MOLECULAR MODELING OF FULLERENE

Conducător Doctorand:Prof. Dr. Mircea V. Diudea Monica Ştefu

ContentsContents

1. FULLERENE MODELING

2. OPERATIONS ON MAPS

3. PERIODIC FULLEROIDS

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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.

Nature (London) , 1990, 347, 354-358. 2

Isolated Fullerenes

N = 60, 70, 76, 78, 82 and 84C78 C76 C70

C84 C82

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• 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.

BASIC RELATIONS IN POLYHEDRA

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.

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Euler’s Theorem on Polyhedra

• v – e + f = χ = 2 – 2g (3)

• χ = Euler’s characteristic• v = number of vertices, • e = number of edges,• f = number of faces,• g = 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.

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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

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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.

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• Stellation of C20 (Dodecahedron)

St(Dodecahedron) Dodecahedron (C20)

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Dual: examples

OctahedronCubeTetrahedron

Du(Tetrahedron) = TetrahedronDu(Cube) = Octahedron

Du(Du(M))=M

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Dual: examples

IcosahedronDodecahedron

Du(Dodecahedron) = Icosahedron

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Truncation – example

Tr(Octahedron) Octahedron

C60 = TR(C20) Truncation operation Icosahedron

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COMPOSITE OPERATIONS

Leapfrog, Quadrupling, Dual of the stellation of a medial, Capra

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.

LEAPFROG

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Le((Pentagonal face):

• Dual of a triangulation is always a cubic net.• Relations in the transformed map are:

Le(M): v’ = dv = 2ee’ = 3ef ‘ = f + v

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Le((M): examples

C60 = Le(C20) Icosahedron = Du(C20)

Dodecahedron (C20)

C60 = Le(C20) = Tr(Du(C20)) = Tr(Icosahedron)

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Schlegel version of Le(M): example

C60 = Le(C20) Dodecahedron (C20)

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Q((M): examplesQuadrupling of a Pentagonal face

Quadrupling of a Cube

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CAPRA• The Capra operation is realizing as follows:

1. - Put two points of degree two on each edge of the map2. - 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 centredvertex

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

Ca(Cube) – Schlegel projection 3. Triangulation of the center vertex

2. A centered vertex on each face1. Two points on each edge

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• The transformed parameters are:

000000 2)12( fsevvdv ++=+=

000 23 fsee +=

00 )1( fsf +=

Ca(Dodecahedron) = C140

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Truncation of the Cubeoctahedron

Tr(Cubeoctahedron) COT4 (side)

Cubeoctahedron = Me(Cube)

3. PERIODIC FULLEROIDS

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EXTENDED ARCHIMEDEAN CAGES

COT8; N = 96COT4; Tr(cubeoctahedron) N = 48 (top)

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The Leapfrog, Quadruple and Capra operations on Tr(cuboctahedron) COT4

Cad (COT4) ; (top) Q(COT4) ; (top) Le(OT4); (top)

Cad(COT4); N =336 (side)Q(COT4); N = 192 (side) Le(COT4); N = 144 (side)

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Operations on Tr(cuboctahedron) COT8 without the polar circle

Cad (COT8) ; (top) Q(COT8) ; (top) Le(COT8) ; (top)

Cad(COT8); N =336 (side) Q(COT8); N = 192 (side) Le(COT8); N = 144 (side)

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FULLERENE COUPLING

C(60-5),5-Z[10,0]C(60/2),5-A[10,0]

C60 (top) C60 (side)

Fullerene C60 and two derived caps. 24

(b) C110,5-Z[10,3]-[7,6,7] = C140; k = 5 (a) C72,6-A[12,8]-[6] = C168; k = 6

2(C(60-5),5-Z[10,0]) + ZC6[10,3]; k = 52(C(60/2),5-A[10,0]) + AC6[10,8];k=6

Tubulenes of a-series (a) and peanut z-series (b).

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PERIODIC FULLEROIDS

C216,6-Z[12,1]-[7]-3 ; k = 6 C144,6-Z[12,1]-[7]-2; k = 6

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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)

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C140(5,7),7-H[14,1]-[7]-2 2 (CN[7,5,75,7,5,7] - k) - 2k

CC55,,CC77 PERIODIC CAGESPERIODIC CAGES

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Diudea’s cage C260(5,7),5-H[10,1]-[7]-6Tetramer C252(5,7),7-H[14,1]-[7]-4

Periodic C5C7 cages

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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)

(6, 7)

(8, 9)

(10)

(3) (4, 5)

(1, 2)

Formulas* for k = 5; 7

55 2)1(2 trkf k ++=

)1(2 55,5 trke k ++=

)23(2 77,5 trke k ++= )12(27,7 −= rke k

55,5,5 2ktv k = )12(2 77,5,5 trkv k ++=

)1(27,7,5 += rkv k)1(27,7,7 −= rkv k

)12(4 += rkNk

otherwiseand,if1 zerokst s == 30*

77 22 tkrf k +=

SOFTWARE

• TOPOCLUJ 2.0 - Calculations in MOLECULAR TOPOLOGYM. V. Diudea, O. Ursu and Cs. L. Nagy, B-B Univ. 2002

• CageVersatile 1.1 - Operations on mapsM. Stefu and M. V. Diudea, B-B Univ. 2003