In the 1920s, when quantum mechanics was in its infancy, chemists and solid state physicists had little choice but to ma
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Table of contents :
CONTENTS
PREFACE TO THE PAPERBACK EDITION
PREFACE
1. CLASSICAL METHODS
2. CRYSTAL FIELDS
3. THE nj SYMBOLS
4. CONFIGURATIONS OF TWO ELECTRONS
5. CONTINUOUS GROUPS
6. SENIORITY
7. FRACTIONAL PARENTAGE COEFFICIENTS
8. CONFIGURATIONS OF MORE THAN TWO EQUIVALENT ELECTRONS
Appendix I. Radial Integrals for Hydrogenic Eigenfunctions
Appendix 2. The Coefficients (UL\U'L' + f) and (WU\W'U' + f)
References
Index
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
PRINCETON LANDMARKS IN MATHEMATICS AND PHYSICS Nonstandard Analysis, by Abraham Robinson General Theory of Relativity, by P.A.M. Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riemannian Geometry, by Luther Pfahler Eisenhart The Classical Groups, by Hermann Weyl Topology from the Differentiable Viewpoint, by John W. Milnor Algebraic Theory of Numbers, by Hermann Weyl Continuous Geometry, by John von Neumann Linear Programming and Extensions, by George B. Dantzig Algebraic Theory of Numbers, by Hermann Weyl Operator Techniques in Atomic Spectroscopy, by Brian R. Judd
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY BRIAN R. JUDD With α new preface by the author
PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Copyright © 1998 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex First published in 1963 by McGrawHill Book Company, Inc. Reprinted by arrangement with the author Library of Congress CataloginginPublication Data Judd, Brian R. Operator techniques in atomic spectroscopy / Brian R. Judd. p. cm.—(Princeton landmarks in mathematics and physics) Originally published: New York : McGrawHill, 1963. Includes bibliographical references and index. ISBN 0691059012 (pbk.: alk. paper) 1. Atomic spectroscopy—Mathematics. 2. Calculus, Operational. 3. Continuous groups. I. Title. II. Series. QC454.A8J82 1998 539.7Ό287—dc21 989450 Princeton University Press books are printed on acidfree paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First printing, in the Princeton Landmarks in Mathematics and Physics series, 1998 http://pup.princeton.edu Printed in the United States of America 1 2 3 4 5 6 7 8 9
10
CONTENTS Preface to the Paperback Edition
ix
Preface
xi
CHAPTER 1. CLASSICAL METHODS
1
11.
Introduction
12. 13. 14. 15. 16. 17. 18. 19.
The Centralfield Approximation The Perturbation Potential Angularmomentum Operators The 3j Symbols Determinantal Product States Matrix Elements The Configuration sd Deficiencies in the Method
CHAPTER 2
CRYSTAL FIELDS
21. The Contribution to the Hamiltonian 22. Finite Groups 23. The Octahedral Group 24. The Rotation Group in Three Dimensions 25. Eigenfunctions and Operators 26. The WignerEckart Theorem 27. Tensor Operators 28. An External Magnetic Field 29. Neptunium Hexafluoride
CHAPTER 3. THE nj SYMBOLS 31. 32. 33. 34. 35. 36.
Three Angular Momenta The 6j Symbol The 9j Symbol Relations between the nj Symbols for η > 3 Mixed Tensor Operators Matrix Elements of X9^
1 2 5 6 9 13 15 17 21
25 25 27 31 34 36 39 41 44 46
54 54 57 60 62 69 70 ν
Vi
CONTENTS
CHAPTER 4. CONFIGURATIONS OF T W O ELECTRONS 41. 42. 43. 44. 45. 46. 47.
76
Introduction Reduced Matrix Elements . . . . The Coulomb Interaction The SpinOrbit Interaction An External Magnetic Field The Magnetic Hyperfine Interaction Magnetic SpinSpin Interaction
76 77 79 81 84 85 87
CHAPTER 5. CONTINUOUS GROUPS
95
51. 52. 53. 54. 55. 56. 57. 58. 59. 510. 511. 512. 513. 514.
Introduction Lie's Theory of Continuous Groups Definitions The Commutators of Tensor Operators The Metric Tensor The Root Figures Representations The Young Tableaux Casimir's Operator The Characters of Representations The Kronecker Product of Two Representations Branching Rules Classification of the Terms of/" Irreducible Representations as Quantum Numbers
CHAPTER 6. SENIORITY 61. 62. 63. 64.
Doubletensor Operators The Commutators of Doubletensor Operators The Seniority Number The LaportePlatt Degeneracies
.
.
.
.
95 96 100 101 106 107 113 119 123 126 131 136 142 147
154 154 156 158 162
CHAPTER 7. FRACTIONAL PARENTAGE COEFFICIENTS. 166 71. 72. 73. 74. 75. 76. 77. 78.
Introduction The Factorization of the Coefficients (Ω; ω\Ώ) Orthogonality Relations The CFP (p*SL\ Jp3SL) The Coefficients (UfL + f\UTL) Reciprocity The Coefficients (WU + f\WU) The Coefficients (^1BiS + 2 HfS)
166 168 171 171 174 179 180 182
CONrENTS
vii
79. Matrix Elements 710. Explicit Formulas for the CFP
CHAPTER 8. 81. 82. 83. 84. 85. 86. 87. 88. 89.
185 189
CONFIGURATIONS OF MORE THAN TWO EQUIVALENT ELECTRONS
193
Introduction Generalized Tensors Applications of the WignerEckart Theorem SpinOrbit Interaction Matrix Elements of V The Coulomb Interaction H1 The Configuration / s Crystal Field Effects Hyperfine Structure in PuI
193 194 196 198 203 204 210 214 216
A p p e n d i x I. Radial Integrals for Hydrogenic Eigenfunctions Appendix 2.
The Coefficients (UL\U'L' + / ) and (WU]WU'
+/).
.
225
.
227
References
233
Index
237
PREFACE TO THE PAPERBACK EDITION
The original publication of this book took place at a time when rareearth elements were becoming more readily available and when computing methods were beginning to make it possible to study felectron configurations in detail. The acceleration of these develop ments has enabled relatively small effects, such as those coming from the magnetic interactions between the f electrons, to be examined. We have now reached a point where large matrices can be diagonalized without too much trouble, and the effects of configuration inter action in the analyses of rareearth and actinide spectra are now routinely represented by multielectron operators. However, the ba sic theory of angular momentum, as developed by Racah and others, remains intact, and it is probably still best exemplified for the/"shell by the topics that were originally chosen for that purpose. The rotation group in three dimensions lies at the heart of angu larmomentum theory. In describing its generalizations, the text can still play a useful role. Modern computing techniques easily cope with multielectron operators, but it is advantageous to define such op erators in terms of the irreducible representations of suitably chosen Lie groups. This aids in the construction of orthogonal operators whose strengths, as measured by a set of adjustable parameters, are mini mally correlated. The matrix elements of such operators can be evalu ated by the generalized form of the WignerEckart theorem. It is fre quently necessary to calculate the coupling coefficients for the Lie groups, and this calls for a familiarity with the topics covered in the later chapters of this book. Particle physicists, whose interests lie in groups other than the ones introduced for the f shell, may well find something to interest them here. Of course no subject stands still. Diagrammatic methods now go far beyond the merely picturesque representation of coupled angular momenta. An advance in another direction stems from second quan tization, in which operators acting on η electrons at a time are repre sented by coupled products of η creation and η annihilation operaix
χ
PREFACE TO THE PAPERBACK EDITION
tors. A development of these ideas leads to the notion of quasispin, which enables all the properties of the seniority number υ to be ex pressed in the language of angularmomentum. However, such elabo rations of the theory, no matter how elegant, might well be consid ered unnecessary distractions by the many chemists who, perhaps surprisingly, have formed a major component of the readership. Their interest in efficient methods of understanding spectra often take pre cedence over mathematical style. Some readers appear to have found the problems at the ends of the chapters difficult. My intent was to indicate the range of the theory to a casual reader as well as to provide an obstacle course for the serious student. Some of the problems hint at the directions taken in the last thirtyfive years in forming what is now a vast literature on the theory of rareearth and actinide spectroscopy. The book was written at a time when only a few of the excited states of the rareearth and actinide ions had been identified, but at least the direction to take in future research seemed clear. For the reprint, my thanks go to Princeton University Press for their interest and for the oppor tunity to correct some typographical and other minor errors. Some were brought to my attention by penciled corrections in library cop ies. However, I can assure all those who have put modulus signs around some of the symbols m in equations (13) and (14) that those equations are correct as they stand. Finally, I thank my many col leagues and friends who have made my life with the f shell such a happy one. Brian R. Judd
PREFACE
Between 1942 and 1949, Racah published four papers in the Physical Review under the general title Theory of Complex Spectra. Before the appearance of these papers, the standard method for finding expressions for the energies of the terms of a complex atom was to follow a set of rules formulated in 1929 by Slater. In attempting to improve on this prescriptive approach, Racah introduced a number of concepts that have since proved to have widespread applications. In the second of his four papers, he defined a tensor operator; in the third, he developed the relation between a spectroscopic term and its parents and defined the coefficients of fractional parentage; in the fourth, perhaps the most remarkable of the series, he applied the theory of continuous groups to the problem of finding the term energies of configurations of equivalent f electrons. Although immediately relevant to atomic spectroscopy, these papers made a greater impact on nuclear physics. It is in this field that further applications of the theory of continuous groups have been made, principally by Jahn, Flowers, and Elliott. The comparative indifference of atomic spectroscopists is not difficult to understand, however. Most of the theoretical problems connected with their work could be handled by the classical techniques described by Condon and Shortley in their book TAe Theory of Atomic Spectra, a volume seemingly insusceptible of improvement. In addition, the frequent occurrence in the new algebra of what are now known as 6j symbols, often tedious to evaluate, served as an effective barrier to all but the most determined experimentalists. The use of continuous groups, viewed from behind the added hurdle of fractional parentage, appeared remote indeed. Two factors are changing this attitude. In the first place, many 6j symbols have now been tabulated; calculations in which tensor operators are used are no longer irksome, and the advantage of this method over the conventional approach becomes obvious. Second, the increasing availability of elements in the rareearth and actinide series has stimulated research into the microwave and optical spectra xi
xii
PREFACE
of not only the free atoms but also the ions situated at lattice sites in crystals. A rareearth or actinide atom, whether neutral or ionized, very often possesses several electrons outside closed shells, a characteristic feature being a configuration including a number of f electrons. For electrons with such high angular momenta, the methods of Condon and Shortley are often extremely involved and compare very unfavorably with the powerful and elegant techniques of Racah. Several elementary texts on the theory of tensor operators are now available. However, the theory of continuous groups is not readily accessible in a convenient form. In attempting to remedy this deficiency, I have taken the opportunity to review the theory of tensor operators with particular emphasis on those aspects of greatest interest to atomic spectroscopists. Throughout, I have had in mind the experimentalist who wishes to grasp the principles of the theory and to learn how to apply them to physical problems, rather than the mathematician, whose interests seem to be well catered for by such writers as Lie, Cartan, and Weyl. I have included a large number of problems both in the text and at the end of the chapters to emphasize the physical interpretation of the mathematics. Most problems are drawn from configurations of the type /", where Racah's techniques show to their best advantage; needless to say, I have not attempted anything approaching a review of the optical and microwave spectra of the rareearth or actinide atoms. This book has evolved from a course of lectures given at Berkeley in 1960. The interest of the audience in the applications of the theory and the encouragement of Professor W. A. Nierenberg were greatly appreciated. My obligation to the pioneering work of Professor Racah will be apparent in the pages that follow; in particular, I am indebted to him and to the editors of the Physical Review for permission to include a number of tables that originally appeared in that journal. I wish to thank Professor G. F. Koster, Dr. F. R. Innes, and Dr. B. G. Wybourne for reading the manuscript and suggesting a number of improvements. My own interest in applying the theory of continuous groups to problems in atomic spectroscopy was aroused by lectures given by Dr. J. P. Elliott at Oxford in 1956, and to him a special word of thanks is due. Brian R. Judd
1 CLASSICAL METHODS
11
INTRODUCTION
All but the lightest atoms are dynamic systems of great complexity. To analyze the properties of such systems, certain assumptions and approximations must first be made. The immediate aim is to simplify the mathematics, but it is important to be guided by physical considerations; for the purpose of the analysis is not just to account for the properties of a particular atom as closely as possible but also to gain insight into its structure and to discern features that are shared by other atomic systems. After the assumptions and approximations have been made, their implications must be rigorously worked out. The conventional methods for carrying out these two steps are described by Condon and Shortley'i in their classic work The Theory of Atomic Spectra. Although the original assumptions remain virtually unchanged today, the second step in the analysis, often the longer and more difficult of the two, has been changed almost beyond recognition by the pioneering efforts of Racah. An understanding of the more direct approach of Condon and Shortley to problems in atomic spectroscopy is nevertheless essential for an appreciation of the later developments of the theory. We shall therefore begin by outlining the fundamental assumptions, and in the working out of these we shall use those techniques of Condon and Shortley that are of the greatest value at the present time, when alternative and more powerful methods are often open to us. t References to the literature are at the end of the book.
1
2
OPERATOR TECHNIQUES
12
THE CENTRALFIELD APPROXIMATION
IN ATOMIC
SPECTROSCOPY
At the outset an atom is visualized as a point nucleus of infinite mass and charge Ze surrounded by N electrons, each of mass m and charge The nonrelativistic Hamiltonian of such a system is
In this expression, is the distance of electron i from the nucleus, its distance from electron j. The second summation in H prevents a separation of the variables in the Schrodinger equation; yet it is too large to be dropped and treated later by perturbation theory. The approximation that is made is to suppose that the potential in which an electron moves can be reproduced by a function The approximate Hamiltonian is therefore
The difference equation
E is the perturbation potential.
can be separated by choosing a set of functions
The Schrodinger
and writing
On deleting the superfluous references to the specific electrons, we find that the resultant equations are all of the type (11) The only difference between Eq. (11) and the Schrodinger equation for the hydrogen atom lies in the presence of in place of The normalized solutions for bound states can thus be written in the form (12)
CLASSICAL METHODS
3
where the spherical harmonics Ytm are defined by
with The radial function depends on the central potential the angular part of is identical to that for a hydrogenic eigenfunction. The symbolk can be regarded as standing for the set of quantum numbers As in the case of the hydrogen atom, the number of nodes (not counting the origin) in is however, the permitted energies are degenerate with respect to mi only, and not to I as well, as they are in the hydrogen atom. The spin of the electron is taken into account by multiplying by either a or /3, corresponding to the two possible orientations along the z axis. Although the substitution for implies that are functions of the coordinates of electrons 1, 2, 3, . . . , respectively, it is clear that any permutation of the electrons with respect to the functions also leads to a solution of the Schrodinger equation. Because of the existence of spin, each of the N\ products of the type must be multiplied by N spin functions, one for each electron. Of all the possible linear combinations of these grand products that can be constructed, only those which are totally antisymmetric with respect to the simultaneous interchange of the spatial and spin coordinates of any pair of electrons are acceptable as eigenfunctions that can occur in nature. This is the mathematical expression of the Pauli exclusion principle; it limits the basic states to determinantal product states. An example of such a state for an atom comprising three electrons is
(15)
Fortunately it is not necessary to write out determinants of this kind in full, since all the relevant information is provided by the diagonal terms. Using plus and minus signs for the functions a and respectively, corresponding to we can simplify to (16)
4
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
The brackets denote a determinantal product state. of such states follow from Eq. (15); for example,
The properties
(17)
When the general theory of determinantal product states is being studied, it is often convenient to incorporate the spin quantum number into the set and we shall use the symbol K to represent the complete set of four quantum numbers. Furthermore, if for example, then will be written for Expanding the general determinantal product state, we obtain
where the sum extends over all permutations P of the electrons' coordinates among the states defined by the quantum numbers and where p is the parity of the permutation; that is, p is if the permutation is even and — 1 if it is odd. Since the eigenvalues are degenerate with respect to and the eigenvalues are distinguished by the list of symbols
This sequence is said to specify the electronic configuration. Traditionally, the letters s, p, d, f , g, . . . are used for the values 0, 1, 2, 3, 4, . . . of I) the number of times a particular symbol occurs is indicated by a raised suffix. A closed shell results when the suffix is equal to This corresponds to the occurrence of all the symbols for a given n and I. The occupation of all the quantum numbers of spatial and spin orientation gives a closed shell an inert character, and it is usually unnecessary to write out the quantum numbers explicitly in a determinantal product state. This is a useful simplification, since closed shells are a common feature of configurations that are energetically lowlying. If the electrons outside closed shells are equivalent, that is, if they possess the same values of n and I, these quantum numbers may also be dropped from the determinantal product state. For example, the electron configuration of is
CLASSICAL METHODS
Only three 4/ electrons are not in closed shells. functions can therefore be written as
The basic eigen
{32Ϊ}, {312}, { 0  Ϊ  3 } , etc.
