The enumeration of heterofullerenes

THEOCH 5317

Journal of Molecular Structure (Theochem) 453 (1998) 1–6

The Enumeration of heterofullerenes F. Zhang*, R. Li, G. Lin Department of Mathematics, Xiamen University, 361005 Fujian, People’s Republic of China Received 7 February 1997; accepted 3 July 1997

Abstract This paper uses Po´lya’s Theorem and the generalized character cycle index to count theoretically the number of possible positional isomers and chiral isomers of the heterofullerene, taking advantage of the symmetry of molecular graphs. A unified formula for specific patterns as well as one for general cases is presented. Furthermore, for a heterofullerene, the enumeration of its equivalence classes under its automorphism group is solved by Burnside’s Lemma. 䉷 1998 Elsevier Science B.V. All rights reserved. Keywords: Group; Orbit; Fullerene; Positional isomer; Chiral isomer

1. Introduction Since the discovery of fullerenes as stable structures of carbon clusters, scientists have been interested in whether similar types of clusters exist. As a result, some breakthroughs have been achieved. Guo et al. [1] deduced, by laser vaporation supersonic cluster beam studies, that if several carbon atoms in C 60 were replaced by boron atoms, C 60- nB n was formed. Miyamoto et al. [2] calculated the electronic structure of hypothetical fcc C 59B as a simple example. K. Esfarjani et al. [3] calculated the electronic structure of C 58BN heterofullerenes in the solid fcc phase (Fm3¯m). Liu et al. [4] studied the electronic structures of substitutional doped fullerenes C 59B, C 59N, C 58B 2 C 58N 2 and C 58BN using a tight-binding approximation method. Bowser et al. [5] studied the stability and structure of C 12 B 24N 24. Xia et al. [6] * Corresponding author. E-mail: [email protected]

also studied B 30N 30. A cluster experiment even suggests a possibility of B 36N 24 with the same soccer-ball-shaped cage structure, [7] although no confirmation has been reported yet. It is likely that many more fullerenes could be composed of the same structure, by replacing the carbons with nitrogens and borons or, possibly, some other elements. This proves the importance of enumerating the possibilities of heterofullerenes. K. Balasubramanian [8–13] has done a lot of work on methods for isomer counting of heterofullerenes and of polysubstituted fullerenes, especially, using the generalized character cycle index (GCCI). Mathematically the isomer counting of polysubstituted fullerene is essentially the same as that of heterofullerene. Y. Shao and Y. Jiang [15] discussed hydrogenated C 60. Furthermore, Fowler et al. [14] also studied the fullerene cages. In this paper we give more general expressions for these counting with more elements substituted. For our purpose we need first to introduce come ideas in combinatorics.

0166-1280/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S 01 66 - 12 8 0( 9 7) 0 02 8 5- 6

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F. Zhang et al./Journal of Molecular Structure (Theochem) 453 (1998) 1–6

2. Burnside Lemma, Po´lya Theorem and GCCI We assume that the reader has the background of group theory and can otherwise refer to any applied group theory textbook for the concepts and basic results about groups [17,18]. Now let G be a group acting on the set V = {1,2,3,…,n}. That is, G is regarded as a permutation group on V. Then any element g of G is a permutation of the set V. Consequently g can be uniquely written as a product of disjoint cycles g = (i1 i2 …is )…(j1 j2 …jt ) We denote the image of i under the permutation g by gi. For 1 ⱕ k ⱕ n, let j k(g) and denote the number of eye of length k in g. We call the polymonial ZG (z1 , z2 , …, zn ) = =

1 ∑ zj1 (g) zj2 (g) …zjnn (g) lGl g僆G 1 2 n 1 ∑ ∏ zkjk (g) lGl g僆G k = 1

the cycle index of G, and the plynomial PxG (z1 , z2 , …, zn ) = =

1 ∑ x(g)z1j1 (g) z2j2 (g) …znjn (g) lGl g僆G n 1 ∑ x(g) ∏ zjkk (g) lGl g僆G k =1

the generalized character cycle index (GCCI), where x(g) is the linear character of the irreducible representation of G. In this paper we use two special cases: One is the antisymmetric representation, that is x(g) = {

1, if g is a proper rotation, − 1, if g is an improper rotation;

and the other when x is 1 for all g. We list the following theorems about the orbits of a group G (equivalence classes under group action, that is, Ga, Gb,…,Gc, where a, b,…, c 僆 V are the representatives of the orbits) without proof Burnside’s Lemma. The number of orbits of G on the set V is N (G) =

