Computation of the Cartesian Coordinates of Buckminsterfullerene Peter semi ETH-Zentrum, CHN H32,8092, Zurich, Switzerland Since its discovery, buckminsterfullerene (Cso)has been a t the focus of growing interest. In the year 1985 Kroto e t al. ( 1 ) reported the first spectroscopic evidence of a novel type of elemental carhon consisting of C60 molecules for which a highly symmetric structure was proposed. Five years later Kratschmer et al. (2) developed a n efficient method to synthesize in a single step. Modifications of this procedure suitable for student laboratory programs have been r e ~ o r t e di n this Journal (3. 4). Beaton (5)recentlv portrayed u tc,chniqu(: fur the construction of threeDilDer. d~menilonnlmodels of Ci. and C7, ." with . . '1110 ~rt'sf'nt work shows how the cargsian coordinates of buckiinsterfullerene can be computed with any calculator that can take roots. The carhon framework can he considered a s a truncated icosahedron ulth a circumscribed sphere of radius r,:, for the icosahedron. The positiuns ol'the vertices of the icosahedron must be computed a s a provisional result. The cartesian coordinates of 10 of the 12 vertices of the icosahedron can he computed a s follows.
v
r2)
Yk= - sin k6
Figure 1. Projection of buckminsterfullerene on the plane
for k = 1, 2, 3, ..., 10, where 6 = 36". The two remaining vertices are located on the z axis a t -r12 and r12fork = 11 and k = 12. From the coordinates of two vertices of the icosahedron connected by edges, the cartesian coordinates of two carbon atoms can he computed as follows:
r, = zj + h(Zh - Z,)
(2)
where
where r5 denotes the bond lengths in the pentagons, and re represents the length of the bonds connecting pentagons. David et al. (6) obtained from a structure determined by X-ray diffraction average values of 1.391 A for re and 1.455 A for r5.From r5 and re the radius r12of the circumscribed sphere of the icosahedron can he computed as follows.
'Present address: lngenieurschule Zurich, Abteilung f"r Aiigemeinen Maschinebau, 8004 Zurich, Switzerland.
302
Journal of Chemical Education
where 5 = r5 + %r6. Using the experimental bond lengths reported in ref 6, we obtain r12= 4.090 A. In buckminsterfullerene the centers of the carbon atoms are located on a sphere with radius r ~which , can be computed from r5 and re a s follows:
From the above experimental values for r5and re we obtain rB = 3.548 A. Upon truncation of the icosahedron the 20 original faces with triangular shape are converted to hexagons with alternating lengths of the sides. For each of the 12 vertices a face with the shape of a regular pentagon is formed. Because the vertices of the icosahedron have been numbered in eq 1, the same numbers can he used to lahel the corresponding regular pentagons. Figure 1 shows a schematic projection of buckminsterfullerene on the plane in which the carbon atoms have been placed on five concentric circles. The pentagons are labeled with numbers as indicated above. There should he 12 pentagons labeled from 1to 12 in Figure 1.Pentagon number 12 formed by the five atoms on the outermost circle is considered, however, to face down, and no lahel is shown. The 60 carhon atoms illustrated as small halls have been labeled with numbers following a spiral numbering scheme recently proposed by Taylor (7).In this system the carhon atoms i n a fullerene are numbered consecutively from one cap of its longest
The Combinations of jand kTo Be Used in eq 2 axis of symmetry through to the To Obtain the Coordinates of Atom Number n. other cap. Figure 1can be used to decide which vertices of t h e icosahedron a r e connected The spiral numbering system proposed by Taylor ( 3has been used for the carbon atoms e d c e s a n d t h e number of t h e atom located with eq 2 for appropriate combinations of the indices j a n d k. T h e table below shows the combinations of j and k to be used i n eq 2 to obtain the coordinates of atom number n. Literature Cited
c.;
1. Kroto, H. W:Heath, J.R.:O'Btien, S. Curl, R. F:Smalley, R. E. Nature 1985,318, 1 6 2
."". >e.,
2. Krltsehmer. w:Lamb, L.D.:Fortiropoulos, K: Huffman, D. R. Norun 1940,347,35435R. 3, Iacoe. D. W.: Potter W. T.: Teereia. D. J . Chzm. Educ. 1992,69.663. 4. Cmig. N. C.: Gee, G. C.;Johnson, A. R. J . ?Arm Pdvr 1992 fill,F,%RRR . ~ ~ 5. Beaton, J. M. J. Chem Educ. 1992, 69, 610~~~
posed by Taylor can be applied to other fullerene~,such as Cn,, C m etc.
Volume 72 Number 4 April 1995
303