(19)
The numbers specify values of mi, it being understood that (nl) = (4/) for all three electrons. 13
THE PERTURBATION POTENTIAL
The Hamiltonian E gives rise to a series of energy levels called con figurations. Usually they are highly degenerate; there are, for exam ple, 364 determinantal product states of the type (19), and this is for a configuration in which all but three electrons are in closed shells. The degeneracy is partially removed by the perturbation potential
»=1
t>; = l
The effect of the first summation is to shift a configuration as a whole; since we shall be interested almost entirely in the structure of a single configuration, these terms can be dropped and there remains N
H1 = £
£
(110)
Corrections to the simplified Hamiltonian H are also represented by contributions to the perturbation potential. The nonrelativistic restriction can be removed for a single electron moving in a central potential represented by U(r) by using Dirac's equation. If the average velocity of the electron is small compared with c, the velocity of light, the relativistic corrections can be represented by additional 2 terms in the nonrelativistic Hamiltonian. For our purposes the most important of these is fts · (grad U X p)/2m2c2
(111)
where ρ is the momentum of the electron. The vector s stands for the spin angular momentum of the electron; its ζ component s, pos
S
6
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
sesses the eigenvalues +¾ and — £ for states defined by the symbols + and — introduced in Sec. 12. Since U is a function solely of r,
and (111) becomes
xdU An grad U =  Tr dr dU * s.\ 2m2c2r dr
(112)
where 1, the orbital angular momentum of an electron in units of h, is defined by 1 = (r χ p)/h. For a manyelectron atom, (112) is replaced by H2 = X Hn)B1 · Uwhere
£(r)
(113)
Λ2 dU 2m 2 c 2 r dr
This expression for H2 is equivalent to that given by Condon and Shortley for the spinorbit interaction. For free atoms and ions, the combination H1 + H2 is usually by far the largest contribution to the perturbation potential. Under its influence each configuration is broken up into levels. As a urst approach to calculating the effect of Hi + H2 in detail, it is assumed that the eigenvalues E' are sumciently well dispersed to permit each configuration to be examined separately. The problem is to apply the perturbation Hi + H2 to a degenerate energy level, taking as basic states those determinantal product states which, for a single sequence (nili)(n2l2) · · • (ΠΝΙΝ), run over all possible values of m, and mt. The direct method is to label columns and rows by the deter minantal product states, evaluate the matrix elements of Hi + H2 and solve the secular determinant. This procedure is feasible for only the most elementary configurations; in the case of Nd 3+ I/ 3 , for example, 364 X 364 = 132496 matrix elements have to be evaluated. Apart from being tedious, this method gives no insight into the struc ture of the configuration, gives no labels for distinguishing the eigen values, and takes no advantage of the frequent recurrence of identical roots which arise when solving the secular equation. 14
ANGULARMOMENTUM OPERATORS
Before we can be in a position to remedy the deficiencies in the direct approach to the problem, we must have some knowledge of the prop
CLASSICAL METHODS
7
erties of angular momentum operators. The fundamental commuta tion relations for the components Jx, Jy, and J1 of an angularmomen tum vector J are [Jx1Jy] = iJ, where
[Jy1Jz] = iJx [A1B] = AB  BA
[J11Jx] = iJv
(114)
The orthonormal states, in the notation of Dirac, 3 are distinguished by the two quantum numbers J and M. As is well known/ the commutation relations (114) lead to the equations JZ\J,M) = M\J,M) P\J,M) = J(J + l)\J,M) J+\J,M) = e*[J(J + 1)  M(M + 1)]*J, M + 1) J\J,M) = *  * [ / ( / + 1)  M(M  1)]ψ, MI)
,, ^
,« ;
The symbols J+ and j, whose most elementary basis is provided by the spin functions a and /3, corresponding to ms = + ! . When J is halfintegral, the factor e~2*iMj is — 1 , showing that, under a rotation by 2π about the ζ axis, \J,Mj) becomes multi plied by — 1 . I t is clearly impossible to decompose SD/ into repre sentations of O, since the latter possess the property that rotations by 2π and 0 are equivalent. On the face of it, this implies that when J is halfintegral we cannot construct linear combinations of the states \J,Mj) that display octahedral symmetry. To overcome this difficulty, O is augmented with an element S that commutes with all elements of the group, obeys the equation S2 = S, and possesses the property that, if the element (R formerly satisfied (R" = U, it now satisfies (R" = 8. Groups that have been extended in this way are called double groups. In general they do not have twice as many classes as the original group, since a class consisting of rotations through π is often not doubled. The characters of the double octa 11 hedral group, as found by Bethe , are given in Table 23. Some branching rules for R3* 0 when J is halfintegral are set out in Table 24. T A B L E 23
Γι T2 T3 Tt Ti
Γβ
Γ, Γ8
T H E CHARACTERS O P T H E D O U B L E OCTAHEDRAL G R O U P
C1
C1
C2
C3
C4
C4
C6
C1
1 1 2 3 3 2 2 4
1 1 2 3 3 2 2 4
1 1 0 1 1 0 0 0
1 1 2 1 1 0 0 0
1 1 1 0 0 1 1 1
1 1 1 0 0 1 1 1
1 1 0 1 1
1 1 0 1 1
(2)i (2)*
(2)1 (2)i
0
0
CRYSTAL FIELDS TABLE
24
BRANCHING RULES FOR HALFINTEGRAL
39
J
~l > ~t
re > rs
~t > ~i > ~I >
~¥>
2·6
r7 + rS re + r7 + rs re re
+ 2rs + r7 + 2rs
THE WIGNER·ECKART THEOREM
A knowledge of the transformation properties of eigenfunctions and operators finds its most useful application in the evaluation of matrix elements such as Suppose that the following conditions are satisfied: 1. The p functions cf>i (i = 1, 2, . . . , p) form an orthonormal basis for the irreducible representation rp of a group g. 2. The q functions 1/;; (j = 1,2, . . . ,q) form an orthonormal basis for the irreducible representation rQ of g. 3. The T functions fh (k = 1,2, . . . ,r) form an orthonormal basis for the irreducible representation r R of g. 4. Under the operations of g, the q operators h; (j = 1,2, . . . ,q) transform in an identical manner to the basis functions 1/;;. In detail, if 1/;; + 1/;; and h; + h; under an operation at" of the group, where
L Tz;(O')l/;z h; = L rz;(O')h z
1/;; =
I
then
I
for any O'. The qr functions I/;;(Jk form a basis for the representation rQ X r R of g. Corresponding to the decomposition
rQ X r R =
Lcsrs
s
sets of linear combinations of the products I/;j(Jk can be constructed, each set forming an orthonormal basis for an irreducible representation of g. We write (219) PI(rsf3) = L (rQj;r R kj r s,8l)l/;j(Jk j,k
where l runs from 1 to 8, for the 8 basis functions of an irreducible representation r s of g. The symbol ,8 serves to distinguish equivalent
40
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
irreducible representations that arise when Equations (210) are examples of Considerable freedom exists in the actual choice of the orthonormal basis functions for a given r s ; once the choice has been made, the coefficients in are fixed. Care is taken to ensure that the functions transform in an identical manner to the functions fa for all /J. Owing to the unitarity of the transformation (219), (220) where Taking precisely the same linear combinations of the quantities as are taken of the functions we may construct functions that transform in an identical manner to the functions In analogy with
We multiply both sides of this equation by and integrate over the entire coordinate s p a c e . T h e only surviving terms in the sum are those for which moreover it is also possible to show that the integrals (221)
are independent of i and can therefore be written as
Hence (222)
This equation constitutes a general statement of the WignerEckart theorem, so named after its original proponents. 1417 A formal proof of the condition finite group with g elements. The equations
can be readily constructed for a
indicate that
According to , the sum over a vanishes for all i, I, m, and n if IV and Ts are inequivalentirreducible representations; it follows that, for the integral to be nonzero, The condition and the invariance of the integrals (221) can be derived in a similar fashion with the aid of
CRYSTAL FIELDS
41
Sometimes other bases for and or other operators that transform according to can be found. We use a superscript to distinguish quantitiesthat possess identical transformation properties: for example, and are two sets of functions that separately transform according to the equation
Equation (222) is now written (223) The sum over /3 comprises cp terms, where cp is the number of times occurs in the decomposition of Hence every set of pqr integrals for a particular choice of a, b, and c can be expressed as a sum over at most cp linearly independent sets of integrals. Put another way, coefficients can be found such that (224) for all i, j, and k, where the sum extends over at most cp terms, provided that the sets of pqr integrals for different u, v, and w are linearly independent. Equation (224) is usually a more convenient form for the WignerEckart theorem if the coefficients have not been tabulated. Needless to say, if all matrix elements vanish. 27 TENSOR OPERATORS The most familiar application of the WignerEckart theorem is to the group The functions and Bk are replaced by (yKMK\ and respectively, where K and K ' are any pair of angularmomentum vectors of the same kind (for example, J and J') In order that the WignerEckart theorem may be applied, the operators must transform in an identical fashion to some set of states where This is guaranteed if the operators, which we now write as transform like the spherical harmonics or, what is equivalent, like the quantities
42
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
The operators , where are said to form the components of a tensor operator of rank k. According to the transformation properties of an operator under rotations about the z axis are completely determined by the commutation relations of the operator with For the operators and to transform in an identical fashion under any rotation, it is necessary only that the commutation relations of with and be the s a m e a s those of The only part of J that does not commute with is the orbital angular momentum 1; on making the substitutions
it is straightforward to prove
Accordingly, the quantities properties as the harmonics relations
possess the same transformation if they satisfy the commutation (226)
These equations were taken by Racah 1 8 as the starting point for the theory of tensor operators. From Eq. (213) it is clear that D * does not occur more than once in the decomposition of the Kronecker product Equation (222) therefore runs
where A is independent of . and The coefficient on the right is simply a YC coefficient; on converting it to a symbol and expanding the description of A, we get
The final factor on the right is called a reduced matrix element. The representation occurs in the decomposition of if the triangular condition is satisfied, that is, if a triangle can be formed
CRYSTAL
FIELDS
43
having sides of lengths K, k, and If the triangular condition is not satisfied, the symbol automatically vanishes. Each term in being a sum of spherical harmonics, is a tensor operator. If is treated as a perturbation, the firstorder approximation requires the evaluation of matrix elements for which For this case, we can construct operators that involve and and that transform like the spherical harmonics consequently the matrix elements of these operators are proportional to those of the tensor operators The following are examples of such operator equivalences for a configuration of equivalent electrons:
B y separating out the operator equivalent factors and become pure numbers, independent of the radial eigenfunction. T h e expressions on the right of are most easily obtained b y replacing by in the explicit expressions for the corresponding symbols; an alternative method is to take the appropriate spherical harmonics and substitute and for x, y, and z, respectively. D u e regard must be paid to the fact that and unlike x, y, and z, do not commute. Thus, in setting up an operator to reproduce the eigenvalues of we must replace with
not with
whose eigenvalues are
but
44
OPERATOR TECHNIQUES /N ATOMIC SPECTROSCOPY
Repeated use of the commutation relations (115) reduces this to an operator the eigenvalues of which are K\K
+ I ) 2  K(K + 1)/3
but the calculation is very tedious. The introduction of the operator equivalences (228) to (230) is due to Stevens. 1 0 Examples of their use in treating the hyperfine interactions have been given by Ramsey. 1 9 Since the calculation for any tensor operator Tglk) must reduce ultimately to an integral between singleparticle states, the triangular condition on I, k, and I indicates that the number of terms that need be included in H3 is limited. For / electrons, all tensor operators Tq(k) for which k > 6 have vanishing matrix elements, and the expansion for H'3 given in Eq. (25) is sufficient for all configurations of the type / n . I t is obvious t h a t there is no need to include terms of odd parity in Hz if all matrix elements are taken between states of a single configura tion. In addition, all components Tqm of a tensor operator T(*> share the same operator equivalent factor, since the latter, being a product of a reduced matrix element and some factors involving K, is independent of q. These considerations, taken together, show that the three operator equivalent factors α, β, and y are the only ones that have to be evaluated when crystal field effects in configurations of the type f" are being studied. Stevens 1 0 has tabulated α, β, and γ for the ground levels of all triply ionized rareearth atoms. 28
AN EXTERNAL MAGNETIC FIELD
To illustrate the power of the WignerEckart theorem even in the weaker form (224), we consider the effect of an external magnetic field H on the energylevel system of a rareearth ion in a crystal lattice. Dirac's equation indicates that for a single electron the appropriate addition to the Hamiltonian is /3H · (1 + 2s), where β = eh/2me is the Bohr magneton. As a first (and extremely good) approximation for the manyelectron case, we therefore include Ht = X /SH · (I1 + 2s H3y> H4, which obtain for rareearth ions, indicate that H4 can be treated as a perturba tion on the energy levels whose eigenfunctions are of the type \f"ySLJTRt), where t distinguishes the various functions that together form a basis for Γ Β . If the symmetry at the nucleus of the rareearth ion is octahedral, TB denotes an irreducible representation of the
CRYSTAt FIElDS
45
group O. Suppose we want to find the effect of an external magnetic field on a doublet T3. The quantity L + 2S is a vector, and under simple rotations its three components transform among themselves like the coordinates (x,y,z) of a point. Thus, for the group O, the components of L + 2S transform according to T 4 . From Table 21 it is easy to prove T 4 X T3 = T 4 + T 6 . The absence of T 3 in the decomposition immediately reveals that all matrix elements of L + 2S between states transforming like T 3 vanish; hence the doublet remains degenerate (to first order) when an external magnetic field is applied. Examples of Eq. (224) with Cp = 1 are easy to find. The states \J,Mj) form a basis for the irreducible representation 3Dj of R3. A vector transforms like SDi, and SDi X SD/ contains in its decomposition the representation SD^ once. Thus the matrix elements of the com ponents of L + 2S must be proportional to those of the corresponding components of any other vector. Since the states are labeled by J, it is very convenient to choose J for this role. Equation (224) now runs (JMj]L, + 2S3]JMj)
= g(JMj]J,\JMj)
(232)
for all Mj, j , and M'j. The symbol g is called the Lande g factor and depends on the detailed nature of the states. Equation (232) corre sponds to a = u, b j£ v, c — w in Eq. (224); the subscript j on L, S, and J stands for x, y, or ζ and corresponds to the same symbol in Eq. (224). Instances where a 9* u, c ^ ν are not difficult to find. For example, the two eigenstates ]S' = \, M's = + £ ) form a basis for the irreducible representation Γβ of 0. The two basis functions for any other doublet of the type T 6 can be chosen to be in a onetoone corre spondence with the components M's = ±1; and since Γβ occurs once in the decomposition of T 4 X Γβ, the matrix elements of /3H · (L + 2S) can be reproduced by (r6)/p6 is combined with Eqs. (26). It is clear from Fig. 22 that the spherical harmonics of the fourth rank are of greater relative importance for Cs2PaCl6 compared with NpF6. However, the values of I are close enough for us to include Axe's data in Fig. 23 without grossly misrepresenting the situation. The agreement is improved by raising the T6 level in the limit of H2 = 0, as indicated in Fig. 22.
CRYSTAL FIELDS TABLE 25
51
PARAMETERS TOR TWO OCTAHEDRALLT COORDINATED IONS
(AU values are in cm 1 .)
AS(T*)
AeV)
Cs2PaCl6
NpF 6
888 41.9 1490
5738 540.5 2405
FIG. 23 The energy levels for a single f electron in an octahedral crystal field are plotted for various relative strengths of H2 and Hz. On the left, H3 = 0; the arrangement of energy levels in Fig. 22 for NpF 6 is taken as the right extreme. Strictly, the levels for Cs2PaCIe should be based on a slightly different right extreme, corresponding to the arrangement in Fig. 22 for Cs2PaCU; however, the situations are sufficiently similar to permit both sets of observations to be represented on the same diagram.
52
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
PROBLEMS 21. Prove that the subgroup of O comprising the permutations
is Abelian and that each element forms a class by itself. Show that the group is isomorphic with the group of planar rotations that send a square into itself and that the angles of rotation corresponding to the four permutations are », and Show that the functions form four onedimensional bases for irreducible representations of and construct the character table. Obtain the branching rules for the decomposition of irreducible representations into irreducible representations in the form
22. Prove that and that
23. Eight charges are each a distance p from the nucleus of an ion and lie at the vertices of a cube. Prove that the contribution to the Hamiltonian can be written in the form of where
The cerium ion Ce8+, which possesses a single / electron outside closed shells, can be inserted in the lattice of CaF2 so that its immediate neighbors are eight fluorine ions lying at the vertices of a cube. According to the experiments of Dvir and Low,26 the two lowest energy levels are of the types r  and r 8 and almost exactly coincide. Show that if the limiting cases ( l ) f = 0 ,  7 b > —a, and (2) a = 6 = 0 are excluded, where a and b are defined by Eqs. (233), f must satisfy
24. The ion possesses the linear structure The free neptunium atom has seven electrons outside a radon core, four of which are lost in bonds with the oxygen atoms, while two go in forming the ion. On the assumption that secondrank spherical harmonics are the most important in the expansion of the electric potential produced by the axial column of negative charge, show that the seven degenerate orbital states of the remaining / electron
CRYSTAL FIELDS
53
break up into four components, and find their separations. Show that the splittings produced by a small spinorbit coupling are in the ratio 3 : 2 : 1 : 0 . Prove that the effect of a magnetic field on the lowest doublet can be reproduced by the spinHamiltonian with i 0.05, and theory. 28
The experimental values27 for crystals of containing small amounts of Np substituted for Suggest reasons for the deviations between experiment
3 THE ni SYMBOLS
31
THREE ANGULAR MOMENTA
We have seen that the coefficient of in the expansion of can be expressed as an algebraic function of and m. This gives a general solution to the problem of passing from , quantization. It is clear from Sec. 16, however, that in general the transition to quantization involves the study of coupling schemes of several angular momenta. As a first step toward putting the manyelectron problem on an algebraic footing we therefore investigate the coupling of three angular momenta and to give a resultant, j. It is not enough to write for a state of the system, since it is easy to see that such a state cannot be unambiguously expressed as a linear combination of the states However, on specifying the resultant of and a unique reduction can be found:
But we can equally couple 54
and
to form
adding the resultant
THE
to
SYMBOLS
55
In this case
The transformation which connects the states in these two coupling schemes can be written as
It can be seen that the transformation coefficients are independent of m by operating on both sides of Eq. (33) with Substituting (31) and (32) into (33), and equating the coefficients of we get
We multiply both sides of this equation by
and sum over becomes
and
In view of
the righthand side
56
OPERATOR
TECHNIQUES
We now define the
IN ATOMIC
SPECTROSCOPY
symbol by the equation
Thus
The phase factors can be combined to give
Equation (35) can be thrown into a more symmetrical form by interchanging columns 1 and 2 of the first 3j symbol on the right, columns 2 and 3 of the second, and reversing the signs in the lower row of the third. This introduces the phase factor
Since
Equation (35) now reads
The symmetry is displayed better if the following substitutions are made:
THE ni SYMBOLS
57
We get, finally,
in agreement with Eq. (2.20) of Rotenberg et al.6 32 THE 6j SYMBOL Equation (36) expresses a 6j symbol as a sum over 3j symbols. Since the latter themselves involve a summation, direct substitution through Eqs. (120) and (122) gives a very complicated expression for the 6j symbol. However, since the 6j symbol is independent of we can set thereby reducing the summations in three of the four 3j symbols in Eq. (36) to a single term. After an appreciable amount of algebraic manipulation, the 6j symbol reduces to a summation over a single variable,
where
The four triangular conditions for the nonvanishing of the are contained in the symbol in the following way:
The
symbols
symbol is related to the W function of Racah18 by the equation
58
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
From Eq. (37) it is clear that the 6j symbol is invariant under any permutation of the columns, e.g., (38) It is also invariant under an interchange of the upper and lower arguments in each of any two of its columns, e.g., (39) The absence of phase factors in equations like (38) and (39) makes the 6j symbol a more convenient quantity to handle in actual calculations than the W function. Regge28 has noted that
and this property has been examined in detail by Jahn and Howell.80 Strikingly different 6j symbols are thereby related, e.g.,
Any 6j symbol whose arguments do not exceed 8 can be found from the tables of Rotenberg et Sometimes it is convenient to retain the detailed form of the 6j symbol, particularly if one of the arguments is small. Cases where the smallest argument is or 1 occur frequently, and their values are set out below:
THE
no;
SYMBOLS
{~ ~ 1 ~} C
2a)(s  2b)(s  2c 1)2c(2c
+ +
(S  2b  1)(s  2b)(s  2c + 1)(s  2c = (1)8 [ (2b 1)(2b 2)(2b 3)(2c  1)2c(2c
+
2(S
+ 1)(s 
= (1)8 [ 2b(2b
+ 1)(2b + 2)(2c 
{~ c ~ 1 b ~ I}
+
+
{~ ~ ~} = (1)8
+ +
59
1)]t 1)
2)]t
(310)
+ 1)
+
+
+ +
2[a(a 1)  b(b 1)  c(c I») [2b(2b 1)(2b 2)2c(2c 1)(2c 2»)1
+
+
where, in each equation, s = a + b + c. Sums over products of 6j symbols can be obtained by considering various recoupling procedures and using Eq. (34).
