Advances in Heterocycling Chemistry. Vol. 32

Advances in Physical Organic Chemistry

ADVISORY BOARD W. J. Albery, FRS University of Oxford A. L. J. Beckwith The Australian National University, Canberra R. Breslow Columbia University, New York L. Eberson Chemical Centre, Lund H. Iwamura Institute for Fundamental Research in Organic Chemistry, Fukuoka G. A. Olah University of Southern California, Los Angeles Z. Rappoport The Hebrew University of Jerusalem P. von R. Schleyer Universität Erlangen-Nürnberg G. B. Schuster University of Illinois at Urbana-Champaign

Advances in Physical Organic Chemistry Volume 32

Edited by D. B E T H E L L Department of Chemistry University of Liverpool P.O. Box 147 Liverpool L69 3BX

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This book is printed on acid-free paper. Copyright 䉷 1999 by ACADEMIC PRESS All Rights Reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-1998 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 0065-3160/98 $30.00

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Contents

Preface

vii

Contributors to Volume 32

ix

Perspectives in Modern Voltammetry: Basic Concepts and Mechanistic Analysis

1

JOHN C. EKLUND, ALAN M. BOND, JOHN A. ALDEN A N D RICHARD G. COMPTON 1 2 3 4 5 6 7 8 9

Introduction 2 General concepts of voltammetry 6 Cyclic voltammetry 27 Hydrodynamic voltammetry 44 Microelectrodes 63 Sonovoltammetry 69 Theoretical modelling 83 A comparison of voltammetric techniques 96 Current and future direction of voltammetry 104 Appendix 110 References 113

Organic Materials for Second-Order Non-Linear Optics J. JEN S WOL FF

AND

121

¨ DIGER WORT MANN RU

1 Introduction 122 2 Basics of non-linear optics 124 3 Quantum-chemical basis for second-order polarizabilities 136 4 Non-linear optical susceptibilities and experimental methods to evaluate ␹(2) and ␤ 153 5 Optimization of second-order polarizabilities: applications to real molecules 168 6 Conclusion 206 Acknowledgements 208 References 208 Tautomerism in the Solid State TA DA SHI S UGAWARA 1 Introduction

AND

219 I S AO TA K A S U

219 v

vi

CONTENTS

2 Proton tautomerism in an isolated system 222 3 Proton tautomerism in the solid state 229 4 Photochromism and thermochromism derived from proton tautomerism 244 5 Photochemical hole-burning 250 6 Dielectric properties derived from proton tautomerism in crystals 7 Concluding remarks 261 Acknowledgements 261 References 262 The Yukawa-Tsuno Relationship in Carbocationic Systems YUHO TSUNO

AND

252

267

MIZUE FUJIO

1 Introduction 267 2 Applications of the Yukawa-Tsuno equation 272 3 Yukawa-Tsuno correlations for benzylic sololyses generating carbocations 276 4 Carbocation formation equilibria 315 5 Yukawa-Tsuno correlations for electrophilic addition of 322 6 Structure–reactivity relationship in polyarylcarbocation systems 7 Stabilities of carbocations in the gas phase 343 8 Theoretically optimized structures of carbocations 362 9 Reaction mechanisms and transition-state shifts 365 10 Concluding remarks 378 Acknowledgements 379 References 379

334

Author Index

387

Cumulative Index of Authors

405

Cumulative Index of Titles

407

Editor’s preface

Thirty six years after the appearance of the first volume of Advances in Physical Organic Chemistry, the subject reaches the last year of the century in good shape. New methodologies and exciting discoveries continue to appear both in purely academic research areas, but also increasingly in applied areas such as the development of novel materials with useful chemical and physical properties. The broad definition of physical organic chemistry used in the series has allowed the inclusion in this volume of reviews on topics that would scarcely have been regarded as appropriate when Victor Gold first saw the need for the series. Thus, the importance of organic electrochemistry as a branch of physical organic chemistry, first recognised in this series in Volumes 5 (1967) and 12 (1976) and discussed in several subsequent reviews, is revisited in a fairly fundamental way with an account of developments in the underlying technique of voltammetry and their implications for studies of chemical kinetics. The relationship between organic structures and nonlinear optical properties is dealt with in a thorough and truly physical organic way in the second contribution, which emphasises that both the physical and organic components of the research must be done to the highest standards if results are to be meaningful. Recent advances in the study of the role of hydrogen bonding on structure in the organic solid state are reviewed in the third contribution. In the final section of this volume, Professor Yuho Tsuno, after a distinguished career in physical organic chemistry spanning more than forty years, has contributed to a review of the part played in the study of the relationship of structure and reactivity by the four-parameter equation that bears his name. I want to express my thanks to all the contributors for the time, effort and enthusiasm that has gone into their reviews and my belief that the scientific community will find this volume just as useful as they have found previous ones in the series. With the coming of the millennium, Advances in Physical Organic Chemistry will have a new Editor. After being associated with the editorial side of the series since 1975, first as Associate Editor, then as Co-Editor and finally, after the sad, premature death of Victor Gold in 1985, as Editor, it seemed to me that it was an appropriate time to pass on the baton to a younger pair of hands. Academic Press have been fortunate to secure the services of Professor Thomas Tidwell of the University of Toronto to lead the series into the next century. I leave the series confident that he will command the support of the whole physical organic chemical community and further extend its appeal. My thanks go to the members of my Editorial Advisory Board, the ever-changing editorial staff at Academic Press, the many contributors who, vii

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PREFACE

over the years, have coped patiently with my comments, queries and grammatical eccentricities, but most of all to the readers who have loyally supported and used the series from its inception and by their citations made it the success that it is today.

Contributors to Volume 32

John A. Alden Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK Alan M. Bond Department of Chemistry, Monash University, Clayton, Victoria 3168, Australia Richard G. Compton Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK John C. Eklund Department of Chemistry, Monash University, Clayton, Victoria 3168, Australia Mizue Fujio IFROC, Hakozaki, Higashi-ku, Fukuoka 812-81, Japan Tadashi Sugawara Department of Basic Science, College of Arts and Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan Isao Takasu Department of Basic Science, College of Arts and Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan Yuho Tsuno

IFROC, Hakozaki, Higashi-ku, Fukuoka 812-81, Japan

J. Jens Wolff Ruprecht-Karl Universita¨t, Heidelberg, Im Neuenheimerfeld 270, D-69120 Heidelberg, Germany Ru¨diger Wortmann Physikalisch Chemie, Universita¨t Kaiserslautern, Erwin-Schro¨dinger-Strasse, D-67663 Kaiserslautern, Germany

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Perspectives in Modern Voltammetry: Basic Concepts and Mechanistic Analysis JOHN C. EKLUND AND ALAN M. BOND Department of Chemistry, Monash University, Clayton, Australia AND

JOHN A. ALDEN AND RICHARD G. COMPTON Physical and Theoretical Chemistry Laboratory, Oxford University, Oxford, UK

1

Introduction 2 Historical aspects 2 Scope of the review 3 Basic definitions 4 The two major classes of voltammetric technique 4 Evaluation of reaction mechanisms 6 2 General concepts of voltammetry 6 Electrodes: roles and experimental considerations 8 The overall electrochemical cell: experimental considerations 12 Presentation of voltammetric data 14 Faradaic and non-Faradaic currents 15 Electrode processes 17 Electron transfer 22 Homogeneous chemical kinetics 22 Electrochemical and chemical reversibility 25 3 Cyclic voltammetry 27 A basic description 27 Simple electron-transfer processes 29 Mechanistic examples 35 Examples of complex mechanisms commonly encountered in organic electrochemistry 38 Examples of electrode reaction mechanisms consisting of extensive combinations of E and C steps 42 4 Hydrodynamic voltammetry 44 Rotating-disc electrodes 46 Channel electrodes 48 Wall jet electrodes 52 Electron-transfer processes 53 Combinations of electron transfer and homogeneous chemical steps 55 5 Microelectrodes 63 General concepts 63 1 ADVANCES IN PHYSICAL ORGANIC CHEMISTRY VOLUME 32 0065-3160/98 $30.00

Copyright 䉷 1999 Academic Press All rights of reproduction in any form reserved

J. C. EKLUND ET AL.

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Mass transport to microelectrodes 64 Microelectrodes and homogeneous kinetics 66 Microelectrodes and heterogeneous kinetics 68 Convective microelectrodes 69 Sonovoltammetry 69 The effect of ultrasound on electrochemical processes 69 Sonovoltammetric experiments: practical considerations 70 Mass transport effects: a simple description 71 Sonotrodes 77 Sonovoltammetry: mass transport effects – further aspects 80 Electrode cleaning and activation 81 Electrode kinetics 82 Coupled homogeneous chemical reactions 82 Theoretical modelling 83 The four components of an electrochemical model 83 Analytical solutions 85 Numerical solutions 86 Finite difference simulations 88 Two-dimensional simulations 93 Chemical kinetics 94 Boundary conditions 94 Current integration 95 Optimization 95 A comparison of voltammetric techniques 96 A quantitative comparison of the kinetic discrimination of common electrode geometries at steady state 97 Steady-state vs. transient experiments 102 Current and future directions of voltammetry 104 Instrumentation 104 Electrodes 105 Voltammetric simulations 108 Investigations in alternative chemical environments 109 Appendix 110 References 113

Introduction

HISTORICAL ASPECTS

The transfer of electrons between species in the solution phase and solid electrodes was known to result in interesting reaction pathways as long ago as the early 19th century, the pioneer in this field being undoubtedly Michael Faraday (James, 1989). However voltammetric techniques have only become popular since the 1940s when instrumentation required to conduct such experiments became readily available. The early studies invariably used a linear sweep DC technique. In the last 50 years, a wide range of techniques of interest in this review have emerged. Thus, Sevcik (1948) reported the first cyclic voltammetric studies, while the fifties and sixties saw the common use of hydrodynamic techniques such as rotating-disc electrode voltammetry (see for

MODERN VOLTAMMETRY

3

example, Hogge and Kraichman, 1954). The seventies and early eighties then witnessed the widespread use of microelectrodes (see for example, Wightman, 1981), initially under near steady-state conditions, and subsequently, when advances in instrumentation occurred, extremely fast-scan voltammetric techniques were introduced (see for example, Howell and Wightman, 1984). Recent studies have reported combinations of voltammetry, microelectrodes and convective transport (see for example, Compton et al., 1993a) and voltammetry in the presence of ultrasound (see for example, Compton et al., 1997a). Thus at the end of this century a wide range of voltammetric techniques are available which utilize DC type waveforms.

SCOPE OF THE REVIEW

This review describes a range of voltammetric techniques based on the use of inherently simple DC waveforms (linear, cyclic or staircase). The description of the techniques begins at a level which presupposes only limited prior knowledge and is suitable for those unfamiliar with the application of voltammetric techniques for the elucidation of electrode reaction mechanisms. As a result, the first two sections of this review aim to provide a sound fundamental basis to build upon when more advanced concepts are described in later sections, thus serving to initiate the novice and refresh the memory of the more experienced electrochemist. The article focuses on typical solutionphase reaction mechanisms encountered in organic and organometallic electrochemistry. For details of important bioelectrochemical processes the reader is advised to consult such texts as Dryhurst and Niki (1988). For details on surface-related processes such as adsorption and corrosion as well as solid state voltammetry and AC, square-wave and pulsed techniques, the interested reader is directed to more general texts (such as Bard and Faulkner, 1980; The Southampton Electrochemistry Group, 1990; Bockris and Kahn, 1993; Brett and Oliveira-Brett, 1993; Bruce, 1995; Kissinger and Heineman, 1996). In addition, there are a number of review articles available in this series on topics related to those described here (Fleischmann and Pletcher, 1973; Eberson, 1976; Parker, 1984; Savéant, 1990). In the remainder of this introductory section, the basic definitions of terms associated with voltammetry are presented and the scope of applications of the technique are summarized. Section 2 provides a basic general introduction to the practical and theoretical concepts of voltammetric experiments of interest, while Sections 3–6 describe the voltammetric techniques: cyclic, hydrodynamic, microelectrode and sono-voltammetry. A range of examples of mechanistic studies is given for each technique. There is obviously a wealth of such examples in the recent literature and it is impossible to be fully comprehensive. Therefore, we focus primarily upon studies from our own laboratories, but a sample of excellent illustrative examples from the recent literature is also provided. Section 7

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summarizes the general principles associated with the simulation of voltammetric responses, while Section 8 compares the relative merits of the voltammetric methods considered. In addition there is an appendix of the symbols and abbreviations used at the end of this review.

BASIC DEFINITIONS

Voltammetric techniques considered in this review involve monitoring the current when a time-dependent potential is applied to an electrochemical cell. The measured current results from frequently complex combinations of heterogeneous (in which an electron is transferred at the solution– electrode interface) and homogeneous processes (which occur in the solution phase). Comprehensive information concerning a particular electrode reaction mechanism of interest can be obtained from examining how the current varies as a function of time and the applied electrode potential. For kinetic studies, it is the variation of this current–potential response as a function of time that is commonly crucial for the qualitative and quantitative determination of a reaction mechanism. The processes that are probed by voltammetry occur at or in the region of working electrode(s). Thus these electrode(s) are the electrode(s) of critical interest. The reference electrode merely provides a fixed reference potential and the counter electrode completes the electrical circuit (see Section 2). A host of processes and species may be probed by voltammetric techniques. Some examples are given in Table 1 of which the first four are considered in this review.

THE TWO MAJOR CLASSES OF VOLTAMMETRIC TECHNIQUE

As noted above, the time element, which is critical for kinetic studies, may be introduced in two ways. Steady-state voltammetry In this form of voltammetry, the concentration distributions of each species in the electrode reaction mechanism are temporally invariant at each applied potential. This condition applies to a good approximation despite various processes still occurring such as mass transport (e.g. diffusion), heterogeneous electron transfer and homogeneous chemical processes. Theoretically it takes an infinite time to reach the steady state. Thus, in a practical sense steady-state voltammetric experiments are conducted under conditions that approach sufficiently close to the true steady state that the experimental uncertainty of the steady-state value of the parameter being probed (e.g. electrode current) is greater than that associated with not fully reaching the steady state. The

MODERN VOLTAMMETRY

5

Table 1 A range of processes that may be probed by voltammetric techniques. Process type

Example (reference)

Redox chemistry

Reduction of pyridine and benzene-substituted n-alkyl esters (Webster et al., 1996). For further examples refer to texts such as Pombeiro and McCleverty (1993) Reduction of ␣-substituted acetophenones (Andrieux et al., 1997) Photochemical halide expulsion from halogenated aromatic radical anions (Compton et al., 1995a) The oxidation of Mo(C5H5)2Cl2 (Compton et al., 1995b) Electron-transfer processes in metalloproteins (Bond, 1994) Trace metal ions (Wang, 1985)

Homogeneous reactions coupled to electron transfer Photoelectrochemical Sonoelectrochemical Bioelectrochemical Quantitative determination of trace species in solution Transfer of species between immiscible phases Adsorption/desorption/ deposition Electroplating Dissolution

Ru(bpy)2⫹ 3 transfer between water and 1,2-dichloroethane (Ding et al., 1996) Absorption/desorption of hydrogen on platinum electrodes (Will and Knorr, 1960). Adsorption of palladium onto platinum (Attard et al., 1994) Copper electroplating (Pletcher, 1982) Dissolution of calcite in the presence of sulphuric acid (Booth et al., 1997)

time-scale of a near steady-state process is determined by the rate at which material reaches the electrode surface. This time-scale may be varied in a number of ways: • Altering the convective rate of transport, e.g. by changing the rotation frequency of a rotating-disc electrode. Experiments in which the convective rate of transport can be altered are known as hydrodynamic techniques. • Decreasing the size of the electrode so that the rate of radial diffusion of material to the electrode surface is enhanced as is the case for microelectrodes. • Applying ultrasound to the electrochemical system of interest. This forms the basis of the sonovoltammetric approach. Transient voltammetry In these experiments, a potential perturbation to the working electrode is applied to the system of interest and the resulting current response is measured as a function of time. Transient techniques include cyclic, linear

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sweep, square-wave, pulsed, AC, etc. voltammetries (see for example Bard and Faulkner, 1980). In the former two cases the potential at the working electrode is scanned in a linear (or staircase) fashion and the current is continuously monitored. The temporal aspect arises from the rate at which the potential is ramped, known as the scan-rate, ␯. When the potential is swept in only one direction the technique is known as linear sweep or staircase voltammetry. If the potential is swept in one direction, and then reversed this technique is known as cyclic voltammetry. Cyclic voltammetry is the prime transient technique discussed in this review.

EVALUATION OF REACTION MECHANISMS

The kinetics of voltammetrically relevant heterogeneous and homogeneous reactions may be examined by varying the critical time parameter of the experiment and monitoring its effect on some voltammetric feature (e.g. half-wave potential, see Section 4 for definition) associated with the process being investigated. The general procedure for obtaining quantitative kinetic data related to an electrode reaction mechanism using a voltammetric technique is schematically shown in Fig. 1. The basic concept is that the experimental voltammetric data are collected and a mechanism for the electrode reaction mechanism is postulated. The proposed mechanism may be theoretically simulated by solving the appropriate mathematical problem. Satisfactory agreement between experiment and theory is used to suggest a quantitative description for a particular mechanism, but as with most kinetic studies ideally the identity of proposed reaction intermediates must be confirmed by an independent technique, e.g. a spectroscopic technique. It is inherently dangerous to assume the structure of a reaction product or intermediate solely on the basis of a voltammetric response.

2

General concepts of voltammetry

As is the case with all experimental methods, the application of voltammetry requires a knowledge of the relevant techniques and protocols that are most appropriate for the particular system being investigated. In order to assist the novice user of the technique to choose an optimal procedure for their particular circumstances, a brief survey of all the relevant facets of the technique that need to be considered when undertaking a voltammetric experiment and interpreting the resulting voltammogram are presented below with reference to stationary solutions and electrodes in the absence of any form of forced convection. Experiments involving additional forms of mass transport will be discussed in Sections 4, 5 and 6.

MODERN VOLTAMMETRY

No

7

Yes

Fig. 1 Schematic diagram representing the process of examining an electrode reaction mechanism using voltammetric techniques.

J. C. EKLUND ET AL.

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ELECTRODES: ROLES AND EXPERIMENTAL CONSIDERATIONS

As alluded to in the introduction a typical voltammetric experiment utilizes three types of electrode. Working electrode (Fig. 2) This is the electrode at which the reaction of interest takes place, e.g. the simple one-electron oxidation–reduction processes given in equations (1) and (2). ⫹ ⫺ A→ ← A ⫹ e (oxidation)

(1)

⫺ B ⫹ e⫺ → ← B (reduction)

(2)

Typically these electrodes are fabricated from an inert and electrically conducting material. Common examples would range from liquid mercury to solid platinum and some forms of carbon (i.e. glassy carbon or graphite). Mercury electrodes (Bond, 1980) are used in the form of dropping electrodes in which the surface is continuously renewed or a hanging mercury drop electrode. Recently diamond film electrodes have been utilized for studies that require wide potential windows (Tenne et al., 1993). Typically, the solid

Fig. 2 Schematic diagram of a typical disc working electrode.

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electrode materials are sealed into a non-conducting support such as glass or Teflon (see for example Koppenol et al., 1994) to form a disc electrode which may have a radius in the millimetre down to micrometre range. The quality of the voltammogram obtained depends on a number of factors related to the nature of the working electrode • The integrity of the surface as measured by its cleanliness, smoothness and reproducibility of preparation. Failure to ensure these ideal conditions may result in high, non-reproducible background currents. Usually electrodes are polished using a systematic method to achieve a high-quality surface using alumina or diamond paste as the polishing material (Cardwell et al., 1996). • The integrity of the seal between the electrode material and the inert, non-conducting supporting mantle and the electrode and mantle surfaces being flush (see Fig. 2). A poor seal or a recessed/protruding electrode results in noisy, non-reproducible voltammograms. • The electrode is present in a vibration free environment and natural convection is minimized by, for example, thermostatting the reaction medium or working in a constant temperature room (⫾1⬚C).

Reference electrodes This electrode provides a fixed reference couple against which the potential of the working electrode is measured, e.g. the aqueous silver/silver chloride reference electrode utilizes the couple ⫺ AgCl(s) ⫹ e⫺ → ← Ag(s) ⫹ Cl (aq)

(3)

to meet these requirements. In order to satisfy the necessary criteria, a reversible redox couple is utilized in the reference electrode half-cell reaction. The potential of a reversible reference electrode is thermodynamically defined by its standard electrode potential, E0 (see for example Compton and Sanders, 1996, for further discussion). Currently, the most commonly used reference electrode in voltammetric studies is the silver/silver chloride electrode (3), which has overtaken the calomel electrode (see for example Bott, 1995) for which the reaction is (4). ⫺ 1/2 Hg2Cl2(s) ⫹ e⫺ → ← Hg(l) ⫹ Cl (aq)

(4)

For both these reference electrode types, the potentials are accurately known relative to the standard hydrogen electrode (SHE) for which the chemical reaction is (5). The SHE is based upon a high surface area platinum black-coated electrode in contact with hydrogen gas (one atmosphere

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pressure) and protons (unit activity) (for definition, see Atkins, 1994 or Compton and Sanders, 1996) and is defined to have a potential of 0.000 V: 1 H⫹(aq) ⫹ e⫺ → ← 2– H2(g)

(5)

The SHE is no longer routinely used due to safety concerns. Other forms of reference electrode exist: • Hg/Hg2SO4 electrodes are used for experiments conducted in the absence of chloride ions. • Ag/Ag⫹ electrodes are commonly used for experiments involving conditions when AgCl is soluble, e.g. when acetonitrile is used as a solvent. • Ag or Pt wire quasi-reference electrodes are used for experiments conducted in the absence of deliberately added electrolyte or in exotic solvents where no established reference electrode couple exists. In order to minimize the risk of contamination from the salt present in the reference electrode (e.g. KCl or NaCl), the reference electrode may be separated from the electrolytic solution and working electrode by a salt bridge. Care must be taken to avoid precipitation of insoluble salts at salt bridge interfaces, e.g. K⫹ and ClO⫺ 4 . Physical separation of the salt-bridge solution from that in the reference electrode and that in the electrochemical cell is achieved by a low porosity device such as a glass sinter or a membrane (see Fig. 3). A salt bridge also helps to minimize problems associated with liquid junction potentials (Eljp). Such potentials in combination with the ohmic potential term resulting from the presence of uncompensated resistance in the electrochemical cell (Ru) may alter the potential applied between the working and reference electrode (Eapp), so that the measured potential (Ecell) is given by (6), Ecell ⫽ Eapp ⫹ IRu ⫹ Eljp

(6)

where I is the electrode current. The liquid junction potential term arises from differences in the composition of the working and reference solutions. As a consequence, ions may diffuse across this solution-solution interface resulting in a potential gradient and the additional Eljp term shown in (6). The salt bridge can be used to minimize this term by containing a high concentration of a salt whose constituent ions have similar ionic mobilities. Thus the liquid junction potentials at the reference/salt bridge and salt-bridge/working solution interfaces will have similar magnitudes but opposite polarities and will therefore cancel each other out. The liquid junction potential term is complicated if an aqueous reference is used in conjunction with an organic working solution, since a knowledge of the free energies of transfer of the ions between the two solvents is required if corrections are to be made to

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Fig. 3 Schematic diagram of a typical Ag/AgCl reference electrode and salt bridge.

compensate for this term. Typically, liquid junction potentials have magnitudes of tens of mV (Bard and Faulkner, 1980). With respect to the IRu potential loss, the Ru term primarily results from the resistance of the solution between the working electrode surface and the reference electrode. Thus, to minimize the IRu term high cell currents, large working electrode/reference electrode separations and low conductivity electroactive solutions must be avoided. For microamp cell currents, the IRu term usually is in the millivolt range for typical organic solvent electrolyte combinations e.g. CH3CN/0.1 M (C4H9)4NPF6. Highly resistive salt bridges and very low porosity reference electrode frits also should be avoided. In voltammetric studies, it is now common to measure the reversible potential, Ef0, for the oxidation of ferrocene (Fc) to the ferricinium cation (Fc⫹) at the working electrode, versus the reference electrode actually used, and subsequently to correct potentials to the Fc/Fc⫹ scale (Gritzner and Kuta, 1984; Connelly and Geiger, 1996). This approach also helps to overcome problems with junction potentials.

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Counter/auxiliary electrodes Typically, an auxiliary electrode consists of a large surface area piece of platinum (wire or gauze) or carbon (disc or rod) placed directly into the test solution. Since current flows through the counter electrode, it must have a sufficiently large surface area relative to the working electrode to prevent limitation of the current flowing in the total circuit. The current measured in a voltammetric experiment flows between the working and counter electrode. In order to prevent contamination with products formed at the counter electrode or reaction of these products with the electroactive solution of interest, a salt bridge may be used in conjunction with the counter electrode, e.g. this is essential in bulk electrolysis experiments.

THE OVERALL ELECTROCHEMICAL CELL: EXPERIMENTAL CONSIDERATIONS

Figure 4 shows a typical electrochemical cell, which contains the following features: • The three electrodes are all in close proximity with the working electrode being centrally placed. It is crucial that the tip of the reference electrode is in close proximity to the surface of the working electrode in order to minimize contributions of IRu drop to the applied potential. This may be efficiently achieved using a Luggin capillary (Fig. 4). • The cell contains an inlet and outlet for an inert gas which displaces electroactive oxygen from the electroactive solution. Typical gases include high-purity nitrogen and argon. • The solution volume is typically in the range of 5–20 ml. • The concentration of the electroactive species of interest is typically in the range 0.1 to 5 mM. The upper concentration limit is chosen to minimize IRu, the contribution of the migration current due to the electroactive species and electrode adsorption effects associated with the presence of high concentrations of electroactive species. At concentrations below 0.1 mM, background current terms may start to become significant relative to the Faradaic current. • A high concentration of electrolyte (⬎0.1 M) is deliberately added to the solution to minimize the solution resistance, reduce transport of ions associated with the electroactive species due to migration and to establish a well-defined double layer. Thus the electrolyte has to be of high purity, dissociate substantially in the solvent of interest, consist of anions and cations that are hard to oxidize and reduce (in order to provide as wide a potential window as possible to examine the reactions of interest) and introduce no undesirable reactivity problems. • If organic solvents are utilized, they may need to be thoroughly dried since

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Solution (ca. 20 ml)

Fig. 4 Schematic diagram of a general electrochemical cell.

water may alter the electrode reaction mechanism or restrict the potential window over which electrode reaction mechanisms can be examined. • In the absence of constant temperature conditions within the laboratory, the electrochemical cell may need to have a water jacket if temperaturesensitive measurements are required. The electrochemical cell is coupled to a three-electrode potentiostatted form of instrumentation. If a two-electrode (working and reference) system were to be used, the current would have to flow through the reference electrode, thus risking instability in the reference potential. Furthermore, in a two-electrode system, the IRu drop could be substantial. In contrast, in the three-electrode potentiostatted system, the current is forced to flow through the counter electrode, thereby avoiding problems with the reference electrode. Additionally, much of the IRu drop is compensated by the potentiostat circuitry (Macdonald, 1977), which drives the potential between the working and counter electrode to a value which compensates the majority of this IRu potential loss. However, the use of a potentiostat does not remove all of the IRu drop, since uncompensated resistance remains due to solution resistance between the tip of the reference and working electrodes, and

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from resistance inherently present in the working electrode and electronic circuitry. In order to verify that the entire system is working correctly, frequent measurements of a well-defined reversible voltammetric redox couple are strongly recommended. Suggested redox couples include the oxidation of ferrocene, Fc (7) in most organic solvents (i.e. acetone, acetonitrile and dichloromethane) containing 0.1 M of a typical organic electrolyte (e.g. (C4H9)4NClO4), ⫹ ⫺ Fc → ← Fc ⫹ e

(7)

or the reduction of potassium ferricyanide (8) in water/0.1 M KCl. 4⫺ ⫺→ Fe(CN)3⫺ 6 ⫹ e ← Fe(CN)6

(8)

Data obtained using these reference couples may be compared with analytical solutions for the peak current, limiting current or voltammetric waveshape (see Sections 3 and 4). Non-compliance of experiment and theory is indicative of malfunctioning instrumentation, poor experimental design (i.e. an unacceptably large uncompensated solution resistance) or a faulty electrode.

PRESENTATION OF VOLTAMMETRIC DATA

Voltammograms are graphical representations of the current (I) dependence of the electrode reaction of interest as a function of the potential difference (E) applied between the working and reference electrodes. In this review, two main voltammetric shapes will be encountered. Figure 5(a) shows the asymmetric peak-shaped response obtained under conditions of cyclic voltammetry at macroelectrodes. Voltammograms obtained under these conditions are characterized by oxidation–reduction peak potentials and peak currents. The relevant theory will be discussed in more detail in Section 3. Figure 5(b) shows the sigmoidal shaped response encountered in steady-state hydrodynamic and microelectrode voltammetry. Voltammograms of this kind are characterized by the limiting current and half-wave potential (see Sections 4 to 6). Figure 5 represents an ideal reversible one-electron transfer process in the absence of IRu drop or capacitative charging current, although in real experiments contributions to the response from both these terms are unavoidable. Figure 6 shows the effect of uncompensated resistance for both transient and steady-state voltammograms, whilst Fig. 7 shows the influence of double layer capacitance on a cyclic voltammetric wave. Note that for steady-state voltammetric techniques only very low capacitative charging

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Fig. 5 Voltammetric shapes commonly encountered: (a) asymmetric peak-shaped response (e.g. cyclic voltammetry) and (b) sigmoidal-shaped response (e.g. steady-state hydrodynamic voltammetry).

currents are expected. Both IRu drop and double-layer capacitance introduce distortions which must be taken into account when comparisons with theory are made. Many theoretical treatments only encompass the Faradaic current and neglect IRu drop and capacitance. Cyclic voltammograms can be presented in an alternative format to that shown in Fig. 5 by using a time rather than potential axis, as shown in Fig. 8. The equivalent parameters in steady-state voltammetric techniques are related to a hydrodynamic parameter (e.g. flow-rate, rotation speed, ultrasonic power) or a geometric parameter (e.g. electrode radius in microdisc voltammetry).

FARADAIC AND NON-FARADAIC CURRENTS

It is implied in the above discussion that the monitored current is made up of two components. (a) The Faradaic current. This component is associated with the transfer of electrons resulting from oxidation–reduction of the electroactive species of interest in solution. (b) The capacitative charging current. As a result of the layer of oppositely charged supporting electrolyte ions being adjacent to the electrode surface (see for example the Southampton Electrochemistry Group, 1990), there is in effect a capacitative arrangement in an electrochemical cell, which charges and discharges whenever the electrode potential is

J. C. EKLUND ET AL.

16

Fig. 6 Effect of uncompensated resistance on (a) a cyclic voltammetric response and (b) a steady-state hydrodynamic voltammogram.

Fig. 7 Effect of double-layer capacitance on a cyclic voltammetric response.

changed. As a result, a capacitative charging current is generated which is directly proportional to the scan-rate as shown in (9). I/A ⫽ 1/A dq/dt ⫽ C⬘ dV/dt ⫽ C⬘ ␯

(9)

C⬘ is the capacitance of the double layer per unit area (usually in the tens of ␮F cm⫺2 range); the other symbols are defined in the appendix. C⬘ is assumed to be independent of electrode potential in the above expression; this is a broad simplification. It can be seen from (9) that voltammetric experiments will be limited in the scan rate which can be used since for high scan rates the information associated with the Faradaic current may be swamped by the presence of an unacceptably large capacitative charging current. This problem may be

MODERN VOLTAMMETRY

17

Fig. 8 (a) Usual current–potential voltammetric presentation format for a cyclic voltammogram and (b) alternative current–time presentation.

minimized by using microelectrodes (Section 5) which have small areas (and hence lower overall capacitances) or by employing steady-state conditions (as described in Sections 4 and 6).

ELECTRODE PROCESSES

A general electrode reaction consists of a number of steps: • Mass transport of material to and from the electrode surface. • Heterogeneous electron transfer. • Homogeneous chemical reactions coupled with electron transfer.

Mass transport In the voltammetric experiment, the amount of electroactive material reaching the working electrode and the ability to alter the rate at which material reaches the electrode are crucial for the determination of potential electrochemical mechanisms. There are three major pathways or modes of mass transport by which electroactive material in solution may reach an electrode (see schematic Fig. 9). Thus the total current (It) consists of a contribution from three sources as given by (10), It ⫽ Id ⫹ Im ⫹ Ic

(10)

where Id is the current associated with diffusional processes, Im is the current associated with migration processes and Ic is the current associated with convective processes.

J. C. EKLUND ET AL.

18

Fig. 9 Modes of mass transport of material to the electrode surface.

Diffusion. This involves movement of species in solution due to a concentration gradient and is governed by Fick’s two laws. For onedimensional diffusion (Fick, 1855a,b) the laws are expressed by (11), Law 1: Jd ⫽ ⫺ D

⭸[A] ; ⭸x

Law 2:

⭸2[A] ⭸[A] ⫽D ⭸t ⭸x2

(11)

or more generally for diffusion in more than one dimension by (12), Jd ⫽ ⫺ D(ⵜ[A])

(12)

where the flux, J, is related to the electrode current, I, by the expression: I ⫽ nAFJ

(13)

As can be seen, the magnitude of the observed peak or limiting current is dependent upon the value of D, which is a function of the solvent medium, the molecular weight of the electroactive species, the temperature, the electrolyte and the molecular dimensions. Table 2 shows typical values of D under a range of conditions. At macroelectrodes, semi-infinite linear or one-dimensional diffusion is appropriate. For microelectrode geometries, the nature of diffusion is more complex, as significant diffusion occurs in more than one dimension (Section 5). Migration. This results from motion of charged species due to an electrical potential gradient. Thus, charged electroactive species and the electrolyte ions

Table 2 Typical values of diffusion coefficients. Electroactive species Ferrocene Ferrocene (C6H13)4N(S2Mo18O62) p-BrC6H4NO2 Cr(CO)6 Ferricyanide Proton

Solvent/electrolyte system Dichloroethane/0.1 M (C4H9)4NClO4 Acetonitrile/0.1 M (C4H9)4NClO4 Acetonitrile/0.1 M (C4H9)4NClO4 Acetonitrile/0.1 M (C4H9)4NClO4 Acetonitrile/0.1 M (C4H9)4NClO4 1.0 M KCl (aq.) 1.0 M KCl (aq.)

T (K) 295 295 293 293 293 293 293

Reference Kadish et al. (1984) Kadish et al. (1984) Bond et al. (1995) Compton and Dryfe (1994) Compton et al. (1993b) Von Stackelberg et al. (1953) Woolf (1960)

D (cm s⫺1) 1.4 ⫻ 10⫺5 2.4 ⫻ 10⫺5 5.0 ⫻ 10⫺6 2.0 ⫻ 10⫺5 1.6 ⫻ 10⫺5 7.6 ⫻ 10⫺6 7.7 ⫻ 10⫺5

J. C. EKLUND ET AL.

20

may migrate and contribute to the migration current. The total migrative flux (Jm) is related to the sum of the migration fluxes for each charged species. For all the charged species, i, present the migration flux due to a potential gradient (⭸E/⭸x) is given (see for example; Bard and Faulkner, 1980) by (14), Jm ⫽

冘

⭸E ⫺zi F Di[i] ⭸x RT all species i

(14)

or more generally for multi-dimensional transport by (15), Jm ⫽

冘

⫺zi F Di[i](ⵜE) RT all species i

(15)

where z is the charge on the electroactive species, i. Obviously, for a neutral species, migration may be neglected. For charged species the magnitude and sign of the migration current are determined by the charge on the ion, zi. In the absence of convection, if an anion is being reduced it will diffuse towards a negatively charged working electrode but will migrate to the positively charged counter electrode which effectively decreases the magnitude of the total current (i.e. the current in the absence of electrolyte will be smaller than in the presence of electrolyte). If a cation is oxidized at a positively charged electrode the current measured will decrease from the diffusion only value and Fig. 10 shows an idealized example under microelectrode steady-state

Fig. 10 Steady-state sigmoidal-shaped voltammograms obtained in the presence and absence of electrolyte for the oxidation of A2⫹ to A3⫹ (adapted from Oldham, 1992).

MODERN VOLTAMMETRY

21

conditions with and without added electrolyte for the one-electron oxidation of species A2⫹ to A3⫹ (which has a counter anion X⫺) (Oldham, 1992). However, in most voltammetric experiments a large excess of supporting electrolyte is utilized and, thus, there will be a negligible contribution of migration to the overall appearance of the voltammogram, as almost all of the migrational transport is associated with the supporting electrolyte. Convection. This is the movement of solution as a whole caused by mechanical forces. The flux due to convection (Jc) in one direction, solution velocity, vx, is given by (16), Jc ⫽ [A]vx

(16)

or more generally, for a velocity vector, v, by (17). Jc ⫽ [A] . v

(17)

Working electrodes which have material reaching them by a form of forced convection are known as hydrodynamic electrodes. There is a wide range of hydrodynamic electrodes: rotating-disc electrodes (Albery and Hitchman, 1971), in which the electrode rotates at a fixed frequency and ‘‘sucks up’’ material to its surface, and channel electrodes (Compton et al., 1993c), over which the electroactive species flows at a fixed volume flow rate, are the primary ones used in the work described in this review (Section 4). Each of the mass transport terms can be combined to give a general mass transport equation describing the temporal variation of each species in the electrode reaction mechanism i.e. (18) for species, A. ⭸[A] zA F Dⵜ([A]ⵜE) ⫽ Dⵜ2[A] ⫺ v . ⵜ[A] ⫹ RT ⭸t

(18)

Note the opposite signs for the convection and diffusion terms as the concentration gradients resulting from each of these processes are in opposite directions. Since migration of the electroactive species of interest can be suppressed by the use of an excess of inert supporting electrolyte, only diffusion and convection are usually of interest to the mechanistic electrochemist. Diffusion of material to the electrode may be controlled by altering the concentration of the electroactive species in solution and by changing the electrode size from macro-dimensions (where, in effect, diffusion is one-dimensional) to micro-dimensions, which may support two- or threedimensional diffusion (as described in Section 5). Many electrochemical experiments are conducted under diffusion-only conditions. However, the rate of mass transport may be most easily changed by adjusting the convective transport element.

J. C. EKLUND ET AL.

22 ELECTRON TRANSFER

The voltammogram for a simple oxidative electron transfer process (19)1 A

E0f , k0, ␣

→ ←

B ⫹ e⫺

(19)

can be described in terms of the three parameters shown: • E0f , the reversible formal potential (V vs. reference electrode). We avoid the use of E 0, the standard electrode potential, because most standard electrochemical experiments are not conducted under conditions of unit activity and standard, temperature and pressure. • k0, the heterogeneous charge transfer rate constant (cm s⫺1) measured at E0f . This determines how far the half-wave potential (E1/2) or peak potential (see Section 3) is removed from E0f . For fast reactions (e.g. k0 ⫽ 0.1 cm s⫺1) E1/2 ⬃ E0f . Table 3 contains examples of typical k0 values for different systems. • ␣, the charge-transfer coefficient. It is a dimensionless parameter and it can be thought of in terms of the reaction coordinate diagram shown in Fig. 11. ␣ predominantly effects the shape and not the position of the voltammetric response, typically it has a value of 0.5. The relationship between all these parameters and the current is given by the Butler–Volmer equation (20), I ⫽ nFAk0([A]x⫽0 exp[(1 ⫺ ␣)(E ⫺ E0f ) nF/RT] ⫺ [B]x⫽0 exp[⫺␣(E ⫺ E0f ) nF/RT])

(20)

where [A]x⫽0 and [B]x⫽0 represent the electrode surface concentrations of A and B. The full theoretical description of a voltammogram is obtained by combining (20) with the appropriate mass transport equation [i.e. the appropriate version of (18)]. In the case of fast electron transfer kinetics, the theoretical expression becomes equivalent to that obtained by combining the Nernst equation with the mass transport equation. An alternative theory to the Butler–Volmer theory for electron transfer is provided by the Marcus–Hush theory (Marcus, 1968; Hush, 1968) which assumes a potential-dependent ␣. Since in most cases ␣ is essentially independent of potential, use of the simpler Butler–Volmer equation is usually adequate. HOMOGENEOUS CHEMICAL KINETICS

It is common for homogeneous chemical reactions to accompany the electron-transfer step. Thus, an electrochemical reaction mechanism may 1

Note that charges, if any, on species A and B in (19) and subsequent equations are omitted for simplicity.

Table 3 Typical values of k0. One-electron transfer process

Solvent/electrolyte system

Oxidation of ferrocene

Acetonitrile/0.1 M (C4H9)4NClO4/ platinum electrode Acetonitrile/0.6 M (C2H5)4NClO4/ gold electrode Acetonitrile/0.6 M (C2H5)4NClO4/ platinum electrode 5.6 M HClO4 (aq.)/platinum electrode 0.1 M NaClO4 (aq.)/mercury electrode 0.5 M K2SO4 (aq.)/glassy carbon electrode 0.1 M (C3H7)4NClO4/DMF/ mercury electrode 0.3 M NaClO4 (aq.)/glassy carbon electrode

Reduction of anthracene Reduction of benzoquinone Reduction of Co(H2O)3⫹ 6 Reduction of Co(NH3)3⫹ 6 Reduction pf Ru(NH3)3⫹ 6 Reduction of cyclooctatetraene Reduction of Eu3⫹

Temperature (K)

Reference

k0 (cm s⫺1)

295

Kadish et al. (1984)

0.09

298

Howell and Wightman (1984)

3.5

298

Howell and Wightman (1984)

0.39

275

Hale (1971)

1.8 ⫻ 10⫺7

298

Hale (1971)

5 ⫻ 10⫺6

293

Marken et al. (1995)

0.055

298

Hush (1968)

0.0087

293

Marken et al. (1995)

0.008

J. C. EKLUND ET AL.

24 ␣ > 0.5

␣ = 0.5

␣ < 0.5

Fig. 11 Reaction coordinate diagrams for simple heterogeneous electron transfer processes at an electrode held at a potential of E0f for a range of differing values of ␣.

consist of a combination of heterogeneous electron transfer and homogeneous chemical reaction steps, each with their own individual rate constants. If the product, B, of reaction (19) undergoes a first-order chemical reaction with rate constant, k, as shown in (21), k

B → C

(21)

then the full process is given by the reaction scheme represented in (22): E

A

E0f , k0, ␣

→ ←

B ⫹ e⫺

(22a)

k

C

B → C

(22b)

This mechanism is denoted as an EC mechanism (Testa and Reinmuth, 1961; Bott, 1997). Thus homogeneous kinetic terms may be combined with the expressions for diffusion and convection [i.e. a modified version of (18)] to give the temporal variation of the concentration of a species in an electrode reaction mechanism. In order to model the voltammetric response associated with this mechanism, a knowledge of E0f , ␣, k0 and k is required, or deduced from a theoretical–experimental comparison, and the set of concentration– time equations for species A, B and C must be solved subject to the constraints of the Butler–Volmer equation and the experimental design. Considerable simplification of the theory is achieved if the kinetics for the forward and reverse processes associated with the E step are fast, which is a good approximation for many organic reactions. Section 7 describes the approaches used to solve the equations associated with electrode reaction mechanisms, thus enabling theoretical simulation of voltammetric responses to be achieved.

MODERN VOLTAMMETRY

25

A common form of electrode reaction mechanism studied is the ECE mechanism: E C E

E 0f,1

⫺ A → ← B⫹e k1

(23a)

B → C

(23b)

⫺ C → ← D⫹e

(23c)

E0f,2

0 0 , Ef,2 and k1, the voltammograms Depending on the relative values of E f,1 associated with an ECE mechanism consists of two resolved one-electron transfer processes (Fig. 12a) or a single overall two-electron transfer process (Fig. 12b) or intermediate situations. In the above oxidative mechanism D could be formed by the homogeneous disproportionation (DISP) step (24)

DISP

k2

B⫹C → A⫹D

(24)

0 0 if Ef,1 is more positive than Ef,2 . If the DISP step (24) does not occur on the time-scale of the voltammetry, the mechanism is ECE, if the DISP step occurs, but is not rate limiting, the mechanism is defined as being DISP 1 and if the DISP step is rate limiting the mechanism is termed DISP 2. In general, if there are two or more E steps then thermodynamically favoured cross-reactions may occur. Other examples of mechanisms containing combinations of E and C steps will be discussed in Sections 3–6. The chemical step may result from external stimulation such as light (Compton et al., 1993c), e.g. the photochemical CE mechanism (25), or magnetic fields (Ragsdale et al., 1996), and CE processes have also been investigated in the presence of ultrasound (Compton et al., 1997a).

C E

h␯

A → B ⫺ B → ← C⫹e

(25a) (25b)

ELECTROCHEMICAL AND CHEMICAL REVERSIBILITY

Finally, a term that should be clearly defined and one that is often used haphazardly is that of reversibility. One must make a clear distinction between electrochemical reversibility and chemical reversibility. • Electrochemical reversibility. This term is related to the kinetics of electron transfer at the electrode surface. For a facile electron transfer, equilibrium is achieved rapidly and the system is defined as being electrochemically reversible; effectively both the forward and reverse electron transfers are

J. C. EKLUND ET AL.

26

(a)

(b) Fig. 12 Peak-shaped and sigmoidal-shaped voltammograms associated with an oxidative ECE process (a) E0f ,1 < E0f ,2 and (b) E0f ,1 ⬎ E0f ,2.

rapid. For the process described by (19) under conditions of electrochemical reversibility, Nernst’s equation (26) applies: E ⫽ E0f ⫹

冢

RT [B]x⫽0 ln [A]x⫽0 nF

冣

(26)

This stems directly from the fact that the electron transfer kinetics for the forward and reverse processes are so facile that equilibrium is attained at each potential applied in the time-scale of the particular experiment. Thus an electron transfer may be termed electrochemically reversible at a scan rate of 50 mV s⫺1, but irreversible at 1000 V s⫺1. The term is therefore a practical rather than absolute one and is dependent upon the time-scale of the electrochemical measurement. • Chemical reversibility. This refers to the stability of the species associated in an electron transfer step to chemical decomposition. Therefore, if in (19) species B decomposes irreversibly as it is formed from the one-electron transfer process, the whole process would be described as being chemically irreversible. However, if the chemical step associated with the decomposition of B was sufficiently fast in both directions on the time-scale of the

MODERN VOLTAMMETRY

27

voltammetric experiment, then the whole process would be termed chemically reversible. These concepts related to electrochemical and chemical reversibility can be demonstrated by considering the EC mechanism described by (19) and (27). k1

B → ← C

C

(27)

k⫺1

• At extremely fast scan-rates the electron transfer step is electrochemically irreversible and the C step is outrun. This would be described as an electrochemically irreversible process. • At moderate scan rates the C step is outrun, but the E step is now reversible. This would be described as an electrochemically reversible process. • At slow scan rates, the E step remains reversible and the C step is irreversible. This would be termed as an electrochemically reversible and chemically irreversible process. • At very slow scan rates the E and C steps are now reversible. This would be described as an electrochemically and chemically reversible process. Some authors would define the above processes as Ei, Er, ErCi and ErCr respectively (Bard and Faulkner, 1980). However, for simplicity, we have chosen not to use this more precise notation in this review. 3

Cyclic voltammetry

A BASIC DECRIPTION

This is undoubtedly the voltammetric technique most widely used by non-specialist electrochemists. We will initially give a very general basic introduction to this method as many of the concepts introduced here apply to the techniques discussed in Sections 4–6. In these experiments, the potential at the working electrode is swept at a fixed scan rate (␯) from an initial value of E1 to a second potential E2. On reaching E2, the direction of the sweep is reversed and, when the potential returns to E1, the scan may be halted and again reversed or allowed to continue to a third potential E3. A typical potential–time profile is illustrated in Fig. 13(a), and Fig. 13(b) shows the coordinate system used to describe a cyclic voltammetric experiment. The scan rate is represented by the magnitude of the slope of the potential–time plot; by convention ␯ is always said to be positive. The expressions given in (28) and (29) describe the potential applied [E(t)] at the working electrode as a function of time: Forward Sweep:

E(t) ⫽ E1 ⫹ ␯t

(28)

Reverse Sweep:

E(t) ⫽ ⫺ E1 ⫹ 2E2 ⫺ ␯t

(29)

J. C. EKLUND ET AL.

28

(a)

Electrode (b) Fig. 13(a) Potential–time profile for a typical CV experiment, (b) coordinate system for a cyclic voltammetric experiment.

These equations assume that the initial scan direction is positive as normally will be the case when studying an oxidation process. In (28) and (29) it is also assumed that the scan rate is the same in both the initial and reverse sweep directions which need not always be the case (the scan rate may be increased in the reverse scan in order to outrun homogeneous chemical steps associated with species formed by heterogeneous electron transfer in the forward scan).

MODERN VOLTAMMETRY

29

The scan rate may range from a few mV s⫺1 (Bard and Faulkner, 1980; Southampton Electrochemistry Group, 1990) to a million V s⫺1 (Wipf and Wightman, 1988; Amatore et al., 1989; Tschunky and Heinze, 1995). The lower scan-rate limit is restricted by the effects of natural convection which arise from the build up of density gradients in the solution resulting from such factors as poor thermostatting or mechanical vibration. Natural convection adds to the rate of mass transport of material to the electrode surface and thereby causes the experiment to deviate from the diffusion-only regime. The upper scan-rate limit is restricted by capacitative charging, as discussed in Section 2, which at very fast scan rates may mask the current associated with the Faradaic process. In cyclic voltammetric experiments, the sole form of mass transport to the electrode surface is diffusion, and in the case of large (millimetre dimensions) electrodes the diffusion of material to the electrode occurs in the single dimension perpendicular to the electrode surface. As will be discussed in Section 5 the situation is more complex for electrodes of smaller dimensions.

SIMPLE ELECTRON-TRANSFER PROCESSES

Initially a simple reversible one-electron oxidation process is examined [see (19)], such as the oxidation of ferrocene to the ferricinium cation in acetonitrile/0.1 M (C4H9)4NClO4 (Sharp et al., 1980; Kadish et al., 1984). In (19), initially only A is present in solution. At the usual macrodisc electrode (radius in the millimetre range), material reaches the electrode by linear diffusion which is perpendicular to its surface (x-direction), and the concentrations of A and B may be obtained as a function of time by solving Fick’s second law of diffusion as applied to species A and B, (30) and (31). ⭸2[A] ⭸[A] ⫽ DA ⭸x2 ⭸t

(30)

⭸[B] ⭸2[B] ⫽ DB ⭸t ⭸x2

(31)

However, the problem is subject to a number of boundary conditions which are defined in Table 4 (the symbols are described in the appendix). The time variation of the electrode potential is given by (28) and (29). Details of the solution of (30) and (31) are beyond the scope of this review, although a general approach for solving voltammetric problems is discussed in Section 7. The cyclic voltammogram shown in Fig. 14 is obtained for the reversible system described in (19) on scanning from an initial potential (E1) which is

J. C. EKLUND ET AL.

30

Table 4 Boundary conditions for a simple reversible one-electron oxidation process examined by cyclic voltammetry. Time coordinate

Spatial coordinate

Species A

Species B

Reason for boundary condition Initially, only A is in solution At large distances from the electrode the concentrations of A and B tend to their original values The rate at which A diffuses to the electrode must equal the rate at which B diffuses away The Nernst equation for an electrochemically reversible system may apply at the site of electron transfer, i.e. the electrode surface

t⫽0

x⭓0

[A] ⫽ [A]0

[B] ⫽ 0

t⬎0

x→⬁

[A] ⫽ [A]0

[B] ⫽ 0

t⬎0

x⫽0 (electrode surface)

DA(⭸[A]/⭸t)x⫽0 ⫽ ⫺DB(⭸[B]/⭸t)x⫽0

t⬎0

x⫽0

ln ([B]x⫽0 /[A]x⫽0) ⫽ nF/RT(E ⫺ E0f )

considerably less positive than E0f to a value considerably more positive than E0f (E2), and back to E1. The shape shown in Fig. 14 arises for the following reasons. On scanning the potential from E1 to more positive values, the concentration of A at the electrode surface ([A]x⫽0) drops as A is converted into B; this results in an increased concentration gradient of A at the surface of the electrode, and thus the diffusional flux of A to the electrode increases. The flux of material A to the electrode surface is directly related to the electrode current by the expression (32). I ⫽ nFAD

冢 ⭸x 冣 ⭸[A]

(32) x⫽0

As the potential approaches E0f , [A]x⫽0 decreases even further. Thus, the flux of A to the electrode continues to increase, causing the current to rise. However, eventually [A]x⫽0 reaches zero and the flux of A cannot change any further. Under the conditions of the CV experiment, once [A] x⫽0 ⫽ 0 the Nernst diffusion layer (the distance from the electrode at which concentration changes in A are associated solely with the electrolysis mechanism and the resulting diffusion is the only form of mass transport) begins to relax further into the solution as the diffusion process tries to equalize the concentrations of A and B

MODERN VOLTAMMETRY

31

IPox

Ired P

Ered P

EPox

Fig. 14 Typical cyclic voltammogram obtained for a reversible one-electron oxidation process.

throughout the solution. Consequently, at very positive potentials the flux of A drops and thus the current is seen to decrease at potentials more positive than the peak oxidative potential (EPox) to give the characteristic asymmetric shape associated with a cyclic voltammogram. On reversing the scan, initially there is a high concentration of B at the electrode surface which decreases as B is oxidized back to A. However, at potentials sufficiently negative of E0f , [B]x⫽0 will return to zero and the same asymmetric peak-shaped response is observed on the reverse sweep as in the forward sweep. Consideration of the above discussion and examination of Fig. 14 shows that there are a number of important parameters in cyclic voltammetry. ox • The peak potentials Ered P and EP . For a reversible process at 25⬚C the peaks will be separated by 56/n mV [where n is the number of electrons red will be transferred; in the example given in (19), n ⫽ 1] and Eox P and EP independent of scan rate. • The half-wave potential, E1/2. The value of E1/2 is related to the peak potentials by the expression (33).

E1/2 ⫽

red (Eox P ⫹ EP ) 2

(33)

J. C. EKLUND ET AL.

32

If A and B have equal diffusion coefficients, E1/2 is identical to the formal reversible potential (E0f ). Otherwise, E1/2 is related to E0f by the expression (34). E1/2 ⫽ E0f ⫹ (RT/nF) ln (DA/DB)1/2

(34)

For most redox couples, E1/2 only differs from E 0 by a few millivolts. • The peak currents Ired and Iox P P (note the baselines from which they are measured in Fig. 14). For a reversible process at 25⬚C, the value of the peak current, in amps, is given by (35). (For symbols definition and their units see the appendix.) 5 3/2 red 1/2 1/2 Iox P ⫽ ⫺IP ⫽ ⫾(2.69 ⫻ 10 )n ADA ␯ [A]0

(35)

This expression applies regardless of the diffusion coefficients of A and B. The increase in current with scan rate may be explained by the fact that as the scan rate increases, less time is available for the Nernst diffusion layer to relax into the solution by diffusion. Consequently, as the scan rate increases, the rate of change of concentration of A at the electrode surface increases, resulting in a greater flux of A to the electrode surface and hence a larger observed electrode current. As noted in Section 2, when the electron-transfer kinetics are slow relative to mass transport (rate determining), the process is no longer in equilibrium and does not therefore obey the Nernst equation. As a result of the departure from equilibrium, the kinetics of electron transfer at the electrode surface have to be considered when discussing the voltammetry of non-reversible systems. This is achieved by replacement of the Nernstian thermodynamic condition by a kinetic boundary condition (36). DA

冢 ⭸x 冣 ⭸[A]

et ⫽ kox [A]x⫽0

(36)

x⫽0

That is, for an irreversible electron-transfer process, the rate-limiting step over a wide range of potentials is the electron-transfer step rather than diffusion. et is related to the electrode potential and the standard rate The constant kox constant, k0, by the Butler–Volmer equation described above. Use of the Butler–Volmer equation and Fick’s laws of diffusion enables the voltammetric response of an irreversible process to be understood. A typical voltammogram associated with an irreversible oxidative oneelectron-transfer process is shown in Fig. 15. A number of differences from the reversible case may be noted. • There is no reverse peak because the reverse electron-transfer process does not occur at a measurable rate.

MODERN VOLTAMMETRY

33

IPox

Fig. 15 Typical cyclic voltammogram obtained for an electrochemically irreversible one-electron oxidation process.

• The peak current, in amps, at 25⬚C is given by the expression (37) (see appendix for symbol definitions and units). 1/2 1/2 5 1/2 ADA ␯ [A]0 Iox P ⫽ (2.99 ⫻ 10 )n(␣na)

(37)

Note that na refers to the number of electrons transferred in the rate-limiting electron-transfer step and n alludes to the total number of electrons transferred, whilst the significance of ␣ is discussed in Section 2. Comparison of (37) and (35) under equivalent conditions reveals that the peak current for an irreversible process is lower than the equivalent value for a reversible one. This feature emerges because the kinetics of the electron transfer are relatively slow in the irreversible case, so that during the course of the potential scan, diffusion has more time to relax the concentration gradient of A at the electrode surface. Consequently, when [A]x⫽0 ⫽ 0, the flux of A to the electrode surface is lower than for the equivalent reversible case, resulting in the occurrence of a decrease in the value of the peak current. The sluggish kinetics of electron transfer also broaden the voltammogram. This feature also results in the peak potential being shifted to a significantly more positive potential compared to the formal potential (E0f ) for the electron-transfer process.

J. C. EKLUND ET AL.

34

• The peak potential is a function of scan rate, unlike the case for a reversible process when the peak potentials are independent of scan rate. As the scan rate increases, the voltammetric peak becomes wider. Thus, the peak oxidation potential shifts to more positive potentials as the scan rate increases. Obviously, therefore there must be an intermediate case in which the kinetics of both the forward and reverse electron-transfer processes have to be taken account of. Such systems are described as being quasi-reversible and as would be expected, the scan rate can have a considerable effect on the nature of the cyclic voltammetry. At sufficiently slow scan rates, quasi-reversible processes appear to be fully reversible. However, as the scan rate is increased, the kinetics of the electron transfer are not fast enough to maintain (Nernstian) equilibrium. In the scan-rate region when the process is quasi-reversible, the following observations are made. • The separation of the forward and reverse peaks (⌬EP) is larger than the value of 56/n mV associated with a reversible process at 25⬚C. Importantly, ⌬EP increases with increasing scan rate and the value of the standard rate constant for the electron-transfer process, k0, may be calculated from the separation of the peaks in a quasi-reversible process (Bard and Faulkner, 1980), provided voltammograms are corrected for solution resistance effects (see below). • The peaks become broader as the scan rate increases, and the peak current is below the value expected for a reversible electron-transfer process. red is equal to one for a • Generally, the ratio of the peak currents Iox P /IP quasi-reversible system. It can be seen that the relative rates of electron transfer and the potential scan rate may crucially determine whether voltammograms are observed to be reversible, irreversible or quasi-reversible. Matsuda and Agabe (1955) proposed the (k0, ␯) regimes in (38), (39) and (40) in order to define whether an electron-transfer process will be observed to be reversible, quasi-reversible or irreversible: Reversible: Quasi-reversible: Irreversible:

k0 ⭓ 0.3␯1/2 cm s⫺1 0.3␯1/2 ⬎ k0 ⬎ (2 ⫻ 10⫺5) ␯1/2 cm s⫺1 k0 ⭐ (2 ⫻ 10⫺5) ␯1/2 cm s⫺1

(38) (39) (40)

Care should always be taken when interpreting the results of cyclic voltammetric experiments to ensure that the effects of the double-layer capacitance and uncompensated solution resistance are considered (see Section 2). Peak currents should be corrected for the baseline capacitative charging current (for example by running a background voltammogram in the absence of the electroactive species), and as the charging current is

MODERN VOLTAMMETRY

35

proportional to scan rate, such background subtractions of this current are crucial at fast scan rates. Uncompensated solution resistance (Ru) causes the peak-to-peak separation (⌬EP) to increase and the peaks to broaden in a cyclic voltammogram and this may make a reversible system appear quasireversible. In the presence of uncompensated resistance, the value of the applied potential will differ from the actual value by IRu. Modern potentiostats are capable of compensating for the majority of the effects of this solution resistance, by feeding back an additional potential, equal to IRu, to the applied potential.

MECHANISTIC EXAMPLES

The following examples show the power of cyclic voltammetry to interrogate the mechanisms of electrode reactions, even when they consist of a combination of heterogeneous (electron-transfer) and homogeneous chemical steps. Combinations of single-electron transfer and homogeneous chemical steps: EC and CE processes A common mechanism encountered in voltammetry is the EC mechanism where the species generated at the electrode surface by an electron-transfer process undergoes some form of homogeneous chemical reaction. An example examined by cyclic voltammetry which illustrates the features of an EC mechanism involves the oxidation of the 18-electron manganese (I) organometallic species fac-Mn(CO)3(dpm) where dpm is the bidentate phosphine ligand Ph2PCH2PPh2 (Bond et al., 1977, 1978). This species undergoes one-electron oxidation to the iso-structural facial 17-electron manganese (II) cationic species. However this cationic species is not in its most stable structural form (Mingos, 1979) and as a result fac-[Mn(CO)3(dpm)Cl]⫹ rapidly converts to the meridional isomeric form of the cation via a first-order reaction. Examination of the cyclic voltammetry (Fig. 16) for the oxidation of this fac-Mn(I) species in acetonitrile/0.1 M (C4H9)4NClO4 at slow scan rates (⬍500 mV s⫺1) reveals a partially reversible one-electron oxidation process having an oxidative peak potential of ⫹1.48 V (vs. Ag/AgCl) and reductive peak potential of ⫹1.35 V; however the peak currents for the oxidation and reduction processes are not equal. The voltammogram is not fully reversible because on the time-scale of this experiment (scan rate ⫽ 500 mV s⫺1) some of the fac cation (fac⫹) has isomerized to the mer form (mer⫹) by the time the potential is swept through the reversible potential for the fac0/fac⫹ couple. Confirmation of this statement is found by continuing the reverse (reductive) part of the potential sweep through the fac0/fac⫹ couple and noting that a new

J. C. EKLUND ET AL.

36

Fig. 16 Cyclic voltammogram obtained for the EC electrochemical oxidation of fac-Mn(CO)3dpmCl (1 mM) in acetonitrile/0.1 M (C4H9)4NClO4 at a platinum electrode. Scan rate, 500 mV s⫺1.

reduction feature is observed at a potential of ⫹0.95 V, which can be shown to correspond to the reduction of the mer⫹ ⫺ Mn(II) cation to the mer0 ⫺ Mn(I) species. If a second cycle of the potential is made (Fig. 16), a new oxidative feature is observed at a potential of ⫹1.02 V corresponding to the oxidation of the mer0 to the mer⫹ cationic form. Thus, the mechanism at scan rates ⬍500 mV s⫺1 is described by the EC scheme in (41) and (42). E

⫹ ⫺ fac0 → ← fac ⫹ e

(41)

C

fac⫹ → mer⫹

(42)

As was observed on the reverse and second scans, the mer species may red undergo a reversible one-electron oxidation (Eox P ⫽ ⫹1.02 V, EP ⫽ ⫹0.95 V) as in (43). E

⫹ ⫺ mer → ← mer ⫹ e

(43)

At very high scan rates the fac0/fac⫹ couple appears to be reversible, with red ⫹ 0 Iox P ⫽ ⫺IP ; concomitantly, the mer /mer couple disappears. Thus, at very ⫹ ⫹ short time domains, the fac /mer isomerization step is outrun and the oxidation process becomes a single one-electron transfer process.

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37

The kinetics of the fac⫹/mer⫹ isomerization step can be determined quantitatively from the scan-rate dependence of the oxidation process. Both theory and experiment show that the peak potential corresponding to the oxidation of the fac0 species (Eox P ) shifts to less positive potentials as the scan rate is increased. This occurs because the oxidation charge-transfer process is electrochemically reversible. Under these circumstances, the isomerization step following the charge transfer removes the product and causes the equilibrium position to move to the right in (41), which effectively facilitates the oxidation step. Consequently, at low scan rates, when the isomerization step is important, the oxidation process requires a lower thermodynamic driving force in order to occur and hence a less positive potential is observed. If the electron-transfer (E) step had been irreversible, the isomerization reaction would have no effect on the voltammetric response since the C step would not be rate determining and no kinetic data could be obtained. A reaction in which the oxidized or reduced form of a compound isomerizes via a first-order process on the voltammetric time-scale is common for a wide range of organometallic and organic compounds (for example: Bard et al., 1973; Bond et al., 1986, 1988, 1992). An example from the field of organic chemistry involves the reduction of diethyl maleate to its radical anion which then isomerizes to the diethyl fumarate anion, again an overall EC mechanism (Bard et al., 1973). There is a wide range of examples of other EC mechanisms such as the reduction of the antibiotic chloramphenicol in which a nitro unit (–NO2) is reduced to a hydroxylamine (–NHOH) (E step) which rapidly converts into a nitroso (–NO) species (C step) (Kissinger and Heineman, 1983). The kinetics of the C step are not always first order or pseudo-first order. A second-order reaction will produce qualitatively similar effects to those described above. However, the relative magnitude of the reverse peak current associated with the E step and hence the extent of reversibility and the shift in peak potential will depend on the concentration of the electroactive species for an EC2 mechanism. A process of this type will have a reversible E step at low concentrations or fast scan rates and an irreversible E step at high concentrations or slow scan rates. An example of an EC2-type reaction (Bond et al., 1983, 1989) is the electrochemical oxidation of cobalt (III) tris(dithiocarbamates) (Co(S2CNR2)3) at platinum electrodes in dichloromethane/0.1 M (C4H9)4NPF6 [equations (44) and (45)]. E C2

⫹ Co(S2CNR2)3 → ← [Co(S2CNR2)3] 2[Co(S2CNR2)3]⫹ → [Co2(S2CNR2)5]⫹ ⫹ 12–S ⫹

(44) (45)

1– 2

Typical cyclic voltammograms for this oxidation process are illustrated in Fig. 17. Note how the peak current associated with the reduction of the cobalt (IV)

J. C. EKLUND ET AL.

38

cation drops relative to the peak height of the corresponding oxidation of Co(III) to Co(IV) as a function of increasing concentration. Similarly, phenothiazine may be oxidized to the cation radical species which then dimerizes forming the 3,10⬘-diphenothiazinyl species (Tsujino, 1969). The product of the electron-transfer step may react, via a second-order process, with a species in solution to form a new product. An example of this type of mechanism involves the reduction of anthraquinone and its derivatives in the presence of oxygen (Jeziorek et al., 1997). To understand quantitatively an EC and EC2 process, the concentration and scan-rate dependence of the associated cyclic voltammograms is matched with theory deriving from the mass transport/kinetic equations for each species. An electroactive species may not be initially present in solution, but may be formed by a reaction of the bulk solution species. This class of electrochemical mechanism is described as a CE mechanism in the Testa and Reinmuth notation. An example of a CE mechanism involves the reduction of ethanoic acid in aqueous solutions, in which protons released by the acid in (46) are reduced reversibly in (47) at the electrode surface (see for example: Vielstich and John, 1960a,b; Daniele et al., 1996). C E

⫺ ⫹ CH3COOH → ← CH3COO ⫹ H 1 H⫹ ⫹ e⫺ → ← 2–H2

(46) (47)

For a typical CE process, the peak current may vary with scan rate in a complex manner. The peak current initially increases with scan rate (varying linearly with ␯1/2) when the C step is reversible. However, as the scan-rate time-scale becomes faster than the kinetic time-scale associated with the C step, the peak current will start to decrease with scan rate as the kinetics of the chemical step are outrun and when the amount of electroactive product that can be oxidized/reduced at the electrode surface becomes limited. Eventually at scan rates that are much faster than the kinetic time-scale associated with the C step, a sigmoidal shaped response is observed when the current is no longer diffusion controlled, and is limited by the kinetics of the C step. Under these conditions the rate constant for the C step may be determined by a simple analytical expression (Saveant and Vianello, 1963, 1967a,b; Nicholson and Shain, 1964).

EXAMPLES OF COMPLEX MECHANISMS COMMONLY ENCOUNTERED IN ORGANIC ELECTROCHEMISTRY

ECE, DISP and EC⬘ mechanisms A very common electrode reaction mechanism encountered in organic electrochemistry is the ECE mechanism. As discussed in Section 2, with this mechanism, a species generated by heterogeneous electron transfer at the

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39

I IPox

Fig. 17 Cyclic voltammograms obtained for the EC2 electrochemical oxidation of Co(S2CNR2)3 (0.1–5.0 mM) in dichloromethane/0.1 M (C4H9)4PF6 at a platinum electrode. Scan rate, 100 mV s⫺1; note each voltammogram has its current normalized to its peak oxidative current.

electrode surface undergoes some form of homogeneous chemical reaction to form a product that may transfer a further electron at the electrode surface. A typical example of an ECE mechanism is provided by the reduction of halonitrobenzenes at platinum electrodes in a range of organic solvents (for example: Holleck and Becher, 1962; Nelson et al., 1973; Bento et al., 1993; Compton and Dryfe, 1994). At slow cyclic voltammetric scan rates, p-iodonitrobenzene (p-NO2C6H4I) exhibits two one-electron reduction processes in DMF/0.1 M (C2H5)4NClO4. The first reduction process is chemically irreversible and the second reduction exhibits a reverse oxidation peak, which suggests this process has some degree of chemical reversibility (Lawless and Hawley, 1969). The form of the cyclic voltammogram is identical to that shown in Fig. 12a for a general ECE process. The final product of this overall two-electron reduction process was found to be the radical anion of nitrobenzene as evidenced by EPR spectroscopy. A simplified form of the ECE mechanism proposed for the reduction of p-iodonitrobenzene is in (48)–(50), where HS is an H atom source. E

⭈⫺ p-NO2C6H4I ⫹ e⫺ → ← [p-NO2C6H4I] HS

(48)

C

[p-NO2C6H4I]⭈⫺ → p-NO2C6H5 ⫹ I⫺

(49)

E

⭈⫺ p-NO2C6H5 ⫹ e⫺ → ← [p-NO2C6H5]

(50)

J. C. EKLUND ET AL.

40

On continuous cycling of the potential through the two reduction processes, the magnitude of the current associated with the first reduction process gradually drops whereas that due to the second process increases. This behaviour is expected for an ECE process in which the p-iodonitrobenzene is reduced irreversibly to the nitrobenzene radical anion. Further, the addition of iodide anions to the solution decreases the rate of formation of nitrobenzene as evidenced by noting that the rate at which the peak current associated with the p-iodonitrobenzene reduction decreases less rapidly on successive scans through both reduction processes. The dependence on the iodide concentration in bulk solution suggests that the C step of the ECE mechanism actually consisted of processes (51) and (52), ⭈ ⫺ [p-I-C6H4NO2]⭈⫺ → ← C6H4NO2 ⫹ I HS

C6H4NO2 → C6H5NO2

⭈

(51) (52)

i.e. the dissociation of the p-iodonitrobenzene was a pre-equilibrium step prior to the rate-limiting abstraction of a hydrogen atom to form nitrobenzene. Obviously, it is possible that the chemical step in an ECE mechanism is second order i.e. an EC2E mechanism. The oxidation of 3-mercaptopropionic acid (RSH) shown in (53)–(55) provides such an example (Forlano et al., 1997): E

⭈⫹ ⫺ RSH → ← RSH ⫹ e

(53)

C2

2RSH⭈⫹ → RS–SR ⫹ 2H⫹

(54)

E

⫺ RS–SR → ← oxidized products ⫹ e

(55)

Cyclic voltammetric simulation techniques were used to examine quantitatively the kinetics of the dimerization step and the rate constant was found to be in the range of (1.7–2.0) ⫻ 103 mol⫺1 dm3 s⫺1. As discussed in Section 2, the second electron transfer step in an ECE mechanism may be competing with a homogeneous electron transfer reaction in solution. This alternative mechanism was described as either an overall DISP 1 or DISP 2 mechanism. It is very difficult to distinguish between a DISP 1 process and an ECE process in which the product of the C step is oxidized at a less positive potential or reduced at a less negative potential than the first electron transfer step as both mechanisms result in a single one- to two-electron voltammetric wave that have similar dependences on scan rate, concentration and electrode area etc. Furthermore, both processes may occur in the one system. An example of a mixed ECE/DISP 1 mechanism involves the reduction of 2-chloroquinoline (ArCl) in acetonitrile/0.6 M (C2H5)4NClO4. Fast-scan cyclic voltammetry (Wipf and Wightman, 1988) at microdisc electrodes was used to probe the rapid kinetics associated with this

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41

mechanism and kinetic parameters were determined by simulating the voltammetric response associated with the reduction of this species as a function of scan rate. Two reduction processes were observed. The first process was associated with the mixed ECE/DISP 1 mechanism shown in (56)–(59). E

⭈⫺ ArCl ⫹ e⫺ → ← ArCl

E0f ⫽ ⫺1.93 V (vs. SCE)

(56)

C

ArCl⭈⫺ → Ar⭈ ⫹ Cl⫺

k ⫽ 6 ⫻ 105 s⫺1

(57)

DISP

Ar ⫹ ArCl

E

Ar⭈ ⫹ e⫺ → Ar⫺

⭈

⭈⫺

→ ← ArCl ⫹ Ar

⫺

K⬎1 ⫻ 10

5

E0f ⬎ ⫺1.00 V

(58) (59)

The product of the second electron transfer step may then accept a proton from the solvent medium to form quinoline which can then be reduced at a more negative potential to give the second reduction wave observed in the cyclic voltammetry of 2-chloroquinoline. Many other examples of mixed ECE/DISP 1 mechanisms exist such as the reduction of 2,2-dinitro-propane, -cyclopentane and -cyclohexane in DMF (Ruhl et al., 1992). After heterogeneous electron transfer at a solid electrode, the resulting reduced/oxidized species may then transfer an electron homogeneously to or from another molecule in solution to regenerate the starting electroactive species. Such a mechanism is described as EC⬘, where C⬘ represents a catalytic step. A typical example of such a reaction involves the oxidation of phenylamine, N,N-diethylphenylamine, histidine and histamine by ferricyanide electrogenerated at a platinum working electrode in basic aqueous/ethanolic mixtures; see (60) and (61) (Rashid and Kalvoda, 1970). E C⬘

3⫺ ⫺ → Fe(CN)4⫺ 6 ← Fe(CN)6 ⫹ e kf

4⫺ ⭈⫹ Fe(CN)3⫺ 6 ⫹ RNH2 → Fe(CN)6 ⫹ RNH2

(60) (61)

Rashid and Kalvoda examined this reaction using cyclic voltammetry by measuring the current enhancement for the electro-oxidation of potassium ferricyanide on addition of the amine. Using working curves derived by Nicholson and Shain (1964) relating the ratio of the peak current measured in the presence and absence (i.e. the diffusion-controlled peak current for oxidation of ferricyanide) of the amine to the parameter kf RT/nF␯ for an EC⬘ mechanism, the kinetic parameter, kf, could be calculated. Electron transfer followed by protonation A very common combination of E and C steps observed in organic electrochemical experiments involves a one-electron reduction step to form an anion radical which is then followed by protonation of the anion radical to give a stable uncharged organic compound. A typical example is the reduction of

J. C. EKLUND ET AL.

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catechol violet (CV) at hanging mercury drop electrodes (Abdel-Hamid, 1996). Cyclic voltammetry shows that at low pH (⬍7) a single two-electron wave is observed, while at higher pH(⬎8), two one-electron waves are observed. Simulation of the voltammetric response at low and high pH using commercial software packages (Gosser and Zheng, 1991) showed that at low pH the reduction pathway followed an ECEC, first-order mechanism (62)–(65), the rate-determining step being the second C step (65). E

⭈⫺ CV ⫹ e⫺ → ← CV ⫹H⫹

(62)

C

CV⭈⫺ → CVH⭈

(63)

E

⫺ CVH⭈ ⫹ e⫺ → ← CVH

(64)

C

⫹H⫹

CVH⫺ → CVH2

(65)

At high pH values, the first electron transfer process occurs without protonation so that a single irreversible one-electron process is initially observed. The resulting catechol violet radical anion is then reduced (66) at a more negative potential in an EC reaction (67) (resulting in the observation of a second reduction wave), in which the dianion is doubly protonated: E C

2⫺ CV⭈⫺⭈ ⫹ e⫺ → ← CV ⫹2H⫹

→ CVH2 CV2⫺ ←

(66) (67)

Many other examples exist for this form of reaction such as the reduction of 4-nitropyridine to hydroxylamine via 4-nitrosopyridine in aqueous media (Laviron et al., 1992) which has been described as proceeding via an overall ECE mechanism (Nadjo and Savéant, 1973).

EXAMPLES OF ELECTRODE REACTION MECHANISMS CONSISTING OF EXTENSIVE COMBINATIONS OF E AND C STEPS

The reduction of substituted derivatives of Buckminster fullerene, C60, provide an excellent example of the use of cyclic voltammetry to probe electrode reaction mechanisms. C60 is well known to undergo a series of reversible one-electron reduction processes forming a sequence of anions of successively increasing charge. In a suitable solvent, C60 may undergo six one-electron reductions (for example see Dubois et al., 1992; Krishnan et al., 1993). The substituted fullerene bipyridyl-C61 (R2C61) shown in Fig. 18 may have four possible isomers (Paolucci et al., 1995). Each pair of isomers either has the CH3C(Bipy) unit on the junction between two adjacent six-membered rings ([6,6] isomers) or between a five-membered ring and a six-membered

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43

ring ([5,6] isomers). Furthermore, each pair of isomers may have the bridgehead atoms bonded (closed form) or not bonded (open form). Cyclic voltammetry of the [5,6]-open form recorded in THF/0.05 M (C4H9)4NPF6 showed seven main reduction processes. The similarity of the voltammetry to free C60 suggested that the electron-transfer processes were primarily fullerene based. However, it was noted that under some conditions, the third reduction peak had been split into two overlapping peaks, even though the total charge passed for the third process always was attributable to the transfer of one electron. At low scan rates, the height of the second of the two overlapping peaks, associated with the third reduction, is greater than that of the second peak. In contrast, at scan rates greater than 400 mV s⫺1, or at low temperatures, only the first process is observed for the third electron transfer. The data associated with the first three reduction processes were interpreted in terms of the mechanism described in (68)–(71). E

⫺ [5,6]-open R2C61 ⫹ e⫺ → ← ([5,6]-open R2C61)

(68)

E

2⫺ ([5,6]-open R2C61)⫺ ⫹ e⫺ → ← ([5,6]-open R2C61)

(69)

E

3⫺ ([5,6]-open R2C61)2⫺ ⫹ e⫺ → ← ([5,6]-open R2C61)

(70)

C

([5,6]-open R2C61)3⫺ → ([6,6]-closed R2C61)3⫺

(71)

Thus, since the third reduction process of the [6,6]-closed form occurs at a more negative potential than that of reduction of the [5,6]-open form when formed at the electrode surface, its trianion is spontaneously oxidized at the electrode to the dianion only to be subsequently reduced when more negative potentials are reached and thereby giving rise to the splitting of the third reduction peak. That is, the second of the two peaks associated with the third reduction process is attributable to the reduction (72). E

3⫺ ([6,6]-closed R2C61)2⫺ ⫹ e⫺ → ← ([6,6]-closed R2C61)

(72)

The existence of the isomerization reaction was confirmed by examining the voltammetry of the synthesized [6,6]-closed form in an equivalent THF solution and noting that the peak potential associated with the third reduction was identical to that associated with the more negative of the peak potentials of the split third reduction wave of the [5,6]-closed form. The value of the rate constant for the chemical isomerization step was determined by comparing experimental voltammograms with simulated voltammograms derived using the postulated mechanism. A similar result was observed in a recent study that observed a splitting of the second reduction process of a doubly-bridged N-substituted imino[60]fullere in 1,2-dichlorobenzene (Zhou et al., 1997).

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Fig. 18 Structures of substituted C60 compounds discussed in text.

4

Hydrodynamic voltammetry

As discussed in Section 2 material may reach the electrode surface by diffusion or convection. In cyclic voltammetry at a stationary electrode, and assuming that migration can be neglected, diffusion is the sole form of mass transport. However, material may additionally be transported to the electrode by convection. This genre of voltammetry, where convection is a dominant form of mass transport, is described as hydrodynamic voltammetry. The focus in Section 4 will be on the use of rotating disc and channel electrodes in studies

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45

of electrode mechanisms. The use of wall jet electrodes also will be briefly covered, although the reader is directed to more comprehensive texts for further information on this form of electrode (see for example Powell and Fogg, 1988). Hydrodynamic voltammetric techniques have the major advantage of being steady-state techniques (see Section 1). Consequently, it is easy to measure limiting currents and half-wave potentials (see below for their definition) as a function of the convective parameter (i.e. flow rate, electrode angular velocity) in the absence of significant problems arising from capacitative charging currents. The potential profile associated with hydrodynamic techniques usually takes the form of a linear sweep between two potentials in which the oxidation or reduction processes of interest occur. As for cyclic voltammetry, the gradient of the ramp represents the scan rate. However, for steady-state techniques, the scan rate used must be sufficiently slow to ensure that the steady state is attained at every potential during the course of the voltammetric scan. The upper value of the scan rate that may be used under the steady-state regime is therefore restricted by the rate of convective mass transport of material to the electrode surface. The faster the rate of convective mass transport the faster the scan rate that may be used consistent with the existence of steady-state conditions. With hydrodynamic voltammetry, it is the time parameter associated with the rate of convection that is critical in the examination of the kinetics associated with electrode reaction mechanisms. This term plays an analogous role to that of the scan rate in cyclic voltammetric experiments. The importance of this time parameter can be seen by examining an ECE mechanism [see (23a)–(23c)]. This mechanism is probed in hydrodynamic voltammetry by examining the effective number of electrons transferred, Neff, as a function of mass transport of material to the electrode. Neff, which will vary between one and two for an ECE mechanism, gives an indication of the competition between loss of intermediates into the bulk solution and the second heterogeneous electron-transfer step. For rapid rates of convective mass transport, Neff tends to a value of one, because the intermediates B and C are swept away from the electrode into bulk solution before the second E step can occur (see Fig. 19). In contrast, at very low rates of mass transport, Neff tends to two, as B and C remain in the vicinity of the electrode for sufficient time to allow C to undergo an electron-transfer process at the electrode surface. Thus, in an ECE process, the homogeneous kinetic process competes with mass transport of material to and from the electrode. In order to probe the kinetics of the C step in an ECE process fully, the voltammetric response must be measured over a sufficiently wide range of mass transport rates so that Neff varies between one and two. For particularly rapid processes, this requirement implies that very fast rates of mass transport are required in order to avoid Neff being equal to two at all transport rates.

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Fig. 19 Schematic representation of an ECE reaction mechanism at a hydrodynamic electrode.

Conversely for slow reactions, low rates of mass transport will be required to achieve significant deviations from Neff equalling one. Consequently, it can be appreciated that it is a study of the competition between the rates of mass transport and chemical kinetics that leads to the quantitative determination of electrode reaction mechanisms in hydrodynamic voltammetry. Importantly, for each hydrodynamic technique, there is one assessable convective transport parameter that directly relates to the kinetic time-scale.

ROTATING-DISC ELECTRODES

The rotating-disc electrode (RDE) consists of a disc electrode, made from a suitable working electrode material (see for example Albery and Hitchman, 1971; Compton and Harland, 1989; Albery et al., 1989), surrounded by a non-conducting sheath (usually PTFE). The complete electrode assembly is constructed so that the sheath and electrode are flush (Fig. 20). Typically, the disc electrode faces downwards into solution (although inverted electrodes have been developed (Bressers and Kelly, 1995) and is rotated around an axis perpendicular to and through the centre of the disc. Under these conditions, a well-defined flow pattern distribution is established, as illustrated in Fig. 21; in effect, solution is sucked towards the electrode and then flung outwards. The general mass transport equation for this form of electrode, in the absence of homogeneous chemical kinetics, is given by (73), ⭸2 [A] ⭸2 [A] 1 ⭸[A] 1 ⭸2 [A] ⭸[A] ⫹ 2 ⫹ ⫹ ⫽D r ⭸␪2 r ⭸r ⭸z2 ⭸r2 ⭸t

冤

冤

⫺ vr

⭸[A] v␪ ⭸[A] ⭸[A] ⫹ ⫹ vz r ⭸␪ ⭸r ⭸z

冥

冥 (73)

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Fig. 20 Schematic representation of a rotating-disc electrode.

where r, z and ␪ are cylindrical polar coordinates (see Fig. 20), and vr, vz and v␪ are the respective components of the solution velocity. Under the usual operating conditions where the flow profile is symmetric about the central axis, [A] is independent of ␪; thus ⭸[A]/⭸␪ and ⭸2[A]/⭸␪2 are equal to zero and, since 1/r ⭸[A]/⭸r can never be infinite, ⭸[A]/⭸r must also be equal to zero. If the electrode radius is significantly smaller than that of the sheath, vz will be independent of r and may be calculated (Cochran, 1934). In addition, under steady-state conditions the concentration profile is invariant with time so that ⭸[A]/⭸t ⫽ 0. Therefore, under these conditions, (73) simplifies to (74). ⭸2 [A] ⫺0.51␻3/2 v⫺1/2 2 ⭸[A] z ⫽ ⭸z2 ⭸z D

(74)

The following experimental conditions must be met in order to ensure compliance with the above theory: • The electrode rotates in a single plane perpendicular to the axis of rotation. • The frequency of rotation is stable with respect to the time required to conduct a voltammetric scan. • The electrode rotation frequency is sufficiently low to prevent turbulent flow, typically in the range 4–50 Hz.

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Fig. 21 Convective flow profile associated with a rotating-disc electrode.

The crucial parameter which controls the time-scale over which electrode reactions are examined at a RDE is the electrode angular velocity, ␻, which is related to the rotation frequency, f, by ␻ ⫽ 2␲f.

CHANNEL ELECTRODES

Figure 22 shows a schematic diagram of a channel electrode (ChE), which consists of an electrode embedded in the wall of a rectangular duct through which solution is made to flow under well-defined laminar steady-state conditions (Compton and Unwin, 1986; Cooper and Compton, 1998). Mass transport of material to (and from) the electrode is both by convection and by diffusion, and for the ChE the time-dependent convective–diffusion equation (75) describing the mass transport may be obtained (Leveque, 1928). ⭸[A] ⭸[A] ⭸[A] ⭸2 [A] ⭸2[A] ⭸2[A] ⭸[A] ⫺ vx ⫹ vy ⫹ vz ⫽D 2 ⫹ 2 ⫹ 2 ⭸x ⭸y ⭸z ⭸x ⭸y ⭸z ⭸t

冤

冥 冤

冥

(75)

The flow is treated as being two-dimensional in the x–y plane. On entering the channel, the solution velocity profile is essentially plug flow. However, the effect of friction at the walls causes retardation of the solution flow in the x-direction (Fig. 23). After a distance le (the entry length) from the entrance, the hydrodynamic layers from each wall merge, and the flow regime established is laminar in form (Compton and Coles, 1983; Albery and Bruckenstein, 1983) in which separate

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49

Fig. 22 Schematic representation of a channel electrode.

Fig. 23 Convective flow profile associated with a channel electrode.

layers (laminae) of solution have characteristic velocities reaching a maximum (v0) at the centre of the channel. The parabolic shape of the ultimate velocity profile is therefore given by the velocity components in (76),

冤

vx ⫽ v0 1 ⫺

(y ⫺ h)2 ; vy ⫽ vz ⫽ 0 h2

冥

(76)

where v0 is the velocity of the solution in the centre of the channel. Further, for macroelectrodes one may simplify (75), by assuming that diffusion in the x- and z-directions is negligible (Levich, 1962), to obtain (77). D

(y ⫺ h)2 ⭸[A] ⭸2 [A] ⫽0 ⫺ v 1 ⫺ 0 ⭸y2 ⭸x h2

冤

冥

(77)

J. C. EKLUND ET AL.

50

Fig. 24(a) Channel flow systems: gravity-fed.

The velocity of the solution in the centre of the channel is related to the solution flow rate, Vf, by the expression (78). 4 Vf ⫽ h dv0 3

(78)

It is this flow rate, Vf, or centre line velocity, v0, parameter that is critical in determining the time domain over which chemical processes can be monitored. Generally, in order to develop laminar flow over the electrode, le should be sufficiently long, and ideally have a value given by (79), le ⫽ 0.1hRe where Re is the Reynolds number (80). v0 h Re ⫽ v

(79)

(80)

Laminar flow is generally achieved for Re ⬍ 2000. Laminar flow is obtained either by a gravity-fed flow system (Compton et al., 1993c) or by a pressurized flow system (Rees et al., 1995a,b) as shown in Fig. 24. In the gravity-fed flow

MODERN VOLTAMMETRY

Fig. 24(b) Channel flow systems: pressurized fast-flow system.

51

52

J. C. EKLUND ET AL.

system, the inlet reservoir is at a greater height than the outlet and the solution may flow through a variety of differing diameter capillaries. Thus, the flow rate is determined by the difference in the height of the solution in the inlet reservoir and the level of the outlet tip as well as the diameter of the capillary through which the solution flows. For typical cell heights (2h) of 0.04 cm and cell widths (d) of 0.6 cm, flow rates in the range 10⫺4–10⫺1 cm3 s⫺1 are readily attainable using this apparatus. A pressurized flow system (Rees et al., 1995a,b) has been recently developed which is designed to force solution through a flow cell so as to induce velocity gradients at the electrode surface which are very much higher. Flow is achieved by applying a large pressure at the inlet reservoir end of the system, whilst maintaining the outlet at atmospheric pressure; otherwise, the design is similar to the gravity-fed system. Values of v0 as large as 75 m s⫺1 can be obtained using this apparatus (h ⫽ 0.01 cm, d ⫽ 0.2 cm), allowing determination of homogeneous kinetic parameters as high as 105 s⫺1 from steady-state measurements (Rees et al., 1995a).

WALL JET ELECTRODES

The wall jet electrode (WJE) has attracted considerable attention in analytical applications of voltammetry (see for example Brett et al., 1995, 1996). In this electrode configuration, a high, fixed velocity jet of fluid is fired, through a nozzle of diameter, a, directly towards the middle of a disc electrode (radius ⫽ r1), whose centre coincides with that of the nozzle (Fig. 25). The solution thus impinges upon the electrode surface and is circulated outwards towards the extremities of the electrode surface, but the recirculated solution can never reach the electrode a second time.

Fig. 25 (a) Schematic representation of a wall jet electrode and (b) convective flow profile associated with a wall jet electrode.

