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McGRAW-HILL SERIES IN ADVANCED CHEMISTRY Senior Advisory Board W. Conard Fernelius Louis P. Hammett Editorial Board David N. Hume Gilbert Stork Edward L. King Harold H. Williams John A; Pople Dudley R. Herschbach BAIR Introduction to Chemical Instrumentation BALLHAUSEN Introduction to Ligand Field Theory BENSON The Foundations of Chemical Kinetics BmMANN Mass Spectrometry DAVIDSON Statistical Mechanics DAVYDOV (Trans. Kasha and Oppenheimer) Theory of Molecular Excitons DEAN Flame Photometry D JERASSI Optical Rotatory Dispersion ELmL Stereochemistry of Carbon Compounds FITTS Nonequilibrium Thermodynamics HELFFERICH Ion Exchange HILL Statistical Mechanics HINE Physical Organic Chemistry KIRKWOOD AND OPPENHEIM Chemical Thermodynamics KOSOWER Molecular Biochemistry LAITINEN Chemical Analysis PITZER AND BREWER (Revision of Lewis and Randall) Thermodynamics POPLE, SCHNEIDER, AND BERNSTEIN High-resolution Nuclear Magnetic Resonance PRYOR Mechanisms of Sulfur Reactions ROBERTS Nuclear Magnetic Resonance ROSSOTTI AND ROSSOTTI The Determination of Stability Constants STREITWmSER Solvolytic Displacement Reactions WIBERG Laboratory Technique in Organic Chemistry Introduction to LIGAND FIELD THEORY Co~J .CARL J.BALLHAUSEN Professor of Chemistry University of Copenhagen, Denmark McGRAW-HILL BOOK COMPANY, INC. New York San Franci.sco Toronto Lontlon INTRODUCTION TO LIGAND FIELD THEORY Copyright © 1962 by the McGraw-Hill Book Company, Inc. All Rights Reserved. Printed in the United States of America. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Librar11 of Congreaa Catalog Card Nu:mber 62-13206 03580 Preface I have tried to give an introduction to that field of chemistry which deals with the spectral and magnetic features of inorganic complexes. It has been my intention not to follow the theory in all its manifestations, but merely to describe the basic ideas and applications. This has been done with an eye constantly aimed at the practical and experimental features of the chemistry of the complex ions. The· book is thus pritnarily intended for the inorganic chemist, but it is true that, in order to follow the exposition, a course in basic quantum mechanics is needed. Simple examples are nearly always used to illustrate the arguments, but the quoted experimental evidence must of necessity be limited. Ne\terthe less, in the last chapter an attempt has been made to cover most of the important work so far performed that lies within the scope of the book. However, the field is advancing so rapidly that a complete survey would be outdated before long. Since I am a chemist writing for chemists, my emphasis and notation will probably appear clumsy to the physicists primarily responsible for the theory. For this I make no excuse. Elegant derivations and condensed notation are in my opinion not desirable in an introduction to a field. Nothing is more dangerous than to force every observation into a fixed framework of ideas. I have tried tQ present the case for the ligand field theory as it is. now understood. It is my personal view that there are really only a few places where we need· to revise part of the theory in order to understand the sundry phenomena. · It must always be remembered, however, that the ligand field theory offers only a model of nature, with all the inherent limitations of modeis. I want particularly to thank, among many others, Dr. Andrew D. Liehr, Mellon Institute, Dr. Max Wolfsberg, Brookhaven National Laboratories, and Dr. Harry B. Gray, Columbia University, for numerous discussions and for help with the manuscript. I am also greatly indebted to Mrs. Lise Seifert for her assistance in preparing the manuscript. Finally, I want to thank the editors of Annual Review of Physical Chemistry for per mission to draw from the 1956 paper written by the late Prof. W. Moffitt and myself. Carl J. Ballhausen Contents Preface . v Chapter 1 INTRODUCTION . 1. 1-a. History of complexes ·1 1-b. Theories of bonding 2 1-c. History of the crystal field approach. 3 References · 5 Chapter B THEORY OF ATOMIC SPECTRA 7 .,. 2-a. Orbitals and states . 7 2-b. Atomic wave functions 10 2-c. The raising and lowering operators 11 2-d. Matrix elements 15 2-e. Two-electron operators . . . . . . 17 2-f. Evaluation of the matrix elements (abll/ru!cd) 19 2-g. Term energies . 21 2-h. General remarks on the method . 25 2-i. Spin-orbit coupling in a hydrogen-like system . 25 2-j. Spin-orbit coupling in a many-electron case 27 2-k Absolute term intervals 29 2-l. Zeeman splitting 30 2-m. Selection rules 32 References 33 Chapters SYMMETRY 34 3-a. Concept of symmetry operators 34 3-b. Nomenclature of sy!Jlmetry operators 36 ~ 3-c. Representations • 37 3-d. Important point groups occurring in inorganic complexes. 41 3-e. Representations and wave functions . 43 3-/. The direct product . 46 3-g. Double groups . 49 3-h. The Eulerian angles 54 References 56 Chapter 4 THE CRYSTAL FmLD THEORY: I. FIELDS OF OCTAHEDJ,UL SYMMETRY . 57 4-a. Formalism . 57 4-b. Octahedral fields 60 vii viii CONTENTS 4-c. Single d electron in a cubic field 62 4-d. Weak fields . 69 4-e. Strong fields. 74 4-f. Fields of intermediate strength 80 4-g. Computation aids . 