provide brief comments and glosses in order to avoid unnecessary misunderstandings: (1) Mills and Nixon based their interpretation on the Kekul(! time averaged oscillating model of benzene. This picture is obsolete, since it was conclusively shown subsequently that benzene had a single-well potential. The idea of rapidly equilibrating Kekul~ forms requires double-well potential, which has been abandoned in the meantime. Therefore, the MN-effect does not exist and this anachronistic structural proposal can be safely discarded [16]. There are two misconceptions in this line of thought. Firstly, the MN-effect is an experimental fact independent of its interpretation, which in turn could be right or wrong. The latter was indeed the case in the original Mills-Nixon milestone paper. Secondly, the MN-phenomenon cannot be reduced to a structural feature only, although fusion of small rings involves some significant geometric changes too. It is intuitively clear that fusion of (highly) strained ring(s) and an aromatic moiety inevitably leads to a change of a number of physical and chemical properties of the latter.

(2)

A double-well shape of the potential energy surface of benzene and its annelated derivatives would imply that even small perturbations would cause large changes in populations of the valence tautomers each confined in its own potential well. Therefore, small perturbations would have far reaching consequences. In contrast, a single-well potential is robust thus being less sensitive to various perturbations. Concomitant with the single-well picture, the characteristic features of small ring annelated benzenes like tris-cyclobutabenzene do not exhibit any appreciable bond alternation: specifically, the bond lengths in the central ring of this compound are 1.40 and 1.38 A. This small difference can be neglected [16]. In another study Boese et al. [17] discussed the matter claiming that bond alternations induced in benzene by annelation with small rings, if present at all, are very minor ( tobs > 'l:eq(R),

(4)

then R is associated with a kinetically stable compound as . W h a t constitutes a kinetically stable c o m p o u n d depends therefore on the interplay of three different time scales. Furthermore, both Xeq(R)and Xesc(R) strongly vary with temperature. In principle, it would therefore be necessary to proceed as follows: For a given temperature T, choose some region R of the energy landscape, calculate Xcq(R;T) and Xesc(R;T)and determine, whether R is locally ergodic and on what time scale compared with relevant observation times. Repeat this for all possible regions, until all kinetically stable regions are found. For each R, the structure of the corresponding metastable c o m p o u n d is given by the statistical mechanical average over all configurations i e R. If desired, one can also compute the local free energy F(R) for each region R and find the thermodynamically stable one R* through minimisation ofF(R). Obviously, this procedure is very expensive computationally a7 . In some cases, a shortcut is possible, if the region R is separated by energy barriers of height EB (measured with respect to the m i n i m u m of R) from the rest of the energy landscape. Then, one often finds an Arrhenius law for the escape time 'l:esc(R) ": exp (+EB/kBT),

(5)

which m e a n s that Xesc(R) grows exponentially with Ifr. Since in most compounds Xcq(R) does not increase exponentially with decreasing temperature, R will usually be kinetically stable below some (possibly very low) temperature. Thus, at low temperatures, local m i n i m a will often be associated with locally ergodic regions as leading to the following recipe for the prediction of metastable compounds: First one finds the local m i n i m a of the hypersurface of the potential energy, called and the measurement time is long enough, one can often replace such a particular time-average of the observable by an average over a whole ensemble of copies of the system that are weighted according to the equilibrium (Boltzmann) distribution. If such a replacement is possible, one says that the system is ergodic. For systems that exhibit metastability, global ergodicity on all time scales does not hold any more, and one speaks of "broken ergodicity''23. Often, local ergodicity replaces global ergodicity, but even this property does not necessarily hold in all systems, e.g. spin glasses might not be locally ergodic on any time scale. a6 For observationtimes tobs< Xeq(R),thislocalergodicityis usuallybroken. a7 F(R) can be calculatedin several ways: from Z(R), ifthis quantity is available(cf.section3), from p(R) based on a very long M C / M D simulation, or by computing the free energy difference AF(R--#0)with respect to a reference system with known free energy F0: FfR) = F0- AF(R-~0).The most common methods used to calculatefree energy differencesare "free energy integration", "free energy perturbation"and "finitetime variation";an overview together with discussionsof computationallyefficientimplementations isgiven in the references24"26. a8 Clearly,there can existadditionallocallyergodicregions,which are separated from the rest of the energy landscape by entropicinstead of energeticbarriers.As an example, picture a valley that is surrounded by high mountains with only one very narrow exit at ground level (no energetic barrier!).In N dimensions, the time to find this exitby walking randomly inside the valleycan be much longer than the equilibrationtime within the valley.

108

"structure candidates". This is followed by the determination of the energy barriers EB around these minima, in order to judge their kinetic stability. A method to calculate energy barriers and local densities of states is described in section 3. Finally, one can determine the thermodynamically stable region R* by computing F(R) for each kinetically stable region R and finding the minimum ofF(R). At high temperatures, the identification of locally ergodic regions of the energy landscape will be considerably more difficult, however, since the physically relevant regions can be considerably more extended, enclosing many local minima of the potential energy hypersurface, e.g. if T is above the transition temperature of a second order phase transition, or in a range, where supercooling is possible. Also, entropic barriers would be expected to play a much more prominent role t h a n at low temperatures. So far no simple characteristic for identifying locally ergodic regions has been suggested for these cases, and the very detailed investigation of the whole energy landscape outlined above appears to be required.

3. T H E LID- AND T H E T H R E S H O L D - A L G O R I T H M The approach presented here 2~ 21, 27 concentrates on achieving as complete as possible a description of regions of the energy landscape close to deep-lying minima. Starting from such a minimum, xo, the pocket R(L,x0) in configuration space t h a t can be reached from the starting point without crossing a prescribed energy lid L is searched - exhaustively for discrete systems (the "lid method") 2~ or statistically via random walks for continuous energy landscapes (the "threshold algorithm") 21. In the case of the lid method, all the states within the pocket and the connections among them are noted. This procedure is repeated for an increasing sequence of energy lids up to a highest value, Lmax, and yields both the number of states and the n u m b e r of local minima with energy E accessible from xo using paths below a given lid, n(E;L,x0) and m(E;L,x0), respectively. This procedure is repeated for all the local minima x present within the pocket. Thus, the local density(ies) of states, i.e. the density(ies) of states restricted to this pocket R(L,x) in state space, g(E;L,x), together with the energy barriers between the local minima within the pocket can be determined. Based on these results, the statistical mechanical properties of the system can be studied, as long as it remains within the prescribed region of phase space a9 , since the local partition function Z(R) for R(L,x) follows directly from the local density of states Z(R) =

~ i ~ R(L,x)

exp(- Ei/kaT)= ~

g(E;L,x) exp(-E/kBT)

(6)

E_ Li, no special problems arise by using the overlap procedure for boot-strapping. However, if such a new region becomes suddenly accessible at Lk, a l a r g e r n u m b e r of low-lying s t a t e s below Lk-1 are now r e p r e s e n t e d in the s a m p l i n g n(E;Lk,x0) t h a n h a d been available for n(E;Lk.l,X0). Thus, one h a s to correct for this effect when using the overlap procedure, often a non-trivial task.

a l 0 This would require an appropriate discretization of the configuration space, which is in most cases not feasible computationally. a l l moveclass = set of neighbouring configurations in phase space that can be reached from a given point with one step of the random walk. A physically reasonable moveclass does allow only those moves of the system that might occur during the regular time-evolution of the system, e.g., small displacements of atoms in a solid, etc. If the only goal is to reach the global minimum as fast as possible, it is of course useful to try to improve the optimisation algorithm by including non-physical moves, avoiding or "eliminating" metastable minima in the process. Obviously, for our purposes such a "high-efficiency" moveclass is a double-edged sword. Thus, during the optimisation (cf. sections 4 and 5) we have included only those non-physical moves that remove or exchange whole ions in the simulation cell, since these operations will only in very special cases lead to the elimination of realistic sub-optima, and even these moves are no longer allowed when we investigate the kinetic stability of the structure candidates. a12 If it is possible to calculate the matrix of second derivatives for each minimum analytica!ly, one can find the normal modes and use their density of states to determine the missing normalization factor for each minimum.

110 The combination of all the local densities of states, g(E;L,xi), represents a lumped picture of the pocket. This description is intermediate between the overall density of states for the whole pocket g~;Lmax) and the exhaustive description of every microscopic detail of the energy landscape within the pocket one can achieve in the discrete case. With this information, it is possible 21 to construct a transition matrix M(T) in the lumped configuration space t h a t allows the simulation of the evolution of the system for temperature T, and thus yields estimates for ~ ( R ) and xesc(R).

4. S T R U C T U R E P R E D I C T I O N A T LOW T E M P E R A T U R E S 4.1. G e n e r a l a s p e c t s For the r e m a i n d e r of the chapter, we will concentrate on the case presumably most i m p o r t a n t for crystalline compounds, where kinetically stable structures are associated with local minima of the energy landscape. The determination of all the local minima of the energy landscape requires the use of global optimisation methods. Since one single run of a global optimisation procedure typically involves hundreds of thousands or millions of energy evaluations, and it is in general necessary to perform hundreds of such runs in order to acquire sufficient statistics on the distribution of deep-lying minima, any reduction in the number of function evaluations involved in the calculation of the energy is of great importance. It is therefore usually not feasible to perform ab-initio calculations of the energy, and one has to simplify the energy calculation by using empirical effective potentials for the interaction between the atoms instead. These potentials have to reflect correctly both the local and the long-range i n t e r a c t i o n a m o n g the atoms/ions: The atoms m u s t not be allowed to overlap too much, and, except via electrostatic forces, they m u s t not interact directly at long distances. Additional terms will enter t h a t reflect the local bonding situation of the atoms, and here usually a "selection" takes place, depending on whether we want to stress e.g. the covalent, ionic or metallic character of the hypothetical compound. Furthermore, we use simulation cells with periodic boundary conditions, in order to deal with the large n u m b e r of atoms in a solid. Since we are mainly interested in crystalline compounds, this choice is preferable over the open boundaries of a cluster. Note t h a t these periodic boundary conditions are not very restrictive, since size, shape and s y m m e t r y of the simulation cell can be freely varied during the m a n y global optimisation runs necessary to achieve a sufficient statistic. In addition to the cell parameters, a multitude of other p a r a m e t e r s can be varied during the optimisation, e.g. the location of the atoms, their degree ofionisation (if we are dealing with ionic systems), and even the n u m b e r of atoms a13 and the composition of the a13 Instead of varying the number of atoms arbitrarily during a given run, it has often proven to be more efficient to keep the composition fixed instead, and to repeat the optimisation with all the other promising compositions afterwards. Since this restriction makes the addition and removal of atoms rather awkward, fixing the composition usually implies fixing the number of atoms, too. Therefore, we also need to repeat the runs for a given composition with different numbers of formula units in the simulation cell. Our observations have shown that if e.g. the number of formula units is doubled, the same minima are still present, but that many additional minima appear, of course. As pointed out in the introduction, the number of possible metastable periodic structures appears to be limitless.

111 system within the cell. The initial configurations for the optimisations usually consist of a cell of ca. 10 times the volume of all the atoms t a k e n together, with the (neutral) atoms placed at random positions within the cell. While simple empirical potentials can produce m a n y promising structures, the precise values of e.g. their cell parameters or the exact locations of the atoms in the cell are not to be expected. Such details would require to perform another optimisation round for e.g. fixed composition, etc., using a more realistic potential. Preferably, one would want to employ either highly refined (semi)empirical potentials 28"32 o r some ab-initio method3335, a14 After the optimisation stage has been finished, the candidate structures with the lowest ground state energy will be analysed with respect to their stability and their local density of states using the methods described in section 3. The final

result of the whole procedure would now consist of an overview over a large number of preferred compositions, and possible structure types and structure elements that would be expected to occur under certain conditions in the investigated chemical system, together with estimates of the relative ground state energies, the kinetic stability and the local density of states of the most promising ones. The last information allows the tentative identification of thermodynamically stable and metastable structures.

4.2. Specific optimisation algorithms While it is a well-known fact t h a t one can devise more and more refined optimisation algorithms based on more and more detailed knowledge of the energy surface of the system, we have so far used only one family of algorithms, which are based on the stochastic simulated annealing algorithm 36 introduced by Kirkpatrick et al. 37 and Cerny 38. The great advantage of this method lies in the relative ease of implementation, the very general applicability independent of the specific optimisation problem, and the great freedom in the choice of a moveclass. Simulated annealing is based on the Metropolis (Monte-Carlo) algorithm 39, which implements a weighted random walk through configuration space. Starting from a current configuration 'T', a neighboring configuration "i+l" is chosen at random according to a set of rules (the "moveclass"). If the energy Ei+l is below or equal El, the move is always accepted, i.e., "i+l" becomes the new current configuration. Else, the move is only accepted with probability exp(-(Ei+l-Ei)/C), where C is a control p a r a m e t e r of the random walk. Thus during a sequence of such MCsteps, the system can climb over barriers of the energy surface. It can be shown that in the long-time-limit (t -~ oo) for an ergodic system the probability p(i) of visiting state ( = configuration) 'T' is given by the Boltzmann-distribution for the system at a t e m p e r a t u r e a15 given by T = C/kB. F u r t h e r m o r e , in analogy to the annealing procedure of a real material, we can reduce the control p a r a m e t e r C down to zero, and expect t h a t the system will end up, with a high probability, in However, it would be very time-consuming to perform a global optimisation, especially in the latter case. But, since usually many promising structure candidates have already been accumulated at this point, one would restrict oneself to essentially local minimisations for each structure type. a15 From this we can conclude that, in certain circumstances, we may interpret T as an actual temperature. a14

112 some deep-lying minimum. It can be shown that during such a simulated annealing run, an ergodic system will reach the global minimum 36, 40, 41, at least for t ~ oo. Clearly, the most important parameters of this optimisation procedure are the temperature program, i.e., the rule according to which C is decreased to zero, and the moveclass. While a considerable effort has been devoted to the developm e n t of efficient general t e m p e r a t u r e p r o g r a m s 4~ 42-46, the choice of a good moveclass still remains an open question, since it is highly problem dependent. For our explorations of the energy landscapes of solids, the best results seem to be gained from a moveclass where 60% - 80% of the moves are devoted to movements of single atoms, ca. 10% each to the change of ionic charges and the addition/removal/exchange of atoms, while the remaining ones should be used to adjust the size and shape of the simulation cell. However, the selection of the moveclass should also take the size of the system, the goals of the current optimisation, and the design of the temperature program into account. Often it might even be useful to change the moveclass as a function of temperature. As far as the t e m p e r a t u r e program is concerned, reasonable results have been achieved using Tn = TOfn (n = 0...nmax), with m MC-steps between temperature updates. Here, TO, f, nmax and m should be chosen according to the size of the system, the moveclass and the objective of the current optimisation run. The latter refers to the fact t h a t different strategies have to be chosen, if one wants to find as deep a m i n i m u m as possible for a given a m o u n t of computer time or whether one tries to gain a general overview over the statistical distribution of local minima and their accessibility a16 . We have employed two algorithms for the purpose of local optimisation: a stochastic quench, which corresponds to a simulated annealing algorithm with C = 0, and a steepest descent algorithm with a line search option 47. While the stochastic quench can be used with the full moveclass including charge transfer and atom exchange, the gradient descent is only useful when only the cell p a r a m e t e r s and the atom positions need to be adjusted.

4.3. Specific e m p i r i c a l p o t e n t i a l s The choice of the empirical potential for the global optimisation obviously depends very strongly on the system one investigates. So far, we have investigated two classes of systems, noble gases 4s and their mixtures 49, and binary 5~ and ternarySl, 54 ionic compounds, since relatively simple two-body potentials are already able to capture many qualitative and also some semi-quantitative aspects of the energy hypersurface of the compound in the derivation of effective energy functions for such test systems. Here we will only discuss the ionic systems.

a16 An extreme option consists of essentially replacing the global optimisation with many local optimisations, where it is hoped that the initial configurations are chosen such that all relevant regions of the configuration space are covered. This requires that either the barrier structure is simple enough that quench runs are not always stranded in high-lying minima even when starting from random initial configurations, or that the starting configurations are already so close to the best sub-optima that only some final adjustments are necessary to reach the desired minimum configurations. The latter case applies, if realistic initial states are constructed by some special algorithm, or if the initial configurations are based on the results of earlier optimisation runs.

113 The approximate potentials for the description of ionic systems consist of three terms, a screened Coulomb term, exp(-ar)/r, a repulsive term, (r -n, n = 12, usually), and an attractive dispersion term (r-6). If the damping factor in the Coulomb-term is not present, i.e., a = 0, the energy function is evaluated using the summation method suggested by deLeeuw55.:

Vij(rij) =

qiqjexp(-txrij) + 13~j -aiij 4xeorij ~rij ! ~rij I

(7).

In addition to these two-body terms, the energy function contains "one-body terms" E0(i) (the ionisation energy or the electron affinity, respectively), a term pv' when allowing volume changes, where p is the pressure and v' the volume per atom, and the chemical potential g(i), the latter being relevant, if the number of atoms is allowed to change during the optimisation. Thus, the energy function per atom we use takes the form E = 2 1 .~. Vij(rij) + 1 .~ Eo(i)(+ pv')(+ 1 .~ g(i)) l~j

1

(8).

1

Three major routes for determining the parameters in the empirical potentials are available, depending on the amount of pre-knowledge about the system. If the participating atoms have already been studied extensively in the context of molecular studies using ab-initio methods, it is sometimes possible to derive effective two- and three-body potentials based on these theoretical results 5658. Experience shows, however, that so far this path does involve considerable effort, taking months of investigation. Furthermore, the potentials cannot always be transferred with the expected accuracy from the molecular environment to the one in the solid state. Thus, one is usually forced to fit the parameters to some experimental observations, either to the properties of some specific known compound(s) or according to average "atomic" properties of the participating atoms, e.g. ionic radii and deformabilities of the ions. The first option encounters two basic problems, however. For one, if no compound of the system is known beforehand, obviously no fit can be performed. Secondly, if already a compound is available experimentally, one can very easily prejudice the model by using only this substance for fitting, and thus suppressing alternative minima of the (true) energy hypersurface. In contrast, the use of "atomic" properties averaged over many different compounds and local atomic environments should result in somewhat more unbiased effective potentials a17 . The disadvantage lies in the lack of specificity of the potential, of course. Because of this possible trade-off of accuracy vs. generality, one will choose the fit procedure according to the specific circumstances of the calculation. In both cases, but especially when using average atomic parameters, one should therefore repeat the optimisation runs varying the potential about the "best" values chosen originally a18 .

a17 Recently, the use of parameters averaged over many binary compounds has been proposed by Bush et al. 59 a18 For an in-depth discussion of the many aspects involved in choosing an empirical potential for the purpose of structure prediction using global optimisation methods, see ref. 51.

114 5. EXAMPLES Let us now discuss some representative and illuminating examples of structure prediction for the case of ionic compounds. Both binary and ternary compounds have been investigated; the latter including mostly unknown or at least not-yet-synthesized compounds. The general form of the potential was as given in section (4.3), with p = 0 and usually without chemical potentials. Both the case a = 0, making an Ewald-summation necessary, and the case a > 0 ( = screened Coulomb potential) have been used in constructing the effective potential. Average ionic radii, and information about the polarizability or "hardness" of the ions, were used to establish starting values of the free parameters in the potential - n, s and ~ij. First a large number of binary systems were investigated 5~ using long simulated annealing runs that would be expected to reach a very good sub-optim~lm of the energy hypersurface (> 106 MC-steps). For most of these runs, the composition of the system and the number of atoms within the simulation cell was kept fixed. Initially, electron transfer between randomly chosen pairs of atoms/ions was allowed. Of course, once the preferred ionic charges had been established as part of the global optimisation procedure, additional studies of the system using ions with fixed charges were admissible. The results are listed in table 1. We note that in nearly all instances the best sub-optimum found during the simulated annealing run turned out to be either the structure seen experimentally to be the preferred one or the structure that agreed with the radius-ratio-rule. Detailed Study of Binary Systems. While the above results clearly show that it is possible to determine very good sub-optima of the energy hypersurface of binary ionic compounds using effective potentials based on characteristic atomic quantities, it was felt necessary to investigate in more detail the energy hypersurface of some particular systems. We have chosen NaC1 as a representative ABsystem 5~ for AB2-systems MgF2, MgCI2, CaF2 and CaCI2.53 The parameters in the effective potential were varied slowly from each set of optimisation runs to the next, in order to study the robustness of the candidate structures. Each such set of runs consisted of 14 or 20 global optimisations of 105 to 3x105 simulated annealing steps, the only difference being the initial value of the random number generator. Thus, for each system, several hundred global optimisation runs were performed. Since it had already been established during several sets of optimisation runs where charge transfer was allowed that the best sub-optima occurred when the "fully" ionised ions were present (Na§ Mg2§ Ca2§ F-, CI-), the ionic charges were fixed at these values, and the moveclass consisted of movement/exchange of the ions, and variation of the size and shape of the simulation cell. Na/Cl. A large number of structure candidates corresponding to local minima of the energy hypersurface were found in this system. They have been listed in table 2. Two of these structures were judged to be of special interest, the "5-5"variant (figure 1), consisting of trigonal bipyramids of C1--ions around the Na+-ions, and a "NiAs"-type variant a19 . For each of these, the energy barriers separating them from other local minima were determined using the threshold algorithm.

Recently, Martin and Corbett have succeeded in synthesizing an unprecedented monohalide, LaI, exhibiting the NiAs-structure-type60.

a19

115 Table 1 B e s t r e s u l t s of s i m u l a t e d a n n e a l i n g r u n s including a c o m p a r i s o n w i t h t h e observed s t r u c t u r e s . D u r i n g t h e optimisation, m a n y a d d i t i o n a l s t r u c t u r e s , c o r r e s p o n d i n g to local m i n i m a of t h e e n e r g y h y p e r s u r f a c e , h a v e b e e n f o u n d in t h e s e systems. Input

S t r u c t u r e (sim.ann.)

S t r u c t u r e (observed)

Na- C1

NaCl-structure

NaCl-structure

Cs- C1

CsCl-structure

CsCl-structure

Li- F

NaCl-structure

NaCl-structure

Na- F

NaCl-structure

NaCl-structure

K- F

NaC1/CsCl-structure

NaCl-structure

Rb - F

NaCl/CsCl-structure

NaCl-structure

Cs- F

NaC1/CsCl-structure

NaCl-structure

Ba- O

NaCl-structure

NaCl-structure

Li- I

NaCl-structure

NaCl-structure

Sr- O

NaCl-structure

NaCl-structure

Ni - O

NaCl-structure

NaCl-structure

Ca- O

NaCl-structure

NaCl-structure

Mg- O

NaCl-structure

NaCl-structure

Ca- F

fluorite-structure

fluorite-structure

Mg- F

rutile-structure

rutile-structure

Li - O

antifluorite-structure


K- O



Sr- Cl

fluorite-structure

fluorite-structure

Ca- C1

CaC12-structure

CaCl2-structure

Sn - O

rutile-structure

rutile-structure

Ti- O

rutile-structure

rutile-structure

Na- O



Si- O

cristobalite-, t r i d y m i t e - s t r u c t u r e

e.g. quartz-, tridymite-, cri s to bali te- s t r u c t u r e

While t h e NiAs-type s t r u c t u r e exhibited a b a r r i e r in excess of 0.13 eV/atom (T = 1300 K), t h e e n e r g y b a r r i e r for the "5-5"-structure w a s f o u n d to be 0.02 eV/atom (T = 200 K). T h u s , t h e l a t t e r one would not be expected to survive a t h i g h t e m p e r a t u r e s , unless it w e r e stabilized in some fashion a2o . A n o t h e r point to note is the high a20 Here, one should note that for a set of optimisation runs, where large amounts of excess chlorine atoms were present, the "5-5"-type structure with the excess chlorine located within the channels occurred as the major stable alternative to the phase separation into one region filled with neutral chlorine and a second one containing the standard NaCl-structure. Thus, one might expect that in the presence of some third component that could fill the channels in the "5-5"structure, it might be stabilized.

ll6 Table 2 Overview over the m i n i m a found for the s y s t e m Na - Cl with composition 1:1. The t e r m s "dense/open s t r u c t u r e s " refer to a r r a n g e m e n t s of different coordination polyhedra without/with large channels or cavities in the structure. coordination n u m b e r

structure-types/

occurence (in %)

of Na by C1

structure-elements

4

anti-PtS-structure, sphalerite

5

edge-connected trigonal bipyramids

5

edge-connected square pyramids

3.0

6

NaCl-structure

37.2

6

NiAs-structure

20.0

6

anti-NiAs-structure

4.3

6

other structures

1.8

7

monocapped p r i s m s

0.3

8

CsCl-structure

0.1

mixed

"dense" structures

12.0

mixed

"open" s t r u c t u r e s

9.0

5.5

edge-connected t e t r a h e d r a 6.8

robustness of the deepest local _mimma with respect to quite large variations of the p a r a m e t e r s in the potentialS~ a21 In order to gain some i n s i g h t with respect to general aspects of the global optimisation for large pressures, we have considered p r e s s u r e s of 1 GPa, 10 GPa, 100 GPa a n d 103 GPa by adding a t e r m pv' to the energy function a22 , where p is the pressure, and v' the volume per atom: H = E + pv'. By m i n i m i s i n g the e n t h a l p y H, we have found t h a t for pressures up to 10 GPa, the NaCl-structure was the deepest m i n i m u m , while above 100 GPa seven-fold and eight-fold (CsCl-structure) coordinations of Na by C1 are preferred compared to the six-fold ones. This agrees qualitatively w i t h the e x p e r i m e n t a l observation t h a t the h i g h - p r e s s u r e modification of NaC1 probably shows a CsCl-structure above 29 GPa 61. AB2 Systems. Since l a y e r e d s t r u c t u r e s are i m p o r t a n t v a r i a n t s in AB2 systems (e.g. the CdI2 a n d CdC12 families of structures), it was d e e m e d n e c e s s a r y to always use a Coulomb potential without a d a m p i n g term. The method chosen to perform the full Coulomb s u m was the one suggested by de Leeuw 5s. Again, the resulting structure candidates in general did not depend on the exact values of the effective potential. The m o s t common s t r u c t u r e types found are s u m m a r i z e d in table 3. We note t h a t these include m a n y commonly found s t r u c t u r e s like a n a t a s , a21 Additional tests, whether there were differences between using a damping factor or an Ewald-type summation for the electrostatic terms, showed that the distribution of local minima encountered during the optimisations did not change significantly when using a damping factor (a > 0). a22 Since we are searching for local minima of the potential energy at T = 0 K, additional terms proportional to kBT In(V) that need to be considered in simulations at finite temperature vanish. Analogously, the influence of external electric or magnetic fields can be included 22.

117

Figure 1: 5-5-structure candidate for NaCl: Na ( b l a c k ) a n d C1 (white) are coordinated bipyramidally. Ions are not drawn to scale. Average Na-Cl-distance 2.7

A. fluorite, but also layer structures like CdI2. Although the energies of the latter structure candidates usually are higher than the best sub-optima, it is a pleasant surprise that these layer structures were competitive even in a purely ionic picture. Using the threshold algorithm (cf. section 3), we computed the barrier structure of the system MgF2 (z = 2) on the tree graph level 27 (figure 2). We see that the second deepest local minimum, the anatas structure, is separated by quite high barriers from the global minimum, the rutile structure. Thus, one would suspect that MgF2 in the anatas structure might well turn out to be kinetically stable, while e.g for the structure "VII", exhibiting monocapped F--prisms about the Mg2+-ions, the energy barrier separating it from the rutile-minimum is so small that '~II" will not be expected to exist for finite temperatures. AB3 Systems. Finally, as an example for AB3 systemsS! ' 52 the energy landscape of the alkalimetal-nitrides Li3N, Na3N, K3N, Rb3N and Cs3N were explored. During the initial (long) optimisation runs with charge transfer allowed, it was found t h a t in the first two cases the full oxidation states (Li +, Na +, N 3-) were reached easily, while the latter three did not succeed in completely crossing the barriers on the energy hypersurface separating the "fully" ionised from the "partly" ionised configurations. Both Li3N and Na3N exhibited several interesting candidate structures as good sub-optima: the Li3N- and the Li3P-structure known to occur in nature, and a sheared "SrO3"-sub-structure of the SrTiO3-perovskite structure, both in the cubic and hexagonal variant, shown in figure 3. Based on these results, it appears reasonable to suggest that Na3N, if it can be synthesized eventually, will crystallize in one of these four structure types. Subsequent energy calculations for these structures using the Hartree-Fock-type ab-initio program CRYSTAL92 s4 showed t h a t their energies were quite close to each other. Concerning the kinetic stability, calculations with the threshold algorithm showed

118

AL 0 I CN=4 layered

structures I

CN = 4,5 I -5.400

_ CN = 5,6

-5.600

_

-5.800

_

Vl-d

"6.000

V14

VI-c -6.200

VII

_

Vl-a

Vl-b

-6.400

Figure 2: Treegraph of the barrier structure for the MgF2- system (z = 2). Circled regions indicate regions containing m a n y local minima (energies in eV/amm): VI-a = rutile (E = -5.35), VI-b = a n a t a s (E =- 5.20), VI-c = half-filled NaCl-structure (E = -5.03), VI-d = CdI2-structure (E =-5.93), VI-e = structure consisting of trigonal prisms (E =- 5.93), VII = structures consisting of mono-capped prisms (E =-6.12). The shaded region contains several r a t h e r deep minima with structures consisting of trigonal bipyramids (n = 5), quadratic pyramids (n = 5) and prisms (n = 6) (always MgFn-coordination polyhedra). t h a t the two "SrO3"-sub-structures were separated from each other by a r a t h e r small barrier (= 0.1 r while the other barriers all exceeded 1 r

119 Table 3 Overview over the most i m p o r t a n t structure-types/elements seen in the systems MgF2, CaF2, MgC12 and CaC12. The polyhedra listed always refer to the coordination of the cation by the anions. The frequency of occurrence during optimisation runs is given in percent. MgF2

MgC12

CaF2

CaC12

t e t r a h e d r a (e.g., HgI2-structure, ZnO2-

0

31

3

4

sub-structure of SrZnO2) trigonal bipyramids connected via

6

1

0

0

20 37

7 45

5 9

16 0

structure-elements/ structure-types

edges and corners CdC12/CdI2-structure rutile anatas

2

0

0

0

CaC12-structure

0

0

0

44

other structures based on octahedra

7

10

0

10

prisms connected via

3

0

0

2

edges and corners monocapped prisms connected via

21

0

15

22

edges and corners fluorite

0

0

66

0

other structures

4

6

2

2

Additional optimisations were performed using K+, Rb+, Cs+, and N3--ions for the initial configuration. The observed structures ranged from the ReO3-structure for Cs3N and a cube-like variant of the Li3N-structure for Rb3N and Cs3N to elevenfold (so-called "tetragonal close packing") and twelve-fold (like Na3N) coordinations of the N3--ions by the cations in K3N. One should note the decrease in the coordination n u m b e r with increasing size of the cation, reminiscent of the radius ratio rule. Ternary Ionic Systems: Ca/Ti/O. One of the examples of t e r n a r y ionic compounds 51 was the system Ca-Ti-O. Of course, there exists a large range of compositions t h a t allow for "full" ionisation (Ca2+, Ti4+, 0 2-) of the system: CanTimOo with p = n + 2m. While in binary compounds the composition corresponding to the "fully" ionised case is established nearly automatically during a general global optimisation run, there exist now a number of equally valid options t h a t correspond to deep local minima separated by rather high energy barriers. For the test runs, the three cases n = 2 and rn = 1, n = 1 and m = 2, and n = 3 and m = 1 were chosen besides the simplest one, n = 1 and m = 1, i.e. CaTiO3. The potential depended only on the average ionic radii, with the assumption t h a t the ions were of medium hardness, and the Coulomb-potential contained a damping factor. In half of the relatively short (2x105 steps) runs for the case (n = 1,m = 1), the best sub-optima found were perovskite-type structures, while in only 20% of the

120

Figure 3a: Sheared SrO3-substructure of the hexagonal perovskite structure as model for the hypothetical compound Na3N. N3--ions are surrounded by Na+-ions forming cubic close packed distorted anti-cube-octahedra. Na+-ions are depicted as black spheres (not drawn to scale). Average Na-Na- and Na-N-distance about 2.6/I~; (space group PmmnZ, no. 59).

Figure 3b: Sheared SrO3-substructure of the cubic perovskite structure as model for the hypothetical compound Na3N. N3--ions are surrounded by Na+-ions forming cubic close packed distorted cube-octahedra. Na§ are depicted as white spheres (not drawn to scale). Average Na-Na- and Na-N-distance about 2.6/~; (space group P42/mmc, no. 131).

121 runs the system ended in a configuration t h a t was not "fully" ionised (Ca2+Ti3+O(02-)2). This agrees well with the experimentally observed 5 structure of CaTiO3, which is considered to be a distorted perovskite-structure. In addition, we estimated the heat of formation at T = 0 K with respect to educts CaO and TiO2, which was found to be about 130 kJ/mol. Considering the highly simplified model for the potential energy, this compares reasonably well a23 with the e x p e r i m e n t a l l y found value 62, 90 IO/mol at standard conditions. For the other three compositions we have investigated, the best sub-optimal structures were "fully" ionised, too. However, so far only some "layered" structures of Ca2TiO4, related to the structure 63 of Ca2SnO4, were found to be t h e r m o d y n a m i c a l l y stable (about 40 kJ/mol) with respect to dissociation into CaO and TiO2. Ca/Si/Br. As a final example, we will discuss the determination of structure candidates for an unexplored t e r n a r y system, Ca/Si/Br appearing to be well-suited for this purpose: F r o m the participating components, one would conclude t h a t a hypothetical t e r n a r y compound of e.g. the composition Ca3SiBr2 should be close enough to an ionic compound (Ca2+, Si4-, Br) in order to allow the use of the simple potential function we have been testing so far. F u r t h e r m o r e , both b i n a r y compounds (Ca2Si, CaBr2) are known, and it is therefore possible to fit the p a r a m e t e r s in the two-body potentials to their properties. Last, but not least, the system has not yet been subject to intensive a t t e m p t s to synthesize such a t e r n a r y compound, leading to the expectation of eventual experimental verification of the predictions. During the testruns needed for the fitting of the parameters, the binary (ionic) compounds proved to be stable local minima. Therefore, we did not precede the long global optimisation r u n s by optimisations involving charge transfer. Instead, the ionic charges were fixed from the outset; and these ions were then, as usual, placed at r a n d o m positions within a large simulation cell. We have concentrated 54 on the composition Ca3SiBr2. Each simulated annealing r u n involved several million optimisation steps, where up to four formula units ( = 24 ions) per simulation cell were used. The structure candidates belonging to two of the best sub-optima found are shown in figures 4a and 4b. The structure in figure 4a can be derived from a NaC1type structure, where the "Na"-sublattice is occupied by the Ca 2§ ions, while the "cr'-sublattice contains the Si4- and Br ions in such a way t h a t the Si4- ions are far away from each other while still allowing for a relaxation of the inevitable distortions due to the somewhat different sizes and hardnesses of the spheres representing Si4- and Br- ions. In figure 4b, an alternative structure is shown, derived from the CsCl-type. Again, the anions occupy the Cl-sublattice, while the cations are located on the sites of the Cs-sublattice. Since no obvious way exists to judge the quality of these structure candidates (at least until the compound has been synthesized!), the ground state energies of these sub-optimal structures were also calculated using the Hartree-Fock-type program CRYSTAL9234 (no global optimisation runs, of course). It turned out t h a t these calculations could not decide either, which structure candidate should actually be preferred: EHF(NaCI-typc) - EHF(CsC1type) = -0.04 eV/atom compared to E(NaCl-typc) - E(CsCl-typc) = -0.04 eV/atom for the simple potentials. a23 Note that our energy of formation with respect to the binary compounds is calculated at T = 0 K, p = 0 Pa, while the experimental value is valid at standard conditions. Since the resulting difference in enthalpy is basically given by the integral over the specific heat, and the contribution from the binary compounds will to a large degree cancel the contribution from the ternary compound, the rough comparison we perform appears to be reasonable.

122

Figure 4a: Structure candidate for Ca3SiBr2 analogous to the NaCl-structure. Si4-ions lie in the center of shaded octahedra formed by Ca2+-ions, Br-ions within the white ones.

Figure 4b: Structure candidate for Ca3SiBr2 analogous to the CsCl-structure. Si4-ions lie in the center of shaded cubes formed by Ca2+-ions, Br-ions within the white ones.

123 6. C O N N E C T I O N S TO E A R L I E R S T U D I E S OF T H E E N E R G Y S U R F A C E OF C O M P L E X SYSTEMS In the field of solid state theory, there exists a long tradition of attempts e4 to understand the structure of solids, be they crystalline or amorphous, of macroscopic size or microscopic like clusters. It has always been clear that (meta)stable structures of solids at low temperatures can be identified by determining the minima of the energy hypersurface of the chemical system (as function of atomic coordinates), with the exception of certain systems dominated by quantum effects. Especially in the early days of solid state physics 65, 66, much effort was devoted to the calculation of cohesive (= ground state) energies of already known compounds 17, and the derivation of elastic constants by variation of the lattice parameters. With the development of computers and fast algorithms, new tools became available to solid state theory. From the molecular dynamics studies of Alder and Wainwright 67 modelling hard sphere liquids to the ab-initio molecular dynamics of the Car-Parrinello algorithm 35 able to perform local optimisations of the structure of simple solids 6s runs a clear path of steady improvement. This work has been paralleled by developments in the theory of molecules and their structure 69. It is by now standard procedure to employ some local optimisation routine during ab initio studies of molecules. A joining of these two strands of research has come about in the investigation of clusters 7~ which can often be viewed either as large molecules or as nanoscopic solids. While the clusters studied so far are often too small to really justify being viewed as extended solids, the work on such mesoscopic systems has had a profound effect on the way one approaches the analogous problem of structure prediction in solids as discussed in this chapter. For two facts in common to dusters and solids have been brought home very strongly 73, 84-86 : the number of possible structures of a cluster/solid characterized by being local minima of the energy hypersurface is very large, and it is not clear at all how to decide on the "best" or "typical" one without studying the energy landscape in great detail. This involves both the calculation of the depth of the local minima, the study of the density of states associated with these minima, and finally the determination of the barrier heights of the saddle points connecting the different structures a24 . While the importance of such saddle points has always been appreciated in the study of molecular reactions 87-91, this has been perhaps less so when dealing with solids, where usually only one structure has been considered at a time. The justification for this traditional procedure lies in the kinetic stability of solids, i.e., the thermodynamic behaviour of a solid may depend only on the small region of the energy landscape around some (meta)stable structure (cf. section 2). It has therefore been perfectly reasonable to concentrate on the local environment of some already known or expected interesting structure, and to perform local optimisations of some energy function. Such local optimisations 35, 68, 92, 93 have been performed with varying degrees of restrictions on the number of atoms, symmetry of the structure, and size of the cell parameters, a25 But if one tries to predict hypothe-

a24 Thus we would expect that e.g. the lid/threshold algorithms will also prove useful in the study of the energy landscape of clusters. a25 In this context, one should also mention the modelling of proteins94 and of amorphous structures56, 95. In the latter case, one aims for a disordered arrangement of the atoms: thus

124 tical compounds without any a-priori information, the whole energy landscape needs to be explored requiring the use of both global optimisation algorithms and methods like the threshold algorithm. An interesting application of the use of global optimisation methods in the structure determination of solids from powder diffraction data - as opposed to the apriori structure prediction described in this chapter - has been introduced in recent years by Pannetier, Newsam, Freeman, Catlow and others 31, 96-101. In their work, it is generally assumed that some experimental information about the compound is already available, e.g. as a X-ray powder diffractogram. Thus, the size and shape of the unit cell, the composition of the compound, the ionic charges, and the number of formula units present in the cell are assumed to be known. It "only" remains to determine the exact positions of the atoms within this cell, a task that often tends to be quite involved, especially if only data from X-ray or neutron powder diffraction experiments are available. Imposing the above constraints, the authors perform global or local minimisations of special cost functions that combine some energy terms based on effective potentials with penalty terms reflecting e.g. Pauling's bond-valence rule 98 or the requirement of an even distribution of ions within the unit cell 97. If the penalty terms and the effective potential are chosen appropriately a26 , the minimisation produces a number of reasonable structure candidates, which then can serve as input either of a structure solving program or a refinement optimisation using a more realistic energy function. In the extreme case, the cost function is solely constructed from penalty terms and equals a "figure of merit" that reflects a-priori knowledge of the typical bonding arrangements in the solid 1~

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Z.B. Maksi6 and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modern Modelling of the Chemical Bond Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

129

Polarizability and Hyperpolarizability of Atoms and Ions David M. Bishop Department of Chemistry, University of Ottawa, Ottawa, Canada K1N 6N5

In 1927 Pauling published his ground-breaking paper: "The Theoretical Prediction of the Physical Properties of Many-Electron Atoms and Ions: Mole Refraction, Diamagnetic Susceptibility, and Extension in Space." The historical setting of this work is recounted and the flavour of the early days of the 'new' quantum mechanics is recalled. Pauling's key ideas concerning the calculation of mole refraction, a quantity which is directly related to a species' polarizability, are analyzed. More recent developments in the theory of polarizabilities are examined and their extensions to higherorder polarizabilities, both static and frequency-dependent (dynamic), are given. Particular attention is paid to the hydrogen atom and the helium, neon, and argon isoelectronic series, all of which were treated in Pauling's original paper.

1. Historic Times Linus Pauling arrived in Munich on April 20, 1926[ 1]. This was very much a case of being in the right place at the right time. He was going to spend one year at the Institute of Theoretical Physics which at the time was directed by Arnold Sommerfeld. For this purpose he had been awarded one of the very first fellowships of the John Simon Guggenheim Memorial Foundation. It was the right place because Sommerfeld was a bridge between the 'old' and the 'new' quantum mechanics: he had spent much time trying, only with partial success, to patch up Bohr's old quantum theory. His Institute was one of the nerve centres in Germany for the development of the revolutionary ideas of Heisenberg (a former student of Sommerfeld) and Schr6dinger (whose seminal paper was published in 1926). Prepublication documents circulated throughout the Institute and, since things were moving so rapidly, this was of great importance. Sommeffeld did not contribute much to the new theory but was a strong supporter and a guiding influence on many young scientists at the time; Heisenberg once said "I learned mathematics from Born and physics from Bohr. And from Sommerfeld I learned optimism." Max Born in G6ttingen and Niels Bohr in Copenhagen directed two of the other 'nerve centres' of the epoch. How Pauling came to be in Munich in 1926 is an interesting story not without its element of skullduggery. He had left from the California Institute of Technology, better known as Cal Tech, where he had received his Ph.D. with summa cum laude in June of 1925. When he had begun his Ph.D. in the fall of 1922 Cal Tech, as such, was only four years old; it had previously been a small manual training college called Throop College. With a lot of money and a lot of talent, it was already on its way to becoming a major

130 scientific institution. One of the talents was the Chairman of the Chemistry Department, Arthur A. Noyes, another was the Chairman of the Physics Department, Robert A. Millikan. It was Noyes who assigned Pauling to Roscoe G. Dickinson to carry out x-ray crystallographic studies for his thesis. Dickinson, himself, was the first person to receive a Ph.D. in chemistry from Cal Tech. As well as doing this, Pauling also worked for a while with Peter Debye, who was a visiting professor. In fact, Cal Tech was no backwater; other visitors, while Pauling was there, were Born, Sommerfeld, Ehrenfest, Franck, Bohr, and Einstein. On top of this, the faculty boasted the two renowned theoreticians Carl Eckart and Paul S. Epstein, whose name will re-occur shortly, as well as the great physical chemist Richard C. Tolman. It was Tolman who was to introduce Pauling to quantum theory. Had Pauling not gone to Europe he still would have been in a young and thriving intellectual environment. The reason he did go was the doing of Noyes. Pauling had shown early brilliance and was now what Variety magazine would call a hot property. He had been given a National Research Council grant to do post-doctoral studies at Berkeley where Gilbert N. Lewis, another giant of the day, was the head of the chemistry department. Noyes was determined not to lose Pauling to Lewis and persuaded Pauling to stay on for a few more months, which Pauling did. Noyes used this breathing space to good effect. He practically 'arranged' that one of the first Guggenheim Fellowships, given for study abroad, should go to Pauling. There was only one snag: the Guggenheim would not be approved until April 1926, consequently, Noyes offered Pauling Cal Tech funds for his travel and living expenses until that time came. Noyes was smart enough to realize that this was an offer Pauling could not refuse. This was the time when great things were happening, and happening fast, in Europe. Pauling did not refuse, he relinquished his NRC scholarship and set out for Germany. For Noyes, Europe was a safe distance from Berkeley and his scheming paid o f f - in the fall of 1927 Pauling returned to Cal Tech and stayed there for thirty more years. One of the first things Pauling did in Munich was to discover a mathematical error in a paper by Sommerfeld's assistant Gregor Wentzel. Pauling wrote this up for Zeitschriftfar Physik[2]. Sommerfeld was sufficiently impressed to say that he would submit Pauling's next paper to the Proceedings of the Royal Society (London). Sommerfeld had just recently been appointed a foreign member of this most prestigious body and articles in their Proceedings could only be submitted by a member. Pauling's 'next paper' was the one which is the focus of this chapter[3]. It was received in London on New Year's Day 1927. In telling this story it is apparent how much was new, flesh and young in 1926: Pauling, Cal Tech, Roscoe, the Guggenheim Fellowships, quantum theory, even Arnold Sommerfeld as far as the Royal Society was concerned!

131

2. The Paper

"The Theoretical Prediction of the Physical Properties of Many-Electron Atoms and Ions: Mole Refraction, Diamagnetic Susceptibility, and Extension in Space" has all the Pauling hallmarks: quick off the mark, broad-sweeping in range, intuitive approximations, and so nearly right. We will be concerned only with the 'mole refraction' part of his article. The mole refraction (R) is related to the refractive index (n) by R =

n 2 -1 n2+lV.,,,.

(1)

= P~176

(2)

where Vmis the molar volume. Since n 2 -1 n2+2

3eo

where po is the number density, eo is the permittivity of vacuum and ~(co) is the dynamic dipole polarizability for light of frequency co, we can write R

=

4Jr N AO~(0)) 3

4~re0

(3)

if the SI convention is used. There is thus a direct proportionality between R and the polarizability c~; usually the frequency dependence is small and o~(o~) is assumed to be the same as the static quantity, o~, defined for co ---) ~,. For this reason a discussion of the mole refraction part of Pauling's paper is just as equally a discussion of the dipole polarizabilities of atoms and atomic ions. The dipole polarizability also expresses the second-order effect of an electric field on the energy levels of an atom or molecule. We can write, for an atom (which has no dipole moment and therefore, in general, no first-order effect): E

= E ~-~:r/72-...

(4)

where subscripts identifying which energy level are omitted. Since the Stark effect describes the phenomenon of shifts in spectral lines due to an electric field, it is apparent that it is governed by the polarizabilities of the species under investigation. The correct prediction of the Stark effect for the hydrogen atom (or of its polarizability) was one of the earliest triumphs of the new quantum mechanics. Prior to 1926, there had been a problem with the H atom polarizability formula as given by Epstein[4]: it gave the wrong results! The formula was based on Sommerfeld's extension of Bohr theory and it was Sommerfeld, himself, who recognized[5] that it was incompatible with the observations made by Takamine and Kokubu[6] in 1919 at the Mount Wilson Observatory. The shift of the whole splitting pattern of H to the red was confirmed but not its magnitude. It is not surprising, then, that when the galley sheets of Schrtidinger's first paper[7] started circulating, there would be a race to see if his new theory could clean up this embarrassing p r o b l e m - after all, this was only the H atom! The finish was almost a dead-heat between Gregor Wentzel[8] in Munich (received date: 18.6.1926), Ivar Waller[9] in Copenhagen (received date: 21.6.1926), and Paul

132 Epstein[10] in Pasadena (submitted dates: 24.7.1926 and 29.7.1926). A sense of the drama is given by the opening paragraph of Epstein's letter to N a t u r e "The theory of atomic oscillations recently advanced by Schroedinger is of extraordinary importance since it throws a new light on the problems of atomic structure and, at the same time, offers a convenient practical method for calculating the Heisenberg-Born intensity matrices. It seemed desirable to apply it to as many special cases as possible. A complete theory of the Stark effect in hydrogen was, therefore, developed."

In all three cases, the essential second-order energy shift for a field strength was given by -

-

~

n4(17n

16Z 4 mee 2

2

- 3 m 2 -9n~ + 19)F 2

(5)

Here Pauling's notation has been used for the quantum numbers, namely m = n2-n,

n3 = n-l-nl-n

2,

OO

OH

E K - Eo

159

which we recognize as the standard expressions of time-independent perturbation theory. However, although conceptually simple and transparent, these expressions are not particularly useful for practical calculations of molecular properties since a complete diagonalization of the Hamiltonian matrix is required. In practice, therefore, the molecular properties are calculated in a different manner, which does not require the diagonalization of the Hamiltonian matrix.

4

Explicit expressions for electric and magnetic properties

Having considered the general expressions for first- and second-order molecular properties, we now restrict ourselves to properties associated with the application of static uniform external electric and magnetic fields. For such perturbations, the Ha miltonian operator may be written in the manner (in atomic units) 1 B .dm + -~ E [B2r~- (B .ri) 2]

H (F,B)= H0- F-de

(52)

i

where the summation is over all electrons and where we have introduced the electric and magnetic dipole operators de

=

- Eri i

dm

=

+ EZIRI

(53)

I

1

1

r, X p , i

s

(54)

i

In Eq. 53, the first summation is over all electrons and the second summation over all nuclei. The positions of the electrons are given by r i and the charges and positions of the nuclei by ZI and R I , respectively. In Eq. 54, the Pi and si are the conjugate momenta and spins of the electrons. We have also introduced the operators

i

for the total orbital and spin angular momenta of the electrons. Note that, in the clamped-nuclei approximation, there are no nuclear contributions to the magnetic dipole operator.

160 Having set up the Hamiltonian, we may calculate the first- and secondorder properties in the eigenvector representation. For the permanent electric and magnetic dipole moments, we obtain

-

(0 Idol o) - - ~ (0 Ir, I 0> § ~ ZzRz i

m

-

(0[dm]0)--

(57)

I

0 ~L+S

0

(58)

Whereas the permanent electric dipole moment vanishes for molecules belonging to certain points groups (e.g., for all molecules that possess a center of inversion), the permanent magnetic dipole moment vanishes for all closedshell systems. To see how the vanishing of the magnetic dipole moment comes about, we first note that S [cs) - 0

(59)

since the closed-shell state ]cs) is a singlet. Next, we note that for all realvalued electronic states ]real> (such as all closed-shell states), the expectation value of any imaginary Hermitian operator (such as the orbital angularmomentum operator) is identically equal to zero (real [LI real) -- 0

(60)

and the angular momentum is said to be quenched. Let us now consider the second-order molecular properties. The static electric dipole-polarizability tensor is given by the expression

- 2 ~ (0 lde[ K> (K IdOl 0> K>o E K - E0

(61)

and is nonnegative for the electronic ground state. For the magnetizability tensor, we obtain

r - __14 ~ (01~1 - r,r~l o) + 2 ~ i

K>o

0 Im(Yzlml)if m < 0

(3)

Here, N is a normalization factor, chosen such that f I Sire ]2 dgt = 1. The original justification for constructing hybrids was that the radial behavior (in position space) of the constituting functions is rather similar, and that they are energetically near-degenerate. These conditions are fulfilled exactly for the eigenfunctions of a one-particle Coulomb system, the hydrogen-like ions with nuclear charge Z. We will therefore illustrate a few concepts on those. The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and "Lorentzian", respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively: RH(r)

--

2n(n + l)!

-~-l-~

(4)

215

N

R.(p)

=

(-i)~

p

2 Z ( n - l - 1)' l!(Z/n)t+2(4p) t+x p t+x 7r(n -4-l)! -((--'~n)2;p~)l+2 "'n-l-1

(p2/zjn 2) p -t- (Z/Tt) 2

(5)

The momentum-space expression was first given by Podolsky and Pauling [3] in 1929. Note, that for any real position function RH, the corresponding momentum radial function RH will be either purely real or purely imaginary, depending on whether the angular part of the orbital is even or odd (see also [8]). The factor ( - i ) l in Eqn.(5) has, e.g., the consequence that s-type and p-type functions do not "mix" in momentum space, which leads to hybrids that have a different nodal structure. 3. H Y B R I D S I N M O M E N T U M 3.1.

Hybrids

SPACE

of the spa-type

The most commonly encountered hybrids in organic chemistry are linear combinations of s and p-type orbitals. Depending on the linear coefficients of the real p-functions in the three Cartesian directions, the resulting set of hybrids will be oriented at different solid angles from each other. For example linear combinations of the form 1/2(s :t:px :t:py-t-pz) will yield hybrids with a tetrahedral angle among them. This basic geometry will be retained in momentum space, since the Fourier-transform (1) is a direction-preserving, unitary transformation. However, the fact that even and odd contributions to the positionspace hybrid are transformed separately (via cosine and sine transforms) into real and imaginary parts of the momentum-hybrid, means that the resulting densities (i.e., the square magnitudes) are inversion symmetric with respect to the origin, which is obviously not the case in position space. Therefore, an arrangement of orbitals in a point group G in position space, will lead to an arrangement in the point group G' = G x Ci in momentum space; this is the direct product of G with the inversion-symmetric group (see [7] and [9]). For the following considerations, the orientations of the hybrids are not relevant, and we therefore mix only pz orbitals with s-functions, resulting in an orbital that points along the polar axis z in both spaces. This takes the following form for the n = 2 shell:

_ ~

+

-

[2-

+

co

0]

=

,

(6)

S i~r(1 + a) r

= -~

(r

In position space, r 1 + v/-acos 0 r - - 2 Z(a cos 2 0 - 1)"

+ x/~r

- 16 Z s/2 [4p 2 - Z 2 - 4 ix/-dZpcosO] 7 r ~ + a ) ( Z 2 + 4p2) 3

(7)

is real and its nodal surface is defined by

(8)

In contrast, the real part of the momentum hybrid ~,~ has a spherical node (at p -

~ r ~ } _ Z/2), whereas the imaginary part has a planar one (at 0 - 7c/2). This means

216 ~a

that the subspace, where r - 0, is one-dimensional. It is, in fact, a circle of radius p = Z / 2 in the xy-plane 2, with center at p = 0. Coulson and Duncanson [4] obtained an analytical expression for a momentum hybrid of a C-atom, constructed from STO's that shows similar structure. Later, Cooper and Loades gave contour plots of several similar hybrids in momentum space [6]. However, neither one did comment on this interesting feature. It is widely known that total momentum densities for atoms are not always monotonically decreasing [10]. In fact the degree of non-monotonicity is dependent on the degree of p-population in an atom. This fact is visible as well in the shape of s p a hybrids in momentum space.

Figure 1. Surface plot of the orbital densities for sp a hybrids in momentum space. The hybrids are based on the hydrogenic wave functions. The three plots pertain to a = 1, 2 and 3, respectively. The hybrids point in the z-direction. A section through the density in the xz-plane is displayed.

In Fig.(1), we display the momentum density contributions of commonly encountered hybrid orbitals, obtained from hydrogenic eigenfunctions with Z = 1. The figure shows surface plots of the densities for s p a in the xz-plane for a = 1, 2 and 3. It may be seen that, while the s p hybrid exhibits a maximum at p = 0, greater p-contributions flatten this maximum out, leading to a plateau for s p 2, and finally a saddle point for s p 3. All of these densities feature two points in the xz-plane where the density vanishes exactly. They are situated on the x-axis, as sections along that axis demonstrate clearly. We show those in Fig.(2). Independently of the mixing coefficient a, those "nodal points" occur at x = -1-1/2 on each equatorial axis. They are the intersection of the aforementioned nodal circle with the displayed plane. To further assess the extremal structure of the densities corresponding to our hybrids, we also display the curvature of the density along the polar axis in Fig.(3). The analytical expression for these curves is 2Throughout the article, we use the notation x, y, z for the Cartesian components of the momentum vector p, and 0, r for its angular spherical coordinates. This is done to avoid excessive subscripts, and confusion with p-orbitals.

217

027r~p(ff) I

8192(a

-

2)

p=o = Z57c2( a + 1)

OP2z

(9)

Therefore, clearly the sp2-hybrid is the limiting case, for which the transition from maximum to saddle-point occurs, independently of the nuclear charge Z. We note that in the paper of Cooper and Loades [6], this distinction was not found for the momentum hybrids. Using Gaussian orbitals, with exponents tailored for the carbon atom, the origin is a saddle point for all three hybrids. For the hybrids that we have constructed, maxima will occur for any a > 2 at

Z Pmax -- ~-

(6 - 10a + 2x/i " 26a + 25a 2) 1/2

(10)

As we should expect, increasing the amount of p character, by increasing the value of a, will shift the maxima of the momentum distribution, to higher values of p. This fact well illustrates Epstein and Tanner's H y b r i d Orbital Principle, which states that "increased p character in an s - p type hybrid orbital results in increased density at high momentum". [11] In addition, one should contrast the appearance of the plots from Fig.(1) with our more familiar position space representation of these sp a hybrids, as shown in Fig.(4). There, we always see two distinct maxima [6], the sharper of which is located right at the origin.

3.2. Hybrids involving d-orbitals For most of organic chemistry the description in terms of sp a hybrids is sufficient for a qualitative picture. However, if the coordination numbers involved are greater than 4, as is the case for the majority of compounds involving transition metals, d-hybridization has to be taken into account. Since the m-quantum number of a d-function influences not

Figure 2. Sections through the orbital momentum densities displayed in Fig.(1) along the x-axis with z = 0, i.e. perpendicular to the main axis of the hybrid. The three plots pertain to a = 1, a = 2 and a = 3, respectively. Note that each density vanishes at two points on the x-axis.

o-1I

o-1I

0.08

01I

0.08

0.08

0"02 I -1

-0.5

oo '

-

-

.

oo

oxs

1

~

5

'oo

218

Figure 3. Second derivative of the orbital momentum density, 027r~p(ff)/Op~, in the z-direction, along the polar axis. Note that for a = 1 (left plot), this quantity is negative around p = 0, for a = 2 (middle plot) it is exactly zero, indicating a plateau, and for a = 3 (right plot) it is positive, denoting a saddle point.

ioo

-

2oo~

Figure 4. The sp, sp 2, and sp 3 hybrid density functions in the xz-plane of position space. As is often the case, orbitals that are quite different from one another in momentum space, can appear very similar in the corresponding position space representation.

only the orientation, but also the shape of the constituting atomic orbitals, we have to distinguish several cases. In the following, we focus on two of them; the first are hybrids that are directed along the z-axis, such as the ones in an octahedral complex, which have the general form 3sp~dbz2. The second lies in the xy-plane, and is exemplary of a hybrid that would be used to describe bonding in square-planar compounds. These hybrids are of the form 2 spxax2_y~ a-b . Other combinations are possible, but will in general show similar structural features.

219 The first class takes the following form (for the n = 3 shell) in momentum space:

~3sp~d2-a,b=

x/'l +la + b (~3s (#) + k/cd~3p~ (#) + k / ~ 3 d 2 (#))

18 3x/3--Z~ (81v/-2p4

--

6Z2p 2 [5~/-2 + 4x/b (3 cos20 - 1 ) ] + v/2Z 4)

(7rV/1 + a -t-b) (Z 2 + 9p2) 4 i 432pcosOv/Z7a

+

(Z 2 - 9p 2)

(~rv/1 -'~ a --}-b) (Z 2 nt- 9p2) 4 "

(11)

In the above equation, the hybrid is clearly broken down into a real part (second line), and an imaginary part (third line). We have found it convenient to analyze these two parts of the hybrid separately because of the earlier mentioned property that the real part of a hybrid will not "mix" with the imaginary part when computing expectation values and densities. The real part of Eqn. (11) arises from the mixing of s- and d-contributions. It has roots on either side of the xy-plane, on two closed C-rotation-symmetric surfaces, that are

Figure 5 9 Nodal surfaces of a s~3d ~'z 2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the z-axis. The intersection of the two types of nodes are two circles around the z-axis. The right-hand plot displays a cut through the xz-plane. Note that the (polar) z-axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.

1IX f .. ...............~.5

"'\ z /,," ....................

-0.5

220 the solutions of a quadratic equation in p2. Figure (5) shows these surfaces (left plot), as well as a section through the zz-plane (right plot). The central sphere in the left plot and the circle in the right plot are nodes of the imaginary part of the hybrid. This imaginary part (third line of E q n . ( l l ) has nodal surfaces which consist of the xy plane (0 = 7r/2, not shown in the plots), and a sphere centered at p = 0 with radius Z/3. As ~a b a result, the density (i.e. the square magnitude of Cs;d,z), vanishes on a pair of circles at pz = =i=z [3 ( 1 - ~ r 1,2 with radius gz [ 2 / 3 + x/~/6v/~l U2 These circles are the intersection of the roots o t the real and the imaginary parts.-This nodal behavior is in clear contrast to the one in position space, where nodal surfaces of a rather complex shape are observed.

Figure 6. Surface plots of the momentum density corresponding to two different hybrids of the 3 b form sp~dz2. For the left plot, b = 2 (as in octahedral hybrids), for the right one, b = 5 (maximum bond strength). The surface plots are sections through the xz plane. Note that they both exhibit four points where the density vanishes exactly.

Fig.(6) shows the momentum densities corresponding to two different hybrids of the type. The first one is a sp3d 2 orbital as is encountered in octahedral complexes, the second one is a "maximum-bond hybrid" sp3d 5. The basic features are rather similar, although somewhat more strongly pronounced in the one with the greater d-component. The plots show a section through the densities in the zz-plane. Particularly for the sp3d 5 hybrid, a set of 4 "holes" parallel to the x-axis can be observed, arising from the aforementioned circular nodes. To show these "holes" more clearly, we have plotted in Fig.(7) circular sections through the densities, passing through the xz-plane with radius 1/3. The graphs show the momentum density as a function of the polar angle 0 in units of 7r. Note that the density is maximal inthe z-direction (0 = tTr;t = 0, 1,2), and in x-direction (0 = tTr;t = 1/2, 3/2), s -pz%~ a ..tb

221 Figure 7. Circular sections through the momentum densities displayed in Fig.(6). The plots display the value of the density along a circle of radius 1/3 in the xz-plane, as a function of the polar angle 0 -- tr. The nodal points are clearly visible.

, / 0

0.5

1 t

;0

I 5

k,o .

as would be expected and is also observed in position space. However, at angles of 00 and

[

7 r - 00 with 00 = a r c c o s ( ~ 2 ), the density goes exactly to zero, since this is the angle under which the real and imaginary nodes intersect. This angle is not dependent on Z, but depends weakly on b. It varies between 7r/2 for b = 1/2 and ~ / 2 - a r c s i n ( 1 / x / ~ ) for b ~ e~. The second class of spd-type hybrids that we will treat here are situated in the equatorial plane, i.e. at 0 = ~-/2. They consist of a linear combination of s, px and d~2_y~, orbitals, and have proven useful in describing the bonding in square-planar complexes. Their form (in momentum space) is:

Csp~d~2_y2-ab' (P) = X/'i +la + b (~3s(~[~) -3t-v/a~3p~ (]~) -+- v/b~3dz2-y2 (g)) 18Z 5/2 (x/~ (81p 4 + 2 4) - 622p 2 (12~/bcos(2r

5v/-6))

(~x/1 + a + b) (Z 2 + 9p2) 4 i 432 px/-Z-~ sin 0 cos r (Z 2 - 9p 2)

+

(~x/'l + a + b ) (Z 2 +

9p2) 4

(12)

Note that the imaginary part is essentially the same as in Eqn.(11), but the real part differs in shape. However, the consequence is a qualitatively different shape of both the nodal surfaces for the real part, and the nodal curves for the density. Figs.(8-10) show various aspects of the resulting density contribution for a 3sp~dx2_y2

222 Figure 8. Plots of the nodal surfaces in a sp~dx2_y2 hybrid orbital (Z = 1) in the momentum representation. The left plot shows the surface due to the real part (i.e., s and d contributions) only, whereas the right one combines it with the planar and spherical nodal surfaces characteristic of the imaginary (i.e., p-) component.

orbital, as this is the one used to describe square-planar complexes. Since the real part of Eqn.(12) is a linear combination of spherically symmetric s-contributions (with 2 spherical nodes) and "Rosetta-shaped" d-functions the resulting nodal surface is somewhat "donutshaped" around the y-direction. It is displayed in the left plot of Fig.(8). On the right-hand side we combined this surface with the planar and the spherical node of the imaginary part, which are due to the hybrids' p-contributions. Cuts through the xy-, xz- and yz-planes may serve to clarify the topology further (see Fig.(9)). The features in the two planes that are cut by the donut (xy and yz), are rather similar to the ones encountered earlier for the axial hybrids s P-a'b z t t z2. However, in the plane of the ring (xz) it differs considerably, and shows no intersections. All curves are rather contained within each other. Note also that the yz-plane is in itself a nodal plane of the imaginary part, and that therefore the cuts arising from the real part are nodes of the density. The intersections of the circle and the "outer" curves in the right plot are in fact intersections of 3 nodal surfaces, 2 from the imaginary and one from the real part. The lower right plot of Fig.(9) shows the nodes of the density (i.e., the intersections in 3Dspace). The two closed curves in the y z-plane arise from intersections of the donut-shaped node of the real part with the planar node of the imaginary one, whereas the other curves are intersections of the spherical node of the imaginary part with the donut, close to its "hole". The explicit expressions for the curves displayed in the lower right plot can be derived. They are: Z

p--~(V/4-x/~sin20•

;r

•

(13)

223 Figure 9. Traces of the nodal surfaces in the xy-, xz- and yz- planes for the 8p2dx2_y2 hybrid. It can be seen that while the basic features in two of the planes (xy and yz) are rather similar to the one observed in 8-ad pz b.2 type hybrids, the situation in the xz-plane is completely different. The lower right plot shows the nodal lines that are the intersections of various surfaces displayed in Fig.(8).

9f /'/l(,~ .... ',, -~ t '~ ;~176176

\.._ .....~

x

........./

-0"5t -1

1Y z

0.5

/

(o2

z \ . . . . . .

4

x

O.

-0.

t,.,

-0"5 I -1

r

2

2arcc~

V~sin2 0

;p=

It is rather surprising that such simple linear combinations will produce such complicated topologies in momentum space. Fig.(10) shows a surface plot of a section of the momentum density in the xy-plane, where the density is accumulated (left). The seemingly monotonous distribution exhibits on closer inspection a good deal of fine structure: first, there are the aforementioned "holes" in the vicinity of the nodal lines; secondly, the apparent maximum reveals itself on an enhanced scale (right plot), to be a saddle point that is minimal in the x-direction.

224 2 Figure 10 9 Surface plot (left) of the momentum density corresponding to a spxdz2_y2 hybrid orbital. The section displayed lies in the my-plane. Although the overall features seem to be rather weak, a complicated nodal structure is observed, and the origin is a saddle point (right). The right plot shows the momentum density on the x-axis.

49.25

-0.03

-0.02

-0. Ol

0

0. Ol X 0 . 0 2

0.03

For all momentum densities, the origin of momentum space is necessarily a critical point. This arises from the inversion center and the requirement of continuity. However, the topology at that point may vary considerably for the hybrids considered here. We pointed out already in the previous section that the spa-hybrids do not always exhibit a pair of off-center maxima. For hybrids containing d-functions, the picture is further complicated 9 To obtain a clear idea, it is best to obtain the diagonal elements of the Hessian matrix of the density Hij = 02rr~(p-)/OpiOpj in Cartesian coordinates 9 From their sign it may be inferred what type of critical point is observed. For the sp~dbz2-type hybrids, this yields

(Oq27r~bOq27rrOq271-~b)(2~/~ 11 2V/~ 0p

'op--7'0p

-

V27rr

=

-

r ( 8 a - 33)

'

-

l18a '

4V/~ -

11)

(15)

-

(16)

where r = 46656/Z57r2(1 + a + b) is a common pre-factor. The eigenvalue structure in Eqn.(15) indicates that p = 0 is a minimum in z-direction, whenever b < 2a 2

lla 121 --~+--~

and

a > 11/8,

(17)

These conditions are independent of Z, since the latter determines only the spatial extent, but not the basic topology. From Eqn.(15) we also infer that d-contributions tend to flatten out the minimum at the origin (see also Fig.(6)) by moving the maxima towards higher momenta, whereas

225 Figure 11. Map for the topology of the momentum density at the origin of momentum space p 0, for hybrids containing d-functions. The left plot is for functions of the Spzadbz2-type, the right one for the Spzadb~2_y2-type. The axes are the mixing coefficients a and b, respectively, and the lines drawn separate regions of different topology. =

35

60I

30'

5O

:o

ring

~~t

/

i0_

25' saddle

20 b 15

rlng

minimum

.

I0

l

5 maximum ,

O'

.0,0

~~ '

'

i

,

J ---~, .... 2

a

saddle ~

3

,

~1 . . . .

p-contributions sharpen the minimum, although not deepening it. In the x-direction (or 121 more generally, perpendicular to the polar axis), p = 0 is a maximum for b < - U , i.e. in all cases of practical interest. The left plot in Fig.(ll) shows the regions of different possible topologies at p - 0 in the (a, b) plane. Note that for "reasonable" hybrids, a < 3, and the origin in momentum space is therefore either a saddle point or a maximum. Rings or minima would only occur with very low relative s-contributions. Note that the Laplacian (16) (as the sum of second derivatives) at the origin does not depend on the d-contributions. They merely serve to "rearrange" the density, whereas p-components will make the Laplacian less and less negative. For the other type of d-containing hybrids, the spxdx2_v2 hybrids, we have

op~'

Op~'

Op~ v~r

~-

'

~ (Sa - 33)

11' -11/ (19)

where T is the same as in Eqn.(15). This means, the density is always a maximum in the z-direction at p = 0 for this type of hybrid. In the x-direction (which is the main axis of the hybrid), it is minimal if b 11/8,

(20)

and in the y-direction it is maximal as long as b < 121/24 ~ 5.0417, i.e. up to and including spxads~2_y2 hybrids The Laplacian is the same as for the previous type, which is

226 a trivial consequence of the fact that it depends (for p = 0) only on the radial part of the constituting orbitals. Note that our example (sp~dx2_y2) has a saddle point of the momentum density at the origin. This fact can not be seen at the resolution of Fig.(10), since the saddle is extremely shallow (the second derivative is "only" 11664(5- 2v/g)/Tr 2 ~ 911.4. The right-hand plot in Fig.(11) shows the relevant section in the (a,b) plane with the proper designations for the type of critical point occurring at the origin. Note that a cage-structure is impossible since the hybrid is always concentrated in the xy-plane. Table 1 Moments of the sp a hybrids in both coordinate and momentum spaces. (r ~) n--

2

1 (3+a)Z 2 12 l+a

n--

1

1Z

128 a+2 15 ZTr(l+a)

6+5a

16 Z(3+4a) 45 7r(l+a) ,,,

4 39+7a

3 Z2(l+a)

n--O n--1

z(i+~)

n--2

6z2(t+~)

n--3

30 Z3(l+a)

n--4

240 Z4(l+a)

5a+7

1 2 ~Z

11+7a

12+7a

15 7r(l+a)

1_z4(37+7~) 48

l+a

4. M o m e n t s of the H y b r i d Orbitals

It is a simple matter to derive expressions for the moments of the hybrid orbital densities. In momentum space, the expressions will take the form, f pnrr~(g)dg = f ({a})Z n

(21)

where f ({a}) is a function of the mixing coefficients ( one mixing coefficient for the 8p a hybrids, two mixing coefficients for the sp~d b hybrids, etc.). The analogous expression in position space has an inverse dependence on the atomic charge, Z.

/ r'~pr

" - g ({a}) Z-~

(22)

where g ({a}) is also a function of the mixing coefficients, although different from the f function in Eqn. (21).

227

Complete expressions for the hybrid charge density moments, in both position and momentum space, are given in Tables 1 and 2. The former table contains results for the sp a hybrids while the latter gives results for the spad b hybrids. Table 2 Moments of the spa d b hybrids in both coordinate and momentum spaces.

(p')

( 2) except for the n-alkanols (-20.178 + 0.40 kJ/mol) and n-alkyl bromides (-20.241 + 0.118 kJ/mol). Our more recent analyses [9] using yet newer data, show little change but they do demonstrate the sensitivity of the regression equation results to the chosen experimental data

306 (see examples in Table 1). Analysis of series for which data are available only for nc < 4 typically gives disparate slopes [10]. Table 1. Constants from the linear regression homologous series (kJ mol") Homologous Series

C,r

analysis of equation

3 for several

standard error in

a

AH~ n-RH n-ROH n-RSH n-RC1 n-RBr

C4-C12, C,6, C,8 C4-C,0, C12, C16 C4-C7, Clo C4, C5, C8, C12, C18 C4-Cs, C,2, C,6

-20.63 -20.14 -20.46 -20.83 -20.23

+ 0.05 + 0.04 + 0.14 + 0.05 _+0.08

-43.20 -194.73 -7.03 -71.34 -26.78

+ 0.53 + 0.40 + 0.95 + 0.59 + 0.70

0.71 0.46 0.65 0.63 0.79

In the least squares analyses of eq. 3, the individual enthalpies were weighted inversely as the squares of the experimental uncertainty intervals. In all cases, r 2 > 0.9999. The s t a n d a r d errors were generated from the unweighted enthalpies, nr is the n u m b e r of carbon atoms in the compound. A composite, or universal, methylene increment is useful for estimating enthalpies of formation for compounds not belonging to an evaluated series. For the purpose of interpolating the enthalpy of formation of a member of an evaluated series, the unique methylene increment for that series should be used. We have done this on numerous occasions to estimate a value for a presumably incorrect experimental enthalpy of formation. Extrapolation is another matter. Calculated values of the slope and intercept in eq. 3 are quite sensitive to the input data and therefore the enthalpies from even a moderately long extrapolation are affected by the choice of data. We wonder whether, once nc becomes large enough, the influence of the substituent Z wanes and the methylene increments for various homologous series become truly Universal and identical to the n-alkane increment. Ref. 8 showed that at least up to nc = 16, the influence of the hydroxyl group in n-ROH on the slope of eq. 3 is not diminished. The success of empirical additivity schemes [11, 12] is based on the recognition t h a t the contributions to the enthalpy depend not only on the atoms and bonds present but also on their particular grouping, for example as CH3, CH2, or CH. Thus, popular and successful estimation schemes use p a r a m e t e r s which are proportional to the number of groups of atoms in the molecule, implicitly or explicitly using for all series a methylene increment derived from alkanes. The enthalpy changes resulting from the presence of heteroatoms requires appropriately derived bond energy terms. If there are any real differences in the

307 methylene increment from various series, they are subsumed in this term. Additional terms correct for steric interactions, for example. Recently, a series of articles [13] has described methods for accurately (< 4.184 k J mo1-1) reproducing enthalpies of formation from ab initio MO and statistical mechanical energies together with empirically-derived bond and group equivalents. Equation 4 shows the relationship between the enthalpy of formation and the electronic, thermal, and structural energy components devised by the authors. hH~ - H F E

+ TOR + POP + 4RT + ZBE + ZGE

(4)

HFE is the Hartree-Fock energy calculated at the HF/6-31G*//HF/6-31G* level of theory; TOR is the energy stored in low-lying vibrational/rotational states; POP is the energy associated with higher-energy conformations; 4RT is the sum of the translational and rotational freedom of the non-linear molecule (6/2 RT) and the PV equivalent for converting energy to enthalpy (1 RT); and BE and GE are the bond and group equivalents, respectively. The assigned value of TOR increases by 0.00067 hartree per carbon (1 hartree = 2625.46 k J moll), beginning with ethane or methanol (TOR = -0.00067). POP is typically evaluated by molecular mechanics calculations [12]. Unlike TOR, POP becomes linear with m at different values of m depending on the homologous series. For alkanes, the constant POP increment (0.00056 hartree per carbon) begins at nc = 4 (n- butane) and for alcohols at nr = 3 (n-propanol). Thus, both POP and TOR are proportional to larger n~ in both series. Calculated electronic energies are available for alkanes [13d, 14] and alcohols [13b] n~ = 1-6. A linear regression of HFE (hartree) vs. no produces for n-alkanes (no = 4-6) HFE (n-RH) + .000061 - [(-39.034695 + 0.000043) 9m] -(1.159595 + .000219) and for n-alkanols (m = 3-6) HFE (n-ROH) + .000061 = [(-39.034686 + .000027) 9m] -(76.00640 + .000127) In both cases, r 2 = 1.0 with variation in the correlation coefficient in the twelfth decimal place. As with the experimental enthalpies of formation, m e t h a n e and methanol deviate from linearity. Within the uncertainty limits of the regression, the electronic methylene increments are identical for the n-alkanes and nalkanols. If the analysis is performed for nc = 2-6 for both homologous series, the slopes are not statistically identical. The sum of the above energy components and the constant 4RT (0.00382 hartree) must linearly correlate with the number of carbon atoms in the compound for n-alkanes (m > 4) and for n-alkanols (no > 3) and again, the slopes must be identical:

308 ZE - ( H F E + TOR + POP + 4RT)= ((x', nr

13'

(5)

The linear regression results are, for alkanes n~ - 4-6 ZE (n-RH) + .000061 = [(-39.033465 + .000043) 9m ] - (1.159545 + .000219) and for alcohols nr = 3-6 ZE (n-ROH) + .000057 = [(-39.033459 + .000026) 9m] - (76.004902 + .000120) To the extent that errors in the statistical mechanical assumptions are negligible and these computed energies accurately reflect nature, it appears the experimentally different enthalpy-of-formation methylene increments for nalkanes and n-alkanols are not due to their electronic or thermal energy components. The bond and group energy contributions to equation 4 can only be calculated empirically, by subtracting ZE from AH~ In Ref. 13d, the hydrocarbon BE and GE parameters were determined from a least squares fit of a wide variety of structurally different alkanes. Because those terms are r e d u n d a n t for an homologous series, we define the n-alkyl series term (ST) according to equation 6 AH~

ZE + ST

(6)

Because both AH~ and ZE are linear with respect to m, so also is ST. The results from the weighted least squares analysis [15] are for alkanes (nc - 4-6) ST (n-RH) + .000232 - [(39.025552 + .000164) 9ne] + (1.1433195 + .000832) and for alcohols (m = 3-6) ST (n-ROH) + .000182 = [(39.025781 + .000081) 9no] + (75.930875 + .000377) We might have thought, because the experimental AH~ methylene increment slopes are different for the two series while the ZE methylene increment slopes are identical, t h a t the methylene increment slopes calculated from ST would be non-identical. The slopes from the linear regressions of ST vs. nc for the two series are just within each other's standard errors. This statistical sameness of all slopes [16] is an artifact because, as it must, combining the ZE and ST methylene increments for each homologous series reproduces the non-identical slopes from equation 3 for the compounds of relevant no. From the slope/intercept constants generated above and eq. 6, the calculated enthalpy of formation (hartree) for an alkane with n~ = 4-6 is

309 AH~ = [(-.007913 + .000170) ,, m] - (.016226 + .000860) and for an alcohol with n~ = 3-6 is AH~ - [(-.007678 + .000085) 9x] - (.074027 + .000396) The s t a n d a r d errors reported for the constants in the two equations above were calculated as the root mean square of the component s t a n d a r d errors and this propagation of uncertainty in the combined equations causes the slopes to barely overlap, whereas there is no overlap of the slopes calculated from eq. 3. We cannot hope to use these equations to calculate accurate enthalpies of formation for alkanes or alcohols with nc > 6 for the same reason we cannot use eq. 3 and the same restricted data range: the calculated results do not reproduce the experimental results due to the sensitivity of the calculation to the slope/intercept. We said earlier t h a t a and ~ from equation 3 are affected by the enthalpy-offormation data used to generate these constants. Historically, the values for various homologous series changed only slightly as newer and additional experimental data were amassed. But subjective, albeit experienced, judgment is exercised when authors choose the "best" enthalpy [11] or the "selected" values [3] to recommend in a compilation. Other than the enthalpy values themselves, the uncertainty intervals also affect the regression constants in the weighted least squares analyses. Calculations [7, 1 lb, 17] show the precision of measuring heat evolution in combustion calorimetry may be about +0.13 k J mo1-1 per C atom. Using the experimental enthalpies of formation [P] for those alcohols and alkanes discussed above, we can produce statistically identical slopes for these series by changing only slightly (both raising and lowering < 0.3 k J tool-1) four of the seven uncertainty intervals. The slopes still seem uncomfortably non-parallel but at least we recognize that we are not considering invariant values. We note t h a t the slope from the least squares analysis of the HFE's of straight-chain aldehydes [7cc] (m = 4-6) is not identical to that from either the n-alkanes or the n-alkanols, primarily due to the extremely small statistical s t a n d a r d error (~10 la) associated with the profound linear fit of the aldehydes. H F E (n-RCHO) - (-39.034740 9no)- 74.847980 We have come full circle. Until there is a method whereby empirical constants are eliminated from our calculations, or until enough experimental data are accumulated, we can only speculate about the convergence of methylene increments from different homologous series.