we obtain the equation ~ Ul,U2iS)j2S,jl Ud2)j12,jS,j) (Ud2)j12,j8,jljhU2i3)j~3,j) = ~U23,j~3) 112
which leads to
A second equation involving 6j symbols can be obtained from the recoupling scheme
12i
We write (U2ia)j23,iI,jl for (jl,(j2is)j2a,jl in the second transformation coefficient. This introduces the phase factor (I)1.+i,';, owing to Eq. (121). We also write the righthand side as
60
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
the phase factor in this case being
33
gives
THE 9j SYMBOL
A transformation of great importance connects states in LS and jj coupling. The transformation coefficient is
or, in more general terms,
This can be written as
The adjacent parts of neighboring transformation coefficients in this sum are identical; we have merely dropped superfluous quantum numbers to make the connection with 6j symbols more apparent. Using Eq. (34), we find the sum to be
On defining the 9j symbol through the equation
we obtain, with a rather more symmetrical set of arguments in the
THE nj SYMBOLS
61
9j symbol, the equation
The writing
symbol can be expressed in terms of
symbols by
A similar expression can be found for On putting the two parts together and using Eq. (122), the transformation coefficient is obtained as a sum over six 3j symbols. Since the transformation coefficient is independent of m, the effect of summing over this variable is equivalent to a multiplication by Using Eq. (313), and transforming to a new set of arguments, we obtain the remarkably symmetrical expression
The symmetry properties of the 9j symbol may be readily found from this formula. An odd permutation of the rows or columns multiplies the 9j symbol by ( —1)B, where R is the sum of the nine arguments of the symbol. An even permutation produces no change of phase, and neither does a transposition. Curiously enough, Regge's repre
62
OPERATOR TECHNIQUES /N ATOMiC SPECTROSCOPY
sentation of the 3j symbol possesses the same symmetry properties (see Prob. 16). 34
RELATIONS BETWEEN THE nj SYMBOLS FOR η > 3
By considering recoupling schemes, any number of equations involving various nj symbols can be established. The underlying symmetry in these equations, which, as in Eqs. (311) and (312), is not immediately apparent, can be emphasized by a careful choice of notation. A deeper appreciation of the equations can be obtained by extracting from the nj symbols within a sum all those triangular conditions that do not involve an index of summation. Every condition can be repre sented geometrically by the junction of three branches, each branch being labeled by one of the three angularmomentum quantum num bers involved in the triangular condition. The coupling diagrams characterize the sums and make the symmetry properties obvious. A great deal of attention has been paid to the topological properties of such diagrams, particularly in connection with the celebrated fourcolor problem; however, the theory 3 1 appears to have been insuffi ciently developed to assist us materially in our analysis of the prop erties of the nj symbols. An appreciable number of relations between various nj symbols have appeared in the literature. In order to include most of them here, their derivations, which in virtually all cases are quite straight forward, will not be given in detail. Factors of the type 2x + 1 occur frequently, and the abbreviation [*] = 2a; + 1
(316)
is made. Since we still require brackets for other purposes, it is to be understood that Eq. (316) is valid only when there is a single (possibly subscripted) symbol enclosed within the brackets. We begin by rewriting Eqs. (311) and (312),
5
2>{: i;} [it :1 Sf »»> 2 > *  » H : i ;){ί ί ; )  ( ; : ; ) (3i8) X
The coupling diagrams are given in Figs. 31 and 32. Special cases of Eqs. (317) and (318) have been given by Elliott and Lane. 3 2 On setting ¢ = 0 and insisting in turn that the second 6j symbol in both
THE nj SYMBOLS
63
a
q
P
c
FIG.31 The coupling diagram corresponding to Eq. (317).
FIG. 32
The 6j symbol.
sums does not vanish, we obtain, with the aid of Eq. (310),
2: and
[x](I)x
2:
{~ ~
[xl {:
~
;} = ll(p,O) ( 1)a+b{[a][bll!
;}
= (_1)2 a Hb
(319) (320)
x
The BiedenharnElliott sum rule,33.34 which can be derived by recoupling four angular momenta, may be regarded as an extension of Eq. (318):
2:
[x] (  I)x+v+q+r+a+b+c+d+e+1 {:
b d
;} {~
x
=
{~
d
f q a
:} {~ fa ~} q ~} {: b :}
(321)
The coupling diagram is given in Fig. 33. If the lower row in the third 6j symbol in the sum is changed from bar to abr, and if the odd
FIG. 33
The BiedenharnElliott sum rule.
64
OPERATOR TECHNIQUES IN ATOMIC
SPECTROSCOPY
multiple of x in the phase factor is eliminated, the equation becomes equivalent to the definition (314) of the 9j symbol. It is convenient to write down at the same time a double sum over the products of two 9j symbols; that it is equal to a 9j symbol can be proved either by recoupling considerations or by algebraic manipulations with the 6j symbols.
The coupling diagram is given in Fig. 34 in two topologically equivalent forms. The first exhibits a threefold axis of symmetry; the second can be inscribed on a Mobius strip in such a way that no two
FIG. 34 Two topologically equivalent coupling diagrams for the 9j symbol. That on the left exhibits a threefold axis of symmetry; that on the right can be inscribed on a Mobius strip.
branches cross each other. Higher nj symbols can be easily generated by inserting extra segments in the strip; for example, the 12j symbol35 36 is defined by
THE ni SYMBOLS
65
An equivalent definition is Ae] r q\ [d a g f ) [c
b
ρ
s
s
h b] τ Λ a e χι
(324)
Equations (323) and (324) possess the same couphng diagram, of course, and it is given in Fig. 35. A second 12j symbol, for which
FIG. 35 kind.
The 12j symbol of the first FIG. 36 kind.
The 12j symbol of the second
no special notation has yet come into general use, can be regarded as an extension of Eq. (321):
Σ
, , ία b χ) ic d x\ ie f \c d V)U ί β) \g h
[X1
•I
χ) ig rj \a
(
h χ b s
d ρ
q e\ (h r c bUs g
e\ b\
a
f
y) [a f
y)
(325)
The couphng diagram, drawn in Fig. 36, can be inscribed on an untwisted strip. 37 The orthonormality of the transformation coefficients of Sec. 33 leads to the equation
Σ
/τ
My) \d ρ
f
JM
in
g y\\d c b) \p'
f
2Jl
q y\ = Kp,p')b(c,c')/[p)[c) (326) c' b)
and the couphng diagram is shown in Fig. 37. The following equa tion, which is often useful, is equivalent to the BiedenharnElliott
66
OPERATOR TECHNIQUES JN ATOMIC SPECTROSCOPY
sum rule:
26« Jl: ί ;) ρ, respec tively, such that the sets form the operators for the respective invariant subgroups 9 and 3C of 3C. Then and
Cf/ = 0
ρ < ρ,τ>
Cp/ = 0
σ > ρ, τ < ρ
Cp/ = 0
ρ
ρ
It follows that (512)
ρ
and the elements of 9 a n d 3C commute. The group 3C is called the direct product of 9 and 3C and is written as either 9 X KC or 3C X g. 54 THE COMMUTATORS OF TENSOR OPERATORS
The matrix of u/ for R3 (see Sec. 52) shows that for this group Xi
= 2/ a
25
A 2 = z
zr
X3
= X3
yr
dz dy dx dz dy dx It is apparent that these expressions are proportional to the orbital angularmomentum operators; hence Eq. (58) for β 3 must be equiva lent to the commutation relations for the components of an angularmomentum vector. Now a vector is a tensor of rank 1, and it occurred to Racah to investigate the commutation relations of tensor operators 68 of higher rank. If it can be established that the tensor operators satisfy equations of the type (58), then we have at our disposal the infinitesimal operators of groups other than β 3 in a very convenient form. Since it is intended to apply group theory to configurations of electrons that are equivalent, we restrict our attention to configura tions of this kind by introducing the tensor operators vη'0 = (2ft + I)* δ(η,η') 8(1,1')
(513)
The factor (2fc + I)* on the righthand side is to be noted. These tensors must of course satisfy the usual commutation relations with respect to the total angular momentum J of the system. Their com mutation relations with respect to one another can be obtained as
102
OPERATOR
follows.
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
It is clear that
Hence
It is obvious that differs from this result only in the phase factor which must be introduced in reversing the first two columns of the 3j symbol. We conclude that
If configurations of more than one electron are being examined, it is more appropriate to use the tensors V(*>, defined by
where the sum runs over the electrons. Since the two tensors and become equivalent when the summation reduces to a single term, that is, when the system under examination involves the coordinates of a single electron, we shall in future use the more general forms To obtain their commutation relations, it is necessary only
CONTINUOUS
GROUPS
103
to replace and respectively. It is apparent that Eq. (514) is of the same form as Eq. (58), and hence the tensor operators can be regarded as the infinitesimal operators of a group. The triangular conditions on the 6j symbol indicate that we can restrict our attention to the components for which and The operators of Eq. (57) can be written as
Here, as elsewhere, we make the summation explicit when the covariant or contravariant character of the indices is no longer obvious. Under the influence of an operator of this type, the eigenfunction becomes
where Because of the linear independence of the 3j symbols, any function which differs infinitesimally from the function can be reached by a suitable choice of the parameters hence the components can be regarded as the infinitesimal operators of the full linear group dimensions. This group is, however, rather too general for many purposes, and it is convenient to limit the transformations to the subgroup which preserves the orthonormality of the eigenfunctions. The equation
is satisfied if the infinitesimal parameters obey the condition (515) The matrix is unitary, and consequently the group is called the unitary group in dimensions, The infinitesimal operators are the same as those for the full linear group. The group is not simple: the infinitesimal operator commutes with every and hence by Eq. (511) forms an invariant subgroup. According to Eq. (515), must be imaginary; so to the first order of smallness the operator __ > can be written
104
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
as where is real. The only significance of the operator is that it leads to a change of phase in the eigenfunctions. Since, to first order, the determinant of is
the effect of dispensing with is to restrict the transformations to those whose determinants of The group formed by the tensor operators is called the unimodular or special unitary group in dimensions,
Subgroups of are not difficult to find. If ki and fc2 of Eq. (514) are odd, then the factor ensures that only those tensor operators for which k3 is odd appear in the summation. According to Eq. (510), the tensor operators of odd rank can be thought of as the infinitesimal operators of a group. Now
This expression vanishes if k is odd; hence the operations of the group leave invariant the expression (516) If the functions ty are broken up into their real and imaginary parts by writing and then the equation
(517) which is in keeping with
CONTINUOUS GROUPS
105
our phase convention, allows I to be written as
The leading term ziz2 exists only if m can assume the value zero. Although it has been implied that I is the azimuthal quantum number of the electron, care has been taken not to assume that I is integral in the derivation of Eqs. (514) and (516). If indeed I is integral, and the invariant is
It is easy to prove from the unitarity of the matrices with elements that
is also invariant, and hence so is the distance between two points of the type
The group whose infinitesimal operators are the tensor operators of odd rank is therefore called the rotation, or proper real orthogonal group, in dimensions, On the other hand, if I is halfintegral,
Weyl61 has given the name "symplectic" to groups that leave invariant bilinear antisymmetric f o r m s . T h u s , if I were halfintegral, the components of odd rank would play the role of the infinitesimal operators of the symplectic group in dimensions, This group is of importance in the jj coupling extreme, where the singleparticle eigenfunctions are of the type in which j is halfintegral. We shall have occasion to introduce a symplectic group in another context in Sec. 62. If in Eq. (514), then only. The corresponding group, which is a subgroup of has the three tensor operators and as its infinitesimal operators and is nothing else but the rotation group in three dimensions,
106
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
In the case of f electrons, an apparently unique event occurs. The tensor operators of odd rank are and and we should expect commutators of the type to contain components of all of them. Very remarkably, however,
and the commutator contains components of and only. Moreover, can be expressed as a factor times ,, and the commutator can give rise to only. Consequently the 14 components of and form the infinitesimal operators for a subgroup of which is called It obviously contains R s as a subgroup. To indicate the succession of groups and subgroups, we write (518) For / electrons, (519) These groups are available for studying the transformation properties of the manyelectron eigenfunctions. 55 THE METRIC TENSOR The symmetric tensor defined by (520) is called the metric tensor. The structure constants that appear in Eq. (514) may be directly substituted in this equation and the sum over the product of the 3j symbols carried out. On replacing the Eq. (520) becomes symbols + 6(^)»7ο 0, we obtain, in the oblique coordinate scheme associated with the operators (535),
+ (el 
R = 2(el  es)
e2)
corresponding to the point (2,1). The coordinates defining K are (Ut + 2, U! + 1). From Eq. (536), gil =
(Z
~)
and the eigenvalues of Casimir's operator are given by K!  R! = gii(K;Kj  RiRj) = (U1 2
+ u!! + UlU2 + 5Ul + 4u2)/12
(554)
510 THE CHARACTERS OF REPRESENTATIONS
Two important topics in the theory of continuous groups have still to be considered: these are the Kronecker product of two representations and the branching rules for determining how an irreducible representation of a group breaks up when the transformations are limited to a subgroup. For finite groups, these problems are most readily solved by introducing the character of a representation, and an analogous procedure is available to us here. As basis functions for a representation of a continuous group, we may choose the quantities u('Y,m), each of which corresponds to a definite weight m = (ml,m!, . . .). The symbol 'Y is required to distinguish functions possessing the same weight [such as Jd! lD,O) and Jd 2lG,0) for the irreducible representation (20) of R6]' The equations SaU('Y,m) = (1
+ 8a"X..)u('Y,m)
=
L C('Y,mj'Y',m')u('Y',m')
r' ,DI'
CONTINUOUS GROUPS
127
define matrices C that form the natural extension of a matrix representation for the elements of a finite group. In the calculation of the characters of a representation of R3, we are able to simplify matters by choosing the axis of rotation to be the z axis, so that the matrices become diagonal. Since the operators are shift operators, the assumption that C is diagonal restricts our attention to infinitesimal operators of the type which commute with all elements of the group. The equations (555) enable the diagonal elements of C to be rapidly found; however, it is at once apparent that a symmetry of the weights about the origin, such as is displayed in Fig. 54, produces entries on the diagonal which, when the trace is formed, cancel in pairs. The character of an irreducible representation of a group for which this occurs is equal to the dimension of the representation and is useless for the purposes we have in mind. This difficulty can be overcome by considering finite rather than infinitesimal transformations. The effect of n successive operations can be reproduced by the operator
Equation (515), combined with the fact that is a linear combination of tensor operators for which implies that all quantities Saj are purely imaginary. In the simultaneous limits we may suppose that the product becomes , where is real. At the same time,
The sum over j is implicit. To make the algebra easier, it is sometimes convenient to relabel the parameters For we write We note that the operator exp is the generalization of exp ) of Sec. 25. Thus, for Rn+i, we find
OPERATOR TECHNIQUES IN ATOMIC
128
SPECTROSCOPY
where , and z are defined as in Eqs. (517). generalizations to of Eqs. (215). It is now a to compute the For the irreducible
These are the simple matter character xrepresentation
where the sum runs over all weights of the representation Thus, for the irreducible representation (11) of
FIG. 59 Under single operations of Weyl's group S for Ri, the weight can be carried into the weights
wnere
i
The expressions for the char
Weights that require an even number
acters of
of operations to be reached are represented by open circles.
( 5 . 5 6 ) a r e geometric series and can b e e a sily summed [see Eq. (212)].
corresponding to Eq.
