Fullerenes 4
Mircea V. Diudea
Faculty of Chemistry and Chemical Engineering
Babes-Bolyai University
400028 Cluj, ROMANIA
diudea@chem.ubbcluj.ro
1
Contents
1. kfz-Tubulenes
2. (5,6,7)kfz-Tubulenes
3. (5,7)kfz-Tubulenes
2
• In simple Hückel theory,1 the energy of the i th
π -molecular orbital is calculated on the grounds
of A(G )
Ei = α + βλi
EHOMO – ELUMO = gap
• Semiempirical approaces:
Heat of Formation HF (kcal/mol)
1. E. Hückel, Z. Phys., 1931, 70, 204.
3
π -Electronic Structure
Relation
Gap shell
symbol
1
λN/2 > 0 ≥ λ N/2+1
≠0
properly
closed
PC
2
λ N/2 > λ N/2+1 > 0
≠0
pseudo
closed
PSC
3
0 ≥ λ N/2 > λ N/2+1
≠0
meta
closed
MC
4
λ N/2 = λ N/2+1
0
open
OP
4
Building Classification
A capped nanotube we call here a tubulene
N Cap Spiral sequence:
6k
4k
k 6k (56)k- A[2k,n]
k 5k 7k (56)k- A[2k,n]
k 5k- Z[2k,n]
3k
13k /2 k (56)k/2(665)k/2- Z [3k,n]
11k
9k
12k
11k
k 6k (56)k (65)k - Z[2k,n]
k (56)k/2(665)k/2(656)k/2 7k- Z [2k,0]((5,6,7)3)
k (56)k/2(665)k/2 63k/2 (656)k/2 7k- Z [2k,0]((5,6,7)3)
k 5k 7k 52k 7k - Z[2k,n]((5,7)3)
Class
fa -tubulenes
ta -tubulenes
tz -tubulenes
fz –tubulenes
kfz –tubulenes
kfz -tubulenes
kfz –dvs
kfz –tubulenes
5
Introduction
TU(6,3)Z[c,n] = zigzag (c/2, 0)
TU(6,3)A[c,n] = armchair (c/2, c/2)
6
kfz –Tubulenes
peanut-shaped
7
Caps of kf z -Tubulenes
C11k (k 6k (56)k (65)k −Z[2k ,0]) ; k = 5
C9k(k (56)k / 2 (665)k / 2 (656)k / 2 −Z[2k,0]); k = 6
8
Caps of kf z -Tubulenes
C12k (k (56)k / 2 (6 6 5)k / 2 6k (656)k / 2 −Z[2k ,0]) ; k = 6
C11k ( k 5k 7 k 52 k 7 k − Z [ 2 k ,0]) ; k = 7
9
((5,6)3)kfz-Tubulenes
10
Peanut kf -Tubulenes
Coalescence of C60: step 1
Geodesic projection
(sp3 [2+2] Cycloadduct)
1. Y. Zhao, R. E. Smalley, and B. I. Yakobson, Phys. Rev. B, 2002, 66,195409.
11
Peanut kf -Tubulenes
Coalescence of C60: step 9 (sp2 peanut dimer)
C120(5 65(5,6)5(6,5)57575-Z[10,1])
Geodesic projection
2. Y. Zhao, R. E. Smalley, and B. I. Yakobson, Phys. Rev. B, 2002, 66,195409.
12
Coalescence of C60: step 19 (sp2 fa-tubulene)
tubulene
C120(5 65 (5,6)5 -A[10,6])
Geodesic projection
13
Energetic Properties
PM3 Energy curve for the C60 coalescence pathway1
1700
HF(kcal/mol)
1600
1500
1400
1300
1200
1100
0
5
10
15
20
SW Steps
1. M. V. Diudea, Cs. L. Nagy, O. Ursu and S. T. Balaban, Fullerenes,
Nanotubes Carbon Nanostruct., 2003, 11, 245-255.
14
Energetic and Spectral Properties
Peanut kf -tubulenes CN (k-Z[2k,n]) ; for n = 1, dimers
Cage
Sym
CN(k 6k(56)k(65)k7k7k -
PM3
HF/at.
