![Page 1: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/1.jpg)
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. .
[email protected]@chem.ubbcluj.ro
![Page 2: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/2.jpg)
ContentsContents
1.1. Other Covering Toroidal Nets.Other Covering Toroidal Nets.2.2. Energy CalculationsEnergy Calculations3.3. Spectral DataSpectral Data
![Page 3: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/3.jpg)
(4,4) Tori(4,4) Tori
![Page 4: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/4.jpg)
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.
![Page 5: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/5.jpg)
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
![Page 6: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/6.jpg)
Rhomb tiledRhomb tiled latticelattice
(4,4) R(4,4) R--expanded netexpanded net(4,4) R(4,4) R--patternpattern
![Page 7: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/7.jpg)
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]
![Page 8: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/8.jpg)
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
![Page 9: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/9.jpg)
((4,8)3) Tori((4,8)3) Tori
![Page 10: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/10.jpg)
((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
![Page 11: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/11.jpg)
((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.
![Page 12: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/12.jpg)
((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.
![Page 13: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/13.jpg)
((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)
![Page 14: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/14.jpg)
((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)
![Page 15: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/15.jpg)
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 <
![Page 16: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/16.jpg)
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
![Page 17: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/17.jpg)
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 >
![Page 18: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/18.jpg)
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
![Page 19: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/19.jpg)
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 ==
![Page 20: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/20.jpg)
((5,7)3) Tori((5,7)3) Tori(pentaheptites)(pentaheptites)
![Page 21: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/21.jpg)
((5,7)3)SP Tori((5,7)3)SP Tori
![Page 22: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/22.jpg)
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
![Page 23: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/23.jpg)
((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.
![Page 24: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/24.jpg)
((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.
![Page 25: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/25.jpg)
((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]
![Page 26: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/26.jpg)
((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
)
![Page 27: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/27.jpg)
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
![Page 28: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/28.jpg)
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)
![Page 29: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/29.jpg)
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
)
![Page 30: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/30.jpg)
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
![Page 31: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/31.jpg)
Other ((5,7)3) ToriOther ((5,7)3) ToriH/VH/V embeddingembedding isomers are possible.isomers are possible.
![Page 32: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/32.jpg)
((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)
![Page 33: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/33.jpg)
((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)
![Page 34: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/34.jpg)
((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)
![Page 35: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/35.jpg)
((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
![Page 36: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/36.jpg)
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
![Page 37: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/37.jpg)
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]
![Page 38: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/38.jpg)
Phenylenic ToriPhenylenic Tori
![Page 39: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/39.jpg)
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)
![Page 40: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/40.jpg)
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)
![Page 41: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/41.jpg)
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)
![Page 42: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/42.jpg)
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)
![Page 43: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/43.jpg)
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)
![Page 44: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/44.jpg)
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
![Page 45: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/45.jpg)
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
![Page 46: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/46.jpg)
Naphthylenic ToriNaphthylenic Tori((4,6,8)3)((4,6,8)3)
![Page 47: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/47.jpg)
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)
![Page 48: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/48.jpg)
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)
![Page 49: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/49.jpg)
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)
![Page 50: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/50.jpg)
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)
![Page 51: Tori 2 - Babeș-Bolyai Universitychem.ubbcluj.ro/~diudea/cursuri si referate/tori 2.pdf · 2005-05-05 · Tori 2 Mircea V. Diudea a Faculty of Chemistry and Chemical Faculty of Chemistry](https://reader033.vdokument.com/reader033/viewer/2022042211/5eb18a5f68fef020eb35f028/html5/thumbnails/51.jpg)
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)