Physical organic chemistry

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PHYSICAL

ORGANIC CHEMISTRY SECOND EDITION

NEIL ISAACS

I

35" Z(c 2(0 oh

Physical organic chemistry Second edition

Physical organic chemistry Second edition

Neil

S. Isaacs

Senior Lecturer in Chemistry, University of Reading

LONGMAN

Addison Wesley Longman Edinburgh Gate

Harlow Essex

CM20

2JE, England

© Longman Group UK Limited 1987 © Longman Group Limited

This edition

1995

All rights reserved; no part of this publication may be reproduced, stored in any retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd., 90 Tottenham Court Road,

London VV1P 9HE. First published

1987

Second edition 1995 Reprinted 1996 British Library Cataloguing in Publication

A

catalogue entry for this

title is

available

Data

from the British Library.

ISBN 0-582-21863-2 Library of Congress Cataloging-in-Publication data Isaacs, Neil S., 1934Physical organic chemistry/Neil S. Isaacs. 2nd ed. cm. p. Includes bibliographical references and index. ISBN 0-470-23456-3 (U.S.) ISBN 0-582-21863-7.

—

—

1.

Physical organic chemistry.

I.

Title.

QD476.I846 1995 547.1 '3—dc20 Typeset by 15 in 10 on 12 pt Monotype Times 569 Produced by Longman Singapore Publishers (Pte) Ltd. Printed in Singapore

94-32137

CD3

Through doubting we come to questioning and through questioning we come to the truth. Peter Abelard, Paris, 1122

Seek for simplicity

— and then distrust

Alfred North Whitehead (1861-1947)

it.

Foreword

to first edition

Physical organic chemistry, the study of the underlying principles and rationale of organic reactions,

period of development,

over eighty years of age. During this

is

much has been

learned which

is

now

enshrined

within the permanent fund of chemical knowledge. At the same time the

new techniques

process of refinement of chemical theory continues,

developed and viewpoints

decade becomes resolved offering

shift

their emphasis.

in another.

A

crucial issue of

are

one

This then underlies the reason for

another text on the subject of physical organic chemistry,

continuing the series of accounts which began with the notable and

book of

still

of 1940 written by Professor Hammett.

same title It is hoped that the present work will help to fill the increasingly large gap between present knowledge and practice and the status of the subject as useful

the

treated in earlier texts. In particular, the last decade has witnessed the

increasing

use

of sophisticated

instrumentation,

nuclear

particularly

magnetic resonance which can probe the structures and even the shapes of molecules in solution. Other trends have been the adoption throughout

branch of the subject of computational techniques including molecular orbital theory both of the simple Huckel type and also at high

every

and of molecular mechanics. These aids

levels

to

understanding are

increasing in importance as the reliability of the results

is

improved and

computers become more available to chemists. The trend is likely and computer graphics (cover design) as an aid to making educated guesses as to molecular properties seems likely to make a major as fast

to continue

contribution to (as

As a

Woodward

result of this,

more towards

the

put

it)

'the

armamentarium of the

our understanding of chemical processes

chemist'.

is

shifting

framework of quantum mechanics. The present

text has

been written with the object of presenting to the senior undergraduate, graduate student and research worker an account of the more important organic reactions including both the traditional evidence

—

—for

it

is

a

on observation and inference and modern approaches. Considerable amounts of data have been included since a firm grasp of a subject is better aided by perusal of collected information than by single representative values. Information up to 1986 is included. Chapters subject dependent

1

to 9 deal with underlying principles of reaction pathways, of the physical

forces

which shape bonding between atoms and of the changes of bonding

mi

.

FOREWORD TO

FIRST l-.DITIOM which are chemical reactions. Chapters 10 to 16 describe present knowledge and understanding of the various reaction types which make up organic chemistry and discuss the ingenious techniques which have been devised for mechanistic investigations.

Space rather than choice has prevented the

inclusion of certain topics including the organic chemistry of sulphur,

phosphorus, silicon and metals, further

book

Gratitude

is

contents and J.

B.

to

now

of great importance but requiring a

justice.

extended to those colleagues

who have

Lambert,

for his help

do them

L.

K.

who have

read and criticized this

text,

advised

University of Reading, November, 1986.

the

notably Professors

Montgomery and N. Turro, and

on the photochemical chapter.

me on

to

Dr

A. Gilbert

Foreword The end of effort in

to second edition

the twentieth century

marks approximately one century of

attempting to understand the basis of chemical reactivity and the

pathways of reactions of organic compounds. The result can be viewed with some satisfaction in that broad principles have been established and the mechanisms of almost all reactions can now be said to be understood in modest detail. The subject has advanced in the eight years since the first edition was published. In particular, the availability of yet more powerful computers has permitted reaction pathways of processes such as Diels-Alder reactions to be mapped by computation with increasing accuracy and the properties of transition states and inaccessible molecules to be studied. Even a limited number of solvent molecules may be included in the computations which, whatever the precision, has greatly enhanced understanding and increased confidence in results inferred from experimental measurements. Single electron transfer routes have revealed unexpected aspects of what were considered well-understood reactions such as nitration. Linear Free Energy Relation-

detailed

ships, increasing in sophistication,

continue to contribute powerfully to

and the experimental measurement of electronic transmission. The theory and practise of chiral induction has come under increasing scrutiny following the economic importance of asymmetric reactivity theory

synthesis while the involvement of metals in organic chemistry has reached the point which

makes organometallic chemistry a

subject of a size

and

complexity to warrant separate treatment and too great to be included

book of this size. Acknowledgements of any improvements in this volume are due to the interest and helpful criticism of readers of whom, in particular, I would like to thank Professors Senning and Lund, University of Aarhus, and Professor Williams, University of Durham, for careful reading of the

within a

manuscript.