Weyl 62 has shown that analogous simplifications can be carried out for the characters of any semisimple group. His formula is (557) where K and R are defined as in Eqs. (553) and (552), and where
The operators (R„ are elements of the group S, mentioned in Sec. 57, and they carry the point to the positions that can be reached by any number of reflections in the hyperplanes through the origin perpendicular to the roots. The parity p is or according as the number of reflections is even or odd. For corresponding to the point in the weight space. The points that are linked by the operations of S are with any combination of signs; they are shown in Fig. 59. For the group it is obvious that the effect of reflecting any point M = £ OA i
CONTINUOUS
GROUPS
129
in the hyperplane perpendicular to the root is simply to reverse the sign of the coefficient a*. Reflection in the hyperplane perpendicular to either can be found by writing
In the first case, we note that
and so the effect of the reflection is to interchange the coefficients o< and and to reverse their signs. In the second case, and are interchanged, but their signs are not reversed. Thus the operations of the group S for carry M into those positions which are given by permuting the coefficients o„ with all possible changes of sign. With this preliminary point disposed of, we can proceed to calculate for The positive roots are all of the type and
The first member of
If we denote by det and Zcth column is into the form
The function
The character of
corresponding to
the determinant whose element in the j'th row the entire sum of terms can be engineered
corresponding to the irreducible representation is very similar,
is simply
If, for exam
130
pIe, l
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
= 2 and
(W1W2)
x(ll)
=
= (11), then 1
e5iB" 12 
e 5iB " /2
e 3iB " /2 _
e 3iB " 121
e5iB'/2
e5iB'/2
e8iO'/2
_
e8iB'/2
e 3iB "/2 
e 3iB "/2
e3iO'/2
e3iO' /2
e iB " 12 eiO' /2
_ _
e iB " 12 eiB' /2
I
_
_
(560)
1
It is straightforward to show that this equation is identical to Eq. (556). The group SU 21+1 is somewhat easier to deal with. We find 2/+1
R
= J..J ~
(l
+1
j)e;
)=1
and the leading term of HR) is exp
[i
r
21+1
(l
+1
j)w(I+H)]
;=1
The effect of reflecting a point :M = (M I,M 2, • • .) in the hyperplane perpendicular to the root ei  ek is simply to interchange the components M. and M k ; consequently ~(R) =
detlexp [i(l
+1
k)w(l+ Hl ll
and x(X~X~ . . . X' 21+1
)  detlexp riel + 1  k detlexp [i(l + 1 
+ ).~)w(l+H)]1
(561)
k)W(I+l j)lI
The character of the matrix representing the identity element is nothing else but the dimension of the representation. Equations such as (561) may be used to compute the dimension D(ltl2 . . . ) as a function of the components of the highest weight (hl2 . . .); however, the procedure for doing this requires some care, since the direct substitution w (1+ 1;) = 0 gives the indeterminate expression 0/0. In Eq. (561) we may first set w(!+H) = (j  l)w, and both determinants become alternants, e.g.,
+ 1  k + ).~)w]1 {exp [i(l + 1  k + X~)w]
detlexp [i(j  l)(l
n
2/+1
">"1
 exp [i(l
+1
h
+ A~)W]}
CONTINUOUS GROUPS
In the limit
131
this expression goes over into
Hence By taking the general formula (557) as his starting point, Weyl62 obtained the expression (563) for the dimension of the irreducible representation semisimple group. For the irreducible representation for example,
of any of ,
and Thus
Similarly, the dimension (W1W1W3) of is found to be
of the irreducible representation
511 THE KRONECKER PRODUCT OF TWO REPRESENTATIONS Suppose that the two sets of functions u(y,m) and form bases for the respective irreducible representations ) of a continuous group. Since the operators Hi must behave like differential operators,
132
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
from Eq. (555). The products u(y,m)u'(y',m') form a representation of the group of dimension D(ZiZ2 · · ^)D(I[I2 ' ' ' ) ; the weights of the representation consist of all possible sums of the weights of the representations (Z1Z2 · · ") and (I1I2 · · ·)· If diagrams such as Fig. 54 are available for many irreducible representations, it is a simple matter to form the Kronecker co O Ocoproduct (ZiZ2 · · ·) X (1% ' ' ')· Suppose, for example, that the decomposition of the Kronecker product (10) X (10) for the group Ri is required. The weights of (10) are (1,0), (0,1), (  1 , 0 ) , FiG. 510 By superposing the five ( 0 ,  1 ) , and (0,0); they form a point array (1,0), (0,1), (1,0), fivepoint array in the weight (0,1), and (0,0) on every point of space. If we take an identical itself, we obtain the weights of the array of points [corresponding to representation (10) X (10), shown the second representation (10)] and, here. The decomposition by translating the array, superpose (10) X (10) = (20) + (11) + (00) its center successively on each of the points of the first array, we follows from Fig. 54. obtain the arrangement of points shown in Fig. 510. Since this can be produced by the superposition of the arrays for (20), (11), and (00), we conclude that the representation (10) X (10) decomposes into (20), (11), and (00): (10) X (10) = (20) + (11) + (00) Equation (564) furnishes a dimension check: 5 X 5 = 14 + 10 + 1 Kronecker products of representations of R3 can be dealt with very simply by this procedure. The weights of the representation SDz, are (L), (L  1), . . . , (L);those of £> s are (S), (S  1), . . . , (S). If the first array of weights is translated in the onedimensional weight space so that its center lies in turn on each weight of the second set, we obtain a linear array extending from (S + L) to (S — L). I t is not difficult to see that this array can be formed by adding the weights of the representations SDs+L, £>S+LU . . . , 3Ds—x,; consequently these are the irreducible representations that occur in the decomposi
CONTINUOUS GROUPS
tion of representations
133
This result is equivalent to Eq. (213). For the i and Eq. (567) reduces to
In general, the number of times the irreducible representation occurs in the decomposition of is given by the coefficient in the equation
To solve for the coefficients, we adapt a method due to Weyl.61 First, the substitution (557) is made for Second, is replaced by
of the irreducThe sum runs over all the weights ible representation a weight that occurs more than once [for example, (0,0) of the representation (20) of gives rise to as many equal terms in the sum as its multiplicity. The lefthand side of Eq. (568) becomes
Since the array of weights group S,
where
is invariant under the operations of the
is any function of m'.
Hence
Because of the range of values over which the components can run, the symbol is not necessarily admissible as an irreducible representation of the group. It might turn out, for
134
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
example, that, for a particular weight the component is negative. In cases such as this, the explicit expression for the character must be consulted to reexpress in an acceptable form. Usually it is necessary only to reorder the columns of a determinant, though occasionally the signs of a column need to be reversed. For for example, the equations
are readily derived. Their generalization to Ru+i can be obtained without difficulty from Eq. (559). If columns h and k of the determinant are interchanged, we get (570) As an illustration we consider the Kronecker product of the representations (11) and (10) of The five weights of (10) are Using Eq. (569), we obtain But Thus
I
and and
The dimension check is Equation (569) can be applied to other groups with equal ease. The orbital states of a single electron with azimuthal quantum number I form the basis for the irreducible representation [100 • • • 0] of and also for the irreducible representation of The possible arrangements of the symbols in the parentheses give the weights of the representation. Consequently,
If
are zero.
then the terms in the sum for which This is because the corresponding determinants
135
CONTINUOUS GROUPS
in the numerator on the righthand side of Eq. (561) possess two equal columns. Hence (A~A~ .•. A~/+l) X (2l[l]I, _[l]l, = (A~' 1, A~', . . . , A~;+l)
+
" "I , A2, + (,,,
"" . • . , Ak11 Ak
... , _[l]I)
+ .. . + 1,
. . . ,A2"1+1 )
+
(571)
where A;' = A;  [1]1 and X~' < A~I' If we suppose that the basis functions for (A~A~ . . . A~I+I) involve the coordinates of n electrons, the bases for the representations on the right of Eq. (571) involve the coordinates of n + 1 electrons. With the aid of Eq. (543), we find that Eq. (571) is equivalent \0 [AIA2 . . . A2/+I] X [10 . . . 0] = [AI 1, A2, . . . ,A2I+l] [AI, A2, .•. , Akl, Ak
+
+
+ '" + 1, . . .
, A21+1]
< Akl.
Thus
Only those representations occur for which Ak [2210 . . . 0] X [10 . . . 0] = [3210 . . . 0]
+ [2220
. . . 0]
+ [22110
+
. . . 0]
An alternative way of representing this decomposition is as follows: (.572)
The shapes on the right can be obtained simply by adding a single cell to the shape [221] in all permissible ways; the condition Ak < AkI eliminates the irregular shape
As might be anticipated from the extreme simplicity of Eq. (572), a set of rules can be constructed for finding the Kronecker product of two irreducible representations of U 2/+ I • No derivation of the prescription will be given here. Apart from a few changes in notation, we shall simply quote Littlewood's70 formulation, which, although derived in connection with the properties of the symmetric groups Sn, can be immediately taken over to the group U 2/+1 because of the
136
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
correspondence established in Sec. 58. We begin with a definition. Consider the partition , where If among the first r terms of any permutation of the n factors of the product
the number of times Xi occurs the number of times x2 occurs the number of times occurs, etc., for all values of r, this permutation is called a lattice permutation. Littlewood's recipe can now be stated in the following form: The shapes appearing in
are those which can be built by adding to the shape cells containing the same symbol a, then m cells containing the same symbol etc., subject to the two conditions: 1. After the addition of each set of cells labeled by a common symbol we must have a permissible shape with no two identical symbols in the same column. 2. If the total set of added symbols is read from right to left in the consecutive rows of the final shape, we obtain a lattice permutation of For example,
(573) To interpret an equation such as this in terms of irreducible representations of we have merely to replace the shape by the representation The dimension of the latter depends on I; from Eqs.(543) and (562), we find that the dimension check on Eq. is
512
BRANCHING RULES
On restricting a group of transformations to a subgroup, irreducible representations of the group decompose into irreducible representa
CONTINUOUS GROUPS
137
tions of the subgroup. Tables 22 and 24 are examples of such decompositions. Our present aim is to find analogous tables for the groups that occur in the sequences (518) and (519). The most direct method is to make use of the formulas for the characters of the representations, in complete analogy with the procedure for decomposing the irreducible representations of finite groups. This involves finding the minimum number of restrictions on the parameters that ensure that the operator of the group reduces to an operator of the subgroup: under these restrictions the character of the irreducible representation breaks up into a linear combination of characters of the subgroup and thereby determines the decomposition of However, if tables of Kronecker products of irreducible representations of both the group and the subgroup are available, this rather tedious procedure can be avoided. As is shown in Sec. 513, a chain calculation can be set up to determine the branching rules from the known decompositions of the Kronecker products. Before illustrating these two methods in detail, it is convenient to describe the somewhat atypical reduction The branching rules are particularly simple: each irreducible representation becomes the irreducible representation of where is related to by Eq. (543). We observe here that all irreducible representations of of the type where a is any integer such that become the same irreducible representation of This implies that when is limited to for example, the following equivalence holds: (574)
To illustrate the direct method for finding branching rules, we consider first the reduction The operator exp the sum of whose eigenvalues determines the character of a representation of a group, is given by exp i for and exp
138
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
say, for
:. If we set
, the two become equivalent; for
The last step follows from Eq. (538). To find how the irreducible representation of breaks up when it is considered as a representation of we have merely to replace by in the expression for and reduce the resultant function to the form (575) Every term in the sum is the character of an irreducible representation of and therefore the values of L that appear immediately determine the representations For example, on putting and in either Eq. (556) or Eq. (560), we get
Thus, under the reduction the irreducible representation (11) of breaks up into and 5>i. This can be concisely expressed We could, of course, have anticipated this result, by since the terms 3P and ZF of d2 are used in Sec. 57 to form basis functions for the representation (11). The convention adopted here, which replaces by the spectroscopic symbol for L, will be frequently used in subsequent work. The reduction can be dealt with in an analogous fashion. The methods of the previous paragraph can be used to show that the operator
CONTINUOUS
GROUPS
139
reduces to if the substitutions are made; hence, by substituting in ) and expressing the resultant function as a sum of the form (575), the decomposition of any irreducible representation of can be obtained (see Prob. 54). The simplicity of the expression for the character of an irreducible representation of , makes it comparatively easy to decompose a particular representation of or into irreducible representations of The method is unsuited to study the reduction because of the complexity of and Fortunately, the general solution to the problem of decomposing irreducible representations of into irreducible representations of can be described by a number of simple rules, and we can thereby avoid having to construct the characters explicitly. All the irreducible representations of for which determinantal product states serve as bases satisfy the inequality in these cases the rules, as stated by Littlewood,70 can be adapted as follows: We draw the shape corresponding to the representation and perform the successive operations: 1. Leave the shape untouched. 2. Delete the two cells at the feet of the two columns, if this is possible. 3. Take the new shape, and delete the two cells at the feet of its two columns. 4. Continue in this way until a shape possessing a single column is obtained or, if the two columns of the original shape are of equal length, until no cells remain. Parentheses rather than brackets are used to describe all the shapes that can be obtained by this process, and the resulting symbols are interpreted directly as the irreducible representations of into which decomposes. Thus the representation [2210000] of Ui corresponds to the shape [221], which, under the above operations, becomes successively
Accordingly,
140
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
The shape [111] already possesses but one column; hence
Since irreducible representations of are described by only I numbers, it is not possible to carry out the procedure if more than I symbols are nonzero. We notice, however, that if we take each of the determinants of Eq. (561) in turn, interchange rows 1 and rows 2 and 21, rows 3 and etc., and then interchange columns 1 and columns 2 and 21, etc., we get
The reduction
is accomplished by setting
under which
On replacing
we obtain
Comparison with Eq. (561) shows that this is nothing else but consequently t h e t w o irreducible representations and break up into the same set of irreducible representations of This in turn implies the equivalence
where a and b are constants which ensure that the entries in the brackets are integers. If we set and we get, using Eq. (543),
It is interesting to note that the two shapes corresponding to these irreducible representations can be fitted together to form a rectangle
CONTINUOUS GROUPS
.
141
For d electrons, for example, (578)
If we use Eq. (542), this equivalence tells us that the states of with form bases for the same irreducible representations of as do the states of with in fact, it establishes the familiar correspondence between electrons and holes. For i gives consequently Similarly, There remain a number of representations of that cannot be decomposed by these methods. An example is [2211100] of In cases such as this, we perform the operations 1 to 4 above, using parentheses to describe the resulting shapes. Thus Murnaghan71 has given modification rules for converting nonstandard symbols such as (22111) to admissible representations of the appropriate rotation group. When the number of 2's is not greater than I, the rules can be concisely summarized by the formula
where
For example, Equation (579) can be applied only to those cases for which all other nonstandard symbols can be avoided by a judicious use of Eq. (577). For / electrons, the reductions and remain to be considered. The operators , for are as follows: .
On writing the operator reduces to a linear combination of the operators (535). The characters for
142
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
decompose into sums of the characters X(U1U2) for G2. With a considerable amount of algebraic manipulation, we arrive at the result X(W1W2Wa) = 2:[x(i  k, j
+ k) + xU 
k  1, i  j)] (580)
where the sum runs over all integral values of i, j, and k satisfying Wa ~
k
~ Wa
The equations
+
and
X(U1U2) =  X(U2  1, U1 1) X(U1, 1) = 0 x(u1,2) = X(U1  1,0)
are used to remove characters X(U1U2) whose arguments do not give admissible irreducible representations of G2 ; they can be obtained by methods similar to those that led to Eq. (570). The reduction G2  ? Rs can be accomplished by writing cpl = 38 and cp2 = 28. 513
CLASSIFICATION OF THE TERMS OF
r
It is stated in Sec. 57 that the totality of states of the configuration l" for a given Sand M s form a basis for a single irreducible representation of U 21+1. Since the representation does not depend on M s, the labeling of states by [X 1X2 • • • X21+1] is equivalent to specifying S. The actual correspondence is given in Eq. (542). The terms that occur in the configuration l" are very easy to find when n < 3, and hence the branching rules for the decompositions of a few representations of U 21+1 to irreducible representations of Rs can be rapidly obtained. It will now be shown how a chain calculation can be set up to determine the branching rules for other representations of U 21+1. The general method will become clear if we take the special case of l = 3. From the configurations f O, p, and p, the following branching rules for the reduction U 7  ? Ra are readily derived: [0000000]  ? [1000000]  ? [2000000]  ? [1100000]  ?
S F S P
+D +G +I +F +H
For conciseness, we shall frequently omit plus signs and arrows in tabulations such as this. By using the corresponding partitions to label the irreducible representations of U 21+1, the necessity of writing out many zeros in the brackets can also be avoided. Suppose that
CONTINUOUS GROUPS
143
we select any two of the four representations of Ui listed above and form their Kronecker product. The irreducible representations of R3 contained in this product can be found by making use of the known branching rules for the two representations of Ui and then applying Eq. (213) repeatedly. The representations of R3 obtained by this procedure give the structure of the collection of irreducible representa tions of U7 into which the Kronecker product decomposes and hence provide information about branching rules. For example, [11] X [1] must decompose to those irreducible representations of R3 contained in the Kronecker product (P + F + H) XF. However, P XF = D + F + G FXF=S+P+D+F+G+H+I HXF=D+F+G+H+I+K+L
and
xD
=
Hence
[111] + [21] »
SPD3F3G3H2PKL
A superscript to a symbol indicates the number of times the corre sponding irreducible representation occurs. It is clear that, if the decomposition of either [111] or [21] were known, then that of the other would follow immediately by subtraction. The problem of finding the irreducible representations of R3 into which [111] decomposes is equiva lent to that of finding the terms of / 3 for which S = f. The latter can be solved by using the elementary techniques of Sec. 16, and the required terms are 4S, *D, Ψ, 4G, and 4 J. Thus [111] * SDFGI and so
[21]
1
+PD FWHVKL
The decomposition of [21] gives the terms of / 3 for which S = ^; we notice that there are two 2H terms, in agreement with the conclusion arrived at in Sec. 16. The method used above to find the decomposition of [111] is essentially alien to the present approach. Fortunately, the branching rules for all the remaining irreducible representations [λιλ2 · · · λ?] of Uj for which λι < 2 can be calculated without having to resort to such a device again. As an illustration, we calculate the terms of f*.