84 4-h. Descent in symmetry . • . 87 4-i. Equivalence of ti, and p electrons 89 4-j. The spectrochemical series 91 Appendix 1 93 Appendix 2 95 Appendix 3 96 References 97 Chapter 5 THE CRYSTAL FIELD THEORY: 11. FIELDS OF TETRAHE- DRAL AND OF LOWER SYMMETRY 99 5-a. Tetragonal fields 99 5-b. Trigonal fields . 103 5-c. Cis, trans and rhombic fields . 106 5-d. Tetrahedral fields 108 Appendix 1 111 References 112 Chapter 6 SPIN-ORBIT COUPLING 113 6-a. Importance of spin-orbit coupling 113 6-b. Spin-orbit coupling for one d electron in octahedral fields. 114 6-c. Spin-orbit coupling for dn-configurations in octahedral fields . 120 6-d. Spin-orbit splittings calculated by the method of Abragam and Pryce 124 6-e. g factors in an octahedral field 127 6-f. g factors in a tetragonal or trigonal field 131 6-g. The spin Hamiltonian . 137 6-h. Magnetic susceptibilities 139 Appendix 1 149 Appendix 2 149 References 150 • Chapter 7 MOLECULAR ORBITALS 152 7-a. General discussion . 152 7-b. Bonding scheme for an octahedral complex. 159 7-c. Estimation of wave functions in an MO scheme 163 7-d. Band intensities in parity allowed transitions 170 Appendix 1 174 Appendix 2 177 References 178 Chapter 8 VIBRONIC INTERACTIONS 180 8-a. Vibrational spectra. 180 8-b. Absorption band intensities for centrosymmetric complexes 185 8-c. Dichroism . 192 8-d. Jahn-Teller configurational instability . 193 8-e. Experimental Evidence of the Jahn-Teller Effect. 205 References 208 CONTENTS ix Chapter 9 .:.OME FURTHER ASPECTS 211 9-a. The Faraday effect . 211 9-b. Optical rotatory dispersion 214 9-c. "Sandwich" compounds 217 9-d. Stability of complex ions 221 References 224 Chapter 10 ELECTRONIC STRUCTURES OF SELECTED INORGANIC COMPLEXES 226 Complezes Containing Sd Electrons . 227 10-a. Scandium 227 10-b. Titanium 227 10-c. Vanadium 228 10-d. Chromium. 235 10-e. Manganese. 245 10-/. Iron. 251 10-g. Cobalt . 255 10-h. Nickel . 261 10-i. Copper . 268 Complezes Containing 4d and 5d Electrons. 273 10-j. Niobium 273 10-k. Molybdenum 274 10-l. Technetium 275 10-m. Ruthenium. 275 10-n. Rhodium 276 10-o. Palladium 277 10-p. Tungsten 277 10-q. Rhenium 278 10-r. Osmium 279 10-s. Iridium. 281 10-t. Platinum 282 References 283 Index. 293 CHAPTER I Introduction 1-a. History of Complexes The name "complex" was first used in the chemical literature late in the nineteenth century. Thanks primarily to the pioneering works of C. W. Biomstrand and S. M. J!Zlrgensen, t a series of compounds for which no explanation could be given in terms of the classical valence picture was characterized. These two investigators and their pupils were especially interested in the compounds containing either trivalent chromium, tri valent cobalt, or divalent platinum and the seemingly irrational products obtained when these ·elements were treated with ammonia, chlorinei etc. Most baffling was the existence of isomers; e.g., the compound CoCla·4NH1 could exist in two forms, a violet one and a green one. In retrospect it is, of course, easy to see why the above-mentioned workers failed to reach an understanding of these and similar phenomena; "stereochemistry" was at that time a completely new field. The complete failure of the classical valence picture tO account for the chemistry of what we now call the transition groups was first realized by Werner (1893). Proceeding mostly on the basis of work performed by Jfllrgensen, he proposed, for instance, that the yellow "luteo salt" Co (NHa)eCla was built in the shape of a regular octahedron, the Co(III) ion being placed in the center and the six ammonia molecules being located at the corners of the polyhedron. Werner introduced two different kinds of valence forces in order to account for the chemistry of these compounds: the "primary valence," which ~quals three in the luteo salt, and the "second ary valence," which equals six in this example. In other words, the com pound should, in a modern language, be formulated as [Co(NHa)e]HCl,&-. The explanation of the previously mentioned puzzle is then clear; the green and violet compounds [Co(NH ),Cl2]Cl are the cis- and trans-dichloro 3 tetrammine cobalt(III) chloride. The study of the isomers of the platinum(II) complexes further led Werner to postulate a square planar configuration for these compounds, an assumption which modern X-ray work has proved to be correct. Time has thus shown that Werner's view on the stereochemistry of the complex t See, for example, Ref. 1 for an account of the early development of the theory. 