Reading, November, 1994

Contents

Foreword

to first edition

Foreword

to

vii

second edition

ix

Symbols and abbreviations

xxiii

Mechanistic designations

xxvii

Models of chemical bonding 1.1

1.2

1

Covalency and molecular structure 1.1.1 The valence bond (VB) model 1.1.2 The molecular orbital (MO) model Approximate molecular orbital theory 1.2.1 The Hiickel molecular orbital (HMO) method 1.2.2 Properties of Hiickel molecular orbitals 1.2.3 The relationship between and VB models 1.2.4 Advanced methods Properties of covalent bonds

MO

MO

1.3

3

3

4 13

26 27 31

1.3.1

Bond

1.3.2

Interbond angles

34

1.3.3

Force constants

35

1.3.4

Bond and molecular dipole moments Molecular and bond polarizabilities Bond dissociation enthalpies (BDE) Group additivities to bond enthalpies

36

1.3.5

1.3.6 1.3.7

1.4

1

2

lengths

Intermolecular forces 1.4.1

1.4.2

31

39

43

45 54

Electrostatic forces

54

Ion-pairs

58

1.4.3

Short-range intermolecular forces

63

1.4.4

The hydrogen bond

67

1.4.5

Charge-transfer complexes

74

1.4.6

Crowns, cryptates, calixarenes and cyclodextrins

76

Problems References

xn

.

COM IMS Hydrogen bonding Donor-acceptor interactions Thermodynamic measures of solvation 5.4.1 Free energies of solution and transfer functions 5.3.3

5.3.4

5.4

5.5'

199

200 202

5.4.2

Activities of solutes

202

5.4.3

'Solvation' in the gas phase

204

The effects of solvation on reaction Solvent effects on rates 5.5.1

rates

and

equilibria

208

5.6.3

on physical properties on solvent-sensitive reaction Scales based on spectroscopic properties

5.6.4

Scales for specific solvation

5.6.1

Scales based

5.6.2

Scales based

205

207

Empirical indexes of solvation

5.6

199

208 rates

214 216

220

5.7

Relationships between empirical solvation scales

223

5.8

The

223

use of solvation scales in mechanistic studies

5.8.1

Multiparameter solvation analysis

226

Problems References

6

Acids and

bases, electrophiles

6.1

6.2

and

235

nucleophiles

Acid-base dissociation strengths of oxygen and nitrogen acids 6.2.1 The effect of pressure on acid-base dissociation

The

235 237

240

KA

240

6.3

Linear free-energy relationships

242

6.4

Rates of proton transfers

6.5

Structural effects

6.2.2

6.5.1

6.6

6.7

The

interpretation of

243

on amine protonation

Linear free-energy relationships

carbon acids 6.6. 1 The measurement of weak acidity Factors that influence carbon acidity Acidities of

— R and — I groups

6.7.

Electronic effects of adjacent

6.7.2

Stabilization by

6.7.3

s-Character of carbon hybridization

6.7.4

Aromaticity

d-orbitals

243 245

246 248 249 249 250 250 251

6.8

252

6.9

255

Rates of ionization of carbon acids Gas-phase acidity and basicity 6.10 Theories of proton transfer 6.11 Highly acidic and highly basic solutions 6.11.1 Highly acidic solutions 6.11.2 Highly basic media 6.12

Nucleophilicity and electrophilicity 6.12.1

Measurement of and basicity

257

259 260 265 265

nucleophilicity: nucleophilicity

266

CONTENTS 6.12.2

Hard and

soft acids

and bases:

267

270

6.12.3 Nucleophilicity scales

The

relationship between nucleophilicity

and 274 276

nucleofugacity 6.12.5 6.12.6

6.13

The 'a-effect' Ambident nucleophiles

The measurement

xv

frontier orbital

interactions

6.12.4

.

277

280

of electrophilicity

6.14

Bronsted relationships

6.15

The

in nucleophilic reactions

Leffler index

280 282

Problems References

7

287

Kinetic isotope effects

7.1

Isotopic substitution

287

7.2

Theory of isotope effects: the primary effect Transition-state geometry Secondary kinetic isotope effects 7.4.1 'Inductive' and 'steric' isotope effects

288

302

7.6

Heavy atom isotope The tunnel effect

7.7

Solvent isotope effects

7.3

7.4

7.5

effects

296 301

304 307 308

7.7.1

Fractionation factors

7.7.2

Solvent isotope effects in mixed isotopic solvents:

7.7.3

295

the proton inventory technique

310

Examples of solvent isotope

312

effects

Problems References

S

Steric

319

and conformational properties 8.1

8.2

The origins of steric strain Examples of steric effects upon

319 reactions

322

8.2.1

Ortho

8.2.2

F-strain effects

324

8.2.3

Bond-angle strain

325

8.2.4

Steric inhibition of resonance

326

8.2.5

Steric acceleration

327

effects

enhancement of resonance

8.2.6

Steric

8.2.7

Calculation of steric

effects: the

Measurement of steric effects upon 8.3.1 The Taft-Ingold hypothesis

328

molecular

mechanics method 8.3

322

328 rates

331

332

xn

.

(

\ IS

Other stenc parameters 8.3.3 Examples of steric LFER Conformational barriers to bond rotation

333

8.3.2

8.4

8.4.1

8.5

337 338

Spectroscopic detection of individual conformers

compounds compounds

341

8.4.2

Acyclic

342

8.4.3

Cyclic

346

Rotations about partial double bonds

350

Group V elements

351

8.5.1

Inversion at

Chemical consequences of conformational isomerism: the Winstein-Holness-Curtin-Hammett principle Problems

8.6

352

References

9

Homogeneous 9.1

369

catalysis

Acid and base catalysis 9.1.1

9.1.2 9.1.3

9.1.4

and general catalysis Mechanisms of acid catalysis Methods of distinguishing between Al and A2

371

reactions

376

Specific

Law

379

9.1.5

Interpretation of the Bronsted coefficients

381

9.1.6

Nucleophilic catalysis

384

9.1.7

Potential-energy surfaces for proton transfers

385

9.1.8

Solvent isotope effects

389

9.1.9

Electrophilic catalysis

The mechanisms

of

some catalysed

Substitutions i- to a carbonyl group

9.2.2

Keto-enol equilibria Hydrolyses of acetals, related

390 392

reactions

9.2.1

9.2.3

9.3

374

Linear free-energy relationships; the Bronsted Catalysis

9.2

369

392 394

ketals, orthoesters

and

compounds

397

9.2.4

Dehydration of aldehyde hydrates and related

9.2.5

compounds The formation

398

9.2.6

hydrazones Decarboxylation

9.2.7

Acid-catalysed alkene-alcohol interchange

400

9.2.8

Some

401

9.2.9

Rate-limiting proton transfers

398 of oximes, semicarbazones

acid-catalysed rearrangements

Catalysis by non-covalent binding 9.3.1

Host-guest interactions

and 399

407 409 411

Problems References

I

CONTENTS 10

Substitutions at saturated carbon

418

10.1.1

Nucleophilic substitution (S N 2)

418

10.1.2

The bimolecular

10.1.3

Solvolytic reactions

10.1.4

Measurement of solvent participation

422 433 435

10.1.5

Kinetic isotope effects

438

10.

