11

Fullerenes 3Fullerenes 3

MirceaMircea V. DiudeaV. Diudea

Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400028400028 ClujCluj, ROMANIA, ROMANIA

diudea@chem.ubbcluj.rodiudea@chem.ubbcluj.ro

22

ContentsContents

1.1. Periodic CagesPeriodic Cages

2.2. fafa--TubulenesTubulenes

3.3. tata--TubulenesTubulenes

4.4. fzfz--TubulenesTubulenes

5.5. kfzkfz--TubulenesTubulenes

6.6. (5,6,7)(5,6,7)kfzkfz--TubulenesTubulenes

7.7. (5,7)(5,7)kfzkfz--TubulenesTubulenes

33

•• In simple In simple HückelHückel theory,theory,11 the energy of the the energy of the i i ththππ --molecular orbital is calculated on the grounds molecular orbital is calculated on the grounds of of AA((GG ) )

EEii = = αα ++ ββλλiiEEHOMOHOMO –– EELUMO LUMO = gap= gap

•• SemiempiricalSemiempirical approacesapproaces::

Heat of Formation HF (kcal/mol)Heat of Formation HF (kcal/mol)

11. E. Hückel,. E. Hückel, Z. Phys.Z. Phys., 1931, , 1931, 7070, 204., 204.

44

OPOPopenopen00λλ NN/2/2 = = λλ NN/2+1/2+144

MCMCmetametaclosedclosed≠≠ 000 0 ≥≥ λλ NN/2/2 > > λλ NN/2+1/2+133

PSCPSCpseudopseudoclosedclosed≠≠ 00λλ NN/2/2 > > λλ NN/2+1/2+1 > 0> 022

PCPCproperlyproperlyclosedclosed≠≠ 0 0 λλNN/2/2 > 0 > 0 ≥≥ λλ NN/2+1/2+111

symbolsymbolshellshellGapGapRelationRelation

ππ --Electronic StructureElectronic Structure

55

A capped nanotube we call here aA capped nanotube we call here a tubulenetubulene

NN Cap Cap SpiralSpiral sequence: sequence: ClassClass

66k k k k 66k k (56)(56)kk-- AA[2[2kk,,nn]] fafa --tubulenestubulenes44k k k k 55k k 77k k (56)(56)kk-- AA[2[2kk,,nn]] ta ta --tubulenestubulenes

33k k k k 55kk-- ZZ[2[2kk,,nn]] tztz --tubulenestubulenes1313k k /2/2 k k (56)(56)kk/2/2(665)(665)kk/2/2-- ZZ [3[3kk,,nn]] fzfz ––tubulenestubulenes

1111k k k k 66k k (56)(56)k k (65)(65)kk -- ZZ[2[2kk,,nn] ] kfkfzz ––tubulenestubulenes99k k k k (56)(56)kk/2/2(665)(665)kk/2/2(656)(656)kk/2 /2 77kk-- ZZ [2[2kk,0],0]((5,6,7)3) ((5,6,7)3) kfzkfz --tubulenestubulenes

1212k k k k (56)(56)kk/2/2(665)(665)kk/2 /2 6633kk/2 /2 (656)(656)kk/2 /2 77kk-- ZZ [2[2kk,0],0]((5,6,7)3) ((5,6,7)3) kfz kfz ––dvsdvs1111k k k k 55k k 77k k 5522k k 77k k -- ZZ[2[2kk,,nn]]((5,7)3) ((5,7)3) kfzkfz ––tubulenestubulenes

Building ClassificationBuilding Classification

66

fafa --TubulenesTubulenes

77

fa fa ––TubulenesTubulenes11

CC168(168(66 6666(5,6)(5,6)66--A[12,8])A[12,8]); ; CC22Cap CCap C36(36(66 666 6 ((5*)5*)66--A[12,0])A[12,0])

CCNN((kk 66kk(5,6)(5,6)kk-- A[2A[2kk,,nn])]);; NN = 12= 12kk + + pp

1. M. V. Diudea, Stability of tubulenes, Phys. Chem., Chem. Phys., 2004, 6, 332-339

88

IntroductionIntroduction

TU(6,3)A[TU(6,3)A[cc,,nn]] == armchair armchair ((c/c/22, , c/c/22))TU(6,3)Z[TU(6,3)Z[cc,,nn]] = = zigzag zigzag ((c/c/22, 0), 0)