1 ∑ lFix(g)l, lGl g僆G

where Fix(g) = {ilgi = i,i 僆 V} is called the fixed points of g. Now let A and C be two sets with lAl = n and lCl = m. We can regard A as the elements of a molecule and C as m other elements to be substituted. If we substitute a boron, for instance, for a carbon, the action can be seen as coloring the carbon with the color boron. Thus any substitution would be regarded as a coloring of the set C with the color set A. Denote C A = {f lf : A → C is a mapping from A to C}. Clearly lC Al = m n. Actually C A is the set of all the colorings of C with the colors A. Further, let G be a permutation group acting on A which naturally induced an act [denoted by (G, C A)] on the Set C A through gf = fg − 1 . We have Theorem 1 (Po´lya’s Theorem). Let F be the set of all orbits of (G, C A), then lFl = ZG (m, m, …, m)

Theorem 2. Let C = {c1 , c2 , …, cm }k1 + k2 + … + km = n, where k i are non-negative integers, i = 1,2,…,m. Denote N(k 1, k 2,…, k m) to be the number of orbits in C A which map exactly k i elements to ci …i = 1; 2; …; m†. Then m

m

m

i=1

i=1

i=1

ZG ( ∑ xi , ∑ x2i , …, ∑ xni ) =

∑

k1 + k2 + … + km = n

N (k1 + k2 + … + km )xk11 xk22 …xkm m ,

namely, N (k1 + k2 + … + km ) is the coefficient of xk11 xk22 …xkm m : Theorem 3 (Generalization of Po´lya’s Theorem). k Substituting ∑m i = 1 xi for z k and in the GCCI, i = … 1,2, ,n, we get the chiral generating function (CGF) m

m

m

t=1

t=1

t=1

CGF = PxG ( ∑ xi , ∑ x2i , …, ∑ xni ) To enumerate all the possibilities of the heterofullerene structures, we have to consider the rotation group of the fullerene, and its whole automorphism (point) group to enumerate the number of chiral isomers. Interpreting the aforementioned problem as one of graph theory, the substitution of elements can be as a colouring of the vertices of the corresponding graph

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F. Zhang et al./Journal of Molecular Structure (Theochem) 453 (1998) 1–6

such that every rotation, proper or improper, of the graph retains the, same colouring.

the following table of the elements of the point group except for those of I.

3. The enumeration of heterofullerenes In this section to begin with, we determine the rotation group of the fullerrence C 60, that is, the icosahedral group I k and , then we proceed to get the whole point group of it. We will calculate the possibilities of different positional isomers using the Po´lya’s counting theorem, and the number of chiral isomers using the generalzation of the Po´lya’s Theorem. See the following table. All the rotations can be divided into the following types:

R 0: R 1: R 2: R 3:

Improper rotations

Number of elements

PI PC 2 PC 3 PC 5

1 15 20 24

the corresponding term in the cycle index z30 2 z41 z28 2 z10 6 6 z10

Improper rotations and their corresponding terms in the GCCI Consequently the GCCI [11] is PxG (z1 , z2 , …, zn ) =

1 60 20 4 28 (z + 14z30 2 + 20z3 − 15z1 z2 120 1

10 6 + 24z12 5 − 20z6 − 24z10 )

I: C 2:

C 3:

C 5:

Rotations

Number of elements

Identity p-rotations that keeps the common edge of 2 hexagons fixed 2 3p-rotations that keeps a pair of antipolar hexagons fixed 2 5p-rotations that keeps a pair of antipolar pentagons fixed

1 15

The corresponding term in the cycle index z60 1 z30 2

20

z20 3

24

z12 5

Rotations and their corresponding terms in the cycle index This is because the identity permutation is expressed as a product of 60 1-cycles; the p-rotation is expressed a product of 30 2-cycles and the 23protation is expressed as a product of 20 3-cycles and the 25p-rotation is expressed as a product of 12 5-cycles. Thus the cycle index of the icosahedral rotation group acting on C 60 is ZG (z1 , z2 , …, zn ) =

1 60 20 12 (z + 15z30 2 + 20z3 + 24z5 ) 60 1

To get the whole point group of C 60, simply multiply I k with the inversion operator P and bracket them together. It is the set S n = {I k, PI k}. Hence we get

