Generation of Heptagon-Containing Fullerene Structures by

Computational Methods

XiaoyangLiu

ThesissubmittedtothefacultyoftheVirginiaPolytechnicInstituteandStateUniversityinpartialfulfillmentoftherequirementsforthedegreeof

MasterofScienceIn

Chemistry

HarryC.Dorn,ChairLouisA.MadsenEdwardF.Valeev

November18,2016Blacksburg,VA

Keywords:Fullerene,TNT-EMFs,SpiralAlgorithm,Geometry,StructureGeneration

Generation of Heptagon-Containing Fullerene Structures by Computational Methods

Xiaoyang Liu

ABSTRACT

Sincethediscoverythreedecadesago,fullerenesaswellasmetallofullereneshave

beenextensively investigated.However,almostallknown fullerenes followthe

classicaldefinition,thatis,classicfullerenesarecomprisedofonlypentagonsand

hexagons. Nowadays, more and more evidence, from both theoretical and

experimentalstudies,suggeststhatnon-classicalfullerenes,especiallyheptagon-

containing fullerenes, are important as intermediates in fullerene formation

mechanisms. To obtain fundamental understandings of fullerenes and their

formationmechanisms,newsystematicstudiesshouldbeundertaken.Although

necessary tools, suchas isomergeneratingprograms,havebeendeveloped for

classical fullerenes, none of them are able to solve problems related to non-

classical fullerenes. In this thesis, existing theories and algorithms of classical

fullerenes are generalized to accommodate non-classical fullerenes. A new

programbasedon thesegeneralizedprinciples isprovided forgeneratingnon-

classical isomers. Along with this program, other tools are also attached for

acceleratingfutureinvestigationsofnon-classicalfullerenes.Inaddition,research

todateisalsoreviewed.

Generation of Heptagon-Containing Fullerene Structures by Computational Methods

XiaoyangLiu

GENERALAUDIENCEABSTRACT

Thethesisdescribesthegenerationofheptagon-containingfullerenestructures

basedongeneralizedspiralalgorithm.Detailsofalgorithmsanddesignsare

discussed.Thethesisishelpfulforpeoplewhoareworkingonfullerene

geometrystudies.Thealgorithmsandprogramsprovidedbythisthesiscouldbe

agoodstartforotherinvestigations.

Forthosewhoarenotfamiliarwithprogramming,theprogramprovidedbythis

thesiscanalsobeusedtogenerateandanalyzeheptagon-containingstructures.

In addition, this thesis can also be helpful for people working on areas that

geometryanalysesofmoleculesarenecessary.

iv

TableofContents1. Background:................................................................................................................................11.1 CurrentEvidenceofHeptagonsinFullerenes,Metallofullerenes,andNanotubes.........................................................................................................................................................11.2 LaSc2N@Cs(hept)-C80:FirstDiscoveredHeptagon-ContainingMetallofullerene 51.3 TheRoleofHeptagon-ContainingFullerenesinFullereneFormation..................91.3.1 Heptagon-containingFullerenesinCageShrinkage.................................................91.3.2 Heptagon-containingFullerenesinCageGrowth...................................................161.3.3 Conclusion................................................................................................................................18

2 FullereneTopology................................................................................................................192.1 FullereneDual..............................................................................................................................192.2 PointGroupsofFullerenes.....................................................................................................21

3 GeneralizedSpiralProgram:..............................................................................................223.1 Euler’sPolyhedraFormula.....................................................................................................233.2 SchlegelProjectionandSpiralGraph.................................................................................253.3 SpiralAlgorithm..........................................................................................................................283.4 GeneralizedFullereneFormationProgramBasedonSpiralAlgorithm.............313.5 ResearchforSingle-Heptagon-ContainingFullerenesintheRangeC30-C38StructuresandSymmetries....................................................................................................................353.6 ResearchforTwo-Heptagon-ContainingFullereneswithHighSymmetries...39

4 Conclusions................................................................................................................................405 References:................................................................................................................................426 Appendix:...................................................................................................................................44A.MethodsforconstructingfullerenecagesfromtheoutputofGspiralprogram........44B.Atlasofsingle-andtwo-heptagon-containingfullerenes...................................................46C.GeneralizedSpiralProgram(Gspiral.cc)......................................................................................46

v

TableofFiguresFigure1Theregularfullerenemotif(left)andheptagon-containingmotif(right)...................................................................................................................................................1Figure2ThechemicalperforationreactionofC60.......................................................2Figure3TwoviewsofthestructureofLaSc2N@Cs(hept)-C80asdeterminedbyasingle-crystalX-raydiffraction.......................................................................................6Figure4ThestructuresofLaSc2N@Cs(hept)-C80andLaSc2N@Ih-C80....................8Figure5ThemechanismofthelossofC2fromC60...................................................10Figure6TherepresentationofthepathwayforthelossofC2carbidefromCs-C86Cl28cage.........................................................................................................................12Figure7ThetransformationfromtheasymmetricfullerenecageC1(51383)-C84tosymmetriccages...........................................................................................................14Figure8Thetop-downmechanismforthefullereneformationinvolvinglow-symmetricfullerenes........................................................................................................15Figure9ThepossiblemechanismsfortheC2carbideincorporationintoafullerene..............................................................................................................................17Figure10ThemechanismofthefullerenegrowthfromC68toC70involvingtheC2incorporationsandtheStone-Walesrearrangements.........................................18Figure11Planarand3Dembeddingoftwofullereneisomers,Ih-C20andIh-C60................................................................................................................................................19Figure12Thedecisiontreeforfiguringoutthepointgroupforanyfullereneisomer..................................................................................................................................22Figure13ThestructuresofIh-C60(left)andC1-C36(right)withoneheptagonalface.......................................................................................................................................24Figure14SchlegelprojectionofIh-C60........................................................................26Figure15Thethreevalidsequencesandcorrespondingspiralgraphs..............30Figure16Themainstructureandtheaccessorycomponentsofthegeneralizedspiralprogram...................................................................................................................35Figure17Themotifsofheptagoncontainingfullerenesthatdeterminetherelativeenergies................................................................................................................38Figure18ThestructuresofaseriesoftwoheptagoncontainingfullereneswithD7handD7dsymmetriesalternately..............................................................................40

1

1. Background:

1.1 Current Evidence of Heptagons in Fullerenes, Metallofullerenes, andNanotubes

Thischapterdescribestheinvestigationsrelatedtoheptagonalfaces.Theyarethe

foundations for my research on heptagon-containing fullerenes. These

experimental data reveal the fact that heptagons exist in fullerenes and

metallofullerenes,andaffectalotforthepropertiesofthemolecules.Myresearch

isfocusedonthestudiesofheptagon-containingfullerenesandmetallofullerenes.

Figure1Theregularfullerenemotif(left)andheptagon-containingmotif(right).

The fact that all known classical fullerenes are comprised of pentagons and

hexagons1attractscloseattentionsfromsciencecommunities.In1991,Smealet

al. discovered from the TEM image that the terminations of nanotubes,2

polyhedralcaps inmostcases,mightbe firstly transformedintoconicalshapes

beforetheclosure.Iijimaetal.3reportedthattheconeanglewasaround20°,and

2

concludedthattheconversionfromthecone-likegrowthtothecylindricalgrowth

wascausedbydefects,commonlythoughtastheincorporationofheptagonsand

pentagons,inthehexagonal1networks. Thediscoveryofheptagonsinnanotubes

strongly implies that heptagons may also exist in fullerene cages, due to the

similaritiesofpropertiesbetweenthesetwocarbonallotropesandthefactthat

they can be synthesized under the same condition.4 In 1995, Wudl’s group5

reportedaperforationreaction(Figure2)thatbroketwoadjacentbondsinC60

and for the first time, the reaction introduces open faces, rings larger than

hexagons, to fullerene cages. The reported reaction is the combination of two

reactions.Oneistheazidereaction betweenC60andazides6andanotheristhe

self-sensitizedphotooxygenation7oftheproductsoftheazidereaction.Inthefirst

step, C60 reacts with azides, and produces N-methoxyethoxymethyl (MEM)-

substituted[5,6]azafulleroid.ThenaC-Cbondisbrokenandaheptagonalfaceis

formed in the fullerene cage. In addition, the reactions make the C=C bonds

adjacent to the N atom more reactive to photooxygenation. Thus,

photooxygenation, in the second step,breaksadditionaloneor twobondsand

introducesfaceslargerthanheptagonsinthefullerenecage.

