Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties
Abstract
:1. Introduction
2. Combinatorial Methods: Generalized Character Cycle Indices, Sheehan’s Method, Euler’s Totient Function and Character Based Enumerations
3. Applications to Polyarenes Enumeration & Spectroscopy
3.1. Enumeration of Hetero-Substituted Polyarenes and Polysubstituted Polyarenes
- Case (1): m even, n odd, D: protons:
- Case (2): m even, n even, D: protons:
- Case (3): m odd, n odd, D: protons:
- Case (4): m odd, n even, D: protons:
- Case (1): m even
- Case (2): m odd
- Case (1): m even, n odd, protons:
- Case (2): m even, n even, protons:
- Case (3): m odd, n odd, protons:
- Case (4): m odd, n even, protons:
3.2. Applications to 13C, Proton NMR and Multiple Quantum NMR of Polyarenes
(n + 4)/2 if n is even.
α48 + 5α47β + 109α46β2 + 1467α45β3 + 16398α44β4 +142945α43β5 +1024059α42β6 +6137527α41β7 + 31453488α40β8 + 139767835α39β9 + 545091767α38β10 + 1882967013α37β11 + 5805816362α36β12 +16077455055α35β13 + 40193661777α34β14 + 91105252497α33β15 + 187904675706α32β16 + 353702280690α31β17 + 609154167250α30β18 + 961821475230α29β19 + 1394641595644α28β20 + 1859521084850α27β21 + 2282140209534α26β22 + 2579809646726α25β23 + 2687302591938α24β24 +…
α24 + 3α23β + 31α22β2 + 181α21β3 + 934α20β4 + 3597α19β5 + 11395α18β6 + 29007α17β7+ 61698α16β8 + 109298α15β9 + 164110α14β10 + 208474α13β11 + 226150α12β12 +…
4. Applications to Nanotubes: Enumerations & Chirality
- (a)
- all vertices of the top/bottom layers are colored with a single color (white) while the vertices of the central layers are colored with different colors;
- (b)
- all vertices of the central layers are colored with a single color (white) while the vertices of the top/bottom layers are colored with different colors;
- (c)
- all vertices of the central layers and vertices of the top/bottom layers are colored with different colors chosen from a single set of colors without making any distinction between vertices of the central layers and vertices of the top/bottom layers.
- (d)
- all vertices of the central layers are colored with colors chosen from one set while all vertices of the top/bottom layers are colored with colors chosen from a different set of colors thus making a distinction between vertices of the central layers and vertices of the top/bottom layers.
5. Applications to Tessellations of Kekulenes, Nanobands, C60 Polymers, Spectroscopy & Topology
6. Conclusions, Helical Structures, Fullerene Polymers and Other Structural Derivatives & Future Perspectives
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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n | M | Name | Formulae |
---|---|---|---|
1 | 6 | Kekulene | C48H24 |
1 | 7 | Septulene | C56H28 |
1 | 8 | Octulene | C64H32 |
1 | 9 | Nonulene | C72H36 |
2 | 6 | 2-Kekulene | C90H30 |
2 | 7 | 2-Septulene | C105H35 |
2 | 8 | 2-Octulene | C120H40 |
2 | 9 | 2-Nonulene | C135H45 |
… | … | … | … |
n | N | ncir-m-cycloarene | Cm(n2+4n+3)Hm(n+3) |
Coefficient | Vector | Coefficient | Vector | Coefficient | |
---|---|---|---|---|---|