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53

Fig. 26 Voltammogram obtained for a simple reversible one-electron transfer process at a hydrodynamic electrode. ELECTRON-TRANSFER PROCESSES

A reversible one-electron transfer process (19) is initially examined. For all forms of hydrodynamic electrode, material reaches the electrode via diffusion and convection. In the cases of the RDE and ChE under steady-state conditions, solutions to the mass transport equations are combined with the Nernst equation to obtain the reversible response shown in Fig. 26. A sigmoidal-shaped voltammogram is obtained, in contrast to the peak-shaped voltammetric response obtained in cyclic voltammetry. There are two critical parameters that are measured in a steady-state voltammogram: • The limiting or mass transport limited current Ilim. As soon as the potential is reached when [A]x⫽0 ⫽ 0, the current reaches a fixed limiting current value that is determined by the mass transport of material to the electrode surface. Under these conditions, material is continuously replenished at the electrode surface by convection, in contrast to the situation in a CV where depletion occurs and a peak-shaped response is observed. Table 5 gives the analytically derived expressions for the limiting currents obtained at the three electrode types discussed in this section. • The half-wave potential, E1/2. For a reversible redox couple in which the oxidized and reduced species have very similar diffusion coefficients, the half-wave potential approximates to the formal electrode potential Ef0. For

J. C. EKLUND ET AL.

54

Table 5 Expressions for the limiting current obtained at a range of hydrodynamic electrodes (see appendix for meaning of symbols). Electrode RDE ChE WJE

Convective flow parameter Angular rotational velocity, ␻ Volume flow rate, Vf Volume flow rate, Vf

Expression for limiting current, Ilim 0.62nFAD2/3 v⫺1/6 [A]0 ␻1/2 0.925nFD2/3 [A]0 w(Vf x2e/h2 d )1/3 3/4 1.59nFD2/3 v⫺5/12 [A]0 a ⫺1/2 r3/4 1 Vf

a reversible electron-transfer process, E1/2 will not vary with the rate of convective mass transport to the electrode surface. For a reversible electron-transfer process, the Tafel relationship corrected for mass transport holds in the central region of the voltammogram (Brett and Oliveira-Brett, 1993). Therefore, for a reversible one-electron process, a plot ⫺1 ) will have a slope of 59/n mV per decade at of E ⫺ E1/2 versus log10(I ⫺1 ⫺ Ilim 25⬚C. As for all voltammetric techniques, sluggish electron-transfer kinetics require the application of an additional potential (overpotential) to drive the electron-transfer process at the same rate as for the equivalent reversible process. Thus, the observed voltammogram is broadened relative to that found for a reversible process (see Fig. 27, in which a reversible process with an identical half-wave potential is shown for comparison). Note that the limiting current is identical to that observed for a reversible case, provided a potential sufficiently positive (oxidation) or negative (reduction) of the half-wave potential is applied. The Tafel gradient for a totally irreversible process at 25⬚C is given by 120/n (if ␣ ⫽ 0.5) mV (Brett and Oliveira-Brett, 1993). It should be noted that, as in the case of homogeneous kinetics, competition exists between heterogeneous electron transfer and transport of material to and from the electrode surface. Hence, as the rate of convective mass transport increases, an initially reversible electron-transfer process may become quasi-reversible and finally at very high rates of mass transport, irreversible. This is entirely analogous to the dependence of the response on scan rate in cyclic voltammetry. A number of studies have been conducted using fast rates of convective mass transport in order to probe the kinetics of heterogeneous processes (see for example Rees et al., 1995b; Macpherson et al., 1995). One example utilized the fast-flow system (Rees et al., 1995b) to examine the reduction of benzoquinone at platinum electrodes in acetonitrile solutions. The heterogeneous kinetics could be probed by examining the variation of the mass transport corrected Tafel plots as a function of the flow rate (over the range 10⫺2 to 3.5 cm3 s⫺1) of solution over the electrode surface. A value of 0.30 cm s⫺1 was obtained for the standard heterogeneous rate constant by comparing the results with numerical or analytical theory. This value was in

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55

Fig. 27 Voltammogram obtained for an irreversible one-electron transfer at a hydrodynamic electrode (note a reversible one-electron process is also illustrated with equal half-wave potential for comparison).

good agreement with the value obtained using the transient CV technique (Howell and Wightman, 1984). As discussed in Section 3, solution resistance and capacitative charging can play a significant role in voltammetry. Due to the typically low scan rates and steady-state conditions, capacitative charging presents a relatively small problem in hydrodynamic techniques. Solution resistance affects the appearance of the voltammogram in much the same way as a decrease in the rate of heterogeneous electron transfer. That is, uncompensated resistance broadens the voltammetric response observed because an additional potential term (the product of the electrode current and the solution resistance term) is present (Coles et al., 1996). Thus, great care has to be taken to ensure that resistance artefacts are accounted for when assessing the kinetics of heterogeneous processes using hydrodynamic and indeed all other voltammetric techniques. Note, however, that the limiting current value is unaffected by the solution resistance or slow electron-transfer kinetics.

COMBINATIONS OF ELECTRON TRANSFER AND HOMOGENEOUS CHEMICAL STEPS

As noted earlier, varying the rate of convective mass transport of material to the electrode surface allows the elucidation of reaction mechanisms via

56

J. C. EKLUND ET AL.

Fig. 28 Examination of (a) CE and (b) EC mechanisms using hydrodynamic voltammetry.

monitoring the dependence of a particular experimental parameter (such as Neff) on mass transport. Comparison of the experimental result with theory derived for a particular reaction mechanism provides quantitative detail concerning the kinetics. In the rest of this section, an overview of results obtained for a range of chemical processes examined using hydrodynamic voltammetric techniques illustrates how complex reaction processes may be probed using such methods. Theoretical details describing how voltammetric data can be analysed to give quantitative results are introduced in Section 7. CE and EC processes The CE and EC mechanisms illustrate how examination of the mass transport dependence of Ilim and E1/2 can provide vital quantitative insights into the nature of these processes. For a CE mechanism, the electrode product of interest is formed via an initial chemical reaction. Consequently, the measured limiting current will directly correlate with the amount of electroactive product formed on the time-scale of the experiment. Thus, sufficiently slow rates of mass transport result in complete conversion of bulk material to electroactive product and under this condition the limiting current will be identical to that calculated from the expressions described in Table 5 for a simple electron transfer process (see Fig. 28a). As the electrode angular velocity (␻) or flow rate (Vf) increases, less of the material reaching the electrode will have converted into

MODERN VOLTAMMETRY

57

the final electroactive product resulting in a current that is lower than the predicted current for a simple electron transfer process. Examination of the limiting current dependence on the convective mass transport rate therefore allows the determination of the kinetic parameters associated with the C step. As discussed in Section 3, a common class of CE mechanism results from the dissociation of weak organic acids, such as ethanoic acid [see (46) and (47)]. In these systems, the reduction of H⫹ resulting from dissociation of the acid may be monitored. Both rotating disc (Vielstich and John, 1960a,b) and wall jet electrodes (Martin and Unwin, 1995) have been used to examine this kind of process. For the RDE, a plot of Ilim /␻1/2 versus Ilim was utilized to determine the forward and reverse rate constants for the dissociation of ethanoic acid in aqueous solutions. Martin and Unwin utilized a microjet electrode that could generate high rates of convective flow transport to examine the dissociation of monochloroethanoic acid. RDE studies revealed that the dissociation rate was so fast that limiting currents associated with a simple one-electron transfer process were obtained at all angular velocities. However, at high flow rates with a microjet, the current was significantly lower than that expected for the simple one-electron reduction of protons. In this study, a similar relationship to that described above for the RDE was utilized to examine the kinetics of acid dissociation reactions and it was established that the rate constant for the dissociation of monochloroethanoic was two orders of magnitude greater than for ethanoic acid. A major advantage of the channel electrode technique is its ability to be utilized in conjunction with other techniques (e.g. photochemical or spectroscopic). This advantage has been utilized effectively in the investigation of the photoelectrochemical oxidation of fac-Mn(CO)3dpmCl in acetonitrile (Compton et al., 1993d). As discussed in Section 3, in the absence of irradiation, this complex undergoes a one-electron oxidation process forming the [fac-Mn(CO)3dpmCl]⫹ cation (Fig. 29). Upon irradiation of the channel electrode surface by 385 nm light, a new oxidative feature is seen at a less positive potential than obtained for the parent oxidation wave. The half-wave potential for this new photoproduct corresponds to that expected for oxidation of mer-Mn(CO)3dpmCl. Consequently, it was postulated that the electrode reaction mechanism in the presence of light was CE in nature (81), the C step involving the photo-isomerization (81a) of the fac species. hv

(81a)

C

fac → mer

E

→ mer ⫹ e mer ← ⫹

⫺

(81b)

A value of 0.07 s⫺1(light intensity ⫽ 40 mW cm⫺2) was obtained for the first-order rate constant associated with the C step by examining the flow-rate dependence of the limiting current associated with the mer oxidation wave.

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J. C. EKLUND ET AL.

Fig. 29 ChE voltammogram for oxidation of fac-Mn(CO)3dpmCl (1.42 mM) in acetonitrile/0.1 M (C4H9)4NClO4 while being irradiated at 390 nm. Solution flow rate is 10⫺3 cm3 s⫺1.

In other photoelectrochemical studies, the oxidation of Cr(CO)3(arene) compounds in acetonitrile was examined using the channel electrode technique (Compton et al., 1993b,e). A combination of in situ EPR, fluorescence spectroscopy and voltammetry showed that the arene and, in some cases, CO were ejected on photolysis and the final chromium compound was either Cr(CO)3(CH3CN)3 or Cr(CO)2(CH3CN)4, which subsequently underwent a one-electron oxidation at the electrode surface. For an EC mechanism with a reversible electron-transfer step, the following chemical kinetics will cause the equilibrium associated with the electrontransfer step to shift to the right side. This causes the half-wave potential associated with the E step to move to a less negative potential in the case of a reduction process and to a less positive potential in the case of an oxidation

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59

process (see Fig. 28b). Measurement of the dependence of the half-wave potential on the rate of mass transport of material to the electrode surface is used to probe quantitatively the kinetics associated with the homogeneous chemical process. An example of this kind of EC process which has been examined by channel electrode voltammetry involves the photoelectrochemical reduction of the crystal violet dye in acetonitrile (Compton et al., 1988a). In the absence of irradiation, crystal violet (CV⫹) undergoes a simple reversible one-electron reduction. Upon irradiation, the half-wave potential associated with the steady-state voltammogram due to the reduction of CV⫹ shifts to more positive potentials, but the magnitude of the transport limited current is unchanged as would be expected for a photo-EC mechanism (82). E C

→ CV⭈ CV⫹ ⫹ e⫺ ← hv

CV⭈ → CV⭈* → products

(82a) (82b)

Measurement of the variation in half-wave potential with flow rate over the channel electrode and comparison with simulated EC theory gave a first-order rate constant for the photoelectrochemical decomposition of CV⭈ of 2.1 s⫺1 (␭, 406 nm; light intensity, 35 mW cm⫺2). ECE and DISP mechanisms For ECE or DISP mechanisms, the parameter usually measured as a function of the convective mass transport parameter is the effective number of electrons transferred, Neff, which for two single-electron transfer steps varies between one and two as described above. A typical example of an ECE mechanism probed with the channel electrode technique involves the dehalogenation of p-bromobenzophenone (p-BrC6H4COPh) (Rees et al., 1995a). After an initial reduction of p-BrC6H4COPh to the radical anion (83a) bromide is ejected (83b) to form a ␴ radical (⭈C6H4COPh) similar in nature to those formed on reduction of halonitrobenzenes (Section 3). This species then abstracts a hydrogen atom from the solvent/supporting electrolyte medium (HS) forming benzophenone (PhCOPh) which is reduced to its radical anion in the second electron transfer step (83c). (83a)

E

→ p-BrPhCOPh⭈⫺ p-BrPhCOPh ⫹ e⫺ ←

C

p-BrPhCOPh⭈⫺ → PhCOPh ⫹ Br⫺

(83b)

E

→ PhCOPh⭈⫺ PhCOPh ⫹ e⫺ ←

(83c)

HS/k

The kinetics associated with halide expulsion are rapid, and, in order to measure the rate constant associated with the C step, fast rates of convective

J. C. EKLUND ET AL.

60

flow had to be utilized. A value of 7 ⫻ 105 s⫺1 was measured for the rate constant via an analysis of the dependence of Neff on flow rate. An ECE mechanism examined with the rotating disc electrode technique involved the pyridination of 9,10-diphenylanthracene (DPA) in acetonitrile/0.2 M tetraethylammonium perchlorate (Manning et al., 1969). Initially, 9,10-diphenylanthracene undergoes a reversible one-electron oxidation (84a) to form the radical cation, which reacts with pyridine (Py) to form an adduct (84b) which is in turn oxidized (84c) at a less positive potential than the initial diphenylanthracene molecule. This reaction sequence leads to formation of a dication that may further react with pyridine to form the final products. E

→ DPA⭈⫹ ⫹ e⫺ DPA ←

(84a)

C

DPA⭈⫹ ⫹ Py → DPAPy⭈⫹

(84b)

E

DPAPy

⭈⫹

→ DPAPy ←

2⫹

⫹e

⫺

Py

→ Products

(84c)

The effective number of electrons transferred, Neff, was monitored as a function of the electrode angular velocity and the equivalent response was simulated (Feldberg, 1969), which allowed the pseudo-first-order rate constant for the reaction of the anthracene cation with an excess of pyridine to be determined. More recent studies (Ahlberg and Parker, 1980; Parker, 1980, 1983) have suggested that the mechanism is more complex than that suggested by this ECE description. The rate law for reaction of DPA⭈⫹ was found to deviate significantly from its simple second-order reaction with pyridine at DPA concentrations below 10⫺4 M. Parker (Ahlberg and Parker, 1980; Parker 1980) suggested that DPA⭈⫹forms an initial ␲ complex with pyridine, this species reacting with DPA⭈⫹ to form DPA and a doubly-charged cationic complex. This complex then undergoes an internal reorganization forming DPAPy2⫹ which combines with pyridine to form the final products. Simulation of the dependence of Neff on flow rate for a mixed ECE/DISP 1 mechanism (Compton et al., 1991a) allowed the quantitative determination of the mechanism for the photochemical expulsion of iodide from 1-iodoanthraquinone (AQI) in acetonitrile. E

→ AQI⭈⫺ AQI ⫹ e⫺ ← HS/k

(85a)

C

AQI⭈⫺ → AQH ⫹ I⫺

(85b)

E

→ AQH⭈⫺ AQH ⫹ e⫺ ←

(85c)

k2

(85d)

DISP

AQH ⫹ AQI⭈⫺ → AQH⭈⫺ ⫹ AQI

In this example, it was shown that the rate of iodide expulsion was sensitive to the wavelength of the light incident upon the electrode surface. The AQI⭈⫺

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61

radical ion UV/visible absorption spectrum exhibits two bands at 565 and 417 nm. Analysis of the voltammetric data showed that the rate of iodide expulsion (85b) was over seven times slower per photon absorbed for the 417 nm band as compared to the 565 nm band. This was suggested to result from the differing nature of the electronic transitions, the 565 nm band being associated with a spin allowed ␲ to ␲* transition and that at 417 nm with spin forbidden (␲, ␲*) to (␲*, ␲*) transition. In a DISP 2 mechanism the second-order disproportionation step is rate limiting (see Section 2). An example of such a process involves the photoreduction of the dye fluorescein in basic aqueous solutions at mercury electrodes (Compton et al., 1988b). The photoreduction of benzophenone (86) and fluorobenzophenone in acetonitrile also proceeds via a DISP 2 type mechanism as verified by channel electrode voltammetry (Leslie et al., 1997). The rate-limiting step is electron transfer (86c) between photoexcited radical anion and the initial anionic species formed on electron transfer at the electrode surface. This process is further complicated by significant conproportionation (86e) and quenching of the benzophenone excited state (86f). Reduction:

→ PhCOPh⭈⫺ PhCOPh ⫹ e⫺ ←

Excitation:

PhCOPh⭈⫺ → PhCOPh⭈⫺*

Disproportionation:

PhCOPh⭈⫺ ⫹ PhCOPh⭈⫺* → PhCOPh ⫹ PhCOPh2⫺ (86c)

Product formation:

PhCOPh2⫺ → Products

h␯

(86a) (86b)

(86d)

Conproportionation: PhCOPh ⫹ PhCOPh2⫺ → PhCOPh⭈⫺ ⫹ PhCOPh⭈⫺ (86e) Quenching:

PhCOPh⭈⫺* ⫹ PhCOPh → PhCOPh⭈⫺ ⫹ PhCOPh (86f)

Examples of more complex mechanisms As discussed in Section 3, the EC⬘mechanism is a common electrode reaction route. An example of an EC⬘ reaction studied by RDE techniques is (87), the iodide mediated reduction of iodate anions in acidic aqueous media (Beran and Bruckenstein, 1968). Iodide is formed at the electrode surface through reduction of iodine in the aqueous solution. E

→ 2I⫺ I2 ⫹ 2e⫺ ←

(87a)

C⬘

⫹ 5I⫺ ⫹ IO⫺ → 3I2 ⫹ 3H2O 3 ⫹ 6H

(87b)

J. C. EKLUND ET AL.

62

Analogously with EC⬘ data obtained for cyclic voltammetric experiments, the ratio of the limiting current for the reduction of iodine in the presence and absence of iodate was compared to that derived theoretically for an EC⬘ mechanism. This analysis of the data allowed elucidation of the reaction mechanism and quantification of kinetic parameters. Compton et al. (1990a) examined the mediated reduction of t-butyl bromide (tBuBr) by the photochemically excited radical anion of tetrachlorobenzoquinone (TCBQ) in acetonitrile solution using a channel electrode. Under dark conditions, the reduction of TCBQ proceeded via a simple reversible one-electron transfer process in the presence of tBuBr. On photo-excitation of the radical anion of TCBQ, the limiting current associated with its formation was enhanced suggestive of the EC⬘ mechanism (88). E

→ TCBQ⭈⫺ TCBQ ⫹ e⫺ ←

(88a)

C⬘

TCBQ⭈⫺ ⫹ tBuBr → TCBQ ⫹ tBuBr⭈⫺

hv, k

(88b)

C

t

fast

BuBr⭈⫺ → Products

(88c)

Analysis of Neff (i.e. limiting current in the presence of light divided by the equivalent current in the absence of light) as a function of flow rate using working curves derived through numerical simulations of a general EC⬘ mechanism at a channel electrode enabled the mechanism to be quantitatively confirmed, the value of k being measured as 1.3 ⫻ 103 mol⫺1 cm3 s⫺1. The oxidation of cobalt (III) tris(dithiocarbamate) (CoL3) in acetonitrile provides an interesting example of an electrode reaction that had to be probed using a combination of cyclic and channel electrode voltammetry over a wide range of experimental conditions (i.e. CoL3 concentration, flow rate, scan rate, etc.) in order to confirm unambiguously that the mechanism was EC2C (89) in nature (Alden et al., 1998). E

→ [CoL3]⫹ ⫹ e⫺ CoL3 ←

(89a)

C2

2[CoL3]⫹ → [Co2L5]⫹ ⫹ oxidized ligand

(89b)

C

[Co2L5]⫹ ⫹ 2CH3CN → CoL3 ⫹ [CoL2(CH3CN)2]⫹

(89c)

Using either cyclic or channel electrode voltammetry in isolation over a relatively narrow range of concentrations failed to lead to detection of the final C step. However, analysis of how Neff varied with flow rate or scan rate revealed that more than one electron was transferred at slow rates of mass transport, as would be expected through the regeneration of CoL3 in the C step. Interpretation of the variation of Neff with scan rate or flow rate enabled the accurate determination of the mechanistic parameters associated with this mechanism.

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63

Finally, an excellent example of the power of channel electrode voltammetry to probe complex reaction mechanisms is provided by the photoelectroreduction of 4-chlorobiphenyl (CBP) using the radical anion of 9,10-diphenylanthracene (DPA⭈⫺) as a mediator (Leslie et al., 1996). Analysis of this mechanism follows the typical pattern described above in which Neff is examined as a function of the rate of mass transport and then kinetic parameters are measured by comparing theoretical data for a specified mechanism with that obtained experimentally. Analysis of the kinetic data over a range of concentrations of DPA and CBP strongly suggested that CBP and DPA⭈⫺ formed a charge transfer complex, resulting in greater catalysis of the decomposition of CBP at lower CBP concentrations. The overall mechanism (90) was found to consist of the five steps (90a)–(90e).

DPA

← DPA⭈⫺ DPA ⫹ e⫺ →

(90a)

DPA⭈⫺ → DPA⭈⫺*

hv

(90b)

DPA⭈⫺* → DPA⭈⫺

(90c)

⫹ CBP → DPA ⫹ Products

(90d)

⭈⫺*

DPA⭈⫺* ⫹ DPA → DPA⭈⫺ ⫹ DPA

5

(90e)

Microelectrodes

GENERAL CONCEPTS

Microelectrodes (also referred to as ultramicroelectrodes) are, as the name implies, tiny electrodes which possess at least one dimension that is sufficiently small that the mass transport regime is a function of size (Montenegro, 1994). In practice, the upper limit of this small dimension is approximately 20 ␮m. At larger sizes natural convection (see Section 2) is likely to cause interference with measurements which must be made under diffusion-only conditions. Although microelectrodes have long been used for in vivo measurements of neurotransmitters where their small size and currents are essential advantages (Adams, 1976), only since the pioneering work of Fleischmann et al. (1984) and Wightman (1981) have the merits of these electrodes been recognized for the study of fast homogeneous and heterogeneous kinetic processes. The now mature field of microelectrode voltammetry is covered by many reviews of which the recent ones by Aoki (1993) and Montenegro (1994) are recommended to readers interested in this subject. Attractive features of microelectrodes relative to conventionally sized electrodes include increased current density, reduced charging currents and reduced ohmic drop (see Section 2). The last of these permits experiments to

64

J. C. EKLUND ET AL.

Fig. 30 Schematic diagram showing microdisc (a), band (b), hemi-cylinder (c) and ring (d) electrodes.

be conducted in very resistive media, particularly, for example, non-polar solvents (Peña et al., 1987) or solutions containing an absence or near absence of supporting electrolyte (Bond et al., 1984). Microelectrodes exist in a variety of geometries, the most important of which are microdisc electrodes. Microbands, cylinders and rings are other possibilities whilst the microsphere or hemisphere is often used to aid theoretical development since the rate of mass transport is invariant over the electrode surface. The different possibilities are illustrated in Fig. 30.

MASS TRANSPORT TO MICROELECTRODES

Traditionally, macroelectrodes operating under diffusion-only conditions such as mercury drops are characterized by one-dimensional mass transport in which diffusion takes place normal to the electrode surface (see Section 2). In contrast, with microelectrodes, such mass transport is found only at short times after electrolysis is initiated when the diffusion layer is small compared to the shortest dimension of the microelectrode. At subsequent times, the rate of mass transport varies locally over the electrode surface with the edges receiving a greater current density owing to the possibility of convergent diffusion as is illustrated in Fig. 31 for the case of a microband electrode. It is this additional convergent diffusion which leads to the significantly enhanced

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65

Fig. 31 A schematic diagram of a microband electrode. The arrows represent the directions of diffusion to the electrode.

mass transport as compared to large electrodes. As a result of this non-uniform diffusion to the electrode surface, the current will not scale with the electrode area. For the case of a microdisc electrode convergent diffusion leads to a steady-state limiting current given by (91). Ilim ⫽ 4nFDre [A]0

(91)

This shows that the limiting current scales with the electrode radius, re, reflecting the non-uniformity of the current density given by (92) (Crank, 1975): J ⫽ (2/␲)nF[A]0 D/(re2 ⫺ r2)1/2

(92)

Equation (92) suggests that the current density should approach infinity near the disc edges; in practice, effects associated with finite electrode kinetics preclude this. Although the microhemisphere is not an electrode of practical importance, the transition between linear and convergent diffusion can be illuminated if the current following a potential step at such an electrode (93) is considered. Ilim ⫽ nFA[A]0 /re ⫹ nFAD1/2 [A]0 /(␲t)1/2

(93)

Here A is the electrode area (A ⫽ 2␲re2). At short times the second term dominates and the mass transport is Cottrellian [I ⬀ t⫺1/2 (Bard and Faulkner, 1980)] whilst at longer times the current tends to the steady-state value predicted by the first term. Note, however, that microsphere and hemisphere electrodes are atypical microelectrodes in that their symmetry dictates that the current density is uniform over the electrode surface; for the microelectrodes shown in Fig. 30 this is not the case as edge mass transport dominates. Implicit in the above is the notation that current–voltage curves measured at macroelectrodes for all but fast voltage scan rates are characterized by a mass transport limited current plateau rather than a current peak as in linear sweep voltammetry at a planar electrode of larger than micro dimensions.

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Fig. 32 Diagrams showing current–voltage curves measured at a microdisc electrode at scan rates corresponding to the limits of (a) convergent diffusion and (b) planar diffusion.

Figure 32 shows a typical microelectrode voltammogram for an electrochemically reversible system under near steady-state conditions. Of course at very fast scan rates the behaviour returns to that of planar diffusion and a characteristic transient-type cyclic voltammetric response is obtained as the mass transport changes from convergent to linear diffusion.

MICROELECTRODES AND HOMOGENEOUS KINETICS

Consideration of Fig. 32 implies that chemical information may be extracted from microelectrode experiments either via steady-state measurements or via transient, often cyclic voltammetric, approaches. In the former approach, measurements are made of the mass transport limited current as a function of the electrode size – most usually the electrode radius for the case of a microdisc electrode. This may be illustrated by reference to a general ECE mechanism depicted by (23a)–(23c) where k is the rate constant for the C step. Kinetic and mechanistic information may be gleaned through examination of the effective number of electrons transferred, Neff, as a function of microdisc radius. Figure 33 shows that Neff varies between 1 and 2. The former limit corresponds to the case of fast mass transport (small radius) since B is lost to

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Neff

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Fig. 33 A working curve showing the relationship between Neff and the dimensionless parameter (kr2e/D).

bulk solution before it can be transformed into C whilst the latter is reached for slow mass transport (large radius) where B is nearly completely transformed into C near the electrode surface. The dimensionless parameter in (94) kre2/D ⫽ (re2/D)/(k⫺1)

(94)

dictates whether the kinetics are ‘‘fast’’ or ‘‘slow’’. The term (re2/D) gives an approximate measure of the time taken to move out of the diffusion layer of the microdisc whilst the term (k⫺1) corresponds to the time taken for appreciable amounts of B to transform into C. Examination of the theoretically generated working curve suggests that for microdiscs of radii 1–10 ␮m, lifetimes of B in the range 0.1–100 ms should be amenable to study in the manner suggested using steady-state measurements. A thorough summary of steady-state microelectrode voltammetry studies which provide information on homogeneous chemistry appears in the review by Montenegro (1994). The dynamic range of microelectrode experiments may be extended by the use of fast-scan cyclic voltammetry. At sufficiently fast scan rates, the mass transport approximates to that seen for conventional cyclic voltammetry although caution is advisable under intermediate conditions (Alden et al., 1997). For these experiments a major advantage of a microelectrode resides in its intrinsically small area which leads to a correspondingly reduced capacitance of the electrode/solution interface associated with its double layer. It is this latter quantity which gives an upper limit to the scan rates accessible if a Faradaic signal (arising from electron

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transfer between the electrode and solution phase species) is not to be masked by a capacitative charging arising from movement of ions into or from the double layer. The interfacial capacitance scales directly with the electrode area so that there is a clear advantage in employing microelectrodes. In practice, useful measurements can be made with fast-scan cyclic voltammetry at microdisc electrodes to probe lifetimes of unstable species which approach the nanosecond regime. This requires voltage scan rates of hundreds of kilovolts per second! An elegant example of this type of study concerns the electroreduction of the 2,6-diphenylpyrylium cation in acetonitrile solution (Amatore and Lefrou, 1991) using a 10-␮m platinum disc electrode. Electrochemically, the cation is reversibly converted into the radical by a oneelectron reduction as shown in Scheme 1.

Scheme 1

The resulting radical can irreversibly dimerize to form a species which displays no electroactivity under the conditions examined. Figure 34 shows three voltammograms measured at scan rates between 75 and 250 kV s⫺1. The fastest scan rate shows the reoxidation of the radical on the return scan whereas with slower scan rates this is progressively lost as the sweep time becomes comparable with the time taken for the radical to dimerize. Interpretation of the current peak data in terms of an EC2 mechanism permitted the deduction that the dimerization rate constant was 2.5 ⫻ 109 M⫺1 s⫺1 corresponding to a half life of 20–50 ns under the conditions studied.

MICROELECTRODES AND HETEROGENEOUS KINETICS

The increased rate of mass transport associated with shrinking electrode size means that electrode processes which appear electrochemically reversible at large electrodes may show quasi- or irreversible electrode kinetics when examined using both steady-state and transient mode microelectrode methods. The latter represents a powerful approach for the determination of fast heterogeneous electrode kinetics. Rate constants in excess of 1 cm s⫺1 have been reported (Montenegro, 1994).

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Fig. 34 Voltammograms (with background correction) for the reduction of 2,6diphenylpyrylium perchlorate at a 10 ␮m platinum disc electrode. The following scan rates were used: (a) 250, (b) 150 and (c) 75 kV s⫺1. The substrate concentration was 10 mM. Data adapted from Amatore and Lefrou (1991).

CONVECTIVE MICROELECTRODES

It is clear from the above discussion that to access faster homo- and heterogeneous processes, increased rates of mass transport are required and that this can be achieved either by shrinking the electrode size or by incorporating convective mass transport. Two recent developments exploit both these features in the same experiment. First a wall jet tube microelectrode has been introduced in which a jet of electrolyte is incident on a microelectrode (Macpherson et al., 1994). This has the effect that the electrode becomes uniformly accessible and extends the microelectrode dynamic range. Alternatively, a miniature channel electrode has been described (Rees et al., 1995a,b) in which very high flow rates, of up to 10 cm3 s⫺1, are utilized in conjunction with microband electrodes as small as 1 ␮m. Homogeneous rate constants as high as 105 s⫺1 are readily measurable from these steady-state experiments. Both the wall tube and the high-speed channel techniques are well suited for the study of ultrafast kinetics and have approximately similar dynamic ranges (see Section 8).

6

Sonovoltammetry

THE EFFECT OF ULTRASOUND ON ELECTROCHEMICAL PROCESSES

Sonochemistry is concerned with the effect of ultrasonic waves on chemical reactivity (Mason, 1991) and is an area of rapidly growing importance in a diversity of applications. Ultrasound has a frequency above that which is audible

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to the human ear (usually 16–18 kHz for adults). A distinction is usually made between ‘‘power ultrasound’’ of frequency up to 18 kHz and ‘‘diagnostic ultrasound’’ of frequency between 1 and 10 MHz, the contrast arising from the much greater sound energy which can be transmitted into a system at the lower frequencies. Ultrasound cannot influence chemical reactions through direct coupling since the frequencies used are too low. Instead, influence is exerted through the phenomenon of cavitation. This arises because as a sound wave is transmitted through a liquid there is alternately compression and expansion of the structure of the latter. Under sufficiently extreme conditions this phenomenon leads to the breakdown of the liquid structure and the formation of cavitation bubbles which are most likely to arise at ‘‘weak spots’’ associated with the presence of tiny suspended particles or gas nuclei. In subsequent compression cycles, these cavities can collapse with the release of large amounts of energy giving rise to high local temperatures and pressures which then exert an effect on the solution chemistry. Ultrasound is known for its capacity to promote heterogeneous reactions (Ley and Low, 1989) mainly through greatly increased mass transport, interfacial cleaning and thermal effects. In addition, homogeneous chemical reactions have been reported to be modified (Suslick et al., 1983; Luche, 1990; Colarusso and Serpone, 1996); for example the sonochemical generation of radical species in aqueous media is important in environmental detoxification (Kotronarou et al., 1991; Serpone et al., 1994). The benefits of coupling ultrasound to electrochemical processes have long been recognized and studied (see, for example, the review by Compton et al., 1997a), especially in the context of specific applications such as electroplating (Walker, 1990), the deposition of polymer films (Akbulut et al., 1986; Oswana et al., 1987), the production of active metal particles (Durant et al., 1995) and sonoelectrosynthesis (Connors and Rusling, 1984; Degrand, 1986, 1987; Chyla et al., 1989; Mason et al., 1990; Matsuda et al., 1994; Durant et al., 1996). These and other applications have been thoroughly and carefully reviewed (Walton and Phull, 1996). Recent experimental advances have allowed the interpretation of sonoelectrochemical phenomena from a physical and mechanistic standpoint. In particular, the principles underlying voltammetric experiments conducted in the presence of ultrasound are now much better appreciated, and situations where the incorporation of ultrasound is beneficial have been identified. Fundamental aspects of sonoelectrochemistry have been reviewed (Compton et al., 1997a,b). In this section we focus almost exclusively on sonovoltammetry.

SONOVOLTAMMETRIC EXPERIMENTS: PRACTICAL CONSIDERATIONS

Section 2 described a number of general concepts surrounding the experimental design of a general voltammetric experiment; this section focuses upon how

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this design is modified for sonovoltametric experiments. Various possibilities exist for introducing ultrasound into chemical and electrochemical reactions (Mason and Lorimer, 1988; Mason, 1991) ranging from whistle reactors to ultrasonic horn transducers. The most popular source of ultrasound in voltammetric experiments is an ultrasonic immersion horn probe, although ultrasonic baths are also encountered. Ultrasonic baths will be familiar from their everyday use in the laboratory where they are commonly used for cleaning surfaces and to aid dissolution. A bath essentially comprises a number of transducers of fixed frequency, commonly 20–100 kHz, attached beneath the physical exterior of the bath unit. Baths typically deliver ultrasonic intensities between 1 and 10 W cm⫺2 to the reaction medium. For sonovoltammetry (or sonoelectrosynthesis) the bath may be filled with distilled water and a conventional electrochemical cell is placed inside the bath at a fixed position (Walton et al., 1995) so that the cell is electrically isolated from the sound source. Alternatively, the internal metal casing of the bath can be coated so that the full volume is available to use as an electrochemical cell (Huck, 1987). For both arrangements results can be highly sensitive to positioning and/or cell geometry effects. An ultrasonic horn transducer consists of a transducer unit attached to a horn (rod) usually made from titanium alloy and which has a length a multiple of half-wavelengths of the sound wave. For the commonly encountered 20-kHz horn this corresponds to 12.5 cm. The horn is then partially inserted into the fluid medium of interest and intense ultrasound is generated at its tip so that, for adequately large intensities, a cloud of cavitation bubbles is visible. This arrangement permits significantly higher ultrasonic intensities (10– 1000 W cm⫺2) to be applied than are achievable with a bath. An often-adopted sonovoltammetric design is that shown in Fig. 35 built around a conventional three-electrode cell and which allows the ultrasound intensity and the distance between the horn and electrode to be continuously varied at a fixed ultrasound frequency of typically 20 kHz. This arrangement is much less sensitive to the shape and dimensions of the electrochemical cell than when a sonic bath is utilized. A further and important point of contrast is that the direct contact of the (metallic) horn with the electrochemical system may dictate the use of a bipotentiostat to control its electrical potential relative to that of the reference electrode (Marken and Compton, 1996). Alternatively, the horn may be electrically isolated (Huck, 1987; Klima et al., 1994). A significant merit of the design shown in Fig. 35 is that the mass transport characteristics may be empirically but reliably established. It is to this essential topic we next turn. MASS TRANSPORT EFFECTS: A SIMPLE DESCRIPTION

Probably the most important consequence of introducing ultrasound into a voltammetric experiment is the increase, often dramatic, of the rate of mass

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SONIC HORN

Transducer

Reference electrode Counter electrode

Resistance thermocouple

Inlet for degassing

Ti tip

d

cooling coil Pt disc

Fig. 35 A thermostatted sonovoltammetric cell in which a sonic horn is located a fixed distance, d, from a disc electrode.

transport of material to and from the electrode surface. Figure 36 shows voltammograms obtained for the oxidation at a 2 mm platinum disc electrode of a 2 mM solution of ferrocene, Fc, in acetonitrile/0.1 M (C4H9)4NClO4 (Compton et al., 1996a) using the cell geometry shown in Fig. 35. Under ‘‘silent’’ conditions, the familiar cyclic voltammetric response of the electrochemically and chemically reversible one-electron oxidation to the ferricinium cation (7) is observed. With the application of power ultrasound experimentally imposed as in Fig. 35, two major changes become apparent. First, substantially higher currents flow. Second, rather than the current passing through a peak, a sustained (‘‘limiting’’) current is observed at high

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Fig. 36 Voltammograms obtained at 25⬚C for the oxidation of 2 mM ferrocene in acetonitrile/0.1 M (C4H9)4NClO4 at a 2-mm diameter platinum disc electrode using a scan rate of 20 mV s⫺1 recorded either (a) under ‘‘silent’’ conditions, or (b) under insonation (20 kHz, 50 mW cm⫺2) using the experimental arrangement of Fig. 35 with d ⫽ 4.0 cm.

positive potentials. Both these observations emphasize the greatly enhanced mass transport conditions further underlined by the calculation that rotating the platinum disc at 200 Hz in the absence of ultrasound would effect a similar change. Further inspection of Fig. 36 shows noise superimposed on the average limiting current suggestive of turbulent flow and/or cavitating bubbles at the interface. Empirical investigations (Compton et al., 1996a) showed the average limiting currents recorded under conditions such as used in the studies illustrated in Fig. 36 to be increased as the horn-to-electrode separation was reduced and/or as the ultrasound intensity was increased. With these two variables held fixed, and neglecting at present the case of microelectrodes (see above), a series of investigations using acetonitrile as solvent showed the limiting current (Ilim) to scale directly as the electrode area (A), and with both the concentration of the electroactive species ([A]0) and diffusion coefficient (D) of the electroactive species. This suggested the parameterization for the limiting current to be as in (95). Ilim ⫽ nFDA[A]0 /␦

(95)

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Fig. 37 The diffusion layer at an insonated electrode.

The quantity ␦ can be interpreted as the mean thickness of a diffusion layer at the electrode surface as schematically depicted in Fig. 37. In this simple picture, the electrode is separated from the turbulent bulk by a laminar sub-layer inside of which the concentration of the electroactive species depletes from the bulk value to that at the electrode surface across the (physically smaller) diffusion layer. Within this model the effects of the ultrasonic intensity and the horn-to-electrode separation emerge through their effects on the size of ␦. If (95) is used to estimate values for the diffusion layer thickness obtained for sonovoltammetry in acetonitrile, values of the order of a few micrometres are obtained – much smaller than encountered in conventional voltammetry under silent (stationary) conditions unless either potential scan rates of hundreds of mV s⫺1, or more, are employed or alternatively steady-state measurements are made with microelectrodes with one or more dimensions of the micrometre scale (Compton et al., 1996b). Further experiments have been conducted to confirm whether or not the presumed diffusion layer and its thickness, ␦, as estimated from (95) corresponds to physical reality. First AC impedance spectroscopy has been used to find the frequency response of the real and imaginary components of the cell impedance and compared with the theoretical prediction for diffusion across a thinned diffusion layer. At very high AC frequencies, where the AC perturbation had insufficient time to probe to the edge of the diffusion layer, effectively the response expected for semi-infinite diffusion was seen (‘‘Warburgian behaviour’’). At lower AC frequencies, as expected, the cell impedance was greatly reduced in the presence of ultrasound. Moreover, not only was the quantitative behaviour as predicted theoretically

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for a thinned diffusion layer but the resulting values of ␦ gave good agreement with corresponding values estimated through (95) under steadystate conditions. These results, at least for acetonitrile solutions and for the ultrasonic intensities adopted, were consistent with the thinned diffusion layer model (Compton et al., 1996a). Potential-step experiments have also been conducted in which the electrode potential was jumped from a value insufficient to induce current flow to one corresponding to a mass transport limited current under steady-state conditions. Again, the results were qualitatively consistent with the thinned diffusion layer model and again gave quantitative results comparable with steady-state conditions (Compton et al., 1996a). Studies have also been conducted on electrode processes where homogeneous chemical reactions are coupled to heterogeneous electron transfer(s). In this example, the reductive dehalogenations of 3- and 4-bromobenzophenone and of o-bromonitrobenzene (denoted as ArBr) dissolved in N,N-dimethylformamide solution were studied (Compton et al., 1996b). The one-electron reductions of these compounds result in the formation of the corresponding chemically reactive radical anions as shown in (96), where HS denotes the solvent/supporting electrolyte system. Ar-Br ⫹ e⫺ → [Ar-Br]⭈⫺

(96a)

[Ar-Br]⭈⫺ → Ar⭈ ⫹ Br⫺

(96b)

HS

Ar⭈ → Ar-H Ar-H ⫹ e

⫺

(96c)

→ [Ar-H]

⭈⫺

(96d)

The radical anion fragments, eliminating bromide ions and forming a ␴ radical, Ar⭈, which abstracts a hydrogen atom from its environment forming a neutral dehalogenated aromatic product which may be reduced further at the electrode surface. This constitutes an overall ECE-type mechanism (see Section 2). Figure 38 details the experimental results for the 3-bromobenzophenone system (Compton et al., 1996b). The behaviour under silent conditions – Fig. 38(a) – is consistent with the above scheme with step (96a) occurring with a peak potential of ca. ⫺1.55 V (vs. SCE) and step (96d) corresponding to the chemically reversible process at more positive potentials. In the presence of ultrasound, mass transport limited currents rather than peaks are seen for these processes. Moreover, as the mass transport is increased by shrinking the horn-to-electrode separation, the size of the current associated with the second step becomes smaller relative to that of the first step. Figure 38(e) shows the observed current as a function of the reciprocal of the diffusion layer thickness, ␦⫺1. It can be seen that there is a smooth transition between nearly two-electron behaviour at lower rates of mass transport (thicker diffusion layers) and almost one-electron behaviour at

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Fig. 38 The reduction of 0.5 mM 3-bromobenzophenone in DMF/0.1 M (C4H9)4NClO4 solution at a 3-mm diameter glassy carbon disc electrode. (a) Cyclic voltammogram measured under silent conditions at a scan rate of 50 mV s⫺1. (b)–(d) Sonovoltammograms obtained with 25 W cm⫺2 intensity ultrasound at 27, 15 and 8 mm horn-toelectrode separations respectively. (e) Plot of sonovoltammetric limiting currents vs the reciprocal diffusion layer thickness. The solid lines show the theoretical expected behaviour for simple one- and two-electron processes respectively whilst the dotted line corresponds to that for an ECE mechanism with a rate constant of 600 s⫺1.

faster mass transport (thinner diffusion layers). The dotted line corresponds to the calculated behaviour for the effective number of electrons transferred, given by (97), Neff ⫽ 2 ⫺ {[tanh(␦2 k/D)]/(␦2 k/D)}1/2

(97)

in an ECE process at a uniformly accessible electrode (Karp, 1968) where D is the mean diffusion coefficient of the mechanistically important species and k is the (first-order) rate constant characterizing the chemical step. For the theoretical curve shown in Fig. 38(e), a value of k ⫽ 600 s⫺1 was used, and is seen to give excellent agreement with the experimental data. Moreover, this

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Fig. 38e

value is in close agreement with that deduced from variable sweep rate cyclic voltammetry experiments conducted in the absence of ultrasound (Nadjo and Savéant, 1971). Results obtained for 4-bromobenzophenone and obromonitrobenzene also gave results consistent with a thinned diffusion layer model and values for the corresponding rate constants in close agreement with those measured independently under silent conditions. These kinetic results are interesting in that they are consistent with the physical reality of the thinned diffusion-layer model introduced above. Moreover it is evident that sonovoltammetry enables fast rate constants to be measured under steady-state conditions at conventionally dimensioned electrodes; otherwise these would only be accessible via transient measurements such as fast-scan cyclic voltammetry or using steady-state microelectrode methodology. Mass transport effects are further considered in due course after pausing to describe an alternative electrode geometry for sonovoltammetry.

SONOTRODES

An interesting alternative to the cell geometry shown in Fig. 35 is the so-called sonotrode approach resulting from the pioneering work of Reisse and

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Fig. 39 Schematic diagram showing a platinum sonotrode.

Fig. 40 Three experimental approaches to sonovoltammetry: (a) face-on, (b) side-on, and (c) sonotrode geometries.

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Fig. 41 Acoustic streaming at an immersion horn.

co-workers (Reisse et al., 1994) in which the tip of the 20-kHz titanium alloy horn serves both as an electrode and as a source of ultrasound. Alternatively, other metals may be employed through the insertion of electrically isolated metal discs into the horn tip as shown in Fig. 39 (Compton et al., 1996c). The former arrangement is popular for electrosynthesis whilst the latter is possibly better suited for sonovoltammetry. In both cases the attraction is the yet further enhanced mass transport in comparison with the face-on mode, and mean diffusion layer thicknesses on the sub-micron scale have been measured. The probable semi-conducting nature of the surface layer of the titanium horn has been exploited for photoelectrochemical purposes (Compton et al., 1997c). Figure 40 summarizes the different geometries employed in sonovoltammetry including the ‘‘side-on’’ approach (Eklund et al., 1996). In this latter case a ‘‘flow over a flat plate’’ model gave good agreement with experiment assuming solution velocities of ca. 100 cm s⫺1 were obtained in solution. These were attributed to acoustic streaming (Marken et al., 1996a), as shown in Fig. 41. These observations prompt a further consideration of mass transport effects.

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Fig. 42 Time evolution of a bubble collapsing near an electrode surface.

SONOVOLTAMMETRY: MASS TRANSPORT EFFECTS – FURTHER ASPECTS

Platinum sonotrodes fabricated by embedding a platinum disc in the tip of a 20 kHz horn produced probably the most enhanced limiting currents observed in sonovoltammetry (Marken et al., 1996a; Compton et al., 1996c) with diffusion layer thicknesses of less than 1 ␮m. Moreover, under these conditions, the magnitude of the limiting current was found to scale with the two-thirds power of the diffusion coefficient, D2/3, rather than directly with D. It is a general feature of hydrodynamic electrodes (Brett and Oliveira-Brett, 1993), with turbulent flow (Barz et al., 1984) as well as laminar, that the presence of significant convection gives rise to limiting currents with this functional dependence. In addition, a similar relationship has been observed for the case of the face-on geometry in aqueous solution using relatively larger ultrasound intensities (Hill et al., 1996). This suggests the need to refine the empirical mass transport model presented above to allow for the significant convection effect, both from acoustic streaming and from microjets resulting from cavitational collapse (Fig. 42); Perusich and Alkire (1991a,b) have modelled the mass transport resulting from microjets and predict a D2/3 dependence. Both convective and cavitational features are evident if the data in Fig. 43 are considered. These data relate to the one-electron reduction of Ru(NH3)3⫹ in aqueous 0.1 M KCl. Comparison of Fig. 43(a) and Fig. 43(b) 6 shows the expected change from silent to sonovoltammetry. The time-resolved data in Fig. 43(c) permit the limiting current to be dissected into an apparently steady-state component and a transient component. The former may be attributed to the effects of a thinned diffusion layer and to acoustic streaming whilst the latter reflects cavitational collapse at the electrode/solution interface (Fig. 42). At least for the conditions used in recording the data given in Fig. 43, consideration of the average current suggests that quantitatively, for the case of millimetre-dimensioned electrodes, the convective contribution may be dominant. The contribution of bubbles to sonovoltammetric currents may be seen best

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Fig. 43 (a) Cyclic voltammogram measured at a scan rate of 200 mV s⫺1 for the reduction of 1 mM Ru(NH3)3⫹ in aqueous 0.1 M KCl. (b) Sonovoltammogram 6 obtained in the presence of 33 W cm⫺2 ultrasound with a 1 mm platinum disc electrode in a face-on geometry with a horn-to-electrode distance of 7 mm. (c) Time-resolved current signal recorded as for (b) at potentials of 0.1 and ⫺0.5 V (vs. SCE). Data adapted from Marken et al. (1996a).

using microelectrodes, as first noted by Degrand and co-workers (Klima et al., 1995) and subsequently by Birkin (Birkin and Silva-Martinez, 1995). These experiments may offer much useful and detailed information about the statistics and dynamics of interfacial cavitation (Birkin and Silva-Martinez, 1996, 1997).

ELECTRODE CLEANING AND ACTIVATION

A notable benefit of the asymmetric collapse of cavitation bubbles at solid/liquid interfaces is the resulting cleaning effect which may be beneficially exploited in the context of the voltammetry of adsorbing or passivating systems. Various workers have reported electrode damage and/or roughening in the presence of ultrasound. Thus insonation of glassy carbon electrodes leads to their activation (Zhang and Coury, 1993) and AFM images of ultrasoundinduced surface modifications have been described (Marken et al., 1996b). For the case of platinum electrodes, substantial roughening was found, both

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through AFM visualization and by means of the fractal dimension of the electrode surface as interrogated through AC impedance spectroscopy (Compton et al., 1994). Similar data have been reported for gold electrodes (Compton et al., 1997b). The benefits of electrode erosion in preserving electrode activity have been seen in the cases of electrogenerated polymers (Madigan et al., 1994), where the ultrasound ‘‘punches’’ holes through the film, the deposition of reduced methylviologen, which can otherwise passivate an electrode surface under silent conditions (Benahcene et al., 1995), and the oxidation of Cr(CO)6, where insonation counters electrode poisoning (Compton et al., 1994). A major future application of sonovoltammetry may well lie in the field of electroanalysis where the ability to maintain electrode activity in ‘‘dirty’’ or otherwise passivating media may extend the range of applicability of such procedures. Reports of the benefits of insonation in anodic and adsorptive stripping voltammetry are just beginning to appear (Marken et al., 1997a; Matysik et al., 1997; Agra-Gutierrez and Compton, 1998).

ELECTRODE KINETICS

Electrochemical reversibility as opposed to quasi- or irreversibility requires electrode kinetics which are fast compared to the rate of mass transport to the electrode (Fisher, 1996). It follows that systems which display reversibility under silent conditions may deviate from this limit under insonation. Accordingly, just as microelectrodes assist the measurement of electron transfer kinetics, so insonation of large (millimetre scale) electrodes gives rise to mass transport coefficients of the order of 0.1 cm s⫺1 so that quite fast electrode kinetics may be measured from steady-state voltammograms. This has been exploited by various workers (Huck, 1987; Marken et al., 1995; Jung et al., 1997) under various sonovoltammetric regimes. For the case of simple one-electron transfer processes, similar results were usually found in comparison to alternative techniques (microelectrodes, fast-scan cyclic voltammetry, etc.) implying that the effect of ultrasound on electrochemical reversibility was indirect, and exclusively through modification of the mass transport, rather than by any influence on the rate of electron transfer. Multi-step processes require further study (Jung et al., 1997) whilst studies of electrode kinetic effects in bubble microjets have been presented (Birkin and Silva-Martinez, 1997).

COUPLED HOMOGENEOUS CHEMICAL REACTIONS

As seen above, just as insonated electrodes may be used to determine quite fast heterogeneous kinetics, so they may be used to probe coupled

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homogeneous kinetics. It is interesting that for the dehalogenation reactions described above, similar rate constants emerge from sonovoltammetry as from transient measurements under silent conditions. This implies that there is no sonochemical enhancement of the coupled chemical reactions in the electrode process; the influence of ultrasound is merely indirect through mass transport enhancement. This is not always the case; for example, it is possible that sonolysis of the solvent system can modify electrode processes and it is well known that OH⭈ radicals are generated from aqueous systems (Mason and Lorimer, 1988) and these may react with electrogenerated intermediates. In other cases specific sonochemical enhancements appear to have been observed (Degrand, 1986; Chyla et al., 1989; Mason et al., 1990; Walton et al., 1994).

7

Theoretical modelling

The difficult part of a voltammetric experiment is extracting the chemical information from the current. Apart from very simplistic analysis (such as the visual interpretation of voltammograms to count the relative number of electrons transferred in each process), the measured current cannot be directly interpreted. As with many experiments, a model must be constructed to predict the current for a given set of conditions and a postulated chemical mechanism. An electrochemical model is concerned with the concentration distributions of chemical species (and possibly the potential distribution if a background electrolyte is not used). If the concentration distributions of all the chemical species can be simulated, the current flowing at the working electrode may be calculated by integrating the concentration gradient at the electrode surface to give the total flux.

THE FOUR COMPONENTS OF AN ELECTROCHEMICAL MODEL

How does one go about constructing such a model? Three pieces of information are required to define the experimental system: kinetics, mass transport and experimental technique. Kinetics The rate of electron transfer and its potential dependence can be described by the Butler–Volmer equation (20) (see Section 2). An electron transfer often initiates a cascade of homogeneous chemical reactions by producing a reactive radical anion/cation. The mechanism can be described mathematically by a rate equation for each species; these form part of the electrochemical model. The rate law of the overall sequence is probed by the voltammetric experiment.

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Mass transport Each of the three mass transport components may be described mathematically, as discussed in Section 2. The effect of all three modes of mass transport may be summed giving the partial differential equation (PDE) (98), zA F ⭸c Dⵜ(cⵜE) ⫽ Dⵜ2 c ⫺ ␯ . ⵜc ⫹ RT ⭸t

(98)

where c is the normalized concentration of species A (c ⫽ [A]/[A]0). This equation describes how the amount of species A at a given point (the concentration) varies through time due to diffusion, convection and migration. This is the second major component of the electrochemical model. It depends only on the electrode geometry, the symmetry of which defines the Laplacian (ⵜ2) operator. Assuming a background supporting electrolyte is used in excess to eliminate migration effects from the experiment, only a convective– diffusion equation is necessary to describe the mass transport. Experiment The experimental technique controls how the mass transport and rate law are combined (and filtered, e.g. by removing convective transport terms in a diffusion-only CV experiment) to form the overall material balance equation. Migration effects may be eliminated by addition of supporting electrolyte; steady-state measurements eliminate the need to solve the equation in a time-dependent manner; excess substrate can reduce the kinetics from second to pseudo-first order in a mechanism such as EC⬘. The material balance equations (one for each species), with a given set of boundary conditions and parameters (electrode/cell dimensions, flow rate, rate constants, etc.), define an I–E–t surface, which is traversed by the voltammetric technique. If all of these are known, the concentration distributions of the species throughout the experiment may be described mathematically as a set of simultaneous partial differential equations. The way these equations are perturbed during the course of the experiment and the boundary conditions required to solve them may also be deduced from these three pieces of information. Solution method The last component of the model is a method to solve this system of (simultaneous partial differential) equations, often as a function of time as the concentration distributions evolve during the experiment. The difficulty of solving these systems depends on the complexity of the material balance

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equations and whether they are linked to each other by the kinetic terms. The main considerations for the complexity of the system are: • Is the PDE spatially one dimensional or multi-dimensional (e.g. CV is one dimensional; ChE is two dimensional)? • Is the PDE to be solved at steady state or in a time-dependent manner? • Can migration effects be ignored? • Are the PDEs coupled by the homogeneous kinetic terms? • Are the PDEs coupled at the electrode surface by heterogeneous kinetics (i.e. are electron transfers fast)? • Are the kinetic steps first or second order (giving rise to linear or non-linear PDEs)? • Are kinetics (or is flow) fast, resulting in a stiff system of equations (i.e. requiring small time steps so as to avoid large numerical errors)? For the simpler cases where a low (usually one) dimensional linear PDE may be solved in isolation, the system may be analytically tractable. For anything more than model problems at most practical electrode geometries, numerical methods are currently the only way by which the equation systems may be solved.

ANALYTICAL SOLUTIONS

For a simple electron transfer [see (1), (2)], it is possible to solve the diffusion equation analytically at steady state, as described for a microdisc by (91) and for a spherical electrode by (99). Ilim ⫽ 4␲nFD[A]0 re

(99)

For hydrodynamic electrodes, in order to solve the convective–diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with

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more complex mass transport, it is necessary to resort to numerical methods.

NUMERICAL SOLUTIONS

These all rely on discretization of a continuous quantity, usually an integral or a derivative, which may then be approximated by operating on the array of discrete values using the simple (⫹, ⫺, ⫻, ⫼) operators available in a computer’s instruction set. Network approach It is possible to represent the entire electrochemical system including the instrumentation (potentiostat, etc.) as a single electrical circuit. The solution is usually spatially discretized into a network of resistance elements (see for example Coles et al., 1996). Double-layer charging can also be incorporated into these models by defining each element to contain a capacitor as well as a resistor. Finite-difference method The space/time over which the problem is formulated is covered with a mesh of points, often referred to as ‘‘nodes’’. At each point, the derivatives in the material balance equation are approximated as differences of the concentrations at the given and surrounding points. This leads to a set of linear equations (based on a five-point stencil in two dimensions – each node is related to its four nearest neighbours) which can be solved to give the solution to the PDE. The methods are well suited to simulations in rectangular regions, which is often compatible with an electrochemical cell. These are by far the most popular methods for electrochemical simulations and will therefore be the focus of the remainder of this section. Finite-analytic method (Chen, 1984) The space is divided up into a mesh of (rectangular) elements. In each element it is assumed that the coefficients of the derivatives in the PDE may be treated as constants. If suitable boundary conditions are specified, the PDE can be solved in the local element using a separation of variables. The value in the particular element is related to that of its neighbours through the boundary conditions, leading to a set of linear equations based on a nine-point stencil in two dimensions (i.e. each element is related to its four nearest and four next-nearest neighbours) (Jin et al., 1996a,b).

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Orthogonal collocation (Villadsen and Stewart, 1967) This is an efficient technique based on polynomial curve fitting. Points are ‘‘selected’’ along the coordinate where the polynomial will fit the solution exactly. These points are the roots of an orthogonal set of polynomials. Derivatives in the material balance equation are replaced with the derivatives of the polynomial. Using some matrix algebra, the coefficients of the polynomials may be eliminated from the equations leaving a linear system of equations for the steady-state case, or a system of ordinary differential equations (ODEs) for a time-dependent simulation. The high efficiency of the method more than offsets the computational effort required to integrate the ODE at each collocation point. See Britz (1988) or Speiser (1997) for more details.

Finite-element methods The PDE is replaced by a quadratic functional (i.e. function of functions; well known for a large class of problems). The space is discretized into a number of elements (usually triangles). In each element, the concentration distribution is approximated by the weighted sum of a set of trial (interpolating) functions of the nodal values. The sum of these weighted interpolation functions forms the approximate solution to the problem and is substituted into the quadratic functional. This is minimized with respect to each unknown weighting factor by setting derivatives to zero. The functional then breaks up into a sum of integrals over each element which leads to a set of linear equations. One major advantage of finite element methods over finite difference methods is the way they naturally incorporate non-uniform meshes. They can therefore be applied to problems with a complex geometry (Stevens et al., 1997), for example elevated and recessed electrodes (Ferrigno et al., 1997), and, in principle, simulation of rough electrodes. On the downside, finite element methods are more complex to program, especially when simulating chemical steps, and result in a linear system of equations which is not neatly banded.

Commercial simulators Over the past few years, a few commercial electrochemical simulation packages have appeared; most are based on the finite difference method or orthogonal collocation. These packages are currently capable only of simulating mass transport in a maximum of one spatial dimension and are therefore practically restricted to modelling voltammetry at large planar and rotating disc electrodes. Speiser (1997) has made a thorough assessment of these in his recent review.

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Working curves and surfaces For many mechanisms, the steady-state E1/2 or Neff value is a function of just one or two dimensionless parameters. If simulations are used to generate the working curve (or surface) to a sufficiently high resolution, the experimental response may be interpolated for intermediate values without the need for further simulation. A free data analysis service has been set up (Alden and Compton, 1998) via the World-Wide-Web (htttp://physchem.ox.ac.uk:8000/wwwda/) based on this method. As new simulations are developed (e.g. for wall jet electrodes), the appropriate working surfaces are simulated and added to the system. It currently supports spherical, microdisc, rotating disc, channel and channel microband electrodes at which E, EC, EC2, ECE, EC2E, DISP 1, DISP 2 and EC⬘processes may be analysed.

FINITE DIFFERENCE SIMULATIONS

Consider a spatial coordinate, x, divided into NK points at a distance ⌬x apart; k is used to index a particular point (k ⫽ 1 . . . NK). The concentration gradient at point k in the grid depicted in Fig. 44 may be represented in three ways as shown in (100)–(102). dc uk ⫺ uk⫺l ⫽ ⌬x dx dc uk⫹1 ⫺ uk⫺l ⫽ dx 2⌬x dc uk⫹1 ⫺ uk ⫽ dx ⌬x

(100) (101) (102)

where uk represents the (normalized) concentration at the point k. The first, known as upwind differencing, centres the derivative at point (k ⫺ 1–2). The last (downwind differencing) centres the derivative at (k ⫹ 1–2). Only the central difference spanning both points either side results in a derivative centred at point k. If either upwind or downwind differencing are used to represent the first derivative, the offset introduces a second-order error known as numerical dispersion, which acts as an artificial viscosity or diffusion term. The upwind and downwind differences may be combined to give expression (103) for the second derivative which is centred at point k. dc d2 c dx ⫽ dx2

冨

dc dx ⌬x ⫺

(k⫹1/2)

冨

(k⫺1/2)

⫽

uk⫹1 ⫺ 2uk ⫹ uk⫺1 (⌬x)2

(103)

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Fig. 44 (a) Upwind, (b) downwind and (c) central finite differences. The grey nodes indicate those used in formulating the finite difference.

Consider a typical time-dependent mass transport equation, such as (104) for a microdisc electrode. ⭸2 c ⭸2 c 1 ⭸c ⭸c ⫽D ⫹ 2⫹ ⭸z2 ⭸r r ⭸r ⭸t

冤

冥

(104)

If this is approximated using (central) finite differences it becomes (105),

冤

t⫹1 t uj⫺l,k ⫺ 2uj,k ⫹ uj⫹l,k uj,k⫺l ⫺ 2uj,k ⫹ uj,k⫹l ⫺ uj,k uj,k ⫹ ⫽D ⌬t (⌬z)2 (⌬r)2

⫹

1 uj,k⫺l ⫺ uj,k⫹l 2⌬r k⌬r

冥

(105)

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t where uj,k denotes a concentration at point (j, k) and at time t. The equation may be written in the general form (106), t⫹l t ⫺ uj,k uj,k ⫽ Aj,k uj,k⫺l ⫹ Bj,k uj⫺l,k ⫹ Cj,k uj,k ⫹ Dj,k uj⫹l,k ⫹ Ej,k uj,k⫹l ⌬t

(106)

where Ajk . . . Ejk are known coefficients [D/(⌬z)2 etc. for the microdisc electrode]. There is a choice as to whether the concentrations on the right-hand side are chosen to be at t or t ⫹ 1. If concentrations at the old time (t) are used, we have an explicit equation (107). t⫹l t t t t t t ⫽ uj,k ⫹ ⌬t[Aj,k uj,k⫺l ⫹ Bj,k uj⫺l,k ⫹ Cj,k uj,k ⫹ Dj,k uj⫹l,k ⫹ Ej,k uj,k⫹l ] (107) uj,k

This can be solved as it stands; the concentration values at the new time are generated from those in the neighbourhood at the old time directly from this equation. On the other hand one could choose to represent the concentrations on the right-hand side of (106) as at (t ⫹ 1) in which case the resulting equation is implicit (108). t⫹l t⫹l t t⫹1 t⫹l t⫹l ⫽ uj,k ⫺ ⌬t[Aj,k uj,k⫺l ⫹ Bj,k ut⫹l uj,k j⫺l,k ⫹ Cj,k uj,k ⫹ Dj,k u j⫹l,k ⫹ Ej,k u j,k⫹l] (108)

This linear system of simultaneous equations may be written in matrix form as (109), b ⫽ [M]x

(109)

where the unknown, x ⫽ ut⫹l, b ⫽ ut (the known vector of concentrations) and [M] is a matrix of coefficients (composed of Ajk . . . Ejk for all j and k). This is not as straightforward as the explicit case, so why bother? The implicit equations are unconditionally stable, whereas the explicit equations break down if the time step is too large. Simulations based on implicit equations are therefore more efficient. The Crank–Nicolson method is a mixture of the implicit and explicit schemes which is less stable but more accurate than the fully implicit scheme (see Britz, 1988). A set of simultaneous equations analogous to those necessary for the fully implicit method must be solved; the matrix [M] has exactly the same structure in either case. Standard numerical methods may be used to solve the linear system; a number of these are listed in Table 6. Some of these, such as the Thomas algorithm and FIFD, take advantage of the structure of [M]. The linear solver is the critical engine behind the simulation. To illustrate this, consider that simulation of more challenging electrode geometries, such as the microband

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Table 6 Some methods for solving linear systems of equations. Gaussian elimination (LU factorization)

Thomas algorithm

Fast implicit finite difference

Gauss–Seidel iteration Successive overrelaxation (SOR) The strongly implicit procedure (SIP)

Multigrid

Preconditioned Krylov subspace

An automated way of directly solving a large set of simultaneous equations, by adding multiples of one equation to another until one equation can be solved, then back-substituting. The time taken is proportional to the square of the number of equations. For a tridiagonal linear system, Gaussian elimination simplifies to a simple algebraic factorization followed by back-substitution. The time taken is linearly proportional to the number of equations. A submatrix is formed at each point on the grid, relating the concentration of each species to the others (kinetically). The material balance equation for all the species may be written with this submatrix down the diagonal – resulting in a block tridiagonal matrix. This may be solved using a matrix version of the Thomas algorithm which requires each submatrix to be inverted (by LU factorization). A simple ‘‘textbook’’ iterative method for solving linear systems – not very efficient when compared with the methods below. The Gauss–Seidel method can be accelerated by calculating the next approximation and then deliberately ‘‘overshooting’’ it. Chebyshev acceleration (polynomial extrapolation) may also be used to improve the rate of convergence. This is an iterative method that calculates the next set of values by direct elimination. A ‘‘small’’ matrix [N] is added to the coefficient matrix [M] so that [M ⫹ N] is easily factored with much less arithmetic than performing elimination on [M]. An iteration parameter controls the ‘‘amount’’ of N added. The method is more economical and the convergence rate is much less sensitive to the iteration parameter than SOR. Subroutines for 2d and 3d SIP may be found in the NAG library (D03EBF and D03ECF). Multigrid (MGRID) methods are very efficient iterative methods for solving linear systems, requiring CPU time which is linearly proportional to the number of equations in the system. They work by approximately solving (‘‘smoothing’’) the problem on a fine mesh, then averaging the errors onto a coarser mesh and (fully) solving for a ‘‘correction term’’ which is then interpolated back up on to the finer mesh and added onto the approximate solution. The process of going up and down through several grid levels (known as a V-cycle) is repeated until the errors fall below the desired tolerance. The NAG library contains a multigrid subroutine, D03EDF. A very flexible group of iterative methods useful for solving sparse linear systems of equations, again available in the NAG library (section F11). A matrix [M] of any structure can be accommodated, yet (unlike Gaussian elimination) only the non-zero elements are operated on. This makes them a good choice for a ‘‘general’’ simulator where different sparsity patterns arise depending on the electrode geometry and mechanism.

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channel flow cell, may require a mesh of 1000 ⫻ 1000 points, resulting in a system of a million equations. If one were to use simple Gaussian elimination to solve such a system, one would be effectively inverting a matrix with 1012 elements requiring 12 days of CPU time (on an SGI Indigo2) and 1.5 ⫻ 107 Mb of memory (using 8-byte double-precision floating-point values)! If one used the multigrid subroutine in the NAG library, the same system could be solved in around a minute (requiring 109 Mb of memory). Most of the work done to date has focussed on systems where mass transport can be described in one spatial dimension (planar, spherical cylindrical and rotating-disc electrodes). In this case, an implicit finite difference discretization results in a linear system with a tridiagonal coefficient matrix, which may be solved directly using the Thomas algorithm. Bieniasz (1997) has produced a commercial package, ELSIM, which allows simulation of a wide range of voltammetric possibilities in one spatial dimension through the use of an ‘‘Electrochemical Simulation Language’’. The concentration profile may be independent of time and vary in more than one dimension; thus a two-dimensional or three-dimensional spatial problem results. Occasionally a system is encountered where rapid convection occurs perpendicular to the electrode surface so that diffusion is negligible in that coordinate. By rearranging equations (76) and (77) and normalizing the concentration, the mass transport to a ChE at steady state is given by (110). D

⭸2 c ⭸c ⫺ vx ⫽0 ⭸y2 ⭸x

(110)

This may be discretized as (111), uk ⫽ uk⫹l ⫺ (␭uj⫺l,k⫹l ⫺ 2␭uj,k⫹l ⫹ ␭uj⫹l,k⫹l)

(111)

(where ␭ ⫽ D/vx) which may be represented by the matrix equation (112) b ⫽ [Tri]x

(112)

(where [Tri] is a tridiagonal matrix) which may be solved using the Thomas algorithm. This space-marching method is known as the backwards implicit method, first applied to channel flow simulations by Anderson and Moldoveanu (1984). The method has been applied to a wide range of systems by Compton and co-workers for both channel (Compton et al., 1988c) and wall jet (Compton et al., 1990b) electrode geometries. The time-dependent mass transport equation (113) ⭸c ⭸2 c ⭸c ⫽ D 2 ⫺ vx ⭸y ⭸x ⭸t

(113)

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may be discretized as (114), t⫹l t t⫹l t⫹l t⫹l t⫹l uj,k⫹l ⫹ ␭V uj,k ⫽ ut⫹l j,k⫹l ⫹ ␭V u j,k⫹l ⫺ ␭D (u j⫺l,k⫹l ⫺ 2u j,k⫹l ⫹ u j⫹l,k⫹l)

(114)

(where ␭V ⫽ ⌬t . vx and ␭D ⫽ ⌬t . D). If this is solved for all k at t ⫹ 1, then, at t⫹l t ⫹ 2, etc., it still conforms to the matrix equation (109) since both uj,k and t uj,k⫹l are known and may be used to form b. This is known as the transient or time-dependent BI method (Fisher and Compton, 1991).

TWO-DIMENSIONAL SIMULATIONS

Most of the practical electrode geometries (microdisc, microband, channel, wall jet) require simulation of two spatial dimensions. Although a few early simulations used a simple explicit method (Britz, 1988), its relative inefficiency is compounded in multiple dimensions. Two ways of adding some implicit character to multidimensional simulations have been adopted: 1. The hopscotch algorithm (Gourlay and McGuire, 1971) works by solving every other node explicitly, then solving the nodes in between implicitly. The overall algorithm is unconditionally stable and thus time steps of any size may be used, but the accuracy of the explicit method is barely improved upon (Britz, 1988). The method was introduced into electrochemistry by Shoup and Szabo (1984) for microdisc simulations and was subsequently adopted by other electrochemists (Magno and Lavagnini, 1995). Feldberg (1987) points out that the method has disadvantages when simulations involve boundary singularities, for example in the simulation of potential step chronoamperometry. 2. The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a ‘‘share’’ of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981; Heinze and Störzbach, 1986) and has subsequently been adopted by others (Taylor et al., 1990; Fisher et al., 1997). The extra accuracy and efficiency of an implicit method may offset the CPU overhead of solving the linear system using an iterative method rather than the Thomas algorithm (as is used in hopscotch and ADI). This is especially true if only the steady-state solution is required (Alden and Compton, 1996a). SOR (Gavaghan, 1997), SIP (Compton et al., 1995c), multigrid (Alden and Compton, 1996b) and preconditioned Krylov subspace (Alden and Compton, 1997b) methods have all been used for this purpose. Of these, multigrid

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methods are the most efficient by a long way (Alden and Compton 1996b, 1997b).

CHEMICAL KINETICS

Chemical reactions couple the matrix equations for each species so they cannot be solved independently. The easy way around this is to approximate the kinetic terms explicitly (using concentrations at the old time), for example in an ECE mechanism species C is made from species B. The finite difference equation for species C could therefore use the concentration of species B from the previous time step as in (115). t⫹l C t⫹l C t⫹l t u ⫹ kB uj,k ⫽ Cuj,k ⫺ ⌬t[Aj,k Cut⫹l j,k⫺l ⫹ Bj,k u j⫺l,k ⫹ Cj,k uj,k

C t j,k

C t⫹l ⫹ Dj,k Cut⫹l j⫹l,k ⫹ Ej,k u j,k⫹l]

(115)

This approximation breaks down at high rate constants requiring ever smaller time steps to keep the simulation stable as the rate constant increases (Alden and Compton, 1997a). Rudolph’s FIFD method (Rudolph, 1991), essentially a block form of the Thomas algorithm, is an elegant way around this for simulations in one spatial dimension. It allows an arbitrary number of kinetic relationships to be simulated and forms the basis of the commercial package DigiSim (Rudolph et al., 1994). Alternatively, all the species can be collected together in a huge composite linear system. Britz (1996) solved this using direct Gaussian elimination and noted that this method would not scale at all well if applied to large problems. Recently Alden and Compton (1997a,b) have shown that such a system may be solved in a very efficient manner using preconditioned Krylov subspace methods and has formulated a scheme for the simulation of an arbitrarily complex reaction composed of first- and second-order homogeneous reactions and electrochemical couples. In order to cope with non-linearities arising from second-order kinetics in an implicit simulation, Newton’s method may be used for which the Jacobian may be derived analytically by differentiating the kinetic terms (Rudolph, 1992; Alden and Compton, 1997b).

BOUNDARY CONDITIONS

At each ‘‘edge’’ of the simulation space, a boundary condition must be supplied. The common ones are as follows. 1. No flux (Neumann boundary): Used where there is a physical barrier to mass transport (such as the wall of the cell) or downstream of a

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hydrodynamic electrode. Applies to the electrode surface for electroinactive species. 2. Bulk concentration (Dirichlet boundary): Used to represent ‘‘edges’’ of the simulation space where the solution has not been perturbed by electrolysis or ‘‘incoming’’ solution in a hydrodynamic cell. 3. Zero concentration (Dirichlet boundary): Used to represent the electrode surface under mass transport limited electrolysis. 4. Nernst or Butler–Volmer equation (Neumann boundary): Used to define the concentration ratio at the electrode surface when electrolysis is not transport limited. The Neumann boundaries (which involve derivatives) are converted into finite difference form and substituted into the finite difference equations for the nodes in the specified region (e.g. above the electrode surface). The Dirichlet conditions (which fix the concentration value) may be substituted directly.

CURRENT INTEGRATION

Once the concentration profile has been obtained, all that remains is to sum the concentration gradients at the nodes above the electrode to give the total flux from which the current may be calculated. The most basic strategy is simple rectangular integration (across the KE points that span the electrode); for a hemispherical electrode, the flux is given by (116).

冕

J⫽D

0

re

⌬r ⭸c . 2␲r . dr ⬇ 2␲D ⌬z ⭸z

冘 KE

(ul,k ⫺ u0,k)

(116)

l

However, more accurate integration schemes such as Trapezia, Simpson’s rule and Romberg integration (Press et al., 1992; Gavaghan, 1997) may give more accurate results for a given mesh. Cubic spline fitting has also been used (Lavagnini et al., 1991).

OPTIMIZATION

As already mentioned, two-dimensional simulations can be very CPU and memory intensive, so techniques which can improve the efficiency (i.e. accuracy for a given number of nodes) are important. Using higher-order Taylor expansions to represent derivatives is a possibility, but the only area where a significant improvement in accuracy may be achieved is in the evaluation of the flux when calculating the current (Britz, 1988; Gavaghan, 1997). A very intuitive improvement is somehow to move more of the finite

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Fig. 45 Coordinates for the simulation of a microdisc electrode: (a) real space (r, z); (b) Michael et al.’s (1989) conformal mapping; (c) Amatore and Fosset’s (1992) closed-space conformal mapping.

difference nodes into regions where the concentration gradient is steep. This is commonly achieved by transforming one or more coordinates and solving the system in the transformed (conformal) space. An exponentially expanding grid in the coordinate normal to the electrode surface usually offers a significant improvement in the simulation accuracy. The choice of function along the electrode surface is less straightforward; a Fermi–Dirac function has been used for simulating microdisc electrodes (Taylor et al., 1990) and a tanh function has been used for the channel electrode (Pastore et al., 1991). Conformal mappings which transform the problem into a natural space are even better, since these often remove boundary discontinuities which are major impediments on the simulation accuracy (Gavaghan, 1997). Deakin et al. (1986) developed a conformal mapping in oblate spheroidal coordinates for the concentration distributions of species near the surface of a microband electrode. This was later extended to double- and triple-band arrangements (Fosset et al., 1991a,b). An analogous transformation was used by Michael et al. (1989) for the microdisc electrode, shown in Fig. 45. This has been refined to a closed-space mapping shown in Fig. 45(c) (which is better for steady-state simulations) by both Amatore and Fosset (1992) and Verbrugge and Baker (1992). Conformal mappings which transform the problem into natural space (Fig. 45a) have also been used for hydrodynamic electrodes; the Hale transformation for the rotating-disc electrode (Hale, 1963; Compton et al., 1988d) and a convection-based transformation for the wall jet (Ball and Compton, 1998).

8

A comparison of voltammetric techniques

Sections 3–6 described four voltammetric techniques that may be used in order to analyse a variety of electrode reactions. This section aims to compare the merits of these forms of voltammetry with regard to the mechanistic

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analysis of processes consisting of combinations of heterogeneous and homogeneous chemical steps. The practical, theoretical and kinetic accessibility aspects of each technique will be focussed upon.

A QUANTITATIVE COMPARISON OF THE KINETIC DISCRIMINATION OF COMMON ELECTRODE GEOMETRIES AT STEADY STATE

The experimental technique and electrode geometry should be selected to match the kinetic time-scale (the time domain over which a chemical process occurs, e.g. 1/k, where k is a first-order rate constant) of the reaction being studied. This is achieved by varying the rate of mass transport via convection, electrode size/shape or potential scan rate. Table 7 shows the characteristic time-scale parameter for a number of common electrode geometries which reach a steady state. The quadratic

Table 7 Characteristic time-scales for common electrode geometries. Electrode geometry Sperical Sonotrodes Microdisc Rotating disc1* Wall-Jet2* Micro-Jet3* Channel* *

Peclet number (Ps)

Characteristic time-scale (tc) re2 D ␦2 D re2 D

None Unknown None Ps ⫽ Sc . R3/2 e CWJE r3/4 e D CMJE re3 Ps ⫽ D 3 Vf xe Ps ⫽ 2 Dd h Ps ⫽

冢 冣

2

re2 (Ps)⫺2/3 D r2e (Ps)⫺2/3 D r2e (Ps)⫺2/3 D xe2 (Ps)⫺2/3 D

Using a first-order approximation for the convection term. For the rotating-disc electrode the Reynolds number is given by Re ⫽ r2e ␻/␯; the Schmidt number, Sc ⫽ ␯/D. 2 For the Wall-Jet electrode the hydrodynamic constant CWJE is given by CWJE ⫽ [125M 3/(216␯5)]1/4. The constant M is given by M ⫽ k4c Vf3/(2␲3 r2jet); kc is an experimentally determined constant. 3 For the microjet electrode the hydrodynamic constant CMJE is given by CMJE ⫽ [1.51(H/a)⫺0.054 (U/a)1/2]3 ␯⫺1/2; a is the nozzle diameter, H is the distance from the nozzle to the electrode; U is related to the volume flow rate by Vf ⫽ ␲(d/2)2 U. 1

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dependence of the time-scale on the electrode radius is the reason that microelectrodes are so significant. Note the similar nature of the expressions for hydrodynamic electrodes where only the magnitude of the Peclet numbers (the Peclet number is the ratio of mass transport by convection to that by diffusion) differs. It can be seen that as the Peclet number increases (i.e. rate of convective transport increases) a faster kinetic time-scale may be accessed. Dimensionless homogeneous rate constants, K, and time parameters, ␶, may be formulated in terms of the characteristic time-scale, tc, using (117) and (118) respectively, K ⫽ k . tc

(117)

␶ ⫽ t/tc

(118)

where k is the homogeneous rate constant and t is the time. The dimensionless homogeneous rate constant, K, is the unique parameter on which the steady-state Neff or E1/2 value depends. The so-called ‘‘working curve’’ of E1/2 or Neff vs. log K completely defines the steady-state behaviour for a particular mechanism at a specific electrode geometry (assuming a first-order approximation for convection when considering hydrodynamic electrodes). It has long been argued (Compton and Unwin, 1986) that electrode geometries that are non-uniformly accessible should have a greater inherent kinetic discrimination than uniformly accessible electrodes, since in the former the kinetic time-scale changes across the electrode surface, which should lead to a ‘‘stretching’’ of the working curve. Comparison of working curves for a wall jet and a rotating-disc electrode for an ECE reaction (Compton et al., 1991b), seems to support this argument. However, Unwin and Compton (1988) showed that the inherent kinetic discrimination of rotating-disc and channel electrode was virtually identical for first-order processes, although they postulated that this situation might change for a second-order process. The recent availability of working curves and surfaces for a range of common mechanisms at a number of electrode geometries (Alden and Compton, 1997a) allows a broad quantitative comparison of the kinetic discrimination of common electrode geometries for both first- and secondorder homogeneous processes. Table 8 shows the approximate range of time-scales and rate constants (for ECE and EC2E reactions) that may be measured by steady-state voltammetry using various electrode geometries. The range of rate constants was calculated from the values of the dimensionless rate constant which gives values of 1.1 and 1.9 for Neff from a working curve for each geometry. These values are generally agreed upon as suitable limits between which a kinetic process is voltammetrically ‘‘visible’’ (Amatore and Savéant, 1978). The calculations are based on the following (typical) experimental parameters:

Table 8 A comparison of the kinetic time-scales accessible with steady-state voltammetry using common electrode geometries. Electrode geometry

Range of time-scales (tc) accessible

Range of log dimensionless rate constant (K)

Range of rate constants (k) which can be measured

ECE

EC2E

ECE/s⫺1

EC2E/mol⫺1cm3s⫺1

Hemispherical Microdisc Rotating disc Wall jet Microjet

400 ␮s–5 s 400 ␮s–5 s 0.2–10 s 1 ms–100 s 10 ␮s–5 ms

3.82 3.93 2.47 3.03 *

4.31 4.47 3.71 3.93 *

2 ⫻ 10⫺3–2 ⫻ 105 6 ⫻ 10⫺3–7 ⫻ 105 9 ⫻ 10⫺2–1 ⫻ 103 9 ⫻ 10⫺5–5 ⫻ 105 *

6 ⫻ 103–2 ⫻ 1012 1 ⫻ 104–6 ⫻ 1012 6 ⫻ 104–2 ⫻ 1010 4 ⫻ 101–8 ⫻ 1012 *

Channel Conventional Microband Fast flow Sonovoltammetry at large electrode

0.1–10 s 3 ms–0.5 s 10 ␮s–10 ms 1 ms–0.1 s

2.45 2.75 2.45 2.46

3.43 3.47 3.43 *

1 ⫻ 10⫺2–5 ⫻ 102 2 ⫻ 10⫺1–1 ⫻ 106 20–4 ⫻ 106 4–1 ⫻ 105

1 ⫻ 104–4 ⫻ 109 3 ⫻ 105–4 ⫻ 1011 1 ⫻ 107–3 ⫻ 1013 *

*No working curves/surfaces available.

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• A typical diffusion coefficient of 1 ⫻ 10⫺5 cm2 s⫺1 for a non-aqueous solvent was used, together with a concentration of 1 ⫻ 10⫺6 mol cm⫺3. Kinematic viscosities for common solvents are in the range 1 ⫻ 10⫺3– 1 ⫻ 10⫺2 cm2 s⫺1. • Commercially available microdisc electrodes of radii 0.6–70 ␮m may be used for steady-state measurements without problems associated with natural convection. Dimensionless rate constants for spherical and microdisc electrode were interpolated from the working curves of Alden and Compton (1997a). • Hemispherical electrodes are experimentally realized using hanging mercury drops for macroelectrodes and mercury-coated microdisc electrodes for microhemispheres. The lower radius limit is thus governed by the microdisc radius; the upper limit has been chosen as 70 ␮m above which natural convection becomes significant. • For the RDE, the operating range of rotation frequency is between approximately 1 and 50 Hz and a typical radius is 0.25 cm. Dimensionless rate constants were interpolated from working curves generated from a fully implicit simulation using preconditioned Krylov subspace methods (Alden, unpublished work). • In the case of the WJE, experimental flow-rates are in the range 1 ⫻ 10⫺3–1 cm3 s⫺1 with a typical jet diameter 0.3 mm impinging on an electrode of radius 0.1–1 cm. Dimensionless rate constants were interpolated from the working curves simulated using the backwards implicit method, in agreement with those of Compton et al. (1990b). Macpherson et al. (1994, 1995) have recently miniaturized the WJE to a uniformly accessible microelectrode system (termed a MicroJet electrode). Flow rates are reported in the range 2 ⫻ 10⫺3–5 ⫻ 10⫺2 cm3 s⫺1 through a 30–60 ␮m radius nozzle at distances varying from tens to hundreds of micrometres from the microdisc electrode. • For the ChE, the following typical parameters were used: d ⫽ 0.6 cm; 2h ⫽ 0.06 cm; w ⫽ 0.4 cm; xe ⫽ 0.1–0.4 cm; Vf ⫽ 1 ⫻ 10⫺3–0.3 cm3 s⫺1. The smallest microband which could be fabricated reliably (xe ⫽ 1 ␮m) was used. Dimensions for the fast-flow cell are: d ⫽ 0.2 cm; 2h ⫽ 0.01 cm; w ⫽ 0.15 cm. This can accommodate electrodes of 1–100 ␮m and flow rates of 1 ⫻ 10⫺2–2.5 cm3 s⫺1. Dimensionless rate constants were interpolated from the surfaces of Alden and Compton (1998). • For sonovoltammetry, for a typical diffusion coefficient (1 ⫻ 10⫺5 cm2 s⫺1) the diffusion layer is reduced to around 1–10 ␮m, corresponding to a typical ultrasound power range of 10–60 W cm⫺2. Using a simple diffusion-layer model, the time-scale may be calculated from tc ⫽ ␦ 2/D. The results are summarized graphically in Figs 46–48. The overall rate constant ‘‘window’’ (Fig. 48) of each geometry is the product of the range of

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Fig. 46 Time-scales accessible by steady-state voltammetry at common electrode geometries.

kinetic visibility at a particular geometry (Fig. 47) and the range of time-scales that can be accessed (Fig. 46). It is clear from Fig. 47 that the hydrodynamic electrodes have a narrower kinetic ‘‘window’’ (i.e. less inherent kinetic discrimination) than diffusion-only systems, but convection allows faster time-scales to be accessed so the effect is more than offset. Comparing the hydrodynamic electrodes, the ECE data support the observations of Compton et al. for the wall jet and Unwin and Compton for the rotating disc. The conventional channel using a macroelectrode and fast-flow channel both operate in the limit of negligible axial diffusion. The enhanced kinetic discrimination of the channel microband electrode arises from using a small microband at slow flow rates thus incurring a significant amount of axial diffusion. As discussed in Section 6, sonication may be used to increase the rate of mass transport at steady state, essentially by providing convection to a macroelectrode. The time-scale of 1 ms is comparable with the smallest time-scale attainable using a wall jet electrode. Both methods provide a way of achieving microelectrode-like mass transport rates at large electrodes, though the well-defined hydrodynamics of the wall jet make this the better choice if enhanced mass transport is the sole objective. The complexity of the modelling method may influence the choice of geometry. This is summarized in Table 9, together with suggested finite difference methods which should provide efficient simulations.

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(a)

Amount of kinetic discrimination (log KECE range)

(b)

Amount of kinetic discrimination (log KEC2E range)

Fig. 47 Kinetic ‘‘visibility’’ at common electrode geometries for (a) an ECE process and (b) an EC2E process.

STEADY STATE VS. TRANSIENT EXPERIMENTS

In linear sweep or cyclic voltammetry, high scan rates can be used to shorten the time-scale below the steady-state time-scale arising from diffusion or convective diffusion. Typically scan rates of 10–3000 Vs⫺1 are used, corresponding to time-scales of 1–10 ␮s, although in fast-scan CV they may reach

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(a)

(b)

103

log k

log k

Fig. 48 Range of rate constants that can be measured by steady-state voltammetry at common electrode geometries for (a) an ECE process and (b) an EC2E process.

1 ⫻ 106 V s⫺1, giving time-scales of 10 ns (Wipf and Wightman, 1988). Note the time-scale accessible by cyclic voltammetry is given by the expression RT/F␯ (Bard and Faulkner, 1980). The price paid for trying to access shorter time-scales is precision. Analogous problems arise with precision when steady-state methods are pushed to their limits (e.g. turbulent flow conditions associated with extremely

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Table 9 The ‘‘complexity’’ of the mass transport at various electrode geometries together with suggested methods for efficient finite difference modelling. Geometry

Diffusion

Convection

Planar Spherical Disc Band Sonovoltammetry

1D 1D 2D 2D 1D

⬜

Rotating disc Channel Microband channel Wall jet

1D 1D 2D 1D

⬜ 储 储 储⬜

Suggested modelling methods CN/FIFD CN/FIFD ADI/(MGRID)/PKS ADI/(MGRID)/(PKS) Uniform diffusion layer model (see Section 6) CN/FIFD BI (BI-FIFD) ADI/MGRID/PKS BI (FIFD-BI)

储 ⬅ parallel to electrode surface; ⬜ ⬅ towards electrode surface; CN ⬅ time-marching Crank– Nicolson; MGRID ⬅ Multigrid; BI ⬅ space marching fully implicit or Crank–Nicolson; BIFIFD ⬅ a space-marching FIFD method; modelling methods in parentheses have not yet been applied to that particular geometry.

high flow rates). Often in transient methods, such as chronoamperometry or cyclic voltammetry, the current changes very rapidly with time; a small error in the measurement of the time at which the current is sampled could introduce a large error in the calculated rate constant. This is particularly an issue when using transient methods with microelectrodes to probe very fast time-scales. The response of the recording device (oscilloscope, A/D converter, etc.) must be significantly faster than the experimental time-scale. Transient methods are also complicated by capacitive currents after large jumps in, or rapidly scanning, the potential. Some of the practical considerations relating to the methods in this review are outlined in Table 10. 9

Current and future directions of voltammetry

This review has examined four major forms of DC voltammetry and their scope for application in kinetic studies. Current developments in the area can be broadly divided into four main categories: modification of voltammetric instrumentation, development of new electrode designs and materials, the application and improvement of voltammetric simulations and investigations in alternative chemical environments. We now examine a number of examples of developments in these four areas. INSTRUMENTATION

As discussed in Section 3, the upper scan-rate limit of a cyclic voltammetric experiment is critical for the examination of fast chemical kinetics and much

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recent attention has been devoted to this area. Wipf and Wightman (1988) used deconvolution techniques in the Fourier domain to remove instrumental distortion of voltammograms obtained at scan-rates in the 105–106 V s⫺1. This approach has also been followed by Andrieux et al. (1988) who used such techniques to separate the Faradaic current from the capacitative charging current. Amatore and Lefrou (1992) have developed potentiostats with the capability of on-line compensation of IRu, which is a hindrance to obtaining quality voltammetric data at fast scan rates. Commercial potentiostats are currently available that claim that scan rates in the kilovolt range can produce reasonable voltammetric data, and it is envisaged that such equipment will become more widely available. In addition, commercial voltammetric equipment is becoming more and more advanced with a range of both DC and AC techniques available to the operator on the one piece of equipment (see, for example, Kissinger, 1994). As discussed in Section 4, instrumentation is being developed that allows access to faster kinetic studies through high rates of convective mass transport. Examples include a fast flow cell (Rees et al., 1995a,b), a microjet electrode (Martin and Unwin, 1995) and a rotating microdisc electrode (White and Gao, 1995). ELECTRODES

Dimensions A wealth of papers have been published recently that describe techniques for the fabrication of smaller and smaller microelectrodes, thus enabling the analysis of more rapid kinetic time-scales. Wong and Xu (1995) describe the fabrication of 100 nm carbon disc electrodes in which high-pressure methane is forced through a micropipette tip and pyrolysed at the outlet forming the carbon electrode surface. Tschunky and Heinze (1995) describe the preparation of shielded micrometre-diameter microelectrodes that pick up vastly reduced levels of stray capacitance associated with fast-scan experiments. Laser micromachining has been used to generate arrays of carbon-ink microelectrodes (Seddon et al., 1994). Demaille et al. have fabricated self-assembled spherical gold ultramicroelectrodes at the end of very fine glass capillaries (Demaille et al., 1997). Lithographic techniques have been used by a variety of workers to fabricate microelectrodes (see for example Compton et al., 1993a and Reimer et al., 1995). In addition, commercially available electrodes of progressively diminishing dimensions are becoming available through a range of companies. Arrays of microelectrodes are being developed which may have individually addressable elements. For example, Kakerow et al. (1994) have produced arrays with as many as 400 individually accessible working microelectrodes in a chip with centimetre dimensions. These advances coupled with the

Table 10 Practical considerations for the methods discussed in this review. Parameter

CV

Micro

Hydrodynamic

Precision

Time errors; capacitive currents at high scan rates

Reduced IRu drop; significant errors in electrode dimensions

Modelling

Time-dependent; high dimensional working surfaces

Two-dimensional diffusion requiring sophisticated solvers

Hybrid methods, e.g. spectroscopic detection or photoactivation

A relatively large cell is required to maintain bulk concentration – may present practical difficulties

Reduced cell size is compatible with most apparatus. May pick up electrical noise from other equipment

Flow profile is perturbed by electrodes which are not flush or smooth Generally steady state. Apart from RDE, models are two-dimensional due to convection EPR, spectroscopic and photochemical methods are easily incorporated into the small, transparent channel flow cell. RDE is less versatile

Sono Noisy results due to inherent turbulence of the system Generally steady state, uniformly accessible

Sensitive apparatus may be damaged by ultrasound or pick up a lot of noise

Electrode: practical considerations

Static solution means that passivating films may deposit on the electrode. Electrode is easily removed and cleaned, however. Electrolysis products may build up in the bulk solution. Natural convection may become significant at low scan rates

Apparatus considerations

Ohmic drop compensation necessary at high scan rates

Difficult to fabricate desired electrode dimensions accurately – electrode elevation and recession is much more significant. Lithographically fabricated microbands are very fragile, cannot be cleaned mechanically. Natural convection may interfere with diffusion-only responses of larger electrodes (⬎10 ␮m) Small currents require high amplification and shielding to eliminate noise

If electrode becomes passivated and mechanical cleaning is necessary, the cell must be disassembled (introducing cell height or electrode–jet distance variation). For ChEs by-products formed at the counter electrode cannot reach the working electrode

Electrode surface must be resistant to ‘‘erosion’’ by sonication. Electrode passivation is reduced by cavitational cleaning.

