Notes on molecular orbital calculations 1961 roberts

Molecular O r b i t a l Calculations B O O K S B Y JOHIC' D ROBERTS B a s i c Organic C h e m i s t r y , P a r t I Nuclear Magnetic Resonance, Applications to Organic Chemistry An Introduction to the Analysis of Spin-Spin Splitting i n High-Re solution Nuclear Magnetic Resonance Spectra Notes on Molecular O r b i t a l ~ a l c u l aions t John D Roberts P r o f e s s o r of Organic C h e m i s t r y California Institute of Technology I l l u s t r a t e d by the Author 1961 THE BENJAMINICUMMIMGS PUBLlSHING COMPANY, INC ADVANCED BOOK PROGRAM Reading, Massachusetts London Amsterdam Don Mills, Ontario Sydney Tokyo NOTES ON MOLECULAR ORBITAL CALCULATIONS First printing, 1961 Second printing, with corrections, 1962 Third printing-Twelfth printing, 1963-1 970 Thirteenth printing, 19 74 Fourteenth printing, 1978 The publisher wishes to express his appreciation to Dr Roberts, who, in addition to writing the manuscript, prepared all the illustrations The author was also responsible for the editing and composition; final pages, ready for the camera, were typed at the California Institute of Technology under Dr Roberts' supervision International Standard Book Number: 0-8053-8301-8(paperback) Library of Congress Catalog Card Number: 61-1 81 59 Copyright @ 1961 by W A Benjamin, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, W A Benjamin, Inc., Advanced Book Program, Reading, Massachusetts, 01867, U.S.A Manufactured in the United States of America ISBN 080538301-8 ABCDEFGHIJ-AL-78 Certificate of Recordation THIS IS TO CERTIFY THAT THE ATTACHED DOCUMENT WAS RECORDED IN THE COPYRIGHT OFFICE ON THE DATE AND IN THE PLACE SHOWN BELOW THIS CERTIFICATE IS ISSUED UNDER THE SEAL OF THE COPYRIGHT OFFICE DATE OF RECORDATION Jan85 VOLUME PAGE VOLUME PAGE OFFICIAL SEAL Acting ~e@st;?r of Copyrights and Assistant Librarian for Copyright Services Certificate of Recordation C-762 November 1984-15,000 THE BEMJAMIN/CUMMINGS PUBLISHING COMPANY 2727 SAND HILL ROAD MENLO PARK, CALIFORNIA 94025 (415) 854-6020 ASSIGNMENT OF COPYRIGHT KNOW ALL MEN BY THESE PRESENTS, t h a t for good and val ua5l e c o n s i d e r a t ~ o n the unders igned The Benjami n/Cummi ngs Pub1 ish ing Company, Inc , the c o p y r i g h t p r o p r i e t o r o f a book e n t i t l e d : NOTES ON MOLECULAR ORBITAL CALCULATIONS does hereby a s s i g n and t r a n s f e r t o : JOHN D ROBERTS his/her h e i r s , executors, a d m i n i s t r a t o r s and assigns, a l l o f the r i g h t , t i t l e and i n t e r e s t o f t h e undersigned i n and t o the c o p y r i g h t i n said book, and any and a l l renewals, extensions and/or continuations t h e r e o f , and i n and t o any and a l l causes o f a c t i o n h e r e t o f o r e o r h e r e a f t e r accrued o r accruing f o r i n fringement o r threatened o r a l l e g e d infringement o f s a i d copyright, renewals, extensions o r c o n t i n u a t i o n s Said copyright was r e g i s t e r e d by the Register of Copyrights i n t h e United States o f America as published on: 13 November, 1961 Copyright R e g i s t r a t i o n No.: ~556620 1.N WITNESS THEREOF, t h e undersigned has executed t h i s agreement i n t h e United States o f America as an instrument under seal t h i s 13th day o f December, 1984 - The BENJAMI N/CUMM I NGS PUBLl SHl NG CO MP A NY , BY &'I.?p_., d.,-"- I t s V i c e President and t k h e r