Group Theory in Chemistry and Spectroscopy A Simple Guide to Advanced Usage

ISBN-10: 048645035X
ISBN-13: 9780486450353
Edition: 2006
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Description: A handbook on group theory for chemists and experimental physicists who use spectroscopy and require knowledge of electronic structures, this volume is constructed around detailed examples. It features a new appendix on recent developments and is  More...

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Book details

List price: $24.95
Copyright year: 2006
Publisher: Dover Publications, Incorporated
Publication date: 8/18/2006
Binding: Paperback
Pages: 464
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 1.342
Language: English

A handbook on group theory for chemists and experimental physicists who use spectroscopy and require knowledge of electronic structures, this volume is constructed around detailed examples. It features a new appendix on recent developments and is appropriate for advanced undergraduates and graduate students. Boris S. Tsukerblat is affiliated with Israel's Ben Gurion University of the Negev.

Preface
Errata
Additional Notes
Symmetry Transformations and Groups
Symmetry transformations
Definition of a symmetry operation
Rotation operation. Symmetry axes
Reflection operation. Symmetry planes
Improper rotation. Rotoreflection axes
Inversion
Summary
Multiplication of symmetry operations. Commutativity
Interrelation between symmetry elements
Symmetry axes
Axes and planes
Definition of a group
Problems
Point Groups and Their Classes
Equivalent symmetry elements and atoms
Classes of conjugated symmetry operations
Rules for establishing classes
Point groups
The rotation groups C[subscript n]
Groups of rotoreflection transformations S[subscript 2n]
The groups C[subscript ch]
The groups C[subscript nv]
The dihedral groups D[subscript n]
The groups D[subscript nh]
The groups D[subscript nd]
The cubic groups (T, T[subscript d], T[subscript h], O, O[subscript h])
Continuous groups
Crystallographic point groups
Rules for the determination of molecular symmetry
Problems
Representations of Point Groups
Matrices and vectors
Definition of a matrix
Matrix multiplication
Multiplication of block-diagonal matrices
Matrix characters
Matrix form of geometrical transformations
Group Representations
Reducible and irreducible representations
Irreducible representations of the cubic group
Atomic orbitals and the effect of symmetry operations
Transformation of p orbitals under the cubic group
Transformation of d wavefunctions under the group O
Basis functions and irreducible representations
Properties of irreducible representations
Character tables
Structure of tables
Polar and axial vectors
Complex-conjugate representations
Groups with an inversion centre
Systems of notation
Problems
Crystal Field Theory for One-Electron Ions
Qualitative discussion
Schrodinger equation and irreducible representations
Splitting of one-electron levels in crystal fields
Formula for reduction of representations
Splitting of the p level in tetragonal, trigonal and rhombic fields
Characters of rotation groups
Classification of one-electron states in crystal fields
Splitting of the d level in cubic fields
Splitting of the d level in low-symmetry fields
Representation-reduction tables. External fields
Problems
Many-Electron Ions in Crystal Fields
Quantum states of a free atom
Classification of levels in crystal fields
Classification method for the LS scheme
Parity rule
Reduction tables for representations of the full rotation group
Strong-crystal-field scheme
The direct product of representations
Definition of the direct product
Characters of the direct product
Decomposition of a direct product into irreducible parts
Clebsch-Gordan coefficients
Wigner coefficients
Two-electron terms in a strong cubic field
Energy levels of a two-electron d ion
Nonrepeating representations
Configuration mixing
Many-electron terms in a strong cubic field
Classification of three-electron terms
Wavefunctions of three electrons
Many-electron wavefunctions
Energy levels
Correlation diagrams
Tanabe-Sugano diagrams
Problems
Semiempirical Crystal Field Theory
Crystal field Hamiltonian
Wigner-Eckart theorem for spherical tensors
Spherical tensors
Matrix elements of tensor operators
Projection operators
Spherical tensors in point groups
Projection operator method
Euler angles, and irreducible representations of the rotation groups
