Silver / Loebl Irreducible Tensor Methods

An Introduction for Chemists
1. Auflage 2013
ISBN: 978-1-4831-9181-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 246 Seiten, Web PDF

Irreducible Tensor Methods: An Introduction for Chemists explains the theory and application of irreducible tensor operators. The book discusses a compact formalism to describe the effect that results on an arbitrary function of a given set of coordinates when that set is subjected to a rotation about its origin. The text also explains the concept of irreducible tensor operators, particularly, as regards the transformation properties of operators under coordinate transformations, and, in a special way, the group of rotations. The book examines the systematic construction of compound tensor operators from simple operators to classify the behavior of any operator under coordinate rotations. This classification is a significant component of the irreducible tensor method. The text explains the use of the 6-j and 9-j symbols to complete theoretical concepts that are applied in irreducible tensor methods dealing with problems of atomic and molecular physics. The book describes the matrix elements in multielectron systems, as well as the reduced matrix elements found in these systems. The book is suitable for nuclear physicists, molecular physicists, scientists, and academicians in the field of quantum mechanics or advanced chemistry.

Autoren/Hrsg.

Silver, Brian L.

Loebl, Ernest M.

Inhaltsverzeichnis

1;Front Cover;1
2;Irreducible Tensor Methods: An Introduction for Chemists;5
3;Copyright Page;6
4;Table of Contents;9
5;Dedication;7
6;Preface;15
7;Introduction;17
8;PART I;21
8.1;Chapter 1. The Rotation Operator;21
8.1.1;1.1 COORDINATE ROTATIONS;21
8.1.2;1.2 THE EULER ANGLES;22
8.1.3;1.3 THE INFINITESIMAL ROTATION OPERATOR;25
8.1.4;1.4 TRANSFORMED FUNCTIONS;27
8.1.5;1.5 THE ROTATION OPERATOR FOR ONE AXIS;28
8.1.6;1.6 THE ROTATION OPERATOR;30
8.1.7;1.7 SOME MISCONCEPTIONS;31
8.1.8;1.8 ROTATIONS IN SPIN SPACE;31
8.1.9;1.9 AN EXAMPLE;32
8.1.10;1.10 THE INVERSE ROTATION OPERATOR;32
8.1.11;1.11 ROTATION OF FUNCTIONS;32
8.1.12;1.12 ROTATION OF OPERATORS;33
8.1.13;1.13 COMMENTS ON THE ROTATION GROUP;33
8.1.14;1.14 COMMENTS ON LIE GROUPS;34
8.1.15;1.15 CONVENTIONS;35
8.2;Chapter 2. The Wigner Rotation Matrices;36
8.2.1;2.1 THE ROTATION MATRICES;36
8.2.2;2.2 QUESTIONS OF PHASE;38
8.2.3;2.3 THE FORMS OF D(1/2) AND D(1);40
8.2.4;2.4 PROPERTIES OF THE ROTATION MATRICES;42
8.2.5;2.5 THE TRANSFORMATION OF COMPONENTS OF TENSORS;45
8.2.6;2.6 ANOTHER LOOK AT D(1/2) ;47
8.2.7;2.7 CONVENTIONS;48
8.3;Chapter 3.
The Coupling of Two Angular Momenta;50
8.3.1;3.1 INTRODUCTORY EXAMPLES;50