10.1.7

The structures of intermediates The phenomenon of 'return'

10.1.8

Rearrangement

10.1.9

The

1

.6

reaction, S N 2

—the S N

1

spectrum

in

S N 1 reactions

criteria for return

'special' salt effect:

an ion exchange

in

445

10.1.10 Structural effects 10.1.11

Leaving-group

upon

447 449

ionization

effects

10.1.12 Bridgehead systems

451

10.1.13 Linear free-energy relationships

451

10.1.14 Intramolecular assistance in ionization

455

10.1.15 Activation parameters

457

10.1.16

The S N 1

460

reactions

10.1.17 Aliphatic S N 2 reactions in the gas phase

461

Electrophilic substitutions at saturated carbon

463

10.2.2

The S E 1 mechanism The S E 2 mechanism

10.2.3

Electrophilic substitution via enolization

10.2.1

463

10.3

Nucleophilic displacements at a vinyl carbon

10.4

Electrophilic displacements at an aromatic carbon 10.4.1

10.4.2

10.5

464 468 469 473 474

Timing of bond-breaking and making general mechanism for electrophilic

The

aromatic substitution

475

10.4.3

The nature of

477

10.4.4

Kinetic isotope effects

481

10.4.5

Kinetics of S E 2-Ar reactions

481

the electrophilic reagents

10.4.6

Structural effects on rates

10.4.7 10.4.8

The ortho-para selectivity ratio, The nature of the intermediate

10.4.9

Ipso attack

10.4.10

The

485 s

p

=

(2f /fp

)

MO interpretation of aromatic reactivity

Nucleophilic substitution at an aromatic centre

10.5.3

The addition-elimination pathway (S N Ar-Ad, E) The unimolecular mechanism The aryne mechanism (E-Ad)

10.5.4

Nucleophilic substitution via ring opening: the

10.5.1

10.5.2

10.6

440 442 443

an

ion-pair

10.2

xvii

418

Substitution reactions at carbon

10.1

.

S N (ANRORC) route Nucleophilic substitutions at carbonyl carbon 10.6.1

Basic hydrolysis of carboxylic esters

491

493 495 495 498 498 503

504 506 507 511

xnu

.

CONTENTS 10.6.2

Acidic hydrolysis of esters

10.6.3

Stereoelectronic factors in the decomposition of

519

the tetrahedral intermediate

521

10.6.4

Other mechanisms

522

10.6.5

Hydrolysis of amides, acyl halides and

for ester hydrolysis

anhydrides

529

10.6.6

Properties of tetrahedral intermediates

533

10.6.7

Nucleophilic catalysis in carbonyl substitutions

536

Problems References

//

551

Elimination reactions

11.1

Base-promoted eliminations in solution Kinetic criteria of mechanisms 11.1.1 11.1.2 Structural effects on rates of elimination

.2

555 556

11.1.3

Kinetic isotope effects

562

11.1.4

Variation of the base-solvent system

566

11.1.5

Competition between elimination and substitution

568

11.1.6

Orientation in product formation

572

11.1.7

Stereochemistry of E2 reactions

574

11.1.8

Frontier orbital considerations

579

11.1.9

Elcb reactions

579

11.1.10 Ester hydrolysis 1 1

551

by the

Elcb mechanism

Intramolecular pyrolytic eliminations (the Ej reactions)

581

58

11.2.1

Ester pyrolysis

582

11.2.2

The Chugaev reaction Amine oxide, sulphoxide and selenoxide

585

pyrolyses

586

Pyrolysis of alkyl halides

587

11.2.3

11.2.4 11.3

a-Eliminations

588

11.4

Oxidative eliminations

589

11.4.1

Oxidations of alcohols by chromium (VI)

590

11.4.2

The Moffatt oxidation

592

Problems References

12

599

Polar addition reactions

12.1

Electrophilic additions to alkenes 12.1.1

Kinetics

12.1.2

Effect of structure

12.1.3

Isotope effects

600

600 602 607

LUNikl

I

Orientation

111.5

The nature of the intermediates in Ad^

reactions

122 Miscellaneous additions 1121 Hydroboration Addition with ring dosure; haWactomzation

113.1

Michad

1132

Carbonyi additions Additions to heteroomniknes

114 Frontier

r.

?

,

tarn

610

MJ 613 617

Addition of carbocabons

Additions to dienes, alkynes and alenes 113 Nudeopmlic additions to mnhipk bonds

123.3

.

am

121.4

1222 1113 1114

.

HI

addition

orbital considerations

Of 631

-yl

wtmilul km

115.1

Examples

04

1252

Stereochemistry

01

via aditilion/rimmihua

632

,.e....:

i'.t'iZ.

EH

MB Mi Ml Ml

01 04 04 01 01 Mi

02

0) _ -:

:

:«

:•-

xx

.

CONTENTS 14

P tricyclic

701

reactions

14.1

Classification of pericyclic reactions

701

14.2

The theory

702

14.2.1

of pericyclic reactions

Conservation of orbital symmetry: correlation

diagrams

703

The frontier orbital concept 14.2.3 The aromaticity concept 14.2.4 Suprafacial and antarafacial geometries Thermal cycloadditions: their scope and characteristics 14.3.1 The Diels- Alder reaction 14.3.2 Stereo- and regiospecificity in Diels-Alder 14.2.2

14.3

14.5

14.3.3

Retro Diels-Alder reactions

14.3.4

The nature of

14.3.5

Related six-electron cycloadditions

Thermal

707 707 709 711

715

reactions

14.4

705

the Diels-Alder transition state

721

723

725

2) cycloadditions

727

14.4.1

Cycloadditions of cumulenes

728

14.4.2

Two-step cycloadditions

732

14.4.3

(2

(2

-I-

+

2) Cycloreversions

734

1,3-Dipolar cycloadditions

736 740

14.6

Electrocyclic reactions

14.7

Cheletropic reactions

742

14.8

Sigmatropic reactions

748

14.8.1 Concertedness in sigmatropic rearrangements Acid catalysis of the Diels-Alder reaction

751

14.9

753

Problems References

//

767

Reactions via free radicals

15.1

The generation of 15.1.1

15.1.2

15.2

15.3

radicals

767

Primary processes Secondary routes

772

The detection of

radicals

15.2.1

Direct observation

15.2.2

Indirect

15.2.3

By chemical

methods characteristics

Reactions of radicals

768

773 773

779 784 788

Radical coupling

788

15.3.2

Displacement (abstraction, transfer) reactions

790

15.3.3

Additions to 7t-systems

793

15.3.4

Fragmentation of radicals

797

15.3.5

Radical rearrangements

798

15.3.6

Radical cyclization reactions

801

15.3.7

Linear free-energy relationships

804

15.3.8

Electron transfer reactions

806

15.3.1

CONTENTS 15.4

15.5

Factors influencing the reactivities of radicals

.

