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

by Tsukerblat, Boris S.

• ISBN13: 9780486450353
• ISBN10: 048645035X
• Format: Paperback
• Copyright: 2006-08-18
• Publisher: Dover Publications

Summary

This handbook on group theory is geared toward chemists and experimental physicists who use spectroscopy and require knowledge of the electronic structures of the materials they investigate. Accessible to undergraduate students, it takes an elementary approach to many of the key concepts. Rather than the deductive method common to books on mathematics and theoretical physics, the present volume introduces fundamental concepts with simple examples, relating them to specific chemical and physical problems.

Table of Contents

Preface	xi
Errata	iv
Additional Notes	xvi
1 Symmetry Transformations and Groups	1	(18)
1.1 Symmetry transformations	1	(10)
1.1.1 Definition of a symmetry operation	1	(1)
1.1.2 Rotation operation. Symmetry axes	1	(4)
1.1.3 Reflection operation. Symmetry planes	5	(1)
1.1.4 Improper rotation. Rotoreflection axes	6	(4)
1.1.5 Inversion	10	(1)
1.1.6 Summary	11	(1)
1.2 Multiplication of symmetry operations. Commutativity	11	(2)
1.3 Interrelation between symmetry elements	13	(4)
1.3.1 Symmetry axes	13	(2)
1.3.2 Axes and planes	15	(2)
1.4 Definition of a group	17	(1)
Problems	18	(1)
2 Point Groups and Their Classes	19	(38)
2.1 Equivalent symmetry elements and atoms	19	(2)
2.2 Classes of conjugated symmetry operations	21	(2)
2.3 Rules for establishing classes	23	(3)
2.4 Point groups	26	(21)
2.4.1 The rotation groups Cn	26	(3)
2.4.2 Groups of rotoreflection transformations S2n	29	(1)
2.4.3 The groups Cnh	30	(3)
2.4.4 The groups Cnv	33	(4)
2.4.5 The dihedral groups Dn	37	(2)
2.4.6 The groups Dnh	39	(3)
2.4.7 The groups Dnd	42	(3)
2.4.8 The cubic groups (T, Td, Th, O, Oh)	45	(2)
2.4.9 Continuous groups	47	(1)
2.5 Crystallographic point groups	47	(1)
2.6 Rules for the determination of molecular symmetry	48	(4)
Problems	52	(5)
3 Representations of Point Groups	57	(38)
3.1 Matrices and vectors	57	(5)
3.1.1 Definition of a matrix	57	(1)
3.1.2 Matrix multiplication	58	(1)
3.1.3 Multiplication of block-diagonal matrices	59	(2)
3.1.4 Matrix characters	61	(1)
3.2 Matrix form of geometrical transformations	62	(4)
3.3 Group representations	66	(3)
3.4 Reducible and irreducible representations	69	(2)
3.5 Irreducible representations of the cubic group	71	(11)
3.5.1 Atomic orbitals and the effect of symmetry operations	71	(3)
3.5.2 Transformation of p orbitals under the cubic group	74	(3)
3.5.3 Transformation of d wavefunctions under the group O	77	(4)
3.5.4 Basis functions and irreducible representations	81	(1)
3.6 Properties of irreducible representations	82	(4)
3.7 Character tables	86	(7)
3.7.1 Structure of tables	86	(1)
3.7.2 Polar and axial vectors	87	(1)
3.7.3 Complex-conjugate representations	88	(2)
3.7.4 Groups with an inversion centre	90	(2)
3.7.5 Systems of notation	92	(1)
Problems	93	(2)
4 Crystal Field Theory for One-Electron Ions	95	(26)
4.1 Qualitative discussion	95	(2)
4.2 Schrödinger equation and irreducible representations	97	(3)
4.3 Splitting of one-electron levels in crystal fields	100	(20)
4.3.1 Formula for reduction of representations	100	(1)
4.3.2 Splitting of the p level in tetragonal, trigonal and rhombic fields	101	(5)
4.3.3 Characters of rotation groups	106	(2)
4.3.4 Classification of one-electron states in crystal fields	108	(3)
4.3.5 Splitting of the d level in cubic fields	111	(4)
4.3.6 Splitting of the d level in low-symmetry fields	115	(2)
4.3.7 Representation-reduction tables. External fields	117	(3)
Problems	120	(1)
5 Many-Electron Ions in Crystal Fields	121	(38)
5.1 Quantum states of a free atom	121	(2)
5.2 Classification of levels in crystal fields	123	(4)