Figure2ThechemicalperforationreactionofC60.Reproducedfromref42.

Copyright1995AmericanChemicalSociety.

NMEM

NMEM

NMEM

OO

O

OO2

hv

1 2 3

3

Anotherchemicalperforationthatintroducedaneleven-atomopenfacetoC70was

reportedbyBirkettetal.inthesameyear.8Thefinalproductisobtainedintwo

steps:Inthefirststep,chlorineatomsinC70Cl109aresubstitutedbybenzenesand

thenafullerenederivative,C70Ph8,isobtained.TheproductbelongstotheCspoint

groupandphenylgroupsareisolatedbetweeneachother.Secondly,C70Ph8was

oxidizedautomaticallyalongwiththebreakingoftheC-Cbondintheopenairand

aneleven-ringwasintroducedintothecage.8Thechemicalperforationreaction

provestheexistenceoflargerringsinfullerenesandprovidesachemicalmethod

for producing fullerene derivatives with polygons besides pentagons and

hexagons,althoughthisreactiondoesn’tcreateaheptagonalface.

Theoretical approaches were also made on heptagon-containing non-classical

fullerenesinseveralstudies.Fowleretal.10appliedtwosemi-empiricalmethods,

QCFF/PI11 (quantum consistent force field/π) and DFTB12 (density functional

tightbinding),tosystematicallyinvestigatethestabilitiesofall426C40isomers,13

includingbothclassicalfullerenesandthosewithoneorseveralheptagonalfaces.

The results show that the stabilities of heptagon-containing fullerenes are

reasonably similar to classical fullerenes. However, there is no heptagon-

containing isomer that is more stable than all classical ones.10 Furthermore,

Fowleretal.calculatedtheenergeticpenaltyfortheintroductionofoneheptagon

inafullerenecage,andthepenaltyisaround90-150kJ/molesuggestedbyDFTB

calculation.10Basedontheseresults,Fowleretal.proposedageneralizedisolate

pentagonrule(GIPR),whichgovernsclassicalaswellasheptagon-containingnon-

classical fullerenes, for predicting the relative stabilities of fullerene isomers.

According to this rule, the relative energy of a fullerene is determined by two

factors,e55ande57,whicharethenumberofpentagonadjacenciesandthenumber

4

of heptagon-pentagon adjacencies, respectively.10 So finding the most stable

fullereneisomerissimplifiedtominimizee55andtomaximizee57simultaneously.

Itishighlypossible,accordingtothetrends,thataparticularheptagon-containing

non-classicalfullerenemayexceedthestabilitiesofallclassicalfullerenesofthe

samesize.Later,Ayuelaetal.1calculatedenergiesofallclassicalisomers14,15ofC62

and those with one heptagon computationally. Heptagon-containing fullerene

isomerC62,withe55=3ande57=5,wasfoundtobethemoststableoneamongall

2385 isomers examined.1 Six calculation methods11,16-18 were employed to

confirmtheresults,andtheywereingoodagreementregardingrelativestabilities.

What’smore, the favored non-classical fullerene cage can be produced by the

insertion of a C2 carbide on the hexagonal face of the Ih-C60, themost famous

knownclassicalisomer,andthentransformedtotheC62isomerthroughtheStone-

Walesrearrangement.1TheconversionfromtheIh-C60tothestableC62fullerene

isomerviathefavoredheptagonalC62isomershowedthatheptagon-containing

fullerenesmightplayagreatroleinfullerenetransformations.

Martsinovich et al.19 reported the isolation and the characterization of C58

fullerene derivatives incorporating heptagons. They obtained the fullerene

derivativesbyheatingthemixtureofC60andfluorinatingreagentsinvacuum,20

suchastransitionmetalfluorides.ThereactionyieldedC60F18andfluorinatedC58

that accounted for around80%and20%, respectively.The crudeproductwas

separatedandpurifiedbyhighpressureliquidchromatography(HPLC),andtwo

C58derivatives,C58F17CF3andC58F18,wereobtained.19Themassspectrometryand

the fluorine nuclear magnetic resonance (19F-NMR) spectroscopy19 suggested

both of these fullerene derivatives had a heptagonal face.Martsinovich et al.19

pointedouttheremarkablestabilitiesofC58fullerenederivativeswereduetothe

5

presenceof fluorineatoms,which released the strainof the fullerene cagesby

introducingsp3hybridizedcarbonatoms.Theyalsoproposedahypothesis21that

C58F17CF3andC58F18wereformedbythelossoftheC2carbidefroma6:5bondof

C60, thebondsharedbyahexagonandapentagon.Thedataobtainedfromthe

synthesis of the derivatives strongly supported the hypothesis that heptagon-

containingfullerenescouldbetransformedfromclassicalfullerenesandactaskey

intermediatesinfullerenetransformations.19

Heptagons, according to the facts disscused above, are proven to exist in

graphenes,fullerenesandmetallofullerenes.Theseheptagon-containingfullerene

species are relatively stable and as a result, are promising candidates as

intermediatesof fullerene formationpathways.So, investigationsonheptagon-

containingfullerenesareimportanttorevealthesecretsofthefullereneformation

mechanisms.

1.2 LaSc2N@Cs(hept)-C80:FirstDiscoveredHeptagon-ContainingMetallofullerene

LaSc2N@Cs(hept)-C80 is the first synthesized metallofullerene isomer with a

heptagonal face. Before the discovery of this isomer, heptagon-containing

fullerenesareonlypredictedby theoriesandcalculations.Obtainingheptagon-

containing fullerene isomers in laboratory is amilestone for studies related to

heptagon-containingfullerenes.

A large portion of non-classical fullerenes, especially heptagon-containing

fullerenes, has been predicted to be competitive with classical fullerenes in

stabilitiesbasedontheoreticalapproaches.22Theconclusion indicatesthat it is

highly possible to isolate heptagon-containing fullerenes, especially thosewith

exohedral or endohedral stabilizations. However, all non-classical fullerenes

6

obtainedinlaboratoryaremodifiedexternallywithhalogenatoms,23andthose

withoutexohedralmodificationswereabsentuntilrecently.24Variousconjectures

havebeenmadeonwhyexohedralderivationscanstabilizeheptagon-containing

fullerenes.Oneofthemsuggeststhattheexohedralmodificationscanstabilizethe

pentagonadjacencydefectsbytransforminghalogenatedcarbonatomsfromsp2

to sp3 and thereby reducing strains. Similarly, metal ions or clusters inside

fullerenes can also stabilize the fused pentagonmotifs and then the fullerene

cages.25-27In2015,Zhangetal.reportedthefirstheptagon-containingendohedral

fullerene,LaSc2N@Cs(hept)-C80.24Itwasobtainedusingarc-dischargetechnique

andcharacterizedbyX-raydiffraction.ThestructureofLaSc2N@Cs(hept)-C80is

illustratedinFigure3.Twoclassicalisomers,LaSc2N@Ih-C80andLaSc2N@D5h-C80,

werealsosynthesizedalongwiththeheptagon-containingone.

Figure3TwoviewsofthestructureofLaSc2N@Cs(hept)-C80asdeterminedbya

single-crystalX-raydiffraction.Theheptagonalfaceandthetwopentalenemotifs

arehighlightedinred.Reproducedfromref.24.Copyright2016JohnWiley&Sons,

Inc.

7

AsFigure3shows,themetalcluster,LaSc2,isplanarwiththemetalionspointing

to the adjacent pentagons.24 The structure is in good agreement with the

assumptionmentionedabovethatmetalionscouldstabilizethefusedpentagon

motifs. Theheptagonal face is curled, in contrast, pentagons andhexagons are

planar,24whichisconsistentwithusualobservations.

