MOLECULAR MODELING OF FULLERENE

MOLECULAR MODELING
OF FULLERENE
Conducător
Prof. Dr. Mircea V. Diudea
Doctorand:
Monica Ştefu
Contents
1. FULLERENE MODELING
2. OPERATIONS ON MAPS
3. PERIODIC FULLEROIDS
1
C60 – First Syntheses
•
Kroto, H.; Heath, J. R.; O’Brian, S. C.; Curl, R. F.; Smalley, R. E.
Sussex University (UK) & Rice University (USA),
Buckminsterfullerene C60 isolated from self-assembling
products of graphite heated by plasma.
Nature (London) , 1985, 318, 162-163.
•
Kraetschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D.
R., Solid C60: a new form of carbon. C60 isolated in
macroscopic amount by arc vaporization of graphite.
2
Nature (London) , 1990, 347, 354-358.
Isolated Fullerenes
N = 60, 70, 76, 78, 82 and 84
C70
C76
C82
C78
C84
3
BASIC RELATIONS IN POLYHEDRA
•
First theorems on the graph counting (Euler)1,2
∑d ( dvd ) = 2e
(1)
∑s ( sfs ) = 2e
(2)
where vd and fs denote vertices of degree d and s-sized faces,
respectively.
1. Euler, L. Solutio Problematis ad Geometriam Situs Pertinentis. Comment.
Acad. Sci. I. Petropolitanae 1736, 8, 128-140.
2. King, R. B., Applications of Graph Theory and Topology in Inorganic
Cluster and Coordination Chemistry,
CRC Press, 1993.
4
Euler’s Theorem on Polyhedra
•
v – e + f = χ = 2 – 2g
•
•
•
•
•
•
•
•
•
χ = Euler’s characteristic
•
v
e
f
g
=
=
=
=
(3)
number of vertices,
number of edges,
number of faces,
genus ; (g = 0 for a sphere; 1 for a torus).
A consequence of Euler’s law:
A sphere can not be tessellated only by hexagons.
Fullerenes need 12 pentagons (preferably isolated ones) for
closing the cage. f5 = 12 and f6 = v/2 – 10
In the opposite, a tube and a torus allow pure hexagonal nets.
5
Schlegel projection
•
A projection of a sphere-like polyhedron on a plane is called a
Schlegel diagram.
•
In a polyhedron, the center of diagram is taken either a vertex,
the center of an edge or the center of a face
C60
The Schlegel projection of C60
6
2. Operations on Maps
•
•
•
A map, M is a combinatorial representation of a closed
surface.1
Several operations on a map allow its transformation in new
maps (convex polyhedra).
Platonic polyhedra: Tetrahedron, Cube, Octahedron,
Dodecahedron and Icosahedron
1. Pisanski, T.; Randić, M. Bridges between Geometry and Graph Theory.
In: Geometry at Work, M. A. A. Notes, 2000, 53, 174-194.
7
• Stellation of C20 (Dodecahedron)
Dodecahedron (C20)
St(Dodecahedron)
8
Dual: examples
Du(Tetrahedron) = Tetrahedron
Du(Cube) = Octahedron
Du(Du(M))=M
Tetrahedron
Cube
Octahedron
9
Dual: examples
Du(Dodecahedron) = Icosahedron
Dodecahedron
Icosahedron
10
Truncation – example
Octahedron
Icosahedron
Tr(Octahedron)
Truncation operation
C60 = TR(C20)
11
COMPOSITE OPERATIONS
Leapfrog, Quadrupling, Dual of the stellation of a medial, Capra
LEAPFROG
Leapfrog, Le is a composite operation that can be achieved in two
ways:
Le(M) = Du(St(M)) = Tr(Du(M))
Le(M) is always a trivalent graph.
Within the leapfrog process, the dualization is made on the
omnicapped map. Le rotates the parent n-gonal faces by π/n.
12
Le(Pentagonal face):
Dual of a triangulation is always a cubic net.