144
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
The Kronecker product
+
χ
corresponds to (SDFGI) X F = SP 3 D 8 F 6 G 4 H 4 Z 3 K 2 LM According to Eq. (577), the decompositions of [1111] and [111] are identical; hence [1111] » SDFGI and so [211] +PWFHPHWLM These decompositions give the terms of/4 for which S = 2 and S = I, respectively. To find the singlets, we use
+
X
+
to which corresponds 3
(PFH) X (PFH) =
6
9
3
3
S PWF G H^PK L MN
On subtracting the parts that arise from [1111] and [211], we get [22] >
i
2
2
SW FG*H PKL N
There is no difficulty in extending this procedure to other configura n tions of the type / . The method for finding branching rules that is described in Sec. 512 treats each particular reduction as a selfcontained problem, to be solved by using the characters of the group and the subgroup. Once the branching rules for a few reductions have been found, how ever, they can often be used with great effect to find others. To illustrate this approach, we show how the branching rules for ϋ7 —> JS7 and U7 —* R% can be combined to find those for R7 —• R3. Suppose, for example, that the irreducible representations of R3 into which (220) decomposes are required. It is necessary only to compare the
CONTINUOUS GROUPS
145
decompositions and with and to obtain the result
We can sometimes use a knowledge of the dimensions of irreducible representations to obtain branching rules. Suppose that we are interested in the reduction We draw up a table for and list the dimensions of the representations of in order of increasing magnitude. D(wiw2wa3)
{wiwiws)
1 7 21 27 35
(000) (000) (100) (100) (110) (200) (111)
L
sS
F F PFH DGI SDFGI
The dimensions of the representations of are calculated from Eq. (566) and tabulated in order of increasing magnitude. D(uiUi)
(UiUt) (uiut)
1 7 14 27 64
(00) (00) (10) (11) (20) (21)
The only representation of with dimension 1 is corresponding to an S state; hence The representation (100) has a dimension of 7 and could decompose either into seven representations of the type (00) or simply into (10). The former case can be excluded, because it implies that, under the reduction Therefore Similarly, (110) is of dimension 21; since an S state is not included in its decomposition, it can break down into or into The first possibility can be discarded, since we have already shown that and (110) does not decompose
146
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY T A B L E 51
B R A N C H I N G R U L E S FOR T H E R E D U C T I O N UI —» Ri
Configuration
Dimension
[X1X2 · · ·]
(WlIC 2 IO 3 )
P P P
1 7 21 28 35 112 35 210 196 21 224 490 7 140 588 490 1 48 392 784
[01 [1] [H] [2] [111] [21] [IUl] [211] [22] [11111] [2111] [221] [111111] [21111] [2211] [222] [1111111] [211111] [22111] [2221]
(000) (100) (HO) (000)(200) (111) (100)(210) (Ul) (110)(211) (000)(200)(220) (HO) (111)(211) (100)(210)(221) (100) (111)(210) (110)(211)(221) (000)(200)(220)(222) (000) (110)(200) (111)(211)(220) (100)(210)(221)(222)
P P P P P
T A B L E 52
B R A N C H I N G R U L E S FOR THE R E D U C T I O N RI•
D(WiWiWa)
(wiWiW»)
(UiUi)
1 7 21 27 35 105 189 168 378 294
(000) (100) (110) (200) (111) (210) (211) (220) (221) (222)
(00) (10) (10)(11) (20) (00)(10)(20) (11)(20)(21) (10)(11)(20)(21)(30) (20)(21)(22) (10)(11)(20)(21)(30)(31) (00)(10)(20)(30)(40)
G2
into three F states. We can continue in this manner and unambiguously obtain the decompositions of (11), (20), and (21). By means of the various techniques described above, the branching rules for all the groups of interest to us can be found. The results for the reductions Uj —» Rj, RJ —> G2, and G2 —> #3 are given in Tables 51 to 53. The original derivation of these decompositions is due to Racah.69 Branching rules for the reductions Ui —> G2,
CONTINUOUS GROUPS T A B L E 53
BRANCHING R U L E S D(UiUi)
1 7 14 27 64 77 77 189 182
FOR THE
REDUCTION 62 —»
Ri
L
(ttlMi)
(00) (10) (H) (20) (21) (30) (22) (31) (40)
147
S F PH DGI DFGHKL PFGHIKM SDGHILN 1 1 1 PDF GmI K LMNO 1 1 1 SDFG HI KL MNQ
Ui —* Rz, and R: —» β 3 can be easily obtained from these tables and are therefore not written out explicitly. In Table 51, only those representations of Uy corresponding to the first half of the / shell are given; the decomposition of the remainder can be obtained by means of the equivalence (577). 514
IRREDUCIBLE REPRESENTATIONS AS QUANTUM NUMBERS
In Sec. 51 we initiated a search for symbols 7 to distinguish terms of the same kind in configurations of the type ln. The aim of the grouptheoretical approach is to replace 7 by the irreducible repre sentations of various groups, and we are now in a position to appraise this method. For configurations of fewer than three equivalent elec trons, the symbols SMSLML are sufficient to define the states; as a modest test of the grouptheoretical approach, we investigate the 3 classification of the 17 terms of / . From Tables 51 to 53, the following scheme is obtained: [λ,λ2 · · ·]
(IUiU) 2 Wa)
(W1M2)
L
[111]
(111)
[21]
(100) (210)
(00) (10) (20) (10) (H) (20) (21)
S F DGI F PH DGI DFGHKL
It is at once apparent that no two terms share the same set of quantum
148
OPERATOR TECHNIQUES /N ATOMiC SPECTROSCOPY
numbers; hence for/ 3 the states can be completely defined by writing \P(wiWiWi)
(UIU2)SMSLML)
The irreducible representations [λιλ2 • • ·] of U7 have been dropped because they are equivalent to specifying S. I t can be seen that no two identical irreducible representations of R7 occur in the decom position of any of the representations of U7 listed in Table 51; simi larly, no two identical irreducible representations of G2 occur in the decomposition of any of the representations of R7 listed in Table 52. The grouptheoretical classification of the terms of / n would be com plete if no two identical irreducible representations of K3 occurred in the decomposition of any of the representations of G 2 ; however, it can be seen from Table 53 that this is not true for (31) and (40). In these cases a symbol τ must be included in addition to the irreducible representations of R7 and G2 in order completely to define the states. Thus, on making the abbreviations (WiW2W3) = W
(UiU2)
=
U
n
the states of / can be unambiguously written as \f"W
UTSMSLML)
For the configuration p", the quantum numbers SMSLML are sufficient to define the states completely; for dn, the classification according to the irreducible representations of R& separates all terms of the same kind (see Prob. 56). For gn, however, the only useful grouptheoretical labels are the irreducible representations of B 9 , and these are quite inadequate for such complex configurations. We shall not consider this shortcoming here, since only a few very simple configurations involving g electrons have so far been reported. The explicit construction of a state defined by grouptheoretical labels is usually a straightforward, though sometimes a tedious, matter. Suppose, for example, that 3
/ (210)(11) W, Ms = i , ML = 5) (6>
is required. Since V is an infinitesimal operator of the group G2, it must be diagonal with respect to U; consequently 2
(/3(210)(21) »L,i,8Vy«/»(210)(ll) # , i , 5 ) = 0 Since
(581)
(/3(210)(21) 2 L,^,8 = {332}*
it is a simple matter to express the matrix element in terms of the parameter χ of Sec. 16. The condition (581) gives χ = » , and the
CONTINUOUS GROUPS
149
required state is just the linear combination of determinantal product states given in Eq. (129). In fact,
PROBLEMS 61. Prove the Jacobi identity
and show that it leads to the following equation for the structure constants: 62. Prove that the operations of G2 leave invariant the trilinear antisymmetric form
in the notation of Sec. 54. (See Racah. 68 ) 63. It can be seen from Fig. 511 that the projection of the weights of the representation (20) of on a line making an angle with an axis gives a
FIG. 511
The diagram for Prob. 53.
pattern of points that is identical to the combined system of weights of and of Rt. Justify this method of obtaining branching rules, and use it to prove that for the reduction and tion
64. Prove that the number of terms with a given S and L in the configure is given by the coefficient TILS in the equation
where
150
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
and where the functions P and D are defined in terms of the quantities
as follows:
except when in which case or when in which case . (This result is due to Curl and Kilpatrick.72) 55. Prove that, for the reduction the irreducible representation (wwO) decomposes as follows:
tion
56. Derive Table 54 for the reduction (See Jahn.73) TABLE 5  4
BRANCHING RULES FOR THE REDUCTION
Configuration
Dimension of ^mens'on [XIXS • IA1A2 • •• •]J
d° d d1 di
[x,x [XiXs2 • • •]•]
1 1
[0] [0] [1]
((1100))
[11] [11] [2]
((11) ID
[[111] 111] [21]
((11) ID
[[ 1U1 l1l1]]
((1100))
15 1 10 0 40
d dl 4
5
d6
(W1W2) (wiwi)
5 10 10
d» d3
TABLE 5  5
and Table 55 for the reduc
((00) 00)
((0000))((2200)) ((1100))((2211))
45
[211]
((1111))((2211))
50
[22]
((0000))((2200))((2222))
[[11111] Hill]
((00) 00)
1 1 24
[2111]
((1111 X ) (2200))
75
[221]
((1100))((2211))((2222))
BRANCHING RULES FOR THE REDUCTION
D(wiwt) D(wiwi)
(wiios) (WLLOS)
L L 88 D D PF PF DG DG PDFGH PDFGH SDFGI SDFGI
1 1
((00) 00)
5
(10)
10 14
11 1 )) (( 1 (20)
3 35 5
((21) 21)
3 35 5
((22) 22)
CONTINUOUS GROUPS
151
11'7. The tensors v(i)(ll') satisfy the equation (1"'lIv(l)(ll')lIl") = 8(1,1"')8(1',I")[kjl
Prove
kl) (1)'1"'+1"1'',
qa
k, kl} v X[ &(1',1")( _1)k,H,+t,+I+I'+I"+I'" {k' 1'" 1 I"
9,
B(ll"') ,
(',)(ll"')
{k' kl I" I'
k.} v (l,)(l"l')] l 90
Show:
1. That the (21 + 1)1 operators v,(l)(ll), the (2l' + 1)1 operators vqa)(l'l'), together with the 2(21 + 1)(21' + 1) operators vq(l)(ll') and v,(·)(l'l), can be regarded as the infinitesimal operators of the group U., where s = 2(1 + I' + 1). 2. That the l(21 + 1) operators vqa)(ll) with odd k, the 1'(21' + 1) operators vq(·)(l'l') with odd k, together with the (21 + 1)(21' + 1) operators
can be regarded as the infinitesimal operators of the group R,. 6S. Suppose that the configurations d", d,,18, and d,,281 are separated by energies that are sufficiently small compared with the separations of the terms to make it necessary to consider the configurations as a single entity. Show that the states may be classified by using the irreducible representations of the groups in the sequence
Investigate the properties of the irreducible representations verify the formula
D('I., •• ) ... (.1  .,
+ 1)(., 
••
+ 2)(", 
'"
("1"'''')
of
R.,
and
+ 1)(.1 + ., + 3)(.1 + •• + 2) X (.,
+ •• + 1)/12
Obtain the scheme given in Table 56 for the reduction U. + R.. Prove that, under the reduction R. + R a, the decompositions of ("1"2"') and ("'''2  ".) are identical. Derive the branching rules of Table 57 for the reduction R. + R •. Deduce that states of the configurations d", d"'s, and d"z,. are uniquely defined by writing
(This method of classifying the states of mixed configurations is due to Elliott,"
who obtained many of the results quoted here and in the preceding problem.) 69. Prove that R. is a semisimple group, but not a simple group. Show that, under the reduction R. + R the irreducible representation ("1"') of R. "
152
OPERATOR TECHNIQUES IN ATOMIC TABLE 56
BRANCHING RULES FOR THE REDUCTION
Dimension
[XiX,s • • •]•] [XiX
1 6 6 15 21 20 20 70 105 105 84 210 210 35 189 175
TABLE 57
SPECTROSCOPY
[0] [0] [1] [1] [11] [2] [111] [111] [21] [211] [22] [2111] [221] [221] [21111] [2211] [222]
(vivtvt) (ripjri) (000) (000) (100) (100) (110) (000)(200) (111X11 (111)(11  1) (100)(210) (100)(210) (110)(211)(21  1) (000)(200)(220) (000)(200)(220) (111)(11  1)(210) (100)(210)(221)(22 (100)(210)(221)(22  1) (110)(200) (110)(200) (110)(211)(21  1)(220) (000)(200)(220)(222)(22  2)
BRANCHING RULES FOR THE REDUCTION D(vivtvi) D( vivira)
(pinn) (»I>'2I'i)
1 6 15 10 10 20 64 45 84 70 35
(000) (000) (100) (110) (HO) (111) (200) (210) (211) (220) (221) (222)
(VJIW (wiwt)S) (00) (00) (00)(10) (00)(10) (10)(11) (11) (11) (00)( 10)(20) (00)(10)(20) (10)(11)(20)(21) (10)(11)(20)(21) (11)(21) (20)(21)(22) (20)(21)(22) (21)(22) (22)
decomposes according to the rule
where (v) denotes the irreducible representation t The weights and used to define a representation of are the natural analogs of those of Prob. 58 and correspond exactly to the scheme used, for example, by L. C. Biedenharn [J. Math. Phys. 2, 433 (1961)]. The weights p and q of G. Ya. Lyubarskii ["The Application of Group Theory in Physics," Pergamon Press, New York (I960)] are related to vi and by the equations
CONTINUOUS GROUPS
153
610. Show that the dimension of the irreducible representation of Spi* is given by
Derive the branching rules of Table 58 for the reduction results have been given by Flowers.75) TABLE 58
BRANCHING RULES FOR THE REDUCTION
D(ffi4 SD and
of Prob. 19 can be replaced by
(111)(20), (210)(20), a n d (210)(21), r e s p e c t i v e l y .
512. Prove that the dimension of the irreducible representation that labels the terms of with spin S is
where
(These
6 SENIORITY
61
DOUBLETENSOR OPERATORS
In the preceding chapter, the spin and orbital parts of a manyelectron eigenfunction have been treated as far as possible as two separate entities, impinging on each other only at the point where their respective tableaux are constructed to be adjoint. As a preliminary step toward studying the relationship between spin and orbit in greater detail, we introduce a type of operator having welldefined properties with respect to these quantities. The operator comprising the components , is said to be a double tensor if it behaves as a tensor of rank k with respect to the total spin angular momentum and as a tensor of rank k with respect to the total orbital angular momentum. In detail,
These equations serve as a basis for the development of the theory of double tensors; however, their resemblance to Eq. (226) is so marked that it is easy to see how the theory of ordinary tensor operators is to be generalized. For example, the WignerEckart theorem for double tensors is
SENIORITY
155
As a second illustration, we give the reduced matrix element of a double tensor T ( ' H acting only on part 1 of a system,
This equation can be considered to be a generalization of Eq. (337). Suppose that is a tensor operator which acts in the spin space of a single particle and for which (63) We see that quantities
is the analogue of
, defined in Eq. (513).
The (64)
clearly form the components of a double tensor, and so do the manyelectron operators (65) Owing to Eqs. (513), (63), and (64), the reduced matrix elements for are given by (66) An application of the WignerEckart theorem to matrix elements of between singleelectron states yields (67) For all applications of the formulas in this paragraph, and K is either 0 or 1; however, it is convenient to retain the general notation to emphasize the analogy with the corresponding orbital quantities. It is to be noticed that is not in general the same as , since the latter includes terms of the type
(68) The commutation relations satisfied by the operators serve to determine which groups are relevant for classifying the orbital parts
156
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
of the nelectron eigenfunctions. In our attempt to remove the somewhat arbitrary barrier between spin and orbit, it is natural to examine the commutators of the double tensors However, it should be pointed out in passing that these tensors enjoy an importance in their own right. The spinorbit interaction and part of the hyperfine interaction are treated later (Sees. 79 and 89) as components of double tensors; that this is possible is evident from a glance at Eqs. (64) and (65). The doubletensor formalism also allows us to treat twoparticle operators involving both spin and orbital variables. By a process of recoupling, the matrix elements of an operator of this kind can be expressed as a sum over weighted products of the matrix elements of double tensors. This method is illustrated in Prob. 62 at the end of the chapter. For extending the procedure to evaluate matrix elements of twoparticle operators between coupled groups of equivalent electrons, the reader is referred to the appendixes of a paper by Innes and Ufford.38 62
THE C O M M U T A T O R S OF DOUBLETENSOR OPERATORS
By methods analogous to those of Sec. 54, it is straightforward to derive the commutation relations
This equation, which is the generalization of Eq. (514), is equally valid if the substitutions
are made.
The number of components of this kind is
and they can be taken to be the infinitesimal operators of the group corresponding to the unitary transformation (610) The nelectron eigenfunctions are totally antisymmetric with respect to the interchange of any two electrons, and they form
SENIORITY
157
the basis for the irreducible representation
of Equation (562) can be readily adapted to find the dimension of this representation; the result is
where evaluates to
and zero otherwise.
