Multi-Shell Polyhedral Clusters

447 Pages·2018·46.589 MB·english

by  Mircea Vasile Diudea

Mircea Vasile Diudea
Multi-Shell
Polyhedral Clusters
MirceaVasileDiudea
DepartmentofChemistry
FacultyofChemistryandChemicalEngineering
Babes-BolyaiUniversity
Cluj-Napoca,Romania
ISSN1875-0745 ISSN1875-0737 (electronic)
CarbonMaterials:ChemistryandPhysics
ISBN978-3-319-64121-8 ISBN978-3-319-64123-2 (eBook)
DOI10.1007/978-3-319-64123-2
LibraryofCongressControlNumber:2017951136
©SpringerInternationalPublishingAG2018
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Nanoworld is the worldseen at the size of 10(cid:1)9 m; searching matter at thisdepth
startedsince1970whenEijiOsawahadenouncedtheconjecturethatthetruncated
icosahedroncouldbeamolecule,latercalledC .Then,in1985,Kroto,Curl,and
60
SmalleygotspectralevidencethatC ,whichshowsasinglepeakin13C-NMR,isa
60
real molecule. They were awarded the Nobel Prize in 1995 for this historical
discovery. Macroscopic synthesis of C came later, in 1990, by the work of
60
Kraetschmerandcollaborators.Iijimareportedin1991thesynthesisofnanotubes;
the period after these pioneering discoveries is commonly called the “Nanoera.”
Developmentofcomputersandtechnologyenabledresearchersandindustrytogo
furtherinresearchandapplications,promotinganexplosivedevelopmentofelec-
tronics, optoelectronics, telecommunications, education, etc. Thereafter, the most
important event (for the actual book) was the recognition of quasi-crystals as
ordered, nonperiodic matter, the class to which the multi-shell clusters belong.
DanShehtmanwastheNoblePrizewinnerfortheseresultsin2011...thenthebook
wasstartedtobewritten...
Topologyisthemathematicalstudyofshapes;themulti-shellclustersconcerned
hereinarereferredtoassetsofshapes,arranged,inanabstractspace,inincreasing
rank (as Egon Schulte proposed in 1980), rather than in the geometrical higher
dimensional space. Cluster models representing primary atomic arrangement are
neededtounderstandtheactualstructureandthentheundergoingtransformations,
both in concept and experimental realization and in the computational treatment.
However, there is little reference to crystallographic entities, e.g., real crystal
networksandquasi-crystals.Also,thisbookdoesnotprovideallpossiblestructures
of a given set of restrictive conditions; it rather gives chosen, representative
examples. This book about multi-shell clusters could be more inspiring for archi-
tects or visual artists in making monumental, artistic works, by its aesthetic
message.
Thestructureofthisbookisasfollows:
AnintroductiontotheChemicalGraphTheoryismadeinthefirstchapter.Itisa
description through the eye of a chemist of the basic notions of Graph Theory:
definitions,topologicalmatricesandindices,countingpolynomials,etc.
Chapter2describessomeofthemostimportantoperationsonmapsthatenabled
thedesignofthemulti-shellclusters,asisshowninthefollowingchapters.
In Chap. 3, rigorous definitions in polyhedra and polytopes of higher rank are
given with a view to helping in the effort of counting structural elements and
naming and extracting mathematical and physicochemical properties of multi-
shell clusters. Some examples of polytope realization are given at the end of this
chapter.
Chapter 4 deals with the complexity and methods of investigation and charac-
terizationofmulti-shellclusters,suchascentralityindexcountedonlayermatrices
and the ring signature index, calculated on rings around each vertex/atom of the
cluster. Theory about these descriptors is given as well as case studies providing
dataontopology,definedonconnectivityratherthangeometry.
From Chaps. 5–9, the topological study is directed to multi-shell clusters clas-
sifiedaccordingtothepointgroupsymmetryoftheparentPlatonicclusters,usedas
seedsinthedesignofmorecomplexclusterswiththeaidofmapoperations.
Chapter10speaksabout chiralmulti-tori,spongystructures,thecomplexityof
whichisgivenbythehighgenussurfaceinwhichtheyareembedded.
Chapter 11 opens a gate to the spongy hypercubes, developed on the Platonic
solids.Thedesignedstructureswerecharacterizedbytopological(figure)counting
andbyOmegaandClujcountingpolynomials.
Finally,Chap.12providesaboundtotherealworldbyenergycomputation,in
anattempttofindmulti-shellcluster(orcorrespondingnetworks)candidatestothe
statusofrealchemical/mineralclusters.
Chapters2,5–10,and12haveAtlassectionsthatdetailthediscussedstructures;
the number of these figures is listed in separate files, in each chapter, while the
figurenumberisassociatedwiththenameofclusterswithinallthetext,tables,and
figuresincluded,foraneasieridentification.