310 4. H O M O L O G O U S SERIES: CYCLOALKANES Consider now a different but formally homologous series composed solely of the repeating methylene groups: CH2, (CH2)2, (CH2)3, (CH2)4. . . . (CH2),. In the current context, we take species with n>2 to mean the unsubstituted cycloalkanes and not alkane-l,o)-diyls or -diradicals. It is quite clear that the series cannot commence with the triatomic molecule CH2 since neither singlet nor triplet methylene contains tetracoordinate carbon atoms. Whether or not the alternate beginning entry, CH2CH2, qualifies as a member is more contentious. While "cycloethane" is essentially never used as a synonym for ethylene [18] and certainly lacks tetracoordinate carbon, there is a certain, now largely historic appeal [19] for describing C2H4 and other olefins in terms of bent or banana bonds instead of using a and ~ orbitals [20]. Olefinic properties of the third entry, (CH2)3, are quite pronounced [21] and, encouraged by a/u descriptions of cyclopropane [19b, 22], suggest the possibility of a "continuum" of molecular properties with increasing n beginning with n > 2. We may consider the numerical difference between the gas phase enthalpies of formation of a given (CHg), and that of n strainless CH2 groups (the methylene increment derived from the n-alkanes) as the strain energy~ SE, of the cycloalkanes, (CH2),. However, unlike the previous series, it is well-recognized that there is no constant increment with increasing n [23]. Nonetheless, the energetics of cycloalkanes are still relatively well-understood and the qualitative features of the variation of their strain energies as a function of ring size is a component of the customary education of contemporary organic chemists. The strain energies of the cycloalkanes C9-C~7 are: 93.7, 115.1, 110.8, 26.6, 0, 26.1, 40.4, 52.6, 51.7, 47.2, 17.0, 21.4, 12.4, 7.6, 7.9, 14.1 kJ mol 1. The near, but fortuitous, equality of the strain energy of cyclopropane and cyclobutane (the archetypical carbocyclic three- and four-membered ring compounds, a.k.a. 3MR and 4MR) has entertained and educated numerous theoretical and physical organic chemists [24]. The occasional complication for at least moderate sized n has been rectified: we recall the earlier, and now corrected, discrepancy for cyclotetradecane that arose because of a faulty measurement of the enthalpy of sublimation of this n - 14 species [25]. We reiterate that had bond energies been strictly additive, strain energies would vanish because the total bond energy for the cycloalkane with a given value of n would precisely equal n(D(C-C)) + 2n(D(C-H)), and the enthalpies of formation of (CHg), would vary linearly with the parameter n. 5. H O M O L O G O U S SERIES:

SATURATED POLYCYCLIC HYDROCARBONS Saturated, polycyclic (alicyclic) hydrocarbons are composed of three tetracoordinate building blocks: quaternary >CCH- and secondary -CH2-. As we address the question of homologies for these species, we must accept that there are idiosyncrasies arising from the varying sizes of the component

311 rings. This is unavoidable because of the above enunciated idiosyncrasies for the related single rings species, the cycloalkanes. Bond energy additivity is unequivocally violated as the total bond energy does not depend solely on the number of hydrogens and the number of carbons. It is clear that there is a reasonably large range of enthalpies of formation, and hence of total bond energies, for any stoichiometry C,Hb we choose [26]. For example, consider the series cyclohexane {1}, trans-decalin {2}, and transsyn-trans-perhydroanthracene {3}: (CH2)6(CH)o, (CH2)8(CH)2, (CH2)lo(CH)4 with their increasing numbers of secondary and tertiary carbons and 6-membered rings (6MR). The members of this series differ by a constant [(CH)2(CH2)2] structural unit.

{1}

{2}

{3}

Since their respective enthalpies of formation (-123.4 • 0.8, -182.1 • 2.3 and 243.2 • 3.8 k J mol") linearly correlate (r 2 = 0.9999) with the number of such structural units, it follows that the reaction (CH2)6 + (CH2),o(CH)4 -+ 2 (CH2)8(CH)2

(7)

is thermoneutral within the reported error bars. Based on the regression constants (slope = -59.4 • 0.7 kJ mol-', intercept = -123.4 • 1.5) the enthlapy of formation for trans-syn-trans-syn-trans-perhydronaphthacene is predicted to be -302.7 • 1.0 k J mol". Another series of 6-membered ring species is cyclohexane, adamantane {4}, diamantane {5}: (CH)o(CH2)6, (CH)4(CH2)6, (CH)8(CH2)6 with their increasing numbers of tertiary carbons and 6-membered rings. Note that this series is not homologous in the sense we have used previously, in that the [(CH)4] increase between members of the series occurs in a disconnected manner.

{4}

{5}

Their enthalpies of formation (-123.4 • 0.8, -134.6 + 2.3, -145.9 + 2.7 k J mol") also show a linear dependence on the increasing number of >CH- groups (slope = -2.8 + 0.1, intercept = -123.4 • 0.1, r 2 = 0.99999). Equivalently, the reaction

312 (CH~)6 + (CH~)6(CH)8 -+ 2 (CH2)6(CH)4

(8)

is thermoneutral within the reported error bars. The above enthalpies of formation allow one to estimate the enthalpies of formation of both -CH2- and >CH- groups in these polycyclic hydrocarbons. Perhaps the simplest is to define the former enthalpy as 1/6 of the enthalpy of formation of cyclohexane. The result, -20.6 kJ mol-', is very similar to the nalkane methylene increment. Accepting this value and assuming additivity of increments in the first series, the enthalpy of formation for the >CH- group is found to be -9.3 k J mol". This value is much more negative (that is, stabilizing) than that derived for the second series, -2.8 kJ mol-'. In the latter series, each additional >CH- group results in formation of an axial substitutent bond to a cyclohexane ring with its concomitant gauche interactions; in the former series each additional >CH- group results in an equatorial substituent bond with no additional gauche interactions. The difference, 6.5 k J mol-', represents the gauche interactions and whatever additional strain accompanies the formation of the rigid adamantane and diamantane cages. In comparison, the 12.9 k J mol-' enthalpy difference between cis- and trans-decalin represents three additional gauche interactions between the rings. 6. T E T R A H E D R A N E AND [ I . I . I I P R O P E L L A N E Let us now consider tetrahedrane {6} and [1.1.1]propellane {7}. Both species have numerous 3MRs or cyclopropane rings. We thus expect both compounds to be highly strained. Do these hydrocarbons belong to any homologous series? Are they to be understood as avatars for the study of yet more wondrous strained species? Are they, as it were, alicycles resplendent in glorious isolation despite the considerable attention paid to both tetrahedrane [27] and [1.1.1]propellane [28] by computationally oriented theoretical chemists?

{6}

{7}

In what follows, we will estimate the enthalpies of formation of tetrahedrane and of [1.1.1]propellane by assuming, perhaps disingenuously, that these species are "normal" and they belong to various "homologous" series that contain one or the other of these two hydrocarbons. Then, using the enthalpies of formation for the earlier members of these series, we will "predict" the desired gas phase enthalpies of formation. Although we remember the often given admonition that it is easier to interpolate than to extrapolate, we will nonetheless look at several

313 different series to derive the desired numbers. Agreement between the methods would suggest normalcy for the tetrahedrane and the propellane while significant disagreement would suggest that these species are unique. 6.1 T e t r a h e d r a n e

Tetrahedrane is certainly to be found amongst any collection of consummately strained polycyclic hydrocarbons. While the "2-dimensional" cyclopropane is composed of three carbons and a single strained 3-membered ring, the "3dimensional" tetrahedrane is made of four carbons and a number of strained cyclopropanes as faces. As befits relative availability and seemingly stability as well, there is yet only one thermochemically characterized tetrahedrane, the tetra-t-butyl derivative [29], for which both condensed and gas phase enthalpies of formation are available [30]. The first approach to tetrahedrane starts with cyclopropane and bicyclobutane with their enthalpies of formation of 53.3 + 0.6 and 217.1 + 0.8 kJ mol 1. However, tetrahedrane, is not the next compound. Cyclopropane has three carbons, all of which are secondary, and together compose one 3-membered ring (3MR). Bicyclobutane has four carbons, two of which are secondary and two of which are tertiary, all together composing two 3-membered rings. What is the next compound if not tetrahedrane? Consider the individual counts of the number of secondary carbons, tertiary carbons, and 3MRs in cyclopropane and bicyclobutane. We have three and two secondary carbons; linearity says the next species would have but one. We have no and two tertiary carbons; the next species would have four. We have one and two 3MRs; the next species would have three. In contrast, tetrahedrane has four tertiary carbons and no secondary carbons that compose its three 3MRs. We identify the one secondary carbon in the hypothetical [(CH2)(CH)4] as the above "strainless methylene increment" and its enthalpy of formation as -20.6 kJ mo1-1 and the other four tertiary carbons compose tetrahedrane. If these compounds form an homologous series, then the enthalpy-of-formation difference between adjacent members should be equal, that is AHf[(CH2)2(CH)2]- AHf[(CH2)3] = {AHf[(CH)4] + (-20.6)} - AHf [(CH2)2(CH)2]

(9)

From this approach, the predicted value of the enthalpy of formation of gaseous tetrahedrane is ca. 402 kJ mo1-1. At the risk of ignoring the consequence of any vestige of homoaromaticity, as well as any other interactions among double bonds and/or 3MRs, the second putative homologous series proceeds from tetrahedrane through benzvalene {8}, semibullvalene {9}, and triquinacene {10}; from a count of no, one, two, and three cyclopentenes with their defining double bonds; and from four, six, eight, and ten carbons. In each of the four cases, there is a central tertiary >CH- affixed to three other tertiary >CH- groups. These latter >CH- groups are joined by no, one, two,

314 and three -CH=CH- finks and by three, two, one, and no direct >CH-CH< bonds, respectively. ..

{8}

{9}

{10}

The (non-combustion calorimetrically) experimentally determined enthalpies of formation for benzvalene [31], semibullvalene [32], and triquinacene [33] are 363, 308, and 224 k J mo1-1 respectively. Assuming linear dependence of enthalpy of formation on any of these "counts" (r 2 = 0.98), we derive a value for the enthalpy of formation of tetrahedrane of 437 kJ mol 1. The equivalent nonstatistical derivation is to assume thermoneutrality for the reaction (CH)8 + (CH)6 --> (CH)lo + (CH)4

(10)

i.e., semibullvalene + benzvalene ~ triquinacene + tetrahedrane, which corresponds to an enthalpy of formation of tetrahedrane of 447 kJ mo1-1. Yet another route to tetrahedrane proceeds through prismane {11} and cubane {12}. Homology is documented in carbon counts: tetrahedrane, prismane and cubane have four, six, and eight tertiary carbons. Recalling that the strain energies of 3- and 4- membered rings are nearly the same, homology is also implicitly asserted by the number of these "microrings". Tetrahedrane has three 3MRs, the tetracyclic prismane has three 4MRs and one 3MR (or alternatively, and quite enthalpically indistinguishable, two apiece 4MRs and 3MRs), and the pentacyclic cubane has five 4-membered rings (4MRs): the number of microrings increases linearly from three to four to five.

{ll}

{12}

The necessary enthalpy of formation of cubane [34] is 622 k J tool 1, and that of prismane [35] is taken as ca. 570 kJ mo1-1. Linear extrapolation gives the enthalpy of formation of tetrahedrane as ca. 518 kJ mol 1. Alternatively, in the absence of a measured enthalpy of formation data for prismane, we assert thermoneutrality for the formal reaction

315 (CH)8 + SE[(CH2),] -+ 2 (CH)4

(11)

(where SE is the strain energy when n is either 3 or 4) and thus derive the enthalpy of formation of tetrahedrane as ca. 370 k J mol 1. The final approach discussed here for estimating the enthalpy of formation of tetrahedrane is to assume the reaction 2 (CH)2(CH2)2 -~ (CH)4 + (CH2)4

(12)

is thermoneutral. The use of this approach was encouraged, in part, from the name bicyclobutane for the species on the left and tetrahedrane (tricyclobutane) and cyclobutane (monocyclobutane) for the two species on the right. In other words, bond energy additivity is supplemented by equality for the ring count, 2 + 2 = 3 + 1. From these assumptions, an enthalpy of formation for tetrahedrane of 2(217.1) - 28.4 = 406 k J mol 1 is deduced. Consider the values we have "derived" for tetrahedrane: 402, 437, 447, 520, 370 and 406 k J mo1-1. The mean value is 430 + 51 k J mol 1. The large standard deviation for the five values suggests that the thermochemistry of tetrahedrane is incompatible with simple modeling and linear extrapolation from homologous series. This conclusion is perhaps not a surprise, but it is nonetheless a disappointment. Our values are also much lower than the internally more consonant values of 561, 554 + 23 and 535 + 4 k J mol 1 derived from the analysis of high level quantum chemical calculations [36]. We thus suggest, admittedly a posteriori, the presence of superstrain [37], i.e. the presence of strain or additional destabilization beyond that of the component rings. The inclusion of the strain energy for a fourth 3MR, a ca. 115 k J tool 1 correction (the strain energy of cyclopropane), results in an amended value of 545 + 51 k J mo1-1. After all, it is the acute 60 ~ C-C-C angle, of which there are 12 (or four 3MRs) in tetradedrane, that is the defining feature of cyclopropane, tetrahedrane and other species with 3MRs. Nonetheless, we still seem to estimate on the low side for the enthalpy of formation for tetrahedrane. It would appear that tetrahedrane stands alone: this species may be considered a hydrocarbon hermit. 6.2 [1.1.1] P r o p e l l a n e

[1.1.1]Propellane is exquisitely strained with its three 3-membered rings as well as two carbons having the intriguing structural feature of an "inverted tetrahedron" [38]. However, unlike tetrahedrane, [1.1.1]propellane seemingly lacked the "need" of bulky (or even any) substituents [39] in order to be successfully isolated [40] and thermochemically characterized by condensed and gas phase enthalpies of formation [41]. We will attempt to derive its enthalpy of formation before comparing it with its experimental value. The first conceptual approach assumes that the reaction enthalpy of its formal synthesis from cyclopropane and 2,2,3,3-tetramethylbutane, equation 13,

316 is the same as that of the formal synthesis of bicyclobutane from cyclopropane and 2,3-dimethylbutane, equation 14. 3 (CH2)3 + (CH3)3C-C(CH3)3 ---) (C)2(CH2)3 + 3 CH3(CH2)2CH3

(13)

2 (CH2)3 + (CH3)2CH-CH(CH3)2 --> (CH)2(CH2)2 + 2 CHa(CH2)2CH3

(14)

Reaction 14 is endothermic by 37 kJ mol 1, and provides additional documentation for the earlier enunciated superstrain [37] for bicyclobutane. Accepting this value as a correction term for equation 13 gives our first estimate of the enthalpy of formation of gaseous [1.1.1]propeUane, namely 349 kJ mol". We may recognize reactions 13 and 14 as part of a homologous series in which the next and last reaction is (CH2)3 + CH3CH2-CH2CH3 --~ (CH2)2(CH2)~ + 1 CH3(CH2)2CH3

(15)

This "non-reaction" reaction is precisely thermoneutral. Linear extrapolation from equations 14 and 15 suggests that equation 13 should be endothermic by 2.37 kJ mol" and so the enthalpy of formation of [1.1.1]propellane would thus be expected to equal 386 kJ mol". The next conceptual approach for the enthalpy of formation of [1.1.1]propellane assumes that the enthalpy of the formal propellane synthesis from cyclopropane and neopentane, equation 16, is the same as that of the formal synthesis of bicyclobutane from cyclopropane and isobutane, equation 17. 3 (CH2)3 + 2 C(CH3)4 --> (C)2(CH2)3 + 2 CH3CH2CH3 + 2 CH3(CH2)2CH3

(16)

2 (CH2)3 + 2 CH(CH3)3 --> (CH)2(CH2)2+ 2 CH3CH2CH3 + 1 CH3(CH2)2CH3

(17)

Equation 17 is endothermic by 44 kJ mol", and is consonant with the earlier 37 kJ mo1-1 attributed as the superstrain in bicyclobutane. Accepting this value as a correction for equation 16 gives our next estimate for the enthalpy of formation of gaseous [1.1.1]propellane, 326 kJ mol 1. We recognize equations 16 and 17 as part of a homologous series involving two and one molecule of n-butane as an additional product. The remaining reaction involves zero molecules of n-butane as a product: (CH2)3 + 2 CH2(CH3)2 --> (CH2)2CH2+2 CH3CH2CH3 + 0 CHa(CH2)2CH3

(18)

This admittedly "nonreaction" reaction is, of course, precisely thermoneutral. Linear extrapolation suggests that reaction 16 should be endothermic by twice 44 kJ mo1-1 and so the enthalpy of formation of [1.1.1]propellane should be 370 kJ mol-i. Our last conceptual approach assumes the reaction enthalpy of the formal [1.1.1]propellane synthesis from 1,1-dimethylcycloprop ane and two

317 cyclopropanes, equation 19, is the same as that of the formal synthesis of bicyclobutane from methylcyclopropane and cyclopropane, equation 20 2 (CH2)2C(CH3)2 + 1 (CH2)3 ~ (C)2(CH2)3 + 2 CH3(CH2)2CH~

(19)

2 (CH2)2CHCH3 + 0 (CH2)3 -~ (CH)2(CH2)2 + 1 CH3(CH2)2CH3

(20)

Equation 20 is endothermic by 43 k J mo1-1, and by being nearly identical to the earlier 37 and 44 kJ mo1-1 for other reactions involving bicyclobutane, provides additional documentation of the superstrain of bicyclobutane. Accepting this value for the second reaction gives our penultimate estimate of the enthalpy of formation of gaseous [1.1.1]propellane of 331 kJ mol -~. We may recognize equations 19 and 20 as part of a homologous series: the remaining reaction is 2 (CH2)2CH2 +-1.(CH2)3 ~

1 (CH2)2(CH2)I + 0 CHa(CH2)2CH3

(21)

This "nonreaction" reaction is precisely thermoneutral. Linear extrapolation suggests that reaction 19 should be endothermic by twice 44 k J mo1-1 and so the enthalpy of formation of [1.1.1]propellane should be 375 k J mol 1. The predicted values for the enthalpy of formation of gaseous [1.1.1]propellane are 349, 386, 326, 370, 331 and 375 kJ mo1-1 , resulting in an average value of 356 • 25 k J mo1-1. These values conceptually "divide" into two categories: 349, 326, 331; and 386, 370, 375, depending on whether the superstrain of bicyclobutane is included once or twice. The measured value of the enthalpy of formation of [1.1.1]propellane [41] is 355 kJ mol 1, suggesting that the superstrain be included some 1.5 or 3/2 times. Maybe this last noninteger multiple is sensible: bicyclobutane has two -CH2- groups affixed to the central CC spoke, while [1.1.1]propellane has three such -CH2- groups. However, given the additionally destabilizing feature of two so strongly inverted tetrahedral carbons in [1.1.1]propellane, we would have thought this species be considerably more strained than it is. Our success suggests [1.1.1]propellane is quite sensible, if not also normal, and is a hydrocarbon harbinger for further understanding of strained species. Yet [1.1.1]propellane has fooled us before [42] and so we are chastened, cautious, and remain curious about this multi-ring hydrocarbon.

REFERENCES 1. 2.

3.

M. Pomerantz and J. F. Liebman, Tetrahedron Lett. (1975) 2385. H. Basch and T. Hoz, in The Chemistry of Organic Germanium, Tin and Lead Compounds (ed. S. Patai), Wiley, Chichester, 1995, and references cited therein. Unless otherwise cited, all enthalpies of formation are from the compendium J. B. Pedley, R. D. Naylor and S. P. Kirby, Thermochemical Data of Organic Compounds (2nd Ed.) Chapman & Hall, New York, 1986.

318

.

.

.

~

9.

10. 11.

12.

13.

14.

S. W. Slayden and J. F. Liebman, in Supplement A3: The chemistry of the double-bonded functional groups, (Ed. S. Patai), Wiley, Chichester, 1997. a) R. L. Montgomery and F. D. Rossini, J. Chem. Thermodynamics, 10 (1978) 471; b)J. F. Liebman, J. A. Martinho SimSes, and S. W. Slayden, Structural Chemistry, 6 (1995) 65. E. J. Prosen, W. H. Johnson and F. D. Rossini, J. Res. Natl. Bur. Stand., 37 (1946) 51. J. D. Cox and G. Pilcher, Thermochemistry of Organic and Organometallic Compounds, Academic Press, London and New York, 1970. P. Sellers, G. Stridh, and S. Sunner, J. Chem. Eng. Data, 23 (1978) 250. a) J. F. Liebman and S. W. Slayden, in Molecular Structure Research (Eds. M. Hargittai and I. Hargittai), JAI Press, Greenwich, CT, in press; b) S. W. Slayden, J. F. Liebman, and W. G. Mallard, in The chemistry of functional groups, Supplement D: The chemistry of halides, pseudo-halides and azides Vol. 2, (Eds. S. Patai and Z. Rappoport), Wiley, Chichester, 1995; c) J. F. Liebman, K. S. K. Crawford, and S. W. Slayden, in The chemistry of functional groups, Supplement S: The chemistry of sulphur-containing functional groups, (Eds. S. Patai and Z. Rappoport), Wiley, Chichester, 1993; d) S. W. Slayden and J. F. Liebman, in The chemistry of functional groups, Supplement E: The chemistry of hydroxyl, ether and peroxide groups Vol. 2, (Ed. S. Patai), Wiley, Chichester, 1993. See, for example J. F. Liebman, M. S. Campbell, and S. W. Slayden, in Supplement F2: The chemistry of amino, nitroso, nitro and related compounds, (Ed. S. Patai), Wiley, Chichester, 1996. a) Cox and Pilcher (ref. 7) reviewed the various modern bond energy schemes and assess their strengths, weaknesses, and equivalencies. Some of the schemes have been updated since then. b) See, for example, N. Cohen and S. W. Benson in The Chemistry of alkanes and cycloalkanes, (Eds. S. Patai and Z. Rappoport), Wiley, Chichester, 1992. c) An extension of the group method to include substructures is detailed in ref. 3. Enthalpies of formation may be calculated with parameters derived from molecular mechanics force fields. See U. Burkert and N. L. Allinger, Molecular Mechanics, American Chemical Society, Washington, D. C., 1982; N. L. AUinger, Y. H. Yuh, and J-H. Lii, J. Amer. Chem. Soc., 111 (1989) 8551; N. L. Allinger, X-F. Zhou, and J. Bargsma, Theochem-J. Molec. Structure, 118 (1994) 69. a) L. R. Schmitz and Y. R. Chen, J. Comput. Chem., 15 (1994) 1437 and references therein; b) N. L. AUinger, L. R. Schmitz, I. Motoc, C. Bender, and J. K. Labanowski, J. Am. Chem. Soc., 114 (1992) 2880; c) L. R. Schmitz, I. Motoc, C. Bender, J. K. Labanowski, and N. L. Allinger, J. Phys. Org. Chem., 5 (1992) 225; d) N. L. Allinger, L. R. Schmitz, I. Motoc, C. Bender, and J. K. Labanowski, J. Phys. Org. Chem., 3 (1990) 732; e) K. B. Wiberg, J. Org. Chem., 50 (1985) 5285. W. C. Herndon, Chem. Phys. Letts, 234 (1995) 82.

319 15. In the least squares analyses, the individual ST energies were weighted inversely as the squares of the experimental enthalpy-of-formation uncertainty intervals only. (The uncertainty intervals for the combined ZE are identical for each data point.) The standard errors in the regression equations were generated from the unweighted enthalpies of formation. 16. For comparison, 39.025936 hartree is the bond energy methylene increment found in Ref. 13d. It was derived from the alkanes and then transferred to functionalized alkanes. 17. D. R. Stull, E. F. Westrum, and G. C. Sinke, The Chemical Thermodynamics of Organic Compounds, Krieger, FLA., 1987. 18. See, for example, A. Greenberg and J. F. Liebman, Strained Organic Molecules, Academic Press, New York, 1978, pp. 43-44. 19. a) L. Pauling, J. Am. Chem. Soc., 53 (1931) 1347; b) C. A. Coulson and E. T. Stewart. in The Chemistry of the Alkenes (ed. S. Patai), Wiley, New York, 1964. 20. See, for example, R. P. Messmer and P. A. Schultz, Phys. Rev. Lett., 57 (1986), 2653. 21. See, for example, J. B. Conant, pp. 1-42, and D. Cremer, R. F. Childs and E. Kraka, pp. 339-410, in The Chemistry of the Cyclopropyl Group Vol. 2, (ed. Z. Rappoport), Wiley, Chichester, 1995. 22. D. Cremer, E. Kraka and K. J. Szabo, in The Chemistry of the Cyclopropyl Group Vol. 2, (ed. Z. Rappoport), Wiley, Chichester, 1995. 23. E. L. Eliel and J. J. Engelsman, J. Chem. Educ., 73 (1996), 1203. 24. See, for example, K. Wiberg, in The Chemistry of the Cyclopropyl Group Vol. 1 Part 1, (ed. Z. Rappoport), Wiley, Chichester, 1995 and J. F. Liebman, in The Chemistry of the Cyclopropyl Group Vol. 2, (ed. Z. Rappoport), Wiley, Chichester, 1995. 25. J. S. Chickos, D. G. Hesse, S. Y. Panshin, D. W. Rogers, M. Saunders, P. M. Uffer and J. F. Liebman, J. Org. Chem., 57 (1992), 1897. Also see ref. 23 for a salutary note. 26. Godleski, P. v. R. Schleyer, E. Osawa, and T. Wipke, Progr. Phys. Org., Chem., 13 (1983), 63. 27. M. N. Gloukhovtsev, S. Laiter and A. Pross, J. Phys. Chem., 99 (1995) 6828 and numerous references cited therein. 28. See T. Kar and K. Jug, Chem. Phys. Lett., 256 (1996) 201 and numerous references cited therein. 29. G. Maier, Angew. Chem. Intl. Ed., 27 (1988) 309. 30. Unpublished measurements of the enthalpy of combustion by M. Mhnsson, and of the enthalpy of sublimation by C. Riichardt, H.-D. Beckhaus and B. Dogan, cited in ref. 29. 31. N. J. Turro, C. A. Renner, T. J. Katz, K. B. Wiberg and H. A. Connon, Tetrahedron Lett., 46 (1976) 4133. 32. K. Hassenriick, H.-D. Martin and R. Walsh, Chem. Rev., 89 (1989) 1125. 33. J. F. Liebman, L. A. Paquette, J. L. Peterson and D. W. Rogers, J. Am. Chem. Soc., 108 (1986) 8267.

320 34. We accepted the value recorded in ref. 3 as measured by B. D. Kybett, S. Carroll, P. Natalis, D. W. Bonnell, J. L. Margrave and J. L. Franklin, J. Am. Chem. Soc., 88 (1966) 626 although we acknowledge the alternative, group increment-assisted experimental value of D. R. Kirklin, K. L. Churney and E. S. Domalski, J. Chem. Thermodyn., 21 (1989) 1105 that suggests the former value is some 40 kJ mo1-1 too low. 35. D. W. Rogers, F. J. McLafferty, W. Fang and Y. Qi, Struct. Chem., 4 (1993) 161. 36. Quantum chemical calculations: a) with derived group equivalents: K. B. Wiberg, J. Comput. Chem., 5 (1984) 197; b) with BAC-MP4 corrections: Carl F. Melius, personal communication to the authors; c) at the generally chemically accurate G2 level: Gloukhovtsev, et al., op. cit., ref. 27. 37. J. F. Liebman and A. Greenberg, Chem. Rev. 76 (1976) 311. 38. K. B. Wiberg, G. J. Burgmaier, K. W. Shen, S. J. LaPlaca, W. C. Hamilton and M. D. Newton, J. Am. Chem. Soc., 94 (1972) 7402. 39. P. Kaszynski and J. Michl, in The Chemistry of the Cyclopropyl Group Vol. 2, (ed. Z. Rappoport), Wiley, Chichester, 1995. 40. K. B. Wiberg and F. H. Walker, J. Am. Chem. Soc., 104 (1982) 5239; K. Semmler, G. Sziemies and J. Belzner, J. Am. Chem. Soc., 107 (1985) 6410. 41. K. B. Wiberg, W. P. Dailey, F. H. Walker, S. T. WaddeU, L. S.Crocker and M. Newton, J. Am. Chem. Soc., 107 (1985) 7247. 42. Liebman and Greenberg, op. cir., pp. 344- 7.


321

Pauling's Legacy: Modern Modelling of the Chemical Bond Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

Stabilization and Destabilization Energies of Distorted Amides Arthur Greenberg' and David T. Moore ~b Y}epartment of Chemistry, University of North Carolina at Charlotte, Charlotte, NC 28223, USA bPresent address: Department of Chemistry, University of North Carolina at Chapel ~ North Carolina 27599-6293, USA

Chapel

1. INTRODUCHON 1.1. Chemical Implications of Strained Amides and Lactams

The amide functional group plays a fundamental role in organic chemistry, biochemistry, medicinal and polymer chemistry [ 1]. Despite the great wealth of information, only recently has there grown a body of_d.~t_aconcerning the structures and energies, and their consequences for i,eaofivity and biological activity, of distorted amide linkages [2]. That such distortion introduces unug~ reactivity has long been recognized [2], for example, in the enhanced reactivity of beta-lactams (2azetidinones, 1) [3] and the great difficulty in isolating all but a few dpha-lactams (aziridinones, 2) [2]. In 1938, Imkes [5] recognized the relevance of Bredt's rule and predicted that small bridgehead bicyclic lactams ("BBLs") slg~d be very reactive. This is consistent with the subsequent lack of success in isolating (unsubstituted) 2-quinuclidone (1-azabicyclo[2.2.2]octan-2-one, 3) [6]. O

N17 1

~ 2

. 3

Dtaing the 1960s Hall attempted to apply this reactivity using highly polymerizable bridgehead lactams [7]. In addition to 1-3, other compounds with distorted amide linkages include stericallyhindered species [8], trans-caprylolactam [9], cyclic dipeptides [10], and twistane lactams [11]. The Bredt's rule analogy is very robust and derives from the structural similarities between the planar olefm linkage and isoelectro.nic, (ideally) planar amide linkage [2,7,12,13]. The high condensedphase rotational barrier (A G*) about the C-N bond (ca 20 kcal/mol; see later discussion pertaining

322 to AH*) [2] in amides is usually attributed to its partial double bond (4A4C), a l t l ~ interpretation has been challenged and modified [ 14,171

O9 -Nc; /

:O~/C-N(

\

4A

411

4C

l.Z. n i o ~ e ~ ImpUcaaous The first determination of an experimental heat ofeondmsfion of a strained amide or laetmn, that ofmethylpenidllin (5), was reported by KB. Woodward and NBS collaborators in 1949 [18]. It indicated a strain energy in the 4-membered ring some 6 keal/mol h i / ~ than in the model monoeyelic beta-laetmns (6). This added strain is, in part, resporm%le for the biological activity of the penicillins. Over twenty years later, Sweet and Dahl used X-ray data to correlate increased laetam distortion in beta-laetam anffoioties with enhanced biological activities [19].

r ;Ozlq 5

6

In the realm of protein chemistry, Ramadumdran [20] emphasized the need to abandon the assumption of absolute planarity in peptiOe linkages. It is recognized that binding with an enzyme suains and activates substrates [21 ]. For peptidases and proteases this involves distortion of the scissile peptide (amide) linkage and specific distortions may lead to specific stereochemical consequences [22]. Another i m ~ biolosic~ topic ~ to l~fide distoaion is the r ~ disc,ovary of peptidyl-prolyl cis-trans i s o ~ ("rotamase"), an enzyme which catalyzes isomedzafion of c e r t ~ cis- and trans-peptides [23-27]. It plays a crucial role in autoimmunosuppression. Experimental evidence suggests that the transition state has a highly distorted peptide linkage [23,24]. The class of proteins tenned "immunophilins" appears to play a role in protein folding at the peptidyl-proline linkage and this is considered to be the rato-detcrmining step [25-27]. Molecular modelling studies of the structure of the active site of rotamase or for design of rotamase inhibitors must inck~e amide distortion.

323 1.3. Effects of Distortion on AcM/Base Properties

Severe distortion of an amide linkage leads to important chemical consequences. Hydrolysis of the 2-quinuclidone 7 was formalto be 10s faster than in model unstrained amides [28]. In contrast to the exceedingly weak basicifies ofunstrain~ amides and lactams, 2-quinuclidones such as 7-10 [28-33] behave as aminoketones and are only slightly weaker than simple amides or ~ [2931]. Consistent with this behavior is the observation that 2-quinuclidones such as 10 are protonated on nitrogen (unstrained amides and lactams are O-protonated) and also alkylate on nitrogen to produce 10-CH3+ for example [32,33]. This raises the very interesting question: what are the degrees of distortion that mark the boundary between N- and O-protonation of amides? This fundamental question of bonding and energetics may have biological in~lications. The pH profiles for hydrolysis of 2-quinuclidones have been employed in studies which model proteolysis [34]. There are no published experinaental gas-phase proton at~ties for distorted amides and laetams.

7

8

9

10 10-CH3+ 11 Our fully-op"mnized 6-31 G* ab initio molecular orbital calculations support a strong preference for N-protonation of 3 while indicating that N- and O-protonation are competitive in 1-azabicyclo[3.3.1]nonan-2-one (11) [35-37]. More r~ent results on other systems will be described. 1.4. Defming Distortion of the Amide Linkage

Dunitz and grmlder [38] defined three independent distortion parameters for the amide linkage: ~ , p y r a m i ~ o n at nitrogen; ~c, the p y r a m i ~ o n at carbon; and x, torsion about the C-N bond. R e c o ~ that ~c is usually quite small they suggested that a plot of distortion energy versus ~ and x (Figure 1) would be valuable for conformational studies [38]. Brown employed a simpler two-parameter distortion model ignoring ~c [39]. Summaries of experimental and calculational distortion parameters in distorted lactams have been published [2,13,35,39,41,42]. There are also two interesting papers which employ the Cambridge Crystallographic Database to examine the interrelationships between amide distortion parameters [41] as well as between parma~ers in all amides [43].

324

A

/

t

9O ~0 6O

0

I0 20 30 40

50 60

~N

Figure 1. Schematic representation of plot of distortion energy of amides as a fin~on OfgN and x. 1.5. Larger Bridgehead Bicydir Lactams: Are They Hyperstable? The discussion to this point has f ~ on smaller species in which nitrogen is constrained to be highly pyramidal and the N-CO bond is highly twisted. Larger bicyclic ~ exemplified by the 3.3.3 (12) (BOLDFACE DENOTES THE BRIDGE CONTAINING THE CARBONYL GROUP) and 4.3.3 molecules (13 and 14), are as yet unknowrL We have somewhat atbiumily termed systems of 3.3.3 and higher (no bridges smaller than 3) "larger" because they allow nitrogen to be planar or very nearly so [44,45]. Indeed, the entire amide linkage in these systems can approach planarity [36]. Since the model amine has already "paid the price" for planarity (i.e. including the ca 6 kcal/mol N inversion barrier), the resonance energy (RE) eal~dmed according to equation 1 may be higher than for normal aeyclic amides or unstrained lactams. These larger bridgehead bicyclic lactams may thus be "hyperstable". This idea is directly related to the large b r i d g ~ bicyclic olefms which have been termed "hyperstable" since they have reduced strain

(1) RE(13) = AH~4.3.3 amine) + AH~4.3.3 ketone) - AI~4.3.3 alkane) - AI~4.3.3 lactam) relative to the corresponding alkane and, thus, manifest reduced enthalpies of hydrogenation [46,47]. For example, the 4.4.3 b f i d g ~ alkene (15) analogous to 16 is calculated to have a (negative) olefin strain (OS) of-16.8 kcal/mol which can be taken as a measure of its hyperstability

[47].

12

13

14

15

16

325 2. BACKGROUND 2.1. Energetics of Distorted Lactams

The goal of generating a plot of distortion energy versus geometric distortion parameters for amides based upon experimental and/or high-level calculational dat__a(e.g. Figure 1) has yet to be achieved. X-ray structures for bridgehead lactams 17-24 [ 13,40,48-50] and for some distorted acyclic amides [41] have been published. While there are numerous published X-ray structures for beta-lactams, 1,3-diadamantylderivative 25 [51] is the only such alpha-lactam. Ph

R~--~O 17

18

R

O~ 19

0

0

20

21

O

-Q 22

23

24

25

Energy data (heats of combustion, formation, hydrolysis) do not exist for any strained amidr or lactam except for combustion data for 5 and 6, as noted earlier, and heats of hydrolysis for "highenergy" amides such as 1-acetylimidazole[52] as well as amides and dipeptides susceptibleto enzyme-catalyzed hydrolysis. This is explained by the extraordinary difficulty in obtaining wxa~ate (i.e. 99.98%) combustion calorimetry data and the generally slow hydrolyses of unstrained amide linkages except for those amenable to catalysisby carboxypeptidase [53]. In contrast, highly reactive lactams such as 10 should be amenable to he~ of hydrolysis studies. Since heats of hydrolysis are usually two orders of magnitude smaller than heats of combustion, the ~ a c y required to obtain useful data is about 98-99%. Wiberg has employed this approach to great advantage in assessing the energetics of various carbonyl compounds [54,55]. Although unstrained amides and lactams hydrolyze too slowly for useful thermochemical measurements, ~ of hydrolysis have been indirectly determined by Wadso for such compounds, e.g. n-butylacetamide, by measuring the enthalpy of aminolysis, using n-butylamine, o f ~ c anhydride [56]. Direct heats of hydrolysis were, however, determined for more activated amides such as diacetylamideunder conditions allowing for direct comparison with unstrained, simple amides [57].

326 2.2. Bonding in Lactams: Is There Still A Role for Resonance? Resonance theory [ 1] has been s u ~ in explaining a wide variety of physical, chemical and spectroscopic properties of amides and laotams [2,41,58]. The plmuuity of the amide linkage, its high rotational barrier, low basicity, tendency for O-protonation, the short C-N bond length, stability toward hydrolysis, low IRc,arbonyl fiequency, ~SNand S3Ccla~cal shifts [58], and UV absorption properties have all been traditionally explained using resonance theory. A different model has been advanc~ by researchers [ 14-17,59] who proposed that the C-N s/gma bond in the planar structure is more polar covalent (stronger and shorter) than the C-N bond in the transition state and is, thus, the origin of the stabilization and high rotational barrier characteristic of amides. researchers correctly predicted that there is little variation in C=O bond length as a fia~on of C-N twisting apparently at odds with the resonance concept. X-ray studies confirm this prediction but the changes, s n ~ as they are, follow the trend predicted by resonance theory [35]. Penin [60] has criticized the atoms-in-molecules approach of these researcla~ particularly for highly electronegative atoms. Our ab m/t/o molecular orbital calculations (6-31G'//6-31G') predict Casymmetryfor 2quinuclidone (3) [35,36]. Furthermore, the c a l ~ e d structures [35,36] for 2-quinuclidone (3) and 1-azabicydo[3.3.1]normn-2-one (11) are consistent with the classical resonance model. The N-CO bond in 3, which lacks amide resonmr,e, is calculated to be the longest such bond reported while the C=O bond is short. In contrast, for the less strained 11, the N-CO bond is slightly longer than for unstrained amides and the C=O bond is somewhat longer than in 3. The variation of C=O bond length between lactams differing significantly in distortion is ratlmr small [14]. These details will be described later in this ~ . In collaboration with T. Darrah Thonm, Oregon State University, we employed gas-phase electron spectroscopy for chemical analysis (ESCA) to assess the nature of charge on N and O in distorted lactams [61a]. An earlier study of N-amnmnioimidates (solid state) found an excellent correlation between N~. shifts and the catbonyl fiequencies as prediO~l by resormn~ theory [62,63]. For l-azabicyclo[3.3, l]nonaw2-one (II), a low value for the ionization energy of Nl, compared to the planar 1-n-butylpyrrolidone (26) supports reduced positive charge on the nitrogen CH~TI2CH2CH3

l

O 1

26 27 in 11 consistent with reduced resonance in this twisted lactam While the differences are ~ they are outside the experimental error (0.05 eV). Similarly, the Oz, ionization energy in 11 is higher than in 26 due to reduced resonance in the former. The N~. core energy in 1,3-di-tert-butylaziridinone (27) is lower and the O1. core energy in 27 is higher than the r162 nux/el values in 1,3-di-tertImtylazifidinone [61]. 1,3-Di-tert-buWlazifidinone(27) is calculated to have a pyramidal nitrogen (as does the diadamantyi derivative [51 ]) and this is consisteat with its UV photoelectron spectrum

327 [64,65]. Our attempts at obtaining gas-phase ESCA data for the two 2-quinuelidone derivatives 7 and 10 were unsuccessful possibly due to thermal decomposition upon heating the sample to improve volatility [61]. {It should be noted that 6, 6, 7,7-tetramethyl-2-quinuelidone (10) is fairly unreactive with nucleophiles due to aerie hindrance [66]}. More recently, we [61b] have obtained results supporting these studies by examining the core potentials of the planar ground states and rotational transition states of formamide and dimethylaeetamide. The simplest explanation of the data is loss of the contribution of 4C with concommitant increased contributions from 4A and 411 [61b]. 2.3. Proton Aff'mities of Bridgehead Bicydic Lactams: N vs O

Unstrained amides and lactams protonate and alkylate on oxygen. This is often explained through reference to resonance structures like 4C alttrmgh the real point is the greatly enhanced resonance in O-protonated (alkylated) amides relative to the neutral amide [71]. In contrast, 2-quinuclidones such as 10 protonate and alkylate on nitrogen since they are really ketoarnines [32,33]. The ab initio calculafional results summarized in Table 1 predict that the 2.2.2 laetam (a trw~cyclohexene analogue), and the 3.3.2, 3.2.2 and 3.2.2 systems (all trans-cycloheptene analogues) protonate exclusively at nitrogen. One can see in this table the decrease in proton a t ~ t y at nitrogen as this atom approaches planarity and the increase in proton aifmity at oxygen as the N-CO twist decreases allowing enhanced pi overlap across this linkage. The 3.3.1 system, a trans-cyclooctene analogue, favors protonation at N by only 1.9 kcal/mol. However, the 3.3.2 system, also a transcyclooctene connected at the 1- and 5-positiom by a two-carbon bridge rather than a one-carbon bridge favors protonation at O. The two-carbon bridge "cuts a bit more slack" to the transcyclooctene-like ring. Interestingly, this prediction reeeives some support from experimental observations by Werstiuk et al [37] of the dichotomy outlined in Scheme 1, although it is not yet clear whether the very slow reaction is under thermodynamic or kinetic control [37]. Scheme 1 CH3OTf days

r

22

CH3OTf days CH3 24

328 Table 1. Calculated 6-31G'//6-31(3" proton ag~nities 0ccal/mol) for N and O on bridgehead bicyclic employing corrected zero-point vibrational energies and thenml factors (298 K) along with the difference (in kcal/mol) favoring N-protonatiotL (See Reference 36 for further details.) Lactam N-Prot0n Affmity ,, O-Proton Aff'miW Difference Favgrine N 2.2.2 (3) 228.9 206.2 +22.8 3.2.2 (30) 224.7 213.6 +11.1 3.2.2 (31) 223.6 214.4 +9.1 3.3.2 (32) 224.7 215.1 +9.6 3.3.1 (11) 219.0 217.6 + 1.4 3.3.2 (33) 214.1 221.2 -7.1 3.3.3 {12) 218.1 221.6 -3.5 2.4. State of Caiculational Studies of Distorted Amide IJnkages With the exception of a significant body of cakml~onal _a~_a_on beta-laOams [67], until recently almost no c~doAational work had been done on distorted laOams [2]. Moletmlar mechanics (MM) studies which treat distorted amides are based upon spectroscopic pmmnete~ which are only suitable for small distortions. MM studies of significantly distorted lactams are, thus, restricted due to the small body of suucawal d~_~referred to earlier and the complete absence ofthennochemical data for suitable bemhmadr compounds. Nevertheless, molecular mechanics calculations have been applied to selected bridgehead lactanx% partiodafly in the [rL3.1] series [13,36,40,48], with reasonable results for these relatively mildly sUain~ m o l ~ . Recently, a study appeared in which ab initio d~t_awere employed to parameterize molecular modumics for amide distortion [68]. More remains to be done in this are& ~ remains, ofccmrse, the question as to what level of theory is required. MNIN) calodafions on 1,3-di-tert-lmtylaziridinone (27) and 1-az~icyclo[3.3.1]nonan-2one 0 D [64,65], the series IS-20 [13,40] and 22 and 24 [37] imiicXe N-CO bond lengths that are naw,h too long [13,35]. We have employed ab initio MO calctflations to study amides. It is known that small basis sets are not appropriate since they do not properly treat N-inversion barriem [69]. We find that the 6-31G* basis set provides very good a g r ~ with experimental smmmral and energetics data, and have applied it sucxessfidly for full optimization of the b f i d g ~ laozms [35,36] and their protonated forms as well as to reprcxka~ the stmOamd properties [70,71] of a/pha-lactams [51,70-73]. Of course, although synthetic studies cited in this chapter and related synttmfic work on bridgehead laoams have been published [74-79], muc&work has yet to be done including a p p r ~ to the larger " h ~ l e " systems.

3. COMPUTATIONAL STUDIF~ 3.1. Molecular Mechanics: Our MM2 computations were perfommd on a Silicon Graphics Indigo2 xZ platform using the confonmfional analysis trackage from version 3.1 of the SPARTAN c o r o n a l chemistry

329 software [80a]. The molecular mechanics computatiom were undertaken with two pfitmry goals in mind. First, we wanted to carry out a low level conformer search to find the global minimmn, or lowest energy conformer ~ ) , before proceeding on to higher levels ofcalculafiom. Second, we wanted to evaluate the performance of MM2 when applied to bicyclic bridgehead laoams. The smaller members of our series have quite unusual geometries and we were curious about how well MM2 was paramete6z~ for such structures. We also started with three larger BBLs (16,28,29) in our series that were later discarded to conserve computational time.

The second question was answered ahnost immediately. When a starting structure close to the experimental g e o ~ for the 2.2.2 (3) BBL was submitted to the MM2 program a message was returned stating that certain ~ e r s were missing and the program terminated. Upon investigation, it was discovered that SPARTAN's version of MM2 had no parameters for a pyramidal nitrogen atom attached to a planar (sp2) carbon atom. The most obvious and certainly the most rigorous way to solve this problem was to derive new parameters from literature _dat_aor high-level ab initio computations. Unfortunately, neither alternative was viable since literature _d~t__aare scarce and the high-level ab initio calculations would be dependent on MM2 or semi-en~irical results for reasonable starting structures. In lieu of rigorous treatment, two 'quick-fix' methods were employed to get some results. The first involved systenmfic parameter duplication. Each missing l:aUmneterfor sp3nitrogen/sp2 carbon was supplied by copying the corresponding parameter for sp2 nitrogel~/Sp 2 carbon. The intended effect was to force MM2 to treat distorted amide bonds as if they were undistorted. While this was crude to the extreme, at least the computations would proceed. The second approach was to use starting structures with a planar amide linkage. This was done by starting with a planar linkage and building the rest of the bicyclic cage around it, checking only that connectivities were correct. For the smaller laOams, this often resulted in comically distorted starting ~ e s , but the MM2 program acc~ted and optimized them. Structures were submitted using either or both of the above methods, and no significant variations were detected. For both large and small ~ the energies were within ~ . 0 0 2 kcal/mol, bond lengths were within iO.001 A, and bond angles and torsional angles were within :~0.05~ ~ errors are certainly within ~ resolution, and are atmlmted to rounding or precisionerrorswithin the computation. All further moleodar mechanics calculations were carried out using both of these crude "patches." Now that structures could be op"mniz~ some questions arose concerning the conformer searches. In version 3.1 of the SPARTAN soi~ware package, the conformer searching algorithm employed the method of Osawa for rings. While it is quite clear from theory how the search should proceed when applied to a mono~clic system, the implementation for bicyclic systems in unclear.

330 Once a starting bond has been selected, how does the algorithm dmose which atoms to l m ~ e ~ Simple tests on bi~do[4.4.3]tridec~me showed that the conformers found depended on which bridge contained the selected bond, and furthermore, when more than one bridge was selected, still other conformers arose for each bicyclic species in our study: one set from each single bridge search, one set from each c~nbination of two bridges, and one set from seardfing all three bridges at the same time. Further tests were carded out in an attanpt to disom~ w h e t ~ there was ~ ~ ~y ~ ~ which bridge would give the lowest-em~ confomzr every time. The results were i n c o m e . T I ~ seven sets of cmftmne~ were producaxi for ~ h m a i m S . We had ~ to use the "'Cadxmyi Substitmion N ' ~ Atom Replacemm~ (COSNAR) [36] (see ~ 2) isodesmic nmdel, so lowest energy confmmets were n e e t ~ not only for tl~ sedes oflactams, but for eacJa nmdd k~one, andne and alkane as well. In all,40 stmlin8 ~ were used, and 7 setsof conformers were generated from each starting struc~a'e. Fortunately, the computational time required for ew,h search was very small: the longest calculation took only one half-hour and most were considerably ~ . Still, well over 2000 total ccmfcmners were found and their energies and sUre,ares catalosed. The positive ~ for all ofthis bookkeeping was that we cotdd be fairly that we had ~ the lowest energy ccmformer in every case. At the very least, we were c r a m that we ~ not reasonably have been any more ~ or rigorous in our approac~ Table 2 lists these lowest energy conformation for each C O S N A R ~ ~ e used. Resonance energies were computed using the COSNAR m,xld ( ~ 2). Rec~ that the A Hf~ of a molecule may be comtmted from the sum of the Benson [81 ] increments and the strain energy. Thus the COSNAR resomnce energy may be viewed as having two parts (Schen~ 2): one arising ~om the difference in Benson increments and the other arising ~om the difference in strain energies.

~ 2

RE=Z* GI + Z * strain Z * indicates that the given quamifies for each compound are summed with our isodesmic sign convention, for exangfle: COSNAR APPROACH: E * GI - Z GI(amide) - Z GI(amine) - Z GI(ketone) + Y.GI (alkane) = 17.33 kc~mol (NOTE: This value is 18.2 kc~l ~ ~ ditfe~mt c o l o n s [36]) Because the COSNAR apprmeh ~ e s only those atoms in the amide linkage, it is easy to see that the tkmon resonance t a m is constant for an bicycac ~ at 17.33 kcal/mol. The loweste~ergy conformation strain ene~es were used to compute the COSNAR strain tmn, which was added to the Benson value to give a resonance energy value. The results are give in Table 3, and it is clear that they are too high, especially for the smaller BBLs. The A H ~ ) ' s for each ~ as computed for the Z GI's and the correo~ MM2 strain ene~es are also listed in Table 3.

331 Table 2 LEC Strain Energies for BBLs and COSNAR Model Compounds Computed with SPARTAN MM2 module (__allencr~es in kcal/mol~ System

Amide

~

K~onr

Amine

2.2.2 3.2.2 3.2.2 3.3.1 3.3.1 3.3.2 3.3.2 3.3.3 4.3.3 4.3.3 4.4.3 4.4.3 4.4.4

31.29 27.46 26.13 17.24 30.03 24.34 29.86 29.94 32.71 31.50 36.02 39.80 37.79

19.59 24.27

20.38 24.70 23.34 19.55 22.92 28.83 28.60 36.13 46.18 43.99 50.64 52.23 50.,41

21.77 27.15

(3) (31) (30) (11) (33) (32) (12) (13) (14) (16) (2g) (29)

18.28 29.99 37.35 48.71 54.90 56.55

20.68 32.61 38.86 46.27 47.26 49.04

In the 2.2.2 case, the x-system of the caJbonyl group is almost completely peqxadicular to the nitrogm lone-pair. This suggests that the resomtr, stabilization slmuld be very close to zero. However, the raw COSNAR resonance energy from M 2 is 8.60 kcaVmol. It is likely that this error arises because the moletxfle is simply too distorted for the MM2 parameter set to handle. Recall that we forced the MM2 algorithm to use undistorted N-CO ~ no matter how distorted the actual lactam was. Therefore, we assumed a resonance of zero in the 2.2.2. case, and applied the same linear correction (-8.60 kcal/mol) to all of the other resonance energies; these are the corrected resonance energies in Table 3. It should be noted that this is probably an overcorrection for the larger BBLs since they are more like the acyclic amides from which the pmmneters were developed. Even so, the 4.3.3 (13) and 4.3.3 (14) lactams are sligl~ hyperstable with respect to the COSNAR reference resonance energy of 17.33 kcaVmol. This mccmraged us to proceed on to higher levels ofcongmtatiorL It is also interesting to note that the 4.3.3 BBLs appear to represent a resonance energy maximum. This last point is examined in more det~ later. Some structural data for the BBLs are listed in Table 4, along with the corrected COSNAR resonance energies. These point to some obvious problems with the MM2 stmoa~es. First of all, the N-CO bond lengths show no sensitivity to increased resomnc~ (the carbonyi bond length is also invarim~ but this is ~ e d form previous experimental and computational studies [ 14,15,35,36]). This is not too s u r p ~ however, since variations in resoruur, e involve electronic effects and MM2 is based only on models from classical physics. Also, the N-CO bonds are too short for the smaller laotams. For example, the N-CO bond is calculated by ab initio (6-31G*) methods at 1.433 A in the 2.2.2 (3) BBL. This error is also not unexpected since the undistorted parameters for this bond

332 Table 3 Raw and Corrected M 2 COSNAR resonance energies for BBL series (all energies in kcal/mol) System LEC E_*.strain RE(raw)" RE(corr.) b A I-I~g~orr. r 2.2.2 (3) 8.73 -8.60 0.0 -27.1 3.2.2 (31) -0.11 -17.45 -8.83 -35.8 3.2.2 (30) 0.10 -17.23 -8.63 -37.2 3.3.1 (11) -4.71 -22.04 -13.71 -46.1 3.3.1 4.71 -12.63 -4.02 -33.7 3.3.2 (33) -7.12 -24.45 -15.85 -43.9 3.3.2 0 2 ) -1.36 -18.69 -10.08 -38.3 3.3.3 (12) -7.69 -25.02 -16.42 -43.2 4.3.3 (13) -11.03 -28.36 -19.76 -45.3 4.3.3 (14) -10.05 -27.38 -18.78 -46.5 4.4.3 (16) -6.99 -24.32 -15.72 -46.9 4.4.3 (28) -4.79 -22.12 -13.52 -43.1 4.4.4 (29) -5.11 -22.44 -13.84 -50.0 a) added Benson resonance energy term (17.33 kcal/mol) b) assumed xero resonance in 2.2.2 lactam - subtracted 8.60 k ~ m o l from raw RE's c) includes same 8.60 kcal/mol correction to strain energy used for RE's were used in all cases. Another problem is that the pyramidalization of the cad~nyl carbon appears to be quite high in all of the lagtams. Dunitz and note that this parameter is nearly always ,

(4)

< ~1~ >

/2/being the Hamiltonian operator. Throughout this Chapter, we use the usual nonrelativistic Born-Oppenheimer Hamiltonian in which no spin operators occur and the nuclear positions are considered fixed. The variational energies calculated using functions (1) and (3) are shown in Fig.1 (a,b) as functions of the internuclear distance R: the Heitler-London function (1) correctly predicts a stable molecule, with roughly the observed bond length and a dissociation limit corresponding to two isolated hydrogen atoms. It is therefore basically satisfactory: it is considerably improved by admitting the two 'ionic' functions (2) but the common value of the parameters (a, b) is only about 0.12 at equilibrium geometry, falling to zero for R large. The function (1), corresponding to a single 'covalent structure', is thus adequate and the so-called 'ionic structures' (2) play a minor role: the bonding in H2 is essentially covalent. The association of covalent bonds with pairing of the spins of electrons on different atoms is the fundamental

367 premise on which the whole of valence bond (VB) theory is based. The alternative choice of signs in (1), corresponding to parallel-coupled spins, yields a triplet excited state whose energy curve indicates repulsion at all distances.

E/eh

/ -1.0

f

b

Figure 1. H2 energy curves (a) Heitler-London (b) Molecular orbital (c) Coulson-Fischer

-1.1

(d) Experimental

- 1.2

,I

oo

l

t

20

t

40

|

I

so

R/.o

It will be recalled that the approach of molecular orbital (MO) theory starts, on the other hand, from an 'independent-particle model' (IPM) in which both electrons occupy th~ same 'bonding MO', r - X A zr- X B , similar to the one used [4] for the hydrogen molecule ion, H +. The bonding MO is in fact the approximate wavefunction for a single electron in the field of the two nuclei; and allocating two electrons to this same MO, with opposite spins, yields the 2-electron wavefunction

=

3a).

(5)

The spins must be paired in this case, to satisfy the Pauli principle, the spatial factor r (rl)r (r2) being already symmetric. In the VB function, the electrons are allowed to 'change places', i.e. to be associated with either of the two nuclei A and B, by using the symmetrized product XA(rl)xB(r2) + XB(rl)xA(r2); but in the MO approach each electron is 'automatically' shared between the n u c l e i - because its orbital contains terms coming from both centres. In MO theory, then, bonding is not associated with exchange of electrons between differently localized AOs, but rather with the fact that the MOs themselves extend over more than one centre - the MOs are essentially delocalized over the centres and the bonding is associated with the

368 presence of each electron in the region where the AO of one atom overlaps with that of another. It is clear that the MO function (5) may be expanded in terms of the AOs XA, XB, to give ~'~MO : ~ ( A - B ) + ~ ( A - B +) + ~ ( A + B - ) , (6) a mixture of VB structures in which covalent and ionic terms all have the same weight. It is often said that the Heitler-London function (1) fails by leaving out the ionic structures and that the MO function (3) fails by giving them too much importance. The failure of the MO function is the more drammatic: it yields an energy curve (Fig.l) which behaves quite incorrectly away from equilibrium geometry and therefore offers no prospect of estimating a dissociation energy as the difference E ( c c ) - E ( R e ) ) . This failure of the MO approximation is quite general: without drastic correction it is incapable of describing systems in 'bond-breaking' geometries.

-

The Heitler-London function was improved by considering alternative 'electron configurations', with both electrons on one centre or the other, and then using the twoparameter function (3). In just the same way, the MO function (5) can be improved by adding configurations in which one or both electrons are promoted to the antibonding MO, r = XA -- XB, which has a nodal plane (r = 0) midway between the nuclei. The (symmetric) spatial functions for such configurations, which are loosely described as 'singly-excited' and 'doubly-excited', are respectively r162

+ r162

r162

By attaching the (antisymmetric) spin factor, for paired spins, we obtain 'configurational functions' (CFs) of singlet type; and these may be mixed with the 'reference function' (5), used as a first approximation to the ground state, to yield a variational function -

~1

+ A ~ 12

22

11 -~- ~ I I 11'

(7)

where subscripts and superscripts indicate the occupied MOs before and after electron promotion. When the parameters in the variation functions (3) and (7) are optimized, by minimizing the variational energy approximation (4) for every internuclear distance, a greatly improved energy curve is obtained (Fig.1 (c)), either expansion giving exactly the same results. This represents the best possible approximation that can be obtained from the basis provided by the orbitals XA, XB. In more general terminology, the orbitals used may be the first 2 members of a basis set, {X1, X2, .. Xi, . . . Xm}, comprising m 'basis functions' (not necessarily AOs), and the approximation obtained represents the 'basis set limit'. It appears that alternative expansions of the wave function, assigning the electrons to either AOs or MOs, can be mathematically equivalent - provided all possible configurations are admitted and the CFs used satisfy appropriate symmetry requirements (e.g. coupling of spins to desired spin multiplicity, 2S + 1). This conclusion is in fact correct and very general: the approach

369 being followed, originally due mainly to Slater, is nowadays known as 'configuration interaction' (CI) and provides the most widely used tool for electronic structure calculations. The choice of orbitals, used in defining the electron configurations, is immaterial provided they are linearly-independent combinations of the given basis functions - for example, MOs built up in LCAO approximation. The above considerations suggest a third type of expansion, proposed by Coulson and Fischer [5] and later by Mueller and Eyring [6], in which the MOs are replaced by 'semi-localized' orbitals of the form CA -- X A -[- ,~XB,

(8)

CB -- X B -~ I-tXA 9

In the case of H2, symmetry of the molecule suggests that # -- A; and for a small positive value of A the two orbitals will each show a distortion towards the other centre, becoming 'egg-shaped'. A generalization of the covalent structure function (1) is then

qiCF = (r162 + r162

/3C~).

(9)

The remarkable property of this function is that, when optimized by varying A at each internuclear distance, it exactly reproduces curve (c) in Fig.1 - even though there is now only one term in the CI expansion! In other words the bonding in the hydrogen molecule is purely 'covalent', with no participation of 'ionic structures'. This result should warn us against attaching too much 'objective' significance to the covalent and ionic structures used in the VB interpretation of molecular structures: the 'structures' are only terms in an expansion of the wavefunction and the relative importance of different terms depends entirely on the definition of the orbitals from which they are constructed. By expanding (9) in terms of the AOs XA, XB it is a simple matter to derive the form (3) and to obtain an expression for the coefficient a(= b) in terms of the parameter ~(= #) in (8): the ionic structures, when working in terms of conventional 1-centre A Os, may thus be implicitly included if we distort the orbitals in order to give them some 2-centre character. Before turning to many-electron molecules, it is useful to ask Where does the energy of the chemical bond come from? In VB theory it appears to be connected with 'exchange' of electrons between different atoms; but in MO theory it is associated with 'delocalization' of the MOs. In fact, the Hellmann-Feynman theorem (see, for example, Ch.5 of Ref.[7]) shows that the forces which hold the nuclei together in a molecule (defined in terms of the derivatives of the total electronic energy with respect to nuclear displacement) can be calculated by classical electrostatics, provided the electron distribution is represented as an 'electron density' P(r) (number of electrons per unit volume at point r) derived from the Schrhdinger wavefunction 9. This density is defined (using x to stand for both space and spin variables r, s, respectively) by P(r)

N ] ~I/(x, X2, ... XN)II/*(X, X2, ' 9x N ) d s d x 2 . . , J

dxN

(10)

370 and, more correctly, represents the probability/unit volume of finding an electron (no matter which) at point r. In this definition, the wavefunction (which may be exact) is assumed antisymmetric and the spin variables are eliminated in the integrations. The factor N arises because, whichever electron is at point x, integration over the other N - 1 variables must give an identical result ( ~ * being symmetric). The significance for chemistry of the Hellmann-Feynman theorem is obvious: the nuclei in a molecule may be regarded as held together, against the repulsion from their positive charges, by the attraction they feel towards any concentration of electron density between them. This interpretation provides a basis for most qualitative discussions of chemical bonding - see, for instance, Ref.[8]). An immediate application is provided by a comparison of the wavefunctions used above for the hydrogen molecule: all are constructed using the first two functions of a basis set X1, X2, ... Xm and a little consideration shows that all such approximations lead to an electron density of the form

P(r) = Z P~x~(r)x*s(r) ~,~

(11)

- a 'bilinear form' in which only the numerical coefficients, Prs, differ between one approximation and another. With just two basis functions, the hydrogen ls orbitals on the two centres, the electron density takes the form (all quantities being real) P(r) = P A A X A ( r ) 2 + P B B X B ( r ) 2 + 2 P A B X A ( r ) x B ( r ) ,

(12)

in which the first two terms are ls-type densities on centres A and B, with numerical 'weight factors', while the third term is an 'overlap density' concentrated mainly in the bond region and appearing with a weight factor 2PAB. The electron density is thus not just the sum of two hydrogen-like densities: the density in the region between the hydrogens may be considerably enhanced whenever 2PAB is large and positive. The importance of the overlap density, which we associate with bonding, is thus proportional to the 'bond order' P A B - a term used extensively in early applications of MO theory but equally applicable in a wider context. Indeed, the matrix P, which collects all the coefficients Prs in (11), is often called the 'charge and bond order matrix'. The off-diagonal elements are (at least formally) the bond orders for pairs of mutually overlapping orbitals; the diagonal elements may be called 'charges' because they measure the amounts of electronic charge associated with the corresponding orbitals. For a general many-electron molecule, equation (11) may be re-written in terms of normalized orbital and overlap densities dr(r) = xr(r)xr(r),

drs(r) = S r s x r ( r ) x s ( r )

(13)

(S~ being the overlap integral for Xr, Xs) and then takes the form P(r) - Z r

qrdr(r) + Z r<s

qrsdrs,

(14)

371 where (in the usual case of real orbitals and coefficients) qr = P ~ ,

(15)

q~s = 2S~sPrs.

Since integration over all space of the electron density P(r) must yield the total number of electrons, it follows that f P ( r ) d r - Z qr + Z qrs r r<s

-

N

(16)

and this gives a convenient means of visualizing how the electron density is spread out over the orbital and overlap regions [9]. In fact, the q's are nowadays usually referred to as 'electron populations' and this analysis of the density, further elaborated by Mulliken [10], is called "electron population analysis". When a population analysis is performed for the various wavefunctions used above, for the hydrogen molecule, it is found that the overlap populations are always positive (indicating a flow of electron density into the overlap region between the nuclei) but that the predicted enhancement of the density in the bond region depends on which function is used: for example, the MO and VB functions give bond populations qaMO

=

2Sab V/(1 + Sab)'

VB

=

2S2b V/(1 -~-S2b) "

(17)

The MO value is too large, by comparison with the value resulting from the best function used so far, while the VB value is a little too small. On the other hand, for the triplet state obtained by interchanging the signs in (1), the bond population becomes strongly negative: this result means that, with parallel coupling of the spins, electron density is 'scooped out' from the orbital overlap region and 'piled up' on the a t o m s - giving a very graphic interpretation of bonding and antibonding character. 3. CLASSICAL VB THEORY: PERFECT-PAIRING AND RESONANCE The 'classical' VB theory, as developed largely by Pauling and his many collaborators during the 'thirties and 'forties, is a direct qualitative generalization of the ideas introduced by Heitler and London in their hydrogen molecule calculation and its extension to m a n y - e l e c t r o n systems. This approach, which preceded Slater's introduction of determinants, rested upon group-theoretical foundations (see Appendix 3 of Ref.[7] for a brief summary of the required results). General treatments of spin coupling and of group theory may be found in the books by Pauncz [11] and Wigner

[12].

372 3.1 S y m m e t r y C o n s i d e r a t i o n s The wavefunction (1), antisymmetric under exchange of space-spin variables of the two electrons, was conveniently factorized into a product of space and spin factors: anti-parallel coupling of the spins (antisymmetric spin factor) then implied symmetry of the spatial factor and a consequent enhancement of electron density in the bond region. Such a factorization is unique to a 2-electron system: for an N-electron system, a Pauli-compatible function must have the form ~(xl,x2,... XN)= E O ~ ( r l , r 2 , . . .

rN)O~(Sl, S2,... SN),

(18)

t~

where the functions ~ all correspond to the same energy, while the O~ are spin functions for the same spin eigenvalues (S, Ms), each of the two sets providing a particular irreducible representation of the group of N! permutations of electron indices. This means, for example, that under the permutation operation/5, any O~ is mixed with its partners according to

Po~ - ~ O~D(P)~.

(19)

A

The set of spatial functions {O~}, which may be either exact degenerate eigenfunctions of the Hamiltonian o p e r a t o r / : / o r approximations to them, must then provide the 'associate' representation b in whicht (using ep for the parity factor, +1 for/5 even or odd)

D,~,~ : E op~b(P)~, = E ep,~D(P)~, A

(20)

A

The matrices D(P) with elements D(P)~ form a matrix representation of the group of permutations; and this representation is characterized by the labels S, Ms which indicate the total spin (S) and its z-component (Ms). From (19) it follows that, when an orthonormal set of spin eigenfunctions is available, the matrix elements D(P)~ may be expressed as D ( P ) ~ - < o~,IPlO,~ > (21) and this shows that the representation label Ms is redundant, since 'stepping-up' or 'stepping-down' the spin z-component in (21) leaves the matrices unchanged. The representation matrices are therefore commonly indicated by D s ( P ) . The index distinguishes the linearly independent spin eigenfunctions of given S, for any chosen Ms, which are mixed among themselves under the permutations; and the number of such functions is the dimension of the representation. t Strictly, the matrix whose elements appear in (20) should be IS(/b), which is the 'dual' (transposed inverse) of D(P). In the present context, however, the representations are real-orthogonal and the two matrices coincide; consequently IS(P) = eD(P).

373 In principle, it is easy to calculate the electronic energy for a wavefunction of type (18). With the assumptions that the spin functions are orthonormal and that H is spin-free, the energy expectation value is < E

> < e, le>, >

m

~,~,,~ < r162

> < O,~lO,,, >

which at once reduces to

and finally to E = < O lf/L < 'I',~lr

>

(any a),

>

(22)

the last step following because (~ being a degeneracy index) the same term is simply repeated in both numerator and denominator, for each value of ~. 2

%. ()

1

2

3

4

5

6

7

8

N (number of electrons)

Figure 2. The branching diagram. In order to make use of (22) it is necessary to decide upon a particular O~, by choosing an appropriate 'spin-coupling scheme'; and then to construct its 'partner' (I)~. The standard irreducible representations Ds(/5) are carried, as in (19), by the 'branching diagram' (BD) functions indicated in Fig.2. But with one exception these provide no simple association with chemical bonding: the exception is the function set up by following the bottom path, in which the spins are coupled successively in pairs, possibly followed by a parallel-coupled sequence to give any desired total spin. Although Weyl and Rumer [13] had already explored the mathematical properties of such functions, it was Pauling [14] who first fully recognized their immediate applicability to problems of chemical bonding and gave simple graphical rules for making energy calculations. To obtain such rules it is necessary to set up orbital approximations to the formally exact eigenfunctions ((I)~) of the Hamiltonian.

374 The natural starting point is an orbital product gtk,,k2 .... kN = Ck, (rl)r

(23)

CkN,

in which Ck. is the orbital occupied by electron n and no orbital appears more than twice, in order that the Pauli principle may be respected. It is convenient to put all doubly occupied orbitals first and (as their corresponding electron spins must always be coupled up-down-up-down ... as in the bottom path in Fig.2) to make no further explicit reference to them. For purposes of defining and counting the independent spin eigenfunctions, N will thus correspond to the number of singly-occupied orbitals, whose electron spins are to be coupled to total spin S. The bottom-path spin function will be denoted by 01 (~ = 1) and it remains only to generate from (23) a combination of permuted products, namely 01, which will serve as a partner to O1 in (20). The bottom-path function is O 1 ( 8 1 , 82, ...

SN) -" 0(81, 8 2 ) 0 ( S 3 , 84)... 0 ( S 2 g - 1 , 82g)Ol(82g+l)... OL(SN),

(24)

in which there are g paired-spin factors, with O(si, sj) = [a(si)~(sj)-~(si)a(sj)]/v~, the remaining spins being parallel-coupled: it is a spin function with S = Ms 1 (N -- 2g) . A standard method of generating a function of given 'symmetry species', behaving like (I)~, is to sum all the permuted products /Sfl, formed from (23) by making the N! index permutations, with suitable coefficients. This recipe is well known_ in group theory [11] and embodied in the 'Wigner operator' (for representation Ds in (20)) 1

p~ =

-if(.

~ r

(25)

P which may be applied to an arbitrary function gt(rl,r2,.., rN) to 'project out' a component which will behave under permutations exactly like the function (I). in (20). Thus (dropping the superscript S, which will be fixed) 1

epDs(P),~Pf~(rl, r2, ... rN)

p ~ t ( r l , r 2 , ... r N ) = (f~v~ ~

\N!]

'~ 0n,

(26)

P

where ' ~ ' means "transforms like". In these equations, the normalizing factor contains 'Wigner's number' (2S + 1)N! ~ N + S + I ) W ( . ~1 N - S ) !

,

(27)

which is the number of linearly independent spin functions of given N, S and coincides with the number of distinct BD paths leading from the origin to the desired end point

375 with spin S. Evidently the simplest way of obtaining 01 is to put A = tr = 1 in (25) and to use (dropping the unessential normalizing factor) O(rl, r2, ... r N ) = E e p D s ( P ) 1 1 P ~ ( r l , P

r2, ... r g ) .

(28)

For future reference we note that the operators defined in (25) possess the properties P~Pt,~ = ~ t , P ~ ,

(29)

and that consequently p l l p l l = p l l i.e. pll is i d e m p o t e n t . The function (28) is a direct generalization of the spatial factor in the Heitler-London function (1): it derives from g orbital pairs (r162 (r162 associated with anti-parallel coupled spins; and each term in the sum will also contain a factor that refers to the un-paired orbitals (if any) whose spins will be parallel-coupled to give a state of total spin S. According to the Heitler-London-Pauling interpretation, it represents a perfectpairing a p p r o x i m a t i o n to the wavefunction of a molecule in which there are covalent bonds indicated by kl - k2k3 - k4 etc. and, for a non-singlet state, one or more 'unpaired electrons' which may endow the system with free-radical character. Clearly, different functions of type (28) will result from different choices of the paired orbitals ( k l - k2), ( k 3 - k4), ... etc. and hence, notionally, with different ways of allocating chemical bonds. All such functions possess the same Pauli-compatible symmetry (n = 1) and may thus be mixed to give, according to the variation principle, an improved approximation to an exact eigenfunction of the spin-free Hamiltonian. In some situations, a single 'structure' may be a good first approximation, giving a good estimate of the energy of a state of given S: this perfect-pairing scheme is then likely, as Pauling suggested, to give a good account of the actual chemistry of the molecule. And indeed most saturated molecules fall into this category, the bonding being well described in terms of overlapping pairs of singly occupied valence orbitals on the constituent atoms. In other cases, however, a single structure may be totally inadequate because there are several acceptable bonding schemes. The classic example is, of course, the benzene molecule. Here, disregarding all except the most loosely bound electrons (i.e. the r electrons, which are empirically associated with the special properties of aromatic systems), the problem is one of associating three (r) bonds with six carbon atoms. If the r-type AOs are denoted by r r ... r two obviously acceptable bonding schemes (cf. (24)) arise from ~ I --" r162162162162162 and ~ I I = r162162162162162 these are represented by the Kekuld structures in Fig.3,

Figure 3. Kekuld and Dewar structures for benzene.