PM3
Gap
Z[2k, n])
Spectral Data
λN/2
λN/2+1
Gap
Shell
130; 5; 1
D5d
10.625
6.198
0.018
-0.048
0.066
PC
140; 5; 2
D5h
10.959
5.731
-0.002
-0.002
0
OP
150; 5; 3
Ci
11.336
5.4419
0.029
0.029
0
OP
144; 6; 1
D6d
10.303
6.120
0.018
-0.036
0.054
PC
156; 6; 2
C6h
10.349
5.513
-0.026
-0.028
0.002
MC
168; 6; 3
D6d
11.927
5.114
-0.023
-0.023
0
OP
168; 7; 1
-
13.065
6.073
0.018
-0.015
0.033
PC
182; 7; 2
-
12.669
5.358
0.021
0.021
0
OP
196; 7; 3
-
-
-
-0.012
-0.012
0
OP
15
Peanut 1 kf –Tubulenes CNN’ (k-Z[2k,n])
C168(6 66(5,6)6(6,5)676666676)
C168(6 66(5,6)6(6,5)676(5,7)3(7,5)376)
1. D. L. Strout, R.L. Murry, C. Xu, W.C. Eckhoff, G. K. Odom, and G. E. Scuseria,
Chem. Phys. Lett. 1993, 214, 576-582.
16
Periodic Fulleroids
C N ( k 6k (56) k ( 6 5) k 7 k − Z[ 2 k ,1]− r ) ; k = 5; r = 4
17
Periodic Counting Types
Periodic Fulleroid Typing Theorem. For a
periodic fulleroid, of formula CN(k 6k (56)k (65)k 7k −Z[2k,1]−r)
the number of faces, edges, and vertices of
various types can be counted as functions of
the repeating unit r and polar ring size k.
1. M. V. Diudea, Periodic fulleroids. Int. J. Nanostruct., 2003, 2(3), 171-183
18
Periodic Fulleroid Counting Types
k = 4, 5, 6, 7
f4k = 2t4 *
(1)
f5k = 2kr + 2t5
(2)
f6k = 2k(r+1) + 2t6
(3)
f7k = 2k(r-1) + 2t7
(4)
19
Periodic Fulleroid Counting Types
e46k = 2k t4
(5)
e56k = 2k (3r + 2 + t5)
(6)
e57k = 4k(r-1)
(7)
e66k = 2k (r+2+t6)
(8)
e67k = 2k (r-1+t7)
(9)
e77k = 4k (r-1)
(10)
20
Periodic Fulleroid Counting Types
v466k = 2k t4
(11)
v566k = 2k (2r + 3 + t5)
(12)
v567k = 4k (r-1)
(13)
v577k = 2k (r-1)
(14)
v666k = 2k t6
(15)
v667k = 2k t7
(16)
v777k = 2k (r-1)
(17)
Nk = 12kr
(18)
tp = 1 if k = p, and zero, otherwise
21
Semiempirical and Spectral Data for Periodic Fulleroids
CN(k-r) ; r = 1, monomers;
monomers r = 2, dimers
PM3
PM3
Spectral Data
Cage
N
Sy
HF/atm
Gap
λN/2
λN/2+1
Gap
Shell
1
C4-1
48
D4d
17.937
6.457
0.5681
-0.1386
0.707
PC
2
C4-2
96
D4d
13.445
6.371
0.0184
-0.0108
0.029
PC
3
C4-3
144
S8
13.554
6.125
0.0403
0.0403
0
OP
4
C5-1
60
Ih
13.512
6.594
0.6180
-0.1386
0.757
PC
5
C5-2
120
D5d
10.625
6.198
0.0185
-0.0483
0.067
PC
6
C5-3
180
-
10.943
5.918
-0.029
-0.029
0
OP
7
C6-1
72
D6d
12.845
6.290
0.5637
-0.1386
0.702
PC
8
C6-2
144
D6d
10.303
6.120
0.0184
-0.0361
0.054
PC
9
C6-3
216
D6d
10.508
5.827
0.0383
0.0383
0
OP
10
C7-1
84
-
12.432
6.162
0.6068
-0.1386
0.745
PC
11
C7-2
168
-
11.075
6.073
0.018
-0.015
0.003
PC
22
Tubulene (left) and peanut z -tubulenes (mean)
corresponding to the multi-peanut (C60)n (right)
23
HOMO eigenvalues of multi peanut
z -tubulenes (C60)4
0.2
0.15
0.1
λ 0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
-0.05
-0.1
no. necks
24
((5,6,7)3)
kfz-Tubulenes
25
((5,6,7)3) kfz-peanut Tubulene
C108(6 (5,6)3(6,6,5)3(6,5,6)376-Z[12,0])
((5,6,7)3) covering
C126(6 (5 6)3 (6 6 5)3 (6 5 6)3 76 (6 5 6)3 69 (6 65)3 (6 5)3 6)
26
((5,6,7)3) kfz-peanut Tubulene
Corrugated tubulene
C 204( 6 (5 6)3 ( 6 6 5)3 ( 6 5 6)3 76 − Z [12,0]−r ) ; r = 4
27
Semiempirical PM3 data for periodic peanut cages
and their relatives; for comparison, C60 is included.
