Group theory applied to chemistry

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Theoretical Chemistry and Computational Modelling For further volumes: www.springer.com/series/10635 Modern Chemistry is unthinkable without the achievements of Theoretical and Computational Chemistry As a matter of fact, these disciplines are now a mandatory tool for the molecular sciences and they will undoubtedly mark the new era that lies ahead of us To this end, in 2005, experts from several European universities joined forces under the coordination of the Universidad Autónoma de Madrid, to launch the European Masters Course on Theoretical Chemistry and Computational Modeling (TCCM) The aim of this course is to develop scientists who are able to address a wide range of problems in modern chemical, physical, and biological sciences via a combination of theoretical and computational tools The book series, Theoretical Chemistry and Computational Modeling, has been designed by the editorial board to further facilitate the training and formation of new generations of computational and theoretical chemists Prof Manuel Alcami Departamento de Qmica Facultad de Ciencias, Módulo 13 Universidad Autónoma de Madrid 28049 Madrid, Spain Prof Otilia Mo Departamento de Qmica Facultad de Ciencias, Módulo 13 Universidad Autónoma de Madrid 28049 Madrid, Spain Prof Ria Broer Theoretical Chemistry Zernike Institute for Advanced Materials Rijksuniversiteit Groningen Nijenborgh 9747 AG Groningen, The Netherlands Prof Ignacio Nebot Institut de Ciència Molecular Parc Científic de la Universitat de València Catedrático José Beltrán Martínez, no 46980 Paterna (Valencia), Spain Dr Monica Calatayud Laboratoire de Chimie Théorique Université Pierre et Marie Curie, Paris 06 place Jussieu 75252 Paris Cedex 05, France Prof Arnout Ceulemans Departement Scheikunde Katholieke Universiteit Leuven Celestijnenlaan 200F 3001 Leuven, Belgium Prof Antonio Laganà Dipartimento di Chimica Università degli Studi di Perugia via Elce di Sotto 06123 Perugia, Italy Prof Colin Marsden Laboratoire de Chimie et Physique Quantiques Université Paul Sabatier, Toulouse 118 route de Narbonne 31062 Toulouse Cedex 09, France Prof Minh Tho Nguyen Departement Scheikunde Katholieke Universiteit Leuven Celestijnenlaan 200F 3001 Leuven, Belgium Prof Maurizio Persico Dipartimento di Chimica e Chimica Industriale Università di Pisa Via Risorgimento 35 56126 Pisa, Italy Prof Maria Joao Ramos Chemistry Department Universidade Porto Rua Campo Alegre, 687 4169-007 Porto, Portugal Prof Manuel đez Departamento de Qmica Facultad de Ciencias, Módulo 13 Universidad Autónoma de Madrid 28049 Madrid, Spain Arnout Jozef Ceulemans Group Theory Applied to Chemistry Arnout Jozef Ceulemans Division of Quantum Chemistry Department of Chemistry Katholieke Universiteit Leuven Leuven, Belgium ISSN 2214-4714 ISSN 2214-4722 (electronic) Theoretical Chemistry and Computational Modelling ISBN 978-94-007-6862-8 ISBN 978-94-007-6863-5 (eBook) DOI 10.1007/978-94-007-6863-5 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2013948235 © Springer Science+Business Media Dordrecht 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my grandson Louis “The world is so full of a number of things, I’m sure we should all be as happy as kings.” Robert Louis Stevenson Preface Symmetry is a general principle, which plays an important role in various areas of knowledge and perception, ranging from arts and aesthetics to natural sciences and mathematics According to Barut,1 the symmetry of a physical system may be looked at in a number of different ways We can think of symmetry as representing • the impossibility of knowing or measuring some quantities, e.g., the impossibility of measuring absolute positions, absolute directions or absolute left or right; • the impossibility of distinguishing between two situations; • the independence of physical laws or equations from certain coordinate systems, i.e., the independence of absolute coordinates; • the invariance of physical laws or equations under certain transformations; • the existence of constants of motions and quantum numbers; • the equivalence of different descriptions of the same system Chemists are more used to the operational definition of symmetry, which crystallographers have been using long before the advent of quantum chemistry Their balland-stick models of molecules naturally exhibit the symmetry properties of macroscopic objects: they pass into congruent forms upon application of bodily rotations about proper and improper axes of symmetry Needless to say, the practitioner of quantum chemistry and molecular modeling is not concerned with balls and sticks, but with subatomic particles, nuclei, and electrons It is hard to see how bodily rotations, which leave all interparticle distances unaltered, could affect in any way the study of molecular phenomena