376 where three other schemes (the Dewar structures) are also indicated. In such cases, in the terminology of the 'thirties, the state of the molecule is best represented as a 'resonance mixture of alternative VB structures'; and the (spatial) wavefunction of correct symmetry is written as (~ = CI~I + CII~II + CIII~)III -~-... CVI~VI,

(30)

where the ~' s are projected, according to (26), from the corresponding orbital products. It must be stressed that there is a unique association between the order in which the orbitals appear in a given gt and the pairing scheme which specifies a Weyl-Rumer (WR) function: whichever orbitals appear in the 1st and 2nd places have their spins paired, similarly for those in the 3rd and 4th places, and so on. Unless otherwise stated, the indices a, A, .. will from now on refer exclusively to W R functions and thus distinguish the different orbital products from which they are generated. It is now time to indicate how the qualitative ideas of VB theory can be put on a quantitative footing. The fundamental premises are simply (i) that valency is associated with singly-occupied orbitals on the constituent atoms of a molecule, more tightly bound electrons being assigned to doubly-occupied orbitals of a 'core'; and (ii) that the valence electrons may be well described using resonance mixtures of alternative spin-paired structures. For the moment, following Pauling and his contemporaries, we explore the implications of (ii) - noting only that (i) represents a remarkable result of empiricism, intuition and faith, to be fully justified only decades later. 3.2 C a l c u l a t i o n of t h e e n e r g y Let us consider first the single structure @(1 - 2 , 3 - 4 , . . .

) = pll~(rl,

r2,.., rN),

(31)

in which it is assumed that the spins of electrons in r r are paired, also those in r r and so on for g such pairs, and that the remaining N - 2g spins are parallelcoupled to a resultant spin S - ~1 ( N - 2g). The variational energy expression (22) (always for functions of type (26) with A - ~ - 1) readily reduces on using (29) et seq. Thus, noting that pll may be 'passed through' the operator /:/ (which is invariant under electron permutations) to give p21 ( - pl~) working on the righthand function, we find

E - M ~ epDs(P)ll < ~ [ / / [ P ~ >,

(32)

P where M is simply a normalizing orbital products. In the early days drastic a p p r o x i m a t i o n s - for this large number of 1- and 2-electron

factor and the matrix element involves only two of VB theory, no progress could be made without expression contains N! terms, each involving a integrals. The approximation made by Pauling

377 and others was that the orbitals could be considered roughly orthogonal i.e. that overlap integrals < Cries > could be neglected for s r r" this results in a drammatic simplification, only two types of contribution remaining. These are the so-called 'Coulomb integral'

Q ---< ~1/2/I~ > : J r162

/:/r162

drldr2.., dry,

(33)

which arises from the identity permutation, and g.. =
= < r162162162 >,

(34)

which arisesfrom the singleinterchange/Srs. The evaluation of (32) willthen proceed nicely, provided a formula can be found for the numerical coefficientDs(Pr,)11 which has not so far been considered. The evaluation of the required coefficient proceeds directly from (21)" for, in general,

Ds(h.)11 : < e, lPle, >=< e lef, >,

(35)

where @~ is a spin function obtained by permuting the spin variables according to the index permutation )5. As Rumer first pointed out [13], the overlap integral on the right (in which the function O~ is not in general a branching diagram function) is easy to evaluate by expanding (24), applying the permutation, and counting the number of terms that exactly match those in the un-permuted function. Rumer's graphical construction involves representing the spin indices by points on a circle (Fig.4) and linking those that refer to the paired spins, any remaining points (shown as bold dots) referring to the parallel-coupled spins:

6

5

Figure 4. Rumer diagrams and a superposition pattern. thus Fig.4a shows a function with two spin pairs, 1-2 and 3-4, and spins 5,6 parallelcoupled; Fig.4b indicates the effect of the spin exchange/523; and Fig.4c shows the result of superimposing the two diagrams. The value of the matrix element for any two WR functions is obtained [14] by inspection of their 'superposition pattern', as depicted in Fig.4c: in the most general case [15], the pattern may contain any number of 'islands' and 'open chains' (E-chains and O-chains connecting an Even or Odd number of points) and the required formula is

- ~5(-1)'"~'2 (n'~'-g),

(36)

378 where u ~ = number of arrow reversals to achieve head-head, tail-tail matching n ~ = number of islands in pattern g

= number of paired spins in each function E

g ~ = 1 (no E-chains), or 0 (otherwise) It should be noted that the use of arrows (a -+ fl), rather than simple links, is in general necessary to ensure the correct phase (• of the result. Two superimposed arrows in a superposition pattern form an island, while an unlinked point counts as an O-chain. It is now a simple matter to derive from (32), using (35) and (36), an expression for the energy associated with a single structure: the final result (first derived by Dirac [16] in 1929)is

E=Q+ ~ Kr8 } ~ (rs) paired

K~- ~

(rs) uncoupled

K,.~,

(37)

(~) parallel

where (rs) refers to any pair of orbitals (counted once only) and "uncoupled" means that r, s are neither paired with each other nor parallel-coupled. The energy formula (37) formed the basis for a vast amount of work, at a semiquantitative level, in which the Coulomb and 'resonance' integrals (Q, Krs) were regarded as disposable parameters and assigned numerical values, characteristic of particular molecules and types of bond, in order to rationalize observed properties such as heats of formation and molecular geometries. Before considering the theoretical validity of this approach, however, we generalize to the case in which the perfect pairing approximation is inadequate and it is necessary to admit resonance among a number of VB structures associated with alternative plausible assignments of the chemical bonds. In the general case, the wavefunction will be expanded in the form (cf.(30)

- E

et seq) (38)

t~

where the 'best' expansion coefficients c~ will follow (according to standard variation theory) on solving a set of 'secular equations' which may be expressed in matrix form as

Hc = EMc.

(39)

The coefficients are collected in the column c, while the square matrices have elements H,~,x =
,

M ~ = < ~,~[~x > .

379 Here it should be noted M contains only the overlap integrals between the structures and describes the 'metric' of the 'vector space' which they span. With the same assumptions as in the derivation of (37) from (32), the expression for the matrix element H ~ reduces first to H,~:~ = QDs(I),r

(~)

where I denotes the identity permutation and, as before, the coefficients of Q and K~s are expressible as matrix elements according to (21) in which a,A now label WR functions. The effect o f / 5 , working on the righthand function, is to modify the superposition pattern for (I)~ and (I)~, and by applying (35) (with 111 replaced by aA I) it is possible to evaluate the required matrix element. The result takes its simplest form for singlet structures (S - 0) and then reduces to H,~ - M,~[Q + E 1

-

Krs

(r, s same island, odd number of links apart)

(40)

E

Krs

(r, s different islands)

2E

Krs

(r, s same island, even number of links apart)],

where M~)~= ( - 1 ) v ~ 2 (n'~-g).

(41)

The matrix element rules (40) and (41), which apply to the superposition pattern for any two structures(I)~, (I)~ for a singlet state, were first given by Pauling [14] in 1933. They were to form the basis of nearly all semi-empirical applications of VB theory to polyatomic molecules during the next few decades. Before considering the defects of the theory in the classical form just presented, it is worth noting that all the previous results may be embodied in an elegant "vector model", introduced by Dirac [16] and van Vleck [17]. This model refers formally to a spin-only system, whose Hamiltonian contains only spin operators and numerical parameters: the matrix elements of this model Hamiltonian, taken between pure spin functions, coincide with those obtained using the rules given above and thus lead to exactly the same secular equations and approximate energy eigenstates. The parameters in the model Hamiltonian are the coulomb and resonance integrals already defined and its form is simply i-Is = Q' - 2 E

K i j S ( i ) . s(j),

( Q ' - Q - ~1 E(~j) Kij).

(42)

(ij) The validity of (42) rests upon the well known Dirac spin-exchange identity, /hij = 1 ~[1 + 4S(i). S(j)], which is easily confirmed by testing the effect of each side on the

380 four spin products a(si)a(sj), a(si)fl(sj), etc.; and it is clear that (42) may be used with any choice whatever of spin functions, not only those of Weyl-Rumer form. 4. THE RISE AND FALL OF CLASSICAL VB THEORY The simple mathematical machinery developed in the last Section was used to great effect, throughout the 'thirties and 'forties, in applications ranging from the prediction of molecular geometries and the estimation of heats of formation to the detailed discussion of bond properties and even to the theoretical study of chemical reactions. In all these areas Pauling was a leading pioneer. To illustrate such applications and to introduce other important developments during this period, one example must suffice, namely that provided by the water molecule.

Example: The electronic structure of H2 0 The water molecule is visualized as arising from the reaction of an oxygen atom in its 3p ground state with two 2S hydrogens. If the electron configuration of the oxygen is indicated as O[ls22s22p22px2py], then the ls 2 represents an 'inner shell', the 2s 2 and 2p 2 represent two 'lone pairs', and the remaining 'valence electrons' will have their spins parallel-coupled (Hund's rule) so that the corresponding (positive) Krs in (42) will enter with a negative sign. If the two hydrogens approach as indicated in Fig.5 there will be two possible coupling schemes for the spins of the 4 valence electrons, namely (I) with pl,H1 and p2, H2 paired; and (II) with pl, P2 and H~,H2 p a i r e d both structures contributing to the observed singlet ground state of the molecule.

0 r

P~

Z

,

(43)

of which the first term has nothing to do with 'exchange' being a one-electron integral, which contains kinetic energy and potential energy associated with the overlap distribution XAXB for an electron in the field of the nuclei: it is this term that is associated with chemical bonding, being large and negative, while the 'true' exchange term < XAXBIglXBXA > is essentially positive. To put S ~ 0 is to throw away the chemical bond! 9 The number of possible VB structures, particularly when those of 'long-bonded' and 'ionic' types are admitted, rapidly becomes astronomical for all but the smallest molecules: and, in situations where (even with carefully chosen hybrids) the validity of perfect pairing cannot be preserved, the technical difficulties of calculation appeared to be insuperable.

383 In summary, the major dilemma of VB theory in its classical form was that its beautifully simple concepts, which served many generations of chemists and were embodied in countless elementary textbooks on valency, could not be reconciled with the exigencies of actual calculation - at least with the means then available. Other methods, more easily adapted to ab initio development, began to overshadow, and eventually to totally eclipse, the area of quantum chemistry in which Pauling played such a major role. 5. MODERN VB THEORY During the last twenty years, there has been a resurgence of interest in VB theory and in the possibility of re-casting the basic approach in a form more amenable to ab initio calculation. Several distinct approaches have emerged, all of them (with present-day computational facilities) capable of competing with the more conventional methods, which normally have their roots in the independent-particle model (IPM) and are usually based on the use of molecular orbitals (MOs) and configuration interaction (CI). The new methods that result will be referred to collectively as forms of "modern VB theory", their common features being that (i) they start from the usual spinfree Born-Oppenheimer Hamiltonian (Sect.l); (ii) they use localized orbitals that are recognizably 'atomic' in character and usually overlap very substantially; (iii) where possible, they use spin eigenfunctions related to those of classical VB theory (those of WR type); and (iv) they proceed in a completely non-empirical way, using the techniques of variation theory, evaluating all one- and two-electron integrals and throwing nothing away. Some of these methods build the wavefunction from spinorbitals and employ Slater determinants, so as to exploit the familiar mathematical machinery developed during the last sixty years: but here we prefer to emphasize the classical VB approach and to show that the matrix element rules used by Pauling and his contemporaries are quite sufficient for making ab inito calculations, provided the multiple exchange integrals are all included. Before doing so, however, it is worth asking why overlap could not be eliminated from the start by employing, instead of the usual AOs, orthogonal linear combinations of them - for which the matrix element rules in their simple form would be rigorously valid. 5.1 V B t h e o r y w i t h o r t h o g o n a l orbitals 1

Soon after Lowdin [21] introduced his well known ' S - 2 prescription' for constructing an orthonormal set from an arbitrary set of AOs, modifying the original set as little as possible in a 'least-squares' sense, Slater [22] suggested that VB theory might be validated by introducing such orbitals and thus eliminating all the problems connected with neglect of overlap. At the same time, however, he repeated the Heitler-London calculation on H2 and reached a completely negative conclusion: with orthogonalized AOs, the single 'covalent' structure predicted no bonding at a l l - the energy curve indicating strong repulsion at all distances!

384 The reason for Slater's disappointing result becomes immediately clear on making a population analysis of the wavefunction [23]; when orthogonalized AOs are adopted, the bond population is found to be strongly negative at all distances. With hydrogen ls AOs, the population values at the equilibrium distance are qa = qb 0.638, qab 0.724, indicating a significant enhancement of electron density in the bond region; but on orthogonalizing the AOs the corresponding electron-pair function gives qa = qb 2.309, qab = --2.618, indicating that electron density is 'scooped out' of the bond region and 'piled up' close to the nuclei. In this way the 'formal' result - that the exchange integral (43) is positive and therefore gives no chemical b o n d i n g - receives a 'physical' interpretation through the charge density and the Hellmann-Feynman theorem. =

-=

=

Of course, on admitting CI by adding ionic structures it should be possible to reproduce the best energy curve shown in Fig.l; but the interpretation of the wavefunction will then be unconventional. In fact [23], using the orthogonal AOs, the best function is found to contain covalent and ionic structures with coefficients 0.781 and 0.442 respectively; and the bonding arises from the large matrix elements through which they interact. It is possible to extend the approach to many-electron molecules [24], deriving matrix elements between all possible types of covalent, ionic and muliply ionic structures, without making any of the approximations made in classical semi-empirical VB theory. The conclusions are general: covalent structures alone cannot account for bonding when they are constructed from orthogonalized AOs, the electron density in the overlap regions being essentially negative until ionic (i.e. 'charge-transfer') structures are admitted. One viable way of making fully ab initio VB calculations is thus to employ orthogonalized AOs: however, although early work [25] showed great promise (in calculations on both ground and excited states), the number of structures that must be admitted for a many-electron system rapidly becomes vast. For this reason, in spite of later ab i n i t i o verification [26] of the preliminary results, the approach has been little used and will not be considered further.

5.2 T h e " n i g h t m a r e of t h e inner shells" Before turning to VB methodology proper, it is essential to justify one basic premise underlying all early work, namely that attention can be confined to the 'valence electrons' alone: for VB theory in all current forms must confront a "N! problem" arising from the need to include all permutations in the calculation of matrix elements- and if inner-shell electrons are to be explicitly included this problem becomes insuperable. The approximate 'separability' of a general N-electron system into physically distinguishable subsystems A,B,..., each referring to its own group of N A , N B , ... electrons, is a familiar idea which permeates the whole of quantum chemistry: the idea is based on a wealth of experimental evidence and has been fully discussed elsewhere [27]. Briefly, a quantum system may be described as 'separable' when its wavefunction

385 may be represented, with high precision, as an antisymmetrized product of the form ~ ( x l , x2, ... X g ) = MA[OA(Xl,... XNA)OB(XNA+I, ... XNa+Ns)... ].

(44)

Here .~ denotes the usual antisymmetrizer, for all electrons, while in general OR is a 'group function' for the group of NR electrons in subsystem R. No exact wavefunction can be written in the form (44): for neither electrons nor basis functions (the global set from which ~ is constructed) can be uniquely partitioned among the subsystems (A,B,...R,...). The Pauli principle is clearly satisfied, since no particular electron labels are associated with any given subsystem; but the possibility of separating the orbital basis into subsets, which can each describe one subsystem with high accuracy, raises delicate questions of overcompleteness. To avoid such problems it is necessary to truncate the function space associated with each group of electrons, so as to eliminate excessive overlap between the functions of different groups. When the truncated sets describing different groups are mutually orthogonal, the subsystem wavefunctions are said to be strong orthogonal; and the wavefunction ansatz (44) may then be used to break down the energy of the whole system into terms for the separate groups, supplemented by pairwise interaction terms. In the present context of a 'core-valence' separation, there will be only two groups, C and V, say, containing core and valence electrons, repectively, and the energy expression takes the form E = Ec + E v + E c v = E c + E [ ~, (45) where E c depends only on the core function Oc and may be calculated without reference to the valence electrons. In the second form, E~~ (= E v + E c v ) contains the whole of the interaction between the electrons of the two groups and thus represents the energy of the valence electrons in an effective field provided by the core: this is the quantity considered in all the semi-empirical applications of classical VB theory - but now it may be defined and, in principle, calculated quite rigorously. The reduction is well known [28] and will be indicated only briefly. The electron density (10) is the so-called 'diagonal element' of a more general quantity, the (spinless) one-electron density matrix, P(r, r'), defined in exactly the same way except that the variables in ~ . carry p r i m e s - which are removed before the integrations. The reduction to (11), in terms of a basis set, remains valid, with a prime added to the variable in the starred function. For a separable wavefunction, the density matrices for the whole system may be expressed in terms of those for the separate electron groups; in particular, for a core-valence separation, !

!

P ( r l ; rl) = Pc(r1; r~) + P v ( r l ; rl),

(46)

which means of course that the total electron density may be obtained simply by superimposing core and valence contributions. When the analogous two-electron quantities are separated in a similar way, the expression for E~/ff in (45) takes a very

386 transparent form:

(47) = < OvI

\i'-1

s

+ ~

i, "--

g(i,j)

I(I)v >,

and this refers explicitly only to the N v valence electrons. The effective Hamiltonian for a single valence electron in the field of the core is

s

= h(i) + [Jc(i)- ~gc(i)],

(48)

where the coulomb and exchange operators (J,/~) have the general effect

3C(1)r

r (49)

/~c(1)r r

= / g(1, l')Pc(rl; r~)r

being an arbitrary spatial function.

The generality and importance of the above results cannot be overemphasized. The wavefunction for the valence electrons may be optimized by variation of (I)y alone, using the effective Hamiltonian in (47) with appropriate orthonormality constraints. In practice this means that, for a function built up from orbitals {r } of the valence space, it is only necessary to replace matrix elements < Crlh]r > by

< r162

> - < r162

> + ~ P,~[< r162162162 > - 89< r162162162 >], (50) i,j

where superscript c indicates the core orbitals and density. It is also clear that by interchanging core and valence labels in the preceding equations, it must be possible to optimize the wavefunction for the core in the effective field due to the presence of the valence electrons. In that way, by using core functions of relatively simple form (e.g. of SCF type), it is perfectly feasible to make an iterative optimization of the whole N-electron wavefunction, with little more difficulty than making a VB calculation on the valence electrons alone. This type of constrained optimization, in which the core and valence functions are allowed to 'float' (instead of being largely predetermined by an a priori choice of basis sets), is essential in order that the resultant wavefunction may give a realistic account of electronic properties and their dependence on molecular geometry.

387

5.3 V B t h e o r y w i t h n o n - o r t h o g o n a l orbitals Let us suppose that a good core function has been constructed and turn attention to the valence-electron system, the number (Nv) of electrons to be considered by VB methods now being relatively modest. It should be noted that the valence system may include a number of bonds and lone pairs and that the term 'core' is used in a general sense to include all other electrons e.g a number of different atomic inner shells or, in the context of organic chemistry, even the electrons of a strongly bonded 'framework' providing the effective field in which the 1r-electrons move. The 'separation' contemplated thus depends on the nature of the problem considered. The simplest possible approach to the construction of the VB wavefunction is essentially that outlined in Section 3; it is also perhaps the oldest, being very close to that proposed by Serber [29] in 1934. Briefly, one starts from a set of orbital products (using henceforth N in place of Nv) ~

= Ckl r

CkN,

(51)

electron labels assumed in natural order 1, 2, ... N, as usual, and projects from each product a function of appropriate symmetry as in (28). Each of the projected functions corresponds to a VB structure with links k i --+ k2, k3 -+ k4, ... and simply by changing the order of the orbitals one finds a whole set of VB structures. Noting that a will label the various orbital sequences klk2.., kg (t~p being the orbital in the pth place), a typical structure is

~

(52),

= p~iC~,~

where Pii is the Wigner operator (25) with n = A = 1. A general VB wavefunction then takes the form (38), namely @ = ~-~'~c~@~, where the coefficients are to be determined from a set of secular equations as in (39). The maximum number of linearly independent structures for a given configuration of N singly-occupied orbitals is given in (27) These structures may be chosen in various ways (see, for example, Ref.[7] p.245) and are relatively few in number; thus, for a singlet state of the benzene r-electron system (Fig.3), with N = 6, there are 5; and for N - 10 (e.g. naphthalene) there are 32. The structures all belong to the same orbital configuration specified by the choice of orbitals in the product (51), but there is no difficulty in dealing with multi-configuration wavefunctions in which, for example, 9 ~ is generated from an entirely different orbital product ~"~L = r r r To anticipate the use of multi-configuration functions, it is convenient to use K, L to denote orbital sets ki, k2, ... and ll, 1 2 , . . . , without reference to order, reserving n, A to distinguish the different coupling schemes within a configuration. Thus, = pa

,

=

(53)

will be spin-coupled functions from configurations K and L and the general matrix

388 element will reduce to (54) where the second step follows as in the derivation of (32). On inserting the expression for pll in (54), the general matrix element becomes 1

H~KL --< Ogl/:/IoL >=

~y.

E epDs(15)11
.

(55)

P There is a similar result, on removing the/:/, for the overlap matrix element

MKL.

The evaluation of (55) is in principle very simple. The quantity < f~gl/~l/~f~L > involves only an orbital product (f~g) on the left and a permuted orbital product /5~tL on the right: it therefore reduces in terms of one- and two-electron integrals (< Ck, lhlr >, < Ck,r162 >, where for example the primed index l~ is that of the orbital into which ri is sent by the permutation/5; each integral being multiplied by a chain of overlap integrals. The other factor in (55) is trivial; it is given by (36) in terms of the Rumer-Pauling superposition pattern for the coupling schemes indicated by ~ and A. For up to about N = 10 the calculation presents no problems, depending only on the availability of efficient algorithms for generating the permutations, handling the superposition patterns, and accumulating contributions to the matrix elements. No storage of intermediate data is required and the first calculations of this kind [30] were indeed performed on a small PC. For N > 10 the direct sum over all permutations rapidly becomes prohibitive; but by exploiting the properties of the symmetric group [31] further progress can be made and calculations for up to about 20 valence electrons become feasible.This is a promising field of development. The last step in the calculation of a high-quality VB wavefunction, is the variational optimization of the orbitals. The procedure to be adopted for this purpose tends to be specific to the particular type of wavefunction and method of calculation, but two main approaches may be distinguished: (i) one in which equations defining a stationary point on the energy surface are set up and solved (as, for instance, in SCF theory); and (ii) one in which the parameters that determine the orbitals are systematically varied (by standard methods of non-linear optimization theory [32]) until a minimum is found. The procedures of type (i) may be efficient when practicable, but require heavy programming and are of limited generality; those of type (ii) tend to be more costly in computing time but are simple and completely general, requiring (at worst) only a means of evaluating the energy. This problem lies outside the scope of the present chapter.

389 5.4 C o n n e c t i o n with other m e t h o d s Although modern VB theory is by now well established, in various forms, it should be remembered that the foundations were all laid in the Pauling era. The first true multi-configuration theory was, in fact, that proposed by Serber [29]: it was capable in principle of ab initio implementation and was the precursor of several group theoretical approaches, mostly based on the use of BD functions, which we mention before indicating the connection with Slater methods.

Branching diagram and related methods Serber's procedure may be related to the approach of Sect.5.3 as follows. Instead of projecting symmetry-adapted functions (all for a - 1) from a variety of orbital products ft~ = klk2.., kg, one may use a single product (with the orbitals of the configuration in a standard order) and then apply the more general Wigner operators (25)" the resultant functions O(') = P-u will all transform like the basis function O. and for different values of # will provide a full set of linearly independent functions associated with different BD paths. The matrix elements that will determine their mixing will then be, for a single configuration,

where the first step follows because p ~ may be passed from bra to ket provided the operator is replaced by its adjoint (with order of the indices inverted); and the second follows from (29). On expanding the operator and putting a - A = 1, it follows that (apart from normalization) < r

> - Z < ftl/}lPf~ > Ds(/5)u ~' P

(56)

with a similar expression for the overlap matrix element. The matrices in the secular equation (39) thus take the form

H = ZHpDs(P), P

M = E MpDs(P),

(57),

P

where the numerical coefficients are Hp - < ~I/:/[P~ > and < ftl/~Ft >. This result was given in Serber's first paper; the multiconfiguration form, in which a configuration index (K, L) is added to each gt to give matrices in block form, followed in the second paper. The difference between (56) and (55) is important: (56) requires explicit knowledge of N! matrices, whose evaluation involves non-trivial algorithms [32] (see Ch.7 of

390 ref.[ll]), for the representation provided by the BD functions; (55) requires only the l 1-element of each matrix, for a pair of WR functions, whose evaluation is trivial. It is not surprising that Serber's method was never pursued in actual computations; but the same general approach, often with sophisticated algorithms for generating the representation matrices, underlies a number of current methods. The main developments in this general area, largely due to Goddard, Gerratt, Gallup and their many collaborators, are well reviewed elsewhere [33],[34],[35].

Methods based on Slater determinants As in the discussion above, let us use the operator (25) to extract, from an arbitrary f~ a component behaving like (I)~,namely

~(~) = p~f~ = ~

epDs(P)~P~2 ~ r

(57)

P (omitting the trivial normalizing factor): there will be several such functions depending on which column (A) of the matrix D s ( P ) Has been taken. The fundamental result (18) may now be used to construct a totally antisymmetric wavefunction with spin included. This will be, for any choice of A,

p

and, by making use of the representation property (19), the quantity in parentheses may be replaced by PO~. Finally, then, ~ ( x l , x 2 , . . . X N ) = A[ft(rl, r2,.., rN)O~(Sl, S2,... 8N)

(58)

will be a fully antisymmetric space-spin function, formed by applying the antisymmetrizer (with the conventional normalization such that ~2 _ A) A = (N!) -1 E cpP P

(59)

to the product ft(9~, for any chosen spin eigenfunction. Since Oh is a general BD spin eigenfunction and may be expanded in terms of products such as a(sl)~(s2).., a(SN), it is clear that ~ in (58) is a spin eigenfunction formed as a linear combination of Slater determinants, the first one being ICtc~ r CNOLNI-i.e. an antisymmetrized product of spin-orbitals. It is thus perfectly possible to perform VB calculations using standard Slater methods and this approach has been used quite widely, first in the early work [36] and more recently in [35] and in the work of Balint-Kurti, van Lenthe, and their collaborators [37]. Unfortunately, the use of Slater determinants does not lead to any great reduction in computational problems, because (i) the numbers involved may be rather large; and

391 (ii) the matrix elements of the Hamiltonian between determinants of non-orthogonal orbitals are not easily reduced, Slater's simple rules being replaced by those of LSwdin [38], which require the evaluation of vast numbers of cofactors of an overlap matrix. In such calculations a key role is therefore played by algorithms connected with the computation and management of cofactors [39].

Some general comments The principal methods in current use differ mainly in (i) their use of group-theoretical or Slater procedures; (ii) whether or not the orbitals are optimized; and (iii) whether one or more orbital configurations are employed. In the group-theoretical approach, single-configuration VB functions were first used extensively by Goddard [40], who succeeded in formulating conditions for a stationary value of the energy associated with a single VB structure, in the form of a pseudo-eigenvalue equation which could be solved iteratively to determine the optimum orbitals. The resultant 'GVB theory' has been widely used, with considerable success. The multi-structure generalization (still for one orbital configuration) was provided by Gerratt and co-workers [33], who used the name 'spin-coupled VB theory' to indicate that the spin coupling was also optimized (through the coefficients with which different structures were allowed to mix). As BD functions are employed these methods do not lead to wavefunctions with an immediate interpretation in terms of the (WR) structures of classical VB form; and to make contact with Pauling's interpretations an a posteriori transfomation is usually necessary.

Multi-configuration VB theory was developed in an entirely different way by Gallup and his collaborators [34], who employed a large number of configurations as an alternative to optimizing the orbitals. This approach bears some resemblance to that used in Sect.5.3, being based on a group theoretical projection operator; but the projected functions correspond to the top path in the branching diagram and again the 'structures' that result have no immediate connection with those of classical VB theory. On the other hand, the approach developed in Sect.5.3 leads directly to wavefunctions expressed in terms of the structures used by Pauling, Wheland, and their contemporaries; there is no limitation to a single configuration (so ionic structures can also be admitted); and the orbitals may be optimized by straightforward numerical methods. Methods based on the use of determinants, particularly those of Balint-Kurti and co-workers [37], share the advantages of simplicity and flexibility; and in spite of a certain lack of mathematical elegance they also readily admit orbital optimization and the use of multiconfiguration wavefunctions. All the methods referred to have reached a certain level of technical perfection and may be used in ab initio VB calculations on molecules with up to, say, 15 electrons in the 'active space' i.e. outside a 'core'. Which one is employed is therefore largely a question of personal taste.

392 6. SOME ILLUSTRATIVE APPLICATIONS The aim of this Section is to present the results of some typical VB calculations, all made using the method described in Section 5.3. In all cases, the basis chosen was of modest quality (usually gaussian 'double-zeta' for the valence orbitals) in order to facilitate comparison with other calculations reperted in the literature (e.g.'full-CI'), for which use of a more extended basis would not have been feasible. The molecular geometries employed range from equilibrium to virtually complete dissociation. Special emphasis is placed on (i) the validity of a core/valence separation, with and without 'freezing' of the core orbitals; (ii) the quality of a perfect-pairing approximation, where appropriate, and the need for resonance mixing as the geometry changes; (iii) the effect of various constraints during the optimization of the orbitals - and the way in which they affect the qualitative picture of the origin of the bonds. 6.1 T h e w a t e r m o l e c u l e Equilibrium geometry is assumed and a contracted gaussian basis is used [41]. The molecule is dissociated by symmetric stretch, energies being calculated at bond length intervals of A R = 0.2Re up to R = 6.0Re (where dissociation is effectively complete). A standard closed-shell SCF calculation is used for comparison; the ls and 2py (normal to the plane, define the core to be used in the 'frozen core' approximation. The VB calculations are performed with just two covalent structures, these being sufficient (according to the qualitative discussion in Sect.4) to describe dissociation in which the oxygen is left in its triplet ground state. Curve (a) in Fig.6 refers to the frozen core approximation; Curve (b) shows the effect of core o p t i m i z a t i o n - which is evidently significant, as the core returns to its freeatom form. Another significant feature of both curves is the growing importance of the second structure, in which the spin pairing in the bonds is replaced by that between the hydrogen atoms and within the oxygen atom; when ~he bonds are stretched to several times their normal length the structure coefficient ratio approaches 1:2. It is easily verified that the corresponding combination of W R spin eigenfunctions then reproduces the 'top' BD function, in which the spins on the oxygen atom are tripletcoupled (as are those on the hydrogens), the two triplets being coupled to a total spin zero. Examination of the orbitals shows that, as dissociation proceeds, the oxygen hybrids shrink back into pure 2p orbitals and the atom thus reverts to its normal 3p ground state. The 'local' triplet coupling of the hydrogen spins indicates a growing repulsion. These findings verify completely the qualitative ideas current in the 'thirties. The calculations reported so far are based on unconstrained mixing of all valence functions: as a result, the optimized orbitals differ greatly from those pictured by Pauling, which - although usually hybrids - were strictly monocentric in character. The optimized forms resemble more closely the Coulson-Fischer orbitals of Sect.2, being 'distorted' AOs which result in considerably increased overlap in the bond regions. In this general context, such AOs have been referred to as 'overlap-enhanced

393 orbitals' [42]. A calculation in which the mixing is constrained, so that every orbital remains essentially monocentric (except for a small contamination arising from orthogonalization against the core), gives very poor results: the overlap enhancement resulting from free inter-atomic mixing is thus vital to the success of an approximation based on covalent structures alone. On the other hand, it should be possible to retain a description in classical VB language by admitting polar structures, exactly as in refining the Heitler-London function for H2. As indicated in Sect.2, the admission of polar structures can well describe bonding even when 'covalent' structures alone fail to do so. The bonding then results from the strong interaction between structures that differ by a single electron transfer across a bond, and is interpreted pictorially in terms of 'electron hopping'. By including 8 ionic structures of this type, and optimizing the multi-configuration function that results, the situation is completely restored: the resultant energy curve is almost coincident with Curve (b).

0.0

-75 6 -

SCF -75.7

-02

-

Q

C 'i'"

-04

-75 8 b c

~ -0.6 w

-75.9

rUJ --76.0

d

-1.0

-76.1

-76.2

i

-0.8 (*)

oo

,

,b

,

2;

"

D/De

s'o

,

Figure 6. PE curves for H20

4;

-,

-1.2

o.o

,b

2,b

D/De

sb

,o

5o

Figure 7. Reduced PE curves for H20.

For comparison purposes, Curve (c) shows the results of an accurate 'coupled-cluster' calculation (obtained using Gaussian 94), while (d) shows those of a full CI calculation [43] (for electrons outside a frozen core), in which about 250 000 Slater determinants were employed. It is well known that the slow convergence of large CI calculations is due to the difficulty of approximating the short-range correlation which inhibits the approach of two electrons are r12 --+ 0. Most of the difference between Curves (b) and (d) must result from this correlation error, which is apparently not strongly dependent on geometry variation. It is revealing to compare potential energy curves by introducing a 'reduced' curve [44]: the reduced PE curve results from a plot of Ered(X) -- E(x) - E(oo) E(c~)- E(1)'

x = R/R~,

(60)

394 which goes from - 1 at equilibrium to 0 at infinity. The reduced curve obtained in this way from Curve (b) is compared with that from the coupled-cluster and full-CI results in Fig.7. Evidently the general shape of the two-structure VB energy curve is in excellent agreement with that obtained from a full-CI study. 6.2 M e t h y l l i t h i u m The nature of the carbon-metal bond is still not well understood: there is, for example, no accepted and unambiguous interpretation of calculated wavefunctions even for the simplest organometallic compound, methyllithium. Pauling pioneered the use of electronegativity scales in the discussion of the ionic character of bonds: lithium is highly electropositive and is expected to form strongly polar, even almost purely ionic, bonds with more electronegative elements. Observations of dipole moments, solubilities, and electric conductivity of solutions all support this conclusion: but theoretical predictions in the literature range from almost purely covalent to almost completely ionic. The disagreement arises from the diversity of the methods of calculation employed and can only be fully resolved by translating Pauling's arguments into their ab initio counterparts. As in the above discussion of the water molecule, the simplest VB procedure would be to fully optimize the orbitals in a perfect-pairing representation of the wave function: but that would obviously preclude any interpretation of the bonding in terms of covalent-ionic resonance. Again, the only way of obtaining a classical VB interpretation of the C - L i bond would be to ' force' the admission of ionic structures by excluding the mixing of carbon and lithium orbitals. When this is done [43], the classical structures that show the carbon-lithium bond as C - Li, C - Li +, and C+Li -, appear with weights* 0.635, 0.376. and-0.011, respectively. The predicted 38~163 ionic character is strikingly close to the 43~163 estimated by Pauling on the basis of his electronegativity values (1.0 and 2.5, respectively) for lithium and carbon. This molecule has been fully discussed [45], both in its equilibrium geometry and over the whole range of distances for breaking of the carbon-lithium bond. Again, as dissociation proceeds the geometry and orbital forms change drastically: the carbon hybrids go from roughly tetrahedral at equilibrium to trigonal planar in the CH3 fragment. The results provide a coherent account of the behaviour of the system, in the language of classical VB theory but with the support of ab initio calculation. 6.3 L i t h i u m fluoride The bond in lithium fluoride is normally regarded as highly ionic, the Pauling electronegativities being roughly 1.0 and 4.0 for Li and F, respectively; and this raises the question of whether the molecule will dissociate into neutral atoms or into ions. As in previous examples, it must be possible to obtain a satisfactory energy curve by using * The appropriate definition, for non-orthogonal structures, is W K -- ~-~L CKMKLCL where MKL is the overlap of structures K, L. The sum of the weights is then unity, but the weights close to zero may become slightly negative.

395 a single covalent structure, provided the orbitals are optimized with no constraints: for the two orbitals of the bond could then delocalize over both centres as in the Coulson-Fischer calculation [5]. But this would offer no simple connection with the ideas of classical VB theory, and it therefore seems worthwhile to impose constraints as in Sect.6.2. Two types of calculation are made: first using a single covalent structure, with two electrons in a fluorine 'lone pair' and two in the bond pair F - L i , free mixing of all valence functions being allowed; secondly, a calculation of 'covalent plus ionic' type (three structures) but with the constraint of no inter-atomic mixing. In the first case, the bonding is formally covalent but the orbitals are free to move from one centre to the other during optimization; in the second case the orbitals remain monocentric and the structures then retain their classical interpretation. The core/valence separation is used throughout, only 4 electrons remaining in the valence shell, but the core function is optimized as in Sect.6.1. The results of the second calculation, with monocentric orbitals, are slightly inferior to those of the first: they can be improved by adding more structures but the results of the three-structure calculation give such a clear picture of the bond breaking that further refinement is unnecessary.

-106.2

-

1.0

(a)

(b)

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4

a)

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J

J -107

0

2

~

~,

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.~

~ R (bohr)

-~

~

~J

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3

,;,

.~

~

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R (bohr)

Figure 8. PE curve (a) and structure weights (b) for LiF Fig.8 shows the energy variation during bond breaking; and also shows how the weights of the three structures change in the process. The results for equilibrium geometry are in good accord with classical expectations: on the basis of his electronegativity scale, Pauling would have predicted a bond with very nearly 80% ionic character. The ab initio value obtained in the present work is 95%, the weights of the three structures being 0.050, 0.950,-0.030 for F - Li, F - Li +, F + Li-, respectively.

396 As the bond is stretched, the covalent structure becomes more important, but the weight of F - Li + remains above 80% until about R - 5.0 bohr: at this point, where dissociation is well advanced, the weights would suggest a 20% probability of separation into singlet-coupled neutral atoms, with an 80% probability of finding two separate ions. Energetically, however, the covalent coupling is favoured: at R - 9 . 0 bohr, where dissociation is virtually complete, the lowest-energy state contains only 1.4% ionic character and describes simply the neutral atoms with singlet coupling between the electrons of their singly occupied valence orbitals. The first excited state, with 98.6% ionic character, represents the pair of ions, whose energy is about 0.24 hartree higher than that of the neutral pair. As in Sect.6.1, comparison with SCF and full-CI curves (the latter, even with frozen ls cores, involving almost 10 million Slater determinants), testifies to the adequacy of the description in terms of three VB structures, the reduced energy curve being in excellent agreement with that from the full-CI calculation. Two states are indicated in Fig.8 to reveal an 'avoided crossing' in which the structure weights for L i - F and Li+F - change abruptly within a very short interval; this is not evident when only one covalent function, based on delocalized orbitals, is employed. 6.4 B e n z e n e a n d its ions

This Chapter could hardly close without reference to resonance in organic chemistry, where its impact has been so enormous. The classic example provided by the benzene molecule was first studied using ab initio VB methods by Gerratt and Raimondi [46], who performed calculations on the 7r-electron system using all five spin-coupled structures (Fig.3) with full optimization of both orbitals and mixing coefficients. Their results gave an impressive demonstration of the fact that, with orbital optimization, a few covalent VB structures could account for almost the whole of the energy obtained from a large CI calculation. With just 6 2p. orbitals as the r-electron basis (inner-shells and a-bonds providing the usual frozen core), the 5 covalent structures reproduced almost perfectly the results of the 'full-CI' calculation which included all 175 covalent and ionic structures: in fact, two Kekul6 structures alone accounted for more than 90% of the 7r-electron correlation energy. The forms of the optimized orbitals were reminiscent of those found by Coulson and Fischer: although strongly localized around each carbon centre, each was polarized towards its neighbours to give a set of overlap-enhanced orbitals (OEOs). Without overlap enhancement the 5 covalent structures alone cannot match even the results of a simple SCF MO calculation. The accuracy of a single-configuration VB description of a typical ~r-electron system, which confirms the validity of the classical picture, is not peculiar to benzene. Similar calculations [47] on the naphthalene molecule yield OEOs of similar form and again give an accurate description of the electronic structure in terms of resonance among a few 'principal structures'. On turning, however, to ionization and attachment processes the one-configuration picture is no longer sufficient, even at a qualitative

397

level. In view of the importance of such processes in organic chemistry, where the ease with which a conjugated molecule can accept or donate a r electron is of great interest in theories of reactivity, it is worth indicating briefly what modifications are necessary. The most important feature of ionization and attachment processes, from the standpoint of VB theory, is the increase in the number of structures that must be included in the wavefunction. Thus, for benzene, the number of covalent structures is 5, all belonging to the configuration with 6 singly-occupied orbitals and differing only in spin-coupling schemes. As already noted, this l-configuration approximation, fundamental to many forms of VB theory, can often give excellent results. But on removing one electron the number of orbital configurations increases; for benzene, for example, there will be s i x - according to which orbital contains the 'hole'. In general, corresponding structures may be indicated by adding a + at the position of the empty orbital and a dot at the position of the orbital whose electron remains unpaired. Thus, in benzene, for each position of the +, the spins may be coupled in 5 linearly independent ways; and this gives 30 structures in all. Similar considerations apply to the negative ion, where 30 structures must be admitted, the + of the positive ion being replaced by a - to indicate the doubly occupied orbital. All such structures are 'covalent' in the sense that no electron-pair link has been replaced by a ( + , - ) pair - as would be the case in forming a conventional ionic structure by an internal electron transfer. To ensure comparability with the neutral molecule calculations, it is therefore necessary to include all 30 covalent structures. A full account of the results of such calculations, for benzene and pyridine, is available elsewhere [48]. The approach of Sect.5.3 is applicable without change and leads to wavefunctions and energies much better than those based on (open-shell) SCF theory. What is more important, however, is the simple interpretation of the wavefunction in terms of a few energetically favoured structures. The weights of these structures appear to govern the chemistry of the ion, the position of the 'dot' being important for radical attack, and the position of the + or - for attack by charged species. There seems to be little doubt that further work along such lines will provide ab initio support for many of the intuitive principles formulated in the Pauling-Wheland era. CONCLUSION It remains only to draw together the main conclusions from this work. They may be stated very briefly as follows: 9 The main concepts and principles used so effectively by Pauling and his contemporaries are just as valid now as they were sixty years ago: chemical bonds may be well described by pairing the spins of the electrons in singly occupied valence orbitals (strong bonds being associated with strong overlap) to obtain a single VB structure; and when more than one structure is intuitively acceptable the description must be extended to admit resonance i.e. the wavefunction must be written as a

398 linear combination of structures corresponding to alternative pairing schemes. 9 Fully ab initio variational calculations, using a wavefunction consisting of VB structures and with optimization of both orbitals and structure coefficients, may be carried out by a variety of methods: these range from the direct 'spin-free' approach, in which only the permutation symmetry of the wavefunction is used (as in the 'preSlater' era), to methods in which the structures (with spin factors included) are expanded over determinants of spin-orbitals. 9 The long-established practice of 'separating' the wavefunction into a product of core and valence functions is entirely satisfactory, provided a strong-orthogonality condition is imposed: the electrons of the core provide an effective field for the valence electrons and vice versa and this permits separate optimization (in rotation) of core and valence functions. In this way VB methodology may be applied where most appropriate, notably to the valence electrons of molecules involved in bond-breaking processes, while simpler approximations may be used to represent the 'passive' groups of electrons (e.g. atomic cores) which undergo only minor changes. The limitations on the calculation are then set by the number of 'active' electrons (those of the VB group) and in this way the VB approach becomes viable for comparatively large molecules. 9 Optimization of the orbitals, no matter how large their mutual overlap may become, is central to any VB calculation based on a small number of structures: orthogonalization of the orbitals removes all the problems of VB theory but at the same time removes the bonds! With free mixing of the basis functions on all centres, a few covalent structures can often give an excellent description of electronic structure of the molecule: the optimal orbitals are then invariably overlap enhanced, the OEO overlaps commonly exceeding 0.9, and ionic structures are often of negligible importance. This is the situation in single-configuration forms of VB theory (e.g. [33]), where all spin coupling schemes may be admitted but only one set of occupied orbitals. 9 Closer contact with VB theory in its classical form can be restored by imposing a constraint of monocentricity on the orbitals, permitting free mixing of basis functions on the same atom but not on different atoms: this is indeed in the spirit of Pauling's original use of hybridization. In this case covalent structures alone are insufficient to give an accurate wavefunction, but this may be remedied by admitting a small number of ionic structures - those which arise from electron transfer between bonded atoms. Each singly ionic structure will have one empty orbital and one doubly-occupied, and will belong to a new configuration with its own set of pairing schemes. For many purposes the use of such constraints is an advantage: the concept of 'covalent-ionic resonance' plays an important descriptive r o l e - and chemistry would be the poorer without it. The limited CI associated with the ionic structures leads to ab initio wavefunctions of similar (or better) quality than those provided

399 by a single configuration of OEOs; but, more importantly, it leads to a transparent interpretation, in classical terms, of what is actually happening in many chemical processes. The few examples in Sect.6 speak for themselves: they show clearly that, in spite of magnificent advances in 'computational chemistry', the use of a handful of VB structures can provide a decent alternative to calculations employing many millions of Slater determinants. In the much quoted words of C. A. Coulson, they offer 'primitive patterns of understanding' that are at the heart of chemistry and not easily found in other ways. REFERENCES [1] Heitler, W. and London, F. (1927). Z. Phys. 44, 455. [2] Pauling, L. and Wilson, E. B. (1935). 'Introduction to Quantum Mechanics'. McGraw-Hill, New York. [3] van Vleck, J. H. and Sherman, A. (1935). Rev. Mod. Phys. 7, 167. [4] Pauling, L. (1928), Chem. Rev. 5, 173 [5] Coulson, C. A. and Fischer, I. (1949). Phil. Mag. 40, 386. [6] Mueller, C. R. and Eyring, H. (1951). J. Chem. Phys. 19, 1495. [7] McWeeny, R. (1993). 'Methods of Molecular Quantum Mechanics 2nd ed.'. Academic, London. [8] Deb, B. M. (ed.) (1981). 'The Force Concept in Chemistry'. Van Nostrand, New York. [9] McWeeny, R. (1951a). J. Chem. Phys. 19, 1614; McWeeny, R. (1951b). J. Chem. Phys. 20, 920; McWeeny, R. (1952). Acta Crystallogr. 5, 463. [10] Mulliken, a. (1955a). J. Chem. Phys. 23, 1833; Mulliken, R. (19555). J. Chem. Phys. 23, 2343. [11] Pauncz, R. (1979). 'Spin Eigenfunctions: Construction and Use'. Plenum. New York. [12] Wigner, E. P. (1959). 'Group Theory and its Application to the Quantum Mechanics of Atomic Spectra'. Academic, New York. [13] Weyl, H. (1956). 'The Theory of Groups and Quantum Mechanics'. Dover, New York (translated from the German edition of 1928); Rumer, G. (1932). Gottingen Nachr. p.377. [14] Pauling, L. (1933). J. Chem. Phys. 1,280. [15] Cooper, I. L. and McWeeny, R. (1966). J. Chem. Phys. 45, 226.

400 [16] Dirac, P. A. M. (1929). Proc. Camb. Phil. Soc. 25, 62. [17] van Vleck, J. H. (1934), Phys. Rev. 45, 405. [18] Wheland, G. W. (1965). 'Resonance in Organic Chemistry'. Elsevier, Amsterdam. [191 Huckel, E. (1930). Z. Phys. 60, 423. [20] Pauling, L. (1931), J. Amer. Chem. Soc. 53, 1367 [21] Lowdin, P.-O. (1950). J. Chem. Phys. 18, 365. [22] Slater, J. C. (1951). J. Chem. Phys. 19, 220. [23] McWeeny, R. (1954). Proc. R. Soc. Lond. A223, 63. [24] McWeeny, a. (1954). Proc. R. Soc. Lond. A233, 306. [25] McWeeny, R. (1955). Proc. R. Soc. Lond. A227, 288. [26] Campion, W. J. and Karplus, M. (1973). Mol. Phys. 25, 921. [27] McWeeny, R. (1997). In 'Quantum Systems in Chemistry and Physics', Proceedings of a European Workshop (ed. Maruani, J.). Kluwer, Dordrecht (in press). [28] McWeeny, R. (1959). Proc. a. Soc. Lond. A253, 242; McWeeny, a. (1960). Rev. Mod. Phys. 32, 335. [29] Serber, a. (1934) Phys. Rev. 45,461; Serber, R. (1934). J. Chem. Phys. 2,697. [30] McWeeny, R. (1988). Int. J. Quantum Chem. 34, 25; McWeeny, R. (1990). Int. J. Quantum Chem. Symp. 24, 733. [31] Zhang, Q. and Li, X. (1989). J. Mol. Struct. 198, 413; Li, J., Wu, W. and Zhang, Q. (1993). Chin. Sci. Bull. 37, 2243; Li, J. and Wu, W. (1993). Theor. Chim. Acta 89, 105. [32] Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T. (1989). 'Numerical Recipes'. Cambridge University Press, New York. [33] Gerratt, J. (1971). Adv. At. Mol. Phys. 7, 141; Cooper, D. L.. Gerratt, J. and Raimondi, M. (1987). Adv. Chem. Phys. 69, 319 [34] Gallup, G. A. (1973). Adv. Quantum Chem. 16, 229; Gallup, G. A., Vance, R. L., Collins, J. R. and Norbeck, J. M. (1982). Adv. Quantum Chem. 16, 229. [35] Raimondi, M., Simonetta, M. and Tantardini, G. F. (1985) Comput. Phys. Rev. 2, 171. [36] Miller, J., Friedman, R., Hurst, R. and Matsen, F. A. (1957) J. Chem. Phys. 27, 1386; Balint-Kurti, G. G. and Karplus, M. (1968). J. Chem. Phys. 50, 478; Raimondi, M., Simonetta, M. and Tantardini, G. F. (1972). J. Chem. Phys. 56, 5091.

401 [37] van Lenthe, J. H. and Balint-Kurti, G. G. (1980) Chem. Phys. Lett. 76, 138; Verbeck, J. and van Lenthe, J. H. (1991). J. Mol. Struct. (Theochem) 229, 115. [38] Lowdin, P.-O. (1955). Phys. Rev. 97,1474. [39] King, H. F., Stanton, R. E., Kim, H., Wyatt, R. E. and Parr, R. G. (1967). J. Chem. Phys. 47,1936. For more recent work see Figari, G. and Magnasco, V. (1985) Mol. Phys. 55, 319; Amovilli, C. (1997) In 'Quantum Systems in Chemistry and Physics', Proceedings of a European Workshop (ed. Maruani, J.). Kluwer, Dordrecht (in press). [40] Goddard, W. A. (1967). Phys. Rev. 157, 73; Phys. Rev. 157, 81; Goddard, W. A. (1968). J. Chem. Phys. 48, 450; J. Chem. Phys. 48, 5377. [41] Dunning, T. H. (1970). J. Chem. Phys. 53, 2823. [42] McWeeny, a. and Jorge, F. E. (1988). J. Mol.Struct. (Theochem) 169, 459. [43] Bendazzoli, G. (1997- private communication) [44] Sosa, C., Noga, J., Purvis, G. D. and Bartlett, R. J. (1988). Chem. Phys. Lett. 153, 139. [45] Wu, W. and McWeeny, R. (1995). J. Mol.Struct. (Theochem) 341,279. [46] Cooper, D, L., Gerratt, J. and Raimondi, M. (1986). Nature 323, 699. [47] Sironi, M., Cooper, D. L., Gerratt, J. and Raimondi, M. (1989). J. Chem. Soc., Chem. Commun., 1989, 675. [48] McWeeny, R. (1996). Chemical Physics 204, 463.




403


Advances in Many-Body Valence-Bond Theory D. J. Klein Texas A&M University-Galveston Galveston, Texas 77553-1675

The area of valence-bond theory as advocated by Linus Pauling has come to be more widely accepted as encompassing interesting novel many-body characteristics, and several many-body techniques have been apparently successfully developed to greatly enhance the theory's computational amenability.

1. INTRODUCTORY SURVEY The idea of valence structures goes back to classical chemistry. And indeed it was then at the heart of chemistry, though the theory was largely of a qualitative nature. There are both brief surveys [1,2] encompassing this history, as well as whole books on the subject [3]. With the advent of quantum mechanics Heitler and London [4] treated the valence bond in H2 in a highly suggestive manner, and shortly thereafter Rumer [5] and more especially Pauling [6,7] extended the theory so as to manifest the general chemical relevance. Ultimately this Valence-Bond (VB) theory as refined to a qualitative resonance-theoretic form linking closely to classical chemical-bonding ideas played a central role in Pauling's masterwork [8] The Nature of the Chemical Bond, which addressed essentially the full variety of chemical structures. But an alternative Molecular-Orbital (MO) approach eclipsed the VB approach, in so far as quantum-chemical research was concerned (say from the 1940s onward for a few decades). There were a variety of reasons for this, including perceived theoretical conceptual difficulties and supposed predictive failures - and perhaps most seriously the MO approach seemed much more computationally amenable. All this history is somewhat more fully discussed (from several points of view) in a couple review articles [2] as well as the first two chapters of [9] Valence-Bond Theory and Chemical Structure. These reviews describe through the period of this eclipse VBtheoretic work which was continued by a (prestigious or perhaps stubborn) minority of researchers (including Daudel, Hartmann, Simpson, Kotani, McWeeny, L6wdin, Matsen, McConnell, Coulson, Oosterhoff, Simonetta, Thorson, Goddard, Gallup, Messemer, & Gerratt though most of these researchers had much diversified other interests too). Indeed these noted reviews in [2] and [9] indicated a sort of renaissance of VB-theoretic interest in the 1980s, and included several points: * Significant progress was made on semiempirical theories especially as regards a

404

variety of applications [10,11]. * Incidentally many of the earlier criticisms of VB theory were themselves critiqued [12], and often found to be somewhat wanting -indeed occasionally even specious. * In the ab initio realm a number of accurate perfect-pairing computations for novel circumstances were performed [13], relations to classical pictures often were explicated [14,15], and yet further highly accurate ab initio computations including "resonance" were [16,17] made feasible and found to be highly accurate, perhaps most notably for benzene. * And finally resonating VB theory was proposed [18] as crucial in understanding hightemperature superconductivity, whereafter the ideas and solution techniques were vastly developed, in a many-body context. Here the focus is to be on many-body VB-theoretic techniques, which now have advanced so incredibly much since the mid-1980s, primarily because of work promoted in connection with high-temperature superconductivity. Of course there were earlier works in this general area, including some by Pauling [19], but such earlier works were fewer, were spread out over a longer period, and were pursued to less high accuracy. Such initial early work is here to be mentioned in context with each type of many-body technique described in later sections. Section 2 sets out some nomenclature and clarifies some oft overlooked points of crucial interest for VB theory. Section 3 identifies crucial aspects of many-body theory, which is here viewed to be such as to allow treatment of systems of sizes including infinite - and some hopefully clarifying discussion is given. Section 4 surveys each of several many-body techniques in the context of VB theory, identifying recent work, while also indicating some of the earlier work on each technique. To reasonably limit the focus here it the survey is primarily of many-body solution techniques as applied to a particular VB model, the covalent-space Pauling-Wheland VB model, represented by the Heisenberg spin Hamiltonian ~' = ,T_,~<jJ~j2si~ j where s k is the spin operator for electron (or site) k and J~jis an exchange parameter of non-ferromagnetic sign (i.e., here J~j>__0). Most simply the non-zero J~j are limited to nearest-neighbor i-j and these are all taken to be identical in value (J~j=J for i-j). There are other representations of this Hamiltonian, e.g., in terms of permutations (on spin indices, or on spin-free electron indices, or perhaps most fundamentally on orbital-site indices). This effective Hamiltonian serves as a relatively "simple" non-trivial example of an explicitly correlated model, serving as a test-bed for different general many-body schemes. But too there are some techniques seemingly specifically designed for this model, and these then (e.g., "classical" molecular field theory) are to be less emphasized here. Many of the methods are believed (and indeed often partially so tested) to be applicable beyond the "simple" Heisenberg model- but even within this limited focus (of the Heisenberg model) the recent literature is immense, so that what is mentioned here is selective and brief.

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2. VB THEORY: BASES, MODELS, & RESONANCE As a first step some of the nomenclature and framework should be clarified. A VB basis consists of configurations of spin-paired localized orbitals, at least for the overall spin singlet case, while for higher overall spins there are also additional unpaired spins. Typical VB spin-pairing (or Rumer) diagrams in correspondence with the basis elements for the case of six electrons in six orbitals (as for the six singlyoccupied atomic rT-orbitals of benzene) appear as

] ",,.

1.

[

"i"

and display an evident close analogy to the primary classical valence structures for benzene, especially if augmented with the rest of the spin-pairing diagram for the oorbitals. Rumer [5] focused on the singlet case with orbitals singly occupied, it being recognized that the extension to doubly occupied orbitals is trivial. Too Pauling [6] early pointed out the ready extension to the overall spin doublet case, and the approach taken hinted at further extension, e.g., as done in Simonetta's group [20]. But it is to be emphasized that there are other possibilities - anticipated to be most significant in cases when the VB structures as formalized by Rumer, Teller, & Weyl [21] do not correspond so closely to classical VB structures. That is, this Rumer basis is built in terms of orbitals around a cycle, and in many cases a chemical structure may have little to do with a single cycle. More important should be spin-pairing patterns involving pairing between nearer pairs of localized orbitals, independently of a relation to the formal basis of Ruiner et al [21]. Granted a VB basis, a semiempirical model represented on such a basis often is described as a VB model. Of course such models can be represented on any basis spanning the same space, but the semiempirical integral approximations are (usually) motivated from considerations in terms of the VB basis elements. One such case is the VB model of Pauling and Wheland expressed on the covalent space of configurations built from singly-occupied orbitals. As already noted the model so developed turns out to be essentially the Heisenberg spin Hamiltonian ~{, which often is expressed on a basis of what is essentially Slater determinants of atomic orbitals, and novel approaches involving bases of products of "spin-waves" are not uncommon in physics. Working with representations on such alternative bases can lead to wave-functions and results seemingly rather far divorced from classical-chemical connection and interpretation. A variety of VB models arise, with possibly the most natural hierarchy [22] indicated in fig. 1. The hierarchy of models occur in the column on the left and corresponding methods of solution are indicated on the right (though in some cases the methods have been little explored to date and are then identified with a "question mark") - the abbreviation CI refers to "configuration interaction". The Pauling-Wheland

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nonorthogonal-AO covalent + ionic VB model

complete CI cluster expansions

1 st ~estfiction

complete CI cluster expansions

primitive covalent VB model 1 st

orthogonalization complete CI Anstitze Neel state Green's functions cluster expansions etc.

Pauling-Wheland VB model 2nd

i

os

otion

Pauling-Wheland resonance model

-

y

"

y

complete CI resonance-theory Ansatz

2 nd orthogonalization

Hemdon-Simpson model

complete CI conjugated-circuits theory

3 rd iestrictio n

complete CI resonance Ansatz

nonorthogonal Clar-structure model 3 rd orthogonalization

Hemdon-Hosoya model

complete CI ,resonance Ansatz

Fig. 1 - Hekarchical scheme for VB models and their solutions.

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model in this figure is the already noted Heisenberg spin Hamiltonian ~- primary historical differences in physics and chemistry being the representation on different bases and application to different systems. The Pauling-Wheland resonance theory model entails a restriction to a subspace with a basis dominated by VB diagrams solely with neighbor spin-pairing (i.e., it is restricted to Kekule structures, which in general cannot all be included in what is usually termed the Rumer basis). In the column on the left of the figure a systematic derivational approach [22] is indicated, with successive steps entailing either: restrictions to ever smaller spaces; or orthogonalizations of natural (initially non-orthogonal) bases for these spaces. It should be noted that the restrictions to a subspace should for greater accuracy fold in the effects of the complementary subspace through higher orders of degenerate perturbation-theoretic considerations. The orthogonalization of the Pauling-Wheland resonance model gives [23,24] the "Herndon-Simpson" model, which also has been termed [24] the "harddimer" model. When supplemented with a suitable wave-function Ansatz this yields [25] that which is known as the "conjugated circuits" model [26], and which has been much studied. The top and bottom models in this sequence really have been rather less studied though. Perhaps too it could be noted that older criticisms concerning nonorthogonality or size-consistency "catastrophes" are connected with certain intermediate derivational steps or models, but such criticisms are often largely irrelevant or misleading, as discussed elsewhere [12,22]. The nomenclature resonating VB theory best should be reserved for treatments in terms of VB states, especially the more chemical corresponding to more local spinpairing patterns. And when such local spin-pairing is made manifest that should be of special note (at least in this review), particularly for the many-body case. Just VB theory (without the adjective resonating) is presumably more general, but engenders an even greater degree of ambiguity in current usage: often this phrase is interpreted as simply indicating the use of a spin-adapted many-electron basis built from spinprojected products of localized orbitals; or sometimes it is interpreted simply to be a method which treats a VB model; or sometimes it merely indicates a suitably locally formulated method of solution.

3. MANY-BODY THEORY It is appropriate too to clarify some nomenclature and aspects of many-body theory. Indeed here (and what seems often to be the usage elsewhere) "many-body" is applied to a theory if it is applicable in practice to infinite systems. That is, if a theory's computational "difficulty" scales with the number N of electrons as N a, it is to be a many-body theory only if a=0, at least for some standard sorts of systems, say with translational symmetry. Since it is often said that the number of 2-electron integrals f fXa'(1)'Xb'(2)'v(1,2)'Xc(3)'Xcj(4)'dT(2)dT(2) in molecular SCF theory is -N 4 (the number of atomic orbitals Xj being ~N), it is perhaps worthwhile to consider how one can view SCF theory as a "many-body" theory. First realizing that each Xi has an amplitude that falls exponentially fast away from its atomic

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center, one sees that the integrals become rapidly very small (and presumably negligible) unless centers a and c are close (whence X~'(1)'X~(1) need not be very small) while also b and d are close. Thence two factors of N are eliminated from the -~N4 integrals. Next rather than dealing with a bare Coulomb interaction, one realizes that the longer electron-electron interactions are going to be largely canceled by the corresponding electron-nucleus and nucleus-nucleus interactions. That is, with the usage of a consequent "dressed" interaction v(1,2) (depending also on a,b,c,d), the quantum mechanical corrections are larger only when the pair of centers a,c is not too far from the pair of centers b,d (it being understood that this is not to preclude an ionic substance for which a classical Coulombic part persists and is to be separately treated in a classical Madelung type of calculation). Thus the number of remaining nonnegligible integrals is -N (with a proportionality "constant" which is larger the greater the desired accuracy, and which best should be changed to assess the degree of accuracy). Next upon invocation of translational symmetry, sets of ~N orbitals become equal, and the last factor of N is eliminated. Finally beyond the integral evaluation there is a Fock-matrix diagonalization step, which for a general M• matrix has a cost -M 3 - but though there are -N orbitals translational symmetry again reduces it to ~N finite-size matrices of which only a finite number at representative wave-vectors are diagonalized (the various quantities of interest being continuous functions of wavevector). Thence the "many-body" character is manifest. As a further clarifying point of discussion it is of value to describe why a common computational technique, namely that of (bare) "configuration interaction", does not qualify as a "many-body" method (within the current definition). Basically the problem has to do with the number ~ of configurations needed for a size-extensive improvement (i.e., an improvement -N in total energy or other size-extensive quantity). As is wellknown the traditional approach to CI via matrix diagonalization has a computational cost -N 3, though if instead of seeking all the eigensolutions only one (say the ground state) is sought, then the cost may be reduced [27] to perhaps almost as little as ~N. Thus the whole question reduces to how an appropriate number of configurations N depends on the number N of electrons. Now wave-functions (like probabilities) are multiplicative in terms of independent circumstances such as excitations in distant parts of a large molecule, polymer, or solid. Thence allowing approximate independence of distant excitations to allow for physically plausible corrections there results an exponential dependence of N on N - which then means CI is very far from being a many-body technique, though it may be useful on sufficiently small systems. Indeed a related point, namely that excited configurations come to dominant in complete CI, was early realized in connection with VB theory (where "excited" configurations might be identified as "ionic" configurations), and this was suggested as some sort of fundamental conceptual problem in VB theory, though really the "same" argument applies to SCF theory. That is, it was not realized that the quality of a local description restricted in a local subspace depends not on the probability that all local regions be so confined, but rather on the probability that the local part of a configuration lie in the local subspace associated to the restricted description independently of the rest of the system. Rather surprisingly the resolution of such misunderstandings was made [28,29] in the mid 1950s in the context of MO theory, though the corresponding misunderstandings in the context of VB theory continued for some time. The well-known non-size-extensivity of

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CI limited to double-excitations above the SCF wavefunction turns out to have a VB analogue with VB diagrams limited to no more than one long bond beyond what is otherwise a Kekule structure- and such has been suggested a few times some decades back apparently without understanding the defect - though this seems to have had little widespread effect on the acceptance or rejection of VB theory. Anyway global CI on very large systems is not a (viable) many-body technique, nor is finitely excitationlimited CI, while however SCF-based (Moeller-Plesset) perturbation theory is wellknown [29] to be a many-body technique. It is to be emphasized that many-body theory is of interest for finite systems as well as infinite ones. First such techniques enable a greater range of finite systems to be treated. Moreover, with many-body techniques well in hand size-dependent errors in quantum-chemical computations should then too be well in hand. Perhaps a main perceived problem with VB theory was that from early on it was framed in terms of a CI approach amongst the various VB basis states. And the number N of relevant VB structures (beyond a single one for certain cases) can increase exponentially with N. A focus on covalent structures only, or more restrictively on Kekule structures only, just diminishes the exponent in the exponential dependence rather than changing the functional form of the dependence. Thus though the CI approach for finite (molecular) systems has been developed [30-.32] explicitly for the VB approach, it is really something more that is needed to make VB theory more generally viable -i.e., what is (and always has been) needed is many-body VB theory.

4. MANY-BODY TECHNIQUES FOR VB MODELS Now it is appropriate to discuss various possible many-body VB approaches, so many of which have been so actively developed in physics in the last decade. Of especial interest are the approaches when the resonating VB structure of the soughtafter state is evident in the computations. Primarily in application to the PaulingWheland VB model (or equivalently to the Heisenberg spin Hamiltonian, for the circumstance of antiferromagnetically signed exchange) the different approaches are considered separately in the following. 4.1. Configuration Interaction Though it has been emphasized that complete CI (for direct application to the large system limit) is not a "many-body" technique there is a simple way to view something somewhat the same as a many-body scheme. One simply takes the results for VB-model computations for a suitable sequence of computable finite systems and makes some intelligent extrapolation to the desired infinite system. Evidently it would be better nomenclature to describe such a many-body approach as extrapolated CI and it is an important approach, with the (seemingly exponentially) ever increasing computing power available to facilitate ever more nearly complete sequences from which to extrapolate. Indeed the extrapolation step can be sufficiently sophisticatedly developed that the extrapolation step is emphasized - e.g., as with some "cluster and moment" schemes and the "renormalization-group" schemes, to be separately considered later in sections 4.3 and 4.7. Often cyclic boundary conditions are used to

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make convergence more rapid, a standard ploy since Bonner & Fisher's extrapolations [33] (including extrapolations for temperature-dependent thermodynamic expectations) and it is perhaps worthwhile to mention a simply implementable idea which enhances convergence even further: adjust [34] the boundary conditions for (apparent) maximum rate of convergence. Thus CI is of much value, it incidently treating the interaction between the considered configurations perfectly correct for any directly treatable (isolated) molecular systems which can also be of much interest. As a consequence there are significant developments of various techniques for CI solutions to VB models, using either a Rumer basis [30] or a Young-Yamanouchi basis [31] or (what is in essence) a Slater-determinant basis [32]. Perhaps it should be noted that though the Rumer basis is non-orthogonal (and some researchers have thereby asserted this engenders great difficulty for extensive CI) Ramesesha & Soos [30] have demonstrated (in practice) notable amenability, when a complete Rumer basis is used. To enable treatment of larger systems another idea [35] selects through numerically determined "importance estimates" a subset of the full CI basis, perhaps extrapolating the results as a function of an importance parameter for the acceptance of configurations, or making some perturbative correction for the less significant configurations.

4.2. Many-Body Perturbation Theory A simple approach for the Heisenberg (or equivalently the Pauling-Wheland VB) model takes the corresponding Ising model as a zero-order description, with Neel-like spins up or down located in so far as possible on neighboring sites. Indeed Hartmann [36] early considered this approach, exhibiting the chemically nicely interpretable analytic results arising in second order, and Malrieu's group [37] has extended and similarly much used this method. Moreover, in the context of interest in high-T superconductivity this approach has further been pushed to higher order [38] to give highly accurate results for extended systems of interest. But [39] there are reservations characterizable in terms of the structural circumstances for which Neel-state or resonating-VB descriptions are more suitable, it being that such "zero-order" descriptions are qualitatively different: * the Neel-like description is best for high-coordination-number systems and little "frustration" (such circumstance being common for many of the 3-dimensional structures of more conventional solid-state physics); whereas * the resonating-VB description is best for low-coordination-number systems which admit many Kekule structures (this case being that of traditional benzenoid chemistry considered by Pauling and Wheland [7,40]). Indeed for the Neel-based theory to work best it is better to have a bipartite system (i.e., a system with two sets of sites all of either set having solely only members from the other set as neighbors). Of course, when there is a question about the adequacy of the zero-order description questions about the (practical) convergence of the perturbation series arises. But for favorable systems these [38] or closely related [41] expansions can now be made through high orders to obtain very accurate results. One can also imagine a perturbation expansion based on the resonating VB limit as zero-order. This has been considered [23,24] for some general circumstance. But for the covalent space for the n-networks of benzenoids the different Kekule structures would all be degenerate, and the degenerate perturbation theory can be neatly

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described to give rise to a new effective Hamiltonian on this subspace, thereby giving rise to the Pauling-Wheland and Herndon-Simpson models of figure 1. 4.3. Cluster and Moment Methods In these schemes complete computations are made for different fragments of the network and the results for these fragments are incorporated into a systematic extrapolation method. One energy cluster-expansion scheme [42] expresses a system energy for any fragment as a sum over contributions from all subfragments, with these different subfragment contributions obtained by ("MObius") inversion from all fragments up to a certain size. The energy for a large fragment then is approximated as a sum over subfragments truncated at a given size, and it may be shown that the result is accurate up to a corresponding order of a suitable many-body perturbation theory, and moreover quite accurate results are obtainable [43], if sufficiently large subfragments (say of up to 10 sites) are utilized. The idea extended to a degenerate perturbationtheoretic version of this provides a means for deriving models, such as VB models [37,44], as of figure 1. A scheme termed the "connected moments" method [45] makes use of what also may be identified as CI matrix elements for such different subfragments, and likewise yields quite accurate results [43,46]. In using these various schemes (as well as many-body perturbation theory) it is appropriate to follow the total system energy estimates as a function of order of approximation, so as to obtain an estimate of the remnant error. Indeed these methods have somewhat the flavor of "local" perturbation-theoretic methods (though with partial contributions from higher orders). Finally such methods apply for other expectations, and related techniques are [47] standard for the computation of statistical thermodynamic quantities. 4.4. Spin-Waves and Green's Functions Early on this type of approach was developed, and for a few decades appears to haw.= been viewed as the approach of choice, though the preference has become very much less clear, especially with the low-coordination-number (lower-dimensional, sometimes frustrated) systems of interest in the solid-state community during the last decade. Often there is some sort of transformation from the spin-operator representation to a second contest (usually bosonic) creation/annihilation-operator formalism, whence expression in wave-vector space and solution via a conventional looking SCF decoupling follows. Rather frequently however such transformations [48] append a non-physical space to achieve the ordinary creation/annihilation-operator formalism, with which there is nothing formally wrong so far as the representation is concerned. But typically in the (SCF-like) solution phase components of the adjunct non-physical space are mixed into the considered approximate "solutions". A recent type, of such treatment [49] is via so-called "slave bosons", and apparently the approach can be carried through to yield presumably rather accurate results. But often it seems [39] that such methods are Neel-state based and are best applicable to systems with higher coordination number with little "frustration". One rigorous transformation [50] of the spin-Hamiltonian representation to a fermionic creation/annihilation-operator formalism avoids such an auxiliary non-physical space, but is usually restricted to the particular case of a linear chain (or something fairly close to this) if a reasonable looking Hamiltonian is to be obtained (i.e., with limitation to 2- or 3- or 4-particle interactions).

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Also some other recent treatments [51] rather broadly seem to fall into the present general category.

4.5. Wave-Function Cluster Expansions There are at least three types of cluster expansions, perhaps the most conventional simply being based on an ordinary MO-based SCF solution, on a full space entailing both covalent and ionic structures. Though the wave-function has delocalized orbitals, the expansion is profitably made in a localized framework, at least if treating one of the VB models or one of the Hubbard/PPP models near the VB limitand really such is the point of the so-called Gutzwiller Ansatz [52]. The problem of matrix element evaluation for extended systems turns out to be somewhat challenging with many different ideas for their treatment [53], and a neat systematic approach is via Cizek's [54] coup/ed-c/ustertechnique, which now has been quite successfully used making use [55] of the localized representation for the excitations. Another type of scheme for the purely covalent models is based on a zero-order Neel-state description. Indeed two different such expansions were proposed a few decades back. The first by Hulthen [56] & Kasteleyen [57] (later rediscovered [58]) involves different multiplicatively cluster-expanded weightings for each determinant of atomic orbitals. The second by Vroelant & Daudel [59] (also later discovered several times [60]) is essentially a standard cluster expansion based on a 0-order determinant of atomic orbitals with different spins assigned to the alternant subsets of sites of a bipartite network. Each scheme is recognized to give rise to a type of statistical mechanical problem, with the wave-function overlap corresponding to a stat-mechanical partition function and the wave-function energy expectation corresponding to a statmechanical local correlation-functional derivative of the partition function. Indeed the Hulthen-Kasteleyn Ansatz gives rise [58,61] in the lowest non-trivial order to the ordinary Ising model, so that the problem is susceptible to exact (variational) treatment in 1- and 2-dimensions. For higher orders for the Ansatz of Vroelant & Daudel the coupled-cluster technique again seems quite powerful [62]. An as yet incompletely explored point is [61] that for the lowest non-trivial order of the Hulthen-Kasteleyn Ansatz as applied to the linear chain in a variational format rather directly gives rise to a whole set of orthonormal "excited" cluster-expanded states which may in turn be used as a new (explicitly correlated) basis for the problem. Also a cluster expansion based on the set of Kekule structures is possible [63], and indeed (in the nomenclature used here) evidently yields the first suggested manybody resonating VB solution scheme (earlier many-body approaches seeking solutions to VB models without much attention to the chemically appealing local spin pairing). This scheme in its lowest order with spin pairing constrained to nearest neighbors (i.e., Kekule structures) has now been rather widely studied [64], and extensions beyond nearest-neighbor spin pairing evidently can be made to a limited extent for modest improvement [65] or substantially further for quite high accuracy [66]. Further there are yet some other seemingly exotic wavefunction Ans~tze which might be classified as cluster expansions [67]. These involve phrases such as "flux phase", "spiral phase", and "commensurate flux phase".

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4.6. Monte-Carlo Computations This general type of approach in a few different versions has over the last decade become especially powerful, though in most versions it really is (like CI) for finite systems - it is just that significantly larger systems are treatable than via CI so that there is less ambiguity and somewhat higher accuracy in extrapolation to infinite systems is possible. For the case of a nearest-neighbor model ~ on a bipartite network there is a way [68] of assigning a phase to the spin-product basis so that the groundstate is "nodeless" (on this signed basis), and the Monte Carlo approach has a special effac, ious possibility for application, which now has been [68] much developed and successfully used. Another approach [58,66] avoids these (nearest-neighbor and bipartiteness) assumptions and replaces them by a high-order wave-function cluster expansion Ansatz whose relevant matrix elements are treated via a modification of the Metropolis-et-al algorithm [70] - the results achieved being capable of high accuracy. Finally also there are other working Monte Carlo schemes which are free of all the previous assumptions (of nearest-neighborness and bipartiteness and of wave-function Ansatz), such also achieving high accuracy [71].