Matrices of irreducible representations of point groups
Basis functions of irreducible representations of point groups
Crystal field effective Hamiltonian
Rules for construction of invariants
Energy levels and wavefunctions
Low-symmetry and conformations of octahedral complexes
Problems
Theory of Directed Valence
Directed valence
Classification of directed [sigma] bonds
Hybrid tetrahedral bonds
Inequivalent hybrid bonds
Site-symmetry method
Classification of hybrid [pi] bonds
Construction of hybrid orbitals
Problems
Molecular Orbital Method
General background
Group-theoretical classification of molecular orbitals
Illustrative example
Ammonia molecule
Tetrahedral molecules: formulation of method
Cyclic [pi] systems
Transition metal complexes
Sandwich-type compounds
Superexchange in clusters
Many-electron states in the molecular orbital method
Molecular terms
Cyclic [pi]-system terms
Terms of transition metal complexes
Magnetic states of dimeric clusters
Problems
Intensities of Optical Lines
Selection rules for optical transitions
Interaction with an electromagnetic field
Selection rules
Optical line polarization for allowed transitions
Polarization dichroism in low-symmetry fields
Wigner-Eckart theorem for point groups
Polarization dependence of spectra for allowed transitions
Approximate selection rules
Two-photon spectra
Selection rules for two-photon transitions
Polarization dependence of two-photon spectra
Effective dipole moment method
Effective dipole moment
Intensities of spectral lines
Problems
Double Groups
Spin-orbit interaction
Double-valued representations
The concept of a double group
Classes of double groups
Character tables of the double groups
Reduction of double-valued representations
Problems
Spin-Orbit Interaction in Crystal Fields
Classification of fine-structure levels
One-electron terms in a cubic field
One-electron terms in low-symmetry fields
Many-electron terms
Spin-orbit splitting in one-electron ions
Wavefunctions of fine-structure levels
Spin-orbit splitting of p and d levels in a cubic field
Selection rules for mixing of S[Gamma] terms
Shifts in the fine-structure levels
Fine structure of many-electron terms
Effective spin-orbit interaction
Symmetric and antisymmetric parts of the direct product
Selection rules for real and imaginary operators
Fine structure of optical lines
Intensities and selection rules
Deformation splitting, two-photon transitions
Problems
Electron Paramagnetic Resonance
Magnetic resonance phenomena
The spin Hamiltonian
Zero-field splittings
Zeeman interaction
Hyperfine interaction for spin multiplets
Electric field effects
Linear electric field effect
Quadratic electric field effect
Combined influence of electric and magnetic fields
Effective Hamiltonian for non-Kramers doublets
Effective Hamiltonian for the spin-orbit multiplet
Problems
Exchange Interaction in Polynuclear Coordination Compounds
The Heisenberg-Dirac-Van Vleck model
Spin levels of symmetric trimeric and tetrameric clusters
Trimeric clusters
Tetrameric clusters
Calculation of spin levels in the Heisenberg model
Structure of the exchange Hamiltonian matrix
Example of calculation of spin levels
The 6j- and 9j-symbols
Application of irreducible tensor method, recoupling
Group-theoretical classification of exchange multiplets
"Accidental" degeneracy
Spin-orbit multiplets
Conclusions from the group-theoretical classification
Non-Heisenberg exchange interactions
Paramagnetic resonance and hyperfine interactions
Classification of multiplets of mixed-valence clusters
Problems
Vibrational Spectra and Electron-Vibrational Interactions
Normal vibrations
Degrees of freedom. Normal coordinates
Classification of normal vibrations
Construction of normal coordinates
Selection rules for IR absorption and combination light scattering
Electron-vibrational interactions
Jahn-Teller effect
Jahn-Teller theorem
Adiabatic potentials
Optical-band splitting in the static Jahn-Teller effect
Vibronic satellites of electronic lines
Polarization dependence of the vibronic satellite intensity
Electron-vibrational interaction in mixed-valence clusters
Problems
Characters of Point Groups
Matrices of Irreducible Representations of Selected Point Groups
Basic Functions of Irreducible Representations of Selected Point Groups
Decomposition of Products of Representations
Effective Hamiltonians for Non-Kramers Doublets
References
Additional References
Index

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