1 | 48 0 0 | 103561517895 | 37 9 2 | 88851068896152860 | 29 10 9 |
5 | 47 1 0 | 383177575979 | 36 10 2 | 234243726928693460 | 28 11 9 |
109 | 46 2 0 | 1254034278774 | 35 11 2 | 546568695999885020 | 27 12 9 |
1467 | 45 3 0 | 11184050659015080 | 29 14 5 | 1135181137598290860 | 26 13 9 |
16398 | 44 4 0 | 21622497914498712 | 28 15 5 | 2108193541021176180 | 25 14 9 |
142945 | 43 5 0 | 37839371323512066 | 27 16 5 | 3513655901417556180 | 24 15 9 |
1024059 | 42 6 0 | 60097825014257646 | 26 17 5 | 5270483851893361530 | 23 16 9 |
6137527 | 41 7 0 | 86807969439172982 | 25 18 5 | 7130654622923896350 | 22 17 9 |
31453488 | 40 8 0 | 114221012397502010 | 24 19 5 | 8715244538987104690 | 21 18 9 |
139767835 | 39 9 0 | 137065214861482908 | 23 20 5 | 9632638700903207990 | 20 19 9 |
545091767 | 38 10 0 | 150119044840161540 | 22 21 5 | 257668099806453820 | 28 10 10 |
1882967013 | 37 11 0 | 5364479296722 | 36 6 6 | 655882435174711976 | 27 11 10 |
5805816362 | 36 12 0 | 27588742194948 | 35 7 6 | 1475735479246515564 | 26 12 10 |
16077455055 | 35 13 0 | 120700745906895 | 34 8 6 | 2951470957321010124 | 25 13 10 |
40193661777 | 34 14 0 | 455980568919225 | 33 9 6 | 5270483852702315676 | 24 14 10 |
91105252497 | 33 15 0 | 1504735876402143 | 32 10 6 | 8432774162750027376 | 23 15 10 |
187904675706 | 32 16 0 | 4377413392313808 | 31 11 6 | 12122112859683551010 | 22 16 10 |
353702280690 | 31 17 0 | 11308317935237736 | 30 12 6 | 15687440169664356522 | 21 17 10 |
609154167250 | 30 18 0 | 26096118180492360 | 29 13 6 | 18302013532416736122 | 20 18 10 |
961821475230 | 29 19 0 | 54056244829095960 | 28 14 6 | 19265277401095105380 | 19 19 10 |
1394641595644 | 28 20 0 | 100904990136381816 | 27 15 6 | 1609893249573173544 | 26 11 11 |
1859521084850 | 27 21 0 | 170277170927824422 | 26 16 6 | 3488102040361648164 | 25 12 11 |
2282140209534 | 26 22 0 | 260423908201122666 | 25 17 6 | 6707888538605476404 | 24 13 11 |
2579809646726 | 25 23 0 | 361699872642264870 | 24 18 6 | 11499237494286660144 | 23 14 11 |
2687302591938 | 24 24 0 | 456884049357215480 | 23 19 6 | 17632164157359925680 | 22 15 11 |
191 | 46 1 1 | 525416656969664460 | 22 20 6 | 24244225715985790086 | 21 16 11 |
4349 | 45 2 1 | 550436497519708228 | 21 21 6 | 29948749413533412462 | 20 17 11 |
64927 | 44 3 1 | 137943703250100 | 34 7 7 | 33276388237117085060 | 19 18 11 |
713757 | 43 4 1 | 586260729113235 | 33 8 7 | 7266879251045544564 | 24 12 12 |
6136471 | 42 5 1 | 2149622651528715 | 32 9 7 | 13415777076559133640 | 23 13 12 |
42952525 | 41 6 1 | 6878792458628112 | 31 10 7 | 22040205198015185064 | 22 14 12 |
251570847 | 40 7 1 | 19385687790199632 | 30 11 7 | 32325634287639864600 | 21 15 12 |
1257841915 | 39 8 1 | 48464219421777720 | 29 12 7 | 42427395004047433380 | 20 16 12 |
5450620065 | 38 9 1 | 108112489398538680 | 28 13 7 | 49914582355035448050 | 19 17 12 |
20712315283 | 37 10 1 | 216224978713510800 | 27 14 7 | 52687614710209402350 | 18 18 12 |
69668620785 | 36 11 1 | 389204961579264336 | 26 15 7 | 23735605596070004280 | 22 13 13 |
209005757019 | 35 12 1 | 632458062469307646 | 25 16 7 | 37298808793141434264 | 21 14 13 |