To find out the impact of heptagonal faces on the stability, they applied a

computational study based on DFT and made comparisons with the classical

isomerssynthesizedsimultaneously.ThecalculationsuggeststhattheCs(hept)-

C80cageistheleaststableone,177kJ/mollessstablethanIh-C80and83kJ/mol

lessthanD5h-C80.24Theencapsulationofnitrideclusterssignificantlyincreasethe

stabilityofCs(hept)-C80isomer,butprovidesonlyaminoreffectontheIh-C80and

D5h-C80 isomers. Therefore, the stability of LaSc2N@Cs(hept)-C80 ranks in the

middle, 34 kJ/mol higher than LaSc2N@D5h-C80 and 27 kJ/mol lower than

LaSc2N@Ih-C80. The result reveals that the thermodynamic stability is not the

dominantfactorinthesynthesisofheptagon-containingfullerenes,ortheyieldof

LaSc2N@Cs(hept)-C80 should be much higher. The unexpected high

thermodynamicstabilityofLaSc2N@Cs(hept)-C80togetherwiththelowyield,five

timeslowerthanLaSc2N@D5h-C80,indicatesthesynthesisofheptagon-containing

fullerenes is under kinetic control. They assume the kinetic instability of the

heptagonalEMFscomefromthetrendofrearrangingtootherisomersandthelow

energy barrier for the rearrangements. In this case, LaSc2N@Cs(hept)-C80 is

closelysimilartoLaSc2N@Ih-C80andcanbetransformedtotheclassicalfullerene

through a single step of C2 segment rearrangement. The transformation is

illustratedinFigure4.

8

Figure 4 The structures of LaSc2N@Cs(hept)-C80 and LaSc2N@Ih-C80. (a) The

structure of LaSc2N@Cs(hept)-C80. (b) The structure of LaSc2N@Ih-C80. The C2

segmenthighlightedingreenisthebondrotatedinthetransformationandthe

heptagonalfaceandfacesarounditaremarkedinred.(c)LaSc2N@C78withtheC78

(22010)cage.ThecageisobtainedfromCs(hept)-C80afterlosingoneC2fragment.

TheremovedC2segmentisshownas“ghost”atomsandthenewfusedpentagons

formed after the removal of the C2 segment is highlighted in black. (d) The

structureofLaSc2N@C3s(hepta)-C82,whichcantransformtoLaSc2N@Cs(hept)-C80

aftertheremovaloftheyellowC-Cbond.Reproducedfromref.24.Copyright2016

JohnWiley&Sons,Inc.

Therearrangements,additionorremovalofaC2segmentshowninfigure4(c)

and(d),providepossibleformationpathsofLaSc2N@Cs(hept)-C80.Inconjunction

9

with the result that LaSc2N@Cs(hept)-C80 can easily transform to more stable

LaSc2N@Ih-C80,LaSc2N@Cs(hept)-C80actsasanintermediatefortheformationof

LaSc2N@Ih-C80fromLaSc2N@Cs(hepta)-C82.

Asmentionedabove, Zhang et al.28 reported that asymmetric fullerenes could

converttohigh-symmetricfullerenesandrepresentedamechanisticlinkbetween

largeandsmallcages.Theasymmetricfullerenes,as investigationssuggest,are

instable in kinetic. Considering the fact that most of heptagon-containing

fullerenesare low-symmetricandevenasymmetric,kinetic instabilitiesmaybe

causedbythelowsymmetries.

1.3 TheRoleofHeptagon-ContainingFullerenesinFullereneFormation

Investigations on heptagon-containing fullerenes are essential to gain

comprehensive understandings of fullerene formations, since heptagon-

containing fullerenes are predicted as important intermediates of fullerene

formations.Inthisthesis,IprovidethegeneralizedspiralalgorithmandGspiral

program to calculate heptagon-containing fullerenes. These structures are

necessary to study the roles of heptagon-containing fullerenes in formation

mechanisms exclusively. Understanding proposed formation mechanisms are

required to push my research forward and to add functions for studies of

formationmechanismsinmyprogram.

1.3.1 Heptagon-containing Fullerenes in Cage Shrinkage

10

Fullerenesandmetallofullereneshavebeenstudiedinmanyaspectsformorethan

thirtyyears.However,theformationmechanismsstillremainunknown.28Most

recently,theevidenceanddataconvergeatonepoint,29non-classicalfullerenes,

especially heptagon-containing fullerenes.Heptagon-containing fullereneshave

been highly considered as the intermediates of fullerene transformations and

formations.Theyarepromisingcandidatesforconnectingfullerenesofdifferent

sizes. So the investigations focused on heptagonal fullerenes will make great

contributionstothefullereneformationresearch.Inthissection,Iwilldescribe

the heptagon-containing fullerene’s role in the fullerene shrinkage, one of the

mostimportantcomponentsofthetop-downmechanism.

In1993,Eckhoffetal.30proposedaC2loss31mechanismbasedonnumerousC2

fragmentation studies. The mechanism30 (Figure 5), involving the heptagon-

containingintermediatesandtheStone-Walesrearrangement,32agreedwithall

experimentalandtheoreticaldata.Theshrinkageoffullerenesactuallyprovidesa

possible path for fullerene transformations between two cages differing two

carbon atoms. This mechanism, although proposed to explain the C2

fragmentation of C60, has beenwidely accepted and applied for demonstrating

fullereneconversions,duetoitsrationalityandcoherencetoexperimentaldata.

Figure5ThemechanismofthelossofC2fromC60.(A)AmotifoftheC60cage.

(B)Thestructureoftheintermediateincorporatingaheptagon.(C)Thefinal

loss of C2 S-W rearrangement

A B C

11

structureafterthelossoftheC2carbideandtherearrangement.Reproduced

formref58.Copyright19932016ElsevierB.V.

In2010,Ioffeetal.23reportedthetransformationfromtheclassicalfullereneC86

totheheptagon-containingnon-classicalfullerenederivativeC84Cl32bythelossof

C2 and the chlorination. The observation strongly supported the C2 loss

mechanism30describedabove.TheC86wasfirstsynthesized,andthenchlorinated

by VCl3 at 250°C for 4-5 days.23 The products were characterized by X-ray

diffraction as well as NMR.33 The characterizations show that therewere two

fullerenederivatives,Cs-C86Cl28 andC84Cl32,34 in the finalproducts.The second

product, although not the major one, is very interesting because of the

incorporationofa7-memberedring.ThestructureofC84Cl32,comprisingthirteen

isolated pentagons,was closely related to Cs-C86Cl28, themajor product of the

chlorination reaction. The relationship between these two products strongly

suggestedthatC84Cl32wasformedviatheshrinkageofCs-C86Cl28bylosingtheC2

carbide from the 5:6 bond position.23 Then, Ioffe et al. proposed a detailed

mechanism(Figure6)toillustratethelossofC2.

12

Figure6TherepresentationofthepathwayforthelossofC2carbidefromCs-

C86Cl28cage.Reproducedfromref60.Copyright2010JohnWiley&Sons,Inc.

The shrinkage of Cs-C86Cl28 is not an isolated phenomenon. Under the same

condition,C90Cl24cantransformtoheptagon-containingC88Cl22/24bylosingaC2

carbide.35Inthiscase,theC90cagewas,surprisingly,anisomerfollowingtheIPR,36

whichwastypicallythoughtverystableandhardtoshrink.Sotheshrinkageof

fullerenesbylosingtheC2carbidesisextremelybroadandcanoccurinalmost

everyfullerenecage.

In 2013, the Dorn group28 reported a remarkable discovery that asymmetric

metallofullerene cages could transform to symmetric ones by the fullerene

shrinkage. They also proposed a breakthrough hypothesis, and the hypothesis

Cl Cl

Cl Cl

CCl2

CCl2Cl

a

b1

b2

b3

c1

c2

c3

chlorinetransfer

chlorinetransfer

chlorine transfer

+C2Cl2 or C2Cl4

Cl Cl

Cl Cl

Cl Cl

13

connected the fullerene shrinkage pathwith the top-down fullerene formation

mechanism. The asymmetric metallofullerenes, Y2C2@C1(51383)-C84 and

Gd2C2@C1(51383)-C84,werepreparedbyelectric-arcsynthesis.Y2C2@C1(51383)-

C84wassynthesizedwith13CenrichedgraphiterodandY2O3;Gd2C2@C1(51383)-

C84 was synthesized with normal carbon rods and gadolinium oxide.37,38 The

productswereseparatedbyHPLCandhadalongretentiontime,indicatingthe

existence of pentalene motifs.39 In addition, Gd2C2@C1(51383)-C84 was

characterizedbythesingle-crystalanalysis,andY2C2@C1(51383)-C84,sinceitwas

enrichedby13C,wasidentifiedby13CNMR.28Theanalysisconfirmedbothofthe

twoproductshadC1symmetry,thelowestsymmetryafullerenecouldhaveand

thought as asymmetric. Then, at high temperature, the asymmetric

metallofullerenecageslostaC2carbide19andshrunktothehighlysymmetricC82

cages,40 surprisingly. Zhang et al., based on the observations, proposed a

transformationmechanism (Figure 7), involving thewidely accepted C2 loss30

mechanismandtheStone-Walesrearrangement.32Asdiscussedabove,23,30,35the

intermediates between two adjacent fullerenes are heptagon-containing

fullerenes.1

14

Figure7ThetransformationfromtheasymmetricfullerenecageC1(51383)-C84

tosymmetriccages.Manyfamousandstablefullerenesisolatedareinvolved.

