Irreducible Tensor Methods
An Introduction for Chemists
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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.
Table of Contents
Preface
Introduction
Part I
Chapter 1 The Rotation Operator
1.1 Coordinate Rotations
1.2 The Euler Angles
1.3 The Infinitesimal Rotation Operator
1.4 Transformed Functions
1.5 The Rotation Operator for One Axis
1.6 The Rotation Operator
1.7 Some Misconceptions
1.8 Rotations in Spin Space
1.9 An Example
1.10 The Inverse Rotation Operator
1.11 Rotation of Functions
1.12 Rotation of Operators
1.13 Comments on the Rotation Group
1.14 Comments on Lie Groups
1.15 Conventions
Chapter 2 The Wigner Rotation Matrices
2.1 The Rotation Matrices
2.2 Questions of Phase
2.3 The Forms of D(1/2) and D(1)
2.4 Properties of the Rotation Matrices
2.5 The Transformation of Components of Tensors
2.6 Another Look at D(1/2)
2.7 Conventions
Chapter 3 The Coupling of Two Angular Momenta
3.1 Introductory Examples
3.2 The Vector-Coupling Coefficients
3.3 A Comment on Phase
3.4 The Evaluation and Properties of the VC Coefficients
3.5 The 3-j Symbol
3.6 Evaluation of the 3-j Symbols
3.7 The Clebseh-Gordan Series
3.8 Two Useful Integrals
3.9 Regge Symmetries
3.10 The ^Coefficient
3.11 A Final Comment
Chapter 4 Scalars, Vectors, Tensors
4.1 Vectors
4.2 Cartesian Tensors
4.3 Irreducible Spherical Tensors
4.4 Irreducible Cartesian Tensors
4.5 Irreducible Tensor Fields
4.6 Scalars
Chapter 5 Irreducible Tensor Operators
5.1 Definition of Irreducible Tensor Operators
5.2 An Example
5.3 Racah's Commutation Relations
5.4 Scalar and Vector Operators
5.5 A Lie Group
5.6 The Construction of Compound Irreducible Tensor Operators
5.7 Scalar Operators
5.8 Standard Basis Vectors
5.9 Another Phase Convention
5.10 Comment on Contragredience
5.11 Adjoint Tensor Operators
Chapter 6 The Wigner-Eckart Theorem
6.1 Introduction
6.2 Proof of the Wigner-Eckart Theorem
6.3 Comments on and Consequences of the Theorem
6.4 Parity
6.5 Selection Rules
6.6 Sum Rules
6.7 Comment on Point Groups
Chapter 7 The 6-j Symbol
7.1 Introduction
7.2 Recoupling
7.3 Properties of the 6-j Symbol
7.4 Invariance of the 6-j Symbol
7.5 Regge Symmetries
7.6 A Warning
Chapter 8 The 9-j Symbol
8.1 Definition of the 9-j Symbol
8.2 Properties of the 9-j Symbol
8.3 The Recoupling of Operators
8.4 Invariance of the 9-j Symbol
Chapter 9 The Matrix Elements of Irreducible Tensor Operators
9.1 Introduction
9.2 Derivation of the Basic Formula
9.3 The Reduced Matrix Elements of ITOs
9.4 Double-Tensor Operators
9.5 Comments on the Basic Equations
Part II
Chapter 10 The Coulomb Interaction
10.1 The Spherical Harmonic Addition Theorem
10.2 The Coulomb Splittings for p2
Chapter 11 Spin-Orbit Coupling
11.1 The Matrix Elements of the Spin-orbit Hamiltonian
11.2 The Spin-orbit Energies for the 3d2 Configuration
Chapter 12 The Magnetic Dipole-Dipole Interaction
12.1 The Dipole-Dipole Hamiltonian
12.2 An Example
Chapter 13 Spin-Spin Couplings
Chapter 14 The Electronic Zeeman Interaction
Chapter 15 Operator Equivalents
15.1 Operator Equivalents
15.2 Off-Diagonal Operator Equivalents
Chapter 16 Real Tensorial Sets in R3-Cartesian Tensors
Chapter 17 Some Multipole Expansions
17.1 Introduction
17.2 Plane Waves
17.3 Electronic Multipole Moments
17.4 The Parity of the Multipole Operators
Part III
Chapter 18 Racah Algebra for Point Groups
18.1 Introduction
18.2 Questions of Phase
18.3 Basis Functions
18.4 Coupling Coefficients for Point Groups
18.5 The V Coefficients
1806 Dihedral Groups
18.7 A Further Comment on Phase
18.8 The W Coefficients
18.9 The X Coefficient
Chapter 19 Operators and Matrix Elements
19.1 Irreducible Tensor Operators
19.2 The Wigner-Eckart Theorem
19.3 Matrix Elements and RMEs of Compound Tensor Operators
19.4 Double-Tensor Operators
19.5 The RME of a Double-Tensor Operator
19.6 Spin-Orbit Coupling
Chapter 20 Spinor Groups
20.1 Introduction
20.2 V and W Coefficients for O*
20.3 The Wigner-Eckart Theorem
20.4 An Example
20.5 Bases for Repeated Representations
Chapter 21 Matrix Elements in Multielectron Systems
21.1 Introduction
21.2 Coefficients of Fractional Parentage
21.3 Values of CFP
21.4 Matrix Elements in Many-Electron Systems
Chapter 22 Reduced Matrix Elements in Multielectron Systems
22.1 Introduction
22.2 Spin-Independent One-Electron Operators
22.3 Spin-Dependent One-Electron Operators-Spin-Orbit Coupling
22.4 Unit Tensors
Part IV
Chapter 23 Spin-Orbit Coupling in a Low-Spin d5 Complex
Chapter 24 Further Examples of Spin-Orbit Coupling
24.1 Spin-Orbit Coupling in Three Open Shells
24.2 Spin-Orbit Coupling for a Dihedral Group
Chapter 25 Electric Dipole Transitions in a Tetrahedral Complex
Chapter 26 Second Quantization
26.1 Operators
26.2 Reduced Matrix Elements
Chapter 27 Photoelectron Spectra of Open-Shell Molecules
Part V
Chapter 28 Vector Fields
28.1 Introduction
28.2 The Transformation of Vector Fields under Rotations
28.3 Eigenvectors of the Rotation Operator for a Vector Field
Chapter 29 Light
29.1 Multipole Expansion of Polarized Light
29.2 The Coherency Matrix
Chapter 30 Light Scattering
References
Index
Product details
- No. of pages: 246
- Language: English
- Copyright: © Academic Press 1976
- Published: January 1, 1976
- Imprint: Academic Press
- eBook ISBN: 9781483191812