562707643953 | 34 13 1 | 930085385883855630 | 24 17 7 | 52218332309646863976 | 20 15 13 |
1366575493563 | 33 14 1 | 1240113847767543320 | 23 18 7 | 65272915386631793610 | 19 16 13 |
3006465804441 | 32 15 1 | 1501190447237652600 | 22 19 7 | 72952081902429966390 | 18 17 13 |
6012931267794 | 31 16 1 | 1651309491932965436 | 21 20 7 | 55948213191205903656 | 20 14 14 |
10964756631510 | 30 17 1 | 2418325503348480 | 32 8 8 | 74597617584478171920 | 19 15 14 |
18274593959490 | 29 18 1 | 8598490570300620 | 31 9 8 | 88584670884029471910 | 18 16 14 |
27892800895190 | 28 19 1 | 26655320766887340 | 30 10 8 | 93795533874004088610 | 17 17 14 |
39049920855330 | 27 20 1 | 72696329114759460 | 29 11 8 | 94490315606290645536 | 18 15 15 |
50207040787046 | 26 21 1 | 175682795387902440 | 28 12 8 | 106301605056802235190 | 17 16 15 |
59335593422274 | 25 22 1 | 378393712685968980 | 27 13 8 | 112945455375981823980 | 16 16 16 |
64495210129142 | 24 23 1 | 729759303148399260 | 26 14 8 | ||
97657 | 44 2 2 | 1264916124783420252 | 25 15 8 | ||
1427470 | 43 3 2 | 1976431445242048080 | 24 16 8 | ||
15343647 | 42 4 2 | 2790256157185981770 | 23 17 8 | ||
128856651 | 41 5 2 | 3565327312427067330 | 22 18 8 | ||
880512171 | 40 6 2 | 4128273729405627230 | 21 19 8 | ||
5031358420 | 39 7 2 | 4334687416503640196 | 20 20 8 | ||
24527846695 | 38 8 2 | 29617022995256180 | 30 9 9 |
Coefficient | Vector (x, y, z) | Coefficient | Vector (x, y, z) |
---|---|---|---|
1 | 24 0 0 | 4292790 | 16 4 4 |
3 | 23 1 0 | 13731492 | 15 5 4 |
31 | 22 2 0 | 34329060 | 14 6 4 |
181 | 21 3 0 | 68650260 | 13 7 4 |
934 | 20 4 0 | 111559140 | 12 8 4 |
3597 | 19 5 0 | 148738030 | 11 9 4 |
11395 | 18 6 0 | 163616810 | 10 10 4 |
29007 | 17 7 0 | 41190876 | 14 5 5 |
61698 | 16 8 0 | 96108684 | 13 6 5 |
109298 | 15 9 0 | 178483956 | 12 7 5 |
164110 | 14 10 0 | 267723414 | 11 8 5 |
208474 | 13 11 0 | 327216106 | 10 9 5 |
226150 | 12 12 0 | 208237164 | 12 6 6 |
49 | 22 1 1 | 356962872 | 11 7 6 |
519 | 21 2 1 | 490830354 | 10 8 6 |
3573 | 20 3 1 | 545356070 | 9 9 6 |
17785 | 19 4 1 | 560936856 | 10 7 7 |
67443 | 18 5 1 | 701168970 | 9 8 7 |
202149 | 17 6 1 | 788825460 | 8 8 8 |
[λ] | A1g | A1u |
---|---|---|
120 0 | 1 | 0 |
119 1 | 10 | 0 |
118 2 | 430 | 200 |
117 3 | 13140 | 10270 |
116 4 | 358580 | 327000 |
115 5 | 8076302 | 7805200 |
114 6 | 153228940 | 151186610 |
113 7 | 2485281890 | 2472015520 |
112 8 | 35049847010 | 34972263360 |
111 9 | 435895574270 | 435487149890 |
110 10 | 4837163423172 | 4835187982608 |
109 11 | 48366143812740 | 48357338386240 |
108 12 | 439304123944130 | 439267535249120 |
107 13 | 3649522483044260 | 3649380289048000 |
106 14 | 27892495987962020 | 27891975424030480 |
105 15 | 197106031753154350 | 197104231299671570 |
104 16 | 1293505382317706270 | 1293499470458272960 |
103 17 | 7913200546609255430 | 7913182073007147040 |
102 18 | 45281066731050988200 | 45281011600933676850 |
101 19 | 243087762638665331530 | 243087605202533277680 |
100 20 | 1227593019502352645116 | 1227592588139680180808 |
99 21 | 5845680586016149192080 | 5845679450165058565230 |
98 22 | 26305561521626686809770 | 26305558641388642521880 |