This expression
which is precisely the number of totally antisymmetric functions that can be constructed from n electrons and functions The investigation of the subgroups of proceeds in a similar manner to the study of the subgroups of , described in Sec. 54. On discarding from the set of infinitesimal operators for the group is limited to the special unitary group From the operators for we select the set comprising and It is clear from Eq. (69) that the commutator of any two components of this set involves members of the set only. Moreover, the components of form the infinitesimal operators of a subgroup, and so do the three components of That both these subgroups are invariant can be seen by setting and _ in Eq. (69); for the triangular conditions on the 6j symbols restrict to 1 and to k, and hence the factor (611) is zero. It followsfrom Sec. 53 that the group with infinitesimal operators and is the direct product and by limiting to this subgroup, we render independent the transformations in the orbital and spin spaces. Of course, we could have separated these two kinds of transformations without imposing the unimodular condition on but if the operator is included in the infinitesimal operators of both invariant subgroups, the group is not strictly a direct product. Under the reduction _ the coefficients in Eq. (610) factorize according to and the irreducible representation
decomposes into a
158
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
number of product representations of the type where denotes an irreducible representation of , and , the shape adjoint to denotes an irreducible representation of This method of combining spin and orbital functions is described in detail in Sec. 58, and we have reached the point where we can use the results of Sec. 54 to decompose further. So far the introduction of groups whose infinitesimal operators are double tensors has added very little to our appreciation of the grouptheoretical classification of states. However, there exists an alternative and more interesting way of decomposing It can be seen from Eq. (69) that the double tensors for which is odd form the infinitesimal operators for a subgroup of for if both and are odd, then must be odd or else the factor (611) vanishes. From the WignerEckart theorem,
The sum for terms of is even, and hence if is odd, the triangular conditions for the nonvanishing of the 3j symbols cannot be satisfied. Consequently the operations of the group leave invariant the XS state of namely, the bilinear antisymmetric form
The group is therefore the symplectic group in
dimensions,
From the collection of components of .. for which is odd, we select the set j and In a manner similar to that used in reducing to it can be shown that is a subgroup of Spti+z. It is therefore apparent that the sequence is an alternative to
63
THE SENIORITY NUMBER
Rather than repeat the analyses of Chap. 5 for the particular case of the group Spu+t, we shall simply state most of the results that are
SENIORITY
presently required. integers
The representations of satisfying
159
> are defined by
The branching rules for the reduction can be obtained by considering the characters of the representations of these two groups; however, like the rules for they can be expressed by means of a simple prescription. The decomposition of of is particularly easy to describe; we merely write down n ones followed by , ones followed by zeros, etc., and enclose the sets of symbols in parentheses,
The last representation in the collection is (100 • • • 0) if n is odd and (000 • • • 0) if n is even. If n exceeds the equivalence which follows from Eq. (577), must first be used. The rules for decomposing any representation of into representations of can be obtained from the book by Littlewood70; they have been given explicitly by Flowers.76 Tables for such as Table 51 allow the branching rules for the reduction to be rapidly written down. For / electrons, for example,
etc. The representations of are defined by the total spin S, and the corresponding multiplicity, , isindicated by the prefixes to the irreducible representations of With the aid of the decomposition (612), the branching rules for the reduction can be easily found by a process of subtraction. They are given in Table 61 for All the representations that are used to classify the states of are of the type To specify a representation of this kind it is necessary only to give the
160
OPERATOR TECHNIQUES /N ATOMIC SPECTROSCOPY
number of ones it contains; this number, defined formally by σ
ν=Σ < * is the seniority number and is listed in Table 61. Since we may in general decompose any representation W = (WiW2 Wi) of R21+1 into irreducible representations of R3, we can speak of the seniority number, or simply the seniority, of a given term of I". The decomposition represented by (612) shows that the terms of Z" break up into sets, each set labeled by a value of v. For example, for Z2, there are two sets, corresponding to υ = 2 and 0; for Z8, there TABLE 61
BRANCHING RULES FOR THE REDUCTION Spu —> SUt V
(σισ 2 · · · ai)
2S+lJf
0 1 2 3 4 5 6 7
(0000000) (1000000) (1100000) (1110000) (1111000) (1111100) (1111110) (1111111)
HOOO) '(100) »(110) «(200) *(111) ! (210) «(111) «(211) »(220) »(110) 4 (211)»(221) '(100) «(210) »(221) »(222) 8 (000) "(200) 4 (220) 2 (222)
X Bj
are also two, corresponding to υ = 3 and 1; for Z4, there are three, and so on. I t is clear that every set of terms labeled by the same value of υ must have an identical structure, since the reduction of £p4!+2 to its subgroups is independent of n. Identical sets of this kind, characterized by a fixed seniority v, occur in the configurations 2 4 2I", Z"+ , . . . , Z '+ "; hence the seniority of every term belonging to a given set is equal to the number of electrons comprising the con figuration in which the set first makes its appearance. I t was in 76 this sense that the concept of seniority was first introduced by Racah. Now W and S uniquely specify v; that this is so for the configurations n / can be seen from Table 61. Hence, if a collection of terms char n n acterized by a given W and S occur in the configurations Z", Z ', Z ", . . . , the seniority of every term in the collection is equal to the smallest integer of the set n, n', n", . . . . For example, the seniority 2 3 1 of the term (100)(10) F of/ is 1, because (100)(10) Ψ occurs in / . It is not difficult to establish the connection between ν and the weights Wt. A given irreducible representation W = (wiw2 · • · Wi) of Ru+i occurs in the decomposition of various representations [λιλ2 · · ·]
SENIORITY
161
of of these, the one corresponding to the shape with the smallest number of cells is If numbers c and d are defined by the equations
then it follows from Eq. (542) that the spin associated with theshape is Hence the seniority of the terms labeled by W and is given by the number of cells in
Since it follows that Of course, the spin of the terms labeled by a given W need not be id, since the modification rule (579) may have been used in the process of decomposing representations of to those of In this case,
and Both possibilities are included in the formulas (613) which not only determine v for a given W and S but also enable the irreducible representations to be constructed from the numbers v and S. [Equations (613) are due to Racah.69] The grouptheoretical classification of states described in Sec. 514 provides the set of quantum numbers (614) to define a state of The symbol is required when n exceeds 2 to distinguish terms that possess the same WSL classification; for / electrons it may be replaced by The introduction of seniority gives the alternative set (615) Both schemes have their advantages: W displays the grouptheoretical properties of the representation; on the other hand, v enters more
162
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
naturally into quantitative calculations. For example, Eq. (550) for the eigenvalues of Casimir's operator for simplifies considerably if v and S are used rather than the weights Wi. We obtain
for either of the two possibilities for d given in Eqs. (613). 64 THE LAPORTEPLATT DEGENERACIES In Sec. 43 are given the energies £J( 2 S + l L) of the terms of P as linear combinations of the integrals If these integrals are regarded as variable parameters completely at our disposal, then we can evidently choose them to make the energies of any three terms coincide. If, for example, we take the ratios (617) we find However, on calculating the energies of the remaining terms of the configuration, using the ratios (617), we find, surprisingly, that three of the singlets coincide, The energy of the remaining term is given by
Unexpected degeneracies of this kind were first noticed by Laporte and Piatt, 7 7 who showed that a large number of terms in configurations of the type coalesce for a single choice of ratios. The energies of the terms of can be obtained from the effective Hamiltonian
To study the LaportePlatt degeneracies, we drop , replace the remaining integrals by their counterparts and use Eqs. (617) The schemes (614) and (615) are not the only ones available. If we are prepared to abandon the quantum number S, for example, thenwe may use the subgroup of whose infinitesimal operators are . The states of ln can now be classified by the set of quantum numbers where has the same significance here as in Prob. 59, and is an additional classificatory symbol.
SENIORITY
to eliminate .
and
163
The resulting expression is
Provided that all calculations are performed within the configurations the tensors and for even k differ only through their reduced matrix elements; hence we can replace by in (618) provided that the factor
is included. form
The expression (618) now assumes the strikingly simple
This suggests that an investigation be made of the operator
The form of since
prompts us to examine Casimir's operator for
There are terms in the double sum on the extreme right, every one of which satisfies equations of the type
(619) The eigenvalues of are given by Eq. (551), and it is straightforward to deduce that the eigenvalues of (620) are Of course, contains terms in the sum for which k is odd. them, we use Eq. (547) and write
(621) To exclude
164
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
The eigenvalues of are given in Eq. (616), and we use Eq. (619) again to evaluate the eigenvalues of the double sum on the right. The eigenvalues of are found to be (622) Since , the eigenvalues of 3 are obtained by subtracting (622) from (621). The result is
The striking feature of the expression for the eigenvalues of is that L does not make its appearance, in spite of the fact that the tensors from which is built act on the orbital variables. The degeneracies in may now be readily understood, since each group of degenerate terms corresponds to a particular pair of values for S and v. Thus, on making the substitutions in (623) and multiplying the result by we obtain an energy of 30^2 for the terms and of in agreement with the initial calculation. The general form of (623) allows us to extend the results to other configurations; for , the terms fall into three sets, with relative energies as follows:
Much of the preceding discussion may appear somewhat hypothetical, since the ratios (617) that are necessary for the degeneracies are so far removed from the ratios (411) that give good agreement with the experimental data for However, for d electrons there exists a physical interpretation for the degeneracies.
The matrix elements of the tensors for which k is odd vanish if taken between states of the configuration ; moreover, the matrix elements of all tensors of rank greater than 4 vanish for d electrons. Hence the matrix elements of are identical, apart from a constant additive term, to those of
SENIORITY
where, in analogy to the notation of Sec. 43, the vectors r< and The sum
165
is the angle between
although divergent,78 behaves under integration like where is the delta function of Dirac;3 it can therefore be regarded as the angular part of the function Hence the LaportePlatt degeneracies correspond to interelectron forces of infinitesimally short range. This is not true in general; for example, the numerical coefficients in the effective Hamiltonian (618), corresponding to / electrons, are not proportional to PROBLEMS
62. Show that
*
(See Innes.") 63. Use the formula for given in Prob. 510 to prove that the dimension of the irreducible representation corresponding to a seniority number is given by
Check some of the decompositions of Table 61 by means of this equation.
7 FRACTIONAL PARENTAGE COEFFICIENTS
71 INTRODUCTION In the grouptheoretical classification of states, the set of quantum numbers lnSMsLML is augmented by the irreducible representations of certain groups, rather than by the nondescript symbol 7. The immediate advantage is that many states can be uniquely specified without going to the lengths of calculating the particular linear com binations of determinantal product states to which they correspond; however, if we are obliged to construct these linear combinations in order to evaluate the matrix elements of the various operators in the Hamiltonian, nothing is gained. The problem of circumventing the explicit introduction of determinantal product states has been given an elegant solution by Racah, 7 6 who elaborated an idea of Bacher and Goudsmit. 7 9 The principle, which is very simple, is most clearly described if the sets of quantum numbers defining states of the configurations ln, ln~l, and I are abbreviated as follows: ySMsLML =Ω ySMsLML = Ω msmi = ω It is understood that Ω refers to a state of ln, Ω to a state of Z" 1 , and ω to a singleelectron state. Since every determinantal product state 166
FRACTIONAL
PARENTAGE COEFFICIENTS
167
for an nelectron system is a sum of terms, each of which involves an eigenfunction of the nth electron, we may write (71) The subscript n to w makes it clear that the final state refers to the nth electron. Owing to the equivalence of the electrons, the matrix element of a singleparticle operator
between two states of
the configuration l" is n times the corresponding matrix element for that is Making use of Eq. (71), we get
(72) The evaluation of the matrix elements of for singleelectron states is trivial, and the problem reduces to finding the coefficients and . We shall assume that the phases of the states of the configurations and I have been chosen to ensure that the coefficients are real; hence
The obvious way of extending the method to twoparticle operators of the type tron
is to factor out the eigenfunctions of elec
as well as those of electron n.
The product eigenfunctions
of these two electrons can be expressed as linear combinations of the states of
; hence we may write (73)
where n and
denote states of
leads at once to
and
respectively.
The relation
168
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
Although this result is of value, there exists a more convenient way of evaluating the matrix elements of twoparticle operators. Of the components comprising of them do not involve the nth electron. Hence
The expansion (71) is used to give the equation
For many cases of interest, is a scalar operator, and all terms in the sum vanish unless and share a common set of quantum numbers Equation (75) relates the matrix elements of G for to those for Zn_1, thus necessitating the construction of a lengthy chain calculation for configurations of large n. This disadvantage is offset by the fact that the coefficients are precisely the ones that occur in the calculation of the matrix elements of singleparticle operators; we are therefore able to avoid the evaluation of the coefficients altogether. In view of these remarks, our attention is directed to the coefficients , which, in detail, are written (76) For a given term of the terms for which the coefficients (76) do not vanish are called the parents of . Conversely, for a given term the terms for which the coefficients do not vanish are called the daughters, or offspring, of Although the calculation of these coefficients may not be easy, it has, at any rate, to be done only once. The results are then available for the construction of the matrix elements of any singleparticle or twoparticle operator. 72
THE FACTORIZATION OF THE COEFFICIENTS
Under the operations of the group states J2) transform according to the irreducible representation . Similarly, the states transform according to and the states w) transform according to ftj. Equation (71) can therefore be interpreted as combining the basis functions of the Kronecker product so that the resulting functions form a basis for from
FRACTIONAL
PARENTAGE COEFFICIENTS
169
this point of view the coefficients are examples of the quantities introduced in Sec. 26. To take advantage of this correspondence, we introduce a theorem of Racah.69 It is quoted here without proof. We denote the irreducible representations of a group 8 by A and those of a subgroup of by B. The symbol b is used to denote a particular basis function of the irreducible representation B. It is supposed that the basis functions for every A of can be taken directly as basis functions for the various irreducible representations . . into which A decomposes, without going through the process of constructing linear combinations. We may therefore label a particular basis function for the irreducible representation A of by ApBb; the index is required when two or more identical irreducible representations B occur in the decomposition of A and serves to make the specification complete. If we suppose that A occurs not more than once in the decomposition of and B not more than once in the decomposition of Racah's theorem can be concisely expressed by the equation
The coefficients of t h e t y p e appearing on the left decompose the Kronecker product into irreducible representations A of similarly, the first coefficient on the right of Eq. (77) is a transformation coefficient connected with the decomposition of the Kronecker product into irreducible representations B of The final coefficient is a quantity independent of bi, b2, and b. Racah's theorem is first applied to and its subgroup The symbols 0 are not required, and the second factor on the right of Eq. (77) becomes
which can be simply written as
without loss of information. for representations of into a spin part
The symbols B in the first factor stand hence the factor itself breaks up
which is nothing else but the VC coefficient
170
OPERATOR
TECHNIQUES
IN ATOMIC
SPECTROSCOPY
and an orbital part
Repeated applications of Racah's theorem give
It is convenient to recombine two of the factors and write
No information in the description of the states is lost by this contraction. The symbol v is preferred to W because, as is shown later, the coefficients on the right of Eq. (78) can be expressed as simple algebraic functions of n, I, v, S, and The single symbol I is used here and in other coefficients to denote the relevant quantum numbers for a single electron. The successive factorizations can be summarized by the formula
The part remaining after the two VC coefficients have been factored out defines the quantities
which are called fractional
parentage coefficients or coefficients of frac
tional parentage, frequently abbreviated to cfp. For f electrons, the groups used for classifying the states are augmented by and a further factorization can be performed. In detail,
The factorizations exhibited in Eqs. (710) and (711) substantially reduce the amount of tabulating that has to be done.
FRACTIONAL PARENTAGE COEFFICIENTS
171
If the group SP4Z+2 is used to classify the manyelectron states, the factorization (79) still results; however, the final factor arises, not through a contraction of the type represented by Eq. (78), but instead in virtue of (Inlii 73
+ ljlnv)(iiS + ljvS)
= (InliiS
+ lj }lnvS)
(712)
ORTHOGONALITY RELATIONS
In view of the orthonormality of the nelectron states, we have
r (n;wjn)(n;wjn') = 8(n,n')
g,w
In full detail, 2:(ln 1WMM sLML;lm.mzjlnW~SM sLML) X (InlWMMsLML;lm8mdlnW'eS'M~L'M~) = 8(W, W') 8(~,n8(S,S')8(M s,M~)8(L,L')8(M L,M~)
(713)
The sum runs over W, ~, S, M s, L, ~h, m., and mz. The distribution of the quantum numbers among the four coefficients of Eq. (79) gives each factor a special responsibility in ensuring that Eq. (713) is satisfied. On making the substitution (79), the sums over Ms , m., ML, and mz can be immediately carried out and give the factor 8(S,S')8(M s,M~)8(L,L')8(ML,M~)
The occurrence of the remaining delta functions is guaranteed if we insist that the equations
r (W~L + ljW~L)(W~L + r +
ljW'eL) =
8(W,W')8(~,n
(714)
L.~
and
(InliiS
lj}lnvS)2
= 1
(715)
ii.S
For I electrons, Eq. (714) is replaced by the pair
are satisfied.
r
L,f
and
+ Ij UrL)(UTL + II U'r'L) = 8(U,U')8(r,r') r (wiT + IIWU)(WU + IIW'U) = 8(W,W')
CUTL
(716) (717)
f)
74
THE CFP (p2SLI Jp3SL)
The advantage of the factorizations indicated in Eqs. (710) and (711) is that the tables of the separate factors are much smaller and more manageable than the single table that would be required to list the
172
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
complete cfp. In the preparation of these smaller tables, we naturally discard methods that give complete cfp in preference to those which determine the entries directly. However, as a first example of the evaluation of some cfp, it is convenient to introduce a method of the first kind; although extremely tedious for all but the most elementary configurations, it is much easier to describe than other more powerful and sophisticated techniques. The objection that the calculation gives complete cfp instead of their component factors can be nullified by choosing configurations of the type the groups and are identical, and the coefficient reduces to at most a phase factor. We write for the complete cfp and consider in detail the case of The starting point is Eq. (71). We make the substitutions and and factor out the YC coefficients,
Suppose we insist that the expression on the right be antisymmetrical with respect to the interchange of any two electrons. It is obviously antisymmetrical with respect to the interchange of electrons 1 and 2; to test for the antisymmetry between electrons 2 and 3, we make the expansion
and couple electrons 2 and 3 by writing
The VC coefficients are converted to 3j symbols, and Eq. (36) is used twice—once in the orbital space and once in the spin space. We obtain
FRACTIONAL
PARENTAGE COEFFICIENTS
173
The antisymmetry with respect to the interchange of electrons 2 and 3 is guaranteed if is even; hence, if is odd, the equation
must be valid. Let us choose the daughter of are and Putting in Eq. (718), we get
Similarly, for we obtain
The forbidden terms
corresponding to the forbidden term
Equations (719) and (720) determine the ratios, one to another, of the three cfp; the equation
fixes their magnitude. The procedure can be repeated for the remaining terms of namely, and The results are collected in Table 71. An arbitrary choice of phase has been made for each row, but the relative signs of the entries in a row are not at our disposal. TABLE 71
THE
Parents (terms of p2)
Offspring (terms of pz)
>S
(terms of p»)
*S 2p * S
*Z>
CFP
lg
0
O
(#)»
0
1£>
1
0
1
0
p 3p
174
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
75 THE COEFFICIENTS i The method described in the previous section can be extended to other configurations. For the tables of cfp are much larger than Table 71, but not impossibly so. However, there are so many terms in configurations of the type that the simple approach we have adopted for finding the cfp is quite impracticable. The table for , for example, has 119 rows and 73 columns. A method for constructing tables of the separate factors that go to make up the complete cfp has been described by Racah.68 The special properties of the configurations need some discussion, and we shall therefore illustrate Racah's method by selecting states from configurations of this type; at the same time it will become clear how to proceed in other cases. The first factor of the complete cfp of Eq. (711) is (721) The procedure we follow in order to evaluate these coefficients consists of a rather elaborate chain calculation, moving in the direction of increasing the dimension of the irreducible representation At the same time as the coefficients (721) are found, tables of the reduced matrix elements
are constructed. As will be shown in Sec. 85, these matrix elements are independent of S, W, and n; we accordingly drop these descriptions and write simply
For the moment, we may regardthis simplification as an extension to of the familiar result that , the set of infinitesimal operators of is proportional to [see Eq. (538)] and consequently independent of S, W, n, U, and , In order to obtain equations involving the coefficients (721), we break into two parts. The first, acts only between parent states, while the second, links states of the added electron. The combination
FRACTIONAL
PARENTAGE COEFFICIENTS
175
is expressed as a sum over matrix elements between coupled states,
The sum runs over Since is an infinitesimal operator for only those terms in the sum contribute for which The coefficients are now factorized, e.g.,
Next, both sides of Eq. (722) are multiplied by
and sums over and are carried out. The quantum numbers and can be removed from the resulting equation by using the WignerEckart theorem, and, on putting we get
A more convenient form for this equation is obtained bymultiplying both sides by and summing over and Owing to Eq. (716), we get the important result
A companion to this equation can be easily derived by using Eq. (716) again. It is
We begin the chain calculation by setting Making use of Eqs. (79) and (710), we easily prove
Our immediate interest lies, not in the complete cfp, but rather in the
176
OPERATOR TECHNIQUES /N ATOMIC SPECTROSCOPY
factor ((00)5 + f\(lO)F).