that only depend on these internal distances Hence, the purpose of the book will be to come to terms with the subtle metaphors that relate our macroscopic intuitive ideas about symmetry to the molecular world In the end the reader should have acquired the skills to make use of the mathematical tools of group theory for whatever chemical problems he/she will be confronted with in the course of his or her own research A.O Barut, Dynamical Groups and Generalized Symmetries in Quantum Theory, Bascands, Christchurch (New Zealand) (1972) vii Acknowledgements The author is greatly indebted to many people who have made this book possible: to generations of doctoral students Danny Beyens, Marina Vanhecke, Nadine Bongaerts, Brigitte Coninckx, Ingrid Vos, Geert Vandenberghe, Geert Gojiens, Tom Maes, Goedele Heylen, Bruno Titeca, Sven Bovin, Ken Somers, Steven Compernolle, Erwin Lijnen, Sam Moors, Servaas Michielssens, Jules Tshishimbi Muya, and Pieter Thyssen; to postdocs Amutha Ramaswamy, Sergiu Cojocaru, Qing-Chun Qiu, Guang Hu, Ru Bo Zhang, Fanica Cimpoesu, Dieter Braun, Stanislaw Walỗerz, Willem Van den Heuvel, and Atsuya Muranaka; to the many colleagues who have been my guides and fellow travellers to the magnificent viewpoints of theoretical understanding: Brian Hollebone, Tadeusz Lulek, Marek Szopa, Nagao Kobayashi, Tohru Sato, Minh-Tho Nguyen, Victor Moshchalkov, Liviu Chibotaru, Vladimir Mironov, Isaac Bersuker, Claude Daul, Hartmut Yersin, Michael Atanasov, Janette Dunn, Colin Bates, Brian Judd, Geoff Stedman, Simon Altmann, Brian Sutcliffe, Mircea Diudea, Tomo Pisanski, and last but not least Patrick Fowler, companion in many group-theoretical adventures Roger B Mallion not only read the whole manuscript with meticulous care and provided numerous corrections and comments, but also gave expert insight into the intricacies of English grammar and vocabulary I am very grateful to L Laurence Boyle for a critical reading of the entire manuscript, taking out remaining mistakes and inconsistencies I thank Pieter Kelchtermans for his help with LaTeX and Naoya Iwahara for the figures of the Mexican hat and the hexadecapole Also special thanks to Rita Jungbluth who rescued me from everything that could have distracted my attention from writing this book I remain grateful to Luc Vanquickenborne who was my mentor and predecessor in the lectures on group theory at KULeuven, on which this book is based My thoughts of gratitude extend also to both my doctoral student, the late Sam Eyckens, and to my friend and colleague, the late Philip Tregenna-Piggott Both started the journey with me but, at an early stage, were taken away from this life My final thanks go to Monique ix Contents Operations 1.1 Operations and Points 1.2 Operations and Functions 1.3 Operations and Operators 1.4 An Aide Mémoire 1.5 Problems References 1 10 10 10 Function Spaces and Matrices 2.1 Function Spaces 2.2 Linear Operators and Transformation Matrices 2.3 Unitary Matrices 2.4 Time Reversal as an Anti-linear Operator 2.5 Problems References 11 11 12 14 16 19 19 Groups 3.1 The Symmetry of Ammonia 3.2 The Group Structure 3.3 Some Special Groups 3.4 Subgroups 3.5 Cosets 3.6 Classes 3.7 Overview of the Point Groups Spherical Symmetry and the Platonic Solids Cylindrical Symmetries 3.8 Rotational Groups and Chiral Molecules 3.9 Applications: Magnetic and Electric Fields 3.10 Problems References 21 21 24 27 29 30 32 34 34 40 44 46 47 48 xi xii Contents Representations 4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia 4.2 Character Theorems 4.3 Character Tables 4.4 Matrix Theorem 4.5 Projection Operators 4.6 Subduction and Induction 4.7 Application: The sp Hybridization of Carbon 4.8 Application: The Vibrations of UF6 4.9 Application: Hückel Theory Cyclic Polyenes Polyhedral Hückel Systems of Equivalent Atoms Triphenylmethyl Radical and Hidden Symmetry 4.10 Problems References What has Quantum Chemistry Got to Do with It? 5.1 The Prequantum Era 5.2 The Schrödinger Equation 5.3 How to Structure a Degenerate Space 5.4 The Molecular Symmetry Group 5.5 Problems References Interactions 6.1 Overlap Integrals 6.2 The Coupling of Representations 6.3 Symmetry Properties of the Coupling Coefficients 6.4 Product Symmetrization and the Pauli Exchange-Symmetry 6.5 Matrix Elements and the Wigner–Eckart Theorem 6.6 Application: The Jahn–Teller Effect 6.7 Application: Pseudo-Jahn–Teller interactions 6.8 Application: Linear and Circular Dichroism Linear Dichroism Circular Dichroism 6.9 Induction Revisited: The Fibre Bundle 6.10 Application: Bonding Schemes for Polyhedra Edge Bonding in Trivalent Polyhedra Frontier Orbitals in Leapfrog Fullerenes 6.11 Problems References Spherical Symmetry and Spins 7.1 The Spherical-Symmetry Group 7.2 Application: Crystal-Field Potentials 7.3 Interactions of a Two-Component Spinor 51 52 56 62 63 64 69 76 78 84 85 91 95 99 101 103 103 