Colorspresentthemotifschangedinthetransformation.Adaptedfromref28.

Copyright2013MacmillanPublishersLimited.

The experiment strongly suggested the top-down fullerene formation

mechanism,28sinceseveralwell-knownhigh-symmetricfullerenes14wereformed

from larger asymmetric fullerenes in the top-down path. The asymmetric

fullerene cages are called “missing links”,28 and are the outcomes of giant

fullerenes’shrinkageaswellastheprecursorsofhigh-symmetricsmallfullerenes.

Heptagon-containing fullerenes, acting as the intermediates in the fullerene

shrinkagepathway,connecttwofullerenesdifferingtwocarbons.Ingeneral,the

big picture of top-down formationmechanism includes three sections.28 First,

15

giantfullerenesareformedfromgrapheneasdiscussedinthepreviouschapters.

Second,asymmetricfullerenesareproducedfromgiantfullerenesbythelossof

C2 carbides. Third, small symmetric fullerenes are created by the shrinkage of

asymmetricfullerenesfollowedtheStone-Walesrearrangements.30,41

Figure8Thetop-downmechanismforthefullereneformationinvolvinglow-

symmetricfullerenes.(a)Primitivegiantfullerenesareformedfromthecoiling

ofthegrapheneontheedge.(b)Thepathwayofthefullereneshrinkage.Inthis

process,giantfullerenestransformtolow-symmetricfullerenesorcalled

“MissingLink”firstandthensymmetricfullerenesareformedviatheshrinkage

of“MissingLinks”.(c)Themechanismforfullereneshrinkage.Heptagon-

16

containingfullerenesandtheStone-Walesrearrangementareinvolved.

Reproducedfromref.28.Copyright2013MacmillanPublishersLimited.

1.3.2 Heptagon-containing Fullerenes in Cage Growth

Recently, although more and more evidence from laboratory and even outer

space42 strongly suggests the top-down fullerene formation mechanism,28 the

bottom-up mechanism are still accepted because there is no direct evidence

dismissingthebottom-upmechanism.Somehypotheseswereproposedtoexplain

experimental observations based on the fundamental idea of the bottom-up

mechanisms.43 Coincidentally, heptagon-containingnon-classical fullerenes, the

proposedintermediatesintop-downtheories,arealsothoughttoplayimportant

rolesinbottom-upmechanisms.44

In2001,Terronesetal.44reportedafullerenegrowthmechanismbackingbottom-

upfullerenetheories,basedontheobservationsandtheprovenhypotheses.Ithas

been widely accepted, based on the data accumulated by investigations on

heptagon-containingfullerenes,thatnon-classicalfullerenes,especiallyheptagon-

containing fullerenes, are comparable in stability1 with classical fullerenes in

manycases.SotheyproposedseveralC2incorporationmechanisms44(Figure9)

involving heptagon-containing fullerenes. The incorporation may occur on

hexagonal faces, edgesof twoadjacenthexagonal facesor adjacentheptagonal

faces,andyieldsthemotifscontainingpentagons,hexagonsaswellasheptagons.

Thecarbideincorporationmechanism,actingasanimportantpartofthefullerene

growth,demonstratedhowsmallfullerenestransformedintolargefullerenesin

detail.

17

Figure9ThepossiblemechanismsfortheC2carbideincorporationintoa

fullerene.(a)and(b)involveonlyclassicalfullerenes.(c)and(d)showthe

incorporationonheptagonalface,andaclassicalfullereneandaheptagon-

containingfullereneareyielded,respectively.(d)indicatestheincorporation

reactingonhexagonalfaceofaclassicalfullereneandaheptagon-containing

fullereneisgenerated.

Wangetal.,45basedontheC2carbideincorporationmechanismdiscussedabove,

reported a complete fullerene growth path (Figure 10) to illustrate the

transformationfromC68toC70.Moreover,theyalsocalculatedtheenergybarrier

foreachstep,andfoundtheenergyneededforthegrowthisrelativelylow.The

fullerenegrowthpathway involved theC2 carbide incorporationaswell as the

Stone-Walesrearrangements,andsurprisingly,thearrangementsrequiredmore

energies than theC2 incorporation.45 It suggests that theC2 incorporations are

very likely to occur since the Stone-Wales rearrangement has been proved by

(a) (b) (c)

(d) (e)

18

theoreticalandexperimentalevidence,althoughithasnotbeendirectlyobserved

yet.45

Figure10ThemechanismofthefullerenegrowthfromC68toC70involvingthe

C2incorporationsandtheStone-Walesrearrangements.Thebarrierenergyis

calculatedforeachstepandshownonthearrows.Adaptedfromref69.

Copyright2012AmericanChemicalSociety.

1.3.3 Conclusion

Heptagon-containing fullerenes are repeatedly proposed as intermediates for

bothfullereneshrinkageandgrowthmechanisms,keypartsofthetop-down42and

thebottom-up43fullereneformationmechanisms,respectively.Investigationson

heptagon-containingfullerenesareessentialtofigureoutwhetherthefullerene

shrinkageorgrowthistrueandfurthermore,determinewhetherthetop-downor

19

thebottom-upformationmechanismiscorrect.Tohaveadeepunderstandingof

fullerene formation, we need to do more intensive investigations in this

overlookedarea.

2 FullereneTopology

2.1 FullereneDual

Sincemy program is developed to analyze and generate fullerene geometries,

fullerenetopologytheoriesareinvolvedinthedesignofmyprogram.Fullerene

geometricanalogs,orduals,havemanyusefulproperties,andthetransformations

betweenfullereneisomersandtheircorrespondingdualsarestraightforward.So

Iusefullerenedualstodocalculationsinmanysteps.

Ingeometry,eachpolyhedronhasanassociateddual.Theverticesofapolygon

correspondtothefacesofitsdual,andviceversa.Accordingtothisdefinition,all

polyhedraareassociatedintopairsasduals.46Forexample,thedualofacubeis

octahedronandthedualoftheIh-C60fullereneisadodecahedron.(Figure11)

Figure11Planarand3Dembeddingoftwofullereneisomers,Ih-C20andIh-C60.(a)

Ih-C20,ofwhichthedualisicosahedron.(b)Ih-C60,whosedualisdodecahedron.

Reproducedfromref32.Copyright2014JohnWiley&Sons,Inc.

20

Therearemanygoodpropertieswhenpolyhedraarestudiedinpairofduals.First,

apolyhedronhasthesamenumberofedgesasitsdual,duetoEuler’spolyhedra

formula.Thedualoperationmakesexchangeofthefactors,fandv.Accordingto

theformulae=v+f+2,e,thenumberofedges,remainsunchanged.Anotherkey

propertyofdualsisthatdualoperationconservesthepointgroupsymmetry.A

polyhedron and its dual belong to the same point group. For example, an

icosahedron and a dodecahedron are duals, and both of them have the Ih

symmetry.Thismakesiteasiertofigureoutthepointgroupofafullerene,since

mostfullerenesstructuresaremuchmorecomplexthanitsdual.Tofindoutthe

symmetry group of a fullerene,we can construct the dual of the fullerene and

figureoutthepointgroupofthedual.

As stated in the definition of fullerenes, carbon atoms are sp2 hybridized. The

hybridization typerequireseachcarbon isconnectedwithother threecarbons

andsurroundedbythreefaces.Sinceinthedualoperation,verticesaremapped

todual’sfacesandfacestodual’svertices,wecaneasilyprovethatafullerene’s

dualisapolyhedroncontainingonlytriangularfaces,calleddeltahedron.Thedual

ofafullerenewithnverticesinvolvesntriangularfacesandn/2+2vertices.A

vertexof thedual ism-valent if the corresponding fullerene facehasmedges.

Obviousl, a fullerene’s dual ismuch easier to be constructed, transformed and

analyzedthanthefullereneitself.Inmostcases,whenwewanttosolveatopology

problemrelatedtoafullerene,wefirstconstructitsdualandapplyoperationson

itsdualandthenreconstructthefullerenestructure.

