Symmetry of Crystals and Molecules

457 Pages·2014·8.262 MB·English

by  Mark Ladd

SYMMETRY OF CRYSTALS AND MOLECULES
(cid:2) (cid:3)
TheorthorhombicspacegroupFddd Fd2d2d2 ;originon222,at−1/8,−1/8,−1/8from1¯.
Symmetry
of Crystals
and Molecules
MARK LADD
FormerlyHeadofChemicalPhysics,UniversityofSurrey
“Athingissymmetricalifthereissomethingyoucandotoitsothatafteryou
havefinisheddoingit,itlooksthesameasbefore.”
HermannWeyl(1983):Symmetry
3
3
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Foreword
The observation that stone-age people produced objects with a measure of
symmetry is not as unexpected as often made out. The opposite would actu-
ally be much more surprising. Totally asymmetric objects in the real world
are extremely rare, if not altogether non-existent. Some objects are admit-
tedly more symmetrical than others, and this rule applies without exception.
Perfectlysymmetricalobjectsarethereforeasrareasthetotallyasymmetrical.
The most perfectly fashioned sphere or the most exquisite crystal contains
some blemish when examined in sufficient detail. The distinction drawn by
chemistsbetweenchiralandachiralmoleculesismeaningless.Iconsideritall
a matter of resolution; any molecule has a degree of chirality between zero
andone.
Thepowerfulconceptofsymmetryappliedwithsuchgoodeffectinscience
isanabstractidealizationwithprecisedefinitiononlyinthesenseoflogic.Like
the absolute zero of temperature or the relativistic limit to material velocity
it can be described mathematically, but remains physically unreachable. The
mathematicaltoolthatdescribessymmetryisknownasgrouptheory.Interms
ofKlein’sfamousErlangenprogramanygeometryischaracterizedbyagroup
oftransformationsunderwhichitspropositionsremainvalid.Inthissensethe
Euclideangroupismorerestrictivethantheprojectivegroup,andbothofthese
are less general than the continuous topological group. Topology allows any
amountofstretchingorcompression,withouttearing;inthismostgeneralcase
thepopularnotionofsymmetrydisappears,butremainsembeddedinthegroup
structure.
Toproduceamonographonatopicaswideassymmetryneedssomecour-
age. In doing so Mark Ladd wisely stipulated at the outset that the book is
restrictedtocrystalandmolecularstructure,forwhichheisperhapsuniquely
qualified. Molecular structure in this context strictly refers to the nuclear
framework as revealed in crystallographic analyses. The appropriate sym-
metry group in this case is the three-dimensional discrete Euclidean group,
commonlyreferredtoasspace-groupsymmetry.
As a seasoned crystallographer, albeit of forgotten vintage, I do not find
much new in this presentation, but I marvel at the comprehensive detail and
rigour.Inthedaysofvisuallyestimatedx-rayintensitiesandBeevers–Lipson
strips,crystalgeometryandspace-groupsymmetrywereexperiencedasprac-
ticalnecessities.Itisnotclearwhereandhowthenewgeneration,whotwiddle
a few knobs to produce a crystal structure, acquires this know-how. Working
through the numerous examples could well provide the necessary hands-on
experience. This is where I see this book becoming essential reading, if not
prescribed for intensive study. Too many papers are being published around
dubioussymmetryarguments.
vi Foreword
Up to a point the analysis is based on Newton’s approach that relies only
onconclusionswhichcanbeformulatedmathematically—withouttheneedof
hypotheses.Inoticeinparticularhowtheauthorwiselyrefrainsfromoffering
physicalinterpretationsforobservedaspectssuchaschirality,opticalactivity
andothersymmetryrelatedphenomena,identifiedandmanipulatedpurelyas
point group variables. Unfortunately, the temptation to reinterpret molecular
symmetryintermsofhybridorbitals,aconceptpopularinquantumchemistry,
wasclearlyirresistible.
ItiscommonchemicalpracticetovisualizeLewistypeelectronpaircova-
lent interaction in terms of real orthogonal classical functions, called atomic
orbitals.Thesefunctions,suchastheso-calledp ,p andp orbitalsareoften
x y z
confusedwithquantummechanicalwavefunctions,whichtheyresembleafter
a fashion. However, these functions cannot be described by quantum num-
bers,conflictwiththeexclusionprinciple,donotrepresentsurfaceharmonics,
which are complex and hence cannot be used to describe electronic orbital
angularmomentumnorspin.Atomicorbitalsareroutinelyusedinlinearcom-
bination, in the same way as Euclidean unit vectors, to visualize directed
covalent interaction, inferred from crystallographic analyses, to derive point
groupsymmetryofmolecularspecies.Theseconceptsare,admittedly,noteas-
ily avoided, being so thoroughly entrenched in chemical group theory, but a
wordofwarningisperhapsnotmisplaced.Fromthesymmetrypointofview,
real p and p functions, which represent linear vibrations, can never be con-
x y
sideredequivalent totherotationdefined byacomplex function. Itisnotthe
pointgroupdescriptionofmoleculesthatisatissue,butthewayitisusedto
representhybridorbitals.Beingoffundamental importance, itcouldserveas
anencouragementforthereadertomakeanindependentassessment,oneway
ortheother.
ProfessorJanBoeyens
UniversityofPretoria
Preface
The study of crystal and molecular symmetry is often considered difficult by
