Group Theory Applied to Chemistry

276 Pages·2013·3.07 MB·English

by  Arnout Jozef Ceulemans (auth.)

Theoretical Chemistry and Computational
Modelling
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Modern Chemistry is unthinkable without the achievements of Theoretical and Computa-
tional Chemistry. As a matter of fact, these disciplines are now a mandatory tool for the
molecularsciencesandtheywillundoubtedlymarkthenewerathatliesaheadofus.Tothis
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reticalChemistryandComputationalModeling(TCCM).Theaimofthiscourseistodevelop
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Arnout Jozef Ceulemans
Group Theory
Applied to
Chemistry
ArnoutJozefCeulemans
DivisionofQuantumChemistry
DepartmentofChemistry
KatholiekeUniversiteitLeuven
Leuven,Belgium
ISSN2214-4714 ISSN2214-4722(electronic)
TheoreticalChemistryandComputationalModelling
ISBN978-94-007-6862-8 ISBN978-94-007-6863-5(eBook)
DOI10.1007/978-94-007-6863-5
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To mygrandson Louis
“Theworldissofullofanumberofthings,
I’msureweshouldallbeashappyaskings.”
RobertLouisStevenson
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
andmathematics.AccordingtoBarut,1 thesymmetryofaphysicalsystemmaybe
lookedatinanumberofdifferentways.Wecanthinkofsymmetryasrepresenting
• theimpossibilityofknowingormeasuringsomequantities,e.g.,theimpossibility
ofmeasuringabsolutepositions,absolutedirectionsorabsoluteleftorright;
• theimpossibilityofdistinguishingbetweentwosituations;
• theindependenceofphysicallawsorequationsfromcertaincoordinatesystems,
i.e.,theindependenceofabsolutecoordinates;
• theinvarianceofphysicallawsorequationsundercertaintransformations;
• theexistenceofconstantsofmotionsandquantumnumbers;
• theequivalenceofdifferentdescriptionsofthesamesystem.
Chemistsaremoreusedtotheoperationaldefinitionofsymmetry,whichcrystallo-
graphershavebeenusinglongbeforetheadventofquantumchemistry.Theirball-
and-stickmodelsofmoleculesnaturallyexhibitthesymmetrypropertiesofmacro-
scopicobjects:theypassintocongruentformsuponapplicationofbodilyrotations
about proper and improper axes of symmetry. Needless to say, the practitioner of
quantumchemistryandmolecularmodelingisnotconcernedwithballsandsticks,
butwithsubatomicparticles,nuclei,andelectrons.Itishardtoseehowbodilyro-
tations,whichleaveallinterparticledistancesunaltered,couldaffectinanywaythe
studyofmolecularphenomenathatonlydependontheseinternaldistances.Hence,
thepurposeofthebookwillbetocometotermswiththesubtlemetaphorsthatre-
lateourmacroscopicintuitiveideasaboutsymmetrytothemolecularworld.Inthe
endthereadershouldhaveacquiredtheskillstomakeuseofthemathematicaltools
ofgrouptheoryforwhateverchemicalproblemshe/shewillbeconfrontedwithin
thecourseofhisorherownresearch.
1A.O. Barut, Dynamical Groups and Generalized Symmetries in Quantum Theory, Bascands,
Christchurch(NewZealand)(1972)
vii
Acknowledgements
The author is greatly indebted to many people who have made this book possi-
ble: to generations of doctoral students Danny Beyens, Marina Vanhecke, Nadine
Bongaerts,BrigitteConinckx,IngridVos,GeertVandenberghe,GeertGojiens,Tom
Maes, Goedele Heylen, Bruno Titeca, Sven Bovin, Ken Somers, Steven Comper-
nolle, Erwin Lijnen, Sam Moors, Servaas Michielssens, Jules Tshishimbi Muya,
andPieterThyssen;topostdocsAmuthaRamaswamy,SergiuCojocaru,Qing-Chun
Qiu,GuangHu,RuBoZhang,FanicaCimpoesu,DieterBraun,StanislawWalçerz,
WillemVandenHeuvel,andAtsuyaMuranaka;tothemanycolleagueswhohave
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,IsaacBersuker,ClaudeDaul,HartmutYersin,MichaelAtanasov,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
manuscriptwithmeticulouscareandprovidednumerouscorrectionsandcomments,
but also gave expert insight into the intricacies of English grammar and vocabu-
lary. I am very grateful to L. Laurence Boyle for a critical reading of the entire
manuscript,takingoutremainingmistakesandinconsistencies.
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
Jungbluthwhorescuedmefromeverythingthatcouldhavedistractedmyattention
from writing this book. I remain grateful to Luc Vanquickenborne who was my
mentorandpredecessorinthelecturesongrouptheoryatKULeuven,onwhichthis
bookisbased.Mythoughtsofgratitudeextendalsotobothmydoctoralstudent,the