Allthemeritsoffullerenedualsarebasedontheprerequisitethateachfullerene

vertex is uniquely associated to three jointly connected vertices of the

21

corresponding dual. In other word, the correspondence between a fullerene’s

verticesanditsdual’sfacesmustbeexclusive.Thisrequiresthatthestructureof

afullerene’sdualcontainsnoseparatingtriangle.Aseparatingtriangleisdefined

by three vertices of a deltahedron and separates the deltahedron into two

deltahedra.Itmayappearwhenmorethanthreeverticesofthedualaremutually

adjacent.Oneof themostdiscussedexamples is thedualofprism,a triangular

bipyramidwithaseparatingtriangle.Luckily,itcanbestrictlyprovedthatduals

ofclassicalfullerenesorheptagon-containingfullerenescannothaveseparating

triangles. However, for other non-classical fullerenes, especially triangle-

containingorsquare-containingfullerenes,thisshouldbeconsidered.

2.2 PointGroupsofFullerenes

Thecalculationsofpointgroupsareanimportantextensioncomponentinmy

program.Thereareseveralwaystoidentifythepointgroupofaparticular

fullereneisomer.Inthisparagraph,Iprovidetheonethatcanbedesignedasa

computeralgorithmandallpossiblepointgroupsforheptagon-containing

fullerenesareincludedinmyprogram.

Afullereneisomerismarkedusingitspointgroup.Forexample,anisomerofC60

that belongs to the icosahedral group (Ih) ismarked as Ih-C60. This is because

isomersbelongingtothesamepointgrouparequitesimilarinmanyaspects.47A

pointgroup isa setof symmetriesanda fullerene isomerbelongs toa specific

pointgroupiftheisomerhasallsymmetriesofthatpointgroup.Althoughthere

aremanypointgroupsforthree-dimensionstructures,fullerenesbelongtoonlya

smallportionofthem.Thisisbecausefullerenesarecomprisedofpolygonalfaces

and these faces limit symmetry operations to a small number. For a classical

22

isomer comprising pentagons and hexagons, the possible n-fold rotational

symmetriesareC1,C2,C3,C5andC6.48Theintroductionoffacesbesidespentagons

andhexagonsmayexpandthesetofpossiblepointgroups.Theidentificationofa

fullerene’spointgroupcanbemadebasedonthesymmetrypointgroupdecision

tree.49 (Figure12)Programsalso follow the same idea to figureout thepoint

groupsof fullerene isomers.Firstly,wecanget theorderof the isomer’spoint

groupandthendeterminetheexactgroupwiththesameorder.

Figure12Thedecision tree for figuring out the point group for any fullerene

isomer.Reproducedfromref.49.Copyright2014JohnWiley&Sons,Inc.

3 GeneralizedSpiralProgram:

Inthischapter,Idescribemyprogramforanalyzingfullerenesindetail.Idesigned

thegeneralizedspiralalgorithmbasedonaspiralalgorithmandimplementedit

inClanguage.Theprogramalsocontainscomponentsforcalculatingtopological

propertiesofheptagon-containingfullerenes.

23

The generations of valid fullerene isomers are required for almost all kinds of

fullereneresearch.It isnothardtogenerateclassical fullerenessincethereare

severalprogramsavailable.50However,noneofthemcanbeusedtogeneratenon-

classical fullerenes, and the lack of non-classical fullerene generation program

actuallylimitsfurtherinvestigationsonnon-classicalfullerenes.Inthischapter,I

willintroduceageneralizedspiralprogram,aprogrambasedonspiralalgorithm

butcanbeusedtogeneratenon-classicalfullerenes.Beforethedemonstrationof

thegeneralizedprogram,Ifirstdescribenecessarydefinitions,theoriesandalso

thespiralalgorithm.

3.1 Euler’sPolyhedraFormula

Euler’stheoremisoneofthebasictheoriesappliedforfullerenestructure

calculations.Inmyalgorithm,Iusethistheoremtodeterminethenumbersof

facesbelongingtoeachcategory.

Euler’spolyhedraformula,namedafterthegreatmathematicianLeonhardEuler,

introduces the relation among the numbers of vertices, faces and edges of a

polyhedron. The relation, which reveals the fundamental properties of a 3-

dimension polyhedron, is expressed by a simple formula: v – e + f = 2. In the

formula,v,eandfarethenumberofvertices,thenumberofedgesandthenumber

offaces,respectively.Afullereneisomer,leavingoutthechemicalpropertiesand

thedetailsofthegeometricstructures,canbeabstractedasapolyhedron,49and

obeys Euler’s polyhedra formula. Considering the fact that all carbons of a

fullerenearesp2hybridizationandeachcarbon-carbonbondisformedbetween

twoatoms,thenumberofedgesis3n/2forafullereneCn.Sothetotalnumberof

facesisn/2+2.Inaddition,thenumberofedgesisequaltothesumofedgesofall

24

facesdividedby two,becauseeveryedge issharedby two faces.Basedonthis

relation, we have another equation: !"#= 𝑒𝑑𝑔𝑒𝑠𝑜𝑓𝑎𝑙𝑙𝑓𝑎𝑐𝑒𝑠 . A classical

fullereneiscomprisedoftwocategoriesoffaces,pentagonsandhexagons.Sowe

can calculate the numbers of pentagons and hexagons in a classical fullerene,

whichare12andn/2–10,respectively.Fornon-classicalfullerenes,ifwegivethe

numberandthetypeoffacesotherthanpentagonsandhexagons,wecanalsoget

thenumbersofallkindsoffaces.Tomakeitclear,Iwillgivetwoexamples,one

for the classical fullerene C60 and another for the non-classical one heptagon-

containingfullereneC36.ThetwofullerenesareshowninFigure13.

Figure13ThestructuresofIh-C60(left)andC1-C36(right)withoneheptagonal

face.

ForC60,n=60,andthelinearsystemisasfollowing:

𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑏𝑒𝑡𝑤𝑒𝑒𝑛𝑓𝑎𝑐𝑒𝑠𝑎𝑛𝑑𝑒𝑑𝑔𝑒𝑠:6789:#

= 𝑛𝐸𝑢𝑙𝑒𝑟>𝑠𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑜𝑛𝑓𝑜𝑟𝑚𝑢𝑙𝑎:

𝑝 + ℎ − "!= 2

p:#ofpentagonsh:#ofhexagon…….….(1)

Settingnequalto60andsolvingthelinearsystem,wecangettheresult:

25

𝑝 = 12ℎ = 20

Fornon-classicalfullerenes,thelinearsystemcanbeobtainedsimilarly.

𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑏𝑒𝑡𝑤𝑒𝑒𝑛𝑓𝑎𝑐𝑒𝑠𝑎𝑛𝑑𝑒𝑑𝑔𝑒𝑠:6789:8:7

#= 36

𝐸𝑢𝑙𝑒𝑟>𝑠𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑜𝑛𝑓𝑜𝑟𝑚𝑢𝑙𝑎:𝑝 + ℎ + ℎ𝑝 − #9

!= 2

hp:#ofheptagons……………………………..(2)

SincethereisonlyoneheptagoninvolvedinC36,hp=1andtheresultforthelinear

systemis:

𝑝 = 13ℎ = 6

3.2 SchlegelProjectionandSpiralGraph

Schlegelprojectionandspiralgrapharethefoundationsofspiralalgorithms.All

the spirals and structure generations are developed based on these two-

dimensionalgraphs.Inmynewalgorithm,Ialsousethesetwotechniques.

Oneof the important featuresof fullerenes is the3-dimensionstructure,which

however, brings difficulties for the isomer representations and the designs of

generationalgorithms.Todescribea3-dimensionstructure,especiallypolyhedra,

an adjacency matrix is commonly used, due to the fact that structures are

determined by vertex connection relationships. In graph theory, an adjacency

matrix isa squarematrixof sizenandn is thenumberofvertices, specifically

carbon atoms of fullerenes. The value of each entry, entry (i, j) for example,

indicateswhethercarbonatomsiandjareconnected.Sincethereareonlytwo

conditions, we can simply use 1 and 0 represent situations of connected and

unconnected, respectively.Basedon thedefinition,each fullerene isomerhasa

correspondingadjacencymatrix.