a l Manager, ".J ALLAN M WYLDE I NC Preface F o r practicing organic chemists the simple, linearcombination-of-atomic-orbitals (LCAO), molecular- orbital method p e r m i t s useful calculations of semi- empirical electronic energies of unsaturated molecules with no m o r e than high school algebra x4 Anyone who can find the roots of - 5x2 + 4x = graphically, analytically, o r -b y successive substitutions can obtain the energy levels and calculate the IT-electron energy of bicycle[ 1.1 01 butadiene bicyclobutadiene If in addition he can solve x4 - 4x2 = 0, then he c a n c o m p a r e bicyclobutadiene with cyclobutadiene and predict what changes the 1, bond would make i n the a- electron energies With no m o r e advanced mathematics, one can compute bond o r d e r s , charge distributions, and reactivity p a r a m e t e r s for both freeradical and polar p r o c e s s e s The r e s u l t s may be crude, but they a r e often highly suggestive; t h e r e i s no excuse for a mode r n organic chemist not to be able to u s e the LCAO method The notes that make up this book have been used f o r many y e a r s a t the California Institute of Technology to introduce s e n i o r s and graduate students to the elements of the simple LCAO method A fairly l a r g e number of e x e r c i s e s a r e i n t e r s p e r s e d i n the text to i l l u s t r a t e important points i s recommended that these b e solved a s encountered It Some of the problems a r e hoped to be suggestive of possible r e s e a r c h problems in the field These Notes a r e not intended a s a complete c o u r s e of study and shouldibe supplemented by the r e f e r e n c e works listed v in the Bibliography recent literature introduction No attempt has been made to survey the The purpose has been to provide a practical As a r e s u l t ho appropriate acknowlkdgment to either the priority of ideas o r to their development has been given This s e t of notes would never have been written without the generous contributions of P r o f e s s o r W G McMillan and Dr V Schomaker to the a u t h o r ' s education i n the subject m a t t e r C a m e r a copy was p r e p a r e d by M r s Allene Luke with the a i d of M i s s Joy Matsumoto JOHN D ROBERTS Table of Contents P r e f a c e v ATOMIC ORBITAL MODELS Hydrogen-like Orbitals Bond Formation Using Atomic Orbitals Hybrid Bond Orbitals 11 Bond Orbitals for Atoms Carrying Unshared Electron P a i r s .14 Orbitals for Multiple Bonds 15 MOLECULAR ORBITAL CALCULATIONS ELEC- 23 The Wave Equation + 23 Molecular Orbitals The LCAO Method The Overlap Integral S 1J The Coulomb Integral a 31 The Resonance Integral P 32 Energy Levels of H z @ 33 Bonding and Antibonding Orbitals 36 The Hydrogen Molecule 39 Localized Bonds 40 Ethylene by the LCAO Method 42 Butadiene E, .43 Butadiene Resonance Energy 47 The Butadiene Wave Functions 48 TRONIC ENERGY LEVELS BOND ORDERS FREE-VALENCE INDEXES 53 The Mobile Bond Order p 53 1J The Free- Valence Index % 56 Charge Distributions qi 58 Self-consistent Fields 59 AND CHARGE DISTRIBUTIONS vii Heine, V 141 Howden, M E H 87 Hllckel, E 75,127 Huisgen, R 131 Hund, F , , J o r d a n , D 140 K e t e l a a r , J A A 140 Kimball, G E 72, 128, 140 Kopineck, H J 83, , Langeland, W.E 128 Lee, C C 89 Lefebvre, R 140 L e w i s , G N Linnett, J W 141 Liu, C T 128 Longuet-Higgins, H C , , 98, 105, 110, 128 Lothrop, W C 128 Maccoll, A 128 McMillan, W G , J r 130 