Note that x ⬅ 1 is also one of the characters of the linear irreducible representations of S n. With this character the GCCI can be used to count the number of orbits under the whole point group S n [3]. In this case we get PxG (z1 , z2 , …, zn ) = P1G (z1 , z2 , …, zn ) =

1 60 (z + 16z30 2 120 1

4 28 12 10 6 + 20z20 3 − 15z1 z2 + 24z5 − 20z6 − 24z10 )

We are now ready to enurnerate the heterofullerenes. From the above discussion the problem is reduced to the colouring of the correspoding graph of 60 vertices. Let A denote the vertex set of the graph representing the 60 carbons, and C = {c 1, c 2,…, c m} the set of all possible different elements to be substituted. The substitution is represented by a mapping f from A to C, i.e. a colouring on A. Consequently the number of all different positional isomers is that of the different orbits of the icosahedral group acting on the power set A C. By Po´lya’s theorem, this number equals ZG (m, m, …, m) =

1 60 (m + 15m30 + 20m20 + 24m12 ), 60

where m = lCl. Further, if we are interested in a certain pattern (k 1, k 2,…, k m) where k i is the number of c i elements i = 1, 2,…, m, then the number of different positional isomers can be obtained by expanding the

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F. Zhang et al./Journal of Molecular Structure (Theochem) 453 (1998) 1–6

For the case of counting the number of chiral isomers, we apply the method to the GCCI

expression into m

m

m

i=1

i=1

i=1

ZG ( ∑ xi , ∑ x2i , …, ∑ xni ) =

∑

k1 + k2 + … + km = n

N (k1 + k2 + … + km )xk11 xk22 …xkm m

and N(k 1, k 2,…, k m), the coefficient of xk11 xk22 …xkm m , is the number we want. Take m = 3 as an example. By counting we get "  3  60 3 3 3 1 2 … n ∑ xi ZG ∑ x i , ∑ x i , , ∑ x i = 60 i = 1 i=1 i=1 i=1       # + 15 =

1 60

+ 15 + 20 + 24

30

3

∑

i=1

x2i

+ 20



3

∑

i=1

20

x3i

+ 24

3

∑

i=1

12

x4i

 + 14  ×

∑

i=1 3

4

∑

i=1

 + 15

i=1 3

 + 20

x2i

∑ xi

 ×

30

3

∑

i=1

6 # =

x10 i

i=1

 ∑

∑

∑

12! 5i1 5i2 5i3 x x x i1 !i2 !i3 ! 1 2 3

− 20

∑

10! 6i1 6i2 6i3 x x x i1 !i2 !i3 ! 1 2 3

∑

6! x10i1 x10i2 x10i3 i1 !i2 !i3 ! 1 2 3

1 (17363244548449800 + 641277000) 60

1 17363245189726800 = 289387419828780: 60

 1 60! 30! + 14 × − 15 120 4!10!46! 2!5!23!  4! 28! ∑ × ∑ i2 + i3 = 4 2!i2 !i3 ! i2 + i3 = 26 2!i2 !i3 ! 4! 28! ∑ + 2!i2 !i3 ! i2 + i3 = 27 1!i2 !i3 !

∑

i2 x + i3 = 28

3

i=1

60!

− 15

10

∑ x6i

i1 + i2 + i3 = 60 i1 !i2 !i3 !

+ 24

i1 + i2 + i3 = 12

∑

12 x5i

− 24

xi11 xi22 xi33

30! 2i1 2i2 2i3 x x x i1 !i2 !i3 ! 1 2 3 20!

∑

3

i=1

 − 20 ×

20! 3i1 3i2 3i3 x x x i1 !i2 !i3 ! 1 2 3

i1 + i2 + i3 = 30

∑ xi



28

60

3

i=1

+ 24

x3i

∑



∑

∑

i=1

3

"

20

3

∑ x2i

1 120

1 = 120

+ 20

i1 + i2 + i3 = 20

∑

i=1

∑ xi , ∑

xni

30! 2i1 2i2 2i3 x x x i1 !i2 !i3 ! 1 2 3

i1 + i2 + i3 = 30

i2 + i3 = 4

i=1



3

x2i , …,

∑

i1 + i2 + i3 = 60

× 52 × 53 × 11 × 57 × 29 × 59 + 150 × 26 × 27 × 7 × 29

+

3

+ 14

If we want to know the number of possible positional isomers of the pattern N 4B 10C 46, for instance, it is the coefficient of the term x 4x 10x 46, namely,   1 60! 30! 1 + 15 × = (47 × 49 × 50 × 51 60 4!10!46! 2!5!23! 60

=

3

60! i1 i2 i3 x x x i1 !i2 !i3 ! 1 2 3

∑

× 30) =

 PxG

1 3i2 3i3 x3i 1 x2 x3

i1 + i2 + i3 = 20 i1 !i2 !i3 !