Flow-regulation apparatus must be calibrated. Cell must ideally be designed to ensure laminar flow in order to simplify modelling. RDE rotation must have stable frequency and be axially symmetric

Horn must be electrically insulated, requiring a bipotentiostat

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development of data collection techniques will lead to access to shorter kinetic time-scales. Materials New electrode materials are beginning to emerge. Xu et al. (1997) describe the development of boron-doped diamond electrodes. The properties of these electrodes are critically dependent upon the level of boron doping; at low levels, the material is an insulator and at high levels the material is semimetallic. Such electrodes have been fabricated using vapour deposition techniques. Major advantages of such electrodes include the exhibition of very low background charging currents (an order of magnitude lower than glassy carbon), enabling them to be used in conjunction with fast scan rates, and the access to a wide potential window in aqueous media [ca. ⫹1.0 V to ⫺1.7 V (vs. SCE)] (Xu et al., 1997). Porous electrodes are gaining popularity especially in sensing applications. Matsuda et al. (1993) describe the fabrication of microporous gold films (pores had a diameter of around 100 nm) from an alumina template. A carbon composite electrode in which carbon is dispersed in a silica matrix has been described by Tsionsky et al. (1994). One of the inherent problems associated with any heterogeneous technique such as voltammetry is the reproducibility of the properties and nature of an electrode surface. Traditionally, studies in which this was crucial utilized a hanging mercury drop, or dropping mercury electrode which ensured a continuously renewable surface. Cardwell et al. (1996) have described improved techniques for polishing electrodes which will go some way towards providing more reproducible electrode surfaces. Reproducibility may be assisted by the development of disposable electrodes (for example, Wang and Chen, 1994) that have developed disposable enzyme microelectrode array strips for glucose and lactate detection. VOLTAMMETRIC SIMULATIONS

Simulation in two dimensions still represents a significant challenge. New expanding grid functions and conformal mappings need to be developed to improve the convergence of these systems. Certain geometries, such as a microdisc electrode in a channel flow cell, require a full three-dimensional simulation. Tait et al. (1993), Booth et al. (1995) and Fisher et al. (1997) have conducted simulations on a model system, though limitations in memory and CPU forced these to be conducted with as few as 20 nodes in one coordinate. Numerical analysts are trying to develop new linear solvers which will combine the flexibility of the preconditioned Krylov subspace methods with the speed of multigrid methods. These would allow simulations involving a large number of interrelated chemical species in a minimal amount of CPU time and threedimensional simulations of real systems to be tackled at a higher resolution. In

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addition, theoretical calculations will become more advanced and enable more chemical phenomena to be considered, e.g. adsorption effects, surface-bound chemical reactions and migration currents (Oldham, 1992; Meunier-Prest and Laviron, 1996; Aixill et al., 1997; Bieniasz, 1997). In addition, with the development of electrodes with dimensions approaching the nanometre regime, the sizes of the electrical double layer and diffusion layer will become comparable. Whereas the simulation techniques described in Section 7 have assumed no coupling of these layers, this will no longer be appropriate in the case of nanoelectrodes, and the resulting development of more sophisticated theories and resultant models will be required. It is envisaged that artificial intelligence will gain popularity in its application to electrochemical problems. Recently genetic algorithms have been applied to a wide variety of chemical areas (for a general text consult Cartwright, 1993) such as the indexation of crystal lattices (Tam and Compton, 1995), molecular conformation determination (McGarrah and Judson, 1993) and design of molecules for applications as therapeutic drugs (Glen and Payne, 1995). It is believed that, ultimately, packages may be available which will enable an optimized mechanistic determination to be made without operator intervention using genetic algorithmic or simulated annealing techniques (see, for example, Shaffer and Small, 1997).

INVESTIGATIONS IN ALTERNATIVE CHEMICAL ENVIRONMENTS

The bulk of the work described in this review was conducted in the solution phase and many fundamental and applied voltammetric studies have focussed upon this area. However, recent fundamental mechanistic work has deviated from this traditional domain. Kulesza and Faulkner (see, for example, Kulesza et al., 1991; Kulesza and Faulkner, 1993) have described voltammetry conducted upon solid materials such as polyoxotungstate molecules and report similar voltammetric phenomena to occur as are observed in the solution phase providing the molecules contain some form of mobile ion and solvents of crystallization. Bond et al. have described the voltammetry of a range of organometallic microcrystals mechanically attached to an electrode surface and placed in an aqueous electrolytic medium (Bond et al, 1993; Bond and Marken, 1994; Shaw et al., 1996). In fact, recent studies by Eklund (Eklund et al., 1999) have shown that photoelectron transfers occur between molecules mechanically attached to an electrode surface in an analogous manner to the solution phase. Thus, it is envisaged that following upon the success of these initial investigations more and more voltammetric studies will be conducted upon solid materials. Similarly, voltammetric investigations are beginning to emerge in supercritical fluids (Olsen and Tallman, 1994) and emulsions (Marken et al., 1997b). In all other areas of physical chemistry, experiments are conducted without

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added electrolyte. The presence of electrolyte complicates the experiment, modifies the solution medium and may play a role in the electrode mechanism. Consequently, voltammetrically derived mechanistic data may not be directly comparable with information obtained using other techniques. Improvements in instrumentation and electrode design should make it possible to reduce the amount of deliberately added electrolyte to the millimolar level and eventually it may be possible to avoid the need to add electrolyte at least for the case where the electroactive species are charged. At the moment only microelectrode techniques are widely used with minimal electrolyte but in the future we suggest that all the techniques described in this review may be conducted in the presence of very low levels of supporting electrolyte. If the voltammetric data may be obtained with minimal, or even in the absence of, added electrolyte the results may be of interest to a wider range of chemists and application of voltammetry in a range of media (e.g. wastewater, seawater, oil) will become more widely embraced.

Appendix: list of symbols and abbreviations A a [A] [A]0 [A]x⫽0 ADI AFM BI BIFD C ChE CN CV C⬘ CWJE CMJE c D DA d DISP DMF E E

electrode area nozzle diameter of wall jet electrode concentration of a general chemical species, A bulk concentration of a general chemical species, A concentration of a general chemical species, A, at the electrode surface alternating direction implicit atomic force microscopy backwards implicit backwards implicit finite difference chemical step (Testa and Reinmuth, 1961) channel electrode Crank–Nicolson (method for solution of differential equations) cyclic voltammogram/voltammetry double-layer capacitance per unit area wall jet electrode hydrodynamic constant microjet electrode hydrodynamic constant normalized concentration of a species i.e. for A, cA ⫽ [A]/[A]0 diffusion coefficient diffusion coefficient of general chemical species, A channel cell width disproportionation dimethylformamide electrode potential electron-transfer step (Testa and Reinmuth, 1961)

MODERN VOLTAMMETRY

Ecell Eapp Eljp E0 E0f Eox P Ered P E1/2 EPR F FIFD f Fc Fc⫹ H h I Ilim IPox IPred It Id Im Ic J Jd Jm Jc j K KE k k k0 ket red et kox le MGRID Neff NK n

111

electrode potential of an electrochemical cell applied electrode potential liquid junction potential standard electrode potential reversible formal potential peak oxidative potential in a cyclic voltammogram peak reductive potential in a cyclic voltammogram half-wave potential of a steady-state voltammetric wave electron paramagnetic resonance Faraday constant (96 485 C mol⫺1) fully implicit finite difference electrode rotation frequency ferrocene ferrocene cation distance from nozzle to electrode in a wall jet electrode channel flow cell half-height electrode current mass transport limited current peak oxidative current in a CV peak reductive current in a CV total electrode current diffusional electrode current migrational electrode current convectional electrode current flux diffusional flux migrational flux convectional flux spatial coordinate used in finite difference grids dimensionless normalized rate constant total number of points along the electrode surface in a finite difference grid general term for a homogeneous rate constant (first or second order) spatial coordinate in finite difference grids standard heterogeneous rate constant reductive heterogeneous electron-transfer rate constant oxidative heterogeneous electron-transfer rate constant lead in length to channel electrode multigrid effective number of electrons transferred in an ECE or DISP process number of points in the x-coordinate on a finite difference grid number of electrons transferred in an electron-transfer process

112

na Ps PDE PTFE q R Ru Re r r re r1 RDE Sc SCE T t t tc U uk

uj,k

Vf v vr v␪ vx vy vz v0 WJE w x x xe y

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number of electrons transferred in the rate-determining step of a multi-electron transfer process Peclet number (ratio of convective to diffusive transport) partial differential equation polytetrafluoroethylene charge universal gas constant (8.314 J K⫺1 mol⫺1) uncompensated solution resistance Reynolds number distance from the centre of a disc electrode co-ordinate parallel to a disc electrode (rotating-disc electrode or microdisc electrode) radius of an electrode radius of a wall jet electrode rotating-disc electrode Schmidt number saturated calomel electrode temperature time temporal coordinate in voltammetric simulations characteristic time-scale of voltammetric experiments related to volume flow rate of a microjet electrode (⫽Vf /␲a2) normalized concentration of a general species in an electrode reaction mechanism at a point k in a one-dimensional finite difference grid normalized concentration of a general species in an electrode reaction mechanism at a point j,k in a two-dimensional finite difference grid volume flow rate in a channel flow cell kinematic viscosity of a fluid general solution velocity solution velocity in r-direction (for a rotating disc) solution velocity in ␪-direction (for a rotating disc) solution velocity in x-direction in a ChE solution velocity in y-direction in a ChE solution velocity in z-direction in a ChE solution velocity at the centre of the channel unit (y ⫽ h) wall jet electrode width of a band electrode general spatial coordinate coordinate along the ChE length of a band or macroband electrode in the direction of the flow general spatial coordinate

MODERN VOLTAMMETRY

z x y zi ␣ ⌬r ⌬t ⌬z

␦ ␭ ␭v ␭D ␪ ␶ ␯ ␻ ⵜ2

113

spatial coordinate perpendicular to the ChE perpendicular distance from a rotating or microdisc electrode spatial coordinate perpendicular to a rotating or microdisc electrode spatial coordinate across a ChE charge on species i charge-transfer coefficient distance between grid points in the r-direction in voltammetric simulations time spacing between time counters in voltammetric simulations distance between grid points in the z-direction in voltammetric simulations mean thickness of a diffusion layer parameter used in voltammetric simulations (D/vx) parameter used in voltammetric simulations (⌬t . vx) parameter used in voltammetric simulations (⌬t . D) angular coordinate in rotating-disc voltammetry normalized/dimensionless time potential scan rate rotating-disc electrode rotation speed Laplacian operator

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Organic Materials for Second-Order Non-Linear Optics J. JENS WOLFF Organisch-Chemisches Institut der Ruprecht-Karls-Universität, Heidelberg, Germany AND

RU¨DIGER WORTMANN Physikalische Chemie der Universität, Kaiserslautern, Germany

1 2

3

4

5 6

Introduction 122 Basics of non-linear optics 124 Introduction 124 Maxwell’s equations 125 Constitutive relations 125 Linear optics: linear susceptibility (␹(1)) 126 Non-linear optics: non-linear susceptibilities (␹(n)) 128 Molecular polarizabilities 133 Quantum-chemical basis for second-order polarizabilities 136 Perturbation theoretical expressions 136 Molecular design and two- and three-level contributions 138 Molecular orbital calculations for second-order polarizabilities of ␲ systems 141 Qualitative trends for ␤ in simple model systems: two-centre model system; one-dimensional NLO-phores 143 Three-centre model system (C2␯ and D3h symmetry) 145 Local field factors and effective polarizabilities: the reaction field model 148 Non-linear optical susceptibilities and experimental methods to evaluate ␹(2) and ␤ 153 Macroscopic susceptibilities and molecular polarizabilities 155 Experimental determination of molecular second-order polarizabilities 161 Optimization of second-order polarizabilities: applications to real molecules 168 ␴ Systems and one-dimensional ␲ systems 168 Two-dimensional (2D) NLO-phores: 1D and 2D architecture 196 Conclusion 206 Acknowledgements 208 References 208 121

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Introduction

Over the past 15–20 years, the non-linear optical (NLO) response of organic molecules has been of wide interest to organic and physical chemists, as well as physicists and chemical engineers. This interest arises from the possible use of such organic materials. In principle, photonics or the hybrid technology optoelectronics, both based on non-linear optical phenomena, can supersede electronics in communication technology. These technologies allow many operations that are currently effected with electron conduction to be replaced by operations based on light conduction. In practice, the advantages would be tremendous, and many promises have been made in this area. Amongst them is a substantially increased rate of data transfer, combined with a high signal-to-noise ratio even over large distances, and miniaturization of some devices. These promises fit well into the current vogue that cherishes the construction of an ‘‘information highway’’ as the key element of future society. Organic materials are believed to have advantages over more traditional inorganic ones for reasons beyond their ease of production and low cost. The bulk properties of organic materials arise from the vectorial addition of molecular properties that are purely electronic in origin. Their response to an applied AC electric field is very fast (femtoseconds). However, the NLO response of inorganic ferroelectric crystals like LiNbO3 rests at least in part on the displacement of ions. The inherent response time of such displacements is much higher and lattice vibrations are also stimulated. Thus, highly undesirable acoustic ringing is created (Bosshard and Günter, 1997). However, it is a truism in materials science that the pathway from vision to reality or from an idea to a marketable product is hardly ever as straightforward as it may seem. The field of organic materials for non-linear optics is no exception. Many problems are encountered when it comes to the translation of a molecular property into a bulk property. It has transpired that some of these problems are not easy to solve with classical NLO-phores* that mostly belong chemically to the class of merocyanine dyes. New design strategies for organic molecules and their respective bulk structures, crystals or oriented composite materials like polymers, are needed. Only a more fundamental understanding of these issues will allow rational optimization of molecular and bulk properties. This review attempts to survey second-order molecular polarizabilities of organic molecules. Emphasis will primarily be given to the underlying physical principles of non-linear optics (e.g. molecular topology), and to the establishment of structure–property relationships. We will largely remain paradigmatic and will not attempt an exhaustive compilation of data because several good review articles and books exist that incorporate tabulations of numerous *The term ‘‘NLO-phore’’, meaning, ‘‘bearer/producer of non-linear optics’’ is preferred here to the often-used ‘‘NLO-chromophore’’.

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results from the primary literature [for reviews and books, see Williams (1984); Butcher and Cotter (1990); Prasad and Williams (1991); Wagnie`re (1993); Burland (1994); Dalton et al. (1995); Denning (1995); Long (1995); Marks and Ratner (1995); Verbiest et al. (1997); Wolff and Wortmann (1998); for multi-author works and conference proceedings that in most cases also contain tutorial chapters see Chemla and Zyss (1987); Hann et al. (1989, 1991, 1992); Marder et al. (1991); Zyss (1994); Bosshard et al. (1995); Lindsay and Singer (1995); Twieg and Dirk (1996); Munn et al. (1997); Nalwa and Miyata (1997)]. Rather, a critical survey of existing design strategies and their respective success is attempted. Also, on citing the relevant literature we have not attempted to be comprehensive, but restrict ourselves to leading references and those that have been omitted or ‘‘forgotten’’ in the oft-cited body of references in the realm of organic NLO-phores. A rational optimization of NLO properties – a truly interdisciplinary effort – requires an understanding of the physical principles that govern the tensorial properties of molecular polarizabilities and hyperpolarizabilities. The behaviour of a NLO material cannot be described by a single scalar ␤ value. In addition different unit systems coexist in the literature and the choice of a different standard value for quartz combined with the use of a different convention can ‘‘increase’’ experimental results by a factor of 10. There is also no such thing as ‘‘the NLO performance’’ as figures of merit may vary with different applications. Hence, the synthetic organic chemist would be ill-served if we avoided mathematics altogether. We will try, however, always to explain the qualitative meaning of the sometimes formidable looking equations in Sections 2–4. On the other hand, the physicist may be bewildered by the plethora of chemical structures in Section 5. Unfortunately, chemical knowledge and intuition eschew the rigour of clear definition. It is hoped that the critical approach tried here will help to remove some of the roadblocks on the way to the implementation of organic materials for NLO. The incorporation of organic molecules into ordered bulk structures – crystals, poled polymers, or other host–guest systems – can only be hinted at. These areas are much less amenable to a theoretically well-founded presentation: a true prediction of crystal structures, let alone the often stressed ‘‘crystal engineering’’ (Desiraju, 1995; Wolff, 1996), is not within near reach, despite the ever-growing number of entries in the Cambridge Structural Database (Allen et al., 1994) and increasing levels of sophistication in computational approaches (Gavezzotti and Filippini, 1996). It has been pointed out that the molecular design strategies to induce a crystal structure suitable for NLO applications fail as often as they succeed (Twieg and Dirk, 1996). Recently, based on the observations of conformational polymorphism in hydrazones (Bernstein, 1992) and on co-crystallization approaches that take advantage of hydrogen bonding (Bernstein et al., 1995; Whitesides et al., 1995), progress has been made (Follonier et al., 1997; Wong et al., 1997). But even given a known and efficient structure, the growth of molecular crystals of good quality is still

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quite a challenge (Hulliger, 1994). On the other hand, incorporation of NLO-phores into polymers and poling to give stable ordered structures is also largely an empirical process. Due to commercial interests, important details are not always disclosed. An introduction to the phenomena of NLO will be given first (Section 2), followed by the evaluation of molecular second-order polarizabilities by theoretical models that both allow their rationalization and the design of promising molecular structures (Section 3). It will be necessary to develop different models for molecular symmetries, but the approach will remain the same. NLO effects and experiments used for the determination of molecular (hyper)polarizabilities will be dealt with in Section 4. Finally, experimental investigations will be dealt with in Section 5, followed by some concluding remarks.

2

Basics of non-linear optics

INTRODUCTION

The electromagnetic field of light can interact with matter through its electrical and magnetic component. As known from classical molecular spectroscopy, the interaction with the magnetic field component is usually weaker in diamagnetic organic molecules by orders of magnitude (cf. the weak magnetic dipole allowed, but electric dipole forbidden n␲* transition in acetone). Magnetic effects will therefore be largely neglected for the discussion of NLO effects. However, they give rise to a number of magneto-optical and magneto-chiral effects even in closed-shell (diamagnetic) molecules (Wagnie`re, 1993; Deussen et al., 1996; Elshocht et al., 1997). Non-linear optics is concerned with the interaction of one or more electromagnetic radiation fields (light) with matter to produce a new field that differs in phase, frequency, polarization or direction, etc., from the initial field(s). The intensity of the incident light is proportional to the square of the electric field amplitude, E. The alternating electric field E of light displaces the charged particles – dipoles, ions, atomic nuclei, and electrons – in the material. A dipole is thus induced, and the induced dipole per volume is called the polarization P. For the frequencies of interest here, which are in the near infrared range or above, only the electrons of an organic material can follow the rapid reversal of the field direction. In an optically linear material the polarization P is linearly proportional to the electric field strength E. This condition no longer holds for optically non-linear materials and a number of non-linear optical effects such as second harmonic or third harmonic generation arise. In the following, these qualitative statements are developed in a more rigorous fashion. The discussion begins with the classical foundation of electromagnetism, Maxwell’s equations. Expressions for linear and non-linear

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susceptibilities will be developed, first for bulk materials, then for molecules. The importance of a consistent definition of macroscopic susceptibilities as well as molecular polarizabilities will be emphasized. Recommended SI units are used throughout.

MAXWELL’S EQUATIONS

All macroscopic aspects of the statics and dynamics of electromagnetic field in the presence of material media are described by Maxwell’s equations. The differential form of these axioms in the International System of Units (SI) or rationalized MKS system (Cohen and Giacomo, 1987; Lide, 1991) is given in (1)–(4). ⵜ ⭈ D(r, t) ⫽ ␳(r, t)

(Gauß’s law)

ⵜ ⭈ B(r, t) ⫽ 0 ⭸B(r, t) (Faraday’s law) ⭸t ⭸D(r, t) ⫹ J(r, t) (Ampe`re’s law) ⵜ ⫻ H(r, t) ⫽ ⭸t ⵜ ⫻ E(r, t) ⫽ ⫺

(1) (2) (3) (4)

Maxwell’s equations form a set of partial differential equations. For a given position vector (units in parentheses) r (m) and a time t (s), they couple the dielectric displacement vector D (C m⫺2), the charge density ␳ (C m⫺3), the magnetic induction vector B (T; T ⫽ V s m⫺2), the electric field strength E (V m⫺1), the magnetic field strength H (A m⫺1), and the total current density J (A m⫺2). Equation (1) relates the electric charge distribution to the dielectric displacement, (2) precludes the existence of magnetic monopoles, (3) describes the creation of an electric field by time-varying magnetic induction, and (4) the creation of a magnetic field by an electric current. The charge density ␳ and the current density J may be regarded as the sources of the electromagnetic fields.

CONSTITUTIVE RELATIONS

Material media respond in a characteristic way to the presence of electric and magnetic fields. This response is expressed by the so-called constitutive or material relations (5) and (6). D(r, t) ⫽ ␧0 E(r, t) ⫹ P(r, t)

(5)

B(r, t) ⫽ ␮0 H(r, t) ⫹ ␮0 M(r, t)

(6)

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P is the dielectric polarization vector (dipole moment per volume, C m m⫺3 ⫽ C m⫺2) induced by the electric field and M the magnetization vector (magnetic dipole moment per volume, A m⫺1) induced by the magnetic field. The constants in (5) and (6) are the vacuum permittivity ␧0 ⫽ 8.85419 ⫻ 10⫺12 C V⫺1 m⫺1 and the vacuum permeability ␮0 ⫽ 4␲ ⫻ 10⫺7 V s2 C⫺1 m⫺1. It is implied by (5) and (6) that the response of the medium is purely local (dipole approximation). The media which are of interest for this review do not contain macroscopic charge and current densities, hence ␳ ⫽ 0 and J ⫽ 0, and they are not magnetized, so that M ⫽ 0. Then Maxwell’s equations and the constitutive relations may be combined to yield the following coupled partial differential equation between the electric field E and the dielectric polarization P: ⵜ ⫻ ⵜ ⫻ E(r, t) ⫽ ⫺ ␮0

⭸2 冸␧0 E(r, t) ⫹ P(r, t)冹 ⭸t 2

(7)

In the next section we will discuss the solution of this equation for the case of linear optics where the constitutive relation defines P as a linear function of the electric field.

(1)

LINEAR OPTICS: LINEAR SUSCEPTIBILITY (␹

)

Optically linear media are characterized by a linear response of the medium (i.e. the charged particles therein) to the electric field. We consider an alternating electric field at position r which varies sinusoidally with time according to (8), E(t) ⫽ E ␻ cos(␻t)

(8)

where E ␻ is the field amplitude, ␻ ⫽ 2␲␯ the circular frequency, and ␯ the frequency. In the electric dipole approximation, the dielectric polarization P(t) is created by local response in the medium. It is then given by (9), P(t) ⫽ ␧0 ␹ (1) (⫺␻; ␻) ⭈ E ␻ cos(␻t)

(9)

where the linear susceptibility ␹ (1)(⫺␻; ␻) characterizes the first-order (linear) response of the medium and is frequency-dependent. The argument in parentheses describes the nature of this dependence. Two waves interact with each other through the medium. The frequency of the resulting wave is stated first, then the frequency of the incident wave(s). In general, ␹ (1) is a second-rank tensor, that is, a 3 ⫻ 3 matrix. This tensor can always be diagonalized by transformation into a main axes system. In an isotropic

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medium ␹ (1) may be represented by a scalar quantity ␹ (1), because only one independent component remains. For such an optically linear and isotropic medium (where ⵜ ⭈ E ⫽ 0) one obtains with (7) and the identity: ⵜ ⫻ ⵜ ⫻ E ⫽ ⵜ(ⵜ ⭈ E) ⫺ ⌬E the wave equation (10). ⌬E(r, t) ⫺ ␮0 ␧0 冸␹ (1) (⫺␻; ␻) ⫹ 1冹

⭸2 E(r, t) ⫽ 0 ⭸t 2

(10)

Possible solutions of this differential equation in weakly absorbing media include damped plane waves (11) travelling in the ⫹z-direction, E(r, t) ⫽ E ␻ exp(⫺ 1–2 ␣␻ z) cos(␻t ⫺ k␻ z)

(11)

where ␣␻ is the natural absorption coefficient and k␻ the magnitude of the wave vector at circular frequency ␻. ␣␻ and k␻ are related to the imaginary and real parts of the linear susceptibility, respectively, and to the refractive index n␻ and speed of light c␻ in the medium by equations (12)–(14).

␻ Im{␹ (1) (⫺␻; ␻)} c␻ n␻

(12)

n␻ ⫽ 兹1 ⫹ Re{␹ (1) (⫺␻; ␻)}

(13)

␻ n␻ ␻ 2␲n␻ ⫽ ⫽ c␻ c0 ␭0

(14)

␣␻ ⫽

k␻ ⫽

Here, c0 ⫽ 2.99792458 ⫻ 108 m s⫺1 is the speed of light in vacuum and ␭0 the vacuum wavelength. For non-absorbing (transparent) media, far from resonances, ␹ (1) is a real quantity. It is then related by (13) to a property better known in chemistry, the refractive index n␻ of the material. A high refractive index, n␻, is therefore an expression of a high linear susceptibility. For optical frequencies as provided by light in the UV–visible range, ␹ (1) is also related to the relative permittivity (dielectric constant), ␧r, because Maxwell’s relation, ␧r ⫽ n2, holds. It is intuitively understandable that the linear relationship (9) may no longer hold at high electric field strengths E ␻. The linear response – reminiscent of Hooke’s law – implies a harmonic potential for the displacement of the charged particles (electrons). In optically non-linear media, anharmonic polarization terms gain importance. This occurs at high electric field strengths as provided by high light intensities. The intensity I ␻ (irradiance, W m⫺2) transported by a plane electromagnetic wave of amplitude E ␻ is given by (15). I ␻ ⫽ 1–2 ␧0 c0 n␻ 兩 E ␻ 兩2

(15)

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For example, E ␻ of a pulsed Nd:YAG laser with an irradiance of I ␻ ⫽ 1 MW cm⫺2 is about 3 ⫻ 106 V m⫺1. This electric field is still small compared to typical intramolecular fields acting on electrons and nuclei. The strength of the electric field arising from a proton at a distance of Bohr’s radius, a0, is about 5 ⫻ 1011 V m⫺1. Nevertheless, the field strengths produced by high-power lasers are large enough to give rise to a variety of non-linear optical processes in the medium.

NON-LINEAR OPTICS: NON-LINEAR SUSCEPTIBILITIES

(␹ (n))

Optically non-linear media are characterized by a non-linear response of the medium to the electric field. For simplicity in the following we assume the media to be homogeneous. Thus the response does not depend on the position within them and the argument r can be omitted. We consider the general case of the medium’s local response when exposed to an electric field which results from a superposition of a static and an optical component given by (16). E(t) ⫽ E 0 ⫹ E ␻ cos(␻t)

(16)

The dielectric polarization (17) created in the non-linear medium as a response to the field (16) contains two linear terms of the form given in (9) but also a number of additional non-linear terms. The frequencies of the latter differ in part from those of the input fields. P(t) ⫽ P 0 ⫹ P ␻ cos(␻t) ⫹ P 2␻ cos(2␻t) ⫹ P 3␻ cos(3␻t) ⫹ . . .

(17)

Given to the third order in the electric field as defined by (16), the results for the Fourier amplitudes P 0, P ␻, P 2␻ and P 3␻ are found to be represented by (18)–(21). P 0 ⫽ ␧0 ␹ (0) ⫹ ␧0 ␹ (1) (0; 0) ⭈ E 0 ⫹ ␧0 ␹ (2) (0; 0, 0):E 0 E 0 ⫹ 1–2 ␧0 ␹ (2) (0; ⫺␻, ␻):E ␻ E ␻ ⫹ ␧0 ␹ (3) (0; 0, 0, 0):E 0 E 0 E 0 ⫹ 3–2 ␧0 ␹ (3) (0; ⫺␻, ␻, 0)⯗ E ␻ E ␻ E 0 ⫹ . . . (18) P ␻ ⫽ ␧0 ␹ (1)(⫺␻; ␻)⭈E ␻ ⫹ 2␧0 ␹ (2)(⫺␻; ␻, 0): E ␻E 0 ⫹3␧0 ␹ (3)(⫺␻; ␻, 0, 0)⯗ E␻E0E0 ⫹ 3–4 ␧0 ␹ (3) (⫺␻; ⫺␻, ␻, ␻)⯗ E ␻ E ␻ E ␻ ⫹ . . . P

2␻

⫽

1 – ␧0 ␹ (2)(⫺2␻; ␻, ␻): E␻ E␻ ⫹ 3– ␧0 ␹ (3)(⫺2␻; ␻, ␻, 0)⯗ E␻ E␻ E0 ⫹ . 2 2

P 3␻ ⫽ 1–4 ␧0 ␹ (3)(⫺3␻; ␻, ␻, ␻)⯗ E ␻ E ␻ E ␻ ⫹ . . .

(19) . . (20) (21)

The prefactors in (18)–(21) result from trigonometric identities such as cos2(␻t) ⫽ 1–2(1 ⫹ cos(2␻t)) and cos3(␻t) ⫽ 1–4(3cos(␻t) ⫹ cos(3␻t)) and from intrinsic permutation symmetries (see p. 131). (For a detailed discussion of conventions for non-linear optical susceptibilities, see Butcher and Cotter,

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1990.) The ␹ (n) in (18)–(21) are the nth-order susceptibilities. They are tensors of rank n ⫹ 1 with 3(n ⫹ 1) components. Thus, a second-order susceptibility, ␹ (2), is a third-rank tensor with 27 components and the third-order susceptibility, ␹ (3), a fourth-rank tensor with 81 components. The number of independent and significant elements is (fortunately) much lower (see p. 131). The second-order susceptibility ␹ (2) vanishes in centrosymmetric media. Practical use of second-order NLO is therefore confined to highly ordered non-centrosymmetric solids, solid solutions, Langmuir–Blodgett layers, or liquid-crystalline materials. Unfortunately, the majority of achiral organic crystals adopt centrosymmetric space groups (Wilson, 1990, 1993; Brock and Dunitz, 1994). In principle, the problem could be solved by employing pure enantiomers which can only crystallize in a non-centrosymmetric space group. However, this in itself does not guarantee an arrangement of the molecules to achieve high values of ␹ (2). Hence, much work has been devoted to orient promising NLO-phores in polymer matrices by application of strong electric fields (Dalton et al., 1995; Lindsay and Singer, 1995; Marks and Ratner, 1995). Many of the different susceptibilities in (18)–(21) correspond to important experiments in linear and non-linear optics. The argument in parentheses again describes the kind of interacting waves. Two waves interact in a first-order process as described above in (9), three waves in a second-order process, and four in a third-order process. ␹ (0) describes a possible zerothorder (permanent) polarization of the medium; ␹ (1)(0;0) is the first-order static susceptibility which is related to the relative permittivity (dielectric constant) at zero frequency, ␧r(0), by (22).

␹ (1)(0; 0) ⫽ ␧r (0) ⫺ 1

(22)

␹ (2)(0; 0, 0) and ␹ (3)(0; 0, 0, 0) are second- and third-order static polarizabilities. ␹ (2)(0; ⫺␻, ␻) describes a process known as optical rectification, i.e. the creation of a static polarization by the interaction of two photons in a second-order non-linear optical medium. ␹ (1)(⫺␻; ␻) is the linear optical susceptibility which is related to the refractive index n␻ at frequency ␻, (13). The Pockels susceptibility ␹ (2)(⫺␻; ␻, 0) and the Kerr susceptibility ␹ (3)(␻; ␻, 0, 0) describe the change of the refractive index induced by an externally applied static electric field. ␹ (3)(⫺␻; ⫺␻, ␻, ␻) describes the dependence of the refractive index on the intensity of an AC field (laser light) and is related to phenomena such as self-focusing or degenerate four-wave mixing (DFWM). The susceptibility ␹ (2)(⫺2␻; ␻, ␻) describes frequency doubling, usually called second harmonic generation (SHG). ␹ (3)(⫺2␻; ␻, ␻, 0) describes the influence of an external electric field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric-fieldinduced second harmonic generation (EFISHG). Finally, ␹ (3)(⫺3␻; ␻, ␻, ␻) is

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the susceptibility for frequency tripling, called third harmonic generation (THG). The most general situation is the non-linear coupling of n fields with frequencies ␻1, ␻2, . . . ␻n to a sum frequency ⍀ ⫽ ␻1 ⫾ ␻2 ⫾. . . ⫾ ␻n through an nth-order susceptibility ␹ (n)(⫺⍀; ⫾ ␻1, ⫾ ␻2, . . . ⫾ ␻n). Two important second-order NLO effects of this more general case are sum frequency generation (SFG), ⍀ ⫽ ␻1 ⫹ ␻2, and difference frequency generation (DFG), ⍀ ⫽ ␻1 ⫺ ␻2 . Units in non-linear optics: SI and cgs systems According to (18)–(21) the product of an nth-order susceptibility with (n ⫺ 1) electric field components must be dimensionless. The SI unit of an nth-order susceptibility ␹ (n) is thus (m V⫺1)n⫺1. Traditional cgs units (electrostatic units, esu), however, are still sometimes used. The general relation (23) relates esu and SI units for susceptibilities (Butcher and Cotter, 1990)

␹ (n)(SI)/␹ (n)(esu) ⫽ 4␲/(10⫺4 c0)n⫺1

(23)

Kramers–Kronig relations: links between real and imaginary parts of susceptibilities Equations (18)–(21) were given for the case of real susceptibilities. However, they have to be treated as complex quantities if the frequency is close to or within the region of an optical transition in the medium. An example in the domain of linear optics was given in (12) where the imaginary part of the first-order susceptibility, ␹ (1)(⫺␻; ␻), was related to the absorption coefficient, ␣␻, of the medium. An example from non-linear optics is the technique of electro-optical absorption measurements (EOAM, p. 167) where the UV– visible absorption is studied under the influence of a static electric field. In EOAM, the imaginary part of the third-order susceptibility, ␹ (3)(⫺␻; ␻, 0, 0), is studied. The technique is used to gain information on molecular ground and excited state dipole moments, as well as the magnitudes and directions of the corresponding transition dipoles. The real and the imaginary parts of a susceptibility are in certain cases coupled through Kramers–Kronig relations (Butcher and Cotter, 1990) such as (24). Re{␹(␻⬘)} ⫽

2 P ␲

冕

⫹⬁

0

d␻

␻ Im{␹(␻)} ␻2 ⫺ (␻⬘)2

(24)

The equation holds for ␹ (1)(⫺␻; ␻) and thus relates the absorption coefficient (imaginary part) to the refractive index (real part) of the medium.

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It also holds (Kuball and Galler, 1967) for ␹ (3)(⫺␻; ␻, 0, 0) and relates EOAM (imaginary part) to the Kerr susceptibility (real part). For more general discussions of Kramers–Kronig relations in NLO see Hutchings et al. (1992). Symmetry relationships The number of independent tensor elements of ␹ (n) (i.e. ones assuming values different from each other), is drastically reduced through permutation symmetries. Some of these are always applicable under the usual experimental conditions, whereas others require special conditions that are not necessarily met. Spatial symmetry often leads to a substantial reduction in the number of non-zero tensor elements. Intrinsic permutation symmetry and overall permutation symmetry. We assume that the response of the medium has reached a steady state. Then the susceptibilities are invariant with respect to all n! simultaneous permutations of the inducing fields ␻i and their Cartesian indices ␣i. This property is called intrinsic permutation symmetry and occurs naturally because in the steady state the response must be independent of the ordering of electric fields E␻1 . . . E␻n. For the example of a third-order susceptibility relationships (25) hold. (3) (3) ␹␮␣ (⫺⍀; ␻1, ␻2, ␻3) ⫽ ␹␮␣ (⫺⍀; ␻2, ␻1, ␻3) 1 ␣2 ␣3 2 ␣1 ␣3 (3) ⫽ ␹␮␣2 ␣3 ␣1 (⫺⍀; ␻2, ␻3, ␻1) ⫽ . . .

(25)

Intrinsic permutation symmetry was already used in (18)–(21) so that only one of the equivalent terms occurs in the equation. Far from resonances of the medium and in the limit ⍀, ␻1, ␻2, . . . ␻n → 0 the permutation symmetry of (25) can be extended to include the first Cartesian index ␮ and the induced frequency ⍀. This property, if at least valid to a good approximation, is called overall permutation symmetry. Kleinman symmetry (index permutation symmetry). Far from resonances of the medium where dispersion is negligible, the susceptibilities become to a good approximation invariant with respect to permutation of all Cartesian indices (without simultaneous permutation of the frequency arguments). This property is called Kleinman symmetry (Kleinman, 1962). It is important in the discussion of the exchange of power between electromagnetic waves in an NLO medium. In many cases approximate validity of Kleinman symmetry can be used effectively to reduce the number of independent tensor components of an NLO susceptibility. Spatial symmetry. Fortunately, for media with symmetry properties, many of the tensor components are zero and not all of the non-zero components are

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independent. The non-zero components of ␹ (2) and ␹ (3) of all crystal classes and isotropic media are tabulated in many textbooks (e.g. Butcher and Cotter, 1990, and references therein; for susceptibilities of higher order see Popov et al., 1995). Here we quote only the results for an isotropic homogeneous system (symmetry class ⬁⬁m or Kh) such as an amorphous polymer, a glass, or a liquid. These systems are centrosymmetric, that is, invariant to inversion. Since the parity of the induced polarization is ungerade, all components of ␹ (2) (and of all other susceptibilities ␹(2n) of even order 2n) are zero. The third-order susceptibility ␹ (3) of an isotropic medium has 21 non-vanishing tensor components of which only three are independent. The general symmetry relations between these can be written using Kronecker ␦ symbols in the compact form (26) (3) (3) (3) (3) ␹␮␣␤␥ ⫽ ␹ZZXX ␦␮␣ ␦␤␥ ⫹ ␹ZXZX ␦␮␤ ␦␣␥ ⫹ ␹ZXXZ ␦␮␥ ␦␣␤

(26)

with the additional linear relationship (27). (3) (3) (3) (3) ␹ZZZZ ⫽ ␹ZZXX ⫹ ␹ZXZX ⫹ ␹ZXXZ

(27)

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, ␹ (3)(⫺␻; ␻ ,0, 0), intrinsic permutation symmetry in the last two (3) (3) indices holds, ␹ZXZX ⫽ ␹ZZXX . The most general Kerr susceptibility of an (3) isotropic medium therefore has only two independent components, ␹ZZZZ and (3) (3) ␹ZZXX. Likewise, the EFISHG susceptibility ␹ (⫺2␻; ␻, ␻, 0), important for the evaluation of second-order molecular polarizabilities in solution (see pp. (3) (3) and ␹ZXXZ , 158 and 162), has only two independent components, ␹ZZZZ because of intrinsic permutation symmetry in the second and third indices. Other important systems are uniaxial isotropic systems, because the widely studied poled polymers belong to this symmetry class (⬁m or C⬁␯). ␹ (2) of such systems has seven non-vanishing components of which four are independent, (2) (2) (2) (2) (2) (2) (2) ␹XXZ ⫽ ␹YYZ , ␹YZY ⫽ ␹XZX , ␹ZYY ⫽ ␹ZXX and ␹ZZZ . For the SHG susceptibility (2) ␹ (⫺2␻; ␻, ␻) the number of independent components reduces to three because of intrinsic permutation symmetry in the second and third index. If the uniaxial system is created by poling of an isotropic system by an external electric field, e.g. a poled polymer or liquid, then to first order in the applied field, EZ0, the number of independent components of ␹ (2)(⫺2␻; ␻, ␻) is only two (Kielich, 1968). It is thus equal to the number of independent components of ␹ (3)(⫺2␻; ␻, ␻, 0) because of (28). (2) (3) 0 ␹␮␣␤ ⫽ ␹␮␣␤ Z EZ

(28)

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Contracted notation for second-order susceptibilities. Traditionally the SHG non-linearity of crystals is described by a tensor d which is related to the SHG susceptibility by (29) (Zernike and Midwinter, 1973). d(⫺2␻; ␻, ␻) ⫽ 1–2 ␹ (2)(⫺2␻; ␻, ␻)

(29)

Because of the intrinsic permutation symmetry of the SHG susceptibility the shorthand notation in (30) was defined. d␮1 ⫽ d␮11; d␮2 ⫽ d␮22; d␮3 ⫽ d␮33; d␮4 ⫽ d␮23 ⫽ d␮32; d␮5 ⫽ d␮13 ⫽ d␮31; d␮6 ⫽ d␮12 ⫽ d␮21; ␮ ⫽ 1, 2, 3.

(30)

An alternative quantity is also traditionally used to characterize the Pockels non-linearity. Here an electro-optical tensor r is defined which is related to the Pockels susceptibility ␹ (2)(⫺␻; ␻, 0) by equation (31) (Singer et al., 1987), r␮␣␤(⫺␻; ␻, 0) ⫽

⫺2 (2) ␹␮␣␤ (⫺␻; ␻, 0) n2␮ (␻)n2␣ (␻)

(31)

where n␮(␻) is the refractive index for a plane wave of frequency ␻ polarized in the ␮ direction. A contracted notation, similar to the one for d, is here used in the form (32) (Yariv and Yeh, 1984). r1␤ ⫽ r11␤; r2␤ ⫽ r22␤; r3␤ ⫽ r33␤; r4␤ ⫽ r23␤ ⫽ r32␤; r5␤ ⫽ r13␤ ⫽ r31␤; r6␤ ⫽ r12␤ ⫽ r21␤; ␤ ⫽ 1, 2, 3.

(32)

MOLECULAR POLARIZABILITIES

We consider now the NLO response of a molecule to an electric field. The resulting equations will be found to be analogous to the ones derived for a bulk medium. Instead of bulk susceptibilities ␹ (n), however, molecular polarizabilities of nth order appear. For the latter, by convention, the lower-case Greek letters in ascending order (␣, ␤, ␥, . . .) are used. Again, an electric field of the form defined in (16) is used. Similar to the macroscopic polarization (17), the expression (33) for the molecular dipole moment p(t) contains linear and non-linear terms. p(t) ⫽ p0 ⫹ p␻ cos(␻t) ⫹ p2␻ cos(2␻t) ⫹ p3␻ cos(3␻t) ⫹ . . .

(33)

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J. J. WOLFF AND R. WORTMANN

The Fourier amplitudes p0, p␻, p2␻ and p3␻, to the third order in the electric field (16) are then given by (34)–(37). p0 ⫽ ␮ ⫹ ␣(0; 0) ⭈ E 0 ⫹ 1–2 ␤(0; 0, 0):E 0 E0 ⫹ 1–4 ␤(0; ⫺␻, ␻):E ␻ E ␻ ⫹ 1–6 ␥(0; 0, 0, 0)⯗E 0 E 0 E 0 ⫹ 1–4 ␥(0; ⫺␻, ␻, 0)⯗E ␻ E ␻ E 0 ⫹ . . . ␻

␻

p2␻ ⫽ 1–4 ␤(⫺2␻; ␻, ␻):E ␻ E␻ ⫹ 1–4 ␥(⫺2␻; ␻, ␻, 0)⯗E ␻ E ␻ E 0 ⫹ . . . p

3␻

⫽

(34)

0

p ⫽ ␣(⫺␻; ␻) ⭈ E ⫹ ␤(⫺␻; ␻, 0):E E ⫹ 1–2 ␥(⫺␻; ␻, 0, 0)⯗E ␻ E 0 E 0 ⫹ 1–8 ␥(⫺␻; ⫺␻, ␻, ␻)⯗E ␻ E ␻ E ␻ ⫹ . . . ␻

1 ␻ ␻ ␻ –– 24 ␥(⫺3␻; ␻, ␻, ␻)⯗E E E

(35) (36) (37)

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n! are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit ␻ → 0. ␤ values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B*) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth ‘‘phenomenological’’ convention (X) is converted to the T convention by multiplication by a factor of 4. Units To add to the confusion noted for conventions of polarizabilities, both cgs and recommended SI units for linear and non-linear optical polarizabilities coexist in the literature. We strongly advocate the use of SI units. The SI unit of the electric dipole moment is C m (Cohen and Giacomo, 1987). Thus, consistent SI units of an nth-order polarizability are defined as: C m(m V⫺1)n ⫽ C mn⫹1 V⫺n, cf. (34)–(37). Conversions from the SI to the esu system for the dipole moment, the first-, second-, and third-order polarizability, are given in (38)–(41).

␮, 10⫺30 C m ⫽ 0.2988 ⫻ 10⫺18 esu ⫽ 0.2988 Debye (D)

(38)

␣, 10⫺40 C m2 V⫺1 ⫽ 0.8988 ⫻ 10⫺24 esu

(39)

␤, 10

3

esu

(40)

␥, 10⫺60 C m4 V⫺3 ⫽ 8.078 ⫻ 10⫺36 esu

(41)

⫺50

Cm V

⫺2

⫽ 2.694 ⫻ 10

⫺30

Some authors have used units for the polarizabilities which differ by a factor of ␧0 from those quoted here. For example, the units 10⫺40 m4 V⫺1 are

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135

occasionally used for ␤ (Nicoud and Twieg, 1987a; Bosshard et al., 1995). It should be noted that these units do not conform to the ones required by the SI system. Symmetry properties The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability ␣(⫺␻; ␻) is a symmetric second-rank tensor like ␹ (1). Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, ␣xx, ␣yy and ␣zz. If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like ␹ (2), ␤ is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media; effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of ␤ (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of ␤ is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)–(44). C2␯ symmetry: ␤xxz, ␤xzx, ␤zxx, ␤yyz, ␤yzy, ␤zyy, ␤zzz C3h symmetry: ␤yyy ⫽ ⫺␤yyz ⫽ ⫺␤yzy ⫽ ⫺␤zyy, ␤zzz ⫽ ⫺␤yzz ⫽ ⫺␤zyz ⫽ ⫺␤zzy D3h symmetry: ␤zzz ⫽ ⫺␤yzz ⫽ ⫺␤zyz ⫽ ⫺␤zzy

(42) (43) (44)

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, ␤rst ⫽ ␤rst, holds for the second-order polarizability ␤rst(⫺2␻; ␻, ␻) in the second and third index. Kleinman symmetry, i.e. permutation symmetry in all Cartesian indices (cf. p. 131), generally holds only in the limit ␻ → 0. In ␲-conjugated organic molecules, almost invariably only few elements are significant, because ␤ is strongly influenced by the molecular topology. If conjugation extends to two dimensions, as in planar donor–acceptor substituted ␲ systems, the ␤ components perpendicular to the molecular plane (yz plane) are generally very small. Thus, for a planar conjugated molecule with C2␯ symmetry, ␤xxz, ␤xzx and ␤zxx can usually be neglected. If conjugation is further reduced to one dimension, as in elongated donor–acceptor substituted ␲ systems, often only one significant component of ␤, ␤zzz, remains. Its

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direction parallels the one of the long molecular axis (z axis). Such systems have therefore been dubbed ‘‘one-dimensional’’ (see below and p. 238). Quantitatively, this is the largest class of NLO-phores investigated so far.

3

Quantum-chemical basis for second-order polarizabilities

In this chapter, the molecular second-order polarizability will be related to electro-optic properties like dipole moment in the ground and electronically excited states, the corresponding transition dipoles and transition energies. Any theoretical description of second-order molecular polarizabilities has to take into account the way charge redistribution occurs within the molecule when a time-dependent electric field is applied. We will begin by considering how molecular wave functions are altered by the presence of such a field. Electronically excited states with charge transfer (CT) character may mix with the ground state. It is important to note that an electronic transition does not in fact occur, the electron cloud is just polarized along the direction given by the CT. We will show that virtual transitions between two and three levels are important for second-order polarizabilities. We then discuss cases of special molecular symmetry and constitution that often occur for experimentally characterized NLO-phores. Calculations on simple model systems then demonstrate on which observable molecular parameters ␤rst depends. These calculations allow one to derive criteria for the rational design of NLO-phores. Finally, we consider condensed phases like solutions and attempt to show their influence on ␤.

PERTURBATION THEORETICAL EXPRESSIONS

The molecular polarizabilities can be interpreted quantum mechanically by using the methods of time-dependent perturbation theory. Under the influence of the electric field, the molecular ground state (兩g典) is changed by admixture of excited states (兩l典, 兩m典 . . .). Collections of such expressions are available in the literature (Ward, 1965; Orr and Ward, 1971; Bishop, 1994b). A comprehensive treatment has also been given by Flytzanis (1975). Here, we only quote the results for the linear optical polarizability ␣(⫺␻; ␻) and the second-order polarizability ␤(⫺2␻; ␻, ␻). The linear optical polarizability may be represented by the sum of two-level contributions (45).

␣rs(⫺␻; ␻) ⫽

冘

␣(lg) rs (⫺␻; ␻)

(45)

l⫽g

They arise from ground state 兩g典 and excited state 兩l典 and are given by (46). lg lg (lg) ␣(lg) (␻ ) rs (⫺␻; ␻) ⫽ ␮r ␮s ⍀

(46)

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Here, ␮rlg ⫽ 具l 兩 ␮r 兩 g典 is the Cartesian component r of the electric transition dipole moment between states 兩g典 and 兩l典 and ⍀(lg) is a dispersion function given by (47),

⍀(lg) (␻) ⫽

2␻lg ប(␻2lg ⫺ ␻2)

(47)

with ␻lg as the circular frequency of the electronic transition between 兩g典 and 兩l典. The second-order polarizability can be written as a sum of two-level contributions and a double sum of three-level contributions in the form (48) (Wolff et al., 1997),

␤rst(⫺2␻; ␻, ␻) ⫽

冘

(lg) ␤rst (⫺2␻; ␻, ␻) ⫹

l⫽g

冘冘 l⫽g

(mlg) ␤rst (⫺2␻; ␻, ␻)

(48)

m⬎l

where the two-level and three-level terms are (49) and (50). (lg) (ll) ␤rst (⫺2␻; ␻, ␻) ⫽ ⌬␮rlg ␮slg ␮lg1 (␻) ⫹ (⌬␮rlg ⌬␮slg ␮lgt ⫹ ␮rlg ⌬␮lgt ␮slg)2 ⍀(ll)(␻) t ⍀ (49) (mlg) lg 1 (ml) ␤rst (⫺2␻; ␻, ␻) ⫽ ␮rml(␮smg ␮lgt ⫹ ␮mg (␻ ) t ␮s ) ⍀ lg ml mg ml mg 2 (ml) ⫹ ␮ r (␮ s ␮ t ⫹ ␮ t ␮ s ) ⍀ (␻ ) lg 2 (ml) ⫹ ␮rmg(␮sml ␮lgt ⫹ ␮ml (␻ ) t ␮s ) ⍀

(50)

The dispersion functions 1⍀(ml) and 2⍀(ml) [including the degenerate form 1⍀(ll), ⍀(ll) in (49)] account for dispersive enhancement by one- and two-photon resonances and are given by (51) and (52).

2

1

⍀(ml)(␻) ⫽

2(␻lg ␻mg ⫺ ␻2) ប2(␻2lg ⫺ ␻2)(␻2mg ⫺ ␻2)

2

⍀(ml)(␻) ⫽

2(␻lg ␻mg ⫹ 2␻2) ប2(␻2lg ⫺ ␻2)(␻2mg ⫺ 4␻2)

(51)

(52)

Similar equations serve as a basis for semi-empirical quantum chemical computations by the ‘‘sum-over-states’’ (SOS) method (Dirk et al., 1986; Kanis et al., 1992, 1994; Tomonari et al., 1997). Quite often, however, only very few excited states contribute to the observed NLO response (Tomonari et al., 1993). This is especially true for the ‘‘one-dimensional’’ NLO-phores to be dealt with in the next section, where to a good approximation only one excited state needs to be taken into account.

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MOLECULAR DESIGN AND TWO- AND THREE-LEVEL CONTRIBUTIONS

Two-level contributions in one-dimensional molecules The second-order polarizability tensor ␤ of elongated ␲ conjugated organic molecules with strong donor and acceptor groups at the terminal positions usually exhibits only one significant component ␤zzz (called a ‘‘diagonal’’ component because it is on the diagonal of ␤). It is parallel to the molecular dipole axis z. Chemically, p-nitroaniline and donor–acceptor substituted (‘‘push–pull’’) polyenes are prototypes of this class of compounds. The perturbation induced by the substitution leads to a low-lying CT band between the ground-state 兩g典 and the excited state 兩a典.

Scheme 1 Characteristic parameters for one-dimensional NLO-phores.

The CT band is characterized by a large change of the electric dipole ag moment upon excitation, ⌬␮ag z , and a large transition dipole, ␮z . As a consequence, terms associated with states 兩g典 and 兩a典 dominate in the perturbation theoretical expression (48)–(50) for ␤ and three-level contributions may be neglected (but see below pp. 143 and 145). Only the (ag) remains significant to a good approximacorresponding two-level term ␤zzz ag ag tion. Since both ⌬␮z and ␮z are parallel to the molecular z axis, the resulting second-order polarizability is called ‘‘one-dimensional’’ (1D). The results for the SHG and the EO polarizabilities and ␤(⫺2␻; ␻, ␻) and ␤(⫺␻; ␻, 0) are (53) and (54), (ag) (ag) ␤zzz (⫺2␻; ␻, ␻) ⫽ ␤zzz (0)

(ag) (ag) ␤zzz (⫺␻; ␻, 0) ⫽ ␤zzz (0)

␻4ag (␻ ⫺ ␻ )(␻2ag ⫺ 4␻2)

(53)

␻2ag(3␻2ag ⫺ ␻2) 3(␻2ag ⫺ ␻2)2

(54)

2 ag

2

(ag) (0) is the two-level contribution to the second-order polarizability where ␤zzz extrapolated to zero frequency (infinite wavelength), ␻ → 0, and is given by (55).

(ag) (ag) (ag) ␤zzz (0) ⫽ lim ␤zzz (⫺2␻; ␻, ␻) ⫽ lim ␤zzz (⫺␻; ␻, 0) ⫽

␻→0

␻→0

ag 2 6⌬␮ag z (␮ z ) (ប␻ag)2

(55)

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(ag) Thus, the intrinsic ␤zzz (0) in one-dimensional ␲ systems depends on the square of the transition dipole, the change in dipole moment, and the square of the inverse transition energy (the HOMO–LUMO energy gap). However, artificially high values will occur due to dispersion enhancement if the frequency of the incident light or its second harmonic is close to the transition frequency (␻ag ⫽ ␻, or ␻ag ⫽ 2␻), (53) and (54). Unfortunately, these one- and two-photon resonance enhancements are technically only of limited use because the material just absorbs the fundamental or the second harmonic and the energy is dissipated. The region of anomalous dispersion at higher frequencies than the two-photon resonance, however, is of potential interest because it naturally supports the phase-matching condition (Cahill and Singer, 1991). In this way high conversion efficiencies may be achieved, provided that the material exhibits sufficient transparency. Phase matching in SHG is otherwise difficult to achieve because of the refractive index dispersion present in any material. Thus the fundamental and second harmonic are refracted differently. It should also be noted again that the equations given are valid only off-resonance. So, while extrapolation to the zero frequency limit (ag) (0) seems straightforward, the extrapolated (␻ → 0) to give the intrinsic ␤zzz values are frequently too high, as subsequent measurements at higher fundamental wavelengths have shown in a number of cases when strong resonance enhancement was present at lower fundamental wavelengths.

Two-level contributions in C2␯ symmetric molecules If a further donor or acceptor group is added to the one-dimensional ␲ systems above, but C2␯ symmetry (as for p-nitroaniline) is retained, molecules such as 3,5-dinitroaniline result. Formally, a doubly degenerate CT from the donor to the two acceptors occurs. The one-dimensional approximation is no longer valid and the ‘‘off-diagonal’’ tensor elements ␤zyy, ␤yzy and ␤yyz may become significant. These arise from low-lying CT bands with transition dipoles ␮ag y perpendicular to the molecular C2 axis (z axis). The two-level contributions to the SHG polarizability for off-diagonal elements can be written in the form (56) and (57),

␻2ag (␻2ag ⫺ ␻2) ␻2ag(␻2ag ⫹ 2␻2) (ag) (ag) ␤yzy (⫺2␻; ␻, ␻) ⫽ ␤yyz (⫺2␻; ␻, ␻) ⫽ ␤(ag) (0) zyy (␻2ag ⫺ ␻2)(␻2ag ⫺ 4␻2) (ag) (ag) ␤zyy (⫺2␻; ␻, ␻) ⫽ ␤zyy (0)

(56) (57)

with the abbreviation (58). (ag) (ag) (ag) ␤zyy (0) ⫽ lim ␤zyy (⫺2␻; ␻, ␻) ⫽ lim ␤yzy (⫺2␻; ␻, ␻) ⫽

␻→0

␻→0

ag 2 2⌬␮ag 2 (␮ y ) (ប␻ag)2

(58)

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J. J. WOLFF AND R. WORTMANN

Scheme 2 Characteristic parameters for two-dimensional NLO-phores with C2␯ symmetry.

[Note that the static limit given in (58) is equal, but the frequency dispersion is different for the two groups of tensor elements in (56) and (57).] Three-level contributions in one-dimensional, C2␯ and D3h symmetric molecules Three-level contributions may be significant for both diagonal and offdiagonal tensor elements. The three-level contributions to the diagonal tensor element ␤zzz(⫺2␻; ␻, ␻) arise when the molecule shows three transition dipoles between the ground state 兩g典 and two excited states 兩a典 and 兩b典, all being bg ab parallel to the molecular z axis, i.e. ␮ag z , ␮z and ␮z . For C2␯ symmetry these terms can only arise if all three states 兩g典, 兩a典 and 兩b典 are symmetric with respect to the molecular C2 axis. To simplify the expressions we assume that the excited states are close in energy (close to degeneracy) so that an average – , can be used and the difference between the excitation excitation frequency, ␻ frequencies, ␻ba, can be neglected, as indicated in (59). – and ␻ – Ⰷ␻ ⬇0 ␻ag ⬇ ␻bg ⬇ ␻ ba

(59)

With these approximations one obtains (60) from (48)–(52). –4 ␻ (bag) (bag) ␤zzz (⫺2␻; ␻, ␻) ⫽ ␤zzz (0) – 2 2 –2 (␻ ⫺ ␻ )(␻ ⫺ 4␻2)

(60)

The three-level contribution extrapolated to zero frequency is proportional to the product of three transition dipole moments (61). (Note the absence of any terms associated with dipole differences, ⌬␮.) (bag) (bag) ␤zzz (0) ⫽ lim ␤zzz (⫺2␻; ␻, ␻) ⫽

␻→0

ba bg 12␮ag z ␮z ␮z (ប␻ag)(ប␻bg)

(61)

In C2␯ symmetric molecules three-level contributions also arise for the off-diagonal tensor elements ␤zyy, ␤yzy and ␤yyz but the symmetry of the excited

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states is different: first, the transition dipole between the ground state 兩g典 and excited state 兩a典 is perpendicular to the C2 axis ␮ag y ; second, the transition dipole, ␮bg , between ground state 兩g典 and excited state 兩b典 is parallel to the C2 axis. z Then, the transition dipole between the excited states 兩a典 and 兩b典 is perpendicular to the C2 axis, ␮ba y . With the approximations (59) one obtains (62), – 2 (␻ – 2 ⫹ 2␻2) ␻ (gab) (gab) (62a) ␤zyy (⫺2␻; ␻, ␻) ⫽ ␤zyy (0) – 2 – 2 ⫺ 4␻2) (␻ ⫺ ␻2)(␻ –2 ␻ (gab) (gab) (gab) ␤yzy (⫺2␻; ␻, ␻) ⫽ ␤yyz (⫺2␻; ␻, ␻) ⫽ ␤zyy (0) – 2 (␻ ⫺ 4␻2)

(62b)

(bag) where ␤zyy is given by (63). (bag) (bag) ␤zyy (0) ⫽ lim ␤zyy (⫺2␻; ␻, ␻) ⫽

␻→0

ba bg 4␮ag y ␮y ␮z (ប␻ag)(ប␻bg)

(63)

MOLECULAR ORBITAL CALCULATIONS FOR SECOND-ORDER POLARIZABILITIES OF ␲ SYSTEMS

The equations derived for two- and three-level contributions to the secondorder polarizability can be used to derive design strategies for the optimization of second-order molecular polarizabilities. As shown in the section above, the NLO response of molecules is controlled by three molecular parameters: transition dipoles between ground and excited states, ␮lg and ␮lm, dipole differences between ground and excited states, ⌬␮lg, and transition energies between ground and excited states, ប␻lg and ប␻lm. Here we concentrate on ␲-conjugated systems and the analysis of the transition dipoles and dipole differences associated with ␲␲* transitions. In the simplest case, only the first HOMO–LUMO (S0–S1) transition is taken into account. This model is adequate for the one-dimensional NLOphores treated in the next section, but it breaks down for multiple combinations of donor and acceptor groups in conjugation. A three-level model for two-dimensional NLO-phores is therefore described later (p. 145). Although the general arguments are not fundamentally changed, the numerical accuracy of these models can be increased by more elaborate computational methods as has been shown mainly for the one-dimensional case (Kanis et al., 1994; Brédas, 1995). The methods have been integrated into common packages for quantum mechanical calculations. In the LCAO method, the molecular orbitals (MO) ␺k of a ␲ system are represented by (64) as a linear combination of atomic orbitals (AO) ␾␮ (usually p-orbitals perpendicular to the molecular plane).

␺k ⫽

冘 ␮

ck␮ ␾␮

(64)

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The closed-shell singlet ground state 兩g典 of a ␲ system with n doubly occupied MO is given by the Slater determinant (65), – – – 兩g典 ⫽ 兩␺1 ␺1 ␺2 ␺2 . . . ␺n ␺n兩

(65)

– where ␺k and ␺k denote MO occupied by two electrons of ␣ and ␤ spins, respectively. Excited singlet states 兩a典 and 兩b典 in which one electron is promoted from MO ␺k to either MO ␺l or to MO ␺m are given by the linear combination of Slater determinants (66) and (67). 兩a典 ⫽ 兩b典 ⫽

1 兹2 1 兹2

– – (兩 . . . ␺k ␺l . . . 兩 ⫹ 兩 . . . ␺l ␺k . . . 兩)

(66)

– – (兩 . . . ␺k ␺m . . . 兩 ⫹ 兩 . . . ␺m ␺k . . . 兩)

(67)

The transition dipoles and dipole differences between states 兩a典 and 兩g典 can be expressed in terms of matrix elements between the MO and applying the ZDO (zero differential overlap) approximation as in (68) and (69),

␮ag ⫽ 具a兩 ␮ 兩b典 ⫽ 兹2 具␺k兩 ␮e(1) 兩␺l典 ⫽ ⫺ e兹2

冘 ␮

cl␮ ck␮ R␮

冘

⌬␮ag ⫽ 具a兩 ␮ 兩a典 ⫺ 具g兩 ␮ 兩g典 ⫽ 具␺l兩 ␮e(1) 兩␺l典 ⫺ 具␺k兩 ␮e(1) 兩␺k典 ⫽ ⫺ e

(68) (c2l␮ ⫺ c2k␮)R␮

␮

(69) where ␮e(1) ⫽ ⫺er(1) is the dipole operator for electron 1 and R␮ is the position vector of atom ␮ of the ␲ system; e is the elementary charge. Analogous relations hold for state 兩b典 and other excited singlet states. The transition dipole between two excited singlet states (66) and (67) is given by (70).

␮ba ⫽ 具b兩 ␮ 兩a典 ⫽ 具␺m兩 ␮e(1) 兩␺l典 ⫽ ⫺ e

冘

cm␮ cl␮ R␮

(70)

␮

These equations are used in semiempirical quantum chemical calculations of non-linear optical polarizabilities by applying perturbation theoretical expressions [the so-called sum-over-states (SOS) method]. Here we use them to derive some qualitative and very general trends in a few simple model systems. To this end we concentrate on the electronic structure, i.e. on the LCAO coefficients. We do not explicitly calculate the transition frequencies. This is justified for the qualitative discussion below since typical transition energies

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of ␲-conjugated organic molecules correspond to wavenumbers in the range 12 500–40 000 cm⫺1 (␭ ⫽ 800–250 nm). Thus the transition energies vary only in a very limited interval. On the other hand, the differences in dipole moments and transition dipoles are much more pronounced, because they can vary between zero and the maximum values which are possible for a given conjugation length.

QUALITATIVE TRENDS FOR ␤ IN SIMPLE MODEL SYSTEMS: TWO-CENTRE MODEL SYSTEM; ONE-DIMENSIONAL NLO-PHORES

For a true 1D NLO-phore, only one component of ␤ is significantly different from zero. Structurally, this requirement is fulfilled to a good approximation by an extended linear ␲-conjugated chain that is substituted at the terminal positions with a donor and an acceptor group. Until recently, it has been an almost universally held view that this molecular design defines the archetypal NLO-phore. It was based on initial experimental evidence, and its theoretical foundation is the so-called ‘‘two-state’’ or ‘‘two-level’’ model (Oudar, 1977; Oudar and Chemla, 1977). The simplest model consists of two centres, one donor (D) and one acceptor (A), separated by a distance l and contains two electrons. Here we consider this simple system to illustrate some general relations between charge transfer, transition intensities and linear as well as non-linear optical polarizabilities. We will show below that the electro-optic parameters and the molecular polarizabilities may be described in terms of a single parameter, c, that is a measure of the extent of coupling between donor and acceptor. Conceptually, this approach is related to early computations on the behaviour of inorganic intervalence complexes (Robin and Day, 1967; Denning, 1995), Mulliken’s model for molecular CT complexes (Mulliken and Pearson, 1969) and a ‘‘two-form/two-state analysis’’ of push–pull molecules (Blanchard-Desce and Barzoukas, 1998). There are only two MO, ␺1 and ␺2, that can be constructed by a linear combination of the two AO ␾1 and ␾2 of the basis. The normalized MO can be represented as functions of a single parameter, c, which can vary between 0 and 1 (c 僆 [0, 1]) by (71) and (72).

␺1 ⫽ c␾1 ⫹ 兹1 ⫺ c2 ␾2

(71)

␺2 ⫽ 兹1 ⫺ c2 ␾1 ⫺ c␾2

(72)

There are only two states to be considered in the calculation of ␤: the ground state, 兩g典, where the MO ␺1 is doubly occupied (configuration ␺21), and an excited state, 兩a典, where one electron is promoted from ␺1 (HOMO) to ␺2 (LUMO) (configuration ␺11 ␺12). The doubly excited configuration ␺22 does not

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ag Fig. 1 Left: Transition dipole moment ␮ag z and dipole difference ⌬␮z in units of (el). (ag) Right: First- and second-order polarizabilities, ␣(ag) and ␤ , in units of (el)2/(ប– ␻) and zz zzz – 3 2 (el) /(ប␻) , respectively, as function of the parameter c2.

contribute to ␤ since the transition dipole to the ground state is zero. Regardless of the chemical nature of the atomic centres 1 and 2, the MO will always have the general structure of (71) and (72) and depend on a single LCAO parameter c. Increasing c is equivalent to increasing the donor/acceptor strength: for c ⫽ 0, centre 2 is the donor, the HOMO is completely localized at centre 2 and the LUMO is completely localized at centre 1; for c ⫽ 1 the situation reverses and the HOMO is completely localized at centre 1 and the LUMO at centre 2. With (71) and (72) it follows from (68) and (69) that the transition dipole between states 兩g典 and 兩a典 is (73) and the dipole moment difference is (74). 2 ␮ag z ⫽ 兹2elc 兹1 ⫺ c 2

⌬␮ ⫽ el(1 ⫺ 2c ) ag z

(73) (74)

Again, e is the elementary charge. Hence, the maximum value (el/兹2) of the transition dipole in this system is reached for c2 ⫽ 0.5, while the maximum absolute values el of the dipole difference, el, are observed at c2 ⫽ 0, 1. These limiting cases hold for any linear conjugated system. They cannot be exceeded for a given conjugation length l. The dipole difference (74) vanishes when the transition dipole (73) is maximized. These trends are illustrated in Figs 1 (left) and 2. Charge transfer upon excitation, i.e. ⌬␮ ⫽ 0, inevitably compromises the magnitude of the transition dipole. This is a rule that can be generalized to include more complex ␲ systems. Expressions (46) and (53) allow us also to analyse the impact of the LCAO parameter c on the first-order and second-order polarizabilities, ␣ and ␤. We assume that the excitation energy does not vary with c as discussed above and

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Fig. 2 Illustration of the electron densities in MO ␺1 and ␺2 as well as transition densities for three values of the parameter c2 of the linear, two-electron DA system. The area of the circles is proportional to c2.

given in (59). The results are displayed in Fig. 1 (right). The two-level contributions ␣(ag) (46) and ␤(ag) (53) are zero at c2 ⫽ 0, 1 because at these points the transition dipole moment ␮ag vanishes. The value of ␤(ag) is also zero at c2 ⫽ 0.5 but there it is the dipole difference ⌬␮ag z that vanishes. There are two extrema at c2 ⫽ 0.211 and 0.789, where ␤(ag) reaches its positive and negative maximum value, with the sign of ␤ determined by the sign of ⌬␮ag z . THREE-CENTRE MODEL SYSTEM (C2␯ AND D3h SYMMETRY)

The simplest system exhibiting two-dimensional polarizabilities belongs to the point group C2␯ and consists of three centres. These may include either one donor (D) and two equivalent acceptors (A), DA 2, or two equivalent donors (D) and one acceptor (A), D2A. For simplicity, it is assumed that the three centres occupy the corners of an equilateral triangle of side length l (cf. Scheme 2). Then, if the three centres are equivalent, the system assumes point group symmetry D3h. There are three MO, ␺1, ␺2 and ␺3, that can be constructed by linear combination of the three AO ␾1, ␾2 and ␾3. As in the one-dimensional case above, the normalized MO can be represented as functions of a single parameter c 僆 [0, 1] by (75)–(77).

␺1 ⫽ ␺2 ⫽

c 兹2 1 兹2

(␾1 ⫹ ␾3) ⫹ 兹1 ⫺ c2 ␾2

(75)

(␾ 1 ⫺ ␾ 3 )

(76)

␺3 ⫽ 兹(1 ⫺ c2)/2 (␾1 ⫹ ␾3) ⫺ c␾2

(77)

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bg ba ag Fig. 3 Left: Transition dipole moments, ␮ag y , ␮z and ␮y , and dipole differences, ⌬␮z bg and ⌬␮z , in units of (el). Right: Two-level and three-level contributions to the second-order polarizability tensor components, ␤zzz and ␤zyy, in units (el)3/(ប– ␻)2, as a 2 2 function of the square of the parameter c . The vertical line at c ⫽ 2/3 indicates the point where the C2␯ case turns into the D3h limit.

Note that ␺2 is independent of c. For a two-electron system, three states have to be considered in the calculation of ␤. The configuration of the ground state 兩g典 is ␺21 and there are two excited states, 兩a典 and 兩b典, with configurations ␺11 ␺12 and ␺11 ␺13. Again, no doubly excited configurations contribute to ␤ since the transition dipole to the ground state is zero. The transition 兩a典 ← 兩g典 is polarized perpendicularly to the C2 axis (z axis), while the transition 兩b典 ← 兩g典 is polarized parallel to this axis. The corresponding transition dipole moments and dipole differences are easily calculated from (68)–(70) and are displayed in Figs 3 and 4. Equation (53) again allows us to analyse the dependence of the second-order polarizability ␤ on the LCAO parameter c. We also assume once more that the excitation energy does not vary with c (see p. 144) and that both – [see (59)]. transition energies can be approximated by a mean value of ប␻ The results for the parallel transition 兩b典 ← 兩g典 for the three-centre system closely resemble those for the two-centre CT system. The dipole difference, ⌬␮bg z , decreases linearly from the positive maximum to the negative minimum value and passes through zero at c2 ⫽ 0.5. The absolute value of ⌬␮bg z , however, is reduced to (兹3/2)el because of the triangular geometry. The transition dipole ␮bg z follows a curve similar to the one of the linear CT system; the maximum (兹3/8)el occurs at c2 ⫽ 0.5. Hence, the behaviour of ␤zzz is also very similar to that of the two-centre system. Note that there are only (bg) , to ␤zzz. two-level contributions, ␤zzz For the perpendicular transition 兩a典 ← 兩g典, the results are different from the two-centre system. The dipole difference ⌬␮ag z decreases linearly from (兹3/2)el to zero while the corresponding transition dipole ␮ag y increases from zero to its

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Fig. 4 Illustration of the electron densities in MO ␺1, ␺2 and ␺3 as well as transition densities for c2 ⫽ 0.2 of the C2␯ symmetric two-electron DA2 system.

maximum value of el/兹2. This can be understood as follows. For c2 ⫽ 0, the HOMO is completely localized at centre 2. Upon excitation, electron density is transferred from centre 2 to centres 1 and 3; thus, the molecular dipole moment is increased. For c2 ⫽ 1, however, the electron density at centre 2 is zero. As a consequence, no charge is redistributed along the z axis, i.e. the difference of dipole moments is zero. The transition dipole ␮ag y , on the other hand, is maximized since the transition density is completely localized at centres 1 and 3. The excited-state transition dipole ␮ba y shows essentially trends which are opposite to those observed for ␮ag y because in ␺3 the electron density at centre 2 changes in an opposite sense compared to ␺1. (ga) According to (55) and (58) the two-level and three-level contributions, ␤zyy (gab) and ␤zyy , to the off-diagonal tensor element ␤zyy can be understood in terms of products of transition dipoles and dipole differences. Remarkably, the off-diagonal element is optimized at c2 ⫽ 0.5, where the diagonal element vanishes. This shows that both types of ␤ components may not be optimized simultaneously. At c2 ⫽ 2/3 the system reaches the limit of D3h symmetry where all three centres are equivalent. The corresponding MO are displayed in Fig. 5. In this case, ␺1 shows a symmetric electron density distribution and ␺2 and ␺3 constitute substates of a twofold degenerate MO of symmetry e. The symmetry requirement of (44) is clearly fulfilled. It is interesting to note that the two- and three-level contributions exhibit the same dependence, the latter being a factor of two larger. It is even more interesting to note that the dipolar two-level contribution does not vanish in the D3h limit. This is in apparent contradiction to the current opinion according to which ‘‘the two-level quantum model is irrelevant in the context of octopoles [such as D3h

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Fig. 5 Illustration of the electron densities in MO ␺1, ␺2 and ␺3 as well as transition densities for the D3h symmetric two-electron three-centre system (c2 ⫽ 2/3). ␺2 and ␺3 are substates of a twofold degenerate MO of symmetry e.

symmetric molecules] owing to the cancellation of all vectorial quantities, including in particular ⌬␮’’ (Zyss and Ledoux, 1994).

LOCAL FIELD FACTORS AND EFFECTIVE POLARIZABILITIES: THE REACTION FIELD MODEL

In condensed media consisting of molecules, the intermolecular forces such as permanent and induced dipole interactions are generally small compared to intramolecular chemical binding forces. Therefore, the molecular identities and properties are conserved to a certain extent. They nevertheless differ significantly from those of an isolated molecule in the gas phase. Therefore, both in linear and non-linear optics the question arises of how to relate molecular to macroscopic properties. More specifically, how do the individual permanent and induced dipole moments of the molecules translate into the macroscopic polarization of the medium? The main problem is to determine the local electric field E L acting on a molecule in a medium which differs from the average macroscopic field E (Maxwell field) in this medium. In the reaction field model (Onsager, 1936), a solute molecule is considered as a polarizable point dipole located in a spherical or ellipsoidal cavity in the solvent. The solvent itself is considered as an isotropic and homogeneous dielectric continuum. The local field E L at the location of the solute molecule is represented by (78) as a superposition of a cavity field E C and a reaction field E R (Boettcher, 1973). E L(t) ⫽ E C(t) ⫹ E R(t)

(78)

This relation has been used for both static and optical fields in the derivation of the well-known Clausius–Mosotti and Lorenz–Lorentz equa-

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tions (Boettcher, 1973). It has recently been generalized to include NLO experiments (Wortmann and Bishop, 1998). The frequency-dependent local and cavity fields The local field can be decomposed into Fourier components with amplitudes E L⍀, E C⍀ and E R⍀ related through (79), E L⍀ ⫽ E C⍀ ⫹ E R⍀

(79)

where the frequency ⍀ adopts all values involved in the NLO process under consideration, e.g. ⍀ ⫽ ␻, 2␻ for SHG and ⍀ ⫽ 0, ␻, 2␻ for EFISHG. The cavity field is related to the macroscopic field in the medium (‘‘Maxwell field’’) by (80), E C⍀ ⫽ f C⍀ E ⍀

(80)

where f C⍀ is the cavity field factor at frequency ⍀. The reaction field is related to the total (permanent and induced) dipole moment p⍀ of the solute molecule at frequency ⍀ by (81), E R⍀ ⫽ f R⍀ p⍀

(81)

where f R⍀ is the reaction field factor. The factors f C⍀ and f R⍀ have been calculated in the literature for spherical and ellipsoidal cavities (Brown, 1956; Boettcher, 1973). For a spherical cavity with radius a, they are given by (82) and (83), f C⍀ ⫽

f R⍀ ⫽

3␧⍀ 2␧⍀ ⫹ 1

2(␧⍀ ⫺ 1) 4␲␧0 a3(2␧⍀ ⫹ 1)

(82)

(83)

where ␧⍀ is the relative permittivity of the solvent at frequency ⍀. For later use, we introduce the abbreviation (84) in which the factor F R⍀ F R⍀ ⫽

1 1 ⫺ f R⍀ ␣sol(⫺⍀; ⍀)

(84)

describes the coupling of the solute dipole to its environment. The

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polarizability ␣sol is defined in (91) below. For a pure liquid (84) can be simplified to (85) F R⍀ ⫽

(␧⍀ ⫹ 2)(2␧⍀ ⫹ 2) 9␧⍀

(85)

and the product f C⍀ F R⍀ becomes equal to the commonly applied Lorentz factor L⍀ (86). L⍀ ⫽ f C⍀ F R⍀ ⫽

(␧⍀ ⫹ 2) 3

(86)

In the case of a static field, the macroscopic relative permittivity ␧0 has to be used in (82) for the cavity field factor, while the optical relative permittivity extrapolated to infinite wavelength ␧⬁ can be applied to estimate the static polarizability ␣(0; 0) in (84). In this way the Onsager–Lorentz factor for a pure dipolar liquid is obtained (87). f C0 F R0 ⫽

␧0 (␧⬁ ⫹ 2) (␧⬁ ⫹ 2␧0)

(87)

Solute polarizabilities The linear and non-linear polarizabilities of a molecule in solution differ from those of the isolated molecule in the gas phase since the molecular properties are modified by solute–solvent interactions. Some of these interactions are present even in the absence of externally applied static or optical fields. For molecules with a non-zero dipole moment ␮g in the electronic ground state the dominant interaction is usually due to the reaction field contribution E R␮0. The molecular dipole moment polarizes the solvent environment and thus generates a polarization field which interacts with the solute. This field is given by (88) (Boettcher, 1973; Wortmann and Bishop, 1998). E R␮0 ⫽ f R0 F R0 ␮g

(88)

In the case of a pure dipolar liquid, (88) simplifies to (89) (Wortmann et al., 1997) E R␮0 ⫽

(␧0 ⫺ 1)(␧⬁ ⫹ 2) g ␮ 6␲␧0 a3(2␧0 ⫹ ␧⬁)

(89)

For molecules with a symmetry axis, the permanent dipole ␮g will coincide with this axis. Since the field E R␮0 is associated with the permanent dipole

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moment of the solute, it has well-defined components in the molecule-fixed frame and may be considered as a perturbation to the solute molecule. Thus the permanent dipole moment ␮g,sol of the solute is larger than the dipole moment in the gas phase due to a reaction-field-induced component as represented in (90).

␮g,sol ⫽ ␮g ⫹ ␣(0; 0) ⭈ E R␮0 ⫹ . . .

(90)

Likewise, it is convenient to define solute polarizabilities in which the molecular polarizabilities are combined with the additional contributions induced by E R␮0. For the linear polarizability one obtains (91),

␣rrsol(⫺␻; ␻) ⫽ ␣rr(⫺␻; ␻) ⫹ ␤rrz(⫺␻; ␻, 0)EzR␮0 ⫹ . . .

(91)

where z is the dipole axis and ␣sol(⫺␻; ␻) may be regarded as the first-order polarizability of the molecule under the influence of its own static reac0 tion field E R␮ . Similarly, we obtain (92) for the second-order solute polarizabilities. sol ␤rst (⫺2␻; ␻, ␻) ⫽ ␤rst(⫺2␻; ␻, ␻) ⫹ ␥rstz(⫺2␻; ␻, ␻,0)EzR␮0 ⫹ . . .

(92)

An analogous relation holds for ␤sol(⫺␻; ␻, 0). The solute–solvent interactions are treated here on the level of dipolar reaction fields. Such effects are increasingly taken into account in quantum chemical calculations on linear and non-linear optical properties of molecules (Karelson and Zerner, 1992; Willets and Rice, 1993; Bishop, 1994a; Di Bella et al., 1994; Tomasi and Persico, 1994). Therefore, the solute polarizabilities represent a natural level at which theoretical and experimental results should be compared. Expressions (90)–(92) allow us to identify the major terms that are missing when results of NLO solution experiments are compared with theoretical calculations for molecules in the gas phase. It turns out that the contributions induced by the reaction field are substantial and often even larger than those of the isolated molecule (Mikkelsen et al., 1993). Effective dipole moment and polarizabilities We have shown above how the reaction field model can be used to estimate solute–solvent interactions in the absence of external fields. Now we introduce effective polarizabilities that connect the Fourier components of the induced dipole moment (33) with the macroscopic fields in the medium. In the linear case, the Fourier component p␻ induced by an external optical field can be represented by the product of the macroscopic field amplitude E ␻ and an – (⫺␻; ␻) using (93). effective first-order polarizability ␣ – (⫺␻; ␻) ⭈ E ␻ p␻ ⫽ ␣

(93)

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In the reaction field model expression (94) is obtained for the effective linear optical polarizability.

␣– (⫺␻; ␻) ⫽ f C␻ F R␻ ␣sol(⫺␻; ␻)

(94)

– (⫺␻; ␻) is involved in several linear optical experiments The polarizability ␣ including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor f C␻ and by the reaction field factor F R␻. For typical media the magnitude of the product f C␻ F R␻ is of the order of 1.3–1.4. In the case of a pure liquid this product simplifies to the Lorentz factor L␻, (86), and (94) simplifies to (95)

␣– (⫺␻; ␻) ⫽ L␻ ␣sol(⫺␻; ␻)

(95)

which is valid only under the very restrictive assumption that the system is a pure liquid consisting of molecules with an approximately isotropic polarizability tensor. The usual situation encountered in NLO experiments in condensed media, however, is a binary system consisting in general of a solute with large and anisotropic polarizability immersed in a solvent with a considerably smaller polarizability. Application of the Lorentz factor is then clearly inadequate since it implies equal polarizability volumes of solute and solvent molecules. It is also important to note that the cavity and reaction field factors in (94) correct only for differences between the local and the macroscopic field. This does not yield the polarizability of the isolated molecule, ␣, but rather the solute polarizability, ␣sol, which contains a contribution induced by the static reaction field E R␮0, (89). In the non-linear case, the Fourier component p2␻ induced by an external optical field E ␻ can be represented in terms of an effective second-order – polarizability ␤(⫺2␻; ␻, ␻) by (96), p2r ␻ ⫽

1 4

冘

– ␤rst(⫺2␻; ␻, ␻)Es␻ E␻t

(96)

st

– where ␤(⫺2␻; ␻, ␻) is given by (97) (Wortmann and Bishop, 1998). – ␤rst(⫺2␻; ␻, ␻) ⫽ F R2␻(F R␻ f C␻)(F R␻ f C␻) ␤sol rst (⫺2␻; ␻, ␻)

(97)

This polarizability is measured by electric-field-induced second-harmonic generation (EFISHG). Again, local field corrections for the optical fields do not yield the second-order polarizability ␤ of the free molecule but rather the solute polarizability ␤sol which contains a contribution induced by the static

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reaction field. No cavity field factor f C2␻ occurs in (97). Thus for a pure liquid (97) cannot be simplified with (86) to the commonly used correction (98), – ␤(⫺2␻; ␻, ␻) ⫽ L2␻ L␻ L␻ ␤(⫺2␻; ␻, ␻)

(98)

where Lorentz local field factors occur symmetrically for all frequency arguments of ␤. Comparison of (97) and (98) shows that use of the Lorentz factor at 2␻ would imply the presence of a cavity field at this frequency. This, however, is physically unrealistic since in an SHG experiment no external field of this frequency is applied. Use of the Lorentz factor for the generated harmonic may thus lead to substantial systematic underestimation of experimentally derived second-order polarizabilities since the cavity field factor f C2␻ ranges between 1.17 and 1.27 for typical organic liquids with ␧ ⬇ n2 of 1.8 to 2.7 (Wortmann and Bishop, 1998).

4

Non-linear optical susceptibilities and experimental methods to evaluate ␹ (2) and ␤

In this section, we investigate the relations between the macroscopic susceptibilities and the molecular polarizabilities. Consistent microscopic interpretations of many of the non-linear susceptibilities introduced in Section 2 will be given. Molar polarizabilities will be defined in analogy to the partial molar quantities (PMQ) known from chemical thermodynamics of multicomponent systems. The molar polarizabilities can be used as a consistent and general concept to describe virtually all linear and non-linear optical experiments on molecular media. First, these quantities will be explicitly derived for a number of NLO susceptibilities. Physical effects arising from ␹ (2) will then be discussed very briefly, followed by a survey of experimental methods to determine second-order polarizabilities. The basis of NLO-effects arising from susceptibilities of second order, ␹ (2), is the interaction of three electric fields with a material. The practical implementation of optical devices requires strong, coherent and monochromatic radiation and hence, laser technology. Not all of the interacting fields need to be optical fields, however. In devices that make use of the Pockels effect, an externally applied electric field is used to alter reversibly the refractive index of a material. In a second harmonic generation (SHG) process two photons of circular frequency ␻ can be transformed into one photon of frequency 2␻. SHG is the NLO effect used most for the evaluation of ␤-tensor elements in solution. A graphical presentation is commonly used to exemplify the SHG process and will also be used here as a first illustration of relations between microscopic and macroscopic NLO response. Consider an unsymmetrically substituted ␲ system like p-nitroaniline. It is intuitively understandable that

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Fig. 6 Graphical representation of SHG with p-nitroaniline [1] as an example of a 1D NLO-phore. Regions of charge depletion (䊉) and increase (䊊) for the S0–S1 electronic transition have been calculated by the PPP method.

the polarization in the donor and acceptor directions is different. For comparison, the calculated changes in charge are shown in Fig. 6 for the first optically allowed transition, which has a pronounced CT character. An oscillating charge will also emit radiation, but the emitted light will not only contain the frequency of the fundamental wave, as a Fourier analysis shows. In addition to a static component, the second harmonic is also present at double the frequency of the fundamental. In analogy, sum frequency and difference frequency generation may be understood as the interaction of two fields with different frequencies. SHG has been advocated as a means for upconversion of laser light that is easily accessible only for comparatively long wavelengths. This requires the development of transparent materials, a condition that has not been

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satisfactorily fulfilled so far (cf. p. 187). Now that the construction of commercially useful blue lasers on the basis of GaN is expected to be feasible within the near future, this demand has somewhat diminished.

MACROSCOPIC SUSCEPTIBILITIES AND MOLECULAR POLARIZABILITIES

Now we study in detail how the macroscopic susceptibilities are related to the molecular properties. A thorough understanding of these relations is essential for both the rational design of molecular NLO materials as well as the experimental determination of the molecular electric properties. Models for the interpretation of macroscopic susceptibilities in terms of molecular dipole moments and polarizabilities usually assume additive molecular contributions (Liptay et al., 1982a,c). Thus, an nth-order susceptibility ␹ (n) can be represented by (99) as a sum of terms that are proportional to concentrations cJ (moles per cubic metre, mol m⫺3) of the different constituents J of the medium.

␹ (n) ⫽

1 ␧0

冘

␨(n) J cJ

(99)

J

are It follows with (18)–(21) that the units for the quantities ␨(n) J C mn⫹1 V⫺n mole⫺1, i.e. those of an nth-order polarizability per mole. Therefore, we refer to ␨ (n) as an nth-order molar polarizability of the J constituent J. These quantities have to be calculated on the basis of a specific molecular model and appropriate local field corrections have to be taken into account. To simplify the notation, we will drop the index J in the following. A summation according to (99) is implied if the system consists of more than one constituent. The calculation of the molar polarizabilities, ␨(n), often involves statistical mechanical averaging over orientational distributions of the molecules. An important example is the distribution function w caused by dipole orientation in an externally applied static electric field E 0 because it describes the process of electric poling of NLO-phores. To second order in the field, the dipolar contributions to this (normalized) function are given by (100), w⫽1⫹

1 – 1 – –g 0 2 –g 2 0 2 – – ⭈ E ) ⫺ (␮ ) (E ) ] ⫹ . . . (␮g ⭈ E 0) ⫹ 2 2 [3(␮ 6k T kT

(100)

– where ␮g is an effective dipole moment of the ground state 兩g典 defined by (101). – ␮g ⫽ f C0 F R0 ␮g

(101)

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For a pure dipolar liquid the prefactors in (101) reduce to the Onsager– Lorentz factor (87). We use a double bar (–) for the effective dipole moment in the energy expression (100). Note that the dipole moment (101) is different from the effective solute dipole moment (102) which results from (84), (88) and (90)

␮–g ⫽ ␮g,sol ⫽ F R0 ␮g

(102)

In the following we present explicit relations for the molar polarizabilities for a number of important macroscopic susceptibilities. These equations will be used subsequently as a basis for the experimental determination of molecular polarizabilities. Refractive index of a liquid or liquid solution [␹ (1)(⫺␻; ␻)] The first-order susceptibility ␹ (1)(⫺␻; ␻) of a molecular liquid can be represented by (99) and the molar first-order polarizabilities of the form (103),

␨(1)(⫺␻; ␻) ⫽

NA Tr[␣– (⫺␻; ␻)] 3

(103)

where the Tr denotes the trace of the molecular first-order polarizability and NA is Avogadro’s constant (104). Tr[␣] ⫽ ␣xx ⫹ ␣yy ⫹ ␣zz

(104)

– was defined in (93). The relation between ␹ (1) The effective polarizability ␣ and the refractive index n was given in (13). Concentration-dependent measurements of the refractive index yield experimental information about the optical polarizability of the molecules. Static permittivity of a liquid or liquid solution [␹ (1)(0;0)] The first-order susceptibility of a molecular liquid ␹ (1)(0; 0) can be represented by (99) and the molar first-order polarizabilities are of the form (105),

␨(1)(0; 0) ⫽

冦

冧

NA 1 – – ␮g ⭈ ␮–g ⫹ Tr[␣– (0; 0)] 3 kT

(105)

where terms up to first-order in (100) are used in the averaging. The effective dipole moments in (105) have been defined in (101) and (102). Expression (22) states the relation between ␹ (1)(0; 0) and the static relative permittivity ␧r.

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Concentration-dependent measurements of this quantity yield the permanent dipole moment in the electronic ground state. The static first-order polarizability ␣(0; 0) in (105) can be estimated from refractive index measurements of ␣(⫺␻; ␻), (103). Second-order non-linearities of a molecular crystal [␹ (2)(⫺␻; ␻, 0) and ␹ (2)(⫺2␻; ␻, ␻)] The second-order susceptibility ␹ (2) of a molecular crystal can be represented in terms of the macroscopic molar polarizabilities ␨(2) by the relation (106),

冘冘 N

(2) ␨␮␣␤ (⫺2␻; ␻, ␻) ⫽

NA 2N p⫽1

– Rp␮r Rp␣s Rp␤t ␤rst(⫺2␻; ␻, ␻)

(106)

r,s,t

where the concentration c ⫽ N/(NA V) is determined by the number N of molecules in the unit cell volume V. Analogous relations hold for ␨(2)(⫺␻; ␻, 0) and ␤(⫺␻; ␻, 0). The Greek and Latin indices refer to Cartesian tensor components in the laboratory and molecule-fixed frame, respectively. The summation index p counts the N different (equivalent and non-equivalent) positions of the molecules in the unit cell. Rp is the transformation matrix between the molecule-fixed frame and the laboratory frame at position p. The factor 1–2 in (106) arises from the different prefactors of ␹ (2) and ␤ in (20) and (36). Second-order non-linearities of a poled polymer [␹ (2)(⫺␻; ␻, 0) and ␹ (2)(⫺2␻; ␻, ␻)] A second-order susceptibility ␹ (2) may be created in a polymer system. At a temperature Tpol, chosen to be slightly above the glass transition temperature TG of the polymer, an external static electric field of magnitude EZpol is applied (the electric field was chosen to define the z axis of the laboratory frame). Then, the polymer is cooled to room temperature (below TG) and the externally induced orientation of the dipolar chromophores becomes frozen. The maximum second-order susceptibility that can be produced by this poling cycle can be calculated by statistical-mechanical averaging with (100) to first order in the poling field. Assuming Kleinman symmetry, the results for the two independent components are given by (107) [cf. (28)]. (2) (2) ␨ZZZ (⫺␻; ␻, 0) ⫽ 3␨ZXX (⫺␻; ␻, 0) ⫽

NA EZpol – – – ␮ g ⭈ ␤ ␯(⫺␻; ␻, 0) 10kTpol

(107)

Analogous relations hold for ␨(2)(⫺2␻; ␻, ␻) and ␤(⫺2␻; ␻, ␻). The quantity ␤␯ in (77) is the so-called ‘‘vector part’’ of the second-order polarizability which

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the orientation distribution projects out of the third-rank tensor ␤ onto the dipole moment vector. Assuming the validity of Kleinman symmetry, the component ␤r␯ of the vector part can be defined by (108).