99

Rules of thumbRules of thumb

•• LeapfrogLeapfrog rulerule11 AA--LERLER

NNLeLe = = 60 + 660 + 6mm == 3(20 3(20 ++ 22mm);); ((mm ≠ 1)≠ 1)

In In fafa-- tubulenestubulenes CCN N ((kk 66kk(56)(56)kk--A[2A[2kk,,nn])]) ((PC PC ))

NNLeLe = 12= 12k k ++22k k x x 33mmmm = 0, 1, 2,…, (= 0, 1, 2,…, (kk = 4 to 7)= 4 to 7)

1. P. W. Fowler and J. I. Steer, 1. P. W. Fowler and J. I. Steer, J. Chem. SocJ. Chem. Soc.,., Chem. CommunChem. Commun., 1987, 1403., 1987, 1403--14051405.

1010

Rules of thumbRules of thumb

•• Armchair cylinder ruleArmchair cylinder rule 1 1 ACRACR

fafa--tubulenestubulenes CCN N ((kk 66kk(56)(56)kk--A[2A[2kk,,nn])]) ((PC PC -- NBONBO))

N N = 12= 12kk + + 22k k ((11+3+3mm))mm = 0, 1, 2,…, (= 0, 1, 2,…, (kk = 4 to 7)= 4 to 7)

1. P. W. Fowler, 1. P. W. Fowler, J. Chem. SocJ. Chem. Soc.,., Faraday TransFaraday Trans., 1990, 86, 2073., 1990, 86, 2073--20772077.

1111

PM3 Heat of formation (per atom) of open and capped PM3 Heat of formation (per atom) of open and capped AA--nanotubes A[nanotubes A[cc,,nn] (at ] (at cc constant)constant)

0

5

10

15

20

25

30

35

40

45

2 4 6 8 10 12

n

Ene

rgy

(kca

l/mol

)

A-open

A-capped

Stability of TubulenesStability of Tubulenes

1212

Stability of TubulenesStability of Tubulenes

0.00

0.10

0.20

0.30

0.40

0.50

0.60

2 4 6 8 10 12 14 16 18 20

n

HO

MO

-LU

MO

gap

(| |)

faTubulenes

Periodicity of HOMOPeriodicity of HOMO--LUMO gap LUMO gap vsvs. tube . tube nn--dimension of dimension of fafa--tubulenes; tubulenes; kk = 6. = 6. The points for LER (1), ACR(2) and PSC(3) in decreasing order ofThe points for LER (1), ACR(2) and PSC(3) in decreasing order of their gap. their gap.

1313

tata --TubulenesTubulenes

1414

ta ta ––TubulenesTubulenes11

CC96(96(66 55667766(5,6)(5,6)66(6,5)(6,5)6677665 5 666)6) ((CC22))22CC24(24(66 55661166-- A[12,0])A[12,0]) + + A[12,4]A[12,4]

CCN N ((kk 55kk 77kk (56)(56)kk--A[2k,A[2k,nn])]) ; ; NN = 8= 8kk ++pp

1.1. M. V. Diudea, Stability of tubulenes, M. V. Diudea, Stability of tubulenes, Phys. Chem., Chem. Phys.Phys. Chem., Chem. Phys., , 2004, 6, 3322004, 6, 332--339339

1515

fa fa --TubulenesTubulenes

Geodesic ProjectionGeodesic ProjectionCC96(96(66 6666(5,6)(5,6)66(6,6)(6,6)66(6,5)(6,5)6666666) 6) ((DD66dd))

fa fa ––TubulenesTubulenes from from ta ta --TubulenesTubulenes byby SWSW isomerizationisomerization 11

1. A. J. Stone and D. J. Wales, 1. A. J. Stone and D. J. Wales, Chem. Phys. LettChem. Phys. Lett., ., 19861986, , 128128, 501, 501--503.503.