26

Although it is convenient to convert a fullerene structure to its corresponding

adjacencymatrix,itishardtofigureoutwhetheranadjacencymatrixisavalid

representation for a fullerene isomer. So projection,51 converting 3-dimension

structuretocorrelative2-dimensiongraph,isanalternativemethod.Forfullerene

cages, abstracted as convex polyhedra, Schlegel projection,52 named after a

famousmathematicianVictorSchlegel,ismostwidelyusedforstructureanalysis.

Schlegel projection is a projection from through a point beyond one of faces,

usuallythepointbeyondthefaces locatedatthenorthpole.If twoverticesare

connectedinthepolyhedra,thenthereisasegmentjoiningpointscorresponding

to these two vertices in the projection graph. The entire projection graph is

dividedbythesesegmentsintosubdivisions,whichconservethesamenumberof

edges as the corresponding faces in the ployhedra. Figure 14 illustrates the

approachofSchlegelprojectionofIh-C60.

Figure14SchlegelprojectionofIh-C60.Reproducedfromref.51.Copyright2014

JohnWiley&Sons,Inc.

Spiralgraph,14asitsnameimplies,isaspiralstripthatreacheseveryfaceofthe

Schlegelimageforexactlyonetime.Thespiralstripstartsatanarbitraryfaceand

then reaches all other faces following a specific rule to get a continuous spiral

curve.Therulerequiresthatatanypointthenextfacetobereachedistheone

27

sharing an edgewithboth the current face and the first face that is still open.

Accordingtothedefinitionabove,itisobviousthattheremayexistmorethanone

validspiralgraphforagivenSchlegelgraph.Asdiscussinginthelastchapter,the

relationbetween the fullerene isomerand theSchlegel graph isone-to-one. So

eachfullereneisomermayalsohavemorethanonespiralgraph.Thefirstquestion

iswhatthehigherlimitis?Accordingtothedefinitionofspiralgraph,ifthefirst

threefacesaregiven,thespiralstripwillbedetermined.Sothetotalnumberof

possiblespiralscanbecalculatedinthisway:first,pickanarbitraryfaceandstart

atthisface.Second,eachedgeofthestartingfaceissharedwithanotherfaceand

allthesefacescanbethenextoneforthespiralstrip.Supposethestartingfacehas

𝐹edges,thenthereare𝐹possibilitiesforthechoiceofthesecondface.Thirdly,to

findthethirdface,therearetwodirectionsfromthesecondface,clockwiseoranti-

clockwise.Sotheformulatocalculatethehigherlimitofallpossiblespirallinesis

asfollowing:

𝐹K×2"

KMN

inwhich,nisthenumberoffacesandFiisthenumberofedgesofithface.

Notallpossiblespiralsencodethefullerenegraphsuccessfully.Ifthespiralenters

afacethatisnotthelastoneandissurroundedbyfacesalreadytraversed,the

spiralwillfailtowindthefullerenegraph.14Intheworstcase,allpossiblespirals

failandthereisnovalidspiralforthefullerenegraph.Actually,therearefullerene

isomersthatcannotbeencodedbyspirals,howevertheproportionisextremely

tiny.Moreimportantly,thesmallestfullerenewithoutaspiralisC380,53afullerene

isomerthatistoolargetobeconsideredinalmostallinvestigations.Inmostcases,

thespiralisstillthemostcommonmethodtoencodeafullerenegraph.

28

3.3 SpiralAlgorithm

Inthischapter,Idescribetheoriginalspiralalgorithm.Thisspiralalgorithmisfirst

designedandappliedformakingcircuitdiagraminelectronicengineering.Then,

itisextendedtootherareasincludingchemistry.

As discussed in the last chapter, fullerenes smaller than C380 can always be

encodedusingspiralstrips.Conversely,wecanalsogenerateaspiralstripfirst

andthenconvertittothecorrespondingfullerenegraph.Wenoticethattheorder

offacesreachedalongthespiralpathdeterminesaspiralstripuniquely.Ifweuse

asinglenumbertorepresentaface,forexample5representsapentagonand6

representsahexagon,wewill geta sequenceofnumbers,whoseorders in the

sequence give the position of their corresponding faces along the spiral path.

AccordingtoEuler’spolyhedraformula,thenumbersofeachkindoffacescanbe

calculated based on the number of carbons. We can generate all possible

sequences based on this result. For example, the classical Td-C20 with 12

pentagonsand4hexagonsshouldberepresentedbyasequenceoftwelve5sand

four6s.Itisnotdifficulttogenerateasetofallpossiblesequences.Allsequences

correspondingtovalidspiralsarecontainedintheset,duetothefactthatevery

validspiralcanberepresentedbyasequence.

The number of all possible sequences also has a higher bound, which can be

calculatedbasedonthepermutationtheory.Let’susetheclassicalfullereneCnas

anexample.AccordingtoEuler’spolyhedraformula,thereare12pentagonsand

n/2-10hexagons,andinotherwords,therearetwelve5sandn/2-106sinthe

sequence.Sothereareatmost:

𝑛2 + 2 !

12! 𝑛2 − 10 !

29

possible sequences. The higher bound determines the time complexity of the

fullerene generation algorithm. The time complexity, which indicates the

efficiency of the algorithm, can be improved, but the improvement is not

significant.54Moredetailswillbediscussedinthenextchapter.

OncewehavegottenthesetSofallpossiblesequencesatthispoint,thenextstep

is to find a subset Sv containing all sequences that can wind up a fullerene

successfully.Each time,wepicka tentative sequence from the setSandcheck

whetheritbelongstoSv.Thewaytocheckasequenceistoconstructafullerene

graphfromthesequencefollowingthespiralcriteria.Asmentionedinprevious

chapters,thespiralcriteriarequirethenewfaceaddedshouldbeconnectedtoits

direct precursor and the first open face, in which open mean the face is not

surroundedbyotherfaces.Ifwecangetafullerenegraphinthisway,thenthe

sequenceisvalid.

Theconstructionprocedurecanalsobeconsideredintheviewoftheconstruction

offullereneduals.Sinceeachcharacterinthesequencerepresentsafaceofthe

fullerene,thecharacteralsocorrespondstoavertexofthefullerenedual.Dueto

thepropertiesoffullereneduals,theassociationoffullereneanditsdualisastrict

onetoonerelation.Ifthesequencecanconstructadualsuccessfully,itisalsoa

validsequenceforafullereneisomer.

Althoughwecanusethemethoddemonstratedabovetocheckwhetheraspiral

strip is valid, theproblem is thatmore thanone spiralmay relate to the same

fullereneisomer.Ifweusetheallpossiblesequencestogeneratefullereneisomers,

there will be duplicated ones. To eliminate repeated spirals and make one

fullerene isomercorresponds toexactlyone spiral strip,weneed to selectone

spiral fromall spirals to represent the corresponding fullerene isomer. Sincea

30

spiralstripisexpressedasasequenceofnumbers,wecanusethesequencewith

the smallest value as the representation, called the canonical sequence.As the

example illustrated inFigure15, threespiralscanwindupTd-C28successfully.

Thesequencesofspiralsare6555556565555556,6555565655555

556and5556565655555556.It isobviousthatthelastonehasthe

smallestvalueamongthethreesequencesanditisthecanonicalsequenceforTd-

C28.

Figure15Thethreevalidsequencesandcorrespondingspiralgraphs.(3)isthe

canonicalsequence,whichhasthesmallestvalue.

Another advantage of the spiral algorithm is that the symmetries can also be

calculatedsimultaneously. In thesymmetry theory, there isasimplerelation,14

whichis:

𝑁Q = 𝑁R|𝐺|

31

Ntisthetotalnumberofspiralsunwoundsuccessfully.Nsisthenumberofvalid

symmetry-distinctspirals.Gistheorderofthepointgroupofthefullereneisomer,

whichisakeyfactortodeterminethesymmetry.Forexample,Ih-C60has120valid

spirals, so Nt = 120. Among all these spirals, we can find there is only one

symmetry-distinctspiral,whichmeansNs=1.Theisomer’spointgrouporderis

120.Thereisaproblem,asmentionedabove,thatthespiralmayfail.However,

theresultdoesnotcountonthis.Forexample,theD2-C28isomercanbeunwound

successfully in164spiralsandforty-oneofthosearesymmetry-distinct.Sothe

pointgrouporderofthisisomeris4,whichpointstoD2.

3.4 GeneralizedFullereneFormationProgramBasedonSpiralAlgorithm

ThischapterdemonstratesthewayIdesignthegeneralizedalgorithm.Although

theprovidedprogramincludesseveralcomponentsandvariousalgorithm,the

generalizedspiralalgorithmisthemostimportantone.