105 107 108 112 112 113 114 115 117 122 126 128 134 138 139 144 148 150 155 156 159 160 163 163 167 170 254 H Solutions to Problems The exciton states on both chromophores are interchanged by the twofold axis and can be recombined to yield a symmetric and an antisymmetric combination, denoted as A and B, respectively One has: |ΨA = √ |Ψ1 + |Ψ2 |ΨB = √ |Ψ1 − |Ψ2 The corresponding transition dipoles are oriented along the positive y- and negative z-direction, respectively: μA = μB = √ √ α 2μ 0, cos , 2μ 0, 0, − sin The dipole-dipole interaction is given by cos α V12 = 4π R12 α (9) For α < π/2, the dipole orientation is repulsive As a result, the in-phase coupled exciton state |ΨA will be at higher energy than the out-of-phase |ΨB state Finally, we also calculate the magnetic transition dipoles, using the expressions from Sect 6.8: iπν iπνμ α mA = √ (r1 × μ1 + r2 × μ2 ) = √ R12 sin (0, 1, 0) 2 iπν iπνμ α mB = √ (r1 × μ1 − r2 × μ2 ) = √ R12 cos (0, 0, 1) 2 These results are now combined in the Rosenfeld equation to yield the rotatory strength of both exciton states: RA = πνμ2 R12 sin α H Solutions to Problems 255 πνμ2 R12 sin α This result predicts a normal CD sign, with a lower negative branch (B-state) and an upper positive branch (A-state) [16] This is a typical right-handed helix, corresponding to a rotation of the dipoles in the right-handed sense when going from chromophore to chromophore along the inter-chromophore axis In the S-conformation the sign of α will change, and the CD spectrum will be inverted 6.5 The direct square of the e-irrep in D2d yields four coupled states: RB = − e × e = A1 + A2 + B1 + B2 (10) The corresponding coupling coefficients are given in the table below This table is almost the same as the table for D4 in Appendix F, but note that B1 and B2 are interchanged Such details are important, and therefore we draw again a simple picture of the molecule in a Cartesian system Both in D4 and in D2d , the B1 and B2 irreps are distinguished by their symmetry with respect to the Cˆ axes D2d E×E x x y y A1 a1 √1 √1 x y y x A2 a2 B1 b1 B2 b2 0 − √1 0 √1 − √1 √1 √1 2 √1 0 In the orientation of twisted ethylene, as indicated in the figure below, the directions of these axes are along the bisectors of x and y In contrast, in the standard orientation for D4 they are along the x and y axes, while the bisector directions coincide with the Cˆ axes, and hence the interchange between B1 and B2 256 H Solutions to Problems Note that the two-electron states are symmetrized, except the A2 combination The symmetrized states will combine with singlet spin states, while the A2 state will be a triplet One thus has: 1 A1 = √ x(1)x(2) + y(1)y(2) √ α(1)β(2) − β(1)α(2) 2 = √ (xα)(xβ) + (yα)(yβ) 1 B1 = √ (xα)(yβ) + (yα)(xβ) B2 = √ − (xα)(xβ) + (yα)(yβ) A2 = (xα)(yα) The A1 and B2 states are the zwitterionic states, while the B1 and A2 states are called the diradical states It is clear from the expressions that in both cases the two radical carbon sites are neutral The zwitterionic states are easily polarizable though 6.6 The carbon atoms form two orbits The pz orbital on the central atom is in the center of the symmetry group and transforms as a2 The three methylene orbitals are in C2v sites, transforming as the b2 irrep of the site group, i.e., they are antisymmetric with respect to σˆ h and symmetric with respect to σˆ v The induced representation is b2 C2v ↑ D3h = a2 + e (11) The SALCs are entirely similar to the hydrogen SALCs in the case of ammonia; this implies, for instance, that the component labeled x is symmetric under the vertical symmetry plane through atom A It will be antisymmetric for the twofold-axis going through atom A since the relevant orbital is of pz type: |Ψa = √ |pA + |pB + |pC |Ψx = √ 2|pA − |pB − |pC |Ψy = √ |pB − |pC The a2 orbitals interact to yield bonding and antibonding combinations at √ E = α ± 3β Since the graph is bipartite, the remaining e orbitals are neces- H Solutions to Problems 257 sarily nonbonding and will be occupied by two electrons The direct square of this irrep yields symmetrized A1 and E states and an antisymmetrized A2 state The expressions for these states are obtained from the coupling coefficients for D3 in Appendix F: 1 A1 = √ x(1)x(2) + y(1)y(2) √ α(1)β(2) − β(1)α(2) 2 = √ (xα)(xβ) + (yα)(yβ) 1 Ex = √ (xα)(yβ) + (yα)(xβ) Ey = √ − (xα)(xβ) + (yα)(yβ) A2 = (xα)(yα) Note that the distinction between zwitterionic and diradical states does not hold in this case Formally, TMM can be described as a valence isomer between three configurations in which one of the peripheral atoms has a double bond to the central atom and the other two sites carry an unpaired electron 7.1 In a cube the d-shell also splits in eg + t2g , but the ordering is reversed Explicit calculation of the potential shows that the splitting is reduced by a factor 8/9: cube =− octahhedron 7.2 Perform