xxi

810

15.4.1

Radical stability

811

15.4.2

Polar influences

814

15.4.3

Solvent effects on radical reactions

817

15.4.4

Steric effects in radical reactions

817

15.4.5

Frontier-orbital considerations

821

The stereochemistry

824

of radicals

Problems References

16

837

Organic photochemistry

16.1

Excited electronic states

837

16.1.1

Absorption of

16.1.2

and horizontal excitation Spin multiplicity: singlet and triplet Sensitization and quenching

16.1.3

16.1.4

light

by molecules

837 838

Vertical

states

Techniques of photochemistry Photochemistry of the carbon-carbon double bond 16.1.5

16.2

16.3

16.4

839

840 844

844

16.2.1

Geometrical isomerization

844

16.2.2

846

16.2.3

Photochemical pericyclic reactions The di-7r-methane rearrangement

16.2.4

Photoadditions to alkenes

852

Photoreactions of carbonyl compounds

851

853

16.3.1

Carbon-carbon bond cleavage

854

16.3.2

Cycloadditions

856 857

Photochemistry of aromatic compounds 16.4.1 Photosubstitutions at the aromatic ring

858

16.4.2

The photo-Fries rearrangement

859

16.4.3

Valence isomerization

859

16.4.4

Photocycloadditions

861

16.4.5

Photo-oxidations with oxygen

864

Problems References

Index

871

Symbols and abbreviations

A,

a

Coulomb

a

Bronsted coefficient

a

Taft solvation parameter

fi

resonance integral (energy unit)

fi

Bronsted coefficient

fi

Taft solvation parameter

fi

Bohr magneton

r

parameter

y

activity coefficient (molal units)

T

,

!

elimination (reaction type) ethyl electrostatic, polarization, steric

exchange and charge transfer energy

constants

a generalized electrophile

ES

(in structures)

£A £T £N

Arrhenius activation energy

£R EC ERE ESR exo, endo

enzyme, enzyme-substrate complex

empirical solvation parameter

Edwards

nucleophilicity parameter

reaction field effective

concentration

empirical rate equation electron spin resonance

(

= electron paramagnetic resonance, EPR, PMR)

stereochemistry with reference to a component of structure related to 'boat' cyclohexane:

SYMBOLS

AND

ABBREVIATIONS

.

xxv

exo side endo side

F F

& J v/o' J mi J p)

force

parameter

field effect

Swain-Lupton

parameter

field effect

partial rate factor (relating to ortho, meta, para positions of a substituted

benzene) 7t»

/n

/ G

AG

tr

G\, GB 9

H HM

Kirkwood

electrostatic solvation functions

(mole fraction units)

activity coefficient

Gibbs free

free

energy

energy of transfer

constants in the Bronsted catalysis law

gyromagnetic ratio enthalpy (heat)

Coulomb

integral

resonance integral

Hamiltonian operator Atfat

HMO HOMO AH

{

H

,

H|, /fR , //A

,

//_

standard heat of atomization

Huckel molecular orbital highest occupied molecular orbital

standard heat of formation acidity functions

Edwards

nucleophilicity parameter

Planck's constant /

+ 1, "I

indicator ratio

inductive effect (electron-donating, electron-withdrawing) acidity function (see

K K

H

)

Kelvin (temperature scale) equilibrium constant acid dissociation constant

Michaelis constant specific rate constant: k u k 2 , k 3 ,

and

successive stages of a reaction

bimolecular, termolecular K Tt\

.

.

.

...

are used both to denote rates of also to distinguish unimolecular,

processes

relative rate constant

rate constants for reaction of isotopic species containing

kc a k s iv kA f*.

,

,

He

rate constants for

components of

a generalized hydrogen isotope

LUMO

lowest unoccupied molecular orbital

M

MO m

D respectively

Boltzmann constant

L L L + ,L. /

H,

solvolysis

Avogadro's number (i.e.

H,

D

or T)

localization energy

nucleophilicity coefficient in

Grunwald-Winstein equation

molar mass (molecular weight) molecular orbital

meter

xxvi

.

AND ABBREVIATIONS

SYMBOUi

m

%

mes

N N,N + Nu:,

Nu":

NAD (NADH) NGP

polarity coefficient in

methanesulphonate

Newton

(unit of force)

nucleophilicity parameters

a generalized nucleophile

nicotinamide adenosine dinucleotide (reduced form)

neighbouring-group participation

n

undex nucleophilicity parameter unshared pair (electrons, MO) an integer

P

dipole

n n n-

Pe

P PKIE P

refractive

moment

total polarizability

empirical solvation parameter

primary kinetic isotope total

phenyl

Pr

propyl

paranitrophenyl pressure (vapour pressure) partial (n)

Q Q

+

effect

bond order

Ph Pnp P

R,

Grunwald-Winstein equation

(-S02 Me)

bond order

partition function

constant in McConnell's equation

charge

q

electric

q

integer in

q

heat

R

gas constant

Woodward-Hoffmann

rule

R—

a generalized unit of structure, usually an alkyl group

-R

resonance

R

effect (electron-donating,

electron-withdrawing, respectively)

molar refraction

Swain-Lupton resonance

effect

constant

correlation coefficient

Woodward-Hoffmann Yukawa-Tsuno constant integer in

rule

overlap integral

S

ASf AS

entropy standard entropy of formation

tr

standard entropy of transfer

S s

empirical solvation parameter

selectivity

empirical solvation parameter

s

symmetric substitution (nucleophilic, electrophilic)

s

SOH

SOMO

substrate (in enzymic reaction schemes)

a generalized protic solvent singly-occupied molecular orbital

MECHANISTIC DESIGNATIONS syns

T t

TFA Tos t-Bu

U

stereochemical designation; on the same side

time trifluoroacetic acid

Me—v^ C—

p-toluenesulphonyl, tertiary butyl, total internal

y

— SO2O —

(CH 3 ) 3 energy

V

vibrational

quantum number

volt

valence

bond

v

volume

t;

velocity of reaction

w w

work

X

an electron-donating (resonance) substituent,

X

empirical solvation scale

x x Y-

concentration, mole fraction

y Zr\

anti-)

temperature

function in Bigeleisen equation

V

xxvii

coefficient in nucleophilicity correlation (see n)

u

V VB

(cf.