1 2 INTRODUCTION TO LIGAND FIELD THEORY molecules was completely right, but it is only in recent years that an under standing of why the complexes form and behave as they do has been reached. The conception of a coordination or complex molecule has in modern times been extended to be any compound containing a central ion surrounded by a cluster of ions or molecules. Such a compound may be more or less chemi cally stable; S04-, for instance, is a very stable complex, whereas Ni(NH3)6++ is not. It is customary to call the ions or molecules surrounding the central ion "ligands." The central ion or atom may accommodate a certain num ber of ligands, and this number is called the "coordination number." In sor, the case of for example, it is four. In some cases a central atom or ion exhibits a different coordination number toward different ligands. A well-known example is Ni(II), because this ion can be either hexa- or tetracoordinated. We say that Ni(II) has the characteristic coordination numbers six and four. The question of the actual coordination number of a certain ion is usually investigated in the solid state by means of an X-ray analysis and in solution by means of an analysis of the consecutive complexity constants. This latter subject has in modern times been especially investigated by J. Bjerrum. Since it is not intended to follow the subsequent development of the history of the coordi nation compounds here, however, we shall refer the reader to Bailar's book1t for a broad outline of the subject. We shall in this book be concerned only with the electronic structures of the transition series, i.e., the elements having unfilled d or f electronic shells. These are the first transition series (Sc--+ Cu), where the 3d shell is being filled up; the second transition series (Y--+ Ag), where the 4d shell is being filled up; and the third transition series involving 5d electrons (La--+ Au). In addition, we have the rare earths (Ce-+ Lu) and the actinides (Th--+ ), where the 4f and the 5f shells, respectively, are being filled. We shall be primarily interested in the chemistry and electronic struc tures of the elements containing d electrons. Furthermore, since most wor.k has been done with the elements of the first 3d transition series, nearly.all.of the examples given in the following chapters are taken from there. It must, however, be realized that this limitation is of a practical rather than of a theoretical nature, because the theory which we shall outline applies to complexes of all the transition elements. 1-b. Theories of Bonding The all-important question for the coordination compounds of the transi tion metals is this: How does one describe and characterize the bonding between the central ion and the ligands in terms of some electronic theory? In modern times three methods have been used to solve the problems of the nature of these bonds and to account for the other properties of the com plexes. They are: t Superscript numbers are those of references listed at the end of the chapter. INTRODUCTION 3 1. The molecular-orbital method 2. The valence-bond theory 3. The crystal or ligand field theory Until recently, most chemists working with the complexes of the transi tion metal ·ions have been mainly interested in the application of the valence-bond theory as exemplified by Pauling2 in his famous book "The Nature of the Chemical Bond." Special emphasis was there laid upon the magnetic properties of the complexes, and a seemingly successful theory was built upon those features. However, more than twenty years have passed since Van Vleck3- 5 demonstrated the superiority of the crystal field approach in the discussion of the magnetic properties of inorganic complexes. Now, it must at once be said that for the complexes under discussion both the valence-bond picture and the crystal field picture can be considered as a specialization of the molecular-orbital method. 6• 7 Indeed, the most useful approach to these compounds is now called the ligand field theory, which is really nothing more than a hybridization of the ideas of Bethe6 and Van Vleck3•4 with those of Mulliken.7•6 Thus the best features of both the valence-bond picture and the crystal field theory are incorporated in the ligand field theory, and it is this theory with which we shall be mostly concerned. As we shall not in this book follow the historical line of development, it is perhaps of some value to scan briefly through the most important papers from which the present theory has emerged. 1-c. History of the Crystal Field Approach The basic idea of the crystal field theory, namely, that the metal ion in the complexes is subjected to an electric field originating from the ligands, is due to Becquerel8 (1929). The same year saw this proposal formulated into an e~xact theory by Bethe. 6 · In a now classic paper, Bethe investigated, by means of symmetry concepts, how the symmetry and strength of a crystalline field affect the electronic levels of the gaseous metal ions. In doing so, he laid down the foundation for all further work in this field. Nearly simultaneous with the work of Bethe was the work of Kramers.9 In 1930, the latter succeeded in proving the very important result that the electronic levels in molecules containing an odd number of electrons must remain at least twofold degenerate, provided that no magnetic field is present. This so-called "Kramers degeneracy" is again closely related to the existence of the "double groups" (Bethe). The first application of the new theory to chemistry was made by Van Vleck (1932). By realizing that the quenching of the "orbital momentum" would be a consequence of the crystalline field model, he succeeded in explaining why the paramagnetism of the complexes of the first transition ° series corresponds to a "spin-only" value.1 Furthermore, the crystalline

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