Thebookincludespersonalresearchresultsoftheauthor,inconnectionwithhis
activity within the Topo Group Cluj, Romania. It is addressed to students and
researchers in the interdisciplinary field of Chemistry, Physics, and Mathematics
aswellastoarchitectsandvisualartists.Hin-filesofthestructuresillustratedinthis
bookaredeposedonline,atwww.esmc.ro,availableonrequest.
I was aided in this effort by my younger colleagues, Dr. Csaba L. Nagy and
Dr. Atena Pirvan-Moldovan, Faculty of Chemistry and Chemical Engineering,
“Babes-Bolyai”University,Cluj,Romania,withquantumchemicalandsymmetry
calculation, figure design, and error checking, while writing the book, which I
highlyappreciate.ManythanksareaddressedtoDr.AttilaBende(Molecularand
Biomolecular Physics Department, National Institute for R&D of Isotopic and
MolecularTechnologies,Cluj,Romania),Dr.BeataSzefler(DepartmentofPhys-
icalChemistry,FacultyofPharmacy,CollegiumMedicum,NicolausCopernicus
University, Bydgoszcz, Poland), Dr. Zahra Khalaj (Department of Physics,
Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran), and Dr. Igor
Baburin(TechnischeUniversita¨tDresden,TheoretischeChemie,Germany)fora
fruitfulcollaboration.
Cluj-Napoca,Romania MirceaVasileDiudea
January22,2017
Contents
1 BasicChemicalGraphTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 BasicDefinitionsinGraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TopologicalMatricesandIndices. . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 AdjacencyMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 DistanceMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 DetourMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 CombinatorialMatrices. . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 WienerMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.6 ClujMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.7 Distance-ExtendedMatrices. . . . . . . . . . . . . . . . . . . . 11
1.2.8 WalkMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.9 ReciprocalMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.10 LayerandShellMatrices. . . . . . . . . . . . . . . . . . . . . . 14
1.3 TopologicalSymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 OperationsonMaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Duald. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Medialm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Truncationt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 PolygonalMappingpn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Snubs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Leapfrogl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Quadruplingq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.8 Septuplingsn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 DefinitionsinPolytopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 n-DimensionalStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 AbstractStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Posets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 VertexFigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.3 AbstractPolytope. .. . . . .. . . . .. . . . .. . . . .. . . . .. 46
3.4 PolytopeRealization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 P-CenteredClusters. . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Cell-in-CellClusters. . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 24-CellandItsDerivatives. . . . . . . . . . . . . . . . . . . . . 51
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 SymmetryandComplexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 EulerCharacteristic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 TopologicalSymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 CentralityIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 RingSignatureIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 RingSignatureinaTranslationalNetwork. . . . . . . . . 61
4.4.2 RingSignatureinSpongyStructuresofHigherRank. . . 63
4.4.3 RingSignatureinSpongyHypercubes. . . . . . . . . . . . 66
4.4.4 TruncationOperation. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 PairsofMapOperation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 SmallIcosahedralClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 SmallCages:SourceofComplexClusters. . . . . . . . . . . . . . . . 77
5.2 TruncatedMPIcosahedralClusters. . . . . . . . . . . . . . . . . . . . . 78
5.3 ClustersbyMedialOperation. . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 ClustersofHigherRank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 LargeIcosahedralClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1 SmallComplexClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 IcosahedralClustersDerivedfromtheC Seed. . . . . . . . . . . . 127
45
6.3 ClustersofDodecahedralTopology. . . . . . . . . . . . . . . . . . . . . 132
6.4 ClustersofIcosahedralTopology. .. . . . . . . . . .. . . . . . . . . .. 133
6.5 RhombDecoratedClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 ClustersofOctahedralSymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.1 SmallClustersasSeedsforComplexStructures. . . . . . . . . . . . 187
7.2 ClustersDecoratedbyOctahedra. . . . . . . . . . . . . . . . . . . . . . . 188