.

parameter in Bunnett equation

a fractional

amount

(0




/

C=C

?

\

C-Nu

y

n'u

'Nu

+ :Nu"

X

Srn1> S rn 2; 1-electron reduction followed by S N 1 or S N 2 sequences S ON l, S ON 2; 1-electron oxidation followed by S N 1 or S N 2 sequences

A-S N 2 A-S N 2 \

\_

H*

±y

H

/

~m~ Nu— C/„. +H 2

C-jO

Nu:

H

«

Al (hydrolytic processes) r\

S

+ +

H-6 *=> slow

SH A2

SA:B

products

(hydrolytic processes)

SH + + H 2

slow

products

Eliminations:

El

H \

H

Nu

c-c„

/I

i'R'

o-

slow

\ /I

,'

**

Nu

V'R'

R

r :B

H

R'

\

R' /

C=CN

i

S N products

R

/

,

MECHANISTIC DESIGNATIONS

.

xxxi

E2 (E2H) "rf-

H

H

If

E2C

Cnu

is

like

Nu

E2H

but the

'soft'

/

/

\

BH* Nu ,

C=C

C-C

,v\

\

base interacts both with

C

and

H

6-

B

H ,\V

Nu 6-

Elcb

f

BH+

:B

\

zA

-

slow

>

\

Nu

z

C-t-C

/-^

H

A

A

Co

-

o=x

\

/

/

\

C=C

+

\

HO -X [X =

Additions:

Ad E 2 \

/

/

\

-

C=C

) E

+

-Qu

C-C,, v Nu

/

^-:Nu"

/

C=C

«

C,

\

N, S, Se]

+ :Nu

_

xxxil

.

SYMBOLS

AND

IHBR1

llAlloW

Ad N 2 Z

r/

\

c=c

—

-

vc-c)*Z

—

7

\i / c-c, .

\ (H

:Nu

+ )

VC-0

VC =ni

VC-OH

-

(H*)

( :Nu"

Ad H 2

H r-

Br

/TAbBr

H

Br

H^Br Ad-E: a sequence

of addition followed by elimination, the net result being

substitution

_H

H

HH

r

N

H

H ^ RCH0H

"H^:B CN

H J

Me3N+

CN

Me3

CN

N+

Me 3N

H

RCHOH

>=< H

E-Ad: a sequence

of elimination followed by addition

substitution. See arynes (Section 10.5.3).

CN

resulting

in

MECHANISTIC DESIGNATIONS

z B:

H

Nu

Nu

-

BH

:,

r

.

xxxiii

H

'Nil'

Nu'

Nu'

Cf

^Nu'

slow

H

H

:B

o»

Nu":

Carbonyl and related substitutions:

B Ac 2

(an

Ad N -E

sequence)

slow

*V

P

i:

II

V

Nu :Nu

:

V

^nu Nu

Nu'

f

^Nu' :Nu~

AAc2 OH

O +

^

OH

+

II

O II

I

H ^

^

^Nu

Nu

^ Cx

V Nu

Nu

(

Nu' :Nu'

AA1 1

:Nu"

,

H*

(anS N l process)

O

R II

R

SOH -

-rf

V

V

II

** R SOLV

+

R

OH

OH SOH

I

OH

R I

I

^

R i

R

R

R3COS

SOLV

O + R3COS + H+

.v.v.v/r

.

SYMBOLS

AND

ARRRl A

A A1 2

I

I

//\

\

(an S N 2 process)

O

V'H H

H

V

H

% H

OH I

C

+

CH3OS + H +

AAc l r

II

V

OR

+

I

^

C't

£or

C

^

V V_

+ROH o

^

x

+

os

H+

j Models of chemical bonding

Covalency and molecular structure

•

Properties of covalent bonds

/./

Approximate molecular

•

Intermodular

orbital theory

•

forces

Covalency and molecular structure

An

understanding of chemical reactivity begins with an understanding of

chemical bonding, the forces which render certain aggregates of atoms (i.e.

the familiar molecules)

that chemical reactions

more

—changes

in

bonding

—

1-3

on this basis may be approached and

stable than others.

It is

a rational and consistent theory of organic chemistry devised. milestones in the understanding of bonding recognition of the electron-pair covalent in 1919,

still

be quoted. The

Two

first,

the

4

bond by Lewis and by Langmuir 5

provides a model for the description of molecular structure

adequate for most purposes and following

may

text.

According to

will

be extensively employed in the

this concept,

valence electrons are shared so

and are regarded as essentially localized in the internuclear space. For the first row elements of which organic compounds are almost entirely composed, this is the octet 2 6 2 (2s 2p ); for hydrogen, Is The second leap in understanding was made by the introduction of quantum mechanics to chemistry following the molecular orbital description of bonding in the hydrogen molecule by 6 Heitler and London, in 1929. This approach superseded the concept of localized electrons and paved the way to quantitative understanding of bonding, the satisfactory calculations of bonding energies, optimum bond lengths and geometries. It will be necessary to turn to these methods, despite the necessity of somewhat lengthy computation, when the need as to create filled shell configurations

,

arises to consider specific

.

molecular orbital properties

(for

example, in

the theory of pericyclic reactions, Chapter 14). Nonetheless,

quantum

2

.

\

1618

1

xk

X i

414

^

.

//

T?

12

.

Monii.s o/

(

m \iK

i/.

ho\pi\(,

Table 1.1

(Continued)

T,

System

r

4> 2

¥

3

4>

4

4\

V6

1126

2175

V-

16

-2175

-1126

17

-2-414

-0618

-0618

0-414

1-618

1618

18

-2-334

-1099

-0-274

0-594

1-374

1-740

-2-414

-1-732

0-414

1000

1000

1-732

20

-2000

-1-247

-1-247

0-445

0-445

1-801

1-801

21

-2101

—1-259

-1000

1000

1-259

2101

-1-848

—

414

-0-765

0-765

1-414

1

-1-931

—1-414

—0-518

0-518

1-414

1-931

-1-970

—1-285

-0-684

0-684

1-285

1-970

-2053

-1-209

-0-570

0-570

1-209

2053

-2-000

-1000

-1000

1-000

1000

2-000

fl z

19

3

P*l

25

26

!