ThefirstfullerenegenerationprogramisprovidedbyFowlerandManolopoulos

in their book “AN ATLAS OF FULLERENES”.14 Following the first generation

program, several programswere released and improvements have beenmade

bothonthegenerationalgorithmandtheprogramdesign.Theseprogramsare

more efficient and provide many powerful capabilities, such as calculating

coordinatesanddrawingstructuregraphs.14,49,51,55However,theideasofthese

programsareallbasedonthespiralalgorithm.Nowadays,non-classicalfullerenes,

especially heptagon-containing fullerenes, become the hot spot of fullerene

research.Moreandmoreinvestigationsshowthatheptagon-containingfullerenes

play significant roles in fullerene formations. A programwith the capability of

generatingnon-classicalfullerenesisurgentlyneeded.

32

Thefirstmodificationismadeonthecalculationofthenumbersoffacesbelonging

todifferentcategories.Intheoriginalspiralprogram,whichhandlestheclassical

fullerenes, we can obtain the numbers of pentagons and hexagons by simply

applying the formulas mentioned in last chapter. A classical fullerene with n

carbonatomscontainsexactly12pentagonsandn/2-10hexagons,respectively.

Withtheintroductionofunusualfaces,polygonsbesidespentagonsandhexagons,

weaddtheeffectsofthesefacesintotheEquation1.Forexample,anon-classical

fullerenewithheptagonssatisfiesthefollowingequations:

𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑏𝑒𝑡𝑤𝑒𝑒𝑛𝑓𝑎𝑐𝑒𝑠𝑎𝑛𝑑𝑒𝑑𝑔𝑒𝑠:5𝑝 + 6ℎ + 7ℎ𝑝

3 = 𝑛

𝐸𝑢𝑙𝑒𝑟>𝑠𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑜𝑛𝑓𝑜𝑟𝑚𝑢𝑙𝑎:

ℎ𝑝 + 𝑝 + ℎ −𝑛2 = 2

wherep,handhpstandforthenumbersofpentagons,hexagonsandheptagons,

respectively.Obviously, the solutions of hp, p andh depend on each other. By

eliminating one unknown at one time, we can find the relations between two

arbitraryunknowns:

ℎ𝑝 = 𝑝 − 12

ℎ + 2ℎ𝑝 =𝑛2 − 10

2𝑝 + ℎ =𝑛2 + 14

Oneoftheinterestingconclusionsisifoneheptagonisintroduced,thenumberof

pentagonswillincreasebyone.Forexample,oneheptagon-containingfullerenes

havethirteenpentagons,whileaclassicalfullerenecontainstwelvepentagons.

The first component of the generalized spiral program, a function output all

possible sequences, is designed based on the formulas mentioned above and

permuting. To make the comments clearer, I used an example, C60 with one

33

heptagon, to demonstrate the details of this process. The fullerene contains

thirteenpentagons,oneheptagonandeighteenhexagons.Sothereare:

32!13! 18! 1!

possibilities for the sequences. In this case, we can firstly choose fourteen

positions for the heptagon and pentagons, and then choose one position from

thesefourteenpositions.Obviously,

32!13! 18! 1! =

3214

141

Similarly, we can apply this method, choosing positions for faces of different

categoriesstepbystep,onmorecomplexsituations.

Thesecondcomponentoftheprogramisdesigntocheckwhetherasequenceis

validforconstructingafullerenegraph.Sinceclassicalfullerenesinvolvemerely

pentagonsandhexagons,theoriginalspiralprogramonlyneedtoconsidertwo

kinds of faces. To enable the program to generate non-classical fullerenes, the

design should be modified to accommodate polygons of other categories. As

mentionedabove,althoughthespiralprogramisspecificallydesignedforclassical

fullerenes,thespiralalgorithmisnot.Thegeneralideacanalsobeappliedonnon-

classical fullerenes. So we can generalize the program by setting parameters

dynamicallybasedonthetypesoffaceswithoutchangingthealgorithm.

The lastcomponentof theoriginalspiralprogramunwindsthefullerene’sdual

andcheckswhethertheinputsequenceisacanonicalsequence.Atthemeantime,

this subroutine calculates the point groups of fullerene isomers based on the

symmetry decision tree mentioned above. As commonly known, classical

fullerenes are restricted to 28 possible point groups,56 which simplifies their

identification.Similarly,wecanalsofindlimitationsfornon-classicalfullerenes.A

34

polygoncanonlyintroducenewn-foldrotationsthatithastothefullerenes.For

example, heptagons introduce C1 and C7 and squares introduce C1, C2 and C4.

According to this rule, the possible number of point groups of non-classical

fullerenesisstillinreasonablerangewhichwecanhandlebyprograms.

Considering that most quantum calculations of fullerenes use coordinates of

atomsasthestartingpoint,Iaddthenewfunctionthatcangeneratecoordinate

files along with the structure generation process. We can first obtain the

connectionsofatomsandthenapplymathematicsonthemtogetthecoordinates.

Itistoodifficulttocalculateatomconnectionsdirectlyfromthefullerenegraph.

However,asdiscussedinlastchapter,wecanfirstconvertafullerenegraphtoits

dual and then calculate the face connections of the dual, which are atom

connectionsofthefullerene.Otherfunctionssuchasfile formattransformation

arealsoaddedintotheprogram.Theaimofthenewprogramistosolveproblems

oftopologiesandmathematics,andhelpuserstofocusonpropertiesoffullerenes

respecttochemistry.

35

Figure16Themainstructureandtheaccessorycomponentsofthegeneralized

spiralprogram

3.5 Research for Single-Heptagon-Containing Fullerenes in the Range C30-C38StructuresandSymmetries

The aim of designing the generalized spiral program is for investigations of

heptagon-containing fullerene. In thischapter, Igiveapreliminary idea for the

applicationsofmynewprogram.

Accordingtotheoriginalspiralprogram,thefirst,namelythesmallestclassical

fullereneisC20andthereisonlyoneisomer,whichbelongstoIhpointgroup.57

The isomer is comprised of 12 pentagons and as suggested by the IPR, it is

unstablebecauseoftheadjacentpentagons.Similarly,smallfullerenes,fromC20

toC58,arealsopredictedasunstable,duetothesamereason.AlthoughC20andC36

havebeenreportedthattheycouldbeobtainedasacovalentlybondedcluster-

36

assembledmaterialinthesolidstate,58,59noneoffullerenesinthisrangehasbeen

synthesizedintheformofemptyfullerenecagesorendohedralfullerenes.Asfor

thesingleheptagoncontainingfullerenes,thefirstisomerisC30withC1symmetry.

ThenumbersofisomersintherangeofC30-C48arelistedinTable1.

#ofclassicalisomers #of1heptagonisomer

C30 3 1

C32 6 2

C34 6 8

C36 15 16

C38 17 42

C40 40 89

C42 45 199

C44 89 388

C46 116 763

C48 199 1429

Table 1 The numbers of classical isomers and single-heptagon-containing

fullereneisomersintherangeofC30-C38.

Obviously, there are more possible isomers for single-heptagon-containing

fullerenes than classical ones. It can be explained that the introduction of one

heptagonal facebringsmorepossibilities for the sequencepermutations.More

possiblepermutationsalwaysmeanmorevalidsequence,althoughitcannotbe

provedstrictly.

37

Oneofthereasonswhyfullerenesinthisrangeareinterestingisthatthenumber

ofisomersisnottoolargeandwecanapplycomprehensivestudiesonallisomers

tofindfundamentalprinciples.Asweallknowforclassicalfullerenes,therelative

energiescanbeestimatedbasedonthecagetopologicalstructures.Inmoredetail,

therelativeenergiesaredeterminedbyagroupofparticularmotifs.Wangetal.60

reported their investigations on the relation between cage topologies and

endohedral metallofullerenes in 2015. According to their research, the total

energiesofEMFs,𝐶!"Z (2n=68-104,q=0,-2,-4,-6)arecalculatedatSCC-DFTB

level and fitted in termsof sevenmotifs. Theydemonstrated that the energies

calculated from the fitting equationwere satisfactory compared to the results

obtainedbythecalculationsbasedonDFT.Duetotheresult,therelativeenergies

canbeproducedbysimplycountingthenumbersofsevenkeystructuralmotifs.

Itcanbeusedtoselectstablefullereneisomersbeforethequantumcalculations

andexperiments.