1616

HOMOHOMO--LUMOLUMO gap gap vsvs. the tube . the tube nn--dimension dimension of of fafa-- and and tata--tubulenestubulenes; ; kk = 6= 6

0.00

0.10

0.20

0.30

0.40

0.50

0.60

1 3 5 7 9 11 13 15 17 19

n

HO

MO

-LU

MO

gap

(| |)

faTubulenes

ta-Tubulenes

1717

Novel armchair cylinder rulesNovel armchair cylinder rules 11

1.1. TA TA cylinder rulecylinder rule TACRTACR : : CCN(kN(k 55kk 77kk(5,6)(5,6)kk--A[2k,n])A[2k,n]) ((PCPC))

N N = 8= 8kk + 2+ 2kk(4+3(4+3mm));; mm = 0, 1, 2,…, (= 0, 1, 2,…, (kk = 4 to 7)= 4 to 7)

or or NN = 12= 12kk + 2+ 2kk((22+3+3mm); ); mm = 0, 1, 2,…, = 0, 1, 2,…,

22. TNA . TNA cylinder rulecylinder rule TNACRTNACR : : ((PCPC--NBONBO))

N N = 8= 8kk + 2+ 2kk(11+3(11+3mm));; mm = 0, 1, 2,…, (= 0, 1, 2,…, (kk = 4 to 7)= 4 to 7)

oror NN = 12= 12kk + 2+ 2kk((33+3+3mm); ); mm = 2, 3,…= 2, 3,…

1. M. V. Diudea, Periodic fulleroids. 1. M. V. Diudea, Periodic fulleroids. Int. J. Nanostruct.Int. J. Nanostruct.,, 20032003, , 22(3), 171(3), 171--183183

1818

DiscussionDiscussion

TACRTACRm = 4m = 4

PSCPSCTNACRTNACRm = 3m = 3

…TACRTACRm = 0m = 0

--tata CCN N ((kk 55kk77kk(5,6)(5,6)kk--A[2A[2kk,n]),n])

PSCPSCACRACRm = 4m = 4

LERLERm = 4m = 4

…PSCPSCACRACRmm = 0= 0

LERLERm=0m=0fafa CCN N ((k k 66

kk(5,6)(5,6)

kk-- [2[2kk,,nn])])

200190180…807060N (k =5)

1919

zz --TubulenesTubulenes

2020

tztz --TubulenesTubulenes

2121

tz tz ––TubulenesTubulenes11

; ; DD66hhtztz--Cap Cap k = 7k = 7 ])6,12[56(108 6C Z−

CCN N ((k k 55kk--Z[2Z[2kk,,nn]] -- Parents of Parents of LERLER fafa--tubulenes of tubulenes of seriesseries C C 33N N ((k k 66kk (56)(56)kk --A[2A[2kk,,nn]]

])0,2[5(3C

kZkk k −

1. Z. Slanina, F. Uhlik, and L. Adamowicz, Z. Slanina, F. Uhlik, and L. Adamowicz, J. Mol. Graph. Modell.J. Mol. Graph. Modell., , 2003, 2003, 2121, 517, 517––522.522.

kk, 5, 5k k , [(, [( 6)6)kk]]n n , 5, 5k k , , kk

2222

Energetic and Spectral PropertiesEnergetic and Spectral Properties

OP00.4140.4141.28217.569D2h846;411

OP00.4140.4144.53917.639C2726;310

OP00.4140.4141.98919.774D2h606;29

OP00.4140.4142.47221.015C6486;18

OP00.4140.4144.14523.492D6366;07

OP00.3280.3282.84220.904C1805;56

OP00.3190.3193.08121.660C1705;45

OP00.3000.3003.49622.713C2605;34

OP00.2710.2715.15923.746C2v505;23

OP00.2260.2265.13025.892C2405;12

OP00.2710.2715.41328.939C2v305;01

ShellGAP(|β| units)

λN/2+1λN/2HF/at.(kcal/mol)

Spectral DataPM3GAP(eV)

PM3Sym.NCagek; n

Data for Data for tztz--tubulenestubulenes CC66k k ((k k 55kk--Z[2Z[2kk,,nn]]

2323

DiscussionDiscussion

Stability of tubulenes:Stability of tubulenes:

a.a. strain of the cap (strain of the cap (σσ frame) frame) b.b. ππ --electronic structure electronic structure

2424

DiscussionDiscussion

•• In In tztz--tubulenes, the tubulenes, the tztz--cap , cap , kk = 5; 6 is not = 5; 6 is not suitable for capping suitable for capping ZZ--tubes, despite their tubes, despite their mutual fitting. mutual fitting.

•• ReasonsReasons: (: (aa) the ) the positive curvaturepositive curvature induced by induced by the pentagons, not relaxed by surrounding the pentagons, not relaxed by surrounding heptagons, as in case of heptagons, as in case of tata--tubulenes. tubulenes. ((bb) the ) the open shellopen shell electronic structure of the electronic structure of the parent fullerene(s), hereditary in the parent fullerene(s), hereditary in the corresponding corresponding tztz--tubulene(s). tubulene(s).