On making the substitutions
U = (00)
W=U=
(10)
L =3
in Eq. (716), the sum over L and f reduces to a single term, corre sponding to L = 0. Hence ((0O)S+/1(1O)F)2 = 1 We may take either the positive or the negative root, since the phases of the remaining factors are still at our disposal. Here, as elsewhere, we make Racah's choice of phase, and the table of coefficients deriving from the parent (00)»Si comprises the single entry ((0O)S+/1(1O)F) = + 1 From Eq. (513), ((1O)FIlF(WII(IO)F) = (H)*
(725)
The next set of coefficients to consider is that corresponding to the parent (10)F. The decomposition (10) X (10) = (00) + (10) + (11) + (20) indicates that the offspring are (00)S; (10)F; (H)F, H; and (2O)D, G, I. By putting η = 2 in Eq. (71) we get (fl*F\}pSL)
= 1
(726)
The factors we are interested in satisfy ((1O)F+/ICfL)* = 1 The phases of the coefficients are not entirely arbitrary; for Eq. (726) must be satisfied, and the remaining factors of the complete cfp, whose TABLE 72
T H E COEFFICIENTS ((1O)F +/117L)
U
L
(00) (10) (H)
S F P H D G I
(20)
«10)F+f\UL) 1 1 1 1 1 1 1
phases are still at our disposal, are independent of L. This means that all the coefficients ((1O)F + /  UL) for a given U must have the same sign. The actual choices made are indicated in Table 72. We
FRACTIONAL
PARENTAGE COEFFICIENTS
177
maintain our program for finding the reduced matrix elements of V (5) by using Eqs. (724), (337), and (338) to derive the following results:
(727)
(728) We pass on to the parents corresponding to i. The decomposition shows that, if the coefficients are not to vanish, U must be (10), (20), L among the restricted set we are or (21). A number ofu coefficients L considering can be seen to be zero by PnotingHthat 3DL< occurs in the decomposition of only if Those which remain * * to be calculated are indicated by asterisks in Table 73; according F E N T R I E S FOR (10) TABLE 7  3 PATTERN OP * (20) D L * U
(21) (10) (20)
(21)
G L I D F F D G G IH D K p L G H K L
*
*
0 P
H1 * * * * * *
*
** ** *
0 0 * 0 * 0*
11 * *1 *1
0
1
0
1
0
1
to Eq. (716), the sum of the squares of the entries in every row must be unity, and this enables us to insert four l's beside the zeros. In doing this, four arbitrary phase factors are fixed. To proceed further, we turn to Eq. (723). The symbols can be dropped, and only one summation, that over remains. We now pick a set of quantum numbers UL for which the coefficients
178
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
are known; for example, we may take UL = (21)H. becomes
Equation (723)
«II)H,J,HIIV(5) II (11)L',J,L') = «21)HIIV(6)1I(21)L')«21)L'1(1l)L'
+ f)
(729)
The matrix element on the left can be evaluated by means of Eqs. (337) and (338), «11)H,j,HII V(5) II (11)L',j,L')
= (I)L'+1{11(2L'
+ 1)1 1
{i,
5
L'} «11)HII(V(5»),II(1l)L')
3
5
+ llO(L',5)(2L' +
1)1
(i
~ ~'}
The reduced matrix elements of (V(5»), that are required are given by (727), and the results of the calculation are entered in Table 74. TABLE
74
THE MATRIX ELEMENTS
«11)H,f,HlIV(6)1I(11)L',J,L')
L'
£' P H
D
F
(¥)t (tilt
(/~)l
G
(U)t
(.!Ji)l (1·l68l ) l
For L' = 2, £I = 1, Eq. (729) becomes (¥)t = «21)HIIV(5)1I(21)D)«21)DI(11)P
+ f)
(730)
and for L' = 2, £I = 5, we get (M)!
= «21)HIIV(5)11(21)D)«21)DI(1l)H
+ f)
(731)
These equations determine the ratio of the two coefficients on their extreme right; since the sum of the squares of these coefficients must be unity, we have and
«21)DI(1l)P «21)DI(1l)H
+ f) + f)
=  (U)! = (N)!
Table 74 can be used again to find the coefficients for U = (21), L' = 3, 4; for each L' an arbitrary choice of phase is made. Thus, for the part of Table 73 corresponding to U = (21), six such choices are made in all.
FRACTIONAL PARENTAGE COEFFICIENTS
179
The six entries of Table 73 still outstanding can be found by using the orthogonality properties of rows with the same L. How ever, the phases of ((H)L + /(20)L) are not at our disposal. We have already decided to take ((H)H +/1(20)7) to be + 1 , and in using the equation ((20)7(ll)7i + /)((ll)/i,/,JF 1 or c(WW'W") > 1 occur frequently. For example, c((20)(20)(20)) = 2; hence we may write (/6(210)(20)5LF/e(210)(20)5Z/) = 4ι(/3(210)(20)2Ζ,7/3(210)(20)2Ζ/) + 4 2 (/ 3 (lll)(20) 4 LF/ 3 (lll)(20)*L')
(87)
for k, L, L' = 2, 4, 6 (in any combination), provided that the two sets of matrix elements on the right are linearly independent. The reduced matrix elements of U (4) , defined by U ( i ) = Vw[fc]i, have been tabulated 88 for /3, so we have only to use Eq. (752) twice to find A1 and A2, and the remaining matrix elements follow at once. The proportionality factors in equations such as (86) and (87) are often very simple; for example, A = (i)*
A1 = A, = I
In some cases they can be obtained without resorting to detailed calculations. For example, it can be shown for the group 0, c((t>)0>)(200 · · · O)) = 1
198
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
where (v) stands for the symbol (11 · · · 10 · · · 0) in which ν ones appear. Hence a constant A' must exist such that
(pv&L\\W\\l"veS'L') = A'(l»vtSL\\WW\\lti'S'L') (88) where κ + k is odd. The factor A' is immediately seen to be + 1 by setting κ = 0, fc = 1, ξ = ξ', S = S', L = U and using Eq. (758). Equation (88) was first obtained by Racah 7 6 by different methods. 84
THE SPINORBIT INTERACTION
The contribution H2 to the Hamiltonian is easier to treat by the methods of the previous section than the Coulomb interaction Hx, and for this reason we prefer to examine it first. From Eq. (755), we see that the matrix elements of H2 depend on the reduced matrix elements of W)(t/)(H00000)) are given in Tables 81 to 83; the first two tables have been derived by McLellan. 8 9 The selection rule AL — 0, ± 1 on H2 obtained in Sec. 44 can be restated by saying that all matrix elements vanish if C(3DI,3L)Z,'SDI) = 0. Every zero in Tables 81 to 83 implies the vanishing of a collection of matrix elements; for example, c((ll)(21)(ll)) = 0, and therefore n
n
(f W(2l)SL\\W^\\f W'(U)S'L')
= 0
(89)
CONF/GURATJONS OF MORE THAN TWO EQU/VAlENT ElECTRONS TABLE 82
199
THE NUMBERS C(WW(UO)) W
W (000) (000) (100) (110) (200) (111) (210) (211) (220) (221) (222)
(100)
(200)
(HO)
1 1
1
1 1 1
1 2 1
1
1
(211)
(220)
1 1 1 1 2 1 1 1
1
(221)
(222)
1 1 1 1
1 1 1
1 1
1 1
1
1 1 1 1 2 1
1 1 1
1 1 1
THE NUMBERS C((V)(V')(1100000)) (»')
M (0)
(D
(2)
(3)
(4)
(5)
(6)
(7)
1
(0) (2) (3) (4) (5) (6) (7)
(210)
1
TABLE 83
(D
(in)
1
1 1
1
1
1
1
1 1
1 1
1 1
1
1 1
1
for all n, W, W, S, S', L, and JJ. In the tables90 of the matrix ele ments of H2 for P, 32 zeros occur that are not covered by the selection rules AL, Δ/S = 0, + 1 ; of these, 8 are examples of Eq. (89). The zeros in Table 82 can be interpreted in a similar way, but the analo gues of Eq. (89) appear to be less useful. (For example, none of the remaining 24 zeros for/ 3 are accounted for by these equations.) The entries in Table 83 form a regular pattern, and we can deduce that all matrix elements of H2 vanish unless Av = 0, ± 2. When dUU'Ol)) = 1, parameters A (independent of L, L', r, and τ') can always be found such that n
(f»WUrSL\\ W^W f W
U'T'S'L')
= A(J*'W"
UTS"L\\W™\\fn'W"'U'r'S'"L')
200
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
for all . and It is interesting to observe at this point that like W (11) , is part of a generalized tensor whose classification according to the groups R 7 and (?2 is(110)(11); hence, when all matrix elements of are proportional to those of Unless , the proportionality factor is so the only useful result is
from Eq. (758). when
The delta functions give the selection rules , The vanishing of (811)
when illustrates this. Similar remarks can be made when example, if
For
Thus when , spinorbit matrix elements are zero unless diagonal with respect to , and L. The vanishing of
is an example of this kind. [The vanishing of the matrix element (811) is not an example of this kind because The terms of maximum multiplicity of a particular configuration all belong to a single representation Moreover,
for all n [except in the trivial case We find from Eqs. (755) and (812) that the diagonal matrix elements of have a dependence on J and L given by
which accounts for the familiar fact that the multiplet splittings of terms of maximum multiplicity can be matched, so that all the levels of any J coincide, by displacing the centers of gravity of the multiplets. This property, which is illustrated in Fig. 81 for the quartet terms
CONFIGURATIONS
OF MORE THAN TWO EQUIVALENT ELECTRONS
201
FIG. 81 In the limit of LS coupling, the multiplets of 1 with can be displaced so that the energies of all levels with the same J coincide. A similar property holds for any set of terms of I" belonging to an irreducible representation W of for which
of , can also be derived from the equation given in Prob. 77. The generalization of Table 83 to electrons of azimuthal quantum number I is straightforward, and we find We can therefore write
for
even.
The explicit expression (814)
has been given by Racah7® (see Prob. 83).
202
OPERATOR TECHNIQUES /N ATOMiC SPECTROSCOPY
When c(UU'(U)) or c(WW'(110)) is 2, every set of matrix elements between states labeled by U and U' or W and W can be expressed as a linear combination of two independent sets, in analogy with Eq. (87). From Tables 81 and 82, it can be seen that this occurs only when U= XJ' or W = W; hence one of the sets can be taken to be the matrix elements of V(1>. To ensure that the other set is not a simple multiple of this set, it is necessary only that a matrix element off diagonal with respect to L should be nonzero. Thus, since c((21)(21)(ll)) = 2, (f»W(21)SL\\W(1»\\fnW'(2l)S'L') = 46(/3(210)(21)2L17/3(210)(21)2L') + A7(P (210)(21)*LV«/»(210)(21)»L') u
(815) 2
The matrix element of W< > between the states (210)(21) (? and (210)(21)2F of/3 can be shown to vanish, apparently accidentally, by using Eq. (756). Since the matrix of V(1> is diagonal with respect to L, it follows from Eq. (815) that all matrix elements of H2 between states of the type (2I)G and (2I)F in any configuration /" are zero. Powerful though techniques of this kind are, they do not account for all the zeros that appear in tables of H2 (see Probs. 84 and 85). T A B L E 84 Hi
ORDER or
CONSTRUCTING M A T R I X E L E M E N T S OP
FOR S O M E L E V E L S O P f*
(210) (21)
'Fi
(210) (20) 'D1
6 1 1 5 1 5
1 2 2 3 0 3
(100) (10)
(100)( 10)'Fi (210)(20)*Di (210)(21)'!)! (111)(20)'£>, (210)(21)*F, (Hl)(IO) 6 Fi
FOR W H I C H J
6
(in)
= I
Di
(20) 1 D1
(210) (21) 'F 1
(111) (10) "F1
1 2 2 3 2 3
5 3 3 4 0 0
1 0 2 0 2 3
5 3 3 0 3 4
To conclude this section, we indicate how advantage can be taken of the existing90 spinorbit matrices for / 3 in order to minimize the labor in constructing some matrix elements of H2 for / 6 . We select the levels of / 6 for which J = 1,>S>2, L > 2 ; b y choosing these special cases, the ground is prepared for the detailed illustration of the hyperfine structure of PuI given in Sec. 89. Table 84 indicates the order of constructing the matrix elements. Apart from the zeros, which correspond to vanishing matrix elements, the numbers in the table show which matrix elements derive from a common equation;
CONFIGURATIONS
OF MORE THAN TWO EQUIVALENT ELECTRONS
203
for example, the first entries are made by performing the summation (756) once and then using
where 85.
is independent of U and L.
TABLE
85
THE
MATRIX
The results are given in Table
ELEMENTS
)
FOR W H I C H
(All entries must be multiplied by (100)
(210)
(210)
(111)
(210)
(10)
(20)
(21)
(20)
(21)
'F,
6 Z>I
(W0)(10)'F, (210)(20)6DI (210)(21
5D
(1U)(20
YDi
(210)(21)6FI (111)(10)*F,
85

)
(
W
 ( H ) *
5 — TTT 5(33)1/21
6Dt
'F i
( i n ) (10) 6
FI
(¥)*
5(14)1/84
5(33)1/21 95 T T 5(462)1/42
0
(if)*
 ( f ) i
( A V ) i
5(14)1/84 5(462)1/42 
A 0 0
0
(«)» 0 2
(•?)* (xVff)i 0
 w 2 s
MATRIX ELEMENTS OF V vanish. But it c a n b e proved that, under these conditions, some matrix elements of are nonzero; indeed,
and T h e r e f o r e S i m i l a r arguments to those which led to now yield thus completing the proof that, for the matrix elements of , the set of symbols can be omitted from the description of the states. 86
THE C O U L O M B INTERACTION H1
It has been seen that the tensor whose scalar component is proportional to the spinorbit interaction, can itself be interpreted as a component of a generalized tensor. The situation for twoparticle operators is more complex. If we express Hi in terms of the equivalent operator
then we are confronted with the difficulty that the scalar products (v (210)(21) 5 D (111)(20)®D
In constructing this table, we can augment
SOME M A T R I X
ELEMENTS
(100X10 yF (100X10)'*'
SD (210)(20 (210)(20))'D
(210)(21)»D
(lll)(20) t 2) (111)(20)«2)
 Ww * ( i f a ) }l 3(f£)l 0
335(2)1/196 15( 66)1/98 15(66)1/98 0
 3 ( fff ) l 15(66)1/98 15(66)1/9 8 405(2)1/196 0
0 00 00 85(2)1/28 85(2)1/28
CONFIGURATIONS
OF MORE THAN TWO EQUIVALENT ELECTRONS
Eq. (754) with equations of the type introduced in Sec. 83. example, and we find
219
For
Since we know that constants A i and A2 independent of L and L' can be found such that since the two sets of matrix elements on the right are linearly independent. It turns out, rather surprisingly, that Moreover, the entry in Table 88 involving the state can also be included in the general formula (838) Some understanding of simplifications of this kind can be obtained by studying the symmetry properties of the corresponding determinantal product states;104 a grouptheoretical interpretation is yet to be given. Equation (838) embraces all the quintets of and all the terms of except The vanishing of between states of different seniority in corresponds to the vanishing of V(2) between states of different spin in Since all the matrix elements of are known in it is a trivial matter to write down the matrix elements of within the quintets of Matrix elements of this type other than those given in Table 88 would be needed in a study of the hyperfine structure of To complete the calculation, we use
The following 9j symbols are required:
220
OPERATOR TECHNIQUES
IN ATOMIC
SPECTROSCOPY
We find and hence P, defined in Eq. (837), is given by
Thus the nuclear moment of 0.021 nuclear magneton, which was estimated on the assumption of a pure 7 Fi level, should be corrected to 0.061 nuclear magneton. The simple treatment given here has made allowance for only one excited level; and although its effect is makes this certainly the most important, the small value of quantity very sensitive to the influence of other levels. If, for example, these additional perturbationsreduce by only a third of the amount that those due to do, which is by no means unreasonable, then the figure of 0.061 nuclear magneton must be more than doubled. It is probable that the remaining discrepancy between the experimental results of Hubbs et al. and Bleaney et al. could be resolved along these lines. PROBLEMS 81. Justify the existence of a constant A, independent of V, L, in the equation
and
where Prove and generalize the equation to relate matrix elements of V between states of maximum S in the configurations
82. Prove that for irreducible representations
where the sum runs over all integral values of x, y, and z satisfying
for (See Construct Use all this 88. Racah.") equation Prove a table that, in ,for and conjunction when with i isand even, theprove factorization (712) to derive Eq. (814).