thosemeetingitforthefirsttime.Ibelievethattherearetwoparticularreasons
forthisproblem.First,crystalsandmolecules,unlikethemorefamiliarplane
figures,arethree-dimensionalbodiesanditcanbedifficulttotakeinsimultan-
eously all parts of such a body and see them in relation to the whole. In two
dimensional figures, such as a rectangle or a regular hexagon, it can be done
readilyandweappreciatethesymmetrywithoutdifficulty.Thesecondreason,
not unrelated to the first, lies in the unfamiliarity of three-dimensional rela-
tionships, notwithstanding we encounter daily the three-dimensional spatial
aspectsofourspace-timecontinuum.
The purpose of this book is to present crystal and molecular symmetry in
a straightforward and practical manner. It evolves from lecture courses given
overmanyyears,atundergraduateandpostgraduatelevels,tothosepursuing
crystallography, chemistry and materials science. It makes liberal use of ste-
reoviewsandcomputeraidstounderstandingthetextualmaterial.Mostofthe
stereoscopic illustrations were drawn with the program PLUTO, by courtesy
of Dr WDS Motherwell, University of Cambridge, and aids to stereoviewing
are discussed in an appendix. Each chapter concludes with a set of problems
anddetailedtutorialsolutionsareprovided.Computerprogramsrelevanttothe
textandtheproblemshavebeendevised,andareavailableviathepublisher’s
website,<www.oup.co.uk/companion/ladd>.
Itisinformativetostudyeverydayobjectsinrelationtosymmetry:aglass
andabeerjug;achairandatable;atiledfloorandabrickwall;aDalmatian
and a Dobermann. In this way, symmetry can become clearer—it is not only
about crystals and molecules. Two notations for symmetry are in use, the
Hermann–Mauguin and Schönflies notations; each has a place in crystal and
molecularsymmetry,andeachisdiscussedindetail.
The study of the symmetry of crystals and molecules may be begun either
with lattices or with a description of the external symmetry of crystals. The
formerhasadvantagesinashortlecturecourseaimedatgettingtogripswith
X-ray diffraction. However, a morphological approach has been chosen for
this book because, in this way, the book should have a more general appeal
anditspacealittleslower.Timeisneededinordertoassimilatetheconcepts
of symmetry and to consolidate them into a working knowledge of the sub-
ject. It is hoped that this book will be helpful to all those meeting symmetry
forthefirsttime,whatevertheirspecialization,andwillpreparethereaderfor
studyofthecurrentdefinitivetextsonsymmetry,the Internationaltablesfor
crystallography,VolumesAandA1.
Pre-knowledgeisconfinedmainlytomathematicsandphysicalsciences,but
no more than would be studied at A-level. A little more mathematics, along
viii Preface
lines indicated, is an advantage in certain contexts, and appropriate guidance
isprovidedinappendicesinordernottointerrupttheflowofthetext.
Themainnotationusedinthebookisindicatedinthefrontmatter.Inaddi-
tion, equations are indicated in the form Eq. (1.2) wherever they occur, or as
Eq.(A1.2)ifinAppendixA1;forcompactness,particularlyinmatrices,aneg-
ativenumber,–1,forexample,isfrequentlywrittenas1;symmetryelements
arewrittenas2orm,thecorrespondingsymmetryoperationsoroperatorsas
2ormandthematricesrepresentingtheseoperationsas2orm.
I would like to express my thanks to those authors and publishers for
permission to reproduce the figures which bear an appropriate acknowledge-
ment. In particular, many illustrations have been taken from four particular
sources:(a)StructuredeterminationbyX-raycrystallography.4thed.Kluwer
Academic/PlenumPublishing,2003;(b)Crystalstructures:latticesandsolids
instereoview,1999;(c)Structureandbondinginsolidstatechemistry,1979,
both Ellis Horwood Ltd, UK; (d) International tables for X-ray crystallo-
graphy,Vol.1,1965;(e)Internationaltablesforcrystallography,Vol.A;and
(f)SymmetryaspectsofM.C.Escher’speriodicdrawings,thelatterthreefrom
the International Union of Crystallography (IUCr). Where used, the copy-
right holders will be acknowledged as ‘Reproduced by courtesy of Springer
Science+BusinessMedia,NewYork(cid:2)c KluwerAcademic/PlenumPublishing’
for(a),‘ReproducedbycourtesyofWoodheadPublishing,UK’for(b)and(c),
and‘ReproducedbycourtesyofIUCr’,followedby(1),(A)or(E),for(d),(e)
and(f),asappropriate.
It is my great pleasure to acknowledge Professor Mike Glazer of the
ClarendonLaboratory,UniversityofOxford,forbothreadingthisbookinits
draftstageandofferingvaluablesuggestions.MythanksalsogotoProfessor
Jan Boeyens of the University of Pretoria for writing the Foreword and for
timely comments. I am indebted to Sönke Adlung, Senior Commissioning
Editor,PhysicalSciences,OxfordUniversityPressforencouragingmetopro-
ceedwiththisbook,thepublicationofwhichcoincideswiththeInternational
Year of Crystallography, celebrating 100 years of X-ray crystallography,
to Jessica White, Assistant Commissioning Editor, Physical Sciences and to
Erin Pearson, Production Editor, Law and Academic Science & Medicine,
Oxford University Press, for advice and technical assistance at all stages of
thepreparationofthebook.
MarkLadd
Bramshott,2014
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