7.3 ClustersDecoratedbyDodecahedra. . . . . . . . . . . . . . . . . . . . . 189
7.4 RhombDecoratedOctahedralClusters. . . . . . . . . . . . . . . . . . . 190
7.5 CubicNetTransforming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8 TetrahedralClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.1 SmallTetrahedralClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 TetrahedralClustersofHigherRank. . . . . . . . . . . . . . . . . . . . 248
8.3 TetrahedralClustersDerivedFromAda20. . . . . . . . . . . . . . . . 248
8.4 TetrahedralHyper-structuresDecoratedwithOnly
Dodecahedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 C RelatedClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
60
9.1 StructuresDerivedfromtheClusterP32@dC .33. . . . . . . . . . 281
60
9.2 StellatedClusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
9.3 C RelatedStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
750
9.3.1 DualsofC andRelatedStructures. . . . .. . . .. . . .. 287
750
9.3.2 MedialsofC andRelatedClusters. . . . . . . . . . . . . 287
750
9.3.3 TruncatedC andRelatedClusters. . . . . . . . . . . . . . 290
750
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
10 ChiralMulti-tori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
10.1 DesignofChiralMulti-tori. . . . . . . . . . . . . . . . . . . . . . . . . . . 335
10.2 DodecahedronRelatedStructures. . . . . . . . . . . . . . . . . . . . . . 338
10.3 CubeRelatedStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
10.4 TetrahedronRelatedStructures. . . . . . . . . . . . . . . . . . . . . . . . 341
10.5 C RelatedStructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
60
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
11 SpongyHypercubes.. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. 363
11.1 SimpleToroidalHypercubes. . . . . . . . . . . . . . . . . . . . . . . . . . 363
11.2 ComplexToroidalHypercubes. . . . . . . . . . . . . . . . . . . . . . . . 365
11.3 TubularHypercubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
11.4 SpongyHypercubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.5 TruncationofHypercube. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 370
11.6 CountingPolynomialsinHypercubes. . .. . . .. . . .. . . .. . . .. 372
11.6.1 OmegaPolynomial. . . . . . . . . . . . . . . . . . . . . . . . . . 372
11.6.2 ClujPolynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
12 EnergeticsofMulti-shellClusters. . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.2 C Aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
20
12.3 Hyper-graphenesbyD Substructures. . . . . . . . . . . . . . . . . . . 389
5
12.4 Hyper-graphenesbyC Units. . . . . . . . . . . . . . . . . . . . . . . . . 389
60
12.5 C AggregateswithTetrahedralandIcosahedralSymmetry. . . 393
60
12.6 C Networkby[2+2]Cycloaddition. . . . . . . . . . . . . . . . . . . . 396
60
12.7 ComputationalMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Abbreviations
d Dual
l Leapfrog
m Medial
t Truncated
CO Cuboctahedron
DCO Dualcuboctahedron
ID Icosidodecahedron
hP Halfpolyhedron
MC Medialcube
LDCO Leapfrog(ofdual)ofcuboctahedron
MDCO Medial(ofdual)ofcuboctahedron
MMC Medial(ofmedial)ofcube
RCO Rhombicuboctahedron
RID Rhombicosidodecahedron
SC Snubcube
SD Snubdodecahedron
TC Truncatecube
TCO Truncatecuboctahedron
TID Truncateicosidodecahedron
TMC Truncate(ofmedial)ofcube
TO Truncateoctahedron
TT Truncatetetrahedron
XAYb FigureatthebottomofthemainfigureXAY
Chapter 1
Basic Chemical Graph Theory
Graph Theory applied in Chemistry is called Chemical Graph Theory. This inter-
disciplinarysciencetakesproblems(likeisomerenumeration,structureelucidation,
etc.) from Chemistry and solve them by Mathematics (using tools from Graph
Theory,SetTheoryorCombinatorics),thusinfluencingbothChemistryandMath-
ematics.Partitioningofamolecularpropertyandreconstructingitfromfragmental
contributions is one of the main tasks of this theory. For further discussion, some
basicdefinitionsinGraphTheoryareneeded.
1.1 Basic Definitions in Graphs
AgraphG(V,E)isapairoftwosets,VandE,V¼V(G)beingafinitenonemptyset
and E ¼ E(G) a binary relation defined on V (Harary 1969). A graph can be
visualized by representing the elements of V by points/vertices and joining pairs
ofvertices(i,j)byanedge/bondifandonlyif(i,j)2E(G).Thenumberofvertices
inGequalsthecardinalityn¼jV(G)jofthisset.Thetermgraphwasintroducedby
Sylvester (1874). There is a variety of graphs, some of them being mentioned
below.
Apathgraphisanon-branchedchain.Atreeisabranchedstructure.Astarisa
setofverticesjoinedinacommonvertex.Acycleisachainwhichstartsandendsin
oneandthesamevertex(Fig.1.1).
A complete graph K is the graph of with any two vertices are adjacent. The
n
number of edges in such a graph is n(n(cid:2)1)/2. Fig. 1.2 illustrates the complete
graphswithn¼1–5.
Inabipartitegraph,thevertexsetVcanbepartitionedintwodisjointsubsets:
V [V ¼V(G); V \V ¼∅ such that any edge (i,j)2E(G) joins V with V
1 2 1 2 1 2
(Harary 1969; Trinajstic´ 1983; Diudea 2010). A graph is bipartite if and only if
allitscyclesareeven(Ionescu1973).Ifanyvertexi2V isadjacenttoanyvertex
1
j2V thenGisacompletebipartitegraph,K ,withm¼|V |andn¼|V |andthe
2 m,n 1 2