1

848

APPROXIMATE MOLECULAR ORBITAL THEORY

i.2

.

ij

Non-alternant

Alternant Even

Odd

Even

./

Odd

/•

J^v

-V^ 1.2.2

Properties of

The following

Hue ke I

molecular orbitals

generalizations can be made.

Alternant hydrocarbon systems (AHs) have

(a)

MOs

symmetrically

E = a on the energy scale. A further division may be made and even-numbered systems. An even AH will have an equal number of bonding and antibonding orbitals only, while an odd AH will have, in addition, one non-bonding (x = 0, E — a). Non-alternant systems will not have a symmetric arrangement of orbitals cf. fulvene, arranged about into odd-

MO

Fig. 1.2. (b) Electron densities of even AHs are 1000 on each carbon. Odd AHs must have a cationic or anionic charge or an odd electron; the charge or odd electron density for linear systems is equally divided between starred atoms (the larger set); non-linear odd AH ions will have an

unequal charge distribution with

57%

as, for

example, the benzyl cation or anion

of charge on the benzyl carbon,

8)

e.g.

7.

Non-AHs

will

in

show a marked disparity of charge at each carbon (for example, which means the compound has a dipole moment, in accordance with

general

observations.

Original matrix 1

4

2 1

1

1

1

1

1

1

i+ T 3

i+ T 3

i+ T 3

Pentadienyl cation

i

4

.

MODELS OF CHEMICAL hOMUM. Eigenvalues over eigenvectors v

= -1-732 *,

-1000

-0-000

+1-000

+1-732

*2

^3

*4

*5

c,

0-289

c2

0-500

-0-500 -0-500

c3

0-577

c4

0-500

c5

0-289

0-577

0-500

0-289

0-000

-0-500

-0-500

-0000

-0-577

-0000

0-577

0-500 0-500

-0000

0-500

-0-500

0-577

-0-500

0-289

unoccupied

occupied

Charge on:

C,:l- [(0-282 2 C 2 :l- [(0-5 2 x C 3 - [(0-577 2 :

1

2)

+

(-0-5 2 x2)] (-0-5 2 x 2)] =

x

2)

+

x 2)

+

(0)]

=

= 0-32 00

0-334

0.57+

Origina matrix 1

Jo

1 1

1

p'

1

0.M4+ l)

1

1

1

1

0.144+

1

1 1

1

1

Benzyl cation

1

7

Eigenvalues over eigenvectors

x= -

V

9.**

-2-101 i

+ 0-000

+1000

+1-259

+2101

^2

^3

**

*5

*6

^7

0000 0000

-0116 0-354

-0-500 -0-500

0-337

0-562

Lo-354

0-354

-0116

0-500

0-238

c2 c3

0-500 0-406

c*

0-354

c5 c6

0-406

.

-1000

-0-397 -0-500

c,

c7

-1-259

oc cupied

-0-756

0-000

0-397

0-238

-0000

-0-500

-0-500

0-378

-0-000 -0-500

0116

0-406

0000

0-500

0-354

-0-354

0000

-0-378

-0000

-0562

0-337

0-500

0-000 0-378

-0-500

0-354

-0-354

0-500

0116

0-406

unoccupied

Original matrix 1 1

1

1

1

1

1

Fig. 1.2

Example

of

Hiickel calculations

0.0731 1

1 1

1

APPROXIMATE MOLECULAR ORBITAL THEORY

i.2

.

//

Eigenvalues over eigenvectors

x=

-2-115

-1000

-0-618

+ 0-254

+ 1-618

+ 1-861

*2

^3

^4

^5

*«

¥,

6

'((oztf--

c3

0-429

-0-500 -0-500 -0-000

c4

0-385

0-500

-0-602 -0-372

c5

0-385

0-500

0-372

0-280

-0-602

-0153 -0153

c6

0-429

0000

0-602

-0-351

0-372

0-439

c,

0-247

,

0-523

'430

0000 0000

0-749

0-000

0-357

-0190

-0-664

-0-351

-0-000 -0-372

0-280

0-602

occupied

0-439

unoccupied

Fig. 1.2 (cont.)

conjugated hydrocarbons (annulenes) can be divided into

(c) Cyclically

two

sets

(i)

by two

different approaches:

Even-numbered conjugated

have a symmetric

MOs,

set of

(£ = a + 2/?) while the others occur in energy though of different symmetry).

Odd-numbered

example in benzene (10), and highest having energies degenerate pairs (pairs of the same

rings, as for

the lowest

rings, necessarily

with ionic or radical character, have

MO

the lowest-energy at (a + 2/?), the others being arranged in nonsymmetric pairs upwards. The cyclopentadienide anion, 9, is an example.

4-J-

4-f-

ft"

The roots of carbons

=

the secular equations for the annulenes (total

2k), are given

= —2

x

a + 0.619/3 a + 2P

by Eq.

cos(ln/k),

number

of

[1.5]:

(ln/k) is in radians

[1.5]

where /

=

0,

+1, +2.../c.

Hence, energies of

E = t

+

a

MOs

are given by Eq. [1.6]:

[1.6]

2/? cos(ln/k).

A more

fundamental division from the viewpoint of chemistry is made between annulenes with (4m + 2) 7r-electrons and those with An electrons, n being a positive integer including 0. The {An + 2) series, having (ii)

2, 6, 10, 14, 18

.

.

.

delocalized 7i-electrons, have

all their

bonding orbitals

16

.

MODELS

(>t

i

Hl-.MH

AL ho\ni\(.

Geometrical construction of the

MOs

of annulenes: inscribe

a regular polygon of the

appropriate ring size

a

in a

2p

20 and with one apex lowermost. Then the circle of radius

energy levels (eigenvalues) at the apices of the

1

\

H-H

lie

polygon.

For example:

Benzene

Cyclopenladiene

filled.