4.7. Renormalization-Group Techniques Here too as formulated for the ground state there are different versions of this powerful technique, which for the present type of models seems best formulated in a real-space framework. The earliest version nicely reviewed [72] shortly after its initial introduction [73] renormalizes a local block (or group) of spins which mutually interact with one another so form to a spin doublet for a block which in turn is coupled via say degenerate perturbation-theoretic calculated interactions to its neighboring blocks - the blocks are identified as "new sites" for a renormalized model and the whole process iterated. This renormalizes a nearest-neighbor Heisenberg model to a new nearestneighbor Heisenberg model, and the iteration process can be analytically continued (toward infinity where the renormalized interblock coupling diminishes toward 0). The whole process now [74,75] has been much applied to 2-dimensional networks. The resullts for the simplest (first-order degenerate perturbation-theoretic) form [72,73,74] give variational upper bounds to the ground state but can be made somewhat more accurate [75] with the sacrifice of this bound. But for what seem typically to be even higher accuracy results there is the numerical "density-matrix" renormalization-group scheme [76]. This scheme keeps a notably more complex structure for each of the new sites obtained from a block, so that at each step the renormalized model really needs to be treated anew - thus one does not rigorously arrive at the infinite limit, but the number of original sites encompassed in the computation goes up exponentially with the number of renormalization steps, so that the technique is simply repeated till the persite changes at subsequent steps is driven down comparable to the numerical "noise" in various utilized computer algorithms. This scheme has been found [76,77] to be highly accurate for a variety of correlated 1-dimensional models. 4.8. Miscellany There are a few other solution schemes. There is [78] novel work paying close attention to sets of states incorporating explicit "resonating VB" character. Another simple approach (which many do not acknowledge as a many-body scheme) entails

4]4

Pauling's general idea [8] that just counting Kekule structures is relevant- perhaps even for semiquantitative work, so that there is a large body of chemical work making such enumerations, e.g., reviewed in [79]. Indeed with focus on the many-body case of extended metallic systems Pauling [80] has in recent years continued with this enumeratively based work. Finally mention should be made of the Bethe Anstaz wavefunctions [56,81], which solve ~ (with nearest-neighbor interactions and all such J~j=J) exactly for the linear chain - but this method seems very difficult to extend to higher dimensions or even to general linear chain systems (though investigations continue making more limited extensions- see, e.g., ref. [82]).

5. OVERVIEW AND PROSPECTS Evidently there are now a number of different types of working many-body techniques for the solution of VB models, and amongst these a couple are categorizable as resonating VB techniques. Rather interestingly for many-body computations (as developed so far for models) there appears little to differentiate in terms of computational difficulty between the use of the VB-based or (explicitly correlated) MO-based pictures. This conclusion of appearance contrasts with statements often made even during the last decade, though usually these are made in the context of ab initio work where the requisite computations are more challenging (most especially) in terms of the evaluation and manipulation of integrals. Such past oft proclaimed statement of "conventional wisdom" might say: "Though there may be conceptual advantages to VB theory its computational implementation is much more difficult." Here it is contrarily contended that once comparisons to simple SCF computations are foregone and instead comparisons are made in terms of an explicitly correlated sizeconsistent many-body MO-based theory, computational difficulties in the MO- and VBbased approaches seem more comparable (and perhaps even sometimes the "same" in some fundamental computational sense). Notably the same techniques such as CI, cluster expansion, Monte Carlo, and renormalization-group turn out often to be the methods of choice for either VB or MO pictures. Also it is interesting how Cl-like manipulations are for high accuracy so intimately involved in so many of the techniques: CI extrapolation, clusters and moments methods, wavefunction cluster expansion, and renormalization-group techniques. Interestingly yet further is how many of the techniques are so intimately related to stat-mechanical techniques for the evaluation of partition functions and consequent expectations: many-body perturbation theory, Green's function techniques, cluster and moment methods, wavefunction clusterexpansions, Monte-Carlo approaches, and renormalization-group methods. But with some further elaboration the present view contrasts with yet another occasionally proclaimed type of statement of "conventional wisdom": "In proceeding to ever higher-order terms in the MO- and VB-based theories convergence of the final results should occur (so that general MO-based computational effaciousness occurs)." Here again we take a contrary view. First the parenthetic statement is no logical direct consequence even if the non-parenthetic assertion was to be granted - whether or not

415

the, computational effaciousness turns out to be comparable likely depends on a detailed comparison of methodologies, and likely also depends on the particular systems treated. Particularly if the many-body limit is taken first, then computations via the, two approaches need not converge toward the same limit. E.g., the two zero-order wave-functions may fall into two different "universality" classes, say with different exclusionary types of long-range orders which are not quenched through extension to higher orders. Such ideas are implicit in a number of studies: * in Lieb, Schultz, & Mattis' [83] emphasis of the occurrence of (non-physical) longrange interactions in typical 0-order SCF (model) Hamiltonians; * irk Thorson & coworkers' [84] findings of qualitative distinctions in the second-order reduced density matrices for resonance-theoretic and SCF-type wave-functions; * in Anderson's proposal [85] of the possibility of a novel type of long-range order in a resonating VB description (of the Heisenberg model ground state on the triangular lattice); and * irk the occurrence of a long-range spin-pairing order [86] in resonance-theoretic (cluster-expanded) wave-functions, this ordering being seemingly complementary to Neel-ordering. That is, building up corrections (via the usual many-body techniques) often seems not to be able to shift a wave-function between universality classes associated to different long-range orders. But of course even for finite systems there also are questions of rates of convergence and ease of interpretation of results - and in this case it would seem that whatever preferences there might be would depend on the particular system under consideration. It seems that there is much promise in and for VB theory, even for many-body VB theory. There are very many techniques, results, and ideas now more-so in the physics literature and much of this should be applicable in a more chemical context. Only an outline of the extensive literature on techniques has been attempted here, and it seems likely that some of these techniques are susceptible to much further improvement. But also it should be mentioned that there are extensive recent theorematic results, e.g., as a sample of more than a hundred pages worth may be noted in [87], some of which may ultimately prove of chemical or materials-science interest. There are [88] other (somewhat differently flavored and more mathematically detailed) reviews of the area from the point of view of physics. Surely VB theory is an important area pioneered by Linus Pauling. Evidently this legacy is becoming more widely appreciated - and a great number of fundamental results and powerful computational methods have been developed within the last decade or so. Bringing chemical relevance out of this flood of work seems an exciting prospect. Presumably there soon will be many chemical and physical applications, such as I think would especially please Pauling. Acknowledgement is made to the Welch Foundation of Houston, Texas.

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423



Ab Initio Valence Bond Description of Diatomic Dications"

Harold Basch', Pinchas Aped', Shmaryahu Hoz" and Moshe Goldberg b "Department of Chemistry, Bar llan University, Ramat Gan 52100, Israel bResearch & Development Directorate, Ministry of Defense Hakiryah, Tel Aviv 61909, Israel ABSTRACT

The electronic structure description of the He22+, NF 2+ and

022+ diatomic

dications has

been explored using ab initio multi-structure valence bond self-consistent field theory in an extended atomic+polarization gaussian basis set. The ground state wave functions are expressed as linear combinations of covalent and ionic bonding forms. In the formally doubly-bonded NF 2+ and triply-bonded O22+, each of the covalent and ionic bonding forms has a multi-configuration expansion. In all three dications, the covalent set by itself has all the qualitative bonding and barrier features of the full ground state curve, superimposed on the coulomb repulsion between atomic monocations. Nonetheless, covalent-ionic mixing is substantial and required for quantitative accuracy. With these properties, the diatomic dications seem to be well described as an ordinary chemical bond plus electrostatic repulsion; the approach taken by Pauling.

Dedicated to Linus Pauling

424 1. Introduction

Diatomic dications are esoteric, high energy molecules which, in the best of cases, are diflficult to characterize experimentally. Already in 1933 Pauling(~) described the energy dissociation curve of the simple dication He22§ using a straightforward valence bond (VB) mixing of a covalent and two ionic configurations. The long range (internuclear distance R> 1.3A) part of the energy interaction curve was found to fit the point charge electrostatic repulsion interaction (--l/R) inherent to the He+:He § covalent configuration. The existence of a metastable energy minimum at short range (R~--0.75A) behind a potential barrier hindering dissociation was attributed to resonant covalent-ionic (He~

2§ and He2+H~ mixing. The detection of He22§ has been reported experimentallyr

although

no spectroscopic constants have been obtained. However, consistent with his general description of the electron pair (single) bond (4), Pauling considered the helium diatomic dication to be fully described as an ordinary chemical bond plus electrostatic repulsion. Hurleyr

adopted this picture of diatomic dications and predicted the energy interaction

curves and spectroscopic constants of a number of them, both in the ground and electronically excited states, by scaling the potential curves of the isoelectronic neutral species and adding the electrostatic repulsion. One source of fascination for the dications has been the suggestion to use He22§ for example, as a source of propulsion energyc~ This application is based on the combined thermodynamic instability of the equilibrium geometry, showing a large exothermic dissociation energy, and kinetic metastability due to a barrier to dissociation. That this unusually shaped energy dissociation curve, showing a barrier and exothermicity, can be described as a superposition of an ordinary chemical bond curve plus electrostatic repulsion has been emphasized by Jonathan, et al~s), and Senekowitsch and co-workers ~176 The latter workers also found, as had Hurley~6), that the scaling procedure using the isoelectronic neutral diatomics generally gives dication energy curves that do not agree quantitatively with experiment or the highest level ab initio curves. Interestingly enough, the addition of a mutual cation-cation polarization potential(l~ to the coulomb repulsion contribution brings the final curves into near-perfect agreement with the best available property values. An alternative model description of dication energy curves has been given by several groups r 17). Based directly on the VB method, this model attributes the origin of the barrier in the ground state energy curve of diatomic dications AB 2§ to an avoided crossing between a purely repulsive covalent

425 (A§ +) energy curve and an attractive (A2+B~ curve (with B more electronegative than A), where the attraction is due to the strong polarization interaction between A 2§ and B. This model of interacting non-orthogonal VB structures has been quantified within a semiempirical framework by Radom and co-workers (~4'15). The major criticism of the avoided crossing model at long range is that it seems to require an unphysically large coupling between asymptotically well-separated covalent and ionic structure curves to accurately locate the energy barrier to dissociation. However, this claim has been refuted ~

At short range, the avoided crossing model goes into less detail, but it seems to imply a

total lack of conventional covalent (A§ +) bonding. An attractive component in the shorter R region superimposed on the coulomb repulsive curve for the A§ § covalent interaction in diatomic dications within a VB framework has also been observed in ab initio calculations Os'~9~. The attractive component could be interpreted as an actual bonding interaction or as a polarization induced stabilization. Given the tendency to short equilibrium internuclear distances for dications, where the polarization interaction would be severely damped, the former mechanism is likely the more dominant. This returns us to the original Pauling VB dication model ~ which has a (weakly bound) covalent (A§ § configuration interacting strongly with the higher energy ionic A2+B~ configuration having a minimum in the region of the ground state energy minimum. This general situation has also been found in the VB description of a number of (single) bond energy dissociation curves in neutral species which involve at least one electronegative dissociating atom like fluorine ~2~

In such cases, the covalent-ionic resonance interaction determines

the binding energy. In other cases, the covalent curve itself can be strongly bound. Examples of both kinds of binding situations have been discussed in dications ~24"25). A covalent-ionic curve crossing is not unique to dications; it has also been found in the SiH3-F energy curve, for example, but without a resultant barrier~2~ In this paper we will explore the forms of the VB covalent and ionic curves for Hez2+, O22+, and NF 2§ to see how they fit the models of dication binding that have been proposed, and where they fit in the spectrum of parameters that are used to classify bonding systems according to these models. The method of choice for this analysis is ab mitio multi-structure VB self consistent field (MSVBSCF) theory using nonorthogonal orbitals (27"29). In this method each electronic configuration, defined as a particular assignment of electrons to VB orbitals, is expressed as a set of one or more VB structures which differ from each other in their spin coupling. The VB structures are written as linear

426 combinations of determinental functions of the spin-orbitals. The nonorthogonal orbitals, which are usually atom or fragment localized, are expanded in the familiar atom-centered gaussian basis sets. The orbitals are divided into active and passive sets. The active set of orbitals can have different occupancies in different configurations. The passive orbitals have a fixed occupancy of two electrons each in all the configurations. All the VB calculations were carried out using the TURTLE set of computer codes, obtained from Dr. J. H. van Lenthe of Utrecht University, Holland ~176 The results on He22+ have been partly reported previously~19), but those for 022+ and NF 2+ are completely new here. 2. He2 2+

As noted above, the helium diatomic dication has been well studied ~

The standard

three configuration VB wave function for the "singly-bonded" (He,-Heb) 2§ consists of a linear combination of configurations (1) to (3): (He.+:Heb+)

[Is.(I)lSb(2) + lSb(1)ls.(2)]

(1)

(He.2+)(Heb ~)

1s~'(1)lsb'(2)

(2)

(He,~ (I-Ieb2+)

1s,'( 1) 1s,'(2)

(3)

where only the unnormalized spatial part of (1) is shown. At the simplest theory level the same set of atom localized orbitals are used for both the covalent [(1)] and ionic or electron transfer [(2) and (3)] configurations, giving ls,'=ls, and lsb'=lsb. This will be called level I. A significant increase in accuracy can be achieved if the covalent and ionic structures have different optimized VB orbitals. This will be called level II, where l s,' and l sb' are allowed to VBSCF optimize as different

orbitals

than

ls,

and

1~

respectively,

to

suit

their

"ionic"

charge

distributiont~9~'29'3~'33'34). The tailored ionic orbitals have been called, "breathing orbitals "t29~). The rational for using tailored orbitals for different VB structures is that the spatial extent of an orbital in an ionic configuration can be substantially different than in a covalent configuration. The use of different orbitals in different configurations is equivalent to a limited configuration interaction calculation with an expanded space of orbitals used to construct more configurations. The nonorthogonal property of the VB orbitals allows the tailored orbital optimization as a natural part of the VBSCF process; something that would be much more difficult using orthogonal molecular orbitals (MO). The He atom ls,, lsb, ls,' and lsb' orbitals were expanded using the (10'6 p) gaussian primitive basis set optimized for the He(3P) state from Partridge's

427 compilation~

The atom basis set was contracted [61111/3111] and augmented by two single

primitive d-type functions (5 components) with exponents of 3.500 and 1.000 O9). Since the VB orbitals are atom localized they don't individually obey inversion symmetry; the homonuclear diatomic point group symmetry is obtained by having Is, and l Sb (and Is,' and l sd) symmetrically equivalent. Analogously, the ls designation really means o in local diatomic symmetry, which includes expansion in all the s-, p- and d-type basis functions of local o symmetry. The resulting energy dissociation curves for the level II ~E8+ electronic ground state, and separately calculated covalent (He+:He § configuration are shown in Figure 1. Here, as noted earlier, the energy curve for the He+:He § covalent configuration is not purely coulombic repulsive (---l/R), as has been assumed in the avoided crossing model~

but shows a 0.80 eV energy

minimum at short internuclear distances (tK---0.711A) which is the barrier to dissociation at R~--1.034A. The attractive part of the covalent configuration energy profile can be interpreted as a stabilizing charge-induced dipole interaction superimposed on the monotonically coulomb repulsive He+:He+ curve, as discussed above~176 Alternatively, and more likely, the minimum should be interpreted as mainly "normal" two center spin-coupled covalent bonding between the He § cations. The stability of such a bond would be enhanced by the calculated close approach of the two He § ions which would allow penetration attraction between the electrons on each ion and the other nucleus. The total level II calculated barrier energy (Eb) for the ground state curve, defined as the energy difference between the minimum (R~--0.726A) and barrier maximum (R~---1.115/lt) distances, is 1.00 eV. Thus the covalent curve by itself shows --80% of the barrier energy. The remaining 0.2 eV of the barrier energy comes from covalent-ionic mixing or resonance energy, even though at the equilibrium bond distance the ionic-covalent energy gap is --20 eV. In fact, the _

covalent-ionic resonance interaction contributes --0.82 eV to get from the purely covalent configuration energy to the level II ground state energy near the equilibrium He-He distance, and this gap decreases to --0.63 eV at -~Rb. Thus, covalent-ionic mixing in the He22§ dication is relatively large(~7'~s). The calculated exothermicity from the energy minimum to the monocation asymptotes is ---9.34 eV(19) for the level II ground state and --10.17 eV for He+:He § alone. In summary, He22§ has a strongly bound covalent configuration energy curve which shows all the

428 features of the full ground state energy curve. However, the total binding energy is determined essentially by covalent-ionic mixing. The results obtained here can also be used to improve the avoided crossing model, besides indicating the need for adding an effective polarization potential to the coulomb repulsive covalent curve. The large covalent-ionic mixing in the semiempirical avoided crossing model ~4'~5) has been calculated using the Wolfsberg-Helmoltz formula~ for the covalent-ionic resonance interaction

(I-Ii2); HI2 = KGl2[Hll + Hz2]

(4)

where Hl~ and 1-122 are the diagonal Hamiltonian matrix elements for the covalent and ionic configurations, respectively, and G~2 is the overlap integral between them. Equation (4) can be inverted to solve for values of K at each internuclear distance. Using the level II VBSCF calculated values ofH~, Hz2 and G~2, K is found to be a near constant, with an average value of --1.215, and slowly increasing in value from K=1.195 at R=0.65A to K=1.235 at R=I.50A. He22+ may not be the most representative example of a chemical system. An analogous analysis of the SiH3-F energy dissociation curve shows K--1 throughout the whole interfragment distance range (26).

3. 022+ The oxygen diatomic dication has been studied extensively, both experimentally and theoretically(5 6 37 57) The equilibrium internuclear distance (R~) has been measured at 1.073A(49), " "

"

.

which is probably the shortest O-O distance known(45). The best estimate for the dissociation energy (D~) to two ground state oxygen atom monocations is ---90 kcal/mole (45'47'5t'52'57), where the minus sign indicates exothermicity. The O-O distance at the barrier maximum (R~) is relatively sensitive to the theory level and has been estimated to be ~1.64A (52'57). On the other hand, the computed height of the barrier (Eb) seems to be stable to calculational improvements at --80 kcal/mole ~45'52), measured from R~ to P,q,. These bond distance and energy values delineate the energy dissociation curve of 022+ to its ground state atomic monocation asymptotes. 022+ is isoelectronic and isovalent with N2 and both have a formal triple bond. The 14 electrons in the dication are divided into a six electron active space and an eight electron passive space. The latter consist essentially of the doubly-occupied ls and 2s atomic orbitals on each center. In molecular orbital theory the electronic configuration of the active or triple bond part of

429 the wave function is written as, (~2~4, or more explicitly, (~z271;x2/1;y2 in cartesian representation, where each doubly-occupied o and n bonding MO is spin-paired. Here, Oz, nx and ny are appropriate symmetry combinations of the z, x and y basis functions on each atom center (O, and O0. The VB description of the active space starts from the atom localized orbitals and bonding is achieved through spin pairing on the interatomic level. The basic spatial configuration is: z, lzblx, lxbly~lybl

(5)

with each active space VB orbital singly-occupied. This configuration also has the correct ground state spatial symmetry. Multiplying configuration (5) is a six spin product function with three (z and three 13 spins. Twenty determinants can be constructed that distribute these six spin functions among the six orbitals to give Ms=0. Such a product function of six spins can belong to electronic states with total spin S=0,1,2 and 3. The ground electronic state of 022+ is a singlet spin state (S=0) and there are only five linearly independent combinations of the twenty Ms=0 determinants that belong to S=0. Therefore, configuration (5) and the appropriate spin functions give rise to five linearly independent VB structures, each with its own variationaUy determined coefficient in a multi-structural expansion of the ground state wave function. Among these five VB structures arising from configuration (5) there is only one structure that has simultaneous two center spin pairing between z, and zb, x, and xb, and y, and Yb, and can be considered to represent the pure covalent triple bond; (Z,:Zb)(X,:Xb)(y,:yb). This structure is listed as number 1 in Table 1. The other 4 structures belonging to configuration (5) also have elements of only one spin-paired interatomic covalent bond each, with the remaining coordinate pairs having parallel spins each. The particular spin eigenfunctions used in this study, as the default option in TURTLE (3~ are called Ruiner functions ~

A descriptive listing of the occupancies, configurations and structures is given in

Table 1. Additional configurations to (5) that belong to the ground state ~Eg symmetry and maintain an equal number of electrons on each center (a covalent distribution) can be constructed by not restricting the occupancy of each of the active set of VB orbitals to one electron, while still maintaining an equal number of cz and [3 spins. Another way of looking at this is to consider "promotions" of electrons starting from configuration (5) that create doubly-occupied orbitals among the active set, but maintain an equal number of electrons on each center. This latter restriction to covalent configurations requires that two electrons be "promoted" simultaneously,

430 for balance. If the additional constraint is imposed that, overall, no more than two electrons can occupy x, y or z orbitals each on both centers combined, then the set of configurations 2 to 7 in Table 1 is obtained. As is clear from Table 1, configurations 2 to 7 have only a single two center spin-paired covalent bond each and only one structure per configuration. Configurations 1 to 7 (structures 1 to 11) represent the covalent part of the electronic ground state of 022+. The ionic or charge transfer configurations are considered next. In these, two electrons are distributed among the three active orbitals on one center and four electrons are distributed among the three orbitals on the other center; again using three ct and three 13 spins, and with the restriction of not having more than two electrons in any given direction (x, y or z). The resultant configurations 8 to 13 for 02+0 ~ are also shown in Table 1. Configurations 8 to 10 have four unpaired spins, which give rise to two linearly independent S=0 structures each formed from six determinants. Structures 12 to 17 have two covalent type spin-paired bonds each. On the other hand, configurations 11 to 13 (structures 18 to 20) have no covalent bonds between the two centers. There is an equivalent set of opposite direction ionic 0002+ electron occupancies labeled configurations 14 to 19 (structures 21 to 29) which are also listed in Table 1. Finally, two electron ionic or charge transfer configurations, 03+0 - and O'O 3+, for the ground electronic state of O2z+ can also be constructed using the same set of spin and symmetry restrictions described above. These give rise to configurations 20 to 22 (structures 33 to 35) for the former, and configuration 23 to 25 (structures 33 to 35) in the latter case. As shown in Table 1, these configurations each show a single spin-paired bond between the centers. VBSCF calculations were carried out using the Huzinaga 9~5p oxygen atom basis set t59) contracted [5111/311 ]. Single sets of s- and p-type diffuse gaussians were added to the atom set on each center with exponents 0.0862 and 0.0637, respectfully, taken from an even-tempered extension of the respective valence functions. In addition, two d-type gaussian functions (5 components each) were used with each oxygen atom having exponents 1.050 and 0.3000. All the active and passive VB orbitals were atom localized, except as indicated below, and were expanded in the full set of basis functions appropriate to the local o or g diatomic symmetry of each VB orbital. As noted in the discussion on He22+, we can expect the oxygen orbitals for the covalent structures (1 to 11) to be different from those of the (singly) ionic structures (12 to 20 and 21 to

431 29); and certainly expect the orbitals of the two sets of ionic structures [02+00 and 0002+] to be different from each other. Here, the situation is more complicated than in the single bond case, like He22§ where there is only one configuration and corresponding structure of each bonding type; covalent and two ionic. Ideally, perhaps, each structure in Table 1 should have its own set of VB orbitals. However, that would be too difficult and the approach adopted here was to use a different complete (passive+active) set of VB orbitals for all the structures belonging to a given bonding type. Thus, altogether 30 VB orbitals are used; ten each for the covalent and the two (singly) ionic bonding types. For comparison purposes, multi-structure VBSCF calculations were also carried out using a common set of (ten) passive and active VB orbitals for all 35 structures in Table 1, including the doubly-ionic structures. This theory level is labeled level I. Level II uses a different set of orbitals for each different bonding type, as described above, but encompasses only the 29 structures representing the covalent and (singly) ionic bonding types in Table 1. The O22§ energy dissociation curve was generated pointwise as a function of O-O distance (R), and the extremum point (energy minimum and transition state) bond lengths and energies were obtained by quadratic or cubic fitting of the bracketing points. The resulting energy dissociation curves are shown in Figure 2. The calculated values of the equilibrium and barrier top distances, as well as the dissociation and barrier height energies, are tabulated in Table 2. These are also compared with experiment and previous results. Two more levels of calculation were carried out, but only for points in the immediate neighborhood of the energy minimum and maximum. Level III adds the two electron transfer configurations 20 to 25 (structures 30 to 35) to the level II list, where the doubly-ionic structures have the same set of VB orbitals as the corresponding direction singly-ionic structures. Thus in Level III, structures 12 to 20 and 30 to 32 have the same set of ten VB orbitals, while structures 21 to 29 and 33 to 35 share their own, different set of ten VB orbitals. The covalent structures (1 to 11), of course, also have their own set of passive+active VB orbitals. Level IV builds on level III by allowing delocalization mixing among the passive VB orbitals on different centers for all the bonding types. As noted above, the doubly-occupied passive orbitals represent essentially the l s and 2s atomic orbitals on each oxygen center and are not shown in Table 1. The delocalized mixing of these inner electrons is not expected to be highly significant, but was examined. In the level IV calculations the six active VB orbitals shown in Table 1 remain atom localized.

432 All four levels of calculation (I to IV) have exactly the same covalent structure asymptotic energy. Therefore, the progression from level I to level IV represents an increase in accuracy level of the calculations, as shown by the decreasing exothermicity (Table 2) measured from the respective equilibrium distances to the common O § + O § ground state dissociation limit. Due to the nonorthogonality of the VB orbitals and, consequently, of the VB structures, the weight of each structure i (Wi) is calculated from the VBSCF structure expansion coefficients (Ci) using the formula~6~

Wi = Ci ~ SijCj J

(6)

where S~j is the overlap integral between structures i and j. Figure 2 shows the 022+ ground state energy dissociation curve for the level I and level II calculations. As discussed above, the 27 structure representation of the VB wave function (Table 1) divides into covalent and ionic bonding types. The energy dissociation curve for the covalenttype structure set alone (structures 1 to 11 in Table 1) can be calculated as a separate covalent only (O+:O§ ground state wave function. Analogously, one set of ionic structures (12 to 20) can also be used to VBSCF a separate, purely ionic O2§ ~ curve. These are also shown in Figure 2. The O§ § curve is seen to have a shape which closely mimics the level I and level II curves. Thus, in Table 2, an energy minimum is found at R~=l.039A and a barrier at th,=1.466A, with a barrier height energy (Eb) of 77.5 kcal/mole for the O+:O§ bonding type. The P~ and Eb values are not so very different from the corresponding parameter values calculated at levels I and II. In general in Table 2, P~ and Rb are seen to increase with the level of calculation and the O§ § calculated values for these distances fall into this trend at the low accuracy end. The calculated exothermicity for O§ § is 154.3 kcal/mole, which is 39% greater than the level II value; the most appropriate for comparison. An especially interesting quantity is the barrier height energy (Eb) at Ih, relative to the energy minimum at P~, which is calculated to be 77.5 kcal/mole for O§ § compared to 70.6 to 71.9 kcal/mole for levels I to IV. Thus the O§ § curve alone has the qualitative and almost the quantitative features of the full covalent+ionic ground state energy curve. Figure 2 shows that the ionic group curve, O~+O~ is asymptotically fiat, going to the 0 2++O~ dissociation limit. The curve falls rapidly as R gets smaller and reaches a deep minimum at R-1.00A. The individual energy curve behaviors of the covalent and ionic structure groups,

433 combined with the increasing covalent-ionic resonance interaction as R decreases, leads to the calculated increases in R~ and R~, and decreases in Eb and Dc in going from the O+:O§ level to level II. As noted previously, R~ seems to be particularly sensitive to the level of calculation. The combined weight of the ionic group structures, as calculated from equation (6) for the level II wave function, is found to uniformly decrease with R. Thus, at R =l.05A the combined weight of the covalent structures at level II is 45%, while at R=I.60A it is 74%. Thus, although the covalent structure group curve by itself already shows approximately the qualitative features of the full structure set curve at level II, covalent-ionic mixing is strong and quantitative accuracy requires taking into account the interaction between the already strongly bonding covalent configuration set and the more strongly attractive ionic set. A comparison between the levels I to IV results in Table 2 for the different distance and energy parameters with experiment and previous high level results shows improving agreement with increasing level of calculation. Probably, the most significant defect in the VBSCF calculations reported here is in the basis set which lacks f-type functions. The importance of f-type basis functions in O 22+ to the quantitatively accurate calculation of energy properties has been demonstrated previously~45'52'57). 4. N F :+

The NF 2§ diatomic dication has been studied both experimentally,Fe• L o

"k

k

o

t

L/I L L

L 1

Ix/'x

2

3

Figure 5. Valence bond structures for high-spin [Fe(L)6] 2+, with two spin-paired t2g electrons of Fe 2+ not indicated. L = H20. lone-pair electron from each of two ligands into two bonding Fe-O MOs, (OFeO d2sp 3 + kn), as is indicated in the latter structure. Resonance between the 15 VB structures of type 2 provides a VB equivalence for the delocalized MO description of this high-spin hypoligated complex. In the Pauling theory of hypoligated complexes, outer sp3d 2 (d = 4dx2-y 2 and 4dz2) hybrid AOs are used to form six Fe-O electron-pair (~-bonds [27]. However the ((~*FeO')l((~*FeO") 1 --) (4d) 2 promotion energy is considerable, and therefore the Pauling description should represent an excited configuration for the complex. Formosinho and Arnaut [28] have also used three-electron bonds in their descriptions of the electronic structures of transition metal complexes in order to calculate bond-orders of transition states when these systems are involved in electron-transfer reactions.

5.3 F+-type colour centres In alkaline-earth oxides CaO and MgO, F-type defect centres (i.e. anion vacancies with trapped electrons) are responsible for many of the luminescence bands which have been observed [29]. The defects produced by elastic collisions act as traps for free electrons [29]. The trapping may be described as the donation of one or more electrons to a Mg 2+ site which is adjacent to the anion vacancy. The donation of one electron into the magnesium 3s AO, which overlaps with the doubly-occupied 2p(~ AO of an adjacent O 2-, enables an (Mg 9 O)- threeelectron bond to be formed via the (M9 + ~2-) (__) ( ~ g 6-) resonance. The odd electron occupies the antibonding MO (~*MgO -- k*3SMg - 2p(~o MO, and may be dolocalized throughout the crystal via the transfer of it from one antibonding (~*MgO MO into another. (The two electrons of each Mg2+O 2- component may be accommodated in the bonding MO (~*MgO - 3SMg + k2p(~o.) This description of the

458 bonding and electron delocalization may be compared with t h a t for the (H20)solvated electron [6], for which the odd electron occupies an antibonding (~*OH MO of H20-, to give VB structures of the type (HO 9 H)'. The odd electron maybe delocalized into vacant antibonding (~*OH MOs of adjacent H20 molecules [6,30]. 5.4 n - t y p e s e m i c o n d u c t o r s If silicon is doped with arsenic, an n-type semi-conductor is obtained. A

Si-Si electron-pair bond is replaced by an As-Si t h r e e - e l e c t r a bond (As 9 Si), oe

which is equivalent to resonance between the Lewis structures (As

9

Si) and

(As Si). The odd-electron of (As 9 Si) occupies the antibonding (~*AsSi MO. Under the influence of an electrical potential, electron conduction may be initiated via the transfer of the odd electron from the (~*AsSi MO into an overlapping antibonding ($*SiSi MO, as is indicated in the VB structures 3 and 4 of Figure 1 [30]. The odd electron is then transferred into a second (~*SiSi MO, to establish the conduction process. 5.5 C o n d u c t i o n in a l k a l i m e t a l s in the solid state # Pauling has described the process of electron conduction in solid state alkali metals in terms of electron transfer from one metallic orbital into another [31]. The metallic orbitals were a s s u m e d to be valence-shell p AOs. An alternative approach to the phenomenon replaces the p AOs with diatomic antibonding (~*s MOs [32] in the simplest treatment. Because of the existence of a small ionization potential for the alkali metal, the tetraatomic [((~s)l] [((~s)2((~*s)1] configuration with a one-electron bond and a three-electron bond should be energetically close to the [((~s)2] [((~s)2] configuration with two electron-pair bonds. Under the influence of an electrical potential, electron conduction may proceed via one-electron delocalizations, as indicated in structures 5-7 of Figure 1. 6. I N S T A B I L I T Y OF T H R E E - E L E C T R O N B O N D S

Hfickel MO theory with the AO overlap integral Sab included generates an instability for the (homonuclear) three-electron bond when Sab exceeds 0.3 [33], and heteronuclear three-electron bonds are also overlap destabilized under compression. The following speculation, which utilizes the overlap instability of three-electron bonds, may have relevance for the provision of a VB formulation for aspects of the mechanism for electron conduction in the high Tc superconductor YBa2Cu307. One formulation of an A type layer of the high Tc s u p e r c o n d u c t o r YBa2Cu307 involves a ...(CuO)(CuO)+(CuO)(CuO)+... arrangements of copper and oxygen ions [30]. Each (CuO) component involves a three-electron bond, which arises from the overlap of the singly-occupied 3dx2-y2 AO of Cu 2+ with a doubly-occupied 2p(~ AO of O 2-, to give a ((~CuO)2((~*CuO)1 configuration. Each #Although these metals are not paramagnetic, it is convenient to include this phenomenon here.

459 (CuO) + component involves a ((~CuO)2 electron-pair bond configuration. The electron-spins are indicated in structure 1 of Figure 6, and each (CuO)(CuO) + component involves a five-electron four-centre bonding unit, as is also present in an n-type semiconductor (see above) and the solvated electron as H502- [30]. Lattice vibrations of the type indicated in structure 2 involve a compression of the three-electron bonds and expansions of the electron-pair bonds. The resulting instability of the three-electron bond would lead to the transfer of electrons from the singly-occupied antibonding MOs of the (CuO) components into vacant antibonding MOs of the (CuO) + components, as is indicated in 2, to generate (g)2(~,)1 •

((~)2(G,)0

((~)2((~,)1

Cu ,~ 0

Cu • 0

O

•

Cuo

0

f

(O)2((~,)1

0

--->

0 ~

((~)

)0

Cu

0

((~)2(g,)0

0

Cu

+-

+-

0

-->

Figure 6. Valence bond structures, lattice vibrations and electron transfer for .. .(CuO)(CuO)+(CuO)(CuO)+...-'...(CuO) 4- (CuO)(CuO) 4- (CuO).. . .

460 s t r u c t u r e 3. This process leads to a speculation t h a t a m e c h a n i s m for superconductivity m i g h t be associated with this process if a unidirectional flow of electrons is able to be established, i.e. if the next vibrational mode is able to generate structure 4 rather t h a n structure 2. At any stage of a vibration, the electronic wavefunction may be expressed as qJ = tl/2 + ~ttl/3 [30]. Conduction via the solvated electron could also involve this type of mechanism [30]. Because the (CuO) components of s t r u c t u r e 1 are n o n - n e i g h b o u r s , superexchange via i n t e r a c t i o n of 1 w i t h t h e VB s t r u c t u r e for (CuO)+(CuO)(CuO)(CuO) + is needed to couple antiferromagnetically the spins of their antibonding electrons. In contrast the B layers of V S a 2 C u 3 0 7 m a y be formulated as (CuO)(CuO)(CuO)(CuO), and therefore the extent of overlap between neigbouring O*CuO MOs should be sufficient to provide the primary contribution to the antiferromagnetic coupling of their electrons spins. It should be stressed that the mechanism for high Tc superconductivity is usually considered to involve the formation of Cooper pairs, and the speculative conduction mechanism of Figure 6 has not given consideration to the formation of these pairs. 7. THE T H R E E - E L E C T R O N B O N D WITH F O U R OR MORE AOs

The equivalence t h a t exists between the MO and VB resonance descriptions of the three-electron bond is only exact when one AO per centre is used to accommodate the electrons. This has been discussed in some detail by Murrell and Ralston [34], and elaborated further in VB calculations for a variety of systems by Hiberty and coworkers [8(a),35]. However, when four or more AOs are used to accommodate the three electrons, the VB structure XII or XIII may still be associated with an orbital wavefunction, and an extended VB-MO equivalence may be developed [36]. We shall use H2" to demonstrate this result for the simplest case, which involves the use of four AOs to accommodate the electrons. The H-atom and H--ion AOs will be designated a and b, and a' and b' respectively. The resulting S = MS = 1/2 spin wavefunction for resonance between the Lewis VB structures (H H)- and (H H)- is given by eq.(5). W(VB) = W(I~

(H

I~l)- + W(I~I

H)-=

la'aa'f~ba[

+ [aab'ftb'a]

(5)

When an a' electron of the (a')2(b) 1 configuration for the VB structure ~!)- is delocalized into the AB bonding MO ~la'b = a' + kb, to generate the

three-electron bond VB structure (H 9 H)-, one S = M S = +1/2 spin wavefunction for the three electrons is given by eq.(6), WI(a'b~a'b)

= 2 1 a ' a b a ~ a ' b f t l - la'abft~ga'bal - la'ftba~tla'bal

= 21a'aba~ga,bf31-I(a'ab[~ + a'ftba)~ga,bal = 2{(a')l(b)l(s

(6) (7)

= 1, M s = 1 ) } { ( ~ a ' b ) l ( s = 112, M S = -1/2)} -{(a')l(b)l(s

= 1, M S = O ) } { ( ~ a ' b ) l ( s = 1/2, M S = +1/2)}

(8)

461 (9)

= 31a'abaa'ftl + 3kla'ababftl

which is equivalent to each of eqs.(7)-(9). For each of the two ( a ' ) l ( b ) 1 configurations of eq.(8), the electron spins are parallel (S = 1). Because eq.(6) involves an S - 1/2 spin-state, the spins of the two electrons of (a')l(b) 1 must be opposed to t h a t of the ~/ab electron. In eq.(9), the two Slater d e t e r m i n a n t s represent (H H)- and (H H)- VB structures with (a')2(b) 1 and (a')l(b) 2 configurations respectively. When a b' electron of the (a)l(b') 2 configuration for the VB structure (H H)- is delocalized into the bonding. MO ~llb'a - b' + ka, to generate the three-electron bond VB structure (H 9 H)-, the resulting S = M S = 1/2 spin wave-function may be expressed according to eq.(10), (10)

WI(b'a~1b'a) - 31b'aaab'ftl + 3klb'aaaaftl

The linear combination of eq.(11) ~ I = q[I(a'b~ga'b)

(11)

- q[I(b'a~b'a)

= 3{la'(~baa'~l-Ib'aaab'f,I + k(la'ababf, I-Ib'aa(Za[tl)}

(12)

= 3(la'ab(Z~a,b~l

(13)

+ laab'a~b,a~l )

=-3(l~a,b(~1*a,b(Z~lla,b~l

+ I~b,aa~*b,aa~b'a~l)/(1

+ kk*)

(14)

represents a 2E-u state. Eq.(14) provides the MO equivalent to a VB t r e a t m e n t for the three-electron bond when four AOs are used to accommodate the three electrons. The antibonding MOs ~/*a'b - k*a' - b and ~*b'a - k * b ' - a are orthogonal to the bonding MOs ~lla'b and ~ b ' a respectively. The S - M S - 1/2 configuration of eq.(15) Wii(a'b~ta'b)

= l a ' a b ~ l l a ' b a l - la'[3ba~la'b(Zl

(15)

= I(a'ab[ ~ - a'[tba)~a,bal

(16)

= { ( a ' ) l ( b ) l ( s = M S = O)}{Olla'b)l(s = 1/2, M S = +1/2)}

(17)

=

(18)

[a'abaa'f~l-kla'abab[~l

may also be constructed. When k = 1, it is orthogonal to the W I ( a ' b ~ a ' b ) of eq.(9), and involves opposed (S = 0) spins for the two electrons of the (a')l(b) 1 configuration in eq.(17). Similar types of properties apply to the W I I ( b ' a ~ b ' a ) of eq.(19) Wii(b'a~ll b 'a ) - I b ' ( Z a ~ b , a ( Z l -

Ib'~ac~b,aal

(19)

462

= Ib'CLaab'f~l- k l b ' ~ f ~ l It may then be deduced t h a t W I I ( a ' b W a ' b ) expressed according to eq.(21), 9 ii(a'b~ra,b)

(20) + WII(b'aWb'a)

may be

+ Wii(b'aagb,a ) (21)

= (l(I)a'bC~*a'bCtO*a'bftl + ](~b,aaO*b,aad~*b,a[~[)/(1 + kk*)

thereby generating a 2Z+g s t a t e . In eq.(21), d~a'b = k*a' + b , d~*a'b = a' - kb, (~b'a = k*b' + a and O*b,a = b' - ka are sets of orthogonal bonding and antibonding MOs. F u r t h e r development of this type of approach - in p a r t i c u l a r with double-zeta ( a ' ) l ( a " ) 1 and !b')l(b") 1 (and larger basis set) configurations for the H-anions of the (~1 H)- a n d (~1 H)- VB structures - is provided in ref. [36]. Bond polarity when A and B are non-equivalent atoms may also be introduced. In the remainder of this paper, we shall restrict our attention to the two AO formulations of the three-electron bond.

8. T H R E E - E L E C T R O N BONDS AND INCREASED-VAI~ENCE STRUCTURES FOR FOUR-EI~ECTRON THREE-CENTRE BONDING Three-electron bonds are usually associated with paramagnetic systems. However it is not generally appreciated t h a t three-electron bonds may be i n c o r p o r a t e d into the VB s t r u c t u r e s for d i a m a g n e t i c s y s t e m s . This incorporation involves the spin-pairing of the unpaired ~r*ab electron of the three-electron bond structure A 9 S with the unpaired electron of a second radical species Y when the orbitals for the two unpaired electrons overlap [46,37,38]. If the odd-electron of Y occupies the AO y , the singlet (S = 0) spin wavefunction for the four electrons is that of eq.(22), (22)

q~(Y-~'*ab) = ly[~abCtVab[~abCtl + lyaW*ab[]altabaVab~l

=-(1 + kk*){ly~aOta~bal + lyaa[~aab[~l + k(ly[~aab~bal =-(1 + kk*)0PxvII +

+ lyaa~bab[~l)} (23)

which is equivalent to eq.(23), where ~ X V I I a n d ~ X V I I I are the S = 0 spin wavefunctions for the Lewis structures XVII and XVIII.

463

9

O0

Y

A

\0

B

0 ~ 0

Y

XVII

A

O0

9

O0

9

B

Y

A

B

XVIII

XIX

In s t r u c t u r e XVII, a "long" or formal bond links the Y and B atoms. When these atoms are non-adjacent, the overlap between the y and b AOs is extremely small, and the Y-B bond then has negligible strength. Often workers do not indicate the presence of this formal bond. When this bond is omitted, VB structure XIX is obtained, which is designated as a singlet/spin-paired diradical s t r u c t u r e . Inclusion of the formal bond leads to the "long-bond"/formalbond/Dewar/designations for structure XVII. Whichever designation is used to describe s t r u c t u r e X V I I or XIX, this type of structure is usually omitted from qualitative VB descriptions of bonding, and Pauling rarely gave consideration to it. However the results of both semi-empirical [39] and ab initio VB calculations from a variety of laboratories [40-44] indicate t h a t often Lewis structures of this type m a y m a k e significant contributions to the ground-state resonance scheme. This is especially the case when the familiar "Kekul~" Lewis structures of type X V I I I involve formal charge separation, (as occurs in 1,3-dipolar molecules for example), and at least one of the Dewar structures does not. An MO three-centre bond index has been introduced by Giambiagi, de Giambiagi and coworkers [45]. These workers have calculated values of this index (IAc, with A and C as terminal atoms), for a variety of systems t h a t involve fourelectron three-centre bonding units, and have concluded that three-centre bonding indices have appreciable values if and only if there exists a "long" or secondary bond between a pair of non-adjacent atoms. Thus the (STO-6G) value of 0.5812 for the IAC of the 1,3-dipolar molecule N20 [45] is appreciable; this result is in accord with the existence of substantial contributions of "long-bond" (i.e. singlet-diradical) Lewis s t r u c t u r e s to the g r o u n d - s t a t e resonance scheme for this molecule [39(c),40]. Because the three-electron bond structure is equivalent to resonance between structures Xa and Xb, it follows from eq.(3) t h a t resonance between the Lewis structures XIX (or XVII) and XVIII is equivalent to use of the VB structure XX. In XX, a thin Y-A bond line is used to indicate t h a t the Y-A bond-number in this structure is fractional, i.e. its bond-number is less than the value of unity that obtains for the Kekul~ Lewis structure X V I I I [4-6,46]. (The fractionality is a consequence of the absence of a Y-A bond in structure XVII [4-6,46].) Valence bond s t r u c t u r e XX is an example of an increased-valence structure for a fourelectron three-centre bonding unit [4-6,37,38]. It may always be generated from the Lewis structure of type XVIII by delocalizing a non-bonding b electron into the bonding MO ~gab, as is indicated in XXI --~ XX, 9

---) XXI

y

9

.

AoB XX

t h e r e b y incorporating the three-electron bond into the VB s t r u c t u r e for a triatomic diamagnetic bonding unit. When the p a r a m e t e r k for ~lab is chosen variationally, the one-electron delocalization indicated in XXI always stabilizes the

464 increased-valence structure XX relative to the Kekul6 Lewis structure XVIII [46]. 9. I N C R E A S E D - V A L E N C E

STRUCTURES AND MULLIKEN-DONOR-

ACCEPTOR COMPLEXES

If Y and A : 13 are respectively an n-electron donor (D) and a sacrifical electron acceptor (A) as defined by Mulliken [47], t h e n the Mulliken-type wavefunction of eq.(24) (24)

WN = ~g(D,A) + k~g(D+-A-)

for the ground-state of the (D...A) complex is equivalent to the wavefunction W(f)'A) for the donor-acceptor complex, in which a one-electron bond is formed between the donor and the acceptor [48]. In eq.(24), the S = 0 spin wavefunctions Ig(D,A) and lg(D+-A ") are given by eqs.(25) and (26) respectively.

~g(D,A) = lyay~gabCZ~gab~l

(25)

~(D+=A-) = ly~g*aba~gabB~tabal + lya~g*ab~gaba~ab~l

(26)

For ~g(D,A), the associated VB structure is the "no (D,A) bond" structure XXII. The dative structure (D+-A ") is obtained from XXII via the delocalization of a Y electron of the donor into the vacant A-B antibonding MO Ig*ab - k * a - b of the acceptor. The delocalization proceeds according to XXIII --+ XXlV --~ XX, to generate the increased-valence structure XX for (D+-A').

OO Y A tB XXII

OO~ Y A~B---->~ XXIII

~,.g]

--->~

A.B

XXIV

When an electron is delocalized from the y orbital of the donor into the Y-A bonding MO ~ y a = Y + la, according to XXV --+ XXVI, 9

At B

---> Y " A : B

465 the VB structure XXVI is generated for the (D 9 spin wavefunction is given by eq.(27). ~(D-A)-

complex. The resulting S = 0 (27)

lyft~yaa~abft~abal + lya~yaft~aba~abftl

Valence-bond s t r u c t u r e XXVI is an example of an increased-valence s t r u c t u r e when LMOs are used to accommodate the three (fractional) bonding electrons [38]. Elsewhere, it has been deduced t h a t the A-atom valence is able to exceed unity in each of the VB structures XX [46] and XXVI [38]. Increasedvalence structures of the type XXVI will be used in the discussion of SN2 reactions which is provided in the next section. (To simplify the spin formulation in this section, we have restricted our a t t e n t i o n to the use of the bonding MO ~ a b = a + kb to accommodate the electrons of the A ; B electron-pair bond. Formulations with Coulson-Fischer [49] type MOs - for example, ~l'ab = a + k'b and ~"ba = b + k"a - are provided in refs. [30,38,50]. For the VB structures of Figure 7 below, it will be assumed t h a t the latter MOs are appropriate.)

10. I N C R E A S E D - V A L E N C E S T R U C T U R E S A N D SN2 R E A C T I O N S

9he NU ;

R

enera ize

aso is

acemoot roaction

+ "}~(-) is usually formulated according to Scheme A of Figure 7, in

which a pair of electrons is transferred in concert from the nucleophile NU(') to the s u b s t r a t e R ; X, and s i m u l t a n e o u s l y the pair of R-X bonding electrons is t r a n s f e r r e d in concert from the s u b s t r a t e to the leaving g r o u p ~ ( - ) . A n alternative formulation proceeds according to Scheme B of Figure 7 [30, 38,50], in which the initial step involves the delocalization of one electron from the nucleophile into a Nu-R bonding MO , to form the VB structure 5 for a reactantlike complex with a one-electron Nu-R bond. Electronic reorganization proceeds via one-electron delocalizations to generate VB s t r u c t u r e 6 for a product-like complex with a one-electron R-X bond. Decomposition of the product-like complex via a one-electron transfer generates NU ~ R + ")~(') as products. Consideration has been given elsewhere to the nature of the reactant-like and product-like complexes at the conclusion and commencement of the reaction respectively [30,38,50]]. It has been d e m o n s t r a t e d [38] t h a t s t r u c t u r e 5 generates s t r u c t u r e 8 at the conclusion of the reaction, and t h a t structure 6 generates s t r u c t u r e 9 at the commencement of the reaction, as is indicated in Figure 7. It has also been deduced [38] t h a t the initial step for the reaction m u s t involve a one-electron transfer from the nucleophile to the substrate, r a t h e r t h a n a concerted two-electron transfer. Structures 8 and 9 of Figure 7 involve a three-electron bond for either

(NuR)-or ( R X ) - . F o r

these species, the less-satisfactory s t r u c t u r e s I

and

9

9

(NuIR)Q

(R--" X)- have been used in ref.51 instead of (Nu 9 R)- and (R 9X)-.

In refs. [30,38,50], the state correlation diagrams with linear connections between structures 5 and 8, and 6 and 9, are schematic only, and the formation of reactant and product complexes as possible intermediates is not then indicated.

466

+ R

---> 9 [Nu ..... R ...... .X](-)

----> Nu ." R

+X(-)

3 Scheme A

"N'u(~')+ R ~ X

.% ~[(.u 9R

~)

~

(.u

,~~)]~-)

:.

5

Nu

6

.

R

+

X(')

d

7

Scheme B

~(~u. ~)~-) + 5

(.u .~

8

9 ~)/-/

~.u

+ (~

6

9

~)(-)

9

Figure 7. Valence bond representations for gas-phase SN2 reaction.

11. T H R E E - E L E C T R O N CENTRE BONDING

B O N D S AND F I V E - E L E C T R O N

THREE-

Five-electron three-centre bonding units involve the distribution of five electrons amongst three overlapping AOs that are located on three atomic centres. The n-electrons of triatomic systems with 19 valence-shell electrons provide examples of these types of bonding units when only valence-shell pn AOs are used to accommodate the electrons. Another example is provided by the indirect interactions of two nitrogen atoms via a hydrogen atom in medium-ring bicyclic compounds [23]. We consider here symmetrical systems, for which there are three canonical Lewis structures, XXVII-XXIX, XO

Y

XO

X

A

B

XXVII

XO

+--> Y

X

A XXVIII

OX

B

~

X

C~V

Y

'A XXIX

OX

B

467 in which the odd electron is located in either a y or an a or a b AO [52,53]. The AO wavefunctions for these structures are those of eq.(28) ~KVII

=

lyaaabO~y~a~l, ~ K V I I I

=

lyaaabO~y~b~l, WXIX = lyaaabaa[~b[~l (28)

and resonance between structures XXVII-XXIX is equivalent to the formation of the linear combination of eq.(29). = )~(WXXVII + WXXIX) + WXXVIII (29) in which ~, is a variational parameter. This resonance is also equivalent to resonance between the three-electron bond structures XXX and XXXI. x

x

ox

Y~

xo

B

x

x

x

AoB

XXX

,~

x

YoAoB

XXXI

XXXII

The wavefunctions for the latter structures are those of eq.(30).

WXXX -lyc~aaba(Xa + y/2)[tb[~l, WXXXI = [Yaaabayi~O~a + b/2)~l

(30)

In turn, ~ = WXXX + ~FXXXI is also equivalent to: (a) The canonical MO (CMO) configuration of eq.(31), ~CMO

(31)

- Kl~)l(~02a~)3(z~)1~)2[~l

in which the canonical MOs are given by eq.(32). Ol = ~

+ (y + b)/2,

02 = y - b ,

03 = ~ * a - (y + b)/2

(32)

(b) The localized MO (LMO) configuration of eq.(33): (33)

T L M O -lyc~aaba(y + ~ ) ~ ( b + ~)~1 with two non-independent three-electron bonds

(lyaaa(y + ka)~ .... I and

I.... a(Xb(~(b + )~a)~l), for which the associated VB structure is XXXII.

Thus five-electron three-centre bonding units have three different types of VB representations, the wavefunctions for which are equivalent [52,53]. As an example, we display them in Figure 8 for the five pi electrons of s y m C102, when it is assumed that these electrons only occupy pn AOs. Spin-pairing of the ~)3 odd electrons of two C102 molecules generates C1204, for which the a s y m isomer

468

O0

O0

O0

:~/~'x~.._~ .:~S~~~:..~/s'~ dxO

o,,;

x

1

2

O0

(+)

O0

X "

O0

"X 0

~d'. ,,.

~

6

5

(.+.)

..

-:o~c~'~..

.. ~~'~ ~' (-;iO~

,90

X

4

"

3

" X

~

(-1/2) 7

"x

x

"

8

Figure 8. Valence bond structures for C102, with no participation of chlorine 3d AOs as hybridization functions.

o~_o/.?.

{,o. 9 '~

.o7'

9

-~ 9 '--~ 1

2

Figure 9. Increased-valence structures for OC1OC102 [53].

469 OC1OC102 has been identified recently [54]. The resulting (equivalent) VB representations [53] involve either nine Lewis structures, or four increasedvalence structures, each of which involves two three-electron bonds, or one increased-valence structure with four three-electron bonds. The latter increasedvalence structure is displayed in Figure 9, together with an increased-valence structure in which the locations of the three-electron bonds have been interchanged [53].

12. THREE-ELECTRON BONDS AND INCREASED-VALENCE STRUCTURES FOR EXTENDED SIX-ELECTRON FOUR-CENTRE BONDING If two three-electron bond structures are juxtapositioned as in VB structure XXXII, so that their antibonding ~*ab and ~*cd MOs overlap, x

Ao

x

0

B

C •

0

9

A

XXXII

9

9B

,e

C

9

9D

XXXIII

the odd-electrons t h a t occupy these MOs may be spin-paired to generate the increased-valence structure XXXIII for a six-electron four-centre bonding unit with the S = 0 spin wavefunction of eq.(34) [55]. ~XXXIII-

[~aba~abft~cdCt~cdft~*aba~*cdft[ + Illtaballlabftlltcdalllcdftlll*cdOhll*abfti

= (1 + kk*)(1 +

k'k'*)(laa~ab[~Vcdad[tbacftl + IVabaaftda~cdftcabftl)

(34) (35)

This VB structure is equivalent to resonance between the Kekul~-type Lewis structure XXXIV and the Dewar-type Lewis structures XXXV-XXXVII, O0

0 ~ 0

A

B

C

O0

9

O0

D

A

B

XXXIV oo

9

O0

A

B

C

XXXVI

~0

O0

C

D

XXXV ~0

D

0~

A

O0

O0

B

C

~

9

D

XXXVII

and it m a y also be generated from structure XXXIV via the one-electron delocalizations that are indicated in structure XXXVIII [55].

470

9

o ~ O

=A*B

X1~glII

X]C~vIII

C~% . .b~.~ (+)

No

(-)

9

C*D

O%Nt, %(+0.5) (-0.s/:~

1

:o7.~"

3

,, o,x. /

:5-

4

/

:oy5

,0

~N 9

0

0

C+)

N,~>_)

N"

f:o:. 9

o~

+

"

7

(4

O N

:oX

N

x o:

Figure 10. Construction of increased-valence structures for NO2 and N204 from Lewis and three-electron bond structures.

471 Example of increased-valence structures, and the Lewis structures and three-electron bond structures from which they may be derived, are displayed in Figure 10 for NO2 and N204. These two molecules involve four-electron threecentre and five-electron three-centre bonding units. N204 also possesses six-electron four-centre bonding units that are components of a ten-electron six-centre bonding unit. Because of the inclusion of Dewar-type as well as the Kekul~-type s t r u c t u r e s in the Lewis structure resonance scheme, the increased-valence structures are more stable than are the familiar Kekul~-type Lewis structures from which they are derived, provided t h a t the one-electron bond polarity parameters, are chosen variationally. Therefore as discussed already in Section 8, a better (i.e. lower energy) VB description of the bonding may be obtained when increased-valence s t r u c t u r e s r a t h e r t h a n only the component Kekul~-type structures, are used to provide VB reoresentions of electronic structure. On several occasions [55a,c], it has been demonstrated that, when A and D, and B and C are pairs of equivalent atoms, with equivalent overlapping AOs involved in the six-electron four-centre bonding unit, the wavefunction for increased-valence s t r u c t u r e XXXIII is equivalent to the covalent (AB---CD) component of the four-centre canonical MO configuration of eq.(36), ~1 (MO) =

I~lC~l [~2cz~213~t3cz~t3131

(36)

in which the canonical MOs are given by eq.(37). ~1

=a+d+L(b+c) =s3+L~l

~ 2 = a - d + k ( b - c) = s4 + ks2

(37)

~3 = L*(a + d ) - (b + c) = L*s3 - Sl ~ 4 = k * ( a - d) - (b - c) = k ' s 4 -

s2

This r e s u l t is obtained as follows [55] via a series of u n i t a r y transformations of the ~/land ~1/3 canonical MOs of Ol(MO), to give eq.(40).

(38)

~ I ( M O ) = 1~1~113~2~213~3a~3131 =

](s3 + ~Sl)a(s3

+ ~Sl)13(s4 + ks2)~

+ ks2)~5(~,*s3 - Sl)a(~*s3 - Sl)131

= I~'1 [Sl~Slf~(s4 + ks2)~(s4 + ks2)~s3~s3 ~1 = K'I [(s3 + k s l ) a ( s 3 + ksl)~5(s4 + k s 2 ) a ( s 4 + k s 2 ) ~ ( k * s 3 - S l ) a ( k * s 3 - Sl)131 =I~ ~~(~ab+~1cd)~(~1ab+~cd)~(~1ab-~1cd)~(~ab-~1cd)~(~*ab+~*cd)~(~*ab+~1*cd)~

=K1 [~taba~llabfS~llcda~llcd~(~ll*ab+~l*cd)~

=Kl(l~ab~ab~cda~Ccd~*aba~*cd~l

+V*cd)13[

+ I~aba~ab~cda~cd~*cda~*ab~l

472 + IXllabCtXlIab~XtlcdaXltcdfJXll*aba~*abf3l + Ixtlaba~abfJXtlcdCt~cd~Xll*cdCt~*cd~l)

(39)

= Kl(Wcov + Wion)

(40)

in which the LMOs of eq.(41) ~ a b - a + kb, ~ll*ab = k * a - b, ~llcd = d + kc, "~l*cd = k * d -

c

(41)

are two-centre A-B and C-D bonding and antibonding MOs. The qJcov of eq.(40) corresponds to the sum of the first two determinants of eq.(39), and is equivalent to the wavefunction for increased-valence structure XXXIII. The MO configuration of lowest energy that will interact with ~l(MO) is the t~2(MO) of eq.(42) O2(MO) = I~l/1a~l/113~2a~213~4a~4131

(42)

in which two electrons of ~1 (MO) have been excited from ~3 into ~4. To simplify the presentation here, ~ is set equal to k. Unitary transformations of the ~2 and ~4 MOs of ~2(MO) then generates eq.(43). O2(MO) -

I~fflr

=IC 2 1 ( ~ a b + ~ l c d ) a ( ~ a b + ~ c d ) f J ( ~ a b - ~ c d ) a ( ~ a b - ~ c d ) f J ( l l l * a b - ~ * c d ) a ( ~ * a b - ~ l t * c d ) ~ l = K21~abCt~ltab[JXllcdOt~llcd~(~*ab

- ~*cd)Ct(~ll*ab - Ilt*cd)~l

= K2(qJion - Wcov)

(43)

Configuration interaction (CI), via eq.(44), gives eq.(45), 9 ( C I ) - CIOI(MO) + C202(M0)

(44)

= (C1K1 - C2K2)Wcov + (C1K1 + C2K2)Wion

(45)

and with C2 < 0 when C1 > 0 for the lower energy linear combination [55], the importance of Wcov is increased relative to qJion. Therefore the dominant contributor to the (S = 0 spin) MOCI wavefunction for a symmetrical six-electron four-centre bonding unit involves two three-electron bond configurations with their odd-electrons spin-paired. Additional excited configurations that generate induction and dispersion interactions contribute to eq.(45) when ~, ~ k. These are examined in refs.[55(a)],

473 but they are not needed here. For an analysis of the MOCI wavefunction of a nonsymmetrical six-electron four-centre bonding unit, see ref. [56]. 13. T H R E E - E L E C T R O N BONDS AND INCREASED-VALENCE S T R U C T U R E S FOR CYCLIC SIX-ELECTRON FOUR-CENTRE B O N D I N G

In the increased-valence structures of Figure 9 for OC1OC102, there are cyclic six-electron four-centre bonding units, and consideration will now be given to the three-electron bond theory for them when these bonding units have either D4h or D2h symmetry. Examples of these systems [57] are provided by (a) the It electrons of S42+; (b) (i) the rt electrons of S2N2 and (ii) six g electrons that are involved in the intermolecular bonding between the monomers of the OC1OC102, I42+, $6N42+ and [(C2F5)2Se2122+ dimers. It is convenient to define the four-centre MOs according to eq.(46), Igl = a + ~ ~2 = a

+ c + ~d

- c , ~3 = b

~4 = n*a

- b

= ~tab + ~ c d

= ~ad

+ ~lbc

- d

+ n* c - d

(46) = ~*ab

+ I]l*cd = 1]l*ad + ~ l * b c

These MOs are canonical MOs for $2N2 and the central C1OC10 moiety of OC1OC102, when K is chosen variationally. For the six-electron four-centre components of the remaining species, ~: = K:* = 1 in the canonical MOs ~1 and /]/4, and/]/2 - / g 3 replace/]/2 and I]/3 as canonical MOs.

A~

I

A.--~

i

DoC

D~

XXXVIiI

XXXlX

Because ~/4 is antibonding with respect to each pair of adjacent atoms, the lowest-energy configuration is given by eq.(47) Ol(MO) ~ I ~ g l a ~ l ~ / 2 a ~ g 2 ~ / 3 ~ 3 ~ l -

](~gl)2(~2)2(~/3)21

(47)

By a series of unitary transformations of the occupied MOs, Ol(MO) of eq.(47) may be transformed to give to eq.(48) O l ( M O ) - / C 1 ( 1 ( ~ 1 ) 2 ( ~ 2 + K~F3)2(~*~F2- ~3)21 + ](~l)2(K*/g2 + ~3)2(~/2 -

~g3)21)/2

474 = K' 1(l(~ab + ~cd)20]lab - ~cd)2(~*ab - ~*cd)2[

+ I(~ad + ~/bc)2(lltad - ~lbc)2(~*ad - ~*bc)2l)/2 = K1 ([(~ab)2(~cd)2(~*ab- Ilt*cd)2l + I(~ad)2(~bc)2(~*ad_ ~*bc)21]2 - K1 (-q~cov + qJion- W'cov + ~F'ion)/2

(48)

in which the ~Fcov, Wion, ~F'cov and W'ion are defined according to eq.(49).

qJcov-

[~abagtab~cdCt~cd~*aba~*cd~l+l~abagtab~cda~cd~*cda~*ab~l

qJion -

Illtaballtab~lltcdalllcd~lll*ab allt*ab~ l+ IVabaVab~VcdaVcd~V*cdaV*cd~ l

W'eov =

IgtadaWad~bc aWbc~*adagt *bc~ l+ I~ada~ad~bc a~bc~* bcagt*ad~ l(4 9 )

W' i on -

I~adaYad~YbcaYbc~Y* aday *~ l+ lyadaYad~YbcaYbc~Y* bcaY *bc~ l

The qJcov and W'cov, each of which involves two three-electron bond configurations, are the wavefunctions for the increased-valence structures XXXVlII and XXXIX. Configuration interaction via eq.(50), W(MOCI) = CI~I(MO) + C2q~2(MO) + C3q~2(MO)

(5O)

with ~2(MO) and ~3(MO) defined according to eqs.(51) and (52) 9 2(MO) = I(yl)2(~2 + Kalt3)2(~4)21

(51)

= (l(~ab + ~cd)2(~ab -~cd)2(~*ab + ~*cd)21 = K2(qJcov + qJion ) r

1(~1)2(~2- K:~3)2(~4)21

(52)

= I(gtad + II/bc)2(ll/ad - Iltbc)2(llt*ad -~*bc)2[ = K3(W'cov + W'ion ) The W(MOCI) of eq.(50) is then equivalent to eq.(53) qJ(MOCI) = {(-C1K1 + C2K2)~Fcov + (C1K1 + C2K2)qJion + (-C1K1 + C3K3)qJ'cov + (C1K1 + C3K3)W'ion)}]2

(53)

475 for which C2 < 0 and C3 < 0 when C1 > 0. Thus according to eq.(53), the MOCI wavefunction of eq.(50) increases the importance of Wcov and W'cov relative to Wion and W'ion. As does eq.(45), eq.(53) indicates that spin-pairing of the unpaired electrons of two three-electron bond structures is the primary process that occurs when their orbitals overlap.

14. T H R E E - E L E C T R O N RESONANCE

BONDS

AND

COVALENT-IONIC

There are two types of covalent-ionic resonance. These involve electronpair bonds(A B A- B +e->A + B- = A ' B A ~ and threeelectron bonds (A B ~-> A" B + = A " B) respectively[58]. The origins of the rotation barrier for N204 and the antiferromagnetism of Cu(II) carboxylate dimers provide well-studied examples of the latter [58-60]. For each case, a sixemmm~

9

O0

ee

9

R

o

o

c

Figure 11. Relevant AOs for rotation barrier and antiferromagnetism studies of N204 and Cu(II) carboxylate dimers.

--'"x

Z

"

"

.,

eO

1

o**No.

2

**N 3

5o

0o/

4

~

Figure 12. Relevant ONNO components of primary Lewis-type VB structures needed for a VB rationalization of the origin of the rotation barrier for the D2h isomer of N204.

476 electron four-centre bonding unit is present [6,55]. The relevant AOs are displayed in Figure 11. It has been calculated that resonance [58,59] between the covalent (NO2-NO2) and the ionic (NO2+NO2 - and NO2-NO2 +) structures of the types 1-4 of Figure 12, are primarily responsible for the planarity of the D2h isomer of N204.This type of resonance establishes stabilizing O-O three-electron bond 9

Gig

oo

interactions of the type ( 0 0 O" 0 + - 0 9 O) in the planar conformer via a non-zero value of the 0 - 0 overlap integral for the cis oxygen AOs. Because the value of this overlap integral is zero in the perpendicular conformer, no cis 0 - 0 overlap stabilization of this conformer occurs. For the Cu(II) carboxylate dimers, resonance between the covalent

Ol

-"

o0

Oo

|

o0

|

S S

~Cu

Cuo"

Cu-

1

Oo,, Cul

2

lO

"-.

".Cu

|

Oo9

~ oCu

(~)

tCu 7

oo 9 (~

Cut

~ Q 6

"o

Cu"

oCu

ot

8

ol

~i|

4

to

Cut

oO

Cu:

3

oo 9

:Cu

to |

9 oCu"

tCu 8

Figure 13. Primary Lewis-type VB structures required for a VB rationalization of the origin of the antiferromagnetism of Cu2(RCO2)4,Ln.

477 (CuO--- CuO) and ionic (CuO+CuO" and CuO-CuO +) VB structures (Figure 13) of the types 1-4 for the S = 0 spin state, and 5-8 for the S = 1 spin state both generate an O-O three-electron bond. However whereas the covalent structures 1, 3, 5 and 7 are almost degenerate, the energies for the S = 1 spin ionic structures 6 and 8 lie well above those for the S = 0 spin ionic structures 2 and 4. As a consequence, the stabilization energy that arises from covalent-ionic resonance is less for the S = 1 spin state than it is for the S = 0 spin state, and therefore an antiferromagnetic alignment of electron spins occurs in the ground-state [60]. Another example of covalent-ionic resonance is provided by VB studies of N-N dimers of HNO [61]. It has been calculated t h a t the barrier to rotation around an N-N bond involves a substantial contribution from three-electron bond ..

covalent-ionic resonance of the type

-'-

(--> ~4+---N-), as well as from

electron-pair bond covalent-ionic resonance ( ~ 4 ~ N ~ N = - ' - N + ~ N+----N-).'" The results of recent MO studies are considered to be in accord with this VB study [62]. For HCONH2, the development of some N-C three-electron bonding ( N - - C ~-> N+---C -) in the planar conformer has been calculated to make a contribution to the rotation barrier of this molecule [63]. 15. C O N C L U S I O N S The concept of the three-electron bond was introduced in Pauling's 1931 papers [1]. Although Pauling did not have the three-electron bond in mind, he considered [64] t h a t sometimes he was most proud of his first 1931 paper [1], which "changed the nature of chemistry in a significant way" [64]. As has been demonstrated here, the incorporation of three-electron bonds into mainstream VB theory provides a significant change in the m a n n e r in which qualitative VB descriptions of electron-rich molecules are formulated. The above review of a selection of aspects of three-electron bond theory, with particular attention given to increased-valence structures, provides a further demonstration of the aphorism that the "three-electron bond probes the ultimate limit of valence for electron-rich molecules" [65]. REFERENCES 1. L. Pauling, J. Am. Chem. Soc. 53 (1931) 1367, 3225. 2. L. Pauling, The Nature of the Chemical Bond, 3rd Edition, Cornell University Press (1960) Chapter 10. 3. (a) T. Kiang and R.N. Zare, J. Am. Chem. Soc. 102 (1980) 4024; (b) A.R. Gregory and V. Malatesta, J. Org. Chem. 45 (1980) 122 ; (c) P.M.W. Gill and L. Radom, J. Am. Chem. Soc. 110 (1988) 4931. 4. R.D. Harcourt, Qualitative Valence-Bond Descriptions of Electron-Rich Molecules: Pauling "3-Electron Bonds" and "Increased-Valence" Theory; Lecture Notes in Chemistry, Volume 30, (Springer, Berlin, 1982). 5. R.D. Harcourt, Chem. & Eng. N e w s , 56, October 3, p.5 (1988). 6. R.D. Harcourt, in Valence Bond Theory and Chemical Structure, (Eds. D.J. Klein and N. Trinajsti~, Elsevier, Amsterdam, 1990) p. 251. See also ref. 7 below for some other accounts of this approach.

478 7(a) A. van der Putten, A. Elzing, W. Visscher, E. Barendrecht and R.D. Harcourt, J. Mol. Struct.(Theochem) 180 (1988) 309. (b) S.J. Formosinho, in Theoretical and Computational Models for Organic Chemistry, (Edts. S.J. Formosinho, I.G. Csizmadia and L.G. Arnaut, Kluwer, Dordrecht, 1991) p. 159. (c) T.M. KlapStke and A. Schulz, Quantenmechanische Methoden in der Hauptgruppenchemie, Spektrum 1996, p. 200. 8.(a) P.C. Hiberty, S. Humbel and P. Archirel, J. Phys. Chem. 98 (1994) 11697 ; (b) P.C. Hiberty, S. Humbel, D. Danovich and S. Shaik, J. Am. Chem. Soc. 117 (1995) 9003 . For some other recent references, see also (c). D.K. Maity, H. Mohan, S. Chattopadhyay, and J.P. Mittal, J. Phys. Chem. 99 (1995) 12195 and refs. 17 and 22 therein. (d) Y. Deng, A.J. Illies, M.A. James, M.L. McKee and M. Peschke, J. Am. Chem. Soc. 117 (1995) 420, and refs. 1-32 therein. (e). M.A. James, M.L. McKee and A.J. Illies, J. Am. Chem. Soc. 118 (1996) 7376. 9. L. Pauling, Chem. Revs. 5 (1928) 173. 10. K. Ruedenberg, Revs. Mod. Phys. 34 (1962) 326. 11. R.D. Harcourt, Am. J. Phys. 56 (1988) 660. 12. (a) M.J. Feinberg, K. Ruedenberg and E.L. Mehler, in Advances in Quantum Chemistry, (Ed. P.-O. LSwdin, Academic Press, New York) 5 (1970) 27. (b) M.J. Feinberg and K. Ruedenberg, J. Chem. Phys. 54 (1971) 1495; (c)K. Ruedenberg, in Localization and Delocalization in Quantum Chemistry, Volume 1, (Eds. O. Chalvet et al., Reidel, Dordrecht Holland, 1975) p. 223. 13. M.J. Feinberg and K. Ruedenberg, J. Chem. Phys. 55 (1971) 5804. 14. W.O. Kermak and W. Robinson, J. Chem. Soc. (1923) 432. 15.E.B.R. Prideaux, Chem. and Ind. 42 (1923) 672. See also R.D. Harcourt, J. Chem. Soc. Faraday Trans. 88 (1992) 1119 for VB alternatives to double and triple bonds that involve one-electron bonds. 16. J.W. Linnett, (a) J. Amer. Chem. Soc. 83 (1961) 2643. (b) The Electronic Structures of Molecules (Methuen,London,1964). (c) Sci. Prog.(Oxford) 60 (1972) 1. 17. R.D. Harcourt and D. Jordan, Specul. Sci. Tech. 3 (1980) 77, 612. 18. (a) R.A. Firestone, Tetrahedron, 33 (1977) 3009. (b) R.F. Langer, J.E. Trenholm and J.S. Wasson, Canad. J. Chem. 58 (1980) 760. (c) W.B. Jensen, Canad. J. Chem. 59 (1981) 807. (d) G. Leroy in Advances in Quantum Chemistry, 17 (1985) 1. (e) B.J. Duke, J. Mol. Struct.(Theochem) 152 (1987) 319. (f) R.D. Harcourt and R.D. Little, J. Am. Chem. Soc. 106 (1984) 41 and references therein 19. See for example: (a) J.W. Linnett and coworkers as referenced in ref. 19(b). (b) R.D. Harcourt and A.G. Harcourt, J. Chem. Soc.Faraday Trans. II, 70 (1974) 743. (c) D.M. Hirst, J.Chem. Soc. Faraday Trans. II, 73 (1977) 422. (d) C. Amovilli, R.D. Harcourt and R. McWeeny, Chem. Phys. Letts. 187 (1991) 494. 20. J. A. Pople, Quart. Revs. 11 (1957) 273. 21. (a) J.W. Linnett, J. Chem. Soc. (1956) 275. (b) Canad. J. Chem. 36 (1958) 54. (c) M. Green and J.W. Linnett, J. Chem. Soc. (1960) 4959. 22. R.D. Harcourt, J. Chem. Ed. 62 (1985) 99. 23. R. Alder, Tetrahedron, 46 (1990) 682. 24. S.A. Chaudhri, H. Mohan, E. Anklam and K.-D. Asmus, J. Chem. Soc. Perkin 2 (1996) 383 and refs. 1-36 therein. 25. T. Clark, J. Am. Chem. Soc. 110 (1988) 1672. 26. (a) R.D. Harcourt and G.R. Scollary, Inorg. Nucl. Chem. Letts. 11 (1975) 821. (b). R.D. Harcourt, ref.4, Chapter 5. 27. L. Pauling, ref. 2, Chapter 5. 28. S.J. Formosinho and L.G. Arnaut, J. Photochem. Photobiol. A: 82 (1994) 11.

479 29. K.J. Caulfield and R. Cooper, J. Am. Ceramics Soc. 78 (1995) 1054. 30. R.D. Harcourt, J. Mol. Struct. (Theochem) 229 (1991) 39. 31. L. Pauling, (a) ref.2, p. 400; (b) J. Solid State Chem. 54 (1984) 297.(c) L. Pauling and Z. S. Herman in in Valence Bond Theory and Chemical Structure, (Eds. D.J. Klein and N. Trinajstic~, Elsevier, Amsterdam, 1990) p. 569. 32. R.D. Harcourt, J. Phys. B, 7 (1974) L41. 33. (a) N.C. Baird, J. Chem. Ed. 54 (1977) 291. (b) N.C. Baird, Pure Appl. Chem. 49 (1977) 223. Applications of three-electron bond theory to N-H bond dissociation energies are described in refs. (a) and (b). (c) R.D. Harcourt, Aust. J. Chem. 31 (1978) 199. (d) R.D. Harcourt, ref. 4, Chapter 3 and refs. therein. 34. J.R. Murrell and J. Ralston, J. Chem. Soc.Faraday Trans. II, 70 (1974) 2004. 35. C. Byrman and J.H. van Lenthe, Int. J. Quantum Chem., 58 (1996) 351. 36. R.D. Harcourt, J. Phys. Chem. A 101 (1997) 2496. Replace (a) (1 + kk*) with (1 + kk*) -1 in eq.(1), and (b) "used 7'' with "used 18'' in the 2nd column. Insert "H2like" prior to "internuclear separations" in line 14 of the first paragraph on p.2500. 37. R.D. Harcourt, (a) Biopolymers 11 (1972) 1551. (b) J. Mol. Struct. 12, 1 (1972) 1, (corrig. 13 (1973) 585). 38. R.D. Harcourt, Int. J. Quantum. Chem. 60 (1996) 553. 39. R.D. Harcourt, (a) Theor. Chim. Acta 6 (1966) 131. (b) Int. J. Quantum Chem. 4 (1970) 173. (c) R.D. Harcourt and J.F. Sillitoe, Aust. J. Chem. 27 (1974) 691. 40. R.D. Harcourt and N. Hall, J. Mol. Struct. (Theochem) 342 (1995) (corrig. 59, 369 (1996) 217), and refs. 15, 31, 36-39, 61, and 66-70 therein.. 41. R.M. Parrondo, P. Karafiloglou, R.R. Papparlado and E. S~inchez, J. Phys. Chem. 99 (1995) 6461. 42. W.B. Floriano, S.R. Blaszkowski and M.A.C. Nascimento, J. Mol. Struct. (Theochem) 335 (1995) 51. 43. R.G.A.R. Maclagan, Aust. J. Chem. 41 (1988) 527. 44. P.C. Hiberty in Valence Bond Theory and Chemical Structure, (Eds. D.J. Klein and N. Trinajstic, Elsevier 1990) p. 221. 45. M.S. de Giambiagi, M. Giambiagi and F.E. Jorge, Z. Naturforsch. A, 39 (1994) 1259 and refs. therein. 46. R.D. Harcourt, J. Mol. Struct. (Theochem) 300 (1993) 245. 47. R.S. Mulliken, J. Chim. Phys. 61 (1964) 20. 48. R.D. Harcourt, Aust. J. Chem. 28 (1975) 881. 49. C.A. Coulson and I. Fischer, Phil. Mag. 40 (1949) 396. 50. R.D. Harcourt, (a) J. Mol. Struct. (Theochem) 253 (1992) 363. (b) New J. Chem. 16 (1992) 667. (c) J. Mol. Struct. (Theochem) (1997) in press. See also ref. 30, and R.D. Harcourt and R. Ng, J. Phys. Chem. 97 (1993) 12210, (corrig. 98 (1994) 3226) for application to the exchange reaction X~ RY --->XR + Y: 51. (a) S. Shaik and A.D. Pross, Accts. Chem. Res. 16 (1983) 343. (b) S. Shaik and A.D. Reddy, J. Chem. Soc. Faraday Trans. 90 (1994) 1631. (c). S. Shaik and P.C. Hiberty in Advances in Quantum Chemistry, 27 (1995) 99. 52 .R.D. Harcourt, J. Chem. Soc. Faraday Trans. 87 (1991) 1089. 53. R.D. Harcourt, J. Phys. Chem. 97 (1993) 1351. 54. A. Rehr and M. Jansen, Inorg. Chem. 31 (1992) 4740. 55. R.D. Harcourt, (a) J. Am. Chem. Soc. 102 (1980) 5195, (corrig. 103 (1981) 5623). (b) J. Phys.Chem. 95 (1991) 6916. (c) Croat.Chem.Acta, 64 (1991) 399. 56. R.D. Harcourt, T.M. KlapStke and P.S. White, Inorg. Chim. Acta (1997) in press. 58. R.D. Harcourt, Chem. Phys. Letts. 218 (1994) 175 and refs. 18 and 20 therein.. 59. R.D. Harcourt and F.L. Skrezenek, J. Phys. Chem. 96 (1990) 1351.