Mann,D E 60 M a z u r , R H 89 Moffitt, W E 128 M o s e r , C 140 Mulliken, R S 30, 33 Nagata, C 96 Namiot, A J 129 Olah, G 140 P a u l i , W Pauling, L 1, 1, 94, 98, 136, 140 Penney, W.G 128 P i t z e r , K S 140 P o r t r i d g e , M.A 140 P u l l m a n , A 128, 133, 140 P u l l m a n , B 133, 140 Regan, C M 127 R i c e , F 140 Rieke, C 33 Ringold, H J 127 R o b e r t s , J D 77, 87,89, 99, 127, 128, 131 R o b e r t s o n , A P 54 ROmming, C 134 Rushbrooke, G S 60, 128 Schmeising, H N 55 S c h o m a k e r , V 128,130 Semenow, D A 89 Shingu, H 96 S i l v e r , M S 89 Simonetta, M 87, 88 Sixma, F L J 99 S l a t e r , J C 11 Staab, H 140 S t r e i t w i e s e r , A , Jr 99, 101, 127, 141 Syrkin, Y.K 129, 140 T e l l e r , E 140 Ugi, I 131 W a l t e r , J 72, 128, 140 W a s e r , J 128 Wheland, G W 60, 73, 94, 98,99, 128, 130, 140 White, W N 89 Winstein, S 87, 88 Woods, W G 87 Yonezawa, T 96 Subject 1ndek Acetylene, atomic o r b i t a l m o d e l of, 17- 18 Acrolein, LCAO t r e a t m e n t of, 79 AH ( s e e Alternant h y d r o c a r b o n s ) Ally1 r a d i c a l , nonbonding o r b i t a l of, 74 r e s o n a n c e energy of, 11 cu ( s e e Coulomb i n t e g r a l ) Alternant hydrocarbons, definition a n d e x a m p l e s of, 60 r e a c t i v i t y of, 94, 96 Anthracene, calculated bond d i s t a n c e s of, 54-55 r e a c t i v i t y of, i n Diels- Alder r e a c t i o n , 102-103 Antibonding o r b i t a l s , 36-38 Approximate LCAO calculations, 105-1 14 A r o m a t i c c h a r a c t e r , 75-76, 127-130 Atomic energy l e v e l s , Atomic o r b i t a l s , u s e i n bond formation, wave functions f o r , 25 Azulene, prediction of r e a c t i v i t y of, 94 Bent bonds, 88-8'9 Benzene, approximate calculation of E T l of, 104 atomic o r b i t a l m o d e l of, 20 bond o r d e r s of, 54 145 f r e e valence index of, 58 r e s o n a n c e energy, 48 Benzenediazonium ion, LCAO t r e a t m e n t of, 134-135 Benzyl r a d i c a l , energy l e v e l s and electronic configuration of, 106 f r e e valence index of, 58 NBMO of, 105-107 p ( s e e Resonance i n t e g r a l ) Bicyclobutadiene, LCAO t r e a t m e n t of, 120-126 m o l e c u l a r d i a g r a m of, 126 Bicyclobutonium cations, 87 Biphenylene, calculated reactivity of, in Diels- Alder reaction, 103 Bond angles, of acetylene, 18 of ethylene, 42-43 and o v e r l a p i n t e g r a l s , 88-89 with o r b i t a l s , p r e d i c t i o n of, of of o r b i t a l s , 12-13 o r b i t a l s , 13 of s~~o r b i t a l s , 13-14 Bond lengths, and bond o r d e r s , 53-54 Bonding o r b i t a l s , 36-38 Bond o r d e r s , p , of benzene, 54 1J of bicyclobutadiene, 124-125 and bond lengths, 54-55 calculation of, 53- 54 c h e c k on calculations of E 'IT with, 55 definition of, 53 and f r e e valence index, 56-57 of small- ring hydrocarbons and r a d i c a l s , 128-130 1,3-Butadiene, approximate calculation of E 'IT of, 110 a t o m i c o r b i t a l model of, 19-20 bond d i s t a n c e s of, 53 bond o r d e r s of, 53 c h a r g e distribution calculated f o r , 59 e n e r g y l e v e l s of, 43-46 f r e e valence indexes of, 57 frontier- electron t r e a t m e n t of, 96-97 group t h e o r y applied t o , 61-63 LCAO t r e a t m e n t of, 43-52 m o l e c u l a r d i a g r a m of, 59 self- consistent field of, 59 Cz c h a r a c t e r table, 71 C2 o p e r a t i o n s , 64-65,70-71 C a r b a n i o n s , small- ring types of, 129-130 Carbon, e