2525

DiscussionDiscussion

•• Comparing the Comparing the tztz--tubulene and tubulene and tata--tubulene , the following tubulene , the following average strain energy values were found (in average strain energy values were found (in kcal/mol): kcal/mol): tz tz ; [14.59 (cap); 4.74 (; [14.59 (cap); 4.74 (ZZ--tube); 8.43 (global)]tube); 8.43 (global)]ta ta ; [10.53 (cap); 4.71 (; [10.53 (cap); 4.71 (AA--tube); 6.89 (global)] tube); 6.89 (global)] Clearly, the Clearly, the tztz--cap is more strained in cap is more strained in tztz--tubulenes than in tubulenes than in tata--tubulenes, while the joining tubulenes, while the joining tubes show a comparable strain. tubes show a comparable strain.

])5,12[56(96 6C

Z−

)657)56()65(756(96 666666C

2626

DiscussionDiscussion

PM3 HFPM3 HF values:values:•• tztz--tubulenes ( tubulenes ( kk = 6, and = 6, and nn = 0 to 4) range from = 0 to 4) range from

23.5 to 17.5 kcal/mol;23.5 to 17.5 kcal/mol;•• tata--tubulenes show lower values (14.5 to 11.8) tubulenes show lower values (14.5 to 11.8) •• Keeping in mind the average strain of 8.257 and Keeping in mind the average strain of 8.257 and

HF of 13.5 for HF of 13.5 for CC6060, it seems that the , it seems that the differencedifferencecould have an could have an electronic origin:electronic origin: the the open open ππ--electronic shellelectronic shell..

•• These data confirm the above suppositions These data confirm the above suppositions about the stability of about the stability of tztz--tubulenes.tubulenes.

2727

fzfz --TubulenesTubulenes

2828

fz fz --TubulenesTubulenes

CC168(168(66 (5,6)(5,6)33(6,6,5)(6,6,5)33--Z[18,5])Z[18,5]); ; CCiiCap Cap CC39(39(6 6 (5,6)(5,6)33(6,6,5)(6,6,5)33))--Z[18,0])Z[18,0])

CCNN((kk (5,6)(5,6)k/2k/2(6,6,5)(6,6,5)k/2k/2--ZZ [3[3kk,,nn]]));; NN = 13= 13kk + + pp

2929

Design of Design of TubulenesTubulenes

CC78 78 ; ; (78: 1) (78: 1) DD33CC78 78 ; ; (78: 4)(78: 4)11 DD33hh ; ; ((66 (5 6)(5 6)33 (6 6 5)(6 6 5)33 6699 (5 6 6)(5 6 6)33 (6 5)(6 5)33 6)6)

F Z F Z ––Tubulenes; Tubulenes; CapCap DerivationDerivation

1. P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes, Oxford University Press,Oxford, U.K., 1995.

3030

nn = even= evennn = odd= odd

odd e v e n

Geodesic projection of the repeat units ofGeodesic projection of the repeat units ofCCNN((k k (5,6)(5,6)kk/2/2(6,6,5)(6,6,5)kk/2/2-- ZZ [3[3kk,,nn]]))

3131

Spectral PropertiesSpectral Properties

HOMOHOMO--LUMO gapLUMO gap of of twintwin oddodd--even series even series CCN(N(k k (5,6)(5,6)k/2k/2(6,6,5)(6,6,5)k/2k/2-- Z[3Z[3kk,,nn]]))with alternating with alternating PCPC and and PSCPSC shell structures shell structures

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8

n

HO

MO

-LU

MO

gap

( )

k=6, PSC, odd

k=6, PC, even

3232

N N = 13= 13kk + + 33kmkm ;;mm = = 11, , 2,…; 2,…; kk = 4, 6, 8,s = 4, 6, 8,s

-- The The capcap CC39(39(6 6 (5,6)(5,6)33(6,6,5)(6,6,5)33))--ZZ[18,0])[18,0]) is is CC6060--derivablederivable..