97 23 | 112084563868390564619870 | 112084556825344707148480 |
96 24 | 453008439719843093324185 | 453008423084819935550720 |
95 25 | 1739552395580465510782712 | 1739552357586305907062080 |
94 26 | 6356056802607579048149000 | 6356056718581715136103840 |
93 27 | 22128493997304209628417280 | 22128493817196173268087480 |
92 28 | 73498212093520559324283920 | 73498211718937259300046560 |
91 29 | 233166741598000966442884040 | 233166740841488289311513600 |
90 30 | 707272449109079345066275200 | 707272447624069094082039400 |
89 31 | 2053371625707638595138215800 | 2053371622872379689408576320 |
88 32 | 5710939832691184420347395710 | 5710939827422078705100951680 |
87 33 | 15229172884919521763241151730 | 15229172875382280514199523400 |
86 34 | 38968765907621805546447563950 | 38968765890797737924153754120 |
85 35 | 95751824795384727236237065030 | 95751824766445595988072443312 |
84 36 | 226080697423665918179837712780 | 226080697375100662367959790680 |
83 37 | 513264286027563834692713658510 | 513264285948013123131287760080 |
82 38 | 1121077256300109181440804823910 | 1121077256172862735028690940680 |
81 39 | 2357136795263310515884594697290 | 2357136795064474082599295897160 |
80 40 | 4773202010358707257612086114170 | 4773202010055058999556359513408 |
79 41 | 9313564898191311432444888679940 | 9313564897737983810337193741600 |
78 42 | 17518372070312058850710272631220 | 17518372069650191620702994780120 |
77 43 | 31777512127415150509871599803740 | 31777512126469853524471252843840 |
76 44 | 55610646222809929010711390106200 | 55610646221488836306052022553040 |
75 45 | 93920202509422472913833055244500 | 93920202507615427898786969026968 |
74 46 | 153130764960751689179624148240700 | 153130764958331858395428437922800 |
73 47 | 241099502278311827209649491672420 | 241099502275138843239010973758880 |
72 48 | 366672159714557209135122371215500 | 366672159710482373202785259933120 |
71 49 | 538783581620959439539367888026550 | 538783581615833385238761111253600 |
70 50 | 765072685901281807653064962794306 | 765072685894964033815883701208344 |
69 51 | 1050099764962022784801615163283710 | 1050099764954393034471851047572500 |
68 52 | 1393401611199060099344586013233020 | 1393401611190030163150578065198920 |
67 53 | 1787760557764276241571586760097890 | 1787760557753801778758977126918480 |
66 54 | 2218147358706984761747143844517710 | 2218147358695075055554709343916900 |
65 55 | 2661776830447873212340258083935022 | 2661776830434598506818581481178240 |
64 56 | 3089562392483687387382849613409250 | 3089562392469181632858390724145920 |
63 57 | 3468982335419907262008024739287820 | 3468982335404366657697752965321340 |
62 58 | 3768032536748931605487595238564080 | 3768032536732607374058506770897600 |
61 59 | 3959627411498706582271887678528880 | 3959627411481893631161976216665600 |
60 60 | 4025621201690294701317661934726204 | 4025621201673315584661128579342944 |
[λ] | A1g | A1u |
---|---|---|
126 0 | 1 | 0 |
125 1 | 11 | 0 |
124 2 | 473 | 220 |
123 3 | 15253 | 11910 |
122 4 | 436673 | 398560 |
121 5 | 10348033 | 10004800 |
120 6 | 206573169 | 203879874 |
119 7 | 3527122728 | 3508834432 |
118 8 | 52385696704 | 52274232236 |