480

60. (a) R.D. Harcourt, F.L. Skrezenek and R.G.A.R. Maclagan, J. Am. Chem. Soc. 108 (1986) 5403. (b) R.D. Harcourt, ref. 4. 61. R.D. Harcourt, F.L. Skrezenek and B.G. Gowenlock, J. Mol. Struct. (Theochem) 284 (1993) 87. See ref. 58(b) for some corrections. 62. R. Glaser, R.K. Murmann and C.L. Barnes, J. Org. Chem. 61 (1996) 1047. 63. R.H. Flegg and R.D. Harcourt, J. Mol. Struct. (Theochem) 164 (1988) 87. 64. G.G. Kauffman and L.M. Kauffman, J. Chem. Ed. 73 (1996) 29. 65. F. Williams, Chem. Eng. News, 57 (1989) J a n u a r y 9, p.2. 66. M.L. McKee, J. Am. Chem. Soc. 117 (1995) 1629.

ADDENDUM:

VB REPRESENTION

F O R 2 N O + 0 2 ---) 2 N O 2

On the basis of an ab initio MO study of the potential energy surface for N204, McKee [66] has provided computational evidence for the existence of a new N204 isomer (ONOONO) with 1A symmetry in the C2 point group. In Figure 14, a VB representation is provided for the formation and decomposition of this isomer via the m e c h a n i s m NO + NO + 0 2 - - > O N . . . O 2 . . . N O ( t s ) ---> O N O O N O -~ ONO .... ONO(ts) --->2NO2 for the gas-phase oxidation of NO [66]. The formation of the ON...O2...NO occurs primarily via the spin-pairing of the antibonding n*x and n*y electrons of ground-state 02 with the antibonding n* electron of each groundstate NO. The three-electron bond VB structures of Figure 4 are used to represent the r e a c t a n t s . The electronic reorganization leads to the formation of the increased-valence structures of type 5 of Figure 10 to represent the NO2 products.

o " O -~~ N o "

:O~N" 0

"

oO

,-,o ~,.,jo

-')

. O m~ m m"N "

9

0

O0

~,~.% ,,~"

_ o , - , ~ -9 %,-,o o%J ~o

_Q-~o,-,___.~,o o%i ~..lo---.~

"-

- -

/\ .N~Oo 9

oo

~

oN~Oo

9

.,$.

oo

;O~N

9 (,-oN

~

O0 9

TM

/

"Q O ~ N "

/ gO

oO ~

+

o0"

~

"0

O O

oNtO

t O0

0 ~

k

.N---O.~ O0

Figure 14. VB representation for gas-phase oxidation of NO to NO2. Convenient 2-dimensional VB representations are used here. The bond-lengths implied by these VB structures are mainly in qualitative accord with those reported in ref. 66.

Z.B. Maksi6 and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modern Modelling of the Chemical Bond

Theoretical and Computational Chemistry,Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

481

Valence bond description of ~r-electron systems J. Paldus a* and X. Li aDepartment of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada We overview our valence bond (VB) approach to the 1r-electron Pariser-Parr-Pople (PPP) model Hami!tonians referred to as the PPP-VB method. It is based on the concept of overlap enhanced atomic orbitals (OEAOs) that characterizes modern ab initio VB methods and employs the techniques afforded by the Clifford algebra unitary group approach (CAUGA) to carry out actual computations. We present a sample of previous results, as well as some new ones, to illustrate the ability of the PPP-VB method to provide a highly correlated description of the ,r-electron PPP model systems, while relying on conceptually very simple wave functions that involve only a few covalent structures. 1. I N T R O D U C T I O N The pioneering paper by Heitler and London [1], providing the first genuine explanation of "the nature of the chemical bond" based on "new" quantum theory, as well as their subsequent work explaining basic valency rules [2] and the crucial role played by the symmetry of the wave function as implied by Pauli's exclusion principle [3], are rightfully considered as the birth of the new field of quantum chemistry. This work decidedly pointed the way to a proper physical understanding of chemical binding, unmistakably indicating that chemical bonds are fundamentally electric in nature and relegating the proponents of the "chemical force" to obscurity, and thus stimulated a fervent search for theoretical methods that would enable at least a qualitative handling of molecular systems with more than one or two electrons. Only a few months after the publication of Heitler and London's paper, these results were reported by Van Vleck at the 1928 St. Louis meeting of the American Chemical Society (cf. [4]). Concluding his splendid expos6 on the "New Quantum Mechanics", he asks: "Is it too optimistic to hazard the opinion that this is perhaps the beginnings of a science of 'mathematical chemistry' in which chemical heats of reaction are calculated by quantum mechanics just as are the spectroscopic frequencies of the physicist?", and confidently reckons: "Of course the mathematics will be laborious and involved, and the results always successive approximations. The theoretical computer of molecular energy levels must have a technique comparable with that of a mathematical astronomer. The quantum mechanics is still very young, and surely it will ultimately be applied further than the hydrogen molecule" [5]. This prophecy was amply fulfilled already a decade *Also st: Department of Chemistry and Guelph-Waterl0o Center for Graduate Work in Chemistry, Waterloo Campus, University of Waterloo, Waterloo, Ontario N2L 3(31, Canada

482 later, as witnessed by the very first textbook bearing the title of "Quantum Chemistry" [6], even though its full development had to await the modern computer era. One of the young scientists at the time who was very clearly aware of these developments and of their potential, and who became one of the key players by greatly contributing to the rapid expansion and success of this new field, was our 'hero of honor', Linus Pauling. In the same volume of Chemical Reviews where Van Vleck's beautiful review appears, we also find a superb review by Pauling [7] of the work on H2 and Hu + by Burrau [8], Heitler, London [1] and others, containing several original developments of his own. It was only a short step from here that led Pauling to apply these ideas to a multitude of molecular systems and to co-develop a general approach usually referred to today as the valence bond (VB) or Heitler-London-Pauling-Slater (H-L-P-S method). Although the appropriateness of the label H-L-P-S was questioned [9], and the important contributions by other scientists, notably those by Kimball and Eyring [10], Rumer [11], Weyl [12], Wheland [13] and others are undeniable, Pauling's work certainly stands out for its thoroughness, generality and both its systematic and exhaustive pursuit of the subject. In his classical series on "the nature of the chemical bond" that first appeared in the Journal of American Chemical Society [14] and later in the newly started Journal of Chemical Physics [15], he quickly progressed from one- and three-electron bonds to the concept of resonance in conjugated systems, while at the same time producing (with E.B. Wilson) an excellent textbook on quantum mechanics [16] that introduced generations of chemists and physicists to the subject, and apparently holds the record for continuous publication without modification by its publisher [9]. A very important ingredient in those developments was Pauling's concept of hybridization, originally referred to as "change in quantization" or "8-p quantization breaking through bond formation" [14,17]. Similar ideas, though in a more limited form [18], were later presented by $1ater [19], who referred to hybrids as "concentrated bond functions". Almost simultaneously with the VB approach (or, in fact, what we would now call the AO based minimum basis set full configuration interaction method, as formulated by $1ater [20]), the molecular orbital (MO) method was developed by Hund [21], LennardJones [22], Herzberg [23], Mulliken [24], and others (see [25] for an early review). Initially, however, this theory merely provided labels for various electronic states of diatomics (via united and separated atom parentages) that were required in the analysis of their spectra, and was not regarded as suitable for quantitative or semi-quantitative calculations of binding energies, particularly when the MOs were represented as linear combinations of AOs (LCAO approximation) [25]. At the same time, the relationship and the relative advantages of VB and MO theories were becoming well understood, as were the possible avenues for their improvement (cf., e.g. [26]), but in view of the drastic simplifications that were made when evaluating the relevant integrals and the necessary computational limitations of the time, neither method was regarded as "any too good" [25]. At a later date, however, it was the MO approach that carried the day. The main reason for its success was not only its conceptual simplicity, but also the mathematical convenience of the orbital orthogonality that drastically simplified any quantitative computations. Only relatively recently, and thanks to our better understanding of suitable AO-like basis sets [30], has the VB approach, with its appeal to a chemist's intuition and its direct link with Lewis' electron pairing ideas started to enjoy a definite renaissance

483 (see, e.g. [27-30]). Present day computing technology enables one to carry out very accurate calculations of various molecular properties, at least for relatively small systems. Nonetheless, one invariably runs into limitations when trying to extend the available methodology and software to larger and larger molecules. In this respect we have to keep in mind that already simple qualitative and semi-quantitative considerations, pioneered by Pauling and his contemporaries, generated a wealth of information and provided understanding of numerous chemical phenomena. This was particularly the case for organic molecules with conjugated double bonds, whose many interesting properties derive from their ,r-electronic structure. These developments culminated by the introduction of the Pariser-Parr-Pople (PPP) and related model Hamiltonians [31], which continue to play an important role to this day, particularly for large and extended systems, and which we consider next. 2. P P P - T Y P E

HAMILTONIANS

Despite the tremendous progress in quantum chemical methodology and especially in computer technology, alluded to in the Introduction, the complexities of molecular electronic structure pose a continual challenge to theoretical chemists. Even though we can presently perform very accurate and reliable calculations to obtain numerous properties of smaller molecules, the exigencies posed by larger systems involving heavier elements beyond the first two rows of the periodic table, especially the transition metals, as well as various organic or biologically important molecules, are likely to remain with us for some time. Some authors even questioned the appropriateness of current ab initio approaches for very large systems of biological significance on epistemological grounds [32]. It must be emphasized that in fact a very similar situation characterizes experimental investigations, since very different experimental techniques are employed, and different questions are asked, when investigating simple diatomics on the one hand and complex systems, such as chlorophyl, proteins, fullerenes, etc. on the other. Even if one could resolve and measure, for instance, all the individual to-vibrational lines in the electronic spectra of large systems involving, say, more than 50 carbon atoms, the usefulness of such information is debatable. Likewise, our theoretical inquiries call for different models, approximations and techniques when handling different types of problems. Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ab initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). In the ab initio case, the relevant finite-dimensional one-electron space is spanned by a minimum, double-zeta, double-zeta plus polarization, etc., basis set and further approxi-

484 mations are then invoked at the N-electron level. In contrast, semiempirical Hamiltonians involve free parameters representing various matrix elements of certain parts of the ab initio Hamiltonian in an unspecified, hypothetical minimum basis set, whose value is determined either through calibration by experiment or by employing additional simplifying assumptions when their number is large. These models can thus be regarded as interpolation or extrapolation schemes based on the general form of the simplest possible quantum mechanical description. One of the most successful semiempirical Hamiltonians that also served as a prototype for many other models is undoubtedly the PPP model of planar ,r-electron systems with conjugated double bonds [31]. Originally designed as a generalization of the tight-binding Hiickel Hamiltonian that would account more realistically for the Coulomb interelectronic repulsion, it eventually evolved into a very useful tool enabling the prediction and rationalization of molecular geometries, electronic, NMR, and ESR spectra, chemical reactivity and other properties of chemical interest. It also served for numerous theoretical developments, even though on a smaller scale than simpler models as represented by the Hubbard, Ising, Heisenberg, and various spin-Hamiltonians, whose simplicity often enables a thorough mathematical analysis without any recourse to numerical computations. These simple models can even provide an understanding of certain puzzling features of the more complex PPP model [34,35], and the similarity in the structure of corresponding wave functions can be very striking [36], even though they may be inadequate for quantitative predictions concerning the excited state manifold. Thus, in contrast to the Hubbard Hamiltonian, the PPP model can provide a realistic description of low lying excited states of planar conjugated systems [31,37]. Moreover, in view of the tight-binding character of its one-electron part, the PPP Hamiltonian is often invariant under topological transformations defining the so-called alternancy symmetry [38]. This approximate symmetry property is absent from any ab initio Hamiltonian, yet provides a good quantum number, and thus effective selection rules, that are particularly useful in multiphoton, circular dichroic and other spectroscopies [39-42]. The key premise of the PPP ,r-electron models, later heuristically extended to allelectron models (CNDO, MNDO, etc.), is that of the zero-differential overlap (ZDO) [43], which drastically reduces the number of required two-electron integrals, leaving only those of the Coulomb type "/m. = (Pvle2/r12]Pv) 9 In contrast to the Hubbard model which considers only on-site interactions, 7.~ = 7..6m, the PPP model accounts for the long-range nature of the Coulomb force. To avoid a multitude of adjustable parameters, one chooses a suitable approximation scheme for these integrals. Approximating the on site ~,. integral via the so-called I-A approximation [44], following earlier developments by Goeppert-Meyer and Sldar [45], one evaluates the remaining Coulomb integrals -y~,~ using a suitably modified Coulomb Law. Presently, the most often used approximation is that of Mataga and Nishimoto (MN) [46], modifying the simple point charge Coulomb interaction so that finite on-site interactions 7 ~ = I s , - A., given by the difference of the valence state ionization potential (I.) and electron affinity (A v), are reproduced when both sites coalesce, i.e.

7m(Rm)

=

e'l(a. +

=

2e2/(7..

with R m, designating the intersite separation.

(1) Thus, only the nearest neighbor one-

485 electron resonance integrals flv~ are left as free parameters. For hydrocarbons one usually employs the spectroscopic value flw = ]5 = -2.4 eV. When the C-C separation significantly differs from the standard equilibrium bond length do = 1.4 ~, as when exploring various distorted structures, we employ the so-called "Mulliken magic formula" for the resonance integrals fl(R); (cf. Eq. (6) of [47]). We can thus conveniently express the standard PPP Hamiltonian in a modified (particle number preserving) second quantized form H. =

+ l~,b'

1

-

(2)

I$,V

where E.,, designate the orbital unitary group generators [33] (or the so-called replacement operators [48]) associated with the hypothetical 2p. carbon AO basis IX~,). Furthermore, z~,v = av -

~

2~7v~;

zv~ =

{fl~,~ 0

if ~u,v are nearest neighbors otherwise, '

(3)

where ct~, is a so-called (one-electron) Coulomb integral, usually approximated by the corresponding valence state ionization potential I~,, and Z~, is the number of 1r-electrons contributed by the/~-th atomic site. For hydrocarbons (Z, = 1, a , = a) we can define the energy scale so that ct~, - a - 0, 1 /J,v

the first sum extending over nearest neighbors only [33]. We also recall that the second quantization realization of U(n) generators has the form E,,. =

xLx

. ,

(5)

o"

where the creation (annihilation) operator

X~(X~,,,)

is associated with the spinorbital

=

As already mentioned, this simple model is capable of describing, among other things, the main features of visible and near-UV electronic spectra for a multitude of 1r-electron systems. The usual implementation of this model relies on the MO formalism, obtaining first the self-consistent field (SCF) MOs, followed by a very limited singly-excited CI (SCI), at least when considering the low lying, electric dipole allowed transitions (the so-called a, p and fl bands [37]). It was later realized, particularly when information concerning dipole forbidden transitions became available thanks to multiphoton or lowenergy electron impact spectroscopies [39-42], that one has to go beyond the TammDancoff or SCI approximation if one is to obtain at least a correct ordering of the low-lying excited states. Particularly for the so-called alternant systems (whose structural formula is a bigraph, see e.g. [38]), whose states may be labelled by an additional quantum number (the so-called plus "+" or minus " - " states of Pariser [49]) providing additional selection rules, a qualitatively different behavior is found for states of "+" and " - " parity when introducing the correlation effects beyond the Tamm-Dancoff level of approximation: the singlet minus states undergo almost an order of magnitude larger lowering when the effect

486 of doubly excited configurations (via SDCI) is introduced as compared to the singlet plus states (and, similarly, though to a lesser degree, the triplet plus states are more affected than the triplet minus states) [50]. While this behavior may seem puzzling at the MO level, it is easy to understand from the VB viewpoint [50]. It has been well known since the early days of quantum chemistry [25] that the MO formalism overestimates the contribution of ionic structures at the expense of covalent ones: already for a simple homopolar diatomic, such as H2, the MO wave function consists of an equal mixture of the covalent and ionic structures, and the weight of the former is further diminished when going to polyatomic molecules. As a result, the role of doubly excited configurations is to reintroduce the correct balance by emphasizing the weight of covalent structures. Now, one can show [50] that in alternant systems the VB covalent structures are always of the minus type for singlets and of the plus type for triplets, while the ionic structures can be of either kind. Consequently, in view of the superfluity of ionic terms at the uncorrelated MO level, it will be the singlet minus states (and, correspondingly, the triplet plus states) that are most strongly influenced by the SDCI, since there are no covalent structures of the singlet plus or triplet minus type. The effect is smaller for triplet states, since they are reasonably correlated already at the SCI level. Thus, simple VB considerations enable us to understand the behavior of the MO model and to formulate simple qualitative rules describing the role of correlation effects. We shall see in the following that a simple VB picture can, in fact, provide very efficient and highly correlated results for the PPP model in general. 3. P P P - V B M O D E L As already indicated, the overemphasis of covalent structures in the VB wave function that is built from AOs, and of ionic structures in the MO wave function, as represented by a single antisymmetrized product of LCAO MOs, has been known since the inception of both approaches [25], as well as the way - at least in principle - how to remedy these shortcomings by proceeding towards the full CI (FCI) or FVB limit, where both procedures coalesce, thereby yielding the exact solution in the finite dimensional subspace defined by the chosen AO set. In this latter procedure, the coefficients associated with distinct MO configurations or VB structures (both covalent and ionic) are varied independently, relying on the variation principle, and the transition from one formalism to the other is achieved using a similarity transformation between the two distinct basis sets spanning the same N-electron space. It is thus not difficult to realize that one could go a long way towards the FVB limit if, during the construction of VB wave functions for relevant covalent structures, one would employ instead of pure, strictly localized AOs, their linear combinations, not unlike the LCAO MOs, while preserving their AO character by allowing only a moderate "delocalization" to neighboring sites. This idea was first exploited by Coulson and Fischer [51] in their study of the hydrogen molecule, using the Heitler-London or VB-type trim wave function in which each AO contained an admixture of the AO on the other nucleus, the mixing parameter being treated as a variational parameter. This idea was later employed by Goddard and collaborators [27,52] in their generalized VB (GVB) method, by Cooper, Gerrat, Raimondi and oth-

487 ers in their spin-coupled VB method [28,53], or its mnlticonfigurational generalizations [54,551 and other "modern VB" approaches [56-581, as wen as by McWeeny [30] who also provided the most convincing and pedagogical illustration of this idea using the example of the H2 dimer [59]. The relevant LCAO-type AOs, whose admixture or mixing coefficients are optimized so that the VB wave function, as represented by the most important covalent structure(s), gives the lowest energy, are usually referred to as the overlap enhanced atomic orbitals (OEAOs). It is essential that these orbitals be nonorthogonal. As a consequence, the N! problem, typical of VB approaches, is still present. Nonetheless, finding the optimal OEAOs leads to highly correlated wave functions involving only a small number of covalent structures that possess all the essential bonding characteristics of the ground or low lying excited states. The quantitative chemical accuracy, if desired, may then be achieved with a moderate VB expansion involving a few hundred up to a few thousand additional structures constructed from the same OEAO set [28]. The key to the success of the spin-coupled VB is thus undoubtedly the exploitation of a flexible LCAOtype AO basis set, whose mnlticenter orbitals are suitably delocalized in the vicinity of each site. We have recently exploited this idea in the context of the PPP model with considerable success [60-65] (see also [66,67]). To preserve the simplicity of the model, we minimize the number of mixing parameters defining the OEAOs. Thus, starting from the hypothetical AOs [Xu) of the PPP model, we construct the OEAOs [r by admixing to each AO [Xu) only the AO's [Xv) on the nearest neighboring sites. Moreover, unless the bond lengths involved are very different, we assume all mixing coefficients to be the same. Thus, in its unnormalized form we write

Ir

=

Ix.> +

~(~u)

Ix ),

(6)

where the symbol ",,~" indicates that the sum extends only over the nearest neighbors of /,. Such an OEAO basis set is referred to as the {bl} basis [60]. The unnormalized form has the advantage that the relative mixing parameter is not dependent on the number of nearest neighbors. The value of the mixing parameter e is then determined by minimizing the total energy. A higher approximation may be achieved by employing two-parameter OEAOs which result from the admixture of the next nearest neighbor AOs (mixing parameter #) as well as the nearest neighbor (mixing parameter r ) ones. Such a basis set is referred to as the {bl, 2} basis, and we can similarly define {bl, 3}, {bl, 2, 3}, and other basis sets. However, it turns out that little is gained in this way [60,61], and that in most applications the {bl} basis is perfectly satisfactory. An important feature of a successful semiempirical approach is the transferability of parameters from one system to another. Although it may sometimes be advantageous to employ different parameters for different homologous classes or states of different multiplicity, etc., the ideal semiempirical parametrization should remain as universal as possible. We found that the {bl} basis is satisfactory in this regard, being transferable from one system to another regardless of the system's topological character (alternant vs. nonalternant systems), spin multiplicity (singlets, doublets, triplets, etc.), or electric charge (neutral species vs. ions). The optimal value of e ranges from ,,~ 0.25 for cyclobutadiene

488 to ~ 0.34 for benzene, the average value being ~ 0.31 [60]. In fact, optimal ~ values for most systems range from 0.30 to 0.32. Moreover, the energy change that results when using the average value ~(ave) = 0.31 rather than the optimal one, ~(opt), for the mixing parameter ~, never exceeds 0.13 eV (the value for cyclobutadiene) and on the average amounts to less than 0.01 eV. We must emphasize that in the PPP-VB calculations referred to above, we employed all the covalent structures for a given system. However, these results remain practically unchanged when only Kekul6 structures are used. Merely in cases when only one Kekul6 structure exists is it advisable to employ one or more additional non-Keknl6 structures. Let us also mention that the actual computations are based on the Clifford algebra unitary group approach (CAUGA) [48,68] (of. also [69]) and employ the Clifford algebra realization of the VB Rumer-Weyl basis [66,70]. We first construct the CAUGA VB states or structures using the just described OEAO basis set(s). The original version of our codes transforms these VB states into CAUGA states in terms of the orthonormal PPP AO basis, since the action of U(n) generators in this basis is very simple. A subsequent and more efllciene algorithm, based on Eqs. (23)-(26) of [70], which uses the fact that the action of the standard "orthogonal" U(n) generators E ~ , defining the P P P Hamiltonian (2) or (4), on the nonorthogonal OEAO basis, say {bl}, simply produces the mixing coefficient in lieu of the Kronecker delta [60]. After determining the result of the action of H, on a given VB structure, we calculate the overlap with the other structures expressed in terms of bispinors or Slater determinants. We also exploit the fact that the overlap matrix is in block diagonal form relative to the distinct orbital occupancies, so that the similarity transformation to the orthogonal basis may be easily carried out block by block. The codes can employ any number of VB structures, both covalent or ionic, up to and including FVB. The latter is often helpful in evaluating the performance and reliability of the PPP-VB approach that is invariably based on a highly truncated N-electron VB basis, usually restricted to only K~kul6 structures. Our codes are also capable of carrying out a population analysis of the resulting VB wave functions in terms of structural weights or bond orders. 4. A P P L I C A T I O N S The primary objective of most applications carried out so far was to assess the performance of the PPP-VB method for diverse alternant and nonalternant ~r-electron systems of aromatic, nonaromatic or antiaromatic character, both electrically neutral and charged. The main emphasis was on ground states of different spin multiplicity, even though some preliminary calculations were also carried out for excited states. The PPP-VB codes were also employed to provide the approximate three- and four-body connected cluster components for the so-called VB-corrected coupled cluster (CC) approach [71]. In the following, we briefly point out the most important aspects of the PPP-VB method and illustrate them with a few typical results. 4.1. C o r r e l a t e d g r o u n d s t a t e s , basis t r a n s f e r a b i l i t y a n d t h e role of ionic structures To assess the effectiveness of the PPP-VB method in describing the correlated ground states of various 1r-electron systems, we have examined a set of 70 relatively small molecules

489 with 4 to 8 1r-electrons, for which we can easily generate the exact FCI or FVB wave functions and energies [60]. It is instructive to first consider the simplest possible examples of aromatic, antiaromatic, and nonaromatic systems represented by benzene, cyclobutadiene (CBD), and linear polyenes (say trans-butadiene and all-trans-hexatriene), respectively. Benzene, epitomizing aromatic systems, distinguishes itself by its high stability, symmetry and reactivity, while CBD exemplifies antiaromatic systems, is highly unstable and its most symmetric square configuration undergoes the Jahn-Teller distortion, resulting in the equilibrium geometry with highly pronounced bond length alternation (see, e.g. [73]). The nonaromatic polyenes then display a moderate bond length alternation which decreases with the increasing size of" the system (see Sect. 4.3). To examine the equilibrium geometry, we also have to consider the a-framework (see also Sec. 4.3). This is simply achieved at this level of approximation by assuming a strictly additive character for individual bonds, so that

(7)

=

i

where the i-th bond relative a-energy contribution AE~,~(Pu) as a function of the bond length P~ is approximated by a suitable harmonic or 3rd order anhaxmonic oscillator =

-

+

-

R,)

.

(8)

The parameters R.,~2 and ~3 are then obtained by fitting the experimental geometry and the totally symmetric stretching mode frequencies of benzene (cf. [47,61,65]). Thus, invoking the usual a-lr separability of the PPP model, the relative total energy is given by the sum AE (t~ = AE~ + A E , .

(9)

No angular deformation potential is needed for our type of problems, in which only a small bond length alternation is assumed with the C-C-C angles preserved. Designating the longer and shorter bond lengths by d+ and d_, respectively, we assume that d+ + d_ = 2d0 (taking do - 1.444/~ for CBD and 1.4/~ in all other cases). We can then characterize the distorted structures by a single parameter A, 1

a = ~(d+ - d_).

(10)

Considering now the PPP-VB model with only two covalent structures (Kekul$ structures for cyclic systems), we find that the resulting potential energy curves as a function of the distortion parameter A are almost identical with the exact FCI or FVB curves (see Fig. 4 of [61]). Only for benzene do we obtain a symmetric, equidistant geometry (A = 0), all other systems displaying bond length alternation (A ~ 0). The largest distortion and the smallest stabilization energy (relative to the symmetric, undistorted structure) are found for CBD (A ~ 0 . 0 5 - 0.06/~, AE ('t~b) ~ 0.025 eV), while for linear polyenes the distortion is smaller (A ~ 0.04/~) and the stabilization energy larger (AE('t~b)/bond ~ 0.05 eV). These are very reasonable values considering the simplicity

of the model (cf. [61]).

490 We also examined [60] the amount of the correlation energy that is recovered by a simple PPP-VB method involving only one or two Keknl~ structures, considering the whole range of the coupling constant (i.e., varying fl from 0 to 5 or 10 eV). Of course, different coupling constants require different optimal mixing parameters e (clearly e -+ 0 as fl --+ 0 since each covalent structure represents the exact solution in this fully correlated limit [33]). For CBD, the {bl} basis already leads to very precise wave functions, and the error in the correlation energy never exceeds 0.1%. For benzene with the spectroscopic parametrization (fl = -2.4 eV), we recover about 76% of the correlation energy using the {bl} basis and over 90% with any 2-parameter basis. The results become progressively poorer as fl increases (for fl = - 1 0 eV, we recover over 83% of the correlation energy using the {bl, 3} basis). Thus, as expected, MO and VB approaches complement one another, becoming exact in each other's limiting cases (MO in the uncorrelated Jill--+ oo limit and VB in the fully correlated fl -+ 0 limit). Remarkably enough, the VB approach works reasonably well even in the weakly correlated limit (where MO is exact), and the mixing parameter e changes very slowly, especially in the physical region of the coupling constant. This is a good indication of the transferability of the OEAO basis from one system to another, thus avoiding reoptimization in each case. Indeed, for a class of 20 typical systems we found that for the physical values of the coupling constant, all optimal values of e fall within the interval 0.25-0.35, the most extreme boundary values corresponding to CBD and benzene. For most systems this range is much smaller, from 0.30 to 0.32, with an average value e(ave) ~ 0.31. Most importantly, when employing e(~ve) = 0.31, the maximum energy deviation relative to the optimized value is only 0.13 eV (for CBD). Excluding CBD, the average energy difference for the 20 systems considered is only 0.01 eV, indicating that the average OEAO basis is transferable with a high degree of accuracy. Consider, finally, the role of other than Kekul~ structures. For benzene, the energy lowering due to the Dewar structures is very small (usually less than 1% of the correlation energy). The ionic structures are most important when the simplest OEAO basis {bl} is used, and significantly less with two-parameter basis sets. The energy improvement is largest for the so-called "asymmetric" structures (see [60]), while the effect of symmetric ones is marginal (~ 1%). Clearly, a very different situation is encountered for the excited states (see See. 4.4). We can thus conclude that the PPP-VB method, employing a simple one- or twoparameter OEAO basis and relying on a few covalent (mostly Kekul~ type) structures, provides an excellent approximation that accounts for a large part of correlation effects. 4.2. Spin properties To explore the applicability of the PPP-VB method to high spin states we investigated conjugated 1r-electron systems of various topology and character [59]. We were interested in these states not only because of their importance in materials science (see, e.g. [7375]), but also from the methodological viewpoint, since in contrast to the MO based approaches, which generally require a different methodology when investigating the low spin closed shells and high spin open shells, we can use the same PPP-VB method for all cases.

For alternant Kekul6 hydrocarbons (i.e., those having at least one Kekul6 structure)

491 with an even number of sites, the transferability of the OEAO basis is once again large: the average value of ~ for a set of 19 hydrocarbons that we examined was 0.303 for singlets and 0.312 for triplets. Thus, the earlier found (see Sec. 4.1) optimal value e(ave) _ 0.31 can be safely used in both cases. As expected by either Longuett-Higgins' [76] or Ovchinnikov's [77] rule, all these systems have a singlet ground state. Moreover, the singlet-triplet splittings are well reproduced quantitatively already with the average {bl} basis and a few important covalent structures. In most cases, the difference between the exact FVB and PPP-VB splittings amounts to less than 0.05 eV, the largest error occurring for benzene (0.28 eV) and ethylbenzene (0.12 eV). When the two-parameter {bl, 3} basis is used for benzene, the error reduces to 0.04 eV. Similarly for alternant Kekul6 ,r-systems with an odd number of sites, we find the average mixing parameter for doublets and quartets to be 0.309 and 0.311, respectively. Thus, using an average value of 0.31 leads to negligible changes in doublet-quartet separations (less than 0.01 eV). Longuett-I-Iiggins' and Ovchinnikov's rules apply again (doublet ground state) and the exact FVB splittings are well reproduced by the PPP-VB method. Both singlet-triplet and doublet-quartet splittings for these systems are significant, exc ~ n g 1 eV in most cases (see [64] for details). Very similar results are also found for non-Kekul6 alternant hydrocarbons. In this case, however, it is the high spin state that has the lowest energy and the singlet-triplet or doublet-quartet separations that are much smaller, ranging between 0.05 and 1 eV. Again, a simple PPP-VB method, involving only the most important covalent structures and the average {bl} basis (e(ave) = 0.31), provides an excellent approximation in the 12 cases that we examined [64]. The most important structures are always those involving the maximal number of double bonds (i.e., Kekul6-like). When truncating the set of these maximally covalent structures one must ensure that any two adjacent sites are It-bonded in at least one retained structure, lest the bonding be r priori biased by truncation. Another interesting test case is provided by alternant non-Kekul6 hydrocarbons having the same number of starred and nonstarred atoms, as represented by a prototypical tetramethyleneethane (TME) molecule. Here Ovckinnikov's and Longuett-Higgins' rule give different predictions for the multiplicity of the ground state (i.e., singlet and triplet, respectively). The actual calculations show that all systems which we examined [64] have a singlet ground state, even though the singlet-triplet separation is very small (about 0.1 eV). Once again, the PPP-VB results are very close to the FVB ones. The proximity of the low and high spin states in these systems is easily understood from the VB viewpoint: the transition from singlet to triplet involves the breaking of a weak, long "bond" between nonadjacent sites in their leading structures. For nonalternant systems (involving cycles with an odd number of sites) Ovchinnikov's rule does not apply. By analogy with alternant systems we can expect, however, that the existence or nonexistence of at least one Kekul6 structure will play an important role: Kekul6 systems should have a low spin ground state. We examined about a dozen of such systems involving 3, 5 or 7 membered rings. The average mixing parameters for the low and high spin states are 0.316 and 0.320, respectively. Again, using an overall average value of 0.31 has a minimal effect on the computed energies and the PPP-VB values are close to the FVB ones. The singlet-triplet (or doublet-quartet for systems with an odd

492 number of sites) separations are quite large for Kekul6 systems (ranging from 0.8 to 3.6 eV) and the above indicated rule holds in all cases that we examined (a low spin ground state). On the other hand, non-Kekul6 nonalternant systems have a high spin ground state and the singlet-triplet separations are much smaller (0.1 to 0.4 eV). We can thus conclude that states of different spin multiplicity (singlets, doublets, triplets, quartets, etc.) of very diverse ,r-electron systems (Kekul6 or non-Kekul6, alternant or nonalternant, aromatic, nonaromatic or antiaromatic) can be satisfactorily described by the PPP-VB method with a severely truncated set of covalent or maximally covalent structures using the same simple OEAO basis set {bl}. In contrast, the MO description requires a different handling of closed and open shell cases and the amount of correlation recovered in states of different multiplicity may be rather unbalanced. 4.3. E l e c t r o n delocalization, r e s o n a n c e a n d b o n d l e n g t h a l t e r n a t i o n The very concept of resonance originated in VB theory and was extensively exploited in qualitative explanations of stability and other properties of ,r-electron systems [78], surviving vitriolic attacks by some Soviet philosophers of the time. Recent re-examination of these ideas [61,65,79-82], based on new semiempirical and ab initio results, shows that the concept of resonance or ,r-electron delocalization is very subtle and prone to misinterpretation when improperly isolated. When we split the relative total energy of benzene (or, in fact, of any cyclic polyene CNHN w i t h a nondegenerate ground state, N = 4v + 2, v = 1,2,--.) into its ~r and lr components, Eq. (9), we observe that it is the ~-energy that stabilizes the symmetric, equidistant structure of a regular polygon, while the ,r-energy component favors the distorted, bond length alternating geometry (see, e.g. [41]). This is easy to understand when we compare the ,r-electron component with, for example, a chain of N = 2n H atoms [83]: the optimal regular polygonal structure will preferably dissociate into n H2 molecules rather than into 2n H atoms, the energy gain being n-times the dissociation energy of H2. Likewise, the *r-electrons prefer a bond length alternating geometry, "crystalizing" to form a Wigner lattice. Now, for N = 6 or even N = 10, the ~r-energy takes over and a regular polygonal equilibrium geometry results. For large cyclic polyenes (and, similarly, for an infinite linear polyenic chain) it is the 1r-component that will dominate, leading to bond length alternation. The magnitude of this alternation as measured by A, Eq. (10), increases with increasing N and stabilizes as N -+ co at A ~ 0.049 /~ (similar results are obtained with MO based approaches as well [41,84,85]). This value agrees reasonably well with the experimentally observed bond length alternation in all-trans-polyacetylene (/~(exp) ,~ 0.052/~) films [86]. The corresponding stabilization energy (AE('tab)/N ~, 0.05 eV) is also reasonably close to the correlated ab initio calibrated result (0.08 eV) [87] (cf.,

[85]). These facts, however, do not imply that the *r-electron resonance energy (for different definitions see, e.g., [88]), associated with *r-electron delocalization, should be regarded as "a byproduct of the a-imposed geometric symmetry" (see [?9] for this and similar statements). To isolate this effect we have to split AE,~ into its "localized" 1r-electron component A E (LvB), given by the sum of the ~r-electron energies of non-interacting ethylenic fragments, and its true "delocalization" or resonance energy A E (P'zs), i.e. ~,E(t~

-- ~ E r -~- h ~ = -- ~,~r + A~(LVB) -~- A~(RBS).

(11)

493 When we plot these various components as a function of the distortion parameter A, we find that each component AE(=Lvs) and AE(~P~s) changes much more appreciably with increasing A than does their sum ABe, or in fact AE.. Of course, the least rapid change is found for the total energy AB (t~ due to the opposing tendencies of A E . and AE~ (and, AE~(P~s) and AE~(LvB), see Fig. 2 of [65]). This analysis shows the essential stabilizing role of the resonance energy, which must be sufficiently strong to overcome the dimerization tendency of AE~ Lw~. Indeed, (AE~ + &E~Lvs~) by itself would produce a bond length alternating structure even for benzene. In fact, for larger cyclic polyenes, the resonance energy is not sufficiently strong to overcome the dimerization tendency of ( A E . + AE(LVS)) and the bond length alternation sets in, as already pointed out. 4.4. E x c i t e d s t a t e s

So far we carried out only a very preliminary study of the excited states [62] for four typical ,r-electron systems: CBT, benzene, hexatriene and naphthalene. In all cases we found that the PPP-VB method can successfully describe the low lying excited states using only a small subset of VB structures. For states of a predominantly covalent character, the same OEAO basis as for the ground state can be used. However, for higher lying states with significant ionic character, a re-optimized OEAO basis is preferable. Thus, for example, using an optimized basis of {bl} quality and only the most important singly ionic structures (in addition to covalent ones) for benzene, we find for the 1B2~, tB2o, tBx~ and 1Bl~ excitation energies values of 3.64, 5.93, 6.90 and 7.09 eV, which differ by less than 2% from the exact FCI values of 3.70, 5.96, 6.77 and 6.99 eV, respectively. Similarly, for all-trans-hexatriene the difference between the PPP-VB and FCI excitation energies are always smaller than 2% (for transitions to both singlet and triplet states). In the naphthalene case, we considered only covalent type transitions and again all errors in the excitation energies relative to FCI never exceeded 4%. We thus believe that the PPP-VB approach represents a viable and useful alternative even for the excited states which, to this day, were invariably treated by MO methods. Particularly in photochemical processes, the insight afforded by the VB formalism may be very useful. 4.5. V B c o r r e c t e d coupled cluster m e t h o d The MO based single reference coupled cluster (CC) approach truncated at the paircluster level (CCSD) [89] is well known to provide a reliable description of correlation effects in nondegenerate ground states (see, e.g. [90]). The size-extensive character of this approach makes it a method of choice when considering small and medium size systems. However, since the MO reference often becomes quasidegenerate when considering nonequilibrium geometries, the negligibility of the 3- and 4-body connected clusters which is essential for a satisfactory performance of the CCSD method - no longer holds. Consequently, the CCSD description deteriorates in such cases or even completely breaks down [91,92]. For accurate calculations, 3-body clusters should be accounted for even when no quasidegeneracy is present (see, e.g. [90,93]). Unfortunately, an explicit account of these clusters (CCSDT method) is computationally very demanding. Since the electronic Hamiltonian involves at most two-body interactions, the energy is fully determined by 1- and 2-body cluster components. The latter can then be obtained by solving the CCSD equations. These equations, however, arise through decoupling of

494 the rest of the full CC chain by neglecting the 3- and 4-body connected dusters. Thus, were we to know these cluster components from some independent source, the CCSD equations corrected for the contributions of 3- and 4-body clusters would yield the exact energies. Of course, the exact values of these higher order clusters can only be found by cluster analysis of the exact (i.e., FCI) wave function. For practical applications, on the other hand, a reasonable estimate of these clusters is perfectly satisfactory. Such an estimate can be obtained from wave functions that account, at least approximately, for these clusters, such as the UHF [92,94,95], VB [67,71], CAS-SCF [96,97], etc. wave functions. This is precisely the idea behind our VB corrected CCSD method [67,71], since even very simple VB wave functions, involving only a few covalent structures, can often provide a good estimate of the desired higher order clusters in view of the complementary character to MO methods (and in spite of their inability to provide good 1- and 2-body cluster components). We have tested this idea using PPP model Hamiltonians, obtaining the 3- and 4-body clusters by cluster analyzing simple PPP-VB wave functions. We carefully explored a number of typical systems for the whole range of the coupling constant. Clearly, it is the highly correlated limit (/3 ~ 0), where the standard CCSD fails, but where PPP-VB works best, that we obtain excellent results. Perhaps the most severe test of the proposed scheme was encountered in our study of the radicaloid mode of benzene dissociation or, equivalently, a recombination of two allylic radicals [67]. Since the separated limit involves open shell subsystems, standard CCSD approaches an incorrect channel. However, when VB corrected CCSD is used, we obtain practically the FCI result. 4.6. Ionization p o t e n t i a l s and electron affinities As a final illustration of the PPP-VB method we present some new results on the ionization potentials (IPs) and electron afHnities (EAs) of cyclic polyenes C~rH~ with an odd number of sites N - 2n + 1, n - 1, 2,-... We compute these quantities as the difference between the energy of the appropriate ion and of the parent molecule. The IPs and EAs obtained with the PPP-VB method employing both the optimal and average {bl} basis are compared with exact FCI results (wherever available) in Fig. 1 and Table 1. The degeneracy (indicated by the type of the pertinent symmetry species) of relevant states is also given in Table 1. We note that the symmetry of the lowest state, for either the neutral or charged polyenes, resulting from our simple VB approach involving only N covalent structures, is always the same as that obtained with the FCI method: the ground state of neutral polyenes is always degenerate (E-type symmetry species), while the ground states of cations and anions alternate between the nondegenerate A-type states and degenerate E-type states (see Table 1). All PPP-VB calculations reported here employ N covalent structures involving n double bonds and either an empty, singly occupied or doubly occupied remaining site. For greater clarity, the standard PPP-VB results using the average mixing parameter e(~*e) - 0.31 are connected via straight lines in Fig. 1. Again, we find that results obtained using optimized and average OEAOs are almost indistinguishable, and are very close to FCI results, as shown for N _~ 9. Note that for N -- 11 the dimensions of the relevant FCI problems are 60 984 for the anion and cation, and 104 544 for the neutral system.

495

I

I

i

I

I

i

I

_

'

9

IP (FCI)

IP (VB, E (~

" t

5

r

IP (VB, ~(ave))

i

:~ /~

\

> v

~

',,/ \

4

LU

t

o

E A (FCI)

A -

EA (VB, ~ (opt)) EA (VB, r

"

,-',, \

s

~

/_\

/.,~

--

,,

I 11

I 13

I 15

3

2

I 3

I 5

I 7

I 9

N

Figure 1. The IPs and EAs obtained with the PPP-VB method, employing both the optimal (opt) and average (ave) {bl} OEAO basis and involving N VB structures, are compared with the exact FCI results.

We observe a typical alternating behavior of the IPs and EAs for these systems. The magnitude of these "oscillations" decreases quite rapidly with an increasing number of sites in the cycle N. Clearly, this behavior reflects the stability of Hfickel (4n + 2) systems having a nondegenerate ground state (note that these states always belong to the A-type symmetry species) and large resonance energies. Thus, while C3H3 will readily ionize to form C3Hs +, the next member C5H5 prefers to accept an additional electron to form the anion CsHs- having the same ,r-electron sextet as benzene. Of course, as the size of the cycles increases, the stabilizing resonance energy becomes smaller and smaller (el. Sec. 4.3) thus making the facility of the positive and negative ion formation less noticeable. 5. C O N C L U S I O N S The above presented examples amply demonstrate that even the simplest version of the proposed PPP-VB scheme, employing a single parameter OEAO {bl} basis, represents a viable, very efficient and transparent method that provides highly correlated description of low lying states of ,r-electron systems. Moreover, the required {bl} basis is readily

496

Table 1 A comparison of the exact FCI total ~r-electron energies E(FCI) (in eV) with those obtained by the PPP-VB method, employing either the optimal [E(VB, e{op0)] or average [E(VB, e(~))] OEAO {bl} basis set and the minimum number g of VB structures, for the ground states of cyclic polyenes CNHN with an odd number of sites N, N = 2n + 1, and of their ions. The degeneracy (Deg.) of these states is indicated by the type of the symmetry species involved (E for doubly degenerate and A for totally symmetric nondegnerate ones). System

N

CNHN

CNH +

3

5

7

9

11

13

15

Deg. E(FCI) E(VB,e (~ E(VB,e (~))

E -4.28 -4.21 -4.12

E -8.51 -8.36 -8.36

E -12.41 -11.99 -11.99

E -16.20 -15.72 -15.72

E

E

E

-19.31 -19.31

-22.90 -22.89

-26.44

Deg.

E -3.95 -3.79 -3.79

A -10.47 -10.21 -10.21

E -12.86 -12.17 -12.17

A

E

A

E(VB,e (~ E(VB,e (~))

A -2.74 -2.74 -2.73

-17.10 -17.05

-19.56 -19.54

-23.77

Deg. E(FCI) E(VB,e (~ E(VB,e (ave))

E 1.77 1.77 1.78

A -6.64 -6.51 -6.47

E -8.61 -8.06 -8.05

A -14.26 -13.57 -13.56

E

A

E

-15.84 -15.84

-20.35 -20.30

-23.02

E(FCI)

CNHN

transferable from one system to another without any appreciable loss of accuracy, so that one can safely employ the same universal value of the mixing parameter, e(ave) = 0.31, for all ~r-electron systems. At the same time, the VB wave functions employed are very compact, involving only a few covalent structures, mostly of the Kekul~ type. The key to the success of this approach in accounting for a large portion of the correlation effects lies in the use of OEAO basis sets. Indeed, it is well known that strictly localized AO basis sets yield VB wave functions (even for the ground states at the FVB level) in which the singly ionic ortho-polar structures predominate over the classically important covalent structures of the Kekul~ type [98]. This is particularly the case for truncated VB expansions: when the expansion involves only covalent and singly ionic structures, the latter ones outweigh the former ones by almost an order of magnitude, and even some meta-polar ionic structures are more important than the covalent Kekul~ structures [98]. When only covalent structures are employed, the resulting energy lies above the SCF energy [98]. Thus, the choice of an appropriately delocalized AO basis is crucial for the success and practical usefulness of the VB approach. At the same time it is essential that such a basis is nonorthogonal, since it is well known that the so-called VB approaches, employing quasilocal yet orthonormalized AO basis sets, lead to very slowly convergent VB expansions.

497 When designing computational algorithms it is thus important to suitably handle the well known N! problem arising from the orbital nonorthogonality (see, e.g. [99]). For the PPP Hamiltonian considered here, this problem can be partially avoided by employing the CAUGA formalism and by reverting to the original effective orthonormal PPP basis. Nonetheless, even more efficient algorithms could be developed by following some of the ideas that are exploited in current ab initio VB approaches. It must also be emphasized that our work focusses on ground states and low lying excited strates that involve primarily covalent structures. This is in fact the case for most existing applications of the VB method, be they at the semiempirical or ab initio level. The higher lying excited states may be of ionic character, or may at least involve a large ionic component. Such states correlate with purely ionic structures in the fully correlated limit (cf. [100]). A proper description of such states will necessitate the consideration of both covalent and ionic structures, and possibly the reoptimization of OEAO basis sets employed. We must recall here that in the MO description, we must consider doubly excited configurations in order to obtain at least a qualitatively correct ordering of these states. A thorough study of such states is thus desirable and their VB description, even at the semiempirical level, should be beneficial for a better understanding of the chemical nature of these states. ACKN OWLED G EMENTS Continued support (J.P.) by NSERC is gratefully acknowledged. REFERENCES

1. 2. 3. 4. 5. 6.

7. 8. 9.

10. 11. 12. 13. 14. 15.

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503

T h e s p i n - c o u p l e d d e s c r i p t i o n of a r o m a t i c , a n t i a r o m a t i c a n d nonaromatic systems David L. Cooper a, Joseph Gerratt b and Mario Raimondi c aDepartment of Chemisty, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, United Kingdom bSchool of Chemistry, University of Bristol, Cantocks Close, Bristol BS8 1TS, United Kingdom CDipartimento di Chimica Fisica ed Elettrochimica, Universit~ di Milano, Via Golgi 19, 20133 Milano, Italy The correlated motion of ~ electrons in a range of aromatic, antiaromatic and nonaromatic systems is investigated using modern spin-coupled theory. In the particular case of benzene, the resulting wavefunction, which is superior to the simple MO description, resembles closely the classical VB picture, except for small (but crucial) deformations of the orbitals. It is argued, by reference to a multiconfiguration approach which subsumes both of the simple MO and VB models, that we should feel comfortable with switching between the two modes of description, according to which leads most directly to correct predictions for the particular problem at hand. The bonding in benzene is contrasted with that in cyclobutadiene and cyclooctatetraene, at various geometries. In the spin-coupled model, the distinguishing feature of antiaromatic species is the occurrence of triplet coupling of electron pairs. Of course, the singlet ground states of both of these molecules distort from the idealized high-symmetry geometries to nonaromatic situations with olefinic bonds. 1. I N T R O D U C T I O N

Organic chemists tend to be pragmatists when faced with rival MO and VB descriptions of molecular electronic structure. Many will use whichever model seems most convenient for the problem at hand. MO descriptions are widely employed in frontier orbital approaches, as in the Woodward-Hoffmann rules, and tend to be favoured when predicting excited states or photoelectron spectra. On the other hand, it is customary to represent reaction mechanisms in terms of resonance between classical VB structures with single, double etc. bonds (plus any unpaired electrons or lone pairs) and then to indicate by means of 'curly

504 arrows' the supposed electron reorganization. The attitude fostered by m a n y text books is that MO theory is in some sense 'more fundamental' t h a n the VB model, and some organic chemists admit to a feeling of unease when using VB-based arguments in discussions with quantum chemists. A somewhat ambivalent view of the relative merits of MO and VB approaches is certainly apparent for aromatic molecules. It is common to use interchangeably terms such as 'delocalization energy' and 'resonance energy'. Students are taught about the ~ electrons of benzene moving more-or-less independently of one another in delocalized orbitals, and the relative stabilities of 'aromatic' and 'antiaromatic' systems are typically discussed in terms of the simple Hfickel 4n+2 rule. On the other hand, the VB model proves to be exceptionally useful when predicting the outcome of electrophilic substitution reactions, for example. Considerations of resonance between Kekul~ (and other) structures and ideas as to the relative energies of Wheland intermediates typically lead to very straightforward predictions of correct products. The purpose of the present account is to discuss some of the key findings for ~-electron ring systems, of modern valence bond theory, in its spin-coupled form. The approach incorporates from the outset the chemically most significant effects of electron correlation, but it retains a simple, clear-cut visuality. Spin-coupled calculations have been performed for a wide range of ~-electron aromatic, antiaromatic and nonaromatic ring systems. Molecules studied include benzene [1-10], cyclobutadiene [11], cyclooctatetraene [12], six-membered ring heterocycles [3,13] and five-membered ring heterocycles [14-16], o-benzyne [17], fused ring systems [18,19,20], various inorganic molecules [4,21,22], rings with methylene substituents [11,18,23], oligomers [16,18,23], and certain unusual species, such as bicyclic 1,6-methano[10]annulene, which features a nonplanar aromatic system [24]. In addition to studying ground states, we have carried out calculations on excited states, of ionization potentials, and for electrophilic substitution reactions. Reviews of our earliest work on benzenoid aromatic molecules and on antiaromatic systems are available elsewhere [25,26]. For each system, we take account of electron correlation for the ~ electrons, but not for the (~ framework. However, the ~ orbitals for aromatic systems lie to a considerable extent within the space of the o orbitals and not well outside it, as is commonly assumed. This might seem to bring into question the fundamental concept of o-~ separation. However, the ~ orbitals are a great deal more polarizable than the ~ orbitals, and so it can be argued that the ~ system provides a large proportion of the response of the system to chemical and other influences. As a result, much of the chemistry of aromatic systems can be understood by considering only the ~ electrons. An important outcome of all these spin-coupled calculations is the consistency of the descriptions. In particular, a simple and highly-visual model emerges for the behaviour of c o r r e l a t e d ~ electrons in all of the aromatic molecules that we have studied. These ~-electron systems are well described in terms of fairly localized, nonorthogonal, singly-occupied orbitals. The special stability of such systems arises in the spin-coupled model from a profoundly q u a n t u m mechanical

505 phenomenon, namely the mode of coupling of the electron spins, as is shown by the magnitudes of the resonance energies, and not from any supposed delocalization of the orbitals. The descriptions of these systems are, to all intents and purposes, unaltered by the inclusion of ionic structures or of additional electron correlation into the wavefunction. It is important to dispense with the received wisdom t h a t MO theory is in some sense more fundamental than VB approaches. On the other hand, it is certainly not our intention to argue that the MO description is somehow 'wrong'. In the particular case of benzene, we quantify to what extent the conventional MO and VB models can be considered reliable approximate representations of a particular type of multiconfigurational wavefunction that is more sophisticated t h a n those obtained from either approach. We conclude that we should not have any serious qualms about switching between the MO and VB representations, according to the nature of the particular problem being addressed. 2. S P I N - C O U P L E D W A V E F U N C T I O N S

The ab initio spin-coupled wavefunction for an N-electron system is based on a single product of singly-occupied orbitals, r162 In general, these orbitals are all distinct and nonorthogonal. The spin-coupled orbitals are expanded in an atomic basis set, much as in MO theory. This orbital product is combined with an N-electron spin function | corresponding to total spin S and projection M, and the configuration is antisymmetrized. The total spin function is expanded in the full spin space and it is fully optimized variationally simultaneously with the spin-coupled orbitals. This is usually done without the imposition of any constraints that would alter the total wavefunction. The spin-coupled wavefunction incorporates from the outset the chemically most significant effects of (nondynamical) electron correlation and describes properly all dissociation pathways. J u s t as in correlated descriptions based on MO theory, it is neither practical nor desirable to include electron correlation for all of the electrons in large systems. In common with other strategies, the orbital space is partitioned into 'inactive', 'active' and unoccupied or 'virtual' subspaces. Electron correlation is incorporated only for the 'active' space, which corresponds to that part of the electronic structure which interests us most. A convenient representation of a general spin-coupled wavefunction takes the form:

WSC

=

A{(P~(p22~ ~"(P,21,O~;~)r162162

N OSM}

(1)

in which A is the antisymmetrizer. The spin function O2p~represents perfect pairing of the spins of the 2n inactive electrons, which are accommodated in the n doubly-occupied orbitals ~,. Conditions that do not alter the total wavefunction are normalization of the k~, and r orthogonalization of the ~i amongst themselves, and orthogonalization of the % to the r We may reorder the spin-

506 coupled active orbitals, as long as we also make the corresponding changes to the total spin function. On the other hand, the wavefunction is not invariant to linear transformation of the r so that the form of the spin-coupled solution represents a unique outcome of the variational procedure for the given choice of active space. The spin-coupled model represents the proper generalization to many-electron systems of the early VB work of Heitler and London [27], and of Coulson and Fischer [28]. A h a l l m a r k of the spin-coupled approach is the direct optimization of nonorthogonal orbitals without preconceptions as to their form or degree of localization, and without constraints on the overlaps between them or on the associated mode of coupling of the electron spins. A key feature is the expansion of the total spin function for the active electrons as a linear combination of all allowed modes of spin coupling. There are in fact many different, useful ways of constructing complete sets of linearly independent N-electron spin functions, as is described, for example, in the book by Pauncz [29]. Spin bases t h a t prove especially convenient for carrying out spin-coupled calculations include the Kotani and Rumer schemes, although we have recently explored also the increased efficiencies that may be achieved with character-projected spin functions. The Serber basis also proves to be very useful as an interpretational tool. It is important to stress that provided we use the full spin space, we may transform easily between these various representations [30]. We may also choose to reorder the orbitals, if this aids interpretation. In essence, the Kotani scheme corresponds to the successive coupling of the spins of individual electrons, according to the usual rules for combining angular momenta. The Rumer scheme, which was widely used in classical VB treatments, corresponds instead to coupling (in arbitrary order) singlet-coupled pairs of electrons and any unpaired spins. The Serber scheme involves the successive coupling of singlet- or triplet coupled pairs of electrons. The Rumer basis turns out to be particularly useful for interpreting the total spin functions for aromatic systems. In the case of N=6 and S=0, there are just five linearly independent modes of spin coupling [29], which may be represented as in Figure 1, in which an arrow i-~j signifies a factor in the total spin function of 2-~((z(i)~(j)-(~(j')~(i)). The similarity to Kekul~ and para-bonded structures for benzene is obvious. The spin-coupled approach, which combines an accurate description of molecular electronic structure with a highly visual picture of the bonding, has now been applied to a very wide range of problems and numerous review articles are available [25,26,31-34], including details of the various computational algorithms. A subsequent chapter is concerned with the n a t u r e of the hypercoordinate bonding to main group elements, as in molecules such as PF 5 and SF~. The spin-coupled description may be refined further, without altering the key features, by means of the incorporation of additional configurations in a nonorthogonal configuration interaction calculation, which we term the spincoupled valence bond or SCVB method. This leads to very high accuracy for

507 ground and excited states, while retaining a compactness t h a t aids u n a m b i g u o u s interpretation.

1 6

1 2

5

1

6

2

6

5

3

5

1 2

1

6

2

5

6

2

5

4

4

4

4

4

R1

R~

R3

R4

R5

Figure 1. Rumer diagrams for N=6 and S=O.

3. B E N Z E N E We have recently carried out new calculations for benzene [8,10], at the geometry shown in Figure 2. All the calculations were performed using MOLPRO [35] or our own codes, as appropriate. The s and p basis functions for C and the s basis functions for H were taken from correlation consistent pVTZ basis sets [36], and these were augmented with polarization functions with exponents de=0.8 and p.=l.0, so that the C/H basis consists of (10s5pld/5slp) Cartesian Gaussians contracted to [4s3pld/3slp]. The MO energy level diagram is shown in Figure 3 with arrows $$ to denote the occupancy in the restricted Hartree-Fock (RHF) configuration, a22,el4.

b2g 139.64 pm H6

108.31pm

1~ /H2

H5/C5~

3\H3

0.10

0.04

e2,,

0.10

,90

,90

~

1.96

H4

a2u

Figure 2. Geometry and atom labelling for benzene (D,~,,).

Figure 3. MO energy level diagram for benzene, annotated in various ways.

508 A '6 in 6' CASSCF calculation [37] was performed for the ~ electrons, keeping the thirty-six • electrons in an (optimized) closed-shell core. Unlike the spincoupled model, this multiconfigurational description involves all possible distributions of the six x electrons in six ~ orbitals. The nondynamical correlation energy retrieved by this wavefunction amounts to -73 millihartree (see Table 1). The n u m b e r s shown alongside each level in Figure 3 are the CASSCF n a t u r a l orbital occupation numbers, some of which show significant deviations from the R H F values of 2 and 0. The weight of the configuration a~e~ in the full CASSCF wavefunction is 88.4%. By definition, the RHF calculation retrieves none of the correlation energy incorporated in the CASSCF calculation.

Table 1 Energies calculated for benzene. Calculation CASSCF SC SC (f.c.) RHF

E/hartree

(E-Ec~)/millihartree

-230.8368216 -230.8293315 -230.8293314 -230.7640556

0 7.49 7.49 72.77

Proportion of

(Ec~-ERHF)

100% 89.7% 89.7% 0%

In common with all other wavefunctions of 'full CI' form, CASSCF wavefunctions are i n v a r i a n t to general nonsingular linear t r a n s f o r m a t i o n of the active orbitals, including n o n u n i t a r y transformations t h a t result in nonorthogonal orbitals [8,10,38,39]. As an alternative to, say, the n a t u r a l orbital representation, we m a y exploit this invariance to generate a r e p r e s e n t a t i o n of the CASSCF wavefunction in which the d o m i n a n t component takes spin-coupled form. We m u s t stress t h a t the wavefunction and the total energy are not changed in a n y w a y by this procedure. It proves straightforward to find an alternative r e p r e s e n t a t i o n of the full CASSCF wavefunction in which a spin-coupled-like component has an overlap with the total wavefunction t h a t exceeds 0.995. The form of the orbitals and the mode of spin coupling [8,10] are very similar indeed to those t h a t we describe later, based on fully-variational spin-coupled calculations. Indeed, this d o m i n a n t spin-coupled-like component has an energy expectation value which lies within 0.1 millihartee [8] of the fully-variational spin-coupled result. Spin-coupled calculations were carried out with an active space corresponding to the six ~ electrons and a frozen core t a k e n directly from the CASSCF calculation. The resulting total energy, labelled SC(f.c.) in Table 1, retrieves 89.7% of the correlation energy incorporated in the CASSCF calculation. A fullyvariational calculation (labelled SC in Table 1), in which we optimize also all of the doubly-occupied (~ core orbitals, gives a further energy i m p r o v e m e n t on the order of only 0.1 microhartree. We find six s y m m e t r y - e q u i v a l e n t C(2p~) orbitals, each associated with a given carbon atom, but exhibiting some deformation

509 towards the neighbouring C atoms on each side, as is shown in Figure 4. These distortions are no larger than those that we have seen for the ~ bonds in alkenes. Numbering the orbitals according to the C atoms with which they are associated, the symmetry-unique overlap integrals are (~1]~2)=0.524, (~1]~3)=0.029 and (q~, I (~4)=-0.157.

Figure 4. Various representations of spin-coupled orbital r for benzene. Left: contours in the horizontal plane l b o h r above the molecular plane. Centre: contours in a vertical mirror plane. Right: a representative isosurface (3-D contour). In the Rumer basis [29] (see Figure 1), the total spin function corresponds to weights of 40.6% each for the two Kekul~ structures (R1,R4) and of 6.3% each for the three para-bonded ('Dewar') structures (R2,R3,R~). These values are very close to those given many years ago by Pauling [40] in his original, and much simplified, classical VB calculation, and discussed by Coulson [41] in his text book. The difference in energy between one of the Kekul~-type structures (R 1) and the full spin-coupled wavefunction (with the full spin space) can reasonably be termed the resonance energy. Using this procedure, we obtain a value of 83.5 k J mo1-1. The remaining -10% of the nondynamical correlation energy not recovered by the spin-coupled wavefunction is due primarily to the omission of doubly-ionic spin-coupled configurations. The contributions from singly-ionic configurations are smaller, as is to be expected for wavefunctions based on fully-optimized nonorthogonal orbitals. Of the various singly-ionic spin-coupled configurations, those with charges in meta positions are the most important [10]. The spin-coupled-like component of the CASSCF wavefunction, the SC(f.c.) wavefunction and the fully-variational spin-coupled wavefunction, which are all exceedingly similar to one another, vindicate the familiar description of benzene in terms of two Kekul~ and three para-bonded structures. A crucial difference,

510 however, is the small degree of deformation of the orbitals towards neighbouring atoms. One consequence of these deformations is t h a t the addition of 'ionic configurations', in which one or more of the valence orbitals is doubly-occupied, leads only to a very modest further improvement in the wavefunction. On the other h a n d , if one insists on strictly localized orbitals, as in classical valence bond theory, then the weights of the covalent structures decrease dramatically, at the expense of significant contributions from the plethora of possible ionic structures [42,43]. Similarly, in an exact representation of a '6 in 6' CASSCF wavefunction in t e r m s of orthogonal localized molecular orbitals, Hirao et al. [44] found weights for the Kekul6 structures of 7.8% each and for the para-bonded structures of 2.6% each, so t h a t 76.6% of the benzene ground state wavefunction is a p p a r e n t l y ionic! All of these observations reinforce our preference for the spin-coupled description of benzene. One of the great strengths of an RHF mode of description is of course the ease with which one m a y obtain first estimates of the relative energies of excited states. In the particular case of benzene, a variety of low-lying states arise from 2 3 2 3 configurations such as a2 ele2, and a2 e,gb2g. However, it is difficult to see a p r i o r i w h y some of these should be covalent valence states, w h e r e a s others are p r e d o m i n a n t l y ionic in character and some low-lying states are Rydberg states. The spin-coupled valence bond (SCVB) method has been used to study all the singlet and triplet valence states and n=3,4 Rydberg states of benzene below the first ionization limit [5]. The numerical accuracy provided by these very compact wavefunctions compares very favourably indeed with the most extensive correlated M O - C I calculations in the literature. We find t h a t '~-only' correlation affords an excellent description of the covalent valence states. The s a m e is true of the Rydberg states, provided t h a t the a core is derived from a calculation on the cation. On the other hand, a proper description of the ionic states requires some account to be t a k e n of g - ~ correlation effects. The accuracy to which covalent states of benzene can be described w i t h ~-only correlation is a further justification for invoking (~-~ separation for the various aromatic, a n t i a r o m a t i c and nonaromatic ring systems t h a t we consider here. We have not addressed the question of w h e t h e r it is primarily the a electrons or the electrons of benzene t h a t drive the preference for a high s y m m e t r y structure [45]. Similarly, we do not investigate here a n y 'bent bond' solutions, based on mixing g and = orbitals [46]. L The spin-coupled descriptions of n a p h t h a l e n e and azulene resemble those for benzene except, of course, t h a t orbitals associated with bridging carbon atoms now show a three-way distortion [18,19]. Analogous descriptions arise also for heterocycles with five- and six-membered rings [3,13-16]; w h e n there are two spin-coupled n orbitals for a given heteroatom, one of t h e m adopts a tightly localized form w h e r e a s the other m a y exhibit significant delocalization onto the neighbouring atoms in the ring. The weights of the different modes of spin coupling, and the computed resonance energies, are consistent with the traditional organic chemistry views of these systems. Simple m o d e r n VB e s t i m a t e s of the ionization potentials [14,21] are at least as good for the lowest

511 states of the ion as those derived from Koopmans' theorem, while the higher ones appear to be considerably more reliable. In contrast to the heterocyclic systems, we have found that various inorganic systems, such as borazine, boroxine, N2S2, and perfluorocyclophosphazenes, are much closer to being zwitterionic species than they are inorganic analogues of benzene [4,21,22]. Spin-coupled studies of Wheland intermediates formed by ring protonation of benzene, phenol and benzonitrile provide ab initio support [7] for the usual qualitative VB arguments used to discuss the energetics and selectivity of aromatic electrophilic substitution reactions. Analogous calculations have also been performed for the reaction between benzene and a methyl cation [9]. 4. C Y C L O B U T A D I E N E

A classic example of an antiaromatic system is of course square-planar cyclobutadiene (D4h), for which the ~-electron MO energy level diagram is shown in Figure 5. The RHF configuration a~ue~ implies states of 1A~, 3A2g, B~ and ~Bu symmetries. According to Hund's rules, the triplet state (3Au) ought to lie lowest. For this state, the square-planar geometry is stable with respect to geometric distortions, but it turns out that this is not the ground state. At this geometry, the state of 1B2g symmetry lies lower by more than 40 kJ mo1-1. Furthermore, this singlet ground state is subject to a second-order Jahn-Teller distortion, such that the equilibrium geometry for cyclobutadiene is based on a rectangular ring with two shorter C=C double bonds and two longer ~ single bonds. The reasons for the breakdown of Hund's rule for some systems with four-fold symmetry, such as square-planar cyclobutadiene, have been explored by various authors [47,48].

b2u

eg

a2u

Figure 5. MO energy level diagram for square-planar (D4h) cyclobutadiene. Straightforward applications of the spin-coupled approach confirm the relative energies and the preferred geometries of cyclobutadiene in these singlet and triplet states [11], without the requirement for multiconfigurational descriptions,

512 as in the analogous MO treatments. Of greater significance for the present account is the physical picture of the electronic structure and bonding revealed by these calculations. The aromaticity of benzene is linked, in spin-coupled theory, to the particular mode of coupling of the electron spins, and so it seems reasonable to suppose t h a t the orbital descriptions of D4/' cyclobutadiene and of D6h benzene could be fairly similar, but for these to be associated with very different modes of spin coupling. To a first approximation, this indeed turns out to be the case. With benzene-like orbitals ordered a,b,c,d around the ring, the symmetry requirements of an overall 1B~ state are such that the electron spins associated with each diagonal (a/c and b/d) must be strictly triplet coupled. These two triplet subsystems combine to a net singlet. A characteristic feature of antiaromatic situations in spin-coupled theory is the presence of such triplet-coupled pairs of electrons.

Figure 6. Symmetry-unique spin-coupled orbitals for the 1B1" state of square-planar C4H4. Those for the :~A2~ ~ excited state are very similar indeed. Contours are shown in the horizontal plane l b o h r above the molecular plane. This is not quite the end of this particular story. In the very special case of electron spins which are strictly triplet coupled, the spin-coupled wavefunction is not altered by taking the sums and differences of the corresponding orbitals: (a+c), (a-c), (b+d) and (b-d), neglecting normalization. The term 'antipairs' has been coined for such an alternative mode of description. It turns out that this orbital representation already corresponds to the required 1B~ symmetry at the square-planar geometry, so that there are now more free parameters in the spin space: the relevant spins need no longer be exactly triplet coupled. We find that

513 the variational calculations for cyclobutadiene exploit this extra degree of freedom, so that the converged spin-coupled solutions for square-planar geometry adopt this antipair form [11], as is shown in Figure 6. The orbitals associated with a given diagonal remain close to a+c and a - c (neglecting normalization), and the relevant electrons spins are almost exactly triplet coupled. The difference in energy between the antipair and localized orbital descriptions is exceedingly small and we may choose to regard them as almost equivalent. The key feature, in either mode of description, is the presence of triplet-coupled pairs of electrons. As the molecule distorts towards its equilibrium geometry, the spin-coupled orbitals rapidly adopt the characteristic shapes and associated pattern of spin coupling expected for two separate C=C double bonds. Starting from square-planar cyclobutadiene, the formal replacement with CH 2 groups of H atoms on opposing diagonals leads to DMCB (see Figure 7), with six electrons. We find that the spin-coupled description of the 3B2u ground state possesses one antipair [11], across the diagonal labelled a/b. The dominance of the relevant mode of spin coupling can be seen most easily by expressing the total spin function in the Serber basis. However, there is also significant triplet character in the two exocyclic C=C bonds, consistent with experimental EPR measurements. The lowest singlet state of DMCB is found to possess lAg symmetry [11], rather than the 1B2u symmetry expected from Htickel theory, but the planar geometry is no more than a transition state for the formation of a nonplanar bicyclic system via the development of a long transannular bond. The coupling together of multiple DMCB triplet units [18], as in BBB (see Figure 8), which features antipairs across the diagonals a/b and a'/b" in separate rings, opens up interesting possibilities for antiferromagnetic polymers.

a

a

a'

b

b

b'

Figure 7. DMCB or 2,4-dimethylenecyclobutane-l,3-diyl.

Figure 8. BBB or bismethylenebiscyclobutylidene.

We have also studied the bicyclic planar structure, with eight ~ electrons, that arises formally from the fusion of benzene and cyclobutadiene rings. Our interest here was in determining the extent to which the particular electronic structure features of the two separate rings might persist. We found [20] that the aromaticity of the distorted benzene ring prevails in the singlet ground state of benzocyclobutadiene, with an essentially isolated double bond in the smaller ring.

514 5. CYCLOOCTATETRAENE Spin-coupled calculations at the idealized Dsh geometry of cyclooctatetraene reveal a description dominated by triplet coupling of pairs of electrons [12], as anticipated earlier. Expressing the total spin function in the Serber basis [29], we find that the mode made up only of triplet-coupled pairs is responsible for 75% of the total. We find that the x orbitals for this antiaromatic system (see Figure 9) adopt localized forms that resemble closely those shown in Figure 4 for benzene, rather than the antipair representation shown for cyclobutadiene in Figure 6. On distorting the regular octagon to another idealized geometry, with alternating shorter and longer sides (D4h symmetry), the triplet-coupled pairs disappear, and we observe instead orbitals and a mode of spin coupling that is characteristic of an alkene (nonaromatic system). The equilibrium geometry for cyclooctatetraene is in fact nonplanar, namely a tub structure (D2d). The spincoupled description of the x-like electrons in this nonaromatic system (see Figure 9) corresponds to four essentially localized olefinic bonds [12].

Figure 9. Symmetry-unique spin-coupled orbitals for cyclooctatetraene. Left: idealized Dsh geometry (contours in the horizontal plane lbohr above the molecular plane). Right: analogous representation for the nonplanar equilibrium geometry (Dz~).

6. CONCLUSIONS Aromatic molecules play a central role in organic chemistry and, although a somewhat fuzzy concept that eludes definition in terms of clear-cut experimental

515 and/or theoretical criteria [49], aromaticity certainly continues to be of great utility at a qualitative level. We have concentrated in the present account on the archetypal aromatic system, benzene, and considered also two classic examples of antiaromatic/nonaromatic systems, namely cyclobutadiene and cyclooctatetraene. Analysis of a CASSCF description of benzene that takes account of (nondynamical) electron correlation for the ~-electron system proves to be especially informative. Simple MO theory, in the form of the RHF configuration 2 4 a2ue ~, accounts for 88.4% of this wavefunction and (by definition) recovers none of the nondynamical correlation. A modern valence bond representation, in the form of a spin-coupled-like component, has an overlap with this wavefunction in excess of 0.995 and it recovers 89.7% of the nondynamical correlation energy. It is clear also from the corresponding fully-variational spin-coupled calculations that the spin-coupled wavefunction is numerically superior to the standard RHF description. On the other hand, the magnitudes of these various numbers suggest that we should not discard the MO description. Instead, we may confidently make predictions using either model, depending on which one appears to be more convenient. We should, of course, be very wary of any situation for which the two models appear to lead to conflicting conclusions. The ab initio spin-coupled description of the correlated ~-electron system in benzene corresponds directly to resonance between Kekul~ and para-bonded structures, built from a single product of nonorthogonal orbitals. These spincoupled orbitals resemble closely those in the covalent-only classical VB model except for small, but crucial, deformations of the C(2p~) functions. This vindicates the continued use of such descriptions in organic chemistry, but we should bear in mind that it is the small degree of delocalization onto neighbouring centres that precludes the necessity of incorporating further (ionic) configurations. Analogous descriptions arise for a wide range of other aromatic species. It is now well established that the simple Htickel 4n+2 rule exaggerates the differences between 'aromatic' and 'antiaromatic' ~-electron systems. At the relevant idealized geometries, the characteristic feature of the spin-coupled description of antiaromatic molecules is the occurrence of essentially tripletcoupled pairs of electrons, whether in antipair or localized orbital form. It is this simultaneous unfavourable coupling of the electron spins, which is suggestive of diradical character, that discourages bonding interactions. Of course, the actual singlet ground state geometries of cyclobutadiene and cyclooctatetraene correspond to nonaromatic situations, with olefinic bonds. However, we have found that antipairs do persist at the equilibrium geometries of a number of systems. For a wide range of aromatic, antiaromatic and nonaromatic systems [1-26], the spin-coupled model provides highly visual, but accurate, descriptions of the motion of correlated ~ electrons in terms of nonorthogonal orbitals and the dominance of particular patterns of spin coupling. A striking feature is the simplicity and consistency of the descriptions that emerge.

