Introduction to Ligand Field Theory

305 Pages·014.36 MB·English

by  Carl J. Ballhausen

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