␤r␯ ⫽

冘

␤rss.

(108)

s

The linear electro-optic effect arising from ␹ (2) describes a change in the refractive index of the material linearly proportional to an externally applied electric field. This effect is of technological importance in high-frequency modulation of optical signals. It cannot only be observed in permanently poled polymer systems (Burland et al., 1994; Dalton et al., 1995; Lindsay and Singer, 1995; Marks and Ratner, 1995) but also in inorganic or organic crystals (Bosshard et al., 1995). UV–visible absorption is not as critical as for SHG since typical optical communication wavelengths are in the near infrared at 1.3 and 1.55 ␮m. Still, the low-energy tails of the intense CT bands of the NLO molecules may represent a primary source of the absorptive losses observed (Moylan et al., 1996). Alternatively, higher overtones of IR vibrations may also lead to losses. Third-order non-linearity ␹ (3)(⫺2␻; ␻, ␻, 0) of a poled liquid or liquid solution The susceptibility ␹ (3)(⫺2␻; ␻, ␻, 0) is determined in the EFISHG experiment [electric-field-induced second harmonic generation; see below (Levine and Bethea, 1974, 1975)]. In order to measure the two independent components (3) (3) ␹ZZZZ and ␹ZXXZ of this susceptibility, the experiment can be performed under two polarization conditions, the incident IR photons being polarized parallel and perpendicular to the externally applied field (Wortmann et al., 1993). For theoretical treatments see also Andrews and Sherborne (1986), Wagnie`re (1986) and Andrews (1993). A concentration series finally yields the molar polarizabilities (Kleinman symmetry assumed) through (109), (3) (3) ␨ZZZZ (⫺2␻; ␻, ␻, 0) ⫽ 3␨ZXXZ (⫺2␻; ␻, ␻, 0)

⫽

冥

冤

NA 1 – – ␮–g ⭈ ␤␯(⫺2␻; ␻, ␻, 0) ⫹ ␥– s(⫺2␻; ␻, ␻, 0) 30 kT

(109)

where ␥s is the ‘‘scalar part’’ of the third-order polarizability defined by (110).

␥s ⫽

冘冘 r

s

␥rssr.

(110)

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– The effective polarizability ␤ was defined in (97). The general result for planar C2␯ symmetric molecules (42) without the assumption of Kleinman symmetry is shown in (111) and (112), (3) ␨ZZZZ ⫽

NA – – – – ␮–zg(3␤zzz ⫹ ␤zyy ⫹ ␤yzy) 90kT

(111)

(3) ␨ZXXZ ⫽

NA – – – – – ␮zg(␤zzz ⫹ 2␤zyy ⫺ ␤yzy) 90kT

(112)

where the contribution of ␥s was neglected because it is small in general. Expressions (111) and (112) allow the experimental determination of the difference ␤zyy ⫺ ␤yzy (Wortmann et al., 1993). If Kleinman symmetry is present, this difference is zero. Kerr susceptibility of a liquid or liquid solution [␹ (3)(⫺␻; ␻, 0, 0)] The quadratic effect of an externally applied field on the refractive index n is described by the third-order susceptibility ␹ (3)(⫺␻; ␻, 0, 0) (Kerr suscep(3) (3) tibility). The two independent components ␹ZZZZ and ␹ZZXX can be interpreted in terms of molar polarizabilities. The results for C2␯ symmetric molecules with only one significant component ␤zzz of the second-order polarizability are expressed in (113) and (114), (3) ␨ZZZZ (⫺␻; ␻, 0, 0) ⫽

冤

冥

(113)

冤

冥

(114)

NA 6 –g – 4 –g 2 – ␮z ␤zzz(⫺␻; ␻, 0) 2 2 (␮z ) ␦␣ z(⫺␻; ␻) ⫹ 90 3k T kT

(3) ␨ZZXX (⫺␻; ␻, 0, 0) ⫽

NA ⫺2 – g 2 – 2 –g – ␮z ␤zzz(⫺␻; ␻, 0) 2 2 (␮z ) ␦␣ z(⫺␻; ␻) ⫹ 90 3k T kT

where small terms arising from the anisotropy of the static polarizability ␣(0; 0) and an isotropic contribution ␥(⫺␻; ␻, 0, 0) have been neglected. ␦␣z is the anisotropy of the first-order polarizability and is given by (115).

␦␣z ⫽ ␣zz ⫺ 1–2 (␣xx ⫹ ␣yy)

(115)

For typical NLO-phores the (␮g)2 ␦␣ term is much larger than the ␮g ␤ term. The molar polarizabilities (113) and (114) were recently used to derive a molecular figure of merit (FOM) for NLO-phores for organic photorefractive (PR) materials with low glass-transition temperature (Wortmann et al., 1996). Rational design of NLO-phores based on this FOM led to organic PRmaterials with unprecedented performance (Würthner et al., 1997).

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Absorption coefficient of a liquid solution [Im{␹ (1)(⫺␻; ␻)}] In the region of an absorption band of the medium, the first-order susceptibility ␹ (1)(⫺␻; ␻) has to be treated as a complex quantity, the real and imaginary part determining the refractive index (13) and the absorption coefficient of the medium (12), respectively. The imaginary part of the molar first-order polarizabilities is given by (116), Im{␨(1)(⫺␻; ␻)} ⫽

NA – (⫺␻; ␻)}] ⫽ ln(10)c ␧ n␻ ␧(␻) Tr[Im{␣ 0 0 3 ␻

(116)

where ␧(␻) is the molar decadic absorption coefficient of the solute determined by Beer’s law. The factor ln(10), i.e. the natural logarithm of 10, is due to the conversion between the natural absorption coefficient (12) and the decadic coefficient ␧. The real and the imaginary part of ␨(1)(⫺␻; ␻) and ␹ (1)(⫺␻; ␻) are related by a Kramers–Kronig transformation (24). Integration of the absorption spectrum over an isolated transition 兩a典 ← 兩g典 yields the magnitude of the transition dipole ␮ag according to (117).

冕

2␲ 2 N A ␧(␻) d␻ ⫽ 兩 ␮ag 兩2 3 ln(10)hc ␧ 0 0 ␻ band

(117)

The square of the transition dipole occurs in the two-level contributions ␣(ag) and ␤(ag) to the first-order and second-order polarizability and is important for their optimization.

Electrochromism of a liquid solution [Im{␹ (3)(⫺␻; ␻,0,0)}]: electro-optical absorption measurements (EOAM) The quadratic effect of an externally applied field E 0 on the absorption coefficient is described by the imaginary part of the third-order susceptibility ␹ (3)(⫺␻; ␻, 0, 0). E 0 influences the molar decadic absorption coefficient of the solute. The absorption coefficient in the presence of the field ␧E is a quadratic function of the applied field strength (118),

␧E(␻) ⫽ ␧(␻)[1 ⫹ L(␻)(E 0)2 ⫹ . . .]

(118)

where the quantity L is a measure of the relative change of ␧ induced by E 0. L depends on the frequency as well as on the angle between the polarization vector of the incident light and the applied field. A detailed microscopic interpretation of the electrochromic effect in liquid solutions has been given (Liptay and Czekalla, 1960; Labhart, 1967). The results are written here with those of the molar polarizabilities introduced above. The relation between the

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traditional quantity L and the molar polarizabilities introduced above is found to be (119). Im{␨(3)(⫺␻; ␻, 0, 0)} ⫽ 1–3 ln(10)c0 ␧0 n␻

L(␻) ␧(␻) 1 ⫽ –3 L(␻) Im{␨(1)(⫺␻; ␻)} ␻

(119)

For an isolated electronic transition between a ground state 兩g典 and an excited state 兩a典, the molar polarizabilities can be represented by (120) and (121). (3) Im{␨ZZZZ }⫽

冤

NA –g –g 2 – – 2[3(m ⭈ ␮ ) ⫺ (␮ ) ] Im{␨(1)} 90k2 T 2 ⫹

(3) Im{␨ZZXX }⫽

6 – g ⭈ ⌬␮ – ag) ⫹ 2(m ⭈ ␮ – g)(m ⭈ ⌬␮ – ag)]1 d Im{␨(1)} [(␮ ប d␻ kT

冥

(120)

冥

(121)

冤

NA – – ⫺[3(m ⭈ ␮ g) ⫺ (␮ g)2] Im{␨(1)} 90k2 T 2 ⫹

6 –g – ag –g – ag 1 d – – – – ⭈ ⌬␮ ) ⫺ (m ⭈ ␮ )(m ⭈ ⌬␮ )] Im{␨(1)} [2(␮ ប d␻ kT

A number of usually small terms related to the ground-state polarizability, the transition polarizability and the square of the dipole difference have been neglected in (120) and (121). Complete expressions have been given (Liptay, 1974; Wortmann et al., 1992). Neglecting changes of the polarizability upon excitation, the effective dipole difference in solution in (120), (121) is given by (122). – ⌬␮– ag ⫽ f C0 F R0 ⌬␮ag

(122)

The real and imaginary parts of ␨(3)(⫺␻; ␻, 0, 0) and ␹ (3)(⫺␻; ␻, 0, 0) are related (Kuball and Galler, 1967) by a Kramers–Kronig transformation given in (24).

EXPERIMENTAL DETERMINATION OF MOLECULAR SECOND-ORDER POLARIZABILITIES

The linear and non-linear polarizabilities of organic molecules are usually determined from measurements of macroscopic susceptibilities of liquid solutions. Classical examples are the measurements of the refractive index, n, or the relative permittivity of pure organic liquids and their interpretation by the well-known Lorentz–Lorenz and Clausius–Mosotti equations. These

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measurements yield the trace of the linear polarizability ␣ and the groundstate dipole moment, ␮g, of the molecules. The underlying theory of dielectric polarization has been presented comprehensively (Boettcher, 1973). Alternatively, the more general expressions (103) and (105) could be applied (Liptay et al., 1982a). In general, it is not possible to study compounds in the form of pure liquids. Instead, concentration series of dilute solutions of the molecules in inert solvents have to be investigated. A molar polarizability at infinite dilution can be derived by a linear regression analysis. This extrapolation scheme has been carefully analysed (Liptay et al., 1982a,c) for the case of linear susceptibilities and a general scheme to derive molar polarizabilities presented. A consistent interpretation requires consideration of corrections arising from the concentration dependence of the local field factors. This is also true for non-linear optical experiments as has been discussed for EFISHG measurements (Steybe et al., 1997). Two methods are available for the evaluation of NLO properties in solution, electric-field-induced second harmonic generation (EFISHG) and hyper-Rayleigh scattering (HRS). Both methods yield complementary information about linear combinations of ␤ elements. Use of polarized fundamental light allows further information about molecular symmetry and tensor elements to be extracted. Electric-field-induced second harmonic generation (EFISHG) The EFISHG phenomenon, i.e. SHG under the influence of an externally applied electric DC field, was first observed with calcite (Terhune et al., 1962). It was later used for the study of pure liquids and gases and solutions (Mayer, 1968; Hauchecorne et al., 1971; Levine and Bethea, 1974, 1976; Ward and Bigio, 1975; Miller and Ward, 1977; Shelton, 1990). Today, EFISHG experiments on liquid solutions (Singer and Garito, 1981; Kajzar et al., 1987; Singer et al., 1989; Cheng et al., 1991b; Bosshard et al., 1992) belong to the standard techniques for determination of second-order polarizabilities ␤(⫺2␻; ␻, ␻) of organic molecules. A solution of the chromophore is poled by an external electric field; thus the statistical centrosymmetry of solutions is broken. Hence, the method is not applicable to molecules without a dipole moment and for electrically-conducting materials like salts. The solution is irradiated with an intense laser pulse, and a small fraction of it is frequency-doubled. Typical lasers are Q:switched Nd:YAG lasers operating at 1064 nm or higher. The coherently generated SHG signal is detected by a sensitive photomultiplier. Variation of the optical path length through the solution leads to a fringe pattern. The amplitudes of the fringes are related to the EFISHG susceptibility ␹ (3)(⫺2␻; ␻, ␻, 0) and a concentration series yields the molar polarizabilities ␨(3)(⫺2␻; ␻, ␻, 0) given in (109). Note again that EFISHG is a third-order process because two fundamental alternating fields, and an additional static electric field, interact with each other. Two molecular

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contributions occur in (109): a temperature-independent contribution through ␥, and a temperature-(orientation-) dependent one through ␤. The former is usually small and therefore neglected. In principle, EFISHG measurements conducted at different temperatures can be used to separate both terms (Lebus, 1995); experimentally, however, this is technically very difficult. Evaluation of EFISHG measurements also requires the application of local field corrections and the knowledge of the molecular ground-state dipole moment ␮g. The latter can be determined by permittivity measurements or electro-optical absorption (EOAM; see pp. 160 and 167). Sometimes ␮␤ values are quoted if the molecular ground-state dipole is not known, and also because it is this product that defines the figure of merit for the electro-optic effect. EFISHG measurements are usually made relative to a standard material, most often quartz. The SHG susceptibility of quartz, however, has been the subject of much debate (Roberts, 1992). An older and often-used value is d11 ⫽ 0.5 pm/V at 1064 nm (Kurtz et al., 1979). The most recent value (Mito et al., 1995) is about 40% lower than the older value and seems to be supported by a comparative HRS/EFISHG study (Kaatz and Shelton, 1996) which concludes d11 ⫽ (0.30 ⫾ 0.02) pm/V. Other values are also being used. Besides a clear statement of which ␤ convention has been chosen, published results of EFISHG measurements should therefore always indicate the quartz reference used in the calibration. EFISHG yields projections of the ␤ tensor on the direction of the molecular dipole moment (z-axis). Hence a specific linear combination of elements is obtained and not a unique ␤-value that is sufficient to characterize the molecular second-order NLO response. This is a serious limitation of the technique; some components of ␤ may be large but will not show up in the experimental results because their projection on the direction of the molecular ground-state dipole is zero. However, the use of polarized incident light with polarization directions parallel and perpendicular to the externally applied electric field allows the extraction of further information on the ␤ tensor. For planar molecules conjugated in the yz plane, components with contributions of the x direction may be safely ignored. Two linear combinations, 1␤z and 2␤z, of tensorial elements may then be determined (Wortmann et al., 1993), (123) and (124): 1

␤z ⫽ ␤zzz ⫹ ␤zyy

(123)

2

␤z ⫽ ␤zzz ⫹ ␤yzy

(124)

Hyper-Rayleigh scattering (HRS) The HRS set-up (Terhune et al., 1965; Clays and Persoons, 1991, 1992; Clays et al., 1994) is in principle similar to the EFISHG experiment, but no electric

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field is applied. Hence, the statistical centrosymmetry of the solution is not broken and HRS relies on instantaneous local deviations (‘‘fluctuations’’) of the orientational distribution from the isotropic average. It is thus an incoherent process where the SH light has to be integrated over a certain solid angle. Hence, the SHG efficiency with the HRS set-up is very low [10 photons for a pump pulse of 2 ⫻ 1015 photons have been reported for a 0.1 M solution of p-nitro-N-methylaniline (Morrison et al., 1996)] and care must be exercised to exclude extraneous contributions to the observed signal. All compounds with appreciable solubility may be investigated, provided they are stable under the irradiation conditions of the experiment. This includes salts and molecules without permanent dipole moment. A concentration series is required to separate solute and solvent contributions to the HRS signal. The apparatus constant requires again the application of some standard. Ultimately, HRS measurements are also referenced against quartz as in EFISHG, but this cannot be done directly. Initially, the solvent has been used as an internal reference (Clays and Persoons, 1991, 1992). However, the ␤ values of common solvents are very small and large errors in their determination have been noted unless sources of error (like scattering from small undissolved particles) were carefully excluded. The value for methanol has been found to be five times lower than initially reported as shown by subsequent measurements on carefully purified solvents (Morrison et al., 1996). It is therefore advisable to use a standard solution of p-nitroaniline as an external reference (Morrison et al., 1996; Wortmann et al., 1997). The SHG detected in the HRS experiment is a quadratic function of the incident intensity. The molar contributions to the HRS signal are thus proportional to the rotational averages 具␤2典 of the second-order polarizabilities. HRS thus complements the information obtained from EFISHG. Again, use of polarized light allows the determination of two different linear combinations of tensor elements. Thus the HRS experiment is usually carried out in a 90⬚ geometry for two different polarization conditions where the incident laser beam is propagating in the X-direction and is linearly polarized perpendicular (Z) to the scattering plane (XY). The frequencydoubled photons are detected in the Y direction with parallel (Z) or perpendicular (X) polarization. Although signal-to-noise ratios suffer with respect to unpolarized detection, polarization-dependent measurements yield information on the dimensionality of ␤ (Heesink et al., 1993; Kaatz and Shelton, 1996). Molecular symmetry manifests itself in the two averages, 具␤2典ZZ and 具␤2典ZX, obtained for parallel (Z) and perpendicular (X) polarization of the scattered 2␻ (SH) photons relative to the polarization (Z) of the ␻ (fundamental) photons and the scattering plane (XY). The ratios of signals observed – called depolarization ratios – can be substantially different for different molecular symmetries because different numbers of tensor elements are significant (Bersohn et al., 1965; Cyvin et al., 1965). For example a combination of polarization-dependent HRS and EFISHG measurements was

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used (Wolff et al., 1997) to determine all ␤ components of a series of planar C2␯ symmetric NLO-phores without the assumption of index permutation symmetry (Kleinman symmetry). For this case the corresponding rotational averages are given by (125) and (126) 具␤2典ZZ ⫽

1 (24␤2yzy ⫹ 24␤yzy ␤zyy ⫹ 6␤2zyy ⫹ 24␤yzy ␤zzz ⫹ 12␤zyy ␤zzz ⫹ 30␤2zzz) 210 (125)

1 具␤2典ZX ⫽ (16␤2yzy ⫺ 12␤yzy ␤zyy ⫹ 18␤2zyy ⫺ 12␤yzy ␤zzz ⫹ 8␤zyy ␤zzz ⫹ 6␤2zzz) 210 (126) These expressions simplify considerably for higher symmetries where the number of independent ␤ components is lower. For a linear system with only one significant component ␤zzz the depolarization ratio, ␳, defined as ⫽ 具␤2典ZZ /具␤2典ZX is expected to be equal to 5. For C3h symmetric molecules the relations (125) and (126) simplify with (43) to (127) and (128). 具␤2典ZZ ⫽

48 2 (␤yyy ⫹ ␤2zzz) 210

(127)

具␤2典ZX ⫽

32 2 (␤yyy ⫹ ␤2zzz) 210

(128)

The same relationships hold for D3h symmetric molecules when ␤yyy is set to zero (44). For both C3h and D3h symmetry, the depolarization ratio is expected to be ␳ ⫽ 1.5. However, the converse need not be true: a depolarization ratio of close to 1.5 does not prove a point group with a threefold rotational axis. The molecule may be conformationally flexible; then HRS detects a superposition of 具␤2典 of the different conformers. A case in point is 1,3,5-trinitro-2,4,6-triisopropylaminobenzene which exists as an interconverting mixture of strongly distorted boat and twist-boat forms (Wolff et al., 1993), with approximate CS and C2 symmetry, respectively, but still shows a depolarization ratio of close to 1.5 (Verbiest et al., 1994; Wortmann et al., 1997). Even in the case of conformational homogeneity it may still strongly deviate from the ideal symmetry provided the impact of conformational distortion on the electronic properties is not great. In a parametric light-scattering experiment, two photons of different frequencies ␻1 and ␻2 are incoherently scattered at the sum frequency ␻1 ⫹ ␻2. The polarization dependence of this experiment yields information on the two-dimensionality of ␤ (Verbiest and Persoons, 1994).

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Pitfalls of the HRS method While the HRS technique seems to be more straightforward both in experimental setup and evaluation of the results several systematic errors need to be pointed out that in the past have led to claims of unrealistically high second-order polarizabilities. With very high fundamental intensities (often preferable in order to increase signal-to-noise ratios), dielectric breakdown occurs and spurious increases of SHG intensities of up to a factor of three have been reported (Stadler et al., 1996a). This may be recognized from the non-quadratic dependence of the alleged SHG signal on I(␻). More importantly, due to the incoherent nature of HRS other processes may also contribute to the signal observed. It may contain significant or even dominant contributions from fluorescence excited by multi-photon processes. Failure to recognize fluorescence leads sometimes to grossly overestimated values of ␤. Until about 1995/6, the issue of fluorescence had not been addressed for the HRS technique. Claims of unusually high NLO activities without specifically stating how fluorescence was taken into account may therefore be regarded with suspicion. Thus, several claims of record hyperpolarizabilities (Laidlaw et al., 1993; Zyss et al., 1993; Dhenaut et al., 1995) had to be retracted, or were at least questioned (Laidlaw et al., 1994; Flipse et al., 1995; Kaatz and Shelton, 1996; Morrison et al., 1996). A quadratic dependence of the alleged SHG signal on I(␻) and a linear dependence on solute concentration alone is not a sufficient guarantee for a true SHG process. Also note that a shift to higher fundamental wavelengths (e.g. from ␭ ⫽ 1064 nm even to ␭ ⫽ 1907 nm) may be a safeguard against two-photon, but not three-photon, fluorescence. Fluorescence may be discriminated from a true SHG signal by the following features: • SHG has a very narrow spectral distribution but fluorescence emission usually has quite a broad one. • SHG has a very short rise and fall-off time in contrast to the typical nanosecond time-scale of fluorescence. • SHG intensity scales with the square of the pump intensity. Hence, the signal intensities obtained with a sequence of bandpass filters with decreasing spectral width (Song et al., 1996; Wortmann et al., 1997) may be used to extrapolate to infinite spectral width. Alternatively, but at the cost of signal-to-noise ratio, a scanning monochromator (Kaatz and Shelton, 1996; Song et al., 1996) can be employed. Lasers with short pulse durations (picoseconds) can be used to distinguish between the different time-scales (Flipse et al., 1995; Morrison et al., 1996; Noordman and van Hulst, 1996). Finally, the ratio of signal intensities using a wide range of laser pump intensity may be used to distinguish the two-photon SHG process from a three-photon fluorescence, but this method is obviously not capable of eliminating two-photon fluorescence (Stadler et al., 1996a).

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Electro-optical absorption measurements (EOAM) In EOAM (Liptay, 1969, 1974, 1976; Baumann, 1976; Liptay et al., 1982c; Bublitz and Boxer, 1997), a static electric field is used to orient molecules in a dilute liquid solution. The UV–visible absorption is then studied with light that is linearly polarized either along the field direction, or perpendicular to it. EOAM were established in the early 1960s as a method for the determination of excited-state dipole moments of molecules (Liptay and Czekalla, 1960; Labhart, 1967). The measured signal arises from a combination of two factors. Since the molecules are partially aligned in the direction of the applied field, the first effect is the electrodichroic effect. Since the molecular dipole moment upon excitation is also changed by the field, the second is the Stark effect (band shift). The electric-field-induced change of the molar absorption coefficient or the molar polarizability given in (120) and (121) is recorded for a number of frequencies within the absorption band. Then a multilinear regression analysis based on the absorption spectrum and its derivatives yields the magnitudes of the electric dipole moment ␮g, the dipole difference ⌬␮ag between the ground state, 兩g典, and excited state, 兩a典, and the direction of the electric transition dipole ␮ag in the molecule-fixed coordinate system. EOAM have also been carried out in thin polymer films (Havinga and van Pelt, 1979; Blinov et al., 1994), Langmuir–Blodgett layers (Blinov et al., 1992), and crystals (Slawik and Petelenz, 1992). While the method is predominantly used to investigate charge-transfer processes during electronic excitation, it may be recalled that EOAM allows the determination of the crucial parameters in the two-level equations of (53)–(58). It may therefore also provide valuable information on second-order polarizabilities. This was first demonstrated when the results of polarizationdependent EOAM and EFISHG experiments for ␤ of p-nitroaniline and 3,5-dinitroaniline were compared (Wortmann et al., 1993). This method allows the estimation of ␤ and is superior in this respect to estimates on the basis of solvatochromism (Paley et al., 1989; Paley and Harris, 1991; Würthner et al., 1993). Good agreement between ␤ values independently derived from EOAM and EFISHG was observed in many cases (Blanchard-Desce et al., 1995, 1997b). The EOAM method has been used extensively as a screening tool (Würthner et al., 1993, 1997; Boldt et al., 1996; Blanchard-Desce et al., 1997a; Steybe et al., 1997; Wolff et al., 1997). Similar experiments were also performed later (Bublitz et al., 1997a,b).

Solid-state methods: Kurtz powder test A quick semi-quantitative check for the efficiency of NLO crystals is the Kurtz method (Kurtz and Perry, 1968) where the SHG intensity of a powdered material with defined grain size is measured against a standard, most often urea or, again, quartz (both of which have quite moderate efficiencies).

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Systematic collections of organic SHG powder test data have been compiled (Nicoud and Twieg, 1987b; Nalwa and Miyata, 1997). 5

Optimization of second-order polarizabilities: applications to real molecules

␴ SYSTEMS AND ONE-DIMENSIONAL ␲ SYSTEMS The theoretical considerations that were developed in Section 3 for one- and two-dimensional NLO-phores merely required the incorporation of donor and acceptor groups coupled to each other. No assumptions were made concerning the structural molecular equivalent of the LCAO coupling parameter, c, between donor and acceptor. In the one-dimensional molecules to be dealt with first, a single donor–acceptor pair is present. For this case, the consequences of the different extent of coupling on molecular polarizabilities were calculated on the basis of a single electronic transition with chargetransfer character (p. 139). The crucial parameters ⌬␮ag, ␮ag, and ⌬Eag (ប␻ag) cannot be optimized independently, and the second-order polarizability was shown to pass through two extrema in the regime of fairly strong coupling. As a consequence of the molecular architecture, in 1D systems only a single, diagonal component, ␤zzz, is dominant. The following conclusions had been drawn for the optimization of 1D NLO-phores (cf. also Scheme 1): • Since the square of the transition dipole, ␮ag, is a measure for the intensity of the transition, the latter should be highly electric dipole allowed. In other words, the (integral) extinction coefficient of the transition should be high. • The difference between ground-state and excited-state dipole moments, ⌬␮ag, should be high. Since dipole moment equals charge times distance, there should be strong charge transfer over a long distance from a donor to an acceptor. • The gap between ground and excited state, ⌬Eag, usually identified with the HOMO–LUMO gap, manifests itself in a low transition energy. This is reflected in a bathochromic shift in the UV–visible spectrum. So, in principle, all that remains to do now is to translate these requirements into a real chemical structure. However, the chemist’s role in the optimization is not merely that of a craftsman. Chemical systems are still much too complex for a thorough theoretical treatment, and chemical intuition and knowledge are required not just for the synthesis of chemical structures whose properties have already been anticipated. The 1D model developed above is quite crude in the following respects. • The influence of only one electronic transition between donor and acceptor has been taken into account; intrinsic effects of any intervening ␴ or ␲

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system are only indirectly reflected through the degree of coupling between the donor and acceptor. While this is a very good approximation for nearly all of the closed-shell molecules that will be dealt with in this section, the second-order polarizabilities of real molecules will always partly arise also from higher-order transitions which are neglected in the one-dimensional treatment. At (often) the expense of physical understanding, they are included in numerical computational procedures. • It should be noted that the surrounding medium has a great influence on the NLO properties (see pp. 148 and 183). Computations for the gas phase are not expected to be quantitatively accurate. The one-dimensional model is clearly inappropriate for most molecules with more than one pair of donor and acceptor in conjugation. They will be dealt with later. A more elaborate theoretical model for these two-dimensional NLO-phores that takes into account three-level contributions was developed in Section 3 (p. 145). • Molecules in the condensed phase polarize each other, a phenomenon well known from such phenomena as solvatochromism. Accordingly, all other parameters being equal, the molecular second-order polarizabilities also depend on the surrounding medium. As solvatochromism in the molecules considered here is a manifestation of a change in the energy of the CT transition, it can in principle be accommodated by changing the coupling parameter accordingly. Despite these shortcomings it will become clear that in the one-dimensional NLO-phores treated in this section, which display a wide range of seemingly disparate chemical structures, the crude model works surprisingly well. Thus, as a consequence of the validity of the two-state model, their second-order polarizabilities in principle ‘‘reduce to p-nitroaniline’’. The reader may even gain the impression that the efforts to improve on the hyperpolarizabilities of even the simplest and most easily accessible ␲ systems (like p-nitroaniline) have been futile. It is true that an efficiency–transparency trade-off exists: At a given wavelength of absorption (related to ⌬E) a maximum value for the second-order molecular polarizability per volume element exists which is not tremendously different from that of very basic unoptimized ␲ systems. However, for applications like the electro-optical effect, a bathochromic shift of the UV–visible absorption is tolerable so that to strive for maximum hyperpolarizabilities is a viable quest. Furthermore, molecular structures with the same intrinsic second-order polarizabilities may differ substantially in their chemical stabilities and their abilities to be incorporated into ordered bulk structures. In the following discussion of 1D NLO-phores, the molecules will be classified according to the mode of coupling between the donor and acceptor: through space (donor–acceptor complexes), through a ␴ system, and through a ␲ system. The last is by far the largest class of NLO-phores investigated. Several caveats should be observed throughout this and the following sections:

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(i) Confusion in the realm of NLO arises both from the use of different conventions for the power expansion (5) and from different values for the standards used to calibrate the measurements (absolute value of quartz, internal/external reference). Values may differ by a maximum factor of 10. We have not always undertaken the sometimes difficult task of finding out which convention was adopted and (even more arduous) which value for quartz was used. Therefore, we have quoted second-order polarizabilities in terms of ␤0 of p-nitroaniline in dioxane (␭max ⫽ 354 nm, ␤0 ⫽ 13.5 C m3 V⫺2, T convention, relative to quartz d11 ⫽ 0.5 pm/V at 1064 nm). p-Nitroaniline is a truly one-dimensional NLO-phore with one significant component as has been verified experimentally by depolarized EFISHG (Wortmann et al., 1993). If different standards and conventions are taken into account, the values measured by different groups are quite consistent. Note that the intrinsic ␤0 of p-nitroaniline depends on the solvent, even when normalized for the solvatochromic shift of the CT absorption. We have chosen the lowest intrinsic ␤0; it is higher by a factor of 1.6 in very polar solvents (see p. 183). Also note that ‘‘␤ values’’ from HRS measurements of molecules with several significant tensor elements will not allow a true comparison of ␤zzz. (ii) Systematic sources of error in the experimental determinations are often overlooked or ignored. Hence, experimental values are sometimes grossly overestimated (in principle, underestimation might also occur in the case of resonance, but seems empirically very rare). The HRS experiment (see pp. 163 and 166) is especially notorious in this respect because there are many sources of spurious signals from multiphoton processes. (iii) It is also common but incorrect to use the frequency dispersion expression [(51), (52)] of the two-state model in order to extrapolate to static ␤0 values in cases where considerable absorption of the second harmonic is observed. The two-state model is not applicable for these cases of strong resonance enhancement. Strictly speaking, not even semiquantitative statements about ␤ are possible. However, as a large body of data, especially on organometallic systems, could not be cited at all under these strict conditions, we will use these values, but insert a ‘‘⬇’’ before them. (iv) A direct comparison of values obtained with EFISHG and HRS is possible only in 1D NLO-phores with one dominant tensor element of ␤. Both methods gauge different combinations of tensor elements (see Section 4). (v) The evaluation of differences in performance for different molecules has to take into account of the different molecular sizes. They may be approximated roughly by the respective molecular masses. We have computed such ‘‘reduced’’ values for the examples quoted. It is to be

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understood that use of a lighter counterion with ionic compounds, and a lighter ligand in organometallic compounds may improve on these values somewhat. A fair comparison would also consider the differences in (␭max)2 because transparency is a desirable property. For several examples we have performed this second ‘‘reduction’’ although ␭max is only a rough characterization of a CT band whose shape may vary considerably. Coupling through space: charge-transfer complexes Coupling of a donor and an acceptor through space is weak in most cases, and maximum NLO efficiency cannot be achieved. While the energy gap, ⌬E, can be quite low (as in the classical example of the intensely coloured quinhydrone), the CT transition is broad and of low extinction. This translates into a low scalar value of ␮ag. Also, the change in dipole moment, ⌬␮ag, is not very pronounced. There are further impediments to the successful use of such aggregates. In intermolecular complexes it is difficult to control the relative orientation of donor and acceptor, and in solution an equilibrium between the complex and its constituents is established. Furthermore, the outcome of electrical poling in a polymer structure is not predictable. True intramolecular complexes of the correct mutual orientation of donor and acceptor belong to the next class of compounds because they are coupled at least through a ␴ framework. In general, they are more difficult to synthesize and probably for that reason have received little attention with the exception of inorganic intervalence complexes. Computations have shown that certain D–A complexes in idealized geometries [2] may show a large change of dipole moment for the CT transition (Di Bella et al., 1993). They also show the complexes to be inferior to conjugated ␲ systems like p-nitroaniline: comparable NLO response is achieved only at the expense of considerable bathochromic shifts and more than double the molecular weight. Experimental investigations are unfortunately lacking (but see [39] in Table 4, Fig. 8, and Scheme 15). An unexpectedly high NLO response was claimed for 1,8-diarylnaphthalenes [3] on the basis of HRS measurements (Bahl et al., 1995). As the two interacting phenyl rings adopt conformations almost at right-angles with respect to the naphthalene ring, it was assumed that the donor and acceptor portions are

Scheme 3

NLO-phores with charge transfer through space.

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coupled only through space (Cozzi et al., 1995). Unfortunately, sources of spurious HRS signals were not eliminated, and subsequent measurements on one representative of this class of compounds showed the HRS signal to be due almost entirely to multiphoton fluorescence (Stadler et al., 1996a).

Donor and acceptor coupled through a ␴ system More efficient coupling of donor and acceptor is provided by an intervening ␴ system. The NLO response of ␴ systems may be calculated simply by the addition of bond increments (Prasad and Williams, 1991), akin to the classical computation of molar refractivities (v. Auwers, 1935; Sutton, 1946; Le Fe`vre, 1965) where the refractive index of an unknown compound is computed on the basis of comparable bond increments. This also holds for the combination of archetypal donor and acceptor groups, like amino and nitro functionalities. Thus, in most cases, the D–A combination does not offer specific advantages over D–D or A–A arrays. It may be concluded that coupling between the functionalities is still weak and contributions from the bridge that are not included in the model dominate the NLO response. The performance may be improved when either the intrinsic hyperpolarizability of the functional groups employed is increased, or else when the coupling efficiency of the bridge is improved. Both approaches have been followed. Heavy elements have many core electrons of high potential energy that are easy to polarize. Hence, high refractive indices (n␻) and therefore high linear polarizabilities, ␣, are associated with their presence in molecules. In comparable structures, an increased linear polarizability may be intuitively suspected to lead to an increased non-linear polarizability. The homologous series of haloforms shows this to be the case (Table 1) (Karna and Dupuis, 1990; Samo´c et al., 1992; Kohler et al., 1993). Unfortunately, the maximum values of ␤ achievable with ␴ systems even of very high molar refractivities are too low for practical applications. In addition, the incorporation of ␴-bonded heavy elements tends to impart chemical (thermal, photochemical) instability on the molecule. Replacement of the conventional C–C bonds in the bridge with Si–Si bonds also somewhat strengthens the NLO response. The ionization energy of the Table 1 NLO properties of haloforms (EFISHG at 1064 nm). CHF3 n 具␣典 [10⫺40 C m2 V⫺1] ␤ [10⫺50 C m3 V⫺2] 具␥典 [10⫺60 C m4 V⫺3]

– 3.13 ⫺0.04 0.02

CHCl3

CHBr3

CHI3

1.4486 9.49 0.07 0.22

1.6005 13.17 0.22 0.50

1.75/1.95 20.07 1.0 2.5

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Scheme 4

173

NLO-phores with CT through a ␴ system.

bridge is reduced (Miller and Michl, 1989), hence both the coupling becomes better and the intrinsic response of the bridge is improved. However, no efficient CT coupling between donor and acceptor groups is achieved as has been demonstrated several times for silicon-bridged donor–acceptor substituted benzenes (Mignani et al., 1991; van Hutten et al., 1996). The most successful molecule [4] is shown although this already incorporates fairly large ␲ systems. Through-bond coupling of a D–A substituted pair of ␲ systems is best optimized through rigid, special geometries of the ␴ system, as was demonstrated many years ago both computationally and experimentally for the phenomenon of spiro-conjugation (Spanget-Larsen et al., 1990; Gleiter et al., 1994 and references therein). Indeed, a CT band can be detected in D–A-substituted spiro-conjugated molecules. Unfortunately, this band again has a low oscillator strength, and the NLO response of spiro-conjugated molecules is therefore not very high (Maslak et al., 1996). Again, the most efficient structure [5] is shown although the second-order polarizability arises in part from the local response of the donor–acceptor combination on its ‘‘right-hand’’ side.

One-dimensional (1D) ␲ systems Efficient coupling between a donor and an acceptor group is provided through a ␲ system. The high anisotropic polarizability of ␲-electrons in general guarantees a much better performance in comparison to ␴ systems. It should be strongly emphasized that it is the ␲ system as a whole, including the conjugated donor and acceptor groups, that is responsible for the secondorder polarizabilities observed. There is no independent ␲ system on to which donors and acceptors are grafted. It is a deeply ingrained practice in chemistry formally to disassemble chemical structures into an independent ␲ system that is substituted with functional groups that can attract or release electron densities. This concept, though clearly inappropriate, is successful for the

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qualitative rationalization of CT absorptions and we will also not completely do away with it. It can be traced back to early heuristic concepts about the colour–constitution problem that were developed before the advent of quantum chemistry (Dähne, 1970). Colour was thought to arise from combinations of a construction set consisting of chromogens (the ␲ system in modern language), auxochromes and anti-auxochromes (acceptors and donors). The contribution of the ␲ system by far exceeds the one arising from the ␴ framework. The latter is therefore assumed to be negligible in the following discussion in accordance with the two-state model. Classification of 1D NLO-phores: direction of charge transfer. The calculations on a simplified 1D model system in Section 3 have shown that ␤ exhibits an extremum both in the regions of positive- and of negative-charge transfer between donor and acceptor on electronic excitation, and that it passes through zero for vanishing charge transfer. It is thus convenient to sort NLO-phores according to the direction of charge transfer. In compounds of Class I the dipole moment is strongly enhanced in the excited state. They show a positive solvatochromism, i.e. the absorption is bathochromically (red)shifted in polar solvents, because the excited state with a high-dipole moment becomes more stabilized. Thus the HOMO–LUMO gap is decreased (Liptay, 1969; Reichardt, 1988, 1994). A strong solvatochromism is an indication of a large difference in dipole moment between ground and first excited state, ⌬␮ag. Therefore, the determination of solvatochromism has been used as a means to estimate ␤zzz (Paley et al., 1989; Würthner et al., 1993; Twieg and Dirk, 1996). p-Nitroaniline [1] is the paradigmatic representative for Class I. Most studies of NLO properties have dealt with Class I compounds because it is more densely populated with common chemical structural elements. Compounds of Class II are centrosymmetric cyanine-like structures and necessarily do not show either second-order polarizabilities or solvatochromism [very small solvatochromism is observed experimentally (Reichardt, 1988)]. Some representatives of Class III, mostly pyridinium betaines like Dimroth’s and Reichardt’s dye [whose transition energies, ET(30), have been used to establish a solvent polarity scale (Reichardt, 1994)], have also been studied. They show a negative solvatochromism because charge is transferred back to the donor upon excitation. Structures and substitution patterns of bridge elements in 1D ␲ systems: conjugation efficiency. The electronic requirements for optimum molecular second-order polarizabilities devised on the basis of the two-state model (see pp. 143 and 168) and the technical requirements to translate molecular properties into stable bulk materials partially coincide with the requirements for dyes in classical domains of application, e.g. in textile dyeing and colour

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Scheme 5

175

The three classes of 1D NLO-phores and their characteristic parameters.

photography. ‘‘Good’’ molecular 1D NLO-phores optimized along the guidelines of the two-state model almost inevitably belong to these known classes of dyes, especially stilbenes and merocyanines. Fine-tuning of donor and acceptor groups then allows optimization of second-order polarizabilities. Intensive work in these areas over more than a century has arrived at guidelines to achieve high tinctorial strengths (high ␮ag), high thermo- and photo-stability, and incorporation into synthetic polymers. From early on, the colour–constitution problem (the relationship between structure and transition energies ប␻ag) has also been intensely investigated (Fabian and Hartmann, 1980; Gordon and Gregory, 1983; Zollinger, 1991). This is especially true for linear donor–acceptor-substituted chains, due to the extensive theoretical work by Dähne (Dähne, 1978, 1991; Dähne and Moldenhauer, 1985; Dähne and Hoffmann, 1990) and others (Fabian, 1980) on the polymethine state. It has also been emphasized (Twieg and Dirk, 1996) that a very close structural analogy exists between NLO-phores advertised as completely new developments and dyes already in use two decades ago for electrochromic applications (e.g. for visualizing biological voltage differences) (Kamino, 1991). The simple example of the isomeric nitroanilines (Levine, 1976; Glania, 1996) will be mentioned first, especially since the performance of the p-isomer will be used as a benchmark in what follows. Furthermore, it shows that the relative positions of donor and acceptor substituents on a ␲ system have a

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Table 2 Electro-optical properties of the first optically allowed transition (EOAM) and NLO properties (EFISHG at 1064 nm) of the isomeric nitroanilines in p-dioxane (Glania, 1996). Parameter

o-nitroaniline [6]

␭ag (nm) 398 ␧max (m302 mol⫺1) 530 ␮gz (10⫺ C m) 13.6 ⫾ 1.0 ⫺30 ␮ag C m) 10.1 z (10 ⫺30 ␮ag (10 C m) 0 y ⌬␮z (10⫺30 C m) 14.4 ⫾ 2.2 1 ␤z (10⫺50 C m3 V⫺2) 12.5 ⫾ 2.4 2 12.5 ⫾ 2.7 ␤z (10⫺50 C m3 V⫺2) 1 ␤z (relative, at 1064 nm) 0.46 1 ␤z (relative, dispersion corrected) 0.35

m-nitroaniline [7]

p-nitroaniline [1]

371 170 15.5 ⫾ 1.1 3.5 5.0 29.7 ⫾ 3.8 7.3 ⫾ 1.3 8.3 ⫾ 1.5 0.27 0.24

354 1590 20.8 ⫾ 0.4 16.4 0 30.7 ⫾ 3.4 27.2 ⫾ 2.6 27.6 ⫾ 3.0 ⬅1 ⬅1

Scheme 6

great influence on its electro-optic parameters. All of the nitroanilines belong to Class I. The p-isomer has superior properties to its o-/m-congeners, a behaviour that is consistent with chemical intuition: Kekulé resonance forms can be written only for the CT in the o- and p-isomers. The difference between ground- and excited-state dipole moments, ⌬␮z, is much greater in the p-case because the CT between donor and acceptor occurs over a larger distance. m-Isomers often have the undesirable properties of highest ␭max and lowest ␧ of the CT transition (Fabian and Hartmann, 1980). An inferior performance is thus also consistent with the expectations on the basis of the two-state model. However, m-nitroaniline is not a simple 1D system as can be inferred from the different projections 1␤z and 2␤z as well as additional HRS measurements (Glania, 1996). Furthermore, the belief that m-substitution leads to inferior properties is too dogmatic as will be shown later (p. 204). The intrinsic second-order polarizabilities of simple benzene derivatives are not sufficient for applications. Thus, the bridge between donor and acceptor has been expanded to give higher ␤zzz. The bridge almost invariably consists of a carbon backbone. Organometallic compounds may or may not incorporate them; they will be dealt with

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Scheme 7

177

Selection of donor and acceptor groups, and bridge elements.

separately later. Only carbon can form long unsaturated chains with itself that are stable. A long chain guarantees that CT occurs over a large distance, hence ⌬␮ag is large. Also the intensities of CT transitions generally increase with conjugation length (higher ␮ag). In principle, the isoelectronic combination of boron and nitrogen could also be employed instead of C⫽C double bonds, but the compounds show decreased conjugation efficacy. The resonance form that shows ␲⫺␲ conjugation has zwitterionic character, and boron nitride is colourless (high HOMO–LUMO gap) in contrast to graphite. In addition, these compounds are often difficult to synthesize and have lower chemical stabilities. The bridge can include ethylene or ethyne units with sp2- and sp-hybridized carbon atoms, respectively. It may also include unsaturated alicyclic and heterocyclic arenes. The latter are classified according to their ␲-electron excess or deficiency, for which we have used the symbols ␲⫺ and ␲⫹, respectively. They may or may not be charged. Bridges formed from sp2-hybridized carbon atoms also allow the incorporation of elements that sustain ␲ bonding. In open chains, nitrogen is used almost exclusively but in heterocyclic structures, sulphur and oxygen may also be incorporated. Occasionally, selenium and even tellurium have been used in this respect, but apparently show not much improvement over their S analogues (Blenkle et al., 1996). In general, conjugation efficiency is highest with chains composed entirely of sp2 C⫺C bonds. However, these chains are flexible and thus conformationally inhomogeneous. In addition, they are unstable with respect to other photochemical (like cis⫺trans isomerization, [2 ⫹ 2]cyclization) or thermal processes (radical initiated or electrocyclizations). The chemical and conformational stability may be increased by rigidifying the carbon backbone. Alkyl substitution leads to allyl(1,3) strain (Hoffmann, 1989), a severe steric interaction between a Z substituent and the substituents at the adjacent saturated carbon atom. Thus, Z and the smallest group, in the present case hydrogen, will lie in one plane. The conformational minimum becomes deeper

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Scheme 8 Conformational instability of chains from sp2 carbon atoms and strategies to avoid it. Some basic combinations of the stilbene and polyarene type.

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and the rotational barrier is heightened. Methyl substitution is encountered in natural carotenes and NLO-phores derived from them (Blanchard-Desce et al., 1995; Hendrickx et al., 1995). Note that the values given in the former paper are a factor of five too high, because an erroneous reference value for methanol was used (Morrison et al., 1996). Incorporation into ring structures is a concept well known from cyanine chemistry to improve on stabilities (Slominskii et al., 1975; Heilig and Lüttke, 1987; Tyutyulkov et al., 1991) and has been used in NLO (Cabrera et al., 1994). Unfortunately, the synthesis becomes more intricate in many cases. In addition, more rigid molecules are also more prone to (sometimes undesirable) fluorescence, and become even more coloured. The conformational (but not chemical) stability can also be increased by the use of chains formed from triple bonds. However, the second-order polarizabilities are not competitive if longer chains are used; carbon in the sp-hybridized state is more electronegative and a worse coupling element than in the sp2 state. Both conformational and chemical stabilities can be improved by the incorporation of aromatic units. By the same token, the electronic structure of the ␲ system is also changed drastically; specifically the fairly high chemical stabilization of arenes in comparison to open chains, their ‘‘aromaticity’’ (Minkin et al., 1994), also attenuates ␲ conjugation. In addition, oligophenyls are twisted due to steric interaction between the o-hydrogen atoms. The twisting can be overcome by annulation, as in the fluorenes shown, but the NLO response is not improved (Cheng et al., 1991b; van Walree et al., 1997b). Electron-rich heterocycles, especially thiophene, are more efficient coupling elements than phenyl groups as has been demonstrated experimentally (Mignani et al., 1990; Cheng et al., 1991a; Jen et al., 1993; Würthner et al., 1993; Varanasi et al., 1996; Steybe et al., 1997). Chemical intuition would assign this to a lowering of aromaticity in this ␲-excessive heterocycle – six ␲ electrons are distributed over five centres – and an additional donor character. It should be noted that a further heterosubstitution with nitrogen in five-membered rings (Dirk et al., 1990) lowers the ␲-electron excess, because it is more electronegative than carbon. In six-membered rings, substitution gives electron-deficient heterocycles directly. Chemical intuition would also place ␲-excessive heterocycles at the donor end, and ␲-deficient ones at the acceptor end. This has been confirmed by semi-empirical computations (Albert et al., 1997). The best compromise between stability and second-order polarizability is offered by a combination of chain and arene elements (stilbenoids in Scheme 8). Scheme 8 shows some basic combinations. Donors and acceptors in 1D ␲ systems. Both the structure of the bridge and the ‘‘strength’’ of donor and acceptor groups decide on the extent of coupling. These strengths may be conveniently defined in terms of the Hammett ␴ values of classical correlation analysis, which have been split also into ‘‘electrostatic’’, ‘‘resonance’’ and even more contributions (Charton, 1987). Especially suitable for NLO are obviously the resonance constants

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Scheme 9 Second-order polarizabilities of some benzene derivatives relative to p-nitroaniline. An entry ‘‘5.1/3.6 (1.8; 498)’’ signifies a static value (␤0) of 5.1 ⫻ pnitroaniline, a value of ␤0/M of 3.6 ⫻ p-nitroaniline corrected for the difference in molecular weights, and a value of 1.8 ⫻ p-nitroaniline if ␤0/M is further corrected for the difference in (␭ag)2 . ␭ag is given in parentheses (498 nm).

␴⫺R/␴⫹R that are believed to gauge ␲ conjugation. Scheme 7 shows some basic donor and acceptor groups. In general, the palette is broader with the latter [including the more exotic phosphonates (Hutchings et al., 1994), the phosphine oxides (Kott et al., 1995) and even the diazonium group (Kang et al., 1995)]. Scheme 9 is a comparison of the efficacy of several donor– acceptor combinations for benzene derivatives. Only values normalized to p-nitroaniline in dioxane are given. Note that with the tricyanovinyl acceptor in the first column the ␲ system has also been quite enlarged. The values in parentheses have been corrected both for molecular mass and different ⌬E of the CT transition and should therefore reflect only the influence of the transition dipole ␮ag, and the change in dipole moment, ⌬␮ag, a procedure that has been suggested previously to give a good basis for a fair comparison of the merits of NLO-phores (Twieg and Dirk, 1996). Note that the differences to p-nitroaniline are not tremendous. Larger ␲ systems will be dealt with below. Since ␤ passes through an extremum for Classes I and III, there must be an optimum combination of donor and acceptor strengths for a particular bridge (Marder et al., 1994a,b). Arenes and stilbenes generally belong to the weakly coupled side of Class I, so increase of donating or accepting strengths increases the NLO response as can be gleaned from Scheme 9 (Cheng et al., 1991a,b). This may no longer be true for more strongly coupled systems that are close to the cyanine limit; consider the paradigmatically low response of the zwitterion [10] below (Staring, 1991). Scheme 10 also shows some successful combinations [8], [9], [11] for Classes I and III. Conflicting results, however, have been reported for the zwitterion [11] (Cross et al., 1994; Szablewski et al., 1997).

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Scheme 10 Representative NLO-phores for the three classes of different donor– acceptor coupling.

It should be pointed out that thermostability of donor and acceptor groups has rarely been addressed. For example, it is known from classical textile dyeing that benzylic C⫺H bonds and C⫺H bonds in the ␣-position with respect to a heteroatom are to be avoided (because they are weak and susceptible to radical attack). Therefore, it is understandable that molecules with diphenylamino donors are regularly more stable than their congeners with dialkylamino groups (Moylan et al., 1993; Jen et al., 1997). Likewise, nitro groups are photoreducible in some media and often impart thermal instability to the molecule. They may be replaced by sulphones that lead to hypsochromic shifts and decreased second-order polarizabilities (Beecher et al., 1993). Structural counterparts of classification: bond length alternation, vibrational polarizabilities. The strength of electron-donating and -accepting capabilities is expected to have an influence also on the geometric structure of the ␲ system as well as on the vibrational properties of the bridge. Relationships between intensities in IR, Raman and hyper-Raman spectra and non-linear optical properties of organic molecules have been investigated (Bishop, 1998; Bishop et al., 1995; Champagne, 1996; Del Zoppo et al., 1996; Lee and Kim, 1997). On a more intuitive level, the correspondence between vibrational (IR)

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Scheme 11 Alleged dominant resonance forms for the three classes of different donor–acceptor coupling.

frequencies and bond order could also be used. It would be expected that there would be equalized bond lengths for cyanines of Class II, and unsymmetrical structures showing different alternations for Classes I (small charge transfer in the ground state) and III (strong charge transfer). Thus, the bond-length alternation (BLA) in linear chains of D–Asubstituted compounds has been advocated to be a crucial indicator for the performance of NLO-phores (Bourhill et al., 1994; Marder et al., 1994b; Gorman and Marder, 1995). It is defined as the mean difference between the lengths of formal C⫺C and C⫽C bonds. Its degree is a consequence of the extent of coupling between donor and acceptor end group. Unperturbed polyenes and molecules with total charge transfer have maximum BLA while it should vanish for the cyanine limit. It may be of some practical use if ␤ values could be estimated from structural parameters directly although the predictive power of this concept for the development of NLO-phores ab initio has been questioned (Twieg and Dirk, 1996). However, the results of experimentally determined crystal structures do not always support this view. For example, cyanines should belong to the centrosymmetric Class II, and no BLA should be observed. For this case, vanishing ␤ follows. While bisdimethylheptamethincyanine bromide does almost remain centrosymmetric in the crystal (Kulpe and Schulz, 1978), it nevertheless shows a substantial BLA of 0.055 Å. For this value, BLA model calculations predict a secondorder polarizability close to the maximally attainable value.

Scheme 12 Selected bond lengths for Me 2 N(CH) 7 NMe 2⫹ Br ⫺ ⫺ ⫹ Me2N(CH)7NMe2 Cl (in parentheses) (Kulpe and Schulz, 1978).

and

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It should be noted that, on the basis of quantum mechanical computations and theoretical considerations, BLA has been questioned as a reliable means to estimate the extent of delocalization in a ␲ system (Schütt and Böhm, 1992). The influence of solvent on the second-order polarizabilities of pnitroaniline. The ‘‘strength’’ of a donor–acceptor combination does not define an invariable geometry and charge separation. The medium has a great influence on these parameters. The influence of solvent on the second-order polarizabilities can be neatly demonstrated with p-nitroaniline. An extensive EFISHG study exists which also includes measurements of ground-state dipole moments (Stähelin et al., 1992). The strong positive solvatochromism of p-nitroaniline has been known for a long time and may also be gleaned from Table 3. Several conclusions can be drawn from Table 3. First, in accordance with the two-state model, ␮␤z, ␤0 and ␤z all increase with decreasing HOMO–LUMO gap. Second, the intrinsic second-order polarizability of p-nitroaniline is increased by two-thirds when the solvent is changed from p-dioxane to methanol or N-methylpyrrolidone, even when the values are corrected for the differences in (⌬E)2. As we have adopted the value for p-nitroaniline in dioxane as a standard, it should therefore be noted that molecules that truly surpass the best performance of p-nitroaniline should have a second-order polarizability of ⭓1.7 p-nitroaniline (dioxane). As a third conclusion, there is a poor correlation between ␤zzz and the static reaction field E R␮0 as predicted by (91). This is in part due to the fact that the bulk static dielectric constant, ␧0 in (89), differs from the microscopic dielectric constant. For example, p-dioxane has long been known for its anomalous solvent shift properties (Ledger and Suppan, 1967). Empirical microscopic dielectric constants can be derived from solvatochromism experiments, e.g. ␧0 ⫽ 6.0 for p-dioxane, and have been suggested to improve the estimation of the reaction field (Baumann, 1987). However, continuum models can only provide a crude estimate of the solute–solvent interactions. As an illustration we try to correlate in Fig. 7 the transition energies of p-nitroaniline with those of a popular solvent polarity indicator with negative solvatochromism. The aberrant behaviour of p-nitroaniline can be ascribed to the hydrogenbonding capabilities of p-nitroaniline as the comparison with the solvatochromism of its N,N-dimethyl derivative reveals (Kamlet and Taft, 1976). While the solvatochromism of these compounds in non-hydrogen-bond accepting solvents is well correlated, the bathochromic shift of p-nitroaniline becomes much stronger in accepting solvents. Deviations of up to 27 kJ mol⫺1 from the value expected on the basis of the correlation for non-coordinating solvents have been noted, quite a substantial fraction of the total shift in transition energy of close to 71 kJ mol⫺1 for p-nitroaniline. These observations illustrate the limited adequacy of continuum approaches in both the modelling

Table 3 Second-order polarizabilities of p-nitroaniline [1] in different solvents (Stähelin et al., 1992). Solvent

p-Dioxane Chloroform Tetrahydropyran Ethylacetate 1,2-Dimethoxyethane Tetrahydrofuran 2-Methoxyethylether Dichloromethane Acetone N-Methylpyrrolidone Methanol Dimethylformamide Acetonitrile

␧r

␭max (nm)

CT energy (kJ mol⫺1)

ET (30)* (kJ mol⫺1)

␮␤z (relative)

␤0 (relative)

␤0 [rel.; corr. for (␭max)2]

2.21 4.81 5.61 6.02 7.20 7.58 7.63 7.77 20.7 32.0 32.63 36.71 37.5

354 348 360 358 366 362 368 350 368 386 370 382 366

338 344 332 334 327 330 325 342 325 310 323 313 327

151 164 152 160 160 157 162 170 177 177 232 181 191

⬅1 0.94 1.40 1.33 1.32 1.33 1.63 0.92 1.66 2.28 1.71 1.92 1.59

⬅1 1.06 1.33 1.36 1.12 1.26 1.36 1.05 1.47 1.95 1.79 1.56 1.68

⬅1 1.10 1.29 1.33 1.04 1.20 1.26 1.07 1.36 1.64 1.64 1.34 1.57

*Transition energies for the solvatochromic pyridinium betaine ET(30) [recalculated from the literature values given in kcal mol⫺1 (Reichardt, 1994)].

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Fig. 7 Comparison of transition energies of ET(30) and p-nitroaniline in different solvents (data from Table 3).

of solvatochromism and local field corrections as noted already by Onsager himself (Onsager, 1936). Solvatochromism and, hence, second-order polarizabilities in ␲ systems substituted with strong donors and acceptors depend on both electrostatic field effects and specific interactions like hydrogen bonding (Reichardt, 1988). Computations that combine semiempirical and molecular mechanics procedures also show substantial (up to 60%) contributions of hydrogen bonding to the computed solvatochromic shifts (Gao and Alhambra, 1997). Influence of solvent and electric fields on the electronic structure of 1D NLO-phores. The class to which an NLO-phore belongs is largely defined through the electron-donating and -accepting strengths of the endgroups. However, as the previous paragraph has shown, the NLO response is quite solvent-dependent. As the D–A substituted ␲ systems as a whole are easily polarizable, one may expect a change in geometric structure (BLA) when a strong external static electric field is applied, because this modifies the donor and acceptor strengths (Scheibe et al., 1976). Such a field can, in principle, already be provided when the surrounding medium (solvent) is changed. It had been mentioned above (p. 167) that solvatochromism has been used as a simple screening tool for ␤. From solvatochromism, i.e. the measurement of ␭ag in different solvents, ⌬␮ag can be estimated if ␮g is known from dielectric measurements (105). Then ␤ may be estimated assuming validity of the two-state model (55). p-Nitroaniline remains a Class I compound over the whole gamut of solvents, but crossover between the different types may be observed with molecules closer to either end of the cyanine range (Class II). Crossover will

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be easiest if cyanines of Class II are to be switched to regions I or III. The experimental evidence in solution experiments is not always unequivocal, however, because cyanine chains may undergo conformational changes and may also form aggregates. Cyanines of even moderate chain lengths may be desymmetrized in the solid state through counterion effects (Dähne and Reck, 1995) and also in solution through the same mechanism, conformational isomerism and/or aggregation (Tyutyulkov et al., 1991; Tolbert and Zhao, 1997). Desymmetrization of formally centrosymmetric squaraine dyes may be observed in thin Langmuir–Blodgett films and, correspondingly, SHG response has been reported (Ashwell, 1996). In the extreme, crossover from Class I to Class III may be effected. This entails change from positive to negative solvatochromism, a phenomenon that has been under controversial debate for some time (Reichardt, 1994). For many of the molecules for which inversion of solvatochromism was claimed, the experimental and theoretical evidence is not unequivocal. For Brooker’s dye [16], inversion of solvatochromism is observed only in very polar solvents (Brooker et al., 1951). These can be protic (alcohols) and form strong hydrogen bonds to the betainic phenolate. The species is thus changed chemically; full protonation of pyridinium-betaines is known to lead to a hypsochromic shift. The crossover can also be observed in dipolar–aprotic solvents. However, these solvents are very difficult to free from protic impurities to a level that is satisfactory for the measurements of optical properties. The latter are carried out at quite low concentrations of the molecule of interest. Thus, conflicting results of NMR relaxation rate and ␤ measurements on one side and apparent inversion of solvatochromism has been noted for some zwitterions of the pyridinumphenoxide type like [15] (Runser et al., 1995): inversion of solvatochromism was noted, but the sign of ␤ remained unchanged. Inversion of solvatochromism has also been claimed for dicyanovinylamino-substituted polyenes [18] (Dehu et al., 1995) in contrast to results reported by previous investigators (Krasnaya et al., 1976; Scheibe et al., 1976; Slominskii and Radchenko, 1977). The most convincing report for change of sign in solvatochromism comes from Maas and coworkers who introduced the stronger guanidine donor instead of a simple amine (Maas and Feith, 1985; Feith et al., 1986; Weber and Maas, 1988). On the basis of bond lengths observed in an X-ray analysis [17] belongs to Class III, and inversion of solvatochromism is observed in solvents of intermediate polarity. Inversion in the sign of ␮␤z has been reported to occur for [13], [14] (Bourhill et al., 1994) that belong to the class of merocyanine dyes with barbiturate acceptors; these have been well studied in the realm of NLO (Ikeda et al., 1989, 1991; Cahill and Singer, 1991; Moylan et al., 1993) and before as charge-shift chromophores for biological applications (Twieg and Dirk, 1996). Theoretical modelling of the influence of solvent on the absorption and structure of D–A substituted ␲ systems has been used early on. The simplest and most ancient model is a valence bond (VB) approach where resonance

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Scheme 13 Molecules for which inversion of solvatochromism has been claimed.

structures belonging to Class I and III are mixed (Förster, 1939). It can reproduce crossover between different classes when an arbitrary coupling parameter is varied (Simpson, 1951; Barzoukas et al., 1996). More refined approaches model the influence of a solvent by the application of an electric field which has been provided by point charges (Scheibe et al., 1976; Gorman and Marder, 1995). The field strength is obviously the crucial parameter in this approach; inversion of solvatochromism may be produced at high field strengths regardless of which level of theory is used. Onsager’s continuum theory (Onsager, 1936) has also been employed to model electric fields created from solvents. The molecule has been placed in a cavity of spherical (Albert et al., 1996), elliptical (Dehu et al., 1995) or arbitrary (Allin et al., 1996) shape. The results obtained depend critically on the size and shape of the cavity chosen. An approach closer to actual solute–solvent interactions has been taken by Monte Carlo methods that use a combination of semi-empirical and molecular mechanics procedures (Gao and Alhambra, 1997). Inversion of solvatochromism was unfortunately not investigated. Selection of 1D NLO-phores studied: the efficiency–transparency tradeoff. In Schemes 9 and 10 some comparisons of second-order polarizabilities were made. It is apparent that within Classes I or III large ␲ systems hold more promise than small ones. Some basic structural types are compared in Scheme 14 (see also Schemes 9 and 15). For a given wavelength of absorption, ␤ is higher for stilbenes and oligoarenes than for simple benzene derivatives

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Scheme 14 Comparison of some basic 1D structures. Data have been taken from several sources (Cheng et al., 1991a,b; Stiegman et al., 1991; Würthner et al., 1993; Steybe et al., 1997; van Walree et al., 1997a,b).

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Fig. 8 Efficiency–transparency trade-off for 1D NLO-phores (cf. Table 4, Scheme 15). The figure of merit is relative to the one of p-nitroaniline [1] (⬅1).

because the CT occurs over a larger distance, so both ⌬␮ag and ␮ag are higher. For the NLO-phores studied the conjugation length is limited by steric attenuation of conjugation, and not by intrinsic factors. Clearly, there is an improvement if the conjugation length is increased but it is not dramatic. We next show a ‘‘masterplot’’ that for reasons of consistency contains data only from one source. Values of ␤0 were calculated from EOAM data. There is most definitely a positive correlation between ␭max and the maximum ␤zzz. Also, an inverse correlation between the transition energy, ប␻, and ␤zzz predicted by the two-state model holds if the maximum attainable values for one particular transition energy are considered. There are many compounds, however, that fall much below this line. They are more cyanine-like (close to Class II, low ⌬␮) and combine low transition energies with low second-order polarizabilities. They are, unfortunately, often omitted in similar diagrams found in the literature which show only a selection of the more successful structures specifically optimized for NLO applications. The transition energy (‘‘HOMO–LUMO gap’’) was not calculated in the model systems in Section 3, but, as a rule of thumb, for sterically unhindered systems ប␻ often correlates inversely with both the conjugation length and the magnitude of the transition dipole (Fabian and Hartmann, 1980) as predicted

Scheme 15.

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by the free-electron-in-the-box model. An intrinsic problem of this type of one-dimensional molecular architecture lies in the high UV–visible absorbance of long conjugated structures that is problematic for a number of applications in NLO. While high absorption in the visible range of the spectrum is tolerable for the electro-optic effect, it is not acceptable, for example, for SHG. This dilemma is known as the ‘‘efficiency–transparency trade-off’’. One possible way out is the use of two-dimensional NLO-phores to be dealt with below. Organometallic one-dimensional ␲ systems The use of molecules incorporating metals in both low- and high-oxidation states has been frequently advocated to increase both ␤ and chemical stability (Long, 1995). Unfortunately, few compounds live up to these promises although the number of molecules investigated is still relatively small in comparison. Many organometallic compounds are just not robust enough against air, water or temperature to be incorporated into technically useful devices. Even if they are, a large number of transition metal complexes show quite bathochromic, metal-centred (ligand field) d–d absorptions in the visible. In contrast to metal → ligand CT (MLCT) transitions, they do not contribute to the NLO response observed (Kanis et al., 1992) but dramatically worsen the efficiency–transparency trade-off. Finally, the coordination number of metals is high. Thus, metals in complexes in low oxidation states often carry triphenylphosphine ligands that are necessary to impart chemical stability and/or the proper donicity on them. However, they are ‘‘idle’’ in the sense of NLO and drastically increase the molecular weight and volume of the compound. Thus, the efficiency–transparency ratios and efficiency–molecular volume ratios of organometallic compounds are often not satisfactory in comparison with ‘‘organic’’ molecules. Two basic approaches have been taken. The first consists in grafting organometallic donor and acceptor groups, such as ferrocene [as in [84] (Calabrese et al., 1991)] ruthenium derivatives [as in [83] (Whittall et al., 1996)] and tungsten carbonyl, instead of their ‘‘organic’’ counterparts on ␲conjugated chains. Quite successful in this respect, although not truly organometallic, are zwitterions based on borate donors and ammonium acceptors [86] (Lambert et al., 1996) and Lewis acid complexation as in [85] (Kammler et al., 1996). The second approach tries to make use of structural motifs stemming from the realm of inorganic chemistry. Classical coordination complexes like [87] and [88] (Thami et al., 1992; Di Bella et al., 1997) were investigated first. Recently, several organometallic intervalence complexes [89]–[94] (Behrens et al., 1996; Hagenau et al., 1996; Morrison et al., 1996; Coe et al., 1997; Hendrickx et al., 1997) have been reported to show high second-order polarizabilities on the basis of HRS at 1064 nm. In all of the successful cases except [94] very

Table 4 Transition wavelength, ␭ag, ground-state dipole, ␮g, transition dipole, ␮ag and dipole differences, ⌬␮ag, associated with low-lying were estimated on the basis of (55). CT bands for a series of 1D NLO-phores (from EOAM). Second-order polarizabilities ␤(ag) 0 Structure [1] [8] [22] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

Reference

␭ag (nm)

␮g (10⫺30 C m)

␮ag (10⫺30 C m)

⌬␮ag (10⫺30 C m)

␤(ag) 0 (10⫺50 C V⫺2 m3)

␤(ag) 0 /molecular mass (M) relative

(a) (a) (b) (c) (d) (a) (b) (a) (a) (a) (a) (e) (a) (a) (a) (a) (f) (f) (f) (f) (f) (f) (f) (c) (c) (c)

354 654 426 391 350 435 457 450 594 515 578 537 408 398 506 543 300 380 510 628 376 392 382 415 514 539

21 40 25 21 3 27 32 23 30 38 49 7 22 17 26 29 21 27 38 38 26 32 20 23 36 49

16 31 26 23 16 19 30 26 29 26 24 12 21 20 23 28 18 20 25 29 20 20 20 25 32 32

30 49 73 31 12 18 44 50 61 11 0 13 37 42 55 52 31 48 41 90 45 35 52 32 30 4

15 306 132 39 6 19 128 104 275 30 0 8 41 40 113 183 14 42 101 454 39 33 46 52 122 17

⬅1 9.3 4.7 1.3 0.4 0.8 4.0 2.5 6.5 1.2 0 0.2 1.7 1.3 4.2 6.0 0.5 1.7 3.3 11.7 1.5 1.0 2.3 1.3 2.7 0.4

[54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]

(c) (c) (c) (c) (c) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g) (g)

412 457 536 529 520 452 466 479 484 499 490 557 583 600 610 396 426 450 471 480 444 503 534 560 569 484 514 531 542

22 28 47 33 43 26 27 27 27 27 26 29 28 29 30 22 22 24 25 26 32 33 34 35 37 23 26 23 25

25 28 32 30 34 25 30 34 36 43 31 36 38 41 46 22 23 27 34 39 25 28 32 34 39 27 30 29 33

20 17 6 13 6 75 83 91 95 104 26 44 74 96 141 38 52 67 74 84 28 48 68 78 107 39 63 73 92

32 42 27 50 30 142 243 367 447 719 90 265 541 881 1671 43 75 153 290 455 53 143 296 422 815 104 228 265 458

1.0 1.2 0.6 0.9 0.7 4.2 6.6 9.3 10.6 16.0 2.0 5.7 10.9 16.8 30.3 1.8 2.8 5.2 9.0 13.0 2.0 4.9 9.3 12.2 21.8 3.0 6.0 6.5 10.6

EOAM were carried out in dioxane at 298 K, except for [39] (in CCl4). (a) Steybe et al. (1997); (b) R. Wortmann, unpublished results; (c) Würthner et al. (1997); (d) Lebus (1995); (e) Liptay et al. (1982b); (f) Boldt et al. (1996); (g) Blanchard-Desce et al. (1997b).

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Scheme 16

Organoelement structures for second-order NLO.

strong resonance enhancement was present. A number of molecules investigated is shown in Scheme 17. The truly metallic intervalence complexes often suffer from low extinction coefficients: despite the low energy gap of the transition, the extent of the electronic coupling between donor and acceptor over the framework chosen remains comparatively small. EFISHG measurements (1910 nm) on donor–acceptor substituted phenylporphyrins like [95] have shown moderate second-order polarizabilities (Suslick et al., 1992). However, HRS (without checks for fluorescence; an EFISHG measurement has been announced in a footnote to this report) gives very high absolute values for the metallized ethynylogues [96] (LeCours et al., 1996). Surprisingly, ␤ was also reported to be frequency-independent for the Zn porphyrin [96a] in the region where the second harmonic is absorbed. Calculations (Priyadarshi et al., 1996) overestimate the values for the phenyl porphyrins investigated by Suslick et al. by a factor of 3–4, but underestimate the reported values for the metallized ethynylogue [96] by an order of magnitude. Open-shell systems The overwhelming majority of organic NLO-phores studied are closed-shell systems. Organic neutral or ionic carbon-centred radicals are very frequently

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Scheme 17

Organometallic intervalence complexes for second-order NLO.

Scheme 18

Porphyrin structures for second-order NLO.

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Scheme 19 An open-shell organic NLO-phore.

unstable with respect to dimerization to give closed-shell species. ␲ Conjugation and/or steric interactions may prevent this happening, as shown by the intensive research on organic magnetic materials, but molecular topologies suitable for NLO are rare. Open-shell ␲ systems are easier to construct either from functional groups that are stable in radical form, like nitroxides, or from transition metal ions. Both types have been advocated as a promising class of NLO-phores on the basis of semi-empirical SOS calculations (Di Bella et al., 1996). Experimental results on a nitroxide radical [97] were disappointing (Nicoud et al., 1990); ␤0 was inferior to the reference compound, and a low-extinction, broad-band absorption up to 800 nm was present in the radical. Inorganic complexes may show more promising results.

TWO-DIMENSIONAL (2D) NLO-PHORES: 1D AND 2D ARCHITECTURE

␲ Systems with only one significant component of ␤ have just been discussed, and it was justified to quote only ‘‘␤ values’’ although ␤ is a tensor of third rank. In this chapter we will discuss molecules where two or more low-lying CT transitions of different polarization substantially contribute to ␤. This can occur if the HOMO and/or LUMO is degenerate, as may occur for molecular symmetry of C3 or higher. It is sufficient, however, if two excited states of CT character are close in energy. In both cases, off-diagonal elements attain significance and ␤ becomes two-dimensional (2D). The term ‘‘2D’’ is defined solely with respect to the two-dimensional properties of ␤, independent from molecular topology. The modulus of ␤, given by (129), 储␤储 ⫽

冪冘␤

2 ijk

(129)

i,j,k

is intrinsically higher for most 2D NLO-phores in comparison to their 1D counterparts, where only one element is significant. It should be noted, however, that the magnitude of the modulus should not be used as a comparative figure of merit, because not all tensor elements may be used simultaneously for an NLO application with linearly polarized light. On p. 145, a three-centre model system of C2␯ symmetry was described and it was shown that D3h symmetry was included as a limiting case. Several basic templates [98]–[105] are shown in Scheme 20. Only one linear combination of tensor elements is obtained both in

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Scheme 20

197

Structural templates for dipolar NLO-phores of C2␯ symmetry and higher, and for non-dipolar NLO-phores of symmetry C3 and higher.

unpolarized EFISHG (p. 162), which yields a projection of ␤ on the direction of the molecular dipole moment, or HRS, which yields a quadratic average of ␤. The evaluation of structure–property relationships, however, requires the determination of all tensor components. Hence, in order to unravel all four significant tensor elements for planar molecules of C2␯ symmetry, the combination of polarization-dependent EFISHG and HRS, as well as EOAM are needed. This array of techniques is available only in few groups and it is of course more tedious to implement. Therefore, few studies have addressed true structure–property relationships for the general case of C2␯ symmetry. The four significant components become of equal absolute magnitude for the limiting case of D3h symmetry where the molecular dipole moment vanishes. In this special case, an HRS measurement is sufficient to evaluate the second-order polarizability. This class of molecules will therefore be considered first. An important caveat should always be borne in mind: molecular structures drawn on paper are deceptive. True molecular symmetry is often lower than the maximum one attainable. In the absence of valid information on molecular conformations in solution, the interpretation of HRS data is therefore not unambiguous because the number of significant elements is different with each subgroup of D3h. A depolarization ratio in agreement with the expected value [1.5 in this case, cf. (127), (128)] corroborates the assumed molecular symmetry, but does not prove it (see below). Non-dipolar 2D NLO-phores: general properties and optimization strategies Molecules of D3h symmetry were included as a limiting case in the treatment of three centre interactions of C2␯ symmetrical model systems (p. 145). A more

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general group theoretical analysis (Jerphagnon, 1970; Maker, 1970; Chemla et al., 1975; Zyss, 1993) also shows that for molecules of C3 symmetry and higher, with the exception of octahedral point groups, ␤ does not vanish. Eight components are significant for C3h symmetry, five for C3␯, four (of equal absolute magnitude) for D3h, and one (␤xyz) for tetrahedral point group symmetry, where excited states become triply degenerate [see (42)–(44)]. The second-order polarizabilities of this class of compounds have been attributed to the octopolar charge distribution (Zyss and Ledoux, 1994), but it should be emphasized again that the origin of NLO activity is not the permanent octopole moment of these molecules. Rather, it is associated with the importance of two excited states close in energy. Molecules without dipole moment should be intrinsically more transparent than their dipolar congeners because they show little, if any, solvatochromism. The mutual polarization of solvent and solute is small because the solute has only higher moments of electric field distribution. Second, the HOMO–LUMO gap is widened. Hypsochromic shifts have been noted in this respect already in the infancy of organic dye chemistry. For example, ‘‘inversion of auxochromes’’ (Kauffmann, 1920; Wizinger, 1961) occurs when malachite green (␭max ⫽ 621 nm) is compared with crystal violet (␭max ⫽ 590 nm). Pariser-Parr-Pople (PPP) calculations already are capable of reproducing this effect (pp. 143ff in Fabian and Hartmann, 1980). On the contrary, bathochromic shifts are found for purely donor- or acceptor-substituted ␲ systems. It has frequently been asserted (Zyss and Ledoux, 1994) that planar molecules of threefold symmetry are more likely to crystallize in noncentrosymmetric space groups. This assertion rests on the assumption that antiparallel (i.e. centrosymmetric) orientation is favoured in dipolar molecules because no net dipole results, while there is no reason for such a preference with non-dipolar molecules. However, no statistical confirmation for this claim has been given. Furthermore, the underlying intuitive correlation between the magnitude of the molecular dipole and the statistical probability of crystallization in a centrosymmetric space group has been challenged on the basis of a Cambridge Structural Database (CSD) analysis (Whitesell et al., 1991). Unfortunately, crystals are probably the only bulk structure with promise for applications of non-dipolar molecules. While dipolar 1D systems can be ordered by static electric fields in polymer matrices, this is not possible for non-dipolar molecules. The situation is exacerbated by the fact that the requirements for ordering are more stringent than in the 1D case. Poling with optical fields has been demonstrated for a triphenylmethane dye akin to crystal violet (Fiorini et al., 1995), but no indication of long-term stability was given. The value of ␤ is now directly proportional [(61), (63)] to a product of three transition dipoles between the ground state, 兩g典, and the excited states 兩a典 and 兩b典. The free-electron model predicts that the transition dipoles ␮ag and ␮bg are magnified by the extension of the conjugation of the ␲ system. Indeed, as a

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rule of thumb, all other parameters being equal, the transition probability also increases with the size of the ␲ system (Fabian and Hartmann, 1980; Gordon and Gregory, 1983; Zollinger, 1991). This should also be true for ␮ba although this quantity is not easily accessible by experiment. The synthetic challenge of 2D NLO-phores can be quite daunting for the organic chemist. Unlike in the case of 1D dipolar systems – which have been intensively investigated in other areas of dye chemistry, e.g. for textile, photographic or (biological) staining applications – the promising basic structures have in most cases not even been prepared. Also, an increase in the size of the ␲ system in 2D molecules requires extension of conjugation in two dimensions instead of just one. Inevitably, the synthetic chemist has to struggle with problems of performing the same reaction on one molecule up to three times in one step (which looks easy on paper, and can be extraordinarily difficult in practice), with steric hindrance, and low solubilities. There are problems from the physics side, too. Large, rigid ␲ systems – desirable to optimize the cube of transition dipoles – in general are also candidates for high fluorescence efficiencies. Now, for molecules without a dipole moment only HRS can be used to evaluate ␤. If the pitfalls noted on p. 166 for HRS are not addressed, and luminescence arising from other multiphoton processes in the region of the observed second harmonic not rigorously excluded, very high, but spurious, signals may result. Exceptionally high second-order polarizabilities for notoriously fluorescent molecules (like complexed ruthenium) have thus been repeatedly published in premier journals. Most of them have been retracted, refuted or drawn into question (Flipse et al., 1995; Morrison et al., 1996; Stadler et al., 1996a; Wortmann et al., 1997). Fluorescence can be reduced by the internal or external heavy atom effect, but also by employing carbonyl compounds like acetone as solvents. Examples of non-dipolar NLO-phores In general, the trisubstituted types [102] and [103] are easier to access synthetically, and it is probably mainly for that reason that most studies have dealt with these templates. If the central unit is carbon, the structures are charged, most often the positively charged [103] was studied. Ionic species can have definite advantages over neutral molecules if crystals are to be used as NLO materials. First, ionic species may be easier to obtain in crystals of good optical quality and large sizes. Second, while a promising molecular NLOphore structure can be retained, different crystal structures may still be produced just by the exchange of counterions. Tedious chemical modifications of basic structures may then be avoided. Steric hindrance between phenyl substituents can effectively reduce conjugation. This is most pronounced for triphenylcarbenium ions – like the well-known triphenylmethane dyes (cf. [115]) – and triphenylamine derivatives (cf. [113]). Bridging of phenyl substituents of course alleviates this

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problem, but as the corresponding compounds are not commercially available, few of them have been studied. The hexasubstituted type [101] often suffers even more from steric hindrance, which in this case not only prevents effective conjugation but also makes synthesis difficult. Tetrahedral structures [105] have been studied occasionally; they are moderately effective only when the centre element is not carbon, but tin or phosphorus. From (61) and (63), as well as the experience with 1D NLO-phores, it may be deduced that small ␲ systems, including simple benzene derivatives, are not likely to give high second-order polarizabilities through either two- or three-level contributions. The corresponding transition dipoles are just too small. This is borne out by experiment. Several simple nondipolar NLO-phores of the tetrahedral type (methane, tetrachloromethane, hexamethylenetetramine) were investigated in the 1960s, in the solid, liquid and gaseous state (for reviews, see Nalwa and Miyata, 1997; Wolff and Wortmann, 1998). Small molecules like guanidinium, tricyano- and trinitromethane anions, or tris-acceptor-substituted benzenes were amongst the first compounds to be characterized when the area was revived in the 1990s. Tricyanomethane anion has also been the subject of a careful computational study; the two- and three-level contributions were evaluated at a high level of ab initio theory (Luo et al., 1996). The discrepancies in comparison with the reported experimental values are largely due to an erroneous value for the internal reference used in the HRS experiment. As was to be expected, low NLO responses in solution were noted (Zyss and Ledoux, 1994). The same holds true for small neutral heterocyclic ␲ systems (Wolff and Wortmann, 1998). The most successful efficiency–transparency ratio for a ‘‘small’’ molecule is probably achieved with the cross-conjugated cyanine [106] (Stadler et al., 1996b; Wortmann et al., 1997), perhaps not too surprising because cyanines are known for their high polarizabilities. However, [106] still suffers from attenuated conjugation caused by steric hindrance (Dale et al., 1987). This can be obliterated in higher vinylogues (Dale and Eriksen, 1988; Reichardt et al., 1995) and the cross-conjugated pentamethine [107] indeed shows higher second-order polarizabilities – as always, at the expense of a considerable red shift in the UV–visible spectrum (Wortmann et al., 1997). Threefold donor–acceptor-substituted benzene derivatives like [109] (Ledoux et al., 1990) or [110] (Verbiest et al., 1994; Wolff et al., 1996b; Wortmann et al., 1997; Wolff and Wortmann, 1998) show better performance; for [109], only powder data and computational results are available. Both are of the hexasubstituted type, but strong intra- and inter-molecular hydrogen bonds provide for planarity. The discrepancy (Brédas et al., 1992) between the observation of a moderate powder SHG efficiency of [109] and the published – (Cady and Larson, 1965) centrosymmetric crystal structure (P1) has been resolved. The powder consists of two polymorphs, with the second one adopting the close to optimal space group P31 (Voigt-Martin et al., 1996, 1997).

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Scheme 21

201

Non-dipolar NLO-phores of the cross-conjugated cyanine and hexasubstituted benzene type (cf. [101], [103]).

Since this is the minor constituent in the powder, the pure active polymorph would probably have responses about 10–20 times that of urea. As pointed out before, hexasubstituted benzenes are sterically congested molecules and the combination of three nitro and three alkylamino groups as in [111] leads to ‘‘benzene’’ rings that are widely distorted from planarity (Wolff et al., 1996a) so these molecules no longer belong to the class of non-dipolar NLO-phores of threefold symmetry. Unfortunately, D3h symmetry is often assumed for these compounds in order to facilitate computations (Dehu et al., 1998), although this does not even hold for the minimum structure of [109] computed for the gas phase (Baldridge and Siegel, 1993). The interpolation of phenylalkynyl units between the central benzene core and the donor and acceptors groups should both alleviate steric strain and improve on the magnitude of transition dipoles. For example, this is borne out by X-ray crystallography and UV–visible spectroscopy for [112], but NLOproperties have not been evaluated so far (Wolff, Wortmann et al., unpublished results, 1998). Larger ␲ systems based on the sterically encumbered oligophenyl type give as disappointing results as in the 1D series, as shown by a series [113] of triphenylamine and pyrimidine derivatives (Stadler et al., 1995, 1996c). The same holds true for simple triphenylmethane dyes, as demonstrated by extensive investigations on crystal violet [115] (Kaatz and Shelton, 1996; Morrison et al., 1996; Noordman and van Hulst, 1996) which showed that

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Scheme 22 Non-dipolar NLO-phores of the trisubstituted types [102] and [103].

previous studies (Zyss et al., 1993; Verbiest et al., 1994) had substantially overestimated second-order polarizabilities. With respect to p-nitroaniline, the absorption of crystal violet is drastically red-shifted, but its ␤0 is only roughly the same if the different molecular weights are considered. Triphenylcyclopropenylium salts may be formally regarded as homologues of triphenylmethane dyes in which steric hindrance is alleviated. A reduced second-order polarizability slightly in excess of p-nitroaniline has been reported for [116] (Verbiest et al., 1994). Purely acceptor-substituted benzene derivatives also alleviate steric strain, but at the expense of efficiency, as [114] shows (Stadler et al., 1996b). The same holds true for neutral large ␲ systems that lack a combination of genuine donors and acceptors, like the tris(ethynylpyridyl)benzene [117] (Noordman and van Hulst, 1996) and organometallic complexes formed from sym-trialkynylbenzene (Whittall et al., 1997). Introduction of suitable combinations leads to quite high second-order polarizabilities. Extended electrondeficient, donor-substituted heterocycles with some rotational freedom have been fully characterized (Wortmann et al., 1997). Completely rigid congeners are too fluorescent to be characterized successfully (Wolff and Wortmann, 1998). The properties of the s-triazine [118] can again be improved by the

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Scheme 23

203

Non-dipolar NLO-phores from nitrogen heterocycles.

interpolation of alkynyl groups that obliterate the steric interaction of the biphenyl type. Accordingly, [119] shows the highest unquestioned ‘‘reduced’’ ␤0/M for a non-dipolar NLO-phore measured sufficiently far from resonance. Some sub-phthalocyanines, especially [120], have been reported to show exceptionally high ␤ values (Sastre et al., 1996). The authors ascribe the SHG measured at ␭ ⫽ 670 nm solely to the band at ␭max ⫽ 580 nm and accordingly perform a dispersion correction only on the basis of the corresponding transition energy. Two weaker bands at ␭max ⫽ 660 and 700 nm were not included in this correction. Tetrahedral structures have been studied occasionally (Lequan et al., 1994; Lambert et al., 1998; Wolff and Wortmann, 1998). Only one tensor element, ␤xyz, is significant. The conjugation between the substituents through the central element seems to be unsatisfactory, despite the enhancement by through-space coupling in [122] and possibly also [121]. Spiro-conjugation in D2d molecular symmetry does improve interaction to some extent as shown on p. 173 (cf. [5] in Scheme 4).

Scheme 24

Non-dipolar NLO-phores of the tetrahedral type.

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Dipolar 2D NLO-phores: general properties, optimization strategies and examples Multiple CT leading to three level contributions and the possibility of ordering through electrical poling is combined with dipolar 2D NLO-phores. The establishment of structure–property relationships for this type of molecule, reviewed recently (Wolff and Wortmann, 1998), is still in its infancy because application of a single analytical method is clearly inadequate to unravel the combination of different tensor elements. It is convenient to keep the number of numerically different tensor elements as low as possible and to study planar molecules of C2␯ symmetry. Out of the seven ␤ components that are significant for this case, only five are independent. In addition, the components in the x-direction, defined as perpendicular to the molecular plane (y, z) are negligibly small, so only four components remain: ␤yyz ⫽ ␤yzy, ␤zyy and ␤zzz. A combination of polarization-dependent EFISHG, HRS and EOA measurements (Wortmann et al., 1993; Moylan et al., 1996; Wolff et al., 1997) allows all to be determined. Even those experimental studies of dipolar 2D NLO-phores that do not attempt a determination of individual tensor components have focused on C2␯-symmetric molecules (Nalwa et al., 1991; Bosshard et al., 1996; Wong et al., 1996). This also holds for accompanying theoretical studies (Nalwa et al., 1995; Tomonari et al., 1997). The first comprehensive study to address 2D behaviour was made with 3,5-dinitroaniline [123]. The two degenerate CT from the amino to the two nitro groups couple to give CT transitions with two mutually orthogonal polarization directions. Two bands with polarization directions perpendicular, and parallel to the direction of the ground-state dipole along the C2 axis can be observed (Liptay et al., 1967; Wortmann et al., 1993). The ‘‘perpendicular’’ band has the lower transition energy. The special advantage of this situation becomes clear if frequency doubling via the ‘‘perpendicular’’ band is considered. Suppose we have a perfectly ordered system, and a fundamental polarized perpendicularly with respect to the C2 axis. The second harmonic will be polarized in the direction of the C2 axis which is the direction of the intramolecular charge transfer. Hence, it cannot be reabsorbed by the ‘‘perpendicular’’ transition! The transition along the C2 axis, however, which does have the correct polarization for absorption, is quite hypsochromically shifted. Two pictures of the net change in charge for the S0–S1 and the S0–S2 transition as calculated by a simple PPP model demonstrate that for both transitions the change in dipole moment is along the molecular axis. Therefore, this is also the polarization direction of the second harmonic. Of technological significance is also the possibility of non-critical phase-matching. This is due to the different frequency dispersion of the twoand three-level contributions to the tensor elements. ␤zzz on one side [(53), (60)] and ␤zyy, ␤yzy, ␤yyz on the other side [(56), (57), (62)] show different frequency dependence. Kleinman’s symmetry condition (p. 131) may also be

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Scheme 25 Degenerate CT in 3,5-dinitroaniline leads to transitions with parallel and perpendicular polarization, but both ⌬␮ag and ⌬␮bg are parallel to the symmetry axis. Regions of charge depletion (䊉) and increase (䊊) for the first two electronic transitions have been calculated by the PPP method.

violated. The same arguments apply in principle also to tetra-substituted benzene derivatives of C2␯ symmetry. As is to be expected from non-dipolar 2D NLO-phores, the ␲ system of simple benzene or naphthalene derivatives is too small to show practically useful NLO responses (Wortmann et al., 1993; Wolff et al., 1997). Especially the ‘‘perpendicular’’ transition is of quite low intensity as indicated by the smaller double-pointed arrow in Scheme 25. Therefore, the tri- and tetrasubstituted blueprint structures [98]–[100] (Scheme 20) were systematically elongated in one, two or all three directions by the interpolation of phenyl-ethynyl bridges. The use of triple bonds eliminates possible conformational isomerism. The trisubstituted type [124]–[127] is shown. By a combination of EOAM and polarization-dependent EFISHG and HRS, all four independent tensor components were evaluated (Wolff et al., 1997). Results are given in Table 5. The dipole moment ␮ lies parallel to the z axis, the y axis within the molecular plane and the x axis perpendicular to the molecular plane. The ␤ values (at 1064 nm) are in units of 10⫺50 C m3 V⫺2. Elongation in one direction gives an essentially one-dimensional NLOphore, because the transition with perpendicular polarization is no longer detectable experimentally (PPP calculations indicate the presence of such a band with low oscillator strength); the efficiency is 3.8/1.6(1.3) ⫻ pnitroaniline despite the m-substitution. Compound [127] is no longer a purely 1D system although the ‘‘perpendicular’’ long wavelength absorption is also missing [for ␤zzz: 2.9/0.74(0.79) ⫻ p-nitroaniline]. If, however, the ␲ system is extended in only two directions, molecules with surprisingly high

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Table 5 ␤ Components of [124]–[127] (Wolff et al., 1997). Parameters

[124]

[125]

[126]

[127]

␭max ␤zzz ␤zyy ␤yzy ⫽ ␤yyz

383, 327 123 ⫾ 22 6 ⫾ 29 4 ⫾ 15

361 16 ⫾ 4 67 ⫾ 4 72 ⫾ 10

393, 327 17 ⫾ 4 34 ⫾ 4 31 ⫾ 5

340 73 ⫾ 15 50 ⫾ 11 50 ⫾ 18

Scheme 26.

off-diagonal elements are obtained that by far exceed the magnitude of the diagonal components. Polarized EFISHG has been performed also on [128] for R ⫽ H (Wortmann et al., 1993). 2D behaviour is apparent from the different values. Incorporation of [128] for R ⫽ alkyl into bulk structures has been studied (Nalwa et al., 1991). Several other systems have been characterized by only a single experimental technique, amongst them organometallic species like [129] (Whittall et al., 1997). In dihydropyran [130] (Moylan et al., 1996) the perpendicular band is also missing and two transitions with parallel polarization contribute to the NLO activities observed. Some other cross-conjugated structures of the type [131] have been characterized for their third-order polarizabilities (Bosshard et al., 1996). Computations have been performed for the tetra-substituted type [98] (Nalwa et al., 1995; Tomonari et al., 1997) at the semi-empirical and ab initio level.