516 REFERENCES

1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22

D.L. Cooper, J. Gerratt, and M. Raimondi, Nature 323 (1986) 699. J. Gerratt, Chem. in Brit. 23 (1987) 327. D.L. Cooper, S.C. Wright, J. Gerratt, and M. Raimondi, J. Chem. Soc. Perkin Trans. 2 (1989) 255. D.L. Cooper, S.C. Wright, J. Gerratt, P.A. Hyams, and M. Raimondi, J. Chem. Soc. Perkin Trans. 2 (1989) 719. E.C. da Silva, J. Gerratt, D.L. Cooper, and M. Raimondi, J. Chem. Phys. 101 (1994) 3866. G. Raos, J. Gerratt, D.L. Cooper, and M. Raimondi, Chem. Phys. 186 (1994) 233. G. Raos, J. Gerratt, P.B. Karadakov, D.L. Cooper, and M. Raimondi, J. Chem. Soc. Faraday Trans. 91 (1995) 4011. T. Thorsteinsson, D.L. Cooper, J. Gerratt, and M. Raimondi, Theor. Chim. Acta 95 (1997) 131. G. Raos, L. Astorri, M. Raimondi, D.L. Cooper, J. Gerratt, and P.B. Karadakov, J. Phys. Chem. A, in press. D.L. Cooper, T. Thorsteinsson, J. Gerratt, and M. Raimondi, Int. J. Quant. Chem., in press (Proceedings of the 37th Sanibel Symposium). (a) S.C. Wright, D.L. Cooper, J. Gerratt, and M. Raimondi, J. Chem. Soc. Chem. Comm. (1989) 1489. (b) ibid, J. Phys. Chem. 96 (1992) 7943. P.B. Karadakov, J. Gerratt, D.L. Cooper, and M. Raimondi, J. Phys. Chem. 99 (1995) 10186. P.B. Karadakov, M. Ellis, J. Gerratt, D.L. Cooper, and M. Raimondi, Int. J. Quant. Chem., in press. D.L. Cooper, S.C. Wright, J. Gerratt, and M. Raimondi, J. Chem. Soc. Perkin Trans. 2 (1989) 263. P.C.H. Mitchell, G.M. Raos, P.B. Karadakov, J. Gerratt, and D.L. Cooper, J. Chem. Soc. Faraday Trans. 91 (1995) 749. M. Sironi, A. Forni, M. Raimondi, D.L. Cooper, and J. Gerratt, to be published. P.B. Karadakov, J. Gerratt, G. Raos, D.L. Cooper, and M. Raimondi, Isr. J. Chem. 33 (1993) 253. G. Raos, J. Gerratt, D.L. Cooper, and M. Raimondi, Chem. Phys. 186 (1994) 251. M. Sironi, D.L. Cooper, M. Raimondi, and J. Gerratt, J. Chem. Soc. Chem. Comm. (1989) 675. P.B. Karadakov, J. Gerratt, D.L. Cooper, M. Raimondi, and M. Sironi, Int. J. Quant. Chem. 60 (1996) 545. M. Raimondi, M. Sironi, J. Gerratt, and D.L. Cooper, to be published. J. Gerratt, S.J. McNicholas, P.B. Karadakov, M. Sironi, M. Raimondi, and D.L. Cooper, J. Am. Chem. Soc. 118 (1996) 6742.

517 23 24 25 26 27 28 29 30 31 32 33

34 35

36 37 38 39 40 41 42

43 44

G. Raos, J. Gerratt, D.L. Cooper, and M. Raimondi, Mol. Phys. 79 (1993) 197. M. Sironi, M. Raimondi, D.L. Cooper, and J. Gerratt, J. Mol. Struct. (THEOCHEM) 338 (1995) 257. D.L. Cooper, J. Gerratt, and M. Raimondi, Top. in Curr. Chem. 153 (1990) 41. D.L. Cooper, J. Gerratt, and M. Raimondi, Chem. Rev. 91 (1991) 929. W. Heitler and F. London, Z. Phys. 44 (1927) 455. C.A. Coulson and I. Fischer, Phil. Mag. 40 (1949) 386. R. Pauncz, Spin eigenfunctions: construction and use, Plenum, New York, 1979. P.B. Karadakov, J. Gerratt, D.L. Cooper, and M. Raimondi, Theor. Chim. Acta 90 (1995) 51. D.L. Cooper, J. Gerratt, and M. Raimondi, Adv. in Chem. Phys. 69 (1987) 319. D.L. Cooper, J. Gerratt, and M. Raimondi, Int. Rev. Phys. Chem. 7 (1988) 59. J. Gerratt, D.L. Cooper, and M. Raimondi, in Valence bond theory and chemical structure, ed. D.J. Klein and N. Trinajsti6, Elsevier, Amsterdam, 1990; pages 287-349. J. Gerratt, D.L. Cooper, P.B. Karadakov, and M. Raimondi, Chem. Soc. Rev., in press. MOLPRO is a package of ab initio programs written by H.-J. Werner and P.J. Knowles, with contributions from J. AlmlSf, R.D. Amos, A. Berning, M.J.O. Deegan, F. Eckert, S.T. Elbert, C. Hampel, R. Lindh, W. Meyer, A. Nicklafi, K. Peterson, R. Pitzer, A.J. Stone, P.R. Taylor, M.E. Mura, P. Pulay, M. Schlitz, H. Stoll, T. Thorsteinsson, and D.L. Cooper. T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007; the pVTZ basis was taken directly from the MOLPRO library. (a) H.-J. Werner and P.J. Knowles, J. Chem. Phys. 82 (1985) 5053. (b) P.J. Knowles and H.-J. Werner, Chem. Phys. Lett. 115 (1985) 259. T. Thorsteinsson, D.L. Cooper, J. Gerratt, P.B. Karadakov, and M. Raimondi, Theor. Chim. Acta 93 (1996) 343. T. Thorsteinsson and D.L. Cooper, Theor. Chim. Acta 94 (1996) 233. L. Pauling, J. Chem. Phys. 1 (1933) 280. C.A. Coulson, Valence, 2nd. edition, Clarendon, Oxford, 1961; chapter 9. (a) J.M. Norbeck and G.A. Gallup, J. Am. Chem. Soc. 95 (1973) 4460. (b) ibid, J. Am. Chem. Soc. 96 (1974) 3386. (c) G.A. Gallup and J.M. Norbeck, J. Am. Chem. Soc. 97 (1975) 970. G.F. Tantardini, M. Raimondi, and M. Simonetta, J. Am. Chem. Soc. 99 (1977) 2913. K. Hirao, H. Nakano, K. Nakayama, and M. Dupuis, J. Chem. Phys. 105 (1996) 9227.

518 45 46 47 48 49

P.C. Hiberty, D. Danovich, A. Shurki, and S. Shaik, J. Am. Chem. Soc. 117 (1995) 7760. P. Schultz and R.P. Messmer, Phys. Rev. Lett. 58 (1987) 2416. H. Kollmar and V. Staemmler, Theor. Chim. Acta 48 (1978) 223. G.A. Gallup, J. Chem. Phys. 86 (1987) 4018. P.J. Garratt, Aromaticity, John Wiley, New York, 1986; chapter 11.


519

Pauling's Legacy: Modern Modelling of the Chemical Bond Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.

Aromaticity and its Chemical Manifestations Kenneth B. Wiberg Department of Chemistry, Yale University, N e w Haven, CT 06520 1. HISTORICAL PRELUDE In the 18th century, a number of naturally occurring compounds were isolated and described as "aromatic" because of their distinctive odor. 1 When the structural theory of organic chemistry was developed in the 19th century, it became apparent that most of these compounds were benzene derivatives. As a result, they became k n o w n as aromatic compounds, in contrast to aliphatic compounds. The benzene derivatives presented an enigma to structural chemists in that although the benzene rings had three double bonds, they underwent substitution rather than addition when treated with reagents such as bromine and nitric acid. No adequate explanation for their behavior was presented prior to the d e v e l o p m e n t of q u a n t u m mechanics. In the early 1930's, two explanations were presented. One was by Pauling making use of valence bond theory, 2 and the other was by E. H/ickel making use of molecular orbital theory. 3 Pauling's use of valence bond theory had a direct connection with the types of structures commonly used by organic chemists, and was relatively easy to understand, provided one did not delve too deeply into its details. The basic postulate was that compounds having n-electron systems that can be described by more than one structure will be stabilized by "resonance" and will have a lower energy than any of the contributing structures. Thus, for benzene one would write

()

--O'-- (D--"@'--"@

where the more important structures are the first two because they have a lower energy than the "Dewar" structures. This type of formulation was easily extended to reactions. 4 Thus, considering the electrophilic substitution of anisole, one might write: +

+

G; "-- C;

+

520 and the donation of n-electron density to the o and p positions would account for the preference for o,p substitution. Of course, it is now recognized that this type of stabilization would by itself lead to reduced reactivity, s and it is the corresponding interactions in the transition state that lead to the observed reactivity and position of substitution. The alternative was molecular orbital theory which was first applied to benzene by HLickel. Here, the energy levels were found to be -2~

13

la

,,,,

The total n-energy is 4xJ3 + 2x(2~) or 8 13, whereas that of three ordinary double bonds would be 3x2J3 or 613. Thus, benzene is stabilized by 213. This model was not cast in the terms usually used by organic chemists, and thus its adoption required considerable time. However, it led in an obvious way to the 4n+2 rule for aromaticity which has been a mainstay of organic chemists interested in the subject. It led to the prediction that the following ions would be stabilized in a fashion similar to that for benzene.

All of them are now known to be unusually stable species. 6 On the other hand, the following would be expected to have little (if any) stabilization and high reactivity because of the half-filled non-bonded orbitals.

,, ,,

(=')

In the case of the square conformation of cyclobutadiene, the energy levels would be -2~ 0

13

x

;

....

A

A

521 Here, two electrons go into the doubly degenerate non-bonding level leading to an unfilled level and high reactivity. The Hiickel n-energy w o u l d be 2x2~ or 4~, the same as that for two unconjugated double bonds. Hence the prediction that it will not be stabilized. It is k n o w n that cyclobutadiene is highly reactive and that the preferred conformation is rectangular in order to break the degeneracy of the highest occupied molecular orbitals. 7 Cyclopentadienyl cation has not been prepared despite many studies, 8 and cyclooctatetraene adopts a "tub" conformation in order to minimize interaction between the ~-orbitals. 9 Valence bond theory, in the terms defined by Pauling, is not able to account for the 4n+2 rule, and the properties o f cyclobutadiene and cyclooctatetraene. It has been suggested that the problem with these molecules is the strain associated with the bond angles in the planar structures, m However, this was shown to be incorrect by the observation that the addition of two electrons to cydooctatetraene leads to the planar dianion. It is only recently that it has been recognized that cyclic permutations must be included in order to properly treat cyclic systems via valence bond theory. 1~ One of Pauling's few failures in structural theory is his nonrecognition of the problems associated with the 4n molecules. 2. VALENCE BOND VS MOLECULAR ORBITAL THEORY: In 1927, Heitler and London carried out a calculation for the hydrogen molecule using what has become known as valence bond theory. ~2 Each electron of the pair could be assigned to nuclei corresponding to wavefunctions of the type W1 = (~a(1)~)b(2) and 92 = (~a(2)(~b(1) In the first, electron 1 is in the ls atomic orbital ~)a and electron 2 is in (~b. In the second, the electrons are reversed. The two wave function when considered separately lead to the same energy. If we wish to form the hydrogen molecule, it is necessary to take linear combinations of the above, and the two new wavefunctions become (neglecting the normalization constant): V + = ~a(1)~b(2) + ~a(2)~b(1) V- = ~a(1)~b(2) - ~a(2)~b(I)

where the first is for the bonding state and the second is for the antibonding state. The binding energy of a hydrogen molecule calculated using the bonding wave function is 86.7 kcal/mol with a bond length of 0.743/k (the experimental values are 109.5 kcal/mol and a bond length of 0.740A). ~a

522 In molecular orbital theory, molecular orbitals are formed by linear combinations of atomic orbitals, and the bonding molecular orbital is ~a + %.14 The molecular wave function is the product of the wave functions for the two electrons or [~a(1) + (~b(1)][~a(2) + %(2)]. When the energy is calculated using this wavefunction, it is found to be 80.0 kcal/mol with a length of 0.732A. In this case, the valence bond result is slightly more satisfactory than the molecular orbital result, but neither is really satisfactory. If we multiply out the MO wavefunction, we obtain V = {a(1)~b(2) + ~a(2)~b(1) + ~a(1)~a (2) + %(1)~b(2) It can be seen that the first two terms are the same as the valence bond wavefunction, and there are an additional two terms. The first two are c o m m o n l y called the covalent terms because each has the electrons associated with both centers. The final two are k n o w n as ionic terms because each places two electrons at o n e center (i.e. H § H ~ and H H*). Valence bond theory generally neglects ionic terms of this type, whereas in MO theory the covalent and ionic terms are treated equally. A better result can be obtained if the proportion of covalent and ionic terms can be adjusted. This can be done by mixing the doubly excited state with the g r o u n d state MO wavefunction. The new wavefunction is then V = [~)a(1) + ~b(1)][4)a(2) + r

+ a[~a(1)- %(1)][~a(2)- %(2)]

The effect of introducing the second terms can best be seen by multiplying out the wavefunction giving = (l+a)[ (~a(1)(~a(2) + (~b(l~b(2)] + (l-a)[ (~a(l~b(2) + (~a(2)(~b(1)] It can be seen that if a = 0, this becomes the MO wavefunction and that if it is -1, it is the Heitler-London wavefunction. With a=-0.59, a binding energy of 92 k c a l / m o l with a b o n d length of 0.748./k is obtained. 15 This is a significant i m p r o v e m e n t . The calculated binding energy may be further improved by using larger basis sets and extensive configuration interaction, and can be made to closely agree with the experimental values. 16 The purpose of the above digression is to indicate that neither VB or MO theory will give close to an exact answer unless configuration interaction is included in the calculation. It is reasonable to believe that both VB and MO will give exactly the same result if both large basis sets and extensive configuration interaction are employed. The reality is that we know how to push MO theory to near the limit, but we do not k n o w how to do VB calculations in a direct way. One n o r m a l l y

523 carries out a MO calculation at the HF level (including no configuration interaction), and then converts the MO wavefunction into VB wavefunctions. 17 Limited correction for electron correlation is generally effected by using Goddard's Generalized Valence Bond model (GVB) 18 or some variant of it. Additional configuration interaction would lead to mixing of the localized valence bond units, and would destroy the valence bond type of representation. It should be recognized that both the molecular orbitals and the valence bond counterparts are just mathematical constructs that facilitate the calculation of the properties of a molecule. They have no physical meaning. The physically meaningful quantity from any of these calculations is the charge density distribution that may be derived from the total wavefunction. From this distribution one can obtain all of the measurable ground state properties of a molecule, including the energy, dipole and higher electrical moments, etc. 19 3. MANIFESTATIONS OF "AROMATIC" STABILIZATION Several quantities have been used to obtain a measure of aromatic stabilization. They include: a. substitution rather than addition b. stabilization energy c. bond lengths d. NMR ring currents e. spectroscopic properties As noted above, the first definition of "aromaticity" was in terms of substitution rather than addition. This is certainly true for many benzene derivatives. However, it must be used with some care since thiophene is by most criteria about as "aromatic" as benzene, but when treated with chlorine or bromine it gives an addition product. The latter is, however, the kinetically controlled product, for when heated or treated with base it loses hydrogen halide and gives the 2halothiophene. 2~ Compounds such as anthracene and phenanthrene, which are recognized as having considerable resonance stabilization, also undergo addition reactions. Thermochemical stabilization is probably the most generally applicable of the simple criteria for "aromaticity". Pauling made use of heat of combustion of benzene and a set of average bond energies to derive a resonance energy of 37 kcal/mol, m The most useful measure of this quantity is derived from the heats of hydrogenation (kcal/mol) obtained by Kistiakowsky and his coworkers: 22

+6

-26

-28

524 The first step in the reduction is endothermic, accounting for the difficulty in hydrogenating benzene. A m i n i m u m value of the stabilization is given by the difference between the first and last steps in the above sequence, or 34 kcal/mol. It is believed that 1,3-cyclohexadiene is stabilized by about 2 kcal/mol, leading to the commonly stated 36 kcal/mol resonance energy of benzene. The resonance energies, or perhaps better, stabilization energies, are really not wholly satisfactory for the bond lengths in cyclohexene and benzene are not the same. There is an additional term, the "compression energy" required to make the bond lengths the same. Simpson has provided a useful way in which to consider the formation of benzene from the Kekule structures. 23 The square of a wavefunction represents a structure:

~/12 =

~2 2 =

and the energy of the Kekule structures will be given by E1 = J ~1 H~ ~a dr

E2 =~ V2 H~ V2 dl:

where H ~ is the appropriate Hamiltonian operator. The two energies are, of course, the same. In order to convert the Kekule structures to the geometry appropriate for benzene, the bond lengths must be changed. This may be considered as a perturbation. Thus

H=H~ where V is the perturbation operator. Here, this operator will lead to the following

Vll = J ~aV~ld~ = J ~l/2V~2dl: and

V12 = J ~1V~2 dl; Then since E 1 = E2, we write a secular determinant of the form E 1 + Vll - E

V12 =0

V12 and

E1 + V l l - E

525

E = E 1 + Vll + V12 The relationship between these quantities is shown in Figure 1. The compression energy may be estimated to be about 15 kcal/mol from a calculation of the change in energy with a change in bond lengths for benzene. 24 The structures having the same bond lengths mix to form a ground state and an excited state. The excited state that has the correct symmetry is found at 260 nm corresponding to 110 kcal/mol above the ground state. Thus, V12 = 55 kcal/mol. The estimated thermochemical resonance energy is then about 40 kcal/mol, in remarkable agreement with the observed value. 2s

-V. 2

compression , energy v" I

vertical stabilization energy

I

thermochemica ~

stabilization energy

Excited state

k +V12

Ground state _1_.

Figure 1. Relationship between vertical and thermochemical stabilization energies and the compression energy. This treatment may give us confidence in the use of the thermochemical stabilization energies of benzenoid compounds. Good values of the heats of formation of many of these compounds in the gas phase are not available. Therefore it is useful to examine some calculated values. The total energies of some annelated benzenes at the B3LYP/6-311G** level are given in Table 1. The zero-point energies of isomers in this series are essentially constant 26 and therefore differences in energy between isomers may be directly compared with experimental values. The experimental difference in energy between anthracene and phenanthrene is well reproduced. The calculated energy difference between naphthacene and chrysene is larger than the experimental difference, but the latter has considerable uncertainty

526 because of the difficulty in measuring the heats of sublimation at room temperature for compounds with high melting points. Table 1. Calculated energies, B3LYP/6-311G** Compound Be n z e n e Naphthalene Anthracene Naphthacene Pentacene Phenanthrene Chrysene Picene a. b. c. 1,

Energy a -232.30855 -385.98493 -539.65518 -693.32294 -846.98953 -539.66320 -693.33905 -847.01581

AEb

aHf r 19.7+0.2 35.9-!-_0.3 55.220.5 69.9+_2.2

-5.0 -10.1 -16.5

49.6+0.4 64.5+1.6

a a H f b Stab. E. per CC bond 36 12.0 60 12.0 80 11.5 99 11.0 117 10.6 5.9-!-_0.7 85 12.2 5.4~.7 109 12.1 133 12.1

Total energies are given in Hartrees, the other energies are given in kcal/mol. Relative energy with respect to the isomeric linearly annelated hydrocarbon. Pedley, J. B. "Thermochemical Data and Structures of Organic Compounds," Vol. Thermodynamics Research Center, College Station, TX, 1994.

Using the experimentally determined stabilization energies for benzene and naphthalene (36 and 60 kcal/mol respectively), one may derive the following expression for the stabilization energy of the benzenoid hydrocarbons based on the ab initio calculations: Stab. Energy = 627.5(E T + nCHX38.70853 + nCx38.11054) where E T is the calculated total energy, 627.5 is the conversion factor for Hartrees to kcal/mol, nCH is the number of CH groups and n c is the number of quaternary carbons. The energies thus derived are summarized in Table 1. It can be seen that the stabilization energy per C-C bond is constant for benzene, naphthalene and the [n]-phenacenes, but decreases with increasing number of benzene rings with the linearly annelated arenes. It is known that whereas benzene and naphthalene are relatively unreactive, the remaining compounds have much higher reactivity toward electrophiles and 1,2- or 1,4-additions. In the case of benzene, addition across a double bond will cause a 34 kcal/mol loss of stabilization (36 less 2 kcal/mol for the butadiene fragment) and addition to naphthalene will cause a 22 kcal/mol loss of stabilization (i.e. 60 k c a l / m o l less 38 kcal/mol for the remaining styrene unit). On the other hand, addition across the 9,10 positions in anthracene will only lead to an 8 kcal/mol loss

527 of stabilization (80 less 2x36) and addition to phenanthrene will only lead to a 13 k c a l / m o l (85 less 2x36) loss of stabilization. Thus, despite comparable stabilization energies per double bond, the latter two compounds would be expected to be m u c h more reactive than the former. With pentacene, the stabilization energy is less than that of the two naphthalene units formed by 1,4 addition across the central ring, and this corresponds to its high reactivity. On the other hand, the isomeric picene w o u l d have an energy change comparable to that of phenanthrene, thus accounting for its greater stability. The 4n cyclic polyunsaturated compounds have been of much interest. A B3LYP/6-311G** calculation for cyclobutadiene gave an energy of -154.71851, and using equation 1, the apparent stabilization energy is -73 kcal/mol. This must, however, be corrected for the strain present in the three-membered ring which should be on the order of 30 kcal/mol. Thus, the n system is destabilized by about 40 kcal/mol. This fits in well with the concept of "antiaromaticity" in which it is suggested that destabilization occurs with the 4n molecules. 27 In the case of planar cyclooctatetraene, the calculated energy is -309.64811, indicating that it is destabilized by 13 kcal/mol. This should be close to the strain associated with the 135 ~ C-C-C angles in the ring, and thus in this molecule the n system is neither stabilized nor destabilized. Addition of two electrons leads to a filled HOMO level, and now the ion is planar despite the strain associated with this geometry. There are several types of aromatic systems in addition to the ones described above. On example is azulene, a member of a class of nonalternant conjugated hydrocarbons. 28 As a result of the different bridging pattern as compared to benzene, the n-energy levels are considerably shifted leading to light absorption in the visible region, and a purple color, as well as reduced n-electron stabilization. 4. SIGMA CONTRIBUTION TO THE GEOMETRY OF BENZENE Most recent studies have concluded that the regular hexagonal geometry of benzene results from the t~ bonds rather than the n bonds. This is reasonable since the c~ C-C bonds are stronger than the n bonds by about 20 kcal/mol. One w o u l d expect the c bonds at each carbon would have the same hybridization if at all possible, and thus in the absence of any other overriding factor, benzene should adopt its observed geometry in order to minimize its c~ energy. The structural preference for the n-bonds is not as clear. Berry noted that the antisymmetric C-C stretching mode, which effectively interconverts K e k u l e type structures, had an unusually low frequency, and concluded that the n-system preferred a Kekule-like structure. 29 Similar conclusions have been derived from

528 other studies of benzene. 3~ However, the most recent study has concluded that the a and n electrons both prefer the hexagonal structure. 31 The group of simple annelated benzenes in Table 2 provide a c o n v e n i e n t context for examining the role of the n-electrons in determining bond lengths. 24 With benzene, naphthalene and anthracene for which good structural data are available, there is very good agreement between the calculated and observed C-C bond lengths. These lengths vary over a rather wide range, 1.339-1.451 /L It is possible to calculate bond indices from the ab initio wavefunctions using the m e t h o d developed by Fulton and Mixon. 32 Further, the total bond index may be separated into its c~ and n components. In the case of benzene, the C-C ~ bond order is 0.425, and the c~ bond index is 0.965. ~3 It is found that whereas the n components cover a wide range, from 0.21 to 0.52, the c~ components only vary from 0.92 to 0.98. Therefore the c~-bonding is relatively insensitive to bond lengths, and the latter is mainly controlled by the ~terms. They are related to the bond lengths as shown in Figure 2.

0.60 0.50 ,-70 o

0.40

-

0.30

-

o

"1o c-.

o c'~

0.20

.--,.,

r-,

0.10 0.0 -0.10

1.35

'

'

'

I

1.40

'

'

'

'

i

'

1 45.

'

'

'

I

1.50

'

'

'

'

1.55

length Figure 2. Relationship between the n-bond index and the C-C bond length for benzenoid hydrocarbons. It has been suggested that equalization of bond lengths is one characteristic of aromatic compounds. 34 However, the data in Table 2 shows that this is not a requirement for stabilization, and also shows that the n-system does have a large effect on the structure.

529 5. MAGNETIC PROPERTIES It is known that benzene and the other well recognized aromatic compounds have an unusually large diamagnetic susceptibility. 35 This is presumably due to the ring current that can be induced in these compounds in the presence of a magnetic field. 36 These ring currents will lead anisotropic susceptibilities with the tensor component normal to the aromatic ring being much larger than the in-plane components. 37 It is interesting to note that in contrast to the above, the antiaromatic compounds are calculated to have significant paramagnetic susceptibilities. 38 Although the susceptibility is a useful criterion, it is not so easily measured experimentally. One of the more useful criteria for "aromatic" character is derived from n m r chemical shifts. It is known that the protons of benzene are found at a lower field than ordinary olefinic protons, and it has been attributed to the ring current in the n-system which will reinforce the applied field at the protons. This has found confirmation in the observation that protons placed over an aromatic ring will be shifted upfield. 39 A particularly striking effect is found with the planar cyclooctadecanonaene in which the outer hydrogens have an unusually large downfield shift (8 9.28) and the inner hydrogens have a remarkably large upfield shift (8 -2.99).40 H H H H

H H

H

H H Schleyer has suggested that the calculated magnetic shielding at the center of a ring might be a useful indicator of aromatic character. 4~ Here, it was found that a plot of the shielding against the aromatic stabilization energies for a series of fivemembered ring heterocycles gave a good linear relationship. Cydopentadiene gave a very small shielding, and it went to -19 ppm with the well stabilized cyclopentadienyl anion. The introduction of a BH into the ring, which should lead to a 4e antiaromatic system, was calculated to give a +17 ppm (i.e., paramagnetic) shielding. Many aromatic and antiaromatic compounds were included in this study, and it provides the best justification for the use of NMR chemical shifts in studying

Table 2. Calculated and observed structures of acenes. Compound

Method

Benzene

Theor

1.394

IRa

1.390

n-Index

0.425

Naphthalene (1) Theor Xray

Anthracene (2)

b

1.375

C

1.415

D

E

F

G

H

J

K

1.425(1) 1.378(1) 1.421(1) 1.426(1)

n-Index

0.302

0.519

0.325

0.287

Theor

1.429

1.367

1.424

1.443

1.398

Xray

1.434(1) 1.369(1) 1.431(1) 1.441(1) 1.403(1)

ED^

1.437(4) 1.397(4) 1.422(16)1.437(4)1.392(6)

n-Index

0.268

0.557

0.286

0.245

0.380

1.433

1.364

1.429

1.450

1.390

1.409

1.450

n-Index

0.253

0.573

0.269

0.224

0.416

0.336

0.222

Theor

1.437

1.361

1.434

1.454

1.385

1.415

1.453

1.401

n-Index

0.245

0.585

0.257

0.213

0.436

0.314

0.210

0.370

1.413

1.378

1.406

1.380

1.413

1.424

1.434

1.356

Phenanthrene (5)Theor

i

1.431

1.422(3) 1.381(2) 1.417(4) 1.412(8)

Naphthacene (3) Theor

Chrysene ( 6 )

1.420

B

ED'

b

Pentacene (4)

A

VI

1.457

Xray'

1.428(9) 1.374(17) 1.386(14)1.399(15)1.412(8)1.416(8) 1.450(7) 1.341(10) 1.468(10)

n-Index

0.330

0.486

0.336

0.484

0.341

0.316

0.241

0.592

0.213

Theor

1.415

1.376

1.407

1.378

1.416

1.425

1.425

1.361

1.452

1.430

1.415

1.428

1.363

1.394

1.381

1.409

1.409

1.421

1.368

1.468

1.428

1.401

f

Xray

Picene (7)

n-Index

0.320

0.497

0.325

0.495

0.329

0.308

0.260

0.564

0.231

0.267

0.362

Theor

1.414

1.376

1.414

1.379

1.415

1.424

1.427

1.360

1.452

1.433

1.4Mg

n-Index

0.323

0.494

0.349

0.491

0.332

0.310

0.254

0.572

0.225

0.260

0.347

a. Pliva, J.; Johns, J. W. C.; Goodman, L. J. MoZ. Spectrsoc. 1991, 248, 427.

b. Brock, C. P.; Dunitz, J. D.; Hirshfeld, F. L. Acta Cryst. 1991,B47, 789. c. Ketkar, S. N.; Fink, M. J. Mol. Struct. 1981,77, 139. d. Ketkar, S. N.,; Kelley, M.; Fink, M.; h e y , R. C. J. Mol. Struct. 1981, 77, 127. e. Kay, M. I.; Okaya, Y.; Cox, D. E. Acta Cyst. 1971,B27,26. f. Cruickshank, D. W. J.; Sparks, R. A. Proc. Roy. Soc. (London) 1960,A258, 270.

g. L = 1.446 (0.253), M = 1.420 (0.290), N = 1.366 (0.535).

1

2

3

4

532 aromaticity. Some interesting observations were that benzene and n a p h t h a l e n e have the same shielding at the center of their rings (-11.5 ppm), and that the center ring of anthracene has greater shielding at its center (-14.3 ppm), and the opposite is true for the outer rings (-9.4 ppm). The magnetic criterion is, however, not a unique indicator of aromaticity. The linear p o l y m e t h i n i u m ions: 42 + (CH3)2N .-- (CH=CH)n-- CH=N(CH3)2 meet most of the requirements. Unlike the linear polyenes that have only a weak conjugation between the double bonds, and have alternating single and double bonds, these ions have essentially equal C-C bond lengths. The ~ electrons are delocalized in the same m a n n e r as with benzene, i.e. one n electron per C-C bond, and just as with benzene, the electronic spectra can be predicted using the free electron model. Although fitting most of the criteria for aromaticity, these ions cannot have significant magnetic properties. The latter requires a cyclic conjugated system. 6. ORIGIN OF THE STABILIZATION OF BENZENE. Although it is easy to demonstrate that benzene and other "aromatic" systems are stabilized, it is not as easy to determine the exact origin of the stabilization. Both valence bond and molecular orbital theories can provide a formalism for "explaining" the stabilization, and the latter can quantitatively account for the energy of benzene and its low reactivity. However, they do not provide a physical model for the stabilization. The latter must come from a consideration of the electron density distribution, for that alone determines the energy of a molecule. Dewar and Schmeising in 1959 provided a simple explanation for the stabilization of benzene which is probably very close to the correct answer. 43 They pointed out that the essential difference between a Kekule structure and benzene itself is that in the former the n-electrons are paired, whereas in the latter they are arranged one n-electron per bond. This delocalization of the n-electrons will reduce their electron repulsion, and result in a net decrease in the total energy. One may come to this conclusion by considering either the VB or the MO representations for benzene. 7. HETEROCYCLIC AROMATIC SYSTEMS. The replacement of CH groups of benzene by nitrogen leads to the azines. Pyridine, pyrazine and pyrimidine have essentially the same stabilization as benzene. Thus, hydrogen transfer reactions between 1,3-cyclohexadiene and the

533 azine to give benzene and a dihydroazine are calculated to be close to t h e r m o n e u t r a l . 44 A m o n g the diazines, only pyridazine, having a N=N bond, has reduced stabilization. The n-electron populations are found to be somewhat greater at the more electronegative nitrogens than at the carbons, but this appears to have little effect on the n-electron stabilization. The replacement of -CH=CH-of benzene by oxygen or sulfur to give furan and thiophene again leads to a 6 n-electron system. The higher electronegativity of oxygen leads to a concentration of n-electrons at oxygen, and significantly reduced nstabilization. Thus, in many ways it acts as an enol ether, and it readily undergoes the Diels-Alder reaction. On the other hand, thiophene with a sulfur that has about the same electronegativity as carbon, has stabilization and properties similar to that of benzene. 45 The replacement o f - C H = C H - o f benzene by BH would lead to a 4 n-electron system which should be antiaromatic, and it has been calculated to have a paramagnetic shift at the center of its ring, in contrast to the diamagnetic shift characteristic of the 6 n-electron systems. 41 8. SUMMARY. Aromaticity may be considered as the stabilization of a molecule that results from x-electron delocalization in a closed shell system. It has a number of chemical manifestations such as an endothermic first hydrogenation energy, a tendency toward bond length equalization, enhanced diamagnetic susceptibilities and characteristic nmr chemical shifts. However, as noted above, none of these are really unique. Polycyclic aromatic compounds having significant n-electron stabilization often have a variety of bond lengths. Linear aromatic systems cannot have special magnetic properties since such properties result from cyclic delocalized systems. In discussing resonance, Pauling stated "The theory of resonance in chemistry is essentially a qualitative theory, which like the classical structural theory, depends for its successful application largely upon a chemical feeling that is developed through practice. "46 The same is true with aromaticity.

534 References:

1 Aromatic is defined as a substance "characterized by a fragrant smell, and usually a by a warm, pungent taste." (Merriam-Webster New International Dictionary). Reviews: Garratt, P. J. "Aromaticity," Wiley, NY 1986. Minkin, V. J.; Glukhovtsev, M. N.; Simkin, B. Y. "Aromaticity and Antiaromaticity: Electronic and Structural Aspects," Wiley, NY 1994. 2 Pauling, L. J. Chem. Phys. 1933, 1,280. Pauling, L.; Wheland, G . W . J . Chem. Phys. 1933, 1,362. Pauling, L.; Sherman, J. I. Chem. Phys. 1933, 1,679. 3 H/ickel, E, Z. Physik. 1931, 70, 204. H/ickel, E. Z. Elektrochem. angew, physik. Chem. 1937, 42, 827. 4 Ingold, C. K. "Structure and Mechanism in Organic Chemistry," Cornell Univ. Press, Ithaca, NY 1953, p. 238. 5 When two or more significant resonance structures may be written for a molecule, it will be stabilized with respect to the basic structure. As a result, the energy required to reach the transition state will be increased unless there are corresponding interactions in the transition state. The latter will be the case for electrophilic substitution on anisole. 6 C5H5-: Bordwell, F. G.; Drucker, G. E.; Fried, H. E. J. Org. Chem. 1981, 46, 632. C7H7+: Doering, W. v. E.; Knox, L. H. J. Am. Chem. Soc. 1954, 76, 3203. C8H8-2: Katz, T. J. J. Am. chem. Soc. 1960, 82, 3784. 7 This is an example of Jahn-Teller distortion: Jahn, H. A.; Teller, E. Proc. Roy. Soc. 1937, A161, 220. 8 Breslow, R.; Mazur, S. J. Am. Chem. Soc. 1973, 95, 584. 9 Bastiansen, O.; Hedberg, L.; Hedberg, K. J. Chem. Phys. 1957, 27, 1311. 10 Wheland, G. "The Theory of Resonance and its Applications to Organic Chemistry." Wiley, NY. 1944, p. 94. 11 Kawajima, S. J. Am. Chem. Soc. 1984, 106, 6496. Maynau, D.; Malrieu, J.-P. J. Am. Chem. Soc. 1982, 104, 3029. Mulder, J. J. L.; Oosterhoff, L. J. Chem. C o m m u n . 1970, 305, 307. 12 Heitler, W.; London, F. Z. Physik. 1927, 44, 455. 13 Wang, S. C. Phys. Rev. 1928, 31,579. 14 Coulson, C. A. Trans. Faraday Soc. 1937, 33, 1479. 15 Weinbaum, S. J. Chem. Phys. 1933, 1,593. 16 In this simple example, configuration interaction serves mainly to correct for the use of a minimal basis set, and allows correct dissociation. When m o r e complete basis sets are used, configuration interaction serves to correct for electron correlation. 17 Cf. Schultz, P. A.; Messmer, R. P. J. Am. Chem. Soc. 1993, 115, 10943.

535 18 Bobrowicz, F. W.; Goddard, W. A., UI, in "Methods of Modem Electronic Structure Theory. Vol. 3. Modern Theoretical Chemistry." Schaefer, H. F., UI, Ed. Plenum Press, NY 1977. ~9 Parr, R. G.; Yang, W. "Density-Functional Theory of Atoms and Molecules," Oxford Univ. Press, NY, 1989, p. 51ff. 20 Coonradt, H. L.; Hartaugh, H. D. J. Am. Chem. Soc. 1948, 70, 1158. Blicke, F. F.; Burckhalter, J. H. J. Am. Chem. Soc. 1942, 64, 477. 21 Pauling, L. "The Nature of the Chemical Bond," 3rd Ed, Cornell Univ. Press, Ithaca, 1960, p. 193. 22 Kistiakowsky, G. B.; Ruhoff, J. R.; Smth, H. A.; Waughan, W. E. J. Am. Chem. Soc. 1935, 57, 876; 1936, 58, 237, 146. 23 Simpson, W. T, J. Am. Chem. Soc. 1953, 75, 597. 24 Wiberg, K. B. J. Org. Chem. 1977 to be published. 25 It should be noted that there is not general agreement on how stabilization energies should be calculated. Cf. Chestnut, D. B.; Davis, K. M. J. Comput. Chem. 1997, 18, 584. 26 Cioslowski, J.; Liu, G.; Martinov, M.; Piskorz, P.; Moncrieff, D.J. Am. Chem. Soc. 1966, 118, 5261. 27 Breslow, R. Acct. Chem. Res. 1973, 6, 393. 28 Coulson, C. A.; Longuet-Higgins, H. C. Proc. Roy. Soc. London 1947, A192, 16. 29 Berry, R. S. J. Chem. Phys. 1961, 35, 2253. 30 Shaik, S. S.; Bar, R. Nouv. J. Chim. 1984, 8, 411. Hiberty, P. C.; Shaik, S. S.; Lefour, J.-M.; Ohanessian, G. J. Org. Chem. 1985, 50, 4657. Shaik, S. S.; Hiberty, P. C.; Lefour, J.-M.; Ohanessian, G. J. Am. Chem. Soc. 1987, 109, 363. Shaik, S. S.; Hiberty, P. C.; Lefour, J.-M.; Ohanessian, G. J. Phys. Chem. 1988, 92, 5086. Hiberty, P. C.; in Topics in Current Chemistry, Gutman, I.; Cyrin, S. J. Eds. Springer: New York, 1990, Vol. 153, p. 27. Jug, K.; Koster, A. M. J. Am. Chem. Soc. 1990, 112, 6772. 31 Glendening, E. D.; Faust, R.; Streitwieser, A, Vollhardt, K. P. C; Weinhold, F. J. Am. Chem. Soc. 1993, 115, 10952. Cf. I-Iiberty, P. C.; Ohanessian, G.; Shaik, S. S.; Flament, J. P. Pure Appl. Chem. 1993, 65, 35. 32 Fulton, R. L. J. Phys. Chem. 1993, 97, 7516. Fulton, R. L.; Mixon, S. T. J. Phys. Chem. 1993, 97, 7530. 33 The n-bond index is less than 0.5 because there are also small 1,3 and 1,4 contributions. 34 Julg, A.; Francois, Ph. Theor. Chim. Acta 1967, 7, 249. Cf. Minkin, et. al., ref. 1. 35 Pauling, L. J. Chem. Phys. 1936, 4, 673. Dauben, H. J., Jr.; Wilson, J. D.; Laity, J. L. J. Am. Chem. Soc. 1968, 90, 811; 1969, 91, 1991. Dauben, J. J., Jr. "Diamagnetic

536 Susceptibility Exhaltation as a Criterion of Aromaticity," in "Non-Benzenoid Aromatics," Snyder, Ed., Vol 2, Academic Press, NY, 1971. 36 Pople, J. A. J. Chem. Phys. 1956, 24, 1111 37 Benson, R. C.; Flygare, W. H. J. Am. Chem. Soc. 1970, 92, 7523. Hurter, D. H.; Flygare, W. H. Top. Curr. Chem 1976, 63, 1976. 38 Schleyer, P. v. R.; Jiao, H. Pure and Appl. Chem. 1996, 68, 209. 39 Cf. Vogel, E.; Roth, H. D. Angew. Chem. Int. Ed. Engl. 1964, 3, 228. Gaoni, Y.; Malera, A.; Sondheimer, F.; Wolovsky, R. Proc. Chem. Soc. 1964, 397. Boekelheide, V.; Phillips, J. B. J. Am. Chem. Soc. 1967, 89, 1695. 40 Baumann, H.; Oth, J. F. M. Helv. Chim. Acta 1982,65, 1885. 41 Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N. J. R. J. Am. Chem. Soc. 1996, 118, 6317. 42 Dauben, H. J., Jr.; Feniak, G. unpublished results. Cf. Feniak, G. Ph.D. Thesis, University of Washington, 1955. Wiberg, K. B. "Physical Organic Chemistry," Wiley, NY 1965, p. 9. D~me, S.; Hoffmann, K. Prog. Phys. Org. Chem. 1990, 18, 1. 43 Dewar, M. J. S.; Schmeising, H. N. Tetrahedron 1959, 5, 166. 44 Wiberg, K. B.; Nakaji, D.; Breneman, C. M. J. Am. Chem. Soc. 1989, 111, 4178. 45 Gronowitz, G., Ed. "Thiophene and Its Derivatives," Wiley/Intersciene, NY 1985. 46 Ref. 21 p. 220.


537

H y p e r c o o r d i n a t e b o n d i n g to m a i n g r o u p e l e m e n t s : t h e s p i n - c o u p l e d p o i n t of v i e w David L. Cooper a, Joseph Gerratt b and Mario Raimondi c aDepartment of Chemisty, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, United Kingdom bSchool of Chemistry, University of Bristol, Cantocks Close, Bristol BS8 1TS, United Kingdom CDipartimento di Chimica Fisica ed Elettrochimica, Universit~ di Milano, Via Golgi 19, 20133 Milano, Italy Modern valence bond theory, in its spin-coupled form, has been used to investigate the nature of hypercoordinate bonding to main group elements. The systems that have been studied in this way include: halides, oxides and oxohalides of phosphorus, sulfur and chlorine; xenon fluorides; SiH~ and CH~; 1,3-dipoles; oxohalides of hypercoordinate nitrogen; fluorophosphoranes; and YXXY dihalides and dihydrides of dioxygen and disulfur. The bonds involving hypercoordinate atoms tend to be highly polar. We find no significant qualitative differences between the hypercoordinate nature of first-row, second-row and noble gas atoms in appropriate chemical environments, nor between the descriptions of the bonding in hypercoordinate and so-called 'normal octet' molecules, except for some differences in bond polarity. We suggest that the 'octet rule' be demoted in favour of the democracy principle: almost all valence electrons can participate in chemical bonding if provided with sufficient energetic incentives. Simple concepts of atomic size and of electronegativity differences prove to be of particular utility in qualitative descriptions. We find no evidence for the utilization of d functions as valence orbitals, or to support notions of p~-d~ back-bonding. 1. I N T R O D U C T I O N

In spite of all the theoretical evidence, accumulated over many years, it is still commonplace for students to be taught that the existence of hypercoordinate molecules such as SF~ and PF 5 relies on the utilization of d orbitals to 'expand the octet'. Indeed, even models based on d2sp3, dsp 2 and dsp 3 hybrid orbitals or p~-d~ back-bonding are still in use to describe hypercoordinate bonding to second-row elements. Of course, the consensus view that has emerged from most of the

538 reliable ab initio investigations published in recent years is that d functions act as polarization functions for second-row atoms, compensating for the inflexibility of s/p basis sets, albeit to a somewhat greater extent t h a n for first-row atoms. See, for example, Refs. 1-11, and references therein. Certainly, it is not justified to regard these d functions as valence orbitals. There are m a n y texts that make the point very clearly t h a t the bonding in a molecule such as SF, has very little to do with the availability of d atomic orbitals, but this is normally done in the context of MO theory, whereas the general ideas of utilizing d orbitals are much more closely allied with the ideas of classical valence bond theory. This, perhaps, is one of the reasons for the continued survival of such models. The purpose of this Chapter is to describe various calculations which have been performed using modern valence bond theory, in its spin-coupled form, resulting in a useful aide memoire which we term the democracy principle. We argue that there are no significant qualitative differences between the hypercoordinate nature of first-row, second-row and noble gas atoms in appropriate chemical environments. 2. d - O R B I T A L P A R T I C I P A T I O N V E R S U S D E M O C R A C Y

The basic methodology that we have used to study the chemical bonding to hypercoordinate main-group elements is much the same as t h a t described in an earlier Chapter: "The spin-coupled description of aromatic, antiaromatic and nonaromatic systems". The spin-coupled wavefunctions for the systems considered here take the form

~'~

=

2 2 20~:: A_ { ~,~...~,, O~O~... ONOo~}

(1)

in which the ~ are distinct, singly-occupied, non-orthogonal orbitals for the N 'active' electrons, O~ is an N-electron eigenfunction of S~ and Sz for S=O (and thus M=0), and A is the antisymmetrizer. The doubly-occupied orbitals r accommodate the 'inactive' electrons and O~'~is the corresponding perfectly-paired spin function. The total wavefunction is invariant to normalization of the orbitals, to orthogonalization of the q); amongst themselves, as well as to orthogonalization of the active orbitals to the inactive ones. On the other hand, once the division into inactive and active spaces has been made, the converged spin-coupled wavefunction is a unique outcome of the variational procedure: it is not invariant to general linear transformations of the 0~The basic strategy adopted for 'normal octet' and hypercoordinate molecules XY,, was first to carry out a standard closed-shell RHF calculation and then to localize the orbitals according to the population or overlap criterion introduced by Pipek and Mezey [12]. In all cases, it was straightforward to identify localized molecular orbitals (LMOs) associated with particular X ~ Y bonds: Visual inspection of the bond LMOs for various phosphorus and sulfur halides revealed no evidence for the active participation of d orbitals. For example, not

539 only are the P - - F LMOs in PF 3 very similar to those in PFs, but there is little change on excluding d functions from the calculation. Furthermore, contrary to the expectations of the standard dsp 3 hybridization model of the bonding in PFs, it was difficult to distinguish P--Fa~ from P--Fe, , as is demonstrated in Figure 1.

P-Fax

P-Feq

;,---, \ i af-x~

I

F

F

P - - F eq

F

P-Fax

/'Z:'-.

; ,".--,: ; tl

tJ--

t]

I

I I l lLLI~Y,-,,,

,'

no dp

no

@

Figure 1" Localized molecular orbitals for PF 5. In the subsequent spin-coupled calculations [9], the ~, were optimized as linear combinations of all the LMOs which correspond to X--Y bonds, plus all the virtual orbitals. This scheme is entirely equivalent to expanding the ~, in the full b~sis atomic basis set, except that it maintains the orthogonality between the active orbitals and the inactive space, which consists of all the other doublyoccupied MOs. The spin function O0N was fully optimized in the full spin space. The active orbitals were thus fully optimized without constraints on their form, on the degree of localization, on the overlaps between them, or on the mode of coupling the electron spins. Nevertheless, we found for each molecule t h a t the optimized spin-coupled orbitals consist of pairs, each clearly associated with a particular two-centre bond, and with predominantly singlet coupling of the electron spins. For example, we show in Figure 2 the pair of spin-coupled

540 orbitals, ~1 and ~2, associated with one of the S---F bonds in SF 6. The first of these orbitals is a two-centre function: it comprises the combination of an spX-like hybrid from sulfur with some 2p character from fluorine. The second orbital is largely a 2p function on fluorine. Spin-coupled orbitals ~3-~ can be obtained from ~ and ~2 by symmetry operations of the molecular point group. It is clear from the parentage of the r that the S---F bonds in SF 6 bonds are very polar. Perfect-pairing dominates the total spin function, with a contribution in excess of 99%. The dominant orbital overlaps (-0.8) occur of course within the pairs that describe each bond, but members of different pairs are not orthogonal: for example, the overlap between different sulfur+fluorine hybrids is -0.3. The form of the spin-coupled orbitals, the overlaps between them and the mode of spin coupling change very little if d functions are excluded from the calculations. Perhaps not surprisingly, we could find no evidence to support a traditional d2sp 3 model of the bonding.

1

r

Figure 2: Symmetry-unique spin-coupled orbitals for SF 6. The spin-coupled description of the bonding in PF 5 resembles that in SF6, with each bond described by the overlap of a phosphorus+fluorine hybrid, split almost equally between P and F, and a distorted F(2p) orbital. The bonds are clearly highly polar. It was difficult to discern differences between the various phosphorus+fluorine hybrids or fluorine orbitals in equatorial and axial positions. In the standard dsp :~ hybridization model of PF~, the three equatorial bonds, based on P(sp2), are supposed to be somewhat different from the two axial bonds, based on P(pd). There is no evidence for this in the spin-coupled calculations or for significant involvement of d orbitals in the bonding. It is useful to bear in mind, contrary to the expectations of the dsp 3 model, that the two sets of bond lengths are very similar, differing by no more t h a n expected from different steric repulsions, and that the two sets of 19F chemical shifts are very similar.

541 Calculations of this type have also been performed for PX n (n=3,5) and SX, (n=2,4) fluorides and chlorides [9]. Much the same basic picture emerges for all of these systems, whether 'normal octet' or hypercoordinate, with the variations in the amount of central-atom character in the two-centre spin-coupled orbital reflecting the polarity of the particular bond. Analogous descriptions were found to apply for XeF,, (n=2,4) and for SiX~ ions (X=H,F). The outcome of our own various numerical experiments with exponents of d basis functions are consistent with the findings of many earlier studies. Simple qualitative notions of significantly polar bonds are certainly of greater utility t h a n supposed d orbital participation in hybridization schemes. Optimum d exponent(s) for a given second-row element, as well as the energy improvement per bond, change very little from 'normal octet' to hypercoordinate systems. Furthermore, the nature of the attached groups and the expansion of the octet have less effect on d function exponents than does the effective nuclear charge of the second-row atom which bears them. From the point of view of the total energy, it can be much more efficient to put d functions on attached electronegative atoms than on the second-row atom. The (small) utilization of d functions on the central atom tends to diminish with increasing quality of s/p basis set. Based on our findings for all of the systems we have studied, we assert the

democracy principle: Almost all valence electrons can participate in chemical bonding if provided with sufficient energetic incentives. The 'constitution' of this principle is founded on the principle of minimizing the total energy, and it is ultimately this last criterion alone which determines how m a n y electrons a particular atom will utilize in chemical bonding. Nevertheless, we can identify features which are likely to be favourable, such as polar bonds which shift density away from a central atom, especially if the formal number of bonds is high. Differences in electronegativity can be a useful first guide to the possible existence of a particular hypercoordinate species. Of course, the sizes of atoms may be such that it is not possible in some cases to cram sufficient electronegative atoms around the central atom. This is one of the reasons why hypercoordinate bonding is likely to be less common for first-row atoms. However, it is clear from the chemistry of xenon, for example, in which the highest formal oxidation states are achieved in oxides and oxofluorides, that doubly-bonded oxygen atoms are at least as effective partners in hypercoordinate bonding as are fluorine atoms. One consequence of this line of reasoning is that in sterically limited situations, such as around first-row atoms, hypercoordinate molecules are more likely to feature at least some X=O highly polar double bonds t h a n larger numbers of X - - F bonds. Concentrating on the key aspects of size and electronegativity, it is easy to make various straightforward predictions/rationalizations, albeit some of them with more t h a n a little hindsight. Fluorine and, to a lesser extent, chlorine are

542 sufficiently electronegative relative to phosphorus for the formation of PF~ and PCI~, but nitrogen is too small for the formation of a stable NF~ molecule. However, the formal replacement of two N u F bonds by N=O should significantly reduce the steric crowding, such that it would be surprising if F3NO did not feature hypercoordinate bonding. We return to this issue in Section 3.2. Sulfur is more electronegative t h a n phosphorus, and it seems that fluorine can still form sufficiently polar bonds for species up to SX,, but that chlorine cannot. Even SC14 is generally formulated as SCI:~CI-, rather t h a n as a hypercoordinate species analogous to SF 4. Moving instead one to the left of phosphorus, it should come as little surprise t h a t silicon should form SiF~ and SiFt-. Indeed, even SiH~, and various derivatives, are stable intermediates that can be studied, for example, in flowing afterglows [13]. One of the models that has been proposed for the bonding in SiH~ invokes resonating axial bonds, one based on Si(3pz) and the other on overlap of H(ls) with an antibonding (~* orbital from the equatorial Sill 3 unit, with delocalization of the fifth valence electron into the equatorial S i - - H bonds [14]. The spin-coupled calculations [9], on the other hand, reveal a mode of bonding t h a t is analogous to that in SiF~ or PF~, and the descriptions of the axial and equatorial bonds S i u H are very similar to one another. Given the relative electronegativities of carbon and hydrogon, it is easy to rationalize why CH~ should be no more t h a n a high energy transition state in certain reactions. In the spin-coupled description of a molecule such as SF 6, the sulfur atom contributes six equivalent, nonorthogonal spX-like hybrids which delocalize onto the fluorine atoms. Each of these two-centre orbitals overlaps with a distorted F(2p) function and the perfect-pairing spin function dominates. Of course, using only 3s, 3px, 3p~ and 3pz atomic orbitals, we can at most form four linearly independent hybrid orbitals localized on sulfur, with a m a x i m u m occupancy of 8 electrons, as in the octet rule. However, the six sulfur+fluorine hybrids which emerge in the spin-coupled description are not linearly dependent, precisely because each of them contains a significant amount of F(2p) character. It is thus clear t h a t the polar n a t u r e of the bonding is crucial. Unlike classical valence bond theory, the spin-coupled approach does not presuppose the form of the orbitals or constrain them to be one-centred. Instead, each orbital is allowed to delocalize onto other centres as much or as little as is necessary to minimize the total energy. Numerous studies have shown t h a t a u g m e n t i n g the spin-coupled configuration with those in which one or more active orbitals is doubly-occupied, has a very modest effect on the total energy. The reference configuration remains dominant, so that the essential physical picture is unchanged. Even a very small utilization of basis functions from other centres can correspond in classical VB terms to the significant utilization of very large n u m b e r s of ionic structures. In the case of the spin-coupled description of the bonding SF6, the corresponding classical VB wavefunction involves resonance between the vast n u m b e r of possible ionic structures. There is nothing intrinsically wrong with such a description, except t h a t we believe it to be unnecessarily complicated.

543 In addition to asserting the democracy principle, that all valence electrons can in principle be used in bonding, we suggest that the much-loved octet rule should be demoted. The bonding in hypercoordinate species does not differ in any significant qualitative fashion from the bonding in 'normal octet' molecules. Furthermore, there are of course plenty of examples, ranging from carbenes to much of the chemistry of heavier main group elements ('inert pair effect'), for which the central atom does not achieve an octet. Of course, there is a great deal of evidence to support the notion that it can be favourable to achieve a formal count of four electron pairs around a central atom, often arranged in a pseudotetrahedral fashion, and so we should retain some sort of 8-electron rule. We should not, however, see such an 8-electron arrangement as some sort of norm, from which deviations have to be explained in terms of additional effects.

3. HYPERCOORDINATE BONDING TO FIRST-ROW ATOMS 3.1.1,3-dipoles Examples of 1,3-dipoles include diazoalkanes, nitrones, carbonyl ylides and fulminic acid. Organic chemists typically describe 1,3-dipolar cycloaddition reactions [15] in terms of four out-of-plane '~ electrons' from the dipole and two from the dipolarophile. Consequently, most of the interest in the electronic structure of 1,3-dipoles has been concentrated on the distribution of the four electrons over the three heavy atom centres. Of course, a characteristic feature of this class of molecules is that it presents awkward problems for classical valence theories: a conventional fashion of representing such systems invokes resonance between a number of zwitterionic and diradical structures [16-19]. Much has been written on the amount of diradical character, with widely differing estimates of the relative weights of the different bonding schemes. From various comparisons of the bond lengths of 1,3-dipoles with those of diatomic species, many authors have commented that the geometries of 1,3-dipoles appear to be consistent with hypercoordinate bonding at the central heavy atom. For example, the very short bond lengths between the heavy atoms in CH2N 2 suggest fully-formed C=N and N~-N multiple bonds. Similarly, the experimental bond lengths in N20 are very close to the values expected for an N - N triple bond and for an N=O double bond. It is tempting to represent this molecule as N-N=O but, according to Pauling [20], 'this formula suggests that the nitrogen atom can form five covalent bonds, which is not true'. Of course, this statement was linked to the octet rule and it seems worthwhile to challenge its validity. Indeed, there have been several attempts to explain how a nitrogen atom might acquire an apparent valency of five, including suggestions of resonance between 'increased-valence' structures built from undistorted atomic orbitals [21]. The mode of bonding which emerges from spin-coupled descriptions on various of 1,3-dipoles and related molecules [22-24] turns out to be closely analogous to the one that we have since established for second-row atoms. Calculations have

544 been performed at many levels, but always result in the same basic picture. For diazomethane, for example, spin-coupled calculations have been performed explicitly for the four 'out-of-plane' ~ electrons, accommodating the inactive electrons in doubly-occupied orbitals taken from either an RHF or an appropriate CASSCF calculation, or optimizing them simultaneously with the active orbitals. In addition, the active space has been increased in various ways, treating explicitly also the 'in-plane' ~ system and/or the (; bonding in the heavy atom backbone and/or the nonbonding electrons on the terminal nitrogen and/or the C - - H bonds. In all cases, we employed the full spin space and we did not impose any symmetry requirements on the active orbitals. The description of the out-ofplane x system changes remarkably little with the level of calculation.

7T2

71"1

i I t I

I 11~.-..~'~\\ % I . tt~\',.. I I \ ~ I 1,1~":~ll I I I

H ~'""'

~" N

,,:_-'_='=',.,". " \x

"" . . . .

"

11

71"3

H C

7T4

HC -.

-.__s

/

Figure 3a: Spin-coupled orbitals for the out-of-plane ~ system of CH2N~. The four spin-coupled orbitals for the out-of-plane ~-electron system of diazomethane are illustrated in Figure 3a as contours in the (;v mirror plane, perpendicular to the molecular plane. Each of these orbitals (~1-n4) takes the form of a deformed 2p~ function, slightly distorted towards one of the neighbouring centres, with one orbital on each of the terminal heavy atoms and two on the

0 H

H

N

N

N

N

L

0 H

H

H

,-.

\

c

N

-

a):.; I

-

:I

3550 donor 3500 3450

1 n

b) 236o --acceptor

/

E

o. 2340

/ 1

Z

o >

2320

2300

1

2

3

4

n

Fig. 4. Calculated s t r e t c h i n g frequencies of a) C-H and b) C - N bonds in linear a r r a n g e m e n t s of (HCN)n, n = l - 5 . Donor refers to the l e f t m o s t molecule ( a s s u m e d to be of the lowest frequency) and acceptor to the r i g h t m o s t (highest v) in NCH"(NCH)n_2..NCH. D a t a derived at SCF/[53/3] level in [65].

5.4. Energy Components J u s t as in the case of the dimers discussed above, it is possible to decompose the e n e r g y of a cluster of n molecules so as to extract i n f o r m a t i o n about the u n d e r l y i n g cause of the cooperativity t h a t is observed. The condensed s t a t e of w a t e r serves as p e r h a p s the m o s t ubiquitous s i t u a t i o n w h e r e cooperativity

587 exists, and is of greatest interest to biology and Pauling's predictions about the physiological importance of H-bonds. The water molecules in ice are arranged in what can be described as hexamers, as illustrated in Fig. 5. Note that this structure is not of the purely sequential type where all six molecules act as both donor and acceptor: one molecule (#5) serves as double donor and another (#4) as double acceptor. / O s ~

..

.N

ss

.., ,,.' "

d

I

~4 . . . . . . . . . .

03

"., ,,,, .,. ,,. ,,. ,,.

Fig. 5. Proton donor and acceptor characteristics in water hexamer examined in [66].

The two and three-body interaction energies in this water hexamer are listed in Table 6. The two-body term is defined as the interaction energy computed for any pair of subunits, in the absence of any others, and in the geometry adopted in the oligomer. Three-body terms refer to the total interaction energy of any given triad of subunits, minus the sum of the three two-body interactions present in this same triad. In essence, the three-body term is similar to the cooperativity parameters described in Section 5.2, except t h at the geometries are not reoptimized for each of the monomer, dimer, and trimer, but are instead all frozen in the structure of the oligomer. Along with the total interaction energies listed in the first column of data in Table 6, this term (two or three-body) is decomposed in the following columns into its electrostatic, exchange, polarization, and charge transfer components, using the same formalism as mentioned in Section 3. Beginning our discussion with the two-body terms in the upper part of Table 6, the results for all adjacent molecules are identical to the data for the 12 pair in the first row of the table. This similarity arises because all adjacent pairs constitute a single H-bond; the concept of double donor or acceptor is only meaningful within the context of three or more molecules. The interaction energy amounts to -2.8 kcal/mol. The attractive electrostatic energy is canceled by the exchange repulsion; polarization and charge transfer energies are both attractive. For all nonadjacent pairs, the EX, POL, and CT terms are quite small, leaving only electrostatic energy in the pairwise interactions. Because of the long-range character of the ES term, there are significant contributions even from molecules on opposite ends of the ring, e.g. 1-4 or 3-6. The signs of these nonadjacent pairwise electrostatic energies can be understood on the basis of the orientations of the particular molecules. For example, molecules 3 and 5 have H atoms pointed at one another, leading to the repulsive 3-5 term. When summed together, the two-body terms amount to -19.2 kcal/mol, less attractive by 4 kcal/mol than the full interaction energy in the hexamer (see

588 last row). With respect to the individual components, the electrostatic term is by n a t u r e fully additive, so the sum of two-body terms is equal to the full ES e n e r g y of the h e x a m e r . The exchange is very n e a r l y additive, with a discrepancy of only 0.1 kcal/mol. The sum of two-body polarization and charge t r a n s f e r components are each about 2 kcal/mol less attractive t h a n the full components in the hexamer.

Table 6 Components (kcal/mol) of two and three-body interactions computed for the water hexamer illustrated in Fig. 5. Data from [66]. AEelec

ES

EX

POL

CT

2-body terms 1-2 1-3 2-4 3-5 4-6 1-5 1-4 2-5 3-6 SUM (]~AE2)

-2.8 -1.2 -0.6 0.8 0.4 -0.6 02 0.2 -0.4 -19.2

-13.1 -1.1 -0.6 0.8 0.4 -0.6 0.2 0.2 -0.4 -81.0

14.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 87.0

-1.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -8.4

-2.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -16.8

3-body terms 1-2-3 2-3-4 3-4-5 4-5-6 1-3-5 2-4-6 1-24 1-3-4 1-4-5 2-5-6 SUM (EAE 3)

-1.4 -1.0 0.8 1.2 0.0 0.0 -0.1 -0.1 0.2 -0.2 -3.8

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 -0.1 0.2 0.0 0.0 0.0 0.0 0.0 0.0 -0.1

-0.8 -0.6 0.5 0.4 0.0 0.0 0.0 -0.1 0.1 -0.1 -2.2

-0.6 -0.4 0.4 0.5 0.0 0.0 0.0 0.0 0.1 0.0 -1.4

]~AE2 + ZAE 3

-23.0

-81.0

86.9

-10.6

-18.3

total in hexamer

-23.4

-81.0

86.9

-10.9

-18.5

The three-body t e r m s are listed in the lower part of Table 6. The first several entries r e p r e s e n t triplets of consecutive molecules a r o u n d the ring; these interactions can be of either sign. Repulsive terms are associated with triplets like 3-4-5 and 4-5-6 t h a t contain either a double-donor or double-

589 acceptor; others are attractive. The ES contributions are identically zero and the exchange Components are quite small. The three-body terms are composed of similar amounts of polarization and charge transfer components. It is worth noting that some of the three-body terms, e.g. 1-2-3, are of larger magnitude t h a n certain pairwise interactions, particularly those between nonadjacent pairs. Nonconsecutive triplets may contain no adjacent pairs, as in 2-4-6, or one adjacent pair, e.g. 1-2-4. In the former case, the three-body energies are less than 0.1 kcal/mol; the latter are all less than 0.2 kcal/mol. The total of all three-body interactions is -3.8 kcal/mol, as compared to -19.2 kcaYmol for the sum of all two-body interactions. When added together, the total of all pairwise and three-body interactions comes within 0.4 kcal/mol of the total interaction energy of-23.4 kcaYmol in the hexamer. With respect to the individual components, there is very little nonadditivity in ES or EX. The total nonadditivity of some 4 kcal/mol is approximately equally divided between POL and CT. 6. SUMMARY It is remarkable that many of the ideas formulated over fifty years ago by Linus Pauling about hydrogen bonds remain valid today, after accumulation of r e a m s of q u a n t i t a t i v e d a t a from both e x p e r i m e n t a l and theoretical perspectives. The correlation between the strength of the H-bond and the electronegativity of the atoms involved has been amply confirmed, with evidence suggesting a greater sensitivity to the nature of the donor than to the acceptor. Pauling's notion t h a t H-bonding is largely an electrostatic phenomenon has also received affirmation; in fact, a purely Coulombic analysis can frequently predict with good accuracy the angular aspects of a given complex. Certain key geometric and spectroscopic characteristics of Hbonds, already considered by Pauling, bear strong relationships to the strength of the interaction, and can be used as indicators of the H-bond strength in the absence of energetic data. Since Pauling's early discussion of the issue of multiple H-bonds, much work has elucidated the quantitative aspects of the cooperativity and calculations have determined its underlying causes. The aforementioned relationships between bond strength and geometric and spectroscopic properties remain valid in chains of H-bonds and these quantities bear a direct relationship with the degree of cooperativity. REFERENCES

1. L. Pauling, The nature of the chemical bond (Cornell University Press, Ithaca, NY, 1940). 2. T.S. Moore and T.F. Winmill, J. Chem. Soc., 101 (1912) 1635. 3. W.M. Latimer and W.H. Rodebush, J. Am. Chem. Soc., 42 (1920) 1419. 4. D.W. Michael, C.E. Dykstra and J.M. Lisy, J. Chem. Phys., 81 (1984) 5998.

590 5.

M.J. Frisch, J.E. Del Bene, J.S. Binkley and H.F. Schaefer, J. Chem. Phys., 84 (1986) 2279. 6. J.E. Del Bene, J. Chem. Phys., 86 (1987) 2110. 7. I~A. Peterson and T.H.J. Dunning, J. Chem. Phys., 102 (1995) 2032. 8. M. Quack and M.A. Suhm, Theor. Chim. Acta, 93 (1996) 61. 9. C.L. Collins, K. Morihashi, Y. Yamaguchi and H.F. Schaefer, J. Chem. Phys., 103 (1995) 6051. 10. M.W. Feyereisen, D. Feller and D jk. Dixon, J. Phys. Chem., 100 (1996) 2993. 11. G. de Oliveira and C.E. Dykstra, J. Mol. Struct. (Theochem), 337 (1995) 1. 12. Y.-B. Wang, F.-M. Tao and Y.-K. Pan, J. Mol. Struct. (Theochem), 309 (1994) 235. 13. S. Saeb~, W. Tong and P. Pulay, J. Chem. Phys., 98 (1993) 2170. 14. D.M. Hassett, C.J. Marsden and B.J. Smith, Chem. Phys. Lett., 183 (1991) 449. 15. F.-M. Tao and W. Klemperer, J. Chem. Phys., 99 (1993) 5976. 16. S.M. Cybulski, Chem. Phys. Lett., 228 (1994) 451. 17. F.-M. Tao and W. Klemperer, J. Chem. Phys., 103 (1995) 950. 18. A. Karpfen, P.R. Bunker and P. Jensen, Chem. Phys., 149 (1991) 299. 19. Z. Latajka and S. Scheiner, Chem. Phys., 122 (1988) 413. 20. D.D.J. Nelson, G.T. Fraser and W. Klemperer, J. Chem. Phys., 83 (1985) 6201. 21. M.J. Frisch, J.A. Pople and J.E. Del Bene, J. Phys. Chem., 89 (1985) 3664. 22. Z. Latajka and S. Scheiner, J. Chem. Phys., 84 (1986) 341. 23. J.W.I. van Bladel, A. van der Avoird, P.E.S. Wormer and R.J. SaykaUy, J. Chem. Phys., 97 (1992) 4750. 24. J.G. Loeser, C.A. Schmuttenmaer, R.C. Cohen, M.J. Elrod, D.W. Steyert, R.J. Saykally, R.E. Bumgarner and G.A. Blake, J. Chem. Phys., 97 (1992) 4727. 25. G. Alagona, C. Ghio, R. Cammi and J. Tomasi, Int. J. Quantum Chem., 32 (1987) 207. 26. R. Taylor, O. Kennard and W. Versichel, J. Am. Chem. Soc., 105 (1983) 5761. 27. R. Taylor and O. Kennard, Acc. Chem. Res., 17 (1984) 320. 28. P. Murray-Rust and J.P. Glusker, J. Am. Chem. Soc., 106 (1984) 1018. 29. A.C. Legon and D.J. MiUen, Faraday Discuss. Chem. Soc., 73 (1982) 71. 30. A.C. Legon and D.J. Millen, Acc. Chem. Res., 20 (1987) 39. 31. A.C. Legon and D.J. Millen, Chem. Soc. Rev., 16 (1987) 467. 32. M.T. Carroll, C. Chang and M.F.W. Bader, Mol. Phys., 63 (1988) 387. 33. J.B.O. Mitchell and S.L. Price, Chem. Phys. Lett., 154 (1989) 267. 34. J.T. Brobjer and J.N. Murrell, J. Chem. Soc., Faraday Trans. 2, 79 (1983) 1455. 35. A.D. Buckingham and P.W. Fowler, J. Chem. Phys., 79 (1983) 6426. 36. A.D. Buckingham and P.W. Fowler, Can. J. Chem., 63 (1985) 2018. 37. A.P.L. Rendell, G.B. Bacskay and N.S. Hush, Chem. Phys. Lett., 117 (1985) 400. 38. V. Magnasco, C. Costa and G. Figari, J. Mol. Struct. (Theochem), 169 (1988) 105. 39. V. Magnasco, C. Costa and G. Figari, Chem. Phys. Lett., 160 (1989) 469.

591 40. S.M. Cybulski and S. Scheiner, J. Phys. Chem., 93 (1989) 6565. 41. G. Chalasinski, M.M. Szczesniak, P. Cieplak and S. Scheiner, J. Chem. Phys., 94 (1991) 2873. 42. K. Morokuma and K. Kitaura, in: Chemical Applications of Atomic and Molecular Electrostatic Potentials, ed. P. Politzer and D.G. Truhlar (Plenum, New York, 1981) p. 215. 43. K. Morokuma and K. Kitaura, in: Molecular Interactions, ed. H. Ratajczak and W.J. Orville-Thomas Vol. 1 (Wiley, New York, 1980) p. 21. 44. R.F. Frey and E.R. Davidson, J. Chem. Phys., 90 (1989) 5555. 45. S.M. Cybulski and S. Scheiner, Chem. Phys. Lett., 166 (1990) 57. 46. A.E. Reed and F. Weinhold, J. Chem. Phys., 78 (1983) 4066. 47. S.J. Harris, K.C. Janda, S.E. Novick and W. Klemperer, J. Chem. Phys., 63 (1975) 881. 48. F.A. Baiocchi and W. Klemperer, J. Chem. Phys., 78 (1983) 3509. 49. W.J. Stevens and W.H. Fink, Chem. Phys. Lett., 139 (1987) 15. 50. R.M. Badger and S.H. Bauer, J. Chem. Phys., 5 (1939) 839. 51. G.C. Pimentel and A.L. McClellan, The Hydrogen Bond (Freeman, San Francisco, 1960). 52. C. Laurence, M. Berthelot, M. Helbert and K. Srafdi, J. Phys. Chem., 93 (1989) 3799. 53. E.E. Tucker and E. Lippert, in: The Hydrogen Bond. Recent Developments in Theory and Experiments, ed. P. Schuster, G. Zundel, andC. Sandorfy Vol. 2 (North-Holland Publishing Co., Amsterdam, 1976) p. 791. 54. H. Koller, R.F. Lobo, S.L. Burkett and M.E. Davis, J. Phys. Chem., 99 (1995) 55. H. Eckert, J.P. Yesinowski, L.A. Silver and E.M. Stolper, J. Phys. Chem., 92 (1988) 2055. 56. R. Kaliaperumal, R.E.J. Sears, Q.W. Ni and J.E. Furst, J. Chem. Phys., 91 (1989) 7387. 57. Z. Gu, C.F. Ridenour, C.E. Bronnimann, T. Iwashita and A. McDermott, J. Am. Chem. Soc., 118 (1996) 822. 58. I. Olovsson and P.-G. JSnsson, in: The Hydrogen Bond. Recent Developments in Theory and Experiments, ed. P. Schuster, G. Zundel, andC. Sandorfy Vol. 2 (North-Holland Publishing Co., Amsterdam, 1976) p. 393. 59. X. Duan and S. Scheiner, Int. J. Quantum Chem., QBS, 20 (1993) 181. 60. J.E. Del Bene, W.B. Person and K. Szczepardak, Mol. Phys., 89 (1996) 47. 61. Z. Latajka and S. Scheiner, Chem. Phys. Lett., 174 (1990) 179. 62. R. Thijs and T. Zeegers-Huyskens, Spectrochim. Acta A, 40 (1984) 307. 63. C. Ceccarelli, G.A. Jeffrey and R. Taylor, J. Mol. Struct., 70 (1981) 255. 64. G. Chalasinski and M.M. Szczesniak, Chem. Rev., 94 (1994) 1723. 65. M. Kofranek, H. Lischka and A. Karpfen, Chem. Phys., 113 (1987) 53. 66. J.C. White and E.R. Davidson, J. Chem. Phys., 93 (1990) 8029.



Pauling's Legacy: Modem Modelling of the Chemical Bond

593


Molecular Similarity and Host-Guest Interactions Paul G. Mezey Mathematical Chemistry Research Unit, Department of Chemistry and Department of Mathematics and Statistics University of Saskatchewan, 110 Science Place, Saskatoon, SK, Canada, S7N 5C9

ABSTRACT

Quantum chemical treatment of host-guest interactions can be approached from a new perspective provided by the extension of electron density shape analysis methods to large systems. Host-guest interactions are manifested in the changes of electron densities detectable in the composite host-guest system, as compared to the electron densities of the individual, non-interacting host and guest molecules. The similarities and dissimilarities between the electron densities of interacting and non-interacting molecules provide quantum chemical descriptors of host-guest interactions. Some of the computational techniques relevant to such analyses are reviewed.

594

INTRODUCTION

Host-guest interactions are fundamental in enzyme actions and in many other biochemical processes [1-10]. In most instances, host-guest interactions involve special aspects of molecular similarity [11-26] and complementarity: interfacing between intermediate-size molecular regions. The range of features where a formal match between the shape properties and electronic properties is required [27] falls between the truly local interactions between individual atom pairs or small molecular fragments and the global interactions of extensive regions of molecules. This intermediate aspect of the range of interactions in host-guest problems provides both a challenge and a fertile ground for novel ideas of representing molecules and their electron density clouds. The approach followed in this contribution is motivated by a rather trivial observation: molecular properties are determined by the electron density distribution [27]. With the introduction of techniques suitable for the generation of ab initio quality electron densities for large molecules such a proteins, the range of electron density analysis has been extended from small systems to molecules of virtually any size. Host-guest interactions often involve a large host molecule and a usually smaller guest molecule, hence the role of large molecule electron density analysis is of special importance in this field. Of course, in most instances, only a small part of the large host molecule participates directly in the host-guest interactions, yet the entire host is often required in order to provide the proper framework and geometrical constraints for the interactions.

595 The first computations of ab initio quality electron densities of large molecules such as proteins have been based on the Additive Fuzzy Density Fragmentation (AFDF) recently [31,32].

principle [28-32], a technique that has been reviewed

The first

AFDF

approach was the MEDLA

method

(Molecular Electron Density "Loge" Assembler, or Molecular Electron Density "Lego" Assembler method)

of Walker and Mezey [33-38], where a pre-

calculated, numerical electron density fragment database is used. The entries in this database are custom-made electron density fragments, obtained from ab initio computations for small molecules which contain the molecular fragment within a local environment reproducing its actual environment within the macromolecule. A more advanced AFDF approach, the ADMA method (Adjustable Density Matrix Assembler method) [39-41]

is based on a density matrix

database and the actual construction of a macromolecular density matrix. This technique, also reviewed in part in ref.[31 ], is suitable for the rapid computation of various additional molecular properties besides electron densities. According to detailed test calculations, [33,34,37], the AFDF method produces

nearly 6-31G**

ab initio

quality electron densities for large

molecules, and is superior to conventional ab initio computations using the smaller standard basis sets. The tests included detailed comparisons of electron densities obtained for the amino acid ~-alanine [33], the model peptide system of glycyl-alanine [34], the reproducibility of the electron density of a H-bond in a helical tetrapeptide [34], the reproducibility of a non-bonded interaction between a sulfur atom and a phenyl ring in a molecular fragment from the pentapeptide metenkephalin [34], the reproducibility of aromatic rings and substituent effects in a series of aromatic molecules [37]. The numerical tests relied on direct, point-by-point comparisons of three-dimensional density grids,

596

and also integrated similarity measures [37] including the Carb6 quantum similarity index.

FROM

FUNCTIONAL

GROUPS

TO EXTENDED

MOLECULAR

REGIONS

Two important tools of electron density modeling and shape analysis are the concepts of molecular-isodensity contour, MIDCO G(K,a) and the associated

density domain DD(K,a) [27]. Here the nuclear arrangement (also called the nuclear configuration) is denoted by K, whereas the electron density threshold is denoted by a. Each MIDCO G(K,a) is the collection of all those points r of the three-dimensional space where the electron density molecule M of conformation K domain

p(K,r)

of the

is equal to the threshold a. The density

DD(K,a) is the collection of all points r where the electron density

p(K,r) is greater than or equal to the threshold a. Using formal notations,

G(K,a) = { r : p(K,r)= a },

(1)

DD(K,a) = { r : p(K,r) _>a }.

(2)

and

The MIDCO

G(K,a)

can be regarded as the boundary surface of the density

domain DD(K,a).

Quantum chemical functional groups have been defined as fuzzy electron density fragments (AFDF fragments) associated with a family of nuclei fk,

597 where there exists some density threshold a such that this family

fk is

separated from the rest of the nuclei of the molecule by the corresponding MIDCO

G(K,a).