5.2.1 Classification method for the LS scheme	123	(1)
5.2.2 Parity rule	124	(1)
5.2.3 Reduction tables for representations of the full rotation group	125	(2)
5.3 Strong-crystal-field scheme	127	(4)
5.4 The direct product of representations	131	(9)
5.4.1 Definition of the direct product	131	(1)
5.4.2 Characters of the direct product	131	(2)
5.4.3 Decomposition of a direct product into irreducible parts	133	(1)
5.4.4 Clebsch–Gordan coefficients	134	(3)
5.4.5 Wigner coefficients	137	(3)
5.5 Two-electron terms in a strong cubic field	140	(4)
5.6 Energy levels of a two-electron d ion	144	(5)
5.6.1 Nonrepeating representations	144	(1)
5.6.2 Configuration mixing	145	(4)
5.7 Many-electron terms in a strong cubic field	149	(8)
5.7.1 Classification of three-electron terms	149	(1)
5.7.2 Wavefunctions of three electrons	150	(3)
5.7.3 Many-electron wavefunctions	153	(1)
5.7.4 Energy levels	153	(1)
5.7.5 Correlation diagrams	154	(1)
5.7.6 Tanabe–Sugano diagrams	155	(2)
Problems	157	(2)
6 Semiempirical Crystal Field Theory	159	(22)
6.1 Crystal field Hamiltonian	159	(1)
6.2 Wigner–Eckart theorem for spherical tensors	160	(3)
6.2.1 Spherical tensors	160	(1)
6.2.2 Matrix elements of tensor operators	160	(3)
6.3 Projection operators	163	(10)
6.3.1 Spherical tensors in point groups	163	(2)
6.3.2 Projection operator method	165	(2)
6.3.3 Euler angles, and irreducible representations of the rotation groups	167	(1)
6.3.4 Matrices of irreducible representations of point groups	168	(2)
6.3.5 Basis functions of irreducible representations of point groups	170	(3)
6.4 Crystal field effective Hamiltonian	173	(7)
6.4.1 Rules for construction of invariants	173	(2)
6.4.2 Energy levels and wavefunctions	175	(3)
6.4.3 Low-symmetry and conformations of octahedral complexes	178	(2)
Problems	180	(1)
7 Theory of Directed Valence	181	(18)
7.1 Directed valence	181	(2)
7.2 Classification of directed a bonds	183	(4)
7.2.1 Hybrid tetrahedral bonds	183	(3)
7.2.2 Inequivalent hybrid bonds	186	(1)
7.3 Site-symmetry method	187	(2)
7.4 Classification of hybrid π bonds	189	(5)
7.5 Construction of hybrid orbitals	194	(3)
Problems	197	(2)
8 Molecular Orbital Method	199	(24)
8.1 General background	199	(2)
8.2 Group-theoretical classification of molecular orbitals	201	(4)
8.2.1 Illustrative example	201	(1)
8.2.2 Ammonia molecule	202	(1)
8.2.3 Tetrahedral molecules: formulation of method	203	(2)
8.3 Cyclic π systems	205	(4)
8.4 Transition metal complexes	209	(2)
8.5 Sandwich-type compounds	211	(3)
8.6 Superexchange in clusters	214	(2)
8.7 Many-electron states in the molecular orbital method	216	(5)
8.7.1 Molecular terms	216	(1)
8.7.2 Cyclic π-system terms	217	(2)
8.7.3 Terms of transition metal complexes	219	(1)
8.7.4 Magnetic states of dimeric clusters	219	(2)
Problems	221	(2)
9 Intensities of Optical Lines	223	(22)
9.1 Selection rules for optical transitions	223	(7)
9.1.1 Interaction with an electromagnetic field	223	(1)
9.1.2 Selection rules	224	(3)
9.1.3 Optical line polarization for allowed transitions	227	(2)
9.1.4 Polarization dichroism in low-symmetry fields	229	(1)
9.2 Wigner–Eckart theorem for point groups	230	(3)
9.3 Polarization dependence of spectra for allowed transitions	233	(1)
9.4 Approximate selection rules	234	(1)
9.5 Two-photon spectra	235	(5)
9.5.1 Selection rules for two-photon transitions	235	(3)
9.5.2 Polarization dependence of two-photon spectra	238	(2)
9.6 Effective dipole moment method	240	(4)
9.6.1 Effective dipole moment	240	(2)
9.6.2 Intensities of spectral lines	242	(2)
Problems	244	(1)
10 Double Groups	245	(10)
10.1 Spin–orbit interaction	245	(1)
10.2 Double-valued representations	246	(3)
10.2.1 The concept of a double group	246	(1)
10.2.2 Classes of double groups	246	(1)
10.2.3 Character tables of the double groups	247	(2)
