Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf ·...

Tori 2Tori 2

Mircea V. DiudeaMircea V. Diudea

aa Faculty of Chemistry and Chemical Faculty of Chemistry and Chemical EngineeringEngineering

BabesBabes--Bolyai University, Bolyai University, 400028400028 Cluj, Cluj, RomaniaRomania. .

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

ContentsContents

1.1. Other Covering Toroidal Nets.Other Covering Toroidal Nets.2.2. Energy CalculationsEnergy Calculations3.3. Spectral DataSpectral Data

(4,4) Tori(4,4) Tori

Square tilledSquare tilled latticelattice

(4,4) pattern(4,4) pattern

1. M. V. Diudea, A. Graovac, Generation and Graph-Theoretical Properties of C4-Tori. Commun. Math. Comput. Chem. (MATCH), 2001, 44, 93-102.

Square tiledSquare tiled torus and torus and LeLe dv.dv.

Le Le ((4,4)[12,36]) = ((4,8)3)R[24,72]((4,4)[12,36]) = ((4,8)3)R[24,72]NN = 1728 = 1728

(4,4)[12,36]; (4,4)[12,36]; NN = 432= 432

Rhomb tiledRhomb tiled latticelattice

(4,4) R(4,4) R--expanded netexpanded net(4,4) R(4,4) R--patternpattern

Rhomboidal Rhomboidal (4,4)(4,4) toritori

(4,4)VR[12,40](4,4)VR[12,40](4,4)HR[12,40](4,4)HR[12,40]

Rhomb tiledRhomb tiled torus and torus and LeLe dv.dv.

Le Le ((4,4)HR[12,36]) = ((4,8)3)S[24,72]((4,4)HR[12,36]) = ((4,8)3)S[24,72]NN = 1728 = 1728

(4,4)HR[12,36]; (4,4)HR[12,36]; NN = 432= 432

((4,8)3) Tori((4,8)3) Tori

((4,8)3)S((4,8)3)S netnet

((4,8)3)S((4,8)3)S net (expanded)net (expanded)((4,8)3) pattern((4,8)3) pattern

((4,8)3)S Tori((4,8)3)S Tori

According to the According to the cutting constructioncutting construction procedure, the procedure, the ((4,8)3)HS[((4,8)3)HS[cc,,nn]] isomer contains, on dimension isomer contains, on dimension ““cc ””, half , half of the number of (4,8) pairs in of the number of (4,8) pairs in ((4,8)3)VS[((4,8)3)VS[cc,,nn].].

Conversely, the number of (4,8) pairs in Conversely, the number of (4,8) pairs in ((4,8)3)VS[((4,8)3)VS[cc,,nn]], , on dimension on dimension ““n n ””, is half of that in , is half of that in ((4,8)3)HS[((4,8)3)HS[cc,,nn]]. .

In other words, the In other words, the ““HH”” embedding isomer will expand, embedding isomer will expand, after optimization, in a horizontal plane after optimization, in a horizontal plane ((ii..ee., perpendicularly to the z axis) while the ., perpendicularly to the z axis) while the ““VV”” isomer, isomer, in a vertical plane. in a vertical plane.

((4,8)3)S[((4,8)3)S[cc,,n n ] Tori] Tori

((4,8)3)VS[20,100]((4,8)3)VS[20,100]((4,8)3)HS[20,100]((4,8)3)HS[20,100]

Toroidal ((4,8)3)Toroidal ((4,8)3)SS structures, in structures, in HHSS ((““horizontalhorizontal squaresquare””) ) and and VVSS ((““verticalvertical squaresquare””) embeddings, respectively.) embeddings, respectively.

((4,8)3)S Tori((4,8)3)S Tori

((4,8)3)VS ((4,8)3)VS (detail)(detail)((4,8)3)HS ((4,8)3)HS (detail)(detail)

((4,8)3)R[((4,8)3)R[cc,,n n ] Tori] Tori

((4,8)3)R[24,72]; ((4,8)3)R[24,72]; NN = 1728 (= 1728 (uniqueunique embedding)embedding)

Distance in ((4,8)3)S[Distance in ((4,8)3)S[cc,,n n ] Tori] Tori

Tori ((4,8)3)HS[Tori ((4,8)3)HS[cc,,nn];]; 0 mod(0 mod(cc,4),4)(1) Case: ; true ((4,8)3)HS[(1) Case: ; true ((4,8)3)HS[cc,,nn] torus; ] torus; cc = 4= 4pp; ; nn = 2= 2rr