l e c t r o n i c configuration, 4, hybridization of, 15 o v e r l a p i n t e g r a l s f o r , 29-30 promotion energy of, 15 r e s o n a n c e i n t e g r a l f o r , 34 Carboniurn ions, calculations of e n e r g y of, 87-89 r e a c t i v i t y of, 87, 100-101 small- ring types of, 12 9-1 30 C h a r a c t e r t a b l e s , C2, 71 Dzv, 66 C h a r g e distributions, qi, of benzyl cation and anion, 106 of bicyclobutadiene, 125 of 1,3-butadiene, 59 definition of, 58 in diazoazide configurations, 139 f o r p y r r o l e , 80 and r e a c t i v i t y , 94-95 C h e m i c a l r e a c t i v i t y ( s e e Reactivity) Coefficients of wave functions, f o r bicyclobutadiene, 122-124 calculation of, 34, 49-51, 122-124 Configuration interaction, 116 C o r r e l a t i o n , e l e c t r o n , 17 Coulomb i n t e g r a l s , cr, f o r c a r b o n i u m ions, 101 c o r r e c t i o n f o r nonself- consistent fields, 60 definition of, 31 i n r e a c t i v i t y p r o b l e m s , 92, 95 C r o s s i n g of energy p r o f i l e s , 93, 103 Cyclobutadiene, d e g e n e r a t e o r b i t a l s of, 74 dipositive ion of, 74 e n e r g y l e v e l s and e l e c t r o n i c configuration of, 73 substituted types of, 128-1 30 CycloHctatetraene, a t o m i c o r b i t a l m o d e l of, 21 nonplanar s t r u c t u r e of, , Cyclopentadienyl anions, 76 Cyclopentadienyl r a d i c a l , g r o u p t h e o r y applied to, 70-71 Cyclopropenium c a t i o n s , 76, 129 d- Orbitals, 78 DE.,, ( s e e Resonance e n e r g y ) Degeneracy, accidental, Degenerate orbitals, of cyclobutadiene, 74-75 p r o p e r t i e s of, 74-75 7-Dehydronorbornyl, cation, calculation of energy of, 87 Delocalization p r o c e d u r e s f o r calculation of r e a c t i v i t i e s , 100101 D i a z o a z i d e s , 77 - c yclization of, 13 13 LCAO t r e a t m e n t of configuration of, 13 Diels- Alder r e a c t i o n , 102-103 Di-2-xylylene, f r e e valence index of, 58 ET, approximate calculations of, 110-1 12 of benzyl radical, 106 calculation of, for benzenediazonium ion, 134- 13'5 of bicyclobutadiene, 120-12 f r o m bond o r d e r s , 55 for butadiene, 43-46 of diazoazide configurations, 13 1-139 for ethylene, 42 of organic azides, 135-136 of cyclobutadiene, 73 of various radicals, 11 Electron correlation, 117 Electron p a i r s , Electron probability, Electron spin, Electronic charges ( s e e Charge distribution) Electronic configurations of a t o m s , Electronic energy, sample calculations f o r H3, 118-120 Electrons, unpaired, Ethylene, atomic orbital model of, 18 f r e e valence index of, 58 LCAO treatment of, 42-43 localized bonds of, 41-42, 43 wave functions of, 42 Exclusion principle, , si( s e e F r e e valence index) 4n + Rule, 75-76, 127-130 F r e e radical reactivity, f r o n t i e r electron methods f o r , 97 F r e e radicals, ET values f o r , 111 small- ring types of, 129-130 ( s e e a l s o Benzyl radical) F r e e radical reactivity, f r e e valence index and, 57-58, 94-96 F r e e valence index, g i , of bicyclobutadiene, 125 f r o m bond o r d e r s , 56-57 of 1,3-butadiene, 57 definition of, 56 and reactivity, 57-58, 94, 96 of small- ring hydrocarbons and radicals, 128-130 of trimethylenemethane, 56, 58 f o r typical substances, 58 Frontier- electron method, 96-97 F r o n t i e r orbital, 97 Ground state predictions of reactivity, 94 Group theory, cyclopentadienyl r a d i c a l determinant and, 70 naphthalene