8.3.2;3.2 THE VECTOR-COUPLING COEFFICIENTS;53
8.3.3;3.3 A COMMENT ON PHASE;55
8.3.4;3.4 THE EVALUATION AND PROPERTIES OF THE VC COEFFICIENTS;56
8.3.5;3.5 THE 3-j SYMBOL;58
8.3.6;3.6 EVALUATION OF THE 3-j SYMBOLS;59
8.3.7;3.7 THE CLEBSCH–GORDAN SERIES;60
8.3.8;3.8 TWO USEFUL INTEGRALS;63
8.3.9;3.9 REGGE SYMMETRIES;65
8.3.10;3.10 THE V COEFFICIENT;66
8.3.11;3.11 A FINAL COMMENT;66
8.4;Chapter 4. Scalars, Vectors, Tensors;67
8.4.1;4.1 VECTORS;67
8.4.2;4.2 CARTESIAN TENSORS;67
8.4.3;4.3 IRREDUCIBLE SPHERICAL TENSORS;68
8.4.4;4.4 IRREDUCIBLE CARTESIAN TENSORS;69
8.4.5;4.5 IRREDUCIBLE TENSOR FIELDS;69
8.4.6;4.6 SCALARS;70
8.5;Chapter 5. Irreducible Tensor Operators;71
8.5.1;5.1 DEFINITION OF IRREDUCIBLE TENSOR OPERATORS;71
8.5.2;5.2 AN EXAMPLE;74
8.5.3;5.3 RACAH'S COMMUTATION RELATIONS;75
8.5.4;5.4 SCALAR AND VECTOR OPERATORS;75
8.5.5;5.5 A LIE GROUP;77
8.5.6;5.6 THE CONSTRUCTION OF COMPOUND IRREDUCIBLE TENSOR OPERATORS;77
8.5.7;5.7 SCALAR OPERATORS;82
8.5.8;5.8 STANDARD BASIS VECTORS;83
8.5.9;5.9 ANOTHER PHASE CONVENTION;83
8.5.10;5.10 COMMENT ON CONTRAGREDIENCE;83
8.5.11;5.11 ADJOINT TENSOR OPERATORS;85
8.6;Chapter 6. The Wigner-Eckart Theorem;86
8.6.1;6.1 INTRODUCTION;86
8.6.2;6.2 PROOF OF THE WIGNER–ECKART THEOREM;87
8.6.3;6.3 COMMENTS ON AND CONSEQUENCES OF THE THEOREM;89
8.6.4;6.4 PARITY;91
8.6.5;6.5 SELECTION RULES;92
8.6.6;6.6 SUM RULES;93
8.6.7;6.7 COMMENT ON POINT GROUPS;94
8.7;Chapter 7. The 6–j Symbol;95
8.7.1;7.1 INTRODUCTION;95
8.7.2;7.2 RECOUPLING;96
8.7.3;7.3 PROPERTIES OF THE 6-j SYMBOL;99
8.7.4;7.4 INVARIANCE OF THE 6-j SYMBOL;102
8.7.5;7.5 REGGE SYMMETRIES;102
8.7.6;7.6 A WARNING;102
8.8;Chapter 8. The 9–j Symbol;103
8.8.1;8.1 DEFINITION OF THE 9-j SYMBOL;103
8.8.2;8.2 PROPERTIES OF THE 9-j SYMBOL;104
8.8.3;8.3 THE RECOUPLING OF OPERATORS;106
8.8.4;8.4 INVARIANCE OF THE 9-j SYMBOL;107
8.9;Chapter 9. The Matrix Elements of Irreducible Tensor Operators;108
8.9.1;9.1 INTRODUCTION;108
8.9.2;9.2 DERIVATION OF THE BASIC FORMULA;108
8.9.3;9.3 THE REDUCED MATRIX ELEMENTS OF ITOs;111
8.9.4;9.4 DOUBLE-TENSOR OPERATORS;113
8.9.5;9.5 COMMENTS ON THE BASIC EQUATIONS;116
9;PART II;119
9.1;Chapter 10. The Coulomb Interaction;119
9.1.1;10.1 THE SPHERICAL HARMONIC ADDITION THEOREM;119
9.1.2;10.2 THE COULOMB SPLITTINGS FOR p2;121
9.2;Chapter 11. Spin-Orbit Coupling;124
9.2.1;11.1 THE MATRIX ELEMENTS OF THE SPIN-ORBIT HAMILTONIAN;124
9.2.2;11.2 THE SPIN–ORBIT ENERGIES FOR THE 3d2 CONFIGURATION;126
9.3;Chapter 12. The Magnetic Dipole-Dipole Interaction;129
9.3.1;12.1 THE DIPOLE-DIPOLE HAMILTONIAN;129
9.3.2;12.2 AN EXAMPLE;130
9.4;Chapter 13. Spin-Spin Couplings;133
9.5;Chapter 14. The Electronic Zeeman Interaction;136
9.6;Chapter 15. Operator Equivalents;139
9.6.1;15.1 OPERATOR EQUIVALENTS;139
