Group theory in chemistry and spectroscopy : a simple guide to advanced usage / Boris S. Tsukerblat

Includes bibliographical references and index.

Chapter1. Symmetry Transformations and Groups -- 1.1 Symmetry transformations -- 1.1.1 Definition of a symmetry operation -- 1.1.2 Rotation operation. Symmetry axes -- 1.1.3 Reflection operation. Symmetry planes -- 1.1.4 Improper rotation. Rotoreflection axes -- 1.1.5 Inversion -- 1.1.6 Summary -- 1.2 Multiplication of symmetry operations. Commutativity -- 1.3 Interrelation between symmetry elements -- 1.3.1 Symmetry axes -- 1.3.2 Axes and planes -- 1.4 Definition of a group -- Chapter2. Point Groups and Their Classes -- 2.1 Equivalent symmetry elements and atoms -- 2.2 Classes of conjugated symmetry operations -- 2.3 Rules for establishing classes -- 2.4 Point groups -- 2.4.1 The rotation groups Cn -- 2.4.2 Groups of rotoreflection transformations S2n -- 2.4.3 The groups Cnh -- 2.4.4 The groups Cnv -- 2.4.5 The dihedral groups Dn -- 2.4.6 The groups Dnh -- 2.4.7 The groups Dnd -- 2.4.8 The cubic groups (T, Td, Th, O, Oh) -- 2.4.9 Continuous groups -- 2.5 Crystallographic point groups -- 2.6 Rules for the determination of molecular symmetry -- Chapte3. Representations of Point Groups -- 3.1 Matrices and vectors -- 3.1.1 Definition of a matrix -- 3.1.2 Matrix multiplication -- 3.1.3 Multiplication of block-diagonal matrices -- 3.1.4 Matrix characters -- 3.2 Matrix form of geometrical transformations -- 3.3 Group representations -- 3.4 Reducible and irreducible representations -- 3.5 Irreducible representations of the cubic group -- 3.5.1 Atomic orbitals and the effect of symmetry operations -- 3.5.2 Transformation of p orbitals under the cubic group -- 3.5.3 Transformation of d wave functions under the group O -- 3.5.4 Basis functions and irreducible representation -- Properties of irreducible representations -- 3.7 Character tables -- 3.7.1 Structure of tables -- 3.7.2 Polar and axial vectors -- 3.7.3 Complex-conjugate representations -- 3.7.4 Groups with an inversion centre -- 3.7.5 Systems of notation -- Chapter4. Crystal Field Theory for One-Electron -- 4.1 Qualitative discussion -- 4.2 Schr��dinger equation and irreducible representations -- 4.3 Splitting of one-electron levels in crystal fields -- 4.3.1 Formula for reduction of representations 4.3.2 Splitting of the p level in tetragonal, trigonal and rhombic fields -- 4.3.3 Characters of rotation groups -- 4.3.4 Classification of one-electron states in crystal fields -- 4.3.5 Splitting of the d level in cubic fields -- 4.3.6 Splitting of the d level in low-symmetry fields -- 4.3.7 Representation-reduction tables. External fields -- Chapter5. Many-Electron Ions in Crystal Fields -- 5.1 Quantum states of a free atom -- 5.2 Classification of levels in crystal fields -- 5.2.1 Classification method for the LS scheme -- 5.2.2 Parity rule -- 5.2.3 Reduction tables for representations of the full rotation group -- 5.3 Strong-crystal-field scheme -- 5.4 The direct product of representations -- 5.4.1 Definition of the direct product -- 5.4.2 Characters of the direct product -- 5.4.3 Decomposition of a direct product into irreducible parts -- 5.4.4 Clebsch Gordan coefficients -- 5.4.5 Wigner coefficients -- 5.5 Two-electron terms in a strong cubic field -- 5.6 Energy levels of a two-electron d ion -- 5.6.1 Nonrepeating representations -- 5.6.2 Configuration mixing -- 5.7 Many-electron terms in a strong cubic field -- 5.7.1 Classification of three-electron terms -- 5.7.2 Wavefunctions of three electrons -- 5.7.3 Many-electron wavefunctions -- 5.7.4 Energy levels -- 5.7.5 Correlation diagrams -- 5.7.6 Tanabe Sugano diagrams -- Chapter6. Semi empirical Crystal Field Theory -- 6.1 Crystal field Hamiltonian -- 6.2 Wigner Eckart theorem for spherical tensors -- 6.2.1 Spherical tensors -- 6.2.2 Matrix elements of tensor operators -- 6.3 Projection operators -- 6.3.1 Spherical tensors in point group -- 6.3.2 Projection operator method -- 6.3.3 Euler angles, and irreducible representations of the rotation groups -- 6.3.4 Matrices of irreducible representations of point groups -- 6.3.5 Basis functions of irreducible representations of point groups -- 6.4 Crystal field effective Hamiltonian -- 6.4.1 Rules for construction of invari -- 6.4.2 Energy levels and wavefunctions -- 6.4.3 Low-symmetry and conformations of octahedral complexes -- Chapter7. Theory of Directed Valence -- 7.1 Directed valence -- 7.2 Classification of directed a bonds -- 7.2.1 Hybrid tetrahedral bonds -- 7.2.2 Inequivalent hybrid bonds -- 7.3 Site-symmetry me -- 7.4 Classification of hybrid π bonds -- 7.5 Construction of hybrid orbitals -- Chapter8. Molecular Orbital Method -- 8.1 General background -- 8.2 Group-theoretical classification of molecular orbitals -- 8.2.1 Illustrative example -- 8.2.2 Ammonia