Cage
Sym.
HF/atom
(kcal/mol)
Gap
(eV)
S/atom
(kcal/mol)
C144( 6 (5 6)3 ( 6 6 5)3 6k (5 6 6)3 76 − Z [12,0])
D3d
11.911
4.729
5.530
C 144 ( k
S6
11.666
4.049
5.343
C 144 ( 6 ( 5 6 ) 3 ( 6 6 5 ) 3 6 k ( 5 6 6 ) 3 7 6 ( 6 6 5 ) 3 6 6 ( 665 ) 3 ( 65 ) 3 6 )
C3
11.789
4.266
5.436
C 126 ( 6 ( 5 6 ) 3 ( 6 6 5 ) 3 ( 6 5 6 ) 3 7 6
C3v
12.349
4.796
5.848
C108( 6 (5 6)3 ( 6 6 5)3 ( 6 5 6)3 76 −Z [12,0]−2)
D3d
12.953
4.870
6.493
C108( 6 (5 6)3 ( 6 6 5)3 ( 6 5 6)3 76 −Z [12,0]−3)
D3d
12.878
4.502
5.687
C108( 6 (5 6)3 ( 6 6 5)3 ( 6 5 6)3 76 −Z [12, 0]−4)
D3d
12.681
4.484
5.321
Ih
13.512
6.593
8.257
( 56 ) k / 2 ( 6 6 5 ) k / 2 6 k ( 6 5 6 ) k / 2 7 k − Z [ 2 k , 0 ])
( 6 5 6 )3 (5 6 6 )3 ( 6 5 )3 6 )
C60
28
POAV – Strain Energy
In the POAV1 theory1,2 the π-orbital axis vector makes
equal angles to the three σ-bonds of the sp2 carbon:
θp = θσπ - 90o
SE = 200(θp )2
120 - (1/3) Σθij
pyramidalization angle
strain energy
deviation to planarity
1. R.C. Haddon, J. Am. Chem. Soc., 112, 3385 (1990).
2. R.C. Haddon, J. Phys. Chem. A, 105, 4164 (2001).
29
Periodic Counting Types
Periodic ((5,6,7)3) Covering Typing Theorem.
For a periodic ((5,6,7)3) covering, of local signature: t5j(0, 4,
1); t6j(2, 2, 2); and t7j(1, 4, 2), j = 5, 6, 7, the number of faces,
edges, and vertices of various types composing its associate
graph, can be counted function of the repeat parameter r and
ring size k of the (equivalent) tube cross section
1. M. V. Diudea, Periodic fulleroids. Int. J. Nanostruct., 2003, 2(3), 171-183
30
Periodic Counting Theorem
Periodic embedding C-j -rm-jm- C ; k = 4, 6, 8
k = 4, 6, 8
Cap
Repeat unit
fk = 1
f5 = k
f5 = 3k/2
junction
f7 = k
f6 = 2k
f6= 5k/2
31
Periodic Counting Theorem
Cap
Repeat unit
Junction
e5,k = k /2
e5,6 = 13k /2
e5,6 = 4k
e5,7 = k /2
e5,7 = k
e6,k = k /2
e6,6 = 2k
e6,6 = 3k
e6,7 = 4k
e7,7 = k
e6,7 = 2k
32
Periodic Counting Theorem
Cap
Repeat unit
v5,6,k = k
v5,6,6 = 3k
v5,6,6 = 11k/2
v5,6,7 = 2k
v5,6,7 = k
v6,6,7 = k
v6,6,7 = k/2
v6,7,7 = k
v6,7,7 = 2k
N = 2k(4r +1)
33
((5,7)3) kz-peanut
Tubulenes
34
Rearrangement of (4, 6) pairs to (5, 5) ones
by SW edge rotation1
C84(7 5777(4,6)777577)
C84(7 577751477577); (Ci )
1. A. J. Stone and D. J. Wales, Chem. Phys. Lett., 1986, 128, 501
35
(5, 7) PERIODIC CAGES
In silico “dimerization” process
2 (C11k(k 5k7k52k7k) - 2k ; k = 7
C20k(k 5k(7k52k7k)25kk)
36
„Tetramer“ C252
37
Periodic Counting Types
((5,7)3) Periodic Cages Typing Theorem.