10.3 Reduction of double-valued representations	249	(4)
Problems	253	(2)
11 Spin–Orbit Interaction in Crystal Fields	255	(22)
11.1 Classification of fine-structure levels	255	(4)
11.1.1 One-electron terms in a cubic field	255	(2)
11.1.2 One-electron terms in low-symmetry fields	257	(1)
11.1.3 Many-electron terms	258	(1)
11.2 Spin–orbit splitting in one-electron ions	259	(8)
11.2.1 Wavefunctions of fine-structure levels	259	(4)
11.2.2 Spin–orbit splitting of p and d levels in a cubic field	263	(1)
11.2.3 Selection rules for mixing of SΓ terms	264	(1)
11.2.4 Shifts in the fine-structure levels	265	(2)
11.3 Fine structure of many-electron terms	267	(5)
11.3.1 Effective spin–orbit interaction	267	(2)
11.3.2 Symmetric and antisymmetric parts of the direct product	269	(1)
11.3.3 Selection rules for real and imaginary operators	270	(2)
11.4 Fine structure of optical lines	272	(2)
11.4.1 Intensities and selection rules	272	(1)
11.4.2 Deformation splitting, two-photon transitions	273	(1)
Problems	274	(3)
12 Electron Paramagnetic Resonance	277	(22)
12.1 Magnetic resonance phenomena
12.2 The spin Hamiltonian	278	(6)
12.2 1 Zero-field splittings	278	(4)
12.2.2 Zeeman interaction	282	(2)
12.3 Hyperfine interaction for spin multiplets	284	(2)
12.4 Electric field effects	286	(5)
12.4.1 Linear electric field effect	286
12.4.2 Quadratic electric field effect	284	(6)
12.4.3 Combined influence of electric and magnetic fields	290	(1)
12.5 Effective Hamiltonian for non-Kramers doublets	291	(5)
12.6 Effective Hamiltonian for the spin–orbit multiplet	296	(1)
Problems	297	(2)
13 Exchange Interaction in Polynuclear Coordination Compounds	299	(56)
13.1 The Heisenberg–Dirac–Van Vleck model	299	(3)
13.2 Spin levels of symmetric trimeric and tetrameric clusters	302	(3)
13.2.1 Trimeric clusters	302	(2)
13.2.2 Tetrameric clusters	304	(1)
13.3 Calculation of spin levels in the Heisenberg model	305	(15)
13.3.1 Structure of the exchange Hamiltonian matrix	305	(1)
13.3.2 Example of calculation of spin levels	306	(2)
13.3.3 The 6j- and 9j-symbols	308	(7)
13.3.4 Application of irreducible tensor method, recoupling	315	(5)
13.4 Group-theoretical classification of exchange multiplets	320	(21)
13.4.1 "Accidental" degeneracy	320	(1)
13.4.2 Spin–orbit multiplets	320	(16)
13.4.3 Conclusions from the group-theoretical classification	336	(3)
13.4.4 Non-Heisenberg exchange interactions	339	(2)
13.5 Paramagnetic resonance and hyperfine interactions	341	(1)
13.6 Classification of multiplets of mixed-valence clusters	342	(11)
Problems	353	(2)
14 Vibrational Spectra and Electron–Vibrational Interactions	355	(40)
14.1 Normal vibrations	355	(7)
14.1.1 Degrees of freedom. Normal coordinates	355	(2)
14.1.2 Classification of normal vibrations	357	(3)
14.1.3 Construction of normal coordinates	360	(2)
14.2 Selection rules for IR absorption and combination light scattering	362	(2)
14.3 Electron–vibrational interactions	364	(2)
14.4 Jahn–Teller effect	366	(4)
14.4.1 Jahn–Teller theorem	366	(1)
14.4.2 Adiabatic potentials	367	(3)
14.5 Optical-band splitting in the static Jahn–Teller effect	370	(6)
14.6 Vibronic satellites of electronic lines	376	(2)
14.7 Polarization dependence of the vibronic satellite intensity	378	(1)
14.8 Electron–vibrational interaction in mixed-valence clusters	379	(14)
Problems	393	(2)
Appendix I Characters of Point Groups	395	(16)
Appendix II Matrices of Irreducible Representations of Selected Point Groups	411	(12)
Appendix III Basic Functions of Irreducible Representations of Selected Point Groups	423	(7)
Appendix IV Decomposition of Products of Representations	430	(3)
Appendix V Effective Hamiltonians for Non-Kramers Doublets	433	(4)
References	437	(5)
Additional References	442	(3)
Index	445