the matrix multiplication and verify that the product matrix is of Cayley–Klein form The multiplication is not commutative: a1 −b¯1 b1 a¯ × a2 −b¯2 b2 a¯ = a1 a2 − b1 b¯2 −a¯ b¯2 − a2 b¯1 a1 b2 + a¯ b1 a¯ a¯ − b¯1 b2 (12) 7.3 The double group D3∗ contains 12 elements In Table 7.5 we have listed the six representation matrices for the elements on the positive hemisphere The Cˆ 2A axis is along the x-direction, Cˆ 2B is at −60◦ and Cˆ 2C is at +60◦ The derivation of the multiplication table and the underlying class structure (see Table 7.6) is based on a straightforward matrix multiplication 258 H Solutions to Problems ℵCˆ 2A ℵCˆ 2B ℵCˆ 2C Multiplication table for the double group D3∗ D3∗ Eˆ Cˆ Cˆ 32 ℵCˆ ℵCˆ 32 ℵ Cˆ 2A Cˆ 2B Cˆ 2C Eˆ Cˆ Eˆ Cˆ Cˆ Cˆ Cˆ 32 ℵCˆ 32 Eˆ ℵ ℵ ℵCˆ ℵCˆ Cˆ 2A Cˆ C Cˆ 2B Cˆ A Cˆ 32 ℵCˆ Cˆ 32 ℵCˆ ℵ Eˆ Cˆ Cˆ ℵCˆ 32 ℵCˆ 32 ℵCˆ 32 Eˆ ℵCˆ Eˆ Cˆ 2C ℵCˆ 2A ℵCˆ 2B ℵCˆ 2C ℵCˆ 2B ℵCˆ 2C ℵCˆ 2A Cˆ 2B ℵCˆ 2A Cˆ 2B ℵCˆ 2C Cˆ 2A C B Cˆ Cˆ Cˆ A ℵCˆ B ℵ Cˆ A ℵCˆ Cˆ 2 ℵCˆ Cˆ B ℵ Cˆ ℵ Cˆ A Cˆ ℵCˆ Cˆ 2B Cˆ C Cˆ 2B Cˆ C ℵCˆ 2C Cˆ A ℵCˆ 2A ℵCˆ 2B ℵCˆ 2C ℵCˆ 2A ℵCˆ 2B ℵCˆ 2C ℵCˆ 2B Cˆ 2C ℵCˆ 2A 2 3 ℵCˆ 2C ℵCˆ 2A Cˆ 2B Cˆ 2C Cˆ 2A ℵCˆ 2B ℵCˆ 2B Cˆ 2C ℵCˆ 2A Cˆ 2B ℵCˆ 2C Cˆ 2A ℵCˆ ℵCˆ Cˆ Cˆ ℵ Eˆ Cˆ 2C Cˆ 2A ℵCˆ 2A ℵCˆ B 2 ℵCˆ 2B Cˆ 2C ℵCˆ 2C ℵCˆ 2A 2 ℵCˆ 2C Cˆ 2A ℵCˆ 2B Cˆ 2C Cˆ 2B ℵCˆ 2A ℵCˆ 2B ℵCˆ 2C Cˆ 2A Cˆ 2B Eˆ Cˆ ℵ ℵCˆ Cˆ 32 2 ˆ ˆ ˆ C ℵ C3 ℵC Eˆ 3 ℵCˆ 2B ℵCˆ 2C ℵCˆ ℵCˆ 2C Cˆ 2A Eˆ ℵCˆ 2A Cˆ 2B ℵCˆ 32 Cˆ B Cˆ C Cˆ 2 ℵCˆ 32 Cˆ ℵ Cˆ ℵCˆ 32 Eˆ Cˆ ℵCˆ Eˆ ℵCˆ 2A Cˆ C ℵCˆ 32 ℵCˆ ℵ Cˆ Cˆ 32 ℵCˆ Eˆ Cˆ ℵ Cˆ ℵCˆ ℵCˆ 32 ℵ 3 7.4 The action of the spin operators on the components of a spin-triplet can be found by acting on the coupled states, as summarized in Table 7.2 As an example, where we have added the electron labels and for clarity: Sx | + = Sx |α1 |α2 = = Sx |α1 |α2 + |α1 Sx |α2 |β1 |α2 + |α1 |β2 = √ |0 i Sy | − = − √ |0 These results can be generalized as follows: Sz |MS = MS |MS (Sx ± iSy )|MS = (S ∓ MS )(S ± Ms + 1) |Ms ± The action of the spin Hamiltonian in the fictitious spin basis gives then rise to the following Hamiltonian matrix (in units of μB ): HZe |0 |+1 0| g⊥ √1 (Bx + iBy ) g⊥ √1 (Bx − iBy ) +1| g⊥ √1 (Bx − iBy ) g|| Bz −1| g⊥ √1 (Bx + iBy ) −g|| Bz 2 |−1 2 We can now identify these expressions with the actual matrix elements in the basis of the three D3 components, keeping in mind the relationship be- H Solutions to Problems 259 tween the complex and real triplet basis, as given in Eq (7.39) One obtains: 1 0|HZe | + = − √ A1 |H|Ex + iEy = √ −a + d + i(−b − c) 2 1 0|HZe | − = √ A1 |H|Ex − iEy = √ a + d + i(b − c) 2 ±1|HZe | ± = x|H|x + y|H|y ± i x|H|y − y|H|x = ±f From these equations the parameters may be identified as follows: a=0 b = −g⊥ By c=0 d = g⊥ Bx e=0 f = g|| Bz The Zeeman Hamiltonian does not include the zero-field splitting between the A1 and E states This can be rendered by a second-order spin operator, which transforms as the octahedral Eg θ quadrupole component: HZF = D D 2S˜z2 − S˜x2 − S˜y2 = S˜z2 − S˜ 3 One then obtains D=3 7.5 The action of the components of the fictitious spin operator on the Γ8 basis is dictated by the general expressions for the action of the spin operators on the S = 32 basis functions It is verified that the spin-Hamiltonian that generates the Jp part of the matrix precisely corresponds to Hp = Jp B · S˜ The fictitious spin operator indeed transforms as a T1 operator and has the tensorial rank of a p-orbital However, as we have shown, the full Hamiltonian also includes a Jf part, which involves an f -like operator To mimic this part by a spin Hamiltonian, one thus will need a symmetrized triple product of the fictitious spin, which will embody an f -tensor, transforming in the octahedral symmetry as the T1 irrep These f -functions can be found in Table 7.1 and are of type z(5z2 − 3r ) But beware! To find the corresponding spin operator, it is not sufficient simply to substitute the Cartesian variables by the corresponding spinor components, i.e., z by S˜z , etc.; indeed, while products of x, y, and z are commutative, the products of the corresponding operators are not Hence, when constructing the octupolar product 260 H Solutions to Problems of the spin components, products of noncommuting operators must be fully symmetrized For the fz3 function, this is the case for the functions 3zx and 3xy , which are parts of 3zr As an example, the operator analogue of 3zx reads 3zx → S˜z S˜x S˜x + S˜x S˜z S˜x + S˜x S˜x S˜z One then has for the operator equivalent of 3z(x + y ): S˜z S˜x S˜x + S˜x S˜z S˜x + S˜x S˜x S˜z + S˜z S˜y S˜y + S˜y S˜z S˜y + S˜y S˜y S˜z = 3S˜z S˜x2 + S˜y2 + i (S˜x S˜y − S˜y S˜x ) = 3S˜z S˜x2 + S˜y2 − S˜z where we have used the commutation relation for the spin-operators: Sx Sy − Sy Sx = i Sz The octupolar spin operator will then be of type Hf = 3 gf Bz S˜z3 − S˜z S˜ + S˜z + Bx S˜x3 − S˜x S˜ + 5 5 + By S˜y3 − S˜y S˜ + S˜y 5 μB S˜x In order