# 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