6

Conclusion

There are so many publications in the realm of organic molecules for non-linear optics that this review has focused on second-order polarizabilities. The translation of these properties into bulk structures could only be hinted at and devices could not be mentioned at all. We have tried to develop a formal description of tensorial properties that is consistent with the SI system and would like to suggest to groups working in the area to adopt it in order to

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Scheme 27

207

C2␯-symmetric two-dimensional NLO-phores for which only combinations of tensor elements have been evaluated experimentally.

avoid further confusion. Every paper should indicate the ␤ convention employed and state a standard value for p-nitroaniline determined by the method used in order to facilitate comparison of the results of different groups. HRS measurements should in addition always detail how fluorescence was taken into account. A serious shortcoming in many papers in the area of non-linear optics (and materials science in general!) is the complete absence of an experimental part concerning ‘‘classical organic chemistry’’. Identity and purity of the compounds investigated are thus impossible to check. The optimization strategies and trade-offs in conventional one-dimensional (1D) NLO-phores are now well established and the interest in this area has probably been shifted away from chromophore design and structure–property relations to the problems of material implementation. However, 1D NLOphores and the two-state model are only limiting cases of the more general two-dimensional (2D) behaviour which comprises both two- and three-level contributions. We would like to emphasize that especially dipolar molecules with multiple charge transfer offer new properties that are useful for practical device implementation. Their optimization, still in its infancy, is probably synthetically not as straightforward as for one-dimensional NLO-phores where (though rarely credited) information both on synthesis and physical properties from other areas of dye applications could be exploited. It may just be this extra challenge which could make work in this area more interesting for the organic chemist who is not intrigued by the fine-tuning of known basic

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structures, and also to the physical chemist who has grown weary of seeing properties reduced to p-nitroaniline. In addition to improved second-order polarizabilities and/or transparencies in some cases, new ways to circumvent the efficiency–transparency trade-off and to achieve non-critical phasematching are provided. More systematic studies will be needed in order to explore the full potential of this fairly new approach, and many structures are possible targets.

Acknowledgements The authors gratefully acknowledge financial support of their own research in the area from the Deutsche Forschungsgemeinschaft, the Volkswagen-Stiftung and the Fonds der Chemischen Industrie. Parts of this review were written during a sabbatical at UW Madison, Wisconsin, USA, and J.J.W. would like to thank the faculty and staff for their kind hospitality.

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Tautomerism in the Solid State TADASHI SUGAWARA AND ISAO TAKASU The University of Tokyo, Graduate School of Arts and Sciences, Tokyo, Japan 1 2

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Introduction 219 Proton tautomerism in an isolated system 222 The proton-tunnelling mechanism in tautomerism 222 The potential profile of the tautomerization of malonaldehyde 225 Molecular symmetry vs. shape of proton potential 227 Photo-induced tautomerization of salicylates 228 Proton tautomerism in the solid state 229 Tautomerization of naphthazarin and tropolone in crystals 230 Tautomerization of carboxylic dimers in crystals 233 Double-proton transfer in porphyrins 236 Multiple proton transfer in pyrazoles 240 Photochromism and thermochromism derived from proton tautomerism 244 Photo-enolization of 2-methylisophthalaldehyde 244 Photochemical vs. thermochemical behaviour of salicylideneanilines 246 Photochromic behaviour of dinitrobenzylpyridines 249 Photochemical hole-burning 250 Dielectric properties derived from proton tautomerism in crystals 252 Tautomerization of naphthazarin and 9-hydroxyphenalenones 253 Dielectric phase transitions of squaric acid 258 Dielectric response of hydrated p-phenylenebis(squaric acid) 259 Concluding remarks 261 Acknowledgements 261 References 262

Introduction

The term of ‘‘tautomerism’’ originated from a combination of tautos (Greek: ‘‘identical’’) and isomerism, and it is a notion introduced by Laar (1885) to designate the reversible structural isomerism consisting of sequential processes: bond cleavage, skeletal bond migration and bond reformation. A typical example is keto–enol tautomerism, which involves proton migration as seen in the case of a tautomerization of acetylacetone [1a]. The proton tautomerism involves two equivalent enol forms [1b, 1c] of acetylacetone. This degenerate proton tautomerism is coupled strongly with proton transfer along the hydrogen bond. Valence isomerism of bullvalene [2] (Doering, 1963; Schröder, 1963) proceeds through degenerate Cope rearrangements. Since 219 ADVANCES IN PHYSICAL ORGANIC CHEMISTRY VOLUME 32 0065-3160/98 $30.00

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such valence isomerism involves only ␲-bond switchings, it is excluded from this review in order to concentrate on the discussion about proton tautomerism. Proton tautomerism occurs readily in solution, because there are no restrictions in fluid media on the conformational changes required for the tautomerization to take place. On the other hand, in the solid state, the mobility of molecular skeletons and protons is strictly limited. This is a main reason why tautomerizations are rarely observed in the solid state. One of the important factors for the feasibility of proton tautomerism in the solid state is the presence of an intra- or intermolecular hydrogen bond along which proton transfer takes place, accompanied by a switching of ␲-bonds. The presence of the hydrogen bond, however, is not sufficient for observing proton tautomerism in the solid state. In fact, the tautomerization is likely to be frozen on the basis of the dipole–dipole interaction between tautomers as discussed in this review. The crucial factor is the shape of the potential profile along the reaction coordinate of the proton transfer. If the proton potential is almost symmetrical in respect of the saddle point, the proton transfer occurs even in the solid state as described in the following sections. Proton tautomerism in the solid state is an ideal system to investigate cooperative phenomena concerning proton transfer, because tautomeric molecules are connected with each other in terms of dipolar and/or hydrogen-bonding interactions. Tautomerization plays a vital role in biological systems as well. For example, multi-hydrogen bonds between base pairs construct the DNA double helix. When the adenine (A) base, for example, is UV irradiated, A is converted to its tautomer (A⬘), and the resultant tautomer A⬘ is misread as guanine (G) and is paired with cytosine (C) (Fig. 1) instead of thymine. Such a miscoding can become the origin of mutation (Watson and Crick, 1953; Goodman, 1995). Recently, tautomerism has attracted considerable attention because of its potential application in molecular devices, the concept of which was originally

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Fig. 1 Mechanism of mutation in a DNA double helix. (a) Regular recognition. (b) Misread recognition.

proposed by Carter (1982). One of the representative models of such devices is a memory function device using a hydrogen-bonded tautomeric system, such as 2-methyl-9-hydroxyphenalenone [3] (Haddon and Stillinger, 1982). This compound exists as an equilibrium mixture of two non-equivalent tautomeric forms. If one of the tautomers can be converted into the other by UV irradiation, and the converted isomer can be read out, the system may work as a molecular memory device. The aim of this review is to describe the essence of proton tautomerism, focusing on the potential profile of proton transfer along the reaction coordinate of the tautomerization in the solid state. In Section 2, proton tautomerism in an isolated system is discussed based on spectroscopic

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investigations and on theoretical calculations (de la Vega, 1982; Kosower and Huppert, 1986; Barbara et al., 1989). In Section 3, dynamic properties of tautomerism in the solid state, especially in crystals, are described, together with a brief introduction to instrumental methods appropriate for the investigation of proton dynamics (Fyfe, 1983). In Sections 4, 5 and 6, some representative physical properties related to proton tautomerism, such as photochromism, thermochromism, photochemical hole-burning (Exelby and Grinter, 1965; Friedrich and Haarer, 1984) and dielectric properties related to proton tautomerism are described, and the potential utility of the phenomenon is discussed.

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Proton tautomerism in an isolated system

Before describing the physical properties derived from proton tautomerism in the solid state, the intrinsic nature of proton tautomerism in isolated systems, such as in the gas phase or in a dilute solution, is discussed.

THE PROTON-TUNNELLING MECHANISM IN TAUTOMERISM

The feasibility of proton tautomerization depends on the shape of the potential profile and on the height of the potential barrier along the reaction coordinate. Consider the one-dimensional potential profile of a degenerate proton tautomerism along the hydrogen bond produced by combining two independent single-well potential curves for the vibrating OH group (Fig. 2). In the classical model, a particle never crosses the potential barrier, unless the vibrational energy level exceeds the barrier height. Proton transfer, however, is of a wave character due to the extremely light mass of the proton. Accordingly, a proton can penetrate the barrier and reach the well on the other side. This phenomenon is called ‘‘proton tunnelling’’, and it causes a

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Fig. 2 (a) Two independent single-well potential curves for the vibrating OH groups. (b) The double-well potential curve for the hydrogen-bonded OH group (⫺OH···O⫽ vs. ⫽O···HO⫺). The solid curve indicates a symmetric combination of the two individual proton wave-functions and the broken curve is for an antisymmetric combination. The tunnelling splitting (⌬0) is also shown.

splitting of vibrational energy levels as a result of the interference of proton wave functions (Fig. 2). The magnitude of the tunnelling splitting (⌬0) corresponds to the difference in eigenvalues between the symmetric and the antisymmetric combination of wave functions, and it is determined by the degree of overlap of wave functions penetrating the potential barrier. Thus the tunnelling splitting is a function of the height of the potential barrier Eo and the proton tunnelling distance Ro. The smaller Eo or Ro is, the larger

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proton tunnelling coordinate

Fig. 3

Diagram of a laser-induced fluorescence spectrum of 9-hydroxyphenalenone [6] in a neon matrix. The dashed arrow indicates a forbidden transition.

becomes the tunnelling splitting. Malonaldehyde [4], tropolone [5] and 9-hydroxyphenalenone [6] are representative tautomeric molecules which undergo a reversible proton transfer in the gas phase or in solution, and a significant contribution from proton tunnelling is indicated by spectroscopic measurements as described below. An electronic energy-level diagram determined from the laser-induced fluorescence spectrum of 9-hydroxyphenalenone [6] in a neon matrix is shown in Fig. 3. An emission line from the lowest vibrational level in the S1 state is found to be split into a doublet. The energy difference between these two lines

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corresponds to the magnitude of the tunnelling splitting (⌬0) in the ground state (S0), and ⌬0 is determined to be 69 cm⫺1 (Bondybey et al., 1984). It should be noted that the tunnelling splitting (⌬0*) in the S1 state is even larger than that in S0, because the potential barrier for the proton tunnelling in the excited state is lower. The values of ⌬0 in the ground state of malonaldehyde and tropolone in the gas phase have been determined to be 21 cm⫺1 (Bondybey et al., 1984) and 0.99 cm⫺1 (Tanaka et al., 1999), respectively. The difference in the tunnelling splittings between these compounds can be interpreted in terms of the difference in bond angles of the five- [5] or six- [4,6] membered ring incorporating the hydrogen bond. Such an angular dependence of the efficiency of proton transfer in hydrogen-bonded systems has been suggested on the basis of theoretical calculations (Scheiner, 1994).

THE POTENTIAL PROFILE OF THE TAUTOMERIZATION OF MALONALDEHYDE

Since malonaldehyde [4] is the most extensively examined system among reversibly tautomeric molecules, it is appropriate to draw a complete potential energy surface (PES) for the proton tautomerism. From the laser spectroscopic studies, the tunnelling splitting of malonaldehyde is estimated to be 21 cm⫺1, that of the deuterated compound is 3.0 cm⫺1 (Baughcum et al., 1981, 1984; Turner et al., 1984). This large isotope effect on the splitting is explained as follows. Since the zero-point vibrational energy level of the O⫺D bond is lower than that of the O⫺H bond, the relative barrier for the tunnelling is higher for the deuterio-compound. Therefore the tunnelling splitting for the deuterio-compound becomes significantly smaller than that of the protiocompound. The barrier of tautomerization of the protio-compound was also estimated to be 6.6 kcal mol⫺1 from the observed value of tunnelling splittings (Baughcum et al., 1984). The mechanism of the proton transfer has been investigated theoretically as

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Fig. 4 PES of malonaldehyde [4] obtained by the SCF calculation as a function of r1 and r2. The bold line above indicates the tunnelling path and the lower line the minimum energy path.

well. A calculation on the tunnelling probability of the one-dimensional model with a symmetrical double well potential was performed by de la Vega (1982). The calculated ratio of the tunnel splittings between the protio- and deuterio-compounds is, however, smaller by two orders of magnitude than the experimental one. The origin of this disagreement has been elucidated by Bicerano et al., (1983) and Carrington and Miller (1986) as follows. The transfer of a hydrogen atom of malonaldehyde is accompanied by a number of small-amplitude motions, including the stretching of O⫺H, C⫺O, and the bending of C⫺O⫺H and O⫺C⫺C bonds. At the moment when the O···O distance becomes shorter owing to the relevant vibrational motions, the proton can tunnel with higher probability. This is called dynamic shortening. The molecular skeleton of the equilibrium geometry [4a] and that of the transition state [4b] are obtained by SCF calculation (Shida et al., 1989). Under such circumstances, a multidimensional PES is required to describe the reaction coordinate of the tautomerization. For example, a self-consistent calculation of the three-dimensional PES of the proton transfer in malonaldehyde was performed by Shida et al. (1989, 1991), varying not only two O···H distances (r1, r2), but also the O···O distance (r3). The potential curve is portrayed as a function of r1 and r2 in Fig. 4, r3 being optimized to minimize the potential energy at each point (r1, r2) on the surface. The change of the O···O distance implicitly includes the contribution to the proton transfer of

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vibrational modes other than the OH stretching. Along the minimum energy path in Fig. 4, the distances r1 and r2 change significantly, coupling with stretching and/or bending vibrations, and the proton transfer proceeds by way of the saddle point. The potential barrier is calculated to be 6.3 kcal mol⫺1. Even for the tunnelling mechanism, the trajectory exhibits downward curvature relative to the diagonal line, suggesting a contribution from dynamic shortening. This phenomenon is called vibration-assisted proton transfer.

MOLECULAR SYMMETRY VS. SHAPE OF PROTON POTENTIAL

The introduction of a substituent into tautomeric molecules, in particular 3-hydroxyenone derivatives, affords important information on the relationship between the molecular symmetry and the potential profile of the proton transfer along the intramolecular hydrogen bond. If a methyl group is substituted at the 2-position of 9-hydroxyphenalenone, the tautomers [3a, 3b] become non-equivalent as described in Section 1. The more complicated case is 2-methylmalonaldehyde [7], or 5-methyl-9-hydroxyphenalenone [8], where a methyl group is substituted in the plane bisecting the molecule. Although these compounds seem to afford a symmetrical proton potential, it is not true when the conformation of the methyl group is taken into account. The top and

side views of [7] are shown in Fig. 5. In the conformer shown in Fig. 5(a), one of the C⫺H bonds is perpendicular to the ␲-plane. Since two tautomers of this conformation are equivalent, the proton potential should be symmetrical. The proton wave functions, therefore, are likely to interfere, leading to a significant tunnelling splitting. On the other hand, in the conformer shown in Fig. 5(b), one of the C⫺H bonds is eclipsed with the ␲-nodal plane; the two tautomers are non-equivalent. Thus the proton potential becomes unsymmetrical and the proton wave functions localize. In fact, the tunnelling splitting of 2methylmalonaldehyde is about ten times smaller than in malonaldehyde. The proton tunnelling in [7] is considered to be coupled with hindered

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Fig. 5 Symmetric and asymmetric double minimum proton potentials derived from the rotation of methyl group of 2-methylmalonaldehyde [7].

rotation of the methyl group. Moreover, rotation of a methyl group is known to occur also through a tunnelling process at cryogenic temperatures, as long as the PES associated with the rotation is reasonably symmetrical. In such a case, the two tunnelling motions have been claimed to occur cooperatively (Busch et al., 1980) on the basis of an analysis of microwave spectroscopic data. The substituent effect on the antiferroelectric phase transition of 9-hydroxyphenalenones [6] is discussed in Section 5 on the same basis. Substituent effects of halogen atom(s) on the tautomerism of tropolone have been studied in detail (Sekiya et al., 1990, 1994; Tsuji et al., 1992, 1993). A substitution pattern to lower the molecular symmetry leads to an unsymmetrical proton potential. Thus tunnelling splittings are not observed. On the other hand, significant tunnel splitting is observed in symmetrically substituted derivatives. When the magnitude of tunnel splitting is examined more closely, the dependence of the tunnel splitting on the kind of substituted group and on the substitution sites is observed even in derivatives with a symmetrical substitution pattern.

PHOTO-INDUCED TAUTOMERIZATION OF SALICYLATES

Even though a proton potential for tautomerization is unsymmetrical, the tautomerization can be promoted by thermal or photochemical excitation. For

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Fig. 6 Energy diagram of the keto and enol forms of methyl salicylate [9] in the ground and in the first excited states.

instance, proton transfer in methyl salicylate [9a] is initiated by UV irradiation (Fig. 6). The photo-excited methyl salicylate [9a*] is isomerized to the enol form [9b*] in the S1 state, and the latter relaxes to afford the enol form [9b] in the ground state. Since the enol form is unstable compared to the keto form in the ground state, the former is converted back into the latter spontaneously. Nagaoka (1988; Nagaoka and Nagashima 1990) proposed a simple qualitative theory for predicting the feasibility of tautomerization through photo-excitation. In the first singlet excited state, a node emerges together with formal unpaired electrons in the ␲-conjugated system, causing a localization of ␲-electrons [10b]. When the bond switching takes place coupled with proton transfer, the formal unpaired electrons vanish due to the recovery of the ␲-conjugation [10c]. This convenient rule can be applied to photochemical intermolecular double proton transfer in the hydrogen-bonded dimer of 7-azaindole [11a,b,c].

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Proton tautomerism in the solid state

Even though rapid tautomerization is observed in an isolated system, it is not necessarily the case in the solid state, in which the symmetrical proton potential is sometimes deformed by an environmental effect. Therefore, the

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tautomerization in the solid state is likely to be frozen at lower temperatures. Accordingly, it is interesting to see how tautomerization processes in the gas phase or in solution are modulated in the crystalline environment. The crystal structure of the enol form of dibenzoylmethane [12] was investigated by X-ray diffractometry (Williams, 1966; Hollander et al., 1973; Jones, 1976; Etter et al., 1987; Bertolasi et al., 1991). Since, in crystals, the molecular structure of the enol form is non-planar, the tautomers become energetically non-equivalent. While a normal C⫽O bond length is 1.25 Å and that of C⫺O is 1.33 Å, the bond lengths determined for the C⫽O and C⫺O bonds in [12] are 1.273(4) Å, 1.311(4) Å, respectively. These bond lengths are regarded as populationally weighted ones, resulting from a rapid averaging process between [12a] and [12b]. The equilibrium constant was determined to be 0.66 in favour of [12b] (Vila et al., 1990). TAUTOMERIZATION OF NAPHTHAZARIN AND TROPOLONE IN CRYSTALS

Naphthazarin [13] is known to afford polymorphic crystal structures (Cradrick and Hall, 1971). Among them, in the polymorphic form C, both the C⫽O and

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C CP/MAS nmr spectra of solid naphthazarin [13].

the C⫺O bond lengths are 1.295 Å. This value suggests that two energetically equivalent tautomers are exchanging rapidly. The polymorphic form C undergoes a phase transition at TC ⫽ 110 K, and the tautomerization is frozen at lower temperatures. The C⫽O and C⫺O bond lengths then become 1.246(3) Å, 1.343(3) Å, respectively, exhibiting a distinct bond alternation (Herbstein et al., 1985). Rapid tautomerization also takes place in polymorphic form B at higher temperatures than 120 K. A solid-state CP/MAS nmr spectrum of the polymorphic form B of naphthazarin at room temperature shows only three signals, assignable to C(9,10), C(2,3,6,7) and C(1,4,5,8), due to the fast intramolecular proton exchange (Fig. 7). At 110 K, the tautomeric process is frozen on the nmr time scale, and the signals are resolved into five peaks (Shiau et al., 1980). This spectral change is coupled with a structural phase transition as well. In the low temperature phase, the tautomeric process is completely frozen as a result of dipolar interactions between tautomeric pairs.

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Fig. 8 13C nmr spectrum of tropolone. (a) In solution. (b) In solid; 13C CP/MAS. (c) Chemical-exchange network in tropolone mapped out in a 2-D 13C CP/MAS experiment.

Tropolone [5] is another example in which the dynamical behaviour of the tautomerism has been investigated in detail. In solution, tropolone undergoes a rapid tautomeric exchange, giving a four-line spectrum (Fig. 8a), assignable to C(4,6), C(5), C(3,7) and C(1,2) at room temperature (Weiler, 1972). On the other hand, a solid-state CP/MAS spectrum at ambient temperature (Fig. 8b) shows seven resonance lines, suggesting that the tautomeric process is much slower than the nmr time scale. The reason for freezing of the tautomerization is ascribed to the formation of dimers [5c] through bifurcated hydrogen bonds in crystals (Shimanouchi and Sasada, 1973). When the crystals are heated above the melting point of 60⬚C, the spectrum becomes almost identical to that in solution. This spectral change suggests that the dimers are dissociated and

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the rapid tautomeric exchange starts to occur in the fused state. The slow tautomeric exchange of tropolone in crystals was recently reinvestigated by the 2-D 13C CP/MAS technique, and the activation energy of the tautomerization was found to be 26 kcal mol⫺1 (Szeverenyi et al., 1983). A more detailed mechanism of tautomerization of tropolone in the solid state has been discussed based on results obtained by a rotor-synchronized MAS 2D exchange 13C nmr technique (Titman et al., 1992). This may prove a useful general method to investigate detailed chemical dynamics in the solid state. Tautomerization of hetero-analogues of tropolone, such as 2-aminotroponimines [14] (Goldstein and Trueblood, 1967; Hexem et al., 1982) and [15] (Machiguchi et al., 1992) has been also investigated in the crystalline state.

As seen from the above examples, intermolecular dipolar interactions and intermolecular hydrogen bonds are the major factors in the freezing of tautomeric exchange in the crystalline state.

TAUTOMERIZATION OF CARBOXYLIC DIMERS IN CRYSTALS

Many carboxylic acids form dimers hydrogen-bonded between the carboxy groups (Leiserowitz, 1975), and such carboxylic acid dimers in crystals are some of the most extensively studied systems for the investigation of proton tautomerism. When one of the carboxyl protons is transferred to the carbonyl oxygen, the new O⫺H group, in turn, becomes a proton donor. Thus proton transfer in a carboxylic acid dimer is recognized as a simultaneous ‘‘doubleproton transfer’’ along two hydrogen bonds [16].

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⌬E

Fig. 9 Potential curve of the double-proton transfer between two configurations of carboxylic dimers. ⌬E denotes the energy difference between two configurations.

The deuterium isotope effect on the rate constant of the double-proton transfer in the solid state can be used as a criterion to judge whether the double-proton transfer proceeds in a simultaneous or in a stepwise manner. The difference between ratios of rate constants kHH/kHD and kHD/kDD depends on the mechanism of the double-proton transfer. If a double-proton transfer occurs in a stepwise manner, the slower step in the kHD process, that is, a deuteron transfer, becomes rate-determining. Consequently the ratio kHH/kHD should be large. On the other hand, since deuteron transfers are ratedetermining in both kHD and kDD processes, the ratio kHD/kDD is expected to be small. By contrast, if a proton transfer and a deuteron transfer occur simultaneously, the ratios of kHH/kDD and kHD/kDD should be of the same order. The observed ratios for benzoic acid dimers in crystals were found to be kHH/kHD ⫽ 23, kHD/kDD ⫽ 10 at 15 K (Sto¨ckli et al., 1990). Accordingly, the double-proton transfer is considered to occur concertedly in benzoic acid dimers [16]. The symmetry around the dimer is, however, usually low in crystals. The degeneracy of configurations (a) and (b) in Fig. 9, therefore, is removed by an environmental effect, which is sometimes called a ‘‘site-splitting’’ effect. The presence of two non-equivalent configurations in crystals has been confirmed by X-ray and neutron diffraction experiments. Although the

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C6H5COOH

C6D5COOH

Fig. 10 Temperature dependence of the nuclear spin lattice relaxation times of benzoic acid [16] crystals. Upper set, C6H5COOH; lower set, C6D5COOH. The solid lines are obtained from the theoretical calculation taking into account phonon-assisted tunnelling.

energies of these two configurations are different, the relative energies fluctuate around the local minima, because each molecule in the tautomeric pair can exchange its internal vibrational energy with surrounding molecules through intermolecular vibrations of the crystal lattice. Proton tunnelling occurs efficiently only when two configurations become isoenergetic through the above mechanism. Thus, this type of tunnelling is called phonon-assisted tunnelling. The nuclear spin lattice relaxation times of the acidic protons of crystals of carboxylic acids were determined by 1H nmr spectroscopy (Meier et al., 1982; Nagaoka et al., 1983). In general, a proton nuclear spin experiences the local magnetic field created by the nuclear spins of protons and other magnetic nuclei (I ⫽ 0) around it. Since the magnitude of the local field fluctuates with the frequencies of migrating protons in the neighbourhood, its fluctuation causes a relaxation of the proton nuclear spin in the crystal. Accordingly, one can evaluate the rate of proton transfer from the relaxation time of the proton nuclear spin. It is to be noted that the T1 vs. temperature plot deviates considerably from the symmetrical V-shaped curve expected from a classical model of thermal vibrations between the two potential minima (Fig. 10). Skinner and Trommsdorff (1988) rationalized this discrepancy on the basis of a proton tunnelling mechanism within the carboxylic acid dimer at lower temperatures. The phonon-assisted tunnelling rate of the carboxylic dimer with a hydrogen bond distance in the range of 2.6–2.7 Å is evaluated to be k0 ⫽ 108–109 s⫺1. It is found that the deuteration of the phenyl group of benzoic acid affects the relaxation times considerably (Fig. 10). This means

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that the probability of the double proton transfer in the benzoic acid dimer is related also to the vibrations of the aromatic CH bonds. This result provides support for the aforementioned phonon-assisted mechanism. Horsewill et al. (1994) examined the hydrostatic pressure effect on the proton transfer in crystals of a carboxylic acid dimer. Under a hydrostatic pressure, the distance of hydrogen bonds becomes shorter, and this is accompanied by a decrease in the potential barrier to proton transfer. The temperature dependence of the rate of the proton transfer turns out to be of a non-Arrhenius type. The influence of phonon-assisted tunnelling becomes evident as the external pressure increases, especially at lower temperatures.

Recently, tautomerism of a hetero-analogue of a carboxylic acid dimer, N,N⬘-diarylamidine dimer [17], in the crystalline state has also been investigated using 15N CP/MAS nmr (Ma¨nnle et al., 1996). A similar tendency concerning phonon-assisted proton transfer was observed.

DOUBLE-PROTON TRANSFER IN PORPHYRINS

Another example of tautomerism which involves a double-proton transfer is tautomerization of porphyrins. Mechanisms for the double-proton transfer of porphine [18], the parent compound of porphyrins, in solution have been discussed for a long time. A point of special interest is whether the double-proton transfer occurs in a synchronous or stepwise manner. There are two structural isomers (cis and trans forms) of porphine (Fig. 11), although only the trans form has been detected by nmr spectroscopy. The energy of the trans form is calculated to be 3–7 kcal mol⫺1 lower than that of the cis form (Merz and Reynolds, 1988; Smedarchina et al., 1989). Theoretical investigations on the potential energy surface of porphine also revealed that the barrier height of the direct conversion from trans to trans is considerably higher than the barrier height from trans to cis (Fig. 12).

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Fig. 11 Double-proton transfer in porphine [18].

The rate constants of the intramolecular double-proton transfer of 15Nlabelled porphine crystal have been determined by 15N CP/MAS nmr spectroscopy, and the HH/HD/DD kinetic isotope effects have also been obtained. The observed kinetic isotope effects for porphine at 273 K are kHH/kHD ⫽ 17, and kHD/kDD ⫽ 1.9 (Braun et al., 1994). Judging from the large difference of the isotope effect, the double-proton transfer is considered to proceed by the stepwise mechanism. The observed tendency is different from that of the tautomerization of carboxylic acids. The stepwise mechanism is also observed in solution. Arguments over the tautomerization of porphine in crystals has not been resolved yet. The distribution of protons on the nitrogen atoms of porphine turned out to be disordered in an early X-ray study (Webb and Fleischer, 1965). A subsequent reinvestigation of the tautomerism of porphine by X-ray diffractometry, however, revealed that the protons are localized on nitrogen atoms as the trans form (Chen and Tulinsky, 1972; Tulinsky, 1973), while the

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Fig. 12 Potential energy surface of porphine [18].

solid-state 13C nmr data suggest that the protons are rapidly migrating between the nitrogen atoms (Frydman et al., 1988). A hint to rationalize this apparent contradiction has been obtained by 1H nmr relaxation time (T1) measurements (Frydman et al., 1989, 1991). The relaxation time data are interpreted as being governed by skeletal rotational processes in the crystals. If a skeletal rotation of 90⬚ occurs coupled with the proton transfers, the protons seem to be localized on the same nitrogen atoms (Fig. 13). Although there are still some arguments about this mechanism (Braun et al., 1994), the co-rotation model seems to explain the apparently conflicting data obtained by different instrumental methods. Tautomerization of porphyrins in crystals exhibits diverse patterns of behaviour. There are polymorphic crystal structures of tetraphenylporphyrin (TPP) [19]; the tetragonal and the triclinic forms (Silvers and Tulinsky, 1967; Codding and Tulinsky, 1972; Tulinsky, 1973). In the tetragonal form, X-ray

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double-proton transfer

239

90° rotation

Fig. 13 Co-rotating model of the proton tautomerism of porphine [18]; combination of proton transfer and 90⬚ rotation.

diffraction data suggest that protons are disordered over the four nitrogen atoms of the porphyrin ring. The solid-state 15N and 13C-nmr spectra of the tetragonal form have also been interpreted on the basis of the rapid migration model (Wehrle et al., 1987; Frydman et al., 1988). On the other hand, in the triclinic form, the experimental results have been interpreted as indicating that the protons are localized on the nitrogen atoms of one of the trans forms. The localization of protons in the triclinic form may be brought about by the non-equivalence of two trans forms. Such a non-equivalency may be derived from the lack of a high local symmetry around the tautomeric molecules in crystals (Fig. 14). Rapid double-proton transfer of other porphyrin derivatives [20] and phthalocyanines [21] has been investigated by 15N or 13C CP/MAS spectroscopy (Limbach et al., 1984; Kendrick et al., 1985; Schlabach et al., 1992). It is to be noted that the mechanism of tautomerization of dibenzotetraazaannulenes [22,23] varies depending on the substitution patterns (Limbach et al., 1984, 1987). In the disubstituted derivative [22], the double-proton transfer was found to occur by a simultaneous mechanism, while in the tetrasubstituted derivative [23], the coexistence of concerted and stepwise mechanisms has been observed.

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Fig. 14 Schematic drawing of the potential energy surface of TPP [19] crystals. Protons are delocalized in the tetragonal crystal, while they are localized in the triclinic crystal.

MULTIPLE-PROTON TRANSFER IN PYRAZOLES

N1-Unsubstituted pyrazole derivatives [24] undergo proton interconversion between two tautomeric forms [24a, 24b]. They are known to form several kinds of intermolecularly hydrogen-bonded crystals. The hydrogen-bonding schemes are classified into four patterns, hydrogen-bonded dimer [25] (Baldy et al., 1985; Smith et al., 1989; Aguilar-Parrilla et al., 1992a,b), trimer [26] (Llamas-Saiz et al., 1994; Foces-Foces et al., 1994; Aguilar-Parrilla et al., 1995), tetramer [27] (Llamas-Saiz et al., 1993; Beagley et al., 1994) and hydrogenbonded infinite chain [28] (Berthou et al., 1970; La Cour and Rasmussen, 1973;

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Elgureo et al., 1995). Among these hydrogen-bonded crystals, proton transfer is detected only in cyclic hydrogen-bonded types [25–27]. The ratios of rate constants for the triple-proton transfer in protio- and deuterio-compounds [26] are determined to be kHHH/kHHD ⬃ kHHD/kHDD ⬃ kHDD/kDDD ⬃ 3.6, and kHHH/kDDD ⬃ 47 ( ⫽ 3.63) at 300 K, indicating that the logarithms of the rate constants change linearly upon deuteration of the N-H group (Aguilar-Parrilla et al., 1997). In the case where the triple-proton transfer occurs concertedly, the process can be interpreted by the two-site model (Fig. 15). In the transition state, three protons should be located in the middle between two adjacent nitrogen atoms. This interpretation is supported by ab initio calculations on the triple-proton transfer system (de Paz et al., 1997).

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Fig. 15 Double minima potential of the triple-proton transfer in a hydrogen-bonded trimer of pyrazole [26].

In contrast, proton transfer is not detected in crystals consisting of hydrogen-bonded chains [28]. Since proton transfer along the chain is tightly coupled with ␲-bond switching, it is an energy-consuming process to move all the protons simultaneously along an infinite hydrogen-bonded chain.

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Photochromism and thermochromism derived from proton tautomerism

When a substrate A is photo-isomerized to substrate B by UV irradiation, the absorption spectrum of A changes to that of B. Furthermore, substrate B may be transformed back to substrate A reversibly by irradiation at a different wavelength or through a thermal process (Scheme 1). Such a phenomenon is designated as photochromism. A change of colour by thermal excitation is called thermochromism.

Scheme 1.

This section describes the characteristics of the photo- and thermochromism derived from proton tautomerization in crystals, using the photo- and thermochromic behaviour of 2-alkylisophthalaldehydes, salicylideneanilines, and dinitrobenzylpyridines as typical examples.

PHOTO-ENOLIZATION OF 2-METHYLISOPHTHALALDEHYDE

Colourless crystals of 2.5-dimethylisophthalaldehyde [29] change to red when placed in sunlight. The red colour fades when the sample is kept in the dark for more than five minutes. The back reaction is accelerated by warming the sample in the dark. This whole process occurs repeatedly at least for several cycles. Similar photochromic behaviour is observed in 2-methyl-5isopropylisophthalaldehydes [30] (Raju and Krishna Rao, 1987). It is claimed

colourless R ⫽ Me [29], i-Pr [30]

red

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Scheme 2.

that, in general, the distance across which hydrogen transfer takes place should be shorter than 2.72 Å for the migration to occur (Bondi, 1964; Scheffer and Dzakpasu, 1978; Taylor and Kennard, 1982; Trotter, 1983; Appel et al., 1983). Furthermore, the oxygen atom and the migrating hydrogen atom are expected to be located on the same plane (Lewis, 1978; Padmanabhan et al., 1986, 1987). According to X-ray diffraction data (Kumar and Venkatesan, 1991), the intramolecular separation between the hydrogen atom of the methyl group and the oxygen atom of the carbonyl group attached at the ortho position of [29] or [30] is 2.58 or 2.54 Å, respectively. Thus, these crystals of [29] and [30] satisfy the above condition. The mechanism of the photochromic reaction has been explained on the basis of results of flash photolysis experiments on 2-methylacetophenone [31] in solution (Scheme 2) (Haag et al., 1977). The photo-enolization occurs in the triplet state through hydrogen abstraction by the carbonyl oxygen from the methyl group, followed by intersystem crossing. The enol form is isomerized back to the starting material thermally. It is intriguing that the kinetic stability of the enol form of 2-alkylisophthalaldehydes [29] and [30] in the crystalline environment is much higher than that of 2-methylbenzaldehyde in solution. It is claimed that the kinetic stability of the enol form is generally higher in the solid state. The tendency may be explained as follows. The structural change from the enol form to the keto form is restricted in the crystalline environment. Besides, the enol has a chance to form an intermolecular hydrogen bond with a neighbouring molecule in crystals so as to enhance the kinetic stability.

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PHOTOCHEMICAL VS. THERMOCHEMICAL BEHAVIOUR OF SALICYLIDENEANILINES

Salicylideneaniline [32] and its derivatives are known to exhibit photochromic behaviour. The colour of the crystal of salicylideneaniline changes from pale yellow to red, and the red colour changes back to the original one in a thermal process. On the other hand, other derivatives of salicylideneaniline show only thermochromic behaviour. It is usually the case that a photochromic crystal never exhibits thermochromism, and vice versa.

According to X-ray diffraction data for the photochromic crystal of 2-chloro-N-salicylideneaniline [33, X ⫽ H, Y ⫽ 2-Cl] (Bregman et al., 1964a), ring B is twisted by ca. 51⬚ from the plane consisting of ring A and the C⫽N bond (Scheme 3). The twisted form suppresses thermal proton transfer from the OH group to the imino nitrogen. It also prevents close packing in crystals, leaving enough space around the enol form (Fig. 16). On UV irradiation,

Scheme 3.

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Fig. 16 Crystal structure of photochromic 2-chloro-N-salicylideneaniline [33].

photo-induced proton transfer occurs accompanied by the cis–trans isomerization of the C⫽C bond, affording the trans-keto form responsible for the red colour (Scheme 3). The transformation proceeds presumably through the cis-keto form as an intermediate (Cohen and Schmidt, 1962; Nakagaki et al., 1977; Barbara et al., 1980; Hadjoudis, 1981; Higelin and Sixl, 1983; Turbeville and Dutta, 1990). It is argued that, in some cases, the isomerization occurs around the C⫽N bond to afford the cis-zwitterion (Lewis and Sandorfy, 1982; Yuzawa et al., 1993). Incidentally the photochromic behaviour of anils cannot be observed in fluid media. This is because, even though isomerization around the C⫺N bond occurs photochemically, the reverse isomerization occurs readily in a thermal process through the inversion of the nitrogen atom. Thus, fading of colours takes place rapidly in fluid media. In thermochromic crystals, on the other hand, the planar form of N-5-chlorosalicylideneaniline [34; X ⫽ 5-Cl, Y ⫽ H] (Bregman et al., 1964b) is packed tightly with an interplanar distance of 3.4 Å (Fig. 17). Consequently, in the planar anils, the proton transfer from the phenolic OH to the imino nitrogen can take place thermally. The two forms, the OH form and the NH one, are thermally equilibrated, and as the temperature increases, the contribution of the cis-keto form becomes larger (Scheme 4). In order to increase proton donor and acceptor abilities, halogen atoms have been introduced into the phenolic ring of salicylideneaniline, and the phenyl ring was replaced by a pyrenyl group [35] (Inabe et al., 1991). According to the X-ray crystal structure of [35], the perchloro phenolic moiety is stacked with the pyrenyl moiety through charge-transfer interaction. The intramolecular hydrogen-bond distance is 2.53 Å and it is much shorter than in the parent anils. The differential Fourier map shows that the hydrogen is almost in the middle between the oxygen and nitrogen atoms (Fig. 18). The

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Fig. 17 Crystal structure of thermochromic N-5-chlorosalicylideneaniline [34].

Scheme 4.

result suggests that the hydrogen bond between the OH group and the imino-nitrogen becomes stronger by virtue of the chemical modification. In other words, the unsymmetrical proton potential is modified to an almost symmetrical one with a low potential barrier. Then the resultant proton potential becomes suitable for observing proton transfer along the hydrogen bond. Kawato et al. (1985, 1986, 1994a) prepared t-butyl derivatives of thermochromic anils [36] and they found that these compounds exhibit photochromic behaviour in crystals. Introduction of bulky substituents is considered to provide enough space for the cis–trans isomerization to occur.

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Fig. 18 Differential Fourier map showing the location of a hydrogen of the OH group in N-tetrachlorosalicylidene-1-aminopyrene [35].

When thermochromic anils are incorporated into host crystals of deoxycholic acid or its derivatives, the incorporated anils have been found to exhibit photochromic behaviour as well (Koyama et al., 1994; Kawato et al., 1994b). The incorporated anils are evidently allowed to isomerize photochemically as in the case of derivatives of anils carrying bulky substituents.

PHOTOCHROMIC BEHAVIOUR OF DINITROBENZYLPYRIDINES

Photochromic behaviour based on proton tautomerism of dinitrobenzylpyridines [37] in crystals has also been reported (Chichababin et al., 1925; Margerum et al., 1962). The colourless crystal became coloured upon UV irradiation. The reaction scheme is thought to be as depicted in Scheme 5. One of the benzylic protons (CH form) of the starting compound migrates to the nitro group on UV irradiation to afford an aci-nitro form (OH form) which, in turn, is converted to the ␣-pyridonoid form (NH form) as a result of proton transfer. The latter compound eventually regenerates the starting material (CH form) (Sixl and Warta, 1985).

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Scheme 5.

5

Photochemical hole-burning

One of the potential applications of the phenomena derived from proton isomerism is hole-burning (Gutiérrez et al., 1982; Friedrich and Haarer, 1984). Some organic dyes involve a partial structure of the hydrogen-bonded tautomer as a pigment. When absorption spectra of such types of dyes are measured in a polymer matrix or in a rigid glass, the half bandwidth is several orders of magnitude wider than that of the intrinsic bandwidth of the dye, reflecting inhomogeneity in the local environment around the tautomeric molecules in rigid matrices. The broadening of the bandwidth in matrices is called an inhomogeneous broadening (⌬␻i). The broadened band consists of many narrow bands, the bandwidth of which is called a homogeneous broadening (⌬␻h) (Fig. 19). If the 0⫺0 transition of the Q band of tetraphenylporphyrin at around 650 nm is photo-irradiated using a laser with a narrow half bandwidth, the tautomeric pigment undergoes a photo-induced structural change in its hydrogen-bond arrangement, and the change is considered to be fixed at cryogenic temperatures. As a result, a sharp hole is created in the spectrum. This phenomenon was found by two Russian groups independently in 1974 (Kharalamov et al., 1974; Gorokhovskii et al., 1974). The potential utility of the phenomenon as an information storage system was proposed by Castro et al. (1978).

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Fig. 19 Principle of photochemical hole-burning. Irradiation at the inhomogeneously broadened band of a tautomeric molecule in a matrix creates a hole due to photo-induced tautomerization.

Phthalocyanines and porphyrins are representative materials for hole-burning experiments (Völker and Macfarlane, 1979). Gorokhovskii photolysed phthalocyanine [21] dispersed in n-octane at 5 K by a ruby laser at the 0⫺0 transition of 694 nm, and he succeeded in creating a hole with a half bandwidth of 0.7 cm⫺1. Photo-irradiation of tetraphenylporphyrin [19] in a phenoxy resin [39] (Sakoda et al., 1988) causes a hole which remains even at temperatures higher than 77 K. A similar result is obtained by irradiation of a water-soluble porphyrin derivative [38] in aqueous polyvinyl alcohol (Horie et al., 1988; Furusawa et al., 1989). These results suggest that rigid protic matrices play a significant role in maintaining a hole at higher temperatures. One of the main concerns in photochemical hole-burning is how to elevate the critical temperature for hole endurance. Incidentally, the photochromic compounds described in the last section cannot be used for photochemical hole-burning experiments. This is because the hole-burning experiment requires a level-selective transition in order to

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create a sharp hole in the absorption spectrum. Such a level-selective transition can be possible only when the electronically excited species has a bound electronic state with discrete vibrational levels. In the case of the photo-induced proton transfer of porphyrins, for example, the structural change during the excitation process is subtle. Consequently, the potential curve of the excited state is similar to that of the ground state. On the other hand, in the case of photochromic materials, the structure of the isomerized compound changes drastically. The potential curve in the excited state is, therefore, much deformed or sometimes becomes even pre-dissociative. Accordingly, the selectivity in the photo-excitation concerning the vibrational levels is diminished, and the bandwidth of the generated hole is enforced to be very broad.

6

Dielectric properties derived from proton tautomerism in crystals

If a material can be electrically polarized under an external electric field and the polarization inverts in response to the alternating electric field, the material is designated as a dielectric. The origins of the polarity inversion are classified as shown in Fig. 20: (a) electric polarization, which is caused by the displacement of electrons relative to the nucleus; (b) ionic polarization caused

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Fig. 20 Mechanisms of polarity inversion in a dielectric. (a) Electric polarization; (b) ionic polarization; (c) orientation polarization.

by the relative displacement of negatively and positively charged ions; (c) orientation polarization caused by orientational motions of ions or polar molecules. When the polarity of ions or polar molecules inverts independently according to the above mechanism(s), the dielectric property of the material is called paraelectric. When a paraelectric material exhibits a spontaneous polarization at lower temperatures and, moreover, its net polarization responds to the external electric field, it is called a ferroelectric. On the other hand, a paraelectric whose net polarization is cancelled at lower temperatures, is called an antiferroelectric. Orientational motions of ions or polar molecules are, however, restricted in most crystals. Only when they can librate in the crystal, does the crystal exhibit orientation polarization.

TAUTOMERIZATION OF NAPHTHAZARIN AND 9-HYDROXYPHENALENONES

From the viewpoint of the dielectric properties of organic molecular crystals, the proton tautomerism of the s-cis form of 3-hydroxyenone [40] is intriguing, because the horizontal component of the dipole moment of a tautomeric

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molecule inverts, coupled with a proton transfer along the intramolecular hydrogen bond. Since the inversion of the dipole moment corresponds to the polarity inversion of a dielectric material, such a crystal composed of tautomeric molecules can be regarded as a dielectric.

The mechanism of the polarity inversion of tautomeric molecules is totally different from the orientation polarization of conventional organic dielectrics, such as camphor and poly(vinylidene fluoride), and the dielectric response of this new type of dielectric should be much faster. Furthermore, a significant contribution of the proton-tunnelling mechanism to the proton tautomerism is frequently observed. Consequently, the dielectric property derived from proton tautomerization should have a high chance of being related to quantum phenomena. Since the proton tautomerism of naphthazarin [13] or 9-hydroxyphenalenone [6] can be investigated precisely in an isolated system, it is intriguing to see how these tautomeric molecules behave as dielectrics in the solid state. Incidentally, if the tautomers keep the centrosymmetry during the tautomerization process, the dielectric response cannot be detected, because the changing directions of the dipole moments coupled with the tautomerization, even though they invert, cancel. The temperature dependence of the dielectric constant of the B-phase crystal of naphthazarin was measured by Mochida, Izuoka and Sugawara (unpublished data). The dielectric constant increases gradually on lowering the temperature, but it starts to decrease at temperatures lower than TC, which is the transition temperature of the structural phase transition. The tendency is explained as follows. In the temperature range above TC, the dipole moments of naphthazarin molecules are randomly inverting through the thermal tautomerization process. When an external electric field is applied, some of the inverting dipole moments tend to respond to the external field. With lowering temperature, the responding fraction of the inverting dipole moments increases in accord with the Curie law. The tautomerization is, however, halted at lower temperatures than TC. In the lower temperature phase, the dipole moments of the tautomeric molecules are ordered in an antiparallel manner. The antiparallel ordering of the dipole moments corresponds to the antiferroelectric phase transition of the dielectric material. The tendency is consistent with that observed by CP/MAS 13C nmr spectroscopic measurement as described in Section 3.

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Fig. 21 Temperature and frequency dependence of the AC dielectric constant of 9-hydroxyphenalenone [6] in a crystal. The thermally activated pattern indicates the predominant contribution of the orientation polarization to the dielectric constant. (a) 10 kHz, (b) 100 kHz, (c) 300 kHz, (d) 1 MHz, (e) 3 MHz, (f ) 10 MHz.

Dielectric measurements on 9-hydroxyphenalenone [6] were performed as well (Mochida et al., 1994a). The temperature dependence of the dielectric constant is totally different from that of naphthazarin, and the dielectric response starts to increase at temperatures higher than 255 K (Fig. 21). This temperature is, in fact, the structural phase-transition temperature from crystal phase I to II. According to X-ray crystallographic analysis (Svensson and Abrahams, 1986), the molecular arrangement is perfectly ordered in phase I, but it is disordered in phase II with reference to the orientation of the long molecular axis. The observed dielectric behaviour of 9-hydroxyphenalenone [6] is thought not to be derived from the tautomeric process, but from a librational motion due to dynamic disorder in the crystal phase II (Fig. 22). Although the tautomerization is considered to take place in phase II, the contribution to the dielectric response is hidden by the larger contribution derived from the orientation polarization. In order to suppress the dynamic disorder of the parent compound in the crystal, the 5-methyl and 5-bromo derivatives of 9-hydroxyphenalenone [8], [41] were prepared. X-Ray crystallographic analysis revealed that the molecular arrangement is completely ordered and that the symmetry around

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Fig. 22 Librational motion of 9-hydroxyphenalenone [6] in the crystal. The arrow shows the dipole moment of the molecule.

the tautomers is reasonably high. Moreover, the intramolecular O···O distances between the carbonyl and the enolic oxygens of [8] and [41] are as short as 2.51 and 2.49 Å, respectively. The CP/MAS 13C nmr spectrum of [8] and [41] shows that a rapid kinetic exchange between two equivalent tautomers takes place in the crystals. The temperature dependence of the dielectric constant of [8] exhibits a Curie-type behaviour, which is similar to that of naphthazarin. The result suggests that this dielectric response is derived

from proton tautomerism. It is also found that the dielectric constant of [8] decreases at lower temperatures than 41 K due to an antiferroelectric phase transition. In contrast with the result of [8], the temperature dependence of the dielectric constant of [41] is very small and the dielectric constant stays almost the same even at cryogenic temperatures (Fig. 23a). The difference in the temperature dependence of the dielectric constant is also rationalized by the proton potential along the intramolecular hydrogen bond. As discussed in the case of 2-methylmalonaldehyde [7] in Section 2, the proton potential of 5-methyl-9-hydroxyphenalenone is considered to be unsymmetrical, when the conformation of the methyl group is taken into account (Busch and de la Vega, 1986). As a result, proton tunnelling is less

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257

Fig. 23 Temperature dependence of the AC dielectric constant at 10 kHz of 5-bromo-9-hydroxyphenalenone crystals: (a) temperature-independent dielectric constant of the protio-compound; (b) deuteration-induced phase transitions of deuteriocompound.

likely to occur, leading to localization of the proton along the hydrogen bond. The tautomerization is thus quenched more easily at cryogenic temperature based on the antiferroelectric interaction between tautomeric molecules. On the other hand, the proton potential of the 5-bromo compound is exactly symmetrical with reference to the reaction coordinate of the tautomerization. Consequently, the proton transfer can proceed through the tunnelling mechanism. This is the reason why the paraelectric behaviour is maintained even at 4 K. The suppression of the antiferroelectric phase transition may be derived from a quantum tunnelling effect. Such paraelectric behaviour can be regarded as ‘‘quantum paraelectricity’’, which is a notion to designate the phenomenon that (anti)ferroelectric phase transitions are suppressed even at cryogenic temperatures due to some quantum-mechanical stabilization, proton tunnelling in this case. This interpretation was proved by the deuteration of the OH group. The temperature dependence of the dielectric constant of the deuterated sample shows a Curie-type behaviour and the antiferroelectric phase transitions were observed at lower temperatures (Fig. 23b). The origin of the dual phase transitions of the crystal of [41] at 22.8 K, and 34.5 K has been discussed in detail (Mochida et al., 1994b; Noda et al., 1994; Moritomo et al., 1995). The heat capacities of protio- and deuterio-forms of 5-bromo-9-hydroxyphenalenone were also measured. The heat capacity of the protio-compound is larger than that of the deuterio one in a certain low-temperature region. This tendency can be rationalized by assuming extra levels derived from the tunnelling splitting. From the excess heat capacity, the tunnelling splitting was deter-

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mined to be 64 cm⫺1 (Matsuo et al., 1998). The value of the tunnelling splitting is consistent with that evaluated in the gas phase (Section 2).

DIELECTRIC PHASE TRANSITIONS OF SQUARIC ACID

Squaric acid [42], which belongs to a family of oxocarbons (Serratosa, 1983; Seitz and Imming, 1992), consists of two s-trans forms of 3-hydroxyenone [43] structures perpendicular to each other, and it shows extremely strong acidity (pKa1 ⫽ 0.54) as an organic acid (West, 1980). Crystalline squaric acid is the only organic substance whose dielectric property has been shown to be

derived from tautomerism accompanied by intermolecular proton transfer. When the tautomerization of squaric acids takes place cooperatively accompanied by the intermolecular proton transfer, the dipole moments invert concertedly (Fig. 24). In the crystal, a sheet structure, consisting of a two-dimensional hydrogenbonded network, is formed (Fig. 24). Within the sheet, the dipole moment of each molecule points in the same direction. This can be regarded as a ferroelectric two-dimensional arrangement. The directions of the gross dipole moments of adjacent sheets, however, are antiparallel to each other (Semmingsen and Feder, 1974). X-ray analysis at room temperature indicates that the protons are localized along the hydrogen bonds and they are not disordered. The hydrogen bond distance is 2.54 Å which is fairly short, but the hydrogen bond can still be classified as a type having a double-well potential. Although the dielectric constant of the squaric acid crystal is small at room temperature, it increases suddenly to become 400 at temperatures higher than 370 K. This tendency suggests that the tautomerization does not occur at room temperature, but it starts to take place accompanied by intermolecular proton transfer above 370 K. The large value of the dielectric constant in the paraelectric phase suggests that the tautomerization takes place in a cooperative manner. The dielectric behaviour of the deuterium substituted crystal has also been studied. The TC is drastically raised to about 500 K. This large isotope effect supports the interpretation that the dielectric response of squaric acid is derived from proton (deuterium) tautomerism. Similar isotope effects on

TAUTOMERISM IN THE SOLID STATE

259

Fig. 24 Sheet structure in the crystal of squaric acid [42] and the cooperative inversion of dipole moments associated with intermolecular proton transfer.

dielectric phase transitions have been observed in many inorganic hydrogenbonded dielectrics represented by potassium dihydrogen phosphate. Application of high external pressures influences the transition temperature to the antiferroelectric phase (Yasuda et al., 1979; Samara and Semmingsen, 1979). The TC becomes lower as the applied pressure increases. Under an ultra-high pressure of about 3 GPa, the antiferroelectric transition itself disappears and the high dielectric constant of ca. 200 is maintained even at cryogenic temperatures (Moritomo et al., 1991). Since Raman diffraction measurements under 3–4.5 GPa revealed that squaric acid exists still as an alternating bond form, the tautomerization coupled with intermolecular proton transfer occurs even at low temperatures (Moritomo et al., 1990).

DIELECTRIC RESPONSE OF HYDRATED p-PHENYLENEBIS(SQUARIC ACID)

In order to construct an organic hydrogen-bonded dielectric which operates under ambient conditions, one-dimensional hydrogen-bonded crystals have been explored. One of the successful examples is 1,4-phenylenebis(squaric acid) (PBSQ) [44] (Takasu et al., 1996), in which two squaric acid moieties are connected by a p-phenylene unit. PBSQ crystallizes in a hydrated form, PBSQ ⭈ 2H2O, and the crystal is characterized by a layered structure, consisting of one-dimensional hydrogenbonded chains of PBSQ molecules (Fig. 25). The hydrogen-bond distance is very short (2.48 Å), even compared with that of squaric acid (2.55 Å). The incorporated water molecules form dimers, and they are located perpendicular

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to the layer. The O···O distance within the water dimer is only 2.45 Å, which is substantially shorter than that in ice crystals (O···O distance 2.76 Å). Thus the dimeric water molecules are considered to be bridged by a proton, existing as [45]. This means that PBSQ exists as a mono-deprotonated species [46], as a result of donation of a proton to the dimeric water molecules. It is certain that the hydrogen bond configuration of PBSQ does not change before and after the proton transfer. Consequently, the hydrogen bond potential in this crystal is symmetrical and the barrier to proton transfer should be low. The crystal structure of PBSQ ⭈ 2H2O is, therefore, almost ideal for observing reversible intermolecular proton transfer.

Fig. 25

Crystal structure of the hydrated crystal of p-phenylenebis(squaric acid); PBSQ ⭈ 2H2O.

TAUTOMERISM IN THE SOLID STATE

261

The dielectric response of PBSQ ⭈ 2H2O derived from tautomerization is observed under atmospheric pressure and at ambient temperature. Furthermore, the dielectric constant turns out to be almost temperature-independent in the temperature range 4–300 K. When PBSQ was deuterated, the dielectric constant obeyed the Curie law, and an antiferroelectric phase transition was observed at 30 K. This result is strong supporting evidence for a significant contribution from the tunnelling mechanism to the dielectric response of the hydrogenous sample.

7

Concluding remarks

Proton tautomerism is a unique phenomenon in which proton transfer occurs along a hydrogen bond and this is associated with ␲-bond switching. Although tautomeric processes have been usually studied in solution, recent progress in laser spectroscopy associated with molecular beam techniques enables us to obtain information on tautomeric processes in the gas phase, affording proof for the wave character of the proton migration along the hydrogen bond. The probability of the proton transfer becomes high when the distance between the proton donating and accepting atoms is shortened by means of vibrational distortions. This situation is somewhat similar to that of electron-transfer phenomena. Investigation of the tautomerism in the solid state has also been explored by means of X-ray diffractometry, high-resolution solid-state nmr spectroscopy, dielectric measurements, etc. Therefore, it is intriguing to examine why proton tautomerism appears differently in the solid phase, depending on the modulated shape of the potential profile of the proton transfer. It is often the case that proton tautomerism is frozen in the solid state, even though the rapid tautomerization takes place in the isolated system. Such a situation is brought about by a significant deformation of the proton potential due to the local asymmetry around the tautomeric molecules in the crystal. Thus it is crucial to design crystal structures so as to obtain a symmetrical proton potential for proton transfer along the hydrogen bond. Proton tautomerism in the solid state has drawn attention from the aspect of application as well. Further investigations in related areas, such as photochromism, thermochromism, photochemical hole-burning and hydrogen-bonded dielectrics, may open a horizon of protonic molecular devices.

Acknowledgements T.S. gratefully acknowledges the outstanding contributions of his collaborators especially Drs Akira Izuoka, Tomoyuki Mochida, Yoshinori Tokura and

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Yutaka Moritomo. The authors also are grateful for a Grant-in-Aid for Scientific Research on new Program Grant No. 05NP0301 and also on Priority Areas (10146103) from the Ministry of Education, Science, Sports and Culture in Japan.

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The Yukawa–Tsuno Relationship in Carbocationic Systems YUHO TSUNO AND MIZUE FUJIO Institute for Fundamental Research of Organic Chemistry, Kyushu University, Fukuoka, Japan 1 2 3

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Introduction 267 Applications of the Yukawa–Tsuno equation 272 Yukawa–Tsuno correlations for benzylic solvolyses generating carbocations 276 Benzylic systems 276 Aryl-assisted solvolyses 295 Solvolysis of 1-arylethenyl sulphonates forming vinyl cations 303 Highly electron-deficient carbocation systems 304 Carbocation formation equilibria 315 Triarylmethyl cations 315 Benzhydryl cations 319 1,1-Diarylethyl carbocations 322 Yukawa–Tsuno correlations for electrophilic addition of alkenes 322 Hydration of arylalkenes 322 Bromination of arylalkenes 326 Structure–reactivity relationship in polyarylcarbocation systems 334 Conformations of carbocations 334 Reactivity–conformation relationship 337 Stabilities of carbocations in the gas phase 343 Structural effects 343 The resonance demand parameter 355 Theoretically optimized structures of carbocations 362 Reaction mechanisms and transition-state shifts 365 Extended selectivity–stability relationships 365 Ground-state electrophilic reactivity of carbocations 366 SN2 reactions of 1-arylethyl and benzyl precursors 372 Concluding remarks 378 Acknowledgements 379 References 379

Introduction

In physical organic chemistry, one of the fundamental concepts is the structure–reactivity relationship, among which the Hammett equation is perhaps the most important and fundamental one. In aromatic electrophilic 267 ADVANCES IN PHYSICAL ORGANIC CHEMISTRY VOLUME 32 0065-3160/98 $30.00

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substitution reactions and nucleophilic substitution reactions in the side chain, the Hammett ␴ scale clearly fails to correlate linearly the effects of substituents. This situation is exemplified by benzylic solvolyses of [1], where a positive charge which can be delocalized by the ␲-system of the ring is generated as in [1C⫹] (Scheme 1). For the quantitative treatment of substituent effects in such reactions, Brown proposed (Brown and Okamoto, 1957) a new Hammett-type structure– reactivity relationship, the Brown equation (1), in terms of substituent constant ␴⫹ instead of ␴ in the original Hammett equation. log(k/k0)

or

log(K/K0) ⫽ ␳⫹ ␴⫹

[1] [2] [3] [4] [5]

(1)

[1C⫹]

R1 ⫽ R2 ⫽ CH3 R1 ⫽ R2 ⫽ XC6H4 R1 ⫽ C(CH3)3; R2 ⫽ CH2C(CH3)3 R1 ⫽ CH3; R2 ⫽ CF3 Scheme 1

[2C⫹] [3C⫹] [4C⫹] [5C⫹]

Solvolysis of benzylic precursors.

The electrophilic substituent constants, ␴⫹, given in Table 1, were defined by a set of apparent substituent constants, i.e. (1/␳) log(k/k0), derived from the solvolysis rates of ␣,␣-dimethylbenzyl(␣-cumyl) chlorides [2] (Scheme 1) in 90% aqueous acetone at 25⬚C. For the definition, the reaction constant ␳⫹ ⫽ ⫺4.54, based exclusively on meta and ␲-electron withdrawing (␲-EW) para substituents, was applied. The use of the Brown equation as a probe of reaction mechanism is essentially based on the alternative use of substituent parameters ␴ and ␴⫹; the better correlation with one of the reference scales, i.e. ␴⫹ in this analysis, indicates closer similarity in the mechanism or in the structure of the transition state to that of the reference reaction, solvolysis of ␣-cumyl chlorides [2]. While the broad applicability of the Brown treatment is widely appreciated, this ␴⫹ treatment has the inevitable limitations of a single reference parameter relationship. The substituent effects in these systems can be more generally described by the Yukawa–Tsuno (Y–T) equation (2). log (k/k0)

or

– ⫽ ␳(␴o ⫹ r⌬␴ – ⫹) log (K/K0) ⫽ ␳␴ R

(2)

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Table 1 Substituent parameters.a Substituent (CH3)2N NH2 OH p-OCH2CH2-md p-MeO-m-Me MeO MeS 3,4,5-Me3 3,4-Me2 PhO 2-Fl Me t-Bu p-MeO-m-Cl p-MeO-m-CN Ph p-MeS-m-Cl F p-MeS-m-CN Cl Br COOMe 3,4-Cl2 CF3 CN NO2

␴po ⫺0.43 ⫺0.36 ⫺0.16 ⫺0.19 ⫺0.18 ⫺0.100 0.12 ⫺0.262 ⫺0.193 0.063 0.00 ⫺0.124 ⫺0.150 0.22 0.48 0.039 0.44 0.20 0.68 0.281 0.296 0.46 0.59 (0.54)g 0.670 0.810

␴m

␴p⫹ b

⫺0.15 ⫺0.15 0.122

⫺1.73 ⫺1.46 ⫺0.98 ⫺0.94 (⫺0.98)c ⫺0.88 ⫺0.80 ⫺0.60 (⫺0.53)c ⫺0.449 ⫺0.38 ⫺0.53 ⫺0.49 ⫺0.311 ⫺0.250 ⫺0.47f ⫺0.21f,h ⫺0.20 ⫺0.29f ⫺0.07 ⫺0.04f,h 0.115 0.150

0.05h 0.155 ⫺0.138e 0.25 ⫺0.069

0.352 0.373 0.391 0.49 0.615 0.710

0.42f 0.62 0.67 (0.67)

–⫹ ⌬␴ R ⫺1.30 ⫺1.1 ⫺0.82 ⫺0.75 ⫺0.70 ⫺0.70 ⫺0.72 ⫺0.187 ⫺0.187 ⫺0.59 ⫺0.49 ⫺0.187 ⫺0.100 ⫺0.69 ⫺0.69 ⫺0.24 ⫺0.73 ⫺0.264 ⫺0.72 ⫺0.166 ⫺0.146 0.00 ⫺0.166 0.00 0.00 0.00

Fujio et al. (1987a, 1988). bTaken from Brown and Okamoto (1958); Stock and Brown (1958), unless otherwise noted. cBrown et al. (1977b). d2,3-Dihydrobenzofuran-5-yl. e3,5-Dimethyl. fBased on ␣-cumyl solvolysis, Fujio et al. (1984). gApplying the ␴p⫹ value rather than ␴po. hSlightly solvent dependent. a

Here ␴o is the normal substituent constant which does not involve any additional ␲-electronic interaction between the substituent and the reaction – ⫹ is the resonance substituent constant measuring the capability for centre; ⌬␴ R ␲-delocalization of ␲-electron donor substituents and is defined by ␴⫹ ⫺ ␴o – Equation (2) for a reaction giving a set of apparent substituent constants ␴ can be rewritten in –⫺ o ⫹ o the form, (␴ ␴ ) ⫽ r(␴ ⫺ ␴ ), where r is constant for the reaction regardless of substituents. As – the increment of any ␴ from ␴o should be a reasonable measure of the resonance capability of the respective substituents, this proportionality represents a linear resonance energy relationship. The original form (Yukawa and Tsuno, 1959, 1965) using ␴ instead of ␴o in (2) has the same significance since the proportionality relation holds for the resonance increment (␴⫹ ⫺ ␴o) or – – (␴⫹ ⫺ ␴). The definition of resonance substituent constants ⌬␴R⫹ by any set of (␴ ⫺ ␴o) is arbitrary, and the definition of the r scale is also arbitrary. While the definition of the r ⫽ 0 scale by ␴o is theoretically most preferable (Yukawa et al., 1966), the definition of the scale for r ⫽ 1.00 by ␴⫹ is only for practical convenience. 1

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Y. TSUNO AND M. FUJIO

(Yukawa et al., 1966).1 Values are listed in Table 1. The parameter r is characteristic of the given reaction, measuring the extent of resonance demand, i.e. the degree of resonance interaction between the aryl group and the reaction site in the rate-determining transition state. The Y–T equation (2) has been used extensively in studies of electrophilic substitution in the aromatic ring, and aliphatic nucleophilic substitution and related reactions forming a carbocation or a carbocationic (electron-deficient) centre at the conjugative position (mostly benzylic position) in the side chain. In aromatic substitution, rates of halogenations were correlated with higher r values than unity (Yukawa et al., 1966). Extensive use of the Y–T equation by Eaborn and his coworkers demonstrated significant variations of r value in aromatic substitution, particularly, with r values lower than unity; these observations have been comprehensively reviewed by Norman and Taylor (1965). The present authors were not involved directly in the further development of this field and therefore these reactions will not be covered in this review. However, neighbouring aryl-assisted reactions are mechanistically related to Friedel–Crafts alkylation, and ␤-aryl assisted solvolyses will be a particular concern in this review. The present review will thus deal only with reactions involving carbocationic transition states, in most cases forming intermediates having carbocationic centres at the conjugative position in the side chain. The Y–T equation (2) introduced the concept of the varying resonance demand of reactions into substituent effect analysis. Nevertheless, the correlation results were always compared with those of the Brown equation (1) implying constant resonance demand. Although some doubts have been expressed as to the necessity of an additional parameter (Johnson, 1978), in the general application of (2), e.g. to more than 150 reactions quoted in this review, the r value changes widely with the reaction. It is not limited only to values lower than unity (0 ⬍ r ⬍ 1), as found for the ␣-cumyl system, but in many cases it is significantly higher than unity (r ⬎ 1). This fact clearly indicates that the Brown ␴⫹ scale [i.e. (2) when r ⫽ 1.0] does not reflect the highest extreme of

2 For the effects of ␲-electron withdrawing substituents in nucleophilic reactions forming anionic reaction centres, the same Y–T treatment as with (2) of ␲-electron donating substituents has also been developed (Yukawa and Tsuno, 1965; Yukawa et al., 1966) in the form of equation (2a),

– log (k/k0) or log (K/K0) ⫽ ␳(␴o ⫹ r⫺ ⌬␴R⫺)

(2a)

– where ⌬␴R⫺ is the resonance substituent constant defined as ␴⫺ ⫺ ␴o, and r⫺ is the resonance reaction constant for p-␲-EW substituents. Equation (2a) was applied to various nucleophilic resonance reactions and equilibria, to give excellent Y–T correlations with widely varying r⫺ values (Tsuno, 1974). More than 40 reactions analysed with (2a) involve nucleophilic aromatic substitution reactions of halobenzenes (Bunnett et al., 1954; Greizerstein et al., 1962; Porto et al.,

THE YUKAWA–TSUNO RELATIONSHIP

271

benzylic resonance exaltation but is merely a single point on the r scale. The r scale permits evaluation of the nature of the transition state, and has been widely applied to the assignment and interpretation of reaction mechanisms (Tsuno and Fujio, 1996). The application and generality of the Y–T relationship have been reviewed by Shorter (1978, 1982), and by Johnson (1973), and further developments of the Y–T analysis were reviewed recently by the present authors (Tsuno and Fujio, 1996).2,3 The Y–T equation has indeed two selectivity parameters r and ␳. The ␳ remains as the reaction constant as in Hammett’s original definition measuring the susceptibility of the reaction to polar effects. When a more precise definition is attempted, this usually refers to the development of charge on the side-chain reaction centre. The magnitude of ␳ should therefore be a measure of the magnitude of the charge developed and of the extent to which it is able to interact with the substituents. In the absence of conjugation of reaction centre and ring, this interaction must be electrostatic, involving the partial charge developed at various points in the molecule through inductive and resonance effects. When the side chain is capable of conjugation with the ring, the interaction may include cross-conjugation. It is here that ␳ and r will be influenced by the same structural factors. The difficulties encountered in using the analysis of substituent effects in solvolyses as a mechanistic probe mostly arise from the mechanistic involvement of the solvent (Shorter, 1978, 1982; Tsuno and Fujio, 1996). Consequently, the behaviour of benzylic carbocations in the gas phase should be the best model for the behaviour of the solvolysis intermediate in solution (Tsuno and Fujio, 1996). The intrinsic substituent effects on the benzylic cation stabilities in the gas phase have also been analysed by equation (2), and they will be compared here with the substituent effects on the benzylic solvolysis reaction. In our opinion, this provides convincing evidence for the concept of varying resonance demand in solvolysis. Finally, we shall analyse the mechanisms of a series of benzylic solvolysis reactions by using the concept of a continuous spectrum of varying resonance demand.

1966; Fujio et al., unpublished), elimination reactions of ␤-arylethyl halides and tosylates (Tsuno, 1974; Fujio et al., unpublished), the phenyl ester hydrolyses (Ryan and Humffray, 1966, 1967; Humffray and Ryan, 1967, 1969), pKa values of phenols, thiophenols, anilinium ions and dimethylanilinium ions (Yukawa et al., 1966; Tsuno, 1974) and pKa values of sulphonanilides (Yoshioka et al., 1962). All these correlations are not directly related to the systems involved in the present review and they will be reviewed separately elsewhere. 3 As the resonance reaction constant (resonance demand parameter) in equation (2a) for nucleophilic reactions is symbolized as r⫺, the resonance demand parameter in equation (2) should be symbolized as r⫹. However, we are dealing with only electrophilic reactions and equilibria, so that the resonance demand parameter is simply represented as r in equation (2) deleting superscript ⫹.

272

2

Y. TSUNO AND M. FUJIO

Applications of the Yukawa–Tsuno equation

For correlation analysis, the Y–T equation can be applied directly by least-squares analysis of the dual parameter relationship. However, improvement of the correlation is not of much interest since the increased number of parameters in the Y–T equation should inevitably improve the precision. Particularly in the investigation of reaction mechanisms, insufficient improvement could be more important for indicating the involvement of mechanistic complexity. The correlation will be frequently modified by various extraneous factors. We therefore illustrate the behaviour of substituent effects in several typical benzylic solvolyses in terms of the Y–T relationship. Thus the pKR⫹ values of symmetrically trisubstituted triarylmethanols [3] (see Scheme 1; Deno and Schriesheim, 1955; Deno and Evans, 1957) give a reasonably good correlation with ␴⫹ rather than with ␴ constants, but it is suggested that the Y–T equation results in a better correlation than the simple Brown ␳⫹ ␴⫹ treatment (Yukawa and Tsuno, 1959). The behaviour of substituents in this equilibrium is illustrated by the so-called Y–T plot in Fig. 1, in which the log K/K0 values for [3] are plotted against the corresponding ␴⫹ and ␴o values. The line segments between ␴⫹ and ␴o values for ␲-electron donor (␲-ED) p-substituents correspond to the ⌬␴–R⫹ values which reflect the resonance capabilities of these substituents; the absence of resonance capabilities of m-substituents and ␲-electron withdrawing (␲-EW) p-substituents are reflected in coalesced points for their ␴⫹ and ␴o values. The ␳m correlation line can therefore be determined unambiguously and used as a rigid reference for the Y–T analysis. The ␴⫹ plots (open circles) of ␲-ED p-substituents consistently deviate downward from a reference correlation line defined by the m-substituents. The ␳ line of the Y–T correlation should be the line which divides the line segments, ␴⫹ ⫺ ␴o, of all substituents, even though these differences for m-substituents and ␲-EW p-substituent groups may be essentially zero, at a constant internal ratio corresponding to the r value of the system. Thus, the Y–T correlation line covering 25 orders of magnitude in equilibrium constant can be defined as a unique line intersecting all line segments for ␲-ED p-substituents at such a constant ratio of 0.76. The excellent correlation results by the least-squares method are given in Table 2. While the Brown ␴⫹ constants also give good linearity, the Y–T correlation is still appreciably better than the Brown one for all the sets. For the range of only p-␲-ED substituents excluding msubstituents, (2) provides a correlation of comparable precision with a slightly larger r value of 0.88. Nevertheless, it should be noted that the correlations using these two r values and also even with r ⫽ unity are practically equal as regards the precision. The confidence of the r value in the Y–T correlation based on the data set of this system relies heavily upon the ␳ value given by the meta correlation. The second example is the correlation of rate constants for the solvolysis of

THE YUKAWA–TSUNO RELATIONSHIP

273

␳ = -11.5 r = 0.76

Fig. 1 The Y–T plot for pKR⫹ values of symmetrically substituted triphenylmethanols – given by Y–T (3) at 25⬚C: 䊊, plots against ␴⫹; 䊉, plots against ␴o; 䊐, plots against ␴ equation with r ⫽ 0.76; 䊏, plots of substituents having invariant ␴ for the three scales.

274

Y. TSUNO AND M. FUJIO

Table 2 Yukawa–Tsuno correlations for typical benzylic reactions. No. System (solventa, temp.) 1.

2.

3.

sym. Trisarylmethanol, pKR⫹ c all substituents Y–T equation Brown equation m- and p-EW p-␲-ED Y–T equation Brown equation ␣-t-Bu-␣-Neop-benzyl-OPNBd Solvolysis (50E,a 75⬚C) all substituents Y–T equation Brown equation m- and p-EW p-␲-ED Y–T equation Brown equation ␣-Me-␣-CF3-benzyl-OTse Solvolysis (80E,a 25⬚C) all substituents Y–T equation Brown equation m- and p-EW p-␲-ED Y–T equation Brown equation

␳

r

Rb

SDb

nb

⫺11.48 ⫺9.87 ⫺10.98 ⫺10.13 ⫺9.30

0.764

0.9987 0.9975 0.9971 0.9983 0.9982

0.412 0.550 0.233 0.362 0.377

13 13 4 9 9

⫺3.378 ⫺2.881 ⫺3.366 ⫺3.346 ⫺2.735

0.779

0.9996 0.9897 0.9997 0.9996 0.9820

0.034 0.157 0.026 0.027 0.164

13 13 5 9 9

⫺6.287 ⫺7.535 ⫺6.241 ⫺6.291 ⫺7.792

1.388

0.9998 0.9903 0.9994 0.9996 0.9750

0.070 0.493 0.064 0.072 0.563

28 28 10 21 21

0.883

0.769

1.389

Solvent, 50E, a (v/v) mixture of 50% aqueous–50% EtOH, 80E; a (v/v) mixture of 20% aqueous–80% EtOH. bR, correlation coefficient; SD, standard deviation; n, number of substituents included in the correlation. cCorrelation (overall ␳) for three aryl-substituents, Yukawa et al. (1966); data taken from Deno and Evans (1957), Deno and Schriesheim (1955), Deno et al. (1955, 1959). d Fujio et al. (1997c). eMurata et al. (1990a). a

␣-t-butyl-␣-neopentylbenzyl p-nitrobenzoates(OPNB) [4] (see Scheme 1; Fujio et al., 1997a). The results of the correlation analysis are summarized in Table 2, and the behaviour of substituents in this reaction is illustrated by the Y–T plot in Fig. 2. Although the range of reactivity change brought about by the substituents covers only four orders of magnitude in reactivity in this set, a linear relationship is observed for m- and ␲-EW p-substituents, sufficing to define the ␳m correlation as a reference common to both the ␴⫹ and the Y–T analyses. The ␴⫹ plots (open circles) of p-␲-ED substituents consistently deviate downward from the reference m-substituent correlation line. The ␳ correlation line can be defined by dividing all the line segments, ␴⫹ ⫺ ␴o, at a constant internal ratio of 0.78. The r value of this system thus defined is essentially identical to the value estimated from the pKR⫹ value of symmetrically substituted trityl cations. It is worthy of note that in this case the same Y–T correlation, especially the same r value, is obtained based only on ␲-ED p-substituents, without the need to use m-substituents. Furthermore, the resonance demand of this system can be differentiated from the value of unity predicted for the ␴⫹ correlation by the precision of the fit. The confidence in

THE YUKAWA–TSUNO RELATIONSHIP

275

␳ = -3.38 r = 0.78

Fig. 2 The Y–T plot for solvolysis of ␣-t-butyl-␣-neopentylbenzyl p-nitrobenzoates [4] at 75⬚C: r ⫽ 0.78. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1997a). Copyright 1997 Chemical Society of Japan.

the r value given by this set contrasts sharply with the rather low confidence limit of the above set of compounds [3]. This is not attributable to the inherent nature of the system but to the nature of the substituent set. In Fig. 2, the points dividing the resonance line segments at the constant ratio, i.e. the r value, can be found most easily from the plots of two substituents of similar –) but very different reactivity (i.e. of similar apparent substituent constants ␴

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Y. TSUNO AND M. FUJIO

– ⫹ values, e.g. p-MeO-m-Cl and p-Me substituents. Furthermore, we can find ⌬␴ R much more easily both intersecting points and the slope of the correlation line from the set of p-MeO-m-Cl, p-Me and p-MeS-m-Cl points, or from the set of p-Me, p-MeS-m-Cl and p-F substituents. From the comparison with Fig. 1, it is evident that the most important requirement is the entire randomness of the – and ⌬␴ – ⫹, of the p-ED substituents involved. reactivity order, i.e. ␴ R Criticism of the Y–T equation has arisen primarily from the relatively small change in the value of r in most benzylic solvolyses. A definite answer to such criticism will be provided by exploring the substituent effects on systems whose r values differ significantly from unity. An appropriate model reaction having a distinctly higher resonance demand (r value) can be found among highly electron-deficient carbocation systems (Murata et al., 1990a,b). Such an example is the solvolysis of 1-aryl-1-(trifluoromethyl)ethyl tosylates [5] (see Scheme 1). This system has already been shown to be far beyond the correlative ability of the Brown ␳⫹ ␴⫹ equation but to be excellently correlated in terms of the Y–T equation. In the Y–T plot of this reaction (Fig. 3), m-substituents and ␲-EW p-substituents covering five orders of magnitude in reactivity fall on a single straight line, while the ␴⫹ plot (open circles) of the ␲-ED p-substituents deviate upward from the correlation line. The Y–T plot – scale with r ⫽ 1.39 contrasts sharply with the poor (squares) against the ␴ linearity of the plot against ␴⫹; i.e. the Y–T correlation line divides all the resonance line segments (␴⫹ ⫺ ␴o) of p-␲-ED substituents by an external ratio of 1.39, which embraces also all m- and p-␲-EW substituents. The results of the analysis of varying resonance demand in solvolyses of a series of substrates are summarized in Table 3. The precision of the fit to (2) is generally found to have a standard deviation (SD value) of ⫾0.04 to 0.10 in log k/k0, depending on the magnitude of the ␳ value. This is comparable to SD values found for the meta correlation with a single set of ␴ values, and also with similar SD values found in correlations when the solvent or the leaving group is changed. Thus, a precision index SD value of 0.015–0.020␴ units is generally taken as an appropriate reference level of acceptable conformity to the Y–T equation.

3

Yukawa–Tsuno correlations for benzylic solvolyses generating carbocations

BENZYLIC SYSTEMS

Most benzylic solvolyses generating relatively stable carbocations belong to a category to which the Brown ␴⫹ constants are effectively applicable. Extensive data on ␣,␣-dialkylbenzyl solvolyses are available from Brown’s original studies, and a wide set of benzylic substituent effects were included in Johnson’s (1978) compilation of Brown ␴⫹ correlations. Although all these

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Fig. 3 The Y–T plot of solvolysis of ␣-CF3-␣-CH3-benzyl OTs [5]: r ⫽ 1.39. For interpretation of symbols, see Fig. 1. Redrawn from the data in Murata et al., 1990a.

data are reasonably treated by the Brown equation implying r ⫽ 1.0, almost all the data sets contained only p-MeO and p-Me as resonance-sensitive substituents. A broad applicability of the Brown ␴⫹ ␳⫹ equation has been demonstrated for solvolyses of an extensive series of tertiary ␣,␣-dialkylbenzyl p-nitrobenzoates (Johnson, 1978), and Brown and his coworkers (Brown et al., 1977a,d, 1978) have reported the ␴⫹ analysis of the substituent effects on the

Table 3 Yukawa–Tsuno correlations for benzylic solvolyses. Systema

Solvent,b temperature

␳

r

␣,␣-Me2-benzyl-Cl ␣,␣,Me2-benzyl-OPNB

(90A, 25⬚C) (80A, 25⬚C)

␣-Et-␣-Me-benzyl-Cl ␣,␣,(i-Pr)2-benzyl-Cl ␣-Me-␣-(c-Pr)-benzyl-OPNB 7-aryl-7-norbornyl-L

(50E, 25⬚C) (50E, 75⬚C) (90A, 25⬚C) (80E, 25⬚C) (80A, 25⬚C) (70D, 25⬚C)

⫺4.59 ⫺4.928 ⫺5.056 ⫺5.19 ⫺4.055 ⫺4.69 ⫺4.88 ⫺2.94 ⫺5.29 ⫺5.67 ⫺5.54

1.00 (1.00) 0.946 1.01 0.978 1.04 1.01

exc. p-CF3 syn-7-aryl-anti-7-norbornenyl-L exc. p-Me2N exc. p-Me2N ␣-t-Bu-␣-Me-benzyl-Cl ␣-t-Bu-␣,o-Me2-benzyl-OPNB ␣-t-Bu-␣-(i-Pr)-benzyl-OPNB ␣-t-Bu-␣-Neop-benzyl-OPNB ␣,␣-(t-Bu)2 benzyl-OPNB fenchyl-OPNB benzyl-OTs X ⭓ p-MeS-m-CN bilinear anal.c kC-Mech. kN-Mech. X ⭓ p-halogens benzyl-Cl X ⭓ 2-naph. X⭓H ␣-Me-benzyl-Cl

R

SD

n

(1.00) 0.85 0.762

0.9994 0.9995 0.9979 0.9979 0.9993 0.9995 0.996 0.997 0.9974 0.9994

0.119 0.118 0.105 0.083 0.079 0.11 0.17 0.52 0.55 0.32

10 10 9 9 17 14 3 6 6 5

⫺2.17 ⫺2.36 ⫺4.28 ⫺2.78 ⫺3.08 ⫺3.378 ⫺2.26 ⫺2.18

1.23 (1.0) 0.91 0.70 0.68 0.779 0.28 1.06

0.99999 0.999 0.9986 0.998 0.998 0.9995 0.994 0.9997

0.010 0.11 0.088 0.0648 0.059 0.034 0.07 0.03

4 4 22 8 12 13 12 7

1.28 1.294 (0.0) 1.29 1.108

0.9995

0.040 0.075

17 33

8

(AcOH, 25⬚C) (97T, 0⬚C)

⫺5.23 ⫺5.187 ⫺1.296 ⫺5.23 ⫺6.780

0.9994 0.999

0.044 0.077

14 18

9 9

(50A, 45⬚C) (97T, 45⬚C) (80A, 45⬚C) (90A, 45⬚C) (97T, 45⬚C) (AcOH, 50⬚C)

⫺5.59 ⫺6.897 ⫺4.97 ⫺4.97 ⫺6.47 ⫺4.559

1.254 1.244 1.15 1.132 1.11 1.11

0.9970 0.9980 0.9993 0.9996 0.998 0.9996

0.104 0.122 0.06 0.06 0.12 0.048

13 21 25

10 10 11 11 12 13

(70D, 25⬚C) (80A, 45⬚C) (50E, 75⬚C) (50E, 75⬚C) (50E, 75⬚C) (50E, 75⬚C) (50E, 75⬚C) (80A, 25⬚C)

21 25

Reference 1, 25 2, 25 2 3 3 4 4 2 5, 25 5 25 6 3 7 7 7 3

␣-t-Bu-benzyl-OTs 2,2-Me2-indanyl-Cl ␣-t-Bu-o-Me-benzyl-OTs ␣-t-Bu-o,o⬘-Me2-benzyl-OTs ␣-(Me3SiCH2)-benzyl-OCOCF3 ␣-(Me3SiMe2Si)-benzyl-Cl (1-arylcyclopropyl)CH2-OTs ␣,␣-Ph2-ethyl-OPNB 1-(p-c-PrC6H4)-1-Ar-ethyl-OPNB mono-substituted trityl-Cl exc p-MeO 9-arylfluorenyl-Cl, exc. p-MeO ␣-Me-benzyl-OAc, pyrolysis ␣-Me-benzyl-OCOOMe, pyrolysis ␣-Me-benzyl-OBz, pyrolysis ␣-(Me3SiCH2)-benzyl-OAc, pyrolysis kSi pathway kE pathway

(80A, 25⬚C) (80E, 25⬚C) (97T, 0⬚C) (80A, 25⬚C) (80A, 25⬚C) (80A, 25⬚C) (90D, 25⬚C) (40E, 25⬚C) (80E, 25⬚C) (80A, 25⬚C) (80A, 25⬚C) (EtOH-Ether, 25⬚C) (90A, 25⬚C) 600 K 650 K 600 K (xylene, 202⬚C)

⫺5.543 ⫺5.650 ⫺7.03 ⫺5.81 ⫺5.50 ⫺5.11 ⫺3.044 ⫺3.85 ⫺1.55 ⫺3.105 ⫺2.384 ⫺2.39 ⫺2.468 ⫺3.394 ⫺3.399 ⫺0.650 ⫺0.687 ⫺0.766

1.093 1.106 1.09 1.14 1.01 1.02 1.099 1.16 0.11 – – 1.041 0.542 0.717 0.765 1.018 0.960 1.26

0.9997 0.9996 0.9995 0.9995 0.9995 0.999 0.9999 0.9992 0.996 0.9987 0.9998 0.9951 0.9994 0.9983 0.9961 0.9941 0.9984 0.9977

0.060 0.067 0.11 0.11 0.081 0.06 0.017 0.055 0.041 0.09 0.026 0.144 0.038 0.075 0.084 0.028 0.025 0.036

31 30 11 11 8 5 7 5 8 3 3 7 6 7 6 16 7 8

⫺2.60 ⫺2.10

0.931 0.734

0.9999 0.9999

0.037 0.018

4 4

12 12 14 15 16 16 17 17 13, 18 19 19 20 21 22 23 23 24

a Partial correlation, X ⭓ or X ⭐ ; correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; substituents of which the ␴⫹ are within the range of ␴⫹ values for substituents X1 to X2. excluding X; excluding substituent X. bAqueous organic solvent, xM; a (v/v) mixture of (100 ⫺ x)% aqueous and x% organic solvent M, where M is E ⫽ EtOH, A ⫽ acetone, D ⫽ dioxane, T ⫽ TFE. c Analysed by the non-linear least-squares method using equation (5); see text.

References: 1, Brown and Okamoto (1957, 1958). 2, Brown et al. (1977a,b,c). 3, Nakata et al. (1999). 4, Fujio et al. (1993c). 5, Gassman and Fentiman (1970). 6, Fujio et al. (1994); Nakata et al. (1999). 7, Fujio et al. (1991b, 1997c). 8, Fujio et al. (1990c). 9, Fujio et al. (1990b). 10, Fujio et al. (1991a). 11, Tsuno et al. (1975); Fujio et al. (1984). 12, Tsuji et al. (1990). 13, Fujio et al. unpublished. 14, Tsuji et al. (1989). 15, Fujio et al. (1992c). 16, Fujio et al. (1991c). 17, Shimizu et al. (1991a,b). 18, Roberts and Watson (1970). 19, Brown and Ravindranathan (1975). 20. Nixon and Branch (1936). 21, Eaborn et al. (1961). 22, Taylor et al. (1962). 23, Smith et al. (1969). 24, Watanabe et al. (1992). 25, the Brown ␳⫹ ␴⫹ correlation.

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solvolyses of 1-aryl-1-cycloalkyl systems. The substituent effects in the tertiary systems of ␣,␣-dialkylbenzyl precursors are correlated to a good approximation with r ⫽ 1.0, and the ␳⫹ value varies appreciably depending on the polar effect of the ␣-substituents. Gassman and Fentiman (1970) used the Brown equation to investigate participation by a remote ␲-bond in solvolysis reactions. They compared substituent effects in the solvolyses of 7-aryl-7-norbornyl [6], and syn-7-arylanti-7-norbornenyl [7] systems. The saturated bicyclic system [6] gave a linear

[6]

[7]

␴⫹ correlation, i.e. r ⫽ 1.0 in the Y–T correlation (2), pointing to the ordinary kC mechanism of tertiary benzylic solvolysis, whereas the latter [7] gave a clearly bisected plot against ␴⫹, indicating competition from the neighbouring C⫽C assisted (k⌬) mechanism (Fig. 4). For this behaviour, Brown and coworkers (Brown and Peters, 1975; Brown et al., 1977a) proposed the term ‘‘tool of increasing electron demand’’ in which the electron supply in the transition state is related to the reduced ␳-value; e.g. ␳⫹ ⫽ ⫺5.3 for [6] to ⫺2.30 for [7] with participation. This tool was further applied to distinguish the non-classical–classical solvolysis mechanisms in 2-norbornyl and 2-norbornenyl systems. The solvolyses of either exo- or endo-2-aryl-2-norbornyl-OPNBs gave good Brown ␴⫹ correlations with nearly identical ␳⫹ values, ⫺3.83 for exo- and ⫺3.75 for endo-isomers, which are identical to that of 1-arylcyclopentyl-OPNB (Takeuchi and Brown, 1968; Brown and Takeuchi, 1968). Electron demand at the electron-deficient centre of the two systems should be the same, and obviously ␴-participation cannot be a significant factor in the predominant exo-substitution in these derivatives. In the solvolysis of 2-aryl-2-norbornenylOPNB [8], there is also no difference in the ␳⫹ values between exo- and endo-isomers which are ⫺4.21 and ⫺4.17, respectively (Brown and Peters, 1975). Similarly, in 2-arylbenzonorbornen-2-yl OPNB [9] (Brown et al., 1969) and its 6-methoxy derivatives (Brown and Liu, 1969), the increasing electron demand cannot be detected at all. It is therefore remarkable that the solvolysis of exo-2-aryl-fenchyl-OPNB [10] gives an r value close to 1.0, and nevertheless, a distinctly lower ␳⫹ value of ⫺2.27, the lowest extreme of ␳ values for tertiary dialkylbenzyl solvolyses (Fujio et al., unpublished).

