Irreducible Tensor Methods

Irreducible Tensor Methods

An Introduction for Chemists

1st Edition - January 1, 1976

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  • Author: Brian L. Silver
  • eBook ISBN: 9781483191812

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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


    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



Product details

  • No. of pages: 246
  • Language: English
  • Copyright: © Academic Press 1976
  • Published: January 1, 1976
  • Imprint: Academic Press
  • eBook ISBN: 9781483191812

About the Author

Brian L. Silver

About the Editor

Ernest M. Loebl

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