Hosoya polynomialHosoya polynomial

(1)(1)

Wiener indexWiener index

(2)(2)

)2(

)1(,...,1)2(2

)12(,...,3

)1(,...,1)2(

)2(2

2)12(,...,1

,)44()14(,4

,)4()1(,1),(

pr

pkkrr

rpkk

pkkp

kpp

ppkk

k

x

xkpxppx

xkmxmxmxiH

+

−=+

−=

−=+

+−=

+

+−+−++

−+−++=

]∑∑

∑ ∑−

=

−

=

−

=

−

=+

+++−+−++

+⎢⎣

⎡+−+−+=

1

1

12

3

12

1

1

1)2(2]2,4[)3)8,4((

)2()2)(44(2)14(4

)2)(4(2)1(4

p

k

r

pk

p

k

p

kkppkrpHS

rpkrkprppk

kpkmpmkmprW

nc <

Distance in ((4,8)3)S[Distance in ((4,8)3)S[cc,,n n ] Tori] Tori

Where is defined by the following recursion: Where is defined by the following recursion:

(3)(3)

for for

With (3), relation (2) becomes:With (3), relation (2) becomes:

(4)(4)

(5)(5)

km

⎪⎩

⎪⎨

⎧

===

+=+⎟⎠⎞

⎜⎝⎛⋅

=2,51,30,0

83/89

32sin32 2/1

tiftiftif

skk

mk

π

2,1,0and,3 =+= ttsk

)136(3

16 222]2,4[((4,8)3)H −++= prrprpW rpS

96/)16624( 222],[((4,8)3)H −++= cnncncW ncS

Distance in ((4,8)3)S[Distance in ((4,8)3)S[cc,,n n ] Tori] Tori

Tori ((4,8)3)HS[Tori ((4,8)3)HS[cc,,nn];]; 0 mod(0 mod(cc,4),4)

(2) Case: ; ((4,8)3)HS[(2) Case: ; ((4,8)3)HS[cc,,cc] = V((4,8)3)S[] = V((4,8)3)S[cc,,cc]]

Hosoya polynomial:Hosoya polynomial:

(6)(6)

Wiener index:Wiener index:

(7)(7)

(8)(8)

(3) Case: ; (3) Case: ; ((4,8)3)HS[((4,8)3)HS[cc,,nn]; this case turns to ((4,8)3)VS[]; this case turns to ((4,8)3)VS[nn,,cc]]

nc =

ppk

kppppk

kpk

kpkp

pppk

kk

xxkpxppx

xkmxmxmxiH5

)1,...,(1)4(4

)14,...,(3

)1,...,(1)2(

)2(2

2)12,...,(1

,)44()14(,4

,)4()1(,1),(

+−+−++

−+−++=

−=+

−=

−=+

+−=

)131(3

32)( 23 −= pppW

96/)1631()( 23 −= cccW

nc >

nc >

Distance in ((4,8)3)S[Distance in ((4,8)3)S[cc,,n n ] Tori] Tori

Tori ((4,8)3)VS[Tori ((4,8)3)VS[cc,,nn]; 0 mod(]; 0 mod(nn,4); ,4); cc = 2= 2pp; ; nn = 4= 4rr..

CaseCase::

Hosoya polynomial:Hosoya polynomial:

(6)(6)

Wiener index:Wiener index:

(7)(7)

(8)(8)

cn 2=

ppk

kpkp

pppk

kk

xxkm

xmxmxiH3

)1(,...,1)2(

)2(

22)12(,...,1

,)8(

)2(,1),(

+−+

+−++=

−=+

+

−=

)19436(6

)( 23

−= pppW

96/)16109(2)( 23 −= cccW

Distance in ((4,8)3)S[Distance in ((4,8)3)S[cc,,n n ] Tori] Tori

On domains where no close formula is given, the On domains where no close formula is given, the following relations are useful:following relations are useful:

]2,2/[)3)8,4((],[)3)8,4(( ncVSncHS WW =

]2,2/[)3)8,4((]2/,2[)3)8,4((],[)3)8,4(( cnVSncHSncVS WWW ==

((5,7)3) Tori((5,7)3) Tori(pentaheptites)(pentaheptites)