lateSamEyckens,andtomyfriendandcolleague,thelatePhilipTregenna-Piggott.
Bothstartedthejourneywithmebut,atanearlystage,weretakenawayfromthis
life.
MyfinalthanksgotoMonique.
ix
Contents
1 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 OperationsandPoints . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 OperationsandFunctions . . . . . . . . . . . . . . . . . . . . . . 4
1.3 OperationsandOperators . . . . . . . . . . . . . . . . . . . . . . 8
1.4 AnAideMémoire . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 FunctionSpacesandMatrices . . . . . . . . . . . . . . . . . . . . . . 11
2.1 FunctionSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 LinearOperatorsandTransformationMatrices . . . . . . . . . . . 12
2.3 UnitaryMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 TimeReversalasanAnti-linearOperator . . . . . . . . . . . . . . 16
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 TheSymmetryofAmmonia . . . . . . . . . . . . . . . . . . . . . 21
3.2 TheGroupStructure . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 SomeSpecialGroups . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 OverviewofthePointGroups . . . . . . . . . . . . . . . . . . . . 34
SphericalSymmetryandthePlatonicSolids . . . . . . . . . . . . 34
CylindricalSymmetries . . . . . . . . . . . . . . . . . . . . . . . 40
3.8 RotationalGroupsandChiralMolecules . . . . . . . . . . . . . . 44
3.9 Applications:MagneticandElectricFields . . . . . . . . . . . . . 46
3.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
xi
xii Contents
4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Symmetry-AdaptedLinearCombinationsofHydrogenOrbitalsin
Ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 CharacterTheorems . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 CharacterTables . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 MatrixTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 ProjectionOperators . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 SubductionandInduction . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Application:Thesp3 HybridizationofCarbon . . . . . . . . . . . 76
4.8 Application:TheVibrationsofUF . . . . . . . . . . . . . . . . . 78
6
4.9 Application:HückelTheory . . . . . . . . . . . . . . . . . . . . . 84
CyclicPolyenes . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
PolyhedralHückelSystemsofEquivalentAtoms . . . . . . . . . . 91
TriphenylmethylRadicalandHiddenSymmetry . . . . . . . . . . 95
4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 WhathasQuantumChemistryGottoDowithIt?. . . . . . . . . . . 103
5.1 ThePrequantumEra . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 TheSchrödingerEquation . . . . . . . . . . . . . . . . . . . . . . 105
5.3 HowtoStructureaDegenerateSpace . . . . . . . . . . . . . . . . 107
5.4 TheMolecularSymmetryGroup . . . . . . . . . . . . . . . . . . 108
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 OverlapIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 TheCouplingofRepresentations . . . . . . . . . . . . . . . . . . 115
6.3 SymmetryPropertiesoftheCouplingCoefficients . . . . . . . . . 117
6.4 ProductSymmetrizationandthePauliExchange-Symmetry . . . . 122
6.5 MatrixElementsandtheWigner–EckartTheorem . . . . . . . . . 126
6.6 Application:TheJahn–TellerEffect . . . . . . . . . . . . . . . . . 128
6.7 Application:Pseudo-Jahn–Tellerinteractions . . . . . . . . . . . . 134
6.8 Application:LinearandCircularDichroism . . . . . . . . . . . . 138
LinearDichroism . . . . . . . . . . . . . . . . . . . . . . . . . . 139
CircularDichroism. . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.9 InductionRevisited:TheFibreBundle . . . . . . . . . . . . . . . 148
6.10 Application:BondingSchemesforPolyhedra. . . . . . . . . . . . 150
EdgeBondinginTrivalentPolyhedra . . . . . . . . . . . . . . . . 155
FrontierOrbitalsinLeapfrogFullerenes . . . . . . . . . . . . . . 156
6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7 SphericalSymmetryandSpins . . . . . . . . . . . . . . . . . . . . . 163
7.1 TheSpherical-SymmetryGroup . . . . . . . . . . . . . . . . . . . 163
7.2 Application:Crystal-FieldPotentials . . . . . . . . . . . . . . . . 167
7.3 InteractionsofaTwo-ComponentSpinor . . . . . . . . . . . . . . 170