Forsingle-heptagon-containingfullerenes,wecanfindananalogueofthisenergy-

motif rule. By inspecting the energies and structural motifs of corresponding

single-heptagon-containing fullerenes, we know that the relative energies are

determined by two categories of motifs, heptagon based and pentagon based

motifs.(Figure17)Theheptagon-basedmotifsstabilizethefullerenesandonthe

otherhandthepentagon-basedmotifsdestabilizethefullerenes.However,tofind

anaccurateformulabasedonthesemotifs,moreworkstillneedstobedone.

38

Figure 17 The motifs of heptagon containing fullerenes that determine the

relativeenergies.(a)showspentagonbasedmotifs,whichdecreasethestability.

(b)Anexampleofheptagonbasedmotifs,whichincreasethestability.

Otherthanfindingtheenergy-motifrule,wecanalsofindtherelationbetween

single-heptagon-containing fullerenes and classical fullerenes by applying

comprehensivestudiesontherangeofC30-C38.Asfarasweknow,smallfullerenes

are not stable, due to the strain effects and the fused pentagons.61 For one

heptagon-containing fullerenes, fused pentagons may appear around the

heptagonalfaceandthesemotifswillstabilizethefullerenes.Thequestionisifthe

introductionofoneheptagonmakesthesefullerenesmorestable.Thecomparison

betweenthesingle-heptagon-containingfullerenesandclassicalfullerenesofthe

same size shows that single-heptagon-containing fullerenes are not better in

stabilitythanclassicalones.Thereasonmaybecomplex.Themotifs,heptagons

surrounded by pentagons, are preferred because the electron distribution

enhances the aromaticity.However, theywill alsobring extra strains for small

39

fullerenes.Oneofthesatisfactoryexplanationsisthatthestructuralstrains,rather

thanthearomaticity,isthedominantfactoroftherelativestabilities.62

Anotherkeyobservationforsingle-heptagon-containingfullerenesisthatmostof

them have low symmetries. The single heptagonal face limits the symmetry

operationsthatcanbeappliedonthecages.Symmetriesareoneoftheessential

characteristics of fullerenes. Although the role of symmetries has not been

understood thoroughly, more and more investigations point out that the

symmetries of fullerenes have close relationship with the properties, such as

kineticstabilities.Furtherresearchisneededtorevealthemysteryofsymmetries.

3.6 ResearchforTwo-Heptagon-ContainingFullereneswithHighSymmetries

Ascommentedinthelastchapter,single-heptagon-containingfullerenesbelong

to low symmetric point groups and most of them are even asymmetric. The

simplest way to obtain high symmetric heptagon-containing fullerenes is to

introduce another heptagonal face.63 Based on the two-heptagon-containing

isomersgeneratedbythegeneralizedspiralprogram,someoftheseisomersare

foundtohavehighsymmetries,suchasD7handD7v.Moreinteresting,thereisa

seriesofisomers,withsymmetriesofD7handD7valternately.Thesecagesstartat

C28 and formed by adding fourteen carbon atoms each time. As illustrated in

Figure18,fullerenesbelongingtothisgrouparecylinder-like,whosepentagons

are connectedwith the two heptagons. These structures are suggested by the

generalizedisolatedpentagonrules,1aswellastheenergy-motifrulediscussed

above, tobepreferred in thermodynamicstabilities.However, thesestructures

alsobringextrastrainsifthecagesarelargeenough.

40

Figure18Thestructuresofaseriesoftwoheptagoncontainingfullereneswith

D7handD7dsymmetriesalternately.

Anotherreasonwhythesefullerenesareessentialisthattheirunusualstructures

are good certifiers for the template synthesis64 ofmetallofullerenes. Template

synthesis is widely applied in many areas, especially organic and inorganic

synthesis.65Oneofthemostsuccessfulexamplesisthetemplateapproachforthe

synthesisofcrownethers.66Thesynthesisofagivencrownetherisbasedona

metalatomoftherightsize.Forexample,18-Crown-6canbesynthesizedusing

potassium as the template and 15-Crown-5 can be synthesized using sodium.

Similarly,metallofullerenescanalsobesynthesizedbythesametechniqueandthe

templatesarechosenbasedofthesizeoftargetisomers.Fullerenesbelongingto

thegroupmentionedabovehavethesamediameterbutdifferentheights.Sothe

corresponding templates also vary in lengths.We can select a series of metal

clustersandcalculate theenergiesofcorrespondingmetallofullerenes tocheck

whetherthetemplatesynthesisisfeasible.

4 ConclusionsWith the program provided in this thesis, it is possible now to generate non-

classical fullerene isomers and to assign point group for each isomer. The

41

generalizedspiralprogramalsoinvolvesthefunctionthatoutputcoordinatesof

carbonatoms.Fromthesecoordinates,itisconvenienttodrawthestructuresas

well as to obtain input files formost quantum chemistry software. A group of

scripts are also provided to generate input files for common software

automatically. These tools save users from solving unfamiliar topological and

mathematic problems. They facilitate the investigations on non-classical

fullerenesandhelpscientistsfocusonchemicalaspects.Fortheconvenienceof

further modifications, the theories behind the generalized spiral program are

demonstrated.

Based on the generalized spiral program, potential investigations on single-

heptagon-containing fullerenes and on two-heptagon-containing fullerenes are

proposed.Bytraversingtheenergiesofsingleheptagoncontainingfullerenesand

associatingtheenergiestoagroupofmotifs,wecanfitasimplelinearformulato

estimate relative energies of heptagon-containing fullerenes without doing

quantum calculations. Most of single-heptagon-containing fullerenes are

asymmetric, so another heptagonal face is introduced to form high-symmetric

fullerenes.Thefirsttwo-heptagon-containingfullereneisC28withD7dsymmetry.

Interestingly,thenwecanconstructaseriesofhigh-symmetriconesthathaveD7d

and D7h alternately, by adding 14 carbons each time. These isomers are good

certifiers for template synthesis of metallofullerenes because of their unique

structures.

In short, this article provides necessary programs and demonstrates related

topologicaltheoriestosimplifythepreparationsforinvestigationsofnon-classical

fullerenes.Allprogramsmentionedareattachedinappendixalongwithexamples.

42

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6 Appendix:

A.MethodsforconstructingfullerenecagesfromtheoutputofGspiralprogram

Thespiralsequenceisthefingerprintofitscorrespondingfullerenestructure.It

hasbeenapprovedinChapter3.4thatthereisaone-to-onerelationshipbetween

45

fullerene’s structure and its spiral sequence. So I demonstrate the way to

reconstructfullerenestructuresbasedonspiralsequences.

Thespiralsequenceisthefingerprintofitscorrespondingfullerenestructure.It

hasbeenapprovedinChapter3.4thatthereisaone-to-onerelationshipbetween

fullerene’s structure and its spiral sequence. So I demonstrate the way to

reconstructfullerenestructuresbasedonspiralsequences.

The reconstructionof a fullerene structure canbedone in severalways. I first

provideastraightforwardmethodthatcanbeimplementedusingmostchemistry

structuremakingsoftware.

1. Pick one ring each time from the ring sequence and put it in the position

indicatedbyspiralalgorithm,whichisdiscussedinchapter3.3.

2.Usetheautomaticstructureadjustfunctionofthesoftware,thenthe2-Dmap

(fullerenegraph)willbecome3-Dstructureanditisagoodstartforcalculations.

Figure 1 Reconstruction of fullerene structure based on sprial sequence. The

numbersinthecenteroffacesindicatetheorderinspiralsequence.Thelastface

inthesprialsequencehasthesameshapeastheborderofthespiralgraph(left

figure).

46

Ihaveverifiedthereconstructionmethodbyagroupoffigureslistedinappendix

usinggaussview.

Another way to reconstruct the fullerene structures are based on adjacency

matricesthataregeneratedbymyprogram.Severalopensourceprogramsare

available for this convertion. In this thesis, EMBED50, developed by Gunnar

Brinkmann,isapplied.ThealgorithemoftheEMBEDcodeistoembedeachvertex

onsphericalsurface.ThecommandforapplyingEMBEDisasfollowing:

embed<infile>outfile

TheinfileistheadjacencymatrixcalculatedbytheGsprialprogramandtheoutfile

istheCartesiancoordinates.

B.Atlasofsingle-andtwo-heptagon-containingfullerenes

C.GeneralizedSpiralProgram(Gspiral.cc)