They are therefore closed-shell molecules of more than usual and are denoted aromatic, benzene being the prime example

stability,

(10,11).

a

-

a+

ff

2/3

10

W2

*1

v3

11

The An

on the other hand, have two electrons sharing a pair of non-bonding MOs. They are therefore of low derealization energy and stability and their chemistry is that of highly reactive mol ecule s. They are denoted antizqrqmatic. The transient molecule cyclobutadiene, 12, exemplifies this type. (The Hiickel energies would indicate zero resonance energy for cyclobutadiene and the term 'non-aromatic' is series,

MO

o-

+;:+

sometimes applied. Refined calculations suggest lower

still

the designation 'anti-aromatic'

is

+

2 (aromatic) molecules.

appropriate.)

It is

utterly different It

is

Hence

a remarkable fact that

wave equation rationalize chemistry of benzene and cyclobutadiene.

the highly approximate solutions to a

(d)

to be even

while the ring currents of circulating 7r-electrons in An systems

are in the opposite sense to those of An 12

its stability

the

possible to include heteroatoms within the scope of simple

HMO theory. While the calculations performed above concern implicitly all-carbon systems, the

MO

The calculated energies for the same form as those for ethene; however, a systems.

homoatomic molecule N 2 would take the and (1 would take different

same solutions are appropriate

for all

1.2

APPROXIMATE MOLECULAR ORBITAL THEORY

values from those appropriate to carbon. This suggests that

Coulomb and resonance

differentiate the

The computation

is

ij

we could

integrals in the interaction

matrix, heteroatom systems could be treated. This

empirical way.

if

.

is

achieved in a purely

as before, with the following adjust-

ments.

The Coulomb

+

(a

SP),

where

the heteroatom

integral appropriate to „„, = 275

nm

(pigment of tomatoes)

Q.

Why

radical a

has the electron spin resonance (ESR) spectrum of the benzyl

predominant

weak subsidiary

triplet splitting

of 1:2:1 ratio together with

some

splitting?

SOMO a

somo—

CH

Va

Electron population 2

Va

c

c

a

-0.756

0.57

o

0.378

0.142

m

0.0

0.0

P

0.142

-0.378

(SOMO = singly-occupied molecular orbital)

A. Splitting of the

ESR

signal

is

due to coupling of the unpaired electron

spin with those of neighbouring nuclei, in this case protons. Both the

and the separation of the lines depends upon the proximity of We need to know, therefore, the distribution of the unpaired electron. Now this resides in the orbital ^4 and the coefficients intensity

the coupled spins.

2

of that orbital are the only data required; c gives the electron distribution.

One can

see that the unpaired electron spends

benzylic carbon and the triplet splitting

is

57%

of

its

time on the

due to the two equivalent

protons there. Weaker and more closely spaced lines are due to splitting

by the ring protons in 2- and 4-positions near which the odd electron spends the remainder of the time.

1.2.)

The relationship between

Both these approaches

MO and VB

models

20

an extreme picture structures emphasize the de-

at their simplest tend to give

of molecular structure. Molecular orbital

i.2

APPROXIMATE MOLECULAR ORBITAL THEORY

.

27

—

and the lack of correlation between electron motions the due to calculation of the properties of each without regard to the occupancy of the others, the 'one-electron approximation'. Valence bond theory, on the other hand, emphasizes the localized nature of el ectron pairs and the extreme correlation of electronic motions implicit in^THngle VB structure. The truth lies somewhere in between. This can be shown diagrammatically by linking corresponding states, as in Fig. 1.4. A more precise theory would predict energy levels to be somewhere between the two extremes in accordance with experimental observations of, for example, photoelectron spectroscopy, which can probe the energies of valence electrons in a molecule. localized nature

MO

latter

MO

1.2.4

Advanced

Hiickel

MOs

MO

methods

6.21-23

are highly approximate and

alternant hydrocarbons so that

much more

limited

in

application

to

exact solutions to the wave

equation of a molecule are needed for the investigation of energy levels

and

reactivity.

The

Fig. 1.4

ab

starling, point for

the Hamiltonia n or en erg y operator

,

initio

calculations

is

the same,

which contains several terms—

Diagram showing

correlation between molecular states as described

MO and VB

by simple

models.

MO

VB

states

states

electronic kinetic energy, electron-nuclear attractions, electron-electron

—

and nuclear-nuclear repulsions all of which are summed over all possible combinations of nuclei and electrons. Simplification can be achieved by prescribing a particular molecular geometry and by limiting the basic set of AOs used. For first row elements, 2s and 2p orbitals are seldom sufficient and the participation of 3s, 3p and 3d orbitals may be included. Each AO is given explicit mathematical form which is computationally convenient and no separation into o- and 7r-types is made initially. Solutions of the wave equation are then obtained iteratively so as to minimize the energy of the system, for which the aid of a very large computer is essential. Programs such as GAUSSIAN 80 are widely available so that computations at this level are within the capabilities of

it

.

.\f(>/>/-/.\

OF

(

HI Ul(

H()\MX(,

\L

many

Programs of this type are under continual revision and are becoming more and more accurate as computers become more powerful. A further constraint on acceptable solutions is the requirement that

non-specialists.

MOs

are either symmetrical or antisymmetrical with respect to

symmetry elements of the molecule. Having minimized the energy with one particular geometry, the calculation may be repeated with small changes in bond lengths and angles until a true minimum is reached which should correspond to the real molecule. The resulting MOs look quite different from those to which one is accustomed from the application of localized

bond models.

All are spread over the

whole molecule and may

be designated a- or 7i-types according to whether they possess an axis or a plane of symmetry. For a molecule such as methane, Fig.

1.5a, the

four

MOs

appear as four combinations for the localized C-H a-bonds, three which of are of equal energy and higher than that of the fourth. Experimental measurements show there are two distinct energy levels; this is

not apparent from the localized bond model.

will suffice for the present; others will

Ethene 12

be introduced

in later chapters.

of which six are bonding and are illustrated together with their

in

MOs

The lowest

energies.

concerned the

few other examples

(Fig. 1.5b) has a total of 12 valence electrons: therefore there are

MOs

actual

A

are

too deep-seated to be

chemical reactions. The important ones for

this

much

purpose are

HOMO and LUMO — the frontier orbitals— which look much like the MOs.

conventional n and n* here the whole set of relevant to

its

the energies

MOs

would be too space-consuming to show most of them are not most purposes attention will be drawn to

It

for a large molecule;

chemistry, so for

and electronic distributions of the

such as acetone (Fig. 1.5c) for instance, has a

frontier orbitals.