This definition reflects the limited autonomy and separate

identity of functional groups within molecules, just as the existence of MIDCOs separating the nuclei of two different molecules placed within a short distance of one another reflects the autonomy and separate identity of the two molecules. Consider a macromolecule M of some nuclear configuration K. For an electron density threshold a, the family of functional groups

F1, F2, . . . , Fm

(3)

containing the nuclear families

fl, f2, . . . , fm

(4)

are characterized by the corresponding density domains

DD l(a,K), DD2(a,K), . . . , DDm(a,K)

(5)

which, by definition, must appear as separate entities. Using the additive, fuzzy density fragmentation (AFDF) approach of the macromolecular density

pM(r),

the fuzzy fragment electron density

contributions

PFI(r), PF2(r), . . . PFi(r),... PFm(r),

(6)

598 can be regarded as the representations of the "share" of each functional group F i within the total macromolecular electron density p M(r). If the nuclear sets fl, f2 . . . .

, fm contain all the nuclei of the molecule,

then one may reconstruct the electron density pM(r)

of macromolecule M by

a simple superimposition of the fuzzy fragment densities PFl(r), PF2(r),

999

P F i ( r ) , . . . PFm(r), of the actual family of functional groups:

pM(r) = ]~i PFi(r) 9

(7)

This approach can be easily generalized for much larger molecular moieties, for example, to entire reactive regions of a macromolecule, representing the "host" region of the molecule, or to the reactive region of the "guest" molecule. Furthermore, the interacting host-guest structure itself can be regarded as a formal "region" of a supermolecule, and the electron density shape analysis can be carded out accordingly. If one drops the condition that there must exist a density domain that contains all the nuclei involved in the molecular moiety considered, then the treatment can be made very general, and applied to virtually any molecular moiety. The given macromolecule M of the specified nuclear configuration K can be regarded as a composite of several regions, where each individual region might be much more extensive than individual functional groups, and where the requirement for "autonomous" density domains for each contributing molecular region no longer applies. For a specified electron density threshold a, the regions

R1, R2, . . . , Rr

(8)

599 contain the nuclear families

fl, f2, . . . . f r ,

(9)

respectively. The additive, fuzzy density fragmentation (AFDF) method applies to the fuzzy regional electron density contributions

PRl(r), PRz(r), . . . PRa(r),... PRr(r),

(10)

which collectively reproduce the macromolecular electron density pM(r).

The

individual regional densities pRj(r) within the total macromolecular electron density pM(r) can be regarded as the quantum chemical representations of the electron density "share" of each region Ri. Following a treatment similar to that applied to functional groups, we assume that the nuclear sets macromolecule

M.

fl, f2 . . . .

, fr contain all the nuclei of the

We may reconstruct the electron density p M(r)

of

macromolecule by superimposing the fuzzy regional densities p Rl(r), PRz(r), 9-- P Ri(r) . . . . pR~(r):

pM(r) = ]~i PR~(r) 9

(11)

The shape analysis and shape comparisons of electron densities of molecular regions provide information relevant to their interactions. In the next ,b

section a brief review of a shape analysis method is given.

600 E L E M E N T S OF E L E C T R O N DENSITY SHAPE ANALYSIS

The complete, three-dimensional shape of fuzzy electron densities can be described in detail using the Shape Group Methods, (SGM), reviewed in detail in a recent monograph [27]. Here, only a brief summary of this method will be given.. The application of SGM to molecular electron densities is based on the comparison of local curvatures of a range of MIDCOs to a range of reference curvatures, involving several steps. First, two ranges are selected: a range of electron density thresholds a and a range of reference curvatures b. For each pair of values

a

and

b within these ranges,

each MIDCO

G(K,a)

is

partitioned into local curvature domains relative to each value b, specifying whether the MIDCO G(K,a) is convex, concave, or of the saddle type relative to this curvature b.

In practice, the local curvature of a MIDCO

surface

G(K,a) at each point r is characterized by a local curvature matrix called the local Hessian matrix. Depending on the local relative convexity, the points are classified into curvature domains of the types,

D0(b),

D l(b)

or D2(b),

respectively. This is accomplished by comparing the local canonical curvatures (the eigenvalues of the local Hessian matrices) at each surface point r to the reference curvature b; a point r of G(K,a) is assigned to a Do(b), D l(b) or D2(b) curvature domain, if none, one, or two (respectively) of the eigenvalues of the local Hessian matrix of the surface at point curvature parameter

b.

r

are smaller than the

The computational problem is simplified by the fact

that for the identification of all the topologically different patterns of curvature domains only a finite number of (a,b) pairs need to be considered.

601

The next step involves a truncation of MIDCO surfaces according to the various curvature domains identified on them. For each (a,b) pair of values, all curvature domains D~t(b) of a specified type removed from the MIDCO G(K,a),

~t (usually, the type l.t = 2) are

and a truncated surface

G(K,a,~t) is

obtained. Note that for the whole range of parameter values a and b of each molecule M, only a finite number of topologically different truncated surfaces are obtained. In the final step, the shape groups of the entire molecular electron density distribution are computed. By definition, the shape groups are the algebraic homology groups of the truncated surfaces, which are invariants within each topological equivalence class of these surfaces. The ranks of these homology groups are the Betti numbers, serving as a set of numerical shape descriptors for the entire range of MIDCOs G(K,a) of the molecule M. For electron density analysis in three dimensions, using two-dimensional MIDCO surfaces, there are three types of shape groups, one for each of the dimensions zero, one, and two. The associated Betti numbers are b01x(a,b), bl~t(a,b), and b2~t(a,b), where the truncation type Ix is also specified. The results of the shape group analysis can be summarized using the Betti numbers. The distribution of various values of Betti numbers bP~t(a,b) as a function of the density threshold a and curvature parameter b is represented by various (a,b)-maps. Discretized versions of (a,b)-maps, in the form of shape matrices

~7~(a,b), serve as numerical shape codes for the molecules.

The total number of elements in the shape code matrix M(a, b) is

t = nanb '

(12)

602

where in this discretized version of the (a,b)-map, na and n b are the number of grid points for parameters a and b, respectively. Shape similarity between two molecules or molecular regions A and B can be expressed using the following shape-similarity measure:

s(A,B) = m[b~(a,b),A,/~(a,b),B ] [ t.

(13)

Here m[/~(a,b),A, ~(a,b),B ] is the number of matches between corresponding elements in the two shape code matrices

M(a,b),A and

~(a,b),B

of the

two molecules or molecular fragments A and B, respectively.

ELECTRON

DENSITY

INTERACTING

ANALYSIS

REACTIVE REGIONS

OF

ISOLATED

AND

OF M O L E C U L E S

The approach we shall follow in the study of host-guest interactions involves the comparison of electron density distributions of molecular regions in the presence and also in the absence of interactions. In this context, some aspects of molecular similarity analysis are applied, evaluating the similarities between two electron densities"

one where these regions are taken in isolation, and

another that includes the interactions between the host and guest regions of the molecules involved. The first model, describing the isolated region approach, can be derived easily from the AFDF principle. Molecular regions are described by fuzzy electron densities analogous to densities of complete molecules, and the local shape analysis of regions follows the same principles as the shape analysis of

603

complete molecules. In order to emphasize the fact that molecular regions are involved, we shall use the terminology "region isodensity contour" (RIDCO) surface, replacing the term "molecular isodensity contour" (MIDCO) surface. The notation

R

is used for the actual region selected for study and M'

denotes the rest of the macromolecule M. The rest of the molecule,

M', may

be composed from several regions, R 1, R2, .. 9 Rr-1, and without restriction on generality, the actual region R, the subject of our study, is assumed to correspond to the last region in the series, R = Rr. If the influence of the rest of the molecule on a molecular region R is unimportant, or, if one is interested in the electron density of a given region in the absence of interactions with other parts of the molecule, then the actual molecular density region R can be regarded as a separate entity. In this case, it is meaningful to consider RIDCO contours for R where the density threshold a

is compared only to the actual regional density

pR(r). In this model, the

RIDCOs are not influenced by the additional density contributions from the rest of the molecule M. In this model, a "non-interacting"

RIDCO for a region R of a molecule

M = RM' is defined as follows:

GR~I'(a) = { r" pR(r) = a, pR(r) > PRk(r), k=l .... m-1 }.

Alternatively, one may define a "non-interacting"

RIDCO

(14)

using the

following relations:

GR~VI'(a) = G R ( a ) n { r" pR(r)>--PRk(r), k=l,...m-1 },

or

(15)

604

GRXM'(a) = GR(a) \ { r" 3 kr { 1,...m-1 }" pR(r) < PRk(r) }.

(16)

These two alternative definitions, (15) and (16), are equivalent to the definition given by eq. (14). These definitions, as well as some of the properties of "non-interacting"

RIDCOs are analogous to those of "non-interacting"

functional groups, discussed in ref. [31]. The interpretation of

RIDCO GR\M'(a) is simple: for a region R in

macromolecule M=RM', GRLM'(a) is the set of all those points r where the electron density contribution

pR(r)

of region

R is dominant among all

regional electron densities PRk(r) within the macromolecular electron density pM(r). The standard Shape Group Method is applicable for the analysis of the entire series of non-interacting thresholds

RIDCOs,

for a whole range of density

a, with the provision of an additional domain type representing the

connection of region R to the rest of the molecule within the actual RM' system. This additional domain type D_ 1 is defined as

D-1 (GRkM'(a)) = { r" r e GR(a), 3 k~ { 1.... m-1 }" pR(r) < PRk(r)}. (17)

On the actual, non-interacting

RIDCO

surface

GR\M'(a)

only the

boundary AD_I (GRXM'(a)) of this additional domain can be found:

AD_I (GR~I'(a)) = { r" r e GR~M'(a), 3 k'a { 1,...m-1 }" pR(r) = PRk,(r), pR~(r) > PRk(r), k=l .... m-1 }.

(18)

605

The domain D-1 (GR\M'(a))

itself exists only on the intact GR(a) contour

surface. For a macromolecule RM', a typical domain D-1 (GRLM'(a)) appears only as a formal cover over a hole of the non-interacting RIDCO G RLM'(a). Following the treatment of non-interacting functional groups [31], simpler representation of region

R in molecule

RM'

a

is obtained if the

comparisons of regional electron densities are not carried out for each region, but the composite M' of all the remaining regions R1, R 2 , . . . ,

Rm-1

is

compared to the regional density pR(r). Using the notation PlVr(r) for the composite density of all the remaining regions,

PlVr(r) = PR~(r)+ PRz(r)+ ... + PRm.~(r),

(19)

the corresponding non-interacting RIDCO surfaces GRhEM'(a) are defined as

GR~M'(a) = { r: PR(r) = a, pR(r) > Plvr(r) }.

(20)

Using this approach, new local domain types appear at those locations of the molecular electron density where the region R connects to the rest M' of the molecule RM'= M:

D-1 (GR~M'(a)) : { r : r e GR(a), pR(r) < PlVr(r)}.

(21)

For the purposes of shape analysis, the boundaries of these additional domains are of importance. These boundaries are

AD_I (GR~M:(a)) = { r : r e GRkZM'(a), pR(r) = PMT(r)}.

(22)

606 By determining all points r

where p R(r) = PlVr(r), these boundaries can be

found with relatively little computational effort. In some instances, the interactions of various molecular regions in a macromolecule RM' or in a composite "supramolecule" of a host-guest system are of interest.

In these cases, a local shape analysis of the "isolated" RIDCO

surfaces GR(a) is no longer sufficient, and the study of the interactions requires the calculation of new density contours. A suitable definition for "interactive" RIDCO surfaces can be given as follows:

(23)

GR(M')(a) = {r- pR(r) + PlVr(r) = a, pR(r) > PlVg(r)}.

On these

GR(M')(a)

correspond to some formal

surfaces there are no domains which would covers

of holes in the RIDCO

GR(M')(a),

however, for consistency with the notations used in the case of non-interacting RIDCOs, the formal boundaries of the holes on

GR(M')(a) are denoted by

AD_ 1 (GR(M') (a))"

AD_I (GR(M')(a)) : { r" r e GR(M')(a), pR(r) = pM'(r)}.

(24)

The computation and shape analysis of the interactive RIDCOs

of a

region R in a macromolecule RM' requires the determination of additional contours. This implies that the shape analysis of interacting

RIDCOs is

computationally more expensive than that of the non-interactive

RIDCOs

GRLM'(a). For the suggested approach to the study of host-guest interactions, the shape analysis of both types of RIDCOs is required.

607 SIMILARITY

SHAPE

MEASURES

AND

DISSIMILARITY

MEASURES IN THE STUDY OF HOST-GUEST INTERACTIONS

Consider the interacting host-guest complex as a single supermolecule M, and denote the region of interaction in this complex by Rhg. The corresponding reactive regions of the isolated host and the isolated guest molecules are denoted by Rh, and Rg, respectively. The regional, fuzzy electron densities associated with the three regions, Rhg, Rh, and Rg, are denoted by

PRhg(r), P Rh(r),

and P Rg(r), respectively. From the latter two regional densities PRh(r), and PRg(r),

one may construct the composite density PRh§

of a formal,

superimposed but non-interacting host-guest region Rh+g,

PRh§

= PRh(r) + PRg(r).

(25)

The Shape Group analysis can be carded out for all these regional electron densities using both the non-interacting and the interacting RIDCO formalisms. We shall use the following notations: subscripts and superscripts nil, ni2, and i correspond to the choices of non-interacting RIDCOs G R\M'(a), noninteracting RIDCOs GRLEM,(a),

and the interacting RIDCOs

GR(M')(a),

respectively. Accordingly, the shape analysis can be carried out for contour surfaces following either one of the conventions nil, ni2, and i, leading to the shape code matrices i~(a,b),R,nil,

respectively.

M(a,b),R,ni2,

and

I~(a,b),R,i,

(26)

608

For each of these options, the similarity measures Snil(Rhg,Rh+g ) = m[/~(a,b),Rhg, nil,/~(a,b),Rh+g, nil] [ t,

(27)

Sni2(Rhg,Rh+g ) = m[~v~(a,b),Rhg,ni2, ~,~(a,b),Rh+g,ni2] / t,

(28)

si(Rhg,Rh+g) = m[~JI(a,b),Rhg,i, M(a,b),Rh+g, i] [ t,

(29)

and

and the associated dissimilarity measures dnil(Rhg~h+g)= 1 - Snil(Rhg,Rh+g)

(30)

dni2(Rhg,Rh+g)= 1 - Sni2(Rhg,Rh+g)

(31)

di(Rhg,Rh+g)= 1 - si(Rhg,Rh+g)

(32)

and

give indications of the extent, range and various details of the host-guest interaction. Among these measures, si(Rhg,Rh+g) and di(Rhg,Rh+g) reflect the most detail of the interactions between the host and guest regions, taking into account the influence of those molecular regions which participate only indirectly in the actual host-guest interaction. The greater the value of dissimilarity measure di(Rhg,Rh+g ), the greater the shape change induced by the host-guest interaction. Additional detail can be found in the shape code matrices 1~(a,b),Rhg, i and

i~(a,b),Rh+g,i themselves; their comparisons can be carded out with

special focus on low or high density ranges, considering a specific range of the

609 curvature parameter b. Whereas most shape changes in host-guest interactions are expected in the low density ranges of the fuzzy regional charge distributions, if significant changes occur in the high density ranges, this is a sign of exceptionally strong host-guest interactions.

Such details are not directly

available from the dissimilarity measure di(Rhg,Rh+g), however, one may diagnose such instances by a direct comparison of high density ranges within the shape code matrices M(a,b),Rhg, i and

~4~(a,b),Rh+g,i.

SUMMARY

The electron densities of local regions of both small and large molecules can be studied in detail using some of the macromolecular quantum chemical computational techniques developed recently. The shape analysis of host-guest systems and the comparison of the electron densities of interacting and noninteracting molecular regions provide measures and detailed descriptions of these interactions.

REFERENCES

[1]

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[2]

M. Karplus and J.A. McCammon, Annu. Rev. Biochem. 53, 263 (1983).

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610

[4]

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[9]

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[10]

P.-L. Chau, and P.M. Dean, J. Computer-Aided Molecular Design, 8, 513, 527, 545 (1994).

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[12]

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[13]

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[16]

E. Besal6, R. Carb6, J. Mestres, and M. Sol~, Foundations and Recent Developments on Molecular Quantum Similarity, in Topics in Current Chemistry, Vol. 173, Molecular Similarity, ed. K. Sen

(Springer-Verlag, Heidelberg, 1995). [17]

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[21]

R.P. Bywater, Quantitative Measurement of Molecular Similarity Using

Shape Descriptors, R. Carb6, Ed.; Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Kluwer Academic Publ.- Dordrecht, The Netherlands, 1995, pp 113-122. [221

M.A. Johnson, and G.M. Maggiora, Eds., Concepts and Applications of

Molecular Similarity, Wiley, New York, 1990. [23]

C.-D. Zachman, M. Heiden, M. Schlenkrich, and J. Brickmann, J.

Comp. Chem. 13, 76 (1992). [24]

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Chem. 14, 1290(1993). [25]

D.L. Cooper, and N.L. Allan, Molecular Similarity and Momentum

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R. Carb6, Ed.; Molecular Similarity and Reactivity: From

Quantum Chemical to Phenomenological Approaches; Kluwer Academic Publ.: Dordrecht, The Netherlands, 1995, pp 31-55. [26]

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Kohonen Neural Network, J. Devillers, Ed.; Neural Networks in QSAR and Drug Design; Academic Press: London, 1996, pp 209-222. [27]

P.G. Mezey, Shape in Chemistry: An Introduction to Molecular Shape

and Topology, VCH Publishers, New York, 1993. [28]

P.G. Mezey, "Density Domain Bonding Topology and Molecular Similarity Measures". In K. Sen, Ed., Topics in Current Chemistry, Vol. 173, Molecular Similarity, Springer-Vedag, Heidelberg, 1995.

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P.G. Mezey, "Methods of Molecular Shape-Similarity Analysis and Topological

Shape Design". In P.M. Dean, Ed., Molecular Similarity

in Drug Design, Chapman & Hall- Blackie Publishers, Glasgow,

U.K., 1995. [301

P.G. Mezey, "Shape Analysis of Macromolecular Electron Densities", Structural Chem., 6, 261 (1995).

[31]

P.G. Mezey, "Functional Groups in Quantum Chemistry", Advances in Quantum Chemistry, 27, 163 (1996).

[32]

P.G. M ezey, "Local Shape Analysis of Macromolecular Electron Densities".

In J. Leszczynski, Ed. Computational Chemistry:

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Pauling's Legacy: Modem Modelling of the Chemical Bond Theoretical and Computational Chemistry, Vol. 6

613

9 1999 Elsevier Science B.V. All rights reserved.

Chemical Bonding in Proteins and Other Macromolecules Paul G. Mezey Mathematical Chemistry Research Unit, Department of Chemistry and Department of Mathematics and Statistics University of Saskatchewan, 110 Science Place, Saskatoon, SK, Canada, S7N 5C9

ABSTRACT

In highly folded macromolecules, such as most globular proteins, the complex pattern of non-bonded interactions between various molecular fragments plays an important stabilizing role. The change of this pattern in the course of various conformational rearrangements may enhance or hinder the conformational processes. The AFDF (additive fuzzy density fragmentation) method of ab initio quality electron density computations for proteins is a tool that provides new insight into these interactions. Based on such calculations, in this contribution a new model, the "Low Density Glue" (LDG) bonding model, is presented that approaches these interactions from a global perspective.

614

1.

INTRODUCTION

Linus Pauling's unprecedented successes were almost always based on a clear recognition of the essential in seemingly very complex problems, leading to simple yet effective solutions. This is not a simple path to follow. In this study an attempt is made to discuss some bonding features common to many proteins. A family of new computational methodologies provides the means for a new approach to study these bonding problems, however, a clear simplification in the Pauling tradition requires additional insight. Electron density is the basis of chemical bonding [1],

and density

functional theory has established that electron density determines the energy contents of molecular arrangements [2,3] as well as other molecular properties [4-10]. Whereas these facts clearly indicate the central role of electron density, for large molecules, electron density analysis faces many difficulties. Proteins show structural complexities which involve local order and important regularities, as well as equally significant irregularities and seemingly accidental, disordered features. Chemical bonding in proteins exhibits a whole range of features not found in small molecules, yet these very features appear essential in the roles proteins play in biochemistry.

There are many open

problems concerning the relative roles of these features and the collective effects of formal non-bonded interactions. Until recently, detailed theoretical studies of global bonding features of proteins were hindered by the lack of detailed enough experimental or theoretical electron density maps for proteins. crystallographic

X-ray

Recently,

structure determination methods for proteins have

improved dramatically, but even with these improvements, the resolution of the observed electron density often leaves some uncertainties concerning the precise location of hydrogen nuclei. For a detailed experimental analysis of chemical

615

bonding, electron densities of higher experimental resolution than those currently available are required. However, recent progress in macromolecular quantum chemistry and advanced molecular modeling has provided some new computational tools which can be used to address some of these questions of macromolecular bonding. The introduction of

AFDF

(additive fuzzy density fragmentation)

methods for the study of functional groups [ 11-13] and the computation of ab initio

quality electron densities for proteins and other macromolecules have

naturally led to a renewed search for regularities in the quantum chemical descriptors of chemical bonding in large systems.

In this contribution a

particular aspect of chemical bonding is discussed that becomes especially important in large, self-interacting systems. The interactions between various segments of a protein which are separated by several amino acids along the polypeptide chain but which fall within a short geometrical distance from one another due to the actual folding pattern, are expected to contribute to the stability of the protein conformation to a significant degree. The importance of formal "non-bonded" interactions in proteins is well recognized, however, these interactions are usually considered as local effects, between well-defined structural entities, such as hydrogen bonds between a given pair of electronegative elements formally "sharing" a proton. In view of the latest results of electron density calculations of proteins and other large molecules, an alternative approach is suggested. In this new model, the "Low Density Glue"

(LDG) bonding model, the formal non-bonded interactions

often merge into broader interactions between interfaces that are extensive and blur the distinction between local interactions conventionally considered as separate effects.

616

0

MACROMOLECULAR QUANTUM CHEMISTRY BASED ON ADDITIVE FUZZY DENSITY FRAGMENTATION (AFDF)

The

AFDF

family of methods and the quantum chemical treatment of

functional groups have been reviewed recently [11].

Here only a brief

introduction of the notations and a summary of the basic results will be given. At each point

r, the electronic density p(r,K) of a molecule of nuclear

conformation K can be computed by the Hartree-Fock-Roothaan-Hall SCF LCAO ab initio method. Using a basis set tp(K) of atomic orbitals tpi(r,K) (i=l,2,...,n) and the nxn dimensional density matrix P(tp(K)), the electronic density p(r,K) is obtained as

n

n

p(r,K) = E E Pij(tp(K)) tpi(r,K) tpj(r,K).

(1)

i=l j=l

Electron density decreases exponentially with distance that suggests that an Additive Fuzzy Density Fragmentation (AFDF) approach can be used for both a fuzzy decomposition and construction of molecular electron densities. The simplest AFDF technique is the Mulliken-Mezey density matrix fragmentation [12,13],

that is the basis of both the Molecular Electron Density Loge

Assembler (MEDLA) [14-17] and the Adjustable Density Matrix Assembler (ADMA) [ 18-21] macromolecular quantum chemistry methods. Within the general Mulliken-Mezey AFDF approach, the set of nuclei of the molecule M are classified into m mutually exclusive families

617

fl, f2,- -., fk,. --, fm,

and for each AO basis function q)i(r,K) and nuclear family fk a membership function mk(i) is defined:

1 if q)i(r) is centered on one of the nuclei of set fk, mk(i) = { 0 otherwise.

(2)

In terms of some wij and wji weighting factors fulfilling the condition

w ij + wji = 1,

wij, wji > 0,

(3)

the elements Pkij(q)(K)) of the n x n fragment density matrix pk(q)(K)) for the k-th fragment are given by

Pkij(tp(K)) = [mk(i)w ij+ mk(j) wji ] Pij(9(K)).

(4)

In the simplest case of the Mulliken-Mezey AFDF approach, the

wij = wji = 0.5

(5)

choice is used, that follows the spirit of the population analysis scheme of Mulliken. The k-th additive fuzzy density fragment pk(r,K) is defined as

618

n

ok(r, K) =

n

]~ E Pkij(~K)) ~(r,K) ~j(r,K), k=l,2,...m. i=l

(6)

j=l

As it can be easily verified, the

AFDF

fragment density matrices

Pk(q0(K)) as well as the density fragments pk(r,K) are strictly additive:

in

P(~K)) =

]~ pk(q)(K)),

(7)

k=l and in

p(r,K)=

]E pk(r,K).

(8)

k--1

If the molecule M is small, then a conventional Hartree-Fock computation followed by the application of the

AFDF

approach allows one to study the

shape and the interactions of local moieties of M in detail. If the molecule M is large, then a conventional Hartree-Fock computation is no longer feasible, however, the fuzzy electron density fragments of the large "target"

molecule

M

still can be computed indirectly using the AFDF

approach. For each nuclear family fk

of M, a small parent molecule Mk

can be designed, where M k contains the same nuclear family

fk with the

same local arrangement and surroundings as is found in the large target molecule M. The fuzzy density fragmentation can be carried out for the small parent molecule M k, resulting in a fuzzy density fragment p k(r,K) corresponding to the nuclear set fk , and other density fragments which are not used for the

619

macromolecular study. By repeating this procedure for each nuclear family fk of M, the fuzzy fragments

p l(r,K), p 2 ( r , K ) , . . . , p k ( r , K ) , . . . , pm(r,K)

obtained from a set of m small "parent" molecules

M1, M 2 , . . . , M k , . . . , Mm,

can be combined and used to construct the electron density p (r,K) of the large target molecule M. The fragment densities themselves can be used for local analysis of the macromolecule M. The

MEDLA

(Molecular Electron Density "Loge" Assembler, or

Molecular Electron Density "Lego" Assembler) method of Walker and Mezey [14-17] was the first implementation of the simplest version (5) of the AFDF approach.

The MEDLA method is based on a numerical electron density

fragment database of pre-calculated, custom-made electron density fragments 9k(r,K), and a subsequent numerical construction of the molecular electron density using eq. (8). According to detailed tests [14,15,17], the MEDLA method generates ab initio quality electron densities for large molecules near the 6-31G** basis set level that has been used for the construction of the fragment density databank.

For the first time, ab initio quality electron

densities have been computed for several proteins, including crambin, bovine insulin, the gene-5 protein (g5p)of bacteriophage M13, the HIV-1 protease monomer of

1564 atoms, and the proto-oncogene tyrosine kinese protein

1ABL containing 873 atoms.

620

The requirement of a numerical databank and some of the problems associated with the grid alignment of combined numerical density data are circumvented in a more advanced application of the AFDF approach relying directly on the fragment density matrices

Matrix Assembler

(ADMA)

density matrix P(~p(K))

method [18-21] generates a macromolecular

that can be used for the computation of a variety of

molecular properties besides densities.

Pk(cP(Kk)). The Adjustable Density

ab initio

quality

macromolecular electron

In electron density computations the accuracy of

macromolecular density matrix P(q~(K))

the

ADMA

corresponds to that of a MEDLA

result of an infinite resolution numerical grid. The construction of the macromolecular density matrix is the simplest if the fragment density matrices pk(q~(Kk) ) obtained from small parent molecules M k fulfill the following mutual compatibility requirements: (a) The local coordinate systems of AO basis sets of all the fragment density matrices pk(q~(Kk) ) have axes that are parallel and have matching orientations with the axes of a common reference coordinate system defined for the macromolecule. (b) The nuclear families used in the fragmentation of both the target and the parent molecules are compatible in the following sense: each parent molecule Mk

may

families

contain only complete nuclear families from the sets of nuclear fl, f2 . . . .

, fk . . . .

, fm,

present in the large target molecule M.

Within each parent molecule additional nuclei may be involved in order to provide linkages to dangling bonds at the peripheries of these molecules. A simple similarity transformation of a fragment density matrix pk(q~(Kk)) using a suitable orthogonal transformation matrix T(k) of the AO sets, and an appropriate choice of nuclear families fk for the various fragments within the

621

macromolecule M and within the "coordination shells" of parent molecules Mk can always ensure the fulfillment of these conditions. The AFDF approach enhanced with these mutual compatibility conditions is referred to as the MC-AFDF approach. The number of AOs in the nuclear family fk of the target macromolecule M is denoted by n k. For each pair (fk, fk') of nuclear families a quantity Ck'k is defined: 1, if nuclear family fk' is present in parent molecule Mk Ck'k = { 0 otherwise,

(9)

An AO q0(r) is denoted by the symbol q0b,k'(r) if its serial number b in the AO set nk' { qOa,k'(r) }

(10)

a=l of nuclear family fk' is emphasized. The same AO q0(r) is denoted by q0jk(r), if its serial index j in the basis set npk { qoik(r) }

(11)

i=l of the k-th fragment density matrix Pk(q0(Kk)) is emphasized, where the total number of these AOs is npk,

622

m

(12)

npk - ]~ Ck'k n k ' . k'=l

The same AO

r

is denoted by r

if its serial index

y in the AO set

n

{ q~x(r) }

(13)

x=l of the density matrix P(K) where for each AO

of the target macromolecule

q~a,k'(r) = ~ k ( r ) - q~x(r)

M is emphasized,

the index x is determined by

the index a in family k' as follows:

k'- 1

x -

x(k',a,f) - a + Z

(14)

n b,

b=l

The last entry f in x(k',a,f) indicates that k' and a refer to a nuclear family. In order to be able to determine the index x from the element index i and serial index

k of fragment density matrix

Pk(tp(Kk)), three quantities are

introduced for each index k and nuclear family fk" for which Ck"k ~ 0 9

k

l!

a'k(k",i) = i + ]~ nb Cbk ,

(15)

b=l

k ' - k'(i,k) = min {k"" a'k(k",i) < 0 },

(16)

623

and (17)

ak(i) = a'k(k',i) + nk'.

The AO index x - x(k,i,P) in the density matrix P(K) of target molecule M

depends on indices i and k and can be expressed using index k' and the

function x(k',a,f)

x-

(18)

x(k,i,P) = x(k',ak(i),f),

where the last entry P in the index function x(k,i,P)

indicates that k and i

refer to the fragment density matrix pk(q)(Kk)). Using only the nonzero elements of each (usually rather sparse) fragment density matrix

pk(q)(Kk)),

the macromolecular density matrix

P(K)

is

assembled by an iterative procedure,

Px(k,i,P),y(k,j,p)(K)- Px(k,i,P),y(k,j,P)(K) + Pkij(Kk)

Since the parent molecules

Mk

(19)

are of limited size, the entire procedure

depends linearly on the number of fragments and on the size of the target macromolecule M. The macromolecular density matrix P(K)

is also a sparse matrix that

simplifies its storage and subsequent computations. Using the macromolecular AO basis (that is stored as a list of appropriate indices referring to a standard list of AO

basis sets),

the macromolecular electron density is computed

according to eq. (1). Using the ADMA method, approximate macromolecular

624 forces and other properties expressible in terms of density matrices can be computed for virtually any molecule, providing a computationally viable approach to macromolecular quantum chemistry. One aspect of this field is discussed in the next section.

3.

" L O W DENSITY GLUE" (LDG) BONDING IN PROTEINS

In large, folded chain molecules an important, low-density component of self-interactions contributes to the structural features of the molecule in a fundamentally different way than formal chemical bonds usually assigned to pairs of atomic nuclei.

These low-density contributions are more widely

distributed within the macromolecule, cannot be assigned to individual atom pairs and are better described as interactions between larger structural elements, such as chain fragments. The range of electron density that is typically involved belongs to the fuzzy, peripheral electron density cloud, and the mutual interpenetration of these fuzzy "clouds" can be used to detect these interactions. Alternatively, one may consider this low-density cloud as a formal, diluted "glue", that still has a non-negligible role in holding the various molecular structural entities together. It is natural to use fuzzy set methods [22-26] for the study of these fuzzy electron distributions [27]. The formal "bodies" of molecules do not have boundaries and the actual shape of molecules is determined by the fuzzy electron distribution. Realistic models describing molecular shapes and chemical bonding must reflect this natural fuzziness [27].

625

One approach involves the following question: to what extent do various points r of the three-dimensional space belong to a single, isolated molecule X? Such problems are typically addressed using fuzzy sets [22-27]. Consider a spatial domain D containing the nuclei of molecule X, and let Pmax denote the maximum value of the electron density within D:

(20)

Pmax = max {p(r), r ~ D}.

In terms of

Pmax,

a fuzzy membership function ~ x ( r ) is defined for points

r of the space, expressing the "degree" of their belonging to molecule X:

(21)

l-tx(r) - p (r)/Pmax-

In a similar spirit, if molecule X is not isolated, then a point r may belong to several different molecules to different degrees, that can also be expressed using fuzzy membership functions. In particular, the total electron density p(r) at some point r can be regarded as a sum of electron densities p y ( r ) attributed to individual molecules X

X

and

Y

and

and Y, respectively. In this case,

the fuzzy membership functions of various points r molecules

px(r)

with respect to the two

are determined by the relative magnitudes of the

individual electron densities px(r)

and P Y(r)-

In the general case, if

PxI(r), Px2(r) . . . .

p x j ( r ) , . . . PXm(r),

are the electron density contributions of individual molecules

(22)

626

(23)

X1, X2, . . . Xi, . . . Xm,

respectively, from a molecular family L of several molecules, then these individual electron densities can be used to represent the "share" in the total electron density of the molecular family L and to define the appropriate fuzzy membership functions The

"share" PXi(r) of each individual molecule Xi

as a part of the

complete electron density, can also be considered in the absence of all other molecules, as a separate, individual fuzzy object. Within some domain

DXj

of the space containing all the nuclei of

molecule Xi the maximum value Pmax,i of the electron density

PXa(r) is

Pmax,i = max { Pxi(r), r s DXi }.

(24)

We may select a point rmax, i where this maximum density value P max,i is realized for the given molecule"

Pxa(rmax,i) = Pmax,i 9

A fuzzy membership function for points

r

(25)

of the space belonging to

molecule Xi (regarded now in the absence of all other molecules) is

~txj(r) = PXj(r)/Pmax,i 9

(26)

By contrast, if all other molecules of the family L are also considered to be present, then each molecule may have some partial "claim" for each point r of

627

the three-dimensional space, that is, the actual "degree of belonging" of a point r to a given molecule X i is influenced by the electron density contributions PXl(r), PX2(r), . . . PXa(r), 9 9 9PXm(r) of all molecules. In this case, the total electron density pL(r) of the molecular family X 1, X2, 9 9 9 Xj . . . .

X m taken at point r is of importance,

pL(r) = Ej pxj(r),

(27)

since individual fuzzy memberships are defined relative to this total density. In particular, the fuzzy membership function ~tXi,L(r) for points r of the space belonging to molecule Xi of the family L is given by

l-tXi,L(r) = ~xi(r) [Pmax,i / pL(rmax,i)]-

(28)

The ratio [Pmax,i / pL(rmax,i)] used in this expression is a scaling factor that provides a proportional treatment of the actual density contributions from various molecules of the family L. An alternative, equivalent expression for the fuzzy membership function txXJ,L(r) is givenby

laXa,L(r) - laXi(r) [Pmax,i / pL(rmax,i)]

= [PXj(r)/Pmax,i ] [Pmax,i / pL(rmax,i)]

= px~(r) / pL(rmax,i) 9

(29)

628 These fuzzy electron density membership functions properly describe the mutual interpenetration of fuzzy electron density clouds within the molecular family L, and it provides a description of how molecules share some common regions of space. The fuzzy, low density glue aspect of bonding in proteins can be modeled using a rather simple approach. In tightly folded arrangements of chain molecules, the formal space filling characteristics are manifested by the merger of electronic density clouds between molecular parts which are not linked directly by formal chemical bonds. Of course, these mergers also contribute to bonding and to the stability of the actual folded pattern.

For proteins and large polypeptides the accumulated

computational experience suggests [15,16]

that the AFDF electron densities

exhibit such non-bonded mergers at many locations within the molecule occurring approximately at the same density threshold am. The recognition of this trend is likely to assist in the search for stable conformations of proteins, in the study of side chain arrangements and in predicting folding patterns. The simplest implementation of this idea is the Self-Avoiding MIDCO approach, proposed as a simple, approximate method for conformation analysis of biopolymers [16]. Consider a threshold value am that corresponds to the onset of most "non-bonded" mergers of MIDCOs G(K, am). Using the AFDF methods, the electron density can be computed for a family R of a large number of nuclear configurations K of the macromolecule. One may test the MIDCOs G(K, am+Aa), G(K, am), and G(K, am-Aa)

for the selected threshold value

am and density increment Aa for each nuclear configuration K of family R. The self-avoiding MIDCO method identifies those nuclear configurations K from this family R which show favorable "non-bonding" interactions:

629

A given configuration K is selected if the non-bonded mergers of density contours which appear for the lower density MIDCO G(K, am-Aa) do not appear yet in the higher density MIDCO G(K, am+Aa). In practice, a value of am falling within the range [0.003 a.u., 0.005 a.u.] of density thresholds and a density increment of

Aa -- 0.001 a.u. appear as a

suitable choice. It is important to realize that the Self-Avoiding MIDCO approach is not a fuzzy set version of a hard surface contact model. If

various parts of a

macromolecule are placed side by side, then the electronic density

charge

clouds mutually enhance each other due to their partial overlap, resulting in an actual shape change of these electron density clouds. The various MIDCOs G(K,a)

experience significant swelling due to this overlap. The merger of the

local parts of the MIDCO actually occurs at a point r outside of each individual MIDCO

that would fall on the

part without the presence of the other

MIDCO part. Consequently, this rather simple, Self-Avoiding MIDCO method incorporates some aspects of the non-bonded interactions resulting in a shape change of the MIDCO surfaces. A more precise and also more detailed analysis is possible if one considers the "Low Density Glue" (LDG) part of the electron distribution, defined as the object

LDG(K, am, Aa) = DD(K, am-Aa) \ DD(K, am+Aa),

where

DD(K, am-Aa)

and

DD(K, am+Aa)

(30)

are the density domains

associated with the MIDCOs G(K, am-Aa) and G(K, a m+Aa), respectively. Using the formalism of fuzzy set theory [26], the "low density glue" object

630

LDG(K, am, Aa) can be thought of in terms of a-cuts, where two cuts are made, with the two values of o~ - am-Aa and o~ - am+Aa. Typically, the object LDG(K, am, Aa) contains at least one hollow interior cavity which is the high density range of the molecule, describing most of the nuclear neighborhoods and the pattern of conventional chemical bonds. For the purposes of our current study, the geometrical and topological features exhibiting multiple connected features are of special interest. The mergers of electron density clouds due to non-bonded interactions are manifested in the fact that the object

LDG(K, am, Aa) is multiply connected.

In fact, the simplest type of connectedness property is the most relevant: arcwise multiple connectedness, or 1-connectedness. In algebraic topology, the one-dimensional homotopy group, also called the fundamental group of the object, provides a concise description of

1-

connectedness. Here we are able to rely on an analogy with another chemical problem that has been already studied in some detail [28], and provides a homotopy group alternative to the homology group characterization of molecular shape [29]. An important application of homotopy groups in chemistry is the study of interrelations among reaction mechanisms. If an upper bound A

is

taken for energy, then the part F(A) of the potential energy hypersurface E(K) that falls below this energy bound represents all molecular species and all their interconversion processes (reactions) which are accessible below this energy bound.

The family of energy-dependent reaction mechanisms which are

realizable below this energy bound A

form an algebraic group, the one-

dimensional homotopy group I-II(F(A)) of the potential energy hypersurface level set F(A), A

for

that is, the potential surface truncated at the given upper bound

energy.

This group

I-I I(F(A))

is the fundamental group of the

631

truncated potential energy hypersurface, and it is referred to as the fundamental

group of reaction mechanisms at the given energy bound A [29]. The fundamental group 1-II(LDG(K, am, Aa))

of the "low density glue"

part of the macromolecular electron density is defined analogously to the fundamental group of potential energy hypersurface level set F(A), as the onedimensional homotopy group of the object LDG(K, am, Aa). One-dimensional homotopy groups describe the patterns of loops which are not contractible into one another within the object LDG(K, am, Aa), hence I-[I(LDG(K, am, Aa)) describes the arcwise-connectedness of LDG(K, am, Aa). Note that arcwise-connectedness differs from connectedness manifested in the contractibility of spherical surfaces; as it has been pointed out above, the "low density glue" part of the macromolecular electron density has a hollow interior, hence not all spherical surfaces within

LDG(K, am, Aa)

are

contractible to a point. For example, the inner wall of the cavity can be regarded as a topological sphere, and this sphere cannot be contracted to a point without "leaving" the body of object LDG(K, am, Aa). Typically, within an object LDG(K, am, Aa) of a macromolecule M,

there are two types of spherical

surfaces, those deformable into the interior wall of the cavity and those contractible to a point. These surfaces and their homotopy equivalence classes are described by the two-dimensional homotopy group I-[2(LDG(K, am, Aa)) of the object LDG(K, am, Aa). This is a relatively simple group, a free group with a single generator. In most cases, the types of one-dimensional loops with homotopicaily different contractibility properties within LDG(K, am, Aa) are more numerous, and the fundamental group generators.

1-II(LDG(K, am, Aa))

has a large number of

632

Similarities in the low density bonding contributions in proteins can be studied by comparing the fundamental groups I-I I(LDG(K, a m, Aa)). If for two proteins, or for two different folding patterns K and K' of the same protein the two fundamental groups, as abstract groups agree,

I-I I(LDG(K, am, Aa)) = I-II(LDG(K', am, Aa)),

(31)

then their low density bonding contributions exhibit a well-defined similarity. In this case, the two conformations, K and K' are regarded LDG-homotopically equivalent, that can be expressed in the notation

K---- K'

(32)

Further specifications are possible even if the two fundamental groups do not agree, by considering the group-subgroup relations among all possible LDG fundamental groups for the given proteins. The problem is fully analogous to the

characterization of the family of fundamental groups of reaction

mechanisms [29] that has been given in terms of a lower semilattice.

The

derivation will not be repeated here; the same method can be adapted for the fundamental groups

I]I(LDG(K, am, Aa)).

A hierarchy of the

LDG

fundamental groups, organized into a lower semilattice, provides the basis for comparisons of the low density bonding patterns in proteins.

633

REFERENCES

[1]

L. Pauling, The Nature of the Chemical Bond, Comell Univ. Press, Ithaca, 1960.

[2]

P. Hohenberg, and W. Kohn, Phys. Rev., 136, B864 (1964).

[3]

W. Kohn, and L.J. Sham, Phys. Rev., 140, A1133 (1965).

[4]

R.G. Parr, Proc. Natl. Acad. Sci. USA, 72, 763 (1975).

[5]

M. Levy, Phys. Rev. A, 26, 1200 (1982).

[6]

A. Becke, Phys. Rev. A, 33, 2786 (1986).

[7]

P. Politzer, J. Chem. Phys., 86, 1072 (1987).

[8]

D. R. Salahub, Adv. Chem. Phys., 69, 447 (1987).

[9]

E.S. Kryachko, and E.V. Ludena, Density Functional Theory of ManyElectron Systems, Kluwer, Dordrecht, 1989.

[~o]

T. Ziegler, Chem. Rev., 91, 651 (1991).

[11]

P.G. Mezey, "Functional Groups in Quantum Chemistry". Advances in Quantum Chemistry, 27, 163-222 (1996).

[12]

P.G. Mezey, "Shape Analysis of Macromolecular Electron Densities", Structural Chem., 6, 261 (1995).

[13]

P.G. Mezey, "Density Domain Bonding Topology and Molecular Similarity Measures". In K. Sen, ed., Topics in Current Chemistry, Vol. 173, Molecular Similarity, Springer-Verlag, Heidelberg, 1995.

[14]


[15]


[16]

P.D. Walker, and P.G. Mezey, J. Math. Chem., 17, 203 (1995).

[17]

P.D. Walker, and P.G. Mezey, J. Comput. Chem., 16, 1238 (1995).

[18]

P.G. Mezey, J. Math. Chem., 18, 141 (1995).

634

[19]

P.G. Mezey, "Molecular Similarity Measures of Conformational Changes and Electron Density Deformations", Advances in Molecular Similarity, 1, 89 (1996).

[20]

P.G. Mezey, Int. J. Quantum Chem., 63, 39 (1997).

[21]

P.G. Mezey, Int. Rev. Phys. Chem., in press (1997).

[22]

L.A. Zadeh, Inform. Control, 8, 338 (1965).

[23]

L.A. Zadeh, J. Math. Anal. Appl., 23, 421 (1968).

[24]

A. Kaufmann, Introduction glla Th~orie des Sous-Ensembles Flous, Masson, Paris, 1973.

[25]

L.A. Zadeh, "Theory of Fuzzy Sets". In Encyclopedia of Computer Science and

[26]

Technology, Marcel Dekker, New York, 1977.

G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice-Hall, Englewood Cliffs, NJ 1995.

[27]

P.G. Mezey, "Fuzzy Measures of Molecular Shape and Size", in Fuzzy Logic in Chemistry, Ed. D.H. Rouvray, Academic Press, San Diego, 1997, pp 139-223.

[28]

P.G. Mezey, Potential Energy Hypersurfaces, Elsevier, Amsterdam, 1987.

[29]

P.G. Mezey, Shape in Chemistry: An Introduction to Molecular Shape and Topology, VCH Publishers, New York, 1993.

635

Figure 1. HIV Protease monomer electron density by the AFDF method. J. Math. Chem. 17, 203 (1995).

636

Figm'e 2. AFDF electron density of Proto-oncogene Tyrosine Kina:se Protein 1ABL. Drug Discovery Today, 2,132 (1997).



637


Models for Understanding and Predicting Protein Structure Dale F. Mierke Gustaf H. Carlson School of Chemistry, Clark University, 950 Main Street, Worcester, Massachusetts, 01610, USA email: [email protected] Department of Pharmacology and Molecular Toxicology, University of Massachusetts, Medical Center, 55 Lake Avenue, North, Worcester, Massachusetts 01655, USA

Introduction The generation of protein sequences (or equivalently the DNA sequences which provide the protein sequence) is accelerating at a great pace. In the foreseeable future the complete human genome will be sequenced. The challenge for the protein chemist is to assimilate and utilize this information [ 1]. Given the strong correlation between structure and function, the question is therefore the determination of structure from the primary protein sequence. Over the years great advances have been made in the experimental determination of protein structure. In the field of nuclear magnetic resonance (NMR), the increasing magnetic field strength, additional radio-frequency channels and pulse-field gradients, and the use of isotopic-enriched proteins including enrichment of 13C, 15N, and partial deuteration have expanded the range of proteins that can be investigated, including the size of the protein and the conditions (pH, temperature) under which they can be investigated [2]. The data analysis, the slowest step in the structure determination, has been greatly facilitated by advances made in automation (signal identification and assignment), a product of greater computational power and algorithm development. In x-ray crystallography advances have been made in both the collection and analysis of the data (cryo-diffraction, determination of phase), also a beneficiary of greater computer power and the development of novel algorithms. However, the generation of suitable crystals is a bottleneck in the rate at which protein structures can be determined. Given the difference in the current rate of generation of protein sequences and the experimental determination of structure, there is a vital role for theory. In addition, the experimental methods listed above are not generally suitable for the analysis of proteins within membrane environments, an area of investigation in which theoretical methods have made the most progress. A large number of transmembrane receptors (proteins which pass through the cellular membrane, often a number of times), vital for signal transduction from the extracellular domain into the cell, have been identified. In addition, an ever increasing

638

number of proteins associated with cellular membranes through palmitoylation or other lipid moiety have been isolated. In this chapter, some methods currently employed in the prediction of tertiary structure from the protein sequence will be highlighted. This is not intended to be a comprehensive review of the literature, a goal that would require an entire book of this size and would quickly become outdated. Instead, the aim is to provide some of the basic tenets of the theories and methods utilized with a few select references to the original literature. Hopefully this will serve as a starting point to delve into this very exciting and ongoing field of research.

Background The goal of protein-structure prediction is to derive the tertiary structure of the protein (defined as the manner in which the protein is bent or folded in three dimensions) given the sequence of amino acids (referred to as the primary structure). In between the primary and tertiary structure is the secondary structure which consists of regularly recurring arrangements of the protein chain in one-dimension (i.e., a-helices and 13-sheets).

"

\

.o3 ') i

Figure 1: Definitions of primary (left), secondary (center), and tertiary structures (right).

639 This nomenclature may also describe the sequence of events in the folding process: the primary sequence adopts secondary structural elements which then fold into the correct tertiary structure. Pauling first postulated that the hydrogen bond played a large role in the folding process. The importance of the hydrogen bond in stabilizing the secondary-structural elements, or-helices and [3-sheets, was quite clear from his earlier work [3-5].

C.....~

(-i

")

Figure 2" Illustration of the hydrogen bonding networks in the secondary structural elements (left) or-helixand (right) 13-sheet. It was thought that the folding of the protein was a rearrangement to find the fold with the maximum number of hydrogen bonds with the correct geometry. And indeed, analysis of typical protein structures provides a large number of hydrogen bonds both within and between the secondary structural elements. However, the pendulum was about to swing away from the hydrogen bond and towards something called the hydrophobic effect [6]. The typical protein sequence contains a large number of hydrophobic amino acids. When these amino acids reach a specific concentration the protein collapses into a globular conformation, similar to the critical micellar concentration of detergents. This state is the result of a collapse of the protein driven by the minimization of the interaction of the hydrophobic amino acids with the aqueous environment. This condensed state is fluid in that the polymer is not in one well-defined conformation but sampling many different conformations until eventually the native conformation with the lowest free energy is found. This, usually ill-defined, state is referred to as the molten globular state. The manner in which

640 the protein quickly locates the correct fold, the global minimum, in the presence of the large number of possible configurations within the molten globule is commonly known as Levinthal's paradox [7]. Great insight into the sequence of events leading to the correct fold has been obtained from experiments involving the controlled unfolding of proteins. The classic experiment by Anfinsen illustrated the reversibility of protein folding with regeneration of full enzymatic activity [8]. If the induced unfolding is completely reversible then an understanding of the unfolding pathway will shed insight into possible folding pathways. The folding is induced by changes in the conditions such as pH and/or temperature or by the addition of denaturing agents including urea or gaunidinium chloride. Ideally the intermediates along the folding/unfolding pathway would be structurally characterized. However, due to their instability this is often not possible (although recently some denatured structures have been characterized by NMR [9], not an easy feat given that in the unfolded state most nuclei experience similar magnetic environments, greatly reducing the signal resolution). One very facile method to probe the intermediate states is with hydrogen exchange [10]. This method utilizes the fact that the amide protons of the protein chain are slightly acidic and the rate of exchange can be altered by pH. At various stages during the unfolding process, deuterium oxide (2H20) is used in place of water and therefore all amide protons that are exposed to and exchanging with the solvent at that time will be replaced with a deuteron. The quantity of deuterons that the protein possesses after refolding can be readily determined by mass spectrometry; the specific location of the sites of exchange within the protein must be determined by nuclear magnetic resonance [10]. The first basic tenet of protein-structure prediction is that the amino acid sequence, the primary structure, contains all of the information required for the correct folding of the polymer chain. This is a first approximation which clearly ignores the role of environment on the induction of structure or the action of chaperone proteins which assist the m vivo folding process. The wide variety of structural motifs that have been observed for proteins is derived from only twenty different monomers (amino acids), many of which are structurally quite similar (i.e., isoleucine and leucine vary only in branching of the butyl side chain). However, there are many cases in which the substitution of amino acids with structurally similar residues (so-called conservative substitution) will lead to a protein that will not properly fold. Studies involving deletion of even small portions of the termini of the protein sequence provide similar results. On the other hand there are proteins related through evolution with as little as 20% sequence identity which adopt similar three-dimensional structures. Therefore the information encoded in the primary sequence is specific for one protein fold, however, there are numerous other sequences, only remotely related at first glance, which will produce the same fold.

Methods

There are many different methods to attack the problem of predicting protein structure. If the protein is a member of a family of closely related proteins, the structure can be postulated

641 through homology analysis. If this is not the case, one can envision a sequential approach in which the secondary elements are predicted from the primary sequence and then different topological orientations of these elements are examined and judged using some energyfunction description of the protein. To reduce the number of topological arrangements the search could be biased towards the arrangements previously observed in proteins of known structure. Other methods are based on the observation that protein structures are tightly packed with almost no vacant space in the interior or core of the protein. Such a tight packing of the amino acids then suggests that the arrangements of the polypeptide chain can be restricted to the points of a lattice, greatly reducing the number of possible orientations. These methods do not differentiate between secondary and tertiary structure nor depend on previously determined protein structures.

HomologyModeling By far and away the prediction method with the greatest accuracy is homology modeling. If there exists a protein sequence, with a known three-dimensional structure, that has a similar primary sequence then the level of confidence that the protein in question will adopt a similar fold is quite high. A model can be built using the known sequence and the resulting structure analyzed for energetically favorable (and unfavorable) contacts (e.g., charge-charge interactions described by a Coulombic interaction). Despite the specificity of the primary sequence for a particular fold discussed above, a sequence identity of only 30% is sufficient for this method to work. More sophisticated methods are available, that utilize not only the primary sequence but the functional properties of the protein and therefore greatly expand the range of homologous proteins [11 ]. Homology modeling has been used with great success in the class of transmembrane protein receptors [12]. Simply given the presence of seven regions of hydrophobic amino acids, model tertiary structures can be constructed. In many instances these models have provided insight for experimental data and for accurate predictions of effects of mutations of specific amino acids [13]. This will be discussed in greater detail below. For homologous sequences, the variations will be for the most part limited to the exterior portion; the central, mainly hydrophobic, core of the proteins will be very similar [14]. Of course, there will be variations, elongation of an or-helix or 13-sheet, to accommodate for specific sequence differences, but these will usually be minor. The greatest differences are therefore found in the hydrophilic loop regions which connect the elements which make up the core of the protein. These segments containing polar and charged amino acids are irregular in structure but certainly not structure-less. Data bases of loop structures have been created and the prediction of the correct loop conformation is usually made with a level of confidence equal to that of the presence of secondary structural elements [ 15,16]. A great deal of information can be obtained from the primary sequence simply by mapping out the hydrophobic and hydrophilic nature of the amino acids. Numeric scales for the hydrophobicity of the amino acids have been developed [ 17,18]. By calculation of a running average of this scale, the identification of transmembrane or membrane-associated regions can be readily identified. This is particularly true for proteins with one region that passes

642 through the cellular membrane, such as the epidermal growth factor receptor. The identification of a single stretch of 18-25 hydrophobic amino acids clearly indicates the transmembrane region. For proteins with multiple transmembrane domains, it is not necessary to have exclusively hydrophobic amino acids; a pair of amino acids with opposite charges may be present in the lipophilic environment of the membrane. Therefore a search for amphipathic or-helices must be undertaken. Amphipathic helices have well-defined hydrophobic character, the hydrophobic face which would project towards the membrane/lipid environment, and a hydrophilic face, which would project out into the aqueous phase or towards the core of a helix bundle. Often times the distinction is not clear and there are regions of mixed hydrophobic/hydr0philic character. Graphically this can be realized with a helical-wheel representation in which the amino acid side chains project out, at 100 degree intervals, from the view along the long, helical axis.

,.._j

Figure 3: Helical wheel representation illustrating the hydrophilic (boxed amino acids) and hydrophobic (circled amino acids) character of an amphipathic helix. All of the hydrophobic residues are on one face of the helix (bottom half of the figure), while the hydrophilic residues are on the top half. The hydrophobic/hydrophilic nature of the different helical faces is clearly illustrated by this projection. The amphipathic character of a segment of a protein can be calculated by use of a Fourier series and calculation of a hydrophobic moment [ 18].

M- {[

sin(jo )] 2 + [ZI~ cOS(jr'O)]2} 1/2

643 The hydrophobic moment gives an orientation of the a-helix with respect to the lipid environment and the aqueous interface of the membrane surface for single a-helices or the hydrophilic core formed from a cluster of transmembrane a-helices (seven and twelve helices are quite common). Other methods, conceptually along similar lines, have produced power spectra which are cleaner, facilitating the identification of the lipophilic moment [ 12]. Now that the location of the helices within the primary sequence of the protein, as well as the relative orientation of each helix with respect to the lipid environment have been identified, the only task that remains is mapping the sequence onto the three-dimensional arrangement of the helices. In the seven-transmembrane receptor field the standard has been the structure of bacteriorhodopsin, which does not couple to a G-protein, determined by electron cryomicroscopy [ 19]. More recently a low-resolution electron crystallographic image of bovine rhodopsin, which does couple to a G-protein, has appeared in the literature [20]. These structures illustrate that the helices are not aligned parallel through the membrane but are titled at angles forming binding pockets within the core and between the helices themselves. The sequences identified above are mapped onto the helices obtained from these experimentally determined topologies. The hydrophobic moments are then used to rotate the helices about the long axis to maximize the fit with respect to the lipid environment. Based on the transmembrane helices of the bacterial reaction center, look up tables on the environmental preference of amino acids have been developed [12]. The table containing values for the three different environments found in these receptors (i.e., lipid environment, hydrophilic core, or helix-helix interface) allows for fine tuning of the helical orientation. Models for a large number of seven transmembrane-helix proteins have been determined. Many of these models have been successfully used to provide atomic insight into the genetic manipulation of the receptors.

Secondary Structure Prediction Every since Pauling described the existence of well-defined secondary structural elements [4,5], a great deal of work towards the accurate prediction of the location of these elements has been carried out. Given the accurate location of the a-helices, B-sheets, and turns the only remaining task would be the correct folding of these elements. This is still a formidable task, as highlighted below, but the number of available conflgurational folds is greatly reduced in this manner. The accuracy of these methods is currently hovering about 60-70 %. These methods are based on the premise that a protein segment of a specified number of residues has a unique conformation (i.e., the secondary element is derived from local interactions) which can be identified in a database of known protein structures [21,22]. It was shown that the length of the segment is important for the success of this procedure. Too short of a segment, and there would be no common structural features: a segment consisting of five residues (pentapeptides) was shown to be insufficient [23]. Using too long of a segment would leave one with too few examples in the database, and therefore the preference for a secondary structure would not be well-defined. This general approach lends itself to the use of neural nets [24-26]. Using neural nets with the inclusion of evolutionary information of the protein sequences has produced a method with greater than 70% accuracy [27].

644 Once the secondary structural elements have been identified, the task of putting the "pieces" of the "puzzle" together must be undertaken. Although for many questions of biological or pharmaceutical interest the prediction of the secondary structural elements is sufficient. If a particular structure-activity relationship requires a helix-loop-helix motif, the prediction of secondary structural elements without regard for the topological arrangement allows one to ascertain if the new protein will be a likely candidate for further investigation. If the mapping of the secondary elements is not sufficient, it is necessary to proceed to combine the "pieces" into a tertiary structure. The first step, as alluded to above, is the development of possible loop conformations which connect the regions of secondary structure. The loops which do not fit into the well-defined category of or-helices or ~3-sheets have been fairly well characterized using the data base of proteins for which the three-dimensional structure is known [ 15,16]. The identification of specific loop conformations provides insight into the possible orientations, or at least provides limitations on the possible orientations, of the various secondary structural elements. The second step is then analysis of the array of amino acids within the secondary structural elements with attention to the environment in which the amino acids would be found. It is clear that a cluster of hydrophobic amino acids would not likely be projecting into the aqueous solution, and more likely projecting into the core of the protein. This analysis provides additional restrictions to the number of possible arrangements in which the secondary structural elements may be found. Another approach is to map the arrangement of secondary structural elements onto the known tertiary structures of other proteins. Currently, approximately one hundred unique protein folds have been identified. There is some question as to if this is an upper limit. If this is indeed the case, then the protein of unknown structure must adopt a known topological fold. The secondary structural elements are mapped onto the template of the different known protein structures. The best fits, as judged by the environmental factors (solvent accessibility) of the individual amino acids, are then further analyzed as probable folds. This procedure is referred to as threading the secondary elements into three-dimensional structures [28].

Primary to Tertiary Prediction Many different methods have been developed which skip over the prediction of secondary structural elements and proceed directly from the primary sequence to the protein fold [2933]. The most straightforward again involves the concept of threading. From a database of known protein structures, templates are created. These templates contain the relative topological arrangement of all the different secondary structural elements. The loops between them have been removed to allow for variability in the number of amino acids (again differences between homologous proteins are usually located in the loop regions). The original sequence is not retained but instead at each location of an amino acid a place holder is created. The protein sequence with unknown structure is then threaded through this template like a string of beads, with each bead representing an amino acid that is located in a place holder or within the variable loop domains. Then an energy or a measure of the fit is calculated and the one with the lowest value is deemed the most probable structure.

645

One advantage is that the template and test protein do not need to be of similar lengths. A very good fit could be identified for the N-terminal portion of a very long test sequence by a much shorter template. Large proteins often adopt different structural domains with identifiable folds. Likewise, a short test sequence could adopt a fold that utilizes only a small portion of the template. This rather straightforward sounding method avoids the problems associated with the identification of secondary structure elements. The assumptions are that most protein folds have already been identified and therefore the unknown structure of the test protein will most likely resemble a fold within the database. It is clear that a novel protein fold will not be identified by this method. Other problems are in the flexibility of amino acid replacement. There are many instances in which a large amino acid in the test sequence replaces a smaller one in the template. This could be energetically costly, while in nature these replacements occur with a few rotations about side chain dihedral angles or very small adjustments of the backbone dihedral angles [29]. The calculation of the energy or the fitting of the test sequence in the fold of the template is no easy matter. The utilization of a full force field with complete atom representation does not properly discriminate between the different folds [31]. This seems to be related to an energy surface that is too fine and the presence of numerous local minima. In its place a potential function based on a statistical analysis of known protein structures has been developed [34]. The pair-wise penalty function provides a pseudo-energy based on the number of times the specific interaction has been observed in known protein structures. This function provides amino acid-amino acid interactions as well as a measure for the solvent exposure of each amino acid [34]. Another similar approach is to use the known three-dimensional structures to create look-up tables which contain the most favorable environmental parameters of each amino acid. The parameter sets are created in terms of secondary structure, hydrogen bonding pattern, solvent accessibility, and local presence of polar atoms [30,32]. In this manner the three-dimensional information is encoded into a one-dimensional string. A comparison is then made of the test protein sequence with this one-dimensional string. If the test sequence is similar, a model fold can be created for further analysis.

Energetic Force Fields In contrast to the methods mentioned above which differentiated between primary and secondary structure or utilized a data base of known protein structures, there is the possibility of utilizing one of the many potential energy functions which have been shown to accurately reproduce many features of proteins, including thermodynamics and molecular motions. The potential energy force fields vary in specific details, mainly depending on the target molecule for which they were developed. A very typical energy force field is shown below.

646 bonds U

.._

+

I k ( R - R0) 2 2

Bonds

_1 k (0 - 00) 2 2

Angles

dihedrals

+

Y~ _1 k [cos(n~ + n) + 1] 2

+ ~

cLqj

+ ~

A _ B

Coulombic

8r

r6

Dihedrals

Lennard-Jones

rl2

Figure 4: Typical potential energy force field for a protein. A simple minded, "brute force," approach would entail the generation of the configurational fold that would minimize the potential energy given by this or similar energetic function. The problem resides in the vast number of conformations (or degrees of freedom) available to the protein. Assuming an average of four dihedral angles per amino acid and that each of these dihedral angles have six possible values, produces over 1200 available states. This is for each amino acid. Clearly a systematic sampling of all of the available states is not possible: the protein folds too fast for this to occur. How a protein is able to accomplish exactly this feat is at the essence ofLevinthal's paradox [7]. With current computational power, the utilization of such a force field in a molecular dynamics or simulated annealing approach, can sample time scales of up to a few nanoseconds not nearly sufficient for the folding of the protein chain (typically on the order of seconds). Given the correct fold, molecular dynamics simulations can provide great insight into small fluctuations about the global configuration, but the full atom representation is not suited for searching the large number of possibilities available to a protein starting from the primary sequence. Therefore, if such simulations are going to be useful for the ab initio searching of global folds, simplifications must be made. It should be noted that full atom representations have been used to examine the unfolding of small peptides with the hope of gaining insight into protein unfolding [35,36]. Reduced Atom Representation One mode of simplification concerns the representation of the polymer itself. The backbone of the protein polymer is illustrated below.

647

et Ci+l.

1

Nil. 1

""~fNi /

~Ci_ 1

CiV_l

Q-i Figure 5: Schematic of protein backbone.

The first simplification is the reduction of the side chain atoms to a single point. Often this point is attached to the position of the beta-carbon (denoted 13C in Figure 5). The radius of the point is defined by the relative size of the side chain. The point can be given charges to mimic the acidic or basic nature of the residue. By variation of the non-bonded parameters the point representation of the side chain can be treated as hydrophobic or hydrophilic. This simplification can reduce the number of atoms by up to 70% depending on the sequence of the protein. Removal of the side chain dihedral angles greatly decreases the number of degrees of freedom available to the protein. However, this comes at a cost of losing many of the fine structural features of the different side chains; these simply cannot be differentiated with the non-bonded parameters of a single point. Given the planarity of the amide linkage between amino acids, the distance between all of the alpha-carbons is 3.8 A. A second simplification is to maintain a constant distance of 3.8 A between the alpha-carbons (denoted otC in Figure 5)and to remove the amide (denoted as HN) and carbonyl (denoted as C'-O') functionalities. The combination of these two simplifications reduces the peptide chain to two points per amino acid. It is not a great leap to reduce the polymer chain to one point with a distance of 3.8 A between each point. The problem with this second simplification is the loss of the hydrogen bonding capability.

648 Although, hydrogen bonding does not seem to play a vital role in the folding of the protein chain, the importance in the stabilization of the final fold has not been questioned. Therefore, the removal of the capability to form hydrogen bonds may cause problems: not in the analysis of the folding process itself, but in the identification and proper calculation of the energy during the sampling of the available states. Recently a series of polymers in which the amino acid side chain was attached to the amide nitrogen rather than to the alpha-carbon were synthesized. These polymers referred to as peptoids do not fold in the same manner as the corresponding peptides. This is evidence that the hydrogen bonding is indeed important for the definition of the global minimum energy structure. It therefore seems imperative to maintain at least three atoms for an accurate representation of the protein chain: NH point to represent the amide group, the 13-carbon point with appropriate charge/character (size, polarity) to reproduce the amino acid side chain, and a CO point for the carbonyl group. Both the NH and CO points are able to form hydrogen bonds. We have recently developed a force field for this reduced atom representation for use in a simulated annealing molecular dynamics simulation [3 7]. The method we use is a hybrid distance geometry approach with a refinement using a molecular dynamics force field with the reduced atom representation. The distance geometry method utilizes the approach of metric-matrix distance geometry as described by Havel and Crippen [38]. The three points for each amino acid are defined from a starting structure of undefined conformation but with accurate bond distances and bond angles. The NH and CO point are taken as the geometric point of the amide and carbonyl groups, respectively. The standard backbone dihedral angles, t~ and W, defined as CO-HN-CB-CO and HN-CB-CONH, respectively, are completely free to rotate. The upper and lower distances are calculated using standard geometry by assuming free rotation. The distances are then further refined by utilization of the law of triangular inequality, resulting in tighter upper and lower distance bounds. A pair of atoms is then randomly chosen and a distance between their upper and lower limits is randomly chosen, a process defined as random metrization [39]. Once the exact distance between one pair of atoms is chosen, the upper and lower distances between the other atoms can be further tightened by re-applying the triangular inequality law. Another atom pair is chosen randomly and the process is repeated producing a real, symmetric matrix of distances. The Eigenvectors associated with the largest Eigenvalues from the diagonalization of this matrix can be used as the principal axes for the Cartesian coordinates of the conformation fulfilling the chosen distances. There are a number of advantages to using the distance geometry approach. The generation of the distance matrix is completely general; no conformations are excluded. In addition, since the molecular constitution is described by distances one is not limited to three spatial dimensions. Utilization of Eigenvectors associated with the four largest Eigenvalues produces coordinates, which are consistent with the tIn'eedimensional protein, in four-dimensions. Higher dimensionality has been used by a number of different groups to simplify the searching of conformational states [40,41]. A threedimensional object can "tunnel" through itself in four dimensions. Casting an object of Npoints into N-1 space allows for the calculation of the global minimum in one step of energy minimization. The subsequent reduction of the dimensionality followed by energy minimization should produce the global minimum [41 ].

649

The resulting four-dimensional structures are then refined with a simple molecular dynamics force field which is solely based on distances (i.e., Coulombic potential, Lennard-Jones nonbonded potential) and therefore is fully consistent with the higher dimensionality. After the simulated annealing protocol, the Cartesian coordinates of the structure is converted back to a real, symmetric matrix which is then diagonalized. The Eigenvectors associated with the three largest Eigenvalues are then used as the principal axes in the generation of the threedimensional Cartesian coordinates. The three-dimensional structure is then further refined by addition of all of the atoms and utilization of a force field for the full atomic representation. Similar procedures have been reported by other research groups [42].

Reduced Conformational Space~Lattice Models Another approach to simplify the protein chain is to reduce the conformational space allowed to the protein. The argument is simply that one of the major forces in protein structure is the formation of a core, a hydrophobic core, in which the side chains are tightly packed with no free volume. It is indeed postulated to be one of the first steps in protein folding. If this is the case, the core of the protein can be simplified to the points on a lattice [43-47]. A lattice, commonly 27 points are utilized, reduces the number of possibilities that need to be examined. Each amino acid is treated as either a single point, the alpha-carbon, or includes both the alpha- and beta-carbons, and assigned to one of the points of the lattice. The searching can be carried out with simulated annealing, molecular dynamics or Monte Carlo algorithms. Given the reduced atom representation and the lattice restricting the locations, one can address Levinthal's paradox since each and every possible configuration can be examined. One recent study [44], used a protein of known structure. They examined every possible conformation and whether it lead to the correct protein fold, in essence the kinetics of the folding process. Surprisingly, they found no correlation between the presence of secondary structural elements and finding the correct fold. Instead the important feature that lead to the correct fold was the presence of the native state as a well-defined energy minimum. The correct energy landscape of configuration is the determinant feature for the correct folding. The idea of the folding process being driven by the available energy landscape has been discussed in the terms of protein folding funnels [48-50]. The folding polymer chains have been liken to glasses; below a certain critical temperature the protein exhibits glass-like properties. When the simulation is below this temperature, no folding occurs. Above this temperature, there is sufficient energy available to the system for the chain to locate the global minimum and adopt the correct tertiary structure. Such simulations are shedding new light onto the protein folding problem.

650

Conclusions In this contribution I have tried to highlight some of the various approaches currently being employed in the field of protein structure prediction. Of course, all of the different methods nor all of the contributions from the many groups working in the field could be covered. However, I hope that the reader will find this as a starting point, with the many references, for delving into this exciting and rapidly expanding area of research. It is clear that there is still a long way to go before the tertiary structure can be accurately predicted directly from the primary sequence.

Acknowledgments The author would like to thank Dr. Maria Pellegrini and Eduaro Mercurio for fruitful discussions and reading of the manuscript.

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651 12. Donnelly, D., Overington, J. P., Blundell, T. L. "The prediction and orientation of ahelices from sequence alignments: the combined use of environment-dependent substitution tables, Fourier transform methods and helix capping rules" Prot. Engng. 1994 7, 645-653. 13. Schwartz, T. W., Rosenkilde, M. M. "Is there a 'lock' for all agonist 'keys' in TM7 receptors" Trends Pharm. Sci. 1996 17, 213-216. 14. Janin, J., Chothia, C. "Domains in proteins: definitions, location and structural principles" Methods Enzymol. 1985 115, 420-430. 15. Sibanda, B. L., Thornton, J. M. "13-haripin families in globular proteins" Nature 1985 316, 170-174. 16. Branden, C. Tooze, J. "Introduction to protein structure", Garland Publishing, Inc., New York, 1991. 17. Kyte, J., Doolittle, R. F. "A simple method for displaying the hydropathic character of a protein" J. Mol. Biol. 1982 157, 105-132. 18. Eisenberg, D., Weiss, R. M., Terwilliger, T. C. "The hydrophobic moment detects the periodicity in protein hydrophobicity" Proc. Natl. Acad Sci. USA 1984 82, 140-144. 19. Henderson, R., Baldwin, J. M., Ceska, T. A., Zemlin, F., Beckmann, E., Downing, K. H. "Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy" J. Mol. Biol. 1990 213, 899-929. 20. Schertler, G. F. X., Villa, C., Henderson, R., "Projection structure of rhodopsin" Nature 1993 362, 770-772. 21. Chou, P. Y., Fasman, G. "Prediction of protein conformation" Biochemistry 1974 13, 222-245. 22. Goldman, N., Thorne, J. L., Jones, D. T. "Using evolutionary trees in protein secondary structure prediction and other comparative sequence analyses" J. Mol. Biol. 1996 263, 196208. 23. Kabash, W., Sander, C. "Dictionary of protein secondary structure: pattern recognition of hydrogen bonded and geometrical features" Biopolymers 1983 22, 2577-2637. 24. Holley, H. L., Karplus, M. "Protein secondary structure prediction with a neural network" Proc. Natl. Acad. Sci. USA 1989 86, 152-156. 25. Rost, B., Sander, C. "Combining evolutionary information and neural networks to predict protein structure" Proteins: Struc. Funct. Genet. 1994 19, 55-72. 26. Bohr, H., Bohr, J., Brunak, S., Cotterill, R. M. J., Lautrup, B., Norskov, L., Olsen, O. H., Peterson, S. B. "Protein secondary structure and homology by neural networks" P FEBS Letters 1988 241, 223-228. 27. Rost, B., Sander, C. "Prediction of protein secondary structure at better than 70% accuracy" J. Mol. Biol. 1993 232, 584-599. 28. Rost, B. "Fitting 1-D predictions itno 3-D structures", in Protein Folds, A distance-based approach", Bohr, H., Brunak, S., Eds., CRC Press, 1996, pp. 132-151.

652 29. Ponder, J. W., Richards, F. M. "Tertiary templates for proteins: use of packing criteria in the enumeration of allowed sequences for different structural classes" J. Mol. Biol. 1987 193, 775-791. 30. Bowie, J. U., Ltithy, R., Eisenberg, D. "A method to identify protein sequences that fold into a known three-dimensional structure" Science 1991 253, 164-170. 31. Jones, D. T., Taylor, W. R., Thornton, J. M. "A new approach to protein fold recognition" Nature 1992 358, 86-89. 32. Johnson, M. S., Overington, J. P., Blundell, T. L. "Alignment and searching for common protein folds using a data bank of structural templates" J. Mol. Biol. 1993 231, 735-752. 33. Michie, A. D., Orengo, C. A., Thornton, J. M. "Analysis of domain structural class using an automated class assignment protocor' J. Mol. Biol. 1996 262, 168-185. 34. Sippl, M. J. "Calculation of conformational ensembles from potentials of mean force: an approach to the knowledge-based prediction of local structures in globular proteins" J. Mol. Biol. 1990 213, 859-883. 35. DiCapua, F. M., Swaminathan, S., Beveridge, D. L. "Theoretical evidence for destabilization of an Gt-helix by water insertion: molecular dynamics of hydrated decaalanine" J. Am. Chem. Soc. 1990 112, 6768-6771. 36. Brooks, C. L. "Molecular simulations of peptide and protein unfolding: in quest of a molten globule" Curr. Op. Struc. Biol. 1993 3, 92-98. 37. Mierke, D. F., Melcuk, A., Pellegrini, M. in preparation. 38. Crippen, G. M.; Havel, T. F. In Distance Geometry and Molecular Conformation, Research Studies Press LTD., Somerset, England; John Wiley, New York, 1988. 39. Havel, T. F. "An evaluation of computational strategies fir use in the determination of protein strucutre from distnace constraints obtained by nuclear magnetic resonance" Prog. Biophys. Molec. Biol. 1991 56, 43-78. 40. Crippen, G. M. "Conformational analysis by energy embedding" J. Comp. Chem. 1982 3, 471-476. 41. Purisima, E. O., Scheraga, H. A. "An approach to the multiple-minima problem by relaxing dimensionality" Proc. Natl. Acad. Sci. USA 1986 83, 2782-2786. 42. Aszodi, A., Taylor, W. R. "Folding polypeptide alpha-carbon backbones by distance geometry methods" Biopolymers 1994 34, 489-505. 43. Skolnick, J., Kolinski, A. "Dynamics monte carlo simulations of a new lattice model of globular protein folding, structure and dynamics" J. Mol. Biol. 1991 221, 499-531. 44. Covell, D. G. "Lattice model simulations of polypeptide chain folding" J. Mol. Biol. 1994 235, 1032-1043. 45. Sali, A., Shakhnovich, Karplus, M. "Kinetics of protein folding. A lattice model study of the requirements for folding to the native state" J. Mol. Biol. 1994 235, 1614-1636.