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Fig. 4 The Brown ␳⫹␴⫹ plot of solvolyses of 7-aryl-7-norbornyl-OPNB [6] (䊉) and syn-7-aryl-7-anti-norbornen-5-yl OPNB [7] (䊊). Reproduced with permission from Gassman and Fentiman, 1970. Copyright 1970 American Chemical Society.

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It was shown that the ␳⫹ values observed in the solvolysis of ArC(c-Pr)MeOPNB [11] and the 2,2-dimethylcyclopropyl analogue are ⫺2.78 and ⫺2.05, respectively, compared with ⫺4.72 for ArCMe2 [2]-OPNB; it should however be noted that these correlations (Brown et al., 1977b) did not involve any p-␲-ED substituents to prove the need of ␴⫹ (or r ⫽ 1.0) for this system. The Brown correlation of 1-cyclohexenyl-1-methylbenzyl-OPNB [12] is also less certain for the same reason, and the cross-conjugation, as well as the steric inhibition of conjugation, should be taken into account in these cases. 1,1-Diphenylethyl-OPNB [13] was also found to give a linear Brown ␴⫹ correlation with ␳⫹ ⫽ ⫺3.32 (Brown et al., 1977c); this is hard to understand because of the evident steric loss of coplanarity of the aryl ring with the carbocation centre in the transition state and will be discussed further below.

Although many uncertain results were involved in these data, it appears evident that the resonance demand of solvolyses of tertiary dialkylbenzyl precursors may be, to a good approximation, essentially 1.0, irrespective of the vast reactivity change caused by the ␣-substituent perturbation (cf., Taylor, 1979; Johnson, 1978). As far as the tertiary benzylic solvolyses are concerned, any structural and mechanistic perturbations are reflected only in the variation of the ␳ parameter. The ␳ value for a reaction series is a parameter of intermolecular selectivity and can change sensitively with the reactivity (or the stability of transition state). This behaviour is often referred to as adherence to the reactivity–selectivity relationship (RSR), where the selectivity (S) may vary inversely with the intrinsic reactivity of members of a reaction series, as formulated in equation (3),

␳ ⫽ S log k0 ⫹ constant

(3)

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Fig. 5 Selectivity–reactivity relationship, ␳ vs. log k plots, for solvolyses of ␣,␣dialkylbenzyl OPNB in 80% aqueous acetone at 25⬚C: plots of ␣,␣-dialkylbenzylOPNB carrying dialkyl groups structurally unspecified, otherwise specified; 䊊, benzyl-OPNBs; squares denote 2-norbornyl derivatives, 䊐 exo- and 䊏 endo-isomers; 䊉, benzyl-OPNBs having conjugative ␣-substituents.

The RSR behaviour of the tertiary benzylic systems treated here is shown in Fig. 5. While there is considerable scatter of the plots, we can see a general trend of inverse-linear dependence of the selectivity upon reactivity. Furthermore, there are two separate correlations practically for solvolysing systems, shown by open and closed symbols; in more detail, we can find independent RSR correlations for closely related series of compounds having a common structural perturbation, e.g. the exo- and endo-series fall on separate correlation lines. The variation in the ␳ values in a closely related series of reactions can be related to the cationic charge developed at the reaction centre in the transition state. Thus, a corollary of the RSR is that the coefficient S should be a measure of whether the transition state occurs early or late on the reaction coordinate as a result of any structural perturbation of the energy of the transition state as well as of the intermediate in the reaction series. The S coefficient in an RSR should be intimately related to the ␣ exponent in the extended Brønsted equation (cf., Johnson, 1980).

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Secondary benzylic solvolyses The solvolyses of secondary benzylic compounds have been found to give a concave or bisected rather than a simple linear correlation by the Brown ␴⫹ treatment (1). The solvolysis of ␣-phenylethyl chlorides [14] in 80% aqueous acetone gives an excellent linear correlation (R ⫽ 0.999) according to (2) with an r value of 1.15 (Tsuno et al., 1975), whereas strongly electron-withdrawing (s-EW) groups deviate upward from the correlation line; cf., Shiner et al., 1968. Although the exalted r value as well as the concave Brown ␴⫹ plot may be attributed to nucleophilic solvent participation for the region of EW substituents in nucleophilic solvents, the finding of a strict linear correlation with the same r value in the less nucleophilic aqueous TFE (Tsuji et al., 1990) argues against the importance of solvent nucleophilicity in this case. Liu et al. (1982) noted that most systems favoured by the Y–T equation are secondary substrates in nucleophilic solvents and such systems may be subject to nucleophilic solvent participation. [14] R1 ⫽ H; R2 ⫽ CH3 [15] R1 ⫽ H; R2 ⫽ t-Bu [ ]-L L : Cl, OPNB, OCOR, etc. (shown only when necessary)

Any SN1–SN2 mechanistic complication should be absent in the solvolysis of ␣-t-butylbenzyl tosylates [15], which have a neopentyl-type structure (Tsuji et al., 1990). Indeed, the substituent effect in the solvolysis is accurately described by (2) with an r value of 1.09 which differs from the value r ⫽ 1.0 for the ␣-cumyl chlorides solvolysis. Based on the linearity of the correlation of the substituent effects on the solvolyses of [14] and [15] in 80% aqueous acetone, an SN1–SN2 mechanistic duality is also unlikely to be the cause of the exalted r value observed in the solvolysis of [14]. The slightly lower r value for the solvolysis of system [15] than for the ␣-methyl analogue [14], is presumably due to incomplete coplanarity of the aryl group with the cationic p-orbital in the transition state of [15] (Tsuji et al., 1990). In the solvolysis of 2,2-dimethylindan-1-yl chlorides (cf. Table 3), the vacant p-orbital developed at the benzylic position is in a proper stereoelectronic conformation to overlap the benzene ␲-system and the r value is practically identical with that observed for the solvolysis of [14]-Cl (Fujio et al., 1992c). Consequently, the resonance demand for the SN1 solvolysis of secondary ␣-alkylbenzyl systems must be appreciably and intrinsically higher than that for the solvolysis of tertiary ␣,␣-dialkylbenzyl systems. Participation of a neighbouring electron-rich group will occur more significantly for a developing secondary carbocation centre than a tertiary one. The solvolysis of 1-arylhexen-5-yl chloride [16] proceeds via ordinary

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k⌬

kc k⌬'

Scheme 2.

solvolysis (kC) and double bond assisted (k⌬ and/or k⬘⌬) pathways (Scheme 2); the k⌬ mechanism should be predominant in the deactivated range of substituents (Mihel et al., 1979), and hence should result in a bilinear Y–T plot, in the same way as observed for [7]. For the ED substituent range where nucleophilic solvent assistance also is unimportant, the substituent effect is identical to that of [14] with r ⫽ 1.15, while the saturated analogue of [16] shows excellent linearity when correlated with the substituent effects on the solvolysis of [14]. A similar participation for neighbouring silyl groups may be anticipated. Shimizu et al. (1990, 1991a,b) analysed the substituent effects of the silyl analogues of [14] with ␣-R ⫽ CH2SiMe3 [17] (Scheme 3) or SiMe2SiMe3; both systems gave good Y–T correlations with r values higher than 1 and with appreciably lower ␳ values than those of the corresponding carbon analogues. The r value identical with that of [15] indicates that despite the significant

Scheme 3.

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Y. TSUNO AND M. FUJIO

␤-silicon rate-accelerating effect of five orders of magnitude in both systems, the ␤-Si-participation (or the k⌬ process) does not seem to be significant at the transition state.

Scheme 4.

The pyrolysis of ␣-phenylethyl acetates at 600 K (Scheme 4) was studied earlier by Taylor et al. (1962) and found to give an excellent Brown ␴⫹ correlation with a very small ␳⫹ value of ⫺0.66. Other esters, benzoate and carbonate, also give linear ␴⫹ correlations with small ␳ values of ⫺0.7 to ⫺0.9 (Smith and Yates, 1965; Smith et al., 1969). Essentially the same substituent effect correlation was also found for the pyrolysis of 1-aryl-2-phenylethyl acetates (Smith et al., 1961). These small ␳ values were attributed to a small degree of charge separation in the transition state for ester pyrolysis. Similar behaviour was observed in the elimination of 1-arylethyl esters [14]-OCOR (Glyde and Taylor, 1975, 1977). The gas-phase pyrolysis of 1-arylethyl chlorides also gave a linear ␴⫹ correlation with a ␳⫹ ⫽ ⫺1.4, suggesting a more highly polarized transition state for halide pyrolysis (Bridge et al., 1968). Pyrolysis of the ␤-trimethylsilyl derivatives [17]-OAc in xylene at 202⬚C was found to proceed along two pathways; a desilylation pathway (kSi) to form styrene and an elimination pathway (kE) to give ␤-silylstyrene (Watanabe et al., 1992). The Y–T equation (2) applies to both pathways providing a correlation with ␳ ⫽ ⫺2.60 and r ⫽ 0.93 for the kSi pathway in a simple SN1 or E1 mechanism and competitive ␳ value of ⫺2.0 with r ⫽ 0.73 for the kE pathway which involves a synchronous elimination mechanism as in Scheme 4. The silicon effect can be observed as an enhanced selectivity in both pathways, indicating an appreciable charge separation, regardless of involvement of silyl-bond fission in the rate-determining step. While the r values may be appreciably less reliable than normal, the difference in the selectivities of the two pathways appears to be significant. Solvolyses with low r-values The tertiary ␣,␣-dialkylbenzyl systems generally show, to a good approximation, a characteristic r value of unity. However, when the ␣-substituent becomes bulkier, the r value decreases and sometimes substantially; the ␣,␣-diisopropyl derivative shows r ⫽ 1.0 (Fujio et al., 1993c), while the

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␣-methyl-␣-t-butyl derivative [18] (Fujio et al., 1994) and [4] show slightly reduced values, and the o-methyl derivative of [18] shows an even more reduced value (Nakata et al., 1999). The decrease in the r parameter in the congested dialkylbenzylic series can be understood if we assume that the aryl group in the transition state has not attained complete coplanarity with the [18] R1 ⫽ CH3; R2 ⫽ t-Bu [19] R1, R2 ⫽ t-Bu [ ]-L L : Cl, OPNB, etc. (shown only when necessary)

developing carbocation centre at the reaction site. Varying r values caused by the steric bulk of ␣-alkyl groups in this series may be regarded as arising from steric restriction of ‘‘intrinsic resonance demand’’ of the carbocation. Thus a diminished value of r ⫽ 0.7 was observed for the ␣-i-propyl-␣-t-butylderivative (Fujio et al., 1997c). A well-known example is the solvolysis of ␣,␣-di-t-butylbenzyl p-nitrobenzoates [19] having two bulky t-butyl groups at the reaction centre. The application of (2) affords an excellent linear correlation (Fig. 6), with ␳ ⫽ ⫺2.26 and r ⫽ 0.28 (Fujio et al., 1991b). This r value is comparable to the 0.27 obtained from the correlation according to (2) of the pKa values of benzoic acids on which the Hammett ␴ scale is based. The o-methyl derivative of [19] (Lomas and Dubois, 1978) gives a slightly more reduced r value of 0.18. The solvolysis of 4-methylbenzobicyclo[2.2.2]octen-1-yl triflates [20] would

[20] be an excellent model of such a system where any exalted ␲-delocalization interaction should be completely prohibited. The carbocation orbital developed at the bridgehead of the bicyclic skeleton is rigidly orthogonal to the benzo-␲-orbital, and in addition, any backside attack by a nucleophile is prohibited. Logarithmic rates are in fact correlated directly with conventional ␴o parameters giving ␳ ⫽ ⫺2.18 (Fujio et al., 1992b). Thus the substituent effect on this solvolysis can be referred to by the so-called resonance-unexalted ␴o reactivity with r ⫽ 0.0, being the lowest limit of exalted ␲-delocalization. The degree of steric inhibition of resonance depends on the bulk of the ␣-alkyl substituents. Rotation of the aryl ring to increase the overlap between

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␳ = -2.26 r = 0.28

Fig. 6 The Y–T plot of solvolysis of ␣,␣-di-t-butylbenzyl OPNB [19] in 50% aqueous EtOH (50E) at 75⬚C: r ⫽ 0.28. For interpretation of symbols, see Fig. 1. Redrawn from the data in Fujio et al. (1991b).

its orbitals and the empty sp2 orbital would be impeded by steric interactions between aryl and ␣-substituents (Scheme 5). The efficiency of resonance interaction should be proportional to cos2 ␪ where ␪ is the dihedral angle between the two overlapping p-orbitals as in (4), r ⫽ rmax cos2 ␪

(4)

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Scheme 5.

where rmax is the intrinsic value of r for the coplanar tert-benzylic system, and may be reasonably set to r ⫽ 1.00 (Nakata et al., 1996). The dependence of r upon the dihedral angle is best understood by assuming a carbocation model for the structure of the transition state for a series of ␣,␣-dialkylbenzylic precursors. This point will be discussed again in the section on gas-phase carbocation chemistry. It is worthy of note that the increased steric congestion at the ␣-position reduces significantly not only the r value but also the ␳ value (cf. Table 3). Interpretation of the decrease of the latter parameter caused by loss of coplanarity is not straightforward. While there seems to be an implication here that the Y–T analysis does not completely separate inductive and resonance contributions, the reduced ␳ values for these sterically hindered systems are not dependent upon the reduced contribution of the resonance effect; e.g. the Y–T ␳ value for [19] is identical with the ␳m value. A typical case of a significant reduction in ␳ value to ⫺2.18, in a sterically congested system, is the solvolysis of fenchyl-OPNB [10] which is sterically quite similar to ␣,␣-di-t-butylbenzylOPNB [19] having an identical ␳ value ⫺2.26 (Table 3). However, the r value of 1.06 for the former contrasts sharply with the diminished value of 0.28 for the latter. This distinct difference in the resonance contribution should reflect the coplanarity of the aryl ring in the transition state of carbocation-like structures [19C⫹] and [10C⫹]; the aryl coplanarities, i.e. the dihedral angles ␪, for carbocations [19C⫹] and [10C⫹] are estimated to be ⬃14⬚ and ⬃60–69⬚, respectively, by ab initio MO optimization (Fujio et al., 1997a,b,c). Benzyl solvolyses The solvolysis of benzyl tosylates [21], R1, R2 ⫽ H in the general structure [1] in Scheme 1, is well known as a typical case where a significant mechanistic shift occurs when the substituent is changed. As a mechanistic transition occurs clearly in this solvolysis, no single linear relationship is expected for the whole range of substituents. Figure 7 shows a Y–T plot for the solvolysis in 80% aqueous acetone (Fujio et al., 1990b,c), in which the bisected (bilinear) correlation giving different ␳ values for the regions of ED and EW substituents was ascribed to a changeover in mechanism from SN1 (DN ⫹ AN) for the former range of substituents to SN2 (ANDN) for the EW substituents. Closer inspection reveals that the plot against ␴⫹ values (open circles) is neither linear nor bilinear but displays a significant split pattern of apparently

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Y. TSUNO AND M. FUJIO

Fig. 7 The Y–T plot of solvolysis of benzyl tosylates [21] in 80% aqueous acetone at 25⬚C; r ⫽ 1.29. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1990b). Copyright 1990 Chemical Society of Japan.

parallel curvatures with significant gaps. In contrast, the correlation versus the – scale with r ⫽ 1.29 gives a linear plot over a 104 reactivity range for Y–T ␴ substituents more reactive than p-MeS-m-CN, and this straight-line correlation is connected to the plot of lower ␳ value for the EW region of substituents. The solvolysis of the chlorides also gives a similar correlation with an identical

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r value for the unimolecular mechanism for the ED region of substituents, while the correlation for the bimolecular mechanism lies appreciably higher than that of the tosylates (Fujio et al., 1991a). The solvolysis in the less nucleophilic 97% aqueous TFE gives a straight-line correlation with an identical r value of 1.27 covering a wider range of substituents down to the m-halogens (Tsuji et al., 1989; Fujio et al., 1990b). The enhanced r value assigned for the dissociative (DN) mechanism of this solvolysis is free from the mechanistic complexity involving nucleophilic solvent assistance in the region of EW substituents. The measured rate constant kt of the solvolysis should be a sum of the rate constants, kC ⫹ kN, for unimolecular and bimolecular processes, respectively. In practice, substituent effects on these competitive processes can be directly analysed by using a non-linear least-squares program based on the assumption of independent Y–T correlations for both kC and kN processes as in equation (5), o

–⫹

o

–⫹

kt ⫽ kCH 10␳C(␴ ⫹rC⌬␴R ) ⫹ kNH 10␳N(␴ ⫹rN⌬␴R )

(5)

This treatment resulted in consistent results for both tosylate and chloride sets. Because of a significant difference in the ␳ values 兩 ␳C 兩 Ⰷ 兩 ␳N 兩, a significant effect was observed only on the kC correlation for the ED region of substrates. Benzhydryl solvolyses Numerous solvolytic studies on diarylmethyl derivatives have been carried out under a variety of conditions. The analysis of substituent effects in the solvolysis of the monosubstituted chlorides [22] was reported earlier (Yukawa and Tsuno, 1959; Yukawa et al., 1966). The purpose of this analysis is to clarify the effect of a fixed substituent Y in one ring on the substituent effect of the variable substituents X on the second ring. Three extensive sets of kinetic data for the solvolysis of X, Y-disubstituted benzhydryl systems under fixed conditions have been reported: one for the ethanolysis (Nishida, 1967), one for the chloride hydrolysis in 85% aqueous acetone at 0⬚C (Fox and Kohnstam, 1964) and one for the bromide hydrolysis (Mindl et al., 1972; Mindl and

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Vecera, 1973); none of these systems involves a sufficient number of substituents for our purpose. The necessary additional data have now been supplied (Fujio et al., unpublished). Schade and Mayr (1988) later demonstrated that various sets of solvolysis rates, determined under different conditions, especially in different solvents, are linearly correlated with each other, and by using these linear correlations, the solvolytic data obtained under different conditions were converted into a standard set in ethanol at 25⬚C; this data set does not appear to be more extensive than ours as regards variety of substituents, but it provides evidence for the linearity between multiple substituent effects in ethanolysis and those in any other solvolyses, especially in the standard set in 85% aqueous acetone at 0⬚C. The Y–T equation has been applied for a series of Y sets of data, i.e. set of reactivity changes by varying substituent X when substituent Y is held constant, and the results are summarized in Table 4. Benzhydryl solvolyses (in Table 4) show Y–T correlations with similar resonance demand and the magnitude of ␳ is significantly dependent on the EW effect of the second aryl substituents. Inspection of Fig. 8 suggests that the discrepancy in the additivity relationship for multiple substitutions is most serious when the first substituent is strongly ED or EW (s-ED, s-EW). A transition state substituted by a s-EW group is more electron deficient than it is in the unsubstituted (Y ⫽ H) series, so that the electron demand in such a transition state must be more than that in the parent system. Consequently, the ␳-value for the Y ⫽ p-NO2 series is larger than that for the Y ⫽ H series. On the other hand, when the Y-substituent is s-ED, such as p-methoxy, then a smaller ␳ results for the series. From these arguments, it is apparent that a change in the extent of electron deficiency in the transition state must be a function of the ED or EW power of the first substituent. This effect can be expressed quantitatively by (6) and (7). log

冢k 冣 ⫽ ␳ (␴–) kXY

HH

Y

X

⫹ log

冢k 冣 kY

H

–) ⫹ ␳ (␴ –) ⫽ ␳Y (␴ X H Y

(6) H

(7)

The ␳Y dependence upon the second term at the right-hand side in (6) implies –) in (7) a RSR of type (3), and the ␳Y dependence upon the second term (␴ Y implies a non-additivity relationship, which can be rewritten in the form (8).

␳Y ⫽ q(␴–)Y ⫹ ␳H

(8)

The non-additivity coefficient q should have a closely similar physical significance to the RSR coefficient S in (3). It should be noted that the Y–T correlations of any Y sets, i.e. for each series with a fixed Y substituent, in

Table 4 Yukawa–Tsuno correlations for benzhydryl solvolyses. Systema (X ⫽ Y) Ar2CH-Cl Ar(p-MeOC6H4)CH-Cl p-PhO ⬎ X ⬎ m-CN Ar(p-MeC6H4)CH-Cl Ar(Ph)CH-Cl X: s-␲-ED exc. s-␲-ED Ar(p-ClC6H4)CH-Cl Ar(m-ClC6H4)CH-Cl X⭓H exc. s-␲-ED Ar(p-NO2C6H4)CH-Cl Ar2CH-Cl (X ⫽ Y) Ar(Ph)CH-Cl X; s-␲-ED Ar(m-ClC6H4)CH-Cl Ar(3,5-Cl2C6H3)CH-Cl

Solvent,b temperature (85A, 0⬚C) (85A, 0⬚C) (85A, 0⬚C) (85A, 0⬚C) (85A, 0⬚C) (85A, 0⬚C) (85A, 0⬚C) (EtOH, 25⬚C) (EtOH, 25⬚C) (EtOH, 25⬚C) (EtOH, 25⬚C)

␳ ⫺8.736c ⫺2.67 ⫺2.93 ⫺3.71 ⫺4.763 ⫺5.286 ⫺4.459 ⫺4.17 ⫺5.160 ⫺5.426 ⫺4.808 ⫺5.099 ⫺8.62c ⫺4.23 ⫺4.81 ⫺4.47 ⫺4.37

r

R

SD

n

1.020 1.11 0.985 1.16 1.112 1.28 1.116 1.34 1.191 1.178 1.147 1.375 1.013 1.077 1.013 1.18 1.38

0.9999 0.998 0.9994 0.993 0.997 0.9964 0.9991 0.998 0.9967 0.9969 0.9995 0.9972 0.9993 0.9976 0.9976 0.9993 0.998

0.075 0.09 0.03 0.23 0.130 0.13 0.048 0.22 0.147 0.120 0.037 0.195 0.100 0.139 0.096 0.083 0.15

10 13 7 17 22 8 14 12 19 15 11 14 9 25 8 18 6

a Partial correlation, X ⭓ or X ⭐; correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; substituents of which the ␴⫹ are within the range of ␴⫹ values for substituents X1 to X2. exc. X; excluding substituent X. bSolvent, 85A; a (v/v) mixture of 15% aqueous and 85% acetone. cCorrelation overall ␳ for two equivalent aryl substituents.

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Fig. 8 The plot of the log(k/k0) for the solvolysis of benzhydryl chlorides in 85% aqueous acetone (85A) at 0⬚C against the sum of the ␴⫹ values: the reference correlation (␳⫹) line was drawn for the plots (䊉) for the symmetrically disubstituted (X ⫽ Y) series, and typical concave correlations for Y ⫽ p-MeO and Y ⫽ p-NO2 series. Data taken from Fox and Kohnstam (1964) and Fujio et al. (unpublished).

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295

Table 4 are all satisfactory but not accurately linear enough at the correlation precision level. Although the change of ␳Y value for the Y series is not strictly linear against ␴–Y for fixed substituents Y, it may be concluded without doubt that the ␳Y – of the value tends to change in an inverse linear way with the magnitude of ␴ Y fixed Y substituents. This behaviour is indeed what would be envisaged for the RSR in this system, and has often been referred to as an example of RSR behaviour. The extension of the RSR as regards transition state structures is of particular importance in providing the basis for the Hammond–Leffler postulate or the extended Brønsted equation both of which lead to the prediction of RSR behaviour (cf., Ruasse, 1993).

ARYL-ASSISTED SOLVOLYSES

Structural effects The ␤-aryl-assisted solvolysis, e.g. of neophyl brosylates [24]-OBs, should be an appropriate model reaction for cases where the resonance demand is low since it gives an r value significantly less than unity (Tsuno and Fujio, 1996). The solvolysis of [24] is considered to proceed through a rate-determining aryl-assisted transition state which leads to the tertiary carbocation (Fujio et al., 1987a, 1990a). The values of log k for the acetolysis of [24] give a substantially better correlation with ␴⫹ than with ␴ constants (Yukawa and Tsuno, 1959), and the Y–T equation (2) results in a better correlation than the simple Brown ␳⫹␴⫹ treatment; this has been corroborated recently by the application of a more extensive data set (Fujio et al., 1987a, 1988, 1990a). The behaviour of substituents in the reaction of [24] is illustrated by the Y–T plot in Fig. 9. The ␴⫹ plots (open circles) of p-␲-ED substituents consistently

Scheme 6

Solvolytic process of ␤-arylalkyl precursors.

296

Y. TSUNO AND M. FUJIO

Fig. 9 The Y–T plot for acetolysis of neophyl brosylates [24] at 75⬚C; r ⫽ 0.57. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1990a). Copyright 1990 Chemical Society of Japan.

deviate downward from the reference m-correlation line. The line segments between ␴⫹ and ␴o values for p-␲-ED substituents which reflect the resonance capabilities of these substituents, are placed randomly with respect to the order of reactivity changes. The Y–T correlation line can be defined as a unique line dividing all the resonance line segments ␴⫹ ⫺ ␴o for ␲-ED p-substituent groups (and also for substituents displaying no resonance effect,

THE YUKAWA–TSUNO RELATIONSHIP

297

i.e. having values of ␴⫹ ⫺ ␴o that are essentially zero), at a constant ratio of characteristic r value. The substituent effect on the rate should reflect only that on the aryl-assisted ionization step and therefore the observed r value of 0.57 should be characteristic of this step. This exalted r value of ca. 0.6 can be rationalized in terms of a direct ␲-interaction between the aryl ␲-system and the ␤-carbocation centre at the rate-determining transition state. Most solvolyses of ␤-arylalkyl arenesulphonates involve a mechanistic complication arising from a concurrent solvent-assisted (kS) process, as shown in Scheme 6 (Schadt et al., 1978; Fujio et al., 1987b,c, 1992a; Goto et al., 1989). The measured rate constant kt of the solvolysis should be the sum of the rate constants for the aryl-assisted k⌬ and the -unassisted kS processes. In practice, substituent effects on these competitive processes can be directly analysed by using a non-linear least-squares program based on the assumption of independent Y–T correlations for both k⌬ and ks processes, i.e. equation (9). The treatment has been simplified by taking rs ⫽ 0 for the ks process of this reaction. o

–⫹

o

–⫹

␳⌬(␴ ⫹r⌬ ⌬␴R ) ⫹ kSH 10␳S(␴ ⫹rS ⌬␴R ) kt ⫽ kH ⌬ 10

(9)

However, it is instructive to apply a classical method of analysis. The Y–T plot for the acetolysis of ␤-arylethyl tosylates [23] is shown in Fig. 10 (Fujio et al., 1987b). In this reaction, only compounds carrying s-ED substituents react predominantly by the aryl-assisted k⌬ process. Clearly, we see the same pattern of Y–T plot for the reactive substrates carrying s-␲-ED substituents which react predominantly by the k⌬ process as for the acetolysis of [24]. The Y–T – ⫹ line segments at a correlation line passes through the points dividing the ⌬␴ R constant internal ratio r ⫽ 0.63 and collapses into a single smooth correlation curve for the r-independent EW substituents reacting by the kS mechanism. The same behaviour has been observed for the solvolyses of threo-3-aryl-2butyl brosylates [25] (Fujio et al., 1992c) and (1-arylcyclobutyl)methyl brosylates [26(m ⫽ 3)] (Roberts, 1974, 1976; Fujio et al., 1996). Rates of these solvolyses were previously reported to correlate linearly with ␴⫹ without a significant break. While these studies have not withstood close scrutiny, it is serious because of implying the operation of only the k⌬ mechanism without a concurrent kS process. The simple Brown ␳⫹␴⫹ treatment appears to be incapable of providing a correct interpretation of these mechanistic details.

(CH2)m [25]

[26] m ⫽ 2,3,4

298

Y. TSUNO AND M. FUJIO

Fig. 10 The Y–T plot for acetolysis of 2-phenylethyl tosylates [23] at 115⬚C; r ⫽ 0.63. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1987b). Copyright 1987 Chemical Society of Japan.

The Y–T correlations of ␤-aryl-assisted solvolyses are summarized in Table 5. ␤-Aryl-assisted solvolyses characteristically show r values of 0.5 to 0.6. The solvolysis of (1-arylcyclopropyl)methyl tosylates [26(m ⫽ 2)] (Roberts, 1968; Roberts and Watson, 1970; Fujio et al., 1996), an analogue of the neophyl system, gave a linear Y–T correlation with an extremely low r value of 0.13 and

THE YUKAWA–TSUNO RELATIONSHIP

Scheme 7

299

Solvolysis processes of 2,2-diarylethyl tosylates [27(X,Y)].

a low ␳ value. This correlation suggests a different mechanism from the ␤-aryl-assistance mechanism; the effect is consistent with that of unassisting ␤-aryl substituents in the transition state. Fujio et al. (1996) concluded the cyclopropyl-assisted mechanism for this reaction, not on the basis of only this correlation but of the highly enhanced reactivity, the solvent effect and product analysis. The Y–T correlations for the acetolysis of 2,2-diarylethyl tosylates [27] in Table 5 are instructive, since the nature of the phenonium transition state may be modified by the presence of the non-participating aryl group, keeping the mechanism essentially the same (Goto et al., 1991; Fujio et al., 1993a,b). The acetolysis of differently disubstituted diarylethyl systems [27(X,Y)] proceeds through two competitive assisted pathways either by the X-substituted phenyl Y (kX ⌬ ) or by the Y-substituted phenyl group (k⌬ ), as shown in Scheme 7. The effects of substituents on the symmetrically disubstituted diarylethyl tosylates, [27(X ⫽ Y)], can be described accurately in terms of the Y–T – scale relationship with ␳ ⫽ ⫺4.44 and r ⫽ 0.53. The Y–T plot against the Y–T ␴ with an appropriate r of 0.53 gives an excellent linear correlation for the whole set of substituents, indicating a uniform mechanism for all of them. When Y ⫽ X, the overall solvolysis rate constant kt corresponds to the sum of the Y rate constants, kX ⌬ ⫹ k⌬ , and hence kt cannot be employed directly in the Y–T analysis. The acetolysis of monosubstituted diphenylethyl tosylates gave a non-linear Y–T correlation, which is ascribed to a competitive X-substituted H aryl-assisted pathway kX ⌬ and the phenyl-assisted k⌬ pathway. By application of an iterative non-linear least-squares method to (9), where the terms k⌬ and H kS are now replaced by kX ⌬ and k⌬ , respectively, the substituent effect on kt can X be dissected into a k⌬ correlation with ␳⌬ ⫽ ⫺3.53, r⌬ ⫽ 0.60, and an

Table 5 Yukawa–Tsuno correlations for aryl-assisted solvolyses. Reaction

neophyl-OBs [24] 2-arylethyl-OTs [23]b 2-aryl-1-methylethyl-OTsb threo-3-aryl-2-butyl-OBs [25]b trans-2-arylcyclopentyl-OTs 2-aryl-2-(CF3)ethyl-OBs

Solventa

(1-arylcyclobutyl)methyl-OBsb (1-arylcyclopentyl)methyl-OTs (1-arylcyclopropyl)methyl-OTs 2,2-diarylethyl-OTs [27(X ⫽ Y)] 2-Ar-2-(m-ClC6H4)ethyl-OTs 2-Ar-2-(3,5-Cl2C6H3)ethyl-OTs 2-Ar-2-PhCHCH2-OTsb

AcOH AcOH AcOH AcOH 97T AcOH 80T AcOH 97T 80A AcOH AcOH AcOH AcOH

2-Ar-2-(p-MeOC6H4)ethyl-OTs

AcOH

Temperature

75⬚C 115⬚C 100⬚C 75⬚C 75⬚C 130⬚C 100⬚C 55⬚C 45⬚C 25⬚C 90.1⬚C 90.1⬚C 90.1⬚C 90.1⬚C 90.1⬚C

k⌬ Correlation

␳⌬

r⌬

⫺3.83 ⫺3.96 ⫺3.53 ⫺3.32 ⫺3.55 ⫺3.12 ⫺4.22 ⫺3.27 ⫺4.18 ⫺1.61 ⫺4.44 ⫺3.60 ⫺3.54 ⫺3.53 ⫺0.88 ⫺0.67

0.57 0.63 0.54 0.56 0.41 0.77 0.63 0.55 0.40 0.13 0.53 0.62 0.66 0.60 0.00 0.21

ks Correlation ␳s

⫺0.19 ⫺0.81 ⫺1.07

⫺1.16

R

SD

0.999

0.038 0.035 0.025 0.042 0.06

0.997 0.997 0.997 0.999 0.995 0.998 0.9994 0.9996

0.05 0.07 0.05 0.077 0.037 0.026 0.03

n Reference

29 27 21 24 7 4 4 17 7 6 16 14 13 21

1 2 3, 4 3 5 6 6 7 5,8 5 9 10 10 11c d

0.998

0.016

10

10

Aqueous organic solvent, xM; a (v/v) mixture of (100 ⫺ x)% aqueous and x% organic solvent M, where M is E ⫽ EtOH, A ⫽ acetone, and T ⫽ TFE. Analysed by the non-linear least-squares method based on equation (9). cCorrelation for the substituted-aryl assisted pathway. dCorrelation for the unsubstituted-phenyl assisted pathway. a

b

References: 1, Fujio et al. (1987a, 1990a). 2, Fujio et al. (1987a; 1990a,b,c). 3, Fujio et al. (1992a). 4. Goto et al. (1989). 5, Fujio et al. unpublished. 6, Okamura et al. (1997). 7, Fujio et al. (1996). 8, Roberts and Arant (1994). 9, Fujio et al. (1993a). 10, Goto et al. (1991). 11, Fujio et al. (1993b).

THE YUKAWA–TSUNO RELATIONSHIP

301

aryl-X-unassisted correlation for the phenyl-assisted kH ⌬ mechanism correlated with ␴o having ␳H ⫽ ⫺0.88. The ␳⌬ and r⌬ values for the effects of the assisting aryl substituents are quite close to those for [24], whereas the low ␳H value with the unexalted ␴o constants for the non-assisting aryls is compatible with a remote ␤-aryl effect. In the acetolysis of 2-aryl-2-(3,5-dichlorophenyl)ethyl tosylates [27(X,3,5-Cl2)], the s-EW 3,5-dichlorophenyl group does not compete in an aryl-assistance pathway with any other aryl groups carrying more s-ED substituents than p-Cl. A Y–T correlation with r ⫽ 0.66 for 13 substituents in the range down to p-chloro was obtained, demonstrating that a single X-substituted phenyl-assisted pathway takes place. Similarly, for the acetolysis of 2-aryl-2-(m-chlorophenyl)ethyl tosylates [27(X,m-Cl)], equation (2) gives a linear correlation with r ⫽ 0.63 for substituents more ED than H. This may reflect the Y–T correlation for the X-substituted aryl-assisted pathway. The r and ␳ values are comparable with those for the aryl-assisted k⌬ pathway of the monosubstituted diphenylethyl system [27(X,H)] and also with those for other k⌬ solvolyses, e.g. neophyl [24] brosylates, suggesting a close similarity of the aryl-assisted mechanisms. On the other hand, the acetolysis of 2-aryl-2-(p-methoxyphenyl)ethyl tosylates [27(X,p-MeO)], probably proceeds uniformly through the p-methoxyphenylassisted pathway. There is a linear Hammett correlation against ␴o (or ␴), attributed to the effect of the non-assisting-aryl-X substituents in the p-methoxyphenyl-assisted mechanism. The ␳ value for the symmetrical bis-arylethyl series [27(X ⫽ Y)] appears too small as compared with that of the neophyl system with the single aryl group. This can be accounted for by the fact that only one of the two ␤-aryl groups participates in the rate-determining aryl-assisted transition state, while the other one affects the unassisted mechanism, and both effects must be additive. Consequently, when a ␳ value of ⫺0.8 is applied for the latter effect, the Y–T correlation (10) is obtained for the aryl participation. This correlation is practically identical to those for the k⌬ processes of Y-fixed [27(X,Y)] systems. – ⫹) ⫺ 0.8␴o ⫽ ⫺3.6(␴o ⫹ 0.61⌬␴ – ⫹) log(k/k0)⌬ ⫽ ⫺4.44(␴o ⫹ 0.53⌬␴ R R

(10)

In the 2,2-diphenylethyl system [27(X,Y)] all the r⌬ values are within a narrow range of 0.63 ⫾ 0.03 and tend to increase only slightly as the substituent in the unassisting aryl becomes more EW. Consequently, the r value of 0.6 can be referred to as the resonance demand characteristic of the ␤-aryl-assisted mechanism. The aryl group participation process is mechanistically a Friedel-Crafts alkylation. The r value of the Friedel-Crafts alkylation by alkyl carbocations was found to be significantly lower than that of protonation or of halogenation of aromatic substrates (Yukawa et al., 1966). Olah interpreted the low r value in terms of an earlier transition state, i.e. less advanced aryl–C⫹ bond formation with r ⫽ 0.6 at the transition state prior to formation of the

302

Y. TSUNO AND M. FUJIO

full-bond intermediate with r ⫽ 1.0 (Olah et al., 1972a,b). However, see later discussion on p. 355. Additivity relationship of substituent effects For the uniform applicability of equation (2) with either a non-unit or a non-zero r value, Johnson (1978) suggested that the reaction conforming to the Y–T equation with r ⫽ 1.00 may be a two-step process, involving a preequilibrium K1 step followed by a rate-determining k2 step where one of K1

k2

[A] J [B] → [C] → Products

(11)

these steps correlates with ␴o (or ␴) and the other with ␴⫹ as in equation (12).

冤

log(k/k0) ⫽ ␳1 ␴o ⫹ ␳2 ␴⫹ ⫽ (␳1 ⫹ ␳2) ␴o ⫹

␳2 –⫹ ⌬␴ R ␳1 ⫹ ␳2

冥

(12)

This is indeed formally equivalent to the Y–T equation (2) and it is therefore capable of reproducing exactly the observed excellent Y–T correlations. A conceivable multistep mechanism of ␤-aryl-assisted solvolysis is one involving a pre-equilibrium ionization to the initially formed non-conjugative cation (referred to as the ␴o step) followed by a rate-determining arylrearrangement step (referred to as the ␴⫹ step) as shown in (13). By the use of equation (12), the ␳1 and ␳2 values are unambiguously assigned for both steps irrespective of any assumption regarding the nature of the steps.

K

k⌬

(13)

While the pre-equilibrium scheme may be an important source of an intermediate r value, any given reaction with an intermediate r value cannot necessarily be said to proceed by the single rate-determining step mechanism or the pre-equilibrium, two-step mechanism. The reaction scheme for aryl-assisted solvolysis in (13) is of course plausible except for the question of whether or not the second step is slow enough to be rate determining. However, in practice, this k⌬ process should not be observed in the solvolysis of ␤-phenylethyl tosylates, since the pre-equilibrium dissociation step into a primary carbocation cannot compete with the kS process (the SN2 mechanism).

THE YUKAWA–TSUNO RELATIONSHIP

303

Nevertheless, although Johnson’s suggestion of the pre-equilibrium scheme cannot be general, his approach is highly significant since it points out the particular importance of the additivity relationship of the Y–T equation. The unification of substituent parameter scales in terms of varying r leads to a unique additivity relationship (14) of substituent effects for the system of k ⫽ k1 , k2 , . . . kj: –⫹ 兺 [log(k/k0)]j ⫽ 兺 ␳j ␴o ⫹ 兺 ␳j rj ⌬␴ R

(14)

This relationship is actually observed in the form (15) or (15a), –⫹ 兺 [log(k/k0)] ⫽ ␳⬘ ␴o ⫹ ␳⬘ r⬘ ⌬␴ R ⫹ o – 兺 [log(k/k )] ⫽ ␳⬘(␴ ⫹ r⬘ ⌬␴ ) 0

R

(15) (15a)

where ␳⬘ ⫽ 兺 ␳j and r⬘ ⫽ 兺 ␳j rj/兺 ␳j. Equations (15) and (15a) allow, in theory, the assignment of any reasonable rj value for each step. Thus, for the overall pre-equilibrium mechanism above, the overall substituent effect can be represented as a single linear Y–T correlation with an apparent r⬘ value. It is of great importance that equation (14) is applicable also for the simultaneous contribution of more than two substituent effects on a single elementary reaction step. We have already discussed the additivity of the substituent effects on the solvolysis of bis-arylethyl tosylates [27(X ⫽ Y)]. The simple additivity inherent in the Y–T relationship makes (2) a useful analytical tool for the elucidation of reaction mechanisms. However, caution should be exercised in order to avoid misleading interpretations.

SOLVOLYSIS OF 1-ARYLETHENYL SULPHONATES FORMING VINYL CATIONS

(16)

Grob and coworkers (Grob and Cseh, 1964; Grob and Pfaendler, 1971) reported that the solvolysis rate constants for 1-arylethenyl bromides [28]-Br in 80% aqueous ethanol at 120⬚C correlate with Brown’s ␴⫹ substituent constants for a limited set of substituents (NH2, OMe, MeCONH and H) with ␳⫹ ⫽ ⫺5.1. Stang et al. (1977) reported substituent effects on the solvolysis of the trifluoromethanesulphonates [28]-OTf in 80% (v/v) aqueous ethanol at 75⬚C. Kobayashi et al. (unpublished) studied the solvolysis of several derivatives of [28]-OTs in 80% (v/v) and 30% (v/v) aqueous ethanol, and

304

Y. TSUNO AND M. FUJIO

extrapolated all the data to a tosylate reactivity set in 80% aqueous ethanol, assuming that the difference in the leaving group does not affect seriously the sensitivity of the substituent effect. When the Y–T equation was applied to the combined substituent data set, it gave a correlation (Table 6) – ⫹). The large negative ␳ value indicates that a log (k/k0) ⫽ ⫺4.10 (␴o ⫹ 1.16⌬␴ R large cationic charge is developed at the benzylic position in the transition state and is consistent with generation of a vinyl cation as the solvolysis intermediate. The r value of 1.16 for the system may suggest that ␲delocalization of positive charge into the aryl ␲-system in the 1-phenylethenyl cation is comparable with that in the ␣-phenylethyl cation. The solvolysis rate constants of 1-aryl-2-methylpropenyl sulphonates, extrapolated to the tosylate solvolysis in 80% aqueous ethanol at 25⬚C, give an excellent Y–T correlation with r ⫽ 1.06, as shown in Fig. 11. The solvolysis of (E)-1-aryl-3,3-dimethylbutenyl bromides also gives an excellent correlation with r ⫽ 1.02. Vinyl cation [28C⫹] can be generated also by the acid-catalysed hydration of phenylacetylene. Application of the Y–T equation to the available data on the hydration of phenylacetylenes in acetic acid–water–sulphuric acid gives r ⫽ 0.87 and ␳ ⫽ ⫺4.5. While the magnitude of ␳ is as large as that obtained for the solvolytic generation of the vinyl cations, and is consistent with the ratedetermining formation of 1-phenylvinyl cation (Noyce and Schiavelli, 1968; Allen et al., 1982a), the r value is appreciably lower than the value for the solvolytic reaction. The attenuation of r may reflect an early transition state, but the large ␳ value is inconsistent with this interpretation. A reasonable interpretation is that based on the pre-equilibrium, two-step mechanism (Scheme 8). In order to obtain the r value of 1.16 for the second step of the generation of the vinyl cation, the pre-equilibrium complexation should have a ␴o-dependence with a ␳ value of ⫺1.1.

Scheme 8. HIGHLY ELECTRON-DEFICIENT CARBOCATION SYSTEMS

Appropriate model reactions having distinctly higher resonance demand r values are provided by systems forming highly electron-deficient carbocations (Murata et al., 1990a,b); as an example, Fig. 3 demonstrates the Y–T

Table 6 Yukawa–Tsuno correlations for solvolyses of arylalkenyl compounds and hydration of arylacetylenes. Systema Solvolysis CH2⫽C(OTs)Ar Me2C⫽C(OTs)Ar t-BuCH⫽C(Br)Ar Hydration Ar-C⬅CH Ar-C⬅C-Ge(Et)3

Solvent,b temperature

␳

r

R

SD

n

Reference

(80E, 75⬚C) (80E, 25⬚C) (30E, 25⬚C)

⫺4.10 ⫺4.974 ⫺5.79

1.16 1.057 1.02

0.9995 0.9997 0.9991

0.074 0.064 0.115

7 8 6

1 2 2

AcOH, H2O⫺H2SO4 AcOH, H2O⫺H2SO4

⫺4.477 ⫺3.264

0.867 0.76

0.9990 0.9989

0.066 0.050

13 13

3 4

Partial correlation, X ⭓ or X ⭐; a partial correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; the substituents of which the ␴⫹ are within the range of those for substituents X1 to X2. exc. X; excluding substituent X. bAqueous organic solvent, xM; a (v/v) mixture of (100 ⫺ x)% aqueous and x% organic solvent M, where E is M ⫽ EtOH. a

References: 1, Kobayashi et al. (unpublished). 2, Matsumoto (1997). 3, Bott et al. (1951); Allen et al. (1982a). 4, Bott et al. (1964).

Y. TSUNO AND M. FUJIO

log (k/k0)

306

Fig. 11 The Y–T plot of the solvolysis of phenyl-2-methylpropenyl sulphonates in 80% aqueous EtOH at 25⬚C; r ⫽ 1.06 (Matsumoto, 1997). For interpretation of symbols, see Fig. 1.

THE YUKAWA–TSUNO RELATIONSHIP

307

Table 7 Substituent effects for extremely electron-deficient systems. Systema

Solvent

Brown equationb

ArCH-PO(OEt)2 円

OMs ArCH-PS(OEt)2 円

Slopec

R 0.998 0.999

⫺4.15

0.881 1.12 1.32 0.726

MeOH TFE

⫺6.70 ⫺10.7 ⫺12.1 ⫺5.69 ⫺10.3 ⫺8.0 ⫺7.2

0.921 0.778

0.9999 0.9995

1 2 2 3 4 5 5

TFE

⫺10.1

⫺6.1

0.918

0.993

6

97HFIP

⫺10.3

1.123

0.9998

(⫺2.99)

0.69

0.985

f

0.45

0.9988

1.05 1.020 1.046 1.08 0.982

0.993 0.994 0.997

TFE TFA TFA EtOH

AcOH

⫺7.15

OMs ArCH(CF3)OTs

Reference

␳⫹b w

␳D⫹b ArC(CN)Me-OMs ArC(CF3)2-OTs ArC(CN)CF3-OTs Ar-␣-ketoNB-OTfd ArC(CF3)Me-Br ArC(SO2Ph)Me-OMs ArC(SOPh)Me-OMs

log k–log k equationa,c

⫺2.99 ⫺6.7 TFA 97HFIP ⫺9.1 ⫺9.8 TFE AcOH ⫺10.1 aq. EtOH ⫺11.9

⫺9.7

e f

0.998

7 8

0.998

Most data are taken from Murata et al. (1990b), otherwise see text. b␳D⫹ and ␳w⫹ are the Brown ␳⫹ values for ED and EW substituents, respectively. cA slope of logarithmic rate constants against the log(k/k0)5 for the solvolysis of ArCMe(CF3)OTs [5] in the corresponding solvent. d2arylbicyclo[2.2.1]heptan-3-on-2-yl triflates. eFor the range of substituents more reactive than 3,4-Me2. fFor the range of substituents less reactive than 3,4-Me2. a

References: 1, Gassman and Guggenheim (1982). 2, Allen et al. (1983b, 1986). 3, Creary (1981). 4, Liu and Wu (1984). 5, Creary et al. (1987). 6, Creary and Underiner (1985). 7, Creary and Mehrsheikh-Mohammadi (1986). 8, Allen et al. (1983a).

correlation for the rates of solvolysis of 1-aryl-1-(trifluoromethyl)ethyl tosylates [5]. This system has already been shown to be far beyond the correlative ability of the Brown ␳⫹␴⫹ equation but excellently correlated in terms of the Y–T equation. Although several sets of rate data for extremely electron-deficient systems having two powerful EW ␣-substituents are available in the literature, most are not sufficient for more precise analysis by the Y–T equation. Most studies so far of solvolyses of this class have been based on the Brown ␳⫹␴⫹ equation (1) because of the small number of substituents, and it has been suggested that the remarkably high ␳⫹ values in the range of ⫺10 to ⫺12 are a characteristic feature of these highly electron-deficient carbocation-forming reactions (Allen et al., 1983a,b, 1985; Creary et al., 1987). However, as summarized in Table 7, all these solvolyses show significantly non-linear Brown ␴⫹ correla-

308

Y. TSUNO AND M. FUJIO

tions with ␳D⫹ ⬵ ⫺10 for the range of ED substituents while ␳⫹W ⫽ ⫺5 ⬃ ⫺6 for the region of EW substituents (Murata et al., 1990b). Since the non-linear relationship rules out the operation of a single mechanism for the whole range of substituents, all these solvolyses should involve a mechanistic change with a break in the ␴⫹ plot in the vicinity of the unsubstituted derivative. Murata et al. (1990b) pointed out that all but one of the 19 relevant data sets available gave good linear relationships (R ⬎ 0.99) when plotted against values of log(k/k0)5 for the solvolysis of [5] in the corresponding solvents, and none shows a significantly higher slope than unity (Table 7). This simple linearity suggests that r values as well as ␳ values must be very similar for all these systems. The solvolysis of ArCH(OMs)PS(OEt)2 (Creary and MehrsheikhMohammadi, 1986) is the only case which shows a clear break in the plot against log(k/k0)5, and it evidently indicates a significant thio group participation for deactivating substrates. No mechanistic change seems to take place in – scale with any of the other reactions. An electrophilic substituent parameter ␴ a high resonance demand (r ⬵ 1.39) is required for a proper description of the substituent effects in these extremely electron-deficient benzylic systems. The solvolyses of benzylic substrates carrying a strongly EW ␣-substituent (Table 8) generate highly electron-depleted carbocations (Tsuno, 1986; Allen and Tidwell, 1989; Murata et al., 1990b) and are expected to have a highly exalted resonance demand. In Fig. 3, the ␴⫹ plot (open circles) of the p-␲-donor substituents deviates upward from the ␳m correlation line. The – scale with r ⫽ 1.39 contrasts sharply linear Y–T plot (squares) against the ␴ ⫹ with the poor linear plot against ␴ . The strong destabilization of the carbocationic transition state by the ␣-CF3 group provides a high demand for positive charge delocalization into the ␣-aryl ␲-system. The solvolysis of 1-aryl-2,2,2-trifluoroethyl tosylates, structure [5] (Scheme 1) with R1 ⫽ H, also gives a good Y–T correlation with an exalted r value of 1.53 (Murata et al., 1990b). The replacement of one ␣-methyl in cumyl [2] by CF3 brings about an increase in r of 0.39. Similarly, replacement of the methyl in ␣-phenylethyl [14] by CF3 gives the same increment in the resonance demand. The replacement of the two methyls in [2] by hydrogen, though not very effective, gives an appreciable r increment of 0.28 in the benzyl system [21]. Solvolyses of electron-deficient carbocationic systems carrying an ␣-carbonyl group also gave good linear Y–T relationships while the r-enhancing effect seems less significant (Takeuchi et al., 1987). The particularly high r value of 1.38 permits the p-nitro- or 3,5-dichlorobenzhydryl systems ([22] with Y ⫽ p-NO2 or 3,5-Cl2; Table 4) to be included in this category. The electron deficiency caused by the s-EW ␣-substituents is clearly reflected in the enhanced r value. The ␣-trifluoromethyl-diarylmethyl system It is well known that the kinetic effects of substituents on two aromatic rings are not additive. The presence on the aromatic ring of a substituent capable of

Table 8 Yukawa–Tsuno correlations for solvolyses of highly electron-deficient benzyl systems. Systema ␣-CF3-benzyl-OTs ␣-Me-␣-CF3-benzyl-OTs Ar-␣-ketoNB-OTf c Ar-␣-ketoNB-OTf c Ar(Ph)C(Cl)⫺COPh ArCH(OMs)-PO(OEt)2 ArCH-PS(OEt)2 円

OMs

X⬎H X⬍H

Ar2C(CF3)-OTs Ar(p-MeOC6H4)C(CF3)-OTs X ⬎ p-MeO-m-Cl X ⭐ p-Me Ar(p-PhOC6H4)C(CF3)-OTs X ⭓ p-MeS-m-Cl X ⭐ p-Me Ar(p-MeC6H4)C(CF3)-OTs X ⬎ p-MeS-m-Cl p-Cl ⭐ X ⭐ p-MeS-m-Cl X⭐H Ar(Ph)C(CF3)-OTs X ⭓ 3,4-Me2 X ⬍ 3,4-Me2 Ar(m-ClC6H4)C(CF3)-OTs X ⭓ p-Me X ⬍ p-Me Ar(3,5-Cl2C6H3)C(CF3)-OTs X ⭓ p-Cl

Solventb, temperature

␳

r

R

SD

n

Reference

(50E, 25⬚C) (80E, 25⬚C) (80T, 25⬚C) (AcOH, 25⬚C) (EtOH, 25⬚C) (97T, 25⬚C) (TFE, 25⬚C) (AcOH, 25⬚C)

⫺6.05 ⫺6.29 ⫺7.06 ⫺5.99 ⫺4.63 ⫺4.20 ⫺7.97 ⫺2.88

1.53 1.39 1.415 1.18 1.24 1.09 1.10 (1.84)

0.9994 0.9998 0.998 0.9992 0.990 0.999 0.999 0.996

0.082 0.070 0.20 0.189 0.30 0.12 0.17 0.22

15 28 19 5 10 9 9 7

1 2

(80E, 25⬚C) (80E, 25⬚C)

⫺4.52 ⫺4.41 ⫺4.15 (⫻2)d ⫺3.94 ⫺1.71

1.37 (0.84) 1.19 1.26 1.01

0.9995 0.9988 0.9994 0.999 0.997

0.078 0.099 0.12 0.09 0.06

4 4 14 6e 7

(80E, 25⬚C)

⫺4.01 ⫺2.04

1.26 (1.07)

0.998 0.9998

0.10 0.02

7e 8

9

(80E, 25⬚C)

⫺6.26 ⫺3.66 ⫺2.91

1.40 1.21

0.999 0.994 0.999

0.10 0.08 0.05

5 6 4

9

(80E, 25⬚C)

⫺6.08 ⫺4.33

1.45 1.26

0.996 0.999

0.12 0.074

6 12

8 8

(80E, 25⬚C)

⫺6.19 ⫺4.81

1.57 1.41

0.996 0.999

0.19 0.05

7 6

10 10

(80E, 25⬚C)

⫺5.95

1.69

0.998

0.18

13

10

3 4 5 6 7

8 9

a Partial correlation, X ⭓ or X ⭐; a partial correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; the substituents of which the ␴⫹ are within the range of those for substituents X1 to X2. exc. X; excluding substituent X. bAqueous organic solvent, xM; a (v/v) mixture of (100 ⫺ x)% aqueous and x% organic solvent M, where E is M ⫽ EtOH, A ⫽ acetone, D ⫽ dioxane, T ⫽ TFE, HFIP ⫽ hexafluoroisopropanol. dCorrelation ␳ for single aryl substituents. eIncluding estimated log k/k0 for H and p-Me of E-conformation, see text.

References: 1, Murata et al. (1990b). 2, Murata et al. (1990a). 3, 2-arylbicyclo[2.2.1]heptan-3-on-2-yl triflates; Creary (1981). 4, Creary (1981). 5, Takeuchi et al. (1987). 6, Creary and Underiner (1985). 7, Creary and Mehrsheikh-Mohammadi (1986). 8, Fujio et al. (1997b). 9, Fujio et al. (1999a). 10, Fujio et al. (1997c).

310

Y. TSUNO AND M. FUJIO

electron donation by conjugation alters the charge distribution at the transition state in such a way that the second substituent in the other ring then interacts with a charge which differs from that which would have prevailed in the absence of this interaction.

[29(X,Y)]

[29C⫹(X,Y)]

Substituent effects on the solvolyses of 1,1-diphenyl-2,2,2-trifluoroethyl series [29] have been analysed on the basis of the Y–T equation (Fujio et al., 1997a,b, 1999a), and the results are summarized in Table 8. The solvolysis of symmetrically disubstituted series [29(X ⫽ Y)] has been shown to correlate with the Y–T equation with excellent precision (Fig. 12). [29(X,Y)] with a s-EW Y ⫽ 3,5-Cl2 also gives a linear correlation, even though no more substituent deactivating than m-Cl is involved. Whereas the linear correlation for the whole substituent range observed in the symmetrical diaryl series [29(X ⫽ Y)] indicates the absence of any mechanistic change with the substituent change, the r value of 1.2 as well as the ␳ value of ⫺4.2 appears to be even smaller than the values expected for highly electron-deficient systems, such as [5] or ␣-CF3-benzyl systems. On the other hand, both the r and ␳ values in the Y ⫽ m-Cl and 3,5-Cl2 series appear to have a reasonable magnitude. The monosubstituted diphenyl series [29(X,H)] did not give a single linear Y–T correlation but a bilinear correlation with significantly different ␳ and r values for the substituent ranges more ED than 3,4-dimethyl and less ED than p-methyl as shown in Fig. 13. Whereas the series [29(X,Y)] with a fixed aryl, Y ⫽ p-Me, H and m-Cl, afford slightly bisected Y–T plots, the r values for the ED substituent range are all highly enhanced; values are 1.40, 1.45, 1.57 and 1.69 for the series Y ⫽ p-Me, H, m-Cl and 3,5-Cl2, respectively. A ceiling to the magnitude of the resonance demand in benzylic solvolyses of highly electron-deficient systems appears to be an r value of 1.7 obtained for the series Y ⫽ 3,5-Cl2. Both ␳ and r values are relatively lower for the range of EW substituents in all the sets. The substituent effect of [29(X,p-PhO)] is shown by the Y–T plot in Fig. 14. All the ␴⫹ points (open circles) of ␲-ED p-substituents deviate significantly in the direction of rate enhancement from the ␳m correlation line of ␳ ⫽ ⫺2.0. Only for the range of substituents less reactive than p-Me (w-ED class), is a precise Y–T correlation obtained, with a ␳YT essentially identical to the ␳m. On the other hand, for more electron-donating substituents than p-Me (s-ED class

THE YUKAWA–TSUNO RELATIONSHIP

311

Fig. 12 The Y–T plot for solvolysis of symmetrically substituted 1,1-diaryl-2,2,2trifluoroethyl tosylates [29(X ⫽ Y)] in 80% aqueous EtOH (80E) at 25⬚C; r ⫽ 1.19. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1997a). Copyright 1997 Chemical Society of Japan.

312

Y. TSUNO AND M. FUJIO

Fig. 13 The Y–T plot for solvolysis of 1-aryl-1-phenyl-2,2,2-trifluoroethyl tosylates [29(X,Y ⫽ H)] in 80% aqueous EtOH (80E) at 25⬚C; r ⫽ 1.45 for p-ED and 1.26 for w-␲-ED groups. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1997b). Copyright 1997 Chemical Society of Japan.

313

log(k/kH)

THE YUKAWA–TSUNO RELATIONSHIP

␳ ⫽ -4.01 r ⫽ 1.26

Fig. 14 The Y–T plot for solvolysis of 1-aryl-1-(4-pheoxyphenyl)-2,2,2-trifluoroethyl tosylates [29(X,Y ⫽ p-PhO)] in 80% aqueous EtOH at 25⬚C; r ⫽ 1.26 for p-␲-ED and 1.07 for w-ED groups. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1999a). Copyright 1998 John Wiley & Sons.

314

Y. TSUNO AND M. FUJIO

substituents), a separate Y–T correlation with ␳ ⫽ ⫺4.0 and r value of 1.26 is obtained. Similarly the Y–T equation fails to correlate the substituent effect on the series [29(X,p-MeO)] (Table 8), resulting in bilinear correlations (not shown), one with r ⫽ 1.26 for the five s-ED class substituents and another with a distinctly lower ␳ value for the range of substituents less ED than a p-methyl group. Liu and coworkers (Liu et al., 1991, 1992) have suggested that both the Brown and the Y–T correlations showed equal ability to predict rates for the series [29(X,Y)] with Y ⫽ p-MeO and p-PhO, and concluded that the observed superiority in linear Y–T plots might not reflect the reliability of the Y–T relationship, but might merely be a computational artifact. Before discussing the predictive ability, we note our conclusion that the Y–T equation is unable to correlate substituent effects on reactions in these series with a single linearity, but that it is capable of indicating the non-linearity of the substituent effect of the series. From general applications, the criterion for acceptable conformity of Y–T correlations can be and/or should be set at a much higher precision level (smaller SD ⬵ 0.1) than that of Brown ␳⫹␴⫹ correlations (Fujio et al., 1999). The Y–T equation gave a slightly improved SD value of ⫾0.24 for the whole set of [29(X,p-PhO)] compared with SD ⫽ ⫾0.32 for the Brown correlation. The large SD value indicates clearly that the reactivity set is far from conforming to the Y–T equation and conforms even less well to the Brown equation. Even though it is often not easy to deduce the significance of non-linearity from correlation analysis, the non-linearity is important as a probe for indicating any change in the interaction of substituents. The deviation from the linear Y–T correlation can be related to the deviation of the two aryl groups from coplanarity. Since the observed r value for [29(X ⫽ Y)] is reduced due to reduced coplanarity, the resonance demand of the (hypothetical) planar system of [29(X ⫽ Y)] should be higher than r ⫽ 1 in the ␣-cumyl system [2]. The exact linear substituent effect for [29(X ⫽ Y)] indicates a conformation in which the two aryl groups in this series have similar deviations from coplanarity with the developing carbocation centre in the transition state; both aryl groups in the series [29(X ⫽ Y)] are equally effective in stabilizing the rate-determining transition state by delocalization of the positive charge. A bilinear correlation indicates the occurrence of a change in the interaction mechanism of the substituents within the range of substituents involved. It is reasonable that the X-substituent effect changes owing to the change in coplanarity from the conformer where both X- and Y-substituted phenyls are equally twisted to a conformer where the Y-phenyl is coplanar and X-phenyl is more twisted (or vice versa). The behaviour of the series [29(X,Y)] with Y ⫽ p-Me appears to be even more typical of this series and is consistent with the presence of three conformations (see below). Whereas the correlation is satisfactory only for the range of w-ED class X-groups from p-Me to p-Cl,

THE YUKAWA–TSUNO RELATIONSHIP

315

either the stronger ED or the EW class X-groups deviate from the correlation line; this is also attributable to the deviation of X- and Y( ⫽ p-Me)-phenyls from coplanarity. Consequently, the Y–T equation is incapable of directly correlating the effects of the full range of X-substituents in any fixed Y series of [29(X,Y)], except for the symmetrically substituted series [29(X ⫽ Y)]. The bilinear correlation obtained in the unsymmetrical series with X ⫽ Y indicates a change in the substituent interaction mechanism within the range of substituents involved. The non-linear correlations of substituent effects in this system can be related to the substituent-induced change in conformation of the transition state.

4

Carbocation formation equilibria

In the same way as changes in reactivity reflect the nature of the transition state, a change in equilibrium constant corresponds to a change in the thermodynamic stability of the carbocation intermediate. For example, substituent effects on the basicities of arylcarbonyl derivatives ArCOR provide a reference for the formation of ␣-hydroxycarbocations (17).

(17)

The Y–T equation correlates the basicity changes for all the members of this series except for benzophenones, where R ⫽ C6H5 in [30(R)], which shows significant deviations of X ⫽ p-MeO and p-HO groups. The r value changes widely with the ␣-R groups, from 1.16 for [30(H)] to 0.39 for [30(NH2)], depending mainly upon the resonance effect of the R groups (Table 9). It is more remarkable that the ␳ values are distinctly smaller throughout the series than those for ␣-alkyl or ␣,␣-dialkyl benzyl solvolyses.

TRIARYLMETHYL CATIONS

The pKR⫹ values of polyarylcarbinols, [3]-OH (Scheme 9) and [22]-OH, should be important reference processes for thermodynamic substituent effects related to the rate-determining dissociation (DN) step in SN1 solvolyses. The pKR⫹ values for symmetrically trisubstituted triarylmethanols

Table 9 Yukawa–Tsuno correlations for carbocation formation equilibria. System pKR⫹ sym-Ar3COH Ar2(Ph)COH pKR⫹ sym-Ar2CHOH pKR⫹ mono-Ar(Ph)CHOH pKR⫹ Ar2CHOH pKR⫹ Y⫽H exc. X ⭐ m-Me Y ⫽ p-MeO exc. X ⫽ p-MeO Y ⫽ p-Me exc. p-MeO Ar-CO-H pKBH⫹ Ar-CO-CH3 pKBH⫹ Ar-CO-OH pKBH⫹ Ar-CO-NH2 pKBH⫹ Ar-CO-Ph pKBH⫹ exc. p-OH, p-OMe, p-OEt ␣,␣-diarylethylene pKR⫹ X⫽Y Y⫽H exc. p-MeO Y ⫽ p-MeO exc. p-MeO

⫺␳

r

R

SD

n

Reference

11.48b 7.511c 9.246c 3.81

0.762 1.052 1.068 1.21

0.999 0.9986 0.9956 0.9963

0.40 0.544 0.42 0.135

12 5 7 9

1

3.97 4.972 2.24 2.288 3.410 3.406 1.715 2.156 1.14 1.20 1.16 1.17

1.11 1.003 (1.45) 0.981 1.194 1.025 1.16 0.762 0.54 0.39 0.71 0.66

0.9953 0.9992 0.9976 0.9988 0.9988 0.9974 0.9958 0.9973 0.9930 0.9919 0.9906 0.9932

0.142 0.054 0.114 0.057 0.100 0.089 0.104 0.067 0.049 0.059 0.080 0.048

13 7 7 6 6 5 7 15 13 12 17 14

6.68c 3.35 3.47 2.20 2.40

0.92 (1.10) 0.69 (1.13) 0.30

0.9966 0.9966 0.9982 0.9939 0.996

0.24 0.19 0.104 0.19 0.12

6 7 6 6 5

1 1 2 1 2

3 4, 5 6 7 8 9

a X ⭓ or X ⭐; a partial correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; substituents of which the ␴⫹ are within the range of X1 to X2. exc. X; excluding substituent X. bCorrelation overall ␳ for three equivalent aryl substituents. cCorrelation overall ␳ for two equivalent aryl substituents.

References: 1, Deno and Evans (1957), Deno and Schriesheim (1955), Deno et al. (1955, 1959), Bagno and Scorrano (1996). 2, Mindl (1972), Mindl and Vecera (1971, 1972). 3, Yates and Stewart (1959). 4, Stewart and Yates (1958). 5, Yukawa et al. (1972). 6, Stewart and Yates (1960). 7, Edward et al. (1960). 8, Mindl and Vecera (1970). 9, Goethals et al. (1978).

THE YUKAWA–TSUNO RELATIONSHIP

317

KR⫹

Scheme 9.

– scale [3(X ⫽ Y ⫽ Z)] give a completely linear Y–T correlation against the ␴ with an r value of 0.76 for the substituent range from p-dimethylamino to p-nitro (Yukawa et al., 1966). On the contrary, less satisfactory Y–T correlations are obtained for the pKR⫹ values for monosubstituted triphenylmethanols [3(X,H,H)] or for the log(k/k0) values for the solvolysis of the corresponding chlorides, in which the s-␲-ED p-methoxy substituent exhibits a higher r value than that for the other w-ED groups. In the monosubstituted [3C⫹(X,H,H)], apparently only the s-␲-ED-substituted aryl tends to enter into coplanarity with the cationic orbital, to exert its maximum resonance effect. The substituent effects in various unsymmetrical series of [3C⫹(X,Y,Z)] are schematically shown in Fig. 15, where pKR⫹ values for various series are – with an r value of 0.76, in reference to the linear plotted against the sum of ␴ – ␴ plot for the symmetrical (trisaryl) series [3(X ⫽ Y ⫽ Z)]. All the unsymmetrically substituted series show significantly concave (quadratic) correlations, which intersect with the linear plot of the [3C⫹(X ⫽ Y ⫽ Z)] series as tangent at the point of X ⫽ Y ⫽ Z. This implies that the ␳ value of any series based on the tangent at the point (X ⫽ Y), where the varying X substituent is the same as the fixed Y substituent, should be identical to the ␳ value for the symmetrical series [3C⫹(X ⫽ Y ⫽ Z)]. The bisaryl-series [3C⫹(X ⫽ Y, Z ⫽ H)] gives a reasonably good linear correlation with a slightly increased r value. However, the Y–T correlation for the monosubstituted series does not result in a satisfactory fit. It is well known that the three aryl rings in the triarylmethyl cation [3C⫹(X,Y,Z)] are twisted out of coplanarity with the vacant orbital at the ␣-carbon by steric interaction. The propeller conformation of the symmetrical [3C⫹(X ⫽ Y ⫽ Z)] equally prevents any aryl groups from exerting their maximum stabilizing effect on the carbocation. It should be noted that in the polyphenylcarbocation system, the energy loss associated with the deviation of one phenyl ring from coplanarity should be largely compensated by the energy gain achieved by synchronous rotation towards increased coplanarity of the other ring.

318

Y. TSUNO AND M. FUJIO

⌺␴(r ⫽ 0.76) Fig. 15 Non-linear substituent effects on the pKR⫹ for triarylmethanols. The – with r ⫽ 0.76. non-linearity is shown schematically as the plots against sum of Y–T ␴ 䊉, trisaryl (X ⫽ Y ⫽ Z) carbocations (as reference); 䊊, bisaryl (X ⫽ Y; Z ⫽ H); 䊐, monoaryl (X; Y ⫽ Z ⫽ H); 䉭, bis-(p-MeO)-trityl ions (X; Y ⫽ Z ⫽ p-MeO) series. Data taken from Deno and Schriesheim (1955), Deno et al. (1955, 1959), Deno and Evans (1957), Ritchie (1986).

THE YUKAWA–TSUNO RELATIONSHIP

319

A Taylor expansion for log kX values with varying X substituents is shown in (18). The second derivative in this expression plays an important role in the case of a curved correlation; concave correlations can be approximated in – with an appropriate r terms of the regressional power series expansion of ␴ X scale. ⭸(log kX) – ⭸2(log kX) – 2 ␴X ⫹ – – )2 (␴X) ⫹ . . . ⭸␴X ⭸(␴ X ⭸␳ – 2 – – ⫹ ␳ (␴ – )2 ⫽ ␳X ␴X ⫹ – (␴X) ⫹ . . . ⫽ ␳X ␴ X XX X ⭸␴X

log(kX /kH) ⫽

(18)

The coefficient ␳XX represents the deviation from a simple linear free energy relationship and is related most obviously to the transition state shift arising from the perturbation by the substituents X (O’Brien and More O’Ferrall, 1978). Simply from the viewpoint of correlation analysis, the concave plots of the substituent effects of equivalently disubstituted (bisaryl) series [3C⫹(X,Y,Z)] with X ⫽ Y; Z ⫽ H, can be treated using equation (18) in terms – with r ⫽ 0.76, to give an excellent correlation with ␳ ⫽ ⫺8.29 and of ␴ X X ␳XX ⫽ ⫺0.117 for two X groups; R ⫽ 0.9999 and SD ⫽ 0.141. This gives a better correlation than the Y–T correlation. The negative value of ␳XX indicates a concave correlation, but any perturbation to cause a change in ␳X should be small. The significant curvature of the correlation for the monosubstituted diphenyl series may also be treated using (18) in terms of the – scale (O’Brien and More O’Ferrall, 1978). same ␴ X

BENZHYDRYL CATIONS

The substituent effect on the equilibrium pKR⫹ forming diarylmethyl carbocations plays an important role in the extended Brønsted relationship (19) by giving the extreme limit of the substituent effect on the transition state of the corresponding benzhydryl solvolysis. log k ⫽ ␣ log KR⫹ ⫹ C

(19)

The pKR⫹ values of diarylmethyl cations [22C⫹] were basically those determined according to the same method as for the triarylmethyl cations in aqueous sulphuric acid solutions by Deno and coworkers (Deno et al., 1955; Deno and Schriesheim, 1955; Deno and Evans, 1957). For reasons of solubility, a wide set of pKR⫹ values in aqueous acetic acid solutions were determined by Mindl and coworkers (Mindl and Vecera, 1971, 1972; Mindl, 1972). Unfortunately, however, the pKR⫹ values, determined by means of the protonation– dehydration method for diarylmethanols, could only be obtained for more stable derivatives and not for those less stable than the unsubstituted

320

Y. TSUNO AND M. FUJIO

derivative. This causes inevitable difficulties in the treatment of the substituent effects in this system especially in the less ED Y series. The monosubstituted diarylmethyl series [22C⫹] with Y ⫽ H can be correlated with the Y–T equation to give an r value (ca. 1) similar to the value for benzhydryl chloride hydrolysis; on more careful inspection, the correlation seems to be appreciably concave, although the extent of non-linearity is not very certain due to the lack of EW substituents. The series Y ⫽ p-Me gives a similar correlation with a slightly lower ␳ value. The Y ⫽ p-MeO series gives a clearly concave Y–T correlation with a significantly reduced ␳ value. For these correlation results, Mindl and Vecera (1971, 1972) examined the non-additivity of substituent effects and the selectivity–stability relationship in this system. While these analyses were not altogether successful, the behaviour of the pKR⫹ substituent effect appears to resemble closely the behaviour of the kinetic (log k) substituent effects in benzhydryl solvolyses. It is therefore remarkable that there is an excellent linear relationship (20) between log k for the hydrolysis and pKR⫹ for the diarylmethyl series over the range from X ⫽ Y ⫽ H to X ⫽ Y ⫽ p-MeO (Fig. 16). log k ⫽ 1.16 log KR⫹ ⫹ const (R ⫽ 0.996 and

(20)

SD ⫾ 0.36)

Here log k values are for solvolyses in aqueous 85% acetone at 0⬚C and log KR⫹ for the corresponding cations in aqueous sulphuric acid–acetic acid solutions. Although the value of the coefficient 1.16 in (20) does not have as direct a physical significance as the ␣-exponent in the extended Brønsted equation (19) because the reaction, solvents and temperature are different, there is still a good linear rate–equilibrium relationship for benzhydryl carbocation formation; the overall correlation embraces clearly concave partial correlations with varying slopes for the respective Y series. The whole pattern of –, should be essentially identical (with only the substituent effects, pKR⫹ vs 兺 ␴ – for the ordinate scale being slightly different) to that of log (kXY /kHH) vs 兺 ␴ solvolyses shown in Fig. 8. The two aryl rings in the benzhydryl cation [22C⫹] are also known to be twisted equivalently from the cationic sp2 plane by ca. 16.0⬚ according to ab initio optimization at the RHF/6-31G* level. While equivalent rotation of the X-aryl ring must be the cause of the reduced resonance demand, especially in the case of equivalently disubstituted benzhydryl cations [22C⫹(X ⫽ Y)], the symmetry will be destroyed by replacing one phenyl ring by a ring substituted by an ED Y-substituent; the Y-substituted phenyl ring will become more nearly coplanar with the sp2 cationic carbon and this will enhance the ␲ delocalization. The electronic demand will therefore be reduced on the other,

THE YUKAWA–TSUNO RELATIONSHIP

321

Fig. 16 A rate–equilibrium plot of log k for the solvolyses of benzhydryl chlorides in 85% aqueous acetone (Fox and Kohnstam, 1964; Fujio et al., unpublished) against pKR⫹ for benzhydrols (Mindl and Vecera, 1971, 1972; Mindl, 1972).

X-substituted phenyl ring which in turn will be enforced to be further out of the plane. In unsymmetrically substituted cases, the stronger cation-stabilizing aryl ring should be less twisted from the plane of the carbocation centre and the other ring should be twisted even more out of the plane. It is noteworthy that the substituent changes on the pKR⫹ value of disubstituted trityl cations [3C⫹(X,Y,H)] are precisely linear against pKR⫹ values of [22C⫹(X,Y)]; conformational effects in [3C⫹(X,Y,H)] where an aryl substituent is fixed as Z ⫽ H should be essentially the same as those in the benzhydryl cations where one of the ␣-substituents remains constant as H.

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1,1-DIARYLETHYL CARBOCATIONS

Simple application of either the Brown or the Y–T equation to the substituent effects on pKR⫹ for ␣,␣-diphenylethylenes [31(X,Y)] (in Table 9) is not necessarily successful (Goethals et al., 1978). The difficulties arise mainly from stereochemical factors; the RHF/6-31G* optimization of the 1,1-diphenylethyl carbocation [31C⫹] provides a propeller geometry shown in Fig. 17. The phenyl rings are twisted from the plane of the sp2 carbocation centre by 23.4⬚ and 34.0⬚ (Fujio et al., unpublished). This difference is probably due to the steric non-equivalence of the methyl group. The correlations, and especially the r values, must be intimately related to the dihedral angle between the substituted phenyl ring and the plane of the carbocation. In fact, the symmetrically substituted series exhibits a linear Y–T correlation with ␳ and r values intermediate between those for symmetrically substituted triarylmethyl and diarylmethyl cations. In the unsymmetrical (X ⫽ Y) series, the p-MeO group appears to deviate from the correlation, as indicated by the SD values, and a lower r value should be assigned for these systems. The non-linear Y–T correlations with rather lower r values for protonation of diarylethylenes probably result from the conformation of the carbocation intermediates and nearly similar results are also observed in bromination. Both reactions will be considered together in a later section (p. 334).

5

Yukawa–Tsuno correlations for electrophilic addition to alkenes

HYDRATION OF ARYLALKENES

Electrophilic addition reactions are some of the most important processes (21) involving a cationic transition state or intermediate. The acid-catalysed hydration of arylalkenes such as styrene [32] is typical of such processes

Fig. 17 The structure of 1,1-diphenylethyl cation, optimized at RHF/6-31G* level (Fujio et al., unpublished).

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(E⫹ ⫽ H⫹) and has been extensively studied by Tidwell’s group (Tidwell, 1984; Allen and Tidwell, 1989). [21] [32] [33] [34] [35]

R1 ⫽ H; R2, R3 ⫽ H R1 ⫽ Me; R2, R3 ⫽ H. R1 ⫽ MeO; R2, R3 ⫽ H R1 ⫽ CF3; R2, R3 ⫽ H

[32C⫹] [33C⫹] [34C⫹] [35C⫹] (⫽[5C⫹])

The proton addition to the double bond leading to a benzyl carbocation is the rate-determining step in the hydration of styrene. Accordingly, the substituent effect is expected to be correlated with the Y–T equation (2). Correlations for the hydration of ␣-substituted styrenes are shown in Table 10. Both ␳ and r values vary significantly with the nature of the ␣-substituents; the r value changes from 1.0 for ␣-CF3-styrenes [35] to 0.42 for ␣-MeOstyrenes [34]. The ␳ value also changes in parallel with the decrease in the r value. These changes in the correlation are reasonably interpreted, at least qualitatively, in terms of the stability of carbocation intermediates. The ␳ value of ⫺5.4 for the ␣-CF3 series [35] is higher than that of the [32] and/or ␣-Me series [33] and is distinctly greater than the value of ⫺2.26 for the ␣-MeO series [34] and ⫺2.58 for the ␣-OPO(OEt)2-styrene series. This is of course the expected result because of the much greater stabilizing effect of the ␣-oxy substituent especially by charge delocalization. The ␳ value for the ␣-Me [33] or ␣-CF3 [35] series is lower in the hydration than that observed in solvolysis of the corresponding esters, but the value for the ␣-MeO series [34] appears comparable or even higher than that for the hydrolysis of acetophenone acetals (Loudon and Berke, 1974) or for the pKBH⫹ of acetophenones (Table 9). On the other hand, the r values for any series of styrene hydrations are definitely lower than those of the corresponding solvolyses. At this stage we are unable to give a definite answer to the question regarding the origin of this significant attenuation of the resonance demand. An earlier transition state appears a reasonable explanation but is difficult to accept for many reasons (see below); a low ␳ value implies an earlier transition state. A pre-equilibrium scheme might also be plausible; given the ␳ value of ⫺1 to ⫺1.5 for pre-equilibrium complexation, the carbocation formation step may have a reasonable ␳ value. Tidwell has noted that the precision of most of the correlations of styrene reactivities is only fair. The correlation of the reactivity requires a choice of the acidity at which to make the comparison. The acidity dependences of the reactivity of these alkenes are not identical, and extrapolations through the

Table 10 Yukawa–Tsuno correlations for hydrations of arylalkenes and the related reactions. System Styrene

␣-Me-styrene, exc. p-COOH ␣-OPO(OEt)2-styrene ␣-CF3-styrene ␣-(MeO)-styrene Hydrolysis, acetophenone acetals Dehydration, ArCH(OH)-CH2Ph

H2O-H2SO4 TFA-CCl4 (1:1) TFA-CCl4 (1:4) H2O⫺H2SO4 H2O⫺H2SO4 H2O⫺H2SO4 H2O⫺H2SO4 5D(AcOH) 29.9⬚C 5D(H3O⫹) MeOH (H3O⫹) 5E-H2SO4

␳

r

R

SD

n

Reference

⫺3.69 ⫺6.29 ⫺5.50 ⫺2.92 ⫺2.577 ⫺5.36 ⫺2.263 ⫺2.548 ⫺2.211 ⫺1.719 ⫺4.37

0.71 0.59 0.67 0.735 0.797 1.00 0.415 0.475 0.359 0.578 0.817

0.9932 0.9958 0.9965 0.9970 0.9985 0.9987 0.9959 0.9977 0.9991 0.9996 0.9984

0.21 0.225 0.223 0.146 0.101 0.176 0.118 0.101 0.053 0.022 0.16

11 9 7 6 6 5 7 6 6 8 6

1 2 2 1 3 4 1 5 5 6 7

References: 1, Chwang et al. (1977). 2, Allen et al. (1982a,b). 3, Attia et al. (1977); Frampton et al. (1972). 4, Koshy et al. (1979). 5, Loudon and Berke (1974). 6, Toullec and El-Alaoui (1985). 7, Noyce et al. (1968).

THE YUKAWA–TSUNO RELATIONSHIP

325

acidity scales are necessary in order to arrive at a common basis for comparison. It appears that the extrapolations that must be employed in most cases cited are one of the causes of the low precision. Tidwell and coworkers (Chwang et al., 1977; Koshy et al., 1979) focused their interest on the reactivity–structure relationship of the hydration rates. Additivity of substituent effects of the four substituents in the ethylene has been studied. As the interaction of substituents with the positive charge in the intermediate [36C⫹] is analogous to that in the electrophilic aromatic substitution intermediate, the substituent effects on the hydration of [36] may

be correlated reasonably by the Brown ␴p⫹ parameters for ␣-substituents R1, R2 and by ␴m⫹ for ␤-substituents R3, R4. A satisfactory reactivity–structure relationship was given by (22), at least for ␣-substituents. log k ⫽ ␳ 兺 ␴⫹(Ri) ⫹ C

(22)

The utility of rate correlations such as (22) depends on the additivity of substituent effects. Whereas the correlation of ␤-substituents requires a different ␳ value, this may be treated as total effects of three ␣-substituents in the developing intermediate [36C⫹], R1, R2 and –CHR3R4. In practice a rather good correlation covers 22 orders of magnitude in reactivity without any clear evidence of curvature. Thus Tidwell stated (Koshy et al., 1979) that the Y–T correlation was unnecessary for the analysis of a large body of hydration data. Non-additivity of substituent effects has been proposed as a criterion for the operation of the RSR so the linearity argues against its applicability in this system. In a description of transition states by structure–reactivity coefficients (Jencks and Jencks, 1977), two alternative types of behaviour were discussed. In ‘‘Hammond’’-type reactions the more endothermic reactions have later transition states, whereas ‘‘anti-Hammond’’ behaviour is characterized by an adjustment of the transition-state structure to take advantage of favourable substituent effects. These results illustrate that different systems can display quite different behaviour in linear free energy correlations. Thus, in alkene protonations, such correlations cover vast ranges in reactivity with only modest changes in sensitivities, while in solvolytic reactions the selectivity ␳ varies depending on the electron supply at the electron-deficient centre (Johnson, 1978).

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BROMINATION OF ARYLALKENES

Alkene brominations have been widely studied by the Dubois and Ruasse group, and the aryl substituent effects and their dependence upon the alkene structures have been precisely analysed, using the Y–T equation as well as other structure–reactivity relationships. The results have been summarized by Ruasse in her review article (Ruasse, 1993). The results of our analysis by the Y–T equation and related equations are summarized in Table 11. Arylalkene bromination is a typical electrophilic addition to form an ␣-carbocation, but markedly non-linear structure–reactivity relationships were observed for brominations of styrene [32], trans-stilbene [37] and ␣-methylstilbene [38] (Ruasse and Dubois, 1972, 1974; Ruasse et al., 1978; Ruasse and Argile, 1983). Some of these curvatures could not be interpreted directly by the Y–T equation but some were related to a mechanistic changeover based on a multipathway scheme (Ruasse, 1990). Three pathways leading to the C␣⫹ and C␤⫹ carbocations and to the bromonium ion in the bromination of trans-stilbene (Ruasse and Dubois, 1972, 1974) are shown in Scheme 10.

[37]

Scheme 10.

A markedly non-linear structure–reactivity relationship (Fig. 18) indicates that the three pathways can compete. For a single s-ED substituent, X ⫽ p-MeO for example, only the C␣⫹ path is followed. A w-ED or EW X-group in one ring associated with ED Y group in the other ring favours the C␤⫹ cation path and two EW groups in both rings favour the bridged intermediate. Due to the mechanistic complexity, it is not possible to obtain strictly quantitative data; only qualitative characteristics of the competition between the three paths can be described. Because our main interest is only in the C␣⫹ pathway, we may apply an analysis by the non-linear least-squares method for two competing pathways using (23), which is essentially the same as (5) (see above). o

–⫹

o

–⫹

kt ⫽ k␣H 10␳␣(␴ ⫹r␣ ⌬␴R ) ⫹ k␤H 10␳␤(␴ ⫹r␤ ⌬␴R )

(23)

Table 11 Yukawa–Tsuno correlations for bromination and the related addition reactions of arylalkenes. Systema (MeOH, 25⬚C) Styrene (exc. p-MeO) bilinearb

␣-C⫹ pathc Br⫹ pathd

Styrene (AcOH, 25⬚C) (exc. p-MeO) ␤-methylstyrene bilinearb ␣-C⫹ pathc Br⫹ pathd ␣-methylstyrene ␣-OCH3-styrene stilbene bilinearb ␣-C⫹ pathc ␤-C⫹ pathe ␣-methylstilbene ␣,␣-diarylethylene X ⫽ Y ␣,␣-diarylethylene Y ⫽ H (exc. p-MeO) ␣,␣-diarylethylene Y ⫽ p-MeO (exc. p-MeO) m⫽2 m⫽3 (exc. p-MeO)

␳

r

R

SD

n

Reference

⫺3.83 ⫺3.904 ⫺4.514 ⫺3.42 ⫺4.61 ⫺4.72

1.08 0.774 0.983 (0.00) 1.25 1.00

0.9978 0.9988

0.147 0.086 0.079

12 11 12

1

0.9973 0.9958

0.23 0.23

8 7

1

⫺4.414 ⫺3.350 ⫺4.247 ⫺1.85 ⫺5.54 ⫺4.43 ⫺1.42 ⫺4.571 ⫺6.95f ⫺3.50 ⫺3.50 ⫺2.20 ⫺2.217 ⫺3.502 ⫺3.414 ⫺3.419

0.927 (0.00) 0.743 0.0 0.9 0.922 (0.0) 0.823 0.538 0.799 0.690 0.486 0.340 0.396 1.175 0.960

0.057

9

0.087

8

0.15

12

0.063 0.18 0.077 0.065 0.061 0.061 0.051 0.08 0.01

8 9 10 9 9 8 6 6 5

0.9994

0.9997 0.9960 0.9987 0.9984 0.9983 0.9979 0.9992 0.9991 1.000

1 1 2 3 4 5 5 5 6 6

Exc. X; excluding substituent X. bAnalysed by the non-linear least-squares method by equation (23). cThe non-linear least-squares correlation for the ␣-C⫹ pathway (see text). dThe non-linear least-squares correlation for the bromonium ion pathway (see text). eThe non-linear least-squares correlation for the ␤-C⫹ pathway (see text). fCorrelation overall ␳ for two equivalent aryl substituents. a

References: 1, Ruasse et al. (1978). 2, Ruasse and Dubois (1984). 3, Ruasse and Dubois (1972). 4, Ruasse and Argile (1983). 5, Dubois et al. (1972); Hegarty et al. (1972). 6, Hegarty and Dubois (1968).

Y. TSUNO AND M. FUJIO

log(k/kH)

328

Fig. 18 The reactivity–structure relationship for the bromination of ring-substituted stilbenes [37] in MeOH at 25⬚C. For interpretation of symbols, see Fig. 1. The curvature shows the X dependence of the competition between bromonium ion kBr and carbocation kC pathways. The graph is drawn from the data of Ruasse et al. (1978). Copyright 1978 American Chemical Society.

329

log(k/kH)

THE YUKAWA–TSUNO RELATIONSHIP

Fig. 19 The reactivity–structure relationship for the bromination of ring-substituted styrenes [32] in MeOH at 25⬚C: competition between the bromonium ion kBr and carbocation kC pathways. For interpretation of symbols, see Fig. 1. The graph is drawn from the data of Ruasse et al. (1978). Copyright 1978 American Chemical Society.

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Y. TSUNO AND M. FUJIO

The plot for stilbenes in Fig. 18 is that for the correlation according to (23); as a matter of fact, the correlation for the C␣⫹ pathway should not be affected by any competing paths because their ␳ values are small. For styrene [32] bromination, (23) can be applied more appropriately; the curvature of the plot (Fig. 19) is not very significant when compared with that in stilbene bromination. Styrene bromination proceeds predominantly via a C␣⫹ carbocation intermediate and is accompanied by a bromonium ion pathway which becomes increasingly competitive as X becomes more EW; the second term in (23) should reflect the bromonium ion pathway, whereas the C␤⫹ carbocation is too unstable to compete with the formation of the C␣⫹ carbocation. Exactly the same correlation is obtained for ␤-methylstyrene, Ar-CH⫽CH(CH3). The bromination of ␣-methylstyrenes [33] involves exclusively a highly stable tertiary benzylic carbocation intermediate without competition from either C␤⫹ or the bromonium intermediate regardless of the ring substituent (Ruasse et al., 1978). The Y–T equation can be applied without difficulty to give an r value slightly lower than those for styrene [32] itself or stilbene [37]. –-relationship shows that For X-substituted ␣-methylstilbenes [38], the linear ␳␴ only the tertiary bromocarbocation is involved (Fig. 20). However for – Y-substituted ␣-methylstilbenes [38], there is a sharp breakdown in the ␳␴ plot, which has been interpreted (Ruasse and Argile, 1983; Argile and Ruasse, 1983) as indicating a change in the mechanism from the secondary to the tertiary carbocation pathway (Scheme 11) when Y changes from an ED to an EW group.

[38] Scheme 11.

Despite the difficulties arising from competing pathways, the substituent effects on the C␣⫹ pathway can be estimated quite successfully and the correlations obtained (Table 11) are reasonably interpreted in comparison with those for the uncomplicated systems. Simple styrenes [32] where ␣-R ⫽ H, irrespective of the ␤-substituents R, give a constant ␳ value of ⫺4.45 and constant r of 0.93. All tertiary ␣-methylstyrene [33] series, irrespective of ␤-substituents, give a similar ␳ value (⫺4.4 ⫾ 0.2) but a slightly lower r value of 0.74–0.82. For the trans-stilbene series [37], Ruasse assumed that the balance between polar and resonance effects does not differ from that in the reaction defining ␴⫹, i.e. that there is no need to use a Y–T equation.

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Fig. 20 The Y–T plot for the bromination of X-substituted ␣-methylstilbenes [38] in MeOH at 25⬚C (data taken from Ruasse and Argile, 1983); r ⫽ 0.82. For interpretation of symbols, see Fig. 1.