to identify the parameter correspondence, let us work out the action of this operator on the quartet functions As an example for a magnetic field along the z-direction, the matrix is diagonal, and its elements (in units of μB ) are given by 3 45 Hf ± = ±gf Bz − + 2 20 1 ± Hf ± = ∓ gf Bz 2 10 ± =± gf Bz 10 By comparing these elements to the results in Table 7.8 we can identify the parameter correspondence as Jf = − gf 10 (13) References Mulliken, R.S., Ramsay, D.A., Hinze, J (eds.): Selected Papers University of Chicago Press, Chicago (1975) Cotton, F.A.: Chemical Applications of Group Theory Wiley, New York (1963) Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory Oxford University Press, Oxford (1970) Boyle, L.L.: The method of ascent in symmetry I Theory and tables Acta Cryst A 28, 172 (1972) Fowler, P.W., Quinn, C.M.: Theor Chim Acta 70, 333 (1986) Griffith, J.S.: The Theory of Transition-Metal Ions Cambridge University Press, Cambridge (1961) Boyle, L.L., Parker, Y.M.: Symmetry coordinates and vibration frequencies for an icosahedral cage Mol Phys 39, 95 (1980) Qiu, Q.C., Ceulemans, A.: Icosahedral symmetry adaptation of |J M bases Mol Phys 100, 255 (2002) Butler, P.H.: Point Group Symmetry Applications, Methods and Tables Plenum Press, New York (1981) 10 Herzberg, G.: Molecular Spectra and Molecular Structure III Electronic Spectra and Electronic Structure of Polyatomic Molecules Van Nostrand, Princeton (1966) 11 Fowler, P.W., Ceulemans, A.: Symmetry relations in the property surfaces of icosahedral molecules Mol Phys 54, 767 (1985) 12 Fowler, P.W., Ceulemans, A.: Spin-orbit coupling coefficients for icosahedral molecules Theor Chim Acta 86, 315 (1993) 13 Stone, A.J.: A new approach to bonding in transition-metal clusters Theory Mol Phys 41, 1339 (1980) 14 Ceulemans, A., Vanquickenborne, L.G.: The epikernel principle Structure and Bonding 71, 125–159 (1989) 15 Longuet-Higgins, H.C.: The symmetry group of non-rigid molecules Mol Phys 6, 445 (1963) 16 Kobayashi, N., Higashi, R., Titeca, B.C., Lamote, F., Ceulemans, A.: Substituent-induced circular dichroism in phthalocyanines J Am Chem Soc 121, 12018 (1999) A.J Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and Computational Modelling, DOI 10.1007/978-94-007-6863-5, © Springer Science+Business Media Dordrecht 2013 261 Index Symbols 10Dq, 168, 169 -enantiomer, 138, 139 quartet, 184, 190 π -modes, 78 σ -mode, 78 A Abel, 27 Abelian, 27, 32, 41, 47, 56, 62 Abragam, 182, 190 Absorption spectra, 144 acac− , 138 Aldridge, 84, 101 Allene, 46 Alternant, 97 Altmann, ix, 16, 19, 72, 101, 164, 190 Ammonia, 21–24, 28–30, 32, 34, 36, 39, 43, 52, 53, 55, 56, 59, 68, 72, 75, 76, 103, 104, 110–112, 250, 256 Ammonia Dynamic symmetry, 110–112 Permutation symmetry, 28–30, 36 Point group, 21–24, 32, 34, 43, 72, 104 SALC, 52–56, 59, 68, 75, 76, 250 Angular momentum, 10, 89, 133, 134, 146, 165, 185, 188, 190, 246 Annulene, 87 Antiprisms, 40, 43 Antisymmetrization, 119, 121, 123, 125, 183, 219, 257 Archimedean solids, 93 Archimedene, 102 Aromaticity, 87 Associativity, 24 Atkins, 191, 261 Atomic population, 85 Automorphism group, 27, 92, 95, 98 Azimuthal coordinate, 165 B Balabanov, 134, 161 Barut, vii Barycentre rule, 146, 169, 170, 186 Basis set, canonical, 65, 107, 217 Bending modes, 65, 83 Benfey, 36 Benzene, 41, 42, 85, 87, 99, 249 Bernoulli, Berry, Berry phase, 10 Bersuker, ix, 128, 134, 161 Bethe, 176, 179, 235 Beyens, ix, 121, 160 Biel, 149, 161 Bilinear interaction, 135, 136, 172 Binary elements, 180 Bipy, 138, 140, 144, 161 Bisphenoid, 41 Bleaney, 182, 190 Boggs, 134, 161 Bohr magneton, 89 Bond order, 85 Born-Oppenheimer condition, 110–112 Boundary condition, 110 Boundary operation, 152 Boyle, ix, 39, 49, 211, 215, 218, 261 Bra function, 116, 173 Bracket, 5, 8, 11, 12, 58, 67, 68, 114, 117, 126, 183 Branching rule, 218 Braun, ix, 144, 161 Bruns, 93 Buckminsterfullerene, 38–40, 49, 157, 158 A.J Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and Computational Modelling, DOI 10.1007/978-94-007-6863-5, © Springer Science+Business Media Dordrecht 2013 263 264 Bunker, 110, 112 Butler, 117, 160, 165, 190, 218, 261 C C60 , 38–40, 49, 102, 157, 158, 190 Calabrese, 40, 49 Cauchy theorem, 154 Cayley, 26, 27, 29, 171–173, 176, 178, 179, 189, 257 graph, 26, 27, 93 theorem, 29, 173 Cayley–Klein parameters, 171–173, 176, 178, 257 Centrosymmetry, 45 Ceulemans, 72, 93, 121, 139, 149, 152, 157, 170, 188 Character, 17, 40, 56–62, 64, 69, 70, 75–77, 84, 115, 119, 128, 135, 140, 152, 165, 167, 169, 179, 235 string, 58, 60–62, 64, 70, 115, 135 table, 57–60, 62, 64, 70, 100, 116, 176–178, 235 theorem, 51, 58, 63, 70, 75, 78, 115, 116, 120, 152, 178, 179 Charge-transfer (CT) transitions, 139 Chatterjee, 149 Chemical bonding, 153 Chemical shift, 91 Chibotaru, ix, 93, 102 Child, 191, 261 Chirality, 21, 46 Chromophore, 138, 147, 255 Circular dichroism, 46, 113, 138, 139, 141, 