653 46. O'Toole, E. M., Panagiotopoulos, A. Z. "Effect of sequence and intermolecular interactions on the number and nature of low-energy states for simple model proteins" J. Chem. Phys. 1993 98, 3185-3190. 47. Socci, N. D., Onuchic, J. N. "Folding kinetics of proteinlike heteropolymers" Jr. Chem. Phys. 1994 101, 1519-1528. 48. Leopold, P. E., Motal, M., Onuchic, J. N. "Protein folding funnels: kinetic pathways through compact computational space" Proc. NatL Acad. Sci. USA 1992 89, 8721-8725. 49. Onuchic, J. N., Wolynes, P. G., Luthey-Schulten, Z., Socci, N. D. "Towards an outline of the topography of a realistic protein folding funnel" Proc. NatL Acad. Sci. USA 1995 92, 3626-3630. 50. Bryngelson, J., Onuchic, J. N., Socci, N. D., Wolynes, P. G. "Funnels, pathways and the energy landscape of protein folding" Proteins: Struc. Func. Genet. 1995 21, 167-195.




655


Possible sources of error in the c o m p u t e r simulation of protein structures and interactions J.M. Garcia de la Vega, a J.M.R. Parker, b and S. Fragac aDepartamento de Quimica Fisica Aplicada, Universidad Autonoma de Madrid, 28049 Madrid, Spain bAlberta Peptide Institute and Department of Biochemistry, University of Alberta, Edmonton, AB, Canada T6G 2S2 CDepartment of Chemistry, University of Alberta, Edmonton, AB, Canada T6G 2G2

1. INTRODUCTION The significance of the results to be obtained from the simulation of protein structures and interactions, abetted by the availability of software packages and of powerful workstations with superb graphic capabilities, is encouraging a staggering proliferation of published material. Unfortunately, the quality of the predictions may be affected by a number of possible sources of error so that, unless the information obtained from the simulation is contrasted with experimental data, the danger exists that the unaware user of a software package may be offering a discussion based on a biased and/or wrongly characterized structure. That is, 'it is important to be aware of the quality of the parameters in use' [ 1]. The purpose of this chapter is to highlight where those sources of error may be found in the simulation procedure and to that end we must examine its components. In this connection some comments regarding the terminology are in order. A model is either 'a description or analogy used to help visualize something that cannot be directly observed' or 'a system of postulates, data, and inferences presented as a mathematical description of an entity' while to model is 'to produce a representation or simulation of' and simulation is 'the imitative representation of the functioning of one system or process by means of the functioning of another', such as in 'a computer simulation of [2]. In that broad sense, modelling and simulation might be taken as being interchangeable and in the literature these two designations are often used as synonyms. In order to avoid confusion, however, we would prefer to consider the adoption of a model as one of the components of a simulation. That is, we will adopt the following scheme: A simulation is performed, within the framework of a theory, for a chosen model, the calculations being carded out by a given method, with those approximations that may be required for practical reasons and using the appropriate numerical techniques. Thus, if we restrict ourselves to those simulations that have a clear quantum-chemical component, in the simplest case one is basing the procedure on perturbation theory, with a model in which the protein is represented by a collection of point particles (i.e., the atoms), with some distance constraints in an attempt to account for the existence of bonds, using an energy minimization method with appropriate computational techniques, the final

656 characterization of the resulting structure being done in terms of atomic coordinates or dihedral angles, etc. The role of perturbation theory is simply to suggest the use of 1/Rexpansions for the evaluation of the interaction energy between non-bonded atoms. The actual expressions of the expansions may be obtained purely from theoretical results but also with some semiempirical considerations. [We will omit in this discussion the existence of additional terms in the potential energy functions as well as those functions obtained in a simplified form from experimental data, although some mention will be made of the latter.] What are then the deficiencies in such a procedure? Perturbation theory is not really applied as such, the model of point particles is rather poor, the 1/R-expansions contain illdefined parameters, the energy minimization is affected (in order to reduce the computing costs) by the practical consideration of a distance cut-off threshold in the evaluation of the interaction energy, and the routine characterization of the structure in terms of its dihedral angles may be misleading. Most of these deficiencies are also present in the more sophisticated Monte Carlo and Molecular Dynamics simulations. The model cannot be improved, given the size of the proteins, and the need for a compromise between accuracy and computing time in the evaluation of the potential energy may be avoided with more computing power. Consequently we will centre our attention on the errors associated with the parameters in the potential energy function and the characterization of the structure. 2. DEFICIENCIES OF POTENTIAL ENERGY FUNCTIONS A detailed account of the derivation of potential energy functions (PEF) has been published recently [3] and therefore only those points of interest for the present purpose will be considered here. The discussion will be centred on those theoretical and semiempirical PEF, which use 1/R-expansions for the non-bonded interactions. The Coulombic term of the electrostatic interaction between two non-bonded atoms, A and B, is given by qAqB/RAB, in terms of their effective charges, qA and qB, and where RAB denotes their separation; in PEF obtained by the fitting of theoretical results, this term may be affected by an appropriate constant. The problems associated with the use of effective charges are many: (a) The effective charges are not physical observables. They are artifacts of an analysis of the electron density obtained from the wave function for the molecule under consideration. As such they depend on the quality of the wave function (i.e., the method used for its determination and whatever approximations have been introduced) as well as on the population analysis adopted for their definition. The original population analysis, proposed by Mulliken [4], has been widely used but the effective charges obtained with it do not reproduce the electrostatic potential. Over the years a great variety of schemes have been developed for the description of the electronic density distribution, in terms of charges, multipole moments, etc. [5 - 41] as well as for the determination of effective charges from electrostatic potentials [42- 59]. (b) The size (i.e., the number of atoms) of a protein precludes at this moment the possibility of obtaining the corresponding wave function. Therefore, in order to be able to proceed with the simulation, recourse must be made of the use of approximate values for the effective charges. An approximation consists of the use of average effective charges for the various classes of atoms [60], obtained from the results of calculations for the individual amino acids.

657 (c) As a rule, when using average charges, the total charge of the individual amino acids will not be reproduced and a renormalization of the charges will be needed. The resulting charges will depend on the renormalization procedure adopted [61,62]. (d) When a peptidic chain is being constructed from the individual amino acids, the formation of the peptidic bond involves the removal of an OH-group from one of the amino acids and of a H-atom from the other [3]. As a rule, the total charge of the OH-group and the charge of the H-atom will not cancel each other and the resulting system will be affected by a total non-vanishing charge. The electric neutrality of the system must then be restored by a new renormalization of the charges. The construction of the peptidic chain may be performed in different ways, with a corresponding effect on the final charges. On one hand one could complete first the peptidic chain before any attempt at optimization is made, needing only one renormalization of the charges. One could, however, proceed with some partial optimization after each peptidic bond is formed, in which case as many renormalizations are required as there are peptidic bonds. It is also possible to construct the peptidic chain from preassembled fragments, which could have been built up in either of the two ways mentioned above. The result is that the final effective charges will be different depending on the path chosen. The values presented in Table 1 illustrate these differences. The differences will be, as a rule, small but might have an appreciable effect on the overall interaction energy. It is not possible to predict a priori what that effect will be because the changes in the effective charges may lead to either a cooperative effect or to an accidental cancellation in the summation Table 1 Renormalized effective charges for the amino N in the natural amino acids a

(b)

(c)

(d) ,,

(b)

(c)

(d)

ala

-0.557

-0.585

-0.570

leu

-0.544

-0.529

-0.528

arg

-0.569

-0.585

asn

-0.558

-0.561

-0.571

lys

-0.580

-0.565

-0.562

-0.547

met

-0.570

-0.553

-0.548

asp

-0.555

-0.546

-0.533

phe

-0.570

-0.548

-0.571

cys glu

-0.567

-0.554

-0.541

pro

-0.536

-0.507

-0.527

-0.557

-0.548

-0.534

ser

-0.589

-0.559

-0.571

gin

-0.559

-0.548

-0.542

thr

-0.564

-0.541

-0.549

gly

-0.589

-0.566

-0.594

trp

-0.570

-0.549

-0.554

his

-0.533

-0.512

-0.527

tyr

-0.574

-0.553

-0.556

ile

-0.544

-0.528

-0.530

val

-0.541

-0.529

-0.531

,

,,

(a) The starting average effective charge used was -0.554 [3]. The values in this table have been obtained with the software package maPSI (S. Fraga and S.E. Thornton, Department of Chemistry, University of Alberta, Edmonton, AB, Canada T6G 2G2), using one of the existing renormalization schemes [61]. (b) In the isolated amino acids, as given by Fraga et al. [3]. (c) In the polypeptide ala-arg-asn-asp-cys-glu-gln-gly-his-ile-leu-lys-met-phe-pro-ser-thr-trptyr-val, with renormalization after each peptidic bond is formed. (d) In the same peptide as above, but constructed from three fragments, with 7, 6, and 7 residues, respectively. For each fragment, renormalization was performed after the formation of each peptidic bond.

658 EZ qAqB/RAB. Just as an example of the possible effect for one term we may consider the interaction between the backbone N of the glycine and proline residues in the peptide in Table 1, constructed directly from the residues or from preassembled fragments (columns c and d). The differences (in absolute value) in the Coulombic energy, when using the effective charges in column (c) versus those in column (d) of Table 1, are 12.1, 9.1, 7.3, and 3.6 kJ, at 3, 4, 5, and 10 A of separation. It seems highly improbable that the final effective charges obtained in this fashion will be identical to the ones that would be derived, adopting the same population analysis, from the protein wave function determined at the same level of approximation as used for the individual amino acids. It is not possible to decide which set of effective charges is more correct, but one must conclude that the effective charges may be affected by an uncertainty and that the simulation may be biased. It must be pointed out, however, that if the fitting of the 1/R-expansion (from the theoretical results) were to be performed using renormalized average charges for the individual amino acids, then the only deficiency left is the one discussed in (d). The preceding list of possible deficiencies is not complete yet. The conformation of the protein under study evolves as the simulation proceeds: the bond angles and lengths within each residue may change and the relative separations and orientations of non-adjacent residues will change. In a proper quantum-chemical calculation such changes, which imply a change in the interactions, would result in a change in the electron density distribution in the protein, which would be reflected in a change in the effective charges (determined, of course, by the same population analysis adopted initially). This deficiency (a) will introduce a new bias in the evolution of the conformation of the protein along the simulation path; (b) may be particularly harmful in docking procedures in which a molecular association is formed, with strong interactions between the two parmers: (c) will also be present when using simplified expansions with fixed coefficients obtained from experimental data for conformations different from those that appear along the simulation path. In order to remedy this situation it has been suggested [54,63-79] that the simulation should be complemented with a quantumchemical component, but attention must be paid in such a case to the quality of the latter. That is, a quantum-chemical calculation at a low level of approximation will not necessarily correct the deficiency. 3. CONFORMATIONAL CHARACTERIZATION Changes in the dihedral angles, t~ and V, which represent about 1/10 of the total degrees of freedom in protein structures, are responsible for most of the large-scale movements in proteins [80]. Consequently, conformational searches are often performed through variation of those dihedral angles, with fixed geometry (i.e., fixed bond angles and distances), and the resulting structures identified by a listing of the final dihedral angles. Variation of either a single ~ or V angle of a peptide conformation (with fixed geometry) produces a large global conformational change. However, conformafional search methods, which vary two or more angles simultaneously, allow for local movement without large changes in the global conformations [81 ]. Since the C0t(i) - C'(i) and N(i+ 1) - Ca(i+ 1) bonds are nearly parallel, a cooperative change of v(i) and O(i+ 1) may be a possible source of local movement without a corresponding large global distortion [82,83]. Using rigid geometry it is not immediately apparent that such a movement will not change the global conformation, since those two bonds are not collinear but displaced by approximately 1 A. If they were colinear, a change A~(i) together with a change At~(i+l) = - A~(i) would maintain the global conformation. This change results in a rotation of the peptide plane by the same increment

659 given to ~g(i), and therefore this cooperative variation of ~(i) and ~(i+ 1) will be denoted as peptide plane rotation (PPR).(Such a procedure is similar to the one involving the use of a virtual or pseudo dihedral angle [84 - 85].) This PPR procedure has been tested in a decapeptide with a starting regular conformation corresponding to a fight-hand o~-helix. Eight additional structures were generated in each case by changing the values of xg(5) and t~(6) by 40 ~ increments, with the restriction that Ate(6) = -Axg(5), maintaining fixed all the remaining dihedral angles t~, ~ at their values in the original conformation; that is, the structures are identified by the values -160~ ~ -120~176176 20~ ~ 0~ ~ 40~ ~ 80~ ~ (= 80~176 120~ ~ (= 120~ ~ of the xg(i)/t~(i+l) angles. Figure 1 illustrates the results obtained; the maximum rmsd is 1.08

Figure 1. Conformations of nine fight-hand t~- helices. The starting conformation is presented at the top of the left column. The other structures are those with the values -80%20 ~ -120~ ~ -160~ ~ and 160~ ~ (on the left column) and 120~ ~ 80~ ~ 40~ ~ and 0~ ~ (on the fight column) for the ~(5)/~(6) angles. See the text for details. and the maximum change in the C~t(1) - Ca(10) distance is 1.2 A. A similar test has been carded out for a left-hand o~-helix, Type I and II 13-turns, and an extended structure, with the similar result that PPR allows for local conformational changes without large global conformational changes.

660 Next, Pancreatic Trypsin Inhibitor (5PTI) was considered because of its characteristics (i.e., a well-defined X-ray structure, with a conformation restricted by three disulfide bonds). The peptidic bond between residues 27 and 28 was selected for the test because residue 28 forms part of a loop with irregular conformation and is not involved in H-bonding. As in the preceding cases, eight structures were again generated with changes of 40 for ~(27) and ~(28), with the restriction A~(28) = -A~(27). The procedure was carried out with InsightlI (Biosym) minimizing the structure using dihedral angles restraints and a tether force for residues 1 - 23 and 32 - 51. This procedure maintains the original X-ray structure for those two regions and allows for flexible geometry optimization of the 24 - 31 region. Figure 2

Figure 2. Superimposed conformations of nine structures of Pancreatic Trypsin Inhibitor (5PTI), including the original X-ray structure. The remaining eight structures have been generated as described in the text, with application of the PPR method. shows, superimposed, the final structures, as well as the original one. The values obtained for the bond angles and distances are consistent with the ranges observed in high-resolution X-ray determination of several PTI structures.

661 The above conformations describe local movements with little global changes. The point to be emphasized, however, is that a characterization of those conformations by a listing of their ~, V values for each residue would seem to suggest that very different conformations were being considered when, in fact, they are very similar when analyzed from the point of view of a PPR. 4. CONCLUSIONS The cautionary tone in part of this chapter has been adopted on purpose, in order to emphasize our opinion that the quality of computer simulations, using PEF with fixed coefficients, cannot be ascertained except by comparison with experimental information. However, once that comparison has been made, the simulation results may be extremely useful. That is why a close collaboration between experimental and theoretical researchers is strongly recommended. The problem regarding the characterization of a structure in terms of its dihedral angles is of a different nature. In fact, the interest of the evidence presented lies in the result that a situation may arise such that a structure is rejected, even though the global conformation is essentially correct, simply because PPR has not been taken into account. The PPR has also implications in conjunction with the thermodynamic hypothesis [86,87], the Levinthal paradox [88], the use of redundant conformations in conformational searches, and the possibility of an enhanced search procedure (projected angle method). These points lie outside the scope of the present work and will be discussed in detail in future work [89]. REFERENCES

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57.


Z.B. Maksi6 and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modern Modelling of the Chemical Bond

665


T h e nature o f V a n der W a a l s b o n d Grzegorz Chatasifiski a, Malgorzata. M. Szcz~niak b, and Stawomir M. Cybulski c aDepartment of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland bDepartment of Chemistry, Oakland University, Rochester, Michigan 48309, United States of America CDepartment of Chemistry and Biochemistry, Miami University, Oxford, Ohio 45056, United States of America

1. I N T R O D U C T I O N Linus Pauling has had a lasting impact on two areas of the theory of molecular interactions [1,2]. He was the first to recognize the importance of weak interactions, such as hydrogen bonding to biology, and he pioneered the theoretical investigations of these interactions. His second powerful idea was the role of molecular shape in such diverse areas as crystal packing and a complementarity of enzyme-substrate interactions. By his own admission Pauling never used a computer in his studies [3]. In a recent reminiscence he said: "I am sure that if I had been relying on a computer to make most of the calculations, some of these ideas, which have in fact turned out to be important, would not have occured to me." By contrast, today's applications of quantum chemistry to these systems are almost exclusively carried out on a computer, taking advantage of unprecedented advances in solving the Schr6dinger equation. The computational methods have become reliable enough so as to provide us with potential energy surfaces, which are suitable for simulations of the behavior of molecular clusters. The wealth of numerical data generated in this way enhances the need for simple and intuitive rationalizations, which were the hallmark of Pauling's work. In this contribution we intend to show that such a program can be most appropriately accomplished by dissecting the interaction between molecules into a few basic components. These components, electrostatic, induction, dispersion and exchange energies, are as important conceptually to the theory of weak interactions as the notions of ionic, covalent, and metallic bonding were to Pauling's approach to chemical bonding. We will demonstrate that any sensible modeling of intermolecular forces must rely on these four basic components. The primary concept of the theory of intermolecular interactions is the intermolecular potential energy surface (PES) [4]. The intermolecular force is defined as the negative gradient of this energy. In the quantum mechanical setting, the PES has its origin in the BornOppenheimer approximation as a potential energy for the motion of nuclei. In the case of molecular interactions, it is usually reasonable to treat the interacting species as rigid bodies in their equilibrium geometries. Such an approach provides the PES, which depends only on the intermolecular degrees of freedom. If the intermolecular forces are strong enough to cause a

666 sizable deformation of the monomers, the intramonomer degrees of freedom should also be included. The crudest approximation is to separate these two effects into the interaction of deformed monomers and the effect of monomer geometry relaxation. A rigorous approach should solve the equations for the nuclear motions on the PES, which parametrically depends on the inter- and (at least some) intramonomer degrees of freedom. At either level of complexity the ultimate task of theory is to interpret and predict the experimental measurements, such as the dissociation energies into free monomers, intra- and intermolecular vibrational frequencies in Van der Waals complexes, integral and differential cross sections for scattering experiments. In all such measurements the effect of the interaction is gauged with relation to relaxed monomers. 2. FUNDAMENTAL I N T E R A C T I O N ENERGY C O M P O N E N T S As the understanding of chemical bonding was advanced through such concepts as covalent and ionic bond, lone electron pairs etc., the theory of intermolecular forces also attempted to break down the interaction energy into a few simple and physically sensible concepts. To describe the nonrelativistic intermolecular interactions it is sufficient to express them in terms of the aforementioned four fundamental components: electrostatic, induction, dispersion and exchange energies. Typical closed-shell molecules feature an uneven distribution of charge density. This charge distribution may be described by permanent multipole moments, or in the simplest approximation, by point charges distributed over the molecules. When two molecules are far apart the moments interact via the Coulomb law, giving rise to a long-range part of the PES (the one, which decays as some inverse power of the intermolecular distance, R-n). This interaction is essentially exhibited in three different forms, primarily derived by London [5] (cf. also Refs. 6 and 7). First, it is the direct electrostatic interaction between multipole moments. Second, since the interaction perturbes the involved species, the multipoles are created and/or modified. These modifications give rise to what is usually referred to as the induction interaction energy. There is also a third type of interaction, called the dispersion interaction, that molds the long range shape of the potential. Whereas the first two interactions may be viewed as clearly related to classical electrodynamics, the dispersion energy emerges because of the quantum mechanical nature of the molecular world. Its semiclassical model, consisting of the interaction of "instantaneous multipoles" due to the fluctuating positions of electrons, may serve only as a simplified visualization. In fact, the dispersion energy is an electron correlation effect that takes place in the area between the interacting monomers. Bringing molecules closer to the Van der Waals minimum, roughly defined by the Van der Waals radii, generally leads to two additional effects. One is an alteration of all three interactions discussed earlier. Indeed, electrostatic, induction and dispersion effects can no longer be modeled by a multipole expansion. This is because the electron clouds begin to overlap giving rise to some exponential damping of the long-range interactions. This overlap brings about a secondary modification, which may easily be built into the classical models. A primary effect on the PES, however, is due to the appearance of a new, purely quantummechanical factor. It has been known under the name of the exchange repulsion or HeitlerLondon (HL) exchange energy. It is due to such quantum mechanical considerations as the delocalization of the electrons and their indistinguishability. It corresponds to "resonance"

667 integrals in Pauling's description of chemical bonds and is related to the Pauli exclusion principle. [8-11] Among closed shell molecules the fully occupied monomer orbitals repel rather than invite the electrons of the partner. This effect of blocking the electronic space has been related to such notions as atomic radii and molecular shapes - the concepts to the development of, which Pauling significantly contributed. It means that in the first approximation the atoms and molecules may be viewed as surrounded by rigid spheres or contours, which make impenetrable borders for partner monomers. In a better approximation, the shapes of monomers are soft, featuring an exponential rise and decay with R. The simplest picture of the intermolecular interaction in the complex is, thus, the following: the monomers stick together because they are glued by electrostatic forces in the form of direct electrostatic energy, the induction energy and the dispersion energy. They cannot collapse onto each other because of the exchange effect (the Pauli principle), which prevents the occupied orbitals from overlapping. This picture may be simulated in a variety of ways, from a very simple one (a direct electrostatic model restrained by hard spheres, which envelope molecules [12,13]), through combined ab initio, model and semiempirical approaches [14-16] to the most sophisticated based on the rigorous solutions of the Schr6dinger equation, in particular by means of the symmetry adapted perturbation theory (SAPT) [17-20]. SAPT, while preserving the backbone of the classical perturbation theory approach of London [5], supplements it with refinements and couplings of the electrostatic, induction, dispersion and exchange terms. This results in a beautiful theory, which blends mathematical rigor with the conceptual simplicity of common intuitions. Before proceeding any further, we will briefly outline the ab initio theory of intermolecular interactions.

3. AB INITIO A P P R O A C H TO I N T E R M O L E C U L A R F O R C E S All the information that is needed to describe a particular intermolecular interaction in mathematical and physical terms is included in the Schr6dinger equation for a system under consideration HW=EW

(1)

where H is the total Hamiltonian of the Van der Waals complex, and W and E are its wave function and energy, respectively. Unfortunately, except for a few systems, solving the Schr6dinger equation is a very difficult task demanding ingenious approximations and state-ofthe-art computational techniques. The first important approximation known as the BornOppenheimer approximation separates the motions of nuclei and electrons. Light electrons are assumed to instantly follow any infinitesimal change of the nuclear positions. This approximation brings about the crucial concept of the potential energy surface (PES) [4,15]. In general, one may visualize nuclei moving due to forces exerted on them by this potential field. In the case of intermolecular interactions, as long as monomers are kept rigid, we can view them as "moving" with respect to each other upon the intermolecular PES. In other words, the PES provides a playground for the interactions of molecules. Knowledge of a PES not only means the understanding of how two molecules interact, but also provides us with a means of simulating and predicting the properties of dimers, trimers and large aggregates of atoms and molecules.

668 The calculation of PESs is most easily accomplished by evaluating the interaction energy, Eint, which is defined as the difference between the energy of the dimer, EAB, and the energies of the monomers, E A and E B Eint = EAB - E A - E B

(2)

The procedure chosen to calculate Eint must ensure that electronic energies of the dimer and monomers are evaluated in a consistent manner [19,21-24]. It should be stressed that this requirement is absolutely crucial, as no method at present can in practice yield EAB, E A and E B energies with an absolute error smaller than Eint. Therefore, Eq. (2), which defines the interaction energy does not offer so simple a computational approach as might be expected at first glance. Two notorious inconsistencies to be alleviated in practice are: basis set inconsistency (same basis set expansion or numerical grid for A, B, and AB must be used, otherwise the basis set s u p e r p o s i t i o n error (BSSE) arises [21,23,24]) and the size inconsistency (a theory to describe AB must guarantee a correct dissociation into A and B, at the same level of theory [25]). Even if the algorithm is correct, it has the disadvantage of giving no direct insight into the nature of the interaction.Therefore, from the very beginning of quantum mechanics, scientists were inclined to solve the Schrrdinger equations through a perturbation procedure that would provide the interaction energy directly and would offer insights into its physical nature and functional form. Originally, the fundamental components related to the Coulomb interaction of permanent, induced, or instantaneous multipole moments were recovered with the aid of the Rayleigh-Schrrdinger (RS) perturbation theory [5-7]. The introduction of exchange effects occurred independently, and originated from the variational theory of the chemical bond formulated by Heitler and London. The development of a unified, rigorous treatment proved difficult because of the symmetry problem related to the indistinguishability of electrons. The classic RS formalism assigns electrons to either of two monomers, and this approximation ("polarization approximation") works as long as the monomers stay far apart [26]. In the intermediate and short-range, though, the antisymmetry of the total dimer wavefunction cannot be efficiently recovered by means of the RS perturbation theory, which becomes divergent. Attempts to deal with the increase-of-symmetry problem were eventually successful and led to a variety of SAPT formalisms [ 11,18,27]. Along with the rapid development of many-body techniques to cope with electron correlation, SAPT today provides us with a rigorous framework, as well as a detailed description of quantum theory of intermolecular forces. The details of SAPT are beyond the scope of the present work. For our purposes it is enough to say that the fundamental components of the interaction energy are ordinarily expanded in terms of two perturbations: the intermonomer interaction operator and the intramonomer electron correlation operator. Such a treatment provides us with fundamental components in the form of a double perturbation series, which should be judiciously limited to some low order, which produces a compromise between efficiency and accuracy. The most important corrections for two- and three-body terms in the interaction energy are described in Table 1. The SAPT corrections are directly related to the interaction energy evaluated by the supermolecular approach, Eq.(2), provided that many body perturbation theory (MBPT) is used [19,28]. Assignment of different perturbation and supermolecular energies is shown in Table 1. The power of this approach is its open-ended character. One can thoroughly analyse the role of individual corrections and evaluate them with carefully controlled effort and desired

669 Table 1 Decomposition of two- and three-body supermolecular (S-MPPT) interaction energies. The contents of S-MPPT terms is described and the leading SAFF terms are indicated in square brackets. S-MPPT

SAFF

Physical interpretation Two-body

AESCF

AE(2)

Electrostatic energy between SCF monomers Exchange repulsion between SCF monomers

AESCF def

Mutual polarization restrained by exchange [13ind,rJ

0 t3( disp

Dispersion energy arising between SCF monomers (2nd order)

(20).

Electrostatic-correlation energy (2nd order). Intra-monomer correlation correction to e~1~ (2) AEdet

1. D e f o r m a t i o n - i n t r a - c o r r e l a t i o n .

e(22)

['~ind,r]

2. Deformation-dispersion. re (30) 1j t disp_ind rt(2)

Ar-,exch

1

(20) 1 9 Exchange-dispersion r[Eexch_dispJ

2. Exchange-intra-correlation AE(3), AE(4), etc.

Higher-order correlation corrections as for AE(2) Three-body

AE SCF

eeI~xch F

1. SE component: single exchanges between monomers 2. TE component: all monomers are involved in the exchange - (20)

e(30)1

SCF-deformation nonandditivity 113ind,r, ind,r I - (20)

1. Exchange-dispersion nonadditivity [Eexch_disp] 2. Exchange-intra-correlation nonadditivity

AE(2)

e(22) 3. Deformation-intra-correlation nonadditivity [Wind,r]

4. Deformation-dispersionnonadditivity trEdisp_ind (30) 1j AE(3 )

..(30) e-disp

1. Dispersion nonadditivity, accounts for triple-multipole terms 2. Higher-order correlation corrections as for AE(2)

AE(4), etc.

Higher-order correlation corrections as for AE(3)

670 precision. The price is involvement of many arbitrary choices and approximations, which at some point may render the method cumbersome and subjective. A proliferation of corrections with an increasing order of the perturbation treatment and additional approximations to circumvent a divergent character of theory may pose problems. Another shortcoming is the limitation to intermediate and long-range interactions. Fortunately, all these drawbacks may be overcome by using the relationship to the supermolecular energies [19,28]. The supermolecular interaction energies defined in Eq.(2) have the advantage of being free from such problems and represent the most reliable interaction energy values for adjusting the final potential. Only a concerted use of the direct supermolecular approach and SAPT may lead to very accurate interaction energies. It also leads to a deeper understanding of the origin and behavior of the components to the interaction energy. Below, we will describe some very recent advances.

3.1

Exchange repulsion versus molecular shape

"The Dutch physicist J.D. van der Waals found that in order to explain some of the properties of gases it was necessary to assume that molecules have a well def'med size, so that two molecules undergo strong repulsion when, as they approach, they reach certain distance from one another. [...] It has been found that the effective sizes of molecules packed together in liquids and crystals can be described by assigning Van der Waals radii to each atom in the molecule. The Van der Waals radius defines the region that includes the major part of the electron distribution function for unshared [electron] pairs." Cf. Fig. 1.A [2]. Strong repulsion described by Pauling in his "General Chemistry" textbook [2] originates from the electron exchange effect that arises between the electron clouds of interacting monomers. The quantum mechanical approach allows us to link the concept of the effective size of molecules or a molecular shape to the exchange effect. We may visualise this shape in terms of the exchange repulsion, which is sensed by a rare gas (RG) atom moving around a molecule. Several researchers have attempted to define alternative pictorial representations [29,30]. In Figs. 1.B and 2 we show the representation proposed by our group [31,32]. In the displayed diagrams, a contour is drawn in polar coordinates that were used to define the motion of a probing RG atom around the molecule. However, the distance R is replaced by the value of the exchange-repulsion energy. In that manner, the contour exhibits local decreases and increases of repulsion, which indicate local concentrations and depletions of electron density in the diffuse region. Several examples of such plots for different molecules are in Figs. 1.B and 2. Let us examine the contour for the C12 molecule in Fig 1.B. How does this compare with the drawing in Pauling's book? Both pictures reveal depletions in the middle of the C1-CI bond. However, the Van der Waals radii approximation fails to predict a characteristic flattening of the electron distribution at the chlorine ends along the internuclear axis. One can see in this and all other drawings that the assumption of a spherical distribution around an atom within a molecule is not justified. The deviation from a spherical distribution may be small, but it is sufficient to influence the shape of the PES, and to determine the equilibrium structures of Van der Waals complexes with non-polar species for, which induction effects are negligible. The PES of RG-C12 exhibits two kinds of minima, one for a T-shaped and one for a coUinear from. The PES for C1F has three minima, and two of them (at the C1 side and the T-shaped one) are due to a reduction of the repulsive effect, cf. Fig.2. The global minimum corresponds to the collinear form RG-C1F. The PESs of RG-HF and RG-HC1 systems feature two collinear minima, and it is clear from Fig. 2, that the one at the

671

Figure 1. A) A chlorine molecule, illustrating the difference between Van der Waals radius and covalent radius (from Pauling's book [2]). B) The exchange repulsion contour of C12, obtained for the He-C12 complex, and defined by two polar coordinates, measured from the center of mass: (Energy, O). The contour is the image of the chlorine molecule shape, detected by a rare-gas atom [32]. C) Relief map of the negative Laplacian of the charge density, -V2p, of the chlorine molecule. One can notice three depletions of the electron charge density: in the region perpendicular to the bond, and along the interatomic axis, for the colinear approach at the both ends.

672

F

F

H

Ct

C

Cl

0

H

Figure 2. The exchange repulsion contours for several molecules, obtained for interactions with rare-gas atoms, and defined by two polar coordinates measured from the center of mass: (Energy, O) [31,32]. The contours are the images of molecules' shapes, probed by structureless atoms. In contrast to plots that show isoenergetic regions, these contours reveal an enhanced anisotropy. Convex and concave regions indicate, respectively, the areas of increased and reduced exchange repulsion.

673 halogen atom end must be due to a reduction in repulsion. The RG-CO complexes are skew T-shaped, again in agreement with the depletion visible in Fig 2. In general, any non-polar species (such as methane, methyl group, N 2, etc.) will be preferably hosted in niches within the exchange repulsion, which are displayed by these contours. Finally, it is important to add that the regions of reduced (or enhanced) repulsion may be directly related to the regions of depletion (or concentration) observed in the diffuse part of the Laplacian of electron density plots proposed by Bader [33]. You may see this in Fig.2, where the negative of V2p for C12 has been drawn. In such a way, scrutiny of-V2p indicates the locations where we may expect reduced repulsion, and which are thus the most favorable sites for a nucleophilic partner to attach. It is important to stress that the analysis of the Laplacian must carefully distinguish between the relative charge concentration in short range and the relative charge concentration in the diffuse region - as they sometimes do not coincide, and only the latter is relevant. In quantitative modeling of PESs the description of the molecular shape as a superposition of atomic components remains an attractive approach, but it is clear from the earlier discussion that it must be extended to accommodate two important factors. The atomic shape is not a rigid, but rather a soft, exponentially decaying electronic charge cloud. In addition, it should be anisotropic with the anisotropy depending not only on the atom itself, but also on its partner in the chemical bond. 3.2

Dispersion as the intermonomer correlation effect The traditional view of dispersion effect introduces the notion of instantaneous multipoles. Such multipoles arise, when for an infinitesimal portion of time, electrons are frozen in their positions, and thus become oriented with respect to positively charged nuclei. A momentary multipole at, say, monomer A, is created, which induces a momentary multipole at monomer B. The resulting electrostatic interaction defines the dispersion effect. In this way fluctuations of negative electron charges of A become correlated with the electron charges of B. It is clear that this effect may be related to the intermonomer electron correlation phenomenon. This simple picture, however, is reasonable as long as we assume that the electrons of A and B do not mingle with one another, or more precisely, do not penetrate each other's occupied space. Mathematically, it means that the condition ra+rb'...t, ~ ,i rT',Cl~ ,

,

- 180

70 BE h

,

,

,

- 120

'

'

B)

,

,

e x c , - , ~ ,

-60

I

'

'

0

I

'

'

,

'

~

50

'

,

,

,

120

I

'

'

I

180

'

'

exchSE

~/'~

+'~~..~ dee

30

-

,

Total

~

r

,

60

O, deg

1

A r2HF

,

exch

L'

n.~_ /

10

~- "+ * -~'~'.~

/7 f'" ~-tt

-10

--=-*-- +----,-- ---'U-. o ,

- 180

,

I

- 120

,

,

d isp

'.~'x,x~ .,.

_

t

.

" "*I

-60

,

exchTE ~

I

0

,

,r

-.-- *--' --- -,-----*--

-*" ,

I

60

,

+

1

120

,

,

1

I

180

O, deg Figure 10. The dependence of the three-body components upon the in-plane rotation of: A) HC1 in the Ar2HCI cluster, B) HF in the Ar2HF cluster. The following abreviations have been used: "exch" - eexch,rL."def" _ z-X~defA'~SCF;"disp" - "disp,~(30)""Total" - AEMP3; SE and TE denote single-exchange and triple exchange respectively.

693

by HC1 on two Ar atoms. The SE exchange nonadditivity was modeled by the interaction between the exchange-induced quadrupole moment on Ar 2 and the permanent moments of the HC1 molecule. Later, Ernesti and Hutson [80] introduced a number of modifications of the Ar2HF model by recognizing other mechanisms. For example, the interaction in the Ar dimer subsystem involves the dispersion effect, which induces a quadrupole moment of opposite sign to the exchange-quadrupole one. In addition, the induction effects in the trimer, due to the presence of HF rather than HC1, required a more accurate treatment. The nonadditive potential obtained in Ref. [80] was successfully employed to predict the change in the red shift of the HF stretch by Bali6 and collaborators [81].

Ar2C02 In this cluster the three-body effect was detected via the observation of the asymmetric stretching frequency of CO 2 by Sperhac et al. [82]. Recent ab initio calculations confirmed the experimental predictions [83]. The exciting aspect of this cluster is that the nonadditive effect on the stretching frequency may be obtained directly with a very good accuracy. The reason is the well defined structure of the Ar2CO 2 cluster, shown in Fig. 11

O Ar ~

,-.~

()

., ,...

" "

II

""t~F

Figure 11. The T-shaped configuration of the Ar2CO 2 cluster. As long as the two Ar atoms are held in equivalent equatorial positions, the interaction with each of them should, in the pairwise additive approximation, result in the same incremental shifts of the asymmetric stretch of CO 2. In reality, a minute nonadditivity of shifts amounting to 0.042 cm-1 was observed by Sperhac et al. when the second Ar atom was added. Rak et al. [83] used a one dimensional model to calculate the ab initio estimate of three basic nonadditive components in the v=0 and v=l levels of this stretch (see Table 3). The three-body dispersion interaction affects the v=0 and v=l levels more than the other two terms. The effect of induction nonadditivity is about an order of magnitude smaller. Even smaller is the exchange nonadditivity. Interestingly, its SE and TE constituents are large, but quite accurately cancel one another. These results are sensible. Ar2CO 2 in the T-shaped configuration is predominantly dispersion bound and the induction effect should play a secondary role only. The signs of TE and SE terms are also easy to predict by means of the models outlined at the beginning of this Section: exchange-quadrupole interaction and distortion due to orthogonalizing, respectively. A different picture emerges when we analyse the effects of the three-body terms upon the

694 Table 2 Comparison of nonadditive terms in Ar2HC1 and Ar2HF. All values are in laEh. Three-body term

Ar2HC1

HL

8exch

Ar2HF

4.8

HL 8exch,SE

24.9

27.2 a)

44 .7 b)

-23.5 a)

- 19.8 b)

(30) ind,r

I 1.5 16.5

30.8 36 1

AEscF

16.3

55.7

AE(2)

8.3

-10.0

~(3o) disp

30.6

21.5

HL 8exch,TE AESCF def

AE(3) 26.3 AEMP3 50.9 a)Approximate calculation from Ref. [79]. b)Accurate calculation from Ref. [72].

20.2 65.9

Table 3 The effect of the three-body contributions upon the frequency shift of the CO 2 antisymmetric stretch (in ~tEh).

Three-body term

<E>v=o

<E>v=l

<E>v=l-<E>v=O

8exc hIlL HL SE Eexch,

0.774 10.849

0.629 10.864

-0.145 0.015

HL E Eexch,T

10.077

- 10.234

-0.158

(30) ind,r

4.376

4.815

0.438

AF~SCF def AFt(2)

1.542 2.321

1.970 2.254

0.428 -0.067

8(30) disp

27.170

27.069

-0.101

Sum:

HE +1~(30) + ..(30) ind,r e'disp Sum: EeI'~xch+AEdef SCF +AE(2)+ ~disp ..(30) s

Experiment [82]

0.192 0.125 0.192

695 frequency shift, i.e. the difference of v=l and v=0. Although the dispersion term is the largest, it differs very little for the the v=0 and v=l states states, and so do the SE and TE terms. The three-body induction term strongly differentiates between the two states, and thus has the most effect upon the shift. The dramatic change in induction effect can be rationalized in terms of the appearance of the dipole moment when CO 2 is deformed along the asymmetric stretching coordinate.

5.2 Water trimer: induction nonadditivity In the water trimer induction nonadditivity provides a dominant contribution, which effectively overshadows all the other terms. Its mechanism is simple. For instance, in a cyclic water trimer the multipoles of A inductively alter the multipoles at B, which, in turn, inductively alter the multipoles at C, which then alter those on A, and so on, until the selfconsistency is reached. Various formulations of this simple model were implemented in the simulations since the 1970s [84-87,63,64,50]. To include the many-body induction effects of point charges interacting with a set of polarizable atomic centers the following classical electrostatics equation is solved iteratively E

1 Fo poI'--2E ~l'i i i

(20)

where [.I,i is the induced dipole on center i and F ~ 1 is the electric field at center i arising from all other fixed charges in the cluster. The induced dipole moment ILl,i and the total electric field at the polarizable center j are evaluated self-consistently from the following expressions:

~l,i=o~iF i

(21)

and

o Fi=Fi + E TijlLtj j~-i

(22)

where ~i is the polarizability of the center (e.g. atom) i and Tij is the dipole-dipole interaction tensor closely related to the interaction matrix desribed earlier, cf. Eq.(10). Eqs.(20-22) SCF neglect the exchange effects arising in tA. . x127 X - , d e f . Yet, they fairly well approximate the latter over a wide range of configurations [45]. This approach has been successfully incorporated into the nonadditive molecular dynamics simulations [88]. The neglect of other nonadditive effects exchange and dispersion - appears to be justified for low energy configurations. They become important, however, for repulsive geometries, e.g. at barriers separating superimposable structures [45]. An ab initio model of nonadditive effects in water, which also includes SE and TE components of the exchange repulsion component, the dispersion term and perhaps some other, secondary terms should be investigated in the near future. So far the effect of nonadditivity in water has been studied in the context of various structural properties, vibrationally averaged structures, O-H frequency shifts [89], zero-point energies, rotational constants, cluster predissociation dynamics and tunneling splittings [58].

696 The importance of three-body effects in the determination of macroscopic properties has also been studied. Recently, nonadditive molecular dynamics simulation of water and organic liquids has been performed [86].

6. SUMMARY The nature of Van der Waals binding may be described in terms of four basic types of interactions: electrostatic, induction, dispersion and exchange. These interactions are useful to classify and understand the physical origin of intermolecular potentials, and the probable structures of Van der Waals complexes. In this context, they play a similar role to the concepts of covalent and ionic binding in strong chemical interactions. The fundamental interaction energy constituents are included in the rigorous quantum theory of weak molecular clusters. They can be calculated with a desired accuracy, and represented with analytical forms that are related to a variety of simple and physically sensible models. Today, both the ab initio calculations and the potential modeling can be performed for small- and medium-sized molecules, and provide reliable intermolecular PESs for a wide range of mutual intermolecular orientations. Future calculations should also include the intramonomer degrees of freedom, and incorporate them into the final potential form. Very little has been done in this area so far but the tools are already available. All these advances will be accompanied by simulations of the dynamics and vibrational averaging, with the ultimate goal of bridging the chasm between our understanding of the electronic structure of atoms and molecules, and macroscopic character of matter.

8. A C K N O W L E D M E N T S We thank Dr. Piotr Cieplak for reading and commenting on the manuscript, and Professor Richard Bader and Dr. Todd Keith for providing us codes, which draw the Laplacian of the electron density. Gaussian 92 codes [90] were used for electronic structure calculations. Support by KBN through the Department of Chemistry, University of Warsaw, within the Project BST/532/23/97 and by the National Science Foundation (Grant no. CHE-9527099) are gratefully acknowledged. The Interdisciplinary Center of Modeling, University of Warsaw is acknowledged for the computational grant.

REFERENCES 1. L. Pauling, The Nature of The Chemical Bond and the Structure of Molecules and Crystals; An Introduction to Modern Structural Chemistry, 3rd ed., Cornell University Press, Ithaca, N.Y., 1960. 2. L. Pauling, General Chemistry, 3rd ed., W.H. Freeman, San Francisco, 1970. 3. L. Pauling in: The Chemical Bond. Structure and Dynamics, A. Zawail (ed.) Academic Press, San Diego, 1992. 4. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.

697

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Z.B. Maksi6 and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modem Modelling of the Chemical Bond

701


The nature of the chemical bond in metals, alloys, and intermetallic compounds according to Linus Pauling Zelek S. Herman* Herman Scientific Consulting, 521 Del Medio Avenue, #107, Mountain View, CA 94040 USA

ABSTRACT A review of the unsynchronized-resonating-covalent-bond theory of metals in presented. Key concepts, such as unsynchronous resonance, hypoelectronic elements, buffer elements, and hyperelectronic elements, are discussed in detail. Application of the theory is discussed for such things as the atomic volume of the constituents in alloys, the structure of boron, and superconductivity. These ideas represent Linus Pauling's understanding of the nature of the chemical bond in metals, alloys, and intermetallic compounds.

1

INTRODUCTIONt

A metal is a substance that possesses several of the following properties: It is a solid at room temperature; it is opaque to light, and when polished it is a good reflector of light, having a peculiar appearance, called metallic luster; it is a good or fairly good conductor of heat and electricity, being malleable (capable of being hammered into thin sheets) and ductile (capable of being drawn into wires). Gold, for example, is so malleable that it can be hammered into foil so thin that it is transparent to visible light. Among elemental metals, mercury is exceptional in being liquid at room temperature although the melting point of gallium is 29.8 ~ and it is a liquid from this temperature until its boiling point, 2905 ~ *Editor's note: For over 14 years, Dr. Herman was Professor Linus Pauling's collaborator at the Linus Pauling Institute of Science and Medicine in Palo Alto, California. Dr. Herman and Pauling's long-time assistant, Dorothy Bruce Munro, are the co-compilers of The Publications of Professor Linus Pauling, which is published on the Internet at http://charon.lpi.org/-zeke. tThis section is based upon the introduction to a book bearing the title of this chapter that was begun by Linus Pauling several years before his death but never completed.

702 Some elemental metals and many intermetallic compounds are brittle, not malleable or ductile. Borderline substances, showing metallic properties to a decreased extent, are called metalloids or semiconductors. Probably the best criterion for distinguishing a metal and a metalloid or semiconductor is the temperature coefficient of thermal and electrical conductivity. With increase in temperature, the thermal and electrical conductivity of a metal decreases, whereas that of a metalloid or semiconductor increases. Six elemental metals are mentioned in the Old Testament (e.g., Numbers 31:22): gold (Hebrew: zahav), silver (kesef, also the Biblical Hebrew word for money), copper (nechosheth, often translated into English as brass, which is an alloy of copper and zinc and may, in fact, not have been known in the time of Moses, or bronze, an alloy of copper and tin, which was known in the time of Moses), iron (barzel), tin (b'deel), and lead (ofereth). The ancient Greeks also recognized mercury. About seventy-five elemental metals are now known. Many metals become superconducting at low temperatures, that is, their electrical resistance is essentially zero. However, copper, silver, and gold, which are among the best electrical conductors, do not become superconducting at any known temperature. Superconductors are generally classified as being one of two types: Type I superconductors completely repel an external magnetic field at temperatures below the superconducting transition temperature (a phenomenon known as the complete Meissner effect) whereas type II superconductors do not completely repel an external magnetic field at temperatures below the superconducting transition temperature but instead go through a so-called vortex state. Two forms of elemental tin are known. One, gray tin, which has a diamond structure with tetrahedral bonding, is a metalloid. White tin, which has a body-centered tetragonal structure, is metallic. Experimental measurements indicate that the two forms of tin lie very close to each other in energy, with gray tin being more stable at lower temperature. The phase transition between the two forms occurs at 13 ~ at temperatures below 18 ~ white tin slowly changes to gray tin. At very low temperatures (< -40 ~ their conversion is sometimes so rapid that metallic tin objects fall into a powder of gray tin; this phenomenon has been called the "tin pest". Nevertheless, white tin becomes a type I superconductor, with the value of the superconducting transition temperature equal to 3.722 K. Another interesting property of white tin is that when rods of it are bent, it makes a distinctive sound, known as the "tin cry". This is due to the breaking of the microcrystals in the highly crystalline structure of white tin. An alloy is a metallic substance formed by melting together or otherwise mixing two or more elements, at least one of which is a metal. Many alloys are intermetallic compounds, with atoms of different elements in a well-defined ratio. Often the phase exists with a range of compositions. Such a phase may be called a solid solution or crystalline solution. Most metals are mutually soluble in the liquid state, showing a single liquid phase over the entire composition range, even when the melting points of the individual elemental metals are very different. There are some exceptions, however. Two liquid phases are observed for lead and zinc, lead and gallium, lead and iron, silver and nickel, silver and

703 chromium, and some other binary alloys. Presumably the factors that lead to the formation of two liquid phases are that the bonds between unlike atoms are not stronger than the average of bonds between the like atoms, and that the structures of the two liquids are sufficiently different as to make it difficult for the molten alloy to assume an intermediate structure. The phase diagrams for the binary-metal systems show great variety [1]. One extreme is illustrated by Ag-Au, for which there are found a complete series of solid solutions with the cubic close-packed structure, and solidus and liquidus curves extending smoothly between the melting points 961.5 ~ for Ag and 1064.4 ~ for Au. Another simple system is Mg-Sn, with three phases at room temperature: nearly pure Mg and nearly pure Sn, and a well-defined compound, Mg2Sn. This compound has the fluoride structure, and its composition might be considered the expected one for bivalent magnesium and quadrivalent tin. Another system in which well-defined compounds occur is Ag-Sr. Here there are four intermediate phases, AgsSr (781 ~ AgsSr3 (760 ~ AgSr (.-~ 680 ~ and Ag2Sr3 (665 ~ It is not possible to interpret these formulas in terms of the usual valences of the elements, in contrast to the situation for inorganic compounds of metals and nonmetals. Another example of the lack of correlation with the periodic table is provided by compounds of the alkali metals with cadmium. In the Li-Cd system there are the compounds LiCd (549 ~ LiCda (370 ~ and Li3Cd (272 ~ Two compounds, NaCd~ (384 ~ and Na2Cdll (364 ~ occur in the Na-Cd system, and in the other three systems only one compound, KCdI3, RbCdi3, and CsCdi3, is found [2]. Compounds formed by the alkali metals are rather similar.

2

QUANTUM MECHANICS OF METALS

AND THE NATURE

The quantum mechanical treatment of molecules can be essentially classified into two main types. The first, molecular orbital theory, as developed by Mulliken, Slater, Pauling, and many others, has enjoyed great success owing to its relative ease of computational implementation. Most people do not associate the name of Linus Pauling with molecular orbital theory. However, Linus Pauling, in fact, in a paper with the title "The Application of the Quantum Mechanics to the Structure of the Hydrogen Molecule and Hydrogen Molecule-ion and to Related Problems," published in the June 1928 issue of Chemical Reviews, introduced the notion that the Pauli exclusion principle can be satisfied by constructing a determinant of spin-orbit functions [3,4]. This determinant is now known as the "Slater determinant" and plays a central role in molecular orbital theory. The alternative to molecular orbital theory, valence-bond theory, as developed by Heitler, London, Slater, and Pauling, is not as easily amenable to computational implementation. Consequently, it has not been employed to any great extent in the detailed computational investigation of molecular systems. Nevertheless, scientists, when visualizing the chemical bond in

704 molecules, still think of the conceptually simple framework provided by the valence-bond method. Furthermore, in a note published in 1932, John C. Slater pointed out that, when each method is refined, the molecular-orbital treatment of molecules and the valence-bond treatment of molecules each converge to the same result [5]. An analagous situation exists for the quantum mechanical treatment of metals, alloys, and intermetallic compounds. Band theory, the commonly employed quantum mechanical theory of the electronic structure of metals, alloys, and intermetallic compounds, started with the discussion by Pauli of the temperature-independent paramagnetism of the alkali metals [6]. This theory was further developed by Sommerfeld, Houston, Eckert, Bloch, Frenkel, Slater, Mott, Jones, Wilson, Bethe, Seitz, Kittel, and many others to the point where band theory provides a good understanding of many of the properties of metals in terms of a calculational method that yields numerical results in good agreement with experiment [7-20]. In the band theory of metals, the outer electrons are treated as nearly free, so that they can move through the system under investigation in the way described by Bloch functions, which account for the periodicity of the system, and interact with the atomic ions arranged in closest packing or some other structure. The local-density-approximation [21] calculations of Moruzzi, Janak, and Williams [22] concerning the electronic properties of metals are characteristic of the considerable success achieved by band theory in its numerical application to the study of the physical properties of particular metals. Simply put, in band theory a substance will display metallic electrical conductivity if the valence band is not completely filled by the valence electrons (and not separated by a large gap from the unoccupied conduction band) since the application of an electric field to the substance will cause the excitation of electrons into states in the conduction band, so that they move through the system. Nevertheless, similar to the situation for the molecular orbital theory of compounds, a number of theoretical and practical problems remain concerning the application of band theory, for example, possible linear dependence of the basis set, lack of convergence, and the need to take explicit account of electron correlation. More importantly, band theory does not provide a conceptually simple way to visualize the nature of the chemical bond in metals. An alternative to the band theory is the unsynchronized-resonating-covalent-bondtheory of metals, alloys, and intermetallic compounds developed by Linus Pauling and some of his coworkers, initially from an empirical investigation of the saturation magnetic moment of the first-row transition metal alloys [23] and later derived theoretically on statistical grounds [24-38]. The basic premise of the unsynchronized-resonating-covaJentbond-theory of metals, alloys, and intermetallic compounds is that the electrons in such systems, as in other substances, are bound to atoms and occupy atomic orbitals. The outer electrons may interact in such a way as to form covalent bonds between the atoms. However, in a metal, alloy, or intermetallic compound, unlike the situation for other substances, the number of bond positions is larger than the number of bonds, leading to resonance of the bonds among the alternate positions and resulting in electrical conductivity under the influence of an applied electric field. Electronic correlation is built into the theory. One of the salient features of the unsynchronized-resonating-covalent-bond-theory of metals, alloys, and intermetallic compounds is that, on average, 0.72 of an orbital per

705 atom must not be occupied either by a bonding electron or an unshared pair of electrons in order for the unsynchronized resonance that confers metallic properties on a substance to occur. This 0.72 of an orbital per atom has been given the appellation the metallic orbital. Moreover, a substance will display metallic character if it has a metallic orbital available for electrons to move through the substance under the influence of an applied electric field. One of the lasting practical results of treating metals in this model has been the tabulation of atomic radii and interatomic distances in metals [39-42]. Another interesting application of the unsynchronized-resonating-covalent-bond-theory of metal is its use in the elucidation of the to the structure and properties of elemental boron and the boranes [43].

3

THE METALLIC

ORBITAL

In 1938, Linus Pauling concluded from an investigation of the physical properties of the metals that, in a sequence such as K, Ca, Sc, Ti, V, Cr, the number of bonding electrons increases monotonically from 1 to 6, remains constant at 6 from Cr to Ni (except for Mn, which has an anomalous structure), and then begins to decrease. He published a curve, now known as the Slater-Pauling curve, of the saturation ferromagnetic moment per atom for the alloys containing chromium through copper in the first-row transition metals [23]. This curve, shown in Figure 1, indicates that the saturation magnetic moment rises to a value of ~ 2.4 Bohr magnetons for Fe-Co alloys, with the maximum value occurring at approximate composition Fe72Co28. Then this value decreases to zero for the alloy with approximate composition Ni44Cu~6. By assuming that each of the six bonds in these alloys requires an orbital, he concluded that, for the transition metals, only 8.28 of the nine 3d, 4s, and 4p valence orbitals are occupied by bonding electrons, ferromagnetic electrons, or electron pairs, and consequently, that 0.72 of an orbital per atom, on average, is without any apparent use. Earlier, in 1928, Heisenberg, had discussed the spin-polarization of covalent bonds as the mechanism of interatomic interaction leading to ferromagnetism [44]. Ten years later, in 1948, Pauling realized that this apparently unused atomic orbital has an important function [24, 27, 30]. Consider a metal consisting of N identical atoms M with valence v, that is, each atom forms v covalent bonds with adjacent atoms. Now, if the number of bond positions were greater than the number of bonds, then the bonds could resonate from one position to another only synchronously, with pairs of bonds changing positions simultaneously:

706

S

S

J

S

s

S

J J

b J J

J

J

S %

Cr

Mn

Fe

Co

Ni

I Cu

COMPOSITION

Figure 1. The Slater-Pauling curve displaying saturation ferromagnetic moment for the first-row transition metal alloys. This figure shows a comparison of experimental values (solid curves) and predicted values (dashed lines) of the saturation ferromagnetic moment per atom, in Bohr magnetons, for Fe--Co, Co-Ni, and Ni-Cu alloys. The short vertical lines indicate change in crystal structure. When the Zener contribution is taken into account, the slope of the dashedline from FeT2Co2s to Ni44Cu56 changes from -1, as shown, to -1.11.

707

M

M

M

M

M

M

M

M

S YNCHRONO US RESONANCE

However, the principle of approximate electroneutrality [45, 46] allows for the occurrence of M + and M-, with valences v - 1 and v + 1, respectively. Therefore, under the condition that there is an available orbital, unsynchronous resonance, involving the shift of a single covalent bond about an atom from one position to another, can then occur:

M

M

M

M-

M

M

M+

M

UNS YNCHR ONO US RESONANCE

In order for unsynchronous resonance to occur, the atoms M + and M ~ must have an unoccupied orbital available so that they can accept an additional bond. M- does not require such an unoccupied orbital because the electroneutrality principle rules out its accepting an additional bond, which would convert it to M 2-. Accordingly, the structural requirement for a system to possess metallic character is that the fraction of the atoms M + and M ~ have available an unoccupied orbital, called the metallic orbital. The average value of 0.72 orbital per atom for the metallic orbital, as deduced from the Slater-Pauling curve, implies that, with unsynchronous resonance of the covalent bonds, the metal consists of 28% M +, 44% M ~ and 28% M-. As will be seen in the statistical theory described in the following section, there exist far more unsynchronized resonating structures per atom than there are synchronized resonating structures. Associated with this increase in the number of resonating structures is an increase in stability for the system, with the increased resonance stabilization energy being approximately proportional to the number of additional resonating structures per atom for unsynchronous resonance, less 1. One is consequently led to conclude that unsynchronized resonance of the covalent bonds between the atoms in metallic systems occurs

708 because of the increased stability resulting from the large number of resonance structures associated with such resonance. Furthermore, the electrical conductivity of metals is a direct consequence of unsynchronous resonance in that the bonds resonate with electronic frequencies as the positive and negative charges pivot from atom to adjacent atom under an applied electric field, as illustrated schematically for the case of lithium metal in Figure 2. The frequency of such pivoting motion is determined by the resonance energy, which is comparable in magnitude to the bond energy and is only about one order of magnitude less than the binding energy of a valence electron to the atom. Linus Pauling first discussed this explanation of metallic conduction in 1948 [25, 27]. Other characteristic properties of metals, such as high thermal conductivity, high ductility and malleability, and negative temperature coeffficients of electrical conductivity can similarly be rationalized in terms of the unsynchronized resonance of covalent bonds. Thus, for example, the negative temperature coefficient of the electrical conductivity is a result of thermal agitation temporarily lengthening some bonds and shortening others. This process interferes somewhat with the resonance of the bonds, which does not occur as frequently between non-equivalent positions as between equivalent, or equienergetic, positions. This explanation for the negative temperature coefficient of metallic electrical conductivity agrees with the usual one, involving scattering of the electrons by phonons. Similarly, unsynchronous resonance accounts for the high ductility and malleability of metals because of the many more pivoting positions accompanying unsynchronous resonance than those accompanying synchronous resonance. Another important feature of the unsynchronized-resonating-covalent-bond theory of metals is that it accommodates the Zener theory of the interatomic interaction that results in ferromagnetism [47]. From Figure 1 it is seen that the saturation magnetic moment of the Fe-Co-Ni-Cu alloys displays nearly linear behavior, over part of its course, as a function of the number of outer electrons in the alloy, although the slope of the negative line is approximately 11% greater in magnitude than that expected from the electron number. It is here that the Zener theory comes into play, for Zener proposed that, under the influence of the atomic magnetic moments of the atoms, a pair of electrons involved in the covalent bond may be decoupled to produce two electrons with spins oriented either parallel or antiparallel to the ferromagnetic moment of the crystal and engaged in the formation of one-electron bonds between atoms. The Zener theory of uncoupling of electron spins by the atomic magnetic moments is similar to the uncoupling of electron spins by an external magnetic field that was proposed by Pauli to account for the temperature-independent paramagnetism of the alkali metals. If it is accepted that the saturation magnetic moment is increased in value by 11% over the atomic value by this uncoupling, then the observed value for iron, 2.22 Bohr magnetons, can be decomposed into a value of 2.00 Bohr magnetons for the atomic moment and 0.22 Bohr magnetons for the one-electron bonds, the other 5.78 of the total valence of 6 being electron-pair bonds. Figure 1 further shows that the value of the saturation magnetic moment becomes 0 at the composition Ni44Cu56, corresponding to 10.56 outer electrons

709

-IANODE

Li

Li

Li

Li

Li -

I

I

[

I

I

Li

Li

Li

Li

Li.

Li +

Li

Li

Li

Li

Li

Li

I

I

I

I

Li

Li

Li

Li-

Li

Li +

Li

Li

Li -

Li

Li

I

I

I

Li

Li

Li

Li

Li

Li

Li

Li

..... Li

Li

' Li

Li

Li

I

I

Li

Li -

Li -

Li

Li

Li

Li

Li

Li

Li

Li

Li

, Li

CATHODE

Li +

.... Li

Li +

"-Li

I Li

Li +

Figure 2. Diagram illustrating motion in lithium metal of a negative charge (an electron) from the cathode to the anode by successive pivoting resonances of a covalent bond.

710 per atom. Of these, 6 are involved in covalent-bond formation, using 6 valence orbitals. This leaves 4.56 electrons per atom as electron pairs, occupying 2.28 orbitals. Thus, 8.28 orbitals of the 9 available are occupied, leaving 0.72 metallic orbital per atom to engage in unsynchronous resonance. Other ways of estimating the number of metallic orbitals per atom, such as the comparison of the observed interatomic distance in white tin and gray tin, or the oxidation numbers of the atoms in the superconducting copper-oxide materials [38], yield less accurate values of about 0.7 metallic orbital per atom.

4

THE DETAILED ANALYSIS OF THE STATIST I C A L T H E O R Y OF U N S Y N C H R O N I Z E D RESONANCE OF COVALENT BONDS

In a footnote to his 1949 paper entitled "A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds," Linus Pauling gave an example of a simple statistical treatment to derive the metallic orbital [27]. Nevertheless, it took him three and one-half decades to publish the detailed statistical treatment [34-36], which is given in the following. Let us consider a crystal composed of N identical atoms, each with covalence v and ligancy L. The number of ways of distributing the v N / 2 positions is given by:

W =

(LN/2)!

[(L - v) N/2]! [vN/2]!

'

(1)

if we exclude multiple occupancy. Using Stirling's approximation

t! = (2rt) 1/2 ( t / e ) t

(2)

,

then the number of ways w = W 1/N in which bonds are arranged around each atom is found to be: LL/2 w-

v,,/2 ( n -

v)CL-,,)/2

"

(3)

Let us further consider the number of resonance structures, u, in which n bonds are arranged about an atom with average valence v in the crystal. This number of resonance structures per atom is proportional to the probability given by the binomial distribution,

711

with proportionality constant w, or

u(L,v,n) = w

v '~ ( L - v) (L-'~) L! L i nl ( L - n)!

(4)

"

For the case of synchronous resonance, n = v, and the number of resonance structures per atom, us~.~, becomes v "/2 ( L - v) (L-v)/2 L] vsy.~h = LL/2 v! ( L - v)!

(5)

"

It is interesting to note that, for L = 4 and v = 2, eqn. (5) gives R ~n(3/2) for the residual entropy of ice; this value differs by only 1% from that given by calculations not involving the approximations made here [48]. We must now consider two classes of metals, hypoelectronic metals, and hyperelectronic metals [29]. A hypoelectronic metal is one composed of atoms in which the number of outer electrons is less than the number of outer orbitals, and a hyperelectronic metal is one composed of atoms in which the number of outer electrons is greater than the number of outer orbitals. For a metal composed of hypoelectronic atoms, the number of bonds n can take the values v - 1, v, and v + 1, corresponding to M +, M ~ and M-, respectively, and eqn. (4) yields the following expression for the number of unsynchronized resonance structures per atom:

Uhy~=

v ~'12 ( L - V ) (L-v)/2 L! LL/2vI(L-v)I

[

L-v

v

[L-v+l

+l+v+i]

]

"

(6)

A comparison of eqns. (5) and (6) reveals that the term in square brackets in eqn. (6) is the ratio of the number of unsynchronized resonance structures per atom to the number of synchronized resonance structures per atom for a hypoelectronic atom. Given the reasonable assumption that the energy corresponding to an unsynchronized resonance structure is the same order of magnitude as that for a synchronized resonance structure, the energy of a crystal composed of hypoelectronic atoms is lowered considerably via unsynchronized resonance. Therefore, one predicts that every element with an extra orbital to serve as the metallic orbital should be a metal. With a single possible exception, namely boron, which will be discussed in a succeeding section, this prediction is borne out. In Table 1 the number of unsynchronized resonance structures per atom for hypoelectronic metals with various values of the ligancy L and valence v are given. These are also shown in Figure 3, from which it is seen that a maximum in the number of unsynchronized resonance structures per atom for hypoelectronic metals occurs at v = L/2. As will be

712

Table 1. Number of unsynchronized resonance structures per atom as a function of valence v and ligancy L for hypoelectronic metals. L\ v 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2.00 2.50 2.92 3.29 3.62 3.93 4.21 4.47 4.72 4.96 5.19 5.40 5.61 5.81 6.01

80

2

3

4

5

6

7

8

2.50 3.50 4.49 5.48 6.47 7.45 8.44 9.42 10.40 11.39 12.37 13.35 14.33 15.32

2.92 4.49 6.25 8.18 10.26 12.49 14.85 17.35 19.97 22.71 25.56 28.52 31.58

3.29 5.48 8.18 11.37 15.08 19.28 24.00 29.21 34.93 41.16 47.89 55.13

3.62 6.47 10.26 15.08 21.00 28.10 36.44 46.10 57.12 69.57 83.51

3.93 7.45 12.49 19.28 28.10 39.19 52.81 69.21 88.66 111.41

4.21 8.44 14.85 24.00 36.44 52.81 73.73 99.91 132.03

4.47 9.42 17.35 29.21 46.10 69.21 99.91 139.65

NUMBER' OF' -

j ' t r.. 0 U I~11i,4, I ~11lt,,,~ IT...,

STRUCTURES PER ATOM

60

I

!

!

i

t L=14 .

40

i

....

I~

20

.

.

.

I

"r L=12 '

.~

z L=11 _

!

L=.10

0 0

2

4

6 8 VALENCE

10

Figure 3. The number of resonance structures per atom for hypoelectronic metals ([:]) and for hyperelectronic metals (A) as a function of unit increase in valence v and ligancy L. Note that the maximum for each L occurs at v - L / 2 .

713

discussed later, this fact is of paramount importance in explaining the structure of such metals. For the case of hyperelectronic metals, that is, a substance composed of elements for which the number of outer electrons is greater than the number of outer orbitals, not including the metallic orbital, the statistical treatment is somewhat more complicated [34]. Let us first consider the valence v of such a metal. The neutral atoms M ~ form z bonds, and the ions M + and M- form z + 1 bonds. Denote the fractions of M +, M ~ and M - by y, x, and y, respectively. Then from eqn. (4) the ratio of the number of neutral atoms to the number of ions, x / y , is given by

x = u(L,v,z) = (L-v)(z+l) 2y u (n, v, z + 1) v (L - z)

(7) '

subject to the constraints

x + 2y-

1

(8)

and

v - x z + 2y (z + 1)

.

(9)

Eqns. (7)-(9) have one solution in the range z < v < z + 1:

v-

z + [ f ( f + 1)]1/2

_

_

f

(10)

,

where

f-[z(L-z)][L-2z-1]

.

(11)

Cu, Ag, and Au have z - 5 and L - 12, so that eqns. (10) and (11) yield the value v - 5.4965 for these metals. In the cases of Zn, Cd, and Hg, each atom has six nearest neighbors and six neighbors more distant, so that z - 4 and 6 < L _< 12, resulting in 4.5585 >_ v >_ 4.4888. For white tin, with z - 2 and four nearest neighbors and six somewhat more distant neighbors (4 _< L _< 6), eqns. (10) and (11) yield the range 2.5359 >_ v >_ 2.4949 for the value of the valence. For all of the hyperelectronic metals, v = z + 1/2 when the ligancy L - 2z + 1, corresponding to half bonds. In all of these cases, v is very nearly equal in value to z + 1/2, so that it is reasonable to assume half-integral values of

714 Table 2. Number of unsynchronized resonance structures per atom as a function of valence v and ligancy L for hyperelectronic metals. L\ v 1 2 2.40 3 2.88 4 3.29 5 3.66 6 4.00 7 4.32 8 4.61 9 4.88 10 5.14 11 5.39 12 5.62 13 5.85 14 6.07 15 6.28 16 6.48

2

3

4

5

6

7

8

2.88 3.84 4.83 5.83 6.82 7.82 8.81 9.81 10.81 11.80 12.80 13.79 14.79 15.79

3.29 4.83 6.59 8.51 10.60 12.83 15.20 17.70 20.32 23.06 25.92 28.88 31.94

3.66 5.83 8.51 11.71 15.40 19.59 24.28 29.46 35.15 41.33 48.01 55.18

4.00 6.82 10.60 15.40 21.29 28.33 36.59 46.14 57.04 69.33 83.08

4.32 7.82 12.83 19.59 28.33 39.30 52.74 68.91 88.06 110.43

4.61 8.81 15.20 24.28 36.59 52.74 73.35 99.08 130.62

4.88 9.81 17.70 29.46 46.14 68.91 99.08 138.08

the valence, as was done on an empirical basis in 1949 [27]. With this assumption that v = z + 1/2, the statistical treatment is the same as for hypoelectronic metals, except that the factor 21/2 must be introduced to account for the fact that there now exist two kinds of atoms (M + and M-), forming z + 1 bands. These differ in that M + does not have an unshared electron pair whereas M- does have one. Under these conditions, the equation for the number of unsynchronized resonance structures per atom for hyperelectronic metals is: 21/2 L! v (*-0/2

( L - V)(L-v+l)/2

VhV~,. = LL/2 ( V - 1/2)! ( L - v + 1/2)!

v(L-v+l/2) 1 + (v + 1 / 2 ) ( L -

] v)

(12) "

Similar to the situation for hypoelectronic metals, the ratio of the number of unsynchronized resonance structures per atom to the number of the synchronized resonance structures per atom is given by the expression in square brackets in eqn. (12). Values of Vhum. are given in Table 2 and shown graphically in Figure 3. For small v and L, the values of Uh~r,,,. are slightly less than those of vhupo; the opposite is true for large v and L. Again, Figure 3 shows that, for each value of L, a maximum in the unsynchronized resonance structures per atom for hyperelectronic metals occurs at v - L/2, as in the case for hypoelectronic metals. The preceding discussion applies to situations in which all the bonds have the same length. For a crystal in which an atom forms two kinds of bonds, that is, L1 bonds with bond number n~ - - V l / L 1 and L2 bonds with bond number n 2 - - v2/L2, eqn. (6) takes the

715 form

l']hypo - C

v+lE

max(n,L1)E

n=v--1

i=min(O,n-L2)

9

?3li (L -

~-~ (L2 - v2) L2-n+i v2

i! (L1 - i)!

(n -- i)! (L2 - n + i)!

(13) '

where

C-

L , L2[ L L~ v~ ~ ( L 1 -

v,) L'-v~ L L~ v~ 2 ( L 2 -

v2) g~-*~ ],/2

(14)

.

For a crystal containing hyperelectronic atoms with two different bond lengths, the number of unsynchronized resonance structures per atom becomes

- c

~~

,,+1/2

,~x(,-,,L~)

,=,,-,/2

i=,m,~(o,,-,-n=)

Z

Z:

i (L1 - i) L~-i

~'

i! ( L 1 -

i)!

v'~ -~ (L2 - v2) L~-'~+i (n-

i)! (L2 - n + i)!

'

(151

where C-

2 '/2 L, L2[ L L' v~'x ( L 1 - Vl) Lx-vl L L2 v~ ~ (L2 - v2) L~-~'2 ]1/2

.

(16)

When V l / L 1 - v 2 / L 2 - v / L , eqn. (13) reduces to eqn. (6) and eqn. (15) reduces to eqn.

(12).

5

CALCULATION OF THE NUMBER OF METALLIC ORBITALS PER ATOM FROM THE STATISTICAL THEORY OF THE UNSYNCHRONIZED RESONANCE OF COVALENT BONDS

From the Slater-Pauling curve for the saturation magnetic moment of the first-row transition metal alloys (Figure 1), it was found empirically that the number of metallic orbitals per atom has the value 0.72, corresponding to 28% M +, 44% M~ and 28~ M-. Based on the statistical treatment discussed in the preceding section, it is now possible to deduce this value on purely theoretical grounds [36].

716

First, the amount of metallic orbital per atom in a metal is given by the ratio of M + and M ~ to M +, M ~ and M- since M + and M ~ require an extra orbital for unsynchronized resonance to occur, whereas M- does not have this possibility according to the principle of electrical neutrality [45]. Moreover, the numbers of M + and M- must be equal and their sum equal to half of M + + 2 M ~ + M-. For a hypoelectronic metal with valency v and ligancy L, the theoretical value of w, the amount of metallic orbital per atom, is

w-

c+ 1 2c+ 1

(17) '

where

c--~

l ( v L-v) v+ 1 + L-v+

(18)

1

"

Therefore,

w-

4Lv - 4v 2 + 3L + 2 6 L V - 6v 2 + 4L + 2

.

(19)

The calculated values of this quantity are 0.684 _< w _< 0.707 for L - 12, 1 _< v _< 6, and all other values of L and v corresponding to known hypoelectronic metals. For hyperelectronic metals, the theoretical number of metallic orbitals per a t o m is given by

-

x+l

(20)

2

where x, the amount of M ~ is found from the simultaneous solution of the following three equations:

x-

d-

[ d ( d - 1)] ~/2

d-

(z+l)(L-z-l) L-2z-1

,

(21)

(22)

717

and

v-

z + 1- x

,

(23)

where z is the number of bonds formed by M ~ z + 1 is the number of bonds formed by M + and by M - , and M +, M ~ and M - occur in the amounts y, x, and y, respectively. Now let us consider the borderline composition Ni~Cul_~ at which the saturation ferromagnetic moment of the alloys of the first-row transition elements is zero. Eqn. (19) yields a value of co = 0.6842 for Ni, with L = 12 and v = 6. For the hyperelectronic metal Cu, with L = 12 and v = 5, eqn. (21)-(23) give x = 0.5035 and v = 5.4965, so that eqn. (20) yields co = 0.7518, somewhat larger than the value of co = 0.6842 for the hypoelectronic metal Ni. To obtain the value of co from the composition Ni~Cu~_~ at which the saturation ferromagnetic moment must be zero requires two steps. First, we consider the weighted mean of the values of co for Ni and Cu:

co - 0.6842c~ + 0.7518(1 - c~)

.

(24)

Second, we obtain another equation for co as a function of a by summing the number of available bonding orbitals. Of the nine outer orbitals (3d54s4pa), six are used, with six electrons, to form bonds. The alloy Ni~Cul_~ has 11 - a outer electrons, of which six are bonding electrons and 5 - c~ occupy 2.5 -c~/2 orbitals as unshared pairs. Therefore, 8 . 5 a / 2 orbitals are used for occupancy by unshared electron pairs, leaving 0.5 + c~/2 of the nine available orbitals to serve as the metallic orbital:

-

0.5

+

.

(25)

The simultaneous solution of eqns. (24) and (25) for co yields a = 0.444 and co = 0.722. The former value is in excellent agreement with the observed composition Ni44Cu56 at the foot of the Slater-Pauling curve, and the latter value is essentially the same as the empirically deduced value of 0.72 for the average number of metallic orbitals per atom. It should be emphasized that the derivation of co given here is completely theoretical, and as far as can be determined, the conventional band theory has never been employed to predict the composition Ni44Cu56 at which the saturation ferromagnetic moment of the first-row transition metal alloys has the value zero.

718

6

THE CRYSTAL STRUCTURES OF THE METALS AND THE MAXIMUM VALUES OF THE METALLIC VALENCE

Many physical properties of metals display an approximate correlation with the number of valence electrons of the periodic sequences of the elements. Lothar Meyer was the first to publish, in 1870, a graph of atomic volume as a function of atomic weight [49]. Rather than atomic volume as a function of atomic weight, it is interesting to consider the reciprocal of the atomic volume as a function of atomic number. With appropriate units, the reciprocal of the atomic volume is, in fact, the atomic density, or molarity, of the element in its pure state. In Figure 4 the molarity of the principal allotropes of the elements is shown plotted as a function of the atomic number z at STP [50]. According to the unsynchronized-resonating-covalent-bond theory of metals, the number of unsynchronized resonance structures per atom increases with increasing ligancy for each value of the valence, as shown in Figure 3. The increase in the number of resonance structures implies an increase in stability, so that a metal will have the structure with the maximum possible ligancy if other factors are not important. For over half of the elements, including most of the hypoelectronic metals, the crystal structure at STP is cubic or hexagonal closest packing, for which each atom has L = 12. Indeed, ligancy 12 is the maximum value allowed for atoms of the same size. For 15 metals, the cubic body-centered structure is the stable one. This A2 structure may also be considered to conform to the above prediction since, in this structure, each atom has eight nearest neighbors and six others further away, so that the effective ligancy is close to 12. In Figure 4 the dip in the curve for the left side of the first-row transition metals occurs for Mn, whose principal aJlotrope has an anomalous structure (complex body-centered cubic packing), which may be associated with Mn having a half-filled d shell. Since the maximum number of unsynchronized resonance structures occurs at valence L / 2 , or bond number n = v / L = 1/2, the maximum value of L for an elementary metal is 12, implying a maximum value of the metallic valence equal to 6. This value, or one only slightly larger, is what is found for the transition metals. In Figure 5 the portion of the second long row of the periodic table has been redrawn using different units for the ordinate. For the portion of this curve with increasing slope, the metallic valence increases from a value of 1 for Rb to a maximum of 6 for Mo to Pd.

v =

For the hyperelectronic metals, there exist unshared electron pairs, as well as valence electrons, in the outer shell. Since these unshared electron pairs on adjacent atoms strongly repel one another, the maximum ligancy should according be reduced. This is observed in metallic tin, gallium, and some other hyperelectronic metals. From Slater's treatment of molecules involving covalent bonds [51], it is found that the exchange energy providing most of the stability of a molecule occurs with the opposite sign and the factor 2 for the interaction energy of electron pairs on adjacent atoms [52]. This destabilizing effect reduces the maximum value of the ligancy in the hyperelectronic metals. Precise calculation of

719

Molarity

A

ClO

9 ea8

v

0

g ..z

o

N~

9 % 9

.

g

r~

9 go

J

9

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