mircea v. diudeachem.ubbcluj.ro/~diudea/cursuri si referate/fullerenes3.pdf · 3 • in simple...
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Fullerenes 3Fullerenes 3
MirceaMircea V. DiudeaV. Diudea
Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400028400028 ClujCluj, ROMANIA, ROMANIA
[email protected]@chem.ubbcluj.ro
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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
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•• 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.
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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
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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
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fafa --TubulenesTubulenes
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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
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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)
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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.
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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.
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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
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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.
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tata --TubulenesTubulenes
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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
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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.
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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
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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
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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)
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zz --TubulenesTubulenes
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tztz --TubulenesTubulenes
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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
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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]]
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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
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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).
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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
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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.
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fzfz --TubulenesTubulenes
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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
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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.
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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]]))
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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
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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
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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]]))
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Diudea’sDiudea’s cagecage CC260(k 5260(k 5kk(7(7kk552k2k77kk))rr55kkk); k = 5; r = 6k); k = 5; r = 6
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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
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FullerenFullerenee CC7070 DD55hh -- Dual Dual && MedialMedial
CC7070
DuDu(C(C7070)) MeMe(C(C7070))
CC7070
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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
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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
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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}
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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}
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[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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4242
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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4343
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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4444
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.