143–145, 147, 159, 161, 261 Circular polarization, 19, 46, 113, 132, 138, 144, 145, 152, 153, 158, 159 Clar, 157–159 Class, 32–34, 45, 57–59, 62, 64, 70, 71, 75, 93, 102, 178–180, 189, 257 Clebsch-Gordan (CG) coefficient, 117, 118, 121, 122, 128, 183 Closed shell, 87, 143 Closure, 24, 33, 61, 69, 136, 137 Clusters, 101, 113, 161, 261 Commutator, 8, 133, 147, 245 Compernolle, ix, 170, 190 Condon, 174 Condon–Shortley convention, 174 Cone, 43, 44 Conical intersection, 132 Conjugation, 32, 158 class, 32 complex, 14–17, 118, 121, 167, 180, 181, 183, 247 Index Conrotatory, 136, 137 Contact term, 140, 144 Continuity condition, 175 Coordinate system, 10, 38, 78, 138, 139, 141, 145, 147 Cartesian, 10, 38, 138 D2 setting, 217 tetragonal, 93–95, 108, 131–133 trigonal, 37 Coordination compounds, 37 Coset, 30–33, 45, 71–73, 248, 252 Cotton, 191, 261 Coulomb interaction, 109 Coulson, 84, 97, 101, 102 Coulson–Rushbrooke theorem, 97 Coupling channel, 128 Coupling coefficients, 117–119, 121, 122, 124, 128, 130, 131, 135, 139, 142, 143, 159, 168, 173, 174, 183, 185, 186, 221–235, 241, 243, 255, 257, 261 exchange symmetry, 122 Crystal-field potential, 169 Crystallography, 40 Cube, 35, 37, 39, 54, 93, 99, 150, 169, 170, 189, 249, 257 Cuboctahedron, 170 Curie principle, 104, 105 Curl, 38, 49 Cvetkovi´c, 84 Cyclobutene, 136 Cyclohexadiene, 137 Cylinder, 34, 40–43 D Day, 139 Day and Sanders model, 139, 140 Degeneracy, 56, 62, 70, 98, 100, 102, 103, 105, 106, 128, 132, 161, 180–183 Deltahedron, 154, 157 Determinant, 15, 16, 45, 54, 83, 125, 126, 159, 164, 171, 172, 246 unimodular, 171 Diatomic, 43 Dihedral, 40, 62, 159, 194, 247 Dipole, 7, 89, 103–105, 113, 127, 139–145, 159, 186, 253, 254 induced, 104 moment, 89, 103–105, 127, 140, 144, 159 Dirac, 11 Dirac notation, 11, 67, 133 Direct square, 119–121, 129, 165, 177, 184, 188, 253, 255, 257 Dish, 102 Index Dish Archimedene, 93 Disrotatory, 137 Dissymmetry, 104 Distortion modes, 130, 133 Dodecahedrane, 38, 49 Dodecahedron, 35, 36, 38, 39, 99, 156, 157, 170 Domcke, 132, 161 Donor–acceptor interactions, 139, 140 Doob, 84 Double group, 176–178, 180, 184, 189, 235, 257, 258 Dual, 7, 10, 35, 39, 153, 154, 156, 157, 170 Dynamic symmetry, 111, 112 E Edge representation, 151–156 Edmonds, 165, 190 Eigenfunctions, 53–56, 61, 81, 85, 89, 90, 97, 103, 105–107, 124, 132, 161 Eigenvalues, 19, 54, 81, 83, 84, 86, 87, 91, 94, 97, 102, 105, 107, 108, 188, 189 Electric, 7, 21, 43, 46, 47, 69, 104, 113, 127, 146, 148, 153, 205, 206 see Stark effect crystal field, 69 dipole, 7, 104, 113, 127 field (E), 42, 47, 96, 151–155, 158, 159 symmetry breaking, 69, 205, 206 Electron diffraction, 83 Electron precise, 155 Enantiomers, 160 Equivalent electrons, 124, 125 Euclid, 39 Euler equation, 54, 150, 251 Euler theorem, 113, 152, 161 Excited state, 41, 135, 145, 148 Exciton, 145, 146, 148, 160, 254 F Face representation, 151 Fagan, 40, 49 Faraday effect, 2, 46 Ferrocene, 40, 41, 112, 252 Fibre bundle, 148, 149, 157, 158 Flint, 134 Fowler, ix, 93, 102, 152, 156, 157, 161, 211, 221, 235, 261 Franck-Condon principle, 144 Fries, 157–159 Frobenius, 56, 72, 76, 93, 148, 249 Frontier orbitals, 138, 156, 157 Fullerenes, 156, 157, 159, 161, 190 265 Function space, 7, 10, 12–16, 23, 51–53, 55–57, 59–61, 63, 64, 66–68, 71, 78, 82, 93, 97, 106, 113, 114, 134, 149, 163, 165–167, 170, 181, 182, 184, 247 G Gauge, 51, 88–90 Genealogical tree, 29, 30, 36, 69 GFP protein, 43 Gilmore, 165, 190 Graph, 26, 27, 36, 84, 92, 95–98, 256 automorphism, 26, 27, 92, 95, 98 bipartite, 97, 98, 256 Great Orthogonality Theorem (GOT), 63 Griffith, 106, 112, 117, 160, 175, 184, 185, 190, 215, 218, 221, 235, 261 Group see Lie groups see point groups Abelian, 27 alternating, 28, 36, 40, 152 cyclic, 27, 28, 31, 39, 41, 42, 44, 62, 86, 87, 100, 108, 165, 181, 252 definition, 1, 28, 32, 98, 108, 109, 111, 128, 176 double, 19, 40, 62, 126, 175–180, 184, 189, 190, 235, 257, 258 generator, 27, 30, 39, 41, 45, 63, 107, 123 halving, 45, 48, 164, 252 orthogonal, 35, 63, 163, 164, 172 permutation, 25, 28, 29, 73, 108–112, 123, 126, 148, 165, 252 symmetric, 27, 29, 36, 42, 55, 58, 62, 76, 93, 100, 104, 123, 148, 150–152, 165, 177, 247, 251, 256 unitary, 57, 58, 163, 171 H Haake, 180, 190 Halevi, 136, 161 Half-integral momentum, 134 Hamiltonian, 8, 17, 18, 53, 84, 85, 89, 92, 103, 105–110, 113, 129, 131, 133–135, 141, 147, 170, 171, 173–175, 181–190, 258, 259 Heath, 38, 49 Helicity, 138, 144 Hemisphere, 176, 179, 257 Hermitian, 19, 170, 183, 247 Hessian, 80, 81, 83, 84 Hexadecapole, ix, 168, 169 Hilbert space, 8, 53 Hilton, 152, 161 Hoffmann, 136, 155 HOMO, 98, 160 266 Homomorphism, 11, 14, 172, 173, 175 Hückel theory, 84, 87, 106 Hydrocarbon, 85, 95, 97 I Icosahedron, 35, 38–40, 154, 170, 217 Indistinguishability, 110, 123 Induction, 32, 51, 69, 71–76, 78, 93, 96, 99, 148–150, 158, 207, 208, 210–214, 249 Integral, 12, 84, 114, 115, 127, 134 hopping, 