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Y. TSUNO AND M. FUJIO

It is noteworthy that the substituent effect on bromination of ␣-methoxystyrene [34] gives a particularly small ␳ value of only 1.85. Even more striking is the complete absence of conjugation between the ring substituent and the cationic centre. These results are the most important argument in Dubois’ interpretation of the mechanism (Ruasse, 1993). The extremely small 兩 ␳ 兩 value and the absence of conjugation between X and the cationic centre were attributed to an early transition state, as required by the highly resonance stabilized intermediate, and to an increase of the substituent–charge distance due to charge delocalization in the methoxy group. Changes in charge distribution, superimposed on this Hammond effect, may reinforce the attenuation of the ring-substituent effects. In addition to its kinetic effect (transition-state shift), the enhancement of the stability of the intermediate has a thermodynamic effect on the ␳ parameter since resonance with the ␣-methoxy group diminishes the influence of the substituents on the formation of the carbocation. Conclusively, attenuation of ␳ for ␣-methoxystyrene bromination results from a combination of shift in the transition-state position, a lengthening in the charge–substituent distance and resonance saturation.

In the bromination of ␣,␣-diarylethylenes [31], resonance effects cannot be fully developed at the transition state, since the aromatic rings are not in the same plane as that of the developing carbocation centre (Dubois et al., 1972), the structure of which is closely similar to that of the carbocation [31C⫹] shown in Fig. 17 (Fujio et al., unpublished). Accordingly, application of (2) should give lower r values and in some cases non-linear correlations, arising mainly from stereochemical factors (cf. Table 11). The geometric dependence of substituent effects should also be similar to that in the protonation equilibria but is statistically not very significant; this will be discussed in more detail later. Elegant examples of the geometry dependence in the bromination of diarylethylene were provided by Hegarty and Dubois (1968). Cycloalkadienes [39] and [40] show good Y–T correlations of distinctly different r values with essentially the same ␳ value (Table 11). In [39], the fused aromatic ring is approximately coplanar with the double bond, within 10⬚, and the free aryl ring must be twisted out of the plane of the double bond by ca. 60⬚. In [40] the fused aryl ring in benzocycloheptadiene is twisted out of the plane of the double bond, allowing the free aryl ring to become planar and

THE YUKAWA–TSUNO RELATIONSHIP

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more effectively conjugated with the ethylenic double bond. These facts as well as the loss of coplanarity in the protonation of diphenylethylenes should be taken into account in the interpretation of the Y–T correlations (Table 11) of substituent effects on the bromination of diarylethylenes [31]. A similar detailed analysis was carried out for disubstituted diarylethylenes [31(X,Y)] by Dubois and coworkers (Dubois et al., 1972; Hegarty et al., 1972), who assumed that the two aryls in unsymmetrically substituted [31(X ⫽ Y)] are in the same conformation as in [40], in which the more ED aryl is coplanar with the carbocation centre and the less ED aryl is twisted out of coplanarity. The effects of the coplanar aryl substituents were approximated by ␴⫹ and those of the twisted groups by ␴ constants. The same kind of treatment was applied to clarify the effects of a fixed Y substituent on the correlation of effects of varying X substituents, demonstrating a significant change in the ␳Y values which are linear when plotted against ␴⫹ of the Y substituents. Furthermore, it was pointed out that the pattern of change of ␳Y values was similar to that in the solvolysis of benzhydryl chlorides. It should be pointed out that these results have played an important role in the later extension of the Dubois–Ruasse analysis of RSR behaviour (Dubois et al., 1984). Reactivity–selectivity relationships (RSR) in the bromination of styrenes The progress of the reaction at the transition state, ␣, is usually obtained from the coefficients of the extended Brønsted equation (19) or of other rate–equilibrium relationships which compare substituent effects on kinetics and thermodynamics. Using (24) the ␳ values can also express this position if ␳k for rates and ␳e for equilibria of the same elementary step are available.

␳k ⫽ ␣␳e

(24)

The RSR of equation (3) should be applied to both rates and equilibria of the same process. Thus the change in ␳ value can also be attributed to the early–late movement of a transition state along the reaction coordinate, which can be related to the destabilization of the reactant or stabilization of the product or intermediate. In the bromination in methanol of several styrenes, X-C6H4C(R␣)⫽CH2, of differing reactivities (Ruasse et al., 1978), the reactivity increase corresponds roughly to an attenuation of both polar and resonance selectivities, ␳ and ␳r, respectively. The reactivity–selectivity relationship based on the data of Table 11 is approximately linear for ␳r but not for ␳. The curvature of the ␳R vs. log(k0)R plot was attributed to the change in the charge distribution due to charge delocalization in R␣, but the ␳r vs. log(k0)R plot was linear since ␳r should be independent of the charge distribution. This led to the conclusion that the problem of the balance between the transition-state position and the

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thermodynamic contribution in the RSR behaviour should be approached via ␳r and not ␳ (Ruasse and Dubois, 1984). In our opinion, however, it is not easy to substantiate this conclusion; in practice, it is not possible to estimate quantitatively the relative magnitude of these contributions from the data of Table 11 alone.

Structure–reactivity relationships in bromination For the structure–reactivity correlation of alkene hydration (Koshy et al., 1979), Tidwell claimed the utility of an extended Brown ␴⫹ equation (22), where ␴p⫹ and ␴⫹m constants are used to describe the effect of ␣- and ␤-substituents respectively. An extended form of (22) has been used to analyse kinetic substituent effects on bromination of alkenes [36], ⌽R␣C⫽CR␤R⬘␤, where ⌽ is a conjugatively ED group and R is alkyl (Bienvenue-Goetz and Dubois, 1981). Equation (25) derived from the data for bromination in methanol involves two interactive terms expressing the interaction between the two ␣-substituents and between ␤ and ␣ substituents, respectively. log k ⫽ ⫺7.7兺(␴p⫹)␣ ⫺ 13.7兺(␴⫹m)␤ ⫺ 7.0(␴p⫹)␾ (␴p⫹)␣ ⫺ 5.8兺(␴p⫹)␣ 兺(␴⫹m)␤ ⫹ 1.64 (25) This equation describes satisfactorily the rates of 30 alkenes covering a reactivity range of 8 log units. Whereas (22) does not imply any interaction terms, i.e. the ␳-values, ␳␣ and ␳␤, are constant throughout the whole reactivity range, a more detailed analysis shows slight variations in ␳ values for the hydration (Table 10). The different behaviour of these two electrophilic additions, both of which go through carbocations has been interpreted in terms of differences in transition state positions.

6

Structure–reactivity relationship in polyarylcarbocation systems

CONFORMATIONS OF CARBOCATIONS

As described above, although the pKR⫹ values for symmetrically trisubstituted – triarylmethanols [3(X ⫽ Y ⫽ Z)] give a linear Y–T correlation against a ␴ scale with an r value of 0.76 (Yukawa et al., 1966), clearly non-linear Y–T correlations are obtained for the pKR⫹ values for monosubstituted triphenylmethanols [3(X,H,H)] or any non-equivalently substituted series [3C⫹(X ⫽ Y ⫽ Z)]. In any polyarylmethyl carbocation the molecule adopts a

THE YUKAWA–TSUNO RELATIONSHIP

335

propeller conformation which results from a compromise between the need to overcome steric repulsion, and achieving a minimal twist angle in order to favour the delocalization of positive charge into the rings. For unsymmetrically substituted carbocations the twist angles of the various rings are likely to be unsymmetrical and the deviation from coplanarity of the rings related to the relative cation-stabilizing capabilities (mainly ␲-delocalizing powers) of the variable and fixed substituents. This kind of conformational dependence has been observed for many polyaryl systems. However, no explicit consideration has been given to its implications for the interpretation of correlations even in a qualitative manner. For this type of analysis, the trityl ion series [3C⫹] involves less than a sufficient number of substituents. Although the benzhydryl solvolysis series has sufficient substituent sets, the change in rotation appears too small to estimate the effect quantitatively. In practice, the solvolyses of ␣-trifluoromethyl-diarylmethyl tosylates [29C⫹(X,Y)] best illustrate this analysis. Conformation of ␣-CF3-diarylcarbocations [29C⫹(X,Y)] The optimized structures at the 6-31G* level, for typical carbocations [29C⫹(X,Y)] are shown in Fig. 21 (Fujio et al., 1999). In the parent carbocation [29C⫹(X ⫽ Y ⫽ H)], the two phenyls are rotated significantly from coplanarity with the sp2 carbocation centre; rotation angles are 23⬚ and 38⬚ (MP2/631G*). Essentially the same structure was obtained for the symmetrically di-pmethoxy-substituted ion [29C⫹(X ⫽ Y ⫽ MeO)]. On the other hand, the optimized structure of the mono-p-methoxy [29C⫹(X,p-MeO)] has the phenyl ring twisted by 53⬚ while the anisyl is coplanar with ␪MeO ⬵ 0⬚. This conformation is 3.3 kcal mol⫺1 more stable than the doubly twisted conformation; the opposite conformation with the phenyl coplanar and the anisyl twisted (␪H ⫽ 0⬚ and ␪MeO ⬎ 50⬚) is a further 9 kcal mol⫺1 less stable. The symmetrically disubstituted precursors [29(X ⫽ Y)] react through a transition state structurally close to the diaryl-carbocation [29C⫹(X ⫽ Y)] of a preferred propeller shape denoted here as the E-conformer where both aryls are equivalently twisted from coplanarity with the reaction centre (␪X ⫽ ␪Y ⬵ 30⬚). When the substituents X are more ED than Y, the cation takes a conformation (denoted as the P-conformer) where the variable X-phenyl is coplanar (␪X ⫽ 0); in this case, the substituent X should then exert its maximum resonance effect on the solvolysis rate. On the other hand, when the variable X substituents are less activating than the fixed Y-phenyl, the Y-phenyl maintains coplanarity while constraining the X-substituted aryl ring to be more twisted; we denote this structure of X-aryl as the T-conformation (␪Y ⫽ 0 and ␪X ⬎ 50⬚).

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Y. TSUNO AND M. FUJIO

Fig. 21 The structures of ␣-trifluoromethyl-␣,␣-diphenylmethyl cations, optimized at the RHF/6-31G* level. From top, unsubstituted, p,p⬘-dimethoxy, mono-p-methoxy and p-methoxy-3,5-dichloro cations. Reproduced with permission from Fujio et al. (1999). Copyright 1998 John Wiley & Sons.

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337

REACTIVITY–CONFORMATION RELATIONSHIP

The ␣-trifluoromethyl–diarylmethyl system Non-linear correlations of [29(X,Y)] for the respective Y series should arise as a result of conformational change. Not only for the parent compound [29(X ⫽ Y ⫽ H)] but also for the substituent range in any series where the X substituents are capable of being ␲-ED as strongly as the Y group, the preferred conformation should be similar to an equivalently twisted one (Econformation). In [29(X,p-MeO)], the conformation for the range of s-ED class X-substituents from p-OCH2CH2-m, p-MeO to p-MeS-m-Cl should be closely related to the geometry of the E-conformation, and the r and ␳ values should be identical to those for [29(X ⫽ Y)] which are characteristic of substituent effects associated with the E-conformation. However, the substrates whose variable X-substituents are capable of distinctly less ␲delocalization than the fixed aryl-substituent Y ⫽ p-MeO should have the T-conformation. Thus, the partial correlation for the s-ED class X substituents may be related to the effect of X-aryl substituents in the E-conformation, and that for the w-ED class substituents may be related to the effect of X-aryls in the T-conformer. The r value for the T conformation based on only w-ED and EW groups should be less reliable. On the other hand, the Y–T correlation for the E-conformation cannot be determined unambiguously, while this conformation evidently requires a separate correlation independent of those for the other conformers. The series [29(X, 3,5-Cl2)] where the X substituents involved are all in the P-conformation, gives a single linear Y–T correlation with a high r value of 1.6–1.7; the r value for the X-aryl substituent effect is responding fully to the resonance demand of the system, [29(X, 3,5-Cl2)] where the fixed aryl with Y ⫽ 3,5-Cl2 is twisted by ca. 60⬚. The assignment of conformations gives a qualitative interpretation of the split pattern of the schematic plot in Fig. 22, and the correlation parameters are summarized in Table 12, in relation to the assigned conformers. A typical plot shown in Fig. 23, is the Y–T correlation of the series [29(X,p-Me)] which comprises three separate linear plots related to the respective conformers. The substrates with X ⫽ p-Me and the same class substituents should have the geometry of the E-conformation and those with the s-ED class X-substituents should have the P-conformation, while substrates carrying more EW substituents (X) should have the T-conformation. The r value of the E-conformers with Y ⫽ p-Me should be close to that of the series with X ⫽ Y, and the r value for the P-conformation must be even higher. Any Y set should have essentially the same form of substituent effect correlation attributable to the three conformations. The Y ⫽ p-MeO set should also have its P-conformer correlation for more ␲-ED X-groups than p-MeO, and the set with Y ⫽ 3,5-Cl2 also should have its T-conformer correlation, though no data are yet at hand. Another important feature of what has been observed in this system is the

Y. TSUNO AND M. FUJIO

log(K/KH)

338

– with Fig. 22 Schematic substituent effect, the plot of log(k/k0) against sum of ␴ r ⫽ 1.19, illustrating the non-additivity behaviour in the solvolyses of 1,1-diaryl-2,2,2trifluoroethyl tosylates [29(X,Y)] in 80% aqueous EtOH (80E) at 25⬚C: The reference correlation (␳) line is defined based on the points (䊉) for the series X ⫽ Y of [29(X,Y)]. The points given by 䉭 are referred to Y ⫽ p-MeO, 䊐 to Y ⫽ p-PhO, and 䊊 to Y ⫽ p-Me; ␳-lines for s-ED substituents are also drawn for series Y ⫽ H, m-Cl and 3,5-Cl2, respectively. Data taken from Fujio et al. (1997a,b; 1999a).

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Table 12 Reactivity–conformation relationship in the ␣-CF3-diarylmethyl solvolysis.a System (Y) range of Xb Ar2C(CF3) OTs, Y ⫽ X X : p-MeO ⬃ m-Cl Y ⫽ p-MeO X ⬎ p-MeO-m-Cl X ⭐ p-Me Y ⫽ p-C6H5O X ⭓ p-MeS-m-Cl X ⭐ p-Me Y ⫽ p-Me X ⬎ p-MeS-m-Cl p-MeS-m-Cl ⬎ X ⬎ H X⭐H Y⫽H X ⬎ p-MeS-m-Cl X : 3,4-Me2 ⬃ p-Cl X ⬍ m-Cl Y ⫽ m-Cl X ⭓ p-Me X : 3,5-Me2 ⬃ p-Cl X ⫽ m-Cl Y ⫽ 3,5-Cl2 X ⭓ p-Cl X ⫽ 3,5-Cl2

Conformationc ␪X (␪Y)

␳

r

SD

n

E (␪X ⬵ ␪Y ) 23 ⬃ 38⬚

⫺4.15d

1.19

0.12

14

E 23 ⬃ 38⬚ T 53⬚ (8⬚)

⫺3.94 ⫺1.71

1.26 1.01

0.09 0.06

6e 7

E 23 ⬃ 38⬚ T 53⬚ (8⬚)

⫺4.01 ⫺2.04

1.26 (1.07)

0.10 0.02

7e 8

P 10⬚ (49⬚) E 23 ⬃ 38⬚ T 45⬚ (15⬚)

⫺6.26 ⫺3.66 ⫺2.91

1.40 1.21

0.10 0.08 0.05

5 6 4

P 8⬚ (53⬚) E 23 ⬃ 38⬚ T 45⬚ (16⬚)

⫺6.1 ⫺4.33

1.45 1.26

0.12 0.07

6 12

P 8⬚ (53⬚) ⫺6.19 P ⬃ E 15⬚ (45⬚) ⫺4.81 E 23 ⬃ 38⬚ ⫺4.2 (tangent)

1.57 1.41

0.19 0.05

7 6

P 6⬚ (59⬚) E 23 ⬃ 38⬚

1.69

0.18

13

⫺5.95 ⫺4.2 (tangent)

Taken from Fujio et al. (1997a,b, 1999). bPartial correlation, X ⭓ or X⭐; correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; substituents of which the ␴⫹ are within the range of X1 to X2. exc. X; excluding substituent X. cConformations of carbocations to be formed; E: two aryl rings are equivalently twisted. P: X-phenyl being coplanar while fixed Y-phenyl twisted. T: X-phenyl twisted while fixed Y-phenyl being coplanar. ␪X, (␪Y); torsion angles of two aryl-rings, see text. dCorrelation ␳ for single aryl substituents. eIncluding the log k/k0 values estimated for X ⫽ H and p-Me in the E-conformation, see text. a

characteristic change in ␳ value for different classes of X substituents within an individual Y set and also for the same class of X substituents between Y series. Clearly the ␳ values are also related to the conformation of the intermediate carbocation, i.e. to the angular deviation of the X-substituted aryl from coplanarity in the carbocation; the s-ED class X-substituents in any series, regardless of the Y substituents, gave the same high ␳ value, ⫺6.1, which is characteristic of an extremely electron-deficient carbocation system; this ␳ value should be related to the effects of X in the coplanar aryl ring in the P-conformer. On the other hand, equivalent class X-substituents for fixed Y

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␳ ⫽ -6.26 r ⫽ 1.40

␳ ⫽ -3.65 r ⫽ 1.21

␳ ⫽ -2.91

Fig. 23 The Y–T plot for solvolysis of 1-aryl-1-(p-methylphenyl)-2,2,2-trifluoroethyl tosylates [29] in 80% aqueous EtOH at 25⬚C. For interpretation of symbols, see Fig. 1. Reproduced with permission from Fujio et al. (1999). Copyright 1999 John Wiley & Sons.

THE YUKAWA–TSUNO RELATIONSHIP

341

in any Y series are found to give a constant ␳ value of ⫺4.2, which is related to the X-phenyl ring twisted by ␪ ⬵ 30⬚ in the E-conformer. The same ␳ value was obtained for the symmetrical series [29(X ⫽ Y)]. Whereas the variation of ␳ value in solvolyses is often ascribed to the shift of the transition state coordinate, this coordinate shift does not appear to be important for the whole range of substituent change, e.g. from bis-(pmethoxyphenyl) to bis-(m-chlorophenyl) in the series [29(X ⫽ Y)]. The conformation of the intermediate carbocation and the angles of rotation of the two aryl rings from coplanarity must be related mainly to the relative resonance capabilities of both X- and Y-aryl substituents. In the parent carbocation [29C⫹(X ⫽ Y ⫽ H)], the two identical phenyl groups are twisted equally out of the plane of the central C⫹ (see above). This symmetry will be destroyed by introducing into one phenyl an ED Y substituent, causing the Y-substituted phenyl ring to become more coplanar with the sp2 cationic carbon and causing the ␲-delocalization to be enhanced. This will reduce the electronic demand on the other X-phenyl ring which in turn will be forced further out of the plane. The benzhydryl system The conformational assignments may be applied in the same way to the pKR⫹ values of benzhydrols, but unfortunately no quantitative information was obtained because of limited substituent data in the respective Y sets. The substituent effects on the solvolyses of benzhydryl chlorides (Table 13) are treated even more precisely with equation (2) than those on the pKR⫹ values and they can be interpreted in the same way based on the conformational assignment. Because of a smaller rotation of 16⬚ in the propeller conformation (E-conformation), the conformation dependence of the substituent effect correlations is not very serious in this system. Nevertheless, the non-linear behaviour should be similar to that in the trityl carbocation system. From comparison of the plots in Figs 8 and 15, the plot for Y ⫽ p-MeO for the benzhydryl cation should be related to the correlation for the T-conformation and that for Y ⫽ p-NO2 should be related to the P-conformer correlation. The difference in the slopes gives no clue as to the intrinsic selectivity (␳) of this system. Ruasse (1993) pointed out that substituent effects (Mindl and Vecera, 1971, 1972; Mindl, 1972) on the equilibria for forming benzhydryl cations, by analogy with those obtained in the solvolyses of chlorides (Nishida, 1967), can be analysed in terms of the selectivity–reactivity relationship (3). The values of the selectivity coefficient S ⫽ 0.34 (thermodynamic) and S ⫽ 0.52 (for the solvolysis), show elegantly that the transition-state shift is significant in this process.

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Table 13 Conformational dependence of substituent effects in benzhydryl solvolyses.a Systemb X⫽Y all substituents Y ⫽ p-MeO all substituents p-Me ⬎ X ⬎ m-CN Y ⫽ p-Me all substituents X: s-␲-ED ⬎ p-MeO-m-Cl X: m,p-alkyl ⬃ p-Hal. X: m-Cl ⬃ p-NO2 Y⫽H all substituents X: s-␲-ED X: m,p-alkyl ⬃ m-Hal. X: m-Cl ⬃ p-NO2 Y ⫽ p-Cl all substituents X: s-␲-ED X: m,p-alkyl ⬃ p-Hal. X: m-Cl ⬃ p-NO2 Y ⫽ m-Cl all substituents X: s-␲-ED X⬎H X ⬍ p-MeO-m-CN Y ⫽ p-NO2 all substituents exc. p-NO2

Conformerc E T P E T P E T P E T P P P⬃E P

␳

r

⫺4.368d 1.02 ⫺2.67 1.11 ⫺2.93 0.99 ⫺3.71 1.16 ⫺5.2e –f ⫺4.11 1.10 ⫺2.61 ⫺4.763 1.11 ⫺5.29 (1.28)f ⫺4.459 1.12 ⫺3.25 ⫺4.17 1.34 ⫺6.1e –f ⫺4.41 1.11 ⫺3.3 ⫺5.16 1.19 ⫺5.60 (1.38)f ⫺5.43 1.18 ⫺4.81 1.15 ⫺5.10 1.38 ⫺5.682 1.282

SD

n

0.04 0.09 0.03 0.23

10 13 7 17 3 8 6 22 8 14 3 12 3 7 2 19 8 15 11 14 13

0.05 0.130 0.13 0.048 0.22 0.07 0.147 0.12 0.120 0.037 0.195 0.152

Solvolyses of benzhydryl chlorides in 85A at 0⬚C (cf. Table 4); Fujio et al., unpublished results. Partial correlation, X⭓ or X⭐; correlation for the range of substituents of which the ␴⫹ values are more negative or less negative than given substituents. X: X1 ⬃ X2; substituents of which the ␴⫹ are within the range of ␴⫹ values for substituents X1 to X2. exc. X; excluding substituent X. c Conformation estimated for the carbocation intermediate, cf. footnote c in Table 12. dCorrelation (␳) for single aryl substituents. e␳⫹ value, assuming r ⫽ 1.0. fIndefinite value because of essentially constant ⌬–␴R⫹ values of all substituents included. a

b

The ␣,␣-diarylethyl system The twist angles of the aryl groups from coplanarity in the propeller conformation of ␣,␣-diarylethyl cation [31C⫹] are comparable with those of [29C⫹(X,Y)]. The dependence of the selectivity parameters upon conformation therefore should be as significant as in the above [29C⫹(X,Y)] system. The substituent effects on the protonation equilibrium of diarylethylenes [31(X,Y)] can be interpreted by assigning a preferred conformation to the cations. It also seems evident in this system that the ␳ value for the P-conformers does not vary so significantly with a series of Y substituents. Any processes involving ␣,␣-diarylethyl cation [31C⫹] as an intermediate should reflect the dependence of selectivity upon conformations. Small ␳ values for the solvolyses of 1,1-diphenylethyl-OPNB [13(X,Y)] cannot be interpreted by the ‘‘tool of increasing electron demand’’.

THE YUKAWA–TSUNO RELATIONSHIP

343

Table 14 Calculated ␳ values for the variation of substituent X when substituent Y is held constant in the bromination of diphenylethylenes. Substituent (Y) p-MeO p-Me m-Me m-MeO p-Br m-Cl m-Br m-NO2 H

␳

R

␴⫹ (Y)

⫺2.27 ⫺3.03 ⫺3.42 ⫺3.69 ⫺3.67 ⫺4.08 ⫺4.05 ⫺4.65 ⫺3.57

0.999 0.996 0.999 0.999 0.999 0.999 0.994 0.999 0.999

⫺0.78 ⫺0.311 ⫺0.066 ⫺0.048 0.150 0.399 0.405 0.674 0

Substituents X used H, p-NO2, m-NO2, m-Br, m-Cl, m-Me H, m-NO2, m-Br, m-Me p-MeO, p-Me, H, m-Me, m-F m-MeO, m-Br, H p-MeO, p-Me, H p-Me, m-Cl, H p-MeO, p-Me, m-MeO p-MeO, p-Me, H p-MeO, p-Me, m-Me, H, p-F, p-Cl, p-Br, m-F, m-Cl, m-NO2

Data taken from Hegarty et al. (1972).

The correlation results for the bromination of diarylethylenes [31(X,Y)] summarized in Table 14 also involve the same serious problem. The ␳ value increases significantly as the fixed substituent Y becomes more EW. This behaviour is indeed what is expected for the quantitative reactivity–selectivity relationship. However, in Table 14, the range of variable substituents X involved in the correlation of the respective Y sets is evidently different from set to set. The correlation for the Y ⫽ p-MeO set giving ␳ ⫽ ⫺2.3 should be referred to as the correlation for the T-conformation where X is more EW than Y, correlations for Y ⫽ p-Me, H and p-Br sets giving ␳ ⫽ ⫺3.6 may be referred to the E-conformation, and those for Y ⫽ m-Hal, especially p-NO2, refer without doubts to the P-conformation. The variation of ␳ value cited in Table 14 demonstrates nothing other than the dependence of the selectivity ␳ upon the propeller conformation of the diaryl carbocations. While there is no doubt regarding the importance of RSR in the mechanistic studies, these results lead to the conclusion that the RSR, or most of the non-additivity behaviour of ␣,␣-diarylcarbocation systems which have been cited as best examples of quantitative RSR, may indeed be only an artifact.

7

Stabilities of carbocations in the gas phase

STRUCTURAL EFFECTS

The mechanistic involvement of the solvent is quite often an important cause of misunderstanding substituent effects in benzylic solvolyses. An effective approach to overcome this difficulty and a contribution to a general theory of substituent effects in benzylic system, which can be directly compared with theoretical results, is to investigate the behaviour of carbocations in the gas

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phase, a medium free of solvent participation and other complicating factors. The intrinsic stabilities of benzylic carbocations have now become available from ion cyclotron resonance mass spectroscopic determinations in the gas phase (Mishima et al., 1989c,d). The relative stabilities of carbocations can be estimated from the free energy changes of the ion–molecule proton transfer equilibria of the corresponding alkenes (26).

(R ⫽ CH3, Et, t-Bu, H, CF3)

(26)

The substituent effect on the stability of ␣-cumyl cations [2C⫹] based on the proton transfer equilibria of ␣-methylstyrenes [33] (Mishima et al., 1989d) can be correlated directly with the ordinary set of solution ␴⫹ values (Fig. 24). Unexpectedly, the correlation covering the substituent range from p-NMe2 to 3,5-(CF3)2 is excellent, and there is no difficulty in defining the gas-phase ␴⫹ scale for r ⫽ 1.00 by these gas-phase stabilities of substituted [2C⫹] cations (Mashima et al., 1989c). The relative basicities ⌬⌬G(CC)H⫹ of ring-substituted styrenes [32] to give 1-arylethyl cations [14C⫹] (Mishima et al., 1989d), and of ␣-CF3-styrenes [35] to give ␣-CF3-␣-arylethyl cations [5C⫹] were determined by the same procedure (Mishima et al., 1989a, 1996b). For these ⌬⌬G(CC)H⫹ sets giving benzylic cations, the Y–T relationship is applicable for analysing the substituent effects on gas-phase carbocation stabilities in the same manner as in the solution phase. –⫹ ) ⫺␦⌬G⬚ ⫽ ␳G(␴o(G) ⫹ rG ⌬␴ R(G)

(27)

–⫹ In (27), ␴o(G) is a normal substituent constant and ⌬␴ R(G) is a resonance ⫹ o substituent constant defined as (␴ ⫺ ␴ )G; rG is a resonance demand parameter, and ␳G is a susceptibility parameter of the system in the gas phase. The gas-phase substituent parameter values are given in Table 15. The results of analysis based on the gas-phase Y–T equation (27) using gas-phase substituent constants given in Table 15 are summarized for proton-transfer equilibria of ethylenes and acetylenes in Table 16. As an example, the Y–T plot for ␣-CF3-␣-arylethyl cations [5C⫹] in Fig. 25 may be compared with the plot for the corresponding solvolysis in Fig. 3. The ⌬⌬G(CC)H⫹ for the ␣-neopentyl-␣-t-butyl series [4C⫹] has been found to correlate linearly with the gas-phase Y–T equation (27) with rG ⫽ 0.81 (Fujio et al., 1997a). ␣-t-Butyl-␣-methylbenzyl cations [18C⫹] also showed a reduced rG value of 0.89 (Mishima et al., 1992b), a value which is close to r ⫽ 0.91

345

log (K/K0)

THE YUKAWA–TSUNO RELATIONSHIP

Fig. 24 The plots of gas-phase stabilities of substituted ␣-cumyl cations [2C⫹] against Brown’s ␴⫹ in solution. Reproduced with permission from Tsuno and Fujio (1996). Copyright 1996 The Royal Society of Chemistry.

derived from the solvolysis; the o-methyl derivative series of [18C⫹] also showed a good correlation with a more significantly reduced rG value of 0.70 (Nakata et al., 1999). The ⌬⌬G(CC)H⫹ values of ␣-trimethylsilylstyrenes, Ar(Me3Si)C⫽CH2, are found to be close to the values of ␣-methylstyrenes [33], and the Y–T correlations appear to be identical with each other (Mishima et al., 1992b).

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Table 15 Substituent constants used for the Yukawa–Tsuno analysis of gas-phase substituent effects. Substituent p-NMe2 p-NH2 p-OCH2CH2-m p-OMe p-OMe-m-Cl p-OMe-m-F p-OMe-m-CN p-SMe p-SMe-m-Cl p-SMe-m-CN p-OH p-t-Bu 3,4-Me2 p-Me 3,5-Me2 m-Me

␴o(G)

–⫹ ⌬␴ R(G)

⫺0.43 ⫺0.19 ⫺0.19 ⫺0.10 0.22 0.22 0.47 0.04 0.25 0.60 ⫺0.05 ⫺0.27 ⫺0.24 ⫺0.13 ⫺0.28 ⫺0.12

⫺1.30 ⫺1.00 ⫺0.75 ⫺0.70 ⫺0.72 ⫺0.72 ⫺0.73 ⫺0.73 ⫺0.73 ⫺0.73 ⫺0.50 ⫺0.17 ⫺0.21 ⫺0.20 0.00 0.00

Substituent p-F p-Cl m-F m-Cl 3,5-F2 p-COCH3 p-CO2Me p-CHO m-CF3 p-CF3 m-CN p-CN m-NO2 p-NO2 3,5-(CF3)2 H

␴o(G)

–⫹ ⌬␴ R(G)

0.20 0.20 0.39 0.36 0.65 0.17 0.14 0.43 0.50 0.56 0.69 0.73 0.73 0.80 0.98 0.00

⫺0.17 ⫺0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Taken from Mishima et al. (1995, 1996c).

Vinyl cations The relative stabilities of vinyl cations also can be given by the gas-phase basicity ⌬⌬G(CC)H⫹ of alkynes, namely as the standard free energy change for the protonation reaction (28).

(28) Application of the gas-phase Y–T equation (27) to the stabilities of 1-arylvinyl cations [28C⫹] gave a linear correlation with high precision covering the whole range of substituents up to p-amino with an rG of 1.18 and a ␳G of ⫺10.2 as shown in Fig. 26 (Mishima et al., 1992b, 1996a). The ␳G value is similar to those for the equilibrium formation of the ␣-cumyl [2C⫹] and ␣-phenylethyl cations [14C⫹], and the rG value is identical to that evaluated from the solvolysis data (Kobayashi et al., 1993; Matsumoto et al., 1998). The basicities of 1-aryl-3,3-dimethylbutynes gave a ␳G value of ⫺9.42 and an rG value of 1.03 reflecting the stabilities of the ␤-t-butylvinyl cations (Matsumoto, 1997; Matsumoto et al., 1998). The standard free energies of the equilibria (28) of m- and p-substituted 1-phenyl-3,3,3-trifluoropropynes gave a large rG value of 1.38 and a ␳G value of ⫺9.93, with a significant stability reduction of 7.2 kcal mol⫺1 by the ␤-CF3 group (Matsumoto et al., 1995). Although the ␳

Table 16 Yukawa–Tsuno correlations for gas-phase basicities of arylalkenes to form benzylic cations. Arylalkenes

␣-CF3-styrene Styrene ␣-Me-styrene 2-Ph-2-butene 3-Ph-2-pentere ␣-t-Bu-styrene ␣-t-Bu-styrene(o-Me) ␣,␤-(t-Bu)2-styrene ␣-(Me3Si)-styrene Phenylacetylene 1-phenylpropyne ␤-t-Bu-phenylacetylene ␤-CF3-phenylacetylene ␣-CF3-benzyl-Cl Benzyl-Cl ␣-phenylethyl-Cl ␤-phenylethyl-Br ␤-phenylpropyl-Br Styrene ⫹ Me3Si⫹

Benzylic cation R1

R2

CF3 Me Me Me Et Me Me Np Me H2C⫽C⫹ H(Me)C⫽C⫹ (t-Bu)CH⫽C⫹ (CF3)CH⫽C⫹ CF3 H Me phenonium⫹ Me-phenonium⫹ ⫹ SiMe3

Me H Me Et Et t-Bu t-Bu t-Bu Me3Si

䉭

H H H

Stability ⌬G(CC) or ⌬G(C⫹)a

␳Gb

rG

R

SD

n

Reference

184.4 193.9 199.1 199.5 199.8 198.0 197.2 196.8 198.4 191.6 193.0 196.8 184.4 182.6 186.6 193.9

10.0 10.1 9.5 9.4 9.5 9.1 6.64 6.7 8.9 10.2 10.1 9.42 9.93 10.6 10.3 10.0 12.6 11.9 5.76

1.41 1.14 1.00 1.00 0.98 0.89 0.70 0.82 (1.0) 1.18 1.12 1.03 1.38 1.53 1.29 1.14 0.62 0.61 0.41

0.998 0.9994 0.9996 0.9997 0.999 0.9986 0.997 0.997 (0.996) 0.999 0.9997 0.999 0.998 0.999 0.999

0.4 0.29 0.28 0.15 0.27 0.32 0.25 0.29 0.3 0.18 0.36 0.50 0.38 0.35

18 27 20 8 9 12 8 9 5 16 10 9 8 17 25

0.998 0.995 0.997

0.34 0.70 0.10

13 15 8

1 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 15 17

a Relative stability of the respective parent carbocations estimated from proton-transfer or halide-transfer equilibria, in kcal mol⫺1. b␳G values in log unit of log(K/K0) for the gas-phase ionization are obtained by multiplying the ␳ values of the gas-phase stabilities ⌬⌬G(CC) by a factor of 1000/2.303RT.

References: 1, Mishima et al. (1989a, 1996b). 2, Mishima et al. (1989c). 3, Mishima et al. (1989d). 4, Mishima et al. (1994). 5, Mishima et al., unpublished. 6, Mishima et al. (1992e). 7, Nakata et al. (1999). 8, Mishima et al. (1992b). 9, Mishima et al. (1992e, 1996a). 10, Kobayashi et al. (1993). 11, Matsumoto (1997); Matsumoto et al. (1998). 12, Matsumoto et al. (1995). 13, Mishima et al. (1990a, 1997). 14, Mishima et al. (1987, 1995). 15, Mishima et al., unpublished. 16, Mishima et al. (1990b); Mustanir et al. (1998). 17, Mishima et al. (1992a).

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Fig. 25 The Y–T plot of gas phase stabilities of substituted ␣-methyl-␣-trifluoromethylbenzyl cations [5C⫹]; r ⫽ 1.39. For interpretation of symbols, see Fig. 1. Data taken from Mishima et al. (1996b).

THE YUKAWA–TSUNO RELATIONSHIP

349

Fig. 26 The Y–T plot of gas phase stabilities of m- and p-substituted ␣-phenylvinyl cations [28C⫹]; r ⫽ 1.18. For interpretation of symbols, see Fig. 1. Reproduced with permission from Mishima et al. (1996b). Copyright 1996 Chemical Society of Japan.

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value is comparable to those for related vinylic carbocations and benzylic carbocations, the rG value of 1.38 is remarkably larger than that of the 1-phenylvinyl cation, indicating an enhanced ␲-interaction between the positive charge and ␲-ED ring substituents engendered by the ␤-CF3 substituent. The stabilities of carbocations for which the appropriate alkene precursors are not available were determined from the gas-phase halide-transfer equilibria (29) of the corresponding benzylic halides (Mishima et al., 1995).

(29) When comparisons are possible, a ⌬⌬G(Cl) ladder for the chloride-transfer equilibria (29) of benzylic halides can be superimposed upon the corresponding ⌬⌬G(CC)H⫹ ladder for proton transfer (26). Thus a wide set of relative gas-phase stabilities of carbocations can be built up based on the same scale. The relative stabilities of substituted benzyl cations [21C⫹] are correlated by equation (27) with a high resonance demand parameter rG ⫽ 1.29 (Mishima et al., 1987, 1995). The linear correlation for the whole range of substituents down to the 3,5-(CF3)2 group (Fig. 27), contrasts with the concave Y–T plot (Fig. 7) of the solvolytic reactivities of [21]. Note that the rG value for the gas-phase stabilities of [21C⫹] is identical with the r value assigned for the SN1 solvolysis of [21] tosylates; hence, the r value of 1.29 must be an intrinsic index inherent in [21C⫹], rather than a correlational artifact of a non-linear relationship for the complex solvolysis mechanism. The series of ␣-CF3-benzyl cations [40C⫹] derived from the chloridetransfer equilibrium (29) gives an excellent linear correlation for the full range of substituents down to 3,5-F2 (Fig. 28) (Mishima et al., 1990a, 1997).

␣-Hydroxy cations Significantly low r values have been observed in the protonation equilibria (pKBH⫹ values) of benzoyl compounds (Mishima et al., 1988, 1990c, 1996c). The trend in the r values mentioned above predicts that a stabilized carbocation will not require a large ␲-delocalization of the positive charge. Substituent effects on the gas-phase basicities (⌬G(CO)H⫹) of the aromatic carbonyl compounds [30(R)], ArCOR (30) have been studied.

(30)

THE YUKAWA–TSUNO RELATIONSHIP

351

Fig. 27 The Y–T plot of gas-phase stabilities of substituted benzyl cations [21C⫹]; r ⫽ 1.29. For interpretation of symbols, see Fig. 1. Reproduced with permission from Mishima et al. (1995). Copyright 1995 Chemical Society of Japan.

Here the positive charge in the conjugate acid ions [30C⫹], ArC⫹(R)OH, is stabilized by the ED hydroxyl group linked to the benzylic carbon, and the stability of the ion is regulated by the electronic effect of the R group (Mishima et al., 1986a,b). The application of (27) to these substituent effects using the gas-phase substituent constants listed in Table 15 provides excellent correlations. Figure 29 demonstrates the linear Y–T correlation for ␣-hydroxy-␣-phenylethyl

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Y. TSUNO AND M. FUJIO

Fig. 28 The Y–T plot of gas-phase stabilities of substituted ␣-trifluoromethylbenzyl cations [40C⫹]; r ⫽ 1.53. For interpretation of symbols, see Fig. 1. Data taken from Mishima et al. (1997).

cations covering the substituent range from p-NMe2 to 3,5-(CF3)2; note that some basic substituents do not give correct pKBH⫹ values in strong acid solution reflecting protonation on the substituent. For the system with R ⫽ CF3 (27) gave an rG of 1.20, a value higher than that (0.82) for the R ⫽ CH3 system (Mishima et al., 1990c, 1996c).

THE YUKAWA–TSUNO RELATIONSHIP

353

Table 17 Yukawa–Tsuno correlations for gas-phase stability of benzylic cations and for the solvolysis of the corresponding substrates. R1

CF3 CF3 H Me Me Me i-Pr Me Me Np i-Pr H2C⫽C⫹ H(Me)C⫽C⫹ Me2C⫽C⫹ (t-Bu)CH⫽C⫹ (CF3)CH⫽C⫹ phenonium⫹

R2

Me H H H Me Et i-Pr t-Bu t-Bu (o-Me) t-Bu t-Bu

Gas-phase stability

Solvolysis reactivity

⌬G(CC)a

⫺ ␳G b

rG

⫺ ␳c

r

184.4 182.6 186.6 193.9 199.1 199.5

10.0 10.6 10.3 10.1 9.5 9.4

1.41 1.53 1.29 1.14 1.00 1.00

6.29d 6.05d 5.23 5.45 4.93 5.05 4.88d 4.70 3.39 4.11 3.79 5.30d

1.39 1.53 1.29 1.15 1.00 1.04 1.01 0.91 0.70 0.78 0.66 1.16

4.91d 6.0e

1.07 1.02

5.04f

0.63

198.0 197.2 196.8

9.1 6.64 6.7

0.89 0.70 0.82

191.6 193.0

10.2 10.1

1.18 1.12

196.8 184.4

9.42 9.93 12.6

1.03 1.38 0.62

Relative stability of the respective parent carbocations estimated from proton-transfer or halide-transfer equilibria, in kcal mol⫺1. b␳G values in log unit of log(K/K0) for the gas-phase ionization are obtained by multiplying the ␳ values of the gas-phase stability ⌬⌬G(CC) by a factor of 1000/2.303RT. cThe Y–T ␳ values corrected to those at 25⬚C in 80% aqueous acetone, otherwise noted. d␳ value in 80% aqueous ethanol at 25⬚C. e␳ value in 30E. f␳ value in AcOH. a

An rG value of 0.50 for the 1-hydroxy-1-methoxybenzyl cation is reasonably attributed to the large stabilization of the positive charge by the hydroxyl and methoxy groups linked directly to the positive carbon centre. Thus, the rG value seems to decrease in the increasing order of ED ability of the R group(s). This trend is qualitatively consistent with that observed for the ordinary benzylic carbocation system [1C⫹], PhC⫹(R1)R2 where R1 and/or R2 ⫽ H, Me, CF3. The gas-phase basicities towards Me3Si⫹ cation can be determined for carbonyl bases by the ICR method. The substituent effect on the Me 3Si⫹ basicity of acetophenone can be described by (27) with r ⫽ 0.70, a value slightly lower than that for the H⫹ basicity (Mishima et al., 1992c). Determination of the gas-phase basicity of styrene bases towards Me3Si⫹ yields an interesting result. Application of (27) to the substituent effects provides an r value of 0.41 and ␳ value of ⫺5.76, excluding the p-MeO and p-MeS substituents which are attacked by Me3Si⫹ in preference to attack on the C⫽C double bond (Mishima et al., 1992a). Although the r value may be less reliable since s-ED substituents are not involved in the correlation, it is clear

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Y. TSUNO AND M. FUJIO

Fig. 29 The Y–T plot of gas-phase stabilities of substituted ␣-hydroxy-␣-methylbenzyl cations [30C⫹]; r ⫽ 0.82. For interpretation of symbols, see Fig. 1. Reproduced with permission from Mishima et al. (1996c). Copyright 1996 Chemical Society of Japan.

THE YUKAWA–TSUNO RELATIONSHIP

355

that both r and ␳ values are distinctly lower than those for H⫹ basicity. This correlation suggests that a conceivable structure of the Me3Si-adduct ion is a partially bridged siliconium ion [17Si⫹] rather than the classical open structure of the ␤-Me3Si-␣-phenylethyl cation [17C⫹]. Phenonium ions The gas-phase stability of the ethylenephenonium ion [25C⫹] and its ring-substituted derivatives has also been determined by the bromide-transfer equilibria (31) of ␤-arylethyl bromides (Mishima et al., 1990c).

(31)

The reaction gave an excellent Y–T correlation with ␳G ⫽ ⫺12.6 and rG ⫽ 0.62 (Fig. 30). The linear correlation which covers the entire substituent range is in contrast to the concave plot (Fig. 10) observed for the solvolysis of [23]. Essentially the same result has been obtained for the ␣-methylphenonium ion (Mustanir et al., 1998; Mishima et al., unpublished). The magnitude of ␳G for the phenonium ions is significantly larger than that observed for benzylic carbocations, but is comparable to that obtained for the stabilities of benzenium ions (Mishima et al., 1989b). It should be particularly noted that the rG value of 0.62 is identical with the r⌬ value observed for the corresponding solvolysis which proceeds through the k⌬ mechanism via a phenonium ion intermediate.

THE RESONANCE DEMAND PARAMETER

The Y–T equation has been used to analyse the substituent effects on carbocation formation equilibria in the gas phase. These correlations are compared with those for the kinetic substituent effects in the corresponding solution phase solvolyses in Table 17 and substituent effects on thermodynamic basicities of carbonyl groups in both phases are compared in Tables 18 and 19. At first glance, it is very surprising that the rG values for the gas-phase stabilities of the cations are, without exception, identical in magnitude with the r values of the corresponding benzylic SN1 solvolyses (cf. Table 17). Furthermore, in comparing the effects of ring substituents on the basicities of benzoyl compounds [30(R)], Ar-COR, in solution and in the gas phase (Table

356

Y. TSUNO AND M. FUJIO

Fig. 30 The Y–T plot of the stabilities of gas phase phenonium ions; r ⫽ 0.62. For interpretation of symbols, see Fig. 1. The graph is redrawn from the data of Mishima et al. (1990a).

19), the r value is also unchanged regardless of a large external stabilization of the positive charge through solvation, while the ␳ value is sensitive to solvent stabilization (Mishima et al., 1988, 1990c, 1996c). The same r value for the thermodynamic carbonyl basicities in both phases is understandable

Table 18 Yukawa–Tsuno correlations for gas-phase basicities of carbonyl compounds to form ␣-hydroxybenzylic cations. Ar-CO-Y Ar-CO-CF3 Ar-COOH Ar-COMe Ar-COOMe Ar-CONMe2 Ar-COMe ⫹ Me3Si⫹

R1

R2

⌬G(CO)a

⫺␳Gb

rG

R

SD

n

Reference

CF3 H Me MeO OH Me

OH OH OH OH NMe2 OSiMe3

184.4 192.1 197.3 195.5 213.8

8.39 8.47 8.39 8.69 8.39 8.02

1.20 1.06 0.82 0.50 0.23 0.70

0.996 0.999 0.999 0.998 0.996 0.998

0.52 0.30 0.31 0.26 0.4 0.3

16 20 24 19 15 14

1 2 2, 3 2, 4 5 6

a Relative stability of the respective parent carbocations estimated from proton-transfer equilibria, in kcal mol⫺1. b␳G values in log unit of log(K/K0) for the gas-phase carbonyl basicity are obtained by multiplying the ␳ values of the gas-phase basicities ⌬⌬G(CO) by a factor of 1000/2.303RT.

References: 1, Mishima et al. (1990c). 2, Mishima et al. (1996c). 3, Mishima et al. (1986a). 4, Mishima et al. (1986b). 5, Mishima et al. (1988). 6, Mishima et al. (1992c).

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Y. TSUNO AND M. FUJIO

Table 19 Yukawa–Tsuno correlations for the stability of benzylic cations, from the carbonyl basicities in the gas and solution phases. R1

CF3 H Me MeO OH OH OH

R2

OH OH OH OH OH NMe2 NH2

Gas-phase basicity

Solution-phase basicity

⌬G(CO)a

⫺ ␳G b

rG

⫺␳

r

184.4 192.1 197.3 195.5

8.39 8.47 8.39 8.69

1.20 1.06 0.82 0.50

2.00 1.994 2.156

1.2 1.163 0.762

213.8

8.39

0.23

1.144

0.537

1.20

0.39

Relative stability of the respective parent carbocations estimated from proton-transfer equilibria of the corresponding carbonyl bases, in kcal mol⫺1. b␳G values in log unit of log(K/K0) for the gas-phase carbonyl basicity are obtained by multiplying the ␳ values of the gas-phase basicities ⌬⌬G(CO) by a factor of 1000/2.303RT. a

insofar as the r value, the intramolecular selectivity parameter of resonance, is insensitive to external stabilization by solvation. The r value should be a measure of the positive charge stabilization through the ␲-delocalization interaction with the aryl ␲-system in the incipient carbocation or the carbocationic transition state and will vary significantly with the ␣-R-substituents. Mishima et al. (1990c) found that the r values for all these systems [1C⫹], ArC⫹R1R2, can be correlated using (32) in terms of both – ⫹ and inductive polar ␴ o substituent constants of R1 and R2, resonance ⌬␴ R p – ⫹ ⫹ 1.28 rG ⫽ 0.45 兺 ␴po ⫹ 0.40 兺 ⌬␴ R

(32)

– ⫹ ⫽ ⌬␴ – ⫹(R1) ⫹ ⌬␴ – ⫹(R2). As a general where 兺 ␴po ⫽ ␴po(R1) ⫹ ␴po(R2) and 兺⌬␴ R R R trend, the system possessing the ␣-substituent R with the greater ED ability appears to have a smaller r value; thus the positive charge formed at the benzylic position must be stabilized in a complementary manner by ␲delocalization into the aryl ␲-system and into ␣-substituents. The r value reflects the share of the aryl group in the overall charge delocalization and should be inherent for the structure of the carbocation, depending only upon the ␣-substituents R1, R2. The identity of r values for the gas-phase carbocation stabilities with those for the corresponding solvolyses provides important information regarding the general nature of the transition state as well as the intermediate of solvolytic process. In benzylic solvolyses forming carbocations, the r value for the transition state of ionization, rk, is identical with the corresponding rG for the gas phase carbocation formation. It is reasonable to assume that the thermodynamic r

THE YUKAWA–TSUNO RELATIONSHIP

359

value for the carbocation intermediate rt is also identical with the intrinsic rG of the gas phase carbocation, even though no rt values are available for most solvolyses. Consequently, the resonance demand of solvolytic ionization process remains the same both for the intermediate and for the cationic transition state, regardless of the position along the reaction coordinate or the amount of cationic charge accumulated at the transition state. These important perturbations at the transition state will be reflected almost exclusively in the ␳ value, implying precise applicability of the extended Brønsted relationship to the solvolytic ionization processes. On the other hand, although the rG values for most cations (cf. Tables 16 and 17) were obtained from alkene basicities ⌬G(CC)H⫹ which correspond to the kinetic protonation of the C⫽C double bond in the solution phase, those rG values are nevertheless identical with the r values for the corresponding solvolyses rather than with those for the C⫽C protonations or acid-catalysed hydrations which are noticeably smaller than that obtained for the solvolytic processes. Accordingly, the discrepancy between the r values for hydration and the gas-phase basicities leads us to the conclusion that, in contrast to SN1 solvolysis, the structure of the transition state of acid-catalysed hydration is appreciably different from that of the corresponding stable cationoid intermediates with respect to ␲-delocalization of the positive charge. Steric loss of resonance interaction The decrease in the r parameter, as in the case of sterically congested systems, can be understood if we assume that the aryl group has not attained coplanarity with the sp2 reaction site. Rotation around the Ar⫺C bond which increases the overlap between the aryl ␲-orbital and the vacant sp2 orbital of the carbocation should be impeded by steric interactions between the aryl ring and the ␣-substituents (Fujio et al., 1994, 1997a). This interpretation of the conformation dependence of resonance demand in terms of the delocalization interaction is more understandable when it applies to the benzyl carbocations (intermediates) than to solvolysis transition states. The gas-phase stabilities of ␣-t-butyl-␣-methylbenzyl cation [18C⫹], its o-methyl derivative series and the ␣-neopentyl-␣-t-butyl series [4C⫹] have been found to give good correlations with significantly reduced r values (Mishima et al., 1992e; Nakata et al., 1999). Empirical r values for these sterically twisted benzylic cations in the gas phase appear to be identical to the values of the corresponding solvolysis transition states (Table 17). The structure of the carbocation should be a satisfactory model for the solvolytic transition state as well as the intermediate immediately following it. The theoretically calculated dihedral angles of optimized structures are, even though slightly dependent on the basis sets, in good agreement with the ‘‘experimental’’ torsion angles ␪expl of the solvolysis transition states as well as of the gas-phase carbocations obtained by using (4), as shown in Fig. 31

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Y. TSUNO AND M. FUJIO

t-Bu,t-Bu

t-Bu,t-Bu (p-MeO)

t-Bu,Me (o-Me) t-Bu,i-Pr t-Bu,Me t-Bu, Np

Fig. 31 The identity plot of the theoretical angles ␪calcd vs. experimental angles ␪expt of rotation of the aryl ring from the plane of ␣-carbocation. For the ␣,␣-di-tbutylcarbocation, the experimental ␪expt value was in a better accordance with the ␪calcd estimated for the p-MeO derivative. Data taken from Nakata et al. (1996).

(Nakata et al., 1996, 1999). The agreement between the theoretical dihedral angles ␪calc and the empirical ␪expl of the twisted benzylic cations confirms that the observed decrease in the r value should be ascribed to a loss of resonance interaction caused by deviation from coplanarity of the carbocation centre and the benzene ␲-system. Taking into account the angular dependence of the r values [see (4)], the Y–T equation can now be rewritten as (33), – ⫹] ⬘ [␴o ⫹ (r cos2 ␪)⌬␴ log(k/k0) ⫽ ␳G R

(33)

THE YUKAWA–TSUNO RELATIONSHIP

361

where r is the intrinsic resonance parameter of the carbocation in which the aryl group lies in the same plane as the sp2 reaction centre. This may be regarded as a conformation–reactivity (or –stability) relationship for carbocation reactions where the aryl coplanarity regulates the reactivity (or stability). It provides a useful tool for assessing the conformation of the transition state leading to the carbocation. Finally the generality of (33) lends theoretical support for our characterization of the parameter r in the Y–T equation (2) as the resonance demand parameter reflecting the degree of ␲-delocalization interaction between the aryl group and the reaction site. Stability–resonance demand relationship The rG value significantly increases as the stability of the parent carbocation decreases, while the ␳G value remains rather constant for a series of benzylic carbocations. The rG values are plotted against the relative alkene basicities ⌬G(CC)H⫹ corresponding to the unsubstituted (parent) carbocations in Fig. 32. It is remarkable that the resonance demand parameters characterizing the stabilities of this series of cations are linearly related to the intrinsic gas-phase stabilities of the parent carbocations. This indeed is the most significant result of the Y–T analyses summarized in Table 16 (or 17). For the different ␣-substituted benzyl cations the linear relationship between rG and alkene basicities ⌬G(CC)H⫹ is described by equation (34) rG ⫽ 0.0261⌬⌬G(CC)H⫹ ⫹ 1.00

(34)

where the ⌬⌬G(CC)H⫹ values are the alkene basicities of the parent carbocations relative to that of ␣-cumyl cation; the slope of 0.0261 appears only for the correlation of sp2-hybridized benzylic carbocation series. Both sp- and sp2-carbocation systems are included in a single linear relationship (34). The estimated rG values, based on the ⌬G(Cl) scale for the chloride-transfer equilibria, can also be included in relationship (34) with a minor calibration for the scale. In the carbocation forming equilibria (26), (28) and (29), the ⌬G values are essentially determined by the stabilities of the carbocations, i.e. the conjugate acid form, without important contributions from the stabilities of the neutral conjugate bases. The rG values for the carbocations in Table 16 should properly be related in the form of (34) with the intrinsic stability ⌬G(C⫹) rather than to the alkene-basicity scale ⌬G(CC)H⫹ of the parent carbocations (X ⫽ H) of the respective series. On the other hand, there is no simple linear relationship in the PhC⫹(OH)R system between the rG values and the carbonyl-basicity values ⌬G(CO)H⫹ of the unsubstituted derivatives (Table 18). It is likely that the effect of stabilities of the conjugate bases is more important in the carbonyl-basicity equilibria than in the C⫽C basicity equilibria. In fact, the rG value obtained from the basicity of acetophenones can be correlated using (34) with the ⌬G(CC)H⫹ value

Y. TSUNO AND M. FUJIO

Resonance demand (r)

362

Fig. 32 The resonance demand r vs. stability relationship for gas-phase cation stabilities. Reproduced with permission from Matsumoto et al. (1995). Copyright 1995 Societa` Chimica Italiana.

estimated for the enol form, C6H5C(OH)⫽CH2, but not with the carbonylbasicity ⌬G(CO)H⫹. 8

Theoretically optimized structures of carbocations

The identity of the empirical r value for the solvolysis transition state with that for the corresponding gas-phase cation indicates that the structures of the transition states in benzylic solvolyses should be essentially constant and that

THE YUKAWA–TSUNO RELATIONSHIP

363

the Leffler–Hammond rate–equilibrium relationship is strictly applicable in all the cases. Intuitively, this can be better understood on the basis of a simplified model of the delocalization interaction of overlapping orbitals when it applies to a benzyl carbocation intermediate rather than for the solvolysis transition state. Thus, the carbocation structure may be taken as a model for the solvolysis transition state as well as for the solvolysis intermediate. The characteristic change of the r value in the solvolysis reaction of benzylic precursors and for the corresponding carbocations should provide important information concerning the solvolysis transition state. The r value, reflecting the ␲-delocalization within the cationic species, appears to remain essentially the same in solution as in the gas phase, and the charge delocalization in the transition state of the solvolytic ionization should be close to that in the carbocation intermediate. Advanced ab initio molecular orbital calculations can be used to find the underlying relationship between quantum chemical quantities and experimental r values, and the relation between r values and theoretical indices provides a basis for the physical meaning of the r parameter (Nakata et al., 1996, 1998, 1999). The charge distribution at the respective atomic positions should be regarded as reflecting the demand of the ␣-cationic centre of benzylic carbocations for ␲-electron delocalization from those atomic positions and should be directly related to the r value. There is linearity between the charge populations at the para and/or the ortho position and at the C␣ position, while there is no significant change in the population at the meta position. Furthermore, slightly sophisticated quantities, the difference in the charge populations between para and meta positions, ⌬q⫹(para) ⫺ ⌬q⫹(meta), and the sum of charge populations at aromatic positions 兺 ⌬q⫹ are also mutually proportional, and all these quantities give linear correlations when plotted against r values. This trend of charge distribution in the benzylic cations is consistent with what was envisaged for the varying resonance demands. The differential charge population, ⌬q⫹(para) ⫺ ⌬q⫹(meta), can be considered as a parameter of intramolecular selectivity, measuring the classical resonance effect. The sum of charge populations at the aromatic positions 兺 ⌬q⫹ should also be an intramolecular selectivity parameter of the aryl group with respect to resonance or, more exactly, ␲-delocalization, and intuitively appears to be the most appropriate index of the resonance demand inherent in the benzylic carbocation. The good linearity between rG values and 兺 ⌬q⫹ is shown in Fig. 33. Theoretical structural indices other than charge populations, especially the bond lengths or bond orders of CAr⫺C␣ bonds, may also be a measure of reactivities or intermolecular selectivities of the benzylic cation series. The Wiberg bond order of the CAr⫺C␣ bond, PAr, ␣, is the simplest theoretical quantity measuring charge delocalization into the aryl ring and representing most closely the concept of resonance demand. The CAr⫺C␣ bond length decreases in a consistent way with increase in the bond order, in parallel with

364

Y. TSUNO AND M. FUJIO

t-Bu,H

i-Pr,i-Pr t-Bu,Np

t-Bu,Me

t-Bu,i-Pr

t-Bu,t-Bu

r value Fig. 33

Sum of Mulliken charge populations at the aromatic positions (RHF/6-31G*) vs. r values for benzylic cations. Data taken from Nakata et al. (1996).

the varying charge delocalization in the benzylic cation systems, which is in turn consistent with the varying degree of resonance stabilization. Indeed, the rG values for a series of benzylic cations can be linearly correlated to a good approximation with the bond orders of benzylic carbocations. This linearity implies that the r parameter should also be an intermolecular selectivity parameter, when it is compared for a series of benzylic cations. While there is no reason to assume that the resonance demand is precisely linear against the degree of overlap between the two orbitals, the behaviour of r is completely

THE YUKAWA–TSUNO RELATIONSHIP

365

consistent with what is envisaged for the resonance demand, indicating that any index for the varying resonance interaction between C␣ and the aromatic moiety changes in parallel with the change in the empirical r value. These theoretical considerations reveal that the empirical r values are intimately related to the theoretical indices of the structures of benzylic carbocations derived from resonance theory. The coefficient r in the Y–T equation can thus be replaced as a first approximation by a set of theoretical quantities, e.g. increment of bond orders (PAr, ␣)H as in (35) or sum of charge populations in the aryl ring (兺 ⌬q⫹)H, for the parent carbocations (X ⫽ H) as in (36). – ⫹] log(k/k0) ⫽ ␳[␴o ⫹ 2.6(PAr, ␣ ⫺ 0.96)H ⌬␴ R ⫹ o ⫹ – log(k/k ) ⫽ ␳[␴ ⫹ 3.0(兺 ⌬q ) ⌬␴ ] 0

H

R

(35) (36)

This provides theoretical support for the concept of varying resonance demand, which reflects the varying degree of ␲-overlap interaction between the benzylic p-orbital and the benzene ␲-system.

9

Reaction mechanisms and transition-state shifts

The currently accepted description of transition-state structural variations in terms of perturbations of free-energy surfaces leads to the expectation that changes in reactant structure which increase the rate of a reaction almost always make the transition state ‘‘earlier’’ in the sense of being more ‘‘reactant-like’’ (Hammond behaviour).

EXTENDED SELECTIVITY–STABILITY RELATIONSHIPS

The identity of r values for solvolysis reactivities and the gas-phase stabilities of the corresponding carbocations implies the generality of the extended Brønsted relationship or Hammond–Leffler rate–equilibrium relationship for benzylic solvolyses, i.e. (37a,b), ⭸ log(k/k0)k ⫽ ␣ ⭸ log[(KR⫹)/(KR⫹)o]e

(37a)

⭸ log(k/k0)k ⫽ ␣G ⭸ log[(KR⫹)/(KR⫹)o]G

(37b)

where logarithmic quantities on the left-hand sides are substituent effects on solvolysis rates and those on the right-hand side are substituent effects on equilibria in solution (37a) and on the stabilities of the corresponding carbocations in the gas phase (37b). Both kinetic and thermodynamic – scale with an r substituent effects in (37) are correlated by a common ␴

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Y. TSUNO AND M. FUJIO

parameter characteristic of a given parent benzylic carbocation. This leads to an important conclusion, that the resonance demand of the SN1 solvolytic transition state as well as the intervening intermediate should, in principle, be inherent in the intrinsic resonance demand of the parent carbocation.

GROUND-STATE ELECTROPHILIC REACTIVITY OF CARBOCATIONS

The generally observed identity of the r value for solvolysis reactivity and gas-phase stability ⌬⌬G(C⫹)H of the corresponding carbocation leads to an important prediction concerning the solvolysis transition state. In a typical (limiting) two-step SN1 mechanism with a single dominant transition state, the r values of transition states for the various nucleophile–cation reactions should be essentially controlled by the intrinsic resonance demand of the intermediate cation; the substituent effect should be described by a single scale of –) with an r value characteristic of this cation. In a substituent constants (␴ recent laser flash-photolysis study (Das, 1993) on the recombination of stable trityl and benzhydryl cations with nucleophiles and solvents, McClelland et al. (1986, 1989) have treated the substituent effects on solvent-recombination processes by (2). Triarylmethyl carbocations The rate constants for reactions of highly stable triphenylmethyl and diphenylmethyl cations with various ionic and neutral nucleophiles have been measured (Gandler, 1985; McClelland et al., 1986) in aqueous acetonitrile and discussed from the view point of a reactivity–selectivity relationship. kh

Ph3COH ⫹ H⫹ J Ph3C⫹ ⫹ H2O

(38a)

kw

KR⫹ ⫽

[R⫹] kh ⫹ ⫽ [ROH][H ] kw

(38b)

The rate constants kw for the solvent-recombination process of the carbocations [3C⫹(X,Y,Z)] were determined by the use of the azide clock method (Richard et al., 1984; Richard and Jencks, 1984a,b,c; McClelland et al., 1991) and the rate constant kh of the forward reaction was derived using (38b) as kh ⫽ kw KR⫹ (McClelland et al., 1989, 1991). While ordinary Hammett-type relationships were found to be inapplicable to the substituent effects on solvent recombination, there is a rate–equilibrium correlation for all available data on triarylmethyl cations, shown as the linear log kw vs. pKR⫹ plot, in Fig. 34 with a slope of 0.64. Such a relationship was earlier suggested by Arnett and Hofelich (1983) and Ritchie (1986). The correlation of kw with the ␴⫹ scale was

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Fig. 34 A rate–equilibrium relationship, log kw vs. pKR⫹, in the ionization of trityl system.

found to show a significant deviation from linearity for the s-␲-ED groups in the monosubstituted trityl carbocations, and McClelland et al. (1989, 1991) noted that the deviations were in the unusual direction of enhancing resonance interactions for better ␲ donors than those defined by the ␴⫹ values. The use of the Y–T equation did not provide much help. On the other hand, the rate constant for the forward reaction kh displayed a much better correlation. This behaviour indicates that on reacting a triarylmethyl cation with water, the resonance and polar effects in the cation fail to parallel one another as the reaction proceeds. McClelland suggested that a simple explanation might be that, because of the partial bond to the nucleophile, in the transition state the fractional positive charge which still resides on the Ar3C is localized on the central carbon, attenuating the resonance interaction more than the polar interaction. We have discussed the significant dependence of the substituent effect upon the coplanarity of aryl rings in [3C⫹(X,Y,Z)]. As the excellent rate– equilibrium relationship shown in Fig. 34 indicates, the effect of the aryl conformation on kw and KR⫹ must be similar for both terms. For the equilibria,

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each series with a fixed Y substituent was affected in a different way by the change in the aryl conformations; the monosubstituted [3C⫹(Y, Z ⫽ H)] series is affected most significantly by the P-E-T conformation change which gives a seriously concave plot of substituent effects. This should be an important cause of the ambiguity of McClelland’s correlational results. Among these series, however, only the symmetrically trisubstituted series [3C⫹(X ⫽ Y ⫽ Z)] does not suffer from these conformational effects and this gives an excellent Y–T plot (Fig. 1). Thus we have applied the Y–T treatment to the kw of this symmetric series to give a linear correlation with ␳ ⫽ 7.49 and r ⫽ 0.69 displaying an excellent precision (SD ⫽ ⫾0.22 and R ⫽ 0.9996). The correlation for the kh( ⫽ kw KR⫹) substituent effect was also obtained as the difference of two Y–T correlations for the pKR⫹ and kw processes. The results are shown schematically in Scheme 12.

␳ ⫽ ⫺3.99, r ⫽ 0.80

␳ ⫽ 7.49, r ⫽ 0.69

␳ ⫽ 11.48, r ⫽ 0.76 Scheme 12.

The results are appreciably different from those based on the analysis of the reactions of monosubstituted triphenylmethyl cations (McClelland et al., 1989). Obviously, this is due to the difficulties of correlating the substituent effects in a simple manner. The preferred results on the trisarylmethyl cation lead to the conclusion that both the forward process of the kc ionization and the reverse process of the solvent-recombination step of the carbocation with – various nucleophiles can be described to a good approximation by a Y–T ␴ scale with an r value of the transition state which is essentially identical with the intrinsic r value of the thermodynamic stabilities. However, we will consider the triarylmethyl system further following similar analysis of ␣-arylethyl cations. Reactions of ␣-arylethyl carbocations The behaviour of ␣-arylethyl cations has been analysed by Richard and coworkers (Richard et al., 1984; Richard and Jencks, 1984a,b,c) using the azide clock technique. Combination of the rate constants for the reaction of the carbocations with water and the acid-catalysed cleavage of 1-phenylethanols

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provides estimates of the equilibrium constants for carbocation formation (Scheme 13) and a picture of the development and distribution of the positive charge as the transition state can again be approached from both directions. There is a considerably larger relative contribution of resonance when the transition state is reached by reaction of solvent with the carbocation than when it is reached by departure of a leaving group from a 1-phenylethyl derivative from the solvolysis direction.

␳ ⫽ ⫺3.7, ␳r ⫽ ⫺4.2 ␳ ⫽ ⫺1.1 ␳ ⫽ ⫺6.9, ␳r ⫽ ⫺9.0 ␳ ⫽ ⫺4.8, ␳r ⫽ ⫺4.2

␳ ⫽ 3.2, ␳r ⫽ 4.8

␳ ⫽ ⫺8.0, ␳r ⫽ ⫺9.0 Scheme 13.

It is of particular interest for us that the r value for the transition state for the solvent recombination reaction is clearly different from that for the intermediate, implying that the reverse approach to the transition state should lose much of the resonance stabilization of the intermediate ion. This is not in accord with results from the triphenylmethyl system. More data should be accumulated before a better understanding can be achieved. For the recombination of ␣-phenylethyl cations with strong nucleophiles, Richard and Jencks (1984a,b,c) obtained good Y–T correlations with the same r value of 1.15 as observed for the solvolysis (Fig. 35). The ␳ value of ⫺2.7 for the bimolecular substitution reaction of azide ion with 1-phenylethyl derivatives is significantly more positive than the value of ␳ ⫽ ⫺5.7 for the solvolysis reaction. This shows that there is a smaller development of positive charge in the transition state for the reaction of azide ion than for solvolysis. It is consistent with a coupled concerted reaction with a transition state in which positive charge development at the benzylic carbon is neutralized by bonding to azide ion. Rate–equilibrium correlations Returning to the triarylmethyl carbocations, we saw that the relationship between log kw and pKR⫹ has been investigated (Arnett and Hofelich, 1983;

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log k

370

Fig. 35 The Y–T plots for the reactions of ring substituted 1-phenylethyl chlorides in 20% acetonitrile in water at ionic strength 0.8 (NaClO4) at 25⬚C; r ⫽ 1.15. Reproduced with permission from Richard et al. (1984b). Copyright 1984 American Chemical Society.

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Ritchie, 1986; McClelland et al., 1989, 1991) for a wide set of triarylmethyl cations, and that there is a reasonably linear correlation encompassing the entire set of triarylmethyl carbocations over 16 pKR⫹ units with a small amount of scatter (Fig. 34). The implication of this behaviour is that despite a change in the cation stability, there is a little change in the apparent position of the transition state, at least as revealed in this rate–equilibrium relationship. Changes in reactant structure which increase the rate of a reaction almost always will make the transition state ‘‘earlier’’ in the sense of being more ‘‘reactant-like’’ (Hammond behaviour). The enormous change in thermodynamic stability of carbocations [3C⫹(X,Y,Z)] must bring about a Hammond shift in the transition state coordinate. The transition state shift can be measured by the ␣-exponent of the extended Brønsted equation (19) or in the present case by the magnitude of the ␳ value. The position of the transition state might have been expected to move along the reaction coordinate with substituents, but the application of the Y–T equation to the kw process provides a single ␳ value indicating a fixed position of the transition state. If we assume that slight curvature exists in the plots of kw and/or kh against pKR⫹ which may be attributed to a variation in the ␳ value (but not r value) – with with substituents, we can apply the More O’Ferrall equation (18) using ␴ r ⫽ 0.76 for the kw reaction to provide the following excellent correlation – ⫺ 0.505(␴ –)2. Thus, the apparent ␳ (SD ⫽ 0.075, R ⫽ 1.000); log(k/k0)w ⫽ 6.28␴ w value clearly increases as the substituent becomes more ED, and the tangent ␳w values at the respective substituents are obtained as given in Scheme 14. The corresponding ␳h values at respective substituents are given as the differences between the equilibrium ␳e( ⫽ ⫺11.48) and ␳w values. The ␳h in Scheme 14 varies in the opposite way to ␳w with substituents. The variation of

Scheme 14.

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either ␳w or ␳h can be related more directly to the shift of the transition state coordinate. According to (24), the Brønsted coefficients ␣ and/or (1 ⫺ ␣) are derived from ␳h /␳e or ␳w /␳e for the respective substituents. Scheme 14 is related to the potential energy profile in the vicinity of the transition state in the dissociation/recombination equilibria of [3C⫹(X ⫽ Y ⫽ Z)] given by the Brønsted-type analysis, the rate–equilibrium – scale for the equilibrium. The potential relationship in terms of a fixed Y–T ␴ energy surface should be essentially non-crossing, and consistently, the assumption that the Hammond shift in the transition-state coordinate is reflected exclusively in the ␳ value but not in the r value can be applied to systems involving an appreciable shift in the structure of the solvolytic transition states. The profile in Scheme 12 for the same system given by the direct analysis based on the Y–T equation involves an important disagreement with that in Scheme 14, as regards the transition-state coordinate. The assumption of the same r value for the kw or kh and KR⫹-substituent effects in the More O’Ferrall treatment implies that the non-linearity in substituent effects should be attributed exclusively to the shift of transition-state coordinate with substituents; consequently, the r values assigned to the kw and kh steps in the direct Y–T analysis should be regarded as averages over the Hammond shift of the transition-state coordinate, implying failure of the Y–T equation. Of course both these treatments should be related to the extreme cases, and the discrepancy in r and/or ␳ is in fact mainly due to the effect of the p-Me2N group. Because of the enormous variation in the thermodynamic stability of [3C⫹(X ⫽ Y ⫽ Z)], i.e. a change of KR⫹ over 1016 in the equilibrium with substituents from p-Me2N to H, a Hammond shift of the transition state should be inevitable in this system. The failure of the precise rate–equilibrium relationship in this system must be caused by the failure of the Y–T correlation for the rate process but not for the equilibrium (cf. McClelland’s conclusion for monosubstituted series [3C⫹(Y, Z ⫽ H)], above). Although the Hammond shift of the transition state is clearly important, the change in ␳ value caused thereby may not generally be significant in the solvolysis process. Consequently, we still wish to emphasize that in spite of the wide spectrum of observed r values, the intrinsic resonance demand is an inherent property of the carbocation to be formed and remains essentially the same in the intermediate and in the transition states of the related reactions of the intermediate carbocation. This is the basis of mechanistic analysis based on the Yukawa–Tsuno relationship.

SN2 REACTIONS OF 1-ARYLETHYL AND BENZYL PRECURSORS

As frequently seen in the solvolyses of primary or secondary benzylic precursors, the mechanism of substitution is quite complex, and the fundamen-

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tal question of whether SN1 and SN2 reactions remain distinct at the borderline or gradually merge is still controversial (Katritzky et al., 1981; Katritzky and Musumarra, 1984; Katritzky and Brycki, 1988). The ion pair mechanism and SN2 (intermediate) mechanism (Bentley and Schleyer, 1976; Bentley et al., 1981) bring about further complication, and it is often very difficult to distinguish some of these proposed mechanisms from each other. The problem is particularly acute for solvolysis reactions because of the problem of distinguishing between first- and second-order reactions involving the solvent. The concentration of the solvent cannot be varied without changing the reaction conditions.

Scheme 15.

The Menschutkin reaction of benzyl tosylates [21]-OTs with dimethylanilines or pyridines in acetonitrile generally proceed by a second-order bimolecular SN2 mechanism for most ring-substituted compounds; the plot of kobs vs. [Nu] passes through the origin within experimental uncertainty (Yoh et al., 1989). However, for the reactions of strong ED derivatives under the same conditions, it was found that there was a significant intercept (i.e. a first-order component) in the kobs vs. [Nu] plots; represented by (39) the intercept is a constant of the benzyl substrate independent of the amine nucleophiles, indicating a concurrent reaction zeroth-order in amine (Kim et al., 1995, 1998). kobs ⫽ k1 ⫹ k2[Nu]

(39)

This evidently indicates the duality of SN1 and SN2 mechanisms or competitive unimolecular and bimolecular processes, and the scheme of overall reaction is illustrated in Scheme 15 (Kim et al., 1995, 1998). The corresponding Menschutkin reaction of 1-arylethyl bromides [14]-Br

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was also found to show more clearly this duality of SN1 and SN2 mechanisms (Lim et al., 1997). The reaction scheme is identical with Scheme 15 with the substrate [21]-OTs replaced by [14]-Br. The enhanced stability of the secondary benzylic cation [14C⫹] relative to [21C⫹] accelerating the SN1 mechanism and the steric deceleration of the SN2 mechanism by the ␣-methyl in [14] bring about closer competition of the two mechanisms for a wider range of substituents in [14]. Now we can examine separately the substituent effects on the respective mechanisms of competing unimolecular and bimolecular processes (cf., Lee et al., 1988). In Fig. 36, log k1 for the unimolecular reaction and log k2 for the bimolecular – with r ⫽ 1.15. The dependence of the S 1 rate one are plotted against Y–T ␴ N on the aryl substituents is clearly far greater than that of the SN2 rate. Although the unimolecular SN1 rate constants could be obtained only for substituents more ED than H, the substituent effect on the unimolecular process can be described by the Y–T equation (2) with a ␳ value of ⫺5.0 and –(r ⫽ 1.15) for the an r value of 1.15. On the other hand, the plot of log k2 vs. ␴ bimolecular (ANDN) process gives a smooth curve. The correlation for the former unimolecular process is identical with that for the corresponding solvolyses of 1-phenylethyl chloride in solvolysing solvents (Tsuno et al., 1975; Fujio et al., 1984). The concave plot for the k2 correlation may be interpreted in terms of a change in ␳ associated with the so-called mechanistic shift as the substituent changes. Application of the Y–T equation fails to give any linear plot but yields a poor correlation with a ridiculously large r value, indicating that the curved correlation should not be attributable to an enhanced resonance effect. – for the bimolecular process can be analysed The plot of log k2 against ␴ – with r ⫽ 1.15 according to the More O’Ferrall equation (18) in terms of ␴ – – 2 (R ⫽ 0.995 and SD ⫽ ⫾0.08); log (kX /kH) ⫽ ⫺1.54␴ ⫹ 0.74(␴) . The negative ␳ value must indicate the accumulation of positive charge at the central carbon in the transition state, i.e. the preferential bond fission (DN) over bond formation (AN) in the pentavalent SN2 transition state. From the quadratic correlation for the bimolecular mechanism, the tangent ␳ values at respective – ; thus the ␳ values substituents can be estimated by ␳X ⫽ ⫺1.54 ⫹ 2 ⫻ 0.74␴ X are as highly negative as ⫺2.88 at p-MeO, ⫺1.54 at H, ⫺0.99 at m-Cl and diminish to ⫺0.34 for the p-nitro substituent. The Menschutkin reaction of benzyl tosylates [21] shows quite similar behaviour. The effect of substituents on the unimolecular process can be correlated by the Y–T equation with r ⫽ 1.3, and that on the bimolecular – with r ⫽ 1.0, with a good process can also be given by (16) in terms of ␴ – precision of R ⫽ 0.997 and SD ⫽ ⫾0.047 (n ⫽ 35); log (kX /kH) ⫽ ⫺1.44␴ – 2 ⫹ 0.54(␴) . The bimolecular process of the Menschutkin reaction of benzyl [21] bromides with dimethylanilines in methanol also shows a similar concave correlation of the substituent effect to that described above.

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␴ (r ⫽ 1.15) Fig. 36 The Y–T plots for the Menschutkin reaction of 1-arylethyl bromides with pyridine in acetonitrile at 35⬚C; r ⫽ 1.15. 䊊, k2 for the bimolecular process; 䊐, k1 for the unimolecular process. Reproduced with permission from Lim et al. (1997). Copyright 1997 Pergamon Press.

For better understanding of the mechanism of a particular reaction, a two-dimensional More O’Ferrall–Jencks (MOFJ) reaction coordinate diagram is most widely and conveniently used; in the case of benzylic nucleophilic substitution reactions, the diagram requires separate coordinates for cleavage of the C⫺L bond (ordinate) and formation of the C⫺Nu bond to the

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nucleophilic reagent (abscissa). In most cases, the structure–reactivity coefficients, ␳, ␤ and ␣, closely related indices concerning the strength or the order of C⫺L and C⫺Nu bonds, are used to construct an energy contour diagram with the transition state and reaction coordinate (Jencks, 1985). Whereas only ␳ values for the substrate reactivities are available for the present set of Menschutkin reactions, changes in the structure–reactivity parameter ␳ should be related to the movement of the transition state. The MOFJ diagram of the Menschutkin reactions is constructed in terms of the bond orders of the C⫺Br and C⫺Nu bonds, which are theoretically estimated by ab initio MO optimization (RHF/6-31 ⫹ G*) of the transition state [42] (Hori et al., 1998; Fujio et al., unpublished). It is interesting to speculate on the changes in the MOFJ reaction coordinate

[41] [42] diagram (Fig. 37) that occur as the mechanism for the Menschutkin reactions changes from unimolecular DN ⫹ AN to bimolecular ANDN. The transition state for the bimolecular reaction for methyl chloride is plotted at the central position on the diagonal dotted line, which is related to the reaction coordinate for a single-stage displacement from reactants to products; this transition-state coordinate implies simultaneously increasing bond order to the nucleophile and decreasing bond order to the leaving group, and without accumulating a significant amount of charge at the central carbon. The position of the transition state of the bimolecular reaction of 1-arylethyl chlorides [14]-Cl significantly deviates in a perpendicular direction from the central position of methyl chloride towards the left upper corner for the carbocation. Most importantly, as the substituent becomes more ED, the coordinate moves in a diagonal direction closer to the corner of carbocation. The transition state for the unimolecular reaction lies on the reaction coordinate along the vertical edge (ordinate) of the diagram close to the left-hand-side upper corner. The reaction coordinate for the bimolecular reaction of the p-MeO derivative runs from the reactant (left-hand-side bottom) almost vertically along the edge, via the transition state and along the upper horizontal edge to the right-hand corner; the coordinate appears very similar to that of unimolecular reaction except for the absence of intermediate. The EW derivatives, e.g. p-nitro-[14], have a transition state towards the centre

377

C–Cl bond order

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N–C bond order Fig. 37 More O’Ferrall–Jencks diagram for the Menschutkin reactions of 1phenylethyl and benzyl chlorides with pyridine. The structures of transition states were optimized by ab initio MO calculation (RHF/6-31⫹G*). 䊊, substituted 1-phenylethyl chlorides with pyridine; 䊐, benzyl chlorides with pyrindine; 䊏, with 4-nitropyridine; 䉫, methyl and 䉭, ethyl chlorides with pyridine (Fujio et al., unpublished).

of the diagram in which there is appreciably increased bonding of nucleophile and leaving group to the central carbon, and the reaction coordinate should be close and almost parallel to the diagonal (dotted line) coordinate of the completely synchronous, concerted displacement. The bimolecular reaction of benzyl chlorides displays similar behaviour, though the transition-state coordinate varies with substituents to a smaller extent but lies spread in a vertical direction and the reaction coordinate of the bimolecular reaction lies at the centre of the diagram. We assume the magnitude of ␳ to be a measure of the charge accumulated at the reaction centre in the transition state, and the more negative ␳ value implies more advanced bond fission compared with bond formation in the

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transition state. A more negative ␳ should be related to a looser transition state where an increased amount of carbocationic charge is accumulated on the benzylic moiety. The transition state should shift in the perpendicular direction to become more carbocationic as the substituent becomes more ED. It is noteworthy that in these typical displacement reactions, the r value, which is close to the intrinsic resonance demand of the corresponding carbocation, remains constant regardless of a significant shift of the transition-state position or of the mechanistic change with substituent perturbation, and that the mechanistic shift is apparently reflected only in the variation of ␳ value.

10

Concluding remarks

We hope that this article has demonstrated the wide applicability of the Y–T equation for quantitative descriptive purposes, the estimation and prediction of reactivities, as well as its potential for analysing reaction mechanisms and for elucidating the structure of transition states. The Y–T equation has proved to be one of the most fundamental and essential equations for understanding structure–reactivity relationships in chemical reactions using the language of the physical organic chemist. In the 1950s, when this empirical relationship was developed, the basic theory used by organic chemists to discuss reaction mechanisms was resonance theory or Hückel molecular-orbital theory. Therefore it may be necessary to reinterpret the classical language in terms of more recent ones based on the most advanced theory of organic chemistry. Indeed the word ‘‘resonance’’ is rather unclear in the language of delocalization theory. Although complete translation to the language of advanced theories of chemical reactivity will need the help of theoretical chemists, the underlying relationship does not appear to need any important correction. The behaviour of carbocations in the gas phase should be the ideal model for reactivity changes in solvolytic reactions involving carbocation intermediates, and, furthermore, the behaviour should be subject to analysis by advanced MO calculation. The substituent effects on stabilities of gas-phase carbocations have been correlated with the Y–T equation, in exactly the same manner as in the solution phase. Perhaps the most important observation is the identity of r values for solvolytic reactivities and the corresponding carbocation stabilities in the gas phase, suggesting r to be a fundamental quantity related to the structure of cations. Extraordinarily good linear relationships between stability and resonance demand parameter r in gas-phase carbocations are quite suggestive of the nature of r. The greatest usefulness of this empirical relationship in mechanistic studies is in the simplicity of this linear structure–reactivity relationship. This equation will without doubt be applied to a wider range of reactions and to the investigation of more detailed mechanisms.

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Acknowledgements The authors are grateful to Professors M. Mishima, S. Kobayashi and M. Sawada who carried out important parts of the investigations, and to Professors R. More O’Ferrall, K. Nishimoto, Z. Rappoport, J. P. Richard and T. T. Tidwell, for research cooperation. We gratefully acknowledge the tremendous contributions of all our coworkers, whose names are cited in the references. Thanks are extended to colleagues both here and abroad for their help in providing data and encouragement. One of the authors (Y.T.) is especially grateful to Professor Herbert C. Brown for his leadership in this field and generous encouragement over 40 years, and to Professor John Leffler for introducing him to the field of the Leffler–Hammond approach to the transition-state structures. The authors would particularly like to express their sincere appreciation to Professor Zvi Rappoport and Professor R. More O’Ferrall for critically reading the manuscript and for stimulating discussions and valuable suggestions in the preparation of this review.