determinant and, 64-70 simplification of determinants with 61-72 ( s e e a l s o Symmetry operations) H3, calculation of binding energy of, 118-120 Hamiltonian energy o p e r a t o r , 24 Heteroatoms, Coulomb integrals f o r , 78-79 LCAO treatment of molecules with, 77-80 resonance integrals f o r , 78 Heterocyclic molecules, LCAO treatrhent of, 77-80 Hexaazabenzene, 7 Homoallyl cations, 87-89 HUckells r u l e , 75-76, 127-130 Hund's r u l e , , , Hybridization, 11-14 i n acetylene, 17-1 i n benzene, 20-21 i n ethylene, 18-1 i n molecules with unshared p a i r s, 14-1 Hydrogen, electronic configuration and energy of, 39-40 Hydrogen atom, 2-3 Hydrogen molecule ion, energy levels and wave functions f o r 33-36 Hydrogen sulfide, bond angles of, 11 Identity operation, 64 Interelectronic repulsion, 4, 39-40, 11 and configuration interaction, 116 Kopineck's tables, 83-84 LCAO method, definition of, 25-26 LCAO treatment, of benzenediazonium ion, 134-135 of bicyclobutadiene, 120-126 of butadiene, 43-52 of diazoazide configurations, 13 1-139 of ethylene, 42-43 of H3,1 18-1 20 of hydrogen molecule, 39-40 of hydrogen molecule ion, 35-37 of organic azides, 135-136 of small- ring hydrocarbons and r a d i c a l s , 127-1 30 validity of, 104, 11 Localization procedures, 98-100 approximate method f o r , 13-114 Localized bonds, 40-41 Methyl radical, free- valence index of, 58 Mobile bond o r d e r (see Bond o r d e r ) Molecular diagrams, of azulene, 94 of bicyclobutadiene, 126 of small- ring hydrocarbons and r a d i c a l s , 12 of 1,3-butadiene, 59 Molecular orbitals, concept of, 25-26 Multiple bonds, orbitals f o r , 15-1 NAH ( s e e Nonalternant hydrocarbons) Naphthalene, approximate calculation of substitution in, 113-1 14 approximate calcblation of E of, 113 IT calculated bond distances of, 54-55 group theory applied to, 64-70 NBMO ( s e e Nonbonding molecular orbitals) Nitrogen, atomic orbital model f o r , 16 Coulomb and resonance integrals for, 78 LCAO treatment of diazoazide configurations, 131-1 39 organic azides, 135-136 promotion energy of, 15 Nodes, in butadiene wave functions, 52 in molecular wave functions, 38 Nonalternant hydrocarbons, definition and examples of, 60 Nonbonding molecular orbitals, of benzyl radical, 105-107 calculation of coefficients f o r , 107-109 f o r calculation of localization energies, 13-1 14 of cyclobutadiene, 74 definition of, 74 E f r o m , 110 7f Nonorthogonal wave functions, Nonplanar systems, 82-90 Nonself-consistent fields, 59-60, 15 heteroatoms and, 79 Normalization factor, 35 Normalized wave functions 24-25, 35 Nucleophilic reactivity, 97 O r b i t a l hybridization, 11-14 O r b i t a l s , antibonding, 36-38 a t o m i c , 2-4 bonding, 36-38 degenerate, hydrogen- like, 1-4 nonbonding, 74, 105-107 o v e r l a p of, , EJ bond f o r m a t i o n with, s, 2-3 - bond f o r m a t i o n with, Orientation i n a r o m a t i c substitution, 91-100 calculation of, 13-1 14 Orthogonal wave functions, 29 Overlap, i n benzene , in cyclotlctatetraene, 21 of hybrid o r b i t a l s , 14 of and p o r b i t a l s , of of of o r b i t a l s , 12 =' o r b i t a l s , 13 z3 orbitals, 14 O v e r l a p i n t e g r a l , calculation of, 82-85 f o r bent bonds, 88-89 f o r c a r b o n a s a function of hybridization and d i s t a n c e , 29-30' definition of, 8-2 of nonplanar p o r b i t a l s , 82-85 Oxygen, Coulomb and r e s o n a n c e i n t e g r a l s f o r , 