84 overlap, 114, 115, 127 resonance, 84 Intensity, 140, 143, 144, 148 Intra-ligand (IL) transitions, 138, 144, 145, 147 Inversion, 2, 17, 18, 34, 40, 42, 44–47, 61, 62, 109–112, 167, 180, 251, 252 Irrep, see representation, irreducible Isolobal analogy, 155 Isomorphism, 28, 172 Isotope shift, 83 J Jahn, 128 Jahn–Teller effect, 70, 128 Judd, ix, 134, 161 K Katzir, 103, 112 Kinetic energy, 80, 88, 89, 109 Klein, 27, 171–173, 176, 178, 179, 189, 257 see Cayley four-group, 27 Kobayashi, ix, 44, 49, 261 Köppel, 132 Kramers’ degeneracy, 106 Kronecker delta, 12, 13, 65 Kroto, 38, 49 L Lagrange theorem, 31 Lagrangian, 80 Lanthanides, 69 Le Bel, 36 Leapfrog, 156, 157, 159 Lie groups, 165, 190 SO(3), O(3), 35, 163, 164, 167, 172, 173, 175, 204, 210, 237 SU(2), U(2), 171–173, 175, 179, 189, 246 Ligand orbitals, 140, 144, 145 Ligator, 138–140 Lijnen, ix, 121, 160, 170, 190 Linear dichroism, 138, 139, 161 Linearly polarized, 46 Lipscomb, 38, 48 Index London, 88 London approximation, 89 Longuet-Higgins, 109–112, 261 Lulek, ix, 149, 161 LUMO, 98, 158 M Magnetic, 21, 41, 42, 46, 47, 63, 69, 87–91, 104, 110, 113, 127, 146–148, 153, 159, 170, 173, 180, 181, 184, 187–189, 205, 206, 253, 254, 260 see Faraday see London dipole, 89, 113, 127, 159, 253, 254 field (B), 47 flux, 91 symmetry breaking, 41, 69, 205, 206 Mallion, ix, 84, 97, 101, 102 Malone, 40, 49 Manifold, 100, 124, 125, 135, 168, 188 Manolopoulos, 156 Martins, 170 Mass-weighted coordinates, 80, 82 Matrix, 3, 4, 6, 7, 10–16, 19, 22, 23, 45, 51–54, 56, 59, 60, 63, 65, 73–75, 77, 80, 81, 83–85, 90, 92, 97, 98, 106–108, 114, 115, 126, 127, 129, 131, 135, 159, 164, 165, 167, 170–175, 179, 180, 182, 186–189, 245–247, 251, 253, 257–260 adjacency, 84, 85 circulant, 85 complex conjugate, 13, 14, 16, 63, 171, 173, 174 diagonalization, 54, 84 element, 14, 22, 23, 54, 59, 64, 73, 74, 83–85, 92, 108, 115, 116, 126–132, 135, 137, 140, 142, 143, 147, 164, 172, 173, 175, 183, 253 orthogonal, 16, 19, 63, 163, 172, 245, 247 trace, 59, 60, 64, 83, 115, 127, 167, 170 transposed, 10, 16, 171 unitary, 11, 14–16, 19, 57, 107, 127, 163, 171, 179, 181, 246 Matsuda, 44, 49 M(CO)3 fragments, 155 Melvin, 149, 161 Methane, 76 Mexican hat potential, 131, 132 Mingos, 156, 161 Miura, 93, 102 Molecular-symmetry group, 109 Monopole, 153 Mulliken, 62 Mulliken symbols, 62, 191 Index Multiplication table, 22–29, 31, 47, 177, 189, 247, 257, 258 Mys, 188, 190 N Neumann principle, 104 Nordén, 138 Normal modes, 81, 100, 129, 250 O O’Brien, 38, 49 OCAMS, 161 Octahedron, 35, 37, 70, 95, 96, 107, 138, 148, 149, 154, 159, 168, 170, 187–189, 216 Octupole, 186 Odaba¸si, 174 Ojha, 95, 102 O’Leary, 84, 101 Omnicapping, 101, 157 Opechowski, 178 Opechowski theorem, 178 Operator action on a function, 4, 51, 52, 181 action on a point, action on an operator, 1, 66 anti-linear, 11 congruence, 46 idempotent, 69 inverse, 18, 23, 171 inversion, 2, 17, 18, 109, 110, 112 ladder, 65, 67, 108, 112 linear, 11–13, 16–18, 53, 105, 114, 133, 146, 147, 180, 188, 253 projection, 65, 67, 86, 100 proper and improper, vii, 163, 165 rotation, 2, 8–10, 16, 23, 53, 86, 112, 133, 134, 176, 180, 245, 246, 251 rotation-reflection, 40 spin, 125, 134, 170, 173, 180, 181, 184–186, 188–190, 259, 260 Orbitals, 4–7, 11, 23, 41, 51–53, 55, 59, 60, 63, 76–78, 84–89, 94, 95, 98–101, 107, 108, 115, 117, 123, 136–145, 149, 155–157, 159–161, 165, 168, 169, 204, 211, 249, 250, 253, 256 d, 78, 95, 107, 108, 165, 168, 204 f, 78, 149 molecular, 4, 51, 77, 84, 85, 95, 98, 99, 101, 149, 156, 161, 211, 249 Order of a group, 32, 33, 59, 114 Organo–transition–metal complexes, 78, 107, 138 Orgel, 138, 161 Orthonormality, 15, 16, 116, 118, 135 Orthorhombic, 34, 40, 71, 131–133 267 P Paquette, 38, 49 Parity permutational, 28, 36, 110, 118 space, 167, 182 Parker, 215, 218, 261 Partial derivative, 129 Partitioning, 30, 31 Pauli exclusion principle, 125 Pauling, 77 Pekker, 134, 161 Permutation group, 126, 165 Perturbation theory, 135 Phase convention, 108, 167 Phillips, 191, 261 Platonic solids, 34–36, 153, 154, 170 Point groups, 34–37, 39, 41–45, 47, 62, 117, 121, 181, 192 C2 , Ci , Cs , 27, 29, 30, 34, 39, 45–47, 72, 75, 94, 104, 192, 205, 206, 211–214, 247, 249, 251–253 C3v , 44, 62 Cn , Cnh , Cnv , 27, 41, 43, 44, 62, 72, 76, 90, 95, 151, 192, 195, 196, 206 D2 , 27, 176, 208, 211, 217, 218, 247 D2h , 34, 38, 39 D6h , 41, 42, 87, 100, 199, 212, 249, 251 Dn , Dnd , Dnh , 40, 45, 90, 194, 198, 199, 206 I , Ih , 35, 36, 39, 40, 93, 99, 120, 205, 209, 210, 214, 218, 237, 248–250 O, Oh , 35–37, 39, 70–72, 78, 93–98, 116, 118, 122, 149, 184, 185, 205, 209, 210, 214 S2n , 42, 62 T , Td , Th , 35–40, 45, 47, 62, 76, 99, 150–152, 181, 205, 208, 209, 213, 220, 249 Polarization function, 149 Polyhedrane, 155 Polyhedron, 48, 93, 99, 148, 150–154, 248 Potential energy, 80, 89, 110 Prisms, 40, 43 Product antisymmetrized, 119, 121, 125, 219 direct group, 45, 46, 176 direct