1. M. V. 1. M. V. DiudeaDiudea, , StudiaStudia Univ. “BabesUniv. “Babes--BolyaiBolyai””, 2003, 48, 31, 2003, 48, 31--40.40.

fz fz --TubulenesTubulenes

F Z F Z --TubulenesTubulenes CCN N ((k k (5,6)(5,6)k/2k/2(6,6,5)(6,6,5)k/2k/2--ZZ [3[3kk,,nn]]);); PCPC

LeapfrogLeapfrog rule, written for rule, written for zigzag cylinderszigzag cylinders, , ZZ--LERLER

3333

Energetic and Spectral PropertiesEnergetic and Spectral Properties

PC0.3731-0.04700.32605.21013.083C6h22812; 215

PC0.4563-0.07740.37895.62914.865D6d19212; 114

PC0.256400.25646.18418.147C6h15612; 013

PC0.4409-0.07210.36885.11510.715C5h19010; 212

PC0.5173-0.08440.43295.51211.875D5d16010; 111

PC0.5453-0.08740.45795.99812.518C5h13010; 010

PC0.3731-0.04700.32605.1129.390C4h1528; 29

PC0.4563-0.07740.37895.49710.158D4d1288; 18

PC0.256400.25645.73011.453C4h1048; 07

PC0.4409-0.07210.36885.19510.381Cs1146;26

PC0.5173-0.08440.43295.56411.015D3d966; 15

PC0.6333-0.11760.51576.08312.294Cs786; 04

ShellGap(|β| units)

λN/2+1λN/2HF/at.(kcal/mol)

Spectral DataPM3Gap(eV)

PM3Sym.NCagek; n

Data for the Data for the ZZ--tubulenestubulenes CCN N ((k k (5,6)(5,6)kk/2/2(6,6,5)(6,6,5)kk/2/2-- ZZ [3[3kk,,nn]]))

3434

Diudea’sDiudea’s cagecage CC260(k 5260(k 5kk(7(7kk552k2k77kk))rr55kkk); k = 5; r = 6k); k = 5; r = 6

3535

SOFTWARESOFTWARE

•• TOPOCLUJ 2.0TOPOCLUJ 2.0 -- Calculations in Calculations in MOLECULAR TOPOLOGYMOLECULAR TOPOLOGY

M. V. M. V. DiudeaDiudea, O. , O. UrsuUrsu and Cs. L. Nagy, Band Cs. L. Nagy, B--B Univ. 2002B Univ. 2002

•• CageVersatileCageVersatile 1.11.1Operations on mapsOperations on maps

M. M. StefuStefu and M. V. and M. V. DiudeaDiudea, , BB--B Univ. 2003B Univ. 2003

3636

FullerenFullerenee CC7070 DD55hh -- Dual Dual && MedialMedial

CC7070

DuDu(C(C7070)) MeMe(C(C7070))

CC7070

3737

33××{20};4{20};4××{10};{10};11××{5};{5};

33××{10};1{10};1××{5};{5};11××{2};{2};22××{20};3x{10};{20};3x{10};CC7070 -- DD55hh

EdgesEdgesFacesFacesVerticesVerticesGraGraphph

EEququivalenivalence Classes ofce Classes of CC7070

3838

Peanut dimers Peanut dimers CC140140;; topologytopology11

1. 1. CsCs. L. . L. NagyNagy, M. , M. StefuStefu, M. V. , M. V. DiudeaDiudea, A. , A. DressDress, , andand A. A. MüllerMüller, , CroatCroat. . ChemChem. . ActaActa, , 20042004, , 7878, 000, 000--000.000.

ab cde

aa bb

cc dd ee

3939

CC2(702(70--5),55),5--H[10,1]H[10,1] = = CC140140 ((CC11))

1. Cs. L. Nagy, M. Stefu; M. V. Diudea and A. Dress, A. Mueler,1. Cs. L. Nagy, M. Stefu; M. V. Diudea and A. Dress, A. Mueler, CC7070 Dimers Dimers --energetics and topology, energetics and topology, Croat. Chem. ActaCroat. Chem. Acta, , 20042004, , 7878, 000, 000--000000

Peanut dimers Peanut dimers CC140140;; topologytopology11

vertex orbits:vertex orbits: 4{10}; 5{20}4{10}; 5{20}

4040

Peanut dimers Peanut dimers CC140140;; topologytopology

DuDu((CC140140);); face orbits:face orbits:MeMe((CC140140);); edge orbits: edge orbits: 9{10}; 6{20}9{10}; 6{20}[5]{2}; 2{10}; [6] 4{10}; [7] {10}[5]{2}; 2{10}; [6] 4{10}; [7] {10}