((5,7)3)SP Tori((5,7)3)SP Tori

A. ((5,7)3)SPA. ((5,7)3)SP Net; signatureNet; signature (1,3)(1,3)

((5,7)3)((5,7)3) pattern (expanded)pattern (expanded)((5,7)3)((5,7)3) patternpattern

((5,7)3)SP((5,7)3)SP Nets Nets -- PentaheptitesPentaheptites

A A ((5,7)3)SP((5,7)3)SP ““spiralspiral”” covering can be derived:covering can be derived:-- from a square (4,4) net by switching and deleting from a square (4,4) net by switching and deleting appropriate edges appropriate edges -- from a hexagonal (6,3) net by from a hexagonal (6,3) net by StoneStone--WalesWalestransformation.transformation.

Local signature: Local signature: (1,3)(1,3)

The The ““SPSP”” specification differentiates this type of covering specification differentiates this type of covering (with the (5,7) pairs disposed in a spiral, in optimized (with the (5,7) pairs disposed in a spiral, in optimized lattice) from other, nonlattice) from other, non--spiral, ((5,7)3) nets.spiral, ((5,7)3) nets.

((5,7)3)SP((5,7)3)SP Tori Tori –– Cutting procedureCutting procedure

H/VH/V embeddingembedding isomers are possible.isomers are possible.

In ((5,7)3)In ((5,7)3)HSPHSP[[cc,,nn] tori, each (5,7) pair takes exactly ] tori, each (5,7) pair takes exactly four squares in the parent (4,4) net, so that four squares in the parent (4,4) net, so that cc/4 such /4 such pairs lie around the tube. The pairs lie around the tube. The nn--dimension is, in this dimension is, in this case, preserved. case, preserved.

Conversely, in ((5,7)3)Conversely, in ((5,7)3)VSPVSP[[cc,,nn] tori, the constant ] tori, the constant dimension is dimension is cc, while the number of pairs around the , while the number of pairs around the torus equals torus equals nn/4./4.

((5,7)3)SP((5,7)3)SP ToriTori

((5,7)3)HSP[20,64]((5,7)3)HSP[20,64]((5,7)3)VSP[6,56]((5,7)3)VSP[6,56]

((5,7)3)((5,7)3)HSPHSP ToriTori

MM+ energy/atom MM+ energy/atom vsvs. . nn in tori HC5C7 [8,in tori HC5C7 [8,nn]] (top) and tubes TUHC5C7 (top) and tubes TUHC5C7 [8,[8,nn] (bottom); the difference between pairs of points is the ] (bottom); the difference between pairs of points is the strainstrainenergyenergy of the of the Torus closureTorus closure

10121416182022242628

10 20 30 40 50 60 70 80 90 100 110

n

Ene

rgy

(kca

l/mol

)

ππ--Electronic structure Electronic structure ofof ((5,7)3)SP((5,7)3)SP ToriTori

Two cases were observed:Two cases were observed:OpenOpen--shellshell, with, with (zero bandgap)(zero bandgap)and and nn/2 /2 degenerate orbitals atdegenerate orbitals at value, for tori ofvalue, for tori ofseries series HSP[8,HSP[8,nn]]..InIn case of case of VSP[8,VSP[8,nn]] series or in tori having series or in tori having cc > 8> 8 the the degenerate orbitals appear at , p>1.degenerate orbitals appear at , p>1.

Pseudo closedPseudo closed--shellshell, with (non, with (non--zero zero

bandgap) and bandgap) and nn/2/2--1 degenerate orbitals at value, 1 degenerate orbitals at value,

for tubes of series for tubes of series TUHSP[8,TUHSP[8,nn]]..