12! 5i1 5i2 5i3 x1 x2 x3 i1 + i2 + i3 = 12 i1 !i2 !i3 !   4! i1 i2 i3 x1 x2 x3 − 15 ∑ i1 + i2 + i3 = 4 i1 !i2 !i3 !   28! 2i1 2i2 2i3 x1 x2 x3 × ∑ i1 + i2 + i3 = 28 i1 !i2 !i3 !

− 24

i1 + i2 + i3 = 10

i1 + i2 + i3 = 6



And the coefficient of the term x 4x 10x 46, for example, is the number of the chiral isomers with pattern N 4B 10C 46 that is

 28! 1 17363244001521600 = 144677033346005: = !i2 !i3 ! 120

F. Zhang et al./Journal of Molecular Structure (Theochem) 453 (1998) 1–6

5

For the number of orbits under the whole point group S n, we simply note that ZIh − PxSn = P1Sn . Hence

fullerene has only rotations, the number of orbits of K on the set V is

P1Sn = 289387419828780 − 144677033346005

N (K) =

= 144710386482775 is the number we want.

4. The enumeration of equivalence classes of heterofullerenes For a heterofullerene, how to determine the number of its orbits (equivalence classes under its automorphism group) is another important problem from the view point of quantum chemistry, NMR and multiple quantum NMR spectroscopy. In a previous paper Y. Lin and one of the present authors point out that this type of problem can be solved by Burnside’s Lemma [16] as well as by Po´lya’s Theorem. And it seems that Burnside’s Lemma is much simpler. In order to solve this type of problem, we need to consider the automorphism group of the heterofullerene molecule. In general the rotation group is a subgroup of the automorphism group. In our case the automorphism group of the fullerene is S n = {I h,PI h} where P is the inversion operator. In fact, other than the elements of I h, and, the inversion P leaves the C 60 molecule invariant. Generally speaking the automorphism group of a heterofullerene is a subgroup of S n. In order to find the number of orbits of a heterofullerene H, first we need to find out its automorphism group. In fact each element of S n whose effect is to map every heteroatom to the same type of heteroatom (e.g. N → N, B → B) form the automorphism group of H. We can check all the elements of S n one by one to determine the subgroup. Then we can use Burnside’s Lemma to calculate the number of orbits of H. Fortunately the fullerene structure has a very decided property that any rotation except for the identity of its automorphism group has no fixed piont in V. Thus if the subgroup contains only rotations, each term Fix (g) in the formula of the Burnside’s Lemma will be 0 except for Fix(I) = n. Hence Theorem 3. If the automorphism group K of a hetero-

lV l 60 = lKl lKl

To illustrate our approach we give a few examples. The first example is when we replace one carbon with a nitrogen. The automorphism group contains two elements: the identity and the reflection with respect to the unique plane containing the nitrogen and other three carbons. So by Burnside’s Lemma, the number of orbits of the heterofullerene is [(1=2)(60 + 4)] = 32, including the nitrogen which itself forms a singleton orbit. The sme result is reached if we use Po´lya’s Theorem. When we fix the nitrogen and substitute another element boron, say, we get 31 possible isomers of the pattern C 58XY for there are altogether 31 carbon orbits on C 58X. The second example is when we replace two antipolar carbon atoms with two nitrogen atoms. In this case the. automorphism group has four elements, namely, the identity, a reflection with respect to the unique plane containing the two element and other two carbons, a p-rotation about the axis perpendicular to this plane and the product of the reflection and the rotation. Hence by Burnside’s Lemma the number of orbits of the heterofullerene is [(1=4)(60 + 4)] = 16. The third one is when we substitute three nitrogen atoms for three carbon atoms on a face of a hexagon which are alternating on the 6-cycle, then we get C 3 as its automorphism group. And by Theorem 3, the number of orbits of the resulting heterofullerene is (60=3) = 20.

Acknowledgements We thank professor Qianer Zhang for his information and helpful discussion.

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