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Author Index Numbers in italic refer to the pages on which references are listed at the end of each chapter Aaron, J. J., 332, 333, 380 Abboud, J. I. M., 376, 382 Abdel-Hamid, R., 42, 113 Abrahams, S. C., 255, 265 Adachi, T., 269, 279, 374, 380 Adams, R, N., 60, 63, 113, 118 Agabe, Y., 34, 118 Agra-Gutierrez, C., 82, 113 Agren, H., 200, 213 Aguilar-Parrilla, F., 240, 242, 262, 263, 264, 265 Agullo´-Lopez, F., 203, 214 Ahlberg, E., 60, 113 Aixill, W. J., 109, 113 Akasaka, I., 279, 289, 291, 300, 381 Akbulut, U., 70, 113 Akkermans, R. P., 70, 79, 80, 81, 82, 116, 118 Alain, V., 167, 193, 209 Alald, A., 5, 114 Albert, I. D. L.,179, 187, 208 Albery, W. J., 21, 46, 48, 113 Alden, J. A., 50, 52, 54, 62, 67, 69, 85, 88, 93, 94, 98, 100, 105, 109, 113, 119 Alder, A. D., 238, 239, 263 Alhambra, C., 185, 187, 211 Alkire, R. C., 80, 119 Allen, A. D., 304, 305, 307, 308, 323, 324, 379 Allen, F. H., 123, 208 Allin, S. B., 187, 208 Almlöf, J., 137, 215, 226, 265 Althoff, O., 179, 209 Altieri, L., 270, 384 Amatore, C. A., 29, 68, 69, 95, 96, 98, 105, 114, 116, 117, 118, 119 Ambidge, I. C., 307, 379 Ancian, B., 316, 322, 381 Anderson, J. L., 92, 114 Andrews, D. L., 158, 208 Andrieux, C. P., 5, 105, 114 Annunziata, R., 172, 210

Aoki, K., 63, 114 Appel, W. K., 245, 265 Arant, M. E., 300, 384 Argile, A., 326, 327, 328, 329, 330, 331, 333, 379, 380, 384 Ariga, T., 344, 346, 347, 353, 383 Arima, K., 346, 347, 350, 351, 355, 383 Arita, N., 297, 300, 381 Arnett, E. M., 366, 369, 379 Asamitsu, A., 259, 265 Ashwell, G. A., 123, 211 Ashwell, G. J., 186, 208 Atkins, P. W., 10, 114 Atobe, M., 70, 118 Attard, G. A., 5, 114 Attia, S. Y., 324, 379 Axon, T. L., 180, 210 Baba, N., 108, 118 Bagley, F. D., 286, 384 Bagno, A., 316, 379 Bahl, A., 171, 208 Baker, D. R., 96, 120 Baldo, M. A., 38, 116 Baldridge, K. K., 201, 208 Baldy, A., 240, 262, 265 Balegroune, F., 173, 213 Ball, J., 96, 114 Banait, N., 366, 367, 368, 371, 382 Banjoko, O., 286, 380 Barbara, P. F., 222, 226, 247, 262, 265 Barbu, E., 150, 164, 166, 199, 200, 202, 216 Bard, A. J., 3, 6, 11, 20, 27, 29, 34, 37, 65, 103, 105, 114, 116 Barghout, R., 19, 57, 58, 115, 116 Barnik, M. I., 167, 209 Bartlet, J., 96, 117 Bartlett, P. D., 305, 379 Barz, F., 80, 114 Barzoukas, M., 143, 167, 173, 186, 187, 191, 193, 208, 209, 213, 214, 215 387

388

Bassoul, P., 191, 215 Baughcum, S. L., 225, 262, 265 Baumann, W., 167, 183, 193, 208, 213 Baumert, J. C., 167, 174, 214 Bax, A., 233, 265 Beagley, B., 240, 262 Becher, D., 39, 117 Bechgaard, K., 124, 210 Becker, J., 155, 162, 167, 213 Beckmann, S., 167, 179, 188, 193, 215 Bedworth, P. V., 167, 181, 209, 212 Beecher, J. E., 181, 208 Beeston, M. A., 54, 100, 118 Behrens, U., 191, 208 Benahcene, A., 82, 114 Benewalenskaja, S. V., 249, 262 Bentley, T. W., 373, 379 Bento, M. F., 39, 114 Beran, P., 61, 114 Beratan, D. N., 194, 214 Bergmann, E. D., 327, 333, 343, 380, 382 Beriot, C., 39, 114 Berke, C., 323, 324, 382 Berkheim, H. E., 274, 316, 318, 380 Bernard, C., 71, 81, 117 Bernstein, C., 80, 114 Bernstein, J., 123, 208 Berry, J. P., 324, 379 Bersohn, R., 164, 208 Berthou, J., 240, 262 Bertolasi, V., 230, 262 Bethea, C. G., 158, 162, 213 Bethell, D., 60, 115 Betterton, K. M., 181, 186, 214 Bicerano, J., 226, 262 Bieniasz, L. W., 92, 109, 114 Bienvenue-Goetz, E., 334, 379 Bigio, I. J., 162, 216 Birkin, P. R., 81, 82, 114 Bishop, D. M., 136, 149, 150, 151, 152, 153, 181, 209, 216 Bittner, R., 159, 167, 193, 217 Bjorklund, G. C., 167, 174, 214 Bjørnholm, T., 124, 210 Blanchard-Desce, M., 143, 167, 179, 180, 187, 193, 208, 209, 213 Blann, W. G., 231, 265 Blenkle, M., 177, 209 Blesa, M. A., 40, 117 Blinov, L. M., 167, 209 Bloor, D., 123, 180, 210, 211, 215 Bockris, J. O’M., 3, 114

AUTHOR INDEX

Boettcher, C. J. F., 149, 150, 162, 209 Böhm, M. C., 183, 214 Boldt, P., 167, 177, 193, 209 Bölger, B., 164, 211 Bolocan, I., 165, 216 Bond, A. M., 3, 5, 8, 9, 19, 35, 37, 57, 58, 62, 64, 108, 109, 113, 114, 115, 116, 117, 119, 120 Bondi, A., 245, 262 Bondybey, V. E., 225, 262 Bonelli, R. A., 270, 381 Booth, J., 5, 19, 82, 108, 114, 115, 116, 118 Borcic, S., 285, 382 Borgault, M., 166, 210 Borowicz, P., 161, 216 Bosshard, C., 122, 123, 135, 158, 162, 204, 206, 209, 211, 216 Bott, A. W., 9, 24, 115 Bott, R. W., 305, 379, 380 Bourhill, G., 166, 171, 172, 180, 182, 186, 191, 199, 208, 209, 212, 213, 215 Boutton, C., 124, 210 Bowen, C. J., 373, 379 Boxer, S. G., 167, 209 Bozec, H. L., 166, 210 Branch, G. E. K., 279, 383 Brandl, S., 200, 201, 202, 215 Branger, C., 203, 212 Brasselet, S., 203, 214 Bräuchle, C., 159, 166, 167, 171, 172, 177, 191, 193, 199, 200, 201, 202, 203, 208, 209, 212, 215, 217 Braun, J., 237, 238, 239, 262, 264 Bredas, J. L., 141, 179, 180, 182, 186, 187, 200, 201, 209, 210, 211, 213, 215 Bregman, J., 246, 247, 262 Bressers, P. M. M. C., 46, 115 Brett, A. M. O., 52, 82, 115, 118 Brett, C. M. A., 52, 54, 80, 82, 115, 118 Brevet, P. F., 5, 87, 116 Bridge, M. R., 286, 380 Brienne, J., 200, 212 Brieux, J. A., 270, 381, 384 Britz, D., 87, 90, 93, 94, 95, 115 Brock, C. P., 129, 209 Brooker, L. G. S., 186, 209 Brown, H. C., 268, 269, 277, 279, 280, 282, 380, 385 Brown, M. S., 249, 264 Brown, W. F., 149, 209 Bruce, P. G., 3, 115

AUTHOR INDEX

Bruckenstein, S., 48, 61, 113, 114 Brus, L. E., 222, 247, 262 Brussaard, H., 191, 208 Brust, M., 105, 116 Brycki, B. E., 373, 382 Bublitz, G. U., 167, 209 Buckingham, A. D., 134, 209 Buddenbaum, W. E., 284, 384 Budnik, U., 200, 214 Bull, S. D., 109, 118 Bunnett, J. F., 270, 380 Burke, L. D., 71, 120 Burkhard, O., 155, 162, 213 Burland, D. M., 123, 134, 158, 159, 183, 184, 209, 215, 216, 217 Burst, R., 173, 215 Bury, P. C., 108, 120 Busch, J. H., 228, 256, 262 Butcher, P. N., 123, 128, 130, 132, 209 Cabrera, I., 179, 209 Cady, H. H., 200, 209 Cahill, P. A., 139, 186, 209 Cai, Y., 181, 212 Calabrese, J. C., 191, 210 Cammann, K., 105, 117 Cano, F. H., 240, 242, 262, 263, 264, 265 Cardwell, T. J., 9, 108, 115 Carpenter, A. K., 39, 119 Carrington, T., 226, 262 Carter, F. L., 221, 262 Cartwright, H. M., 109, 115 Castiglioni, C., 181, 210 Castro, A. J., 270, 384 Castro, G., 250, 262 Cativiela, C., 240, 262 Cesar, A., 200, 213 Chamberlain, M. C., 191, 210 Champagne, B., 181, 210 Chandrasekhar, J., 179, 216 Chang, H. S., 316, 380 Chang, N.-L., 123, 208 Chang, S.-M., 314, 382 Chapelle, F., 82, 117 Charra, F., 198, 211 Charton, M., 179, 210 Chauchard, E., 166, 203, 212 Che, C., 307, 379 Chemla, D. S., 123, 143, 198, 210, 214 Chen, B. M. L., 237, 262 Chen, C. J., 86, 115 Chen, C. T., 194, 215

389

Chen, H.-I., 314, 382 Chen, J., 109, 118 Chen, Q., 108, 120 Cheng, L.-T., 162, 179, 180, 182, 186, 188, 191, 194, 209, 210, 213, 215 Chiang, Y., 304, 305, 324, 379 Chichababin, A. E., 249, 262 Chin, D. N., 123, 216 Chiu, P.-F., 314, 382 Cho, B. R., 166, 215 Chopra, A., 173, 213 Choy, M. M., 163, 212 Chwang, W. K., 324, 325, 380 Chyla, A., 70, 71, 83, 115, 120 Cinquini, M., 172, 210 Claramunt, R. M., 240, 263 Clarke, L., 83, 120 Clays, K., 123, 124, 163, 164, 165, 179, 186, 187, 200, 202, 210, 211, 216 Cochran, W. G., 47, 115 Codding, P. W., 238, 262 Coe, B. J., 191, 210 Cohen, E. R., 125, 134, 210 Cohen, M. D., 247, 262 Colarusso, P., 70, 115 Cole, J. M., 180, 215 Coles, B. A., 48, 50, 52, 54, 55, 59, 61, 69, 79, 86, 105, 115, 116,119 Colton, D., 83, 120 Colton, R., 19, 35, 37, 57, 58, 62, 109, 113, 114, 115, 116 Compton, R. G., 3, 5, 9, 10, 19, 21, 23, 25, 39, 46, 48, 50, 52, 54, 55, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 72, 73, 74, 75, 79, 80, 81, 82, 85, 86, 88, 92, 93, 94, 96, 98, 100, 105, 108, 109, 113, 114, 115, 116, 117, 118, 119, 120 Connelly, N. G., 11, 116 Connors, T. F., 70, 116 Cooper, J. A., 48, 50, 52, 59, 69, 105, 108, 115, 116, 119 Cooper, J. B., 9, 117 Cotter, D., 123, 128, 130, 132, 209 Cotts, P. M., 159, 217 Coury, L. A., 81, 82, 118, 120 Coy, S. L., 225, 265 Cozzi, F., 172, 210 Cradrick, P. D., 230, 262 Crank, J., 65, 116 Creary, X., 307, 308, 309, 380 Crick, F. H. C., 220, 265 Cross, G. H., 180, 210, 215

390

Cseh, G., 303, 381 Curtin, D. Y., 231, 263, 265 Cyvin, S. J., 164, 210 Czekalla, J., 160, 167, 213 Dähne, S., 174, 175, 186, 210 Dairokuno, T., 297, 300, 381 Dale, J., 200, 210 Dalton, L. R., 123, 129, 158, 210 Daltrozzo, E., 185, 186, 187, 214 Daniele, S., 38, 116 Daniels, F., 109, 114 Das, P. K., 366, 380 Davies, C. L., 108, 115 Davies, C. W., 93, 108, 117 Davies, D. H., 286, 380 Davies, R. E., 123, 198, 208, 216 Davies, S. G., 19, 50, 52, 58, 59, 69, 105, 109, 115, 116, 118, 119 Day, P., 143, 214 de Jonge, R., 166, 199, 211 de la Vega, J. R., 222, 226, 228, 256, 262 de Maeyer, L.,163, 210 de Paz, J. L. G., 242, 263 Deakin, M. R., 96, 116 Decius, J. C., 164, 210 Degrand, C., 70, 71, 81, 83, 116, 117 Dehu, C., 179, 186, 187, 201, 210, 211 Del Zoppo, M., 181, 210 Delplancke, J,-L., 70, 79, 116, 119 Demaille, C., 105, 116 DeNardin, Y., 159, 167, 193, 217 Denning, R. G., 123, 143, 164, 166, 179, 191, 199, 201, 210, 212, 213 Deno, N. C., 272, 274, 316, 318, 319, 380 Desiraju, G. R., 123, 210 Detzer, N., 158, 159, 163, 167, 170, 204, 205, 206, 216 Deussen, H. J., 124, 210 DeVoe, R. G., 159, 217 Dhenaut, C., 166, 202, 203, 210, 214, 217 Diaz, L. E., 238, 239, 263 Dı´az de Villegas, M. D., 240, 262 Diaz-Garcia, M. A., 191, 203, 210, 214 DiBella, S., 151, 171, 191, 196, 210, 211 Diederich, F., 204, 206, 209 Dietz, F., 179, 186, 215 DiMagno, S. G., 194, 212 Ding, J, Q., 19, 23, 29, 117 Ding, Z., 5, 116 Dirk, C. W., 123, 137, 162, 174, 175, 179, 180, 182, 186, 211, 215

AUTHOR INDEX

Dobson, P. J., 3, 105, 115 Doering, W. von E., 219, 263 Döpp, D., 245, 264 Dörr, F., 185, 186, 187, 214 Doucet, J. P., 316, 322, 332, 333, 380,381 Dreier, T., 200, 216 Drewer, H., 105, 117 Drost, J. K., 179, 212 Dryfe, R. A. W., 19, 21, 25, 39, 50, 52, 54, 59, 69, 93, 105, 108, 115, 116, 119 Dryhurst, G., 3, 116 Dubois, D., 42, 116, 118 Dubois, J. E., 287, 326, 327, 328, 329, 330, 332, 333, 334, 379, 380, 382, 384 Dueber, T. E., 303, 384 Duerst, G. N., 225, 262 Duesler, E. N., 231, 263, 265 Duggan, P. J., 180, 211 Dunitz, J. D., 129, 209 Dupuis, M., 172, 212 Durant, A., 70, 116 Durst, T., 181, 208 Dutta, P. K., 247, 265 Dykovskaya, L. A., 250, 263 Dyl, D., 38, 117 Dzakpasu, A. A., 245, 264 Eaborn, C., 279, 305, 379, 380 Eberlein, W., 204, 213 Eberson, L., 3, 116 Edstrom, K., 29, 119 Edward, J. T., 316, 380 Effenberger, F., 167, 174, 179, 188, 193, 215, 217 Eggers, M. D., 307, 380 Eisenträger, T., 167, 193, 209 Eklund, J. C., 3, 5, 19, 21, 23, 25, 50, 57, 58, 62, 70, 71, 72, 73, 75, 79, 80, 82, 109, 113, 114, 115, 116, 118, 120 El-Alaoui, M., 324, 385 Elflein, O., 204, 213 Elgureo, J., 240, 242, 262, 263, 264, 265 Elich, K., 161, 216 Elshocht, S. V., 124, 211 Endo, Y., 225, 265 English, J. H., 225, 262 Eriksen, O. I., 200, 210 Ermer, S., 158, 204, 206, 214 Ernst, R., 234, 235, 264, 265 Essex-Lopresti, J. P., 191, 210 Etter, M. C., 230, 263 Evans, D. H., 41, 119

AUTHOR INDEX

Evans, W. L., 272, 274, 316, 318, 319, 380 Exelby, R., 222, 263 Fabian, J., 175, 176, 179, 186, 189, 198, 199, 211, 215 Fabre, O., 79, 119 Farmworth, K. J., 240, 262 Farsari, M., 180, 210 Faulkner, L. R., 3, 6, 11, 20, 27, 29, 34, 65, 103, 109, 114, 118 Faure, R., 240, 262 Feagins, J. P., 198, 216 Feder, J., 258, 265 Feil, D., 173, 215 Feiner, F., 171, 201, 208, 215 Feith, B., 186, 211, 213 Feldberg, S. W., 37, 60, 93, 94, 114, 116, 119 Fentiman, A. F., Jr., 279, 280, 281, 381 Fernando, D. R., 109, 114 Ferretti, V., 230, 262 Ferrigno, R., 87, 116 Fetterman, H., 123, 129, 158, 210 Feyereisen, M. W., 137, 215 Fick, A., 18, 117 Filippini, G., 123, 211 Finnin, B, C., 108, 120 Fiorini, C., 198, 211 Fisher, A. C., 3, 19, 57, 58, 60, 62, 82, 87, 93, 98, 105, 108, 115, 116, 117, 120 Flannery, B. P., 95, 119 Fleischer, E. B., 237, 265 Fleischmann, M., 3, 63, 64, 114, 117, 119 Flipse, M. C., 166, 179, 188, 199, 211, 215, 216 Flörsheimer, M., 123, 135, 158, 209 Fluder, E. M., 228, 262 Flytzanis, C., 136, 211 Foces-Foces, C., 240, 242, 262, 263, 264, 265 Fogg, A. G., 45, 119 Follonier, S., 123, 211 Fontana, A., 82, 117 Forlano, P., 40, 117 Förster, T., 187, 211 Fort, A., 167, 186, 187, 191, 193, 208, 209, 214, 215 Fosset, B., 96, 114, 117 Fost, J., 105, 119 Fox, J. R., 291, 294, 321, 380 Fragala´, I. L., 171, 191, 196, 210 Frampton, R. D., 324, 380

391

Francois, H., 70, 79, 116, 119 Franssen, O., 188, 215 Fréchet, J. M. J., 181, 208 Frey, M. H., 233, 263 Friedli, A. C., 180, 213 Friedrich, J., 222, 250, 263 Friedrich, S., 239, 263 Frisch, H. L., 164, 208 Frydman, B., 238, 239, 263 Frydman, L., 238, 239, 263 Fuhr, G., 105, 119 Fujii, T., 279, 284, 374, 385 Fujimoto, S., 259, 265 Fujio, M., 269, 271, 274, 275, 276, 277, 279, 280, 284, 286, 287, 288, 289, 290, 291, 295, 296, 297, 298, 299, 300, 304, 307, 308, 309, 310, 311, 312, 313, 314, 335, 336, 338, 339, 340, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 363, 364, 373, 374, 375, 376, 380, 381, 382, 383, 385 Fujishima, A., 8, 120 Fujita, M., 229, 264 Fujiyama, R., 274, 276, 277, 304, 307, 308, 309, 383 Fulian, Q., 93, 108, 117 Funatsu, K., 269, 295, 297, 298, 300, 380, 381 Fünfschilling, J., 172, 214 Furusawa, A., 251, 263 Fyfe, C. A., 222, 231, 263, 265 Gaines, S., 191, 210 Galler, W., 131, 161, 212 Gandler, J. R., 366, 381 Gao, J. L., 185, 187, 211 Gao, L., 200, 216 Gao, X. P., 25, 105, 119, 120 Garci´a, J. I., 240, 262 Garcia, M. B. Q., 52, 115 Garito, A. F., 162, 215 Garrard, N., 64, 119 Garreau, D., 105, 114 Gassman, P. G., 279, 280, 281, 307, 381 Gavaghan, D. J., 93, 95, 96, 117 Gavezzotti, A., 123, 211 Geiger, W. E., 11, 116 Geletneky, C., 159, 217 Geskin, V., 201, 210 Ghosn, R., 123, 129, 158, 210 Giacomo, P., 125, 134, 210

392

Gilli, G., 230, 262 Gilli, P., 230, 262 Girault, H. H., 5, 87, 93, 96, 105, 116, 119, 120 Glania, C., 137, 150, 158, 159, 163, 164, 165, 166, 167, 170, 175, 176, 179, 188, 193, 199, 200, 202, 204, 205, 206, 209, 211, 215, 216 Gleiter, R., 173, 211, 215 Glen, R. C., 109, 117 Glezer, V., 108, 120 Glukhovtsev, M. N., 179, 213 Glyde, E., 286, 381 Godt, A., 181, 208 Goethals, G., 316, 322, 381 Göldenitz, J., 167, 193, 209 Goldstein, P., 233, 263 Golesworthy, R. C., 279, 380 Gompper, R., 200, 201, 202, 215 Goodale, J. W., 70, 120 Goodman, M. F., 220, 263 Goovaerts, E., 180, 215 Gordon, D. M., 123, 216 Gordon, H. M., 162, 215 Gordon, P. F., 180, 211 Gordon, T. F., 175, 199, 211 Görlitz, G., 191, 212 Gorman, C. B., 180, 182, 187, 211, 213 Gorokhovskii, A. A., 250, 263 Gosser, D. K., Jr., 42, 117 Goto, M., 269, 274, 276, 277, 279, 289, 290, 291, 295, 296, 297, 298, 299, 300, 304, 309, 380, 381, 383 Gourlay, A. R., 93, 115 Grabaric, B. S., 35, 114 Grabaric, Z., 35, 114 Grabowska, A., 161, 216 Graf, F., 235, 264 Graff, R., 186, 214 Graham, E., 188, 215 Grahn, W., 171, 177, 208, 209 Gramlich, V., 123, 216 Grandjean, D., 173, 213 Granger, M. C., 108, 120 Grant, D. M., 238, 239, 263 Gray, D., 180, 210 Greaves, C. R., 92, 100, 115 Gredel, F., 165, 200, 201, 216 Green, J. C., 191, 210 Greenhill, H. B., 37, 114 Gregory, P., 175, 199, 211 Greizerstein, W., 270, 381

AUTHOR INDEX

Grinter, R., 222, 263 Gritzner, G., 11, 117 Grob, C. A., 303, 381 Gross, H., 200, 216 Grosser, T., 43, 120 Groth, P., 200, 210 Guan, H.-W., 194, 212 Guggenheim, T. L., 307, 381 Gun, G., 108, 120 Günter, P., 122, 123, 135, 158, 162, 204, 206, 209, 211, 216 Haag, R., 245, 263 Haarer, D., 222, 250, 262, 263 Haase, W., 167, 209 Habu, M., 228, 265 Haddon, R. C., 221, 225, 262, 263 Hadjoudis, E., 247, 263 Hadziioannou, G., 173, 215 Hagan, C. R. S., 82, 118 Hagan, D. J., 131, 211 Hagenau, U., 191, 208, 211 Hagimoto, K., 163, 213 Hale, J. M., 23, 96, 117 Hall, D., 230, 262 Hamaguchi, H., 247, 265 Hamamoto, K., 271, 385 Hann, R. A., 123, 211 Hapiot, P., 105, 114 Hardwick, R., 249, 264 Hargrove, R. J., 303, 384 Harland, R. G., 46, 115 Harms, K., 200, 214 Harper, A. W., 123, 129, 158, 210 Harris, J. M., 167, 174, 214 Hartmann, H., 175, 176, 189, 191, 198, 199, 211, 212 Hartter, D. R., 324, 383 Hasegawa, T., 233, 264 Hashimoto, K., 8, 120 Hauchecorne, G., 162, 211 Havinga, E. E., 167, 211 Hawley, M. D., 39, 118 Hayakawa, F., 279, 285, 384 Hayashi, S., 235, 264 Hazama, K., 300, 384 Healy, D., 180, 210 Heck, J., 191, 208, 211 Heesink, G. J. T., 164, 211 Hegarty, A. F., 327, 332, 333, 343, 380, 382 Heilig, G., 179, 211

AUTHOR INDEX

Heineman, W. R., 3, 37, 117 Heinz, J., 29, 93, 105, 117, 120 Hendrickson, A. R., 37, 114 Hendrickx, E., 124, 179, 180, 186, 187, 191, 208, 210, 211, 212 Henning, J., 239, 264 Herbstein, F. H., 231, 263 Heseltine, D. W., 186, 209 Hexel, J. G., 233, 263 Hickey, S. J., 87, 120 Hidaka, H., 70, 119 Higelin, D., 247, 263 Hill, H. A. O., 80, 117 Hillenbrand, D., 137, 165, 167, 201, 204, 205, 206, 216 Hintsche, R., 105, 119 Hirota, N., 235, 264 Hirsch, A., 43, 120 Hirst, J., 93, 116 Hockless, D. C. R., 202, 206, 216 Hofelich, T. C., 366, 369, 379 Hoffmann, H., 173, 211 Hoffmann, K., 175, 210 Hoffmann, M. R., 70, 118 Hoffmann, R. W., 177, 211 Hogge, E. A., 3, 117 Hollander, F. J., 230, 263 Holleck, L., 39, 117 Hong, Q., 5, 115 Honjyo, H., 225, 265 Horaguchi, T., 300, 384 Hori, K., 376, 382 Horie, K., 251, 263 Horn, H., 159, 217 Horsewill, A. J., 236, 263 Hoshino, N., 247, 263 Houbrechts, S., 123, 180, 191, 202, 206, 210, 212, 216 Howard, J. A. K., 180, 215 Howell, J. O., 3, 23, 55, 117 Huck, H., 71, 82, 117 Hulliger, J., 123, 124, 135, 158, 209, 211 Humffray, A. A., 271, 382, 384 Humphrey, D. G., 109, 116 Humphrey, M. G., 191, 202, 206, 216 Huppert, D., 222, 263 Hush, N. S., 22, 23, 117 Hutchings, D. C., 131, 211 Hutchings, M. G., 180, 211 Hutchinson, F., 67, 113 Iga, K., 248, 263

393

Igarashi, T., 248, 263 Ikeda, H., 186, 211, 212 Ikegami, S., 280, 380 Ikenaga, K., 344, 346, 347, 383 Imahori, H., 373, 382 Imashiro, F., 235, 264 Imming, P., 258, 265 Inabe, A., 258, 264 Inabe, T., 247, 263 Inoue, H., 344, 346, 347, 348, 349, 350, 351, 352, 356, 383 Inuishi, Y., 259, 265 Irngartinger, H., 165, 173, 200, 201, 211, 216 Ishiki, M., 248, 263 Itai, S., 344, 347, 348, 349, 383 Ito, M., 70, 119 Ito, S., 228, 265 Iwamoto, M., 251, 264 Izuoka, A., 255, 257, 258, 264, 265 Jagerovic, N., 240, 242, 262. 263 Jahn, D. A., 230, 263 James, F. A. J. L., 2, 117 Jaruzelski, J. J., 274, 316, 318, 319, 380 Javed, T., 83, 120 Jeffery, J. C., 191, 210 Jen, A. K.-Y., 123, 129, 158, 179, 181, 210, 212, 216 Jencks, A., 325, 382 Jencks, W. P., 325, 366, 368, 369, 370, 376, 382, 384 Jenneskens, L. W., 166, 179, 188, 199, 211, 215, 216 Jensen, H. J. A., 151, 213 Jeon, S.-J., 166, 215 Jeon, Y.-M., 180, 212 Jeoung, S. C., 166, 215 Jerphagnon, J., 163, 198, 210, 212 Jeuell, C. L., 302, 384 Jeziorek, D., 38, 117 Jiang, Z. Q., 245, 262 Jiménez, P., 240, 264 Jin, B. K., 86, 117 Joergensen, P., 151, 213 John, D., 38, 57, 120 Johnson, C. D., 270, 271, 276, 277, 302, 325, 382 Jones, C. C., 46, 113 Jones, M. T., 42, 116 Jones, P. G., 171, 208 Jones, R. D. G., 230, 263

394

Josse, D., 173, 213 Judson, R. S., 109, 118 Jundt, D., 167, 174, 214 Jung, C. G., 82, 117 Kaarli, R. K., 250, 263 Kaatz, P., 123, 135, 158, 163, 164, 166, 201, 209, 212 Kadish, K. M., 19, 23, 29, 42, 116, 117, 118 Kahn, S. U. M., 3, 114 Kajzar, F., 162, 212 Kakerow, R., 105, 117 Kalvoda, R., 41, 119 Kamata, J., 308, 309, 385 Kamino, K., 175, 212 Kamlet, M. J., 183, 212 Kammler, R., 191, 212 Kanagasabapathy, V. M., 307, 366, 367, 368, 371, 379, 382 Kanatomi, H., 248, 249, 263 Kang, C. H., 345, 347, 353, 357, 383 Kang, H.-K., 374, 382 Kang, S. H., 180, 212 Kang, T.-I., 166, 215 Kanis, D. R., 137, 141, 191, 212 Kapon, M., 231, 263 Karelson, M. M., 151, 212 Karna, S. P., 172, 212 Karp, S., 76, 117 Kato, K., 300, 384 Katritzky, A. R., 373, 382 Katz, H. E., 162, 179, 211, 215 Kauffmann, H., 198, 212 Kauranen, M., 123, 124, 211, 216 Kawabe, Y., 186, 211 Kawasaki, K., 186, 211, 212 Kawato, T., 248, 249, 263 Kelly, J. J., 46, 115 Kenkel, J. V., 37, 114 Kendrick, R. D., 239, 263, 264 Kennard, O., 123, 208, 245, 265 Kevekordes, J. E., 37, 114 Keyes, G. H., 186, 209 Kharalamov, B. M., 250, 263 Khundkar, L. R., 188, 215 Khursid, M. M. T., 240, 265 Kielich, S., 132, 212 Kikukawa, K., 344, 346, 347, 383 Kim, D., 166, 215 Kim, H.-J., 274, 275, 289, 309, 310, 311,

AUTHOR INDEX

312, 313, 314, 335, 336, 338, 339, 340, 344, 381 Kim, H.-Y., 374, 382 Kim, K., 180, 212 Kim, K. S., 181, 212 Kim, S.-H., 279, 291, 297, 298, 299, 300, 373, 374, 375, 381, 382 Kimura, Y., 233, 264 King, L. A., 162, 179, 211, 215 Kinishita, C., 279, 285, 384 Kintzinger, J. P., 186, 214 Kirby, J. A., 279, 286, 384 Kirsch-DeMesmaeker, A., 70, 79, 116, 119 Kirtman, B., 181, 209 Kissinger, P. T., 3, 37, 105, 117 Klein, O., 242, 262, 263 Kleinman, D, A., 131, 212 Klemperer, W. G., 109, 118 Klima, J., 71, 81, 117 Knittel, P., 324, 325, 380 Knöpfle, G., 123, 162, 209, 211 Knorr, C. A., 5, 120 Knutson, D., 270, 380 Kobayashi, S., 279, 286, 287, 299, 300, 302, 346, 347, 353, 359, 362, 381, 382, 383, 384 Kobayashi, T., 247, 264 Koehler, C., 105, 119 Koga, K., 346, 347, 362, 382 Köher, M., 237, 238, 262 Kohguchi, H., 225, 265 Kohler, D., 172, 212, 214 Kohno, K., 258, 264 Kohnstam, G., 291, 294, 321, 380 Kominami, K., 251, 264 Koppenol, M., 9, 117 Körnich, J., 191, 208 Koshihara, S., 259, 264 Koshy, K. M., 324, 325, 334, 379, 380, 382 Kosower, E. M., 222, 263 Kotronarou, A., 70, 118 Kott, K. L., 180, 212 Koyama, H., 248, 249, 263 Kraft, S., 150, 164, 166, 199, 200, 202, 216 Kraichman, M. B., 3, 117 Krämer, P., 137, 150, 158, 159, 163, 164, 165, 166, 167, 170, 174, 179, 188, 193, 199, 200, 202, 204, 205, 206, 209, 214, 215, 216, 217

AUTHOR INDEX

Krasnaya, Z. A., 186, 212 Krause, J., 105, 117 Kreis, R., 234, 265 Kresge, A. J., 304, 305, 324, 379 Kress, R. B., 231, 263 Krishna Rao, G., S., 244, 264 Krishnan, V.,42, 118 Krog, D., 124, 210 Kuball, H. G., 131, 161, 212 Kubota, T., 271, 385 Kucherov, V. F., 186, 212 Kuchta, B., 123, 214 Kulesza, P. J., 109, 118 Kulpe, S., 182, 212 Kumar, V. A., 245, 264 Kumbhat, S., 81, 118 Kundshi, B., 249, 262 Kuo, M.-Y., 284, 382 Kuroki, K., 251, 263 Kuroki, Y., 257, 264 Kurtz, S. K., 163, 167, 212 Kustanovich, I., 238, 263 Kusuyama, Y., 279, 284, 374, 385 Kuta, J., 11, 117 Kutner, W., 42, 116, 118 Kuwamura, T., 279, 286, 381 Kuwanao, J., 70, 119 La Cour, T., 240, 264 Laar, C., 219, 264 Labhart, H., 160, 167, 212 Labre, P., 82, 114 Lacasse, R., 42, 118 Lagier, C. M., 230, 265 Laidlaw, W. M., 164, 166, 179, 191, 199, 201, 212, 213 Laing, M. E., 96, 115 Lalama, S. L., 133, 215 Lamaty, C., 284, 384 Lambert, C., 191, 203, 212 Lancelot, C. J., 297, 384 Lang, W., 155, 162, 193, 213 Langeveld-Voss, B. M. W., 124, 211 Längle, D., 137, 150, 164, 165, 166, 167, 199, 200, 202, 204, 205, 206, 216 Larcombe-McDouall, J. B., 240, 265 Larsen, J. P., 55, 86, 115 Larson, A. C., 200, 209 Lasarev, G., 109, 114, 116 Lassere, F., 63, 117 Lavagnini, I., 38, 93, 95, 116, 118 Laviron, E., 42, 118

395

Lawless, J. G., 39, 118 Laynez, J., 240, 264 Le Fèvre, J. W., 172, 212 Lebus, S., 158, 159, 161, 163, 167, 170, 173, 179, 193, 204, 205, 206, 209, 212, 213, 216 LeCours, S. M., 194, 212 Lederer, P., 60, 115 Ledger, L. M. B., 183, 212 Ledoux, I., 148, 162, 166, 177, 180, 191, 196, 198, 200, 202, 203, 209, 210, 211, 212, 214, 217 Lee, H.-W., 374, 382 Lee, I., 374, 382 Lee, J., 25, 119 Lee, J. Y., 181, 212 Lee, V. Y., 181, 186, 214 Lefrou, C., 29, 68, 69, 105, 114 Lehman, M. S., 231, 263 Lehn, J.-M., 167, 179, 200, 209, 212 Leigh, P. A., 3, 105, 115 Leiserowitz, L., 233, 246, 247, 262, 264 Leising, F., 179, 213 Lequan, M., 203, 212 Leslie, T. M., 187, 208 Leslie, W. M., 61, 63, 118 Leung, D. S., 158, 204, 206, 214 Leung, Y.-K., 324, 379 Leveque, M. A., 48, 118 Levich, V. G., 49, 118 Levine, B. F., 158, 162, 175, 213 Lewis, E. S., 245, 264 Lewis, J. W., 247, 264 Ley, O., 108, 120 Ley, S. V., 70, 118 Li, G., 200, 216 Lide, D. R., 125, 213 Lilly, M. N., 279, 380 Lim, C., 373, 374, 375, 376, 382 Lima, J. L. F. C., 52, 115 Limbach, H.-H., 236, 237, 238, 239, 240, 242, 262, 263, 264, 265 Lin, Y.-S., 314, 382 Lindsay, G. A., 123, 129, 158, 213 Liptay, W., 155, 160, 161, 162, 167, 174, 193, 204, 213, 216 Lisec, T., 105, 119 Lister, T. E., 108, 120 Liu, K.-T., 280, 284, 307, 314, 380, 382 Liwo, A., 38, 117 Llamas-Saiz, A. L., 240, 242, 263, 264 Lomas, J. S., 287, 327, 333, 343, 382

396

Lomax, A., 37, 114 Long, N. J., 123, 191, 213 Looser, H., 167, 174, 214 Lo´pez, C., 240, 263 Lorimer, J. P., 70, 71, 83, 115, 118, 120 Loudon, G. M., 323, 324, 382 Lovejoy, S. M., 158, 204, 206, 214 Low, C. M. R., 70, 118 Luche, J. L., 70, 118 Lukaszuk, K., 159, 167, 193, 217 Lum, K. K., 279, 286, 384 Lumpkin, R. S., 187, 208 Lundquist, P. M., 159, 217 Luneau, I., 247, 263 Luo, Y., 200, 213 Lüttke, W., 179, 211 Luz, Z., 233, 265 Lyznicki, E. P., Jr., 324, 379 Maarsman, A. W., 179, 188, 216 Maas, G., 186, 211, 213, 216 Maccoll, A., 286, 380 Macdonald, D. D., 13, 118 Macfarlane, R. M., 250, 251, 262, 265 Machiguchi, T., 233, 264 Maciel, G. E., 233, 265 Macpherson, J. V., 54, 69, 100, 118 Madeiros, M.J., 39, 114 Madigan, N. A., 82, 118 Maeda, Y., 299, 300, 381 Maerschalk, C., 79, 119 Maes, J., 202, 206, 216 Magallanes, J. F., 40, 117 Maggini, M., 42, 119 Magno, F., 38, 93, 95, 96, 116, 118, 119 Mah, Y., 62, 113 Mahon, P. J., 37, 62, 113, 114 Maker, P. D., 162, 163, 198, 213, 215 Malagoli, M., 180, 215 Malinski, T., 19, 23, 29, 117 Mammen, M., 123, 216 Man, H.-T., 179, 209 Mann, D. R., 37, 114 Mann, T. F., 37, 114 Manning, G., 60, 118 Männle, F., 236, 264 Manoli, Y., 105, 117 Marcaccio, M., 42, 119 Marcar, S., 69, 100, 118 Marcus, R. A., 22, 118 Marder, S. R., 123, 162, 167, 179, 180,

AUTHOR INDEX

181, 182, 186, 187, 188, 191, 209, 210, 211, 212, 213 Margerum, J. D., 249, 264 Marken, F., 3, 23, 25, 70, 71, 74, 75, 79, 80, 81, 82, 109, 114, 116, 117, 118, 119 Marks, T. J., 123, 129, 137, 141, 151, 158, 171, 179, 187, 191, 196, 208, 210, 211, 212, 213 Marsman, A. W., 166, 179, 188, 199, 211, 215, 216 Martin, R. D., 57, 105, 118 Martin, R. L., 37, 114 Maruyama, Y., 247, 263 Maslak, P., 173, 213 Mason, D., 96, 115 Mason, T. J., 69, 70, 71, 72, 73, 75, 83, 116, 118, 120 Mathias, J. P., 123, 216 Matschiner, R., 137, 150, 159, 164, 165, 166, 167, 193, 199, 200, 202, 204, 205, 206, 209, 216, 217 Matsuda, H., 34, 108, 118 Matsuda, K., 70, 118 Matsumoto, T., 305, 306, 346, 347, 362, 382, 383 Matsuo, T., 258, 264 Matsushita, H., 249, 263 Matysik, F., 82, 118 Matysik, S., 82, 118 Mayer, M. G., 162, 213 Mayne, C. L., 238, 239, 263 Mayr, H., 292, 384 McAleer, J., 93, 96, 120 McCarthy, T. D., 50, 52, 59, 69, 105, 119 McClelland, R. A., 366, 367, 368, 371, 382 McCleverty, J. A., 5, 119 McComb, I.-H., 158, 204, 206, 214 McConald, P. J., 236, 263 McCormick, M. J., 35, 114 McGarrah, D. B., 109, 118 McGuire, G. R., 93, 117 McMahon, R. J., 180, 212 Meerholz, K., 159, 167, 193, 203, 212, 217 Mehlhorn, A., 179, 186, 215 Mehrsheikh-Mohammadi, M. E., 307, 308, 309, 380 Meier, B. H., 234, 235, 264, 265 Meier, U., 123, 211 Meijer, E., 124, 211 Membrey, F., 316, 322, 381

AUTHOR INDEX

Meredith, G. R., 162, 179, 180, 188, 193, 210, 215 Merz, K. M., Jr., 236, 264 Metzler, M. R., 19, 58, 115, 116 Meunier-Prest, R., 42, 109, 118 Meutermans, W., 240, 264 Meyer, M., 105, 117 Meyer, R., 234, 265 Meyers, F., 180, 182, 186, 200, 209, 213, 215 Meyrueix, R., 179, 213 Michael, A. C., 96, 118 Michael, H., 307, 379 Michl, J., 173, 213 Midrier, L., 167, 193, 209 Midwinter, J. E., 133, 217 Mignani, G., 173, 179, 213 Mihel, I., 285, 382 Mikkelsen, K. V., 151, 213 Miles, F. B., 324, 383 Miller, C. K., 162, 213 Miller, L. J., 249, 264 Miller, R. D., 158, 173, 181, 186, 209, 213, 214 Miller, W. H., 226, 262 Mills, G., 70, 118 Mindl, J., 291, 316, 319, 320, 321, 341, 382 Mingos, D.M. P., 35, 119 Miniewicz, A., 123, 214 Minkin, V. I., 179, 213 Mishima, M., 269, 274, 276, 277, 279, 280, 286, 287, 289, 290, 291, 295, 296, 297, 298, 299, 300, 304, 307, 308, 309, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 362, 363, 364, 373, 380, 381, 382, 383 Mita, I., 251, 263 Mitani, T., 247, 263 Mito, A., 163, 213 Mitoseriu, L. C., 52, 115 Miyamoto, T., 279, 287, 288, 381 Miyata, S., 123, 168, 200, 204, 206, 214 Mocak, J., 9, 108, 115 Mochida, T., 255, 257, 258, 264 Moe, H., 270, 380 Moir, J. E., 37, 114 Mokwa, W., 105, 117 Moldenhauer, F., 175, 210 Moldoveanu, S., 92, 114 Moninot, G., 42, 116, 118

397

Montenegro, M. I., 39, 63, 67, 68, 114, 119 More O’Ferrall, R. A., 319, 383 Mori, A., 228, 265 Mori, N., 259, 264 Mori, R., 228, 265 Mori, T., 251, 263 Morin, F. G., 238, 239, 263 Morimoto, H., 274, 275, 289, 309, 310, 311, 312, 313, 314, 335, 336, 338, 339, 340, 344, 381 Moritomo, Y., 255, 257, 259, 264, 265 Morrison, I. D., 164, 166, 179, 191, 199, 201, 213 Morten, D. H., 373, 379 Mosher, H. S., 249, 264 Moss, E. T., 240, 262 Moubarski, B., 109, 114 Mount, A., 46, 113 Moylan, C. R., 158, 159, 173, 181, 186, 204, 206, 213, 214, 217 Muir, R. J., 307, 379 Muller, J., 167, 193, 209 Mulliken, R.S., 143, 214 Munn, R. W., 123, 214 Murata, A., 269, 274, 276, 277, 279, 295, 304, 307, 308, 309, 374, 380, 381, 383 Murphy, M., 71, 120 Murr, B. L., 284, 384 Murray, K., 109, 114 Mustacich, R. V., 123, 129, 158, 210 Mustanir, 346, 347, 350, 352, 354, 355, 356, 357, 383 Musumarra, G., 373, 382 Nadjo, L., 42, 77, 119 Nagakura, S., 247, 264 Nagaoka, S., 229, 235, 264 Nagaosa, Y., 109, 114 Nagashima, U., 229, 247, 263, 264 Naito, T., 251, 263 Nakagaki, R., 247, 264 Nakagawa, Y., 80, 117 Nakajima, K., 204, 206, 214 Nakamoto, K., 297, 298, 299, 300, 381 Nakamura, H., 347, 383 Nakamura, J., 247, 264 Nakamura, K., 204, 206, 214 Nakashima, K., 287, 381 Nakata, K., 274, 279, 280, 284, 286, 287, 289, 345, 347, 359, 360, 363, 364, 381, 383

398

Nalwa, H. S., 123, 168, 200, 204, 206, 214 Namboothiri, I. N. N., 179, 216 Nei, L., 5, 116 Nelson, R. F., 39, 119 Nerenz, H., 177, 209 Neta, P., 41, 119 Nicholson, R. S., 38, 41, 119 Nicoud, J. F., 135, 168, 196, 214 Niki, K., 3, 116 Nishida, S., 291, 341, 383 Nishimoto, K., 279, 287, 289, 345, 347, 359, 360, 363, 364, 383 Nishimura, Y., 228, 265 Nixdorf, M., 173, 211 Nixon, A. C., 279, 383 Noda, Y., 257, 264 Nomura, H., 279, 287, 345, 347, 359, 360, 363, 381, 383 Nonaka, T., 70, 118 Noordman, O. F. J., 166, 201, 202, 214 Norman, R. O. C., 270, 383 Northing, R. J., 96, 115 Nowacka, M., 38, 117 Nowlan, V. J., 324, 379 Noyce, D. S., 304, 324, 383 Nunzi, J.-M., 198, 211 O’Brien, M., 319, 383 Ohe, M., 274, 279, 280, 287, 289, 359, 381 Ohno, A., 300, 384 Ohshima, Y., 225, 265 Ohta, M., 300, 384 Ohta, N., 229, 264 Okamoto, H., 247, 263 Okamoto, K., 308, 309, 385 Okamoto, Y., 268, 269, 279, 380 Okamura, M., 300, 384 Okaniwa, K., 247, 263 Okusako, Y., 297, 300, 381 Olabe, J. A., 40, 117 Olah, G. A., 302, 384 Oldham, K. B., 20, 21, 109, 119 Oliveira-Brett, A. M., 54, 80, 115 Olivieri, A. C., 230, 238, 239, 263, 265 Olsen, S. A., 109, 119 Onsanger, L., 148, 185, 187, 214 Ookubo, N., 137, 204, 206, 215 Opella, S. J., 233, 263 Orlandi, G., 42, 119 Orlovic, M., 285, 382 Orr, B. J., 136, 214

AUTHOR INDEX

Ortiz, R., 167, 209 Osajima, E., 279, 285, 384 Osaki, K., 246, 262 Ossowski, T., 38, 117 Oswana, S., 70, 119 Otsu, T., 279, 284, 381 Oudar, J. L., 143, 198, 210, 214 Oyama, K., 324, 379 Padmanabhan, K., 245, 264 Page, R. D., 37, 114 Page, S. D., 5, 71, 72, 73, 75, 82, 116, 120 Paley, M. S., 167, 174, 214 Palto, S. P., 167, 209 Pan, F., 123, 204, 211, 216 Pangborn, A., 181, 208 Paolucci, F., 42, 119 Parker, V. D., 3, 60, 113, 118, 119 Pastore, P., 95, 96, 118, 119 Patel, K., 8, 120 Paul, I. C., 231, 263, 265 Payne, A. W. R., 109, 117 Payne, R. M., 5, 115 Peaceman, D. W., 93, 119 Pearson, W. B., 143, 214 Pelizzetti, E., 70, 119 Peña, M. J., 64, 119 Perry, J. W., 180, 182, 186, 188, 209, 213, 215 Perry, K. J., 188, 215 Perry, T. T., 167, 212 Persico, M., 151, 215 Personov, R. I., 250, 263 Persoons, A., 123, 124, 163, 164, 165, 166, 179, 180, 186, 187, 191, 200, 201, 202, 203, 206, 208, 210, 211, 212, 216 Perusich, S. A., 80, 119 Petelenz, P., 167, 215 Peters, E. N., 277, 279, 280, 282, 380 Peterson, H. J., 274, 316, 318, 380 Peterson, M., 29, 119 Petit, M. A., 191, 215 Petrier, C., 82, 114 Petukhov, V. A., 186, 212 Pfaendler, H. R., 303, 381 Pfluger, E., 29, 114 Phull, S. S., 70, 71, 83, 120 Pierce, B. M., 180, 182, 200, 209, 213 Pierrot, M., 240, 262, 265 Pilgrim, M., 19, 120 Pilkington, M. B. G., 61, 92, 115 Pinson, J., 105, 114

AUTHOR INDEX

Pipin, J., 181, 209 Pivonka, P., 291, 382 Pletcher, D., 3, 5, 39, 114, 117, 119 Poga, C., 159, 217 Polla, E., 285, 382 Pombeiro, A. J. L., 5, 119 Ponzini, F., 172, 210 Popov, S. V., 132, 214 Porter, R. D., 302, 384 Porto, A. M., 270, 384 Posposil, J., 279, 286, 384 Powell, F. E., 45, 119 Prasad, P. N., 123, 172, 214 Prato, M., 42, 119 Press, W. H., 95, 119 Preˆtre, P., 123, 135, 158, 162, 209 Price, R., 5, 114 Prieto, F., 109, 113 Pritchard, R. G., 240, 262 Priyadarshi, S., 194, 214 Prokof’ev, E. P., 186, 212 Prout, K., 5, 115 Puccetti, G., 180, 196, 211, 214 Puglisi, V. J., 37, 114 Qian, W. J., 86, 117 Rachford, H. H., 93, 119 Radchenko, I. D., 186, 215 Raffia, S., 42, 119 Ragsdale, S. R., 25, 119 Raju, B., 244, 264 Ramamurthy, V., 245, 264 Rao, C. G., 269, 277, 279, 282, 380 Rao, V. P., 179, 212 Rao, Y.-H., 164, 208 Rappoport, Z., 297, 298, 299, 300, 346, 347, 362, 381, 382, 383 Rase, J., 167, 193, 209 Rashid, A., 41, 119 Rasmussen, S. E., 240, 264 Rathna, A., 179, 216 Ratner, M. A., 123, 129, 137, 141, 151, 158, 171, 179, 187, 191, 196, 208, 210, 211, 212, 213 Rauch, J. E., 164, 210 Ravindranathan, M., 269, 277, 279, 280, 282, 380 Rebane, L. A., 250, 263 Rebbitt, T. O., 5, 70, 74, 75, 82, 116, 118

399

Reck, G., 186, 210 Reddy, D. P., 94, 119 Reed, B. L., 108, 120 Rees, N. V., 50, 52, 54, 59, 69, 105, 119 Rehm, T., 155, 162, 167, 193, 213 Reichardt, C., 174, 184, 185, 186, 200, 214 Reimer, K., 105, 119 Reinhoudt, D., 165, 200, 202, 216 Reinmuth, W. H., 24, 110, 120 Reisner, A., 171, 208 Reisner, G. M., 231, 263 Reisse, J., 70, 79, 116, 119 Rentzepis, P. M., 247, 262 Rérat, C., 240, 262 Reverdy, G., 82, 114 Reynolds, C. H., 236, 264 Rheingold, A. L., 173, 213 Rho, M. M., 277, 380 Rice, J. E., 134, 151, 159, 183, 184, 215, 216, 217 Richard, J. P., 366, 368, 369, 370, 384 Richards, P., 83, 120 Rieker, A., 43, 120 Rikken, G., 162, 179, 180, 188, 210 Ritchie, C. D., 318, 366, 371, 384 Robello, D. R., 181, 208 Roberts, D. A., 163, 214 Roberts, D. D., 279, 297, 298, 300, 384 Robin, M. B., 143, 214 Robinson, J., 63, 64, 114, 117 Rollett, J. S., 93, 117 Rosenbaum, M., 324, 379 Rospert, M., 105, 117 Ross, R. A., 286, 380 Rossomando, P. C., 238, 263 Rothenberg, M. E., 366, 368, 370, 384 Roullier, L., 42, 118 Roux, M. V., 242, 263 Rowe, W. F., 225, 262 Roy, D., 324, 325, 334, 382 Ruasse, M. F., 295, 326, 327, 328, 329, 330, 331, 332, 333, 334, 341, 379, 380, 384 Rudolf, M., 94, 119 Rueda, M., 109, 113 Ruhl, J. C., 41, 119 Ruiter, A. G. T., 164, 211 Rumpel, H., 239, 264 Runser, C., 167, 186, 187, 208, 209, 214 Rusling, J. F., 70, 116 Ryan, J. J., 271, 382, 384

400

Saeki, Y., 279, 286, 287, 289, 297, 298, 299, 300, 359, 360, 363, 364, 381, 383 Saika, A., 235, 264 Saito, E., 249, 264 Sakaguchi, S., 276, 304, 307, 308, 309, 383 Sakai, T., 186, 211, 212 Sakizadeh, K., 373, 382 Sakoda, K., 251, 264 Samara, G. A., 259, 264 Sambrotta, L., 238, 263 Samo´c, A., 172, 214 Samo´c, M., 172, 214 Samson, S., 179, 191, 211, 213 Samuel, I. D. W., 166, 198, 210, 211 Samyn, C., 165, 200, 202, 216 Sanders, G. H. W., 9, 10, 81, 82, 115, 116, 118 Sandorfy, C., 247, 264 Santos, J. H., 9, 108, 115 Sasada, Y., 232, 265 Sasaki, K., 228, 265 Sastre, A., 203, 214 Saunders, L. L., 198, 216 Savage, C. M., 162, 163, 215 Savéant, J. M., 3, 5, 38, 42, 77, 98, 105, 114, 119 Sawada, M., 269, 270, 271, 274, 279, 284, 291, 301, 316, 317, 334, 373, 374, 385 Schade, C., 292, 384 Schadt III, F. L., 297, 384 Schaefer, H. F., 226, 262 Schäfer, G., 200, 214 Schanne, L., 193, 213 Scheffer, J. R., 245, 262, 264 Scheibe, P., 185, 186, 187, 214 Scheiner, S., 225, 264 Scherer, G., 239, 264 Schiavelli, M. D., 304, 383 Schilling, M. L., 179, 211 Schlabach, M., 237, 238, 239, 262, 264 Schleyer, P. von R., 297, 373, 379, 384 Schmälzlin, E., 203, 212 Schmid, S., 202, 206, 216 Schmidt, G. M. J., 247, 262 Schmidt, R., 245, 264 Schnakenberg, U., 105, 119 Schneider, S., 185, 186, 187, 214 Schreiber, M., 204, 206, 209 Schriesheim, A., 272, 274, 316, 318, 319, 380 Schröder, G., 219, 265

AUTHOR INDEX

Schubert, P. F., 70, 120 Schuld, T., 191, 211 Schulz, B., 182, 212 Schulz, G., 200, 216 Schütt, J., 183, 214 Schwesinger, R., 167, 193, 209 Scorrano, G., 42, 119, 316, 379 Seddon, B. J., 105, 119 Seitz, G., 258, 265 Seki, Y., 297, 298, 380 Sekiya, H., 228, 265 Semmingsen, D., 258, 259, 264, 265 Sens, R., 159, 167, 193, 217 Seo, E. T., 39, 119 Serbutoviez, C., 123, 196, 211, 214 Serpone, N., 70, 115 Serratosa, F., 258, 265 Seto, C. T., 123, 216 Seto, N. O. L., 324, 379 Sevcik, A., 2, 119 Shaffer, R. E., 109, 119 Shain, I., 38, 41, 119 Shao, Y., 105, 119 Sharp, M., 29, 119 Shaw, S. J., 109, 119 Shea, K. J., 123, 129, 158, 210 Sheik-Bahae, M., 131, 211 Shelton, D. P., 134, 162, 163, 164, 166, 201, 212, 214, 216 Sherborne, B. S., 158, 208 Sheu, H.-C., 314, 382 Shi, H. S., 86, 117 Shi, Y., 123, 129, 158, 210 Shiau, W.-I., 231, 263, 265 Shibata, T., 308, 309, 385 Shibuya, Y., 269, 279, 374, 380 Shida, N., 226, 265 Shimanouchi, H., 232, 265 Shimizu, H., 259, 265 Shimizu, N., 279, 285, 286, 350, 355, 356, 357, 383, 384, 385 Shimoni, L., 123, 208 Shiner, V. J., Jr., 284, 384 Shorter, J., 271, 384 Shoup, D., 93, 119 Shu, C.-F., 284, 382 Siebrand, W., 236, 265 Siegel, J. S., 172, 200, 201, 208, 210, 212 Silk, T., 61, 63, 118 Silva-Martinez, S., 81, 82, 114 Silvers, S. J., 238, 265 Simanek, E. E., 123, 216

AUTHOR INDEX

Simkin, B. Y., 179, 213 Simon, J., 191, 203, 212, 215 Simpson, W. T., 187, 215 Singer, K. D., 123, 129, 133, 139, 158, 162, 186, 209, 213, 215 Sixl, H., 247, 249, 263, 265 Skiebe, A., 43, 120 Skindhøj, J., 180, 213 Skinner, J. L., 235, 265 Skul’bidenko, A. L., 179, 215 Slawik, M., 167, 215 Slominskii, Y. L., 179, 186, 215 Small, G. W., 109, 119 Small, R. D., 133, 215 Smedarchina, Z., 236, 265 Smeets, W. J. J., 179, 216 Smith, G., 70, 83, 115 Smith, G. G., 279, 286, 384, 385 Smith, J. A. S., 240, 262, 265 Smith, Z., 225, 262, 265 Sobrados, I., 240, 264 Sohn, J. E., 123, 132, 162, 213, 215 Song, N. W., 166, 215 Soula, G., 173, 213 Southampton Electrochemistry Group (1990), 3, 15, 29, 120 Spackman, R. A., 55, 86, 115 Spanget-Larsen, J., 173, 215 Spangler, C. W., 179, 180, 188, 210 Speiser, B., 87, 120 Spek, A. L., 179, 191, 208, 216 Spiess, H. W., 233, 265 Spreiter, R., 204, 206, 209 Stadler, S., 166, 171, 172, 177, 191, 199, 200, 201, 202, 208, 209, 212, 215 Stähelin, M., 172, 183, 184, 212, 214, 215 Stammers, M. A., 164, 166, 179, 191, 199, 201, 213 Stang, P. J., 303, 384 Staring, E. G. J., 180, 215 Stearn, G. M., 59, 92, 115 Steenken, S., 366, 367, 368, 371, 382 Steier, W. H., 123, 129, 158, 210 Stein, J., 200, 214 Stephenson, B., 286, 380 Stephenson, G. R., 191, 211 Stevens, N. P. C., 87, 120 Stevenson, S. H., 162, 179, 180, 188, 210 Stewart, R., 316, 380, 385 Stewart, W. E., 87, 120 Steybe, F., 167, 179, 188, 193, 215 Stiegman, A. E., 179, 180, 188, 210, 215

401

Stillinger, F. H., 221, 263 Stock, L. M., 269, 385 Stöckli, A., 234, 265 Störzbach, M., 93, 117 Strojek, J. W., 108, 120 Stucky, G. D., 123, 213 Stytsenko, T. S., 186, 212 Suba, C., 186, 214 Sugawara, T., 255, 257, 258, 259, 264, 265 Sumi, K., 259, 265 Suppan, P., 183, 212 Suslick, K. S., 70, 120, 194, 215 Susuki, T., 279, 289, 290, 291, 300, 381 Sutter, K., 123, 135, 158, 209 Sutton, L. E., 172, 215 Svensson, C., 255, 265 Svirko, Y. P., 132, 214 Swain, G. M., 108, 120 Swann, M., 180, 210 Swanson, S. A., 181, 186, 214 Szablewski, M., 180, 210, 215 Szabo, A., 93, 119 Szeverenyi, N. M., 233, 265 Tadjer, A., 179, 186, 215 Taft, R. W., 183, 212 Tagawa, H., 248, 263 Tait, R. J., 108, 120 Tajammal, S., 240, 262 Takada, T., 137, 204, 206, 215 Takahashi, C., 163, 213 Takahashi, H., 247, 259, 264, 265 Takasu, I., 259, 265 Takemura, T., 229, 264 Takeno, T., 233, 264 Takeshita, H., 228, 265 Takeuchi, K., 280, 308, 309, 380, 385 Tallec, A., 5, 114 Tallman, D. E., 109, 119 Tam, K. Y., 109, 120 Tam, W., 162, 179, 180, 188, 191, 210 Tamura, I., 257, 264 Tanaka, H., 108, 118 Tanaka, K., 70, 119 Tanaka, K., 225, 265 Tanaka, T., 225, 265 Taniguchi, H., 346, 347, 353, 382, 383 Tardivel, R., 5, 114 Tardy, C., 5, 114 Tashiro, M., 302, 384 Taylor, G., 93, 96, 120

402

Taylor, R., 245, 265, 270, 279, 282, 286, 381, 383, 384, 385 Tedesco, V., 62, 113 Templeton, D. H., 230, 263 Tenne, R., 8, 120 Terao, T., 235, 264 Terasaki, T., 355, 383 Terhune, R. W., 162, 163, 215 Terzian, R., 70, 119 Testa, A. C., 24, 110, 120 Teukolsky, S. A., 95, 119 Tevesov, A. A., 167, 209 Thami, T., 191, 203, 212, 215 Therien, M. J., 194, 212, 214 Thiébault, A., 186, 214 Thomas, P. R., 180, 215 Thornton, A., 180, 215 Tidwell, T. T., 304, 305, 307, 308, 323, 324, 325, 334, 379, 380, 382, 385 Tiemann, B. G., 180, 182, 186, 209, 213 Tipping, A. E., 240, 262 Titman, J. J., 233, 265 Toiron, C., 240, 262 Tokunaga, E., 287, 381 Tokura, Y., 255, 257, 259, 264, 265 Tolbert, L. M., 186, 215 Tolmachev, A. I., 179, 215 Tomasi, J., 151, 215 Tomonari, M., 137, 204, 206, 215 Toome, V., 19, 120 Toppare, L., 70, 113 Torres, T., 203, 214 Toullec, J., 324, 385 Treptow, B., 150, 164, 166, 199, 200, 202, 216 Trommsdorf, H. P., 235, 250, 262, 265 Trotter, J., 245, 265 Trueblood, K, N., 233, 263 Tsao, M.-L., 314, 382 Tschunky, P., 29, 105, 120 Tsionsky, M., 105, 108, 116, 120 Tsuji, T., 228, 265 Tsuji, Y., 269, 274, 279, 280, 284, 287, 288, 289, 291, 295, 359, 381, 385 Tsujino, Y., 38, 120 Tsuno, Y., 269, 270, 271, 272, 274, 275, 276, 277, 279, 280, 284, 285, 286, 287, 288, 289, 290, 291, 295, 296, 297, 298, 299, 300, 301, 304, 307, 308, 309, 310, 311, 312, 313, 314, 316, 317, 334, 335, 336, 338, 339, 340, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354,

AUTHOR INDEX

355, 356, 357, 358, 359, 360, 362, 363, 364, 373, 374, 375, 380, 381, 382, 383, 384, 385 Tulinsky, A., 237, 238, 262, 265 Turbeville, W., 247, 265 Turner, P., 225, 265 Twieg, R. J., 123, 135, 137, 158, 159, 167, 168, 174, 175, 180, 181, 182, 186, 204, 206, 211, 214, 215, 217 Tykwinski, R. R., 204, 206, 209 Tyley, G. P., 98, 115 Tyutyulkov, N., 179, 186, 215 Ujita, H., 228, 265 Underiner, T. L., 307, 309, 380 Unwin, P. R., 48, 54, 57, 69, 96, 98, 100, 105, 115, 118, 120 Urbanczyk-Lipkowska, Z., 230, 263 Uschmann, J., 173, 215 Usui, S., 344, 346, 347, 350, 351, 383 Uzan, R., 332, 333, 380 v. Auwers, K., 172, 215 Vallat, A., 42, 118 Van, T. C., 166, 202, 217 van der Linden, J. G. M., 191, 208 van Hulst, N. F., 164, 166, 201, 202, 211, 214 van Hutten, P. F., 173, 215 van Pelt, P., 167, 211 Van Steveninck, R. F. M., 109, 114 van Walree, C. A., 166, 179, 188, 199, 211, 215, 216 Vandercammen, J., 79, 119 Varanasi, P. R., 179, 216 Vecera, M., 291, 316, 319, 320, 321, 341, 382 Vega, S., 238, 263 Veldman, N., 191, 208 Venkatesan, K., 245, 264 Verbiest, T., 123, 124, 165, 166, 200, 202, 211, 212, 216 Verbrugge, M. W., 96, 120 Vetterling, W. T., 95, 119 Vianello, E., 38, 119 Vielstich, W., 38, 57, 80, 114, 120 Vijayaraghavan, D., 236, 263 Vila, A. J., 230, 265 Villadsen, J. V., 87, 120 Villaeys, A., 191, 215 Vincent, E. J., 240, 262 Vogel, E., 237, 238, 262

AUTHOR INDEX

Voigt-Martin, I. G., 200, 216 Völker, S., 251, 265 Von Stackelberg, M., 19, 120 Voss, B., 191, 208 Vu, T., 109, 114, 116 Wagner, B., 105, 119 Wagner, P. J., 245, 263 Wagnière, G., 158, 216 Wagnière, G. H., 123, 124, 137, 211, 216 Walker, R., 70, 120 Waller, A. M., 59, 62, 92, 100, 115 Waller, D. N., 70, 79, 80, 82, 116 Waller, G. A., 109, 113 Walsh, C. A., 158, 209 Walsh, L., 245, 262 Walsh, P. K., 222, 262 Walter, J. N., 19, 58, 109, 114, 115 Walters, M. K., 93, 108, 115, 117 Walton, D. J., 70, 71, 72, 73, 75, 83, 115, 116, 118, 120 Walton, D. R. M., 305, 380 Wang, C. H., 194, 212 Wang, J., 5, 120 Wang, J. H., 70, 120 Ward, J. F., 136, 162, 214, 216 Warta, R., 249, 265 Watanabe, S., 279, 285, 286, 384, 385 Watanabe, T., 204, 206, 214 Watson, D. G., 123, 208 Watson, J. D., 220, 265 Watson, T. M., 279, 298, 384 Wawer, I., 236, 264 Way, D. M., 19, 109, 114 Webb, L. E., 237, 265 Weber, H.-M., 186, 211, 216 Webster, R. D., 3, 120 Wedd, A. G., 19, 108, 109, 114, 116 Wehning, D., 155, 162, 167, 193, 213 Wehrle, B., 237, 238, 239, 240, 262, 263, 264, 265 Weidenberg, H., 204, 213 Weiler, L., 232, 265 Wellington, R. G., 3, 5, 60, 93, 105, 115, 116 Wenseleers, W., 180, 215 West, R., 258, 265 Wetzel, W. H., 279, 286, 385 Weyrauch, T., 169, 209 Whitaker, C. M., 180, 212 White, A. M., 302, 384 White, H. S., 25, 105, 119, 120

403

Whitesell, J. K., 198, 216 Whitesides, G. M., 123, 216 Whittall, I. R., 191, 202, 206, 216 Whyte, T., 37, 114 Wichern, J., 167, 177, 193, 209 Wightman, R. M., 3, 23, 29, 40, 55, 63, 96, 103, 105, 116, 117, 118, 120 Will, F. G., 5, 120 Willand, C. S., 181, 208 Willets, A., 134, 151, 216 Williams, D. E., 230, 265 Williams, D. J., 123, 162, 172, 181, 208, 213, 214, 216 Wilson, A. J. C., 129, 216 Wilson, E. B., 225, 262 Wilson, R. B., 231, 263 Wilson, R. J., 198, 216 Winard, R., 70, 116 Wipf, D. O., 29, 40, 103, 105, 120 Wirtz, J., 245, 263 Wizinger, R., 198, 216 Wolff, J. J., 123, 137, 150, 164, 165, 166, 167, 199, 200, 201, 202, 203, 204, 205, 206, 216 Wolfrum, H., 250, 263 Wondenberg, R. H., 166, 199, 211 Wong, D. K. Y., 105, 120 Wong, H., 191, 208, 211 Wong, K. Y., 179, 212 Wong, M. S., 123, 204, 216 Woolf, L. A., 19, 120 Wortmann, R., 123, 137, 149, 150, 152, 153, 158, 159, 161, 163, 164, 165, 166, 167, 170, 173, 174, 179, 188, 193, 199, 200, 202, 203, 204, 205, 206, 209, 213, 214, 215, 216, 217 Woznicki, W., 38, 117 Wright, W. V., 327, 333, 343, 382 Wu, T.-R., 307, 314, 382 Würthner, F., 159, 167, 174, 179, 188, 193, 217 Xu, J., 108, 120 Xu, L. Y. F., 105, 120 Yakimanski, A., 200, 216 Yamada, Y., 204, 206, 214 Yang, J.-S., 314, 382 Yannoni, C. S., 239, 263, 264 Yap, G. P. A., 173, 213 Yariv, A., 133, 217 Yasuda, N., 259, 265

404

Yates, B. L., 286, 384 Yates, K., 316, 380, 385 Yeh, P., 133, 217 Yoh, S.-D., 373, 374, 375, 382, 385 Yonetani, K., 249, 263 Yoon, H. N., 179, 209 Yoshioka, M., 271, 385 Young, V. A., 324, 380 Yudin, S. G., 167, 209 Yukawa, Y., 269, 270, 271, 272, 274, 279, 284, 291, 295, 301, 316, 317, 334, 373, 374, 385 Yurrtas, K., 70, 113 Yuzawa, T., 247, 265 Zalkin, A., 230, 263 Zerbetto, F., 42, 119, 236, 265

AUTHOR INDEX

Zerbi, G., 181, 210 Zerner, M. C., 151, 212 Zernike, F., 133, 217 Zhang, H., 81, 120 Zhang, Z. X., 86, 117 Zhao, X., 186, 215 Zheludev, N. I., 132, 214 Zheng, F., 42, 117 Zhou, J., 43, 120 Ziari, M., 123, 129, 158, 210 Zimmermann, H., 238, 239, 264, 265 Zollinger, H., 175, 199, 217 Zschokke-Gränacher, I., 172, 212, 214 Zuliani, P., 181, 210 Zyss, J., 123, 148, 162, 166, 173, 177, 180, 196, 198, 200, 202, 203, 209, 210, 211, 212, 213, 214, 217

Cumulative Index of Authors

Ahlberg, P., 19, 223 Albery, W. J., 16, 87; 28, 139 Alden, J. A., 32, 1 Allinger, N. I., 13, 1 Anbar, M., 7, 115 Arnett, E. M., 13, 83; 28, 45 Ballester, M., 25, 267 Bard, A. J., 13, 155 Beer, P. D., 31, 1 Baumgarten, M., 28, 1 Bell, R. P., 4, 1 Bennett, J. E., 8, 1 Bentley, T. W., 8, 151; 14, 1 Berg, U., 25, 1 Berger, S., 16, 239 Bernasconi, C. F., 27, 119 Bethell, D., 7, 153; 10, 53 Blackburn, G. M., 31, 249 Blandamer, M. J., 14, 203 Bond, A. M., 32, 1 Bowden, K., 28, 171 Brand, J. C. D., 1, 365 Bra¨ndstro¨m, A., 15, 267 Brinkman, M. R., 10, 53 Brown, H. C., 1, 35 Buncel, E., 14, 133 Bunton, C. A., 22, 213 Cabell-Whiting, P. W., 10, 129 Cacace, F., 8, 79 Capon, B., 21, 37 Carter, R. E., 10, 1 Chen, Z., 31, 1 Collins, C. J., 2, 1 Compton, R. G., 32, 1 Cornelisse, J., 11, 225 Crampton, M. R., 7, 211 Datta, A., 31, 249 Davidson, R. S., 19, 1; 20, 191 Denham, H., 31, 249 Desvergne, J. P., 15, 63

de Gunst, G. P., 11, 225 de Jong, F., 17, 279 Dosunmu, M. I., 21, 37 Eberson, L., 12, 1; 18, 79; 31, 91 Ekland, J. C., 32, 1 Emsley, J., 26, 255 Engdahl, C., 19, 223 Farnum, D. G., 11, 123 Fendler, E. J., 8, 271 Fendler, J. H., 8, 271; 13, 279 Ferguson, G., 1, 203 Fields, E. K., 6, 1 Fife, T. H., 11, 1 Fleischmann, M., 10, 155 Frey, H. M., 4, 147 Fujio, M., 32, 267 Gale, P. A., 31, 1 Gilbert, B. C., 5, 53 Gillespie, R. J., 9, 1 Gold, V., 7, 259 Goodin, J. W., 20, 191 Gould, I. R., 20, 1 Greenwood, H. H., 4, 73 Hammerich, O., 20, 55 Harvey, N. G., 28, 45 Hasegawa, M., 30, 117 Havinga, E., 11, 225 Henderson, R. A., 23, 1 Henderson, S., 23, 1 Hibbert, F., 22, 113; 26, 255 Hine, J., 15, 1 Hogen-Esch, T. E., 15, 153 Hogeveen, H., 10, 29, 129 Huber, W., 28, 1 Ireland, J. F., 12, 131 Iwamura, H., 26, 179 Johnson, S. L., 5, 237 Johnstone, R. A. W., 8, 151 Jonsa¨ll, G., 19, 223 Jose´, S. M., 21, 197 405

Kemp, G., 20, 191 Kice, J. L., 17, 65 Kirby, A. J., 17, 183; 29, 87 Kitagawa, T., 30, 173 Kluger, R. H., 25, 99 Kochi, J. K., 29, 185 Kohnstam, G., 5, 121 Korolev, V. A., 30, 1 Korth, H.-G., 26, 131 Kramer, G. M., 11, 177 Kreevoy, M. M., 6, 63; 16, 87 Kunitake, T., 17, 435 Kurtz, H. A., 29, 273 Ledwith, A., 13, 155 Lee, I., 27, 57 Le Fe`vre, R. J. W., 3, 1 Liler, M., 11, 267 Long, F. A., 1, 1 Lu¨ning, U., 30, 63 Maccoll, A., 3, 91 Mandolini, L., 22, 1 Matsson, O., 31, 143 McWeeny, R., 4, 73 Melander, L., 10, 1 Mile, B., 8, 1 Miller, S. I., 6, 185 Modena, G., 9, 185 More O’Ferrall, R. A., 5, 331 Morsi, S. E., 15, 63 Mu¨llen, K., 28, 1 Nefedov, O. M., 30, 1 Neta, P., 12, 223 Nibbering, N. M. M., 24, 1 Norman, R. O. C., 5, 33 Nyberg, K., 12, 1 Okamoto, K., 30, 173 Olah, G. A., 4, 305 Page, M. I., 23, 165 Parker, A. J.., 5, 173 Parker, V. D., 19, 131; 20, 55 Peel, T. E., 9, 1

406

Perkampus, H. H., 4, 195 Perkins, M. J., 17, 1 Pittman, C. U. Jr, 4, 305 Pletcher, D., 10, 155 Pross, A., 14, 69; 21, 99 Ramirez, F., 9, 25 Rappoport, Z., 7, 1; 27, 239 Reeves, L. W., 3, 187 Reinhoudt, D. N., 17, 279 Ridd, J. H., 16, 1 Riveros, J. M., 21, 197 Robertson, J. M., 1, 203 Rose, P. L., 28, 45 Rosenthal, S. N., 13, 279 Ruasse, M.-F., 28, 207 Russell, G. A., 23, 271 Samuel, D., 3, 123 Sanchez, M. de N. de M., 21, 37 Sandstro¨m, J., 25, 1 Save´ant, J.-M., 26, 1 Savelli, G., 22, 213 Schaleger, L. L., 1, 1 Scheraga, H. A., 6, 103 Schleyer, P. von R., 14, 1 Schmidt, S. P., 18, 187

CUMULATIVE INDEX OF AUTHORS

Schuster, G. B., 18, 187; 22, 311 Scorrano, G., 13, 83 Shatenshtein, A. I., 1, 156 Shine, H. J., 13, 155 Shinkai, S., 17, 435 Siehl, H.-U., 23, 63 Silver, B. L., 3, 123 Simonyi, M., 9, 127 Sinnott, M. L., 24, 113 Stock, L. M., 1, 35 Sugawara, T., 32, 219 Sustmann, R., 26, 131 Symons, M. C. R., 1, 284 Takashima, K., 21, 197 Takasu, I., 32, 219 Takeuchi, K., 30, 173 Ta-Shma, R., 27, 239 Tedder, J. M., 16, 51 Tee, O. S., 29, 1 Thatcher, G. R. J., 25, 99 Thomas, A., 8, 1 Thomas, J. M., 15, 63 Tonellato, U., 9, 185 Toullec, J., 18, 1 Tsuno, Y., 32, 267

Tu¨do¨s, F., 9, 127 Turner, D. W., 4, 31 Turro, N. J., 20, 1 Ugi, I., 9, 25 Walton, J. C., 16, 51 Ward, B., 8, 1 Watt, C. I. F., 24, 57 Wentworth, P., 31, 249 Westaway, K. C., 31, 143 Westheimer, F. H., 21, 1 Whalley, E., 2, 93 Williams, A., 27, 1 Williams, D. L. H., 19, 381 Williams, J. M. Jr, 6, 63 Williams, J. O., 16, 159 Williams, R. V., 29, 273 Williamson, D. G., 1, 365 Wilson, H., 14, 133 Wolf, A. P., 2, 201 Wolff, J. J., 32, 121 Wortmann, R., 32, 121 Wyatt, P. A. H., 12, 131 Zimmt, M. B., 20, 1 Zollinger, H., 2, 163 Zuman, P., 5, 1

Cumulative Index of Titles Abstraction, hydrogen atom, from O⫺H bonds, 9, 127 Acid solutions, strong, spectroscopic observation of alkylcarbonium ions in, 4, 305 Acid–base behaviour in macrocycles and other concave structures, 30, 63 Acid–base properties of electronically excited states of organic molecules, 12, 131 Acids and bases, oxygen and nitrogen in aqueous solution, mechanisms of proton transfer between, 22, 113 Acids, reactions of aliphatic diazo compounds with, 5, 331 Acids, strong aqueous, protonation and solvation in, 13, 83 Activation, entropies of, and mechanisms of reactions in solution, 1, 1 Activation, heat capacities of, and their uses in mechanistic studies, 5, 121 Activation, volumes of, use for determining reaction mechanisms, 2, 93 Addition reactions, gas-phase radical directive effects in, 16, 51 Aliphatic diazo compounds, reactions with acids, 5, 331 Alkyl and analogous groups, static and dynamic stereochemistry of, 25, 1 Alkylcarbonium ions, spectroscopic observation in strong acid solutions, 4, 305 Ambident conjugated systems, alternative protonation sites in, 11, 267 Ammonia, liquid, isotope exchange reactions of organic compounds in, 1, 156 Anions, organic, gas-phase reactions of, 24, 1 Antibiotics, ␤-lactam, the mechanisms of reactions of, 23, 165 Aqueous mixtures, kinetics of organic reactions in water and, 14, 203 Aromatic photosubstitution, nucleophilic, 11, 225 Aromatic substitution, a quantitative treatment of directive effects in, 1, 35 Aromatic substitution reactions, hydrogen isotope effects in, 2, 163 Aromatic systems, planar and non-planar, 1, 203 Aryl halides and related compounds, photochemistry of, 20, 191 Arynes, mechanisms of formation and reactions at high temperatures, 6, 1 A-SE2 reactions, developments in the study of, 6, 63 Base catalysis, general, of ester hydrolysis and related reactions, 5, 237 Basicity of unsaturated compounds, 4, 195 Bimolecular substitution reactions in protic and dipolar aprotic solvents, 5, 173 Bromination, electrophilic, of carbon–carbon double bonds: structure, solvent and mechanisms, 28, 207 13

C NMR spectroscopy in macromolecular systems of biochemical interest, 13, 279 Captodative effect, the, 26, 131 Carbanion reactions, ion-pairing effects in, 15, 153 Carbene chemistry, structure and mechanism in, 7, 163 Carbenes having aryl substituents, structure and reactivity of, 22, 311 Carbocation rearrangements, degenerate, 19, 223 Carbocationic systems, the Yukawa–Tsuno relationship in, 32, 267 Carbon atoms, energetic, reactions with organic compounds, 3, 201 Carbon monoxide, reactivity of carbonium ions towards, 10, 29 Carbonium ions (alkyl), spectroscopic observation in strong acid solutions, 4, 305 Carbonium ions, gaseous, from the decay of tritiated molecules, 8, 79 Carbonium ions, photochemistry of, 10, 129 407

408

CUMULATIVE INDEX OF TITLES

Carbonium ions, reactivity towards carbon monoxide, 10, 29 Carbonyl compounds, reversible hydration of, 4, 1 Carbonyl compounds, simple, enolisation and related reactions of, 18, 1 Carboxylic acids, tetrahedral intermediates derived from, spectroscopic detection and investigation of their properties, 21, 37 Catalysis by micelles, membranes and other aqueous aggregates as models of enzyme action, 17, 435 Catalysis, enzymatic, physical organic model systems and the problem of, 11, 1 Catalysis, general base and nucleophilic, of ester hydrolysis and related reactions, 5, 237 Catalysis, micellar, in organic reactions; kinetic and mechanistic implications, 8, 271 Catalysis, phase-transfer by quaternary ammonium salts, 15, 267 Catalytic antibodies, 31, 249 Cation radicals in solution, formation, properties and reactions of, 13, 155 Cation radicals, organic, in solution, and mechanisms of reactions of, 20, 55 Cations, vinyl, 9, 135 Chain molecules, intramolecular reactions of, 22, 1 Chain processes, free radical, in aliphatic systems involving an electron transfer reaction, 23, 271 Charge density–NMR chemical shift correlation in organic ions, 11, 125 Chemically induced dynamic nuclear spin polarization and its applications, 10, 53 Chemiluminescence of organic compounds, 18, 187 Chirality and molecular recognition in monolayers at the air–water interface, 28, 45 CIDNP and its applications, 10, 53 Conduction, electrical, in organic solids, 16, 159 Configuration mixing model: a general approach to organic reactivity, 21, 99 Conformations of polypeptides, calculations of, 6, 103 Conjugated, molecules, reactivity indices, in, 4, 73 Cross-interaction constants and transition-state structure in solution, 27, 57 Crown-ether complexes, stability and reactivity of, 17, 279 Crystallographic approaches to transition state structures, 29, 87 Cyclodextrins and other catalysts, the stabilization of transition states by, 29, 1 D2O–H2O mixtures, protolytic processes in, 7, 259 Degenerate carbocation rearrangements, 19, 223 Deuterium kinetic isotope effects, secondary, and transition state structure, 31, 143 Diazo compounds, aliphatic, reactions with acids, 5, 331 Diffusion control and pre-association in nitrosation, nitration, and halogenation, 16, 1 Dimethyl sulphoxide, physical organic chemistry of reactions, in, 14, 133 Diolefin crystals, photodimerization and photopolymerization of, 30, 117 Dipolar aprotic and protic solvents, rates of bimolecular substitution reactions in, 5, 173 Directive effects in aromatic substitution, a quantitative treatment of, 1, 35 Directive effects in gas-phase radical addition reactions, 16, 51 Discovery of the mechanisms of enzyme action, 1947–1963, 21, 1 Displacement reactions, gas-phase nucleophilic, 21, 197 Double bonds, carbon–carbon, electrophilic bromination of: structure, solvent and mechanism, 28, 171 Effective charge and transition-state structure in solution, 27, 1 Effective molarities of intramolecular reactions, 17, 183

CUMULATIVE INDEX OF TITLES

409

Electrical conduction in organic solids, 16, 159 Electrochemical methods, study of reactive intermediates by, 19, 131 Electrochemical recognition of charged and neutral guest species by redox-active receptor molecules, 31, 1 Electrochemistry, organic, structure and mechanism in, 12, 1 Electrode processes, physical parameters for the control of, 10, 155 Electron donor–acceptor complexes, electron transfer in the thermal and photochemical activation of, in organic and organometallic reactions, 29, 185 Electron spin resonance, identification of organic free radicals by, 1, 284 Electron spin resonance studies of short-lived organic radicals, 5, 23 Electron storage and transfer in organic redox systems with multiple electrophores, 28, 1 Electron transfer in the thermal and photochemical activation of electron donor–acceptor complexes in organic and organometallic reactions, 29, 185 Electron-transfer reaction, free radical chain processes in aliphatic systems involving an, 23, 271 Electron-transfer reactions in organic chemistry, 18, 79 Electron-transfer, single, and nucleophilic substitution, 26, 1 Electron transfer, spin trapping and, 31, 91 Electronically excited molecules, structure of, 1, 365 Electronically excited states of organic molecules, acid-base properties of, 12, 131 Energetic tritium and carbon atoms, reactions of, with organic compounds, 2, 201 Enolisation of simple carbonyl compounds and related reactions, 18, 1 Entropies of activation and mechanisms of reactions in solution, 1, 1 Enzymatic catalysis, physical organic model systems and the probolem of, 11, 1 Enzyme action, catalysis by micelles, membranes and other aqueous aggregates as models of, 17, 435 Enzyme action, discovery of the mechanisms of, 1947–1963, 21, 1 Equilibrating systems, isotope effects on nmr spectra of, 23, 63 Equilibrium constants, NMR measurements of, as a function of temperature, 3, 187 Ester hydrolysis, general base and nucleophilic catalysis, 5, 237 Ester hydrolysis, neighbouring group participation by carbonyl groups in, 28, 171 Exchange reactions, hydrogen isotope, of organic compounds in liquid ammonia, 1, 156 Exchange reactions, oxygen isotope, of organic compounds, 2, 123 Excited complexes, chemistry of, 19, 1 Excited molecules, structure of electronically, 3, 365 Force-field methods, calculation of molecular structure and energy by, 13, 1 Free radical chain processes in aliphatic systems involving an electron-transfer reaction, 23, 271 Free radicals, identification by electron spin resonance, 1, 284 Free radicals and their reactions at low temperature using a rotating cryostat, study of, 8, 1 Gaseous carbonium ions from the decay of tritiated molecules, 8, 79 Gas-phase heterolysis, 3, 91 Gas-phase nucleophilic displacement reactions, 21, 197 Gas-phase pyrolysis of small-ring hydrocarbons, 4, 147 Gas-phase reactions of organic anions, 24, 1 General base and nucleophilic catalysis of ester hydrolysis and related reactions, 5, 237

410

CUMULATIVE INDEX OF TITLES

H2O–D2O mixtures, protolytic processes in, 7, 259 Halides, aryl, and related compounds, photochemistry of, 20, 191 Halogenation, nitrosation, and nitration, diffusion control and pre-association in, 16, 1 Heat capacities of activation and their uses in mechanistic studies, 5, 121 Heterolysis, gas-phase, 3, 91 High-spin organic molecules and spin alignment in organic molecular assemblies, 26, 179 Homoaromaticity, 29, 273 Hydrated electrons, reactions of, with organic compounds, 7, 115 Hydration, reversible, of carbonyl compounds, 4, 1 Hydride shifts and transfers, 24, 57 Hydrocarbons, small-ring, gas-phase pyrolysis of, 4, 147 Hydrogen atom abstraction from O⫺H bonds, 9, 127 Hydrogen bonding and chemical reactivity, 26, 255 Hydrogen isotope effects in aromatic substitution reactions, 2, 163 Hydrogen isotope exchange reactions of organic compounds in liquid ammonia, 1, 156 Hydrolysis, ester, and related reactions, general base and nucleophilic catalysis of, 5, 237 Interface, the air–water, chirality and molecular recognition in monolayers at, 28, 45 Intermediates, reactive, study of, by electrochemical methods, 19, 131 Intermediates, tetrahedral, derived from carboxylic acids, spectroscopic detection and investigation of their properties, 21, 37 Intramolecular reactions, effective molarities for, 17, 183 Intramolecular reactions of chain molecules, 22, 1 Ionic dissociation of carbon–carbon ␴-bonds in hydrocarbons and the formation of authentic hydrocarbon salts, 30, 173 Ionization potentials, 4, 31 Ion-pairing effects in carbanion reactions, 15, 153 Ions, organic, charge density–NMR chemical shift correlations, 11, 125 Isomerization, permutational, of pentavalent phosphorus compounds, 9, 25 Isotope effects, hydrogen, in aromatic substitution reactions, 2, 163 Isotope effects, magnetic, magnetic field effects and, on the products of organic reactions, 20, 1 Isotope effects on nmr spectra of equilibrating systems, 23, 63 Isotope effects, steric, experiments on the nature of, 10, 1 Isotope exchange reactions, hydrogen, of organic compounds in liquid ammonia, 1, 150 Isotope exchange reactions, oxygen, of organic compounds, 3, 123 Isotopes and organic reaction mechanisms, 2, 1 Kinetics and mechanisms of reactions of organic cation radicals in solution, 20, 55 Kinetics of organic reactions in water and aqueous mixtures, 14, 203 Kinetics, reaction, polarography and, 5, 1

␤-Lactam antibiotics, the mechanisms of reactions of, 23, 165 Least nuclear motion, principle of, 15, 1 Macrocycles and other concave structures, acid–base behaviour in, 30, 63 Macromolecular systems in biochemical interest, 13C NMR spectroscopy in, 13, 279

CUMULATIVE INDEX OF TITLES

411

Magnetic field and magnetic isotope effects on the products of organic reactions, 20, 1 Mass spectrometry, mechanisms and structure in: a comparison with other chemical processes, 8, 152 Matrix infrared spectroscopy of intermediates with low coordinated carbon, silicon and germanium atoms, 30, 1 Mechanism and structure in carbene chemistry, 7, 153 Mechanism and structure in mass spectrometry: a comparison with other chemical processes, 8, 152 Mechanism and structure in organic electrochemistry, 12 1 Mechanisms and reactivity in reactions of organic oxyacids of sulphur and their anhydrides, 17, 65 Mechanisms, nitrosation, 19, 381 Mechanisms of proton transfer between oxygen and nitrogen acids and bases in aqueous solutions, 22, 113 Mechanisms of reaction in solution, entropies of activation and, 1, 1 Mechanisms of reaction of ␤-lactam antibiotics, 23, 165 Mechanisms of solvolytic reactions, medium effects on the rates and, 14, 10 Mechanisms, organic reaction, isotopes and, 2, 1 Mechanistic analysis, perspectives in modern voltammetry: basic concepts and, 32, 1 Mechanistic applications of the reactivity–selectivity principle, 14, 69 Mechanistic studies, heat capacities of activation and their use, 5, 121 Medium effects on the rates and mechanisms of solvolytic reactions, 14, 1 Meisenheimer complexes, 7, 211 Metal complexes, the nucleophilicity of towards organic molecules, 23, 1 Methyl transfer reactions, 16, 87 Micellar catalysis in organic reactions: kinetic and mechanistic implications, 8, 271 Micelles, aqueous, and similar assemblies, organic reactivity in, 22, 213 Micelles, membranes and other aqueous aggregates, catalysis by, as models of enzyme action, 17, 435 Molecular recognition, chirality and, in monolayers at the air–water interface, 28, 45 Molecular structure and energy, calculation of, by force-field methods, 13, 1 Neighbouring group participation by carbonyl groups in ester hydrolysis, 28, 171 Nitration, nitrosation, and halogenation, diffusion control and pre-association in, 16, 1 Nitrosation mechanisms, 19, 381 Nitrosation, nitration, and halogenation, diffusion control and pre-association in, 16, 1 NMR chemical shift–charge density correlations, 11, 125 NMR measurements of reaction velocities and equilibrium constants as a function of temperature, 3, 187 NMR spectra of equilibriating systems, isotope effects on, 23, 63 NMR spectroscopy, 13C, in macromolecular systems of biochemical interest, 13, 279 Non-linear optics, organic materials for second-order, 32, 121 Non-planar and planar aromatic systems, 1, 203 Norbornyl cation: reappraisal of structure, 11, 179 Nuclear magnetic relaxation, recent problems and progress, 16, 239 Nuclear magnetic resonance, see NMR Nuclear motion, principle of least, 15, 1 Nuclear motion, the principle of least, and the theory of stereoelectronic control, 24, 113

412

CUMULATIVE INDEX OF TITLES

Nucleophilic aromatic photosubstitution, 11, 225 Nucleophilic catalysis of ester hydrolysis and related reactions, 5, 237 Nucleophilic displacement reactons, gas-phase, 21, 197 Nucleophilicity of metal complexes towards organic molecules, 23, 1 Nucleophilic substitution in phosphate esters, mechanism and catalysis of, 25, 99 Nucleophilic substitution, single electron transfer and, 26, 1 Nucleophilic vinylic substitution, 7, 1 O⫺H bonds, hydrogen atom abstraction from, 9, 127 Organic materials for second-order non-linear optics, 32, 121 Oxyacids of sulphur and their anhydrides, mechanisms and reactivity in reactions of organic, 17, 65 Oxygen isotope exchange reactions of organic compounds, 3, 123 Perchloro-organic chemistry: structure, spectroscopy and reaction pathways, 25, 267 Permutational isomerization of pentavalent phosphorus compounds, 9, 25 Phase-transfer catalysis by quaternary ammonium salts, 15, 267 Phosphate esters, mechanism and catalysis of nucleophilic substitution in, 25, 99 Phosphorus compounds, pentavalent, turnstile rearrangement and pseudoration in permutational isomerization, 9, 25 Photochemistry of aryl halides and related compounds, 20, 191 Photochemistry of carbonium ions, 9, 129 Photodimerization and photopolymerization of diolefin crystals, 30, 117 Photosubstitution, nucleophilic aromatic, 11, 225 Planar and non-planar aromatic systems, 1, 203 Polarizability, molecular refractivity and, 3, 1 Polarography and reaction kinetics, 5, 1 Polypeptides, calculations of conformations of, 6, 103 Pre-association, diffusion control and, in nitrosation, nitration, and halogenation, 16, 1 Principle of non-perfect synchronization, 27, 119 Products of organic reactions, magnetic field and magnetic isotope effects on, 30, 1 Protic and dipolar aprotic solvents, rates of bimolecular substitution reactions in, 5, 173 Protolytic processes in H2O–D2O mixtures, 7, 259 Protonation and solvation in strong aqueous acids, 13, 83 Protonation sites in ambident conjugated systems, 11, 267 Proton transfer between oxygen and nitrogen acids and bases in aqueous solution, mechanisms of, 22, 113 Pseudorotation in isomerization of pentavalent phosphorus compounds, 9, 25 Pyrolysis, gas-phase, of small-ring hydrocarbons, 4, 147 Radiation techniques, application to the study of organic radicals, 12, 223 Radical addition reactions, gas-phase, directive effects in, 16, 51 Radicals, cation in solution, formation, properties and reactions of, 13, 155 Radicals, organic application of radiation techniques, 12, 223 Radicals, organic cation, in solution kinetics and mechanisms of reaction of, 20, 55 Radicals, organic free, identification by electron spin resonance, 1, 284 Radicals, short-lived organic, electron spin resonance studies of, 5, 53 Rates and mechanisms of solvolytic reactions, medium effects on, 14, 1 Reaction kinetics, polarography and, 5, 1 Reaction mechanisms in solution, entropies of activation and, 1, 1

CUMULATIVE INDEX OF TITLES

413

Reaction mechanisms, use of volumes of activation for determining, 2, 93 Reaction velocities and equilibrium constants, NMR measurements of, as a function of temperature, 3, 187 Reactions in dimethyl sulphoxide, physical organic chemistry of, 14, 133 Reactions of hydrated electrons with organic compounds, 7, 115 Reactive intermediates, study of, by electrochemical methods, 19, 131 Reactivity indices in conjugated molecules, 4, 73 Reactivity, organic, a general approach to: the configuration mixing model, 21, 99 Reactivity–selectivity principle and its mechanistic applications, 14, 69 Rearrangements, degenerate carbocation, 19, 223 Receptor molecules, redox-active, electrochemical recognition of charged and neutral guest species by, 31, 1 Redox systems, organic, with multiple electrophores, electron storage and transfer in, 28, 1 Refractivity, molecular, and polarizability, 3, 1 Relaxation, nuclear magnetic, recent problems and progress, 16, 239 Selectivity of solvolyses and aqueous alcohols and related mixtures, solvent-induced changes in, 27, 239 Short-lived organic radicals, electron spin resonance studies of, 5, 53 Small-ring hydrocarbons, gas-phase pyrolysis of, 4, 147 Solid-state chemistry, topochemical phenomena in, 15, 63 Solids, organic, electrical conduction in, 16, 159 Solid state, tautomerism in the, 32, 129 Solutions, reactions in, entropies of activation and mechanisms, 1, 1 Solvation and protonation in strong aqueous acids, 13, 83 Solvent-induced changes in the selectivity of solvolyses in aqueous alcohols and related mixtures, 27, 239 Solvent, protic and dipolar aprotic, rates of bimolecular substitution-reactions in, 5, 173 Solvolytic reactions, medium effects on the rates and mechanisms of, 14, 1 Spectroscopic detection of tetrahedral intermediates derived from carboxylic acids and the investigation of their properties, 21, 37 Spectroscopic observations of alkylcarbonium ions in strong acid solutions, 4, 305 Spectrocopy, 13C NMR, in macromolecular systems of biochemical interest, 13, 279 Spin alignment, in organic molecular assemblies, high-spin organic molecules and, 26, 179 Spin trapping, 17, 1 Spin trapping and electron transfer, 31, 91 Stability and reactivity of crown-ether complexes, 17, 279 Stereochemistry, static and dynamic, of alkyl and analogous groups, 25, 1 Stereoelectronic control, the principle of least nuclear motion and the theory of, 24, 113 Stereoselection in elementary steps of organic reactions, 6, 185 Steric isotope effects, experiments on the nature of, 10, 1 Structure and mechanisms in carbene chemistry, 7, 153 Structure and mechanism in organic electrochemistry, 12, 1 Structure and reactivity of carbenes having aryl substituents, 22, 311 Structure of electronically excited molecules, 1, 365 Substitution, aromatic, a quantitative treatment of directive effects in, 1, 35 Substitution, nucleophilic vinylic, 7, 1

414

CUMULATIVE INDEX OF TITLES

Substitution reactions, aromatic, hydrogen isotope effects in, 2, 163 Substitution reactions, bimolecular, in protic and dipolar aprotic solvents, 5, 173 Sulphur, organic oxyacids of, and their anhydrides, mechanisms and reactivity in reactions of, 17, 65 Superacid systems, 9, 1 Tautomerism in the solid state, 32, 219 Temperature, NMR measurements of reaction velocities and equilibrium constants as a function of, 3, 187 Tetrahedral intermediates, derived from carboxylic acids, spectroscopic detection and the investigation of their properties, 21, 37 Topochemical phenomena in solid-state chemistry, 15, 63 Transition states, the stabilization of by cyclodextrins and other catalysts, 29, 1 Transition-state structure in solution, cross-interaction constants and, 27, 57 Transition state structures, crystallographic approaches to, 29, 87 Transition-state structure in solution, effective charge and, 27, 1 Transition state structure, secondary deuterium isotope effects and, 31, 143 Transition-state theory revisited, 28, 139 Tritiated molecules, gaseous carbonium ions from the decay of, 8, 79 Tritium atoms, energetic reactions with organic compounds, 2, 201 Turnstile rearrangements in isomerization of pentavalent phosphorus compounds, 9, 25 Unsaturated compounds, basicity of, 4, 195 Vinyl cations, 9, 185 Vinylic substitution, nuclephilic, 7, 1 Voltammetry, perspectives in modern: basic concepts and mechanistic analysis, 32, 1 Volumes of activation, use of, for determining reaction mechanisms, 2, 93 Water and aqueous mixtures, kinetics of organic reactions in, 14, 203 Yukawa–Tsuno relationship in carborationic systems, the, 32, 267