117 9 | 686407979303 | 685795703012 |
116 10 | 8028934612915 | 8025849284552 |
115 11 | 84659644031796 | 84645309793480 |
114 12 | 811283921490655 | 811221894648266 |
113 13 | 7114189337715709 | 7113938271431488 |
112 14 | 57421136296522190 | 57420179480527816 |
111 15 | 428742633734011054 | 428739188570867700 |
110 16 | 2974395957907605578 | 2974384184884777864 |
109 17 | 19246072541004160475 | 19246034252371767360 |
108 18 | 116545605020781110873 | 116545486120454041778 |
107 19 | 662469593565669880140 | 662469240221433288120 |
106 20 | 3544211884087688220237 | 3544210876693600321224 |
105 21 | 17889830252498885495425 | 17889827492074590075716 |
104 22 | 85383277805294190143784 | 85383270521181826240644 |
103 23 | 386080901136593900270904 | 386080882599144920683392 |
102 24 | 1656930517051376784683269 | 1656930471481634923631114 |
101 25 | 6760276470775467874719927 | 6760276362433949041317696 |
100 26 | 26261073896914647611843439 | 26261073647477690543603952 |
99 27 | 97263236471662895274416233 | 97263235914956931417312702 |
98 28 | 343895014286357966923112919 | 343895013080626398572379156 |
97 29 | 1162127978543417947038420093 | 1162127976007021782564392832 |
96 30 | 3757547129116693561369533113 | 3757547123929752463745097544 |
95 31 | 11636274977618676741474605088 | 11636274967299175559427516672 |
94 32 | 34545191334481858494269819958 | 34545191314493091129392663952 |
93 33 | 98401454094672584680155613591 | 98401454056952557263117170000 |
92 34 | 269156918536182307562667604533 | 269156918466792562463927930652 |
91 35 | 707498185837591596996314883276 | 707498185713084020804419188984 |
90 36 | 1788398191930007116460908717629 | 1788398191711979048251416225608 |
89 37 | 4350157764075290579543044835397 | 4350157763702510856253493800128 |
88 38 | 10188527394682682072242746308316 | 10188527394060064806948032860284 |
87 39 | 22989497710884551633808458024272 | 22989497709868307190203644545008 |
86 40 | 50002157520879567765636859278504 | 50002157519257903388142469358316 |
85 41 | 104882574311653493200042912232931 | 104882574309122653904387904539904 |
84 42 | 212262352772955598204815146300663 | 212262352769091372968466328442032 |
83 43 | 414652037974188760093034977579044 | 414652037968414603187921841451176 |
82 44 | 782184526177360507348845920120703 | 782184526168914032231473122983532 |
81 45 | 1425314025477099466617348186776881 | 1425314025465000766616408261522640 |
80 46 | 2509792088337944983451863878699918 | 2509792088320970669250957680677753 |
79 47 | 4271986533338402240952986091515568 | 4271986533315071146968456514703232 |
78 48 | 7030977836115966289850178437041004 | 7030977836084542036709394169335632 |
77 49 | 11192168800343594818507662954115382 | 11192168800302112216589382096345792 |
76 50 | 17235939952524034981288300451784128 | 17235939952470353920645281657293988 |
75 51 | 25684930125323990978539590227007264 | 25684930125255882540509198550606740 |
74 52 | 37045572296133623677675596842206368 | 37045572296048886307617780234533304 |
73 53 | 51724006602141374421656431169602456 | 51724006602037978956693506504856184 |