012/2/ >= +NN λλ

2/Nλ

pN −2/λ012/2/ ≥> +NN λλ

12/ −Nλ

Energy (EHT) Energy (EHT) per atom per atom vsvs. . nninin ((5,7)3)HSP((5,7)3)HSP ToriTori

EEHT = -0.0074Ln(n ) - 2.4815R2 = 0.9796

-2.516

-2.514

-2.512

-2.510

-2.508

-2.506

-2.504

-2.50210 30 50 70 90 110

n

Ene

rgy

(au)

Trend of Trend of EHTEHT gap gap

inin ((((5,7)3)HSP[8,5,7)3)HSP[8,nn] ] Tori atTori at nn > 40> 40

EEHT = 0.0001x2 - 0.0202x + 0.8468R2 = 0.9569

0

0.05

0.1

0.15

0.2

0.25

0.3

30 40 50 60 70 80 90 100 110

n

Ene

rgy

(eV

)

Spiral code ofSpiral code of ((5,7)3)((5,7)3)SPSP ToriTori

The The spiral codespiral code ofof ((5,7)3)((5,7)3) SPSP ToriTori ::

[ ] )4/(/4/4 5)(7,7)5( :PH nccS

[ ] )8/(/2/2 5)(7,7)5(:PV nccS

Other ((5,7)3) ToriOther ((5,7)3) ToriH/VH/V embeddingembedding isomers are possible.isomers are possible.

((5,7)3) Tori((5,7)3) Tori

((5,7)3)V[12,102]; ((5,7)3)V[12,102]; NN = 1224; = 1224; Local signature: Local signature: (1,3)(1,3)

((5,7)3) Tori: ((5,7)3) Tori: ((((5, 5, 7)3)VA7)3)VA TorusTorus

krNrkrkZN kkk 8;30;6;T)]0,2[757( 2 ===

−−

Local signature: Local signature: (2,4)(2,4)

((5,7)3) Tori((5,7)3) Tori

((5,7)3)TR[12,24]; N = 432((5,7)3)TR[12,24]; N = 432;; Local signature: Local signature: (2,4)(2,4)

((5,6,7)3) Tori((5,6,7)3) Tori

krNrkrkZN kkk 8;30;6;T)]0,2[7)566()665(( 2/2/ ===

−−

For a ((5,6,7)3)VA[For a ((5,6,7)3)VA[pp,,qq] torus, derived from a Z[] torus, derived from a Z[cc, , nn] torus] torusby SW edge rotations, (by SW edge rotations, (pp; ; qq) = () = (cc/4; /4; nn/4) and the repeat unit /4) and the repeat unit rr = = qq

Periodic ((5,6,7)3) Covering Typing Theorem.For a periodic ((5,6,7)3) covering, of For a periodic ((5,6,7)3) covering, of local signature: tlocal signature: t5j5j(0, 4, (0, 4, 1); t1); t6j6j(2, 2, 2); and t(2, 2, 2); and t7j7j(1, 4, 2), j = 5, 6, 7,(1, 4, 2), j = 5, 6, 7, the number of faces, the number of faces, edges, and vertices of various types composing its associate edges, and vertices of various types composing its associate graph, embedded in the torus, can be counted function of the graph, embedded in the torus, can be counted function of the repeat parameter r and ring size k of the (equivalent) tube repeat parameter r and ring size k of the (equivalent) tube cross sectioncross section

Periodic Periodic CountingCounting TypesTypes

1. M. V. 1. M. V. DiudeaDiudea, Periodic , Periodic fulleroidsfulleroids. . Int. J. Int. J. NanostructNanostruct..,, 20032003, , 22(3), 171(3), 171--183183

Total no. atomsTotal no. atomsEdgesEdges

VerticesVerticesFacesFaceskk = 4, 6,= 4, 6,……

krf =5

krf 26 =krf =7

kre 46,5 = kre =7,5

kre 26,6 = kre 47,6 =kre =7,7

krv 36,6,5 = krv 27,6,5 =

krv =7,6,6 krv 27,7,6 =

krN 8=

Tiling counting formulas for toroids, of periodic Tiling counting formulas for toroids, of periodic ((5,6,7)3)((5,6,7)3)covering, covering, with general formulawith general formula TTN N ((566)((566)

kk/2/2(665)(665)

kk/2/277

kk--Z[2k,0Z[2k,0--r]r]

Phenylenic ToriPhenylenic Tori

Phenylenic ToriPhenylenic Tori

((4,6,8)3)HPH[12,48]; ((4,6,8)3)HPH[12,48]; NN = 576= 576;; (0,2,2); (2,0,4); (2,4,2)(0,2,2); (2,0,4); (2,4,2)

Phenylenic ToriPhenylenic Tori

((4,6,8)3)VPH[12,144]; ((4,6,8)3)VPH[12,144]; N N = 1728= 1728;; (0,2,2); (2,0,4); (2,4,2)(0,2,2); (2,0,4); (2,4,2)

PhenylenicPhenylenic ToriTori

HPHXHPHX--netnetHPHHPH--netnet

.