A

ketone

HOMO (constructed largely

from an unshared pair) in which the electron density is mainly on oxygen mainly located on carbonyl carbon. Electron-donating and a capacity is expected to be at oxygen and electron-accepting capacity at carbon, and this is the reactivity pattern observed. Acrolein (Fig. 1.5d) shows a large lobe of its both at carbonyl carbon and at ^-carbon in accordance with the principle of homology making both of these

LUMO

LUMO

electrophilic centres.

Semi-empirical

MO

methods

(SCF) treatment involves inclusion of terms which contain three and four AOs (three- and four-centre integrals) known as differential overlap. A three-centre term, for instance, would signify interaction between the overlap region of AOs and k. Such terms are troublesome to evaluate and many computational methods adopt Full self-consistent field

i,

simplifying procedures yielding

approximation. The

method ignores

all

CNDO

MO

information of varying degrees of

(Complete Neglect of Differential Overlap)

such terms. The

of Differential Overlap) includes

INDO). Such methods

;'

INDO

some

version (Intermediate Neglect

terms, as does

are computationally easier

MINDO (Modified and can

yield satis-

.

i.2

APPROXIMATE MOLECULAR ORBITAL THEORY

.

29

£/eV

-tt -H-J4-

+f

-O-MI" -0.9320

Methane

£/kJmor' (kcalmol

V

7

-1

+627 (150)

if§&

- 100o

+

^** r-% -1340

-H-*s

(-320) (-349)

44-Vi * 4-fV,

-1680 (-400)

-2050 (-490)

-t-f^i

-2650 (-633)

Fig. /.5

MOs

and

Examples of

LUMO);

c,

all valence electron ab initio MOs. a, Methane; b, ethene (bonding acetone (frontier orbitals); d, acrolein (frontier and adjacent MOs).

)

.

MODI

U

OF

(

HEMIC

1/

BONDING H

r^>

t

£/kJmol (kcal ')

'

1

1

(»

f

+ 795

'O

(190)

*.2

-836

H

*H>

It

(-200)

It

•

H

1

-1050 (-250)

1

1

H

/H

*i

V^^H^Th ITCO

H Acetone

(frontier orbitals)

£/kJmor' (kcalmol

I

-1 )

I

LUMO

h--%c^c

— -5a

,

-1050

~*^(-250)

c

-1210

^i^-1250 c \

H

F/g. /.5

(Cominued)

^it H

Propenal (acrolein) (frontier orbitals)

-300)

*-V

H

Cfl

a /

,ci

HnQ \\r U H

H

,C, CI

H H )=,#* Br— / I— Group

Ionicity/%

1-78

39

0-96

18

0-76

14

0-46

8

p/D

\:=o

2-25

— OCH — ON0

3

81

2

273

(a)

c— C— C— C— C— CI C— Br C— C—

Ionicity/%

0-89

17

0-49 1-43 1

43

9 30

61

11

0-41

7

011

1-8

003

0-5

Group

— C==N —CONH — S0 —

p/D

3-44 3 4

2

425

2

The negative end of the dipole will be on the atom with The bond dipole contribution, p, is given by P/D

(c)

p/D

contributions

Group

(b)

Bond

=

(x A

-

the greater

x-

[1-9]

Xb).

The

ionicity of the bond, i.e. the percentage contribution of the ionic + terms, is given by B~, in structure,

VB

A

Ionicity

=

16p

+

3-5p

Molecular dipole moments, 1.8);

they are the vector

2

p,

sum

[1.10]

.

may of the

be obtained experimentally (Table

bond moments, a

fact

which

may

be of use in structure determination. All molecules which are centroare and conversely all those with p ^ symmetric must have p =

i.j

PROPERTIES OF COVALENT BONDS

.

#

Table 1.8 Selected values of

Compound

p/D

CHHal 3 Hal = F Hal = CI

1-6

Compound

p/D

experimental molecular dipole moments. Values are for the gas

phase but often

differ significantly if

measured

in solution

or by

Hal Hal

different workers.

MeX

=

Br

10 10

=

I

0-8

X X X X X X X X X X X X X

MeCOX X =F X = C1 X = Br X = H X = OMe PhX X=F X = C1 X = Br X =1 X = N0 2

2-96 2-71

2-45 2-68 1-77

1

35

1-75

= = = = = = = = = = = = =

F

1-8

C1 Br

1-94

I

1-6

OH

171

OMe

1-3

1-79

SMe SH

1-45 1

26 32

SeMe

1

CN

40

N0 NH

3-4

2

1

2

SCN

29

3-34

1-7 1-7

40

a

ci

s

u

n

!

I

k^J

ortho

2-5

ortho

4-63

meta

1-72

meta para

3-4

para

2-6

not centrosymmetric. For example, dichloromethane has a molecular

moment, p = 1-6 D; this clearly excludes a planar centrosymmetric structure; on the other hand, vector addition of the bond dipoles would predict a value p= 1-2 D, showing a discrepancy due both to experimental error and the approximate nature of the additivity relationdipole

ship.

Dipolar properties give

rise to electrostatic forces

repulsion between molecules and forces

r.j.j

by which reactions are

make an important

of attraction or

contribution to the

initiated; see Section 1.4.

Molecular and bond polari^abilities

38-40

Electrons, being charged particles, respond to an external electric

field.

Since the bonding electrons in a molecule have a degree of mobility, the charge distribution will be affected by an electric field which creates

an induced dipole present.

The

in addition to

any permanent dipole which

direction of the response in the molecule

to diminish the electric field gradient

phenomenon produce

is

and

is

known

is

may

be

such as to tend

as polarizability.

The

of great importance in chemistry since ionic reagents

in their vicinity

very large

field

gradients which can induce dipoles

4o

.

MODELS

01

CHEMICAL BONDING neighbouring molecules and thereby bring about additional attractive forces which may lead to reaction. Many other physical properties depend in

upon

polarization; the most familiar

wave

of a light

is

the refraction of light.

is

accompanied by an

The passage

oscillating electric field at right

angles to the direction of propagation which produces a corresponding

nearby molecule. This interaction reduces the

oscillating dipole in a

velocity of propagation of the wave, which

index,

n,

medium

of the material

polarized the molecule, the higher polarizability

visible

Polarizability,

light).

induced dipole produced by unit

=

gradient

=

2

[