9.6.2;15.2 OFF-DIAGONAL OPERATOR EQUIVALENTS;141
9.7;Chapter 16. Real Tensorial Setsin R3—Cartesian Tensors;144
9.8;Chapter 17. Some Multipole Expansions;148
9.8.1;17.1 INTRODUCTION;148
9.8.2;17.2 PLANE WAVES;148
9.8.3;17.3 ELECTRONIC MULTIPOLE MOMENTS;149
9.8.4;17.4 THE PARITY OF THE MULTIPOLE OPERATORS;152
10;PART III;153
10.1;Chapter 18. Racah Algebra for Point Groups;153
10.1.1;18.1 INTRODUCTION;153
10.1.2;18.2 QUESTIONS OF PHASE;154
10.1.3;18.3 BASIS FUNCTIONS;154
10.1.4;18.4 COUPLING COEFFICIENTS FOR POINT GROUPS;155
10.1.5;18.5 THE V COEFFICIENTS;158
10.1.6;18.6 DIHEDRAL GROUPS;161
10.1.7;18.7 A FURTHER COMMENT ON PHASE;161
10.1.8;18.8 THE W COEFFICIENTS;162
10.1.9;18.9 THE X COEFFICIENT;164
10.2;Chapter 19. Operators and Matrix Elements;167
10.2.1;19.1 IRREDUCIBLE TENSOR OPERATORS;167
10.2.2;19.2 THE WIGNER–ECKART THEOREM;168
10.2.3;19.3 MATRIX ELEMENTS AND RMEs OF COMPOUND TENSOR OPERATORS;169
10.2.4;19.4 DOUBLE-TENSOR OPERATORS;171
10.2.5;19.5 THE RME OF A DOUBLE-TENSOR OPERATOR;173
10.2.6;19.6 SPIN–ORBIT COUPLING;174
10.3;Chapter 20. Spinor Groups;177
10.3.1;20.1 INTRODUCTION;177
10.3.2;20.2 V AND W COEFFICIENTS FOR O;178
10.3.3;20.3 THE WIGNER–ECKART THEOREM;179
10.3.4;20.4 AN EXAMPLE;180
10.3.5;20.5 BASES FOR REPEATED REPRESENTATIONS;182
10.4;Chapter 21. Matrix Elements in Multielectron Systems;183
10.4.1;21.1 INTRODUCTION;183
10.4.2;21.2 COEFFICIENTS OF FRACTIONAL PARENTAGE;184
10.4.3;21.3 VALUES OF CFP;188
10.4.4;21.4 MATRIX ELEMENTS IN MANY-ELECTRON SYSTEMS;188
10.5;Chapter 22. Reduced Matrix Elements in Multielectron Systems;191
10.5.1;22.1 INTRODUCTION;191
10.5.2;22.2 SPIN-INDEPENDENT ONE-ELECTRON OPERATORS;192
10.5.3;22.3 SPIN-DEPENDENT ONE-ELECTRON OPERATORS–SPIN–ORBIT COUPLING;194
10.5.4;22.4 UNIT TENSORS;196
11;PART IV;199
11.1;Chapter 23. Spin-Orbit Coupling in a Low-Spin d5 Complex;199
11.2;Chapter 24. Further Examples of Spin-Orbit Coupling;201
11.2.1;24.1 SPIN–ORBIT COUPLING IN THREE OPEN SHELLS;201
11.2.2;24.2 SPIN–ORBIT COUPLING FOR A DIHEDRAL GROUP;203
11.3;Chapter 25. Electric Dipole Transitions in a Tetrahedral Complex;206
11.4;Chapter 26. Second Quantization;209
11.4.1;26.1 OPERATORS;209
11.4.2;26.2 REDUCED MATRIX ELEMENTS;212
11.5;Chapter 27. Photoelectron Spectra of Open-Shell Molecules;215
12;PART V;217
12.1;Chapter 28. Vector Fields;217
12.1.1;28.1 INTRODUCTION;217
12.1.2;28.2 THE TRANSFORMATION OF VECTOR FIELDS UNDER ROTATIONS;218
12.1.3;28.3 EIGENVECTORS OF THE ROTATION OPERATOR FOR A VECTOR FIELD;225
12.2;Chapter 29. Light;229
12.2.1;29.1 MULTIPOLE EXPANSION OF POLARIZED LIGHT;229
12.2.2;29.2 THE COHERENCY MATRIX;232
12.3;Chapter 30. Light Scattering;235
13;References;239
14;Index;243
15;Physical Chemistry;247