72 54 | 69923194110294214696088434473985680 | 69923194110170466866523001472260968 |
71 55 | 91535817744376799116378300971826720 | 91535817744231511058933713108728176 |
70 56 | 116054340354469299983499331537508704 | 116054340354301950607134075303190832 |
69 57 | 142522874119515507211110040589336016 | 142522874119326379203129178324865584 |
68 58 | 169553074383553918092169927083932224 | 169553074383344188394338933483678944 |
67 59 | 195417102679343529229105477101092448 | 195417102679115305059831028491385312 |
66 60 | 218215764658594707645979394768009504 | 218215764658350990916595981027726176 |
65 61 | 236102302745360626194077729265514496 | 236102302745105211277102095497067808 |
64 62 | 247526607716907796799690373591361056 | 247526607716645097621485771818723552 |
63 63 | 251455601490191198999846211143625568 | 251455601489926026768523441574344800 |
TI | AHK(2) | RK(3,3) |
---|---|---|
Zagreb-1 | 2520 | 2520 |
Zagreb-2 | 3366 | 3366 |
Randić | 177.8786 | 177.8786 |
Atom-bond connectivity | 322.9188 | 322.9188 |
Padmakar Ivan | 168,480 | 168,480 |
Wiener | 1,140,348 | 1,185,720 |
Hyper-Wiener | 12,876,702 | 14,183,508 |
Schultz | 589,684 | 6,134,528 |
Gutman | 7,622,832 | 7,934,112 |
Harary | 5635.002 | 5565.1216 |
Balaban | 0.3237 | 0.3120 |
Mostar | 80,256 | 79,920 |
Szeged | 10,338,816 | 10,325,272 |
Harmonic Szeged | 66,834.8034 | 67,043.7870 |
Edge Partitions | (22)18(23)252(33)198 | (22)18(23)252(33)198 |
Vertex Partitions | 641228 | 24488 |
Spectral Pattern | 1 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 2 1 1 2 | 112421154.154211124 |
2 2 1 1 1 2 2 1 1 1 2 2 1 2 1 2 1 2 2 1 | ||
1 2 2 1 1 2 2 1 1 2 1 2 2 1 2 1 2 2 1 1 | ||
2 2 1 1 2 2 1 1 2 2 1 2 1 2 1 1 1 1 2 2 | ||
2 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 1 | ||
2 2 1 2 1 1 1 2 2 1 2 2 1 2 1 1 2 1 2 | ||
2 1 2 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 | ||
2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 | ||
2 1 1 1 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1 | ||
1 2 2 1 2 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 | ||
2 2 1 2 1 2 1 2 2 1 1 1 2 2 1 1 1 2 2 2 | ||
1 1 2 1 2 1 2 1 2 1 2 2 2 1 1 1 2 2 1 | ||
Spectral Difference | 0.009612 | |
RMS Spectral Difference | 0.013128 | |
HOMO-LUMO | 0.755β a | 0.744β |
Eπ | 524.4053β | 524.4139β |
13C-NMR | 32 signals | 92 signals |
Proton NMR | 14 signals 1:1:1:1:2:2:2:2:2:2:2:2:2:2 | 38 signals 1:1:1:1:2:2:2:2:2:2:2:2:2:2: 2:2:2:2:2:2:2:2:2:2: 2:2:2:2:2:2:2:2:2:2: |
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Balasubramanian, K. Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties. Symmetry 2022, 14, 34. https://doi.org/10.3390/sym14010034
Balasubramanian K. Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties. Symmetry. 2022; 14(1):34. https://doi.org/10.3390/sym14010034
Chicago/Turabian StyleBalasubramanian, Krishnan. 2022. "Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties" Symmetry 14, no. 1: 34. https://doi.org/10.3390/sym14010034
APA StyleBalasubramanian, K. (2022). Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties. Symmetry, 14(1), 34. https://doi.org/10.3390/sym14010034