78 p Bonds, angles of, 8-11 p O r b i t a l s , calculation of o v e r l a p of, 82-85 T o v e r l a p of, 16 p, ( s e e Bond o r d e r ) 1J P a u l i exclusion principle, , P e n t a l e n e , approximate calculation of E Tr of, 12 P e n t a z o l e s ( s e e Diazoazides) ~r Bonds, 16 i n acetylene, 17 i n benzene, 20-2 i n butadiene, 20 i n ethylene, 18 i n nitrogen, 17 T-Electron energy ( s e e E ) IT IT-Overlap, 16 P r o d u c t s t a b i l i t i e s , and r e a c t i v i t y , 102-103 P r o p y l c h l o r i d e , r e a c t i v i t y of, 100-101 P s e u d o a r o m a t i c compounds, 128-1 P y r i d i n e , LCAO t r e a t m e n t of, 78 P y r r o l e , c h a r g e distributions calculated f o r , 80 qi ( s e e C h a r g e distribution) Radicals (see F r e e radicals) R a t e p r o b l e m s ( s e e Reactivity) R e a c t i o n r a t e s ( s e e Reactivity) Reactivity, a p p r o x i m a t e calculations of, 113-1 14 a r o m a t i c substitution, 13-1 14 of azulene, 94 c a r b o n i u m ion, 87, 100-10 a n d c h a r g e d i s t r i b u t i o n s , 94-95 delocalization p r o c e d u r e f o r calculation of, 100-101 i n Diels- Alder r e a c t i o n , 102-103 a n d f r e e valence index, 95 LCAO t r e a t m e n t of, 88-89, 91-104 localization p r o c e d u r e f o r calculation of, 98-100 p e r t u r b a t i o n m e t h o d s f o r , 95-98 p r o b l e m s of, 91-93, 104 Resonance energy, of benzene, 48 of benzyl r a d i c a l , 106 of bicyclobutadiene, 12 of 1,3-butadiene, 47 of cyclobutadiene, 73 of small- ring hydrocarbons and r a d i c a l s , 128-130 Resonance integral, calculation a s function of d i s t a n c e , 82 carbon, a s a function of distance, 33 definition of, 32 a s function of angle, 32 and overlap integral, 82 S ( s e e Overlap integral) 1J Secular determinant, 28 Self- consistent fields, definition of, 59 a Bonds, u-IT i n t e r a c t i o n s , 11 Single bonds, Singlet s t a t e , 74, 127-130 93- Bonds, i n acetylene, 17-18 =-Orbitals, 12-13 ='-orbitals, 13 93-0rbitals, 13-14 S y m m e t r y operations, DzV s y m m e t r y , 63-66 two-fold a x e s , 63-66, 70-71 Tetraazacyclobutadiene, 77 T r a n s i t i o n s t a t e , 92-93 Trimethylene methane, f r e e valence indexes of, 56, 58 Triphenylmethyl chlorides, ionization of, 10 T r i p l e bond, atomic o r b i t a l s f o r , 16-18 T r i p l e t s t a t e , 128-130 of ~ ~ c l o b u t a d i e n e73 , 156 Tropylium cations, 76 Unshared electron p a i r s , bond formation of a t o m s with, 14-1 Variation method, 27-28 and configuration interaction, 116 Wave equation, 23-25 Wave functions, of bicyclobutadiene, 12 1-124 of 1, 3-butadiene, 48-52 derivation through group theory, 67-69, of ethylene, 42 f o r hydrogen molecule ion, 34-35 normalized, 24-25 Water, atomic orbital model of, 9, 10 bond angles in, 9-1 hybridization in, 14-1 ... Author 1961 THE BENJAMINICUMMIMGS PUBLlSHING COMPANY, INC ADVANCED BOOK PROGRAM Reading, Massachusetts London Amsterdam Don Mills, Ontario Sydney Tokyo NOTES ON MOLECULAR ORBITAL CALCULATIONS. .. ARTICLES ON LCAO 127 Appendix I11 GENERAL BIBLIOGRAPHY 140 Name Index 142 SubjectIndex 145 CALCULATIONS Chapter Atomic Orbital Models MOLECULAR ORBITAL and valence bond calculations. .. Chemistry An Introduction to the Analysis of Spin-Spin Splitting i n High-Re solution Nuclear Magnetic Resonance Spectra Notes on Molecular O r b i t a l ~ a l c u l aions t John D Roberts P r o f e