For a periodic cage with ((5,7)3) decoration, the
number of faces, edges, and vertices of the various
types can be counted as functions of the
repeating unit r and polar ring size k.
1. M. V. Diudea, Periodic fulleroids. Int. J. Nanostruct., 2003, 2(3), 171-183
38
Periodic Cages ((5,7)3) Counting Types
k = 5, 7
f5k = 2k (r+1) + 2t5 *
f7k = 2kr + 2t7
(1)
(2)
39
Periodic Cages ((5,7)3) Counting Types
e55k = 2k (r + 1 + t5)
(3)
e57k = 2k (3r + 2 + t7)
(4)
e77k = 2k (2r - 1)
(5)
40
Periodic Cages ((5,7)3) Counting Types
v555k = 2k t5
v557k = 2k (2r + 1 + t7)
(7)
(8)
v577k = 2k (r + 1)
(9)
v777k = 2k (r - 1)
(10)
Nk = 4k (2r +1)
(11)
tp = 1 if k = p, and zero, otherwise
41
Semiempirical and spectral data
for ((5,7)3) periodic cages
(5,7) Cage
Sym
CN(k 5k(7k52k7k)r5kk)
PM3
HF/at
PM3
GAP
λN / 2
ΛΝ/2
Spectral
λ N / 2 +1 Data
ΛΝ/2+1
GAP
Shell
1
60 ; 5; 1
Ci
21.158
5.623
0.3797
0.2290
0.1507
PSC
2
100; 5; 2
Ci
18.906
5.592
0.2785
0.2785
0
OP
3
140; 5; 3
-
-
-
0.2979
0
0.2979
PSC
4
84; 7; 1
Ci
16.249
4.538
0.2452
0.2311
0.0141
PSC
5
140;7; 2
Ci
15.828
5.114
0.2231
0.2231
0
OP
6
196; 7; 3
-
-
-
0.2199
0.0155
0.2044
PSC
42
((5,7)3) Periodic Cages
• The ((5,7)3) periodic cages tend to
isomerize to the more stable fa-tubulenes
43
C260 (5,7) Cages1
Fowler’s C260 cage
Dress’ C260 cage
1. G. Brinkmann and A. Dress and , Fantasmagorical fulleroids, MATCH Commun.
Math. Comput. Chem., 1996, 33, 87-100.
44
Diudea’s cage C260(k 5k(7k52k7k)r5kk); k = 5; r = 6
45
SOFTWARE
• TOPOCLUJ 2.0 - Calculations in
MOLECULAR TOPOLOGY
M. V. Diudea, O. Ursu and Cs. L. Nagy, B-B Univ. 2002
• CageVersatile 1.1
Operations on maps
M. Stefu and M. V. Diudea, B-B Univ. 2003
46
Conclusions
• Construction of tubulenes, by various
capping of armchair and zigzag
nanotubes, was presented.
• Periodicity of their constitutive topology
was evidenced by typing enumerations.
Analytical formulas were given.
47
Conclusions
• The π-electronic structure of the modeled
cages showed a full pallet of shells, with a
clear relationship skeleton-electronic
structure.
48
Conclusions
• Semiempirical calculations support the idea that new,
relatively stable molecules, with various tessellation, may
candidate to the status of real molecules.
• The strain energy (by POAV1) show such structures as
relaxed, in comparison with C60.
49