CONFIGURATIONS
OF MORS
THAN TWO EQUIVALENT ELECTRONS
221
84. Use Eq. (69) to prove
for all
where
denotes the term of
for which
Hence show that Deduce that for all
, S, and
86. By considering the matrix elements of the commutator
between the states
and
prove that
for all 86. Prove that
if Check that this equation is consistent with the results of Prob. 81. 87. Obtain the following analogue to Eq. (838):
where 88. The terms of the halffilled shell are separated into two classes; for all members of the first, is odd, while for those of the second, it is even. Prove that there are no nonvanishing matrix elements of between terms of different classes if is odd and none between terms of the same class if is even. 89. Verify the formula
The lowest configuration of Eul is and the ground level is Indicate how contributions to the term Al • J in the effective Hamiltonian for the hyperfine interaction can arise through admixtures of the levels and in the ground level, and prove that, near the LS limit, the contribution to A is
where E t and are the energies of and above respectively. 810. For d electrons, the parameters of Condon and Shortley1 are related
222
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
Prove that the Coulomb interaction within a shell of equivalent d electrons can be replaced by the operator eeEo
where
and
+ elEl + elE'
L (v,(OI • v;!OI) i>i el = L [7(v,(OI. vPI) + 2(Vi(21 • V,(I,) + 2(V,(4) • vp»] i>i e! = L [18(v,(21 • v,i
with
and Show that eo, el, and e. are the components of generalized tensors and may be labeled by WL = (00)0, (00)0, and (22)0, respectively, where W denotes an TABLE
89
THE NUMBERS
c(WW'(22»
W' W (00)
(10)
(11)
(20)
(21)
(22)
             (00) 1 (10) (11) (20) (21) (22)




1

1
1
1
1 1

1 1 1
1 1 1 2 1
1 1
1 1 1
irreducible representation of R i • Prove that both eo and el are diagonal in a scheme in which the states are taken to be of the type Id"VSMsLML) and that their matrix elements are given by tn(n  1) and 7(n  v) 2
respectively. the operator
+ v(v + 2)
_ S(S
4
+ 1)
Obtain the values of c(WW'(22» given in Table 89.
Show that
0' = 7(V(11)'  3(V(31)2
is the scalar component of a tensor transforming according to (22) of R 6, and deduce that, with the possible exception of W = (21), parameters A independent of L exist such that
CONF/GURAT/ONS OF MORE THAN TWO EQUIVALENT ELECTRONS 223 where W = (voiwt). Prove that, for all representations W and W [including the case of W = W = (21)], the factorization (d»WSL\et + 0'IdTTSL) =
B(d»,WW'S)(WML)\W)
can be carried out. 811. Extend the techniques of Sec. 86 to double tensors, and construct two operators whose descriptions in terms of irreducible representations of Spu, R1, G2, and Rt are (1111000)(220)(22)5 and (2200000)(220)(22)S respectively. Show that the eigenvalues of these operators for the seven terms of P are proportional to those of the operators ei + Ω and Ω, respectively, in the notation of Sec. 86. Construct the table of numbers c((v)(v')(i)), and prove that the selection rules on e% + Ω are given by Av = 0, ± 2 , ± 4 Prove also that all matrix elements of e> + Ω within the manifolds of states for which ν = 6 or 7 are zero, and show that the dependence of the matrix elements (/WrSLIe 3
+a\pv'UVSL)
on U, U', τ, τ', S, and L is the same for all values of n. Use the orthogonality property of states characterized by the labels (1111000) and (2200000) to derive the equation 7i[S][L](f»vWUrSLIe3 + a\pvWUrSL)(
UL\0\ UL) = 0
where the sum runs over W, U, τ, , 14, 15, 147, 197, 201, 209, 210 /«, 122, 123, 144, 210 /*, 210, 221 /«, 24, 194, 210220 P, 24, 221 g',90 g", 148 1 I , 7691 Contact interaction, 85, 87 Continuous groups, 96 representation for, 113118, 147149 Conway, J. G., 84, 211, 212, 217, 234, 236 Coulomb interaction, 5, 7981, 204212, 222, 223 Coupling, of four angular momenta, 63
Direct product, 101, 157 Double group, 38 Double tensor operators, 154158 commutation relations for, 156 Dreitlein, J., 235 Dvir, M., 52, 234
LS, 21, 82, 93, 94 of three angular momenta, 54 Coupling diagrams, 6269, 74, 75, 92 Crystalfield parameters, 47, 51 Crystallographic group (see Group) Curl, R. F., 150, 235
Eckart, C , 233 Edmonds, A. R., 7, 78, 233, 234 Effective Hamiltonian, 162 Eigenfunctions, hydrogenic, 80, 81, 90, 211, 212, 225, 226 radial, 3 Eisenhart, L. P., 235 Eisenstein, J. C , 49, 23. Electron spin, 3, 48 Elliott, J. P., 62, 63, 65, 67, 151, 192, 234236 Equivalence, operator, 43, 46 Equivalent factor, operator, 43, 91, 214 Euler angles, 35, 97, 98 Europium atom, 221 Europium bromate, 215, 216 Europium chloride, 215, 216 Europium ethylsulfate, 213, 215, 216 Europium ion, 24, 194, 211216 External magnetic field, 44, 84, 85 Hamiltonian for, 44, 84
Dagys, R., 235 Daughters, 168 Degeneracies, LaportePlatt, 162165 Degeneracy, accidental, 36 Determinantal product states, 35, 1418, 2124 Dicesium zirconium hexafluoride, 50 Dieke, G. H., 194, 234, 236 Dimension of representations, 130, 131 for G1, 147 for β 6 , 131 for R,, 151 for Ri, 131 for Spt„ 153 for Sp 4 I +5 , 165 for SUa+I, 131 for Ua+i, 153 Dirac, P. A. M., 5, 7, 165, 233
Fano, U., 234 Feenberg, E., 233 Fermi, E., 85, 235 Fifteen,? symbol, 75 Figures, root, 109112 Finite group, 28 Flowers, B. H., 153, 234, 235 Fractional parentage coefficients, 170192 for d", 191 explicit calculation of, 182185, 189, 190 for/», 174182 for p 8 , 171173 tabulation of factors, 227232 Fred, M., 234 Freed, S., 213, 216, 234, 236 Fried, B., 90, 235
ii, 2i
/NDEX
239
Fronsdal, C , 235 Full linear group, GLsi+i, 103
Gruber, J. B., 211, 212, 236 Gruen, D. M., 234
g factor, Land6, 45, 84, 85 G 2 group (see Group) Gaunt, J. A., 78, 234 Gerstenkorn, S., 217, 236 Glembockij, J., 226 Godparent, 190 Goldstein, H., 97, 235 Goodman, G. L., 234 Goudsmit, S., 166, 235 Griffin, J. J., 233 Griffith, J. S., 205 Group, Abelian, 28, 100 Cartan's notation for, 112 continuous, 96 crystallographic, C1, 52 O, 3134, 45, 205 double, 38 finite, 28 full linear, GLa+i, 103 G2, of Cartan, 106, 111, 112, 118, 124, 126, 131, 146149, 196, 223 rotation, R,, 3436, 97, 98, 105, 223 Rt, 151, 152, 162, 189 K5, 108, 113118, 131, 132 R„ 151 R1, 106, 131, 223 R„ 148 B 2 I x I. 105, 111, 112, 124, 125, 195 S, of Weyl, 116, 128, 129 special unitary, SUi, 157 SiJ 3 , 110, 118 SUiW, 104, 109, 112, 118, 124, 125 SUiW, 157 symmetric, St, 33, 121 Sn, 122, 135 symplectic, Sp 4 , H l Spiy, 105, 111, 112 Spu+2, 158, 223 unitary, Us, 150 Ue, 151 Ui, 117, 148 Uii+i, 103, 109, 116118, 123 l/41+ί, 156, 157, 195 (See also Branching rules; Characters; Dimension of representations; Kronecker product)
Hall, G. R., 234, 236 Hamermesh, M., 235 Hamiltonian, approximate, 2 for configuration / 2 , 76, 77 effective, 162 for external magnetic field, 44, 84 for ion, in crystal lattice, 25, 214 in cube of charges, 52 in octahedron of charges, 27 for magnetic hyperfine interaction, 85, 221 for magnetic spinspin interaction, 87 for quadrupole interaction, 91 spin, 45, 53 for spinorbit interaction, 6 for spinotherorbit interaction, 93 Hayes, W., 46, 234 Helium atom, 89 Hellwege, A. M., 234 Hellwege, K. H., 215, 216, 234, 236 Heroux, L., 236 Hobson, E. W., 235 Hope, J., 234 Horie, H., 67, 93, 234 Howell, K. M., 58, 234 Hubbs, J. C , 216, 218, 220, 236 Hund's rule, 212 Hutchison, C. A., Jr., 234 Hydrogenic eigenfunctions, 80, 81, 90, 211, 212, 225, 226 Hyperfine interaction, magnetic, 8587, 216221 Hypergeometric function, 226 Identity, Jacobi, 149 Infinitesimal operator, 99, 103, 195, 218 Innes, F. R., 66,89,92,156,165,234,235 Integrals, Slater, 80, 90, 162, 226 Interaction, Coulomb, 5, 7981, 204212, 222 magnetic hyperfine, 8587, 216221 quadrupole, 91 spinorbit (see Spinorbit interaction) spinotherorbit, 93, 94 spinspin, 8791, 94, 223
240
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
Interval rule (see Landi interval rule) Invariant subgroups, 100
Low, W., 52, 234 Lyubarskii, G. Ya., 152
Jacobi identity, 149 Jahn, H. A., 58, 191, 234236 Jeffries, C. D., 234 Johnsen, TJ., 236 Jucys, A., 235 Judd, B. R., 233, 236
McLaughlin, R. D., 83 McLellan, A. G., 198, 236 Magnesium ion, 91 Magnetic field, external, 44, 84, 85 Margolis, J. S., 234 Marrus, R., 236 Marvin, H. H., 93, 235 Matrix elements, reduced, 42, 77, 78, 155, 186188, 203 for tensor operations, 72, 73 Mauza, E., 205 Metric tensor, 106, 107 Metropolis, N., 13, 233 Mixed tensor operators, 6973 Mobius strip, 64, 65, 74, 75 Modification rule, 141, 161 Moore, C. E., 233 Multiplet, 9 Murnaghan, F. D., 141, 235
Kahle, H. G., 215, 216, 236 Kaiser, W., 194 Kamimura, H., 205 Kilpatrick, J. E., 150, 235 Klinkenberg, P. F. A., 234 Konig, D., 234 Korostyleva, L. A., 216 Koster, G. F., 45, 234 Kronecker product, 30, 34, 131136 for G2, 197, 198, 205 for O, 34 for R3, 36, 42, 132, 143145, 198 for Ri, 132, 134, 222 for K7, 199, 205, 220 for Si, 122 for Spn, 199 for Spu+t, 197 for SUa+i, 134136 for !/«+ι, 134136, 143, 144 for 1/41+2, 168, 169
Lammermann, H., 217, 236 Lando g factor, 45, 84, 85 Lando interval rule, 82, 89, 91, 93 deviations from, 94, 214 Lane, A. M., 62, 234 Lanthanum ion, 76 Laporte, 0., 162, 235 LaportePlatt degeneracies, 162165 Lattice permutation, 136 Lee, W., 235 Level, definition of, 6 Lie, S., 96, 99, 235 Littlewood, D. E., 135, 136, 235 Llewellyn, P. M., 234, 236 Lomont, J. S., 28, 233 Loudon, R., 236
Neodymium crystals, 210 Neptunium hexafiuoride, 4653 Neptunyl ion, 52, 53 Nielson, C. W., 227 Nierenberg, W. A., 236 Ninej symbol, definition of, 60 relation to other nj symbols, 61, 6469 use in recoupling tensor operators, 87 Nutter, P. B., 227 Obi, S., 93 Offspring, 168 Operator equivalence, 43, 46 Operator equivalent factor, 43, 91, 214 Operators, Casimir's, 123126, 162, 163, 208 doubletensor, 154158 infinitesimal, 99, 103, 195, 218 rotation, 37, 127 shift, 7, 115 (See also Quantization) singleparticle, 16, 167 tensor (see Tensor operators) twoparticle, 16, 17, 167, 168, 225
INDEX
OrdSmith, R. J., 234 Orton, J. W., 91, 235 Owen, J., 91, 235 Pake, G. E., 233 Parents, 168 Partition, 119 Pauli exclusion principle, 3, 119 Permutation, of electrons, 3 lattice, 136 Piatt, J. R., 162, 235 Plutonium atom, 24, 194, 216220 Plutonium ion, 216, 217 Plutonyl ion, 76 Positive roots, 115, 116 Praseodymium ion, 76, 83, 84, 90, 211 Product, direct, 101, 157 Product states, determinantal, 35, 1418, 2124 Pryce, M. H. L., 49, 234, 236 Quadrupole interaction, 91 Quantization, change of, 9,1315, 22, 24 Racah, G., 1, 42, 84, 96, 100, 101, 117, 124, 125, 149, 160, 161, 166, 169, 174, 176, 179, 180, 190, 191, 198, 201, 204, 206208, 210, 220, 227, 234, 235 Radial eigenfunctions, 3 Radial integrals, 77, 225, 226 Ramsey, N. F., 44, 234 Reciprocity, 179, 180 Recoupling of tensor operators, 8588, 205 Redmond, P. J., 189, 236 Reduced matrix element, 42, 72, 73,77, 78, 155, 186188, 203 Regge, T., 22, 23, 58, 61, 233, 234 Reilly, E. F., 210, 236 Representation, 28, 29, 35 for continuous groups, 113118, 147149 (See also Branching rules; Characters; Dimension of representations; Kronecker product) Ridley, E. C , 90, 235
241
Root figures, 109112 Roots, 108 positive, 115, 116 Rose, M. E., 234 Rotation group (see Group) Rotation operators, 37, 127 Rotenberg, M., 13, 57, 58, 80, 233 Runciman, W. A., 84, 210, 234, 236 RussellSaunders (LS) coupling, 21, 82, 93,94 Rutherford, D. E., 122, 235
S group of Weyl, 116, 128, 129 SainteLague, A., 234 Salpeter, E. E., 233 Samarium atom, 194, 211 Samarium ion, 194 Sancier, K. M., 234 Sarup, R., 194, 234 Satten, R. A., 209, 234, 236 Sayre, E. V., 213, 216, 234, 236 Schaack, G., 236 Scheffers, G., 235 Schouten, J. A., 112, 235 Schur's lemma, 29 Schwartz, C , 77, 234 Schwinger, J., 234 Selection rules, 87, 89, 198, 223 Seniority, 160162, 223 Sequences of subgroups, 104106, 157, 158 Shape, 119 Sharp, R. T., 23, 233 Shift operator, 7, 115 (See also Quantization) Shortley, G. H., 1, 17, 21, 76, 80, 90, 93, 210, 221, 233, 235 Singleparticle operator, 16, 167 Sixj symbol, definition of, 56 relation to other nj symbols, 57, 60, 6473 special values of, 58, 59, 74 symmetry properties of, 57 Slater, J. C , 17, 233 Slater integrals, 80, 90, 162, 226 Space, weight, 108 Special unitary group (see Group) Spedding, F. H., 234 Speiser, A., 233
242
OPERATOR TECHNIQUES IN ATOMIC SPECTROSCOPY
Spherical harmonics, addition theorem for, 26, 79 as basis functions, 35 definition of, 3 use in defining tensor operator, 41 Spin of electron, 3, 48 SpinHamiltonian, 45, 53 Spinorbit interaction, 6, 8184, 198203 coupling constant for, definition of, 18 Hamiltonian for, 6 Spinotherorbit interaction, 93, 94 Hamiltonian for, 93 Spinspin interaction, 8791, 94, 223 magnetic, Hamiltonian for, 87 Standard tableau, 119 Stapleton, H. J., 234 Statz, H., 45, 234 Stevens, K. W. H., 24, 44, 46, 233 Structure constants, 99101 Subgroups, 35, 100 invariant, 100 sequences of, 104106, 157, 158 Sum rules, 6367, 75 Symmetric group (see Group) Symmetry properties, of six/ symbol, 57 of three/ symbol, 12, 23 Symplectic group (see Group) Tableaux, 119123, 135 Tanabe, Y., 67, 205, 234 Tensor, metric, 106, 107 Tensor operators, commutation relations for, 42, 102, 151 commutators of, 102, 156 definition of, 41, 42 double, 154158 generalized, 194196, 218, 223 mixed, 6973 recoupling of, 8588, 205 reduced matrix elements for, 72, 73 regarded as infinitesimal operators, 103, 104, 108, 157 Term, 8 Thorium ion, 76, 83, 84 Three/ symbol, definition of, 12 relation to six/ symbol, 57, 73 special formulas for, 22, 78
Three/ symbol, symmetry properties of, 12, 23 Trees, R. E., 77, 91. 234 Triangular condition, 42, 62, 89 Tsang, T., 234 Tubis, A., 226 Twelve/ symbol, 64, 65, 74, 75 Twoparticle operator, 16, 17, 167, 168, 225 Ufford, C. W., 66, 92, 93, 156, 234 Unitary group («ee Group) Uranium ion, 76, 83, 84 Uranyl rubidium nitrate, 216 Uspalis, K., 226 Vanagas, V., 226 Van der Waerden, B. L., 112, 235 Van Vleck, J. H., 93 Vectorcoupling coefficients, 10 calculation of, 1012 formulas for, 12, 23 (See also Three/ symbol) Vizbaraite, J., 235 Weight, 114, 116, 152, 161, 192 Weight space, 108 Weinetock, B., 234 Weyl, H., 29, 107, 116, 128, 133, 233, 235 Wigner, E. P., 28, 233 WignerEckart theorem, 3941 for doubletensor operators, 154, 158 for Gt, 196, 197, 207 for O, 4446 for Ri, 41, 42, 73, 86 Wood, D. L., 194 Wooten, J. K., 13, 233 Worcester, J. L., 236 Wybourne, B. G., 84, 210, 227, 234, 236 Yamanouchi, T., 122, 235 Yanagawa, S., 93, 235 Young, A., 119, 122 Young, D., 234 Young tableaux, 119123, 135 Zvironaite, S., 235