representation, 125 multiplicity, 121, 128, 184 scalar, 10–12, 14, 58, 59, 63, 80, 115, 118, 127, 128, 154, 174, 245 symmetrized, 120, 121, 125, 177, 184, 219, 252, 253, 259, 260 Pseudo Jahn–Teller effect (PJT), 134, 136, 137 Pseudo-doublet, 188 268 Pseudo-scalar representation, 152 Pythagorean tradition, 35 Q Qiu, ix, 218, 261 Quantum chemistry, vii, 8, 103, 104, 106, 108, 110, 112, 149 Quartet spin state, 184, 188 Quinn, 211, 261 R Rank, 168, 259 Reciprocity theorem, 72, 93, 148 Reduced matrix element, 127, 128, 142, 173 Representation determinantal, 126 faithful, 23, 175 ground, 72–74, 126 irreducible, 55–59, 62, 63, 65, 68, 106, 107 mechanical, 148, 149, 154, 155, 211 positional, 73, 74, 148–150, 211 pseudoscalar, 152, 153, 156 Reversal, 11, 16–18, 63, 91, 109, 110, 129, 180–184 space, 18, 63, 109, 110, 181–183 time, 11, 16–18, 63, 91, 110, 129, 180–184 Rhombohedral, 168, 169, 189 Right-thumb rule, 90 Ring closure, 136, 137 Rodger, 138, 161 Rodrigues, 174 Rosenfeld equation, 254 Rotation, 1–10, 16, 22, 23, 32, 34, 40–45, 48, 52, 53, 61, 78, 83, 86, 87, 90, 95, 107, 109, 111, 112, 133, 134, 139, 151–154, 163, 164, 167, 172, 175, 176, 179, 180, 189, 197, 235, 245, 246, 248, 249, 251, 255 bodily, 52, 61, 111, 112 matrix, 3, 6, 7, 10, 16, 23, 45, 52, 53, 83, 163–165, 167, 172, 175, 179, 180, 245, 251 operator, 2, 8–10, 16, 23, 53, 86, 112, 133, 134, 176, 180, 245, 246, 251 optical, 36 pole, 2, 3, 164, 175, 176 Rotatory strength, 148, 254 Rouvray, 97, 102 Ru(bipy)3 , 144 Ruthenocene, 40, 41 S Sachs, 84 SALC, 53, 55, 59, 65, 67, 76, 77, 86, 152 Index Salem, 84, 101 Samuel, 93, 102 Sanders, 139 Satten, 189, 190 Schäffer, 39 Schrödinger stationary equation, 17, 106 time-dependent equation, 17 Schulman, 93, 102 Schur, 56, 61, 63, 107 Secular equation, 54, 81, 132, 186 see Zeeman Selection rule, 65, 115, 128–130, 135, 137, 142 Shapere, 5, 134 Shimanouchi, 81, 101 Similarity transformation, 32, 72 Singleton, 40, 62 Site symmetry, 75, 76, 93, 94, 148, 149, 249 Smalley, 38, 49 sp3 -hybridization, 76, 77, 150 Spherical harmonics, 165–168, 218 Spin, spinor, 170, 171, 173–175, 177, 180, 183–185, 259 Spin-orbit coupling, 175, 186, 235, 261 Splitting scheme, 71, 121 Stabilizer, 30, 33, 39, 72 Standard fibre, 149 Stanger, 93, 102 Stark effect, 46 Stereo-isomers, 70 Stone, 261 Stretching modes, 82, 83 Subduction, 29, 51, 69–73, 75, 87, 165, 167, 168, 184, 207–210, 212, 214, 218, 237, 249 Subelement, 73–75 Subgroup, 29–34, 36–40, 43, 45–48, 69–72, 101, 128, 164, 176, 248, 251, 252 Subporphyrin, 44 Subrepresentation, 65, 68, 126 Sum rule, 75, 136, 148 Symmetry see operators see point groups active definition, 1, 7, 10, 22, 46, 130, 132 breaking, 30, 34, 41, 69, 70, 128, 136, 205, 206 dynamic, 41, 111, 112 hidden, 95, 98, 106 spherical, 34, 35, 40, 117, 149, 163–166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 204, 205 Index T Takeuchi, 44, 49 Tangential modes, 78 Teller, 128 Tetragonal compression, 133 Tetragonal elongation, 132 Tetrahedron, 35–37, 76, 77, 99, 150–152, 154 Tight-binding model, 84 Time reversal, 11, 16–18, 63, 180–183 Time-even, time-odd, 181, 182 Topology, 150 Trace, 57, 59–61, 64, 68, 83, 109, 115, 119, 120, 127, 165–167, 170 Transfer term, 140–144 Transition dipole, 140, 144 Translation, 10, 17, 18, 36, 43, 48, 78, 82, 109, 154, 158, 249 Triangular condition, 128 Triphenylmethyl, 95–98, 106 Trischelate complex, 177 Trivalent polyhedron, 93, 154 Troullier, 170 Truncation, 157 Twisted cylinder, 43 Two-well potential, 132 269 Van’t Hoff, 36 Vector, 3, 7, 10–15, 41, 43, 52–54, 60, 63, 77, 79, 80, 87–90, 104, 106, 140, 142, 146, 163–165, 170–176, 247 axial, 41 polar, 43, 104 potential, 80, 87–89, 146 row versus column, 7, 10, 14, 53, 63, 247 Vertex representation, 150 Vibrational modes, 81, 131, 149 Vibronic interaction, 160 Vollhardt, 93, 102 W Walỗerz, 188 Wales, 156, 161 Walsh diagram, 3, 132, 165 Wigner, 17, 19, 113, 117, 126–129, 135, 136, 140, 142, 160, 173, 181, 185, 190 Wigner–Eckart theorem, 113, 126–129, 135, 136, 140, 173, 185 Wilczek, 5, 134 Woodward–Hoffmann rule, 136 Wunderlich, 38 Wylie, 152 U Uni-axial, 43 Unit element, 14, 22, 24, 27–30, 32, 38, 44, 59, 61, 69, 152, 164, 172, 175, 176, 245–247, 251 Uranium, 78, 80, 83 Y Yarkony, 132 Yersin, 144 Yu-De, 81 Yun-Guang, 81 V Vanquickenborne, ix, 139, 161, 261 Z Zeeman interaction, 170, 182, 186 ... number A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given A.J Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and... with the usual convention in chemistry textbooks In this script the part of the observer is played by A.J Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and Computational... the xz plane as shown in the figure We attach labels A, B, C to distinA.J Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and Computational Modelling, DOI 10.1007/978-94-007-6863-5_3,
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