molecule -- 8.2.3 Tetrahedral molecules: formulation of method -- 8.3 Cyclic π systems -- 8.4 Transition metal complexes -- 8.5 Sandwich-type compounds -- 8.6 Superexchange in clusters -- 8.7 Many-electron states in the molecular orbital method -- 8.7.1 Molecular terms -- 8.7.2 Cyclic π-system terms -- 8.7.3 Terms of transition metal complexes -- 8.7.4 Magnetic states of dimeric clusters -- Chapter9. Intensities of Optical Lines -- 9.1 Selection rules for optical transitions -- 9.1.1 Interaction with an electromagnetic field -- 9.1.2 Selection rules -- 9.1.3 Optical line polarization for allowed transitions -- 9.1.4 Polarization dichroism in low-symmetry fields -- 9.2 Wigner Eckart theorem for point groups -- 9.3 Polarization dependence of spectra for allowed transitions -- 9.5 Two-photon spectra -- 9.5.1 Selection rules for two-photon transitions -- 9.5.2 Polarization dependence of two-photon spectra -- 9.6 Effective dipole moment method -- 9.6.1 Effective dipole moment -- 9.6.2 Intensities of spectral lines -- Chapter10. Double Groups -- 10.1 Spin orbit interaction -- 10.2 Double-valued representations -- 10.2.1 The concept of a double group -- 10.2.2 Classes of double group 10.2.3 Character tables of the double groups 10.3 Reduction of double-valued representations Chapter11. Spin Orbit Interaction in Crystal Fields 11.1 Classification of fine-structure levels 11.1.1 One-electron terms in a cubic field 11.1.2 One-electron terms in low-symmetry fields 11.1.3 Many-electron terms 11.2 Spin orbit splitting in one-electron ions 11.2.1 Wave functions of fine-structure levels 11.2.2 Spin orbit splitting of p and d levels in a cubic field 11.2.3 Selection rules for mixing of SΓ terms 11.2.4 Shifts in the fine-structure levels 11.3 Fine structure of many-electron terms 11.3.1 Effective spin orbit interaction 11.3.2 Symmetric and antisymmetric parts of the direct product 11.3.3 Selection rules for real and imaginary operators 11.4 Fine structure of optical lines 11.4.1 Intensities and selection rules 11.4.2 Deformation splitting, two-photon transitions Chapter12. Electron Paramagnetic Resonance 12.1 Magnetic resonance phenomena 12.2 The spin Hamiltonian 12.2 1 Zero-field splittings 12.2.2 Zeeman interaction 12.3 Hyperfine interaction for spin multiplets 12.4 Electric field effects 12.4.1 Linear electric field effect 12.4.2 Quadratic electric field effect 12.4.3 Combined influence of electric and magnetic fields 12.5 Effective Hamiltonian for non-Kramers doublets 12.6 Effective Hamiltonian for the spin orbit multiple Chapter13. Exchange Interaction in Polynuclear Coordination Compounds 13.1 The Heisenberg DiracVan Vleck model 13.2 Spin levels of symmetric trimeric and tetrameric clusters 13.2.1 Trimeric clusters 13.2.2 Tetrameric clusters 13.3 Calculation of spin levels in the Heisenberg model 13.3.1 Structure of the exchange Hamiltonian matrix 13.3.2 Example of calculation of spin levels 13.3.3 The 6j- and 9j-symbols 13.3.4 Application of irreducible tensor method, recoupling 13.4 Group-theoretical classification of exchange multiplets 13.4.1 "Accidental" degeneracy 13.4.2 Spin��orbit multiplets 13.4.3 Conclusions from the group-theoretical classification 13.4.4 Non-Heisenberg exchange interactions 13.5 Paramagnetic resonance and hyperfine interactions 13.6 Classification of multiplets of mixed-valence clusters Chapter14. Vibrational Spectra and Electron Vibrational Interactions 14.1 Normal vibrations 14.1.1 Degrees of freedom.

Normal coordinates 14.1.2 Classification of normal vibrations 14.1.3 Construction of normal coordinates 14.2 Selection rules for IR absorption and combination light scattering 14.3 Electronvibrational interactions 14.4 Jahn Teller effect 14.4.1 Jahn Teller theorem 14.4.2 Adiabatic potentials 14.5 Optical-band splitting in the static Jahn Teller effect 14.6 Vibronic satellites of electronic lines 14.7 Polarization dependence of the vibronic satellite intensity 14.8 Electron vibrational interaction in mixed-valence clusters

Group theory inferences are applied to analyze the results of practically all spectroscopic methods used in organic and inorganic chemistry. This book is a manual for experimentalists and the fundamental concepts are introduced by way of examples, solving particular chemical and physical problems. Contents: Symmetry transformations and groups; point groups and their classes; representations of point groups; crystal field theory for one-electron ion; many-electron ions in crystal fields; semitempiric crystal field theory; theory of directed valency; methods of molecular orbitals; intensities of optical lines; double-groups; spin-orbit integration in crystal fields; electron paramagnetic resonance; exchange interaction in the polynuclear coordination compounds; vibrational spectra and electron-vibrational interactions.