. .

.

Signature: (0,2,2); (2,0,4); (2,4,2)Signature: (0,2,2); (2,0,4); (2,4,2)

Phenylenic ToriPhenylenic Tori

((4,6,8)3)HPHX[12,96]; ((4,6,8)3)HPHX[12,96]; N N = 1152= 1152;; (0,2,2); (2,0,4); (2,4,2)(0,2,2); (2,0,4); (2,4,2)

Phenylenic ToriPhenylenic Tori

((4,6,8)3)VPHX[12,96]; ((4,6,8)3)VPHX[12,96]; N N = 1152= 1152;; (0,2,2); (2,0,4); (2,4,2)(0,2,2); (2,0,4); (2,4,2)

ππ--Electronic Structure of Phenylenic [Electronic Structure of Phenylenic [cc,,nn] Tori] Tori

pCpC------HPHHPHsheetsheet

OpOp11nn/3/3c c = 2 mod 4= 2 mod 4nonnon--alternantalternant

VPHXVPHX

MMnn/3+1/3+1nn/3+1/3+1c c = 0 mod 4= 0 mod 4alternantalternant

VPHXVPHX

OpOp11cc/3/3n n = 2 mod 4= 2 mod 4nonnon--alternantalternant

HPHXHPHX

MMcc/3+1/3+1cc/3+1/3+1n n = 0 mod 4= 0 mod 4alternantalternant

HPHXHPHX

OpOp11cc/2/2n n --oddoddnonnon--alternantalternant

VPHVPH

MMcc/2+1/2+1cc/2+1/2+1n n --evenevenalternantalternant

VPHVPH

OpOp11nn/2/2c c --oddoddnonnon--alternantalternant

HPHHPH

MMnn/2+1/2+1nn/2+1/2+1c c --evenevenalternantalternant

HPHHPH

ShellShellNBONBO--NBO+ NBO+ cc ((nn) ) Net TypeNet Type

TorusTorus

Phenylenic [Phenylenic [cc,,nn] Tori; Topology] Tori; Topology

VPHXVPHX

HPHXHPHX

VPHVPH

HPHHPHRing Spiral CodeRing Spiral CodeSeriesSeries

2/])864[( 3/ nc

3/])684[( 2/ nc

2/])684[( 3/ nc

3/])864[( 2/ nc

Naphthylenic ToriNaphthylenic Tori((4,6,8)3)((4,6,8)3)

Naphthylenic ToriNaphthylenic Tori

((4,6,8)3)HNP[25,60]; ((4,6,8)3)HNP[25,60]; N N = 1500= 1500;; (0,2,2); ((0,4,2);(1,3,2)); (2,6,0)(0,2,2); ((0,4,2);(1,3,2)); (2,6,0)

Naphthylenic ToriNaphthylenic Tori

((4,6,8)3)VNP[12,120]; ((4,6,8)3)VNP[12,120]; N N = 1440= 1440;; (0,2,2); ((0,4,2);(1,3,2)); (2,6,0)(0,2,2); ((0,4,2);(1,3,2)); (2,6,0)

NaphthylenicNaphthylenic ToriTori

HNPXHNPX--netnetHNPHNP--netnet

.

. .

.

Signature: Signature: HNP/VNP; (0,2,2); ((0,4,2);(1,3,2)); (2,6,0)HNP/VNP; (0,2,2); ((0,4,2);(1,3,2)); (2,6,0)HNPX/VNPX ; HNPX/VNPX ; (0,4,0); (1,3,2); (0,8,0)(0,4,0); (1,3,2); (0,8,0)

The distance degree spectrum of HNP 20,The distance degree spectrum of HNP 20,nn tori tori (normalised by (normalised by nn))

DDS of HNP 20,n series

0

50

100

150

200

250

300

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

k

d(G

,k)

Distance degree fingerprint of a HNPX/VNPX torusDistance degree fingerprint of a HNPX/VNPX torus(embedding(embedding isomers representing one and the same object)isomers representing one and the same object)

DDF of HNPX [16,120] & VNPX [12,160] Tori

0

10000

20000

30000

40000

50000

60000

70000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

k

d(G

,k)