• Relations in the transformed map are:
•
Le(M): v’ = dv = 2e
e’ = 3e
f‘ = f+v
13
Le(M): examples
Dodecahedron (C20)
Icosahedron =
Du(C20)
C60 = Le(C20)
C60 = Le(C20) = Tr(Du(C20)) = Tr(Icosahedron)
14
Schlegel version of Le(M): example
Dodecahedron (C20)
C60 = Le(C20)
15
Q(M): examples
Quadrupling of a Pentagonal face
Quadrupling of a Cube
16
CAPRA
•
The Capra operation is realizing as follows:
1. - Put two points of degree two on each edge of the map
2. - Put a vertex in the center of each face of M and make (1, 4)
connections, between the center and the new two-valent vertices.
3. - The last simple operation is the truncation around the centred
vertex
It rotates the parent s-gonal faces by π/2s.
•
The sequence above discussed is illustrated in the following:
. . . .
. . . .
E2(M)
Pe(E2(M))
. .
. .
Tr(Pe(E2(M)))
17
Example: Ca-operation in Cube and its Schlegel version
1. Two points on each edge
2. A centered vertex on each face
3. Triangulation of the center vertex
Ca(Cube) – Schlegel projection
18
•
The transformed parameters are:
v = (2d 0 + 1)v0 = v0 + 2e0 + s0 f 0
e = 3e0 + 2s0 f 0
f = ( s0 + 1) f 0
Ca(Dodecahedron) = C140
19
3. PERIODIC FULLEROIDS
Cubeoctahedron = Me(Cube)
Tr(Cubeoctahedron) COT4
(side)
Truncation of the Cubeoctahedron
20
EXTENDED ARCHIMEDEAN CAGES
COT4; Tr(cubeoctahedron) N = 48
(top)
COT8; N = 96
21
The Leapfrog, Quadruple and Capra operations
on Tr(cuboctahedron) COT4
Le(COT4); N = 144 (side) Q(COT4); N = 192 (side)
Le(OT4); (top)
Q(COT4) ; (top)
Cad(COT4); N =336 (side)
Cad (COT4) ; (top)
22
Operations on Tr(cuboctahedron) COT8 without the
polar circle
Le(COT8); N = 144 (side) Q(COT8); N = 192 (side)
Le(COT8) ; (top)
Q(COT8) ; (top)
Cad(COT8); N =336 (side)
Cad (COT8) ; (top)
23
FULLERENE COUPLING
C60 (side)
C(60/2),5-A[10,0]
C60
(top)
C(60-5),5-Z[10,0]
Fullerene C60 and two derived caps.
24
2(C(60/2),5-A[10,0]) + AC6[10,8];k=6
2(C(60-5),5-Z[10,0]) + ZC6[10,3]; k = 5
(a)
(b) C110,5-Z[10,3]-[7,6,7] = C140; k = 5
C72,6-A[12,8]-[6] = C168; k = 6
Tubulenes of a-series (a) and peanut z-series (b).
25
PERIODIC FULLEROIDS
C144,6-Z[12,1]-[7]-2; k = 6
C216,6-Z[12,1]-[7]-3 ; k = 6
26
The C60,5-A[10,18]-[6] tubulene (left hand side) and peanut z-tubulenes (mean
side) corresponding to the periodic, multi-peanut (C60)4 (right hand side)
27
C5,C7 PERIODIC CAGES
2 (CN[7,5,75,7,5,7] - k) - 2k
C140(5,7),7-H[14,1]-[7]-2
28
Periodic C5C7 cages
Tetramer C252(5,7),7-H[14,1]-[7]-4
Diudea’s cage C260(5,7),5-H[10,1]-[7]-6
29
C5,C7 Periodic Cages Typing Theorem.
For a periodic cage C5,C7 the number of faces, edges, and vertices
of the various tapes can be counted as functions of the repeating
unit r and cycle size k (Table)
Formulas* for k = 5; 7
f 5k = 2k (r + 1) + 2t 5
f 7 k = 2 kr + 2t7
e 5, 5 k = 2 k ( r + 1 + t 5 )
(1, 2)
(3)
e5,7 k = 2k (3r + 2 + t 7 )
e7,7 k = 2k ( 2r − 1)
(4, 5)
v5,5,5k = 2kt 5
v 5, 5, 7 k = 2 k ( 2 r + 1 + t 7 )
(6, 7)
v5,7,7 k = 2k (r + 1)
v7,7,7 k = 2k ( r − 1)
(8, 9)
N k = 4k (2r + 1)
* ts = 1
if s = k , and zero otherwise
(10)
30
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