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Irreducible Tensor Methods
An Introduction for Chemists
1st Edition - January 1, 1976
Author: Brian L. Silver
Editor: Ernest M. Loebl
9 7 8 - 1 - 4 8 3 1 - 9 1 8 1 - 2
Irreducible Tensor Methods: An Introduction for Chemists explains the theory and application of irreducible tensor operators. The book discusses a compact formalism to describe… Read more
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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.
PrefaceIntroductionPart 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 EquationsPart 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 OperatorsPart 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 TensorsPart 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 MoleculesPart 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 ScatteringReferencesIndex