4141

[5] 1{4};2{8}; [6] 5{4};3{8}; [7] 1{8};[5] 1{4};2{8}; [6] 5{4};3{8}; [7] 1{8};1x{2};6x{4};23x{8};1x{2};6x{4};23x{8};7x{4};14x{8};7x{4};14x{8};c140efc140ef

[5] 3{4};1{8}; [6] 5{4};3{8}; [7] 2{4};[5] 3{4};1{8}; [6] 5{4};3{8}; [7] 2{4};3x{2};45x{4};3x{8};3x{2};45x{4};3x{8};35x{4};35x{4};c140eec140ee

[5] 6{2};2{4}; [6] 9{2};5{4};1{6};[5] 6{2};2{4}; [6] 9{2};5{4};1{6};[7] 1{2};1{6};[7] 1{2};1{6};2x{1};102x{2};1x{4};2x{1};102x{2};1x{4};70x{2};70x{2};c140dfc140df

[5] 6{2};2{4}; [6] 10{2};4{4};1{8};[5] 6{2};2{4}; [6] 10{2};4{4};1{8};[7] 4{2};[7] 4{2};2x{1};98x{2};3x{4};2x{1};98x{2};3x{4};68x{2};1x{4};68x{2};1x{4};c140dec140de

[5] 8{2};1{4}; [6] 9{2};5{4};1{6};[5] 8{2};1{4}; [6] 9{2};5{4};1{6};[7] 2{2};1{4};[7] 2{2};1{4};2x{1};104x{2};2x{1};104x{2};68x{2};1x{4};68x{2};1x{4};c140ddc140dd

[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[7] 1{2};2{4};[7] 1{2};2{4};9x{2};48x{4};9x{2};48x{4};4x{2};33x{4};4x{2};33x{4};c140cec140ce

[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[7] 1{2};2{4};[7] 1{2};2{4};2x{1};100x{2};2x{4};2x{1};100x{2};2x{4};66x{2};2x{4};66x{2};2x{4};c140cdc140cd

[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[5] 3{2};4{4}; [6] 1{2};8{4};1{6};[7] 3{2};1{4};[7] 3{2};1{4};2x{1};94x{2};5x{4};2x{1};94x{2};5x{4};66x{2};2x{4};66x{2};2x{4};c140ccc140cc

[5] 3{2};3{4}; [6] 10{2};7{4}; [7] 3{2};[5] 3{2};3{4}; [6] 10{2};7{4}; [7] 3{2};2x{1};96x{2};4x{4};2x{1};96x{2};4x{4};70x{2};70x{2};c140bdc140bd

[5] 3{2};3{4}; [6] 7{2};7{4};1{6};[5] 3{2};3{4}; [6] 7{2};7{4};1{6};[7] 3{2};[7] 3{2};2x{1};87x{2};7x{4};1x{6};2x{1};87x{2};7x{4};1x{6};66x{2};2x{4};66x{2};2x{4};c140bcc140bc

[5] 3{2};3{4}; [6] 10{2};7{4};[5] 3{2};3{4}; [6] 10{2};7{4};[7] 1{2};1{4};[7] 1{2};1{4};2x{1};102x{2};1x{4};2x{1};102x{2};1x{4};64x{2};3x{4};64x{2};3x{4};c140bbc140bb

[5] 1{2};2{10}; [6] 4{10}; [7] 1{10};[5] 1{2};2{10}; [6] 4{10}; [7] 1{10};9x{10};6x{20};9x{10};6x{20};4x{10};5x{20};4x{10};5x{20};c140aac140aa

FacesFacesEdgesEdgesVerticesVerticesGG

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ConclusionsConclusions

•• Construction of tubulenes, by various Construction of tubulenes, by various capping of armchair and zigzag capping of armchair and zigzag nanotubes, was presented. nanotubes, was presented.

•• Periodicity of their constitutive topology Periodicity of their constitutive topology was evidenced by typing enumerations. was evidenced by typing enumerations. Analytical formulas were given.Analytical formulas were given.

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ConclusionsConclusions

•• The The ππ--electronic structure of the modeled electronic structure of the modeled cages showed a full pallet of shells, with a cages showed a full pallet of shells, with a clearclear relationship skeletonrelationship skeleton--electronic electronic structurestructure..

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ConclusionsConclusions

•• Semiempirical calculations support the Semiempirical calculations support the idea that new, relatively stable molecules, idea that new, relatively stable molecules, with various tessellation, may candidate to with various tessellation, may candidate to the status of real molecules.the status of real molecules.